Encyclopedia of Nanoscience and Nanotechnology 9781588830661, 1588830667, 6006507007508

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Encyclopedia of Nanoscience and Nanotechnology Volume 10 Number 1 2004 1

Soft and Hard Magnetic Nanomaterials Juli¨¢n Gonz¨¢lez; Oksana Chubykalo; Jes¨²s M. Gonz¨¢lez Sol-Gel Derived Nanocrystalline Semiconductor Oxide Gas Sensors

27

Satyajit Shukla; Sudipta Seal Solid Lipid Nanoparticles and Nanostructured Lipid Carriers

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R. H. M¨¹ller; M. Radtke; S. A. Wissing Solid-State Synthesis of Carbon Nanotubes

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Hua Chun Zeng Sonochemical Synthesis of Nanomaterials

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S. Sundar Manoharan; Manju Lata Rao Stealth and Biomimetic Core-Corona Nanoparticles

83

Ruxandra Gref; Patrick Couvreur Stilbenoid Dendrimers

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Herbert Meier; Matthias Lehmann Strain Effects in Manganite Nanostructured Thin Films

107

W. Prellier; B. Mercey; A. M. Haghiri Gosnet Structural Characterization of Single-Walled Carbon Nanotubes

125

Daniel E. Resasco; Jose E. Herrera Structure of Mesoporous Silica

149

Wuzong Zhou Structure of Nanoclusters by High-Resolution Transmission Electron Microscopy

161

J. Urban Structures of Epitaxial Quantum Dots

175

X. Z. Liao; J. Zou; D. J. H. Cockayne Superhard Nanocomposites

191

Gregory M. Demyashev; Alexander L. Taube; Elias Siores Superplasticity in Nanoceramics

237

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M. Pal; D. Chakravorty Supramolecular Coordination Polymers

247

Jing-Cao Dai; Zhi-Yong Fu; Xin-Tao Wu Surface Functionalization of Semiconducting Nanoparticles

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Marie-Isabelle Baraton Surface Nano-Alloying

283

Ruxandra Vidu; Nobumitsu Hirai; Shigeta Hara Synthesis of Covalent Nanoparticles by CO2 Laser

301

Nathalie Herlin-Boime; Martine Mayne-L'Hermite; C¨¦cile Reynaud Synthesis of Inorganic Nanowires and Nanotubes

327

S. Sharma; H. Li; H. Chandrasekaran; R. C. Mani; M. K. Sunkara Synthesis of Metal Chalcogenide Nanoparticles

347

Jun-Jie Zhu; Hui Wang Synthesis of Nanomaterials Using Microemulsion Process

369

Suresh C. Kuiry; Sudipta Seal Synthetic Nanoinorganics by Biomolecular Templating

381

S. Behrens; E. Dinjus; E. Unger Techniques in Electrochemical Nanotechnology

393

P. Schmuki; S. Maupai; T. Djenizian; L. Santinacci; A. Spiegel; U. Schlierf Theory of Semiconductor Quantum Devices

411

Rita Claudia Iotti; Remo Proietti Zaccaria; Fausto Rossi Thermal Conductivity of Semiconductor Nanostructures

425

Alexander A. Balandin Transition Metal Acetylide Nanostructures

447

Thomas M. Cooper Transition Metal Nanocluster Assemblies

471

K. Sumiyama; T. Hihara; D. L. Peng; S. Yamamuro Transition Metal Atoms on Nanocarbon Surfaces

509

Antonis N. Andriotis; Madhu Menon Transition Metals-Based Nanomaterials for Signal Transduction

519

Bamaprasad Bag; Parimal K. Bharadwaj Transport in Self-Assembled Quantum Dots

537

S. W. Hwang; Y. J. Park; J. P. Bird; D. Ahn Transport in Semiconductor Nanostructures

547

H. Le¨®n; R. Riera; J. L. Mar¨ªn; R. Rosas UHV-SPM Nanofabrication

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G. Palasantzas; J. Th. M. De Hosson; L. J. Geerligs 595

Unconventional Nanolithography Kahp Y. Suh; Dahl-Young Khang; Y. S. Kim; Hong H. Lee

613

Vacuum Nanoelectronics Ning-Sheng Xu

639

Visible-Light-Sensitive Photocatalysts Hiromi Yamashita; Masato Takeuchi; Masakazu Anpo

655

X-Ray Characterization of Nanolayers Dirk C. Meyer; Peter Paufler

681

X-Ray Microscopy and Nanodiffraction S. Lagomarsino; A. Cedola X-Ray Photoelectron Spectroscopy of Nanostructured Materials

711

J. Nanda; Sameer Sapra; D. D. Sarma Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

725

Fiorenzo Vetrone; John-Christopher Boyer; John A. Capobianco Zinc Oxide Nanostructures

767

Chun-Hua Yan; Jun Zhang; Ling-Dong Sun Copyright © 2004 American Scientific Publishers

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Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Soft and Hard Magnetic Nanomaterials Julián González Facultad de Química, Paseo Manuel de Lardizabal 3, 20018 San Sebastián, Spain

Oksana Chubykalo, Jesús M. González CSIC, Cantoblanco, Madrid, Spain and UCM-RENFE-CSIC, Las Rozas, Madrid, Spain

CONTENTS 1. Introduction 2. Soft Nanocrystalline Materials 3. Hard Nanocrystalline Materials 4. Micromagnetic Numerical Simulations 5. Conclusions Glossary References

1. INTRODUCTION Magnetic materials today constitute, from the standpoint of the size of their world market and in the narrow concurrence with the semiconducting materials, one of the major groups of functional materials. They are involved in a broad range of technologies, from electromechanics to those related to information recording. Although it is difficult to find a common point underlying this large set of applications and devices, the occurrence of hysteresis in their response to an applied magnetic field could be considered the most representative feature of the magnetic materials phenomenology. Magnetic hysteresis can be described by two quantities, namely, the remanence and the coercive force. For a given set of phases, present in a particular material and characterized by particular values of saturation magnetization, order temperature, and magnetocrystalline anisotropies, the coercivity and remanence depend on many different extrinsic parameters, from the phase morphology (crystallites size and shape) to the distribution of defects present in them and, particularly, the characteristics of the intergranular and interphase couplings. The basic consequences of this are the possibility of optimizing and, in some cases, tailoring the properties of a material for a particular application, during the last century, most of the research efforts in the field

ISBN: 1-58883-066-7/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

were concentrated on the control of the micro- and nanostructures of a relatively reduced set of relevant phases. It also is important to state that, whereas the coercivity of the technologically used magnetic materials covers six decades (from ca. 5 × 10−6 T up to ca. 5 T), the remanence values are bound by the value of the saturation magnetization, and the whole range of technologically relevant magnetic materials varies from 0.5 T up to ca. 2 T. The introduction in the technologically relevant magnetic materials of structural correlation lengths of the order of the nm has several important consequences. First, and in the particular case of the nanoparticulate and the nanocrystalline materials, it results in a significant increase of the surface(grain boundary)-to-(particle/grain)volume atomic ratio. Since the moments present at the surfaces and grain boundaries are characterized by a co-ordination different from that corresponding to the bulk materials, the local values at these regions of the magnetization, order temperature, exchange constant, and anisotropy can be significantly different from those corresponding to bulk-like regions and largely influence and even rule the global behavior of the system. Second, the reduction of the crystallite size crucially influences, through the reduction of the absolute number of defects present inside the structurally coherent regions, the global value of defect sensitive properties as, very relevantly, the coercivity. Finally, and most importantly, the nanostructuration brings about the problem of the interphase coupling at length scales comparable with the magnetic correlation lengths, i.e., the exchange and dipolar correlation lengths, given the width of a domain wall in a bulk and planar uniaxial systems, respectively. Since the coupling is largely ruled by the characteristics of the exchange interactions at the grain boundaries, and since, for the time being, the control of those properties could only be achieved heuristically, it is not exaggerated to state that the main goal of the present research on the magnetic properties of nanostructured materials could be the achievement of better control of the magnetic properties of the intergranular regions.

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (1–26)

2 To support this statement, we will focus on the discussion of the implications of the particularities of the intergranular coupling on the hysteretic behavior of nanocrystalline materials. First, we will cover the case of the (mainly single phase) reduced magnetocrystalline anisotropy materials, in which the dipolar correlation length frequently can be much larger than the crystallite size (ca. 15 nm), thus resulting in an extremely soft behavior linked to the occurrence of exchange-induced averages of the local anisotropy. The second section will be dedicated to the demagnetization process of high anisotropy, single- and multiphase nanocrystalline materials characterized by exchange correlation lengths comparable with or smaller than the crystallite size. In this case, the goal is either the reduction of the intergranular coupling (single-phase materials) aiming at the increase of coercivity or the achievement of large remanences linked to the occurrence of strong coupling between hard and soft grains, the latter having dimensions comparable with their exchange correlation length. We will end this chapter by reviewing the state of the art of the micromagnetic modeling, a numerical technique allowing both the analysis of systems for which (as it is the largely majority case today) there are not experimental data on the local magnetic properties and the implementation of elements of device design. This last section also will include a discussion of the influence on the performance of magnetic recording media of the control of the intergranular exchange.

2. SOFT NANOCRYSTALLINE MATERIALS 2.1. Introduction to the Soft Magnetic Materials Nanocrystalline materials, obtained by devitrification of the precursor amorphous alloy, displaying soft magnetic character (high magnetic permeability and low coercivity), have been the subject of increasing attention from the scientific community, not only because of their potential use in technical applications but also because they provide an excellent setting in which to study basic problems in nanostructures formation and magnetism [1–8]. In fact, these materials provide a crucial point in opening up new fields of research in materials science, magnetism, and technology, such as metastable crystalline phases and structures, extended solid solubilities of solutes with associated improvements of mechanical and physical properties, nanocrystalline, nanocomposite and amorphous materials that, in some cases, have unique combinations of properties (magnetic, mechanical, corrosion, etc.). Technological development of the fabrication technique of the amorphous precursor material and studies of the structure, glass formation ability, and thermodynamics, and magnetism of amorphous alloys were intensively performed in 1960s and 1970s. These aspects have been analyzed extensively in few review papers and books [9–11]. Most commercial and technological interests have been paid to soft amorphous and nanocrystalline magnetic materials. Initially, it was believed that ferromagnetism could not

Soft and Hard Magnetic Nanomaterials

exist in amorphous solids because of a lack of atomic ordering. The possibility of ferromagnetism in amorphous metallic alloys was theoretically predicted by Gubanov [12] and the experimental confirmation of this improbable prediction was the main cause of the sudden acceleration of research on amorphous alloys from about 1970 onward, this onrush of activity was due both to the intrinsic scientific interest of a novel and unexpected form of ferromagnetism and also to the gradual recognition that this is the key to the industrial exploitation of amorphous ferromagnetic alloys. The amorphous alloy ribbons obtained by the melt-spinning technique have been introduced widely as the soft magnetic materials in the 1970s. Their excellent magnetic softness and high wear and corrosion resistance made them very attractive in the recording head and microtransformer industries. In contrast with the flood of work on magnetic behavior, the study of electrical transport (i.e., magnetoimpedance effect) is very recent and is making significant progress. Conventional physical metallurgy approaches to improving soft ferromagnetic properties involve tailoring the chemistry and optimizing the microstructure. Significant in the optimizing of the microstructure is recognition that a measure of the magnetic hardness (the coercivity, Hc ) is roughly inversely proportional to the grain size D for a grain size exceeding ∼01 to 1 m (where the grain size exceeds the domain wall thickness). In such cases, grain boundaries act as impediments to domain wall motion, and, thus, finegrained materials usually are magnetically harder than large grain materials. Significant developments in the understanding of magnetic coercivity mechanisms have lead to the realization that for very small grain size D < ∼100 nm [13–21], Hc decreases rapidly with increasing grain size. This can be understood by the fact that the domain wall, whose thickness exceeds the grain size, now samples several (or many) grains so that fluctuations in magnetic anisotropy on the grain-size length scale are irrelevant to domain wall pinning. This important concept suggests that nanocrystalline and amorphous alloys have significant potential as soft magnetic materials. In this section, we explore issues that are pertinent to the general understanding of the magnetic properties of amorphous and nanocrystalline materials. As the state of the art for amorphous magnetic materials is well developed and much of which has been thoroughly reviewed [11, 22–24], we will concentrate on highlights and recent developments. The development of nanocrystalline materials for soft magnetic applications is an emerging field for which we will try to offer a current perspective that may well evolve further with time. The development of soft magnetic materials for applications requires the study of a variety of intrinsic magnetic properties as well as development of extrinsic magnetic properties through an appropriate optimization of the microstructure. As intrinsic properties, we mean microstructure insensitive properties. Among the fundamental intrinsic properties (which depend on alloy composition and crystal structure), the saturation magnetization, Curie temperature, magnetic anisotropy, and magnetostriction coefficient are all important. In a broader sense, magnetic anisotropy and magnetostriction can be considered as extrinsic in that,

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2.2. Microstructural Characterization

–1

1W·g

An important part of the recent developments corresponding to nanostructured materials is related to those obtained by controlled crystallization, either by annealing the amorphous single phase or by decreasing the cooling rate from the liquid of metallic systems. Typically, in these nanostructures, precipitate sizes range between 5 and 50 nm embedded in an amorphous matrix with nanocrystal volume fractions of 10 to 80%, which means particle densities of 1022 to 1028 m−3 . We present the most relevant aspects of the nanocrystallization process of Fe-based nanocrystalline alloys as a two-phase system, namely -Fe or -Fe(Si) grains embedded in a residual amorphous matrix, which, being ferromagnetic, results in a material with extremely good soft magnetic properties. The microstructural analysis of the primary crystallization of Fe-rich amorphous alloys usually has been done by using conventional techniques such as differential scanning calorimetry (DSC), X-ray diffraction (XRD), transmission Mössbauer spectroscopy (TMS), and transmission electron microscopy (TEM). In this way, through the combined structural analysis of these techniques, useful information, such as the dependence of the microstructure upon time and temperature of treatment, can be obtained. The kinetics of metastability loss of the disordered system above glass transition, i.e., under less than equilibrium conditions, is a key subject, because it provides new opportunities for structure control by innovative alloy design and processing strategies. Several examples include soft and hard magnets and high-strength materials [30–32]. Most studies focus on the crystallization onset as a measure of kinetic stability under heat treatment and recognise the product phase selection involved in nucleation and the role of competitive growth kinetics in the evolution of different microstructural morphologies [33, 34]. Differential scanning calorimetry has become quite effective as a means of studying the nature of nanoscale structures and their stability. It has been established that the initial annealing response allows one distinguish between a sharp onset for a nucleation and growth or a continuous grain growth of pre-existing grains [35]. Kinetic

data on the transformation often are obtained from this technique. Figure 1 shows the DSC curves of the as-prepared amorphous alloy (Fe735 Cu1 Nb3 Si175 B5 , trademark Finemet), as well as of that of alloy previously annealed for 1 hour at 703 K and 763 K, respectively [36]. For the as-prepared alloy, the calorimetric signal shows some relaxation before the exothermic nanocrystallization process, as well as with the qmc Curie temperature of the amorphous phase (TC ≈ 595 K). When the sample has been annealed at 703 K, relaxation is no longer apparent in the calorimetric signal, the Curie temperature is shifted about 15 K, to higher values, and the nanocrystallization process is slightly advanced in temperature (2 K). On further increasing the annealing temperature, the calorimetric signal shows no clear changes in the Curie temperature of the annealed sample with respect to annealing at 703 K. However, a clear shift of the nanocrystallization onset toward higher temperatures (40 K) can be observed, as well as a significant decrease of its area [36]. Consequently, DSC measurement permits evaluation of the maximum temperature to prevent partial crystallization during previous heating and the maximum heating rate to control the temperature of the sample by analyzing the transformation peak connected to the primary crystallization. Calorimetric measurements, however, give information about microstructural development. Microstructure determination by the use of several techniques (XRD, TMS and TEM) allows us to complete the understanding of the mechanisms of the primary crystallization. Transmission Mössbauer spectroscopy has the main advantage of giving local information of an active element (Fe nuclei in these alloys). Because Fe is present both in the nanocrystalline precipitates and in the disordered matrix, TMS provides information on local ordering in both phases, which can be correlated with the changes in the short-range order of the amorphous phase and with the composition of nanocrystalline phase. X-ray diffraction and TEM provide a

Heat flow / W·g–1

for a two-phase material (in aggregate), they depend on the microstructure. A vast literature exists on the variation of intrinsic magnetic properties with alloy composition. Although new discoveries continue to be made in this area, it can be safely stated that a more wide open area in the development of magnetic materials for applications is the fundamental understanding and exploitation of the influence of the microstructure on the extrinsic magnetic properties. Important microstructural features include grain size, shape, and orientation; defect concentrations; compositional inhomogeneities; magnetic domains; and domain walls. The interaction of magnetic domain walls with microstructural impediments to their motion is of particular importance to the understanding of soft magnetic behavior. Extrinsic magnetic properties important in soft magnetic materials include magnetic permeability and coercivity, which typically have an inverse relationship. Thorough discussions of soft magnetic materials are available [25–28, 29].

as prepared after 1 h annealing at 703 K after 1 h annealing at 763 K

600

650

700

750

800

850

900

T (K)

Figure 1. DSC signals of Fe735 Si135 B9 Cu1 Nb3 alloy obtained after heating at 40 K/min. For the as-quenched amorphous alloy (solid line); after 1 h isothermal annealing at 703 K (dashed line) and at 763 K (dotted line). Reprinted with permission from [37], M. T. Clavaguera-Mora et al., Progress in Materials Science 47, 559 (2002). © 2002, Elsevier Science.

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close look at the developed microstructure and permit the characterization of the precipitates, showing their morphology and grain-size distribution. Figure 2(a–c) shows a TEM image of a Finemet alloy after heating at 763 K at three annealing times (3, 10, and 30 minutes), while Figure 2(d) corresponds to the case (b) with increasing contrast, which allows the shape and size of the precipitates to be determined [37]. As can be seen, the microstructure is characterized by a homogeneous, ultrafine grain structure of -Fe(Si), with grain sizes around 10 nm and a random texture, embedded in a still amorphous matrix. The formation of this particular structure is ascribed to the combined effects of elements as Cu (which promotes the nucleation of grains) and Nb, Ta, Zr, Mo,—(which hinders their growth) and their low solubility in -Fe(Si) [38–40]. Nevertheless, the size and morphology of the nanocrystals in these alloys, as well as their distribution, could be analyzed by the application of local probe techniques. These techniques, such as scanning tunneling microscopy (STM) and atomic force microscopy (ATM) provide threedimensional (3D) topographic images at the nanometer level [41, 42] and represent powerful tools to study the surface properties and structures of metals and alloys.

2.3. On the Ferromagnetism in Amorphous and Nanocrystalline Materials There are, when concerning the magnetic order in materials having structural disorder (such as is the case of amorphous and nanocrystalline alloys), some fundamental questions related to the existence of such a well-defined magnetic order. Ferromagnetic interactions of the magnetic materials can be immediately considered as ferromagnetic structures. In this naive idea, the magnetic anisotropy effects have been neglected. Magnetic moments tend to arrange their orientations parallel to each other via exchange interactions; they

(a)

(b)

(c)

(d)

Figure 2. TEM images of Finemet samples after (a) 3 min; (b) 10 min; (c) min annealing at 763 K; and (d) is the case (b) after contrast increasing. Reprinted with permission from [37], M. T. Clavaguera-Mora et al., Progress in Materials Science 47, 559 (2002). © 2002, Elsevier Science.

do this when lying along a magnetic easy axis that is in the same direction at every point in the material. However, if the easy axis orientation fluctuates from site to site, a conflict between ferromagnetic coupling and anisotropy arises. As long as we imagine lattice periodicity, a ferromagnetic structure is a consequence of ferromagnetic exchange interactions, the strength of the anisotropy being irrelevant. In this situation, we are assuming a major simplification, namely, the direction of the easy axis is uniform throughout the sample. With this simple picture, we present crucial questions related to the influence of an amorphous structure on magnetic order. Regarding the magnetic order in amorphous and nanocrystalline materials, we know that it stems from two contributions: exchange and local anisotropy. The exchange arises from the electron–electron correlations. The mechanism of the electrostatic interactions between electrons has no relation to structural order and is sensitive only to overlapping of the electron wave functions. With respect to magnetic anisotropy, it also originated by the interaction of the local electrical field with spin orientation, through the spin-orbit coupling. Therefore, magnetic anisotropy also is a local concept. Nevertheless, the structural configuration of magnetic solids exerts an important influence on the macroscopic manifestation of the local anisotropy. As a consequence, when the local axes fluctuate in orientation owing to the structural fluctuation (amorphous and nanocrystalline materials as examples), calculations of the resultant macroscopic anisotropy become quite difficult. In the case of amorphous ferromagnetic alloys, the usual approach to the atomic structure of a magnetic order connected to a lattice periodicity is not applicable. These materials can be defined as solids in which the orientation of local symmetry axes fluctuate with a typical correlation length l = 10 A. The local structure can be characterized by a few local configurations with icosahedral, octahedral, and trigonal symmetry. These structural units have randomly distributed orientation. The local magnetic anisotropy would be larger in the units with lower symmetry. In general, these units are characterized by fluctuations of the orientation local axis. It is remarkable that with these types of structures, the correlation length, l, of such a fluctuation is typically the correlation length of the structure and ranges from 10 A (amorphous) to 10 nm (nanocrystals) and 1 mm (polycrystals). Fluctuations in the interatomic distances associated with the amorphous structure also should contribute to some degree of randomness in the magnetic interactions of the magnetic moments. Nevertheless, such randomness is expected not to affect the magnetic behavior qualitatively [11, 43]. Moreover, random distribution of the orientation of the easy axis drastically affects the magnetic properties. The random anisotropy model developed by Alben et al. [44] provides a successful explanation of how the correlation length, l, exerts a relevant influence on magnetic structure. The important question is What is the range of orientational correlation of the spins? Let L be the correlation length of the magnetic structure. If we assume L > l, the number of oriented easy axes in a volume L3 should be N = L/l3 . The effective anisotropy can be written as: Keff = K/N 1/2

(1)

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where K is the local anisotropy where strength is assumed to be uniform everywhere. By minimizing the total energy with respect to L, the following expression can be deduced: L = 16A2 /9K 2 l3 

(2)

where A is the exchange stiffness parameter. If we consider A = 10−11 J/m and l = 10−9 m, which are typical values of ferromagnetic metallic glasses [45], L in equation (2) becomes 105 /K 2 . For 3D-based alloys, we can take the value of K corresponding to crystalline samples (∼104 J/m3 ) leading to L around 10−9 m, which is equal to the structural correlation length of an amorphous material. In addition, the random anisotropy model provides the following expression for the average macroscopic anisotropy: K = K 4 l6 /A3

(3)

Equation (3) points out that the macroscopic structural anisotropy is negligible in 3D amorphous alloys K ∼ 10−9 K); this is a consequence of the averaging of several local easy axes, which produces the reduction in magnitude. Special attention has been paid, in the last decade, to the study of nanocrystalline phases obtained by suitable annealing of amorphous metallic ribbons owing to their attractive properties as soft magnetic materials [1, 15, 19–21, 23, 46–48]. Such soft magnetic character is thought to have originated because the magnetocrystalline anisotropy vanishes and there is a very small magnetostriction value when the grain size approaches 10 nm [1, 12, 46]. As was theoretically estimated by Herzer [12, 46], average anisotropy for randomly oriented -Fe(Si) grains is negligibly small when grain diameter does not exceed about 10 nm. Thus, the resulting magnetic behavior can be well described with the random anisotropy model [12, 19, 23, 46–48]. According to this model, the very low values of coercivity in the nanocrystalline state are ascribed to small effective magnetic anisotropy (Keff around 10 J/m3 . However, previous results [19, 21, 49] as well as recently published results by Varga et al. [50] on the reduction of the magnetic anisotropy from the values in the amorphous precursors (∼1000 J/m3 ) down to that obtained in the nanocrystalline alloys (around 300–500 J/m3 ), is not sufficient to account for the reduction of the coercive field accompanying the nanocrystallization. The enhancement of the soft magnetic properties should, therefore, be linked to the decrease of the microstructure– magnetization interactions. These interactions, originating in large units of coupled magnetic moments, suggest a relevant role of the magnetostatic interactions, as well a role in the formation of these coupled units [19, 49]. In addition to the suppressed magnetocrystalline anisotropy, low magnetostriction values provide the basis for the superior soft magnetic properties observed in particular compositions. Low values of the saturation magnetostriction are essential to avoid magnetoelastic anisotropies arising from internal or external mechanical stresses. The increase of initial permeability with the formation of the nanocrystalline state is closely related to a simultaneous decrease of the saturation magnetostriction. Partial crystallization of amorphous alloys leads to an increase of the frequency range, where the permeability presents high values [51]. These high values in the highest

possible frequency range are desirable in many technological applications involving the use of ac fields. It is remarkable that a number of workers have investigated the effects on the magnetic properties of the substitution of additional alloying elements for Fe in the Fe735 Cu1 Nb3 Si135 B9 alloy composition, Finemet, to further improve the properties, e.g., Co [52–56], Al [20, 57, 58], varying the degree of success. Moreover, it was shown in [20] that substitution of Fe by Al in the classical Finemet composition led to a significant decrease in the minimum of coercivity, Hcmin ≈ 05 A/m, achieved after partial devitrification, although the effective magnetic anisotropy field was quite large (around 7 Oe) [59]. The coercivity behavior was correlated with the mean grain size, and a theoretical low effective magnetic anisotropy field of the nanocrystalline samples was assumed in contradiction with those experimentally found in [49, 50, 58]. Although amorphous Fe-, Co-, and Ni-based ribbons are slightly more expensive compared with conventional soft magnetic materials, such as sendust, ferrites, and supermalloys (mostly due to the significant content of Co and Ni), they found considerable applications in transformers (400 Hz), ac powder distributors (50 Hz), magnetic recording as a magnetic heads, and magnetic sensors. The main reason for using amorphous alloys such as soft magnetic materials is a saving of the electric energy wasted by magnetic cores. Besides, the combination of high magnetic permeability and good mechanical properties of amorphous alloys may be used successfully in magnetic shielding and in magnetic heads [51]. Production of about 3 millions heads per year in Japan in the mid-1980s has been reported [51]. The internal stresses, as the main source of magnetic anisotropy in amorphous and nanocrystalline materials, are due to the magnetoelastic coupling between magnetization and internal stresses through magnetostriction. Consequently, these materials are interesting for field- and stresssensing elements because the Fe-rich amorphous alloys exhibit high magnetostriction values (s ≈ 10−5 ) and, therefore, many of magnetic parameters (i.e., magnetic susceptibility, coercive field, etc.) are extremely sensitive to the applied stresses. The discovery of Fe-rich nanocrystalline alloys carried out by Yoshizawa et al. [1] was important owing to the outstanding soft magnetic character of such materials. Typical compositions of the precursor amorphous alloys, which, after partial devitrification, reach the nanostructure character with optimal properties, are FeSi and FeZr, with small amounts of B to allow the amorphization process, and smaller amounts of Cu, which act as nucleation centers for crystallites, and Nb, which prevents grain growth. This effect is provided by Zr in FeZr alloys. After the first step of crystallization, FeSi or Fe crystallites are finely dispersed in the residual amorphous matrix. In a wide range of crystallized volume fraction, the exchange correlation length of the matrix is larger than the average intergranular distance, d, and the exchange correlation length of the grains is larger than the grain size, D. Magnetic softness of Fe-rich nanocrystalline alloys is due to a second complementary reason: the opposite sign of the magnetostriction constant of crystallites and residual amorphous matrix,

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which allows reduction and compensation of the average magnetostriction. Figure 3 shows the thermal variation of the coercive field (Hc ) in a Finemet-type (Ta-containing) amorphous alloy. This behavior is quite similar to that shown in the case of Nbcontaining ones and particularly, evidences the occurrence of a maximum in the coercivity linked to the onset of the nanocrystallization process [60, 61]. Considering the grain size, D, to be smaller than the exchange length, Lex , and the nanocrystals are fully coupled between them, the random anisotropy model implies a dependence of the effective magnetic anisotropy K, with the sixth power of average grain size, D. The coercivity is understood as a coherent rotation of the magnetic moments of each grain toward the effective axis leading to the same dependence of the coercivity with the grain size [16]: Hc = pc


 K 4 D6 K K 4 D6 = pc 1 3 K = 1 3  Js Js A A

(4)

where K1 = 8 kJ/m3 is the magnetocrystalline anisotropy of the grains, A = 10−11 J/m is the exchange ferromagnetic constant, Js = 12 T, is the saturation magnetic polarization and pc is a dimensionless prefactor close to unity. The predicted D6 dependence of the coercive field has been widely accepted to be followed in a D range below Lex (around 30 to 40 nm) for nanocrystalline Fe–Si–B–M–Cu (M = Iva to Via metal) alloys [21, 46, 62–65]. A clear deviation from the predicted D 6 law in the range below Hc = 1 A/m was reported by Hernando et al. [19]. Such deviation was ascribed to effects of induced anisotropy (e.g., magnetoelastic and field induced anisotropies) on the coercivity could be significant with respect those of the random magnetocrystalline anisotropy. As a consequence, the data of Hc D were fitted by assuming a contribution from (i) the spatial fluctuations of induced anisotropies and (ii) Ku  to Hc (i.e., a dependence Hc = a2 + bD6  was found with

a = 1 A/m representing the contribution originating from the induced anisotropies). To investigate the effect of the grain size on coercivity, this dependence of Hc D in alloys treated by Joule heating was obtained. Experimental results on this dependence are shown in Figure 4 [21]. The fitting of this dependence appears to follow, surprising, a rough dependence of the Hc ∝ D3−4 type (the best regression was found fitting the D 3 law). It must be noted that our data of Hc D correspond to a grain size variation between about 10 to 150 nm. As it is well known, an analysis of this Hc D data in terms of the random anisotropy model is only justified if the grain size is smaller than Lex and, hence, could not be applicable (in the framework of the random anisotropy model) to the range grain size above Lex , which results in being only two points of our data in Figure 4 [21]. These points should correspond to a magnetic hardening due to the precipitation of the iron borides. In this case, the random anisotropy model should be applied by taking into account the volume fraction and the different anisotropy of the iron borides. This indicates that K1  should vary as D3 , contrary to the theoretical D6 law. This indicates that Hc is mainly governed by K1 , which varies as D3 , contrary to the theoretical D6 law. This contradiction of the Hc D law between the theory and the experimental has recently been explained by Suzuki et al. [62] considering the presence of long-range uniaxial anisotropy, Ku , which influences the exchange correlation value and length, and yields an anisotropy average given by:  Ku · K1 · D3 1 K = Ku + · (5) 2 A3/2 The second part of (5) corresponds to K1  and if Ku is coherent in space or if its spatial fluctuations are negligible to K1 , this second part ultimately determines the grain size influence on the coercivity. Such influence changes from the D6 law to a D3−4 one when the coherent uniaxial anisotropies dominate over the random magnetocrystalline anisotropy. An additional point in order to justify the Eq. (5)

100 10

4

Hc (A/m)

Hc (A/m)

103

10

102

10

1

Fe73.5Si13.5B9Cu1Ta3 1 Hc - 2.9×10 1 0

10

20

30

40

50

60

Jann (A/mm2)

Figure 3. Evolution of the coercive field, measured at room temperature, as a function of the current density after: (o) 1 min and (__) 10 min of annealing time. Reprinted with permission from [21], N. Murillo and J. González, J. Magn. Magn. Mater. 218, 53 (2000). © 2000, Elsevier Science.

0.1 101

–4

3.35

D

102

D (nm)

Figure 4. Dependence of the coercive force, Hc , with the average grain diameter, D, for the two studied compositions (Fe735 Si135 B9 Cu1 Nb3 and Fe735 Si135 B9 Cu1 Ta3 . Reprinted with permission from [21], N. Murillo and J. González, J. Magn. Magn. Mater. 218, 53 (2000). © 2000, Elsevier Science.

7

Soft and Hard Magnetic Nanomaterials

is connected with the fact that the results of Figure 4 were obtained in samples treated by the Joule heating effect. This kind of annealing could induce some inhomogeneous magnetic anisotropy, which could be responsible for this significant change of the grain size dependence of the coercive field. This argument is supported by the remanence ratio, which is achieved by these samples of values around 0.50. As a consequence, it can be assumed that the presence of more long-range uniaxial anisotropies are larger than the average magnetocrystalline anisotropy K1 . It should also be noted that Eq. (5) can, interestingly, account for the occurrence of dipolar and deteriorated exchange intergrain interaction and thus can be more realistic than the simple anisotropy averaging, since those features are involved in the accomplishment of a nanocrystallization process. An interesting study can be one related with the dependencies of the magnetization in partially crystallized samples. Figure 5 displays such dependencies. In Table 1, the evolution with the annealing time of the average grain size, D, the Si atomic percentage diffused in the Fe crystalline lattice, and the crystalline phase percentage are listed. It has been mentioned that in samples annealed with short times (0.5 to 5 minutes), there was no evidence of crystallization. In samples treated for long times, the Si content was of the order of 20% at and slightly larger in the samples having a larger grain size. On the other hand, the current density dependence of the coercivity for Finemet-type alloys results in a very interesting study (illustrated by the Fig. 6). Such dependencies exhibit a peak of coercivity in nanocrystallized samples (treated for 60–720 minutes). This peak of coercivity occurs above the Curie point of the residual amorphous phase. The intensity and the width of the peak strongly depend on the annealing current density. It must be mentioned that the current density above the Curie point of the amorphous matrix being paramagnetic and with its thickness is high enough to avoid exchange interactions between the grains, the nanocrystalline sample can be magnetically considered

Table 1. Evolution with the annealing time of the percentage of crystalline phase, percentage of Si content inside bcc phase, and average grain size of Fe735 Cu1 Nb3 Si135 B9 amorphous alloy ribbon treated by current annealing at 40 A/mm2 .

Tann (min) % crystalline phase 60 120 300 720

60 65 70 78

% of Si content inside bcc phase

Average grain size (nm)

19 21 21 22

12 15 15 17

as an assembly of isolated or weakly magnetostatic interactive single domain particles. The coercivity behavior can be interpreted in the framework of the two-phase model [47]. At room temperature, the system is soft because the exchange between crystallites is large enough to make the correlation length larger than both the intergranular distance and the grain size. As the current density rises and approaches the Curie point of the residual amorphous matrix, the exchange constant decreases, and some grains start to be weakly coupled. The exchange correlation length decreases and the crystallites progressively start to act as pinning centers. It is interesting to mention the variations on the Curie point with the annealing time (Fig. 6) [62], corresponding to either the remaining amorphous matrix or the nanocrystalline phase. In the temperature range below the Curie temperature of the residual intergranular amorphous phase (TCam , the nanocrystallites are coupled magnetically via the exchange interaction acting over the bcc–amorphous–bcc coupling chain. However, this coupling chain is diminished in the temperature range above TCam , and the nanocrystalline alloys behave as an assembly of isolated magnetic particles in which the magnetically hardest domain configuration is expected. Consequently, we observe a significant increase in Hc in the temperature range above ∼TCam [16]. This effect is a possible disadvantage of the nanocrystalline soft

1.5

80

1.2 60

Hc (A/m)

µ0 Ms (T)

2h

0.9

1h

40

5h

0.6 12 h

20 0.3

10

20

12 h

1h

"as-cast" 0 0

5h

2h

30

40

50

60

J (A/mm2)

Figure 5. Current density dependence of the saturation magnetisation of samples previously treated at 40 A/mm2 with 0, 0.5, 1, 2, 5, 60, 120, 300, and 720 min. Reprinted with permission from [62], J. González, J. Mater. Res. 18, 1038 (2003). © 2003, Materials Research Society.

0

0

10

20

30

40

50

J (A/mm2)

Figure 6. Variations of coercive field with current density of samples previously treated at 40 A/mm2 with 0, 0.5, 1, 2, 5, 60, 120, 300, and 720 min. Reprinted with permission from [62], J. González, J. Mater. Res. 18, 1035 (2003). © 2003, Materials Research Society.

8

2.4. Processing of Nanocrystalline Alloys. Induced Magnetic Anisotropy The magnetization characteristics of Finemet-type nanocrystalline magnets (FeCuNbSiB alloy) similar to those of metallic glasses, also can be well controlled by the magnetic anisotropy induced by field annealing, stress annealing, and stress plus field annealing. Magnetic field annealing induces uniaxial anisotropy with the easy axis parallel to the direction of the magnetic field applied during the heat treatment. The magnitude of the field-induced anisotropy in soft nanocrystalline alloys depends upon the annealing conditions (that is, if the magnetic field is applied during the nanocrystallization process or firstly the sample is nanocrystallized and then submitted to field annealing) [63, 66] and on the alloy composition (relative percentage content of Fe and metalloids) [67]. Nevertheless, this field-induced anisotropy is induced at a temperature range of 300 to 600  C (above the Curie temperature of the residual amorphous matrix and below that of the Curie temperature of the -Fe(Si) grains. Thus, the anisotropy induced during nanocrystallization should primarily originate from the bcc grains. The amorphous matrix has a rather inactive part, since its Curie temperature is far below the typical field annealing temperature. The evolutions of the different types of anisotropy induced in a typical alloy susceptible to being nanocrystallized, as a function of current density (thermal treatment carried out by current annealing technique under action of stress and/or field) are shown in Figure 7. As can be seen, stress and stress plus field induced anisotropies increase with the current density (temperature) up to a maximum value at 45 A/mm2 , which may be related to a maximum of the coercive field. The increase of induced magnetic anisotropy up to 45 A/mm2 could be ascribed to an increase in the intensity of the interactions between the metallic atoms, and, consequently, an increase of the induced anisotropy could be expected. This argument is linked to the internal stress relaxation produced by thermal treatment in the metallic glasses. Similarly, field-induced

300

200 SFA 100

Kmax (J/m3)

magnetic alloys from the application viewpoint. On the other hand, the intergranular residual amorphous phase plays an important role. The presence of the residual amorphous phase is essential to maintaining the metastable thermodynamical equilibrium of the nanostructure. This behavior should be understood, taking into account the compositional change of the amorphous matrix (with progressive lost of Si and Fe with the annealing time), which can significantly change the Curie point of this phase. Unavoidable mixing of atoms at the interface nanocrystal–amorphous gives rise to the formation of thin layers of alloys with unknown compositions. It has been proposed that the exchange penetration is likely to be the main cause of the Curie temperature enhancement of the matrix, but with contributions from a magnetostatic interaction as well as compositional sharp gradients of the interface. Unfortunately, the lack of knowledge about the nature of this interface opens an interesting question related to the coupling between two phases with a large interface area as is the case of these soft magnetic nanocrystalline Fe-base alloys.

Soft and Hard Magnetic Nanomaterials

FA

0

–100 SA –200

–300 34

36

38

40

42

44

46

48

50

jann (A/mm2)

Figure 7. Maximum magnetic anisotropies versus current annealing intensity induced in Fe735 Cu1 Nb3 Si155 B7 samples. Reprinted with permission from [76], C. Miguel et al., Phys. Status Solidi A 194, 291 (2002). © 2002, Wiley–VCH.

anisotropy decreases with the current density down to minimum value. Such minimum is observed again at around 45 A/mm2 . For current densities higher than that of the maximum, induced anisotropy monotonically decreases with current annealing density. Studies on the stress-induced anisotropy [68–76] indicate that resembling behaviors as those in metallic glasses also can be found in Finemet type nanocrystalline magnets. Although the occurrence of this effect has been well confirmed, nevertheless, its origin seems to be not entirely understage at present. Herzer proposed [70] an explanation, claiming that this anisotropy is of a magnetoelastic nature and is created in the nanocrystallites -Fe(Si) grains due to tensile back stresses exerted by the anelastically deformed residual amorphous matrix. The above conclusion seems to be highly probable because of a strong correlation between the stress-induced anisotropy and the magnetostriction of the nanocrystallites found by Herzer [70]. However, Hofmann and Kronmüller [72] and Lachowicz et al. [73] suggested an alternative explanation of the origin of the considered anisotropy. They adapted the Néel’s calculations of atomic pair directional ordering proposed by Néel [77] to the conditions of the investigated material, obtaining a theoretical value of the energy density of the stressinduced anisotropy of the same order of magnitude as that observed experimentally. Consequently, besides the magnetoelastic interactions within the nanocrystallites suggested by Herzer, the directional pair ordering mechanism in -Fe(Si) grains is also a very probable origin of the stress-induced anisotropy in Finemet-type material. The occurrence of dipolar and deteriorated exchange intergrain interaction also should be considered to explain the origin of the stress-induced anisotropy in the nanocrystalline alloys [19, 49]. This leads to a more realistic situation than the simple anisotropy averaging, since those features are involved in the accomplishment of a nanocrystallization process. In this way, the procedure to obtain the weighted average anisotropy nicely proposed by

9

Soft and Hard Magnetic Nanomaterials

eff

The effective magnetostriction, s , observed in nanocrystalline alloys at different stages of crystallization, has been interpreted as a volumetrically weighted balance among two contributions with opposite signs originating from the bcc-FeSi grains (cr s  and residual amorphous matrix (am s ) according to Herzer [42]: cr am eff s = s + 1 − ps

(6)

where p is the volumetric fraction of the crystalline phase. Therefore, assuming negative and positive sign as for the nanocrystalline and amorphous phase, respectively, the varieff ations of s (including the change of sign observed in some nanocrystalline composition) can be interpreted as a consequence of the variations of the p parameter. Although this simple approximation gives the qualitative explanation of the effective magnetostriction in Fe-based nanocrystalline alloys [42], it does not consider many effects occurring in the real materials. More exact calculations take into account that the magnetostriction of the residual amorphous phase is not constant but depends on the crystalline fraction due to the enrichment with B and Nb [61, 78]. Consequently, Eq. (6) can be modified in the form [78]: cr am eff s = s + 1 − ps + kp

(7)

where k is a parameter that expresses changes of the magnetostriction in the residual amorphous phase with evolution of the crystallization. In many cases, even this model does not fit the experimental results, demonstrating that the effective magnetostriction in nanocrystalline material cannot be described by a simple superposition of the crystalline and amorphous components [61]. In the case of the FeZrBCu nanocrystalline system in which the bcc-Fe phase is formed, the model described does not fit the experimental data, even through the am s p dependence as was shown by SlawskaWaniewska and Zuberek in [79, 80]. They considered this to be an additional contribution to the effective magnetostriction, which arises from the enhanced surface to volume ratio describing interfacial effects [79–83]. Therefore, the Eq. (7) of the effective magnetostriction could be approximated by: eff s

=

pcr s

+ 1 −

pam s

+ kp +

pss S/V

(8)

5

0

cr

2.5. Saturation Magnetostriction Behavior

where the last term describes the effects at the interfaces and depends on the surface-to-volume ratio for the individual grain, as well as on the magnetostriction constant ss , which characterizes the crystal-amorphous interface. Equation (8) is the basic dependence, which can be used to interpret the experimental data on the effective magnetostriction in Fe-based nanocrystalline alloys at different stages of crystallization. The composition of the Fe(Si) grains (depending on the annealing temperature) should be considered, giving rise to different values of the magnetostriction constant for the crystalline phase. The appropriate values of cr s can be obtained from the compositional dependence of the saturation magnetostriction in the polycrystalline -Fe100−x Six , shown in Figure 8 [42, 84]. Thus, the first term in the Eq. (8) can be treated as the well-defined one. Figure 9 presents the crystallization behavior and accompanying changes in the magnetostriction of the classical Finemet (Fe735 Cu1 Nb3 Si135 B9 ) alloy published by Gutierrez et al. [85]. The analysis of these data, according to Eq. (8), allows (i) estimation of the contribution from the crystalline phase (see Fig. 8a, where the values of cr were found s from the combined Figs. 8 and 9a), and then (ii) fitting of eff the experimental (s − pcr s ) on p dependence in Eq. (8). The results, both experimental points and the fitted curve (solid line) are shown in Figure 9b. Assuming spherical -Fe(Si) grains, with radius R, the last term of Eq. (8) can be expressed as 3ss /R, and the magnetostriction constant, which describes the interface ss , can be estimated. For the soft nanocrystalline alloys (Finemet and FeZrBCu alloys) [42, 61, 79], R = 5 nm, and, thus, ss has been found to vary in the range 4.5–71 × 10−6 nm. These results are one order of magnitude smaller than values of the surface magnetostriction obtained in multilayer systems. However, investigations of Fe/C multilayers have shown that not only the value but also the sign of the surface magnetostriction constant depends on the structure of the iron layer, and it has been found that for crystalline iron, ss (bcc-Fe) = 457 × 10−6 nm, whereas, for the amorphous iron, ss (am-Fe) = −31 × 10−6 nm [86]. Thus, the value of the interface magnetostriction obtained in the nanocrystalline systems is within the range of the surface magnetostriction constant estimated for thin iron layers. It should be noted that, contrary to Fe/c multilayers, in the nanocrystalline materials, both the crystalline and amorphous phases are

λs [10–6]

Alben et al. [42] strongly depends on the degree of magnetic coupling. This stress anisotropy is induced, as has been noted previously, inside the grains. The maximum value (around 1000 J/m3  is clearly lower than 8000 J/m3 , corresponding to the magnetocrystalline anisotropy of the -Fe(Si) grains; therefore, the origin of the stress anisotropy should be strongly connected to the internal stresses in the FeSi nanocrystals. An interesting question should be one related to the coupling between these two phases with a large interface area such as is the case of Fe-rich nanocrystalline alloys. For this, a deep knowledge about the nature of the interface results are to be determinant. Unavoidable mixing of atoms of the interface gives rise to the formation of thin layers of alloys of unknown composition, which makes this study complicated.

–5

–10 0

5

10

15

20

Si in α-FeSi [at %]

Figure 8. Saturation magnetostriction of the polycrystalline -Fe100−x · Six . Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

10

Soft and Hard Magnetic Nanomaterials

of

λs [10–6]

Si in α-FeSi [at %]

18.0 17.5 17.0 16.5 16.0

a

20

b

15 10 5 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

crystalline fraction

Figure 9. Si content in -Fe(Si) grains (a) and magnetostriction of the Fe735 Cu1 Nb3 Si135 B9 alloy (b) versus crystalline fraction. Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

magnetic, and they are coupled through exchange and dipolar interactions. It must thus be expected that the magnetic interactions in the system can affect the magnetoelastic coupling constant at the grain–matrix interfaces. In addition, the structure and properties of the particular surfaces that are in contact, as well as local strains at the grain boundaries also should be considered. The problem of the surface/interface magnetostriction, however, requires further studies, which, in particular, should include measurements at temperatures above the Curie point of the amorphous matrix, where only ferromagnetic grains should contribute to the effective magnetostriction, simplifying a separation between bulk and surface contributions.

3. HARD NANOCRYSTALLINE MATERIALS 3.1. An Introduction Permanent magnets are devices capable of producing, during a time of the order of several years and without any on-service energy input, a magnetic field whose magnitude is at least in the order of the tens of T. The devices, based on fields produced by permanent magnets, are ubiquitous in modern technology; a good example of this point is that in a modern car, it is typically possible to identify up to 30 different magnets [87]. In most of the cases (the exception are those in which the magnets are used to produce a field gradient), the figure of merit characterizing the device is the so-called energy product of the permanent magnet [88]. If the permanent magnet is used as a part of a magnetic circuit in which the magnetization is essentially uniform, the energy product gives the magnetic energy density, stored in a thin air gap introduced in the circuit and limited by surfaces (the poles) perpendicular to the average magnetization (the energy product also is proportional to the square

of the induction created in the gap). In addition to a sufficiently high energy product, a permanent magnet should exhibit (especially those used in the automotive industry) a weak decrease of the coercive force with the increase of the temperature and good corrosion resistance properties. Modern permanent magnets are based on the so-called hard magnetic phases, that is on phases difficult to demagnetize. A hard phase should present, at the temperatures of interest from the standpoint of the applications [89], (i) a coercive force large enough to preserve the remanence from either on-service or spurious demagnetizing effects (it is possible to show that, in a uniform magnetization circuit, the stability of the remanence is granted by a coercivity larger than half the magnitude of the remanence), (ii) a remanence as large as possible (in a polycrystalline hard material, the remanence value results from the saturation magnetization value and from the degree of macroscopic texture of the grain orientation distribution), and (iii) a magnetic transition temperature high enough so as to be compatible with the temperature increases occurring during the use of the magnet. The optimization of this set of properties can, first of all, be correlated to an adequate choice of the structure and intrinsic properties of the hard phase. The induction of a large coercivity is linked to the occurrence of a large magnetic anisotropy. For the coercivity values required by modern applications, the anisotropy can only be of a magnetocrystalline origin [90]. The magnetocrystalline anisotropy results from the electrostatic interaction between the charge distribution of the atoms bearing the localized magnetic moments responsible for the macroscopic magnetization and the crystalline electric field created by the ions surrounding those magnetic atoms. Requisite for the observation of large magnetocrystalline anisotropy is the existence of a large spin-orbit coupling in the atoms having magnetic moments [91]. This large spin-orbit coupling has, as a consequence, the fact that any modification of the orientation of the moments is linked to a rotation of the charge linked to the orbital part of the total moment and, consequentially, to a variation of the energy of interaction between that charge and the ions in its neighborhood (the relative orientation(s) resulting in a minimum of this interaction energy are called easy axes and those corresponding to maxima, hard axes). Considering this initial requisite, it is possible to identify the complete Rare-Earth series as a group of elements with a potential to either be used as or to form hard phases [92] (the rare-earth elements have large spin-orbit interactions and the Rare-Earth ions exhibit, with the only exception of Gd, large asphericities which make the crystalline field interactions highly dependent on the charge orientation).1 The achievement of a large saturation magnetization is linked to the identification of a highly packed atomic structure in an element or compound formed by atoms bearing the larger possible magnetic moments and having the smallest possible atomic volume. Due to their large atomic volumes, which is not compensated by their often large atomic moments, the rare-earths are, in respect to the elevated The occurrence of uniaxial anisotropies (corresponding to hexagonal, tetragonal, or rhombohedral crystal structures) is an additional requisite for the achievement of elevated coercivities. 1

11

Soft and Hard Magnetic Nanomaterials

magnetization, in a clear disadvantage with respect to the transition metal magnetic elements and, specially, in comparison with Co and Fe (which, in turn, have, due to the very small spin-orbit coupling, a reduced magnetocrystalline anisotropy). The third requisite, that corresponding to the elevated order temperature, also excludes the rare-earths since the small magnitude of the exchange interactions between the elements of the series results in the fact that the rare-earth with the larger Curie temperature is Gd, the most spherically symmetric rare-earth ion, which goes paramagnetic at ca. 300 K, an order temperature incompatible with any practical application. It is thus possible from this discussion to discard any magnetically pure element as a potential hard magnetic phase. A hard phase must, consequently, at least be a binary compound joining in a single structure high anisotropy, magnetization, and order temperature [93]. The intermetallic transition metal–rare earth alloys are thus clear candidates for a hard magnetic behavior and, in fact, hexagonal phases of the SmCo system exhibit coercivities in single-phase, polycrystalline samples of up to 4 T at room temperature (these phases can bear remanences of the order of 1 T, have reasonably good corrosion properties, and can be used up to ca. 650 K without a large deterioration of their hysteretic properties). The main problem with the extensive use of SmCo magnets is related to the limited availability of Co and its high (and highly fluctuant) price. Phases alternative to the SmCo ones and not containing Co are the ordered tetragonal FePt (exhibiting coercivities smaller than those obtained in SmCo) and, more importantly, the ternary tetragonal NdFeB, the phase having the better hard properties achieved up to the moment. In this section, we will review the basic characteristics and behaviors of these hard phases, the links between the demagnetization properties and the phase distribution and morphology, the influence of the demagnetization mode in the optimization of the preparation methods, and will finish with the analysis of the way of overcoming the limitations of the hysteretic properties related to the values taken by the intrinsic quantities: the induction of different types of nanostructures.

3.2. Magnetization Processes and the Phase Distribution On the origin of the possibility of controlling the demagnetization behavior are the particularities of the demagnetization process. In the particular case of a polycrystalline material, demagnetization takes place in a complex way (Fig. 10), involving some (or all) of the stages in the sequence nucleation–expansion–propagation–pinning– depinning [94]. We will describe these stages, as well as their dependence on the extrinsic characteristics of the materials. The nucleation process corresponds to the first (occurring at a smaller demagnetizing field) irreversible departure of the distribution of the magnetic moments present in a certain region (typically a grain) of a material from the configuration associated to the remanent state. That departure, depending on the main phase properties, could involve either the formation of a reversed region limited

Grain boundaries

Applied demagnetizing field Pinned wall

Reversed nucleus

M

Reduced anisotropy region

M

Unreversed region Expanded nucleus Secondary phase non-magnetic precipitate (pinning centre) Propagating wall

Figure 10. Schematic illustration of the intragrain demagnetization sequence (nucleation-nucleus expansion-propagation-pinning stages).

by a domain wall or that of an extended inhomogeneous local magnetization distribution, which corresponds to the onset of the global reversal process and takes place preferentially at (i) points where local demagnetization fields are more intense (as the edges and corners in well-crystallized, polyhedral grains) and (ii) points where the local anisotropy and/or exchange are significantly reduced with respect to their values in a free-from-defects area of the main phase (to be effective, these sites of preferential nucleation should have transverse dimensions of the order of the magnetic moments structures formed upon nucleation). In the most general case, the occurrence, at a given field value, of nucleation does not have as a consequence the complete reversal of the magnetization of the sample. This is so just because the growth in size of the total or partly reversed regions requires the input of energy in the system due to the fact that, in general, the moments in those regions do not point along easy axes (as required to minimize the anisotropy energy) and/or are not fully parallel (which is the configuration minimizing exchange and dipolar energies). It is thus necessary to increase the magnitude of the demagnetizing field in order to balance the increase of the energy of the distribution of magnetic moments required by the growth in size of the localized reversed region from which the global reversal proceeds. This process is denominated by nucleus expansion and takes place, reversibly, up to an applied field value for which the nucleus can steadily grow in size (that field is called the propagation field). In the particular case of the hard magnetic materials, the nucleation and propagation fields can be significantly different. Once the condition of steady expansion of the reversed nucleus is achieved, the magnetization reversal can only be stopped if the structures involved in the propagation (e.g., a domain wall limiting the reversed area) get trapped in regions where they have an energy lower than that corresponding to the previously swept areas. Those pinning centers are associated with secondary phases

12

Soft and Hard Magnetic Nanomaterials

of deteriorated crystallinity regions where the anisotropy and/or the exchange energies are lower than those of the well-crystallized main phase (planar structures as second phase precipitates are especially effective as pinning centers). If a propagating nucleus gets trapped in one of these pinning points, in order to proceed with the reversal process, it is necessary to increase the applied demagnetizing field in the magnitude required to unpin the propagating structure (the depinning field is related to the characteristics of the difference on the anisotropy, exchange and dipolar energies between the pinning center, and the well-crystallized areas). From this discussion, it is clear that the field required to fully reverse the magnetization of a grain in a polycrystalline hard material coincides with the maximum of the nucleation, propagation, and depinning fields. As for the complete reversal of the magnetization of a polycrystalline material, the crucial role corresponds to the intergranular regions (Fig. 11). Those intergranular regions (grain boundaries separating main hard-phase grains) could have a differentiated structure (thus, being a secondary phase) or could just correspond to the main phase structure accumulating defects and additive elements so as to provide the transition between the crystallographic orientations of different main-phase grains. In both cases, the most important points are (i) the occurrence and type of the exchange interactions in the intergranular structure and (ii) the thickness of the grain boundary. These two points will be analyzed in detail in the next section but, for the moment, we can say that: (a) If the main phase grains are perfectly exchange coupled through the grain boundaries, the global coercive force will correspond to the coercivity of the grain having the smallest reversal field (the structures propagating in the reversal of that grain will not have any hindrance to propagate across the sample). This is a particularly undesirable case, since the coercivity is linked to the region in the material having the most deteriorated magnetic properties. (b) If the main phase grains are partially exchange coupled (either if they are uniformly coupled through an Strongly exchange coupled grains

Partly exchange coupled grains

Exchange decoupled grains

Grains coupled through a low anisotropy intergranular phase

Wall pinned inside a low anisotropy grain boundary

Freely propagating wall

Partly pinned wall

Wall pinned outside a high anisotropy grain boundary

Figure 11. Domain wall propagation across differently exchange coupled grains.

intergranular exchange constant that is a fraction of that of the main phase, case A, or if they are nonuniformly coupled, case B), the increase in volume of the reversed regions will find either pinning centers (case A, stopping the walls inside the grain boundaries) or propagation barriers (case B, stopping the grains outside the grain boundaries) at the grain boundaries. The global coercivity will be a convolution of the distribution of grain reversal fields and of the depinning fields of the grain boundary regions (this is a particularly complex case since the reduced exchange at the grain boundaries also can influence the distribution of nucleation fields). (c) If the main phase grains are fully exchange decoupled, the global coercivity directly results from the distribution of grain reversal fields. This case is, in principle, preferred for the optimization of the hard materials since, as we will see in the next section, it is compatible with the state of the art about the control of the properties of the intergranular regions and ensures the achievement of coercivities directly related to the structure and properties of the main phase. It is clear that the detailed knowledge of the actual demagnetization mechanism taking place in a concrete material is the key to the achievement of a relevant optimization of its hysteretic properties and that the basic mechanism for that purpose is the control of the phase distribution and morphology. This information is, nevertheless, quite elusive to simple and straightforward analysis and, usually, can only be partially obtained. To that purpose, the most commonly analyzed data are those corresponding to the temperature dependence of the coercive force [95], due to the availability of models correlating in simple terms the magnetic properties and some microstructural features. The conclusions of those models [96, 97], initially proposed by Brown, and basically adequate to describe nucleationruled magnetization reversal processes, can be summarized in Eq. (9) Hc T  = HK T  − Neff Ms T 

(9)

where HC T  is the temperature dependence of the coercivity, and Neff are parameters fitting the experimental coercivity data, HK T  is the experimental temperature dependence of the anisotropy field, and MS T  is the experimental temperature dependence of the magnetization. Equation (9) simply states that at all the temperatures, the coercive force can be obtained from the anisotropy field of the main phase (the maximum observable coercivity in that phase) multiplied by a factor (lower than the unity) that contains all the sources of deterioration of the local coercivity and decreased in the local demagnetizing field at the site of the reversal onset. The fitting parameter can be factorized on the contributions of the texture of the grains of the main phase, the occurrence of either complete intergranular exchange or perfect grain decoupling, and the local reduction of the anisotropy related to the occurrence of poor crystallinity or soft regions. As for the Neff parameter, the local demagnetizing factor, it can be larger than the unity since it does not necessarily describe fields originated by uniform magnetization distributions [98].

Soft and Hard Magnetic Nanomaterials

Despite its simplicity (it reduces the complex influence of the phase distribution to only two parameters), Eq. (9) almost universally describes the temperature dependence of the coercivity [99] and is very useful to analyze the behavior of a series of samples prepared under similar conditions. A typical example of this type of study is the analysis of the influence of additive elements on the grain decoupling that (provided other influencing factors are maintained constant) should correspond to an increase of with the increase of the decoupling. Also, the achievement of better crystallinities, resulting in better defined grain edges and corners can be correlated easily to the increase of Neff . The occurrence of pinning–depinning processes can be quite unambiguously identified from the measurement (usually carried out in highly textured samples) of the angular dependence of the coercivity [100]. If the main phase grains in these samples are well exchange coupled, the presence of depinning walls can be pinpointed from the observation of a monotonous increase of the coercivity with the increase of the angle % formed by the direction of the saturation remanence and the applied demagnetizing field. That increase is related to the increase of the pressure exerted over the pinned walls by the applied field that is proportional to 1/ cos %. In contrast with the reversal mechanisms linked to pinning, a nucleation-ruled reversal usually exhibits an angular dependence of the coercivity characterized by an initial decrease with the increase of %, a minimum and a further increase, a behavior resembling (but, in general, not mimicking) the angular dependence of the coercivity predicted for systems demagnetizing according to coherent rotation mechanisms. Finally, in what concerns the analysis of the magnetic properties, the measurement of the relaxational properties (those related to the thermally activated magnetization reversal) can yield and estimate (the so-called activation volume) of the size of the region involved in the onset of the reversal process [101]. That estimate can be useful to try to identify the characteristics of the points at which the nucleation occurs but, generally, the discussions based on the measurement of the magnetic viscosity are not very conclusive.

3.3. Hard Nanocrystalline Materials The most consolidated reason to reduce the size of the grains of the main phase in a hard magnetic material is related to the previously discussed influence of the defects on the promotion of the magnetization reversal. This point is clear if the occurrence in the material of a uniform density of defects is assumed. In this case, the reduction of the grain size will result in a reduction of the number of defects per grain, and, regarding the nucleation processes, the lower the number of defects, the larger the nucleation field. Typical average grain sizes of materials optimized following this idea are in the range of 10 to 100 nm. The nanocrystalline structure is induced by means of four main techniques: (i) Mechanical alloying and mechanical grinding. These two techniques are characterized by the introduction in the starting materials (either pure elements in the

13 case of the alloying or a prealloyed material in that of the grinding) of a large amount of mechanical energy that results in a large increase of their free energy (linked both to large deformations and to a high degree of intermixing). From that high-energy state, the material can decay (either spontaneously or after a low temperature treatment) to lower energy configurations, as those corresponding to an extended solid-state solution, metastable phases, and stable phases. Mechanical alloy has been widely used to produce NdFeB and, to a lower extent, SmCo. It is an inexpensive technique, resulting in a powdered material having typical grain sizes of the order of 10 nm that allows to process large amounts of materials and whose only drawback is the impossibility of producing highly textured magnets. (ii) Rapid quenching from the melt. This preparation technique usually involves the quenching of a quasiamorphous material and an ulterior treatment that allows crystallizing the main phase in grains of the order of 50 to 100 nm. The mechanical alloying mainly has been used to prepare NdFeB and does not result in macroscopically anisotropic materials. It is, nevertheless, possible in this case to produce a high-quality textured material by hot mechanic working of the quenched materials (these textured samples are not nanostructured). (iii) Physical vapor deposition techniques used to grow thin and thick hard magnetic films. Although, there is a large variety of physical vapor deposition techniques (laser ablation, evaporation, molecular beam epitaxy, etc.), the most promising and widely used is sputtering. Generally speaking, the nanostructure is induced either by room temperature deposition of a quasi-amorphous phase, followed by a high temperature thermal treatment, or by direct deposition at moderately high substrate temperatures. The most relevant hard phase deposited by using sputtering has been NdFeB [102], although, recently there is a lot of activity on SmCo-based phases [103]. (iv) Electrodeposition. Although limited in the achievable compositions (basically CoNiP, CoP, and FePt), this is by far the technique that allows the higher film deposition rates and also that allows an easier implementation [104]. As in other cases, it is possible to deposit amorphous films that afterward are crystallized or, directly and depending on the composition of the film and deposition bath, nanocrystalline films. In addition to the increase of coercivity linked to the reduction down to the nm scale of the main hard-phase grains, the induction of nanocrystallinity in permanent magnet materials has, as a consequence, the increased influence on the demagnetization processes of the grain boundary regions. As we will discuss, that influence can bring about the possibility of major optimizations of the properties of these materials. Let us consider first a single-phase nanocrystalline material: a high anisotropy phase for which the domain wall width, L, is larger than the grain boundary thickness, d, but smaller than the average grain size, D. Typical examples of

14

Soft and Hard Magnetic Nanomaterials

that kind of material are the NdFeB-based alloys obtained by controlled crystallization of an amorphous precursor. The crystallization process used to prepare those samples usually is implemented by rapidly heating the precursor material up to the treatment temperature (typical heating rates are in the range of the tens of thousands of degrees per minute), keeping them at that temperature for a few minutes and cooling them down at a rate of variation of the temperature comparable to that used during the heating process. This treatment strategy aims at the enhancement of the crystallite nucleation process and, simultaneously, to the reduction of the grain growth (both basic crystallization mechanisms typically have independent kinetics) and results in a very fine and homogeneous nanostructure, characterized, in the case of the NdFeB-based materials, by average grain sizes of the order of a few tens of nm and, also, by very clean (free from defects) grain boundaries, with a typical thickness of the order of a few interatomic distances. The rapid heating also can avoid a significant precipitation of unwanted secondary phases (as, mainly, -Fe in the case of the considered system). In fact, the control of the heating rate during the crystallization process seems to be crucial since, for instance, in the case of the Ndx Fey B100−x−y alloys, the coercive force measured in the as-crystallized material varies from a value close to zero, corresponding to heating rates of the order of 10  C/minute up to a value of 4 T in the optimally treated sample, that crystallized by using a heating rate of 15 × 104  C/minute [105]. The relevant point is that the presence in these materials of very thin grain boundaries, free from defects to a large extent, result in the occurrence of strong intergranular exchange coupling. Thus, in the boundary region limiting two grains having different easy axes orientations, a structure of magnetic moments allowing a gradual transition between those easy axes directions is formed (Fig. 12). Those domain wall–like structures have two relevant effects on the hysteretic parameters. On the one side, and taking into account that the transition structures have a typical transverse dimension of the order of L and are larger than the grain boundary thickness, they result on a remanence, Mr , enhancement (with respect to the value corresponding •

Composite material (Mr >> and Hc and Hc 5. The precipitate is then peptized in deionized water with high pH (>11), which is obtained by adding appropriate amount of NH4 OH. The resulting sol is suitable for thin-film formation.

(10)

This precipitate is separated using a centrifuge and washed several times using deionized water to remove the H+ and

3.5. Controlling Growth of SnO2 Nanocrystallites Two approaches have been reported to control the growth in the size of the nanocrystallites during the high-temperature exposure [16, 78]. The first approach [78] is based on replacing the surface OH group with another functional group that does not condense as OH groups and that could eventually form the “pins” at the grain boundaries during high-temperature exposure. Hexamethyldisilazane (HMDS; [Si(CH3 )3 ]2 NH) is a OH scavenging reagent because it replaces the OH groups in the SnO2 gel to form noncondensing methyl siloxyl group, and when decomposed in air above 300  C, it transforms to extremely small SiO2 particles, which could serve as the “pinning” for the SnO2 particles. When heated at 150  C for 1 h, HMDS vaporizes and reacts with the surface OH groups by following reaction: 2Sn

OH + SiCH3 3 2 NH → 2Sn

SiCH3 3  + NH3 (11)

The methyl siloxyl group decomposes after firing at 200– 500  C for 1 h in air, producing Sn O Si bond on the surface, which is not condensing as the surface OH groups. Moreover, the process produces extremely tiny SiO2 particles over SnO2 nanoparticles, which act as “pinning sites” for the grain boundary movement during densification, thus avoiding excessive growth at higher temperature. In the second method [16], the alkaline Sn(OH)2 sol is subjected to the hydrothermal treatment in an autoclave at 200  C for 3 h. Such hydrothermally treated sol particulates are more resistant to grain growth as compared to untreated sol.

32

Sol–Gel Derived Semiconductor Oxide Gas Sensors

4. CHARACTERIZATION OF SOL–GEL COATED SnO2 THIN FILMS Typical SEM micrograph of nanocrystalline SnO2 thin film coated on a Pyrex glass substrate is shown in Figure 4 [77]. The SnO2 thin film appears to be relatively smooth. At some locations, few cracks are observed on the film surface, which are characteristic features of the sol–gel dip-coated thin films. Typical broad-scan XPS spectrum, within the B.E. range of 0–1100 eV, obtained for the nanocrystalline SnO2 thin film dip-coated on the Pyrex glass substrate is shown in Figure 5a [77], which primarily shows the presence of Sn and O on the glass surface after the dip-coating process. Typical narrowscan analysis of Sn (3d) spectra, within the B.E. range of 480–500 eV, is presented in Figure 5b. Sn 3d5/2 B.E. level of 485.8 eV is observed in Figure 5b, which is in between the Sn 3d5/2 B.E. values of 484.9 eV and 486.7 eV reported for pure Sn and SnO2 , respectively [79]. Hence, the observed Sn 3d5/2 B.E. value of 485.8 eV is attributed to the presence of Sn-oxidation state less than +4. The O:Sn relative atomic ratio of ∼1.6 is calculated from the survey spectrum and is in agreement with the Sn 3d5/2 B.E. value. The nonstoichiometric O:Sn ratio suggests the presence of oxygen vacancies [80], which is responsible for the n-type semiconductor property of the film. Typical TEM images, obtained from the FIB-milled TEM sample of the nanocrystalline SnO2 thin film dip-coated on the Pyrex glass substrate, are shown in Figure 6 at different magnifications [77]. The thickness of the nanocrystalline SnO2 thin film can be directly measured via these micrographs. Various regions corresponding to Pt, Au-Pd (both originating from the FIB-milling procedure), SnO2 (originating from the sol–gel dip-coating process), and the glass substrate are marked appropriately. The SnO2 thin film appears to be smooth and highly continuous. No cracks are visible within the film, although slight variation in the film thickness is noted in these images. The average SnO2 thin-film thickness is observed to be within the range of ∼100–150 nm. Typical AFM images of the nanocrystalline SnO2 thinfilm sol–gel dip-coated on the Pyrex glass substrate, at low and high magnifications, are presented in Figure 7a and 7b, respectively [77]. The nanocrystalline SnO2 thin film is observed to be made up of nanoparticles having nearspherical shape and uniform particle size distribution. Very dense packing of nanoparticles is noted in these micrographs. The average nanoparticle size is estimated to be

(a)

(b) Sn 3d3/2 O 1s

C 1s 1200

1000

800

600

400

200

0

500

495

B.E. (eV)

490

485

480

B.E. (eV)

Figure 5. Broad-scan (a) and narrow-scan (b) XPS analysis of SnO2 thin-film gas sensor [47].

within the range of 15–20 nm and is comparable with the resolution limit of AFM determined by the AFM-tip radius. Typical HRTEM images obtained from the FIB-milled TEM sample are presented in Figures 8a and b at different magnifications. Selected-area electron diffraction (SAED) pattern obtained from the center of the nanocrystalline SnO2 thin film is presented in Figure 8c [77]. From these images, it is obvious that the sol–gel dip-coated SnO2 thin film is highly dense and nanocrystalline in nature. The nanocrystalline SnO2 thin film–glass substrate interface is visible in Figure 8a. No porosity or cracks are visible at the interface, which suggests very adherent nature of the sol–gel dipcoated nanocrystalline SnO2 thin film. Porosity and cracks are not observed within the nanocrystalline SnO2 thin film at nanoscale level, indicating very dense packing of the SnO2 nanocrystallites within the thin film, which is in agreement with the AFM analysis. The average SnO2 nanocrystallite size of ∼6–8 nm is measured from these HRTEM images. The nanocrystallite size distribution is observed to be very narrow. SAED pattern, Figure 8c, shows continuous and diffused ring patterns, which suggests the nanocrystalline nature of SnO2 thin film. The variation in the average nanocrystallite size as a function of calcination temperature, within the range of 100–1000  C, reported for the sol–gel derived nanocrystalline SnO2 thin films or powders is presented in Figure 9 [9, 12–14, 21, 22, 25]. It can be observed that, within the calcination temperature range of 100–700  C, the grain growth is marginal and lies within the range of 3–20 nm. However, above 700  C, rapid grain growth is observed. Grain size as high as 300 nm is observed at the calcination temperature of 1000  C. The activation energy for the grain growth, for high (700–1000  C) and low temperature ranges (100–700  C), can be determined from the

(b)

(a) Cracks

Sn 3d 5/2

Sn 3d

Pt

Pt Au-Pd Au-Pd

Au-Pd Au-Pd SnO22

Glass

SnO22

Glass

1 µm

Figure 4. SEM micrograph of SnO2 thin-film gas sensor [47].

Figure 6. TEM micrographs of SnO2 thin-film gas sensor at low (a) and high (b) magnifications [47].

33

Sol–Gel Derived Semiconductor Oxide Gas Sensors 350

(b)

100 nm

125 nm nm

300

Crystallite Size (nm)

(a)

250 200 150 100 50 0 -50 0

Figure 7. AFM micrographs of SnO2 thin-film gas sensor at low (a) and high (b) magnifications [47].

activation energy plot, Figure 10. The activation energy values of 91 kJ/mol and 9 kJ/mol are determined for high and low calcination temperature regions, respectively. Very low activation energy value (9 kJ/mol) observed in the lower temperature range (100–700  C) is attributed to the possible generation of excess oxygen-ion vacancy concentration in the SnO2 nanoparticles, as reported for other oxide particles [81].

5. GAS SENSING CHARACTERISTICS OF SOL–GEL DERIVED NANOCRYSTALLINE SnO2 THIN FILMS 5.1. Mechanism of Gas Sensing When a freshly prepared sol–gel derived nanocrystalline SnO2 semiconductor thin film is exposed to air, physisorbed oxygen molecules pick up electrons from the conduction − band of SnO2 and change to O− ads or O2ads species. Consequently, a positive space-charge layer forms just below the surface of SnO2 particles, which creates a potential barrier between the particles increasing the electrical resistance of the nanocrystalline SnO2 film, Figure 11 [82]. However, when a reducing gas comes in contact with the nanocrystalline SnO2 film, it gets oxidized via reaction with

200

400

600

800

1000

1200

Calcination Temperature (°C)

Figure 9. Effect of calcination temperature on the crystallite size for the sol–gel derived nanocrystalline SnO2 [9, 12–14, 21, 22, 25]. − the surface adsorbed O− ads or O2ads species and subsequently electrons are reintroduced into the electron depletion layer, leading to decrease in the potential barrier. The sensitivity of the nanocrystalline SnO2 thin film is usually determined by the ratio (Ra − Rg )/Rg , where Ra and Rg are the resistance of the sensor in air and reducing gas, respectively. A typical response of the nanocrystalline SnO2 sensor is schematically shown in Figure 12. The figure shows that the sensor exhibits a resistance “Ra ” in air. When the reducing gas comes in contact with the sensor, its resistance decreases rapidly initially, then decreases at lower rate with increasing time and attains a steady-state resistance value (Rg ). The total time taken by the sensor to attain 90% of the steady-state value is known as the “response time” of the sensor. If the air is flown back over the sensor, it retains its original resistance in air. The time required to reach 90% of the original resistance is known as the “recovery time” of the sensor. The gas sensitivity, response time, and recovery time are the three most important parameters of the nanocrystalline gas sensor. During the preparation of the nanocrystalline SnO2 thin film, the sol–gel synthesis parameters are optimized to maximize the gas sensitivity and to minimize the response and recovery times.

5.2. Summary of Reported Results (a)

SnO2

SnO2

(b)

Glass

Various parameters such as nanocrystallite size, calcination and operating temperatures, film thickness, and amount of porosity, have been reported to affect the gas sensitivity of the sol–gel derived nanocrystalline SnO2 thin film. Figure 13 shows the effect of the nanocrystallite size on the 6 5

(c) Ln D

4 3 2 1 0 0.0005

0.001

0.0015

0.002

0.0025

0.003

1/T (K-1)

Figure 8. HRTEM images (a, b) and SAED pattern (c) of SnO2 particles within a gas sensor thin film [47].

Figure 10. Activation energy determination plot for the sol–gel derived nanocrystalline SnO2 . D-crystallite size (nm).

34

Sol–Gel Derived Semiconductor Oxide Gas Sensors _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + _ + + + + + + + _ + _ + + + + + + + + + _ + _ + _ + _ _ + + _ + + + _ _ + + +_ _ + _ + _+ _+ _ _ + _ _ _ _ _

Gas Sensitivity (%)

3000

Air Barrier Potential

Reducing Gas

2500 2000 1500 1000 500 0

0

50

100

150

200

250

300

350

Nanocrystallite Size (nm)

Figure 13. Variation in the gas sensitivity as a function of nanocrystallite size [9, 12–14, 21, 22, 25].

gas sensitivity of the sol–gel derived nanocrystalline SnO2 thin film [9, 12–14, 21, 22, 25]. The gas sensitivity is observed to be enhanced drastically below a nanocrystallite size of ∼10 nm while the gas sensitivity is almost independent of the nanocrystallite size above this size range. The enhancement in the gas sensitivity below ∼10 nm crystallite size has been attributed to the “grain-control” resistance mechanism, which enhances the transducer function of the sensor [83, 84]. A semiconductor gas sensor has two functions [84]. First is the receptor function, which recognizes or identifies a chemical gas component. The receptor function in this case is played by the surface oxygen. If a metal or a metal oxide is added onto the surface as a surface modifier, it may also influence the receptor function. The receptor function of a semiconductor is thus determined by the ability of the semiconductor surface to interact with a reducing gas component. The second function of a semiconductor gas sensor is a transducer function, which transforms the chemical signal into an electrical signal. In this case, the electrons generated via chemical reaction are conducted through the gas sensor material. As these conduction electrons have to overcome the potential barrier induced by the space-charge layer, the microstructure of the semiconductor gas sensor plays an important role to control the transducer function. The magnitude of this potential barrier for the electronic conduction strongly depends on the ratio of the nanocrystallite size D to the space-charge layer thickness L.

Gas In

Gas In

In a semiconductor gas sensor, the nanocrystallites are connected to each other via necks forming aggregates (largesized particles). The large-sized particles are in turn connected to the neighbors via grain boundary contacts. When the nanocrystallite size is very large (D ≫ 2L), the conduction electrons have to overcome the potential barrier induced by the space-charge layer at each grain boundary contact, Figure 14a. In this situation, the resistance of the gas sensor is determined by the resistance offered by the grain boundary contacts, which is independent of the grain size. Hence, when the nanocrystallite size is very large compared to the space-charge layer (D ≫ 2L), the sensitivity of an oxide semiconductor sensor is independent of the nanocrystallite size. As the nanocrystallite size decreases, the space-charge layer penetrates deeper into each of the nanocrystallites. When the nanocrystallite size approaches the space-charge layer thickness (D ≥ 2L), the space-charge layer forms channels at each neck within a particle, Figure 14b. As the conduction electrons must move through these channels, they experience additional potential barrier in addition to the potential barrier existing at the grain boundaries. Under these constraints, the resistance of the gas sensor is determined predominantly by the neck resistance. As the neck size is observed to be proportional to the nanocrystallite

(a)

(b)

Barrier

Potential

Figure 11. A model showing the variation in the potential barrier of the nanocrystalline SnO2 gas sensor due to change in the environment [52].

Potential

Grain Boundaries

Grain Boundary Contacts

Barrier

Necks

(c)

Ra

Potential

Sensor Resistance

Air

Barrier

Rg Gas Out

Gas Out

Time

Figure 12. Schematic diagram describing the variation in the sensor resistance with the change in the environment.

Grain Boundary Contacts

Figure 14. Resistance-control models for the sol–gel derived nanocrystalline SnO2 thin film under different conditions: (a) D ≫ 2L, (b) D > 2L, and (c) D < 2L. D-crystallite size and L-space-charge layer thickness [54].

35

Sol–Gel Derived Semiconductor Oxide Gas Sensors 1800

Gas Sensitivity (%)

Gas Sensitivity (%)

3000 2500 2000 1500 1000 500

1500 1200 900 600 300 0

0 0 0

200

400

600

800

1000

1200

size, the sensitivity of the gas sensor is dependent on the nanocrystallite size for D ≥ 2L. As the nanocrystallite size reduces below twice the spacecharge layer thickness (D < 2L), the entire nanocrystallite becomes depleted of electrons. The space-charge layer then penetrates into the nanocrystallite completely, Figure 14c. The electrical resistance of the gas sensor increases abruptly and is controlled mainly by the grain resistance under this situation. In this region, the ratio of electron concentration in gas (ng ) to that in air (na ) within a nanocrystallite decreases as the distance from its surface increases. As a result, when D < 2L, the sensitivity of the gas sensor increases with decreasing nanocrystallite size in this range. Thus, the nanocrystallite size D relative to the spacecharge depth L is one of the most important factors affecting the sensing properties of a semiconductor oxide gas sensor. For a thin sputtered SnO2 film and sintered powders, the space-charge depth has been calculated to be ∼3 nm at 250  C [85, 86]. Hence, for the sol–gel derived nanocrystalline SnO2 gas sensor, very high sensitivity is theoretically expected just above and below the nanocrystallite size of ∼6 nm. Figure 15 shows the effect of the calcination temperature on the gas sensitivity for the sol–gel derived nanocrystalline SnO2 thin-film sensor [5, 8, 9, 12–14, 17, 21, 22, 25]. The maximum gas sensitivity is observed to be reported at the calcination temperature of 450  C. Above and below this calcination temperature range, the gas sensitivity decreases. The sol–gel derived nanocrystalline SnO2 thin film is always amorphous at room temperature. Hence, the calcination treatment at higher temperature is necessary to crystallize 3000

Gas Sensitivity (%)

2500 2000

100

150

200

250

Film Thickness (nm)

Calcination Temperature (°C)

Figure 15. Variation in the gas sensitivity as a function of calcination temperature [5, 8, 9, 12–14, 17, 21, 22, 25].

50

Figure 17. Variation in the gas sensitivity as a function of film thickness for sol–gel derived nanocrystalline SnO2 film [5, 9, 13, 17, 21].

the film. At the same time, higher calcination temperature results in the grain growth, Figure 9, which may decrease the gas sensitivity. Hence, due to the optimum balance between the amount of crystallization and the degree of grain growth, the maximum sensitivity value is observed at the intermediate calcination temperature of 450  C. Decrease in the film thickness with increasing calcination temperature would also tend to enhance the gas sensitivity. The dependence of the gas sensitivity on the operating temperature, within the range of 100–500  C, is shown in Figure 16 [5, 8, 9, 12–14, 17, 21, 22, 25]. The gas sensitivity increases with increasing operating temperature, reaches the maximum value at 320–350  C, and then decreases with further rise in the operating temperature. The electrical resistance of the sensor is also known to follow similar behavior as a function of operating temperature [82]. Within the lower operating temperature range of 100–320  C, the adsorption of the oxygen ions dominates, which decreases the conduction electron density in the film, which favors the high gas sensitivity. On the other hand, in the higher temperature range, 350–500  C, desorption of the oxygen ions dominates, thus reducing the conduction electron density, which reduces the gas sensitivity. The low activation energy for the reaction between the reducing gases and the adsorbed oxygen ions at higher operating temperatures is also a major factor for increased gas sensitivity at higher operating temperatures. On the other hand, instability in the microstructure at high operating temperatures can lead to reduced gas sensitivity at higher operating temperatures. The variation in the gas sensitivity as a function of the film thickness is shown in Figure 17 [5, 9, 13, 17, 21]. The graph shows that the maximum gas sensitivity appears at the film thickness of ∼110 nm, while above and below this thickness, the gas sensitivity decreases. To explain the thickness dependence, a single-crystal thin-film model has been suggested [87], which excludes the effects of microstructure.

1500 1000

SnO2

Pore

500 0

r 0

100

200

300

400

500

600

Operating Temperature (°C)

Figure 16. Variation in the gas sensitivity as a function of operating temperature [5, 8, 9, 12–14, 17, 21, 22, 25].

2l

Figure 18. Model of SnO2 thin film having straight channel structure [55].

36

Sol–Gel Derived Semiconductor Oxide Gas Sensors 2000

(a)

Maximum Sensitivity (%)

CA/CA,s

(b)

(c) (e)

(d)

H2

1600

1200 CO

800

NO

400

x/l

CH3OH

0

Figure 19. Generalized gas concentration profiles within SnO2 thin film, film having straight channel structure, with increasing function of lk/DK 1/2 (from ‘a’ to ‘e’).

The sensitivity S defined by this model is given as

0

5

10

15

20

25

30

35

Molecular Weight (g/mol)

Figure 21. Variation in the maximum gas sensitivity of the sol–gel derived nanocrystalline SnO2 thin film as a function of molecular weight of various gases [5, 9, 21, 22]. 900

Response Time (sec)

where ns is the density of electrons per unit area that are returned to SnO2 particle surface by the oxidation of gas with the surface-adsorbed oxygen ions (O− ) and nB is the electron density per unit volume in the film bulk. The model predicts that the gas sensitivity would increase with decrease of the film thickness. In contrast to this behavior, for the film thickness less than ∼110 nm, a markedly lower gas sensitivity, which decreased with decreasing film thickness, exists. The amount of porosity in the sol–gel derived nanocrystalline SnO2 thin films has also been reported to decrease drastically below the film thickness of 70–200 nm [8, 88]. Very compact films exhibit low surface areas, and as a result, they offer reduced number of the active sites for the oxidation reactions with reducing gases and would decrease the gas sensitivity with decreasing film thickness below the critical thickness range. The decrease in the gas sensitivity with increase in the film thickness can also be explained based on the following model [88]. Consider a porous film having thickness 2l and represented by a straight channel structure as depicted in Figure 18. Each channel is assumed to be a round pore of radius r and length 2l. The gas molecules diffuse into the pores by Knudsen mechanism from both the sides of the film. The Knudsen diffusion coefficient (DK ) under these

750

450 300 150 0 100

150

200

250

300

350

400

450

500

Operating Temperature (°C)

Figure 22. Variation in the response time of the sol–gel derived nanocrystalline SnO2 thin film as a function of operating temperature [5, 8, 9, 12, 13, 18, 21]. 900 750 600 450 300 150 0 150

175

200

225

250

275

300

Operating Temperature (°C)

Figure 23. Variation in the recovery time of the sol–gel derived nanocrystalline SnO2 thin film as a function of operating temperature [5, 9, 12, 18, 21]. 900

Recovery Time (sec)

1000

800

Sensitivity (%)

600

Recovery Time (sec)

(12)

S = ns /nB t

600

400

750 600 450 300 150

200 0 0

0 25

30

35

40

45

50

55

60

65

70

200

400

600

800

1000

Response Time (sec)

Amount of Porosity (%)

Figure 20. Variation in the gas sensitivity as a function of amount of porosity for the sol–gel derived nanocrystalline SnO2 film [8, 12].

Figure 24. Variation in the recovery time of the sol–gel derived nanocrystalline SnO2 thin film as a function of response time [5, 9, 12, 18, 21].

37

Sol–Gel Derived Semiconductor Oxide Gas Sensors Table 1. The gas sensing properties of the sol–gel derived other nanocrystalline semiconductor oxides. (A)

Sensor

Precursors

Zinc Oxide

Zinc Acetate

Calcination temp. ( C)

Gas (ppm)

Film thickness (nm)

300–500

CH4

4000

Crystallite Operating size temp. (nm) ( C) 300–600

300

Zinc Oxide-Pd2+ Catalyst

Sensitivity (%)

Ref.

750

[63]

1500

Barium Strontium Titanate

—

475

H2 (5–1000)

125

—

250

18–900

[59, 60]

Lead Zirconate Titanate

Lead Acetate Trihydrate, Zirconium Acetylacetone, Titanium Isopropoxide

410

H2 (1000)

160

—

175

1600

[60, 61]

Titania

Titanium Butanol Alkoxide Titanium Butoxide

—

O2 (10−28 –10−3 atm) O2 (0.0496–1 atm)

—

—

300,000

80

640

143

[91]

55

55–60

190

682

[53]

12

450

480

11

400

225

800

Titania-Niobium Titanium Butoxide, Oxide Niobium Ethoxide

600

O2 (10,000)

Titania

600

H2 (500)

600–1000 1,000,000–100,000 [90]

60 Titanium Sulfate, Niobium Chloride

Titania-Niobium Oxide

Disc 5.0 mm diameter, 0.5 mm thick

CO (500)

2233

H2 (500) CO (500) Titania

Tetraethyl Ortho-Titanatate

Titanium Isopropyl Alkoxide Indium Oxide

—

500

570 375

CH3 OH (100 ppm) C2 H5 OH (100 ppm) NO2 (4 ppm) CO (300 ppm)

—

500

H2 O (80%)

300–400

800

CH4 (40) CO (40) C2 H5 OH (45) NH3 (15–1100)

Pellister Type

3–30 nm

400

700

500

1700

400 500 400 400

2110 3666 50 50

8 nm. In this region, the number of conduction paths yields an exponential increase with increasing t. The critical exponent, 2, and the critical thickness, tcele , can be evaluated by fitting the data with the scaling law. The detailed analysis revealed a nonuniversal behavior of the electrical conductivity: the conductivity critical exponent 2 increases from 1.1 to 1.8 by increasing the cluster size from 6 to 13 nm [91, 92]. The 2 value of 1.8 is almost close to those for 3D percolation [201], although 2D growth of deposits is confirmed by TEM observations. This result can be accounted for by the soft-percolation model, which takes into account the wide distribution of connectivity between clusters [192, 193]. This model suggests that the electrical connectivity between clusters becomes wider for larger cluster size. The origin of such wide connectivity for larger clusters is possibly attributed to the heterogeneous interface structures between clusters (e.g., lattice defects or imperfect contacts), associated with the increase in melting temperature relative to the smaller clusters. Raising the temperature of substrate leads to a delay in the onset of the electrical percolation owing to the mutual coalescence of clusters [199]. Magnetic Percolation Magnetic interactions among clusters also extend to grow into a network during deposition. One of the noticeable features is the enhanced stability of ferromagnetism of the assemblies [198, 202]. For clusters whose size is small enough to exhibit superparamagnetism, the magnetic state of the assembly changes from superparamagnetic for isolated clusters to ferromagnetic for networked clusters as deposition proceeds. Figure 12 shows the evolution of magnetic coercivity, Hc , for Co clusters with 35

Coercivity, Hc (kA/m)

30 25 20 15

various sizes as a function of t. It should be noted that Hc rapidly increases with increasing t at the very early stage of deposition and then saturates. These saturated Hc values do not markedly change for thicker films with a 100-nm thickness [165]. This suggests that the magnetic interactions give the assemblies an additional magnetic anisotropy that effectively suppresses superparamagnetism. Another feature of the cluster assemblies can be highlighted by comparing the result of Figure 12 with those of magnetic granular films [202], in the latter of which nanometric magnetic granules are precipitated in nonmagnetic matrices via a heat treatment. In the latter films, the Hc markedly enhanced at around the geometrical percolation threshold, and then rapidly decreases with further increasing the content of magnetic materials [202]. This rapid reduction in Hc is attributed to the change of a magnetization mechanism from rotation to wall motion. Based on the fact that Hc is roughly constant for our thicker assemblies of Co clusters, the magnetization mechanism is still in rotation mode as for the isolated clusters. This implies that the original nature of the clusters is well preserved even after they are assembled.

