Epioptics-7, Proceedings Of The 24th Course Of The International School Of Solid State Physics: Proceedings of the 24th Course of the International School of Solid State Physics 9789812702982, 9789812387103

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EPIOPTICS-7

THE SCIENCE AND CULTURE SERIES - PHYSICS Series Editor A Zichichi European Physical Society Geneva Switzerland Series Editorial Board P G Bergmann J Collinge V Hughes N Kurti T D Lee, K M B Siegbahn G 't Hooft P Toubert E Velikhov G Veneziano G Zhou

1 Perspectives for New Detectors in Future Supercolliders 1991 2 Data Structures for Particle Physics Experiments Evolution or Revolution?, 1991 3 Image Processing for Future High Energy Physics Detectors, 1992 4 GaAs Detectors and Electronics for High-Energy Physics, 1992 5 Supercolliders and Superdetectors 1993 6 Properties of SUSY Particles 1993 7 From Superstrings to Supergravity, 1994 8 Probing the Nuclear Paradigm with Heavy Ion Reactions, 1994 9 Quantum-Like Models and Coherent Effects 1995 10 Quantum Gravity, 1996 11 Crystalline Beams and Related Issues, 1996 12 The Spin Structure of the Nucleon, 1997 13 Hadron Colliders at the Highest Energy and Luminosity 1998 14 Universality Features in Multihadron Production and the Leading Effect, 1998 15 Exotic Nuclei 1998 16 Spin in Gravity Is It Possible to Give an Expenmental Basis to Torsion? 1998 17 New Detectors 1999 18 Classical and Quantum Nonlocality 2000 19 Silicides Fundamentals and Applications, 2000 20 Superconducting Materials for High Energy Colliders, 2001 21 Deep Inelastic Scattering, 2001 22 Electromagnetic Probes of Fundamental Physics, 2003 23 Epioptics-7,2004

EPIOPTICS-7 Proceedings of the 24th Course of the International School of Solid State Physics

Erice, Italy

20-26 July 2002

Editor

Antonio Cricenti Series Editor

A Zichichi

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v

World Scientific

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PREFACE This special Volume of World Scienhfic contains the Proceedings of the 7th Epioptics Workshop held in the Ettore Majorana Foundation and Centre for Scienbfic Culture Ence Sicily from July 20 to 26 2002 The Workshop was the 7th in the Epioptics senes and the 24th of the International School of Solid State Physics Antonio Cncenb from CNR Isbtuto di Struttura della Matena and Theo Rasing from the University of Njimegen were the Directors of the Workshop The Advisory Committee of the Workshop included Y Borensztein from U Pans VII P Chiaradia and R Del Sole from U Roma I1 Tor Vergata 0 Hunden from U Trondheim J McGilp from Tnmty College Dublin W hchter from TU Berlin and P Weightman from U Liverpool Fortyfour scientists from fourteen countnes attended the Workshop The Workshop has brought together researchers from universibes and research inshtutes who work in the fields of (semiconductor) surface science epitaxial growth matenals deposibon and optical diagnosbcs relevant to (semiconductor) matenals and structures of interest for present and anhcipated (spin) electronic devices The Workshop was aimed at assessing the capabilities of state of the art optical techniques in elucidating the fundamental electronic and structural properhes of semiconductor and metal surfaces interfaces thin layers and layer structures and assessing the usefulness of these techmques for opbmizabon of high quality multdayer samples through feedback control dunng matenals growth and processing Particular emphasis has been dedicated to dynamical processes through the use of pump probe techniques together with the search for new opbcal sources Some new applications of Scanning Probe Microscopy to Matenal science and bioloDca1 samples dned and in vlvo with the use of different laser sources have also been presented Matenals of parhcular interest have been silicon semiconductor metal interfaces semiconductor and magnebc multilayers and III V compound semiconductors The Workshop as well as the notes collected in this Volume combined the tutonals aspects adequate to a School with some of the most advanced topics in the field whch better charactenze the Workshop We wish to thank our sponsors the Italian National Research Council (CNR) and the Sicilian Regonal Government for facilitabng a most successful Workshop We are grateful to the Director of the International School of Solid State Physics Prof G Benedek to the Director of the Ettore Majorana Centre Prof A Zichichi and to the Centre staff members for the excellent support organizabon and hospitality provided

Antonio Cricenti

V

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CONTENTS Preface

V

Theory of Surface Optical Properties Rodolfo Del Sole Maurizia Palummo Olivia Pulci

1

The Application of Reflectance Anisotropy Spectroscopy to Organics Deposition G Bussetti, C Goletti, P Chiaradia, T Berzina, E Dalcanale S Nardis A Froiio, R Paolesse, E Mazzone, C Di Natale A D’Amico

21

First-Principles Optical Spectra of Semiconductor Surfaces From One-Particle to Many-Body Approach Maurizia Palummo, Olivia Pulci, Rodolfo Del Sole

29

Ab-lnitio Optical Properties of Bn( 110) and GaN( 110) Surfaces G Cappellini G Satta, M Palummo, G Onida

44

Longitudinal Gauge Theory of Second Harmonic Generation at Semiconductor Surfaces B S Mendoza

52

Optical Second Harmonic Generation and the Characterisation of Low Dimensional Nanostructures on Planar Silicon J F McGilp

62

Nonlinear Optical Properties of Oligothiophene Self-Assembled Monolayers on Gold Substrate T Tamura, E Mishina, Y Miyakita Q -K Yu, S Nakabayashi

80

Second-Harmonic Generation and Photoemission from A1 Quantum Wells on Si( 111)7x7 K Pedersen P K Kristensen J Rafaelsen, N Skivesen T G Pedersen P Morgen Z Li S V H o f i a n n

86

Femtosecond Sum Frequency Generation on Molecular Adsorbates Mischa Bonn

95

VII

Vlll

Fabrication and Characterization of Organic Light Emitting Diodes Based on Alq, S Gambino G Baldacchini S Gagliardr S Loreti R M Montereall A Pace F Michelotti F Onorati

105

Optical Spectroscopy and Scanning Near-Field Optical Microscopy on He+Irradiated Lithium Fluoride Crystals V Mussi A Cricenti, R M Montereali, B Jacquiel; P Moretti E Nichelatti F Somma

111

“Supersohtons” in Infinite Arrays of Plane-Parallel Layers or Coupled Identical Optical Waveguides Igor V Gerasimchuk,Alexander S Kovalev

122

Atomic Force Microscopy as a Tool for Biophysical and Cellular Biology Studies M Girasole, A Cricenti

128

Phase Contrast Radiology in Matenals Science Y Hwu Wen Li Tsar, A Groso G Margaritondo Jung Ho Je

143

Resonant Magneto-Optical Phenomena in Structures With Two-Dimensional Electron Plasma v v Popov

157

Nonlinear Magneto-Optics Principles and Applications for Magnetic Thin Films The0 Rasing

168

Inhomogeneous Luminescence from Exciton-Polanton Modes in Quantum Well T V Teperik V V Popov, N J M Honng

184

Ultrafast Magnetization Switching Dynamics The0 Rasing

190

THEORY OF SURFACE OPTICAL PROPERTIES

RODOLFO DEL SOLE MAURIZIA PALUMMO, OLIVIA PULCI Istztuto Nazzonale per la Fzszca della Matena, Dzpartamento dz Fzszca dell’hzverszta dz Roma ‘Tor Vergata , Vza della Rzcerca Scaentzfica 1 IOU133 Roma, Italy The theory of surface optical properties is reviewed including the solution of light propagation equations at surfaces and methods for electronic structure calcula tions Realistic calculations of surface optical properties within the single particle approximation are discussed GaAs(ll0) ans Si(100) are considered as test cases

1 Introduction

Optical spectroscopy is a widely used tool for surface studies and character ization This is due to the fact that the optical probe does not damage the sample and has the best energy resolution On the other hand, due to its large penetration, it is poorly sensitive to the surface region Hence some tricks have been devised to increase its surface sensitivity Two experimental techniques, Surface Differential Reflectance (SDR) and Reflectance Anisotropy Spectroscopy (RAS) 3, have been developed to this purpose In the former case, the reflectance of the dean surface is measured first, and then that of the surface after chemisorption, the difference, the SDR, a few percent of the reflectance, surely originates at the surface The latter technique is used on the surfaces of cubic materials in this case the bulk contribution to reflectance, described by Fresnel formula, does not depend on the direction of light polarization, while the surface contribution, due to the lower symmetry of the surface, may depend on it Hence, the reflectance difference between measurements carried out with two different light polarizations in the surface plane, the reflectance anisotropy (RA), again a few percent of the total reflectance, is generated at the surface The interpretation of these experiments is generally not straightforward Hence the full potentiality of surface optical spectroscopy, as well as of any other spectroscopy, can be exploited only by a strong interaction of experimental and theoretical work The contribution of theorists to this collaboration can be of three types (1) in cases where two or more structural

