Handbook of Semiconductor Technology (2 Volume Set)

Handbook of Semiconductor Technology Volume 1 Kenneth A. Jackson, Wolfgang Schroter (Eds.)

@WILEY-VCH

Handbook of Semiconductor Technology Volume 1 Kenneth A. Jackson, Wolfgang Schroter (Eds.)

BWILEY-VCH

Weinheim . New York . Chichester . Brisbane . Singapore . Toronto

Editors: Prof. K. A. Jackson The University of Arizona Arizona Materials Laboratory 4715 E. Fort Lowell Road Tucson, A 2 85712, USA

Prof. Dr. W. Schroter IV. Physikalisches Institut der Georg-August-Universitat Gottingen BunsenstraBe 13- 15 D-37073 Gottingen, Germany

This book was carefully produced. Nevertheless, authors, editors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: applied for Deutsche Bibliothek Cataloguing-in-Publication-Data A catalogue record is available from Die Deutsche Bibliothek ISBN 3-527-29834-7

0 WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2000

Printed on acid-free and chlorine-free paper. All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition, Printing and Bookbinding: Konrad Triltsch, Print und digitale Medien GmbH, D-97070 Wiirzburg Printed in the Federal Republic of Germany.

Preface

Rapid scientific and technological developments have enabled microelectronics to transform the computer industry of the sixties into today’s information technology, which is now revolutionizing communications and the information media. Larger than the car business, information technology is one of the industries most impacted by physical research and technology transfer in the 20thcentury. This will continue for at least the first two decades of the 21Stcentury. The two volumes of this Handbook describe the underlying scientific and technological bases of this unique development, with the first addressing the science, and the second the technological framework of the field. Written by experts who have made major contributions to this enterprise, the chapters span from defect physics to device processing to present a panorama of the key steps, models, and visions -in short the evolution - of microelectronics. At the same time, this Handbook can be seen as a carefully written status report, specially valuable to those engaged in the continuing interplay between semiconductor science, technology, and business, and in the creation of new markets, such as sensor arrays, power and high frequency devices, solar cells, and blue lasers. Circuit and systems design, which turn science and technology into end-user products, are not included as separate chapters, because each would need a volume in its own right. However, due to their close connection with process science and technology, they are briefly treated as the need arises. In semiconductors, science frequently develops in close interplay with technology, and fundamental investigations and technological advances cross-pollinate each other in an unprecedented fashion. The miniaturization of the transistor, begun forty years ago, is approaching dimensions, where present concepts appear to break down, and the available characterization methods may no longer function. Other devices, such as solar cells, are now entering the mass market, placing increasing demand on materials quality, process efficiency, and, of course, cost. At present, the most promising approach to addressing these challenges appears to involve fundamental understanding and modeling of highly complex nonlinear solid state phenomena, in short, physically-based, predictive simulation of complex process technological sequences. The first volume places particular emphasis on the concepts and models relevant to such issues. Starting with a description of the relevant fundamental phenomena, each chapter describes and develops the mechanisms and concepts used in current semiconductor research. Experimental details are provided in the text, or summarized in tables and diagrams to the extent needed to illustrate the models under discussion. The Handbook begins with chapters on the basic concepts of band structure formation, charge transport, and optical excitations (chapters 1 and 2), the physics of defects (point defects, impurities, dislocations, grain boundaries, and interfac-

VI

Preface

es) in crystalline semiconductors, particularly Si and GaAs (chapters 3-8, lo), special materials, such as hydrogenated amorphous Si (chapter 9), concluding with semiconductors for solar cell applications, silicon carbide, and gallium nitride (chapters l l - 13). I am very grateful to the contributors who took the trouble to write a chapter for this volume. I thank Prof. Peter Haasen, Prof. Abbas Ourmazd, PD Dr. Michael Seibt, and Prof. Helmut Feichtinger for many useful proposals and critical comments. I also thank Dr. Jorn Ritterbusch and Mrs. Renate Dotzer of WILEYVCH for their advice and very agreeable cooperation. Let me finally quote from a letter of one of the authors (A. 0.):"We wish the reader as much fun with the material as we have had - as much fun, but much less hard work". Wolfgang Schroter Gottingen, April 2000

List of Contributors

Prof. Dr. Helmut Alexander Universitat zu Koln 11. Physikalisches Institut Abteilung fur Metallphysik ZulpicherstraBe 77 D-50937 Koln Germany Chapter 6 Dr. Alain Bourret CEAIDCpartement de Recherche Fondamentale sur la Matikre CondensCe Service de Physique des MatCriaux et Microstructures 17 rue des Martyrs F-38054 Grenoble France Chapter 7 Prof. Dr. W. J. Choyke Department of Physics and Astronomy University of Pittsburgh Pittsburgh, PA 15260 USA Chapter I 1 Prof. Dr. Robert P. Devaty Department of Physics and Astronomy University of Pittsburgh Pittsburgh, PA 15260 USA Chapter 11

Prof. Dr. Helmut Feichtinger Karl-Franzens-Universitat Graz Inst. f. Experimentalphysik Universitatsplatz 5 A-801 0 Graz Austria Chapter 4 Dr. Dieter Gilles Wacker Siltronic AG Johannes-Hess-Str. 24 D-84489 Burghausen Germany Chapter I0 Prof. Dr. Ulrich M. Gosele MPI fur Mikrostrukturphysik Am Weinberg 2 D-06 120 Halle Germany Chapter 5 Dr. Martin L. Green Bell Laboratories Lucent Technologies Murray Hill, NJ 07974-2070 USA Chapter 8 Prof. Dr. Robert Hull University of Virginia Department of Materials Science and Engineering 151 Engineers Way Charlottesville, VA 22903-4745 USA Chapter 8

Vlll

List of Contributors

Dr. Christian Kisielowski National Center for Electron Microscopy Lawrence Berkeley National Laboratory Berkeley, CA 94720 USA Chapter 13 Dr. Joachim Kruger University of California Department of Materials Science and Engineering Berkeley, CA 94720 USA Chapter 1 Prof. Dr. Michel Lannoo Institut Superieur D’Electronique du Nord 41 Boulevard Vauban 59046 Une France Chapter 1 Prof. Dr. Hans Joachim Moller Institut fur Experimentelle Physik TU Bergakademie Freiberg D-09599 Freiberg Germany Chapter 12 Prof. Dr. Abbas Ourmazd IHP Im Technologiepark 25 D-15236 Frankfurt/Oder Germany Chapter 8 Dr. W. D. Rau IHP Im Technologiepark 25 D- 15236 Frankfurt/Oder Germany Chapter 8

Dr. Jean L. Rouvikre CEAIDCpartement de Recherche Fondamentale sur la Matihre Condensee Service de Physique des MatCriaux et Microstructures 17 rue des Martyrs F-38054 Grenoble France Chapter 7 Prof. Dr. Wolfgang Schroter Georg-August-Universitat Gottingen IV. Physikalisches Institut BunsenstraDe 13- 15 D-37073 Gottingen Germany Chapter 10 Dr. P. Schwander IHP Im Technologiepark 25 D- 15236 Frankfurt/Oder Germany Chapter 8 Dr. Michael Seibt Georg-August-Universitat Gottingen IV. Physikalisches Institut Bunsenstralje 1 3 - 15 D-37073 Gottingen Germany Chapter 10 Dr. Robert A. Street Xerox Palo Alto Research Center 3333 Coyote Hill Road Palo Alto, CA 94304 USA Chapter 9

List of Contributors

IX

Prof. Dr. Teh Y. Tan Department of Mechanical Engineering and Materials Science Duke University Durham, NC 27708 USA Chapter 5

Prof. Dr. Rainer-G. Ulbrich Georg-August-Universitat Gottingen IV. Physikalisches Institut Bunsenstrarje 13- 15 D-37073 Gottingen Germany Chapter 2

Prof. Dr. Helmar Teichler Georg-August-Universitat Gottingen Institut fur Materialphysik Hospitalstrarje 3-7 D-37073 Gottingen Germany Chapter 6

Prof. Dr. George D. Watkins Sherman Fairchild Center for Solid State Studies, Lehigh University, 16A Memorial Dr. East Bethlehem, PA 18015-3185 USA Chapter 3

Dr. Jany Thibault CEA/DCpartement de Recherche Fondamentale sur la Matibre CondensCe Service de Physique des MatCriaux et Microstructures 17 rue des Martyrs F-38054 Grenoble France Chapter 7

Prof. Dr. Eicke R. Weber University of California Department of Materials Science and Engineering Berkeley, CA 94720 USA and Lawrence Berkeley National Laboratory Material Science Division Berkeley, CA 94720 USA Chapter 13

Dr. Raymond T. Tung Bell Laboratories Lucent Technologies Murray Hill, NJ 07974-2070 USA Chapter 8

Dr. K. Winer Xerox Palo Alto Research Center 3333 Coyote Hill Road Palo Alto, CA 94304 USA Chapter 9

Contents

1 Band Theory Applied to Semiconductors

M . Lannoo

................

1

2 Optical Properties and Charge Transport . . . . . . . . . . . . . . . . 69 R. G. Ulbrich 3 Intrinsic Point Defects in Semiconductors 1999 . . . . . . . . . . . . 121 G. D. Watkins 4 Deep Centers in Semiconductors H. Feichtinger

....................

167

5 Point Defects, Diffusion, and Precipitation . . . . . . . . . . . . . . . 23 1 T I.: Tan, U. Gosele 6 Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 H. Alexander, H. Teichler

7 Grain Boundaries in Semiconductors . . . . . . . . . . . . . . . . . . 377 J. Thibault, J.-L. Rouviere, A. Bourret 8 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 R. Hull, A. Ourmazd, W D. Rau, I? Schwandel; M. L. Green, R. i? Tung

9 Material Properties of Hydrogenated Amorphous Silicon . . . . . . . 541 R. A. Street, K. Winter 10 High-Temperature Properties of Transition Elements in Silicon . . . . 597 W Schroter, M. Seibt, D. Gilles 11 Fundamental Aspects of S i c W J. Choyke, R. l? Devaty

......................

661

12 New Materials: Semiconductors for Solar Cells . . . . . . . . . . . . 715 H. J. Moller 13 New Materials: Gallium Nitride . . . . . . . . . . . . . . . . . . . . . 77 1 E. R. Webel; J. Kriiger, C. Kisielowski Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

Handbook of Semiconductor Technologj Kenneth A. Jackson, Wolfgang Schrotei CoDvriaht 0 WILEY-VCH Verlaa GrnbH. 2000

1 Band Theory Applied to Semiconductors Michel Lannoo Dkpartement Institut SupCrieur d’Electronique du Nord. Institut d’Electronique et de MicroClectronique du Nord. Villeneuve d’Ascq. France

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 1.1.1 From Discrete States to Bands . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Bloch Theorem for Crystalline Solids . . . . . . . . . . . . . . . . . . . . . 7 1.1.3 The Case of Disordered Systems . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.4 The Effective Mass Approximation (EMA) . . . . . . . . . . . . . . . . . . 10 1.1.4.1 Derivation of the Effective Mass Approximation for a Single Band . . . . . 10 12 1.1.4.2 Applications and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . The Calculation of Crystalline Band Structures . . . . . . . . . . . . . . 14 1.2 1.2.1 Ab Initio Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 1.2.1.1 The Hartree Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1.3 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1.4 Beyond Local Density (the G-W Approximation) . . . . . . . . . . . . . . . 17 17 1.2.1.5 The Pseudopotential Method . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Computational Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.2.1 Plane Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 1.2.2.2 Localized Orbital Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.3 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3.1 The Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . 21 22 1.2.3.2 The Empirical Pseudopotential Method . . . . . . . . . . . . . . . . . . . . 1.3 Comparison with Experiments for Zinc Blende Materials . . . . . . . . 23 23 1.3.1 The General Shape of the Bands . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.1 The Tight Binding Point of View . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1.2 The Empirical Pseudopotential Method . . . . . . . . . . . . . . . . . . . . 26 1.3.2 The k-p Description and Effective Masses . . . . . . . . . . . . . . . . . . 29 1.3.3 Optical Properties and Excitons . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3.4 Ab Initio Calculations of the Excitonic Spectrum . . . . . . . . . . . . . . . 34 1.3.5 A Detailed Comparison with Experiments . . . . . . . . . . . . . . . . . . 34 1.4 Other Crystalline Materials with Lower Symmetry . . . . . . . . . . . . 36 1.4.1 General Results for Covalent Materials with Coordination Lower than Four . . . . . . . . . . . . . . . . . . . . . . 36 1.4.2 Chain-Like Structures Like Se and Te . . . . . . . . . . . . . . . . . . . . . 37 1.4.3 Layer Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.4.4 New Classes of Materials: the Antimony Chalcogenides . . . . . . . . . . . 39

2 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 1.6.1 1.6.2 1.7 1.7.1 1.7.2 1.7.3 1.7.4 1.8

1 Band Theory Applied to

Semiconductors

Non-Crystalline Semiconductors . . . . . . . . . . . . . . . . . . . . . . 44 The Densities of States of Amorphous Semiconductors . . . . . . . . . . . . 44 Numerical Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Dangling Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 The Case of SiO. Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Disordered Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Definitions of the Different Approximations . . . . . . . . . . . . . . . . . 52 The Case of Zinc Blende Pseudobinary Alloys . . . . . . . . . . . . . . . . 54 Systems with Lower Dimensionality . . . . . . . . . . . . . . . . . . . . 57 Qualitative Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 The Envelope Function Approximation . . . . . . . . . . . . . . . . . . . . 59 Applications of the Envelope Function Approximation . . . . . . . . . . . . 61 Silicon Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

List of Symbols and Abbreviations

List of Symbols and Abbreviations lattice parameter basis vector of the unit cell bandgap energy conduction-band and valence-band energy electron charge envelope function reciprocal lattice vector resolvent operator hamiltonian Planck constant elements of the hamiltonian total angular momentum joint density of states wave vector orbital angular momentum optical matrix element effective mass density of states momentum vector momentum of electron and hole matrix element of the momentum operator spin vector kinetic energy transfer matrix Bloch's function potential crystaI volume perturbation coupling constant splitting energy energy of the state n real and imaginary part of the dielectric constant wave length frequency self-energy operator conductivity atomic states wave function volume of the unit cell atomic volume frequency

3

4

ATA a-Si, c-Si CPA DOS EMA EMT EPM EPR ESR ETB EXAFS HOMO LCAO LDA LUMO MCPA TBA UPS VCA VSEPR XPS

1 Band Theory Applied to Semiconductors

average t matrix approximation amorphous and crystalline silicon coherent potential approximation density of states effective mass approximation effective mass theory empirical pseudopotential method valence pair repulsion electron spin resonance empirical tight binding extended X-ray fine structure highest occupied molecular orbital linear combination of atomic orbitals local density approximation lowest unoccupied molecular orbital molecular coherent potential approximation tight binding approximation ultraviolet photoemission spectroscopy virtual crystal approximation valence shell electron pair repulsion X-ray photoemisssion spectroscopy

1.1 General Principles

1.1 General Principles The aim of this chapter is to establish the basic principles of the formation of bands in solids. To do this we start with a one-dimensional square well potential which we consider as a simplified description of an atom. We then bring two such square wells into close contact to simulate the behavior of a diatomic molecule. This can be generalized to include a large number of square wells to illustrate the concept of energy bands corresponding to the one-dimensional model of the free electron gas. These results are shown to be independent of the boundary conditions, allowing us to use Born von Karmann cyclic conditions and to classify the band states in terms of their wave vectors. Such a simple description does not take into account the spatial variations of the true potential in crystalline solids. However, the free electron gas results can be generalized through the use of Bloch's theorem which allows the classification of all energy bands in terms of their wave vectors, the periodic boundary conditions imposing that the allowed energy values be infinitely close together. This does not hold true, however, in nonperiodic systems such as amorphous solids and glasses. However, we show that, in this case, although general considerations impose the existence of bands, these can contain two types of states with localised or delocalised behavior. This represents the fundamental difference between nonperiodic systems and crystalline materials. A final situation of practical interest is the case where a slowly varying potential is superimposed on a rapidly varying crystal potential. Such cases can be treated in a simple and very efficient manner by the effective mass theory (EMT), also termed the envelope function approximation. The best known applications of the EMT correspond

5

to hydrogenic impurities in semiconductors and, more recently, to various kinds of system involving semiconductor heterojunctions, which will be discussed in Sec. 1.7.

1.1.1 From Discrete States to Bands We want to describe here in the simplest way the basic phenomena which occur when atoms are brought close to each other to form a solid. To do this, the attraction potential of the free atom is represented by a one-dimensional square well potential and the main trends concerning its energy levels and wave function are analyzed. We first consider the square well potential of Fig. 1-1, with depth V, and width a. We assume a to be of atomic dimensions (i.e., a few A)and take V, to be large enough for the energy E of the lowest states to be such that E ~ V ,In. such cases these lowest states are close to those of an infinite potential, i.e., their wave functions and energies E,, are given by:

vn

v,, - sin k,, x En

pt2 -kn2 2m

nn --

( n = 1,2, ...) a the allowed values k, of the one-dimensiona1 wave vector k practically corresponding k,,

I" Figure 1-1. One dimensional square well potential of height V , and of width a.

6

1 Band Theory Applied to Semiconductors

to vanishing boundary conditions at x = 0 and a. For a typical atomic dimension of a = 3 A, the distance in energy between the lowest two levels is of the order of 10 eV, typical of the values found in atoms. Of course the values of the excited states cannot correspond to what happens to true three-dimensional potentials, but this will not affect the main qualitative conclusions derived below. Let us now build the one-dimensional equivalent of a diatomic molecule by considering two such potentials at a distance R (Fig. 1-2). We can discuss qualitatively what happens by considering first the infinitely large R limit where the two cells can be treated independently. In this limit, the energy levels of the whole system are equal to those of each isolated well (i.e., E l , E2 ...) but with twofold degeneracy since the wave function can be localised on one or the other subsystem. When R is finite but large, the solutions can be obtained using the first order perturbation theory from those corresponding to the independent wells. This means that the twofold degenerate solutions at E l , E2 ... will exhibit a shift and a splitting in energy resulting in two sublevels of symmetric and antisymmetric character. This shift and splitting will increase as the distance R decreases. Such a behavior is pictured in Fig. 1-3, from R = 03 to the limiting case where R = a. This case is particularly easy to deal with since it corresponds to a single potential well but

r

a

R

0

Figure 1-2. The double well potential as a simple example of a molecule.

1

- - - - - -- - - - _ - - --------_

--- -

/

/ - -

I-

321a

-R

Figure 1-4. Formation of bands for a Kronig-Penney model where a is the well width and R the interwell distance.

of width 2 a instead of a . The consequence is that one again gets the set of solutions Eq. (1-l), but the allowed values of k, are closer together with an interval n / 2 a instead of n/a. This means that the number of levels is multiplied by two as is apparent in Fig. 1-3. The generalization to an arbitrary number N of atoms is obvious and is pictured in Fig. 1-4. At large inter-well separation R, the degeneracy of each individual level E l , E2 ... is N . At closer separation, these Ievels shift and split into N distinct components. When R = a one recovers one single well of width N u , which means that the allowed values of k, are now separated by n / N a . For a crystal, N becomes very large

1.1 General Principles

(or order lo7 for a 1D system) and these allowed values form a “pseudocontinuum”, i.e., a set of discrete values extremely close to one another. The same is true of the energy levels whose pseudocontinuums can either extend over the whole range of energies (the case R = a in Fig. 1-4) or more generally ( R > a) are built from distinct groups of N levels. These groups of N levels are always contained in the same energy intervals irrespective of the value of N . They are called the allowed energy bands, the forbidden regions being called gaps. These qualitative arguments can be readily generalized to realistic atomic potentials in three dimensions. One then gets a set of allowed energy bands which contain a number of states equal to the number of atoms N times an integer. These bands are, in general, separated by forbidden energy gaps.

1.1.2 Bloch Theorem for Crystalline Solids For crystals it is possible to derive fairly general arguments concerning the the properties of the allowed energy bands. Before doing this let us come back to our simplified model of N square wells in the situation R = a where one gets a single well of width N a . We have seen that the states are completely characterized by the allowed values k, of the wave number k which are equal to n xlNa with n > 0. It is to be noted, however, that even this simple model contains complications due to the existence of boundaries or surfaces. It is the existence of these boundaries that lead to solutions having the form of stationary waves sin(kx) instead of propagating waves exp(i kx) which are mathematically simpler (the sine function is a combination of two exponentials). Of course the increased mathematical complexity of the vanishing boundary conditions is not very important for this particu-

7

lar one-dimensional problem, but it becomes prohibitive in real cases where one has to deal with the true surfaces of a threedimensional material. If one only needs information about volume properties it is possible to avoid this problem by making use of Von Laue’s theorem which states that perturbations induced by surfaces only extend a few angstroms within the bulk of the material. This means that for large systems, where the ratio of the numbers of surface to volume atoms tends to zero, volume properties can be obtained by any type of boundary conditions even if they do not seem realistic. In that respect, it is best to use Born von Karman’s periodic conditions in which one periodically reproduces the crystal under study in all directions, imposing that the wave function has the same periodicity. One then gets an infinite periodic system whose mathematical solutions are propagating waves. For instance, the linear square well problem with R = a becomes a constant potential extending from minus to plus infinity. Its solutions can be taken as:

2nn k, =L

( n% 0 integer)

They are mathematically different from Eq. (1-1) but they lead to the same answer for physical quantities. For instance, the interval between the allowed values of k is larger by a factor of two but this leads to the same density of states (number of levels per unit energy range) because the states are twofold degenerate (states with opposite values of k give the same energy). One can also show that the Fermi energy and electron density are identical in both cases.