4. SPECTROSCOPIC CHARACTERIZATION 4.1. Transmission Electron Microscopy Transmission electron microscopy has been widely used to characterize atomic structures and nanoscale morphologies of cluster assemblies. Since electron microscopes operate at high acceleration voltage (usually 100–400 keV), the electron wavelength is much smaller (∼10−3 nm) than the atomic size and interatomic distance of substances. This gives rise to a good atomic resolution with relatively simple operations, and hence, it becomes almost an indispensable technique for nanoscale structural characterizations in the current cluster research. So far, the application of TEM has been largely limited to noble metal clusters such as gold and silver, mainly because of their chemical stability against oxidation. These clusters are usually well isolated on substrates, resulting in a simple image interpretation. Some review articles on the electron microscopic studies on the cluster structures are available [203, 204]. In this section, we shall give a brief overview on TEM observations of individual transition metal clusters and their assemblies, including some results of noble metals and metalloids. To see general principles on TEM, some good literature is provided [205–207].

4.1.1. Bright- and Dark-Field Imaging d = 6 nm

10

d = 8.5 nm 5

d = 13 nm

0 0

5

10

15

Deposition thickness, t (nm) Figure 12. Magnetic coercivity, Hc , for Co clusters with various sizes as a function of t.

Conventional bright-field imaging is the most common TEM mode to see the shape and size of supported clusters on substrates. Historically, the external shape of gas-evaporated ultrafine particles has been extensively studied by the groups in Japan, largely contributed by researchers at Nagoya University [131, 208–215]. They performed serial structure characterizations for particles of transition metals (V, Cr, Mn, Fe, Co, Ni, Cu, etc.) and alkaline metals (Be, Mg, Ca, etc.), whose typical sizes range from submicron to micrometer order. The particles are usually well faceted

484 with distinct crystallographic planes. Even a single element particle exhibits multiple particle shapes, depending on their atomic structures and synthetic processes. They reported, for instance, that bcc metal particles (Fe, V, Nb, Ta, Cr, Mo, W) show a single type of crystal habit: the rhombic dodecahedron with (100) planes and its truncated variants [213]. Fcc metal particles (Ag, Au, Co, Ni, Cu, Pd) were found to exhibit the octahedron and its truncated variants when they are a single crystal [131]. Fcc metals also show characteristic twin-related structures when they are not a single crystal. The typical forms are a pentagonal decahedron and an icosahedron, which have been found in vacuumdeposited Au islands on single-crystalline substrates of alkali halides [125, 134, 216]. There, the pentagonal decahedron and icosahedron consist of 5 and 20 tetrahedral crystallites, respectively, which are packed together with their (111) planes bounded [125, 134]. This arrangement generates a twin boundary at each interface and a particle having such twins is called a MTP. Since the tetrahedra cannot fully fill the entire space in the particle, inhomogeneous lattice strains are locally introduced. A similar twin-related form has also been found in needle-shaped Ge precipitates in an Al matrix [217]. There are also other shapes that are related with decahedron: the star decahedron [218], Marks decahedron [203], and the round decahedron [219]. Thermodynamical consideration based on the Wulff construction [220] was provided to account for the appearance of these various external shapes of unsupported particles. The theory requires a minimization of surface-free energy of the particles, resulting in characteristic crystal habits. However, the experimentally observed structures are not necessarily consistent with these Wulff shapes owing to the significant effect of particle growth kinetics [203]. The results are well summarized in [9]. The bright-field imaging is also useful to characterize overall morphology of cluster assemblies, as well as individual particles. When gas-phase clusters are deposited on a substrate, the geometrical connections between the clusters are developed. The connectivity in an early stage of the deposition can be quantitatively evaluated by image analysis of the bright-field micrographs. Using this method, the morphological evolutions during the assembling process of deposited Co clusters were studied in terms of the percolation theory [91, 92]. Assembling processes of other sorts of clusters (e.g., Bi [221, 222] and Sn [178, 200]) have also been observed. The bright-field observations also revealed the characteristic thermal stability and instability of cluster assemblies. The temperature-dependent observations of the morphological change of Co cluster assemblies showed a severe coalescence behavior between contacting clusters at elevated temperatures (T ≥ 250 K) [199, 223]. Thermally sintered experiments of unsupported FePt clusters in a cluster beam showed that the increased sintering temperature improves the shape and crystallinity of the particles from a poorly crystallined, irregular shape to a regularly faceted, multiply twinned structure [145, 146]. When magnetic clusters having a stable magnetic moment are deposited, they prefer to form chains to reduce the magnetostatic energy [208, 224]. Applying an external magnetic field leads to a strong tendency of forming linear chains along the field

Transition Metal Nanocluster Assemblies

direction. This yields an anisotropic magnetic behavior with respect to the chain direction. It is a hard task to observe a thick deposit assembly by TEM, largely owing to the difficulty in thinning the fragile specimen through which the electrons are sufficiently transmissible. There is an article in which the cross-sectional view of a softly landed Sb cluster film, showing a granular morphology where the incident cluster size is relatively retained [4]. This morphology is fairly well explained by the ballistic deposition model [225], where incident particles are impinged on the surface and stuck to the supported particles after physical contact. Dark-field imaging is a complementary technique against the bright-field imaging. It uses only the diffracted beams instead of the nondiffracted (or direct) beam to form the images. The objective lens aperture is necessary to select the diffracted beams. When a specific diffracted beam is chosen, the resulting dark-field image contains bright contrast regions from which the same Bragg diffraction takes place. This technique is powerful in characterizing the atomic structures of polycrystalline particles and in analyzing the structural heterogeneity of multiple-phase clusters. A good example can be seen in the structural determination of MTPs [125, 134, 216].

4.1.2. High-Resolution Electron Microscopy High-resolution electron microscopy (HREM) is capable of directly imaging local atomic arrangements of clusters. This technique is particularly useful for the structural characterization of nanosized clusters less than 10 nm. Many experiments have reported the frequent appearance of various atomic structures that do not exist in the corresponding bulk substances as equilibrium states. A prominent example is the increased stability of icosahedral and decahedral structures over fcc for smaller clusters. This has been first found in Au clusters [125, 134, 216], followed by other fcc metals such as Ag [226]. High-resolution electron microscopy observation of thermally sintered, unsupported fcc FePt clusters show an increased stability of icosahedral structure as the sintering temperature is increased [145, 146]. Even bcc metals (e.g., Na [227]) has an icosahedral form when the particle size is sufficiently small. Recent computer simulations also support the similar structural transitions from fcc to icosahedral or decahedral structures for various types of potentials as the cluster size is reduced: Lennard–Jones clusters [227–230], Au clusters [231, 232], and Morse potentials [233, 234]. Since the energy difference between the competing forms is very small, slight modifications of surface faceting are sufficient to transform from one structure to another. A recent simulation result pointed out the significance of entropic contribution to the free energy to determine the stable structure of small clusters [235, 236]. Combining the high-resolution imaging and a video recording techniques, in-situ observations of dynamic structural change can be carried out. Structural fluctuation of supported Au clusters (∼2 nm) was first observed by Iijima and Ichihashi [141] and Smith et al. [237] at nearly the same time. They clearly showed the real-time rearrangement of constituting atoms and the transformation between different

Transition Metal Nanocluster Assemblies

forms of the particles. It is interesting to point out that even a 2-nm Au cluster exhibits a distinct crystal habit. The possible reasons for this structural instability have been raised thus far in terms of a Coulomb explosion [238], an Auger cascade [239], a charging effect [141], and an electron beam heating [237, 240]. After these findings, similar experiments have been done using different substances [240, 241]. Another interesting topic of in-situ electron microscopy is the in-situ experiments in electron microscopes. Mori et al. installed small evaporation sources in a TEM column and observed time-resolved alloying phenomena of small clusters [242–245]. They found that small clusters are spontaneously alloyed in a very rapid fashion that cannot occur in the equilibrium state. These in-situ experiments highlighted a marked instability of small clusters even at room temperature.

4.1.3. Electron Diffraction Transmission electron microscopy also provides reciprocalspace diffraction patterns as well as real-space images. Since electrons have very large scattering cross-sections compared to X-rays and neutrons, a small amount of specimens are sufficient in detecting sizable diffraction intensities. The electron diffraction experiments are performed either in a relatively large region using a selected area aperture (the selected area diffraction) or in a nanometer region using a nanosized probe (the nanodiffraction), the latter of which allows us to examine the atomic structure of a single cluster. The selected area diffraction experiments have been applied to various sorts of clusters, resulting in the findings of a variety of nonequilibrium phases. The high temperature phases in an equilibrium phase diagram occasionally appears in small-sized clusters, such as fcc phase for Co clusters [198, 246] (the hcp phase is stable at room temperature) and chemically disordered fcc phase for stoichiometric FePt clusters [145] (the chemically ordered L10 tetragonal phase is stable at room temperature). Entirely different structures from the bulk phases have also been found, like A15 structures for Cr [118], Mo, and W [213] clusters.

4.2. Scanning Probe Microscopy (SPM) Scanning tunneling electron microscope (STM) and atomic force microscope (AFM) are powerful tools in understanding the surface structure and morphology of all kinds of materials [247, 248]. The nanometer scale surface modification is also possible by controlling the bias voltage, the tunneling current, and the distance between tip and surface in an STM system [249]. The surface atoms and clusters are manipulated by application of high bias voltage which induces the electric field intensity of 108 V/cm and the current density of 106 A/cm2 . Using the powerful STM technique, we can also study physical properties of material surfaces in nanoscale, such as local density of states (LDOS), friction coefficients, adhesion force, etc. and a magnetic domain structure using a magnetic tip, that is, magnetic force microscope (MFM). An SPM includes all of these sophisticated techniques [250, 251]. One of the central issues of clusters deposited on the substrates is the intercluster interaction—integrity and interaction between cluster and substrate. The lateral dimensions of the transition metal clusters as imaged by the STM depend

485 on the curvature radius of the tip used. The apparent height of clusters is much closer to the actual cluster diameter or size measured by TEM and mass spectrometry. During the annealing process, surface diffusion of clusters is much enhanced on HOPG substrate, leading to the creation of the fractal-type cluster assemblies [252, 253]. The similar process on the Si(111) surface leads to a surface modification of Si atom sites in close vicinity to clusters up to 850 K. When clusters are deposited on the strongly interacting surface, the cluster shape changes from that of the free cluster in the gas phase. It has been reported that Ag19 clusters are almost spherical in the gas phase probably because the shape is governed by the electronic structure. When Ag19 clusters are soft-landed on a Pt(111) surface, a regular hexagon with a monoatomic height is observed by the STM [254, 255]. The STM measurement of soft-landed Ni9 clusters on Si(111)- 7 × 7 surface indicates that clusters are fixed to the impinged position, while Ni3 clusters migrate on the surface and aggregate into characteristic nanometer-sized structure [256]. Mn clusters, whose size is about 2.6 nm, have been deposited on clean Si(1111)- 7 × 7 and terminated by a C60 monolayer. They are of irregular shape, but grossly deformed upon adsorption onto the surface. There is no evidence of either cluster coalescence or preferential bonding sites [257]. Electron tunneling between a cluster and a substrate gives rise to an interesting effect, namely, the Coulomb blockade [258, 259]. The energy of the electron-transferred cluster is increased by a charging term, Ec = e2 /2C, where C is the capacitance of the cluster, causing a gap for electron tunneling. The steps with voltage widths of e/C (the so-called Coulomb staircase) have been observed in the current-voltage characteristic curve of the nanometer-sized metal clusters such as Au55 and Pt309 stabilized by surfactants (organic ligand shells) [260, 261]. The quantum size effect (QSE), which is the discreteness of the energy levels, is also one of the distinctive properties in nanometer-sized clusters. In some ligand-stabilized metal clusters, an additional structure to the regular Coulomb staircase has been observed and ascribed to quantum-sized levels of the cluster [260]. In an STM system, the tunneling electrons that are emitted from the tip and incident on the sample are followed by the inelastic process which induce photoemission at around the tip-sample region [250, 251]. The photoemission spectra represents a particularly versatile channel of information besides the tunneling current. Their intensity, spectral distribution, angular emission pattern, polarization status, and time correlation are accessible by sensitive optical detection methods [262]. For Ag and Cu clusters, the light emission spectra with about 10-nm resolution can be identified when geometric effects are minimized by using a sufficiently small tip [263, 264]. The occurrence of a distinct emission line from alumina-supported Ag clusters can be explained by the decay of a collective electron oscillation (Mie-plasmon resonance) [265]. This emission lines shift to higher energies and their widths increase with decreasing the cluster diameter, because of the reduced screening of the plasmon oscillation due to the Ag 4d electrons and/or an enhanced electron surface scattering rate in small clusters.

486 The magneto-optical effect, magnetic circular dichroism, has been observed for Co metals with STM-induced luminescence: the degree of polarization to be about 10% [266]. The map of the circular polarization has been directly compared with the STM topography for island-like Ni films, where the highly circular-polarized area is well correlated with Ni particles of 50–70 nm in diameter [267].

4.3. Photoelectron Spectroscopy A great deal of effort has been devoted to the study of photoelectron spectroscopic studies for the supported metal cluster during the last decades [268–274]. A sufficient sensitivity to detect electrons and an intensive light source from a synchrotron radiation have contributed to investigate the electronic structures of the nanometer-sized clusters [275, 276], although the fraction of emitted electrons from nanometer-sized clusters is expected to be very small. Photoelectron spectroscopy is generally divided into two techniques relating to laboratory photon sources—gas discharge lamps leading to an ultraviolet photoelectron spectroscopy (UPS) and soft X-ray sources that are used for an X-ray photoelectron spectroscopy (XPS). In both cases, photoionization cross-sections are sufficiently small to ensure that the photon penetration depth is large relative to the meanfree path for inelastic scattering of the emitted electrons which therefore ensures the surface sensitivity [277]. The size-dependent electronic structure of metallic clusters is quite interesting because their electronic and chemical properties can vary from those of bulk materials [276, 278, 279]. Au clusters, for example, change from their inert bulk state to a chemically active state when they are in a form of small particles [276, 278, 280]. The synchrotronbased, high-resolution photoelectron spectroscopy (PES) measurement has indicated that Mo nanoclusters are inactive even when exposed to 150 L of oxygen at 300–850 K. However, these Mo nanoclusters are oxidized by reaction with NO2 at 500 K to form molybdenum oxides, MoO2 or MoO3 [281]. Changes in the energy position and width as a function of their size have been observed in the core level spectra and the valence and conduction band spectra of the metal clusters [271–275]. Most experimental studies have been carried out for island-like grown films prepared by thermal evaporation of materials [271–273]. The photoemission spectra from the mass-selected supported clusters have recently obtained [274, 275]. The shifts of core levels and valence bands, as well as a narrowing of the bands, are obtained when the cluster size is decreased [271]. The core-level shifts with the cluster size are interpreted as a size dependence of the initial electronic state, that is, a change in the number of valence d electrons. The repulsive Coulomb interaction between core and valence electrons also affects the core-electron binding energies, being sensitive to the valence-electron configuration. Since valence s and p electrons of transition and noble metals are much more diffused than d electrons, the core levels are expected to shift towards lower binding energy with increasing the d electron number. The recent study on mass-selected Mn clusters deposited on an HOPG substrate has shown significant changes in the photoemission line shape of the 3s core level relative to bulk Mn [275].

Transition Metal Nanocluster Assemblies

This result can be interpreted as an increase in the Mn magnetic moment in the cluster assemblies, relative to the bulk Mn metal. It is worthwhile mentioning that the core level shifts observed in supported metallic clusters can stem largely from the charge left on the cluster by electron emission [282]. Ionizing spectroscopy leaves a charge on the clusters, whose contribution to the polarization energy is partially screened by the substrate for the supported clusters. For clusters that weakly interact with conducting substrates [271], the energy shifts in photoemission spectra show that the initial state is much more sensitive to cluster size than the final state. In a Pdn /C system [273], the initial-state energy shifts and cluster charging are found to play a minor role because the observed dependence of the spectra is ascribed to mixing of Pd 4d and graphite $ ∗ electron bands, where the coupling is proportional to the inverse cluster radius. The final one-hole local density of states, however, is modified in a complex way with reducing cluster size because of the increased number of surface and edge atoms. Some experimental results on the same system suggest the cluster charge model. A limited number of photoemission studies have been studied for metallic clusters whose sizes are not well determined. With corroboration of SPM experiment, photoemission spectra have been observed for Pd clusters on carbon substrate where the cluster sizes are less than 1 nm in the mean, indicating a conductance gap in tunneling spectroscopy [283]. Ultraviolet photoemission and bremsstrahlung isochromat (BI) spectroscopic measurements also show the emergence of new states closer to EF with an increase in the cluster size. These results demonstrate the occurrence of a size-dependent, metal-tononmetal transition in Pd clusters.

4.4. X-Ray Spectroscopy Based Upon SOR Using very intense X-ray beams from a synchrotron orbital radiations (SOR) source, oscillations and small humps are well resolved just above and below the X-ray absorption energy edge giving the element-specific geometrical structure (an extended XAFS) and electronic structure (an X-ray absorption near edge structure (XANES)) [284]. In particular, X-ray absorption techniques are so surface-sensitive that they are quite useful in understanding the local structure of dilute samples, such as submonolayer coverages of nanoclusters or nanocluster assemblies deposited onto substrates. Highly polarized, incident X-ray beams also make it possible to understand magnetic states of clusters supported on substrates with detecting helicity changes in the absorption spectra of X-rays passed through magnetic material [285].

4.4.1. XAFS and XANES Spectroscopies The radial distributions derived for the XAFS signals at the Co and Ni K-edges have been obtained for Co and Ni cluster-assembled films, which were prepared by a laser vaporization cluster source and deposited on polyimide, silicon, and glass substrates [286]. The simple scattering of the photoelectron by the neighbors, and neglecting the multiple scattering, distinguishes atoms in the core of the clusters surrounded by 12 first neighbor shell for the first metallic

Transition Metal Nanocluster Assemblies

4.4.2. XMCD Spectroscopy X-ray magnetic circular dichroism (XMCD) in X-ray absorption spectroscopy (XAS) can determine the atomic spin (mS ) and orbital (mL ) moments by applying magnetooptical sum rules to the data [285, 290–292]. In Fe clusters supported on the substrate, the L edge X-ray absorption spectra were observed using circularly polarized X-rays with their angular momentum parallel or antiparallel to the sample magnetization, where the intensities of the L1 and L2 edge spectra of 3d electrons were measured in the magnetic field applied along parallel or antiparallel to the circularly polarized photon spin. Figure 13 shows size-dependence of

0.30

0.25

m L (µB)

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mS+7mT (µB)

neighbors (the first shell) and atoms at the (111) surface surrounded by nine first metallic neighbors on one side (second shell) and by first O neighbors on the other side (third shell). For the second shell, the dilatation of the metal-metal distance is attributed to the interaction of O atoms in the nearsurface region of clusters. The simulation of the Ni and Co cluster assemblies indicates that the number of metallic neighbors (the first and second shells), which is related to the surface-volume ratio, is 10.5–11 on average, showing the cluster sizes ranging from 3–5 nm for Ni and Co cluster assemblies with the truncated octahedron (for large clusters) and multiple twinned shapes. In Co cluster assemblies covered with Au films, the decomposition of the XANES spectra reveals coexistence of metallic Co clusters with about 0.6-nm thick CoO layers [287]. Moreover, the detailed peak analyses of assembled Co clusters with 6–14 nm in diameter indicates that the fcc phase is predominant for 6-nm size clusters and the hcp phase is increased for larger sized clusters. These results are consistent with those obtained by the high-resolution TEM observation. Fe clusters, whose sizes range from 2–20 nm in diameter, have been embedded in noble metal (Ag and Cu) matrices using a ICB cluster source for Fe and a thermal evaporation source for Ag or Cu, and their radial structure function has been estimated from the XAFS spectra measured at the Fe-K edge [59, 60, 288]. In Ag matrices, distorted bcc Fe clusters are formed for an as-prepared state and strainfree bcc Fe clusters after annealing. In Cu matrices, on the other hand, distorted fcc clusters are formed for the dilute Fe concentration region (less than 20 at%) while distorted bcc clusters for the Fe-rich concentration region (more than 60 at%). The decomposition analyses of XANES for Fe clusters embedded in Cu matrices also qualitatively show a continuous reduction in the volume fraction of bcc/fcc between 20 and 60 at% Fe, being in accordance with the spectral analyses of Mössbauer spectra for the same specimens. X-ray absorption fine structure studies have also done for Co clusters embedded in nonmagnetic Ag, Pt, and Nb matrices using both a laser vaporization, inert-gas condensation cluster source and a thermal evaporation source [289]. They confirm that a pure Co core is conserved to be fcc in all cases. When Co is immiscible with the matrix element (Co/Ag), the cluster/marix interface is quite abrupt. Even when Co is miscible with the matrix element, an alloying effect is limited to one or two atomic layers of the cluster/matrix interface at an ambient temperature and it is promoted by annealing at higher temperatures.

487

2.05 2.00 1.95 1.90 1.85 0

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Cluster Size (atoms) Figure 13. Atomic orbital magnetic monent mL and spin term ms + 7mT as a function of cluster size for Fe clusters on HOPG. The filled square and open circle symbols in both figures represent the measurements taken at 7 = 0 and 55 . Lines A and B in the upper figure denote the value measured in the 25-nm MBE film and the typical value of bulk Fe, respectively. Line C in the lower figure indicates the value measured for the MBE film. Reproduced with permission from [296], S. H. Baker et al., J. Magn. Magn. Mat. 247, 19 (2002). © 2002, Elsevier Science.

the components, mS + 7mT and mL for Fe clusters, which were prepared by a gas aggregation source (a thermally vaporization source), size-selected by a quadrupole mass filter and deposited on HOPG substrates installed in a UHV chamber [293–296]. mT is the dipole moment and 7 is the angle between the direction of photon beam relative to the sample normal, where mT = 0 at 7 = 0 and mT = 0 at 7 = 55 (the magic angle) for a single crystal or a sample with a rotational symmetry normal to the surface. Comparing two data in Figure 13, the magnetic dipole moment is opposite in sign to the spin moment, contributing about 5% to the sum for measurements normal to the surface. The spin moment of the smaller size Fe clusters is larger than the bulk value. The dipole moment increases with decreasing cluster size, being about two times larger than the bulk value throughout the size range, while it does not show a significant variation with angle. The increase in both mS and mL contributions to the magnetic moment can be ascribed to the high

488 proportion of surface atoms with a low coordination, which produce a narrowing of the d-band responsible for magnetism and an increase in the density of states at the Fermi level, and reduce symmetry and less effective quenching of orbital magnetism by the crystalline field. At low temperatures, the total magnetic moment, mS + 7mL , in supported Fe clusters is much larger than the bulk value, being ascribed to the enhancement of mS and mL and their parallel alignment owing to the positive spin-orbit coupling in the more than half occupation of d-electrons. However, it is not so large as the free cluster values probably due to the clustersubstrate interaction. The XMCD of Fe L edge spectra have been observed for size-selected Fe clusters as a function of the cluster size between two and nine atoms which were deposited in an ultra-high vacuum on perpendicularly magnetized Ni films, where a soft landing was achieved by precovering the substrate with argon buffer layers to maintain the selected size [297]. In Fe clusters, whose sizes are less than 10 atoms the enhancement of orbital moments is more significant and the ratio of mL to mS reveals an oscillatory size dependence. It has been speculated that the geometrical arrangement and magnetic properties of such small clusters depend on whether they consist of even or odd numbers of atoms. With increasing coverage of unfiltered Fe clusters with an average size of about 600 atoms deposited on HOPG substrates the value of mS + 7mL , moreover, increases slightly indicating the dipole moment decreases with an increase in the cluster density, while mL rapidly decreases due to the increasing proportion of clusters in contact with each other, thus increasing the average atomic coordination [295]. The magnetic remanence estimated by field-dependence of XMCD for the isolated Fe clusters are superparamagnetic even at 40 K, decaying rapidly with increasing temperature, but the magnetic anisotropy is enhanced considerably. Dichrorism is also observed in the 3p angle resolved photoemission spectra of the transition metals taken with linearly or circularly polarized photoemission spectra [298]. The former measures the difference in the spectra in response to reversing the alignment between the in-plane sample magnetization and the linear polarization of the XUV light (MLDAD) and the latter between the outof-plane sample magnetization and angular momentum of circularly polarized light (MCDAD). Since photoemission spectra must be collected in a zero field, the techniques are only applicable to measure the remanence state. At 40 K, the MLDAD signal is zero within experimental error in the isolated Fe cluster. While increasing the coverage, it becomes finite and rapidly increases. This suggests that the interactions between the clusters lead to the coherent shape anisotropy of the films. The remanence becomes larger than the bulk signal, probably due to asperomagnetic configuration in the layered cluster assemblies. When Fe clusters are coated by Co thin films, Fe spin moments are increased, being about 10% larger than those of the noncoated Fe clusters, but the orbital moment reveals no significant change [294, 298]. Since the XAFS intensity of Co-coated Fe clusters is almost the same as the noncoated one, no significant charge transfers between Fe and Co and an alloy effect is confined to the interface. Therefore, the enhancement of spin moments is attributed to an increase in

Transition Metal Nanocluster Assemblies

the Fe valence band exchange splitting induced by the interaction with the matrix. The mT contribution is also reduced by Co coating owing to the reduction in the magnetostatic energy. When Co clusters with 300–9000 atoms are codeposited with Cu atoms on Si (100) substrates, the total magnetic moments are reduced to less than a half value of a deposited Co film, and the orbital moments are enhanced considerably to be comparable to the spin moments [299]. Similar features have been observed for Co clusters formed on Au (111) substrates, under ultrahigh vacuum conditions [300], where the L edge spectra contains no extra peak resulting from chemical core shifts of the p-states and there is little or no contamination of Co atoms. When Co clusters are covered with C films, both the spin and orbital moments are reduced, where the latter reduction is more marked [301]. The Co K-edge spectra also indicate a slight reduction in the orbital moments of 4p-electrons being about 1+5 × 10−3 2B induced by spin-orbit interactions of 3d-electrons on the neighboring sites through the 4p-3d hybridization. Several speculations, such as a structural change from fcc to hcp, antiferromagnetic exchange coupling between Co clusters, etc. have been proposed. The careful observation of Cr-L3 edge absorption spectra of Cr clusters (with the sizes less than 10 atoms) whose size are less than 10 atoms indicate that the line broadening and shift are attributable to bimetallic electronic state at the cluster/matrix interface, geometrical structure change, oxidation of surface atoms, etc. [302]. We need more systematic and unique approaches at arrive at a common interpretation of such environment-sensitive aspects of magnetic transition metal cluster assemblies.

4.5. Nuclear Scattering and Resonance Techniques As discussed in Section 4.1, the thermally activated fluctuation of magnetization in a small particle is described by the an approximate expression for the relaxation time:  = 0 exp kV /kB T 

(16)

where K is a magnetic anisotropy constant, V a volume of a particle, kB the Boltzmann constant, T the absolute temperature, 0 the angular precession frequency in the order of 10−9 s [303]. In small cluster assemblies, we observe a ferromagnetic behavior, for a measuring time scale, m > , while a paramagnetic behavior (superparamagnetism) for m < . For m ≈ , we can detect a ferromagnetic/superparamagnetic transition at a certain temperature, the blocking temperature TB . Using conventional magnetic measurement with m ≈ 1–102 s, TB is of an order of room temperature in small Fe particles whose sizes are 10–20 nm in diameter, depending upon respective anisotropy characteristics. Today, there have been a lot of advanced nuclear scattering and resonance techniques, with which we can choose the observation window, m a very wide range between 10−4 and 10−12 s to understand the detailed mechanism of magnetic relaxation phenomena, as well as their internal crystalline and magnetic structures.

Transition Metal Nanocluster Assemblies

4.5.1. Neutron Scattering Neutron scattering is quite a informative method for material characterization because the neutron cross-section is the sum of a nuclear and a magnetic contribution. In a smallangle, neutron scattering experiment of cluster assembled systems, we may obtain the particle sizes, interparticle distance, spin correlations within the particle, and interparticle spin correlations. However, the neutron-matter interaction is so weak and the penetration depth of neutrons in matter is extremely long, so a large amount of specimens is necessary for measurement [304]. This technique has been mainly applied for small particles precipitated in oxide or polymer matrices which are mass productive. With a decreasing temperature of Fe particles with an average diameter of 2 nm embedded in Al2 O3 matrices, an increase in particle magnetization (an intraparticle term) is dominant down to 100 K, while magnetic correlation between the total spins of neighboring particles (an interparticle term) is dominant below 100 K [305]. Besides the elastic neutron scattering mentioned above, inelastic neutron scattering giving space and time pair correlation of spins is effective in analyzing the spin dynamics, susceptibility, relaxation, etc., where the energy spectra consist in two parts: a central peak (a no-energy-transfer component) and inelastic spectrum (an energy-transfer component) [304]. The energy scale 0.01–100 meV, corresponding to the time scale 10−10 –10−12 , is typical of anisotropy energies for the lowest values and of the exchange energies for the highest ones. The energy line width of an inelastic spectrum has been observed as a function of temperature for Fe particles embedded in Al2 O3 matrices. Above 250 K, the energy line width is temperature-independent, indicating that the magnetic intensity is all contained in the inelastic peak. Below 250 K, it becomes larger and a magnetic intensity occurs in the central peak indicating the slowing down of some magnetic component. There are two characteristic times for the fluctuation of the particle magnetization. A neutron spin-echo-spectroscopy with the time-scale range between 10−11 and 10−7 s has been further observed at different scattering wave numbers to clarify the slow fluctuation in the Fe particles embedded in Al2 O3 matrices [306]. The analyses of both the central and satellite components indicate the following relaxation picture: T ≥ K, there is single-particle fluctuations, while below T ≤ 100 K, the temperature is less than the magnetic anisotropy energy barrier and the relaxation splits into two components: The slow fluctuation is attributable to the local longitudinal component fluctuating between energy minima and the fast one to the transverse fluctuations around the temporary mean orientation in one of the minima. This correlation leads to a more pronounced freezing of spins in space and time in comparison with the noninteracting particle systems.

4.5.2. Muon Spin Rotation Spectroscopy Muon spin rotation spectroscopy is another powerful technique in studying spin dynamics of complicated magnetic materials such as spin glass [307, 308]. The spin polarized muons implanted in a sample occupy interstitial sites, precess in the local magnetic field, and decay into positron and two neutrinos. The time-decay spectrum of positrons gives

489 the muon precession frequencies and the internal magnetic fields of the sample. This technique covers an extremely broad time-scale between 10−11 and 10−4 s, being comparable to that anticipated for superparamagnetic systems. The kinetic energy of incident muons is so high as an order of MeV that they penetrate thin films and small specimens to be implanted in substrates. So far as concerned with spin dynamics of small particles, this technique has first applied to the specimen, in which a small Co particle precipitated in 0.5-mm thick Cu discs [309]. Recently the low-energy muon beam of orders 0.5–30 keV has been available, where the time-scale set by the muon precession and decay is an order of 10−5 s. The low-energy muon rotation has been observed for monodispersed size Fe cluster assemblies prepared in a gas condensation source and embedded in a Ag thin film matrix [310]. Here, the effective Fe concentration was about 0.1% in volume and most of implanted muons stop in the Ag matrix between the clusters. The static distribution of the local fields (dipole fields associated with the magnetic clusters) causes the implanted muons to precess out of step and leads to a decay damping. This experiment indicates that 0 = 1+2 × 10−8 s and K = 2+3 × 105 Jm−3 . The enhancement of K by a factor five is consistent with the enhancement of orbital moments in clusters with the reduced coordination of atoms at the interfaces.

4.5.3. Mössbauer Spectroscopy The Mössbauer effect is a recoil-free emission and resonant absorption phenomenon of
-ray between the excited and ground states of a nucleus embedded in a material [311, 312]. This technique is quite powerful in studying specimens containing a reasonable amount of 57 Fe atoms, because the half-life of a 57 Co radioactive isotope is long (270 days) for conventional laboratory works. Several parameters, such as an isomer shift, a quadrupole splitting, and a magnetic hyperfine field, can be obtained via the data fitting procedure. It is possible to analyze quantitatively the local electronic structure, bonding state, atomic arrangements, magnetic structure, formation phase, phase transition, atomic diffusion, etc. [313]. Relaxation Behaviors Concerned with the magnetic relaxation of Fe nanoparticles, m ≈ 10−8 s of 57 Fe nuclei, corresponding to the Lamor precession period of the magnetic moment of the nucleus in the exited state (the nuclear moment, I = 3/2), which is 14.4 keV higher than the ground state (I = 1/2) [312]. A collapse of a hyperfine split spectra can be detected at T ≈ TB , being much lower than room temperature in nanosize Fe particles. With a wide temperature range-studies of Mössbauer spectroscopy and AC and DC magnetization measurements of bcc Fe particle assemblies, the average particle size was determined to be about 3 nm, which is constitent with TEM observation and the parameters in Eq. (16) were estimated as follows: 0 ≈ 10−10 s and K ≈ 1+2 × 105 Jm−3 [313]. Mössbauer spectroscopy has been observed for noninteracting or weakly interacting magnetic particles [315–318, 386]. A maghemite (
-Fe2 O3 ) nanoparticle is superparamagnetic if isolated. However, interparticle magnetic interactions in magnetic particle assemblies lead to spin-glass-like ordering at low

490 temperatures [316, 386], and the relaxation time decreases with decreasing particle interactions [317–319]. This phenomenon has been often termed superferromagnetism [316]. Spin Canting Characteristics Spin canting of small maghemite particles (
-Fe2 O3 ) [319–328] has been discussed via Mössbauer spectroscopy measurement: nonsaturation behavior of magnetization at 4.2 K even in a magnetic field of 50 kOe is attributable to the existence of random canting of the surface spins caused by competing antiferromagnetic exchange interactions. For small ferrite particles (NiFe2 O4 [329], CoF2 O4 [330]), a noncollinear magnetic structure was proposed by Mössbauer spectroscopy experiments: the ferrite particles consist of a core with the usual spin arrangement and a boundary surface layer with atomic moments inclined to the direction of the net magnetization. Similar effects have also been experimentally discussed with corroboration of numerical calculations [331, 403, 404]. The antiferromagnetic superexchange interaction is disrupted at the surface of the ferrimagnetic oxide crystallites because of missing oxygen ions or the presence of other impurity molecules. Such broken exchange bonds between surface spins lead to surface spin disorder, being compatible with a spin-glass-like behavior at the surface. Such spin canting in nanoparticles has long been considered as a surface effect [320–322, 331, 403, 404]; recent Mössbauer spectroscopy experiments of maghemite particles have shown that spin canting is not a surface effect, but rather a finite-size effect that is uniform throughout the whole volume of the particles [326–328] and, consequently, may cause a substantial decrease of the saturation magnetization of small ferrimagnetic particle systems in comparison to that of the bulk material. Phase Analyses and Surface Oxidations Mössbauer spectroscopy has been extensitvely studied in the understanding of microscopic precipitation processes and structures of Fe and Co particles in Cu and Au matrices prepared via heat treatments from their supersaturated solutions [60, 332]. The application of this technique to Fe clusters embedded in Ag and Cu matrices (granular alloy films prepared by the combination of ICB and thermal evaporation sources) reveals the following results. At room temperature, broad singlet and doublet lines are observed in the dilute Fe concentration region (in a small Fe cluster size region) and a broad sextet line in the condenced Fe concentration region (in a large Fe cluster size region). At 5 K, a broad ferromagnetic sextet line is observed in the whole concentration for Fe/Ag granular alloy films [332], while very broad singlet, doublet, and sextet lines are detectable in the dilute Fe concentration region, unresolved sextet lines in the intermediate Fe concentration region (at around 20 at% Fe) and broad sextet lines in the Fe rich concentration region for Fe/Cu granular alloy films [60, 333]. These results indicate that Fe clusters are always bcc and ferromagnetic in Fe/Ag granular alloy films, while they are fcc and change from ferromagnetic to antiferromagnetic in the Cu rich concentration region and become bcc and ferromagnetic in the Fe rich concentration region for Fe/Cu granular alloy films. Presence of oxide surface layers have been discussed using Mössbauer spectroscopy observation for oxidized Fe particles, passivated Fe nanoparticles, and Fe cluster assemblies

Transition Metal Nanocluster Assemblies

[164, 334–346]. Most of the experimental results showed that oxide surface layers are a nonstoichiometric Fe3 O4 phase and/or a mixture of stoichiometric Fe3 O4 and
-Fe2 O3 phases and they are composed of small crystallites. The spin canting phenomenon mentioned has also been observed in thin oxide layers on metallic Fe cores [334–337].

5. CHARACTERISTIC MAGNETIC AND TRANSPORT PROPERTIES OF CLUSTER-ASSEMBLED MATERIALS 5.1. Magnetic Properties 5.1.1. Theoretical Background Single Magnetic Domain Particles Review of single domain particles includes those by Bean and Livingston [347], Bean [348], Brown [349], Wohlfarth [350], and LesliePelecky and Rieke [351]. When the size of magnetic particles is reduced to a few tens of nanometers, the formation of domain walls becomes energetically unfavorable and each particle becomes a single magnetic domain. The single domain size (dsd ) for spherical particles has been estimated [246]: dsd = 24 (fcc-Co), 40 (hcp-Co), and 24 nm (Fe). Changes in the magnetization can no longer occur through a domain wall motion and instead require the coherent rotation of spins, resulting in larger coercivity (Hc ). In this case, Hc is strongly dependent on temperature (T ). In order to reverse particle spins, it should have enough thermal energy to surmount the activation energy barrier (8E = KV ). At high temperatures, where the particles have higher thermal energy, they require a smaller field to reverse the magnetization. This field is equal to Hc for single-domain ferromagnetic particles having no interaction between them and is given by [352]:

Hc =

    25kB T 1/2 2K 1− Ms KV

(17)

where V is the volume of the ferromagnetic particle, K the magnetocrystalline anisotropy constant, Ms the saturation magnetization, and kB the Boltzmann constant. When T approaches zero, Hc approaches 2K/Ms , that is, Hc# 0 = 2K/Ms . Equation (17) predicts that the value of Hc increases as the cluster size becomes larger. For particles with constant size, there is a temperature, called the blocking temperature, TB , above which the metastable hysteretic response is lost for a particular experimental time, and the particles exhibit a superparamagnetic behavior. For uniaxial particles, TB can be taken as KV /25kB , and hence, we get   1/2  T Hc = Hc# 0 1 − TB

(18)

This T 1/2 dependence has been observed in many nanoparticle systems [156, 353–355].

Transition Metal Nanocluster Assemblies

491

Packing Fraction In order to form a practical magnet or a specimen for studies, the magnetic nanoparticles must be compacted, with or without a nonmagnetic binder or matrix, into a rigid assembly. Then an important variable is the packing fraction, Pc , defined as the volume fraction of magnetic particles in the assembly. According to the simple theory on a random assembly of single-domain particles with uniaxial anisotropy, the magnetic coercivity is expressed as [356]: Hc Pc  = Hc 0 1 − Pc 

(19)

where Hc 0 is the coercivity of an isolated particle (Pc = 0). Some materials obey this simple relation but many do not, because magnetic intercluster interaction generally exists in cluster assemblies [357]. They are more or less strong according to the packing fraction. In any case, long-range magnetic dipolar interaction is always present. If the matrix is metallic, Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction occurs and depends on 1/L3 , where L is the distance between centers of clusters, as in dipolar interaction. When the matrix is insulating, superexchange interaction could exist at short range according to the structure and nature of the matrix and bonding at the cluster-matrix interface. Those intercluster interactions make the problem complicated. Therefore, a more detailed theoretical analysis on the cluster-assembled system is needed. Magnetic Relaxation Phenomenon Magnetization reversal may be realized by changing the applied field or by thermal activation. Time-dependence of the magnetization under a constant field is referred to as magnetic viscosity. The magnetic viscosity due to thermal activation is a general property of all ferromagnetic materials. In bulk materials, the time-scale for magnetic viscosity is so large that magnetic viscosity often cannot be significant. However, in fine particle or cluster systems, magnetic viscosity becomes important. Thermally activated magnetic reversal has been investigated intensively, because it might be a limiting factor for the achievable density of conventional magnetic recording media. Especially, in 1988, Chudnovsky and Gunther [358] predicted that at a low temperature, the magnetic moment can quantum mechanically tunnel through the energy barrier U with a probability of exp −B, where B = U /kB Tco , kB is Boltzmanns constant and Tco is the crossover temperature. Experimentally, the magnetic relaxation phenomena of macroscopic quantum tunneling have been observed in many magnetic systems [359–361]. In a single-domain particle system with a unique value of the anisotropy energy barrier, the magnetization relaxes according to the law M t = M 0 exp −t, where  = 1/ is the transition rate and  is the relaxation time or decay lifetime. In real systems, there is an energy barrier-height distribution resulting from intrinsic and extrinsic anisotropy, which leads to a logarithmic time relaxation [362]: M t = M t0 :1 − S T  ln t/t0 ;

(20)

where S T  is the magnetic viscosity whose temperature (T ) dependence characterizes the relaxation behavior of the system. At a temperature range of Tco < T ≪ TB , where TB is

the blocking temperature for the magnetic moment of the particles, S T  can be written as   dM 1 (21) = n EkB T S T  = − M t0  d ln t where n E is the energy barrier distribution function and E is the mean-energy barrier within the experimental window. If n E is constant in a certain range of energies, the magnetic viscosity will be linear in T over the corresponding temperature range. The temperature independence of the magnetic viscosity below a critical (crossover) temperature TB is considered as evidence of macroscopic quantum tunneling in nature. It is predicted theoretically [363] that the crossover temperature from thermal activation regime to quantum tunneling regime scales with the component of magnetic anisotropy normal to the easy axis, so the tunneling effect should be observable at experimentally accessible temperatures only for materials with high anisotropy constants. Unidirectional Exchange Anisotropy Unidirectional exchange anisotropy (UEA) was first discovered in fieldcooled Co/CoO particles more than 40 years ago and shown to be caused by the strong exchange coupling between the ferromagnetic (FM) Co core and the antiferromagnetic (AF) CoO layer [364]. The typical UEA effect is a marked shift of the hysteresis loop against the applied field, commonly referred to as an exchange bias field, Heb , when field-cooling the sample from temperatures above the Néel temperature TN of the AF to T < TN . This hysteresis loop could be obtained if the energy function was exactly given by E = −Ku cos 7

(22)

where Ku is the UEA constant, and energy minimum occurs at 7 = 0 and energy maximum occurs at 7 = 180 . In the hcp cobalt particle, only a direction along c axis (7 = 0 ) is stable in an infinite field. The phenomena related to the UEA effect have also been studied theoretically [365–368] and experimentally [369– 375], because they are technologically important to domain stabilizers in magnetoresistive heads [376] and spin-valvebased devices [377]. However, the understanding of the UEA effect has not been well understood because it is very difficult to determine the AF spin structures in interfacial layer contributing to Heb . The first simple model [364] dealt with the unidirectional anisotropy by assumption of a perfect uncompensated plane of the AF at the interface and predicted Heb which was two orders of magnitude larger than the observed ones. Mauri et al. [365] explained the small experimental value of Heb by assuming the formation of a domain wall parallel to the interface which dramatically lowers the energy required to reverse the magnetization. Alternatively, Koon [367] predicted a correct value for Heb as a result of a perpendicular orientation between the FM and AF axes, similar to the classical spin-flop state in bulk AF. The recent polarized neutron diffraction experiment has shown that exchange coupling between the Co and CoO layers is apparently responsible for the increased projection of the AF moments perpendicular to the cooling field direction [373]. Although the theoretical models have

Transition Metal Nanocluster Assemblies

492 mainly explained the unidirectional anisotropy and obtained the correct order of Heb , they have predicted no effect on the coercivity Hc . Experimentally, the shifted hysteresis loop is always accompanied by an enhancement of the corecivity, which is much larger than the intrinsic value of the FM core [364] and layer [369]. Quite recently, Schulthess and Butler [368] have made a calculation for CoO/FM films using an Heisenberg model and have shown that there are two coupling mechanisms at work, the spin-flop coupling (being responsible for a large coercivity) and FM-AF coupling through uncompensated defects (accounting for exchange bias field).

5.1.2. Co and Fe Cluster Assemblies For size-monodispersed Co and Fe cluster-assembled films without a nonmagnetic binder or matrix, their magnetic properties have been studied [156, 165, 198, 378]. Hightemperature electrical conductivity measurements [379] and TEM observations [156, 165] indicated that in a cluster size range of 6–15 nm, individual Co and Fe clusters are distinguishable at room temperature (RT). This suggests that those clusters in the cluster-assembled films maintain their original size at RT. The actual thickness estimated from a low-magnification cross-sectional SEM image is about ta = 800 nm, while the effective thickness estimated from the thickness monitor was te = 200 nm, that is, ta is about four times that of te . This implies that the cluster-assembled films are very porous and have a cluster-packing fraction of about 25% (=te /ta = 200 nm/800 nm). On the basis of the aforementioned simple theory (Eq. (19)), such a low-packing fraction leads to a large coercivity. Figure 14 shows the d dependence of Hc at T = 300 K for the present Co and Fe cluster assemblies together with that for fcc Co (solid line) and Fe (dash line) nanocrystals estimated from the following equation [380], using the reported values for the bulk Fe and Co [381]: Hc =

Pc K14 d 6 Ms A3

(23)

103

coercivity, Hc (Oe)

102 101 d6 (fcc Co)

0

10

10–1 10–2

d 6 (Fe) Co cluster assemblies Fe cluster assemblies Nanocrystalline FeCuNbSiB [Ref.380] Co(Fe)-based amorphous alloy

5.1.3. Core-Shell-Type, Oxide-Coated Co and Fe Cluster Assemblies

10–3 1

10

where A is the exchange stiffness constant. Clearly, for the Fe cluster assemblies, their Hc values are larger by an order of magnitude than those of the Fe nanocrystals estimated from Eq. (23) for a given d. On the other hand, in these cluster assemblies, the effect of intercluster interactions strongly affects their magnetic properties. The most striking feature is a superparamagneticto-ferromagnetic transition by increasing the film thickness of Co clusters with d = 6 nm [198, 378]. Using the K value of bulk materials [381], simple estimation from TB = KV /25kB indicates that the superparamagnetic state should appear at RT for Co clusters with d < 17 nm (fcc-Co) and d < 10 nm (hcp-Co) and for -Fe clusters with d < 16 nm. However, the Co and Fe cluster-assembled films with cluster sizes that are much smaller than these critical sizes show a ferromagnetic behavior [156, 165, 198, 378]. Such ferromagnetic behaviors of the small Co and Fe clusters is ascribed to strong interactions between magnetic nanoparticles that can lead to ordering of the magnetic moments and thus an increase of TB of the zero-field cooled magnetization [382–386]. This phenomenon has been termed superferromagnetism and the ordering of the magnetic moments of interacting particles may be calculated by use of a mean field model [382–384]. In samples of ferromagnetic or ferrimagnetic ultrafine particles, the magnetic dipole interaction between pairs of particles may be comparable to the thermal energy and can therefore influence the relaxation behavior. The exchange interaction between surface atoms of two magnetically ordered particles in close contact can also result in significant interaction effects. These interacting factors give results not in accordance with the Néel model for superparamagnetic relaxation but can be explained by the formation of an ordered spin-glass-like state [386]. The superparamagnetic-to-ferromagnetic transition with increasing the cluster-assembled film thickness also implies that the effective magnetic anisotropy constant Keff of the clusters is enhanced in thicker cluster-assembled films. Therefore, attempts were made to calculate the value of Keff from the temperature-dependence of the coercivity for Fe cluster-assembled films according to Eqs. (17) and (18) [156]. The results indicated that the Fe cluster-assembled films follow closely Hc versus T 1/2 law and the fitting is very satisfactory. The value of Keff increases with decreasing the cluster size and is of the order of 106 erg/cm3 . In the case of d = 9 nm, the value of Keff is 3+66 × 106 erg/cm3 . Thus, the experimental value (Keff ) of the anisotropy constant is larger by an order of magnitude than that (5 × 105 erg/cm3 ) of the bulk Fe [381].

100

Cluster diameter, d (nm) Figure 14. Coercivity, Hc , at T = 300 K as a function of the grain diameter d for the Co- and Fe-cluster-assembled films and nanocrystalline materials. The solid and dash lines indicate the results for fcc Co and Fe nanocrystalline materials estimated from Eq. (23), respectively.

Oxide-passivated particles of Co [153, 155, 387–391], CoFe [387, 392, 393], Fe [156, 355, 394–397], and Ni [398, 399] are used to study the UEA effect induced by exchange coupling between the FM metal cores and the AF or ferrimagnetic (FIM) oxide shell layers. For these cluster assemblies, magnetic reversal mechanism and spin structures at a core-shell interface are different from those for simple FM/AF bilayers because of single-domain structure of Co core grains, the

Transition Metal Nanocluster Assemblies

493

small size of cores and shell crystallites, and real interface roughness, leading to some characteristic properties [153, 155, 156, 389–391, 395, 397]. Their coercivity and hysteresis loop shift induced by field cooling were strongly affected by temperature, cluster size, and oxygen gas flow rate during deposition or volume ratio of the core to shell, as well as the sort of metal core and oxide shell. Correlation Between Heb and Hc By selecting cluster size and adjusting the thickness of the oxide shells, large Hc and Heb can be obtained. Figure 15 shows the hysteresis loops of the Co/CoO monodispersed cluster assembly with d = 6 nm prepared at RO2 = 1 sccm after the sample was zero field cooled (ZFC) and field cooled (FC) from 300 to 5 K in a magnetic field, H, of 20 kOe. The direction of H used to measure the loops was parallel to that of the cooling field. For this sample, the thickness of the CoO shell have been estimated to be about 1 nm by direct observation of the high resolution transmission electron microscope, being consistent with the Co core size of about 4 nm estimated from the Langevin fitting to the experimental data above room temperature. The large Heb (= H1FC + H2FC /2) value of 10.2 kOe is detected, which indicates the presence of strong UEA in the present sample. As seen in the inset of Figure 15, Heb increases with increasing the cooling field and almost becomes unchanged when the cooling fields are higher than 10 kOe. On the other hand, the large coercivity Hc (= H1ZFC − H2ZFC /2 ≈ 5 kOe) is also obtained for the ZFC case in which the UEA effect is randomized. Clearly, the enhancement of Hc mainly stems from the UEA effect. The correlation between Heb and Hc have been discussed for the Parmalloy/CoO bilayers [374, 375] as a theoretical extension of Malozemoff’s model [366]. The UEA effect is interpreted in terms of random exchange fields due to roughness and inperfection at the FM and AF interface, giving the correct order of magnitude for Heb . Recently, the correlation

between Heb and Hc for small Co/CoO cluster assemblies have also been discussed experimentally [155]. Figure 16 shows coercivities, Hc and HcFC (= H1FC − H2FC /2), of the ZFC and FC samples at 5 K as a function of Heb for the monodispersed Co/CoO cluster assemblies with d = 6 and 13 nm prepared at different RO2 . Both Hc and HcFC increase with the increase of Heb , indicating the uniaxial anisotropy is compatible with the UEA. It is noteworthy that the value of HcFC is about twice as large as that of Hc at a given Heb value. This fact suggests that a magnetization reversal mechanism of rotation exists and an enhanced uniaxial anisotropy parallel to applied field is induced. Disappearance Temperature of the UEA Effect In the oxide-coated cluster assemblies, a second interesting observation is that the UEA effects often disappear at a temperature (Td ) that are much lower than the Néel temperature (TN = 293 K) of the bulk CoO and the magnetic orderdisorder transition temperatures of the bulk Fe oxides (
Fe2 O3 : ∼1020 K; Fe3 O4 : ∼858 K; FeO: ∼198 K). For the Co/CoO cluster assemblies [153, 155, 390], the UEA effect disappears at Td = 150 − 200 K, while for the Fe/Fe-oxide cluster assemblies [156, 355, 397], it becomes undetectable at about Td = 50 K. For example, Figure 17 shows coercivities, HcZFC and HcFC , of the ZFC and FC sample and exchange bias field Heb as a function of temperature for the oxide-coated Fe cluster assemblies with d = 9 nm prepared at RO2 = 0 and 3 sccm. For the FC sample, HcFC is defined as the average of the absolute value of positive and negative fields. The difference in HcZFC and HcFC becomes more significant with decreasing T for both RO2 = 0 and 3 sccm. Such a bifurcation effect starts at about T = 50 K. Heb also 10 9

0.003

Hc 8

d =6 nm

Coercivity, Hc and Hc (kOe)

T =5 K

0.002

0.001 H FC 1 0.000

H FC 2 H ZFC 1

FC

Heb=10.2 kOe ZFC FC

H ZFC 2 0.002

M (emu)

Magnetization (emu)

RO2=1 SCCM

–0.001

–0.002

0.001 0.000

H 1 kOe 10 kOe 20 kOe 50 kOe

–0.001 –0.002 –40

–20

0

20

40

–40

–30

–20

–10

0

10

20

30

40

7 6 5 d = 6 nm

4 3

d = 13 nm

2 1

H (kOe)

–0.003 –50

FC

Hc

50

Magnetic field (kOe)

O2 gas flow rate (RO2) increases

0 0

2

4

6

8

10

Exchange bias field, Heb (kOe)

Figure 15. Hysteresis loops of the ZFC and FC Co/CoO monodispersed cluster assembly with mean cluster size of d = 6 nm prepared at the O2 gas flow rate RO2 = 1 sccm. The inset shows hysteresis loops after field-cooling the sample in different magnetic fields. Reproduced with permission from [155], D. L. Peng et al., Phys. Rev. B 61, 3103 (2000). © 2000, The American Physical Society.

Figure 16. Coercivities, Hc and HcFC of the ZFC and FC samples at 5 K as a function of the exchange bias field Heb for the Co/CoO monodispersed cluster assemblies with d = 6 and 13 nm prepared at different RO2 . Reproduced with permission from [155], D. L. Peng et al., Phys. Rev. B 61, 3103 (2000). © 2000, The American Physical Society.

Transition Metal Nanocluster Assemblies

494 0.0020

(a)

3000

M (emu)

(Oe) FC

and Hc

FC 1500 RO2 = 3 sccm

0.0000 –0.0005

1st cycle

–0.0010

2nd cycle 3rd cycle

–0.0015

Hc

ZFC

(a)

T=5K

0.0005

ZFC

1000

RO2 = 1 SCCM

0.0010

2500 2000

d = 6 nm

0.0015

d = 9 nm

RO2 = 0 sccm

500

14th cycle

–0.0020

–40000

–20000

0

20000

40000

Magnetic field (Oe) 0 10400

(b) 96

Heb (Oe)

d = 9 nm

1000

Heb (Oe)

800 RO2 = 0 sccm 600

10000 92 9600

RO2 = 3 sccm

Training effect (%)

100

(b)

1200

88 9200

400

1

3

5

7

9

11

13

15

Training cycle numbers

200 0 0

50

100

150

200

250

300

Temperature, T (K) Figure 17. (a) Coercivities, HcZFC and HcFC , of the ZFC and FC samples, and (b) exchange bias field Heb as a function of temperature, T , for the oxide-coated Fe cluster assemblies with d = 9 nm prepared at RO2 = 0 and 3 sccm. Reproduced with permission from [156], D. L. Peng et al., J. Appl. Phys. 92, 3075 (2002). © 2002, American Institute of Physics.

rapidly decreases with increasing T and becomes negligibly small above T = 50 K.

Training Effect It is worth mentioning that the dependence of the exchange bias field Heb on repeated magnetization reversals, namely, the so-called training effect, is a diminution of Heb upon the subsequent magnetization reversals [369]. Figure 18(a) shows typical results of the hysteresis loops measured along the field-cooling direction at 5 K for the monodispersed Co/CoO cluster assembly with d = 6 nm and RO2 = 1 sccm [155]. The successive loops do not coincide with each other and show a decrease in Heb . Figure 18(b) shows the dependence of Heb and the training effect on the training cycle number at 5 K. Here, the training effect was defined as the fraction of the initial value. The decrease of Heb is larger for the second cycle and then become unchanged after further numbers of the training cycles. The training effect is about 89% after the 14th cycle. However, for the monodispersed Fe/Fe-oxide cluster assemblies, the decrease of Heb is very remarkable by increasing the training cycle number, and the training effect is decreased to about 30% after the 13th cycle [156]. This result is clearly different from that of the monodispersed Co/CoO cluster assembly [155]. This suggests that the training effect strongly depends on magnetic properties of the oxide shells, namely, antiferromagnetism or ferrimagnetism.

Figure 18. (a) Successive hystersis loops measured at 5 K along the easy axis after cooling from 300 K in a field of +20000 Oe along the same direction; (b) Heb and training effect as a function of the training cycle number for the monodispersed Co/CoO cluster assembly with d = 6 nm. Reproduced with permission from [155], D. L. Peng et al., Phys. Rev. B 61, 3103 (2000). © 2000, The American Physical Society.

Spin Disorder at Core-Shell Interface With regard to the origin of the enhanced coercivity or uniaxial anisotropy, disappearance of the UEA effect at lower temperatures, and large training effect, spin disorder states and/or spin glass are highly plausible at around the core-shell interface [153, 155, 156, 355, 369, 391, 397]. At first, Mössbauer spectroscopy studies on surface oxidized Fe nanoparticles revealed that the surface shell consisted of very small crystallites and that a large spin canting characterized the oxide phase [400–402]. Secondly, in ferrite nanoparticles, a surface spin disorder has been experimentally discussed [403, 404], being corroborated by numerical calculations. The antiferromagnetic superexchange interaction is disrupted at the surface of the ferrimagnetic oxide crystallites because of missing oxygen ions or the presence of other impurity molecules. Such broken exchange bonds between surface spins lead to surface spin disorder, being compatible with a spin-glass-like behavior at the surface. In addition, another evidence of the spin-glass-like state has also been given by the alternatingcurrent (ac) magnetic measurement for the Co/CoO cluster assembly with d = 6 nm and RO2 = 1 sccm [391]. For this sample, the sharp cusps (peaks at freezing temperature Tf ) in both ZFC and FC magnetization curves are observed. Since the frequency shift in Tf can offer a good criterion for distinguishing a spin-glass-like material from a superparamagnet, the dependence of the position of the cusp (at freezing Tf ) on the frequency of the ac field were measured. The experimental result [391] indicates that Tf depends on the frequency of the ac field and the peak is shifted to the

Transition Metal Nanocluster Assemblies

Macroscopic Quantum Tunneling
MQT  Effect Magnetic relaxation measurements were performed for the CoOcoated Co cluster assembly using the following procedure: first, the sample was cooled from 300 K to a lower temperature in low magnetic field, Ha = 100 Oe; the field was then reversed to Hb = −100 Oe and the variation of the magnetization with time was measured at this temperature. As shown in the inset of Figure 19, the magnetic relaxation follows logarithmic time-dependence (Eq. 20). There is no single exponential time dependence as expected for a collection of identical, noninteracting single-domain clusters aligned in the same direction by a field (i.e., the anisotropy energy barrier is universal throughout the system). This implies a wide distribution of the anisotropy energy, which is ascribed to polycrystalline CoO and a different interfacial state in spite of the narrow cluster size distribution. By least-square fitting of Eq. 20 to the results in the inset of Figure 19, the S value is estimated as a function of temperature and shown in Figure 19. The temperature variation of S at a high temperature range deviates from linearity. However, for 8 < T < 50 K, S varies linearly with T , extrapolating to zero when T = 0, as would be expected for the magnetic relaxation via thermal activation. This indicates that the interaction between the Co cores is smaller than the energy barrier height, probably because the dipole interaction between the Co cores is shielded partially by the AF CoO shell. The other remarkable feature is that the S values are independent of temperature at T ≤ 8 K. Such a nonthermal relaxation character below a few Kelvin has been observed in several nanostructured materials with the broad distribution of sizes or anisotropy energy barriers [359–363, 406], being ascribed to the MQT effect of magnetization. The MQT effect is observable at experimentally accessible temperatures only for materials with high uniaxial anisotropy. Indeed, for this CoO-coated Co cluster assembly, Hc = 5 kOe (see Fig. 16) and the uniaxial anisotropy constant, K ≈ Hc × Ms ≈ 7+2 × 106 erg/cm3 , which is larger than the

0.000108

60

5K 7K 10 K 16 K 18 K 23 K 30 K 35 K 40 K 50 K

M (emu)

0.000107

50

S [–1/M(t0)dM/d(lnt)] (×10–4)

low-temperature direction with decreasing the frequency of measurement. When the frequency varies between @ = 1000 and 1 Hz, Tf is reduced by about 10%: 8Tf /:Tf 8 log @; is about 0.03. These values are the same order as those of the spin glasses and smaller than the values of the superparamagnets [405]. Therefore, the hypothesis of spin disorder at the surface of the oxide crystallites or the interface of the core and oxide shell is applicable for the oxide-coated Fe, Co, and Ni cluster assemblies. According to this hypothesis, the onset of loop shift and bifurcation between HcZFC and HcFC below Td can be ascribed to a freezing of disorder surface spins of the oxide shell crystallites. The presence of such a disordered spin freezing state leads to not only the loop shift but also the large Hc , which is much larger than the intrinsic value of the FM core, because an ideal interface between the FM core and oxide shell should have no effect on the enhancement of Hc . The training effects also further support such a hypothesis of the spin-glass-like state in the interfacial layers between the FM core and oxide shell. The repeated magnetization reversal at high fields makes the interfacial spins change to a new frozen spin state and decreases the net interfacial uncompensated antiferromagnetic magnetization, causing a decrease of Heb and Hc .

495

40

0.000106 0.000105 0.000104

60 K

0.000103

70 K

0.000102

80 K

4

30

6

8

10

ln t (s)

20

10

0 0

20

40

60

80

100

Temperature (K) Figure 19. Magnetic viscosity, S, as a function of temperature for the Co/CoO, monodispersed cluster assembly with d = 6 nm prepared at RO2 = 1 sccm. The inset shows time-dependence of magnetization at different temperatures. Reproduced with permission from [153], D. L. Peng et al., Appl. Phys. Lett. 75, 3856 (1999). © 1999, American Institute of Physics.

bulk value (K = 4+5 × 106 erg/cm3 and 2+5 × 106 erg/cm3 for bulk hcp and fcc Co, respectively) [407]. Therefore, the high crossover temperature from a thermal activation regime to a quantum tunneling regime, Tco = 8 K, is ascribed to the enhanced uniaxial anisotropy due to exchange coupling between the FM Co core and AF CoO shell.

5.1.4. MicroSQUID Magnetometry Studies of Individual Magnetic Nanoclusters As described above, because of the limited sensitivity of conventional magnetic characterization techniques, most of the experimental studies on nanosized particles were carried out on large assemblies of particles, where distributions of particle sizes, shapes, and detects rendered the interpretation quite difficult. From a fundamental point of view, it is very necessary to study the magnetic properties of a single nanocluster. Recently, an ultrahigh sensitivity magnetometry technique based on microSQUID devices has been developed [408] to detect, for the first time, the magnetic signal of one single nanocluster with a size as low as 1000 atoms [409]. Such a challenge is achieved by embedding the Co clusters directly in the metal niobium constituting the superconducting microSQUID loop at low temperature (7 K) to considerably improve the cluster-microSQUID coupling. Thin 20-nm Nb-films with embedded Co-cluster (≈3 nm in diameter, fcc-truncated octahedron) were prepared using the low-energy cluster beam codeposition technique and subsequently an electron beam microlithographed to pattern a large number of microSQUID loops. The concentration of Co-clusters is low enough ( 0+3 sccm, as shown in the next section, the large magneto-resistance effect (spin-dependent, tunnel-type conductivity) was also observed. Further examination on the temperature-dependence of E T  for RO2 > 0+3 sccm shows a % ∝ exp −b/T  behavior [93], which differs from the well-known temperature dependence of % ∝ exp −b/T 1/2  behavior for disordered granular materials [422]. This is ascribed to the uniform cluster size and surface CoO layer (barrier) thickness in the monodispersed Co/CoO cluster assemblies [154]. From the linear part of the plot of log % versus 1/T , the Ec values of the Co core clusters are estimated to be Ec = 5+2, 6, and 5.8 meV for RO2 = 0+35, 0.4, and 1 sccm, respectively, which are in agreement with the calculated value of 5.4 meV using the expression [23]: Ec =

e2 /2$F0 Fd:s/ d/2 + s;, where F is the dielectric constant (12.9 for CoO), F0 = 8+854 × 10−12 F/m, d is the mean diameter of the Co cores (11 nm) and s is the separation between neighboring Co cores (2 nm).

15

4.2 K

10

5 300 K 0 0

0.5

1

Ro2 (SCCM) Figure 22. Variation of (a) resistivity, E, and (b) the absolute value of the magneto-resistance ratio, MR, at 300 and 4.2 K for the CoOcoated Co-cluster assemblies as a function of RO2 . Reproduced with permission from [154], D. L. Peng et al., Phys. Rev. B 60, 2093 (1999). © 1999, The American Physical Society.

different saturation behaviors in magnetization and MR. For core-shell Fe-Fe oxide systems, for instance, the magnetic field dependence of the MR values exhibits no saturation tendency in field up to H = 70 kOe even at low temperatures, disagreeing with the magnetization curves that are of a saturation tendency [427, 435]. This behavior has been interpreted considering spin-disorder surface layers of the oxide grains, interparticle magnetic correlations, and microscopic mechanisms similar to those responsible for the MR in other granular systems.

6. CONCLUDING REMARKS We have started from the incentive that assembling of monodispersed size transition metal clusters is one of the key technologies in fabricating nanostructure-controlled materials. After mentioning several cluster preparation methods, we have dealt with the plasma gas condensation method in detail: the experimental results and theoretical consideration of formation and growth of size-controlled metal clusters, alloy cluster formation, and nonequilibrium structures in clusters. Then, we have described incidence and deposition of clusters onto the substrate, experimental results and theoretical aspects of cluster-diffusion and -migration on the substrate surface, and codeposition to form cluster-embedded or granular materials. We have discussed how individual clusters first form discontinuous small aggregates and later

Transition Metal Nanocluster Assemblies

499

continuous networks in terms of the percolation concept, and a possibility of ordered assembly using patterned substrates. Finally, we have displayed the two physical properties of cluster assemblies, magnetic and electrical properties. In small ferromagnetic clusters, a single-domain structure is favorable due to the serious loss of domain wall energy. When the surface of ferromagnetic metal clusters is oxidized, magnetic coupling between the ferromagnetic core and antiferromagnetic or ferrimagnetic shell crystallites yields both exchange anisotropy and very large coercivity. The magnetic moment is fluctuated and relaxed thermally at high temperatures and quantum mechanically at low temperature. At low temperatures, moreover, electron localization aspects have been observed in slightly oxidized cluster assemblies, whereas tunneling-type-conductivity and magnetoresistance with Coulomb-Blockade type characters are observed in well-oxidized cluster assemblies. We wish to emphasize that the assembling of monodispersed-size clusters is a prerequisite in highlighting the characteristic properties and their temperature-dependence, that is, the clear crossover of

magnetic relaxation and conductivity between the high and low temperature regimes. All the results that are dealt with in this article are summarized in Table 1. This chapter demonstrates that we can prepare sizecontrolled clusters as building blocks and just stand on the entrance to fabricate cluster assembled materials. We are searching new academic and industrial fields to apply small clusters: medical science, drug delivery, catalysis, etc. We should more actively utilize the anomalous properties of much smaller clusters with one or two nanometers in diameter; for instance, atomic moments of Fe, Co, and Ni clusters are 1.5 or 2 times larger than those of their bulk metals [19]. In order to assemble such small clusters without losing their unique characteristics, we have to find stabilization and surface-coating methods other than surface oxidation [436]. Gas-phase syntheses, using vacuum-based systems, are capable in preparing impurity-free and isotropic shape clusters minimizing degradation of unique properties, whereas condensed-phase syntheses will inevitably suffer from inclusion of oxygen and other impurities from surrounding

Table 1. Summary of experimental results for transition metal cluster assemblies.

Elements

Deposition method

Size

Measurement

Ref.

6–13 nm 30 nm 9–20 nm 5–20 nm 2–5 nm 2–6 nm 0.6–1.4 nm 6–40 nm 7–16 nm

TEM Mössbauer spectroscopy Magnetism TEM and magnetism Magneto-resistance Magnetism Magnetism and magneto-resistance Magnetism TEM, SEM, magnetism, and magneto-resistance XMCD

[88, 89] [334, 335] [338] [395] [58, 59] [164] [60, 61] [346, 397] [156, 431]

[388–390] [408] [151] [91, 92, 165, 198]

Cr Fe

PGC GC GC GC SECB LV SECB GC PGC

Co

GC LV LV PGC

5–35 nm 2–5 nm 3 nm 6–15 nm

PGC PGC LV PGC

6–15 nm 9 nm 3 nm 4.5 nm

Magnetism Micro-SQUID, magnetism Giant magneto-resistance TEM, SEM, SAXS, magnetism, and electrical conductivity Magnetism and magneto-resistance TEM and electrical conductivity Micro-SQUID, magnetism TEM and magnetism

Co-Al Co-Pt

PGC PGC

13 nm 5–8.5 nm

TEM and EDX TEM and EDX

[116] [117]

Ni Cu Nb Nb-Ag

PGC PGC PGC PGC

2–4 nm 1 equiv) leads to fluorescence enhancement initially, but undergoes subsequent bimolecular quenching.

N

Red

N

Ox

N Ph HN

H N

Ph HN

N NH

S

S

B

A

17

H N

H N

H N

Red N Ox

S

S N

NH

B

A

18

In case of 19, both Zn(II) and a bulky carboxylate can modulate [127] the PET process. When it is complexed to a Zn(II) ion, the through-bond PET is blocked although the through-space PET is still operational (Scheme 1). As a result, the addition of one equivalent of Zn(II) ion results in a modest enhancement of fluorescence. Upon the further addition of an equivalent of triphenylacetate anion, the through-space PET is blocked as the phenyl groups of the acid position themselves in-between the donor and the acceptor (Scheme 1). This results in the further increase in the emission enhancement. HN

N

H 2N

N

H 2N H2N

Zn 2+

N

X

H 2O, O H -

eT N

eT N

N

N OH 2

H N

Ph 3CC OO

HN

O H N

Zn 2+ N H

Zn 2+ N H

HN

N N

Scheme 1. Pictorial representation of modulation of PET process in the system 19 by both Zn(II) and a bulky carboxylate group. Reprinted with permission from [127], I. Bruseghini et al., Chem. Commun. 1348 (2002). © 2002, Royal Society of Chemistry.

The intrinsic fluorescent ligand 20, exhibits [128, 129] a significant fluorescence enhancement upon the addition of a Zn(II) ion, thus serving as a specific signaling system for this metal ion. Other divalent metal ions quench fluorescence. The fluorophore, 5-chloro-8-methoxyquinoline has been connected to the receptor diaza-18-crown-6 to have the PET system, 21. As a chemosensor, it selectively binds and responds [130] to Cd(II). Besides, Cu(II), Hg(II), Tl(I), and Pb(II) also form stable complexes with the receptor but do not interfere with the determination of Cd(II), provided that concentrations of these metal ions are not much higher compared to that of Cd(II) in the medium. A large enhancement is also observed with Zn(II), Ca(II) and to a lesser extent with Sr(II) and Ba(II). Interestingly, 22, where the receptor is the same but the 8-hydroxyquinoline moiety is attached via C-7, is an effective fluorescence sensor [131] for the Hg(II) ion in the presence of other metal ions including Mg(II). However, if the two NO2 groups are replaced by two Cl groups, the resultant system becomes [132] a chemosensor for Mg(II). CN

N

N

N

R

20 Cl

Cl R=

N

O

O

N

HN

N

N

N

N HN

R

19

O

OC H 3

O

O

21

OC H3

Transition Metals-Based Nanomaterials for Signal Transduction NO 2

NO 2 O

O

N

N

N

N OH

OH

O

O

22

The emission intensity of the indolyl maleimide derivative 23 increases [133] almost 80-fold when it binds (1,4,7,11-tetraazacyclododecane)zinc(II) via supramolecular interactions. The emission enhancement is attributed to significant perturbation of the excited state of 23 upon coordination of the Zn(II) complex. There is a low-lying internal charge transfer (ICT) state with the indole being the donor and the maleimide, the acceptor. This ICT state is in thermal equilibrium with the
-
∗ excited state (Scheme 2). Upon coordination with the macrocycle-bound Zn(II) ion, the maleimide group deprotonates with concomitant weakening of the ICT interaction [134]. This new situation leads to greater quantum yield for the
-
∗ emission. The binding being reversible, it has the potential of acting as a switch. N

H N

O

O

H

+

H 2N NH

N

N

HN

NH

n n=1-5

N

Zn 2+

H

O

N

O

ClO 4

24

25

N

Zn

TE A

2+

Br N

N H

Such sensors that can act in aqueous medium are of practical importance. The same group reported [136] few anthrylazamacrocycles (25) which show enhancement by factors of 20–190 depending upon the system in an aqueous medium. The lipid 26 with an attached pyrene group [137–138] forms mixed vesicles at pH 7.5 with distearoylphosphatidylcholine. In this mixed vesicle, two or more of 26 can occupy adjacent positions. Upon excitation at 346 nm, the vesicles exhibit two distinct emission bands—one at 377 nm as weak, which is attributable to pyrene monomer emission, and the other at 477 nm due to excimer formation. When a Cu(II) salt is added to the vesicles, the metal is bonded to the iminodiacetate moiety and as a consequence, the excimer emission decrease with concomitant increase of the pyrene monomer emission. Thus, a Cu(II) ion does not quench fluorescence; it just induces a reorganization of the lipid molecules such that most of them are separated. The Cu(II)induced changes are fast and reversible and can be used for quantitative analysis of the metal ion present.

-

H N

NH

H OH 2

Br

HN

525

H 3O

+

N H

H

N H

2 ClO 4-

23

The PET system reported by Yoon et al. with partially immobilized polyamine as the receptor (24) act as chemosensors [135] for Hg(II) and Cu(II) ions in aqueous medium. E / eV

E / eV

3.0

ICT

3.0

ES

It has been argued [139] that the key factors in the design of fluorescent signaling systems are (1) rigidization of the constituent parts of the system to reduce metalfluorophore interaction and their preorientation leading to electron decoupling of the fluorophore and the receptor. Based on this design principle, Rurack and Resch-Genger have synthesized a number of fluoroionophores (27–29). In 27, a macrocyclic receptor is virtually electronically decoupled from the fluorophore unit. This compound exhibits [140] a high enhancement of fluorescence in the presence of Cu(II), besides heavy metal ions such as Hg(II) and Ag(I). Other first-row transition metal ions probed did not show any significant enhancement of fluorescence. In 28, when the macrocycle is azacrown ether, the electron transfer is

ES H 3C (H 2C) 17 O

ICT (CH 2) 9 O

2.0

O O

O

O

N

O

2.0

= O

hν /

hν 1.0

hν /

hν

hν

O

(26)

= Cu2+

1.0

0.0

0.0 (i)

(ii) Excimer fluorescence

Scheme 2. Schematic energy level diagrams of an indolyl maleimide derivative 23, (i) in polar solvents and (ii) at low temperature and upon coordination to a Lewis-acidic metal complex. Reprinted with permission from [133], B. K. Kaletas et al., Chem. Commun. 776 (2002). © 2002, Royal Society of Chemistry.

Monomer fluorescence

Scheme 3. Change in fluorescence properties of a phospholipid membrane containing 26 upon Cu(II) complexation. Reprinted with permission from [138], R. Kramer, Angew. Chem. Internat. Ed. Engl. 37, 772 (1998). © 1998, Wiley-VCH, Weinheim.

Transition Metals-Based Nanomaterials for Signal Transduction

526 suppressed [141, 142] and an increase in the fluorescence quantum yield is observed with known quenchers like Pb(II), as the ionophore does not allow the metal ion and the fluorophore to be in communication. With azathia crown, both Ag(I) and Hg(II) show significant fluorescence enhancement. The fluorophore and the receptor in 29 are not properly oriented and electronically decoupled. The receptor shows [140] selectivity for Zn(II) and exhibits fluorescence enhancement by a factor of ∼50.

HN

N N

O2S

SO2

N

pKa1= 4.8

N

NiII

N N

N

N

NiII N OH 2

S S

N F F

pKa2 = 6.7

N

B N

S

S

O2 S N

27 N

O X N

R

N

=

X X X=O,S

R

28

N

N

N

HN

COOH

O

OH

N

O

HO

N

X

N

N

NiII

N

Scheme 4. A schematic representation of detection of geometrical change around Ni(II) in 31 by signal transduction mechanism. Reprinted with permission from [144], L. Fabbrizzi et al., Inorg. Chem. 41, 4612 (2002). © 2002, American Chemical Society.

binding. This pH-controlled attachment/detachment of the pendant arm with concomitant change in the spin-state from high-spin to low-spin can be used in the signal transduction mechanism. In the high-spin state, this system exhibits a very low intensity of emission which is measurably increased in the low-spin state. Solid-state X-ray structural data on these and similar systems will be of utmost importance as these systems represent photonic devices whose emission can be mechanically controlled.

O

29

When (cyclam) is attached to a naphthyl group through a methylene spacer (30), in the format, “receptor-spacerfluorophore,” the emission intensity of the fluorophore with Ni(II) ion as input is found [143] to be dependant on the spin-state of metal ion. When the metal ion is high spin with axial ligation, paramagnetic quenching of the fluorescence is observed even though PET from the N atom to the fluorophore is blocked. However, a square-planar diamagnetic Ni(II) ion exhibits a distinctively higher quantum yield as it cannot effectively quench the fluorescence. This signal transduction mechanism of Ni(II) ion has been applied in a system containing a reinforced macrocycle [144] as the receptor (31), where a naphthylmethyl group and a sulfonamide group are attached. The reinforced macrocycle does not allow axial coordination to Ni(II) from one side. The sulfonamide moiety itself does not show any coordination behavior; it can, however, deprotonate at a relatively low pH value and can then bind the metal ion anchored in the macrocyclic cavity rendering it in a square-pyramidal, highspin state (Scheme 4). Of course, there might be a possibility of hexacoordination with a water molecule involved in

NH

HN

NH

N

30

SO 2 HN

N N

N

NiII N

31

Transition Metals-Based Nanomaterials for Signal Transduction

hν′

x

380

H2 O hν

O NH

e

hν

N H

N

NH O

H N

+ 2OH

H N

N Ni

- 2H

N

+

N

N

32

In a related molecule (33), the same authors have shown that Cu(II) ion can be translocated intramolecularly [147] by changing the pH of the medium. This translocation can be followed by fluorescence using coumarin-343 as an auxiliary fluorophore as shown in 33. A metal ion that enters the cavity of a cryptand is isolated from the surroundings. Therefore, fluorescent sensors in the format, “receptor-spacerfluorophore” built with cryptand as the receptor make the M-F communication negligible, while at the same time M-R interactions become high due to the cryptate effect [148]. Two typical cryptand-based molecular fluorescent signaling systems (34–35) are shown. 471 nm

430 nm N

430 n m N O O O

O NH NH O

O

HO

N O

H N

O

O -2 H

N N

N H

+

H N

N

N N

N

N

N H

O

N N

33 N

N O

O

O

O

O

O

O

O N

N N

34

N

N

O

530

580

Figure 6. Emission spectra of free 34 and it’s Zn(II) complex.

N

N H

H2 O

O

480 wavelength (nm)

II

O

x

430

-

O

-

NiII

Emission intensity (arbitary units)

Translocation of a metal ion from one compartment to another in a reversible and repetitive manner has been achieved [145, 146] in 32 which contains two distinct binding sites. Amide nitrogens being poor donors, a metal ion such as Ni(II) occupies the site away from the anthracene moiety at pH ∼ 7.5 and cannot quench anthracene fluorescence. When the pH is increased to ∼9.5, the amide nitrogens are deprotonated and the metal ion moves to the site closer to the fluorophore causing quenching of fluorescence. The translocation of the metal ion can be repeated many times making it a pH-controlled reversible system.

527

N

N N

35

Compound 34 shows [75, 149] a very efficient PET process from the lone pair on the N atom bound to the anthryl moiety. In the presence of a first-row transition metal ion as

input to 34 in dry THF, a strong fluorescence is observed (Fig. 6). The extent of enhancement depends on the nature of the metal ion as well as the receptor cryptand. The enhancement, however, is at least 100 times with respect to the metal free state. The four N donor atoms at the N4 end of the cavity can bind a metal ion and the strength of this binding is reflected in the emission quantum yield. A metal ion like Co(II), Cu(II), or Zn(II) with tetracoordination, the fluorescence enhancement is maximum due to strong donation of the nitrogen lone pairs. Metal ions like Mn(II), Ni(II), and Fe(III), which prefer higher coordination, also exhibit a high intensity of fluorescence. These metal ions can achieve higher coordination by binding solvent molecules. Zn(II) shows the highest enhancement among the metal ions studied due to its nonquenching nature. The receptor having a poor chemoselectivity, provides an example of an OR logic gate [150, 151] with transition metal ions as input. As the anthracene group is not in communication with a metal ion bound inside the cavity, the emission band does not shift by more than 5 nm in the metal complex with respect to the metal free state. The modular design of fluorescent signaling affords to vary the spacer, receptor, and the fluorophore as well. However, different fluorophores can lead to systems with more complicated emission characteristics. In 35, a carbonyl group is the spacer in place of methylene [152], which leads to changes in the interaction between the lone pair of nitrogen and the anthracene group. The emission spectra of carbonylsubstituted anthracenes has been found to be greatly influenced by the nature of the carbonyl substituent [153, 154]. Thus, although 9-substituted anthryl ketones are virtually nonfluorescent at room temperature in aprotic solvents, due to a suitably placed (n,
∗  triplet level that enhances intersystem crossing, the anthracene amide carbonyls have the (n,
∗ , too high in energy to affect S1 decay and are fluorescent in nature. Earlier studies revealed [153, 155, 156] that in the ground state (So  of 9-substituted carbonyl anthracenes like 9-anthramide and N,N-diethyl-9-anthramide, steric hindrance between the carbonyl group and the ring hydrogens keep the carbonyl group twisted almost 90 out of plane of the anthracene ring, which precludes extensive conjugation between the two. These compounds thus exhibit anthracene-like structured emission. However, the shape as

Transition Metals-Based Nanomaterials for Signal Transduction

528 well as position of the emission spectra of 9-anthramide has been found to be quite solvent dependant [153] due to increasing conjugation between the carbonyl group and the anthracene
system. In contrast, the spectra of N,N-diethyl9-anthramide are solvent independent [157] and show the same structured emission as unsubstituted anthracene due to much higher steric hindrance to rotation of the carbonyl group with respect to the anthracene moiety. The emission spectra of 35 is very similar compared to N,N-diethyl9-anthramide, indicating restricted rotation of the carbonyl group in the molecule. The restricted rotation has two significant effects: (1) 35 does not show any exciplex like 34, and (2) the PET process in 34 is less efficient compared to that in 34 due to unfavorable donor-acceptor orientation [158] in the former. Consequently, the quantum yield of emission does not increase more than 20 times upon addition of a metal ion as input. Compound 36 has [159] three pyrene groups attached to the cryptand receptor via -CH2 CO-linkages. The metal-free compound exhibits the locally excited (LE) emission of the pyrene moiety. The PET from the N atoms to the pyrene is inefficient due to unfavorable orientation of the spacer and the fluorophore. This molecule also shows a broad structureless band, which begins to build up while the LE emission decreases as the concentration of the compound increases in dry THF. This broad emission is assigned as intramolecular exciplex involving a nitrogen lone pair and the fluorophore, which is another efficient deactivation channel of the donor-acceptor pair. Due to an inefficient PET process in 36, the recovery of fluorescence (i.e., enhancement observed) seldom rises by a factor of 10. N O

O

O

O

O

N

O

N

N

N

36

The efficiency of the PET process in the metal-free 37–40 depends on the nature of the spacer [160]. These systems also provide an excellent opportunity to investigate the possibility of intramolecular exciplex and intramolecular excimer formation as a function of the nature of the spacer unit which allows varying degrees of distance of separation of donor and acceptor, besides flexibility of one with respect to the other. The quantum yield for the exciplex emission is greater for 38 compared to 37 as the nitrogen lone pair, and the naphthalene
orbitals are favorably oriented in 38. It has been shown that for naphthalene-(CH2 n amine molecules (n = 1–4), the exciplex emission intensity is maximum for n = 2 suggesting a most favorable “in-line” approach of the lone-pair electrons of the amine and the
orbitals of naphthalene. The quantum yield of fluorescence monomer emission increases significantly in the case

of 37 and 38 upon the addition of a transition metal ion as input. In both cases, the maximum enhancement was found with Zn(II) ion input (∼9 fold for 37 and ∼7 fold for 38) probably due to its nonquenching nature. When an electron withdrawing fluorophore is attached to such a cryptand (41), the PET is found [161] not to be very effective in the metalfree state and the fluorescent enhancement does not show more than 20–30 times in the presence of a first-row transition metal ion. This points to the ineffectiveness of an electron withdrawing fluorophore with cryptand receptors where once inside the cavity, the metal ion is isolated from the surroundings and the fluorescence quenching mechanism is not operative. If two receptors [162] are attached to a single fluorophore as in 42, then both the cavities must be occupied at the same time for fluorescence enhancement leading to an AND gate. Such AND gates are indeed known in the literature [77, 163–171]. With 42, an AND gate can be realized [172] with transition metal ions as inputs. However, in the concentration range for the fluorescence measurements, it will be almost impossible to establish the situation where only one of the cavities will be occupied. A better design for an AND gate will be one where two different types of receptors are connected to a fluorophoric unit.