1

2

models are hypotesized, comparison of calculated with measured optical properties may confirm one model and discard the other ones, (11) if the structural model is well established and a reasonable agreement is found between theory and experiment, the theoretical interpretation helps in iden tifying the origin of the observed structures (for instance, a given peak may be due to transitions between surface states, or alternatively may involve bulk states modified by the surface) (111) strong many body effects, which require a deeper theoretical insight, can occur in some cases After the atomic structure of the surface of interest has been determined by energy minimization or inferred from experiments, three main steps are involved in the theory, 1) the determination of one-electron wave functions, 11) the calculation of the dielectric susceptibility, possibly including many-body effects, 111) the solution of light-propagation equations After the pioneering work of Feibelman 4 , the last problem has been solved in a quite simple way by Bagchi et al for the jellium surface at the end of the seventy’s The solution has been generalized to the case of crystal surfaces by Del Sole in 1981 The surface contribution to reflectance, that is the deviation from Fresnel formulas, can be calculated from the di electric susceptibility of the vacuum-crystal interface This can in turn be obtained, within the one electron or Random Phase Approximation (RPA), from the wave functions These and their energies should be calculated ah initio according to the Green’s function method, using the so called GW approximation for the exchange correlation self energy The electron hole interaction and local-field effects altogether may be included by solving a Bethe Salpeter equation for the interacting electron and hole Calcula tions carried out according to this recipe for some surfaces have yielded excellent agreement with experiments Unfortunately such calculations are too demanding from the computational point of view to yield the optical properties of most surfaces Therefore one must often resort to simpler -yet less accurate methods, allowed by the partial cancellation of self energy, ex citonic and local-field effects The best compromise between speed and ac curacy is to determine the wave functions and energies according t o density functional theory within the local density approximation (DFT LDA), and to describe many body effects through a suitable rigid upward shift of the conduction bands with respect to valence bands, the so-called scissors operator A cruder approximation is to describe the wave functions and energies within semi empirical methods, for instance using semi empirical tight binding l o For its computational speed the tight binding method has been employed in 1986 in the first realistic calculations of surface optical properties I * , while it is now used only for very complex systems, where

3 ab-initio methods cannot be applied Quantitative agreement between theory and experiment has been ob tained in the few cases where calculations of optical properties have been carried out according t o the Bethe Salpeter approach All other methods can yield, at best, qualitative agreement 2 Three-layers m o d e l

After some work done by Drude at the end of the nineteenth century t o determine the ellipsometric response of thin films, the modern history of surface optics begins in 1971 with the model of McInthire and Aspnes (MA) l 3 They modeled the dielectric tensor of the crystal vacuum inter face by an isotropic three-layer model a surface layer of depth d (usually a few Angstroms) and dielectric constant E , is sandwiched between vacuum (dielectric constant 1) and bulk (dielectric constant Eb) This model ne glects the nonlocality and anisotropy of the surface dielectric tensor, and approximates its inhomogeneity by a two- step function The solution of light propagation equations is in this case straightforward, although the resulting formulas are somehow cumbersome In order to simplify them, McInthire and Aspnes made an expansion up t o the first order in (w/c)d, which is small, of the order of 1/100, for visible and ultraviolet radiation the zero-order term, which does not account at all for the surface layer, yields Fresnel formulas of reflectivity 14, while the first-order term gives with great accuracy the surface contribution to reflectance, i e the relative deviation from Fresnel formulas l5 According to McInthire and Aspnes 13, it 1s

AR,/R, for s light and

(1 e

= ~ ( w / c ) ~ c o s ~ ' ~-~ t[b () /E ( t b,

-

I)]

(1)

light polarized perpendicularly to the plane of Incidence)

AR,/R, = 4(w/c)dcosBx Im[[(€b -SZn28)(€,

-Eb)

+€zS%n26'(E,1- € l l ) ] / ( E b -

1)(Ebcos28- S%n2e)](2)

for p light (1 e light polarized within the plane of incidence) Here 0 the angle of incidence and the w dependence of the dielectric functions understood

IS

IS

3 Microscopic calculation of the surface contribution t o reflectance

The solution of light propagation equations at a surface has been given in an elegant and closed form by Bagchi et a1 in 1979 for jellium surfaces,

4

and later generalized to crystal surfaces by Del Sole The electric field is written as the sum of a zeroth order term, corresponding to Fresnel formula, which ignores all surface features, plus a surface correction, of first order in d/X, where d is an estimation of the surface thickness and X is the llght wavelength The results for the reflection coefficients of s and p radiation are

AR,/R,

=~(~/c)cosI~~T AtYy T z [> < /(Eb

-

l)]

(3)

for incident s light, polarized along y, and

AR,/R,

= 4(w/c)cosL9Irn[[(~~ - szn20)
I/(%

-

AcXx> +

1)(tbcos28- szn2L9)],

(4)

for incident p light, polarized in the xz plane (the plane of incidence has been assumed to be coincident with the xz plane) All surface features are >, defined embodied in the quantities < Atxx >, < AtYv> and < At;: as follows

< AE,, >=

/

dz /dz’[t,,(z,

/

dz’[t;:(z, z’) - 6(z - Z ’ ) / E ~ ( Z ) ] ,

2‘) -

to(z) -

for i=x,y, and

< At;:

>= J’dz

(6)

with EO(Z) equal 1 in vacuum ( z < 0), and Eb in bulk ( z > 0) The offdiagonal components of the dielectric tensor mix s and p waves, so that, in the case of incident s wave (p wave), the reflected light has also a p wave (s wave) component, which is however of second order in (w/c)d Therefore such mixing can be neglected in the first-order solution which is of interest here The parameters < At,, >, < Aeyy > and < At;: >, having dimension of length, completely describe surface optical properties In addition to reflectance, also surface effects on light transmission through a slab can be calculated in terms of them l6 Power absorption at the surface is then readily obtained from energy balance, too l8 Ellipsometry can also be obtained in this way l 5 Therefore, even without calculating those parameters from wave functions, a number of experiments can be coherently described in terms of them

’’

5

In the case of the jellium surface the dielectric tensor is diagonal and isotropic in the surface plane Therefore s or p reflectances do not de pend on how the plane of incidence crosses the surface, that is the optical response is isotropic in the surface plane Moreover, the xz and yz com ponents of the dielectric tensor vanish as a consequence of the cylindrical symmetry around z l9 In the case of real crystals, on the other hand, there may be anisotropy also in the surface plane, since < AE,, > is in general different from < AcYy > It is remarkable, however, that the xy element of the dielectric tensor, which has never been assumed to vanish in the present treatment, does not appear in the first order formulas (3) and (4) it gives rise to the second-order mixing between s and p reflected light dis cussed above On the contrary, the dielectric tensor components E,,(z, z’) and eyz(z, z’) contribute to determining the surface response functions, according t o ( 5 ) The fourfold integrals in ( 5 ) are very difficult to evaluate fortunately, eZz(z,2)and cyz(z, z’) vanish at some surfaces because of the symmetry, and in other cases are estimated t o give a small contribution to the reflectance 2o Hence they will be neglected from now on Another point worth mentioning is the frequency range where the present formulation is valid It is apparent from ( 5 ) and (6) that this for mulation breaks down whenever it is impossible to invert E_,(z, z’), that is close t o the frequency of a bulk or surface plasmon Hence thc present formulation for p light is valid only below the frequency of the surface plasmon Instead that for s light, as far as the off-diagonal elements of the dielectric tensor are neglected as discussed above, does not contain < E&? >, and is therefore always valid Finally, having obtained reliable expressions for the surface contribu tions to reflectivity of s and p light, It is worth discussing the validity of the popular MA model Since in the latter the surface dielectric constant E, is a phenomenological parameter, one may hope the MA model to be exact with a suitable choice of cs This is readily seen by comparing the MA formulas (1) and (2) with the corresponding microscopic results, (3) and (4) It is clear that, in order to describe s light reflectivlty, one should choose E , = Eb + < AcYv>Id, on the other hand, for p light, one should choose E , = Eb < AE,, >f d and 1f c s = 1f E b + < AE;: > I d at the same time Since this is not possible in general, we must conclude that the MA model is not able to describe the surface contribution t o the reflectivity of p light The simplest model able to do this is the anisotropic three layer model, where the surface layer is allowed t o be anisotropic, with a diagonal dielectric tensor eft given by

