Epioptics 2000: proceedings of the 19th course of the International School of Solid State Physics: Erice, Sicily, Italy, 19-25 July 2000 9789812810854, 9812810854

Table of contents :
Optical properties of excitons in semiconductor quantum wells and microcavities, L.C. Andreani
optical properties of semiconductor surfaces, G. Chiarotti et al
measurements of second-order optical susceptibility in crystalline and porous silicon, M. Falasconi et al
giant bilayer haemoglobin from earthworm studied by X-ray absorption spectroscopy and atomic force microscopy, M. Girasole et al
SNOM and spectroscopy - a technique beyond the diffraction limit, V. Marocchi and A. Cricenti
theory of second harmonic generation at semiconductor surfaces, B.S. Mendoza
differential reflectivity and angle resolved photoemission of PbS(100), M. Tallarida et al
deep-level defects at a lattice-mismatched InGaAs/GaAs interface, O. Yastrubchak et al
and other papers.

Citation preview

THE SCIENCE AND CULTURE SERIES — MATERIALS SCIENCE Series Editor: A. Zichichi

EPIOPTICS 2000

Editor

Antonio Cricenti World Scientific

EPIOPTICS 2000 Proceedings of the 19th Course of the International School of Solid State Physics

EPIOPTICS 2000 Proceedings of the 19th Course of the International School of Solid State Physics

Erice, Sicily, Italy

19 - 25 July 2000

Editor

Antonio Cricenti Instituto di Struttura della Materia, Roma, Italy

Series Editor

A. Zichichi

V f e World Scientific « •

New Jersey • London • Singapore SinaaDore••Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EPIOPTICS 2000 Proceedings of the 19th Course of the International School of Solid State Physics Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4771-0

Printed in Singapore.

Preface This special Volume of World Scientific contains the Proceedings of the 6th Epioptics Workshop held in the Ettore Majorana Foundation and Centre for Scientific Culture, Erice, Sicily, from July 19 to 25, 2000. The Workshop was the 6th in Epioptics series and the 19th of the International School of Solid State Physics. Antonio Cricenti from CNR- Istituto di Struttura della Materia and Theo Rasing from University of Njimegen, were the Directors of the Workshop. The Advisory Committee of the Workshop included Y. Borensztein from U. Paris VII, P. Chiaradia and R. Del Sole from U. of Roma II-Tor Vergata, 0 . Hunderi from U. of Trondheim, J. Mc Gilp from Trinity College of Dublin, W. Richter From TU Berlin and P. Weightman from U. of Liverpool. Fortyfive scientists from fourteen countries attended the Workshop. The Workshop has brought together researchers from Universities and Research Institutes which work in the fields of (semiconductor) surface science, epitaxial growth, materials deposition and optical diagnostics relevant to (semiconductors) materials and structures of interest for present and anticipated (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 properties of semiconductor and metal surfaces, interfaces, thin layers and layer structures, and assessing the usefulness of these techniques for optimization of high quality multilayer samples through feedback control during materials growth and processing. Materials of particular interest have been silicon, semiconductor-metal interfaces, semiconductor and magnetic multilayer and III-V compound semiconductors. The workshop as well as the notes collected in this Volume combined the tutorials aspect adequate to a School with some of the most advanced topics in the field, which better characterize the Workshop. We wish to thank our sponsors, the Italian National research Council (CNR) and the Sicilian Regional Government for facilitating a most successful Workshop. We are grateful to the Director of the International School of Solid Stete Physics, Prof. G. Benedek and to the Director, Prof. A. Zichichi, and staff members of the Ettore Majorana Centre for the excellent support, organization and hospitality provided. Finally we would like to thank Dr. M. Girasole who assembled and kept track of the various proceedings, to Dr. V. Marocchi, Dr. V. Mussi, Dr. A. Sieni and to D. De Fazio for their timely and efficient assistance in the organization of the Workshop/School. Antonio

V

Cricenti

CONTENTS

Preface

v

Optical Properties of Excitons in Semiconductor Quantum Wells and Microcavities L. C. Andreani

1

Polarizable Bond Model for Reflectance Anisotropy and Second Harmonic Generation for Si(100) Surfaces N. Arzate and B. S. Mendoza

18

Optical Properties of Semiconductor Surfaces G. Chiarotti, P. Chiaradia, C. Goletti and F. Arciprete Near-field Photoluminescence Spectra of Laser Active Point Defects in LiF Films A. Cricenti, R. Generosi, M. Luce, V. Marocchi, S. Bigotta, A. Bonfigli and R. M. Montereali Measurements of Second-Order Optical Susceptibility in Crystalline and Porous Silicon M. Falasconi, L. C. Andreani, M. Patrini, A. M. Malvezzi, V. Mulloni and L. Pavesi Combination of Surface Characterising Technique and X-ray Spectroscopy to Detect the Interaction of a Dihydropyridine Drug with Red Blood Cells M. Girasole, A. Cricenti, A. Congiu-Castellano, A. Arcovito, G. Boumis and G. Amiconi Giant Bilayer Hemoglobin from Earthworm Studied by X-ray Absorption Spectroscopy and Atomic Force Microscopy M. Girasole, A. Cricenti, A. Congiu-Castellano, A. Marconi, C. Davoli, A. Arcovito and G. Amiconi

VII

24

39

46

52

64

viii

Is Reflectance Anisotropy Spectroscopy Useful to Investigate Langmuir-Blodgett Films? C. Goletti, P. Chiaradia, M. Drago, R. Paolesse, C. Di Natale, F. Delia Sala, P. LugliandA. D'amico SNOM & Spectroscopy: A Technique Beyond the Diffraction Limit V. Marocchi and A. Cricenti Near-Field Photoluminescence Spectra of CsPbBr3 Nanocrystals in CsBr-PbBr2 Co-Evaporated Films V. Marocchi, A. Cricenti, G. Gonte, A. De Simone, F. Somma and M. Nikl Theory of Second Harmonic Generation at Semiconductor Surfaces B. S. Mendoza

74

79

94

99

Optical Anisotropy of Clean and Oxidized GaAs(001)-4x2 C. Novello, V. Emiliani, C. Goletti, V. Berkovits and P. Chiaradia

109

Interaction of Homogeneous and Surface Waves with Metal Nanoparticles V. A. Sterligov, P. Cheyssac, S. I. Lysenko and R. Kofinan

115

Differential Reflectivity and Angle Resolved Photoemission of PbS (100) M. Tallarida, A. Cricenti, C. Ottaviani, B. J. Kowalski, E. Guziewicz, A. Szczerbakowz and B. A. Orlowski

121

Spectroscopic-EUipsometrical Characterization of Photon-Surface-Plasmon Coupling in a Multilayer Diffraction Grating O. Yastrubchak and O. Demchuk

128

Deep-Level Defects at a Lattice-Mismatched InGaAs/GaAs Interface O. Yastrubchak, A. Makosa and T. Wosihski

134

Localization of Light Beam in Nonlinear Optical Waveguide Array /. V Gerasimchuk and A. S. Kovalev

139

Optical Properties of Excitons in Semiconductor Quantum Wells and Microcavities L.C. ANDREANI Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica 'A. Volta', Universitd degli Studi di Pavia, via Bassi 6, 1-27100 Pavia, Italy

A few basic properties of excitons in quantum wells and in semiconductor microcavities are reviewed at an introductory level. The first lecture treats electron-hole and center-of-mass quantization regimes in wide quantum wells, electron-hole and center-of-mass localization in narrow quantum wells, and excitons bound to interface defects: emphasis is given on the behavior of the oscillator strength, which marks clearly the crossover between the various regimes. The second lecture deals with planar microcavities and the strong coupling regime of exciton-radiation interaction, three-dimensional microcavities and the modification of spontaneous emission rate (Purcell effect), and the possibility of obtaining a vacuum-field Rabi splitting for quantum dots in three-dimensional microcavities.