Transition Metals-Based Nanomaterials for Signal Transduction

N

N O

O

O

O

HN

HN

N

O

O

N

NH

NH N

N

42

Dendrimers [173–177] with fluorophoric groups at the periphery have been constructed [178–185] for fluorescent signaling by transition metal ions. A fourth generation dendrimer (43) with 32 dansyl units at the periphery shows intense fluorescence in the metal-free state [186]. In presence of a transition metal ion such as Co(II), this fluorescence is quenched signaling the presence of the metal ion. R

R

HN

NH

R R R

NH NH

N

N

R

N

HN

R

N

N

N

R

N

N

R

NH

N

N

N

N

N N

N

R

N

N

R HN

N

N

R

HN NH

NH NH

NH

NH

R

R

R R

R

HN

HN R

R

N

HN NH

NH

N

N

NH R R

R

NH

N

HN

R

R R

N R=

O2 S

43

6. NONLINEAR OPTICAL EFFECTS It must be mentioned at the outset that it is not the intention here to discuss molecules with nonlinear optical activity. Rather the discussion is restricted to the modulation of NLO behavior of organic molecules by metal ions. However, it is imperative to present in a rudimentary fashion, the phenomena of nonlinear optical activity for the interested new readers on the subject. A number of excellent books and articles are available on this subject [186–192]. When a molecule is subjected to an oscillating external electric field (light), the induced change in molecular dipole moment (polarization) can be expressed by a power series in the field strength Ej as in Eq. (1) where pi is the electronic polarization induced along the ith molecular axis, the linear polarizability, the quadratic hyperpolarizability,  the cubic hyperpolarizability, and so on.


pi = ij Ej +

ijk Ej Ek + ijkl Ej Ek El + · · · · (1) j

j≤k

j≤k≤l

(2)

2

N

HN

R

N

N

NH

N

N NH

3

IJKL EJ EK EL + · · · ·

(3)

IJK = Ns F ijk TIJK · · · ·

NH

N

N




J ≤K≤L

NH

NH NH

J ≤K

+

R

NH N

R

For small fields, the quadratic and cubic terms in Eq. (1) can be neglected, so that the induced polarization is proportional to the strength of the applied field, which is the linear optical behavior. However, when a molecule is subjected to an intense electric field such as that of an intense laser light, the second and third terms in Eq. (1) become important and nonlinear optical behavior is observed. Thus, nonlinear optical effects deal with the interaction of applied electromagnetic fields in various materials to generate new electromagnetic fields altered in frequency, phase, or other physical properties. The macroscopic polarization for an array of molecules is given by Eq. (2), where the  n values are the macroscopic susceptibilities. The macroscopic susceptibilities are related to the corresponding molecular terms    by local field factors F , molecular packing density Ns , and appropriate coordinate transformation, TIJK .
1
2 IJ EJ + IJK EJ EK PI = J

R

NH

NH

R R

R

529

The discussions are restricted to quadratic NLO effects which lead to frequency doubling (SHG), frequency mixing, and the electrooptic Pockels effects that are of immense technological importance [193, 194]. At the molecular level, compounds likely to exhibit large values of molecular hyperpolarizability , must have 1. excited states close in energy to the ground state for easy access by visible/infrared light, 2. a large oscillator strength for the transition, 3. a large difference between the ground- and excitedstate dipole moments. Although in a molecule is closely related to bulk nonlinearity  2 in the solid state, large values of do not mean that the molecule when crystallized, will show a high value of  2 . For this, the molecule must crystallize in a noncentrosymmetric space group. It has been a belief for a long time that extended
systems with a considerable molecular dipole character are most promising candidates as second-order NLO materials [192]. Two such dipolar molecules p-NA [196–198] and DANS [199] are illustrated in Figure 7. In the dipolar approach, is associated with only one HOMO-LUMO electronic transition O2N

O 2N

NH2

NMe2

p-NA

DANS

NMe2

NO2 H2N

+C

X-

Me2N

NH2 NO2

O2N NH2 NMe2

Crystal Violet

TATB

Figure 7. Examples of dipolar and 2D molecules with NLO property.

530 of charge-transfer (CT) character [200]. These molecules lack significant off-diagonal components; hence, is termed one-dimensional (1D). A major problem [201] of traditional dipolar chromophores is the nonlinearity/transparency trade-off, whereby the desirable increase in second-order polarizability is accompanied by a bathochromic shift [202] of the electronic transition, leading to reabsorption of the second harmonic light, making them ineffective in frequency doubling applications. Moreover, these molecules are difficult to crystallize in noncentrosymmetric space groups. Strategies have been developed during the past decade to circumvent these drawbacks by extending the CT dimension from one to two or even to three [197]. These two- and three-dimensional (2D and 3D) chromophores [203–206] with C3  D3 , or T symmetry, have several advantages: 1. they are more transparent as the lack of a permanent dipole moment results in negligible solvatochromism, 2. enhanced nonlinearity due to coupling of the excited states at no cost of transparency, 3. greater probability of crystallization in a noncentrosymmetric space group due to very low dipole moment. Based on this principle, several molecules have been designed. Noteworthy among them are 1,3,5-triamino2,4,6-trinitrobenzene (TATB) [207], crystal violet [208–211], tris(4-methoxyphenyl) cyclopropenylium bromide [208], and triazene derivatives [212, 213], 1,3,5-tricyanobenzene derivatives [214], D-
-A functionalized cryptands [215, 216], etc. As NLO materials, organic molecules have certain advantages. They are potentially more versatile due to large values, highly resistant to optical damage, and above all possibilities of designing molecules, highly suitable for SHG applications. In recent years, a number of organic solidstate structures with SHG activity have been synthesized based on crystal engineering principles [217–221]. However, incorporation of metal ions into organic compounds adds a new dimension of study and also introduces many intriguing possibilities [222–229] as NLO chromophores. When an organic molecule showing quadratic nonlinearity can act as a ligand to bind a metal ion, the metal-ligand ensemble can exhibit nonlinear effects quite different from the original molecule. A metal ion plays the role of a transducer by accepting electron density from the ligand moiety, thereby altering polarizability of the latter. The great structural and electronic diversity available in a metal-organic ensemble is likely to introduce into chromophores design strategies, a number of tunable parameters not available with only organic structures. Again, the coordination tendencies of metal ions can be taken advantage of in designing metal-assembled giant structures with NLO activity. With the tremendous advancement in the NLO field, it is anticipated that materials which can modulate or tune the NLO property will find various novel applications. A schematic representation of three different strategies [230] applied for switching of NLO responses in dipolar D-
-A derivatives are given in Scheme 5. The presence of redox-active metal centers provides extensive opportunities for modulation of NLO responses. Coe et al. have shown [231] that dipolar ruthenium(II) ammine complexes of 4,4′ -bipyridinium ligands (44) can

Transition Metals-Based Nanomaterials for Signal Transduction Type I D

π

A

-e D

+e -

+

π

A = A1

π

A = D

π

A

π

D1 A

Type II D

π

A

+e-e -

D

-

Type III D D

π

structural change A

A

A = D

D

Scheme 5. A schematic representation of different strategies for modulating NLO responses in D-
-A molecules. Reprinted with permission from [230], B. Coe, Chem. Eur. J. 5, 2464 (1999). © 1999, Wiley-VCH.

exhibit tunable values, originating from the intense, low-energy MLCT excitations. The d6 Ru(II) behaves as a powerful
-donor and can be readily oxidized to the electron-deficient Ru(III) state, electrochemically or chemically. Visible absorption and HRS signals of this complex featuring [RuII (NH3 5 ]2+ as donor is attenuated upon oxidation with a solution of 30% aqueous H2 O2 and 2 M HCl taken in a 1:1 molar ratio. The complex reverts back to the original Ru(II) state upon reduction with 62% aqueous N2 H4 solution. The difference in between 44 and 44ox is found to be 10-20 fold. Thus, reversible switching of the molecular NLO responses can be achieved through the redox reaction of a donor (D) group. MLCT

donor

H3N

3+

NH3

H3N

2+

Ru

+ N R

N

21

NH3

H3N

acceptor

+e - -e 4+ NH3

H3N

acceptor

3+

H3N H3N

Ru

+ N R

N NH3

acceptor

21ox R = M e , Ph

44

Compounds functionalized with ferrocene units as the electron donor with a conjugated side-arm linked to an acceptor unit have been extensively used for studying SHG in the last decade. Malaun et al. have recently reported [232] an octamethylferrocene donor unit and a nitrothiophene acceptor, linked by an ethylene bridge (45). The octamethylferrocene unit reduces at a lower potential compared to ferrocene. One-electron oxidation of 45 upon treatment with 1 equiv. of ferrocenium hexafluorophosphate in MeCN affords [45]+ (PF6 . The values of 45 and [45]+ determined in CH2 Cl2 at 1064 nm are 316 ± 32 × 10−30 and 25 ± 5 × 10−30 esu, respectively. Thus, compound 45 has SHG efficiency which is about one order of magnitude greater compared to that of [45]+ . The redox interconversion of this

Transition Metals-Based Nanomaterials for Signal Transduction

531

compound thus provides a pathway for modulation of the value of the complex.

I 2ω M

I2 ω M2+

π

A

Fe

A

R

D

π

n

S

O2N

Compound 46 and 47 were found to exhibit [233, 234] very high powder SHG responses: about 220 and 62 times that of area, respectively.

N—CH3

Fe

Fe

NO2 46

47

Houbrechts et al. have shown [235] that D-
-A functionalized chromophores undergo changes in the intramolecular charge-transfer (ICT) properties upon metal ion binding, which is further responsible for the modulation of its secondorder NLO efficiencies. The functionalized crown ether (48) show a value of 38 × 10−30 esu, while its 1:1 metal complexes with Na+ , K+ , and Ba2+ show of 32, 27, and 24 × 10−30 esu, respectively. The largest reduction is obtained for Ba2+ confirming that the bivalent ion is more efficient in attenuating the electron-donor character of the ring nitrogen atoms. The NLO active 1,3-cone functionalized calixarene, where a significant bathochromic shift of 40 nm is also reported [236]. The cation binding improves the ICT which is reflected in the HRS signal intensity enhancement, although a correct determination of could not be possible in the absence of a well-defined isobestic point in the absorption spectra. H3C

O2N

n

M1 ( n = 1) ; B1 ( n = 2 ) ; L1 ( n = 3 )

Iω

45

O

N

O

O

O

Iω

Scheme 6. A sketch for switching of NLO responses in D-
-A functionalized cryptand molecules. R, receptor; D, donor;
, aromatic ring; A, acceptor; n, number of D-
-A units; M2+ , metal ions; I, fundamental light; I2 , second harmonic light; I2  M, modulated second harmonic light. Reprinted with permission from [237], P. Mukhopadhyay et al., J. Mater. Chem. 12, 2786 (2002). © 2002, Royal Society of Chemistry.

as Mg(II) and Ca(II) do not show any change in absorption spectra, since they occupy the upper part of the cryptand core and are thus unable to communicate with the D-
-A chromophore units. Addition of a metal ion such as NiII , CuII , ZnII , and CdII induces an anti-auxochromic shifts in the absorption spectra of the functionalized cryptand molecules, as these metal ions are in direct communication with the D-
-A chromophore units. N O

N O 2N

O

O

N

N

N

NO 2

O

O

O

N

NH

NO 2

N O 2N

HN

N

O 2N

O 2N

NO2

NO 2

49

50

On the gradual addition of metal ion, the absorbance spectra show a well-defined isobestic point, suggesting a 1:1 stoichiometry with the
-A functionalized cryptand. The

CH3 N

M2+

D

1-5

NO2

48

Cryptands having a three-fold symmetry axis [237] passing through the two bridgeheads can act as perfect skeletons for designing D-
-A chromophores (49). Besides, rigidity of the cryptand cavity can be varied by incorporating different groups in the three bridges. Some of the cryptand molecules can be selectively functionalized with different
-A units (50), giving rise to molecules where metal ions can occupy the cavity and bond the N atom attached to the
-A unit, thereby modulating the electrical polarizability of the system (Scheme 6). As the N atom donates its electron density to the metal ion in the cavity, its donating power to the
-A moiety diminishes (Fig. 8). The alkali metal ions such as Li(I), Na(I), and K(I) or the alkaline earth metal ions such

Abs

1-0

3

0-5

0 300

400

500

Wavelength (nm)

Figure 8. Effect of Cu(II) addition on the UV spectra of B1 : (-) metal free (5 × 10−5 M); (– ·· –) Cu(II) (25 × 10−5 M); (- -) Cu(II) (5 × 10−5 M); (– - –) Cu(II) (16 × 10−4 M); (– · –) Cu(II) (5 × 10−4 M); (– · · · –) Cu(II) (8 × 10−4 M); (- - -) Cu(II) (16 × 10−3 M); (— · —) Cu(II) (5 × 10−3 M). Reprinted with permission from [237]. P. Mukhopadhyay et al., J. Mater. Chem. 12, 2786 (2002). © 2002, Royal Society of Chemistry.

Transition Metals-Based Nanomaterials for Signal Transduction

532 mono
-A functionalized M1 exhibits a maximum blue shift in the presence of CuII ions (21 nm) with respect to the metal ion-free cryptand. Similarly, the bis
-A functionalized B1 shows a maximum blue shift in the presence of CuII ions (30 nm). These results indicate that ICT from the amine (donor) to the nitro (acceptor) within the conjugate unit is hindered to the maximum extent as the nitrogen lone pair interacts with the CuII ion. As expected, the value decreases with the increase of metal ion concentration, reaches a minimum corresponding to 1:1 stoichiometry, and then remains unaltered (Fig. 9). Maximum modulation of is achieved with Cu(II) ion. The trisdinitro-substituted cryptand shows no change in the absorption spectra with any of the metal ions as it does not bind a metal ion inside the cavity due to steric as well as electronic reasons. It has been known from our earlier studies [238] on the NiII complex with the unfunctionalized cryptand Lo that the metal ion can be removed from the cavity by adding NaCN as the more stable [Ni(CN)4 ]2− complex. So the addition of NaCN to either of the NiII complexed
-A functionalized cryptands results in the restoration of the position and the absorbance of the metal-free cryptands. Zinc(II) ion has been used [239] as a template to assemble 2,2′ -bipyridine based NLO-phores around it for having dipolar and multipolar NLO-active systems (51–53). The free ligands exhibit intense intraligand charge transfer in the visible region which undergoes a bathochromic shift in the complex due to coordination of the N atoms of the bipyridine moiety to Zn(II) ion. The bathochromic shift depends on the number of coordinated ligands and decreases substantially as the ligand:metal ratio increases. This result is significant and shows that a metal ion can substantially modulate the polarizability of NLO chromophores. This strategy of gathering more than one 1-D unit around a metal ion leading to multipolar systems has been used to fabricate a number of NLO chromophores [240–249]. The increase in the quadratic nonlinear optical responses is due to [250–252] the red shift of the intraligand charge transfer transition upon metal coordination. A metal ion, depending on its electronic configuration, can impose a square 70

HRS Signal

60

50

40

30

20 0

5

1

2

3

4

5

Equivalents of metal ion added

Figure 9. Plot of HRS signal (I2 /I2 2 ) of M1 with different metal ion inputs; M1 + NiII • M1 + ZnII 
 M1 + CdII  M1 + CuII . Reprinted with permission from [237], P. Mukhopadhyay et al., J. Mater. Chem. 12, 2786 (2002). © 2002, Royal Society of Chemistry.

planar geometry and if bound to
-delocalized ligands such as bipyridine or terpyridine, the increase in the nonlinear effects can be substantial [253–255]. Bu2N Bu2N N N

NBu 2

Cl N

Zn

Zn

N

Cl

N N

Bu2N

Bu2N

NBu 2

51

52

Bu 2N

NBu 2

N N N Zn N N N

Bu 2N

Bu 2N

NBu 2

NBu 2

53

7. CONCLUSION A large volume of work has been done on molecular fluorescent probes especially with alkali and alkaline earth metal ions. It is conceivable that new designer systems can be synthesized for fluorescent signaling with transition metal ions circumventing the quenching nature of these metal ions. Supramolecular chemistry should have a very strong influence in such endeavors. Organometallic systems acting as subunits, by virtue of the large number of available coordination sites about the metal center, can be utilized for directing self-assembly processes to design nanosized structures. In particular, 3D metal ions would appear as versatile candidates because of their capability of imposing many different stereochemical arrangements, depending upon their electronic configuration. Easy change of the redox states of these metal ions will add another dimension in designing. The emission characteristics of a fluorophoric system can be modulated by these metal ions for use in sensing and molecular-scale arithmetic. A combination of optical fiber technology and sensory circuits will make the remote detection of toxic ions possible. There are several problems remaining in this area, waiting to be tackled by the enthusiastic chemists. The foremost being systems that can show reversibility of the emission signal ideally without any fatigue. Besides, the emission signal must be recognized by another system separated in space with the help of a “molecular wire.” A completely organic molecule may not be ideal for this purpose. Introduction of a metal ion is likely to improve its characteristics. For this, either the metal ion should translocate within the molecule, or change in its oxidation state be felt by the fluorophore for the molecular system to act as a switch. While such translocation is

Transition Metals-Based Nanomaterials for Signal Transduction

reported to occur chemically, that is, by changing pH or the addition of a complexing anion, completely photochemically reversible systems should be the goal. Such systems will not only be fast in response but the fatigue due to chemical intervention will be absent. Incorporation of transition metal ions into NLO active organic compounds will introduce many new tunable variables as the metal ions can perturb the polarizability characteristics of the organic part. With the advancements made in the technology using nonlinear optical effects, design of NLO chromophores with switchability will enhance their scopes to a great extent.

GLOSSARY Cryptand A macrobicyclic ligand whose preorganized synthesis through ligand topology, binding ability, and structural rigidity determines the stability, selectivity, and properties of their complexes with metal ions. Emission quantum yield The ratio of the number of photon emitted to that of the absorbed. It approaches unity if the radiationless rate of deactivation is much less than the rate of radiative decay. Molecular fluorescence The emission of the excited molecule from a higher excited state to the low-lying ground state after the excitation. The emission occurs at a higher wavelength than that of the absorption. Nanomaterials The nanosized materials that undergo change in physical, chemical, and mechanical behavior in comparison to bulk materials. Nonlinear optics (NLO) The phenomenon of optical behavior of materials with the interaction of laser light. When certain molecules are introduced to an applied electromagnetic field, the induced change in the molecular polarizability can be expressed as a nonlinear function of the applied electromagnetic field. Photoinduced electron transfer (PET) An intercomponent process which is operative between the receptor and the signaling unit. When the signaling unit is a fluorescent fragment, the perturbation of PET modulates the photophysical behavior of the fluorophore and signals the recognition.

ACKNOWLEDGMENT Partial financial support for this work from the Council of Scientific and Industrial Research, New Delhi, is gratefully acknowledged.

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Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Transport in Self-Assembled Quantum Dots S. W. Hwang Korea University, Seoul, Korea; University of Seoul, Seoul, Korea

Y. J. Park Semiconductor Materials Laboratory, Nanodevice Research Center, KIST, Seoul, Korea

J. P. Bird Arizona State University, Tempe, Arizona, USA

D. Ahn University of Seoul, Seoul, Korea

CONTENTS 1. Introduction 2. Interaction Between Two-Dimensional Electron Gas and Self-Assembled Quantum Dots (SAQDs) 3. Tunneling Through SAQDs 4. Transport Through SAQDs Using STM 5. Transport Through SAQDs Using Nanofabrication 6. Selective Growth of SAQDs 7. C-V Measurements 8. Conclusions Glossary References

1. INTRODUCTION Transport through low-dimensional semiconductors has been an important and interesting topic since the advent of two-dimensional electron systems (2DES), such as high quality metal-oxide-semiconductor (MOS) [1] and highelectron-mobility-transistor (HEMT) structures [2]. The

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density of states (DOS) of low-dimensional semiconductors is drastically different from that of three-dimensional ones and the electron-electron interaction frequently plays a pronounced role in electron transport. There have been important benchmarks in the transport study of such lowdimensional systems, including the observation of the quantum Hall effects [3] in 2DES, conductance quantization [4] in one-dimensional electron systems (1DES), single electron tunneling [5] in closed zero-dimensional electron systems (0DES), and finally, quantum interference effects [6] in open 0DES. Even though there has been a lot of effort aimed at developing elaborate methods for realizing low-dimensional systems [7], most 1DES and 0DES for electron transport studies are still fabricated by lithography of MOS and HEMT structures [8]. Figure 1 shows two examples where lowdimensional systems are realized by lithography and pattern transfer. Figure 1a demonstrates the definition of a quantum dot by the so-called split-gate technique [9]. Here, electron beam (EB) lithography and lift-off of metal gates is performed on a HEMT wafer. Negative biases on the gates can deplete the 2DES underneath and form a small, dotshaped region connected to the source and the drain reservoir. Figure 1b shows a silicon-on-insulator wire fabricated by etching the top silicon followed by EB lithography. Further definition of two poly-silicon gates that wrap across the wire, and the application of negative biases to them, can

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (537–545)

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(a) Split gates

(a)

AlGaAs/GaAs

100nm

(b)

SAQDs

Shaped 2EDS GaAs Source

(b) Quantum dot

Si wire Barrier gates Drain

SiO2 Si

Figure 1. (a) Split-gate-definition of a quantum dot. Negative biases on electron-beam defined metal gates on a HEMT wafer can deplete the 2DES underneath, and form a small dot-shaped region. (b) Poly-silicon wrap gate definition of a quantum dot. Two poly-silicon gates wrapping around the etched silicon wire can define a quantum dot tunnel-coupled to the source and the drain reservoir.

define a quantum dot connected to the source and the drain reservoir. Among many interesting approaches to the realization of low-dimensional systems without any ex-situ lithography process, self-assembly inside a growth chamber has achieved much success. The self-assembly is different from regrowth techniques in that low-dimensional structures can be grown in-situ while the regrowth technique still requires ex-situ lithography, except in a few very clever cases [10]. Selfassembled quantum dots (SAQDs) are in-situ, automatically grown, quantum dots that are formed on semiconductor substrates and thus represent a self-assembly of 0DES [11]. High-quality, defect-free, 0-D electronic states are expected to exist in SAQDs, since the formation of the dots is achieved in clean growth chambers at high temperatures (T ). Furthermore, layers of semiconductors can be grown on top of SAQDs for passivation (these are usually called cap layers). We can expect that the quality of SAQDs are much better than that of regrown systems, or other chemically synthesized 1-D and 0-D semiconductor systems without cap layers [12]. The in-situ growth mechanism of SAQDs is strain relaxation of InAs against GaAs surface (the so-called Stranski– Krastanov mode [13]). The basic idea of this approach is that the strain between a thin wetting layer of InAs and the surface of a GaAs substrate can be released by forming small dots of InAs. Figure 2a shows a typical top view of GaAs substrate after the growth of InAs SAQDs. Randomly

Figure 2. (a) A scanning electron micrograph (SEM) of a GaAs surface after the growth of InAs SAQDs; (b) a cross-sectional TEM of SAQDs. The magnified image clearly shows the formation of a crystalline dot.

distributed SAQDs with an approximate diameter of 20 nm are seen. Figure 2b shows a transmission electron micrograph (TEM) of the SAQDs. The magnified TEM image shows clear crystallization of the dot. The typical size of SAQDs ranges from 5 to 50 nm depending on the growth condition and the condition of the substrate, and the typical height is less than 20 nm. Usually, there is a small distribution in the size and height of the SAQDs, even when they are grown on the same substrate. The size of lithographically fabricated dots is limited by the depletion region, and practically, it is difficult to achieve such a small size. Therefore, the Coulomb charging energy and the quantum energy of SAQDs are expected to be larger than those of lithographically fabricated dots. To date, much effort has been focused on the optical characterization of SAQDs [14], and there are many possible optical applications [15]. Optical probing is much simpler in terms of device preparation and can be used to study most of the quantized energy states. However, optical probing differs from transport investigations in that it mostly utilizes interband transitions, while the transport through SAQDs involves intraband processes (i.e., states in the conduction band). Furthermore, transport studies are of direct interest for possible electron device applications of SAQDs. This chapter reviews recent progress in the study of electronic transport in various semiconductor devices incorporating SAQDs. There are many different ways of

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incorporating SAQDs into semiconductor devices, and we group them here into several categories. This categorization is mainly for the convenience of discussion. In Section 2, the discussion will focus on transport studies of various 2DES with neighboring SAQDs. Such transport is expected to reveal the interaction between the 2DES and the layer of SAQDs. Tunneling through SAQDs is an important method for studying the electronic states of SAQDs. Several experimental studies, using tunneling structures, will be reviewed in Section 3. Scanning tunneling microscopy (STM) has been mostly used to observe the shapes of SAQDs. On the other hand, it is possible to utilize STM to drive a tunnel current between the tips and SAQDs. There have been a few experimental investigations of the STM current through SAQDs, and in Section 4, these will be introduced. Since SAQDs are automatically formed during epitaxial growth, as shown in Figure 2a, there is no selectivity in their position. For this reason, most of the transport studies that will be discussed in the following sections have been undertaken on devices with a large number of SAQDs. However, present nano-processing technologies have made it possible to fabricate devices with only a few, or single SAQDs. Furthermore, there has been a lot of effort at controlling the position of SAQDs in the growth stage. Transport studies of SAQDs embedded in nano-devices will be discussed in Section 5, and selective growth of SAQDs will be introduced in Section 6. Capacitance-voltage (C-V ) measurements are a useful tool for determining energy states in these structures since both charging and discharging of SAQDs affects the capacitance. In Section 7, several C-V measurement studies will be discussed. In Section 8, conclusions will be given as well as a discussion of new issues such as transport through coupled SAQDs and its relation to possible realization of quantum computing.

2. INTERACTION BETWEEN TWO-DIMENSIONAL ELECTRON GAS AND SELF-ASSEMBLED QUANTUM DOTS (SAQDs) Figure 3 shows a schematic of the device that is used to study the effect of SAQDs on a 2DES. The basic structure is a HEMT device that also features a layer of SAQDs inserted in the middle of the wafer. The growth of the SAQDs can be performed right after the growth of the GaAs buffer, so that the SAQDs reside at the GaAs/AlGaAs hetero-interface, or it can alternatively be inserted in AlGaAs insulator layer. In the first case, the SAQDs will act as direct scattering centers for the 2DES. In the latter case, the SAQDs can act as remote Coulomb scatterers, which smoothly modulate the potential for the electrons. The device for transport measurements can have the form of a transistor as shown in Figure 3, or it may take the form of a Hall bar. Sakaki et al. studied the effects of InAs SAQDs on the electron transport in a 2DES [16], utilizing a so-called inverted HEMT structure in which the AlGaAs layer was beneath the GaAs layer and the InAs SAQDs were grown in the middle of the upper GaAs layer. They found that the low-temperature electron mobility decreased, while the

Gate

GaAs cap

AlGaAs

SAQDs

Ohmic contacts

GaAs

Figure 3. A schematic HEMT FET with a layer of InAs SAQDs at the hetero-interface.

electron concentration increased with a decrease of the distance between the hetero-interface and the layer of InAs SAQDs. This observation was explained by the potential modulation induced by the presence of the SAQDs. Phillips et al. fabricated an FET with InAs SAQDs adjacent to the AlGaAs/InGaAs/GaAs pseudomorphic quantum well channel (the SAQDs were embedded in the AlGaAs layer) [17]. Distinct steps and a negative differential resistance (NDR) were observed in the current-voltage (I-V ) characteristics of this structure. Such features could be a manifestation of the resonant transport through quantum energy states, and these authors attributed the existence of these discrete quantum states to the bound states in SAQDs. Kim et al. measured the mobility of several GaAs/AlGaAs quantum wells in which InAs SAQDs were inserted [18], and found that this mobility was a function of the dot density. This observation could be explained using a similar argument as in [16]. A 2DES with an InAs SAQD layer at low T provides us with a good system in which we can study the interplay between disorder and quantum interference. Ribeiro et al. reported a metal-insulator transition at zero magnetic field (B) in such a 2DES [19]. A layer of InAs quantum dots were grown near the 2DES and they studied the T dependence of the diagonal resistance (Rxx ) at various electron densities. They found a critical density, below which Rxx increases with decreasing T (in the insulating regime), while above which Rxx decreases with decreasing T (in the metallic regime). Recently, Kim et al. studied high B transport of several HEMT wafers with various different growths of InAs SAQDs at hetero-interfaces [20]. They systematically studied the T dependence of Rxx at high B. The transition is again from an insulating behavior to a metallic behavior and they attributed the transition to the short-ranged Coulomb potential originating from the InAs dots.

3. TUNNELING THROUGH SAQDs Tunneling is the most direct and efficient method to probe quantum energy states in SAQDs. Figure 4 shows a typical tunnel transistor with SAQDs. This particular design has n+ GaAs contact layers, graded GaAs layers, and AlAs barriers on both sides of the GaAs well with InAs SAQDs. In other types of design, however, the SAQDs can be inserted in the AlAs barriers. The whole structure is then etched into a pillar to form a tunnel diode. When the diameter of this pillar is several hundred m, the number of SAQDs participating in the vertical transport is large and the measured current will exhibit the averaged properties of the dots. When the

540

Transport in Self-Assembled Quantum Dots

Contact + n GaAs Graded GaAs AlAs GaAs SAQDs GaAs AlAs Graded GaAs

Gate

n+ GaAs

Contact

Figure 4. An example of tunnel transistor incorporating InAs SAQDs. This particular example shows a AlAs/GaAs/AlAs double-barrier structure with a SAQD layer grown in the middle of the GaAs quantum well. The metal gate deposited at the sidewall can further deplete the diameter of the active region.

diameter is sub-m or several hundred nanometers, however, only a few dots participate in the transport and we usually are able to observe the characteristics of individual dots. In this case, a Schottky gate deposited along the periphery of the cylinder can be used to further squeeze the size of the diode. Figure 5 shows a typical differential conductance-voltage (dI/dV-V ) characteristic measured from the device with a design similar to Figure 4. The data was taken at T = 20 mK and at B = 18 T and it shows a clear conductance peak. Such a conductance peak is the result of tunneling through a discrete quantum state in SAQDs. The size of the diode was several hundred m in this case, and therefore, this characteristic is an averaged behavior of numerous numbers of SAQDs. Itskevich et al. reported the measurement of the current through a single-barrier GaAs/AlAs/GaAs heterostructure diode at low T [21]. The diode had InAs dots embedded in the AlAs layer. Diodes with various diameters ranging from 30 m to 400 m were fabricated and characterized. They observed current peaks and interpreted them as being due to the resonant tunneling through the quantum states of individual quantum dots. The measured current showed oscillations as functions of B and it originated from the DOS of the 2DES in the emitter accumulation layer. In spite of their interpretation, the sample incorporates a large number

dI/dV (S)

0.4

T = 20 mK, B = 8 T Single QD

0.3 0.2 0.1 0.0 0.00

0.05

0.10

0.15

0.20

V (V) Figure 5. A typical dI/dV-V characteristic measured from the device shown in Figure 4. The data was taken at T = 20 mK and B = 18 T. The conductance peak is a result of magneto-tunneling through a quantum state of the dot.

of dots (the density of the dots is 2 × 1011 cm−2  and it is rather difficult to think that each current peak originated from an individual SAQD. A similar structure was studied by Narihiro et al. [22] and a resonant peak in the I-V characteristic was observed. The data was interpreted again as being due to resonant tunneling through SAQDs. The magneto-tunneling was also studied and a systematic change of the current shape as a function of B was observed. This magneto-tunneling data was explained by the shift of the Fermi surface of the emitter. A tunneling structure with SAQDs embedded in the AlAs barrier was used for the careful study of magnetotunneling through SAQDs. The tunneling current split with the increase of the magnetic field [23]. The splitting with the increase of the magnetic field made it possible to estimate the g-factor of InAs QDs. The observed g-factor ranged from 0.52 to 1.6. This contrasts with the value of bulk InAs (−14.8). The authors suggested that this change might be related to the quantized energy states, which arise due to the small size of the dots. Using a similar sample, the authors of [24] performed tunneling spectroscopy on a 2DES by measuring the tunnel current from the 2DES through the 0-D ground state of a single quantum dot embedded in the barrier of a tunnel diode [24]. The data revealed a structure that resembled the Landau level (LL) fan diagram. The authors also observed a sudden onset of the current as a function of the bias, and interpreted this as arising from a Fermi edge singularity. Thus, it was possible to obtain a quasi-particle lifetime from an LL fan diagram. They also observed the shift of the lowest LL to lower energy and they suggested that it originated from the exchange enhancement of the spin-splitting. The diameter of the mesa in this study was 5 m, and it was claimed that most of the dots had ground states above the Fermi energy of the 2DEG at zero bias. The shape of the electronic wave functions in SAQDs were visualized in a magneto-transport study of SAQDs embedded in a tunneling structure [25]. The magnitudes of the tunneling current peaks were measured as a function of B. Since the increase of B corresponded to the shift of the Fermi surface in the k-space, the dependence of the current on B could be converted to the magnitude of the wavefunction in the k-space. Under very high B values, the Fermi edge singularity could be observed in the resonant tunneling from highly doped three-dimensional GaAs electrodes through an InAs quantum dot, again embedded in an AlAs barrier [26]. Clear step-like features in the I-V curve with an undershoot were interpreted as the manifestation of the Fermi edge singularity. The observed peak splitting at high B was thought to be from the Zeeman splitting of the dot, and the observed oscillations of the peak position were considered to be from the Landau quantization. From the observed Zeeman splitting, the g factor value of 0.8 was obtained.

4. TRANSPORT THROUGH SAQDs USING STM Scanning tunneling microscopy is a good tool for studying the transport through SAQDs, since it is possible to achieve access to individual dots on the surface. For the observation

541

Transport in Self-Assembled Quantum Dots

5. TRANSPORT THROUGH SAQDs USING NANOFABRICATION Most of the important transport experiments discussed thus far were undertaken on samples and devices with sizes larger than at least several m. It is only recently that studies of transport through only a few, or single, SAQDs became possible. Intensive nanometer semiconductor processing is usually needed for the fabrication of such devices. Using multiple gating, squeezing of a m-sized tunneling mesa structure has also been demonstrated [30]. This structure utilized four-way gates and local depletion of the dots was achieved by biasing the gates with different voltages (In contrast, Fig. 4 shows an example of one-way gating.). By biasing each gate separately and monitoring the movement of the current peaks, it was possible to relate the current peaks and the particular dot positions. As shown in the schematic of Figure 6a, Jung et al. fabricated two metal electrodes on a wafer incorporating an InAs SAQD layer [31]. It is clear from the scanning electron micrograph of the actual device of Figure 6b that the size of the gap between the two metal electrodes was comparable to the size of a single SAQD and there was a single dot between the two electrodes. Figure 6c shows the I-V characteristics measured between the two electrodes of this structure at several different temperatures. There is a clear negative differential resistance and they attribute this feature to the resonant tunneling of electrons through the quantum states of the single SAQD. Figure 7 schematically shows another interesting nanodevice for the study of transport through a single SAQD [32]. The dot layer is positioned in the electron channel of an n-doped AlGaAs/GaAs HEMT structure. A point-contact is then fabricated by wet-etching and the size of the channel region was such that it contained at most three dots. While this etched point-contact fabricated from a normal HEMT

(a)

(b)

Nano-electrodes

100 nm

GaAs cap layer

SAQDs GaAs substrate

(c)

400

300 I (pA)

of sample surfaces, a feedback loop has to be switched on to maintain the same current level. However, it is also possible to obtain the I-V characteristics between the sample surface and the STM tip by disconnecting the feedback loop and such a technique is called scanning tunneling spectroscopy (STS). Rubin et al. performed the first measurement of current through a single SAQD by STM [27]. The technique was specifically called ballistic electron emission microscopy (BEEM). Their measurement showed clear differences between the STM current measured on the wafer with dots and without dots. Recently, more detailed STS measurements were done [28]. While the gap of 1.4 V was observed in the I-V curve on a bare GaAs surface, the gap of 1.25 V was seen in the I-V curve of a SAQD. This value is much larger than that of 0.4 V usually observed in InAs, and this observation was considered as evidence of strong electron confinement. Millo et al. studied the transport through a single InAs nanocrystal quantum dot by using low-temperature STM [29]. They observed a series of staircases in the STM current and the data were consistent with single electron tunneling through quantum states. They were even able to observe two-fold s state and six-fold p state resonances.

200

132 K 118 K

100 102 K 77 K 0

0

50

100 V (mV)

150

Figure 6. (a) A schematic of the nano-device to study transport through a single SAQD; (b) an SEM of the fabricated device. A single SAQD is located in-between two metal electrodes with the gap of approximately 30 nm [31]; (c) the measured I-V characteristics from the device at various temperatures. Clear NDR can be noticed [31]. Reprinted with permission from [31], S. K. Jung et al., Appl. Phys. Lett. 75, 1167 (1999). © 1999, American Institute of Physics.

showed conductance steps, the point-contact fabricated from the HEMT with SAQDs showed resonance peaks. The resonance peaks were interpreted as the conduction through the p-shells in a single SAQD. The authors also studied the effect of B on the transport and found the shift of resonance peak positions. The shift of the peak position with B was consistent with a simple quantum dot model with a harmonic confining potential. The most standard way of having a small number of dots is fabricating a diode with the smallest possible diameter. Hill et al. fabricated a gated tunnel diode incorporating InAs SAQDs [33]. The diameter of the diode was 0.7 m and the gate bias could further deplete the active area so that they were able to observe current peaks originating from individual dots.

Etched region Gate

SAQDs GaAs substrate

Figure 7. Another type of nano-device which was used to study transport through a single SAQD. A quantum point contact was defined by wet etching from a wafer similar to that in Figure 3.

542

Transport in Self-Assembled Quantum Dots

6. SELECTIVE GROWTH OF SAQDs The works discussed in the previous section all relied on post-fabrication for the position control of SAQDs. After the growth of InAs SAQDs on the whole wafer, pattern definition was performed to select either a few, or even a single SAQDs for transport study. Since it is considered that the main mechanism for the SAQD formation is strain relaxation, it is possible to selectively position those SAQDs by engineering the surface strain of the substrate. Such selectivity in the growth of SAQD was first identified by Tsui et al. [34]. They patterned the substrate into a GaAs mesa, and then made facets surrounded by SiO2 . In this way, they found that SAQDs grew only on the GaAs mesa. In subsequent work [35], the GaAs mesa was reduced to 1 m wide, but the authors still found a selectivity of InAs dot growth. Almost perfect site control was achieved by connecting various ultra-high vacuum equipments [36]. These authors used in-situ EB patterning and Cl2 etching to pattern holes on the substrate, and performed subsequent MBE growth without breaking vacuum. In this study, they found that the SAQDs were preferentially self-organized in the patterned holes. Alternatively, Hyon et al. used the oscillating cantilever of an atomic force microscope (AFM) to carve the substrate [37]. They found that the growth position of the SAQDs strongly depended on the slope of the facet of the carved grooves or holes. They could grow SAQDs selectively in the carved grooves or only on top of the hills formed in-between grooves. Figure 8a shows a onedimensional array of InAs dots grown on top of the carved hill and Figure 8b shows the dots grown only inside the

(a)

z 50n

y

(a) In As

Si

In As

(b)

SiO2

Si

SiO

Figure 9. Selectively grown SAQDs utilizing patterned Si/SiO2 substrates. (a) The growth of SAQDs occurs only on the Si stripes [39]; (b) when the width of the stripes is narrow enough, 1-D arrays of SAQDs can be achieved [39]. Reprinted with permission from [39], B. H. Choi et al., Appl. Phys. Lett. 78, 1403 (2001). © 2001, American Institute of Physics.

grooves. Hahn et al. [38] utilized a SiNx mask to form a GaAs mesa and grow InAs dots. Depending on the shape and the size of the mesa, they were also able to control the position and even the number of dots on the mesa. Similar selectivity could also be achieved using a layer of SiO2 covering a Si surface [39]. The nano-patterned SiO2 was found to successfully block the formation of the InAs SAQDs, and it was therefore possible to grow the dots only on the Si surface. Figure 9a demonstrates that InAs dots are grown only on the region where Si is exposed. As shown in Figure 9b, one-dimensional arrays of the InAs dots can be formed by narrowing the width of the Si stripes. Recently, it was found that QDs can also be grown on an InGaAs/GaAs superlattice template [40]. The template can provide an appropriate strained layer, since it produces strain fields with high uniformity, enabling the flexible control of alignment of QDs without the need of any complicated preprocesses. The superlattice slightly over the critical thickness affects only the driving of adatoms into local strain relaxed regions. Therefore, a strain relaxation from the misfit dislocations gives rise preferably to the nucleation of QDs. The authors of [40] found that there was a dependence of QD ordering on the number of superlattice periods.

100nm 100nm x

(b)

z

y

50n

100nm 100nm x

Figure 8. Selectively grown SAQDs after carving a GaAs surface by the AFM cantilever oscillation. Depending on the shape of the carved sidewall, SAQDs can be positioned (a) on top of the hills [37] or (b) inside the grooves [37]. Reprinted with permission from [37], C. K. Hyon et al., Appl. Phys. Lett. 77, 2607 (2000). © 2000, American Institute of Physics.

7. C-V MEASUREMENTS Even though there is no direct electron transfer in C-V measurements, this technique is still an important transport measurement, since the charging and discharging of SAQDs will provide details on the energy level structure of the dots. The first capacitance measurements of InAs dots were done by Medeiros–Ribeiro et al. [41]. They found peaks in the low T C-V measurement which they suggested originate from charging of the s-state. A subsequent experiment by the same group also resolved the p-state in the capacitance measurements [42]. Positively charged defects were shown to be important in C-V measurements. Belyaev et al. measured CV  and CB for many different samples with SAQDs embedded in AlAs [43]. Their data showed a sign of larger positive charges as the thickness of the AlAs layer increases. Recent interesting C-V measurements utilized a pn junction structure [44]. In contrast to previous experiments, these studies revealed a large amount of charges in the pn structure. Therefore, the

543

Transport in Self-Assembled Quantum Dots

Table 1. Detailed information about the InAs SAQDs used in various transport experiments. The growth conditions, size of the dots, average dot density, and the structure of the wafer embedding the SAQDs are shown. The first column displays the category of the device and the reference number. Type [Ref.]

n (×1010 cm−2 )

d (nm)

h (nm)

T ( C)

t (ML)

2DES [16]

10

10∼20

8

450

≥1.5

2DES [17] 2DES [18]

5 0.25∼0.58

36∼28

8∼4

510 530

2.5 2.15

0.3∼5 0.3 20

28 10

4

525

1.61∼2.15 1.8

450

1.1∼2.2

10 10∼15 30 2∼7

3 3∼4 3

20

5

520∼550 470 520

0.5 nm 2 2.1

Nano [30] Nano [31] Nano [32] Nano [33] C-V [41]

10

5 10 2

electric field was not linear. These authors compared their measurements with the results of self-consistent energy level calculations.

8. CONCLUSIONS Transport in semiconductor devices incorporating SAQDs is an interesting research topic, since the SAQD is a unique structure obtained automatically during crystal growth. Transport studies of these structures have been undertaken in many different ways such as the study of transport in 2DES with a nearby SAQD layer, tunneling transport through SAQDs, STS on individual SAQDs, transport in various nano-devices incorporating a few or a single SAQD, and C-V measurements. Even though the detailed device structures are different in these experiments, the main outcomes from them are probing the electronic energy states of SAQDs with and without B. In this chapter, we tried to cover all the important experimental progress related to these dots. However, a complete coverage of recent

Vertically alignedQDs

inverted AlGaAs/GaAs, InAs QDs in GaAs layer,  doping AlGaAs/InGaAs, InAs QDs in AlGaAs,  doping GaAs quantum well, InAs QDs in GaAs well,  doping AlGaAs/GaAs, InAs QDs in GaAs layer,  doping similar to [18] InAs QDs embedded in AlAs barrier, GaAs contact layers with graded doping similar to [21] similar to [21] similar to [21] InAs QDs embedded in thin GaAs solution phase pyrolitic reaction of organometallic precursors similar to [21], (311)B GaAs substrate InAs QDs in a thin GaAs buffer (MOCVD) similar to [19] similar to [21] InAs QDs with thick layers of GaAs on both sides

530

research would have been impossible since the amount of work is already vast and the field is still actively moving. Finally, Table 1 summarizes all the details related with the SAQDs used in various transport experiments discussed in this chapter. Recently, transport through coupled quantum-dot systems is of interest, since we can study coherent coupling between two 0DES (such a coupled quantum-dot system is called quantum-dot molecule) and such coherence is recognized as an important ingredient in the development of solid-state quantum computing [45]. Recent progress has made it possible to grow vertically aligned stacked SAQDs [46] and to study the transport through such stacked SAQDs. Figure 10 shows an example of stacked SAQDs. It demonstrates six SAQD layers stacked in the (001) direction. The uniformity and alignment of the dots are quite acceptable. Since the dots can be closely stacked, a much larger interaction is expected in this vertically coupled SQADs than the case of the lithographically defined coupled quantum dots, where

3

[001]

T = 20 mK, B = 18 T Stacked QD

[110]

dI/dV (S)

2DES [19] 2DES [20] Tunnel [21, 24, 25] Tunnel [22] Tunnel [23] Tunnel [26] STM [27] STM [29]

Summary of Layer Structure

2

1

0 0.00

0.05

0.10

0.15

0.20

V (V)

Figure 10. A TEM showing six stacked layers of InAs SAQDs.

Figure 11. A dI/dV-V characteristic taken from a diode with two stacked layers of SAQDs. The data were taken at T = 20 mK and B = 18 T.

544 the finite depletion layer again becomes a limiting factor. There have been some optical experiments demonstrating an entanglement/controlled coupling [47] and a measurement of the relaxation time [48] utilizing self-aligned stacked SAQD systems. However, electronic transport studies through coupled SAQDs have been rarely reported. Figure 11 shows an example of the dI/dV-V taken from a diode with stacked layers of SAQDs at low T and high B. It shows abundant peaks and rich features which might be expected to originate from quantum coupling between the dots. Future transport studies of such stacked-dot systems will be very interesting.

GLOSSARY Capacitance-voltage (C-V ) measurement A slowly varying ramp voltage and an AC signal are applied to the electrode deposited on the sample to be investigated. Then the AC current is measured to obtain the capacitance as a function of the applied DC bias. Nano-fabrication Processes for making nano-devices and nano-structures. Quantum dot A tiny piece of semiconductor whose electrons are isolated from nearby reservoirs. Their sizes range from 1000 nm to several nm. Scanning tunneling microscope (STM) By monitoring the tunneling current between the scanning probe with a small radius of curvature and the sample surface, we can obtain the topology of the surface in atomic resolution. Self-assembled quantum dot A quantum dot formed automatically during the crystal growth. Transport Electronic conduction in semiconductors, devices, or quantum dots. Transport measurement can usually reveal the energy spectrum of the system under investigation. Two-dimensional electron gas A sheet of electrons confined at the interface between semiconductor and insulator.

ACKNOWLEDGMENTS This work was supported by the Korean Ministry of Science and Technology through the Creative Research Initiatives Program under Contract No. M10116000008-02F0000-00610, while the work at Arizona State University (ASU) was supported by the Office of Naval Research (N00014-99-1-0326) and the Department of Energy (DE-FG03-01ER45920). The authors thank L. W. Engel in NHMFL, M. H. Son and D. Y. Jeong at iQUIPS for providing us unpublished data. S. W. Hwang thanks Prof. D. K. Ferry and Prof. S. M. Goodnick for supporting the preparation of this chapter during his stay in ASU. He also thanks B. Naser and M. Elhassan for careful reading of the manuscript.

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545 39. B. H. Choi, C. M. Park, S.-H. Song, M. H. Son, S. W. Hwang, D. Ahn, and E. K. Kim, Appl. Phys. Lett. 78, 1403 (2001). 40. T. Mano, R. Nötzel, G. J. Hamhuis, T. J. Eijkemans, and J. H. Wolter, Appl. Phys. Lett. 81, 1705 (2002). 41. G. Medeiros-Ribeiro, F. G. Pikus, P. M. Petroff, and A. L. Efros, Phys. Rev. B 55, 1568 (1997); G. Medeiros-Ribeiro, J. M. Garcia, and P. M. Petroff, Phys. Rev. B 56, No. 7 (1997). 42. B. T. Miller, W. Hansen, S. Manus, R. J. Luyken, A. Lorke, J. P. Kotthaus, S. Huant, G. Medeiros-Ribeiro, and P. M. Petroff, Phys. Rev. B 56, 6764 (1997). 43. A. E. Belyaev, S. T. Stoddart, P. M. Martin, P. C. Main, L. Eaves, and M. Henini, Appl. Phys. Lett. 76, 3570 (2000). 44. R. Wetzler, A. Wacker, E. Schöll, C. M. A. Kapteyn, R. Heitz, and D. Bimberg, Appl. Phys. Lett. 77, 1671 (2000). 45. C. H. Bennet and D. P. DiVincenzo, Nature 404, 247 (2000). 46. G. S. Solomon, J. A. Trezza, A. F. Marshall, and J. S. Harris, Jr., Phys. Rev. Lett. 76, 952 (1996). 47. M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, Science 291, 451 (2001). 48. P. Boucaud, K. S. Gill, J. B. Williams, M. S. Sherwin, W. V. Schoenfeld, and P. M. Petroff, Appl. Phys. Lett. 77, 510 (2000).

Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

Transport in Semiconductor Nanostructures H. León, R. Riera, J. L. Marín, R. Rosas Universidad de Sonora, Sonora, México

CONTENTS 1. Introduction 2. Carrier Description 3. Nonequilibrium Carriers 4. Scattering Mechanisms: Transition Rates 5. Low-Field Transport 6. High-Field Transport 7. Screening of the Scattering Mechanisms 8. Summary Glossary References

1. INTRODUCTION Among the nanostructured materials, semiconductor nanostructures may play a crucial role in the fabrication of very small devices, which in turn may allow a very large integration scale already known as nanoelectronics. Such kind of semiconductor structures can be constructed since high quality performance of modern equipment gives the possibility of controlling the crystal growth process up to a few atomic layers (i.e., with a nanometric precision) as well as the chemical composition, according to the required stoichiometric rules. Several techniques have been developed for this purpose, such as metal-organic chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE), about which a brief explanation can be found in [1]. MBE and MOCVD allow the implantation of donor or acceptor impurities in the desired region of the semiconductor and are known as modulation-doping techniques. The main feature of these structures is that electrons and holes participating in optical and transport phenomena are basically confined to a certain region, whose dimensions

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in one, two, or three directions are of the order of several up to a few tens nanometers. They are called quasitwo-dimensional (Q2D), quasi-one-dimensional (Q1D), or quasi-zero-dimensional (Q0D) systems, respectively, and all of them are also named low-dimensional semiconductor structures or semiconductor nanostructures. Two kinds of such structures can be distinguished: the semiconductor heterostructures and the delta-doped semiconductors, where the electron or the hole confinement can be achieved in different ways, as described in the following examples. Semiconductor heterostructures are grown with well matching materials from the crystallographic point of view, as the ones formed with III–V binary compounds and ternary alloys (e.g., GaAs/Alx Ga1−x As, where x is the fractional concentration of Al atoms per each Ga atom in the aluminum–gallium arsenide matched on gallium arsenide, whose use is widely extended in actual samples). One of the most studied structures is a modulation-doped double heterostructure B/A/B, where donors are placed in material B (e.g., Alx Ga 1−x As) whose conduction-band bottom lies higher than the one of material A (e.g., GaAs). The electrons supplied in the former one are captured in the layer formed by the latter one because of the conduction band offset at the interfaces; therefore, this structure is called a quantum well (QW). The electrons remain trapped in the QW as long as the Fermi level lies below the band offset and scattering processes do not throw them to states with energies above the band offset. In a single heterostructure (SH) of the same materials B/A, n-doped in the same way, the electrostatic interaction between electrons and their partner donors causes a band bending forming a sort of triangular well in the material-A side; thus, electrons are attracted to a very thin layer by the interface. Since the depth of this triangular well is considerably lesser than the one of the rectangular well formed in the double-heterostructure case, electron concentration cannot be too high. There are also other Q2D systems: the inversion layers in metal-oxide semiconductor (MOS) silicon based structures, where the confinement is similar to the one in SHs; the multiple quantum wells (MQWs) B/A/B/A/B, where one can identify a few noncoupled QWs; and the superlattices (SLs)

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (547–580)

548 B/A/B/A/B   , where one can identify a very large (modeled as infinite) number of coupled QWs. The periodicity of a SL in the direction perpendicular to the interfaces gives rise to a set of minibands instead of single bands, which makes it an especially interesting research object whose properties are widely studied but are beyond the scope of this chapter. Modulation-doping techniques allow one to construct delta-doped semiconductors, where donors are placed in a very thin layer inside the bulk material (e.g., GaAs). The intentionally attained high donor concentration makes the impurity levels split into an impurity band, and so electrons of this band can move along the layer, but they cannot escape from its approximate limits because of the ion-impurity attraction. This structure is very similar to a QW and, as in that case, electrons remains trapped in the delta-doped layer as long as the Fermi level, which is higher as the electron concentration is larger, lies below a certain effective energy threshold and scattering processes do not throw them to states with energies above this threshold. This way one has another Q2D electron system. Semiconductor heterostructures B/A are classified according to the relative positions of the valence-band tops. In type I heterostructures the top of the valence band in the material B lies lower than the one in the material A. Then acceptors can be placed in the former to have holes accumulated in the latter one, as in Alx Ga1−x As/GaAs heterostructures, where usually phosphorus atoms supply the holes that are mainly confined to the GaAs layer. In type II heterostructures the top of the valence band in the material B lies higher than the one in the material A. Then acceptors can be placed in the latter one to collect the holes in the former one, as in Si/Si1−x Gex heterostructures, where frequently boron atoms supply the holes that are mainly confined to the Si1−x Gex layers. In type III heterostructures the tops of the valence bands in both materials are approximately at the same level and holes cannot be confined, as in the GaSb/InSb heterostructures [2]. Several techniques are used to get a second confinement direction in the so-called quantum wires (QWIs) or quantum well wires (QWWs). Thus, for instance, once a GaAs based QW is grown, by means of electron beam lithography and dry etching, one conforms barriers in the direction perpendicular to the original layer for obtaining etched rectangular QWIs [3]. Similarly, QWIs are also fabricated on Alx Ga1−x As/GaAs single heterostructure using laser lithography [4]. Another way is to build a series of Alx Ga1−x As/GaAs double heterostructures on nonplanar surfaces by means of a MOCVD or MBE technique to form a V-groove array of QWs [5, 6]. This way at the bottom of each V one has two confinement directions because of the barrier material. These structures are examples of Q1D systems, where freestanding wires are also included. Electrons and holes in semiconductor nanostructures have de Broglie wavelengths greater than or approximately equal to the lengths in confinement directions of the region they are concentrated. Therefore, their motion in any of these directions bears an essentially quantum character, with discrete values of momentum and energy, while in nonconfined directions it is usually treated as a quasiclassical motion, with continuous values of momentum and energy, which means that instead of conduction and valence bands there are two

Transport in Semiconductor Nanostructures

respective sets of subbands. This essential feature makes necessary a reformulation of a vast part of the physics in such structures concerning electrons, holes, phonons, and, in general, the whole list of quasiparticles or collective excitations, as well as their interactions and responses to external perturbations. Theoretical and experimental studies show that the carrier confinement leads to the quantum size effects (QSEs), that is, to peculiar characteristics of these systems, ranging from the density of states through the optical and transport properties, due to the size reduction of the transversal dimensions of the region where electrons and holes are concentrated, which is called the conduction channel, since it plays such a role in quasiclassical transport. It must be pointed out that the carrier confinement is not complete, for always there exists a nonzero probability of finding the carriers out of, although very near, the conduction channel. Transport perpendicular to the interfaces is frequently called vertical transport, because actual heterostructures are grown with interfaces in horizontal planes. This is strictly connected with quantum phenomena, as the resonant tunnel effect, and will not be discussed here. This chapter is concerned with horizontal or parallel transport exclusively, a field of great experimental and theoretical activities. Electrical conductivity, quantum Hall effect, thermoelectric power, Coulomb drag, and many other phenomena attract the attention of thousands of scientists all over the world whose articles are published in dozens of scientific journals. This field could be hardly reviewed in 100 pages, so a specific topic must be selected to introduce the fundamental theory, as well as to give a substantial account of the work concerning this topic basically in the last 20 years. The construction of two-dimensional electron-gas fieldeffect transistors, including the MOS field-effect transistor (MOSFET) and the modulation-doped field-effect transistor (MODFET), opened the door for high-speed semiconductor devices [7]. Really, to obtain high-frequency performance in a transistor, one has to limit the transit time through the device. This can be achieved by either decreasing the gate length or increasing the drift velocity of the carriers. Once the gate lengths have been reduced up to micrometer or submicrometer range, the efforts are devoted to the enhancement of the carrier drift velocity. In these semiconductor nanostructures there are two ways of enhancing the carrier drift velocity: by increasing the carrier mobility at relatively low electric fields or by increasing the applied electric field. The latter exhibits a saturation effect and, furthermore, the drift velocity starts to decrease when the electric field increases beyond sufficiently high values, in a region with negative differential conductivity. Therefore, great interest has been put on enhancing the carrier mobility along the conduction channel by reducing, as far as possible, different carrier scattering mechanisms and, consequently, to make possible the fabrication of high-electronmobility transistors (HEMT’s). Nevertheless, the problems related to relatively high electric field, such as energy dissipation, are also important because, despite the fact that the applied voltages can be very low, the micrometric or submicrometric gate lengths make the electric field very high [8]. The MOSFETs are already widely exploited in logical and

Transport in Semiconductor Nanostructures

memory circuits, but this is not the case with MODFETs for technological reasons [9]. This chapter is mainly concerned with steady-state charge transport along the conduction channel in semiconductor nanostructures, because it is not only of practical interest for nanoscale transistor design but also represents a means for investigating how electrons and holes are scattered by different mechanisms, particularly by phonons, which are important in optical properties to be considered in nanoscale optoelectronic devices.

2. CARRIER DESCRIPTION 2.1. Charge Carriers For electronic purposes in semiconductor technology charge carriers can be electrons or holes, which are as a rule supplied by donor or acceptor impurities respectively. The creation and annihilation of electron–hole pairs is mainly involved in optoelectronic devices. Modulation-doped techniques allow one to place the impurities conveniently to obtain the required characteristics of the device design. In delta-doped semiconductors impurities are in a very thin layer inside the bulk material, which is itself the conduction channel; and in semiconductor heterostructures impurities are in the barrier material, out of the conduction channel. The latter idea is the milestone for the HEMT fabrication. The cleanliness of the growth processes together with the inclusion of an undoped region, called the spacer layer, between the doped region and the conduction channel in semiconductor heterostructures aims to reduce substantially the carrier–ion-impurity scattering, apart of other defects, in order to attain high mobilities. Record mobility values have been attained in Alx Ga1−x As/GaAs SHs, both for electron mobility  = 147 × 106 cm2 V−1 s−1 reported in [10], at very low temperature (T = 01 K) and ordinary areal concentration (nS = 24 × 1011 cm−2 ), as well as for hole mobility h = 12 × 106 cm2 V−1 s−1 reported in [11], also at very low temperature (T = 1 K) and low areal concentration (nS = 08 × 1011 cm−2 ). The study of transport phenomena is carried out, as a rule, in the envelope function approximation of the solid state band theory, namely the effective mass approximation (EMA) for the electrons in the conduction subbands or holes in the valence subbands. The electron and hole effective masses are usually determined experimentally (e.g., from cyclotron resonance experiments) and are considered isotropic, as in direct gap semiconductors with the bottom of the conduction band at the  point of the Brillouin zone (e.g., GaAs). Nevertheless, the theory can be applied to indirect gap semiconductors with anisotropic electron effective masses at or L points of the Brillouin zone, as in the case of Si and Ge respectively, by taking an averaged isotropic value, a usual procedure in many calculations. Electrons attract attention in most of the works, certainly because for a given material they are lighter than holes and, consequently, higher mobilities can be expected, which is, as already remarked, a desirable characteristic for high-speed devices. Electron effective mass is denoted as m∗ , while the electron charge is e < 0. All the derivations for holes in the valence subbands could be accomplished in principle by

549 the substitution of electron values by the pertinent heavy or light hole effective masses m∗hh or m∗lh and bearing in mind that the hole charge is e > 0; besides, one must also take care of the degeneracy at the top of the valence subbands, although it can be suppressed by the strain at the interface due the difference in lattice constants, which is remarkable in Si/Si1−x Ge heterostructures [12]. One deals with acceptor instead of donor impurities in hole transport. Parabolic subbands are considered, despite the fact that nonparabolicity may be important for very narrow QWs [13]. The introduction of distinct subband effective masses for electrons [14, 15] would complicate a little the derivation of the fundamental equations and would lead to nonremarkable different results of measurable quantities. Similarly, the distinction between the effective masses inside and outside the conduction channel would mean a refinement of calculations, which can be left out of consideration in a first approach to the transport phenomena, although it can be easily introduced in the equation for the envelope wavefunction in the confinement direction or plane [16].

2.2. Carrier Confinement For the sake of briefness and compact presentation of theory, the 3D vectors are denoted with capital letters, while 2D or 1D vectors will be denoted with small letters. Thus the position is (1)

R = R + R⊥ = r +


where the component along the conduction channel, parallel to the interfaces in heterostructures, will be simply r = R , while the component perpendicular to the conduction channel, perpendicular to the interfaces in square QWs or cylindrical QWIs, will be simply
= R⊥ . This way one has
= zez

(2)


= yey + zez

(3)

r = xex + yey r = xex

in Q2D and Q1D systems respectively. The choice of Cartesian coordinates, with unitary vectors ex , ey , and ez , allows the desired compact presentation of theory, although cylindrical coordinates are to be chosen in order to take advantage of the symmetry in cylindrical QWIs. The carrier confinement suggests a clear separation in the Hamiltonian of the term related to the motion along the conduction channel and the term related to the confinement direction or plane. Thus the unperturbed one-particle Hamiltonian for electrons is written in the EMA as

where

2
0 = −
 2 +  2  + U 
 H
2m∗ r

r2 =

  + 2 x2 y

r2 =

 x2


2 =

(4)

 z2

(5)

  + 2 y 2 z

(6)


2 =

in Q2D and Q1D systems respectively. The confining potential U 
 is frequently modeled in different ways or calculated self-consistently.

Transport in Semiconductor Nanostructures

550 The corresponding eigenfunctions (i.e., the electron envelope functions) are k r
 = A−1/2  
 expik · r

(7)

where  is the subband label and k is the wavevector for the quasiclassical motion of electrons, which must be understood as  = l = 1 2 3   

k = kx ex + ky ey

 = n l = 1 1 1 2 2 1   

k = kx ex

(8) (9)

with A = S, the normalization area, in Q2D systems and A = L, the normalization length, in Q1D systems respectively. This way, during the motion between two consecutive scatterings, one has a plane wave for electron free motion along the conduction channel, while another wave describes the electron motion perpendicular to the conduction channel in dependence of the confinement model. Consequently, the electron energy is the sum of two terms, E k = E + E =


2 k2 + E 2m∗

(10)

where E = Ek is the kinetic energy, associated with quasiclassical motion along the conduction channel, and E is the subband energy, that is, the electron energy at the bottom of the given subband (note the wavevector is measured from the bottom value too). Such an important characteristic as the density of states (DOS), including a factor 2 because of the spin, is given by "#El k$ =



"l E =

l

"#Enl k$ =



 m∗ S l

"nl E

%
2

&#El k − El $

(11)

n l

=

   L 2m∗ 1/2 &#Enl k − Enl $ %
E n l

(12)

in Q2D and Q1D systems respectively, where the Heaviside unit step function is   1 E > 0 (13) &E = 0 E < 0 and explicitly reflects discontinuities when the total energy crosses subband bottoms. The DOS is not a continuous function of the increasing total energy. It has a staircase form in Q2D systems, with a constant value for each subband, while it diverges as E −1/2 when the total energy leaves the bottom of a given subband  and tends to a value proportional to E−1/2 when it approaches the bottom of the following subband in Q1D systems. These behaviors of the DOS in low-dimensional systems can be understood as the first remarkable QSEs. Furthermore, they clearly indicate that transformation of summations in momentum into integrations in energy can be accomplished only for a fixed subband, namely,   F k = " EF E dE (14)  k



0

where F k is any function depending on subband index and wavevector. This transformation leads to all subsequent QSEs, which can be observed in the behavior of many transport and optical properties of low-dimensional systems. The carrier confinement to the conduction channel has been modeled in different ways, mostly assuming a certain coordinate dependence of the confining potential or calculating it self-consistently. The infinite barrier model (IBM) assumes U 
 = 0 inside and U 
 = outside the conduction channel and leads to analytical expressions for the envelope functions  
, which are strictly confined with no penetration into the barriers, as well as for subband energies E . This way some progress in calculations can be made before computational methods are necessary. Therefore the IBM has been widely employed for rectangular QWs and QWIs or cylindrical QWIs, although in the latter case the zeros of the Bessel functions have to be found numerically, in order to calculate scattering rates, mobilities, and so on when the attention is mainly paid to QSEs due to electron confinement. The finite barrier model (FBM) assumes U 
 = 0 inside and U 
 = U0 = const outside the conduction channel and leads to analytical expressions for the envelope functions  
, which are partially confined with some penetration into the barriers, and also for subband energies E , but they include numerically obtained quantized quantities that appear instead of the wavevectors after the solution of transcendental equations. This way little progress in calculations can be made before computational methods are necessary. Nevertheless, the FBM has been also employed for rectangular QWs or QWIs and cylindrical QWIs in order to calculate scattering rates, mobilities, and so on when more realistic results are desired. The barrier height is frequently taken from empirical formulae, for examples, in Alx Ga1−x As/GaAs heterostructures U0 = 0693 x + 0222 x2 (eV) [17] and for x = 03 one has U0 = 228 meV. The FBM has been employed for rectangular QWs and cylindrical QWIs. But it is not suitable for rectangular QWIs because the variables y and z cannot be separated in this case. Self-consistent (SEL) calculations are often carried out by solving coupled Schrödinger and Poisson equations, 

−


2 2  +U 
  
 = E  
 2m∗


U 
 = Ue 
+Uf 
+Uxc 
 
2 Ue 
 =

(15) (16)

2

4%e #n
−nD 
$ *0

(17)

where Uf 
 is a certain model potential, usually the FBM when appropriate, and Uxc 
 is the exchange and correlation potential, which in turn is taken in different ways (e.g., as in [18]). Frequently calculations are carried out in Hartree approximation [i.e., one sets Uxc 
 = 0]. The donor concentration nD 
 must somehow be modeled inside and outside the conduction channel. Notice that in modulationdoped heterostructures it is very low inside and quite high outside the conduction channel, while in delta-doped semiconductors it is very high inside and very low outside the

Transport in Semiconductor Nanostructures

551

conduction channel. The electron concentration is n
 =

N 

2

 


=1





0

" E dE

(18)

where N is the number of occupied subband and " E is the DOS given in (11) or (12). This way computational methods for numerical calculations are required from the very beginning to find both the envelope function  
, which is partially confined with some penetration into the barriers, and subband energies E . All of this is supposed to give the more realistic results to interpreted experiments. A detailed comparison of mobility calculations with IBM, FBM, and SEL wavefunctions and energies can be found in [19], where the SEL results of [20], obtained with the Hedin– Lundqvist density functional for exchange and correlation, were employed. A triangular potential model (TPM) has been also utilized for electron confinement in one direction. It assumes U z = eFz z, when an electric field Fz is applied to confine the electrons in this direction. This leads to analytical expressions for the envelope functions and subband energies,

   2m∗ 1/3 eFz z − El  (19) l z = Ai
2 e2 Fz2 

1/3 



 2/3

where the parameter b is also chosen to give the correct ground subband energy. A parabolic potential model has been also used for confinement in one direction. It assumes U z = 21 Dz2 , where D is an appropriate constant. This leads to analytical expressions for the envelope functions l z, which are the ones of harmonic oscillators (i.e., related to Hermite polynomials), and for subband energies El . They have been frequently employed for calculations in the so-called parabolic QWs and it could also be taken in one of the confinement directions in a QWIs, if in the IBM is assumed in the other direction.

2.3. Carrier Statistics Different scattering mechanisms, such as electron–phonon or electron–ion-impurity interactions, together with electron– electron interaction are responsible for the randomization of momentum and energy values of the electrons in the conduction channel, in such a way that one can think of a gas of noninteracting electrons, in correspondence with the electron gas model of many-body theory (MBT), whose total charge is neutralized by a uniform background of positive charges associated with the ion cores of the crystal. In order to describe the density of the electron gas a generally used parameter rs must be defined as

(20)

%rs2 =

1 nS a2B

(25)

where Aiz are the Airy functions. They have been employed for calculations in SHs or triangular QWs and could be also employed for one of the confinement directions in rectangular QWIs, when in the other direction an IBM is assumed. Different envelope functions in one confinement direction can be constructed to find the minimum subband electron energy following a variational method. The Fang–Howard variational (FHV) envelope functions for the two lowest subbands are [21]

2rs =

1 nL aB

(26)

El =


2 e2 Fz2 2m∗

3% 2

l+

3 4

 1/2  bz b3 1 z = z exp − 2 2     3 1/2  bz 3b bz exp − 1− 2 z = z 2 3 2 

(21) (22)

where the parameter b is chosen by minimizing the subband energy E1 . It results in b ∝ m∗ e2 nS 1/3 . They have been frequently employed for calculations in SHs and inversion layers as well as in one of the confinement directions in rectangular QWIs, when in the other direction an IBM is assumed. A better fit to the TPM wavefunctions, at least to the ground state, is obtained with new variational wavefunctions [22], 

   b5 1/2 bz exp − 24 2     5 1/2  bz bz b 1− exp − 2 z = z2 24 5 2 1 z = z2

where aB is the Bohr radius, nS is the areal electron concentration in Q2D systems, and nL is the linear electron concentration in Q1D systems. Thus rs is small for a highdensity electron gas and large for a low-density one. In the former case average kinetic energy is much larger than the potential energy due to electron–electron interaction, including exchange and correlation energies, and the electron gas model is expected to be valid [23]. Interacting electrons are considered to be an electron liquid, particularly as in the Fermi liquid theory. Electron gas is ruled by Fermi–Dirac statistics, which means that the equilibrium distribution function is

(23) (24)

f

0

−1  E k − EF +1 #E k$ = exp kB T

(27)

with kB the Boltzmann constant, T the temperature, and EF the Fermi level. The distribution function for the degenerate electron gas can be easily transformed to be written as f0 E =

1 exp#3E − EF $ + 1

(28)

in order to have a clear reference to the considered subband and the kinetic energy. Here 3=

1 kB T

and

EF = EF − E

(29)

Transport in Semiconductor Nanostructures

552 Note that EF is nothing else but the Fermi level measured from the bottom of the subband . The normalization condition in a low-dimensional system reads N N    N = N = " Ef0 E dE (30) =1

=1 0

where N is the total number of electrons in the conduction channel, N is the number of electrons belonging to subband , and N is the number of occupied subbands. From (28) and (30) one obtain the relations between areal nS or linear nL concentrations and Fermi level EF for a degenerate electron gas, N  3%
2 nS = ln 41 + exp#−3EF − El $5 ∗ m l=1  2 2 1/2 N  %
6−1/2 EF − E  n = L 2m∗ =1

(31)

(32)

in Q2D and Q1D systems respectively. Here the so-called Fermi integral was introduced,  E m f0 E dE (33) 6m EF  = 0

which leads to analytical results only in the cases of completely degenerate electron gas, when one has the Heaviside unit step function f0 E = &EF − E

(34)

or nondegenerate electron gas, when one has the Maxwell– Boltzmann distribution function f0 E = exp #−3E − EF $ = exp3EF  exp−3E (35) This way for completely degenerate electron gas one obtains N  %
2 n = EF − El  S m∗ l=1  2 2 1/2 N  %
EF − E 1/2 n = L 8m∗ =1

(36)

(37)

while for nondegenerate electron gas one obtains N  3%
2 nS = exp#37 − El $ ∗ m l=1 1/2  N  3%
2 exp#37 − E $ n = L 2m∗ =1

3. NONEQUILIBRIUM CARRIERS The electron gas can be carried out of equilibrium by the application of an external electric field or by any external action leading to concentration or temperature gradients, as in the case of pulsed lasers. The treatment of nonequilibrium carriers for concrete calculations of transport coefficients, such as mobility, energy-loss rate, or differential thermopower, has mostly been developed in the frame of a quasiclassical theory. This way the quantum problem of finding the transition rates between electron states is as a rule solved by the standard Rayleigh–Schrödinger perturbation theory, mostly limited to the first-order Born approximation. Correspondingly, the statistical problem of finding the nonequilibrium distribution functions is widely treated in the Boltzmann transport equation (BTE) formalism. The many-body effects are then taken into account by the inclusion of the exchange-correlation potential in self-consistent calculations of  
 according to (16), and by means of the screening of the interactions, which scatter electrons from a state into another. The BTE formalism gives not only a clear understanding of transport phenomena, but also a very adequate fitting of experimental data in semiconductor nanostructures, as can be seen in quite a number of works cited. Many works have been devoted to calculations in the frame of a completely quantum approach of MBT, as in [24–26]. Following the Kadanoff and Baym (KBA) formalism, the equation of motion for the nonequilibrium Green functions and the introduction of the Wigner distribution functions lead to the corresponding system of quantum transport equations, known as quantum Boltzmann equations because they are analogous to the BTEs. Thus, the KBA formalism applied to low-dimensional systems takes into account the finite lifetimes of electron states and the local character of the Wigner distribution functions with respect to the transverse coordinate
[27–31], but the obtained results are not very different from the ones obtained following the quasiclassical theory [32]. The socalled memory function approach has also been employed [18, 33]. Finally, a certain dielectric formalism has been developed [34].

3.1. Boltzmann Transport Equations (38)

(39)

in Q2D and Q1D systems respectively. Generally the electron concentration is determined experimentally (e.g., nS from Shubnikov–De Haas oscillations), and it is frequently a controlled parameter in actual experiments. Thus it is the Fermi level that must be calculated. Notice that in all the cases, except for a completely degenerate electron gas in a Q2D system, these are transcendental equations that must be solved numerically when two or more subbands are occupied. On the contrary, when only the lowest subband is occupied one obtains algebraic equations in all the cases, except for a degenerate electron gas in Q1D systems.

The multisubband transport theory was originally developed in [35], although not in connection with semiconductor nanostructures, but with multivalley transport. It is reviewed in [21]. In a frame of a MBT approach, it is clear that in general when intersubband scatterings take place, the density matrix has off-diagonal elements between different subbands. When the level broadening due to intersubband scattering is sufficiently small in comparison with subband energy separations, one can neglect such off-diagonal parts and use a diagonal approximation, as in [32], or simply the BTE formalism. Bearing in mind that the quasiclassical formalism holds exclusively for transport along the conduction channel, it clearly results that in nonequilibrium state the unknown distribution functions f k r t related to different subbands do not depend on the transverse coordinate
. Each of

Transport in Semiconductor Nanostructures

553

these functions is determined by one kinetic equation, which means they must satisfy a system of BTEs,  1 e  + k E · r + F · k f k r t =
If k r t (40) t



where F is the applied electric field (assuming no applied magnetic field) and   r = e + e x x y y r =

 e x x

  e + e k = kx x ky y k =

 e kx x

(42)

(43)

′ k



f krt#1−f′ k ′ rt$W′ kk ′ 

(45)

′ k

with W′ k k ′ ) the transition rate between electron states

 k and ′  k ′ , which includes all scattering mechanisms considered in the studied problem. Bearing in mind the transformation (14) one realize that BTEs are integrodifferential equations. Note the collision integral operator couples equations to each other. Furthermore, the number of equations is taken equal to the number of occupied subbands.

3.2. Steady-State Transport

(46)

coupled each other by the collision integral operators
I given by
If k = Iin k − Iout k

(50)

and the average energy of these carriers in the  subband,

k

E k =

E kf k  k f k

(51)

The current density j and the electron concentration n = nS , the areal concentration in Q2D systems, or n = nL , the linear concentration in Q1D systems, are quantities that can be measured in experiments and include the contribution of all occupied subbands. The well known expression j = envd is still valid in semiconductor nanostructures if a mean drift velocity is defined as d v n v =     n d

(52)

with n the areal (nS = nl ) or linear (nL = nnl ) concentration of electrons belonging to subband . Similarly, the average energy of the low-dimensional electron gas is defined as = E



 E k n





n

(53)

It must be clear that the total electron concentration is

The steady-state transport occurs when the applied electric field is F = const and, consequently, the current density j remains constant (the stability can be broken in some circumstances, as will be quoted). One assumes there exist no concentration or temperature gradients in any direction of quasiclassical motion. Thus the nonequilibrium subband distribution functions f k are both r and t independent and must satisfy the system of BTEs e F · k f k =
If k


v kf k  k f k

k

(41)

where the incoming to and outcoming from state  k particle fluxes are  Iin krt = f′ k ′ rt#1−f krt$W′  k ′ k (44) Iout krt =



vd = v k =

in Q2D and Q1D systems respectively. The right hand side of each equation is the so-called collision integral operator given by
If k r t = Iin k r t − Iout k r t

The constant electric field drives the electron gas with a certain drift velocity, which must be redefined for lowdimensional systems because of the subband character inherent to the energy spectrum of involved carriers. Now the determination of f k is required for calculating the drift velocity of the carriers in the  subband,

(47)

where the incoming to and outcoming from state  k particle fluxes in this case are  (48) f′ k ′  #1 − f k$ W′  k ′  k Iin k =

n=



n

(54)



Once the steady-state transport regime is established and maintained, the momentum and the energy continuously supplied by the field to the carriers are transferred to the surrounding scattering centers with the same average rate. The momentum relaxation time ;M is defined from the momentum balance as eF =

m∗ v k ;M

(55)

while the energy relaxation time ;E is defined from the energy balance as eF · vd =

E k ;E

(56)

′  k

Iout k =



′ k

f k #1 − f′ k ′ $ W′ k k ′ 

(49)

Both of them are usually evaluated for physical considerations in different transport phenomena.

Transport in Semiconductor Nanostructures

554 3.3. Solution of the System of BTEs The solution of the system of BTEs in semiconductor nanostructures is very complicated, except for relatively low electric fields. Therefore, many calculations related to transport phenomena have been carried out by means of Monte Carlo techniques [36–40], which are very powerful procedures for concrete situations. Nevertheless, most of the calculations are based upon approximate analytical solutions, which allow an essentially physical insight in transport properties of these structures and give sufficiently accurate results to fit experiments in a large number of situations. The analytical solutions are quite general in the low-field regime, but this is not the case in the high-field regime, as will be discussed. The theoretical calculations of the works cited in this chapter are always carried out following analytical solutions of the BTEs. Monte Carlo calculations are explicitly indicated. One must recall that in thermodynamical equilibrium momenta and energy are randomly distributed among the electrons due to the different scattering mechanisms present in the semiconductor nanostructures. It must be pointed out that the principle of detailed balance (PDB), which holds exclusively in this equilibrium state, reads

 0 f0 E 1 − f′ E ′  W′ k k ′  0

= f′ E ′ #1 − f0 E$W′  k ′  k

(57)

0 which effectively leads to
If k = 0 when there is no applied electric field nor concentration or temperature gradients. For elastic (EL) scatterings, when electron energy does not change, the PDB reduces to ′ ′ EL WEL ′  k  k = W′ k k 

(58)

while for inelastic (NE) scatterings, when electron energy is increased (+) or reduced (−) in a certain amount
=, the PDB reduces to ′ NE ′ WNE ′  k  k = W′ k k  exp±3
=

(59)

This principle is useful for many derivations, such as the linear approximation to solve the system of BTEs explained in the section devoted to low-field transport.

4. SCATTERING MECHANISMS: TRANSITION RATES The transition rates appearing in the collision integral operators of the BTEs include all the scattering mechanisms relevant for the considered transport phenomena. Electrons in semiconductor nanostructures can be scattered mainly by phonons (PH) of the acoustic and optical branches, ion impurities (IM) inside and outside the conduction channel, interface roughness (IR) in the boundary between materials forming the heterostructures, and alloy disorder (AL) because of the alien atoms in the host material, for instance, aluminum in Alx Ga1−x As instead of arsenic in GaAs. Thus one must understand that  CP W′ k k ′  = W′ k k ′  (60) CP

where formally CP = PH, IM, IR, AL. For a given scattering mechanism the transition rates are calculated in first-order Born approximation and, consequently, the Fermi golden rule (FGR) applies to calculate the corresponding transition rates, CP ′ W ′ k k  =

2%  CP

V ′ k k ′ Q 2av
j Q j

× @E + E − E ′ − E′ ±
=jQ 

(61)

where the subscript j has different meanings according to the considered interaction. It labels the polarization modes in the case of phonon scattering (CP = PH), the individual ionized impurities in the case of impurity scattering (CP = IM), or the individual alien atoms in the case of alloy disorder scattering (CP = AL). It does not appear in the case of interface roughness scattering (CP = IR), for the statistical correlation between the individual scattering centers taken into account, as will be seen. The Dirac delta function indicates energy conservation. Thus, in an elastic scattering, where the total electron energy is conserved, one simply has
=jQ = 0, while in an inelastic scattering the energy gained (+) or lost (−) by the electron is
=jQ . Here the subscript “av” means statistical averaging of the scattering centers. In correspondence with (1) now a 3D wavevector is Q = q + q⊥ , with q⊥ = qz ez

(62)

q⊥ = qy ey + qz ez

(63)

q = qx ex + qy ey q = qx ex

in Q2D and Q1D systems respectively. Screening of the interactions due to the fluctuations of the carrier concentration around the scattering centers must be taken into account. A linear response theory is generally applied to find this effect, whose result is incorporated by means of a screening factor or a matrix dielectric function when transition rates are calculated. Amount the earlier works on this subject one can find a modified Debye model derivation of a screening factor in [41]. The Thomas-Fermi approximation (TFA) and the random phase approximation (RPA) for the matrix dielectric function are widely employed [21]. Thus, the matrix elements for the screened interaction are given by ′ CP Q = Vj ′ k k



′ CP C−1 ′ BB ′ q =jQ V0jBB ′ k k Q

(64)

BB ′

CP ′ where V0jBB Q are the corresponding matrix ele′ k k ments for the bare interaction; and C−1 ′ BB ′ q =jQ  are the elements of the inverted matrix dielectric function. The inversion of a matrix dielectric function is not so simple as it seems at first sight and it is a matter of controversy. Many works avoid this question by making different approximations, particularly the so-called size-quantum limit (SQL) approximation, when only the lower subband is occupied and the intersubband transitions are ignored. The screening of the interactions deserves a special discussion, which can be found in the corresponding section (“Screening of the Scattering Mechanisms”).

Transport in Semiconductor Nanostructures

4.1. Electron–Phonon Interaction Electron–phonon interaction plays a very important role not only in transport phenomena but also in a variety of optical phenomena and, consequently, is the subject of a lot of works concerning semiconductor nanostructures. It is generally accepted that acoustic (AC) phonons keep their three-dimensional character, because the elastic constants of materials forming the heterostructures have quite close values. Furthermore, calculations with bulk AC phonons fit very well experimental data of optical and transport coefficients, so one can find very few articles where the problem of AC phonon confinement is considered [43]. On the other hand, optical (OP) phonons bear an essentially confined character, as shown by Raman scattering experiments in layered structures [44]. First explanations to these results were based upon very simple linear chain models [45–48]. Microscopic models have been proposed to find the dispersion relations both in layered and wire structures [49]. However, most of the works have been devoted to continuum models, maybe due to the fact that long wavelength phonons are the ones mainly involved in electron–phonon interaction. Two kind of approaches could be found: dielectric continuum models (e.g., in [50–55]), which consider the matching of electrostatic potential at the interfaces and do not take account of phonon dispersion, and hydrodynamic models (e.g., in [56–60]), which consider the matching of displacements at the interfaces and take account of the phonon dispersion. These models keep the distinction between transverse and longitudinal polarizations and give rise to interface, shear, and confined modes, the latter with different quantized frequencies, which allow quite good interpretation of optical experiments. The main feature of interface modes is the exponential decay of their amplitudes with increasing distance from the interface on both sides. The continuum models have been considered in many calculations of transport properties with satisfactory results, although the consideration of the phonon confinement does not affect the order of magnitude or the different functional dependence of calculated quantities, such as momentum relaxation rates [58], mobility [32], or power loss [61]. But an essential contradiction arises between these two models, because the imposition of matching conditions of one kind leads to the breakdown of matching conditions of the other kind [62]. Therefore, new models have been developed to overcome this contradiction. That is the case of different dispersive continuum theories developed in [63, 64]. For instance, in the latter there are neither purely transverse nor longitudinal polarizations, except at the center of the Brillouin zone, and very complicated dispersion relations, expressed by means of transcendental equations, are obtained. This model, generally with the restriction that the perpendicular to the interface displacement is zero and after some approximations, has been successfully employed in calculations of various optical properties, with acceptable agreement with experiment. However, very few calculations of transport coefficients have made use of this model, mainly because of the analytically unmanageable expressions appearing in derivations, which entails numerical methods from the very beginning [26]. Phonon confinement has been taken into account in several transport properties calculations, but in contrast with optical properties calculations,

555 it seems that no substantial differences emerge with respect to the consideration of bulk optical phonons. For the sake of a comprehensive and compact presentation of theory, nonconfined AC and OP phonons are considered in this section. Phonon scattering is essentially inelastic, although for low-field regime acoustic-phonon scattering is considered elastic, which will be discussed later. Electrons are scattered by phonon absorption (+) or phonon emission (−); then it is clear that PH+ ′ PH− PH ′ ′ W ′ k k  = W′ k k  + W′ k k 

(65)

according to the FGR (61), where the screened matrix elements are given by (64). The deduction of the interaction Hamiltonian can be found in many textbooks on semiconductor or solid state theory (e.g., in [65, 66]) for bulk phonons. In low-dimensional systems one must take care of separation of the envelope function in the plane-wave part and the confinement part according to (7). The matrix elements for the bare interaction result,    1 1 1/2 PH± ′ V0j k k Q = N + − ± ′ jQ 2 2 × CPH QG′ q⊥  k − k ′ ± q

(66)

where CPH Q is the corresponding coupling constant. Here the phonon occupation number NjQ leads to the Bose–Einstein distribution function nB 
=jQ  after statistical averaging: nB 
=jQ  =

1 exp3
=jQ  + 1

(67)

Because of the broken translational symmetry in confinement directions there appears a form factor for the electron–phonon interaction defined as  G′ q⊥  = ∗′ 
 
 expiq⊥ ·
d
(68) When phonon confinement is considered, instead of expiq⊥ ·
 quite complicated functions can appear even for simple models, as in [67]. Only a very simple confinement model leads to a simple function [58]. The Kronecker delta   1 if k − k ′ ± q = 0 ′ (69) k − k ± q = 0 if k − k ′ ± q = 0

indicates momentum conservation in the plane or in the axis of translational symmetry for Q2D or Q1D systems respectively.

4.1.1. Acoustic–Phonon Scattering We are thoroughly considering electrons close to the bottom of the conduction subbands, so electron–phonon interaction involves long wavelength phonons as the main contributors for the scattering processes. For AC phonons the longwavelength approximation amounts to
=jQ =
uj Q

as

Q→0

(70)

where uj is the sound velocity for the j polarization mode.

Transport in Semiconductor Nanostructures

556 For very low temperatures, T → 0, the Bose–Einstein distribution function is approximated simply by zero, which means there are no AC phonons to be observed. For low temperatures this factor is given by the complete (67) and this is known as the Bloch–Gruneisen regime. For not so low temperatures this factor is nB 
=jQ  ≃

kB T
uj Q

when
uj Q ≪ kB T

(71)

and this is known as the equipartition approximation. The latter is the most employed in calculations, which allows advancement in obtaining some analytical expressions. For instance, in GaAs based heterostructures the equipartition approximation works well for T ≥ 4 K [68]. For mobility calculations the electron–AC-phonon interaction is considered an elastic scattering, which means one must set
=jQ = 0 in the argument of the Dirac delta function of the FGR. But in the equipartition approximation the small phonon energy is retained to give nB 
=jQ  + 1 ≃ nB 
=jQ  ≃

kB T ≫1
uj Q

(72)

Thus, in the latter approximation, the phonon absorption and phonon emission transition rates equal each other and, consequently, one has AC ′ W ′ k k 

=

AC+ ′ 2W ′ k k 

=

AC− ′ 2W ′ k k 

@E′ − E ′ ±
uj Q = @E′ − E ′  ±
uj Q@′ E′ − E ′ 

(74)

1 + 
uj Q2 @′′ E′ − E ′  2

(75)

where E′ = E + E − E′ and [69]

− 

−

′′

′

′

′′

gE @ E′ − E dE = g E′ 

(76) (77)

The elastic approximation, limited only to the first term in the expansion, would prevent AC phonons from participating in electron energy loss, which is important at low temperatures when electrons cannot emit optical phonons. In both polar and nonpolar semiconductors the local elastic deformations of the media, along which lattice waves of the acoustic branch propagate, give rise to small perturbations of the electron energies. This is the electron–ACphonon interaction via the so-called deformation potential (DP) coupling, also named nonpolar acoustic interaction.



2
Dac Q 2Id uL V

1/2

(78)

where Dac is the acoustic deformation potential, Id is the material density, uL is the sound velocity for the longitudinal polarization mode, the only one contributing to acoustic phonon scattering by means of this coupling, and V is the normalization volume. The Dac value must be fixed from experiment (e.g., for GaAs based heterostructures it has been the subject of a quite great controversy [70–72], ranging from 7 to 16 eV depending upon whether the screening of this interaction is taken into account). This question will be reconsidered in sections devoted to mobility and power loss calculations. In polar semiconductors, when the crystals do not have points of inversion symmetry, local elastic deformations are accompanied by local changes in the electrical polarization of the media, which also give rise to small perturbations of the electron energies. This is the electron–AC-phonon interaction via the so-called piezoelectric (PE) coupling, also named polar acoustic interaction. This is essentially an isotropic interaction even in cubic crystals. Nevertheless, for zinc blende crystals an isotropic approximation is given by CjPE Q =



e2 h214
Aj Q 2Iuj *0 V Q

1/2

(79)

where h14 is the only element of the piezoelectric tensor different from zero, *0 is the static lattice dielectric constant, and uj is the sound velocity for the longitudinal (L), j = 1, or transverse (T), j = 2 3, polarization modes. The h14 value has been also controversial, (e.g., for GaAs based heterostructures [68, 73], ranging from 12 to 144 × 107 V cm−1 ). Dealing with bulk phonons a solid angle aver%2 age of this interaction can be made to obtain Aj Q = 64 15 [74], but one can also find Aj Q = 1. Particularly, in a Q2D system formed with heterostructures grown in (0, 0, 1) orientation in crystals with zinc blende symmetry, anisotropy is considered by the factors [75] AL Q =

gE ′ @′ E′ − E ′ dE ′ = −g ′ E′  ′

CDP Q =

(73)

Furthermore, in the matrix dielectric function one also sets =jQ = 0 and one talks about static screening according to (64). For energy loss calculations, the electron–AC-phonon interaction must be considered just a little inelastic or quasielastic scattering, which means that the Dirac delta function in the FGR can be expanded as



For cubic crystals this is an isotropic interaction given by the coupling constant

9qz2 q 4 2 qz2 +

3 q2

and AT Q =

8qz4 q 2 + q 6 4 qz2 + q 2 

3

(80)

which obviously introduces additional complications in calculations, which do not reveal substantial differences with fully isotropic approximations. For wurzite crystals, as one of the crystal phases of the GaN, there are five components of the piezoelectric tensor different from zero, h15 = h24 , h31 = h32 , and h33 , and the isotropic approximation can be found in [42].