8

1 Band Theory Applied to Semiconductors

The results given by Eq. (1-2) can be generalized to real crystalline solids through the use of the Bloch theorem. Such solids are characterized by a periodically repeated unit cell and, with Born von Karman boundary conditions, they have translational periodicity which greatly simplifies the mathematical formulation of the solutions. Let us then consider a three-dimensional crystal for which one has to solve the one-electron Schrodinger equation: H

3P)= E 3w

(1-3)

where r is the electron position vector. If we call aj the basis vectors of the unit cell, the fact that the system has translational symmetry imposes that:

I 3 (r+aj)12= I 3 (r)12

(1-4)

which, for the wave function itself, gives:

3 (r+ aj) = exp (i ‘pj) 3 ( r )

(1-5)

If we now consider a translation by a vector R

R = C mj aj j

mj being integers, we automatically get the condition:

which defines a new function uk (r). The direct application of a translation R to this expression leads to: (1-10)

qk(r + R ) = exp [i k (r + R ) ] uk (r + R)

and a comparison with Eq. (1-8)automati cally leads to: Uk

(r + R ) = uk ( r )

(1-11

i.e., that the function uk (r)is periodic. Equations (1-9) and (1- 11) constitute Bloch’s theorem, which states that the eigenfunctions can be classified with respect to their wave vector k and written as the product of a plane wave exp(ikr) times a periodic part. They are thus propagating waves in the crystal lattice. This is a generalization of the one-dimensional case discussed above. If one injects qk(r)into Schrodinger’s equation Eq. (1-3), then the eigenvalue will become a continuous function E ( k ) of the wave vector k . However, the vector k can only take allowed values chosen in such a way that qk(r)satisfies the boundary conditions, which are:

3k (r + Nj aj>= 3 k (r)

(1-12)

where Nj is the number of crystal cells along

aj . Using Eq. (1-9) and Eq. (1-1 1) this im-

poses the conditions:

The phase factor in Eq. (1-7) is a linear function of the components of R and can be written quite generally under the simpler form k R , leading to:

qk(r + R ) = exp (i k * R ) qk(r)

(1-8)

where k is such that k . aj is equal to (in this expression we have indexed the wave function by its wave vector k). To exploit Eq. (1-S), which is a direct consequence of translational invariance, we can write without loss of generality:

qk(r)= exp (i k - r ) uk (r)

(1-9)

(1-13) where nj is an integer. These allowed values can be expressed more directly with the help of the basis vectors of the reciprocal lattice a? defined by: a?

a, = 2 n

SjL

(1-14)

which leads to: (1-15)

1 . 1 General Principles

This again generalizes the one-dimensional situation described by Eq. (1 -2) to any real crystal in I-, 2-, or 3-dimensions. As a conclusion Bloch’s theorem leads us to write the wave function as a plane wave exp (i k .r ) modulated by a periodic part. Its energy E ( k ) is a continuous function of the wave vector k . This one takes discrete values which form a pseudocontinuum. The allowed energies will then be grouped into bands as discussed more qualitatively in the previous section. A final point is that one can show that the energy curves E ( k ) and Bloch functions qk(r)are periodic in reciprocal space. One then gets the complete information about these quantities from their calculation for k points lying in one period of the reciprocal lattice. From the symmetry properties ( E ( k )= E ( - k ) ) it is better to use the period symmetrical with respect to the origin, called the first Brillouin zone.

1.1.3 The Case of Disordered Systems Different varieties of solids exist which do not exhibit the long range order characteristic of perfect crystals. They differ qualitatively among themselves by the nature of their disorder. A simple case is the alloy system where one can find either A or B atoms on the sites of a perfect crystalline lattice. This kind of substitutional disorder is typical of the ternary semiconductor alloys Gal,A1,As where the disorder only occurs on the cationic sites. We shall deal in more detail with this problem in Sec. 1.6. Other cases correspond to amorphous semiconductors and glasses which are characterized by a short range order. For instance, in a-Si, the silicon atoms retain their normal tetrahedral bonding and bond angles (with some distortions), but there is a loss of long range order. A lot of covalently-bonded systems (with coordination numbers small than 4) can be found in the amorphous or

9

glassy state. All have a short range order and moderate fluctuations in bond length, but in some cases they can have large fluctuations in bond angles. In all these situations, even for the disordered substitional alloy, it is clear that one cannot make use of Bloch’s theorem to classify the band states. We could even ask ourselves if the concept of energy bands still exists. The simplest case demonstrating this point again corresponds to the one-dimensional system with square well potentials. We can simulate a disordered substitutional alloy by considering square wells of width a as before, and of depth VA for A atoms and VB for B atoms, with VA> VB. We also assume that the interwell distance is R = a to get the simplest situation. If the material were purely B then one would get a constant potential which we take as the origin of energies. Thus each time one substitutes an A atom for a B one this results in an extra potential well of width a a depth VA- V,. Two A neighbors lead to a well of width 2 a, and a cluster of M A neighbors gives a well of width M a (Fig. 1-5). The alloy will consist of a distribution of these potential wells separated by variable distances. It is interesting to investigate the nature of the states for such a problem. The simplest case corresponds to energies E > 0, where the states have a propagating behavior and there are solutions at any positive value of E. The existence of boundary conditions due to the fact that the system contains a finite but large number N of atoms

o

f)

P

20

Figure 1-5. Simple representation of a disordered binary alloy with square well potentials of width a for one impurity, 2a for a pair, etc.

10

1 Band Theory Applied to Semiconductors

will simply transform this continuum of states into a pseudocontinuum. This results in a band of “extended states’’ for E > 0. The situation for E < 0 is drastically different. An isolated impurity A, represented by a single well, gives at least one energy level at E , , or even more at E2, E,, etc. The corresponding wave function is localized and decays exponentially as exp (- k Iz I) where k =

Jg.

Clusters of M atoms

lead, as described before, to denser sets of levels in the well since the interval between allowed k values is divided by M . In particular, for a very large cluster, one obtains a pseudocontinuum of levels, the lowest one being infinitely close to the bottom of the potential well. One has thus a statistical distribution of potential wells of varying width giving rise to a corresponding distribution of levels located between E = 0 and E = - (V,- VB). Again this leads to pseudocontinuum of levels. Furthermore, there is the possibility of interaction between the potential wells when these are close enough. This also acts in favor of a spread in energy levels. The states in the potential wells can have a more or less localized character depending on the distance between the wells. We shall discuss this problem later. The conclusion of this simple model is that the occurrence of disorder, at least in somes cases, also leads to the existence of a pseudocontinuum of states, even if Bloch’s theorem does not apply. Of course we have only discussed one particular model. We shall later discuss other cases of disorder or randomness which also lead to the existence of well-defined energy bands. 1.1.4 The Effective Mass Approximation (EM4 Interesting cases that often occur in practice correspond to the application of a po-

tential slowly varying in space to a crystalline solid. Such situations can be handled relatively easily without solving the full Schrodinger equation, by using the so called “effective mass approximation“. One major field of application has been the understanding of hydrogenic impurities in semiconductors (for reviews see Bassani et al., 1974; Pantelides, 1978). More recently the same method, often called “the envelope function approximation”, has been applied to the treatment of semiconductor heterojunction and superlattices as will be discussed in Sec. 1.7.

1.1.4.1 Derivation of the Effective Mass Approximation for a Single Band Let us begin with the simplest case of a crystal whose electronic structure can be described in terms of a single energy band, the solution of the perfect crystal Schrodinger equation:

H , % (r)= E (W % (r)

(1-16)

If a perturbative potential V ( r )is applied to the system we can describe a solution of the perturbed system $J (r)as a linear combination of the perfect crystal eigenstates ( k belonging to the first Brillouin zone), (1-17) and obtain the unknown coefficients by projecting the new Schrodinger equation, (Ho + v> $J 0.1 = E $J (r)

(1-18)

onto the basis states (pk (r).This immediately leads to the set of linear equations:

E(WdW+X

(pk+w=Ea(k) (1-19) At this stage we need to simplify the matrix elements of V,otherwise it is impossible to k‘

(Vkl VI

1 . 1 General Principles

go further, except numerically. We use the fact that the % (r)are Bloch functions, cpk

(r)

( 1-20)

uk (r)

and express the potential matrix element as: (1-21)

((PkIVI Q)k’)=

=J ~

(k’ - k ) . r

(

~

1

U $ ( r ) uk* (r)d3 r

In view of the Bloch theorem the product u$ uk,is a periodic function of r and we can expand it in a Fourier series, 1

uz ( r )ukt ( r )= -

V G

c k , k * ( G )eic.r

(1-22)

where v is the crystal volume and G are the reciprocal lattice vectors. The matrix element Eq. (1-21) can thus be expressed exactly as: ( 1 -23 a) c k , k ’ ( G ) V ( k ’ + G -k) ( q k VI Q ) k ’ ) =

I

G

1

V ( q ) = - I e i q ‘ rV ( r ) d r

(1-23 b)

V

At this level we must make some assumptions about V ( r ) . The first one is that it varies slowly in space (i.e., over distances which are large compared to the size of the unit cell). This means that its Fourier transform decreases very rapidly with the modulus of the wave vector, i.e., that one can neglect terms with G#O in Eq. (1-23) and, also, that only terms with k ’ = k will effectively contribute. We now make the second central assumption of the EMA, that we look for solutions whose energy E is close to a band extremum k,. If this is so, only states with k = k , will have a ( k ) sensibly different from zero in Eq. (1-19). This means that one can rewrite Eq. (1-19) using Eq. (1-23) under the approximate form: ( 1 -24) E ( k ) a ( k ) + xC k o , k o ( 0 ) V ( k ’ - k ) a ( k ’ ) = O k’

11

However, c k o , k o ( o ) has an important property: it is given by the following integral over the crystal volume

c k o , k o ( o ) = s u i 0 ( r )uko(r) d3 r = = J qzo(r)a k o ( rd3) r = 1

( 1 -25)

in view of the fact that the wave functions % (r),are normalized. The final form of the EMA equation is thus (1-26) E ( k ) a ( k ) + xV ( k ’ - k ) a ( k ’ ) = E ~ ( k ) k’

It is interesting to derive a real space equation from this by Fourier transforming Eq. (1-26). To perform this we must take into account the fact that the function a ( k ) is strongly peaked near k,. We thus introduce the following Fourier transform:

F(r)=Cu(k)exp(i(k-ko).r)

(1-27)

k

such that F ( r ) varies slowly in space when a ( k ) only takes important values in the vicinity of k=k,. To get an equation for F ( r ) we multiply Eq. (1 -26) by exp i (k-k,) . r and sum over k , assuming that one makes a negligible error in the potential term by extending the summation over k to the whole space. This leads to the real space equation: { E(k,-i

V,)+ V ( r ) }F ( r ) = E F ( r ) (1-28)

This is a differential equation in which the operator k, -i V, has been substituted for k in the dispersion relation E ( k ) . As we have seen, the function F ( r ) is likely to vary slowly withr(ora(k) #Oonlyfork=ko)so that one can expand E ( k ) to the second order in the neighborhood of k - k,. Calling a the principal axes of this expansion we have: A2 E ( k ) -- E ( k 0 ) + -( k , - k ~ , )(1-29) ~ a 2m,

x

which defines the effective masses ma along direction a.This allows us to rewrite

12

1 Band Theory Applied to Semiconductors

-

scaled parameters e2 e2/e and m --z m *. This leads to a set of hydrogenic levels with

Eq. (1-28) as:

an effective Rydberg

=IE-E(ko)IF(r) which represents the usual form of the EMA equation as derived by many authors (Bassani et al., 1974; Pantelides, 1978). It is interesting to examine the meaning of the function F ( r ) .To do this we start from the expansion Eq. (1-17) of w(r), express the q k ( r ) as in Eq. (1-20) and factorize eiko.r . This gives k

As a ( k ) is peaked near k,, we approximate Uk( r ) by its value at k,, which leads us directly to:

w (r)= F ( r ) (PRO (r)

(1-32)

This means that ( r )can be rewritten as the product of the Bloch function (which varies over a length typically of the order of the interatomic distances) times a slowly varying “envelope function”. The advantage of the EMA is that one directly obtains F ( r ) from a Schrodinger-like equation involving the effective masses.

1.1.4.2 Applications and Extensions The first well known use of the EMA was for hydrogenic impurities in semiconductors. If we treat single donor substitutional impurities, like As in Si or Ge for instance, the excess electron will see an attractive potential roughly given by -e2/&r (where E is the dielectric constant), which one considers as slowly varying. This can stabilize levels in the gap in the proximity of the bottom of the conduction band E,. For a single minimum and an isotropic effective mass one gets an hydrogenic-like equation with

m*e4

which, for 2A2 E2 typical values of m*=0.1 and & = l o ,becomes of order 14 meV, i.e., fairly small compared to the band gap. This result correctly reproduces the order of magnitude found in experimental data. However, to be truly quantitative, the EMA must in many cases satisfy the following requirements: ~

- It must include the effective-mass anisotropy when necessary. This has been done in Faulkner (1968, 1969), one effect being the splitting of p states, for instance. - It must also properly include the valleyvalley interactions when there are several equivalent minima. This can be done, for instance, by first order perturbation theory on degenerate states (since there are as many identical impurity states than there are minima). With these improvements the EMA theory has achieved considerable success for single donor impurities, especially for excited states (see Bassani et al., 1974); Pantelides, 1978) and Table 1-1 for reviews. Only the ground state is found to depart significantly from the predicted levels at this stage of the theory. This is due to the deviations of the potential from its idealized form -e2/&r in the impurity cell. The corresponding correction is known as the chemical shift. It is also possible to treat more exactly the many-valley interactions by recently derived methods described in Resca and Resta (1979, 1980). The case of acceptor states derived from the valence band is more complicated. This is due to the threefold degeneracy of the top of the valence band. This means that (r) must be written as a combination of the

13

1.1 General Principles

Table 1-1. Comparison between theoretical and experimental energy spacing (cm-’) for donor impurities in silicon. The theoretical values are taken from Faulkner (1968,1969). (The spacing between excited states, independent of the ground state position, is more suitable than the observed position of the transition from the ground state to perform a comparison with the theory since this ground state is not hydrogenic.) Transition

Theory

P

As

Sb

2 P, -2 Po 3 Po -2 Pi 4 Po -2 P+ 3 P, -2 P, 4 P* -2 P* 4 f, -2 f, 5 p, -2 f, 5 f, -2 p* 6 Pt -2 P i 3 s -2p* 3 43 -2 Pi 4 s -2P* 4 fo -2 P* 5 Po -2 Pi 5 Po -2 P i 6 ht -2 P,

5.11 0.92 3.07 3.29 4.22 4.5 1 4.96 5.14 5.36 0.65 2.65 3.55 4.07 4.17 4.77 5.22

5.07 0.93 3.09 3.29 4.22 4.51 4.95 5.15 5.32

5.1 1 0.92 3.10 3.28 4.21 4.49 4.94 5.14 5.32

5.12 0.91 3.07 3.29 4.21 4.46 4.92 5.31

2.64 4.08 4.17 4.76 5.52

+cV ( k ’ - k ) q ( k )= E q ( k )

Zhij ( k ) U j ( k )

k’

j

(1-34)

Now one can define slowly vaying functions F;. (r) by the Fourier transformation Eq. (1 -27) from ai(k) with k = 0 and get the generalization of Eq. (1-28):

Z hij ( - iV,)Fj(r)+ V(r)F;( r )= E 6 ( r ) .l

4.70

Bloch states ( ~ , , ~ (with ( r ) n = 1, 2, 3) belonging to each of the three energy branches E, (k).The first part of the derivation proceeds as for a single band and the generalization of Eq. (1-26) becomes:

E , , ( k ) a , ( k ) + C V ( k ’ - k ) a , ( k ’ ) =E a , ( k ) k’

the quantities that can be expanded to second order in k are the elements hjj(k)of a 3 x 3 matrix (6 x 6 if spin orbit is included (Kane, 1956, 1957; Luttinger and Kohn, 1955; Luttinger, 1956)) whose eigenvalues are the E,(k). Considering the a,(k) as the components of a 3-component column vector ( n = 1,2,3) expressed on the basis of the eigenvectors of hij(k),it is advantageous to rewrite Eq. (1-33) using the natural basis states of hij(k).This leads to

(1 -35)

As the h, are of second ord in k , this represents a set of coupled second order differential equations whose solution will lead to the expression of the envelope functions. Finally, in this approximation, the total wave function q (r)becomes, by using the basis set corresponding to h,:

(1-36) jk

which, if only k 2:0 is involved, becomes

(1-33)

(1-37)

This equation is diagonal in n and is apparently a simple to solve as for the single band extremum. However, the difficulty is to proceed further and transform it to a real space equation as in Eq. (1-28). The reason is that one can no longer define the derivatives of E,(k) near the valence band maximum at k = 0. This will be shown in detail in Section 1.3.2, following Kane’s derivation (Kane, 1956, 1957) in which it is shown that

We do not discuss here the application of the method to hydrogenic acceptor states (details can be found in Bassani et al., 1974; Pantelides, 1978). We shall later see its use in quantum wells and superlattices. In such cases, the application of the envelope function approximation is complicated by the problem of boundary conditions which we discuss in Sec. 1.7.

14

1 Band Theory Applied to Semiconductors

1.2 The Calculation of Crystalline Band Structures We have seen that, for crystalline solids, the use of Bloch’s theorem allows us to demonstrate quite generally the existence of energy bands. However, in its derivation we have made the implicit assumption that one could write a Schrodinger equation for each electron taken separately. Of course this is in principle not permissible in view of the existence of electron-electron interactions and one should consider the N electron system as a whole. It has been shown that one can generalize the Bloch theorem to the one particle excitations of crystalline manyelectron systems. However, when this is done there is no exact method available to calculate these excitations (which correspond to the energy bands) in practice. One is left with approximate methods which are all based on the reduction of the problem to a set of separate one particle equations whose eigenvalues are used to compare the experimental one particle excitations. First, we thus give a brief account of most one-electron theories that have been used so far: Hartree, Hartree-Fock, local density, etc. We also discuss recent advances which have allowed to considerably improve the local density results (the so-called “G-W approximation”, which consists in a first order expansion of the electron self energy in terms of the screened electron-electron interaction). Usually, it is not necessary to include core electrons in calculations involving properties of the valence electrons. To achieve this separation one replaces the true atomic potentials by “first principles pseudopotentials” of which we give a short description. All this completely defines the single particle equations and, in Sec. 1.2.2 we present some techniques that can be used for their resolution. These techniques often involve a substantial amount of computa-

tion but they can be applied to simple crystals such as the zinc-blende semiconductors. However, there is a need for simpler empirical methods, either for physical understanding or as simulation tools for more complex systems. We describe two such methods in Sec. 1.2.3: the empirical pseudopotential method (EPM) and the empirical tight binding approximation (TBA).

1.2.1 Ab Initio Theories We describe here some basic methods that lead to approximate single particle equations. We begin with the Hartree approximation which is the simplest to derive and illustrates the general principles that are applied. We then discuss the Hartree-Fock approximation, local density theory, and its recent improvements via the G-W approximation.

1.2.1.1 The Hartree Approximation The full N electron Hamiltonian (for fixed nuclei) can be written: ( 1 -38)

where the hi are independent individual Hamiltonians containing the kinetic energy operator of electron i as well as its attraction by the nuclei. The second term in Eq. ( 1-38) represents the electron-electron interactions, rG being the distance between electrons i andj. If one could neglect these, the problem would be exactly separable, i.e., one could obtain the solution of the full problem by simply solving the individual Schrodinger equations,

hi vfii(ri)= &ni vfii(ri),

( 1 -39)

the full wave function being a product of individual wave functions (if one forgets for the moment the fact that it must be anti-

1.2 The Calculation of Crystalline Band Structures

symmetric) and the total energy being the sum of individual energies. The inclusion of the electron-electron interactions prevents the problem from being separable. However, one can find approximate individual equations by using a trial wave function (in the variational sense) which is of a separated form, i.e., it is a simple product of individual functions. For N electrons this gives

v @ I . ..rN) = n q n i (c)

( 1-40)

ni

The unknown wave functions can be obtained by using the variational method, i.e., by minimizing the average value of H with respect to the qni.This leads to a set of individual equations which are the Hartree equations. However, these can be obtained directly by the following simple physical argument. If the problem can be separated, then the Hamiltonian of electron i will consist of the sum of its kinetic energy operator, its potential energy in the field of the nuclei, and its potential energy of repulsion with other electrons. This leads to the Schrodinger equation:

where the second term in the Hamiltonian represents the average electrostatic repulsion exerted on electron i by all the other electrons. There are as many equations (1-41) as there electrons, i.e., N . Each oneelectron Hamiltonian contains the wave functions of the other electrons which are unknown. One has thus to proceed by iterations until a self-consistent solution is found, i.e., the wave functions injected into the hamiltonian are the same as the solutions of Eq. (1 -41). The Hartree approxima-

15

tion has been used in understanding the basic physics of atoms. It is not refined enough to be used for actual band structure calculations.

1.2.1.2 The Hartree-Fock Approximation The main drawback of the Hartree approximation is that its wave function is not properly antisymmetrized. If one wants to use a trial function corresponding to independent electrons one cannot make use of a simple product of individual wave functions but instead one must consider a Slater determinant of the form

q ( r l ... r N ) =

(1 -42)

The variational method can be applied in exactly the same way as the Hartree method. However, it is not longer possible to get the one-electron equations using a simple argument. The application of this technique leads directly to:

=Eqn,(~.)

( 1 -43)

The result is the Hartree contribution plus a correction factor, the exchange term, due to antisymmetry in the electron permutations. It is important to notice that the qni(ri) are “spin orbitals”, i.e., products of a spatial part multiplied by a spin function (Slater, 1960). This is fundamental in order to maintain the Pauli principle.

16

1 Band Theory Applied to Semiconductors

The Hartree-Fock method is not very easy to apply numerically in view of the complexity of the exchange terms. Its application to covalent solids like diamond or silicon (Euwema et al., 1973; Mauger, Lannoo, 1977) leads to the overall correct shape of the energy bands, but a large overestimation of the forbidden gap. For instance, one gets 12 eV and 6 eV for diamond and silicon respectively compared to the experimental values of 5.4 eV and 1.1 eV respectively. Improvements on the Hartree-Fock approximation can be made including what are called correlation effects.

1.2.1.3 The Local Density Approximation The local density approximation is an extension of the Thomas-Fermi approximation based on the Hohenberg and Kohn theorem (Hohenberg and Kohn, 1964) which shows that the ground state properties of an electron system are entirely determined by the knowledge of its electron density @ (r).The total energy of the interacting electron system can be written: (1-44)

where T represents the kinetic energy, the second term gives the electrostatic interelectronic repulsion, Vex, is the potential due to the nuclei, and Ex,is the exchange correlation energy. A variational solution of the problem (Kohn and Sham, 1965) allows us to derive a set of one particle Schrodinger equations of the form:

with t representing the one electron kinetic energy and @(‘)=

c

I4Jk(‘)I2

(1 -46)

occupied k

The practical resolution of Eqs. (1-45, 1-46) is, as we will discuss later, usually performed by replacing the atomic potentials by pseudopotentials, avoiding the explicit consideration of atomic core states. Originally these pseudopotentials were treated empirically, but now methods exist which allow us to determine them quantitatively from the properties of the free atoms (Hamann et al., 1979).The knowledge of these pseudopotentials plus the local density treatment allows a complete determination of the solutions q k of Eq. (1-45). The wave functions q k can be calculated either using an expansion in plane waves or in orbitals localized on the atoms. Equations (1-45) and (1-46) have to be solved in a self-consistent way. Up to now this formulation has been exact, the problem is that the quantity Exc,and thus V,,, is not known in general. The local density approximation then is based on the assumption that, locally, the relation between Ex, and @(r)is the same as for a free electron gas of identical density, which is known quite accurately. This approximation turns out to give satisfactory results regarding the prediction of the structural properties of molecules and solids. For instance, in solids (either with sp bonds or d bonds, as in transition metals) the cohesive energy, the interatomic distance, and the elastic properties are predicted with a precision better than 5% in general. This remains true for diatomic molecules, except that the binding energy is overestimated by about 0.5 to 1 eV (see Cohen, 1983; Schluter, 1983 for recent reviews on the subject).

17

1.2 The Calculation of Crystalline Band Structures

It is tempting to use the differences between the eigenvalues of Eq. (1-45) as particle excitation energies. This is not justified in general as seen by the predicted values for the energy gap E ~ = E ~ - E “ in semiconductors and insulators, taken as the difference in the energies cC of the first empty state and E~ of the last filled state. The local density value of E~ is always found to be substantially smaller than the experimental (Hamann, 1979). It is equal to 0.6 eV for silicon instead of 1.2 eV and even vanishes for germanium. The origin of the errors cannot be traced to the use of an overly simplified exchange and correlation potential such as LDA. This has been clearly shown by the almost identical results obtained using an improved exchange-correlation potential. The exchange-correlation term thus cannot be reduced to a simple local potential.

