Encyclopedia of Nanoscience and Nanotechnology 9781588830609, 1588830608

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Encyclopedia of Nanoscience and Nanotechnology Volume 4 Number 1 2004 Hydrogen and Oxygen Interaction with Carbon Nanotubes

1

George E. Froudakis Hydrogen Storage by Carbon Nanotubes

13

R. G. Ding; J. J. Finnerty; Z. H. Zhu; Z. F. Yan; G. Q. Lu Hydrogenated Nanocrystalline Silicons

35

K. Shimakawa Impurity States and Atomic Systems Confined in Nanostructures

43

R. Riera; R. Betancourt-Riera; J. L. Marín; R. Rosas Inherently Conducting Polymer Nanostructures

113

Gordon G. Wallace; Peter C. Innis; Leon A. P. Kane-Maguire Inorganic Nanomaterials from Molecular Templates

131

Sanjay Mathur; Hao Shen In-situ Microscopy of Nanoparticles

193

Mark Yeadon In-situ Nanomeasurements of Individual Carbon Nanotubes

205

Zhong Lin Wang Integrated Passive Components

215

Richard Ulrich Interatomic Potential Models for Nanostructures

231

H. Rafii-Tabar; G. A. Mansoori Interfacial Defects in Nanostructures

249

I. A. Ovid'ko Interfacial Effects in Nanocrystalline Materials

267

K. T. Aust; G. Hibbard; G. Palumbo Ion Implanted Nanostructures

283

Karen J. Kirkby; Roger P. Webb Ion Sputtering on Metal Surfaces

295

C. Boragno; F. Buatier de Mongeot; U. Valbusa Ionic Conduction in Nanostructured Materials

309

Philippe Knauth

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Kelvin Probe Technique for Nanotechnology

327

G. Koley; M. G. Spencer Kinesin and Nanoactuators

345

Konrad J. Böhm; Eberhard Unger Kinetics in Nanostructured Materials

359

V. V. Skorokhod; I. V. Uvarova; A. V. Ragulya Kondo Effect in Quantum Dots

379

H. Matsumura; R. Ugajin Laser-Scanning Probe Microscope Assisted Nanofabrication

389

Piero Morales Laser Fusion Target Nanomaterials

407

Keiji Nagai; Takayoshi Norimatsu; Yasukazu Izawa; Tatsuhiko Yamanaka Laser Induced Surface Nanostructuring

421

Yuji Kawakami; Eiichi Ozawa; Isamu Miyamoto Layer-by-Layer and Langmuir-Blodgett Films from Nanoparticles and Complexes

441

Marysilvia Ferreira; Valtencir Zucolotto; Marystela Ferreira; Osvaldo N. Oliveira Jr.; Karen Wohnrath Layer-by-Layer Nanoarchitectonics

467

Katsuhiko Ariga Lead Sulfide Nanoparticles

481

Jochen Fick; Alessandro Martucci Light-Harvesting Nanostructures

505

Teodor Silviu Balaban Light Scattering of Semiconducting Nanoparticles

561

G. Irmer; J. Monecke; P. Verma Localization and Interactions in Magnetic Nanostructures

587

F. G. Aliev; V. K. Dugaev; J. Barna Low-Dimensional Nanocrystals

607

Shu-Hong Yu; Jian Yang; Yi-Tai Qian Low-Frequency Noise in Nanomaterials and Nanostructures

649

Mihai N. Mihaila Low-Temperature Scanning Tunneling Microscopy

667

Wolf-Dieter Schneider Luminescence of Semiconductor Nanoparticles

689

Wei Chen; Alan G. Joly; Shaopeng Wang

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Luminescent Organic-Inorganic Nanohybrids

719

R. A. Sá Ferreira; L. D. Carlos; V. de Zea Bermudez Macroscopically Aligned Carbon Nanotubes

763

Pascale Launois; Philippe Poulin Magnesium-Nickel Nanocrystalline and Amorphous Alloys for Batteries

775

H. K. Liu Magnetic Field Effects in Nanostructures

791

R. Riera; J. L. Marín; H. León; R. Rosas Magnetic Semiconductor Nanostructures

835

Wolfram Heimbrodt; Peter J. Klar Magnetism in Nanoclusters

899

C. Binns Magnetism in Rare Earth/Transition Metal Multilayers

925

V. O. Vas'kovskiy; A. V. Svalov; G. V. Kurlyandskaya Magnetoimpedance in Nanocrystalline Alloys

949

B. Hernando; P. Gorria; M. L. Sánchez; V. M. Prida; G. V. Kurlyandskaya Copyright © 2004 American Scientific Publishers

4/24/2007 11:48 PM

Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Hydrogen and Oxygen Interaction with Carbon Nanotubes George E. Froudakis University of Crete, Heraklio, Crete, Greece

CONTENTS 1. Introduction 2. A Brief Experimental Overview 3. The Need for Theoretical Modeling 4. Ab Initio Approaches in Large Systems 5. Hydrogen Interaction with Carbon Nanotubes 6. Oxygen Interaction with Carbon Nanotubes 7. Conclusions Glossary References

1. INTRODUCTION Since the discovery of carbon nanotubes a lot of experimental and theoretical work has appeared investigating and analyzing their structural and electronic properties [1–3]. Especially single walled carbon nanotubes (SWNTs) were proposed both as microcavities for storage [4–9] and as ideal molecular wires for microelectronic applications [10]. Furthermore the functionalization of the SWNT surface was found to affect both their storage capacity [9] and their electronic properties [10–12]. Hydrogen has been recognized as an ideal energy carrier but has not been used yet to a large extent. One of the major problems is the difficulty of efficient storage. In the beginning metal alloys were tested for storage tanks but even though they have sufficient storage capacity, they are expensive and heavy for commercial production focused on mobile applications. In the recent years, carbon based materials attracted attention due to the discovery of novel carbon nanomaterials like fullerenes, nanofibers, and nanotubes [1–3]. Especially SWNTs, which have diameters of typically a few nanometers, have been suggested as suitable materials for gas storage [4]. Since pores of molecular dimensions can adsorb large quantities of gases, hydrogen can condense to high density inside narrow SWNTs even at room temperature [5]. The high hydrogen uptake of these ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

materials suggests that they can be used as hydrogen-storage materials for fuel-cell electric vehicles [6–8]. Recently it was experimentally proved that the electronic and transport properties of SWNTs are extremely sensitive to their exposure to gas molecules [10–12]. Especially oxygen attracted a lot of interest due to the importance of its interaction with SWNTs both in their synthesis by purification and their functionalization for controlling their electronic properties. Considering that all previous experimental studies of SWNTs have used samples exposed to air, the results of these measurements must be carefully reevaluated.

2. A BRIEF EXPERIMENTAL OVERVIEW A lot of recent experiments are trying to investigate the hydrogen storage in SWNTs. First, in 1997, Dillon et al. reported that SWNTs could store hydrogen [4]. Using temperature-programmed desorption spectroscopy they showed that hydrogen would condense inside SWNTs under conditions that do not induce adsorption within a standard mesoporous activated carbon. Two years later Liu et al. found out that this storage can take place in room temperature [5]. They used SWNTs of 1.85 nm diameter, synthesized by a semicontinuous hydrogen arc discharge method, and found a storage capacity of 4.2 wt% at room temperature and under modestly high pressure; 78.3% of the adsorbed hydrogen can be released under ambient pressure and room temperature. The same year Chen et al. reported that alkali-doped carbon nanotubes possess high hydrogen uptake [9]. They investigated lithium and potassium doped carbon nanotubes and found hydrogen absorption of 14 to 20 wt% between 400
C and room temperature. These values are higher than those of metal hydride and cryoadsorption systems. The stored hydrogen could be released at higher temperatures and the sorption–adsorption cycle can be repeated with minor loss of the storage capacity. After that a lot of experimental work has been performed trying to investigate the hydrogen adsorption in SWNTs and to improve the storage capacity of the tubes by doping them

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (1–11)

2 [13–20]. Nevertheless the experiments on hydrogen storage in nanotube samples have been controversial as reported by the recent review of Ding et al. [8]. Concerning the electronic properties of the nanotubes, Collins et al. reported that they are extremely sensitive to the chemical environment [10]. The exposure to atmospheric air or oxygen dramatically influences the nanotubes’ electrical resistance, thermoelectric power, and density of states, as determined by transport measurements and scanning tunneling microscopy [10]. The most interesting observation of Collins et al. is that these electronic parameters can be reversibly tuned by a surprisingly small concentration of adsorbed gases.

3. THE NEED FOR THEORETICAL MODELING On the other hand, until recently, there was no sufficient theoretical explanation of gas adsorption in SWNTs but only guesses about this procedure. This affected badly both the understanding of the nature of these materials and the improvement of their storage capacity that ended up in a random procedure. The need for theoretical modeling was crucial because even though the experiments can inform us what is happening, only theory allows us to find out why it is happening and what will happen in similar conditions. Concerning the gas adsorption in carbon nanotubes, the theoretical calculations can be extremely useful for understanding the elementary steps of the adsorption procedure and give an insight into the phenomenon.

4. AB INITIO APPROACHES IN LARGE SYSTEMS One of the major headaches in computational material science is how accurate theoretical techniques can be applied in large systems. By “accurate theoretical techniques” we mean quantum chemistry methods, called ab initio, due to the fact that they do not enforce any parameters to the system but solve the Schrödinger equations from first principles. The grater advantage of the ab initio methods is that they can provide structural, electronic, and dynamic properties of the calculating system in high accuracy. On the other hand the computational cost increases dramatically with the number of the electrons in the system. The problem that arises in polyatomic systems is how to compromise the relatively large size of the system and an accurate ab initio method without ending up with a prohibitively large calculation. There are three possible solutions to this dilemma of treating large systems with ab initio methods: (a) the periodic density functional theory model, (b) the mixed quantum mechanics/molecular mechanics model, and (c) the cluster model.

4.1. Periodic Density Functional Theory Model The periodic density functional theory (DFT) model is schematically presented in Figure 1a for a (4,4) SWNT. A part of the system (central part of the tube presented

Hydrogen and Oxygen Interaction with Carbon Nanotubes

Periodic Boundary Conditions

Quantum Mechanics Molecular Mechanics

Molecular Mechanics

Quantum Mechanics

A

Cluster Approach

B

C

Figure 1. Three possible models for treating large systems with ab initio methods. (A) Periodic DFT model simulating a (4,4) SWNT. A part of the system (blue colored carbon atoms) is separated and treated as unit cell for a periodic building of the system. (B) The QM/MM model simulating a (4,4) SWNT. The total 200-atom tube was separated in three cylindrical parts. The inner one was treated with DFT (40 blue color carbon atoms) while the two outer parts with molecular mechanics (brown color carbon atoms). The dangling bonds at the ends of the tube were saturated with hydrogen atoms. (C) The cluster model. A part of the system is separated and treated as individual cluster.

with blue color carbon atoms) is separated and treated as a unit cell. This unit cell is periodically repeated in the space (for the tube in one dimension only) building an infinite system. Implementation of periodic boundaries conditions to the equations solves the mathematical part of the problem. The advantage of this approach is that the total system is treated with ab initio techniques. The disadvantage is that local interactions and defects studied in the unit cell are automatically repeated periodically. In this way local properties become periodic. An example of the use of this approach is presented in Section 5.3.1.

4.2. Quantum Mechanics/Molecular Mechanics Mixed Model The mixed quantum mechanics/molecular mechanics (QM/MM) model is schematically presented in Figure 1b. The basic idea of this method is that the system is divided in two parts: one is treated by ab initio methods and the other by molecular mechanics. The bordering of the two parts is arranged by introducing link atoms. More details for this model can be found in [21]. For the (4,4) SWNT presented in Figure 1b, the total 200atom tube was separated in three cylindrical parts. The inner one was treated with DFT (40 blue color carbon atoms) and the two outer parts with molecular mechanics (brown color carbon atoms). The dangling bonds at the ends of the tube were saturated with hydrogen atoms. The advantage of the QM/MM model is that no periodic constrains are introduced to the systems. The disadvantage is that only a relatively small part of the system is treated quantum mechanically while the rest is used for constraining the boundaries. This method is suitable for studding local properties. An example of the implementation of the QM/MM approach in SWNTs is presented in Section 5.3.2.

Hydrogen and Oxygen Interaction with Carbon Nanotubes

4.3. The Cluster Model In the cluster model (Fig. 1c) a part of the system is separated and treated as an individual cluster. Extra attention has to be given to the dangling bonds that are generated at the braking areas. These dangling bonds have to be saturated properly with the addition of extra atoms (for example hydrogens). In this way the boundary instabilities that arise from QM/MM and the periodicity problems introduced by periodic DFT can be eliminated. The drawback of this method is that only a relatively small part of the system can be used. An example using the cluster model approximation is presented in Section 6.1. It is clearly shown that no one of all these three techniques gives a perfect and unique solution to the problem. All show advantages and disadvantages and the decision as to which one we should choose have to be made according to the problem we face each time.

5. HYDROGEN INTERACTION WITH CARBON NANOTUBES The theoretical simulations in this field can be generally classified in two categories according to the theoretical approximation that they are based on. The first group employs Monte Carlo and molecular mechanics classical algorithms in order to investigate the physisorption of hydrogen in SWNTs while the second uses ab initio or semiempirical quantum techniques for studying mainly the chemisorption of atomic hydrogen in SWNTs. A review of theoretical calculation of hydrogen storage in carbon-based materials can be found in [22].

5.1. Hydrogen Physisorption in Carbon Nanotubes First Darkrim and Levesque in late 1998 using a LennardJones potential performed a grand canonical Monte Carlo simulation of hydrogen storage in a cell of SWNTs and investigated the influence of the tube diameter on the storage capacity [23]. They found that adsorption decreased as the SWNT diameter increased due to the fact that a large part of the volume inside or outside the tube is out of the attractive force range of the solid–gas interaction. They also note the dependence of their results on the intermolecular potential used for the hydrogen–carbon interaction. This is the major disadvantage of classical simulations and points to the need for ab initio calculations in the field. In 1999 Johnson and co-workers using the Silvera– Goldman potential for the H2 –H2 interaction and the Crowell–Brown potential the H2 –tube interaction studied hydrogen adsorption in neutral [24] and positively and negatively charged [25] SWNTs. Their results show that idealized graphitic nanofibers (slit pores) give significantly better performance for hydrogen storage than SWNT arrays. They also underline the importance of the packing geometry of the SWNTs in the storage capacity. In addition, a 0.1 e/C charging of the nanotubes increased the adsorption up to 30%. In the year 2000 Williams and Eklund simulated H2 physisorption in finite-diameter carbon nanotube ropes

3 using grand canonical Monte Carlo [26]. Their simulation clearly shows that small diameter ropes are preferable for hydrogen storage. They also point out an essential difference between models and experiments that has to do with the ideal and “atomically clean” surface of the tubes in the simulation. The great advantage of these classical approaches, except for the luxury to take into account large systems, is the temperature dependence of the simulations. On the other hand they are parameter dependent and cannot provide an insight into the chemical bond as the ab initio methods do.

5.2. Quantum Approaches In 2001 the quantum picture was introduced to the molecular dynamic study of hydrogen in SWNTs either by quantum molecular dynamics algorithms [27] or by minimal ab initio calculations in parts of classical-optimized tube geometries [28]. In the first case, Cheng et al. using the Vienna ab initio simulation package performed a quantum mechanical molecular dynamics simulation of H2 absorption in a trigonal two-dimensional lattice of armchair (9,9) SWNTs [27]. The potential energy surface near the equilibrium point was found to be relatively flat and significant changes of the lattice constants (>0.5 Å) resulted in only small changes in the lattice energy (98 purified purified aB*

L ~ aB*

L 0; Y x = 0 if x < 0]. The impurity position along the growth axis is zi . The x y origin is at the impurity site because all the impurity positions are equivalent in the layer plane; C is the projection of the electron position vector in the layer plane .Cx y/. 0 are In the absence of the impurity the eigenstates of H separable in x y and z,   2 2
k ⊥    0 L k⊥  = L + L k (103) H ⊥ 2m∗e

Impurity States and Atomic Systems Confined in Nanostructures

60 where L labels the quantum well eigenstates (energy L ), that is, the quantum well bound L < V0  and unbound  = k  k . Since the L k   basis is states L > V0  and k ⊥ x y ⊥ complete we may always expand the impurity wavefunction 3loc in the form
 L k   3loc  = cL k (104) ⊥ ⊥ ⊥ L k

The Coulombic potential couples a given subband L, as  ⊥ with all others. The intersubband well as a given vector k coupling (especially the one with the subbands of the quantum well continuum) is difficult to handle. In a quasi-two  ≡ dimensional situation we would like to set cL k ⊥  cL0 k⊥ 0L L0 , that is, to neglect intersubband coupling [67, 68]. This procedure is convenient as the impurity wavefunction displays a separable form:  loc  = 9L0 zC  r3

(105)

 is the solution of the twoThe wavefunction C dimensional Schrödinger equation

pˆ x2 + pˆ y2  =  − L0 C  (106) + Veff C C 2m∗e where Veff is the effective in-plane Coulombic potential, Veff C =

1 −e2  + dz 9L20 z   − C2 + z − zi 2

(107)

which is L0 and zi dependent. A solution of Eq. (106) can be sought variationally, the simplest choice for the ground state being the nodeless one-parameter trial wavefunction  1 2 0 C = exp−C/( (108) (  where ( is the variational parameter. The bound state energy is obtained through the minimization of the function 2e2   −x
2 L0 zi  ( = − xe 2m∗e (2 ( 0  + dz 9L20 z dx + L0 ×  − x2 + (42 z − zi 2

(

  1  = N91 z exp − 3loc r C2 + z − zi 2 (

(111)

where N is a normalization constant, ( is the variational parameter, and attention is focused on the ground bound state attached to the ground quantum well subband 1 . Calculations are less simple than with the separable wavefunction [Eq. (105)]. Comparing the binding energy deduced from Eqs. (105)–(111) one finds, for infinite V0 , that the separable wavefunction gives almost the same results as the nonseparable wavefunction if L/a∗B ≤ 3. This is the range where for most materials the quantum size effects are important. Other variational calculations have been proposed [72, 73]. For example, instead of using a nonlinear variational parameter one uses a finite basis set of fixed wavefunctions  is numerically diagonal(often Gaussian ones) in which H ized. The numerical results obtained by using a single nonlinear variational parameter compare favorably with these very accurate treatments.

3.1.2. Results for the Ground Impurity State Attached to the Ground Subband Figures 9 to 11 give a sample of some calculated results for the ground impurity state attached to the ground subband. Two parameters control the binding energy:

(109)

and the binding energy is

bL0 zi  = L0 − Min L0 zi  (

barrier. At L = 0 the only sensible result is to find a 1s bulk hydrogenic wavefunction corresponding to the barrier-acting material (in this model the latter has a binding energy equal to R∗y ). Once again this state cannot have a wavefunction like that of Eq. (105). If V0 is infinite the result at L = 0 is qualitatively different. The quantum well only has bound levels whose energy separation increases like L−2 when L decreases. The smaller the well thickness, the better the separable wavefunction becomes. At L = 0, one obtains a true two-dimensional hydrogenic problem whose binding energy is 4R∗y whereas ( = a∗B /2. To circumvent the previous difficulties and obtain the exact limits at L = 0 and L =  for any V0 one may use [69–71]

(110)

One should be aware that the decoupling procedure [Eqs. (105)–(110)] runs into difficulties in the limits L →  and L → 0. In the former case the energy difference L0 +1 − L0 becomes very small and many subbands become admixed by the Coulombic potential. Clearly, for infinite L one cannot describe the bulk 1s hydrogenic bound state by a separable wavefunction. In the latter case L → 0 one finds similar problems due to the energy proximity between the ground quantum well subband 1 and the top of the well V0 . Consequently 91 z leaks more and more heavily in the

(i) Thickness dependence of the impurity binding energy: the dimensionless ratio L/a∗B indicates the amount of two-dimensionality of the impurity state. If L/a∗B ≥ 3 or L/a∗B ≤ 0 2 and V0 ≈ 3 eV in GaAs–Ga1−x Alx As the problem is almost three-dimensional. This is either because the subbands are too close L/a∗B ≥ 3 or because the quantum well continuum is too close L/a∗B ≤ 0 2. The on-center donor binding energy increases from R∗y L →  to reach a maximum L/a∗B ≤ 1 whose exact L location and amplitude depend on Vb . Finally it decreases to the value R∗y at L = 0 [72, 73]. If V0 is infinite, the maximum is only reached at L = 0 and has a value of 4R∗y [69]. (ii) Position dependence of the impurity binding energy: the impurity binding energy monotonically decreases when the impurity location zi moves from the center to the edge of the well and finally deep into the

Impurity States and Atomic Systems Confined in Nanostructures

61

0

5

εb / RB*

1

2 10

εb (meV)

1 3 –L / 2

L/2

2 3 4

4 0

5

10

L / aB*

15

Figure 9. Calculated dependence of the on-edge (crosses) and oncenter (circles) hydrogenic donor binding energy versus the well thickness L in a quantum well with an infinite barrier height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

barrier. In Figure 12, it is shown that this decrease is rather slow for zi > L/2; for instance, a donor placed 150 Å away from a GaAs–Ga0 7 As Al0 3 As quantum well (L = 94 8 Å) still binds a state by ≈0.5R∗y which is ≈2.5 meV [70].

0

60

100

L (Å) Figure 11. Dependence of the on-edge hydrogenic donor binding energy in GaAs–Ga1−x Alx As quantum wells versus the GaAs slab thickness L. m∗e = 0 067m0 .  = 13 1. V0 = 212 meV (curve 1); 318 meV (curve 2); 424 meV (curve 3); and infinite (curve 4). Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Bulk GaAs Donor ground state

-6

-8

2

x = 0.1

εb/R*y

ε (meV)

x = 0.2 -10

-12

x = 0.3

1

x = 0.4

V0 = 318 meV

-14

V0 = ¥ 0

-16 1

20

40

60

80

100

120

Number of GaAs monolayers Figure 10. Dependence of on-center hydrogenic donor binding energy in GaAs–Ga1−x Alx As quantum wells versus the GaAs slab thickness L. Vb x = 0 85 [g Ga1−x Alx As − g GaAs]. One monolayer is 2.83 Å thick. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

0

L/2

200

100

300

-zi (Å) Figure 12. Dependence of the hydrogenic donor binding energy in a quantum well versus the impurity position zi (a) in the case of a finite barrier well (V0 = 318 meV) and (b) in the case of an infinite barrier well [69]. There is an interface at −zi = L/2. L = a∗B = 94 8 Å. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

62 3.1.3. Excited Subbands: Continuum The procedure followed for the bound state attached to the 1 subband can be generalized for excited subbands 2  3  as well as for the quantum well continuum. However, it becomes rapidly cumbersome since a correct variational procedure for excited states requires the trial wavefunctions to be orthogonal to all the states of lower energies. For separable wavefunctions [Eq. (105)] this requirement is automatically fulfilled and one may safely minimize L0 zi  ( to obtain a lower bound of bL0 zi . The new feature associated with the bound states attached to excited subbands is their finite lifetime. This is due to their degeneracy with the two-dimensional continua of the lower lying subbands. This effect, however, is not very large [67, 68] since it arises from the intersubband coupling, which is induced by the Coulombic potential: if the decoupling procedure is valid the lifetime of the quasi-bound state calculated with Eq. (105) should be long. For impurities located in the barriers (an important practical topic with regard to the modulation-doping technique), we have seen that they weakly bind a state below the 1 edge. There exists (at least) a second quasi-discrete level attached to the barrier edge (i.e., with energy ≈ V0 − R∗y ). This state is reminiscent of the hydrogenic ground state of the bulk barrier. Due to the presence of the quantum well slab it becomes a resonant state since it interferes with the two-dimensional continua of the quantum well subbands LL < V0 . Its lifetime K can be calculated using the expression 2       2
 −e2
   ˜ r  zi   L k⊥  =   2 2 2K
   C + z − zi  L k⊥ 


2 k⊥2 (112) × 0 V0 − R∗y − L − 2m∗e

Impurity States and Atomic Systems Confined in Nanostructures

difference between the on-center donor ground state (quasi 1s) and the excited states (quasi 2px , 2py ) agrees with the far-infrared absorption and magnetoabsorption data [74]. On-edge donor levels have also been investigated.

3.1.5. Acceptor Levels in a Quantum Well The problem of acceptor levels in semiconductor quantum wells is much more intricate than the equivalent donor problem. This is due to the degenerate nature of the valence bands in cubic semiconductors. In quantum wells this degeneracy is lifted (light and heavy holes have different confinement energies). However, the energy separation between the heavy and light hole subbands is seldom comparable to the bulk acceptor binding energies. Thus many subbands are coupled by the combined actions of the Coulombic potential and the quantum well confining barrier potential. No simple decoupling procedures appear manageable. Masselink et al. [75] used variational calculations to estimate the binding energy of the acceptor level due to carbon (a well-known residual impurity in MBE grown GaAs layers). Their results, shown in Figure 13, agree remarkably with the experiments of Miller et al. [76]. One notices in Figure 13 the same trends versus quantum well thickness as displayed by Coulombic donors (Figs. 10 and 11). The binding energy first increases when L decreases (increasing

where ˜ is the 1s bulk hydrogenic wavefunction of the barrier. If the distance d separating the impurity from the quantum well edge is much larger than a∗B , Eq. (112) simplifies to  3/2    R∗y
2d V0 − L
∗ exp − ∗ ≈ 16Ry PL 2K V0 − L aB R∗y L (113) where PL is the total integrated probability of finding the carrier in the Lth state (energy L  in any of the two barriers of the quantum well. One sees from Eq. (113) that the lifetime K is strongly dominated by the escape processes to the excited subband L whose energy is nearest to V0 − R∗y . As an example let us take a GaAs–GaAlAs quantum well with thickness L = 50 Å, V0 = 0 2 eV and assume that d = 3a∗B (i.e., d ≈ 300 Å). We then get K ≈ 3 × 10−6 s. The quasibound state can thus be considered, to a reasonable approximation, as stationary 
/K ≈ 4 × 10−8 R∗y .

3.1.4. Excited Impurity Levels Attached to the Subband The Schrödinger equation [Eq. (102)] has several bound states below 1 ; their binding energies have been calculated by several groups [72, 73]. The calculated energy

Figure 13. Dependence of the on-center carbon binding energy versus well width in GaAs–Ga1−x Alx As quantum wells. Vb x = 0 15 [g Ga1−x Alx As − g GaAs] is the assumed hole confining barrier height. The open circles are the experimental values obtained by Miller et al. [76]. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Impurity States and Atomic Systems Confined in Nanostructures

tendency to two-dimensional behavior), then saturates, and finally drops to the value of the acceptor binding energy of the bulk barrier. Finally, it should be noted that the relative increase in binding is smaller for acceptors than for donors. This is because the bulk acceptor Bohr radius is much smaller than that of the bulk donor.

3.2. Hydrogenic Impurities within Asymmetric and Symmetric Quantum Well Wires In this section the energy of the ground and first excited states, the binding energy, and oscillator strengths for hydrogenic impurities confined within a cylindrical quantum well wire with a finite-height potential well are studied variationally as a function of the wire radius and of the relative position of the nucleus within the quantum well wire for different barrier-height potential [77]. The trial wavefunction is constructed as the product of the free wavefunction of hydrogen impurity and a simple auxiliary function that allows the appropriate boundary conditions to be satisfied. In this context, the variational method here used constitutes a useful approach to study asymmetric and symmetric quantum systems confined by penetrable potentials.

If the nucleus of the hydrogen impurity is located on the symmetry axis of the cylindrical quantum well wire, the model Hamiltonian for the electron within the quantum well wire can be written in atomic units (
= m∗e = e = 1) as

with

Vb C =



0 V0

0 ≤ C ≤ C0

C0 ≤ C < 

(114)

(115)

 where r = C2 + z2 is the electron–nucleus distance, C is the cylindrical coordinate parallel to xy plane, z is the coordinate along the wire axis, C0 is the wire radius, and V0 is the confining potential barrier. The physical meaning of V0 in this context is to simulate, on the average, the effective potential step created by the composition difference between the quantum well wire and its surroundings. It is well known that the Schrödinger equation for this Coulomb-type potential is not separable in cylindrical coordinates, the natural symmetry of the wire; thus we are forced to use an approximate method to calculate the ground state energy and first excited states for this system. In order to use the variational method to solve approximately the Schrödinger equation with the Hamiltonian given by Eq. (114), we must construct a trial wavefunction 3 with the basic properties listed as follows: 30

finite

3r

→ 0 as r → 

1 53r 3r 5C

continua at C = C0

The function 3 will contain a parameter or a set of parameters that allows us to minimize the ground state energy by imposing that 50 =0 5?i

(117)

where I?i J is the set of parameters mentioned earlier. A possible ansatz wavefunction for this problem is of the form  3i = AC0 − ?C exp−?r 0 ≤ C ≤ C0 31s = (118) 3o = B exp−?r exp−@C C0 ≤ C <  where A, B are normalization constants and ?, @ are the variational parameters involved in the calculation. These functions must satisfy 3i C=C0 = 3o C=C0   53o  53i  = 5C C=C0 5C C=C0

(119) (120)

By imposing condition (120) to the function given by Eq. (118) we have that

3.2.1. On-Axis Hydrogenic Impurity

 = − 1 6 2 − 1 + Vb C H 2 r

63

(116)

@=

? 1 − ?C0

(121)

That is, we need to find only one variational parameter to minimize the ground state energy. Furthermore, when V0 → , ? → 1, @ → , and 3o → 0, as expected. The continuity condition on the boundary, Eq. (119), relates B and A: 

? B = AC0 1 − ? exp (122) 1−? The suitable Hamiltonian for 2p-type excited states is the same that given by Eq. (114), and the variational wavefunctions for 2px , 2py , and 2pz states, when the nucleus of hydrogenic impurity is fixed on the symmetry axis of the wire, are of the form  3i = CC0 − PC      × exp−PrC cos  0 ≤ C ≤ C0 32px = (123)  3o = D exp−Pr     × exp−QCC cos  C0 ≤ C <   3i = CC0 − PC      × exp−PrC sin  0 ≤ C ≤ C0 32py = (124)  3o = D exp−Pr     × exp−QCC sin  C0 ≤ C <   3i = CC0 − PC      × exp−Prz 0 ≤ C ≤ C0 32pz = (125)  3o = D exp−Pr     × exp−QCz C0 ≤ C < 

Impurity States and Atomic Systems Confined in Nanostructures

64 where C and D are normalization constants and P, Q are variational parameters to be determined and are related by Q=

P 1 − PC0

(a)

(126)

and

P D = CC0 1 − P exp 1−P



(127)

The results for the energy of the ground and 2pz states as a function of wire radius and different barrier heights are displayed in Figure 14. For a given value of the finite barrier potential the ground state (excited state) energy increases from −0.5 (−0.125) Hartrees as the wire radius is reduced. These values are characteristic of the “free” hydrogen atom. The binding energy b for the hydrogenic impurity is defined as the ground state energy of the system without Coulomb interaction w , minus the ground state energy in the presence of electron–nucleus interaction 0 ; that is,

(b)

(128)

b = w − 0

The binding energy defined in this way is a positive quantity. In Figure 15a, we display the variation of the hydrogenic impurity binding energy b as a function of wire radius C0 for several values of finite potential barrier.

3.2.2. Off-Axis Hydrogenic Impurity If the nucleus of the atom is located on the x axis, at a distance b from the axis of the wire, the electron–nucleus distance is  r ′ = r − b eˆx  = x − b2 + y 2 + z2  = C2 + b 2 + z2 − 2Cb cos  (129) 20 V0 = 50.0 (hartree) 10.0 2.0 1.0

1.5

Energy

with r the position vector of the electron relative to the origin on the wire axis. Then the suitable Hamiltonian is

2pz states

 = − 1 6 2 − 1 + Vb C H 2 r′

1s states

1.0

(130)

where Vb C is the same as that given by Eq. (115). The trial wavefunction for the ground state energy can then be written as  3i = AC0 − ?C exp−?r ′  0 ≤ C ≤ C0 31s = (131) ′ 3o = B exp−?r  exp−@C C0 ≤ C < 

0.5

0.0

-0.5

-1.0

Figure 15. (a) Binding energy of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of the wire radius and potential barrier height. (b) Binding energy of the hydrogenic impurity confined in a penetrable GaAs–Ga1−x Alx As quantum well wire with the nucleus on the axis as a function of the wire radius and potential barrier height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

0

1

2

3

4

ρ0 Figure 14. Energy of the ground and 2pz states of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of wire radius and potential barrier heights. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

where, as before, A and B are normalization constants and ?, @ are the variational parameters involved in the calculation. Again, ?, @ and A, B are, respectively, related by Eqs. (121) and (122). When the nucleus of hydrogenic impurity is off the symmetry axis, it is not possible to find the wavefunctions, for the 2px and 2py states, that satisfy the orthonormality condition as is required by the method. Therefore we will restrict to the 2pz state only. The suitable Hamiltonian is the same

Impurity States and Atomic Systems Confined in Nanostructures

that given by Eq. (116), and the variational wavefunction is given by  3i =CC0 −PCexp−Pr ′ z 0 ≤ C ≤ C0 (132) 32pz = 3o =Dexp−Pr ′ exp−QCz C0 ≤ C < 

65 (a)

where, as before, C, D are normalization constants and P, Q are the variational parameters involved in the calculation. Again, P, Q and C, D are, respectively, related by Eqs. (126) and (127). In Figure 16, we show the energy of the ground and 2pz states as a function of b/C0  for C0 = 1 0 Bohr and several heights of potential barrier. In Figure 17a, we display the variation of the hydrogenic impurity binding energy b as a function of b/C0  for C0 = 1 0 Bohr and several heights of potential barrier. (b)

3.2.3. Optical Properties To predict the absorption peak due to 1s–2p transitions as a function of the wire radius and confining degree, we have calculated the transition energy between these states as well as their oscillator strengths. The f0 oscillator strength is defined as f0 =

2 1s r · er 2p2 ' 3

(133)

where er is the incident light polarization vector and ' = 2p − 1s is the transition energy between 2p and 1s states. The 1s–2pz oscillator strength f0 (for er  z is shown in Figure 18 as a function of wire radius for several heights of potential barrier, when the nucleus is fixed on the center of the wire. For a given value of the finite barrier potential the oscillator strength increases from 0.139, the characteristic value of the “free” hydrogen atom, as the wire radius is reduced. Finally, in Figure 19, we show the oscillator strength f0 (for er  z) as a function of relative nucleus position b/C0 

Figure 16. Energy of the ground and 2pz states of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radius C0 = 1 0 Bohr as a function of the relative nucleus position and potential barrier height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 17. (a) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radius C0 = 1 0 Bohr as a function of the relative nucleus position and potential barrier height. (b) Binding energy of an off-axis hydrogenic impurity enclosed within a penetrable GaAs–Ga1−x Alx As quantum well wire of radius C0 = 1 0 Bohr as a function of the relative nucleus position and potential barrier height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

for C0 = 2 0, 3.0, and 4.0 Bohrs and several potential barrier heights for the off-axis case. In Figure 14, we show the ground and first excited states energies of the hydrogenic impurity confined within cylindrical quantum dot with the nucleus on the symmetry axis, as a function of C0 for several V0 values. The ground state energy has a similar behavior as in [65], as well as for the hydrogen atom and for electron systems (like H− , He, Li+ , and Be2+ ) within a penetrable spherical box [78]; that is, the energy diminishes as the confinement box size increases (for a given value of V0 ), and for a given size of the box the energy increases as V0 increases. Similar behavior is found by Nag and Gangopadhayay [79] who show graphically qualitative results for heavy holes and electrons within a quantum well wire with cylindrical and elliptic cross-section. They have used the physical constants of the Ga0 47 In0 53 As/InP system.

Impurity States and Atomic Systems Confined in Nanostructures

66 0.6 V0 = ∞ 0.20 0.15 0.10 0.05

Oscilator Strength

0.5

0.4

(hartree)

0.3

0.2

0.1

0.0 0

1

2

3

4

5

6

7

8

9

10

ρ0 (Bohr) Figure 18. 1s–2pz oscillator strength of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of the wire radius and potential height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

This energy behavior can be understood on the basis of the uncertainty principle because when confinement dimensions are reduced the energy increases. If dimensions continue decreasing there will be a point at which the kinetic energy will be greater than the potential energy associated with internal interactions of the system. In an extreme situation of confinement the potential energy is only a perturbation of total energy for a free particle system. For attractive potentials, like Coulomb’s potential, there is a very clear competition between kinetic energy and potential energy. If C0 decreases, then potential energy decreases, but kinetic energy always increases because of the localization of wavefunction. 0.6

Oscilator Strength

ρ0 = 2.0 (Bohr) = 3.0 = 4.0

V0 = 0.2 (Hartree) 0.1

0.5

0.4

0.3

0.2

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b/ρ0 Figure 19. 1s–2pz oscillator strength of an off-axis hydrogenic impurity enclosed within a penetrable quantum well wire of radii C0 = 2 0, C0 = 3 0, and C0 = 4 0 Bohr as a function of the relative nucleus position and potential height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

In Figure 15a, we display the binding energy of a hydrogenic impurity in a quantum well wire with finite potential barrier, as a function of the width of the wire. The results show that, due to the finite confining capacity of the quantum well wire, there is a critical radius for which the electron is no longer confined. Indeed, as the confining barrier increases this critical radius become smaller, as expected from simple physical considerations. For a given value of the finite barrier potential the binding energy increases from its bulk value as the wire radius is reduced, reaches a maximum value, and then drops to the bulk value characteristic of the barrier material as the wire radius goes to zero. This is due to the fact that as the wire radius is decreased the electron wavefunction is compressed thus leading to an enhancement of the binding energy. However, below a certain value of C0 the leakage of the wavefunction into the barrier region becomes more important, and the binding energy begins to decrease until it reaches a value that is characteristic of the barrier material as C0 → 0. This effect has been studied for heavy and light excitons confined within quantum wires [80], for hydrogen impurities within quantum well wire of GaAs1−x Alx As [81, 82], and excitons within quantum well wire in the presence of a magnetic field [83]. In Figure 15b, we show the same results for the quantum well wire of GaAs1−x Alx As for the barrier potentials V0 = 265 0, 53.0, 10.6, 5.3 meV that correspond to the x = 0 36, 0.08, 0.016, 0.008, Al concentrations, respectively. The quantitative comparison of the curves in Figure 15b with the results of [80, 82, 83] show that our variational calculations lead to the same results. The peak in the binding energy occurs for the smallest value of C0 for which the probability of electrons to be found outside the well is not significant; that is, the enhancement in binding energy occurs because the confining potential is forcing the electrons to move only in a smaller space and to spend most of their time closer the nucleus. This strong enhancement of the binding energy has important consequences for optical and transport properties of quantum well wires. In Figure 16, we show the ground and first excited state energies for the asymmetric case. The ground state energy approaches the value calculated in [65] for the infinite potential barrier case, as the potential barrier is increased, and when the size of the quantum well wire becomes infinite and the nuclei is close to the boundary. For a given value of the finite barrier potential the energy of the 2pz excited state is almost independent of the relative position of the nuclei within the quantum well wire, as compared with the variation of the ground state energy. This is due to the orientation of the orbital in which the electron moves; a similar effect has been studied for a hydrogen atom enclosed between two impenetrable parallel planes and for a heavy exciton in a CdS film in [84]. Also, we can note that for C0 = 1 Bohr, the ground state energies calculated for V0 = 1, 2, 10, 50 Hartrees are in exact agreement with the ground state energies when the atom nucleus is on the symmetry axis as we can see in Figure 14. The same is true for the first excited state energies (compare Fig. 16 with Fig. 14). In Figure 17a, we can note that the value of the binding energy for the asymmetric case decreases as the nucleus

Impurity States and Atomic Systems Confined in Nanostructures

approaching the quantum well wire is increased. A similar behavior occurs for the binding and the ground state energy of a hydrogenic impurity placed in a rectangular cross-section quantum well wire of GaAs–Gax Al1−x As [81]. In Figure 17b, we show the same results as in Figure 17a, for the case of a hydrogenic impurity within a cylindrical quantum well wire of GaAs–Gax Al1−x As with C0 = 103 4 Å. The calculations were carried out for barrier potentials V0 = 265 0, 53.0, 10.6, 5.3 meV that correspond to aluminum concentrations x = 0 36, 0.08, 0.016, 0.008. The quantitative comparison of Figure 16b with the results of [81] shows the effect of the geometry on the binding energy. The binding energy for C0 = 1 Bohr is in agreement with the binding energy when the nucleus is on the axis of the wire (compare Fig. 17a with Fig. 15a). In Figure 18, the 1s–2pz transitions of the studied states in the symmetric case are shown. We note a similar behavior to that found in other systems which are confined in regions with different symmetries, for example, the heavy exciton case confined in a KCl ionic sphere for several radii and penetrabilities, in which the excitonic transitions vanish for given sphere sizes, as a consequence of the finite confining potential value, and moreover in that case the absorption peak is shifted to high energies as the sphere radius decreases. This effect has been experimentally observed in SiO2 spheres [84]. In addition, for V0 = , there is a critical radius of quantum well wire for which the absorption has a maximum and then returns to bulk value as the radius continues decreasing. In Figure 19, we show the oscillator strength f0 (for el  z) as a function of relative nucleus position b/C0  for C0 = 2 0, 3.0, and 4.0 Bohrs, with the following potential barrier values: V0 = 0 1 and 0 2 Hartrees for the off-axis case. The maximum in the absorption peak allows us to predict the optimum site for the location of the hydrogenic impurity within the quantum well wire. We can note that, for the same parameter values as those mentioned earlier, the transition intensities are in agreement with the transition intensities of the studied states in the symmetric case (compare Fig. 19 with Fig. 18).

67 either off the center of the sphere or off the axis of the cylinder). Of course, in such cases the corresponding Schrödinger equation is no longer separable.

3.3.1. Applications of the Method In this section, two explicit examples using the variational method application will be described. Both involve the hydrogen atom confined within a domain with impenetrable boundaries. In the first example, we shall consider the atom within a spherical surface where the center of surface and the nucleus of the atom differ by a constant distance a. Referring to Figure 20, the coordinates of the electron of the hydrogen atom with respect to the nucleus (r′ ) and with respect to the center of the confining sphere ( r) are related by (134)

r′ = r − a

The corresponding Hamiltonian can now be written as 1  = −162 − + Vb r H 2  r − a

(135)

where Vb is a confining potential defined as Vb C =



+

r > r0

0

r < r0

(136)

The ground state wavefunction for the free hydrogen atom in spherical coordinates is given as 30 = A exp−r ′ 

(137)

where r ′ is the electron–nucleus distance and A is a normalization constant. Z

3.3. Asymmetric Confinement of Hydrogen by Hard Spherical and Cylindrical Surfaces The variational method is used to calculate the ground state energy of the hydrogen atom confined within hard spherical and cylindrical surfaces, for an atomic nucleus, which is off the center of symmetry of the confining boundary. It is shown that the wavefunction for the free hydrogen atom in spherical coordinates (referred to the center of symmetry of the confining surface) can be used, without further assumptions, to construct the trial wavefunction systematically. The latter is assumed to be the product of the free wavefunction and a simple cutoff function that satisfies the appropriate boundary conditions. In order to show the advantage of the method we present in detail two cases: the hydrogen atom confined within hard spherical and within cylindrical surfaces. The asymmetry of these systems is due to the nucleus of the atom being shifted off the center of symmetry of the confining surface (i.e.,

r′

N+

e-

a r

r0 0

Figure 20. Coordinates for the off-center hydrogen atom relative to the center of the confining sphere.

Impurity States and Atomic Systems Confined in Nanostructures

68 Without loss of generality, we can assume that the nucleus is located on the z axis. The trial wavefunction for the ground state can be written as   = Ar − r0  exp−? r − a 9r

(138)

where r0 is the radius of the spherical confining domain, A is a normalization constant, and  = .r 2 + a2 − 2ar cos /1/2 r − a

(139)

Here,  is the usual polar angle of the spherical coordinates. The results of the variational calculations of the ground state energy with this trial wavefunction are displayed in Table 1. A comparison with the results obtained in [85, 86] is also shown. Figure 21 shows the ground state energy for different sizes of the confining sphere as a function of a/r0 . In the case of cylindrical coordinates, it is obvious that the Schrödinger equation for the Coulomb potential is nonseparable; however, the application of the variational method is still possible in the same context as could be done for spherical coordinates, as we shall see. If the nucleus of the atom is located at the x-axis, at a distance b (as depicted in Fig. 22), the electron–nucleus distance r ′ and the position of the electron relative to the origin on the axis of the cylinder r are related by  r ′ = r′  = r − b

(140)

That is, r ′ = .x − b2 + y 2 + z2 /1/2 = .C2 + b 2 − 2Cb cos  + z2 /1/2 (141) The trial wavefunction can be written as  r − b 9r = BC − C0  exp−?

(142)

 is given where C0 is the radius of the cylinder and r − b by Eq. (141). The ground state energies resulting from the Table 1. Ground state energy for the off-center hydrogen atom within an impenetrable spherical box as a function of the position of the nucleus (relative to the center of the confining sphere) for various radii of the box. r0

a/r0

a

b

c

4 0

0 1 0 5 1 0

3 0

0 1 0 5 1 0

−0 4804 −0 4673 −0 4390

−0 483 18 −0 481 05 −0 473 35

−0 483 15 −0 481 02 −0 473 42

2 0

0 1 0 5 1 0

−0 1198 −0 0235 +0 2022

−0 122 86 −0 066 07 +0 158 52

−0 122 86 −0 068 89 +0 127 51

−0 4209 −0 3910 −0 3223

−0 423 58 −0 413 73 −0 377 19

Figure 21. Ground state energy for an off-center hydrogen atom enclosed within an impenetrable spherical box as a function of the position of the nucleus (relative to center of the sphere), for various radii of the confining box. Note that when a/r0 ≫ 1 and r0 ≫ 1, the energy approaches −1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

variational calculations, using this trial wavefunction, are displayed in Table 2. A comparison with the results from [87] is also shown. In Figure 23, we show the ground state energy as a function of b/C0  for different values of C0 . The trial wavefunction is accomplished by referring the electron–nucleus distance for the free atom to the origin placed at the center of symmetry of the confining surface, without further assumptions. The results obtained by applying the direct variational method to compute the ground state energy of the hydrogen atom enclosed within spherical or cylindrical surfaces show good agreement with more elaborate calculations as can be seen from Tables 1 and 2. Furthermore, the trial wavefunctions are flexible enough to describe the case when the size of the confining surface becomes infinite and the nucleus is close to the boundary. The latter corresponds to the case when the hydrogen atom is close to a plane.

−0 423 58 −0 413 92 −0 378 40

Note: Energy units: Hartrees, distance units: Bohrs. a Results of this subsection. b Results of Gorecki and Byers Brown [85] obtained by boundary perturbation theory. c Results of Brownstein [86] obtained by a variational method in which the trial wavefunction does not satisfy the boundary conditions.

Figure 22. Coordinates for the off-axis hydrogen atom relative to the axis of the confining cylinder.

Impurity States and Atomic Systems Confined in Nanostructures Table 2. Ground state energy for the off-axis hydrogen atom within an impenetrable cylindrical box as a function of the position of the nucleus (relative to the axis of the confining cylinder), for various radii of the box. C0

b/C0

a

1 0

0 0 0 2 0 4 0 6 0 8 0 99

1 5854 1 6544 1 8111 2 0243 2 2305 2 3724

1 385 68 1 553 68 1 794 95 2 036 988 2 203 82

0 0 0 2 0 4 0 6 0 8 0 99

−0 1791 −0 1363 −0 0401 0 1050 0 2704 0 3785

−0 247 82 −0 159 586 0 0059 04 0 2085 54 0 3369 83

0 0 0 2 0 4 0 6 0 8 0 99

−0 4677 −0 4477 −0 4044 −0 3291 −0 1671 −0 0388

−0 487 −0 467 −0 398 −0 199 −0 038

374 144 212 550 415

−0 499 −0 499 −0 498 −0 456 −0 104

997 943 329 063 528

2 0

4 0

10 0

0 0 0 2 0 4 0 6 0 8 0 99

b

−0 4912 −0 4921 −0 4859 −0 4696 −0 3988 −0 1183

Note: Energy units: Hartrees, distance units: Bohrs. a Results of this subsection. b Results of Tsonchev and Goodfriend [87] obtained by expanding the wavefunctions in a basis of functions which depend on the polar angle .

3

Energy (Hartrees)

2

ρ0 (Bohrs) 1.0 2.0 4.0 10.0 20.0

1

0

-1 0.0

0.2

0.4

0.6

0.8

1.0

b / ρ0 Figure 23. Ground state energy of an off-axis hydrogen atom enclosed within an impenetrable cylindrical box as a function of the position of the nucleus (relative to axis of the cylinder), for various radii of the confining box. Note that when b/C0 ≫ 1 and C0 ≫ 1, the energy approaches −1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

69 3.4. Confined Electron and Hydrogenic Donor States in a Spherical Quantum Dot According to hydrogenic effective mass theory, exact solutions and quantum level structures are presented for confined electron and hydrogenic donor states in a spherical quantum dot of GaAs–Ga1−x Alx As [88]. Calculated results reveal that the values of the quantum levels of a confined electron in a quantum dot can be quite different for cases with finite and infinite barrier heights. The quantum level sequence and degeneracy for an electron in a quantum dot are similar to those of a superatom of GaAs– Ga1−x Alx As but different from those in a Coulombic field. There is stronger confinement and larger binding energy for a hydrogenic donor in a spherical quantum dot of GaAs– Ga1−x Alx As than in the corresponding quantum well wires and two-dimensional quantum well structures. The binding energy and its maximum of the ground state of a donor at the center of a quantum well are found to be strongly dependent on the well dimensionality and barrier height. Because the transverse and longitudinal variables do not separate, the impurity states in two-dimensional quantum wells and quantum well wires cannot be solved exactly. Therefore approximation methods should be used. A reasonable trial function is needed to obtain a correct variational state of an impurity in two-dimensional and onedimensional confined systems, and calculated results are more accurate if the coupling effect between the impurity and well potentials is considered using a trial function which has (or a part of which has) correctly both donor and well potential effects [89]. However, for a hydrogenic donor at the center of spherical quantum dots the exact solutions [90] can be obtained. It is interesting not only from a physical point of view but also from a mathematical point of view to compare the solutions and binding energies with those of two-dimensional and one-dimensional systems. In this section the exact solutions and quantum level structures for confined electron and hydrogenic donor states in spherical quantum dots are reported. The dependence of the quantum levels and the binding energies on the dimensionality of quantum wells is also presented. The calculation is based on the effective mass approximation. It has been known to give excellent results for the electronic structure of GaAs–Ga1−x Alx As two-dimensional quantum wells and (AlAs)n /(GaAs)n superlattices if the well width or n is sufficiently large. The limit is estimated to be about 30 Å (n ≈ 10) [91]. Therefore it should also be valid for the GaAs–Ga1−x Alx As quantum dots, as the size (diameter for a ball) is sufficiently large. Based on the facts mentioned earlier, the limit for a ball diameter is also estimated to be the same value and equal to 30 Å. Here we treat the cases where the diameter is larger than the critical size. It is interesting to point out that the maximum quantum confinement of an electron in the GaAs–Ga1−x Alx As quantum ball is already obtained before the diameter approaches the critical value. In addition, polarization and image charge effects can be significant if there is a large dielectric discontinuity between the quantum ball and the surrounding medium [92]. However, this is not the case for the GaAs–Ga1−x Alx As quantum system; therefore we ignore such effects.

Impurity States and Atomic Systems Confined in Nanostructures

70 According to hydrogenic effective mass theory, the electron bound states and their binding energies have been found in two-dimensional quantum wells and quantum well wires. Normally, the effective mass equation is reliable for weakly bound states, and one might worry that the effective mass equation is inappropriate when the binding energy is greatly enhanced in spherical quantum dots of GaAs– Ga1−x Alx As. However, the bandgap of GaAs is 1.4 eV, while R∗y = 5 3 meV. Thus roughly a 100-fold enhancement of the binding energy is necessary before the effective mass equation becomes inapplicable. This difference is much greater than the enhancement seen in the cases considered here, so that the theory is still reliable for the bound states in spherical quantum dots of GaAs–Ga1−x Alx As. Let us for definiteness consider a hydrogenic donor at the center of the quantum dot of radius r0 . The potential due to the discontinuity of the band edges at the GaAs–Ga1−x Alx As interface r = r0 is  V0 r ≥ r0 Vb r = (143) 0 r < r0 where r is the electron–donor distance. The barrier height V0 is obtained from a fixed ratio of the bandgap discontinuity. According to hydrogenic mass theory, the Hamiltonian for the donor is  = −6 2 − 2w + Vb r H r

(144)

It is written in a dimensionless form so that all energies are measured in units of the effective Rydberg R∗y and all distances are measured in units of effective Bohr radius a∗B . w is equal to 1. In order to solve the Schrödinger-like equation  H3r   = 3r  

(145)

3lm r   = 3 l rYlm  

(146)

the wavefunctions of an electron with well-defined values of the orbital l and magnetic m quantum numbers in a spherical symmetric potential, which is the quantum well and Coulomb potential, are written in the form

Substituting Eq. (146) in Eq. (145), we find an equation for the radial function: r2

d 2 3 l r d3 l r  + 2r + .l − Vb r/r 2 2 dr dr  − ll + 1 + 2wr 3 l r = 0

(147)

Using the method of series expansion, we can solve Eq. (147) exactly. It should be noted that zero and infinity are a regular and an irregular singular point of Eq. (147), respectively. In the region 0 < r, we have a series solution, which has a finite value at r = 0 as 3 l r = Ar


l

n=0

n al n r

(148)

where A is a constant, 1 l+1

(149)

"# ! l l nn+2l +1 al n = − 2wan−1 +lan−2

n = 234 (150)

l

a0 = 1

l

a1 = −

and

In the region r0 < r, we can obtain a normal solution [93]. It approaches zero at r =  and is found in the form 3 l r = B exp−Kl rr Cl where Kl = .V0 − l/1/2

N


bnl r −n

(151)

n=0

Cl = −1 +

w Kl

(152)

and l

b0 = 1

l

bn+1 = −Cl −n−1Cl −n+l +1bnl n = 012

/2Kl n+1

(153)

B is a constant. The series appears suitable for numerical computation for large r [93]. However, they are not suitable for r0 if it is small. In order to get exact value at small r0 , we find a solution of uniformly convergent Taylor series in the region r0 < r ≤ Rp , where Rp is a proper point (e.g., Rp ≥ 2a∗B ) for using Eq. (151). For the sake of using the matching conditions at r = Rp to obtain the eigenenergy equation, it is written as 3 l r = C




n=0

cn r − Rp n + D




n=0

dn r − Rp n

(154)

where C and D are constants, c0 and d1 are equal to 1, and c1 and d0 are equal to 0, respectively. Noting that cn and dn are equal to 0 for negative n, the other cn can be determined by the recurrence relation  $ cn = −2Rp n−12 cn−1 + −n−2n−1+ll +1 %  −2wRp +Kl2 R2p cn−2 +2Kl2 Rp −wcn−3 +Kl2 cn−4 % &$ 2 (155) Rp nn−1

and the dn ’s obey a similar recurrence relation. Using the matching conditions at the interface r = r0 and Rp , we can obtain the equation of the eigenenergies l as follows:    W11 0 W13 W14     0 W23 W24   W21 =0  (156)  0 W32 1 0      0 W42 0 1  That is,

W21 W42 W14 + W32 W13  − W11 W42 W24 + W32 W23  = 0 (157)

Impurity States and Atomic Systems Confined in Nanostructures

where W11 = W14 = W21 = W32 = W24 = W42 =

71 3.4.1. Quantum Levels and Binding Energies




n al n r0

n=0 


n=1 


n=0 N


W13 =




n=0

cn r0 − Rp n

dn r0 − Rp n n−1 l + nal n r0

W23 =




n=0

ncn r0 − Rp n−1 (158)

bnl R−n p

n=0 


n=1 N


n=0

ndn r0 − Rp n−1 −n + Cl − Kl Rp bnl Rp−n−1

can be solved numerically. Once the nth eigenenergy n l l is known, the A, B, C, and D [hence 3n r] are known with l l the use of the normalized condition of 3n r. This 3n r depends on the value of l, the quantum well, Coulomb potential, and energy n l. We should point that we have neglected the difference of the electron effective mass between GaAs and Ga1−x Alx As in the Hamiltonian and the matching conditions. If the effective mass difference is considered, similar formulas can be obtained. If there is no Coulomb potential in the Hamiltonian of Eq. (144) (i.e., w = 0), using the same formulas, we l can obtain wavefunction 3n r w = 0 and quantum levels n l w = 0 of an electron in the quantum well. In fact, Eqs. (148) and (151) become the spherical Bessel function and Hankel function if w = 0. The equation of eigenenergies l w = 0 is k0 + K0 tank0 r0  = 0

if l = 0

ikl hl iKl r0 jl−1 kl r0  + Kl hl−1 iKl r0 jl kl r0  = 0 if l ≥ 1

(159) (160)

and kl = .l w = 0/1/2

Kl = .V0 − l w = 0/1/2

A numerical calculation for GaAs–Ga1−x Alx As spherical quantum dots of the r0 between 0 15a∗B and 7 0a∗B with different V0 has been performed. In Table 3, the quantum levels of an electron in a spherical quantum dot with different r0 and V0 have been shown. The levels n l are indicated by two symbols n and l as shown in Eq. (162). n is equal to the number of the root of Eqs. (158) or (159) and (160) in order of increasing magnitude (i.e., n = 1 2 3 and hence n − 1 is the radial quantum number as usual). l is the usual notation (i.e., s p d ). Thus we have 1p, 1d, 2s, 1f levels (states) and so on if the n and l are used as the level notation, and we have 1s, 2p, 3d, 2s, 4f levels, and so on, if the principal quantum number, which is equal to n + l, and l are used as the notation. It is interesting to point out that when V0 approaches infinity  2 (163) n l = Tln /r0

where Tln is the nth root of the lth-order spherical Bessel function. In Table 3, it is shown that the different values of n l are obtained as r0 is equal to 1a∗B and 2 5a∗B , respectively. It is also shown that the values of quantum levels are different between infinite and finite barrier heights. The differences increase as the r0 and finite V0 decrease. There are an infinite number and a finite number of bound states for a spherical quantum dot with infinite and finite barrier heights, respectively. There is no bound state if r0 < Rc = 0 5/V0 1/2 [94]. However, the order of n l is the same for both infinite and finite barriers [i.e., the unique level sequence 1s, 1p2p, 1s3d, 2s, 1f 4f , and so on]. We should note that the level order is different between both cases of a spherical quantum dot and Coulomb field, in which the level order of an electron is 1s, 2s, 2p, 3s, 3p, 3d, and so on if the principal and orbital quantum numbers are used as the level notation. It is because of the lack of the deep attractive region in the vicinity of the center of a spherical quantum dot. For the motion of an electron in a Coulomb field, the quantum levels are only dependent on the principal quantum number np and degenerate with respect to both l (orbital quantum number) and m (magnetic quantum number). The total degree of degeneracy of a quantum level with np is equal to n2p (excluding spin degeneracy). For an electron in a spherical quantum dot, however, Table 3. Quantum levels of an electron in a SQD of GaAs–Ga1−x Alx As.

(161) l

where jl and hl are the lth-order spherical Bessel function and Hankel functions of the first kind, respectively. Then, the same results are obtained if the wavefunctions and quantum levels are calculated with use of the Bessel and Hankel functions. Once n l w = 1 and n l w = 0 are obtained, the binding energy of the corresponding donor states in the spherical quantum dot is given by nb l = n l w = 0 − n l w = 1

(162)

r0 = 1

2.5

nl

1s (1s)

1p (2p)

1d (3d)

2s (2s)

1f (4f )

2p (3p)

V0 =  80 60 40

9 872 7 957 7 702 7 292

20 187 16 225 15 679 14 786

33 212 26 593 25 642 24 045

39 476 31 425 30 191 28,004

48 832 38 919 37 420 34 777

59 676 47 016 44 789

 80 60 40

1 580 1 446 1 427 1 396

3 230 2 958 2 919 2 854

5 314 4 866 4 800 4 692

6 316 5 781 5 702 5 572

7 814 7 151 7 054 6 892

9 548 8 735 8 613 8 409

Note: The notation with the principal quantum number is shown in parentheses. Effective atomic units are used.

Figure 24. Ground state energy (10 ) and first excited state (11 ) energy levels of an electron in a spherical quantum dot versus the dot radius r0 . The top and middle dashed curves represent the levels 11 and 10 of the dot of V0 = , respectively. The solid curves a, b, c, and d represent the levels 11 and 10 of the wells of V0 = 80 and 40R∗y , respectively. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

12

(a)

8

a

(b)

c

4

cd ab

10

6 d

3 4

b

8

2

6

0

4 0

0.2

0.4

0.6 * B

r0 (a )

0.8

1.0

ε1B (Ry*)

the quantum levels are dependent on both n and l and only degenerate with respect to the m. The total degree of degeneracy of a quantum level with n and l is equal to 2l + 1 (excluding spin degeneracy). It is worthwhile to point out that the degeneracy can be lifted in the other kinds of quantum dots. In quantum boxes with circle cross-sections, for example, the degeneracy is lifted partly. Now, we can conclude that the quantum level sequence and degeneracy for an electron in a spherical quantum dot are quite different from those in a Coulomb field, and that this distinguishing feature of levels might cause new phenomena in this type of GaAs–Ga1−x Alx As structure. In Figure 24, the ground and first excited energy levels of an electron in a spherical quantum dot as a function of r0 for an infinite barrier height and two finite barrier heights V0 = 40 and 80R∗y were, respectively, plotted. It is shown that the differences of energy levels between different barrier heights increase as r0 is decreased and that the difference of the first excited state energy is larger than that of the ground state energy for a fixed value of r0 . It is also shown that there are no bound states for a spherical quantum dot with a finite V0 if r0 < Rc , as mentioned earlier. In Figure 25, the binding energies of the ground and first excites states of a donor in a spherical quantum dot as a function of the r0 for three barrier heights V0 = 80, 60, and 40R∗y , respectively, were shown. It is readily seen that as r0 decreases both the binding energies increase continuously until their maxima and then decrease fast. The values of the binding energies can be much larger than those of quantum well wire and a twodimensional quantum well as r0 is smaller. It is interesting to point out the ratio 1b 0/1b 1 increases as r0 increases from some small value. 1b 0 and 1b 1 are almost independent of V0 and respectively equal to 1.192 and 0 576R∗y at r0 = 7 0a∗B . However, the ratio 1.192/0.576 is still much less than 4, which is the limit value of a three-dimensional hydrogenic donor as r0 (approaches infinity). In Figures 24 and 25, it is easily seen that as the r0 decreases the binding energies with respect to different states of a donor in a spherical quantum dot increase until

Impurity States and Atomic Systems Confined in Nanostructures

ε1B (Ry*)

72

a b c

2

d

1 0 1

2

3

r0 (a*B )

Figure 25. Binding energy of the ground state (0B ) and the first excited (1B ) states of a donor in a spherical quantum dot versus the dot radius r0 . The curves a, ab, and b represent 0B of the dot of V0 = 80, 60, and 40R∗y , and the curves c, cd, and d represent 1B of V0 = 80, 60, and 40R∗y , respectively. Arrows indicate the relevant vertical scales. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

their maxima and that the increase of the binding energies is always much less than the increase of the energies of the corresponding states of an electron only confined by the spherical quantum dot, although the binding energies can be much larger than those in the corresponding two-dimensional quantum well and quantum well wire. It means that confinement effects [95, 96] are dominant in the range of r0 . Further, it is also true for the higher excited states. Therefore we can know what kind of quantum-level sequence we will have if the motion of an electron is confined by both a spherical quantum well and a Coulomb field with the same center. The level sequence is similar to that of a three-dimensional hydrogenic donor if r0 is much larger and quantum confinement due to the spherical quantum dot is very weak. However, it is similar to that of the electron in the spherical quantum dot if the quantum confinement of the spherical quantum dot is stronger than that of the Coulomb potential. Based on what we have mentioned, we can understand why the quantum level structure of GaAs– Ga1−x Alx As superatoms [93, 97] is similar to that of an electron in a spherical quantum dot and quite different from those of ordinary atoms and why the electronic structure of the superatoms is dominated by no-radial-node states of 1s, 1p2p, 1d3d, and so on. In Figure 26, the maximum binding energies bmax for the hydrogenic donor ground and first excited states in a spherical quantum dot as a function of the barrier height V0 were plotted. It is shown that the enhancement of the maximum is greater in a spherical quantum dot (quasizero-dimensional) than in the corresponding quantum well wire and two-dimensional quantum well as V0 is increased. This is because of the enhancement of the electron confinement in three dimensions in the spherical quantum dot. In Figure 27, it was shown that the maximum binding energy bmax of ground states of a donor at the center of a different kind of quantum well depends on the well dimensionality and barrier height V0 and presents quasi-two-, one-, and zero-dimensional features of the hydrogenic donor, respectively. It is interesting to note that the mean values of

Impurity States and Atomic Systems Confined in Nanostructures

Figure 26. Maximum binding energies bmax for the hydrogenic donor ground (l = 0) and the first excited (l = 1) states in a spherical quantum dot versus the barrier height V0 . Arrows indicate the relevant scales. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

the maxima of the two-dimensional quantum well [89] and spherical quantum dot are very close to (slightly larger than) the maxima for the quantum well wire [82]. We should point out that the maximum binding energies of higher excited states can also be used to present the dimension features. The radial Eq. (147) was solved and the exact solutions of confined electron and donor states in a spherical quantum dot were obtained. The quantum levels and binding

Figure 27. Maximum binding energies bmax of a donor ground state in a quantum well versus the well dimensionality and the barrier height V0 . For the quasi-one-dimensional case, the dashed lines represent the maximum binding energies of quantum well wires and the solid lines represent the mean values of the maxima of the two-dimensional quantum wells and the spherical quantum dots. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

73 energies of a donor in the spherical quantum dot are calculated numerically. The numerical results reveal that the values of the quantum levels of a confined electron in a spherical quantum dot with a finite barrier height are different from those with an infinite barrier height. The differences increase as r0 decreases. However, the quantum level order is the same for both infinite and finite harrier heights. It is also shown that the quantum level sequence and degeneracy for an electron in a spherical quantum dot are similar to those of a superatom and different from those in a Coulomb field. The quantum level structure of a donor in a spherical quantum dot is similar to that of an electron only confined by the spherical quantum dot as the quantum confinement due to the well potential is stronger than that due to the donor potential. It is useful for understanding the shell model [97] in microclusters. On the basis of the calculated results, the crossover from three-dimensional to zero-dimensional behavior of the donor states in a spherical quantum dot is shown when the radius becomes small. The binding energy of a hydrogenic donor state in the well of GaAs–Gax Al1−x As and its maximum are strongly dependent on the well dimensionality and the barrier height and there is a larger confinement and binding energy of a donor state in a spherical quantum dot than in a quantum well wire and quantum well. Using calculated results of quantum well and quantum well wire, it has been shown that the maxima of the binding energies of hydrogenic donors in quantum wells, quantum well wire, and spherical quantum dots of GaAs–Gax Al1−x As can be used to present, respectively, quasi-two-, one-, and zero-dimensional features of the hydrogenic donor states. Further, it has been found that the maximum of the binding energy of donor ground state in a GaAs–Gax Al1−x As quantum well wire is about half of the summation of the maximum binding energies in the corresponding two-dimensional quantum well and spherical quantum dot. It should be pointed out that impurities could be located anywhere in a spherical quantum dot and that the binding energies will decrease and the level ordering will change as the impurity location shifts to the edge or out of the spherical quantum dot. Based on the exact solutions obtained, the quantum levels and binding energies of a donor located out of the center of a spherical quantum dot can be obtained by use of a variation method. The exact solutions are also useful for the calculation of excited states in a spherical quantum dot, which is a kind of quantum dot. It will be interesting to compare the calculated results about quantum levels and binding energies of impurity and exciton states in a spherical quantum dot with those of other kinds of quantum dots.

3.5. Shallow Donors in a Quantum Well Wire: Electric Field and Geometrical Effects In this section the effects of an external electric field on donor binding energies in quantum well wires with cylindrical and square cross-sections are investigated. A system with a GaAs quantum well surrounded by Alx Ga1−x As potential barriers in the x, y plane has been chosen. The electron is thus free to move in the z direction, in the absence of

Impurity States and Atomic Systems Confined in Nanostructures

74 a Coulomb center binding the electron. A realistic finite potential well model is considered [98]. The behavior of b under an electric field is different for quantum well wires of rectangular and cylindrical crosssection. While the direction of the electric field is immaterial for cylindrical wires, it is very important for wires with rectangular cross-section. It is found that the binding energy of the hydrogenic impurities is a rather sensitive function of the geometry of the wire especially under the influence of an electric field. It is also found that the electric field effects on b are extremely sensitive to the impurity position in or outside the wire.

3.5.1. Theory and Calculation It is convenient to use the Cartesian coordinates for wires of rectangular cross-section and cylindrical coordinates for wires of circular cross-section. The Hamiltonian for the wire of rectangular cross-section, lying along the z direction, is 

2
2 5 52  (164) + 2 + Vb x y H0 = − 2m∗e 5x2 5y where

Vb x y =

   V0      0       V

0

Ly y r0 , r0 being the radius of the dot. The  impurity position is denoted by R. The eigenfunction of the Hamiltonian in the absence of the impurity for the ground state (n = 1 and l = 0) and for the infinite potential well is [134]

(251)

i = −Eg

ρ(ε)

2 e2  = pˆ −  + Vb r H ∗ 2me   r − R

 uf r)  are the periodic parts of the Bloch states where ui r), for the initial and final states. Taking the energy origin at the first conduction subband as depicted in Figure 50, we have for the energy of the initial (first valence level) state

ε n=1

Figure 50. Schematic of some possible absorption transitions in an infinite GaAs–(Ga,Al)As quantum dot of radius r0 = 300 Å with a donor impurity band. The density of state, C, from the positional dependent donor binding energy is shown schematically on the left-hand side. The dependence of the binding energy as a function of the donor impurity position is shown on the right-hand side.

Impurity States and Atomic Systems Confined in Nanostructures

91

the photon vector potential. Following the effective mass approximation, the foregoing matrix element may be written as [140]

with

and

1   pu ˆ i r  Pfi = d r u∗f r X X

8

(255) 6

εb /Ry*

e−l i  C er · Pfi Sfi f H

10

(256)

4

2

Sfi =



 i r  d r ff∗ rf

(257) 0

where X is the volume of the unit cell and ff fi  is the envelope function for the final (initial) state. For the case of  (, the donor impurity we have, for Sfi = Sfi R   r0 2 1/2  ( dr sin2 k10 r N R Sfi = r0 0   × d sin  exp.−( r − R/

0

(258)

For an infinite GaAs quantum dot of radius r0 with one impurity inside, the transition probability per unit time for valence to donor transitions is given by   
2  W  R = W0 S 2 Y ' (259) 2m0 a2B fi where aB is the Bohr radius and Y ' is the step function. In this expression we have for ' and W0 ' =
 − Eg + g r0  R

(260)

W0 =

(261)

4m0 2 a C2 er · Pfi 2
3 B

For a homogeneous distribution of impurities and assuming that the quantum dot radius is much larger than the lattice parameter one has for the total transition probability per unit of time WT  =

3  r0  2 dR W  RR r03 0

(262)

For the numerical computing, we used  = 12 58, m∗e = 0 0665m0 , m∗h = 0 30m0 , where g = 1 424 eV. A schematic representation of a GaAs quantum dot doped with a homogeneous distribution of the donor impurities is shown in Figure 50. The edges for optical absorption from the first valence subband to the donor impurity band are represented by
1 , and to the first conduction subband by Eg . The transition
2 corresponds to absorption to an impurity level associated with donors at the edges of the quantum dot. At the right we also show the density of the states for the impurity band. In Figure 51, we display the donor binding energy as a function of the donor position inside the quantum dot for infinite potential well with different radii. The donor binding energy decreases as the donor position increases reaching a

0.0

0.2

0.4

0.6

0.8

1.0

ri /r0 Figure 51. Donor binding energy as a function of the impurity position for infinite GaAs–(Ga,Al)As quantum dots with distinct dot radii. From top to bottom, r0 = 50, 100, 200, 1000 Å. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

minimum as the donor position is equal to the radii of the  for an quantum dot. The absorption probability W  R infinite quantum dot with one single impurity as a function of
 − g is shown in Figure 52. In Figure 52a, we present the absorption probability for an infinite GaAs quantum dot of radius r0 = 3000 Å. We observe that there is a noticeable peak structure associated with a single impurity located at the center of the dot, which is much larger than the structure associated with a single impurity next to the edge of the dot, meaning that we have essentially reached the bulk limit. Our results for an r0 = 1000 Å and r0 = 500 Å quantum dots are shown in Figure 52b; the structure associated with a single impurity located at the center of the dot is smaller than the structure associated with a single impurity at the edge of the dot. In Figure 53, we display the total absorption probability for an infinite GaAs quantum dot with a homogeneous distribution of impurities. For a quantum dot of r0 = 3000 Å the total absorption probability as a function of
 − g is shown in Figure 53a. An absorption edge associated with transitions involving impurities at the center of the well and a peak related with impurities next to edge of the dot are observed. The peak associated with impurities located next to the edge of the dot is much larger than the peak associated with impurities located next to the center of the dot. This behavior is quite different from that found for GaAs quantum wells and quantum well wires of comparable dimensions. This is a consequence of the quantum confinement and the homogeneous distribution of the impurities in the quantum dot. When the radius of the quantum dot decreases we observe that the peak associated with impurities located next to the center diminishes (see Fig. 53b), which may be understood by means of the behavior of the density of the states as a function of the binding energy (see Fig. 3 in [135]). In this figure, it is seen that for small radii the density of the states

Impurity States and Atomic Systems Confined in Nanostructures

92 ε1

1.6

ε2

(a) 5

(a) 1.2

WT(ω) / 10-1

W(ω) / 102

4

3

0.8

2 0.4 1 0.0 -6

0 -6

-5

-4

-3

-2

-4

-3

-2

-1

(ℏω–εg) (meV)

-1 2.5

(ℏω–εg) (meV) ε1 ε1

-5

ε2

(b)

ε2 2.0

4

WT(ω) / 102

(b)

W(ω) / 104

3

1.5

1.0

2 0.5 1

0.0 -6

-4

-2

0

2

(ℏω–εg) (meV) 0 -6

-4

-2

0

2

(ℏω–εg) (meV) Figure 52. Optical absorption spectra (in units of W0 ), as a function of the
 − g , for valence to donor transitions in infinite GaAs– (Ga,Al)As quantum dots of different radii. r0 = 3000 Å (a), r0 = 1000 Å (full curve), and r0 = 500 Å (dashed curve) (b). 1 and ′1 (2 and ′2 ) correspond, respectively, to the onset of transitions from the first valence subband to the lower (upper) edge of the impurity band. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

is a smooth function which exhibits a well-defined peak at the binding energy associated with on-edge donors. When the radius is very large, the structure of the max Van Hovei like singularity is clearly a peak associated with on-center donors, which corresponds to the bulk limit. For quantum dots of r0 > 1000 Å it is observed that the peak related to on-edge donors is still significant, a situation that is very well reflected in our results. On the other hand, a remarkable difference in the total absorption probability is the absence of the peak structure associated with impurities located at wells for quantum dots with radii less than 500 Å with quantum wells and quantum well wires.

Figure 53. Total optical absorption spectra (in units of W0 ), as a function of the
 − g , for valence to donor transitions in infinite GaAs–(Ga,Al)As quantum dots of different radii. r0 = 3000 Å (a) and r0 = 500 Å (b) for a quantum dot with a homogeneous distribution of impurities, where there is no interaction between them. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

4. NONCONVENTIONAL AND ATOMIC IDEALIZED CONFINED SYSTEMS IN NANOSTRUCTURES 4.1. Ground State Energy of the Two-Dimensional Hydrogen Atom Confined within Conical Curves In this section the variational method is used to calculate the ground state energy of the two-dimensional hydrogen atom confined by impenetrable conical curves [141]. The confinement of the atom in a region limited by two intersecting parabolas as well as within a cone was also considered. The results show the transition of this system to a quasi-one-dimensional hydrogen atom when the curves are open. In the case of closed curves, a transition to a quasizero-dimensional hydrogen atom is observed. These limiting

Impurity States and Atomic Systems Confined in Nanostructures

cases, as well as other intermediate situations, are discussed within this section. In particular, the ground state energy of this kind of system is calculated for different confining situations. Moreover, the behavior of the ground state energy is analyzed, as the confining region reaches an extreme value, that is, when the system becomes quasi-one-dimensional (open curves) or quasi-zero-dimensional (closed curves) as well as other intermediate situations. The results show a very interesting behavior of this property depending upon the confinement region as well as the closed or open character of the curves. The latter would represent an interesting research topic concerning the study of other properties of this system such as its polarizability, pressure due to the boundaries, dipole moment, etc.

4.1.1. Schrödinger Equation In atomic units (
= e = m∗e = 1), the Schrödinger equation for the two-dimensional confined hydrogen atom in orthogonal curvilinear coordinates can be written as 



 5 h2 5 5 h2 5 1 1 + + Ven q1  q2  − 2 h1 h2 5q1 h1 5q1 5q2 h1 5q2  (263) + Vb q1  q2  3q1  q2  = 3q1  q2  where q1  q2  is the orthogonal curvilinear coordinate system in the plane and Ihi J is the scale factor, given by h2i =



5x 5qi

2

+



5y 5qi

2

(264)

Ven q1  q2  is the Coulombic interaction Ven q1  q2  = −

1 rq1  q2 

(265)

while rq1  q2  is the electron–nucleus separation, in this set of coordinates. The confining potential that it will be assumed is of the form  + q1  q2   D (266) Vb q1  q2  = 0 q1  q2  ∈ D where D is a given finite (or infinite) confining domain of the plane. The solution of Eq. (263) must satisfy 3q1  q2  = 0

q ∈ 5D

(267)

where 5D is the boundary of D. The model Hamiltonian for the confined quantum system under study can be written, in atomic units, as  = − 1 6 2 + Ven q + Vb q H 2 q

(268)

where Ven q and Vb q are given by Eqs. (265) and (266), and 6q2 is derived from Eq. (263).

93 At this point, the variational method can be implemented to calculate the ground state energy of the system by constructing the functional ? =

  1 9i∗ − 6q2 + Ven q 9i d 2 q 2 D



(269)

and minimizing it with respect to ?, restricted to 

D

9i∗ 9i d 2 q = 1

(270)

as usual. The area element is given as d 2 q = h1 h2 dq1 dq2 . To this extent, the method described previously can be used to study a confined system for which the interactions with the surrounding medium are not considered (i.e., enclosed within a potential barrier of infinite depth). The next subsection is restricted to deal with the case of a typical one-electron confined system in two dimensions, namely, the case of hydrogen atoms confined within a given region of the plane with different geometries.

4.1.2. Variational Wavefunctions for Different Geometries As is well known, the Schrödinger equation for the unconfined two-dimensional hydrogen atom is separable in polar coordinates (r ); its ground state wavefunction can be written as r  = C0 exp−k0 r

(271)

where C0 is a normalization constant, r is the electron–  nucleus distance, k0 = −20 , 0 = −2 Hartrees is its ground state energy, and the nucleus was assumed to be located at the origin. When the atom in the plane is restricted to a given open or closed domain D,  and 0 must change to fit the new conditions accordingly. If the boundary, 5D of D, is impenetrable for the electron, then  = 0 at 5D. The latter means that due to confinement, new quantization rules must be found; that is, the old “good” quantum numbers used to define the energy of the unconfined system are no longer useful to characterize the energy of the now confined system. Moreover, as the region of confinement was assumed arbitrary (in shape and size), the energy of a given state would depend on a continuous parameter associated with the size (or shape) of the domain D (or its boundary 5D). In the case of the confined two-dimensional hydrogen atom, the approximate (variational) ground state energy would involve the use of a trial wavefunction given by 9q1  q2  = Af qi  q0 exp.−?rq1  q2 /

(272)

where q1  q2  is the system of coordinates compatible with the symmetry of the confining boundary, rq1  q2  is the electron–nucleus distance in these coordinates, f qi  q0  is a geometry-adapted auxiliary function such that f = 0 at qi = q0  qi is a coordinate associated with the geometry of the boundary, and ? is a variational parameter.

y

e Nucleus

2a

x

(a) 0.5 a (Bohrs) 0.5 1.0 2.0

0.0

Energy (Hartrees)

The geometry of the confining boundary allows one to choose f in a quite simple way, namely, similar to its contours. To exemplify the latter, if the assumed confining boundary is a circle of radius r0 , then f = r0 − r (i.e., this function maps all circles of radius 0 ≤ r ≤ r0 , that is, the allowed region for the atomic electron). Similarly, if the confining boundary is an ellipse of “size” Q0 (eccentricity = 1/Q0 ), then f = Q0 − Q and the function would maps all ellipses of size consistent with 1 ≤ Q ≤ Q0 , thus generating the allowed space for the atomic electron in the plane. The definition of f can be done for other confining symmetries accordingly. A description on the flexibility of so-constructed trial wavefunctions, to deal with a variety of confining situations, can be found in [65, 66, 78, 111, 121]. In particular, in [78, 111] other physical properties for one- and two-electron confined atoms are calculated to test the goodness of the wavefunctions. The results show good agreement compared with exact or more elaborate approximated methods. Hence, once that coordinate system q1  q2 ) is chosen, all necessary elements for constructing the trial wavefunction are defined and the procedure outlined in the previous section can immediately be used to calculate and minimize the energy functional. Of course, the transformation equations defining the chosen coordinate system are assumed to be known; indeed, it is usually the case and they can be found elsewhere. It is worth mentioning that the binding energy, defined as the absolute value of the difference between the energies of the electron with and without the Coulombic interaction, has only meaning in the case of closed curves. In the case of open curves, the electron energy belongs to the continuum spectrum and the binding energy is meaningless. Moreover, in the former situation, the binding energy would behave as the total energy, that is, as the confining region becomes smaller or either grows without limit, due to the closed and impenetrable property of the confining boundary. Figures 54a, 55a, 56a, 57a, 58a, and 59a display the confining geometries considered in this section, while Figures 54b, 55b, 56b, 57b, 58b, and 59b show the behavior of their corresponding ground state energies as a function of confining parameter, respectively. The ground state energy of the two-dimensional hydrogen atom confined within an ellipse is displayed in Figure 54b as a function of parameter Q0 (which defines the degree of confinement) for a = 0 5, 1.0, and 2.0 Bohrs, respectively. Notice that for a given Q0 different than 1 and , when a decreases, the major and minor semiaxes also decrease (i.e., the region of confinement is smaller and the energy increases). If Q0 → 1, the two-dimensional hydrogen atom becomes a quasi-zero-dimensional system and the ground state energy goes to  (see Fig. 54a). In the case that Q0 → , the atom is allowed to occupy the whole plane for any value of a, thus recovering the case of an unconfined two-dimensional hydrogen atom, as expected. In Figure 55b, the energy for the ground state of the two-dimensional hydrogen atom confined in a region of the plane limited by a hyperbola (see Fig. 55a) is displayed as a function of U0 (which defines the degree of confinement) for a = 0 5, 1.0, and 1.5 Bohrs, respectively. As can be observed, for U0 = 0, the hyperbola becomes degenerate and their two

Impurity States and Atomic Systems Confined in Nanostructures

-0.5 -1.0 -1.5 -2.0 -2.5 1.5

2.0

2.5

3.0

3.5

4.0

ξ0 (b)

Figure 54. (a) Region of elliptic confinement. Limiting case: Q-zerodimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within an ellipse as a function of the parameter Q0 , for three values of the semidistance between foci a = 0 5, 1.0, and 2.0 Bohr, respectively. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press. (a)

y

y

e

e

x

Nucleus

Nucleus

a

2-D

1-D

0≤η≤1

x a

–1≤η≤0

(b)

Energy (Hartrees)

94

η0

Figure 55. (a) Region of hyperbolic confinement. Limiting case: Q-onedimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a hyperbola, as a function of U0 , for three values of the semidistance between foci a = 0 5, 1.0, and 2.0 Bohr, respectively. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Impurity States and Atomic Systems Confined in Nanostructures y

95 y

(a)

e

e x

Nucleus

Nucleus

1-D

x

(a) 0-D

0.5 8

-0.5 6

Energy (Hartrees)

Energy (Hartrees)

0.0

-1.0 -1.5 -2.0 -2.5 0.5

1.0

1.5

2.0

η0 (b) Figure 56. (a) Region of parabolic confinement. Limiting case: Q-onedimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a parabola, as a function of Q0 . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

branches coincide with the y-axis, so that the confinement region becomes a half plane for any value of a different from zero and infinity. When U0 and a → 0, the coalescence of the two branches of the hyperbola (U0 → 0− and U0 → 0+ ) leads to a quasi-two-dimensional hydrogen atom with its nucleus at the y-axis; the ground state energy approaches −2/9 Hartree. When a → , the whole plane becomes available for the atom, thus leading to the unconfined twodimensional hydrogen atom (see Fig. 55a). Another observation is that for U0 → −1 the whole plane is forbidden (x-axis is excluded), thus resulting in the one-dimensional hydrogen atom with zero energy (see Fig. 55b), which physically means that all excited states belong to the continuous spectrum or to the ionized atom. A similar behavior was found in [142, 143] for a three-dimensional hydrogen atom limited by paraboloidal or hyperboloidal surfaces. Figure 56b shows the energy for the ground state of the two-dimensional hydrogen atom confined in a region of the plane limited by a parabola (see Fig. 56a) as a function of Q0 (which defines the degree of confining). Notice that as Q0 →  the parabola opens and the confinement region decreases until the whole plane is available; thus a transition occurs from a confined two-dimensional system to an unconfined two-dimensional system (see Fig. 56a), whose energy is −2 Hartrees. In the same way, if Q0 → 0 the parabola closes

4 2 0 -2 0

1

2

3

4

r0 (Bohrs) Figure 57. (a) Region of circular confinement. Limiting case: Q-zerodimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a circle, as a function of its radius r0 . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

and the forbidden region increases, confining the system in the −y-axis, thus realizing the transition from a confined two-dimensional system to a confined quasi-one-dimensional system, whose energy approaches zero (see Fig. 56b). In Figure 57b, the ground state energy of the twodimensional hydrogen atom confined in a circular region of the plane is depicted as a function of r0 (radius of the circle in atomic units). As can be observed, for r0 → , the allowed space approaches the whole plane (i.e., the system becomes the unconfined two-dimensional hydrogen atom whose ground state energy is −2 Hartrees). Another observation is that, when r0 → 0, the forbidden circular region becomes a point and a transition of the two-dimensional system to a quasi-zero-dimensional system (see Fig. 57a) with energy  →  occurs. Figure 58b displays the ground state energy of the twodimensional hydrogen atom confined in a region of the plane limited to a cone obtained when  = 0 and 0 ≤ r < , as a function of 0 (polar angle in radians). In the limit when 0 → 0 the cone decreases; then the transition of the confined two-dimensional system to the quasi-one-dimensional system with energy  → 0 occurs, as in the previous situations. When 0 increases to 2, the allowed region grows

Impurity States and Atomic Systems Confined in Nanostructures

96

(a)

(a)

y

e

θ0

Nucleus

e

x

Nucleus

1-D 0-D

(b) 0.0 10 8

Energy (Hartrees)

Energy (Hartrees)

-0.1 -0.2 -0.3 -0.4

6 4 2 0

-0.5 0

1

2

3

4

5

6

7

θ0 (Radians) Figure 58. (a) Region of conical confinement. Limiting case: Q-twodimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a cone, as a function of the opening angle 0 . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

and the system behaves as a quasi-two-dimensional system; the edges of the cone are still a forbidden region (see Fig. 58a). The latter case corresponds to the minimum of the curve at 6.2856 radians (2) where the energy is ≈−2/9 Hartree and coincides with the case of hyperbolic confinement in the limit U0 → 1, a → 0 (see Figs. 55a and 58a), as expected. Figure 59b shows the ground state energy of the twodimensional hydrogen atom confined in a region of the plane given by the intersection of two symmetrical confocal parabolas (see Fig. 59a), as a function of Q0 (root of the distance to the foci, in atomic units). As can be observed, when Q0 → 0 the two parabolas (and their intersections) are closer to the origin. The allowed region is almost a point, the transition of a confined two-dimensional system to a quasi-zero-dimensional one occurs, and accordingly, the energy becomes infinite, as expected. Moreover, in the case when Q0 → , the available space enlarges until the unconfined two-dimensional atom is recovered, that is, the energy approaches −2 Hartrees (see Fig. 59b). The overall results show that the ground state energy of the two-dimensional hydrogen atom confined by an impenetrable potential has two extreme limiting cases, namely: (a) In the case of closed conical curves, a transition to a quasi-zero-dimensional system occurs where the ground state energy becomes infinite, that is, the confinement region approaches a point; (b) in the case of open conical curves,

-2 0

1

2

3

4

η0 Figure 59. (a) Region of confinement given by the intersection of two symmetrical parabolas. Limiting case: Q-zero-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by the intersection of two symmetrical confocal parabolas, as a function of Q0 . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

a transition to a quasi-one-dimensional system occurs in which the atom is confined to a straight line with origin at the nucleus and whose ground state energy becomes zero. Case (b) is closely related to the one-dimensional hydrogen atom discussed in [144–149], except that the binding energy is found to be infinite in [148]. The main difference between the limiting case (b) and the so-called Loudon onedimensional atom [148] rests in the fact that Loudon did not take into account the energy due to the confinement because his model is one-dimensional and our limit case is quasione-dimensional. A discussion of this matter is made in next section. Moreover, since the binding energy of Loudon’s one-dimensional atoms is greater than the rest mass energy of the electron, the treatment of the atom should be done in a relativistic scheme, which renders a large, but finite, binding energy [150]. In summary, the confining geometries studied in this section lead us to conclude following extreme behavior: Circular: r0 → 0

→

Elliptic: Q0 → 1

→

[quasi-zero dimensional (Q-0D)] (Q-0D)

Impurity States and Atomic Systems Confined in Nanostructures

Hyperbolic:

   U0 → 0     U0 → 1

 = a a = 0

 = a

(Q-2D)

(Q-2D)

→0

(Q-0D) Parabolic (two): U0 Q0  → 0  →   0 → 0 →0 (Q-1D) Cone: 0 → 2  ≈ −2/9 (Q-2D)



The behavior of the ground state energy of the confined twodimensional hydrogen atom, when one dimension becomes infinitely small, will be analyzed in this section. If the nucleus of the atom is located at the origin and the confining region is a symmetrical narrow strip of width 2y0 , the Hamiltonian for the system is (in atomic units) (273)

(274)

The corresponding Schrödinger equation can be written as   1 52 1 52 1 − − + Vb y −  3x y = 3x y 2 5x2 2 5y 2 x2 + y 2 (275) where  is the total energy and the wavefunction 3x y must satisfy the boundary condition 3x ±y0  = 0

(276)

As has been pointed out by Jan and Lee [151], as the strip becomes very narrow (i.e., y0 is very small), the fast motion in the y-direction allows the wavefunction to be written, approximately, as 3x y ≈ 9yYx

(277)

Since 3x y must satisfy the boundary condition (276), then 9y ≈ cosk0 y

y0

−y0

−

cos2 k0 y dy

−1

= y0−1

d2 + V x Yx = x Yx eff dx2

where x =  − y and

4.1.3. One-Dimensional Hydrogen Atom, as a Limiting Situation

where Vb y is the confining potential, defined as  0 −y0 ≤ y ≤ y0 Vb y =  otherwise



(280)

is a normalization constant and y = k02 =  2 /4y02 is the kinetic energy of the fast motion in the y-direction due to confinement. Equation (279) can be rewritten as

(Q-1D)

2 2  = − 1 5 − 1 5 + Vb y −  1 H 2 2 5x 2 5y 2 x2 + y 2

where N2 =

U0  a → 0  → −2/9 (Q-2D)      U → −1  →0 (Q-1D) 0

Parabolic (one): U0 Q0  → 0

97

(278)

with k0 = /2y0 . Introducing Eq. (277) into Eq. (275) and multiplying the result by cosk0 y, the integration over y from −y0 to y0 yields the equation    y0 d2 −2 2 2 − 2 +N cos k0 y dy Yx = −y Yx  dx −y0 x2 +y 2 (279)

Veff x = N

2



y0

−y0





−2

x2 + y 2

cos2 k0 y dy

(281)

(282)

A similar equation was found in [151] for a hydrogenic impurity confined within a quantum well wire. The result of averaging the two-dimensional Coulombic attraction over the fast y motion is such that Eq. (281) can be interpreted as Schrödinger equation for the onedimensional atom under the potential Veff x. Moreover, a careful analysis of Veff x in the limit y0 → 0 reveals that this potential has the same behavior as the one-dimensional Coulomb potential, as was also pointed out by Jan and Lee [151] in the case of a three-dimensional hydrogen atom confined within a cylinder whose radius approaches zero. The energy x in this limiting situation becomes the same as that obtained by Loudon [148], that is, −. It can also be noted that when y0 → 0, y →  and thus the total energy for the confined one-dimensional atom  = x + y → 0. The latter fact supports the results obtained in cases when the open confining curve approaches a straight line.

4.2. Geometrical Effects on the Ground State Energy of Hydrogenic Impurities in Quantum dots The effect of nonsphericity of quantum dots on the ground state energy of hydrogenic impurities is studied in the frame of the effective mass approximation and the variational method [152]. The difference in composition of the quantum dot and the host material is modeled with a potential barrier at the boundary of the dot. To make the analysis, two symmetries are considered for the quantum dot: spherical and spheroidal. In this way, the ground state energy is calculated as a function of the volume of the quantum dot, for different barrier heights. The results show that the ground state is strongly influenced by the geometry of the dot; that is, for a given volume and barrier height, the energy is clearly different if the dot is spherical or spheroidal in shape when the volume of the dot is small (strong confinement regime). As the volume of the dot increases (weak confinement regime) the geometry becomes irrelevant, as expected. The confinement of excitons in quantum dots and other microstructures such as quantum wells, quantum well

Impurity States and Atomic Systems Confined in Nanostructures

98 wires, etc. was corroborated experimentally in the past [113, 153–156]. The main feature of this effect corresponds to a frequency shift of the first excitonic peak to higher energy, compared to the mean bulk electronic peak. There have been many theoretical efforts to explain quantitatively this quantum mechanical effect [92, 157–163]. However, in all these works, the dot is assumed to be of spherical shape. This last assumption is not strictly true, since the electron microscopy studies of these systems show microparticles whose shapes vary from quasi-spherical to pyramidal [113]. The aim of this work is to analyze quantitatively the dependence of the ground state energy of a hydrogenic impurity on the geometry of the quantum dot. The latter would be of interest since it might constitute a qualitative study of the behavior of excitons confined within these microstructures. We assume that the impurity is confined within a potential barrier of finite depth, which emulates, on the average, the surrounding medium in which the dot is embedded. The calculation of the impurity ground state energy is performed using a variational approach, which previously was shown to be useful in dealing with this type of confined system in a fairly good fashion [66, 78, 111, 121]. The model Hamiltonian for the impurity is assumed to be valid within the effective mass approximation and we have considered, for the sake of comparison, spherical and spheroidal dots of the same volume and confining barrier. In this way, the ground state energy of the impurity is calculated as a function of the volume and the confining barrier heights.