1 1.1

Excitons in quantum wells Introduction

[1-3]

Excitons are electronic excitations of a crystal which go beyond the independentelectron approximation and represent the bound states of an electron-hole pair. Excitons in semiconductors are usually of the shallow, or Wannier-Mott type: i.e., the binding energy is small compared to the band gap and the average electron-hole radius is much larger than the lattice constant. Wannier-Mott excitons are described with good accuracy by a two-particle effective-mass equation in which the kinetic terms contain the effective masses of electron and hole and the Coulomb attraction is screened by the dielectric constant of the crystal. The binding energy of the ground state, or effective Rydberg, is given by R*=u,*e4/2£,.2 f (where p.* is the reduced effective mass and e, the dielectric constant) and is much smaller than the atomic Rydberg due to the small effective mass and large dielectric constant of semiconductors. Each excitonic level has a quadratic dispersion with the total electron-hole mass. Excitonic effects in bulk semiconductors are observable only at low temperature, when the linewidth due to electron-phonon scattering is smaller than the binding energy. The interaction of excitons with the electromagnetic field is characterized by a dimensionless quantity, the oscillator strength, which is defined as follows: 1

2

/=

^rl^ X c|e-s ; -P^ 0 )| 2 ^

(i)

mQnco' ' where ^¥u (*Pexc) is m e ground (excitonic) state, pi are the electron momentum operators, e is the polarization vector, m0 is the free electron mass and ft) is the frequency of the transition. The oscillator strength of an exciton associated with an allowed interband transition is nonzero only for s states and decreases as ri3, where n is the principal quantum number. The oscillator strength of a bulk excitonic transition is proportional to the crystal volume, so the quantity of physical significance is the oscillator strength per unit volume, which is related to the absorption coefficient integrated over the transition line (provided polaritonic effects can be neglected) [2]. A quantum well is a thin layer of a semiconductor surrounded by thick layers of another semiconductor with a larger band gap. The alignement of the bulk band structures determines the band discontinuity, or band offset, between conduction (or valence) band egdes. The variation of the band egde from one material to the other gives rise to quantum confinement effects; we consider here only type-I structures, in which electrons and holes are confined in the same material. Electronic states in heterostructures (quantum wells, superlattices, and systems of lower dimensionality like quantum wires and dots) can be treated in a phenomenological yet accurate way by the envelope-function method [1]. In this framework, electrons and holes are described by an effective mass equation (possibly a multiband one in the case of band degeneracy) with a confining potential determined by the band offset. A quantum well is characterized by a square-well potential, leading to the particle-in-a-box problem of quantum mechanics. The motion along the growth direction (taken as z) is quantized, but the energy levels have a two-dimensional dispersion En(kn) as a function of in-plane wavevector ku and are called subbands. Excitons in quantum wells are described by the following hamiltonian (Eg is the band gap of the well material and Ve(ze), Vh(zh) are square-well confining potentials) [4]: H = Eg

*2y2 ^2y2 1 — - ^ + Ve(ze) + Vh(zh) 2me ^-mh

2 p — : . e r r\ e 111

(2)

The interplay between quantum confinement and electron-hole attraction gives rise to various physical regimes, which are the subject of this lecture. The oscillator strength of a quantum-well exciton is proportional to the sample surface: only the oscillator strength per unit area has a physical meaning. It can be measured from the absorption probability (a dimensionless quantity) integrated over the excitonic peak [2].

3

In the strict two-dimensional limit (vanishing well width and infinite barrier height) the hamiltonian (2) describes a hydrogenic problem in two dimensions: the binding energy of the ground state is 4R and the effective Bohr radius aB is half the bulk value. Starting from a very wide quantum well and reducing the well width L, the binding energy has a monotonic increase from the bulk value R to the 2D value 4RJ if the barrier heights are taken as infinite [5,6]. However, when the finite band offset is considered, the binding energy reaches a maximum and decreases again towards the three-dimensional value when the well width goes to zero [7]: this is due to the fact that the carrier wavefunctions leak into the barrier material for small well widths. The oscillator strength also has a maximum when the binding energy does. 1.2 Electron-hole versus center-of-mass confinement in wide quantum wells The increase of the binding energy Eb over the bulk value, due to confinementinduced decrease of the electron-hole separation, leads to more stable exciton states and to much more prominent excitonic effects in absorption and photoluminescence. Several calculations of binding energies, oscillator strengths and absorption spectra of quantum well excitons have been published. Most of them refer to a range of well thicknesses of the order of or smaller than the exciton radius aB: in this case electron and hole subbands are separately confined and the exciton can be associated with a pair of subbands. This situation is called strong (or electron-hole) confinement regime. An accurate determination of binding energies and oscillator strengths in the strong confinement regime requires the inclusion of additional physical effects like degeneracy of the bulk valence band and mixing of quantum well subbands (the heavy hole-light hole mixing problem [1]), conduction band nonparabolicity, discontinuity of the effective masses and of the dielectric constant between well and barrier materials. When all these effects are taken into account, the calculated values for the ground state binding energy turn out to be strongly increased [8], in particular the maximum binding energy in GaAs/AlGaAs quantum wells can become larger than the two-dimensional value 4R*: this has been verified experimentally [9]. Calculating absorption spectra in the continuum requires solving the Schrodinger equation in k-space [10]. When the well width becomes larger than the exciton radius a different physical regime is reached. This can be seen by comparing the characteristic energies: Ec=7t2 |/(2mL 2 ) for quantum confinement, R' for the Coulomb attraction. When L~aB quantum confinement dominates, electron and hole subbands are separately quantized with quantization energies Ee, Eh and any pair of subbands gives rise to a quasi-2D excitonic series: the energy spectrum is given by Eexc=Eg+Ee+Eh-Eb. When L»aB, on the other hand, the Coulomb interaction dominates and must be diagonalized first, which means that the excitonic center of mass is quantized as a whole in a thin film [11,12]. The quantization width is L-2D, where D is a dead-layer thickness. In this weak (or center of mass) confinement