1.2.1.4 Beyond Local Density (the G-W Approximation) To correct for the deficiencies of local density in defect calculations, one simple approach has been to use a “scissors” operator (Baraff and Schluter, 1984) which corrects for the band gap error by using a rigid shift of the conduction band states. It was later shown (Sham and Schluter, 1985, 1986; Perdew and Levy, 1983; Lannoo and Schluter, 1985) that this procedure was closer to reality than expected, since the one electron exchange correlation potential of the local density formalism must experience a discontinuity across the gap. This is correctly handled by the scissors operator for bulk semiconductors but the applicability of the scissors operator to the defect levels is still questionable. In particular, in the case of extended defects such as surfaces, it is doubtful that the defect states are correctly obtained. The advantage of this correc-

tion is mainly that it does not add any computational requirements when compared to conventional LDA calculations. A more sophisticated way of improving the density functional theories is to evaluate the electron self-energy operator C(r,r ’, E ) (Lannoo and Schluter, 1985; Hybertsen and Louie, 1985; Godby, 1986). This Ccontains the effects of exchange and correlation. It is non-local, energy-dependent, and non-hermitian. Its non-hermiticity means that the eigenvalues of the new one particle Schrodinger equation, (t+

vest

+ V,)

W n M

+

(1-47)

+ I dr’ z(rir ’ r Erik) q n k ( r ’ ) = Erik qnk(r)

will generally be complex. The imaginary part gives the lifetime of the quasiparticle, and V , is the Hartree-potential. The self-energy operator can be estimated using the G-W approximation (Hedin and Lundquist, 1969). The self-energy is expanded in a perturbation series of the screened Coulomb interaction, W The first term of the expansion corresponds to the Hartree-Fock approximation. Details can be found in Lannoo and Schluter (1985); Hybertsen and Louie (1985); Godby et al. (1986, 1987); Hedin and Lundquist (1969). These early works have been completed by more recent calculations (Rohlfing et al., 1993; Northrup et al., 1991; Northrup, 1993; Blase et al., 1994). The results show a great improvement on the values predicted for the gap of semiconductors as evidenced in Louie (1996).

1.2.1.5 The Pseudopotential Method The full atomic potentials produce strong divergences at the atomic sites in a solid. These divergences are related to the fact that these potentials must produce the atomic core states as well as the valence states.

18

1 Band Theory Applied to Semiconductors

However, the core states are likely to be quite similar to what they are in the free atom. Thus the use of the full atomic potentials in a band calculation is likely to lead to unnecessary computational complexity since the basis states will have to be chosen in such a way that they describe localized states and extended states at the same time. Therefore it is of much interest to devise a method which allows us to eliminiate the core states, focusing only on the valence states of interest which are easier to describe. This is the basis of the pseudopotential theory. The pseudopotential concept started with the orthogonalized plane wave theory (Cohen et al., 1970). Writing the crystal Schrodinger equation for the valence states, ( T + V) Iq) = E

lq)

( 1 -48)

one has to recognize that the eigenstate 1111) is automatically orthogonal to the core states 1 c ) produced by the same potential V. This means that I q) will be strongly oscillating in the neighbourhood of each atomic core, which prevents its expansion in terms of smoothly varying functions, like plane waves, for instance. It is thus interesting to perform the transformation l111)=(1-P)l4

( 1-49)

where P is the projector onto the core states (1-50)

I 111) is thus automatically orthogonal to the

core states and the new unknown 19) does not have to satisfy the orthogonality requirement. The equation for the “pseudostate” 19) is: ( T + V)(1 - P ) ) q ) = E ( l - P ) ) q )

(1-51)

Because the core states Ic) are eigenstates of the Hamiltonian T + V with energy, E,,

one can rewrite Eq. ( -51) in the form

{T+ V+C(E-E,) C

4

(CI}

I d = E Id

(1-52)

The pseudo-wave function is then the solution of a Schrodinger equation with the same energy eigenvalue as 1111). This new equation is obtained by replacing the potential V by a pseudopotential V p s = V + C ( E - E J Ic)(cl c

(1-53)

This is a complex non-local operator. Furthermore, it is not unique since one can add any linear combination of core states to 1 q ) in Eq. (1-52) without changing its eigenvalues. There is a corresponding non-uniqueness in Vps since the modified 1 cp) will obey a new equation with another pseudopotential. This non-uniqueness in Vps is an interesting factor since it can then be optimized to provide the smoothest possible Iq), allowing rapid convergence of plane wave expansions for 19).This will be used directly in the empirical pseudopotential method. Recently, so-called “first principles” pseudopotentials have been derived for use in quantitative calculations (Hamann et al., 1979). First of all, they are ion pseudopotentials and not total pseudopotentials as those discussed above. They are deduced from free atom calculations and have the following desirable properties: (1) real and pseudovalence eigenvalues agree for a chosen prototype atomic configuration, (2) real and pseudo-atomic wave functions agree beyond a chosen core radius r,, (3) total integrated charges at a distance r > r, which agree (norm conservation), and (4)logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives which agree for r > r,. These properties are crucial for the pseudopotential to have optimum transferability among a variety of chemical environments, allowing self-con-

19

1.2 The Calculation of Crystalline Band Structures

sistent calculations of a meaningful pseudocharge density.

1.2.2 Computational Techniques We now will discuss some techniques which allow us to calculate energy bands in practice. They are simply based on an expansion of the eigenfunction of the one-electron Schrodinger equation in some suitable basis functions, plane waves, or localized atomic orbitals. These seem to be, by far, the most commonly adopted methods at this time.

obtained by mixing the plane wave eik only with other plane waves whose wave vector is k + G, where G is any reciprocal lattice vector, and not with all plane waves with arbitrary wave vectors. Of course, we are only interested in low lying states so that we can truncate the expansion of Eq. (1 -54) at some maximum value of IGI, which we label C,,,. The one particle Hamiltonian is thus expressed on this basis as a finite matrix of elements: ’

1.2.2.1 Plane Wave Expansion Plane waves form a particularly interesting basis set for crystalline band structure calculations in conjunction with the use of pseudopotentials. As we shall see below the total pseudopotential can be expressed as a sum of atomic contributions. These consists of a bare ionic part which will be screened by the valence electrons. The resulting atomic pseudopotentials are often assumed to be local, i.e. to be simple functions of the electron position. However, in general, they should be operators having a non-local nature, as discussed later. The best starting point is the expression Eq. (1-9) of the Bloch function in which one makes use of the fact that the function u k ( r ) is periodic and can thus be expanded as a Fourier series: Uk(r)

= cUk(G) eiG.r

(1-54 a)

G

where G are reciprocal lattice vectors. This leads to the natural plane wave expansion for the wave function: (pk ( r )= C U R (G) ei(k+ G ) ’ r

(1-54 b)

G

The gain due to the Bloch theorem is that ) wave vector k any Bloch state ~ ( rwith belonging to the first Brillouin zone is

which readily becomes

(1-55)

A2 2 H G , G ~= ( ~) I k + G 1 &,G* + V G , G * ( ~ ) 2m ( I -56) where the second term is the potential matrix element. In the case where Vis a simple function of r, this matrix element can be reduced to: V G , y ( k )= V(G’-G)

(1-57)

Such plane wave expansions can be used in different contexts. We will later develop an application known as the empirical pseudopotential method (EMP). One can also apply these expansions to first principles calculations. This is formally easy in the local density context where the potential takes a simple form. It becomes more complex, but still can be adapted, in the HartreeFock theory or even in the G-W approximation as used in Hybertsen and Louie (1985, 1986) and Godby et al. (1986).

1.2.2.2 Localized Orbital Expansion In this expansion the wave function is written as a combination of localized orbitals centered on each atom: ly =

c i,a

Cia Pia

(1 -58)

20

1 Band Theory Applied to Semiconductors

where qin is the athfree atom orbital of atom i, at position R j . As each complete set of such orbitals belonging to any given atom forms a basis for Hilbert space, the whole set of qiais complete, i.e., the viaare no longer independent and Eq. (1 -58) can yield the exact wave function of the whole system. In practice one has to truncate the sum over a in this expansion. In many simplified calculations it has been assumed that the valence states of the system can be described in terms of a “minimal basis set” which only includes free atom states up to the outer shell of the free atom (e.g., 2s and 2p in diamond). It is that description which provides the most appealing physical picture, allowing us to clearly understand the formation of bands from the atomic limit. The “minimal basis set” approximation is also used in most semi-empirical calculations. When the sum over a in Eq. (1-58) is limited to a finite number, the energy levels E of the whole system are given by the secular equation: det I H - E S I = O

(1-59)

where H is the Hamiltonian matrix in the atomic basis and S the overlap matrix of elements: Sia,j p = (Pi, I qjp)

(1 -60)

These matrix elements can be readily calculated, especially in the local density theory, and when making use of Gaussian atomic orbitals. The problem, as in the plane wave expansion, is to determine the number of basis states required for good numerical accuracy. An interesting discussion on the validity of the use of a minimal basis set has been given by Louie (1980). Starting from the minimal basis set 1 qia)one can increase the size of the basis set by adding other atomic states Ixi,), called the peripheral states,

which must lead to an improvement in the description of the energy levels and wave functions. However, this will rapidly lead to problems related to overcompleteness, i.e., the overlap of different atomic states will become more and more important. To overcome this difficulty, Louie proposes three steps to justify the use of a minimum basis set. These are the following:

1. Symmetrically orthogonalize the states I cp,) belonging to the minimal basis set between themselves. This leads to an orthogonal set 2. The peripheral states Ixip) overlap It is thus necesstrongly with the sary to orthogonalize them to these I &), which yield new states defined as:

laa). Ian). [xi,)

3. The new states I&) are then orthogonalized between themselves leading to a new set of states lfi,). Louie has shown that, at least for silicon, the average energies of these atomic states behave in such a way that, after step 3, the peripheral states lji,) are much higher in energy and their coupling to the minimal set is reduced. They only have a small (although not negligible) influence, justifying the use of the minimal set as the essential step in the calcuation. The quantitative value of LCAO (linear combination of atomic orbitals) techniques for covalent systems such as diamond and silicon was first demonstrated by Chaney et al. (197 1). They have shown that the minimal basis set gives good results for the valence bands and slightly poorer (but still meaningul) results for the lower conduction bands. Such conclusions have been confirmed by Kane (1976), Chadi (1977), and Louie (1 980) who worked with pseudopotentials instead of true atomic potentials.

1.2 The Calculation of Crystalline Band Structures

The great asset of the minimal basis set LCAO calculations is that they provide a direct connection between the valence states of the system and the free atom states. This becomes still more apparent with the TBA (tight binding approximation) which we shall later discuss and which allows us to obtain extremely simple, physically sound descriptions of many systems.

1.2.3 Empirical Methods Up to very recently, first principles theories, sophisticated as they may be, could not accurately predict the band structure of semiconductors. Most of the understanding of these materials was obtained from less accurate descriptions. Among these, empirical theories have played (and still play) a very important role since they allow us to simulate the true energy bands in terms of a restricted number of adjustable parameters. There are essentially two distinct methods of achieving this goal: the tight binding approximation (TBA) and the empirical pseudopotential method (EPM).

1.2.3.1 The Tight Binding Approximation This can be understood as an approximate version of the LCAO theory. It is generally defined as the use of a minimal atomic basis set neglecting interatomic overlaps, i.e., the overlap matrix defined in Eq. ( 1 -60) is equal to the unit matrix. The secular equation thus becomes det ( H - E Z I = O

( 1 -62)

where I is the unit matrix. The resolution of the problem then requires the knowledge of the Hamiltonian matrix elements. In the empirical tight binding approximation these are obtained from a fit to the bulk band structure. For this, one always truncates the

21

Hamiltonian matrix in real space, i.e., one only includes interatomic terms up to first, second, or, at most, third nearest neighbors. Also, in most cases one makes use of a twocenter approximation as discussed by Slater and Koster (1954). In such a case, all Hamiltonian matrix elements (viaI H I qp) can be reduced to a limited number of independent terms which we can call H a b ( i , j ) for the pair of atoms (i,j) and the orbitals (a,p).On an “s, p” basis, valid for group IV, 111-V, and 11-VI semiconductors, symmetry consideration applied to the two-center approximation only give the following independent terms: Ha,p ( i * j )= Hss(i?j)*Hsu(i,j)y H u s ( i , j ) ( 1 -63) H,, H,, (U (

i

J

3

where H , is strictly zero in a two-center approximation and s stands for the s orbital, othe p orbital along axis i , j with the positive lobe in the direction of the neighboring atom, and n a p orbital perpendicular to the axis i, j . With these conventions, all matrix elements are generally negative. Similar considerations apply to transition metals with s, p, and d orbitals. Simple rules obtained for the H a b ( i , j ) in a nearest neighbor’s approximation are given in Harrison (1980). They are based on the use of free atom energies for the diagonal elements of the tight binding hamiltonian. On the other hand, the nearest neighbor’s interactions are taken to scale like d-2 (where d is the interatomic distance) as determined from the free electron picture of these materials which will be discussed later. For s, p systems, this gives

where d is expressed in A.

22

1 Band Theory Applied to Semiconductors

Such parameters nicely reproduce the valence bands of zinc-blende semiconductors but poorly describe the band gap and badly describe the conduction bands. Improvements on this description have been attempted by going to the second nearest neighbors (Talwar and Ting, 1982) or by keeping the nearest neighbors treatment as it is but adding one s orbital (labelled s*) to the minimal basis set (Vogl and Hjalmarson, 1983). The role of this latter orbital is to simulate the effect of higher energy d orbitals which have been shown to be essential for a correct simulation of the conduction band. The quality of such a fit can be judged from Fig. 1-6, which shows that the lowest conduction bands are reproduced much more correctly. Fairly recently it has also been shown that the replacement of s* by true d orbitals improves the simulation in a striking manner (Priester).

15

c va(r

j,a

- Rj - ra)

(1-65)

where j runs over the unit cells positioned at Rj and a is the atom index, the atomic position whithin the unit cell being given by ra. Let us first assume that the v, are ordinary functions of r, or, in other words, that we are dealing with local pseudopotentials. In that case the matrix elements [Eq. (1-57)] of V between plane waves become:

5

a -5 -1c

W

One advantage of the tight binding approximation is that it provides a natural way of relating the electronic properties of a solid to the atomic structure of its constituent atoms. As will be discussed later, it is also the most appropriate way to calculate the properties of disordered systems. However, for crystalline semiconductors the empirical pseudopotential method seems to be the most efficient way to get a good overall description of both the valence and the conduction bands. The basis of this method is the plane wave expansion of the wave function given by Eq. (1-54) plus the use of a smooth pseudopotential. The matrix to be diagonalized was derived in Eq. (1-56) but now we pay more attention to the potential matrix elements. In EPM one assumes that the selfconsistent crystal pseudopotential can be written as a sum of atomic contributions, i.e., V ( r )=

ia

-15

1.2.3.2 The Empirical Pseudopotential Method

(k+GIVIk+G’)= L

r

X U,K

(1-66)

r

Figure 1-6. Comparison of the sp3 s* description of silicon (- - -) with a more sophisticated calculation (-). The vertical axis represents energies in eV, the horizontal axis the wave vector along symmetry axes in the Brillouin zone. Note that the valence band is practically perfectly reproduced.

where i2 is the volume of the unit cell. Suppose that there can be identical atoms in the unit cell. Then the sum over a can be expressed as a sum over groups /3 of identical atoms with position specified by a second

1.3 Comparison with Experiments for Zinc-Blende Materials

index y (i.e. ra = ryB). Calling n the number of atoms in the unit cell we can write

=C Sp(G’-G)vp P

(G’-G) (1-67)

where Sp(G) and v p (G) are, respectively, the structure and form factors of the corresponding atomic species, defined by

and

5

n vp (G) = - vP ( r) eiG . rd3r 2 !

(1 -68)

In practice, the empirical pseudopotential method treats the form factors va (G)as disposable parameters. In the case where the vp (r)are smooth potentials their transforms vp (G) will rapidly decay as a function of I G I so that it may be a good approximation to truncate them at a maximum value of G,. For instance, the band structure of tetrahedral covalent semiconductors like Si can be fairly well reproduced using only the three lower Fourier components v(lG1) of the atomic pseudopotential. There are thus two cut-off values for I G I to be used in practice: one, GM,limits the number of plane waves and thus the size of the Hamiltonian matrix; the other one, G,, limits the number of Fourier components of the form factors. We shall later give some practical examples. The use of a local pseudopotential is not fully justified since, from Eq. (1.52), it involves, in principle, projection operators. It can be approximately justified for systems with s and p electrons. However, when d states become important, e.g., in the conduction band of semiconductors, it is necessary to use an operator form with a projection operator on the 1 = 2 angular components.

23

1.3 Comparison with Experiments for Zinc-Blende Materials In this section we apply methods which allow us to understand the general features of the band structure of zinc-blende materials. We detail, to some extent, simple models based on tight binding or the empirical pseudopotential. We also discuss briefly the results of the most sophisiticated recent calculations. We then concentrate on the general treatment of the band structure near the top of the valence band, using the k-p perturbation theory as in the work by Kane (1 956; 1957). In the third part, we examine the optical properties of these materials putting particular emphasis on excitonic states. Finally, we give a comparison of predicted band structures with photo-emission and inverse photo-emission data, in the light of what was done on the basis of non-local empirical pseudopotentials.

1.3.1 The General Shape of the Bands In this section we start from two points of view: (i) the molecular or bond orbital model derived from tight binding and (ii) the nearly free electron picture. We show that both give rise to similar qualitative results at least for the valence bands.

1.3.1.1 The Tight Binding Point of View Consider an A-B compound in the zinc blende structure where the atoms have tetrahedral coordination. The minimal atomic basis for such systems consists of one s and three p orbitals on each atom. One could solve the Hamiltonian matrix in this basis set and get the desired band structure directly. However, one can get much more insight into the physics by performing a basis change such that, in the new basis set, some matrix elements of the Hamiltonian

24

1 Band Theory Applied to Semiconductors

will be much larger than the others. This will allow us to proceed by steps, treating first the dominant elements and then looking at the corrections due to the others. This is the general basis of molecular (Harrison, 1973; Lannoo and Decarpigny, 1973) or bond orbital models, two names for the same description. The natural basis change is to build sp3 hybrids of the form

Of course these states will be strongly degenerate since their degeneracy is equal to the number N of bonds in the system. The wave function of the bonding and antibonding states will take the form, for a bond ij connecting two neighbors i and j ,

(1-72)

( 1 -69)

where ‘psi is the s orbital of atom i and qp,G is one of its p orbitals pointing from i to one of its nearest neighbors j . By doing this for each atom, each bond in the system will be characterized by a pair of strongly overlapping hybrids qQ and ‘pji as shown in Fig. 1-7. It is clear that the dominant interatomic matrix elements of the Hamiltonian are given by (cpijlHlqji)=-B

B>O

(1-70)

while the diagonal elements (qQlH I qQ)are equal to the average sp3 energies of the atoms, which we denote E A for atom A and EB for atom B. In the first step we neglect all other matrix elements. The problem is then equivalent to a set of identical diatomic molecules, each one leading to one bonding and one antibonding state of energy:

\

\

CP ij

qji

/

/

Figure 1-7. Pair of sp3 hybrids involved in one bond, as defined in Eq. (1-69).

In the following discussion we take the convention that i is an A atom a n d j a B atom. As there are two electrons per bond, the ground state of the system corresponds to completely filled bonding states and empty antibonding states. This description defines the “molecular” or “bond orbital model” in which the bonding states give a rough account of the valence band and the antibonding states, of the conduction band. This model has been extremely successful in describing semiquantitatively the trends in several physical properties of these materials: ionicity, effective charges, dielectric susceptibilities, average optical gaps, and even cohesive properties (see Harrison (1980) for more details). With such a simple starting point, the formation of the band structure is easy to describe. The inclusion of further interactions which were neglected in the molecular model will tend to lift the degeneracy of the bonding and antibonding states. Exactly the same arguments as those developed in Sec. 1.1 lead to the conclusion that there will be the formation of a bonding band from the bonding states. For this we write the wave function ?,bb as a combination of all $ J ~ , ~ : q b =

ab,ij q b , i j

pairs ij

(1-73)

25

1.3 Comparison with Experiments for Zinc-Blende Materials

The leading correction term in the Hamiltonian matrix will be the interaction between two adjacent bonds which takes one of two values, A, or A,, depending on the common atom. Projecting Schrodinger’s equation on these states one gets the set of equations: ( 1-74) (E- Eb) abij = AA a b f + AB abirj

x

c

i‘+i

jf# j

where the sums are over adjacent bonds having an A or B atom in common. At this stage it is interesting to introduce the following sums

+ A, + AB) Ubij = A,

Si + A,

(E-E~-~AA+AB)S~=ABCS~ jci (1 -77) (E-Eb +A, - 3 A ~ ) s j= d ~ c S i icj

where the sums are over the nearest neighbors of one given atom. Injecting the second Eq. (1-77) into the first one gives

{ ( E - Eb)2 - 4 A ( E - Eb ) - 12 = (A’ - 6’)

x

Si2

2

a2} si = (1 -78)

i2

where now the sum is over the second nearest neighbors of atom i. This set of equations on sublattice A is just the same as what would be obtained for a tight binding s band on an f.c.c. lattice. This leads to the following solutions:

E=Eb+ +2 4

(1 -79)

2

k, a k,a +cos---cos2 2

+

2

2

a being the lattice parameter, and k,, k,, and k, the components of the wave vector along the cube axes. It is clear that v, varies continuously from 12 to -4. The extrema of the bands ( q= 12) are then given by:

E = Eb + 2 4 k 4 4 = Eb

E=Eb+2A+416)

Sj

(1 -76) Summing this either overj or i, one gets two equations:

with A = AA + AB and 6 = 2

( 1-80)

2

+6 4 -24

(1-81)

while for v,= - 4 a gap is opened in the band, its limits being given by

so that one can rewrite Eq. (1-74) as

(EWE,

with

[4 A* + 12 S2 + (A2 - S2) ~

1 ~

(1-82)

Equation (1 -79) gives only two bands while we started with four bonding orbitals per atom of the A sublattice. The two bands which appear to be missing can be obtained by noting that Eq. (1-76) has a trivial solution for which all the Siare zero, with nonzero abij if

E = Eb - 2 4

(1 -83)

This is the equation of a twofold degenerate flat band with pure p character, since all Si and Sjare zero. To summarize the results, one can say that the interaction between bonding orbitals has broadened the bonding level into a valence band consisting, in the present model, of two broad bands and a twofold degenerate flat band. Exactly the same treatment can be applied to the antibonding states by changing Eb into E, and defining A, and A, describing the interactions between antibonding states. This antibonding band will thus represent the conduction band. The resulting band structure is compared in Fig. 1-8 to a more sophisticated calculation from Chaney et al. (1971) (its ’ ~

1 Band Theory Applied to Semiconductors

The advantage of the simple picture we have just detailed is that it can be generalized, as we shall see, to a lot of other situations such as covalent systems with lower coordination (Sec. 1.4) or to non-crystalline and amorphous semiconductors. Furthermore, all tight binding descriptions with parametrized interactions give results which are in good correspondence with those we have derived, with the minor corrections we have mentioned. One such empirical model is the sp3 s* description of Vogl et al. (1983) which leads, for GaAs, to the band structure of Fig. 1-9, which can be shown to be in good agreement with experiments for the valence band and the lowest conduction band.