4.2.1. Brief Description of the Method If Iqi J denotes the orthogonal set of coordinates compatible with the symmetry of the confining boundary, the trial wavefunctions can then be written as 9i q ? = Af q q0  ?0i q ?

q < q0

(283)

and 90 q @ = Bq @

q > q0

(284)

where ? and @ are parameters to be determined, q0 is associated with the size and symmetry of the confining boundary, A and B are normalization constants, 0i is the free system wavefunction,  is a function with the proper asymptotic behavior as q0 → , and f is an auxiliary function that allows the condition 1 590 1 59i = (285) 9i 5q 90 5q at q = q0 to be satisfied. The model Hamiltonian for the confined quantum system under study, in the effective mass approximation, can be written, in atomic units, as  = − 1 6 2 + V q + Vb q H 2 q

At this point, the variational method can be implemented to calculate the energy of the ground state of the system by construing the functional    1 ? @ = 9i∗ − 6q2 + V q 9i d 3 q 2    1 2 ∗ (288) + 9o − 6q + V0 + V q 9o d 3 q 2 and minimizing it with respect to ? @, restricted to   (289) 9i∗ 9i d 3 q + 9o∗ 9o d 3 q = 1 as usual.

4.2.2. Geometry Adapted Trial Wavefunctions In the effective mass approximation, the model Hamiltonian for an hydrogenic impurity (m∗n = ) confined within a quantum dot, in the context of the later section, can be written as  = − 1 6 2 + V q + Vb q H 2 q

where

Vb =

0

q < q0

V0

q > q0

(291)

and V q = −

1 rq

(292)

Here rq is the electron–nucleus separation and we have used effective atomic units. The symmetry of the confining domain determines the form of rq in the set of coordinates compatible with it. In the following, we construct explicitly the trial wavefunctions for the chosen symmetries. First, if we assume that the hole is located at the center of a confining sphere of radius r0 , the trial wavefunction in spherical coordinates can be written as [66] 9i = A exp−?rr0 − ?r

r < r0

(293)

and 9o = B exp−@r/r

r > r0

(294)

The continuity of the logarithmic derivative at r = r0 , given in Eq. (285), allows one relate ? and @ by @=

(286)

where V is the potential that accounts for the internal interactions of the components of the system and Vb is the confining potential. Here we shall assume that Vb has the form  0 q < q0 (287) Vb = V0 q > q0



(290)

?r0 1 − ? + 2? − 1 r0 1 − ?

(295)

in such a way that only a variational parameter needs to be determined, once that the energy functional is minimized in the standard way. If we now consider the exciton confined within the prolate spheroid IQ = Q0

−1≤U ≤1

0 ≤  ≤ 2J

(296)

Impurity States and Atomic Systems Confined in Nanostructures

with the nucleus located at a foci, the electron–nucleus separation can then be written as

(298)

Q > Q0

without Coul. Tail with Coul. Tail 1 V0 = 2.0 (Eff. Ry) R = 1.0 Eff. Bohr

0

(299)

Once again, the boundary condition given by Eq. (285) allows one to connect ? and @ through

-1 0

50

100

Volume (Eff. Bohr3)

(300)

As the reader must be aware, this form for the trial wavefunction satisfies automatically the boundary condition on its logarithmic derivative with respect to U, at Q = Q0 . As in the previous case, only a variational parameter needs to be determined when we use the variational approach. We have to point out that the auxiliary function of the form (q0 − ?q), which was used to construct the trial wavefunction for two symmetries, is flexible enough to describe the situation in which V0 → . In that case ? → 1 and correspondingly @ →  [see Eqs. (295) and (300)] so that the external wavefunction becomes vanishingly small in this limit, as expected. In Figure 60a and b, the energy for the ground state of the hydrogenic impurity confined within spherical and prolate spheroidal quantum dots is displayed as a function of the volume, for V0 = 2 0 and 8.0 effective Rydbergs respectively. In both figures we have assumed an R = 1 effective Bohr for the case of the prolate spheroidal dot. The effect of considering a different geometry can be noticed immediately from these figures, which supports that, for a given volume of the dot and barrier height, the ground state energy shows a dependence on the shape of the confining box that is particularly stronger for smaller sizes of the dot. The effect of including the internal interaction potential (Coulomb term) of the components of the system (electron–nucleus) in both regions (inside and outside the quantum dot) is also shown in the figures. When the Coulomb term is considered in the exterior of the quantum dot, a remarkable difference can be noticed for the smaller barrier height potential and smaller sizes of quantum dot. The latter is due to the fact that the confining capability of the box decreases and correspondingly, the probability of finding the electron in the exterior of the quantum dot is increased (see Fig. 60a). In Figure 60b, this effect is smaller. In Figure 60a, we can also observe a stronger dependence, by including the Coulomb term in the exterior of the quantum dot, for the spheroidal symmetry than for spherical one. Another observation is that, for the spheroidal symmetry, we have considered the nucleus of the impurity placed in a focus not in the origin, but in the latter case, keeping in mind the work of Marín et al. [65], the resulting geometrical effect would be stronger.

(b)

2 Spheroidal dot Spherical dot without Coul. Tail with Coul. Tail

Energy (Eff. Rydbergs)

Q < Q0

and

?Q0 1 − ? + ? @= Q0 1 − ?

Spheroidal dot Spherical dot

Energy (Eff. Rydbergs)

where 2R is the interfocal distance of the spheroid. Following the solutions of Coulson and Robinson [122] for the free hydrogen atom in this coordinate system, the trial wavefunction is of the form

9o = B exp−@Q − ?U

2

(a)

(297)

r1 = RQ − U

9i = A exp.−?Q − U/Q − ?Q0 

99

1 V0 = 8.0 (Eff. Ry) R = 1.0 Eff. Bohr

0

-1 0

50

100

Volume (Eff. Bohr3) Figure 60. Ground state energy for a hydrogenic impurity within quantum dots of a spherical and prolate spheroidal shape as a function of their volume for a fixed value of the barrier height. (a) V0 = 2 0R∗y , (b) V0 = 8 0R∗y . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

In Figure 61a, we show the energy for the ground state of the hydrogenic impurity confined in a spherical quantum dot as a function of the volume, for V0 = 8 0 and 2.0 effective Rydbergs. In the solid curves, the Coulomb term is not included in the external region of the quantum dot while in the dashed ones we have considered it. Figure 61b shows the same curves as in Figure 61a for the prolate spheroidal quantum dot with R = 1 0 effective Bohr. A difference between the curves with and without the Coulomb term in the exterior of the quantum dot can be noticed. This difference is greater for the smaller barrier height potential and in the region of strong confinement regime (small sizes of the quantum dot). This effect is similar to that discussed earlier for Figure 60. In Figure 62, we make a comparison of the ground state energy of the hydrogenic impurity confined within a prolate spheroidal quantum dot, for V0 = 2 0 effective Rydbergs and

Impurity States and Atomic Systems Confined in Nanostructures

100

8

without Coul. Tail with Coul. Tail

6

V0 (Eff. Ry)

2

Energy (Eff. Rydbergs)

Energy (Eff. Rydbergs)

(a)

8.0 2.0 4

2

without Coul. Tail with Coul. Tail R (Eff. Bohr) 0.5 1.0

1

V0 = 2.0 (Eff. Ry) 0

-1

0 1

0

1

2

3

2

3

4

r0 (Eff. Bohrs) (b)

Energy (Eff. Rydbergs)

8

without Coul. Tail with Coul. Tail

6

V0 (Eff. Ry) 8.0 2.0

4 R = 1.0 Eff. Bohr 2

0

1

2

3

ξ0 Figure 61. Ground state energy for a hydrogenic impurity within quantum dots of: (a) spherical symmetry as a function of the dot radius and barrier height, (b) prolate spheroidal symmetry as a function of the eccentricity and barrier height. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

different interfocal distances 2R. The solid curves correspond to R = 0 5 and R = 1 0 effective Rydbergs without the Coulomb term, while dashed ones are for the same values of R but the Coulomb term in the exterior of the quantum dot is included. The latter figure confirms that there is a significant variation of the energy for different interfocal distances of the spheroidal quantum dot and that the difference is greater for the lower value of R and even more significant when the Coulomb term in the exterior of the quantum dot is not included. Thus, the previous results show that a quantitative fitting of the energy-size curves for the impurity (or, qualitatively, for the exciton) to experimental results (for a given value of V0 ) must be taken with caution, independently of the model used to make comparison. This is the case in most of the theoretical approaches dealing with semiconductor crystallites, since a spherical shape is assumed to fit the experimental results.

4

5

ξ0

Figure 62. Ground state energy for a hydrogenic impurity as a function of the interfocal distance and the eccentricity for a fixed value of the barrier heigth. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

4.3. Hydrogen Atom and Harmonic Oscillator Confined by Impenetrable Spherical Boxes The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wavefunctions have been assumed as the product of the “free” solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method.

4.3.1. Direct Variational Approach The exact solution for the free system can be found in any text of quantum mechanics [94]. Indeed, the corresponding energy and wavefunctions are given as n = −

1 n2

n = 1 2

(301)

3nlm r   = Nnl 2r/nl F −n + l + 1 2l + 2 2r/n 

r Y m   (302) × exp − n l

where
= m = 1, Ylm   are the spherical harmonics, F a b z is the confluent hypergeometric function [164], and Nnl is a normalization constant. When we impose confinement on this system, the Hamiltonian is slightly modified and can be written as

where

′ = − 1 6 2 − 1 + V ′ r H 2 r V ′ r =



+ 0

r > r0 r ≤ r0

and r0 is the radius of the confining spherical box.

(303)

(304)

Impurity States and Atomic Systems Confined in Nanostructures

101

The corresponding Schrödinger equation is still separable, but the resulting radial equation  2 d 2 d ll + 1 2 + + − + 2 Rr = 0 (305) dr 2 r dr r2 r

3.3 Exact Variational H.V.

Energy (Rydbergs)

2.2

must be solved with the following boundary condition: Rr0  = 0

(306)

This means that, in order to obtain the energy spectrum, we must find the roots of Eq. (306). The new situation must be tackled in a more complicated way since we must construct a convergent series representation of Rr and then solve it numerically. Alternatively, we solve the same problem approximately, with the aid of the modified variational method discussed previously. We exploit the fact that the solutions of the “free” hydrogen atom are known [see Eq. (302)] in order to choose the trial wavefunctions as ′ 9nlm r   = N r0 − r2?rl F −n + l + 1 2l + 2 2?r

× exp−?rYlm  

(307)

where N is a normalization constant that depends on r0 and ?, as well as on n and l. We can note that 1/n is replaced by ? in this choice for 9 ′ . The reason is that, as a result of confinement, the number n is no longer a good quantum number to specify the state of the system (l is still a good quantum number since symmetry has not been broken). This ansatz gives more flexibility to the variational wavefunctions, allowing for the calculation not only of the ground state energy, but also of excited states, with only one variational parameter. Furthermore, the quantum virial theorem for enclosed systems [165] is satisfied by these functions. In order to show the adequacy of the method, in the following we shall restrict ourselves to states described by nodeless wavefunctions involving different symmetries. When we use Eq. (307) as a trial wavefunction, together with the Hamiltonian given by Eq. (303), the energy can be readily found by minimizing the functional   ′ dK 9 ′∗ H9 (308) X

with respect to ?, within the bounded volume X, restricted to satisfy the constraints implied by  9n′∗ 9m′ dK = 0nm (309)

1.1

2p 3d

0.0

1s -1.1 0

5

15

20

Figure 63. Energy levels of the enclosed hydrogen atom as a function of the radius of the box r0 . Exact and variational calculations are compared (see text). Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

fashion proposed here. Note that, in contrast to the hypervirial treatment, the energy values obtained by the variational method always lie as an upper bound to the exact results for all box sizes, as expected. Incidentally, Fernandez and Castro [165] were the first to obtain the ground state energies shown in Figure 63 variationally. These authors used a trial wavefunction identical to Eq. (307) while analyzing the fulfillment of the quantum virial theorem for enclosed systems by approximate wavefunctions. We now turn our attention to the case of the enclosed harmonic oscillator following a procedure similar to the case of the enclosed hydrogen atom. Once again, the problem of the “free” oscillator is exactly solved and the energy and wavefunctions are given by [94] nl = 2n + l + 3/2

n = 0 1 2

(310)

× exp−r 2 /2Ylm  

(311)

2

3nl r   = Nr F −n l + 3/2 r  l

are the spherwhere we have set
= m =  = 1, ical harmonics, F a b z is the hypergeometric function, and N is a normalization constant. As in the case of the hydrogen atom, when we impose the confinement, the Hamiltonian is modified to give Ylm  

where

2 ′ = − 1 6 2 + r + V ′ r H 2 2

X

This procedure is straightforward and can be done through direct algebraic manipulation. Figure 63 shows the results obtained after minimizing Eq. (308) as compared to the “exact” values obtained by numerically solving Eq. (306) [124] for the ground state and the 2p and 3d excited states. Also shown in this figure are the results due to Fernandez and Castro [166] who used a method based on both hypervirial theorems and perturbation theory [167]. A remarkable agreement can be noticed, showing that the problem can be tackled in the simpler

10

r0 (Bohrs)

′

V r =



+

0

r > r0 r ≤ r0

(312)

(313)

r0 being the radius of the confining spherical box in units of 
/m1/2 . Once again, if we were to solve the problem exactly, the Schrödinger equation is separable and the resulting radial equation  2 2 d ll + 1 d 2 + − r + 2 Rr = 0 (314) − dr 2 r dr r2

Impurity States and Atomic Systems Confined in Nanostructures

102 should be solved with the condition Rr0  = 0

(315)

which would force us to use a numerical treatment. In the modified variational method (as in the case of the confined hydrogen atom) the symmetry of the enclosed harmonic oscillator is not broken by the confinement and l is still a good quantum number but n ceases to be so. To construct the trial wavefunctions, we simply replace r 2 by ?r 2 in the argument of the functions in Eq. (311) and we proceed, as in our previous example, to find the energy of the system for each box size. In Table 5 we compare the exact results obtained from Eq. (315) [124] and those obtained by the application of the variational procedure, for the ground state and the first excited state. For completeness, we also show the results for the ground state, obtained through the hypervirial perturbational method using an 11-term perturbation expansion [168]. Once again, a very good agreement is observed. An important criterion for this selection is the fulfillment of the virial theorem for these systems [165]. At this stage it is important to note that there are several papers in the literature dealing with different techniques to tackle these problems [125]. In this connection, Fernandez et al. [165, 169–171] have done a thorough study of the use of the virial theorem for quantum systems subject to Dirichilet and/or Neumann boundary conditions. As we have mentioned before, in both examples, the agreement of the results obtained by the proposed modification of the direct variational approach is remarkable, in spite of the simplicity of the method. We note that the symmetry of the systems and their confinement were intentionally chosen to be compatible (i.e., both being of spherical symmetry).

4.4. Energy States of Two Electrons in a Parabolic Quantum Dot in Magnetic Field The energy spectra of two interacting electrons in a quantum dot confined by a parabolic potential in an applied magnetic field of arbitrary strength are obtained in this section. The Table 5. Energy levels of the enclosed harmonic oscillator as a function of the radius of the box. Ground state (n = 0, l = 0)

First excited state (n = 0, l = 0)

r0

?

aHV

var

exact

?

var

exact

1 0 1 5 2 0 2 5 3 0 4 0 5 0

0 1310 0 1073 0 1365 0 1935 0 2606 0 3530 0 4085

5 0756 2 5050 1 7648 1 5517

5 1313 5 5265 1 7739 1 5567 1 5105 1 5033 1 5025

5 0755 2 5050 1 7648 1 5514 1 5061 1 5000 1 5000

0 6339 0 3217 0 2402 0 2385 0 2769 0 3645 0 4075

10 3188 4 9169 3 2514 2 6901 2 5337 2 5015 2 5012

10 2822 4 9036 3 2469 2 6881 2 5313 2 5001 2 5000

Note: Exact and variational calculations are compared. Energies in units of
. Radii in units of 
/m1/2 . a Reference [168].

shifted 1/N expansion method is used to solve the effective mass Hamiltonian [172]. The influence of the electron– electron interaction on the ground state energy and its significant effect on the energy level crossings in states with different angular momentum are shown. The dependence of the ground state energy on the magnetic field strength for various confinement energies is presented. The magnetic field dependence plays a useful role in identifying the absorption features. The effects of the magnetic field on the state of the impurity [173] and excitons [95, 159, 161, 174–176] confined in quantum dots have been extensively studied. Kumar et al. [177] have self-consistently solved the Poisson and Schrödinger equations and obtained the electron states in GaAs–GaAlAs for both cases: in zero and for magnetic fields applied perpendicular to the heterojunctions. The results of their work [177] indicated that the confinement potential can be approximated by a simple one-parameter adjustable parabolic potential. Merkt et al. [178] have presented a study of quantum dots in which both the magnetic field and the electron–electron interaction terms were taken into account. Pfannkuche and Gerhaxdts [179] have devoted a theoretical study to the magnetooptical response to far-infrared radiation (FIR) of quantum dot helium, accounting for deviations from the parabolic confinement. More recently, De Groote et al. [180] have investigated the thermodynamic properties of quantum dots taking into consideration the spin effect, in addition to the electron–electron interaction and magnetic field terms. The purpose of this section is to show the effect of the electron–electron interaction on the spectra of the quantum dot states with nonvanishing azimuthal quantum numbers and the transitions in the ground state of the system as the magnetic field strength increases. Here we shall use the shifted 1/N expansion method to obtain an energy expression for the spectra of two confined electrons in a quantum dot by solving the effective mass Hamiltonian including the following terms: the electron– electron interaction, the applied field, and the parabolic confinement potential.

4.4.1. Theory and Model Within the effective mass approximation, the Hamiltonian for an interacting pair of electrons confined in a quantum dot by parabolic potential of the form m∗e 0 r 2 /2 in a magnetic field applied parallel to the z-axis (and perpendicular to the plane where the electrons are restricted to move) in the symmetric gauge is written as = H

 
2 6i2 1 ∗ 2 2
c z e2 − + m L  r + + e i i 2m∗e 2 2  r1 − r2  i=1 (316)

2


where the two-dimensional vectors r1 and r2 describe the positions of the first and the second electron in the x y plane, respectively. Lzi stands for the z-component of the orbital angular momentum for each electron and c = eB/m∗e c, m∗e , and  are the cyclotron frequency, effective mass, and dielectric constant of the medium, respectively.

Impurity States and Atomic Systems Confined in Nanostructures

The frequency  depends on both the magnetic field B and the confinement frequency 0 and is given by 

2 1/2 (317)  = 20 + c 4 The natural units of length and energy to be used are the effective Bohr radius a∗B = 
2 /m∗e e2 and effective Rydberg 2 R∗y =
2 /2m∗e a∗B . The dimensionless constant P =
c /2R∗y plays the role of an effective magnetic field strength. √ Upon introducing the center of mass R √ = r1 + r2 / 2 and the relative coordinates r = r1 − r2 / 2, the Hamiltonian [180] in Eq. (316) can be written as a sum of two separable parts that represent the center of mass motion Hamiltonian, ∗ 2 R = −
6 2 + me 2 R2 +
c LR H R ∗ 2me 2 2 z

(318)

and the relative motion Hamiltonian,

∗ 2 2 r = −
6 2 + me 2 r 2 +
c Lr + e H r z ∗ 2me 2 2 r

(319)

Equation (318) describes the Hamiltonian of the harmonic oscillator with the well-known eigenenergies ncm  mcm = 2ncm + mcm  + 1
 +


c mcm 2

103 to an effective potential, which does not vary with k¯ at large values of k¯ resulting in a sensible zeroth-order classical result. Hence, Eq. (321) in terms of the shift parameter becomes  ¯ ¯ d2 k¯ 2 .1 − 1 − a/k/.1 − 3 − a/k/ − 2 + 2 dr 4r  V r + (322) 3r = r 3r Q where V r =

4.4.2. Shifted 1/N Expansion Method The shifted 1/N expansion method, N being the spatial dimensions, is a pseudo-perturbative technique in the sense that it proposes a perturbation parameter that is not directly related to the coupling constant [181–184]. The aspect of this method has been clearly stated by Imbo et al. [181–183] who had displayed step-by-step calculations relevant to this method. Following their work, here we only present the analytic expressions that are required to determine the energy states. The method starts by writing the radial Schrödinger equation, for an arbitrary cylindrically symmetric potential, in a N -dimensional space as   k − 1k − 3 d2 + V r 3r = r 3r (321) − 2 + dr 4r 2 where k = N + 2m. In order to get useful results from 1/k¯ expansion, where ¯k = k − a and a is a suitable shift parameter, the large ¯ k-limit of the potential must be suitably defined [185]. Since the angular momentum barrier term behaves like k¯ 2 at large ¯ so the potential should behave similarly. This will give rise k,

(323)

and Q is a scaling constant to be specified from Eq. (325). The shifted 1/N expansion method consists of solving Eq. (322) systematically in terms of the expansion param The leading contribution term to the energy comes eter 1/k. from 

k¯ 2 1 r02 V r0  k¯ 2 Veff r = 2 + (324) Q r0 4 where r0 is the minimum of the effective potential, given by

(320)

labeled by the radial (ncm = 0 1 2  and azimuthal (mcm = 0 ±1 ±2 ±3  quantum numbers. The problem is reduced to obtaining eigenenergies nr  m of the relative motion Hamiltonian. The energy states of the total Hamiltonian are labeled by CM and relative quantum numbers, ncm mcm W nr m. The coexistence of the electron–electron and the oscillator terms makes the exact analytic solution with the present special functions not possible.

2 1 2 2  +  r +m c r 4 2

2r03 V ′ r0  = Q

(325)

It is convenient to shift the origin to r0 by the definition x = k¯ 1/2 r − r0 /r0

(326)

and expanding Eq. (322) about x = 0 in powers of x. Comparing the coefficients of powers of x in the series with the corresponding ones of the same order in the Schrödinger equation for one-dimensional anharmonic oscillator, we determine the anharmonic oscillator frequency, the energy ¯ Q, r0 eigenvalue, and the scaling constant in terms of k, and the potential derivatives. The anharmonic frequency parameter is  V ′′ r  1/2 ¯ = 3 + ′ 0 V r0 

(327)

and the energy eigenvalues in powers of 1/k¯ (up to third order) read  nr  m

 1 1 − a3 − a P k¯ 2 + 3 + P1 + 22 = V r0  + ¯kr 4r0 4 r0 0

(328)

The explicit forms of P1 and P2 are given in next section. The shift parameter a, which introduces an additional degree of freedom, is chosen so as to make the first term in the energy series of order k¯ vanish, namely, 

 2 − a 1 k¯ ¯ − nr + =0 2 2 r02

(329)

Impurity States and Atomic Systems Confined in Nanostructures

104 by requiring an agreement between 1/k¯ expansion and the exact analytic results for the harmonic and Coulomb potentials. From Eq. (329) we obtain

140

(330)

where nr is the radial quantum number related to the principal n and magnetic m quantum numbers by the relation nr = n − m − 1. Energies and lengths in Eqs. (321)–(330) are expressed in units of R∗y and a∗B , respectively. For the two-dimensional case, N = 2, Eq. (325) takes the following form:  2r0 V ′ r0  = 2 + 2m − a = Q1/2 (331)

Once r0 (for a particular quantum state and confining frequency) is determined, the task of computing the energy is relatively simple. The results are presented in Figures 64–68 and Tables 6 and 7. The relative ground state energy 00 of the relative motion, for the zero magnetic field case, against the confinement length is displayed in Figure 64. The present results (black dots) clearly show an excellent agreement with the numerical results of [178] (dashed line). In Figure 65, the first low energy levels 00, 10, and 20 of the relative Hamiltonian are presented as a function of the effective confinement frequency , using parameters appropriate to InSb, where the dielectric constant  = 17 88, electron effective mass m∗e = 0 014m0 , and confinement energy
0 = 7 5 meV [180]. The energy levels obviously show a linear dependence on the effective frequency. As the effective frequency  increases the confining energy term dominates the interaction energy term and thus the linear relationship between the energy and the frequency is maintained. This result is consistent with [180]. To investigate the effect of the electron–electron interaction on the energy spectra of the quantum dot, we plotted

120

20〉

εr (meV)

a = 2 − 2nr + 1¯

160

100

10〉

80 60

00〉

40 20 0 0

5

10

15

20

25

30

35

40

ω Figure 65. The low-lying relative states 00, 10, and 20 for two electrons in a quantum dot made of InSb as a function of confinement frequency . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

in Figure 66 the total ground state energy 00W 00 of the full Hamiltonian for independent (solid line) and interacting (dashed line) electrons as a function of the ratio c /0 . The figure shows, as we expect, a significant energy enhancement when the electron–electron Coulombic interaction term is turned on. Furthermore, as the magnetic field increases, the electrons are further squeezed in the quantum dot, resulting in an increase of the repulsive electron–electron Coulombic energy and in effect the energy levels.

103

2.5

102

2.0

ε (R*y)

ε (R*y)

1.5 101

1.0 100 0.5 101 0

5

10

ℓ0 / a*B

0 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

ωc / ω0 Figure 64. The relative ground state energy 00 for the electrons in a quantum dot as a function of confinement length ℓ0 = 
/m∗e 0 1/2 for the zero magnetic field. This section’s calculations: closed circles; Ref. [178]; solid line. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 66. The total ground state energy 00W 00 for two electrons in a quantum dot as a function of the ratio c /0 . For independent (solid line) and interacting (dashed line) electrons. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Impurity States and Atomic Systems Confined in Nanostructures

105 Table 6. The roots r0 determined by Eq. (684) for quantum dot states with nonvanishing azimuthal quantum number (m) against the ratio c /0 .

2.5

2.0

m c /0

m -5

ε (Ry*)

1.5

-4

1 2 3

-3 -2 -1 0

1.0

0.5

0 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

ωc / ω0 Figure 67. The total eigenenergies of the states 00W 0m, m = 0 −1 −2  −5, for two interacting electrons parabolically confined in the quantum dot of size ℓ0 = 3a∗B as a function of the ratio c /0 . Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

The energy level crossings are shown in Figure 67. We have displayed the eigenenergies of the states 00W 0m, m = 0 −1 −2  −5, for two interacting electrons parabolically confined in the quantum dot of size ℓ0 = 3a∗B as a function of the ratio c /0 . As the magnetic field strength increases the energy of the state m = 0 increases while the energy of the states with nonvanishing quantum number m decreases, thus leading to a sequence of different ground states, as reported in [185]. In the interacting system, the higher the angular momentum of the relative motion, the lower the interaction energy. This is caused by the structure of the relative wavefunction: The larger the angular 55 50 45 eV

ε (meV)

40

2m =1
ω 0

35

V

30


ω

me =5 0

0

−1

−2

−3

−4

−5

4 262 3 692 3 188

4 827 3 682 2 943

5 457 4 212 3 398

6 075 4 719 3 825

6 659 5 193 4 229

7 209 5 635 4 586

momentum the larger the spatial extent and therefore, the larger the distance between the electrons [186]. To confirm this numerically, we list in Table 6 the roots r0 of the potential for the interacting electrons in the quantum dot, for states with different angular momenta. At particular values of the ratio c /0 , as the azimuthal quantum number m increases, the root r0 also increases and thus the electron– electron interaction Vee r = 2/r0 , in the leading term of the energy series expression, decreases. In Figure 68, we showed the dependence of the ground state energy on the magnetic field strength for confinement energies:
0 = 6 and 12 meV. For constant values of the magnetic field, the larger the confinement energy, the greater the energy of the interacting electrons in the quantum dot. The spin effect can be included in the Hamiltonian, Eq. (316), added to the center of mass part as a space independent term, and Eq. (318) is still an analytically solvable harmonic oscillator Hamiltonian [180]. We have compared, in Table 7, the calculated results for the ground state energies 00 of the relative Hamiltonian at different confining frequencies with the results of Taut [187]. In a very recent work, Taut has reported a particular analytical solution of the Schrödinger equation for two interacting electrons in an external harmonic potential. The table shows that as 1/ increases the difference between both results noticeably decreases until it becomes ≈1.4 × 10−3 at 1/ = 1419 47. Quantum dots with more than two electrons can also be studied. The Hamiltonian for ne -interacting electrons, provided that the electron–electron interaction term depends only on the relative coordinates between electrons V ri − rj  = e2 /rij , and parabolically confined in the quantum dot, is separable into CM and relative Hamiltonians. The parabolic potential form V ri  = mi 20 ri2 /2, i = 1 2 3  ne , is the only potential which leads to a separable Hamiltonian. The CM motion part is described by

25

Table 7. The ground state energies (in atomic units) of the relative Hamiltonian calculated by 1/N expansion at different frequencies, compared with the results of Taut [187].

20 15

1/

10 0

5

10

15

20

25

30

35

40

45

γ Figure 68. The relative ground state 00 energy versus the magnetic field strength for two different confinement energies. Reprinted with permission from [205], J. L. Marín et al., in “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

4 20 54 7386 115 299 523 102 1054 54 1419 47

1/N expansion

Taut

0 4220 0 1305 0 0635 0 0375 0 0131 0 0081 0 0067

0 6250 0 1750 0 0822 0 0477 0 0162 0 0100 0 0081

Impurity States and Atomic Systems Confined in Nanostructures

106 the one-particle Hamiltonian, Eq. (318), with the electron mass replaced by the total mass M = ne m∗e and the electron charge replaced by the total charge Q = ne e. The relative Hamiltonian part, which involves only the relative coordinates and momenta, has a cylindrically symmetric potential and can be handled by the 1/N -expansion technique. When the confining potential is quadratic, FIR spectroscopy is insensitive to the interaction effects because of + CM and relative motions. The radiation dipole operator i ei ri =  being a pure CM variable, then it does not couple QR, r which contains all the electron–electron interactions. to H The dipole operator then induces transitions between the states of the CM but does not affect the states of the relative Hamiltonian. The eigenenergies for the CM Hamiltonian, Eq. (320), do not change because c in the energy expression remains the same; namely, QB/Mc = eB/m∗e c. Consequently, the FIR absorption experiments see only the feature of the single electron energies. There are only two allowed dipole transitions ('m = ±1) and the FIR resonance occurs at frequencies ± =





c 2

2

 + 20 ± c 2

(332)

Many different experiments on quantum dots have proved the validity of Kohn’s theorem and that the observed resonance frequency of an electron system in a parabolic potential is independent of electron–electron interactions and thus the actual number of electrons in the well, as reported by Wixforth et al. in a very recent review article [188]. In conclusion we have obtained the energy spectra of two interacting electrons as a function of confinement energies and magnetic field strength. The method has shown good agreement with the numerical results of Merkt et al. [178], Taut [187], and Wagner et al. [185]. Our calculations have also shown the effect of the electron–electron interaction term on the ground state energy and its significance on the energy level crossings in states with different azimuthal quantum numbers. The shifted 1/N expansion method yields quick results without putting restrictions on the Hamiltonian of the system.

4.4.3. Calculation of Parameters
1 and
2 The explicit forms of the parameters P1 and P2 are given in the following. Here R∗y and a∗B are used as units of energy and length, respectively

and

, P1 = c1 e2 + 3c2 e4 − ¯ −1 e12 + 6c1 e1 e3 + c4 e23 P2 = T7 + T12 + T16

(333)

(334)

where T7 = T1 − ¯ −1 .T2 + T3 + T4 + T5 + T6 /

T12 = ¯ −2 .T8 + T9 + T10 + T11 / −2

T16 = ¯ .T13 + T14 + T15 /

(335)

with T2 = c1 e22 + 12c2 e2 e4

T1 = c1 d2 + 3c2 d4 + c3 d6

T3 = 2e1 d1 + 2c5 e42

T5 = 6c1 e1 d3 + 2c4 e3 d3

T8 = T10 =

4e12 e2

T4 = 6c1 e1 d3 + 30c2 e1 d5 T6 = 10c6 e3 d5 T9 = 8c4 e2 e32

+ 36c1 e1 e2 e3

24ce12 e4

+ 8c7 e1 e3 e4

T11 =

T13 = 8e1 e3 + 108c1 e1 e3

(336)

12c8 e32 e4

T14 = 48c4 e1 e3

T15 = 30c9 e3

where c, d, and e are parameters given as c1 = 1+2nr

c4 = 11+30nr +30n2r

c8 = 57+189nr +225n2r +150n3r

c5 = 21+59nr +51n2r +34n3r

c6 = 13+40nr +42n2r +28n3r

c9 = 31+109nr +141n2r +94n3r

c7 = 31+78n2r +78n3r

c2 = 1+2nr +2n2r

c3 = 3+8nr +6n2r +4n3r

ej = j /¯ j/2

(337)

di = 0i /¯ i/2

where j = 1 2 3 4 and i = 1 2 3 4 5 6: 32 − a 2 2r0 5 2r0 −1 − 4 = + Q 4 Q 31 − a3 − a 1 − a3 − a 02 = − − 2 4 52 − a 22 − a 04 = − 2 3 2r0 7 2r0 − − 06 = + 2 Q 4 Q

1 = 2 − a 3 = 01 = 03 = 05 =

2 = −

(338)

5. MECHANISM OF TERAHERTZ LASING IN SiGe/Si QUANTUM WELLS Blom et al. [189] presented a theoretical calculation, which showed the formation of resonant states, and explained the origin of the observed temperature dependence of the dc conductivity under low bias voltage. Thus, it was shown that the mechanism of terahertz (THz) lasing is population inversion of the resonant state with respect to the localized impurity state. This is the same mechanism of lasing as in uniaxially stressed p-Ge THz lasers. In an early experiment [190] published in 1992, THz radiation was observed from bulk p-Ge under uniaxial stress and a dc electric field, both applied along the same crystal axis. Eight years later measurements of the radiation spectrum characterized the observed radiation as pulse mode lasing under pumping of a strong electric field [191]. A theoretical explanation of the lasing phenomenon followed almost immediately [192], which proved the formation of resonant states as the required mechanism for achieving population inversion. Shortly thereafter, a tunable continuous

Impurity States and Atomic Systems Confined in Nanostructures

cap layer of 60 nm thickness. Over the cap layer a thin layer of SiO2 typically appears, on which ohmic contacts were installed. The middle of the well was 0 doped with a boron concentration of 6 × 1011 cm−2 . In both the buffer layer and the cap layer, at a distance of 30 nm from the respective QW interface, a 0 layer of boron was doped with a concentration of 3 × 1011 cm−2 . In this system Edef = 31 meV [200]. Using a one band variational approach given in [201], the binding l h energies E1s and E1s of the impurity levels attached to the LHB edge and the HHB edge, respectively, were calculated and both were found to have a value of about 27 meV. One very important feature of the QW system is the interface states between the SiO2 layer and the Si cap layer, which can accumulate almost all holes from the 0 doping in the cap later. This resulted in pinning of the chemical potential at the interface state energy level, which lay in the bandgap at ' ≈ 0 4 eV measured from the valence band top [202]. This charge redistribution inside the sample built up a strong electric field in the QW, which was detected experimentally [197, 199]. The energy level structure in the QW calculated is shown in Figure 69. The zero energy was set at the chemical potential T. The solid curve marks the HHB edge and the dotted curve the LHB edge. The lowest quantization energy levels E1hhb in the HHB and E1lhb in the LHB were indicated, along with the lowest impurity states attached to each band. In the case depicted (T = 4 K), the electric field strength in the l QW was 19 kV/cm, and E1hhb − E1s = 5 meV. The overlap l of the impurity level E1s with the E1hhb 2D subband made it possible to form the resonant state required for THz lasing. Once they proved that a resonant impurity state could be formed in the structure, the theory of population inversion developed for strained bulk p-Ge [195] could be applied directly to explain the origin of the observed lasing in the SiGe/Si QW. In order to understand the transport properties of the SiGe/Si QW system, which is relevant for the electric pumping of carriers into the resonant states, they also calculated the concentration of free holes, pv T , as a function of the temperature, and their results are plotted as the solid curve in the upper panel of Figure 70. The temperature 50 h E1s

0

E1hhb

µ l E1s

HHB

Energy (meV)

wave p-Ge THz laser was realized [193] under weak electric field pumping, accompanied by a complete theoretical interpretation [194]. The theory of population inversion based on the formation of resonant states has been given in detail very recently [195]. Under uniaxial stress the heavy hole band (HHB) in p-Ge lies lower than the light hole band (LHB) by an amount Edef . Dictated by the symmetry properties, one set of impurity levels is attached to the edge of the HHB, and another set is attached to the edge of the LHB. If Edef is greater than the hole binding energy of the lowest impurity level attached to the HHB edge, this impurity level overlaps the Bloch states in the LHB. Resonant states are then formed. Under an external electric field, the impact-ionized holes are accelerated toward the resonant level. With the proper combination of impurity concentration, temperature, electric field, stress, and scattering strength of phonons and impurities, holes have a large probability of occupying resonant states [192, 194, 195]. A resonant level population inversion with respect to those impurity levels which attach to the LHB edge is then formed, and THz lasing occurs [191, 193]. The externally applied uniaxial stress can be replaced by the strain in a QW with lattice mismatch, and a Si/Gex Si1−x /Si QW with x < 0 2 and a boron-doped well was proposed [196, 197]. In such a structure the well is stretched along the growth direction, so the HHB lies above the LHB. When impurities are added to the SiGe/Si QW system, all relevant energy levels are affected by the electric field produced by the charge redistribution in the system. It is plausible that a strong electric field in the QW may favor the formation of resonant states for THz lasing. Such a field can be achieved by employing two aspects of the sample structure. First, the Si buffer layer and the Si cap layer on each side of the well will be 0 doped with boron acceptors. Second, the existence of a thin SiO2 layer on top of the cap layer creates interface states between them. In such a structure THz lasing was indeed detected [198]. Because the carriers in a SiGe/Si QW are electrically pumped, electric transport in the well has been investigated in detail in order to clarify the relevant physical processes [197, 199]. Under a weak dc bias, the temperature behavior of the conductivity $, plotted as ln$ as a function of the inverse temperature 1/T , exhibits two different linear regimes, below and above T ≈ 20 K, respectively, as shown by Figure 1 in [197]. Further experiments on magnetoconductivity and Hall mobility [197, 199] have indicated that the low temperature conductivity may be due to hopping. In the high temperature region, the slope of the ln$ vs 1/T curves suggests an activation energy of about 12 meV for Ge content x = 0 1 and 18 meV for x = 0 15, if indeed the activation process exists. It was suggested in [197, 199] that the possible hole activation is between the two impuh l and E1s attached to each respective band. The rity levels E1s present study has disproved this suggestion. The mechanism of the high temperature conductivity and its relation to the THz lasing are the questions to be answered. In [189] an extensive numerical study of a Si/Ge0 15 Si0 85 /Si QW was performed. The system structure consisted of, in sequence, an n-Si substrate, an i-Si buffer layer of 130 nm thickness, a Ge0 15 Si0 85 well of 20 nm thickness, and an i-Si

107

-50

E1lhb

LHB

-100

-150 50

55

60

65

70

75

80

85

90

x (nm) Figure 69. Energy level structure in the QW calculated at T = 4 K. The position x is measured from the interface between the cap layer and the SiO2 . Reprinted with permission from [189], A. Blom et al., Appl. Phys. Lett. 79, 713 (2001). © 2001, American Institute of Physics.

Impurity States and Atomic Systems Confined in Nanostructures

– 15 – 20 9

10

– 25

Current (nA)

104

103

103

Pumping current (mA)

– 10 1010

E1hhh

Concentration (cm-2)

–5

– µ (meV)

–0

1011

RADIATION 102

2

10

10

10 CURRENT

Radiation intensity (Arb. Units)

108

1

1 103 100

200

400

700

1000

Voltage (Volts) 102 0.00

0.05

0.10

1/T

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0.20

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Figure 70. Upper panel showing the calculated temperature dependencies of the free hole concentration pv T  (solid curve) and the number concentration of charged acceptors in the QW (dotted curve). The dashed curve is the energy separation E1hhb − T. The lower panel displays the measured current as function of the temperature. Upper panel showing the calculated temperature dependencies of the free hole concentration pv T  (solid curve) and the number concentration of charged acceptors in the QW (dotted curve). The dashed curve is the energy separation E1hhb − T. The lower panel displays the measured current as function of the temperature. Reprinted with permission from [189], A. Blom et al., Appl. Phys. Lett. 79, 713 (2001). © 2001, American Institute of Physics.

dependence of pv T  was due to the thermal excitation of holes from the chemical potential into the 2D subbands. This was governed by the energy separation E1hhb − T, which depends strongly on the temperature, as shown in the upper panel of Figure 70. These theoretical findings and the following experimental data will support their conjectured mechanism of THz lasing in the SiGe/Si QW. A Si/Ge0 15 Si0 85 /Si QW sample with the previously specified structure was fabricated with a MBE machine. High voltage pulses of 0.3 Ts duration were applied parallel to the QW via the deposited ohmic contacts. Since the 0 layer of boron in the buffer layer also supplies holes to the substrate, a pn junction type of carrier distribution is formed, which prevents the bypass of current through the substrate. When opposite surfaces parallel to the growth direction were polished to serve as an optical resonator, strong THz radiation was detected at liquid helium temperature. The radiation intensity is orders of magnitude stronger than the intensity of spontaneous emission. In Figure 71 the radiation intensity and the corresponding electric pumping current as functions of the pulse voltage are plotted. The abrupt increase of the current around 100 V marks the onset of impact ionization of the boron acceptors in the QW. A threshold current for the radiation is clearly seen. The characteristic features of the observed THz radiation are exactly the same as those of THz lasing reported recently [203]. This was expected, because the only difference between their sample and the sample in [198, 203]

Figure 71. Radiation intensity and electric pumping current as functions of the pulse voltage. Reprinted with permission from [189], A. Blom et al., Appl. Phys. Lett. 79, 713 (2001). © 2001, American Institute of Physics.

was the concentration of 0 doping in the buffer and cap layers: 3 × 1011 cm−2 in their sample and 4 × 1011 cm−2 in the other ones. At a low dc bias of 2 V the current I flowing parallel to the QW was measured as a function of the temperature. The result is plotted in the lower panel of Figure 70. Similar to the temperature behavior of the free hole concentration pv T , the curve lnI vs 1/T also displays different characteristic features below and above T ≈ 20 K. In fact, in the high temperature region, the slope of the lnI vs 1/T curve is almost the same as the slope of the lnpv  vs 1/T curve. They then concluded that, in this temperature region, the dependence of the current I on 1/T was caused by the temperature dependence of the thermal population in the free hole levels but does not suggest an activation process as stated in [197, 199]. Then, they turned to the transport process in the low temperature region. Their calculated free hole concentration pv T  shown in Figure 70, although small, is non-negligible at low temperatures. However, for such low temperatures and low concentration, the mobility may be very small [204], resulting in a decrease in the current of an order of magnitude. Nevertheless, they cannot rule out completely the possibility of hopping transport for the following reason. The charged impurity ions in both the buffer layer and the cap layer create a long range random potential in the QW. At the same time the spatial variation of the alloy composition produces a short range random potential. Both effects cause fluctuation of the 2D subband edge, and their calculations gave an energy fluctuation of about 5 meV. The transport of holes in the 2D subband would then exhibit hopping behavior with an activation energy of about 2–3 meV. It is important to emphasize that such a hopping process is entirely different from that stated in [197, 199], where the transport is ascribed to the hopping of holes from one acceptor to the other within the 0 layer in the well. Such a process requires partially charged acceptors. In conclusion in [189] Blom et al. calculated the number concentration of charged

Impurity States and Atomic Systems Confined in Nanostructures

acceptors in the QW and plotted it as the dotted curve in the upper panel of Figure 70. At low temperatures, the number concentration of charge acceptors in the QW is far too small to produce a measurable hopping conductivity. It can be emphasized that the mechanism of population inversion through the formation of resonant states is essential for the realization of semiconductor QW THz lasers. A full understanding of this mechanism will allow the design of advanced THz lasers using various lattice mismatched heterostructures with different semiconductors.

GLOSSARY Conduction band Energy band in a crystalline solid partially filled by electrons. Confined system A quantum system in which carriers, atoms or molecules are restricted to exist in one, two or three dimensions of the order of a few nanometers. Effective mass The mass of a particle in a crystalline solid. Gap Forbidden region for the electrons, which it is determined by the difference of energies between conduction and valence bands. Idealized confined systems A non-real isolated quantum system whose size is of the order of few nanometers. Impurity An atom or molecule embedded in a different material. Impurity states The additional electron levels which appears due to the presence of an impurity. MBE, MOCVD and Lithography Methods of synthesis used to the fabrication of the nanostructured materials. Nanostructure Material medium whose size is of the order of few nanometers. Quantum dot A quantum system, in which the carriers are confined in all directions. Quantum well A quantum system, in which the carriers are confined in only one direction, the other dimensions remain free. Quantum well wire A quantum system, in which the carriers are confined in two directions, the other remains free. Valence band Energy band in a crystalline solid completely filled by electrons.

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Encyclopedia of Nanoscience and Nanotechnology

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Inherently Conducting Polymer Nanostructures Gordon G. Wallace, Peter C. Innis, Leon A. P. Kane-Maguire University of Wollongong, Wollongong, Australia

CONTENTS 1. Introduction 2. Synthesis of Inherently Conducting Polymers Nanostructures 3. Unique Properties and Applications of Inherently Conducting Polymers Nanostructures 4. Conclusions Glossary References

1. INTRODUCTION Inherently conducting polymers (ICPs) such as polypyrroles, polythiophenes, and polyanilines (I–III shown in Scheme 1) are extremely useful organic electronic conductors. The applications of these materials have been reviewed recently [1–3], spanning areas as diverse as nanomaterials [4], electrochromics [5], sensors [6], artificial muscles [7, 8], smart membranes [9, 10], platforms for cell culturing [11, 12] and corrosion protection [13]. Many of these applications involve electrochemical switching processes (to be discussed in more detail). In the quest to improve electrochemical switching speeds of ICPs, researchers have recently turned attention to control of the structure at the nanodomain. It is also interesting to note that ICP macrostructures are actually composed of nanodomains of much higher conductivity than the bulk material [14–17] suggesting that nanodimensional control will also improve the bulk conductivity of ICPs.

1.1. What Are ICPs? The synthesis and electrochemical switching properties of ICPs such as polypyrroles (PPy), polythiophenes (PTh), and polyanilines (PAn) have been reviewed in recent monographs [18, 19]. The polymerization can be initiated chemically or electrochemically and involves formation of lower molecular weight oligomers via oxidation. These are then further oxidized

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(at lower potentials than the initial monomer) to form a polymer that eventually precipitates or deposits onto the anode in an electrochemical cell as a critical molecular weight is exceeded: A

+

–

A N H

oxidize

N H

–

(1) n

m

A counterion (A− ) is incorporated during synthesis to balance the charge on the polymer backbone. The anion of the chemical oxidants used provides the dopant A− . Electrochemical oxidation provides greater flexibility in terms of the anion that can be incorporated from the electrolyte used as the polymerization medium. Polypyrroles can be formed from neutral aqueous solutions while acidic conditions are required for aniline solubilization and polymerization. Thiophene polymerization is commonly undertaken in organic solvents due to poor monomer solubility in aqueous solutions.

1.1.1. Polypyrroles The mild oxidation potentials needed to initiate formation of polypyrrole in aqueous solution have enabled the formation of a wide range of polypyrrole structures by simply varying the dopant. Simple metal recognition capabilities are introduced by incorporation of metal complexing groups as dopants [20, 21], or electrocatalytic effects are induced by use of appropriate dopants [22], covalently attached redox sites [23, 24], and/or inclusion of micron-sized metallic particles [25, 26]. Conducting polymers are also known to promote electron transfer into/out of biological entities [27]. Biomolecular/recognition can be introduced by incorporation of antibodies or enzymes or even nerve growth factors into the polymer at the time of synthesis, as reviewed recently [28]. The incorporation of complex biomolecules as dopants can be accomplished while retaining the unique electronic properties of the polymer backbone. For example, the

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (113–130)

114

Inherently Conducting Polymer Nanostructures +

1.8

A–

A–

n

m

m

(I)

N

1.7

S n

Abs

N H

SO3H

+

OH

1.6 1.5 1.4

(II)

1.3 300

H N

500

700

900

1100

Wavelength (nm)

H N

Figure 2. UV-visible spectrum of a PPy HQS film grown galvanostatically onto ITO glass from a solution containing 0.2 M pyrrole and 0.1 M HQS. Reprinted with permission from [188], V. Misoska, Ph.D. Thesis, University of Wollongong, 2002.

+ n A– (III) Scheme 1. For polypyrroles and polythiophenes, n is usually ca. 3–4. For optimal conductivity, there is a positive charge on every third or fourth pyrrole or thiophene unit along the polymer chain. For the conducting emeraldine salt form of polyaniline (III) a radical cation resides at alternate N sites.

Abs.

ultraviolet-visible (UV-vis) spectrum for PPy containing DNA as dopant is shown in Figure 1. This UV-vis spectrum shows the typical polaron/bipolaron 475 nm bands of doped polypyrrole at ca. 500 nm and 900 nm. The exact location of these bands is dependent on the dopant used. As illustrated in Figure 2, when 8-hydroxyquinoline-5-sulfonic acid (HQS) is used as dopant, the lower wavelength peak is even further blueshifted. The dopant also influences the degree of absorption in the near infrared beyond 900 nm and this is attributed to changes in polymer conformation (tight coil versus expanded coil). The polymer conformation has been found to be substrate dependent, with polypyrroles deposited on hydrophilic glass displaying an absorption band at 1180 nm, which is replaced by a free carrier tail extending to 2600 nm for polymers deposited on hydrophobic surfaces [29]. As well as varying the functional properties of ICPs, the dopant plays a key role in determining the electronic (conductivity) and mechanical properties (e.g., tensile strength) of the resultant material. For example, even slight changes in the molecular structure [30] of a range of sulfonated aromatic dopants influence these properties (Table 1). The incorporation of surfactant-like dopants such as dodecylbenzene sulfonate has also proved useful in sol0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 300

ubilizing polypyrrole in organic solvents [31, 32]. The presence of alkyl and alkoxy substituents on the backbone of polypyrrole also increases their solubility in organic solvents. In contrast, polymerization of pyrrole monomers bearing sulfonated substituents has provided water soluble polypyrroles [33]. A very useful recent advance has been the development of a facile route for the modification of preformed polypyrroles containing good leaving group such as N -hydroxysuccinamide [34]. An important feature of these ICP structures is that they are amenable to facile oxidation/reduction processes that can be initiated at moderate potentials. For polypyrrole the oxidation state can be reversibly switched, as shown in Eq. (2). The doped oxidized forms exhibit good electrical conductivity ( = 1–100 S cm−1 , while the reduced forms have very low conductivity ( ∼ 10−8 S cm−1 . The dynamic character of these polymer systems, with chemical, physical, and mechanical properties being a function of applied potential, is intriguing and is the basis of their proposed use in intelligent material systems [19]. + A– N H

n

0

+e– N H

–e–

m

+ A– (Solution) n

m

(2) If the dopant anion (A− ) is small and mobile (e.g., Cl− ), then upon reduction the anion will be efficiently ejected from the polymer [Eq. (2)]. However, extensive studies with polypyrroles have shown that if the dopant is large and immobile (e.g., if A− is a polyelectrolyte such as polystyrene sulfonate), an electrically induced cation exchange process occurs, according to 0

+ A– N H 500

700

900

1100

Wavelength (nm) Figure 1. UV-visible spectrum of a PPy DNA thin flim grown onto indium tin oxide (ITO) glass from a solution containing 0.2 M pyrrole and 0.2% w/v salmon sperm DNA. Reprinted with permission from [188], V. Misoska, Ph.D. Thesis, University of Wollongong, 2002.

+

X+

+e– –e–

n

m

N H

n

A– X+ m

(3) where the cation (X+ ) is incorporated from the supporting electrolyte solution. The effect of the original dopant incorporated as well as the other ions in the electrolyte on the electrochemical switching characteristics of polypyrroles has been clearly demonstrated [30].

115

Inherently Conducting Polymer Nanostructures Table 1. Effect of the counterion on the tensile strength and conductivity of polypyrrole membranes [30]. Membrane

PPy/BSA

PPy/PTS

PPy/EBS

Counterions

PPy/MS

SO3H

SO3Na SO3Na

CH2CH3

60–70

36–47

40–55

40–50

19–20

90–110

90–110

50–70

47–70

50–70

The direct polymerization of thiophene is more complicated in that the monomer oxidation occurs at potentials more positive than required to overoxidize the polymer. Hence monomers with alkyl groups attached to the 3-position [36], bithiophene [37, 38], or terthiophenes [39] are often used as starting materials since they have lower oxidation potentials. To avoid use of organic solvents for polythiophene synthesis, monomers can be dissolved in aqueous solution using surfactants [40] or molecular inclusion compounds such as cyclodextrins [41]. While the inclusion of the extensive range of dopants available with polypyrrole is not available with polythiophenes, some functional dopants have been incorporated [42]. Specific dopants such as the polyether [43] shown in Scheme 2 induce exceptional mechanical properties into polythiophenes, with tensile strengths of the order of 120 MPa readily obtained. The UV-visible absorption spectra obtained (Fig. 3) after polymerization of bithiophene clearly show polaron/ bipolaron absorption bands that are eliminated once the polymer is electrochemically reduced. Given the simple synthetic chemistries available to covalently attach functional groups to thiophene substrates prior to polymerization, this has been the preferred approach to introduce functionality into polythiophenes [44]. In conjunction with our collaborators at Massey University, OSO3 – TBA+

OH CH3

CH3 O

SO3H

70–80

1.1.2. Polythiophenes

CH3

SO3Na

17–23

The redox processes described have a dramatic effect on the physical and chemical properties of the polymer. Conductivity will decrease, anion exchange capacity will diminish [Eq. (2)], cation exchange capacity may increase [(Eq. 3)] and hydrophobicity will be altered in a manner determined by which ion exchange process predominates. The mechanical properties are also influenced by the oxidation state with a greater elongation-to-break usually observed in the reduced materials [35]. The polymer also undergoes dramatic color changes upon redox switching (see Section 1.1.2) which is the basis of electrochromic devices based on ICPs [5].

O

SO3H

CH3

O

O n

CH3

O m

Scheme 2. Structure of the S-PHE used in this work. TBA = tetrabutylammonium.

New Zealand, we have produced a range of substituted terthiophene precursors (Fig. 4), from which functional polymers have been produced [45]. As discussed for polypyrroles, the polythiophenes undergo reversible oxidation/reduction processes (Fig. 5). Oxidation/reduction usually occurs at more positive potentials than for the polypyrroles and is not readily achieved in aqueous solutions [45], presumably due to the hydrophobic nature of the thiophene based polymer. Some workers have attached ether groups to the polymer backbone, increasing hydrophilicity and the rate of switching in aqueous media [46]. Another interesting feature of polythiophenes is that the n-doped state is more readily accessible than with either polypyrroles or polyanilines. This enables the polymer to be rendered conductive at more negative potentials. A number of authors [47, 48] have highlighted the importance of substituents on the polythiophene backbone in determining the accessibility (potential required for reduction) and stability of the n-doped state.

1.1.3. Polyanilines Like polypyrrole, conducting polyaniline (PAn) and its ring-substituted analogs are generally prepared via either chemical or electrochemical oxidation/polymerization of the appropriate aniline monomer in aqueous solution. However, acidic conditions (pH generally 0–1) are required both 0.8 0.7 0.6 0.5

Abs.

Tensile strength (MPa) Conductivity (S/cm)

PPy/NPS

SO3Na

CH3

H3C

CH3

PPy/BS

SO3H

D

0.4

C

0.3 A

0.2 0.1 0 350

B 450

550

650

750

850

950

1050

Wavelength (nm)

Figure 3. UV-vis spectra of PBT/S-PHE composite galvanostatically deposited (1 mA cm−2 for 50 s) onto ITO coated glass from a solution containing 0.2 M 2,2′ bithiophene and 2% S-PHE. (A) Polymer after perparation. (B)–(D) Polymer after application of an applied potential for 60 s in propylene carbonate solution containing TBAP; (B) 0 V, (C) +1.3 V (D) + 1.5 V. Reprinted with permission from [43], J. Ding et al., Synth. Met. 110, 123 (2000). © 2000, Elsevier Science.

116

Inherently Conducting Polymer Nanostructures R

S

NH2

NO2

S

Fe / NH4Cl

S

S

S

S

S

S

S

R = NMe2, OMe, Ph, CN, NO2 bipy bipy Cl Ru Cl N

N

S S

N

N

N

S

S

S

S

S

S

S

S

Ar

O n

O

N

O

S S

Ar

O

O

S

S

S

N H N

H N Ar

Ar

Fe

S S

S

S S

S

S

n=1 n=2

S

S

Figure 4. Styryl terthiophene monomers synthesized.

to solubilize the aniline monomer and to ensure that the conducting emeraldine salt PAn · HA form of the polymer is produced [see structure (III)]. The most common chemical oxidants are ammonium persulfate or ferric chloride, in 2− − which cases HSO− 4 /SO4 or Cl anions are incorporated as the dopant anions A− at the radical cation nitrogen sites along the PAn chains. Using electrochemical polymerization, a much wider range of dopant anions may be incorporated along the polyaniline chains, depending on the electrolyte employed. After deposition on the working electrode, films of the PAn · HA may often be removed as mechanically robust, standalone membranes. On the other hand, chemical polymerization has the advantage of being a simple process capable of producing bulk quantities of PAn · HA powders on a batch basis. In both chemical and electrochemical polymerization the initial and rate-determining step is believed [49–51] to

i1 (µA)

40

20

0

–20

0

0.2

0.4

0.6

0.8

1

1.2

E(V) Figure 5. Cyclic voltammogram of a PBT/S-PHE coated platinum electrode obtained after immersion in a propylene carbonate solution containing 0.5 M LiCIO4 , scan rate = 100 mV s−1 . PBT/S-PHE was prepared galvanostatically (1 mA cm−2 for 3 min) using a solution containing 0.2 M 2,2′ -bithiophene and 2% S-PHE.

be oxidation of the aniline monomer to give the radical cation of aniline. This is followed by coupling of the radicals, predominantly N and para forms, and subsequent re-aromatization to give a dimer. This undergoes more facile oxidation than the aniline monomer, leading to chain propagation and eventual deposition of the emeraldine salt from solution. The nature of the dopant anion incorporated along the growing polymer chain during polymerization has a profound effect upon the morphology [52], conductivity [53], switching characteristics, and solubility of the resulting polyaniline salts. Incorporation of polyelectrolyte dopant anions during either chemical or electrochemical oxidation of aniline has been an area of intense recent interest. These large anions can often be preferentially incorporated even when an acid such as HClO4 is present in large excess [54]. As will be discussed further in Section 3, this can induce water “solubility” onto the resultant emeraldine salt when the anion is a polyelectrolyte such as poly(styrenesulfonate) or polyacrylate [55–57]. This approach has helped in overcoming one of the problems previously associated with polyanilines, namely their insolubility in most common solvents. It has also provided a convenient route to aqueous nanosized dispersions of conducting emeraldine salts (see Section 4). Biological polyelectrolytes such as DNA have also been incorporated into emeraldine salts using this approach [58, 59]. On the other hand, organic solvent solubility has been induced in polyanilines via the incorporation of surfactant-type dopant anions such as dodecylbenzenesulfonate, racemic 10-camphorsulfonate (CSA− ) [60], and dinonylnapthalene sulfonate [61]. Another significant development has been the facile generation of optically active emeraldine salts such as PAn·(+)-HCSA or PAn·(−)-HCSA by the simple expedience of employing chiral acids such as (+)- or (−)-HCSA as the electrolyte during electrochemical polymerization [62, 63]. These chiral conducting polyanilines are believed to preferentially adopt single-handed helical structures for their polymer chains depending on which hand of the dopant HCSA acid is employed. A large range of substituted anilines are available. Polymerization of these has given emeraldine salts whose properties differ significantly from those of the parent unsubstituted PAn · HA salts. For example, the presence of alkyl or alkoxy substituents [64] on the aniline rings imparts enhanced solubility in organic solvents on the resultant polymers. On the other hand, sulfonate groups lead to water solubility, as in poly(2-methoxyaniline-5-sulfonic acid) [65]. Postpolymerization modification of polyanilines has also been pursued to introduce added functionality into the polymer for a variety of applications. For example, treatment of the emeraldine base (EB) or leucoemeraldine base (LB) forms of polyaniline with fuming sulfuric acid has led to water-soluble, sulfonated polyanilines (SPANs) in which 50– 75% of the aniline rings bear sulfonate groups [66]. These latter emeraldine salts are self-doped (i.e., ring-bound sulfonate groups provide the dopant anion for the radical cation nitrogen sites along the chain). An exciting recent development has been the synthesis of poly(aniline boronic acid) [67]. This promises to be a convenient precursor for

117

Inherently Conducting Polymer Nanostructures

the facile synthesis of a wide range of substituted polyanilines that are difficult to synthesize directly from their respective monomers. Doped PAn · HA emeraldine salts are good electrical conductors, with conductivities typically in the range 1– 10 S cm−1 . It is generally agreed that polarons are the charge carriers responsible for this high conductivity. Evidence includes the observation of a strong electron spin resonance signal associated with the radical cation sites in structure (III) [68]. It has also been shown that bipolaron states exist in polyaniline, but these are few in number and are not associated with the conducting regions of the polymer [69]. The attachment of functional groups to the aniline rings generally decreases the conductivity of polyaniline emeraldine salts. This is attributed to steric crowding causing a marked twisting of the polymer backbone from planarity, leading to decreased conjugation along the chains. Structural defects such as those arising from undesirable orthocoupling of aniline radical cations during the polymerization also lead to impaired electrical conductivity. There has been considerable recent interest in enhancing the conductivity of emeraldine salts via their exposure to “secondary dopants” such as m-cresol. This can cause an increase in conductivity of several orders of magnitude. The enhanced conductivity has been attributed to the adoption of an “extended coil” conformation by the polyaniline backbone, with the polarons delocalized along the chains [70]. The UV-visible near-infrared spectra of polyanilines are very sensitive to the polymer chain conformation. For example, when PAn · HCSA emeraldine salts are generated by acid doping the EB form with camphorsulfonic acid in organic solvents such as chloroform, DMSO, or NMP [Eq. (4); See Fig. 8], they typically exhibit three absorption bands in the visible region, as shown in Figure 6. The strong band observed in these cases at 750–850 nm has been assigned as a localized polaron band, while the two bands at ca. 430 and 340 nm are attributed to a second polaron band and a – ∗ (bandgap) band, respectively [71]. These spectral features and the moderate electrical conductivity of such salts (ca. 1 S cm−1 ) are considered diagnostic of a “compact coil” conformation for the polyaniline chains. In contrast, no long wavelength localized polaron band is observed in the visible region for PAn · HCSA salts generated by analogous

acid doping in “secondary dopant” solvents such as m-cresol. These much more conducting salts ( > 100 S cm−1 ) instead exhibit an intense free carrier tail in the near infrared region with an absorption maximum at ca. 2500 nm. These latter features are considered diagnostic of an “extended coil” conformation for the polyaniline backbone: EB + HA −→ PAn · HA

(4)

Circular dichroism (CD) spectroscopy provides a particularly powerful tool for probing the polymer chain conformation in related optically active emeraldine salts. For example, the “compact coil” form of PAn·(+)-HCSA in NMP solvent shows the characteristic bands seen in Figure 7. The strong bisignate CD bands at ca. 795 and 720 nm are associated with the localized polaron absorption at ca. 775 nm for this polymer in NMP (see Fig. 6) [72, 73]. Overlapping bisignate CD bands are also observed at lower wavelengths associated with the other two lower absorption bands seen for PAn·(+)HCSA in Figure 6. The CD spectrum of “extended coil” PAn·(+)-HCSA (not shown) bears no similarity whatsoever to that seen in Figure 7 for the “compact coil” form [74]. Polyaniline contrasts with other ICPs in that it has three readily accessible oxidation states (fully reduced leucoemeraldine, partly oxidized emeraldine, and fully oxidized pernigraniline), as shown in Figure 8 [75]. In addition, reversible protonation/deprotonation equilibria occur for two of these oxidation states. Thus, the emeraldine salt form (ES), which is the only electrically conducting form of polyaniline, is typically dedoped at pH > 4 to give nonconducting EB. Conversely, the reverse acid doping of EB with HA acids [Eq. (4)] provides a convenient route to conducting PAn · HA emeraldine salts with a wide range of acids. The pernigraniline oxidation state of PAn can similarly exist as both a protonated pernigraniline salt or as a neutral pernigraniline base (Fig. 8). On the other hand, the fully reduced leucoemeraldine oxidation state appears to exist only in the neutral base form. Polyaniline can be rapidly and reversibly switched between the various forms shown in Figure 8. The pH and redox switching is accompanied by marked changes in colour. For example, alkaline treatment of green PAn · HA emeraldine salts rapidly generates the blue color of EB. The associated UV-visible spectral changes caused by such alkaline dedoping are shown in Figure 8. Emeraldine base exhibits a characteristic exciton band at ca. 600 nm as well

3 Emeraldine salt

60

2.5 2

20

Ellipticity (mdeg)

Absorbance

40

Emeraldine base

1.5 1 0.5

0 330 –20

380

430

480

530

580

630

–40 –60 –80 –100

0 300

–120

400

500

600

700

800

900

1000

1100

Wavelength (nm)

–140 –160

Wavelength (nm)

Figure 6. UV spectra showing EB–ES. Reprinted with permission from [189], C. Boonchu, Ph.D. Thesis, University of Wollongong, 2002.

Figure 7. CD spectrum of PAn·(+)-HCSA.

680

730

780

118

Inherently Conducting Polymer Nanostructures

N

N

2. SYNTHESIS OF INHERENTLY CONDUCTING POLYMERS NANOSTRUCTURES

N

N

n Pernigraniline Base (Purple)

–2e– –4H+

–

+2e– +4H+ H N +•– A

H N

+OH 2n

–

A number of approaches have been used to produce inherently conducting polymer nanocomponents.

+2e– +2H+

–2e –2H+ H N

H N

N

N n

+HA

Emeraldine Salt (Green)

Emeraldine Base (Blue)

–

–2e +2e

–2e– –2H+

–

H N

H N

+2e– +2H+ H N

H N n

Leucoemeraldine Base (Yellow)

Figure 8. Chemical transitions observed in the oxidation/reduction of polyaniline.

as a – ∗ band at ca. 330 nm [76]. Such alkaline dedoping when performed on PAn · HA films also leads to a dramatic reduction in electrical conductivity, the EB form being an insulator. The reversible redox switching of polyaniline films can also be readily monitored by cyclic voltammetry. For example, the cyclic voltammogram of PAn · HCl in 1 M HCl is shown in Figure 9. In the cathodic sweep, oxidation peaks are observed at ca. 0.2 and 0.7 V (vs Ag/AgCl) that may be attributed to the successive oxidations of leucoemeraldine to emeraldine to pernigraniline. polyleucoemeraldine yellow

light green

polyemeraldine green

1.50

2.1. Use of Steric Stabilizers With a view to improving the processability of inherently conducting polymers, a range of colloidal materials have been produced over the past 10 years. The conventional approach involves chemical oxidation of the monomer in the presence of a steric stabilizer such as PVA to produce stable colloidal dispersions [77–79] with particle sizes typically 10–200 m. Recently this approach has been refined and oxidation of aniline in DMSO in the presence of PVA has been used to produce dispersions containing particles as small as 5 nm [80]. In our laboratory we have developed an electrohydrodynamic approach for the synthesis of inherently conducting polymer colloids [81]. This involves the use of a flow-through electrochemical reactor using a high surface area vitreous carbon electrode. Using appropriate flow-through conditions, stable colloidal dispersions containing 100–200 nm spherical particles are readily produced (Fig. 10). This electrochemical approach allows facile incorporation of a wide range of dopants during colloid formation. For example, protein-containing colloids [82], colloids containing corrosion inhibitors [83], and, more recently, optically active colloids [84] have been produced.

polypernigraniline

blue

violet

increasing pH

1.00

current (mA)

0.50

0.00

–0.50

–1.00

–1.50 –0.20

0.00

0.20

0.40

0.60

0.80

1.00

potential (V)

Figure 9. Cyclic voltammogram of plyaniline (HCI) on a glassy carbon electrode; 1 M HCI(aq); 50 mV/s. The potentials at which structure and color changes occur and the change in the potential of the second redox reaction with pH are shown as well. The second oxidation peak moves to a less positive potential with increasing pH.

200 nm Figure 10. Transmission electron micrograph of colloidal polypyrrole nitrate. Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

119

Inherently Conducting Polymer Nanostructures

More recent work [85] has shown that the use of Tiron (Scheme 3) as dopant enables production of monodispersed polypyrrole particles of 50 nm (Fig. 11). Tiron has previously been shown to act as an electrocatalyst in the formation of polypyrrole [86].

2.2. Micellar Polymerization and Microemulsion Routes A novel approach to the formation of ICP nanoparticles has been developed through the use of micellar polymerization and microemulsion techniques. The advantage of such an approach is that the particle size can be predefined by establishing the appropriate size and geometry of the templating micelle (Fig. 12). For example, formation of polyaniline nanoparticles has been achieved via polymerization in a micelle approach using either sodium dodecyl sulfate (SDS) [87] or dodecylbenzenesulfonic acid [88–90] as the surfactant stabilizer. Particle sizes in the range of 10–30 nm with conductivities as high as 24 S cm−1 have been reported. Poly(3,4-ethylenedioxythiophene) (PEDOT) and polyaniline nanoparticles have also been synthesised via the micellar route with sodium dodecyl sulfate and dodecylbenzene sulfonic acid [91]. Polyaniline nanoparticles were spherical and 20 to 60 nm in diameter and PEDOT was from 35 to 100 nm. The observed conductivities of 20 to 50 S cm−1 (by pellet) were higher than those observed for larger micrometer sized particles prepared in bulk solutions. Control of the micelle size utilized in the formation of polyaniline nanoparticles has been achieved by tailoring the stabilizer to modify the resulting micelle dimensions. Kim et al. [92, 93] have used amphiphilic polymer molecules, hydrophobically end-capped poly(ethylene oxide), and varied the hydrophilic PEO midsection length to control the final micelle size. The micelle structures formed were referred to as flower type, with the aniline monomer and hexane cosolvent interacting with the hydrophobic end groups. The resultant nanostructures ranged from 20 to approximately 300 nm in size depending on the molecular weight of the hydrophilic PEO midsection. Metallic and semiconducting nanoparticles have been routinely produced at the sub-5 nm scale for some time but it has been a challenge to produce polymer nanoparticles smaller than this. Jang et al. [94] have reported the synthesis of polypyrrole nanoparticles at the 2 nm scale via a microemulsion route carried out at 3 C. Pyrrole monomer was initially formed into micelles and then oxidized by ferric chloride. Typical surfactants used in

100 nm

Figure 11. Transmission electron micrograph of aggregated polypyrrole tiron colloid particles (×150 K). Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

the preparation of these sub-5 nm particles were quaternary ammonium based cations such as octyltrimethylammonium bromide, decyltrimethylammonium bromide, and dodecyltrimethylammonium bromide. At room temperature these surfactants produced nanoparticles larger than 10 nm while at 70 C ca. 50 nm particles were observed. Nanoscale patterning of polypyrrole dot structures of 80 to 180 nm diameter in self-organized arrays has also been achieved via a block copolymer surface-micelle templated approach [95]. Using this approach, micelle templates were prepared using the Langmuir–Blodgett (LB) technique to deposit an AB diblock copolymer onto a mica substrate followed by the selective chemical deposition of polypyrrole onto these templating structures. Selective deposition is achieved by exploiting the preference of polypyrrole to deposit onto hydrophobic surfaces rather than hydrophilic substrates. An advantage of this approach is that lithographic patterning techniques can be avoided due to the self-assembly of the micelle via the LB technique. Inverse structures of these PPy dots have also been prepared

Oxidant

OH –

OH

SO3

Monomer containing micelle

SO3–

(IV) Scheme 3. Tiron.

Polymer containing micelle

Figure 12. Schematic illustrating micellar/microemulsion nanoparticle synthesis. Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

120 by extended polymer synthesis in the presence of selfassembled polystyrene cores that are selectively removed by toluene. Selvan et al. [96, 97] utilized a block copolymer micelle of polystyrene-block-poly(2-vinylpyrridine) in toluene exposed to tetrachloroauric acid which was selectively adsorbed by the micelle structure. On exposure of this solution to pyrrole monomer, doped polypyrrole was observed to be synthesized concurrently with the formation of gold nanoparticles as a result of the reduction of the bound AuCl− 4 . The nanoparticles formed had a monodispersed (7–9 nm) gold core surrounded by a PPy shell after annealing at 130 C for 1 hour. Depending on the postsynthesis treatment, different shaped nanoparticles such as spherical, cubic, tetrahedral, and octahedral shapes were formed due to micelle coagulation. Dendritic nanoaggregate structures were also reported to form via a vapor phase polymerisation of the pyrrole monomer onto cast films of the block copolymer. In related work by Zhou et al. [98, 99] metal ion reduction of Au, Pd, and Pt salts with a -conjugated polymer, poly(dithiafulvene) (PDF), capable of electron donating to the metal salts thereby reducing them and itself being oxidized to provide steric and electrosteric stabilization of the nanoparticle. The resultant metal cores of the composite nanoparticles were 5–6 nm in size. Interestingly, the surface plasmon absorption band in the UV-vis spectrum for a metal nanoparticle (Pd, Pt) was significantly redshifted to 550 nm from an anticipated 510–525 nm. This shift was attributed to the electronic interaction of the oxidized PDF with the metallized cores. Using a chemical approach involving inverse microemulsions, it has been shown that small monodisperse polyaniline particles can be produced [100–102]. Inverse microemulsion polymerization of polyaniline by Chan et al. [101] was reported to give particles of 10–35 nm diameter that were spherical and could be synthesized using either the chemical or electrochemical route. The presence of stabilizing surfactant was found to be detrimental to the electrical conductivity of these particles when dried, due to formation of an insulating barrier around the nanoparticles in the bulk. After washing to remove the surfactant, chemically prepared materials exhibited conductivities of up to 10 S cm−1 while for electrochemically prepared materials conductivities as high as 200 S cm−1 were reported. Gan and co-workers have used [102] an inverse microemulsion approach that involves formation of barium sulfate nanoparticles, which are then coated by polyaniline. These composite nanoparticles had a reported conductivity from 0.017 to 5 S cm−1 , with particles ranging in size from 10 to 20 nm. Xia and Wang [103] have used ultrasonication during inverse microemulsion polymerization of polyaniline to produce spherical nanoparticles with diameters of 10–50 nm and conductivities in the order of 10 S cm−1 . The ultrasonication was found to increase the rate of polymerization of aniline which is typically slow when the microemulsion route is used. Rate increases are obtained by acceleration of the heterogeneous liquid–liquid chemical reactions in solution. A secondary advantage is the prevention of aggregation due to particle agitation induced by ultrasonication. Polypyrrole nanotubes have also been synthesised via an inverse microemulsion self-assembly route by Jang and

Inherently Conducting Polymer Nanostructures

Yoon [104]. Nanotubes were synthesised by using bis(2ethylhexyl)sulfosuccinate (20.3 mmol) in hexane (40 mL) which form tube- or rodlike micelles, due to the presence of a double tail on a small hydrophilic head group, when aqueous FeCl3 (1 mL, 9.0 M) was added. The Fe3+ cations are effectively trapped in the center core of the micelle. Addition of pyrrole (7.5 mmol) to this solution then results in interfacial polymerization at the micelle surface, resulting in hollow nanotubes 95 nm in diameter and up to 5 m in length. The electrical conductivity of these nanotubes was up to 30 S cm−1 and was observed to be a function of the FeCl3 oxidant concentration. More interestingly, the conductivity of these materials did not appear to be limited by the presence of the surfactant molecule, which in other emulsion polymerization routes tends to lead to a thick insulating coating that limits the observable conductivity.

2.3. Physical Templates Conducting polymers have been grown within zeolites [105, 106]. With channels as small as 0.3 nm this allows assembly of single molecular wires. Conductivities of 10−9 S cm−1 for such wires contained within zeolites have been reported and this increases to 10−2 S cm−1 when the polymer is extracted from the host structure. Martin and co-workers [107–109] have used controlled pore size membranes as templates to electrochemically grow fibrillar mats of ICPs. Similar structures have also been produced using nanoporous particle track-etched polycarbonate membranes with both polypyrrole [110–113] and polyaniline [114] via chemical and electrochemical techniques. The approach involves oxidation of the monomer within the pores of a template. This is achieved electrochemically as illustrated in Figure 13. The electrode substrate used can be either a conventional flat surface system such as solid gold, platinum, or glassy carbon, resulting in structures such as shown in Figure 14a. Alternatively, more novel electrode substrates such as platinized PVDF membranes with nominal pore size of 0.45 m can be used (Fig. 14b). Carbon nanotubes (CNTs) have been directly synthesized via a template carbonization of polypyrrole on an alumina template membrane (Whatman Anopore) with 0.2 m diameter pores and 60 m in total thickness [115]. Polypyrrole was chemically deposited on and through the membrane

Apply E

Membrane with linear pores attached to an electrode

Increasing

Dissolve

Template

Time

Away Template

Removed

Initiate polymer growth within pores

Polymer growth continues

Figure 13. Assembly of fibrillar ICPs using controlled pore size templates. Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

Inherently Conducting Polymer Nanostructures (a)

(b)

Figure 14. Scanning electron micrographs of fibrillar structure of PPy/pTS on (a) a flat electrode surface and (b) a PVDF membrane. Reprinted with permission from [188], V. Misoska, Ph.D. Thesis, University of Wollongong, 2002.

via a chemical oxidative process. Surface deposited polymer was removed by polishing with alumina powder and ultrasonication. This membrane was then carbonized in an Ar atmosphere and then etched in 48% HF to remove the alumina template. Metal CNTs impregnated with Pt, Pt–Ru, and Pt–WO3 were prepared by treatment with the appropriate metal salt solutions prior to etching. The resulting CNTs were single wall type with diameters ranging from 1 to 5 nm. The application of these nanostructures as electrode substrates for methanol fuel cell was explored, utilizing the presence of dispersed metals such as Pt0 and Ru0 metal and W6+ in the CNT, providing catalytic regions for methanol oxidation. Freestanding nanotubes are formed by the selective dissolution of the templating membrane. Hollow tubes of polypyrrole are initially formed in the templating structure followed by the in-filling of these tubes at longer synthesis times. Electrical conductivities of these structures

121 (e.g., 375 S cm−1 at 20 nm) show an enhancement over that of the bulk material (1.5 S cm−1 ) grown under similar conditions [110]. A similar conductivity enhancement effect based on tube diameter has also been reported by Cai et al. with conductivities up to 103 S cm−1 [116, 117]. Applications of template-assisted growth of aligned polyaniline nanofibril arrays in the area of field emission have been explored by Wang et al. [118]. Nanofibril arrays were prepared by chemically depositing polyaniline into 20– 200 nm pores in anodic aluminum oxide film 30 to 60 m thick that was subsequently etched away with 0.3 M H3 PO4 or by ultrasonication [119]. Field emission studies revealed a low electric field threshold of 5–6 V/ m, with a minimum emission current density of ca. 0.01 mA/cm2 and a maximum emission current density of 5 mA/cm2 . Comparisons to other materials with lower emission current densities, such as diamond and CNTs, were made and the authors noted that aligned nanofibril arrays were attractive due to their high surface area, lower manufacturing cost, and robust mechanical properties. Hollow ICP cigar-shaped nanotubes with sealed ends, synthesized via the track-etched polycarbonate template route, have also been reported by Mativetsky and Datars [113]. These materials exhibited a small drop in conductivity as the diameter decreased from 400 to 50 nm, contrary to the previous reports [110–112]. The small decrease in conductivity of the order of 50 mS cm−1 is believed to result from an increase in the electron scattering within the nanocylinder walls or from the presence of large impedances in the nanostructure. The authors attributed this different dependence on tube diameter by treating the tube as a solid cylinder, resulting in conductivity trends observed by others, or as reported in this instance as a hollow nanostructure which gave these results. Others have shown that the electrochemical properties of polyanilines grown in a template such as sol–gel silica or PVDF are greatly improved [120]. This is attributed to improved order at the molecular level. Other templates used to assist assembly of nanostructured conducting polymers are synthetic opals [121–126] based on polystyrene or silica spheres. The area of synthetic opals is of interest for use as photonic bandgap crystals: materials in which photons of a given energy cannot travel through or propagate in a crystal but rather are reflected by the lattice structure [125]. Of further interest is that opal templates provide a route to establishing high range order in the nanodomain, resulting in high surface-to-volume ratios. Template opal structures (Fig. 15) are preformed from monodisperse spherical colloidal particles that are permitted to self-assemble into close packed arrays via a sedimentation process assisted by either gravity or pressure/microfiltration. After template formation, conducting polymer is formed by infiltration of monomer into the void spaces within the crystal structure, followed by subsequent oxidative polymerization. The final stage is the removal of the templating core to leave an inverse structure from the templating material with essentially the same optical properties as the original host [123] (Fig. 16). Others [121, 122] used a similar templating approach, but prepared self-assembled polystyrene (PS) latex opals onto gold substrates followed by the direct electropolymerization of PPy, PAn, and polybithiophene (PBT)

122

Inherently Conducting Polymer Nanostructures

Monodisperse solution (315nm)

Indium-Tin-Oxide glass slide Room temperature for 90 mins

Scanning Electron Microscopy Synthetic Opal

Figure 15. Self-assembly of opal structures onto electrode substrates via monodisperse colloidal dispersions. Reprinted with permission from [188], V. Misoska, Ph.D. Thesis, University of Wollongong, 2002.

into the interstitial void spaces of the host matrix. Upon removal of the PS, structural shrinkage was observed for PPy and PAn but PBT exhibited no such effect. Using this approach, interchanneling between adjacent template layers was evident, giving evidence for the formation of a threedimensional (3D) macroporous nanostructure. Yoshino et al. [124] utilized SiO2 opal templates followed by the infiltration of soluble preformed poly(3-alkylthiophene) and poly(2,5dialkoxy-p-phenylene vinylene) and finally removal of the SiO2 spheres by HF etching. These nanostructured materials were then investigated for photoluminescence (PL) and electroluminescence properties. A spectral narrowing of the PL and an increase in excitation intensity was observed with respect to the solution form of these polymers.

2.4. Molecular Templates Distinctly different from the templating techniques described previously, this approach relies on the interaction at the molecular scale of the conducting polymer and a templating dopant or surfactant/polyelectrolyte to form a nanoparticle. The molecular template not only functions to direct morphology but it can also impart other properties to the ICP nanoparticle, as will be discussed. Others have shown that molecular templates, such as a polyacrylate film predeposited on a carbon electrode, can be used effectively to create polypyrrole nanowires via electrochemical oxidation of pyrrole [127, 128]. Choi and Park [129] have modified surfaces with cyclodextrins to assist in the electroformation of polyaniline nanowires. Others have utilized the concept of molecular templates to form polyaniline supramolecular rods with electrical conductivity improved by 2 orders of magnitude [130].

Host

infiltration

Removal

Opal Template

ICP infiltrated Opal Template

ICP Inverse Opal

Figure 16. Schematic illustrating preparation of an inverse ICP opal using a synthetic opal template. Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

A template guided synthesis of water-soluble chiral conducting polyaniline in the presence of (S)-(−)- and (R)(+)-2-pyrrolidone-5-carboxylic acid [(S)-PCA and (R)-PCA] has been reported to produce nanotubes [131]. The structures prepared have outer diameters of 80–220 nm with an inner tube diameter of 50–130 nm. It was proposed that the tubular structures form as a result of the interaction of the hydrophobic aniline being templated by the hydrophilic carboxylic acid groups of the PCA in aqueous media during chiral tube formation. The resultant tubes were shown to be optically active, suggesting that the polyaniline chains possess a preferred helical screw. McCarthy et al. [132, 133] reported a template-guided synthesis of water-soluble chiral polyaniline nanocomposites. The nanoparticles were prepared by the physical adsorption of aniline monomer onto a templating poly(acrylic acid) in the presence of (+)- or (−)-CSA, followed by chemical oxidation. Using this approach, optically active nanocomposites of approximately 100 nm diameter were formed. Earlier work by Sun and Yang [134] using polyelectrolytes produced similar nonchiral dispersions in which the polyaniline chain is interwound with a water-soluble polymer by electrostatic forces [135]. Similar work by Samuelson et al. utilized DNA as a chiral template for polyaniline [89]. Shen and Wan [136] have shown that chemical oxidation of pyrrole (by APS) in the presence of -naphthalene sulfonic acid (-NSA) results in the formation of tubules down to ca. 200 nm in diameter. The tubular structures had reasonable conductivity (10 S cm−1 ) and were soluble in m-cresol. In later work [137] it was reported that the morphology was significantly influenced by the concentration of the -NSA. Granular morphologies were present when [-NSA] was less than 0.2 M, while on increasing the -NSA concentration fiber formation becomes the dominant morphology. At -NSA concentrations above 2.9 M, a block-type morphology predominated with an associated drop in conductivity to 2 mS cm−1 . Monomer concentration was shown to have little effect on the resultant morphology and electrical conductivity. The morphological steps in tube formation were described as grain to short tube to long tube. Other morphology types have also been noted using the -NSA anion, namely hollow microspheres ranging from 450 to 1370 nm (with inner wall thicknesses of 20 to 250 nm, respectively) and conductivities of up to 190 mS cm−1 [138]. A similar approach has been extended to polyaniline [139]. The influence of a secondary inorganic acid codopant has also been demonstrated, with conductivities from 10−1 to 100 S/cm [140]. Others [141] have used lipid tubules as templates during chemical oxidation of pyrrole to form nanofibers with diameters between 10 and 50 nm and lengths reaching up to several hundred hundreds of micrometers. The concept of using molecular templates has been taken a step further by Zhang and Wan [142] by the incorporation of 10 nm Fe3 O4 nanomagnet particles into the -NSA nanorods and nanotubes that are 80–100 nm in diameter. These PAni–-NSA/Fe3 O4 nanostructures were observed to exhibit superparamagnetic behavior (i.e., hysteresis loop effects). More importantly, both the electrical conductivity and magnetic properties of these nanoparticles could be

Inherently Conducting Polymer Nanostructures

manipulated by control over the loading level of the nanomagnetic particles. Increased loading of the Fe3 O4 nanoparticle decreased the conductivity from ca. 70 mS cm−1 for no loading to ca. 10 mS cm−1 at 20 wt%, while inducing superparamagnetic behavior in the nanocomposite. Self-assembled polyaniline nanofibers and nanotubes, which also exhibit photoisomerization functionality, have been described by Huang and Wan [143] using azobenzenesulfonic acid (ABSA) as the molecular templating surfactant, dopant, and photoactive agent. Nanostructures formed were similar to those by Wan et al. [131, 136, 138, 140, 142] discussed previously, with diameters of 110–130 nm and fiber lengths of 3–8 m. The photoinduced isomerization, from trans to cis, of the ABSA dopant was observed by UV-vis spectroscopy at 430 nm (n– ∗ transition) after irradiation of the nanocomposites at 365 nm (over 0, 2, 6, 10, and 12 min) by UV-vis spectroscopy from 300 to 800 nm. The photoisomerization of the composite material was observed to be slower than for ABSA due to steric hinderance as a result of the trans-ABSA interacting along the polymer backbone.

2.5. Nontemplated Routes There are a number of synthetic routes that are capable of producing nanoparticles in the absence of a precursor substrate or a molecular template. For example, He and co-workers [144] have described a technique wherein polyaniline is deposited between two nanoelectrodes to form a nanojunction. As the junction is decreased in size to just a few nanometers, abrupt switching characteristics are observed as opposed to the more gradual transition observed during electrochemical switching of polyaniline structures with larger dimensions. A further report from the same laboratory [145] added a new dimension by initiating growth of polyaniline between a scanning tunneling microscope (STM) tip and a gold substrate. The STM tip was modified to allow only a few nm to take part in the growth of the nanowire. During growth the tip and substrate were monitored at 20–100 nm resolution. After growth the nanowire could be stretched by moving the STM tip to produce wires up to 200 nm in length and with diameters as low as 6 nm. These structures had conductivities of about 5 S cm−1 . Electrodeposition of polyaniline containing [C60 ] fullerene as dopant [146, 147] has been shown to result in 2D or 3D fibrillar structures [147, 148]. Diameters from 10 to 100 nm with fibril lengths of up to 3000 nm were observed. The fibrillar network had conductivities in the range 10–100 S cm−1 . Two-dimensional networks have been prepared using pulsed potential techniques while 3D networks were observed to form over longer synthesis times. Via this approach there is apparent control over the density of contact points for the individual nanonetworks. Others [148] have used electron bean lithography to create polythiophene-based nanopatterns withs linewidths down to 50 nm and gaps of 10 mm. They have shown that after exposure to a 50 kV electron beam poly(3octylthiophene) becomes less soluble in chlorobenzene, enabling development of nanotracks that have conductivity of the order of 1 S cm−1 .

123 Electrospinning [149, 150] is another recent nontemplated method. This simple approach is based on the electrostatic fiber spinning of composite fibers of polyaniline with polyethylene, oxide, polystyrene, or polyacrylonitrile. These fibers are formed when a high electric field (5–14 kV) is placed between the tip of a metallic anodic spinning needle loaded with the dissolved polymer solution (0.5–4 wt% PAn and 2–4 wt% host polymer) and an opposing cathode plate separated by 20 cm. The presence of the high electric field results in the surface tension of the polymer-loaded solution at the needle tip to be exceeded, expelling a polymer fiber from the surface toward the opposing cathodic plate. The transit time from the anode tip to the cathode plate is accompanied by a desolvation and drying process in part assisted by the electrostatic charges placed upon the solvent molecules causing electrostatic repulsion. The resulting nanofiber composite is reported to have lengths in the meter range and is collected as an interwoven mesh with large surface-to-volume ratios (∼103 m2 /g). These fibers are ohmic in nature. Fiber dimensions of less than 100 nm have been routinely produced by this technique. More recently [149] fibers of PAn have been directly spun from a 20 wt% solution of PAn (VersiconTM ) in 98% sulfuric acid at 5 kV. The electrospinning method has also been extended to produce continuous polyaniline/polyethylene oxide monofilament nanofibers down to 60 nm diameter at a maximum spin rate of 1130 m/min and a conductivity of 33 S cm−1 [151]. Aligned polymer nanowire structures have been developed using an electrochemical deposition technique to form aligned polyaniline arrays on smooth and textured electrode substrates without the need for a porous templating structure [152, 153]. Oriented polymer nanowires were formed using a preprogrammed constant-current deposition method where the current density was stepped down throughout the nanowire growth. In the initial stage of nanowire growth, 50 nm polymer nuclei were deposited from an aqueous electrolyte containing 0.5 M aniline and 1.0 M perchloric acid at a current density of 0.08 mA cm−2 over 30 min. The current density was subsequently stepped down over a 3 hr period from 0.04 to 0.02 mA cm−2 . Extended polyaniline growth in the initial nucleation phase resulted in the formation of a thick polymer deposit with randomly oriented, thick-branched polymer structures over the surface. Typical nanowire structures had diameters for 50 to 70 nm and lengths of approximately 0.8 m. Complex surface geometries have been demonstrated using this technique where nanowires have been deposited onto ordered silica spheres deposited onto the electrode substrate as well as onto textured surfaces to produce hierarchical structures. The potential for these surfaces in chemical sensing has been demonstrated for H2 O2 detection utilizing iron(III) hexacyanoferrate nanoparticle catalysts supported by the nanowire structure. Polypyrrole nanowire have also been electrochemically synthesized on graphite/paraffin composite electrodes from a 0.2 M phosphate buffer (pH 6.86) electrolyte containing 0.2 to 2.0 M lithium perchlorate and 0.1 to 0.5 M pyrrole [154]. Nanowires were grown by cyclic voltammetry and potential step methods. Nanorod growth is directed by inducing microstructure at the electrode surface. This was achieved by predigesting the graphite rod electrode in

124 boiling 10% hydrochloric acid and then boiling 10% nitric acid for 30 min, respectively, resulting in a porous graphite rod surface which was subsequently in-filled with paraffin at 150 C and finally polished. The resultant electrode surface was covered with nitro-groups to which the pyrrole monomer selectively adsorbs providing a nucleation zone for nanotube formation. The resultant tubes were unaligned and irregularly dispersed across the surface as an entangled mesh with diameters of ca. 50–200 nm. Thin 50 to 100 nm freestanding polypyrrole nanofilms have been synthesized utilizing oxidative interfacial polymerization at a water–chloroform interface [155]. Thicker 3 to 4 m polypyrrole films have been produced via this method [156]. Control over the final film thickness was achieved by removal of the immiscible solvents from above and below the nucleating film after a few minutes of polymerization. Stirring of the interfacial solutions resulted in the inhibition of polymer nucleation at the interface. Thin 5 nm polypyrrole films on mica and graphite surfaces have been formed via a admicellar polymerization route [157]. The thin films were formed due to the surfactant (CTAB and SDS) in solution adsorbing to the substrate surface as a thin film micelle containing the pyrrole monomer, which was subsequently polymerized by ferric chloride solution. On the graphite surface the polypyrrole morphology consisted of thin discs and interlinked islands. On the mica surface the morphology consisted of randomly scattered islands with no continuous film formation evident. Dip-pen nanolithography has been performed on a number of surfaces via a coated atomic force microscope (AFM) tip at the 130 nm scale up to several micrometers using SPAN as the writing ink [158]. Transfer of the conducting polymer ink from the AFM tip to the substrate is achieved via capillary action upon tip contact to the substrate surface with 0.5 nN contact force. The deposited SPAN tracks were found to electroactive. Near-field optical lithography has also been performed to draw nanostructures of poly(p-phenylene vinylene) (PVP) using light delivered from a scanning near-field fiber probe, with a 40 or 80 nm optical aperture, on a scanning nearfield optical microscope AFM device [159]. Polymer features are drawn by exposing the PVP soluble precursor to ultraviolet light (325 nm HeCd laser, 0.2 mW at the sample surface) rendering the exposed precursor insoluble and permitting the dissolution of the remaining polymer. Following chemical conversion for the exposed PVP precursor a fully conjugated polymer can be achieved. Two-dimensional photonic crystal arrays have been demonstrated by this method with pillars 32 nm tall, 200 nm wide (at half height) and a lattice constant of 333 nm (ca. 30 × 30 dot lattice).

Inherently Conducting Polymer Nanostructures

direct incorporation of a wide range of dopants, including in this case the optically active camphor sulfonate ion to produce chiral particles with nanodimensional ICPs incorporated therein (Fig. 17). Applications of conducting polymers to nanosized semiconductor particles, commonly referred to as quantum dots, have also been made to produce nanocomposite materials [162]. These materials are typically spherical semiconductor crystals of 5 to 10 nm in size, such that quantum confinement effects predominate within these structures. At this size, these nanocrystals have special optical and electronic properties resulting from these quantum effects. A prime application of these materials is that tight size control will influence the onset of absorption and the fluorescence wavelength, thereby affecting the color of the observed particles in solution. CdS and Cu2 S quantum dots were coated by mixing into an N -methylpyrrolidinone (NMP) solution containing the EB form of polyaniline. Extended exposure of the EB to these nanoparticles resulted in the oxidation of the EB to the pernigraniline base form. Co-precipitation of the EB and nanocrystals in NMP was achieved by the addition of ethanol. For CdS, electron transfer from the polymer to the nanocrystals is energetically favorable. Similarly, excitons created on the nanocrystals can also be transferred to the polymer resulting in an electron on the nanoparticle and an electron hole in the polymer. This creates a potential to control the performance of photovoltaic devices by altering the nanocrystal’s size and concentration. In another example, Bremer and co-workers [163] polymerized polyaniline onto nanodimensional polyurethane particles dispersed using a steric stabilizer (Fig. 18). We

2.6. Nanocomposites The final approach to be reviewed involves the use of preformed nanosized structures to assist in the formation of ICPs with nanodimensions. For example, Pope and coworkers introduced the concept of using silica nanoparticles to stabilize inherently conducting polymers [160]. We subsequently used this approach to produce polyaniline based silica nanocomposites [161] using an electrohydrodynamic synthesis technique. The electrochemical approach allows

622 900954

80.0 KV

X120 K

50 nm

Figure 17. High magnification transmission electron micrograph of PAn·(+)-HCSA/silica nanocomposites. Adapted with permission from [16], V. Aboutanos et al., Synth. Met. 106, 89 (1999). © 1999, Elsevier Science.