+

6

In order the original isotropic MA model to be correct, it is required that the microscopic dielectric tensor is completely isotropic (which is not verified even at the jellium surface’), and, moreover, according to the last equation, that the spatial average of the inverted susceptibility equals one over the spatial average of the susceptibility, that is, the complete absence of anisotropy and nonlocality is required 4 Realistic calculations

The goal of surface optical-property calculations is t o determine the surface response functions ( 5 ) and (6), and therefore the reflectance and other measurable properties, from the electron wavefunctions of a semi infinite system A problem arises from the fact that electronic calculations for a semi infinite crystal are extremely cumbersome Because of the lack of periodicity along z, the calculation cannot be restricted to a single layer, but should be carried out for a semi-infinite stack of layers along z Some methods have been developed to cope with this problem, where the surface is considered a short-range perturbation on an infinite solid for instance, in a firstneighbor tight-binding scheme, deleting the interactions between atoms in two adjacent layers of an infinite crystal will create two non interacting semi-infinite solids The electron wavefunctions are then found in terms of the Green’s function of the infinite solid and of the surface perturbation Nevertheless the determination of the eigen(sec, for instance, Ref (’I)) values and eigenfunctions is very cumbersome for each value of E, one has to calculate and invert the Green’s function matrix (of the order of 20x20), and look for the particular value of E for which the secular equation is satisfied This must be repeated for each state! On the contrary, the usual expansion methods used in the bulk yield, with a single matrix diagonalization, all eigenvalues and eigenfunctions For this reason, realistic surface calculations are usually carried out in a slab geometry That is, the semi infinite crystal is replaced by a slab if this is thick enough, each surface of the slab should be representative of the surface of the semi-infinite crystal In practice, this is achieved by choosing the slab thickness in the range from 10 to 30 monolayers Let us consider s light reflectance In the slab geometry, equation (3) can be written as 2o

7

whcre the half slab polarizability a;; (w)has been introduced

1 j’

m

03

4 7 r a 3 4 = (1/2)

dz

-M

dz”L$b)(z,z’,W)

-

b ( z - z’)],

(11)

M

dividing by 2 the full-slab polarizability The quantity in the square brack ets on the right-hand side of (11) rapidly approaches zero for z and/or z’ outside thc slab, therefore the twofold integral over z and z’ convergcs with out problems Notice that a;: has dimension of Icngth, due to the twofold z-integral on the right-hand side of ( l l ) ,so that the reflectance (10) is a pure number Within the single particle scheme, the imaginary part of the slab polarizability is related to the transition probability induced by the radiation between slab eigenstates 2o

Im47ra,hC(w)= (4n2e2/m2w2A)CgC,cJp: c(z)]2S[Ec(lc’) - E,(z)

-

tw],(12)

4

where p: .(k) is the matrix element of the y-component of the momentum operator between initial (valence, v) and final (conduction, c) states at thc point in the two dimensional Brillouin Zone, and A is the slab area The real part is computed via the Kramers kronig transform Another ingredient of (10) is the bulk dielectric function E b ( W ) Its imaginary part is analogous to (12), where now eigenstates and eigenvalues refer to the infinite crystal, and the vectors in the summation are three dimensional Again the real part is computed via the Kramers Kronig transform

z

5

Tight-binding calculations

As shown above, the study of the optical response of surfaccs within the single particlc picture amounts to calculating (1) the single particle spec trum, and (11) the transition probability between occupied and unoccupied statcs both for the perfect crystal and for a slab whose surfaces have the bame structure as the one under consideration The first realistic calcula tion of surfacc optical properties has been carried out in 1986 for S1(111)2x1 using a tight binding approach They startcd from Si by Selloni et a1 bulk bands calculated using the tight binding Hamiltonian of Vogl et a1 l o , which involvcs s p and s* orbitals (5 per atom) The s* orbitals mimic the d orbitals, needed to yield a good description of the first conduction band Then they calculated the electronic states and the optical properties of a Si slab of 18 atomic layers along thc [111] dircction, with the surfaces rcconstructed according to Pandey’s chain model 22 Electron wave functions were expanded in the tight binding base mentioned above, according

to

23

8 where n is a band index and

+lz(F‘j are Bloch sums built with the orbitals

41 ?I, l k-(F‘j = N-1/2cEezG

“l(+

g), Here 2 labels a vector

(14)

and the c,l(Z),s are expansion coefficients of the two dimensional Bravais lattice parallel to the surface, and k is a vector in the corresponding two-dimensional Brillouin Zone As it is usual in semiempirical tight binding, the matrix elements of the Hamiltonian between neighboring orbitals (we consider only intraatomic and first-neighbor mnteractions) are fitted t o reproduce bulk energy bands (The fitted matrix elements for group IV and 111-V semiconductors are given by Vogl et a1 lo) Then, when atomic distances at the surface are different with respect t o the bulk case, the interatomic matrix elements of the Hamiltonian are allowed t o vary with distance according t o the l / d 2 rule of Harrison 24 The matrix elements of the momentum operator between eigenstates were found in terms of those between localized orbitals using (13) and (14), the momentum matrix elements between localized orbitals were obtained from the commutator of Hamiltonian and position operator 4

getting

< 1 ~ l ~ >< l ”1 ”~g I H l l ’ ~>]

(16)

The interatomic matrix elements of the position operator r‘ were ne glected, taking advantage of the orthogonality of the orbitals Of course only a few terms are nonvanishing in the sum in (16), since the Hamiltonian couplcs only first-neighbor orbitals Only two parameters are needed, in addition to the Hamiltonian ones, namely the intratomic sp and s*p dipoles, which were fitted t o reproduce at best the bulk dielectric function of silicon < slz)p,>= 0 27A , and < s*lz)pz>= 108A This approach has been applied to a number of clean surfaces, among which S1(111)2x1 si(111)7X7 25, S1(100)2xl 26, the (110) surfaces of 111-V compounds 27, and also to some adsorbate covered surfaces, as S i ( l l l ) / A s ”, GaAs(llO)/Sb 29 and InP(llO)/Sb 30, generally yielding good results An improvement of this method has been made by Chang and Aspnes 31 and by Ren and Chang 32 for the case of GaAs(100) the most important d states (two) are used instead of s*, and the momentum matrix elements

9

are fitted to those calculated within the empirical pseudopotential method It has been seen that including the d states rather than s* is important for higher energy transitions, above 4 eV in GaAs The greatest success of the tight-binding method in the calculation of surface optical properties has been the calculation of the SDR of the very complicate 7x7 reconstruction of Si(111), where ab-initio methods are computationally too demanding to be used In spite of such complexity, the calculated SDR is in very good agreement with the experiment, and all the experimental spectral features up to 3 5 eV are clearly explained in terms of transitions across surface states 25 As a consequence, their behaviour as a function of hydrogen coverage has been interpreted, shedding light on the process of hydrogen adsorption 33 6 Self-consistent pseudopotential calculations

One can improve on the tight binding method by using the selfconsistent electronic structure determined using pseudopotentials with a plane-wave basis set, within the local density approximation to density functional theory A repeated slab approach must be used to deal with the symmetry breaking introduced by the surface and to preserve the tridimensional periodicity of a bulk calculation A superlattice is constructed in this way In this type of calculations, the wavefunction is expanded in plane waves

where the (?s are the vectors of the reciprocal superlattice and the c , ~ ( i ) k representing the oneare expansion coefficients The matrix H z 6 electron Hamiltonian

(c),

H =p2/2m

+ v,,,(q +

S

d 3 + ( ~ ) / i ~ $1-

+ v,m

(18)

in the plane-wave basis, is calculated and diagonalized, obtaining the eigenvalues and the expansion coefficients The second term on the right hand side of (18) is the potential energy of the electron interaction with ion cores, the third term is the electron electron electrostatic interaction (the Hartree term), involving the electron density p(f7, and the fourth term is the exchange-correlation potential, usually obtained from that of the freeelectron gas by means of the Local Density Approximation (LDA) s Since the electron density appears in the Hartree term, as well as in the exchangecorrelation potential, the matrix elements of the Hamiltonian must be cal culated self consistently with the electron density, which is an output of the