1.3.1.2 The Empirical Pseudopotential Method

Figure 1-8. Comparison of the simple model description of diamond (- - -) with a more sophisticated calculation (-). The vertical axis corresponds to energies in eV.

parameters have been adjusted to give overall agreement). It can be seen that it already reproduces the essential features of the valence band. The inclusion of further matrix elements left out from the previous simple picture will have the following qualitative effects: i) the bonding-antibonding interactions lead to a slight repulsion between the valence and conduction bands, and ii) the inclusion of interactions between more distant bonds induces some dispersion into the flat bands.

A major improvement in the detailed description of the bands of tetrahedral semiconductors has been achieved with the use of empirical pseudopotentials. Let us then first discuss its application to purely covalent materials like silicon and germanium. The basis vectors of the direct zinc-blende lattice are a/2 (110), a/2 ( O l l ) , and a/2 (101). The corresponding basis vectors of the reciprocal lattice are 2xla ( l l i ) , 2nla (11 l), and 2 x / a (1 11). The reciprocal lattice vectors G which have the lowest square modulus are the following, in increasing order of magnitude:

-G U

2

[&)G2

2x

000 111 200 220 311

0 3 4 8

(1-84)

11

For elemental materials like Si and Ge there is only one form factor v(G)but we

1.3 Comparison with Experiments for Zinc-Blende Materials

Figure 1-9. Comparison between the sp3 s* band structure of GaAs (-) and the empirical pseudopotential one (- - -). Vertical scale: energies in eV, honzontal scale: k values.

a I

2

0

(

-'

-2

-4

-4

-6

-6

-0

-a

-10

-10

-12

-12

-14

.I4

have seen in Eq. (1.67) that the matrix element of the potential involves a structure factor which is given here by:

S(G)= cos G z

(1-85)

where the origin of the unit cell has been taken at the center of a bond in the (1 11) direction and where z i s thus the vector a/8 ( 1 11). For local pseudopotentials this matrix element ( k + G 1 V I k + G')can be written V ( G ) and is thus given by: V ( G ) = v(G) cos (G*

27

Z)

(1-86)

The structure factor part is of importance since, among the lowest values of 1 GI quoted in Eq. (1-84), it gives zero for 2 n l a (2, 0,O). If one indexes V ( G )by the value tak) ~then only the en by the quantity ( ~ / 2 nG2, values V,, V,, and V , , are different from zero. It has been shown (in Cohen and Bergstresser, 1966) that the inclusion of these three parameters alone allows us to obtain a satisfactory description of the band struc-

ture of Si and Ge. This can be understood simply by the consideration of the free electron band structure of these materials which is obtained by neglecting the potential in the matrix elements Eq. (1-56) of the Hamiltonian between plane waves. The eigenvalues are thus the free electron energies A2/2mIk + G l2 which, in the f.c.c. lattice, lead to the energy bands plotted in Fig. 1-10. The similarity is striking, showing that the free electron band structure provides a meaningful starting point. The formation of gaps in this band structure can be easily understood at least in situations where only two free electron branches cross. To the lowest order in perturbation theory, one will have to solve the 2 x 2 matrix.

L

2m

'

28

1 Band Theory Applied to Semiconductors

True bands

Free-electron bands

Figure 1-10. Correspondence between free-electron and empirical pseudopotential bands, showing how degeneracies are lifted by the pseudopotential.

The resulting eigenvalues are

E ( k )= 2rn

* { [ g . l k + G ’ r - l2k + G r

+ I V(G‘-

G) 12>1’2

I+

(1-88)

whose behavior as a function of k is pictured in Fig. 1-11. The conclusion is that there

is the formation of a gap at the crossing point whose value is 21 V(G’- G)(. Note that for this to occur the crossing point at G’+G k=-must lie within the first Bril2 louin zone or at its boundaries. For points where several branches cross, one will have a higher order matrix to diagonalize but this will generally also result in the formation of gaps. This explains the differences between the free electron band structure and the actual one in Fig. 1-10. The number of parameters required for fitting the band structures of compounds is different in view of the fact that there are now two different atoms in the unit cell with form factors uA(G) and v,(G). The matrix elements V(G) of the total pseudopotential will thus be expressed as (1-89)

V(G)=VS(G)cos (G-z)+iVA(G)sin(G.z)

Figure 1-11. Opening of a gap in the nearly free electron method. The full line corresponds to the two free electron branches, the dashed lines to the two branches split by the potential Fourier component.

where Vs and VA are equal to (vA+ vB)/2and (vA-u,)/2, respectively. The number of fitting parameters is then multiplied by 2, the symmetric components Vi, V;, and V,: being close to those of the covalent materials and the antisymmetric components be-

29

1.3 Comparison with Experiments for Zinc-Blende Materials

ing Vf, V t , and V; since the antisymmetric part of V, vanishes.

1.3.2 The k-p Description and Effective Masses We have already seen for hydrogenic impurity states (Sec. 1.1) that the concept of effective masses near a band extremum is very powerful. This will prove still more important for heterostructures which we discuss later. In any case it is desirable to provide a general framework in which to analyze this problem. This is obtained directly via the k-p method which we present in this section. The basis of the method is to take advantage of the crystalline structure which allows us to express the eigenfunctions as Bloch functions and to write a Schrodingerlike equation for its periodic part. We start from (1-90)

{g+

v}eik.r u k ( r ) = E ( k )eik’ruk( r )

where we have written the wave function in Bloch form. We can rewrite this in the following form:

i

( P + A k ) 2 + V u k ( r ) = E ( k ) u k ( r )(1-91) 1 2 m which is totally equivalent to the first form. To solve this we can expand the unknown periodic part uk(r)on the basis of the corresponding solutions at given point k,, which we label unk0(r): uk ( r )=

C cn ( k )un, kn ( r )

(1-92)

n

The corresponding solutions are the eigenvalues and eigenfunctions of the matrix with the general element (1-93)

We now use the fact that u,,ko is an eigenfunction of Eq. (1.91) for k = k,, with energy E,(k,). This allows us to rewrite the matrix element Eq. (1-93) in the simpler form:

with P n n , ( k d = (un,koi~Iun*,ko)

(1-95)

Diagonalization of the matrix A ( k )given by Eq. (1 -94) can give the exact band structure (an example of this is given in Cardona and Pollak, 1966). However, the power of the method is that it represents the most natural starting point for a perturbation expansion. Let us illustrate this first for the particular case of a single non-degenerate extremum. We thus consider a given non-degenerate energy branch E,(k) which has an extremum at k = k , and look at its values for k close to k,. The last term in Eq. (1-94) can then be considered as a small perturbation and we determine the difference En( k )En(ko)by second order perturbation theory applied to the matrix A ( k ) .This gives A2 En ( k )= E n ( k o )+ -( k - k o ) 2 + 2m A2

c [ ( k- koEn1.(Printk 1~1[En’( k (koko

+2 m

n‘zn

-

-

(1-96) ). Pn’n 1

)

which is the second order expansion near k , leading to the definition of the effective masses. The last term in Eq. (1 -96) is a tensor. Calling 0 a its principal axes, one gets the general expression for the effective masses m:

30

1 Band Theory Applied to Semiconductors

This shows that when the situation practically reduces to two interacting bands, the upper one has positive effective masses while the opposite is true for the lower one. This is what happens at the T point for GaAs, for instance. Another very important situation is the case of a degenerate extremum, i.e., the top of the valence band in zinc blende materials which occurs at k = 0. We still have to diagonalize the matrix Eq. (1-94) taking k , = 0 and, for k = 0, the last term can still be treated by the second order perturbation theory. By letting i a n d j be two members of the degenerate set at k = 0 and I , any other state distant in energy, we now must be apply the second order perturbation theory on a degenerate state. As shown in standard textbooks (Schiff, 1955) this leads to diagonalization of a matrix:

I

Af’(k) = Ei(0)+ -k 2 2m “

J

c$ t

The top of the valence band has threefold degeneracy and its basis states behave like atomic p states in cubic symmetry (i.e., like the simple functions x, y , and z). The second order perturbation matrix is thus a 3 x 3 matrix built from the last term in Eq. (1-98) which, from symmetry, can be reduced to (Kane, 1956, 1957; Kittel and Mitchell, 1954; Dresselhaus et al., 1955)

Lk:

+ M(ky’ + k:)

A2 k 2 It is this matrix plus the term - on its 2m diagonal wich define the h&) matrix of Sec. 1.1.4 to be applied in effective mass theory to a degeneratestate. Up to this point we have not included spin effects and in particular spinorbit coupling, which plays an important role in systems with heavier elements. If we add the spin variable, the degeneracy at the top of the valence band is double and the k ep matrix becomes a 6 x 6 matrix whose detailed form can be found in (Bassani et al., 1974;Altarelli, 1986; Bastard, 1988). One can slightly simplify its diagonalization when the spin orbit coupling becomes large, from the fact that

L S = 112 ( J 2 - L2 - S 2 )

(1-101)

where J = L + S . Because here L = l and S=1/2, J can take two values J=3/2 and J = 112. From Eq. (1-101) the J=3/2 states will lie at higher energy than the J = 1/2 ones and, if the spin orbit coupling constant is large enough, these states can be treated separately. The top of the valence band will then be described by the J =3/2 states leading to a 4 x 4 matrix whose equivalent Hamiltonian has been shown by Luttinger and Kohn (1955) to be:

-Y3

c

kakp

(J,

Jp+ Jp 2

Ja)

1

(1-102)

Nkx k, Lk,’+M(k; +k:) Lk: + M ( k ? +ky’)]

where L, M , and N are three real numbers, all of the form: h2

(Pa)il ( P P)rj

-3 m 2 I E,(o)-E,(o)

(1 -1 00)

Finally, as shown by Kane (1956, 1957), it can be interesting to treat the bottom of the conduction band and the top of the valence band at k = O as a quasi-degenerate

1.3 Comparison with Experiments for Zinc-Blende Materials

system, extending the above described method to a full 8 x 8 matrix which can be reduced to a 6 x 6 one if the spin orbit coupling is large enough to neglect the lower valence band,

1.3.3 Optical Properties and Excitons One of the major sources of experimental information concerning the band structure of semiconductors is provided by optical experiments. In particular fine structures in the optical spectra might reflect characteristics of the band structure. To see this in more detail let us discuss the absorption of the light which is proportional to the imaginary part of the frequency dependent dielectric constant, i.e.,

which can be expressed as an integral over the surface S(hw) in k space such that Ec(k)- Ev(k)= hw. This gives

2 dS Jvc(w)=y J (2n) S ( w ) /vk(Ec(k)-Ev(k))(S (1- 106) It is clear from this expression that important contributions will come from critical points where the denominator of Eq. (1- 106) vanishes. These can originate from k points where one has the separate conditions (1- 107) Critical points of this kind only occur at high symmetry points of the Brillouin zone (such as k = 0, for instance). Other critical points are given by vk(Ec(k) - Ev(k)) = 0

.s(Ec-E,-Ao)d3k

(1-103)

where the optical matrix element is: (1- 104)

x being the wave vector of light. From the

31

(1-108)

The behavior near such Van Hove singularities can be discussed quite generally by expanding E, - E, to the second order in k , around the singular point ko. This formally gives

E, -E, = Eo +

3 a=l

aa(k, - ko,)*

(1-109)

Bloch theorem this matrix element is nonzero only if k' is equal to k + x + G (G is the and the qualitative behavior of the joint denreciprocal lattice vector). As 1x1 is much sities of states Jvc(w)near such a point desmaller than the dimensions of the first Brilpends on the sign of the a,. Typical results louin zone, this means that the transition is are shown in Fig. 1-12, showing that, from vertical within the Brillouin zone (i.e., it the shape of the absorption spectrum, one occurs at fixed k ) . Fine structures in ~ ~ ( 0 can ) infer a part of the characteristics in the due to allowed transitions can be studied by band structures. As an illustration, we show discarding the k dependence of the optical in Fig. 1- 13 the curves E* ( w ) for silicon and matrix element Mv,(k). In a narrow energy germanium in comparison to the predicted 0 ) curves (Greenaway and Harbeke, 1968). range around such a structure, ~ ~ (becomes proportional to the joint densities of Up to now we have only discussed optistates cal transitions between one particle states in n which an electron is excited into the con(1-105) duction band, leaving a hole in the valence band. This is permissible if we treat the electron and hole as independent particles.

32

1 Band Theory Applied to Semiconductors

t

(@I

(a)

at hv = Eg, the energy gap, can show lines

\*

\

*

I I

we

w.

Figure 1-12. Schematic joint densities of states near the critical points for different situations (see text).

However, these quasi-particles have opposite charges and attract each other via an effective potential -e2/Er. This coulombic potential can give rise to localized gap states such as hydrogenic impurities, so that the absorption spectrum, instead of starting

m*e4 1 at E g - - - 2 2 2 ' where m* is a suitable 21i E n effective mass. Again the justification of this proceeds via the effective mass theory, but in a way slightly more complex than for impurities. To find the excitonic wave function, we can write the total wave function of the excited states in the form (Knox, 1963; Kittel, 1963):

a(ke,kh)@(k,,kh)

qexc=

(l-l1')

ke,kh

where the @correspond to the excited states obtained from the ground state by exciting a valence band electron of wave vector kh to a conduction band state k,. To use the effective mass approximation we introduce a two particle envelope function by the Fourier transform: (1-111)

ke,kh

5

Figure 1-13. Experimental and predicted e2(w) for Ge.

33

1.3 Comparison with Experiments for Zinc-Blende Materials

For simple band extrema with isotropic effective masses, it can be shown along the same lines as in Sec. 1.1 that F(r,, rh) obeys the effective mass equation:

{,-+-

~h2

2

Pe 2m,* 2m:

&Ire-rh

Mext

I

E = Eg + -- -2M 2&2A2 n2

= (90I Cpi I V e x c )

(1-1 16)

From the definition of the envelope function Eq. (1-1 11) we see that C a ( k , k )

(1-113)

where M and m* are the total and reduced masses respectively and k is the wave vector for the center of mass motion. From this it is clear that the lowest excited states are those for k = O and these give rise to the hydrogenic lines. It is interesting to determine the oscillator strength for exciton absorption for comparison with one particle transitions. We have seen before that, at a given frequency, the strength of the absorption is determined by the optical matrix element. For many electron states, this element is given by Mexc

=~ v c a(k,k) k

One can separate the center of mass and relative motion in this Hamiltonian in such a way that the total energy becomes:

A2k2 m * e 4 1

tions, i.e., only if k, = k h = k . This matrix element is identical to the Mvc(k) defined above and we take it to be constant over the small range of k involved. We then get

(1-114)

i

k

is equal to F(r,r)d3r where we take re-rh=r. We thus obtain the final result:

IMexc12= IM,,121 I F(r,r)d3 r l2

(1-1 17)

For the simple model we have just considered, the lowest exciton wave function is

where V is the volume of the specimen, R is the center of mass position and a the exciton Bohr radius, thus leading to (1-119) This is to be compared to the one particle spectrum which is given by (1- 120)

where C p i is the one electron sum of the I

individual momenta, % is the ground state, is one of the exciton states whose and qeXc general form is given in Eq. (1-110). One can expand qeXcand express and q ( k h k,) as Slater determinants, in which case Me,, becomes:

-

= C(kh/Plke)a(ke,kh)

where m* is the reduced mass. This sum reduces to the density of states of a 3D electron gas, so that

(1-115)

ke ,kh

We have seen that the one particle matrix elements are non-zero for vertical transi-

It is better for comparison with lMexcI2 to calculate the integrated lM(co) up to a fre-

34

1 Band Theory Applied to Semiconductors

quency o.One thus gets

'a'(Aw - E g ) J

]

312

=6n[

Ao - E,

(1-122)

where E l , is the exciton binding energy in the 1s state. Typical values are El, = 10 meV, Ao - Eg = 200 meV in which case the ratio, Eq. (1- 122), is of the order of 20%.

1.3.4 Ab Initio Calculations of the Excitonic Spectrum Ab initio calculations of ~ ( ofor)semiconductors have recently been reported (Albrecht et al., 1997). They generalize the G-W approximation of Sec. 1.2.1.4 to the case of electron-hole excitations. This requires the calculation of the two-particle Green's function, which obeys the BetheSalpeter equation (Nozibres, 1964; Sham and Rice, 1966; Strinati, 1984). This is equivalent to solving a Schrodinger equation for the electron-hole pairs with complex potential energy meaning that true eigenstates of the system are not obtained. The dominant potential energy term is the dynamically screened electron-hole attraction, while the smaller exchange term remains unscreened. Again, as for GW, the ) ) greatly improved over results for ~ ~ ( c r are those obtained in LDA, for which the main peak is incorrectly described (Albrecht eta]., 1997).

1.3.5 A Detailed Comparison with Experiments We give an account here of some results comparing empirical pseudopotential bands with X-ray photo-emission and inverse

photo-emission results (Chelikowsky et al., 1989). These experimental techniques combined with reflectivity data can yield nearly complete information concerning the occupied and empty states. A comparison of experimental data with theory will then provide a stringent test of the validity of the predictions. The empirical pseudopotential used by Chelikowsky et al. (1989) and Cohen and Chelikowsky (1988), was built from a local and a non-local part. The local part was, as described before, restricting the non-zero components to be V,, V,, V,, and Vll. As emphasized by Chelikowsky et al. (1989), this procedure yields and accurate description of the reflectivity and photo-emission spectra. However, for Ge, GaAs, and ZnSe, non-local corrections to the pseudopotential are necessary to produce similar accuracy. This is caused by d states within the ion core which modify the conduction band structure of these three materials and make the pseudopotential non-local. As discussed before, a simple correction for a specific I-dependent term can be written 4n

VNL( k ,G - G')= -(2 I Qa

+ 1) 4 (COS 0) .

.Id r r 2 K ( ~ M % ) j M )

(1-123)

where x = k + G,cos0 = x - x ' l x x ' , Qa is the atomic volume, P , is a Legendre polynomial, j , is a spherical Bessel function, and V,(r) is the non-local correction. In Chelikowsky et al. (1989) a simple Gaussian form has been used for V,whose parameters are fitted to experiment. The local and non-local parameters are tabulated in Chelikowsky et al. (1989). The cut-off in the plane wave expansion was taken at an energy El = 8 Ry and extra plane waves up to E, = 13 Ry were introduced by the perturbation theory. The corresponding results for Ge and GaAs are compared to experimental

35

1.3 Comparison with Experiments for Zinc-Blende Materials

4 r d\r n-Ge(l11)12xl

p-GaAs( 110) 1x 1

L

L

A

A

r

r

A

A

X

X

w

w

K

K

z

r

z -10

-5

0

5

10

15

r

-10

-5

0

5

10

15

20

Energy relative to Ev (eV)

Energy relative to E, (eV)

Figure 1-14. Comparison between photo-emission measurements and calculated densities of states for Ge: experimental intensities (top), theoretical density of states (middle), corresponding calculated bands (bottom) for which the vertical axis corresponds to the wave vector. The experimental data corresponds to Xray photo-emission spectroscopy (XPS) or to Bremsstrahlung isochromate spectroscopy (BIS).

Figure 1-15. Comparison between photo-emission measurements and calculated densities of states for GaAs.

data in Figs. 1-14 and 1-15. The agreement is fairly good, especially if one notes that both photo-emission and optical data are reproduced with similar accuracy.

At this point it is also interesting to measure the accuracy of the first principles G-W calculations for these materials. This must be done keeping in mind that these calculations are performed starting from local density calculations which, as we have seen, lead to large discrepancies in excitation energies. One result taken from Hybertsen

36

1 Band Theory Applied to Semiconductors

like structures such as Se or Te for which we use the tight binding description, comparing the results with photo-emission data. We briefly discuss the case of lamellar materials. Finally, we consider a class of semiconductors with unconventional bonding, the Sb chalcogenides.

2

Ge

0 -2

F-

-4

1.4.1 General Results for Covalent Materials with Coordination Lower than Four

$ -6 9)

C

W

-8

L

A

r

Wavevector

A

X

5;

Figure 1-16.Comparison between angular resolved photo-emission and the G-W calculation of Hybertsen and Louie (1986) showing typical experimental errror bars.

and Louie (1986) compares the valence band structure with angular resolved photoemission (see Fig. 1-16).

1.4 Other Crystalline Materials with Lower Symmetry In this section other cases of crystalline semiconductors are examined. In these the bonding is more complex than that in zincblende materials which constitute, in a sense, the prototype of covalent or partly ionic bonding. We begin by generalizing the tight binding arguments discussed for tetrahedral systems to cases with lower coordination. We then specifically consider chain-

Let us consider systems where each atom has N equivalent bonds to its neighbors. As with tetrahedral compounds, we want to build a molecular model which provides a simple basis for the understanding of their band structure. On each atom we build N equivalent orbitals which point exactly or approximately towards the nearest neighbors. We consider in all cases an sp minimal basis; this is always possible since one has four basis states from which one forms only Ndirected orbitals, with N c 4. The remaining atomic states are then chosen by taken into account the local symmetry (we shall later see specific examples of how this can be achieved). Once this is done the basic electronic structure follows almost immediately. Again the directed states strongly couple in pairs as in diatomic molecules and form a bonds with a a bonding state and a a * antibonding state. It is this coupling which dominates the Hamiltonian matrix and the cohesive properties. To the lowest degree of approximation, all other states remain uncoupled at their atomic value. The resulting level scheme then consists of N / 2 abonding and N / 2 a* antibonding states plus 4 - N non-bonding atomic states per atom at energies which depend on the specific case under consideration. Of course, these levels are all strongly degenerate and, if the molecular model is meaningful, further inter-

37

1.4 Other Crystalline Materials with Lower Symmetry

actions will lift this degeneracy to form well-defined and separate energy bands.

1.4.2 Chain-Like Structures Like Se and Te One instructive example of a chain-like structure ispicturedinFig. 1-17, whichrepresents an elemental system where each atom forms two equivalent bonds with an interbond angle of 90”. This situation does not exactly represent the crystalline structure of Se and Te but is very close and will help us in understanding the properties of these materials. We will discuss this case from the tight binding point of view, the same qualitative description of the bands being obtained with the pseudopotential approach. Let us begin the tight binding picture by starting from the molecular model. As discussed before we first have to build two directed states pointing towards the neighbors. With the local axes of Fig. 1-17, these are pure p states px and py with the positive lobe oriented towards the neighbors. Then, from symmetry, the other atomic states which do not participate in 0bonds will be the s state and the pz state, perpendicular to the plane of the two bonds. The parameters appropriate to Se and Te are such that roughly Ep -E, = 10 eV, H, = - 2 eV which from Eq. (1-64) means that H s u = - 1.3 eV and H,= - 0.6 eV. The results of the molecular model are pictured in Fig. 1-18 with the corresponding electron

Figure 1-17. Simplified chain-like structure for Se and Te.