125

Inherently Conducting Polymer Nanostructures

Conducting polymer “shell”

(a)

(b)

(c)

(d)

Outer stabiliser layer

Colloidal material “core” (eg. polyurethane) Figure 18. Cor–shell nanoparticles. Reprinted with permission from [4], G. G. Wallace and P. C. Innis, J. Nanosci. Nanotech. 2, 441 (2002). © 2002, American Scientific Publishers.

adapted this approach to produce highly optically active nanoparticles (20 nm) based on polyaniline doped with (+)camphor sulfonate coated onto polyurethane particles [164]. A major disadvantage of using this approach is that these structures commonly aggregate to form raspberry-like structures during the polymerization process [161]. In an attempt to minimize this affect, Xia and Wang synthesized polyaniline nanosilica [165] and nanocyrstalline titanium oxide composites [166] while applying ultrasonic irradiation during the chemical polymerization process. Although aggregation was evident, the resultant nanoclusters of 60 to 120 nm were obtained from the initial 15 to 40 nm SiO2 cores. Similar results were observed for PAn–TiO2 nanocomposites. Other morphologies that resulted from the ultrasonication included thread-like and sandwich-like aggregated structures. Nanocomposites have also been fabricated utilizing aqueous vanadium or iron oxide sols and polyaniline to produce particles characterized by X-ray crystallography as ca. 9 and 13 nm in diameter, respectively [167]. Strong interaction between the inorganic V2 O5 core and its polyaniline shell indicates that the PAn–V2 O5 nanoparticle functions as an intercalation compound, with electron transfer from the unshared electron pair of the nitrogen atom to the V5+ ion evidenced by electron paramagnetic resonance (EPR) spectroscopy. In the case of a Fe3 O4 core, the ferromagnetic character of the Fe3 O4 was the dominating feature of the EPR spectrum. However, infrared spectroscopy indicated that electronic interactions are occurring between the Fe3 O4 core and the conducting polymer shell. Recent work aimed at other objectives may also provide an interesting route to the production of ICP-containing nanocomposites: this is in the use of CNTs as host “particles.” CNTs in their own right possess interesting electronic properties and when combined with other conjugated polymers synergistic effects have been observed [168, 169]. Our studies have focused on the use of aligned carbon nanotubes as templates for ICP deposition. Our recent work in this area [170] has revealed significant improvements when PPy glucose oxidase is coated onto aligned CNTs for use as biosensors (Fig. 19). We have also shown that ICPs can be highly effective dispersants for CNTs, presumably attaching

Figure 19. Aligned carbon nanotubes with a bioactive conducting polymer. (a) Pure CNT array before treatment. (b) Aligned CP-CNT coaxial nanowires; inset shows clear image of single tube coated with PPY. (c) PPY only deposited on the top of CNT surface due to high density of tubes. (d) Polymer formed on both outside of walls and the top of the surface of the CNT array. Reprinted with permission from [170], M. Gao et al., Electroanalysis, 15, 1089 (2003). © 2003, Wiley-VCH.

themselves to the CNT structure. The properties of these novel dispersants of nanomaterials are yet to be investigated. The molecular template concept used to produce polyaniline nanotubes [136] (as discussed) has also been used prepare to sulfonated multiwalled carbon nanotubes [171]. In this instance, the sulfonated nanotube surface serves a dual role acting as an in-situ dopant for the polyaniline as well as providing a supramolecular template onto which the anilinium ions in solution bind. Polymerization therefore occurs predominantly at the sulfonated nanotube surface. The resultant hybrid structure consists of a 10 nm thick polyaniline coating on the CNT surface, providing a novel way of introducing and functionality of to CNTs.

2.7. ICP Surfaces for Facilitating Metal Nanoparticle Growth Utilizing ICPs as potential platforms for supporting nanoscale structures has been investigated to provide novel supported catalysts. The concept of embedded metal nanoparticles was recently presented by Zhou et al. [98, 99]. In this instance the metal nanoparticles were embedded in the polymer matrix and therefore protected from the chemical environment. Wang and co-workers [172] employed a similar approach by chemical reduction of AuCl3 or Pd(NO3 2 by polyaniline powder in either aqueous dispersions or in NMP, a common solvent for the EB form of PAn. Reduction of palladium nanoparticles was slow with EB; hence the polyaniline was initially reduced to the leucoemeraldine base form in order to facilitate reduction of the Pd2+ to its metallic state. The nanoparticle sizes for the Au and Pd were observed to be 20 nm from NMP solution and 50– 200 nm from polyaniline powder. Gold–polyaniline nanocomposites have been prepared by the chemical reduction of

126 HAuCl4 and simultaneous polymerization of aniline to produce 26 nm Au nanoparticles dispersed in the polyaniline powder matrix [173]. Platinum and platinum oxide nanoparticles, 4 nm in size, have been successfully deposited onto polypyrrole/polystyrene sulfonate colloidal dispersions (with dimensions ranging from 30 to 1000 nm) [174] for use as catalysts. Pt(NH3 )4 Cl2 and Na6 Pt(SO3 )4 metal salts were reduced by formaldehyde or H2 O2 respectively in order to form Pt nanoparticles on the ICP colloid surface. These nanocomposite dispersions have been reported to be effective catalysts for oxygen reduction in a proton exchange membrane fuel cell, indicating that the metal catalytic particles are accessible for chemical interaction. Polythiophene has also been investigated as a potential substrate for the formation of catalytic platinum nanoparticles, 2–4 nm in size, for oxygen reduction. The Pt nanoparticles were homogeneously deposited across a predeposited polythiophene film by direct electrodeposition from a solution of 12.5 M H2 SO4 and 4 mM H2 PO4 . The resulting composite was produced in a similar fashion as other electrocatalysts such as carbon-supported platinum.

3. UNIQUE PROPERTIES AND APPLICATIONS OF INHERENTLY CONDUCTING POLYMERS NANOSTRUCTURES As ICPs approach nanodimensions a number of trends become evident which involve either the preservation of the bulk properties of the ICP and/or the enhancement of the properties of the nanomaterial such as conductivity. Some of these influences of nanostructure reported in the literature are discussed. He et al. [144, 145] have created a conducting polymer nanojunction switch with abrupt switching characteristics that are attributed to a single nanocrystalline domain in which the individual polymer strands collectively undergo the usual insulator–conductor transition. Nanoelectronics will be an expanding area of application for ICPs. A recent example has been illustrated for the fabrication of organic photodiodes and light emitting diodes [175]. Fullerene derivatives grafted onto soluble polythiophene, based upon the PEDOT system, have been utilized to form an integrated donor–acceptor polymer system. Nanopatterned electrodes were prepared by using a silicone rubber replica stamp which was used to transfer an organic resist image of an optical grating (280 nm periodicity) on top of a metal film. This film was subsequently etched to form a nanopatterned metal electrode substrate, onto which the polymer was cast forming the electronic device. Short submicrometer/nanoscale distances between the metallic electrodes were important due to the polymer layer having a low charge carrier mobility (exciton), emphasizing the importance of nanofabrication for these devices. Numerous reports have discussed the fact that nanocomponents have inherently greater conductivity than the macroscopic analog [111]. This is not surprising given the model described by Epstein [176] and Kaiser [177] in which the bulk properties of ICPs arise from crystalline (highly

Inherently Conducting Polymer Nanostructures

conductive) nanodomains in a sea of amorphous (less conductive) material. A number of experimental approaches [178] have been used to verify the presence of these domains. Formation of these nanodomains provides us with a unique opportunity to access these regions of high order and subsequently higher conductivity. A challenge in accessing enhanced electrical properties is that many approaches for the formation of nanoparticles require the use of an insulating surfactant. This was noted by Chan et al. [101] where removal of the insulating surfactant required for nanoparticle formation caused the conductivity to increase. Others [179] have shown that dramatic improvements in specific capacitance for polythiophene tubules can be obtained, increasing from 4 C/g polymer for a flat film up to 140 C/g polymer and 148 C/g polymer for micro- and nanotube formations, respectively. The realization of efficient photovoltaic devices based on inherently conducting polymers also depends on the ability to control structure at the nanodimension [180, 181]. Such materials generate electron/hole pairs (excitons) upon irradiation. The ability to generate electricity from these charge pairs depends on our ability to split these charges at appropriate interfaces. Unfortunately, the lifetime of excitons is short and only those formed within 10 nm of an interface will ever reach it. The development of efficient nanostructures is therefore critical to this area. Metallic nanoparticles exhibit well-documented phenomena of surface plasmon absorption processes where the free electrons at the surface of the nanoparticle generate a dipole moment to interact with visible wavelengths [182, 183] when the particle is too small itself to interact with light by Bragg scattering processes. It has been reported that the surface plasmon absorption can interact with an encapsulating -conjugated polymer when it is oxidized [98, 99]. This interaction is reported to be due to the direct electronic communication of the surface electrons of the metal nanoparticle with the encapsulating polymer. The result of this interaction is a redshift of the optical plasmon absorption band with respect to a lone metal nanoparticle. This shift in absorption can be achieved by an aggregation process of the metal nanoparticle, but no such aggregation was evident implying direct electronic interaction with the polymer and metal electron. Xia and Wang [103] elegantly demonstrated the advantages and pitfalls to be encountered as we enter the nanoworld. As particle size decreases the degree of doping and crystallinity increases, with a concomitant increase in conductivity. They also showed crystalline domain sizes ranging from 10 to 40 nm and that as size decreases lattice defects may occur causing the crystalline structure to be broken due to the smaller size. An industrial application of polyaniline utilizing nanodispersions of the ICP into corrosion inhibiting paint latexes has been developed whereby 70 nm particles are dispersed to form a self-organised complex of ultrafine network [184]. For a number of years it has been demonstrated that ICPs can impart corrosion resistance to metal substrates. When these nanoparticles have been melt dispersed with a loading of 1% w/v in paint, the composite material is observed to have similar metallic properties to the bulk

127

Inherently Conducting Polymer Nanostructures

material. This arises due to the nanoparticle forming a network structure permitting electron tunneling via a continuous conducting pathway [185]. A clear advantage of utilizing these nanodispersions is that a similar corrosion inhibitor activity is achieved to that found using larger particle sizes, requiring higher polymer loading, or coated films. Mandal has shown that the use of polyaniline nanocomponents results in extraordinarily low (0.03 vol%) percolation thresholds in polyvinylalcohol [186]. Polypyrrole–iron oxide nanocomposites have been prepared for application as humidity and gas sensors [187]. Improved sensor sensitivity is achieved by increasing the surface area based upon microstructure. By this approach nanoparticles were prepared by the reaction of ferric nitrate and methoxyethanol in the presence of pyrrole to produce a nanostructured aggregate with the appropriate morphology and surface area to function as a sensor. Recently we have used aligned carbon nanotube platforms to fabricate inherently conducting polymer based biosensors with excellent performance characteristics [170].

4. CONCLUSIONS The unique properties of materials brought about by control of structure at the nanodimension have generated much excitement in the research community in recent times. Those of us interested in the modification and efficient utilization of inherently conducting polymers have not escaped the nanotechnology euphoria. Potentially, the benefits to be reaped from control of the ICP structure from the nanoscale to the microdomain and beyond are substantial. The more highly ordered structures obtained are expected to yield novel electronic, electrochemical, mechanical, chemical, and photoelectrochemical properties. Researchers have developed innovative strategies and approaches that enable controlled dimension nanocomponents to be produced. As with all areas of nanotechnology, the challenge is to assemble these nanocomponents into devices or structures that display the benefits of nanodimensional control.

GLOSSARY Dopant An anionic additive incorporated into the matrix of an inherently conducting polymer to provide electrostatic neutrality to the cationic charges on the polymer backbone and to modulate the electrical properties of the polymer. Emeraldine (salt or base) The half oxidized state of polyaniline. When protonated as the salt polyaniline is conducting. As the base the polymer is an insulator or poorly conducting. Inherently conducting polymer A polymeric material which conducts and electrical current by means of its electronic structure, namely via -bond conjugation along the polymer backbone, and has no other conducting fillers added. Conductivity can be modulated by the addition of dopant anions to the polymer matrix and controlling the polymer oxidation level. Leucoemaraldine (base) The fully reduced and insulating form of polyaniline.

Monomer A single repeat unit within a polymer chain structure. Pernigraniline (salt or base) The fully oxidized and non conducting form of polyaniline. Polyelectrolyte A polymer which has either anionic or cationic functionality attached directly to or a as a part of the monomer repeat unit. Anionic or cationic character can be controlled by the number of repeat units, with these functionalities, incorporated into the polyelectrolyte structure. Polymerization The process whereby single monomer units are reacted to form a larger continuous molecular chain structure. Redox Reduction (electron loss) and oxidation (electron gain) processes.

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Encyclopedia of Nanoscience and Nanotechnology

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Inorganic Nanomaterials from Molecular Templates Sanjay Mathur, Hao Shen Institute of New Materials, Saarbrücken, Germany

CONTENTS 1. Introduction 2. Chemical Synthesis of Nanomaterials 3. The Metal-Organic Precursor Approach 4. Methods for Molecule-to-Material Transformation 5. Molecular Clusters as Building Blocks to Nanoparticles and Nanostructured Films 6. Conclusions Glossary References

All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical mathematics of the city of heaven. —Sir Thomas Browne, 1658

1. INTRODUCTION Typically, a nanostructure is a material structure assembled from a layer or cluster of atoms with size on the order of nanometers [1–5]. A number of methods exist for the synthesis of nanostructured materials, including synthesis from atomic or molecular precursors (chemical or physical vapor deposition, gas condensation, chemical precipitation, aerosol reactions, biological templating, etc.), processing of bulk precursors (mechanical attrition, crystallization from the amorphous state, phase separation, etc.), and processes ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

in nature (e.g., biological systems) [6–12]. Although it is general, this definition underlines the fact that a positional control over constituting atomic or molecular units is inherently associated with the synthesis of nanostructured materials. When evaluated dimensionally, the domain of nanoscale structures ( c, it is parallel. Due to its intrinsic approximations, the BH model works well at intermediate ion doses, when erosion produces ripples with a small local slope. The consequences of the BH model have been verified in many experiments on amorphous and semiconductor materials. The ripple rotation at grazing incidence angles has been verified in graphite, glass, SiO2 , diamond, Si, GaAs, and Ge. The dependence of the ripple structure on the ion energy is in the erosive terms vx* y : they depend on the way in which the energy is released in the surface, and then they are related to the penetration depth of the ion, which is a function of energy. Then we expect that the ripple wavelength is a function of the energy. More precisely, we expect that 4x* y ∝ Ep

(10)

Such a behavior has been verified on graphite [39] (p = 1), Si [45] (p = 08), and diamond [40] (p = 1). However, the

BH model foresees a flux dependence of the ripple wavelength (4 ∼  1/2 ) which has been observed only in Si [41]. In the BH model, the behavior of the wavelength with temperature is determined by the surface mobility of the diffusing species. By writing the surface diffusivity D as   Ea (11) D = D0 exp − kB T where Ea is the activation energy for surface self-diffusion, the coefficient K of Eq. (8) depends on the temperature as   E D 38 exp − a (12) K = 20 n kB T kB T and on the wavelength as 4x* y ∝ T

1/2



E exp − a 2kB T



(13)

The effect of surface mobility during ion-beam treatment has been studied experimentally on graphite. The experiment shows that the ripple wavelength increases slowly with temperature, from 70 to 110 nm between 300 and 450 K as expected, due to the increase of the surface mobility. Comparing the results with the BH model, Habenicht obtained an activation energy of Ea = 014 eV, which is rather small compared to others. He ascribes this fact to radiationenhanced diffusion effects that should be taken into account. However, the energy deposition of the ion beam into the near surface region can affect several parameters of the surface mobility, such as the density of diffusing species, the surface free energy, and the activation energy for surface diffusion. The model cannot account for the formation of ripples at low temperature since it considers only a thermal mechanism of smoothing. Additional mechanisms have been proposed to explain the inadequacy of BH model. In particular, Makeev and Barabasi [46] demonstrated that ion sputtering could produce an effective surface diffusion (ESD) that does not imply mass transport. The ESD explains the presence of ripples at low temperatures where the surface diffusion is not activated, and was predicted in the computer simulation of the ripple formation on amorphous carbon [47] and observed in the case of GaAs [48]. Besides its ability to explain most of the experimental results, the BH model fails when the ion dose becomes larger and larger. Cuerno and Barabasi [49] revisited the BH model, introducing two new terms in the equation: /2 h /v  /h /2 h /h = −v0 + 0 +vx  2 +vy  2 /t / /x /x /y  2  2 2y /h 2 /h + x + +K1 2 1 2 h+2x*y*t (14) 2 /x 2 /y The coefficients 2x , 2y depend on parameters such as the penetration depth and the angle of incidence. In this way, the equation becomes nonlinear, and the temporal evolution of the height h shows a transition from a periodic to a fractal behavior, as found in the experiments.

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Ion Sputtering on Metal Surfaces

We have seen that the BH model (and its modifications) is able to reproduce most of the experimental findings in semiconductors and amorphous materials. But when we try to apply it to the metal case, some difficulties arise. First, the spatial periodicity increases in time, both in the diffusive and erosive regime. This effect is out of the model. Second, periodic patterns also appear at normal incidence, while in the model, no regular pattern can grow if  = 0 . Finally, the nanostructures on metals are crucially dependent on temperature (in shape and periodicity). On the other hand, many features have their behavior foreseen by the model: the periodicity depends on the ion flux as  1/2 , at grazing incidence the ripples are always parallel to the ion beam, and so on. In order to explain these differences, we proposed to modify the diffusion terms in Eq. (14) [24]. In fact, while the erosive terms have an expression that is quite general, which depends only on the parameters of the sputtering, the diffusion terms are too simple, as they do not take into account many aspects of the diffusion processes in metals: the existence of different energetic barriers, the directionality of the diffusion, and the existence of different diffusing species. However, a detailed study of the equation and its consequences is still in progress.

evidence of QD fabrication via ion erosion in the case of GaSb (100) [52, 53]. They observed that ion erosion creates dots remarkably well ordered and uniformly distributed. The method of ion erosion may therefore open a new road for nanofabricating large-area surfaces patterned at low cost. But the potential applications are also in other fields of material science. Using ion erosion, it was recently possible to fabricate magnetic wires structures in the case of a Co film deposited on a Cu (100). The experiment shows a remarkable change in the magnetization anisotropy of the film after bombardment by ions. 5 ML of Co have been deposited on a Cu (100) surface at T = 180 K. The sample has been successively bombarded with Ar+ ions at an incident angle of 70 (erosive regime), E = 1 keV; in this condition, the ripples supported on Cu (100) are elongated along the projection of the ion beam on the surface. Finally, ion erosion can produce nanoscale patterned substrates for deposition methods either on metallic substrate or glasses [54]. Rippled glasses are also expected to be good candidates as templates, for instance, for aligning liquid crystals or organic molecules.

6. APPLICATIONS

In this chapter, we reviewed the results on surface nanostructuring by ion sputtering, mainly in the case of metal surfaces. The surface morphology is found to be dependent on surface symmetry if the temperature is high enough and the incidence angle is close to zero (normal incidence). In this diffusive regime, the diffusion of the surface defects (adatom and vacancy clusters) created by the ion impact determines the evolution of the morphology. On the contrary, in the erosive regime (low temperature and grazing incidence), the ripples created are determined only by the ion beam direction. The possibility to pattern metal surfaces on a nanometer scale opens a new route to many applications in nanotechnology since the ion sputtering method can be applied easily and also adapted to large-scale production.

In this section, we illustrate some applications of the ion sputtering in different fields. The advantage of the method is that it is very easy to perform, can be used to nanostructure large samples (several cm2 ), and the results depend on easily controlled parameters, such as ion beam energy, incidence angle, ion species, ion flux and dose, and substrate temperature. Ion erosion has interesting applications in chemistry. Creating and controlling the density of artificial defects may lead to the growth of materials with unusual catalytic properties. In fact, artificially created surface defects can enhance the surface reactivity. One illuminating example of this possibility is the experiment of Costantini and co-workers [50, 51], who recently demonstrated that the dissociation probability of O2 on Ag(100) can be enhanced by orders of magnitude after modification with ion sputtering. The most interesting result is that the reactivity can be tuned by changing the angle of sputtering. The dissociation probability of O2 molecules changes from less than 10−3 on the flat surface to 0.1 on the rippled surface. In perspective, this straightforward result opens up the possibility to control the chemical properties of a surface by modifying in-situ its morphology, that is, by creating different amounts and different kinds of defects, like kinks at step edges, on which the reactivity of the chemisorbed species can be enhanced. The recent observation of similar structures on Rh (110), which were investigated by X ray diffraction, suggests possible applications in catalysis. The fabrication of quantum dots (QDs) is currently a topic of great interest due to the potential applications in optoelectronics. However, the methods available for their fabrication are rather limited because the actual lithographic techniques cannot produce QDs of appropriate size and density for device applications. Facsko et al. gave the first

7. CONCLUSIONS

GLOSSARY Atomic force microscope (AFM) Microscope based on a cantilever with a sharp tip, which is deflected by the forces acting between the surface and the tip. An AFM can resolve atoms on insulating materials. Low energy electron diffraction (LEED) Instrument based on the reflection of electrons from a surface. It gives information on the simmetry of the surfaces. Scanning tunneling microscopy (STM) Microscope based on the tunneling effect, able to resolve atoms on a metallic surface. Secondary ions mass spectroscopy (SIMS) Analytical technique that uses ion sputtering to extract other ions from the surface. These secondary ions are analyzed with a mass spectrometer, in order to get information on the chemical species present on the surface. Spot profile analyzer–LEED (SPA–LEED) A variance of normal LEED, used to observe symmetry on a larger scale.

308

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Ion Sputtering on Metal Surfaces 29. S. van Dijken, D. de Bruin, and B. Poelsema, Phys. Rev. Lett. 86, 4608 (2001). 30. F. Buatier de Mongeot, G. Costantini, C. Boragno, and U. Valbusa, Phys. Rev. Lett. 84, 2445 (2000). 31. C. Teichert, C. Ammer, and M. Klaua, Phys. Status Solidi A 146, 223 (1994). 32. R. M. V. Murty, T. Curcic, A. Judy, B. H. Cooper, A. R. Woll, J. D. Brock, S. Kycia, and R. L. Headrick, Phys. Rev. Lett. 80, 4713 (1998). 33. G. Costantini, F. Buatier de Mongeot, C. Boragno, and U. Valbusa, Phys. Rev. Lett. 86, 838 (2001). 34. M. Navez, C. Sella, and D. Chaperot, Compt. Rend. 254, 240 (1962). 35. X.-S. Wang, R. J. Pechman, and J. H. Weaver, J. Vac. Sci. Technol. B 13, 2031 (1995). 36. H. J. W. Zandvliet, H. B. Elswijk, E. J. van Loenen, and I. S. T. Tsong, Phys. Rev. B 46, 7581 (1992). 37. T. M. Mayer, E. Chason, and A. J. Howard, J. Appl. Phys. 76, 1633 (1994). 38. S. Habenicht, W. Bolse, K. P. Lieb, K. Reimann, and U. Geyer, Phys. Rev. B 60, R2200 (1999). 39. S. Habenicht, Phys. Rev. B 63, 125419 (2001). 40. A. Datta, Y.-R. Wu, and Y. L. Wang, Phys. Rev. B 63, 125407 (2001) 41. J. Erlebacher, M. J. Aziz, E. Chason, M. B. Sinclair, and J. A. Floro, Phys. Rev. Lett. 82, 2330 (1999). 42. D. Datta, S. R. Bhattacharyya, T. K. Chini, and M. K. Sanyal (2001). 43. D. P. Adams, T. M. Mayer, E. Chason, B. K. Kellerman, and B. S. Swartzentruber, Surf. Sci. 371, 445 (1997). 44. R. M. Bradley and J. M. E. Harper, J. Vac. Sci. Technol. A 6, 2390 (1988). 45. J. J. Vajo, R. E. Doty, and E. Cirlin, J. Vac. Sci. Technol. A 14, 2709 (1996). 46. M. A. Makeev and A. L. Barabasi, Appl. Phys. Lett. 71, 2800 (1997). 47. I. Koponen, M. Hautala, and O.-P. Sievanen, Phys. Rev. Lett. 78, 2612 (1997). 48. S. W. MacLaren, J. E. Baker, N. L. Finnegan, and C. M. Loxton, J. Vac. Sci. Technol. A 10, 468 (1992). 49. R. Cuerno and A. L. Barabasi, Phys. Rev. Lett. 74, 4746 (1995). 50. G. Costantini, F. Buatier de Mongeot, S. Rusponi, C. Boragno, U. Valbusa, L. Vattuone, U. Burghaus, L. Savio, and M. Rocca, J. Chem. Phys. 112, 6840 (2000). 51. L. Vattuone, U. Burghaus, L. Savio, M. Rocca, G. Costantini, F. Buatier de Mongeot, C. Boragno, S. Rusponi, and U. Valbusa, J. Chem. Phys. 115, 3346 (2001). 52. S. Facsko, T. Bobek, T. Dekorsy, and H. Kurz, Phys. Status Solidi B 224, 537 (2001). 53. S. Facsko, T. Dekorsy, C. Koerdt, C. Trappe, H. Kurz, A. Vogt, and H. L. Hartnagel, Science 285, 1551 (1999). 54. C. C. Umbach, R. L. Headrick, and K.-C. Chang, Phys. Rev. Lett. (2001).

Encyclopedia of Nanoscience and Nanotechnology

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Ionic Conduction in Nanostructured Materials Philippe Knauth Université de Provence—CNRS, Marseille, France

CONTENTS 1. Introduction 2. Theory of Ionic Conduction in the Bulk 3. Ionic Conduction At and Near Interfaces 4. Nanocrystalline Ceramics and Thin Films 5. Nanocomposites 6. Conclusions Glossary References Electrolytes, as respects their insulating and conducting forces, belong to the general category of bodies; and if they are in the solid state (as nearly all can assume that state), they retain their place presenting then no new phenomenon. Michael Faraday On induction, in “Experimental Researches in Electricity,” Volume 1, p. 1344, Series XII. Taylor and Francis, London, 1839.

1. INTRODUCTION Ceramics have been a part of human culture for thousands of years. Ceramic objects, including bricks, tiles, and pottery, can be traced back to the ancient civilizations of Mesopotamia, Egypt, and Asia. From a materials scientist’s point of view, ceramics can be defined as nonmetallic, dense, polycrystalline solids that are prepared by shaping and high temperature firing, typically around 1000
C. A large part of conventional ceramics is based on alumino-silicates, especially clay. From the 19th century on, the physical and chemical properties of these ionocovalent materials were investigated in earnest. It was then recognized that their electrical properties were quite different from those of the earlier investigated metallic solids. As early as 1833, Michael Faraday reported experiments on ionic conducting inorganic solids: “There is no other body with which I am acquainted, that, like sulphuret of silver, can compare with metals in conducting power for electricity at low tension when hot, but which, unlike them, during

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cooling, loses in power, whilst they, on the contrary, gain. Probably, however, many others may, when sought for, be found” [1]. The large enhancement of the electrical conductivity of silver sulfide Ag2 S with increasing temperature was difficult to interpret at that time, because it was in strong contradiction to the behavior of metals. The first major invention based on solid electrolytes was an electric lightning device using the “Nernst mass,” a mixture of zirconia with rare earth oxides, including the particularly favorable composition 85% zirconia and 15% yttria [2]. Although the Nernst lamp was finally not used extensively in practice—it was replaced in 1905 by the simpler tungsten filament lamp—this discovery gave a major impulse to the development of solid state ionics. Around 1850, Gaugain had already discovered the principle of solid electrolyte fuel cells: “air and alcohol vapor separated by a glass envelope and heated up to high temperature can form a couple capable to develop electricity” [3]. Gaugain obtained similar results with a porcelain tube and with other gaseous fuels. Nearly one century later, after fruitless efforts with liquid electrolytes, Baur and Preiss proposed a great number of solid electrolytes for fuel cells, including glasses, porcelains, clays, and different oxide mixtures, and concluded that “unsurpassed is the Nernst mass” [4], even if degradation problems remained. The role of temperature, doping, and stoichiometry variations influencing the electrical conductivity and ionic diffusivity in solids, often described as “defect chemistry,” was treated in pioneering papers in the 1920s and 1930s. Joffé investigated the ionic conductivity of NaCl, SiO2 , and CaCO3 crystals [5]. Based on these observations, Frenkel postulated the existence of point defects in order to understand the diffusion properties of solids [6]. Wagner and Schottky applied the methods of statistical thermodynamics to the problem of disorder in an ordered binary compound AB in 1930 [7]. Wagner also introduced the term excess electron and electron defect and comprehensively discussed the consequences of electronic and ionic disorder on the electrical conductivity of ionic solids [8]. He had already noticed that these concepts could be used to deduce quantitatively

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (309–326)

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Ionic Conduction in Nanostructured Materials

the dependence of conductivity on the partial pressure of the electronegative component. The main conclusion of these researches is that the electrical conductivity at moderate temperatures of many ionocovalent solids is not due to electronic carriers only but contains a more or less important contribution of mobile ions. The discipline that deals with the preparation, characterization, modeling, and applications of such solid materials supporting ionic motion is called “solid state ionics.” Progress in solid state ionics is related to major technological developments in the domains of energy storage and conversion (batteries and fuel cells) and environmental monitoring and control (chemical sensors). The presence of interfaces, recognized 50 years ago to play a key role in semiconductor science and microelectronics, strongly impacts also the ionic conductivity. The evaluation of multiphase materials, commonly designated as “composites,” containing heterophase boundaries was started after Liang discovered in 1973 an ionic conduction enhancement in a heterogeneous material: dispersion of small alumina particles in a matrix of the moderate ionic conductor LiI leads to 50 times higher ionic conductivity as compared with pure LiI [9]. The conductivity enhancement is a consequence of the formation of space charge regions with enhanced ionic carrier concentration near the interfaces. Similar effects, although somewhat smaller, are expected in polycrystalline materials, where grain boundaries are present. Logically, the enhancement should be particularly significant in systems with a high density of grain or phase boundaries, like in nanocrystalline ceramics and thin films or in nanocomposites. The objective of this chapter is to present the current knowledge on this topic and to review different classes of nanomaterials where an important ionic conductivity has been reported. But before that and for those not familiar with the subject, let us summarize the essential concepts on ionic conduction in the bulk and at, or near, interfaces.

2. THEORY OF IONIC CONDUCTION IN THE BULK The total electrical conductivity
of a solid at a given temperature is the sum of the partial conductivities of ionic and electronic charge carriers:
qi i ci (1)
= i

In this equation, qi is the charge, i is the mobility, and ci is the concentration of the charge carrier i. It is thus apparent that two parameters can be modified in order to increase the conductivity of a solid: the carrier concentration and/or mobility. The concentration of ionic defects may be increased in one of several ways, as shown in Figure 1 for the case of silver chloride AgCl. (1) Deviation from stoichiometry (Fig. 1a and b): a reduction or oxidation of the compound produces simultaneously electronic species and point defects, thereby leading to mixed conduction.

Figure 1. Defect chemistry in the bulk of AgCl. From top to bottom: (a) charge carrier concentration as function of compound stoichiometry, (b) “phase diagram”: composition range of the homogeneous compound, (c) influence of doping by Cd2+ ions, (d) Arrhenius diagram of silver vacancy concentration. The Kröger–Vink nomenclature is explained in Section 2. Reprinted with permission from [48], J. Maier, Nanoionics and soft materials science, in “Nanocrystalline Metals and Oxides—Selected Properties and Applications” (P. Knauth and J. Schoonman, Eds.), p. 84. Kluwer Academic, Boston, 2002. © 2002, Kluwer Academic Publishers.

(2) Doping (Fig. 1c): the addition of aliovalent impurities requires the generation of ionic defects with opposite charge in order to maintain electrical neutrality. (3) Intrinsically disordered solids: many solids pass through an order–disorder transition as the temperature is increased (e.g., point 4 in Fig. 1d or the – transition in AgI, three-dimensional disorder). In other cases, the disorder is limited to disordered

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planes (intercalation compounds, e.g., Na--alumina) or one-dimensional channels (tunnel compounds). Likewise, amorphous phases (inorganic glasses and polymers) present a high intrinsic disorder and, in certain cases, also exhibit ionic conductivity. (4) Formation of space charge regions in the vicinity of interfaces: the space charge forms in response to a plane of opposite charge adsorbed on the surface or segregated at a grain boundary. This phenomenon will be discussed in some detail.

three types are experimentally observed. The predominant disorder type depends mainly on the solid’s crystal structure. (1) Cation Frenkel disorder corresponds to the formation of a cation interstitial plus a cation vacancy:

The ionic mobility depends on a number of factors. The most important is the height of the potential barrier that the ion must overcome in order to pass from one well to an adjacent well. In general, the barrier height depends on a number of factors, including the strain energy that needs to be expended for the ion to “squeeze” through the bottleneck, the polarizability of the lattice, and the electrostatic interactions between the ion and its surroundings. Perhaps the easiest to visualize is the strain energy and one is therefore tempted to assume that solids or interfaces with the highest free volume should exhibit the highest mobility. While this is often the case, as in short circuit diffusion at extended defects (e.g., grain boundaries and dislocations), there are many examples where solids with smaller channels support higher mobilities than those with larger channels, due to polarization effects.

Fr G
is the standard Gibbs free energy of the Frenkel reaction. Obviously, interstitial formation is easier for small ions and/or in relatively open lattices. Therefore, cationic Frenkel disorder is more often found then the corresponding anionic Frenkel disorder (also called anti-Frenkel type), because anions are generally larger. The anion Frenkel reaction can be written

2.1. Defect Chemistry: Kröger–Vink Nomenclature Let us introduce now the commonly used nomenclature for the description of defect chemical reactions, proposed in 1956 by Kröger and Vink [10]. The point defects are considered as dilute species and the solid plays the role of the solvent. Several analogies can be found between intrinsic defect formation and self-dissociation of water. (1) A pair of charged defects is formed, which is responsible for electrical conduction. (2) Defect formation is thermally activated. A mass action law constant using defect activities (or concentrations for dilute species) describes the defect equilibrium. (3) An acidity–basicity concept can be introduced [11]. In the Kröger–Vink notation, the subscript shows the site of a defect; the subscript i stands for an interstitial site. The effective defect charge is written as a superscript, relative to the ideal lattice: a dot 
stands for a positive and a prime ′ for a negative charge. The vacancy is written V. For example, O′′i represents a doubly charged oxygen interstitial ion, V

O is a doubly charged oxygen vacancy. Bulk defect chemical reactions must obey: (1) mass balance, (2) balance of lattice sites, (3) charge balance (global electrical neutrality). The electrical neutrality condition is, however, not respected near interfaces, where deviations can occur. In a binary ionic compound M+ X− , four types of intrinsic ionic disorder can be generated by permutation of the elementary defects, ion vacancies, and interstitial ions, but only

′ MM + Vi = Mi
+ VM

(2)

KFr is the Frenkel equilibrium constant: ′ KFr = cMi
cVM = exp− Fr G
/kT

XX + Vi = Xi′ + VX


(3)

(4)

Due to the small radii of silver and copper ions, silver halides (such as AgCl, see Fig. 1) and copper halides belong to this group, but also certain anion conductors with the relatively open fluorite-type lattice (e.g., ceria or zirconia). (2) Schottky disorder is due to the coupled formation of cation and anion vacancies: ′ + VX
+ MX MM + XX = VM

(5)

MX represents ions, which have been displaced to “new” interfacial sites. KSch is the Schottky equilibrium constant: ′ KSch = cVM cVX
= exp− Sch G
/kT

(6)

Sch G
is the standard Gibbs free energy of the Schottky reaction. One already imagines that this type of disorder will be found mainly in dense crystal lattices. For example, closepacked alkali halides (including NaCl) show Schottky-type disorder. The fourth theoretically possible defect combination, a pair of cation and anion interstitials, is not found in reality, because it is very difficult to create interstitials on both sublattices. (3) In addition to intrinsic ionic disorder by point defects, one must take into account intrinsic electronic disorder by creation of electron–hole pairs, which can be written 0 = h
+ e′

(7)

This process can be thermally activated or due to photons. The excess electrons are in the conduction band, whereas electron holes are located in the valence band of the compound. The temperature-dependent equilibrium constant of this reaction is Kel T = ch
ce′ = A exp−Eg /kT

(8)

The prefactor A contains the effective mass of hole and electron; Eg is the bandgap energy of the compound. Intrinsic electron–hole pair formation and vacancy– interstitial pair creation (Frenkel reaction) can both be represented in level diagrams, like those used in solid state physics. Figure 2 shows this type of diagram for the case

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Ionic Conduction in Nanostructured Materials

local partial free energy

interstitial ionic level~ µ°i regular ionic level

~ -µ° v

Kox is the equilibrium constant and ox G
is the standard Gibbs free energy of the oxidation reaction. Excess electrons and oxygen vacancies are formed in the reduction reaction

µ~Ag+

OO = 0

valance band

~• -µ° h

~ µ° e'

Figure 2. Level diagrams of ionic and electronic disorder in AgCl. The electrochemical potentials i of silver ions and electrons are connected via the chemical potential of silver Ag , in other words the exact stoichiometry of the compound. The indicated prevailing n-type semiconductivity corresponds to a slight Ag excess in AgCl. Reprinted with permission from [48], J. Maier, Nanoionics and soft materials science in “Nanocrystalline Metals and Oxides—Selected Properties and Applications” (P. Knauth and J. Schoonman, Eds.), p. 82. Kluwer Academic, Boston, 2002. © 2002. Kluwer Academic Publishers.

of AgCl. If equilibrium is established, the electrochemical potentials of electrons [e− , i.e., the Fermi level] and ions [(Ag+ )] are connected via the chemical potential of one of the components, here Ag [(Ag)] (i.e., by the exact compound stoichiometry): Ag = Ag+ + e−

(9)

Ag = Ag+ + e−

(10)

(4) “Antisite” disorder describes the interchange of ions between two sublattices. While such exchange between cation and anion sites is not observed in binary ionic systems due to trivial electrostatic reasons, “antisite” disorder can be observed in ternary and higher order compounds in which cations disorder between two different cation sublattices: this is common, for example, in solids with the spinel structure [12]. (5) As one can easily verify in Eqs. (2), (4), and (5), the intrinsic defect reactions do not modify the composition of the exactly stoichiometric solid (the “Daltonide”). In addition to these intrinsic disorder types, one can also observe “extrinsic” disorder due to composition changes of the solid, be it the presence of foreign ions (impurities or dopants) or a nonstoichiometry. The latter is induced by chemical potential changes of one of the components, for example oxygen in the case of oxides. Oxidation or reduction of the material results in deviations from stoichiometry (corresponding to a “Berthollide”) and the formation of both ionic and electronic charge carriers. For example, electron holes and oxygen interstitials are created in a Frenkel-disordered oxide by the oxidation reaction 1 O g + Vi = O′′i + 2h
2 2

(13)



ce′ 2 P O2 1/2 = exp− red G
/kT (14) Kred = cVO

µAg

µ~ econduction band

1

O g + VO + 2e′ 2 2

(11)

Kox = cO′′i ch
2 P O2 −1/2 = exp− ox G
/kT (12)

Kred is the equilibrium constant and red G
is the standard Gibbs free energy of the reduction reaction. As a consequence, the p-type or n-type conductivity increases together with the deviation from stoichiometry. At reduced temperature, impurities (or dopants) will dominate near stoichiometry, while under reducing or oxidizing conditions, defects associated with nonstoichiometry will control the electrical properties of the material. The electrical properties eventually become intrinsic when the temperature increases to sufficiently high values. Kröger and Vink discussed various aspects of stoichiometry deviations in inorganic compounds and developed diagrams that show defect concentrations as a function of the chemical potential of the components. These diagrams can be much simplified under the assumption of only two majority defects, according to the so-called Brouwer approximation [13]. A general discussion of these phenomena is outside the scope of this chapter but can be found in many standard texts [14].

2.2. Ionic Conduction in the Bulk: Hopping Model Ionic conduction in solids [15] is due to thermally activated ion hopping [16, 17]. Assuming Boltzmann statistics, the diffusion coefficient Di is a function of the jump distance a, the characteristic attempt frequency 0 (typically ≈1013 s−1 and the free energy of migration migr G = migr H − T migr S: Di = a2 0 exp− migr G/kT

(15)

The factor  takes into account geometrical and so-called correlation effects. For example, the backward jump has a slightly higher probability than the forward jump, but, on the other hand, cooperative motion can lead to higher diffusion coefficients than isolated jumps. The Nernst–Einstein equation relates the ionic mobility i to the diffusion coefficient (k is Boltzmann’s constant): i = Di qi /kT

(16)

Using Eqs. (1), (15), and (16), the ionic conductivity can be expressed as
ion = qi2 /kT ci a2 0 exp migr S/k exp− migr H /kT (17) A general equation representing the ionic conductivity can thus be written with a prefactor
0 :
ion =
0 /T exp− act H /kT

(18)

Most crystalline and amorphous fast ion conductors (the latter below their glass transition temperature) satisfy this equation. For crystalline fast cation conductors (such as

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Ionic Conduction in Nanostructured Materials

-AgI), the activation enthalpy act H is typically below 0.5 eV; for anion conductors, it is normally larger (0.5–1.0 eV). The activation enthalpy can contain different contributions. (1) If the concentration of the mobile ionic defects is fixed by charged background impurities and dopants,

′ = cYZr !, as in the case of Y-stabilized ZrO2 2cVO the activation enthalpy act H represents only the defect migration enthalpy migr H (cf. curve 2 in Fig. 1d). (2) Assuming a thermally activated defect creation, the carrier concentration ci has a temperature dependence such as Eq. (3) or (6): ci = ci0 exp− form H
/2kT

(19)

In this “intrinsic” case, Hact is the sum of the defect migration and formation enthalpies ( act H = migr H + form H
/2; cf. curve 1 in Fig. 1d). (3) If deviations from stoichiometry are observed, the activation enthalpy can be related to reaction enthalpies, such as the oxidation enthalpy for metal deficient oxides, cf. Eq. (12), or the reduction enthalpy for oxygen deficient oxides, cf. Eq. (14). (4) At low temperatures, defect association phenomena can be observed. These issues were discussed early on by Lidiard [18]. Dreyfus and Nowick investigated such point defect association phenomena in doped sodium chloride [19]. In the case of NaCl, doping with divalent cations requires the formation of sodium vacancies for charge compensation:
′ + VNa + 2NaCl MCl2 + 2NaNa = MNa

(20)

At low temperature, defect association due to Coulomb interactions between the oppositely charged defects begins to predominate:
MNa

+

′ VNa

=


′ MNa " VNa

(21)

(M
Na ,V′Na ) refers to a bound dopant–vacancy pair. In the temperature domain in which association occurs, the conductivity has an effective activation enthalpy ( act H = migr H + ass H/2, where ass H is the association enthalpy; cf. curve 3 in Fig. 1d). There are other important experimental consequences from such association: (1) The formation of vacancy–dopant pairs leads to a weaker overall enhancement of conductivity. (2) The pairs act as dipoles and contribute to dielectric relaxation processes. Haven established the presence of loss peaks due to defect pairing in 1953 [20]. (3) The dopant ion being in the immediate vicinity of a vacancy, it can be expected to diffuse faster than the host cation. In addition to the formation of defect pairs, Lidiard showed that long-range defect interactions also play a role and adapted the Debye–Hückel theory of aqueous electrolytes

to the case of ionic crystals [18]. The resulting activity variations can often be described by a cube root law of the defect concentrations [21]. The electrostatic interactions are also the origin of the frequency dependence of the ionic conductivity and dielectric permittivity of structurally disordered solid electrolytes (cf. the “universal dielectric response”) [22, 23].

3. IONIC CONDUCTION AT AND NEAR INTERFACES An interface is a two-dimensional region separating two parts of a system. The word interface is used in the following synonymous with the term boundary. One can differentiate two main types: (1) Phase boundaries separate two phases with different chemical composition and/or structure; they are found in polyphase materials, also called composites. (2) Grain boundaries separate two grains of identical composition and structure; they are found in polycrystalline materials. At reduced temperature, boundaries can significantly enhance the ionic conductivity of polycrystalline or composite materials and thin films as compared with single crystals of identical composition. However, boundaries can also block ionic charge carriers and reduce the ionic conductivity of a material. To understand this behavior, one has to take into account the anisotropy of boundary properties: transport might be facilitated along an interface or blocked across the interface (parallel and perpendicular effects). For microcrystalline materials, study of bulk and interface effects is experimentally possible by impedance spectroscopy, because the different electrical properties of bulk and interface regions usually lead to a significant difference of the time constants #: # = $/


(22)

In this equation, $ is the dielectric permittivity and
is the electrical conductivity of the respective region. Therefore, bulk and boundary responses can often be clearly separated in the complex impedance plane as function of the signal frequency. For nanocrystalline materials, the overall electrical response is, however, more complex and the theoretical discussion of impedance spectra is not trivial [24]. For a general discussion of impedance spectra of solid ionic conductors, the reader can consult a standard textbook [25]. Let us now review the atomic mechanisms involved in the enhanced ionic conductivity at and near interfaces: one can distinguish enhanced boundary core diffusion and equilibrium (and out-of-equilibrium) space charge layer effects.

3.1. Boundary Core Diffusion High resolution electron micrographs of grain boundaries in oxides (e.g., NiO) show that the disordered region between grains is typically only a few atomic layers thick, not unlike the “grain boundary width” postulated for the interpretation of grain boundary diffusion data in metals (0.5 nm) [26]. The defects create a new periodic interface structure with

314

Ionic Conduction in Nanostructured Materials

regularly repeating structural units. Ordinary boundaries are formed by portions of such special boundaries. The boundary core is a fast diffusion path and can act as a short-circuiting pathway for diffusion at reduced temperature. This mechanism has been known in metallurgy for quite a long time but applies obviously also in ionic materials. The enhancement of the oxygen ion diffusion coefficient, due to surface and grain boundary diffusion, was already recognized in 1960 by comparison of single crystalline and polycrystalline Al2 O3 [27]. More recently, Atkinson showed that the grain boundary diffusion coefficient in nickel oxide NiO is as much as six orders of magnitude larger than the bulk one [28]. Accelerated diffusion in the boundary core can be expected for two reasons: (1) the large defect concentration at interface sites (high percentage of displaced atoms), corresponding to a low defect formation energy, (2) the large excess free volume, which enhances the defect mobility, corresponding to a low defect migration energy. The activation energy for interfacial diffusion is therefore significantly lower than for bulk diffusion, because defect formation and migration need less energy at interface sites. This is the origin of the fast boundary diffusion at low temperature. In the “neutral core model,” the interface core is also held responsible for an ionic conductivity enhancement (see Section 4 for more details). In usual microcrystalline materials, the cross-sectional area of grain boundaries is, however, very small: using an elementary brick layer model, the ratio of cross-sectional areas of grain boundaries (gb) of width d and cubic grains of size L is Agb /Abulk = 2d/L

(23)

The general conclusion from these considerations is that grain boundary diffusion is the main diffusion contribution at low temperature. It leads to a significant enhancement of mass transport in comparison with single crystals, if the grain size is sufficiently small.

3.2. Space Charge Layer Models Boundary core diffusion is, however, not the only and generally not the most important origin for an ionic conductivity enhancement, but space charge effects are. Space charge regions are formed near interfaces in ionic materials to compensate charged defects and dopants, which segregate to surfaces and grain and phase boundaries. Two segregation modes are commonly recognized: one is segregation of overor undersized dopants to the boundary core, due to elastic relaxation, and the second is segregation of charged defects in the space charge regions, due to electrostatic interactions. Space charge effects have been known to be important in colloid systems for some time: Verwey and Overbeek established the electrostatic theory applicable to these systems in 1948 [29]. Grimley and Mott discussed boundary conditions at Ag/AgBr and AgBr/liquid electrolyte interfaces in 1947 [30]. Lehovec calculated the distribution of lattice defects

and the space charge potential at the surface of ionic crystals following a statistical thermodynamic approach in 1952 and already mentioned the implications for ionic conduction [31]. Kliewer and Koehler applied these concepts to the case of NaCl in 1965 [32]. Poeppel and Blakely extended this treatment by taking the density of surface sites into account [33]. Wagner used the space charge layer concept in 1972 to explain conductivity effects in two-phase materials, such as metallic inclusions in a semiconducting oxide or mixtures of two semiconducting oxides [34]. Space charge segregation of dopants was studied in a number of systems, for example, in TiO2 and CeO2 by Chiang and co-workers [35–37]. The first evidence for the importance of space charge effects in solid state ionics was the experimental work of Liang published in 1973. The ionic conductivity of the moderate Li+ ion conductor LiI was enhanced by a factor of almost 50 by dispersing small alumina particles in the ion conductor matrix. A large number of papers confirmed this “heterogeneous doping” effect on other ionic conductor composites, with a major contribution by the group of Saki and Wagner [38]. Similar enhancement effects can be predicted if dislocations are present. However, the number of experimental studies is smaller, given their difficulty: Nowick studied the conductivity of plastically deformed NaCl crystals [39]. Dudney investigated the enhancement of conductivity due to dislocations and stress effects in AgCl composites [40]. The space charge layer theory of ionic conductor systems was largely developed by Maier since 1984 [41]. The fundamental idea is that ionic defects (or dopants) can be trapped at an interface. This corresponds to an interfacial segregation process that can modify the excess (positive or negative) interface core charge ' and the interface potential (. All species with an opposite charge are enriched in the adjacent space charge region (accumulation layer), whereas the species with the same charge are depleted (depletion layer): qi )ci /)' < 0

(24)

A so-called “inversion” layer is observed for a species that is the majority carrier in the bulk but a minority carrier in the space charge region [42]. Space charge effects are relevant for different ionic conducting materials, including ionic conductor–insulator composites, thin films, and polycrystalline solids; the higher the interfacial area in these materials, the higher the expected conductivity effect. (1) Composites: The driving force for trapping can be the presence of a second phase with chemical affinity for a mobile ion. For example, “basic” oxides present many nucleophilic hydroxide surface groups, which can attract and fix cations. The space charge regions are then enriched in mobile cation vacancies, which are often majority ionic charge carriers at low temperature. The space charge layers represent then high conductivity regions that can short-circuit the bulk, if they percolate [43]. An optimized effect is expected when the two phases have a nanometric size. (2) Thin films: The interaction with a gas phase can also lead to a space charge effect, particularly if the solid

315

Ionic Conduction in Nanostructured Materials

ionic conductor is in thin-film form [44]. For example, ammonia is known to form strong complexes with copper ions in aqueous solution. Copper ions can also be trapped at the surface of a solid copper ion conductor in contact with a gas phase containing ammonia. This effect can be used for ammonia gas detection [45]. Large conductivity effects are also expected in ionic conductor heterolayers with small periodicity (cf. Section 4.4). (3) Polycrystalline materials: Ions can be trapped at the grain boundary core, which has a larger free volume than the bulk [46]. Obviously, the trapping effects can be particularly important in nanocrystalline solids, which present a very large grain boundary density. Space charge effects on conductivity depend on whether one considers: (1) transport along or across a boundary, (2) accumulation, depletion, or inversion layers, (3) “Gouy–Chapman” or “Mott–Schottky” situations. However, the impact of space charge layers on conductivity can always be expressed by the relation ) ln
i x /)(x /kT = −qi

(25)

(x is the space charge potential relative to the bulk, modifying the charge carrier concentration in the space charge region. Let us consider now the assumptions and detailed relations for the two cases of complete and partial charge carrier equilibrium.

composite and nanocrystalline ionic conductors, as we will see later. At equilibrium, the electrochemical potentials i of charged species are constant across the interface, but the chemical potentials i change (( is the internal electrical potential): di = di + qi d( = 0

(27)

Using the relation between the chemical potential of the species i and its thermodynamic activity ai , it follows that (28)

kTd ln ai = −qi d(

For small defect concentrations, the activity can be replaced by the concentration and the defect concentration profile in the space charge region can be formulated as function of the bulk carrier concentration ci : ci x /ci = exp −qi (x − ( /kT! = exp −qi (x /kT! (29) The local concentration in the space charge region ci x depends on the difference between the local and the bulk electrical potential, (x and ( , which is in the following consistently taken as ( = 0 to simplify the equations. For positive values of (x , the concentrations of all negative defects are raised by the exponential factor, while those of the positive defects are reduced by the same factor and vice versa for negative values (Fig. 3a). The divergence of

Ci

3.2.1. Complete Equilibrium: “Gouy–Chapman” Model

VAg ′

If all defects are in local thermodynamic equilibrium, the boundary can be described by a model similar to the classical electrochemical model of the electrode–electrolyte interface, outlined around 1910 by Gouy and Chapman [47]. The formation of space charge layers is a consequence of local thermodynamic defect equilibrium: since the boundary core has its own defect chemistry, a defect redistribution between bulk and boundary core leads to space charge layers adjacent to the core. For a Frenkel disordered material M+ X− , the defect reaction can be written:

Agi.

a) Gouy-Chapman Model

0

2λ

x

Ci

A′ .. VO

e′ ′ MM + VS = MS
+ VM

(26)

Here, VS and M
S are respectively an empty interface site and a metal ion trapped at this site. The metal vacancies V′M are distributed in the region adjacent to the interface, which becomes electrically charged (space charge region). It is assumed that the bulk defect thermodynamic properties, such as the standard chemical potential 
or the Frenkel equilibrium constant [cf. Eq. (3)] and also the bulk defect mobility remain unchanged inside the space charge region, up to the boundary core. This is certainly an oversimplification, because it is known that the local structure changes more gradually and not in a steplike manner. However, a similar assumption is made in semiconductor physics and the conclusions obtained by the space charge model are validated by experiments in a number of cases, including

b) Mott-Schottky Model

0

λ*

x

Figure 3. Defect concentration profiles near an interface (at x = 0) with a positive interface charge [positive interface potential (0 ]. The upper part (a) represents the Gouy–Chapman situation, where complete defect equilibrium is established. The defect profiles of depleted silver interstitials and accumulated silver vacancies are observed inside the space charge layer width 2,, where , is the Debye length [cf. Eq. (30)]. The lower part (b) represents the Mott–Schottky case, where the defect equilibrium is only partially established. Here, the acceptor dopant (A′ ) is assumed to be immobile at the temperature of the experiment: its concentration is constant up to the interface. The defect profiles of depleted oxygen vacancies and accumulated electrons are observed inside the space charge zone ,∗ [cf. Eq. (39)]. The depicted situation corresponds to an inversion layer.

316

Ionic Conduction in Nanostructured Materials

defect profiles is observed over a distance proportional to the Debye length , that is conveniently defined, as in semiconductor and liquid electrolyte theory, with respect to the bulk concentration ci of the majority defect ($ is the dielectric permittivity): ,2 = $kT/2qi2 ci

(30)

From this relation, one recognizes immediately that an enhancement of the bulk carrier concentration, by appropriate doping or temperature increase, reduces the Debye length and thus the space charge layer width. Space charge effects are thus low temperature effects and can have a considerable influence on the electrical properties of ionic conductor ceramics with reduced charge carrier concentrations, because the space charge layer width (approximately twice the Debye length) can be orders of magnitude larger than the typical grain boundary core width of a few atomic distances. Let us calculate now the conductivity along a boundary for charge carrier accumulation. This requires integration of the concentration profile of the accumulated carrier, called 1, from the boundary (x = 0) to the bulk:


1 = 1/L



L

0




1 x dx = q1 1 /L



on ionic conduction. Apart from the ionic conductivity enhancement due to the increased interface area, which can easily be extrapolated, an additional conductivity increase can be observed when the Debye length becomes comparable to the grain size. In that case, the space charge regions overlap; in other words, the defect density no longer reaches the “normal” bulk value, even at the center of the particles (Fig. 4). Mathematically; the model of semi-infinite space charge layers breaks down. In the limit of very small grains, local charge neutrality is nowhere satisfied and a full depletion (or accumulation) of charge carriers can occur with major consequences for ionic and electronic conductivity. This is equivalent to a flat-band situation, where the conductivity is homogeneous over the whole nanocrystallite, and corresponds to a change of the standard electrochemical potential. The defect thermodynamic [50], electrical, and mass transport properties then cannot be extrapolated from conventional size scaling laws and become truly grain-size dependent. In this case, the ionic conductivity should not depend linearly on the grain size L. The supplementary enhancement can be estimated introducing a “nanosize” factor g that can be obtained from [48]

L

0

c1 x dx

It is assumed that the defect mobility 1 takes the bulk value also in the space charge region. This total conductivity
1 can be separated into the bulk conductivity
 and the space charge layer conductivity
1sc . For large accumulation of carrier 1, one obtains, using Eq. (30), a square-root concentration dependence for the space charge layer conductivity [48]:


1sc = q1 1 2, c10 c1 1/2 /L = 2$kTc10 1/2 1 /L

g = 4,/L c10 − c1∗ /c10 !1/2

(31)

c1∗ is the defect concentration in the grain center. For a large effect (c10 ≫ c1∗ ), g = 4,/L, and with a grain size L = 0-4,,

L ~ L2 1

(32)

c10 is the concentration in the first layer adjacent to the boundary core x = 0 . The effective concentration of the species 1 is the geometrical mean of bulk and “interface” concentration; the effective thickness of the space charge region is 2,. The Gouy–Chapman model permits a quantitative interpretation of conductivity measurements where complete charge carrier equilibrium is established, for example on composite materials with a matrix of a low temperature ion conductor, such as AgCl [43]. Here, the silver vacancy concentration is enhanced in the space charge region (accumulation layer), due to Ag+ adsorption on the second phase nanoparticles. The simplified distribution topology of the two phases is expressed by geometrical factors [41]. An equation of type (32) can also be used to estimate the conductivity of CaF2 /BaF2 heterolayers [49], as long as the space charge regions can be considered semi-infinite, in other words as long as they do not overlap.

3.2.2. Mesoscopic Effects (Thin Films and Nanostructured Materials) Interfaces in systems with very small lateral dimensions (e.g., extremely thin films, nanocrystalline, or nanocomposite materials) can be expected to have a more important effect

(33)

0

X*

L

0 L2

L1

ζo L ~ L3 ζ

Y” 1 0 ζo

X*

L

0

L

0 λ

L3

L2

L~λ

ζ* 1 0

X* X

L3 L

Figure 4. Left: Defect concentration profiles as function of the particle size L. . is the normalized defect concentration (i.e., divided by the bulk concentration). The two top figures show the macroscopic situation, when the grain size L is much larger than the Debye length , [cf. Eq. (30)]. Separate space charge regions are observed. Bottom: mesoscopic situation of very small particles: L ≈ ,. The space charge regions overlap: the “normal” bulk defect concentration is not reached. Right: Parallel conductances for thin films of various thickness L3 ≪ L2 ≪ L1 . Reprinted wit permission from [48], J. Maier, Nanoionics and soft materials science, in “Nanocrystalline Metals and Oxides— Selected Properties and Applications” (P. Knauth and J. Schoonman, Eds.), p. 96. Kluwer Academic, Boston, 2002. © 2002, Kluwer Academic Publishers.

317

Ionic Conduction in Nanostructured Materials

the conductivity would be enhanced by a supplementary order of magnitude. A major objective of experiments is to check the validity of this prediction by studying systems with small grain size and/or low bulk carrier concentration. Mesoscopic effects due to overlapping space charge regions have for example been observed in very thin heterolayers [49].

⊥ −1 = 1/L 0⊥ 2 = 
2

3.2.3. Partial Equilibrium: “Mott–Schottky” Model The Mott–Schottky model corresponds to the case found in doped silicon at ambient temperature: if a dopant is assumed to be immobile (frozen-in) at the temperature of the experiment, the bulk dopant concentration cD is constant up to the boundary (Fig. 3b). In that case, Poisson’s equation must be integrated with a constant charge density related to the dopant density and the charge of the ionized dopant (' = cD qD ), in the one-dimensional case for an acceptor: d 2 (x /dx2 = −'/$ = cD qD /$

(34)

The required two boundary conditions for this second-order differential equation are determined by the potential in the bulk and the absence of electric field outside the depletion layer: (=0

and

d(/dx = 0

The resistivity across a boundary 0⊥ 2 can be obtained by integration of the concentration profile of the depleted species 2 in the space charge region, using Eqs. (39) and (40):

for x > ,∗

(35)

A solution of Poisson’s equation gives the potential profile in the depletion region [51, 52] x ≤ ,∗ : (x = ,∗ − x 2 cD qD /2$

(36)

The interface potential at x = 0 is 2

(0 = ,∗ cD qD /2$

(37)

One can easily show that the relation between the two potentials is (x = (0 1 − x/,∗ 2

(38)

Given that equilibrium is only partially established, the interface potential (0 is not determined by the thermodynamic boundary conditions alone but can be modified externally, for example, by applying a voltage. This corresponds to so-called Mott–Schottky experiments in liquid electrolyte or to Mott–Schottky semiconductor–metal contacts. The width of the space charge region ,∗ depends on the interface potential (0 and is not exclusively determined by the bulk carrier concentration, from Eqs. (37) and (30): ,∗ = 2$(0 /cD qD !1/2 = 2,qD (0 /kT 1/2

(39)

Even in heavily doped situations, where the Debye length , is very small, ,∗ can be perceptible if the boundary potential (0 is large. Inserting Eq. (36) into Eq. (29) and using Eq. (39), it is easy to calculate, this time for a depleted species called 2, a Gaussian-type concentration profile: c2 x /c2 = exp −q2 /qD x − ,∗ /2,!2 !

(40)

c2 is the bulk carrier concentration of the depleted species.



L

0


2 x −1 dx = 1/Lq2 2



L

0

dx/c2 x (41)

One can notice that ,∗ = 2,q2 (0 /kT 1/2 = 2, lnc2 /c20 !1/2

(42)

c20 is the concentration in the first layer adjacent to the boundary core (x = 0). The separation into the bulk resistivity 0bulk and the grain boundary resistivity 0⊥ gb leads to [48] 0bulk = 1/q2 2 c2 = 1/ q2 2 c20 expq2 (0 /kT ! 0⊥ gb

(43)

∗

= , Lq2 2 2c20 lnc2 /c20 ! = ,∗ / Lq2 2 2c20 q2 (0 /kT !

(44)

The effective thickness of the space charge layer is ,∗ and the effective concentration [2c20 lnc2 /c20 ]. Two electric properties can be used to estimate the interface potential (0 , which is the main parameter influencing the defect chemistry in the space charge region. (1) The grain boundary capacitance (A: sample crosssectional area) is related to the effective grain boundary thickness. By insertion of Eq. (30) into Eq. (39), one easily gets C/A = $/,∗ = $q2 c2 /2(0 !1/2

(45)

This equation is related to the well-known Mott– Schottky equation that allows determination of a dopant density in semiconductors from interfacial capacitance measurements [53]. (2) Using Eqs. (43) and (44), one gets easily the ratio of grain boundary resistivity 0gb and bulk resistivity 0bulk : 0gb /0bulk = expq2 (0 /kT / 2q2 (0 /kT!

(46)

At the reduced measurement temperature generally used for nanocrystalline materials, one can often reasonably assume that dopants are immobile, so that the Mott–Schottky model will be useful for a quantitative interpretation of conductivity measurements on nanoceramics, including the model system of pure and acceptor-doped ceria discussed in Section 4. Furthermore, the Mott–Schottky model was used to interpret the grain boundary resistivity of highly pure zirconia ceramics. In this refractory compound, dopants can still be considered immobile at quite high measurement temperatures [54].

318 3.3. Percolation Models for Composites Although analytical equations, such as Eq. (32), are in good agreement with experimental conductivity data for many ionic conductor composites, these expressions cannot describe the behavior near critical points. Two thresholds exist in ionic conductor/insulator composites. For a small concentration of insulator, the space charge regions around the phase boundaries are isolated in the matrix of the ionic conductor and do not effectively contribute to ionic conductivity enhancement. There exists a first critical concentration, where a continuous network of highly conducting paths extends through the whole sample. With further increase of insulator concentration, the conductivity increases drastically: this is the domain where the analytical equations describe the experimental data satisfactorily. At large volume fractions of insulator, a second critical concentration is attained, where conduction paths become disrupted, because continuous layers of insulator grains are formed (conductor– insulator transition). The conductivity drops sharply after this second threshold. Effective medium theories are unable to describe the behavior near critical points, because property fluctuations are important here. Critical phenomena can, however, be described in the framework of percolation theory, first applied to the problem of ionic conductor/insulator composites by Bunde et al. [55]. However, the assumptions made in their model (i.e., randomly distributed ionic conductor and insulator particles having identical sizes) do not correspond to the experimental evidence. In the experiments, ionic conductor and insulator have usually a different mean grain size and the interfacial interactions lead to the formation of continuous layers of insulator around the ionic conductor grains. Both experimental facts were taken into account in an improved percolative transport model [56, 57]. A particularly important conductivity enhancement is predicted when both phases have nanosized grains: percolation theory was applied to discuss the ionic conductivity of nanocrystalline Li2 O:B2 O3 composites [58]. The percolation models are complementary to the analytical approach; in both cases the physical model is based on the space charge layer concept.

4. NANOCRYSTALLINE CERAMICS AND THIN FILMS The synthesis, processing, and characterization of nanocrystalline ceramics (“nanoceramics”) and thin films belong to the emerging and rapidly growing field called nanotechnology. Substantial progress has been achieved in the last decades in the preparation of ultrafine grained precursor materials [59, 60]. Nanoceramics (i.e., three-dimensional solids composed of crystallites with a mean size below 100 nm [61]) can be prepared from precursor nanopowders by hot-pressing, where shaping and sintering are performed during a unique procedure, typically around 600
C under several thousand bars [62]. Nanocrystalline ceramics can present improved mechanical properties, hardness combined with ductility [63], which

Ionic Conduction in Nanostructured Materials

are outside the scope of this chapter. There seems to be convincing evidence now that the crystallites in nanoceramics present few extended lattice imperfections. Structural investigations of nanocrystalline oxides showed crystallites with a high degree of perfection, separated by sharp grain boundaries without indication of amorphous regions [64]. The high order in the grain interior can be understood by the existence of a large number of grain boundaries, which can act as defect sinks, in atomic proximity. Extended X-ray fine structure studies of nanocrystalline oxides showed that the grain boundary structure appears to be essentially similar to that in conventional microcrystalline materials [65, 66]. The gaslike or glasslike grain boundary structure sometimes postulated in the early literature was not confirmed. Solute segregation at grain boundaries and surfaces in nanocrystalline solids can lead to an apparent solubility enhancement for solutes with low bulk solubility: the large increase of the apparent copper solubility in nanocrystalline CeO2 was attributed to interfacial segregation and correlated with the mean grain size of the ceria particles [67, 68]. The dopant segregation can tune the activity of nanostructured catalysts [69]. Let us review now major studies on ionic conduction in nanocrystalline ceramics and thin films, with emphasis on model investigations. The domain of “nanoionics” is currently expanding at a fast pace, like the whole sector of nanoscience and nanotechnology. A few recent reviews on this topic can be found in [70–76].

4.1. A Model System: Pure and Doped CeO2 Ceria is the most studied nanocrystalline oxide so far, given its importance as solid electrolyte [77] and for oxygen storage [78], when appropriately doped, and for catalysis. Nominally undoped CeO2 nanoceramics [64] remain n-type semiconducting even at high oxygen partial pressure, in contrast to conventional microcrystalline samples that are ionically conducting, due to acceptor doping, and become mixed conducting only under reducing conditions. The large increase of the electronic conductivity, the decrease of ionic conductivity, and reduction enthalpy as compared with microcrystalline samples are striking results, confirmed by several experimental investigations [79, 80]. Similar results were also obtained for CeO2 thin films [81–83]. Grainsize dependent conductivity data of various authors are reported in Figure 5. Two competing models can be used for interpretation: (1) the neutral boundary core model, where charge effects are neglected, (2) the space charge concept, where charge effects are paramount.

4.1.1. Neutral Boundary Core Model In the “neutral core model,” the enhancement of conductivity is explained by the high disorder in the boundary core, due to a significantly lower standard reduction enthalpy

319

Ionic Conduction in Nanostructured Materials -2

log(σ) [S/cm]

-4

-6

-3

-8

-1 0 1

10

100

1.000

10.000

Grain size L [nm] Figure 5. Grain size dependence of the conductivity of nanocrystalline ceria ceramics. The solid line represents the calculated electronic conductivity, based on analytical equations for grain size in the micrometer range, numerical calculations for nanometric sizes. The dotted line represents the calculated ionic conductivity. The symbols stand for electronic conductivity data of various authors. Reprinted with permission from [92], A. Tschöpe, Mater. Res. Soc. Symp. Proc. © Material Research Society.

red H
′ at interface sites, making the reduction reaction much easier: 1

O O = VO + 2e′ + O2 g 2

(13)

One can write the equilibrium constant of the reduction reaction at interface sites [cf. Eq. (14)] as

Kred′ = cVO ce′ 2 P O2 1/2 = A exp− red H
′ /kT (47)

In order to obtain the experimentally observed slope −1/6 in the ln
vs ln P O2 dependence, one has to assume an electrical neutrality relation:

2cVO = ce′

(48)

In other words, the defect concentrations due to the reduction reaction are assumed so high that doping can be neglected, so that the vacancy concentration is not fixed by acceptor impurities, like in conventional samples. One then obtains ce′ 3 = 2A exp− red H
′ /kT P O2 −1/2

(49)

Using Eq. (1), one gets finally the experimentally observed power law: ln
= ln ee 2A 1/3 ! − 1/6 ln P O2 −  hop H + red H
′ /3 kT

(50)

hop H is the small polaron hopping enthalpy (around 0.4 eV in ceria [84]). Using the data of [64], the standard reduction enthalpy is strikingly lower ( red H
′ ≈ 2 eV) than in conventional CeO2 (4.7 eV [85]). This implies a corresponding enhancement of the oxygen deficiency to values of the order 10−6 –10−5 , in comparison with typically 10−9 for microcrystalline samples. This is not unrealistic: the fraction of reduced interface oxygen sites can be estimated from the

interface-to-volume ratio and for a 10 nm mean grain size, only one out of 10,000 grain boundary sites needs to be reduced to dominate the defect and transport behavior of the nanoceramics [64]. Even larger oxygen deficiencies, up to values of 10−3 , were actually measured on nanocrystalline CeO2 , using coulometric titration with oxygen solid electrolyte cells [86]. An exceptionally high oxygen deficiency was also determined in nanocrystalline ceria–praseodymia solid solutions with high Pr content: x > 0-1 in Pr0-7 Ce0-3 O2−x at 640
C [87]. In this system, two factors contribute to the high nonstoichiometry: (1) the easy valence change of praseodymium ions (Pr4+ /Pr3+ , (2) the high interface density. Large deviations from stoichiometry can significantly improve the oxygen storage capacity of nanocrystalline oxides. The existence of many interfaces improves also the oxygen exchange kinetics and the oxygen diffusivity. Coulometric titration experiments using oxygen concentration cells gave high chemical diffusion coefficients (around 10−6 cm2 /s at 600
C) with an exceptionally low activation energy (0.3 eV). These results are also suspected to reflect the presence of a large density of fast diffusion pathways with reduced defect migration energy [88] and can be interpreted in the framework of a “neutral core model.” This conclusion remains, however, to be validated for other solid solutions and by different techniques. In conclusion, the “neutral core model” treats changes of defect thermodynamic and transport properties as solely related to the boundary core and neglects charge effects, so that an electrical neutrality condition, such as Eq. (48), can still be applied.

4.1.2. Space Charge Layer Model Mason and Hwang [89] found a reduced ionic conductivity and an enhanced activation energy for ionic conduction in CeO2 nanoceramics (1.6 eV, about twice the value for microcrystalline CeO2 ) that cannot be understood in the “neutral core model,” because the ionic conductivity should also be enhanced in the boundary core. Tschöpe first proposed a consistent model, where the increase of electronic conductivity and the decrease of ionic conductivity in nanocrystalline ceria is explained by the existence of space charge regions [90]. Kim and Maier checked the consistency of this model using defect thermodynamic considerations [91]. If the bulk defect thermodynamic properties are unchanged inside the space charge zone, the defect concentrations are subjected to the following consistency condition, cf. massaction law (47):



c0 VO c0 e′ 2 = c VO c e′ 2

= A exp− red H
/kT P O2 −1/2

(51)

c0 is the concentration in the first layer adjacent to the boundary core; c is the bulk concentration. Concerning the

320 P O2 and T dependencies, one sees easily that the following equations should be observed locally:

) ln c0 VO /) ln P O2 +2) ln c0 e′ /) ln P O2 = −1/2 (52)

) ln c0 VO /)1/T +2) ln c0 e′ /)1/T = − red H
/k (53)

The excellent fulfillment of these consistency criteria, demonstrated in [91], and the quantitative interpretation of P O2 exponents and activation energies for ionic and electronic conduction are very strong arguments for the space charge model in the case of pure and doped ceria. Very recently, Tschöpe [92] succeeded in determining the grain size dependence of the electrical conductivity of ceria from micro- to nanometer range in excellent agreement with the experiment, as shown in Figure 5, using analytical equations for large grain sizes and numerical calculations for the smallest ones, where analytical equations break down. The large set of experimental investigations and the detailed theoretical analysis permit one to draw already a few general conclusions: (1) A positive boundary charge is observed in ceria, probably due to boundary reduction. Space charge potentials between 0.3 [91] and 0.7 V [93] have been calculated; they depend only weakly on temperature and oxygen partial pressure. This leads to severe depletion of oxygen vacancies and strong accumulation of electrons in the space charge regions. (2) The electron accumulation strongly enhances the electronic conductivity of the nanoceramics, which becomes dominant in nominally undoped ceria. (3) The transition from predominantly ionic to electronic conductivity upon grain size reduction has been confirmed by measurements of the grain-size dependent thermopower [93, 94]. (4) Experiments on very small particle sizes, to investigate quantitatively nontrivial size effects by space charge overlap, remain to be done. The oxygen vacancy depleted space charge regions should be an obstacle to oxygen ion transport across grain boundaries, enhancing the grain boundary resistivity of nanoceramics compared with conventional samples. Experimentally, the opposite is observed: the specific grain boundary resistivity of fully dense CeO2 nanoceramics [64, 95] is orders of magnitude lower as compared with microcrystalline samples. However, it is well known that in most cases, grain boundary blocking is actually due to segregated impurities. One can assume that the dilution of segregants, given the larger grain boundary area in nanoceramics, is the origin of the enhancement of the conductivity across the grain boundary. For a more detailed study of the grain boundary blocking effect, let us turn now to the case of doped zirconia.

4.2. Doped ZrO2 Ca- or Y-doped zirconia is the most prominent crystalline oxygen ion conductor, with paramount importance for potentiometric oxygen sensors and solid oxide fuel cells. The grain boundary resistance in this compound was studied several times. In conventional stabilized ZrO2 , Ca and Si

Ionic Conduction in Nanostructured Materials

segregation has been correlated with variations of the blocking grain-boundary conductivity. A significant enhancement of the blocking grain boundary conductivity was observed at grain sizes below 2 m, because the grain boundary coverage by segregated solutes obviously decreased with decreasing grain size [95]. In nanocrystalline materials, solute segregation occurs on a large grain boundary area and the mean grain boundary concentration of segregated solutes is much lower than in microcrystalline materials with the same solute content. This should further enhance the conductivity across a grain boundary in nanoceramics. Indeed, nanocrystalline tetragonal Y-doped ZrO2 presents a grain boundary conductivity one to two orders of magnitude higher than comparable microcrystalline samples, with a low, grain-size independent activation energy [96]. Both results can be attributed to a reduced grain-boundary concentration of Si, in other words a “dilution” of segregated impurities over the very large grain boundary area. In highly pure Y-doped zirconia, where no siliceous intergranular phases leading to flux constriction are observed, there is now sufficient evidence that the blocking grain boundary resistance is related to oxygen vacancy depletion layers, consecutive to boundary reduction. The oxygen-vacancy depletion layers can be described in the framework of the Schottky–Mott model (see Section 3.2.3), as shown recently [97]. The space charge potential and oxygen vacancy profiles can be calculated quantitatively using interfacial capacitance and resistance data [cf. Eq. (45) and (46)], obtained from impedance spectroscopy. The results are in good agreement although a discrepancy remains concerning the widths of the space charge layers. A particular effort has been devoted to the preparation and study of phase-pure nanocrystalline zirconia samples with tetragonal or monoclinic structure. Nanocrystalline tetragonal ZrO2 was successfully prepared using electrostatic spray deposition [98] and the grain size effect on the tetragonal–monoclinic phase transition was studied by Raman spectroscopy and X-ray diffraction [99–101]. The dopant-size effect on the transport properties was also investigated [102]. In tetragonal Y-doped ZrO2 nanoceramics, the ionic conductivity was similar to that of microcrystalline samples [96]. 18 O diffusion profiles in nominally undoped fully dense nanocrystalline ZrO2 with monoclinic structure showed grain boundary diffusion at deeper penetration [103]. Within the entire temperature range between 450 and 950
C, the grain boundary diffusion coefficient was three to four orders of magnitude higher than the bulk diffusion coefficient, the latter being similar to earlier results on monoclinic samples. The grain boundary diffusion coefficient remained, however, below the diffusion coefficient in coarse-grained cubic Ca- or Y-stabilized zirconia, which is related to a high bulk vacancy concentration [103]. A complete analysis applying the space charge models is certainly worthwhile.

4.3. TiO2 : Anatase and Rutile Titanium dioxide is an oxide of uttermost importance for applications in dye-sensitized solar cells, for photocatalysis or in resistive oxygen sensors, especially in metastable

321

Ionic Conduction in Nanostructured Materials

TiO2 = Ti

i + 2e′ + O2 g Kred =

Ti

i !

′ 2

(54)
′

e ! P O2 = B exp− red H /kT

(55)

Given the large Na+ acceptor concentration in these samples (>1000 ppm), which remains from the precipitation process of the anatase precursor using sodium hydroxide [62], one can assume that the interstitial concentration is fixed by the acceptor impurity concentration 2 Ti

i ! = A′Ti ! . One then gets the P O2 and T dependence of the conductivity: ln
= lnee 2B/ A′Ti ! 1/2 − 1/2 ln P O2 − red H
′ /2kT (56) The calculated standard reduction enthalpy ( red H
′ = 7-8 eV) is well below the value in conventional microcrystalline TiO2 (≈10 eV), indicating a reduced defect formation enthalpy at interface sites. One can notice that the estimated Debye length, taking into account the large acceptor density, is very small so that a “neutral layer model,” disregarding space charge effects, might be sufficient to explain the experimental data in this case. A more complete investigation of nanocrystalline anatase is currently in progress.

4.4. CaF2 and Related Materials Calcium fluoride is a well-known solid F− ion conductor. In nanocrystalline CaF2 ceramics, a largely increased ionic conductivity can be observed [109], as shown in Figure 6.

0 -1

log (σT/Scm-1K)

anatase modification. Furthermore, TiO2 is a model mixed conductor that can be doped with acceptor or donor ions and can present departures from stoichiometry and domains of ionic or electronic conductivity. The energetics of nanocrystalline TiO2 modifications, rutile, anatase, brookite, were recently reviewed [104]; the electrical properties are only partially investigated. Nanocrystalline anatase TiO2 ceramics show a conductivity plateau at high oxygen partial pressures, indicating an uncommon domain of ionic conductivity at reduced temperature (450–600
C) with a plausible activation energy (≈1 eV) [105]. Nanocrystalline rutile [106] ceramics, obtained with the addition of about 1 mol% SnO2 , show a similar plateau with an identical activation energy (≈1 eV) [107]. Predominant ionic conduction in a rutile single crystal was found at a significantly higher temperature (900
C) and with a higher activation energy (1.6 eV) [108]. The different behavior can be attributed to the existence of a large density of interfacial diffusion paths in the nanoceramics, where ionic transport is enhanced with reduced migration energy. A steep increase of electronic conductivity with an unusually large partial pressure exponent (−1/2) is observed at low P O2 [105]. The large pressure dependence is certainly of interest for sensor applications. The −1/2 exponent can be interpreted in a neutral layer model by assuming that the majority ionic defects are not completely ionized, in other words that the concentrated ionic (titanium interstitials) and electronic defects are partially associated. The reduction reaction can for example be written with doubly ionized titanium interstitials:

-2 -3 -4

3

-5

2

-6 -7 1.0

1 1.2

1.4

1.6

1.8

2.0

2.2

2.4

103T-1/K-1 Figure 6. Ionic conductivity of CaF2 . (1) Microcrystalline CaF2 : mean grain size ∼0.2 m. (2) Microcrystalline CaF2 activated with SbF5 . (3) Nanocrystalline CaF2 : mean grain size ∼10 nm. Adapted with permission from [48], J. Maier, Nanoionics and soft materials science, in “Nanocrystalline Metals and Oxides—Selected Properties and Applications” (P. Knauth and J. Schoonman, Eds.), p. 91. Kluwer Academic, Boston, 2002. © 2002, Kluwer Academic.

Conductivity values and activation energy suggest dominating transport in space charge layers. By simply scaling up the space charge effects observed in microcrystalline samples according to the increased interface density, the correct order of magnitude of conductivity is obtained (Fig. 5). A systematic deviation at higher measurement temperatures is related to grain coarsening during the experiments. The activation energy corresponds to the migration energy of fluoride vacancies and suggests an accumulation of F− vacancies in the space charge regions. Given the Na background concentration in the studied samples, the computed Debye length (1 nm) is well below the grain size, so that an observation of mesoscopic space charge effects is not expected; this requires highly pure nanocrystalline samples. Sata et al. succeeded in growing multilayer structures of BaF2 /CaF2 by molecular beam epitaxy on alumina substrates at 500
C [110]. The multilayer period was varied between 16 and 430 nm with an overall thickness of 500 nm (Fig. 7). At low temperature, the activation energy is dominated by BaF2 , in accordance with the larger vacancy mobility in this compound, whereas the activation energy at higher temperature seems to be determined by space charge layer conduction in CaF2 . This indicates a transfer of F− ions from BaF2 to CaF2 rather than phase boundary core segregation of F− ions. The conductivities along the phase boundaries increase linearly with the number of heterojunctions provided that the period is greater than 100 nm. Films with smaller periods exhibit an anomalous conductivity increase. Calculation of the Debye length (15 nm) revealed that this effect is related to an overlap of space charge regions. Such heterostructures, routinely prepared by the semiconductor community, open up the possibility to measure properties reproducibly on a scale normally inaccessible in solid state ionics and to design devices which take advantage of the size and scale of such structures. Other papers on nanocrystalline inorganic ion conductors can be found in [111, 112].

322

Ionic Conduction in Nanostructured Materials 101

σT/ohm-1cm-1 K

100

10-1

10--2

10-3

10-4

10-5 1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

103T-1/K-1 Figure 7. Arrhenius plots of conductivity of BaF2 /CaF2 heterolayers, BaF2 (open triangles) and CaF2 (open squares) thin films. The numbers indicate the BaF2 /CaF2 period. The inset shows the thickness dependence of conductivity at 320
C. One observes an anomalous increase of conductivity for thicknesses comparable to 4 ,, where , is the Debye length. Reprinted with permission from [49], N. Sata et al., Solid State Ionics 154–155, 497 (2002). © 2002, Elsevier Science.

5. NANOCOMPOSITES

Figure 8. Stacking fault arrangement at the Al2 O3 /AgI boundary, which is equivalent to a sequence of subnanometer heterolayers of - and -AgI, leading to pronounced disorder in the cation sublattice. Reprinted with permission from [48], J. Maier, Nanoionics and soft materials science, in “Nanocrystalline Metals and Oxides—Selected Properties and Applications” (P. Knauth and J. Schoonman, Eds.), p. 103. Kluwer Academic, Boston, 2002. © 2002, Kluwer Academic.

5.1. Ceramic and Glass Nanocomposites The development of ionic conducting ceramic and glass nanocomposites started with the discovery by Liang of a conductivity enhancement in a two-phase material and, in the same period, of ionic conducting glasses. Mesoscopic effects highlighting the importance of interfaces in nanocomposites are the stabilization of metastable phases and the variation of phase transition temperatures [113]. Interfacial interaction energies of the order of 0.5 eV in composite materials with alumina are great enough to cause phase transitions in a polymorphic compound, such as AgI. Furthermore, Coulomb interactions at high defect concentrations near interfaces can naturally lead to order– disorder phase transitions. The considerable enhancement of ionic conductivity observed in AgI–Al2 O3 composites, by more than three orders of magnitude [114], is outside the range of conventional space charge effects but can be explained by a nanosized ionic heterostructure -AgI/-AgI with a layer thickness below the Debye length, so that mesoscopic conductivity effects result from overlapping space charge regions with considerable disorder in the Ag+ sublattice. This kind of heterostructure can also be viewed as a succession of stacking faults (Fig. 8). The total AgI exists in the seven-layer polytype form if the alumina concentration exceeds 30 mol% [115]. This striking enhancement of ionic conductivity may also be expected for other structures in which structural units alternate with a typical distance below the Debye length and efforts should be directed to such ionic heterostructures. The study of ionic conductivity in microporous materials, such as zeolites [116–118], that contain tailored channels with nanometric dimensions might become another future highlight.

Glass ceramics, formed by crystalline inclusions inside a glassy matrix, are often nanocomposites, because the size of the crystallites is very small. Dispersed -AgI nanocrystals in AgI-based glass matrices of composition AgI–Ag2 O– Mx Oy (Mx Oy = B2 O3 , GeO2 , WO3 were discovered by Tatsumisago et al. [119–121]. These glass ceramics exhibit high ionic conductivities and low activation energies at room temperature. The larger activation energies observed at lower temperatures were attributed to a positional ordering of Ag+ ions in the microcrystals. The pronounced conductivity increase observed during initial stages of devitrification of some AgI–Ag2 O–Mx Oy glasses depends on the interfacial area between the glass and the low conductivity crystalline inclusions. This suggests the existence of a highly conducting interfacial region, possibly due to a mobility change close to the interface [122]. In the AgI–Ag2 O–V2 O5 system [123], the increase depends on the interfacial area between glass and crystalline Ag8 I4 V2 O7 inclusions. An interface related effect is also responsible for the conductivity enhancement in partially crystallized AgI–Ag2 O–P2 O5 glasses. The absence of conductivity enhancement in the AgI–Ag2 O–B2 O3 system was explained by formation of -AgI microcrystals. When the conductivity of the crystallites is higher than that of the glassy matrix, the interface region is short-circuited. The research activity on glass and ceramic composites slowed down in recent years due to persisting mechanical problems which are difficult to resolve. Other papers on ceramic and glass nanocomposites can be found in [124–138].

Ionic Conduction in Nanostructured Materials

5.2. Polymer Nanocomposites Polymer electrolytes are actively developed as separator material in solid lithium batteries and proton exchange membrane fuel cells (PEMFCs). The search of polymer nanocomposites with high ionic conductivity and good mechanical properties has been pursued in the past decade with the objective to provide flexible, compact solid membranes free from leakage and available in variable geometry. Most PEMFCs rely on the transport of protons through hydrated regions of Nafion, which has become the industry standard virtually by default, or other polymers containing sulfonate groups. Complications arise from the hydrated nature of the polymer electrolytes, because humidification is required to maintain ion conduction and prevent irreversible dehydration damage. This excludes thermal excursions to higher temperatures. However, nanocomposite Nafion + SiO2 membranes can retain water at higher temperatures [139–142] and were found able to operate at 130
C or more. Membranes with nanocrystalline TiO2 were also described [143]. With too much water, on the other hand, degradation of the mechanical properties of the polymer is observed due to swelling. This is the reason for investigations on water-free systems, including polymer–ceramic composites (e.g., Nafion + SiO2 membranes doped with heteropolyacids, such as phospho- or silicotungstic acid) [144–146]. An even larger effort has been devoted to polyethyleneoxide (PEO) based systems for lithium ion batteries, which include: (1) The solid (PEO-LiX) type. It reaches practically useful ionic conductivity values only at 60–80
C. The ionic conductivity is often enhanced in amorphous polymer electrolytes, which behave as rubberlike “soft solids.” (2) The hybrid polymer electrolyte (gel type). It contains organic molecules (“plasticisers”), such as ethylene or propylene carbonate, and is gel-like, losing to some extent the advantages of solid polymer electrolytes: mechanical stability and processing flexibility. (3) The inorganic filler-containing composite type. A simple strategy is to blend conventional polymer electrolytes with dry ceramic powders [147], which can improve ion conductivity and mechanical stability. The polymer composites development started in the 1980s initially to improve the elastic and tensile properties of glasses and ceramics, which are often too hard and brittle to be useful as solid electrolytes. Skaarup et al. [148] made polymer-ceramic composites using a large percentage of Li3 N in doped PEO and insulating polyethylene matrices. Surprisingly, the latter, supposedly isolating, were actually more conducting than the PEO composites. This indicates that the polymer mainly holds the ceramic particles together but does not provide an ionic conduction pathway. Mixtures of Nasicon with doped PEO showed smaller dc conductivity than either the pure ceramic or the pure polymer electrolyte. The considerably enhanced interface resistance was probably due to a poor contact between the two phases [149]. Mixtures of a Li-ion conducting Li1+x Alx Ti2−x (PO4 3 ceramic with an amorphous copolymer ethylene oxide-propylene oxide showed conductivities

323 approaching bulk ceramic values, but the importance of pores was confirmed [150]. Tortet et al. presented a percolation model of proton-conducting composites made with an inert polymer (PPS) and brushite (CaHPO4 2H2 O) [151, 152]. Weston and Steele [153] first mixed PEO–LiClO4 complexes with -alumina powder in order to improve the mechanical strength of the polymer electrolyte. A significant ionic conductivity enhancement was obtained using -Al2 O3 as inorganic filler [154]. An important contribution was made by Croce and co-workers, who first added ′′ -alumina or LiAlO2 to PEO electrolytes in order to improve the mechanical properties of the polymer [155] and later succeeded in preparing a polymer composite with dispersed TiO2 nanoparticles with a largely enhanced conductivity [156, 157]. A significant increase of the ionic conductivity and of the Li ion transference number and a decrease of activation energy were observed in polyacrylonitrile composites with dispersed -Al2 O3 [158]. The discussion of the mechanism of the conductivity enhancement and improvement of Li+ ion transport number is far from being terminated and various explanations have been given. (1) It was attributed to hydroxide groups on the oxide nanoparticle surface [159], providing an additional route for fast Li+ ion transport, possibly involving a space charge effect [160]. In principle, modifications of local defect distribution are expected near any type of interface, but they are more or less important, depending on the chemical affinity between the two phases. (2) It was ascribed to interactions between Lewis acid sites on the nanoparticle surface, probably OH groups, and anions or PEO segments, which help release more free Li+ ions and suppress crystallization of amorphous regions [161–163]. (3) A recent study using differently heat-treated SnO2 nanoparticles as second phase concluded that oxygen vacancies on the oxide surface act as Lewis acid sites [164]. In this model, the oxygen vacancies are assumed to compete with Li ions to coordinate with PEO segments, preventing PEO crystallization on one hand and releasing free Li+ ions on the other. Furthermore, ClO− 4 ions are also bound to oxygen vacancies, decreasing the tendency to ion association. (4) Other structural factors, such as a change of the glass transition temperature, have also been discussed [165]. However, preparation problems, notably poor adhesion, pores, and gaps at the phase boundaries, make the theoretical discussion of conductivity effects in polymer nanocomposites in the present status a difficult task. More preparative and theoretical efforts are definitely needed to better understand and further improve the properties of polymer–ceramic nanocomposites. This is a major challenge that should be met in the future. Other papers on polymer nanocomposites can be found in [166–183].

324

6. CONCLUSIONS After the first decade of research on ionic conduction in nanostructured materials, a few general conclusions can already be drawn. Concerning nanocrystalline materials, the main conclusion so far is that space charge theory seems to provide an adequate interpretation of “bulk” conductivity at least for the most studied model compound, CeO2 . In this case, space charge regions lead actually to a counterintuitive decrease of ionic conductivity due to oxygen vacancy depletion. So far, clear ionic conductivity enhancement has been found in CaF2 -based systems, either nanocrystalline ceramics or heterolayer thin films with small periodicity. In the case of the most prominent oxygen ion conductor, doped ZrO2 , the results are ambivalent. However, the effect of impurity segregation and charge carrier depletion layers on the grain boundary resistance seems now to be reasonably well understood. Concerning nanocomposites, especially polymer-based systems, there is much greater uncertainty on mechanistic details related to less mastered synthesis conditions. This leads to a wide variety of interpretations for conductivity changes, including microstructural effects, like amorphous versus crystalline regions in the polymer, and defect chemical ones, like the existence of cation–anion pairs or cation adsorption on inorganic filler particles. A more careful sample preparation will be necessary before better understanding and predictability of nanocomposite performance can be achieved. Given the importance of polymer nanocomposites in energy storage and conversion, continuing progress in this field can safely be foreseen.

GLOSSARY Interface (synonym: boundary) Two-dimensional transition region in a material separating (a) two phases (e.g. in composites), also called phase boundary, (b) two crystallites with different orientation (e.g. in ceramics), also called grain boundary. The interface core has a width of a few atomic layers only. The presence and density of interfaces greatly influence many materials properties, including the electrical conductivity. Percolation Geometrical model used for the description of disordered or random systems, introducing scaling laws with universal parameters. Near a critical value, property fluctuations become important: e.g. ionic conductor-insulator transitions in solid state ionics. Solid ionic conductor Ionocovalent solid, in which the electrical conduction is essentially due to mobile ions. Space charge layer Region in proximity of extended defects (such as interfaces or dislocations), where electroneutrality is not observed due to the symmetry break. The space charge compensates the interface core charge. These regions can present very different electrical properties than the bulk, due to enhanced or reduced charge carrier concentrations.