4

regime the energy spectrum is approximately given by Eexc=Eg-R*+7t2Y/(2M(L2D)7). The two regimes can be distinguished by the energy of photoluminescence peaks. It is interesting to look at the behavior of the oscillator strength at the crossover from strong to weak confinement. In the electron-hole confinement regime the oscillator strength increases on decreasing well width, due to the reduced separation between electron and hole. On the other hand when the excitonic center of mass is quantized the oscillator strength increases with increasing well width, because it is proportional to the volume occupied by the center of mass. Thus the oscillator strength must have a minimum at the crossover from strong to weak confinement, as measured for CdTe/CdZnTe quantum wells [13] and calculated for CuCl quantum wells [14]. Apparently, no systematic investigations of the crossover regime exist for the thoroughly studied GaAs/AlGaAs system. Anyway, it appears that the oscillator strength is a more significant quantity than the binding energy to characterize the different regimes for the excitonic states. 1.3 Electron-hole versus center-of-mass localization in narrow quantum wells The behavior of exciton states when the well width goes to zero is also interesting and nontrivial. When the well width becomes much smaller than the Bohr radius and the carrier wavefunctions are mostly localized in the barrier region, the singleparticle energies are close to the top of the potential well and it is more useful to think of the electronic states of the barrier as the unperturbed ones, while the "quantum well" acts as a localized, nearly 8-like attractive potential. A localization energy can be defined by writing the energy of, e.g., electron levels as Egb-Eioc, where now Eg is the band gap of the barrier. The physical behavior of the exciton when the well width goes to zero can be discussed in analogy with the quantum well to thin film crossover. As long as the localization energies of electron and hole remain larger than the exciton binding energy, electrons and holes are separately localized and each pair of electron-hole levels gives rise to an excitonic series: the energy spectrum is given by Eexc=Eg -E|0ce-Ei0Ch-Et, and the exciton is still quasi-2D. This can be called a strong (electron-hole) localization regime. However when the localization energy becomes smaller than the binding energy (actually that of the barrier material) the Coulomb attraction dominates and the center of mass of the exciton is localized as a whole. In this new regime of weak (center of mass) localization the energy spectrum is given by Eexc=Egb-R -E|0cexc and the internal exciton wavefunction is that of the bulk barrier material. Let us look at the behavior of the oscillator strength per unit surface. In the regime of separate electron and hole localization the oscillator strength decreases with decreasing well width, since the carrier wavefunctions become more delocalized in the barriers and the in-plane exciton radius increases towards the 3D value. On the other hand when the exciton is 3D but with a localized center of mass

5

the oscillator strength is proportional to the localization length: this increases on decreasing the well width, thus the oscillator strength must also increase. This behavior is analog to the so-called "giant oscillator strength" of excitons weakly bound to neutral impurity states in the bulk [15]. As a consequence, the oscillator strength per unit area has again a minimum at the crossover from electron-hole to center of mass localization for very narrow quantum wells. In order to give a quantitative determination of exciton binding energy and oscillator strength in the various physical regimes, a numerical method has been developed [16] in which the exciton states are expanded in a large non-orthogonal basis consisting of products of gaussians in the electron and hole coordinates along the growth direction and exponentials for the relative in-plane motion. Exciton energies and wavefunctions are obtained by calculating hamiltonian and overlap matrix elements and solving a generalized eigenvalue problem. In Fig.la we show the oscillator strength per unit area in GaAs/Al0.i5Ga085As quantum wells, as a function of well width in a log scale. The behavior described above can be recognized: the oscillator strength has two minima, in correspondence with the transitions from quasi-2D to 3D behavior for very large or very small thicknesses. The crossover from electron-hole to center of mass confinement in large wells takes place at a well width of about 300 A, i.e., at -2.5 aB (the Bohr radius in GaAs is -140 A). The crossover to center of mass localization in narrow wells occurs for L-8 A. The oscillator strength per unit surface has a local maximum in the quasi-2D regime, but can reach larger values for either very wide or very narrow wells. The well width at which the oscillator strength has a minimum in narrow quantum wells depends sensitively on the Al concentration in the barriers. For shallow quantum wells (x~0.01) the crossover to center of mass localization takes place at L-100 A, as it has been measured in magneto-optical experiments [17]. For x=0.35, on the other hand, the minimum is expected to occur between one and two monolayer thickness. A cathodoluminescence measurement on a sample containing GaAs/Al0 35Gao 65As quantum wells from one to eight monolayers has been reported in Ref. [18]. The sample is excited from the cleaved edge with the electron beam and the luminescence is collected in the normal direction. The cathodoluminescence signal as a function of electron-beam position shows a well-defined maximal intensity in correspondence to the excitation of each single quantum well: this allows to deduce the oscillator strength of the lowest excitonic transition, which has indeed a minimum around a thickness of three monolayers.

6

10°

10'

10z

well width (A)

103

°

25

50 we

"

wldth

75

10

°

Figure 1. (a) Oscillator strength per unit area of the lowest heavy-hole exciton in GaAs/Alo i5Ga0 S5As quantum wells; (b) binding energy of the lowest heavy-hole exciton in GaAs/AlxGa!_xAs quantum wells for different Al concentrations [16]. Calculating the binding energy when the well width goes to zero requires taking into account conduction band nonparabolicity and the variation of effective mass and dielectric constant between well and barriers. The binding energy of the lowest direct exciton (the heavy-hole exciton) in GaAs/AIxGa^As quantum wells is shown in Fig. lb. Note that the limiting values for L—>0 depend on the Al concentration, since the parameters of the barrier material differ. The maximum binding energy in GaAs/AlAs quantum wells is about 26 meV and it occurs at a few monolayer thickness: this has to be compared with the bulk binding energy (R =4 meV in GaAs) and the "2D limit" of 4R*. Actually the GaAs/AlAs system becomes type-II for Lr±u)c: the Fabry-Perot condition discretizes the wavevector in the bulk polariton dispersion. In other words, the mixed exciton-cavity modes represent two-dimensional exciton polaritons, which are called cavity polaritons. The above argument is a qualitative one, particularly since it does not take into account damping of the Fabry-Perot mode (expressed by the linewidth ym, which represents the inverse of the escape time of photons) and of the exciton (given by a linewidth yex, which may have homogeneous and inhomogeneous contributions). When radiative coupling together with dampings are considered, two regimes can occur: for a radiative coupling smaller than the dampings the exciton-radiation interaction is in a weak coupling regime, which means that the exciton has a radiative decay but with a modified emission pattern; when exciton-light coupling is larger than damping the strong-coupling regime takes places, in which quasi-stationary cavity polaritons are formed and a Rabi splitting occurs. A semiclassical theory of exciton-cavity polaritons requires solving Maxwell equations for the multilayer structure, which can be done in a transfer-matrix formalism [36,37,38]. The complex mixed-mode energies close to resonance are found to be accurately given by a twooscillator model, in which two states of frequencies (Oex-iYex and u)m-iym are coupled by a matrix element V=(27te2fxy/(£rm0Leff))1/2, where fxy is the oscillator strength per unit area of the quantum-well exciton and Leff is an effective cavity length which takes into account penetration of light in the dielectric mirrors. The precise condition for being in strong coupling is V>lyex-ym I/4; the Rabi splitting at resonance for low damping is 2 |V. The dispersion of cavity polaritons can be measured e.g. by angle-resolved reflectivity or photoluminescence [39]. A Rabi splitting has been observed in III-V microcavities even at room temperature. Cavity polaritons are in fact easier to observe than bulk polaritons and have been the subject of a variety of cw and time resolved investigations in the last years [37,40].