10

calculations This is done by iterating the calculation (usually a few times) until the input and output charge densities are equal The valence and conduction wave functions in which we are interested must be orthogonal to core states this requirement enforces them to have rapid oscillations near the ion cores, so that a large number of plane waves is required In order to avoid this, a pseudopotential is used rather than the true potential V,,,(F,) 23 The pseudopotential equals the true potential outside the cores, and is smoother near the cores, in such a way that core states do not appear among the eigenstates of the Hamiltonian (18) This trick overcomes the orthogonalization of valence and conduction states t o core states the ultimate result is that a smaller mumber of plane waves is required to expand the wave functions of interest, making their com putation easier Many types of pseudopotentials have been defined since the introduction of this concept state of-the art calculations employ nonlocal ( i e different pseudopotentials are used for s, p d atomic orbitals) and norm conserving (1 e the normalization of wave functions is the same as in the case where the true potential is used) pseudopotentials (see, for instance, Bachelet et a1 , Ref (34)) Finally one should mention that using a local energy-independent is based on the density functional the exchange-correlation potential Vzc(f') ory 7 , which is strictly valid only for the ground state of a many electron system Therefore excited states, which are of course essential in determining optical properties, should not be described in this framework, but rather in the many-body Green's function formalism 35 This implies changing Vzc(f')into a non- local energy dependent self energy &(F, F , E ) which makes the calculation much more cumbersome As a matter of fact, energy bands calculated according t o the density functional hamiltonian (18), involving Vz,(f'),are close to the experimental ones the nearly unique shortcoming is that the calculated gap between filled and empty states is underestimated, from 0 5 to 2 eV, going from semiconductors to insulators 37 Since generally has been found that such underestimation is almost con stant for all k points, this shortcoming can be fixed by increasing the gap between filled and empty states by a fixed amount, the so-called scissors operator Once the eigenfunctions and the eigenvalues of the Hamiltonian (18) are known, one has t o determine the absorption strength according t o the golden rule formula (12) The matrix elements of the momentum operator between valence and conduction states are obtained using the expansion

11

4

where n and m are band indices A single sum over G is obtained, exploiting the fact that p’ is diagonal in a plane-wave basis

< z + Glpli + 2 >= 6d 6, (Z + G )

(20)

This contrasts with the double sum over the basis orbitals (lzand l’2of (16)) to be carried out within the tight binding scheme and makes faster the determination of optical properties in some cases (depending on the chosen pseudopotential, the geometry and the number of plane waves) using plane waves, rather than localized orbitals Pseudopotential calculations of the optical properties of GaAs and GaP (110) surfaces have been carried for the first time out by Manghi et a1 2o In order to save computer time, a local pseudopotential was used, containing a few parameters adjusted t o fit the energy levels of the free ion After that, many other calculations have appeared, the first, complete ab initio calculation using norm conserving nonlocal pseudopotentail has been that of Pulci et a1 38, which will be briefly discussed in the next Section

7 Application to G a A s ( l l 0 ) We consider GaAs(ll0) as a first example for the application of the tech niques discussed above The (110) surfaces are the cleavage faces of I11 V compounds No re construction is observed, but a relaxation of the surface occurs the bonds in the first layer rotate in such a way that the anion goes a little bit out of the surface, and the cation goes a little bit inside 39 The rotation angle is close t o 31 degrees RAS experiments on GaAs(ll0) have been carried out by Berkovits et a1 3 , who observed two structures at 2 6 and 2 8 eV respectively, interpreted in terms of transitions between surface states More recent RAS experiments have been carried out by Esser et a1 3o Calculations for this surface have been carried out within both T B and the ab-mitio P P P W method We here discuss the first complete ab initio calculation 38 and the subsequent inclusion of self-energy effects 4o A DFT LDA calculation of electron states has been carried out for repeated slabs of 11 atomic layers each, separated by 7 empty layers, with the surfaces relaxed according t o the model described above The wave functions have been expanded in plane waves, up to a maximum kinetic energy of 15 Rydbergs using a nonlocal norm conserving pseudopotential of Bachelet

12

GaAs(110) _ _ _ LDA

GW

5

/ 1

10:

/-’

Y

x

P

wE

00 :

-1 0 :

x

M

X’

r

Figure 1 Calculated electronic band structure for the GaAs(ll0) surface Light shaded areas represent the bulk projected bands calculated within DFT LDA Dark shaded areas GW bulk projected band structure Dashed lines surface DFT LDA bands solid lines GW surface states

Hamman Schluter type 34 In this basis the Hamiltonian is represented by a 5,000 x 5,000 matrix In Fig 1 we show the bands in the two dimensional Brillouin zone, where the shaded area represents the projected bulk bands All gaps between filled and empty states are underestimated by DFT LDA, which can be corrected by including self energy effects in the calculation This has been done 40 according to the so called GW approximation 3 5 , and the resulting bands are also shown in Fig 1 After this correction, the band energies are in agreement with those obtained from direct and inverse photoemission experiments, for the filled and empty bands, respectively All bulk gaps increase by about 0 6 0 7 eV, while the gaps between surface states may undergo slightly larger corrections, up to 1 eV 40 This fact, which is rather general, suggests that it is possible to avoid the cumbersome self energy calculation for the slab, using the average bulk value for the gap opening This is equivalent to shifting upwards all empty s t a t ~ sby a constant amount This approach is called the scissors operator

13

GaAs(110) Reflectance Anisotropy

p1

1

\ 2

1

1

1

~

1

1

1,

~

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1

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,

,

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,

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15

20

25

30

35

40

45

50

Energy (eV) Figure 2 Theoretical DFT LDA and GW Reflectance Anisotropy Spectra compared with the experimental one 30 After Ref 40

approximation, since it can be described as cutting the band plot within the gap with scissors and shifting up all empty bands by the same amount In Fig 2 we compare the calculated RAS with the experiment by Esser et a1 30 DFT LDA underestimates transition energies, so that the re sulting spectrum, shown in the upper curve, is redshifted with respect t o the experiment, shown in the lower curve The inclusion of self- energy effects fixes the gap problem, so that the outcoming spectrum, shown in the middle curve, is in better agreement with experiment, as far as the energy positions of the spectral features are concerned (This is a complete GW calculation 4 0 , avoiding the scissors operator approximation ) The residual discrepancies are due to the neglect, in the calculation, of the electron-hole interaction, as anticipated in the introduction For a detailed treatment of many-body effects, including self-energy, local-field and electron-hole inter action effects see the paper by Palummo et a1 in these proceedings

14

A noticeable result of all types of calculations l5O' 3s 40for this surface is that only the lowest energy RAS structure, appearing in the experiment close to 2 7 eV, is due to transitions across surface states All other spectral structures are due, at variance with the assumption of Berkovits et a1 3 , to transitions across surface modified bulk states

8 Application to Si(100) We consider as a second example Si(100) This surface is extensively studied both from the experimental and theoretical points of view, due to its technological importance Now is generally accepted that its atomic structure is characterized by the presence of surface dimers along the [ O l i ] direction, which implies at least a (2 x 1) reconstruction, but two other energetically competing reconstructions are possible, within the same top-atom bonding characteristic The alternation of dimer buckling along the [ O l l ] dimer rows leads to a p(2 x 2) reconstruction, while the alternation of buckling angles also along the direction perpendicular to the rows leads to the c(4 x 2) phase For some time it has been controversial if, in the 2x1 reconstruction, the dimers are symmetric or asymmetric, i e buckled Scanning Tunnelling Microscopy (STM) measurements have revealed both symmet ric and asymmetric dimers, mainly asymmetric ones at low temperature, while most dimers appear to be symmetric at room temperature 43 44 45 SDR experiments on the clean Si(100)2x1 surface have been carried out in 1980 by Wierenga et a1 41 in the visible range, and by Chabal et a1 (1983), in the infrared 4 2 Molecular oxygen was used in the former work to saturate the dangling bonds, while hydrogen, oxygen and water were used in the latter work, with similar results The two measurements are shown in Fig 3 altogether Calculations have been first carried out within a tight binding approach for both models 26 the resulting SDR for the asymmetric dimer model is shown in Fig 4, dashed curve in the lower panel The agreement with experiment is quite good, while the symmetric dimer model (not shown here) predicts a strong peak at about 0 6 eV, which has not been seen in the experiment, arising from transitions from dimer 7r bonds to dimer 7rantibonds (this peak moves to 1 5 eV for buckled dimers, since the buckling increases the gap between the two states) An ab initio calculation of the optical properties of this surface has been recently carried out within the Pseudopotential Plane Wave (PPPW) ap proach 46 A kinetic energy cutoff of 15 Rydberg and a repeated slab of 12 atomic laycrs and a vacuum rcgion of 4 empty laycrs are nccded to obtain

15 0 030

0 020

0 010

0000 00

10

20

30

40

50

60

Figure 3 SDR theoretical spectra obtained considering different reconstructions for the clean surface full line (2 x 1) as a dashed line p(2 x 2) dotted line c(4 x 2) The hydrogenated surface is (1x1) with 2 monolayers of H T h e experimental results from Refs 42 and 41 are shown by squares and dots respectively After Ref 46

a good convergence of the optical spectra of Si(100) A scissors opcrator of 0 5 eV has been used t o fix the gap problem This calculation confirms the Tight Binding findings for the SDR spectrum (see Fig 3, where the are compared with the experiments) ab initio calculations of Ref (“6) Therefore optical spectroscopy discards a substantial presence of symmetric dimers in a broad temperature range, from 40 t o 310 K , where the measurements of Chabal et a1 were carried out 42 The large number of