Ionic Conduction in Nanostructured Materials

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Encyclopedia of Nanoscience and Nanotechnology

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Kelvin Probe Technique for Nanotechnology G. Koley, M. G. Spencer Cornell University, Ithaca, New York, USA

CONTENTS 1. Introduction 2. Kelvin Probe Technique: Basic Concepts 3. Experimental Details 4. Applications 5. Summary Glossary References

1. INTRODUCTION The Kelvin probe technique has established itself as an important tool in material characterization. Named after Lord Kelvin, the technique was first demonstrated by him to the Royal Society in 1897 [1]. In his experiment, he demonstrated for the first time the use of a null method to quantitatively measure the work function difference between two dissimilar metals, such as copper and zinc. As the technique became more popular, modifications were made to the original measurement apparatus as well as to the measurement procedure. In 1932, Zisman [2] modified the measurement procedure by mechanically vibrating one plate of the condenser (consisting of two plates of dissimilar metals) and nullifying the contact potential difference (CPD) by an audiofrequency amplifier arrangement. Later, in 1976, Besocke and Berger [3] replaced the mechanical drive and feedthrough with a piezoelectric oscillator and used a lockin amplifier and feedback mechanism to nullify the CPD automatically and continuously. A further modified version of the ac Kelvin probe was demonstrated by Weaver and Abraham in 1991 [4], in which an ac voltage was applied to the probe tip in addition to a dc voltage that was used to nullify the CPD (through measurement of force rather than current as before). In the same year, Nonnenmacher, O’Boyle, and Wickramasinghe reported on Kelvin probe measurements performed in conjunction with atomic force ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

microscopy (AFM) [5]. With this technique the lateral resolution of the potential measurement could be improved quite significantly ( EG > DMF > eth > en > py > THF,whereas the dielectric constants for DEE, benzene, and toluene are much smaller than those polar solvents. The Bi-Tu complexes are more easily formed in polar solvents than in apolar solvents. It is believed that the difference in particle sizes (width and length) could be attributed to the influence of the solvents on the solubility of Bi-Tu complexes and the decomposition reaction rate of the Bi-Tu complexes.

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627

Table 3. Effect of solvents on the synthesis of Bi2 S3 and physical properties of solvents [403]. Dielectric403 constant

Boiling point403 ( C)

Temperature ( C)

Time (h)

eth py EG H2 O

2455 123 3866 801

783 1152 1976 100

140 140 140 140

H 2 Oa H 2 Oa DME THF en DMF DEE Benzene Toluene

801 801 720 758 142 3671 4335 2275 2568

100 100 852 66 1173 153 346 801 1108

150 100 140 140 140 140 140 140 140

Solvent

a

Product

Particle sizes

12 12 12 12

Bi2 S3 Bi2 S3 Bi2 S3 Bi2 S3

6 6 12 12 12 12 12 12 12

Bi2 S3 Bi2 S3 Bi-Tu Bi2 S3 + others Bi2 S3 + others Bi2 S3 + others No reaction No reaction No reaction

30 nm × 500 nm 20–30 nm × 1–3 m 130–200 nm × 3.5 m 300 nm × 2.25 m + short rods 40 nm × 150 nm 40 nm × 80 nm

The Bi2 S3 powders were obtained by the hydrothermal treatment in [316].

The structural analysis shows that the infinite linear chains within the M2 S3 (M = Sb, Bi) crystal structure will play a crucial role in the anisotropic growth characteristics as shown in Scheme 6. In fact, the one-dimensional growth of nanocrystals is actually the outside embodiment of inside crystal structures. However, the autogenous pressure in the solvothermal/hydrothermal process was found to play a key role in the orientation growth of the nanocrystals. The reaction was conducted in a closed solution system, which may be similar to the sealed-tube pyrolysis reaction of Bi(SBn)3 (Bn = CH2 C6 H5  [398]. The solid-liquid interactions between the formed Bi-Tu complexes and the solvent and the decomposition of the complexes under sealed conditions could be responsible for the formation of the Bi2 S3 NWs. In addition, the solvent acts as both reaction medium and dispersion medium so that it will prevent the aggregation of the particles and favor the production of uniform Bi2 S3 powder with good dispersivity in the SDP.

Scheme 6. The XY projection of the structural view of M2 S3 (M = Bi, Sb), showing the infinite chain structure. The modeling was done with the Cerius2 software (Accelrys).

The Bi2 S3 NWs prepared by SDP are much longer than those obtained by the hydrothermal treatment of an alkaline sol (pH = 8.0–10.0) from BiCl3 and Na2 S · 9H2 O, with EDTA as a complexing agent, as shown in Figure 21. The direct mixing of Bi3+ with free S2− with EDTA as complex in aqueous solution will result in spontaneous nucleation and produce a large number of nuclei. Then the nucleation is more accelerated than the growth. Therefore, only smaller particles can be produced. In contrast, the decomposition of the Bi-Tu complexes in the SDP will proceed slowly and produce a smaller number of nuclei in the solution than the direct ion-exchange reaction due to the positionresistance effect of the formed Bi-Tu complexes and its relatively stable property, which would be favorable for the oriented growth of the NWs. Therefore, the growth stage is more accelerated than the nucleation stage, since the system can finish the nucleation stage with a smaller number of nuclei. Sb2 S3 NRs can also be easily produced with high yield by the same procedure [315]. When anhydrous SbCl3 was added to the methanol solution of thiourea, the solution immediately turned yellow, which implies that the Sb3+ thiourea complex is formed in the solution. The formation of such a complex was confirmed by the IR spectrum, UV-vis spectrum, and 1 H NMR spectrum. In the IR spectrum of the Sb3+ -thiourea complex (Fig. 22a), three characteristic absorption peaks at 3370 cm−1 , 3290 cm−1 , and 3184 cm−1 can undoubtedly be assigned to the -NH2 stretching vibration. The -NH2 bands did not shift to lower frequencies with regard to pure thiourea (Fig. 22b), which indicates that a bond from the nitrogen to the metal is not present [404]. Meanwhile, the frequency of the C-N stretching vibration of the complex blue shifted from 1474 cm−1 in pure thiourea to 1511 cm−1 , which approaches the value for a double bond. The blue shift of the C-N stretching vibration implies that thiourea uses the sulfur atom to coordinate with the metal ion in the complex [405]. The C S stretching vibration of the complex at 1413 cm−1 was split into two peaks because of the

Low-Dimensional Nanocrystals

628

Transmission (a.u.)

(a)

(b)

4000

3500

3000

2500

2000

1500

1000

500

Wavenumber (cm-1)

Figure 22. IR spectra of the as-prepared complex (a) and pure thiourea (b). Reprinted with permission from [323], J. Yang et al., Chem. Mater. 12, 2924 (2000). © 2000, American Chemical Society.

formation of a metal-sulfur bond [405]. Another very apparent difference in the IR spectrum between pure thiourea and the Sb3+ -thiourea complex appeared at approximately 1100 cm−1 . The strong absorption peak of pure thiourea at 1084 cm−1 was extremely weakened in the complex. In addition, another band associated with a C S vibration of the complex was red shifted from 729 cm−1 in pure thiourea to 709 cm−1 . The red shift can be attributed to the reduced double-bond character of the C S bond due to the sulfur bonding with the metal. 1 H NMR spectrum presents one broad unresolved peak at 6.96 ppm that is shifted to the high field with regard to pure thiourea. It seems that the increase in electron density around the hydrogen atom is caused by the coordination of the ligand with the metal ion. The crystallization process of the Sb2 S3 NRs from the complex was followed by the XRD technique. The results show that the crystallinity of stibnite has been improved greatly with the prolonged reaction time. Figure 23a shows that no diffraction peaks corresponding to the starting

materials were observed. When the precursor was treated in an autoclave for 2 h at 120  C, amorphous stibnite was obtained, as shown in Figure 23b. When the treatment time was prolonged to 3 h, the violet-red stibnite obtained was poorly crystallized, as shown in Figure 23c. Until the treatment time reached 6 h, the black-gray stibnite obtained was not well crystallized. The crystalline products (Fig. 23d) were identified as orthorhombic stibnite with cell constants a = 11228 Å, b = 11329 Å, c = 3844 Å, which are close to the values in the literature (JCPDS file no. 6-0474). The time-dependent shape evolution of the nanocrystals during the crystallization was followed. When the reaction lasted for 2 h at 120  C, the Sb2 S3 powders obtained were composed of amorphous nanoparticles with irregular shape and an average size of about 65 nm. When the reaction time was prolonged to 3 h, some NRs appeared in the poorly crystallized Sb2 S3 powders. After 6 h all of the initial irregular particles transformed into well-crystallized NRs with an average diameter of 60 nm that is close to the size of the initial amorphous nanoparticles. In this transformation process, many neck-like connections were observed among the adjacent nanoparticles. Meanwhile, in the head or edge of the NRs just formed, there were many obvious bulges formed by different nanoparticles. The results suggest that probably the initial adjacent nanoparticles self-assemble together to form the final Sb2 S3 NRs with the crystallinity improved. The transformation from amorphous nanoparticles to crystallized NRs is much faster in a closed system than that in an open system. When the same solution was refluxed for 6 h in an oil bath, the products obtained were still irregular nanoparticles, as shown in Figure 24a. When the reflux time was prolonged to 10 h, some NRs did appear in the products, although the majority of the latter remained as irregular particles (Fig. 24b). When the initial amorphous nanoparticles were taken out and were washed to remove the adsorbed reactants and by-products, they were still transformed into NRs from the initial irregular nanoparticles as shown in Figure 24c, when they were treated in a closed autoclave for 10 h at 120  C. With increasing reaction temperature, large Sb2 S3 single crystalline NRs with lengths of 5 m and 100 nm in diameter can be synthesized in

(b)

(a)

(c)

Intensity (a.u.)

(a) (b) (c)

(d) (e) 10

20

30

40

50

60

200 nm

250 nm

333 nm

70

2*Theta (2θ)

Figure 23. XRD patterns of the products obtained by solvothermal decomposition at different times at 120  C. (a) The precursor. (b) 2 h. (c) 3 h. (d) 6 h. (e) The standard. Reprinted with permission from [323], J. Yang et al., Chem. Mater. 12, 2924 (2000). © 2000, American Chemical Society.

Figure 24. TEM photographs of the products obtained by (a) refluxing the methanol solution of the precursor for 6 h at 120  C in a flask; (b) refluxing the methanol solution of the initial nanoparticles for 10 h in a flask at 120  C; (c) maintaining the methanol solution of the initial nanoparticles at 120  C for 10 h in an autoclave. Reprinted with permission from [323], J. Yang et al., Chem. Mater. 12, 2924 (2000). © 2000, American Chemical Society.

Low-Dimensional Nanocrystals

629

methanol at 180  C for 12 h. The Bi2 S3 and Sb2 S3 NWs/NRs with high aspect ratios may be useful for important applications. The time dependence of the shape evolution of the nanocrystals during the crystallization controlled by pressure can be illustrated as in Scheme 7. The above solvothermal process can be rapidly extended for the synthesis of other chalcogenide nanomaterials through the choice of suitable reaction conditions and precursors. A recent report shows that Sb2 Se3 NWs with a diameter of 30 nm and lengths as long as 8 m can be synthesized in diethylene (DEG) media at 120–140  C by the solvothermal reaction, with SbCl3 , ammonia, NaSO3 , and Se as reactants [325]. The further optimization of the solvothermal reaction will make it possible to readily synthesize well-defined 1D III–IV nano-building blocks, which could be very useful for the synthesis of other 1D semiconductors based on sulfo salts of bismuth and related compounds and sulfoantimonates of antimony and related compounds.

2.5.3. Growth of Other Metal Chalcogenide Crystals Both solvents and temperatures play important roles in the formation of different phases, their phase transformation, and morphologies of the products. It is possible to selectively synthesize metal chalcogenide nanostructured materials and grow large single crystals with perfect shapes and preferred sizes by choosing the suitable conditions. These materials with different phases and unusual shapes may have interesting catalytic, semiconducting, and magnetic properties. In this section, the influence of solvents on phase, shapes, and reactivity will be discussed with specific examples. Various metal chalcogenide materials can be synthesized under mild conditions by the following solvothermal reaction [318, 372, 373]: solvent

MC2 O4 + E −−−→ ME + 2CO2 ↑ M = Metal E = S
Se
Te

(16)

Table 4 lists experimental conditions for the synthesis of various metal chalcogenide materials and the characterization of the powders. As shown in Table 4, the solvothermal reactions in en proceed more completely than in other solvents such as py and THF, suggesting that the higher reactivity in en can be achieved compared with other solvents. Both solvent and temperature play key roles in the reaction, shapes, sizes, and phases. Solvent in particular has a significant effect on the morphology of the particles as shown in Table 4. The reaction of bismuth oxalate with selenium and tellurium proceeded in a manner similar to that of lead (i)

(ii)

(iii)

(iv)

Scheme 7. Pressure-controlled formation process of the Bi2 S3 and Sb2 S3 NRs.

oxalate. Bi2 Se3 particles with flake-like morphology are single crystal [318]. The influence of solvents on the phase transformation during the solvothermal reaction was well demonstrated by a Ni-S system [377]. A detailed study of the synthesis of various phases of nickel sulfide materials such as NiS, Ni3 S2 , NiS103 , NiS2 , metastable Ni3 S4 , and their phase transformation and the phase transformation under solvothermal conditions was recently reported to be studied with a liquid-solid interfacial reaction between nickel substrate and sulfur, and the reaction Ni2+ ion with sulfur in different solvents such as en, water, ethanol, toluene, and py at lower temperatures (≤200  C) [377]. When the nickel salt was used instead of metal Ni, the reaction proceeded more easily. The main results of the reaction between Ni2+ and S in en at different temperatures are summarized in Scheme 8. Through control of the reaction time, the phase transformation process from NiS (M) to NiS2 (cubic) can be nicely captured. The SEM image in Figure 25a shows that NiS (M) whiskers, with diameters ranging from 1 m and lengths up to several micrometers, and spherical NiS2 particles with a size of 1.2 m were found to coexist in the product obtained at 200  C for 12 h. EDX analysis confirmed that the spherical particles and whiskers in Figure 25 are composed of exact compositions of NiS2 and NiS (M) phases, respectively. The tips of the NiS whiskers tend to bend, “melt,” or dissolve further as shown in Figure 25b, indicating that the formation of the NiS2 phase was at the expense of NiS (M) whiskers, and the transformation process from the NiS (M) phase to NiS2 indeed exists during the reaction with the prolonging of the reaction time, which corresponds to the XRD results. Pure NiS2 single crystals with well-defined dodecahedron crystals and uniform sizes of about 1.5–2.7 m can be obtained by further prolonging the reaction time to 18 h at 200  C as shown in Figure 25c. Figure 25d shows a typical NiS2 single crystal with a well-developed dodecahedron 101 unit polytype shape and a size of 1.8 m. Cubic NiSe2 single crystals with a perfect octahedral shape and a size of 15 m were obtained by a reaction of NiC2 O4 · 2H2 O with Se in en [373]. FeSe2 , CoSe2 crystals can be synthesized by a modified solvothermal process [374]. Various ternary metal chalcogenide NRs such as CdIn2 S4 [346], CuInS2 , AgInS2 [347], AgBiS2 [348], Ag3 CuS2 [349], Cu3 SnS4 [350], and PbSnS3 [351] have been synthesized by solvothermal/hydrothermal approaches under mild conditions. Even though the mechanism of the reaction and the formation of 1D NRs are still not clear, solvothermal/hydrothermal processes have already shown powerful versatilities and capabilities in the controlled solution synthesis of nonoxide 1D nanocrystals, in contrast to previous high-temperature approaches.

2.5.4. Controlled Growth of 1D Chalcogen Nanocrystals with More Complexity Chalcogens are a group of important elements with unique combinations of many interesting and useful properties [406], which are also important reactant sources for the generation of metal chalcogenides, such as for the synthesis of semiconductor materials.

Low-Dimensional Nanocrystals

630

Table 4. Summary of various metal chalcogenides synthesized by a solvothermal process, including experimental conditions such as reactants and mole ratio (M/E) (M = metal source, E = S, Se, Te), solvent, temperature, and time; the phases were detected by X-ray powder diffraction, symmetry, and the particle shape. Reactants (M/E ratio)

Solvent

Reaction conditions

Phase

Symmetry

Shape

CdC2 O4 + S(2:1) CdC2 O4 + S(2:1) CdC2 O4 + S(2:1) CdC2 O4 + Se(2:1) CdC2 O4 + Se(2:1) CdC2 O4 + Se(2:1) CdC2 O4 + Te(2:1)

en py EG en py py en

120–180 C, 12 h 160–180  C, 12 h 160–180  C, 12 h 140  C, 12 h 160  C, 12 h 140  C, 12 h 180  C, 12 h

CdS CdS CdS CdSe CdSe Se + CdSea CdTe

Hexagonal Hexagonal Hexagonal Hexagonal Hexagonal

Rod-like Spherical Spherical Rod-like Spherical

Cubic

Rod-like

PbC2 O4 + S(1:1) PbC2 O4 + S(1:1) PbC2 O4 + Se(1:1) PbC2 O4 + Se(1:1) PbC2 O4 + Te(1:1) PbC2 O4 + Se(1:1)

en py en py en py

120 120 160 160 160 160

PbS PbS PbSe PbSe + Sea PbTe PbTe + Tea

Cubic Cubic Cubic

Square Square Square

Cubic

Square

SnC2 O4 + S(1:1.2) SnC2 O4 + Se(1:1.2) SnC2 O4 + Te(1:1.2)

en en en

240  C, 8 h 180  C, 8 h 170  C, 8 h

SnS SnSe SnTe

Orthorhombic Orthorhombic Cubic

Plate-like Rod-like Spherical

Ag2 C2 O4 + S(1:1) Ag2 C2 O4 + S(1:1) Ag2 C2 O4 + S(1:1) Ag2 C2 O4 + Se(1:1) Ag2 C2 O4 + Se(1:1) Ag2 C2 O4 + Se(1:1) Ag2 C2 O4 + Se(1:1) Ag2 C2 O4 + Te(1:1) Ag2 C2 O4 + Te(1:1) Ag2 C2 O4 + Te(1:1) Ag2 C2 O4 + Te(1:1)

en py THF en py py THF en en py THF

140 140 140 140 140 160 140 180 160 180 180

Ag2 S Ag2 S Ag2 S Ag2 Se Ag2 Se + Sea Ag2 Se + Aga Ag2 Se + Sea Ag2 Te + Sea Ag + Ag2 Tea Ag + Ag2 Tea Ag + Ag2 Tea

Monoclinic Monoclinic Monoclinic Orthorhomic

Spherical Spherical Spherical Plate-like

Bi2 (C2 O4 3 + Se(1:1) Bi2 (C2 O4 3 + Se(1:1)

en py

140  C, 12 h 140  C, 12 h

Bi2 Se3 Bi2 Se3 + Sea

Hexagonal

Plate-like

NiC2 O4 · 2H2 O NiC2 O4 · 2H2 O NiC2 O4 · 2H2 O NiC2 O4 · 2H2 O NiC2 O4 · 2H2 O NiC2 O4 · 2H2 O

en en en en py THF

170 200 220 180 180 170

NiSe2 Ni085 Se Ni3 Se2 NiSe2 NiSe2 NiSe2

Cubic Hexagonal Triclinic Cubic Cubic Cubic

Octahedral Starfish-like Dendritic Spherical Octahedral Octahedral

+ + + + + +

Se(1:2.4) Se(1:1) Se(3:2) Se(1:1) Se(1:2.2) Se(1:2.2)



C, C,  C,  C,  C,  C, 



C, C,  C,  C,  C,  C,  C,  C,  C,  C,  C, 



C, C,  C,  C,  C,  C, 



12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12

8 8 8 8 8 8

h h h h h h

h h h h h h h h h h h

h h h h h h

Monoclinic

From [61, 318, 372, 373, 375]. a Dominant phase in the sample.

en 120-160°C, 12 h S Ni2+

solvent

en

SSP ≤ 200°C

180°C, 12 h

en 200°C, 12 h

en 200°C, 18 h

Ni(en) xS y

Ni(en)xxSSyy Ni(en)

NiS2

+

NiS2

+ NiS nanowhiskers

NiS2single crystals with {101} unit polytype (dodecahedron shape)

Scheme 8. Summary of the results of the reaction of Ni2+ with S in ethylenediamine under different conditions. Reprinted with permission from [377], S. H. Yu and M. Yoshimura, Adv. Funct. Mater. 12, 277 (2002). © 2002, Wiley-VCH.

Trigonal selenium NWs with well-defined sizes and aspect ratios have been synthesized by a solution-phase approach [233, 234]. The first step of this approach involved the formation of amorphous -Se in an aqueous solution through the reduction of selenious acid with excess hydrazine by refluxing this reaction mixture at 100  C: H2 SeO3 + N2 H4 → Se ↓ + N2 ↑ + 3H2 O

(17)

When this solution was cooled to room temperature, the small amount of selenium dissolved in the solution precipitated out as nanocrystallites of trigonal t-Se. Cooling the solution to room temperature will produce a small amount of trigonal Se in the solution, which will act as seeds for the formation of a large amount of trigonal Se NWs during further aging of the solution in a dark place [233]. The 1D

Low-Dimensional Nanocrystals a

631 a

b

b

Seed

1µm

c

Figure 27. SEM images of the Te nanotubes synthesized by refluxing a solution of orthotelluric acid in ethylene glycol for (a) 4 min and (b) 6 min. The white arrow indicates the presence of seeds. Reprinted with permission from [235], B. Mayers and Y. N. Xia, Adv. Mater. 14, 279 (2002). © 2002, Wiley-VCH.

d

Figure 25. SEM images for products of the reaction of Ni(NO3 2 · 6H2 O with S in ethylenediamine. (a) and (b) Coexistence of NiS nanowhiskers with uniform spherical NiS2 nanoparticles obtained at 200  C, 12 h, [Ni2+ ] = [S] = 0005 mol. (c) Uniform and well-developed NiS2 single crystals obtained at 200  C for 18 h, [Ni2+ ][S] = 1:3, [S] = 0005 mol. (d) A typical NiS2 single crystal displayed well-developed {101} polytype with dodecahedron shape and a size of 1.2 m. Reprinted with permission from [377], S. H. Yu and M. Yoshimura, Adv. Funct. Mater. 12, 277 (2002). © 2002, Wiley-VCH.

morphology of the final product was determined by the linearity of infinite, helical chains of Se atoms contained in the trigonal phase. A recent report by Xia et al. shows that single crystalline NWs of Ag2 Se can be synthesized by templating against the trigonal Se NWs [328, 329] at room temperature as shown in Figure 26. In addition, trigonal tellurium NBs, NTs, and nanohelices can be synthesized [235, 236]. The tellurium NTs were synthesized by reflexing a solution of orthotelluric acid in ethylene glycol at ∼197  C [235] as shown in Figure 27. The early stage shows that tubular structures grew from the cylindrical seeds as shown in Figure 27a. A very simple controlled hydrothermal route has been developed for the synthesis of tellurium NBs, NTs, and nanohelices by reaction of sodium tellurite (Na2 TeO3  in aqueous ammonia solution at 180  C [236]. The NBs have thicknesses of about 8 nm with widths of 30–500 nm and lengths up to several hundred micrometers, as shown in Figure 28a. The NBs tend to twist and form helices as shown in Figure 28b. When the NBs were further

A

B

twisted and rolled, NTs formed. In addition, an interesting nanostructure, a “coaxed nanobelt within a nanotube,” was observed, as shown in Figure 28c. The template-rollgrowth mechanism and template-twist-joint-growth mechanism were proposed to explain the formation of such special nanostructures [236]. However, the detailed mechanism still needs to be investigated further. The above novel chalcogen nanostructures can be rapidly used as templates for the synthesis of other, more specialized and more complex 1D and 2D metal chalcogenide nanostructures, which could find interesting applications.

2.6. Solution-Liquid-Solid Mechanism The solution-liquid-solid (SLS) growth mechanism, which is analogous to the well-known VLS mechanism, was first discovered by Buhro et al. [66]. This route is mainly for growing group III–V nanofibers (InP, InAs, GaAs, Alx Ga1−x As, InN) in hydrocarbon solvents at relatively low temperatures (less than or equal to 203  C) [66–69]. More features about this route have been summarized in [67], as shown in Figure 29. The synthesis involved the methanolysis of t-Bu2 In[P(SiMe3 2 ]}2 in aromatic solvents to produce polycrystalline InP fibers (dimensions 10–100 nm × 50–1000 nm) at 111– 203  C [68]. The chemical pathway consists of a molecular component, in which precursor substituents are eliminated, and a nonmolecular component, in which the InP crystal lattices are assembled. The two components working in concert comprise the SLS mechanism. The molecular component proceeds through a sequence of isolated and

a

250 nm

m

30 n

[0

100 nm

1µm

b

500 nm

c

500 nm

] 01

50 nm

Figure 26. SEM (A) and TEM (B) images of uniform nanowires of Ag2 Se that were synthesized through a reaction between the 32-nm nanowires of t-Se and an aqueous solution of AgNO3 at room temperature. Reprinted with permission from [328], B. Gates et al., J. Am. Chem. Soc. 123, 11500 (2001). © 2001, American Chemical Society.

Figure 28. TEM images of (a) a typical tellurium nanobelt, (b) a helical nanobelt, (c) a typical helical nanobelt within a nanobelt-roll nanotube. Reprinted with permission from [236], M. S. Mo et al., Adv. Mater. 14, 1658 (2002). © 2002, Wiley-VCH.

Low-Dimensional Nanocrystals

632 Solution

Liquid

Solid

[t-Bu2ln(µ-PH2)]3 (4) growth direction

protic catalyst 3t-BuH

(InP)n

In(I ) P(In) flux droplet polycrystalline lnP

Al0.03Ga0.97

100 nm

Figure 29. Solution-liquid-solid (SLS) mechanism and a representative Al01 Ga09 As nanowhisker grown by the SLS mechanism. Reprinted with permission from [68], T. J. Trentler et al., J. Am. Chem. Soc. 119, 2172 (1997), and [69], P. D. Markowitz et al., J. Am. Chem. Soc. 123, 4502 (2001). © 1997, 2001, American Chemical Society.

fully characterized intermediates to form the [t-Bu2 In(PH2 ]3 complex. The complex, which is alternatively prepared from t-Bu3 In and PH3 , undergoes alkane elimination catalyzed by the protic reagent MeOH, PhSH, Et2 NH, or PhCO2 H. In the subsequent nonmolecular component of the pathway, the resulting (InP)n fragments dissolve into a dispersion of molten In droplets and recrystallize as InP fibers [68]. This approach has been extended for the synthesis of InN NWs by solution-state thermolysis of the dialkyl(azido)indane precursor [259]. Group IV Si and Ge crystals can be generated by a supercritical fluid solution-phase approach [70]. Recently, Korgel et al. reported a supercritical fluid solution-phase self-assembly approach with a similar SLS mechanism for growing bulk quantities of defect-free Si NWs with diameters of about 4–5 nm and aspect ratios greater than 1000 by using alkanethiol-coated gold nanocrystals as uniform seeds to direct one-dimensional Si crystallization in supercritical hexane [71]. In this process, the sterically stabilized Au nanoparticles were dispersed in supercritical hexane together with diphenylsilane, which underwent decomposition at 500  C and 270 bar. The phase diagram for Si and Au indicated that at temperatures above 363  C, Si and Au will form an alloy in equilibrium with pure solid Si when the Si concentration with respect to Au is greater than 18.6%. Under this condition, the Si atoms dissolve into the sterically stabilized Au nanocrystals until the supersaturation is reached, at which point they are expelled from the particles as a thin nanometer-scale wire. The supercritical fluid medium with high temperature promotes Si crystallization. The Au nanocrystals will be maintained to seed NW growth under supercritical conditions.

2.7. Capping Agent/Surfactant-Assisted Synthesis 2.7.1. Self-Assembly under Hot Conditions Capping agent/surfactant-assisted synthesis has been widely explored for the fabrication of NRs, NTs, and more complex structures. Recently, Alivisato’s group synthesized the elongated CdSe nanocrystallites by injecting a solution of dimethylcadmium and selenium in tributylphosphine into a mixture of hexylphosphonic acid (HPA) and trioctylphosphine oxide (TOPO) at 340–360  C [11, 72]. The surfactant molecules adsorb and desorb rapidly from the nanocrystal surface at the growth temperature, permitting the addition and removal of atoms from the crystallites, while aggregation is suppressed by the presence of (on average) one monolayer of surfactant at the crystallite surface. It is well known that the solubility of crystals increases as the size of the crystals decreases, according to the GibbsThompson law. This law plays an important role in determining the growth kinetics of the nanocrystals. Peng et al. observed that if the monomer concentration in the solution is higher than the solubility of all existing nanocrystals, all nanocrystals in the solution grow and the size distribution narrows. The so-called focusing of size distribution can be exploited for the spontaneous formation of close to monodisperse colloidal nanocrystals and can form a threedimensional orientation. The CdSe NRs with variable aspect ratios can be well achieved by kinetic control growth of the nanoparticles (Fig. 30). The diverse range of observed shapes can be understood as arising from three basic effects: the nanocrystals will eventually tend toward nearly spherical shapes at slow growth rates; rods form at high growth rates by unidirectional growth of one face; and HPA accentuates the differences in the growth rates among various faces. A large injection 50 nm a

b

c

g d

e

f

c -axis 10 nm

Figure 30. TEM images of different samples of quantum rods. (a–c) Low-resolution TEM images of three quantum-rod samples with different sizes and aspect ratios. (d–g) High-resolution TEM images of four representative quantum rods. (d) and (e) are from the sample shown in (a); (f) and (g) are from the sample shown in (c). Reprinted with permission from [11], X. G. Peng et al., Nature 402, 393 (1999). © 1999, Macmillan Magazines Ltd.

Low-Dimensional Nanocrystals

1D-growth

a

b

10 nm

10 nm

Figure 31. HRTEM image (a) of a typical tetrapod-shaped CdSe nanocrystal, looking down the [001] direction of one arm. Lattice spacings confirm that all four arms are of the wurtzite structure. In image (b), we see a tetrapod that has branches growing out of each arm. There are zinc blende layers near the ends of the original arms, and the branches are wurtzite with some stacking faults. Reprinted with permission from [72], L. Manna et al., J. Am. Chem. Soc. 122, 12700 (2000). © 2000, American Chemical Society.

The wurtzite structural characteristics are shown in Scheme 10, showing that all of the atoms on both facets perpendicular to the c axis (unique facets) have only one dangling bond without surface reconstruction [407]. The facets terminated by negatively charged Se atoms and positively charged Cd atoms are the (001) facet and (001) facet, respectively. The negatively charged (001) facet is more or less uncoated, because the ligands in the solution are all electron-donating ligands and should bind exclusively to cationic species. Additionally, without surface reconstructions, any surface Cd atom grown on the (001) facet has to possess three dangling bonds, even if the surface Cd atoms reach a full monolayer. These unique structural features of the (001) facet and the dipole moment along the c axis significantly increase the chemical potential of the unique facets, especially the (001) facet, compared with the others [407]. The above surfactant-driven shape-controlled synthesis strategy was further extended for the synthesis of group III– VI semiconductor NRs for the first time by Cheon’s group

(001)

volume or a very high monomer concentration favors rod growth. Recently, Peng et al. proposed a diffusion-controlled crystal growth model based on their careful observation of the shape evolution of CdSe NRs as shown in Scheme 9 [407]. They believed that a typical temporal shape evolution of CdSe quantum rods occurs in three distinguishable stages. When the Cd monomer concentration in the solution was between 1.4% and 2% of cadmium element by mass, all of the nanocrystals grew almost exclusively along their long axis, and both the aspect ratio and the volume of the crystals increased rapidly. This stage is called the “1D-growth stage.” The second stage, the “3D-growth stage,” occurred when the Cd monomer concentration dropped to between 0.5% and 1.4%. In this stage, crystals grew simultaneously in three dimensions [Fig. 31]. The aspect ratio remained constant, but the crystal volume increased. It should be mentioned that this stage was not observed before [11], probably because of the higher growth rates. The final stage, called “1D-to-2D ripening,” was identified when the Cd concentration was constant at 0.5%. The aspect ratio of the rods dropped noticeably, because the dimension of the crystals increased along the short axis and decreased along the long axis. Nanocrystal volumes and number remained constant, and there was no noticeable net growth or net dissolution of nanocrystals. This indicates that the monomers very likely moved on the surface of a crystal from one dimension (c axis) to the other two dimensions in an intraparticle manner. The 1D-growth stage was confirmed by a reaction whose monomer concentration was maintained in the corresponding 1D-growth range for a longer time through the addition of more monomers to the reaction system at certain time intervals. The long axis of the quantum rods can be further extended from about 35 nm to over 100 nm by the secondary injections, but the short axis remained almost constant, at about 3–4 nm. In this model, each crystal is surrounded by a diffusion sphere as shown in Scheme 9 [407]. The monomer concentration gradient between the bulk solution and the stagnant solution, as well as the diffusion coefficient of the monomers, determines the direction (out of or into the diffusion sphere) and the diffusion flux. The monomer concentration in the stagnant solution maintains the solubility of a given facet by the rapid growth onto or dissolution from the facet.

633

3D-growth 1D/2D-ripening

(100)

Monomer concentration

Scheme 9. A schematic diagram for the proposed mechanisms of the three stages of the shape evolution. The circle in each stage is the interface between the bulk solution and the diffusion sphere. Arrows indicate the diffusion directions of the monomers. The double-headed arrows represent the diffusion equilibrium in the 1D-to-2D ripening stage. Reprinted with permission from [407], Z. A. Peng and X. G. Peng, J. Am. Chem. Soc. 123, 1389 (2001). © 2001, American Chemical Society.

Cd:

Se:

Ligand:

Scheme 10. Schematic structure of CdSe quantum rods in growth. The most stable form of a rod is shown on the left; its (001) facet terminated by Se atoms does not have any ligands. After growing a monolayer of Cd atoms on the (001) (right), this facet is still relatively active compared with the other facets, because the surface Cd atoms on this facet have three dangling bonds. See text for more details. Reprinted with permission from [407], Z. A. Peng and X. G. Peng, J. Am. Chem. Soc. 123, 1389 (2001). © 2001, American Chemical Society.

Low-Dimensional Nanocrystals

634 [254]. Gallium phosphide semiconductor nanocrystals can be synthesized by using thermal decomposition of a single molecular precursor, tris(di-tert-butylphosphino)gallane (Ga(PtBu2 3 , in a hot mixture of amine stabilizers. As in the case of CdSe, the shape of GaP nanocrystals can also be varied from nanospheres to rods with highly monodispersed size distributions by controlling the type and amount of stabilizing surfactants. When only trioctylamine (TOA) was used as a stabilizer, spherical GaP nanoparticles with zinc blende structure were formed. Increasing the stabilizer ratio of HAD to TOA leads to the formation of NRs [254]. The addition of HDA to TOA leads to changes in the shape and crystalline phase of the GaP nanocrystals. Figure 32 shows that wurtzite GaP NRs with a diameter of 8 nm and a length of 45 nm were grown by thermal decomposition of the precursor in TOA solution, which was injected into a mixture of TOA and hexadecylamine (HDA) at 330  C. A low HDA-to-TOA ratio seems to favor the formation of zinc blende nanospheres, but a high concentration of HDA leads to the formation of the wurtzite phase and induces anisotropic growth of the nanocrystals [254]. The steric effects of the stabilizers during crystal growth are illustrated in Scheme 11 [254]. The thermodynamically stable GaP zinc blende structure is a staggered conformation with 111 directions, and the kinetically stable wurtzite structure is an eclipsed conformation with 002 directions (Scheme 11). Kinetic stability of the wurtzite structure is induced by strong dipole interaction of incoming GaP monomers with surface GaP lattice atoms. The conformation of crystal structures is highly affected by changes in the stabilizer, since stabilizers can dynamically bind to the crystal surfaces during the GaP crystal growth. When the highly bulky tertiary amines (e.g., TOA) are used as stabilizers, a staggered conformation is favored, minimizing steric hindrance between these ligands and GaP lattices (Scheme 11, path A), and zinc blende GaP is preferred over wurtzite. In contrast, when an excess amount of less sterically hindered HDA is added to TOA, the rotational barrier between GaP-HDA complexes and GaP lattices is reduced. Therefore, the formation of the kinetically stable wurtzite GaP is now facilitated (Scheme 11, path B) under the kinetic growth regime induced by a high monomer concentration [254].

a

b 002 100 3.26

40 nm

4 nm

Figure 32. Large-area TEM image (a) and a HRTEM image (b) of GaP nanorods. Reprinted with permission from [254], Y.-H. Kim et al., J. Am. Chem. Soc. 124, 13656 (2002). © 2002, American Chemical Society.

a

b

Staggered form

c

Eclipsed form

GaP-HDA complexes

P

HDA

Ga

N

P

C8H17 C6H17

Ga

C4H17

C16H33 P

N

Ga

Path A P P P

H H

Path B P

P

P

Ga Ga Ga Ga P P P P Ga Ga Ga Ga P P P P Ga Ga Ga Ga A: zinc blende

P

staggered conformation

eclipsed conformation

B: wurtzite

Scheme 11. Proposed mechanism for surfactant-driven steric effects on the crystalline phases (a, b) and rod growth (c) of GaP nanocrystals. Reprinted with permission from [254], Y.-H. Kim et al., J. Am. Chem. Soc. 124, 13656 (2002). © 2002, American Chemical Society.

The steric difference between these two stabilizers seems to induce the anisotropic growth of the wurtzite GaP. It is likely that, when wurtzite seeds are formed, sterically bulky TOA selectively binds to the other faces (e.g., 100 and 110 faces) with staggered conformation rather than to 002 faces and blocks growth on these faces. On the other hand, GaP-HDA complexes continuously supply monomers on the 002 faces with high surface energy and therefore promote growth along the c axis (Scheme 11c). In addition, the thermolysis of a monomeric precursor [Zn(TePh)2 ][TMEDA] prepared from Zn(TePh)2 and donor ligand TMEDA in a mixed surfactant trioctylamine-dimethylhexylamine produced ZnTe NRs [312], which could be templated from the rod-like micelles formed in the mixed-solvent system. Currently, this approach has been successfully applied to the synthesis of transition metal NRs such as Co [205], Fe [208], and Ni [210], as well as perovskite BaTiO3 , SrTiO3 NRs [104].

2.7.2. Self-Assembly under Natural/Mild Conditions In contrast to the above capping agent/surfactant-driven synthesis of semiconductor NRs in hot solvent, self-assembly of nanofiber bundles, NTs, NWs, and their structure modulation under natural/mild conditions have emerged recently [25, 72–79]. The reverse micelle reaction medium usually consists of either the anionic surfactant AOT (sodium bis(2ethylhexyl)sulfosuccinate) or nonionic surfactants. Various kinds of 1D inorganic NRs/NWs such as BaCrO4 , [25, 75], BaSO4 [74, 75], BaCO3 [77], and BaWO4 [78], CaSO4 [107], CaCO3 [408], CdS [292, 409], and Cu [410] have been synthesized in reverse micelle media or microemulsions. Other novel nanostructures, such as Ag nanodisc [192], flat CdS triangles [291], and CdS, CdSe NTs/NWs [76], can also be synthesized by this approach. The strong binding interactions between surfactants and inorganic nuclei effectively inhibit the crystal growth and put the spontaneous structure reconstruction and selforganization of the primary nanoparticles under control. Micrometer-long twisted bundles of BaSO4 and BaCrO4 nanofilaments in water-in-oil microemulsions were prepared from the anionic surfactant AOT [25]. The reaction occurs at

Low-Dimensional Nanocrystals

room temperature in unstirred isooctane containing a mixture of Ba(AOT)2 reverse micelles and NaAOT microemulsions with encapsulated sulfate (or chromate) anions. The reverse micelles are about 2 nm in diameter and consist of a spherical cluster of about 10 Ba2+ ions strongly associated with the sulfonic acid headgroups of the surfactant, along with the water of hydration [25]. In contrast, the microemulsions are larger (4.5 nm across) because they contain bulk water (aqueous Na2 SO4 or Na2 CrO4  at a water-tosurfactant molar ratio of w = 10. When mixed together, the two reaction fields interact so that the constituents are slowly exchanged and BaSO4 or BaCrO4 nanoparticles nucleate and grow within the delineated space. With time, other filaments are formed parallel to the original thread to produce a small bundle of coaligned inorganic nanofilaments held together by surfactant bilayers. The locking in of new filaments by surfactant interdigitation generates a bending force in the nonattached segment of the longer primary thread. This results in the coiling of the bundle into a characteristic spiral-shaped structure several hundred nanometers in size that becomes self-terminating at one end because further addition of the primary nanoparticles is prevented by spatial closure. The construction of higher-order structures from inorganic nanoparticle building blocks was successfully demonstrated by achieving sufficient informational content in the preformed inorganic surfaces to control long-range ordering through interactive self-assembly [25]. The NRs have flat surfaces with low curvature so that the hydrophobic driving force for assembly can be strengthened through intermolecular interaction, resulting in the formation of a bilayer between adjacent particles by the interdigitation of surfactant chains attached to nanoparticle surfaces. When the 2− 2+ [Ba2+ ]:[SO2− 4 ] (or [Ba ]:[CrO4 ]) molar ratio is equal to 1.0, remarkable linear chains of individual BaSO4 or BaCrO4 NRs are formed as shown in Figure 33.

635 Semiconductor NTs and NWs have recently been obtained with the use of nonionic surfactants such as t-octyl(OCH2 CH2 x OH, x = 9
10 (Triton-X) and anionic surfactant AOT [76]. NWs of sulfides and selenides of Cu, Zn, and Cd with high aspect ratios can be prepared with Triton 100-X. The results show that it is possible to obtain both NTs and NWs of CdSe and CdS by this surfactant-assisted synthesis. For the synthesis of CdSe NTs, a suspension of cadmium oxide (10 mmol) was prepared in 20 ml of Triton 100-X (∼24 mmol). A solution of NaHSe (NaBH4 /Se in 40 ml of water) was added dropwise under constant stirring to the suspension at 40  C in an argon atmosphere. The resulting mixture was refluxed for 12 h and left overnight. The NTs are generally long, with lengths up to 5 m, as shown in Figure 34. The outer diameter of the NTs is in the 15–20-nm range, and the diameter of the central tubule is in the 10–15-nm range. The wall thickness is therefore around 5 nm. The formation mechanism of the NTs in the presence of surfactant is still not clear. It should be pointed out that the poor crystallinity of the nanostructures obtained by this approach as well as the use of large excess surfactants could restrict the applicability of this approach.

2.8. Bio-Inspired Approach for Complex Superstructures 2.8.1. Polymer-Controlled Crystallization Bio-inspired approaches to the synthesis of inorganic minerals have been a hot research subject [73, 80, 81]. Recently it was shown that so-called double-hydrophilic block copolymers (DHBCs) [81–94] can exert a strong influence on the external morphology and/or crystalline structure of inorganic particles such as calcium carbonate [82–85], calcium phosphate [86], barium sulfate [87, 88], barium chromate [89–91], cadmium tungstate [92], and zinc oxide [93]. Elegant nested calcium phosphate nanofibers were mineralized in the presence of DHBC poly(ethylene oxide)-balkylated poly(methacrylic acid) (PEO-b-PMAA-C12 ), which was synthesized from PEO-b-PMAA by partial alkylation with dodecylamine [411]. Through the control of the pH of the Ca2+ -loaded polymer solution, delicate mesoskeletons of interconnected calcium phosphate nanofibers with starlike, neuron-like, and more complex nested forms can be obtained, as shown in Figure 35 [86].

120 nm

Figure 33. TEM image showing ordered chains of prismatic BaSO4 nanoparticles prepared in AOT microemulsions at [Ba2+ ]:[SO2− 4 ] molar ratio = 1 and w = 10. Scale bar = 50 nm. Reprinted with permission from [25], M. Li et al., Nature 402, 393 (1999). © 1999, Macmillan Magazines Ltd.

Figure 34. TEM image of CdSe nanotubes obtained with Triton 100-X as the surfactant. Reprinted with permission from [76], C. N. N. Rao et al., Appl. Phys. Lett. 78, 1853 (2001). © 2001, American Institute of Physics.

Low-Dimensional Nanocrystals

636 a

b

500 nm

c

0.5 µm

500 nm

Figure 35. TEM images of calcium phosphate block copolymer nested colloids. (a) Star-like form at early stage at pH 3.5. (b) Later stage showing complex central core. (c) Neuron-like tangles produced at pH 5. Reprinted with permission from [86], M. Antonietti et al., Chem. Eur. J. 4, 2491 (1998). © 1998, Wiley-VCH.

Very long BaSO4 and BaCrO4 nanofibers and fiber bundles with remarkable similarity have been successfully fabricated by a polymer-controlled crystallization process at room temperature [88–90]. More complex morphologies of hashemite can be formed in the presence of partially phosphonated block copolymer poly(ethylene glycol)-blockpoly(methacrylic acid) (PEG-b-PMAA-PO3 H2 , PEG = 3000 g mol−1 , 68 monomer units; PMAA = 700 g mol−1 , 6 monomer units) [84]. Figure 36a and b shows SEM images of fibrous superstructures with sharp edges composed of densely packed, highly ordered, parallel nanofibers of BaCrO4 . The TEM micrograph with higher resolution in Figure 35c shows the self-organized nature of the superstructure. Whereas the majority of the fibers appear to be aligned in a parallel fashion, gaps between the single fibers can form but are also closed again. An electronic diffraction pattern taken from such an oriented planar bundle as shown in Figure 36d (right) confirmed that the whole structure

100 µm

100 µm

a

b

420 410

[2

10

] 220

400

210 200

scatters as a well-crystallized single crystal where scattering is along the [001] direction and the fibers are elongated along [210]. The initially formed nanoparticles stabilized by polymers are amorphous with sizes of up to 20 nm, which can aggregate to larger clusters. Evidently, this state of matter is the typical starting point for all types of highly inhibited reactions. The very low solubility product of barium chromate (Ksp = 117 × 10−10 ) shows that the superstructures do not really grow from a supersaturated ion solution but by aggregation/transformation of the primary clusters formed [89]. All structures always grow from a single starting point, implying that the fibers grow against the glass wall or other substrates such as TEM grids, which obviously provide the necessary heterogeneous nucleation sites. The growth front of the fiber bundles is always very smooth, suggesting the homogeneous joint growth of all single nanofilaments with the ability to cure occurring defects, in line with the earlier findings for BaSO4 [88]. The opening angle of the cones is always rather similar, which seems to depend on temperature, degree of phosphonation of the polymer, and polymer concentration [89]. The control experiments show that the higher the temperature, the more linear the structures become. The superstructure developed more clearly, and to a much larger size, when the mineralization temperature was lowered to 4  C, as shown in Figure 37a. A rather lower phosphonation degree (∼1%) of a PEG-b-PMAA is already powerful enough to induce the formation of the fiber bundles and the superstructures, as shown in Figure 37b. Interestingly, secondary cones can nucleate from either the rim or defects onto the cone, thus resulting in a tree-like structures (Fig. 37a). The fact that a cone stops growing once a second cone has nucleated at one spot on the rim shows that the growth presumably is slowing down with time, favoring the growth of the secondary cone. The cone-like superstructures tend to grow farther into a self-similar, multi-cone “tree” structure, which was observed before for barite mineralized in the presence of polyacrylates, but only under very limited experimental conditions [88]. From a structural viewpoint, the hashemite/barite crystals have a mirror plane perpendicular to the c axis, implying that a homogeneous nucleation will always result in crystals with identical charges of the opposite faces, and no dipole crystals can be formed. However, a dipole crystal may be favored

4-10 4-20 2-10 2-20

200 µm

20 µm

0-20 020 -200 220 -210

c

500 nm

d

2-10 2-20

4-20 -400 4-10

a

Figure 36. SEM and TEM images of highly ordered BaCrO4 nanofiber bundles obtained in the presence of PEG-b-PMAA-PO3 H2 (21%, phosphonation degree) (1 g liter−1  ([BaCrO4 ] = 2 mM), pH 5. (a) Two fiber bundles with cone-like shape and length about 150 m. (b) Very thin fiber bundles. (c) TEM image of the thin part of the fiber bundles, inserted electronic diffraction pattern taken along the 001 zone, showing the fiber bundles are well-crystallized single crystals and elongated along [210]. Reprinted with permission from [89], S. H. Yu et al., Chem. Eur. J. 8, 2937 (2002). © 2002, Wiley-VCH.

b

Figure 37. (a) A typical cone-like superstructure contains densely packed BaCrO4 nanofiber bundles in the presence of PEG-b-PMAAPO3 H2 (21%, phosphonation degree) (1 g liter−1  ([BaCrO4 ] = 2 mM, pH 5), 2 days, 4  C; (b) multi-funnel-like superstructures with remarkable self-similarity in the presence of PEG-b-PMAA-PO3 H2 (1%, phosphonation degree) (1 g liter−1  ([BaCrO4 ] = 2 mM, pH 5). Reprinted with permission from [89], S. H. Yu et al., Chem. Eur. J. 8, 2937 (2002). © 2002, Wiley-VCH.

Low-Dimensional Nanocrystals

637

for a heterogeneous nucleation as one end of the crystal is determined by the heterogeneous surface, the other by the solution/dispersion. The crystal has a dipole moment  = Q · l (Q = charge and l = length of the crystal), which increases while the crystal is growing. This implies self-limiting growth so that a new heterogeneous nucleation event on the rim should become favorable after the dipole moment has reached a critical value due to energy minimization [89]. Based on the foregoing experimental observation and understanding, a self-limiting growth mechanism was proposed to explain the remarkable similarity of the superstructures [89] as shown in Scheme 12: (i) At the beginning, amorphous nanoparticles are formed, which are stabilized by the DHBCs (stage 1). (ii) Heterogeneous nucleation of fibers occurs on glass substrates, and the fibers grow under the control of a functional polymer, presumably by multipole field-directed aggregation of amorphous nanoparticles as proposed in [88]. (iii) The growth is continuously slowed down until secondary nucleation or overgrowth becomes more probable than the continuation of the primary growth. The secondary cone will grow as the first ones have done. (iv) The secondary heterogeneous nucleation event taking place on the rim can occur repeatedly, depending on the mass capacity of amorphous nanoparticles in the system. A very recent observation shows that low-molecularweight polyacrylic acid sodium salt serves as a very simple structure-directing agent for the room-temperature, largescale synthesis of highly ordered cone-like crystals or very long, extended nanofibers made of BaCrO4 or BaSO4 with hierarchical and repetitive growth patterns as shown in Figure 38, where temperature and concentration variation permit control of the finer details of the architecture, namely length, axial ratio, opening angle, and mutual packing [90]. The formation of interesting hierarchical and repetitive superstructures is worth exploring further for other mineral systems. Interestingly, single uniform BaCrO4 nanofibers can also be synthesized by a combination of polymer-controlled crystallization process and controlled nucleation by colloidal species producing locally a high supersaturation of both DHBCs and Ba2+ . The addition of a minor amount of a cationic colloidal structure such as PSS/PAH polyelectrolyte capsule (PSS/PAH: poly(styrene sulfonate, sodium salt)/polyallylamine hydrochloride) in the same reaction system can promote the independent growth of a number of

4 µm

1 µm

a

b

Figure 38. Complex forms of BaSO4 bundles and superstructures produced in the presence of 0.11 mM sodium polyacrylate (Mn = 5100), at room temperature, [BaSO4 ] = 2 mM, pH = 53, 4 days. (a) Detailed superstructures with repetitive patterns. (b) Well-aligned bundles. Reprinted with permission from [90], S. H. Yu et al., NanoLetters, in press. © American Chemical Society.

fibers, thus leading to the dynamic discrimination of side nucleation and the resulting altered superstructures. In addition to the formation of the mentioned superstructures under the control of block copolymers, DHBCs can also be used to fine-tune the nanostructural details of other inorganic crystals. Very thin 1D and 2D CdWO4 nanocrystals with controlled aspect ratios were conveniently fabricated at ambient temperature or by hydrothermal ripening under the control of DHBC [92]. The TEM image in Figure 40a shows very thin, uniform CdWO4 NRs/NBs with lengths in the range of 1 to 2 m and a uniform width of 70 nm along their entire length (aspect ratio of about 30). The thickness of the NBs is ∼6–7 nm. The slow and controlled reactant addition by the double-jet technique under stirring maintains the formation of intermediate amorphous nanoparticles at the jets [84] so that nanoparticles are the precursors for further particle growth rather than ionic species. Successive hydrothermal ripening after the double-jet reaction leads to a rearrangement of the rods into 2D lens-shaped, raftlike superstructures with a resulting lower aspect ratio, as shown in Figure 40b. In contrast, very thin and uniform nanofibers with a diameter of 2.5 nm, a length of 100– 210 nm, and an aspect ratio of 40–85 as shown in Figure 40c

200 nm (i)

(ii)

(iii)

(iv)

Scheme 12. Proposed mechanism for the formation of BaCrO4 complexity superstructure with self-similarity by polymer-controlled crystallization. Reprinted with permission from [89], S. H. Yu et al., Chem. Eur. J. 8, 2937 (2002). © 2002, Wiley-VCH.

Figure 39. TEM images and electron diffraction pattern of the separated and very long BaCrO4 nanofibers with high aspect ratio obtained in the presence of a small amount of PSS/PAH capsules (20 l), PEGb-PMAA-PO3 H2 , 1 g liter−1 , [BaCrO4 ] = 2 mM, pH 5, 25  C. (a) A full view of the nanofibers. Reprinted with permission from [91], S. H. Yu et al., Adv. Mater., in press. © Wiley-VCH.

Low-Dimensional Nanocrystals

638 a

b 80 nm

[10

0]

200 nm

c

50 nm

d

50 nm

Figure 40. TEM images of the samples obtained under different conditions. (a) and (b) No additives: (a) pH = 53, double jet, −3 [Cd2+ ]:[WO2− M (final solution), at room temperature; 4 ] = 83 × 10 −3 (b) pH = 53, double jet, [Cd2+ ]:[WO2− M (final solution), 4 ] = 83 × 10 then hydrothermal crystallization: 80  C, 6 h. (c) pH = 53, in the presence of PEG-b-PMAA (1 g liter−1 , 20 ml, double jet, [Cd2+ ]:[WO2− 4 ]= 83 × 10−3 M (final solution), then hydrothermal crystallization: 80  C, 6 h. (d) Direct hydrothermal treatment of 20-ml solution containing −2 equal molar [Cd2+ ]:[WO2− M, pH = 53, at 130  C, 6 h, 4 ] = 83 × 10 −1 in the presence of 1 g liter PEG-b-PMAA-PO3 H2 (21%). Reprinted with permission from [92], S. H. Yu et al., Angew. Chem. Int. Ed. 41, 2356 (2002). © 2002, Wiley-VCH.

can readily be obtained when the DHBC PEG-b-PMAA is added to the solvent reservoir before the double-jet crystallization process, and the mixture is then hydrothermally ripened at 80  C. When the partly phosphonated hydrophilic block copolymer PEG-b-PMAA-PO3 H2 (21%) (1 g liter−1  is added at an elevated temperature of 130  C, even without the use of the double jets but at higher concentrations and coupled supersaturation, very thin platelet-like particles with a width of 17–28 nm, a length of 55–110 nm, and an aspect ratio of 2–4 are obtained by a direct hydrothermal process, as shown in Figure 40d. The nanoparticles display an interesting shape-dependent evolution of the luminescent properties, which may be of interest for applications. Polymer-controlled crystallization in water at ambient temperature provides an alternative and promising tool for the morphogenesis of inorganic nanocrystals and superstructures that could be extended to various systems.

2.8.2. Supramolecular Self-Assembly Supramolecular directed self-assembly of inorganic and inorganic/organic hybrid nanostructures has been emerging as an active area of recent research. The recent advance shows the remarkable feasibility of mimicking a natural mineralization system by a designed artificial organic template.

The supramolecular functional polymer can be directly employed for mineralization template of novel inorganic nanoarchitectures. CdS helices were successfully templated from supramolecular nanoribbons [94]. This novel inorganic nanostructure has a coiled morphology with a pitch of 40–60 nm, which can be rationalized in terms of the period of the twisted organic ribbons. A triblock architecture termed “dendron rodcoils” (DRC) can hydrogen bond in head-to-head fashion through dendron segments and selfassemble into nanoribbons [94]. The mineralization of the helical structures was done in both 2-ethylhexyl methacrylate (EHMA) and ethyl methacrylate (EMA). Figure 41 shows a typical CdS helix with a zigzag pattern and a pitch of 40–50 nm, which was isolated from a 1 wt% gel of the DRC in EHMA, to which a solution of cadmium nitrate in THF had been added prior to exposure to hydrogen sulfide gas [94]. HRTEM studies revealed that the polycrystalline zinc blende CdS is made up of small domains with grain sizes of about 4–8 nm. The results suggest that it is possible to achieve good control over the morphology of the templated product by using extremely uniform, stable, nonaggregated supramolecular objects as templates. A peptide-amphiphile (PA) molecule was designed for the mineralization of PA nanofibers and hydroxyapatite (HAp) nanofibers [412]. This amphiphile molecule can assemble in water into cylindrical micelles because of the amphiphile’s overall conical shape, resulting in alkyl tails packed in the center of the micelle while the peptide segments are exposed to the aqueous environment [412]. The PA molecules were found to self-assemble at acidic pH but disassemble at neutral and basic pH. After self-assembly into nanofibers, the nanofibers were cross-linked by the oxidation of the cysteine thiol groups through treatment with 0.01 M I2 . The crosslinked PA fibers contained intermolecular disulfide bonds, and intact fiber structures were still kept. These cross-linked fibers with negative charged surfaces are able to direct mineralization of HAp to form a composite material in which the crystallographic c axes of HAp are coaligned with the long axes of the fibers. The alignment of HAp nanofibers resembles the lowest level of hierarchical organization of bone [413]. In addition, supramolecular self-assembly using organogelators as template to transcribe inorganic nanostructures has been intensively studied. Organogelators are low-molecularweight compounds that can gelate organic fluids at low

25 nm

Figure 41. TEM micrograph of a typical CdS helix with a pitch of 40–50 nm precipitated in gels of the DRC in EHMA. Reprinted with permission from [94], E. D. Sone et al., Angew. Chem. Int. Ed. 41, 1705 (2002). © 2002, Wiley-VCH.

Low-Dimensional Nanocrystals

concentrations [414–419]. The gelators can gelate in organic solvents to form unique superstructures, which can transcribe inorganic nanostructures [420–429]. A helically structured silica has been successfully templated by a sol–gel transcription in chiral organogel systems [420–426]. The results show that certain cholesterol derivatives can gelate even tetraethoxysilane (TEOS), which can be used to produce silica by sol–gel polymerization [420]. Sol–gel polymerization of gelated TEOS solutions produces silica with a novel hollow fiber structure due to the template effect of the organogel fibers [420–426]. The formation of a three-dimensional network based on fibrous aggregates in organic fluids could be responsible for the formation of the gelling phenomenon. This approach can be extended to the production of other metal oxide nanofibers or NTs with chiral structures. TiO2 fibers were fabricated with the use of an amphiphilic compound, trans-(1R,2R)-1,2-cyclohexanedi(11aminocarbonylundecylpyridinium) hexafluorophosphate as template [427]. The amphiphilic compound containing cationic charge moieties is expected to electrostatically interact with anionic titania species under basic conditions in the sol–gel polymerization process of Ti[OCH(CH3 2 ]4 . Transition-metal (Ti, Ta, V) oxide fibers with chiral, helical, and nanotubular structures can be prepared by the sol–gel polymerization of metal alkoxides, with trans-(1R,2R)- or trans-(1S,2S)-1,2-di(undecylcarbonylamino)cyclohexane as structure-directing agents [428]. A recent report describes well-defined TiO2 helical ribbons and NTs obtained through sol–gel polymerization of titanium tetraisopropoxide [Ti(Oi Pr)4 ] in gels of a neutral dibenzo-30-crown-10appended cholesterol gelator [429]. These chiral helical tubes could have various applications in electronics, optics, and photocatalysts.

2.8.3. Small Molecular Organic Species-Mediated Crystallization In contrast to the previously mentioned polymer-controlled crystallization and supramolecular template synthesis, some small molecular organic species can also exert significant influence on crystal orientation growth. A very recent report by Liu et al. describes unusually oriented and extended helical ZnO nanostructures grown from a synthetic ceramic method, which is surprisingly very similar to the growth morphology of nacreous calcium carbonate [95]. The helical ZnO nanostructures were grown on a glass substrate containing oriented ZnO NR arrays [430], which was placed in a solution containing Zn(NO3 2 , hexomethylenetetramine, and sodium citrate and was reacted at 95  C for 1 day. Figure 42 shows the similarity of SEM images of the helical ZnO nanocolumns (Fig. 42a and b) and the growth tip of a young abalone shell containing oriented columns of aragonite nanoplates (Fig. 42c). In the secondary growth on the helical NRs, aligned and well-defined nanoplates are formed, as in nacre. The side-width growth of the ZnO nanoplates leads to hexagonal ZnO plates that begin to overlap with one another.

639 a

300 nm

b

c

300 nm

5 µm

Figure 42. Comparison of ZnO helical structures with nacre. (a) and (b) High-magnification image of oriented ZnO helical columns. (c) Nacreous calcium carbonate columns and layers near the growth tip of a young abalone. Reprinted with permission from [95], Z. R. Tian et al., J. Am. Chem. Soc. 124, 12954 (2002). © 2002, American Chemical Society.

The organic species were reported to act as simple physical compartments or to control nucleation or to terminate crystal growth by surface poisoning in the biomineralization of nacre [431, 432]. The similarity of the biomimetic structures of helical ZnO rods to nacreous CaCO3 indicated that helical growth might play some role in the formation of organized nacreous calcium carbonate. However, the organic species citrate ion still plays critical roles in the formation of such structures since the citrate ion has a large tendency to inhibit the growth of the (002) surfaces, possibly through selective surface adsorption. The discussed polymer controlled mineralization and small molecular organic species-mediated mineralization could shed new light on the morphology and orientation control of inorganic crystals.

2.9. Oriented Attachment Growth Mechanism Traditionally, crystal growth can be described as an Ostwald ripening process. The formation of tiny crystalline nuclei in a supersaturated medium occurred first and then was followed by crystal growth. The larger particles will grow at the cost of the small ones because of the difference between large particles and the smaller particles of a higher solubility. Penn and Banfield [96–98] confirmed that both anatase and iron oxide nanoparticles with sizes of a few nanometers can coalesce under hydrothermal conditions in a way called “oriented attachment” (Fig. 43). The crystal lattice planes may be almost perfectly aligned, or dislocations at the contact areas between the adjacent particles lead to defects in the finally formed bulk crystals. The so-called oriented attachment was also proposed by other authors for the crystal growth of TiO2 [99] and for micrometer-sized ZnO particles during the formation of rod-like ZnO microcystals [100]. However, until now, no direct evidence has been reported.

Low-Dimensional Nanocrystals

640

100 nm

Figure 43. TEM micrograph of a single crystal of anatase that was hydrothermally coarsened in 0.001 M HCl, showing that the primary particles align, dock, and fuse to form oriented chain structures. Reprinted with permission from [98], R. L. Penn and J. F. Banfield, Geochim. Cosmochim. Acta 63, 1549 (1999). © 1999, Elsevier Science.

A very recent report by Weller et al. provided some strong evidence that ZnO NRs can be conveniently self-assembled from small ZnO quasi-spherical nanoparticles based on the oriented attachment mechanism by the evaporation and reflux of a solution containing 3–5-nm smaller nanoparticles [102]. Previously, self-assembly of nanoparticles capped by ligands was mainly driven by the interactions of the organic ligands rather than by the interaction of the particle cores. ZnO sol with an average particle size of approximately 3 nm as shown in Figure 44a was easily prepared by adding dropwise a 0.03 M solution of KOH (65 ml) in methanol to a solution of zinc acetate dihydrate (0.01 M) in methanol (125 ml) under vigorous stirring at about 60  C. Refluxing of the concentrated solution leads to the formation of rod-like nanoparticles. Prolonging the heating time mainly leads to an increase in the elongation of the particle along the c axis. After refluxing for 1 day, single crystalline NRs with average lengths of 100 nm and widths of approximately 15 nm were formed, as shown in Figure 44b. The authors argued that oxide nanoparticles are very favorable for oriented attachment for two reasons: first, organic ligands, which prevent intimate contact of crystal planes, are usually not needed for stabilization; and second, crystalline fusion of correctly attached particles leads to a gain not only in lattice-free energy but also in the free energy of polycondensation [102]. The spontaneous aggregation and self-assembly of the small nanoparticles into very elongated NRs/NWs was again observed in the case of CdTe when the protective shell of organic stabilizer on the surface of the initial CdTe nanoparticles was removed [103]. The presence of “pearl-necklace” agglomerates as intermediates of the

A

B

5 nm

40 nm

5 nm

Figure 44. TEM images of ZnO. (A) Starting sol. (B) After 1 day of reflux of the concentrated sol. The insets show high-resolution TEM images of individual nanoparticles. Reprinted with permission from [102], C. Pacholski et al., Angew. Chem. Int. Ed. 41, 1188 (2002). © 2002, Wiley-VCH.

nanoparticle-NW transition suggested that the growth mechanism could be related to a special interaction/attraction between the nanoparticles. A dipole-dipole interaction was believed to be the driving force for such self-organization of the nanoparticles [103]. This approach provides a new pathway to high-quality luminescent CdTe NWs, even though the yield is still low. Although there have been only few reports on the selforganization of nanoparticles, this new growth mechanism could offer an additional tool for the design of advanced materials with anisotropic material properties and could be used for the synthesis of more complex crystalline threedimensional structures in which the branching sites could be added as individual nanoparticles, and could even lead to nearly perfect crystals [101, 102].

3. SUMMARY In this chapter, the state of art of the synthesis of lowdimensional nanocrystals and more complex superstructures through the different approaches has been overviewed. The current advances in synthetic methodologies demonstrated that it is possible to rationally design 1D, 2D, and even more complex superstructures. The “hard” approaches, such as template-directed growth method, the VLS mechanism, the VS mechanism, and oxideassisted NW growth routes, have shown their powerful and straightforward pathways to the target 1D nanomaterials and their superstructures with high crystallinity, although they usually require either a carefully and costly predesigned template or rather strict reaction conditions. The “soft” approaches show very promising alternative pathways for the synthesis of 1D nanocrystals. In addition to the strategies of solvent-mediated shape and structure modulation, the soft-hard interface between polymers/surfactants (or organic species) and inorganic crystals is also an area of great interest at present. The current progress indubitably emphasizes that all crystals will probably be amenable to morphosynthetic control by use of the appropriate templating molecules. The emerging new solution routes to 1D NRs/NWs and more complex superstructures, such as the solvothermal process, the SLS mechanism, capping agent/surfactant-assisted synthesis, bio-inspired approaches, and the oriented attachment growth mechanism, open alternative doorways to low-dimensional nanocrystals and more complex superstructures. Among these solution strategies, the solvothermal process shows possibilities and versatility for selective synthesis of various semiconductor nanocrystals with controllable shape, size, phase, and dimensionalities under mild conditions. The successful synthesis of various semiconductor NRs/NWs with the use of a suitable ligand solvent as a “shape controller” indicates that the solvothermal process is indicative of a fruitful and promising route to 1D and 2D semiconductor nanocrystals in the future. The further understanding on the nucleation, crystallization, self-assembly, and growth mechanism of the nanocrystals in these solution strategies should open new rational pathways to prepare various kinds of highly crystalline nanowires/nanorods, nanotubes with well-defined monodispersed, sizes, and aspect ratios, and highly ordered/oriented

Low-Dimensional Nanocrystals

complex superstructures. Furthermore, it is no doubt needed to investigate the relationship between the structural specialties (shape, sizes, phases, dimensionality, and complexity etc.) of the low dimensional nanocrystals and their properties, which could shed light on their novel applications and fundamental properties as building blocks in nanoscience and nanotechnology.

ACKNOWLEDGMENTS S. H. Yu thanks the funding support from the “Hundreds of talent plan” of Chinese Academy of Sciences. We thank the financial support from National Natural Science Foundation of China.

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Encyclopedia of Nanoscience and Nanotechnology

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Low-Frequency Noise in Nanomaterials and Nanostructures Mihai N. Mihaila National Institute of Microtechnology, Bucharest, Romania

CONTENTS 1. Introduction 2. Small is Beautiful but Noisy 3. There is Much Noise at the Bottom 4. When the Electron is the Signal: Low-Frequency Noise in Single Electron Transistors 5. Noise Engineering in Semiconductor Quantum Wires 6. Noise in Point Contacts and Metallic Nanobridges 7. Blinking Quantum Dots Glossary References

1. INTRODUCTION Microscopic particles such as small pollen grains bounce erratically when immersed in water. The discovery of this phenomenon, systematically investigated by Brown [1] but observed long before, was the starting point in the study of fluctuations and noise. Almost eight decades were necessary for the scientists to understand that brownian motion is one of the most subtle macroscopic manifestation of the microscopic world. Fluctuations or noise proved to be very fine tools to access the microscopic world for they allowed the determination of the Avogadro number [2], Boltzmann constant [3, 4], the electron charge [5, 6], and, more recently, the fractional Laughlin charge, e/3, where e is the elementary charge [7]. Since Schottky’s magistral work on the vacuum diode [8], it became clear that the performances of the electronic devices processing the signals are fundamentally limited by their intrinsic noise. Soon after Schottky’s discovery, Johnson found that the brownian motion of the charged carriers generates thermal noise [3]. Both shot noise and thermal noise are unavoidable and act as fundamental limits, ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

especially in the high-frequency domain. During his experiments on thermal and shot noise, Johnson also found that at low frequency the noise spectrum in the vacuum diodes changes inversely to the frequency (1/f noise). For almost 80 years, 1/f noise—as it is called today—proved to be a universal and one of the most difficult to explain phenomena. Besides thermal, shot, and 1/f noise, two other kinds of noise were found in semiconductors and the new solid-state devices: generation–recombination noise and a kind of noise which consists of random switching between two (or more) distinct levels. In large devices, it was called burst or popcorn noise, while in small dimensional structures it is known as random telegraph signal (RTS) noise. The unabated scaling of the device dimensions is systematically approaching the microscopic roots of these low-frequency noise phenomena. This inexorable trend requires the reduction of power supply voltages, which “amounts to reduction in the signal noise level” [9]. The top down approach in the contemporary nanotechnology will encounter some given thresholds where “the noise is the signal,” to quote Rolf Landauer [10] who used this syntagm in a different context, however. In nanoscaled solid-state devices, the signal to noise ratio is close to 1, thus leading us to the vicinity of the fundamental limits. As a consequence, finding where the noise comes from could be a sine qua non condition for the further evolution in the field. It is very interesting that the noise in nanostructures brings back to light some very old and unsolved problems regarding the controversial physical mechanisms of 1/f noise. In this context, the purpose of this contribution is twofold. On one hand, we will briefly review the status of the low-frequency noise research in nanomaterials and some nanostructures/nanodevices. On the other hand, we shall investigate whether the nanoscience could answer old, perennial questions, in the field of low-frequency noise. The very old yet hot problematics of the 1/f noise will be discussed in the light of some new achievements in the nanoscience. It is shown that the very old question regarding the topology of the noise sources (surface or bulk origin) of this phenomenon is clarified, while its microscopic origin

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (649–666)

Low-Frequency Noise in Nanomaterials and Nanostructures

650 cannot be unambiguously answered. Although present even in the quantum dots, the RTS noise as a fundamental component of the 1/f spectrum seems to be a nonsequitur. The enhanced phonon density of states in nanomaterials seems to be a source of higher noise levels, while the surface anharmonicity is proposed as a possible ingredient in the generation of 1/f noise in carbon nanotubes. The low-frequency noise appears to be a powerful tool to characterize nanomaterials and nanodevices.

2. SMALL IS BEAUTIFUL BUT NOISY 2.1. Infrared Catastrophe: From Vacuum Diode to Nanotriode and Carbon Nanotube At high enough frequency, the noise of a vacuum diode is dominated by shot noise [8]. Below a given frequency, Johnson [3] observed that noise intensity starts to increase above shot noise. The noise continuously increases while the frequency decreases (1/f ). No tendency to roll off was found to date. Since its discovery, the phenomenon was found in an astonishing number of physical systems. In solid and solidstate devices, 1/f noise is omnipresent. By analogy with the ultraviolet catastrophe in the theory of blackbody radiation, 1/f noise was called infrared catastrophe. Soon after its discovery, 1/f and burst noise were found to dominate the low-frequency noise of bipolar transistors. In comparison with a vacuum triode, the bipolar transistor was very noisy [11], its performances being much worse than those of a vacuum diode. However, it was not due to the reduced dimensions of the new device but rather to the undeveloped technology. This is not the case for some microelectromechanical systems manufactured by advanced modern technology. For instance, 1/f noise is the dominating noise mechanism in small piezoresistive ultrathin cantilevers [12]. Although a 100 nm thin cantilever features superior performances, its 1/f noise is higher than in a 1 m thick cantilever [11]. Moreover, for the 100 nm thin cantilevers, the noise intensity increases while the volume decreases. The tendency for the noise is to decrease with the total number of the carriers in the cantilever, which is in agreement with the Hooge phenomenological formula [13], SV  = 2 V Nf

noise performances evaluated by Driskill-Smith et al. [15]. Nanopillars of 1 nm diameter or less and having heights up to 15 nm are formed on the cathode surface by a natural lithographic technique. In contrast to the classical vacuum diode, the electrons are emitted from the metallic cathode by the Fowler–Nordheim effect. The emission takes place at nanopillars, in a gas for nanodiodes and in vacuum in the case of the encapsulated nanotriodes. For both nanodiode and nanotriode, the turn-on voltage is about 8 V and does not depend on temperature. A fine structure in the field-emitted current (anode current) vs. gate voltage was observed from room temperature to 20 K. In the nanotriode, this structure does not depend on the time it has been registered or temperature, which suggests that it is an inherent property of the device. It is reminiscent of the universal conductance fluctuations observed in solid-state devices and is the fingerprint of the electron interference between cathode and anode. As can be seen from Figure 1a, when a constant voltage is applied, the emission current shows time fluctuations. Bistable (burst or RTS) noise pulses were observed at room temperature, while at 20 K they disappeared. This behavior is in contrast to what happens in some other physical system such as, for instance, MOS transistors where the burst (RTS) noise becomes dominant at low temperature. When Fourier transformed, the 1/f noise was found to dominate

(1)

where SV is noise spectral density, at a given frequency (f ), of a fluctuating voltage (V ) which develops across the terminals of a resistor when a current is injected in it; N is the total number of either electrons or atoms in the investigated sample (still in dispute). This fact clearly points out that the performances of reduced-dimensionality systems can be limited by noise. An illustrating example in this respect can also be the nanometer-scale electrometer whose ultimate performance at higher frequency is limited by thermal noise [14], whereas for frequencies lower than 103 Hz, the limits are imposed by 1/f noise. In fact, by reducing the dimensions we come closer to the limit imposed by the noise phenomena. Vacuum nanodiodes and nanotriodes have been recently fabricated and their

Figure 1. (a) Current stability of the nanotriode at 300, 77, and 20 K. RTS noise is observed at room temperature. (b) 1/f -like noise spectrum in a nanotriode operating at 6 nA and at 300 K, in the frequency ranges 0.03–10 Hz and 10 Hz–4 kHz. Reprinted with permission from [15], A. G. Driskill-Smith et al., J. Vac. Sci. Technol. B 18, 3481 (2000). © 2000, American Vacuum Society.

Low-Frequency Noise in Nanomaterials and Nanostructures

the noise performance of the nanotriode below about 104 Hz (Fig. 1b). The frequency exponent is about 1.3, but a closer look at the spectrum indicates that this is somehow larger, namely, 1.43. This is mainly due to the presence of the bistable noise at room temperature. The 1/f spectrum crosses the shot noise floor at a corner frequency of about 105 Hz [15], while from the Johnson’s data (from 1925!) [3] for classical triodes, most of the corner frequencies are far less than 105 Hz. Apparently, the nanotriode is much noisier than the classical triode. However, one can compare quantitatively the noise performance of the two devices. For instance, in a vacuum triode (see Fig. 7 from Ref. [3]), at a frequency of about 1000 Hz, “the ratio of the apparent to the actual charge of the electron,” in the terminology used by Johnson himself [3], is about 10. This is nothing but the ratio between 1/f noise intensity and shot noise, 2eI, where e is the electron charge and I = 5 mA is the average current through the triode. With these data, one obtains Sshot = 2eI = 16 × 10−21 A2 /Hz and S1/f = 10 × Sshot = 16 × 10−20 A2 /Hz. From the noise data for nanotriode [15], we have S1/f ≈ 10−24 A2 /Hz, at a frequency of 1000 Hz, for a current I = 6 × 10−9 A. Assuming that noise intensity depends quadratically on current for both devices, one obtains the normalized noise intensity S1/f /I 2 = 64 × 10−16 Hz−1 for the classical triode and S1/f /I 2 = 27 × 10−7 Hz−1 for the nanotriode. Hence, in terms of normalized values, the nanotriode is apparently much noisier (eight orders of magnitude!) than the vacuum triode. However, this difference could be much smaller if the comparison is made between the 1/f noise parameters, , because it depends on the total number of carrier (N ) in the sample. The simplest way out is to compare, as Johnson did [3], the Fano numbers, which is the ratio between the actual 1/f noise and the shot noise intensity: S1/f /Sshot . In this case, at the same frequency (1000 Hz), S1/f /Sshot = 10 for the vacuum triode and S1/f /Sshot ≈ 103 for the nanotriode. Consequently, the very small nanotriode is two orders of magnitude noisier than its classical counterpart. In the same line of argumentation, it came as a big surprise when, recently, Collins and co-workers [47] reported very high 1/f noise levels in single-walled carbon nanotubes. The noise intensity was 4 to 10 orders of magnitude higher than in conventional systems. By conventional systems they meant carbon resistors which are well known for their very high noise levels. Usually, the recommendations are that for low-frequency noise circuitry these resistors should be avoided. In this context, the applicability of the carbon nanotubes for low-frequency noise applications is of great concern: “Unless the excess noise can be somehow suppressed, this certainly calls into question the applicability of SWNTs [single-walled carbon nanotubes] for many low-noise electronic applications,” to cite Collins and co-workers [47]. Although the contacts are suspected to play a role in the noise measurement on carbon nanotube, there are some other groups that reported similar results. Besides increased noise levels, in small dimensional structures unusual noise spectra can occur. If properly interpreted, they could be invaluable in the understanding of the noise mechanisms, some of them still in dispute.

651 2.2. “Forgotten” Spectra: The Case of the MOS Transistor The case of MOS transistor is of particular interest not because this device is very noisy but merely for the fact that it is the most important constituent of the present and future ultralarge scale integrated circuits. In a sense, it can be considered as a paradigmatic example for the evolution of an electron device from microscopic scale to the nanoscale dimensions [16]. Its high noisiness is due to the specific surface conduction mechanism in the channel. McWhorter number fluctuation mechanism [17] due to the tunneling into interface/oxide states was for quite a long time used to explain the occurrence of 1/f noise in this device. Tunneling into a single state generates a Debye–Lorentz spectrum. If the states are uniformly distributed in the oxide and the distribution of their relaxation time ( ) is proportional to 1/ , a 1/f spectrum is obtained by Lorentzian superposition. The noise intensity scales inversely with the channel length (L) [18, 19]. For channel length L < 06 m, the McWhorter theory fails to explain experimental data [18]. However, a Lorentzian superposition is still used to generate the 1/f noise spectrum. This approach is based on the observation that in small MOSFETs, the RTS is the dominant noise source. This is due to a carrier capture into a single trap located around the Fermi level. Experiments performed on very small devices [20–23], usually in the mesoscopic region, revealed that only Lorentzian spectra corresponding to RTS noise are present. The quite general belief is that 1/f noise decomposes into its fundamental components, so in reducing the dimensions what we are seeing is the contribution of each fluctuator [20–24]. Ghibaudo et al. [25] observed that the noise intensity in MOS structures increases by orders of magnitude when the scaling factor increases. Of great concern is also the fact that the noise sample-to-sample dispersion strongly increases (three orders wide) with the scaling factor. At the same time, “a strong evolution of the noise spectra from 1/f to Lorentzian like for devices with the channel lengths from 1 to 0.1 m” [24] was calculated. Nevertheless, the observation of Lorentz spectra in very small devices does not have a unique interpretation. An alternative explanation would be that the mechanism generating 1/f noise does not exist anymore or it is so weak that the RTS noise is dominant. Some examples given in this chapter seem to support this statement. In a very small MOSFET there is a good chance of having no trap in the channel. A related question would be: does 1/f noise disappear when there is no trap in the channel? A partial answer to this question already exists, since the experiments performed by the Grenoble school [25] showed that the residual signal after the subtraction of the RTS noise exhibits a 1/f spectrum. Similar results were obtained long ago by Strasilla and Strutt [26]. In some small MOSFETs, 1/f noise is somewhat hidden. For instance, recently Bu et al. [27] performed noise measurements on nanometric MOSFETs with 100 nm channel length and 25 nm channel width. Although RTS noise was present in the time trace, Figure 2 shows that neither 1/f nor Lorentzian spectra were observed but a 1/f 3 spectrum was found. Such spectra were long searched by van der Ziel’s

Low-Frequency Noise in Nanomaterials and Nanostructures

652

coined for some time in connection with the contribution of the RTS noise to the 1/f noise [20, 33]. Recently, to check the role of the surface states, Kramer and Pease [34] proposed noise measurements on cylindrical surroundinggate nanoscale MOS transistors. The dimension of the channel (few nanometers) could allow single or, possibly, “zero trap” behavior; hence, one can discriminate between different mechanisms of noise in MOSFETs. To date, we are not aware whether these measurements have been performed or not.

3. THERE IS MUCH NOISE AT THE BOTTOM [35] Figure 2. 1/f 3 input-referred noise spectra observed in a MOSFET with an ultranarrow channel at different gate voltages and T = 100 K. Note that at frequencies lower than 100 Hz, the slope of the spectra is even higher. An RTS spectrum is presented for comparison. Reprinted with permission from [27], H. M. Bu et al., Appl. Phys. Lett. 76, 3250 (2000). © 2000, American Physical Society.

Minnesota school [28, 29] who predicted that such a spectrum could be described by the formula SI =

A + B/f 1 + 2f 2

(2)

In this case, 1/f noise and RTS noise go hand-in-hand, in the sense that one modulates the other. A spectrum of this form can be obtained only in the situation in which both phenomena have a common source. Such “forgotten” spectra are rarely discussed although they are not such a very rara avis. For, instance, a 1/f 3 spectrum was observed in a germanium filament by Mapple et al. [30] in 1955 and used by McWhorter himself in his famous paper (see Fig. 1 of Ref. [17]) to exemplify the existence of 1/f noise. Apparently, these results do not support the RTS noise as the fundamental component of 1/f noise. On the other hand, the fact that such a spectrum occurs in nanometer MOSFETs could be a chance to give an answer to this longstanding question. Also, some experiments performed recently at Bell Labs [31] on deep submicron MOS transistors with 100 nm gate length and 40 nm gate oxide revealed that, in comparison with a 0.5 m gate length MOSFET, the 1/f noise intensity increases by about two orders of magnitude and the spectrum spans over seven decades of frequency, with a corner frequency of about 107 Hz. To explain such an extension in the frequency, one needs more Lorentzians, therefore more traps. In direct connection with these results, one can come up with some questions such as: does the number of traps increases by decreasing the channel dimensions? This question has to be answered if credence is to be given to the idea that Lorentzian superposition is the source of the 1/f noise. A possible answer to all these questions would be to look for other noise mechanisms in MOSFETs [32]. In essence, the new approach reveals the role of both surface and bulk phonons in the 1/f noise generation in MOSFETs. Tunneling is not excluded but the noise is associated merely with the phonons which inelastically assist the carriers tunneling into surface states. The idea has also been

3.1. Low-Frequency Noise in Nanocrystalline and Nanoparticle Films In the last half century, noise measurements have been performed on granular structures and discontinuous metal films. It is now well established that granularity and discontinuity are an important source of excess noise in these materials. In fact, it is considered that diffusion of the atoms at the grain boundaries or the tunneling between the metallic islands are the mechanisms responsible for the 1/f noise generation in such systems. If in the case of some classical materials the granular structure was more or less accidental, this particular feature is a desired, intrinsic property of nanomaterials. The study of noise in granular, discontinuous, or grain-boundary-containing materials heralded what is going to happen in nanomaterials. Specific to nanomaterials is the presence of the grain boundaries in a high concentration. Naturally, they strongly contribute to the scattering of conduction electrons, so they could be a source of excess noise. Performing noise measurements on gold nanocrystalline films, with grain diameters between 10 and 60 nm, Ochs et al. [36] found a 1/f noise spectrum. Indeed, the noise intensity was about 5 times larger than in polycrystalline gold films. To our knowledge, this is the first noise measurement on a nanocrystalline film. Also, the grain boundary area per unit volume was found to be 5 times larger in nanomaterials than in the polycrystalline film. Consequently, the authors attributed the increased noise level to the presence of grain boundaries which “occupy a large volume fraction” [36] and, as such, large 1/f noise is to be expected. According to Ochs et al. [36], noise measurement could be a novel tool to investigate the atomic motion at the grain boundaries between nanocrystallites. Another way to understand why in nanocrystalline materials the 1/f noise level is high would be to resort to the microscopic interpretation of the 1/f noise parameter. As is well known, from Hooge’s [13] heuristic formula, the noisiness of a linear physical system can be evaluated by calculating the so-called 1/f noise parameter, . Understanding the physical significance of this parameter has been a longstanding goal for the last three decades. In this respect, it has been recently proposed [37] that a connection between the 1/f noise parameter () and the Eliashberg function, 2 F , could exist,  ∝ 2 F 

(3)

Low-Frequency Noise in Nanomaterials and Nanostructures

where F  is the phonon density of states, 2 is the electron–phonon matrix element, and  is the atomic vibration frequency. The above equation predicts increased noise levels in systems with increased phonon density of states. Apparently, this is in contradiction with the role of the grain boundaries in generating excess 1/f noise. However, recent results demonstrating that the atoms located at the grain boundaries are at the origin of enhanced phonon density of states in nanomaterials show that relation (3) is not incompatible with the role of the grain boundaries. For instance, it has been shown both theoretically [38, 39] and experimentally [40, 41] that in nanomaterials, at both low and high energy sides of the phonon spectrum, there is an enhancement in the phonon density of states. This enhancement is mainly “caused by the high number of grain-boundary atoms” [39], while the “atoms within grains do not noticeably contribute to the additional low-frequency modes” [41]. Moreover, the phonon density of states depends on the size and shape of the grains [41, 42]; namely, F  increases by decreasing grain size. This can explain, at least qualitatively, why in some physical systems the noise intensity increases with the particle size. All of these arguments support the idea that the increased noise levels observed by Ochs et al. [36] in nanocrystalline gold could be associated with an enhanced phonon density of states at grain boundaries. Noise measurements have been used by Otten et al. [43] in order to investigate monocrystalline PbS nanoparticle films. The main goal was to look for an explanation of the “charge transport mechanisms in these structures” [43]. The samples were prepared by statistical deposition of 20 nm diameter PbS nanoparticles “until a nonhomogeneous film with a mean thickness of one monolayer is reached” [43]. As for the substrates, either GaAs or GaAs topped with a SiNx layer for insulation was used. Prior to nanoparticle deposition, white noise has been detected on the SiNx substrate. On GaAs samples, diffusion noise spectra have been found, 1/f , with the slope = 1/2 and 3/2. A problem here would be to explain where the factor 1/2 comes from, because it is specific to one-dimensional diffusion [44]. The weighted noise spectra exhibit maxima at frequencies which shift with the applied voltage. Assuming that the PbS nanoparticles create areas with different conductivity between the electrodes, it results that these areas “modulate the charge transport due to the interparticle potential barrier and the particle–substrate interface” [43]. The thermal agitation of the carrier between nanoparticles is influenced by the applied voltage through the interparticle potential. From the shift of the characteristic frequency, which is due to the different applied voltage [45], the diffusion coefficient can be calculated. Since the GaAs samples are three orders of magnitude more conductive than the SiNx -type ones, the authors conclude “that the main charge transport takes place in the substrate and in the surface” [43]. Hence, the charge carriers located at the PbS nanoparticles randomly move between these and their movement is influenced by applied voltage through the interparticle potential. As is apparent from these results, the conduction phenomena in nanoparticle films could be quite complicated and the noise measurement offers a very fine tool to reveal as well as characterize them. This could prove very useful for device applications.

653 Tungsten trioxide (WO3  nanoparticle films have also been investigated by noise measurements by Hoel et al. [46]. The particles were deposited onto an indium tin oxidecoated glass substrate. Aluminum contacts were deposited on the nanoparticle film to create a capacitor. The authors found that thermal noise calculated from the ac impedance measurements is in very good agreement with the measured value, so the fluctuation–dissipation theorem is valid for this dielectric material. 1/f noise has also been found, especially in the thinnest samples. It has been attributed to the penetrating columns of aluminum in the structure of the nanoparticles. All these results show “that noise measurements can be used for quality assessment of nanocrystalline insulating films” [46].

3.2. Low-Frequency Noise in Carbon Nanotubes: The End of an Old Controversy The noise properties of carbon nanotubes are of great interest because they are perceived as the future building blocks for nanoelectronics and molecular electronics. The first noise measurements on carbon nanotubes have been performed by Collins et al. [47]. Excess 1/f noise has been found in different configurations of SWNTs such as isolated individual tubes, two-dimensional (2D) films, and 3D materials. In a single SWNT, the noise power (SV ) was found to depend quadratically on the dc biasing current. Therefore, as in many other solid-state physical systems, 1/f noise in carbon nanotubes is due to genuine resistance fluctuations. A Hooge-type formula was used to characterize the absolute intensity of the voltage noise power for all 1D, 2D, or 3D morphologies: SV = AV 2 /f . Collins et al. found that the parameter A is a function of the sample resistance: A = 10 × 10−11 R. For a sample with a resistance R = 100 , the noise amplitude is A = 10−9 . When compared with the conventional systems, such as high quality metal films or metal film containing grain boundaries, or damaged by electromigration or ion bombardment, carbon fibers or very noisy carbon composite resistors, the 1/f noise in carbon nanotubes was found to be four to ten orders of magnitude larger! Carbon resistors are well known for their very high noise levels and usually, it is recommended to avoid their use for lowfrequency noise circuitry. If so, the high noise level could strongly hinder the applicability of the carbon nanotubes in low-noise nanoelectronics. When compared with the Hooge formula, A is simply connected with the 1/f noise parameter () by the relation A = /N , where N is the total number of the carrier/atoms in the sample. For a SWNT with N = 104 atoms  ≈ 02, which is two orders of magnitude larger than the value 0.002 postulated by Hooge [13]. The authors do not exclude the possible contact contribution to such large noise, although the measurements on 2D and 3D systems were done in a four-probe configuration. The authors suppose that the source of noise could be the electrical barriers at nanotube– nanotube junctions but this is not valid for single SWNTs. However, in this last case, the measurement was done in a two-probe configuration. Recently, Roumiantsev et al. [48] have investigated noise on partially iron-filled, multiwalled carbon nanotubes

654 (MWNTs). Contacting the nanotube is a fundamental problem, in general, and in the noise measurements especially. In this experiment, the nanotubes were sprayed onto a SiO2 surface and the individual nanotube, with optimal shape and most strategically placed in rapport to the previously deposited Cr/Au contacts, was contacted by focused ion beam (FIB) method. The noise measurements were done in the ohmic region of the I–V characteristic. As in the case of Collins et al. [47], 1/f -like noise spectra were found, with close to unity. In contrast to the Collins et al. [47] results, Roumiantsev et al. [48] observed that the noise intensity is much smaller. Also, the current dependence of the 1/f noise intensity was very different from the quadratic dependence found by Collins et al. [47]. In this case, the noise spectral density depends sublinearly on the injected current: SV ∼ I  , with  between 1 and 1.5 for different samples. According to Roumiantsev and co-workers, “the noise could be generated by a small nonlinear contact resistance at the points of the contacts of FIB metal to the nanotube.” [48]. When measured in a simple FIB stripe, the noise showed a quadratic dependence on the current, SI ∼ I 2 ; therefore the role of the contact is decisive in the noise measurements. It would be of interest to mention here that nonlinear effects in the 1/f noise of a 2D electron gas have also been reported although the noise measurements were performed in the ohmic region of the I–V characteristics [49]. The local deviations from the quadratic law have been attributed to the specific electron– phonon interaction in a 2D electron gas. Since the carbon nanotube is a quantum wire, one can expect nonlinear effects in its 1/f noise even when the contact is perfect and this is due to the specific scattering mechanisms in a 1D electron gas. In addition, it is worth noting that the noise measurements of Collins et al. [47] were done at very small voltages (mV), while Roumiantsev et al. [48] measured at much higher voltages (volts). At these voltages, nonlinear effects can be very important. 1/f noise has been very recently reported by Postma et al. [50] in individual metallic SWNTs. In equilibrium at room temperature, the thermal noise floor measured by Postma in a nanotube with a resistance R = 126 k was in excellent agreement with theory: SV = 4kB T R = 21 × 10−16 V2 /Hz, where the terms have their usual meaning. Under nonequilibrium conditions, when current is injected into the nanotube terminals, the noise spectrum of the voltage fluctuations is 1/f and its intensity depends quadratically on voltage. For the resistance of 126 k, a parameter A = /N = 18 × 10−6 was found to fit the noise data. For a nanotube with a diameter of about 1.4 nm and a 0.5 m length [51], containing approximately 2000 conduction electrons, the 1/f noise parameter is about  = 36 × 10−3 , very close to the value postulated by Hooge 2 × 10−3 . This value is at odds with the value calculated by Collins et al. [47], who found a two order of magnitude larger value. However, if it is to compare the noise data of Postma et al., for the nanotube with the resistance R = 126 k, with those of Collins et al., from the fitting relation A = 10 × 10−11 R of Collins et al. [47], one finds A = 126 × 10−7 , an order of magnitude smaller than the value obtained by Postma. In fact, the experimental values determined by Collins et al. (Fig. 3 from Ref. [47]) are of the order 10−6 ; hence there is no discrepancy between the Postma et al. data and those

Low-Frequency Noise in Nanomaterials and Nanostructures

of Collins et al. in terms of parameter A. The discrepancies in the  values come from the fact that Collins considers N = 104 to be the total number of the atoms in the sample, while Postma considers N = 2000 the total number of the conduction electrons in the sample. Postma et al. also stated that “carbon nanotubes provide the right ingredients to study the origin of 1/f noise in detail” [50]. Indeed, the temperature dependence of the 1/f noise in carbon nanotube seems to be relevant for the physical mechanism of the 1/f noise. It has been recently shown that the frequency exponent ( ) could be affected by the lattice anharmonicity (nonlinearity) [37]. The anharmonicity participation in the 1/f noise can also be inferred from some simple analogies between the temperature dependence of 1/f noise and the thermal conductivity of carbon nanotubes. In the first measurement on the temperature dependence of 1/f noise in a SWNT, Postma et al. [50] reported that in a temperature range from 8 to 300 K, the 1/f noise intensity (A) increases by three orders of magnitude. In a temperature interval ranging from 4.2 to 290 K, Templeton and McDonald [52] found that 1/f noise in conventional carbon resistors did not vary by more than an order of magnitude. The same situation was reported by Fleetwood et al. [53] between 77 and about 400 K. The results of Postma et al. [50] are shown in Figure 3. Similar results were obtained on several samples, the sample-to-sample variation being roughly within one order of magnitude [51]. From Figure 3 one can speculate that there is a tendency in the noise to show a plateau or a maximum around 300 K. In the same temperature range, it has been experimentally demonstrated that the thermal conductivity of multiwalled carbon nanotubes increases by three orders of magnitude and exhibits a maximum at 320 K [54]. Also, Osman and Srivastava [55] calculated that the thermal conductivity of a nanotube with a diameter of 5 nm shows a maximum located at 300 K. It does not depend on chirality but on the diameter. This peak has been attributed to the participation of the phonon–phonon interaction, namely, to the Umklapp processes [54, 55]. If any, a maximum or a plateau in noise close to the temperature where the thermal conductivity is dominated by Umklapp processes would

Figure 3. Temperature dependence of the parameter A = /N = fSV /V 2 for a single-wall carbon nanotube. The inset shows the temperature dependence of the nanotube resistance. The lines are guides to the eye. Reprinted with permission from [50], H. W. Ch. Postma et al., in “Electronic Correlations: From Meso- to Nanophysics” (T. Martin, G. Montanbaux, and J. Trân Thanh Vân, Eds.) EDP Sciences, 2001, France. © 2001, EDP Sciences.