4.1.2. Optical Phonon Scattering Dealing with long wavelength OP phonons, the dispersionless Einstein model is frequently assumed. This amounts to
=jQ =
=j0

as Q → 0

(81)

Transport in Semiconductor Nanostructures

557

where =j0 is the optical phonon frequency at the center of the Brillouin zone for the L-polarization mode (j = 1) and the T-polarization modes (j = 2 3). For sufficiently low temperatures, kB T ≪
=j0 , the Bose– Einstein distribution function is approximated simply by zero, which means there are no optical phonons to be absorbed. For other temperatures this factor is given by the complete nB 
=j0  after (67). The electron–OP-phonon interaction is essentially an inelastic scattering, so in the argument of the Dirac delta function one must set =jQ = =j0 . Furthermore, in the matrix dielectric function one also must set =jQ = =j0 and then talk about dynamic screening according to (64). The screening of this interaction has been frequently ignored (e.g., in mobility calculations for SHs [68] and QWs [32]) or considered as static screening (e.g., in polaron calculations for SHs [76] and QWs [57]), but also the pertinent temperature-dependent dynamical screening has been included in mobility calculations for SHs [77] and QWs [78]. In polar semiconductors the local elastic deformations of the media, along which lattice waves of the optical branch propagate, are accompanied by local changes of the polarization field, which give rise to small perturbations of the electron energies. This is the electron–OP-phonon interaction via the so-called Fröhlich interaction coupling, also named the optical polar interaction. For cubic crystals this is an isotropic interaction, where only the L-polarization mode participates, given by the coupling constant 

2%e2
=LO 1 CLO Q = −i *∗ V Q2

1/2

2Id =0 V

IM ′ ′ V0ill q = Ci2D qG2D ′ k k ill′ q k − k + q

(83)

1/2

(84)

where =0 is the optical phonon frequency and Dop is the optical deformation potential. This interaction is important for intervalley scattering between  and L or X points in the Brillouin zone for III–V compounds and related ternary compounds, where the edge zone LO phonons participate,

(85)

where the coupling constant is

(82)

with * the high-frequency and *0 the static lattice dielectric constants of the material. The commonly found values for GaAs are * = 1253 and *0 = 1082 [78], and for GaN are * = 95 and *0 = 537 [79]. In both nonpolar and polar semiconductors the local elastic deformations of the media, along which lattice waves of the optical branch propagate, give rise to other small perturbations of the electron energies. This is the electron–OP-phonon interaction via the so-called optical deformation-potential coupling, also named then nonpolar optical (NO) interaction. For cubic crystals this is an isotropic interaction given by the coupling constant CNO Q =

The electron–ion-impurity interaction is by far the most important interaction in delta-doped semiconductors and plays the main role in limiting the mobility at low temperatures in semiconductor heterostructures, as will be shown. Impurity scattering arises from the donors (acceptors), which supply electrons (holes) to the conduction channel in the semiconductor nanostructures. This is an essentially elastic scattering, since ions are very much heavier than electrons or holes. The Coulomb interaction between the carriers and the ion impurities, located at Ri = ri +
i , is well known in space representation and must be Fourier transformed to obtain the nonscreened Hamiltonian in a form suitable for calculation of matrix elements. The matrix elements for the bare interaction in a Q2D system are given by

2%e2 −iq·ri e *0 qS

(86)

with ri = xi ex + yi ey and S the normalization area. Because of the broken translational symmetry in the confinement direction there appears the form factor for the electron– impurity interaction as G2D ill′ q =

1 1 1 = − *∗ * *0

2
Dop

4.2. Electron–Ion-Impurity Interaction

Ci2D q =

where =LO is the longitudinal optical (LO) phonon frequency (e.g., for GaAs the most extended value is
=LO = 365 meV [78], while for GaN
=LO = 928 meV [79]). Here an effective lattice dielectric constant is introduced as



and a dispersionless model =0 = =LO . In GaAs the energy difference between  and L valleys is 03 eV, while it is 0.47 eV between  and X valleys, with the Dop values ranging from 4 to 10 × 108 eV cm−1 [80].



l∗′ ze−q z−zi l z dz

(87)

The matrix elements for the bare interaction in a Q1D system are given by IM ′ ′ V0i q = Ci1D qG1D ′ k k i′ q k − k + q

(88)

where the coupling constant is Ci1D q =

2%e2 iqx xi e *0 L

(89)

with L the normalization length. Because of the broken translational symmetry in the confinement plane there appears the form factor for the electron–impurity interaction as  G1D ∗ ′ 
K0 q
−
i  
d
(90) i′ q = where K0 is the modified Bessel function of the second kind and  (91)


−
i = y − yi 2 + z − zi 2

Transport in Semiconductor Nanostructures

558 4.2.1. Background Impurity Scattering The cleanness of the growth processes reduces the concentration of undesired impurities inside the conduction channels to low enough values, which prevents this mechanism from being relevant in transport phenomena in modulationdoped heterostructures, which has been experimentally verified. However, in delta-doped semiconductors the required impurities are in the conduction channels and are the most important scattering centers. In recent years the intentionally delta-doped QWs, inside the conduction channel, have also been investigated [81–84]. In these cases this mechanism is known as the background impurity (BI) scattering. Since impurities are somehow distributed inside the conduction channel, the summation on i to find the transition rate is replaced by an integration. In Q2D systems this is done as     BI Vinn′ k k ′ q2 =

a

−a

i

2  BI ′ q dzi nD zi Vinn ′ k k

(92)

where nD zi  is the profile of volume concentration of donor impurities and 2a is the QW width (for a SH, the lower limit must be zero and a must be understood as a certain effective layer width of the formed conduction channel). If necessary, a similar procedure can be done in Q1D systems, 

BI ′

Vi q 2 = ′ k k

i



B

BI ′ nd 
i  Vi q 2 d
i ′ k k

(93)

where nD 
i  is the local volume concentration of donor impurities and B is the quantum wire cross section.

layer width (for a SH the factor 2 must be omitted). In Q1D systems this is done as  i

i

2

q = 2



a+s+b a+s

BI ′ 2 nD 
i  Vi ′ kk q d
i

2

q dzi (94)

where nD zi  is the profile of volume concentration of donor impurities, s is the spacer layer width, and b is the depletion

(95)

4.3. Alloy Disorder Scattering Ternary compounds are usually fabricated by the substitution of one of the components atoms of a binary compound by atoms of the same group in the periodic table. The alloy disorder is due to the alien atoms in the host material, for instance, aluminum in Alx Ga1−x As instead of arsenic in GaAs. The alien atoms represent almost point defects in the crystal lattice of the host material, so that a certain short-range scattering potential is modeled as the interaction Hamiltonian. The matrix elements for the bare interaction result [86], AL ′ ′ V0i q = C AL qGAL ′ k k i′ q k − k + q

(96)

where the coupling constant is given by the interaction potential DAL with the alien atoms of radius R0 as follows: 4% D R3 3 AL 0

(97)

According to [86] the alloy potential values are the following: for x = 03 in Alx Ga1−x As, one has DAL = 156 eV; for y = 053 in Iny Ga1−y As, one has DAL = 055 eV; and for z = 053 in Inz Al1−z As, one has DAL = 04 or 13 eV. Here the form factor is GAL i′ q = ′ zi  zi G′ q

(98)

where G′ q is given by (68). Since alien atoms are somehow distributed inside or outside the conduction channel (e.g., as in Iny Al1−y As/ Inx Ga1−x As and Alx Ga1−x As/GaAs respectively), the summation on index i, which labels the alien atoms, to find the transition rate according to the FGR is replaced by an integration as  i

RI ′ nD zi  Vinn ′ k k

B∗

CiAL q =

Different doping profiles can be achieved by means of modulation-doped techniques in semiconductor nanostructures. The uniformly doped barrier is mostly found, although delta-doped semiconductor heterostructures, with impurities in the barriers, are also investigated [85]. The remote impurity (RI) scattering of electrons comes from the ionized donors placed out of the conduction channel in modulation-doped heterostructures. The inclusion of a spacer layer aims to reduce this mechanism, but this cannot suppress it completely, because the donors must be sufficiently close to the conduction channel to supply the required electrons efficiently. A SH is doped from one side exclusively. For simplicity symmetrically doped QWs and QWIs are considered to present quite general and compact expressions in this section. Since impurities are somehow distributed outside the conduction channel, the summation on i to find the transition rate is replaced by an integration. In Q2D systems this is done as RI ′

Vinn ′ k k



where nD 
i  is the 2D profile volume concentration of donor impurities and B∗ is depletion region cross section, whose concrete form depends on the symmetry of the QWI cross section and the desired profile for the modulationdoped region.

4.2.2. Remote Impurity Scattering



BI ′ 2

Vi ′ kk q =

AL ′ 2

Vi ′ kk q =

 1 AL ′ 2 N0 x1−x Vi ′ kk q d
i B 2 (99)

where 21 N0 x1 − x is the concentration of scattering centers, which is taken to be constant, N0 is the number of atoms per unit volume of the crystal, and B is the region where alien atoms are located.

Transport in Semiconductor Nanostructures

559

4.4. Interface Roughness Scattering Interface roughness arises from the transition region of one or two monolayer between the materials forming the heterostructure because of the effectively random distribution of alien atoms or, as also happens, it is due to inappropriate selection of growth temperatures and deposition rates leading to growth by islanding in 3D rather than advancing terrace edges in 2D. This way, instead of perfect planar interfaces one has irregular ones. Fluctuations along the interface are random and correlated in a Gaussian manner, with root mean square height 0 and correlation length M0 . This is taken into account in the confining potential to derive the corresponding matrix elements [87], which in the case of hard wall potentials leads to [88] IR ′ ′ Q = CIR qGIR V ′ k k ′ z0  k − k + q

(100)

where to keep the compact form of presentation the coupling constant is understood as    1 2 2 1/2
2 2 2 CIR q = % 0 M0 exp − q M0 2m∗ 4

(101)

(103)

j = OF

where the conductivity O = const. This is the well-known Ohm law. In this case the average energy of the electrons is slightly different from the one of the phonons, which are in thermal equilibrium at the temperature T of the thermal bath. In this regime the energy transfer from the electron system to the phonon system can be disregarded, because this is not so important as the momentum transfer to the different scattering centers, which leads to the resistance of the concrete device.

5.1. Distribution Function in Linear Approximation Dealing with a low electric field a linear approximation in its strength is enough for the nonequilibrium distribution functions related to the different subbands. Thus for solving the system of BTEs one starts from the assumption that f k = f0 k+f1 k and

and the remaining factor as  d d′  z  = GIR ′ 0  dz dz z=z0

One can talk about a low field when experimentally there is a linear dependence between the current density and the applied electric field,

f1 k ≪ f0 k

(104)

0 f k

(102)

with z0 the corresponding interface coordinate. The parameter values depend on the concrete grown interface (e.g., for a given Iny Ga1−y As/Inz Al1−z As heterostructure they have been estimated as 0 = 0283 nm and M0 = 10 nm [88], while for a given Si/Si 1−x Gex heterostructure they have been chosen as 0 = 05 nm and M0 = 10 nm) [12]. Another approach to the interface roughness scattering has been recently developed in [89], where an effective Hamiltonian is derived taken into account the fluctuations 0 r at the interfaces and certain modulated envelope functions in the confinement direction are proposed.

5. LOW-FIELD TRANSPORT Studying the steady-state transport in semiconductor nanostructures it has been experimentally established that in the graphics of the drift velocity as a function of the applied electric field (vd vs F ) three regions can be clearly distinguished: the one of linear dependence, the one of nonlinear dependence and saturation, as well as the region when drift velocity decreases with increasing field [90, 91]. The latter is the region of negative differential resistance (NDR). As in the case of bulk semiconductors, there is a linear response of the system to the applied electric field in a range of strength values very attractive for device design, particularly related to the HEMT performance, as was highlighted in the Introduction. Furthermore, the investigation of different scattering mechanisms can be carried out without a series of disturbing phenomena arising in the nonlinear response regime.

is the equilibrium distribution function and where 1 f k is a linear term in F = F . Particularly in Q2D systems, this can be also understood as the development of f k in Legendre polynomials, whereas the variable angle P is taken to be the one between k and F, up to the second term [92]. The most employed method to solve the BTEs is to set the perturbation to the distribution function in the form e F · k f0 k; k
d 0 e
f E; E = ∗F ·k m dE 

f1 k =

(105)

where ; k or ; E is just a certain quantity with time dimensions, but without any physical meaning at this stage, depending on energy and not on wavevector direction since all the scattering mechanisms are assumed or approximated by isotropic interactions. In the case of elastic scatterings this assumption corresponds to the so-called relaxation time approximation and a closed form is obtained to calculate ; E. But when any inelastic scattering is considered, an iterative procedure is commonly followed [92, 93], on the line originally developed in [94, 95], to find ; E [also denoted as Q E in many works]. Both cases are discussed. This way, for the steady state transport, taking account of the PDB, straightforward manipulations transform the system of BTEs into a system of integral equations,    k′ (106) S′ k k ′  ; E − R ;′ E ′  1= k ′  k ′

where R = cos S with S the scattering angle between k and k ′ for Q2D systems, while R = ±1 for Q1D systems, as well as 0

S′ k k ′  = W′ k k ′ 

1 − f′ E ′  0

1 − f E

(107)

Transport in Semiconductor Nanostructures

560 5.1.1. Elastic Scattering

iterative formula is employed:

For any elastic scattering the electron keeps its total energy 0 0 and f′ E ′  = f E as well as ;′ E ′  = ; E, since both quantities depend on the total energy E k. Because of the delta Dirac function in the FGR, after going from summation in k ′ to integration in E ′ one obtains the actual relaxation time corresponding to subband  for the considered interaction, 



 EL E′ 1 S E E′  R 1 − R = ;EL E ′ R E

1/2 

(108)

 + 1/2 E ′ + ;LO ′ m−1 E′  E ′  R    − 1/2  LO E′ − LO − S E E ;LO + ′  R ; m E − R ′  m−1 E′  E ′  R

S EL EE′ R··· =

R





2%

0

S EL EE′ cos S···dS

(109)

S EL EE′ R··· = S EL EE′ 1···

R

+S EL EE′ −1···

(110)

in Q2D and Q1D respectively. When several elastic scatterings are present, the relaxation time, also called transport lifetime or classical relaxation time, is given by 1 ;ES E

=



1 ;EL E

EL

(111)

5.1.2. Inelastic Scattering For any inelastic scattering the electron does not keep its 0 0 total energy and f′ E ′  = f E as well as ;′ E ′  = ′ ; E, since E k = E′ k . Because of the delta Dirac function in the FGR, after going from summation in k ′ to integration in E ′ for the LO-phonon scattering one obtains 

 + 1/2 E ′ + LO S E E  R ; E − R 1= ;′ E′  E ′  R    − 1/2  LO E′ − − LO (112) + ;LO S E E ′  R ; E − R ′ E′  E ′  R 

LO

+ ′

LO 



± where formally E ′ = E′ ±
=LO is the kinetic energy in the subband, into which the electron is scattered after absorption (+) or emission (−) of energy
=LO . Note that according to (109) or (110) once the summation over R is accomplished, one has a system of algebraic equations. Therefore, this operation must be carried out before any further calculation. The already mentioned iterative procedure is frequently followed to find the quantities ;LO E [96, 97], which in this case cannot be interpreted as relaxation times, since randomization of energy cannot be reached by the exclusive actions of inelastic scattering mechanisms. The following



+ LO S LO E E ′  R ; m E − R



(113)

LO Iterations start from the zero-order iteration ; 0 E = 0, which would correspond to equilibrium, and follow until the step M, when the values converge within the pertinent error. The first-order iteration takes into account scattering out processes exclusively,

where E′ = E + E − E′ is the kinetic energy in the subband, into which the electron is scattered, and formally EL = DP, PE, AL, IR, according to elastic scatterings introduced. Note here that 



1=

1 LO ; 1 E

=

  LO  + LO − S E E E E ′  R ′  R + S

′  R

(114)

and it works well at low temperatures, kB T ≪
=LO , when the scattering-in processes due to phonon absorption can be neglected due to the scarcity of LO phonons according to (67), and those due to phonon emission can also be neglected since only a few electrons have energy
=LO or greater. This is known as the low-temperature relaxationtime approximation. If one considers that most of the electrons have energies very much greater than the phonon energy, the latter is neglected with respect to the former and one simply sets ± ;LO E = ;LO ′ E  to obtain the high-energy relaxationtime approximation:   + 1/2   LO 1 E′ + = S E E′  R 1 − R ;HE E ′ R E   − 1/2   LO E′ − (115) S E E′  R 1 − R + E ′  R It works well at high temperatures, kB T ≫
=LO , when according to the PDB in the form (59), this can be considered an elastic mechanism. This is also known as the hightemperature relaxation-time approximation.

5.1.3. Elastic and Inelastic Scatterings Altogether When both elastic and inelastic scattering must be considered altogether, the same iterative procedure is followed according to the formula 1= +

; m E ;ES E 

′ R

+



′ R



+ LO S LO E E ′  R ; m E − R



− LO S LO E E ′  R ; m E − R





+ E ′ E − E ′ E

1/2 1/2



+ ;LO ′  m−1 E′ 



− ;LO ′  m−1 E′ 

(116)

This situation deserves a brief comment. It is generally argued that owing to inelastic LO-phonon scattering, which cannot randomize energy and so prevent the electron gas from returning to equilibrium, the idea of a relaxation time

Transport in Semiconductor Nanostructures

561

is not valid. That is correct when only this kind of scattering mechanism is present, which is an ideal situation. But when there are also present elastic scatterings, as in real systems, randomization of energy is reached and electron gas returns to the equilibrium state once the external agents are switched off. This is a fact; otherwise one could never reobtain the equilibrium state in samples employed in experimental measurements of transport properties, where optical phonons coexist together with acoustic phonons and other scattering centers. Therefore the quantity ; M E obtained after the required convergence of the iterative procedure actually is the relaxation time, in spite of the fact that no closed form could be derived to find it and only an approximate value can be calculated. The low-temperature relaxation time approximation, taking account of (114), results in 1 ;LT E

=

1 ;ES E

+

  LO  + LO − S EE ′ R+S′ EE′ R

′ R

(117)

while the high-temperature approximation, taking account of (115), results in 1 1 1 = ES + ;HT E ; E ;HE E

(118)

5.2. Scattering Rate and Relaxation Rate Two different quantities are important in the study of transport phenomena: the scattering rate and the relaxation rate, also called the momentum relaxation rate. The former is related to the lifetime of a given electron state, bearing an essential quantum character, while the latter is connected to the mean free pass of an electron in the given state, bearing a quasiclassical character. The scattering rate of the electron state  k takes account of the scattering out processes exclusively and is defined as S k =

 1 W ′ k k ′  = ;s k ′ k ′ 

(119)

where W′ k k ′  includes one or more interaction couplings, according to (60). This is the equivalent of the imaginary part of self-energy in MBT, which gives the level broadening of the energy corresponding to this quantum state. This way the scattering time ;s k, also known as quantum scattering time or quantum relaxation time, is nothing else but the quasiparticle lifetime of MBT. On the other hand, the relaxation rate is defined for elastic scattering mechanisms as     ES 1 E′ = W ′ EE′  1−R (120) M E = ; E ′ k′  E where ; E is the relaxation time, also known as classical scattering time or transport relaxation time, and ES W ′ E E′  includes one or more of the elastic scatterings, depending on the desired result. According to the comment in previous section, when inelastic scatterings act together with elastic scatterings one can also understand the

relaxation rate as M E =

1 1 = ; E ; M E

(121)

where ; E = ; M E is the convergent value found after M iterations of the formula (116). Many works have been devoted to finding the scattering rates or the relaxation rates due to the different scattering mechanisms. Most of the works aim to compare different interaction couplings or to establish the role of screening in these interactions. Bulk phonons as well as GaAs based delta-doped semiconductors and Alx Ga1−x As/GaAs based semiconductor heterostructures are always considered in the works cited; otherwise it is specially remarked. Studying the electron–phonon interaction early works on this subject made use of the so-called momentum conservation approximation (MCA) in QWs, considering that the momentum is conserved in confinement directions in spite of the broken translational symmetry [98, 99]. It is argued that the MCA gives the main contribution to the scattering or relaxation rates and, furthermore, analytical expressions for the transition rates are easily obtained. But computational facilities made it unnecessary and more accurate calculations are usually carried out and reveal that this approximation is not so good [67]. In the low-field regime the low-temperature region, where electrons do not reach the LO-phonon emission threshold, is clearly distinguished from the intermediateand high-temperature regions. A similar situation is found in the high-field regime, bearing in mind the electron temperature. The approximate limits of these regions can be established only after concrete calculations of experimentally measurable quantities and vary from one material to another. The electron–phonon interaction is investigated in [68]. The FSH variational wavefunction of the lowest subband for a SH is taken to calculate scattering and relaxation rates. The anisotropic factors (80) for PE coupling are used and static screening is taken into account by means of the RPA dielectric function, when only the lowest subband is populated and neglecting intersubband scatterings. An interesting comparison between DP and PE couplings is presented, as illustrated in Figure 1. Notice that the role of the PE scattering is more important as the electron energies are smaller as well as the temperature and areal concentration are lower. The LO coupling is also included, ignoring the screening effect. The numerical results for the inverse of −1 the perturbation function ;1M E [there denoted as Q−1 E] are compared with the relaxation time obtained in the lowtemperature and high-energy approximations, as illustrated in Figure 2. The contribution of acoustic phonons is also displayed. Note that the first approximation works at low temperatures well when the electron energies do not reach the LO-phonon threshold, while the second approximation works well at high temperatures when electron energies are larger than twice the LO-phonon energy. The IBM wavefunctions for rectangular QWs are employed in [74] to calculate relaxation rates, where also these couplings are compared (there the isotropic version for PE coupling is assumed); furthermore, it can be noticed that the DP relaxation rate coincides with the DP scattering rate, which as a function of the total energy has the same staircase form of the DOS for a Q2D system and presents the same

Transport in Semiconductor Nanostructures

562 (a) 15.0

(a) 4000.0 ns = 1×1011 cm–2 T = 20 K

DP

3000.0

φ–1 (109 s–1)

9

τ–1 (10 s–1)

10.0

5.0

τ–1 LT T = 20 K ns = 1×1011cm–2

2000.0

PE

τ–1 HE

1000.0

0 0.0

1.0

2.0

3.0

4.0

0.0

5.0

10.0

20.0

30.0

40.0

50.0

60.0

E/EF

E/EF (b) 6000.0 (b)

30.0

τ–1 LT

ns = 1×1011 cm–2 DP

φ–1 (109 s–1)

20.0

9

τ–1 (10 s–1)

T = 40 K

PE

10.0

4000.0 T = 300 K ns = 1×1011 cm–2

φ–1 2000.0 –1

τac 0 0.0

0.0 1.0

2.0

3.0

4.0

5.0

10.0

20.0

30.0

40.0

τ–1 HE 50.0

60.0

E/EF

E/EF Figure 1. Comparison between DP and PE couplings for two values of temperature. Reprinted with permission from [68], T. Kawamura and S. Das Sarma, Phys. Rev. B 45, 3612 (1992). © 1992, American Physical Society.

Figure 2. Numerical results for the inverse of the perturbation function Q−1 E are compared with the relaxation time obtained in the low-temperature and high-energy approximations. Reprinted with permission from [68], T. Kawamura and S. Das Sarma, Phys. Rev. B 45, 3612 (1992). © 1992, American Physical Society.

discontinuities of the DOS for a Q1D system. The screening effect is ignored, since relatively low electron concentrations are considered. Similar calculations are carried out in the KBA formalism with a constant energy level broadening in [97], where one can appreciate that the discontinuities in the DP scattering rate, in both Q2D and Q1D systems, are smeared out because of the finite lifetime of electron states. For a weak piezoelectric material such as GaAs in the conduction channel of a semiconductor heterostructure, both the scattering and the relaxation rates due to the DP coupling are always larger than the ones due to PE coupling, except for very low electron energies, as can be appreciated in Figure 1 taken from [68]. Polar optical phonons always deserve special interest, since they play the main role in transport phenomena at high temperatures and the electron amount. Earlier works on this subject are [100, 101]. For a SH in the just cited [68] the quantity ;1−1M E [there denoted as Q−1 E] is calculated and compared with high-temperature and low-temperature −1 E] and approximations ;1−1 E [there denoted as ;HE −1 −1 ;11 E [there denoted as ;LT E] respectively. In this work screening was ignored. For a QW the IBM electron confinement is assumed in [67] to calculate the scattering rate and compare with the high-temperature approximation result of

the momentum relaxation rate; furthermore, the effect of phonon confinement is also investigated according to a simple model which considers LO phonons completely confined to the conduction channel [57]. Similar calculations, with the same electron confinement model, but a different simple phonon confinement model, are carried out in [58]. Comparison of scattering rates and relaxation rates can be found also for QWs [98, 99]. Screening of this interaction is assumed to be small and ignored in these works, because relatively low electron concentrations are assumed. Phonon confinement is also considered in scattering rate calculations of [102], where importance of electron–interface– phonon interaction is highlighted for narrow QWs. For the widely employed material GaAs in the conduction channel of a semiconductor heterostructure, both the scattering and −1 −1 the momentum relaxation rates ;LT E and ;HE E due to the LO coupling are always larger than the ones due to electron–AC-phonon interaction, via DP and PE couplings together, except for energies below the LO-phonon emission threshold, as can be appreciated in Figure 2 taken from [68]. The electron–ion-impurity interaction is investigated in many works. The difference between the scattering time and the relaxation time is pointed out in [76] for a SH at zero temperature. Important results are presented in [103] for a

Transport in Semiconductor Nanostructures

563

QW, where IBM wavefunctions and energies are employed. Analytical expressions of the momentum relaxation rate are obtained for BI and RI scatterings. Screening is ignored in this work. A detailed investigation is presented in [104] for a delta-doped semiconductor, where SEL wavefunctions and energies are employed. Scattering rates due to electron–ionimpurity interaction are calculated for two occupied subbands. Comparison of screened form factors is done when the RPA, diagonal RPA, and TFA matrix dielectric functions are used. The IBM wavefunctions and energies for a rectangular QWI are used in [105], where also the MCA was applied to obtain analytical expressions for scattering and relaxation rates due to electron–phonon interaction via DP and LO couplings. A similar investigation with the same model is carried out in [106]. In these papers screening is disregarded. The IBM electron confinement for a cylindrical QWI is used in [107, 108] to calculate scattering rates due to acoustic phonons via DP, PE, and LO couplings. In the latter paper the static screening is taken into account by means of the RPA result for the 3D case, when only the lowest subband is occupied and intersubband transitions are ignored. According to these calculations the screening effect for optical phonons is larger than for acoustic phonons, but this may be a misleading conclusion, since a true Q1D dielectric function as well as distinction between static and dynamical screening is required for a more precise evaluation. On the other hand, LO scattering rates are larger than the AC scattering rates, as in the Q2D case in a similar temperature range. The electron–ion-impurity interaction is investigated in many works. Analytical expressions of the relaxation rate are obtained for BI and RI scatterings in [106] for a QWI, where IBM wavefunctions and energies are employed. Screening is ignored in this work.

5.3. Low-Field Mobility Theoretical and experimental works have been widely addressed low-field, linear, and dissipationless transport along the conduction channel. As in the case of bulk semiconductors, the usual transport coefficient to characterize the low-field regime in semiconductor nanostructures is the Ohmic or conductivity mobility , referred to simply as the mobility and given by O = en

(122)

where n = nS is the areal or n = nL is the linear electron concentration in Q2D or Q1D systems respectively. Notice that the mobility is negative for electrons (e < 0) and positive for holes (e > 0); this way the conductivity is always positive. For a given subband it must be calculated as  =

e ; E m∗ 

(123)

where ; E =



k



0 d f E ; E E − dE 0 k f E

(124)

is the statistically averaged value of the quantity ; E, which has the meaning of relaxation time when elastic scatterings are involved but is just a time-dimension quantity when calculations involve LO-phonon scattering exclusively, in accordance with the discussion of (116). Note that for a highly degenerate electron gas, when the equilibrium distribution can be approximated by the Heaviside function, one obtains  ≃

e ; E  m∗  F

(125)

which means that only electrons in a very narrow neighboring of the Fermi line, in Q2D systems, or the Fermi points, in Q1D systems, participate in charge transport. Frequently mobility calculations are carried out in the SQL, considering that only the lowest subband is occupied and neglecting intersubband scattering, which means that everywhere one sets  = ′ = 1, which works quite well for many SHs and narrow QWs or QWIs [109]. The electron concentration cannot be very high in order to ensure the Fermi level lies quite below the bottom of the first excited subband; furthermore, the thermal energy must be low enough to avoid intersubband transitions even after LOphonon absorption processes. For multisubband transport, according to (52), the average mobility results [21, 35]:   n = (126)  n

It must be pointed out that experimental determinations are carried out through Hall effect measurements, when samples are placed in a constant and uniform low magnetic field Bz perpendicular to the conduction channel. The usual quantity measured is the Hall mobility [110], H = ORH =

OFy jx Bz

(127)

where O is the conductivity when no magnetic field is applied and RH is the Hall coefficient, which in turn is determined by Bz , the Hall electric field Fy , and the current density jx along the applied electric field. The Hall mobility is calculated according to an expression similar to (126), but where H  =

e ;2 E m∗ ; E

(128)

The Hall ratio is defined as H /. It varies between 11 and 14 according to several calculations for Alx Ga1−x As/ GaAs based heterostructures [74, 111] in the commonly found ranges of temperatures, electron concentrations, and QW widths. This ratio is expected to be in similar ranges for other materials. Therefore in the usual log–log graphics quite often-calculated mobilities are compared with measured Hall mobilities. Mobility calculations or measurements in the low-field regime can be found in a very large number of articles, where a quite enormous amount of results are reported. An important part of these works is devoted to establishing the role of different interaction couplings in limiting the

Transport in Semiconductor Nanostructures

564 mobility and the relevance of the screening effect. The temperature and concentration dependence, as well as the QW width, the QWI radius, or the spacer-layer thickness dependence, has been widely investigated both in theoretical and experimental works. Electron transport is investigated and bulk phonons are assumed, and the SQL approximation for calculations is justified, in the works cited; otherwise it is specially mentioned. The most studied heterostructures are AlAs/GaAs and Alx Ga1−x As/GaAs, where the supplied electrons or holes are mainly confined to the gallium arsenide. These heterostructures are remarkable by their excellent matching of lattice constants at the interfaces. Pseudomorphic heterostructures, such as Iny Al1−y As/Inx Ga1−x As, where the supplied electrons or holes are mostly accumulated in the gallium indium arsenide, are often investigated. In recent years the Alx Ga1−x N/GaN heterostructures, where electrons and holes are basically concentrated in the gallium nitride, have also attracted the attention of research work. A vast review of the band parameters for III–V compounds and their ternary alloys, as well as the characteristics of frequently fabricated heterostructures, can be found in a recent review [2]. Mobility has been mostly measured in Alx Ga1−x As/GaAs SHs [112–118], including the highest values ever reported [10, 119], which were quoted previously. However, many mobility measurements have been carried out for lateral transport in Alx Ga1−x As/GaAs based QWs and SLs [120– 132] because these types of semiconductor nanostructures have at least two advantages over the SH type: they can have larger electron concentrations and the conduction channel thickness can be well defined, thus allowing the manipulation of these parameters to find the optimal values of mobility and conductivity. The Alx Ga1−x As/GaAs based heterostructures are always considered in the works cited; otherwise it is explicitly indicated. Electron–phonon interaction has attracted the attention of a lot of works. An interesting investigation is presented in [68] for a SH, with the model and approximations already mentioned in the preceding section. The relative role of acoustic phonons, via DP and PE couplings, and LO phonons can be appreciated and the main conclusion concerns the acoustic deformation potential energy, which is taken as Dac = 12 eV after comparison of the calculated with the experimentally determined parameter ac in the well known temperature dependence, 1 1 = + ac T  AC 0

(129)

in the range 4–40 K when the electron–LO-phonon interaction can be neglected. The piezoelectric constant is taken h14 = 12 × 107 V/cm. A similar temperature dependence of the DP-scattering limited mobility had been obtained before for a QW with simple IBM electron confinement and equipartition approximation for phonons in some works, as in [133], where also expressions for PE-scattering limited and LO-scattering limited mobilities can be found, as well as the one when the optical deformation-potential coupling must be considered. The latter coupling was previously investigated in silicon inversion layers, as in [134]. Although the DP coupling dominates in such a weakly piezoelectric

material as GaAs, the PE coupling cannot be disregarded for accurate calculations. A detailed study of the phonon-limited mobility is also presented in [78] for a QW, with the IBM wavefunctions and energies and the LO confinement model already employed to calculate scattering rates [67] and mobilities [32] in order to compare with results when bulk LO phonons are considered. The effect of this phonon confinement reveals as an improvement of mobility values in less than 25% for a QW width about 10 nm, but it can be considerably larger in narrow QWs. The full temperature-dependent static and dynamical screening of electron–AC-phonon and electron– LO-phonon interactions is considered by means of the RPA matrix dielectric function. The relative role of DP, PE, and LO couplings can be appreciated; mobility is mainly limited by AC phonons up to 50 K. The screening effect is important not only for electron–AC-phonon interaction, but also for electron–LO-phonon interaction, where this means an increasing of mobility values larger than 30%. The acoustic deformation potential energy is taken as Dac = 135 eV and the piezoelectric constant is taken as h14 = 144 × 107 V cm−1 . It must be noted that the Dac exact value is quite controversial, as was indicated, but since it must be fixed from experiments it will depend on the models and approximations employed in calculations as well as on h14 , which in turn gives rise to further controversy, as can be seen in [68, 135]. Special attention has been paid to the PEscattering limited mobility in [75, 136] as well as in [137], where the conclusion is drawn that this mechanism dominates over the DP scattering in CdS or ZnO, but not in GaAs, InSb, or InAs based surface layers. Electron–ion-impurity interaction has been widely investigated. Important efforts in semiconductor heterostructures have been put in the cleanliness of the growth process for avoiding BI scattering, considered in [128, 138] for QWs to estimate its relative role in limiting the mobility. The inclusion of a spacer layer aims to reduce the RI scattering; it has been considered in many works, such as [103] and some others cited. Mobility calculations and comparisons with their own experiments can be found in [139], where the role of the spacer layer is carefully investigated. The RI scattering is considered together with electron–phonon interaction via DP, PE, and LO couplings. FHV wavefunctions are employed and for elastic scatterings static screening is taken into account by the screening factor of [41]. The screening of the electron–LO-phonon interaction is ignored. The electron–ion-impurity interaction dominates in the lowtemperature range up to 40 K, where the mobility is increased by more than an order of magnitude when the spacer-layer thickness varies from 0 through 18 nm. There is reported a maximum value  = 15 × 106 cm2 V−1 s−1 at T = 1 K when the spacer thickness is 30 nm. The concentration dependence of the mobility and the contribution of the different scattering mechanisms are also shown. Similar works, where theory and experiments are presented, can be found in [140] for SHs, as well as in [141–143] for QWs. Multisubband transport in semiconductor heterostructures has been investigated in several works (e.g., in [144–146]). The IBM and the FBM electron confinement are used in [147, 148] for a MQW to calculate the RI-scattering limited

Transport in Semiconductor Nanostructures

zero-temperature Hall mobility, investigate its dependence on the spacer-layer thickness, and compare with experiments of [120]. The screening effect is considered by a simple TFA dielectric function. The FHV wavefunctions are employed in [149, 150] for a SH to calculate the low-temperature mobility limited by DP, PE, BI, RI, and AL scatterings. Two-subband calculations and comparison with their own experiments are in [88]. In this work SEL wavefunctions and energies are employed to calculate the two-subband Hall mobility limited by DP, PE, LO, BI, RI, IR, and AL scatterings. Static screening for elastic scatterings is considered in RPA approximation with the finite temperature polarizability according to the Maldague expression, as reviewed in the section devoted to the screening of the scattering mechanisms. Calculated and measured mobilities as functions of the QW width are presented for an Alx Ga1−x As/GaAs based QW and for a pseudomorphic Alx Ga1−x As/Iny Ga1−y As based QW. The role of the different scattering mechanisms is investigated. The IR and AL scatterings give more contributions for narrow QWs. The latter scattering is by far more important for the Alx Ga1−x As/Iny Ga1−y As based QW. There is a region where the mobility decreases as the QW width increases, due to the connection of intersubband scattering. This fall is more abrupt at T = 77 K than at T = 300 K due to the thermal broadening around the Fermi level, which crosses the second subband bottom for a given QW width. After this fall, the mobility increases again with the increasing QW width. In a multisubband model this feature will be repeated as many times as the number of considered subbands, but with a reduction of the fall, as the subband is higher. Such an oscillatory dependence of the measured mobility on the QW width has been reported for a Alx Ga1−x As/GaAs based QW in [151]. The role of the alloy-disorder scattering depends on the extension of the envelope functions in the region where alien atoms are distributed. For instance, in a GaAs/Alx Ga1−x As based heterostructure electrons are mainly confined to the GaAs material; even SEL envelope functions slightly penetrate into the barrier material and, consequently, the AL scattering play a minor role in the electron-state lifetime or in limiting the mobility. Nevertheless, it has been included in calculations of [133, 141, 142], as well as in [152], where the effects of exchange and correlations are incorporated through local field corrections of many-body effects. This estimation is also supported by mobility calculations carried out in [88, 149, 150] cited previously. However, in Iny Ga1−y As/Inz Al1−z As or Iny Ga1−y As/Alx Ga1−x As based heterostructures electrons are mainly confined to the Iny Ga1−y As material and, regardless the chosen model for envelope functions and their penetration into the barrier material, the AL scattering plays an important role in electron-state lifetime and in limiting the mobility. This can be appreciated in [150, 153], for SHs and in [88] for QWs. Mobility calculations for Inz Al1−z As/GaIn, Alx Ga1−x As/GaAs, and Iny Ga1−y As/InP based QWs are presented in [86], where the importance of this mechanism for narrow QWs is stressed. Similar calculations are carried out for bulk Si and for Si/Gex Si1−x in [154], where it is shown that the AL scattering reduces the mobility in an order of magnitude.

565 The interface-roughness scattering depends on the overlapping of the envelope function with the height of interface fluctuations ( 0 ), so it becomes quite important for narrow conduction channels. This can be clearly appreciated in mobility calculations for QWs [88] and conductivity calculations for QWIs [87]. This mechanism has been included in calculations of [155, 156]. This mechanism is also studied in Iny Ga1−y As/Alx Ga1−x As based QWs [157] and SHs [158], where experimental measurements are also reported. Thus, the IR scattering presents a clear dependence upon the transverse dimensions of the conduction channels. The IBM wavefunctions and energies have been widely employed for mobility calculations in rectangular QWIs in [103, 106, 159–161] to investigate the role of BI, RI, DP, and LO scatterings. The latter mechanism is specially considered in [162]. Similar calculations can be found in [74, 137, 163] to investigate the PE scattering in an isotropic approximation. Screening is ignored in all these works. Another sort of IBM electron confinement is used for mobility calculations in cylindrical QWIs in [164, 165] to investigate the role of BI, RI, DP, PE, and LO scatterings. Static screening for a finite temperature is considered for elastic scatterings. The IR scattering is investigated in [166]. Two-subband transport is studied in [167] to investigate RI and IR scatterings. Screening is taken into account in this work by means of the diagonal approximation of the RPA matrix dielectric function; this way the inversion of an eight-index matrix is avoided. The conclusions about the relative roles of the different scattering mechanisms are similar to the ones in Q2D heterostructures. Higher mobility values could be expected if one realizes that, with respect to the Q2D systems, the new direction of quantization imposes an additional restriction to available states for electron scattering. However, comparison depends on the temperature range as well as the selected concentrations and transverse dimensions of the conduction channel. The characteristics of different types of AlN/GaN and GaN/Alx Ga1−x N based heterostructures attract the attention of research work. A special feature on this heterostructures is the possible sheet polarization at the interface depending on the crystal phase of matched materials [168]. The piezoelectric interaction plays an important role in these structures. Hall mobility measurements are presented in [169] for the usual 4–200 K temperature range and considerably high electron concentrations. The experimental values are about  = 28 × 103 cm2 V−1 s−1 up to T = 80 K for nS = 15 × 1013 cm−2 , and it falls to  = 074 × 103 cm2 V−1 s−1 at T = 300 K for nS = 21 × 1013 cm−2 . One of the highest values reported is  = 6 × 104 cm2 V−1 s−1 at T = 005 K for nS = 24 × 1012 cm−2 [170]. The highest Hall mobility at room temperature is reported in [171],  = 2019 × 103 cm2 V−1 s−1 for nS ≃ 1013 cm−2 . Similar measurements are also reported for low temperatures in [172, 173], where the role of the AL scattering is investigated, and in [174], where the maximum intrinsic mobility  ≃ 105 cm2 V−1 s−1 is attained. For very high temperatures, between 300 and 500 K, measured mobilities are reported in [175]. Temperature dependence is experimentally investigated in [176]. Acoustic-phonon limited mobility is calculated in the Bloch– Gruneisen regime, when the phonon population is given by (67), for a GaN based QW in [177], where the results for

Transport in Semiconductor Nanostructures

566 the zinc blende and wurzite crystal phases are compared. Self-consistent calculations are carried out in [178] for GaN/ Alx Ga1−x N and GaN/Inx Ga1−x N heterostructures, where the increased IR scattering is explained by the enhanced concentration of electrons near the interfaces due to their spontaneous sheet polarization, present in different types of nitride interfaces [168]. The Si/Si1−x Gex heterostructures are frequently investigated. The supplied electrons or holes are mainly confined to the Si or to the Si1−x Gex layers respectively. The mismatch of lattice constants is not negligible and they are known as strained layers. Transport in a silicon QW is investigated in [179], where BI, RI, and IR scatterings are considered. Mobility measurements are reported in [180–182], where high values are attained:  = 18 × 104 cm2 V−1 s−1 at T = 77 K and  = 283 × 103 cm2 V−1 s−1 at T = 300 K. Mobility calculations and comparison with their own experiment can be found in [183]. The role of the different scattering mechanisms (RI, DP, and IR) can be clearly appreciated by comparison of the calculated and measured values. For electron transport with a concentration around nS = 2 × 1012 cm−2 in a silicon channel of a Si1−x Gex /Si/Si1−x Gex QW (x = 038), when the QW width is d = 75 nm, the results are presented in Figure 3. Three regions can be observed. At very low temperatures, the RI scattering dominates, between 20 and 80 K, both RI and DP scatterings are important, and for higher temperatures, the DP scattering is the main interaction. The contribution of the interface roughness is certainly negligible. For hole transport with a concentration around nS = 1 × 1012 cm−2 in

the germanium channel of a Si1−y Gey /Ge/Si1−y Gey QW (y = 04), when the QW width is d = 7 nm, the results are presented in Figure 4. Note that in this case the IR scattering is dominant up to 100 K; for higher temperatures the contribution of the DP scattering is also important. Hole mobilities are also calculated in [184] and experimental results are reported in [185, 186] Heterostructures based on II–VI binary compounds and their alloys are also investigated. In a CdS/ZnSe based QW, where electrons are in the cadmium sulfide, for a QW width of about 4 nm and a concentration nS = 15 × 1013 cm−2 , the mobility does not exceed 380 cm2 V−1 s−1 at room temperature [187]. An experimental value of  = 6 × 104 cm2 V−1 s−1 at T = 2 K for nS = 3 × 1011 cm−2 is reported for a CdTe/Cd1−y Mgy Te based QW in [188]. Electron–ion impurity interaction plays the crucial role in limiting the mobility in delta-doped semiconductors, because the delta-doped layer is itself the conduction channel. This means that in these structures the BI scattering is dominant in limiting the mobility and one must expect mobilities lower than in semiconductor heterostructures. But, on the other hand, all the undesirable effects of the interfaces are suppressed, which may be an advantage for certain devices where also considerable high carrier concentrations are required. Among the highest mobilities in a single delta-doped layer are the ones reported in [189], namely  = 126 × 104 cm2 V−1 s−1 at T = 77 K for nS = 3 × 1012 cm−2 and  = 118 × 104 cm2 V−1 s−1 at T = 77 K for nS = 6 × 1012 cm−2 . 107

RI 5

DP

RI

10

106

µ (cm2 V–1 s–1)

µ (cm2 V–1 s–1)

DP

105

IR 104

104

Total

Total 103

10 10

100

100

T (K)

Τ (Κ) Figure 3. Comparison of mobility calculations and measured values, for electron transport with a concentration around nS = 2 × 1012 cm−2 in a silicon channel of a Si1−x Gex /Si/Si1−x Gex QW x = 038, when the QW width is d = 75 nm. Reprinted with permission from [183], S. Madhavi et al., Phys. Rev. B 61, 16807 (2000). © 2000, American Physical Society.

Figure 4. Comparison of mobility calculations and measured values, for hole transport with a concentration around nS = 1 × 1012 cm−2 in the germanium channel of a Si1−y Gey /Ge/Si1−y Gey QW y = 04, when the QW width is d = 7 nm. Reprinted with permission from [183], S. Madhavi et al., Phys. Rev. B 61, 16807 (2000). © 2000, American Physical Society.

Transport in Semiconductor Nanostructures

In this work the mobility in a double delta-doped layer is also investigated as a function of the interlayer distance d. For d = 144 nm a peak mobility  ≃ 100 × 104 cm2 V−1 s−1 at T = 77 K is observed, but it is lower than the corresponding value in the single delta-doped sample for the similar concentration nS = 6 × 1012 cm−2 . Experimental results in single, double, and triple delta-doped semiconductors are also reported in [190–192]. A detailed investigation is presented in [193] for this type of Q2D system, where SEL wavefunctions and energies are employed. The BI-scattering limited mobility at zero temperature is calculated for two occupied subbands and the influence of the static screening through the RPA matrix dielectric function is investigated when up to three empty subbands are taken into account. Comparison between the so-called quantum [;S k is placed in (123) instead of the relaxation time] and transport mobilities is done. Results are in good agreement with experimental data of [194]. Similar calculations, with the same model and approximations, are carried out in [104], where they compare the calculated mobilities when the Thomas–Fermi approximation, the diagonal RPA, or properly RPA matrix dielectric functions are employed. Other mobility calculations in delta-doped semiconductors can be found in [170, 195, 196]. Multisubband calculations, with up to three coupled BTEs, are carried out in [83], where the effect of nonparabolicity is investigated. The intentionally delta-doped QWs, inside and outside the conduction channel, have also been investigated, since due to the band offsets at the interfaces the extra confinement makes a wider conduction channel or parallel channels, which in principle allows one to have higher concentrations. For instance, a delta-doped channel is fabricated in the barrier of a GaAs/Iny Ga1−y As/GaAs QW [85], where mobilities are reported, namely  = 33 × 104 cm2 V−1 s−1 at T = 77 K and  = 055 × 104 cm2 V−1 s−1 at T = 77 K for y = 037. The inclusion of the delta-doped channel inside a QW is most expanded. A GaAs based QW is fabricated with an additional delta-doped channel in the center [81]. Quantum and transport mobilities in such a structure are calculated in [82] including several subbands. Mobility measurements and the concentration dependence are reported at liquid helium temperature in [84], where a GaAs/Iny Ga1−y As/Alx Ga1−x As QW is delta doped in the center.

6. HIGH-FIELD TRANSPORT A nonlinear response of the system to the applied electric field is difficult to investigate bulk materials but has attracted attention in semiconductor nanostructures. Particularly, the size reduction in MOSFETs and MODFETs has to do with the conduction channel or gate length too, varying from a fraction up to several micrometers [7]. This does not lead to QSEs but makes possible the realization of very high electric fields. For instance, applying a gate voltage of 1 V along a conduction channel of length 0.25 m gives a field strength of 40 kV cm−1 , which is a factor of 100 greater than the typical fields to produce hot electrons in bulk materials at room temperature, as stated in an important review on hot electrons presented in [80].

567 One can talk about high-field transport when a nonlinear dependence between the current density and the applied electric field is found experimentally, that is, j = OF F

with F = F

(130)

where the conductivity OF  = const depends on the module of applied electric field, its strength, for the properties of semiconductor nanostructures are assumed isotropic, except of course when the sample is placed in an external magnetic field. In this case the average electron energy is quite larger than the average energies of phonons, which are considered in thermal equilibrium at temperature T ; therefore they are known as hot electrons. In this regime the energy transfer from the electron system to the phonon system has to be carefully considered. Hot electrons are involved not only in the nonlinear dependence between the current density and the applied electric field (130), but also in the saturation of the current values and in the NDR. The latter is connected to the negative differential conductivity, dO dj = OF  + F 2kF′ when  = ′ and = = 0), one has     m∗ S
=nn′ 2 47 ′ 1/2 0 ′ − 1− 1+ ℘ ℰq 
=nn′ 7  = − 2%
2 ℰq ℰq (188)



∗

2

2m L % 2
2  × ln 1 −

℘ 0 ℰq 
=′ 7 ′  = −

1/2

1 (189) ℰq +
=′ 47 ′ ℰq (190) ℰq +
=′ 2

while if ℰq < 27 ′ +
=′ + #27 ′ +
=′ 2 − 
=′ 2 $1/2 (this condition leads to q < 2kF′ when  = ′ and = = 0), one has ∗

0


1 q
 = B = B ′ V where ℱBB ′ ′ q =



1  ′ ℘ ′ q =ℱBB ′ q A  ′ 


q
 =  ×′ V

ℋ′ q
B∗ ′ 
B 
d


(194)

(195)


1 is Notice the action of the screening potential operator V just multiplied by a number V1 . The basic assumption (147) can be written as
q = B = B ′ V
0 q = B + B ′ V
1 q = B (196) B ′ V

Making use of (194), after some manipulation one obtains for the full Fourier transforms in space and time 
q =  (197)
0 q = B = CBB ′ ′ q =′ V B ′ V ′

Here the matrix dielectric function was introduced as

The introduced functions are ℬ′ ℰq 
=′

finite lifetime of quasiparticle states because of the scattering mechanisms, as can be found in [238] for a Q2D system. The matrix element between subbands B and B ′ results in

℘n′ ℰq 
=nn′  = −

mS 2%
2

0

℘′ ℰq 
=′  = 0

(191) (192)

for Q2D and Q1D systems respectively. The zero-temperature polarizability is given by 0

℘′ q= = ℘ 0 ℰq −
=′ EF +℘ 0 ℰq 
=′ EF′  (193) which if  = ′ and = = 0 presents discontinuities at q = 2kF , the Kohn effect reported for Q2D systems [230] and Q1D systems [231]. These discontinuities are smeared out if one takes into account the energy level broadening due to

CBB ′ ′ q = =  − B ′ − B ′  − WBB′ ′ q = (198) where the susceptibility was defined as WBB ′ ′ q = =

1 vq℘′ q =ℱBB ′ ′ q A

(199)

with the polarizability given by (183). Notice the normalization area A = S or the normalization length A = L will disappear from the susceptibility when going from summation into integration in (183) according to (11) or (12) respectively. The form factor for the electron–electron interaction given by  ′ ℱll′ nn′ q = e−q z−z l∗′ zl zn∗ ′ z′  × n z′ dz dz′  ℱll′ nn′ q = K0 q
−
′ B∗ ′ 
B 
 × ∗ ′ 
′  
′ d
d
′

(200)

(201)

for Q2D and Q1D systems, where vq = vq =

2%e2 *q

(202)

2e2 *

(203)

respectively. The obtained result for the matrix dielectric function is known as the random phase approximation in MBT, where it can also be derived in a different way. The major defect of the RPA matrix dielectric function is that all exchanges and correlation effects have been disregarded. However, it gives sufficiently good results for the screening of scattering mechanisms in transport theory both in

Transport in Semiconductor Nanostructures

576 semiconductor heterostructures [68, 78] and delta-doped semiconductors [104, 170, 193]. Some works are devoted to the enhancement of this result in the line of Singwi– Tasi–Land–Sjolander theory, but they are mainly concerned with elementary excitation spectrum of the electron liquid (interacting electron gas).

7.2. Screened Matrix Elements for a Given Scattering Potential The just derived equation (197) can be written in an obvious notation as  0 CBB ′ ′ q =V′ q = (204) VBB ′ q = = ′

which gives each matrix element of the bare scattering potential trough the matrix elements of the screened scattering potential. However, what is required is the inverse relation,  −1 0 V′ q = = C′ BB ′ q =VBB (205) ′ q =

One must be aware that in Q2D systems if the subband index n = 1 2 3     N , then the dielectric function matrix has N 2 × N 2 elements, while in Q1D systems if subband indices n l = 1 2 3     N , then the dielectric function matrix has N 4 × N 4 elements. That is the case of multisubband transport calculations in delta-doped semiconductors [104] and semiconductor heterostructures. However, the just explained method is objected to by some authors [232], who mainly stress that no formal justification is given to cut off the number of considered subbands, even when some empty subbands are included in the dielectric function matrix. They propose a method for a Q2D system, which allows one to obtain the inverted dielectric function C−1 q = z z′ . Although they also limit the number of considered subbands, they provide a formal demonstration of the convergence of this method when the number of considered subbands increases to infinity [232, 240]. This result has not been employed in transport calculations. A completely different approach is followed in [34, 241] to directly obtain C−1 ′ BB ′ q = for a Q1D system, but concrete calculations are not presented.

BB ′

where C−1 ′ BB ′ q = are the elements of the inverted matrix dielectric function. The problem of inverting the RPA matrix dielectric function is not so trivial as it seems at first sight. Notice that one deals with a four-index matrix or an eight-index matrix, Cnn′ ll′ q = Cnn′ ll′ ii′ jj ′ q =

n n′  l l′ = 1 2 3    i i′  j j ′ = 1 2 3   

(206) (207)

in Q2D or Q1D systems respectively. This suggests the use of tensor algebra, but it reduces to relabel the matrix indices once, in the first case, or twice, in the second case, to obtain a manageable two-index matrix. But this must be done in the adequate order to obtain the proper mathematical object ruled by the well-known matrix algebra theory. A detailed discussion of how to invert the matrix dielectric function for a Q2D electron gas was presented in [233], where it is shown how the relabeling of a four-index matrix must be carried out in order to be congruent with two-index diagonal unit matrix. A similar procedure may be repeated for a Q1D electron gas. Many works avoid this question by making different approximations. Concrete calculations are mostly carried out in SQL approximation, when only the lowest subband is considered and the dielectric function is just a number, both in Q2D [149, 150] and Q1D [107, 108, 236] cases. Sometimes strictly 2D or 1D dielectric functions, where no indices appear, have been employed, even for intersubband scattering in Q2D [68, 148] and Q1D systems. Another approximation takes account only of the diagonal terms [167], so that 0 V′ q = = C−1 ′ ′ q =V′ q =

(208)

Finally, in several calculations the matrix dielectric function is actually inverted, of course after limiting the number of considered subbands, which can include some empty ones.

8. SUMMARY Transport in semiconductor nanostructures has been briefly reviewed. Fundamental topics have been selected aiming to give a comprehensive theoretical basis as well as a substantial account of the research work in this area over the last 20 years. The quasiclassical BTE formalism still is widely employed, because it allows a clear understanding and explanation for most of the experimentally determined characteristics of transport properties in these low-dimensional systems. This chapter is focused on steady-state transport because it is particularly important for the investigation of the different scattering mechanisms able to limit the mobility and, consequently, the drift velocity. The latter is directly related to the field-effect transistor design for high-speed devices. The construction of semiconductor nanostructures has stimulated the search for novel materials as well. The SiGe/ Si or SiGe/Ge based heterostructures, as expected, have been widely investigated. But the narrow-gap III–V binary compounds and their ternary alloys, such as the AlGaAs/GaAs based heterostructures, where the record electron and hole mobilities have been reached, have been the most studied low-dimensional systems. Particularly in recent years the wide-gap III nitride binary compounds and their ternary alloys, such as the AlGaN/GaN based heterostructures, have attracted the attention of the researchers. The II–VI binary compounds and their ternary alloys, as the CdMgTe/CdTe based heterostructures, have also been investigated, but not so intensively as the other mentioned materials. The distinction between low-field and high-field transport regimes, as well as the role of the different scattering mechanisms in limiting the carrier mobility, depends on the materials forming the semiconductor nanostructures, their parameters, and the experimental conditions. Many questions remain open in this research field, which is tightly connected to the size reduction and high quality performance of electronic devices.

Transport in Semiconductor Nanostructures

GLOSSARY Acoustic phonon Phonons produced in materials with one atom in each elemental cell. Charged carrier An electron or a hole. Conduction band Energy band in a crystalline solid partially filled by electrons. Effective mass The mass of a particle in a crystalline solid. Elastic scattering A process in which the scattered particle has the same energy before and after event. Electron–phonon interaction The coupling between electrons and the oscillations of ions. Gap Forbidden region for the electrons, which it is determined by the difference of energies between conduction and valence bands. Hall effect Deviation of the movement of charge carriers due to transverse magnetic field. Hole The absence of an electron in the valence band. Inelastic scattering A process in which the scattered particle has different energy before and after event. Mobility Parameter that characterizes the movement of the carriers in conductors. Optical phonon Phonons produced in materials with two (or more) different atoms in each elemental cell. Phonon Collective elemental excitation which describes the oscillations of ions in solids. Quasi-dimensional system A three-dimensional quantum system in which the movement of carriers is restricted in one, two, or three directions of a few nanometers. Scattering rate Sum of transition probability per unit time. Self-consistent field approach A many body theory that considers that particles are moving independently in a mean field due to other particles. Transition rate Transition probability per unit time. Valence band Energy band in a crystalline solid completely filled by electrons.

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Encyclopedia of Nanoscience and Nanotechnology

www.aspbs.com/enn

UHV-SPM Nanofabrication G. Palasantzas, J. Th. M. De Hosson University of Groningen, Groningen, The Netherlands

L. J. Geerligs Delft University of Technology, Delft, The Netherlands

CONTENTS 1. Introduction 2. Lithography on Atomic Resist 3. Vapor–Deposition Nanofabrication 4. Oxide-Based Lithography 5. Current Technology Applications 6. Conclusions Glossary References

1. INTRODUCTION An eminent physicist, Hendrik Casimir, who headed Philips Research (1957–1972) believed that scientific and technological advancement can be described in terms of a sciencetechnology spiral. Although science and technology follow separate streams of development, they interact continuously leading to scientific and technological progress. Clearly, research in the field of microelectronics and materials science provide a good example of such interaction. Indeed, results of scientific research are crucial for new technologies, and technological advances stimulate further scientific progress. In the 1980’s, the development of pattern generation and the fabrication of advanced semiconductor structures made feasible the study of microscale circuits for the first time. Researchers discovered methods to manipulate single electrons in micron-scale circuits and to observe quantum effects when electrons are confined in small spaces. The advancement of scanning probe microscopy (SPM) techniques allowed the miniaturization down to nanoscale of electronic devices thus opening the possibility for further exploitation of quantum mechanical phenomena at room temperature. Indeed, in recent years there has been an outstanding development of new methods for micro- and nanofabrication

ISBN: 1-58883-066-7/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

techniques using SPM [1–15], which will be essential to scientific progress in many areas in physics, materials science, chemistry, and biology. They will form enabling technologies for applications such as nanoelectronics, molecular electronics, micro-optical components, nanoelectro-mechanical systems, catalysis, etc. Advances are strongly aided by the highly engineered and successful lithography techniques that are used in microelectronics. One of the fundamental limits in lithography is imposed by the properties of the resist layer, since for the smallest feature, size one would like the thinnest resist and/or the highest possible contrast. Although today, conventional electron beam lithography is widely used to pattern features with dimensions larger than 10 nm [8], direct writing schemes based on SPM lithography in combination with chemical vapor deposition (CVD) and physical vapor deposition (PVD) schemes of various metals and semiconductors (i.e., for Fe, Pd, Ni, Cd, Si, Al) [1–6, 9–15] have led to feature sizes less than 10 nm. It is hoped that nanoscale-size metal patterns can be used as model systems to study aspects of nanoscale electronics such as single-electron tunneling [2]. An example design of a nanoscale device is shown in Figure 1 which was made by H desorption by the scanning tunneling microscopy (STM) tip on a hydrogen (H)-passivated Si(100)2 × 1 surface. In such a device, in order to observe charging effects in electronic transport measurements (where the object has to by definition be coupled to the environment), the thermal energy has to be lower than the change in charging energy by single electron transfer. Since, at room temperature, the thermal energy is of the order of kB T ≈ 0025 eV (kB the Boltzmann constant and T system temperature in Kelvin), the island size has to be below 10 nm in order that the change in charging energy Ec = q 2 /2C (with C the island capacitance and q its charge) satisfies the requirement Ec > KB T [2]. Under this condition, electrons tunnel between the electrodes and the island one-by-one leading to a well-defined charge state. The discrete flow of charged particles gives rise to the so-called Coulomb blockade.

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (581–594)

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Figure 1. Design of a single electron-tunneling transistor with island size 30 × 30 nm.

Quantum mechanical effects start playing a dominant role in nanometer-size structures, since the number of energy levels is discrete and rather limited [2, 16]. Consider as a simple example the case of a thin metal film of thickness d (400  C [33]), while other metals form upon annealing stable silicides. Co has the advantage that all its silicide phases are metallic [29] while, for example, FeSi2 is usually semiconducting (depending on the corresponding phase) [34]. Figures 10b and 10c shows a sample where 0.13 ML of Co was deposited at RT on a H-passivated Si(001) surface with depassivated lines of about ∼4–5 nm wide after which the surface was annealed at 410  C for 20 sec. The silicidation reaction in the wire area leads to an increase in silicide volume, which leads to a more distinct and compact structure than that formed at RT. The monohydride side remains intact even closely to the wire boundary despite the annealing processes, since H starts to desorb for T ≥ 470  C [10]. (a)

(b)

Notably, differences appear when the wire is formed perpendicular (Figure 10b) and along the dimer rows (Figure 10c). In the second case, a more compact structure with more regular boundaries is found. This finding is in good agreement with the shape and appearance of CoSi2 islands on Si(100) formed at elevated temperature [32], which show edges that are straight on the atomic scale parallel to the dimer rows while they can be rather rough in the perpendicular direction. This is possibly due to different directional surface strain and adatom mobility parallel and perpendicular to the Si dimer rows.

3.3. Palladium (Pd)-PVD Nanofabrication Since the maximum thickness of well-defined silicide features is limited by the selectivity of the deposition process, Mitsui et al. [35] have investigated the dependence of selectivity on deposition temperature. Upon Pd-PVD on an STM-patterned monohydride terminated Si001-2 × 1 surface, it was found that Pd growth selectivity is due to the different diffusion lengths on bare and H-terminated surfaces, and that the selectivity increases with increasing temperature from 500 K to 600 K [35]. This is shown in Figure 11, where palladium selectively deposited on a 6 nm wide strip of bare Si(001) on an H-terminated surface. The nanostructure consists of uniform clusters on the bare strip. The selectivity results from the difference in Pd diffusion lengths on bare versus H-terminated silicon. A crystal structure was observed after a 1000 K anneal [35].

3.4. Aluminum (Al) and Iron (Fe)-CVD Nanofabrication Chemical vapor deposition (CVD) is an alternative to evaporation for material delivery (Figure 12) [35]. In CVD, the growth element arrives at the surface incorporated in a molecular precursor. On appropriately reactive surfaces, the molecule dissociates and deposits the growth element (Figure 12b). Chemical vapor deposition is a popular

(c)

Figure 10. (a) 0.13 ML of Co deposited at RT resulting in wire (patterning parameters as in Fig. 2a) width approximately 8–9 nm. Filled state image acquired with bias −2 V and current 0.12 nA). (b) Co/Si wire deposited at RT and annealed at 410  C for 20 sec with a wire along the dimer rows (filled state image acquired with bias voltage −189 V, current 0.28 nA). (c) Wire perpendicular to dimer rows (filled state image acquired with bias voltage −169 V, current 0.34 nA). For (b) and (c) the patterning was performed with current 2 nA, sample bias 7 V, and writing speed 300 nm/sec. Scan size for all STM images is 50 × 50 nm. Reprinted with permission from [6], G. Palasantzas et al., J. Appl. Phys. 85, 1907 (1999). © 1999, American Institute of Physics.

Figure 11. Palladium deposition and fabrication of nanoscales wires with width less than 10 nm on H-Si0012 × 1 reconstructed surfaces (courtesy E. Ganz).

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(b)

Absorption of CVD precursor Desorption

Dissociation

Diffusion

(c)

Nucleation of metal

(d) 5.0 nm

5.0 nm

Figure 12. (a) Schematic explanation of the CVD process. (b) STM image of an Al nanostructure CVD fabricated on H resist. The line width is less than 3 nm and the height 0.3–0.5 nm (courtesy E. Ganz). Reprinted with permission from [35], T. Mitsui et al., J. Appl. Phys. 85, 522 (1999). © 1999, American Institute of Physics. (c) H-depassivated area prior to Fe CVD by electron beam stimulated desorption with a sample bias of 1.8 V and a line dose of 100 mC/cm. The STM image was taken with sample bias of −23 V and tunneling current 0.15 nA (courtesy T. M. Mayer). Reprinted with permission from [5], D. P. Adams et al., Appl. Phys. Lett. 68, 2210 (1996). © 1996, American Institute of Physics. (d) STM image after Fe CVD growth taken with a sample bias of −23 V and a tip-sample current of 0.15 nA (courtesy T. M. Mayer). Reprinted with permission from [5], D. P. Adams et al., Appl. Phys. Lett. 68, 2210 (1996). © 1996, American Institute of Physics.

method for microelectronics device fabrication because of its low cost, high deposition rate, high purity, good conformal coverage, and potential for strong growth selectivity.

3.5. Al-CVD Mitsui et al. [35] used an Al precursor, dimethylaluminum hydride [DMAH, Al(CH3 )2 ]. This is a promising, new chemical precursor because of its stability, low carbon contamination, and ease of introduction. Al CVD, using DMAH, has been extensively studied over the last few years [4]. Tsubouchi et al. have used DMAH to selectively deliver Al to a silicon surface patterned by electron beam lithography and selective oxidation [5]. The surface was patterned into dihydride terminated and oxidized regions. Under the conditions of their experiment, the DMAH preferentially deposited Al on the hydrogen-terminated surface. However, this is an undesirable negative patterning process, with the Al deposited outside the patterned areas. A variation of this method, which provides a positive patterning process, is rather difficult [36]. An example of Al CVD on H-passivated Si(001) surfaces is shown in Figure 12b. At RT, DMAH is adsorbed intact on the bare Si surface obtained by H desorption. At temperatures above 120  C, Al deposition began. Indeed, clean Al growth was observed for substrate temperatures between 150 and 300  C [35]. By contrast with the earlier work on bare Si, it was found that DMAH does not stick on RT monohydride-terminated Si surface. Furthermore, Al deposition was not observed on monohydride

surfaces up to 300  C. This indicates that DMAH can provide high Al selectivity between bare and monohydrideterminated Si(001) surface at temperatures between 150 and 300  C [35]. It is important to note that there is no Al growth enhancement along the border between the silicon and hydrogen-terminated regions in Figure 12b. Such an enhancement was observed by Shen et al. [3]. It was attributed to free Al atoms on the H-terminated surface diffusing to and sticking on the bare region. For Al CVD, the DMAH deposits Al only on bare Si regions. The work by Mitsui et al. [35] demonstrated that this lithography method combines the high selectivity of the CVD precursor DMAH with the precision of an STMpatterned hydrogen resist, where sizes as small as 2 nm can be achieved. Employing computer control in the patterning phase, this process can be used to fabricate complex structures as shown in Figure 12b. The origin of the strong Al selectivity [35] stems from the fact that the Si0012 × 1 reconstructed surface is composed of pairs of Si atoms, each of which are joined by a strong -bond and a weak -bond. The -bond is only 0.9 eV below the Fermi level, which makes Si0012 × 1 a highly reactive surface [37] on which the DMAH molecules are able to dissociate and deposit Al. In the case of the hydrogen-terminated surface, the -bond is replaced by two strong Si H bonds and the surface is passivated. Thus, the DMAH is unable to chemically bond or dissociate on such a surface.

3.6. Fe-CVD Finally, we shall describe the work of Adams et al. on selective CVD growth of Fe nanostructures using a CVD precursor, Fe(CO)5 [5] on a patterned hydrogen-terminated surface (Figure 12c). Fe is selectively deposited onto the patterned areas by pyrolysis of Fe(CO)5 , as is shown in Figure 12d. The Fe lines formed on the clean Si have a rough surface morphology and are most likely polycrystalline. Scanning tunneling microscopy and heavy ion backscattering spectrometry confirmed that that 2 ML monolayers of Fe are locally deposited during a 60 L dose [5]. Separate experiments, by x-ray photoelectron spectroscopy, indicated very small C and O impurity levels, characteristic of pyrolysis onto Si substrates at elevated temperatures [5]. Fe nucleation during CVD growth is dominated by sitespecific chemical reactivity of the precursor Fe(CO)5 with the Si substrate. Former work has shown that decomposition of the Fe(CO)5 molecule occurs predominantly at Si dangling bond sites [5]. Passivation of the dangling bond sites with hydrogen removes the active site for pyrolysis and effectively raises the activation barrier for nucleation. Notably, the Fe CVD is autocatalytic on Si. For example, the barrier to dissociation of the precursor on an existing Fe cluster is smaller than on the clean Si surface by approximately 0.14 eV on Fe and 0.40 eV on Si [5]. Therefore, once a nucleus is created, it grows rapidly compared to the formation of additional clusters. This highly nonlinear growth rate is advantageous in maintaining the area selectivity of metal deposition on clean Si surfaces [5]. These results for Fe show clearly that nanostructures can be grown by CVD on patterned H-terminated Si surfaces and also for metals

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other than Al. Scanning tunneling microscopy confirms that the adsorbed H is still intact as monohydride in unexposed surface areas and is not disrupted by the Fe(CO)5 .

4. OXIDE-BASED LITHOGRAPHY 4.1. Ni-Oxidation Besides patterning on H-passivated Si surfaces, an electron beam allows direct pattern formation on metal surfaces. This is initially shown in Figure 4 for the case of Ni3 Al surfaces. The apparatus that was used for these studies [7] consists of a UHV (Figure 13b) (base pressure ∼4 × 10−8 Pa) scanning Auger/electron microscope (field emission JEOL JAMP7800F). Under typical imaging conditions (accelerating voltage 10 keV and electron beam current (a)

Electron probe Input lens of hemispherical electostatic analyzer EDS detector

(b) CF objective lens

Differentially pumped ion gun

Backscattered electron detector (topography and composition images seperately displayable) X Secondary electron dtector

I = 24 nA, which will be used for the e-beam induced oxidation, if not otherwise stated) the attained beam spot size is ∼15 nm. The AES measurements were performed on polished sample cross-sections, which were cleaned by Ar-sputtering prior to e-beam exposure. Auger depth profile analysis was performed by low-rate Ar+ sputtering. Oxygen was provided by the UHV atmosphere to initiate chemisorption and direct oxidation under the influence of the e-beam. For Auger maps, it was recorded the ratio (Peak-Background)/Background from the direct spectrum (Figure 13a). The oxidation of Ni3 Al was described [7] with the model of Li et al. [20], which is based on the premise that the incident electrons create additional nucleation sites around which oxide islands grow. In terms of Auger intensities, this model reads of the form [20] IO t = A − A − Bexp−kt − k/e exp−e t − 1 with e the electron beam density flux (cm−2 sec−1 ),  the electron cross-section for the creation of oxide nucleation sites, t the oxidation time, and k an oxidation rate constant. This model describes the oxidation data for e-beam spot sizes d < 30 m (see Figure 14 for d = 5 10 m). A is the saturated Auger intensity and B the intensity at chemisorption. For e-beam sizes of d ≥ 30 m, the oxidation curve changes its shape and reveals a much slower oxidation process. The latter is also confirmed by the Auger map of O (Figure 13) where the spot intensity decays drastically with increasing e-beam size for d > 30 m. The flux e is estimated by the relation e = I/d 2 /4, assuming circular beam spot size of area ≈d 2 /4 (with d the beam diameter) and I the beam current. Depth profile analysis (Figure 15) showed that the corresponding Ni-oxide depth decreases with increasing e-beam size. This also offers an alternative way to monitor Ni-oxide thickness in a nanometer range. Such a process can be strongly relevant for the fabrication of antiferromagnetic/ferromagnetic junctions [25], and lithography techniques on Ni-based surfaces [2, 7], as well as Ni-oxide formation for catalysis studies [17–21]. The oxidation curves for beam sizes of d ≥ 50 m (Figure 14a) indicate an oxidation process which is characterized by a tilted constant slope area, followed by a saturation (see Figure 14a for d = 50 m. The formation of oxide nuclei by the beam is no longer sufficient to dominate the O chemisorption, and therefore its contribution to the Auger intensity. Thus, we obtain [7]: IO t = A − A − Bexp−kt − k/e  × exp−e t − 1 − C exp−kch t

Y

Tilt R

Z

Figure 13. (a) SAM image of O. Data acquired with 100 ms/eV at 40 tilted samples. The typical 15 nm beam spot size was used for SAM imaging. (b) Diagram of the SEM/SAM imaging and data acquisition geometry. Basic components: Secondary electron detector for SEM topography imaging, hemispherical mirror analyzer for Auger imaginganalysis, EDS detector for x-ray bulk microanalysis, ion gun etching for depth profiling, sample stage with its possible motion mechanisms.

(1)

by inclusion of chemisorption characterized by a reaction constant kch . The corresponding fits are shown in Figure 14a for the d = 30 50 m oxidation curves. Saturation is necessary for the fit because it yields the parameter A. The fit for the d = 50 m oxide spot indicates that O chemisorption occurs rather fast with a reaction constant larger than that of the e-beam-induced oxidation. Similar oxidation scenarios have also been observed for other systems with significant affinity to O such as Ni [7, 17–21], Al(111) [38], and Mg (0001) [39], where oxide nucleation occurs long before the saturation of a chemisorbed coverage is reached.

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4.2. Al-Mg-Oxidation

1400

d=5 µm

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d= 10 µ

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2678

3060

Finally, we will examine the case where two (or more) oxidizing elements are present as, for example, in Al-Mg alloy surfaces, where both Al and Mg contribute to the e-beam oxidation process. The present sample surface consisted of a rather random distribution of islands with a bulk composition of (15 ± 1) at % Mg (Mg in solid solution in Al: Al/Mg), where in-between these islands, the corresponding areas have more than double Mg content of (38 ± 1) at % Mg (Al3 Mg2 . or !-phase). The oxidation took place on the Al3 Mg2 phase (Figure 14b). The oxygen curves are shown in Figure 14b and the fits correspond to the more general equation:

t (min)

(b)

IO t = A −

d=5 µm d=10 µm d=20 µm

1200

1200

I (p-p)

IO(p-p)

800

600

O

900 600

Mg

400 300 Al

0

200

0

300 600 900 1200 1500 1800 2100

t (min) 0 200

400

600

800

1000

t (min)

Figure 14. (a) Oxygen peak-to-peak intensity (p-p) vs. oxidation time. Data acquired with 400 ms dwell time (time/eV), 40 tilted samples, and I = 24 nA beam current. The solid lines are fits by the oxidation model. Reprinted with permission from G. Palasantzas et al., J. Vac. Sci. Technol. A 19, 2581 (2001). © 2001, American Institute of Physics. (b) Oxidation curves for oxygen of Al-Mg alloy surface for various e-beam spot sizes d as indicated. The solid lines are the best fits in terms of Eq. (2), and the inset shows an example of the oxidation behavior of all elements (Al, Mg, and O). Reprinted with permission from G. Palasantzas et al., Appl. Surf. Sci. 191, 266 (2002). © 2002, Elsevier Science.

1400 1200

µm d= 30

µm

d=

70

IO(p-p)

50

600

µm

d=

10

d=

1000 800

µm

400 0µ

10

d= m

200

Aj exp−t/#j  − 1/#j e 

j=1

1000

0

N 

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 d (nm)

Figure 15. Depth profile analysis with an e-beam spot size of 5 m and 400 ms dwell time. Reprinted with permission from G. Palasantzas et al., J. Vac. Sci. Technol. A 19, 2581 (2001). © 2001, American Institute of Physics.

× exp−e j t − 1

(2)

with N ≥ 1. After saturation, we
have IO  = A, and at t = 0, we have IO t = 0 = A − j=1 N Aj , which corresponds to the chemisorption saturation. For the present experiment (Figure 2), there are two elements participating in the oxidation process, namely, Al and Mg, which means that N = 2 with indices 1 ≡ Al and 2 ≡ Mg. As Figure 14b indicates, the oxidation kinetics surpasses the O chemisorption regime and fast oxide growth dominates the oxidation process. Such an oxidation scenario has also been observed in other systems with significant affinity to O such as Ni [7, 17–21], Al(111) [38], and Mg (0001) [39], where oxide nucleation occurs long before the saturation of a chemisorbed coverage is reached. At any rate, the Al-Mg alloy oxidation is consistent with the idea that Mg diffusion occurs during the e-beam-induced oxidation of the Al3 Mg2 surface, which is enhanced by the formation of Al-oxide and Mg-oxide upon e-beam exposure and the high affinity of Mg to O. The activation energy of Mg diffusion in Al2 O3 -MgO is QMg ∼ 3699 kJ/mol, which is smaller than that of O diffusion QO ∼438.9 kJ/mol in the same oxide [40, 41], as well as the diffusion prefactor for Mg Mg is much larger than that of O (Do ≈ 47102 cm2 /sec ≫ DoO ≈ 8910−1 cm2 /sec [40, 41]). These values indicate Mg diffusion and thus segregation through the e-beam-induced oxide layer towards the surface with further MgO formation. Notably, on the areas where Mg is in solid solution, only the oxidation of Al occurs almost without any Mg participating in the oxidation process and/or segregating towards the surface region. Besides the intense electron beams (keV range) described previously, atomic force microscopy (AFM) also allows selective oxidation of thin metal films to define narrow metal wires by oxidizing unwanted regions of wide metal wires [42]. This process allowed the fabrication of lateral metal-oxidemetal tunnel junctions by introducing thin regions of oxide in an otherwise continuous metal film. By making these junctions very small, the junction capacitance also decreases [2]. These narrow wires can constitute the building blocks from which more complex devices can be built. Recent examples of such devices include a SET transistor by Matsumoto et al. [43], and a metallic point contact by Snow et al. [44]. In the pioneering work by Snow et al. [2, 42, 44], point contacts and tunnel junctions were formed using the AFM

590 tip to anodise through the cross-section of Al (or Ti) nanometer-scale wires (formed by optical lithography and metal lift-off to pattern 1000-nm-wide × 5–8-nm-thick metal wires connected to contact pads) while in-situ electrical measurements were used as a feedback to guide the anodization. It should be noted that the writing speed and resolution for the AFM-based local oxidation technique is comparable to the results, which have been obtained for the local exposure of a resist layer. However, the local oxidation technique has the important advantage that it works reliably and gives reproducible results (provided that the degree of humidity can be kept constant). Indeed, local oxidation of materials in combination with subsequent pattern transfer has been one of the most successful device fabrication techniques.

5. CURRENT TECHNOLOGY APPLICATIONS 5.1. Effects of Deuterium Desorption on MOS Technology The studies by Shen and Lyding et al. [1, 2, 45], on H desorption by the STM tip on Si(001) surfaces led to the realization that deuterium was about two orders of magnitude more difficult to desorb than H [2, 46]. An indirect application of these results concern metal-oxide-semiconductor (MOS) transistors with regard to hot carrier degradation effects [2, 47]. In MOS transistors, due to molar volume mismatch at the SiO2 /Si interface, not all of the Si ends up bonded to the oxide resulting in dangling bonds that are becoming charged during transistor operation, thus scattering conduction electrons and shifting voltages used in transistor operation. The dangling bonds are passivated by H after annealing the Si-wafers at ∼400  C in H-atmosphere [2]. During transistor operation, however, hot electrons that flow close to the SiO2 /Si interface stimulate H desorption, which leads to reappearance of dangling bonds and transistor degradation. Tests performed by Lucent [47], using deuterium to passivate dangling bonds, indicated increment of the transistor lifetime by an order of magnitude or more [2, 47].

UHV-SPM Nanofabrication

numbers of qubits will be difficult [51]. Therefore, making solid-state architectures [52] with their promise of scalability is very important. In 1998, Kane [53] proposed a solid-state quantum computer design using P nuclei (nuclear spin I = 1/2) as the qubits in isotopically pure Si (I = 0) (Figure 16). One of the major challenges of this design is to reliably fabricate an atomically precise array of P nuclei in Si. Figure 16a shows the process to implement the Kane architecture [48]. This schematic is showing two P qubits in a linear array, incorporated into isotopically pure Si and isolated from surface metal A and J gates by an insulating barrier. Figure 16b shows a possible process to fabricate an array of P qubits in Si. A low defect density Si0012 × 1 surface is H-passivated (Figure 17), where after an STM tip is used to selectively desorb H, exposing Si on an atomic scale permitting only one phosphine (PH3 ) molecule to adsorb at each of the required sites (Figures 18–20). Finally, lowtemperature silicon overgrowth can be used to encapsulate the P array [48]. Although an STM tip has been used for atomic-scale arrangement of metal atoms on metal surfaces [54], rearrangement of individual atoms in a semiconductor system is not straightforward due to the strong covalent bonds involved. Therefore, O’Brien et al. [48] employed the H resist strategy on Si0012 × 1. The requirements for this quantum computer design are very strict. For high-resolution lithography, the Si surface must be atomically flat with a low defect density to allow the formation of a near-perfect resist layer, where one H atom bonds to each surface Si atom (Figure 17). The ability to desorb individual H atoms requires a sharp, large cone angle tungsten tip in order to form desorption sites ∼1 nm in (a)

J-Gates A-Gates

Barrier Silicon 31

P Qubits

(b) Mono-hydride Deposition

5.2. Silicon-Based Quantum Computer The challenge to build a quantum computer has been motivated from the formidable computational power such a device could offer. In fact, Si-based proposals, using the nuclear or electron spin of dopants as qubits, are attractive due to the long spin relaxation times involved, their scalability, and the ease of integration with existing silicon technology. Fabrication of such devices, however, requires atomic scale manipulation, which represents an immense technological challenge. It has been demonstrated by O’Brien et al. [48] that it is possible to fabricate an atomically precise linear array of single phosphorus (P)-bearing molecules, as qubits, on a silicon surface with the required dimensions for the fabrication of a silicon-based quantum computer. A quantum bit or qubit is a two-level quantum system which is the building block of a quantum computer. To date, the most advanced realizations of a quantum computer are qubit ion trap [49] and nuclear magnetic resonance [50] systems. However, scaling these systems to large

PH3 Dosing

H Si

Hydrogen Desorption

Silicon Overgrowth STM Tip

Figure 16. (a) A schematic of the process to fabricate the Kane architecture. Detail of the Kane quantum computer architecture [47] are showing two P-qubits in a linear array, incorporated into isotopically pure Si and isolated from surface metal A and J gates by an insulating barrier. (b) Process to fabricate an array of P-qubits in Si. A low-defect density Si0012 × 1 surface is H-passivated. An STM tip is used to desorb H atoms, exposing Si on an atomic scale permitting only one PH3 -molecule to adsorb at each required site (courtesy J. O’Brien). Reprinted with permission from [48], J. L. O’Brien, et al., Phys. Rev. B 64, 161401 (2001). © 2001, American Physical Society.

591

UHV-SPM Nanofabrication

(c)

(d) Height (nm)

Height (nm)

0.30 0.20 0.12 nm

0.10 0.00

0

1

2 3 4 Distance (nm)

5

0.30 0.20 0.10 0.00

6

0.17 nm

0

1

2 3 4 Distance (nm)

5

6

5

6

Figure 17. (a) Filled-state STM image of a clean, very low defect density Si0012 × 1 surface. This image was acquired at a sample bias of 2.1 V and a tunneling current of 0.4 nA. (b) Fully H-terminated Si0012 × 1 surface. Image acquired at bias 2.5 V and 0.4 nA. The surface is almost entirely 2 × 1 monohydride, with few sites of 1 × 1 dihydride and 3 × 1 trihydride (courtesy J. O’Brien). Reprinted with permission from [48], J. L. O’Brien, et al., Phys. Rev. B 64, 161401 (2001). © 2001, American Physical Society.

Figure 18. Bonding structure of PH3 on Si0012 × 1. (a) Schematic of the c(4 × 2) structure of PH3 bonded to the Si0012 × 1 surface, where red dimers indicate PH3 bonding sites. (b) STM image of this structure acquired at a sample bias of 2.3 V and tunneling current 0.2 nA. (c) STM image of three hydrogen desorption sites of H-Si0012 × 1-H surface. The highlighted regions in (a) and (c) indicate that only one PH3 molecule can adsorb at each desorption site (courtesy J. O’Brien). Reprinted with permission from [48], J. L. O’Brien, et al., Phys. Rev. B 64, 161401 (2001). © 2001, American Physical Society.

(h)

0.30

Height (nm)

(g) Height (nm)

size. These sites will be subsequently exposed to high-purity PH3 gas for the required P atom placement (Figure 16b), avoiding the introduction of any spin or charge impurities that would be fatal to quantum computer operation [53]. In order to allow the adsorption of one PH3 molecule, and therefore only one P atom, it is necessary to desorb an area that exposes less than or equal to two Si dimers as is shown in Figure 18a. This is because PH3 bonds to the Si0012 × 1 surface with a c(4 × 2) surface periodicity (Figure 18b). The sticking coefficient of PH3 molecules on the clean Si surface

0.20 0.12 nm

0.10 0.00

0

1

2 3 4 Distance (nm)

5

6

0.30 0.20

0.17 nm

0.10 0.00

0

1

2 3 4 Distance (nm)

Figure 19. Demonstration of single adsorption through an STMpatterned H resist. STM images of two desorption sites before (a) and after (b) PH3 dosing. The corresponding line profiles (c) and (d) show a characteristic height increase of 0.05 nm. Three desorption sites before (e) and after (f) PH3 dosing and corresponding line profiles (g) and (h). STM mages were acquired at a sample bias of −18 V and tunneling current of 0.4 nA (courtesy J. O’Brien). Reprinted with permission from [48], J. L. O’Brien, et al., Phys. Rev. B 64, 161401 (2001). © 2001, American Physical Society.

is 1. Figure 18c shows three smaller than 1 nm in diameter H desorption sites in a row with a pitch of ∼4 nm on a H-passivated surface. The ability of the H-passivation layer as a barrier to PH3 adsorption is demonstrated by the uniform H coverage after PH3 dosing. As Figure 19a indicates, the bright protrusion at each of the H desorption sites is the signature of the single Si dangling bond, after desorption of just one H atom (in this case from the left side of the dimer: the remaining H on the Si dimer is known to be transient [48] and has been observed to diffuse from one side of the dimer to the other). Figure 19b shows the same area as Figure 19a after exposure to PH3 gas. Analysis of the line profiles in Figures 19c and 19d shows a characteristic increase of ∼0.05 nm in the protrusion height after PH3 dosing [48]. Figures 19d and

592

Figure 20. Three-dimensional view of the PH3 molecule on the Si0012 × 1 surface (courtesy J. O’Brien).

19e show three desorption sites (in a line perpendicular to the Si dimer rows), before and after PH3 dosing. The height increase confirms the adsorption of a PH3 molecule and corresponds to the difference between the exposed Si dangling bond and the adsorbed PH3 molecule. At any rate, O’Brien et al. [48] has answered the critical questions of whether H resist is effective during exposure to PH3 and whether or not PH3 will adsorb to an STM-exposed site sufficiently small to achieve one and only one PH3 molecule at that site. This is an important advance in qubit fabrication for the realization of a scalable Si-based quantum computer.

UHV-SPM Nanofabrication

Figure 21. This schematic shows the direct observation of standingwave patterns in the local density of states of the Cu(111) surface (courtesy D. Eigler). Reprinted with permission from [57], http://www.almaden.ibm.com/vis/stm/. © IBM.