Figure 1-18. Molecular levels for Se and Te, the energy scale being of order 15 eV between the s and u* states.

-20

-16

-12

-8

-4

Energy (eV)

0

4

8

Figure 1-19. Photo-emission results for Se (a) and Te (b) compared to the calculated densities of states: - - - (experiment), -(theory).

population per atom and the nature of the states. The upper valence band is the nonbonding pz band corresponding to the wellknown “lone pair” electrons. Comparison with the photo-emission results of Fig. 1- 19 (Shevchik et al., 1973) shows that the molecular model already gives an essential account of the results.

38

1 Band Theory Applied to Semiconductors

As for covalent systems, we can now study the broadening of the molecular levels into bands. We treat each band separately, which is valid to the first order in perturbation theory. We make use of a nearest neighbors tight binding Hamiltonian as defined in Sec. 1.2.3.1 with four parameters H,,, H,, H,,,, and H, (note that the H , connecting the px, py functions involved in (7 bonds are already included in the molecular model). The case of the s band is the simpler one, with an interaction H,, between nearest neighbors. We are dealing with a system containing two atoms per unit cell (Fig. 1-17) and easily get the s band dispersion relation:

1 I

ka E,(k) = E, f 2 H,, cos 2

(1-124)

The pz lone pair band can be treated in the same way since the pz orbitals are coupled together only via the interaction H., This gives:

1 1

ka E z ( k ) = E p + 2 Hnn cos 2

(1-125)

The broadening of the (7 and (7* bands is slightly more involved. Let us refer the bonding and antibonding orbitals to one sublattice whose atoms are labelled i, their neighbor in the cell being labelled i f . From Fig. 1- 17 the bonding and antibonding states can be written: (1-126)

The energies of these states are +-IHual. With our tight binding Hamiltonian there is no interaction between adjacent bonds but only with second nearest neighbor bonds. Moreover, the x-like bonds do not couple with the y-like bonds giving rise to doubly degenerate bands. It is simple to show that only the H,interaction is involved and one

gets the following dispersion relations: Eb(k)=-IH,l +IH,Icos(ka) (1-127) E,(k) = + 1H, I + IH,, I cos (k a> Of course the linear chain model of Fig. 1-17 does not exactly correspond to the real structure of Se and Te. These Materials are characterized by helicoidal chains with three atoms per unit cell (Hulin, 1966). The interbond angle is 100" for Se and 90" for Te. Under such conditions the basic features of the previous model still remain valid. When the bond angle is not exactly 90", one builds on each atom two symmetrical px and py orbitals in the plane of its two bonds, pointing only approximately towards its neighbors. One then builds a pz orbital perpendicular to the plane and, finally, the s orbital is left uncoupled. This leads to a molecular model exactly identical to the previous one except that the bonding and antibonding levels will now be sin2(8/2), where 8 is the bond at +IH,I angle. The dispersion relation will also be slightly modified with respect to those in the simple model, especially for the lone pair band since the pz orbitals are no longer parallel. However, all the basic qualitative features will remain unchanged. All conclusions of the tight binding description are confirmed by experiments (Shevchik et al., 1973) and also by empirical pseudopotential calculations (Schluter et al., 1974). The main difference is that the weak interaction between chains induces some degree of three-dimensional characters which wash out to some extent the onedimensional divergences of the density of states (Schluter et al., 1974).

1.4.3 Layer Materials It is not possible here to give a complete account of what has been done on layer materials. We thus restrict ourselves to the

1.4 Other Crystalline Materials with Lower Symmetry

case of the crystalline germanium monochalcogenides Ge-Se and Ge-Te for which we can generalize the simple model discussed above for the Se and Te chains. In these material both Ge and Se (or Te) atoms have threefold coordination with interbond angles close to 90". Furthermore, they consist of two-dimensional layers, see Fig. 1-20. We can generalize the molecular model to this kind of system in a straightforward manner. We idealize the situation by considering interbond angles of 90" and build local axes 0 x, y, z, on each atom along these bonds. The natural basis states for the molecular model will be the s states and the corresponding px, py, pz orbitals, each of these having its positive lobe pointing toward one neighbor. In the molecular model the s states remain uncoupled while all p orbitals couple by pairs, leading to abonding and (T*antibonding states. The resulting level structure is shown in Fig. 1-21. The number of electrons per GeSe unit is 10. As there are 2"s" states, 30, and 3 0 " states per unit, one concludes that all s and (T states are filled and these form the valence band. The fundamental gap then takes place between the (T bonding and (T*antibonding states. The level position in Fig. 1-21 have been calculated using the parameter values given in Bergignat et al. (1988) for GeSe. The s level of Se is much lower, followed by the Te s level, and the (T and (T*levels. The resulting valence band compares extremely Ge

Figure 1-20. Simplified layer structure for GeSe.

39

d

+% s.Ge s,se

-%-

_ct_

Figure 1-21. Molecular levels for GeSe, the energy scale between the s and u* levels being of order 15 eV.

well with the XPS measurements for crystalline GeSe, noting that the densities of states of the CJ levels are 3 times larger than for s states.

1.4.4 New Classes of Materials: the Antimony Chalcogenides One major interest of these materials is that they clearly show how band theory can clarify the understanding of the chemical bond in a class of semiconductors with fairly complex structure. This subsection is a summary of what can be found in Lefebvre et al. (1987 and 1988). Tin and antimony atoms have an electronic configuration Sn: [Kr] 4d1°5 s25p2, Sb:[Kr] 4d1'5 s25p3. Some chalcogenide compounds are insulators in which the bonds have strong covalent character while others are characterized by a lone pair, 5s2, which does not take part in the bonding but whose properties are directly correlated to the Sn or Sb coordination and to the structural packing (Gillespie and Nyholm, 1957) (for instance, such a correlation is the basis of chemical valence shell electron pair repulsion VSEPR theories (Gillespie, 1972)). These materials are characterized by a large range of electrical behaviors (insulator, semi-conductor, semi-metal). Their general formula can be expressed as B,Xx, A,BbXx, or A,&,XxIi where the atoms are alkaline or alkaline-earth or T1,

40

1 Band Theory Applied to Semiconductors

Pb for A, Sn or Sb for B, a chalcogen for X(S, Se, Te), and iodine for I. Until recently, systematic analyses of this family of materials have consisted of a determination of their atomic structure and their electrical conductivity and also of Mossbauer experiments performed on 19Sn and on 121Sb.This has allowed Ibanez et al. (1986) to build a simple chemical bond picture using the concepts of “asymmetry” and “delocalization” of the 5s2 lone pair. It is thus interesting to analyze the exact meaning of such notions through the combined use of photo-emission (UPS and XPS) measurements and band structure calculations. We show in the following discussion that this allows us to obtain a coherent picture of the electronic properties of this class of materials and, furthermore, that one can derive a molecular approximation of the full band structure which allows a clear understanding of the physical nature of the distortion experienced by the 5s2 lone pair electron distribution. We consider five representative elements of the Sb family chosen to exhibit the whole range of electronic properties, i.e., SbI,, Sb2Te,, SbTeI, T1SbS2, and Tl,SbS,. Let us first summarize the previous understanding of the electronic properties of these materials. First, X-ray diffraction studies have provided the bond lengths and interbond angles which have been connected to the bonding character (covalent, ionic, or Van der Waals), the distortion of the packing around antimony atoms being attributed to the stereochemical activity of the 5s2 lone pair E(Sb). Mossbauer spectroscopy gives information on the electron distribution around the Sb atom through the isomer shift 6 (directly connected to the 5 s electron density at the nucleus) and the quadrupole splitting. A which reflects the electric field gradient. Using these data plus the electrical properties (conductivity a,

’

band gap E g ) , these materials have been classified into three groups on the basis of their E(Sb) behavior: i) E(Sb) is stereochemically inactive, localized around the Sb nuclei with strong 5s2 character. This corresponds to octahedral surrounding of the Sb atoms and to insulating behavior (ex: SbI,). ii) E(Sb) is stereochemically active as is seen by the distorted surrounding of the Sb atom and the corresponding reduction of the 5 s character at the Sn nucleus, which can be attributed to s-p hybridization. These materials are semiconductors (1.2 eV 5 Eg S 2 eV) with weak conductivity (a= Q-’ cm-l). iii) E(Sb) is again stereochemically inactive, i.e., the Sb environment is again octahedral as in the first group but there is a 5s density loss at the nuclei and these compounds are semi-metals ( E g = 0.1 eV, CT = 10, Q-’ cm-’ ). The lone pair E(Sb) is then considered as partly delocalized (Ibanez et al., 1986). On the basis of these known properties a tentative description of the band structure has been proposed (Ibanez etal., 1986) whose main features are schematized in Fig. 1-22. Fig. 1-22 (1) corresponds to case (i), (2) and (3) to case (ii) with sp hybridization, and (4) to the delocalized 5s2 lone pair. The important point is that such a description assumes that the “5s” band is moving from the bottom to the top of the valence band which is relatively hard to understand on simple grounds. A better understanding of these electronic properties requires a band structure calculation which is difficult to perform in view of the large numbers of atoms per unit cell (between 6 and 24 for the materials considered here). For this reason it is necessary to use empirical tight binding theory whose simplicity allows a full calculation. The

E:

Sb atomic levels

1.4 Other Crystalline Materials with Lower Symmetry

d(Sb)

2

3

tight binding description for these materials is based on the use of an “s-p” minimal basis set and, as usual, on the neglect of interatomic overlap terms. The Hamiltonian matrix elements are taken from Harrison’s most recent set of empirical parameters (Harrison, 1981). However, the situation here is more complex since there are several close neighbor distances. To account for this, we apply Harrison’s prescription to the interatomic matrix elements Hap ( R , ) for atoms which are at the nearest neighbor’s distance R , and determine the other H a p ( R ) by the scaling law:

[ [:

I1

Hap( R )= H a p ( R l )exp - 2.5 - - 1

(1- 128)

valid for R lying between R , and a cut-off distance R, chosen to correctly represent the crystal. The use of an exponential dependence and of the parameter 2.5 is discussed in Allan and Lannoo (1983). Of course, any type of empirical theory has to be tested in several ways and, for this reason, we use XPS-UPS measurements to confirm the nature of the predictions for the valence band densities of states for which tight binding should give a reasonable description. Some characteristic results are reproduced

41

Figure 1-22. Initially proposed densities of states for the antimony chalcogenides, the energy scale being 10 to 15 eV between the lower und upper levels.

4

in Fig. 1-23 where it can be seen that the empirical tight binding calculation correctly gives the number and position of the main peaks of the valence band density of states and also gives the correct valence band width. In principle, tight binding is better suited to the description of valence bands but Table 1-2 shows that the predicted gaps also compare well with the experiments. One can then calculate the number Ns of 5s electrons on the Sb atom which is the basic quantity determining the Mossbauer isomer shift S. Fig. 1-24 shows the linear correlation between the measured 6 and the computed values of N,. This proportionality demonstrates that the band structure determination gives a completely coherent picture of the electronic structure of these compounds. A striking feature of these results is that the loss of 5s electrons is extremely small with a maximum value of about 0.1. The notion of strong or weak 5s character on Sb is thus quite relative as is the notion of delocalization since the band states are always delocalized over the whole crystal. However, if we sum up their contribution to the 5 s population on the Sb atoms we get something that is always close to N s = 2. A final comment that one can make concerning the band structure is that in all cases one finds the 5s Sb density of states at the bot-

42

1 Band Theory Applied to Semiconductors 2 4 , (calc)

0.05

0.10

TI,SbS, TISbS,

0

N 1

2

3

*

Energy (eV)

Figure 1-23. Predicted valence band densities of states for antimony chalcogenides compared to UPS and XPS spectra. The full lines correspond to theory. The vertical axis represents intensities or densities of states in arbitrary units.

Table 1-2. Predicted gaps (Eg,p) compared with experimental gaps (Eg,e ) in eV. Compound

Egr P

Eg.e

SbI, Sb2Te3 SbTeI TISbS2 Tl,SbS,

2.40 0.14 1.32 1.73 2.12

2.30 0.21 1.45 1.77 1.80

tom of the valence band in contradiction with the qualitative chemical picture of Fig. 1-22 that relates the 5s2 delocalization to a shift of this density of states towards higher energies.

Figure 1-24. The Sb Mossbauer isomer shift 6 versus 2-Ns(Nsbeing the number of Sb s electrons) and the number of N of missing neighbors.

At this level, a better understanding can only be obtained using a simple physical model. To do this we idealize the atomic structure by considering that a reasonable first order description of the immediate environment of an Sb atom consists of the perfect octahedron of Fig. 1-25. However, among the 6 possible sites i = 1 to 6, N are taken to be vacant. We then treat such a Sb-M,, unit as a molecule for which we take as basis functions: qs, px, (py, and cp, which are the s and p states of the Sb atom and xiwhich are the “p” states of the existing M atoms that point towards the Sb atom (i.e., their positive lobe is in its direction). The “p” levels of the Sb and M atoms fall in

1.4 Other Crystalline Materials with Lower Symmetry

43

Figure 1-25. Simplified molecular model of the antimony chalcogenides: a) idealized octahedron around one Sb atom, b) the two possible situations along one axis with no or one missing neighbor, c) the resulting schematic level structure.

the same energy range and, to simplify, we take them as degenerate while the s level of the Sb atom is about 10 eV lower, For this reason we first treat the coupling between the p states alone. For a given direction a = X, Y, or Z two situations can occur (see Fig. 1-25b): i) There are two M atoms i(=l, 2, or 3) and j ( = 4, 5 , or 6) . In this case the Sb p state qa only couples to the corresponding antisymmetrical combination (xi- x j ) / j h giving rise to a c~ bonding state and a c ~ * antibonding state while the symmetrical + remains at the p energy. ii) There is only one M atom i. In this case pa couples to xito give again a c ~ * and c~ state but with a smaller splitting.

(xi xi)/*

The resulting level structure is shown in Fig. 1-25c. All these molecular levels have to be filled except for the c ~ * antibonding levels. At this stage the 5s electron population on the Sb atoms is exactly N , = 2. The only factor that allows for a reduction in Ns is the coupling of (ps with the empty antibonding states c ~ * . However, in case i) these are antisymmetrical so that the coupling vanishes by symmetry. This is not true in case ii) where such coupling will exist and will induce a reduction in N,. We then arrive at the conclusion that the total reduction in

N , will be proportional to the number N of missing M atoms, if 0 S N 5 3. Fig. 1-24 b shows that the plot of the chemical shift S versus N also gives a straight line but with a reduced accuracy as compared to Fig. 1 -24a. The constant factor relating 2 - N , to N can be evaluated by the second order perturbation theory and gives a result comparable to Fig. 1-24 a. This means that the simple molecular model contains the essence of the behavior of the 5s lone pair. The combination of XPS measurements and band structure calculations thus leads to a fully coherent description of these materials allowing us, for instance, to understand the trends in the Mossbauer chemical shift related to the 5s electron density at the Sb nucleus. This gives a precise meaning to the empirical notions of asymmetry and delocalisation of the 5s2 lone pair which were previously used in solid state chemistry. Similar considerations were then applied to analyze the meaning of the chemical notion of tin oxidation number in the compounds belonging to the SnS-In,&-SnS, family (Lefebvre et al., 1991). The conclusion is that the difference between the two oxidation states Sn(I1) and Sn(1V) corresponds to a variation of about 0.7 Sn 5s electrons, the SnII atoms always being in strongly distorted sites. The ETB technique

44

1 Band Theory Applied to Semiconductors

is equally helpful to predict trends in the valence states of binary and ternary gallium and arsenic oxides and compare them with photoemission data (Albanesi et al., 1992). A particularly interesting application is the case of lithium insertion in three dimensional tin sulfides (Lefebvre and Lannoo, 1997), where the change in tin oxidation state was related to the fact that the lithium donor turns out to be a defect with negative correlation energy. Finally, the family of tin monochalcogenides from SnO to SnTe was also studied by ETB combined with abinitio LDA calculations, both leading to similar conclusions (Lefebvre et al., 1998).

1.5 Non-Crystalline Semiconductors In the preceding section we have discussed the properties of several crystalline covalently bonded systems with varying coordination numbers. These are usually determined in such a way that each atom satisfies its local valence requirements. In most cases this leads to the 8 - N or octet rule which links the coordination Z to atomic valence through the relation Z = 8 - N for N 2 4. However, there are a lot of exceptions to this rule, for example, the case of crystalline Ge-Se in which Z is 3 for both Ge and Se instead of being respectively 4 and 2. In all these covalently bonded materials the cohesive energy mainly results from the formation of the local bonds between nearest neighbors and not from incomplete filling of a broad band as in metals. This cohesive energy is much less sensitive to variations in interbond angles, and to long range order than to stretching of the covalent bonds. This explains why, under appropriate preparation conditions, most of these materials can be found either in the amorphous or in the glassy state character-

ized by a loss of the long range order. However, as shown by the determination of their radial distribution function, these systems still possess a well-defined short range order similar to what is observed in the crystalline phases. In this section we examine some features of non-crystalline semiconductors. We first consider some elemental amorphous systems like a-Si, a-As, and a-Se and examine possible modifications in the density of states. We then detail the properties of a typical intrinsic defect likely to be present in aSi, i.e., the isolated dangling bond. Finally we make some comments on the electronic structure of more complex systems like SiO,, Ge,Sel-, ...

1.5.1 The Densities of States of Amorphous Semiconductors From our qualitative discussion in Sec. 1.1 remember that it is likely that amorphous semiconductors will give rise to energy bands. In these cases, the pseudocontinuum of states arises no only because there are a large number of atoms but also because there is some disorder inherent to such structures which tends to spread the energy spectrum, leading to band tails. To get a more precise feeling for what happens in the amorphous state, it is necessary to build idealized models which could be mathematically tractable and be considered as reference situations. It is for such reasons that continuous random networks have been developed to model systems like a-Si, a-Ge, at a-SO,. These are constructed by representing atoms and bonds as balls and sticks and connecting them together randomly without loose ends or dangling bonds. Usually such networks lead to a predicted radial distribution function relatively close to the experimental one. However, they remain idealized descriptions since the

1.5 Non-Crystalline Semiconductors

real material can contain clusters, dangling bonds, and other eventual deviations. Assume for a moment that we are dealing with such an idealized lattice for a-Si. The first problem that arises concerns the general shape of the band structure: does it lead to a fundamental energy gap and to the same structures in the valence band as in the crystalline system? An interesting answer to such complex questions can be obtained via simplified Hamiltonians such as those offered by the empirical tight binding theory. This was achieved by Weaire and Thorpe (197 1) on the basis of an Hamiltonian first proposed by Leman and Friedel (1962). This was based on the use of sp3 hybrids and included only two parameters: the intrabond coupling between such hybrids and the coupling between any two hybrids centered on the same atom. Here we shall reproduce the same conclusions by using the more involved Hamiltonian described in Sec. 1.3.1.1 for crystalline Si. We again build bonding and antibonding orbitals which are exactly the same as for the crystal. We treat the broadening of the bonding and the antibonding band separately. Concentrating on the bonding band, we get the analog of Eq. (1-76),

(E - Eb + 2 4 ) Ubij = A(si + sj)

(1- 129)

in view of the fact that we are dealing with an elemental system ( A A= A , = A ) . We directly sum this expression over j and get ( 1- 130)

where the sum overj extends over the nearest neighbors of atom i. Equations (1-130) are the same as those one would obtain for an s band on the same lattice with unit nearest neighbors interactions &Si = C j

sj

(1-131)

45

The eigenvalues of the two matrices are thus related through

E = Eb + 2 A

+ Ae

(1-132)

The Hamiltonian matrix defined by Eq. ( 1- 13 1) is known as the connectivity matrix and general topological theorems (Ziman, 1979) allow us to show that its eigenvalues E must lie in the interval [- 4, + 41 imposed by the coordination number. This means that the energies in the bonding band must lie in the range (1-133) Eb+ 2 4 -4 E S E b + 2 4 + 4 IAl This is the same result as for the crystalline case and thus shows that the bonding band for this idealized model of a-Si must be contained in the same energy interval as for cSi. The same reasoning applies to the antibonding band. Thus the gap between these two bands is at least as large as for the crystal in the same model. Inclusion of the interactions between the bonding and antibonding states can only increase this gap by mutual repulsion of the two bands. The first conclusion of the model is thus that the gap still exists in a-Si. The second question concerns the structures in the valence band. In the crystal some characteristic structures of the density of states are due to the Van Hove singularities which are a signature of long range order, as we discussed above. In the amorphous system these should be washed out as shown by the comparison of the density of states n ( & ) of the connectivity matrix Eq. (1-131) between the crystal and a Bethe-lattice with coordination 4 (Lannoo, 1973). This is effectively what the XPS and e2(co) curves represented in Fig. 1-26 show for c-Si and a-Si. The real atomic structure in a-Si is likely to deviate from the idealized one. To satisfy the local bonding constraints, distortions in bond angles and even in bond

46

1 Band Theory Applied to Semiconductors

(1979) and detailed consideration about localized and extended states can be found in Ziman (1979) and Weaire (1981).

40-

c-Si

30 -

-3 v

w"

20

10

1.5.2 Numerical Computations

I

-

Improvements in computational power have now allowed numerical calculations of the electronic states in amorphous structures. Major progress in this regard has been the derivation of computer-generated random continous networks with tetrahedral

-

0

lengths are likely to occur. These will give rise to band tails. It is thus expected that the gap region will not be free of states since the conduction and valence band tails will overlap as shown schematically in Fig. 1-27. As regards the transport properties, the localized or the delocalized nature of these states is of importance. A commonly adopted view (Ziman, 1979; Weaire, 1981) is that of a mobility edge separating the two types of states as shown in Fig. 1-27. Finally, the situation is complicated further by the existence of defects in the bonding, in particular the isolated dangling bonds which are treated in the following section. A more complete review of amorphous or glassy semiconductors is given in Ziman

EC

EC

Figure 1-27. Schematic density of states in the gap region of amorphous semiconductors.

et al., 1988), the first order Raman spectrum is found to be in very good agreement with the experimental one (Marinov and Zotov, 1996). It also reproduces with good accurracy the experimental radial distribution function of amorphous silicon. Recent theoretical studies of phonons in a-Si also confirm the superiority of this model compared to others (Knief et al., 1998). The electronic structure for the WootenWiner-Weaire model has been calculated (Allan et al., 1998) in the sp3s* ETB approximation (Vogl et al., 1983) and a d-* variation of the tight binding parameters with distance d (Harrison, 1980). Application of the method to a-Si gives rise to large band tailing, which is attributed to a small number of bond angles that deviate greatly from the tetrahedral 109" value. An efficient simulation of hydrogenated amorphous silicon (a-Si :H) was obtained by removing the atoms with the most distorted bond angles and saturating the resulting dangling bonds with hydrogen atoms. The resulting gap and Urbach tails then become comparable with the experimental one (Allan et al., 1998). The same description

1.5 Non-Crystalline Semiconductors

has allowed the study of confinement effects for quantum dots of a-Si-H (Delerue et al., 1996) and to show that doping of a-Si-H results in much larger binding energies of donors and acceptors than in the crystalline material (Allan et al., 1999).