Low-Frequency Noise in Nanomaterials and Nanostructures

suggest that also Umklapp phonons are involved in the noise generation. Even more interesting, since the nanotube is nothing but a rolled surface, we are dealing with the surface anharmonicity which is stronger than bulk anharmonicity. However, it would be quite unusual for the surface anharmonicity to explain such a huge noise, 4 to 10 orders of magnitude more than in carbon resistors, as observed by Collins et al. [47]. Individual and two crossing multiwalled carbon nanotubes were investigated by noise measurements by Ouacha et al. [56]. The nanotube diameter was about 10 nm and the length from 15 to 3 m. At frequencies below 103 Hz, the noise intensity SV  exceeded the thermal noise limit. In the individual nanotube, the spectrum is 1/f 102 , while in the crossed nanotubes the noise intensity is higher and the spectrum is 1/f 156 . A frequency exponent close to 3/2 would indicate a 1D diffusion noise in nanotube [44]. Such an exponent was never observed before in a carbon nanotube and, according to the authors, it “indicates a current dependent coupling mechanism to the measured voltage noise” [56]. In the individual nanotube, the noise intensity depends quadratically on the current, SI ∼ I 198 , while a superlinear dependence has been observed in the crossed nanotubes. The authors consider that this unusual dependence can be assigned to the presence of a small resistance at the junction between the two nanotubes, or, possibly, to the metal–nanotube contacts. On the other hand, the quadratic dependence found in individual nanotubes is a clear indication that 1/f noise comes from conductivity fluctuation. The ratio between the current noise SI  and the shot noise 2eI was calculated, at a frequency f = 104 Hz, and found to be about 2.47 for voltages higher than 50 mV. It shows that the noise intensity is quite close to shot noise “which suggests the presence of an electrical barrier at the nanotube–nanotube junction” [56]. In spite of some disputable problems regarding the noise measurement in such a very difficult to access physical system, the experiments on single wall carbon nanotubes are a beautiful demonstration of how nanoscience can clarify some very old disputes in the field of 1/f noise. For instance, in the case of SWNT there is no bulk atom. “Every atom that constitutes the SWNT is a surface atom” [47]. Hence, 1/f noise in a SWNT is a pure surface effect, a statement which is in glaring contradiction with Hooge’s idea that “1/f noise is no surface effect” [13]. It was the 1/N dependence, where N is the total number of the carriers in the sample, which caused Hooge to state that 1/f noise is a volume effect. It is also a matter of evidence that in carbon nanotubes McWhorter’s [17] number fluctuation model does not work at all. In this context, the surface atomic motion as a microscopic source of 1/f noise comes to the fore, as experimentally demonstrated in very thin (8 nm) discontinuous platinum films [57, 58]. In this case, the role of the surface atomic motion proved to be fundamental. Excess phonon densities of states were found in carbon nanotubes in comparison with graphite [59]. According to the relation (3) between the 1/f noise parameter and the phonon density of states, this could be a cause for the excess noise in nanotubes, although it is difficult to invoke for a quantitative justification. However, it turned out recently that lattice anharmonicity affects the frequency exponent [37]; thus one

655 could extend this observation and suppose that anharmonicity is involved in the 1/f noise of carbon nanotubes, too. If this holds true, the role of the anharmonicity can be much more important in SWNT than in MWNT because the surface atoms feature stronger anharmonicity than bulk atoms. It can explain, at least in principle, the different noise levels between SWNT and MWNT. In this context, we propose surface lattice anharmonicity as a possible ingredient in the generation of 1/f noise. This can be the result of some nonlinear excitations in nanotubes.

4. WHEN THE ELECTRON IS THE SIGNAL: LOW-FREQUENCY NOISE IN SINGLE ELECTRON TRANSISTORS The single electron transistor (SET) or the Coulomb blockade electrometer [60] is considered to be the most probable candidate to replace the MOS transistor although it is very likely that the MOSFET will also be scaled down to the nanometer range [16]. However, as recently pointed out by Ahmed [61], if it is to work with 10 electrons in the channel, due to fluctuations in the number of electrons it would be impossible to distinguish between the two classical 0 and 1 levels. In this context, the advantage of working with a single electron becomes evident. The major obstacle in application of the SET transistor in the “single electronics” is the excess 1/f noise observed in the fluctuating charge of the island. If these noise sources could be eliminated, what remains is the shot noise, which is due to the intrinsic conduction mechanism in SET. Measurements performed by Zimmerli et al. [62] revealed that the shot-noise energy sensitivity of a SET “could approach
(Planck constant) if the source of 1/f noise were eliminated.” The energy sensitivity EN  is defined as the ratio QN2 /2C , where QN is the charge noise and C is the island capacitance. According to Zimmerli et al. [62], 1/f noise in SETs would be “probably associated with the presence of charge traps in the dielectric surrounding the island electrode.” Using a multilayer fabrication process, Vissher et al. [63] obtained SET structures with better noise performance. Although they used amorphous SiO as dielectric layer, well known for its high density of traps and defects, the charge √ noise QN was found to be 4 × 10−4 e/ Hz, an order of magnitude lower than the result of Zimmerli et al. [62]. Verbrugh et al. [64] observed that charge noise intensity increases with the island size. They have also revealed that I–V characteristics of different SET structures are well fitted by a model where the heat flow from the device is limited by the electron–phonon coupling in the island. As for the origin of the 1/f noise in SETs, it is still unclear but experiments performed to date [65, 66] indicate that this is due to fluctuating traps “distributed either in thin dielectric layers (including the barriers) adjacent to the island or, alternatively but more likely, in a volume of the substrate” [65]. According to Zimmerman et al. [66], a “significant modulation of the transistor island charge” could be induced by some noisy defects located outside the tunnel barriers. However, in aluminum single electron tunneling transistors, Tavkhelidze and Mygind [67] have observed the same noise

656 spectrum for fixed bias on both slopes of V Vg , where V is the dc voltage across the device terminals and Vg is the gate voltage. This is a hint that noise could be brought about by the trapping near or inside the junction which limits the tunnel current through the device. Altogether, these results indicate that the substrate whereupon the island is placed is strongly contributing to the noise of the SET. Employing a stacked design which eliminates the effect of the substrate, Krupenin et al. [68] effectively reduced the substrate contribution to the background noise of metallic SET structures. They have also found evidence of a noise source due to the barrier conductance fluctuations but the mechanism producing these fluctuations remained obscure. Besides a gain dependent noise, excess noise has been reported in both Nb and Al SETs [69] but the noise intensity did not follow a quadratic dependence on the voltage; it excludes the association of this noise with the resistance fluctuations. This excess noise was tentatively attributed to the current which heats the SET. The influence of the temperature on the excess noise in SETs is crucial for the high-temperature applications. Kenyon et al. [70] investigated the temperature dependence of noise in metallic SETs between 85 mK and 4 K. They have found that above 1 K, the noise intensity increases quadratically with the temperature. An effect yet to be explained is that below about 0.5 K, the noise intensity does not tend to zero but saturates. Single electron transistors have also been fabricated in crossed [71], freestanding [72], and buckled nanotubes [73]. The Coulomb blockade effect was observed in devices made from both SWNTs [71] and MWNTs. Ahlskog et al. [71] obtained a single electron transistor by crossing two MWNTs of 15 nm diameter. At low temperature, the lower tube is metallic while the upper one, playing the role of gate, is semiconducting. The Coulomb blockade in this device developed only at sub-Kelvin temperatures (150 mK). The authors investigated the transport and noise properties of the lower nanotube. The noise was measured over one period of the gate modulation curve. “The noise measured at the output of the SET had approximately a 1/f power spectrum” [71] and the noise intensity featured an irregular dependence on the gate voltage Vg . When compared to the gain variation, it has been revealed that the noise intensity corresponding to the zero-gain point at a maximum of the current is rather shallow. This indicates the presence of some other noise sources in the nanotube SET, in addition to the charge fluctuations amplified by the SET. This could be a current dependent noise source for the noise measurement done outside the Coulomb blockade (room temperature) revealed that the noise power in MWNT is roughly equal to the high noise level observed by Collins et al. [47] in SWNTs. The input equivalent charge noise sensitivity at √ 10 Hz is about 6 × 10−4 e/ Hz, which compares very well with the typical values for metallic SETs. Trapping centers of charge close to the island or in the tunneling barrier cause the 1/f  ∼ 1 − 2 in SET structures. To reduce noise intensity in SETs one has to avoid contact of the central island with the dielectric material. This solution has been selected by Roschier et al. [72] to realize an ultrasensitive electrometer using a freestanding carbon nanotube as island. A MWNT with a diameter of 14 nm

Low-Frequency Noise in Nanomaterials and Nanostructures

was manipulated by atomic force microscopy to move on top of two gold electrodes. A side gate was used to modulate the current. Noise measurement on this structure evidenced 1/f 2 spectra at both minimum and maximum gain. The charge noise of this freestanding carbon nanotube SET strongly√improved. At a frequency of 45 Hz, a value of 6 × 10−6 e/ Hz has been found, which is comparable with the best metallic SETs. Using the atomic force microscope, Postma et al. [73] created two buckles in a metallic nanotube. The nanotube segment between the two buckles featured Coulomb blockade effects when the nanotube potential was varied by a nearby gate [73]. When measured as a function of gate voltage, at a frequency of 10 Hz and a temperature of 60 K, the noise intensity clearly shaped the peaked IDS vs. gate voltage characteristic, where IDS is the drain-source current. The optimum value √ of the input equivalent charge noise is q = 2 × 10−3 e/ Hz , which reasonably compares with the values for the conventional SET at very low temperature (mK).

5. NOISE ENGINEERING IN SEMICONDUCTOR QUANTUM WIRES A quantum wire (QWR) has one unconfined direction and therefore it is a 1D conductor. As a consequence, the electronic density of states is highly localized and peaked [74], allowing for high electron mobility. This property is extremely useful to realize different nanostructures such as QWR field effect transistors, to give only an example. Progress in this field was somewhat hindered by the photolithographic accuracy. However, this disadvantage has been recently removed by the realization of a self-organized QWR structure, a technique successfully applied to obtain In015 Ga085 As/GaAs quantum-wire field effect transistors with very good performances [75]. To date, the noise performances of the semiconductor QWR-based nanodevices have not been evaluated. Also, the noise data for the quantum wire itself are scarce. However, Sugiyama et al. [76] measured noise in both V-groove and ridge InGaAs QWRs of 10 nm thickness, 100 nm width, and 2 m length. For both QWRs the noise spectrum was 1/f -like. In the V-groove QWR, a bulge in the spectrum was found at around 10 kHz. The estimated 1/f noise parameter at 1 Hz was 21 × 10−3 for the V-groove QWR, while for the ridge one calculations led to a value of 64 × 10−4 . The differences between the 1/f noise parameter have been put down to different factors such as surface scattering, interface roughness scattering, and strain. Due to the one-dimensional phonon confinement and the singular density of electronic states, the quantum wire is a system of particular interest to investigate the 1/f noise mechanisms. The first theory of the 1/f noise in semiconductor QWR has been introduced and elaborated by Balandin and co-workers [77]. Other contributions are due to Bandyopadhyay, Svizhenko, and others [78]. The theory naturally adopts the hypothesis of mobility fluctuation because the trapping–detrapping in such one-dimensional systems is very improbable. Momentum relaxation contributes significantly to mobility fluctuation 1/f noise because it modifies the change in the velocity ("v) of an electron in the collision process. In such a 1D system, an electron can be scattered

Low-Frequency Noise in Nanomaterials and Nanostructures

directly forward or directly backward; thus no sideways scattering is possible. The first process does not considerably change "v, so it contributes weakly to 1/f noise. On the contrary, backscattering, which typically turns the electron around, consistently modifies "v, so its contribution to 1/f noise is important. Quantitatively, the general idea of the theory is to calculate the change in the velocity, "v, of an electron due to a collision process. ("v2 is then averaged in the wavevector space (k-space) over all final scattering states weighted by the scattering rate. The result, ("v2 k, is then averaged to obtain "v2 by ensemble averaging over the initial states assuming, in the case of Balandin et al. [77], that their occupation is described by a drifted Maxwellian distribution. "v2 can also be obtained directly by Monte Carlo simulation, as was done by Bandyopadhyay and Svizehenko [78]. Once "v2

is known, the 1/f noise parameter can be calculated [79] from  = 4/3$ "v2 /c 2 &, where  is the fine structure constant ( = 1/137) and c is the speed of light. The effect of the magnetic field on the 1/f noise parameter was also taken into consideration. Balandin et al. [77] found that the theory gives  values which are “less than or comparable to that found in three-dimensional mesoscopic samples” [77]. From the above considerations, it follows that to control the noise in a quantum wire one has to relax the energy without relaxing momentum. In the case of a bulk sample, if high energy phonons would be involved in the energy relaxation, the momentum would be relaxed too. This is due to the fact that such phonons have a high wavevector; therefore backward scattering is favored and naturally the momentum strongly relaxes. Hence, one has to look for phonons of high energy but small wavevector (k), because they can relax energy without strongly affecting the momentum. In a quantum wire, phonon folding due to quantum confinement allows for the energy relaxation without relaxing the carrier momentum [74]. Consequently, although phonon folding increases the scattering rate, the 1/f noise intensity goes down. It results that 1/f noise suppression in QWR is possible and that phonon folding can be exploited in order to make quieter devices. To this purpose, bulk devices could be replaced with parallel arrays of QWR [78]. At the same time, in the presence of a magnetic field, the noise is strongly reduced especially at lower temperature and low electric field [78]. The explanation for 1/f noise quenching is that magnetic field “suppresses the acoustic phonon mediated backscattering by spatially separating the wavefunctions of oppositely traveling states in a quantum wire” [77]. A disadvantage of the acoustic phonon confinement in a QWR is that it also degrades the mobility and, therefore, affects other parameters of the nanodevices. A solution could be the presence of a magnetic field. In appropriately designed QWRs or quantum wire-based nanodevices, high magnetic fields can be locally produced by inserting magnetic quantum dots of Fe, Ni, or Co [78].

6. NOISE IN POINT CONTACTS AND METALLIC NANOBRIDGES Historically, the first noise measurement in mechanical point contacts (PC) was done by Hooge and Hoppenbrouwers [80, 81]. By using shear deformation in liquid helium,

657 Yanson et al. [82] later fabricated noble metal point contacts with the diameter (d) smaller than the electron mean free path ((). These are much better than the “needle– anvil” pressure-type contact. Since d < (, the contacts are satisfying the condition of ballistic transport regime for electrons. From the I–V curve nonlinearities, the second derivatives d 2 V /dI 2 have been determined which, according to Yanson’s [83] point contact spectroscopy method, are the image of the Eliashberg function (phonon spectrum), 2 F . The noise spectra observed in all contacts were 1/f like. In the case of silver, no similarity was found between the Eliashberg function and the energy dependence of the noise intensity, while for copper there are some similarities but only for energies larger than the optical phonon energy (approx. 30 meV). However, in the case of the gold point contact, there is a good connection between the noise structure and the Eliashberg function, especially for longitudinal phonon peaks (see Fig. 1 of Ref. [82]). The most interesting result found by Yanson and co-workers [82] is an oscillatory structure in the noise which is “approximately periodic in eV and independent of d.c. voltage polarity.” At higher temperature the oscillations are slightly smeared out, indicative of their spectroscopic origin. The authors assigned the structure to the coherent phonon emission, but it was also suggested that Umklapp processes, among others, could be important. The role of Umklapp phonons was later clarified in a paper by Akimenko et al. [84]. To this purpose they have measured noise, at and below 4.2 K, in point contacts of metals with different Fermi surfaces. Point contacts with diameters between 10 and 30 nm have been prepared from Na, Cu, and Sn. For sodium contacts, the point contact phonon spectrum and the noise intensity were determined as a function of energy (eV) in all possible regimes: ballistic regime (both momentum and energy mean free path are much larger than the diameter), diffusion regime (the electrons are elastically scattered by impurities or defects), and the thermal regime (the PC properties are bulklike material). A very clear phonon spectrum was observed in the ballistic regime. At the same time, the noise featured a very well defined fine structure composed of maxima and minima. In the diffusion regime, the phonon spectrum is not essentially modified, while the noise structure is strongly smeared out. Both the phonon spectrum and the noise structure practically disappear in the thermal regime. The phonon spectrum is determined by the energy distribution function, which is the same in both ballistic and diffusion regimes. This feature explains why no differences occur between the phonon spectra in the two regimes. What is essentially different between the two regimes is the momentum distribution function. This is the reason behind the fact that strong differences exist between the noise structure in the two regimes. Therefore, the noise spectra in PC is determined “by the momentum distribution function of electrons and thus are more intimately connected with the electron–phonon scattering dynamics” [84]. By far, the most interesting phenomenon observed in the ballistic regime is the structure of minima and maxima in the noise vs. eV . No correlation between this structure and the differential resistance dV /dI in all PCs has been determined; therefore the structure cannot be associated

Low-Frequency Noise in Nanomaterials and Nanostructures

658 with the resistance change. As before, the noise minima have been correlated with the phonons having wavevectors close to the Brillouin zone boundaries for the correlated emission can explain the deep noise. For sodium and copper, the noise maxima correlate well with the Umklapp phonons which assist the transitions between the Fermi surfaces in the adjacent Brillouin zones. The noise measurements in nanometric point contacts evolved as a very fine tool to investigate the transport properties in materials. Moreover, the results of Akimenko et al. [84] show that the noise spectra are more sensitive to the crystallographic anisotropy than the PC phonon spectra because of the strong anisotropy of the nonequilibrium phonon distribution. This property can be exploited to properly design low noise nanostructures and nanodevices. By the means of electron beam lithography and reactive ion etching, Ralls et al. [85–87] and Holweg et al. [88, 89] patterned a nanohole in a silicon nitride membrane. Evaporating metal onto both sides of the membrane, a nanobridge is formed between the two massive electrodes. A sketch of the nanobridge is presented in the inset of Figure 4. The metallic nanobridge is the microfabricated equivalent of the mechanical point contact but its mechanical and electrical stability is much better. At low temperature, the electron mean free path (() is much larger than the diameter of the nanobridge, thus leading to ballistic transport. This is the situation that was probed by both groups of authors who found high quality point-contact phonon spectra (Eliashberg function) in investigated nanobridges. For a constriction with the radius a, the resistance R can be approximated with the relation [86] R = */2a + 4*(/3a2

(4)

where * is the resistivity. The first term (Maxwell) dominates the resistance in the diffusive regime, while the second one (Knudsen) is dominant in the ballistic regime (( ≦ a).

Figure 4. Eliashberg function (phonon spectrum) of a 15  copper nanobridge, obtained by point-contact spectroscopy at 4.2 K. The spectrum is of high quality for the background signal is almost absent. The inset shows the schematic of the nanobridge, where a is the diameter of the nanoconstriction. Reprinted with permission from [85], K. S. Ralls and R. A. Buhrman, Phys. Rev. Lett. 60, 2434 (1988). © 1988, American Physical Society.

The last term depends only on the geometry of the constriction because * ∼ 1/(. According to the point-contact spectroscopy method [83], the derivative of R vs. applied voltage signal gives the phonon spectrum dR/dV ∼ 2 F , where the terms have their usual meaning. The phonon spectrum of a copper nanobridge, as determined by Ralls et al. [86], is shown in Figure 4. One observes that the background signal is very small and this is the proof that the nanoconstriction is of good quality. Ralls et al. [86–88] performed detailed noise investigations on metallic nanobridges of copper, aluminum, and palladium with resistance ranging from 1 to 100 , corresponding to the constriction diameter of (1.7–17) nm. The temperature dependence of the noise intensity differs from metal to metal. For instance, at temperatures below 150 K, sharp peaks in the noise were detected for copper and aluminum. They were assigned to “strategically placed defects” which give Lorentzian noise spectra. Below about 100 K, extremely rare two-level fluctuators (TLF) were observed in the Al nanobridge; therefore the noise was very small, impossible to measure. However, at about 100 K, the noise increases very sharply, one order of magnitude in about 20 K. In Pd, the metal with the highest melting point, the noise is almost temperature independent, although a close look at the data indicates some broad peaks in the noise, too. The different behavior of noise in the three metals is attributed to the “strong differences in the distribution of defect activation energies” [87]. Searching for the constituents, sometimes called fluctuators [90], of the 1/f noise in metals, Ralls and Buhrman observed time trace noise signals in very small copper nanobridges at T < 150 K. Usually, excess noise over the thermal noise was found only at some temperatures. The authors found that this excess noise is produced by resistance switching between two discrete levels. The amplitude of this random signal is independent of temperature and depends linearly on the applied bias, which points to the genuine resistance fluctuations. The average time ( ) spent in the low or high resistance state is thermally activated: = 0 expE0 /kT , where 0 is the so-called attempt time and E0 is the activation energy. For different fluctuators, the values of 0 were between 10−11 and 10−15 s, clustering around 10−13 s, while the activation energies ranged from 30 to 300 meV. The attempt time value of 10−13 s is characteristic for the atomic vibration in solid. From the amplitude of the resistance fluctuation "R, the change in the cross section "+ ∼ d 2 "R/R, where d is the constriction diameter, ranged from 0.28 to 3.6 Å 2 , values of the order of atomic dimensions. Together with the values of the attempt time, the atomic dimensions of the cross scattering section point to “the individual atomic motion as the origin of this noise” [87]. Consequently, in the model elaborated by Ralls et al. TLF is generated by “reversible motion of a single defect in the constriction region between two metastable states” [87]. The above considerations show that low-frequency noise in nanobridges is not due to defect diffusion in nanoconstriction. TLFs were also observed by Holweg et al. [89] in gold nanobridges. In both copper [86] and gold nanobridges [89], TLFs manifest at lower temperature and low bias voltages. At high voltages, many fluctuators are active and, at even higher voltages, net atomic motion (electromigration) occurs

Low-Frequency Noise in Nanomaterials and Nanostructures

in the sample. A consequence of the electromigration is that the average resistance changes. Different TLFs were observed in the time trace after such an event [86, 89]. The overall sample heating is excluded as a source of resistance fluctuation because of the small background signal observed in the phonon spectrum. Another reason is the asymmetry in the fluctuation rate as a function of the voltage polarity, an effect evidenced in both copper [86] and gold [89] nanobridges. A defect jumping back and forth in a doublewell potential can be a source of two-level discrete resistance switching. Under the effect of the bias, the activation energy E of the defect is modified, E = E0 − ,V

(5)

where , is the electromigration parameter and V is the voltage. It was proved experimentally that for both bias polarities, the defect fluctuation rate increases, supposedly due to the fact that the temperature of the defect Td  is higher than the lattice temperature T . Thus, one has to replace T with Td in the expression of : = 0 $expE0 − ,V /kTd &

(6)

According to Holweg et al. [89], the local heating of the defect above lattice temperature is due to the inelastic scattering with the electrons. Under this assumption, a model, not presented here, was developed by Holweg et al. [89] which allowed for the calculation of Td . With this expression, it was found that the relation (6) fits well the experimental dependence of the fluctuation rate in a gold nanobridge [89] in both lower and higher states of the resistance. Ralls et al. [86] followed another way. First, from the experimental values of 0 , E0 , and , at low voltages V , the measured fluctuation time ( ) in an aluminum nanobridge was converted into an effective temperature of the defect Td . Then, a model was elaborated which took into consideration the fact that the defect not only gains energy from the electrons but it can also lose energy to the lattice. They have found that the experimental data are well fitted by the theory for both bias polarities. The effect of the lattice on the TLFs behavior in nanobridges, at large biases, was also investigated by Kozub and Rudin [91]. In this approach, the TLF is described with the help of the soft double-well potential model. Hence, the TLF behavior was investigated in terms of the interaction between the potential and the nonequilibrium phonons emitted by the ballistic (nonequilibrium) electrons. Calculating the probability for interlevel transition induced by both phonons Wph  and the electrons Wel , they arrived at Wph ≧ Wel , which strongly supports the starting hypothesis. The analytical expression for the temperature of the defect (TLF) was finally found to fit well the bias dependence of the TLF relaxation time for aluminum, gold, and copper nanobridges. An important feature of the model is that it can explain the asymmetry of the TLF relaxation rate on bias polarity by the current flow direction-induced asymmetry in the spatial distribution of the nonequilibrium phonons. This asymmetry can be very different for normal phonons, in comparison with the Umklapp phonons. In the theory of Kozub and Rudin [91], the role of nonequilibrium phonons is essential because they are

659 responsible for the activation of TLFs in nanobridges, hence for noise. Some other observations seem to further support this hypothesis. For instance, in a silver nanobridge, Holweg et al. [88] reported that the rms amplitude of the conductance starts to decrease at a bias voltage of 10 mV, which corresponds to the transverse-acoustic phonon peak in the point-contact phonon spectrum. This decreasing behavior was attributed to the phonon emission by electrons with excess energy eV . Another possible example is taken from Ralls and Buhrman [87]. They observed that noise in small resistance nanobridges—that is, “large” devices—is the same as in the bulk films deposited in the same manner. Large deviations occur for nanobridges of high resistance (small samples). To explain this, it was assumed that more active fluctuators exist in smaller samples and the cause could be the surface tension produced by the extremely small radius of curvature of the nanoconstriction. But a smaller radius can also lower the melting temperature and the activation energies for atomic motion in the nanobridge. As a consequence, they assumed that in small copper nanobridges one could get a maximum in the 1/f noise intensity at a lower temperature than the one found (500 K) by Eberhard and Horn [92] for a bulk copper film. Indeed, as Figure 5 shows, for a 7  (c ∼ 64 nm) nanobridge, a maximum exists at 350 K. Speculating about the origin of this peak, and in connection with the possible relationship (3) between the noise intensity and the Eliashberg function, we have tried to connect the maximum in noise with some specific phonon energies in copper [93]. In Figure 5, curve 1 represents the Eliashberg function for longitudinal phonons obtained by deconvolution of the point contact spectrum of Ralls and Buhrman (Fig. 1 from Ref. [87]) for a 15  nanobridge. Its maximum at about 325 K is rather close to the noise maximum. In a better fit with the noise maximum is an Eliahsberg function (curve 2 in Fig. 5) obtained by Yanson [83]

Figure 5. Comparison between the temperature dependence of the 1/f noise spectral density (SV ) in a 7  copper nanobridge and the different Eliashberg functions: 1—from Ralls and Buhrman [85]; 2—from Yanson [83]. Curve 3 is the bulk phonon spectrum for copper calculated by Kurpick and Rahman [94]. Reprinted with permission from [93], M. Mihaila, in “Quantum 1/f Noise and Other Low-Frequency Fluctuations in Electronic Devices” (P. H. Handel and A. L. Chung, Eds.), Am. Inst. Phys. Conf. Proc. Vol. 466, p. 48, American Institute of Physics, New York, 1999. © 1999, American Physical Society.

660 at 1.8 K for a Cu point contact of 4 . It has a maximum at about 350 K, the better fit most probably due to the fact that the point-contact resistance is closer to the nanobridge resistance. Also shown in Figure 5 is the bulk phonon spectrum of Cu calculated in Ref. [94]. The highest peak, corresponding to the longitudinal phonons (30 meV), is located at the noise maximum. As for the type of defect causing two-level resistance fluctuations, Ralls and Buhrman [87] did not get a definite answer. In any case, a single point defect is ruled out because there is no reason why interstitials, vacancies, or impurity atoms would jump between only two metastable positions. Complex time traces were found and analyzed by Ralls and Buhrman [85–87] in terms of the interactions between defects. In fact, they found that defect interaction dominates the fluctuation dynamics and it is the reason the system samples all times, which is essential for 1/f noise. They conclude that there is actually no “independent defect even in crystalline films with elastic-scattering length of 200 nm” [87]. In this context, Ralls and Buhrman introduced the concept of “defect glass” as an entity composed of the defects involved, by their strong interaction, in the generation of 1/f noise. At room temperature, in both copper [87] and gold [89] nanobridges the resistance still switches but the noise spectrum changes from a Lorentzian, as it is at low temperature, to a 1/f -like spectrum. Since the gold constriction is in the diffusion regime, Holweg et al. [89] calculated, from an estimation of the diffusive resistance, the 1/f noise parameter (). They obtained  = 56 × 10−3 , which is well within the domain found for other physical systems. The authors consider that this is an additional argument that 1/f noise is due to the Lorentzian superposition. However, the frequency exponent of their spectrum is about 1.4, a value slightly outside the usual domain of the 1/f -like frequency exponent 07 < < 13 and rather close to the specific diffusion noise exponent (3/2). A similar situation was presented by Ralls and Buhrman [87] for a copper nanobridge. In this case, the spectrum was also 1/f -like, with ≈ 13. It seems that in the presence of TLFs at room temperature, the frequency exponent is always larger than 1 whereas, at low temperature, a pure TLF noise signal features a Lorentzian spectrum, as demonstrated by Holweg et al. [89]. This transition from a Lorentz spectrum to a 1/f -like one is associated with the activation of many fluctuators. However, for the same physical system, the frequency exponent vs. temperature can be larger or smaller than 1, within the widely accepted values: 07 ≤ ≤ 13. At higher temperature, in the presence of TLFs, the frequency exponent tends to 1 but is consistently larger than 1. It would be interesting to see whether, in the presence of TLFs, the frequency exponent could be slightly lower than 1. The first noise measurement in a quantum point contact (QPC) was done by Li et al. [95]. A 2D electron gas confined at the GaAs/AlGaAs heterointerface was constricted with two point-type gates (see the insert in Fig. 6). Figure 6 shows that the channel resistance (R) has plateaus which are close (within 10%) to the theoretical quantized values (h/2eN , with N = 2, 3, 4, and 5). Below 1 kHz, the current noise spectrum SI f  has a 1/f dependence and is described by the relation SI f  = A/f + SIO , where SIO is a white

Low-Frequency Noise in Nanomaterials and Nanostructures

Figure 6. Dependence of the point contact resistance R vs. Vg and of the parameter A0 vs. Vg . The values of A0 were obtained at three currents (I = 04/ 05/ and 0.6 A) from the fitting relation SI f  = A0 I 2 /f + SIO . The approximate plateaus in R are assigned to the quantized resistance h/2e2 N , where N = 2/ 3/ 4, and 5. Device geometry is shown in the inset. Reprinted with permission from [95], Y. P. Li et al., Appl. Phys. Lett. 57, 774 (1990). © 1990, American Physical Society.

noise background. In general, when the gate voltage Vg  is fixed, the coefficient A ∼ I  , with  = 2 ± 04 for −24 V < VG < −14 V and current I < 06 A. A new coefficient A0 = A/I 2 was defined and represented in Figure 6 vs. Vg , for I = 04/ 05, and 0.6 A. With some exceptions, A0 is independent of current, increases as the channel width goes down (by decreasing Vg , and features peaks between the R plateaus. In this case, the 1/f noise is correlated with the resistance quantization. The authors consider that 1/f noise in the quantum point contact “may result from some processes which are related to the 1D density of states in the channel” [95]. The white noise component SIO was attributed to the trapping and detrapping of the conduction electrons at the GaAs/AlGaAs interface, its intensity being, quite unexpectedly, below the full shot noise. Low-frequency noise spectroscopy has been used to investigate the kinetics of charge transport in quantum point contacts [96, 97]. Dekker et al. [96] observed that low-frequency noise characteristics of quantum point contacts vary from device to device. For instance, in a QPC the noise spectrum was a Lorentzian, while in another one the spectrum was 1/f -like and “weaker by many orders of magnitude” [96]. Although the spectra are very different, in both QPCs the normalized spectral density(SV /V 2 , at low temperature (1.4 K), exhibits strong universal quantum size effects. Figure 7 shows that SV /V 2 vs. G, where G is the conductance in units of 2e2 /h, features sharp minima at the quantized conductance plateaus, where the Fermi energy 1F is between the bottom energies En of two 1D subbands. The maxima in noise occur between the plateaus in G, where 1F = En+1 . Attributing the quantum size effects in noise “to fluctuations in the transmission probability for the subband which is closest to cutoff,” Dekker and co-workers [97] found that SV /V 2 = 1/G2 $3G/31F &2 SEF , where SEF is the spectral density of the fluctuations of Fermi energy. The factor $3G/31F &2 /G2 vs. G is shown at the bottom of Figure 7. It is evident that the universality of the

Low-Frequency Noise in Nanomaterials and Nanostructures

661 are occupied” [98]. The absence of the noise on the conductance plateaus is due to the lack of scattering and this is the expression “of the insensitivity of the quantized resistance to the exact microscopic device parameters” [99]. Hence the quantum constriction “feels” the potential fluctuations induced by the presence of a defects only when the gate voltage value is between two consecutive plateaus. The defect can be located by modifying the gate voltage as done in Refs. [98, 100], but its nature is still not clear. Finally, we mention that some of the issues briefly reviewed here have also been discussed by Kogan [101].

7. BLINKING QUANTUM DOTS

Figure 7. Relative noise spectral density SV /V 2 vs. conductance (G), at 1.4 K, for two quantum point contacts: QPC1 with 1/f noise spectrum (values measured at 100 Hz) and QPC2 with Lorentzian spectrum (values measured on the Lorentzian plateau). Arrows denote upper bounds. Bottom figure: calculated factor [3G/31F &2 /G2 vs. G. Reprinted with permission from [96], C. Dekker et al., Phys. Rev. Lett. 66, 2148 (1991). © 1991, American Physical Society.

quantum size effect in noise comes from the dependence of SV /V 2 on the 1/G2 $3G/31F &2 . The difference between the two noise spectra resides in the fact that the discrete resistance switching (RTS noise) was observed in the QPC featuring a Lorentz noise spectrum. At high temperature (T > 15 K), the RTS signal disappears and the noise spectrum changes its shape from a Lorentzian to a 1/f -like one. Dekker et al. [96] explained the occurrence of the RTS noise by the presence of a single electron trap in the immediate vicinity of the point contact. Random filling and emptying of the trap modulate the electrostatic confining potential and thus the conductance of the QPC. In the presence of a strong external magnetic field, not only did the existing noise minima sharpen but additional minima were found [97]. It is an indication that a residual backscattering exists in the constriction. As in the case of QWRs, the magnetic field could be a possibility to engineer noise in QPC. The observation of Dekker et al. [96] that the disappearance of the RTS noise is accompanied by a transition from a Lorentz spectrum to a less intense 1/f -like spectrum casts some doubt on the idea that RTS noise is the fundamental component of the 1/f noise. The suppression of the RTS noise at the conductance plateaus has also been investigated by Timp et al. [98] and Cobden et al. [99, 100] in a onedimensional wire (channel) using the same split-gate technique to constrict a highly conductive 2DEG. They found that the transition between two adjacent plateaus in the quantized conductance is accompanied by RTS noise. There is no RTS noise on the plateaus which signifies that “RTS is inhibited whenever an integral number of 1D subbands

Quantum dots (QDs) are the analogue of the real atoms because they have atomiclike energy levels. Often called “artificial atoms,” they stand as very attractive candidates for optical memories and high performance zero-dimensional devices such as optical switches, lasers, etc. At the same time the silicon single-electron quantum dot transistor operating at room temperature looks very promising for singleelectron memory [102], while a quantum dot-based sensor was proposed to measure the magnetic fields at micron scale [103], to give just a few examples. However, some of these possible applications could be affected by the fact that the luminescence of the quantum dots randomly switches between discrete levels. These are the so-called blinking quantum dots. The effect is the photoluminescence analog of the electrical RTS noise. This blinking phenomenon manifests in a variety of QDs realized from different materials [103–107]. To fully illustrate how a QD blinks, Figure 8 shows the time trace of the photoluminescence of an InP quantum dot as registered by Pistol et al. [113]. The photoluminescence switching between two or more states was attributed by Nirmal et al. [104] to QD photoionization. Zhang et al. [106] reported the emission intermittences from II–IV ZnCdSe QDs with a diameter of about 120 Å, embedded in a ZnSe matrix. The intensity of the emission lines of the photoluminescence spectra, registered at 40 K and different time, featured sudden increases and decreases from run to run. These fluctuations qualified as “on” and,

Figure 8. Discrete levels in the photoluminescence of an InP blinking quantum dot observed by Pistol et al. [112]. Reprinted with permission from [112], M.-E. Pistol et al., Phys. Rev. B 59, 10725 (1999). © 1999, American Physical Society.

662 respectively, “off” states “reflect the ionization and neutralization of the QDs” [106]. By decreasing the temperature the emission time became very long. This observation suggests an explanation in terms of the thermal ionization of QDs. Ionization, corresponding to the off state, can be produced either by thermal ionization or Auger scattering. In a thermal ionization process, the carriers in QDs are thermally excited over the confinement barrier into the matrix, while in an Auger process, the recombination energy of an electron–hole pair is transferred to a third particle which is ejected to the matrix. The reverse processes correspond to the “on” state. In the model proposed by Efros and Rosen [109], in the “off” state the luminescence is quenched by nonradiative Auger recombination, whereas “the durations of the off periods depends on the ionization rate of the dot via thermal or Auger autoionization” [109]. Temporal instabilities in both ground and first excited state luminescence have been observed by Bertram et al. [107] in a single strain-induced quantum dot realized in a GaAs/(AlGa)As material system. The photoluminescence spectra were extracted at 6.9 K. When registered at a given time, the ground state line was dominant, while after a few seconds, the spectral line of the first excited state took over. The effect was called two color blinking (TCB). The TCB appeared “as beating of the luminescence intensity between the ground and the excited state” [107]. No physical explanation was possible for this phenomenon. Aoki et al. [108] observed that macroscopic InGaAs clusters blink even at room temperature. The lifetimes spent by the blinking signal in both on and the off states superlinearly decreased when the excitation intensity went down. This is clear evidence that RTS noise is a many-carrier effect generated by a photoassisted process, not by a thermal one. It was suggested that the off state could be caused by nonradiative centers such as dislocations located in the InAlAs cluster. Only one from 100–1000 clusters shows RTS noise. Because the estimated average density of dislocations in a cluster was one, it resulted that not all dislocations generated RTS. Another possibility would be to have more than one dislocation in a cluster. It might be of some relevance for the above considerations to mention that burst noise, the equivalent of the RTS noise in larger devices, was quantitatively correlated with the presence of the dislocation clusters in the p–n junctions [110, 111]. Detailed investigations on the photoluminescence RTS noise in InP quantum dots were performed by Pistol et al. [112, 113]. The samples were grown on a InGaP barrier, which was lattice-matched to a GaAs substrate. The QDs were fully strained for they have been capped with a 300 nm InGaP layer. The dots were excited at 7 K with the 488 nm line of an Ar+ laser or the 532 line of a YAG laser [112, 113]. Under these excitations, the QDs continuously emitted light. However, in some of them the intensity switches between two discrete levels [112] (see Fig. 8). One hundred of more than 100,000 dots exhibited RTS noise in the emission intensity. Quantum dot switching between three or even four levels have been found too [113]. Since no switching was observed at zero excitation power, it was concluded that the mechanism is induced by the excitation light. In the on state, the emission intensity vs. excitation power density is linear, while the switching rates scale superlinearly with

Low-Frequency Noise in Nanomaterials and Nanostructures

the power density. Hence, multiple particles are involved in the switching. For power excitation of about 50 W/cm2 , the RTS disappears and the dot irreversibly remains in the on state even at reduced excitation power. The switching rates increase with both excitation power and temperature. The results of Pistol et al. [112] for the emission spectra of a quantum dot in the on state and the off state, at different excitation power densities, are shown in Figure 9. As can be seen, the spectra are similar in both on and off state and there is no line energy shift. State filling is visible in the off state. Pistol et al. [112] observed that if the carriers are generated only in the dot and not in the barrier, RTS is still active; therefore the mechanism involved seems to be the modulation of the radiative recombination instead of the modulation of the carrier capture in the dot. Another interesting observation of Pistol [113] is that in the off state the emission lines are slightly “thiny” than in the on state, which is the signature of a lower electron concentration in the dot when in the off state. The most probable candidate to explain the effect is a competing nonradiative channel associated with a native defect in the neighborhood of the dot. The ways the dot “communicates” with the defect can be different. For instance, electricfield-assisted ionization of the dot due to a nearby charged defect is excluded because, under the influence of an electric field, no Stark shift was found in the case of a switching dot. The defect involved in the RTS noise has to be mobile; otherwise switching turn off under strong illumination conditions cannot be explained. A defect activated via phonon

Figure 9. Photoluminescence spectra of a switching InP quantum dot registered by Pistol et al. [112] for different excitation power densities (P = 05 W/cm2 ). Full lines correspond to the on state and dashed lines correspond to the off state. Reprinted with permission from [112], M.-E. Pistol et al., Phys. Rev. B 59, 10725 (1999). © 1999, American Physical Society.

Low-Frequency Noise in Nanomaterials and Nanostructures

emission (phonon-kick model) seems to fulfill this condition. Phonons could be generated by carrier relaxation in the dot [112]. Phonon-assisted recombination at the defect-induced deep levels would be another possibility [113]. Although the attempt of Pistol [113] to observe emitted infrared radiation failed, Sugisaki et al. [114] found that the blinking rate of a InP QD is sensitive to infrared radiation. In addition to a band-to-band excitation (Ar-ion laser), Sugisaki et al. [114] irradiated the dot with a near-infrared laser (Ti–sapphire) and observed that the switching rate is very sensitive to an infrared energy threshold of 1.5 eV. In addition, new blinking dots were triggered when the surface was intentionally scratched by a needle. They concluded that “deep defect levels with an excitation threshold energy of 1.5 eV” [114] are involved in the generation of InP QDs photoluminescence RTS noise. We mention that the phonon-activated defect model to explain RTS noise in QDs shows close resemblance to the model proposed by Kozub and Rudin [91] for the RTS noise in metallic nanobridges. Also, fluorescence of 27 Å radius (ZnS overcoated) CdSe QDs fluctuates between two distinct levels [115]. Neither the quantum jump model nor the Auger ionization model are able to explain experimental facts [115]. Instead, an exponential distribution of traps within the semiconductor gap reasonably accounts for the probability distribution of the off state. However, for the on state this model fails to give an adequate answer. This is one of the reasons the “observed blinking kinetics is inconsistent with any static model and instead requires fluctuations in the QD environment” [115]. Genuine electrical RTS noise was observed by Peters et al. [116] in a silicon quantum dot of about 35 nm radius, placed in the channel of a deep submicron MOSFET. At liquid helium temperature and around the threshold, roughly equidistant peaks occurred in the conductance vs. gate voltage (Vg ) dependence. These Coulomb blockade oscillations are the signature of a quantum dot in the channel. Keeping the drain current constant and sweeping the gate voltage (Vg , Peters et al. [116] observed that the drain voltage (VD  switches between two or more Coulomb oscillation curves. Maxima and minima were observed in the dependence of the amplitude of this RTS noise vs. gate voltage. The minima occur at the points where the two Coulomb curves cross each other. To provide an explanation for this result, the authors consider that the shift of the Coulomb curves is due to a single fluctuator which produces a fixed fluctuation, "7, of the dot potential 7, and, consequently, fluctuations "G in the dot conductance G. Finally, "G ∼ 3G/3VG ; hence minima in "G should occur where the conductance is zero. This is in good agreement with what has been experimentally found. However, RTS noise manifested only in some “specific gate-voltage ranges.” Outside these voltage intervals, the noise was weaker and featured a 1/f -like spectrum. In the ohmic region, it is a resistance fluctuation effect because its intensity has a quadratic dependence on drain current. The dependence of both normalized drain voltage noise fSVD /VD2 and VD on gate voltage is shown in Figure 10a and b, respectively. The period of the noise variation is half that of the Coulomb oscillations. The noise is strongly suppressed at the conductance peaks and valleys. To explain this behavior, Peters et al. resorted again to the idea of

663

Figure 10. Coulomb oscillations in the drain voltage vs. gate voltage (a) and dependence of fSVD /VD2 vs. gate voltage (b) at two frequencies; the measurements were done at T = 19 K and ID = 6 nA; the factor 1/VD 2 3VD /3VG 2 (times a constant) is presented with the full line. Reprinted with permission from [116], M. G. Peters et al., J. Appl. Phys. 86, 1523 (1999). © 1999, American Physical Society.

a “uniformly fluctuating potential created by a fixed distribution of fluctuators” [116]. Under this assumption, the drain spectrum SVD is directly connected to the gate voltage fluctuation SVg 8 SVD = 3VD /3VG 2 SVg . Finally, the relative drain noise intensity fSVD /VD2 is described by the factor 1/VD 2 3VD /3VG 2 , which is determined from timeaveraged characteristics (Fig. 10a). The authors concluded that if noise is considered “as a result of a complex superposition of an ensemble of fluctuators” its behavior can be described by a model which relies on time-averaged transport properties. In summary, low-frequency noise phenomena in small dimensional systems such as nanotriodes, MOS transistors, single electron transistors, quantum wires, quantum dots, and nanomaterials such as nanocrystalline gold and carbon nanotubes were briefly presented. 1/f noise and RTS noise are the dominating noise phenomena in nanodevices and nanomaterials. However, in nanostructures and nanodevices they could manifest in an unpredictable way, hence the difficulty to control. Small dimensional systems seem to be very noisy while, on the other hand, there is much information in the low-frequency noise. If properly interpreted, the noise can be an invaluable tool to investigate nanomaterials and nanoscaled solid-state devices. Low-frequency noise proves to be a noninvasive, inexpensive, very sensitive, subtle, and fundamental tool to characterize nanomaterials and nanostructures. On the other side, some passages in this work brought some arguments that nanoscience could answer some longstanding questions in the field of low-frequency noise. Some of them such as topology of the 1/f noise sources were in passing discussed in the light of the new phenomena observed in carbon nanotubes. Surface anharmonicity was proposed as a fundamental ingredient in the generation of 1/f noise in carbon nanotubes. Through the noise measurements, the nanoscience and nanotechnology are already

Low-Frequency Noise in Nanomaterials and Nanostructures

664 paying a great tribute to the science from which they were born. And all these seem to be only the beginning. Much more is within reach.

GLOSSARY Generation-recombination noise (g-r noise) Fluctuations originating from the processes of generation and recombination in semiconductors. Noise spectrum Frequency distribution function of a random signal. 1/f noise Fluctuations with a spectrum inversely proportional to the frequency (f). Random telegraph signal (RTS) noise Fluctuation mimicking telegraph signal and consisting of random transitions between two or more discrete levels. Shot noise Fluctuations orginating from the granular structure of the electrical charge. Thermal noise Fluctuations orginating from the random discontinuities in the motion (Brownian, m.n.) of the carriers due to their collisions with the atoms (R. E. Burgess, British J. Appl. Phys. 6, 185 (1955).

ACKNOWLEDGMENTS The author is very grateful to Dr. Henk W. Ch. Postma for communicating some details about his noise measurement in carbon nanotubes. Many thanks are also due to Dr. Ioan Pavelescu and to Andrei-Petru Mihaila for invaluable help in preparing the manuscript.

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Encyclopedia of Nanoscience and Nanotechnology

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Low-Temperature Scanning Tunneling Microscopy Wolf-Dieter Schneider Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

CONTENTS 1. Introduction 2. Experimental 3. Magnetic Adatoms 4. Surface States 5. Supramolecular Self-Assembly 6. Clusters 7. Specific Nanostructures 8. Conclusion Glossary References

1. INTRODUCTION Nearly half a century ago, on December 29, 1959, Richard P. Feynman gave a visionary talk at the annual meeting of the American Physical Society, entitled There is Plenty of Room at the Bottom
1. It was an invitation to enter a new field of physics, the problem of manipulating and controlling things on a very small scale, the nanoscale. Feynman was not afraid to consider the final question as to whether, ultimately—in the great future—we can arrange the atoms the way we want, the very atoms, all the way down! What would happen if we could arrange the atoms one by one the way we want them   The principles of physics do not speak against the possibility of maneuvring things atom by atom. It is not an attempt to violate any laws, it is something, in principle, that can be done, but, in practice, it has not been done because we are too big   When we get to the very, very small world—circuits of seven atoms—we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on a small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics. So, as we go down and fiddle around with the atoms down there, we are working with different laws, and we can expect to do different things. We can manufacture in different ways. We can use, not just circuits, but some system involving the quantized energy levels, or the ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

interactions of quantized spins, etc   At the atomic level, we have new kinds of forces and new kinds of possibilities, new kind of effects   But it is interesting that it would be, in principle, possible for a physicist to synthesize any chemical substance that the chemist writes down. Give the orders and the physicist synthesizes it. How? Put the atoms down where the chemist says, and so you make the substance. The problems of chemistry and biology can be greatly helped if our ability to see what we are doing, and to do things on an atomic level, is ultimately developed—a development which cannot be avoided. In 1981 the vision of Feynman became closer to reality with the invention of the scanning tunneling microscope (STM) by G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel [2, 3]. They describe their invention as follows [3]. The principle of the STM is straightforward. It consists essentially in scanning a metal tip over a the surface at constant tunnel current as shown in Figure 1. The displacements of the metal tip given by the voltages applied to the piezodrives then yield a topographic picture of the surface. The very high resolution of the STM rests on the exponential dependence of the tunnel current on the distance between the two tunnel electrodes, i.e., the metal tip and the scanned surface. Subsequently the STM and its numerous descendents have become the natural tool for investigations on the atomic scale. This instrument has allowed us to perceive individual atoms on surfaces and it was realized that the tip of the STM is a very versatile instrument, not only for imaging, but also for manipulation of individual atoms. The first experiment to show a stunning realization of the atom-by-atom construction of artificial, man-made nanostructures in the spirit of Feynman was performed in 1990 by D. M. Eigler and E. K. Schweizer [4]. Using a scanning tunneling microscope operated at a temperature of 4 K, they were able to position Xe atoms on a single-crystal Ni surface with atomic-scale precision. This capability allowed the authors to fabricate structures of their own design, a linear cluster of 7 Xe atoms and the letters IBM consisting of 35 Xe atoms, atom by atom. The last sentence in their publication The prospect of atomic scale logic circuits is a little less remote indicated the research direction which led to the realization of an

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (667–688)

Low-Temperature Scanning Tunneling Microscopy

668

Pz Vp

Px

δ Py

A C

VT ∆s CU

s

B

JT

Figure 1. Principle of operation of the scanning tunneling microscope. (Schematic: distances and sizes are not to scale.) The piezodrives Px and Py scan the metal tip M over the surface. The control unit (CU) applies the appropriate voltage Vp to the piezodrive Pz for constant tunnel current JT at constant tunnel voltage VT . The broken line indicates the z displacement in a y scan at (A) a surface step and (B) a contamination spot, C, with lower work function. Reprinted with permission from [3], G. Binnig et al., Phys. Rev. Lett. 49, 57 (1982). ©1982, American Physical Society.

atomic switch realized with the STM [5] just one year later. The authors report the operation of a bistable switch which derives its function from the motion of a single Xe atom which is moved by applying a voltage pulse of the appropriate sign reversibly between two stable positions on the STM tip and on a Ni surface. The state of the switch was identified by measuring the conductance across the leads. Most recently, A. J. Heinrich et al. [6] made a breakthrough in realizing atomic-scale circuits: nanometer-scale computation by cascades of molecular motion. They present a scheme in which all of the devices and interconnects required for the one-time computation of an arbitrary logic function are implemented by atomically precise arrangements of CO molecules bound to a Cu(111) surface. The motion of one molecule causes a nearby molecule to hop to a new site, which in turn moves another molecule in a cascade of motion similar to a row of toppling dominoes. They used a low-temperature STM to assemble and demonstrate a three-input sorter that uses several AND gates and OR gates. The most exciting feature of these molecular cascades is their size: a three-input sorter implemented in next generation CMOS technology requires an area of 53 m2 , while the cascade implementation uses only 200 nm2 , an area 260,000 times smaller. Feynman’s dream of put the atoms down where the chemist says, and so you make the substance has also been realized very recently. Hla et al. [7] write in their abstract: All elementary steps of a chemical reaction have been successfully induced on individual molecules with an STM in a controlled step-bystep manner utilizing a variety of manipulation techniques. The reaction steps involve the separation of iodine from iodobenzene by using tunneling electrons, bringing together two resultant phenyls mechanically by lateral manipulation and, finally, their chemical association to form a biphenyl molecule mediated by excitation with tunneling electrons. The procedures presented here constitute an important step towards the assembly

of individual molecules out of simple building blocks in-situ on the atomic scale. These few prominent examples from the nanoscale research laboratory set the scene for the present chapter. We have chosen to review recent advances in low-temperature scanning tunneling spectroscopy and microscopy of atoms, molecules, supermolecules, clusters, and islands on mostly metallic surfaces which have provided new opportunities for investigating locally and individually the geometric, electronic, optical, and magnetic properties of nanostructures on surfaces, created by manipulation with the STM tip (artificial nanostructures) or by self-assembly (diffusion and growth). The emphasis is on the discovery of new phenomena characteristic for the nano- and subnanometer scale. This choice was motivated by the fascination emerging from different aspects of local spectroscopies and microscopies documented by recent research highlights, the observation of, for example, light emission from single molecules and clusters [8–12], surface electron standing wave patterns at steps, quantum corrals, islands, and impurities [13–23], the Kondo effect of single magnetic adatoms and clusters [24–26], the quantum mirage [26], the observation of the Abrikosov flux lattice and the density of states near and inside a fluxoid [27–29], the local pair-breaking effects of magnetic impurities on superconductivity [30], the effect of individual impurity atoms on the superconductivity in a high-temperature superconductor [31], self-assembly phenomena of adsorbed molecules forming supramolecular structures [32–37], the formation of surface-supported supramolecular structures whose size and aggregation patterns are controlled by tuning the noncovalent interactions between individual adsorbed molecules [38], insulators at the ultrathin limit [39], spin-resolved tunneling determining magnetic properties of nanostructures [40–43], single-molecule vibrational spectroscopy [44], singlebond formation with an STM [45], chemical reactions on the nanoscale [7], manipulation of single atoms, molecules and clusters on surfaces including lateral movement, vertical transfer, and dissociation [4, 33, 46–57], nanometer-scale computation by cascades of molecular motion [6]. The cited examples clearly demonstrate that we are now able to construct, to image, to characterize, and, to some extent, to functionalize artificial man-made nanoscale objects. These and related achievements with other scanning probe methods, for example, the atomic force microscope (AFM) [58], made it possible to give a definition for the emerging field of nanotechnology [59]. Nanotechnology is the creation and utilization of materials, devices, and systems through control of matter on the nanometer-length scale, that is at the level of atoms, molecules, and supramolecular structures. The essence of nanotechnology is the ability to work at these levels to generate larger structures with fundamentally new molecular organisations. These nanostructures, made with building blocks understood from first principles are the smallest human-made objects, and they exhibit novel physical, chemical, and biological properties and phenomena. The aim of nanotechnology is to learn to exploit these properties and efficiently manufacture and employ the structures. As STM and STS have now become mature research tools for the nanoworld, there exist already a selection of books and review articles in the literature, treating both technical aspects and scientific subjects [60–70]. Especially

Low-Temperature Scanning Tunneling Microscopy

the proceedings of the series of International Conferences on Scanning Tunneling Microscopy/Spectroscopy and Related Techniques, which were organized in Santiago de Compostela (Spain) in 1986, Oxnard (USA) in 1987, Oxford (UK) in 1988, Oarai (Japan) in 1989, Baltimore (USA) in 1990, Interlaken (Switzerland) in 1991, Beijing (China) in 1993, Snow Mass Valley (USA) in 1995, Hamburg (Germany) in 1997, Seoul (South Korea) in 1999, Vancouver (Canada) in 2001 give an excellent account on the progress made in the field. The interested reader will find a wealth of valuable information in these publications which provide a good starting point to enter the field. The present chapter is organized as follows. In Section 2, typical state-of-the-art setups of a low-temperature STM in ultrahigh vacuum and the different modes for scanning tunneling microscopy and spectroscopy including electrons and photons as probes are outlined. Subsequently, in Section 3, results on single, supported atoms concerning the Kondo effect are presented. In Section 4, the lifetime of surface state electrons is addressed and standing wave phenomena due to the scattering of surface state electrons at islands, steps, corrals, and other quantum well structures on the (111) surfaces of noble metals are reviewed. Section 5 deals with adsorbed molecules and two-dimensional supramolecular self-assembly. Section 6 summarizes the knowledge obtained on individual molecular and metal clusters. Section 7 discusses the dielectric and magnetic properties of nanostructures, with a brief account on vortices of magnetic flux lattices in classic and high-temperature superconductors. Finally, Section 8 presents a short summary and an outlook for the field.

2. EXPERIMENTAL At low temperatures, due to the reduced broadening of the Fermi level of the STM tip and the sample, an energy resolution from the meV down to the eV range in scanning tunneling spectroscopy (STS) is achievable. Moreover, the absence of surface diffusion together with the spatial resolution of the STM enables detailed studies of electronic states on and near single adsorbed atoms, molecules, clusters, and other nanoscale structures [69]. Typical low-temperature ultrahigh-vacuum (UHV) STMs [4, 25, 41, 71–86] operate at a pressure of 10−11 mbar and employ temperatures down to 4 K and 1.3 K. In some of these low-temperature STMs, magnetic fields are applied in the tip-sample region [85–87] and, in addition, luminescence measurements are carried out [87]. More recently, a few new instrumental developments towards the millikelvin range using a dilution refrigerator [88] or a 3 He refrigerator [6, 89, 90] with magnetic fields up to 7 and 11 Tesla in the tunnel junction have been reported. As STM tip material, usually, electrochemically etched W, Pt, or Ir tips were prepared in UHV by heating and rare-gas ion bombardment. Information of the local density of states (LDOS) [91] is generally obtained by measuring the differential conductance dI/dV versus the sample bias voltage V performed under open feedback loop conditions with lock-in detection; see, for example, [13]. Constant-current topographs very close to the Fermi level have been shown to correspond to the LDOS of the sample surface[13]. Moreover,

669 simultaneously spectroscopic (dI/dV ) and constant-current images are acquired [26]. For recording spectra on, for example, semiconductors or ultrathin insulators, over wide voltage intervals, the tip-sample separation is varied continuously during the voltage sweep in different modes, (i) by a linear ramp [92], (ii) by keeping constant either the tunnel current or (iii) the tunnel resistance. In these modes, band onsets naturally appear as peaks [93, 94]. The tip-surface region of an STM emits light when the energy of the tunneling electrons is sufficient to excite luminescent processes [95]. Photons emitted from the tunneling gap are collected by a lens [8, 10] or a parabolic mirror [9] and transmitted through a sapphire viewport. Outside the vacuum chamber these photons are refocused onto an optical fiber leading to a cooled multi-alkali photomultiplier which is operated in a pulse counting mode. The count-rate is recorded by the STM electronics quasi-simultaneously with the acquisition of constant-current topographs for each image pixel [8, 11]. For spectroscopic measurements the light is focused outside the vacuum chamber onto the slit of a UV/visible-grating spectrograph and detected with a liquid nitrogen cooled CCD camera [9, 11, 96, 97].

3. MAGNETIC ADATOMS The STM has been used to perform local spectroscopy with atomic-scale resolution in a study of Fe atoms on a clean Pt(111) surface [98]. Resonances of 0.5 eV width were found in the adatom local density of states above the Fermi level. Characteristic circular standing wave patterns around single adatoms have been detected in spatially resolved spectroscopic STM images [13, 14, 16, 18, 99]. Recently the local pair-breaking effects of a single magnetic adatom on a classical superconductor have been observed with scanning tunneling spectroscopy [30]. In what follows, the effects of magnetic and nonmagnetic adatoms on the LDOS of noble metal surfaces are reviewed. In particular, a depletion of the LDOS near the Fermi energy, which occurs for magnetic adatoms only, is interpreted in terms of Kondo scattering. The interaction of a single magnetic impurity with the conduction electrons of its nonmagnetic metallic host gives rise to unconventional phenomena in magnetism, transport properties, and the specific heat. As the temperature approaches T = 0 K, the local moment is gradually screened by the conduction electrons of the metal host and a manybody nonmagnetic singlet ground state is formed close to the Fermi energy. These low-energy excitations with a characteristic width of  = kTK (TK is the Kondo temperature and k the Boltzmann constant) are known as Abrikosov–Suhl [101, 102] or Kondo resonance [103–108]. High-resolution photoelectron spectroscopy (PES) provided direct experimental evidence of such a resonance [109–111]. However, a direct observation of the Kondo effect from a single magnetic impurity on an atomic length scale was achieved only recently in STM and STS experiments [24–26]. In the first experiment, magnetic Ce adatoms on a singlecrystal Ag(111) surface were chosen to represent a typical Kondo system [24, 107, 110, 111]. Isolated Ce atoms and, for comparison, isolated Ag atoms on Ag(111) were deposited by evaporation from a tungsten filament onto the Ag substrate at T = 5 K. The adatoms appear as protrusions with ≈09 Å and ≈12 Å height for Ag and Ce, respectively,

Low-Temperature Scanning Tunneling Microscopy

670 with typical widths of ≈15 Å in constant-current topographs taken at a tunneling resistance of a few hundred M. Figure 2b displays a dI/dV tunneling spectrum measured on top of an isolated Ce adatom. A depression is observed in dI/dV around the Fermi energy. When the tip is moved laterally, the typical spectrum of the Ag(111) surface state onset (for Au(111) and Cu(111) see [13, 113, 114]) at E  −70 meV reappears (Fig. 2e) [19, 115]. At intermediate distances (Fig. 2c,d) the surface state edge broadens reflecting a decreased hot-hole lifetime due to scattering of surface state electrons at the impurity [134]. The weak spectral features above the onset of the surface state are due to standing wave patterns caused by scattering at the adatoms [15, 19, 99]. These energy-resolved Friedel oscillations [114] are also visible in space [13–16] as weak circular waves around the adatoms (see Section 4). The tunneling spectra of a single adsorbed Ag atom (Fig. 2a) are featureless throughout the range from −200 mV to 200 mV which is attributed to quenching of the Ag(111) surface state [19]. These data indicate that the dip near the Fermi energy EF is indeed characteristic of the magnetic Ce adatom. The authors of [24] offered an explanation for the existence of an antiresonance in the tunneling spectra as opposed to the resonance predicted by theory [116] and measured by photoemission for similar systems [111, 117]. The probing of a localized state immersed in a continuum by the tunneling microscope resembles the spectroscopy of a discrete autoionized state, treated by Fano [118]. When the transition matrix element for the localized state approaches zero, the autoionization spectrum shows an antiresonanceas a consequence of the interference effects at the site with the localized state. At the usual tip-surface distances, the STM is known to probe sp wave functions rather than the confined

(a) 0Å

Ag on Ag(111)

4f core-levels [119]. Thus, the dI/dV spectrum should show an antiresonance instead of a f -spectral peak as observed experimentally. By approximating the Fano dip shape by the inverse non-crossing-approximation (NCA) [117, 120] resonance [24], the lineshape in Figure 3 (solid curve) is obtained, in agreement with the experimental observations (dots). The width of the Kondo antiresonance obtained from the NCA calculation for the single isolated Ce impurity on Ag(111) (Fig. 3 (solid curve)) is  = kTK = 25(5) meV yielding a Kondo temperature of TK = 290(50) K which is considerably lower than the value of 1000 K found for bulk -Ce [107, 111]. This result is consistent with the expected decrease in hybridization with decreasing Ce coordination. These findings and the interpretation within the Fano picture have been corroborated by related STS observations of Co atoms on Au(111) [25, 121] (see Figs. 4 and 5) and on Cu(111) [26]. Recent experiments [122] on Co adatoms on Cu(111) and Cu(100) have detected a scaling of the Kondo temperature with the host electron density at the magnetic impurity. Furthermore, a quantitative analysis of the tunneling spectra revealed that the Kondo resonance is dominated by the volume density of states. The same group observed the scattering phase shift of isolated Co impurities at the surfaces of Ag(111) [123]. A disappearance of the Kondo resonance for atomically fabricated Co dimers (i.e., by manipulation with the STM tip) on Au(111) has been reported and was explained as the result of reduced exchange coupling between Au conduction electrons and ferromagnetic cobalt dimers [124]. Recently, this group investigated the temperature-dependent electronic structure of isolated Ti atoms on Ag(100) [125]. They find that the Kondo resonance is strongly broadened when the temperature is increased, confirming the role of electron-electron scattering as the main thermal broadening mechanism. The interaction of magnetic impurities with electrons confined in one dimension has been studied recently by spatially resolving the local electronic density of states of small Co

(b) 0Å

dl/dV (a.u.)

16

(d) 20Å

(e) 140Å

dl/dV (a.u.)

(c) 10Å

15

Ce on Ag(111)

Ce on Ag(111) –200

–100

0

100

200

Sample Voltage (mV) Figure 2. dI/dV spectra on (a) a single Ag adatom and (b)–(e) on and near a single Ce adatom at T = 5 K. The lateral distance between the tip position and the center of the adatom is indicated. Before opening the feedback loop the tunnel parameters were V = 200 mV, I = 01 nA. Reprinted with permission from [24], J. Li et al., Phys. Rev. Lett. 80, 2893 (1998). ©1998, American Physical Society.

–50

–25

0

25

50

Sample Voltage (mV) Figure 3. dI/dV spectra of a Ce impurity on Ag(111) for an energy range around the Fermi level. Squares: measurement (V = 100 mV, I = 01 nA before opening the feedback loop). Solid line: calculation using the Anderson single impurity model (see text). Reprinted with permission from [24], J. Li et al., Phys. Rev. Lett. 80, 2893 (1998). ©1998, American Physical Society.

Low-Temperature Scanning Tunneling Microscopy

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1.0

Experiment Off Cobalt Atom

Theory

dl/dV (arbitrary units)

0.8

0.6

0.4

On Cobalt Atom

0.2

0.0 –100

–50

0

50

100

Sample voltage (mV) Figure 4. dI/dV of a Co impurity on Au(111) and over the nearby bare Au surface for an energy range around the Fermi level. Dashed line: fit to the data with a modified Fano theory. Reprinted with permission from [25], V. Madhavan et al., Science 280, 569 (1998). ©1998, American Association for the Advancement of Science.

clusters on metallic single-walled nanotubes. The spectroscopic measurements performed on and near these clusters exhibit a narrow peak near the Fermi level which had been identified as a Kondo resonance [126].

Evidence for an orbital Kondo resonance on a transitionmetal surface has been reported recently on an atomically clean Cr(001) surface. STS revealed a narrow resonance 26 meV above the Fermi level which was attributed to the manifestation of an orbital Kondo resonance formed by two degenerate dx z , dy z surface states [127]. A Fano lineshape has been also obtained in recent theoretical treatments of the Kondo effect of a single magnetic adatom [128–131]. A beautiful experiment of a “quantum mirage” has been performed recently, where the Kondo resonance is seen in both foci of an ellipitical quantum corral while the magnetic Co adatom is placed only at one focal position [26]. Evidently, detection and spectroscopy of individual magnetic adatoms open new perspectives for probing magnetic nanostructures. The controlled modification of the electronic structure of a single Mn adsorbate placed within a geometrical array of adatoms on Ag(111) added a new aspect to the spectroscopy of single adatoms [22]. The spectral changes result from coupling between the adsorbate level and surface electronic states of the substrate. The surface state electrons are scattered coherently within the adatom array, influencing the local electronic structure of the single adatom within the array. The dimension and geometry of the adatom array provide a degree of control over the induced changes.

4. SURFACE STATES 4.1. Lifetimes

12Å

2.5

9Å 5Å 2.0

dl/dV (arbitrary units)

3Å 2Å 1.5 0Å –3Å 1.0

–4Å –7Å

0.5

–10Å –12Å

0.0

–100

–50

0

50

100

Sample voltage (mV) Figure 5. A series of dI/dV spectra with the tip held at various lateral spacings from the center of a single Co atom on Au(111). Reprinted with permission from [25], V. Madhavan et al., Science 280, 569 (1998). ©1998, American Association for the Advancement of Science.

Noble-metal surface states have been the focus of substantial experimental and theoretical attention [13–16, 18, 19, 99, 114, 132, 135–143]. Accurate values are known for their binding energies, effective masses, and Fermi wavevectors, including the temperature dependence and their response to various changes in surface conditions [144]. Overall, there is good agreement of theoretical predictions with experimental results, mainly with angle-resolved photoemission spectroscopy (ARPES). However, measured values of the ARPES linewidth  of a surface state hole [145] which is linked to its lifetime  = h/ , despite prolonged investigation by ARPES [115, 146–148], deviated significantly from theoretical predictions. Surface imperfections have been invoked to explain the failure in observing theoretically predicted linewidths [144, 146, 148]. In the case of Ag(111), the experimental linewidth surpassed theoretical values by ∼300%. STM with its ability to characterize surface topology and to identify minute amounts of contamination is ideally suited to address this issue. Defect-free surface areas can be selected for experimentation to avoid scattering from defects. Spectroscopy of the differential tunneling conductance dI/dV versus the sample voltage V , which is related to the local density of states above the surface [91, 119], can be used to probe the surface state on Ag(111). In typical conductance spectra (dI/dV ) measured at low temperatures, the surface state gives rise to a sharp rise in the conductance near −70 meV which reflects the onset of tunneling from the surface state band. In the experiments, the overall slope of the spectra varies to some extent. The variations are presumably caused by differences in tip structure

Low-Temperature Scanning Tunneling Microscopy

672 or chemical composition. In more detailed measurements the width of the onset denoted  is resolved. This width reflects the lifetime of the surface state hole left behind by a tunneling electron [134]. An alternative approach to measuring lifetimes, first indicated by Hasegawa and Avouris [14], is to analyze the spatial decay of electron standing wave patterns near defects [150]. An analytical relation between the experimental width  and the ARPES linewidth  = 2 , where  is the imaginary part of the electron self-energy, is obtained, using approximations which have shown to be very acceptable in the context of surface state spectroscopy [13, 16, 18, 19, 99, 132]. The surface state is modeled by a two-dimensional electron gas with effective mass m∗ and binding energy ES . Lifetime effects are included via a constant self-energy . The density of states at the surface, n!E", then is [151] n!E" = NB + m∗
% − &!E − ES "/2% 2

(1)


cos &!E" = E/ E 2 + 2

(2)

where

NB represents a constant background of bulk states at the surface. A simple expression for the tunneling current can be used if the tip is electronically featureless, the coupling between the tip and the sample is weak, and the energy and momentum dependence of the tunneling matrix elements can be neglected. This yields  I!V " = C n!( + V "
f !(" − f !( + V " d( (3) −

where C is a constant and f the Fermi function. From Eq. (3), the differential conductance is dI  Cm∗   f !E − V + ES " dE = CNB + dV 2% 2 − E 2 + 2

(4)

Equation (4) describes a sharp increase near ES , with a width determined by  as observed experimentally. A Sommerfeld expansion of Eq. (4)—which is reasonable since in the experiment at 5 K thermal broadening of the Fermi edge is small compared to — yields    ≃ % 1 + O !T /"2 (5)

Equation (5) shows that the self-energy can be directly obtained from the experimental width of the onset . Thermal broadening can safely be neglected. More detailed calculations outlined below corroborate this result although the factor % is slightly modified to ≃09% in the case of Ag(111). This smaller value results from the part of the electronic wavefunction in the tunneling barrier region where  = 0. For more realistic calculations of the conductance, multiple-scattering techniques were employed to obtain the sample Green function [152]. Lifetime effects were taken into account via an imaginary self-energy which is constant and restricted to the sample. For a tip modelled by a single atom, using a 4-eV optical potential to give a realistic spectral density [153], conductance spectra are obtained within

the many-body tunnelling theory of Zawadowski and Appelbaum and Brinkman [154].  affects the broadening of the onset of the surface state band. In addition, one observes that  also modifies the relative contribution of the surface state to the conductance above the onset. A comparison of low-temperature STM spectra with these calculations leads to an estimated imaginary self-energy of  ∼ 5 meV. For this value there is, in addition, an approximate agreement of the relative contribution of the surface state to the total conductance for voltages above the onset. The surface state doubles the total conductance in both experiment and calculation. In ARPES, after eliminating experimental factors,  causes the full width at half maximum of the Lorentzian lineshape of the peak associated with the surface state  = 2. McDougall et al. [147] have evaluated the temperature dependence of the width of the Cu(111) surface state. They observed  varying linearly between 75 meV at 625 K and 30 meV at 30 K and compared these data to a perturbative treatment of the electron-phonon coupling [155] with the Debye model and Fermi liquid theory for the electron-electron scattering [156]. Their model provides a quantitative explanation of the observed linear temperature dependence which is attributed to a dominant phonon contribution [147]. Nevertheless, the predicted absolute value  ∼ 14 meV at T = 30 K lies well below the observations. Surface roughness has been invoked to explain this difference [147]. In agreement with this proposal, Theilmann et al. [148] observed a correlation between the photoemission linewidth and the width of LEED spots from the same surface. However, extrapolation to perfect surface order yields  = 43 ± 5 meV at room temperature, leaving a considerable gap to the theoretical estimate of  = 30 meV. For Ag(111), the situation was comparable. Using the ARPES data of Paniago et al. [115]— = 20 meV at T = 56 K—and repeating the calculations of [147] with an electronphonon mass enhancement parameter of 0.13 [155] and Debye energy of 18 meV, one finds  = 5 meV at T = 0 K for states at the surface state binding energy. The phonon contribution increases  at T = 56 K to ∼6 meV. Again, a sizeable difference exists to the ARPES value. The value for  (= /2) of 5 meV represents a significantly lower experimental estimate of  than previously available. New experimental data obtained under improved experimental conditions by Kliewer et al. [157] indicate a yet smaller value of  ∼ 3 meV (see Fig. 6). Along with new, similar STM results for Au(111) and Cu(111), these data are explained by calculations which go beyond previous quasi-free electron gas models and take band-structure effects into account [157]. These calculations show that the contribution of electron-hole scattering for occupied surface states has previously been underestimated and remove the remaining discrepancy between experimental and calculated self-energies of noble-metal surface states. On the other hand, new high-resolution photoemission data [158, 159] achieved now an instrumental resolution comparable to the one of STS, of  = 2 of 6 meV and 23 meV for the surface states on Ag(111) and on Cu(111), respectively. In view of the much larger surface area seen in photoemission, this is a bit surprising. In a recent STS investigation with still improved experimental conditions a

dl/dv (arb. units)

Low-Temperature Scanning Tunneling Microscopy

Cu(111)

Au(111)

Ag(111)

30 mV

23 mV

8 mV –90

673

–70

–50

–700

–500

–300

–500

–400

–300

Sample Voltage (mV)

Figure 6. dI/dV spectra for Ag(111), Au(111), and Cu(111). All spectra were taken at least 200 Å away from impurities and are averages of different single spectra from varying sample locations and tips. Reprinted with permission from [157], J. Kliewer et al., Science 288, 1399 (2000). © 2000, American Association for the Advancement of Science.

value of  = 4.9±0.2 meV has been obtained on Ag(111) which is in the direction of the above argument [160]. An interesting method to determine the energy dependence of surface state lifetimes has been developed recently. This method may be termed lifetime engineering [22, 23], as it is based again on the atom manipulation capabilities of the STM. Both groups arranged artificial arrays of atoms in the form of rectangles (Mn on Ag(111) [22]) or triangles (Ag on Ag(111) [23]). The scattering of the surface state electrons at these adatoms results in complex interference patterns (see next section) which were observed in spectroscopic images and local dI/dV spectroscopy. From a lineshape analysis of dI/dV spectra and multiple scattering calculations, quasiparticles lifetimes near the Fermi level were derived. The experimental results indicated that the electron lifetimes deviate from a !E − EF )−2 dependence and reflect the electronic band structure at the surface as well as the local influence of the adatom array.

Figure 7. Constant current 13 × 13 nm image of a single Fe adatom on the Cu(111) surface. The concentric rings surrounding the Fe atoms are standing waves due to the scattering of surface state electrons with the Fe adatom. Reprinted with permission from [15], M. F. Crommie et al., Science 262, 218 (1993). © 1993, American Association for the Advancement of Science.

4.2. Standing Wave Phenomena

surface state band and a concomitant modification of surface properties associated with the surface state electrons [162–164]. However, to date there has been little attention paid to the systematics of the confinement, especially in the limit of smaller-scale structures. Can one still talk of surface state electrons supported on a structure just three or four atoms wide? These structures are the ones most likely to result in significant depopulation, and exist naturally on surfaces especially during epitaxial growth. In contrast to the corral studies of [15], a recent angle-resolved photoemission study [172] on vicinal Cu(111) surfaces with monatomic

On the close-packed faces of the noble metals some electrons occupy surface states, quasi-two-dimensional states localized at the metal surface and which have nearly freeelectron-like dispersion [161]. In recent years there has been a renaissance of interest in the physics of these electrons, which it is argued play an important role in a variety of physical processes, including epitaxial growth [162], in determining equilibrium crystal shapes [163], in surface catalysis [164], in molecular ordering [165], and in atom sticking [166]. This attention may be largely attributed to the advent of the scanning tunneling microscope (STM), which has enabled direct imaging in real space of electrons in surface states, and their interactions with single adsorbates [13, 17], steps [14, 16], and other structures [15, 16] (see Figs. 7 and 8). Probably the most striking observation has been of the confinement of surface state electrons within artificial nanoscale structures [15], geometrical arrays of Fe atoms positioned with atomic-scale precision using the STM (see Fig. 9), and which have also received theoretical attention [167–171]. Scattering at natural structures such as steps has also been seen to result in lateral localization [16]. Confinement is important as it raises the energies of the surface state electrons, resulting in a depopulation of the

Figure 8. Constant current 50 × 50 nm image of the Cu(111) surface at 5 K. Standing waves in the LDOS with a periodicity of ∼1.5 nm can be seen emanating from monatomic step edges. Reprinted with permission from [13], M. F. Crommie et al., Nature 363, 524 (1993). © 1993, Macmillan Magazines Ltd.

674

Low-Temperature Scanning Tunneling Microscopy

–52 mV

69 mV

140 mV

239 mV

321 mV

–50 mV

70 mV

140 mV

240 mV

320 mV

Figure 10. Upper row: topographic image of an approximately hexagonal Ag island on Ag(111) (area ∼94 nm2 ), and a series of dI/dV maps recorded at various bias voltages (at T = 50 K). Lower: geometry of a hexagonal box confining a two-dimensional electron gas, and the resulting local density of states. Experimental values have been used for the surface state onset E0 , electron effective mass m∗ , and the island size. The calculations include a self-energy of  = 02!E − E0 ". Reprinted with permission from [19], J. Li et al., Phys. Rev. Lett. 80, 3332 (1998). © 1998, American Physical Society.

Figure 9. Circular quantum corral built from 48 Fe atoms on the Cu(111) surface. Average diameter of ring (atom center to atom center) is 14, 26 nm. An electronic interference pattern can be seen inside of the Fe ring. Reprinted with permission from [15, 168], M. F. Crommie et al., Science 262, 218 (1993). © 1993, American Association for the Advancement of Science, and M. F. Crommie et al., Surf. Rev. Lett. 2, 127 (1995). © 1995, World Scientific Publishing Company.

steps spaced ∼15 nm apart has found shifts in surface state energies only a fraction of those expected on the basis of ideal confinement, suggesting the step edges only act as weak and permeable barriers. Thus the validity of confinement as a mechanism of surface state depopulation seemed to be in question. Standing waves on an Ag island at room temperature, identified as resulting from surface state confinement, have been first observed in [16]. Li et al. [19] report on a lowtemperature study of surface state electrons confined to small Ag islands on Ag(111). In contrast to adatom corrals, these are stable structures even at elevated bias voltages in the STM, enabling the observation of standing waves over a wide range of voltages. Complementary electronic structure calculations enabled the authors to extend the investigation down to the smallest island sizes, quantifying the nature of surface state confinement. Differential conductance maps taken above individual islands exhibit strongly voltage-dependent features. Figure 10 shows a typical series of scans taken above an island. For voltages around −65 mV and lower (negative voltage corresponds to tunneling from the sample), the image of the interior of the island is featureless. At higher voltages standing wave patterns of increasing complexity are observed, beginning with a single peak near the center of the island and gradually evolving to multiple rings with sixfold azimuthal modulations, which are partially distorted by deviations of the island shape from hexagonal symmetry. STM measurements of dI/dV are related to the local density of states above the surface [91], and the standing waves may be identified as originating from surface state electrons, confined by the rapidly rising potential at the edges of the island. On the clean surface the surface state disperses upwards from E0 = −67 meV, with an effective mass m∗ = 0.42 [18, 115].

Positioning the STM tip above the center of a particular island and recording dI/dV spectra, a series of peaks is observed, reminiscent of those measured at the center of adatom corrals [15]. The peaks correspond to energy levels of the confined surface state electrons, broadened into resonances by single-particle scattering processes [167, 169], many-body interactions [170], and instrumental effects such as thermal broadening. Electronic structure calculations on small Ag islands on Ag(111), using the multiple-scattering approach described in [170, 173], were performed to model the experimental observations [19]. The potentials were taken from ab initio calculations of the clean surface, which give the binding energy and effective mass of the surface state band as E0 = −72 meV and m∗ = 037. Regular hexagonal islands with sizes up to 631 atoms were considered. As the island size decreases, the calculated levels rise markedly, consistent with a high degree of confinement. To quantify the nature of the confinement the energy levels were analyzed in terms of two-dimensional “particle-in-abox” eigenstates, corresponding to electrons with effective mass m∗ confined within a hexagonal domain of potential E0 by infinitely high barriers [133]. The eigenvalues scale with the inverse of the area of the hexagon, . In atomic units (
= me = e2 = 1) En = E0 +

+n m∗ 

n = 1 2 3   

(6)

Only eigenstates which transform according the A1 representation of the group C6 (6 mm) of the hexagon [175] have a nonzero amplitude at the origin, and for these the lowest few are +1 = 9296 +4 = 4870, and +10 = 1170 [133]. In comparing with the experimental dI/dV data and/or the multiple-scattering calculations, it is not clear a priori what the relevant size of an island is. The confinement is effected by the rapid rise in the potential beyond the edges of the island. However, a sizeable fraction of the surface state extends into the first few layers of atoms and so will not be directly influenced by the island edge. The effective location of the boundary confining the surface state electrons is uncertain, relative to either the measured topological linescan of the island or the actual positions of the edge atoms. To circumvent this difficulty a fit of the results from the scattering calculations with the energies from Eq. (6)

Low-Temperature Scanning Tunneling Microscopy

675

was carried out, assuming the effective boundary lies a constant distance (independent of island size) beyond the positions of the edge atoms. This distance was the only fitting parameter. The scattering calculations were then used to simulate constant-current topographs by evaluating the integrated LDOS above the surface, enabling an identification theoretically of the effective boundary on the topographical cross section. The location identified as the effective boundary of the surface state electrons lies close to the midpoint of the rise in the topological linescan, ∼05 nm beyond the edge atoms. The precise value depends upon the potential in the vicinity of the island edge, for which only an approximate description is available, but which always coincides closely with the midpoint of the rise in the simulated topological linescan. With this definition, remarkable agreement between the energies of the surface state electrons as derived from both experimental dI/dV spectra and the theoretical LDOS was found, compared to the scaling behavior expected of twodimensional electrons. The smallest island contains just 19 atoms, and the smallest measured island is  ∼ 16 nm2 containing ∼100 atoms. These results indicate that as far as the energy levels are concerned, it is valid to talk of surface states confined to nanoscale islands, right down to the smallest island sizes. The confinement corresponds to an infinite barrier, and the area to which the electrons are confined corresponds to a boundary lying a fixed distance beyond the outermost atoms, independent of both energy and the island size. There are deviations from this interpretation of the electron behavior, as witnessed by the finite width of the confined energy levels. The levels should be infinitely sharp for ideal confinement. The width of each energy level stripped of instrumental contributions, En , implies a finite lifetime for the confined electrons, tn ≃
/En , dominated by scattering into other electron states at the island edges [167, 169, 170]. While these processes do not influence the energies of the levels, one potential consequence is that depopulation of the surface state only becomes significant for island sizes smaller than those derived from a two-dimensional analysis. Even at T = 0 each energy level will have a finite width, resulting in partial occupation after the level has risen above the Fermi energy. Assuming a Lorentzian lineshape, the state will be 1/4 full when En lies En above the Fermi level. For an estimate of the significance of this effect, the experimental spectra yield En ≃ 02 !En − E0 "

(7)

Using this relation, the island area at which the lowest level is 3/4 depopulated is 10% smaller than that which yields complete depopulation assuming perfect confinement. Hence the influence of the finite widths is marginal. A more detailed analysis of the level widths is given in [133, 134], incorporating them into the two-dimensional description as a self-energy provides a good model of the confined states. Allowing for the clear deviations in shape of the experimental island from the ideal hexagon assumed theoretically, which (alongside the finite resolution of the STM) results in the more rounded features, there is good agreement between the theoretical and experimental images. The

largest differences arise at the island edges, when experimental dI/dV data are influenced by tip-height variations [18] and because theoretically the background of bulk states is not included. This quantitative study of electron confinement to nanoscale Ag islands on Ag(111) using low-temperature STM and electronic structure calculations confirmed the validity of the “particle-in-a-box” model for confined surface state electrons. The energies conform to the expected scaling behavior down to the smallest of island sizes, and the width of the confined levels has only a marginal effect on the surface state depopulation. For larger island sizes the island edges have the same atomic structure as low-energy steps oriented along 1¯ 10 directions on the (111) surface, and so these conclusions should hold also for confinement to raised terraces and to finger-like structures characteristic of the dendritic formations which occur in the low-temperature growth of Ag/Ag(111) [176]. In addition, these conclusions should also hold for confinement on Cu(111) and Au(111) surfaces, due to the similarities in the band structures of the noble metals. The failure to observe these effects in angleresolved photoemission [172], apart from limits in the instrumental resolution, most likely results from the presence of a sizeable distribution of terrace widths within the spot area of the incident radiation. Subsequently, confinement of Ag(111) surface state electrons had been studied in symmetric and asymmetric resonators formed by two atomically parallel step edges. The local density of states in the resonator was measured by dI/dV spectroscopy and spectroscopic imaging and was analyzed within a simple Fabry–Pérot model. The energydependent reflection amplitudes and scattering phase shifts of the different step edges were determined [20]. A recent investigation of the surface state depopulation on such small Ag(111) terraces showed [177] that with decreasing terrace width, the electronic density in the occupied surface state shifts monotonically towards the Fermi level, leading to a depopulation at 3.2 nm terrace width in quantitative agreement with the above Fabry–Pérot model. These observations also confirm the earlier results obtained on small silver islands [19, 133]. It was shown later that a simple “particle-in-a-box” model is already sufficient to identify all major features in the observed dI/dV maps and in constant-current topographic images of a similar quantum box [160]. These authors obtained atomically resolved constant-current images of a Ag(111) single-crystal surface at a temperature of 5 K, which simultaneously display standing wave patterns arising from electron confinement of surface state electrons to nanoscale terraces. It was shown how the energy-dependent patterns develop from the superposition of allowed wavefunctions in the quantum box. This experiment constitutes a direct verification of the Tersoff–Hamann model for arbitrarily localized tip wavefunctions [160]. It has been predicted [178] and found recently that the standing waves mediate long-range oscillatory interactions between adatoms [179–181]. In the case of Cu on Cu(111), the interaction potential was determined by evaluating the distance distribution of two adatoms from a series of STM images taken at temperatures of 9–21 K [180]. The longrange interaction had a period of half the Fermi wavelength

676 and decayed for larger distances as 1/d 2 . A comparison of three adsorbate/substrate systems, Cu/Cu(111), Co/Cu(111), and Co/Ag(111), essentially confirmed the earlier observations and demonstrated that the long-range interactions depend crucially on the Fermi wavelength of the surface state electrons [181]. The realization of two-dimensional imaging of electronic wavefunctions in metallic single-walled carbon nanotubes represents another highlight in low-temperature STS [182]. The measurements reveal spatial patterns, which can be directly understood from the electronic structure of a single graphite sheet. The authors observed also energy-dependent interference patterns in the wavefunctions and exploit these to directly measure the linear electronic dispersion relation of the nanotube. In the context of standing wave patterns, the observation of the development of one-dimensional band structure in artificial gold chains marks the beginning of manipulation and spectroscopy of local one-dimensional structures on surfaces [183]. The authors used the ability of the STM to manipulate single atoms to build well-defined gold chains on NiAl(110). The electronic properties of the one-dimensional chains are dominated by an unoccupied electron band, gradually developing from a single atomic orbital present in a gold atom. Spatially resolved conductance measurements along a 20-atom chain provided the dispersion relation, effective mass, and density of states of the free electron-like band.

Low-Temperature Scanning Tunneling Microscopy

(a) hcp fcc

O

O– N+

10 nm

1

(b)

(c)

2 hcp

fcc

10 nm

10 nm

fcc

hcp

5. SUPRAMOLECULAR SELF-ASSEMBLY

Figure 11. STM images at 50 K of a reconstructed Au(111) surface with adsorbed NN; (a) 0.7 ML NN. Inset: Structural formula of NN. The dashed line encloses the “exclusion” area resulting from steric repulsion. The distance of the full line from the dashed line indicates the strength of a negative electrostatic potential computed on the dashed line. (b) 0.1 ML NN at 65 K. (c) 0.2 ML NN at 50 K. Inset: 0.2 ML NN at 10 K. Reprinted with permission from [32], M. Böhringer et al., Phys. Rev. Lett. 83, 324 (1999). © 1999, American Physical Society.

Self-assembly of regular arrays of thermodynamically stable nanostructures from appropriately functionalized molecules is a promising approach to future mass-fabrication of nanoscale structures [184, 185]. Molecules adsorbed on surfaces are particularly appealing because their arrangement is directly observable with the scanning tunneling microscope [186–190]. Understanding supramolecular aggregation starting from the basic interactions of the constituent molecules is a prerequisite to eventually control the selfassembly process. The aromatic molecule 1-nitronaphthalene (NN, structure Fig. 11, inset) adsorbed on the reconstructed Au(111) surface [191] was chosen in a recent experiment as structural unit for self-assembly. The confinement of NN to two dimensions introduces a chirality not present in the gas phase. On the surface both enantiomers are present in equal amounts, thus forming a racemic mixture. The Au(111) sample was held at room temperature during the deposition and below 50 K during the measurements. In order to minimize tipinduced motion of the molecules at submonolayer coverages [192, 194, 195], small tunneling currents (I = 10 pA) were used. At coverages between 0.05 and 0.15 ML self-assembled quasi-0D clusters of distinct size and structure appear at the fcc elbows of the reconstruction (Fig.11b). At slightly higher coverage (0.2 ML) identical clusters are observed within fcc domains (Fig. 11c) and, sporadically, within hcp domains and on domain walls. All decamers appear identical in the STM images, except for a mirror symmetry (clusters 1 and 2 in Fig. 11c). They consist of an 8-molecule ring surrounding

a 2-molecule core. Manipulation experiments at decreased tunneling resistance show that the decamers behave like stable “supermolecules” [33]. Therefore, the structure and stability of the decamers is determined by highly specific intermolecular forces while the interaction of these “supermolecules” with the reconstructed substrate and a mutual repulsion at small distances determine their lateral spacing. At medium coverage (0.3–0.75 ML) the growth mode changes to the formation of 1D molecular double chains (Fig. 11a at 0.7 ML) guided by the reconstruction domains or by step edges. At full monolayer coverage 1D and 2D periodic molecular structures coexist on the surface [193]. At a tunneling voltage V = −23 V, intramolecular structure is resolved and the contrast pattern shows the expected handedness [32]. The charge density calculated for the highest occupied molecular orbital (HOMO) of the free NN molecule closely resembles the observed submolecular structure [32]. At lower bias voltage hybridization with Ausubstrate states results in a nearly symmetric appearance of the molecules [196] where the long axis of the ellipsoids coincides with the long axis of the naphthalene core. Molecular-dynamics simulations rationalize the observed supramolecular arrangements [32]. Within each single strand the molecules are arranged in a “head-to-tail” configuration via hydrogen bonds between a negatively charged oxygen atom and the “backside” hydrogen atom at a carbon atom of a neighboring NN molecule. The second strand is rotated by 180 and shifted by half a period such that opposite charges are close to each other. The double chains are positively

Low-Temperature Scanning Tunneling Microscopy

677

charged outside and therefore mutually repel as observed experimentally. Straight, defect-free segments of the double chains consist of exclusively one NN enantiomer and thus represent a 1D conglomerate [32]. Below a critical density of molecules quasi-0D decamers are energetically more favorable than linear double chains with unsaturated hydrogen bonds at their ends [32]. In the decamers a homochiral molecular ring along the periphery is stabilized by a core composed of two molecules with opposite chirality. STM observations and theoretical modeling led to a detailed understanding of the stability and the internal geometry of self-assembled supramolecular structures of the 2D chiral molecule 1-nitronaphthalene on Au(111). Recently, these tools were extended to obtain an understanding of the supramolecular self-assembly and selective step decoration of these molecules on the Au(111) surface [197]. Furthermore, a coverage-driven chiral phase transition from a conglomerate to a racemate was followed in real space [34, 198, 199]. Subsequently, other supramolecular nanostructures have been built by self-assembly on surfaces taking advantage of hydrogen bonding [35–37]. These range from onedimensional supramolecular nanogratings on Ag(111) [35] made of chains of benzoic acid to the mesoscopic correlation of supramolecular chirality in one-dimensional hydrogenbonded assemblies [36] (see Fig. 12). The stereochemical (a)

C N O H

δ-PVBA

O

N

H

O

O H

O H N

O

H O

H O

O

H

O

H

H

N

N

H O

H O

O

O H

N O H

N

O H

H

H

O H

N

(b)

N

λ-PVBA

mirror plane Figure 12. (a) Chirality of PVBA upon confinement to two dimensions. The mirror symmetry is reflected by a dashed line. (b) STM topographs of the two possible supramolecular chiral twin chains from self-assembly of PVBA on Ag(111) (image size 4 × 135 nm). The corresponding models for the energetically favored configurations reveal the underlying chiral resolution (hydrogen bonds indicated by dashes). Reprinted with permission from [36], J. Weckesser et al., Phys. Rev. Lett. 87, 096101 (2001). © 2001, American Physical Society.

effects in supramolecular self-assembly on surfaces, that is, the 1D versus 2D enantiomorphic ordering of these organic molecules, were reported in [37]. In the meantime, a promising new approach towards a controlled self-assembly had been developed for substituted porphyrin molecules adsorbed on a gold surface: the formation of surface-supported supramolecular structures whose size and aggregation pattern were rationally controlled by tuning the non-covalent interactions between individual absorbed molecules [38].