13 2.3 Photonic confinement in three dimensions: the pillar microcavity. spontaneous emission rate (Purcell effect)

Change of

Let us turn to systems with three-dimensional photonic confinement. We shall focus on the cylindrical (or pillar) microcavity structure: this is obtained from a planar microcavity by etching laterally to form a cylinder of dielectric- with radius of the order of a micron (see Fig.4a) . Photonic confinement in such a structure is given by two mechanisms: vertical confinement is provided by the dielectric mirrors of the planar microcavity, while lateral confinement is given by total internal reflection produced by the discontinuity of the average refractive index. Indeed, for III-V microcavities based on GaAs/AlGaAs or InP/InGaAs the refractive indices of the constituent materials are close to each other and of the order of three; an infinite pillar can be viewed as a cylindrical waveguide [41] where the core has the average refractive index of the dielectric and the cladding is air. For a given wavevector kz along the axis, guided modes exist in the frequency region between ck z /n in and ck z /n out , where nin (nout) is the inner (outer) refractive index; these modes have a transverse wavevector which is real in the core region and imaginary in the cladding, i.e., the electric field is oscillating in the core and exponentially decreasing in the outer medium.

Figure 4. (a) Schematic structure of a pillar microcavity; (b) blue shifts of the lowest energy modes in GaAs/AlGaAs pillar microcavities with respect to the reference planar cavity, as a function of pillar radius. Points: experimental results [42]; solid lines: theoretical results including the energy dependence of the refractive index [43].

14

While the full problem of calculating the exact photonic modes in a pillar microcavity is a very complex one, a simple approximation for the lowest modes follows from noting that vertical confinement usually dominates and the vertical dynamics can be separated from the lateral dynamics [42]. In this decoupling scheme, the resonance frequencies of the eigenmodes are approximately given as

^=-JMV^T, nr \\ a J

o)

{ Lc )

where a is the radius of the pillar, Lc is the length of the planar microcavity, and X|„ is a zero of a Bessel function of order /: the pillar microcavity acts as a resonant cavity in the optical region, with discretized vertical and trasnverse wavevector components. The eigenfrequencies (3), or the expressions which result from a more realistic calculations, are those of a photonic dot. The energies of photonic modes in three-dimensional microcavities can be measured from the photoluminescence peaks when the cavity contains a broad-band emitter, since the emission spectrum is concentrated at the energies of cavity modes (a weak continuum emission due to coupling with leaky modes of the waveguide is also present). In the experiment of Ref. [42] the emitter consists in planar array of self-assembled InAs quantum dots placed in a GaAs cavity. In Fig. 4b we show the energies of confined photonic modes as a function of pillar radius. Note that the energy shift on reducing the radius is approximately proportional to 1/a2, as follows from Eq. (3) when vertical confinement dominates. Analogous results have been obtained in Ref. [44] with quantum well excitons in rectangular microcavities. A basic quantum electrodynamic phenomenon which has been recently demonstrated in three-dimensional microcavities is the change of spontaneous emission rate, known as Purcell effect [45]. If the emitter has a well-defined frequency and is in resonance with a single cavity mode u., its emission rate calculated by standard perturbation theory is YSE=FHYO> where Yo is the emission rate without the cavity and

is called the Purcell factor. In this formula, Q^co^Yn is the quality factor and V^=la^(r)r2 is the mode volume defined in terms of the the normalized mode function o^(r) of the electric field. The mode volume is calculated at the position of the emitter; it is usually of the order of the cavity volume. The enhancement of the spontaneous emission rate is maximized if the cavity volume is small, the quality factor is large and the emitter is placed at a maximum of the electric field. The

15

enhanced spontaneous emission in semiconductor microcavities has been recently observed for self-assembled In As quantum dots in cylindrical microcavities [46]. 2.4 Vacuum-field Rabi splitting for quantum dots in three-dimensional microcavities? The last issue we address in this lecture is the possibility of reaching the strongcoupling regime and observing a vacuum-field Rabi splitting for a single quantum dot in a 3D microcavity [25]. The quantum dot can be modelled as a two-level system with a ground state Ig) and an excited state le); the states of the radiation field for a single mode u, can be denoted as lnH), where n^ is the number of photons in the mode. The coupling constant g=(d- E) of the dipole interaction between the quantum dot transition and the cavity mode is calculated as g=(7i;e2f/(eImoV^))1/2, where/is the oscillator strength of the transition and V^ is again the mode volume. The quantum Hamiltonian describing the dynamics of the two-level system interacting with the cavity mode is called the Jaynes-Cummings model. Its basis states are the direct product of quantum dot and radiation states: these can be indicated by Ig.n^) or le,n^. At resonance and in the absence of interaction, the ground state is lg,0), the first excited doublet consists of the degenerate states le,0), lg,l), the second excited doublet consists of le,l), lg,2), etc. The dipole interaction couples the two states of each doublet and produces a splitting 2 g(n^+l)1/2. The "vacuum-field" Rabi splitting is the one in the lowest doublet, for which one has n^=0 and the splitting is 2fig . The above Hamiltonian treatment does not take into account the finite linewidths ya, yCiii of the quantum dot transition and of the cavity mode. Dampings are best treated by a master equation for the density matrix. Within this formalism, the dynamics of the quantum dot-cavity system can be calculated analytically in the case of weak excitation (i.e., when only the ground state and the lowest excited doublet are kept). At resonance, the luminescence spectrum is found to have maxima at complex frequencies given by

a± = coQ-^(ra+rc)±,\g^-\ "* ;-*» | .

(5)

For g>rya-YCi(ll/4 there is a splitting in the real part: this is the vacuum-field Rabi splitting. The condition for being in strong-coupling regime is similar to that for quantum well excitons in planar microcavities, however the physical system has very different behavior since the quantum well exciton is a boson for weak excitation while the quantum dot transition is a fermion. The difference manifests itself in the behavior for increasing excitation intensity: the Rabi splitting of quantum well excitons does not depend on excitation level for weak enough excitation, while the Rabi splitting for a quantum dot transition increases as (%+l) .