1.5.3 Dangling Bonds A lot of physical situations in tetrahedrally coordinated materials involve the rupture of bonds. The simplest well-documented case is the Pb center at the Si-SiO, interface which corresponds to a tricoordinated silicon atom, i.e., to the isolated dangling bond. Such defects are also likely to occur in a-Si. Another well-known situation is the vacancy in silicon (and to a less extent in compounds) where there are four interacting dangling bonds. Let us first shortly recall the basic physical properties of dangling bonds. The simplest description comes from a tight binding picture based on an atomic basis consisting of sp3hybrid orbitals. The properties of the bulk material are dominated by the coupling between pairs of sp3 hybrids involved in the same nearest neighbor’s bond. This leads to bonding and antibonding states which are then broadened by weaker interbond interactions to give, respectively, the valence and conduction bands. In the bonding-antibonding picture, the rupture of a bond leaves an uncoupled or “dangling” sp3 orbital whose energy is midway between the bonding and antibonding states. When one allows for interbond coupling, this results in a dangling bond state whose energy falls in the gap region and whose wave function is no longer of pure sp3 character, but is somewhat delocalized along the backbonds. Experimentally this isolated dangling bond situation is best realized for the P b center, i.e., the tricoordinated silicon atom

47

at the Si-SiO, interface (Poindexter and Caplan, 1983; Caplan et al., 1979; Johnson et al., 1983; Brower, 1983; Henderson, 1984) but it can also occur in amorphous silicon (Jackson, 1982; Street et al., 1983) as well as in grain boundaries or dislocations. It has been identified mainly through electron spin resonance (ESR) (Poindexter and Caplan, 1983), deep level transient spectroscopy (DLTS) (Johnson et al., 1983; Cohen and Lang, 1982) and capacitance measurements versus frequency and optical experiments (Jackson, 1982; Johnson et al., 1985). The following picture emerges: i) The isolated dangling bond can exist in three charge states: positive D+, neutral Do, and negative D-. These respectively correspond to zero, one, and two electrons in the dangling bond state. ii) The effective Coulomb term UeE, i.e., the difference in energy between the acceptor and donor levels, ranges from -0.2 to 0.3 eV in a-Si (Jackson, 1982) to about 0.6 eV at the Si-SiO, interface (Johnson et al., 1983). The ESR measurements give information on the paramagnetic state Do through the g tensor and the hyperfine interaction. Their interpretation indicates that the effective “s” electron population on the trivalent atom is 7.6% and the “p” one 59.4%, which corresponds to a localization of the dangling bond state on this atom amounting to 67% and an “s” to “p” ratio of 13% instead of 25% in a pure sp3 hybrid. This last feature shows a tendency towards a planar sp2 hybridization. Several calculations have been devoted to the isolated dangling bond. However, only two of them have dealt with the tricoordinated silicon atom embedded in an infinite system other than a Bethe lattice. The first one is a self-consistent local density calculation (Bar-Yam and Joannopoulos, 1986) which concludes that the purely electronic value of the Coulomb term (i.e., in the

48

1 Band Theory Applied to Semiconductors

absence of atomic relaxation) is U = 0.5 eV. The second one is a tight-binding Green's function treatment in which the dangling bond levels are calculated by imposing local neutrality on the tricoordinated silicon (Petit et al., 1986). In this way the donor and acceptor levels are respectively ~ ( 0+), = 0.05 eV and ~ ( 0-), = 0.7 eV. Their difference corresponds to U = 0.65 eV, which is in good agreement with the local density result. Both values correspond to a dangling bond in a bulk system and can be understood simply in the following way: the purely intra-atomic Coulomb term is about 12 eV for a Si atom; it is first reduced by a factor of 2 since the dangling bond state is only localized at 70% on the trivalent atom; finally, dielectric screening reduces it by a further factor of E = 10. The final result 6 / ~ gives the desired order of magnitude 0.6 eV. At the Si-SiO, interface, however, the situation becomes different because screening is less efficient. A very simple argument leads to the replacement of E by ( E + 1)/2 so that the electronic Coulomb term for the P, center should be twice the previous value, i.e., U(P,) = 1.2 eV. An extremely important issue is the electron-lattice interaction. There is no reason for the tricoordinated atom to keep its tetrahedral position. A very simple tight binding model (Harrison, 1976) shows that this atom does indeed experience an axial force that depends on the population of the dangling bond state. This is confirmed by more sophistical calculations (Bar-Yam and Joannopoulos, 1986). The net result is that, when the dangling bond state is empty (D+) then the trivalent atom tends to be in the plane of its three neighbors (interbond angle 120"). On the other hand, when it is completely filled (D-) it moves away to achieve a configuration with bond angles smaller than 109" as for pentavalent atoms. Finally, the situation for Do is obviously intermediate

with a slight motion towards the plane of its neighbors. For the three charge states D+, Do, and D-, corresponding to occupation numbers n =0, 1, and 2, respectively, one can then write the total energy in the form E(n, u ) = (1-134) = n Eo + (1/2) U n2- F ( n ) u + (1/2) k u2 where u is the outward axial displacement of the tricoordinated silicon atom, F(n) the occupation dependent force, U the electronelectron interaction, and k the corresponding spring constant which should show little sensitivity to n. We linearize F ( n ) F ( n ) = Fo + FI (n - 1)

(1 -1 35)

an minimize E(n, u) with respect to u to get Emin(n). The first order derivative of Emin(n)at n = 1/2 and n = 3/2 gives the levels &(O, +) and &(-, 0). The second order derivative gives the effective correlation energy:

u,, =u--42 k

(1-136)

Theoretical estimates (Bar-Yam and Joannopoulos, 1986; Harrison, 1976) give Fl = 1.6 eV A-' and k = 4 eV (A2)-' (Lannoo and Allan, 1982) so that F:/k is of the order 0.65 eV. This has strong implications for the dangling bond in a-Si where U,, becomes slightly negative as concluded in Bar-Yam and Joannopoulos (1986) but this result should be sensitive to the local environment. On the other hand, with U= 1.2 eV, the P, center at the Si-SiO, interface would correspond to Ueff=0.6eV, in good agreement with the experiment. The theoretical finding that Ueffis slightly negative for the dangling bond in a-Si leads to an inverted order for its levels, in which case the Do state could never be stable (Bourgoin and Lannoo, 1983). This

1.5 Non-Crystalline Semiconductors

does not agree with the experiment, in which an EPR spectrum which seems characteristic of Do has been observed. One possible reason for this discrepancy is the suggestion that a-Si may contain overcoordinated atoms (Pantelides, 1986) which might be responsible for the observed features. However, recent careful EPR measurements (Stutzmann and Biegelsen, 1989) seem to rule out this possibility, practically demonstrating that dangling bonds indeed exist and with a positive U,, This would mean that theoretical calculations have underestimated the electronic U for reasons which are still unclear. 1.5.4 The Case of SiO, Glasses

We now give a simplified analysis of the valence band structure of these glasses based on an extension of the tight binding arguments developed before. Let us first consider the case of Si02.The molecular model is essentially the one developed by Harrison and Pantelides (1976). The building Si-0-Si unit is shown in Fig. 1-28. Again one builds sp3 hybrids on the Si atoms while on the oxygen atom one keeps the natural sp basis. The oxygen “s” state is by far the lowest in energy and, to

49

the first order, its coupling with other states can be neglected. It will remain atomic-like at its atomic value E,(O). On the other hand, the oxygen p energy E,,(O) is closer to the silicon sp3 energy Eand the interaction of the corresponding states must be taken into account. The molecular states of the Si-0-Si unit of Fig. 1.28 are then built from the two sp3hybrids a and b pointing towards the oxygen atom and the px, p,,, and pz oxygen “p” states. It is clear that pz, being perpendicular to the Si-0-Si plane, will remain uncoupled at this level of approximation giving one state at the atomic value E,,(O). Thus the sp3 states a and b will only couple to px and py via the projection of these states along the axis of the corresponding nearest neighbor direction. All interactions reduce to one parameter ppdefined as the interaction between an sp3 orbital and the p orbital along the corresponding bond. By symmetry px only interacts with (a-b)/* giving rise to strong bonding and antibonding states at energies (1-137)

while py and (a + b)/* lead to weak bonding and antibonding states (1-138)

‘U-

Figure 1-28. Si-0-Si unit for building the molecular model with the two sp3 hybrids of the Si atoms and the three p states of the 0 atom, p: being perpendicular to the plane of the figure.

where 2 8 is the Si-0-Si bond angle. The resulting valence band density of states per Si-0-Si unit is pictured in Fig. 1-29. It consists of delta functions at energies E,(O), ESB, EWB, and E,(O), the

50

1 Band Theory Applied to Semiconductors

Figure 1-29. a) SiOz density of states calculated by Robertson (1983). b) Density of states in the molecular model with a Gaussian broadening of 0.5 eV. - The vertical axis corresponds to densities of states in arbitrary units, the horizontal axis to energies, in units of 5 eV, SB, WB and LP denote strong bonding, weak bonding and lone pair states.

-15

-10

-5

0

weight of each state per Si-0-Si unit being equal to unity. The influence of further interactions can now be analyzed as for pure Si. If we call As and Aw the interaction between strong and weak bonding states belonging to adjacent Si-0-Si units then we can repeat the treatment previously applied to Si simply by replacing A with As or A,. This means that we get densities of states in the strong and weak bonding bands that have exactly the same shape as for Si, consisting of the superposition of a broad and a narrow, almost flat band. This behavior is apparent in the calculated density of states in Fig. 1-29a. These results are in good qualitative correspondence with photo-emission data (Hollinger et al., 1977; Di Stefan0 and Eastman, 1972; Ibach and Rowe, 1974) and more sophisticated numerical calculations (Chelikowski and Schluter, 1977; O'Reilly and Robertson, 1983). Essential information provided by Eqs. ( 1 - 137) and (1-1 38) is that the splitting between the strong and weak bonding bands is a very sensitive function of the Si-0-Si angle 2 8. Any cause of randomness in Bsuch as the existence of a strained SiO, layer is then likely to induce some broadening of

5

10

these two bands and partially fill the gap between them. Note that in an extreme situation where 0 - 90", like in GeSe,, the strong and weak bonding energies become identical, leading to a qualitative change in the shape of the density of states. We are now in the position to discuss qualitatively the behavior of the SiO, layer as a function of composition. Numerical calculations have been performed that are all based on a tight binding treatment combined with a more or less refined cluster Bethe lattice approximation (Lannoo and Allan, 1978; Martinez and Yndurain, 1981). Again the molecular model gives precious information that is confirmed by the full calculation. For this, let us consider the results of Lannoo and Allan (1978) which are pictured in Fig. 1-30. In the molecular model each Si-0-Si bond corresponds to a density of states as given in Fig. 1-29b, while each Si-Si bond corresponds to one bonding state which, for the parameters corresponding to Fig. 1-30, falls at an energy slightly higher than E,(O), the energy of the SiO, non-bonding state. If we start from the SiO, limit each Si-Si bond acts like an isolated defect giving rise to a defect level at

1.6 Disordered Alloys

51

energy. At still higher Si concentrations there will be three and then four Si-Si bonds connected to each Si atom. For reasons discussed above such situations are characterized by a flat band at + 2A and a broad band at lower energies. This regime will thus exhibit a peak of increasing height at the energy of the top of the pure Si valence band. This is exactly what happens in Fig. 1-30. Note also that the strong bonding band has exactly the reverse behavior: the height of its peak and its width both decrease. The last point that is clearly seen in the figure is that, for smaller 0, the separation between the strong bonding states and the weak and non-bonding states is smaller. All these conclusions are in qualitative agreement with experimental observations concerning the valence band of the SiO, systems and of the transition layer at the Si-SiO, interface.

s! h

C

1.6 Disordered Alloys -15

-12

-9

-6

-3

E (eV)

Figure 1-30. Theoretical density of valence band states for SiO, systems for different compositions and different bond angles. The flat parts correspond to the zero of n ( E ) in each case.

the energy of the bonding state of Si, i.e., just above the SiO, valence band. When the concentration of Si-Si bonds increases this defective state will begin to broaden into a band with no defined structure at small concentrations. This corresponds to Si04,3 in Fig. 1-30. It is only when the concentration is high enough for Si chains to appear that the situation changes qualitatively. If, as before, A characterizes the interaction between two adjacent Si-Si bonding states, then the DOS of a Si-Si chain exhibits two divergences at k 2 A from the bonding

Compounds with a well-defined lattice but where there is substitutional disorder on the lattice sites compose a series of important electronic structure problems. This is, for instance, the case of pseudobinary semiconductor alloys like Ga,Al,-,As, In,Gal-,As .. . . Since such alloys are of much importance, here we give an account of the work performed on these systems. However, before doing this we must introduce general methods like the virtual crystal approximation (VCA), the average t-matrix approximation (ATA), and the coherent potential approximation (CPA). Again these methods, which correspond to complex systems, are applied within the framework of the tight binding approximation. To perform such calculations on disordered systems one must use the Green's

52

1 Band Theory Applied to Semiconductors

function formalism. For this we introduce the resolvent operator G ( E )of the system, defined as

where H is the Hamiltonian of the system and E is the energy. One major property of the resolvent operator is that the density of states n (E) can be expressed as 1

n ( ~=)- - Im TrG(E)

n

n(&)= - - Tr Im(G(E)) n;

(1-141)

The advantage is that the quantity (G(E)) becomes statistically homogeneous, i.e., if one has a random alloy on a given lattice, (G( E)) acquires the lattice periodicity, while the original G(E)has not. It is to be noted that (G(E))is not equal to ( E + i r] - (H))-’; instead we define a self-energy operator Z(E)such that one can write ( G ( E ) ) =( & + i r ] - ( H ) - Z ( ~ ) ) - l

We want to present different possible levels of approximation here and illustrate them on a tight binding model of a random alloy. This model will consist of a tight binding s band designed to treat a random alloy A,B,-,, the atomic sites forming a lattice with one site per unit cell. This tight binding Hamiltonian can thus be written: (1-143)

(1-140)

where Tr stands for the trace and Im means imaginary part. When considering a large disordered system, the fact that one takes the trace means that one performs an average over different local situations. One would get the same result by performing an ensemble average, i.e., by considering all configurations that the system could take and weight them by their probability. This means that n ( E ) can also be obtained by performing the average (G(E))of the resolvent operator and writing 1

1.6.1 Definitions of the Different Approximations

(1-142)

The determination of this 2 i s the aim of the following different approximations.

where V is a nearest neighbor interaction taken to be independent of disorder while the on-site terms can take two values; for an A atom, E~ for a B one. Let us begin with the simplest type of approximation: the VCA (virtual crystal approximation). As indicated by its name, this consists of assuming that the average Hamiltonian (H) = fi gives a correct account of the electronic structure. This means that the exact (G(E))is replaced by an approximate expression G(E): G(E)= ( E + i r] - R)-’

( 1 - 144)

From its definition, the expression of R in the simple tight binding model is given by: H=[xE*+(l-X)Eg]C

+ v C 10(1’1

1

Il) (ZI+

(1-145)

11’

which can be solved by using the Bloch theorem. We now want to go a step further and do the proper averaging of (G)over one site while the rest of the crystal is treated in an average field approximation. The first obvious thing to do is to start from the virtual crystal, determine G, then consider the local fluctuations on a given site as perturbations and perform the average (G1Jfor that site,

53

1.6 Disordered Alloys

where G , is the diagonal element of G. Let us illustrate this with our specific model. We thus start from fi defined in Eq. (1- 145) and look at the possible fluctuations at site 1. If site 1 is occupied by an A atom, the on site perturbation will be WA

A comparison between the two expressions allows us to express EATA in the form

(1-151) I

= &A - [x &A + ( 1 - x) EB] = = ( 1 -x) (&A - &B)

(1 - 146)

while for a B atom it will be WB=&B-[x&A+

(1-x)&B]=-x(&A-&B)

I

It is customary to introduce the average t matrix f, equal to the numerator in Eq. (1-151). In this way one gets the usual form of ZATA (see Ziman, 1979, for details):

(1-147) We treat both cases at the same time by assuming that there is a perturbation Wl on site 1 with possible values WA or WB. The diagonal matrix element G f Ican be obtained by applying Dyson's equation (G = G + G WG ) ,

since has translational periodicity. We now say that the average (Gll) with GII given by Eq. (1-148) represents a good approximation. This defines the average t-matrix approximation which can be written

On the other hand, ( G ) is quite generally related to Z(E)through Eq. (1-142). To the same degree of approximation as before, one should thus write that (Gff)ATA is due to the on site perturbation ZATAapplied to site 1. This leads to

(1 -1 52)

This provides a considerable improvement on the VCA, but a definitely better approximation is given by the CPA (coherent potential approximation) which we will now discuss. The CPA uses the same basic idea as the ATA, i.e., it is a single site approximation. However, instead of using an average medium corresponding to the average Hamiltonian fi it is certainly much better to use H+E(&),where the 2 i s the unknown selfenergy. Starting with this, the perturbation at site 1 is now given by W,- E(E),Wf being defined as before. Thus the average ( G f l )is given by (1-153) -

(Glf

)CPA =

Go ( E - X E ) ) 1 - [W, - a & ) 1Goo(&- a&))

But one must note that the self-energy is assumed to be the same on each lattice site so that the average crystal resolvent is simply Goo, where E is shifted by the coherent potential Z(E).On the other hand, for Z to be consistently defined, should be equal to Goo(&-2'(~)).Applying this to Eq. (1-153) one immediately gets the condition ( 1 - 1 54)

[w,- 2(&)1 Goo ( E - a&)) )=O

1 - [% - Z(&)] G&-

Z(&))

54

1 Band Theory Applied to Semiconductors

which defines the expression of ZCpA(&) bond. In such a way the Hamiltonian is defined on a perfectly regular lattice, but (Ziman, 1979). even in this case it is not approximation in view of the existence of several orbitals per 1.6.2 The Case of Zinc Blende atom and of diagonal and non-diagonal disPseudobinary Alloys order. The possibility of performing an MCPA Here we summarize fairly recent calculacalculation is related to the choice of the tions performed on these disordered alloys particular unit cell shown in Fig. 1-3I. As specifically, In, -,Ga,As and ZnSe,Tel-, in the previous sections, use is made of a sp3 (Lempertet al., 1987). This work makes use energy the coupling VIA between two of an extension of the CPA called the MCPA different sp3 hybrids, and the intra bond (molecular coherent potential approximacoupling VfB. As we have seen, such terms tion) which is particularly well adapted to dominate the Hamiltonian for zinc-blende these systems. semiconductors and they are likely to give This case of alloys is of technological the dominant contribution of disorder efinterest and it is important to be able to treat fects. disorder effects accurately. This disorder As explained at the beginning of this seccan be conveniently divided into chemical tion the disordered alloy Hamiltonian will and structural components. The former is related to the different atomic potentials of be replaced by an effective Hamiltonian: the two types of atoms, while the latter is (1-155) Heff(d = (H> + a.9 associated with local lattice distortions, essentially due to differences in bond length. MCPA is based on the assumption that the Such an effect was observed by Mikkelsen self-energy Z(E)is cell-diagonal (the cell and Boyce (1982,1983) in EXAFS (extendbeing defined in Fig. 1-31). It is thus repreed X-ray fine structure) measurements on sented by an 8x8 matrix in the basis of the In,-,GaJs. There the In-As and Ga-As bond lengths were found to vary by less than 2% from their limiting perfect crystal values, despite a 7% variation in the average X-ray lattice constant. This was also found in other zinc blende alloy systems (Mikkelsen, 1984). The electronic structure of these materials is described in the tight binding approximation (extended to second nearest neighbors in the particular calculation of Lempert et al., 1987). The structural problem in an A;-fi;B alloy causes a difficulty since it leads to local distortions with respect to the .....:. ....... ...... ...%. average zinc-blende structure. This is overI Zf ..:...... ..... come by assuming that the atoms lie on the sites of the average crystal but scaling the Figure 1-31. The unit cell in the molecular coherent dominant nearest neighbor interaction to potential approximation, defining the relevant interthe value appropriate to an A'B or A"B actions between sp3 hybrids.

3' >

1.6 Disordered Alloys

sp3 orbitals. With this one can obtain exactly the same expression as in ordinary CPA except that one must replace all quantities W,, ($0 by 8x8 matrices within the same sp3 basis. The consistency condition (Eq. (1- 153)) also applies but with matrix inversion and multiplication. We can rewrite it for our two component system (atoms A' and A") as:

a

(1-156) { [(WA, GO]-' - 1 }-' + + { [(wA,,GO]-' - 1 }-' = 0

a

or, since

WAe

*

(1-157)

+ WAt, = 0, to:

Z(&)=-[wA<-z(&)]. (&

-z(&))

-

(1-158)

. [wA**- Z(&)I

which is the basic consistency condition allowing the determination of the complex self-energy matrix Z(E).Equation ( 1 - 158) is then solved iteratively to get the unknown Z(E)matrix. In practice, advantage can be taken of the tetrahedral symmetry of the unit cell. The eight sp3 orbitals of this cell can be transformed into basis states for the irreducible representations of the T, symmetry group, i.e., A, and T2 in the present case. On the A site this comes back to the s and p atomic orbitals, while on the neighbors this gives the same combinations of the four sp3 orbitals as the s and p on the central atom. The matrix Eq. (1-158) is then split into four independent 2 x 2 blocks, which considerably reduces the difficulties. It is interesting to give the explicit form of the perturbation matrices WA*and WA,,. We have seen that chemical disorder is modeled by three distinct matrix elements ,& , VIA, and V,,, in Fig. 1-31, which for an atom A lead to an 8 x 8 matrix which we label h,. The average crystal is then given

by:

hA = (1 - x) hA! + xhAU

( 1- 1 59)

and the perturbation matrices WA#and WAN are: WA, = hA*- LA = X ( h ~-' h ~ " )

(1 - 160)

They are thus defined in terms of the differences &A, - &A", VIA!- V,A*t, and V2A,Bv2Ar'B

which can be transformed to: (WA*-a-'+(WA"-Z)-'=2Go

(for the compound A;_,A:B)

55

*

We will not discuss here the details of the parametrization scheme used in ref. (Lempert et al., 1987), but instead summarize some of their results. Figures 1-32 and 1-33 give the densities of states, comparing the results of the VCA, the site CPA with chemical disorder only, and the MCPA. Clearly the VCA is unsatisfactory for some of the spectral features, particularly for ZnSeo,,Te0.,. A fairly important physical quantity is the band gap bowing, i.e., the deviation from linearity with concentration of the alloy band gaps. The results are shown in Table 1-3 demonstrating that theory does not give a full account of the experimental value of the bowing effect. The difference could be attributed to several causes: uncertainties in the perturbative tight binding parameters, intercell disorder effects, clustering, etc. In any case, it must be realized that the bowing effect is very small, c0.3 eV. It is also of interest to briefly discuss other calculations performed on these alloys and compare them with the MCPA. The first of them is the bond-centered CPA of Chen and Sher (1981) which includes disorder effects associated with each bond, i.e., in the form of a 2 x 2 matrix in the corresponding bonding-antibonding state basis. This includes structural disorder due to VIA and should be, in principle inferior to the MCPA

56

1 Band Theory Applied to Semiconductors

h

.-C

Q v)

v)

(I)

c m c v)

.c v)

C

X Y

-12

-8

-4

0

"

-12

Energy (eV)

0

-6

6

Energy (eV)

Figure 1-32. Density of states in VCA, CPA and MCPA for In,,G%,,As. For a discussion see text.