5.4. Local Charge Deposition by Proximal Probe Microscopy Proximal probe microscopy can be used to perform local charge deposition in Co nanoclusters embedded within insulating matrices such as SiO2 , and subsequently, to imageinjected charge and charge transport [61]. Charge is injected from the probe tip into the sample by application of a

5.3. SPM Manipulation of Individual Atoms and Molecules Individual atoms on a surface can be manipulated with an STM microscope due to the strong interaction between the tip and surface at a small working distance [56]. By sliding atoms to exactly controlled positions, a structure can be built atom-by-atom as it was demonstrated in the past by Eigler et al. [56] (see also Figure 21). More examples can be found on the IBM website [57]. Figure 21 shows the direct observation of standing-wave patterns in the local density of states of the Cu(111) surface. These spatial oscillations are quantum-mechanical interference patterns caused by scattering of the two-dimensional electron gas of the Fe adatoms and point defects. Surface state electrons on Cu(111) were confined to closed structures (corrals) defined by barriers built from Fe adatoms. The barriers were assembled by individually positioning Fe adatoms, using the tip of a lowtemperature STM. A circular corral of radius 71.3 Angstrom was constructed in this way out of 48 Fe adatoms [56, 57]. Such a technique has been the most spectacular demonstration of scanning probe lithography, which allows the possibility of constructing artificial molecules out of their constituent atoms. Apart from individual atoms, molecules can also be manipulated [58]. The manipulation of atoms and molecules is related to a nanofabrication technique that is often called mechanosynthesis, opening the road towards molecular nanotechnology [59, 60].

Figure 22. (a) Sample and probe tip geometry for proximal probe charging of Co nanoclusters embedded in SiO2 , (b) Electrostatic force microscope images of a Co/SiO2 discontinuous multilayer sample before and after charging at +12 V and −12 V, (c) Sample charge as a function of time, showing the dependence of discharging time on the sign of the injected charge (figures were obtained from http://kesey.ucsd.edu/ group/PROJECTS/., and courtesy E. T. Yu).

593

UHV-SPM Nanofabrication

voltage pulse (Figure 22). Electrostatic force microscopy is then used to image the charged region of the sample. Charge can be injected in a controlled way into areas of radius ∼20– 50 nm. It has been observed that in structures with a thin (∼3–5 nm) sputtered SiO2 layer between the Co clusters and the Si substrate, the discharge time is of several minutes (increasing with increasing SiO2 layer), and are substantially longer for positively charged clusters than for negatively charged clusters. This observation can be explained semiquantitatively as a consequence of the single-electron charging energy of a small cluster (cluster radius ∼15 nm) [61] (Figure 22). The thin film metal/insulator systems, which consist of alternating metallic and insulating layers with the metal layer consisting of a discontinuous assembly of nanoscale clusters, as opposed to a continuous film, can be potentially important in magnetic recording industry because it can display a substantial room-temperature negative magneto-resistance which saturates at low magnetic fields. The observed magneto-resistance arises from spin-polarized tunneling between metallic nanoclusters embedded in the insulating matrix. Such tunneling processes can be highly sensitive to local variations in the structure of both the conducting clusters and the insulating matrix, and it is of considerable interest to probe electrical, magnetic, and transport properties in these structures at the nanometer scale [61].

Quantum computers Computer very fast where the basic operation unit is due to quantum phenomena. Single electron tunneling transistor (SET transistor) Electrons tunnel “one by one” showing a discrete flow of charge. SPM nanofabrication Fabrication of nanometer size structures by use of scanning probe microscopes.

ACKNOWLEDGMENTS We would like to thank Prof. J. R. Tucker and Prof. T.-C. Shen for supplying and giving permission to use the STM images in Figures 7 and 8, useful communication with J. R. Tucker on quantum computing, Prof. E. Ganz for supplying and giving permission to use the STM images in Figures 11 and 12b, Dr. T. M. Mayer for supplying and giving permission to use Figures 12c–12d, Dr. J. O’Brien for supplying and giving permission to use Figures 16–20, Dr. D. Don Eigler for permission to use the web-file image for Figure 21, and Prof. Dr. E. Yu for permission to use his web files for Figure 22. We would also like to thank J. Lyding for useful correspondence, as well as our former collaborators S. Rogge, B. Ilge, and L. Boellaard for their contribution on SPM nanofabrication studies.

REFERENCES 6. CONCLUSIONS The transition from microelectronics to nanoelectronics demands the introduction of new fabrication techniques. Indeed, besides self-organization of molecule structures, nanoscale-pattering techniques can play an important role for achieving the micro → nano transition. Future electronic devices and circuits will strongly depend on the technological ability to fabricate nanoscale insulating and metallic structures. From the examples mentioned previously, it is clear that SPM is emerging as an important new route for nanofabrication with potential technological relevance for modifying surfaces under UHV conditions. Control at nanometer length scales has already opened the exploration of a new class of device physics. Experiments are also in progress at various nanotechnology research groups to connect nanostructures to macroscopic contacts (i.e., Figure 2) to electrically characterize these nanodevices. Moreover, electrical studies of nanometer-scale devices will also open the possibility of studying atomistic process of electromigration phenomena, which constitutes an important issue in today’s integrated circuit technology. Finally, in terms of proximal probes, these studies have provided the possibility of extending applications in information technology, that is, through quantum computer architectures. We should note, however, that besides the SPM nanofabrication techniques described in this article, other techniques also exist as described in [59].

GLOSSARY Lithography Nanodevices ( 450 nm). Visible-light irradiation of the Cr-ion-implanted TiO2 in the presence of NO at 275 K leads to the direct decomposition of NO into N2 , O2 , and N2 O with a good linearity to the irradiation time. The liquid-phase photocatalytic degradation of 2-propanol diluted in water into acetone, CO2 , and H2 O also proceeded on the V-ion-implanted TiO2 catalysts under visible-light irradiation ( > 450 nm). Under the same conditions of visible-light irradiation, these photocatalytic reactions do not proceed on the un-implanted

2.5

B)

2 1.5 1 0.5 0

Acceleration Energy: 150 keV 0

10

20

30

Amounts of V Ions / x10-7mol • g-cat.-1

Figure 5. Effects of the acceleration energy (A) and the amount of implanted ions (B) on the reactivity of V-ion-implanted TiO 2 for the photocatalytic degradation of 2-propanol diluted in water.

Visible-Light-Sensitive Photocatalysts

642 X 10-9 8

Solar beam:

Amount of V ions (Ncm2): (a) 0

6

Transmittance / a.u.

NO Removal / mol•min –1

38.5 mW•cm-2

4

2

(b) 3 x 1016 (c) 6 x 1016

Shift toward longer wavelength (a)

(b) (c)

0 TiO2

Cr/TiO2

V/TiO2

Catalysts

with low acceleration energy (0.2–2 keV), efficient TiO2 thin-film photocatalysts can be prepared on various types of substrates such as porous glass plate and activated carbon fiber under mild and dry conditions [139–143]. In the ICB method, the Ti cluster beam is produced by ejecting the Ti vapor ionized by the electrons emitted from the ionization filament and accelerating the produced ionized clusters by an accelerating electrode. Active TiO2 thin films were produced by impingement of the ionized clusters and oxygen gas onto the support substrate. The advantages of using the ICB method are (1) no contamination because of the dry process in high vacuum, (2) production of highly crystalline films on various substrates without calcination, and (3) wellcontrolled thickness. Characterization of these films by means of ultraviolet– visible (UV–vis), X-ray absorption fine structure (XAFS) spectroscopy, scanning electron microscopy (SEM), X-ray diffraction (XRD), and XPS techniques showed that the TiO2 in transparent thin films having a well-crystallized structure mainly in the anatase phase could be formed on a silica glass plate (PVG) by the ICB method. This TiO2 thin film also exhibited efficient reactivity for the photocatalytic decomposition of NO into N2 and O2 under UV irradiation. The photocatalytic properties and physical properties of the TiO2 thin films are dependent on the film thickness, and higher reactivity can be observed on the thinner TiO2 films. TiO2 thin films prepared on the silica glass plate exhibit higher photocatalytic reactivities than TiO2 powder and catalysts prepared by the conventional impregnation method. Furthermore, UV irradiation of the TiO2 thin films in a diluted aqueous solution of propanol, dichloroethane, or phenol led to the efficient decomposition of these organic pollutants finally into CO2 , H2 O, and HCl. These results have clearly shown that the ICB method is useful in the preparation of transparent TiO2 thin films having efficient photocatalytic reactivity. To improve the electronic properties of the TiO2 thin films to absorb visible light, metal ions (V or Cr ions) were implanted into the TiO2 thin films at high energy acceleration (150 keV) using metal ion implantation. As shown in Figure 7, the UV–vis absorption spectra of these

300

400

500

600

700

Wavelength / nm

Figure 7. UV–vis spectra (transmittance) of the V-ion-implanted TiO2 thin-film photocatalysts.

metal-ion-implanted TiO2 thin films were found to shift toward the visible-light region, depending on the amount of metal ions implanted. They were found to exhibit an effective photocatalytic reactivity for the liquid-phase degradation of propanol diluted in water (Fig. 8) and the decomposition of NO into N2 and O2 at 295 K under visiblelight ( > 450 nm) irradiation, while the un-implanted original TiO2 thin film does not exhibit reactivity. It was clearly demonstrated that the application of such ion beam techniques as the ICB method and metal ion implantation can allow us to prepare TiO2 thin films that can work effectively under visible-light irradiation. Using the transparent TiO2 thin film, which can operate under visible-light and solar beam irradiation, many innovative solutions can be applied to environmental problems (Fig. 9). Mechanism in the Metal-Ion-Implanted TiO2 Metal ion implantation is very effective in modifying TiO2 semiconductor powder and thin film to absorb and utilize visible light. The local structure of implanted metal ions and their role in the modification of the electronic state of TiO2 have been investigated using various spectroscopic techniques. 15

Conversion of 2-propanol / %

Figure 6. Photocatalytic decomposition of NO on the un-implanted original TiO2 and the V- and Cr-ion-implanted TiO2 photocatalysts under outdoor solar beam irradiation.

V ion-implanted TiO2 thin film

10

5

0

un-implanted original TiO2 thin film

light on

0

50

100

150

200

Reaction Time / h

Figure 8. Reaction time profiles of photocatalytic degradation of 2-propanol on the V-ion-implanted TiO2 thin-film photocatalysts under visible-light irradiation ( > 450 nm).

Visible-Light-Sensitive Photocatalysts

643

Room Light

O

(a) Cr / TiO2

O Ti

O

Solar Beam

Intensity / a.u.

Pollutants in Air & Water Photocatalytic Degradation Clean Air & Water

Solar Beam Spectrum

Metal Ionimplanted TiO2

preedge

(b) Cr2O3

O Cr

O Ti

O

O O

FT of k3χ(K) / a.u.

Sun

Normalized Absorption

Visible Light

O

O

O

Cr-O

TIO2 Thin Film on Glass

O

O O

Cr-O-Ti

(A)

Cr-O

Cr-O-Cr

(B)

200 400 600 800 1000

Wavelength / nm

Utilization of Visible Light Efficient TiO2 Thin Film

Metal Ion-implantation Ionized Cluster Beam Method

5960

6000

6040

0

Figure 9. The visible-light-sensitive transparent TiO2 thin film can provide many innovative solutions to environmental problems.

Figure 10 shows the depth concentration profile of the V ions in the V-ion-implanted TiO2 thin film deposited on glass plate where ion implantation was performed at 150 keV. Most V ions were implanted within the bulk of the TiO2 thin films, and the deeper the ions were implanted in the bulk of the TiO2 , the larger the increase in the acceleration energy. XPS measurements of the catalysts did not show any evidence of the presence of implanted metal ions on the TiO2 surface, indicating that implanted metal ions are highly dispersed within the bulk of TiO2 but not on the top surface. The use of metal ion implantation to modify the bulk electronic properties of TiO2 without any change in the structure and properties of the top surface of the TiO2 can be considered as one of the most significant advantages. On the other hand, metal-ion (Cr, V)–doped TiO2 prepared chemically by the impregnation and the sol–gel method could make no shift in the adsorption band of TiO2 . These modifications of electronic properties of TiO2 have been made in different ways, depending on the methods of metal ion doping, and metal ion implantation is the smartest technique to modify TiO2 to absorb visible light. Figure 11 shows the XAFS [X-ray absorption near-edge spectroscopy (XANES) and Fourier transform extended X-ray absorption fine structure (FT-EXAFS)] spectra of the Cr-ion-implanted TiO2 thin film. The analysis of these spectra indicates that in the Cr-ion-implanted TiO2 the Cr ions

2

4

Figure 11. XANES (a, b) and Fourier transforms of EXAFS (A, B) of the Cr-ion-implanted TiO2 thin film after calcination at 723 K (a, A) and Cr2 O3 (b, B) as a reference.

are highly dispersed in the lattice of TiO2 having octahedral coordination. These Cr ions are isolated and substituted with Ti4+ ions in the lattice positions of the TiO2 . On the other hand, the Cr-doped TiO2 catalysts chemically prepared by impregnation as well as by the sol–gel method were found to have a mixture of the aggregated Cr oxides having tetrahedral coordination similar to CrO3 and octahedral coordination similar to Cr2 O3 [144–147]. Figure 12 shows the electron spin resonance (ESR) spectra of the V-ion-implanted TiO2 powder catalysts measured before and after calcination in O2 . Distinct signals, which can be assigned to the reticular V4+ ions, were detected only after calcination at around 723–823 K. These signals due to the reticular V4+ ions have been observed only with the metal-ion-implanted TiO2 catalyst after calcination and can exhibit an efficient redshift in their absorption band. On the other hand, the ESR signals assignable to the reticular V ion have never been observed with V-ion-doped TiO2 prepared chemically by impregnation as well as by the sol–gel method.

a) before calcination

109

b)

[VO2+]

723 K

Intensity / a. u.

6

Distance / Å

Energy / eV

V

106

105 Ti

Si

[VO2+]

C)

[V4+(reticular)]

873 K 3

10

104 0 level

1 0

100

200

300

400

500

2600

3000

3400

3800

4200

[G]

Depth / nm

Figure 10. Depth concentration profile of V ions obtained from SIMS measurements with the V-ion-implanted TiO2 thin film.

Figure 12. ESR spectra of the V-ion-implanted TiO2 photocatalysts before calcination (a) and after calcination in O2 at 723 K (b) and 873 K (c).

Visible-Light-Sensitive Photocatalysts

644 These spectroscopic investigations suggest that the substitution of octahedrally coordinated Ti ions in the bulk TiO2 lattice with implanted metal ions is important for modifying TiO2 to be able to adsorb visible light and operate as an efficient photocatalyst under visible- and/or solar-light irradiation. To prove this mechanism is induced by metal ion implantation, a theoretical approach using the ab initio molecular orbital calculation on the basis of the density functional theory method has been carried out. Because of the capacity of the calculation, the binuclear cluster models having an octahedral coordination similar to the coordination of TiO2 have been applied. In these cluster models, the metal ions having various electric charges (V5+ , V4+ , V3+ ) are substituted for the Ti4+ ion octahedrally coordinated in the lattice position of TiO2 [94]. As shown in Figure 13, the models having V4+ ions Ti4+ –V4+  and V3+ ions Ti4+ – V3+  can have an energy gap of Ti(d)–O(d), smaller than that of un-implanted pure Ti oxide Ti4+ –Ti4+ . The overlap of Ti(d) and V(d) orbitals is observed with these models. These results indicate that the mixing of the Ti(d) orbital of the Ti oxide and the metal(d) orbital of the implanted metal ions in the low electric charge is essential to decrease the energy gap between the Ti(d) and O(p) orbitals of the Ti oxide. Considering this mechanism, it can be proposed that in the metal-ion-implanted TiO2 the overlap of the conduction band due to the Ti(d) of TiO2 and the metal(d) orbital of the implanted metal ions can decrease the bandgap of TiO2 , enabling it to absorb visible light (Fig. 14). The present results indicate that the substitution of Ti ions with the isolated metal ions implanted into the lattice position of the bulk of TiO2 (Fig. 15) is the determining factor for the utilization of visible light and solar beam.

2.1.2. Magnetron Sputtering Although the combination of the ICB method and the metal ion implantation method can be applied to prepare TiO2 thin films that can operate under visible light, it requires Energy gap (eV) Ti(d) ~ O(p):

2.84 eV

3.16 eV

2.40 eV

2.73 eV

0

Ti

V Ti

-1

Ti

Energy / eV

Ti Ti

Ti, V

Ti, V V

Ti, V

-2

Ti, V

V

V Ti, V V

-3 V

-4

O

hybrid of Ti(d) and M(d)

Ti(d)

CB

λ = 400-600 nm

λ = 380 nm

O(p)

VB

O(p)

Figure 14. Energy diagram of TiO2 and metal-ion-implanted TiO2 .

two processes. Recently, it has been found that visible-lightsensitive TiO2 thin-film photocatalysts can be prepared using a conventional RF magnetron sputtering (RF-MS) deposition method [148]. This is an alternative and more practical preparation method that has been successfully applied to create TiO2 thin-film photocatalysts that can initiate various significant reactions effectively under visible-light irradiation. In the preparation of TiO2 thin films using a conventional RF-MS deposition apparatus (Fig. 16), a TiO2 plate with a rutile structure was used as the source material (sputtering target). Moreover, only Ar was used as the sputtering gas without the coexisting O2 as the reactive gas in the process chamber. The sputtered particles, such as Ti4+ and O2− , were accumulated on the substrate surface to form the TiO2 thin films. Figure 17 shows the UV–vis absorption spectra of the TiO2 thin films prepared by the RF-MS deposition method with different substrate temperatures. The thin film deposited at 373 K shows high transparency and interference fringes, indicating that stoichiometric and uniform TiO2 thin films can be prepared. As the substrate temperature is increased, the thin films become more efficient in absorbing visible light. These results clearly indicate that only TiO2 thin-film photocatalysts that have undergone a modification of their electronic properties are actually able to utilize visible light for some photocatalytic reactions. UV-light ( > 270 nm) irradiation of the TiO2 thin films in the presence of NO was found to initiate the photocatalytic decomposition of NO into N2 , O2 , and N2 O. The higher the preparation temperatures, the lower the photocatalytic reactivity observed under UV-light irradiation. This is due to the fact that the thin films prepared at relatively lower temperatures are highly transparent and are able to utilize the incident light more efficiently. On the other hand, TiO2 thin films prepared at relatively higher temperatures (∼873 K) exhibited a slightly yellow coloring. This is clear evidence of the efficient absorption of visible light. Visible-light

O O

-5

O

O

O

-6 OH2 HO H2O

IV

Ti

O

OH2

Ti

O

OH

OH2

OH 4+

OH2

OH

4+

〈 Ti - Ti 〉

HO

H2O

HO 4+ 5+ 〈 Ti - V 〉 H2O OH OH v O OH2 V Ti OH O OH OH2

IV

V

O2-

2-

2-

2-

O

O

OH O

Ti

O OH 4+

O2-

O2-

O2-

O

OH2 4+

〈 Ti - V 〉

OH2

4+

OH HO H2O

Ti4+

3+

〈 Ti - V 〉 OH2 OH III O OH2 V Ti OH O OH2 OH2

Figure 13. Energy level of higher lying molecular orbital obtained by the ab initio molecular orbital calculations with the cluster models of the V-ion-implanted TiO2 photocatalyst.

O2-

Mn+

O

O2-

O2-

O22-

Ti4+

2-

O

O

2-

Figure 15. Local structure of metal-ion-implanted TiO2 . Implanted metal ions Mn+ (Cr3+ –Cr4+ , V3+ –V4+ ) are highly dispersed and isolated in the TiO2 matrix and substituted with octahedrally coordinated lattice Ti4+ with a low oxidized state.

Visible-Light-Sensitive Photocatalysts

645 Heater

Target N

Magnetic Field

S

S

Magnetic Field

(TiO2)

N

N

S

Figure 16. Schematic diagram of an RF magnetron sputtering deposition method.

( > 450 nm) irradiation of these TiO2 thin films in the presence of NO was found to initiate the decomposition of NO into N2 , O2 , and N2 O. As the deposition temperature increases, the photocatalytic reactivity under visiblelight irradiation became higher, reaching a maximum at 873 K. Moreover, the order of the photocatalytic reactivity under visible-light irradiation corresponds to that of the relative intensity at 450 nm in the UV–vis absorption spectra of these catalysts (Fig. 18). These results clearly show that TiO2 thin films prepared at relatively higher temperatures (∼873 K) can work as effective photocatalysts that can absorb and operate efficiently under visible-light irradiation. With this RF-MS technique, TiO2 with a shortage of lattice oxygen was produced in the bulk of the thin film, which can induce the absorption of visible light.

λ > 450 nm

0.4

1

0.5

0 0 300 400 500 600 700 800 900 1000

Relative Intensity at 450 nm of UVVIS absorption spectra

TiO2 thin film

0.8

Yield of N2 formation / µmol•m-2

Substrate (Quartz glass)

Preparation temperature / K

Figure 18. Effects of preparation temperatures on the photocatalytic reactivities for the decomposition of NO under visible-light irradiation and the relative intensity at 450 nm in the UV–vis absorption spectra.

oxidative removal of NOx from the gas phase. The plasmatreated TiO2 , which can be easily produced on a large scale, is expected to have a higher efficiency in utilizing solar energy than the raw TiO2 material. On the other hand, several researchers have reported that Pt-loaded TiO2 photocatalysts prepared by a chemical loading method such as conventional impregnation exhibited photocatalytic reactivity for the degradation of organic compounds under visible-light irradiation [154–160]. In these catalysts, the loaded Pt can partially reduce the TiO2 to generate the lattice oxygen defects, which are similar to those generated by plasma irradiation. Although the detailed mechanism on the absorption of visible light has not been clarified yet, the oxygen defects are probably responsible for the absorption and utilization of visible light.

2.1.3. Plasma Irradiation It has been reported that the reduction of TiO2 by hydrogen plasma treatment creates a new absorption band in the visible-light region and is expected to create photocatalytic activity under visible-light irradiation. The TiO2 semiconductor powder, which has photocatalytic activity in the UV-light region, was treated using microwave and radiofrequency (RF) plasmas [136–138]. The TiO2 was reduced, thus having an oxygen deficiency. The RF-plasma-treated TiO2 absorbed visible light between 400 and 600 nm and showed a photocatalytic reactivity in this region, as measured by the

Transmittance / a.u.

(a)

200

(b) (e)

(c) (d) Efficient absorption of visible light

300

400

500

600

700

800

Wavelength / nm

Figure 17. UV–vis absorption (transmittance) spectra of TiO2 thin films prepared by an RF-MS deposition method. Preparation temperatures (K): (a) 373, (b) 473, (c) 673, (d) 873, (e) 973.

2.2. Chemical Process 2.2.1. Metal-Doped TiO2 and ZnS Photocatalytic water splitting is regarded as artificial photosynthesis. Since the Honda–Fujishima effect using a TiO2 electrode was discovered, numerous researchers have extensively studied water splitting using semiconductor photocatalysts. Several photocatalysts with wide bandgaps, such as NiOx –Ta2 O5 , NiOx –K4 Nb6 O17 , NiO–K2 La2 Ti3 O10 , and NaTaO3 , have been reported to be highly active for splitting water into H2 and O2 under UV irradiation [149–153]. Even in the presence of sacrificial reagents, the well-known photocatalysts that are active under visible-light irradiation are only Pt/CdS for H2 evolution and WO3 for O2 evolution [26–41]. Some modified layered compounds, such as Pt– RbPb2 Nb3 O10 and CdS–K4 Nb6 O17 , have also been reported to show photocatalytic reactivity to some extent even under visible-light irradiation [150, 151]. Transition metal ion doping has been applied by many researchers to modify the electronic structure of photocatalysts to utilize visible light. The doping of a foreign element into an active photocatalyst with a wide bandgap to make a donor or an acceptor level in the forbidden band is one of the ways to develop a new visible-light-sensitive photocatalyst (Fig. 19). There are many reports that metal-ion-doped photocatalysts, such as TiO2 , SrTiO3 , and ZnS, can absorb and utilize visible light. In particular, the effects of doping of Cr ions into TiO2 and SrTiO3 on the photocatalytic

Visible-Light-Sensitive Photocatalysts

646

0 1

O2/H2O

Mn+ (LUMO)

(i) nondoped TiO2

2

1.23 eV

Visible light absorption New level for oxidation (HOMO)

2

UV light absorption

3

VB

O(2p) (stable but too deep)

Figure 19. Modified band structure of visible-light-sensitive photocatalysts.

reaction have been widely studied. Although it is easy to put the color to the semiconducting photocatalyst by metal ion doping, in most cases the doped metal ions operate as charge recombination centers, which leads to a decrease in the photocatalytic reactivity. It has recently been found that TiO2 and SrTiO3 co-doped with Sb5+ and Cr3+ ions showed intense absorption bands in the visible-light region and possessed energy gaps 2.2 and 2.4 eV, respectively [101]. TiO2 co-doped with Sb5+ and Cr3+ ions evolved O2 from an aqueous silver nitrate solution under visible-light irradiation, while SrTiO3 co-doped with Sb5+ and Cr3+ ions evolved H2 from an aqueous methanol solution (Table 2). The activity of TiO2 photocatalysts co-doped with Sb5+ and Cr3+ ions was remarkably higher than that of TiO2 doped with only Cr3+ ions. This was due to the fact that the charge balance was kept by co-doping of Sb5+ and Cr3+ ions, resulting in the suppression of the formation of Cr6+ ions and oxygen defects in the lattice, which should work as effective nonradiative recombination centers between photogenerated electrons and holes (Fig. 20). Table 2. Visible-light-sensitive photocatalysts and their photocatalytic reactivity for water splitting to evolve H2 and O2 from aqueous solution.

Photocatalyst

Bandgap Sacrificial Activity (mmol/h) (eV) reagent H2 O2 Ref.

Conventional type Pt/CdS WO3

2
4 2
8

K2 SO3 AgNO3

850 —

— 65

[100] [100]

Doping type NiOx /In0
9 Ni0
1 TaO4 Pt/SrTiO3 :Cr,Sb Pt/SrTiO3 :Cr,Ta TiO2 :Cr,Sb Cu–ZnS Ni–ZnS Cu–TiO2

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

None CH3 OH CH3 OH AgNO3 K2 SO3 K2 SO3 CH3 OH

17 78 70 — 450 280 1.3

8 — — 42 — — —

[119] [101] [101] [101] [100] [100] [114]

New VB formation type BiVO4 AgNbO3 TaON TaON RbPb2 Nb3 O10 Pt/HPb2 Nb3 O10

2
4 2
86 2
4 2
4 2
6 2
6

AgNO3 AgNO3 AgNO3 AgNO3 AgNO3 CH3 OH

— — — — — 3.2

37 37 158 158 1.1 —

[100] [100] [128] [128] [151] [151]

Solid-solution type Pt/AgInZn7 S9

2
4

940

—

[133]

Pt/In2 O3 (ZnO)3

2
6

Na2 S + K2 SO3 CH3 OH

1.3

—

[135]

Absorbance / arb. units

Potential / V

CB H+/H

(ii) TiO2 doped with Sb (3.45 %) (iii-v)TiO2 codoped with (iii) Sb (1.25 %)/Cr (0.5 %) (iv) Sb (1.25 %)/Ni (0.5 %) (v) Sb (1.25 %)/Cu (0.5 %)

(i)

(iii) (iv)

(ii)

400

(v)

500

600

700

800

Wavelength / nm

Figure 20. Diffuse reflection spectra of (i) nondoped TiO2 , (ii) TiO2 doped with Sb (3.45%), (iii)–(v) TiO2 codoped with (iii) Sb (1.25%)/Cr (0.5%), (iv) Sb (1.25%)/Ni (0.5%), and (v) Sb (1.25%)/Cu (0.5%). Reprinted with permission from [101], W. Kato and A. Kudo, J. Phys. Chem. B 106, 5029 (2002). © 2002, American Chemical Society.

ZnS is one of the active photocatalysts for hydrogen evolution from aqueous solutions in the presence of sacrificial reagents even without co-catalysts such as platinum. Although the conduction band level is high enough to reduce H2 O into H2 , ZnS with a bandgap of 3.7 eV can operate as a photocatalyst only under UV irradiation. Cu2+ - and Ni2+ -doped ZnS photocatalysts, Zn0
0957 Cu0
043 S and Zn0
999 Ni0
001 S, have a pale yellow color and have energy gaps of 2.5 and 2.3 eV, respectively. The absorption bands of these metal-ion-doped ZnS photocatalysts consist of two types of bands. One is a visible-light absorption band due to the transition from doped Cu2+ and Ni2+ to the conduction band of ZnS and the other is a UV absorption band similar in origin to that of ZnS. Under visible-light irradiation, the H2 evolution from aqueous potassium sulfite and sodium sulfide solutions can be observed on these doped Zn0
0957 Cu0
043 S and Zn0
999 Ni0
001 S photocatalysts (Table 2) [102, 103].

2.2.2. TiON and TaON By metal ion implantation, Ti4+ ions in TiO2 can be substituted with implanted metal ions to modify the electronic state of TiO2 to absorb visible light. On the other hand, the substitution of O2− ions in TiO2 with other anions such as N3− and S2− has been applied to modify the electronic state of TiO2 [125, 126]. The doping of N3− ions into TiO2 was carried out by heating a TiO2 sample in the presence of NH3 . Also during the formation of the TiO2 thin film using the sputtering technique in the presence of N2 , N3− -ion-doped TiO2 thin film was produced directly. It has been found recently that this N3− -ion-doped TiO2 (films and powders of TiO2−x Nx ) has revealed an improvement over TiO2 under visible light  < 500 nm) in terms of the optical absorption properties and photocatalytic reactivity such as photodegradation of methylene blue and gaseous acetaldehyde and hydrophilicity of the film surface (Fig. 21). Nitrogen doped into substitutional sites of TiO2 has proven to be indispensable for bandgap narrowing and photocatalytic reactivity, as assessed by first-principles calculations and X-ray photoemission spectroscopy. Instead of nitrogen

Visible-Light-Sensitive Photocatalysts

647

O2-

O24+

O2-

O2Ti

4+

O2-

O2Ti

4+

O2-

@

O2-

Absorbance / %

O2-

O2Ti

80

O2-

O2-

O2-

= N3- or defect 60

TiO2

40

TiO2-xNx 20

0 200

300

400

500

600

700

Wavelength / nm

Figure 21. Experimental optical absorption spectra of TiO2−x Nx and TiO2 films. Reprinted with permission from [125], R. Asahi et al., Science 293, 269 (2001). © 2001, American Association for the Advancement of Science.

ions, the substitution of the oxygen ions of TiO2 with carbon and sulfur ions has also been found to be effective in modifying the electronic state of TiO2 to absorb visible light [126, 132]. A transition metal oxynitride, TaON, and a transition metal nitride, Ta3 N5 , are also novel photocatalysts responding to visible light [128–130]. Ta3 N5 and TaON evolve H2 and O2 by visible-light ( < 600 nm) irradiation in the presence of a sacrificial electron donor and acceptor, respectively, without any noticeable photoanodic or cathodic corrosion, representing candidates for overall water splitting by visible light (Table 2).

3. MIXED OXIDES 3.1. ABO4 -type Binary Oxide The valence band of most oxide semiconductors consists of O(2p) orbitals. The valence band level is about 3 V, which is deep enough to oxidize H2 O into O2 . Because it causes oxide semiconductor photocatalysts to have wide bandgaps, it is necessary to make a valence band consisting of an orbital other than O(2p) to develop a visible-light-sensitive photocatalyst. Recently, a binary oxide having an ABO4 composition, such as BiVO4 and InTaO4 , was found to exhibit photocatalytic reactivity even under visible-light irradiation [116–124]. Bi3+ with a 6s2 configuration is a good candidate for forming such a valence band. The highly crystalline BiVO4 powders with scheelite (monoclinic) and zircon-type (tetragonal) structures were synthesized selectively by an aqueous process. The BiVO4 powder with the scheelite structure showed a high photocatalytic activity for O2 evolution in the presence of a sacrificial reagent (Ag+ ) under visible-light irradiation ( > 420 nm). The zircontype BiVO4 with a bandgap of 2.9 eV mainly showed a UV

absorption band, while the scheelite BiVO4 with a bandgap of 2.4 eV had a characteristic visible-light absorption band in addition to the UV absorption band (Fig. 22). The UV absorption bands observed in the zircon-type and scheelite BiVO4 were assigned to the band transition from O(2p) to V(3d), whereas the visible-light absorption was due to the transition from the valence band formed by a hybrid orbital of Bi(6s) and O(2p) to the conduction band consisting of V(3d). The photocatalytic activity of the BiVO4 powder prepared by the aqueous process was much higher than that of BiVO4 prepared by a conventional solid-state reaction. The highly crystalline powder without significant formation of defects and the decrease in the surface area can be obtained by the aqueous process at room temperature, which led to efficient photoreactivity under visible-light irradiation (Table 2). Furthermore, it has been found that the Ni doping into InTaO4 can form In1−x Nix TaO4 (x = 0–0
2), which can first induce the direct splitting of water into H2 and O2 under visible-light irradiation ( > 420 nm) with a quantum yield of about 0.66% at 402 nm even without the sacrificial reagents (Table 2). The bandgap energy is narrowed with Ni doping from 2.6 (nondoped) to 2.3 eV (0.1 Ni doping). This bandgap change in Ni-doped compounds can be attributed to internal transitions in a partly filled Ni d shell.

3.2. Z-Scheme Reaction This has been studied on water splitting into H2 and O2 using two different semiconductor photocatalysts and a redox mediator, mimicking the Z-scheme mechanism of photosynthesis [161, 162]. It was found that the H2 evolution took place on a Pt–SrTiO3 (Cr–Ta-doped) photocatalyst using an I-electron donor under visible-light irradiation. The Pt–WO3 photocatalyst showed an excellent activity of the O2 evolution using an IO− 3 electron acceptor under visible light (Fig. 23). Both H2 and O2 gases evolved in the stoichiometric ratio (H2 /O2 = 2) for more than 250 h under visible light using a mixture of the Pt–WO3 and the Pt– SrTiO3 (Cr–Ta-doped) powders suspended in NaI aqueous solution. This is the first time that stoichiometric water splitting occurred over oxide semiconductor photocatalysts under visible-light irradiation. A two-step photoexcitation a) Scheelite structure (monoclinic)

Absorbance / arb. units

100

CB VB b) Zircon type structure (tetragonal)

300

400

500

V3d Bi6s O2p

600

Wavelength / nm

Figure 22. Diffuse reflectance spectra of BiVO4 prepared by an aqueous process at room temperature (a) Scheelite structure (monoclinic), (b) zircon-type structure (tetragonal). Reprinted with permission from [100], A. Kudo, J. Ceram. Soc. Jpn. 109, S81 (2001). © 2001, The Ceramic Society of Japan.

Visible-Light-Sensitive Photocatalysts

648

-

H+/H2 O2/H2O

H2O

to develop photocatalyst-included zeolites that can demonstrate their unique photocatalytic properties even under visible-light irradiation.

H2O H2

+

O2

4.1. Metal-Ion-implanted Ti Zeolite Pt/TiO2

(b)

PS1 [H2] PS2 [O2]

-

+

H /H2 Ox/Red

-

Ox

H2O H2

Red

O2/H2O

Red H2O

+

O2

Pt/WO3

+

Pt/SrTiO3(Cr,Ta) 〈I-/IO3-redox mediators〉 Ox

Figure 23. (a) Conventional photocatalytic reaction mechanism (onestep system) and (b) reaction mechanism mimicking Z scheme (twostep system). Reprinted with permission from [162], K. Sayama et al., J. Photochem. Photobiol., A 148, 71 (2002).

mechanism using a pair of I− /IO− 3 redox mediators has been proposed. The quantum efficiency of the stoichiometric water splitting was approximately 0.1% at 427 nm.

4. ZEOLITE PHOTOCATALYSTS The unique and fascinating properties of zeolites involving transition metals within the zeolite framework or cavity have opened new possibilities for many application areas not only in catalysis but also for various photochemical processes [144–147, 163–230]. The transition metal ions in metallosilicate catalysts are considered to be highly dispersed at the atomic level and also to be well-defined catalysts that exist in the specific structure of the zeolite framework. The wellprepared zeolite sample should contain only the isolated metal ions in the framework. This is of great significance in the design of highly dispersed transition metal oxide catalysts such as Ti [163–190], V [191–200], Cr [144–147], and Mo [201–205], which can be excited under UV irradiation by the following charge transfer process: Mn+

h

O2− − → M n−1 +

O− ∗

M  Ti V Cr Mo

These charge transfer excited states, that is, the electron– hole pair states that localize quite near to each other as compared to the electron and hole produced in semiconducting materials, play a significant role in various photocatalytic reactions such as the decomposition of NO into N2 and O2 or the degradation of organic impurities in water, the photooxidation reaction of hydrocarbons, and the photoinduced metathesis reaction of alkanes [144–147, 163–229]. These photocatalytic reactions were found to proceed with high efficiency and selectivity, displaying quite different reaction mechanisms from those observed on semiconducting TiO2 photocatalysts in which the photoelectrochemical reaction mechanism plays an important role. However, the metal oxide moieties included within the framework of zeolite can be activated only under UV irradiation. It is attractive

Although TiO2 has an octahedral coordination, titanium oxide having a tetrahedral coordination (tetra Ti oxide) can be prepared in a silica matrix such as zeolite and mesoporous silica. Ti zeolites with the tetrahedrally coordinated Ti-oxide species showing unique reactivities for various photocatalytic reactions under UV irradiation (220–260 nm) [163–190, 230] are good candidates for efficient and selective photocatalysts. Although these tetrahedrally coordinated Ti-oxide species can exhibit unique photocatalytic reactivity, they can only adsorb and utilize UV light at around 220–250 nm to form the charge transfer excited state as active species. It is also vital to develop Ti zeolites that can operate efficiently under visible-light irradiation. The application of metal ion implantation [3] to modify the Ti zeolites is very interesting. Recently, it has been found that metal ion implantation with V ions has been applied on Ti zeolites and Ti-containing mesoporous silica to design photocatalysts that can operate under visible-light irradiation. The metal ion implantation with V ions on Ti zeolites and Ti-containing mesoporous silica was carried out at 150 eV. Figure 24 shows the effect of metal ion implantation on the diffuse reflectance UV–vis absorption spectra of Ti-HMS mesoporous silica having a tetrahedrally coordinated Ti oxide in the framework. As shown in Figure 24, the V-ionimplanted Ti-HMS can absorb light at longer wavelengths (∼450 nm), while the original un-implanted Ti-HMS absorbs UV light at a wavelength shorter than 300 nm. These results indicate that metal ion implantation is effective for modifying the Ti-HMS to absorb visible light and exhibit the photocatalytic reaction under visible-light irradiation (Fig. 25). These metal-ion-implanted Ti zeolites (V/Ti zeolites) and Ti-containing mesoporous silica (V/Ti mesoporous silica) exhibit efficient photocatalytic reactivity for various reactions such as NO decomposition and partial oxidation of hydrocarbons even under visible-light irradiation in keeping the unique photocatalytic properties of the tetrahedrally coordinated Ti-oxide species. The investigations using XAFS analysis on the local structure of active sites and the molecular orbital calculations on the electronic structure of active O2Ti4+ O

Absorbance / a.u.

(a)

shift

200

250

300

350

O

V O

Implanted V ions (from left to right): 0, 0.66, 1.3, 2.0 (µmol/g-cat)

400

450

500

Wavelength / nm

Figure 24. Diffuse reflectance UV–vis absorption spectra of the V-ionimplanted Ti-HMS.

Visible-Light-Sensitive Photocatalysts

649

N2, O2, N2O

photocatalysts

25

Yields / µmol•g-TiO2-1

20

N2 N2O

V/Ti-HMS

N2 N2O

Ti-HMS

Dark Light ON

15

λ > 390 nm 10

λ > 420 nm

5

0 -1

0

1

2

3

Time / h

Figure 25. Reaction time profiles of the photocatalytic decomposition of NO on Ti-HMS and V-ion-implanted Ti-HMS under visible-light irradiation ( > 390 nm, 420 nm). The amount of implanted V ion: 2.0 mol/g-cat. The yield of N2 ( ,
) and N2 O (, ) formation on V/Ti-HMS; the yield of N2 () and N2 O () formation on Ti-HMS.

sites have revealed that the direct coordination and interaction between the implanted metal ions and the tetrahedrally coordinated Ti-oxide species can modify the electronic state of the Ti-oxide species to absorb and utilize visible light.

4.2. Cr Zeolite Highly dispersed Mo- or Cr-oxide catalysts have been shown to exhibit high activities for various photocatalytic reactions such as the photooxidation reaction of hydrocarbons or the photoinduced metathesis reaction of alkanes [144– 147, 201–203]. Recently, it has been found that the chemically prepared Cr-containing mesoporous silica molecular sieves (Cr-HMS) having tetrahedrally coordinated isolated Cr-oxide (chromate) moieties can operate as efficient photocatalysts for the decomposition of NO and the partial oxidation of propane with molecular oxygen under visible-light irradiation. Cr-HMS mesoporous molecular sieves (0.02, 0.2, 1.0, 2.0 wt% as Cr) were synthesized using tetraethylorthosilicate and Cr(NO3 )3 · 9H2 O as the starting materials and dodecylamine as a template and calcined in a flow of dry air at 773 K. The results of the XRD analysis indicated that the Cr-HMS has a mesopore structure and the Cr-oxide moieties are highly dispersed in the framework of HMS, while no other phases are formed. Cr-HMS exhibited a sharp and intense pre-edge peak in the XANES region, which is characteristic of Cr-oxide moieties in tetrahedral coordination [144–147]. In the Fourier transforms of the EXAFS spectra, only a single peak due to the neighboring oxygen atoms (Cr O) can be observed, showing that Cr ions are highly dispersed in Cr-HMS. The analysis of the EXAFS spectrum of Cr-HMS indicated that tetrahedrally

coordinated Cr-oxide (chromate) moieties having two terminal C O bonds existed as in an isolated state [two oxygen atoms (Cr O) at 1.57 A and two oxygen atoms (Cr O) at 1.82 A]. The ESR technique was also applied to investigate the coordination state of the Cr-oxide species by monitoring the Cr5+ ions formed under UV irradiation of the catalyst in the presence of H2 at 77 K. After photoreduction with H2 at 77 K, the Cr-HMS exhibited a sharp axially symmetric signal at around g = 1
9 (g = 1
880, g∧ = 1
945), attributed to the isolated mononuclear Cr5+ ions in tetrahedral coordination. As shown in Figure 26, the UV–vis spectra of the Cr-HMS exhibit three distinct absorption bands at around 250, 360, and 480 nm, which can be assigned to the charge transfer from O2− to Cr6+ of the tetrahedrally coordinated Cr-oxide moieties. Without a Cr ion, the HMS exhibited no absorption band above 220 nm. The absorption bands assigned to the absorption of the dichromate or Cr2 O3 cluster cannot be observed above 550 nm, indicating that tetrahedrally coordinated Cr-oxide species exists in an isolated state. Cr-HMS exhibited photoluminescence spectra at around 550–750 nm upon excitation of the absorption (excitation) bands at around 250, 360, and 480 nm, respectively. These absorption and photoluminescence spectra are similar to those obtained with well-defined highly dispersed Cr oxides anchored onto Vycor glass or silica [144–147] and can be attributed to the charge transfer processes on the tetrahedrally coordinated Cr-oxide moieties involving an electron transfer from O2− to Cr6+ and a reverse radiative decay, respectively. UV-light irradiation ( > 270 nm) of the Cr-HMS in the presence of NO led to the photocatalytic decomposition of NO and the evolution of N2 , N2 O, and O2 in the gas phase at 275 K. The Cr-HMS also shows photocatalytic reactivity even under visible-light irradiation ( > 450 nm). As shown in Figure 27, the N2 yields increase linearly with the irradiation time. The presence of both Cr-oxide species included within the HMS as well as light irradiation are indispensable for the photocatalytic reaction to take place, and the direct decomposition of NO to produce N2 , O2 , and N2 O occurs photocatalytically on the Cr-HMS. Although the reaction rate under visible-light irradiation is smaller than that under 1.2

O2-

a Kubelka-Munk function

hv ( λ > 390, 420 nm) NO

O2Cr6+

0.8

O O Mesoporous silica

O 2Cr5+

visible light

λ > 450 nm

O2-

O

O

b 0.4

c 0 200

300

400

500

600

700

800

Wavelength / nm

Figure 26. UV–vis spectra of Cr-HMS catalysts: (a) 2.0 wt%, (b) 1.0 wt%, (c) 0.2 wt% as Cr.

Visible-Light-Sensitive Photocatalysts

650 1.5 N2 Selectivity a) 45 %

N2 Yields / µmol • g-cat-1

light on

Zeolite O2-

O21

O2-

Cr6+

Ti4+ O

a) λ > 270 nm

O

O

Visible light sensitive moieties

N2 Selectivity b) 97 %

0.5

V

O

O

Mesoporous silica

b) λ > 450 nm 0 0

1

2

3

4

5

6

7

8

Reaction time / h

Figure 27. Reaction time profile of N2 formation in the photocatalytic decomposition of NO on the Cr-HMS catalyst at 273 K (2.0 wt% as Cr) under UV-light irradiation (a,  > 270 nm) and visible-light irradiation (b,  > 450 nm).

UV-light irradiation, the selectivity for N2 formation (97%) under visible-light irradiation is higher than that of UVlight irradiation (45%). These results indicate that Cr-HMS can absorb visible light and act as an efficient photocatalyst under not only UV light but also visible light and especially Cr-HMS can be useful to form N2 under visible-light irradiation. The addition of NO into the Cr-HMS led to an efficient quenching of the photoluminescence spectrum of the catalyst, its extent depending on the amount of NO added. These results indicate that the charge transfer excited state of the tetrahedrally coordinated isolated Cr-oxide moieties, (Cr5+ O− )∗ , easily interact with NO and this photoexcited species plays an important role in the photocatalytic reaction under UV and visible-light irradiation. On the other hand, light irradiation of the Cr-HMS in the presence of propane and O2 led to the photocatalytic oxidation of propane. The partial oxidation of propane with a high selectivity for the production of oxygen-containing hydrocarbons such as acetone and acrolein proceeds under visiblelight irradiation, while further oxidation proceeds mainly under UV-light irradiation to produce CO2 and CO. The selectivity of partial oxidation production under visible-light irradiation observed at 12% propane conversion is higher than those observed under UV-light irradiation at 26% conversion and even under UV-light irradiation for the shorter reaction time with 11% conversion. These results indicate that the tetrahedrally coordinated isolated Cr-oxide moieties in HMS can exhibit efficient photocatalytic reactivity for the oxidation of propane under visible-light irradiation with a high selectivity for the partial oxidation of propane. These results have clearly demonstrated that the Cr-HMS can absorb visible light and act as an efficient and selective photocatalyst under visible-light irradiation. This photocatalytic system with tetrahedrally coordinated Cr-oxide moieties dispersed on mesoporous silica seems to be a good candidate for converting abundant visible- or solar-light energy into useful chemical energy, as well as the metal-ionimplanted tetrahedrally coordinated Ti-oxide moieties dispersed on zeolite and mesoporous silica (Fig. 28).

Figure 28. Visible-light-sensitive photocatalysts included within the frameworks of zeolite and mesoporous silica.

5. CONCLUSIONS In this review, recent applications to the design of visible-light-sensitive photocatalysts are summarized. Large progress has been made in the design of visible-light-sensitive photocatalysts. Especially, the modification of TiO2 photocatalysts to utilize visible light and the design and development of visible-light-sensitive photocatalysts, which can induce photocatalytic water splitting to produce H2 and O2 efficiently, are the most fascinating research topics. Among the various unique attempts to control the electronic structure of semiconducting photocatalysts, the substitution of metal ions (Ti ions in the case of TiO2 ) with transition metal ions by metal ion implantation or metal doping and/or the substitution of anions (O ions in the case of TiO2 ) with N, S, and C ions (or defects) by chemical methods have been found to be significantly effective (Fig. 29). At the present, many researchers are engaged in applying these techniques to the design of efficient visible-light-sensitive photocatalysts with the assistance of the theoretical molecular orbital calculation and spectroscopic analyses such as XAFS using synchrotron radiation. Furthermore, the application of an advanced dry process such as ion engineering techniques (metal ion implantation, ICB method, magnetron sputtering, etc.) is the new method for the design of unique photocatalysts even in the form of transparent thin films that can operate efficiently not only under UV-light irradiation but

2-

O2O2-

O Metal ion implantation, Metal ion doping

O

O2O

2-

O2O2-

Ti4+ 2-

O

O2-

O2-

O2-

O2-

O Ti4+

O2-

O

2-

O2O2-

Ti4+ 2-

O

2-

O

O2-

2-

O

Mn+: Cr, V, Mn, Fe metal ions etc.

O2O2-

Ti4+

O2-

Mn+

4+

Ti

2-

O2-

O

2-

Substitution of metal ions

2-

O2-

O2-

O

2-

O2-

Ti4+ O2Chemically anions doping, plasma irradiation

O2Ti4+

O2-

O2O2-

O24+

Ti

2-

O O2: N, S anions, C atoms or defects O

2-

O2-

Substitution of anions or addition of defect sites

Figure 29. Attempts to control the electronic structure of semiconducting photocatalysts (TiO2 ) by the substitution of metal ions or anions to design visible-light-sensitive photocatalysts.

Visible-Light-Sensitive Photocatalysts

also under visible-light irradiation. The present research has opened the way to many innovative possibilities, significantly addressing urgent environmental concerns, and can also be considered an important breakthrough in the utilization of solar energy.

GLOSSARY Artificial photosynthesis The set of reactions in which light is harvested as the plants use light to synthesize high-energy molecules. Reactions such as the reduction of CO2 with H2 O and water splitting to produce H2 and O2 can be carried out under visible-light irradiation without plants but using artificial compounds such as semiconductor photocatalysts. Metal ion implantation A variant of the ion beam techniques in which an accelerated metal ion beam has been used to modify the electronic structure of semiconductors. In this method, metal ions are accelerated in the electronic field and injected into the sample target as the ion beam. These metal ions can interact in a different manner with the sample surface, depending on their kinetic energy. In ion implantation, metal ions are accelerated enough to have a high kinetic energy (50–200 keV) and can be implanted into the bulk of samples. On the other hand, metal ions accelerated to have a low kinetic energy (0.2–2 keV) are deposited to form a thin film on the top surface of the sample. With these unique properties of the ion beam techniques, well-defined semiconductor materials and thin films have been developed. Photocatalyst The photocatalyst can exhibit the photocatalytic reaction (photocatalysis), which is a catalytic reaction involving light absorption by a catalyst or by a substrate under light irradiation. Photocatalysis on semiconductors such as TiO2 can be regarded as a light-induced redox process. Under light irradiation with an energy larger than the bandgap energy of the semiconductors, an electron is excited from the valence band to the conduction band, leaving a hole. The electrons formed in the conduction band and the holes formed in the valence band recombine or become trapped in the surface defect sites. These surface trapped holes are powerful oxidants and the trapped electrons can act as good reductants. X-ray absorption fine structure (XAFS) spectroscopy A technique using X-rays with variable wavelengths as probes to investigate the local atomic coordination structure. The nondestructive ex-situ and in-situ measurements can be applied to both amorphous and crystalline samples. Extended X-ray absorption fine structure (EXAFS) spectroscopy and X-ray absorption near-edge spectroscopy (XANES) are variants of the X-ray absorption technique. Zeolite A microporous solid with pore size ranging from approximately 3 to 14 Å. They are composed of aluminosilicates, aluminophosphates, and metal silicates. They have a framework with corner-linked (TO4 ) tetrahedra and a void space that forms the channel, cages, and supercages. The term zeolite is now applied to different materials, such as mesoporous silica (pore size, ca. 20–30 Å), that express zeolitic behaviors such as the selective adsorption of small molecules (molecular sieves), ion exchange ability, large surface areas, and perfectly defined pore sizes.

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Encyclopedia of Nanoscience and Nanotechnology

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X-Ray Characterization of Nanolayers Dirk C. Meyer, Peter Paufler Technische Universität Dresden, Dresden, Germany

CONTENTS 1. Introduction 2. Structural Parameters of Nanolayers 3. Basic Phenomena Due to Interaction of X-Rays with Flat Surfaces and Nanolayers 4. Characterization Methods 5. Nanolayer Materials References

1. INTRODUCTION Thin films behave significantly different from bulk material if the layer thickness is, for example, of the order of several to a hundred nanometers. The influence of surface-near atoms cannot then be neglected depending on the property under investigation. The first few atomic layers are known to deviate notably in atomic, electronic, and magnetic structure from those more distant from the surface. This often includes surface segregation of impurities or surface enrichment of solute species. Moreover, the total number of atoms per unit area is of the order of 4 × 1013
t/a mm−2 for a layer of thickness t and lattice parameter a, (i.e., measuring signals proportional to this figure may become too small to be detected). For the preparation of nanolayers a number of methods are available. However, both synthesis and handling of these layers remain challenging tasks because of the need for high-purity materials and for monolayer precision with layer growth. In addition to that, the great success of nanolayers in the field of microelectronic and optoelectronic devices, for storage of information, as protecting layers, as customized (multilayer) structures, or as coupled systems (e.g., polyelectrolytes/ surfactants or protein/lipids) has increased the demand for novel structural analytical tools. For the characterization of thin films established X-ray analysis methods have been modified and dedicated new techniques were developed which predominantly make use ISBN: 1-58883-066-7/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

of the typical characteristics and quality of these samples. Since 1912 [1] it has been well known that X-ray diffraction methods can provide structural information of solids when wavelengths of the order of interatomic distances are employed. Since the scattering cross section of atoms for X-rays is rather weak, the small number of atoms contained within thin layers may prevent diffraction maxima from being detected. A way around insufficient scattering intensity has been found by using longer paths of X-rays within the specimen (i.e., grazing incidence of X-rays) and higher X-ray photon fluxes (e.g., synchrotron radiation sources). Below a critical angle of incidence c , with respect to the surface, total external reflection occurs (e.g., c = 0223 for Si wavelength = 154 Å [2]) whereby the totally reflected intensities are significantly larger than in the case of specular reflection above c . Exploiting multiple refraction and reflection of X-rays at the surface and the nanolayer/substrate interface in a range of angles of incidence directly above c (typically up to 10 ), sharp interference patterns may be obtained. The refractive index for X-rays can be written as n = 1 −  − i with the dispersion term  ≥ 0
≈76 × 10−6 for Si; wavelength = 154 Å) and the absorption term
≈ 017 × 10−6 for Si). Thus it is obvious that the values of refractive indexes of solids for X-rays differ in a characteristic manner from those of other photon energies. They are always slightly less than 1. This causes high demands concerning the measuring accuracy to proof and use the associated phenomena. The availability of intense, highly polarized and energytunable X-ray photons from synchrotron sources has enabled significant enhancement of experimental possibilities. As an example, in the case of resonant scattering  may change by more than a factor 2 [3]. One can cause this situation by the purposeful choice of the energy of X-rays according to element-specific binding energies of the electrons. For angles of incidence i > c the penetration depth increases and reaches a maximum value at i = 90 of /4 ≈ 103 –104 nm for most materials. Thus, by tuning the angle of incidence the analysis can be accomplished depth-resolvedly. Owing to these properties, analytical methods using X-rays can be applied across the whole variety of materials

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (655–680)

656 irrespective of the chemical composition, microstructure, dimension, and even degree of crystallinity. Depending on the area of application (for an overview we refer to [4]), distinct structural parameters, determinable by means of X-ray analysis methods, are used for the characterization of nanolayers. In Section 2 we intend to briefly categorize these parameters and provide the reader in Section 4 with details of the appropriate method for obtaining them. Examples of current work on X-ray characterization of different types of nanolayer materials are given in Section 5. Section 3 has been included to introduce X-ray methods to those readers who are not familiar with them. Recognizing the vast variety of analytical methods where X-rays are involved (electromagnetic waves with wavelength range ≈1 × 10−2 –30 nm), we restricted ourselves to methods exploiting X-rays as both incident and emitted probes. Excitation by/or emission of electrons, for example, will therefore not be dealt with here.

2. STRUCTURAL PARAMETERS OF NANOLAYERS The structure of nanolayers may be described at different levels of scaling. Here the parameters determinable by means of X-ray analysis methods will be summarized. Scattering of X-rays by atoms is mainly due to the core and the outer electrons (i.e., the electron density distribution in physical space is the structural parameter obtainable at the smallest scale). Also, spin density waves in thin films and multilayers accompanied by strain waves can effectively be studied. It is a very recent development that magnetic moments may also be determined directly by means of X-ray scattering and absorption spectroscopy. By measuring the specular reflectivity with X-rays of energies tuned close to binding energies of electrons, the usually weak magnetic scattering signal may become as large as several percent of the scattering caused by electric interaction. Moreover, it then becomes element specific. Because of the large amount of data involved, simplifications are often used when exploiting experimentally obtained intensities. Thus, disregarding the electron distribution, in the case of crystalline layers the coordinates of atomic nuclei with respect to a unit cell are taken as characteristics of a certain structure. Noncrystalline layers are structurally characterized by pair-correlation functions indicating the radial distribution of atoms. If the nanolayer proves multiphase, then both the structures and the quantities of all crystalline phases (including nanovoids) have to be determined in order to characterize the material properly. Characterization thereby, besides the structural identification of crystalline phases, aims also to obtain size, shape, and distribution of the appropriate particles. Supplementary information about the orientation distribution of the crystallites and their texture is needed, when the layer is polycrystalline. Moreover, strains due to internal incompatibilities or external stress can reduce the symmetry of the distorted material as compared with the equilibrium bulk state, so that the strain tensor serves as another set of parameters. It proved useful to distinguish long-range and short-range distortions, where long range means large compared to the unit cell or crystallite diameter.

X-Ray Characterization of Nanolayers

The understanding of properties at a structural level is essentially facilitated when the roughness of surface and layer/substrate interface is given quantitatively. Finally, the thickness of a nanolayer is the simplest geometric parameter on the nanoscale. Macroscopic material properties, such as the mass density, can be derived partially likewise on the basis of a number of nanoscale parameters.

3. BASIC PHENOMENA DUE TO INTERACTION OF X-RAYS WITH FLAT SURFACES AND NANOLAYERS 3.1. X-Rays and Waves When X-rays interact with atoms or molecules, energy from the electromagnetic wave is absorbed as well as emitted by them. These processes require extensive mathematical and quantum mechanical treatment. Fortunately, the fundamental ideas are comprehensible using the concept of photons approximated by wave packets. A wave packet is localized in space by superposing several wavelengths . Incident waves coming from an external source may in most cases be well approximated by plane waves, whereas waves emitted by atoms within the solid have to be rather dealt with as spherical waves. In a crystalline solid they may interfere forming an outgoing plane wave. When the amplitude of the single scattered wave is small compared with the incident wave amplitude, then the amplitude of the doubly scattered wave is negligible compared with the singly scattered wave. This simplifies the superposition of scattered X-ray photons in a solid considerably. Reflection and refraction of X-rays at discontinuities of the refractive index can be analyzed geometrically using the ray concept, which is applicable so long as the discontinuities encountered by the wave packet during its propagation are large compared with its average wavelength. In this respect X-rays behave like visible light as long as absorption of energy can be neglected.

3.2. Interaction of X-rays with Atoms of Matter The interaction of X-rays with photon energies on the order of 10 keV (“hard X-rays,” usually used for X-ray structure analysis) with atoms of matter is limited almost exclusively to their electrons. For quantitative description the attenuation of photons due to the photoelectric effect and scattering processes are to be differentiated as substantial processes. In the case of photoelectric absorption the irradiating photon of energy E extracts a bound electron. A following interatomic electron transition leads to the emission of a fluorescence photon of a characteristic energy. For quantitative description the linear photoelectric absorption coefficientN 
E is assigned to this process. Coherent and incoherent scattering of photons is described by the total linear scattering coefficient 
E. The coherent scattering process, with which the energy of the photons and their phase characteristics remain unchanged, is the basis of X-ray interference methods (see Fig. 1). The interaction effects can be determined quantitatively by measuring the attenuation of a parallel beam of X-ray

X-Ray Characterization of Nanolayers

657 The atomic scattering amplitude is a function of the scattering vector Q attributed to a scattered photon. Its modulus Q is usually expressed in dependence on the scattering angle  and the photon wavelength by Q=

4 sin
 

(3)

For the case of forward scattering
 = 0  the atomic scattering amplitude corresponds to the number of electrons per neutral atom
f0 = Z. From the viewpoint of classical mechanics the scattering process can be described by a forced oscillation with damping. In extension to the classical model it has to take into consideration that the relativistic mass of the bound electrons increases due to the excitation, whereby the scattering power decreases. This influence can be considered by adding a relativistic correction. The differential equation for the description of a forced oscillation with damping in the model of the harmonic oscillator yields mathematically complex-valued solutions. Thus the atomic scattering has also to be taken as a mathematically complexvalued function. In addition corrections have to be considered, which become substantial for the case of resonance. Energies of resonant interactions correspond to ionization energies of the atoms. The photon gives rise to real and virtual transitions, where it is absorbed and re-emitted. These processes cause the corrections f ′
Q E and f ′′
Q E to the atomic scattering amplitude. It was shown by experiment that their dependencies on Q are mostly small [5]. Thus the atomic scattering amplitude is usually expressed by Figure 1. Basic phenomena due to interaction of X-rays (energies in the order of 10 keV) with flat surfaces and nanolayers. X-ray beams directed toward a smooth vacuum/matter interface at angles of incidence i below a critical angle c will undergo total external reflection (1). For angles of incidence above c at ideally smooth interfaces two secondary beams, specular reflected (reflection angle e = i  and refracted ones, arise (2a). The refracted beam is attenuated in the matter, whereby photoelectric absorption is followed by the emission of characteristic fluorescence photons. Deviations from an ideally smooth curvature cause nonspecular diffuse scattering (2b). If matter exhibits long-range order elastic scattering can cause diffraction phenomena (diffraction angle 2) (3).

photons with incident intensity I0
E in a medium. If these photons penetrate a flat parallel plate of homogeneous isotropic material and thickness t (surface normal of the plate is parallel to the beam direction) the intensity I
E behind the plate is given by the exponential attenuation law I
E = I0
Ee−
Et

(1)

with the total attenuation coefficient 
E = 
E + 
E

(2)

whereby 
E stands for the contributions of photoelectric absorption and 
E stands for elastic and inelastic scattering, respectively. The amplitude of the electromagnetic field of X-ray photons, coherently scattered by an atom of atomic number Z, normalized with respect to a free electron, can be described by means of the atomic scattering amplitude f0 .

f
Q E = f0
Q E + f ′
E + if ′′
E

(4)

The values for the individual terms are available in tabulated form. Usually, numerical values are based on the theoretical computations of atomic scattering amplitudes of Cromer and Liberman [6]. The real part f ′
E of the correction to the atomic scattering amplitude and the imaginary part f ′′
E are interconnected (self-consistent Kramers–Kronig transform pairs). The physical origin of the causal relationship between both parts was described in detail by Toll [7]. Using the example of nickel (Z = 28) typical values and the influence of the distance of photon energy to the absorption edge (ionization energy of a bound state) can be illustrated. The energy of the K-absorption edge of nickel is 8.333 keV. For photons with an energy of 7.500 keV the corrections amount to f ′ = −2076 and f ′′ = 0579, respectively. At an energy of 8.340 keV, just below the absorption edge, the appropriate values are f ′ = −6886 and f ′′ = 3914, while at an energy of 9.000 keV values of f ′ = −6886 and f ′′ = 3914 are to be considered. It can be assumed that for photon energies with significant distance to the absorption edges the amount of corrections die down, whereby f ′′
E exhibits a jumplike behavior when exciting ionization energies.

3.3. Interaction of X-Rays with Interfaces When directing an X-ray beam toward a smooth vacuum/matter interface, like in the case of visible light, specular reflection and refraction (transmission) can be observed as basic phenomena. A deeper understanding and a linkage

X-Ray Characterization of Nanolayers

658 to electrodynamics and wave optics is given by the Huygens– Fresnel principle. In the case of X-rays it is characteristic that a refracted beam can only exist in the medium above a critical angle included by beam and surface plane. For angles of incidence below this critical angle total external reflection occurs. The angle characteristic of the reflected beam has to be described by the reflection law. The intensities of the transmitted radiation are attenuated in the medium according to an exponential law, whereby the attenuation coefficient, besides the path length of the beam, covers contributions of all interactions of X-rays with matter as mentioned. The angle of refraction, characteristic of the material as in the case of visible light, observes Snell’s law. For the intensities of the three partial beams (incident, reflected, and transmitted) found at the interface, conservation of energy is required. Considering the continuity conditions of the components of the field vectors at an interface, as demanded by electrodynamics, the amplitudes of the waves are described numerically by reflection factor r and transmission coefficient t (Fresnel equations). The squares of these amplitudes are normally called “reflectivity” and “transmission,” respectively.

3.4. Quantitative Description of Reflection, Refraction, and External Total Reflection The reflection and refraction of X-rays can be described quantitatively using refractive indices of the media, which form the interface. The refractive index of a medium is a function of the frequency and thus the energy of the photons and is defined through c n= v

(5)

whereby c stands for the speed of light in vacuum and v for the phase velocity of the wave in the medium. For a medium, which contains the chemical elements j with the atomic number Zj , the atomic weight Aj , and the densities j , on the basis of the model of the harmonic oscillator with damping, the refractive index n for X-ray photons of the wavelength is given by n=1−


j Na r0 2 f 2 Aj j j

(6)

whereby Na is Avogadro’s constant, r0 = e2 /mc 2 = 2818 × 10−13 cm is the classical electron radius, and fj is the element specific atomic scattering amplitude. In the case of forward scattering of X-ray photons with energies sufficiently different from the ionization energies of the atoms, the approximation fj ∼ Zj can be used (i.e., in general the atomic scattering amplitude can be estimated by the number of electrons per atom). If the energy of the incident X-ray photons lies in the vicinity of the ionization energy of a species of scattering atoms, contributions of resonant dispersion fj′ and absorption ifj′′ must be considered, as already mentioned. Accordingly, the refraction indexes

have to also be described as mathematically complex variables and can be expressed as n = 1 −  − i

(7)

with the parameters of dispersion = and absorption


j  Na Zj + fj′ r0 2 2 j Aj

=


j ′′ Na r0 2 f  2 Aj j j

(8)

(9)

As already mentioned in Section 1, the values of refractive indexes of solids for X-rays differ in a characteristic manner from those of other photon energies: They are always slightly less than 1 [e.g., for copper (Z = 29) and a photon energy of 8.600 keV we have  = 230 × 10−6 and = 04 × 10−6 ]. Hence for transition of X-rays from vacuum to matter total external reflection occurs below a critical angle of incidence c , which is typically smaller than 1 (e.g., for copper c = 039 at a photon energy of 8.600 keV or ≈ 144 Å). If the angle of incidence i falls short with respect to c instead of a refracted beam an exponentially decaying X-ray wave field (“evanescent wave”) develops, the penetration depth of which is typically 5 nm. For the reflected beam the well known law of reflection applies (i.e., the angle of reflection is equal to that of incidence).

3.5. Deviations from an Ideal Interface Any chemical and structural deviations from an ideal interface lead to deviations from the ideal beam propagation regarded so far. One consequence is loss of intensities of reflected beams accompanied by a redistribution in diffuse scattered contributions. In the case of transition from vacuum into a medium it is directly observable on the vacuum side, thereby allowing for determination of the real structure of the interface as described in Section 4.3.

3.6. Energy Dependent Fine Structure of Absorption and Scattering Since 1920 it has been well known that X-ray absorption spectra of atoms in solids exhibit energy-dependent oscillating fine structures for energies of about 1 eV above the absorption edges [8, 9]. Similar fine structures in scattered intensities were described for the first time in 1956 [10]. The mentioned fine structures are caused by the interaction of photoelectrons, excited in the case of resonant scattering by the interaction with X-ray photons, with the potentials of neighboring atoms. Thus from a measurement of the fine structures as a function of the energy of exciting X-rays, conclusions on the structural short-range order of the resonant scattering atoms can be drawn. The methods of X-ray absorption fine structure (XAFS) and diffraction anomalous fine structure (DAFS) based on these peculiarities have been developed as routine tools for the determination of structural short-range order. The fine structures

X-Ray Characterization of Nanolayers

659

one can imagine to be caused in a simplifying model by the interference of the photoelectron waves emitted by the resonantly scattering atom and backscattered by the neighboring ones. In the case of scattering a virtual photoelectron has to be regarded accordingly. In the quantum-mechanical description one speaks of a structure-dependent change of the transition matrix, which is used for the calculation of the probability of the transition of a bound electron to the state of a free photoelectron. For the quantitative description of oscillating fine structure all possible scattering paths of photoelectrons must be summed up. The sum over all paths without reflections at the neighbors yields smooth portions fs′
E + ifs′′
E which correspond to corrections of the atomic scattering amplitudes mentioned. If the summation runs over all paths with reflections at the neighbors, oscillating contributions fos′
E + ifos′′
E have to be added. The atomic scattering amplitude for atoms with structural ordered environment can therefore be expressed by   f
Q E = f0
Q E + fs′
E + if ′′s
E +
fos′
E + if ′′os
E

(10)

Following [11] the oscillating portions can be approximated by ˜
fos′
E + if ′′os
E = fs′′
E%
E

(11)

with the complex-valued fine structure function ˜ %
E = % ′
E + i% ′′
E

(12)

In the XAFS and DAFS theory physical quantities are usually expressed in dependence on the wave number of the excited photoelectron waves. The wave number K as a function of the energy of the exciting X-ray photons E and the electron binding energy E0 (ionization energy of the regarded bound state) is given by  2m
E − E0   (13) K=
2 The complex-valued fine structure function can be calculated by summing all possible scattering paths of photoelectrons to ˜ %
K =

N


m=1

S02 KR2m

bm
K exp)i
2KRm + *m
K 2

2

+ 2c
K+e−2m K e−2Rm /,
K

corresponds to the XAFS function. Simplifying the approximations of the XAFS function is possible for different cases. The frequency analysis of a measured fine structure, or the numeric fit of a model function to it, allows for the determination of parameters of the structural short-range order of absorbing/resonantly scattering atoms. The backscattering amplitudes and phase shifts are theoretically computed. Further peculiarities of the interaction of X-rays with long-range ordered media can be observed if the exciting radiation is polarized. These effects arise due to transitions of excited electrons into free energy levels which can be strongly directed in a solid. In the case of resonant scattering the atomic scattering amplitude consequently has to be described as a tensor (“anisotropic anomalous scattering”; see pioneering work by Templeton and Templeton in [12]). The availability of synchrotron radiation, in particular the high intensity of radiation emitted by these sources, which is available with circular polarization, made possible the observation of weak effects, to be understood as a subset of resonant scattering phenomena. These effects concern magnetic properties; the assigned interaction process is called “magnetic scattering.” The comparatively strongest effects occur in rare-earth elements and in actinides for M-shells. Magnetic scattering and absorption experiments thus enable experiments yielding singular information about magnetic and electronic structures. For a review of pioneering work in the field of X-ray magnetic circular dichroism, the absorptive counterpart of magnetic resonant scattering, and its basics we refer to [13].

(14)

Here m is the index of a particular scattering path, N is the total number of possible paths, S02 is an amplitude reduction factor due to inelastic losses, Rm is the effective path length of path m (half of the total path length), bm
K is the effective curved wave backscattering amplitude for path m with phase shift contribution *m
K, c
K is the phase shift of photoelectron waves at the emitting atoms, m2 is the effective Debye–Waller factor for path m (thermal and structural disorder), and ,
K is a combined damping factor (mean free path of photoelectron, lifetime of excited state). The imaginary part of the complex-valued fine structure function

4. CHARACTERIZATION METHODS 4.1. X-Ray Diffraction 4.1.1. Introduction The X-ray diffraction (XRD) method involving powder samples was developed only a short time after the fundamental X-ray interference experiments of single crystals by Laue and co-workers in 1912 [1]. After the descriptive explanation of the diffraction of X-rays by crystals as reflections from lattice planes by Bragg in 1913 [15] further pioneering work was done by Debye and Scherrer in 1916 [16]. For an overview concerning the historical development and fundamentals of XRD we refer to [17, 18]. While the main field of application of this method, besides objectives of single crystal structure analysis, is still on the determination of the Bravais lattice type and lattice parameters of unknown materials as well as the phase composition of materials, the interest in determining parameters of the real structure by XRD has continuously increased. A substantial milestone in the second half of the 20th century was the introduction of the Rietveld method [19, 20] combined with dedicated indexing procedures. This was due to a renaissance of XRD by opening the field of ab initio structure determination for powders and polycrystalline samples. In particular the extraction of information from XRD experiments has extraordinarily profited from the rapid development of computers in the past decades. Apart from the possibilities, which resulted from the availability of synchrotron radiation since the 1980s, also XRD with

660

X-Ray Characterization of Nanolayers

conventional X-ray sources could benefit from the development of new instruments. Important innovations were the introduction of parallel beam optics from graded multilayer reflective mirrors (“Göbel mirrors”) [21] and the availability of capillary optics. Additional highlights include the development of position sensitive area and energy resolving detectors and their implementation in XRD setups. Recently particular experiments under nonambient conditions attained much attention. As in the case of other characterization methods for the investigation of thin layers XRD techniques have been modified and dedicated strategies have been developed. If nanolayers are polycrystalline, they are characterized by grain size and shape, phase composition, texture, and residual stress of the constituent crystallites. For single crystalline layers, their orientation relation with respect to the substrate is in the focus of interest. Concerning XRD investigation of the latter group of samples we refer to Section 4.2.

4.1.2. Conventional XRD XRD is undisputed as the standard method for the investigation of crystallographic characteristics of materials. An important advantage is low requirements for sample preparation, whereby in most cases the investigation can be performed nondestructively. However, that would not be sufficient to justify the special importance of the method. Rather the high intrinsic precision is the most outstanding benefit, due to the possible precise knowledge of wavelength as the initial parameter. Since the precision is connected to the diffraction angles of reflection used for the calculation of lattice parameters, the possibility to realize large diffraction angles by choosing the appropriate wavelength is important in comparison to other techniques. X-ray diffraction data restricted to small diffraction angles can provide information on the structure of the scattering objects at lower resolution (i.e., it is sensitive to the cross shapes of molecules, inclusions, or voids). This method has proved to be a powerful probe because of its independence on the structure at atomic scale. So it can be applied to amorphous as well as to crystalline samples. Crucial structural information is derived from anomalous small angle X-ray scattering (ASAXS), because of its ability to provide data related to a certain species separately. As an example, an ASAXS investigation of hydrogenated amorphous Si–Ge alloys deposited by plasma enhanced chemical vapor deposition showed an inhomogeneous distribution of Ge with correlation lengths of about 1 nm [22]. For angle-dispersive XRD experiments (for typical setup see Fig. 2), which are the most common techniques, both radiation from conventional X-ray tubes and radiation emitted by synchrotrons have to be monochromized. While it is possible to employ energy-resolving detectors, which is meaningful particularly when using X-ray tubes, the general way is the monochromization by Bragg reflection of a properly oriented single crystal. For this many techniques using flat or bent crystals or graded multilayer reflective mirrors have been developed. The most common monochromators for use with X-ray tubes are Johannson type collimators with bent curved crystals. They are particularly suitable for established focusing Bragg–Brentano and Seemann–Bohlin

Figure 2. Schematic setup of angle-dispersive X-ray diffraction.

diffractometer geometries. For an overview of the variety of XRD geometries and their individual advantages we refer to [23]. A large spectrum of new XRD applications was opened by the pioneering work of Coster et al. in 1930 [24] which showed that X-ray resonance (“dispersion”) could be useful for studying crystal structures. In the meantime it became obvious that experiments under resonance conditions of interaction of X-rays with atoms of crystals (“resonant anomalous X-ray scattering”) can yield a lot of additional information. Today, resonant X-ray scattering is a routine tool mainly used for overcoming the phase problem in X-ray structure analysis, as well as to achieve a scattering contrast for structures containing chemical elements with similar atomic numbers. Since for these experiments X-rays with defined photon energies according to the ionization energies of the scattering atoms must be available, synchrotrons are the sources of choice for these experiments. Nowadays, resonant scattering is also employed to enhance magnetic scattering. For an exhaustive overview on theory and applications of resonant X-ray scattering we refer to [3]. The experimental setups dedicated to powder samples can be only partly used for the investigation of nanolayers. That is why dedicated strategies were developed, mostly inspired by the low interaction volumes and geometric constraints in the case of nanolayer samples.

4.1.3. XRD of Nanolayers For the investigation of nanolayers grazing incidence diffraction rather than a conventional focusing Bragg–Brentano setup is appropriate (see an example in Fig. 3). The main goal of grazing incidence setups is to increase the fraction of radiation diffracted from the nanolayer which at the same time causes a reduction of fractions scattered by the substrate. The crystalline structure of the layer can be probed in depth by varying the angle of incidence, what is outlined in more detail in Sections 4.2 and 4.3. The volume contributing to the interference phenomena is limited by the coherence length of radiation, which typically is of the order of 1 m.

X-Ray Characterization of Nanolayers

Figure 3. Measured diffracted intensities of a Mo/Si multilayer (40 periods, 2.7 nm Mo + 4.1 nm Si deposited on Si substrate) versus scattering angle. Photon energy was 8905 eV; a fixed angle of incidence of 15 was used. Reflection indices of polycrystalline Mo (W-type of are indicated, while Si layers appeared structure, space group Im 3m) to be amorphous.

Generally, stationary nanolayer specimens have to be irradiated with a beam of small divergence in the diffraction plane (horizontal divergence) at angles of incidence typically below 5 . Although this geometry means a loss of advantages of the focusing setups, the detector can be moved, searching for secondary diffracted intensities in the whole range accessible due to the geometrical constraints. To limit the horizontal divergence of the detector, aperture Soller slit collimators are usually used, often accompanied by secondary crystal monochromators. For nanolayers the same kind of information as in the case of polycrystals can be obtained. Stronger restrictions exist for textured or single crystal layers due to geometric constraints. This is because only those crystallites which fulfill the Bragg condition at the angle settings predicted by the angle of incidence and the detectable range of scattering angles will contribute to the measured signal. Thus the structure parameters can only be determined for a selection of layer volume, which may be due to a fraction on the order of 1% or less, thereby not being necessarily representative of the layer as a whole. The relative uncertainty of the determination of lattice parameters of nanolayers in conventional setups is in the order of 10−3 , whereas changes of parameters can be obtained with a precision of 10−4 .

661 Fig. 4). Thereby the Bragg angle setting is realized by the implemented goniometer. By means of the Eulerian cradle the sample can be rotated independently about two additional axes which are always perpendicular to each other. The definition of the angles * and %, commonly used, can be seen from Figure 4. Typically, the axis of sample rotation about * is chosen to be in parallel with the surface normal of flat nanolayers. Measurement of a pole figure is then performed by the detection of peak intensities of Bragg reflection by rotating the sample through a wide range of * and %. Due to the wide range of tilts the spatial resolution is limited to about 1 mm2 . An improvement can be achieved by appropriate reducing the primary beam cross section by means of capillary optics. For the spatially resolved analysis of nanolayers by means of XRD a motorized x–y–z specimen stage is used in conjunction with the Eulerian cradle. This stage enables a precise specimen translation in three directions for the mapping investigation of parameters. Lattice stress can be analyzed with the same experimental setup as for texture measurements. It is determined by evaluating the shift of Bragg angles of selected reflections in dependence on the Eulerian angles * and % and the changes of widths (more generally the shape) of the diffraction peaks. The influence of residual stress (internal stress with vanishing external forces) on diffraction behavior depends on the specific group to which it belongs. Shifts of the Bragg reflection maxima, caused by changes of lattice spacings in certain directions, are attributed to “macro residual stresses” ranging far beyond the unit cell. Broadening of the Bragg peaks, on the other hand, is attributed to short-range strain fields, which at a given size of the crystallite may be caused by localized lattice imperfections. For an overview of stress analysis and the relationship between X-ray methods and other dedicated techniques we refer to [25–27].

4.1.4. Texture and Stress Measurements Due to the peculiarities of the deposition methods and the special energetic situation of a large free surface, nanolayers often exhibit fiber textures, whereby the fiber axis is spatially close to the surface normal. In this case XRD measurements in symmetrical geometry will show only one significant kind of reflection, if the fiber axis lies in the scattering plane. In all other cases a complete texture analysis has to be done. Conventionally, this is identical to a pole figure measurement. For this purpose dedicated texture goniometers with Eulerian cradles have been developed (see schematic drawing in

Figure 4. Eulerian cradle for X-ray texture and stress analysis, useful also for single crystal structure analysis. The specimen is rotated through three axes (angles 0, %, and 1 with respect to the coordinate system of the goniometer). The detector can be turned around the 0-axis independently (angle 2).

X-Ray Characterization of Nanolayers

662 4.2. Grazing Incidence Diffraction Analysis

Q

4.2.1. Introduction For the investigation of nanolayers, X-ray diffraction schemes with grazing incidence and/or exit angles have attracted particular interest. They also allow for the characterization of very thin surface layers of crystals [28–30] (see Figure 5 for basic setup). In the case of plane and smooth surfaces using the “total external reflection” (TER) effect for grazing X-rays, the penetration of radiation inside nanolayers and crystals covers a range from the order of one nanometer to less than a few micrometers. Thus, generally speaking, grazing incidence of X-rays allows for the study of surface structures with atomic depth resolution. In literature [31] grazing incidence setups are classified into three major types: coplanar “extremely asymmetric diffraction” (EAD), “grazing-incidence diffraction” (GID), and “grazing Bragg–Laue diffraction” (GBL) geometries (see Fig. 6). The first case of coplanar EAD is due to diffraction of lattice planes constituting the Bragg angle with the crystal/nanolayer surface, whereby either the incident or exit beam is grazing [32–35]. For GID the diffraction lattice planes are perpendicular to the surface of the crystal/nanolayer and both the beams are grazing [36–39]. In the third case of GBL a combination of the EAD and GID is realized, involving the diffraction from lattice planes inclined at small angles to the crystal/nanolayer surface normal, so that the corresponding reciprocal lattice vectors (normal of diffraction lattice planes) exhibit small angles to the surface. Under this condition a combination of asymmetric diffraction with either grazing incidence or grazing exit can be realized by a small variation of the angle of incidence or by switching between these two cases in one diffraction experiment [40]. Grazing geometries are widely used in studies of semiconductor crystal surface structures, including epitaxial nanolayers and multilayers. Thereby parameters of real structure, for instance caused by oxidation, ion-implantation, or etching techniques, are the subject of studies for EAD [41–43], GID [44–46], and GBL [47], respectively. For a more exhaustive collection see [31]. Also, charge and spin density waves (e.g. in Cr layers) have been observed making use of strain waves [48]. Recently intensive efforts were made to formulate a general diffraction model covering all grazing geometries. Like in the case of X-ray reflectometry (Section 4.3), quantitative description of X-ray diffraction at grazing incidence

Figure 5. Schematic setup of grazing incidence diffraction at single crystalline samples. Vector h indicates the normal of reflecting lattice planes.

Q

Q αe = αi

αi

Q

Q αb

Q

αe = αi

ω

αi

Q Q αb

Q

αi

αe = αi

ω

Figure 6. Three major types of grazing incidence setups (after [31]). (a) Coplanar extremely asymmetric diffraction, (b) grazing-incidence diffraction and grazing Bragg–Laue diffraction. Wave vectors Q3 Q0 incident waves, Qs specularly reflected waves and Qh diffracted waves. Vector h represents the reciprocal lattice vector according to momentum transfer Q0 → Qh in reciprocal space, being in the surface normal direction of diffraction lattice planes in real space. The angles i and e are defined with respect to sample surface (i and e for specular reflection according to Fig. 1 and b inclination in the case of Bragg reflection). 0 is the angle of incidence for excitation of Bragg reflection of the crystal/nanolayer.

and/or exit is treated by two models, which differ in principle. The first is based on an extended kinematical approach, frequently called the “distorted wave Born approximation” (DWBA) [38, 44]. The second model uses an extended dynamical theory of X-ray diffraction [32, 39, 46, 49]. Both approaches describe the effects of refraction and specular reflection at crystal surfaces and interfaces of nanolayers in

X-Ray Characterization of Nanolayers

the case of grazing X-rays. The kinematical theory of X-ray diffraction is applicable to mosaic crystals and thereby, with respect to characterization of nanolayers, to layers thinner than the X-ray extinction depth. Thus the kinematical theory is limited to cases of weak interaction of incident and diffracted X-rays. Strong interaction, due to high intensity of diffracted peaks, is to be described by dynamical theory of X-ray diffraction. By previous studies a general theory applicable to X-ray diffraction with grazing incidence and/or exit has been accomplished in a matrix form [46, 49–51]. In [31] the matrix dynamical theory was reformulated in a recursion matrix to overcome serious numerical problems with transfer matrices. In addition, [31] presented experimental checks by double-crystal EAD for the case of strained multilayers. Moreover, a linkage to the established scalar models by Parratt [52] and Bartels et al. [53] by reduction of the matrix recursion formulas for cases of X-ray grazing incidence far from the Bragg diffraction condition and for those of Bragg diffraction with no grazing beams was accomplished. The reader can also profit from the discussion on X-ray standing waves in multilayers by Stepanov et al. [31].

4.2.2. Applications Structural information of nanolayers by means of GID can be obtained by searching for deviations from the ideal bulk scattering of extended crystals and attributing these deviations to the layers. This requires in most cases that the bulk is a good single crystal, so that scattering from it is dominated by Bragg peaks (see an example in Figure 7). Additional scattering can then be attributed to the influence of surface or involved nanolayers. This idea is the basis of the so-called “truncation rod analysis.” The designation is based on the concept of rodlike reciprocal lattice points. Since the

Figure 7. Measured diffracted intensities (black line) and simulation (gray line) of a pseudomorphic SiGe layer (100 nm, epitaxially grown on Si substrate, composition of mixed crystal layer is indicated) versus angle of incidence 0. Because of the relatively large distance between 004 Bragg angles of layer and substrate the detector angle 2 was moved, too. Photon energy was 8048 eV; the experiment was done using highly parallel radiation. The oscillations in the reflectivity are caused by the interaction of wave fields of single crystalline substrate and layer existing in the vicinity of the Bragg reflection.

663 method has been extensively used to study surfaces of single crystals and adsorbed monolayers, it was also extended to nanolayers deposited on crystalline substrates (for reviews see [54]). To obtain quantitative information, the truncation rod profiles have to be fitted to theoretical models. Recently GID has been used to study surface freezing in monolayers [55], self-assembled (chemisorbed) films, and different substrates [56–58]. The method has also proved its efficiency in solving the structure of proteins adsorbed to a phospholipid monolayer [59] or in investigating the structural change of lipids due to adsorption of enzymes [60]. A review of GID studies of thin inorganic films can be found in [61].

4.3. X-Ray Reflectometry 4.3.1. Introduction The TER of X-rays from solids with smooth surfaces was first reported 10 years after the discovery of X-ray diffraction in crystals [62]. Since that time on the basis of this phenomenon a very efficient analytical method has been developed. Due to the typically small angles enclosed by X-rays with sample surfaces as a prerequisite for total external reflection of hard X-rays (cf. Section 3.4), there exists an overlap with other methods using the geometries of grazing incidence (see also Section 4.2). This partly leads to ambiguities in the designation of certain experimental techniques. We want to summarize here, under the term “X-ray reflectometry” (XRR), experiments based on the phenomenon of “specular X-ray reflection” (SXR), including the TER and diffuse scattering of X-rays at smooth surfaces/interfaces of samples. When looking at the two phenomena mentioned first, the term “X-ray specular reflection” is frequently used. Diffuse scattering occurs if interfaces are not ideally smooth with respect to the wavelength of X-rays. Experiments using this phenomenon are mostly marked by the synonymous terms “X-ray diffuse scattering” or “X-ray nonspecular reflection.” Fundamental work in this field was carried out in the 1960s. A basis for this is the report by Yoneda [63] on the angle dependent intensity modulation in X-ray diffuse scattering, today usually designated as “Yoneda wings.” In addition, yet another method should be considered here which aims, frequently in the same experiment, at the excitation of fluorescence radiation in the sample. Accordingly, the method is called “total reflection X-ray fluorescence spectrometry” (TRXF). At present a combined use of XRR and X-ray spectrometry in one and the same experiment is in progress, where mostly for multilayer systems new attractive information is to be expected (see Fig. 8 for basic setup). A lot of progress was possible in the few last decades due to the availability of synchrotron radiation, as well as the improved quality of the samples produced by modern preparation methods. Simultaneously the theoretical analysis of that experimental data has been developed. For extended reviews we refer here to [64–67] as examples.

664

Figure 8. Schematic setup of angle-dispersive reflectometry and total reflection X-ray fluorescence analysis.