6. CLUSTERS 6.1. Electronic Properties Whereas the electronic structure of atoms, molecules, and extended solids is quite well explored from both an experimental and a theoretical viewpoint, very little is known about the systematic development of the electronic structure as single atoms are combined to form a solid. There are fundamental questions to be answered about the onset of metallic behavior in small metal clusters or the influence of quantum size effects on the electronic structure of small particles. Therefore, it is of primordial interest to study individually, that is, locally, these objects of “embryonic” condensed matter [200]. An ideal tool for this type of investigation is again offered by scanning probe methods which have been used in three pioneering studies of nanometer-size clusters of Au on GaAs(110) [201], Fe clusters on the same surface [202], and of size-selected Si10 clusters on a reconstructed Au(001) surface [203], which all have been published already in 1989. In the first investigation [201] a characteristic spectrum of bandgap states was observed for the Au particles grown on GaAs. Both donor and acceptor states have been observed and they have been identified with the first and second electron states of the Au-Ga bond, respectively. In the second study, the formation of a local Schottky barrier has been realized by deposition and growth of Fe clusters (9–127 atoms per cluster) on GaAs surfaces in ultrahigh vacuum. STS revealed the onset of the metallic character of the Fe clusters above about 35 atoms per cluster by the observation of a continuum of occupied cluster-induced states at the Fermi energy in the bandgap of the semiconductor [202]. The third experiment [203] represents the first STM/STS study of supported size-selected clusters on a solid surface. The Si clusters were generated by pulsed laser vaporization of a silicon rod in a continuous flow of He buffer gas. The quadrupole mass spectrometer was set to transmit only Si10 clusters and after mass analysis the ions were focused into a low-energy ion beam and deposited on the sample with an energy of approximately 5 eV, well below the threshold for cluster fragmentation. After cluster deposition the sample was transferred to the STM chamber. During deposition, transfer, and measurement, the sample was maintained under UHV conditions. The cluster images were found to depend on the sample bias voltage, and from STS measurements the clusters were found to have a bandgap of about 1 eV. A wide variety of cluster images were observed even though size-selected clusters were deposited. When only Si atoms were deposited on the surface flat islands

Low-Temperature Scanning Tunneling Microscopy

678 were formed. Surprisingly, in spite of these first promising results, up to the present time experiments on size-selected supported clusters with local probes have been scarce. One interesting experiment used the fact that the tipsurface region of an STM emits light when the energy of the tunneling electrons is sufficient to excite luminescent processes [95]. These processes provide access to dynamic aspects of the local electronic structure that are not directly amenable to conventional STM experiments. In order to explore the lateral resolution obtainable in the photon signal, adsorbed C60 molecules on a reconstructed Au(110)(1 × 2) surface were chosen [8]. At sub-monolayer coverage C60 islands with monomolecular height were observed. In the STM image of Figure 13a indi-

A

B

Figure 13. Au(110) surface covered with an annealed monolayer of C60 . Topograph (a) and photon map (b) are represented as gray-scale images. Area: 6.5 nm by 6.5 nm, tip voltage Vt = −28 V, tunneling current It = 44 nA, intensity scale: 800 cps, temperature of tip and sample: 50 K. Reprinted with permission [8], R. Berndt et al., Science 262, 1425 (1993). © 1993, American Association for the Advancement of Science.

vidual C60 molecules are resolved as protrusions. They form an approximately hexagonal array with a corrugation amplitude of 0.1 nm. A comparison of this STM image with the photon emission map measured simultaneously (Fig. 13b) shows that the individual C60 molecules give rise to bright spots in photon emission. The photon emission is maximal when the tip, acting as an electron source, is placed above a molecule. The emitted intensity from intermolecular regions is significantly lower. The lateral extent of the emission spots (0.4 nm full width at half maximum) is smaller than that of the topographic features (0.6 nm). For a few molecules no enhanced emission is observed, indicating that the photon emission channel contains some additional information on C60 not apparent in the topographic image. It was suggested [8] that the molecules interact strongly with the electromagnetic modes of the cavity between tip and sample. In analogy to the case of surface-enhanced Raman scattering (SERS) from molecules in pores of Ag films [100], such a confinement effect may also affect molecular photon emission. The above experiment [8] was the first photon emission measurement where individual molecules self-assembled in nanometer separation were clearly resolved. This technique has again been used recently to measure photon emission spectra of individual alumina-supported silver clusters obtained by evaporation and growth [9]. The light emission] was attributed to the excitation and the decay of local plasmon modes (Mie-plasmon resonances) in these particles [9]. As the cluster size decreases, the resonance shifts to higher energies and the linewidth increases. In the 1.5to 12-nm size-range of the clusters studied, intrinsic size effects are proposed to be at the origin for the observed size dependence of the Mie resonance [9]. Similar investigations were carried out on Ag clusters on Si single-crystal surfaces, where the light emission enhancement was also ascribed to the excitation of plasmon modes [11]. Moreover, coupled plasmon modes were detected in an ordered hexagonal monolayer of silver nanospheres on Au(111) [10]. A very recent finding was the observation of luminescence from metallic quantum wells, a new phenomenon in STM-induced photon emission [96] from metal surfaces. Another recent highlight in STM-induced light emission was the observation of luminescence from individual supported molecules on an ultrathin alumina film [12]. The energy gap of pristine silicon clusters supported on HOPG has been studied recently by STM and STS [204]. The clusters have been obtained by magnetron sputtering of a Si target and deposition on HOPG at room temperature; that is, they were not size-selected but have been individually addressed by the tip of the STM. Clusters with sizes between 0.25 and 4 nm were studied and the size dependence of the bandgap was determined. For clusters below 1.5 nm, gaps up to 450 meV were found, while for larger particles no gaps were recorded. The results were explained in terms of a transformation from diamond to a compact structure occurring at 1.5 nm (about 44 atoms per cluster). For clusters with diamond structure the surface dangling bond density is high, leading to electronic states filling the energy gap. On the other hand, the compact arrangement of the smaller clusters tends to eliminate dangling bonds. Therefore, finite gap values are observed for clusters with less than 44 Si atoms [204]. Interesting STS results on individual non-size-selected

Low-Temperature Scanning Tunneling Microscopy

Pt and Ag clusters produced by a PACIS cluster source and deposited onto HOPG have been obtained recently [205]. The observed spectral structures for Pt have been interpreted in terms of a Fano resonance and those of Ag clusters in terms of three-dimensional electron confinement in analogy to recent spectroscopic findings and interpretations for Kondo systems (Ce atoms and Ce clusters on Ag(111) [24], Co atoms on Au(111) [25], and on Cu(111) [26]), and for two-dimensional electron confinement found in quantum corrals [15, 22, 26] and on islands [16, 19]. Effects of electron confinement have also been observed recently with STS for silver-islands grown in nanopits on HOPG [206]. Recently, STS and STM have been applied to study the onset of the catalytic activity of Au particles grown on titania [207], which appeared to be correlated with the layer thickness of the particles on the surface. In this context, the recent investigations of the catalytic activity of small supported Au clusters on MgO are interesting [208] where the challenge of local investigations with the STM still remains. While inert as bulk material, nanoscale gold particles dispersed on oxide supports exhibit a remarkable catalytic activity. Temperature-programmed reaction studies of the catalyzed combustion of CO on size-selected small monodispersed Aun !n ≤ 20" clusters supported on magnesia, and first-principles calculations, reveal the microscopic origins of the observed unusual catalytic activity, with Au8 found to be the smallest catalytically active size. Partial electron transfer from the surface to the gold cluster and oxygen-vacancy F-center defects are shown to play an essential role in the activation of nanosize gold clusters as catalysts for the combustion reaction. The above presentation of the spectroscopic results on deposited clusters clearly shows that valuable information on these nanosystems can now be obtained by the application of an arsenal of local and nonlocal surface science analysis methods. Therefore, in the near future a much more intense employment of scanning probe techniques such as STM, STS, AFM, and others will beyond any doubt improve considerably the assembly, characterization, and functionalization of size-selected clusters on solid surfaces.

6.2. Geometric Structure The microscopic structure is of primordial interest as it influences most physical and chemical properties of the cluster. The determination of the exact microscopic arrangement of the atoms in a cluster on a surface is an enormous experimental challenge. Diffraction methods, which are perfectly adapted to obtain the positions of the atoms in a material, have been applied for the study of free clusters [209–212]. Two other experimental techniques were employed so far to obtain the geometric structure of clusters on surfaces, STM and infrared spectroscopy (IRS). In principle, STM is able to reveal the exact arrangement of all atoms in a planar cluster. For three-dimensional clusters the positions of the topmost layer of surface atoms can be imaged. In favorable cases, the atomic arrangement on the lateral facets was also obtained. In both cases a minimal cluster size seems to be necessary in order to obtain atomic resolution of the individual atoms of the cluster [213].

679 Infrared spectroscopy can in principle give a more complete picture if all the normal modes in the cluster are obtained. An example for measuring vibrational modes of ligand molecules in a nonlocal mode is given below [214]. When used on the local scale, inelastic electron tunneling spectroscopy (IETS) is especially promising [44]. First attempts for mapping metal clusters on surfaces with atomic resolution were performed in the late 1980s, summarized in [215]. Small two-dimensional platinum and aluminum clusters were imaged on highly oriented pyrolytic graphite (HOPG) with an STM at room temperature. The interpretation of this pioneering work is, however, unconfirmed, as pinning of clusters or contaminations on defects might have influenced the images. Mo et al. [216] showed in a beautiful measurement that the Stranski–Krastanov growth of Ge on Si(001) resulted in a metastable 3D phase consisting of small well-defined hut clusters. These were imaged with atomic resolution and shown to consist of prisms or four-sided pyramids with four [103] facets (see Fig. 14). The group of Henry achieved a complete characterization of the morphology of a 27-atom palladium cluster supported on a cleaved MoS2 single crystal [217]. The three-dimensional shape, azimuthal orientation on the substrate, and arrangement of atoms on lateral facets were determined from atomically resolved images of a palladium cluster grown in UHV on the MoS2 (0001) facet. The resulting in-situ STM image is shown in Figure 15. The cluster clearly consists of two monolayers. The first layer is a regular hexagon with a three-atom wide side and an additional atom attached to the left hexagon side. This layer is composed of 20 atoms. The top layer has a regular hexagonal shape and the sides are composed of two atoms, resulting in a top layer of 7 atoms. This Pd27 cluster is schematically shown also in Figure 15. The observed atomic arrangement is identical to the structure of bulk palladium with a (111) basal plane. From this atomically resolved STM image, in particular from the two-dimensional representation, the relative orientation between cluster and the substrate is obtained in real space, showing that the azimuthal orientation of the particle densePd-atom rows are parallel to the rows of sulfur atoms of the MoS2 surface. So far, this is the only experiment where the number of atoms is directly obtained from the STM image and it seems that the system was well chosen, as for

Figure 14. STM image of a single Ge hut cluster. Perspective plot: scan area 40 × 40 nm. The height of the hut is 2.8 nm. Reprint with permission from [216], Y.-W. Mo et al., Phys. Rev. Lett. 65, 1020 (1990). © 1990, American Physical Society.

Low-Temperature Scanning Tunneling Microscopy

680 (a)

t

ce

a )f

0 10

(11

1) f

ace

t

(

(a) (b)

(b) Figure 15. (a) STM image of an isolated Pd cluster of small size (∼1.5 nm) supported on the (0001) surface of MoS2 . A single atom and a trimer are seen on the top and on the bottom of the figure, respectively. (b) Schematic representation of the cluster containing 27 atoms. Reprinted with permission from [217], A. Piednoir et al., Surf. Sci. 391, 19 (1997). © 1997, Elsevier Science B. V.

other systems atomic resolution of such small clusters was not obtained. Imaging large particles on oxide surfaces with atomic resolution is less system-restricted. Hojrup Hansen et al. [213] reported atomic resolution on Pd particles hundreds to thousands of atoms large. The authors believe that obtaining atomic resolution is also possible for other systems as long as the particle diameter is larger than ∼4 nm and higher than about four layers (≥500 atoms/particle). The Pd particles were grown on a 0.5-nm-thick Al2 O3 film. Figure 16 shows STM images of a Pd particle with atomic resolution. It reveals a (111) layer for the top facet. The measured nearest-neighbor distance is 2.76 ± 0.07 Å, indicating the absence of any strain in the cluster (d = 275 Å; for

Figure 16. Atomic-resolution images of crystalline nanosize Pd clusters. (a) 95 × 95 nm image size. The resolution is kept a few layers down the sides, allowing identification of the side facets. The dots in (b) indicate the atomic positions consistent with a (111) facet. Reprinted with permission from [213], K. Hojrup Hansen et al., Phys. Rev. Lett. 83, 4120 (1999). © 1999, American Physical Society.

Pd(111)). In addition, these results show that for such large particles supported on oxide surfaces it is possible to obtain atomic resolution across the entire cluster surface, although the tunneling conditions at the edge of the cluster change. As the tip-apex moves away from the top layer of the particle when approaching the edge, tunneling occurs between the top layer of the particle and the atoms on the side of the tip. In some cases the authors even report atomic resolution on the largest side facets of the particles and observed a (111) crystallographic orientation. In addition, the particle morphology was characterized by three parameters, which are the height of the particles, the width of the top facets, and the ratio between the side lengths of the top facets.

Low-Temperature Scanning Tunneling Microscopy

681

The smallest observed particle with a crystalline structure had a top facet of 2–3 nm widths and a height of 0.5–1 nm, corresponding to a thickness of 2–4 layers. Furthermore, these measurements revealed quantitative information on the work of adhesion. The observed shapes of the particles were compared with the ones resulting from a Wulff construction based on calculated surface energies. For Pd on Al2 O3 a value of Wadh = 2.8 ± 0.2 J/m2 was reported. In another example [218] the same group synthesized on a Au(111) template single-layer MoS2 nanocrystals with a width of ∼3 nm. The MoS2 nanocrystals were obtained by first growing ∼3-nm-wide Mo particles on the Au(111) template and by subsequent sulfidation in an H2 S atmosphere. Atom-resolved STM images reveal that the small nanocrystals exhibit triangular morphology in contrast to bulk MoS2 . Figure 17 depicts an atomically resolved STM image of such a triangular nanocrystal. The observed protrusions are arranged with hexagonal symmetry and an average interatomic spacing of 3.15 ± 0.05 Å. This is consistent with the interatomic spacing of S atoms in the (0001) basal plane of MoS2 . From the apparent height of only 2.0 ± 0.3 Å the authors conclude that the MoS2 nanocrystals are present as single layers on the Au surface. The triangular shape of the monolayer crystals is in contrast to the expected hexagonal morphology of multilayer MoS2 . This implies that one of the edge terminations is considerably more stable. To answer which edge structure is more stable, high-spatial-resolution images of the edge structures of the triangles were taken. These atomic-resolution images showed that the S atoms at

the edges are out of registry with the S atoms in the hexagonal lattice of the basal plane and that they are shifted by half a lattice constant along the edge. A comparison with DFT calculations revealed that the observed edge structure is only obtained when the stoichiometry on the edges is changed; for example, only one S atom is bound to a Mo edge atom. Thus these atomic-resolution STM images provided insights into the morphology (shape) and edge structure of the MoS2 nanocrystals. The same group observed directly by STM that a single layer of these MoS2 nanoparticles supports one-dimensional electronic edge states (see Fig. 17), which were viewed as one-dimensional conducting wires [219]. For small metal clusters composed of up to a few tens of atoms and supported on metal surfaces, no atomically resolved STM images exist; different cluster sizes can often be distinguished only by small variations in their heights and not by different contrasts in the measured electron density. An interesting method was introduced by Schaub et al. [220] where size-selected Ag19 clusters on a Pt(111) surface were decorated with rare-gas atoms. This leads to a pronounced corrugation in the rare-gas necklace around the cluster, making the rare-gas atoms a sensitive probe to determine cluster size. Figure 18 shows a high-resolution STM image and a linescan across the cluster of a gas phase deposited Ag19 cluster on Pt(111) surrounded by 12 Kr atoms. This situation can easily be obtained as the binding energy of Kr to step edges and around the cluster is enhanced. From the observed number of Kr atoms around the silver cluster and the shape of the ring, the cluster size is deduced. By assuming a silver atom diameter of 2.77 Å, and knowing the Pt(111) lattice constant, and by determining the height of the cluster to be of only one monolayer, the authors showed that a regular hexagon containing 19 atoms fits perfectly the topographic image seen in

(a) Height [Å]

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Figure 17. An atom-resolved STM image (41 × 41 nm) of a MoS2 nanocluster. The grids show the registry of the edge atoms relative to those in the basal plane of the MoS2 triangle. The yellow triangle is the manifestation of a one-dimensional electronic edge state. The inset shows a Wulff construction of the MoS2 crystal. Reprinted with permission from [218], S. Helveg et al., Phys. Rev. Lett. 84, 951 (2000). © 2000, American Physical Society.

Pt(111) orientation

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Figure 18. Ag19 cluster on Pd(111) surrounded by 12 Kr atoms. (a) The image has been filtered to increase the contrast. (b) Linescan across the cluster. (c) Model superimposed to the image. (d) Model. Reprinted with permission from [220], R. Schaub et al., Phys. Rev. Lett. 86, 3590 (2001). © 2001, American Physical Society.

Low-Temperature Scanning Tunneling Microscopy

682 (a)

(b)

[11

Figure 18c. Figure 18d shows that the Kr necklace is sensitive to the form of the cluster and the exact number of silver atoms in the cluster. This is the first experiment where after cluster deposition a one-to-one correspondence between the cluster size in the beam and on the surface was obtained.

0]

7. SPECIFIC NANOSTRUCTURES

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Metal oxides play a crucial role as insulators in many electronic and magnetic devices. As these devices become ever smaller, it becomes more and more important to understand the behavior of ultrathin insulating layers. Recently, the growth of ultrathin insulating NaCl layers on Al(111) [221] and of CoO and NiO layers on Ag(001) [222] has been studied by STM and atomic resolution has been obtained. From the change in image contrast a maximum thickness of three layers for successful imaging was inferred [221]. Based on image contrast as well as on tunneling current versus voltage curves taken on both NiO islands and Ag substrate, the existence of a bandgap has been deduced [222]. For nanostructures of CaF1 and CaF2 on Si(111), chemical imaging of insulators has been demonstrated using STS in combination with a simple model for tunneling across two barriers, the vacuum gap and the insulating film [94]. However, no atomic resolution has been obtained in this case and, for CaF2 , it was only assumed that the full gap is already formed for the first CaF layer on Si(111) [94]. As model system for a metal oxide, Schintke et al. [39] have chosen MgO on silver. The films were grown on a Ag(001) substrate at 500 K by evaporating Mg in an O2 partial pressure of 1 · 10−6 mbar [223]. After deposition of 0.3 ML MgO (Fig. 19a), two-dimensional square islands of 10–15 nm size have nucleated homogeneously on the Ag(001) surface. Some islands are embedded in the silver layer of the upper terrace, due to Ag adatom diffusion [224]. Two different island contrasts (i, ii) near a terrace step (Fig. 19a) are indicative of the position and orientation of the terrace step prior to nucleation. After deposition of about 2 ML MgO (Fig. 19d), the Ag surface is completely covered with MgO, forming terraces of typically 50 nm width and 3D pyramidic islands. Figure 20a shows representative dI/dU spectra measured with the tip positioned above a 1-ML-thick MgO island. At negative sample bias (occupied sample states), the LDOS increases at −4 V, whereas at positive sample bias (unoccupied states), two structures are detected around 1.7 V and 2.5 V. Between −4 V and +17 V, the tunnel current remains finite and the dI/dU spectrum is essentially flat. The intensity of the LDOS peak around 1.7 V observed in STS for 1 ML MgO diminishes for 2 ML and is no more detectable for a MgO film of 3 ML (Fig. 20b). This feature is attributed to MgO-Ag interface states [225]. In contrast, the high local density of unoccupied states around 2.5 eV does not decrease with film thickness and is identified with MgO states [226, 227]. Consequently the onset of the 2.5-eV peak observed in STS (Fig. 20a,b) corresponds to the onset of the MgO(001) empty surface state band [228–231]. Combining the local dI/dU observations of the occupied and unoccupied LDOS on the ultrathin MgO films (Fig. 20)

0]

[11

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Figure 19. Morphology of MgO thin films: STM images at 50 K. (a) 0.3 ML MgO/Ag(001), U = 50 V, I = 10 nA; (b) Ag(001) atomic resolution through an MgO island, U = 30 mV, I = 2 pA (left bottom corner: bare Ag substrate); (c) atomic resolution of the MgO layer (one type of ion is resolved), U = 25 V, I = 50 pA; (d) 2.0 ML MgO/Ag(001), U = 30 V, I = 10 nA. Reprinted with permission from [39], S. Schintke et al., Phys. Rev. Lett. 87, 276801 (2001). © 2001, American Physical Society.

with the conventional surface science measurements on thick MgO films [232], implies that the electronic structure of a MgO(001) single-crystal surface develops already within the first three monolayers. First-principles calculations based on DFT yielded the layer-resolved LDOS as a function of the number n of adsorbed MgO layers (0 ≤ n ≤ 3) [39]. The states at  determine the minimum gap width, in agreement with the electronic structure of MgO [226, 231, 233]. In the gap, the average surface LDOS decays exponentially with the number of adsorbed MgO layers. Increasing the MgO film thickness up to three layers produces a surface bandgap corresponding to that of the five-layer pure MgO slab, which is representative for the MgO(001) surface. Field resonance states influence the STM image contrast [234]. Their modified energetic positions when tunneling through the oxide are used to distinguish between the oxide and the metal. For STM images taken at, for example, 5 V (Fig. 19a), which corresponds to the energetic position of the second Ag(001) field resonance state when tunneling through 1 ML MgO [39], the 1 ML MgO islands appear higher than the Ag-Ag steps. At higher coverage, the apparent MgO-MgO step heights (e.g., Fig. 19d) are found to correspond to their geometric heights when tunneling into MgO states. At a bias voltage of 2.5 V, atomic resolution of the MgO layer is obtained. (Fig. 19c). The observed surface lattice constant has twice the value of the one of MgO; thus only

Low-Temperature Scanning Tunneling Microscopy

(a)

683 through a 1 ML MgO island (Fig. 19b). For bias voltages outside the gap, the islands appear always with bright contrast. Thus, within the first three atomic layers of MgO, a bandgap of about 6 eV develops corresponding to the value found for MgO(001) single crystals. These results do not just constrain the minimum usable thickness for layers of insulating MgO (and, by inference, other wide-bandgap materials). They also suggest that, by carefully controlling the number of dielectric monolayers deposited on a metal substrate, the electronic, magnetic, and chemical properties of the resulting surface could be tuned in a controlled manner. This opens new perspectives for the development of oxide heterostructure-based nanodevices [235].

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Ubias (V) Figure 20. Electronic structure of ultrathin MgO/Ag(001) films as determined by STS-dI/dU spectra on defect-free regions (U—sweep interval, tunneling parameters before opening the feedback loop, linear tip-sample distance variation): (a) tip placed above a 1 ML MgO island: dz = −012 nm/V; 3.0 to 0 V, −60 to 0 V, U0 = −60 V, I0 = 02 nA, dU dz U0 = 30 V, I0 = 02 nA, dU = −016 nm/V; (b) film thickness dependz = dent dI/dU spectra: 3.0 V to 1.0 V, U0 = 30 V, I0 = 10 nA, dU −025 nm/V. Reprinted with permission from [39], S. Schintke et al., Phys. Rev. Lett. 87, 276801 (2001). © 2001, American Physical Society.

one type of ion is imaged. According to theory, the observed protrusions are due to accumulated charge density above the Mg atoms. Within the gap (−4V to +25V), MgO reduces exponentially the probability of tunneling into the substrate. The remaining finite density of Ag states within the gap leads to atomic resolution of the Ag(001) substrate imaged

One of the more remarkable states of a superconductor, the Abrikosov flux lattice, is also accessible to STM. New insights into the nature of type-II superconductivity have been achieved by this local probe of the electronic density of states both inside and outside the flux cores [27–29]. The Abrikosov flux lattice was imaged in NbSe2 by tunneling into the superconducting gap edge. The tunneling conductance into a single vortex core was found to be strongly peaked at the Fermi energy, suggesting the existence of core states or core excitations. Moving laterally away from the core, this peak evolves into a density of states which is consistent with a BCS superconducting gap [27]. Subsequent investigations of the same group [28] revealed a zero-bias peak in the superconducting-vortex-core spectra which splits within a coherence length of the core. Further away from the core these split peaks merge gradually with the gap edge and give a direct local measure of the superfluid velocity. Images of the vortex core states showed an anisotropy which was interpreted as a consequence of the crystalline band structure and the interaction with neighboring vortices of the Abrikosov flux lattice. The local density of states of a superconducting vortex core has also been measured as a function of disorder in the alloy system Nb1−x Tax Se2 [29]. Here the peak observed in the zero-bias conductance at a vortex center was found to be very sensitive to disorder. The peak gradually disappeared with substitutional alloying and at x = 02 the density in the vortex center was found to be equal to that in the normal state. It was suggested that the vortex-core spectra provide a sensitive measure of the quasi-particle scattering time.

7.3. Impurities and Superconductors The competition between magnetism and superconductivity manifests itself in the dramatic reduction of the superconducting transition temperature when magnetic impurities are introduced into a superconductor [236]. Macroscopic planar tunnel junctions doped with magnetic impurities have shown sub-gap features in the density of states [237]. The first direct measurement on the structure of a magnetically induced quasi-particle excitation on the atomic length scale around a single magnetic impurity has only been achieved recently with a low-temperature STM [30]. Tunneling spectra obtained near magnetic adsorbates revealed the presence of excitations within the superconductor’s energy gap which were detected over a few atomic

684 diameters around the impurity at the surface. The details of the local tunneling spectra were rationalized within a model calculation based on the Bogoliubov–de Gennes equations [30]. In the copper oxide high-temperature superconductors the superconductivity is believed to originate from strongly interacting electrons in the CuO2 planes. Substitution of a single impurity atom for a copper atom strongly perturbs the surrounding electronic environment and therefore was used to probe high-temperature superconductivity at the atomic scale. In a very interesting recent experiment [31], a lowtemperature STM had been employed to investigate the effects of individual zinc impurity atoms in the high-temperature superconductor Bi2 Sr2 CaCu2 O8+ . Intense quasi-particle scattering resonances [238] were found at the Zn sites, coincident with a strong suppression of superconductivity within ∼15Å of the scattering sites. Imaging of the spatial dependence of the quasi-particle density of states in the vicinity of the impurity atoms revealed the long-sought fourfold symmetric quasi-particle cloud aligned with the nodes of the d-wave superconducting gap which is believed to characterize superconductivity in these materials.

7.4. Nanoscale Magnetism A long-standing challenge in magnetism was to correlate the structural and electronic properties on the local scale with the magnetic properties. Two different concepts have been used to achieve spin-polarized vacuum tunneling. (i) With magnetic STM probe tips the spin-valve effect [239] is exploited, which relies on the fact that the tunneling conductance between two ferromagnetic electrodes separated by an insulating barrier depends on whether the magnetic moments are directed parallel or antiparallel. This effect has been studied before in planar tunnel junctions [240–242] and has been applied in STM to probe the topological antiferromagnetic order of a Cr(001) surface with a CrO2 tip [243]. (ii) Optically pumped GaAs tips enable spin-polarized vacuum tunneling [244], a technique which had been applied to image the magnetic domain structure of thin Co films [245]. (Most recently, electron spin resonance techniques have been combined with STM to detect radio-frequency spin signals from clusters of a few organic molecules [250].) These experiments were limited by the need to separate topographic, electronic, and magnetic information in the case of magnetic probe tips, and to eliminate thermal or film thickness induced effects in the case of semiconducting tips. These difficulties were overcome by tunneling into the surface state of Gd(0001) [40] which is exchange-split into a filled majority and an empty minority spin-contribution [246, 247]. In analogy to the low-temperature experiments performed with ferromagnet-insulator-superconductor planar tunneling junctions [248, 249], where the quasi-particle density of states of superconducting aluminum is split by a magnetic field into spin-up and spin-down part, Bode et al. [40] used two spin-polarized electronic states with opposite polarization to probe the magnetic orientation of the sample relative to the tip. The authors demonstrated spin-polarized tunneling by measuring the asymmetry of the differential tunneling conductance with magnetically (Fe) coated W-tips

Low-Temperature Scanning Tunneling Microscopy

at bias voltages corresponding to the energetic positions of the two spin-contributions of the exchange-split surface state in an external magnetic field. This method enabled the separation of the electronic and magnetic structure information. By mapping the spatial variation of the asymmetry parameter, they were able to observe the nanomagnetic domain structure of Gd(0001) ultrathin films with a spatial resolution below 20 nm [40]. This group achieved atomic resolution in spin-polarized STM at 16 K by imaging a two-dimensional antiferromagnetic structure within a pseudomorphic monolayer film of chemically identical Mn atoms on W(110) [41]. They show, with the aid of first-principles calculations, that the spin-polarized tunneling electrons give rise to an image corresponding to the magnetic superstructure and not to the chemical unit cell. Subsequently, this group observed a magnetic hysteresis on a nanometer scale in an ultrathin ferromagnetic film, consisting of an array of iron nanowires grown on a stepped W(110) substrate. The microscopic sources of hysteresis, domain wall motion, domain creation, and annihilation were observed with nanometer resolution [42]. Recently, Kubetzka et al. [43] performed low-temperature spin-polarized STM of two monolayers Fe on W(110) using tungsten tips coated with different magnetic materials. With Cr as a coating material, they recorded images with an antiferromagnetic tip. The advantage of its vanishing dipole field was most apparent in external fields. This approach appears to resolve the problem of the disturbing influence of a ferromagnetic tip in the investigation of soft magnetic materials and superparamagnetic particles.

8. CONCLUSION In this review STM and STS experiments at the scale of atoms, molecules, supermolecules, clusters, and other nanostructures have been presented which reveal new and exciting physical and chemical phenomena. The common link is the characterization of the structural and electronic properties of supported nanostructures at atomic-scale spatial and meV-scale spectral resolution. The opportunities of this concept in the emerging field of nanotechnology, where size matters, are evident. For example, a reduction in size often provides an increase in speed for electronic or magnetic devices. Any further advancement towards a future nanotechnology will depend crucially on the precise control, characterization, and ultimately, functionalization, of matter at the atomic and molecular level. This prospect remains an inciting scientific and technological challenge. To cite again R. P. Feynman, there is plenty of room at the bottom    .

GLOSSARY Adatom An atom adsorbed at the surface. Catalytic activity The capacity of a substance to take part in a chemical reaction. Chemical imaging The possibility to obtain a chemical (elemental) contrast in STM images. Cluster Aggregate formed by atoms.

Low-Temperature Scanning Tunneling Microscopy

Confined electronic state An electronic state which is confined in one, two or three dimensions. The confinement gives rise to quantum size effects. Fano resonance Interference between two competing electronic transitions leading to the same final state. Field resonance state A two-dimensional unoccupied electronic state which is localized outside the surface. High-temperature superconductor Superconductor with a high critical temperature, typically a ceramic. Impurity An atom (or adatom) of a species different from the one of the medium. Kondo effect Screening of the magnetic moment of an impurity by the conduction electrons of the medium. Lifetime Laps of time during which an excited state exists, before it decays. Photon emission Inelastic process in which photons are emitted during electron tunneling between tip and sample of an STM. Standing waves Stationary electronic waves, due to constructive interference between waves travelling backward and forward. Superconductor Material for which the following properties are verified below the critical temperature: it is perfect diamagnet, it has no measurable DC electrical resistivity, it presents a gap centered around the Fermi level. Supramolecular self-assembly Self-organization of molecule, giving rise to molecular aggregates. Surface diffusion The capacity of an adatom or a molecule to diffuse at surfaces. It depends on both, the adatom and the surface, as well as on the temperature. Surface state Electronic state existing only at the surface, decaying exponentially into bulk and into vacuum. Ultrathin insulator Insulator film grown on a substrate with a thickness of only few atomic layers. Vortices In superconductors, regions of the material that remain in the normal state, through which the lines of the magnetic field pass.

ACKNOWLEDGMENTS It is a pleasure to acknowledge a very fruitful and stimulating collaboration with S. Abbet, A. Baldereschi, R. Berndt, M. Böhringer, O. Bryant, R. Car, S. Crampin, B. Delley, A. De Vita, R. Gaisch, J. K. Gimzewski, U. Heiz, J. Li, L. Libioulle, F. Mauri, S. Messerli, K. Morgenstern, F. Patthey, M. Pivetta, B. Reihl, A. Sanchez, S. Schintke, R. R. Schlittler, F. Silly, M. Stengel, M. Tschudy, and M. Vladimirova. This work was supported by the Swiss National Science Foundation.

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Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Luminescence of Semiconductor Nanoparticles Wei Chen Nomadics, Inc., Stillwater, Oklahoma, USA

Alan G. Joly Pacific Northwest National Laboratory, Richland, Washington, USA

Shaopeng Wang Nomadics, Inc., Stillwater, Oklahoma, USA

CONTENTS

1. INTRODUCTION

1. 2. 3. 4.

1.1. Quantum Size Confinement

Introduction Fluorescence of Semiconductor Nanoparticles Luminescence Dynamics Photoluminescence of Doped Semiconductor Nanoparticles 5. Electroluminescence of Nanoparticle Light Emitting Diodes 6. Thermoluminescence 7. Cathodoluminescence 8. Magnetoluminescence 9. Upconversion Luminescence and Anti-Stokes Luminescence 10. Photostimulated Luminescence and Medical Imaging of Nanoparticles 11. X-Ray Excited Optical Luminescence 12. Biological Labels and Markers 13. Fluorescence Energy Transfer and Biological/Chemical Sensors 14. Luminescence Temperature Dependence and Temperature Sensors 15. Luminescence Pressure Dependence 16. Luminescent Nanoparticles for Optical Storage 17. Summary Glossary References ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

There has been much recent interest in low dimensional systems such as quantum wells, quantum wires, and quantum dots. The interest of this subject stems from two main desires. The first is the desire to understand the transition from molecular to bulk electronic properties, while the other is the prospect of practical application of these materials to optoelectronic devices, photocatalysts, and chemical sensors. Perhaps the most striking property of nanoscale semiconductors is the massive change in optical properties as a function of size due to quantum confinement. This is most readily manifest as a blueshift in the absorption spectra with the decrease of the particle size. The variation of the energy gap in semiconductors with size may also result in different emission wavelengths for different sizes of nanoparticles. One example is shown in Figure 1 [1] for ZnS-capped CdSe nanoparticles with different sizes which display a fluorescence rainbow of blue–green–orange–yellow–red with the emission maxima at 443, 473, 481, 500, 518, 543, 565, 587, 610, and 655 nm, respectively. The exciton Bohr radius is a useful parameter in quantifying the quantum confinement effects in semiconductor physics. The Bohr radius (aB ) of an exciton in semiconductors may be calculated by [2]

2 1 1 aB = 2 (1) + e me mh where  is the dielectric constant,
is the Planck constant, and me and mh are the electron and hole effective mass respectively. As the particle size is reduced to approach

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (689–718)

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690

Figure 1. Ten different emission colors of different size ZnS-coated CdSe nanoparticles excited with a near-UV lamp. From left to right (blue to red), the emission maxima are 443, 473, 481, 500, 518, 543, 565, 587, 610, and 655 nm, respectively. Reprinted with permission from [1], M. Y. Han et al., Nature Biotechnol. 19, 631 (2001). © 2001, Macmillan Magazines Ltd.

the exciton Bohr radius, there are drastic changes in the electronic structure and physical properties. These changes include shifts of the energy levels to higher energy, the development of discrete features in the spectra, and the concentration of the oscillator strength into just a few transitions (Fig. 2) [2]. A straightforward theoretical model based on the effective-mass approximation [2] can be applied to explain most observations. This model has established two limiting regimes: the weak and the strong confinement regimes. The weak confinement regime occurs when the particle radius is larger than the exciton radius. In this regime, the exciton translational motion is confined and the size dependence of the energy of the exciton can be expressed as Enl = Eg −

R∗y n2

+

2
2 nl

(2)

2MR2

energy

conduction band LUMO bandgap HOMO valence band

molecules

nanoparticles

bulk solids

Figure 2. A schematic model for the energy structures of bulk solids, nanoparticles, and isolated molecules.

where Eg is the bulk bandgap, R∗y is the exciton Rydberg energy,
is the Planck constant, R is the particle size, M = me + mh , and me and mh are the effective mass of the electron and hole respectively. Xnl are the roots of Bessel functions describing the energy states with n being the number of the root and l being the order of the function (1S = 314, 1P = 449, and 1D = 576) [2]. The strong confinement regime occurs when the particle radius is smaller than the exciton radius. In this regime, the individual motions of electrons and holes are independently quantized. In this case, the size dependence of the exciton energy can be expressed by Enl = Eg +


2

2 2R2 nl

(3)

where  is the reduced mass of the electron and hole pair, 1/ = 1/me + 1/mh . In both regimes the main experimental effects of confinement are the appearance of a structured absorption spectrum due to the presence of discrete energy levels and the blueshift of the absorption edge, which is roughly proportional to the inverse of the square of the particle radius. However, some electronic properties such as electron–hole interactions are expected to be modified only in the strong confinement regime. This is due to the increase of the spatial overlap of the electron and hole wavefunctions with decreasing size. As a consequence, the splitting between the radiative and nonradiative exciton states is enhanced largely in the strong confinement regime. Quantum confinement not only causes an increase of the energy gap (blueshift of the absorption edge) and the splitting of the electronic states but also changes the density of states (DOS). Many novel physical properties and potential applications of low-dimensional semiconductors and many of the differences between the electronic behavior of the bulk and of quantum-confined low-dimensional semiconductors are due to their differences in the density of states [3]. Figure 3 shows the variation of the DOS with dimensionality [3]. The dimensionality of the system describes the number of dimensions of free transport of an electron gas; thus boxes are two-dimensional structures, quantum wires are one-dimensional, and quantum dots are considered zerodimensional because electrons are confined in all spatial dimensions. Passing from three dimensions to two dimensions the density of states N E changes from a continuous dependence, where N E ∼ E 1/2 , to a steplike dependence. Thus the optical absorption features are different for the bulk and for the quantum well structure. The optical absorption edge for a quantum well is at higher photon energy than for the bulk semiconductor and, above the absorption edge, the spectrum is stepped rather than smooth, the steps corresponding to allowed transitions between valence-band states and conduction-band states. In addition, at each step sharp peaks appear corresponding to confined electron–hole (exciton) pair states. In the case of lower dimensional systems (quantum dots, nanocrystallites, clusters, nanoparticles, colloids, etc.), the DOS becomes more discrete as the dimensionality decreases, and large optical absorption coefficients are observed [4]. The low-dimensional structure has proven to be very promising in applications to semiconductor lasers, due mainly to the quantum confinement of the

Luminescence of Semiconductor Nanoparticles

691 recombination rate, luminescence efficiency, and the radiative lifetime in materials. The oscillator strength of the free exciton is given by [7]:

3D Bulk Semiconductor

1D Quantum Wire

0D Quantum Dot E Figure 3. Profiles of the density of states of three-dimensional bulk semiconductors, a two-dimensional quantum well, a one-dimensional quantum wire, and zero-dimensional quantum dots. Reprinted with permission from [3], A. P. Alivisatos, J. Phys. Chem. 100, 13226 (1996). ©1996, American Chemical Society.

carriers and the variation of the density of states with dimensionality [5]. The changes in the DOS lead to a change in the gain profile, a reduction of threshold current density, and a reduction of the temperature dependence of the threshold current. Owing to the steplike density of states, high gain with a lower spontaneous emission rate has been realized in a GaAs/AlGaAs GRIN-SCH SQW laser [6]. Thus lowdimensional structured materials are interesting, for both basic research and practical applications.

where m is the electron mass, E is the transition energy,  is the transition dipole moment, and U 02 represents the probability of finding the electron and hole at the same site (the overlap factor). In nanostructured materials, the electron–hole overlap factor increases largely due to quantum size confinement, thus yielding an increase in the oscillator strength. The oscillator strength is also related to the electron–hole exchange interaction that plays a key role in determining the exciton recombination rate. In bulk semiconductors, due to the extreme dislocation of the electron or hole, the electron–hole exchange interaction term is very small, while in molecular-size nanoparticles, due to confinement, the exchange term should be very large. Therefore, there may be a large enhancement of the oscillator strength from bulk to nanostructured materials. The radiative decay lifetime () is closely related to the oscillator strength of a transition by [8]   (5)  = 45 2A /nf

where n is the refractive index and A is the emission wavelength. Thus, the lifetime may be shortened with decreasing size due to the increase of the oscillator strength, f . Higher efficiencies with concomitant shorter decay times thereby make nanoparticles a promising new type of luminescent material.

2. FLUORESCENCE OF SEMICONDUCTOR NANOPARTICLES Typical absorption and fluorescence spectra of CdS nanoparticles are shown in Figure 4 [9]. The sizes of the two samples estimated from the absorption edge are 3.4 nm

II

1.2. Quantum Size Confinement and Quantum Efficiency of Nanoparticles From these arguments, it is clear that nanoparticles may have tunable absorption and emission spectra. In addition, luminescent nanoparticles may have higher quantum efficiency than conventional phosphors, making it possible to design and fabricate more sensitive sensors or more efficient devices. The oscillator strength, f , is an important optical parameter that underlies the absorption cross-section,

(4)

sample II

rel. fluorescence intensity

N(E)

2m E2 U 02


absorbance

2D Quantum Well

fex =

sample I 400

600

800

1000

wavelength [nm] Figure 4. Absorption and emission spectra of CdS nanoparticles. Sample I is 3.4 nm in size and sample II is 4.3 nm. Reprinted with permission from [9], A. Hasselbarth et al., Phys. Lett. 203, 271 (1993). © 1993, Elsevier Science.

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692

12

Relative intensity (a.u.)

1.24nm

9 1.65nm

6 2.28nm

3

0 300

350

400

450

500

550

600

650

Wavelength (nm)

Figure 5. Luminescence spectra of ZnS nanoparticles with average sizes of 1.24, 1.65, and 2.28 nm, respectively. The luminescence exhibits almost exclusively trap state emission, likely because of poor surface passivation. Reprinted with permission from [12], W. Chen et al., J. Appl. Phys. 82, 3111 (1997). © 1997, American Institute of Physics.

particles are not well passivated. The luminescence intensity increases as the size decreases, as the increase in the trapped states results in more luminescence. The trap state emission also shifts to the blue as the size decreases demonstrating that the trap states exhibit quantum confinement behavior. Similarly, the results reported by Hoheisel et al. [13] show that both the excitonic and the trap state emission peaks shift in energy as a function of size in CdSe nanoparticles (Fig. 6), demonstrating that the emission energy of the surface states in these particles is also correlated to the quantum-size effects.

3. LUMINESCENCE DYNAMICS Absorption of a photon places an atom or molecule in a naturally meta-stable excited state. This meta-stable state must ultimately re-emit the energy in one or more of three possible relaxation pathways. The first is radiative, via either fluorescence or phosphorescence. The second pathway is through nonradiative processes such as internal conversion

Radius: 21 Å 16 Å

Intensity (a.u)

(sample I) and 4.3 nm (sample II) respectively. Both of the samples exhibit a sharp excitonic fluorescence band (at 435 nm for the 3.4 nm sample and 480 nm for the 4.3 nm sample), as well as a broad fluorescence band at longer wavelengths arising from the recombination of trapped charge carriers at surface or defect states. Optical excitation of semiconductor nanoparticles often leads to both band-edge and deep trap luminescence. The size dependence of the excitonic or band edge fluorescence has been studied extensively and can be reasonably explained by the effective-mass approximation. The fluorescence process in semiconductor nanoparticles is very complex and most nanoparticles exhibit broad and Stokes-shifted luminescence arising from the deep traps of surface states [10]. Only clusters with good surface passivation show high band-edge emission. The absence of band-edge emission has been previously attributed to the large nonradiative decay rate of the free electrons trapped in these deep-trapped states. As the particles become smaller, the surface/volume ratio and hence the number of surface states increases rapidly, reducing the excitonic emission [11]. Thus, surface states often determine the physical properties, especially the optical properties, of nanoparticles. Often, however, little is known about the physical properties of the surface states. For particles in such a small size regime, a large percentage of the atoms are on or near the surface. Surface states near the bandgap can mix with interior levels to a substantial degree, and these effects may also influence the spacing of the energy levels. Thus, in many cases it is the surface of the particles rather than the particle size that determines the properties. It is, therefore, a major goal to characterize the surface states and to control them through chemical modification. For example, 99% of the atoms are on the surface for a 1 nm sized Si particle [3]. The existence of this vast interface between the nanoparticles and the surrounding medium can have a profound effect on the particle properties. The imperfect surface of the nanoparticles may act as electron and/or hole traps following optical excitation. Thus the presence of trapped electrons and holes can in turn modify the optical properties of the particles. Although there are many reports on the luminescence of nanoparticles, only a small number are dedicated to the size dependence of the fluorescence from surface states. By examining the size dependence of the two main emission features (i.e., the excitonic and the trapped emissions), it is possible to determine to what extent the carriers are confined in the emitting state. If the luminescence and the absorption of the surface states are dependent on size, it indicates that not only the excitons but also the trapped carriers at the surface states are confined by the quantum size effect. If the luminescence of the surface states is not dependent on size, then the trapped carriers are not likely confined by the quantum size effect. Thus knowledge of the surface and defect states response to quantum confinement may be important when attempting to adjust the energy levels of the surface states relative to the intrinsic levels through quantum confinement. The emission spectra of ZnS nanoparticles of different sizes are shown in Figure 5 [12]. No excitonic emission is observed in the ZnS nanoparticles. The luminescence clearly arises from trap states, indicating that the surfaces of the

13 Å

11 Å

Fluorescence

Absorbance

9Å 1.5

2

2.5

3

Photon Energy (eV)

Figure 6. Absorption and emission spectra of CdSe nanoparticles with average sizes of 9, 11, 13, 16, and 21 Å. Quantum confinement results in both the excitonic and trap state emissions clear blueshift with decreasing particle size. Reprinted with permission from [13], V. Hoheisel et al., J. Chem. Phys. 101, 8455 (1994). © 1994, American Institute of Physics.

Luminescence of Semiconductor Nanoparticles

693

(6)

where krad and knon are the radiative and nonradiative rates respectively. The observed luminescence therefore decays exponentially with time, It = I0 exp−t/obs 

(7)

with the observed lifetime obs proportional to 1/kt . Thus, in a purely radiative system such as an isolated atom, the observed lifetime is proportional to the inverse of the radiative rate. In this case, the lifetime is termed the natural radiative lifetime. The natural lifetime may be related to the absorption strength or oscillator strength f by Eq. (5). Thus, a strong transition has a large radiative rate and therefore a short natural lifetime. Most photochemical systems are not purely radiative; therefore the influence of nonradiative transitions must be taken into account. Nonradiative transitions in general serve to deplete the excited state, thus increasing kt . This results in a shortened observed lifetime tobs , which is shorter than the natural radiative lifetime. In some cases, nonradiative energy transfer from a nearby excited molecule or atom results in promotion of a ground state atom or molecule to its excited state. If the energy transfer is fast relative to the spontaneous luminescence rate, the luminescence does not display a single exponential decay but exhibits a rise followed by a decay as the population of the excited level initially increases due to the energy transfer. The rise time of the luminescence may then be related to the energy transfer rate, while the fall time can be related to kt in a straightforward manner. It is important to realize that in the absence of this energy transfer, all excited state luminescence from a single species should display a single exponential decay. In practice, often

Intensity (arb. units)

kt = krad + knon

biexponential or higher order lifetime decays are observed. The existence of a biexponential decay does not, however, imply two different rate processes for a single species but rather the existence of two distinct subsets of species, each with its own unique lifetime. Higher order, multiexponential decays then represent larger numbers of distinct species and often can be modeled as a distribution of species, each with its own decay constant. It is clear that the longer the natural lifetime, the higher the probability for nonradiative processes to deplete the excited state. Nonradiative processes also result in a shortened lifetime; however, the luminescence efficiency similarly decreases. Thus, the goal in luminescent systems is to increase the luminescence efficiency either through increasing the radiative rate (decreasing the natural lifetime) or decreasing the nonradiative rate (increasing the observed lifetime). Nanoparticles may have an increased natural radiative rate and thus enhanced quantum efficiency due to the effects of quantum confinement on the electron–phonon coupling and DOS. However, in nanoparticles, perhaps the single most important contribution to the nonradiative rate is trapping to surface states. Trapping to surface states usually occurs on the subnanosecond timescale [15]. Figure 7 shows luminescence decays from Eu2 O3 nanoparticles encapsulated in the porous host MCM41 [16]. The existence of multiple time regimes is apparent as subnanosecond to millisecond timescale decays are represented. The subnanosecond decays shown in the inset of Figure 7 have been attributed to either species exhibiting surface state trapping of carriers or else to energy transfer to defect states within the MCM-41. Passivation of surface states is therefore a key requirement in increasing the luminescence efficiencies of nanoparticles relative to bulk materials [16].

Intensity (arb. units)

or energy transfer. The third possibility is chemical change, such as the breaking of bonds to form a new chemical species. In a typical experiment, a photon is absorbed taking the atom or molecule from the ground state to an excited state, where the excited state may be a rotationally and/or vibrationally excited molecule in the electronic excited state. Internal conversion is the process where ro-vibrational energy is lost to the surroundings as the molecule cascades down the ro-vibrational ladder. In solids, promotion of an electron from the valence band to the conduction band produces an excited state which may then relax by phonon relaxation to the conduction band edge. This energy is ultimately lost as heat to the surroundings. Internal conversion in excited states is very rapid relative to radiative rates and therefore the system may be assumed to radiate from the lowest energy level of the lowest excited state of like multiplicity. This is known as “Kasha’s rule” and is obeyed a majority of the time [14]. Once in the lowest excited state, there are many possible forms of relaxation, both radiative and nonradiative. The total rate of relaxation is the sum of all rates. In this chapter we will assume that there is no chemical change and that all luminescence originates from the lowest excited state (Kasha’s rule) [14]. Therefore the total relaxation rate (kt ) out of the lowest excited state is given by

Eu2O3 Bulk

Eu/MCM 600 °C

Eu2O3 Bulk

0

200

400

Time (nsec)

Eu/MCM 140 °C

Eu/MCM 600 °C

Eu/MCM 700 °C

Eu/MCM 900 °C 0

1

2

3

4

5

Time (msec)

Figure 7. Lifetime decay curves of the 5 D0 → 7 F2 emission following excitation at 525 nm of Eu2 O3 bulk powder, an Eu2 O3 /MCM-41 mixture, and Eu2 O3 particles in MCM-41 prepared by heat treatment at 140, 600, 700, and 900  C. The inset shows an expanded view of the short-time (8 eV) [62] and the energy levels of the dopant are contained within the energy gap of the host. However, for semiconductor phosphors such as ZnS:Tb3+ or CdS:Eu3+ , the host energy structure must be considered because the host energy gap is less than 5 eV and the dopant energy levels intercept the host energy levels. Thus the energy relaxation from an excited state to a lower state of the dopant is possible via energy levels of the semiconductor host. This is particularly true for nanoparticles in which the energy levels are split. It is well known that the energy splitting between two levels increases with decreasing size [2]. According to Eq. (13), p increases with decreasing size. Therefore multiphonon relaxation from higher levels to lower levels may decrease in nanoparticles. This is actually the famous “photon bottleneck” [63]. According to the energy gap law, this is quite favorable for improving the upconversion efficiency. Summarizing these arguments, in nanoparticles, due to size confinement, the increase of electron–hole wavefunction overlap may enhance the absorbance and luminescence efficiency as well as shorten the emitting state lifetime. Due to the decrease of both the phonon and electric state densities, the electron–phonon interaction is weaker. These are strong reasons to consider doped nanoparticles as a new type of upconversion material with good efficiency and faster response, whether the upconversion is due to energy transfer or two-photon absorption. For these reasons, upconversion luminescence of nanoparticles is becoming a topic under much investigation. The study of nanoparticle upconversion luminescence is relatively new and not much work has been reported; yet strong upconversion luminescence of Mn2+ in ZnS:Mn2+ nanoparticles has been observed [64]. The upconversion emission band excited at 767 nm is redshifted from the photoluminescence emission excited at 300 nm (Fig. 16) [64]. In the nanoparticles, the decay lifetimes of the upconversion emission excited at 767 nm are shorter than the 300 nm excited luminescence lifetimes, while in bulk the two decays are almost identical. When the photoluminescence is obtained by excitation at 383.5 nm, which is the sum energy of two photons at 767 nm, the emission spectra and the lifetimes of the two types of luminescence are almost identical [64]. The power dependence of the photoluminescence is linear, while that of the upconversion emission is quadratic [64]. Based on these observations, two-photon excitation has been identified as the excitation mechanism responsible for the upconversion luminescence of Mn2+ in ZnS:Mn2+ nanoparticles.

Luminescence of Semiconductor Nanoparticles

Absorbance (a.u.)

701

10 nm

a II I

400

450

3.5 nm

500

550

I

Normalized Emission Intensity (a.u.)

Emission Intensity (a.u.)

4.5 nm

III

II

II

I

600

650

700

b

III

c

III

3 nm 500

550

600

650

700

750

Wavelength(nm)

ZnS:Mn/USY

Figure 17. The absorption spectra (a), the photoluminescence emission spectra (b, excited at 350 nm), and the anti-Stokes luminescence spectra (c, excited at 750 nm) of CdTe nanoparticles with average sizes of 3 (I), 5 (II), and 6 nm (III), respectively [67]. bulk 550

570

590

610

630

650

Wavelength (nm) Figure 16. The photoluminescence spectra after excitation at 300 nm (solid), at 383.5 nm (dash), and the upconversion luminescence spectra resulting from 767 nm excitation (dot) of ZnS:Mn2+ bulk and nanoparticles. Reprinted with permission from [64], W. Chen et al., Phys. Rev. B 64, 041202(R) (2001). © 2001, American Physical Society.

Upconversion luminescence is visible to the naked eye in these systems. This reveals that upconversion luminescence of ZnS:Mn2+ can be realized at relatively low power densities. Our recent tests demonstrate that an 808-nm semiconductor quantum well laser module with 40 microwatts (power density of 1 mW/cm2 ) can stimulate upconversion luminescence of ZnS:Mn2+ nanoparticles visible to the unaided eye. Similar upconversion luminescence is observed in ZnS:Mn2+ , Eu3+ nanoparticles. Emissions of both Mn2+ and Eu3+ are observed in the upconversion emission spectra and the upconversion is due to two-photon absorption of Mn2+ and the emission of Eu3+ is due to energy transfer from Mn2+ [65]. Upconversion luminescence has also been reported in undoped semiconductor nanoparticles such as CdS and CdTe nanoparticles [66]. The term anti-Stokes luminescence is frequently used to describe the upconversion luminescence in undoped semiconductor nanoparticles. In general, any luminescence that occurs at frequencies larger than that of the excitation frequency is called anti-Stokes luminescence. Therefore, in this case, upconversion is just one variety of anti-Stokes luminescence. Figure 17a displays the optical absorption spectra of three CdTe nanoparticle samples [67]. The absorption edges of these nanoparticles are blueshifted from the 827 nm bulk CdTe bandgap as

a consequence of quantum size confinement. According to the shift and based on the effective-mass approximation [2], the estimated particle sizes are around 3, 5, and 6 nm, respectively. Their photoluminescence spectra are shown in Figure 17b. The nanoparticles show a pronounced excitonic luminescence band, which shifts to higher energies for smaller sizes. Strong anti-Stokes luminescence is observed from these nanoparticles as shown in Figure 17c with excitation at 750 nm. As in the photoluminescence, the anti-Stokes luminescence is also size dependent, shifting to higher energies for smaller nanoparticles. These results may be reasonably explained by a two-step absorption process via surface states [24, 67], which is actually upconversion luminescence through a two-step absorption.

10. PHOTOSTIMULATED LUMINESCENCE AND MEDICAL IMAGING OF NANOPARTICLES Photostimulated luminescence (PSL) describes the phenomenon in which phosphors release trapped charge carriers when stimulated by infrared or visible light [68]. The trapped carriers recombine with luminescent centers to generate the PSL. Thus, in photostimulated luminescence the stimulation (excitation) wavelength is longer than the emission wavelength in contrast to photoluminescence. Because photostimulable phosphors have the ability to store energy, they are also called storage phosphors. X-ray storage phosphors such as BaFBr:Eu2+ have been widely discussed [68]. The PSL mechanism of BaFBr:Eu2+ involving X-ray irradiation proposed by Takahashi et al. [69] assumes that during X-ray irradiation, the Eu2+ ions are partly ionized into their trivalent charge state (Eu3+ ) and the liberated electrons drift via the conduction band to form F centers. Upon subsequent photostimulation, the electrons are released from the

Luminescence of Semiconductor Nanoparticles

702 F centers into the conduction band and they recombine with Eu3+ ions to produce the photostimulated luminescence of Eu2+ at 390 nm (Fig. 18) [69]. Other recent results indicate that tunneling may be the mechanism of recombination in BaFBr:Eu2+ phosphors [70]. In nanoparticles, the energy scheme can be modified via quantum size confinement. This offers a new way of designing PSL phosphors. When electrons and holes are produced in nanoparticles by photoexcitation, the electrons and holes may de-excite or relax to the lowest excited states and recombine to give luminescence. They also may be trapped by electron or hole traps at the surfaces, interfaces, and/or in a surrounding host. The electrons or holes at traps are in a metastable state. When stimulated by light or by heat, some may be released and recombine to give luminescence (i.e., photostimulated luminescence or thermoluminescence). Controlled charge separation and trapping in nanoparticles are key to the processes of erasable optical storage, sensors, and digital imaging. The PSL of nanoparticles requires not only charge separation and trapping but also the return of the carriers to the nanoparticles. Evidence for charge separation, trapping, and subsequent return has been obtained from pump–probe measurements and photon-gated hole burning (PHB) of nanoparticles [71]. For example, in BaFCl:Sm2+ , the hole-burning process can be described as the photoionization reaction Sm2+ + (trap) → Sm3+ + (trap)−

(14)

If the electrons release from the traps and return to the Sm2+ , the hole will be erased, a process termed hole filling [72]. The hole burning corresponds to the photoionization process in PSL and the hole filling is similar to the photostimulation. Thus, the occurrence of PHB in a system is an indication of PSL, and vice versa. In most systems, hole burning is only observable and stable at low temperatures. Only in a few materials is hole burning at room temperature possible. However, PSL is not as dependent on temperature. For example, in a

BaFClx Br1−x :Sm2+ system, low temperature is necessary to observe PHB, while strong PSL is observed at room temperature [72]. The hole-burning efficiency or the storage density increases by a factor of 5I /5H , where 5I is the inhomogeneous linewidth and 5H is the homogeneous linewidth. This ratio is highly temperature dependent. Phonon-broadening causes 5H to increase with temperature and makes hole burning difficult at high temperature. Phonon broadening has little effect in the PSL process; thus it is unnecessary to work at low temperature. This is one of the advantages of PSL and forms the basic premise for the construction of a doped nanoparticle that exhibits PSL for use in storage devices, sensors, and optical image display systems. Two new and intriguing upconversion materials are Ag and AgI nanoclusters encapsulated in zeolite-Y (Ag/Y and AgI/Y). AgI nanoclusters encapsulated in zeolite-Y show strong luminescence and photostimulated luminescence at room temperature. Figure 19a shows the PL spectra of AgI/Y following excitation at 305 nm [73]. The broad emission band actually consists of two subbands. The first band results from AgI nanoparticle emission peaking at 474 nm and appears as a small but discernable shoulder on the blue energy edge of the broad emission peak in Figure 19a. The second band results from Ag nanocluster emission centered at 510 nm. Ag is known to coexist with AgI nanoparticles and efficient energy transfer is known to occur among them. The luminescence decreases in intensity when the sample is irradiated by ultraviolet light at 254 nm (Figure 19b). This decrease can be partially recovered by exposing the sample to visible light (Figure 19c). After ultraviolet (UV) irradiation for a few minutes, strong PSL can be detected from AgI nanoparticles in zeolite-Y as shown in Figure 20 [73]. PSL excitation is easily stimulated using near-infrared wavelengths. Figure 20 shows that the emission consists almost exclusively of Ag nanoclusters resulting in a narrower band than the PL emission band [48]. Similarly, Ag nanoclusters in zeolite-Y (Ag/Y) are photosensitive and exhibit strong photostimulated luminescence. Similar to AgI/Y, Ag/Y particles show a marked decrease in PL after UV irradiation. This decrease

Conduction band

Excitation at 305 nm

F+

F

PSL

6.5eV

8.3eV 4.6eV 3.2eV

Eu2+

Emission Intensity (a.u.)

a 2eV

c b

Eu3+

Valence band Excitation

300

400

500

600

700

Wavelength (nm)

PSL

Figure 18. A schematic model for excitation and photostimulated luminescence processes in a BaFBr:Eu2+ phosphor. Reprinted with permission from [69], K. Takahashi et al., J. Lumin 31–32, 266 (1984). © 1984, Elsevier Science.

Figure 19. Photoluminescence spectra of AgI/Y before (a) and after (b) UV irradiation at 254 nm for 5 minutes. After exposure to a visible lamp for 5 minutes (c). Reprinted with permission from [73], W. Chen et al., Phys. Rev. B 65, 24554041 (2002). © 2002, American Physical Society.

Luminescence of Semiconductor Nanoparticles

703

PSL Intensity (a.u.)

Excitation at 840 nm

400

450

500

550

600

Wavelength (nm) Figure 20. Photostimulated luminescence spectrum of AgI/Y after UV irradiation at 254 nm for 10 minutes. Excited at 840 nm. Reprinted with permission from [73], W. Chen et al., Phys. Rev. B 65, 24554041 (2002). © 2002, American Physical Society.

is almost completely reversible following irradiation by photons between 650 and 900 nm. Thus both materials may be considered for use as reusable image or digital storage media. Figure 21a displays the PL lifetime decays at 500 nm from AgI/Y following excitation at 305 nm [73]. The inset shows the existence of a short ( -1 . A finite magnetic field dramatically alters this situation, replacing the continuum Eq. (44) by pointlike singularities [Eq. (42)] separated by gaps. Thus, depending on the respective locations of the characteristic electron energy 5 (the Fermi energy for a degenerate electron gas at T = 0 K) and %mnz , the electron gas will behave either like an insulator (5 = %mnz  or a metal (5 = %mnz . Notice that similar effects cannot be found in bulk materials: the carrier-free motion along the field prevents the density of states from vanishing at any energy larger than
c /2 − gc∗
B B/2. Of course, we know that imperfections (impurities, interface defects, etc.) will alter the deltalike singularities of 2m %, rounding off the peaks, adding states into the gaps. However, it remains generally accepted that, if B is large enough (i.e., B ≥ few Teslas in good GaAs–Ga1−x Alx As heterostructures at low temperature), the previous conclusions retain their validity, at least for the following meaning: for energies belonging to a certain bandwidth of finite (yet undefined) extension, the density of states is large and corresponds to conducting states, whereas for other energy segments, the states are localized (i.e., at T = 0 K are unable to contribute to the electrical conduction at vanishingly small electric fields). Some words have to be added concerning the broadening effects. The present state of the art is far from being satisfactory. First, there is, to the authors’ knowledge, no general consensus on the microscopic origin of these effects.

Most likely it is sample dependent: in some samples, charged impurities play a dominant part. In others, interface roughness or alloy scattering (if it exists) may dominate. Second, even if we assume that the disorder potential has a simple algebraic form (e.g., uncorrelated deltalike scatterers), it proves difficult, if not impossible, to obtain precise information on 2m % by any method other than by numerical simulation. The complications are twofold: (a) The unperturbed density of states is highly singular. This precludes the use of non-self-consistent perturbative treatments of the disorder potential. (b) When the required self-consistency is included, calculations are performed up to a finite order in the perturbation approach. By doing so, the nature of the states which occur in the tails of the broadened delta function are ill-treated: a localized level requires an inclusion of the electron disorder potential to an infinite order.

2.2. Magnetic Effect in Quasi-One-Dimensional Systems The energy spectrum of a particle confined within a circle in the presence of an external magnetic field perpendicular to the plane of confinement is studied both exactly and approximately by the quasi-classical formalism (WKB). For pure spatial confinement (without magnetic field) the energy spectrum for states other than the ground state is twofold degenerate, while in the case of pure magnetic confinement the spectrum shows infinite-fold degeneration, typical of Landau states. For both types of confinement, the latter infinite-fold degeneration is lifted due to spatial confinement. Interestingly enough, for a given ratio between spatial and magnetic confining lengths, the magnetic flux is quantized. The conditions for the quantization of the flux are established; the nature and peculiarity of the energy spectrum are also discussed. Basically, the physical properties are implicitly contained in the wavefunction; it is clear that this function is modified whenever the system is restricted on its available space either by its size (spatial confinement) or by an external perturbation (field confinement). In this context, the first attempt at the study of these kind of systems will be the solution of the Schrödinger equation in which the corresponding Hamiltonian addresses in one way or another some of the characteristics of the spatial restrictions [83]. If the boundary which encloses the system is impenetrable, its energy increases as the confinement region decreases in agreement with the uncertainty principle; that is, more localization in space means an increase in the moment and consequently in the energy. If the latter boundary is penetrable then there is a probability of the system to escape and thus if the surroundings possess available states the system can occupy those which will be consistent with its energy and selection rules. One of the simplest systems of this kind is a single particle confined by impenetrable boundaries and perturbed by an external magnetic field. This system was first studied by Rensink [52] to show how the energy spectrum transforms

Magnetic Field Effects in Nanostructures

797

2.2.1. Spatial Confinement Let us consider the two-dimensional confinement of a particle within a circle of radius r0 and kz = 0. The Hamiltonian in this case can be written as

0

(46)

f0 r0 ' = 0


2 #2n m

1 = √ An m J m exp im' 2(

(50)

and  ∗ 1 f0n −m = √ An m J m exp−im' = f0n m 2(

(51)

Due to the properties of the zeroes of J m , the energy levels satisfy 1 m +1

< %0

2 m

< %0

2 m +1

< %0

300

n m

Figure 4. Energy %0 as a function of confinement radius r0 . From bottom to top, the states with n m  = (1,0), (1,1), (1,2), and (1,3) are displayed. Reprinted with permission from [186], J. L. Marín et al., “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

of its momentum and consequently a greater energy). (ii) As the confinement radius is increased, the energy level separation decreases and they coalesce smoothly into a single one, the free particle spectrum. The latter is consistent with the results of Rensink [52], as expected.

2.2.2. Magnetic Confinement Let us now consider the case of a charged particle moving in the plane x y in the presence of a homogeneous magnetic field B
= B0 eˆz , without spatial confinement [i.e., Vb r = 0]. The Hamiltonian for this system can be written as

(49)

2m∗ r02

Here, #n m denotes the nth zero of J m . Note that for m = 0, the energy levels are twofold degenerate with eigenfunctions

1 m

200

(48)

due to the form of the confining potential Vb . The solutions of Eq. (47) have already been studied previously; thus, considering the condition for the wavefunction given by Eq. (48) the energy spectrum is obtained as

%0

100

a (Å)

where %0 is the energy of the particle and f0 r ' is its envelope wavefunction that must satisfy

f0n m

200

0

In polar coordinates r ', the corresponding Schrödinger
0 is given by equation for H     1  2m∗  1 2 r + 2 2 + 2 /%0 − Vb r0 f0 r ' = 0 r r r r '
(47)

=

300

(45)

where the confining potential is given by  0 0 ≤ r ≤ r0 Vb r =  r > r0

n m

α1,0 α2,0 α3,0 α1,1

400

100

2
0 = pˆ + Vb r H 2m∗

%0

500

Energy (meV)

smoothly to that of a free particle spectrum as the magnetic field strength goes to zero. In this context we shall study a similar system but we further analyze the situations raised whenever the typical length introduced by both types of confinement are far apart or nearly equal.

3 m

< %0

···

(52)

The dependence of the lowest energy levels on the confinement radius r0 is displayed in Figure 4. Two main features emerge from this figure: (i) as the confinement radius decreases the energy of each level increases. This fact is consistent with the uncertainty principle (i.e., more localization of the particle implies an increase

where


2
= P H 2m∗

 2 2
2 = pˆ − q A
= pˆ 2 − q pˆ · A
· pˆ + q A
2
− qA P 2 c c c c

(53)

(54)

Now, for any state vector 9

 



·  9 pˆ · A9 = −i
 · A9 = −i
 · A9 +A  
+A
· pˆ 9 = −i
 · A (55)


= 0 is chosen, pˆ · A
=A
· p; ˆ Then, when the gauge  · A

that is, A and pˆ commute and H can be written as 2 2
= pˆ − q A
· pˆ + q A
2 H ∗ ∗ ∗ 2m mc 2m c 2

(56)


= 0, A
is of As a consequence of the symmetric gauge  · A the form  
= 1 B
× r
= 1 B0 −eˆx y + eˆy x = 1 B0 r eˆ' A 2 2 2

(57)

Magnetic Field Effects in Nanostructures

798


where
Lz is the z-component of the angular momentum L.
is If we define the cyclotron frequency c = −qB0 /m∗ c, H now given by 2

1
= −
 2 + 1 c
Lz + m∗ 2c r 2 H 2m∗ 2 8

50 ε0,0 ε0,1, ε1,0 ε0,2, ε2,0 ε0,3, ε3,0

40

Energy (meV)

This choice leads to     1 1  1
−y = B0
Lz = − i
B0 A · pˆ = − i
B0 x 2 y x 2 ' 2 (58)

30

20

(59) 10

This Hamiltonian can be envisioned as that of a two
with oscildimensional oscillator (first and third terms in H lating frequency  = c /2, perturbed by the “potential” c
Lz /2 = 
Lz . That is,

0 0

10

20

30

40

50

B0 (Teslas)


′
=H
osc + H H

(60)


osc flm r ' = %osc flm r ' H

(61)

2.2.3. Spatial and Magnetic Confinement


′ flm r ' = %′ flm r ' =
mflm r ' H

(62)

In this context we shall study the case of a charged particle with both types of confinement considered in the previous cases. In this context, the Hamiltonian can be written as


osc = −
2 /2m∗  2 + 1/2m∗ 2 r 2 and H
′ = 
with H Lz .
H
′ 0 = Moreover, it can be easily shown that /H


osc H
′ 0, a property that allows one to establish that H
, /H ′

Hosc , and H possess common eigenfunctions. In view of the latter, if flm r ' is the complete set of eigenfunctions of
, then H and

Here %osc is the energy of a two-dimensional isotropic harmonic oscillator, which can be found by separating the corresponding Schrödinger equation in polar coordinates, to give %osc =
2l + m + 1

l = 0 1 2   

(64)

F −1nm m + 1 #a2  = 0

and the wavefunctions   Nlm m 1 2 flm r ' = √ r exp − #r 2 2(

× F −l m + 1 #r 2  expim'

(66)

As can be seen, the inclusion of the term Vb r requires that the wavefunction satisfy the boundary condition given by Eq. (48) and this fact prevents l [Eq. (63)] from being a good quantum number as in the previous section. Instead, the boundary condition is now satisfied if [see Eq. (65)]

%lm = %osc + %′ =
2l + m + m + 1 1
 2l + m + m + 1 2 c

2 1
bc = −
 2 + 1 c
Lz + m∗ 2c r 2 + Vb r H 2m∗ 2 8

(63)

The total energy for this system is then given by

=

n m

Figure 5. Energy %0 as a function of magnetic field strength B0 . From bottom to top, the states with l m = (0,0), (0,1), (0,2), and (0,3) are displayed. Note that the states l 0 = l m with m = 0 or negative are infinitely degenerate, while the states l 0 = 0 m with m positive are twofold degenerate. Reprinted with permission from [186], J. L. Marín et al., “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

(67)

where now 1nm represents the nth zero of F . Then the energy for the system can be written as (65)

where # = m∗ c /2
, Nlm is a normalization constant, and F is the confluent hypergeometric function [52, 78]. As can be immediately seen, the energy levels for a given l and m negative or zero are infinitely degenerate since m + m = 0 in such a case. The levels given by Eq. (64) are the wellknown Landau levels [84]. Moreover, as c → 0 (i.e., when the field is vanishingly small), the levels coalesce into a single level corresponding to the free particle, analogous to the previous section. Figure 5 shows the behavior of the energy levels as a function of the strength of the magnetic field B0 .

%nm =

1
 21nm + m + m + 1 2 c

(68)

an expression similar to Eq. (64), but the infinite degeneration of the levels for a given n and m negative is lifted by the spatial confinement because, in spite of the fact that m is still a good quantum number, the zeroes of F depend on its value, in contrast with the case of the preceding section. Hence, the problem can be solved if we find the zeroes of F as a function of r0 for a given value m and c . Alternatively, it is interesting to note that
Lz commutes with all operators appearing in Eq. (66) and then the states of the new situation have a definite value of m. The latter

Magnetic Field Effects in Nanostructures

799

property allows one to construct the total wavefunction as the expansion  

n=0

    #n m r 1 expim' = √ an J m r0 2( n=0 =

 

n=0

ε1,0 ε1,1 ε1,2 ε1,3

400

an f0n m r '

Energy (meV)

fbcm r ' =

500

an nm

(69)

Here we have exploited the fact that radial functions found in Section 2.2.1 represent a complete set of orthogonal functions satisfying the proper boundary condition. Of course this is not the only basis in which the total wavefunction can be expanded, but it would represent an alternative representation of F in terms of J m #n m r/r0  to that of [78]. As can be noted, in the Dirac notation, nm n′ m′  = )nn′ )mm′

(70)

Hence, to find the energy levels for a given value of m, we must diagonalize the matrix with elements  1 1 n m Ann′ = %0 +
mc )nn′ + m∗ 2c n r 2 n′  (71) 2 8 Strictly speaking, the involved matrix is infinite, but in practice we can use one large enough, but finite, depending on the precision required to find the eigenvalues. In the present case we have used a (256,256) matrix, a choice that leads to a precision of about six decimal places. The diagonalization procedure will render the eigenvalues as well as the eigenvectors >an ? and thus we have the complete solution of this problem. In Figure 6, the energy of the lowest states is displayed as a function of the strength of

300

200

100

0 0

100

200

300

a (Å) Figure 7. Energy of the lowest states for a particle confined within a magnetic field B0 = 1 Tesla, as a function of confining radius r0 . From bottom to top, the states with n m = 1 0, (1,1), (1,2), and (1,3) are displayed. Reprinted with permission from [186], J. L. Marín et al., “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

the field for r0 = 20 Bohrs, while Figure 7 shows the same levels as a function of r0 for B0 = 1 Tesla. Before closing this case we can mention three main points: (i) the spatial confinement lifts the degeneration on the Landau levels (see Fig. 6). (ii) The magnetic confinement lifts the degeneracy of levels with positive and negative values of m (see Fig. 7). (iii) Each kind of confinement introduces a typical length; namely, the spatial confinement length is of the order of r0 while  the magnetic confinement length is of the order of lm ≈
c/ q B0  48452B0−1/2 (Bohrs), when q = e and B0 is given in Teslas. In this sense, we can talk about a competition between the two kinds of confinement depending on the ratio of r0 /lm , a fact that will be discussed in the next section.

2.2.4. WKB Approximation for Energy Levels Now we shall study approximately a variety of situations raised depending on the relative size of confining region a and the characteristic length introduced by the magnetic field lm . To do the latter we shall follow the WKB formalism as developed in the book of Morse and Feshbach [85]. In the first place, we note that the extension of one-dimensional WKB formalism to the two- or three-dimensional radial Schrödinger equation is accompanied by the change 1 → m2 (two-dimensional) or 4   1 2 (three-dimensional) ll + 1 → l + 2 m2 −

Figure 6. Energy of the lowest states for a particle confined within a circle of radius r0 = 20 Bohrs, as a function of magnetic field strength. From bottom to top, the states with n m = (1,0), (1,1), (1,2), and (1,3) are displayed. Note the energy scales introduced by the selected confinement regime. Reprinted with permission from [186], J. L. Marín et al., “Handbook of Advanced Electronic and Photonic Materials and Devices” (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

(72)

in the centrifugal barrier. Once the latter is recognized, the formalism can be applied in exactly the same way as in the one-dimensional case. The effective potential in our case is of the form Veff r = 12 r 2 +

m2 r2

(73)

Magnetic Field Effects in Nanostructures

800 For illustrative purposes, this potential is displayed in Figure 8 from which two different situations merge, depending on the size of spatial confinement as compared with magnetic confinement, namely, when the energy is smaller or greater than Veff a = 12 r02 + m2 /r02 . This fact is important since we are interested in analyzing approximately the solution of this problem, say by the quasi-classical method or more precisely by the WKB method. Notice that when the energy is smaller than Veff r0  there are two classical turning points whereas when it is greater there is only one, and hence the application of the WKB approximation is different for each case. When only one turning point exists, the approximate wavefunction can be written as   r 1 (74) qr dr − F r ≈ A cos 4 r1 where A is a normalization constant and r1 is the turning point (lowest zero of qr ), where qr = and #2 =



(75)

#2 − Veff

2m∗ % m∗ c − m
2


Performing the integral of qr , the eigenvalues satisfy the following transcendental equation:    2   # #2 −212 r02 ( 2 4 2 2 2 # r0 −1 r0 −m + −arcsin  21 2 #4 −4m2 12      3 #2 r 2 −2m2 ( = 2( N + (79) +arcsin − m  0 2 4 r02 #4 −4m2 12

In the case when r0  r2 (greatest zero of qr ) it can be easily shown that   3 1 %Nm 
c 2N + m + m + (80) 2 2 an expression which is similar to that of the Landau levels [see Eq. (64)], except that the level spacing is slightly different because of the boundary condition on the wavefunction at r = r0  r2 . Thus in the case when the greatest turning point is equal to the radius of the confining region, the corresponding spectrum is Landau-like and represents an extreme situation because in this case   21  # 4m2 12 r0  r2 = √ 1+ 1− #4 21

(76)

from which one can see that

Additionally, the boundary condition demands that F r0  = 0, which implies that 



a 1 1 qr dr − ( = N + ( 4 2 r1

or



a r1

(77)

  3 ( qr dr = N + 4

(78)

which represents the quantization condition as a function of r0 and the magnetic field through r1 (lowest zero of qr ). 8 r=a Vc = ∞

6

Veff (arb. units)

r1 4

r1

r2

2 r0

0 0

1

2

(81)

3

r (arb. units) Figure 8. Qualitative drawing of Veff r used to illustrate its turning points and minimum.

#2 = 12 r02 +

m2 = Veff r0  r02

(82)

Then when #2 ≥ Veff r0 , only one turning point can exist and the energy spectrum of the system is of Landau type. However, when #2 ≤ Veff r0  two turning points merge and the quantization rule, in the standard WKB approximation, would be  

a 1 qr dr = N + ( (83) 2 r1 but this result will only be valid for r2 ≪ r0 because the corresponding WKB function in the region r2 < r < r0 is exponentially decaying and does not satisfy the boundary condition at r = r0 . Hence only when r2 ≪ r0 will the latter function approximately be consistent with this boundary condition and will the energy spectrum of the system behave Landau-like [i.e., as that given by Eq. (80)]. Whenever the condition r2 ≪ r0 is not fulfilled, the standard WKB approximation cannot be used. Another interesting situation corresponds to the case when there is only one turning point but r0  a (see Fig. 8). At this point Veff has a minimum when a = m /1. Then inserting this value into Eq. (79), we obtain √    − + − √ (V V V 3 − V r0 + − arcsin √ (84) = 2( N + 41 21 4 V+ where V − = #2 − 2 m 1

(85)

Magnetic Field Effects in Nanostructures

801

and V + = #2 + 2 m 1

(86)

Moreover, when #2 ≫ 2 m 1, √ 2 m 1 V− 1− √ + #2 V

(87)

and the approximation arcsin1 − z 

( − 2z1/2 2

z≪1

can be used to obtain  2 
2 ( 2 N + 21 m + 23 1 %Nm = + m
c 2 2 2m0 r0

(88)

(89)

which clearly resembles the energy of a particle in a box “perturbed” by a term related with the eigenvalues of
Lz . Since the minimum of Veff depends on m and the strength of the field, we note also that the infinite degeneration is removed for the level whose value of m is “tuned” by the  field strength to satisfy r0 = m /1. The latter property could not be applied to the ground state since in such a case r0 = 0 and the system would behave as a confined oscillator whose energy levels would be similar to those studied in [86]. Interestingly enough,  if we analyze more closely the situation when r0 = a = m /1 it can be seen that r02 =

or

2

c

m = m ∗ = 2 m 1 m c

e B0 B0 (r02 = m

hc

e

(90)

(91)

The term of the left is just the magnetic flux across the confining circle while the term of the right represents an integer number of times m the quantum flux
c/ e . The latter means that in this situation the magnetic flux is quantized; thus by changing the field strength, the system can be tuned into states in which the magnetic flux√is quantized. It also must be noted that when r0 = a, r0 = 2 m lm , where lm is the magnetic length defined in the previous case. The results obtained in Sections 2.2.1, 2.2.2, and 2.2.3, which are summarized in Figures 4–7, show the same trends as those obtained by Rensink [52]; that is, when the spatial confining radius is much greater than the magnetic length the spectrum approaches smoothly the free particle spectrum, while when it is much less the energy spectrum is a sort of Landau-like spectrum. In between, the spectrum is very rich because the typical infinite-folded degeneration of the Landau-like spectrum is lifted by the spatial confinement. Special attention is necessary for the energy spectrum for which the spatial confining radius coincides with the minimum of the effective potential, since in this case three main features merge: (i) The magnetic flux is quantized for field strengths which are an integer multiple of the inverse of the cross-sectional area of spatial confining [see Eq. (91)].

(ii) The nature of these states is quite different from those of Landau-like or pure spatial confining-like states, since they can only be observed when the spatial confining length is an integer multiple of the magnetic length. In this sense, they may be called “surface states.” (iii) These special or “surface states” are nondegenerate [see Eq. (89)] and their energies are greater than those of the “bulk” states [87] [compare Eq. (68) and (89)]. The aforementioned properties make this system attractive for further study and characterization. It would be interesting to study its energy spectrum and/or some important physical properties when, for instance, the confining potential is finite. The latter would also motivate new experiments, which would lead to some insight on the overall physical properties of these kinds of systems.

2.3. Two-Dimensional Quantum Dot in an External Magnetic Field In the last few years the properties of the fundamental quantum systems, quantum dots, defined in two-dimensional electron gases in semiconductor heterostructures, have become the subject of intensive research because these systems are widely used in many fields of modern technology [88]. In this connection, the problem of creation of quantum computers [89–91], memory devices and electronics (see [92] and references therein), new types of lasers [93–95], and other problems connected with the ultrafast switching and quantum information processing should be mentioned. Quantum dots are the most promising candidates for this kind of technology because they have a discrete atomiclike density of states, due to the strong quantum confinement. Many important results on quantum dot can be found in a great number of review articles (see for example [96, 97]). For all the applications mentioned, the feature of bigger importance is the flexibility of the electronic structure of quantum dot, which is controlled by magnetic and electric external fields, so it is of particular interest to investigate the structure of the states in more detail for different external conditions. Here we discuss the one-electron energy spectra in a twodimensional quantum dot with magnetic confinement. The spectra are calculated under the assumption of two different types of the boundary conditions. In the first case, there is only the spatial confinement due to a nonmagnetic finite potential barrier (we will call spatial confinement); in the second case the spatial confinement is combined with the magnetic spatial confinement.

2.3.1. Equations The Hamiltonian for an electron confined by the finite potential V r
 within a magnetic field B
in a twodimensional quantum dot is given by H =−


2 2 c 1
 + Lz + m∗ 2c r 2 + V r ∗ 2m 2 8

(92)

where r and  are defined in the plane. We have used the same symmetric gauge as that used by Rosas et al. [83].

Magnetic Field Effects in Nanostructures

802 In this work we begin with a more realistic shape of the
instead of that used in [83]. Namely, let the potential V r potential be described by the following expression:
= V r



0

0≤r ≤a

Vc

(93)

r >a

2.3.3. Quantum Dot with Magnetic Confinement The case of a charged particle moving in the homogeneous magnetic field B0 without a spatial confinement was discussed in a previous work [83]. Here we would only like to mention that in this case the Schrödinger equation for a particle of mass m∗ is

In this case the Schrödinger equation can be rewritten in polar coordinates, 

  
2 1   1 2 r + 2m∗ r r r r 2 '2 + i




∗

m 2 2  −  r + E − Vc B = 0 ' 2

(94)

where the frequency  = c /2 was introduced.

2.3.2. Quantum Dot without Magnetic Field If the magnetic field is not included, Eq. (94) can be rewritten as    
2 1 2
2 1  r C + C +E −Vc B = 0 (95) 2m∗ r r r 2m∗ r 2 '2 In this equation E is the energy of the particle and the wave function C must satisfy the boundary condition at r = a: C ′ r ' C−′ r ' = + C− r ' C+ r '

(96)

Here C− is the solution of Eq. (95) for the region 0 ≤ r ≤ a and C+ for r > a and the derivatives are taken with respect to r. However, for the region 0 ≤ r ≤ a we should take the solution of the Schrödinger equation with the magnetic field instead of C− . Let us consider Eq. (95) in the region r > a. In this case the variables are separable and the substitution C = Rr9' yields the set of well-known equations rrR′ ′ − r 2 k 2 + m2 R = 0 2

9 +m 9 =0 ′′

(97) (98)

2m∗ E − V  < 0
2

for r > a

The solution of this equation set is B+ = AKm kreim'

     1 2
2 1  r + 2 2 2m∗ r r r r '  m∗ 2 2  −  r +E C =0 + i
 ' 2

which has the solution  

m %−m− m −1 x Cm = Aeim' e− 2 x 2 F −

m +1Fx 2

(100)

(101)

where r2 2 2lm  = c 2 x=

2 lm =


m∗ 

c

=

c
eB0

E %=


(102)

In the case when we need to take in consideration the spin of the electron we should change the energy E by E +
s B0  in the last expression, and the corresponding degeneracy will be lifted. The energy levels that can be obtained with the boundary condition at r = a for the wavefunctions corresponding to the regions 0 ≤ r ≤ a and r > a are [see Eq. (96)] 1 K ′ r 1 Bm′ r 2  = ∗ m ∗ 2 m Bm r  m Km r

(103)

where Bm r 2  is the solution of Eq. (101) in the region 2 , and Km r is the Macdonald function 0 ≤ r ≤ a, r 2 = 2xlm which is the solution of Eq. (100) for the region r > a.

2.3.4. A More Realistic Potential It is clear that the energy levels obtained from condition (103) are approximations. More strictly speaking, we should substitute in this equation the solution of Eq. (100) for the region r > a with an additional constant −V , the height of the potential barrier, instead of the function Km r [the potential in this case becomes m∗ /22 r 2 + V ]. In this case Eq. (100) takes the form

where m is a separation constant and

k 2 =



(99)

where Km is the modified Bessel functions (or Macdonald functions), and A is a constant. The solution of Eq. (95) in the region 0 ≤ r ≤ a as well as the energy spectrum have been discussed earlier [83]. For this reason we do not need consider it here again (in this region of r we are interested in the wavefunction that corresponds to the case with magnetic field).

C ′′ +

1 ′ %−V −m m2 1 B − 2B + B+ C=0 x 4x 2x 4

(104)

and its solutions can be written as  

m %−V −m− m −1 x Bm+ = Aeim' e− 2 x1 2 F1 −

m +1Fx 2 (105)

Magnetic Field Effects in Nanostructures

803

for the region r > a (in this case we have the relation E − V < 0, and for the internal region the solution is equal to (101),  

m % − m − m − 1 − im' − x2 2

m + 1F x Bm = Ae e x1 F1 − 2 (106) where all variables are the same as in Eq. (105), and % = 1 . In this case the boundary condition is E +
s B
 B ′ + Bm′ − = m Bm − Bm +

(107)

which leads to the transcendental equation for energy:   V + 1 m + 2F x F1 −# m + 1F x # − V 1 F1 −# + 2 1 − #1 F1 −# + 1 m + 2F x1   V × F1 −# + m + 1F x = 0 2

(108)

The results of calculations of the energy spectra that correspond to the boundary conditions of Eq. (107) are presented in Figure 9. The energies are plotted as functions of the parameters of the problem—the magnetic field and the height of the rectangular potential barriers—and all values are shown in relative units. In this figure it can also be observed that for the size of the potential barrier Vc /
 = 1 there are no energy states when the spatial confinement and magnetic confinement are almost the same. The energy states exist when a > lm , and for the case when a < lm the intensity of the magnetic field is such that the electron energy levels are located above the potential barrier and electrons escape from the interior of the quantum dot. Under these conditions the magnetic field can be used in order to tune the electronic states inside the quantum

dot. As should be expected, these results are similar to those calculated in [83], but our results are thought to be more realistic ones. In Figure 10 one can see a more detailed comparison of the one-electron energy spectra for the two cases which correspond to different potentials [boundary conditions (103) and (107)]. The corresponding one-electron energy spectra are presented as functions of external magnetic field. It also is observed that for a potential barrier bigger than the potential barrier of Figure 9 there only exist energy states with a < lm for m = 0 since the state with m = 1 only exists with a > lm ; this only happens for magnetic fields inside the quantum dot. However, with the magnetic fields inside or outside the quantum dot the energy level with m = 0 extends to a/lm = 17 approximately (i.e., from the region with a < lm to the region a > lm ). Thus, with a magnetic field in the exterior of the quantum dot the energy states inside the quantum dot can be manipulated. We can also conclude that the choice of a diamagnetic or paramagnetic material in the exterior of the quantum dot can be used to manipulate or tune the electronic states in the quantum dot for a fixed a and B. In Figures 11, 12, and 13 the variation of state with m = 0 with respect to the potential barrier V r for several values of a/lm , which characterize the relation between the spatial and the magnetic confinements, is shown. In Figure 11, a = 05lm , in Figure 12 a = lm , and in Figure 13 a = 2lm . The variation of the curves means that when the spatial confinement increases the energy states due to magnetic confinement appear for a lower potential barrier V r; on the other hand, if the magnetic confinement decreases the energy states tend to disappear, as one can expect. Also plotted in Figure 13 is the state with m = 1 and we can observe that for barriers lower than that of Figure 11 this state goes out to the exterior of the quantum dot because 5

1.0 4

m=0

m=1

m=0

ε = E/h ω

ε = E/h ω

0.9

0.8

3

2

0.7 m=0 1

0.6

0.6

0.5 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a/Im

a/Im Figure 9. The one-electron energy spectrum for a QD, with the magnetic field only in the inner part of the QD where the confinement potential is V r. The energy is calculated under assumption of boundary condition (107) as a function of the width of the confinement potential a/lm for constant value Vc /
 = 1. Reprinted with permission from [88], A. Lipovka et al., Phys. Low-Dim. Struct. 9/10, 97 (2002). © 2002, VSV.

Figure 10. Solid lines are the same as in Figure 9, but for Vc /
 = 5. Dashed line is the one-electron energy spectrum for the QD, with the magnetic field inside and outside the QD where the confinement potential is V r. The energy is calculated under the assumption of boundary condition (107) as a function of the width of the confinement potential a/lm for constant value Vc /
 = 5. Reprinted with permission from [88], A. Lipovka et al., Phys. Low-Dim. Struct. 9/10, 97 (2002). © 2002, VSV.

Magnetic Field Effects in Nanostructures

804 5

5 m=0

4

ε = E/h ω

ε = E/h ω

4

3

m=1

3

2

2

m=0

1

1

1

2

3

4

5

2

1

U = 2πV0 /h ω

3

4

5

U = 2πV0 /h ω

Figure 11. The one-electron energy spectrum in the QD with the magnetic field only in the inner part of the QD with the confinement potential given by V r. The energy is calculated under the assumption of boundary condition (107) as a function of the depth of the confinement potential Vc /
 for a constant value a/lm = 05. Reprinted with permission from [88], A. Lipovka et al., Phys. Low-Dim. Struct. 9/10, 97 (2002). © 2002, VSV.

it has a larger energy than for m = 0. Thus, the existence of this energy state inside the quantum dot requires bigger potential barriers V r.

Figure 13. The same as in Figure 11 but for a/lm = 2. Reprinted with permission from [88], A. Lipovka et al., Phys. Low-Dim. Struct. 9/10, 97 (2002). © 2002, VSV.

function for infinite and finite confining potentials [98]. For a given value of the magnetic field, the binding energy is found to be larger than the zero field case. This behavior is explained in terms of an average electron–hole separation, which depends on the wire radius and the magnetic field strength.

2.4.1. Infinite Potential Barrier Case 2.4. Exciton Binding Energy in a Quantum Wire in the Presence of a Magnetic Field A calculation of the ground state energy of an exciton confined in a cylindrical quantum wire in the presence of a uniform magnetic field as a function of wire radius is made using a variational approach. It is assumed that the magnetic field is applied parallel to the axis of the wire. The calculations have been performed using a suitable variational wavefunction taken as a product of the appropriate confining confluent hypergeometric functions and a hydrogenic

The Hamiltonian for the interacting electron–hole pair inside a cylindrical wire of radius r0 , with infinite potential barrier at the surface, in the presence of a magnetic field applied parallel to wire axis, is given by
= H

    1 e
2 e
2 1 ˆ ˆ pe + Ae + p − A 2m∗e c 2m∗h h c h −

e2 + Ve 2e Ie  + Vh 2h Ih  H r
e − r
h

(109)

where  1/2


re − r
h = 22e + 22h − 22e 2h cosIe − Ih  + z2

5

and

4

ε = E/h ω

(110)

Veh 2eht Ieh  =

m=0

3



0 

0 ≤ 2e and 2h ≤ r0

2e or 2h > r0

(111)

For a uniform magnetic field we can write 2

  B
× r
eh
eh r
eht = A 2

1

1

2

3

4

5

U = 2πV0 /h ω Figure 12. The same as in Figure 11 but for a/lm = 1. Reprinted with permission from [88], A. Lipovka et al., Phys. Low-Dim. Struct. 9/10, 97 (2002). © 2002, VSV.

(112)

where B
= B e
z . In cylindrical coordinates the magnetic field potential becomes A2 = Az = 0, AI/eh0 = B2eh /2. The inclusion of the Coulombic potential leads to a nonseparable differential equation which cannot be solved analytically. Therefore, it is necessary to use a variational approach to calculate the eigenfunctions and eigenvalues of the Hamiltonian.

Magnetic Field Effects in Nanostructures

805

Following Brown and Spector [46] we take into account the cylindrical confining symmetry, the presence of the magnetic field, and the electron–hole Coulombic potential, by choosing a trial wavefunction for the ground state, which can be written as product of a hydrogenic part and the radial solutions of an electron and a hole in a cylindrical wire in the presence of magnetic field, applied parallel to the wire axis:


= f r

  -e +-h   F −a01e 1F-e   N exp −   2    

0

× F −a01h 1F-h exp/−.
re − r
h 0


r0 B = T  + V  is found The ground state energy H after tedious algebra,     1 1
2 2 T  =
ce a01e + . +
ch a01h + + 2 2 2
∗

(118)

where
∗ is the effective reduced mass and 0 ≤ 2e and 2h ≤ r0

2e or 2h > r0

(113)

V  = −

-eh =

22eh 2#2c/eh0

=

eB22eh


#c/eh0 = ∗ meh c/eh0

2
c



N

−2

dG = −2( d.

%b r0 B = −

(114)

is the cyclotron radius, c/eh0 = eB/m∗eh c is the cyclotron frequency, N is the normalization constant, and . is a variational parameter. Equation (113) satisfies the boundary condition that f 2eh = r0  = 0, while a01/eh0 is the eigenvalue for the electron (hole) in a wire in a magnetic field, calculated numerically from the boundary condition eigenvalue equation. N is given by

 22e 2e d2e 2h d2h exp − 2 G= 2#ce 0 0     22h 22e 2 exp − 2 × F −a01e 1F 2 2#ce 2#ch   22 I0 2.2< K0 2.2>  × F 2 −a01h 1F 2h 2#ch r0



r0


r0 B = Le 1 + 2a01e  + Lh 1 + 2a01h  H  2 + .a∗B + 4a∗B

(115)



G1 =

where 2< 2>  is the lesser (greater) of 2e and 2h ; the particular values for 2< and 2> are determined by electron and hole locations and the integration limits. In Eq. (116), I0 and K0 are the modified Bessel functions of the second kind of order zero. The binding energy %b r0 B for the exciton is defined as the ground state energy of a system without Coulombic interaction, minus the ground state energy %0 r0 B =
r0 B in the presence of electron–hole interaction. H That is,     1 1 +
ch a01h + %b r0 B =
ce a01e + 2 2 − %0 r0 B

(117)

The binding energy defined in this way is a positive quantity.