0 020

- (2x1) LDA 0010

I\ I \ I I

\ \

0 000

-0010

-

-0 020

ENERGY (eV) Figure 4 Comparison of plane wave and T B results for the reflectance anisotropy (defined as the reflectance for light polarized along the dimers minus that for light polarized perpendicularly to the dimers normalized to the average reflectance) (panel a) and for the unpolarized reflectivity (panel b) calculated for Si(100) (2 x 1) After Ref ( 4 6 )

symmetric dimers appearing in STM at room temperature 43 44 45 is probably due to the dynamical flipping of buckled dimers between two equivalent buckling configurations (up-down, down-up), which is time averaged in a STM measurements 47 Differently from the SDR spectra which are qualitatively similar for both approaches (see the lower panel of Fig 4), the RAS, shown in the

17

0 010

0 005

0 000

3

0005

0 010

0 015

0020 1 ENERGY (eV) Figure 5 RA (defined as in Fig 4) theoretical spectra obtained with the different Si(100) reconstructions considered full line (2 x 1) dashed line p(2 x 2) dotted line c(4 x 2) The experimental results 51 scaled up by a factor of 5 are also shown after Ref 46)

upper panel of Fig 4,has opposite signs for most of the energy range of the figure The right result IS, of course, that of the ab-initio PPPW approach, although the TB result coincides with the expectation based on a naive picture of the dimers considered as isolated Szz molecules (greater reflectance is expected for light polarized along the dimers) The discrepancy can be explained on the basis of the omission, in the TB approach, of the direct interaction between atoms belonging to adjacent dimmers In fact, the explanation of this apparently paradoxical sign of the reflectance anisotropy

18

can be found looking at the surface band structure the dispersion of the surface-localized 7r and 7r* bands along the row direction (1 e perpendicu larly to the dimers) is comparable t o the 7r-7r* separation 52 This means that the interaction between adjacent dimers is about as large as the interaction between the p , orbitals of the two atoms in a single dimmer Hence, the correct picture of this surface is that of chains of interacting dimers, oriented in the direction perpendiculr t o the dimers themselves and separated by large valleys through which the dimers interact very weakly PPPW calculations have been carried out 46 also for the low temperature reconstructions, ~ ( 4 x 2 )and ~ ( 2 x 2 ) The agreement between the calculated and measured RAS (shown in Fig 5 ) , the latter taken at room temperature, is satisfactory 46 All spectral features can be explained by a suitable mixing of the three reconstructions A noticeable feature of this surface is that all spectral features up to 3 eV involve surface states as initial and/or final states 9 Conclusion

Thirty years after the formulation of MacInthyre and Aspnes’ three-layers model, a paradigm is now available to carry out microscopic calculations of surface optical properties The DFT LDA scissors operator approach is at present the best compromise between accuracy and a reasonable computational effort Important information about the relevant electronic tran sitions can be obtained in this way, although for a quantitative agreement with experimental data one has t o include in the calculations the electron-hole interaction The semi-empirical tight binding approach with first-neighbor interactions is faster but less accurate, it yields a good de scription, as compared to ab initio results, of the unpolarized reflectance, e g SDR, but not always of the reflectance anisotropy In the latter case, indeed, a very accurate description of wave functions and optical matrix elements is necessary An important point, which has been here discussed with a few examples, is whether or not surface optical spectra are determined by transitions across surface states or not This is clearly the case for photon energies below the bulk gap Clear examples are the infrared peaks observed close to 0 5 eV in SDR for Si and Ge (111)2x1 48 49 As discussed in Section 8, transitions across surface states also dominate the optical spectra of Si(100) up to 3 5 eV, since transitions across bulk states start at 3 eV and are weak below 3 5 eV In the case of GaAs(ll0) discussed in Section 7, instead, surface states are degenerate with bulk bands As a consequence, surface optical structures start at 2 6 eV, well above the direct bulk gap

+

19

(1 5 eV), and RAS as well SDR spectra are dominated by bulk to bulk state transitions However, one cannot infer, as a general rule, that this is always the case when RAS or SDR structures are energetically above the bulk direct gap a counterexample is that of GaAs(lOO)c(4x4) 5 0 , where the termination with a double layer of As makes the surface greatly different from the substrate and leads to strong surface to surface state transitions above the bulk gap In summary, with the exception of the trivial case of photon energy below the bulk gap, the surface or bulk character of a spectral feature in R.AS or SDR cannot be assessed on the basis of its energy only, but needs a thorough theoretical study

References 1 S Nannarone P Chiaradia F Ciccacci, R Memeo, P Sassaroli S Selci and G Chiarotti, Solid State Commun 33,593 (1980) 2 D E Aspnes and A A Studna, Phys Rev Lett 54 1956 (1985) 3 V L Berkovits I V Makarenko T A Minashvih, and V Safarov, Solid State Commun 56 449 (1985) 4 P J Feibelman, Phys Rev B 12 1319 (1975), ibidem 4282 (1975) ibidem 14 762 (1976) 5 A Bagchi R G Barrera and A K Rajagopal Phys Rev B 20 4824 (1979) 6 R Del Sole Solid State Commun 37 537 (1981) 7 P Hohenberg and W Kohn Phys Rev 136 B864 (1964) 8 W Kohn and L J Sham Phys Rev 140 A1113 (1965) 9 M Palummo 0 Pulci and R Del Sole this volume 10 P Voegl, H P Hjalmarson and D J Dow J Phys Chem Solids 44 365 (1983) 11 A Selloni, P Marsella and R Del Sole, Phys Rev B 33,8885 (1986) 12 P Drude, Ann Physik Chem 36,532 (1889) ibidem 865 (1890), ibidem 39,481 (1890), ibidem 43,146 (1891) 13 J D E McInthire and D E Aspnes, Surface Sci 24 417 (1971) 14 J D Jackson, Classacal Electrodynamzcs, John Wiley, New York (1962) 15 R Del Sole, in Photonzc Probes of Surfaces edited by P Halevi Elsevier (1995), page 131 16 R Del Sole Surface Sci 123 231 (1982) 17 L Wolterbeek G F Akkerhuis and A van Silfhout Surface Sci 152/153 1071 (1985) 18 P J Feibelman, Phys Rev B 23 2629 (1981) 19 A Bagchi Phys Rev B 15 3060 (1977) 20 F Manghi R Del Sole A Selloni and E Molinari Phys Rev B 41 9935 (1990) 21 F Bechstedt and R Enderlein, Semzconductor Surfaces and Interfaces Akademie Verlag, Berlin (1988) 22 K C Pandey, Phys Rev Lett 47,1913 (1981), ibidem 49,223 (1982) 23 F Bassani and G Pastori Parravicini, Electronzc states and optacal tran

20

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

42 43 44 45 46 47 48 49

50 51 52

sztzons zn solzds ed R A Ballinger, Pergamon Press (Oxford New York Toronto) W A Harrison Electronzc structure and the propertzes of solzds Freeman San Francisco (1980) C Noguez C Beitia W Preyss, A I Shkrebtii, M Roy Y Borensztein and R Del Sole, Phys Rev Lett 76 4923 (1996) A I Shkrebtii and R Del Sole, Phys Rev Lett 70, 2645 (1993) see Ref (15) page 163 L Reining, R Del Sole M Cini and J Guo Ping Phys Rev B 50 8411 (1994) P Chiaradia, A I Shkrebtii, C Goletti J Wang and R Del Sole Solid State Commun 85, 497 (1993), ibidem 89, 87 (1994) N Esser R Hunger J Rumberg, W Richter, R Del Sole and A I Shkrebtii, Surface Sci 307/309, A 1045 (1994) Y C Chang and D E Aspnes, Phys Rev B 41, 12002 (1990) S F Ren and Y C Chang, Phys Rev B 44, 13573 (1991) C Beitia, W Preyss R Del Sole and Y Borensztein, Phys Rev B 56, R4371 (1997) G B Bachelet D R Hamann and M Schluter Phys Rev B 26,4199 (1982) L Hedin Phys Rev 139 A796 (1965) F Aryasetiawan and 0 Gunnarsson Rep Prog Phys 61 237 (1998) F Bechstedt and R Del Sole Phys Rev B 38 7710 (1988) 0 Pulci, G Onida, R Del Sole, and A I Shkrebtii, Phys Rev B 58 1922 (1998) see Ref ('l), page 251 0 Pulci G Onida, R Del Sole and L Reining, Phys Rev Lett 81, 5374 (1998) P E Wierenga, A van Silfhout, and M J Spaarnay, Surface Sci 99, 59 (1980) Y J Chabal et a1 , J Vac Sci Techno1 A 1,1241 (1983) R M Tromp et a1 , Phys Rev Lett 55, 1303 (1985) R Wiesendanger et a1 Surface Sci 232, 1 (1990) R Wolkow Phys Rev Lett 68 2636 (1992) M Palummo, G Onida R Del Sole and B S Mendoza, Phys Rev B 60, 2522 (1999) A I Shkrebtii R Di Felice, C M Bertoni and R Del sole Phys Rev B 51, R11201 (1995) P Chiaradia A Cricenti, S Selci and G Chiarotti Phys Rev Lett 52, 1145 (1984) G Chiarotti G Del Signore and S Nannarone Phys Rev Lett 2 1 1170 (1968) M Rohlfing, M Palummo G Onida and R Del Sole, Phys Rev Lett 85, 5440 (2000) A Balzarotti, M Fanfoni, F Patella F Arciprete, E Placidi, G Onida and R Del Sole, submitted to Surface Sci R Shioda and J van der Weide, Phys Rev B 57, R6823 (1998) A Ramstad, G Brooks and P J Kelly Phys Rev B 51 14504 (1995)