16

The conditions for having a vacuum-field Rabi splitting require a large coupling constant g, hence a small cavity volume, and small enough linewidths. For In As quantum dots in micro-pillars, as studied in Ref. [46], the optimal conditions are met for a pillar radius a=0.5 urn, when the cavity volume is V^O.13 (urn)3, the quality factor is Q^=2- 103 and the mode width is YC,M=0.6 m eV (the linewidth of the quantum dot transition is about an order of magnitude smaller). The complex energies of Eq. (5) are plotted in Fig. 5 as a function of the oscillator strength. In can be seen that for low oscillator strength there is a splitting in the imaginary part: the upper branch corresponds to the modified quantum dot spontaneous emission rate and gives back the Purcell effect. For high oscillator strength, instead, there is a vacuum-field Rabi splitting in the real part. The strong-coupling regime occurs if the oscillator strength is larger than about 100. For InAs quantum dots the oscillator strength is f=10 (this can be deduced e.g. from the measured lifetime t=1.2 ns), thus we conclude that InAs quantum dots in pillar microcavities are in weak-coupling regime. Although the quality factor can be improved in different structures like microdisk cavities supporting whispering-gallery modes [47], it seems unlikely that a vacuum-field Rabi splitting can ever be observed for this system.

Figure 5. (a) Real and (b) imaginary parts of the energy shifts (Eq. (5)) for the coupling between the lowest mode of a micropillar and the quantum dot transition, as a function of the oscillator strength, for a pillar radius a=0.5 urn and a quality factor 0^=2- 103 [25].

50

100

153

(Krillaiixstergh

Thus we need an emitter with a larger oscillator strength. This can be provided by a quantum well exciton localized at interface defects in narrow quantum wells, as discussed in the previous lecture. Indeed, a localized exciton behaves as a zero-dimensional emitter, like in the assumed model of a two-level system, provided the spacing between energy levels remains larger than the cavity mode linewidth: this condition is largely met for defect radii smaller than 500 A. The oscillator strength of localized excitons in GaAs/AlAs quantum wells is shown in Fig.2 and its behavior as a function of defect radius was already discussed. We see that the oscillator strength can be much larger than the critical value for having a vacuum-field Rabi splitting,

17

thus localized quantum-well excitons embedded in pillar microcavities can be in strong-coupling regime for either small or large defect radius. Since defects with a large radius can to some extent be controlled by the process of growth interruption, using large defects seems a promising possibility for observing a vacuum-field Rabi splitting with a single three-dimensionally confined emitter in a three-dimensional solid-state microcavity.

3

Acknowledgements

It is a pleasure to acknowledge the collaboration with R.C. Iotti, G. Panzarini, and J.M. Gerard, which led to the results presented in this paper.

References [1] Bastard G., Wave Mechanics applied to Semiconductor Heterostructures (Les editions de Physique, Les Ulis Cedex, 1990). [2] Andreani L.C., in Confined Electrons and Photons - New Physics and Devices, edited by Burstein E. and Weisbuch C. (Plenum, New York, 1995), p.57. [3] Yu P.Y. and Cardona M., Fundamentals of Semiconductors (Springer, Berlin, 1996). [4] Gaussian units are used in this paper. To get formulas in SI units, replace e2—>e2/(47t£o), where Eo is the vacuum permittivity. [5] Miller R.C., Kleinman D.A., Tsang W.T. and Gossard A.C., Phys. Rev. B 24, 1134(1981). [6] Bastard G., Mendez E.E., Chang L.L. and Esaki E., Phys. Rev. B. 26, 1974 (1982). [7] Greene R.L., Bajaj K.K. and Phelps D.E., Phys. Rev. B 29, 1807 (1984). [8] Andreani L.C. and Pasquarello A., Phys. Rev. B 42, 8928 (1990). [9] Gurioli M. et al., Phys. Rev. B 47, 15755 (1993). [10] Winkler R., Phys. Rev. B 51, 14395 (1995). [11] Cho K. and Kawata M„ J. Phys. Soc. Jpn. 54, 4431 (1985). [12] D'Andrea A. and Del Sole R., Phys. Rev. B 41, 1413 (1990). [13] Merle d'Aubigne Y. et al., J. Cryst. Growth 101, 650 (1990). [14] Andreani L . C , D'Andrea A. and Del Sole R., Phys. Lett. A 168, 451 (1992). [15] Rashba E.I. and Gurgenishvili G.E., Fiz. Tverd. Tela 4, 1029 (1962) [Sov. Phys.-Solid State 4, 759 (1962)]. [16] Iotti R.C. and Andreani L . C , Phys. Rev. B 56, 3922 (1997). [17] Fritze M. et al., Phys. Rev. Lett. 76, 106 (1996). [18] Andreani L.C. et al., Physica E 2, 151 (1998). [19] Schwabe R. et al., J. Appl. Phys. 77, 6295 (1995).

POLARIZABLE BOND MODEL FOR REFLECTANCE ANISOTROPY AND SECOND HARMONIC GENERATION FOR Si(100) SURFACES

N . A R Z A T E A N D B E R N A R D O S. M E N D O Z A Centro de Investigaciones en Optica., A. C, Leon Guanajuato, E-mail: [email protected]

Mexico

Spectra for reflectance anisotropy and second harmonic generation of clean Si(100) reconstructed surfaces, based on the polarizable bond model, are presented. The crystal is modeled as an array of point-like polarizable dipoles which are considered to be at the middle of every Si-Si bond. The model incorporates the reconstruction of the surface trough the local field effect. Spectra for the reconstructed surfaces 2 x 1 and c(4x2) are presented and are compared with the experiment.

1

Introduction

The use of optical spectroscopic probes to study surfaces and interfaces has gained great interest recently. They can be used out of UHV environments and also are noninvasive, nondestructive, and have a wide spectral coverage as well. Reflectance Anisotropy Spectroscopy (RAS) is one of the linear optical techniques that is used to characterize structural and electronic properties of semiconductors surfaces. l For cubic crystals, the bulk optical response is isotropic, thus anisotropies induced by changes of structure at the surface of the crystal can be observed by RAS. l Going to the nonlinear response second harmonic generation (SHG) is a very sensitive non-linear optical technique which has been successfully applied to test the surface of centrosymmetric media. 2 SHG is based on the fact that the surface and the bulk have different structural symmetry. For media with inversion symmetry SHG is forbidden (within the dipole approximation), in the bulk, but allowed at the surface where the inversion symmetry is broken. These two experimental techniques, RAS 3 , 4 and SHG °'6''>8'9J have been applied to clean and adsorbate covered Si(100) reconstructed surfaces, where the atomic structure is formed by asymmetric buckled surface dimers. 10 On one hand, Shioda and van der Weide4 performed measurements of RAS from highly oriented single-domain (2 x 1) reconstructed Si(100) surfaces where the anisotropy shown was attributed to the dimers. On the other hand, Hofer7 and Dadap et al. 8 made a detailed experimental SHG study of the clean and hydrogen-covered Si(100) double-domain surface as a function of hydrogen coverage and temperature. In order to understand the observed RAS and SHG experimental spectra of surfaces and interfaces we present spectra for Si(100) reconstructed surfaces 18

19

calculated by using the model of polarizable bonds, 11 in which the crystal is viewed as an array of point-like polarizable dipoles where each dipole is located at the middle of every Si-Si bond since the maximum distribution of charge is precisely located here. An explanation of the main parameters of the model is given in section 2. The spectra for the 2x1 and c(4x2) reconstructed surfaces are presented in section 3 and we give conclusions in section 4. 2