Figure 1-33. Density of states in VCA, CPA and MCPA for ZnSeo,,Teo.,. For a discussion see text.

from that point of view. A second approach is the supercell method which models an A&,,A&B zinc-blende pseudobinary alloy as a chalcopyrite crystal (Zunger and Jaffe, 1983). This allows them to perform selfconsistent local density calculations. However, the relation to a disordered alloy is not trivial and, for this reason, a calculation has been performed in Lempert et al. (1987) with the same tight binding Hamiltonian in the supercell approach. We reproduce the results in Figs. 1-34 and 1-35 which demonstrate that the supercell results are much closer to the MCPA than are those of the bond-centered CPA. This shows that the main features of the density of states are primarily related to local bonding properties. We should finally mention the recursion method (Haydock, 1980) which should, in principle, be well adapted to this kind of

problem. It allows a direct determination of the local Green functions. This would be fairly efficient for getting the shape of the density of states, but much less efficient for the determination of band edges which require much more precision. Table 1-3. Comparison of experimental with theoretical values of band gap bowing.

Linear interpolation Deviation from linearity (x = 0.5) VCA Site CPA MCPA Supercell Expt.

530-13Ox

-59

-79 -66 -60 - 123, - 133

3040-330~

+

10 - 5 - 90 - 49

-270

1.7 Systems with Lower Dimensionality

57

(a) MCPA

n

(b) Bond-Centered CPA

(c) Supercell

-12

-8

-4

0

-

(b) Bond-Centered CPA

-

( c ) Supercell

4

-12

1.7 Systems with Lower Dimensionality With the progress of growth techniques it is now possible to build semiconductor structures which have two-, one, or even zero-dimensional character. Such structures are not only important for semiconductor devices but they also exhibit a lot of interesting physical properties. We shall shortly describe some basis aspects of these systems. We begin with their qualitative fea-

-6

0

6

Energy (eV)

Energy (eV) Figure 1-34. Density of states in MCPA, bond-centered CPA, and the supercell approach for Ino.,Gao.,As.

I

Figure 1-35. Density of states in MCPA, bond-centered CPA, and the supercell approach for ZnSe0STe0.5.

tures, then analyse the use of the envelope function approximation, especially the problem of the correct boundary conditions. We then treat the case of quantum wells and superlattices. We also discuss hydrogenic impurities and excitons. Finally we present results obtained for silicon quantum dots.

1.7.1 Qualitative Features Molecular beam epitaxy and other growth techniques have allowed to deposit

58

1 Band Theory Applied to Semiconductors

For a system confined in 3 - d directions and free in the other d directions, &k is the dispersion relation in d dimensions (with k the corresponding wave vector), while E, are the levels resulting from the confine) ment. We can rewrite n ( ~ as (1-163) n(~= ) dE’d(E - E, - E ’ ) a(&’- ~ k )

semiconductors layer by layer. The simplest system one can obtain in this way is the quantum well, corresponding, for example, to a layer of GaAs within bulk GaAlAs (Fig. 1-36). The GaAlAs gap is larger than the GaAs gap, and the conduction and valence band offsets are such that one gets attractive potentials both for electrons and holes. Assuming that one can treat both electrons and holes in an effective mass approximation, this leads to a standard quantum mechanical one-dimensional problem except for the free particle motion in directions parallel to the layer. More recently it has become possible to grow quantum boxes in which the particles are confined in all directions. Let us concentrate on the electrons, for instance, and discuss the general features of the density of states in such systems. By definition this is given by:

x

The quantity

2 a(&’ - &k ) defines the denk

sity of states vd(e’)of the d dimensional free electron gas (only defined for ~’>0).This allows us to express n ( E ) in the form

n ( ~ ) = Yxd ( E - E n ) n

(1-164)

For a free electron gas in d dimensions, it is well known that Y ( E ) = E ~ ‘ * - ’ so that we get, quite generally, n

(1-165)

This is plotted in Fig. 1-37 for bulk systems, quantum wells, wires, and dots, each showing remarkably distinct behaviors. It is also possible to produce gradual transitions between 2D and 3D behavior, for instance. For this, one can increase the width of the layers in which case the E,, will tend to form a pseudocontinuum as discussed in Sec. 1.1. One can also build a superlattice with equidistant quantum wells. When these are close enough, the states E, broaden into 1D bands, the steps in the density of states of Fig. 1-37 become smoother and n (E ) tends to 3D behavior. All these cas-

n Figure 1-36. Conduction and valence band square well potentials for a layer of GaAs in Gal -,Al,As.

E

k

n

&

4

Figure 1-37. Schematic densities of states for: a) bulk material, b) a quantum well, c) a quantum wire,d) a quantum dot.

59

1.7 Systems with Lower Dimensionality

es represent nice examples of general considerations about band structure formation.

1.7.2 The Envelope Function Approximation

rll

1 df continuous. These reduce m * dz to the normal conditions for materials with the same effective mass. They also have the advantage of leading to the conservation of the current probability across the interface, which is a constraint that must necessarily be fulfilled. An illustration of the problem raised when choosing the boundary conditions is provided by a 1D tight binding s band model with diagonal energies U , or UB and nearest neighbors interactions Ic;, or & depending on the material A or B, the coupling between A and B being taken as /3 (see Fig. 1-38). Writing the wave function C, q, , where qnis the “s” orbital of atom tinuous,

The electronic structure problem of these structures is more complex than for bulk materials, especially when space-charge effects are included, in which cases the potential has not only discontinuities but also macroscopic curvature (e.g., n-p heterojunctions). Fortunately, in regions where the potential is slowly varying and where the carriers are close to the band extrema, it is possible to simplify the treatment by using the effective mass approximation discussed previously in Sec. 1.1.4. For a single band extremum the wave function takes the form of Eq. (1-32), i.e., it is the product of the Bloch function at the extremum times a slowly varying envelope function F ( r ) which is the solution of the effective mass equation, Eq. (1-30). This equation can be solved in regions where the potential has no discontinuity, but then one has to match the solutions from both sides of one discontinuity. Let us illustrate this at a simple interface where there is a conduction band discontinuity along the z direction; at z = 0 between materials A on the left and B on the right. Assuming isotropic effective masses mi and m;, the envelope function takes the form ~ ( r=f(z) ) eikll

functions which must satisfy the standard boundary conditions and not their envelope part. It is, however, customary to use the following boundary conditions: f ( z ) con-

(1-166)

where 11 stands for quantities parallel to the interface plane. The matching problem only concerns fA(z) and fB(z). For such a potential in a vacuum, the matching would require continuity of the wave function and of its derivative. However, heref,(z) andf, ( 2 ) are not the true wave functions but envelope functions in materials with different bulk properties. Normally it is the total wave

--

n

n and the Cn are unknown coefficients, one gets the following set of equations:

(E-

uA>

cn ‘6.4

5-1

(Cn-1+ Cn+l)

( E - U A ) CO = P A c-1 + pc1 ( E - 4 3 ) Cl c 2 + PCO ( E - U,) c, = & (C,-I + C,,,)

(1-167)

=A

n22

For bulk states of the infinite materials A or B, the Bloch theorem tells us that the coefficients C, behave as Coexp(ikna) where a is the lattice parameter and the energy bands are given by UA,B+ 2PA,,cos(ka). The extrema of these bands occur at k = 0 or k = d a and the Bloch function at these

.

--_ -2

PA

%PA

-1

% 0

P 2 P B SPB 1

2

.-3

Figure 1-38. The A . B interface for a linear tight binding chain with the intraatomic and nearest neighbors’ parameters.

60

1 Band Theory Applied to Semiconductors

extrema is such that C, = Co (+ l),. To make use of the effective mass approximation for the interface problem we must consider energies falling near an extremum for both materials A and B; this extremum corresponds to UA 2 2PA and UB 2 2/&, and the nature of the extremum is not necessarily the same in both materials. In this case we write cAn

= cAO

6 f A n

CBn

= cB1

&;-'fBn

(1-168)

= +- 1, E~ = 2 1, the sign dependwhere ing on the nature of the extremum. In this case fAn and fBn are expected to be weakly dependent on n and one can express them as the value taken by the continuous envelope functions fA(z) or fB(z) at z = nu. These envelope functions are solutions of the bulk equations, i.e., the first and fourth Eq. (1-167), given by (1-169)

The second and third equation can be written in a similar way: (1- 170) (E-UJfA (0) = &A PAfA (- a ) + BfB(+ a ) (E-UBI~B( a ) = EB /&fB(2a) + P f A ( 0 )

We can inject into these last two equations the analytic continuations fA (+ a ) and fB(0), solutions of Eq. (1-169). This leads directly to f ~ ( + a=) &A

P f~( + a )

As the functionsf,,,(z) are slowly varying, the quantities $ [fA,B ( a ) + fA,B (o)] give:

(1 ) for the "+" sign, the value of the functionsf,,, at the interface, and (2) for the "-" sign, the value of a .fi,B where f' is the derivative. Consideringfandf' as the components of a column vector, one can then formally write:

(1-172) where T is a 2 x 2 transfer matrix given by:

-(-

&AD+

T=-

PA

(1-173) This expression of the boundary conditions shows that the simple boundary conditions used in most calculations are not generally obeyed. In general, one should replace them by the transfer matrix formulation Eq. (1-172) which has been generalized in Ando et al. (1989); Ando and Akera (1989) to realistic band structures. It is interesting to discuss the situations for which the transfer matrix Eq. (1-173) leads to the usual boundary conditions. For instance continuity of the envelope function (fA=fB) requires that T I , = 1 and T,, = 0. This last condition implies that

p2 = 5 PA&. The &A

signs of the different coupling parameters are obviously the same; the requires that EB/EA= 1, i.e., that one is dealing with extrema of the same nature on both sides. If this is realized, the T matrix becomes:

( 1- 174)

1.7 Systems with Lower Dimensionality

The sign of can be absorbed as a change in phase of the function fB (2). In this case, continuity of the wave function is ensured only if &/PA= 1 corresponding to equality of the effective masses for the two materials. If this is so, continuity of the first derivative is automatically fulfilled. This is the only situation for which the usual continuity equations automatically apply. This seems to hold true for GaAlAs/GaAs systems as shown in Ando et al. (1989); Ando and Akera ( 1 989).

1.7.3 Applications of the Envelope Function Approximation The simplest application of the envelope function approximation corresponds to the isolated quantum well for single isotropic band extrema of the same nature in the two materials. This occurs for the conduction band of GaAs-AI,Ga, -,As systems in which the minimum is of I'symmetry in both materials when 0 c x c 0.45. Furthermore, the effective masses are of the same order of magnitude in the two materials. As we have seen, the well and barriers are in the GaAs and GaAlAs parts respectively. In such a situation the normal boundary conditions should apply and one has thus to solve a simple square well problem, the electron mass being replaced by the effective mass m*. The superlattice case is treated as a Kronig-Penney-type model and leads to a similar broadening of the quantum well levels into bands. The corresponding valence band problem under the same conditions is not as simple. As we have seen in the impurity case, it is necessary to solve a set of coupled differential equations which can more or less be simplified after some approximations as discussed in Altarelli (1986); Bastard (1988). Another interesting problem concerns the behavior of hydrogenic impurities in

61

quantum wells. Again the basic case concerns the isotropic band minimum with effective mass m*.In three dimensions the m*e4 binding energy is R* = ___ - 5.8 meV 2h2 E2 in GaAs. The same problem in two dimensions gives a binding energy equal to 4R*. This exact value is of interest in understanding the trends as a function of the quantum well thickness. The first calculation of this problem was performed variationally (Bastard, 1981) for a quantum well bounded by two infinite barriers. The variational wave function was written as v(')=Xl(z)exP[-ud7GGF] (1- 175) where z is the direction perpendicular to the layer, zithe impurity position, @thein plane distance from the impurity, and xl(z) the state of the ground quantum well sub-band. As expected, the resulting ground state binding energy increases from its 3D value as the thickness decreases. This method runs into difficulties in the small thickness limit where effective mass theory should not apply in the z-direction. In this case, it is more appropriate to perform a strict 2D application of effective mass theory where the variational function is taken as a product of the exact Bloch function at the bottom of the lowest sub-band times exp-cxg (Priester et al., 1984). This gives the 2D limit exactly by construction and is valid as long as the binding energy is smaller than the intersub-band separation. Finally, the case of acceptor impurities is extremely complex and we do not discuss it here (references can be found in Altarelli, 1986; Bastard, 1988). It is interesting to look at the exciton states and the corresponding optical absorption in systems with reduced dimensionality. Let us first consider a quantum well and

62

1 Band Theory Applied to Semiconductors

the simple effective mass equation, Eq. (1 - 1 12), for the exciton envelope function. The problem is complicated by the fact that there is confinement in the z-direction. There can still be free motion of the center of mass in the directions x and y parallel to the layer. The ground state obviously corresponds to no such motion and two variational wave functions for this state have been sought (Bastard, 1988): F(Ze, Z h , @ ) = N l

or

Xl,e(ze)Xl,h(Zh)eXP--

Q il

(1 -176)

where Q = J(xe - x h ) 2 + (ye - y h ) * , N~ and N2are normalization constants, ilis the variational parameter and X1e(ze) and X1h(Zh) are the lowest states in each quantum well in the absence of electron hole interaction. With these trial wave functions one gets the result that the binding energy increases when the width L of the well decreases to reach a limiting value of four times the bulk one, as for hydrogenic impurities. It is interesting to calculate the oscillator strength by using the same method as in Sec. 1 . 3 . 2 . Using for x1and x2 a sin ( k z ) form, one readily obtains from Eq. (1 - 117)

For strong confinement, iltends to its 2D limiting value a / 2 so that this ratio becomes equal to I

12

(1-180)

where L is of the order of the interatomic spacing. This ratio can thus become, for G A S , as large as 300 showing that the 2D confined exciton has much more oscillator strength that the bulk one. This is confirmed by experimental observations (Dingle et al., 1974). Let us finally try to extend this to the quantum box. We then consider a spherical box with infinite potential at the boundary r = R. We only consider the limit of strong confinement where the kinetic energy -- terms dominate the electron-hole attraction (they scale respectively as 1/R2 and UR). The latter can thus be included in the first order perturbation theory. At the lowest order, the envelope function for the lowest excitonic state takes the form F(re,rh)=N

sin ( k re ) sin ( k rh) re

(1-181)

rh

with k=Jt/R. One can now evaluate IMo,exc,12for this OD case from the general expression Eq. (1 -1 17). This gives IM0,exc.12

2

=1~vc1

( 1 - 1 82)

(1-178)

Thus the strength of this exciton reIative to the bulk one for the same volume of material given here by V = $ n R 3 is equal to

Thus the relative strength of optical absorption between the quantum well and bulk exciton is, for the same volume V = S L of material, given by

(1-183)

(1-179)

which is the result of Kayanuma (1988). Again the bulk exciton radius is a = 100 A while in the limit of very strong confinement R = 10 or less. This means that the

63

1.7 Systems with Lower Dimensionality

enhancement is still larger than in quantum wells.

1.7.4 Silicon Quantum Dots It is now well accepted that the luminescent properties of porous silicon (Psi) (Canham, 1990) are at least partly due to quantum confinement effects. Indeed this complex material is thought to be composed of nanocrystallites between which electron and holes can tunnel. This means that direct optical transitions which are forbidden in the bulk crystalline material become allowed due to breakdown of the k selection rule. We thus review here the theoretical results for idealized silicon crystallites and compare them with experimental data. We mainly concentrate on the predictions for the band gap versus size (blue shift) and the fine structure of the levels related to exchange splitting. We also consider the case of amorphous Psi. We finally review other predictions for the existence of selftrapped excitons, the radiative and nonradiative recombination rates, and finally the dielectric properties of such crystallites. Let us start with ideal spherical crystallites. Their electronic structure has been calculated using different techniques: the effective mass approximation (EMA), empirical tight binding (ETB), empirical pseudopotential (EPS) and finally the “ab initio” local density approximation (LDA). As discussed before, EMA is a continuous approximation which reduces the crystallite to a finite spherical potential well. All other methods require specification of the atomic structure, which is taken as a spherical silicon atoms cluster with its surface dangling bonds saturated by hydrogen atoms. Figure 1-39 gives what are believed to be the best predictions for the energy gap versus size. This gap is defined as the difference in energy between the LUMO (low-

bulk

I

I

I

I

1/6

1/3

1/2

1

lld (nm-’)

Figure 1-39. Energy gap versus confinement parameter lld for hydrogen terminated clusters: ETB results (Proot et al., 1992; Delerue et al., 1993) (continuous line), EPS (Wang et al., 1994) (X), LDA results of (Delley et al., 1995) (00) and Hirao et al., 1993) (+).

est unoccupied molecular orbital) and HOMO (highest occupied molecular orbital). The strikingly good, agreement between the ETB (Proot et al., 1992; Delerue et al., 1993), the EPS (Wang et al., 1994), and LDA predictions can be noticed (Delley et al., 1995; Hirao, 1993). (Note that the LDA gap values include a rigid shift of 0.6 eV, since this method underestimates the bulk band gap by this amount.) The reason why these three quite different methods give identical results is related to the fact that: i) they produce a bulk band structure of similar quality over an energy range of 20 eV, and ii) they correctly treat the boundary conditions at the cluster surface (Delerue et al., 1996). This is not the case for poorer ETB fits like the sp3s* (Hill et al., 1995) which underestimates the gap while the EMA leads to an overestimation. Figure 1-40 gives the comparison between theory and optical data. For this the calculated gap has to be corrected by the

64

1 Band Theory Applied to Semiconductors

(Calcott et al., 1993; Suemoto, 1993) of some samples. In particular, an onset of a few millielectronvolts is observed. In addition, the lifetime of the luminescence decreases when going from low (4 K) to higher temperatures (100 200 K) in parallel with an increase in the luminescence intensity (Calcott et al., 1993; Zheng et al., 1992; Vial et al., 1993). Both effects have been interpreted on the basis of a "two level" model, the lower state being a spin triplet, the upper one a singlet, and the corresponding splitting being due to excitonic exchange (Calcott et al., 1993; Zheng et al., 1992; Fishman et al., 1993). However, afull ETB calculation, including spin-orbit and the excitonic exchange, has shown that this level structure is not obtained for spherical crystallites (Martinet al., 1994). This is due to the strong degeneracies of near band-gap states for silicon. It is only when considering strongly asymmetric crystallites that some splitting and quenching occurs in such a way that the two level model becomes justified, in agreement with experiment. Figure 1-41 compares the predictions for the exchange splitting [corrected with respect

2.2 -

$ 2.0 3 h

2

1.8 1.6 -

1.4 -

-

PL

Pi

I

'

1.21 " " ' 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Diameter (nm) I

'

Figure 1-40. Compilation (Lockwood, 1994), of optical bandgaps of silicon crystallites and porous silicon samples obtained from optical absorption (empty symbols) and luminescence (full symbols and also the more recent results (Ehbrecht et al., 1997)). The dashed and continuous lines represent the calculated values (Proot et al., 1992; Delerue et al., 1993) with and without the excitonic correction.

electron-hole attraction (Proot et al., 1992; Delerue et al., 1993). A long-standing puzzle is the large difference between optical absorption and previous luminescence energies (much larger than the expected Stokes shift of the order of 25 meV for a gap of 2.6 eV). As shown on Fig. 1-40, the optical absorption data practically coincide with the predictions, while the previous luminescence data compiled in Lockwood (1994) are much lower with a weaker dependence on size. However, one group (Ebrecht et al., 1997) has reported luminescence measurements which are now quite close to the theoretical values. Finally, quite recent combined theoretical and experimental studies conclude that the large Stokes shift can be attributed for oxidized samples to the existence of Si=O bonds at the interface (Wolkin et al., 1999). Although porous silicon is a heterogeneous material on a microscopic scale, a fine structure clearly appears in the excitation spectrum of the visible luminescence at 2 K

__ +

40-

+

9 2 30-

+

v

OM 'f! 20 -

x

+ X

A 0

A

L

.

.3 3

a rn -

10

-

0 1.5

1.7

1.9

2.1 2.3 Gap (eV)

2.5

2.7

Figure 1-41. Excitonic exchange splitting; experimental values: full symbols; calculated values (empty symbols) for different orientations of the asymmetric crystallites (Martin et al., 1994).

1.8 References

to those of Martin et al. (1994) by using unscreened exchange interactions] versus size, with the two sets of experimental values showing pretty good agreement. Complete interpretation of the experimental data for the temperature behavior would require detailed analysis of the population of more excited states. We briefly review here other calculations performed on silicon crystallites:

- Amorphous silicon clusters: Many authors have considered that such clusters are not likely to give rise to quantum confinement and a blue shift of the gap. ETB calculations show, on the contrary, that amorphous hydrogenated silicon clusters give practically the same blue-shift dependence on size as the crystallites (Allan et al., 1997). - Recombination rates: The radiative recombination rates have been calculated (Proot et al., 1993; Delerue et al., 1993) to be 1-10 m s-'. Nonradiative recombination via surface neutral dangling bonds was also shown (Proot et al., 1993; Delerue et al., 1993) to eliminate the luminescence of the corresponding crystallites, whereas charged dangling bonds will luminesce in the infrared, as observed. Finally, Auger recombination is fast, in the nanosecond range, as calculated in Delerue et al. (1995) and Mihalescu et al. (1995). - Self-trapped excitons: Some aspects of broadband luminescence might be ascribed to the possibility of forming selftrapped surface excitons [as calculated in Allan et al. (1 996)] by high energy photo excitation. - Dielectric properties: It has been demonstrated (Lannoo et al., 1995) that the effective dielectric constant of nanometersized crystallites is much smaller than the bulk value.