The application of Reflectance Anisotropy Spectroscopy to organics deposition G Bussetti*, C Goletti* and P Chiaradia*, T Berzina+, E DalcanaleA,S Nardisl, A Froiio’ and R Paolessel, E Mazzone , C Di Natale” and A D Armco”

* Dipartimento di Fisica and Unita INFM Universita di Roma Tor Vergata via della Ricerca Scientlfica I 00133 Roma Italy -k Dipartimento di Fisica and Unita INFM ADipartimentodi Chimica Organica e Industriale and Unita INSTM Universita di Parma Parco Area delle Scienze 17/A 43100 Parma Italy I Dipartimento di Scienze e Tecnologie Chimiche Dipartimento di Ingegneria Elettronica Universita di Roma Tor Vergata 00133 Roma Italy We present the use of Reflectance Anisotropy Spectroscopy (RAS) as a probe to study the deposition of organics We started the study of ordered layers deposited onto disordered substrates by Langmuir Blodgett (LB) and Langmuir Schaefer (LS) techniques nevertheless we believe that the goal is the application of RAS to OMBE (Organic Molecular Beam Epitaxy) to monitor the growth process in real time and in situ similarly to the deposition of inorganic semiconductors After a short introduction we discuss an example testing the RAS sensitivity to the structure of the growing layers in particular we report recent data about the polarization dependence of the optical reflectivity for LS PdCIoOAPporphyrin layers Deposition has been carried out at two values of the surface pressure (n1=30 mN/m n2=10mN/m), corresponding to different layer structures The RAS spectra measured in two cases are successfully explained in terms of the particular morphological characteristics of the layer

When In the nud 80‘s Reflectance Anisotropy Spectroscopy ( U S ) was independently developed by D E Aspnes [l] and by V Safarov [2], its first applications were directed to the study of 111-V sermconductors, to detect bulk anisotropies m the former case, surface states amsotropies in the latter However, only after the first, evident demonstration of its excellent sensitivity to the chermstry of differently terrmnated surfaces, RAS underwent a remarkable expansion [3-41 Those results clearly showed that RAS is able to

21

22

provide detailed, accurate information about the growth of inorganic sermconductors in Ultra-High-Vacuum (UHV), with a nearly one-to one correspondence with Reflected High Energy Electron Diffraction (MEED), without suffering the obvious limtations of electron spectroscopies to the UHV pressure range T h s finding opened the way to the next important success, the use in Metal Oxide Vapour Phase Epitaxy (MOVPE) growth as a real time and zn sztu probe, a role impossible to non-optical spectroscopies [5] Other important applications of RAS followed to clean surfaces of metals and semconductors [ 6 ] ,solid-solid interfaces [7] solid-liquid mterfaces [8], etc More recently, RAS has been applied also to organic layers, showing that spectra are reliably connected to the electronic properties of the deposited molecules and to the morphological characteristic of the layer [9- 131 In this paper we show that RAS is an excellent diagnostic tool for the growth of organic layers, although at the moment the data are lirmted to Langmuir Blodgett (LB) and Langmuir-Schaefer (LS) deposition In fact, a clean and highly controlled deposition technique for organics in UHV has been developed, in analogy to Molecular Beam Epitaxy (MBE) for inorganic sermconductors [ 141 Like MBE, also Organic MBE (OMBE) needs techniques capable to characterize zn situ and in real time the deposited layers the analogy with inorganic deposition shows that optical techniques offer the answer to this need In RAS spectroscopy, the linearly polarized electric field of light is modulated between two perpendicular directions (a and b) of the sample The results are given in terms of the ratio between the measured variation of the sample reflectivity (Ar) and the average reflectivity (r), as a function of the photon wave length

r where r is the complex reflectivity coefficient In the present work only the real part of Ar/r will be presented Experimentally, it is extracted by measuring the second harmonic of Arh signal with a lock-in [ 151 It results that

23

where R" (Rb)is the real reflectance coefficient for a (b, polarized light Usually a and b coincide with well-defined symmetry directions of the system Differently from metals and semconductors, for organic systems the identification of such directions is not straightforward, since they cannot be gathered from the crystalline structure of the material, and must be obtained from the superstructure yielded by growth Experiments were performed using a home-made RAS apparatus All spectra have been recorded in near-normal incidence, in the photon-energy range 1 5 eV- 4 5 eV The samples have been always kept at room temperature, in air The linearly polarized electric field of the light has been modulated between directions a and b shown in the inset of fig (1) aligned with the edges of the substrate This experimental configuration has been chosen measuring the dependence of RAS spectra upon the azimuthal rotation by an angle $ around the axis perpendicular to the substrate (coincident with the incidence direction of light), and searching for the maximum RAS signal amplitude The porphyrin complex used in this work (PdCloOAP, palladium octadecyl porphyrin) has been synthesized according to the procedure described III ref (16) The compounds were dissolved in a mxture of chloroform and hexan (1 2, v/v) in the concentration of 0 33 m g / d The films were deposited with a KSV System 5000 from the surface of pure water prepared with a Milli Q system (resistance of 18 2 MSZ cm) The deposition of the PdCloOAP films was carried out after monolayer compression up to two different values of surface pressure 1) n=30 mN/m (sample #1) ii) n=lO mN/m (sample #2) compressed at the rate of the 10 mm/mn, onto gold-coated glass substrates 10x40 mm2 in size The boundaries of deposited regions were parallel to the short side of substrate (parallel to a ) and the film thickness gradually varied from 0 to 22 monolayers with the increment of 2 monolayers along the b side The width of each region was equal to 3 mm From Atomc Force Microscopy (AFM) experiments performed on simlar samples [17] we know that porphyrin layers, deposited with the chosen values of the surface pressure ll,show different structural arrangements When n=30mN/m, molecules are arranged in horizontal columns formed by stacks of molecules with edge-on orientation relative to the substrate This structure is stable over time When n=lOmN/m, the structure of the deposited layer is time dependent Several days after deposition, AFM shows an ordered array of spirals organized in a two-dimensional lattice with a lattice parameter of several mcrons (see figg 2 and 3 of ref 17)

24 ”””

n=30mN/rn 0

100

-

300

-

200

I

I

I

I

300

400

500

600

I

I0

Wavelength (nm) Fig 1 2xRe(Ar/r) spectra measured at Langmuir Schaefer layers of PdCtoOAP porphynn deposited onto gold (n=30mN/m) Total coverage ranges from 0 to 22 monolayers The absolute sign of the quantity 2xRe(Ar/r)=2 (R Rb)/(R +Rb) is arbitrary In the inset a and b directions (defined above) with respect to the substrate The layers (3 mm large) of different thickness deposited by LS technique have been also sketched

Also in RAS results, we find two distinct, well characterlzed coverage regimes if n=30 mN/m there is a critical coverage 0 = 0, (8 monolayers) 1) below which the line shape shows a large “peak-like ’ structure in the Soret band (-400 nm) region (that is defined as the light absorption of the porphyrin n-system from the fundamental state to the second excited state), augmenting its intensity with coverage Above 0,, the RAS signal shows a large “derivative-like” structure in the Soret band, reducing its intensity with coverage (probably caused by an increasmg surface disorder) (fig (l)), 11)

if n=lO mN/m the spectral line shape remains always the same (apart from a variation in the peak amplitude, due to the different thickness), i e peak-like” as in case i ) when Oting tight-binding (TB) data foi this suiface" 2 Stiuctuial Plopelties of BN(110) a i d GaN(110) Sui faces Deiisitj functional calculation3 have been caiiied out within the local drii5itj appioximatioii (LDA) for tlie exchange aiid coiielation functional' Sepalable, iioiiii-coiiseiviiig >oft peudopotentials liar e been geiieiated nithin tlie scheme of Tioulliei and Maitiiib" Foi Boion, an eneig) cut-off of 55Ry foi the I ~ o h i is h a m oibitals has beeii used Non lineal coie coiiec-