Theory

The polarizable bond model was proposed by Mendoza and Mochan 11 which applied the model to the unreconstructed surface Si(100)lxl and explained the bulk Ei transition of Si in terms of a vertical strain induced by surface reconstruction in accordance with Daumet al. 6 In the following we introduce the main parameters of the model. The microscopic linear and non-linear susceptibility tensors of each bond are written in terms of the linear polarizabiiities which indeed depend on position through their particular bond orientation and their surface or bulk location. We consider each dipole to be represented by a cylindrical anisotropic centrosymmetric harmonic oscillator, whose principal polarizabiiities are ay and a x , where || (_L) is parallel (perpendicular) to the bond. Close to the visible spectral region, we expect that the main contributions to or|| originates in bonding-antibonding transitions, while Qj. is due mainly to transitions involving atomic states with different symmetry. We assume that the latter has larger resonant frequencies than the former, and we approximate a x by a Lorentzian centered at some relative high frequency wx with a weight characterized by a frequency uip, and u>c as a damping parameter,

,

* L H =

{f Wp 2 ( }.

>2.

(1)

LO-± - (OJ + iwcy where we allow the factor / to be proportional to the amount of charge transfer to the upper Si of the dimers. We take OJP to have a nominal value, and then the factor / = 1 is taken for all the dipoles except for the dimer dipole for which / > 1. To obtain a\\(u>), we use the following equation, P{B,u)

=

£

-^t^E(B,u),

(2)

where P(B,u)) is the total bulk dipole moment, E(B,u) is the electric field in the bulk and the e(w) is the experimentally measured bulk dielectric function. Since P(B,LJ) is a function of a|| and a x , Eq. 2 yields an analytical relation between a and e(w), which is a generalized Clausius-Mossoti relation11. Therefore, once uij_, uip and uic are chosen in Eq. 1, we can solve Eq. 2 for a\\ for any

20 T

1

1

i

1

photon energy (eV)

1

1

1

0-31—i

1

1

i

1

r

two photon energy (eV)

Figure 1: RAS and SHG optical spectrum of Si(100) surface. The dashed line is for the i d e a l ( l x l ) surface, the dashed-dotted line is for the strain-relaxed surface, the thin solid line is for to the 2 x 1 reconstructed surface and the thick solid line is for the experiment.

given e(ui) and then we follow the method of Mendoza and Mochan11 to solve the local field equations for the linear and nonlinear dipole moments. Finally, the RAS spectra which involve only the linear dipole moments12 and the SHG spectra which involves both linear and non-linear dipole moments through the second order surface susceptibility tensor are calculated13. 3

Results

The geometry used for the reconstructed 2x1 surface14 correspond to a structure with a dimer of 0.6 A, since the spectrum of this surface showed to be the most optimal structure 15 that resembled the experiment. Figure 1 shows RAS and the SHG spectrum for the Si(100) surface. The theoretical results are compared to experiments performed on single domain surfaces for RAS4, and double domain surfaces for SHG.8 The spectra shown are for the ideal l x l surface, strain-relaxed surface where the last layer of atoms is relaxed inwards by 5% of the interplane distance and for the reconstructed 2x1 surface. It is mention that in these two cases the same values for the frequency parameters of equation (1) are used, with / = 1, for all dipoles (including the dimers). Seeing the RAS spectrum we notice that the anisotropy for the strain-relaxed surface is identically zero, just as the ideal l x l surface is, since the vertical strain introduced in the model, leave the surface, as before, isotropic. The RAS spectrum for the 2x1 surface shows the low frequency experimental peak found at 1.6 but displaced in energy by 0.2 eV upward. This peak is usually assigned to a surface state, 16 which in our theory results from the dimer's local field. The positive experimental peak at 4.3 eV is also qualitatively reproduced but displaced in energy by 0.4 eV. On the other

21 2 0 CO


av,axFor frequencies co»comax eq. (3) becomes:

26 r>

e'(co) = l

"'max

-p

C co'e"(co')dco'

.

(5)

On the other hand, at sufficiently high frequencies electrons behave as free so that: E'(03)=l—\ CO

2 where top =

,

(6)

2 4ne Nne ft SJ

is the square of the plasma frequency. m Here N is the density of atoms, neff the number of electrons per atom (in the given group) and m is the free electron mass. Comparing (5) and (6) one obtains: w

1

™

n

eff= — r ~ i — f co s (a) )dco l 2JT e N JQ

,

(7)

or, for a surface layer of thickness d: '[e"y (co)d]dco

.

(10)

Equations (8) or (9) and (10) are the expected "sum rules" valid for surfaces. In order to relate them to measurable quantities, we summarize the theoretical results of the three-phase model ofAspnes and Mclntyre [7]. Let us call R(d) the reflectivity of the surface when a surface layer of thickness d is present and R(0) the reflectivity of the free surface, and define:

AR_R(d)-R(0) R

R(0)

With hypothesis ii) (e" ox =0), Rox=R(0) and definitions (1) and (11) are identical. Classical electromagnetic theory shows that for di, A. D'Amicoc) Dipartimento di Fisica and Unitd INFM, b)Dipartimento di Scienze e Tecnologie Chimiche, ^Dipartimento di lngegneria Elettronica and Unitd INFM, Universitd di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: Goletti@roma2. infn. it The polarization dependence of the optical reflectivity for sapphyrin layers deposited by Langmuir-Blodgett technique onto a gold substrate has been measured. The experimental results show that characteristic spectra are related to layers of different thickness. An interpretation of the main spectral features in terms of the anisotropic optical properties of the sapphyrin molecule is presented.

1. Introduction Langmuir-Blodgett (LB) deposition is a well established technique for growing organic compounds [1] and is undergoing a novel development because of the increasing importance of organic materials in present electronic and optoelectronic devices [2]. New results have been reported recently about the electronic and morphological properties of LB organic layers, namely of sapphyrin [3] and cadmium arachidate [4]. These studies have demonstrated that the use of spectroscopic techniques, either "ex situ" to characterize the layer or "in situ" to monitor the growth process, is fundamental to understand the growth mechanism. Until now, with a few exceptions, Reflectance Anisotropy Spectroscopy (RAS) has been applied mostly to inorganic semiconductors [5] and metals [6]. In this paper we show that RAS can provide important information also about LB organic layers, similarly to what has been demonstrated for other molecules adsorbed onto metals [7,8]. 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 wavelength: Ar/r=2(r"-r'')/(r"+r6)

74

75

where r is the modulus of the complex reflectivity coefficient p=re' . In a crystal, a and b are chosen so that they coincide with well defined symmetry directions of the crystal. This allows to interpret the detected anisotropies in terms of intrinsic different optical properties of the system along such directions. In the present case, things are more complicated. When organic molecules are deposited onto a substrate, the system possesses no a priori orientations along which the anisotropy could be measured. Nevertheless, we have detected well defined RAS signals. Are they related to real, intrinsic anisotropies of the organic layers?