G1 dG1 /d.

 2 %b r0 B = − .a∗B − 2a∗B

where

(116)


2 2 2e2 G . − 2
∗ H dG/d.

(120)

For computational purposes, we normalize the expression for the binding energy %b r0 B [Eq. (120)] in Rydberg units and define Leh =
c/eh0 /2R∗y exc . In addition, we transform the integral for G in a dimensionless form by letting
r0 B and %b r0 B are 2 = tr0 ; the expressions for H then written as

with

(119)

Therefore,

In Eq. (113) the variable 

4(e2 2 N G H



G1 dG1 /d.

(121) (122)

1     th dth exp −te2 -r0 e F 2 −a01e 1F te2 -r0 e te dte 0 0     × exp −th2 -r0 h F 2 −a01h 1F th2 -r0 h 1

× I0 2.r0 t< K0 2.r0 t> 

(123)

and -r0 /eh0 = r02 /2#2c/eh0 .
r0 B with respect In order to find the minimum of H to . and thus to obtain a lower bound to the ground state the variational method is used. The radial double integration in G1 is performed numerically since there is no analytical method to evaluate it.

2.4.2. Finite Potential Barrier Case For the finite potential barrier case, the potential in the Hamiltonian [Eq. (109)] is taken as zero for 2eh < r0 and V0 for 2eh > r0 . All the other assumptions remain the same. Furthermore, we assume that the electron and hole masses are constant (at their values in GaAs) across the barrier. Again, following Brown and Spector [46] we take into account the cylindrical confining symmetry, the presence of the magnetic field, and the Coulomb potential by choosing a trial wavefunction for the ground state, which can be written as a product of a hydrogenic part and the radial solutions of

Magnetic Field Effects in Nanostructures

806 an electron and a hole in a cylindrical wire in the presence of a magnetic field, parallel to the wire axis,    -e +-h   N exp − F −a01e 1F-e     2      re − r
h    ×F −a01h 1 -h exp−.
       F −a01e 1F-r0 e  - +-h    exp − e  N  ′   U −a01e 1F-r e 2   0   ′     ×U −a01e 1F-e F −a01h 1F-h        × exp−. r
e − r
h          F −a -e +-h 01h 1F-r0 h  f r
e r
h  = N   exp −  2 U −a′01h 1F-r0 h      ′    ×F −a01e 1F-e U −a01h 1F-h       × exp−.
re − r
h         F −a01e 1F-r0 e  F −a01h 1F-r0 h        N  ′   U −a01e 1F-r0 e U −a′01h 1F-r0 h           - +-h   U −a′01e 1F-e  × exp − e   2       re − r
h  ×U −a′01h 1F-h exp−.


IN = 0 ≤ 2e and 2h ≤ r0

r0

2e > r0 and 2h ≤ r0



r0

0

2e d2e



r0 0

%b r0 B = − 2e ≤ r0 and 2h > r0

2e and 2h > r0

at 2eh = r0

d I + IL + IM + IN  d. K

F 2 −a01e 1F -r0 e 

U 2 −a′01e 1F -r0 e 



r0

 × U −a′01e 1F -e K0 2.2e 

r0

0

× F 2 −a01h 1F -h I0 2.2h 

(124)

IKK =



ILL =

F −a01e 1F -r0 e 

(126) IMM =

and

(128)

× U −a′01h 1F -h K0 2.2h

r0



(129)

(131)

IKK + ILL + IMM + INN  dIKK + ILL + IMM + INN /d.

(132)

1   te dte th dth exp −te2 -r0 e 0 0     × F 2 −a01e 1F te2 -r0 e exp −th2 -r0 h   × F 2 −a01h 1F th2 -r0 h I0 2.r0 t< K0 2.r0 t> 

U 2 −a′01e 1F -r0 e 



1

(133)

  te dte exp −te2 -r0 e

1   × U 2 −a′01e 1F te2 -r0 e K0 2.r0 te  th dth 0     × exp −th2 -r0 h F 2 −a01h 1F th2 -r0 h I0 2.r0 th  (134)

  F 2 −a01h 1F -r0 h  1   te dte exp −te2 -r0 e U 2 −a′01h 1F -r0 h 0   × F 2 −a01e 1F te2 -r0 e I0 2.r0 te 

     × th dth exp −th2 -r0 h U 2 −a′01h 1F th2 -r0 h 1

(127)

2h d2h exp−-h 

(130)

1

2

F −a01h 1F -r0 h  r0  2e d2e exp−-e  = 2 ′ U −a01h 1F -r0 h 0

 × F 2 −a01e 1F -e I0 2.2e  2h d2h exp−-h   2

  U 2 −a′01h 1F -h

where

(125)

2

IM




2 2 2e2 IK + IL + IM + IN  . − 2
∗ H dIK + IL + IM + IN /d.

 2 %b r0 B = − .a∗B − 4a∗B

2e d2e exp−-e 

 2

r0

−a′01e 1F -e

We normalize the expression for the binding energy %b r0 B, in units of exciton Rydberg R∗y exc ,

2h d2h exp/−-e + -h 0F 2 −a01e 1F -e 

× F 2 −a01h 1F -h I0 2.2< K0 2.2> 

IL =



The binding energy %b r0 B is calculated as in the infinite potential barrier case, leading to the following expression:

with IK =

2

× I0 2r0 t< K0 2.r0 t> 

while a01eh and a′01eh are the eigenvalues for the ground state of the problem inside and outside the wire for an electron and a hole, respectively. N is given by N −2 = −4( 2

F 2 −a01e 1F -r0 e  F 2 −a01h 1F -r0 h      U 2 −a′01e 1F -r0 e U 2 −a′01h 1F -r0 h



 × 2h d2h exp/−-e + -h 0 2e d2e ×U

where -eh = 22eh /2#2c/eh0 . Equation (124) satisfies the boundary condition 1 fi 1 fo = ∗ m∗i 2eh mo 2eh

and

INN =

× K0 2.r0 th 

(135)

F 2 −a01e 1F -r0 e  F 2 −a01h 1F -r0 h  U 2 −a′01e 1F -r0 e  U 2 −a′01h 1F -r0 h 



    th dth exp − te2 -r0 e + th2 -r0 h te dte × 1 1  ′    2 × U −a01e 1F te2 -r0 e U 2 −a′01h 1F th2 -r0 h × I0 2.r0 t< K0 2.r0 t> 

(136)

The values of the physical parameters pertaining to the material GaAs used in the calculations are given in Table 1 [77].

Magnetic Field Effects in Nanostructures

807

Table 1. Physical parameters pertaining to the material GaAs. m∗e

H

L1

L2

m∗hh x y m∗hh z m∗lh x y m∗lh z

GaAs 0.067m0 12.5 7.36 2.57

0.10m0

0.45m0

0.21m0

0.08m0

For mathematical convenience and without any loss of physical insight, isotropic hole mass is assumed and, therefore, we can write an expression for an isotropic hole mass: 

m∗h

−1

=

−1 1  ∗ −1 2 ∗ + mh z m x y 3 h 3

(137)

The reduced mass for the heavy hole exciton is 00447m0 and that for the light hole exciton is 00449m0 . Since both heavy and light hole exciton masses are close to each other, for the rest of the topic we present results for the heavy exciton. In Figure 14 the variation of the exciton binding energy %b as a function of wire radius r0 for several values of the magnetic field in the infinite potential barrier case is shown. For a given value of the magnetic field the binding energy increases as the wire radius is reduced and diverges as r0 → 0. For small values of the wire radii (r0 ≤ aB ) and for values of the magnetic field considered in this work, the exciton binding energy is relatively insensitive to the variation of the magnetic field, since the confinement effect due to the potential barrier is more significant than that due to the application of the magnetic field. For large values of the wire radii (r0 ≥ 3a∗B ) the binding energy for B = 0 approaches its value in the bulk GaAs (i.e., the exciton Rydberg) and 7

varies with the magnetic field essentially in the same fashion as calculated by Aldrich and Greene [99]. In Figure 15 the variation of the exciton binding energy %b as a function of magnetic field for several values of the wire radii in the infinite potential barrier case is plotted. Again, for small values of the wire radii (r0 ≤ a∗B  the binding energy is relatively insensitive to the variations of the magnetic field; however, for large values of the wire radii (r0 ≥ 3a∗B ) the variation of %b with magnetic field is much more pronounced due to the strong confinement effect of the magnetic field. In Figure 16 the variation of the exciton binding energy %b as a function of the radius of the wire for several values of the magnetic field for a finite potential barrier is plotted. The Al concentration in the barrier material is assumed to be x = 03. For a given value of the magnetic field the binding energy increases from its bulk value in GaAs as the wire radius is reduced, reaches a maximum value, and then drops to the bulk value characteristic of the barrier material as the wire radius goes to zero. This is due to the fact that as the wire radius is decreased the electron wavefunction is compressed thus leading to the enhancement of the binding energy. However, below a certain value of r0 the leakage of the wavefunction into the barrier region becomes more important and, thus, the binding energy starts decreasing until it reaches a value that is characteristic of the barrier material as r0 → 0. For a given value of r0 the binding energy increases as a function of the magnetic field due to the increasing compression of the wavefunction with magnetic field. In Figure 17 the variation of the exciton binding energy as a function of the magnetic field for several values of the wire radius is shown. It is found that for small values of r0 /a∗B ( 13 atoms. In the smallest clusters, however, it continues to predict a moment of about the same size as the larger clusters while the experimental data show the large step up to close to the atomic limit for N = 12. This discrepancy is interesting as it implies that the step is due to the appearance of the orbital moment at close to its atomic value. It seems therefore that the formation of the close-packed icosahedron at N = 13 almost completely quenches the orbital moment and it does not reappear at larger cluster sizes. For N > 13 the moment is mostly due to spin but is still enhanced significantly relative to the bulk value. Fujima [101] predicted that Fe7 clusters with a pentagonal bipyramidal structure have a

Magnetism in Nanoclusters

6.0

909

Fe Atomic limit (d6) = 6 µB

5.5

Fe

5.0 4.5

Magnetic Moment (µB/atom)

4.0 3.5 3.0 2.5 Fe Bulk limit = 2.193µB

2.0

Ni

2.0 Ni Atomic limit (d8) = 5 µB 1.5

1.0

Ni Bulk limit = 0.612µB

0.5

0.0

0

20

40

60

80

100

Cluster size (atoms) Figure 7. Total moments (in B /atom) in free FeN and NiN clusters as a function of N for N < 100 measured by Knickelbein [94, 95] (squares), Billas et al. [92] (triangles), and Apsel et al. [93] (circles). The measured values are compared with calculations by Andriotis and Menon [98] (open circles), Reddy et al. [99] (open circles plus line), and Reuse and Khanna [100] (open squares).

spin moment of 2.86 B /atom for the bulk interatomic spacing but a slight contraction results in a coplanar (i.e., not collinear) arrangement of the moments. The calculations for NiN clusters by Reddy et al. [99] shows a minimum in the moment for N = 16 rather than N = 13 as observed. This is probably due to omission of the orbital moment as in the case of Fe implying again that formation of the icosahedron at N = 13 quenches the orbital moment in Ni clusters though the effect is not as big as in Fe clusters. Reuse and Khanna [100] calculated moments in free Ni clusters for sizes up to N = 13 and found a low moment at this size as well as predicting the icosahedral structure. They did not continue the calculation to larger sizes, however, so the minimum moment predicted by the model may not be at N = 13.

5.2. Antiferromagnetic 3d Transition Metals The small number of atoms in nanoclusters of metals that are antiferromagnetic in the bulk leads to the possibility of an imbalance of the spin sublattices and frustration. Thus a ferrimagnetic or even ferromagnetic ground state may be expected. An initial experimental study of Cr clusters by Douglass et al. [102] using the gradient field deflection technique showed that CrN did not have a detectable net magnetic moment for N in the range 9–31 atoms. The sensitivity of the apparatus led them to conclude that the upper bounds for any net moment varied from 0.77 B /atom

for Cr9 to 0.42 B /atom for Cr31 . This result was in contrast to many early calculations that predicted large and easily measurable net moments for CrN clusters for N up to 51 [103–105], though the expected moments were shown to be highly sensitive to the structure [105]. Lee and Callaway [106] calculated the magnetic moment in Cr9 and Cr15 clusters in a bcc structure as a function of lattice spacing and observed the disappearance of a net moment below a critical value. They found that at the bulk Cr spacing the magnetic moment for both Cr9 and Cr15 clusters was below the detectable threshold of the experiment. A structure-optimized calculation by Cheng and Wang [107] showed a unique growth by dimers up to Cr11 after which the bcc structure is stabilized. They found that the clusters were always antiferromagnetic with a size-dependent atomic moment. Subsequent calculations have investigated the stability of noncollinear moments resulting from frustration [101, 108]. Kohl and Bertsch [108] found that CrN clusters for N ≤ 13 strongly favor noncollinear configurations and Fujima [101] found that with decreasing interatomic spacing in Cr7 clusters, with a pentagonal bipyramid structure, the spin configuration changes from collinear (antiparallel) to coplanar. Recently the field gradient deflection experiment on Cr clusters was repeated by Bloomfield et al. [109] using a more sensitive apparatus and it was found that all clusters studied in the size range N = 8–156 atoms showed a net magnetic moment that varied from almost zero to 1.87 B /atom. Even more interesting was the discovery that at a given cluster size there were always two isomers with different moments. The null result of the earlier experiment was due to the fact that the isomers with the largest moments had a low abundance. For example Cr9 has an abundant isomer with a moment of 0.65 B /atom and a rare one with a moment of 1.87 B /atom. In this latter case, for a collinear alignment of the moments even if the localized atomic moment was close to the free atom value of 5 B (d 5 configuration), at least six of the nine spins must be parallel. The finding of multiple moments at a given cluster size, however, is not necessarily due to isomers with a different atomic structure. An earlier calculation by Lee and Callaway [110] revealed that Cr9 clusters with a bcc structure could exist in several magnetic states with different moments simultaneously. They found that for some atomic spacings four or five states could coexist. The data of Bloomfield et al. [109] may well be an experimental confirmation of this prediction. Figure 8 shows the magnetic moments in free MnN clusters for N = 11–99 measured by Knickelbein [88]. He observed a moment across the size range that attains a maximum value of 1.5 B /atom for N = 15. Deep minima were also observed for N = 13 and N = 19 suggesting an icosahedral and double icosahedral structure at these cluster sizes. Calculations are not available for Mn clusters in this size range but density functional calculations for Mn2 –Mn5 [111] find compact and symmetrical equilibrium geometries but also the existence of isomers whose binding energy is only slightly higher than the ground state structure. In all cases magnetic moments of 5 B were found; that is, the spin configuration is ferromagnetic and the spin moment has the atomic value. The conditions required for noncollinear

Magnetism in Nanoclusters

910 1.0

Mn

Magnetic Moment (µB/atom)

Magnetic Moment (µB/atom)

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

0

20

40

60

80

100

fcc

0.9 0.8

fcc ico

0.7

fcc

fcc fcc

0.6 0.5

bcc bcc

0.4 bcc

0.3

fcc

0.2 0.1 0.0 0

ico

10

20

alignment of the moments in Mn7 clusters with a pentagonal bipyramidal structure were examined by Fujima [101]. He found that as the interatomic spacing relative to the bulk value varies between 0.8 and 1 the spin configuration changes from collinear (antiparallel) to disordered to coplanar.

5.3. Nonmagnetic 3d and 4d Metals The Stoner criterion for itinerant magnetism [112] is D EF I > 1 where D EF  is the density of states at the Fermi level and I is the exchange integral  I = * 2 rK r dr (11) in which K r is the exchange-correlation enhancement of the applied field and * r is proportional to the charge density at the Fermi level. Both D EF  and I are modified at interfaces and in nanostructures leading to the possibility of stabilizing magnetism in small clusters of elements that are paramagnetic in the bulk. Despite several theoretical predictions for permanent magnetic moments in V clusters [103, 113, 114] none have been detected [102]. There is a general decrease in I with increasing atomic number [115] so that magnetism is less favored in 4d and 5d elements and is not found in any bulk elements from these series. Although many 4d and 5d metals have been predicted to show magnetism as monolayers [116] there are only two experimentally proven examples of 4d magnetism, that is, Ru monolayers on graphite [117] and free Rh nanoclusters [56]. The magnetic moment per atom as a function of Rh cluster size is shown in Figure 9 and compared with the unrestricted Hartree–Fock calculations by Villaseñor-Gonzalez et al. [118]. The calculation optimized the structure at each size by determining the optimum bond length for icosahedral, fcc, and bcc atomic arrangement and then observing which of these had the highest cohesive energy. The predicted structure at each point is shown in Figure 9 and it is seen that the most stable arrangement alternates between all three structures. The calculations of the magnetic moment

40

50

60

70

80

90

100

Cluster size (atoms)

Cluster size (atoms) Figure 8. Total moments (in B /atom) in free MnN clusters as a function of N for N = 1–99. Reprinted with permission from [88], M. B. Knickelbein, Phys. Rev. Lett. 86, 5255 (2001). © 2001, American Physical Society.

30

Figure 9. Total moments (in B /atom) in free RhN clusters for N < 100 measured by Cox et al. [56] (open circles). The measured values are compared with calculations by Villaseñor-Gonzalez et al. [118] (filled circles). The most stable structure, for which the calculation was performed, is shown at each cluster size. Reprinted with permission from [118], P. Villaseñor-Gonzalez et al., Phys. Rev. B 55, 15084 (1997). © 1997, American Physical Society.

of the most stable structures at each cluster size reproduce the data reasonably well and in particular reproduce the minima at N = 13, N = 17, and the maximum at N = 19. Interestingly the most stable structure at N = 13 is found to be bcc-like instead of the icosahedra assumed in the case of the 3d transition metals. A 13-atom icosahedral Rh cluster is found to show a magnetization maximum in contrast to the data. The double icosahedron at N = 19 also shows a maximum in agreement with the measurements. The variations in the magnetic moment per atom derive not only from changes in magnitude of the individual atomic moments but also from spin alignments that are sometimes antiferromagnetic. The neglect of the orbital contribution in this type of calculation may, however, change the agreement with the experiment, in which the total (orbital + spin) moment is measured. Cox et al. [56] also attempted to detect magnetism in Pd and Ru clusters in the size range 10–115 atoms but found no measurable moment and set upper limits of 0.32 B /atom for Ru10 clusters and 0.4 B /atom for Pd13 clusters. A tight binding calculation including s, p and d-orbitals of PdN and RhN clusters up to N = 19 showed either small or zero moments for Pd clusters and larger values for Rh clusters [119].

5.4. Rare Earths The character of magnetism in rare earths is very different from that of transition metals because the 4f orbitals, which produce most of the magnetic moment, are screened from the environment by the 6s electrons. The localization of the 4f electrons within screening orbitals generally results in both the orbital and spin moments maintaining their full atomic values in the bulk. The large orbital moment and spin–orbit coupling is responsible for the very high values of the magnetic anisotropy energy observed in rare earths. Magnetic deflection experiments on free Gd clusters [13, 91, 120] at temperatures around 100 K have revealed two types of behavior depending on the number of atoms in the cluster. At some cluster sizes, the deflection profiles show that the cluster magnetization scales with B/T and these

Magnetism in Nanoclusters

are assumed to be superparamagnetic as in the case of transition metal clusters. At other sizes the cluster moment, measured from the deflection profile, does not scale with B/T and in this regime the behavior is ascribed to the moments being blocked so that they are locked to the crystal lattice and undergo the same rotation as the cluster. The simplest interpretation of these observations is that the magnetic anisotropy, which is higher in the rare earths than the transition metals, varies with the number of atoms in the cluster producing a consequent variation in the cluster blocking temperature TB . For example Gerion et al. [91] found that TB is between 45 and 75 K for Gd13 and Gd21 but around 180 K for Gd22 . The different experiments disagree, however, on the cluster sizes that show the different behaviors. Douglass et al. [120] found that all cluster sizes in the range Gd11 –Gd26 displayed locked-moment behavior with the exception of Gd22 , which was superparamagnetic. In contrast, Gerion et al. [91] showed that Gd13 and Gd21 were superparamagnetic while Gd22 was blocked. Tb also shows the locked-moment or superparamagnetic behaviors described above depending on the cluster size [121]. For the superparamagnetic clusters it is possible to extract the total moment per atom from the magnetization vs B/T using Eq. (2) as with the transition metals and in all cases moments/atom significantly lower than the bulk value were found. For example Gd13 and Gd21 clusters have moments of 5.4 and 5 B /atom respectively compared with a bulk value of 7.55 B /atom. This has been explained by a theoretical model that takes into account an Ruderman, Kittel, Kasuya and Yosida (RKKY)-type interaction between the atoms in which the exchange force between nearest neighbors is ferromagnetic but switches to antiferromagnetic between second nearest neighbors [89]. The clusters are found to adopt a hexagonal structure and within a range of the ratio of the ferromagnetic to antiferromagnetic coupling strengths, canting of individual atomic spins away from perfect ferromagnetic alignment is predicted. Thus the moment per atom measured along the applied field will be reduced. The reduction in moment relative to the bulk is also observed in Tb clusters [121], presumably for the same reason.

5.5. Magnetic Ordering Temperature The thermal behavior of the moment has so far been discussed in terms of superparamagnetism where the cluster supermoment is excited over the cluster anisotropy barrier. At sufficiently high temperatures the internal spin alignment will be disrupted leading eventually to normal paramagnetism. Gerion et al. [90] measured the specific heat of free Fe250−290 , Co200−240 , and Ni200−240 clusters in the temperature range 80–900 K. The method involves using an experimental setup similar to that used in measuring the magnetic moments and the perturbation of the beam profile in response to a heating laser is used to determine the specific heat. The clearest results are for Ni200−240 which shows a broad peak centered at 340 K on top of a nearly constant baseline of 6 cal/(mol K)—the classical Dulong and Petite value for bulk Ni. The peak is interpreted as due to the ferromagnetic to paramagnetic phase transition in the clusters

911 and its width is well described by the mean-field approximation. The temperature of the transition is significantly reduced relative to the bulk value of 627 K. The specific heat for the Co200−240 clusters shows a monotonic rise from 5.5 cal/(mol K) at 300 K to 15 cal/(mol K) at 900 K implying that the phase transition is not reachable in the temperature range of the experiment. The behavior of Fe250−290 clusters is more enigmatic showing specific heat values significantly lower than the Dulong and Petite values. A peak at around T = 600 K does occur but is poorly described by mean-field theory. One possibility discussed in the paper is that Fe clusters undergo a magnetic transition between a high-moment and low-moment state with different lattice parameters. Billas et al. [12] observed the magnetic ordering temperatures in FeN and NiN clusters directly by measuring the magnetic moment as a function of temperature up to over 900 K. In all cases they found the magnetic moment was constant above some critical temperature TC N , which was interpreted as the transition temperature for paramagnetism. The magnetization in this state, para , should attain √ a value T =0 / N corresponding to N randomly aligned atomic moments so that in a cluster where N ∼ 100, para is significantly above √ zero. In fact it was found that para was higher than T =0 / N . This was later explained by Pastor and Dorantes-Dávila [122] as due to short-range magnetic order (SRMO) above the transition temperature, which is also observed in bulk magnetic materials. The effect √ of this is to increase the paramagnetic moment to T =0 -/N where - is the average number of atoms in a SRMO domain. The high temperature behavior of rare-earth clusters contrasts with that of the transition metals. The measured ordering temperatures (420 K for Gd21 and over 500 K for Gd13 ) are significantly higher than the bulk value of 293 K. In addition the moment above √ the ordering temperature tends asymptotically to T =0 / N [91] indicating that there is no SRMO above the ordering temperature and the disorder is on the atomic scale. The decay in magnetic moment per atom with temperature found for Gd13 clusters is consistent with the canted spin model [91, 123].

6. MAGNETISM IN EXPOSED CLUSTERS ADSORBED ON SURFACES In this section, the behavior of magnetic clusters deposited on a surface and left exposed to vacuum will be considered. Working in this regime requires that the cluster deposition must be done in-situ and the measurements performed in very good UHV conditions (∼10−11 mbar) to avoid contamination of the clusters. In the case of free clusters, some gas adsorption on the clusters can be tolerated as the mass separation can distinguish the clean clusters from those with gas molecules attached. In deposited samples, most in-situ magnetic measurements have been done with spatially averaged techniques that measure the entire cluster assembly and are unable to discriminate between the behavior of clean and contaminated clusters. There are, however, several experimental advantages in studying adsorbed clusters; for example, it is possible to exert accurate control over the cluster temperature down to low cryogenic values allowing the blocked state to be observed. In addition structural

Magnetism in Nanoclusters

912 techniques such as X-ray diffraction, STM, and TEM can be used as well as more powerful magnetic probes such as XMCD, which is able to determine the spin and orbital components of the total moment. Finally, the interactions between the clusters can be studied by accumulating a sufficient density on the surface.

6.1. Magnetism in Isolated Adsorbed Clusters A natural starting point in describing adsorbed clusters is to review measurements carried out at very low coverages to determine the behavior of the isolated particles. An important question to address is whether the enhanced magnetic moments observed in free transition metal clusters are retained if the cluster is adsorbed on a surface. The earliest reported studies of isolated and exposed magnetic clusters deposited from the gas phase onto surfaces in UHV were of size-selected 2.5 nm (700-atom) Mn clusters at very low coverages on HOPG substrates [124, 125]. Photoelectron spectroscopy using synchrotron radiation was used to probe shallow core levels and showed changes in the lineshape of the Mn 3s photoemission spectra that were consistent with an increased magnetic moment of the atoms in the cluster estimated at about 20% larger than the bulk value. The experiment was insensitive to the spin configuration. More recently the powerful XMCD technique has been utilized in deposited cluster experiments and as well as addressing the magnitude of the moment it is possible to obtain information on the separate contributions of the spin and orbital components. To gain some understanding of the effect of the interaction with the substrate it is useful to consider results from isolated atoms of Fe, Co, and Ni adsorbed on K and Na films obtained by Gambardella et al. [126]. They studied films with coverages as low as 0.2% of a monolayer and used the most accurately measured quantity obtainable by the XMCD technique, that is, the ratio (see Section 3)

were then removed by flash-heating to 100 K. Substantial increases in the orbital to spin ratio and the spin and orbital moments per valence band hole relative to the bulk were observed but uncertainty in nh and the Tz term meant that absolute values were not given. Larger clusters in the size range 180–690 atoms deposited at low coverage on graphite were studied using XMCD by Baker et al. [128]. In this study the value of nh was obtained by comparing the total Fe L2/ 3 absorption cross section of the clusters with that of a thick conventional film deposited in-situ and was found to be the same within experimental uncertainty. Thus nh could be taken as the bulk value of 3.39 [129]. In addition the Tz term was obtained by measuring the dichroism as a function of the incident angle of the X-rays so absolute values of the magnetic moments were obtained and the total moment was found to be about 10% greater than the bulk value with about half of this enhancement coming from an increased orbital contribution. The different datasets for adsorbed Fe atoms [126], small FeN (N = 1–9) clusters [127], and large Fe clusters [128] can be compared directly by plotting the orbital ( Lz ) and spin ( Sz ) angular momenta per valence band hole, nh , as is done in Figure 10. For the adsorbed Fe atoms on K, the valence state was identified as d 7 so nh = 3 [126]. The orbital and spin moments for the isolated atoms shown in Figure 10 were derived from the published R value, assuming that most of the reduction in R is due to a partial quenching of the orbital moment. For the small clusters the published data for the orbital component, Lz /nh , can be entered directly on the plot. The spin term is given as Sz + 7/2 Tz and the pure spin component has been extracted assum1.2

Fe Atomic limit = 1

1.0 0.8

nh 0.6

1.2

Atomic limit

1.0

R=

Lz 2 Sz + 7 Tz

0.4

(12)

to determine how “atomic-like” the adsorbed atoms were. The valence state of each element was determined by modelling the shape of the L2/ 3 absorption edge and was found to be d 7 for Fe, d 8 for Co, and d 9 for Ni. Knowing this the quantities in Eq. (12) can be obtained by Hund’s rules and the atomic value of R (Ratom ) compared with the measured one (Rexp ). The ratio Rexp /Ratom was found to be 100%, 89%, and 63% for Ni, Co, and Fe respectively; that is, Ni has the same moments as in a free atom with the same valence while Co and Fe show a 10% and a 37% decrease respectively. The reductions are due to a partial quenching of the orbital moment but even in the case of Fe the quenching is much less than in isolated Fe impurities embedded in a bulk host. Mass-selected FeN clusters with sizes N = 2–9 adsorbed, at the dilute limit, on Ni thin film substrates have been studied by Lau et al. [127]. For such small clusters it is important to ensure a soft landing and this was achieved by depositing the clusters, decelerated to 2 eV/atom onto Ar multilayers that were physisorbed at 15 K on the substrate. These

Bulk limit

0.8 0.6 0.4

0.2

0.2 0

0

2

4

6

8

10

Cluster Size (atoms)

0.8

Atomic limit for d6 = 0.75 Adatom on K (d7 - atomic limit = 1)

nh

0.6

Atomic limit (d6) 0.6 0.4

0.4 0.2 00

2

0

4

6

8

10

Cluster Size (atoms)

0.2

Bulk limit 0

100

200

300

400

500

600

700

Cluster Size (atoms) Figure 10. Angular momentum values Sz and Lz measured by XMCD for Fe clusters adsorbed on surfaces. Fe1 /K data from Gambardella et al. [126], Fe1 –Fe9 /Ni data from Lau et al. [127], and Fe180 – Fe690 /HOPG data from Baker et al. [128].

Magnetism in Nanoclusters 0.30

bulk µS

0.20 1.0 0.15 0.10 0.05

0.5

bulk µL 0

2.5

µS +7/2µT (µB)

1.5

5

7.5

10

0.0 12.5

Coverage (Å) Figure 11. Orbital (unfilled squares) and spin (unfilled circles) moments for unfiltered Fe clusters (mean size 600 atoms) deposited on HOPG as a function of cluster coverage expressed as the equivalent film thickness. The filled symbols are the values for a mass-filtered deposit with the filter central mass set to 600 atoms. An STM image at a coverage of approximately 0.2 cluster monolayers (4 Å) is shown. Reprinted with permission from [14], K. W. Edmonds et al., Phys. Rev. B 60, 472 (1999). © 1999, American Physical Society.

remanent magnetization. At this temperature, isolated clusters are superparamagnetic and will have zero remanence. The behavior is independent of substrate and for coverages below half a cluster monolayer, the MLDAD signal is zero within experimental error. A small but finite MLDAD signal

3

6.2. Effect of Cluster–Cluster Interactions All the results presented are at the dilute limit when the clusters can be assumed to be noninteracting. As the density on a surface is increased the effect of the interaction between the clusters can be observed. Figure 11 shows how the orbital and spin moments, measured by XMCD, vary with coverage for unfiltered Fe clusters with a mean size of 600 atoms deposited on HOPG [14]. The coverage is expressed as the equivalent film thickness in Å so for clusters of this size a complete monolayer corresponds to ∼20 Å. Also shown on the diagram (filled symbols) are the measured moments for a mass-selected deposit with the filter set to the mean value of the unfiltered distribution. It is evident that the orbital and spin values at low coverage in the filtered and unfiltered deposits are similar. With increasing coverage the orbital moment drops toward the bulk value as the clusters come into contact. The spin term increases but this is due to a decreasing contribution from the (negative) T (dipole) component as the film becomes more bulklike. Figure 12 shows the development with cluster density of the remanence at 40 K of films produced by depositing unfiltered Fe clusters with a mean size of ∼450 atoms on Cu and HOPG substrates [61]. The data were obtained using magnetic linear and circular dichroism in the angular distribution of Fe 3p photoelectrons (MLDAD and MCDAD) that measure, respectively, the in-plane and out-of-plane

2.0

0.25

µL (µB)

ing that Tz is the average of the value in isolated atoms

−1/7 and that in the larger deposited clusters −011/7 [128]. The Lz /nh and Sz /nh values for the larger clusters were obtained from the published orbital and spin magnetic moments and dividing by the value of nh (3.39) measured during the experiment. Whatever the inaccuracies in the assumptions made in the comparison of the different datasets and notwithstanding the different substrates used in the experiments some general conclusions can be drawn from Figure 10 about the size-dependent magnetic behavior of supported Fe nanoclusters. Both the orbital and spin components decrease rapidly with cluster size and with only three atoms/cluster values close to the bulk value are found for both Lz /nh and Sz /nh . This is different from the behavior of free Fe clusters in which moments close to the atomic limit were found up to N = 12. There is evidence for short and long period oscillations in both moments as a function of size. Over a wide size range the total magnetic moment is about 10% larger than the bulk with about half of the enhancement coming from the orbital moment. The adsorbed cluster moments converge with the bulk values for N ≥ 700, as observed in free clusters. Ab initio calculations of the spin moment in small 4d metal clusters with different geometries on Ag(100) were carried out by Wildbereger et al. [130]. While they found a permanent moment in dimers for elements between Mo and Rh in the periodic table, magnetism was less favored as the cluster size increased. In the case of the largest clusters studied (i.e., nine-atom flat islands), only Ru and Rh had stable moments. The Rh moment was 0.6 B , which compares with 0.8 B measured in nine-atom free Rh clusters [56].

913

MLDAD (au)

value for MBE film

2

1

0

MCDAD (au)

1

0 0

1 Coverage

2 (ML)

3

Figure 12. MCDAD (out-of-plane remanent magnetization) and MLDAD (in-plane remanent magnetization) signals as a function of coverage for unfiltered Fe clusters with an average size ∼450 atoms deposited on graphite (open circles) and Cu (filled circles). All measurements were at 40 K. The dense film shows an in-plane anisotropy with a remanence higher than a molecular beam epitaxy-grown film deposited in-situ. An STM image at a coverage of approximately 0.2 cluster monolayers is shown. Reprinted with permission from [61], K. W. Edmonds et al., J. Appl. Phys. 88, 3414 (2000). © 2000, American Institute of Physics.

Magnetism in Nanoclusters

914 1.5

(a) 1.0

Fe1Ag99 T = 50 - 300K

0.5

0 0.2

-0.5

Probability

Magnetic Moment (10-6 Am2)

is observed between 0.5 and 1 ML thickness, followed by a steep rise between 1 and 1.4 ML. No MCDAD signal is visible at 40 K at any coverage above the noise level. As the coverage increases, the interactions between the clusters lead to a stabilization of the magnetization at 40 K. This, however, produces strong out-of-plane demagnetizing fields, and so the remanent magnetization of the sample is in-plane due to the coherent shape anisotropy of the film. At the highest coverage the remanence exceeds that of an Fe film of the same thickness grown in-situ with a conventional evaporator.

-1.0

0.1

0 0

2 4 6 8 Particle Diameter (nm)

-1.5 -3

7. MAGNETISM IN EMBEDDED CLUSTERS

A comprehensive magnetometry study of Fe and Co clusters deposited from the gas phase and embedded in co-deposited Ag films at very low volume fraction was carried out by Binns and Maher [131]. Magnetic isotherms measured in the plane of the surface, at temperatures 50–300 K, of unfiltered Fe clusters embedded with a 1% volume fraction in Ag are shown in Figure 13a (symbols). Each was fitted by a set of Langevin functions [Eq. (2)] with different moment values where the amplitude of each Langevin function was a fitting variable. Ten size bins were used in the range 0.5– 8 nm and the average amplitude as a function of particle size (moment) after fitting all data sets is shown in the inset in Figure 13a. The calculated curves are displayed as lines and the fit is excellent in every case. The size distribution is the usual asymmetric shape and fitting it to a log-normal distribution (inset) gives a most probable cluster size of 2.57 nm with a standard deviation of 1.95 giving a median diameter of 3.0 nm. This distribution is similar to those obtained by direct STM imaging of deposited cluster films. It was pointed out by Allia et al. [132], however, that this procedure is hazardous. In an interacting system it is possible to obtain an excellent Langevin fit with an “apparent size” that is different from the real size and varies with temperature. Three conditions must be met to confirm ideal superparamagnetism with no interactions between the clusters: • The isotherms should display no hysteresis. • The isotherms should scale with H/T . • The fitted size distribution should be independent of temperature. The lack of hysteresis is evident and the other two conditions are demonstrated in Figure 13b, which is the data in Figure 13a replotted against H/T and an inset that shows the invariance with temperature of the median size obtained from the Langevin fits. This sample thus displays perfect superparamagnetism above 50 K.

0

1

1.5

(b)

1.0

2

3

0.5

Fe1Ag99 T=50K T=100K T=150K T=200K T=250K T=300K

0

4 Diameter (nm)

7.1. Isolated Clusters—The Dilute Limit

-1

Field µ0H(T)

Magnetic Moment (10-6 Am2)

Co-depositing clusters and vapor from a conventional deposition source enables one to embed clusters with a known size distribution and a controlled volume fraction in a matrix of another material. The removal of the requirement to work in-situ allows a wide range of measurement techniques to be applied including conventional magnetometry.

-2

-0.5 -1.0

3 2 1 0

-1.5 -0.05

-0.03

-0.01

0

0.01

100 200 300 Temperature (K)

0.03

0.05

µ0H/T (T/K) Figure 13. (a) Magnetization isotherms in the range 50–300 K of Fe1 Ag99 (open squares) compared to fits by Langevin functions (lines) with a size distribution represented by 10 size bins in the range 0.5– 8 nm. The inset shows the average probability of each bin for the optimum fit to curves at temperatures >50 K () and the corresponding lognormal distribution (line) with dmax = 257 nm and  = 195. (b) The same data plotted against H /T showing the scaling predicted by the Langevin functions. The inset shows the median size from the distributions as a function of temperature and demonstrates the invariance of the fitted size vs T required for an ideal superparamagnetic system. Reprinted with permission from [131], C. Binns and M. J. Maher, New J. Phys. 4, 85.1 (2002). © 2002, IOP Publishing.

The magnetic isotherms measured in the plane of the surface at temperatures 50–300 K of unfiltered Co clusters embedded with a 2% volume fraction in Ag [131] are shown in Figure 14a (symbols) along with the Langevin fits using the procedure already described (lines). In this case the fitted median size displayed in the inset in Figure 14b is temperature-dependent below about 150 K. This could either be due to weak dipolar interactions at the slightly higher volume fraction [132] or to the higher anisotropy of the Co clusters relative to Fe requiring a higher temperature to reach the superparamagnetic limit. In either case the superparamagnetic limit is reached at a temperature ∼150 K. This is also demonstrated in Figure 14b in which the same data are plotted against H /T and only the 100 and 50 K curves are visibly separated from the rest. The size distribution obtained by Langevin fits as described above to the curves taken at 150 K and above along with the fitted log-normal distribution is shown in the inset in Figure 14a.

2.0

(a)

915 1.5

Co2Ag98

(a)

T = 50 - 300K

1.5 1.0

0.5

0.2

-0.5 -1.0 -1.5

0.1

0

-2.0 -2

0

2 4 6 8 Particle Diameter (nm)

0

2

4

Field µ0H(T) (b)

1.5

Co2Ag98

-0.5

K=263,000 Jm-3

-1.0 0

(b)

2

1 2 Field µ0H(T)

1

0

T=150K - 300K T=50K, 100K

-1

1.0 0.5

-2

K=774,000 Jm-3

0

1 2 Field µ0H(T)

0 -3

4 Diameter (nm)

-0.5 -1.0 -1.5

3

-0.03

-0.01

-2

-1

0

1

2

3

3

Field µ0H(T)

2 1 0

-2.0 -0.05

3

Co2Ag98 T = 2K

Moment (au)

2.0

0.0 Moment (au)

0

Magnetic Moment (10-6 Am2)

0.5

-4

Magnetic Moment (10-6 Am2)

Fe1Ag99 T = 2K

1.0

Probability

Magnetic Moment (10-6 Am2)

Magnetism in Nanoclusters

0

0.01

100 200 300 Temperature (K)

0.03

0.05

µ0H/T (T/K) Figure 14. (a) Magnetization isotherms in the range 50–300 K of Co2 Ag98 (open squares) compared to fits by Langevin functions (lines) with a size distribution represented by 10 size bins in the range 0.5– 8 nm. The inset shows the average probability of each bin for the optimum fit to curves at temperatures >150 K () and the corresponding log-normal distribution (line) with dmax = 245 nm and  = 181. (b) The same data plotted against H /T showing the scaling predicted by the Langevin functions for data in the range 150–300 K (open squares). The data at 100 and 50 K (line) show a departure from the scaling law. The inset shows the fitted median size as a function of temperature and demonstrates that the apparent size becomes temperature-invariant (the true size) above 150 K. Reprinted with permission from [131], C. Binns and M. J. Maher, New J. Phys. 4, 85.1 (2002). © 2002, IOP Publishing.

For the Co clusters the most probable size is 2.45 nm with a standard deviation of 1.81 giving a median diameter of 2.8 nm. At 2 K most of the clusters in the Fe1 Ag99 sample discussed above are below the blocking temperature and as shown in Figure 15a the magnetic isotherm develops hysteresis [131]. The remanence, Mr , of an assembly of blocked particles reveals the symmetry of the anisotropy axes and their distribution in space. For example uniaxial anisotropy axes randomly distributed over three dimensions give Mr /Ms = 05 but if they are distributed over two dimensions (2D) in the plane of the applied field, Mr /Ms = 071. The equivalent values for cubic anisotropy axes are 0.82 distributed over 3D and 0.91 distributed over 2D. The measured remanence is ≈0.4 and is thus closest to the case for uniaxial anisotropy axes randomly distributed over 3D. In this case the magnetization between saturation and

Figure 15. (a) Magnetization isotherms at 2 K of Fe1 Ag99 sample:

• field sweeping down, () field sweeping up. The inset shows the decay from saturation (•) compared to a calculation (line) assuming a random distribution of uniaxial anisotropy axes [Eq. (13)]. The best fit anisotropy constant is displayed in the inset. (b) As (a) but for Co2 Ag98 . Reprinted with permission from [132], C. Binns and M. J. Maher, New J. Phys. 4, 85.1 (2002). © 2002, IOP Publishing.

remanence is obtained at each field value by minimizing over all alignments of the anisotropy axes the intraparticle energies E1 = KV sin2 1 −  − B cos 1

(13)

where K is the anisotropy constant, V is the particle volume, and and 1 are the angles between the applied field and the anisotropy axis and particle magnetization vector respectively. The inset in Figure 15a compares the curve calculated thus with the data and it is evident that this simple model reproduces the data accurately. So in zero field the system is a collection of static, randomly aligned cluster giant moments each pointing along the local anisotropy axis. The anisotropy constant is a parameter of the fit and optimizes at K = 263 × 105 J m−3 , which is in reasonable agreement with the value of K = 23 × 105 J m−3 obtained by a SQUID measurement of a similar sample [133] and is about 5 times the bulk value. As discussed next a more detailed study of cluster anisotropy by Jamet et al. [75] showed that the magnetocrystalline term is a small component of the total measured value. A uniaxial anisotropy of the individual clusters is expected as it will be produced by any incomplete atomic shell. Only clusters containing magic numbers of atoms are expected to have a cubic anisotropy but even in this case it is likely to be lost due to other processes during deposition such as collision with the substrate.

Magnetism in Nanoclusters

916 The blocking temperature, Tb , of a cluster with a volume V and anisotropy constant K is given by Tb =

KV kB ln exp /0 

(14)

where exp is the measurement time (∼100 sec for dc magnetometry) and 0 is the lifetime due to the natural gyromagnetic frequency of the particles. A measurement of Fe nanocluster with a similar size distribution embedded in Ag yielded 0 = 10−8 s [133] so for the Fe particles having the median size (3 nm) in the Fe1 Ag99 sample, with the anisotropy constant shown in Figure 15a, a value of Tb = 85 K is obtained. At 2 K clusters with a diameter less than 1.4 nm, or about 5% of the distribution shown in the inset in Figure 13a, are unblocked explaining in part the observation of a remanence less than 0.5. The magnetic isotherms were found to be almost isotropic but a slightly higher anisotropy constant was found by fitting Eq. (13) to the out-of-plane data. This was interpreted as a slight shape or stress anisotropy resulting from the deposition onto the surface. The coherent in-plane anisotropy of the cluster film was estimated to be ∼9 × 103 J m−3 [131]. The isotherms at 2 K of the dilute Co cluster sample are shown in Figure 15b. Again the remanence is slightly less than 50% and assuming the clusters are randomly oriented over 3D and have a uniaxial anisotropy the approach to saturation can be modelled using Eq. (13). This is compared to the data in the inset in Figure 15b and the anisotropy constant obtained is over three times larger than that found in the Fe cluster assembly giving a blocking temperature of ∼20 K. The larger blocking temperature will require a higher temperature than Fe clusters to obtain superparamagnetism as already observed. Jamet et al. [75] provided yet more detail on the isolated properties of embedded nanoparticles by using the

-SQUID technique (Section 3) to determine the threedimensional switching field distribution of a single Co nanoparticle containing about 1000 atoms embedded in Nb. The data could be fitted by the Stoner–Wohlfarth model [6] with the anisotropy constants: K1 = 22 × 105 J m−3 along the easy direction, K2 = 09 × 105 J m−3 along the hard direction and a fourth order term, and K4 = 01 × 105 J m−3 . By estimating the contributions from shape and magnetoelastic terms they concluded that the main source of anisotropy in the particle was the surface anisotropy term described by Néel [134] with the fourth order term corresponding to the magnetocrystalline anisotropy. An XMCD study of Co clusters assembled at low volume fraction in Cu matrices was carried out by Eastham and Kirkman [18] who measured an orbital magnetic moment 2.2 times larger than the bulk value—a similar enhancement to that found in exposed Fe clusters (Section 5). By a careful analysis of the orbital to spin ratio as a function of the applied field, they were able to show that the atomic spin direction is determined only by the external field and the surface anisotropy, which again highlights the importance of this term in determining the cluster magnetization. As pointed out earlier, one of the most important advantages of cluster assembly is the ability to prepare granular films of miscible materials and thus magnetic grains can be

studied in a magnetic matrix. Baker et al. [128] used MXCD to measure the orbital and spin moments in mass-selected Fe clusters coated with Co. As discussed in Section 3 XMCD can focus on the orbital and spin values within specific elements and they found that the Co coating increases the total magnetic moment within the Fe particles to a value similar to that found in free Fe clusters. This was due to a large increase in the spin moment as also observed in Fe–Co alloys and within a rigid band model is ascribed to an increase in the majority spin band population by electrons from the Co [135]. In the case of the cluster-assembled film there is not a well mixed alloy although there may be alloying at the interface [16]. In addition, no significant change was observed in the total Fe L-edge X-ray absorption (“white line”) intensity indicating that there is no significant change in the number of Fermi level d-band holes in the Fe. The increase in the spin moment was attributed to an increase in the Fe valence band exchange splitting produced by the interaction with the Co.

7.2. Interacting Cluster Films Developing a detailed understanding of the effect of interactions between clusters embedded at higher volume fractions is important not only to satisfy scientific curiosity but also in order to fully exploit such systems in the development of high performance magnetic materials, for example high-moment films (see next section). In such applications the required volume fraction of nanoparticles is close to or above the percolation threshold and is well into the strong interaction regime. When the interparticle interactions become significant the system displays a rich variety of magnetic configurations resulting from competing energy terms. The dipolar interaction introduces frustration as it is impossible to produce an optimum alignment for every particle. In addition there is frustration resulting from the competition between the interparticle dipolar and exchange terms and the intraparticle anisotropy energy (magnetocrystalline, shape, magnetoelastic, etc.) that requires the magnetization vector to be aligned along specific axes in each particle. A thorough investigation of Fe clusters assembled in Ag matrices at volume fractions from the dilute limit to pure cluster films with no matrix was carried out by Binns et al. [74]. Below the percolation threshold (volume fraction 1 the magnetic correlation length at zero field is Ra , and the magnetic vector in each particle points along the local intraparticle anisotropy axis. Note that in an arrow representation this state would be identical to that in isolated noninteracting particles at absolute zero. This regime is illustrated in Figure 16b. With increasing interparticle exchange (or decreasing intraparticle anisotropy) the configuration becomes a correlated superspin glass (CSSG) in which the magnetization vector in neighboring particles is nearly aligned but the random deviation of the moments from perfect alignment produces a smooth rotation of the magnetization throughout the system with a magnetic correlation length that is a factor 1/32r larger than the particle diameter. This regime is illustrated in Figure 16c. The disordered CSSG state is fragile and application of a small field produces a “ferromagnet with wandering √ axes” [137] with an approach to saturation that follows a 1/ H dependence in three dimensions [137] and a 1/H dependence in two dimensions [136]. These both change to a 1/H 2 dependence above a crossover field [139] Hco = 2A/Ms R2a . Other studies of pure Fe and Ni cluster-assembled films have used the RA model to analyze magnetization data from cluster-assembled films [140–142]. The magnetic configuration could be described as a CSSG in all cases. This is an important find for technological applications as it shows that dense interacting cluster assemblies are magnetically soft. Binns et al. [131] found that at 300 K, films with an Fe volume fraction >70% formed CSSGs but at 2 K, for all volume fractions, Fe cluster assemblies coated with Ag were in the random magnetization state depicted in Figure 16b. Thus at a sufficiently low temperature the anisotropy dominates the exchange coupling and the magnetization at zero field in each nanocluster is forced to lie along the local anisotropy axis. This is the state predicted by the RA model for 3r > 1. The increase in anisotropy at low temperature was ascribed to magnetoelastic stress induced by the different expansion coefficients of the Fe cluster film and the Ag capping layer. This analysis was supported by the observation that uncapped pure cluster samples transferred into the magnetometer in UHV, as described in Section 3, did not undergo the transition to the random cluster magnetization at low temperature and remained in the CSSG state all the way down to 2 K [74, 131]. A summary of the behavior of Fe cluster assemblies embedded in Ag is provided by the phase diagram shown in Figure 17 [74].

Magnetism in Nanoclusters

918 2.6 super-paramagnetic

2.5 interacting super-paramagnetic (ISP)

100

correlated super-spin glass (CSSG)

2.4 2.3 µ (µB/atom)

T(K)

2.2

collective blocking

10

Bulk spin moment

2.1

Systematic error in XMCD measurement

2.0

single particle blocking

Spin moment measured in thick Fe film

1

1.9

0

25

50

75

100

Fe volume fraction (%) Figure 17. Magnetic phase diagram of Ag-capped Fe nanoclusters (3 nm diameter) assembled in Ag matrices.

8. APPLICATIONS OF CLUSTER-ASSEMBLED FILMS In this section examples of technological applications that exploit the flexibility in the control of magnetic properties afforded by assembling films from nanoclusters will be described.

8.1. High-Saturation-Moment Materials One of the important advantages of the cluster deposition technique in the creation of artificial materials is its ability to make granular mixtures of even the most miscible metals. For example a study using extended X-ray absorption fine structure (EXAFS) has shown [21] that for Co clusters embedded in Pt, alloying is limited to a single atomic layer at the interface leaving a pure core of Co within the grains. The same result was found for Co clusters embedded in a Nb matrix [143]. For Fe clusters embedded in a Co film, XMCD data show that the number of Fe 3d holes per atom is indistinguishable from the bulk demonstrating that the grains consist mainly of pure Fe with intermixing confined to the particle surfaces [16]. Creating granular mixtures of miscible materials is important for the creation of new magnetic materials. A prized discovery would be a substance that had a saturation magnetization density greater than conventional Fe70 Co30 alloy (permendur)—the most magnetic material available today. This would be an important find for a wide range of technologies and most immediately for the magnetic recording industry. The conventional alloy has a magnetic moment per atom of 245 B , which is 10% greater than bulk Fe. Free Fe nanoparticles can have magnetic moments up to 54 B per atom in the case of Fe12 (see Fig. 7) but these reduce significantly when the clusters are deposited on a surface (see Fig. 10) and drop further toward the bulk value when the particles are brought into contact to make a film [14]. On the other hand it has been shown that coating the supported Fe nanoclusters with Co restores the magnetic moment back to the free cluster value [128]. The results are displayed in Figure 18, which shows the spin moment

0

100

200

300

400

500

600

700

Fe cluster size (atoms)

Figure 18. Variation of the spin moment with cluster size in exposed Fe clusters on HOPG (filled squares), Co-coated Fe clusters on HOPG (open squares), and the calculated spin moments for free Fe clusters coated with Co [144]. Also shown is the spin moment measured by XMCD in a conventional Fe film deposited in-situ and the accepted spin moment for bulk Fe. Correcting the measured spin moments in the Co-coated clusters by the discrepancy gives the values shown by the filled circles.

in B /atom of exposed (filled squares) and Co-coated (open squares) Fe clusters on HOPG as a function of size. The data reveal that the enhanced spin moments measured in the exposed clusters relative to that in a conventional Fe film deposited in-situ are increased further by about 10% after coating with Co. Also shown in Figure 18 are the values of the spin moments in Fen Co1021−n clusters, composed of a pure Fe core coated with Co, as a function of n calculated by Xie and Blackman [144] (open circles). These are higher than the measured values but as discussed in Section 6, even after accounting for the Tz term, XMCD measurements of spin moments have systematic errors due to uncertainties in the number of valence band holes, nh , and inaccuracies in applying the magneto-optical sum rules [63, 64] to solid state systems. In addition XMCD is insensitive to any spin polarization of the s–p electrons, which is included in the calculation. An estimate of the systematic error in this particular experiment can be obtained by comparing the spin moment measured in a conventional Fe film under the same experimental conditions to the accepted bulk Fe spin moment. Shifting the measured values for the Co-coated Fe clusters by this difference yields the values shown by the filled circles, which agree well with the calculation. This procedure is, however, insecure since all the sources of error (e.g., nh , s–p spin polarization, etc.) are different in the clusters to the bulk. Clearly however, XMCD underestimates the spin moment and coating the Fe clusters with Co substantially increases their spin moment. The orbital moment averages 0.16 B over the size range shown [128] and adding this to the data gives a total magnetic moment of 2.7 B /atom in 100-atom Fe clusters coated with Co. Thus embedded Fe cluster assemblies in Co matrices hold real promise for creating a new material with a record magnetization density. This depends to some extent on the value of the Co moment but with a high volume fraction of the Fe grains, the Co matrix itself will be in the form of

Magnetism in Nanoclusters

nanoscale grains in which moments as high as 23 B have been observed [145]. Interestingly the calculation by Xie and Blackman [144] revealed a further increase in the spin moment if a degree of Fe–Co intermixing at the interface was included. This suggests that a moderate heat treatment of the cluster-assembled Fe–Co films could yield higher moments. As pointed out in the previous section, dense interacting cluster assemblies are magnetically soft and thus would be valuable in applications such as write heads where a strong local controllable field is required.

8.2. High Anisotropy Binary Clusters The ultimate limit of magnetic recording is defined by storage on individual single-domain particles. For example if these were 5 nm particles separated in both directions by 10 nm, the density would be 1000 Gbits/cm2 , or about 1000 × the storage density in modern hard disk drives. Leaving aside the technical problem of making organized arrays of the particles over macroscopic areas or exactly how to read and write there is the more fundamental problem that transition metal particles of this size are superparamagnetic at room temperature. In order to realize this next level of technology it is necessary to create nanoparticles with a sufficiently high anisotropy to be blocked at room temperature. A promising avenue is the use of transition metal/ rare-earth mixtures as reported by Negrier et al. [146] who made SmCo clusters by using a SmCo5 target in their laser evaporation source. Measurements using X-ray photoelectron spectroscopy (XPS) of the deposited films showed that although the stoichiometry of the target was preserved, the Sm was in the form of Sm2 O3 while the Co was pure. The most likely arrangement is that the Sm oxide forms a skin around a pure Co core. Despite the lack of elemental mixing within the clusters, a significant increase in the coercivity was observed relative to pure Co clusters in a Ag matrix with a corresponding increase in the blocking temperature from 25 to 45 K. Further increases can be expected with improved control of the mixing within the cluster by, for example, using sources with two targets. Any recording application using clusters requires that the clusters be organized into regular arrays over macroscopic length scales. Large-scale self-organization remains one of the holy grails of nanotechnology and several promising techniques have been demonstrated in the case of atomic deposition. Recently Perez et al. [147] demonstrated the formation of organized arrays using deposited Au nanoclusters on functionalized graphite substrates that had been patterned using a focused ion beam. A careful study of all the growth parameters showed that it was possible to obtain conditions where every cluster was at a defect site with none in between. Although it is impossible to obtain a condition where every defect contains a single cluster it is possible to have every defect site contain a small group of clusters. If these were magnetic they would be exchange coupled and act as a storage bit.

9. CONCLUSION This chapter has tried to demonstrate the enormous benefit of studying magnetic nanoclusters in understanding the fundamental magnetic behavior of matter. Addressing this

919 borderland between the atom and the bulk requires the development of a deep understanding of the electronic and magnetic properties of materials and how they are influenced by atomic structure. It is also clear that films made from deposited clusters will form an important class of materials in future technologies.

GLOSSARY Amorphous material A solid material in which the atoms are not arranged on a crystal lattice but are distributed randomly, as in a snapshot of a liquid structure. Anisotropy boundary (or barrier) The energy barrier separating two different alignments of the magnetization in a solid. Anisotropy energy The energy associated with a specific alignment of the magnetization. Antiferromagnetic material A material containing magnetically ordered atoms in which there are equal numbers of atoms with their magnetizations pointing in opposite directions so the material presents no external magnetic field. Atomic number The number of protons in an atom of the element showing its position in the periodic table. bcc lattice Body-centered cubic lattice. A crystal lattice consisting geometrically of a stack of cubes in which the atoms are arranged in the corners of the cubes with an additional atom at the center of each cube. Blocking temperature The temperature below which a nanocluster has a static non-fluctuating magnetization. Bond length The distance between atoms in a solid. Cohesive energy The difference in energy between N atoms as isolated atoms and the same atoms bound in a cluster. Correlated super-spin glass (CSSG) A magnetic configuration in a solid built out of nanoclusters in which the magnetization of neighboring clusters is approximately aligned but over a larger distance becomes uncorrelated. Curie temperature The temperature above which the atoms in a ferromagnetic material become magnetically disordered. Density of states The number of allowed (electron) states per unit energy interval. Dimer Two atoms bonded into a pair. Dipolar interaction The classical long-range interaction between magnetic dipoles (see also exchange interaction). Domain boundary The boundary between two magnetic domains in a magnetic material. Electronic shell The allowed electron states contained within a specific pair of allowed values of the principal (n) and angular momentum (l) quantum numbers. The number of electrons in a shell is determined by the number of allowed values of the remaining magnetic (ml ) and spin (ms ) quantum numbers within the specific (n/ l) pair. Extended X-ray absorption fine structure (EXAFS) The ripples in X-ray absorption strength for photon energies greater than an X-ray absorption edge (see X-ray absorption edge) resulting from inter-atomic X-ray scattering.

920 Exchange interaction A short range (atomic scale) magnetic interaction between two atomic electrons arising from the difference in energy produced by reversing the spin quantum number of one of the electrons. The interaction has no classical analogue and at the range of atomic separations is much more powerful that the classical dipolar energy. fcc lattice Face centered cubic lattice. A crystal lattice consisting geometrically of a stack of cubes in which the atoms are arranged in the corners of the cubes with additional atoms at the center of each of the faces of the cubes. Fermi level The energy of the highest filled electron state in a solid. Ferrimagnetic material Magnetic material containing magnetically ordered atoms with their magnetic moments point in different directions but in which there is a net imbalance of the magnetization in one direction. Ferromagnetic material Magnetic material containing magnetically ordered atoms in which the magnetic moments of all the atoms within a magnetic domain are aligned. Fractal Object with a structure that repeats on smaller and smaller scales and which can be defined to have a noninteger dimensionality. Gaint magnetoresistance (GMR) Anomalously large variation in electrical resistance as a function of applied magnetic field found in nanostructured magnetic materials. Hall Probe An instrument using the Hall effect, i.e. the voltage developed across a current-carrying conductor on application of a magnetic field, to measure the magnetization of a material. Hartree–Fock (also tight binding) A scheme for calculating electronic states by summing over atomic orbital basis sets. hcp A crystal structure consisting geometrically of a stack of planes in which each has atoms ordered in a hexagonal structure. Helical undulators An insertion device in an electron storage ring which indices a helical distortion in the electron orbit to produce circularly polarized synchrotron radiation. Hysteresis Process in magnetic materials whereby the magnetization depends not only on the existing applied field but also on the magnetization history of the sample. Icosahedron Twenty-sided body. Particularly stable structure often found in small atomic clusters. Inert (or rare) gas One of the gases He, Ne, Ar, Kr, Xe, Rn containing only filled electronic shells making them particularly unreactive. Isomer Alternative structure containing the same number of atoms. Laser ablation Process where material is evaporated from a target by an incident high power laser. Macroscopic quantum tunneling The process of quantum tunneling, i.e., the transition of a quantum particle through an energy barrier that is forbidden in a classical system, in objects that are much larger than a single atom. Magic number A number of atoms in a cluster that forms a particularly stable structure.

Magnetism in Nanoclusters

Magnetic anisotropy The tendency of the magnetization in a magnetic material to be aligned along specific directions as a result of a minimum in the anisotropy energy. Magnetic correlation length The distance over which the direction of magnetization in a magnetic material is aligned. Magnetic domain Mesoscopic region in a magnetic material over which the atomic magnetic moments are aligned. A magnetic material tends to form domains since this produces a lower energy configuration than having all atoms in the sample magnetized in the same direction. Magnetic isotherms The magnetization versus applied filed curve for a magnetic material obtained at a specific temperature. Magnetic precession The rotation of a magnetic moment in response to an applied magnetic field. Magnetocrystalline anisotropy The magnetic anisotropy due to the crystal structure of a magnetic material producing a minimum in the anisotropy energy along certain crystallographic directions. Magnetoelastic etc anisotropy The magnetic anisotropy arising from strain along a particular direction in the material. Magnetic circular dichroism in the angular distribution (MCDAD) The variation in the intensity of photoemission along a particular direction arising from a change in the relative alignment of the sample magnetization and the angular momentum of the incident radiation. Mesoscopic A length scale that is large compared to atomic dimensions but small compared to macroscopic sizes. A typical mesoscopic size is of the order of 1 m. Magnetic linear dichroism in the angular distribution (MLDAD) The variation in the intensity of photoemission along a particular direction arising from a change in the relative alignment of the sample magnetization and the linear polarization of the incident radiation. Monolayer A single layer of atoms adsorbed on a surface. Monomer A single isolated atom. Monte Carlo simulation A calculation scheme for systems of large numbers of particles where the evolution of the system is followed by allowing random steps of the particles and determining the lowest energy configuration at each step. Nanocluster (or cluster) A small cluster of atoms typically containing 10–1000 atoms (1–3 nm diameter). Nanostructure A generic name for a structure that is a few nm across. Includes nanoclusters, islands formed by depositing atoms on surfaces, monolayers, etc. Neel temperature The temperature above which the magnetic moment of the atoms in an antiferromagnetic material become disordered. Orbital moment The magnetic moment in an atom arising from the orbital motion of the atoms around the atomic nucleus. Paramagnetic material Material displaying no magnetic order of its constituent atoms but which acquires a magnetization in the same direction as an applied magnetic field. Percolation threshold The volume fraction of grains in a granular material at which there is at least one continuous

Magnetism in Nanoclusters

pathway through touching particles from one side of the sample to the other. Photoemission (or photoelectron spectroscopy) The emission of electrons from a surface resulting from the application of ultra-violet light with a photon energy greater than the electron work function. Photon spin The angular momentum of circularly polarized light. Quantum dots Nanostructure whose dimensions are of the same order as the electron wavelength in all three directions. In such devices the electrons are bound to the structure as a whole with corresponding energy levels. Quantum wires A very narrow strip whose width is of the same order as the electron wavelength in two dimensions but is macroscopic in the third dimension. Remanence The magnetization remaining after a magnetic field has been applied to a ferromagnetic material and then removed. Rotational temperature The temperature associated with the rotation of a nanocluster. Saturating (or saturation) field The magnetic field required to produce magnetic saturation in a ferromagnetic material. Saturation A state in a ferromagnetic material in which all the atomic magnetic moments are pointing in the same direction. Shape anisotropy Magnetic Anisotropy produced resulting from any non-spherical shape of a sample. Single-domain particles A piece of magnetic material that is too small (typically 10 nm), the structural state first of all depends on the temperature regime of the deposition. However, in very thin films the thickness plays a key structural role. This is true both for uniform films and multilayers. For the latter, the characteristic size parameter is the thickness of each individual layer. Transmission electron microscopy and X-ray diffraction are the most widespread methods for studying the structure of all kinds of thin films. Figure 1 shows the electron diffraction pattern and the microstructure images in a dark-field regime for a rf-sputtered Si/[Gd(7.5)/Co(3)]4 /Si sample. This formula contains all the important information about the general structure of the multilayered film. It says that this sample has buffer and covered Si layers and Gd layers of thickness LGd = 75 nm alternated with Co layers of thickness LCo = 3 nm. The structure LGd + LCo = Lp is repeated four times, where Lp is a period of the structure. The main feature of this electron diffraction pattern is two very diffused rings, corresponding to the Gd and Co components. The diffused state of the reflections testifies to the amorphous state of these metals. At the same time, one can define the most probable distances between the atoms (0.3 and 0.2 nm, respectively). Dark-field observation enables us to separate the images corresponding to different diffraction maxima, that is, related to mainly Gd or Co (Fig. 1a and 1b).

Figure 1. Electron diffraction patterns and dark-field microstructure images corresponding to different diffraction maximums for an Si/[Gd(7.5)/Co(3)]4 /Si sample. The systematic interatomic distances are 0.3 and 0.2 nm.

Magnetism in Rare Earth/Transition Metal Multilayers

Table 1. R/T multilayer studies. R element

LR < 2 nm

LR > 2 nm

Tb

10, 35, 37, 41, 42–81

Dy Gd

37, 12, 48, 38, 19, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48,

7, 10, 35, 39, 43, 48, 56, 57, 59, 76, 81–86 37, 54, 56, 85, 87 6, 11, 24, 26, 27, 28, 32, 35, 36, 38, 40, 48, 56, 89–123, 132 38, 48, 56 19, 38, 48, 56, 85, 87, 124 48, 56, 127 48, 56, 129 48, 56 48, 56 48, 56 48, 56 48, 56 48, 56 48, 56 48, 56

Pr Nd Sm Y La Ce Eu Ho Er Tm Yb Lu

48, 18, 56, 48, 38, 56, 56, 56 56 56 56 56 56 56 56

54, 56, 87, 88 28, 31, 37, 38, 90–92 56 48, 56, 87 125–127 129

Note: Publications reporting R/T multilayer studies are listed in this table, taking into account the type of rare earth layer (R) (first column) and the structural state of the R layer. It is known that R thickness is an important parameter. In each particular case there is a certain thickness (Lt ) corresponding to the transition from the amorphous to the crystalline state. Lt depends on composition and fabrication conditions but usually it is about 2 nm. Therefore, we have divided all references into two large groups with respect to this parameter. It is clear from the table that this approach has an internal logic: group one (LR < 2 nm) has collected the data about R/T structures with thin Tb layers and perpendicular anisotropy interesting with respect to magnetic recording; group two (LR > 2 nm) has collected data with relatively thick Gd for model study.

The interfaces, zones between different layers, play a very important role in influencing macroscopic magnetic properties. Although there is great scientific interest in studying the interface problem, many questions about their structure and chemical state are still unanswered. The main point of discussion is the level of mixing of the different layers. Low-angle X-ray diffraction is a powerful method for study of interface. This method provides the opportunity to work in a framework of Woolf–Bregg conditions for reflecting atomic planes, which are situated relatively far apart. In multilayers, the surfaces of the layers act as reflecting surfaces. Therefore, the appearance and position of the diffraction maximum of low-angle X-ray dispersion helps to define the presence of the periodic structure and its period. At the same time, the profile of the low-angle diffraction contains even more profound information about the state of the reflecting surfaces. Extracting this information usually includes a modeling process and the comparison of the calculated diffractogram with the experimental results [27]. Figure 2 shows an example of low-angle X-ray diffraction in Cu–K radiation for the Cu/[Gd(7.5)/Co(3)]20 /Cu. One can see up to five diffraction peaks; these confirm to the very high localization level of the layers. At the same time the total diffraction profile depends not only on the perfection of the interfaces but also on their relative volume in the total volume of the material. Because of this fact, there is a tendency for the number of maxima to be reduced as the period of the multilayered structure gets smaller [2, 130]. Nevertheless, the authors of [4] found, for the rf sputtered R/Fe multilayers (R = Gd, Tb, and Dy), that the presence of the layered structure was maintained up to the layer thickness of the atomic size scale, thus providing the experimental basis for their conclusion about the possibility of preventing mixing in the materials of the type mentioned above. The same conclusion was reached in [24], on the basis of numerical analysis of the low-angle X-ray diffraction profiles of thermal deposited Tb/Fe multilayers. These researchers found a good match between the experimental and the calculated curves, assuming wave-shaped interfaces and no mixing. However, many other publications reported significant diffusion between R and T layers on the basis of Auger electron spectroscopy measurements [23, 27, 43–46, 82, 94, 131]. This method allows definition of 5

10

Intensity (arb. units)

The common feature for such different metals is the very small size ( 520 K and the crystallization of Tb, Fe, and Tb–Fe and as a result destruction of the multilayered structure take place at ta ≥ 770 K [49]. Other methods of influencing the interfaces include the use of different substrates and buffer layers, electrical bias of the substrates, and implantation. The structural characteristics of Gd/Co films [27, 95] prepared by molecular beam epitaxy deposition onto an Al2 O3 substrate with a Mo buffer layer were compared with those deposited onto a Si substrate and a Si substrate with a SiN buffer layer. Low-angle X-ray diffraction analysis shows the maximum mobility of Co into Gd for the Al2 O3 substrate, resulting effectively in a reduction of the thickness of the pure Co layer by more than 50%. A difference in the diffusion level was observed for multilayers deposited onto Si substrate: it is higher in the presence of the buffer layer. The influence of the substrate on the properties of the multilayered films is achieved through substrate relief and formation of a special thermal regime during the deposition. In our view, this last factor could play the key role in the above experiment. The influence of the voltage applied to the substrate (electric substrate bias) during the rf sputtering deposition [138] has also been studied by means of low-angle X-ray diffraction of Gd/Fe films. It was shown that the presence of a negative bias of −30 V causes smoother interfaces and smaller crystalline grains. Possibly these two effects have a reciprocal relationship and the interface smoothing is the result of the increased microstructure dispersion. Different effects were reported for the ion implantation treatment [50, 137, 139]. The simplest consists of layer mixing as a result of irradiation, for example, in a Gd/Co sample by Ar ions with energy of 150–300 keV (a dose of 1016 cm−2

[131]. It was found that in multilayered structures with small periods ( Tcomp the external field direction and the magnetic moment of Fe are also parallel. Above the critical field curve lies the area of the angular phase, this phase is described as twisted. The term twisted magnetic phase was not introduced by accident, but specifically to emphasize a special feature of noncollinear magnetic structure in layered materials. In contrast to uniform ferrimagnets, in multilayers the negative exchange interaction between magnetic elements is spread nonuniformly through the multilayer volume: the highest value corresponds to the interfaces and it decays very fast toward the center of the magnetic layer. As a result there is a spatial nonuniformity in noncollinear structures, in which the angle between the local magnetization and an external field decreases from the separating interface toward the center of the layer. Very weak self-exchange interaction and the dependence of the magnetization of R layers on the thickness produce additional complications. The latter is conditioned by the magnetic action of the 3d layer in the area near the interface on the Gd in this area. As a result, the twisted magnetic structure is formed basically in R layers and is characterized by the angular and amplitude dependence of the magnetization. This model was first developed theoretically [172] and then confirmed by resonance X-ray magnetic dispersion [24, 109]. Magnetic phase transitions can be stimulated not only by magnetic field variation but also in a fixed field by temperature variation (see as an example the measured and calculated T -H diagram for [Gd(8.4)/Fe(4.2)]n in [32]). The temperature dependencies of R/T multilayers magnetization are peculiar in this case. Figure 11 shows MT dependencies

M, Gauss

600

3

a

2

400

1 200 0

0

100

200

b

M, Gauss

600

6

400

5 4

200 0

300

0

100

200

300

T, K Figure 11. Experimental (a) and calculated (b) temperature dependencies of the magnetization of [Gd(7.5)/Si(1)/Co(3)/Si(1)]20 . The curves are recorded in the fields H = (1) and (4) 200 Oe, (2) and (5) 500 Oe, (3) 1000 Oe, and (6) 750 Oe. Reprinted with permission from [112], G. S. Patrin et al., JETP Lett. 75, 159 (2002). © 2002, IAPC “Nauka/ Interperiodica.”

935 for a [Gd(7.5)/Si(1)/Co(3)/Si(1)]20 multilayered structure, with in-plane measurements made for the magnetic field of 200–1000 Oe [112]. All curves have local maxima. Their origin seems to be associated with the existence of a noncollinear magnetic structure and the increase of the magnetization within its frame. The low values of critical fields are related to weak interlayer exchange in the R/T structure because of the presence of the nonmagnetic Si spacer. The increase in the magnetic field strength results in the extension of the temperature interval where the anomalous critical points appear basically by extension in the low temperature region. On the high temperature side, the positions of the anomalous critical points are limited by the Gd transition to the paramagnetic state. R/T multilayers show the disturbance of the collinear structure in the magnetic field, that is, the most general property of all low anisotropy ferrimagnets. Special features of multilayered structures are the low field of the magnetic phase transitions and space nonuniformity of the twisted magnetic phase.

3.4. Phenomenological Models Although results of many experiments on the magnetic properties of R/T multilayers have been published recently, their microscopic theory remains to be developed. The situation is easy to understand, because of the influence of factors such as their complicated structural/chemical state and the strong role of technological factors when it comes to the repetition and comparison of experimental data. Meanwhile phenomenological models of interlayer exchange based on mean field theory have been developed. The simplest model supposes nonuniform layer mixing; for Co/Dy see, for example, [37]. According to this model there is a sine-shaped modulation of the composition over the thickness from pure Co to pure Dy. To calculate spontaneous magnetization, the film has to be sectioned into thin sublayers of corresponding composition of amorphous Dy–Co for each one. The calculation corresponding to all layers enables us to calculate the temperature dependencies of the spontaneous magnetization Ms T for different periods of the layered structure. Using the Co/Dy system as a model, Shan et al. [37] demonstrated the nonmonotonous behavior of the calculated dependencies and achieved reasonable agreement with experiments. But, at the same time, this approach should be considered as something of a simplification, because it does not allow adequate description of the field behavior of the magnetic structure. Mixing of the layers is the key assumption of a more complicated theory [173] where the R/T layered structure is treated on the base of a cubic lattice with partial mixing of the different types of atoms in the interface area. In a Heisenberg model framework, using Monte Carlo simulations, it was shown that the temperature dependence of spontaneous magnetization, compensation, and Curie temperatures may depend on the layer thickness and the interface state. The theoretical and experimental results for Tb/Fe structures are in satisfactory agreement, when one takes into consideration the concentration dependencies of the atomic exchange parameters and variation of the Tb atomic magnetic moment.

In contrast, the well-defined layers approach is used in [174] with the iteration method to find the equilibrium magnetization distribution with temperature and applied field as parameters. The film structure is divided into thin sublayers with uniform magnetization in each one as oriented in the plane of the sublayer. The transfer from one sublayer to the next is characterized by changes to the value and orientation of magnetization in a discrete way. Both the first and second sublayers depend on the kind of atoms under consideration (R or T) and the presence of an effective field in each sublayer. In the absence of magnetic anisotropy and selfdemagnetization, the effective field in each sublayer consists of the sum of the external field and the exchange fields of the adjacent sublayers. The exchange fields between layers of the same composition are positive and those of different compositions are negative. The aim is to construct the configuration of the magnetic moments to minimize the energy in the external field and the exchange energy of the system. The search for equilibrium magnetic configuration is carried out by the iteration method. In the initial state there is random orientation of the magnetic moments of all layers. The first iteration consists of fixing of the magnetic moment of a randomly selected sublayer parallel to the effective field and calculation of its value at a certain temperature. The temperature dependence of the magnetization is described by the Brillouin function. Then the whole procedure is repeated for the fixed moment of the first sublayer and repeated until we have achieved parallel orientation of the magnetic moments and local effective fields in all sublayers and their corresponding values at the temperature under consideration. This state corresponds to the minimum of free energy. In the localized layer model [44, 172, 175] efforts have been made to use an analytical approach to describe R/T structures according to molecular field theory. The spontaneous magnetization at T = 42 K and temperature dependence of the Ms were calculated for Gd/Fe multilayers with different periods of the layered structure. The agreement with experimental data is close enough, given the condition of a sharp decay of the magnetic moment of Fe atoms in the films with less than a 15 nm period. This condition seems to be related to the Fe transition to the amorphous state. Among the studies based on the sharp interface approach, we have listed a theoretical study [176]. Here the exchange interaction is assumed not only to be effective at the interface but also to be spread through the volume by exponential decay. Therefore, the authors use an additional model parameter, the characteristic length of the exchange interaction (lex . Temperature dependencies of the spontaneous magnetization for Tb/Co multilayers with different layer thickness and different compensation temperatures have been calculated according to mean field theory. The best agreement of the theory with an experiment was found at lex = 05 nm. In another nonuniform interlayer exchange model [30, 93, 99, 113], assumptions are made about exponential law for the exchange interaction and the iteration method is used for definition of the equilibrium magnetic structure. Figure 12 (curve 1) shows the temperature dependence of spontaneous magnetization for the [Gd(7.5)/Co(3)]20 structure calculated by this method. The result of Ms T

Magnetism in Rare Earth/Transition Metal Multilayers

600

Ms, Gauss

936

400

2

200

1 0

0

100

200

300

T, K Figure 12. Temperature dependence of the magnetization for a [Gd(7.5)/ Co(3)]20 film: circles, experimental results; (1) calculated by the model of nonuniform exchange interaction between layers; and (2) by the model of partial mixing layers. Reprinted with permission from [113], A. V. Gorbunov et al., Phys. Met. Metall. 91, 560 (2001). © 2001, IAPC “Nauka/Inteperiodica.”

calculations for the mixed layers model, but without taking interlayer exchange into account, is shown in Figure 12 (curve 2) for comparison. The first model describes the experimental data (points) better. The studies [30, 93, 99, 113] take into account the influence of different neighbors on magnetic properties of the R and T layers. As a result changes in the compensation temperature of the Gd/Co films with nonmagnetic spacers were calculated (Fig. 10) and anomalies in the temperature dependencies of the magnetization were explained (Fig. 11b). In summary, there are several phenomenological models of R/T magnetism. The majority of them give a satisfactory description of the temperature dependence of spontaneous magnetization and magnetization behavior in the magnetic field. It is difficult to say which one is the best, because all of them operate by adjusting some theoretical parameters, but we can state the general principle, which helps achieve adequate description of the magnetic properties of R/T multilayers, namely, the presence of long range, nonuniform exchange between R and T layers and the reciprocal dependence of the atomic and electronic structures of the adjacent layers.

4. MAGNETIC ANISOTROPY Why do R/T multilayers attract so much attention in the scientific community? The answer is simple: because they may have perpendicular magnetic anisotropy (PMA) [2]. PMA is uniaxial magnetic anisotropy with an easy magnetization axis perpendicular to the plane of the film; that is, the orientation of the magnetization vector is perpendicular to the plane of the layered structure, even without application of an external magnetic field. Because of this property R/T multilayers are considered as a potentially interesting medium for perpendicular recording, including magnetooptical perpendicular recording [177]. Other interesting magnetic anisotropy features of these materials are very high PMA value [41], localization of most anisotropy parts in areas near to the surface of the magnetic layers [62], and the characteristics of the shape anisotropy of R/T multilayered structures [115].

Magnetism in Rare Earth/Transition Metal Multilayers

937

Macroscopic magnetic anisotropy is the manifestation of structural anisotropy, which appears in one way or another in all magnetically ordered systems. The main kinds of structural anisotropy, which may drive magnetic anisotropy are crystalline lattice, preferred orientation of the axes of likeatom pairs, anisotropy elastic deformation, nonuniform distribution of the defects over the grain boundaries, ordered distribution of microscopic structural formations possessing shape anisotropy, and shape anisotropy of the macroscopic objects. All of the structural features mentioned above can appear in thin films. However, such small subjects have an additional very specific feature, namely, surface anisotropy. This occurs when atoms located near the surface are in a specific position. The surface atoms have asymmetrically arranged nearest neighbors. The volume fraction of these atoms and role of the surface anisotropy increase with a decrease in the thickness of the layer. In multilayered systems, the atoms near the interfaces are in the same situation. Therefore, in the majority of publications related to the study of R/T anisotropy, the most attention has been devoted to the structure of the interfaces. We discussed the size parameters of interfaces in Section 1, but more detailed data on the atomic structure of the interfaces are needed to understand the nature of their magnetic anisotropy. One of the sources of such information is the method of extended X-ray absorption fine structure (EXAFS). On the basis of data collected by the EXAFS method for Tb/Fe, the atomic distribution anisotropy in the interfaces can be quantified [178–180]. The main effect is interlayer diffusion with a preference direction perpendicular to the interface plane. As a result there is asymmetry for nearest neighbors around any selected atom: on the side of the terbium layer there are more Tb atoms among the nearest neighbors and there are more iron atoms on the Fe side. Structural anisotropy is connected with a gradient of chemical components, oriented perpendicular to the multilayer plane. By analogy with perpendicular magnetic anisotropy, this is called perpendicular structural anisotropy (PSA). The PSA unit, Kpsa , is the relative difference of probabilities for two atoms of the same kind to be nearest neighbors in the directions parallel and perpendicular to the film plane [180]. Figure 13 shows the experimental and calculated Kpsa dependencies for iron atoms on the period, Lp , of the Tb/Fe layered structure if the thickness of the iron and terbium layers is the same, LTb ≈ LFe . This dependence is nonmonotonous with a maximum corresponding to the period value of 1–2 nm. This result is confirmed by experimental study of PMA. For example, one can compare the dependencies for the anisotropy constant, Ku , on the thickness of 3d layers in R/T structures of various compositions [37]. All Ku LT dependencies have a maximum. Their position is variable on the LT axis depending on the components, but always in the interval from 0.4 to 1 nm, the same as the position of the structural anisotropy maximum. At the same time linear decay of Ku was reported [57] for Tb/Fe structure in the interval of Lp from 0.6 to 5 nm. Although the

K PSA

4.1. Structural Origins of Perpendicular Magnetic Anisotropy

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5

2

1

3

0

1

2

3

4

5

6

7

8

Bilayer period (nm) Figure 13. Comparison of experimental (circles) and calculated (lines) perpendicular structural anisotropy obtained from Fe EXAFS and simulation, respectively. Different curves corresponds to different diffusion degree. Reprinted with permission from [180], Y. Fujiwara et al., J. Magn. Magn. Mater. 177–181, 1173 (1998). © 1998, Elsevier Science.

relationship between PSA and PMA is not contested, PSA is mentioned in other studies [4, 39, 55, 181, 182]. The PMA value was also found to be almost independent of the substrate; that is, mechanical tensions which sometimes arise on the substrate–film boundary have very little effect [181].

4.2. Mechanisms of Magnetic Anisotropy The fact that perpendicular magnetic anisotropy in R/T multilayers depends on the structural magnetic anisotropy does not define the mechanism of magnetic anisotropy. The atomic structure of the interlayer interfaces described above has given rise to two different explanations of the principal governing PMA mechanisms. The first is two-ion anisotropy as a result of different dipole interactions of neighbor atoms (the Neel–Taniguchi and Yamamoto model). The second is single-ion anisotropy, which may occur because of spin– orbital interaction in metals that have anisotropic electron shells. Most research tends to favor the second theory. One of the arguments supporting the second point of view is easy to accept: strong PMA appears only in multilayers that contain anisotropic electron shell ions [58, 83, 84, 183]. The data on the anisotropy of Dy/Fe, Dy/Co, Tb/Co, Gd/Fe, and Gd/Co systems can be found in [37]. In multilayers with anisotropic Dy and Tb ions the perpendicular anisotropy constant is 1 order of magnitude higher than that corresponding to the spherical Gd ion. The authors of the Nd/Fe, Pr/Fe, and Gd/Fe study [38] came to the same conclusion. However, nonspherical electron shell shape, that is, a nonzero orbital ion moment, is not in itself a sufficient condition for the formation of PMA. For instance, the comparative study of Fe/Nd and Fe/Tm multilayers [136] resulted in the fact that PMA was found only in the Fe/Nd multilayer although Tm is also a nonspherical ion.. The complete analysis of all R/Fe spectrum multilayers (R = Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, or Lu) can be found in [56]. Only nonmagnetic Y, La, Yb, Lu, and Gd with zero orbital moment do not lead to single ion anisotropy formation. For other elements there is no prohibition, but PMA was found only in multilayers containing Pr, Nd, Tb, and Dy. Finally, not only is anisotropy of the electron shell important, but also the configuration of the crystalline field of the neighbor ions is important. Perhaps the atomic structure of the interfaces forming the crystalline field depends

938 slightly on the kind of rare earth atoms, but these variations are small and difficult to detect. An even more surprising conclusion was drawn recently [69] on the basis of X-ray magnetic circular dichroism analysis of Tb/Fe multilayers where transformation of the Tb electronic structure was observed as a consequence of a decrease in the layer thickness. It consists of the loss of orbital moment by Tb ions automatically prohibiting their taking part in single ion anisotropy formation. In the Dy/Co study [181], the anisotropy dipole interaction of the atomic magnetic moments approach is used to describe PMA, in the same way as for Gd–Co amorphous films [184, 185]. However, in the multilayered structures the anisotropy carriers are localized in the interfaces, which leads to an almost incredibly high value of the parameter that defines the pair distribution anisotropy.

4.3. Influence of Various Factors on the Magnetic Anisotropy To clarify the nature of the magnetic anisotropy in R/T multilayers, continued detailed investigation of the atomic structure of both magnetic layers and interfaces is needed. The lack of direct experimental data has stimulated many indirect studies of the influence of different factors on the PMA. As a result of the temperature dependence of the Dy/Co multilayer magnetic anisotropy study [54], the PMA value at T = 42 K was found to be many times stronger than that for T = 300 K. The Ku temperature dependence correlates with 2 temperature behavior. This seems to support the singleMDy ion theory of the PMA mechanism. Many attempts have been made to clarify the connection between different characteristics of interfaces and PMA, but it is not easy to make a comparative analysis of different authors’ data because of differences in preparation and experimental methods. The Mössbauer effect has been very useful for obtaining information about the magnetic anisotropy of R/T multilayers. In particular, it was shown that in Nd/Fe multilayers, magnetic anisotropy increases as the interface thickness decreases [186]. The decrease of PMA due to interface roughness was described in [68, 187]. In addition, the conclusion connecting PMA with a very wide mixed interface in the Tb/Fe system was reached [188]. In Dy/Co multilayers, the maximum PMA value was found at 0.6 nm for both Dy and Co layers. In this condition, the ratio of the interface size and the total volume is maximum [54, 87, 189]. It must be kept in mind that the interface structure may depend on the deposition order of the layers. The delicate experiments for the magnetic anisotropy study for twolayered Tb/Fe and three-layered Tb/Fe/Ag films [59–61, 190, 191] were carried out in this context. The interface was considered to be the major source of PMA for Tb deposition onto the Fe layer. The interfaces are sharp and, in this case, Fe has a crystalline structure. Fe deposition onto the Tb layer creates rough interfaces and initiates the amorphization of the layers [92]. The influence of different interfaces on PMA was investigated in the [Tb/Fe/Y]10 , [Y/Fe/Tb]10 , and [Tb/Fe]10 systems [62]. The Y layer was introduced to exclude the Tb on Fe and Fe on Tb boundaries. All systems had more or less pronounced PMA. At temperatures below 100 K, stronger anisotropy was detected for Tb on Fe than

Magnetism in Rare Earth/Transition Metal Multilayers

for Fe on Tb. In the 250 to 300 K temperature range, Fe on Tb related anisotropy was higher. A possible explanation might include the differences in magnetism for the two types of boundaries. At low temperatures strong single ion Tb anisotropy corresponding to the sharp interface dominates. At temperatures higher than that of Tb magnetic ordering (219 K) it decreases, but weaker anisotropy related to mixed boundaries starts to be important. The mixed boundary actually consists of a ferrimagnetic alloy of Tb–Fe with high Neel temperature. The length of this boundary results in low PSA and correspondingly small PMA. Comparison of PMA values at room temperature revealed the lowest Ku value for Tb on Fe, an intermediate value for Fe on Tb, and the highest for the two types of boundaries together because, apart from the one-ion anisotropy in Tb/Fe samples, dipole interaction between Fe layers through the Tb layer may take place. This mechanism was suggested as an explanation for the CeH2 /Fe system anisotropy [193]. The magnetic layers themselves may play a certain role in the formation of magnetic anisotropy. The main point here is the influence of the thickness on the magnetic properties of each layer in multilayered structures. For example, the increase of magnetization and consequently of the demagnetizing field is the result of increased layer thickness in Tb/Fe multilayers [41] together with a decrease in PMA [67]. At the same time, amorphization of the relatively thin layers appears. At room temperature both Fe and Tb lose their magnetic order whereas a decrease in magnetic phase volume leads to a corresponding decrease in the PMA value. Similar ideas were considered for R/Fe (R = Pr, Nd, and Tb) system analysis [48]. There are data about the presence of PMA only in those Tb/Fe structures that do not have pure terbium [43, 44, 194]. When pure Tb layers have been included in a multilayered structure, PMA has been detected at temperatures below 220 K, that is, after the Tb transition in the ferromagnetic state. It is on this basis that the conclusion about PMA induction in the interfaces in the presence of the exchange interaction between Fe layers through the Tb layers has been reached. A similar result has been reported in a study by transmission electron microscopy of pure Tb segregation with Tb layer thickness of more than 1 nm [64]. The occurrence of segregation coincides with PMA disappearance at room temperature. However, there are also data about the presence of PMA in Tb/Fe multilayers with LTb = 2 nm [65]. One interesting feature of the structural transition taking place in R/T structures as the layer thickness changes is the iron crystallization and, as a consequence, an increase in PMA, which is caused by the formation of a sharp interface between the layers [42]. Comparing the multilayer properties formed at two different substrate temperatures of 150 and 300 K, these authors found that iron crystallization took place at 300 K with thinner layers and it was precisely these samples, with a minimum thickness of crystalline Fe, which had the maximum PMA value. The increase of the perpendicular magnetic anisotropy value after heat treatments of Tb/Fe has been explained in other research by the very sharp interface formation [49, 63]. The temperatures and thickness of the layers related to structural changes of multilayers vary from one publication to another because of dependence on varying preparation methods.

Magnetism in Rare Earth/Transition Metal Multilayers

4.4. Enhanced Magnetostatic Effect The majority of the results discussed are related to R/T multilayers with perpendicular magnetic anisotropy. The most important condition of PMA formation is the presence of layers with anisotropic rare earth ions in a multilayered structure. Gd is irrelevant in this case. It makes Gd/T multilayers less attractive for magnetic recording applications. However, due to high Curie temperature and collinear ferromagnetism, Gd/T can be very useful as a model system for studying R/T structures [93, 114, 174, 197, 198]. Specifically, study of the magnetic anisotropy of a layered ferrimagnetic [Gd(7.5)/Co(3)]20 structure was undertaken in a magnetic compensation region of 200 to 300 K including the compensation temperature (Tcomp ≈ 240 K) and no perpendicular anisotropy was detected [115]. In accordance with shape anisotropy, the magnetization was oriented in the plane of the films, but the actual shape anisotropy constant was much higher than 2Ms2 , where Ms is the spontaneous magnetization of the ferrimagnetic structure. Figure 14 shows KT

and 2Ms2 T dependencies in the magnetic compensation region. This discrepancy disappears if we take into account an additional contribution in magnetostatic energy related to the rising of the magnetic charges in the interface areas. Because of a negative exchange interaction between Co and Gd, the surface density of these charges is defined as a sum

1.0

1

0.8

3

Ku (10 erg/cm )

0.6

6

Let us now return to the results of the R/T study at different temperatures. We have discussed the possibility of a significant increase in PMA values at low temperatures, but this is true only for structures with relatively thin layers, where the interfaces play the main role. At greater thicknesses the temperature decrease may cause an opposite effect, even so far as to change the magnetization orientation to the film plane from perpendicular into parallel. Such a reorientation of the spins is brought about by a change in the correlation between the induced perpendicular anisotropy and shape anisotropy. The existence of the reorientation itself and the reorientation temperature both depend on the average chemical composition of the multilayer (correlation of the thickness of the layers) and period of the layered structure [41]. The dependence on the multilayer period to a certain extent can be explained by changes of the interlayer exchange interaction whose role increases with a decrease of the period of the layered structure. Similar results were reported in [195], although in [63, 196] the authors found no dependence of the magnetization orientation on the thickness of the Fe and Tb layers at low temperatures. An additional important factor influencing PMA is the number of periods, n, in multilayer structure [55, 66]. For [Tb(0.42)/Fe(0.75)]n magnetization is oriented in the plane of the layered structure if n varies from 1 to 30. Increasing the number of layers (n) up to 35 causes PMA to occur, but its value shows only very small changes for further increase in n. For the [Tb(0.95)/Fe(1.15)]n structure, PMA appears at n ≥ 5, but the authors do not discuss the possible origin of these variations [55, 66]. We might suppose that the temperature regime of deposition plays an important role because it may vary over the extended period of total deposition time needed for structures with high n value.

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0.4 0.2 0.0

2 200

220 240 260

280 300

T (K) Figure 14. Temperature dependencies of magnetic anisotropy constants measured by a torquemeter (1) and calculated as Ks = 2Ms2 (2), respectively. Reprinted with permission from [115], V. O. Vas’kovskiy et al., J. Magn. Magn. Mater. 203, 295 (1999). © 1999, Elsevier Science.

of the normal components of the magnetizations of the layers, but not as a normal component of total magnetization of the ferrimagnetic Gd/Co structure. A similar enhanced magnetostatic effect was also observed in Gd15 Fe85 /Gd40 Co60 amorphous multilayers [199].

5. ATTENDANT PROPERTIES Magnetoordered substances possess a number of physical properties that are not directly related to the set of inherent magnetic properties, but become available because of the existence of spontaneous magnetization. Usually, they are revealed during magnetization and consist of changes in size (magnetostriction) or electrical resistance (magnetoresistance effect) of the samples or of changes in the characteristics of the refracted or reflected light (magnetooptical effects), the absorption of the electromagnetic field energy (magnetic resonance), etc. Thus, generally there is a possibility of driving the nonmagnetic parameters of these materials and of products made from them using a magnetic field. Very often properties that are precisely attendant on magnetism define the importance of the material for technological applications. The values of the effects described above, as well as those of spontaneous magnetization, are defined to a considerable extent by the chemical composition but may also reflect the structural state. Furthermore, many features of nonmagnetic properties depend on the type of magnetic structure and on features of their magnetic anisotropy. In this section we review very briefly these related phenomena in R/T multilayers.