46

tionz (NLCC), have been taken into account in generating pseudopotentialb and pseudowavefunctions for the B atom13 In the case of GaN an energy cut-off of 40 Rydbeig has been used, and NLCC have been consideied for the Ga atom A lowei eneigj cutoff (15 Rldbeig) i b used foi G a 4 . ~ ( 1 1 0 The cleaved (110) suiface of zincblende mateiial3 ietains the piimitive (1x1) peiiodicity and piezentz a well bnown ielaxation pattern At the buiface layei the anion moJes away froin the bulh, favouiing the p bonding with three iieighbouring cation\, while the surface cation move3 into the bull, The zuifavouiing an sp2 binding with three neighbouiing anions'' face dinier rotation angle w and dinier bond contiaLtion Cg foi the thiee buifaces under study ale given in Tab I in comparison with pieviou5 data For nitride5 (110) suifaces, rotational angle.: are nearly halved with re5pect to GaA5(110), and the bond contractions are appreciable, while they aie negligible foi GaAs This behaviour can be ascribed to the highei bond ionicity of the III-nitiides17 l8

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ENERGY (eV) Figuie 1

Calculated R 4 spe~tiurnof the BN(110) suiface

47

3 Suiface Optical Plopexties i iitliiii tlie slah geoiiietir tlie RA bpectiuiii caii be wiitteii abb

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nlieie R, 15 tlie aTeiage of tlie two ieflectiLit7 bigiials, aiid lialf-~lahpolaiizabilit\ nliose iiiiagiiiaiy pait

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(2) In Eq (1) th i b the bull, dielec tiic fuiictioii14 aiid d is tlie half-slab tliichlie>> In Eq ( 2 ) 4 15 tlie tiaii\\eise aiea of tlie +lab aiid = L y s The c alculatioii of CI. aiid tt iiivolres tlie coiiiputatioii of tlie iiiatiix elements of the iiioiiieiituiii opeiatoi betneen aleiic e aiid coiiduc tioii state5 calculated a t tlie DFT-LDA level In Fig 1 u e iepoit tlie calculated ieflectaiice aiiibotiopi \pectiuiii of tlie BN(110) 5urface In the piebent paper \re asbuiiie that the 1 diiectioii i b paiaIIeI to the zigzag chaiiib of the (110) 5uifaLe (1 e 'r = ( i i o ) , r = (001) 111 a( (oidaiice n i t h the liteiatuie" 15) The RA bpectiuiii, defined tlieiefoie as Rk;oRl is doiiiiiiated at tlie oiiset bv tlie reflectivity aloiig the E diiec tioii, with a fiist pobitibe stiuctuie pobitioiied aiouiid 3eV Tliib f i i b t ztiuc tuie 15 followed b\ a sliaip negative peal, at 5eV In Fig L we iepoit tlie baiiie yuaiitity ohtaiiied in oui calculatioiis foi CTaAb(110) aiid (+aN(110) in (oiiipaiizoii n i t h T B data" Oui aL - initio RA bpectiuiii al5o zlionb a fiibt iiegatile peah i n the GaN(110) case Thib hehaLioui 1) diffeieiit to what happeiib i n Gais(110) Foi t l i i b bkbteiii oiie find5 a fiibt pobitire peal, at lowei eiieigies iii tlie R 4 5pec tiuni Tlie fiist stiuctuie at the R A bpectiuin foi GaAs(l1O) coiiieb fiom buiface-buiface ( 5 5 ) tianbition3 a t 1 point of the SBZ arid is defiiiitely polaiized aloiig the zig-zag chain diiectioiil" 'O See also ieceiit tlieoietical iesults and tlieii coinpaiiboii with expeiiiiieiits in the uoih of Biodeiseii aiid Scliatthe In tlie abe of CTaN(110) the fiibt negative peal, lias been asbigiied 114 Noguez l1 to suiface-bull, ( s b ) coiitiibutioiib folloned at liigliei eiieigieb by zuifac e-uiface ( $ 5 ) tiaiibitioiib Foi these tialibition5 tlie I c oiiipoiieiit of tlie polaiizability is laigei than the 1 oiie, giving iibe to tlie negative btiuctuie 111 tlie R A bpectiuiii at loit erierg) foi GaN(11O) Tlie iiiteiplay betwren sb aiicl 5 5 contiibutioiis yields iii the T B data a negative plateau alm o d 0 5 e I n ide which follons at higher eiieigies tlie f i i b t negative slope In

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ENERGY (eV) Figuie 2 R 4 s p e d i a of the (110) surface foi GaN and GaAs(l1O) obtained in the present woih Results for GaN(110) by Noguez l 1 ale also iepoited foi cornpailson

Fig 3 we iepoit the imaginarvpait of (J)(times 4nd) foi GaN( 110) and BN(110), obtained in the piesent calc ulationb To gain inore inbight in the origin of the RA spectra the diffeient contiibutionq to the tiansitions have been ieported 1 e surface-~uiface(ss),buiface-bulh( d),bulh-suiface( b 5 ) hulh-hull,( bb) A5 i\ clear fioin thib figuie, the pre5ent ub initio iesults for GaN(110) do a5bigii the low eiieigv stiuctuieb of the R 4 5pectiuin to the sb and the 55 tiansition5 No noticeable platenu follows the firbt negative 5lope in the RA bpectiuin i n our calculation5 foi GaN(110) On the othei hand, a diiect coiie5pondence can be diaivn foi the highel eiiergv stiuctuie5 in the RA bpectia between piebent iesultb and the TB ones Thu5 the TB iebultb by Noguez l 1 foi the RA bpectium of GaN( 110) have been coiifiinied heie withiii the ab initio DFT-LDA bchenie In the ca5e of BN(110) the onwt

''

49

150 100 50 0 85 65 45 25 70 50 30 10 35 25 15 5 100

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ENERGY (eV)

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6

7

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9

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10

ENERGY (eV)

Figuie 3 Cali ulated iniagiriaij pait of the half slab polaiizabilitj foi GaN(110) and BN(llU)(timez 4 x d ) arid components of u p l ( d ) foi diffeient tiansitioii~ bolid line ( w ) Left panel GaN(110) Right panel BN(110) L," I (4)dashed line u ;

of the RA bpectiuin is po,iti\e due to 5 5 coiitiibutioiib (Fig 1) At about 5e\ the sign of the R il bpectiuiii change, with 5 s tianbition, doiiiiiiated b j the 1 Loiiiponeiit giving rise to a zliaip iiegatirr btiuctuie Sigiiificaiit coiitiibutioiis fioiii 5 b aiid b5 tiaiisitioiib appeal abole beV \$'e caii asciibe to thebe tiaiibitioiib the secoiid pobitive btiuctuie aboie be1 iii Fig 1 Fioin Fig 1 a i d Fig 3 it 1, c h i that the fiizt po5itire stiuctuie at oiiset oiigiiiateb fioiii ss tiaiisitioiis Thebe tahe place aiouiid the i\r poiiit of the zuiface Biillouiii Zone ( 5BZ)13 In coriespoiideiiLe of that poiiit the miiiiiiiuiii gap betweeii Jd aiid C'3 buiface state, foi BN(110) taheb place Theze ztateb 011 the othei hand, ,lion both an aliiiost flat behavioui aloiig the k' d i i e ~ t i o i i ' ~The fiibt iiegatile stiuctuie iii the R 4 bpectiuiii of BN(110) coiiieb fiom 5 5 tiaiisitioiis aloiig thib diiectioii aiid 111 a iiiiiioi \+a> fiom 56 iieai the F iegioii To buiiiiiiaiize at low eiieigi for G a N ( 110) an iiiteiplav betxeeii $6 aiid 5 5 tiaiisitioiiz doe3 tahe plaLe mith the fiist negative btiuctuie clue to 5 b tianbition5 iiiaiiilj In BN( 110) the fiist pobitire stiuc tuie aiid the folloniiig A a i p iiegatile peal, coiiie fioiii ss tiaiisitioii, The laigei stieiigth of the

r

50

stiuctuies piemit 111 tlie optic a1 zpectia of tlie iiitiides (110) 5uifaces caii br a\( iibrd to the laigei ioriicity (chaige abvmiiieti>) aiid boiid stlength ( c ohe>i\e eiieigy) of tlie iiitiideb with re5pec t to the othei 111-1 coiiipouiids ( r g ~ a 4 2 ) l8~ ' 4 Akiiowledgiiieiits