4.3.2. Specular Reflectivity The basic concepts for the theoretical description of external reflection of X-rays at smooth surfaces and interfaces of multilayers were reviewed in [66, 67] (see Fig. 9 for schematic drawing of the origin of interference phenomena in specular reflectivity of nanolayers). Quite substantial pioneering work was done in the 1950s by Parratt [52]. Parratt’s formalism requires a recursive calculation of the intensities of the reflected beam. In addition several approximate formulas have been proposed. An analytical formula for calculating X-ray and neutron reflection from thin surface films has been presented by [68]. Another analytical expression

Figure 9. Origin of interference phenomena in specular reflection of nanolayers. (a) A nanolayer with homogeneous optical thickness deposited on a thick substrate with smooth surface can cause two sets of reflected beams with definite phase difference (influence of refraction not indicated in the drawings) giving rise to angle/energy dependent interference phenomena for angles of incidence in the vicinity of the critical angle of total external reflection. Analysis of interference characteristics can yield information on layer thickness, density, and interface roughness. (b) In extension to (a) a number of different nanolayers (multilayer) can cause a corresponding set of reflected beams of an appropriate number. In the case of periodic layer stacking in addition to (a) from the positions of reflection maxima in space of wave vectors the period and from the number of secondary maxima the number of periodic layers can be determined.

X-Ray Characterization of Nanolayers

is based on a one-dimensional scattering-length-density profile, using a weighted superposition approximation [69]. As outlined in [64], from comparisons of this formula with Parratt’s recursion formula, the “Born approximation” (BA), and the DWBA, it was concluded that it is the most accurate one. It is valid in the entire range of wave vector transfer, except for the narrow region around the critical angle of total reflection. Another analytical approximation for the calculation of X-ray reflectivities in the case of multilayers is given in [70], yielding quantitative results which agree well with that of the recursive algorithm according to Parratt (deviations of less than 1% in the vicinity of the critical angle of total external reflection). The determination of roughness from SXR, like for the nonspecular case, is based on modelling the roughness profiles. In the case of specular reflectivities this can be done with the introduction of a roughness parameter in the Fresnel reflection coefficient. One parameter is sufficient for the description of the influence of roughness on specular scattering. On the other hand this means a rather coarse modelling of the surface. This is due to the fact that for specular reflectivity the scattering vector is always perpendicular to the interface, thereby testing the almost vertical interface roughness. If the model of the vertical roughness profile of the interface is based on the assumption of a Gaussianlike distribution of ideally smooth interfaces the information obtainable is rather limited to the root mean square of this parameter. A model of Sinha et al. [71] is based on the use of a damping factor for the BA model. An additional linkage to the total reflection X-ray fluorescence analysis is given in [72]. In extension, the modelling of an interface by a series of smooth transition layers has been put forward in [73, 74]. This model is in many cases the comparatively most accurate one and is also useful for the modelling of interdiffusion at interfaces and widely used for the simulation of X-ray specular reflectivities of multilayers. The applicability of different models for the simulation of the roughness can be estimated by the presentation in [64]. Parameters determined from XRR measurements are not affected by an atomic long-range order (i.e., amorphous samples can also be examined). It was Compton who proposed measuring the refractive index by specular reflection in 1922 [75]. Then Kiessig provided pioneering values of the dispersion parameter  [76] and of the thickness [77] of Ni layers 22 nm thick. The thickness was derived from interference between surface-reflected and layer/substrate interface-reflected beams (“Kiessig fringes”). Since thickness of nanolayers (including layers in multilayer stacks and their number), average electron density, or refractive index can be determined directly from the reflection curves, quantitative analysis including determination of roughness parameters is mostly based on constrained least-squares fitting of model curves to the measured ones (see an example in Fig. 10). For an accurate analysis the diffuse scattering has to be taken into account, which also contributes to intensities detected at specular angles. The best solution is to measure these contributions under near-specular conditions and to subtract them from the experimentally determined specular reflectivities. The same applies to possible contributions of

X-Ray Characterization of Nanolayers

Figure 10. Measured (dotted) and calculated (full line) X-ray specular reflectivities of an Fe/Al multilayer (six periods, 4.3 nm Fe + 10.4 nm Al deposited on Si substrate) versus angle of incidence. Photon energy was 7280 eV above the Fe K absorption edge at 7112 eV.

fluorescence radiation excited in the sample. In addition, in each case the systematic influence of sample size must be considered. In particular at the smallest angles only part of the incident beam will hit the sample in the direction of propagation, thus not being available in full for interaction with the sample (also valid for large samples due to the limited detection windows). The thickness of several layers can be determined with an absolute accuracy of about 0.2 nm. An accurate determination of the distribution of mass densities thereby requires additional information. These may be obtained by taking advantage of experiments under anomalous scattering conditions (scattering contrast) or of additional use of neutron scattering. The accuracy of determination of the interface roughness is strongly influenced by the sample quality on the one hand and the availability of a realistic model of the surface on the other. In the case of magnetic films when using resonant interaction polarization effects can be studied or have to be considered [78, 79].

4.3.3. X-Ray Diffuse Scattering Diffuse scattering, or nonspecular scattering, can yield the shape function of the interfaces of nanolayers (“in-plane structure”). For the basics of quantitative modelling of X-ray diffuse scattering we refer to overviews in [64–67]. In contrast to the case of X-ray specular reflectivity the interfaces and the roughness parameters are usually described by means of surface shape functions z
x y (x y-plane parallel to the interface). Important parameters, which can be obtained by modelling of experimental intensities, are on the one hand the correlation length, which is due to the distance in the x y-plane, at which the correlation of structures has decayed with 1/e and on the other hand a parameter due to the texture of the interface (known as the Hurst parameter related to the surface fractal dimension). Thereby this texture can be modelled between extreme situations of jagged to slowly oscillating interfaces. In addition, the separation

665 of compositional and density fluctuations from interfacial roughness can be achieved. For the determination of interface parameters from diffuse scattering different types of angle scans are possible. Usually longitudinal (near-specular), transverse (rocking curve), radial (detector), and full (source and detector moved) scan geometries are differentiated exhibiting individual sensitivities to certain characteristics. As in the case of X-ray scattering from crystals, X-ray diffuse scattering from interfaces at grazing incidence can be modelled, to a certain extent, by a kinematical approach (BA), where comparable restrictions are valid (phenomena like extinction, refraction, and multiple scattering of X-rays are neglected). Thus the range of validity of this approximation is restricted to the angles of incidence and reflection above the critical angle of total external reflection. For the case of multilayers the BA was extended introducing interface roughness by noncorrelated contributions and correlated ones propagating from layer to layer [80]. Further developments and extensions of the BA for the case of multilayers were presented in [81, 82]. The dynamic approach (DWBA) which overcomes the limitations of the BA by considering refraction and multiple scattering of X-rays was also applied for X-ray diffuse scattering [71]. For the simulation of intensities of diffuse scattering from multilayers, for example DWBA approaches as developed by Sinha [83] are used. For an overview of developments in this field we also refer here to [64].

4.3.4. Total Reflection X-Ray Fluorescence Analysis From the measurement of X-ray fluorescence excited in nanolayers in the case of total external reflection information can be obtained which is partially comparable to that from specular reflection. Though this offers the possibility of oversampling, it is more interesting for the experimentalist that from fluorescence unique additional information, in particular the spatial distribution of the individual atomic species, can be obtained. An overview of TRXF experiments can be found in [84]. Like in the case of X-ray specular reflectivity common models for quantitative analysis of TRXF in the case of multilayers are based on the matrix approach. They allow for simulation of X-ray fluorescence intensities from structures with concentration modulation. An important feature of TRXF represents the formation of an “X-ray standing wave field” (XSW) in nanolayers during total external reflection. This standing wave field is formed due to the layered structure and extends also outside the layer. Thereby the X-ray fluorescence excitation is locally connected to the position of the standing wave field which is due to a periodic modulation of the exciting energy. The possibilities which are offered when XSW are combined with other established spectroscopic methods are dealt with in Section 4.7. To sum up a great potential exists in the combination of features of X-ray reflection, diffuse scattering, and X-ray spectroscopy in one and the same experiment.

666 4.4. X-Ray Topography 4.4.1. Introduction Methods which aim at producing images of lattice planes of a crystalline sample by means of Bragg reflected X-rays are summarized under the term “X-ray diffraction topography”. The lattice planes thus assessed may be either bounding planes of the specimen or others inclined to the surface. For pioneering work in this field we refer to [85–90]. In general, X-ray topography gives valuable information on the distribution of strains arising from extended crystal defects or from point defect clusters. When combining topographs of different reflections of the same volume, a spatially resolved picture of the strain distribution may be obtained. Characterization of crystal perfection becomes particularly important if related to the growth history of the sample. For nanolayers note that this information can be obtained depth sensitively, revealing the state of the substrate/layer interface and discovering inhomogeneities within the layers. In the beginning, only individual Laue reflections, i.e. without specification, of wavelength were employed for X-ray topography. Their nonuniformity gives a direct picture of either local variations of the lattice orientation or of the extinction of X-rays. They are still exploited where the Laue method is applied anyway for structure analysis. Later monochromatic X-ray techniques became predominate because of their higher angular resolution and their applicability to polycrystalline samples. For a detailed description of the development of the method cf. [91–93]. Since topographic contrast can be affected by various factors, a number of different geometries and strategies of X-ray topography have been developed. A substantial classification can be made regarding the use of monochromatic or polychromatic radiation, whereby in the latter case much progress was due to the availability of synchrotron radiation. For a quantitative understanding the effect of extinction of radiation has to be considered. If the sample can be classified as an “ideally perfect crystal” there is, in addition to the absorption, attenuation of the X-ray beam in the sample exactly for the situation of Bragg reflection. For a sample to be classified as an “ideally imperfect crystal” the influence of extinction can be neglected. Thus an intensity contrast in a Bragg reflected X-ray beam, which is due to a variation in perfection of the crystal lattice, can occur. It is usually called “extinction contrast” or “diffraction contrast.” As mentioned, image contrast can be due to the misorientation of parts of the sample with respect to the satisfaction of the Bragg condition. This contribution is called “orientation contrast.” The strength of the effect is affected by the range of wavelengths of radiation used and its divergence. A comprehensive treatment of geometry and theoretical foundations of XDT are given in [94, 95]. For an overview of dynamical theory of X-ray diffraction we refer to [96, 97]. The dynamical theory of diffraction is compulsary for perfect crystals. Contrary to the kinematical theory, which deals with the global intensity diffracted by the specimen, the dynamical theory deals with the propagation of the X-rays inside the specimen and investigates the properties of a wave field formed by superposition of all incoming and scattered partial waves. Mathematically, the dynamical theory

X-Ray Characterization of Nanolayers

is entirely based on Maxwell’s equations of electrodynamics. Generally, the integrated intensities of reflections are smaller for perfect crystals than those for imperfect ones where the kinematical theory applies.

4.4.2. Experimental Setup For the classification of experimental setups the terms “extended” and “limited beam methods” are normally used. In “extended beam setups” wide areas of a sample are illuminated, whereby the intensities diffracted by a perfect crystal plate would show a homogeneous intensity distribution. In the most common “integrated wave topography” the incident beam is divergent and/or polychromatic. For the subgroup of “white beam topography” a polychromatic beam of low divergence is required. The set up is then a conventional Laue experiment. Mostly with laboratory sources the “Lang topography” technique [98] is used yielding images which are produced by integration over a spatial distribution of divergent waves on the entrance surface of the crystal. In modern experiments for the enhancement of sensitivity and discrimination of the contributions of background radiation more than one crystal is used in the topographic setup, whereby one crystal can be represented by the sample. If samples are slightly curved due to strain, like wafers with superposed layers or to other effects, a compensation of sample bending can be achieved by slight bending of the first crystal (monochromator). The method is called “curved collimator topography.” In “limited beam setups” the width of irradiating beams is restricted to about 10 m in the direction perpendicular to the scattering plane. After diffraction from an ideally perfect crystal an inhomogeneous intensity distribution is detected, which changes sensitively due to the influence of strain fields.

4.4.3. X-Ray Topography of Nanolayers In conventional X-ray topography an orientation contrast from nanolayers can be detected when using a sufficiently large distance between the sample and image detector (see Fig. 11 for basic setup and Fig. 12 for schematic drawing of origin of image contrast in X-ray topography of nanolayers). For the observation of misorientations usually a minimum layer thickness exists. This is due to the fact that a minimum size of crystal is required to make distorted and perfect parts of a crystal distinguishable by means

Figure 11. Schematic setup of gracing incidence topography of crystalline nanolayers. Vector h indicates the normal of reflecting lattice planes.

X-Ray Characterization of Nanolayers

Figure 12. Illustration of the origin of image contrast in X-ray topography of nanolayers. Vector h indicates the normal of reflecting lattice planes.

of the detected integrated reflection intensities. To overcome this limitation dedicated experimental techniques have been developed. When using double crystal setups, which allow for well defined narrow wavelength and angular bandpasses, the detectable contrast is determined directly by the misorientation. Another strategy is to use the interaction of waves scattered by crystalline nanolayers on thick crystalline substrates (see an example in Fig. 13). If the layer is epitaxially linked to the substrate and the difference of lattice parameters is small, they can contribute to the diffracted intensities simultaneously due to the interference of the wave fields of both crystals. The accomplished interference phenomenon (“Moiré effect”) allows for an extremely sensitive detection of weak strains and the possibility to visualize localized defects. In this connection the investigation of epitaxial nanolayers on thick substrates, to be classified as perfect crystals, in transmission geometries using sufficiently hard X-rays, is possible (“Moiré topography”). As with other X-ray techniques originally developed for investigation of extended crystals also for characterization of nanolayers by means of XDT grazing incident geometries play a key role. Progress in the use of “grazing incidence X-ray diffraction topography” (GIXDT) techniques for the analysis of thin crystalline surface layers has been reported

Figure 13. Topographic image (reflected sample area: ∼12 × 8 mm2  of a pseudomorphic SiGe layer (100 nm, epitaxially grown on Si substrate). Photon energy was 8048 eV; the experiment was done using highly parallel radiation. For exposure the Si 224 reflection, tuned to 60% of maximum intensity, was used. Perpendicular lines are caused by misfit dislocations at the SiGe/Si interface.

667 by [42, 99–102]. Typically grazing incidence angles are in the order of 01 –1 , mostly larger than the critical angles of total external reflection. From GIXDT experiments information about strain and lattice defects, like dislocations, can be obtained. Thereby typical signal depths extend to the order of micrometers and can be tuned to a certain extent by choosing appropriate angles of incidence. However, in very thin layers, like in epitaxial systems with a layer thickness of some monolayers, the smallest tuneable penetration depth can exceed the overall layer thickness. Normally in such cases grazing incident angles below the appropriate critical angles, for which the penetration depth of radiation lies in the order of 1 nm, are used to image near surface structures [103–105]. Since in common geometries polychromatic radiation is used as standard, in [99] the use of polychromatic radiation for the investigation of very thin layers at angles below the critical angle of total reflection was described (“white beam synchrotron radiation total reflection X-ray topography”). From Laue patterns of topographs features of surface layers up to thickness of 1 nm can be derived.

4.5. X-Ray Absorption Fine Structure Analysis 4.5.1. Introduction XAFS spectroscopy has developed to be one of the basic tools for the X-ray analysis of nanolayers. By means of X-ray fluorescence analysis the chemical composition can be determined for atomic numbers ≥5. Small concentrations of elements of ∼10−4 can be detected; typical relative accuracy of the content is in the order of 0.1%. As outlined in Section 3 from oscillating fine structure of linear photoelectric absorption coefficientNs element specific information on parameters characterizing the structural short-range order can be obtained (see illustration of origin of XAFS in Figs. 14 and 15). For exhaustive overviews we refer to [106–108]. The XAFS oscillations can usually be observed at energies upto 1000 eV above an absorption edge. The determination of quantitative structural information from XAFS experiments requires a high resolution measurement of the total attenuation coefficient 
E. Since the XAFS is a weak contribution to the total attenuation of X-rays in matter, 
E has to be determined with an accuracy of more than 10−3 . XAFS experiments require X-rays tuneable in energy with a relative energy resolution of typically ∼10−4 , detectors with noise smaller than 10−3 , and a wide range of linear characteristics of operation. Because of the suitable wide range of X-ray energies/wavelengths synchrotron sources are typically used. Quasi-monochromatic radiation is selected by double crystal monochromators. For experiments with energies in the order of 10 keV for this purpose silicon or germanium single crystals are commonly used.

4.5.2. Transmission and Fluorescence XAFS Experiments For the monitoring of primary X-ray intensities and transmission XAFS experiments ion chambers are commonly used. For fluorescence XAFS measurements, several kinds

668

Figure 14. Basic processes behind XAFS experiments. (a) Promotion of a core electron to the energetic continuum by photoelectric absorption of X-rays. From the viewpoint of quantum mechanics the arising photoelectron can be described as a wave propagating out of the excited atom. (b) Decay processes of the excited states: emission of photons of element-specific characteristic energies (fluorescence) and free Auger electrons (kinetic energy element-specific at a given energy of excitation). For indirect measurement of absorption coefficientNs it is essential that the probability of production of both fluorescence photons and Auger electrons is directly proportional to the absorption probability of X-rays by the atoms.

of detectors are applicable. Mainly in the case of transmission XAFS experiments samples with homogeneous thickness and density (free from pinholes) are required. An optimum sample thickness t for samples with a sufficient concentration of the resonantly absorbing element can be estimated from the condition 
E t ∼ 25 at energies E just above the absorption edge (ionization energy). Typical optimum thickness of concentrated samples is in the order of 10 m; for diluted samples (relative concentrations of not less than 10%) it is around a few millimeters. For XAFS investigations of nanolayers, thick samples, or lower concentrations of resonantly absorbing elements the monitoring of characteristic X-ray fluorescence intensities is the preferred technique (see Fig. 16 for basic setup of fluorescence XAFS and an example of measured florescence XAFS in Fig. 17). The resolution limit for diluted samples is approximately 1 ppm. Since in fluorescence XAFS measurements elastically and inelastically scattered components of exciting X-rays and characteristic fluorescence intensities

X-Ray Characterization of Nanolayers

Figure 15. Illustration of the origin of XAFS: outgoing and (back)scattered portions (excited atom is placed into condensed matter) of final state wave functions. The outgoing and (back)scattered contributions interfere depending on the electron wavelength and the interatomic distance. If the energy of incident (exciting) X-rays is scanned across the ionization energy the kinetic energy of the excited photoelectrons is varied. (a) Case of constructive interference. (b) Destructive interference for the same spatial distribution of the atoms as in (a) but at higher kinetic energy of excited photoelectron waves (lower wavelength). The interference modulates the absorption probability, giving rise to oscillations of the coefficient of photoelectric absorption. Since the oscillation characteristics are determined by the interatomic constellation, structural information can be obtained from its quantitative analysis.

of other lines can also give rise to the radiation of interest, a suitable energy resolution of the detection system is required. In general, two parameters are important for high quality fluorescence XAFS measurements: the solid detection angle and the energy resolution. Only in the case of diluted samples it is suitable to collect as much fluorescence as possible. In all other respects a well defined takeoff angle is demanded in order to allow for correct application of self-absorption corrections. In addition it has to be

X-Ray Characterization of Nanolayers

669 Commonly used energy resolving detectors for absorbed X-ray photons with energy in the order of ∼10 keV allow for energy resolutions of 200 eV and better. Certain electronic energy resolution requires a finite amount of time, limiting the throughput that can be processed. State-of-the-art detectors at the mentioned energies exhibit typical saturation rates of less than 105 absorbed photons per second. In reality at synchrotron sources this upper limit of total intensity, which can be processed by these detectors, often limits the effectiveness of fluorescene XAFS measurements.

4.5.3. Experiment Figure 16. Schematic setup of fluorescence XAFS.

considered that the sometimes dominating radiation contributions, which are elastically scattered by the sample, are not emitted isotropically since the radiation is mostly linearly polarized. Optimal suppression can be achieved by placing the detectors perpendicular to the plane defined by the incident beam and surface normal of the sample. In particular for concentrated samples this also limits the influences of self-absorption expressed by an amplitude damping of the measured fluorescence XAFS. To reduce contributions of other spectral components physical filters can be placed between sample and fluorescence detector. The material of filters is commonly chosen to profit from the well known selective absorption of chemical elements due to their atomic number. With Z being the atomic number of the resonantly absorbing element of the sample, in most cases of characteristic K-radiation a filter of chemical element with Z-1 is useful (commonly called “Z-1 filters”). Due to selective absorption it preferentially absorbs the elastically scattered and K -fluorescence radiation and lets through the K-fluorescence intensities. To avoid contributions of fluorescence radiation from the filter itself, usually a collimating Soller slit is additionally used.

Figure 17. Measured Fe K fluorescence XAFS intensities of an Fe/Al multilayer (30 nm Al + 30 nm Fe + 5 nm Al2 O3 cover layer, deposited on Si substrate) versus photon energies in the vicinity of the Fe K absorption edge at 7112 eV.

The determination of the absorption coefficientN can be done in the simplest case by a transmission arrangement, as described at the beginning of Section 3. When a sample with homogeneous thickness (t ∼ 10 m) and density is arranged between two ionization chambers and the intensity ratio is monitored, the total attenuation coefficient 
E can be calculated directly. The steps for the separation of contributions of the linear scattering coefficient are represented in detail, for example, in [106, 107]. In most cases the simple transmission arrangement is not applicable for the investigation of nanolayers. Above all, this is due to the fact that an investigation of the layers in most cases is meaningful only if adhering to the substrate. Otherwise a relaxation of strain would occur thus changing the properties of the layer. The photoelectric absorption coefficientN may alternatively be determined by taking advantage of other transition processes involved such as Auger electrons, secondary electrons (from several internal electron transitions), and fluorescence photons. Even the photo current can be exploited in certain material systems (e.g., semiconductors) [109]. Notably the interlink between photoelectric absorption and coherent scattering of photons can be used in the case of sample systems with translation symmetry to select spatially a certain part of the atoms for fluorescence (cf. Section 4.6). Due to their small free path length (approx. 2 nm) photoelectrons emitted from surfaces are suitable for investigation of the upper atomic layers of these surfaces (SXAFS surface X-ray absorption fine structure). A clear representation of the characteristics of this method, which requires working under vacuum conditions, is given in [110]. In the case of fluorescence XAFS experiments, the signal depth is given by the penetration depth of the exciting radiation at a given angle of incidence and the absorption of the fluorescence photons on the path to the detector which can be influenced by the detection angle. Using arrangements with small angles in the vicinity of the critical angle of total external reflection, the method can become sensitive to a view monolayers at the surface. The maximum signal depth is limited by the absorption of the radiation and is typically in the order of 10 m. If the samples bear surface qualities making them useful for X-ray reflectometry experiments, there is an additional way to detect the energy dependent fine structure of the absorption coefficientN. It is the energy dependent fine structure of specular reflected intensities which are recorded at angles in the vicinity of the critical angle of total external reflection. The evaluation of fluorescence XAFS measurements, which are commonly used for nanolayers, is described in

X-Ray Characterization of Nanolayers

670 somewhat more detail in [111]. Apart from the fundamentals of quantitative analysis aspects of the planning of experiments can be accomplished on this basis.

4.5.4. Analysis The general approach to an analysis of XAFS spectra implies the intensive use of computers for fast calculations, mostly in an interactive regime. On the basis of the fundamentals described, for instance, in [106, 107] extensive software packages have been developed. One of a widely used code is FEFF [112], enabling ab initio simulation of the XAFS signal to be done. The analysis of the absorption coefficientN in the vicinity of the absorption edge [“near edge structure” (XANES), ∼ 200 eV above the edge; sometimes the term is also used for resonant characteristics in the direct pre-edge range] is a field in itself aimed at primarily determining the local electronic structure [107]. The energetic position of the absorption edge, mostly attributed to the vanishing second derivative of the curve, is usually used as the origin of the photoelectron kinetic energy. XAFS data have to be subjected to Fourier transformation to obtain the atomic distribution in physical space. This procedure needs rather sophisticated modelling. Modern approaches take multiple scattering effects of photoelectron waves into account. The photoelectron emitted is not only backscattered directly by the nearest neighboring atoms (scattering angle 180 ) but also via multiple scattering by more distant neighbors. The theoretical backscattering amplitudes and phase shifts can be taken from tables [113], or calculated, for instance by means of the FEFF code [112]. Amplitudes and phase shifts which are calculated on the basis of complex exchange-correlation potentials [114] already involve the contribution from inelastic photoelectron scattering. Additionally they may be extracted from the XAFS spectrum of a standard sample. The accuracy of the radial distribution depends not only on the model used but also on the distance in physical space taken into account for fitting. For the nearest neighbors typically the interatomic distances can be determined with a precision of 0.01 Å. The relative uncertainty of the number of backscatterers (coordination numbers) and Debye–Waller factors is in the order of 10%. For the next-nearest neighbors and still longer distance correlations the accuracy deteriorates. Thus the correlation within distances around the absorbing atoms up to ∼5 Å can be determined by XAFS to an acceptable precision. The limit of spatial sensitivity lies at distances of 10–15 Å.

4.6. Diffraction Anomalous Fine Structure 4.6.1. Introduction The diffraction anomalous fine structure technique represents a combined spectroscopic, structural, and crystallographic method which has developed to become a routine analysis technique since the 1980s. For a description of surveys of DAFS history, theory, experimental methods, data analysis techniques, and some applications we

refer to [11, 115–117]. As outlined in Section 3, energy dependent diffraction and absorption fine structures are closely related by unitarity and causality, with which the DAFS experimental setups and analysis could profit from the established XAFS tools. For a discussion of certain experimental demands we refer to Section 4.5. Due to its special features DAFS spectroscopy has been applied to nanolayers, too. Because DAFS combines absorption spectrometry with diffraction, it offers distinct advantages compared with conventional XAFS (for illustration see Fig. 18).

a)

b) Figure 18. XAFS and DAFS of a partially ordered layer [(Ga,In)P epitaxially grown on GaAs substrate]. In the case of Ga and In atoms separate shades stand for definite occupation of sites by either Ga or In atoms, whereas mixed ones indicate an occupation at random. (a) Excitation of Ga K fluorescence in the general case involves Ga atoms of both layer and substrate. Especially, volume fractions of the layer, to be distinguished from the rest by another type of crystallographic structure, contribute to the integral fluorescence yield of all atoms of the layer accordingly. Scanning of the energy of incident (exciting) X-rays above the ionization energies (absorption edges) and monitoring of integral fluorescence intensities (fluorescence XAFS) can yield average information of at least all atoms in the layer. (b) Structural short-range order of ordered volume fractions can be determined exclusively by using appropriate Bragg diffraction peaks (superlattice reflections) for DAFS measurements. In the case shown this is possible although the different regions contain the same kind of atoms. Scanning of the energy of incident (exciting) X-rays above the ionization energies (absorption edges) like in (a) and monitoring of Bragg reflection intensities can yield structural information of atoms selective with respect to their assignation to a certain crystallographic structure.

X-Ray Characterization of Nanolayers

4.6.2. Features of DAFS One important enhanced feature of DAFS is its diffraction wave vector selectivity. This means that different spatial regions or components of the sample can be selected for analysis when causing Bragg diffraction peaks at separable locations in reciprocal space. In other words, the structural short-range order of each region can be determined exclusively, by using an appropriate Bragg diffraction peak (see Fig. 18 and example of measured DAFS in Fig. 19). This is also possible if these different regions contain the same atomic species. Wave vector selectivity is also important for the analysis of nanolayers. Individual layers of epitaxially grown strained or compositionally modulated layer systems can be selected, but also even buried monolayers [118] have been subject of exclusive investigation of parameters characterizing structural short-range order. Another example is the quantitative characterization of ordered domains in epitaxially grown (Ga,In)P layers [119]. These ordered regions, representing merely a small fraction (a few percent) with identical chemical compositions as their environment, were selected with the use of superlattice reflections caused by ordering. Experiments for the study of strained epitaxial semiconductors by means of DAFS have been reported in [120]. Another important feature of DAFS spectroscopy is its crystallographic site selectivity. This means that short-range order information for inequivalent sites within the unit cell can be obtained selectively. This is possible even when these sites are occupied by the same atomic species. The specific site information can be obtained by the deconvolution of differently weighted contributions of inequivalent sites to the reflections selected. Thus crystallographic knowledge is necessary for the evaluation. In [11] the site selective DAFS investigation of short-range order of Cu atoms occupying two inequivalent sites in an epitaxially grown YBaCuO (001) layer on a MgO (001) substrate was described. In agreement with the orientation of the layer in symmetrical geometry different orders of 001 Bragg reflection were obtained. Site

Figure 19. Measured DAFS reflection intensities of superlattice reflection (003 reflection, space group R3m; see Fig. 18) caused by partial ordering in a (Ga,In)P layer (1.69 m, epitaxially grown on GaAs substrate) versus photon energies in the vicinity of the Ga K absorption edge at 10,369 eV.

671 selective information could be obtained by the variability of contributions of Cu atoms at inequivalent sites to the different orders of Bragg reflection. In the example of DAFS experiments with epitaxially grown (Ga,In)P layers it was shown that the lack of inversion symmetry, typically for the zinc-blende-type structure, has a significant impact on the DAFS signal [121]. This sensitivity of DAFS on the structure polarity, thus on the directions in the crystal at the same magnitudes of the wave vectors, allows for further contributions of the method to structure analysis. In recent years the enhancement of the relatively weak magnetic dichroism near the absorption edge has been used to provide site-specific local information about the magnetic environment of the resonantly absorbing atom. In the case of multilayers with different magnetic ordering magnetic DAFS makes it possible to distinguish the same atomic species in different layers by means of its magnetic neighborhood. An adequate extension of the information results from the tensor characteristics of the atomic scattering amplitude (see Section 3) which can be tested by means of polarization dependent experiments.

4.6.3. Experimental Setup DAFS experiments commonly require polychromatic radiation, a tuneable monochromator with energy resolution appropriate for XAFS experiments. In addition an automatic goniometer allowing for alignment of the sample and a DAFS detector to record selected reflections at any photon energy of the DAFS scan are required (constant wave vector scans). For DAFS experiments using sharp Bragg reflections as in the case of single crystals an additional dynamic positioning system (e.g., a piezoelectric tilt table) can be used [121]. Special setups do not require monochromators. The possibility of DAFS experiments using polychromatic synchrotron radiation and a single crystal acting as the sample as well as the monochromator was outlined in [122]. Additionally, polychromatic synchrotron radiation in connection with a suitably curved crystal monochromator allows for an energy dispersive setup [123]. In most cases the parallel detection of fluorescence XAFS intensities is necessary, to be used for the extremely sensitive absorption correction of the DAFS signals. The importance of the absorption correction is due to the fact that the DAFS intensities undergo attenuation causing absorption fine structure comparable with a transmission XAFS experiment. Since this fine structure contains averaged short-range order information of all resonantly absorbing atoms of the sample, without correction of these contributions the selectivity of the DAFS analysis discussed previously would be lost. Energy-dependent background contributions (fluorescence of the sample, radiation scattered at slits or diffusely scattered by the sample) can be measured at angle positions near the Bragg angle. For practical use the predetermination of Bragg peak positions (angle of incidence and detection angle) at a couple of relevant energies is useful. During the DAFS measurement the peak positions can be adjusted according to a polynomial function determined on the basis of the grid points.

X-Ray Characterization of Nanolayers

672 4.6.4. Quantitative DAFS Analysis The first step of a quantitative DAFS analysis is the background subtraction. Then an absorption correction of the measured reflection intensities must be made. Since the DAFS analysis is usually based on the kinematic theory of X-ray diffraction in the case of single crystals and strong textures extinction corrections have to be applied. An algorithm for extinction correction, which can be applied to the investigation of nanolayers, is described in [124]. The next steps of a complete analysis are the calculation of the smoothed curve (without the fine structure) of the reflected intensities and the adaptation of the medium energy dependent course of the corrected measured intensities to gain the oscillating part, from which the complex-valued fine structure function can be obtained by applying an iterative Kramers–Kronig algorithm (see [11]). Finally the short-range order parameters have to be calculated by modelling the theoretical fine structure function and then comparing it with the experimental one as extracted from DAFS signals. Alternative means for the DAFS evaluation as well as the procedure to obtain site selective information can be seen in [11]. After computation of the complex-valued fine structure function the DAFS analysis can be done by means of the developed procedures of the XAFS analysis, described to a somewhat larger extent in Section 4.5. For the accuracy of the determinable parameters of shortrange order the same statements as in the case of the XAFS apply. Regarding the scattering volumes a lower limit can only be seen regarding the formation of significant Bragg maxima which are interlinked with the number of coherent scatters. The increasing radiation quality of new generation X-ray sources will also lead to further developments in this regard.

4.7. X-Ray Standing Wave Analysis 4.7.1. Introduction X-ray standing wave fields can be generated when plane wave X-rays undergo reflection or diffraction phenomena. The standard application of XSW experiments is the solution of the phase problem in X-ray single crystal structure analysis and the investigation of lattice position of a certain kind of atom, which can be present in extremely small concentrations (see Fig. 20 for illustration of generation of XSW in the case of Bragg reflection and the origin of depth sensitivity by position dependent excitation of fluorescence intensities). A descriptive representation of appropriate applications can be found in [125]. Since analysis under XSW conditions can allow for extraordinarily sensitive surface analysis and additionally for unique information from thin films and multilayer systems it has been developed to be a powerful tool for the X-ray investigation of nanolayers. A substantial advantage of XSW based analysis is that it does not require vacuum conditions. Due to the nature of X-rays the XSW represent a spatially fixed periodical distribution of electromagnetic field energy. It is remarkable that the spatial period in the case of diffraction at crystals is directly interlinked with the distance of lattice planes. When exciting XSW by using total external reflection at surfaces, the spatial period can be tuned in an extended range. Since

maximum fluorescence

Q0

Qh

a) minimum fluorescence

Q0'

Qh'

b)

Figure 20. Illustration of the generation of XSW in the case of Bragg reflection and the origin of depth sensitivity by position dependent excitation of fluorescence intensities. If the condition of excitation of a Bragg reflection is fulfilled in addition to the vector of incident waves Q0 the vector of diffracted waves Qh can be defined. The XSW represent a spatially fixed periodical distribution of electromagnetic field energy with spatial period directly interlinked with the distance of lattice planes. Simplifying, the creation of the XSW can be imagined by interaction of incident and reflected X-rays, whereby the intensities of the XSW at a particular (equivalent) spatial position can be changed by slightly altering the angle of incidence or the energy of exciting X-rays, respectively. (a) Maxima of XSW (deep colored horizontal regions) according to atomic positions. The correlation between local field strength and absorption/fluorescence excitation causes maximum fluorescence yields from atoms involved. (b) Maxima of XSW in between atomic positions yields lower fluorescence intensities from the atoms considered in (a).

in the case of single crystals the amplitude and the phase of the XSW are derived from the dynamical theory of X-ray diffraction (for an overview see [97]) in complete form, frequently a simple model for a descriptive discussion is used: “the incident and reflected X-rays create the XSW by their interaction.” For a number of applications it is important that the XSW can extend outside of the surface of crystals or nanolayers, where the position sensitive analysis of adsorbates or monolayers becomes possible. The intensities of the XSW at a particular spatial position can be changed by slightly altering the reflection angle, or the energy of exciting X-rays. The position of the XSW can be determined experimentally, whereby the phase is determined relatively. XSW analysis is justified on the possible locally weighted excitation of atoms.

X-Ray Characterization of Nanolayers

673

4.7.2. X-Ray Standing Waves in the Case of Multilayers A complex description is required if both the substrate and layers are to be characterized as single crystallites and the XSW are generated by the excitation of Bragg reflections. An example of this is the epitaxially grown strained layer system. If the lattice parameters of the substrate and layers deviate from each other, the wave field within a given layer cannot be assumed to exhibit the periodicity of its lattice planes correctly. Above all this is the case when the XSW arises primarily from substrate diffraction. For an exhaustive description of the theoretical basis for this situation, which directly presupposes the application of approaches of dynamical theory of X-ray diffraction, we refer to [31]. An interesting field of application for the analysis of nanometer multilayers was opened by coupling the possibilities of the XSW with the method of the fluorescence XAFS. Therefore not the Bragg diffraction of crystalline substrates and layers but instead the total external diffraction and refraction phenomena are used for the generation of the XSW (see Figs. 21 and 22). The advantage of this coupling is that the structural information from fluorescence XAFS can be obtained depth-resolvedly with respect to the interfaces by tuning the XSW to a certain position during the measurement. Summarizing, the method aims at a locally weighted excitation of the atoms to fluorescence, whose energy-dependent XAFS yields information on their structural short-range order. As a prerequisite plane samples and interfaces of good quality are required. All in all the samples have to be suited for characterization by means of X-ray reflectometry; Regarding the demanded accuracies of beam divergence and sample positioning the experimental setup has to fulfill the same requirements as that in the case of X-ray reflectometry, however, for the spectroscopic part a polychromatic X-ray source with a tuneable monochromator is required. With respect to this device the specifications for XAFS experiments (see Section 4.5) have to be considered. The fundamentals of depth-resolved analysis of structural short-range order by combining Fe-K XAFS and XSW, described using the example of the investigation of the neighborhood of Fe atoms in a sixfold Fe/Al nanomultilayer, can be found in [126]. In addition we refer to earlier publications in this field [127, 128]. For these experiments the shift of the position of the electromagnetic field of XSW perpendicular to the interfaces was used. Therefore the angle of incidence was varied in the vicinity of the first order Bragg reflection of the multilayer stack which occurs slightly above the critical angle of total external reflection. For quantitative analysis geometrical and physical parameters of the sample, such as single layer thickness, number of periodic repeated layers, roughness parameters of interfaces, and single layer densities, have to be known. They can be determined by fitting a modelled reflection curve to an experimental one (see Section 4.3). Since XAFS experiments are accomplished in the vicinity of the absorption edges (ionization energies of atoms), the strong influence of the resonant scattering has to be taken into account. The next step is to evaluate the XSW distributions within the multilayer stack for different angles of incidence. By varying the angle of incidence in the vicinity of the first order Bragg reflection

Figure 21. Measured X-ray specular reflectivities (dotted) and Fe K fluorescence intensities (full line) of the Fe/Al multilayer specified in the caption of Figure 10 versus angles of incidence. Photon energy was 7580 eV above the Fe K absorption edge at 7112 eV. The modulation in fluorescence intensities at angles of incidence in the vicinity of the first order Bragg reflection of the multilayer stack at approx. 042 indicates the existence and shift of standing wave fields within the multilayer in this case. Marked positions: (1) maximum of the first order Bragg reflection, (2) inflection point of the course of Fe K fluorescence intensities in the vicinity of (1), and (3) maximum of Fe K fluorescence.

and thus shifting the position (phase) of the standing wave field within the multilayer stack, the contribution of different regions within certain layers to the excited fluorescence intensities can be distinguished. For depth-sensitive analysis a certain number of XSW positions have to be selected whose adjustment can be done in a reproducible manner. Therefore the first order Bragg reflection of the multilayer, the fluorescence maximum and minimum, and the inflection

Figure 22. Calculated components Sx of the Poynting vector S (x direction parallel to interfaces) of the Fe/Al multilayer specified in the caption of Figure 10 versus penetration depth z perpendicular to the interfaces (surface: z = 0 nm) for different standing wave field positions in the vicinity of the first order Bragg reflection of the multilayer stack as defined by Figure 21 [1 (full line), 2 (dashes), and 3 (dots)].

674 point of the maximum course in the vicinity of the Bragg reflection are special positions for this procedure. To hold a selected XSW position fixed during the XAFS measurement, with which the energy of the exciting photons is changed, the angle of incidence has to be adjusted accordingly while varying the energy. The energy of incident X-rays and the reflection angle are related by known equations. Due to the influence of refraction the experimental determination of these angle positions at a limited number of energies is preferred. Then a polynomial function can be determined to adjust the scattering angle while tuning the energy step by step during the XAFS measurement. In general, the energy-dependent fine structure of the photoelectric absorption coefficientN can be extracted from both the fluorescence intensities and from the reflected ones. When the XSW is shifted the contribution of resonantly absorbing atoms and the other (nonedge) atoms to the total attenuation coefficient will change. Therefore the integral field intensities in the individual layers had to be calculated to determine the effective attenuation and photoelectric absorption coefficientNs for the evaluation of XAFS to be discussed. In [126] this was accomplished by using a recursive matrix approach. The attenuation in the Poynting vector component of the electromagnetic field of the XSW in the stack direction can be used to determine the contribution of different layers to the fluorescence and therefore to the structural information determined from XAFS. Its depth distribution can yield the weights for the local contribution to absorption and thus to fluorescence intensities. To characterize the average neighborhood of resonantly absorbing atoms an XAFS measurement with fixed angle of incidence can be performed. In this case the angle of incidence should be chosen due to small reflectivities; thus only weak interactions between incident and reflected waves cause XSWs with negligibly small spatial intensity modulations. The accuracy of quantitative parameters characterizing structural short-range order in the neighborhood of resonantly absorbing atoms achieved by the combination of XSW and XAFS is comparable with that of conventional XAFS measurements (see Section 4.5). Regarding the depth resolution it is difficult to state a general limit, since the sample characteristics are important. Generally a high reflectivity of the multilayer Bragg reflection is desirable, because it is equated to the formation of a significant XSW. Additionally, in the case of multilayers a high homogeneity of characteristic parameters of several layers repeated within the stack is a condition for good results. In case of multilayers dedicated for use as mirrors for soft X-ray applications, exhibiting reflectivities of typically 70% in the energy range of 10 keV, approximate monolayer resolution could be achieved for the average parameters [129].

5. NANOLAYER MATERIALS 5.1. Introduction As already outlined in previous chapters the small number of atoms contained within nanolayers is the main challenge for X-ray characterization. Additionally, the cross section of atoms for X-rays is rather small. In the theoretical

X-Ray Characterization of Nanolayers

description of the interaction of X-rays with matter in Section 3 it was discussed that the total attenuation of hard X-rays with photon energies in the order of 10 keV is mainly caused by elastic/inelastic scattering and photoelectric absorption. Diffraction and reflection/refraction phenomena are in direct connection to scattering while occurrence of characteristic fluorescence radiation is directly caused by photoelectric absorption. In Section 3 it was shown that, in addition, both scattering and absorption are interconnected by causality. Regarding X-ray characterization of nanolayers by means of diffraction phenomena in particular the continuous direct dependence of the atomic scattering amplitude on the number of electrons per atom has to be considered. This causes on the one hand challenges for the characterization of nanolayers of elements with low atomic numbers, thereby including the wide range of organic nanolayers in the growing field of biological applications. On the other hand it is difficult to distinguish scattering contributions of atoms with neighboring atomic numbers. A way out is the use of resonant interactions for X-rays with photon energies in the vicinity of ionization energies of the atoms (cf. Section 3.2). If resonant scattering of photons with energies corresponding to K-absorption edges should be applied, for elements with atomic numbers less than ∼20 the experiments have to be performed under vacuum/inert gas conditions because of significant attenuation of X-rays with these energies in air. This applies to spectroscopic methods, which are based on the detection of fluorescence photons of adequate energies, as well. Also the absorption cross section for atoms with X-rays, beside edgelike jumps at photon energies due to ionization energies of the atoms, depends directly on the atomic number. Thus, also in this case, the largest challenges exist for the investigations of nanolayers of atoms with low atomic numbers. In particular if the interesting atomic species exhibits a small fraction (lower than ∼1 at%), simultaneous excitation of fluorescence radiation of the other components can cause low effect/background ratios, which prevent a characterization in justifiable times. In addition, mostly in the case of nanolayers, the contribution of scattered exciting radiation to the background signal can impose further limitation. Summarizing, limits of X-ray characterization of nanolayers are to be seen in cases of light elements and diluted concentration. To be complete we should mention here the demands for the interface quality and layer homogeneity for X-ray reflection experiments specified already (Section 4). The use of diffraction phenomena requires accordingly ordered volumina. High fluxes and spatial/time coherence of X-rays supplied by synchrotron radiation sources allow for a progressive shift of these limits. In particular with the help of free electron X-ray lasers under construction, in coming decades structural analysis of individual molecules should become possible. In this case it is assumed that, due to the high energy densities, the object of investigation is destroyed after characterization. Also for investigations using conventional synchrotron radiation, in particular in the case of organic specimens, an influence on the structure by the interaction with X-ray photons has to be considered. Apart from a mostly unwanted change of substance nature,

X-Ray Characterization of Nanolayers

radiation damage can limit the structural information to be extracted from the samples. Contributions on this topic can be found in [130]. In the following we will specify some current work on X-ray characterization of nanolayers which, beyond the examples in the preceding sections, is exemplary for the applicability and resolution of the methods for different classes of specimen and focus of investigation. For an overview the cited papers are arranged according to these aspects in Table 1.

5.2. Examples Temperature- and time-resolved investigations of thin organic multilayer films [Fe(II)–polyelectrolyte–amphiphile complex, deposited on silicon by means of the Langmuir– Blodgett technique, single layer thickness ∼5 nm] by means of X-ray scattering were reported in [131]. By using an energy-dispersive setup (geometrical fixed) at a synchrotron radiation source, X-ray reflectivity and in-plane diffraction were recorded simultaneously. After increasing the temperature (273–348 K), transformations of the Langmuir– Blodgett phase could be observed from X-ray reflectivity measurements. Simultaneously detected in-plane diffraction signals indicated changes of the hexagonal arrangement of amphiphilic chains (lattice spacing ∼0.42 nm). The structure of hexadecylamine-urease monolayers at the air–water interface has been investigated by X-ray reflectometry and GID studies [132]. The thickness of the twodimensional crystalline films was determined to be in the order of 0.1 nm (accuracy ∼ ±001 nm). From the experimental results, among other facts it was concluded that hexadecylamine, when spread on top of an urease film at the air–water interface, forms a stable, well-organized structure. The growth of self-assembled monolayers of octadecyltrichlorosilane on oxidized Si(111) substrates from the liquid has been studied in-situ by means of X-ray reflectometry [133]. It was found that the film grows through the formation of islands of vertical molecules. Typical layer thickness were in the order of 1 nm. The generation of large-area X-ray topographs (whole surface of 200-mm-diameter silicon wafers) is described in [134]. By using a horizontally wide monochromatic beam available from a synchrotron radiation source, topographs of the whole surface of the wafers could be obtained with one-shot exposure. From the topographs differences in surface-strain distributions, caused by various steps of silicon wafer manufacturing processes, could be detected. To achieve sufficient sensitivity asymmetric reflections with a glancing angle of 026 (above the critical angle of external total reflection of 008 at the photon energy of 21.45 keV) were used. In [135] an anomalous behavior of the specular reflection of X-rays under the condition of noncoplanar grazing incidence diffraction was studied and used for characterization of a native oxide amorphous film on the surface of a Si single crystal. In general, because of the similarity of the electron densities of layer and “substrate,” the determination of parameters of oxide layers by means of specular reflection in symmetric geometry is very difficult. Under conditions of strongly asymmetric diffraction the authors determined the

675 thickness of the oxide layer with high accuracy (28 ± 01 nm) and stated the sensitivity of the method to be in the order of one monolayer. Reactively sputtered Ta2 O5 nanolayers (thickness ∼22.4 nm) were investigated by means of reflection mode Ta-L3 XAFS investigation at several different glancing angles in the vicinity of the critical angle of external total reflection [136]. Also, here the Ta–O bond lengths of the layers were given with an accuracy of ±0002 nm, while the appropriate coordination numbers were estimated with an absolute accuracy of ±009. In comparison to a polycrystalline -Ta2 O5 sample the Ta–O distances for the amorphous layers were found to be smaller. X-ray topography was used to determine the critical thickness of InGaAs epitaxial layers on GaAs [137]. For this purpose the topographs were taken in-situ in dependence on the temperature applied. An indirect determination of the polarity of ferroelectric nanolayers [PbTiO3 grown on SrTiO3 (001) substrates, layer thickness 10–40 nm] was accomplished by means of XSWS [138]. The XSWs were generated inside the thin epitaxial layer by excitation of Bragg diffraction and were used for determination of the polarity. This succeeds by the polarization-dependent relative sublattice positions of the Ti and Pb ions. The method allows for detection of relative displacements of sublattice layers within the unit cells in the order of 0.01 nm. As outlined in Section 4.2 the GID is established for the investigation of epitaxial layers on single-crystalline substrates. In the past few years related objects, self-assembled quantum dots, have been studied extensively. They can be formed if the amount of deposited material is in the order of one monolayer, typically. At an appropriate coverage dislocation-free coherently strained dots are formed. At higher coverage dislocations start to appear, generating relaxed islands on the surface. By X-ray scattering experiments [139] it was shown that for coherently strained islands (InAs on GaAs) the lattice parameter misfit induces a relaxation gradient inside the dots. Beyond that, an X-ray study of structural changes of selfassembled PbSe quantum dots in PbSe/PbEuTe superlattices [140] should be specified here as an example. From surface-sensitive grazing incidence small angle X-ray scattering (GISAXS) the shape of the freestanding dots could be determined exclusively. Information on shape and chemical composition of the buried dots has been obtained from GID experiments. Another interesting subject accompanying with the growth of strained semiconductor layers is nanowires. When using vicinal substrates, the regularly terraced surfaces can act as templates for subsequent growth of laterally ordered islands; at appropriate conditions wires instead of islands can be formed. In [141] wide-angle coplanar X-ray diffraction studies on appropriate Si/SiGe wire superlattices are reported. To obtain information on the shape and the lateral correlation, also of buried wires, GISAXS for different information depths was used. The elastic stress relaxation in Ga022 In078 As080 P020 quantum wire structures (1% compressively strained) with a wire width of 35 nm and a thickness of 8 nm was investigated by X-ray scattering using asymmetric reflections of the InP

X-Ray Characterization of Nanolayers

676 Table 1. Examples for X-ray characterization of nanolayer materials. Nanolayer class

Material (thickness t)

Characterization method

Focus of investigation

Ref.

Organic multilayers

Fe(II)–polyelectrolyte–amphiphile complex on silicon substrate, t ∼ 5 nm (single layer)

X-ray reflectometry, X-ray diffraction

transitions of the Langmuir– Blodgett phase in dependence on temperature

[131]

Monolayers at air–water interface

hexadecylamine–urease monolayers at the air–water interface, t ∼ 01 nm

X-ray reflectometry, GID

layer thickness and structure formation

[132]

Self-assembled monolayers

octadecyltrichlorosilane on silicon substrate, t ∼ 1 nm

X-ray reflectometry

layer growth mode (in-situ)

[133]

Surface layers

silicon wafers, diameter ∼ 200 mm

X-ray topography

surface-strain distributions

[134]

Surface layers

native oxide amorphous layers on surface of Si single crystals, t ∼ 28 nm

X-ray reflectometry GID

morphological parameters of the oxide layer

[135]

Sputtered amorphous layers

Ta 2 O5 layers on float glass plates, t ∼ 224 nm

XAFS (reflection mode)

structural short-range order parameters

[136]

Epitaxial layers

(In,Ga)As on GaAs substrate, t ∼ 200 nm

X-ray topography

critical thickness (in-situ)

[137]

Ferroelectric layers

PbTiO3 on SrTiO3 substrate, t ∼ 10–40 nm

XSW

polarity

[138]

Quantum dots

InAs on GaAs subtrate

GID

strain relaxation gradient inside the dots

[139]

Quantum dots

PbSe dots in PbSe/PbEuTe superlattices (5 periods, tperiod ∼ 35–65 nm)

GISAXS

shape and chemical composition of freestanding dots

[140]

Quantum wires

wires in/on Si/SiGe superlattices on Si substrate, 20 periods, tSi ∼ 10 nm, tSiGe ∼ 25 nm

GISAXS

shape and lateral correlation of wires (incl. buried ones)

[141]

Quantum wires

(Ga,In)(As,P) wires on InP substrate, wire width ∼ 35 nm, twire ∼ 8 nm

XRD (grazing exit)

elastic stress relaxation in wire structures

[142]

Quantum wires

InAs wires on InP substrates, precursor layer thickness ∼ 25 monolayers

DAFS (grazing beam geometry)

strain and chemical composition inside the wires and close to the interface

[143]

Epitaxial multilayers

XANES/ XAFS Co/Pt on Pt/Mo buffer layers (polarization dep.) on Al 2 O3 substrates, t ∼ 016–107 nm (single layers)

local structure in dependence on thickness

[144]

Quantum wells (strained layer superlattices)

Ge/Si (5 periods) on Si substrate, XAFS (grazing incidence t ∼ 4 monolayers (single layers) fluorescence)

influence of mismatch strain on local structure, interface mixing

[146]

Multilayer gratings (lateral patterned multilayers)

(Ga,In)As/InP on InP substrate, grating periodicity ∼ 1380 nm, grating depth ∼ 432 nm, t ∼ 47 nm (single layers)

grating parameters, vertical composition profile, strain tensor

[147]

GID XRD

substrate (grazing exit) [142]. The long-range periodicity of the wire positions causes narrow periodic peaks in the direction perpendicular to the wire direction. With an analytical calculation presented in [142] the X-ray diffraction

intensities were described quantitatively and are found to show a good agreement with the experimental data. InAs/InP (001) self-assembled quantum wires achieved by deposition of 2.5 monolayers of InAs have been investigated

X-Ray Characterization of Nanolayers

by means of grazing incidence diffraction anomalous fine structure (GIDAFS) [143]. For that reason As K GIDAFS spectra were measured at incidence and detection angles close to the critical angle of total reflection of the substrate. Thus information about composition and strain inside the quantum wires and close to the interface could be obtained. The accuracy of bond length obtained from experimental data was in the order of 0.001 nm. Five epitaxial [Co(t nm)/Pt(1 nm)]30
t = 016–107 nm) multilayers [prepared on Pt/Mo buffer layers on Al2 O3 (11–20) substrates] were subjected to polarization dependent Co K fluorescence XANES and XAFS investigations to investigate the local structure in the neighborhood of the Co atoms [144]. The results have shown that at the given growth temperature (470 K) no significant interdiffusion can be observed at the Co/Pd interfaces. The structure of the Co layer changes depending on its thickness, whereby a thickness of 0.3 nm can be taken as a threshold. The accuracy of experimentally determined bond length (between ∼0.250 and ∼0.265 nm) was in the order of 0.005 nm. As outlined in Section 4.5 in fluorescence XAFS experiments the surface sensitivity can be improved significantly by grazing incidence excitation. In principle submonolayer sensitivity can be achieved this way [145]. By means of grazing incidence fluorescence XAFS the local structure around Ge atoms in Ge/Si monolayer strained-layer superlattices (quantum wells) grown on Si(100) substrate have been studied by [146]. The Ge–Ge and Ge–Si bond lengths have been obtained with an accuracy of ±0.001 nm, from which it could be concluded that mismatch strain is accommodated by both bond compression and bond bending in the Ge4 layer. From coordination numbers, determined with an absolute accuracy of ±0.2, conclusions on a substantial interface mixing could be drawn. Lateral patterned (Ga,In)As/InP multilayers (multilayer gratings) were subjected to analysis of surface shape and the spatial distribution of strain by combined high-resolution X-ray diffraction and GID (coplanar and noncoplanar triple-crystal diffractometry) [146]. Characteristic grating parameters and accuracy determined by GID were grating periodicity
1380 ± 5 nm, grating depth
432 ± 10 nm, thickness of (Ga,In)As layers
47 ± 2 nm, and thickness of InP layers
47 ± 2 nm. These results were compatible with the vertical composition profile obtained by coplanar X-ray diffraction. It was found that elastic strain relaxation causes dramatic deformations of the X-ray diffraction patterns in the measured reciprocal space maps. Different components of the strain tensor could be determined by recording diffraction patterns around different reciprocal lattice points. In [147] a detailed description of the influences of the grating shape, the morphological setup, and the related lattice distortions fields on the X-ray diffraction patterns/maps is given.

5.3. Summary The examples illustrate that by means of X-ray characterization methods structural parameters of nanolayers (see Section 2) can be obtained nondestructively with an accuracy on the picometer scale. As outlined, characterization of

677 nanolayers of elements with lower atomic numbers requires a relatively higher experimental effort. If there is no limitation to the extent of the nanostructures in certain directions, in case of interference methods these parameters are determined by averaging over areas/volumes which are at least limited by the coherence length of the radiation. In reverse this gives a limit for the investigation of correlation length of structures obtainable by means of X-ray methods. As an upper limit the size of 1 m can be considered at usual X-ray sources. This corresponds also to the spot size of X-rays to be achieved with modern optics for a reduction of excitation of samples in two dimensions (local analysis). The morphology of interfaces of nanolayers can be characterized on a scale of 0.1 nm. As illustrated by the given examples, at suitable sample characteristics a reduction of the investigated volume in one direction down to one monolayer can be achieved. Within these volumes element-specific bond lengths can be determined with an accuracy in the order of one picometer and average coordination numbers with an accuracy of ±0.01 atoms in the most favorable case.

ABBREVIATIONS AAS Anisotropic anomalous scattering. ADXD Angle dispersive X-ray diffraction. ASAXS Anomalous small angle X-ray scattering. BA Born approximation. BREFS Bragg reflectivity extended fine structure (=DAFS). CCT Curved collimator topography. CDW Charge density wave. DAD Dispersive anomalous diffraction. DAFS Diffraction anomalous fine structure. DANES Diffraction anomalous near-edge structure. DDAFS Dispersive diffraction anomalous fine structure. DIFFRAXAF Diffraction X-ray absorption fine structure. DWBA Distorted-wave Born approximation. EAD Extremely asymmetric diffraction. EDAFS Extended diffraction anomalous fine structure. EDXD Energy dispersive X-ray diffraction. EXAFS Extended X-ray absorption fine structure. GBL Grazing Bragg Laue diffraction. GID Grazing incidence diffraction; see also GIXD. GIDAFS Grazing incidence dispersive diffraction anomalous fine structure. GISAXS Grazing incidence small angle X-ray scattering. GIXD Grazing incidence X-ray diffraction. GIXDT Grazing Incidence X-ray diffraction topography. MAD Multiple-wavelength anomalous dispersion. NEDAFS Near-edge diffraction anomalous fine structure (=DANES). NEXAFS Near edge X-ray absorption fine structure (=XANES). NSXR nonspecular X-ray reflectivity. NXES Normal X-ray emission spectroscopy.

678 REFLEXAFS Reflectivity extended absorption fine structure (=DAFS). RXES Resonant X-ray emission spectrum. RXMS Resonant X-ray magnetic scattering. SANS Small angle neutron scattering. SAXS Small angle X-ray scattering. SDW Spin density wave. SMAD Simultaneous multiple-wavelength anomalous dispersion. SXAFS Surface X-ray absorption fine structure. SXR Specular X-ray reflectivity. TER Total external reflection. TRXF Total reflection X-ray fluorescence. USAXS Ultrasmall angle X-ray scattering. XAFS X-ray absorption fine structure. XANES X-ray absorption near edge structure. XAS X-ray absorption spectroscopy. XDS X-ray diffuse scattering. XES X-ray emission spectroscopy. XMCD X-ray magnetic circular dichroism. XRR X-ray reflectometry. XSW X-ray standing wave field.

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Encyclopedia of Nanoscience and Nanotechnology

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X-Ray Microscopy and Nanodiffraction S. Lagomarsino, A. Cedola Istituto di Fotonica e Nanotecnologie—CNR, Roma, Italy

CONTENTS 1. Introduction 2. Production of X-Rays 3. Optics for X-Rays 4. X-Ray Microscopy Techniques and Applications 5. X-Ray Nanodiffraction 6. Conclusions Glossary References

1. INTRODUCTION The purpose of this chapter is to give some insight into the field of X-ray microscopy and microdiffraction, which, though initiated decades ago, only in recent years had a strong, and quite explosive, development. The reason is twofold: on one side, the availability of very brilliant X-ray sources, based on synchrotron radiation, fostered practical application of X-ray optics, which were studied theoretically but had a too low efficiency with standard X-ray tubes to attract research beyond the speculative one. On the other side, the “nanoworld” is developing so fast and with so interesting perspectives that many researchers were encouraged to find new and more powerful methods to characterize nanostructures. The large potentialities of X-rays are, therefore, also exploited in this direction. We have to point out that the field of X-ray microscopy and microdiffraction is now so large that it is impossible to give an exhaustive account of all the methods and applications. We will, therefore, try, in the following, to concentrate on works that reached, or are soon likely to reach, submicrometer or nanometer (i.e., below 100 nm) spatial resolution. In this respect, we can speak of nanodiffraction. Good reviews on this subject can be found in the literature [1–3]. A comprehensive state of the art can be found in the proceedings of the Sixth International Conference on X-Ray Microscopy [4]. Strictly speaking, X-rays are already the most powerful microscopy method because, since the twenties, it has been realized that X-rays are diffracted by periodic (crystalline)

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arrangement of matter and are, therefore, able to give structural information with an accuracy of the order of fractions of a nanometer (1 nanometer = 10−9 m). This is the basis of all the crystallography, including the biomolecular crystallography, which brought in the fifties to reveal the DNA intimate structure. However, in crystallography, the sample is homogeneous and of macroscopic dimensions, therefore, the information is averaged on large volumes. With microscopy or nanodiffraction, we mean the ability to distinguish between microscopic portions of the sample that differ for their structural, compositional, or morphological properties from the rest of the sample. We distinguish the terms microscopy and nanodiffraction in the sense that nanodiffraction (based on Bragg diffraction, see Section 1.1.3) probes the structural properties of matter, i.e., those related to its crystalline, periodic atomic arrangement. With microscopy, instead, we probe the compositional or topographical properties, through a map of electron density obtained by means of absorption or phase variation contrast or through a map of element-specific radiation (fluorescence) or photoelectrons emission. It is interesting that the most advanced microscopy technique, which we will mention at the end of this chapter, is based on diffraction phenomena from nonperiodic structures and, thus, represent the ideal link between Bragg nanodiffraction and standard microscopy. For pedagogic purposes, the most important properties of X-rays and their interaction modes with matter will be briefly reviewed in the next paragraph. In Section 2, the main characteristics of the X-ray sources will be mentioned, mainly with the purpose to give an idea about the difference between laboratory and synchrotron radiation sources. In Section 3, the X-ray optics able to produce submicrometer beams will be described, while Section 4 will report some of the principle and more promising microscopy technique based on X-rays, together with some example of applications. Finally, Section 5 will describe nanodiffraction techniques, and Section 6 will lead us to the conclusion of the entry.

1.1. Properties of X-Rays X-rays are part of the electromagnetic (e.m.) wave spectrum, extending in energy from about 250 eV to 50 keV. Due to the simple relation E = h = hc/, where E is the energy; , the Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (681–710)

682 wave frequency; , the wavelength; c, the light speed, and h, the Planck constant; the same range can be expressed in wavelengths between 50 nm and 0.025 nm. The part at lower energy (up to about 2 keV) is called “soft,” the more energetic, “hard.” Some relevant difference between soft and hard X-rays will be put in evidence. X-rays interact with matter essentially via electrons. An isolated electron subjected to interaction with an X-ray plane wave is instantly accelerated and then radiates as a dipole antenna. The strength of the interaction has been established by Thomson and the ratio between the incident amplitude and the scattered one is proportional to the Thomson scattering length (also called classical electron radius) r0 = e2 /mc 2 = 2 82 × 10−13 cm. If the electron is bound to an atom, a form factor (atomic scattering factor) must be considered that takes into account the distribution of electrons in the atom. It can be useful to describe, from a phenomenological point of view, three phenomena that take place when X-rays interact with matter and that are at the basis of the microscopy and nanodiffraction techniques: absorption, refraction, and diffraction [5].

1.1.1. Absorption When an X-ray photon beam having an intensity I0 impinges on a target of thickness t, the intensity I after the target is simply given by: I = I0 e− t , where is called the linear absorption coefficient. The physical phenomenon at the basis of absorption is the photoelectric effect. When an X-ray photon has an energy larger than the binding energy of a given electron shell, the photon is absorbed, and the electrons are ejected with a kinetic energy given in first approximation by the difference between the photon energy and the binding energy. This implies that the absorption coefficient as a function of photon energy is not continuous but has sudden jumps in correspondence of the electron-binding energies of the target elements. The jumps are called absorption edges, and the absorption coefficient before and after the absorption edge can differ from a factor of about 6 to 8 [6]. The absorption edge of the first shell of electrons is called K, the second is L, the third is M, etc. It is important to note that the binding energy and, therefore, the absorption edges, also can have a slight dependence on the valence state of the element (chemical shift). When an electron is ejected, a hole in the corresponding shell is created, which is immediately filled by electrons of the next shell. This gives rise, in turn, to emission of a photon (fluorescence) or to other electrons (Auger electron). In the energy region intermediate between two absorption edges, the absorption coefficient is approximately proportional to 3 . This implies that is very different in the soft and hard regions. To give an example, photons of 25 keV can pass 1 mm of Al with low losses but are completely absorbed by an Al layer 1 micron thick if their energy is 410 eV.

X-Ray Microscopy and Nanodiffraction

wave traveling in medium 1 impinges on the interface at an angle 1 (Fig. 1a). The beam traveling in medium 2 will make an angle 2 related to 1 through the simple formula: cos 1 / cos 2 = n2 /n1 where n1 and n2 are the real parts of the refraction indexes of mediums 1 and 2. By using quantum mechanical theory,  and  for a monatomic material in the X-ray region can be expressed, respectively, as [7]  = 2 N0 /Am r0 Z + f ′ 

The refraction index r of a material can be expressed by r = n − i = 1 −  − i. We recall here that n = r r 1/2 , where r and r are, respectively, the relative (to the vacuum) dielectric constant and the relative magnetic permeability of the material under consideration. Imagine two mediums, 1 and 2, separated by a plane, smooth interface. An e.m.

(1.1)

where N0 is the Avogadro’s number, A is the atomic mass, m is the density, r0 is the classical electron radius, Z is the atomic number of the element, and f ′ is a correction factor related to the rapid variation of the atomic scattering factor for X-rays in proximity of the absorption edges, is the linear absorption coefficient;  is always positive, and this implies that for X-rays, all the materials have a refraction coefficient less than 1. Therefore, if medium 1 is a vacuum (or air) n2 < n1 and the radiation will travel in the material with an angle 2 < 1 , then, as a consequence, there will be an incident angle 1 = c for which 2 = 0. In this case, the incident beam is totally reflected, and only an evanescent wave can enter into the material, c is called the critical angle for total reflection, and it is not difficult to see that c ≈ 21/2 . Therefore, c is proportional to  and to the square root of Z and m .

1.1.3. Diffraction In refraction and reflection, the atoms do not absorb the X-ray photons, but, once excited by them, they reirradiate a X-ray beam of the same energy (in the elastic approximation). The same happens in diffraction. Diffraction is a common phenomenon of an interaction between a wave and an object having dimensions of the same order of magnitude as the wavelength in consideration. Every point of the object, following the principle of Huygens, becomes a source of spherical waves of the same wavelength as the incoming radiation. We can distinguish between Fresnel and Fraunhofer diffraction [8, 9]. If we take a scale factor defined as d 2 /2, where d is the object dimension and  is the wavelength, we can roughly speak of Fresnel diffraction when at least one of the relevant distances source–object (D1 ) or object-detector (D2 )—is smaller or of the same order as d 2 /2. Fraunhofer diffraction will take place when both the relevant distances are D ≫ d 2 /2, and, therefore, plane wave approximation is valid. Bragg diffraction is a special case of Fraunhofer diffraction, if many objects interact (a)

(b) medium 1 α1

1.1.2. Refraction and Total Reflection

and  = /4

medium 1

α2

medium 2

medium 2

Figure 1. Refraction for X-rays: (a) since the refraction index is ), and the diffracted beams will lie on a conical surface of aperture  around the axis, which coincides with the incoming radiation. The projection on a screen perpendicular to the incoming beam will then give rise to a circle.

1.1.4. Coherence An important property of an electromagnetic field, and, therefore, of X-rays, is the coherence. There are two kind of coherence: spatial coherence and time coherence. To define spatial coherence, let us consider two points P1 and P2 that lie on the same wave front of an e.m. wave at time t = 0. Let E1 t and E2 t be the corresponding electric fields at these points. By definition, the difference between the phases of the two fields at t = 0 is zero. If this difference remains zero at any time t > 0, there is a perfect coherence between the two points. If this occurs for any two points of the e.m. wave front, the wave is perfectly coherent (the degree of spatial coherence is 1). Usually, for any point P1 , the point P2 must lie within some finite area (coherence area or length) around P1 to maintain the phase correlation. From the Van Cittert-Zernike theorem [8] it comes out that the radiation field from a primary incoherent source gains coherence during the propagation. Furthermore, the larger the distance is from the source and the smaller the source size is, the larger is the coherence length. The coherence length l can, therefore, be approximately defined as l = L/s, where  is the wavelength; L, the distance source-sample; and s, the transversal source size. To define temporal coherence, consider the electric fields of the e.m. wave at a given point P, at times t and t + #. If, for a given time delay #, the phase difference between

the two field values remains the same for any time t, there is a temporal coherence over a time #. If this occurs for any delay #, the wave is perfectly time coherent (the degree of time coherence is 1). If this occurs for a time delay # 0 < # < #0 , there is a partial time coherence, with a coherence time equal to #0 .

2. PRODUCTION OF X-RAYS It is beyond our scope to give a comprehensive account of the available X-ray sources, but we consider it useful to give some basic ideas, to understand the main differences between laboratory sources and synchrotron radiation sources.

2.1. Production of X-Rays Through Interaction of Electrons with Matter The standard laboratory X-ray tubes are based on the interaction of electrons with matter: an electron beam accelerated to several keV hits a target of a given material and in this way produces strong ionization, with ejection of electrons of the inner shells and production of intense fluorescence whose energy is characteristic of the target element. Moreover, the electrons are decelerated by the bound electrons of the target and, as a consequence, emit a continuous spectrum of radiation (Bremhstrahlung). Therefore, the total spectrum of an X-ray tube is composed of quite narrow peaks of characteristic radiation superimposed on a “white” component. The radiation is emitted in all the directions, but, in general, the target is viewed at a small angle to lower, at least in one direction, the projected dimensions. Typical projected target dimensions are 0 4 × 0 4 mm2 for a “point” focus and 0 04 × 1 mm2 for a “line” focus. Recently, microfocus X-ray generators were produced with a source dimension of a few microns in diameter. The drawback is a low value of current necessary to not damage the target. To have an even smaller X-ray source, electrons can be focused by means of electromagnetic lenses. This is what happens, for example, in electron microscopes. Scanning electron microscopes (SEM) have been suitably modified to hit a thin target at the focus position and to generate an X-ray beam with linear dimensions of the order of 0.2 [11].

2.2. Production of X-Rays Through Interaction of Light with Matter (Plasma Sources) X-rays also can be generated when an intense laser beam interacts with liquid or solid matter, giving rise to a plasma. The plasma then re-emits a spectrum made of several lines, which depend on the target. A high-power laser such as Nd-YAG is generally used. Thin solid foils can be used as targets, but, in this case, debris, i.e., small solid fragments ejected by the target, can damage sensitive components (such as optical elements) positioned close to the plasma. Moreover, the position of the beam on a solid target must be continuously changed, and the target itself must be removed and changed very often. Recently, microscopic liquid-jet or liquid-droplet targets are used, which eliminate the problem of debris and of regeneration. The target is generated

684 by forcing a liquid in a capillary nozzle [12, 13]. Typically, sources use ethanol at room temperature or liquid nitrogen at cryogenic temperatures. In the first case, a C line at 3.37 nm is used, while N lines at 2.88 and 2.48 nm are generated in case of liquid nitrogen sources. Typical source dimensions are of the order of 10–20 .

2.3. Synchrotron Radiation Here we want to give just some basic ideas about the mechanism of production of X-rays with synchrotron radiation (S.R.) and S.R.’s main properties. A much more complete presentation, but still clear and concise, is given by D. Raoux in the proceedings of the school “Hercules” [5]. Synchrotron radiation is the e.m. field radiated by accelerated relativistic charged particles. Electrons or positrons are used. Any accelerated charge radiates an e.m. field that, if the particle has a speed v ≪ c (c light speed), is isotropic around the acceleration. However, if v ≈ c, then the relativistic effects dominate and the radiated e.m. field is sharply peaked in the direction of motion of the particles. The cone aperture is 1/% , where % is the ratio between the particle energy and its rest mass. For electrons or positrons of energy E expressed in GeV, then % = 1957∗ E. Thus, for 6 GeV electrons, the aperture is about 80 rad. To get a radial acceleration, the easiest way is to hold the particles in a circular orbit of a radius of curvature  by means of magnets, which exert the Lorentz force on the charged particles. The synchrotron radiation sources are, therefore, basically storage rings where electrons or positrons have circular orbits, and the radiation is that which originates from bending magnets. The emission spectrum is continous from infrared radiation up to a critical wavelength c , which depends on  and %. The critical wavelength is defined as the value for which half of the total power is emitted at wavelengths shorter than the critical one; c is given by c = 4/3/% 3 . To improve the intensity of emission, insertion devices (I.D.) have been conceived, where the charged particles pass through alternating magnetic poles and are, therefore, compelled to have a zig-zag trajectory. At each wiggle, they emit radiation that is, therefore, enhanced in a 2N factor, where N is the number of poles. These devices are called wigglers or undulators, and the distinction essentially is due to the relation between the angular deviation  at each wiggle and the aperture 1/%. Values of  ≫ 1/% identify wigglers, while values of  ≈ 1/% identify undulators. The radiation emitted by a wiggler is the incoherent sum of the radiation fields emitted by each individual magnet. The spectrum is continuous but shifted at higher energies with respect to that of a bending magnet. Instead, in the undulator regime, the amplitudes of the field radiated at each period of the particle trajectory may interfere, resulting in a periodic radiation field. The spectrum is thus not continuous, but resonances occur at given frequencies (the fundamental and the harmonics). Both in wigglers and undulators, the spectral characteristics depend on the angle with respect to the I.D. axis. In general, a narrow cone around the axis is selected. What is important in S.R. is not only flux, which is defined as the number of photons per unit time in a given band-pass energy, F = ph/s/)/; )/, is conventionally taken as

X-Ray Microscopy and Nanodiffraction

the value of 10−3 . Also taking into account the angular aperture (i.e., the collimation), we can speak of brightness that is the flux per solid angle, Brightness = ph/s/mrad2 /)/. In fact, the size source is of importance, and, therefore, the brilliance becomes the parameter of interest, Brilliance = Brightness/*x *y = ph/s/mrad2 /*x *y /)/, where *x and *y are the transverse source sizes (horizontal and vertical). The emittance of the source  is defined as  = h ∗ v , where the horizontal and vertical emittances h and v are defined as h = *x *x′ , and v = *y *y′ , where *x′ and *y′ are the divergence of the beam in the horizontal and vertical directions, respectively. The figure of merit of a synchrotron is brilliance and emittance. The first should be as high as possible, and the last should be as low as possible. Just to give an idea about numbers, if an X-ray tube has a brilliance of the order of 107 , brilliance at an undulator beamline of third generation S.R. can be as high as 1020 –1021 . Horizontal and vertical emittances at the European Synchrotron Radiation Facility (ESRF) reached values as low as 4 and 0.025 nm, respectively. Time structure also is important, at least for time-resolved experiments. Synchrotron radiation is not continuous but is emitted in bunches corresponding to the electron bunches in the storage ring. Duration of a single bunch is, for example, at ESRF in Grenoble, of 20 picosec. Time between bunches depend on the filling mode of the storage ring. For the single-bunch mode (the one generally used for time-resolved experiments), it is about 3 s at ESRF. Scientists are now planning S.R. sources of the fourth generation, with impressive performance. They are based on the concept of free electron lasers composed of a linear accelerator of electrons operating at several GeV and long undulators (several tens of meters). Radiation produced by such sources should have brilliance 10 orders of magnitude higher than the present most powerful sources and a time structure with pulses of few tens of femtoseconds (i.e., 10−14 s). This should open a new world in many fields, including the one described in this discussion.

3. OPTICS FOR X-RAYS To perform X-ray microscopy or microdiffraction experiments, the X-ray beam must, in general, be conditioned to have the necessary spatial resolution. In principle, a simple pinhole could do this task, but photon flux would be lost. Therefore, focusing X-ray optics must be used to concentrate photon flux in small dimensions, as lenses do in visible spectrum. This paragraph will describe at a basic level some optical elements for X-rays that are used in microscopy and microdiffraction. The focusing problem well known in light optics becomes difficult as soon as X-rays are considered because, at short wavelengths, the radiation is very weakly refracted by materials 1 − n ∼ 10−5 –10−6 , as C. W. Röntgen himself concluded just after having discovered X-rays. The impossibility of using X-ray lenses delayed the beginning of X-ray optics until the development of the dynamical diffraction theory by perfect crystals. Indeed, the use of Bragg diffraction has allowed developing optics based on geometrical focusing by bent or curved elements: crystals, mirrors, or multilayers. These approaches are widely used but, due to the source

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X-Ray Microscopy and Nanodiffraction

size and divergence of laboratory sources, they are capable of reaching a resolution of several microns only. The advent of S.R. sources and, in particular, the high brilliance of modern high-energy synchrotron storage ring facilities (ESRF, Grenoble France; Advanced photon source (APS), Chicago, IL; Spring 8, Harima, Japan; Elettra, Trieste, Italy; Advanced light source (ALS), Berkeley, CA), gave new impulse to research for innovative X-ray optics. Thus, other focusing optics have been thought as striving for submicrometer beam generation. Presently, there are only five types of X-ray optics producing a submicrometer beam: Fresnel zone plates [14] and Bragg-Fresnel optics [15] based on diffraction, refractive lenses [16] based on refraction, Kirkpatrick-Baez mirrors [17] based on total reflection, and waveguides based on standing waves [18]. Capillaries, which are widely used to shrink the X-ray beam to micrometer dimensions, will not be treated here because their efficiency as a submicrometer beam is very low. The Fresnel zone plates are mostly used in the soft X-ray region, while the other optical elements mentioned before are mostly used in the hard part of the X-ray spectrum. This is due to the fact that in the hard region, fabrication of Fresnel zone plates is more difficult than in the soft region, as will be better illustrated in the following discussion. The figures of merit useful to compare the performance of the different optics are: (i) the beam size produced, (ii) the coherence of the beam, (iii) the fabrication complexity, and (iv) the gain. This latter is defined as the ratio between the output flux density over the input flux density: I F s G = exit = exit inc Iinc sexit Finc where F is the flux and s represents the size of the beam; G, that is a dimensional, refers to the performances of a theoretical slit, which reduces the beam from-to the same sizes than the optical element does or, which is the same, it represents the flux density gain and, therefore, an intrinsic property of the optical element. In fact, if the incident beam size sinc increases over the spatial acceptance of the optical element and, therefore, the input flux increases, the gain G remains the same, even if the efficiency decreases.

3.1. Fresnel Zone Plates Although the first zone plate was constructed by Lord Rayleigh in 1871, the first published work was written by Soret in 1875. However, only Wood, in 1898 [19], carried out the first experiments on zone plates by using visible light, starting the development of this new optics. Baez [20] first suggested the focusing applications of the zone plates in the wavelength region between extreme ultraviolet and soft Xrays in 1961.

3.1.1. Theoretical Principles The zone plate is so called because it is constructed following the Fresnel-zone law: a plane opaque screen with a circular aperture of radius d illuminated by coherent radiation (i.e., a plane wave) produces, at a generic distance z from it (Fig. 2), the usual diffraction pattern of lighter and darker rings on a screen normal to the z axis.

z+λ/2

P

d

z

Figure 2. Opaque screen with a circular aperture of radius d. The points on the aperture distant z + /2 from P define the radius of the first Fresnel zone for the point P.

This diffraction pattern is not of interest here, and the analysis will be focused on the diffraction along the z axis alone. The field amplitude W z along the z axis oscillates depending on the d2 /z ratio according to the following relation [8]: W z = Ae

−ikz

1 − e

−iNF

 = −2iAe

−ikz+ 2 NF 

 NF sin 2 (3.1a)


where: NF =

d2 z

(3.1b)

is the Fresnel number. Different regions of the aperture emit radiation, which interferes constructively or destructively depending on the distance z. For any given point P on the z axis, it is convenient to classify the regions on the z = 0 plane (where the aperture is located) depending on the constructive or destructive interference produced in P. These regions are named Fresnel zones. The first Fresnel zone for the point P distant z from the screen is the circle of radius r1 :
2  r12 = z + − z2 ≈ z 2 It identifies the points on the z = 0 plane at a distance smaller than z + /2 from P. The radiation emitted from these points interferes constructively in P since the optical path lengths differ by less than /2. The field generated by this circle in P is easily obtained from (3.1a) by taking d = r1 and therefore NF = 1: W1 = Ae−ikz 1 − e−i  = 2Ae−ikz

(3.2)

so that its amplitude is two times larger than the field amplitude in the absence of the screen. In other words, if the aperture had a radius d = r1 , then the whole radiation passing through it would interfere constructively in P, thus explaining the increased intensity. The second Fresnel zone is a ring of internal radius r1 and external radius given by r22 = z + 2 − z2 ≈ 2z

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X-Ray Microscopy and Nanodiffraction

The field produced at P by the points in this second Fresnel zone is obtained by setting d = r2 in (3.1b) and subtracting the field produced by the points in the first zone: W2 = Ae−ikz 11 − e−2i  − 1 − e−i 2 = −2Ae−ikz

(3.3)

As it is apparent from the negative sign, if the aperture had a radius d = r2 , the radiation from the first Fresnel zone would interfere destructively with the radiation coming from the second one and the intensity in P would vanish. The radius of the generic nth zone is rn =

√

nz

n = 13 2 nmax

(3.4)

where nmax is the total number of zones. It is important to notice that the Fresnel zones are independent on the size of the aperture and they only depend on the position of P and on the wavelength. For a point at distance z, the number NF in Eq. (3.1b) is the number of Fresnel zones contained in the aperture. In particular, if this number is an even integer, then the interference of the radiation from the various zones is destructive and the intensity in P is zero. If NF is an odd integer, then the radiation from one whole zone is unbalanced and the intensity is maximal. The Fresnel zone plate (FZP) (Fig. 3) is a screen of alternately transparent and opaque zones equal in size to the

Fresnel zones. In this way, only the radiation from odd or even Fresnel zones is transmitted through the screen. At the point of observation, the exposed zones produce optical disturbances of the same phase and, therefore, the resultant optical intensity is much greater than that observed in the absence of the screen [see Eq. (3.2) or Eq. (3.3) for the field amplitude coming from the first and second Fresnel zones, respectively]. As a result, the zone plate concentrates the incident intensity in a small region like a lens does.

3.1.2. Focal Distance Illuminating the screen with a plane wave, for m = 0, the zone plate produces spherical waves. If r1 is the radius of the innermost zone, and m is the number of diffracted order, the focal length fm of a zone plate is given approximately [1] by: m r2 k = 2 ⇒ fm = 1 2fm m r1

(3.5)

It is important to notice that the position of the focal point depends on the wavelength. This chromatic aberration is more important than the chromatic aberration in optical lenses, and it is caused by diffraction instead of dispersion as in the case of glass lenses. As a consequence of this aberration phenomenon, a zone plate has to be used with radiation  = cmnmax [1], where c is a constant. having a ) given by ) Theoretical studies have shown that with c = 0 5–1, practically no contrast in the image is lost compared to imaging with quasi-monochromatic radiation [21].

3.1.3. Resolution The width of the outermost zone of a zone plate can approximately be written as drn =

rn 2nmax

(3.6)

The most important characteristic of the zone plate is the resolution ), which can be expressed, according to the Rayleigh criterion [1, 22], as ) = 1 22

drn m

(3.7)

where drn is the width of the outermost zone of the FZP. The smaller the drn , the better the resolution. Therefore, from Eq. (3.6), to obtain a high resolution FZP, a large number n of zones has to be made. It should also be noted that the spatial resolution depends on the wavelength  and that the resolution is inversely proportional to the order.

3.1.4. Focal Spot Size

Figure 3. Scanning electron microscopy picture of a zone plate composed of alternately transparent and opaque zones equal in size to the Fresnel zones.

It must be pointed out that the focal spot size )m of the mth order focus is not exactly equal to ), because there are other factors contributing to the broadening of the beam. Indeed, for an incident source size )s distant L from the FZP, the focal spot size at a distance fm from the FZP can be expressed as  1/2 )m = )2 + )2i3 m + )2c

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X-Ray Microscopy and Nanodiffraction

3.1.6. Blazed Profile

where )i3 m is the magnified source size: )i3 m = )s

fm L

and )c is the chromatic aberration: )c = D

)E E

with D being the diameter of the FZP and )E/E, the resolving power of the monochromator.

3.1.5. Efficiency Another important parameter characterizing a zone plate is the efficiency with which it collects incident photons into its mth order focus. For a FZP of thickness t and laminar zone profiles, the groove efficiency is given by [1] Eeff = with:

1  2 m2



1 + e−245 − 2e−45 cos 5

n = 1 −  − i

5=

2t 

4=



 

(3.8)

(3.9)

where n is the complex refraction index and  is related to the absorption coefficient. The simplest type of FZP is an amplitude plate composed of zones alternately transparent and opaque to X-rays. In this case, the focusing originates mainly from the relatively different absorption of two neighboring zones. If one only considers the absorption in the material of the absorbing zones ( = 0), then the two exponential terms of Eq. (3.8) vanish for sufficiently large t, and one gets theoretical efficiency of an amplitude zone plate, which is Eeff = 10 1% for the first diffractive order. The efficiency can be considerably improved by using Fresnel phase zone plate (FPZP), which focus X-rays through the relative phase change between two neighboring zones. The FPZP can be realized by choosing an appropriate material with a thickness t such that the retardation 5 of the incident wave is  in those zones that are opaque in the amplitude FZP. From Eq. (3.9): 1 t =  2

(3.10)

If absorption is ignored, i.e.,  = 0, the groove efficiency, Eq. (3.8) becomes

Another factor reducing the efficiency of a zone plate comes from the very nature of diffractive optics, which leads to the formation of many diffraction orders. Indeed, a significant fraction of incident photons are delivered to the undiffracted zero order and diffraction orders other than the primary first order. A lower contrast and a reduced usable flux are the consequences. By introducing asymmetries in the grooves [23, 24], it is possible to control the position of the maximum intensity in the order sequence and, therefore, to blaze the optical element. Theoretical research by Tatchyn [25] studied the improvement in focusing efficiency of a zone plate by optimizing the zone profiles. The basic principle of a zone plate with a blazed profile (ZPBP) is that in propagating through a period consisting of two regular Fresnel zones, the rays experience phase delays such that their effective optical path length from the incident surface to the focus is identical. As a consequence, all the rays will constructively interfere at the focus. It may be shown by using Huygens principle or numerical calculation [25–27] that 100% focusing efficiency may be obtained with a parabolic zone profile when the absorption of X-rays is negligible. Since it is difficult to produce zone plates with an optimal blazed profile, a multisteps profile approximating the ideal one has been used to produce ZPBPs. In Figure 4 is represented the SEM of a circular gold ZPBP, produced at IFN–CNR (IFN–CNR) of Rome [24], made of four levels. The calculated focusing efficiency of a 4-step ZPBP can reach 81% by using Be, 73% with Ni, and 50% with Au. The ZPBP in Figure 3 has been tested at APS, and a focusing efficiency of 45% for the linear and 39% for the circular one has been measured and a spot size with full width at half-maximum (FWHM) of 2 2 × 5 has been found. The relatively large spot size produced by a ZPBP is compensated by the high efficiency.

3.1.7. State of the Art and Fabrication Process The construction of the first self-supported gold Fresnel zone plate was performed in 1961 by etching followed by gold microelectrodeposition [14]. This zone plate was about 10- m thick and had 19 open zones, with an outermost zone width of about 17 m.

Eeff = 4m−2  −2 resulting for the first order Eeff = 40 4%. It should be noted, however, that perfectly transparent phase-shift materials do not exist for X-rays, so that a FPZP is always an amplitude-phase element, and this fact causes a decrease in the focusing efficiency and an increase in the background. Taking into account the transmittance T of the supporting membrane of the zone plate, one can define the absolute efficiency Ea as Ea = Eeff T

Figure 4. Scanning electron microscopy picture of a circular gold ZPBP.

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X-Ray Microscopy and Nanodiffraction

Development of advanced lithographic techniques has allowed remarkable progress in zone plate fabrication. Holographic [28] and interferometric [29] lithography produced zone plate patterns with hundreds of zones with an outermost zone width as small as 60 nm. Excellent zone plates have been fabricated by electroplating of nickel into polymer molds [30, 31]. Microzone plates with a minimum outermost zone width of 20, 25, and 30 nm were fabricated [32] with an efficiency of 60%, 71%, and 83% of the maximum theoretical achievable efficiency (9.2%, 16.2%, and 18.0%, respectively). Semiconductors also are commonly used, as they can be structured with high aspect ratios by plasma etching. Germanium lenses optimized for operating at 2.4 keV have been constructed [33] with an efficiency of 60% of the theoretical value (18.3%) and a resolution better then 100 nm. Figure 5 shows the processing sequence of an electronbeam lithography technique for fabricating zone plate with the central stop. The pattern generator is programmed to produce the desired patterns. After development, the zone plate is formed by electroplating gold, nickel, or another suitable metal. Advanced reactive ion etching (RIE) processes can be used to transfer the zone plate patterns with an outermost zone width as small as 19 to 55 nm in Ge and Ta.