2. Experimental Experiments were performed in Rome, using a home-made RAS apparatus. All the spectra have been recorded in near-normal incidence, in the photon-energy range 1.5 eV-5.5 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. From inspection of Scanning Tunneling Microscope (STM) images, the sapphyrin chains resulting after deposition appear preferentially orientated along the direction of immersion (and, successively, extraction) of the sample into the liquid [3]. Measuring RAS spectra at different angles of rotation of the sample around the substrate normal, the higher signal anisotropy was recorded when a and b axes formed angles equal to +45 and -45 degrees with the dipping direction. This sample orientation was fixed for the experiment. The molecule of E2M8-sapphyrin forming the LB film (13,17-Diethyil2,3,7,8,12,18,22,23-octamethylsapphyrin diidrochloride) possesses two optical dipoles in the molecular plane, resulting one perpendicular to the other. Experimental details of LB deposition have been given elsewhere [3]. Here we just mention that two monolayers at a time were expected to be deposited by dipping the gold substrate into the solution and then extracting it. The film thickness was gradually changed from 2 to 20 nominal monolayers, each step in coverage being equal to 2 monolayers. We have already demonstrated that the nominal monolayers are different from the measured true coverage of the substrate [3]. As an example, 8-10 nominal monolayers are necessary to cover completely the gold substrate. Nevertheless, we will refer to the nominal monolayers as a label for the different layers.

3. Results and discussion From STM and Kelvin probe experiments performed on identical samples we know that only after 5 cycles of deposition (corresponding to 10 nominal ML) the gold substrate is completely covered by sapphyin molecules [3]. Also in RAS results, we find two distinct, well characterized coverage regimes: i) below 10

76

nominal ML; ii) above ten nominal ML, that is after the completion of the first sapphyrin monolayer.

200

300

400

500

600

700

800

Wavelength (nm) Fig. 1 - RAS spectra measured at LB layers with increasing nominal thickness of sapphyrin. In the inset, the directions a and b along which the light has been linearly polarized are reported with respect to the geometrical shape of the sample. The absolute sign of the quantity Ar/r is arbitrary. The arrow in bold shows the direction along which the sample has been dipped into the liquid for deposition.

In fig. 1, the curves display the RAS signals measured for coverage equal to, respectively, 0, 10, 12, 16, 18 and 20 nominal ML. When sapphyrin begins to be deposited onto gold, RAS detects a slight signal variation between 300 and 450 nm, showing a weak, large structure augmenting its intensity with coverage. When additional sapphyrin grows onto the first complete layer (=10 nominal ML), a new peak develops at about 480 nm, whose amplitude increases with coverage until it becomes the dominating structure of the whole spectrum. This peak exhibits a clear dependence upon rotation of the sample around the normal to the substrate, having its maximum when a and b axis coincide with the chosen experimental configuration (see above). A straightforward interpretation of these results is that RAS measures an intrinsic anisotropy of the optical properties of the sapphyrin molecule, related to the "in plane" dipoles responsible for the Soret band. From STM measurements [3] we know that, up to the completion of the first monolayer, sapphyrin molecules stand with their plane nearly perpendicular to the gold substrate. An angle of about 70 degrees has been directly measured. Consequently, dipole

77

optical transitions are not favorite because of the adsorption geometry. On the contrary, starting from 12 ML, a probable planar adsorption of the molecules above the first complete monolayer (as in the case of porphyrins [9]) would explain the anisotropy increase, when a and b polarization axes in this case nearly coincide with the in plane dipoles of the molecule. At this stage, however, our conclusions could seem rather speculative. Moreover, the optical anisotropics of a sapphyrin layer are, at our knowledge, unknown. Therefore, we have originally modeled the optical behavior of a sapphyrin layer as follows: a) the optical anisotropy of the sapphyrin molecule has been evaluated by using a semiempirical quantum chemistry approach method (INDO/SCI: Intermediate Neglect Differential Overlap with Spectroscopic parameterization), and its structure has been determined by using ab-initio geometrical optimization. The resulting optical anisotropy is not directly related to the sapphyrin ring itself but it is mainly due to the asymmetric position of the peripheral substituents; b) the optical behavior of our LB samples has been described by the reflectivity signal produced by a three layers system (substrate, sapphyrin, air)[10]: Ar 2cod = (A Ae"sapph - B AE 'sapph) (1). r c A and B are real quantities respectively connected to the dispersive and absorbitive part of the dielectric functions of gold, known from literature [11] (for definition of A and B, see ref.10); c) to evaluate At"sappi, and te'sapPh w e have used a rather crude approximation, modeling the sapphyrin layer with Lorentz oscillators whose parameters (energy position, oscillator strength) were determined from INDO/SCI results. In fig. 2 we have reported the experimental curve corresponding to a coverage of 20 ML, and the curve computed from equation (1), the parameters being the ones resulting from INDO/SCI calculation. It is evident that both curves exhibits the same lineshape, with the exception of the wavelength range below 400 nm, suggesting other effects not considered in the calculus as responsible for the measured anisotropy, for example transitions perpendicular to the plane of the molecule. Further developments of the theoretical approach will clarify this point.

4. Conclusions The experimental results that we have presented clearly show that RAS is helpful to characterize LB organic layers both in terms of coverage and morphology. The strong RAS signal observed near 480 nm is likely related to the in-plane optical anisotropy of the sapphyrin molecule, as tested by new, accurate calculations of the optical properties of a sapphyrin molecule

78

8
«

w\\\ \ \

1-X

°> o

10-3-7

w \

\

v \

\

\ \ \

\\^

6

\ v

(D 10-4

• x=0

\

.

1.36

. 1.40

. I . 1.44

Band gap energy (eV)

v ^ ti

; • x=1%

\\\ \

\

* Y \\

• - x=3% — i — i — i

.

1.32

\

A x=2% 10-5 2

^

f

\\\

GaAs

""

i_ U

y\

/'

O

SI

iA

-

^^ \ \ V \_ \ CJ1 < o.eo w \

1 /

x 1X

g

,-

1 '

• In Ga As

g °- 65

CM

T

X

\ *

i — i — i — i

3.0

i

*D \

i

ED1 Electron traps \

i

* — i — i

3.5 1000/T (1/K)

•

'

4.0

•

i — i — i —

4.5

Figure 2. Temperature dependence of the thermal emission rates (Arrhenius plots) for the EDI electron trap revealed in the GaAs^Sb^/GaAs heterostructures (full points) and InGaAs/GaAs heterostructure (open squares). Inset shows the electronemission activation energy from EDI traps versus band-gap energy at 300 K obtained for the GaAs^Sb^GaAs heterostructures (circles) and for the InGaAs/GaAs heterostructure (squares). Broken line represents constant position of the trap level with respect to the valence-band edge of Ev + 0.74 eV.