T h r autlioib would lihe to thaiil, 0 Pulci a i d R Del sole fol helpful d1>c~i>bioii\ This \toil, ha5 been buppoitecl lij the ENEX Computing Enviioiiiiieiit at C eiitio Ric eic he ENE A C asacc ia (Rome)

Refei eiices 1 J J Pouch b 4 illteroxitz (Eds ) Syntheai5 nnd Propertzea of Boron N i t ) i d t TIan5 Tech Publications 4edermaim~cloif 1330 ) F i o i ~ t i t i t of , Group III Iztricler Ed br J H Edgar, I\aiisas State Unixersit\ (Ll 5 4 ) IN\PE( Institution of Electric Engineer5 1394 3 Group 111 Yitr i d t 5 t m i c o n d u c t o r ~ Compounds P h y a i c a and -1ppliccition>, Edited b x B ( 11 Ovfoid h e n c e Pulhcations London 1338 4 A 1 I Eieinets I\ Talyeinura, H \Lisa D Goldberg 'I Bando L D Blanl, \ \at0 I\ T\ataiiabe Phjs Rex B 5 7 , 5655 (1393) 5 \ Nahaniura, T AIuLai, TvI \enoh 411111 Phvi Lett 64 I G Y T (1994) G Landolt Born\tein Neu 5erie\ 19Y2, Lo1 17a ed 0 Madelung hi \chulz, aiid H Weiss (Be1lin Springer ) I D i a m o n d 5ilicon ( ur bide and Related It zde Bandyrip 5enikconductor 5 edited by J T Gla\\, R M e w e r and N FLijimori, LIRS Symposia Proceedings No 1 G L (Material5 Rewaich 5ocietF Pitt4Iurgh Pa, 1990) h1 J Paisle), Z h t a i J B Poqthill and R F Dabis J Lac 5ci Techno1 4 7 T O 1 ( l 9 W ) (T LIartin, \ \trite J Thornton and H AIorhoc 413111 Phx5 Lett 58 2 3 7 5 (1991) Y R Del bole R t J t c t n n c e Slit' trorcopy Thtor y Photonic Prolws at 'xirfaces Ed )1, P Halexi Ekehier bcience, 4ni5terdam, 1395, pp 1 3 3 1'73 9 R A 1 Dieizel, E I\ LJ (riois Denbity Functional Theory, bpinger, N \1 1990 10 0 Pulci CT Ornda R Del \ole L Reining P h i s Rev Lett 81 5374 (199Y) 11 C Noquez P h > \ Rex B 62, 2GS1 (2000) 1 2 N Troullier and J L illartins Ph15 Re\ B 4 3 , 1993 (1991) 1 3 \ CT Louie 5 Froxen hi L ( ohen P h j s Rev B 26 1738 (19Y2) 14 0 Pulci CT Onida A I 5hbrehtii R Del \ole B 4dolph P h \ \ Rex 5 5 GGY5( 1397) 1 5 F hlanghi R Del \ole 4 belloni E Molinari P h j s Rex B 4 1 9935 (1930) 1 G R M o t t o ( T P brira\ta\a, A ( Ferraz, Surf \ci 426(1999), 75 Y 2 17 4 Filippetti L Fioreritini ( T C appelliiii 4 B05iii P h ~ 5Rex B59 Y 0 1 G (1999) 1 Y U C i o w i e r J Furtliniuller F Bechqtedt Phpq Re\ B58 l T 2 L (1998)

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19 B Adolph, V I Ga\rilenbo I\ Tenelsen, F Bechstedt R Del bole Phv. Rex 53, 9797 (1996) 20 >( Zhu, 5 B Zhang S G Louie h l L Cohen, P h i \ Rev Lett G3 2112 (1989) 31 S Brodersen and \V Schatthe, Phas Rev B 66, 153303 ( 2 0 0 2 ) 23 0 Pulci, G Onida, R Del Sole, A Shbrebtii, Phys Rev B 5 8 , 1922 (1998) 23 G Cappellini, G Satta, M Palummo G Onida,(to be published)

LONGITUDINAL GAUGE THEORY OF SECOND HARMONIC GENERATION AT SEMICONDUCTOR SURFACES

BERNARD0 S MENDOZA Centio de Investigaczonea e n Optzca A P 1 948, Leon, Guanajuato, 37000 hlexnco brnbIQcto ms ,

A theoietical ieb iew of suiface second hainiunic genelation fiom semiconductoi surfaces Lasecl on the longitudinal gauge is piesented The nonlineai susceptibility tensoi 1 is split into t n o teirns one that is ielated t o intei band one election tiansitions arid the othei is related t o intra band one election tiansitions The ecluibalence of this foiniulation t o the tiansbeise gauge appioaLh is shown and the possibility of onfirming it5 numeiiLal accuiaL1 is discussed Also, the calculation of the suiface second haimoniL iadiated intensitj R within the thiee layei model is derivrd nith 1 and R one has a complete desciiption of this fascinating optical phenomena (

1 Introduction Secoiid liaiiiioiiic geiieiation (SHG) has become a poneiful bpectiobcopic tool to stud\ optical piopeities of suifaces and iiiteifaces since it ha3 the advantage of beiiig buiface beasitire For ceiitio>\iiiiiietiic inateiialb iiiLeisioii >\ mmetiv forbids nithiii the dipole appiouiiiatioii, 5HG fiom the bulk but it 15 allowed at tlie buiface, wlieie tlie iiireisioii svmiiietrv 15 bioheii Tlieiefoie, SHG should iiecebbaiilj come fioin a localized surface iegioii 5HG allow5 to study the 5tiuctuial atoinic aiiaiigeineiit and phase trailzition5 of clean and adsorbate covered burfaceb, aiid biiice it 15 an optical piobe, it caii be ubed out of ITHV coiiditioiis aiid is iioii-iiivasive aiid 11011destiucti\ e On tlie expeiiiiiental side, tlie iiew tunable high iiiteiisitv lase1 sybteiiib have made 5HG bpectroscopy ieadil) accesbible and applicable to a uide iaiige of by5teiiib Howerei the theoretical development of tlie field i b \till an oiigoiiig bubject of iebeaich Soiiie iecent advances foi the case of bemiconducting aiid iiietallic bybteiiis have appeared in the liteiatuie M lieie tlie coiifioiitatioii of theoietical models n ith expeiiineiit lias \ ield coirect ph\bical iiiteipietatioiis for the SHG spectra

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52

53

In a pieviou\ aiticle, ’ n e ievieued boiiie of the ieceiit iesults in tlie studj of SHG using tlie tiansveise gauge foi the coupling betueeii the electioiiiagiietic field and the electron In particular, we showed a method to svsteiiiatically investigate the diffeient coiitiibutioiis to the obbeir ed peak> iii SHG l o Tlie approach consisted in the bepaation of tlie diffeieiit contribution> to the nonlinear susceptibility accoidiiig to l w aiid Zw tiansition3 aiid to the surface 01 bull, chaiactei of the state3 among which the tianbition5 take place To coinpleinent above iesults, on this aiticle we ieliew the calculation of the iioiiliiieai busceptibility using tlie longitudinal gauge, aiid h o w that both gauges give, as they should, the saiiie iesult We dibcub5 a pobbible iiumeiical check up on thib eyuivalenc) Also the 50 called thiee-layei-model foi tlie calculation of tlie surface iadiated SH efficiencv 13 pie3ented

2 Longitudinal Gauge calculation of

x

To calculate I within tlie longitudinal gauge, we follow the aiticle by Aveiba aiid Sipe l1 4 more recent derivation can also be found i n Ref [12-13] In tlie length gauge the hamiltoiiiaii 15 given by H=Ho-er

E,

(1)

wIieie Ho = y ’ / ~ m + \ nheie T is the peiiodic cirstal potential aiid the electiic field E = -A/c nitli A tlie vectoi potential Iio ha5 eigenlalues hw,l(k)aiid eigeiivectois Ink > (Bloch btates) labeled by a band index 12 aiid civstal momentum k Tlie I iepieseiitatioii of the Blocli state3 is given h> < rlrik >= e L k l u n k ( r ) wheie , u,k(r) i b cell peiiodic aiid < nklink’ >= d,l,llb(k - k’) The Ley iiigiedient in tlie calculation ale tlie matrix eleiiieiits of the position opeiatoi r , that we write as r = ZVk then14

< nklrlmk’ >= 2

s

dre-2k

l l*, k ( r ) v k ’ e i k ‘ U,,&’(r)

= -zbnmvkJd(k - k’) + d(k - k’)