5.1. Magnetostriction The elastic deformation, which, in the presence of magnetic ordering, occurs in a crystal lattice or in an amorphous structure of a solid-state body, reflects the existence of the exchange and the magnetic interactions between elemental carriers of magnetism. As a result the volume and shape of macroscopic subjects are changed. We say that volume and, accordingly, linear magnetostrictions take place. Linear spontaneous magnetostriction is important for applications, because the elongation or shortening of the magnetic sample

in a certain direction depends on the spontaneous magnetization orientation. The magnetization process takes place by rotation of the magnetization vector in the whole sample or only in part of it, and this causes a rotation of the total or local deformation axis resulting in changes in the sample size in the direction of the applied field. Deformation caused in this way is defined as magnetostriction. Magnetostriction is an even effect; that is the deformation does not depend on the sign of the magnetization. Very high, giant magnetostriction is a characteristic feature of many rare earth magnets. As for single-ion magnetic anisotropy, it is caused by rare earth ions. The magnetization rotation caused by spin–orbital interaction is accompanied by a change in the orientation of the electron orbits of these ions in an intercrystalline field and deformation of the atomic structure and of the sample as a whole. Among the crystalline magnets with giant magnetostriction, a TbFe2 crystalline intermetallic compound is well known. In the thin film state it is usually amorphous but shows relatively high magnetostriction. The main disadvantage in terms of practical applications for this material and for materials of similar composition is their low magnetic susceptibility. Many efforts have been made to decrease magnetic anisotropy while keeping the high magnetostriction of R-based materials. Among other possibilities is that of creating multilayered thin film structures with optimal magnetoelastic properties, for instance, trying to improve the magnetic susceptibility of a Tb–Fe film by introducing soft magnetic material sublayers with high magnetization (like Fe) [200]. The best results were achieved in a [Tb04 Fe06 (4.5)/Fe(6.5)]n system with strong exchange interactions between the elements of the layered structure. After heat treatment at 550 K the magnetostriction in the low field of 200 Oe is 1 order of magnitude higher than that corresponding to single-layer films of the same composition, and this sounds very promising. A similar effect of introducing Fe sublayers has been reported for Tb342 Fe658 films [201]. Figure 15 shows the dependence of magnetostriction, , on the strength of the magnetic field H for single-layer Tb–Fe films and for [Tb3422 Fe658 LTb /Fe(10)]n with different thicknesses of Tb–Fe layers. The authors believe that creation of the layered structure results in a decrease in the strong perpendicular anisotropy, which is typical of single-layer films of this composition and as a consequence facilitates the in-plane magnetization process. The authors have also pointed out that the multilayers have a ferrimagnetic structure that transits in a noncollinear phase in the field of about 1 kOe. The magnetostriction curves H reflect in some way the magnetization process, opening the possibility of using it as a tool for the R/T multilayer study. The same opinion is expressed in relation to the Gd/Fe ferromagnetic resonance study [116], in which the magnetoelastic deformations were found to depend on the deposition methods. Specifically multilayers prepared by DC sputtering have higher magnetostriction than those prepared by rf sputtering. This can be explained by the presence of sharp interfaces in the first type of films.

Magnetism in Rare Earth/Transition Metal Multilayers

Elastic coupling coefficient, b (MPa)

940

a

20 b

15 c

10 d

5 e

0 0

200

400

600

800

1000

H (Oe) Figure 15. Elastic coupling coefficients (magnetostriction) of [Tb–Fe (X nm)/Fe(10 nm)]n multilayers with various Tb–Fe thicknesses: (a) X = 15, (b) 20, (c) 25, and (d) 30 nm and (e) a Tb–Fe single layer film measured in the longitudinal direction of the magnetic field. Reprinted with permission from [201], T. Shima et al., J. Magn. Magn. Mater., 239, 573 (2002). © 2002, Elsevier Science.

5.2. Magnetoresistance Magnetoresistance is the change in electric resistance, R, of a ferromagnetic conductor caused by a magnetic field. This is a galvanomagnetic effect. In magnetically ordered materials it reflects a magnetic contribution to the conductive electron scattering, which depends on the value and orientation of the spontaneous magnetization. Therefore, one may separate the two components. In uniform magnets only the component related to the magnetization orientation is of practical interest, that is, anisotropic magnetoresistance. The magnetic field causes magnetization rotation and changes the electrical resistance of the sample, and thus the effect of the anisotropy of the magnetoresistance takes place. As for magnetostriction, magnetoresistance is an even effect. Some time ago, a new type of magnetoresistive effect, giant magnetoresistance (GMR), was discovered in multilayered and granular magnetic structures [154, 202]. It consists of the change in electrical resistance at a variation of mutual orientation of magnetizations in heterogeneous magnetic structures with ultrathin elements. Let us emphasize that for both magnetoresistance and giant magnetoresistance we are dealing with direct current flowing over the ferromagnetic conductor. The discovery of GMR in Fe/Cr systems was preceded by the discovery and study of exchange coupling between magnetic layers across a nonmagnetic metallic layer [151, 153, 202] (see Section 3.1). The GMR effect occurs in a Fe/Cr system at a certain Cr thickness when antiferromagnetic coupling takes place and adjacent Fe layers have the opposite orientation to the spontaneous magnetization. When an external magnetic field aligns magnetizations in parallel configuration, the multilayer resistance decreases. Although GMR of Fe/Cr is induced by antiferromagnetic coupling, it was shown that antiferromagnetic coupling is not a necessary condition of GMR which can be caused, for example, by hysteresis effects [203, 204]. GMR is due to propagation of spin-polarized conductive electrons through the ferromagnetic films, divided by sublayer or spindependent scattering at the interfaces. In R/T multilayers both the first effect, magnetoresistance, and the second, giant magnetoresistance, take place. R and T layers are not identical in terms of anisotropic magnetoresistance [91, 130, 170] because, compared with

Magnetism in Rare Earth/Transition Metal Multilayers

rare earth metals, 3d metals have much higher anisotropic magnetoresistance. Because of the high specific resistance in R layers, the electric current flowing in a multilayer is pushed out from the R to the T layers as well. Therefore, the magnetoresistance of a R/T system reflects mainly the magnetoresistance of its T layers. It was shown in Section 3.3 that this feature is successfully used for registration and study of the magnetic phase transitions induced by a magnetic field [100, 110, 117]. Moreover, magnetoresistance analysis may yield certain information about the atomic structure [118]. The word giant is not very appropriate for describing the spin-dependent resistance of R/T multilayers because the value of the effect is small, even compared with ordinary anisotropic magnetoresistance. The small value of the effect makes it difficult to study when the geometry of the experiment determines that the electric current and the applied field are parallel to the plane of the film (CIP geometry). If the electric current flows perpendicular to the plane of the film (CPP geometry) and an external field is applied in the plane of the multilayered structure [202] and, therefore, magnetization occurs in the plane of the sample, the GMR is easier to measure because anisotropic magnetoresistance is absent. It is absent because magnetization and the electric current are perpendicular all the time. There is an opinion that resistance changes in the field take place because of deformations of the ferrimagnetic structure caused by the magnetic moments of R and T layers [110]. Figure 16 shows an example of RH dependence for a Gd/Co structure. High fields used in this, as in many other similar studies, cause a paraprocess in the gadolinium subsystem. Changes of the spontaneous magnetization of Gd should influence the RH behavior. The low value of magnetoresistance is a great disadvantage for R/T multilayers in technological applications and makes them noncompetitive with many other magnetoresistive materials. At the same time, it is possible to use R/T structures as a part of more complicated systems showing GMR. Figure 17 shows the magnetometrical and magnetoresistive hysteresis loops for a [Tb(1)/Co(1)]16 /Co(5)Cu(2.5)/Co(5) sample [70]. Here the multilayered [Tb(1)/Co(1)]16 element with in-plane magnetic anisotropy and a high coercive field acts as a pinning layer. The [Tb(1)/Co(1)]16 structure and the adjacent Co layer are

∆R (Ω)

0.000 -0.001 -0.002 -0.003 -30 -20 -10

0

10

20

30

H (kOe) Figure 16. Resistance changes R in CPP geometry for a [Co(10) nm/ Gd(10 nm)]50 multilayer as a function of the applied magnetic field measured at 4.2 K. The initial resistance before applying the magnetic field is defined as R = 0. Reprinted with permission from [110], H. Nagura et al., J. Magn. Magn. Mater. 240, 183 (2002). © 2002, Elsevier Science.

941

Figure 17. Magnetometrical (a) and magnetoresistive (b) hysteresis loops for the [Tb(1)/Co(1)]16 /Co(5)/Cu(2.5)/Co(5) spin–valve structure.

connected to each other by exchange interaction and the multilayered structure provides the stability of the Co layer in the external field. At the same time, another Co layer, separated from the first by a Cu spacer, has the possibility of remagnetization in a small magnetic field. As a result, mutual magnetization of the cobalt layers may be changed by an external magnetic field, that is, GMR, which, in this particular case, could be better described as the spin–valve effect, occurs. The term spin–valve relates to a case where the possibility of separate remagnetization of the layers is caused not by the negative exchange interaction between the layers, but by the difference in the magnetic anisotropy or the coercive field of the layers [203, 205].

5.3. Magnetooptical Kerr Effect Magnetooptical effects are certain changes of optical properties of a medium caused by application of the external magnetic field. In magnetically ordered materials they have special peculiarities due to the existence of spontaneous magnetization. From both basic and practical points of view, the most interesting are the Faraday and Kerr effects. They consist of the rotation of plane polarization at light refraction through (Faraday) or reflection from (Kerr) the magnet. In metals, the light penetration depth is limited by a skin effect and does not exceed 100 nm. Therefore, for multilayers only the magnetooptical Kerr effect (MOKE) is really interesting. The suitably smooth surface of multilayers is an advantage in this case. Different effects are produced depending on the direction and angle of the light incidence. The polar effect corresponds to geometry when magnetization points along the surface; a normal effect results in the rotation of the plane of polarization as a function of the perpendicular component of the magnetization. The longitudinal effect describes the plane polarization rotation for the light inclination onto the surface of the sample when the position of the magnetization is in the plane of the sample. The longitudinal effect can also be used for perpendicular polarization. It is proportional to the magnetization component in the plane of the light incidence. The transverse orientation effect in which the magnetization is perpendicular to the plane of incidence produces little contrast in a visible image [206]. There are a number of reasons for interest in the Kerr effect as observed in R/T multilayers [16, 20, 71, 72, 88, 92, 119, 178, 206–212]. The first is the comparison of the optical properties of multilayers with single-layer analogous

942 material. The authors of [207] have studied the Kerr rotation spectra for uniform amorphous and multilayered Gd/Fe films. The value of the effect is similar in both samples, but the layered structure opens the way for an additional technological possibility of controlling the magnetic and magnetooptical properties [190] useful for applications such as magnetic recording. Both high anisotropic Tb/Fe and low anisotropic Gd/Co films are interesting for the technique of super high resolution [16, 73] (see Section 5.4) as well as for other applications. The second line of interest is based on the Kerr effect using the magnetization process study method [88, 178]. The magnetooptical effect, in contrast to magnetostriction and magnetoresistance, is an odd effect; that is, it contains information about the sign of magnetization. As we mentioned in Section 3.3, the magnetooptic hysteresis loops are more informative than magnetometric loops for studying phase transitions. One may explain this finding by taking into account the fact that T metals have higher plane polarization rotation values than R metals. Figure 10 shows an example of a magnetooptical hysteresis loop observed via the longitudinal Kerr effect for perpendicular polarization. Direct domain structure observation is possible through the Kerr effect, although the number of publications related to R/T multilayers is small [66, 74, 92, 213]. Structural investigations can also be undertaken via the Kerr effect [20, 119]. Elegant, quick, and appropriate ways of detecting different types of coupling in layered magnetic structures by means of the MOKE were presented as the basis for the magnetic domain study [204, 214].

5.4. R/T Structures and Magnetooptical Recording Can you imagine our community without audio and video recorders, CD-ROMs, etc.? They are part of our lives. The two basic methods of recording are magnetic and optical. Here we describe very briefly the general principles of magnetooptical recording and the role of R/T materials. The principle behind any recording is representation of numbers by the state of a bistable physical system; therefore, in the recording process we are dealing with binary numbers. Any number in a binary system can be written as a specific sequence of the elements equivalent to “1” or “0.” In the magnetooptical recording medium we have a rather complicated multilayer structure of different layers, not all of which are magnetic. At the beginning, the homogeneously magnetized medium stores no information. We can then write onto it the required sequence of digits by a process of thermomagnetic writing, that is, cooling in or without a local magnetic field after local heating by a short laser pulse to a temperature close to the Curie temperature. Any zone with initial magnetization direction may represent the information unit “1.” If a field biased in the opposite orientation is present during cooling, a magnetic domain is formed with opposite magnetization orientation to the initial magnetization. This represents the information unit “0.” The size of the domain is determined by the temperature and by the profile of the temperature-dependent coercivity, that is, the magnetic properties of the magnetic film used. The stability of this information unit depends on the type

Magnetism in Rare Earth/Transition Metal Multilayers

of the domain wall between the homogeneous part of the disc and the information unit, on the magnetization, and on the anisotropy energy. The reading process is performed by using the magnetooptical Kerr effect at typical wavelengths of 780 and 820 nm. To obtain the maximum signal, the magnetization should be perpendicular to the film plane (perpendicular recording) [212]. The quality and stability of the writing and subsequent reading depend strongly on the material properties of the magnetooptical films. PMA is required to keep the magnetization of the domains normal to the plane of the film/disc. Therefore, the energy of the demagnetizing field must be compensated for by the energy of the perpendicular anisotropy. This condition can be fulfilled when the magnetization is sufficiently small. It is achieved in modern devices by use of amorphous thin films consisting of R and T metals, coupled antiferromagnetically. The film thickness is on the order of 50 to 100 nm. Tb25 Fe75 is the most usual composition. In closely related compounds Tb and Fe are partially substituted for by other R metals (Dy and Gd) and T metals (Co and Ni), respectively. As mentioned in Section 3.1, different temperature dependencies of the magnetic T and R sublattices of the films normally have a compensation point, where the magnetizations of R and T cancel each other out, and the total magnetization is zero. The compensation point can be chosen close to room temperature by appropriate composition selection. As a consequence of a very small magnetization, the coercivity becomes very large at the compensation temperature and a very high field would be required for changes of the magnetic moments. As a result, a domain structure written at a temperature of about 500 K, close to Curie temperature, is very stable against external fields at room temperatures. In addition to the magnetic layer for magnetooptical processing, the recording disc incorporates a nonmagnetic antireflecting coating, which reduces the polarization component of directly reflected light. To increase the signal-to-noise ratio a magnetooptical double layer (Tb11 Fe64 Co25 /Tb24 Fe65 Co11 [215] or trilayers or quadrilayers with an additional reflecting layer [216] can be used. To increase data access time, the erasing process can be incorporated into a so-called direct overwrite cycle [217]. One solution applies exchange-coupled double layers (ECDLs) as storage media [218]. These ECDLs can be used for magnetic super-resolution [16, 73], where one of the layers works as a magnetooptical mask for the super-resolution. It allows an increase in the density of information recording. One may see from our very short description that a magnetooptical recording medium contains a number of different magnetic and nonmagnetic layers. From this point of view they are similar to R/T multilayers. The composition of magnetic layers in a magnetooptical structure includes R and T elements. R/T multilayers may have suitable magnetooptical and magnetic properties for them to be used in real devices. The basic principles of magnetooptical recording have many points in common with the physics of R/T multilayers. We can say, therefore, that R/T systems could be considered as a model structure or potential candidates for magnetooptical recording.

Magnetism in Rare Earth/Transition Metal Multilayers

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5.5. Magnetic Resonance

6. CONCLUSIONS

Magnetic resonance consists of the intensive absorption of high-frequency field energy by a magnet when the frequency of the oscillations of the external field and oscillations of the magnetic moment of the magnet are equal. In magnetically ordered materials this resonance frequency is defined by the effective field, which reflects all interactions related to magnetization. The components of the effective field are the demagnetizing field, the anisotropy field, and exchange (including interlayer) interactions. This explains a potential interest in ferromagnetic resonance as a method to study R/T magnetism. There are just a few publications on R/T structures. Let us mention the ferrimagnetic Tb/Fe [59], Gd/Fe [116], and Gd/Co [99, 120, 121] multilayers. Interesting results were reported for a [Gd(7.5)/Si(LSi /Co(3)/Si(LSi ] study at various temperatures. The magnetic compensation appeared in this structure in the range of 100 to 300 K. The compensation temperature depends on the thickness of the Si sublayer LSi . To describe ferromagnetic resonance in uniform two-sublattice ferrimagnets, the effective magnetomechanic ratio, eff , is used. This is defined as the difference between magnetizations of the sublattices divided by the difference of the specific mechanical moments of these sublattices. In practice, it is more convenient to use its normalized value, known as the effective g-factor, geff , instead of using eff . Figure 18 shows the temperature dependencies of geff calculated on the basis of experimental resonance field data for the samples with LSi = 03 nm (curve 1) and LSi = 05 nm (curve 2). Dashed vertical lines mark Tcomp . The character of the geff T changes near the compensation temperature and is different at high and low temperatures. There is a break point in the curve corresponding to Tcomp . This can be interpreted in terms of sequential changes in the states of compensation of the magnetic moments and mechanical moments of the Gd and Co layers as the temperature increases. Therefore, there is a general possibility of using phenomenological ferromagnetic resonance theory to interpret the resonance properties of layered ferrimagnetic structures.

Multilayered structures consisting of rare earth and 3d metal layers have been intensively studied for the last 15 years. Different technologies for their preparation such as ultrahigh vacuum thermal evaporation and ion sputtering have been developed. Independently of the preparation method, these subjects may be described as a system of magnetic layers connected by exchange interaction. It is precisely this type of interaction, together with the individual properties of the different layers, that defines the R/T system magnetism. Rare earth (Tb, Gd, and Dy) and 3d (Fe and Co) metals are those used most frequently for the sublayers that define the negative exchange interaction and ferrimagnetic ordering in a system of magnetic layers. The characteristic feature of the temperature dependence of spontaneous magnetization is the existence of magnetic compensation. The quantitative characteristics of the spontaneous magnetization and compensation temperature depend on many parameters, including the sublayer thickness ratio, period of the layered structure, structural state, and the presence of nonmagnetic spacers. The main structural features of R/T multilayers are amorphization, which appears at a relatively low thickness of layers, and the existence of interface areas. The morphology of the interfaces depends on the preparation technology, order of the layer deposition, and subsequent treatments. R/T multilayers with anisotropic Tb or Dy ions show strong perpendicular anisotropy, which has a single ion nature and is caused by anisotropy of the atomic structure of the interfaces. An enhanced self-demagnetization effect takes place in all ferrimagnetic multilayers. It occurs because of formation of demagnetizing fields in the interfaces. The characteristic feature of the magnetization process of R/T structures is a magnetic phase transition and a twisted structure. A number of phenomenological models were developed to describe R/T spontaneous magnetization and changes of the spontaneous magnetization in the external magnetic field. It seems that the most suitable is the model of nonuniform interlayer exchange with reciprocal influence of the sublayers, and charge transfer. Some nonmagnetic properties such as magnetostriction, magnetoresistance, the magnetooptic Kerr effect, and magnetic resonance have been studied with the objective of using R/T multilayers in the future both as a tool for magnetization process research and as a basis for technological applications. We have a relatively clear picture of the nature of magnetism in R/T multilayers today. Major progress in this field of modern physics depends very much on progress in the technology of preparation, but meanwhile valuable work can be done by extending the range of magnetic and nonmagnetic components in layered structures, by precise study of the properties attendant on magnetic ordering, by increasing the reproduction level, and by special study of the compatibility of these materials with the aim of increasing their technological applications and improving miniaturization.

5

geff

4 1

3

2

2 Tcomp

1 100

150

200

250

300

T, K Figure 18. Temperature dependence of an effective g-factor for [Gd (7.5)/Si/Co(3)/Si]20 films with different Si spacer thickness: (1) 0.3 nm; (2) 0.5 nm. Dashed lines indicate the compensation temperatures of the films.

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GLOSSARY Atomic structure The presence/absence of order in the positions of the atoms at precise distances compared with the atomic size. Interfaces Zones between different layers of multilayered structures. Ion sputtering The expulsion of the atoms from the target surface by inert gas ions bombardment. Magnetism in rare earth metals Caused by 4f electrons located in internal electron shells. This leads to localization of magnetic moments on atoms and to the absence of direct exchange interaction in the 4f electron system. Magnetic ordering occurs because of indirect exchange through the conduction electrons. This interaction is oscillating and acts over a long range. It causes low values of magnetic ordering temperatures and aids in the formation of complex magnetic structures. Nanoscale rare earth/transition metal (R/T) multilayers NM/[R(a)/T(b)]n /NM samples. This formula contains all the important information about the general structure of the multilayered film. It says that this film has buffer and covered nonmagnetic (NM) layers and rare earth (R) layers of thickness LR = a nm alternated with transition layers (T) of thickness LT = b nm. The structure LR + LT = Lp is repeated n times, where Lp is a period of the structure. R/T ferrimagnetic structure The magnetic moments of R and T layers play the role which, in a classic ferrimagnet, would be played by R and T, the magnetic moments of the sublattices. Twisted magnetic phases Special feature of a noncollinear magnetic structure in layered materials. In contrast to uniform ferrimagnets, in multilayers the negative exchange interaction between magnetic elements is spread nonuniformly through the multilayer volume. As a result there is a spatial nonuniformity in noncollinear structures, in which the angle between the local magnetization and an external field changes from the separating interface toward the center of the layer. Ultrahigh vacuum thermal evaporation or molecular beam epitaxy (MBE) Film preparation at ultrahigh vacuum of 10−8 –10−10 torr, involving strong local heating of the target, as a rule by a high-energy electron beam created by an electron beam gun.

ACKNOWLEDGMENTS We gratefully thank Jane Kalim for her help with the manuscript and Professor Blanca Hernando for the encouragement to write this chapter. This work was partially supported by Award Rec-005 of USA Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF), MCyT and Universidad de Oviedo in the framework of the “Ramon y Cajal” program, and by the Spanish CICyT under Grants MAT2000-1047 and MAT2001-0082-C04-01. The authors would like to express their gratitude for all authors (J. Teillet, A. Fnidiki, D. García, A. Hernando, M. Vázquez, A. V. Gorbunov, J. P. Andrés, J. L. Sacedon, J. Colino, J. M. Riveiro, Z. S. Zhang, Y. T. Wang, F. Pan,

Magnetism in Rare Earth/Transition Metal Multilayers

N. H. Duo, T. M. Danh, M. Kaabouchi, C. Sella, N. N. Schegoleva, S. M. Zadvorkin, J. M. Barandiarán, N. G. Bebenin, L. Lezama, J. Gutiérrez, D. Schmool, H. Nagura, K. Takanashi, S. Mitani, K. Saito, T. Shima, D. N. Merenkov, A. B. Chizhik, S. I. Gnatchenko, M. Baran, R. Szymchak, G. S. Patrin, D. A. Velikanov, Y. Fujiwara, T. Masaki, X. Y. Yu, S. Tsunashima, S. Iwata, H. Fujimori, and P. A. Savin) and publishers (International Academic Publishing Company MAIK “Nauka/Interperiodica,” American Institute of Physics, Wiley-VCH Verlag GmbH, Elsevier Science, and B. Verkin Institute for Low Temperature Physics and Engineering) for their permission to reproduce the figures from publications.

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Magnetoimpedance in Nanocrystalline Alloys B. Hernando, P. Gorria, M. L. Sánchez, V. M. Prida, G. V. Kurlyandskaya Universidad de Oviedo, Calvo Sotelo, Oviedo, Spain

The Maxwell equations can be easily solved for the case of a sample with cylindrical symmetry. Our wire-shaped sample has a conductivity  and circular permeability  . The magnetic field induced by the drive current is circular, B =  H . The impedance can then be derived in this case by solving Maxwell equations [2]:
 1 J0 ka Z = Rdc ka (2) 2 J1 ka

CONTENTS 1. Impedance 2. Magnetoimpedance 3. Soft Magnetic Nanocrystalline Alloys 4. Magnetoimpedance Effect in Nanocrystalline Alloys Glossary References

1. IMPEDANCE Our interest in impedance is due to the change of this magnitude in a magnetic material under the application of some external parameters like stress, torsion, magnetic fields, and so on. Impedance is not only sensitive to the magnetic properties of a material, but also to its structure. Therefore, it is a good parameter that allows the study of the properties of nanostructured magnetic materials. Our magnetic nanostructured sample is submitted to an ac drive current passing through it. The drive current frequency is f . The sample is now part of an ac circuit, and according to the well-known Ohm’s law, a voltage is induced between the ends of the sample with the same frequency of the drive current, and its value is proportional to the drive current. The proportionality is established by using the complex impedance Z. Vac = Iac Z

(1)

The induced voltage can be measured by an oscilloscope. The impedance Z depends on the drive current frequency, and can be separated into its real and imaginary components (Z = Z ′ + jZ ′′ ). Let us consider the current distribution in the sample’s section. When the drive current frequency increases, the current will be mainly concentrated close to the surface of the sample [1] due to the skin effect. The exact solution to the problem generally depends on the shape of the conducting sample and the drive current. To obtain the solution, the Maxwell equations have to be applied. ISBN: 1-58883-060-8/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

with k = 1 + ja/ and  = c/2 f −1/2 , where j =  −1, and J0 and J1 are Bessel functions of the first and second kind, Rdc is the resistance of the wire at f = 0, and  is the skin penetration depth. The impedance depends on the properties of the material through the skin penetration depth, where  and  appear. The magnetic and structural properties of the material are characterized by these parameters. The action of other external properties, such as a magnetic field, would change these parameters through the change of the magnetic domain configuration or structure. The ac frequency also affects the skin effect: the higher the frequency, the smaller  is, and the narrower the outer shell of the sample is, through which the current flows. In the case of a sample with no circular cross section, the resolution of the Maxwell equations is difficult, although the impedance shows the same dependence on the skin penetration depth, and the behavior of the impedance can be explained qualitatively through the interaction of all of the involved parameters.

2. MAGNETOIMPEDANCE In the last ten years, there has been a considerable interest in transport phenomena due to their possibilities for use in technological applications, such as magnetic recording and magnetic sensors. One of them is the magnetoimpedance (MI) effect concerning large variations in the impedance (both real and imaginary components) of a soft ferromagnetic conductor when submitted to a static magnetic field, with a high-frequency drive current flowing through it. This

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (949–966)

Magnetoimpedance in Nanocrystalline Alloys

950 effect was discovered in FeCoSiB amorphous wires [3–6] and ribbons [7]. The MI effect has been observed in other different Co-rich amorphous ribbons, microwires, and thin films [8–26], and also in nanocrystalline Fe-rich wires, ribbons, and thin films [17 26–56]. As mentioned above, the MI effect has a classical electromagnetic origin, and can be explained by solving the Maxwell equations and Landau–Lifshitz equation of motion [1] for a magnetic conductor when a high-frequency driving current flows along it. The MI effect is a consequence of the impedance Z, dependence on the applied bias field H , and the drive frequency f due to the skin effect. Z can be expressed in terms of real (resistive) and imaginary (reactive) components as Zf  H  = Z ′ f  H  + jZ ′′ f  H . In the MI measurements, an ac drive current flows through the sample. The current amplitude is kept constant during the measurements. An axial magnetic field is applied in the axial direction by a pair of Helmholtz coils. The ac current induces a voltage between the ends of the sample that is measured by a digital oscilloscope and that is proportional to the impedance. The magnetoimpedance ratio is obtained with respect to a maximum field Hmax at which the sample is saturated. A 45 Oe field is high enough to saturate a 10 cm long sample. Z/Z% = 100 ×

ZH  − ZHmax  ZHmax 

(3)

The MI rate behavior depends on the drive current amplitude and frequency, and on other parameters, such as magnetic permeability, magnetostriction, induced anisotropy, and so on. When the axial field is applied, the transversal permeability which appears on the skin penetration depth  decreases, making  increase, and changing the current distribution in the sample section. This produces a change in impedance [57]. The MI depends on several parameters, and can also give information about them, as will be explained in the following sections. The magnetostriction coefficient seems to play an important role since the highest values are obtained at an almost negligible magnetostriction. This is related to the magnetic softness of the sample, as well as to a high permeability value. The magnetoimpedance phenomenon appears at a frequency at which the  is on the order of the sample thickness, at about 104 Hz in the case of 130 m diameter wires, and at a somewhat higher frequency for 20 m thick ribbons. The MI spectrum shows a maximum at a certain frequency, and then decreases, for all type of samples.

3. SOFT MAGNETIC NANOCRYSTALLINE ALLOYS This section is subdivided into four parts, and constitutes a general introduction to the nanocrystalline soft ferromagnetic alloys. The most remarkable aspects related to the preparation methods used to synthesize these compounds, the different treatments applied in order to improve their magnetic response, as well as some generalities concerning the structural and magnetic evolution during the nanocrystallization process are explained.

3.1. Introduction The term “soft magnetic nanocrystalline alloy” is commonly applied to a singular type of material that exhibits a superb soft magnetic response due to its ultralow values of the magnetic anisotropy (excellent research reviews on this topic are, for example, those by Herzer [58] and McHenry et al. [59]). These kinds of alloys are usually obtained by devitrification of Fe-rich metallic glasses, and in the last decade, they have been the subject of an enormous research effort due to their promising applications in a large variety of technological devices. On the other hand, these materials have also opened new and unexplored fields of basic research in magnetism, as well as in solid-state physics in general because the competition between two or more magnetic phases, amorphous and nanocrystalline, coexisting in the same system, gives rise to a very rich scenario for the study of complex magnetic interactions [60]. During the last three decades, nanostructured materials have attracted huge attention, not only because of their unusual structural, electrical, magnetic, and optical properties, different from corresponding noncrystalline precursor alloys, but because they have conditions and techniques of preparation different from the bulk materials as well. Fine-grain structures can be obtained by heat treating an amorphous alloy precursor with a noncrystalline structure. Amorphous alloys inherently have a uniform distribution of constituent elements. The last is a condition suitable for the formation of nanostructured materials, making the glassy alloys promising precursors for controlling the nanoscale structure because of the high degree of freedom for a topological and chemical structure that persists in these amorphous materials [61]. Currently, minority elements are added in a glass-forming amorphous alloy to promote massive nucleation through the addition of insoluble atoms such as Cu, and to inhibit grain growth by adding refractory and stable species such as Nb, Ta, and so on. The addition of these elements has a key effect on the improvement of material properties, such as mechanical strength or corrosion resistance [62]. In the late 1980s and early 1990s, Yoshizawa et al. [63], Herzer [64], and Suzuki et al. [65] obtained, for the first time, these kinds of alloys in a nanocrystalline state, and published their unique soft magnetic behavior results. The interest in these alloys, as already mentioned, lies in the very soft magnetic properties shown, especially in the low-frequency range (megahertz), surpassing the magnetic response shown by their precursor alloys in the amorphous state. These properties include high saturation magnetic polarization (up to 2–2.1 T), quasivanishing macroscopical anisotropy, nearly zero saturation magnetostriction (s ≈ 10−7 ), and therefore, high magnetic permeability ( ≈ 105 ). All of these features combined with the reduction of magnetic losses, in the low-frequency range, originating by an appreciable decrease of the electrical resistivity and coercive field (Hc < 1 A/m) values, shown in the nanocrystalline state, make them good candidates for technological applications such as power transformers, common-mode choke coils, flux-gate magnetometers, or magnetoelastic microsensors and transducers [59].

Magnetoimpedance in Nanocrystalline Alloys

The composition of these nanocrystalline soft magnetic compounds is based on the Fe–M metallic glasses (where M is a metalloid such as B, Si, Al, P, or Ge), largely studied in the 1970s and 1980s, but with the addition of a small amount (less than 15at%) of transition metals (TM) such as Zr, Nb, Cu, Hf, V, Ta, and so on [58 59 63–67]. There are four families of such alloys. 1. Fe–TM–Cu–Si–B alloys: TM is Nb, Ta, V, and so on, the percentage of metalloid atoms is around 20–25at%, the amount of TM is rather low (around 3at%), and the percentage of Fe is not higher of 75at%. These alloys show the lowest values for the coercive field (less than 1 A/m), but due to the high amount of metalloids, and the fact that the magnetic nanocrystalline phase is FeSi, the saturation magnetic polarization Js never exceeds 1.3–1.4 T. The better magnetic response is obtained with TM = Nb, the socalled FINEMET alloys. 2. Fe–TM–B–(Cu) alloys: In this case, TM is Zr, Hf, or Nb, the percentage of metalloids is lower than in the previous group (2–10at%), and the amount of Fe is over 85at%. The nanocrystalline phase is now nearly pure –Fe, giving rise to high values of Js (up to 1.7 T), but the coercivity is about one order of magnitude higher than in group 1) above, reaching values of several amperes per meter, due to the slightly higher saturation magnetostriction. The most studied compounds of this group are FeZrB(Cu) alloys, commonly known as NANOPERM alloys. 3. Fe–Co–TM–B–(Cu) alloys: The composition of these alloys is similar to those of group 2), but Fe atoms are substituted for Co ones in several concentrations, reaching values of Js above 2 T, as for an Fe44 Co44 Zr7 B4 Cu1 alloy. However, the coercivity never decreases below 20 A/m because the ferromagnetic nanocrystalline phase, actually FeCo, has larger anisotropy values due to the presence of Co atoms. These alloys are called HITPERM. 4. Other alloys: There is a huge number of ferromagnetic nanocrystalline alloys in which other metalloids, such as Al, P, or Ge, substitute B and/or Si or, also, Ni is included in the composition, substituting Fe and/or Co atoms, but up to now, none of these has shown better soft magnetic properties than the compounds included in the previous three groups. If the so-called nanocrystalline regime is pursued, these alloys have to be submitted to appropriate heat treatments in order to provide enough thermal energy to favor the structural transition from a disordered metastable glassy structure to a stable and more ordered crystalline one. In other words, the crystallization of the initial amorphous state takes place when the alloy is annealed. The crystallization process takes place in two differentiated steps [68 69]. In the first step, the ferromagnetic Fe or FeSi crystalline phase, with an average grain size of about 10–15 nm, is formed. At this moment, the system is composed of a mixed structure of Fe(Si) nanograins embedded in a remaining amorphous matrix or intergranular region, which becomes more and more inhomogeneous as this first crystallization step continues. Once the first crystallization process has finished,

951 the second step leads to the crystallization of the remaining amorphous matrix, enriched in B and TM, into Fe–B or FeTM(B) phases (with much higher values for the magnetic anisotropy), and the growing of the Fe(Si) nanograins above 20 nm. These two processes overlap in most cases, when Fe–metalloid alloys are submitted to heat treatments [70], and occur for temperatures above 400  C; hence, the nanocrystalline state is never reached, and the soft magnetic properties shown in an amorphous state do not improve during the crystallization process. However, the addition of a small amount of Cu, which acts as a nucleation center for crystallites, accelerates the first step, and takes place at lower temperatures (below 500  C). On the other hand, the effect of the addition of other transition metals, such as those mentioned above, retards the second crystallization step up to temperatures above 700  C, and prevents grain growth. In between both steps, the structure of these materials consists of a large amount of Fe or FeSi nanosized grains (around 70–80% of the sample) surrounded by the remaining amorphous ferromagnetic matrix (less than 30%) [71]. The origin of the excellent soft magnetic properties displayed in this state lies in the nanometer grain size of the Fe(Si) phase, which is lower than the exchange correlation length. As is well known, this critical dimension plays a definite role in magnetic interactions. This behavior has been largely explained in the framework of the random anisotropy model [72] by Herzer [58 64]. Another reason contributes to this magnetic softening: say, some of these systems, especially those of group 1), exhibit near-zero average magnetostriction due to the opposite sign of this parameter for the Fe(Si) nanocrystals and for the amorphous matrix, so no macroscopic magnetoelasic anisotropy is present in the samples.

3.2. Preparation: Composition and Geometry (Thin Films, Ribbons, Wires, and Microwires) As mentioned above, the magnetoimpedance phenomenon consists of the change of total impedance of a ferromagnetic conductor under the application of a magnetic field [7–14]. Therefore, by definition, the MI effect is expected in the sample only under the conditions of the ferromagnetic state, that is, below the Curie temperature, and if it is large enough to show the ferromagnetic behavior. What is large enough? Some time ago, we could answer the question about critical size of the ferromagnetic sample only on the basis of theoretical predictions. But new experimental methods like the micro SQUID setup [73] are available now to study the magnetization behavior of an individual magnetic cluster, for instance, of 1000 atoms, which is about 3 nm in size in the case of cobalt [74]. Magnetic nanostructures are subjects of special interest due to many possible potential applications. One may distinguish between tiny separate nanosized elements showing interesting properties and macroscopic objects having nanosize structural units. The nanosize clusters [73 74], polycrystalline materials with grains of nanosize [26 28 37 44 49 51 53 56 63 64 75–78], or multilayers with layers of nanoscale thickness [16 48 79–82] are examples of such new structures. Although the term “as small as possible” seems most popular for modern applications, there are theoretical and

Magnetoimpedance in Nanocrystalline Alloys

952 technological limits on the size of the elements involved in the functionality for each particular case. Here, we wish to show that, for an MI element, the size limit of the ferromagnetic state lies far below the critical size for the appearance of a strong MI effect. As mentioned in the Introduction, the MI effect in a uniform ferromagnetic conductor is based on the impedance dependence on the classic skin penetration depth  =  f1/2 for the flowing ac current of angular frequency  = 2f . The impedance of a ferromagnetic sheet of thickness d, width b, and length L under the condition b ≫ d ≫  can be written as [12] Z ≈ L1 + j

4f1/2 4b

(4)

where  = 1/ is the electrical resistivity, and  is the low-frequency transverse magnetic permeability. Figure 1 shows the frequency dependence of the skin depth and the impedance for different values of the transverse magnetic permeability. The geometrical parameters of the sheets were: L = 10 cm, b = 0#8 mm, d = 20 m, that is, the b ≫ d condition was satisfied. We are interested in comparing the results of the MI measurements for a nanocrystalline sample (FINEMET ribbons, for example), with an approximation taken in the framework of the classical model (Eq. (2) [1]) for samples of the same geometry. The same parameters for approximation were taken as for experimental samples. The resistivity  = 1#5 $ · m was measured using the four-point method for a dc current of 10 mA. One may see that an increase of the exciting current frequency

1000

=1 = 10 = 100 = 1000 = 100000

FINEMET

Ω)

µ µ µ µ µ

a

800

400

Z(

Skin depth, δ (µ m)

1200

Impedance,

1400

200 0 1 10 Frequency, f (MHz)

600 400 200 0 1

10

Frequency, f (MHz) µ =100

Skin depth, δ(µm)

40

b

µ=1000

30

δ =20 µm

*

20 10

δ =2 µm 0 -20

0

20

40

60

80

Frequency, f (MHz)

*

100

Figure 1. Calculated frequency dependencies of classic penetration depth and the impedance of a conductive sheet with different values of the effective permeability. The measurements of a nanocrystalline FINEMET Fe73#5 Cu1 Nb3 Si13#5 B9 ribbon (inset) are in agreement with the model estimation.

results in a decrease of the calculated penetration depth and an increase of the total impedance. But there is a strong dependence of  and Z values on the magnetic state of the sample: the higher the effective magnetic permeability, the less, for the same frequency, the penetration depth and the higher the impedance variation are. The inset in Figure 1a shows the impedance dependence on the exciting frequency for selected values of the permeability. The calculated data are compared with the results of MI measurements for FINEMET nanocrystalline ribbons of the same geometry; they are in the reasonable agreement. The second condition, d ≫ , for the frequency range under consideration can be satisfied only for the sheets with very high transverse permeability ( = 1000 in Fig. 1b). The dashed line  = 20 m corresponds to the thickness of our nanocrystalline ribbon. When do the skin effect, and consequently the MI effect, appear? When  is less then half the thickness of the sample. Therefore, if we are interested in a small size of the sample, the size condition of the skin effect results in a strong increase of the operating frequency. This is, of course, an undesired complication. Let us illustrate these ideas for the sample with a very high, but realistic effective permeability of  = 1000. Figure 1b shows that, for the 20 m thick element, the skin/MI effect appears at f

0#8 MHz, but for the 2 m element, the operating frequency is about 110 MHz. Two “stars” in Figure 1b prompt this question: “size or frequency?” A high operating frequency on the order of 100 MHz required another level of the electronic part of the device (higher price, complicated circuit, etc.). Therefore, one has to decide up to which size the complication of the sensing process and price of the device can be compensated by the miniaturization. Another important point to mention is that the higher the effective permeability of the magnetic material is, the lower the operating frequency we need for the same size of sensitive element. The size of the sensitive element based on the ribbon geometry material is not acceptable for many applications. Therefore, there were many theoretical and experimental studies of the MI effect in thin-film geometries [1 12 16 79 80 82]. But just a few results were reported for nanocrystalline thin films on the strong dependence of the properties on the technology of preparation [48]. Nie et al. [83] reported that MI of the Mumetal sheets depends very much on the thickness of the samples, being maximum for a thickness of 1 mm and negligible for a 12.5 m thick sheet. This means that, for nanocrystalline materials, apart from the usual dependence of the MI on the geometry of the sample, there are many additional parameters related to nanocrystallization and induced anisotropy features in thin samples. The nanocrystalline materials can be obtained from a metallic glass with the shape of ribbons, wires, or microwires, as described below. Metallic glasses are obtained using a rapid solidification technique. The composition is first chosen and mixed. Afterwards, it is pressed and sintered in an H2 atmosphere, in order to eliminate oxygen and avoid oxidation during the melting process. The next step is to melt the alloy in an induction furnace at 1500  C. The alloys will be used to produce metallic glasses with different shapes. All rapidly quenching techniques are based on a melted alloy falling over a moving surface, and cooling down at about 106 K/s.

Magnetoimpedance in Nanocrystalline Alloys

3.2.1. Ribbons The technique employed first by Duwez [84] is used to obtain ribbon-shaped samples. The melt-spinning technique consists of the injection of the melted alloy over a Cu wheel, which is spinning at a high speed. The Cu is a good thermal conductor, and produces cooling of the alloy at a rate of about 106 K/s. The prepared solid alloy, by using this method, is introduced in the quartz melting pot, which has a hole of about tenths of a millimeter at the bottom. The alloy is molten with an induction furnace at 1300  C and in an Ar atmosphere. The furnace works at 2500 kHz. As the sample contains 80% of metals, the Foucault currents produce the heating and melting of the sample. A higher Ar pressure is applied to produce the expelling of the alloy over the spinning wheel. The thickness, width, and characteristics of the ribbon depend on the speed of the wheel, the Ar pressure, the orifice diameter of the melting pot, its distance and inclination with respect to the wheel, and the alloy temperature at the expelling time. The ribbons obtained with this technique have a thickness on the order of several tenths of microns, and widths between 1 and 3 mm.

3.2.2. Wires Samples with a circular section can be obtained by the meltspinning technique. The production technique was initially developed by Masumoto et al. [85]. The molten alloy is obtained in the same way as explained above. The method consists of injecting the melting alloy in a flow of water, which is turning at a high speed. The water is introduced in a hollow toroid that is spinning, making the water turn and keeping it inside due to the centrifugal force. A rapid transfer of heat freezes the atoms when the metal hits the water, obtaining an amorphous solid. The cooling speed is about 106 K/s. The cooling of the alloy takes place from the outer surface of the alloy, which acquires a cylindrical shape when it falls into the water. Wires with diameters between 100 and 150 m can be obtained by this procedure.

3.2.3. Microwires The microwires have a ferromagnetic nucleus and a glass cover. They are obtained by the Taylor–Ulitovsky technique, which has been modified in the last 40 years [86]. The alloy is introduced in a Pyrex tube, that will be melted with the alloy [87]. The alloy is melted by the furnace as explained above in the other techniques, and the Pyrex is melted due to the contact with the molten alloy. The molten alloy with a glass layer is cooled when it passes a flow of cooling material (water, oil, etc.), and it is collected in a coil that produces an axial stress on the microwire. The microwire diameter varies with the following parameters: furnace potency, distance of the alloy to the furnace, and speed of the coil collecting the sample. The diameters of the nuclei vary from 1 to 25 m, and the glass coat has a thickness between 1 and 10 m. The internal stresses are much stronger than the conventional amorphous wires as a consequence of the presence of the glass coating which, coupled with the magnetostriction, can give rise to strong magnetoelastic anisotropy which determines the magnetization process. Amorphous and nanocrystalline microwires may show a magnetoimpedance effect in a very

953 high range of the frequency from kilohertz up to hundreds of megahertz [88 89]. Nanocrystalline materials have been obtained from an amorphous alloy. Nanocrystals of sizes ranging from 5 to 10 nm and compositions of AlRu, SiRu, NiTi, or CuEr have been obtained in this way. Amorphous alloys of FeB also have been produced using this technique, in order to induce crystals with sizes in the 10 nm range after a suitable annealing technique. Nanocrystals have been obtained in the Fe73#5 Si13#5 B9 Cu1 Nb3 alloy by Yoshizawa et al. [63]. This alloy is produced as an amorphous ribbon, obtaining a nanocrystallized state after an additional treatment above the crystallization temperature. Ultrafine grains of the –FeSi phase and sizes between 10–20 nm are obtained after a suitable treatment. The interest of this composition lies in the high magnetic permeability and soft magnetic properties when the grain size is about 10–20 nm [64]. The magnetic softness is a dramatic function of the grain size. In order to determine the nanocrystalline structure of the material obtained with any geometry, an additional annealing treatment is needed [90]. This annealing can be performed in a conventional furnace, heating the sample above the crystallization temperature. The annealing has also been successfully performed by using the current annealing technique, which consists of the application of a dc current across the sample. The temperature increases in the sample due to the Joule effect, reaching the desired temperature in a very quick way, although a dc magnetic field is induced, and is not homogeneous inside the sample.

3.2.4. Thin Films and Multilayers Magnetic materials in a thin-film shape can be prepared by different methods, such as thermal deposition on a cold substrate, electrodeposition, laser ablation, or sputtering. Sputtering can be provided using alternating or direct current in the inert gas atmosphere (Ar or Kr). The ions of the inert gas move from the substrate (anode) to the target (cathode). As a result of the ion bombardment, some atoms of the target are set free, and then are deposited onto the substrate. If the rate of deposition is high enough (on the order of 0.1 nm/s) and cooling of the substrate is used (by water, liquid nitrogen, or even liquid helium), the deposited film shows an amorphous structure. Apart from the composition of the target, the deposition rate, and the temperature/type of the substrate, the thickness of the film can be a critical parameter in order to obtain an amorphous or nanocrystalline state in as-deposited films. The same as for nanocrystalline ribbons, it is much easier to prepare thin films in a nanocrystalline state using sputtering deposition, followed by conventional or field annealing for nanocrystallization [48]. The thickest single-layer samples usually do not exceed the 5 m limit, even if deposited onto Si because of the separation of the sample on the substrate. In any sputtering system, it is very important to ensure the homogeneity of the deposited samples from the point of view of the composition and thickness. There are many methods to control these parameters and to avoid strong deviations [91–94]. Homogeneity can even be a more critical parameter in the case of the MI thin-film element because it

954 includes not only the magnetic film itself, but also the electrical contacts and other parts of the imprinted circuit. Up to now, we have been addressing a single-layer MI thin-film element. A very sensitive magnetoimpedance recently has been reported to appear in multilayered structures of two very soft magnetic layers separated by a highly conductive layer [12 48 82 95–97]. In such a complicated structure, the homogeneity of the sample properties is very important in order to avoid critical stresses in a three-dimensional structure. The sensitivity of the magnetoimpedance in singlelayer Co-rich thin films typically does not exceed 10%/Oe [98], but it can be on the order of 30%/Oe in the case of a Co-based sandwich with a Cu conductive lead, where a large impedance change appears at lower frequencies. The condition of a strong skin effect is not required in “soft ferromagnetic/conductor/soft ferromagnetic” sandwiches. For a sufficiently large width of the MI sandwich, the contribution of the outer magnetic layers to the total impedance is defined by the external inductance. In this case, the impedance shows linear variation with effective magnetic permeability and frequency. Although there have been many optimistic discussions with respect to the potential of the MI sandwiches to be employed in developing magnetic heads for high-density magnetic recording, finally it seems that they are better suited sensor applications as very sensitive detectors of a small magnetic field. An additional advantage of a thin-film MI structure consists of the possibility to use the thin-plate permanent magnet attached to the substrate to generate the bias magnetic field to achieve a better response. The MI sandwich is composed of an inner conductive lead of a nonmagnetic material (Cu, Ag, Au) and two outer layers of amorphous or nanocrystalline very soft magnetic material. In order to enhance the value of the effect, two magnetic layers and a conductive layer are deposited in the shape of a closed-loop structure. The unidirectional magnetic anisotropy in each magnetic layer of the MI sandwich is formed in the transverse direction, that is, perpendicular to the conductive lead, but in the plane of the magnetic films. It can be done by application of a constant magnetic field, usually on the order of 100 Oe, during the deposition. The MI effect in MI sandwiches is much stronger compared with that in a single-layered thin film of the same thickness. This can be explained taking into account the very weak influence of stray fields in an MI closed-loop structure, being actually part of the more general fact that all mechanisms involved in the magnetization process contribute to the magnetoimpedance effect [99]. The new types of magnetic sensors on the basis of the MI effect in sandwiched thin-film structures were developed recently to be employed as direction, revolution, and position sensors [100]. They show no hysteresis, good linearity, reasonable temperature stability, and very high sensitivity with a detection resolution of 10−3 Oe, that is, higher than any other type of thin-film sensor like an element based on magnetoresistance or the Hall effect. A selfoscillation circuit technology is very useful for construction of the MI-element-based sensor in order to avoid parasitic displacement currents and reflected signals. In oscillation circuits like the Colpitts oscillator, the MI element is employed as inductance, and the MI effect is amplified in a resonant circuit due to a simultaneous change of

Magnetoimpedance in Nanocrystalline Alloys

both the impedance and the current amplitude [101 102]. Another example of possible technological applications of the impedance sandwiches is strain sensor, which detects the change in impedance with applied strain in magnetostrictive thin films [103 104]. Reported in this case are gauge factors ten times larger than those of semiconductor strain gauges.

3.3. Postpreparation Treatments Once the samples are prepared in any of the geometries mentioned in the previous section, two kind of heat treatments, with different main objectives, are used, with the aim of reaching the optimal magnetic state of the particular alloy: first, to nanocrystallize the samples, and second, to obtain a particular anisotropy distribution combining heat treatments with the application of a magnetic field, tensile, and or torsional stresses. These two aspects are explained in the following paragraphs. • Nanocrystallization treatments: The nanocrystalline state is obtained when the as-prepared amorphous samples are conveniently annealed. Conventional heat treatments are performed using a furnace under controlled atmosphere (Ar, He, or vacuum) to prevent oxidation of the samples. The range of the annealing temperatures is in between 450 and 600  C, and time periods on the order of several minutes or hours [58 59]. There is another way to heat the samples that consists of passing an electrical current along the sample, particularly used in the case of ribbon or wire geometries. This method, called flash annealing, current annealing, or Joule heating, is based on the Joule effect [105], and allows us to obtain the sample in a nanocrystalline state in a quicker way (10–100 s) [106 107]. • Induction of anisotropies: Magnetic anisotropy in nanocrystalline soft magnetic materials for MI applications can be induced by traditional methods like annealing in a magnetic dc or ac field [78] or stress annealing [75 76]. Many parameters play an important role in the magnetic anisotropy, and consequently, MI response formation. For example, for FeCuNbSiB ribbons, the Si content could be critical for the field annealing, leading to a decrease or increase of the magnetoimpedance ratio [100]. All treatments below the Curie point usually need to be provided in a magnetic field higher than the saturation one. The temperature, time of exposure, rate of heating and cooling, and specific stress for stress annealing are the important parameters. Both the stress and field annealing can be done in two different ways: annealing of as-cast samples under field or stress, or using previously nanocrystallized samples, and then submitting them to a stress or field annealing. Since the magnetoimpedance depends on the permeability, there are many possibilities for modifying the GMI responses. Apart from the objective of nanocrystallization in order to achieve extra soft magnetic material, one may take into account the value of the transverse induced magnetic anisotropy [44 80]; the anisotropy distribution [44] and thermal or time stability may be the parameters of interest both for the

Magnetoimpedance in Nanocrystalline Alloys

Heat flow (arbitrary units)

amorphous Fe(Si)

220

Fe(Nb)B

>

>

200

First crystallization => Fe(Si)

180

Second crystallization => Fe(Nb)B

160

140 400

450

500

550

600

650

700

750

Electrical resistivity, ρ, (µΩcm)

The conditions that allow the formation of these nanostructured materials depend on the crystallization temperatures, the thermodynamic quantities such as the crystal-interfacial energy and the free-energy driving force for homogeneous versus heterogeneous crystal nucleation, the small or large crystal nucleation frequency at sufficiently low growth rates, and the volume change during the crystallization. It was mentioned previously that the crystallization of these materials takes place in two differentiated steps, which are clearly separated. Figure 2 constitutes a schematic approach to the crystallization process of these kinds of alloys. In the upper plot, two different curves are shown, a differential thermal analysis (DTA) scan, together with a resistivity versus temperature T  curve for a typical FINEMET alloy. Three different pictures representing the internal structure of the alloys in three characteristic structural situations (amorphous, nanocrystalline, and fully crystallized states) are included as insets, in order to clarify the whole picture. Among this, and in the lower part of Figure 2, the X-ray diffraction patterns and hysteresis loops that correspond to the three different structural are also presented. The two exothermic peaks of the DTA curve and the two minima of the T  curve situated around 550 and 700  C

240

>

3.4. Structural Evolution

FINEMET ALLOY

>

study of technological applications and magnetization processes. Magnetic anisotropy in nanocrystalline soft magnetic materials for MI applications can be induced by traditional methods like annealing in a magnetic field [44 75 76]. Usually, it is a dc field applied in the direction of the ribbon or wire axis or in a perpendicular direction [37]. There were attempts to use low- and high-frequency ac [77] fields as well. It is known that ac field annealing leads to a magnetic structure formation with no stabilization of the domain walls, which is a very favorable feature for the MI conditions of the magnetization. Stress annealing is another effective method of anisotropy formation. It was found that, not only can simple uniaxial magnetic anisotropy be formed as a result of stress annealing, but also, a very complicated high-order anisotropy, which results in interesting and unwanted hysteresis of the MI. As mentioned, many parameters play an important role in magnetic anisotropy formation. For example, for FeCuNbSiB ribbons, the Si content could be critical for field annealing to decrease or increase the MI ratio, as well as the cooling conditions [77 78]. All treatments below the Curie point usually need to be provided in a magnetic field higher than the saturation one, and kept at room temperature during the cooling process. The temperature, time of exposure, specific stress for stress annealing, and rate of heating and cooling are very important, not only as each parameter itself, but also, they have a mutual influence, and should be checked together in many cases. Both stress and field annealing could be done in two different ways: annealing of as-cast samples under field or stress, or using previously nanocrystallized samples, and then submitting them to stress or field annealing [37].

955

800

Temperature (º C)

X-ray diffraction

Hysteresis loops

amorphous

nanocrystalline

fully crystallized

Figure 2. Schematic view of the crystallization process for a typical FINEMET alloy, showing the thermal evolution of the electrical resistivity, a DTA curve, as well as X-ray diffraction patterns and hysteresis loops at three characteristic structural stages.

correspond to the beginning of both crystallization steps. In between both peaks, the system is in a nanocrystalline state, that is, a mixture of small grains, of a few nanometers in average size, embedded in the remaining amorphous matrix. The most common experimental technique used to evaluate the degree of crystallization in a given material is obviously X-ray powder diffraction [69] because it is available in most of the research laboratories. This technique allows us to confirm the amorphous state of the as-prepared sample (only broad haloes are present in the diffraction patterns, instead of the characteristic sharp peaks corresponding to the Bragg reflections), as well as the evolution of the structure during the heat treatments. In this way, the values of the most important structural parameters, such as lattice constant and average grain size, can be obtained from the position of the peaks and their widths, respectively. Also, neutron diffraction is a valuable technique that gives information on the whole sample (not only from the surface, as is the case for X-rays), and also permits in situ structural studies of the nanocrystallization kinetics [108]. But these diffraction techniques are not the unique ones that help in the structural evolution study, Mössbauer spectroscopy give detailed and quantitative information on the amount of Fe atoms in each different environment, as well as details of the hyperfine magnetic parameters [69 109 110]. Also, electron

956 microscopies give information about the average grain sizes of the nanocrystallites, and their distribution along the sample surface [62]. Finally, it is worth noting that a relatively new technique, atomic force microscopy, allows us to obtain high-resolution topographic 3-D images of the alloy surface [111], as can be seen in Figure 3a and b for a FINEMETtype alloy in an amorphous and a nanocrystalline state, respectively. The crystallization process of this kind of material can be summarized as follows. In the first step, the soft magnetic Fe or FeSi crystalline phase, with an average grain size around 10–15 nm, is formed, leading to a mixed structure of nanograins embedded in the remaining amorphous matrix or intergranular region, which is becoming more and more inhomogeneous when first crystallization occurs. The composition and structure of the nanograins depend on the alloy composition as well. • For FINEMET alloys, the structure of the nanograins is a BCC–FeSi solid solution (when the amount of Si is less than 12at%) or a DO3 –FeSi (for percentages of Si above 12at%). • For NANOPERM alloys, the structure is BCC–Fe. • For HITPERM alloys, the structure varies, depending on the percentage of Co, and can be a BCC–FeCo ordered () for a nearly equal content of Fe and Co (Co atoms occupy fixed Fe positions) or a disordered (′ ) solid solution for a low Co content (Co atoms occupy Fe atoms randomly). Once the first step is finished, the second step leads to the crystallization of the remaining amorphous matrix into Fe–B or FeMT(B) phases (with high values for the magnetic anisotropy; see the evolution of the hysteresis loops in Fig. 2), and the growing of the Fe(Si) phase. These two processes overlap in most cases, when FeSiB alloys are submitted to heat treatments, and occur for temperatures above 400  C; thus, the nanocrystalline state is never reached, and the soft magnetic properties shown in an amorphous state do not improve during the crystallization process. However, the addition of a small amount of Cu, which acts as a nucleation center for crystallites, accelerates the first step, and it takes place at lower temperatures (below 500  C); meanwhile, the addition of other transition metals, such as those mentioned above, retards the second crystallization step up to temperatures above 700  C, and prevents grain growth. In between both steps, the structure of these materials consists of a large amount of Fe or FeSi nanosized grains (around 70–80% of the sample), surrounded by the remaining amorphous ferromagnetic matrix (less than 30%).

Figure 3. Atomic force microscopy 3-D images of a FINEMET alloy, a: in an amorphous (as-prepared) and b in a nanocrystalline state.

Magnetoimpedance in Nanocrystalline Alloys

3.5. Magnetic Behavior As shown above, nanocrystalline alloys consist of particles or grains with sizes ranging in the nanometer scale up to 20–30 nm embedded in the amorphous matrix-forming nanocomposite. The precipitation of these nanocrystalline particles from the amorphous matrix improves the tensile strength and ductility of the alloy due to the strong reduction of the anisotropy induced by residual internal stresses and, as a consequence of that, the softening of the magnetic properties in the nanostructured state. Metallic Fe–TM–metalloid-based amorphous and nanostructured materials produced by rapid solidification of the melt are the most important ones from a commercial point of view because their soft magnetic properties make these nanocrystalline materials very suitable for use as high-frequency electronic components in magnetic devices or magnetic sensors [9 62 112–115]. Fe-based amorphous ferromagnetic metallic glasses show a larger magnetization than those based on Co, and much larger than those containing Ni. However, iron-rich amorphous alloys generally have fairly high magnetostriction (s ≈ 20 × 10−6 ), limiting their magnetic permeability. The formation of a nanocrystalline Fe-rich alloy can lead to a drastic reduction in magnetostriction and coercivity due to the well-coupled Fe nanocrystallites by exchange interaction, thus favoring easy magnetization and, simultaneously, high permeability. These nanocrystalline alloys are composed primarily of crystalline Fe grains, having at least one dimension on the order of a few nanometers, that are embedded in a residual amorphous matrix. Thus, such materials provide an excellent context in which to study the principles of magnetism in nanometer structures [112 113]. In particular, the so-called soft ferromagnetic alloys (characterized by high values of saturation polarization and low values for the coercive force and core losses) have been the most widely studied, and the typical composition in atomic percent is about 70–80% of Fe and 20–30% of metalloid atoms, such as B, Si, P, and so on [62 113 114]. The origin of the soft magnetic properties lies in the weak magnetic anisotropy presented by these materials in the amorphous state. When the alloys are submitted to temperatures above 450–600  C, crystallization of the system takes place, leading to magnetic hardening of the material due to the appearance of different crystalline phases with high values of the magnetocrystalline anisotropy. However, the addition of a small amount (5–10at%) of transition metals such as Nb, Zr, Cu, Ta, and so on drastically changes the crystallization kinetics. First, the crystallization temperature increases slightly, and second, there is a large interval of temperatures (up to 200  C) in which the average grain size of the principal crystalline phase remains below 15–20 nm lower than the typical exchange correlation length for these systems, which is about l ≈ 35 nm. The magnetic response of the nanostructured material is, at this point, even better than in an amorphous state [61 71 116] and nowadays, typical compositions of prototype Fe-rich nanocrystalline alloys such as Fe73#5 (SiB)22#5 Cu1 Nb3 (FINEMET alloys) or Fe100−x (ZrB)x (NANOPERM alloys) are the softest magnetic materials known, with attractive soft magnetic properties, including

Magnetoimpedance in Nanocrystalline Alloys

values of the coercive force HC below 1 A · m−1 , vanishing magnetostriction (s ≈ 10−7 ) due to the balancing between the opposite contribution of this parameter from the Fe(Si) nanocrystals and from the residual amorphous matrix, magnetic permeability  higher than 105 , and polarization saturation values JS over 1 T [63 117 118]. The change in the surface magnetic domain structure from the as-cast state to the nanocrystalline one, during the annealing of the ferromagnetic samples, can be investigated by using the Bitter technique, where the colloidal ferrofluid is attracted by the Bloch wall’s gradient field, thus revealing the intersection of every Bloch wall with the sample surface [19] Photographs a)–c) in Figure 4 show the evolution of the magnetic domain structure for the Fe73#5 Si16#5 B6 Cu1 Nb3 ribbons from the as-cast state a) to the different annealed states. In all photographs, the longitudinal ribbon axis lies along the horizontal direction. Hence, photograph 4a) reveals an island domain structure on the bright surface of the as-cast sample. The observed islands domain in a) correspond to closure domains resulting from compressive internal stresses, in which the anisotropy direction lies perpendicular to the ribbon plane, and whose narrow laminar widths are between 4.1–5.8 m. These experimental measured values are in good agreement with the theoretical one, around 4.3 m, calculated for the corresponding Landau–Lifshitz domain width [120]. When the annealing temperature is increased up to 425– 475  C, the internal stresses are nearly removed, and nanocrystallization begins to take place, with a mean grain diameter around 12 nm. It can be observed that a stripe domain structure appears in these annealed samples, with a measured domain width around 5.5–6.8 m for the ribbon annealed at 425  C [Fig. 4b)], which exhibits a magnetic hardening, and which diminishes to 2.7 m for the ribbon annealed at 475  C [Fig. 4c)], when the nanocrystallization begins. The latest domain width value measured is in very good agreement with that calculated from the random anisotropy model (2.8 m) [72]. In this late state, all of the

957 anisotropies start to vanish, and the magnetization vector tends to align parallel to the ribbon plane, increasing the transversal magnetic permeability, and therefore improving the magnetic properties of the sample. At higher annealing temperatures, the pattern domain exhibits a much more fine stripe structure, where the easy axis of magnetization may lie between the longitudinal and transverse directions, as can be observed through other techniques [81 38]. The transverse-induced anisotropy in a nanocrystalline FeSiBCuNb alloy could mainly be due to the magnetoelastic anisotropy exerted through annealing, by the FeSi nanocrystallites, that induces a transverse anisotropy via magnetoelastic interaction with tensile back stresses created by the inelastically deformed residual amorphous matrix [37 44 54 121 122]. If this transverse anisotropy is sufficiently large, the domains will orient themselves along the anisotropy axis. Hence, this uniaxial anisotropy can control the magnetization in the ribbon domains, and therefore the high-frequency response of these soft nanocrystalline materials. Figure 5 shows the variation of both magnetostriction and conductivity with the annealing temperature for the Fe73#5 Si16#5 B6 Cu1 Nb3 [FINEMET (B6)] ribbons. It is important to note that near-zero magnetostriction can be achieved in Fe-rich alloys by nanostructuring. In nanocrystalline alloys, the sum of the volume-weighted magnetostrictions of the nanocrystalline grain phase and of the amorphous grain boundary phase can result in a net vanishing magnetostriction, even when the precursor amorphous alloy shows strong magnetostriction. In FINEMET nanocrystalline systems, the positive magnetostriction of the residual amorphous phase (s amorph = 20 × 10−6 ) is balanced by the negative magnetostriction of the growing –Fe(Si) nanocrystalline phase (s FeSi = −6 × 10−6 ) [62 64 113 123]. In our particular composition, FINEMET (B6), a compensated or zero magnetostriction value is obtained after heating at a temperature of about 565  C, measured by the small angle magnetization rotation technique [124]. Heating at higher temperatures gives rise to slightly negative values of magnetostriction, which also lead to the modification of the domain structure that, in this case, results in the appearance of transversal domains [54] On the other hand, it is clear that the conductivity can also influence the impedance and magnetic permeability 30

7000 6500

25

Fe

3

B

20

16.5 6

s

-1

1

6000

15 -1

10

σ (Ω cm)

5

5000

-6

λ (x10 ) s

Figure 4. Pattern domain of Fe73#5 Cu1 Nb3 Si16#5 B6 amorphous and nanocrystalline ribbons. a) As-cast sample. b), c) Annealed at 425 and 475  C, respectively. The scale shows a 0.1 mm division. Reprinted with permission from [54], M. Tejedor et al., J. Non-Cryst. Solids 287, 396 (2001). © 2001, Elsevier Science.

4500 RT

-6

5500

λ (x10 )

σ (Ω cm)

Cu Nb Si

73.5

100

200

300

400

T (ºC)

0 500

600

Figure 5. Evolution of conductivity  and saturation magnetostriction s with annealing temperature for Fe73#5 Nb3 Cu1 Si16#5 B6 ribbons. Lines are guides for the eye.

Magnetoimpedance in Nanocrystalline Alloys

958

4. MAGNETOIMPEDANCE EFFECT IN NANOCRYSTALLINE ALLOYS Another interesting feature, recently discovered in these kinds of nanostructured materials in the form of FeCuNbSiB ribbons, wires, microwires, and films, or similar compositions, is the so-called magnetoimpedance effect, which consists of a strong decrease of the electrical impedance Z of the ferromagnetic sample when submitted to an external static magnetic field, as was mentioned above [11 21 26–39 41–44 48–56 129–136]. In the case of FINEMET materials, and using the classical eddy-current theory, it was found that large MI was associated with the reduction in resistivity during the crystallization of the samples, together with a vanishing magnetostriction [11 28–34 39]. The first fact can be explained by taking into account that a decrease in the electrical resistivity implies an increase of the eddy-current losses, and hence the MI maximum. Moreover, if we consider the approach of the existence in the material of current lines distorted and dragged by magnetic domain walls oriented perpendicular to the current direction, we may take into account the effects of microeddy currents generated by the free displacement of the domain walls on the effective magnetic permeability for these soft ferromagnetic nanocrystalline materials. Therefore, we cannot neglect the influence of microeddy currents on the MI effect [11]. There is a clear relationship among the reduction of the resistivity of the samples, the vanishing of the residual anisotropies accompanied by the strong reduction of the saturation magnetostriction, and the maximum of the magnetic permeability and MI effect response as the crystallization process advances. In fact, and as shown in Figure 6, the maximum MI value is obtained for the ribbon annealed at 575  C when the value of  is 15% lower than the corresponding one for the amorphous precursor, and the transverse magnetic permeability is maximum [39 41]. Similar results are found in samples with other geometries and compositions [21 28 37 48 77]. These results are in agreement with the evolution of the surface magnetic domain structure of the annealed samples and the behavior of the electrical conductivity and magnetostriction shown above. The ribbon of FINEMET (B6) annealed at 575  C, which exhibits the maximum MI effect, is the one that presents the smallest magnetostriction value (s ≈ −3#6 × 10−7 ), which involves vanishing residual anisotropies, and the appearance of a transverse anisotropy

30

Fe

25

∆ Z/Z(%)

of the sample, and therefore its high-frequency response [125 126]. As can be observed in Figure 5, not only is s modified during the successive annealing, but also , where  increased from about 5000 ($ · cm)−1 in the as-quenched state to about 6800 ($ · cm)−1 at the higher annealing temperature of about 575–580  C. This strong increase in the conductivity of the nanocrystalline ribbons (nanocryst is on the order of 15% higher than amorph ) reduces the skin depth, which contributes to an increase of the impedance of the sample, because this fact favors the appearance of both macroscopic eddy currents as well as the microscopic-induced currents due to the free transverse domain wall displacements [125 127 128].

Cu Nb Si

73.5

1

3

16.5

B

500 ºC 575 ºC 600 ºC

6

20

f = 500 kHz 15 10 5 0 0

20

40

H (Oe)

60

80

100

Figure 6. Field dependence of the magnetoimpedance ratio for some annealed Fe73#5 Cu1 Nb3 Si16#5 B6 ferromagnetic ribbons at a frequency of 500 kHz.

induced by the magnetoelastic interaction between the FeSi nanocrystallites formed with tensile stresses of the residual amorphous matrix, resulting in a transversal magnetic permeability maximum. All that, accompanied by a strong reduction in the resistivity of about 15% with respect to the as-cast state, which favors the appearance of both macroscopic eddy currents and the microscopic ones due to the free transverse domain wall displacements, gives rise to a drastic increase of the MI effect at low frequencies of the passing current across these nanostructured materials.

4.1. Research Tool The magnetoimpedance effect can be employed as a research tool to study magnetic properties of amorphous and nanocrystalline materials. It is not only important for the development of new technological applications, but also in the field of basic research in order to provide information about other interesting magnetic parameters of the materials. Due to its dependence on several soft magnetic properties, like anisotropy field, magnetostriction, and magnetization processes, the behavior of these properties can be inferred from the magnetoimpedance dependence on frequency and magnetic field and its value.

4.1.1. Magnetostriction Coefficient The MI effect shows a characteristic behavior with the applied magnetic field. In the case of as-quenched or annealed samples, the magnetoimpedance rate shows a monotonous decrease from a maximum value at H = 0 Oe, as in the case of nanocrystalline samples that have been annealed at a certain temperature in order to obtain their softest magnetic properties. When the sample has a transverse-induced anisotropy, the impedance curve versus the applied field shows a peak at a field on the order of the anisotropy field of the sample [137]. This fact can be used to determine the anisotropy field of the samples, which can provide us with valuable information, for example, to determine magnetostriction coefficients, as explained in the next paragraph. Figure 7 shows the behavior of magnetoimpedance as a function of the applied magnetic field, together with the hysteresis loop for a stress-annealed amorphous ribbon. The behavior of nanocrystalline-annealed samples, with an

959

70

0.7

60

0.6

50

0.5

40

0.4

30

0.3

20

0.2

10

0.1

where a b, and c are constants. There is a relationship between the magnetostriction and the anisotropy field: s =

Js (T)

∆ Z/Z(%)

Magnetoimpedance in Nanocrystalline Alloys

0

2

4

6

8

H(Oe) Figure 7. Magnetoimpedance ratio at 1 MHz versus the applied magnetic field and hysteresis loop obtained at 50 Hz of a Co66#5 Fe3#5 Si12 B18 amorphous ribbon subjected to an annealing treatment of 340  C, followed by a stress annealing at 340  C with a 300 MPa applied tensile stress.

induced transverse anisotropy, is exactly the same. It can be seen how the anisotropy field can be obtained by the maximum of magnetoimpedance. One of the properties that can be deduced from the MI behavior is the saturation magnetostriction coefficient. The magnetostriction can be obtained from the MI effect due to the MI dependence on the applied stress. s has a linear dependence on applied stress [138] s = s 0 − k

(5)

The impedance behavior as a function of the applied bias field depends on the anisotropy field, which can be changed by the application of tensile stresses. A peak of MI appears at the anisotropy field, and its position can be determined. Let us consider the case of a soft magnetic sample with a negative magnetostriction and no induced anisotropy. In the unstressed state of the sample, the impedance drops continuously with the increasing bias field. The application of tensile stresses on this sample would increase the magnetoelastic anisotropy field, and the MI peak would appear at a higher field. This would be the case for nanocrystallized samples, which have a negative saturation magnetostriction coefficient that usually occurs when the sample has been annealed above around 550  C, depending on the composition. When the saturation magnetostriction coefficient is positive, the behavior of the magnetic sample is somewhat different. The value of the anisotropy field could not be determined because the MI effect would have a maximum at H = 0. But if the positive magnetostriction sample has an induced anisotropy, it will have a peak at Hk in the unstressed state, and further application of tensile stresses, which would decrease the anisotropy field, would lead to a smaller value of the anisotropy field, which can be determined [139]. The anisotropy field does not vary linearly with stress, but it can be fitted as a second-order polynomial of the form Hk = a 2 + b + c

(6)



dHk d



(7)

It can be easily verified that the saturation magnetostriction is a linear function of stress, as shown in Eq. (5). From Eqs. (7) and (5), we can obtain s 0 =

0

0

 0 Ms 3

0 Ms b 3

(8)

which is the unstressed value of the saturation magnetostriction coefficient, and k=

ds 20 Ms a = d 3

(9)

which is the measured slope. Using this procedure, we can find the s 0 values of all types of samples, with positive and negative magnetostriction, with an accuracy similar to the well-known SAMR (small angle magnetization rotation) method [124 140]. This method can be applied to all types of samples, but in the case of nanocrystallized samples, the application of stresses might produce the breaking of the sample due to its fragility. Although this method is accurate, some specific problems can be found. The Z/Z − H curves may be broad and not symmetric, it being difficult to correctly estimate the position of the impedance peak.

4.1.2. Nanocrystalline State The MI effect shows a great variation in its value due to structural changes in the magnetic nanocrystalline samples. A correlation between magnetoimpedance and microstructure as a function of the average grain size can be established. The average grain size of the FeSi crystallites in these samples is about 10–15 nm in the studied range of annealing temperatures. Other materials with similar compositions have similar values of grain sizes [141]. The grain size increases strongly after annealing at temperatures of 600  C, approximately, when new crystalline phases nucleate as well. Figure 8 shows the dependence of the MI ratio as a function of the annealing temperature in Fe73#5 Cu1 Nb3 Si16#5 B6 ribbons. The highest value of the MI ratio, close to 60%, is reached for an annealing temperature of 575  C, which corresponds to the softest behavior of this material due to the grain size. The MI temperature dependence allows us to investigate the nanocrystalline state of annealed samples by measuring the magnetoimpedance. This behavior has been seen in ribbon- and wire-shaped nanocrystalline materials [27], showing that the MI rate response depends on the sample structure, and the sample shape influence is not as important. The highest MI rate is obtained when the nanocrystalline sample reaches its softest magnetic behavior, that is, when the FeSi crystallites have grain sizes of about 10 nm.

Magnetoimpedance in Nanocrystalline Alloys

960 2.5

permeability (arb.unit.)

60

∆ Z/Z (%)

50 40 30 20 10 0 450

Fe

Si

73.5

B Cu Nb

13.5 9

1

3

2

as cast 420ºC 460ºC 550ºC 600ºC 640ºC

1.5

1

0.5

0 500

550

600

650

0

10

20

30

40

H (Oe)

T (ºC) Figure 8. Magnetoimpedance ratio as a function of the annealing temperature of the nanocrystalline Fe73#5 Cu1 Nb3 Si16#5 B6 ribbons.

Figure 9. Permeability obtained from the impedance measurements in nanocrystallized wires versus the applied magnetic field.

4.1.3. Permeability Measurement

This phase has a mean grain size of about 10 nm. Finally, an important magnetic hardening occurs for temperatures above 600  C, mainly as a consequence of the precipitation of new Fe–B phases [141]. As the applied magnetic field is increased, the transverse permeability decreases, due to the magnetization of the sample in the axial direction. The behavior of permeability as a function of frequency and other parameters can then be obtained from Eq. (10) for all types of samples, without taking into account the geometrical constant k. Figure 10 collects the permeabilities of nanocrystallized ribbons in arbitrary units as a function of the drive current frequency. For low frequencies (up to 104 Hz), the permeabilities for all of the samples have the same value. For higher frequencies, the nanostructure of the ribbons influences the permeabilities, making them change. The 460  C annealed ribbon has the lowest permeability in all of the studied frequency range. When the annealing temperature is increased, the permeability increases. Up to 4 × 104 Hz, the 575  C annealed sample has the largest permeability. For higher frequencies up to 105 Hz, the highest value corresponds to the 550  C annealed ribbon. A further increase of the annealing temperature leads to a hardening of the magnetic behavior of nanocrystalline samples, and to a decrease in permeability, as can be observed in the abovementioned figure.

where  = 2f is the angular frequency. Note that, due to the form of this transformation, the real component of permeability ′ depends on the imaginary part of impedance, and conversely, the imaginary part of permeability ′′ is obtained from the real part of impedance. The geometrical constant k can be evaluated in the case of a wire-shaped sample by assuming, as a first approximation, a wire homogeneously magnetized in the circumferential direction. By defining the induction L as the ratio of magnetic flux * to the current intensity i L = ,*/,i [142]. Figure 9 shows the case of a wire with composition Fe73#5 Cu1 Nb3 Si13#5 B9 that has been annealed at different temperatures in the range 400–640  C during 1 h in an Ar atmosphere. The wires have a length of 8.5 cm and a diameter of 120 m. The figure shows the behavior of permeability, evaluated as explained above, versus the applied axial magnetic field. The magnetic softening of the sample can be inferred from the permeability behavior. At low annealing temperatures, up to around 500  C, the permeability shows low values, but the largest permeability at H = 0 Oe is obtained at 550  C. The annealing at higher temperatures leads to a further magnetic hardening of the sample. The softest magnetic properties, including the lowest coercive force, are achieved for annealing temperatures close to 550  C, where the nanostructured FeSi phase precipitates.

3.4 3.2

permeability (arb.unit.)

The real and imaginary components of impedance can be obtained by using a lock-in amplifier, which separates the out-of-phase components. The magnetoimpedance ratio and sensitivities for both components of impedance can be calculated in the same way, as Z ′ /Z ′ % and Z ′′ /Z ′′ % for the resistance, and reactance, respectively. The total impedance would be written as Z ∗ = Z ′ + jZ ′′ , where j = −11/2 . When dealing with a magnetic system, it is often advantageous to use the complex inductance formalism, and deal with L∗ instead of Z ∗ since it is directly related to the value of permeability. The transformation from impedance to permeability (through inductance) can be simply written as   kj Z∗ (10) ∗ = kL∗ = − 

3 2.8 2.6 50 kHz 100 kHz 120 kHz

2.4 2.2 2 1.8 450

500

550

600

650

T (ºC) Figure 10. Permeability obtained from magnetoimpedance measurements for annealed Fe73#5 Cu1 Nb3 Si16#5 B6 ribbons as a function of the annealing temperature for three selected drive current frequencies.

Magnetoimpedance in Nanocrystalline Alloys

961

4.1.4. Magnetization Processes

4.2. Technological Applications

The MI effect behavior depends on the magnetization processes that take place in the sample when the ac current is flowing through it, and is subjected to external influences that modify its magnetization. The magnetoimpedance rate spectra give us information about the magnetization processes that take place at different frequencies. The different domain structures of as-quenched and nanocrystalline samples are responsible for their distinguishable MI effects [38]. At low frequencies, the magnetization processes take place by domain wall displacements, but at higher frequencies, the domain walls are not able to follow the ac magnetic field, and the magnetization processes are due to moment rotations. The relaxation frequency fx is the frequency which separates the two main areas of different magnetization processes. The MI spectrum shows a maximum in this area, and fx can be easily obtained. The value of fx has been recently seen to be dependent on the magnetostriction coefficient [143]. This matter is still undergoing research, but could show in the future why the MI effect has larger values when the saturation magnetostriction coefficient is small and negative. The relaxation frequency decreases with an increase of the magnetostriction coefficient for all types of samples, both nanocrystalline and amorphous ones. Figure 11 shows this dependence for a nanocrystalline sample, and a comparison to an amorphous one. The high magnetostriction value and low permeability of the FeCuNbSiB ribbon in the as-quenched state explain that the MI effect is not observed until partial devitrification and a softer magnetic behavior are achieved by annealing. Although all treatments for the amorphous ribbon resulted in a decrease of the relaxation frequency as a function of the magnetostriction coefficient (see arrows in the curves), the opposite tendency is observed for the nanocrystalline ribbons: the relaxation frequency becomes higher after the treatments. And the representation of the relaxation frequency as a function of the magnetostriction coefficient shows the same behavior in all of the samples: it is a monotonous decay. The same behavior has been previously observed in other amorphous ribbons [143].

The earliest work on applications based on the MI effect was devoted to a magnetic field sensor employing amorphous ribbons [144–147]. Active research recently has been done on the development of miniaturized sensors based in MI [12]. The special advantage of the MI effect as detected in amorphous wires in comparison with other magnetic phenomena is that even small fields (below 100 A · m−1 ) can produce a strong impedance variation. In fact, they look very competitive in comparison with magnetic field sensors based on magnetoresistence or the Hall effect. The main applications are related to the influence of the magnetic field. In particular, current sensors and proximity sensors are actually sensor devices measuring the magnetic field. The MI effect strongly depends on the magnetic softness of the sample, including both higher induction and permeability, and for special environmental conditions, materials capable of operating at higher temperatures. As previously mentioned, large circular (wire shape) or transversal (ribbon shape) permeability and a small resistivity and magnetostrictive coefficient are required to observe the MI effect. The magnetic permeability is determined by the chemical composition, crystal structure, microstructure, and shape of the sample. Alloys with small magnetocrystalline anisotropies and magnetostrictions give rise to particularly soft magnetic materials. Amorphous and nanocrystalline magnetic materials are now competitive with SiFe bulk alloys and Fe–Co alloys, and their soft magnetic properties exceed those of the bulk alloys based on Fe, Co, and Fe–Co [59]. The first MI measurements in nanostructured materials were performed in FeSiBCuNb wires devitrified from an amorphous precursor by annealing at different temperatures [28 148]. Optimum soft magnetic behavior was achieved after annealing at about 550  C. Meanwhile, the maximum impedance variation appears for the softest nanocrystalline wire; a larger MI ratio is observed after annealing at a higher temperature (near 600  C), where the resistivity clearly decreases as a consequence of crystallization [13]. We should also remark that the MI effect in nanocrystalline materials (wire or ribbon morphology) [28 39] does not reach values as large as for amorphous Co-rich alloys. This must be associated with the smaller values of the circular or transversal permeability in that nanocrystalline material. It could be possible that the nanocrystalline sample is softer when it is axially magnetized, but what is required in the MI effect is a very large circular or transversal permeability, which involves a circular or transversal and very weak magnetic anisotropy.

1

2.2

0.8

Fe73.5Cu1Nb3Si16.5B6

2

Co66.5Fe3.5Si12B18

1.8 1.6

0.7

1.4 1.2

0.6

f x (MHz)

f x (MHz)

0.9

1 0.5 0.4 -10

0.8 0.6 0

10

20

30

40

50

λ s (x10 ) -7

Figure 11. Relaxation frequency fx relationship with the magnetostriction coefficient of nanocrystalline ribbons, and comparison to amorphous ribbons. The arrows guide the eye and indicate the scales.

4.3. Unique Properties of Nanocrystalline Alloys for Special Applications Giant magnetoimpedace has attracted significant attention in the last years due to possible technological applications in position sensors, rotary encoders, and direction sensors for navigation, current sensors, biomedical and environmental controllers, nondestructive control sensors, and reading heads [7 12 44 48 80 101 135 145–153]. It is possible to modify and adapt the magnetoimpedance responses of appropriate material for each particular application performing specific treatments. Although many of the proposed

962 applications can be realized using very different materials for the same type of application, there are a few of them where amorphous materials are useless due to particular conditions. Recently, high-temperature magnetic measurements were reported, opening the possibility to study both the basic problems of the exchange-coupling process in nanocrystalline materials and the conditions of possible high-temperature applications of the magnetic materials [154 155]. Why do we need sensors for high-temperature applications? It is easy to answer this question by reference to the everyday needs of our life surrounded by modern technology: it would be much easier to control the space and aircraft engines and other technical blocks, nuclear and power station functionality, metallurgical production, and many other potentially dangerous processes in order to avoid a catastrophe or simply little controlled actions. The high-temperature sensing has different basic problems. The first one is the temperature dependence of the saturation magnetization and the Curie point existence [156 157]. It is very difficult to find a magnetic material at elevated temperatures, which can be used at the same time for sensing processes. We say that temperatures from 0 to 180  C correspond, in most general estimations, to the normal functionality interval. There are many amorphous materials with a measurable MI response at room temperature in which MI effect totally disappears at 180  C because of a low Curie temperature (for example, in Fe3 Co64 Cr3 Si15 B12 ribbons with a Curie temperature of 160  C and a crystallization temperature of 570  C [158 159]). Another problem is structural stability. The amorphous state is a metastable one. The closer we are to the crystallization temperature (around 500  C for many amorphous materials), the less stable are the structural features of the amorphous sensitive element. Especially important, if we take into account those facts, is the presence of proper induced magnetic anisotropy, which shows even faster temperature degradation compared with the changes of structural parameters [160 161]. A high-temperature giant magnetoimpedance effect was reported for Fe73#0 Cu1#0 Nb2#5 V1#0 Si13#5 B9#0 nanocrystalline ribbons with measuring temperatures up to 550  C [135]. A maximum relative change of the impedance of about 60% of the MI ratio with a sensitivity of about 20%/Oe in a small field appears at about 320  C. Let us emphasize that the maximum sensitivity of magnetic sensors based on a giant magnetoresistance effect does not exceed 2%/Oe in CoCu multilayers at room temperature. The MI effect of Fe73#0 Cu1#0 Nb2#5 V1#0 Si13#5 B9#0 nanocrystalline ribbons still exists at 500  C (being 5%). The authors explain these outstanding properties by the thermal evolution of the magnetic softness and magnetic anisotropy of the nanocrystalline sample: as the temperature rises, a combination of stress release and magnetocrystalline anisotropy decrease in the amorphous phase further softens the ribbon magnetically, increasing its MI. There were additional attempts to study the MI effect in amorphous and nanocrystalline thin films and MI sandwiches of similar composition (Fe71#5 Cu1 Cr2#5 V4 Si12 B9 ) [150 154]. The MI ratio reaches its maximum of about 140% at a very low characteristic frequency of 4 MHz in FeCuCrVSiB/Ag/FeCuCrVSiB, but the temperature measurements for nanocrystalline films are missing at the moment.

Magnetoimpedance in Nanocrystalline Alloys

4.4. Recent Progress of the Magnetoimpedance Effect in Nanocrystalline Materials In this section, an updated overview covering the very recent works based in the MI effect in nanocrystalline alloys is given. The principles of remote-interrogated stress sensors based on high-frequency magnetic materials showing the inverse magnetostrictive or magnetoimpedance effects has been reported by Ludwig et al. [162]. Fe50 Co50 /Co80 B20 amorphous/nanocrystalline multilayers have been prepared, showing ferromagnetic resonance frequencies up to 5 GHz and good magnetoelastic properties in the gigahertz range. The high-frequency remote interrogation allows the reduction of the sensor antenna size and measurements using high bandwidths. The MI effect in powder samples of Fe-based nanocrystalline Fe73 Cu1 Nb1#5 V2 Si13#5 B9 has been reported for the first time [163]. The magnetic permeability of the powder materials may not be as high as that of thin films or ribbons and the coercivity may not be as small, but they can be conveniently pressed into different shapes such as cylinders and rings, offering less brittleness than nanocrystalline ribbons and wires, and many possible high-frequency applications. A systematic study of the high-temperature MI effect in nanocrystalline Fe73#5 Cu1 Nb3 Si13#5 B9 ribbons with different annealing temperatures has been presented [164]. The discussion is given on the basis of a random anisotropy model [165], to explain the relationship between the thermal dependence of the MI and the magnetic coupling between grains. A small GMI effect was found in nanocrystalline Fe79#5 P12 C6 Mo0#5 Cu0#5 Si1#5 ribbons obtained by annealing the precursor amorphous at temperatures ranging from 350 to 500  C [166]. A review of the amorphous alloy preparation, measurement system, and various annealing techniques used in the MI enhancement can be found in [167]. The effects of Cr substitution, the type of thermal treatments (in a conventional furnace and by current annealing), and the sample length in MI have been reported in Fe73#5−x Crx Cu1 Nb3 Si13#5 B9 (x = 0 and 10) nanocrystalline wires [168], and a similar study of the MI effect with annealing temperature was done in glass-covered CoBSiMn microwires [169]. The domain structure and induced anisotropy influences on the MI are both discussed. Two other recent works contribute to the study of the evolution of the MI effect in some nanocrystalline ribbons, previously submitted to different annealing treatments in an as-cast state: stress annealing on Fe88#02 Zr4#69 Nb2#31 B4#98 ribbons [170], and current annealing on Fe73#5 Cu1 Nb3 V2 Si15#5 B7 [171]. The MI behavior is explained in both papers by taking into account the role of the induced magnetic anisotropy, saturation magnetostriction coefficient, and the microstructural changes developed with the thermal treatments. A correlation among the structural, electrical, and magnetic properties, together with the MI effect response in some heat-treated FINEMET types of different composition and FeZrB ribbons, is given in [172]. High values of the MI ratio up to 130% have been achieved, at a drive frequency of 5 MHz, by subjecting FeZrB ribbons to a current annealing treatment.

Magnetoimpedance in Nanocrystalline Alloys

Table 1 displays a brief summary of some of the investigations mentioned through this chapter, indicating the studied material composition and its main properties related to the MI effect, with the respective references.

963 Finally, we can conclude that these compounds could be excellent candidates for the design and development of new magnetic sensors and high-frequency devices based on nanocrystalline alloys.

Table 1. MI Effect in Nanostructured Medium. Composition Fe4#3 Co68#2 Si12#5 B15 . Co68#15 Fe4#35 Si12#55 B15 FeCoSiB, HyMu
alloy, Fe77#5 Si7#5 B15 , Co68#5 Mn6#5 Si10 B15 ; CoP, Fe73#5 Cu1 Nb3 Si13#5 B9 , NiFe, CoFeB Fe73#5 Cu1 Nb3 Six B22#5−x x = 9 16 FeNiCrSiB, FeNiCrSiB/Cu/FeNiCrSiB, FeCuNbSiB, FeCuNbSiB/Cu/FeCuNbSiB, Fe/Cr nanoscale strutures, FeNi/Cu/FeNi and FeNi/Si/Cu/Si/FeNi, FeNi based multilayers Fe73#5 Cu1 Nb3 Si13#5 B9

Fe73#5 Cu1 Nb3 Si16#5 B6

Properties/remarks Low magnetostriction alloys. Induced anisotropy. MI and magnetoinductive effects. Overview of the MI effect in magnetic materials with different geometries (wires, ribbons, tubes, and thin films is given), including materials with a nanocrystalline structure. Sensitive field and frequency-dependent impedance.

Ref. [4 12 17] [7 8 17]

Magnetic anisotropy distribution and giant magnetoimpedance. Domain wall motion influence on MI. GMI effect in sputtered films and sandwiches at relatively low frequencies, and under the effect of an antisymmetric transverse bias field. Domain structure. Giant magnetoresistance in multilayers.

[53 128]

The correlation between GMI and magnetic anisotropy has been studied in field and stress one- and two-step annealed ribbons. Stressannealed ribbons with nonuniform induced anisotropy. Hysteretic behavior of MI. Magnetic anisotropy in as-quenched and stress-annealed nanocrystalline alloys: creep-induced and magnetoelastic components. High saturation polarization and magnetostriction in as-cast state.

[37 44 75 76 160 161]

[16 20 48 57 79 82 93 94 97]

[49 54 77 78 122 143 157 167 172]

Fe81 B11 Nb7 Cu1

High magnetic permeability, low coercive field, and vanishing magnetostriction in nanocrystalline state. Influence of nanocrystallization on evolution of domains and MI effect. dc and ac field-induced magnetic anisotropy. Ultrasoft magnetic properties such as large MI, incremental permeability, nearly zero coercivity, zero magnetostriction. Ultrasoft magnetic properties of nanocrystalline alloy.

[51]

FeCoNi tubes

Nonlinear magnetoimpedance.

[52 151]

Permalloy/Ag nanoscale multilayers

MI effect in homogeneous permalloy films and multilayers.

[80]

Ni77 Fe14 Cu5 Mo4

MI in commercial Mumetal for different thicknesses.

[83]

FeNiCu

MI of glass-coated microwires up to 200 MHz.

[88]

Co68#25 Fe4#5 Si12#25 B15

Influence of current annealing on magnetic properties and giant magnetoimpedance in glass-covered microwires. Giant magnetoimpedance in amorphous and nanocrystalline multilayers: experiment and theory. MI in sputtered films for Colpitts oscillator-type field sensor.

[89]

Fe84 Zr7 B6 Cu1 Al2

Fe73#5 Cu1 Nb3 Si13#5 B9 /SiO2 / Ti/Cu/Ti/SiO2 /Fe73#5 Cu1 Nb3 Si13#5 B9 (CoFe)80 B20 FeCoSiB FeCoSiB/Cu/FeCoSiB, CoSiB/Cu/CoSiB Fe73 Cu1 Nb1#5 V2 Si13#5 B9 Fe73 Cu1 Nb2#5 V1 Si13#5 B9 Fe92−x−y Zr7 Bx Cu1 Aly x = 2 4 6 8 y = 0 0#5 1 1#5 FeCuCrVSiB Ni77 Fe14 Mo5 Cu4 Fe77#5 Si7#5 B9 , Co67 Fe3 Cr3 Si15 B12

Thin-film magnetic field sensor utilizing MI effect. MI effect in magnetostrictive sandwiches. High-frequency longitudinally driven GMI effect in stress-annealed nanocrystalline ribbons. MI effect in the nanocrystalline alloy system. Magnetic properties and GMI in RF-sputtered films, followed by annealing treatment. Angular dependence of the MI in RF magnetron-sputtered films.

[50]

[95] [98 102] [100 104] [45 134 135] [136] [149 152] [150]

Fe79#5 P12 C6 Mo0#5 Cu0#5 Si1#5

Comparative study of the anisotropy, magnetic properties, and domain structure and MI. MI in nanocrystalline ribbons.

[158–159] [168]

Fe88#02 Zr4#69 Nb2#31 B4#98

MI in stress-annealed nanocrystalline ribbons.

[170]

Fe91 Zr7 B2

MI in current-annealed nanocrystalline ribbons

[172]

964

GLOSSARY Amorphous alloy Alloy with no long range atomic order. Magnetic anisotropy The magnetic properties depend on the direction in which they are measured. Magnetic domain Volume of material in which all atomic magnetic moments are aligned in the same direction. Magnetic permeability The ability of a material to support a magnetic flux density. Magnetoimpedance Impedance change in a ferromagnetic sample when submitted to a magnetic field. Magnetostriction Deformation of a magnetic substance due to its magnetic state. Nanocrystalline alloy Alloy containing crystalline nanograins embebed in an amorphous matrix.

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