137

The principal argument for the assignment of EDI traps to electron states of dislocations is the logarithmic dependence of the amplitude on the filling time of the traps with electrons, observed over several orders of magnitude of that time. Such a nonstandard capture kinetics points out that the arrangement of EDI traps is strongly correlated, so that the Coulomb interaction between electrons captured at the traps limits their population, which is characteristic of dislocations [3]. On the contrary, amplitudes of DLTS peaks related to isolated point defects show a saturation for long filling times. The temperature dependence of the thermal emission rates of electrons from the EDI trap (Arrhenius plots) measured with DLTS in both GaAsSb/GaAs and InGaAs/GaAs heterostructures is shown in figure 2. The concentration of this trap in the region of the epilayer up to about 1 jum from the interface increased with increasing lattice mismatch and we relate the trap to threading dislocations formed in the epitaxial layers of ternary compounds as a result of the lattice mismatch. 3 Discussion and conclusions The activation energies of the EDI trap, evaluated from the slopes of the Arrhenius plots shown in figure 2, decrease with an increase of the Sb or In content in the epilayer similarly as the band-gap energy in the ternary compound decreases. They are presented in the inset in figure 2 versus the band-gap energy of the respective compound. The observed dependence points out that the energy level position of the trap with respect to the top of the valence band remains constant in each material, suggesting that the defect state is composed primarily of the valence band states [2]. Similar dependence of a trap activation energy on the band-gap energy has recently been found by Pal et al. [4] for the electron trap attributed to threading dislocations in MBE-grown InGaAs layers with higher (10 to 30%) In content. The hole trap at Ev + 0.71 eV, which we call HD3, was detected in the DLTS experiment under forward bias injection when the DLTS active region comprised the lattice-mismatched interface. Therefore, we relate the trap to defects associated with the lattice-mismatched interface in the heterostructure. Likely the same hole trap has been recently found by Du et al. [5] in InGaAs/GaAs lattice-mismatched heterostructures with various In content and layer thickness. That trap, labelled H4 by the authors, with the DLTS activation energy between 0.67 and 0.73 eV, has been related to misfit dislocations at the interface by comparing the DLTS spectra

138

in various heterostructures with the distribution of dislocations revealed by means of transmission electron microscopy. References 1. Fitzgerald E.A, Mater. Sci. Rep. 7 (1991) 87 2. Wosinski T., Makosa A., Figielski T. and Raczynska J., Appl. Phys. Lett. 67 (1995)1131 3 . Wosinski T., J. Appl. Phys. 65 (1989) 1566 4. Pal D., Gombia E., Mosca R., Bosacchi A. and Franchi S., J. Appl. Phys. 84 (1998) 2965 5. Du A.Y., Li M.F., Chong T.C., Xu S.J., Zhang Z. and Yu D.P., Thin Solid Films 311 (1997) 7

Localization of light beam in nonlinear optical waveguide array I. V. Gerasimchuk, A. S. Kovalev B. Verkin Institute for Low Temperature Physics and Engineering 47 Lenin Ave., 61103 Kharkov, Ukraine E-mail: gerasimchuk@ilt. kharkov. ua The localized states of light flux propagating in the array of parallel plane nonlinear optical waveguides are investigated. It is shown that the problem is reduced to a model of coupled anharmonic oscillators and all the parameters of such a model are found exactly. Both theoretical and experimental investigations of spatial localization of light beams with a large power were given considerable attention in recent years. The spatial localization of nonlinear optical beam in a few neighboring waveguides in a medium with Kerr nonlinearity was investigated theoretically by A.B. Aceves et al [1]. The numerical simulations proving this result within the framework of discrete nonlinear Schrodinger equation for the field amplitudes in waveguides have been reported. But the interaction of optical waveguides was described by a phenomenological parameter and the origin of nonlinearity in the equation was not discussed in the paper. Later such spatial localization of light flux was observed experimentally [2] and the results were compared with phenomenological discrete model. The aim of our paper is to investigate the localization of nonlinear stationary waves propagating along a system of identical parallel plane nonlinear optical waveguides. Note, that usually the investigations of nonlinear properties of waveguide arrays are performed by using discrete models for the wave amplitudes in waveguides [1-4] which are typically described by phenomenological equations with arbitrary parameters. We take into account the nonlinear Kerr terms only in the waveguides (assuming that the width of waveguides is much smaller than the dis139

140 tance between the adjacent ones) because of the smallness of average field amplitude in the wide regions for a weak waveguides coupling. So we use the restriction of a linear optical medium between waveguides. For the system proposed the equation for slow envelope of nonlinear monochromatic wave E(z,t)

(z

axis is perpendicular to the planes of parallel

waveguides) propagating along the waveguides is an ordinary nonlinear Schrodinger equation:

i

^ at

= -^fs^-2ajy\E\2E,

^ aZ

(1)

jtrrx

where parameter A > 0 for the waveguides. For one waveguide the equation (1) has the following solution for stationary localized beam: E = EQ • e x p ( - e | z | - / « ? ) ,

where e = v - to and En =

(2)

. If we introduce the total "intensity" of optical

flux in the form

W=f\E\2dz

(3)

then it does not depend on the frequency CO : W = 2 / A . But this property is not universal. Let us consider the system of two parallel plane waveguides at the positions z = +a . In this case the problem is reduced to linear wave equation

i

+

dt

r

dz1

=0

with the boundary conditions at waveguides positions

(4)

141 E\

dE

dE

dz

dz

(5)

=E\

ITa-0

ITa+0

= M\E\2E)

(6)

The light flux localized in the waveguide system is described by the solution in the following form: (7)

E

a,P=Aa,?'e±

Ey ={B-e'e:

+

C-e8Z)-e~'°"

(8)

in the regions a (z < -a), /3 (z > a), and y (-a < z < a) . Using the solution (7) and (8), we can rewrite the boundary conditions (5) and (6) as

E

_:*£.(1 + e - * " ) £ '

+-^—(E

E,)=0,

(9)

where i,j = 1,2, i * j , and El2 = E\z = +a) are the field amplitudes in the waveguides. For a weak coupling of the waveguides ( £ a » 1) (large distance between them or strong localization of wave in the waveguides) equations (9) describe the stationary oscillations of two weakly coupled anharmonic oscillators

e2E.-—E;+e2e-2"(El-E.)-0 J ' 2

(10)

Umt=^e-^\E,-Ej\2.

(11)

with a bound energy

It is an important feature of the result obtained that the oscillators' bound energy depends not only on the parameters of our system but on the parameter CO of our solution. Therefore, as for the modeling of the parallel plane waveguide array by

142

a chain of oscillators, the interaction, in fact, is the function of frequency of a monochromatic wave propagating in the system. (In papers [1,2] an effective constant of interaction was involved.) The system of equations (9) allows 3 types of the possible stationary states: the symmetric state (S) with equal fluxes in two waveguides:

(12) V A antisymmetric state (A) with equal fluxes but with opposite phases of the field in the waveguides:

E^-E2=M-(l-e^y\

(A)

(13)

V A

and inhomogeneous state (N) with equal phases but unequal fluxes in two waveguides: /

/ *

*