3.1.8. Hard X-Rays If high spatial resolution is desirable, some technical considerations have to be taken into account. In the soft X-rays spectral region, zone plates with a focal spot approaching 20 nm and a high efficiency have been developed [31– 37]. Nearly all of them in use today are fabricated by using various forms of an electron-beam lithographic technique. These techniques, however, cannot be directly used for producing zone plates for hard X-rays at very high resolution because the thickness required is beyond the fabrication capabilities. Indeed, since the absorption coefficient is inversely proportional to E, thicker opaque zones are required for an amplitude zone plate working at high photon energies. On the other hand, the thickness [Eq. (3.10)] required to obtain a  phase shift in a phase zone plate increases as E increases because  generally is proportional to 1/E 2 . These facts imply technical problems when hard X-rays are concerned. For a phase zone plate working at 8 keV, the required thickness, when using Au for the opaque regions, is 1.5 m. If a resolution of 0.1 m is desirable, an outermost zone width equal to 0.08 m [Eq (3.7)] for

Electron beam exposure

1

PMMA resist Gold plating base SiN window

4

2

First developpment and RIE

5

3

First electroplating

6

Second developpment and RIE

Resist and plating base removal

Figure 5. Steps of the fabrication process of a Fresnel zone plate.

the first order has to be fabricated. An important technical parameter is the aspect ratio defined as K=

t drn

representing the ratio between the thickness and the width of the zones. The K value for the previous requirements is 18.3, which is beyond the capabilities of even state-of-the-art microfabrication facilities with a single step. Several approaches have been developed to effectively obtain the required aspect ratio [38]. One approach used at IFN-CNR consists in growing the FPZP in two steps, duplicating the pattern on another side of a thin membrane on which the first FPZP pattern was made, or on top of it [23, 24]. It is clear that in both cases, it is, in principle, possible to reduce the required aspect ratio for each FPZP pattern by a factor of two. By using this technique, a FPZP working at 8 keV, producing a spot size of 150 nm (FWHM) for the first order and 90 nm (FWHM) for the third one, with a focusing efficiency of 13%, has been constructed and tested [24]. Two other approaches to fabricate FZP for hard X-rays could be mentioned: the sputtered sliced technology and the fabrication of high-aspect linear zone plate by selective etching of Si. In the first approach, layers of different materials (e.g., NiCr and SiO2 ) are deposited by sputtering onto a microwire of borosilicate glass. The wire, reinforced by nickel electroplating, is then sliced perpendicular to its axis and then carefully polished to the desired thickness (a few microns). Measured efficiency is of the order of a few percentages, significantly lower than the theoretical value, but this kind of zone plate fabrication preparation is still promising [39]. In the second approach, a linear zone plate, with a staircase profile to increase efficiency, is fabricated in Si. To obtain the required phase difference, the linear zone plate is tilted to increase the light path length ([40], and page 48 in [4]).

3.2. Bragg–Fresnel Lenses In the previous section, it was remarked that a FPZP allows an efficiency higher with respect to the efficiency obtained with an amplitude FZP. The focusing effect of the FPZP is achieved through the phase retardation of the incident wave by  in those zones that are opaque in the amplitude FZP. Unfortunately, as mentioned earlier, perfectly transparent phase-shift materials do not exist for X-rays, so that such a FZP is always an amplitude-phase element and the focusing efficiency is not as good as theoretically expected. A solution to this problem consists in working in reflection instead of transmission. This situation can be obtained by using the Bragg diffraction of a perfect crystal, where a FZP is grown. This is what a Bragg–Fresnel lens (BFL) does [41–44]: the wave reflected at a Bragg angle by the lower surface of the BFL zone structure gains an additional phase shift as compared to that reflected by the upper surface. The height of the relief has to be adjusted to have a  phase shift. Notice that the intensity of the beams reflected from the lower or the upper surface is determined by the Bragg

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X-Ray Microscopy and Nanodiffraction

condition only, and it is almost unaffected by absorption. Thus, BFL is purely a phase element, and, upon diffraction of the X-ray wave on the BFL zone structure, no amplitude modulation of the reflected wave is observed. Moreover, the BFL phase shift for a certain reflection is independent of the energy, and it is determined by the structure height of the relief only. Diffraction efficiency of the BFL is very close to the theoretical performance and is about 40%. The limiting spatial resolution that can be obtained in BFL is given by the width of the outermost zone of the zone structure, and a fraction of a micron is achievable. A linear BFL on a flat substrate is analogous to a cylindrical lens in visible light optics and produces one-dimensional focusing of X-rays. Since the phase properties do not depend on energy, the same lens can be applied for a wide range of energy determined only by Bragg’s law. A focus spot size of 0.8 m has been observed by Snigirev [44], and a focus efficiency about 35% was measured. The curved crystal BFL proposed by Hartman et al. [45], provides a two-dimensional X-ray focusing and both energy tuning and high flux. In this case, sagittal focusing is achieved by the linear BFL, while meridional focusing is produced by the bent BFL crystal substrate, as illustrated in Figure 6. The corresponding flux enhancement is determined both by the increase of the acceptance of the focusing element and by the increase of the crystal bandwidth due to the deformations introduced by the bending. The minimum focus size in the sagittal direction is determined by the resolution of the BFL (≈0.3 m). The latter can be decreased significantly for high energies, where the depth-related aberration becomes important, by using Laue geometry. The focal length in the meridional plane of the bent crystal in the symmetric case is

FM =

R sin ;B R = 2 4d

where R is the radius of curvature, ;B is the Bragg angle, and d is the d spacing of the reflection used. For out-of-plane (sagittal) focusing, the focal length of the BFL is given by FS =

r12 

Bent linear BFL Slits ESRF bending magnet X-ray source

High resolution X-ray film

Figure 6. Schematic display of the setup used to obtain a twodimensional X-ray focusing with a curved crystal BFL. Reprinted with permission from [45], Y. Hartman et al. Nucl. Instrum. Methods Phys. Res., Sect. A 385, 371 (1997). © 1997, Blackwell Publishing.

where r1 is the innermost zone radius. Then the wavelength o for which the focal lengths are the same is obtained as  d 0 = 2r1 R and, therefore, the focal length is FM =

R0 4d

Because the parameters of BFL structure are fixed, when tuning the energy, the crystal curvature must be changed, and the sample has to be moved to a different focal length FM . A curved crystal BFL has been tested at the optics beamline at ESRF [45] and a two-dimensional focusing of 17.2 keV X-rays was obtained with a focal spot size of 4 5 × 9 1 m2 in vertical and horizontal directions, respectively. The large beam size was compensated by a gain of 300 times in flux density with respect to a flat BFL.

3.3. Compound Refractive Lenses The X-ray refractive lenses are the focusing elements that more directly adapt the methods developed in visible light optics to X-rays. As discussed in the introduction, this attempt was unsuccessful due to the low refraction index and the strong absorption of the X-rays. In 1948, Kirkpatrick and Baez [17] considered the possibility of a refractive lens for focusing X-rays, and they concluded that it was unfeasible. Suehiro, in 1991 [46], discussed how to realize X-ray refractive lenses, and he proposed the development of spherical refractive lenses with high-Z material. However Michette [47] soon objected that high-Z material, necessary for refracting the X-rays to focalize, would have an absorption that would reduce transmission to an unacceptably low value. In 1993, Yang [48] showed that a single refractive lens made of low-Z material would be characterized by a relatively small radius of curvature and a large aspect ratio of the concave surface. These conditions render fabrication of a single refractive lens very difficult or even impossible. Finally, in 1996 Snigirev et al. [16] found a simple and effective procedure to build an X-ray refractive lens with low absorption and reasonable focusing distance. The first principle is that since the X-rays have the refraction index n < 1, a hole (that is, the air contained in the hole) drilled in a low-absorbing material can play the role of a lens in visible light. A simple concave lens fabricated as a cylindrical hole drilled in a low-Z material would focalize an X-ray beam. The low-Z material assures a low absorption, while the concave shape will focalize the refracted beams at a distance: F = R/2

(3.11)

From Eq. (3.11), it is clear that for a feasible radius R = 300 m, the low-Z material, for example, Al ( = 2 8 × 10−6 at 14 keV), will produce a focus at 54 m, which is unacceptable for almost all microfocus experiments. Snigirev et al. [16] built up a compound refractive lens (CRL) consisting of a linear array of many individual lenses

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X-Ray Microscopy and Nanodiffraction

manufactured in a low-Z material (Fig. 7). The lenses, stacked in a row, increase the refractive power; in this case, the focal distance of the array is F = R/2N  For the conditions above, a compound lens with N = 30 single lenses brings the focus to F = 1 8 m, which is widely acceptable. The diffraction limited resolution of the lens *f is defined by an effective lens aperture A: *f = rf /A where rf is the image distance. The real gain of the compound refractive lens, taking into account the source size *0 and the X-ray absorption in the material, is 
A r0 +1 g=a *0 rf where a = exp− Nd, r0 is the source distance, and *1 = *0 rf /r0 ) is the real focus size, defined as the demagnified source size.

3.3.1. Parabolic Refractive Lenses Although these results are very interesting, nevertheless, the focused beam produced by the array of drilled holes did not achieve a submicron dimension. A substantial improvement in the performance of refractive lenses has been obtained by using as a shape, a double paraboloid of rotation [49]. With modern numerically controlled machines, it is possible to generate a rotational parabolic shape within 1 m and a surface roughness below 0.1 m. The main advantages of this new solution are the suppression of the spherical aberrations and the focusing in both directions. By using this new solution, a focal spot size in the submicrometer range can be achieved. Moreover, parabolic profiles can be generated with higher accuracy. Indeed, a parabolic tool always generates a force component normal to the surface, reducing the tendency for microcracks and, hence, for profile errors and surface roughness. In contrast, a sphere generates, at the equator line, only shear forces, without compression. It is also clear that the radius of curvature R and the geometrical aperture 2R0 are decoupled from one another for a parabola, whereas, they are not for a sphere. Therefore, for a parabolic lenses, it is possible to combine a geometrical aperture of 1 mm with a radius of curvature R of 200 m and below. The smaller the value of R, the lower is the focal length F , and the smaller is the number of lenses needed in a stack.

2R

d

Figure 7. A Compound refractive lens consisting of a number N of cylindrical holes placed close together in a row along the optical axis.

By using a stack of 220 Al lenses with a radius of curvature R = 209 m, a focal distance of 328 mm is obtained. By using an incident radiation of 9.71 keV, a spot of 0 480 × 5 17

m has been detected at the focus, with a measured gain of 367 [49].

3.3.2. Silicon Planar Refractive Lenses with Minimized Absorption A recent development of refractive lens fabrication comes from the application of microelectronics technology [50]. Although, in the actual state of the art, a submicrometer beam has not yet been produced by using this technology, nevertheless, the results are promising. The transmission and, therefore, the gain of the CRL, discussed earlier, is limited by: (i) attenuation in the bridges between adjacent individual lenses, (ii) significant attenuation in the outer parts of the CRL, and (iii) roughness of hole surfaces. These limitations can be overcome by using a dedicated refractive profile obtained with modern advanced technologies. The silicon microfabrication technique allows the preparation of a dedicated refractive profile by removing passive parts of material, where phase variation is a multiple of 2 within the lens thickness, the so-called kinoform lenses. A planar parabolic lens with minimized absorption is composed of N pairs of segments cut from a parabolic cylinder. Planar microelectronics technology allows a number of individual lenses to be fabricated onto a single wafer, as with a normal CRL, to shorten the focal distance. The lens with the aperture A = 150 m consists of N = 5 individual lenses, each comprising p = 10 pairs of segments. It has been tested at the undulator (ID22) beamline at ESRF by using beam energy of 18 keV [51]. The focal distance is 80 cm, and the focal spot (FWHM) is 1.8 m with a gain of 19. Taking into account the detector point spread function of 1.2 mm, a deconvoluted submicrometric spot size is obtained.

3.4. Total Reflection Optics: Kirkpatrick–Baez Mirrors An interesting optics to focus hard X-rays is given by grazing incidence total-reflection mirrors. A pair of elliptical mirrors arranged in a crossed mirror geometry (Kirkpatrick– Baez configuration) is able to focus X-rays to submicron size. In Figure 8, a schematic diagram of the system installed at beamline 8C of the storage ring in the Photon Factory (Japan) is shown. The X-ray beam passing through a Si(111) symmetric double crystal monochromator (CM) is focused by the elliptical mirrors. The radiation incident on the first mirror (M1  is vertically focused. The reflected beam from M1 is incident on the second mirror (M2 , which horizontally focuses the beam. To achieve a focused image, it is necessary to satisfy the focus equations at the center of the first and second mirrors: 1 1 2 + = A p1 q1 R1 sin ;1

1 1 2 + = p2 q2 R2 sin ;2

691

X-Ray Microscopy and Nanodiffraction q2

F

p2

M2 M1 CM S

q1

p1

Figure 8. Schematic diagram of the optical system of two crossed Kirkpatrick-Baez curved mirrors to form a demagnified image of a X-ray source.

where p1 , q1 , p2 , q2 are clear, from Figure 8, R1 and R2 are the mirror radii of curvature, and ;1 and ;2 are the angles of glancing incidence for the first and second mirror, respectively. For the simple case of cylindrical mirrors, the following conditions are fulfilled: p2 = p1 + s

q2 = q 1 − s

where s is the distance between mirror centers. When multilayer mirrors are used, it is additionally necessary to satisfy the Bragg equation for each mirror: 2d1 sin ;1 = 2d2 sin ;2 = m where d1 and d2 are the periods of the multilayers, and m is the reflection order. Combining the equations given above, one arrives at the following expression:   K − s
1 p1 d1 R 2 s = 1+ d2 R1 p1 K1 where K1 is the magnification of the first mirror (K1 = q1 /p1 ). In general, the magnification of the second mirror K2 = q2 /p2 is different from that of the first one, and, for the special case of identical radii and periods, one can find [51] 1 K1 = K2 Kirkpatrick and Baez developed this crossed mirror geometry to eliminate astigmatism of a single spherical (or cylindrical) mirror used at the glancing incidence. The spatial resolution of the Kirkpatrick and Baez system was limited mainly by spherical aberration. Therefore, the mirrors with elliptical cylinder shapes instead of spherical or cylindrical ones are used to remove spherical aberration. The beam size at the focusing point obtained by Iida and Irano [52] is 0 7 × 0 9 m2 , with a flux of 7 × 105 ph/s/mA. The advantages, with respect to other focusing optics, are that small aberration optics can be designed, the focusing properties do not depend on the incident X-ray energy, some types of mirror figures can be manufactured with high accuracy, and the working distance from the focusing element to the sample can be relatively long.

Recently, a crossed mirror system has been developed at the ESRF [53]. In this case, two orthogonal Pt-coated silicon substrates are bent into an elliptical shape to fit the experimental condition, such as focal distance and focusing. At about 140 m from the source, the dimensions of the monochromatic beam of 19 keV energy were defined by precise slits 0.2 mm × 0.25 mm wide. This beam was first reflected by a 170-mm long vertically focusing platinumcoated mirror set at 3 mrad grazing incidence. Then it was focused horizontally onto the sample by the 96-mm-long second mirror. The measured FWHM were 0.2 m horizontal and 0.6 m vertical. These values were bigger than both the ideal source image and the diffraction-limited spot sizes. Vibrations were clearly identified as a major contribution to blurring. The gain was estimated to 3 5 × 105 . At Spring-8, (Harima, Japan) a similar system has been installed, and it provides a spot size comparable with that produced in Grenoble. However, the bent system is quite different, since the parabolic shape of the mirrors is obtained during the fabrication process, as schematically illustrated in Figure 9. As can be seen, in this case, a parabolic-shaped mold is prepared and a plane SiO2 plate is pressed and bent against the surface of the mold. Afterward, the bent SiO2 plate is plane polished. When the plate is removed from the mold, it returns elastically to its original shape, and a parabolic surface is finished on the upper (polished) plane. The advantages of this fabrication process are: (i) a compact system due to the absence of the bender system, and a small focal length as a consequence; (ii) small roughness due to the plane polishing instead of a spherical polishing; (iii) smaller slope error than a spherical polishing.

3.5. X-Ray Waveguide In 1974, Spiller and Segmuller [54] demonstrated, for the first time, the propagation of the X-rays in a waveguide composed of a BN film sandwiched between two layers of Al2 O3 , but no flux enhancement was demonstrated. Exploiting previous studies made by Bedzyk et al. [55] on the X-ray standing waves created above a reflecting mirror surface, in 1992, Wang et al. [56] studied the resonanceenhanced effect that takes place inside a two layer system. In 1993, Feng et al. [57] reproposed the three layer structure of Spiller, with different materials (Si–polymide–SiO2 ), and they showed the higher enhancement effect created inside with respect to the two layers structure of Wang. Only in 1995, Feng et al. [58] and Lagomarsino et al. [18] independently proposed the use of this waveguide configuration

Figure 9. Sketch of the fabrication process steps to realize an parabolic mirror from a mold.

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X-Ray Microscopy and Nanodiffraction

for X-ray microbeam production. Since then, different configurations and different layer materials were used to obtain higher and higher efficiency in the submicrometer beam production. Up to now, the most efficient waveguide [59] is composed of Mo–Be–Mo, and it is able to produce an X-ray beam with one dimension of 37 nm, with a gain of 100. Very recently a first attempt of a two-dimensional waveguide able to compress the X-ray beam in two dimensions, has been proposed by Pfeiffer et al. [60]. Even if the actual efficiency is still not competitive with others X-ray optics, significant improvements are expected in the near future. Furthermore, the unique characteristics of the beam produced by the waveguides, which will be illustrated in the following sections, allow experiments that were otherwise unfeasible.

3.5.1. Resonance Effect in a Two-Layer Waveguide

A12

X

R12

A12

R12 T21

θ A23~T12 α1 High Z substrate (Cr) Low Z (C)

D=

Z

 2 sin 1

(3.12)

where 1 is the incident angle at the interface between medium 2 and medium 3, and it differs from the incident angle ; (Fig. 10) because of the refraction in medium 2. It should be remarked that the internal 2–1 interface does not provide total reflection of the incident beam on it. As a consequence, T12 and R21 beams have different amplitudes. Moreover, the phase shift between them is
 
2 2 2d sin 1 ≈ 2d1  )=   
 1/2 2 2d ; 2 − 2c = 

where d is the thickness of medium 2, and 1 = ; 2 − 2c has been used. When the incident angle ; is such that the phase shift ) is a multiple of 2, the constructive interference between T12 and R21 creates an enhancement of the field inside medium 2; a resonance effect takes place. This occurs when ) = m2 ⇒

  d = ≈  1/2 2 m 2 sin 1 2 ; − 2c

(3.13)

where m is a positive integer. When a beam traveling in a medium is reflected at an interface with another medium with a higher refraction index, it acquires a phase shift of  [8]. Therefore, when T12 and R21 interfere constructively, then the R12 and T21 beams interfere destructively. As a consequence, there is a minimum in the reflectivity curve for each ; fulfilling Eq. (3.13), as it is shown in Figure 11.

R

1,0 0,8

Reflectivity R

An X-ray beam impinging on a surface at an angle smaller than a certain critical angle is totally reflected. The interference between the incident and reflected beams creates a standing wave above the surface with a period D, depending on the incident angle and on the wavelength. In the case of a thin film on a substrate, the interference between the reflected beam from the surface and the reflected beam from the substrate gives rise to new effects, e.g., the Kiessig oscillations. Quite relevant in this regard is the occurrence of a resonance effect inside the deposited film if this has an electronic density smaller than that of the substrate. To better analyze this effect, one should consider the structure in Figure 10. The incident, reflected, and transmitted amplitudes of the “elementary” electric field are indicated as Aij , Rij , and Tij , respectively, where “ij” refers to the interface where the radiation impinges. (e.g., Rij represents the amplitude of the field that impinges from medium i on the interface ij, and it is reflected back in medium i). Given a thin layer of a light material (e.g., C) deposited on a layer of a heavier material (e.g., Cr), then the following situation occurs [56]: since the critical angle of the lighter material is less than the critical angle for the heavier material, for any wavelength , there is an interval where the incident angles ; are larger than the critical angle for medium 2 and smaller than the critical angle for medium 3. This interval also is extended by the fact that the field refracted inside medium 2 reaches the interface with medium 3 at an angle 1 < ; (Fig. 10).

For any of such angles ;, the X-ray beam can penetrate in the lighter material but is totally reflected from the heavier one. Therefore, the radiation is trapped inside medium 2. The interference of the incident (A23 ) and reflected (R23  beams creates a standing wave inside medium 2. The period D is given by

130 nm C 0,6

Cr substrate

0,4 0,2

Medium 1 (air) R23

T12 T23

R21

Medium 2

d

Medium 3

Figure 10. Schematic picture of the incident (A) reflected (R) and transmitted (T ) beams at the interfaces of a structure composed of thin film of a light material deposited on a layer of a heavier material.

0,0 0,12

0,14

0,16

0,18

0,20

0,22

0,24

0,26

0,28

Incident angle [deg]

Figure 11. Numerical simulation of the reflectivity curve from the structure represented in the inset. Deep minima occur each time a resonance effect takes place. Calculations are made for an incident energy of 13 keV.

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X-Ray Microscopy and Nanodiffraction

It is interesting to consider the periodicity D of the standing waves created in the film due to total reflection at the 2–3 interface. Since    D= (3.14a) ≈ A 1 = ; 2 − 2c 2 sin 1 21

air

d m

(3.14b)

Figure 12 shows a schematic representation of an X-ray waveguide able to produce at the exit, a submicrometer vertical beam with a horizontally unchanged input condition (planar waveguide). A very thin cap layer has been added on top of the two-layer structure described before. In this example the bottom and top layers are made of Mo, and the intermediate layer is made of Be. At angles more grazing than the critical angle for the air– Mo interface, the incident radiation is totally reflected and only an evanescent wave can travel inside the cap–Mo layer, and it is transmitted in the Be layer (guiding layer). To have maximum efficiency, the cover Mo layer must be very thin (few nanometers) to allow a substantial part of the incoming beam to penetrate. It is important to note that the incident angle 1 at the Be–Mo interface is smaller than the external incident angle ; because of the refraction inside the Be layer. As in the two-layer system described in the preceding discussion, a resonance occurs for the incident angles that satisfy Eq. (3.13). In correspondence of the resonant modes, minima take place in the reflectivity curve, due to the destructive interference out of the guiding layer, as will be discussed. Figure 13 represents the calculated intensities for the first three resonant modes excited in the structure of Figure 12. An important consequence of the guiding process described is that the incident beam impinging on the waveguide is squeezed in one dimension (the vertical dimension in this case) to the thickness of the Be layer, i.e., to fractions ER θ high Z (5 nm of Mo) α1

2α1

high Z (20 nm of Mo) SiO2 substrate

II mode

100

0 III mode 0

3.5.2. Three-Layer Waveguide: Production of Submicrometer X-Ray Beam

low Z (74 nm of Be)

150

50

(3.14c)

and the standing waves have nodes at the interfaces air– medium 2 and medium 2–medium 3. For each ; fulfilling Eq. (3.13) or D fulfilling Eq. (3.14c), a resonant mode is created inside medium 2.

Mo

I mode

From Eq. (3.13), it comes out that in resonance conditions: D=

Be

200

Field intensity

 D≈  1/2 2 2 ; − 2c

250

500

1000

Depth [Angstrom]

Figure 13. Numerical simulation of the field inside the waveguide represented in Figure 10, for the first three resonances.

of a micron. Thus, the X-ray beam is compressed and the flux density is increased by orders of magnitude [18, 58], as in an optical waveguide. As pictorially represented in Figure 12, two beams exit from the waveguide simply considering reflection at the top and bottom interfaces of the guiding layer. The splitting angle between the two beams is 21 and, therefore, the splitting angle increases as the incident angle increases. A waveguide similar to the one represented in Figure 12 has been tested [61] on the microfocus beamline “ID13” at ESRF by using a monochromatic beam at 13 keV. As is clear from Figure 14, the waveguide beam travels parallel to the surface, while the reflected beam travels at an angle ; with respect to it. Figure 15a and b show both the reflectivity and the waveguided intensity propagated through the Be layer of the waveguide as a function of the external incident angle. The dots in the curve represent the points measured when using a pin diode at a fixed position (close to the waveguide exit) vs the incident angle. In Figure 15a, the ordinate represents the reflectivity R of the waveguide normalized to the incident signal. In Figure 15b, the ordinate represents the flux exiting from the waveguide (also normalized to the incident flux). Since the waveguide beam and the reflected beam travel in different directions, separated by ;, a slit in front of the pin diode (Fig. 14), allows to distinguish, at a certain distance from the waveguide, the two beams by simply translating up and down the detector-slit system. The perfect correlation between the angular position of the maxima of intensity at the exit that corresponds to the excited modes (transverse electric [TE] modes∗ ) and the reflectivity minima confirms that only the efficient coupling gives rise to an exiting beam. To analyze, in detail, the structure of the different modes exiting from the waveguide, a charged-coupled device (CCD) camera was positioned at 840 mm from the waveguide, and the images for the different resonant angles were

E

Figure 12. Schematic representation of the X-ray waveguide, which produces a beam with the vertical dimension equal to 37 nm.

∗

Due to the linear horizontal polarization of S.R., the electric field of the incoming beam and of the multiply reflected beams in the waveguide stays normal to the incidence plane. Therefore, only the TE modes propagate [8].

694

X-Ray Microscopy and Nanodiffraction θTE0

θTE 1

t=2s

t=2s

θTE2

θTE3

θTE4

t=2s

t=5s

Reflected beam Monochromator θ Wg beam

x

θ Detector Slits

reflected beam

Waveguide

Figure 14. Schematic representation of the set-up used on ID13 at ESRF to test the waveguide.

recorded. The experiment was carried out on the optics beamline BM5 at ESRF with a 17 keV monochromatic incident beam. The results are reported in Figure 16. The five pictures in this figure represent the first five modes exiting from the waveguide for the resonant angles defined by Eq. (3.13). At the same time, in each picture are recorded the direct, the reflected, and the guided beam. The intensity in the direction of the incident beam must be attributed to the presence of higher harmonics, which are reflected by the channel-cut monochromator and which pass through the substrate. By rotating the waveguide, the direction of the incident beam always remains the same, while the reflected beam goes up as the incident angle increases. The guided beam always appears midway between the incident and the reflected beams since its direction is parallel to the (a)

Reflectivity

0.60

0.40

0.20

(b)

WG transmission [%]

0.15

beam optimized TE0 t = 60 s incident beam direction +

t = 40 s

Figure 16. Images of the first five resonant orders exiting from the waveguide recorded with a Frelon CCD camera with 10 m pixel size at a distance of 840 mm from the waveguide end. On the bottom of each picture, the acquisition time is reported. The inset on the left represents the TE0 mode optimized by using a higher acquisition time.

waveguide surface. This is exactly true for the TE0 , while, for the higher resonances, two beams exit from the waveguide whose median direction is tangential to the waveguide. This is consistent with the pictorial view in Figure 12 of two beams propagating in the waveguide. Except for the first resonance, it is clear from Figure 16 that the angle between the two beams exiting from the waveguide increases as the mode number increases. In the first picture, representing the TE0 mode, instead, only one beam is resolved, because, in this case, the splitting angle is comparable to the divergence of each beam. A detailed study of the guided beam images shows (see the inset of the TE0 mode in Fig. 16) an nonhomogeneous horizontal profile. This speckle structure is due to the presence of unpolished beryllium windows on the beamline, which deteriorate the quality of the incident beam. The same inhomogeneity cannot be appreciated on the incident and reflected images in Figure 16 because of saturation. The interference fringes above the reflected beam are a typical phenomenon in total reflection from rough mirror surfaces [62] and are of no concern in the specific physics of interest here.

3.5.3. Mathematical Representation of the Field Inside the X-Ray Waveguide

0.10

0.05

0.00

guided

The field amplitude E n [63] at a certain distance x from the middle of the nth layer of a multilayer structure, can be calculated by means of the following expression: 0.13

0.14

0.15

0.16

0.17

Incident angle [deg]

Figure 15. (a) Measured reflectivity R of the waveguide structure as a function of the incident angle and (b) measured intensity I exiting from the waveguide terminal in the direction tangential to the guide surface. Both spectra are normalized to the incident signal and are taken at 13 keV photon energy. The line connects the measured points as a guide for the eye.

E n x = En e−ikz z eikx x + EnR e−ikz z e−ikx x

(3.15)

where En and EnR are respectively the incident and reflected field amplitudes in the middle of nth layer. In the particular case of a perfectly symmetric waveguide with symmetrically injected radiation (i.e., in the direction parallel to the waveguide) as occurs for an optical waveguide, the modes inside the guiding layer (e.g., Be) must

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X-Ray Microscopy and Nanodiffraction

be odd or even functions. Indeed, in this particular case, E2R  = E2 eiC , in Eq. (3.15) and the field inside the Be can be written: C

C

C

E 2 x = E2 ei 2 e−ikz z eikx x e−i 2 + eikz z eikx x ei 2  

C C = 2E2 e−ikz z ei 2 cos kx x − 2 

C C 2 sin 1 x − = 2E2 e−ikz z ei 2 cos  2

(3.16)

where 1 is the incident angle at the Be–Mo interfaces and x
zc , the intensity distribution is well described by Fraunhofer diffraction or far field approximation. In this regime, for a coherent source, the intensity distribution becomes the square modulus of the Fourier transform of the field distribution of the source  2  2 E xeikx x dx Iz x = −

where E 2 is the electric field inside the guiding layer and exiting from the waveguide. In the particular case of a symmetric (and symmetrically illuminated) waveguide, the source field distribution E 2 at z = 0 is given by the sin-like and cos-like functions of Eq. (3.17a and b) for x inside the Be layer (−d/2 ≤ x ≤ d/2) and zero otherwise. In this simple case, the Fourier transform, giving the field distribution in the Fraunhofer region, is feasible analytically. For the first (m = 1) TE0 mode, e.g., one obtains  x −ikx x e dx cos Ez x = E2 d − d2
  d2 x −ik sin 1 x = E2 cos dx e d − d2 

+d −d +d −d eiA 2 − e−iA 2 iE2 eiA 2 − e−iA 2 + =− 2 A+ A− 

d 2




(3.20)

where the notation A± = ±/d − k sin 1  has been introduced and where the second equality arises because kx = k sin 1 , with 1 being the angle between the wavevector k and the x = 0 plane. The resulting intensity distribution is

    sin2 A+ d2 sin2 A− d2 + A+2 A−2     2 sin A+ d2 sin A− d2 (3.21) + A+ A− 2

Iz x = Ez x =

E22

Various observations are now in order. The first two terms in the square brackets are sinc functions, giving rise to two main peaks, while the third term is an interference contribution. The main peaks are located at an angle max deter1 mined by the condition A± = 0, and, therefore, sin max =± 1

  =± kd 2d

One can notice that this relation coincides with the resonance condition Eq. (3.13). This observation formally substantiates the pictorial view of Figure 12 according to which two beams (related to the E and E R components of the field inside the Be layer) exit each one forming an angle 1 with the x = 0 plane. The FWHM of each main peak can be simply calculated from Eq. (3.21), and, for a given distance z from the end of the waveguide, it is given by )theo z =

 z 2d

(3.22)

Performing the same treatment leading to Eq. (3.20) for a generic mode in Eq. (3.17), it can easily be demonstrated that Eq. (3.22) stays valid at any resonance order. On the other hand, one finds that the two main peaks are separated by a distance z )xm z = m d which increases with the order m of resonance. For the first order TE0 , the interference term in Eq. (3.21) is positive for x = 0 and, therefore, it represents a constructive interference in this position. This positive contribution around x = 0 has a width of the order of )xint z = z/d. As a consequence for this TE0 mode, the two main peaks cannot be resolved, since they are “joined” by this additional peaky structure in the middle.

3.5.5. Coherence The waveguide is an optical device capable of producing a microbeam with a high degree of coherence. Indeed, the constructive interference that occurs inside the guiding layer in resonance condition necessarily involves only the coherent part of the incident beam. This is clear from the mode structure of the field inside the C layer, which implies strong interference phenomena. However, several effects can degrade this coherence, like roughness and scattering from impurities inside the carbon. An experiment performed at beamline BM5 at ESRF has shown that these effects do not deteriorate the coherence of the guided beam. The experiment consisted in the analysis of the intensity distribution of the waveguided beam in the far field. To this purpose, a CCD camera was used at 840 mm distance from the waveguide end, as it is schematized in Figure 14. Spectra were taken for the waveguide adjusted to the angles at which the resonant modes give rise to an exiting beam. Experimental profiles of the images recorded for the first five modes (TE0 –TE4  are presented in Figure 17, together with the theoretical calculations. These latter are based on Eq. (3.21) and are similar for the higher order modes describing the intensity distribution far away from the waveguide. In particular, at a distance z ≫ zc (zc = d 2 / is the distance of transition between the Fresnel and Fraunhofer diffraction region), the field is the Fourier transform of the field at z = 0 and, therefore, supposing a perfect coherence of the beam at the waveguide exit, the intensity can be written as 2   2 ¯ ikx¯ x¯ d x¯ = E I coh x = Exe −

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X-Ray Microscopy and Nanodiffraction

derived as G=

Iout = Iin

TF Hd/2 F SH

= 2T

S d

(e)

The highest measured gain (G = 100) has been obtained for a waveguide composed of a 74 nm Be layer on a 20 nm Mo and covered with 5.5 nm Mo on silicon substrate, working at 13 keV.

4. X-RAY MICROSCOPY TECHNIQUES AND APPLICATIONS

(d)

In the previous section, we have analyzed, in some detail, the main X-ray optics that can provide submicrometer beams. In the following, an overview of applications is given. In Section 4, microscopy (i.e., imaging) techniques are reviewed. In Section 5, nanodiffraction is taken into consideration. (c)

4.1. Experimental Considerations and Instruments Here we want to describe the basis of the most relevant techniques and instrumentation used in microscopy and nanodiffraction.

(b)

4.1.1. Contact Microscopy

(a)

–3

–2

–1

0

1

2

3

Diffraction angle [mrad]

Figure 17. Comparison of the angularly dependent intensity distribution at 840 mm from the waveguide terminal (dots) with the calculated diffraction pattern (solid line) at 13 keV photon energy for the different modes a) TE0 , b) TE1 , c) TE2 , d) TE3 , and e) TE4 .

where the electric field E, just at the exit, has the sin-like and cos-like form, as in Eq (3.17). A high degree of coherence between the two beams is proved by the good agreement between the experimental result and the calculation for the TE modes and, in particular, for the TE0 one.

3.5.6. Gain Experimentally, we can measure the transmission T of the object as the ratio between the photon flux F ′ at the exit of the waveguide and the incident photon flux F . With the known values S and H for the vertical and horizontal dimensions of the incident beam and with the known FWHM size S ′ = d/2 of the exiting beam, the experimental gain is

The simplest, and more ancient, X-ray microscopy technique is the contact one. The sample is put in tight contact with a sensitive material and is illuminated by an X-ray beam as much collimated as possible. In the first studies, which date back to the beginning of past century [64, 65], the sensitive material was a photographic plate, substituted later by a photoresist layer that allowed for improved resolution [66]. The image is formed through photoabsorption process: the pattern recorded on the photoresist is a map of the variation of a photoabsorption coefficient in the sample. After development, the photoresist, if positive as the poly-methylmethacrylate (PMMA), the most commonly used, is thinner, if the received dose is higher, i.e., in the more transparent regions of the sample. The recorded pattern can then be viewed by optical or electron microscopy and, more recently, by atomic force microscopy. The resolution r that can be reached with contact X-ray microscopy is mainly limited by diffraction effects, if h is thickness of the specimen r ≈ h1/2 , where  is the wavelength. Resolutions of the order of 0.1 are typically reached with this method. Another effect that deteriorates resolution is the penumbra effect due to the angular divergence of the incoming beam.

4.1.2. Projection Microscopy The working principle of the projection microscopy is quite simple: a point source of X-rays, most often generated by a focused electron beam, project the shadow of the sample put at a distance D1 from the source onto a screen at a distance D2 . The magnification M is then simply given by M = D1 + D2 /D1 ≈ D2 /D1 , if, as usual, D2 ≫ D1 . The resolution limit generally is dominated by the source size s. Substantial improvements have been obtained by using a high brightness field gun scanning electron microscope

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X-Ray Microscopy and Nanodiffraction

(FEG–SEM), which allowed a spot as small as 0.2 [11]. As the source becomes smaller and smaller, approaching the ideal limit of a point source, the coherence increases, and Fresnel diffraction must be taken into account. This makes the image more complicated than considering photoelectric absorption as only responsible for the contrast. In fact, the phase of the e.m. wave and its modifications must be considered, but the more complex treatment results, in fact, in most of the cases, in an enhanced contrast, as will be discussed later. The same principle of projection microscopy has recently been applied to the X-ray waveguide, illustrated in Section 3. In this case, the X-ray beam from a S.R. source is compressed in a plane or in a wire through an interferometer effect and comes out from the waveguide with dimensions of the order of few tens of nanometers and a high degree of coherence. Because the high coherence of the beam generated by the waveguide implies necessarily phase contrast effects, an illustration of the potentialities of this method will be given after a discussion on the phase contrast principles.

4.1.3. Full Field and Scanning Microscopy With the advent of S.R. sources, the projection microscopy that uses point source X-rays generated in modified SEM lost much of its importance. In fact, the high brilliance of S.R. allowed more sophisticated and more performant instrumentation. This is the case of full field X-ray microscope, or simply X-ray microscope (XM), and scanning X-ray microscope (SXM). The modern instruments are in great majority based on FZPs, even if the first instrument of this kind was developed by Kirkpatrick and Baez with mirrors [17]. An instrument that served as a model for many others is the Gottingen XM, installed in the Bessy S.R. source in Berlin [67]. In brief, a condenser FZP receives the radiation and focuses it at the object plane (Fig. 18). The radiation can be, in principle, both monochromatic or polychromatic. Since the FZP focalizes different energies at different distances, a pinhole in the object chamber can act as an energy passband filter. The degree of monochromatization is given, in this case, by /) ≈ D/2d, where D is the diameter of the condenser FZP and d, the pinhole diameter. In Figure 18, the instrument at the beamline ID21 of the ESRF

Zone plate objective lens

is depicted, which receives radiation already monochromatized. The central stop of the condenser blocks the direct S.R. A micro FZP, with much finer structures and reduced numerical aperture than the condenser FZP, generates a magnified image of the object in the image plane. When working in the soft X-ray region, the spatial resolution can be as good as few tens of nanometers. In the hard X-ray region, however, the limit is for the moment around 150 nm, mainly due to difficulties in fabricating high resolution FZP for high energies (see Section 3.1). Scanning X-ray microscopes use the high-resolution micro FZP to focalize a monochromatic beam on a submicrometer spot at the sample (Fig. 19). The sample is then mechanically scanned in front of the beam with high accuracy, and the transmitted beam is measured by a high sensitivity detector. The two instruments are complementary and each have their advantages and disadvantages. The micro FZP, which in the XM is behind the sample, has a quite limited efficiency, therefore, only a small percentage of the photons arriving on the sample contributes to the image formation. Moreover, a low efficiency two-dimensional detector such as a photographic plate or a CCD camera must be used. This increases the dose delivered to the sample, which in case of sensitive materials like the biological ones, is quite deleterious. In SXM, instead, no optical elements are behind the sample, and a high efficiency detector can be used. Moreover, secondary effects such as fluorescence radiation can be measured, increasing the quantity of available information. As a counterpart, SXM requires much higher brilliant and coherent sources than the XM to keep the acquisition time reasonable, and, also, the spatial resolution is, in general, worse than that obtainable with XM. Fresnel zone plates are very efficient in the soft X-ray region, but, at higher energies, their fabrication at high resolution becomes more difficult, and S.R. beamlines working in the hard X-ray region equipped with other optical elements such as refractive lenses [16], waveguides [18] or Kirkpatrick-Baez mirrors [17] are in operation.

4.1.4. Bragg Magnifier An alternative way to improve spatial resolution is to use, after absorption from the sample, diffraction from an analyzing crystal set in a strong asymmetric reflection [68, 69]. A sketch is shown in Figure 20. The image magnification M is given by the ratio M = sin / sin  where  and  are, Sample, raster scanned

Spatially resolving detector Crystal monochromator

Sample Crystal monochromator

Zone plate condenser

Beam steering multilayer Undulator

Figure 18. Schematic view of a full field X-ray microscope (from the Web site of Beamline ID21 at ESRF: www.esrf.fr, with permission from J. Susini).

iransmission detector

Aperture Zone plate objective lens

Undulator Fluorescence detector

Figure 19. Schematic view of a scanning X-ray microscope (from the Web site of Beamline ID21 at ESRF: www.esrf.fr, with permission from J. Susini).

699

X-Ray Microscopy and Nanodiffraction Detector

Monochromator

Sample

Analyser

Figure 20. Schematic view of the Bragg magnifier: the beam from the sample impinges on the analyzer crystal at a glancing angle 1 and the diffracted beam leaves the sample at an angle 2 with respect to the surface. The magnification M is given by the ratio: sin 2 / sin 1 .

respectively, the angles that the diffracted and the incident beams form with the analyzing crystal surface. To predict the exact behavior of the analyzing crystal, the dynamical theory of diffraction in asymmetric conditions should be used [70]. If magnification and high resolution is desired in both directions, as is usually the case, two crossed crystals with surfaces in the horizontal and vertical planes must be used. The resolution that can be reached with this system depends not only on the magnification M, but also on the X-ray source geometrical characteristics, and values of the order of 0.2 have been claimed [68, 69]. It is worth noting that this method can work easily for hard X-rays, as in [69], where an energy of about 23 keV has been reached, with magnification up to 100 in both directions. Since the detector, as in projection microscopy, is at a certain distance from the sample and analyser, phase contrast effects due to propagation are visible. This will be discussed later in more detail.

4.2. Modes of Operation and Some Examples of Applications

deals with biological studies. However, increasing attention and more instruments are now dedicated to the hard region of the spectrum, where thicker samples can be examined and tomographic studies (see below) can be carried out. An example is given in Figure 21, which shows a part of a giant chromosome in a wet state, where the contrast and the spatial resolution are really remarkable [71]. It is worth stressing the importance of the fact that with X-ray microscopy, it is possible to view whole cells in their wet state. However, care must be given to the problem of radiation damage. The important parameter in this respect is dose. Dose is defined as the ratio of energy absorbed in the volume illuminated and the mass pertaining to that volume. It is important to keep the dose as low as possible because, after a certain threshold value, sensitive material is damaged. This also is a serious limitation regarding spatial resolution. Calculations show [2, 11] that, independent on the microscopy method in use, the dose D is proportional to d −4 , where d is the minimum feature size detectable (in other words, the spatial resolution). This implies that for biological studies, the ultimate limit in resolution is not given by available flux or optics perfection but by radiation damage. A method that can reduce the threshold dose for radiation damage is to freeze the samples in a state that closely resembles their natural environment (cryomicroscopy). Other methods instead reduce the dose necessary to detect a feature with a given signal–noise ratio. If the radiation damage represents the limit in resolution, enhancing the contrast will also result in improving achievable spatial resolution. One method relies in tagging a probe with nanoparticles of some heavy elements (generally gold) that enhance both absorption and phase contrast [1]. This technique, for example, has been used to immunolocalize proteins or nucleic acids in mammalian cells [72]. This is

In this section, we will give an overview of the principle methods in use in microscopy, together with some examples of applications. These are not intended to be exhaustive, but just to give an idea about the potentialities of the different methods.

4.2.1. Absorption Contrast Photoelectron absorption contrast is the usual way matter is imaged in X-ray radiography. The contrast comes from a nonhomogeneous distribution of matter with a different linear absorption coefficient and/or with different thickness. As discussed in Section 1, the absorption coefficient is not a continuous function of energy (or ) but has discontinuities in correspondence with absorption edges. This is usefully exploited to enhance contrast. The most relevant example is in biological matter where the major components are water and organic C compounds. In the wavelength range (the so-called “water window”) between the absorption edges of oxygen ( = 2 34 nm ⇒ E ≈ 530 eV) and C edge ( = 4 38 nm ⇒ E ≈ 283 eV), the radiation is weakly absorbed by water but strongly absorbed by organic matter, resulting in good contrast. Most important, this allows hydrated and even (initially) living cells to be studied. The problem of radiation damage will be discussed later. The presence of the water window is the main reason why many microscopes work in the soft X-ray region where most of the activity

Figure 21. X-ray image illustrating the distribution of pores on the surface of the nucleus of a human mammary epithelial tumour cell (T4). The nuclear pores were labeled by using primary antibodies to a protein from the nuclear pore complex. The contrast has been enhanced by silver. Magnification: 2400x with 20 nm pixel size. Wavelength: 2.4 nm. (Reprinted with permission from [72], W. Meyer-Ilse et al., J. Microscopy 201, 395 (2000). © 2000, Blackwell Publishing.

700 a very important problem in a post-genomic era, where the main task is to determine function of genes. The same labeling with gold also has been used to study tumoral cell [71]. In [71], the investigators show images of the distribution of nuclear pores on the surface of the nucleus of a human mammary epithelial tumor cell at a pixel resolution of 20 nm (Fig. 21). Even if, at the moment, we are not aware of direct applications in the field of nanobiotechnologies, we are confident that these microscopy techniques will give, in the near future, significant contributions as characterization tools in this important emerging field. Beyond biological studies, X-ray microscopy in absorption mode also has been applied in other fields, for example, characterization of micro- and nanoelectronic materials. The advantages with respect to optical microscopy are not only better spatial resolution but also the possibility to visualize buried layers invisible, both to photons in optical microscopy and to electrons in SEM microscopy, which can only probe surface topography. The open environment accessible to hard X-rays also allows probing devices while active. This is the case, for example, of electromigration (EM) in passivated Cu interconnect structures, where the high values of current density can induce voids in the conducting microstructures or stress [73]. High spatial-resolution (100 nm) studies of EM in passivated Cu interconnections have been carried out in real time [74] with a full field XM. With this technique, it has been possible to record, in real time, an image sequence that allows the study of the time evolution of mass transport that led to formation of voids and interruption of line. Not only was the spatial resolution significantly higher than what could be obtained with optical microscopy, but the X-ray measurements were conducted on passivated layers not accessible to other types of microscopy, therefore, on conditions corresponding to those of the real devices. In environmental science, X-ray microscopy has been used to study intimate structures of colloidal systems in soil. As an example, the comparison between untreated and thermal-treated soil has shown a drastic change in the soil structure with a buildup of larger particles resulting from the collapse of finer particles due to dehydration, which followed the thermal treatment [75].

4.2.2. Magnetic Circular Dichroism Atoms with a net magnetic moment show, in the vicinity of absorption edges, a strong circular dichroism, i.e., a relevant difference in absorption coefficient between the left and the right circularly polarized radiation. This difference also appears if the sample is illuminated with circularly polarized photons and the magnetic moment of the sample is directed parallel or antiparallel to the beam direction. This happens, for example, for the L edges of Fe, Co, Ni, etc. Thus, magnetic domains with up and down magnetization can be visualized by X-ray absorption microscopy with a circularly polarized beam at high spatial resolution (≈30 nm). This is of high value in studying magnetic nanostructures. A distinct advantage of X-ray microscopy is that the images can be recorded at varying external magnetic fields, allowing a detailed study of the switching of magnetic domains. In this study, a matrix of 1 × 1 dots was prepared in a 0.4 nm

X-Ray Microscopy and Nanodiffraction

Fe/0.4 nm Gd × 75 multilayer. Images at different external magnetic fields were taken, showing both the domain structure at nanometer scale and the collective switching behavior [76]. Another advantage of X-ray microscopy with respect to other techniques is the element specificity that can give information about magnetic coupling mechanism. Due to the temporal structure of S.R., time-dependent effects, such as spin dynamics, also can be studied [77].

4.2.3. Phase Contrast Phase contrast arises when the coherence of the beam allows visualization of spatial variations of the real part of the refraction index, which induces changes in the phase of the incident radiation. (Photoelectron absorption is related to its imaginary part, see Section 1.) To understand how phase variations can be imaged, start with a plane wave eikz traveling along the z direction, impinging on a inhomogeneous sample with refraction index rx3 y = 1 − x3 y − ix3 y and thickness t. If we define 5x3 y = 2t/ and

x3 y = 2t/, the sample transmission function qx3 y is defined as qx3 y = ei5x3 y− x3 y , and just after transmission from the sample at z0 , the wavefunction can be expressed as: f x3 y = eikz0 qx3 y. The intensity I1 at z0 is I1 = f x3 y2 = e−2 Therefore, any information about the phase is lost. To have detectable contrast due to phase variations, it is necessary either that the beam, which has interacted with the sample, be superposed to a reference beam or the small angular deviation due to the change of phase be detected. In the first case, the interference between the two beams carrying a definite phase difference gives rise to the contrast. The superposition can be obtained in different ways. Here we will mention three methods: Zernike phase contrast [78], Gabor in-line holography [79], and differential interference [80]. Diffraction from an analyzer crystal can be used in the case where the small angular deviation has to be detected. 4.2.3.1. Zernike Phase Contrast Schmahl [1, 81, 82] described in detail the way the Zernike phase contrast method can be applied in the soft X-ray regime. The basic point is the presence of a phase plate put in the back focal plane of the micro FZP, which modifies the phase of the zero order incoming radiation. This causes interference with the first order diffracted radiation from the FZP in the image plane. Careful calculations have been carried out to evaluate the contrast and the dose delivered to the sample [83]. It turns out that the contrast is higher (and in some cases much higher) than the absorption contrast for nearly all wavelengths [1]. More important, especially for sensitive materials like the biological ones and the macromolecules, there is, in general, a strong reduction in the dose required to detect a feature with a given signal–noise ratio, up to three orders of magnitude at shorter wavelengths. 4.2.3.2. In-Line Holography With the term in-line holography, we mean all the techniques based on the interference buildup due to simple propagation in free space. If we start from the transmission function, which gives the field amplitude just after the object, the field fz at the image plane

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X-Ray Microscopy and Nanodiffraction

after a distance z from the object is given by the Fresnel integral: fz x3 y =

i · eikz  2 2 qX3 Y eiHx−X +y−Y  I/z dX dY z

where coordinates X3 Y refer to the object plane and x3 y to the image plane. Everything can then be developed in Fourier space, but we do not want to go further in mathematical description. The physical meaning is that, because of Fresnel diffraction, an interference field builds up, caused by the superposition of diffracted waves and a reference wave constituted by the direct field not affected by the object to visualize. This type of holography can be implemented both in the soft [84] and in the hard [85–87] X-ray regions and also with laboratory sources [82, 84]. Both a plane wave [84–86] and a spherical wave [11, 87, 88] can be used, the key point being their degree of coherence. In the first case, there is no magnification, therefore, often, the detector limits the spatial resolution, which, at best, is of the order of 1 micron; in the second case (projection mode), a magnified image of the sample is produced, and better resolution can be obtained. In holography, three regions can be distinguished: the near field, the Fresnel (or intermediate) field regime, and the far field [11, 89]. To be in one or another region (whose borderlines are only qualitatively defined) depends on the relative magnitude of the defocus distance D and a length scale defined as DF = d2 /, where  is the wavelength and d is the linear size of the object under illumination. The defocus distance, however, has a different definition for holography with plane waves or with spherical waves. In illumination with plane waves, the defocus distance Dp w . is just the sample–detector distance D2 (Fig. 22). Only for D2 = Dp w = 0, the image is completely free from phase contrast effects. With spherical waves, the defocus distance Ds w = D1 ∗ D2 /D1 + D2 . Note that Ds w approaches Dp w

for D1 ≫ D2 . With this in mind, we can describe the effects that arise in the different regions with an unified treatment where the only variable is the defocus distance. In the nearfield regime D ≪ DF , and the contrast is proportional to z Source

Sample

a)

D2 ≈ D F

D1

Detector

Source Sample

b)

D1

D2

Figure 22. Schematic view of the in-line holography with (a) a plane wave and with (b) spherical wave illumination. The defocus distance in the first case is D ≈ DF , in the second case, is Ds w = D1 ∗ D2 /D1 + D2 .

the second derivative of the phase. The contrast is, therefore, particularly enhanced at the edges, while slowly varying phases do not produce practically detectable contrast. A sharp edge is imaged in the near-field with a characteristic black–white fringe. The contrast also is proportional to D, but quite soon, the image becomes more complex entering in the intermediate regime. It also is worth to point out that  in this regime acts only as a multiplicative factor in the intensity but not in the structure of the fringes [11, 88]. Therefore, polychromatic sources also can be used, and the image contrast will result, in this case, from a weighted sum of the incident spectrum. The image in the near-field regime closely resembles the object, but, to quantitatively determine a projected density, a phase reconstruction process must be carried out. In the approximation of a thin, homogeneous sample, the phase reconstruction can be carried out by using a single defocused image [90]. Figure 23 shows a phase contrast image of latex spheres (a) and the phase retrieved projected density (b) [11]. In the Fresnel region (D ≈ DF ), a more complex system of fringes set up. In [84], is shown an example of a hologram taken in the soft X-ray region ( = 2 4 nm) of zymogen granules, together with its reconstructed image obtained through a computer simulation of the illumination of the hologram with the original reference wave. By using spherical waves and the projection mode, spatial resolution able to visualize nanostructures can be attained. As an example, Figure 24 shows a phase contrast of a gold test pattern on a Si nitride mask 0.3 wide taken with the X-ray waveguide [87], which provides a highly coherent beam, very well approximated, in the vertical plane, by a spherical wave. Three images are shown for different values of D1 and different defocus distances and magnifications. Note how the fringes change from one image to the other, and note also how the defect in alignment, barely visible in the SEM image taken at low resolution, is very clearly seen in the X-ray images. The spatial resolution was determined in this case by simulation and resulted to be in the order of 100 nm, mainly due to a nonideal experimental apparatus. This experiment made use of a planar waveguide (see Section 3.5), which has a strong drawback to allow magnification only in one direction (in the cited example, the vertical one). It has been demonstrated that resonance conditions also can be reached in a two-dimensional waveguide [60], providing a coherent point source in the nanometer range. Its implementation as a very powerful projection microscope is in progress. In the Fresnel region, the phase retrieval procedure is much more complex than in the near field. Several algorithms have been proposed to achieve quantitative information. The one developed by Coene et al. [91] needs at least two images of the same object taken at different defocus distances (however, in general, more than two images are taken at different distances to improve the reliability of the phase retrieval). This phase retrieval procedure also can be used in tomography to get a three-dimensional (3-D) image of the sample. An example of such reconstruction in three dimensions will be presented later, when discussing tomography. The last regime of phase contrast, the far-field or Fraunhofer regime, is that commonly accessed in Bragg diffraction, where constructive interference of atoms give rise to strong peaks. The far-field regime is now of great interest

702

X-Ray Microscopy and Nanodiffraction (a)

10 µm

1m (b)

10 µm

0.5 m (a)

(b)

0.2 m (c)

Figure 24. Phase contrast images of a gold test pattern taken with the X-ray waveguide at three different magnifications. Energy ≈13 keV. Note the different fringe patterns for the three magnifications and the jump due to a fabrication defect.

same order of the outermost zone width. The superposition of the two images gives rise to a pattern consisting of linear fringes with spacing s = A/a, where A is the distance of their back focal planes to the detector, and a is the lateral displacement of the two zone plates. [80]. Images of low-absorbing nanostructures show impressive improvement of contrast with respect to standard zone plates.

Figure 23. a) Images of a cluster of ≈9 latex spheres taken with a projection microscope obtained by a modified SEM. Ta target with 15 keV excitation voltage so that the main X-ray energy corresponds to the TaM lines at 1.7 keV. Distance electron beam focus sample = 933 . Both phase and absorption contrast effects are visible. b) projected density of a) obtained by phase-retrieval method. Reprinted with permission from [11], S. C. Mayo et al., J. Microscopy 207, 79 (2002). © 2002, Elsevier.

because analysis of Fraunhofer diffraction from single, nonperiodic objects could lead to significant improvements in spatial resolution. We will discuss more deeply this subject in the section dedicated to nanodiffraction. It is interesting to note how the borders between microscopy and diffraction lose their rigid character, and the two apparently different methods approach each other to become the expression of the same physical effect. 4.2.3.3. Differential Interference Contrast In this recently developed method [80], interference is created by superposition of two images of the same object created by a zone plate doublet composed by two zone plates very slightly displaced, one with respect to the other. The displacement along the optical axis must be much smaller than the image distance. The lateral displacement must be smaller or of the

4.2.3.4. Analyzer Crystal Analyzer crystals in Bragg diffraction after transmission of the beam through the specimen can be used for two purposes: magnification of the image or detection of small angular deviations. The first case has been mentioned in a previous paragraph with the term Bragg magnifier. As sketched in Figure 20, the incoming beam impinges on the analyzer crystal at a very grazing incident, while the diffracted beam leaves the crystal at a much larger angle. From dynamical diffraction theory [70], the angular acceptance in the grazing angle incidence is quite large, therefore, small deviations of the incoming beam are accepted. This means that the phase contrast present in the magnified images is due essentially to the same mechanism of the in-line holography discussed earlier [92]. If small deviations of the incoming X-ray beam due to sharp variations of phase have to be detected, a geometry with a less grazing incident beam, or even an asymmetry geometry with the incoming beam at a larger angle with the crystal surface than the diffracted beam must be adopted. As a consequence, the acceptance is much narrower, and, at a certain Bragg angle, the deviated beams are not at all diffracted, while, for a slightly different angular position, only the deviated beams are visible. This is the basis of diffraction enhanced imaging [93, 94]. However this technique, because of the lack of large magnification factors, has the spatial resolution limited by the detector resolution (about 1 ).

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X-Ray Microscopy and Nanodiffraction

4.2.4. Fluorescence Microscopy One of the most useful and exploited properties of X-rays is its ability to excite fluorescence radiation characteristic of the element probed and, therefore, carry out a nondestructive chemical characterization of materials. The basic mechanism has been illustrated in Section 1. A chemical mapping of the sample, therefore, can be obtained by simply focusing the X-ray beam with some optics (zone plates, refractive lenses, Kirkpatrick and Baez mirrors, waveguides, etc.) and then measuring an X-ray fluorescence spectrum (or a portion of it), while scanning the sample in front of the beam. An interesting example [95] has been obtained at ESRF (ID22) on single human ovarian adenocarcinoma (IGROV1) cell treated with 5 M of iododoxorubicin. The contour map of the treated freezeddried cancer cell was mapped with a 14 keV polychromatic “pink” excitation. The spatial resolution is essentially that of the optical element used, while the spectral resolution is dictated by the fluorescence detector. The natural evolution of X-ray fluorescence analysis is on one side spectromicroscopy, on the other side, fluorescence tomography. Examples in both fields are given in the following Sections.

4.2.5. Spectromicroscopy In spectromicroscopy, the chemical mapping is not obtained by measuring fluorescence radiation but rather using the large absorption coefficient changes that occur at the absorption edges of elements. Chemical sensitivity is then obtained, recording images with a scanning or a full-field microscope at a number of different energies appropriately selected to differentiate the chemical components of the system. Each image pixel provides a certain absorption spectrum, which is the weighted sum of the absorption spectra of the different components. The entire image sequence can then provide, after careful analysis, the chemical composition at each sample location, or, conversely, the spatial distribution of each element, with the accuracy of the optical system in use [96], which can be as good as a few tens of nanometers. If a single element has to be put in evidence, subsequent acquisition of two images, one just before and one just after the absorption edge of the element, followed by digital subtraction of the two images, can strongly enhance the contrast and give a clear picture of the spatial distribution of that particular element. Applications of spectromicroscopy span in many fields, e.g., bioscience, material science, environmental science, cultural heritage, etc. Many examples can be found in [4]. Here, we just mention studies of protein adsorption sites on polymers [96] and studies on variations of cross-linking of gels [97]. A very interesting feature of X-ray spectromicroscopy is that it is not only sensitive to chemical composition but also to the chemical environment (valence state) [98]. This comes from the fact that different chemical states can change the energy position of the absorption edge. If images are recorded through an absorption edge with sufficient energy resolution, fine changes in the spectrum can give precise information about the speciation of the chemical components and their spatial distribution in a nondestructive way and under different environmental conditions. This is perhaps one of the most

powerful and unique features of X-ray microscopy. Again, interested readers can find many examples in [4].

4.2.6. Tomography Up to now, we have considered the sample as a thin slab, and the density information was limited to an x3 y plane. However, samples of interest are often three dimensional, and the full object reconstruction is, in many cases, the final goal. Soft X-rays are limited in this respect because their penetration in most materials is very shallow. Hard X-rays are, therefore, needed, and it is not by chance that X-ray microtomography has been developed mainly in these last years, where third generation hard X-ray S.R. sources like ESRF in Europe, APS in the United States, and Spring8 in Japan became operative. In X-ray microtomography, absorption images are taken for many different angular positions of the samples around an axis perpendicular to the X-ray beam propagation direction, and the images are then computer processed to reconstruct the 3-D object density distribution. To have reliable results, a complete rotation of 360  should be accomplished in steps as small as fractions of a degree. The simplest way is to use a collimated beam and plane waves, but the resolution, is in this case, limited by the detector to ≈1 . To have better spatial resolution, either focusing optics or a projection geometry can be used. With focusing optics, care must be taken that the sample thickness be smaller than the depth of focus. Another critical aspect for resolution is the mechanical perfection of the rotation axis: wobbles during rotation can introduce severe artifacts and difficulties with proper reconstruction. Tomography can be done with high resolution even with table-top laboratory sources now commercially available. In some cases, even tomography at 80nm resolution on microelectronics integrated circuits (IC) is claimed. Very recent extensions of tomography rely on the possibility to reconstruct the object following its chemical composition (fluotomography) or recording phase contrast images (holotomography). In fluotomography, sections of the samples are taken, measuring the fluorescence spectrum for each section and for each beam position. Difficulties in subsequent analysis are related to the fact that absorption coefficient of both the primary radiation and of fluorescence radiation depends strongly on density distribution, which is the unknown to be found. Special algorithms must, therefore, be developed in order to reliably reconstruct the 3-D composition [99, 100]. A similar problem is encountered in holotomography, where the spatial distribution of the phase and of the absorption coefficient must be reconstructed in 3-D, and, therefore, reconstruction procedures must be followed [101–103], The phase reconstruction algorithm proposed by Cloetens et al. [103] uses three different distances for each phasecontrast image to reconstruct the phase for a single section. This procedure must then be repeated for all the sections. The advantage of holotomography with respect to tomography based exclusively on absorption is that 3-D structures of light materials can be reconstructed. Remarkable examples can be found in the literature. For example, 3-D reconstruction of a polystyrene foam fragment showing the internal structure of the cell can be found in [102].

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With an SXM, instead of measuring the transmitted intensity or the fluorescence, photoelectrons generated by the focused X-ray beam can be detected by suitable electron detectors, allowing accurate spectroscopic information related to composition and electronic state. Alternatively, the sample can be illuminated uniformly by the X-ray beam without any X-ray optics, and the photoelectrons then are detected through a suitable electron optics, which raster the sample at high-spatial resolution. This last can reach even values as high as tens of nm. In both cases, a high vacuum environment is required, and only a thin surface layer is probed. Moreover, if artifacts are to be negligible, flat and slightly conductive samples must be used. Very interesting applications recently have been implemented with this technique in the study of carbon nanotubes [104]. The electronic structure of multiwalled carbon nanotubes has been studied with a spatial resolution of the order of 90 nm with a scanning photoelectron microscope (SPEM) at the Elettra S.R. light source in Trieste (Italy). The microscope is equipped with FZPs and a hemispherical electron analyzer and can work both in spectroscopy mode and in imaging mode. The carbon nanotubes were aligned perpendicularly on a Si substrate. The SEM measurements and SPEM analysis were made on cross sections obtained by cleavage. Figure 25 shows a SEM micrograph. The valence band and the C1s spectra were measured with SPEM on spatially selected regions along the tube axes. The valence band spectra and the C1s photoelectron spectromicroscopy image are presented, respectively, in Figure 26a and b. The locations where the valence band spectra were measured are indicated in the spectromicroscopy image. It clearly appears that valence band spectra from the tips of the nanotubes have a substantially larger spectral intensity in the energy range about 1 eV below the Fermi edge with respect to spectra taken from the nanotube sidewalls. Taking into account a number of considerations not reported here, the authors interpret these results, assuming a higher density of dangling bonds at the spherically curved tips with respect to the defective density at the cylindrically curved sidewalls.

Ambient (vacuum)

Substrate (St) X3 . 500

270201

Figure 25. Scanning electron microscopy of a cross section of the aligned multiwall carbon nanotubes. (Courtesy of S. Suzuki).

(a) 1 2 3 4 5 6 7 8

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4.2.7. X-Ray Photoelectron Microscopy

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Figure 26. a) Valence band photoemission spectra in the vicinity of the Fermi level from spatially selected regions. b) C1s photoelectron spectromicroscopy image, where locations of the measurements are indicated by white spots. Note the difference of spectra taken in the 1 and 7 positions with respect to the other locations, indicating a larger spectral intensity about 1 eV below the Fermi edge. (Courtesy of S. Suzuki).

This example illustrates well the potentiality of a technique that combines high spatial resolution with spectroscopic capabilities.

5. X-RAY NANODIFFRACTION The wavelength of hard X-rays ( ≈ 0 1 nm) is close to the interplanar spacings of crystalline materials. As already mentioned in Section 1, this allows the study of structural properties through Bragg diffraction. In nonhomogeneous samples, or in samples subjected to nonuniform stress, it can be of high interest to probe the local structural properties with high spatial resolution. This can be particularly important in interface problems or in ultra large scale integration (ULSI), where reduction of critical dimensions down to the order of 0.1 and correlated technological steps can introduce unwanted strain field deleterious for device performance. Emerging nanotechnologies also requires development and improvement of structural microprobes. Since nanodiffraction requires hard X-rays, optics such as Kirkpatrick and Baez mirrors [17], waveguides [18], refractive lenses [16], and FZP are used. Tapered capillaries also are used, but for resolutions not better than 1 , and we will not treat them here.

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5.1. Monochromatic Beam We first discuss Bragg nanodiffraction applications, which use a monochromatic beam. Examples with waveguides, FZP and mirrors will be reported.

5.1.1. Waveguides As already mentioned, planar waveguides compress an incident X-ray beam of several tens of microns, reducing one of its dimensions of three orders of magnitude. A narrow line beam of a few tens of nanometers is thus produced. The beam is highly coherent and has a divergence of about 1 mrad. In the phase-contrast paragraph, we presented examples of high-resolution phase-contrast images obtained with this optics. We will discuss its application in nanodiffraction and, in particular, in studies of local strain measurements. To have high strain sensitivity, a high angular resolution is needed, which is incompatible with high angular divergence. However, significant angular divergence is only in the plane of compression (i.e., the vertical one) and the horizontal high collimation of the S.R. source is left unmodified by the waveguide. We now describe in some detail, a nanodiffraction experiment aimed at measuring local strain in Si substrate due to SiO2 isolation micropatterning. Let the sample coordinates be x, y, and z, with the xy plane coinciding with sample surface, x in the vertical direction, y in the horizontal one, and z perpendicular to the surface (Fig. 27). The diffracting plane is yz, and the high spatial resolution of the Waveguide is along x. The projected beam size on the sample surface along y depend on the horizontal slit aperture and the Bragg angle. For typical incident beam size of 50 × 50 2 and Bragg angle of 20 [Si(400) reflection at 13 keV photon energy], the projected beam size along y is of the order of 150 . Along x, the beam size depends strongly on the distance D1 between the waveguide exit and the sample. Due to divergence, the

x y

z

synchrotron radiation

D2

D1

Figure 27. Schematic arrangement of X-ray microdiffraction with the waveguide in the projection geometry. The coherent beam from the waveguide impinges on the sample surface at an incident angle J and is diffracted at an angle 2;. The divergence in the vertical plane allows a magnification M given by (D2 + D1 /D1 ≈ D2 /D1 . The oxide stripes lying in the plane xy can be analyzed with high spatial resolution along x. Diffraction takes place in the horizontal plane where collimation of S.R. beam allows high angular resolution. The diffracted beam is imaged on the CCD detector for each angular position of the rocking curve (see text). (Reprinted with permission from [106], S. Di Fonzo et al., Nature 403, 638 (2000). © 2000, Macmillan Magazines Ltd.

beam size increases about 0.1 every 100 of increasing distance. With such conditions, two modes of operation are possible: the projected mode and the scanning mode. The projected mode is very similar in concept to the projection microscopy as illustrated, with the only difference being that the two-dimensional detector will record the image of the diffracted beam instead of the image of the transmitted one. As in microscopy the magnification M is given by the ratio M = D1 + D2 /D1 , where D1 and D2 are, respectively, the waveguide sample and the sample–detector distances. The projection mode relies on the coherence properties of the beam and requires a two-dimensional detector whose pixel size P is correlated to the spatial resolution r through the simple relation: r = P /M. In the scanning mode, instead, the sample is simply scanned in front of the waveguide exit and diffracted intensities are recorded for each sample position by a standard detector (e.g., scintillator). In this case, the spatial resolution is given by the vertical beam size on the sample, which critically depends on distance D1 due to the beam divergence. Both operation modes have been implemented. The samples consisted of narrow stripes aligned along y. In the first studies, LOCOS (local oxidation of silicon) SiO2 stripes were analyzed, successively shallow trench isolation (STI) structures, more convenient for high-density packaging, were studied. Strain was measured by recording the diffracted intensity as a function of deviation from the Bragg angle [known as a rocking curve (R.C.) or diffraction profile]. The R.C. of a perfect crystal is the convolution of the theoretical Darwin reflectivity, with the instrumental function, which takes into account energy spread, incident beam divergence, reflectivity of a monochromator if any, etc. [70]. A strained lattice will give a R.C. with enhanced intensity or even additional peaks on the low-angle side of the Bragg peak if the strain is tensile (higher lattice parameter) or in the high-angle side in the case of a compressive strain. With our geometry (symmetric reflection), the zz component of the matrix strain along the surface perpendicular has been probed. The spatial localization of strain is straightforward in the case of the scanning mode: a R.C. is recorded for each spatial position of the sample in a vertical scan, and strain information are extracted, analyzing the individual R.C. In the case of projection geometry, the diffracted intensity is recorded on a two-dimensional detector for each angular position of the sample during a R.C., then the images are analyzed to extract a diffraction profile for each vertical small portion of the sample within the allowable spatial resolution. In both cases, a number of diffraction profiles must be analyzed. In the analysis, the penetration of X-rays into the crystal must be taken into account. The diffraction profiles, therefore, reflect a strain information integrated in depth, and a strain depth profile should be obtained eventually. To do this, an analysis based on dynamical diffraction theory for deformed crystals [105, 106] has been carried out, which ideally divides the crystal into thin layers, each with its own strain value. The integrated strain in a LOCOS structure about 5 wide and another about 1 wide are presented in Figure 28, while Figure 29 reports the strain depth profile distribution for three locations a distance 100 nm from each other in the narrow structure [107]. These measurements were carried

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X-Ray Microscopy and Nanodiffraction (a) 2.0

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Lateral position (µm) Figure 28. Spatial variation of depth-integrated strain under SiO2 stripes of different widths w: a) w ≈ 5 , b) w ≈ 1 . Strain is indicated by the length of the variation of reciprocal space vector with respect to the undeformed substrate. Both in a) and b), an inset shows an atomic force microscopy height profile of the Si-oxide structures. Reprinted with permission from [106], S. Di Fonzo et al., Nature 403, 638 (2000). © 2000, Macmillan Magazines Ltd.

out in the projection mode. A narrow (0.2 ) STI structure also was measured in the scanning mode. In this case, the spatial resolution was of the same order of magnitude of the structure, therefore, a single R.C. centered on the structure described its strain. A few words about the complexity of measurements and the comparison between projected and scanning modes: a key point to preserve spatial resolution is the proper alignment between a waveguide beam and

εzz × 104

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stripes along the y direction. A slight misalignment (typically larger than 1 mrad) could seriously deteriorate spatial resolution. The projected mode deserves, in principle (and also in practice), better spatial resolution. Distance D1 is not so critical for resolution and values around few mm are acceptable. However, it requires higher brilliant sources and sensitive two-dimensional detectors. Scanning mode is less demanding on this point of view, but, on the counterside, the distance D1 is much more critical and cannot be larger than a few tens of microns if high spatial resolution is desired. Anyway, in case of polycrystalline materials, the projection mode cannot be applied. We, therefore, used the scanning mode in studies of biomaterials where the aim was to study bone reconstruction at the interface with a Zr orthopedic prosthesis covered with bioglass. Interesting results about structural differences between native bone and bone reconstructed at the interface were found [108]. The method applied in biomaterials can also be applied to study single fibers. In particular, local studies on single cellulose fibers have shown the additional information that can be extracted by using micro- and nanodiffraction, with respect to other conventional techniques like electron microscopy. Cellulose molecules aggregate, forming small crystals called microfibrils (a few nm in diameter). The arrangement of the microfibrils in single fibers (average diameter 20 m) is of great variability, and the main interest concerns the disorder, both inside the microfibrils and in their arrangement in the fiber. In each fibrils of a Kevlar fiber, the polymer chains form hydrogen-bonded sheets. A nanodiffraction experiment [109] on a single Kevlar fiber showed that the sheets present a radial organization around the fiber axis and form pleats along the fiber axis, with a periodicity of 0.5 and an angle of 170 between neighboring pleats. The method described can be applied virtually to any monocrystalline or polycrystalline material, including hard and soft condensed matter. In particular, microdiffraction with waveguides is of particular interest where interface problems are of relevance, such as coatings, damaged surface layers, etc.

5.1.2. Zone Plates An interesting experiment that studied local strain at grain boundaries in colossal magnetoresistive films by using a microbeam generated by a phase zone plate can be found in [110]. The measurements, taken at a spatial resolution of about 0.35 , show that strain relaxation takes place at the grain boundaries, yielding a lattice constant different from that at the interior of the grain, thus supporting magnetic measurement results. Other microdiffraction experiment using zone plates were carried out on Cu interconnects where EM can have deleterious effects on device performance. The measurements, carried out with 0.2 spatial resolution, aimed at study strain close to the voids and hillocks created by EM. A strong influence of Ti adhesion layer on the Cu microstructure has been put in evidence [111].

Depth (µm)

Figure 29. Strain depth profile for three adjacent lateral positions along x (see Fig. 29), distant 100 nm from each other in the 1 wide SiO2 stripe of Figure 34. (Reprinted with permission from [106], S. Di Fonzo et al., Nature 403, 638 (2000). © 2000, Macmillan Magazines Ltd.

5.2. White Beam A completely different approach has been followed by Larson and coworkers [112]. They used a submicrometer white beam produced by Kirkpatrick and Baez mirrors

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to generate a Laue (i.e., transmitted) diffraction pattern from the sample (either a polycrystalline or a monocrystalline one) recorded by a two-dimensional detector. With a white beam, many reflections are excited simultaneously, and the complex pattern is computer analyzed to determine Miller indices of reflections, crystallographic orientation, and stress/strain tensor [113]. To obtain high spatial resolution, a Pt wire 50 thick is scanned in front of the sample at a distance DP t ≪ DD , where DD is the wiredetector distance. The wire is stepped in submicrometer steps, and at each step the pattern is recorded. The difference between two successive steps allows correlation of the individual CCD pixel to the element volume that contributed to this pixel differential intensity. The entire ensemble of the differential intensities allows reconstruction of the map of the individual volumes, which, along the path of the beam, gives rise to the Laue diffraction pattern. The method has been applied to analysis of hot-rolled Al alloy, giving quantitative information about grain sizes and inter and intragranular rotations. In the case of a cylindrically bent Si crystal, it provides depth-resolved measurement of elastic strain tensor. White beam also is used in another recently developed technique, which allows tracking of individual nanocrystals by using diffracted X-rays (diffracted X-ray tracking [DXT] [114]. With this method, it has been possible to follow the dynamics of single DNA molecules [115]. The principle is the following: artificial nanoparticles, consisting of a Si–Mo multilayer were fabricated by using deposition techniques and reactive ion etching to reduce the lateral dimensions of the nanocrystals to about 30 nm [114]. The nanoparticles, dispersed in a gel, were illuminated with collimated white Xrays and diffracted spots from individual nanocrystals were imaged with a CCD camera. The rotational movements of the nanoparticles were then monitored following the displacement of the diffracted spots. In this way, the viscosity of supercooled liquid water at atmospheric pressure could be determined [114]. Even more interesting, single DNA molecules were attached to the nanocrystals, as depicted in Figure 30. In this way, the rotational motion ( and  degrees of freedom, see Fig. 30) of the DNA molecule could be monitored [115];  corresponds to angular displacement , and  to angular displacement J of the diffracted spots. By following the displacement of several diffracted spots, the investigators were able to conclude that the main motion was the  rotation, while the  movement was not detected and determined both the diffusion coefficient and the drift velocity of the nanoparticles. These two parameters essentially characterize the nature of the Brownian motion. It is expected that this technique can, in the future, give important information about biological processes in living cells.

5.3. Diffraction Microscopy A new field “diffraction microscopy” is now emerging where far-field diffraction of nonperiodic objects are analyzed to reach the ultimate resolution limit in X-ray imaging. The starting point is the oversampling concept [116]. In Bragg diffraction from a periodic array, strong Bragg peaks are generated where constructive interference takes place, while destructive interference keeps intensity at very low levels in

White X-rays (SR)

Diffracted X-rays

SH

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Figure 30. i) Schematic drawing of the diffracted X-ray tracking (DXT) method to detect motion of single DNA molecule in aqueous solution. The DXT traces the displacement of the single diffracted X-ray spot from the one-dimensional artificial Si–Mo nanocrystal, which is linked to the single DNA molecule. The diameters of the nanocrystal is about 30 nm and that of the DNA molecule, 2.5–3 nm. In the figure, the two rotational degrees of freedom  and  of the DNA molecule are indicated, corresponding, respectively, to angular displacements  and J of the diffracted spots. ii) The nanocrystals physically adsorbed on the Au–quartz substrate are used as a calibration of the system stability. iii) When the direction of the stacking period of the Si–Mo nanocrystals is parallel to the substrate, the diffraction spots cannot be monitored.

between the peaks. When crystal dimensions become smaller and smaller, the Bragg peaks become broader and broader and increasing intensity is found on Bragg peak tails. At the limit of nonperiodic objects, a continuous diffraction pattern is generated if the specimen is illuminated with coherent light. This continuous pattern can be sampled at frequencies finer than the Nyquist frequency, which corresponds to the inverse of the specimen size. At sufficiently small frequency, the phase can be retrieved by using an iterative algorithm [117, 118]. Experiments were carried out both in the soft [119] and in the hard X-ray region spectra [117, 120]. In [117], the experiment, which reached the record spatial resolution of 8 nm, was carried out at a wavelength of 0.2 nm (corresponding to an energy of about 6 keV) at the Spring-8 synchrotron light source in Japan. The sample, fabricated by electron beam lithography, consisted of two identical single-layered Ni nanostructure patterns separated by 1 in depth and rotated each other by 65 in plane. While in the SEM micrograph, only the upper pattern is visible, due to the small depth of focus of electron microscopy, the X-rays are capable of providing both the top and the bottom patterns, after reconstruction from the diffraction pattern. The microradiograph has a spatial resolution of 8 nm, but the two patterns are overlapped. Three-dimensional reconstruction also was carried out from a limited number (31) of two-dimensional diffraction patterns recorded at different angular setting of the sample from −75 to 75 in 5 steps. However, the spatial resolution of the 3-D reconstruction was of the order of 50 nm. The example reported here shows clearly the potentiality of the method that has, with respect to other forms of microscopy, the advantage of a higher penetration depth, which allows the nondestructive 3-D reconstruction. It is self-evident that this technique can give important contributions to biological

708 science where whole cells could be imaged at high resolution and in the nanoscience field, where, in principle, nanodimensional structures could be imaged nondestructively in three dimensions at atomic resolution.

6. CONCLUSIONS In this discussions, we have tried to give an idea about the state of the art of a research field that, though it has its origins at the beginning of last century, only in these last years started to show all its potentialities. In this respect, a virtuous circle has been set up: the big progress in S.R. pushed for the development in X-ray optics, which now leads to routine experiments in the nanometer range. On the other hand, the achievements reached in this field also pushed researchers to develop powerful and more brilliant laboratory sources that now can profit from the new X-ray optics available to reach deep submicrometer spatial resolution. We, therefore, look forward to more and more achievements and expansion of this field. In particular, X-ray microscopy and nanodiffraction will certainly be able to give significant contributions to nanoscience and nanotechnology. A new frontier is at the horizon: 4th generation S.R. sources such as the free electron lasers with pulses as short as a few tens of femtoseconds and peak brilliances 10 orders of magnitude higher than the present most powerful sources, are now under construction or are planned. The astonishing properties of these new radiation sources will literally open a new world (that of extremely short probes with extremely high brilliance), where experiments that are now impossible will become feasible. Fantasy at work.

GLOSSARY Absorption contrast The contrast in X-ray radiography and microscopy due to spatial variations in the sample of the imaginary part of the refractive index, which causes intensity modifications of the transmitted wavefield. Coherence Property of an electromagnetic field to maintain a specific relation of phase into a given spatial and temporal interval. Coherence length The extension of the spatial region where the X-ray beam can be considered as coherent. Coherent diffraction Diffraction from sampled regions smaller than the coherence length of the incoming X-ray beam. Phase contrast The contrast in X-ray radiography and microscopy due to spatial variations in the sample of the real part of the refractive index, which causes phase modifications of the transmitted wavefield. X-ray fluorescence microscopy X-ray microscopy technique based on the excitation of fluorescence radiation. Each element has characteristic fluorescence lines differing by their energies. X-ray photoelectron microscopy X-ray microscopy technique based on the analysis of photoelectrons generated by the incoming X-ray beam. X-ray spectromicroscopy X-ray microscopy technique based on the large variations of absorption coefficients at

X-Ray Microscopy and Nanodiffraction

absorption edges of the constituent elements. Requires image acquisition at different energies of the incoming X-ray beam. X-ray tomography Three-dimensional reconstruction of the sampled volume obtained with a large number of radiographies, both in absorption or in phase contrast mode, taken at different orientation of the sample with respect to the incoming X-ray beam.

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Encyclopedia of Nanoscience and Nanotechnology

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X-Ray Photoelectron Spectroscopy of Nanostructured Materials J. Nanda Stanford University, Stanford, California, USA

Sameer Sapra, D. D. Sarma Indian Institute of Science, Bangalore, India

CONTENTS 1. Introduction 2. X-Ray Photoelectron Spectroscopy 3. X-Ray Photoemission Spectroscopy Applications to Nanostructured Materials 4. Conclusion Glossary References

1. INTRODUCTION Photoemission spectroscopy has traditionally been a powerful technique for probing the electronic structure in both condensed and gas phase systems [1–4]. Apart from this, it can also provide useful information like elemental composition, chemistry, and oxidation states of surfaces, etc. [3]. This form of spectroscopy assumes an even greater significance in the case of nanomaterials, where the surface plays a dominant role in determining the material properties [5–7]. Despite the inherent limitations in applying this technique, especially for wide bandgap materials, there have been quite a number of studies in literature pertaining to X-ray photoemission spectroscopy (XPS) in nanomaterials [8–12]. Such studies can be broadly classified into two areas: 1) determining the detailed composition, nature of surfaces, and bonding of the nanostructured materials from both qualitative and quantitative analysis of the photoemission core levels; and 2) evolution of electronic structure as a function of size by studying the valence band/levels using both X-ray and ultraviolet photon energies. This article focuses primarily on the X-ray photoemission core level studies of various kinds of nanostructured materials like nanocrystallites (NC), nanotubes, and nanofilms. The organization of this article is as follows: Section 2 gives a brief outline of the various sample preparation techniques for XPS followed by Section 3, which has several subsections listing the XPS studies in a ISBN: 1-58883-066-7/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

wide variety of nanosystems, mainly semiconducting nanoparticle chalcogenides, oxides, metallic nanoparticles, nanotubes, and nanowires. Section 4 is the conclusion. However, we would like to point out that this chapter only discusses those systems where XPS has been used as an important tool for obtaining both qualitative and quantitative information, about the surface and bulk chemical nature and not merely as a characterizing tool used for chemical analysis.

2. X-RAY PHOTOELECTRON SPECTROSCOPY 2.1. Sample Preparation Techniques for XPS Most of the semiconducting NCs, with the exception of those belonging to IV–VI groups, show strong quantum size effect within a size regime of 1–10 nm [5, 6]; in the case of IV–VI NC, such size effects can be observed for sizes that are even larger by a factor of 4 to 8 [13]. Growing these crystallites, using methods of colloid chemistry, has been the most widely used approach. Normally, this is achieved by arresting the growth of the NCs in solution either by capping the surface atoms by organic moieties, which directly bond to the surfaces of the crystallites, or using stabilizers to control their growth [14–16]. There are also reports of growing nanocrystallites within the cavity of micelles or vesicles [17, 18]. Such chemical synthetic procedures often lead to the production of free-standing powder of NCs that are stable for a considerable period of time. The NCs then consist of a semiconducting core, surrounded by an organic shell. Typically, performing photoemission experiments on such materials, which have a wide bandgap material in the core surrounded by an insulating organic layer, leads to the charging up of the sample. This is because, in the case of wide bandgap materials and insulators during the photoemission, photoelectrons are continuously ejected from the sample. However, the charge neutralization does not occur within the

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 10: Pages (711–724)

time-scale of the experiment leading to the accumulation of positive charges on the sample. When this process continues over a period of time and sufficient charge accumulates on the surface, a dynamical equilibrium is reached at a certain potential difference between the sample and the ground, due to the electric field-induced charge compensation. However, the potential difference between the sample and the ground depends on the transport property of the sample and the photon intensity. Subsequent photoelectrons emitted from the sample experience the positive potential on the sample surface resulting in a shift of the kinetic energy of the emitting photoelectron to a lower value and thereby, resulting in an apparent increase in the observed binding energy of the photoelectron signals. In the case of highly insulating samples, the whole surface does not necessarily attain the same steady-state potential giving rise to an inhomogeneous charging. An inhomogeneously charged sample leads to a different extent of binding energy shifts for different parts of the sample, resulting in a broadening of the photoemission signal. Such binding energy shifts, and broadening of the signals make it difficult for any quantitative measurement of the absolute energy positions of the various levels and its comparison between different samples. In view of such problems, there have been a number of techniques reported in the literature for minimizing the effect of sample charging [19, 20]. One such method is to attach the individual nanocrystallites to a conducting surface like gold or aluminum using organic linker molecules like dithiols. The binding molecules place the nanocrystallites at approximately 10 Å from the conducting surface. Such distances supposedly allow the charge to tunnel between the nanocrystal and surface and thus compensate for the charging of the sample. Bowen Katari et al. [8] used this procedure for XPS study of CdSe NCs. The sample for photoemission was prepared by allowing the hexanedithiol to self assemble on the ion-etched gold evaporated on glass slides. These slides were then dipped into a solution of nanocrystals. The surfaces of the NC covalently bonds to the sulfur of the hexanedithiol. Another method for charge compensation in the case of wide bandgap NCs like CdS and ZnS has been reported by Nanda and Sarma [9, 21]. This method consists of dispersing the NC powders uniformly in a graphitic matrix. The graphitic powders form a conducting network on the top of the crystallites and thus, dissipate the charges accumulated on the surface of the NC. In some cases, it was found that the presence of graphite alone was not enough for the removal of the charging. In those cases, a low-energy electron flood gun was used in addition to the graphitic coating to further minimize the charging. The efficacy of this method is demonstrated in Figure 1(a) and 1(b) for the case of bulk ZnS sample. The Zn 2p3/2 core level spectrum of bulk ZnS powder and graphite is shown in the spectrum labeled (1). This spectrum clearly exhibits multiple features spread over a wide energy range. The spectral shape clearly shows the presence of inhomogeneous charging. For example, the most intense peak appears at 1025.7 eV, while there is a broad shoulder near 1022.5 eV binding energy. This is in sharp contrast to the fact that the Zn 2p3/2 signal from pure bulk ZnS is expected to have a single feature. While mixing the sample with graphite does help in reducing the extent of

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

(a)

Zn 2p 3/2

(2)

Sample Charging ZnS (3) (1)

Intensity (arb. units)

712

1020

1025

(b)

1030

Zn 2p 3/2

ZnS

1020

1023

1026

Binding Energy (eV)

Figure 1. (a) Zn 2p3/2 core level spectra of bulk ZnS showing various stages of charging leading to minimal charging position. (b) Similar spectra collected at various time intervals at minimal condition showing the consistency of the method.

charging, this procedure alone was not sufficient in eliminating the charging. A low-energy electron flood gun was therefore used to eliminate the remnant charging of the sample. The presence of the sample charging was monitored by the shift in the peak position as well as by the full width at half maximum (FWHM) of the Zn 2p3/2 spectrum for different conditions of the flood gun as shown in the spectra labeled 2 and 3 in Figure 1a; the narrowest energy width was found to be the no-charging condition. Figure 1(b) shows the charge-compensated Zn 2p3/2 spectra collected at different time intervals during which the experiment was performed. This overlapping Zn 2p3/2 spectra shows the reproducibility of minimal charging states for such wide bandgap samples. Consequently, all the core level and valence band spectra were collected while maintaining such consistent position of the core level.

3. X-RAY PHOTOEMISSION SPECTROSCOPY APPLICATIONS TO NANOSTRUCTURED MATERIALS 3.1. Semiconductor Nanoparticles: Group II–VI 3.1.1. Chemical Composition and Core-Level Chemical Shifts As discussed in the earlier part of the text, significant information about the chemical nature of the surfaces and composition of nanocrystallites can be obtained from the core

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

713

level analysis. X-ray photoemission spectroscopy is commonly used to study compositional analysis of materials. The XPS peak intensity from a single element for a homogenous sample is given by the expression [4]

CdSe

20000

SeO2

(1)

where n is the number of atoms per cm3 , F is the photon flux,  is the photoelectric cross-section,  is the detection angle,  is the mean escape depth, which is a function of the electron kinetic energy, A is the area of the sample probed,

is the efficiency of photoelectrons with full kinetic energy, and T is the transmission factor. Equation (1) can be written as

a) 18000 b)

cls/sec

I = nF A T

22000

c)

16000

d) 14000

n = I/S where

e)

1 nF  T

12000

(2)

S is called the atomic sensitivity factor. Some of these factors, like the photoelectric cross-section, can be calculated theoretically and others depend on the experimental conditions, the design of the spectrometer, and its electronic detection system. The relative atomic composition is then determined by simply dividing each atomic core level peak intensity by the sensitivity factor. However, in the case of nanocrystallites, since the mean escape depths are comparable to the size, the intensities were calculated by multiplying the intensities at each point by an exponential factor, which decreases with depth from the sample surface. Bowen Katari et al. have reported a detailed composition analysis of various sizes of CdSe NC [8]. In their study, the calculated ratio of Cd: Se was 1.02 ± 0.14 and it did not vary systematically with size. The surface of the CdSe NCs of various sizes was chemically passivated by tri-n-octylphosphine oxide (TOPO). The surface coverage of the ligands on the surface of the nanocrystals was determined by comparing the Cd and P peaks. The ratio of Cd/P for various NC sizes fits best to a 1/r curve, implying that the ligands are on the surface. Furthermore, the ratio of P/Cd showed a systematic decrease with an increasing crystallite size showing the correct trend. The percent coverage of ligands on the nanocrystals decreases as the nanocrystal radius increases, reaching saturation coverage of around 20%. Characterization of the surface coverage by alternative techniques like NMR on CdSe NC surfaces also confirms such a trend [22]. These workers also studied the surface of CdSe NC as well as surface modifications upon exposure to the atmospheric conditions. This was performed by carefully studying the Cd and Se core levels of the NC over a period of time. These samples showed an oxidized Se peak after a typical exposure time of around one day. The binding energy of the oxide peak appears at the same position as seen for Se oxide on bulk CdSe. Figure 2 shows the Se 3d core level of different sizes of CdSe NCs. One of the interesting findings of their study was the gradual appearance and disappearance of the oxide peaks as a function of time. The surface properties of the CdSe NC changes drastically by extracting the TOPO-capped crystallites in Pyridine. In this case, Pyridine replaces the ligand in the solution. The XPS survey scans

f)

10000 52

54

56

58

60

62

Binding Energy (eV)

Figure 2. Oxidation of the CdSe nanocrystallite surface as evident from the Se 3d cores: (a) 13.1 NC as prepared; (b–f) after 24, 48, 72, 96, and 120 h in air. The oxide peak re-emerges after a finite time interval. Reprinted with permission from [8], J. E. Bowen Katari et al., J. Phys. Chem. 98, 4109 (1994). © 1994, American Chemical Society.

then show the absence of P. However, the close-up scan of the P region shows a small peak indicating a small amount of P on the nanocrystal surface. The absence of N 1s signal even in the narrow energy window suggested that when the sample was kept under vacuum, pyridine desorbs leaving behind a clean nanocrystal surface. When such samples were exposed under atmosphere, the surface Cd atoms also get oxidized as shown by the Cd 3d core level in Figure 3. Nanda et al. [23] also carried out similar studies on CdSe NC thin films prepared by chemical and electro-chemical 60000

CdSe

CdSe CdOx

55000

CdOx

50000 b)

cls/sec

S=

45000

40000 a)

35000

30000 400

405

410

415

420

Binding Energy (eV)

Figure 3. Oxidation of the CdSe surface: Cd 3d cores (a) 15.9 NC as prepared showing no oxide Cd component. (b) Same size NC after a few weeks under atmospheric condition. Reprinted with permission from [8], J. E. Bowen Katari et al., J. Phys. Chem. 98, 4109 (1994). © 1994, American Chemical Society.

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

depositions [24, 25]. X-ray photoelectron spectroscopy measurements were carried out on these films using a monochromatic Al K radiation at a photon energy of 1486.6 eV. The Cd 3d core level spectrum of CdSe NC films with an average particle size of about 5 and 15 nm exhibited single features as shown in Figure 4a, suggesting only one type of Cd species present in the sample. The 3d level signal has a doublet structure due to the spin-orbit interaction resulting in Cd 3d3/2 and 3d5/2 levels at binding energies of 405.6 and 412.35 eV, respectively. The Se 3d spectra of the corresponding sample are shown in Figure 4b. The experimental Se 3d spectrum could be fitted with three Gaussian doublets with each individual doublet signifying different types of Se species. The three selenium species, in order of increasing binding energies of 53.5, 54.6 and 59 eV, correspond to the bulk, surface, and oxide of Se, respectively [23]. A detailed discussion on these different species will be presented later in the text. Detailed XPS core level studies of metal sulfide, semiconducting NC like CdS and ZnS have been carried out by various groups [9–11]. Nanda et al. have studied extensively the surface structure and bonding of various sizes of CdS and ZnS NC [9, 10]. Nanocrystallites of different sizes were synthesized using the same capping agent, 1thioglycerol, for both CdS and ZnS. By changing the ratio of sulfide to thiol, the NC sizes could be varied. The details of the sample preparation and characterization have been reported elsewhere [26]. The Zn and Cd core level spectra of different sizes of NC and bulk are shown in Figure 5a and b, respectively. When these levels are probed by relatively bulk-sensitive AlK radiation (h = 14866 eV, as in this case), one sees only a single peak feature in each of the spin-orbit components without any second Zn or Cd species, similar to the spectrum of bulk. In the case of NC, the Zn 2p and Cd 3d levels are slightly broadened and also shifted towards lower binding energy. These broadening in NC can arise due to a combination of factors ranging from the intrinsic size distribution of the synthesized NC

3d5/2

Cd 3d

(b)

CdSe-5

XPS

Intensity (arb. units)

XPS 3d3/2

Intensity (arb. units)

(a)

CdSe-5

CdSe-15

CdSe-5g

(1)

(2) CdSe-15g

415

410

(3)

405

Binding energy (eV)

60

58

56

54

52

Binding Energy (eV)

Figure 4. (a) Cd 3d core level spectra of 5- and 15-nm CdSe NC thin films. The spectrum labeled as CdSe-5 and 18 nm were chemically grown whereas CdSe-5g and 18g are electrodeposited on gold. (b) Se 3d core level spectrum of 5nm CdSe NC film. The circles are the experimental curve and the solid line is the overall fit. Dotted lines are the individual Se Components obtained from spectral decomposition as discussed in the text.

ZnS

2p 3/2

(a)

Zn 2p Core level 2p 1/2 ZnS-1.8 nm

ZnS-2.5 nm ZnS-3.5 nm

Intensity (arb.units)

714

Bulk ZnS

1020

1030

CdS

1040

1050

(b)

Cd 3d core level 3d 5/2

3d3/2 CdS-2.2 nm

CdS-4.5 nm

Bulk CdS 402

404

406

408

410

412

Binding Energy (eV)

Figure 5. (a) XPS spectra of Zn 2p core level of bulk ZnS and three NC sizes of 3.5, 2.5, and 1.8 nm, respectively. (b) Cd 3d core level spectra of bulk CdS and two NC sizes of 4.5 and 2.2 nm, respectively.

(about 8–10% in this case) to some degree of uncompensated charging as discussed earlier in the text. The S 2p and S 2s of the NC provide valuable information about the surfaces. The inset in Figure 6a shows the S 2p spectrum of bulk CdS, which can be described by two Gaussians having the same FWHM of 0.44 eV separated by a spin-orbit splitting of 1.2 eV, as shown by the solid line. On the contrary, the S 2p spectra of the NC are broad and appear to have multiple features as illustrated in Figure 6a for 2.2 nm CdS NC. A single Gaussian doublet in this case cannot describe the S 2p spectrum as shown in the figure for 2.2 nm CdS NC. A good fit to the experimental spectrum was obtained with three sets of Gaussian doublets with the 2p3/2 signal centered at 161.8, 162.9, and 163.9 eV, respectively. In each case, the spin-orbit splitting turns out to be 1.2 eV, the same as in bulk CdS. The spectral decomposition procedure as discussed above is further justified by the fact that the experimental S 2s spectrum of the same sample could be fitted well with three single Gaussians with their energy separations and intensity ratios being fixed to those between the three components observed in the case of S 2p, as shown in Figure 6b. The dotted lines in fitting are the spectra from individual sulfur species whose relative abundances are given by the areas under each of these curves labeled as (1), (2), and (3). The inset in this figure shows the S 2s spectrum of bulk CdS showing a single peak feature as expected. Based on the binding energy values of the various sulfur species, one can assign (1), (2), and (3) to the sulfur atoms belonging to the core, surface, and the thiols that cap the surface of the NC. This was further ascertained by recording the core levels using Mg K monochromatic radiation, which has a relatively high surface sensitivity. There was a marginal increase in the surface components compared to the aluminum X-ray source. Similar results were also obtained from XPS core

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

I

Expt. Fit

160

162

164

166

Binding Energy (eV)

Intensity (arb. units)

S 2p ZnS-1.8 nm

Intensity (arb. units)

CdS-2.2 nm

Expt. Fit Error

S2p Bulk CdS

Expt.

Intensity (arb. units)

S 2p

Intensity (arb. units)

(a)

715

Resi.

II I III

II 224

226

228

230

232

Binding Energy (eV) I

III

Fit II

Expt. Fit Resi.

S 2s ZnS-1.8 nm

Resi. III

160

162

164

166

Binding Energy (eV)

Figure 7. S 2p core level spectrum of 1.8 nm ZnS NC showing different sulfur species labeled as I, II, and III, respectively. The experimental spectra are shown by solid circles, the solid lines are the calculated fits. The triangles show the residual of the fit at each point. 160

162

164

166

Binding Energy (eV) Intensity (Arb. units)

(b)

Intensity (arb. units)

S 2s CdS-2.2 nm

Expt.

I

S 2s Bulk CdS

Fit

222

224

226

228

Binding Energy (eV)

Res. II III

222

224

226

228

Binding Energy (eV)

Figure 6. (a) S 2p core level spectrum of CdS–2.2-nm NC showing different sulfur species marked as I, II, III, respectively. Inset shows the S 2p core level of bulk CdS showing only a single sulfur species. The solid circles are the experimental points and the solid lines are the calculated fit. The triangles are the residuals at each point. (b) S 2s core level spectrum of 2.2 CdS NC showing different species of sulfur species labeled as I, II, and III, respectively. Inset shows the corresponding spectrum of bulk CdS showing only a single feature. The solid circles are the experimental points while solid lines are the calculated fit and the triangles show the residual of the fit at each point.

level studies on ZnS NC of various sizes synthesized by a similar procedure [9]. Sulfur species were also noticed in the case of 1.8 nm ZnS NC as illustrated in Figure 7. The inset shows the corresponding S 2s spectrum fitted using a consistent set of parameters as was discussed in the context of CdS NC. In another study, similar to those of [9] and [10], Winkler et al. investigated NC surfaces by recording the core

levels spectra employing high-resolution X-ray photoelectron using a synchrotron source [11]. They also reported various sulfur species in CdS NC of various sizes similar to those reported by Nanda et al. [10, 21]. However, they could vary the surface sensitivity by tuning the incident photon energy and thereby being able to vary the degree of bulk and surface contributions in the core level spectrum. The different NC samples studied were approximately 7, 4, and 2.7 nm in diameter and were capped by mercaptopropanoic acid. Interestingly, the Cd 3d5/2 core level spectra recorded with the maximum surface sensitivity (h = 445 eV) revealed the existence of two components at 406.1 (A) and 406.6 (B) eV, as shown in the inset of Figure 8. These two Cd groups are assigned to the Cd in the bulk (A) and Cd atom bonding to the sulfur atoms capping agent (B). The S 2p spectra of 7 and 4 nm particle sizes showed three features corresponding to the sulfur atoms belonging to the bulk, surface, and capping of the NC, shown in Figure 8(a) and 8(b). However, the S 2p-core level in the case of the 2.7-nm diameter sample could be fitted to only four spin-orbit doublets (labeled I–IV) as shown in Figure 7c. While the origin of the first three components are the same as the corresponding numbered spectral components for 7- and 4-nm sized particles, the component IV is tentatively assigned to S S bonds of the capping thiol group, which is abundant in the case of smaller sized particles due to an increased surface-tovolume ratio. Furthermore, the surface-sensitive S 2p spectra (recorded at h = 203 eV) showed an oxidized peak at a binding energy of 168.8 eV in case of the larger 7- and 4-nm sized particles, whereas this feature was almost negligible in the case of 2.7 nm NC. This is most probably due to the less number of unsaturated S atoms in the case of the smaller NC. It is often required to anneal the chemically prepared NC for minimizing the defects in their surfaces leading to better emission efficiency. However, annealing methods can lead to a considerable change in the surface composition and structure; hence, there is a need to understand in detail

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

716

Intensity. (a. u.)

CdS-NP(d=7.0 nm) S 2p

Cd 3d5/2 hν=445 eV (Ekin=40 eV)

b) hν=203 eV (Ekin=41 eV) I

×2 408 407

406 405 404

Binding Energy (eV)

II

Intensity. (a. u.)

III

×3

a) hν=500 eV (Ekin=338 eV)

I

3.1.2. Particle Size and Structure Analysis from XPS

II III ×2 166

165

164

163

162

161

160

Binding Energy (eV) CdS-NP(d=2.7 nm) S 2p d) hν=203 eV (Ekin=41 eV)

IV I II III α β

×2

c) hν=500 eV (Ekin=338 eV) I IV II III

×2 166

165

164

163

the surface of such systems. The above authors have also reported the thermal behavior and stability of CdS NC on similar sized CdS NC by closely monitoring the core level features of the NC [27]. Their results indicate that thermal annealing of CdS NC samples at 240  C in ultra-high vacuum conditions modifies the surface from being sulfurterminated at normal room temperature to that of Cd. This is possibly due to the desorption of the capping material at these temperatures or/and the segregation of Cd atoms from the interior of the particle to the surface. The characterization and mechanism of nanoscale selfassembly is an important area of current research. X-ray photoemission spectroscopy studies have been performed to characterize the layer-by-layer self-assembly of composite thin films of CdS nanoparticles and alkanedithiol on gold substrate [28]. The layer-by-layer structure was confirmed by an angle-resolved photoemission technique at each composite film preparation step.

162

161

As discussed in the previous section, photoemission spectroscopy can provide important information on the surface composition and structure of NC. The mean escape depth of the photoelectrons from a sample depends on its kinetic energy and is approximately given by d = 0.5(K.E)1/2 Å [1]. For example, the mean escape depth of an S 2p electron for AlK monochromatic radiation is about 18 Å. Since such length scales are comparable to the NC sizes, the core level spectrum contains signal from both the interior and surface of the NC. A quantitative analysis of such a core level spectrum will provide an estimate of the mean NC size. This was first realized by Nanda et al. [10] and they provided such an analysis in cases of CdS and ZnS NC [9, 10]. Here, we briefly summarize the method. As discussed in detail in section 1.2, the S 2p-core level of the NC shows three distinct species (Fig. 5b). The intensity of each component obtained from least-squared fitting of the S 2p spectrum is proportional to the relative number of atoms in each of the corresponding layers. The positions and normalized intensities of the three sulfur species, namely, the core, surface, and capping are listed in Table 1 for CdS and ZnS NC, respectively. Given these experimental intensities, one can also calculate the photoemission core level intensities corresponding to different regions of the NC. Assuming the NC to be spherical, one can model a typical nanocrystallite into a three-layer core-shell structure as shown in Figure 9 consistent with the three-sulfur region as evidenced from the sulfur core level.

160

Figure 8. a–b S 2p XPS of CdS 7.0-nm particle. The photon energy was set (a) to h-500 eV for the bulk sensitive measurements. The three different S 2p doublets labeled I–III are obtained from fitting as described in the text. The inset shows the Cd 3d5/2 spectrum displaying two different species of Cd. The residuals of the fitting results are shown below the corresponding spectral. Reprinted with permission from [11], U. Winkler et al., Chem. Phys. Lett. 306, 95 (1999). © 1999, Elsevier-Science. c–d S 2p core level spectra of 2. 7-nm sized CdS nanoparticles showing four different S species. The residuals of the fitting are shown below each corresponding spectral. Reprinted with permission from [11], U. Winkler et al., Chem. Phys. Lett. 306, 95 (1999). © 1999, Elsevier-Science.

Table 1. Experimentally obtained intensities of the three sulfur species, namely, the core (1), surface (2), and capping (3), in percentage of the total intensity as obtained from the S 2p core level fitting for CdS and ZnS nanoparticles. Sample

Icore

Isur

Icap

CdS–I (4.5 nm) CdS–II (2.2 nm) ZnS–3.5 ZnS–2.5 ZnS–1.8

63.3 49.8 51.6 46.7 32.5

21.0 28.0 30.5 36.9 47.3

15.7 22.2 17.9 16.3 20.2

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

Capping (R2) Surface (R1) Core (R0)

Figure 9. Schematic model of a nanocrystallite showing various regions, namely, the core a radius of R0 , a shell of radius R1 , and capping layers of radius R2 .

The photoelectron intensity from a particular level of an atom is given by 
z dv dI = I0 exp − 

(3)

where dI is the infinitesimal intensity contribution from a volume dv situated at a distance z from the surface. The prefactor I0 depends on the type of atomic level, nature of the sample, and the details of the spectrometer such as the detection efficiency corresponding to a photoelectron of a particular kinetic energy, angle of acceptance, photon flux, etc.  is the mean-free path of the photoionized electron. Integrating the above expression for the infinitesimal intensity contribution over suitable limits, one can obtain the total intensity corresponding to each of the three regions. Although the instrumental details, like the analyzer characteristics, etc., are the same for different regions of the crystallites, the number densities of the sulfur atoms are not the same in the bulk and the capping materials. This could easily be evaluated from the physical parameters of CdS or ZnS bulk and the capping material thioglycerol, as shown below: Thio M CdS IoThio = Thio X CdS = 035 CdS Io M 

(4)

In case of a spherical nanocrystallite, eq. (3), after integration, can be conveniently expressed in terms of spherical polar coordinates as I = Io



R11

R1





0



2

0

exp




 −f r 2 r dr sin  d d 

(5)

where, f r = R2 − r 2 sin2 2 − r cos ; on account of spherical symmetry, integration over  gives only a factor of 2 and the integration over variables r and  are over the suitable limits discussed below. The intensity ratios for the surface to the core and the capping to the core are given, respectively, by:   R  I0CdS R01 0 exp −f r  r 2 dr sin  d Isur =  R   Icore I0CdS 0 0 0 exp −f r  r 2 dr sin  d   R  I0Thio R12 0 exp −f r  r 2 dr sin  d Icap =   R  Icore I0CdS 0 0 0 exp −f r  r 2 dr sin  d

717 These integrals were evaluated numerically for different choices of R0 , R1 , and R2 . The values of R0 , R1 , and R2 were used as adjustable parameters for calculating the intensity ratios given in the above equations, and the set of values that match the experimental intensity ratios best provides one with the size of the core, surface, and the capping layers of the NC. A comparative tabulation of the sizes obtained from this method and the sizes obtained using independent techniques such as X-ray diffraction and transmission electron microscopy (TEM) is given in Table 2 for CdS and ZnS NC, respectively. The sizes of the nanocrystallites estimated from the analysis of the photoemission core level agree reasonably well with those from other methods.

3.2. Semiconductor Nanoparticles: III–V and Other Groups Following the early approaches of [10] and [11], McGinley et al. have most recently studied the surface of colloidally prepared InAs NC using photoelectron spectroscopy with synchrotron radiation [12]. These NC were in the size range of 30–60 Å passivated with trioctylphosphine (TOP). The phosphorous atom of TOP bonds with the surface atoms of the NC. Spectral decompositions of the As 3d and In 4d core levels at different photon energies provide qualitative information about the surface bonding and reconstruction. The core level spectra of In and As were fitted to a minimum number of Voigt functions, using a simplex optimization routine after a proper background subtraction (using a polynomial function in this case, shown in Figs. 2 and 3 of [12]). As illustrated in the figure, each of these spectra can be fitted with three Voigt components indicated by V, S1 , and S2 respectively. The higher kinetic energy component V is a well-resolved, spin-orbit doublet, with a spin-orbit splitting value of 0.69 eV in this case, while the lower kinetic energy components, S1 and S2 , are much broader to show the spin-orbit separation. The typical Gaussian widths of the Voigt function for these two components were 120 ± 005 and 120 ± 002 eV, respectively. The contributions of S1 and S2 to the total core-level signal decrease in the spectra with h = 4833 eV as compared to those with h = 1183 eV, as shown in the figure. With increasing nanocrystal size, the intensities of the surface components (S1 and S2 ) decrease in comparison to the volume component V, consistent with Table 2. Sizes of the various regions in the nanocrystals, CdS–I, CdS– II, and ZnS obtained from the analysis of the photoelectron intensities of various sulfur components and their comparison with other experimental techniques. R0 , R1 , and R2 are the radii of the various regions in the nanocrystals, R0 , R1 , and R2 , as illustrated in Figure 9.

Sample

(6)

(7)

CdS–I (4.5 nm) CdS–II (2.2 nm) ZnS–3.5 ZnS–2.5 ZnS–1.8

Core diameter 2R0 (nm)

Surface layer diameter 2R1 (nm)

Capping layer diameter

Diameter from other experiment (nm)

4.1 2.0 3.5 1.9 1.5

4.4 2.3 4.0 2.3 2.0

4.9 2.7 4.6 2.7 2.4

4.4 2.2 3.5 2.5 1.8

718 the decrease in the surface-to-volume ratio. The component S1 , which is shifted by 0.22 eV from the volume component V, is ascribed to the formation of As As bonds on the surface, similar to the formation of As trimers in the case of clean bulk InAs surfaces [29]. Component S2 is associated with the surface As atoms bonded to the passivating organic TOP. The magnitude of this core-level shift from the volume component V is about 1.4 eV, indicating a significant amount of charge transfer from As to P atoms. The In 4d core level spectra of the NC could also be described with three Voigt components, similar to the As 3d (Fig. 3, [12]). However, in this case, the surface components S1 and S2 are shifted by 0.44 and 2.4 eV with respect to V and the corresponding Gaussian widths were found to be 0.65, 190 ± 010, and 160 ± 010 eV, respectively, for the V, S1 , and S2 . These authors have ascribed the S1 components to a sp2 bonding geometry of the surface In atoms, whereas S2 is identified with In P bonds at the nanocrystal/TOP interface. The origin of the extreme broadening of the surface core level spectra has been ascribed to the possibility of a number of different bonding geometries available for the surface atoms of a multi-faceted NC surface resulting in a distribution of bond lengths, distortions, and strain, which is collectively defined as the surface roughness by the above authors. Such inhomogeneity invariably gives rise to the broadening of the core level. However, such broadening is in addition to the broadening arising due to the distribution of pinning of the Fermi level by defect sites, which would broaden all the core levels to the same degree including the volume component. Recently, Borchet et al. [30] have characterized the core shell nature of ZnS passivated InP NC by taking advantage of the variation of the escape depth on the energy of the photon. The detailed synthetic procedure is described elsewhere [31]. Their results demonstrate that the In atoms are in the core, while Zn is located in the surrounding shell. Using a similar analysis as reported by Nanda et al. [10], they have determined the average thickness of the core, shell, and capping layer. Furthermore, the ratio of the normalized intensities of In to Zn core levels showed an increase with increasing kinetic energy, demonstrating that at lower kinetic energy, the In intensity is screened by the surrounding Zn atoms. If the In and Zn atoms were randomly distributed in the NC as in the case of an alloy, the ratio of the normalized intensities should stay at a constant value determined by their atomic concentrations. Earlier, Hoener et al. had demonstrated a similar shell-core structure in the case of CdSe-ZnSe NC using X-ray and Auger spectroscopies [32]. The normalized XPS and Auger intensity ratios of Cd and Zn cations showed a marked difference in the case of core-shell NC than that of a random mixture of CdSe and ZnSe particles.

3.3. Nanoparticle Oxides Oxides of semiconductors form an important category of materials that have important optoelectronic properties [33]. Nanoparticles of oxides are currently an important area of study as these materials might have tremendous impact in the area of magnetic semiconductors or spintronics, and memory devices [34]. One of the interesting areas in which

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

XPS has been used in these materials is to study their size, spatial distribution, as well as their functionality. This is often manifested as a change in the nature of the surface species of the nanoparticle assemblies, which can then be monitored and characterized by XPS. An example of such is reported by Prabhakaran et al. in the case of Fe2 O3 nanoparticles on the surface of Si [35]. Fe2 O3 nanoparticles were synthesized sonochemically and incorporated into Si wafers, leading to multiple light emissions and multiple functionality all at the same time. At an annealing temperature of 850  C in UHV chamber and the Si wafer treated with Fe2 O3 nanoparticles, the Fe3+ is reduced by the silicon, and desorbs as SiO, resulting in the formation of magnetic nanoparticles consisting predominantly of Fe0 . The reduction of Fe2 O3 to elemental Fe takes place by the following reaction: Fe2 O3 + 3Si −→ 2Fe + 3SiO During the reaction, an XPS core level of Si 2p and Fe 2p were monitored at several temperatures to study the reaction. The Si 2p core level is shown as an inset in Figure 10 showing the disappearance of the oxide contribution at an elevated temperature. The main figure shows the gradual reduction of the Fe2 O3 peak (Fe3+ ) with the temperature showing predominantly Fe0 , which is chemically shifted by

Si 2p 760 °C

Fe 2p

720 °C RT

98

707

104

110

Binding Energy (eV) oxides

760 °C

730 °C

720 °C

500 °C

710.5

700

724

720

RT

740

Binding Energy (eV)

Figure 10. XPS measurement of Fe 2p core level as a function of temperature showing the reduction of Fe2 O3 to Fe on Si (III) surface. The corresponding Si 2p spectra are shown in the inset. Reprinted with permission from [35], Prabhakaran et al., Adv. Mater. 13, 1859 (2001). © 2001, Wiley-VCH.

X-Ray Photoelectron Spectroscopy of Nanostructured Materials (a) Pure SnO2 O 1s hν = 766.25 eV

Counts

300k

200k Surface

Bulk

100k

0

230

232

234

236

238

240

Photoelectron Kinetic Energy (eV)

(b) 0.30

Surface / Bulk Intensity Ratio

3.5 eV towards higher binding energy. The Si wafer with the as-incorporated amorphous Fe2 O3 particles exhibits superparamagnetic behavior, which is characteristic of amorphous nanoparticles [36]. Upon annealing, the samples show soft ferromagnetic behavior. Another interesting XPS study on oxide nanoparticle has been reported by Schmeisser et al., in which they have observed a large dipole moment at the interface of metal semiconductor core, which is manifested in the form of a core level shift [37]. This has been observed in the case of Sn/SnO2 and Ta/Ta2 O5 nanoparticles. The oxidic shell covering the metal cluster produces a high interface dipole moment whose origin is probably due to the charge difference between the metallic core and the substrate. The corresponding shift in the XPS core level is proportional to the strength of the dipole moment, which can vary depending on the size of the core and the oxide layer thickness. The smaller nanoclusters do not show a dipole shift and the Sn 3d core level spectra only shows the metallic (at 484.4 eV) and oxidic (486.7 eV) contribution (Fig. 2 of [37]). However, the larger clusters exhibit new peaks. The oxidic part is shifted to 488.8 eV from its usual position. Thus, the net shift induced by the interface dipole moment amounts to 2.1 eV when compared to the oxidic peak. The Sn 4d of the clusters shows a dipole-induced shift of about 5 eV compared to bare SnO2 (100) single crystals and the valence band of the nanoparticle is shifted by 2.8 eV. McGinley et al. have studied pure and Sb-doped SnO2 nanoparticles using photoemission spectroscopy [38]. Photoelectron spectra with synchrotron radiation were recorded for Sn 3d, Sb 3d, and O 1s core levels. The O 1s core level spectrum of pure SnO2 nanoparticles is shown in Figure 11a. The spectrum could be fitted with two voigt components. With increasing photon energy, the relative contribution of the surface and bulk components changes as shown in Figure 11b. The lower kinetic energy component of the spectrum is due to O atoms in a surface environment and the bulk component, which shifted by +130 ± 005 eV, has the same shift found for surface hydroxyl groups (OH− ) in a study of thin film SnO2 [39]. Hence, it was suggested that the surface hydroxyl groups bond to the surface in a manner similar to that found for the O bridging atoms (O2− ) of the SnO2 (110) surface [40]. It appears that there is no clear surface environment for Sn as the Sn 3d core level spectrum is fitted with only one Voigt function. This primarily suggests that Sn atoms are largely confined to the bulk of the nanoparticle with oxygen atoms remaining largely on the surface and participating in the surface bonding. Furthermore, these workers have addressed the important question of the dopant distribution in the nanoparticle and whether the oxidation state is similar throughout the structure. Figure 12 shows the Sb: Sn 3d3/2 core level intensity ratio as a function of photon energy for samples with two different Sb concentrations, namely, 9.1 and 16.7%. This shows an enrichment of Sb at the surface in SnO2 nanoparticles for both concentrations. A quantitative analysis of Sb 3d core level done for nanoparticles synthesized from precursors containing SbIII and SbV shows that Sb atoms located in the surface have a oxidation state of SbV . The photoemission features are, however, very broad with a Gaussian FWHM of 1 eV when the total experimental

719

Pure SnO2 O 1s Voigt Ratios

0.25

0.20

0.15

0.10

0.05 500

600

700

800

900

1000

1100

1200

Photoelectron Kinetic Energy (eV)

Figure 11. (a) O 1s core level spectrum of pure SnO2 nanoparticles fitted with two Voigt functions. (b) Intensity ratio of the components in (a) as function of photon energy. Reprinted with permission from [38], McGinley et al., Eur. Phys. J. D16, 225 (2001). © 2001, Springer-Verlag.

resolution was set to 240 meV. This highlights the effect of the expected internal faceting on the photoemission core levels. Bullen and Garrett have recently studied surface nature and composition of TiO2 nanoparticle arrays using XPS [41]. Using close-packed polystyrene nanosphere masks, TiO2 arrays were fabricated using a chemical route. The average composition of the particles was confirmed using XPS. Figure 13 shows the Ti 2p spectrum of a nanoparticle array grown on glass after removal of the polystyrene mask. The shape, binding energy, and the spin-orbit splitting of the Ti 2p photoemission spectrum are characteristic of the TiO2 . Comparison with the spectrum of a TiO2 (110) single crystal shown in (a) with that of the nanoparticle array reveals some evidence of Ti3+ species as indicated by a slight low BE shoulder on the Ti 2p3/2 peak. However, it was difficult to determine the percentage of Ti3+ on the surface because of the close proximity of Ti4+ (TiO2 ) and Ti3+ binding energies. The large surface-to-volume ratio and high curvature of these particles suggests that the Ti3+ species may be associated with corner, edge, or terrace defect sites. There have also been a few reports on XPS study on other oxide nanoparticles, for example, ZnO, where it has been mainly used as a characterizing tool [42]. X-ray photoelectron spectroscopy study of nanostructured CeO2 films was reported by Wang et al. [43]. Cerium oxide serves as an important catalyst in many chemical reactions

X-Ray Photoelectron Spectroscopy of Nanostructured Materials

720

(a) TiO2(110) single crystal

Intensity

hν = 766.25 eV

0.35

Ti 2p3/2

Sn 3d3/2 Sn 3d3/2

Ti 2p1/2 200

220

240

260

280

300

Intensity (a. u.)

Photoelectron Kinetic Energy (eV)

Sb 3d3/2 / Sn 3d3/2 Core Level Intensity Ratio

0.30

(b) TiO2 nanoparticles

0.25 Ti3– states

9.1% Sb

0.20

16.7% Sb 470

468

466

464

462

460

458

456

BE (eV) 0.07

Figure 13. XPS comparison of Ti 2p region for a rutile TiO2 (110) single crystal and TiO2 nanoparticle arrays on a glass substrate. The solid circles are the spectrum shown in (a) overlapped to highlight the presence of Ti3+ surface species on the nanoparticle. Reprinted with permission from [41], H. A. Bullen and S. J. Garret, Nano Lett. 2 (2002). © 2002, American Chemical Society.

0.06

0.05

600

800

1000

1200

Photon Energy (eV)

Figure 12. Sb: Sn 3d3/2 core level intensity ratio as a function of photon energy for samples with two different Sb concentrations. Lines are drawn as a guard to the eye. The inset shows a typical wide-range spectrum from which the core level intensities were calculated. Reprinted with permission from [38], McGinley et al., Eur. Phys. J. D16, 225 (2001). © 2001, Springer-Verlag.

and the chemical state of ceria is an important factor that controls the catalytic activity. The films were grown by the electrodeposition method and the average crystallite sizes were in the range of 6–8 nm. These authors have studied the Ce 3d, 4d, O 1s, and valence band XPS of these films at different sintering conditions. Their results indicate that the produced films were nonstoichiometric, with a typical value in the range of 1.90–1.98 for x in CeOx . The photoelectron spectra also showed an increase of Ce4+ concentration at the surface with increasing sintering temperature.

3.4. Metallic Nanoparticles Many interesting optical and electronic properties are observed in the case of nanometer-sized metal particles due to confinement of the free-charge carriers [44]. A particular example is the presence of a strong band in the visible region of the absorbance spectrum of noble metallic particles. This strong band is attributed to the surface plasmon

oscillation modes of the conduction electrons in the particle, which are coupled through the surface to the external applied electromagnetic field. Because of this reason, the optical properties of the nanometer-sized gold, copper, and silver particles have been extensively studied in the recent past [45]. Ultra-fast dynamics of these metal colloids have also been studied in the recent past to understand the basic electron-phonon and electron-electron interactions and their relaxation mechanisms [46]. X-ray photoelectron studies have been performed recently on some of these metallic nanoparticles to study their reactivity and surface properties. Boyen et al. recently reported the unique, oxidation-resistant 55-atom gold cluster [47]. These authors have studied the size dependence of the oxidation behavior of gold nanoparticles using XPS. Generally, bulk gold is highly oxidation-resistant; however, it can form Au2 O3 by electrochemical methods, by exposure to atomic oxygen delivered by molecular dissociation at a hot filament, or radicals provided by oxygen plasma [48, 49]. For this experiment, the Au nanoparticles were synthesized by a micellar technique giving a narrow size dispersion [50]. The samples were then exposed to oxygen plasma in order to remove the ligand shell thus allowing the study of the naked Au cluster of different sizes. Since we are interested in the response of such pure Au nanoparticles to highly reactive oxygen species, a flat Au (111) surface was first treated under identical condition for comparison. The XPS core-level spectra of bulk Au-4f region is shown in Figure 14A for two different take-off angles (having different surface sensitivities). Both spectra reveal

X-Ray Photoelectron Spectroscopy of Nanostructured Materials A Au-4f

metal

Intensity (arb. units)

oxide

µ

60°

0°

92

90

88

86

84

82

80

Binding energy (eV)

B

Au-4f

a 7.9 nm

Intensity (arb. units)

b

2.0 nm

c

1.6 nm

Au65 (1.4 nm)

d

e

1.3 nm

f

10) is one of the approaches used to modify the surface of growing particles. Thus to control the particle size, addition of a surface modifier is necessary as it coats the surface of the particles and provides a barrierlike protection against agglomeration [53]. Europium doped yttrium oxide was synthesized via preparation of a precursor solution by dissolving a known amount of yttria salt and Eu(NO3 3 ·5H2 O in a solvent and was stirred for 2 hours. The modifier solution was prepared by dissolving 10 wt% of surface modifier (1:1 mixture of tween-80 and -alanine, -caprolactum, or emulsogen-OG) with respect to Eu2 O3 /Y2 O3 in 50 ml of aqueous ammonium hydroxide solution (pH > 10) and was stirred for 1 hour at room temperature. The precursor solution was then added to the modifier solution at a controlled rate with vigorous stirring, which converted into gel. The gel was then centrifuged (6000 rpm for 30 minutes) and the aqueous solution was removed by refluxing in toluene. The toluene was removed by evaporation and the resulting white powder was dried in an oven at 60  C for 24 hours. In this study [54] different solvents were used (ethanol and water) in the precursor solution, and it was found that the Eu3+ emission was higher when ethanol was used. Similarly, different precursor salts were studied—yttrium nitrate [Y(NO3 3 ·5H2 O], yttrium acetate [Y(CH3 COO)3 ·4H2 O], and yttrium chloride (YCl3 ·6H2 O)—and it was determined

that the Eu3+ emission intensity decreased with the type of precursors used in the following order: − CH3 COO− > NO− 3 > Cl

These effects on the luminescence and spectroscopy of the Eu3+ ion will be discussed in detail in Section 4. The modifier was shown to play an important role on the crystallite as the particle size decreased in the following order of modifiers used: -caprolactum(181 nm) > -alanine(23 nm > emulsogen-OG(10 nm) The concentration of the modifier was increased from 0 to 10 wt% with respect to Eu2 O3 /Y2 O3 and the particle size was reduced from 6 m to 10 nm [53]. The modifier protects the generated particles by forming a layer and significantly reduces the surface free energy of the particulate matter thus preventing interaction with neighboring particles. The size of yttrium doped nanocrystals can be successfully tailored using chemical wet synthesis in the presence of a modifier in different pH conditions [55]. The size and morphology of the particle are greatly influenced by the pH conditions, where low pH conditions were found to be favorable for obtaining nanocrystals with uniform morphology. Low pH (80  C). Under these conditions, the microgel was in its collapsed form having a relatively small hydrodynamic diameter and an increased charge density. The solution was kept boiling for 1 hour and then filtered. The precipitates were washed with deionized water and dried at 110  C in an oven. To convert the powder from the hydroxycarbonate precursor to the oxide the product was fired at 980  C for 6 hours. Nanometer Y2 O3 :Eu particles were also synthesized by firing a hydroxy carbonate precursor, which was prepared from two aqueous solutions [24]. One solution was composed of the metal nitrates Y(NO3 3 and Eu(NO3 3 while the other was an aqueous Na2 CO3 solution. The two solutions were mixed and stirred for 10 minutes. The precipitate was separated by centrifugation at 3000 rpm for 15 minutes and was dried at 80  C for 24 hours. The precursors were fired at 900  C for 30 minutes in air and cooled rapidly to room temperature yielding particles with diameters of approximately 60 nm. It was also possible to produce nanocrystals of Y2 O3 :Eu3+ by colloidal chemical methods [65] involving precipitation of amorphous spherical precursor material, which was then fired to form the nanocrystalline material. Yttrium and europium chloride salts in water were adjusted to a pH of ∼1 using HCl. An excess of urea, typically 15×, was dissolved in the solution and then heated to >80  C for 2 hours. The urea was slowly decomposed and a burst of particle nucleation was achieved when a certain pH was reached (pH 4–5). The particles continued to grow uniformly until the cation supply was exhausted. The obtained precipitate was washed, flocculated, dried, and fired in air for 3 hours at temperatures greater than 1000  C. The material consisted of agglomerations of nanocrystals with the size of the agglomerates being defined by the drying stage. It was also possible to produce nanocrystals in the 2–10 nm range by colloidal methods which were defect-free. The surface of the nanocrystals can provide nonradiative recombination routes. However, by using a surface-capping agent, these effects can effectively be minimized. To a solution of EuCl3 ·6H2 O in methanol, an equal amount of triocytl phosphine oxide

729

(TOPO) solution was added. The TOPO binds to the surface and has three functions: (i) It prevents particle agglomeration (steric passivation). (ii) It achieves electrical passivation. (iii) The ratio of TOPO to Eu3+ ions may be used to control the particle size. Excitons migrating to the surface can deexcite, both radiatively and nonradiatively via surface dangling bonds, which may act as trap states. TOPO bound to the surface passivates these states and allows only recombination with the Eu3+ 4f shell. The solution was stirred for 10 minutes and the nanocrystals were precipitated by the addition of a controlled amount of methanolic NaOH solution (pH 5.5–6). The precipitation was carried out in a nonaqueous environment in order to reduce the possibility of hydroxide formation. When the colloidal precipitation reaction occurs in an alcoholic solution, the dehydrating properties of the alcohols result in formation of the oxide [66]. A TOPO:Eu3+ ratio of 1:1 resulted in particles in the 4 nm range. Eu-doped yttrium oxide nanocrystals were synthesized using the reverse microemulsion technique, which is a particularly attractive reaction medium in terms of being able to obtain monodispersed nanoparticles with controlled morphology [67]. Reverse microemulsion consists of an aqueous phase dispersed as microdroplets ( is given by N2 t =

N0 A01 A12 I02 .2−1 .1−1 + A12 I0

(12)

where N0 is the initial population of the ion in the ground state ( 0>), and Aij is a characteristic constant involving the oscillator strengths for the transitions from the initial state i> to the final state j>. I0 describes the density of photons in the pump beam while .1−1 and .2−1 are the intrinsic relaxation rates of levels 1> and 2>, respectively. According to Eq. (12), we note that the upconverted luminescence varies quadratically with the pump beam (I0 ) but varies linearly with the concentration of the emitting particle. ETU proceeds according to a scheme in which two ions in close proximity are excited in an intermediate level 1> and are coupled by a nonradiative process in which one ion returns to the ground state 0> while the other is promoted to the upper level 2>. Using the same symbols as in Eq. (12), the population of level 2> can be written as N2 t =

N0 A01 I0 2 =u .2−1 .1−1 2

(13)

where =u is the rate constant for the ETU process. Similar to ESA, in the ETU process the population of level 2> varies quadratically with the density of photons in the pump beam (I0 ) but, at variance with ESA, also varies quadratically with the dopant concentration N0 . PA is produced by absorption from an excited state of the rare earth ion. Thus the pump laser wavelength is resonant with a transition from the intermediate metastable level to a higher excited state. The absorption of the pump photons directly populates the higher excited state; therefore an energy transfer process is responsible for producing the population in the intermediate excited state [126]. In PA, one ion initially in the metastable state produces two ions in this state as a result of photon absorption and subsequent energy transfer. Under the right pumping conditions 2 ions can produce 4 in the metastable state, 4 can produce 8, the 8 can produce 16, etc. The avalanche process requires a minimum pump intensity and is characterized by a pump threshold. If this threshold is not achieved, the upconversion will be inefficient [132].

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

WMPR = C exp−4E

(14)

and 4=−

ln h?

(15)

where E is the energy gap to the next lower level, h? is the highest phonon energy of the material, and the parameters C and 4 are constants and can be derived from the measurements of the fluorescence lifetimes and the calculation of the WEDT , the electric dipole transition rates. To obtain efficient upconversion, selecting a lattice with low phonon energies is essential. In a lattice with high phonon energies, the rare earth ion will predominantly decay from its excited level nonradiatively to the next lower lying level via the emission of phonons. On the other hand, in materials that possess low phonon energies, the probability of multiphonon relaxation is low and thus the ion will mainly decay radiatively via the emission of photons. The efficiency of the upconversion depends also on the spatial distribution of the dopant ions. Increasing the dopant concentration will result in an increased upconversion efficiency. However, this is successful only to a certain extent and cannot be considered as a viable solution to increasing the upconversion luminescence since at higher dopant concentrations, cross-relaxation between rare earth ions quenches the luminescence, thereby decreasing the effect of the increased concentration.

4.5.1. Optical Properties and Upconversion of Y2 O3 :Ho3+ Nanocrystals The optical spectroscopy and upconversion properties of Ho3+ doped nanocrystalline and bulk Y2 O3 , as a function of holmium concentration (0.1, 0.5, 1, 2, 5, and 10 mol%) have also been investigated [134]. Nanosized Y2 O3 crystals doped with 0.1, 0.5, 1, 2, 5, and 10 mol% Ho2 O3 (Y1998 Ho0002 O3 , Y199 Ho001 O3 , Y198 Ho002 O3 , Y196 Ho004 O3 , Y190 Ho010 O3 , and Y180 Ho020 O3 , respectively) were prepared using a solution combustion synthesis procedure. The average crystallite size of the samples was in the range of 10 to 15 nm determined from line broadening of X-ray reflections. For comparison purposes, micrometer sized bulk Y198 Ho002 O3 and Y180 Ho020 O3 samples were prepared by conventional solid state reaction. In the case of all samples under investigation, the authors observed that the room temperature visible luminescence spectra excited at 457.9 nm exhibited four distinct emission bands (Fig. 5): blue emission between 480 and 500 nm corresponding to the 5 F3 → 5 I8 transition; green emission in the region of 530–580 nm attributed to the (5 F4 $ 5 S2 → 5 I8 transition; red emission between 630 and 680 nm corresponding to the 5 F5 → 5 I8 transition; and NIR emission between 735 and 775 nm attributed to the (5 F4 $ 5 S2 → 5 I7 transition. The transition energies were observed to be similar for both the nanocrystalline and bulk samples; hence the crystal field

5

(i)

(ii)

(iii)

(iv)

10 mol%

5 mol%

Intensity x 105 (Arbitrary Units)

The efficiency of the upconversion is a function of the rate of multiphonon relaxation, which is dependent upon the energy gap separating the upper level and the next lower level and the highest phonon energy in the material. The rate of the multiphonon relaxation can be expressed as [133]

751

4

3

2 mol%

2 1 mol%

1 0.5 mol% 0.1 mol%

0 500

550

600

650

700

750

800

850

Wavelength (nm)

Figure 5. Room temperature luminescence of nanocrystalline Y2 O3 : Ho3+ upon excitation at 457.9 nm. (i) 5 F3 → 5 I8 , (ii) (5 F4 $ 5 S2 → 5 I8 , (iii) 5 F5 → 5 I8 , (iv) (5 F4 $ 5S2 → 5 I7 . Reprinted with permission [134], J. A. Capobianco et al., from Chem. Mater. 14, 2915 (2002). © 2002, American Chemical Society.

energy was reasonably similar for the nanocrystalline and bulk samples [134]. A change in the relative intensities of the green and red emission bands was observed between the bulk and nanocrystalline samples. The overall luminescence was also severely reduced by about one order of magnitude in the nanocrystal samples compared to the bulk samples. The authors attributed this type of behavior to an increase in multiphonon relaxation due to adsorbed atmospheric carbon dioxide and/or water [134]. The medium IR spectra of the nanocrystalline Y2 O3 :Ho3+ samples investigated revealed bands around 1500 and 3350 cm−1 , which were attributed to stretching modes of carbonate and hydroxide ions, indicating that the materials have adsorbed atmospheric CO2 and H2 O These adsorbed ions on the surface of the nanocrystalline materials provide vibrational quanta of higher wavenumbers compared to the intrinsic phonons of yttria (≈600 cm−1 ) resulting in an increase of multiphonon relaxations from all excited levels. This larger nonradiative decay probability is thus responsible for the lower emission efficiencies observed in the nanocrystals. Furthermore, the rapid quenching of the (5 F4 $ 5 S2 → 5 I8 transition relative to the 5 F5 → 5 I8 transition in the nanocrystalline samples was also attributed to the presence of these adsorbed atmospheric molecules. The energy gap between the 5 S2 and 5 F5 levels for the Ho3+ ion in Y2 O3 single crystals [135] has been reported to be 2666 cm−1 . As five intrinsic phonons (about 600 cm−1 ) were required in the bulk material to bridge this gap, the 5 S2 level was only slightly affected by multiphonon decay, while in the nanocrystals only two of the high energy carbonate ion

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

phonons available or just one hydroxide high energy phonon were required to span this gap making multiphonon relaxation of (5 F4 $ 5 S2 ) levels much more probable. A rapid quenching of the (5 F4 $ 5 S2 → 5 I8 transition was also observed relative to the 5 F3 → 5 I8 and 5 F5 → 5 I8 transitions in the bulk and nanocrystalline samples as the Ho3+ dopant concentration increased from 1 to 10 mol%. The authors stated that this type of behavior was related to an ion-pair cross-relaxation (energy transfer) process represented by the 5 S2 → 5 I4 and 5 I7 ← 5 I8 transitions. In this energy transfer process a donor Ho3+ ion in the 5 S2 excited state relaxed nonradiatively to the 5 I4 state, while in another simultaneous nonradiative process an acceptor Ho3+ ion in the ground 5 I8 state was excited to the 5 I7 state, thereby quenching the luminescence of the (5 F4 $ 5 S2 ) level. Room temperature emission decay curves of the (5 F4 $ 5 S2 → 5 I8 transition upon 457.9 nm excitation were also reported for both the bulk and nanocrystalline samples [134]. The lifetimes are reported in Table 4. Decay curves for the bulk samples were fitted with a single exponential model. However, a deviation from first exponential behavior was observed by the authors in all decay curves obtained for the nanocrystalline samples. The difficulty in fitting with a single exponential model in the nanocrystalline samples was attributed to a distribution of dopant ions within the individual nanocrystals that are coupled in various degrees to the absorbed surface molecules. This resulted in the dopant ions located close to the surface having faster decay times than those ions located inside the nanocrystals. Hence, a nonexponential decay was observed as a significant portion of the dopant atoms reside near the surface due to the particles’ small size (about 10 nm). Furthermore, the lifetimes of the excited states were found to be significantly faster in the nanocrystal samples than in the bulk samples [134]. For example, the (5 F4 $ 5 S2 → 5 I8 lifetime in the 1 mol% bulk sample was found to be 135 s as opposed to 14 s (first time constant) or 57 s (second time constant) in the similarly doped nanocrystalline sample. This type of behavior was again attributed to the presence of adsorbed species on the surface of the nanocrystals that led to nonradiative relaxation of the excited states. The higher energy phonons that were present in the nanocrystals as opposed to the bulk samples resulted in a lower-order multiphonon process from the (5 F4 $ 5 S2 ) level to the 5 F5 level. The larger rate of Table 4. Decay times obtained from an exponential fit of the room temperature decay curves for the (5 F4 $ 5 S2 → 5 I8 transition upon 457.9 nm excitation.

nonradiative processes in the nanocrystalline samples resulted in a shorter lifetime for the (5 F4 $ 5 S2 ) level. Upon excitation with 646 nm radiation from a dye laser that populated the 5 F5 level, anti-Stokes emissions corresponding to the 5 F3 → 5 I8 and (5 F4 $ 5 S2 → 5 I8 transitions were observed in both the bulk and nanocrystalline samples (Fig. 6). A power study of the upconverted blue and green emission indicated a two-photon upconversion process. The upconversion luminescence was thought to result from an ESA process involving the 5 I7 energy state as the intermediate level (Fig. 7). In the assigned mechanism the laser light brought the Ho3+ ion into the 5 F5 level, which then nonradiatively decayed to the 5 I7 level. An excited state absorption process brought the ion to the 5 F3 level. The ion then emitted through the 5 F3 → 5 I8 transition or nonradiatively decayed to the lower lying levels and the 5 F4 $ 5 S2 → 5 I8 transition occurred [134]. The upconversion process was noted to be far less efficient in the nanocrystal samples which was attributed to an increase of multiphonon relaxations of excited levels due to the presence of the aforementioned adsorbed species on the surface of the nanocrystals. The populations of the intermediate levels from which a part of the excited ions can be reexcited to the upper emitting levels by ESA are substantially impacted by the nonradiative decay rate that increases as the lattice phonon energy becomes higher [136]. The highest available phonon energy not only affects quantum efficiencies of the emitting levels such as 5 F3 and 5 S2 levels but also impacts the upconversion efficiencies, which are primarily determined by populations of intermediate levels such as the 5 I6 and 5 I7 . The excitation populations of intermediate levels 35 (i)

(ii)

10 mol%

30 5 mol% Intensity x 103 (Arbitrary Units)

752

25

20

2 mol%

15 1 mol% 10 0.5 mol% 5

Bulk

Decay time (s)

Y198 Ho002 O3 Y180 Ho020 O3

0.1 mol%

135 60

0

Nanocrystalline

First decay time (s)

Second decay time (s)

Y1998 Ho0002 O3 Y199 Ho001 O3 Y198 Ho002 O3 Y196 Ho004 O3 Y190 Ho010 O3 Y180 Ho020 O3

17 20 14 9 3 2

77 73 57 40 14 4

475

500

525

550

575

Wavelength (nm)

Figure 6. Upconverted emission of Ho3+ -doped Y2 O3 nanocrystals at room temperature upon 646 nm excitation, showing (i) 5 F3 → 5 I8 , (ii) (5 F4 $ 5 S2 → 5 I8 . Reprinted with permission from [134], J. A. Capobianco et al., Chem. Mater. 14, 2915 (2002). © 2002, American Chemical Society.

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

20

5

F2, 5F3

5

F4 S2

Energy × 10 3 (cm-1)

5

15

10

5

F5

5

I4

5

I5

5

I6

5

I7

5

I8

5

0 Ho3+

Figure 7. Mechanism for Ho3+ upconversion in Y2 O3 for 646 nm excitation resulting in excited state absorption via the 5 I7 level.

and the quantum efficiencies of emitting levels of Ho3+ in a host crystal with a higher phonon energy are undoubtedly lower than those in a crystal with a lower phonon energy. Capobianco’s group also observed anti-Stokes emission corresponding to the 5 F3 → 5 I8 $ 5 F4 $ 5 S2 → 5 I8 , and 5 F5 → 5 I8 transitions in the bulk samples upon excitation with 754 nm radiation from a titanium sapphire laser that excites the 5 I4 level (Fig. 8). Though strong luminescence was observed from the bulk samples no upconversion was noted in the nanocrystal samples. As mentioned previously, behavior of this type was attributed to an increase of multiphonon relaxations of excited levels due to the presence of carbonate and hydroxide ions on the surface of the nanoparticles. The (5 F4 $ 5 S2 → 5 I8 transition demonstrated a quadratic dependence on the power of the pump beam indicating that two photons were involved in the excitation process. Again, an ESA process was thought to be the dominant mechanism. In this case there were two distinct ESA processes 5

F2, 5F3

20

5

F4 S2

5

5

Energy × 103 (cm-1)

F5

15

5

I4

5

I5

10

5

I6

5

I7

5

5

I8

0 Ho

3+

Figure 8. Mechanism for Ho3+ upconversion in Y2 O3 for 754 nm excitation resulting in excited state absorption via the 5 I7 level.

753

occurring concurrently that originated from the 5 I6 and 5 I7 levels and could populate the 5 G6 $ 3 K8 , and (5 F4 $ 5 S2 ) levels, respectively. Therefore, the authors proposed the following mechanism. The laser light brought the Ho3+ ion into the 5 I4 level, which then nonradiatively decayed to the 5 I7 level. An excited state absorption process brought the ion to the (5 F4 $ 5 S2 ) levels. The ion then emitted through the (5 F4 $ 5 S2 → 5 I8 transition or could have nonradiatively decayed to the lower lying levels and the 5 F5 → 5 I8 transition occurred. A second ESA process originating from the 5 I6 level was also possible. For the 754 nm excitation wavelength used in this study there was another possible resonant excited state absorption: 5 I6 → 3 K8 $ 5 G6 . It was most likely this ESA process, along with nonradiative relaxation, which was responsible for the weak blue emission from the 5 F3 level.

4.5.2. Optical Spectroscopy and Upconversion of Nanocrystalline Y2 O3 :Er3+ and Y2 O3 :Er3+ , Yb3+ The optical properties of cubic nanocrystalline Y2 O3 doped with 10 mol% Er3+ prepared by combustion synthesis were studied and compared with a microcrystalline Y2 O3 :Er3+ (bulk) of equal concentration synthesized by conventional solid state techniques [137]. The authors measured the diffuse reflectance spectra in the UV and visible regions for both the nanocrystalline and bulk samples. The reflectance spectrum of the nanocrystalline material was composed of a series of relatively sharp features in the visible region, accompanied by an absorption edge at approximately 350 nm (Fig. 9). The sharp bands were assigned to the intraconfigurational f –f transitions from the 4 I15/2 ground state to the excited states of the Er3+ ion, while the edge was assigned to the intrinsic absorption of the yttria host. The reflectance spectrum of the bulk sample was composed of the same f –f transitions. However, it was more resolved and had an absorption edge extending from at least 400 nm. The onset of the intrinsic absorption of the Y2 O3 host, assigned to band-to-band transitions, was shifted toward the blue for the nanocrystals, compared to the bulk sample. The authors believed that the reduced resolution of the spectral features in the reflectance spectrum of the Y2 O3 :Er3+ nanocrystals samples could have been partly due to inhomogeneous broadening induced by the presence of disorder in the material. The visible luminescence spectra of nanocrystalline and bulk Y2 O3 :Er3+ excited at 488 nm, were presented and the features in the 520–570 nm region were assigned to the transition from the thermalized (2 H11/2 $ 4 S3/2 ) excited states to the 4 I15/2 ground state of Er3+ ions in the C2 sites since the f –f electric dipole transitions are forbidden in the C3i sites due to its center of inversion. The overall shape of the emission bands for the two samples were similar except for a slight difference in the relative intensity of the peaks at 520–530 nm (assigned to emission from 2 H11/2 ), which were more intense for the nanocrystals. The authors believed this was caused by the hypersensitivity of the 2 H11/2 → 4 I15/2 transition, whose intensity is strongly influenced by small distortions of the sites accommodating the Er3+ ions. Thus, they suggested that in the nanocrystalline sample, the degree

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

754

Table 5. Decay times of bulk and nanocrystalline Y2 O3 :Er3+ obtained from an exponential fit of the room temperature decay curves for the (2 H11/2 $ 4 S3/2 → 4 I15/2 transition upon 488 nm excitation.

100 (vi)

(v)

(ii) (iv)

(i)

(viii)

80

(vii)

(iii)

% Reflectance

Decay time (s) 60

(ix)

40

20

0 20 0

30 0

40 0

50 0

60 0

70 0

Wavelength (nm)

Figure 9. Diffuse reflectance spectrum of Y180 Er020 O3 nanocrystalline sample. (i) 4 I15/2 → 4 F9/2 , (ii) 4 I15/2 → 4 S3/2 , (iii) 4 I15/2 → 2 H11/2 , (iv) 4 I15/2 → 4 F7/2 , (v) 4 I15/2 → 4 F5/2 , (vi) 4 I15/2 → 2 H9/2 , (vii) 4 I15/2 → 4 G11/2 , (viii) 4 I15/2 → 2 K15/2 , (ix) 4 I15/2 → 4 G9/2 . Reprinted with permission from [137], J. A. Capobianco et al., Phys. Chem. Chem. Phys. 2, 3203 (2000). © 2000, The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

of distortion of the C2 sites, in which the dopant ions were found, was greater than in the bulk sample. Furthermore, sharp peaks in the red region (640–700 nm) of the spectra of the nanocrystalline and bulk materials were observed and assigned to the C2 allowed 4 F9/2 → 4 I15/2 transition. Bettinelli’s group [40] studied the near infrared (NIR) emission of Y2 O3 :Er3+ (10 mol%) nanocrystals prepared via the combustion synthesis. The authors observed strong features at 990 nm and 1570 nm assigned to transitions from the 4 I11/2 (A) and 4 I13/2 (B) excited states to the 4 I15/2 ground state, respectively. Comparison of the NIR emission spectrum of the nanocrystalline material to the bulk showed that the Stark structure of the observed transitions was identical. However, they observed remarkable differences in the relative intensities of the bands originating from the various excited states. The integrated intensity ratio [R = IA /IB ] was distinctly lower for the nanocrystalline material (R = 0242) than for the bulk (R = 0668) and was attributed to different cross-relaxation properties in the nanocrystalline and bulk materials. A similar behavior was also observed between bulk and nanocrystalline Y2 O3 :Nd3+ . The decay times of the thermalized (2 H11/2 $ 4 S3/2 → 4 I15/2 transition were reported for the 10 mol% Y2 O3 :Er3+ bulk and nanocrystalline samples as well as for 1, 2, and 5 mol% nanocrystalline samples (Table 5). The decay times of the nanocrystalline material were shown to be concentration dependent and the variation of the decay times with concentration was believed to be due to the 2 H11/2 + 4 I15/2 → 4 I9/2 + 4 I13/2 cross-relaxation mechanism, which caused significant depopulation of the excited state [137]. Furthermore, the

Bulk Y180 Er020 O3

56

Nanocrystal Y198 Er002 O3 Y196 Er004 O3 Y190 Er010 O3 Y180 Er020 O3

38 12 31 34

decay time of the 10 mol% Y2 O3 :Er3+ bulk sample was more than one order of magnitude longer than the nanocrystal sample with an identical dopant concentration. The authors attributed this behavior to the fact that yttria powders could adsorb carbon dioxide and water from the atmosphere. The authors presented the FTIR spectrum of the Y2 O3 :Er3+ nanocrystalline material (1 mol%), which showed the presence of a band at approximately 1500 cm−1 attributed to the presence of carbonate ions. The adsorption of CO2 is efficient due to the large surface area (64 m2 g−1 ) of the nanocrystalline material prepared by the combustion synthesis technique. The presence of these CO2− ions on the 3 surface yields vibrational quanta of relatively high wavenumbers (1500 cm−1 ) compared to the intrinsic phonons of yttria (having a cutoff wavenumber of 597 cm−1 ), which are the only ones available in the bulk material. The authors state that the energy gap separating the 4 S3/2 state from the lower lying 4 F9/2 state was approximately 2795 cm−1 . Therefore, in the nanocrystalline material, the presence of the vibrational quanta of 1500 cm−1 made multiphonon relaxation of (2 H11/2 $ 4 S3/2 ) much more probable than for the bulk sample, where at least five phonons are required to bridge the gap. The authors also briefly studied the upconverted luminescence following continuous wave excitation (exc = 815 nm) into the 4 I9/2 ← 4 I15/2 transition and observed anti-Stokes emission bands assigned to the 2 H11/2 →4 I15/2 $ 4 S3/2 → 4 I15/2 , and 4 F9/2 → 4 I15/2 transitions centered at 525, 550, and 660 nm, respectively. The spectral band shapes and position were identical to those obtained upon 488 nm excitation and the authors stated that sequential absorption of photons or energy transfer upconversion were the two mechanisms which may have been responsible for the upconversion. The authors extended the spectroscopic investigation [93] by studying the effects of the Er3+ concentration (1, 2, 5, and 10 mol%) on the upconversion emission properties of nanocrystalline and bulk Y2 O3 :Er3+ . Continuous wave excitation with visible (exc = 488 nm) or NIR radiation (exc = 815 nm) produced green emission between 520 and 570 nm ascribed to 2 H11/2 $ 4 S3/2 → 4 I15/2 transition and red emission between 640 and 700 nm from the 4 F9/2 → 4 I15/2 transition (Figs. 10 and 11). The authors compared the spectra of all the samples under investigation and showed that although they were identical and unaffected by the pump wavelength, 488 or 815 nm, there was a notable reduction in the

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications 5 (a) 1 mol% (b) 2 mol% (c) 5 mol% (d) 10 mol%

Intensity x 105 (Arbitary Units)

4

3

(i) (ii)

(d)

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(c)

1 (b)

(a)

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500

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600

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Figure 10. Room temperature luminescence of nanocrystalline Y2 O3 :Er3+ upon excitation at 488 nm. (i) (2 H11/2 $ 4 S3/2 → 4 I15/2 , (ii) 4 F9/2 → 4 I15/2 . Reprinted with permission from [93], J. A. Capobianco et al., J. Phys. Chem. B 106, 1181 (2002). © 2002, American Chemical Society.

overall luminescence when pumping with 815 nm. The authors presented the MIR spectrum of the yttria nanoparticles, which showed bands at approximately 1500 and 3350 cm−1 due to vibrational modes typical of carbonates and hydroxyl ions, respectively. The presence of these 3. 0

755

groups on the surface yields vibrational quanta of relatively high wavenumbers compared to the phonons of pure yttria and increased the efficiency of multiphonon relaxation. and OH− ions In order to reduce the amount of CO2− 3 on the surface of the nanoparticles, the authors attempted different heat treatments on a nanocrystalline Y2 O3 :Er3+ (10 mol%) sample. The nanosample was treated at 800  C for 17 hours and then cooled to room temperature followed by a treatment at 1000  C for 65 hours and then subsequently cooled to room temperature. The MIR spectra of the doped nanoparticles obtained after the heat treatments indicated that the surface contamination was reduced from the overall intensities of the bands at 1500 and 3350 cm−1 . However, the conditions used for the experiments were not sufficient to completely remove the contaminants (Fig. 12). The authors stated that longer heat treatment at higher temperatures was not possible as an aggregation of the nanoparticles could occur, a process in which they combine to form larger particles. In this case, as the spectroscopy of the nanocrystalline material is particle size dependent, a comparison between the luminescence of the heat-treated and non-heat-treated nanocrystalline materials would not be possible. It was noted that in the bulk sample no bands at 1500 and 3300 cm−1 were present. The authors observed a concentration dependent enhancement of the red (4 F9/2 → 4 I15/2 ) upconverted emission following irradiation with 815 nm. In order to get a better idea of the mechanism(s) involved in the upconversion, the authors performed a power dependence study. It was shown that the intensity of the upconverted luminescence, Io , is proportional to some power n of the excitation

(a) 1 mol% (b) 2 mol% (c) 5 mol% (d) 10 mol%

80

(ii)

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Kubelka - Munk

Intensity x 104 (Arbitray Units)

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(a) (b) (c)

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Figure 11. Upconverted emission of Er3+ -doped Y2 O3 nanocrystals at room temperature, showing (i) (2 H11/2 , 4 S3/2 → 4 I15/2 and (ii) 4 F9/2 → 4 I15/2 $ exc = 815 nm. Reprinted with permission from [93], J. A. Capobianco et al., J. Phys. Chem. B 106, 1181 (2002). © 2002, American Chemical Society.

0 1000

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3000

4000

Wavenumbers (cm-1)

Figure 12. Diffuse reflectance spectra of nanocrystalline Y180 Er020 O3 following sequential heat treatment: (a) 800  C for 17 h, (b) 1000  C for 65 h, (c) bulk Y180 Er020 O3 sample shown for comparison. Reprinted with permission from [93], J. A. Capobianco et al., J. Phys. Chem. B 106, 1181 (2002). © 2002, American Chemical Society.

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

756 intensity, Ii , and can be written as

(16)

where n = 1$ 2$ 3$    . The superscript n is the number of photons required to populate the emitting state and was determined from the slope of the graph ln(intensity) versus ln(power). The fitting of the data yielded a straight line with a slope of approximately 2 for the (2 H11/2 $ 4 S3/2 → 4 I15/2 and the 4 F9/2 → 4 I15/2 transitions in all samples under investigation. PA was immediately ruled out as a mechanism of upconversion as no power threshold was observed in the power study. Thus, the authors determined that the upconversion occurred via a two-photon ESA or ETU process. In the ESA and ETU mechanisms (Fig. 13), the laser light brought the Er3+ ion to the 4 I9/2 level, which then nonradiatively decayed to the 4 I11/2 level. After this nonradiative relaxation, either an energy transfer from a neighboring Er3+ ion in the 4 I9/2 state (ETU) or a second photon from the incident laser beam (ESA) brought the ion to the 4 F3/2 level. Alternatively, after the initial excitation, the Er3+ ion could nonradiatively decay down to the 4 I13/2 level. Again, either an energy transfer process from another Er3+ ion in the 4 I9/2 state or a second photon populated the 2 H11/2 level. Emission from the lower lying states can then be observed. The authors noted that if the upconversion occurred only via the above mechanisms, upconversion spectra with identical relative intensities as in the direct emission (exc = 488 nm) would have been expected. This was clearly not the case as an enhancement of the red emission for the 4 F9/2 state with increasing Er3+ concentration was observed. This effect was demonstrated using a graph of the ratio of the integrated intensity of the green emission [(2 H11/2 $ 4 S3/2 → 4 I15/2 ] over that of the red emission [4 F9/2 → 4 I15/2 ], versus 4

F3/2 F5/2

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Figure 13. Mechanisms for Er3+ upconversion in Y2 O3 for 800 nm excitation showing both excited state absorption via the 4 I11/2 or 4 I13/2 levels (left) or energy transfer upconversion from a second Er3+ ion (right).

4 4 Ratio [(2H - 815 nm 11/2, S3/2 ) / F9/2 Ratio (Green/Red) - ]815 nm

8

Integrated Area (2H11/2' 4S3/2/4F9/2)

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Figure 14. Graph of the ratio of the integrated areas of the (2 H11/2 $ 4 S3/2 → 4 I15/2 and 4 F9/2 → 4 I15/2 bands [(2 H11/2 $ 4 S3/2 /4 F9/2 ] versus Er3+ concentration for the nanocrystalline material. Reprinted with permission from [93], J. A. Capobianco et al., J. Phys. Chem. B 106, 1181 (2002). © 2002, American Chemical Society.

Er3+ concentration at both (488 and 815 nm) wavelengths (Fig. 14). The authors observed that as the dopant concentration was increased in the nanocrystalline material, the difference in ratios became less significant and was indicative of presence of an emission enhancement. They postulated that a second mechanism was responsible for populating the 4 F9/2 level only. The laser light (815 nm) excited two neighboring ions to the 4 I9/2 state. One ion nonradiatively decayed to the 4 I11/2 state, while the other decayed to the 4 I13/2 state. A phonon-assisted energy transfer process occurred via two transitions, 4 I9/2 → 4 I13/2 and 4 I11/2 → 4 F9/2 , and resulted in the population of the 4 F9/2 state (Fig. 15). Silver’s group studied the upconverting properties of nanocrystalline Y2 O3 :Er3+ , Yb3+ following HeNe laser excitation at 632.8 nm into the 4 F9/2 state of the Er3+ ion [60] and observed that the intensity of the anti-Stokes luminescence was 1/15 that of the Stokes luminescence. The power dependence study showed that the upconversion occurred via a two-photon process thus populating the emitting level via ETU. The authors also observed that the Stokes and anti-Stokes luminescence lost intensity when Yb3+ was added to the Y2 O3 :Er3+ matrix indicating the presence of a back transfer mechanism from Er3+ to Yb3+ . The authors [61] studied the dependence of the upconversion on the temperature in Y2 O3 :Er3+ nanocrystals and predicted the relative populations of the 4 S3/2 and 2 H11/2 states using a three-level model. The model, comprised of the 4 S3/2 (Level 2) and 2 H11/2 (Level 3) states as well as the ground

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

757

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Figure 15. Energy level diagram of Er ions in Y2 O3 showing the twophoton energy transfer upconversion process responsible for populating the 4 F9/2 level (exc = 815 nm).

I15/2 state, can be written as   I3 E = A exp − 32 I2 kT

WR3 g3 h?3 WR2 g2 h?2

650

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Figure 16. Room temperature Stokes luminescence of nanocrystalline Y2 O3 :Er3+ , Yb3+ prepared via (a) propellant and (b) wet synthetic routes following excitation with 488 nm. (i) 2 H11/2 $ 4 S3/2 → 4 I15/2 , (ii) 4 F9/2 → 4 I15/2 , (iii) 4 I9/2 → 4 I15/2 , (iv) 4 S3/2 → 4 I13/2 . Inset: Room temperature Stokes luminescence of bulk Y2 O3 :Er3+ , Yb3+ . Reprinted with permission from [138], F. Vetrone et al., J. Phys. Chem. B in press.

(17)

where I2 and I3 are the integrated emission intensities of the 4 S3/2 → 4 I15/2 and 2 H11/2 →4 I15/2 transitions, respectively, E32 is the energy gap between the two states, T is the temperature, and k is Boltzmann’s constant. The preexponential factor A is given as: A=

600

Wavelength (nm)

3+

4

550

3+

(18)

where WR2 and WR3 are the radiative transition probabilities of the two transitions, g2 and g3 are the (2J + 1) degeneracies of the two levels, and h?2 and h?3 are the photon energies of the respective transitions from levels 2 and 3 to level 1. They showed that such an analysis for the C2 and C3i sites yielded straight lines and analysis of the C2 site yielded an energy gap (E32 ) of 628 cm−1 with a preexponential factor (A) of 7.6 while the C3i site showed E32 equal to 244 cm−1 with a A equal to 0.5. The upconversion properties of cubic nanocrystalline Y2 O3 :Er3+ and Y2 O3 :Er3+ , Yb3+ prepared by both the combustion and hydrolysis techniques were studied following excitation with 980 nm into the 4 I11/2 ← 4 I15/2 transition of Er3+ and the 2 F5/2 ← 2 F7/2 of Yb3+ [138]. The authors showed that the Stokes emission spectra of the co-doped bulk sample as well as the nanocrystalline samples prepared by the two different techniques following excitation with 488 nm were similar in both intensity and peak shape to their respective singly doped erbium Y2 O3 spectra (Fig. 16). However, the NIR emission spectrum of the bulk Y2 O3 :Er3+ , Yb3+ sample showed emission from the 4 I11/2 → 4 I15/2 transition centered at approximately 1000 nm but also provided evidence that an energy transfer process from the

excited Er3+ ions to the Yb3+ ions in the ground state was present. Following irradiation of the co-doped bulk sample with 488 nm, only the Er3+ ions should have been excited as Yb3+ has only one excited state in the NIR (∼10,600 cm−1 ) and therefore no emission from the Yb3+ ions should have been observed. However, peaks were observed in the NIR emission spectra and attributed to Yb3+ emission and thus the authors assumed that an energy transfer between the Er3+ and Yb3+ ions was operative. The Er3+ ion was initially excited to its 4 F7/2 state with the 488 nm pump photons. A cross-relaxation process of the form (4 F7/2 $ 2 F7/2 → 4 I11/2 $ 2 F5/2 ) occurred following the initial excitation. After the Yb3+ ion was excited to its 2 F5/2 state, it could emit radiatively, as evidenced by the NIR emission spectrum, or it could transfer its energy back to the Er3+ ion exciting it to the 4 F7/2 state once again. It was apparent from the intensity of the 4 S3/2 → 4 I15/2 transition compared to that of the 4 I11/2 → 4 I15/2 and 2 F5/2 → 2 F7/2 transitions that the back transfer from Yb3+ to Er3+ is favored over the radiative 2 F5/2 → 2 F7/2 emission from the Yb3+ ion. The authors noted that the peaks attributed to Yb3+ emission were very weak and barely detected in the nanocrystalline co-doped samples and were attributed to the high phonon energies from adsorbed H2 O and CO2 inherent in this type of material. In the nanocrystalline material, the energy transfer from Er3+ to Yb3+ was severely limited due to the very efficient multiphonon relaxation from the 4 I11/2 to the 4 I13/2 excited states. The high phonon energies significantly reduced the population reservoir in the 4 I11/2 state and since this mechanism involved this state, the process became highly inefficient. The upconversion spectra of nanocrystalline and bulk Y2 O3 :Er3+ , Yb3+ following excitation with 980 nm radiation

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

were presented, and the spectra exhibited three distinct emission bands centered at approximately 530, 560, and 670 nm and correspond to green emission from the 2 H11/2 $ 4 S3/2 and red emission from the 4 F9/2 excited states to the 4 I15/2 ground state of the Er3+ ions, respectively (Fig. 17). In the power study of the upconverted luminescence, the graph of ln(intensity) versus ln(power) showed no inflection point and yielded a slope of n equal to approximately 2 for all samples under investigation. Thus, the authors proposed that two photons were involved in the upconversion mechanism responsible for populating the green and red levels. Following 980 nm irradiation of Y2 O3 , the Er3+ ion was excited to the 4 F7/2 state via two successive energy transfers from the Yb3+ ions in the 2 F5/2 state (Fig. 18). Thus, one Yb3+ ion transferred its energy to an Er3+ ion in the ground state, thereby exciting it to the 4 I11/2 state. This process was followed by a transfer of energy from another Yb3+ ion also in its excited state and resulted in the population of the 4 F7/2 state of the Er3+ ion. Of course, interactions between two Er3+ ions could not be ignored and the authors state that an NIR photon from the pump beam will also excite an Er3+ ion to its 4 I11/2 state. Another Er3+ ion also in the 4 I11/2 state and in close proximity would transfer its energy to the initial ion thereby exciting it to the 4 F7/2 state (Fig. 18). However, following the addition of Yb3+ ions to nanocrystalline Y2 O3 :Er3+ , this process was greatly diminished due to the large absorption cross-section of the ytterbium ions. In the aforementioned mechanism, the authors expected the green emission to dominate the spectrum, as was the case when directly exciting the 4 F7/2 state with 488 nm radiation

Normalized Intensity

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Figure 17. Room temperature anti-Stokes luminescence of nanocrystalline Y2 O3 :Er3+ , Yb3+ prepared via (a) propellant and (b) wet synthetic routes following excitation with 978 nm. (i) 2 H11/2 $ 4 S3/2 → 4 I15/2 , (ii) 4 F9/2 → 4 I15/2 . Reprinted with permission from [138], F. Vetrone et al., J. Phys. Chem. B in press.

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Figure 18. Mechanism for Er3+ upconversion in Y2 O3 for 980 nm excitation showing the energy transfer upconversion between two Er3+ ions and the energy transfer from the Yb3+ to Er3+ ions responsible for populating both the 4 F7/2 and 4 F9/2 states.

since the 4 F9/2 state was populated via multiphonon relaxation from the 4 S3/2 state. The authors observed a change in the relative intensities between the green and red upconverted emissions in the nanocrystalline material in that the red emissions dominated the spectrum unlike the bulk sample where the green emission dominated. The authors proposed that an ion-pair process of the type (4 F7/2 $ 4 I11/2 → 4 F9/2 $ 4 F9/2 ) was responsible for directly populating the 4 F9/2 state and bypassing the green 2 H11/2 and 4 S3/2 states. A slight enhancement of the population in the 4 F9/2 state was also observed in the bulk sample albeit to a lesser extent than in the nanocrystalline material. The ion-pair process occurred via two energy transfer processes and thus should have occurred with equal probability in both the bulk and the nanocrystalline samples having the same dopant concentration. The authors state that this process did not account for the drastic differences in the magnitude of the red enhancement between the bulk and nanocrystalline samples. The magnitude of the red enhancement was slightly different in the two nanocrystal samples prepared by different techniques and so the authors studied the microstructure of each sample determining that the microstructure played only a minor role in the upconversion. The distinguishing factor between the bulk and nanocrystal materials was the and OH− impurities on the surface presence of the CO2− 3 of the nanocrystals. They proposed a mechanism which utilized these high-energy phonons. Following the initial energy transfer from the Yb3+ ion, the large vibrational quanta associated with these impurities allowed for an efficient depopulation of the 4 I11/2 state, which resulted in the subsequent population of the 4 I13/2 state. Since the gap between the 4 I11/2 and 4 I13/2 states is approximately 3600 cm−1 a few highenergy phonons in the nanocrystalline material could easily bridge the gap. In contrast, the bulk material required many more phonons to bridge the same gap as it utilized only the intrinsic phonons of yttria. Following the multiphonon relaxation to the 4 I13/2 state, another energy transfer from the Yb3+ ion in the 2 F5/2 state excited the Er3+ ion directly to the 4 F9/2 state. An excess energy of approximately 1600 cm−1

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

was present, which was dissipated by the nanocrystal lattice to conserve energy (Fig. 18). In order to show that this mechanism was operative, the authors presented the upconversion spectrum of bulk Y2 O3 :Er3+ , Yb3+ following excitation with 1064 nm and observed that 4 F9/2 → 4 I15/2 emission was more intense relative to the (2 H11/2 $ 4 S3/2 → 4 I15/2 emission when compared to the spectrum obtained following excitation with 980 nm. The enhancement in the bulk material occurred because the 4 F9/2 ← 4 I13/2 transition is nearly resonant with the 1064 nm line of the Nd:YAG laser. Since the excess energy in this transition was on the order of about 800 cm−1 , the probability of the process occurring in the bulk material was much greater thereby resulting in an enhancement of the 4 F9/2 population. The infrared-to-visible upconversion of nanocrystalline Y2 O3 :Er3+ , Yb3+ prepared using an emulsion liquid membrane was also studied by Hirai et al. [72] following excitation with 960 nm. The authors observed a predominantly red emission and attributed it to the (4 F7/2 $ 4 I11/2 → 4 F9/2 $ 4 F9/2 ) ion-pair energy transfer process, which bypassed the green (2 H11/2 and 4 S3/2 ) emitting levels. The authors showed the variation of the 4 F9/2 → 4 I15/2 transition intensity as a feed function of the Yb3+ ion by measuring the ratios of Yb concentration to the overall M3+ concentration, BYb/Y + Yb + Er Cf . The red emission was seen to increase slowly from a value of BYb/Y + Yb + Er Cf = 0 to a maximum of BYb/Y + Yb + Er Cf = 008, after which it decreased slowly. Thus, the authors state that the sensitization effect of the Yb ion on the activator Er ion varied with Yb ion concentration. At higher Yb concentrations, the Yb ions may act as trapping centers and dissipate the energy nonradiatively, instead of transferring it to the Er ion. Similarly, the authors studied the variation of the 4 F9/2 → 4 I15/2 transition intensity as a function of the Er3+ ion concentration. They showed that intensity increased rapidly from a starting value of BEr/Y + Yb + Er Cf = 0 to a maximum at BEr/Y + Yb + Er Cf = 01 and then decreased with further increase of Er3+ ions. This behavior may have been due to cross-relaxation or energy migration between activator ions or to quenching sites where the excitation energy was lost nonradiatively. These quenching sites could be impurities or defects within the lattice, which were invariably present.

5. APPLICATIONS Yttrium oxide is a material with a wide range of applications. We have chosen to concentrate on the three main applications of nanocrystalline Y2 O3 : optical and electrical, mechanical (refractories and ceramics), and catalytic. In each case the application will be briefly discussed along with the possible advantages of utilizing the nanosized version of the oxide. In the majority of cases, the nanosized particles exhibit significantly different properties from those of bulk materials.

5.1. Optical and Electrical 5.1.1. Display Devices The visible-light-generating components of emissive, full color displays are called phosphors. Phosphors are composed of an inert host lattice and an optically excited activator, typically a 3d or 4f electron metal. The development

759

of new types of high resolution and high efficiency conventional and planar displays has created a need for optical phosphors with new or enhanced properties. For application in the emerging full color, flat panel display industry, thermally stable, high luminous efficiency, radiation resistant, fine particle size powders are required. The demands of these newer technologies have produced a search for new materials and synthesis techniques to improve the performance of phosphors. High efficiency phosphor materials are the key for development of these new devices [56]. Rare earth containing compounds have long found uses as phosphors in lamps and display devices due to sharp, intensely luminescent f –f electronic transitions. Micrometer sized Y2 O3 :Eu3+ phosphors have been used since the 1970s as the red component in television projection tubes and fluorescent lighting devices [139]. This material has been utilized for some time due to its efficient luminescence under UV and electron beam excitation [65]. The red emission is due to a 5 D0 → 7 F2 transition within the europium impurity. Though Eu3+ is the most common utilized of the rare earth ions in oxide phosphors, it is possible to dope various other rare earth ions such as Er3+ , Tb3+ , Tm3+ , Dy3+ , Ho3+ , Pr3+ , and Sm3+ , either singularly or in combinations, to achieve visible luminescence. Thus, these materials are also possible candidates for use as phosphors in numerous lighting and display devices. The CRT is still the most commonly used display device in the world today. A basic CRT utilizes a heated cathode, two sets of anodes (focusing and accelerating), and steering coils to produce and direct electrons at phosphors coated onto a screen to generate light and thus a picture. The commercial phosphors utilized in the majority of current CRTs are micrometer in size due to problems encountered such as decreased luminescence as one goes to smaller particle sizes. If it were possible to produce smaller spherical particles in the nanometer range with luminescent properties similar or improved over their current micrometer counterparts, then these particles could be easily processed into smaller pixels than those currently used in conventional CRT screens. This opens up the possibility of higher resolution which is desirable for the future of high definition television (HDTV) [94]. The potential for forming a densely packed layer (because of the size and shape of the particles and their self-assembly abilities) should also improve any aging problems encountered with traditional phosphors [94]. Plasma display panels (PDP) are a new generation of large flat panel displays. PDPs are based on multiple microdischarges working on the same basic principles as the aforementioned fluorescent tubes. These PDP modules consist of a cell structure where a combination of gases is confined. Each cell has electrodes that enable gas ionization under an electric field. The UV rays that are created from this plasma gas excite a phosphor layer that converts them into red, green, or blue color. The PDP is the only technology able to provide direct view displays with diagonals in the range 30 to 65 inches, which can be viewed under roomlight or sunlight conditions. The high energy UV photons (147 nm, 8.5 eV) that impinge on the phosphor powders cause reduction in luminous efficiency of the display over time because of radiation damage induced in the material [44]. Thus, oxide phosphors were found to be optimal for

760

Yttrium Oxide Nanocrystals: Luminescent Properties and Applications

PDP devices compared to other conventional sulfide based phosphors which suffer from this type of degradation. It is also hoped that the proposed increase in luminescence in the nanocrystalline oxide phosphors will result in brighter and more energy efficient PDP displays. The field emission display (FED) has been recognized as one of the most promising technologies for flat panel display. The FED functions on principles similar to the CRT as a beam of electrons are utilized to excite the phosphors, which then emit light. However, they differ in that the FED uses multiple electron emitters for each pixel arranged in a grid. This technology eliminates the beam steering system of the CRT and allows screens to be manufactured as thin as 10 mm. Another benefit over CRTs is that the electrons are not created by heat and thus the display does not need to warm up and produces much less unwanted heat. FEDs also have several advantages over the current choice for flat panel displays, liquid crystal devices: they require no back light, are very light, and have a wide viewing angle, a short response time, a very high contrast ratio, and excellent color properties. Recent advances in FED technology have placed a requirement on the development of a corresponding screen technology. An important field of practical application of the FED is the development of phosphors, which show high efficiency and good stability at low-voltage electron excitation and high current density. However, phosphors synthesized through conventional methods are inappropriate for the FED due to reduced cathodoluminescent efficiency, large particle size, and lack of morphological control. Therefore, in recent years luminescent nanostructured materials have become attractive for FED applications. Compared with a conventional CRT screen, an FED operates with lower energy (3–10 keV) but higher current density (1 mA/cm2 ) beams impinging on the phosphors. This requires more luminously efficient and thermally stable materials [44]. Luminous efficiency is defined as the ratio of the energy out (lumens) to the input energy. Due to the low excitation voltage, electron penetration into the luminescent particles is very shallow. This means that the effective luminescent region may be confined near the surface of particles and that the low voltage excitation favors small particles or large specific surface area. As a result, the small size of nanoparticles allows complete penetration by low-voltage electrons for efficient material utilization [67]. In addition, sulfide-based phosphors used in the traditional CRT are known to be rapidly degraded at high current density needed on FED. This outgassing from the highly efficient sulfide-based phosphors has been shown to degrade the cathode tip of the field emitter array and cause irreversible damage [44]. Thus, the study of oxide-based phosphors which are stable at the high current density has progressed in many fields [68]. There is, therefore, a need to develop oxide-based phosphors that display a higher efficiency than the materials conventionally used in CRT screens under low to medium electron accelerating voltages [65]. In the case of oxide-based red phosphors for FED application the most promising material to date is Y2 O3 :Eu3+ particles. Another requirement is on the particle size distribution: there is a maximum and minimum particle size limitation to the powders. For FED applications about five particle layers

are required to achieve optimal light output. Large particles (>8 m) require thicker layers, increasing the phosphor cost and also producing more light scattering. Additionally, the pixel pitch (∼250 m) places a maximum on particle size. It is therefore advantageous that the phosphor particle size be as small as possible as this potentially leads to higher screen resolution, lower screen loading, and a higher screen density [65]. Unfortunately, it is usually found that smaller particles (