65

1.8 References Albanesi, E. A., Sferco, S. J., Lefebvre, I., Allan, G., Hollinger G. (1992), Phys. Rev. B46, 13260. Albrecht, S., Onida, G., Reining, L. (1997),Phys. Rev. B55, 10278. Allan, G., Lannoo, M. (1983),J. Phys. (Paris),44, 1355. Allan, G., Delerue, C., Lannoo, M. (1996), P hys. Rev, Lett. 76, 2961. Allan, G., Delerue, C., Lannoo, M. (1997), Phys. Rev. Lett. 78. 3161. Allan, G.,.Delerue, C., Lannoo, M. (1998), Phys. Rev. B57. 6933. Allan, G., Delerue, C., Lannoo, M. (1999), unpublished. Altarelli, M. (1986), in: Heterojunctions and Semiconductor Superlattices: Allan, G., Bastard, G., Boccara, N., Lannoo, M., Voos, M. (Eds.). Heidelberg: Springer Verlag, p. 12. Ando, T., Wakahara, S., Akera, H. (1989). Phys. Rev. B40, 11 609, Ando, T., Akera, H. (1 989), Phys. Rev. B40, 1 1 6 19. Bachelet, G. B., Hamann, D. R., Schliiter, M. (1982), Phys. Rev. B26, 4199. Baraff, G. A., Schliiter, M. (1994), Phys. Rev. B30, 3460. Bar-Yam, Y., Joannopoulos, J. D. (1986), Phys. Rev. Lett. 56, 2203. Bassani, F., Iadonisi, G., Preziosi, B. (1974), Rep. frog. Phys. 37, 1099. Bastard, G. (1981),Phys. Rev. B24, 4714. Bastard, G. (1988), Wave Mechanics applied to Semiconductor Heterostructures. Paris: Les Editions de Physique. Bergignat, E., Hollinger, G., Chermette, H., Pertosa, P., Lohez, D., Lannoo, M., Bensoussan, M. (1988), Phys. Rev. B37,4506. Blase, X., Zhu, X., Louie, S . G. (1994), Phys. Rev. B49,4973. Bourgoin, J., Lannoo, M. (1983), Point Defects in Semiconductors II, Springer Series in Solid State Science, Vol. 35, Berlin, Heidelberg, New York: Springer Verlag. Brower, K. L. (1983), Appl. Phys. Lett. 43, 1 I 1 1. Calcott, P. D. J., Nash, K. J., Canham, L. T., Kane, M. J., Brumhead, D. (1993), J. Phys.: Condens. Marter 5, L9 l . Canham, L. T. (1990), Appl, Phys. Let?. 57, 1046. Caplan, P. J., Poindexter, E. H., Deal, B. E., Razouk, R. R. (1979),J.Appl. Phys., 50, 5847. Cardona, M., Pollak, F. H . (1966), Phys. Rev. 142, 530. Chadi, D. J. (1977),Phys. Rev. B16, 3572. Chaney,R.C.,Lin,C.,Lafon, E. E. (1971), Phys. Rev. B3, 459. Chelikowski, J . R., Schliiter, M. (1977), Phys. Rev. B 15,4020. Chelikowski, J . R., Wagener, T. J., Weaver, J. H., Jin, A. (1989), Phys. Rev. B40, 9644.

66

1 Band Theory Applied to Semiconductors

Chen, A.B., Sher, A. (1981), Phys. Rev. B23, 5360. Chen, A, B., Sher, A. (1982), J. Vac, Sci. Technol., 21,138. Cohen, M . L. (1983), Proc. Enrico Fermi Summer School, Varenna. Cohen, M . L., Berstresser, T. K. (1966), Phys. Rev. 141, 789. Cohen, M. L., Chelikowski, J. R. (1988), Electric Structure and Optical Properties of Solids. New York: Springer Verlag. Cohen, J. D., Lang, D. V. (1982), Phys. Rev. B25, 5285. Cohen, M . L., Heine, V., Weaire, D. (1970), Solid State Physics: Ehrenreich, H., Seitz, F., Turnbull, D. (Eds.), New York, London: Academic Press, Vol. 24. Delerue, C . , Allan, G., Lannoo, M. (1993), Phys. Rev. B48, 11 024. Delerue, C., Lannoo, M., Allan, G., Martin, E., Mihalescu, I., Vial, J. C., Romestain, R., Miiller, F., Bsiesy, A. (1995), Phys. Rev. Lett. 75, 2228. Delerue, C., Lannoo, M., Allan, G. ( 1 996), Phys. Rev. Lett. 76, 3038. Delley, B., Steigmeier, E. F. (1995), Appl. Phys. Lett. 67, 2370. Dingle, R., Wiegmann, W., Henry, C . H. (1974), Phys. Rev. Lett., 33, 827. Di Stefano, T. H., Eastman, D. E. (1972), Phys. Rev. Lett., 29, 1088. Dresselhaus, G., Kip, A. F., Kittel, C. (1955), Phys. Rev. 98, 368. Ehbrecht, M., Kohn, B., Huisken, F., Laguna, M. A,, Paillard, V. (1997), Phys. Rev. B56, 6958. Euwema, R. N., Wilwhite, D. L., Suiratt, G. T. (1973), Phys. Rev. B 7, 818. Faulkner, R. A. (1968), Phys. Rev., 175, 991. Faulkner, R. A. (1969), Phys. Rev., 184, 713. Fishman, G., Romestain, R., Vial, J. C. (1993), J. Phys. IV3, 355. Gillespie, R. J. (1972), Molecular Geometry. London: Van Nostrand Reinhold, Corn. LTD, p. 6. Gillespie, R. G., Nyholm, R. (1957), Quart. Rev. Chem. Soc., 11, 339. Godby, R. W., Schliiter, M., Sham, L. J. (1986), Phys. Rev. Lett., 56, 2415, and (1987). Phys. Rev. B35, 4 170. Greenaway, D. L., Harbeke, G. (1968), Optical Properties and Band Structure of Semiconductors, Oxford: Pergamon Press. Hamann, D. R. (1979), Phys. Rev. Lett., 42, 662. Hamann, D. R. Schliiter, M., Chiang, C. (1979), Phys. Rev. Lett., 43, 1494. Harrison, W. A. (1973), Phys. Rev. B8, 4487. Harrison, W. A. (1976), Surf: Sci., 55, 1. Harrison, W. A. (1980), Electronic Structure and The Properties of Solids. The Physics of the Chemical Bond. New York: Freeman. Harrison, W.A. (1981), Phys. Rev. B24, 5835. Haydock, R. (1980), in: Solid State Phys: Ehrenreich, H., Seitz, F., Turnbull, D. (Eds.), New York: Academic Press, Vol. 35, p. 215.

Hedin, L., Lundquist, S . (1969), Solid State Physics, 23, 1. Henderson, B. (1984), Appl. Phys. Lett., 44, 228. Hill, N. A., Whaley, K. B. (1995), Phys. Rev. Lett. 75, 1130. Hirao, M., Uda T., Murayama, Y. (1993), Muter. Res. SOC.Proc. 283, 425. Hohenberg, P., Kohn, W. (1964), Phys. Rev., 136 B, 864. Hollinger, G., Jugnet, Y., Tran Minh Duc (1977). Sol. State Comm., 22, 277. Hulin, M. (1966), J. Phys. Chem. Sol., 27, 441. Hybertsen, M. S., Louie, S . G. (1985), Phys. Rev. Lett., 55, 1418. Hybertsen, M. S., Louie, S. G. (1986), Phys. Rev. B34, 5390. Ibach, H., Rowe, J. E. (1974), Phys. Rev. BlO, 710. Ibanez, A., Olivier-Fourcade, J., Jumas, J. C., Philippot, E., Maurin, M. (1986), Z. Anorg. Allg. Chem. 6, 5401541, 106. Jackson, W. B. (1982), Solid State Comm., 44, 477. Johnson, N. M., Biegelsen, D. K., Moyer, M. D., Chang, S . T., Poindexter, E. H., Caplan, P. J. (1983), Appl. Phys. Lett., 43, 563. Johnson, N. M., Jackson, W. B., Moyer, M. D. (1985), Phys. Rev. B31, 1194. Kane, E. 0. (1956), J. Phys. Chem. Sol. I , 82. Kane, E. 0. (1957), J. Phys. Chem. Sol. I , 249. Kane, E. 0. (1976), Phys. Rev. B13, 3478. Kayanuma, Y.(1988), Phys. Rev. B38, 9797, Kittel, C . (1963), Quantum Theory of Solids. New York: Wiley and Sons Inc. Kittel, C., Mitchell, A. H. (1954), Phys. Rev. 96, 1488. Knief, S., Von Niessen, W., Koslowski, T. (1998), Phys. Rev. B58, 4459. Knox, R. S . (1963), Theory of Excitons, Sol. State Phys., Supplt. 5: Seitz, F. and Turnbull, D. (Eds.) New York: Academic Press. Kohn, W.,Sham,L. J . (1965), Phys. Rev. 140A, 1133. Lannoo, M . (1973), J. de Phys., 34, 869. Lannoo, M., Allan, G. (1978), Sol. State Comm., 28, 733. Lannoo, M., Allan, G. (1982), Phys. Rev. B25, 4089. Lannoo, M., Decarpigny, J. N. (1973), Phys. Rev. B 8 , 5704. Lannoo, M., Schliiter, M., Sham, L. J. (1985), Phys. Rev. B32, 3890. Lannoo, M., Delerue, C., Allan, G. (1995), Phys. Rev. Lett. 74, 3415. Leman, G., Friedel, J. (1962), J. Appl. Phys. Supplt. 33, 281. Lefebvre, I., Lannoo, M. (1997), Chem. Matel: 9, 2805. Lefebvre, I., Lannoo, M., Allan, G., Ibanez, A., Fourcade, J., Jumas, J. C., Beaurepaire, E. (1987), Phys. Rev. Lett., 59, 2471. Lefebvre I., Lannoo, M., Allan, G., Martinage L. (1988), Phys. Rev. B38, 8593. Lefebvre, I., Lannoo, M., Olivier-Fourcade, J., Jumas, J. C. (1991), Phys. Rev. B44, 1004.

1.8 References

Lefebvre, I., Szymanski, M. A., Olivier-Fourcade, J., Jumas, J. C (1998), Phys. Rev. B58, 1896. Lempert, R. J., Hass, K. C., Ehrenreich, H. (1987), Phys. Rev. B36, 1 1 1 1 . Lockwood, D. J. (1994), Solid State Commun. 92, 101. Louie, S. G. (1980), Phys. Rev. B22, 1933. Louie, S. G. (1996). Quantum theory of real materials, Chelikowsky, J. R. and Louie, S. G. (Eds.), Kluwer, Boston, p. 83. Luttinger, J. M., Kohn, W. (1955), Phys. Rev. 97, 869. Luttinger, J. M. (1956), Phys. Rev. 102, 1030. Maley, N., Beeman, D., Lannin, J. S. (1988), Phys. Rev. B38, 10611. Marinov, M., Zotov, N. (1996),Phys. Rev. B55, 2938. Martin, E., Delerue, C., Allan, G., Lannoo, M. (1994), Phys. Rev. B50, 18258. Martinez, E., Yndurain, F. (1981), Phys. Rev. B24, 5718. Mauger, A., Lannoo, M. (1977), Phys. Rev. B15, 2324. Mihalescu, I., Vial, J. C., Bsiesy, A,, Miiller, F., Romestain, R., Martin, E., Delerue, C., Lannoo, M., Allan, G. (1995), Phys. Rev. B 51, 17 605. Mikkelsen, J. C. Jr., Boyce, J. B. (1982), Phys. Rev. Lett., 49, 1412. Mikkelsen, J. C. Jr., Boyce, J. B. (1983), Phys. Rev. B28,7130. Mikkelsen, J. C. Jr. (1984), Bull, Am. Phys. SOC.,29, 202. Motta, N. et al. (1985), Sol. State Comm., 53, 509. Northrup, J. E., Hybertsen, M. S., Louie, S. G. (1991), Phys. Rev. Lett. 66, 500. Northrup, J. E. (1993), Phys. Rev. B47, 10032. Nozikres, P. (1964), Theory of Interacting Fermi Systems, Benjamin, New-York. O’Reilly, E. P., Robertson, J. (1983), Phys. Rev. B27, 3780. Pantelides, S . T. (1978). Rev. Mod. Phys., 50, 797. Pantelides, S. T. (1986), Phys. Rev. Lett., 59, 688. Pantelides, S. T., Hamson, W. A. (1976), Phys. Rev. B 13, 2667. Petit, J., Lannoo, M., Allan, G. (1986), Sol. State Comm., 60, 861. Perdew, J., Levy, M. (1983), Phys. Rev. Lett., 51, 1884. Philippot, E. (1981), J. Solid State Chem., 38, 26. Poindexter, E. H., Caplan, P. J. (1983),Prog. Surf. Science, 14, 201. Priester, C., to be published. Priester, C., Allan, G., Lannoo, M. (1983), Phys. Rev. B28. 7194. Priester, C . , Allan, G., Lannoo, M. (1984), Phys. Rev. B29, 3408. Proot, J. P., Delerue, C., Allan, G. (1992),Appl. Phys. Lett. 61, 1948. Resca, L., Resta, R. (1979),Sol. State Comm., 29,275. Resca, L. Resta, R. (1980), Phys. Rev. Lett., 44, 1340. Robertson, J. (1983), Advances in Phys., 32, p. 383. Rohlfing, M., Kriiger, P., Pollmann, J. (1993), Phys. Rev. B48. 17791.

67

Sawyer, J. F., Gillespie, R. J. (1987), Progress Inorg. Chem., 34, 65. New York: Intersciences Publi. J. Wiley and Sons. Schiff, L. I. (1955), Quantum Mechanics, 2nd ed. New York: McGraw-Hill, p. 158. Schliiter, M. (1983), Proc. Enrico Fermi Summer School, Varenna. Schliiter, M., Joannopoulos, J. D., Cohen M. L. (1974), Phys. Rev. Lett., 89, 33. Sham, L. J., Rice, T. M. (1966), Phys. Rev. 144, 708. Sham, L. J., Schliiter, M. (1983), Phys. Rev. Lett., 51, 1888. Sham, L. J., Schliiter, M. (1985), Phys. Rev. B32, 3883. Shevchik, N. J., Tejeda, J., Cardona, M., Langer, D. W. (1973), Sol. State Comm.,12, 1285. Slater, J. C. (1960), Quantum Theory ofAtomic Structure, Vol. 1. New York: McGraw-Hill. Slater, J. C., Koster, G. J. (1954), Phys. Rev. 94, 1498. Street, R. A., Zesch, J., Thomson, M. J . (1983),Appl. Phys. Lett., 43, 672. Strinati, G. (1984), Phys. Rev. B29, 5718. Stutzmann, M., Biegelsen, D. K. (1989), Phys. Rev. B40, 9834. Suemoto, T., Tanaka, K., Nakajima, A., Itakura, T. (1993), Phys. Rev. Lett. 70, 3659. Talwar, D. N., Ting, C. S. (1982), Phys. Rev. B25, 2660. Vial, J. C., Bsiesy, A., Fishman, G., Gaspard, F., HCrino, R., Ligeon, M., Muller, F., Romestain, R., Macfarlane, R. M. (1993), Microcrystalline Semiconductors-Materials Science and Devices, Fauchet, P. M., Tsai, C. C., Canham, L. T., Shimizu, I., Aoyagi, Y. (Eds.) MRS Symposia Proceedings no 283 (Materials Research Society, Pittsburgh), p. 241. Vogl, P., Hjalmarson, H. P., Dow, J. D. (1983),J. Phys. Chem. Sol., 44, 365. Wang, L. W., Zunger, A. (1994), J. Phys. Chem. 98, 2158. Weaire, D. (1981), in: Fundamental Physics ofAmorphous Semiconductors, Springer Series in Solid State Science, Yonezawa, F. (Ed.). New York: Springer Verlag, p. 155. Weaire, D., Thorpe, M. F. (1971), Phys. Rev. B4, 2508. Wolkin, M. V., Jorne, J., Fauchet, P., Allan, G., Delerue, C. (1999), Phys. Rev. Lett. 82, 197. Wooten, F., Winer, K., Weaire, D. (1989, Phys. Rev. Lett. 54, 1392. Wooten, F., Weaire, D. (1987), Solid State Phys. 40, 1. Zheng, X. L., Wang, W., Chen, H. C. (1992), Appl. Phys. Lett. 60, 986. Ziman, J. M. (19791, Models of Disorder. Cambridge University Press. Zunger, A., Jaffe, J. E. (1983). Phys. Rev. Lett., 51, 662.

Handbook of Semiconductor Technologj Kenneth A. Jackson, Wolfgang Schrotei CoDvriaht 0 WILEY-VCH Verlaa GrnbH. 2000

2 Optical Properties and Charge Transport

. .

R G Ulbrich IV. Physikalisches Institut. Georg.August.Universitat. Gottingen. Germany

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.6 2.6.1 2.6.2 2.6.3 2.7 2.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Optical Properties of Bulk Crystals . . . . . . . . . . . . . . . . . . . . . 71 Band Structure and Electron-Hole Pair Excitations . . . . . . . . . . . . . . 72 Dielectric Polarization and Response Function . . . . . . . . . . . . . . . . 74 Correlated Electron-Hole Pairs: Excitons . . . . . . . . . . . . . . . . . . . 78 79 Single-Particle versus Pair Excitations . . . . . . . . . . . . . . . . . . . . Exciton-Photon Coupling: Polaritons . . . . . . . . . . . . . . . . . . . . . 80 Interband Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . . 83 Direct versus Indirect Transitions . . . . . . . . . . . . . . . . . . . . . . . 83 Radiative Recombination Rate . . . . . . . . . . . . . . . . . . . . . . . . 84 Angular Dependence of Interband Transitions . . . . . . . . . . . . . . . . 85 Fundamental Gap Spectra at Low Excitation . . . . . . . . . . . . . . . . 85 Exciton Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Systems with Restricted Dimensionality . . . . . . . . . . . . . . . . . . 89 Electronic Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Quantum Wells. -Wires and -Dots . . . . . . . . . . . . . . . . . . . . . . 92 Optics of Quantum Confined Systems . . . . . . . . . . . . . . . . . . . . 94 Microcavities and Photonic Band Gaps . . . . . . . . . . . . . . . . . . . 94 Charge Transport and Scattering Processes . . . . . . . . . . . . . . . . 95 Momentum and Energy Relaxation of Carriers . . . . . . . . . . . . . . . 98 Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Carrier-Carrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Coherent Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Spin Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Nonlinear Optics and High Field Transport . . . . . . . . . . . . . . . . 109 110 Nonlinear Dielectric Response . . . . . . . . . . . . . . . . . . . . . . . . Carrier distributions far from equilibrium . . . . . . . . . . . . . . . . . . 111 Carrier Dynamics on Ultrashort Time Scales . . . . . . . . . . . . . . . . 112 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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2 Optical Properties and Charge Transport

2.1 Introduction Interest in charge transport and optical properties of semiconductors dates back half a century. The experiments of Shockley and co-workers upon transport of photo-injected minority carriers in germanium and silicon clarified for the first time the three basic phenomena which are common to electrical transport in semiconductors (Shockle y, 1950): (i) carrier energies exceeding kT, (ii) various types and energy dependence of carrier-lattice coupling, (iii) controllable carrier densities. They pointed at the fundamental importance of electronic band structure for charge transport and optical properties of semiconductors, and how both differ from metals. Their work on Ge and Si soon initiated the exploration of III-V- and II-VI compounds. Scientific progress in this field accelerated considerably when the enormous potential and far-reaching practical applications of semiconductor devices became evident in the late 1950s and early 1960s. The last three decades have brought tremendous diversification to semiconductor physics. Starting with the work of Esaki and coworkers in 1970, earlier visionary ideas (Kroemer, 1985) on carrier confinement in and transport through heterostructures with geometrically designed and chemically “engineered” band structures were developed into real devices (for a recent review, see Esaki, 1993). In these years, steady progress in understanding semiconductors in the broader context of solid state physics was accompanied by an unparalleled development of sophisticated growth and processing techniques for semiconducting materials. The refinement of evaporation techniques, notably layer-by-layer deposition of atoms under ultra-high vacuum con-

ditions, as well as photon-, electron- and ion-beam mediated lithographic procedures led ultimately to the fabrication of manmade nanostructures on the atomic scale. Somewhat less spectacular, but equally important, the fabrication of ultrapure and highly perfect bulk crystal material was improved steadily, so that at present two prototype semiconductors - silicon and gallium arsenide - are by far the best-understood crystalline solids. The fabrication of increasingly complex devices with restricted dimensionality on the nanoscale, as well as the availability of well-defined semiconductor bulk materials, have produced new and unforeseen physical phenomena. Quantum size effects in geometrically confined systems play the key role: Their generic wealth is manifest in the unique properties of so-called quantum wells and superlattices, quantum wires, and quantum dots. Carrier ensembles in such systems, and their peculiar transport properties in external electric and magnetic fields, have now become topics of great practical interest. Spectacular new findings have come about in the context of localization and charge transport in semiconductors: the integral quantum Hall effect in 2-d confined carrier systems (von Klitzing et al., 1980; von Klitzing, 1986); strong Coulomb correlation of carrier motion in such systems leading to the fractional quantum Hall effect (Tsui et al., 1982; Stormer, 1984); quantized conductance in laterally confined quasi-ld configurations (Beenacker and van Houten, 1990), the Coulomb blockade in fully confined systems with few electrons (Kastner, 1992), and Aharonov-Bohm-type interference phenomena in mesoscalic ring structures (Yacoby et al., 1995). During the past two decades, the main focus of scientific interest has shifted from quasi-classical, i.e. particle-like transport phenomena in bulk semiconductors to their coherent,

2.2 Optical Properties of Bulk Crystals

i.e. wave-like and phase-sensitive quantum mechanical counterparts in ensembles with few electrons. Closely related is the growing interest in optical properties of structured semiconductors and their widespread applications in optoelectronic devices. The renaissance of the seemingly mature field of classical optics, which had started with the invention of the laser in 1961, continued with key contributions from semiconductor physics: injection lasers, quantum dot lasers, microcavity devices, efficient photon detectors and emitters in the near infrared, optical nonlinearities and bistability, ultrafast optical switching and coherent pulse generation and propagation are examples. Expanding experimental abilities in synchronization and phase control of ultrashort laser light pulses have caused also here a shift in scientific interest: from studies of incoherent carrier population and energy relaxation effects to the coherent control of polarization waves, with descriptions of scattering beyond perturbative schemes, in an effort to understand the dynamics of optically coupled pairs on the microscopic level. Electrical transport and optical properties of semiconductors have been treated accordingly in many review articles and text books. The aim of the present article is to give a comprehensive survey of some of the basic concepts, to illustrate them in specific situations, and to discuss some of the key experimental results. The level of presentation should help the unfamiliar reader to become aware of the state-of-the-art in the field and assist him or her in evaluating the contemporary original literature. Elementary coverage of electronic band structure and optical transitions, the effects of carrier confinement and carrier scattering, and electrical transport will be given to some extent. This article does not cover the aspects of magneto-optics and magneto-

71

transport, but does contain some selected material on nonlinear transport and nonlinear optics in semiconductors. For further information on these topics, the reader is referred to the monographs and the review articles cited at the end of this chapter.

2.2 Optical Properties of Bulk Crystals The optical properties of bulk crystalline semiconductors in the visible part of the spectrum are closely connected with their electronic band structure. All optical spectra reflect essentially the unperturbed energy spectrum En@) of filled electron states below, and empty ones above the Fermi ( r )n. energy, with wavefunctions ~ # ~ , ~Here is the band index, k the electron wavevector. To a certain degree, details of optical spectra in real crystals are modified by the presence of impurities and structural defects. At finite temperatures T > 0, the agitation of thermal lattice vibrations broadens and shifts the optical spectra. Calculations of electronic band structures - at any level of sophistication - use input parameters which are derived from spectroscopic data. In retrospect, practically all important physical quantities related to semiconductor band structures were extracted from empirical spectra of absorption, emission and reflectivity. The most relevant parameters are fundamental and higher band gaps, the magnitude of momentum matrix elements, symmetries of wavefunctions at special points in the Brillouin zone, and joint density-of-state spectra of relevant bands. A common starting point for the discussion of electronic properties of crystalline solids is the spatially periodic crystal potential V&): one assumes a priori that each crystal electron moves in the field of all

72

2 Optical Properties and Charge Transport

nuclei and all other electrons, and that all electrons see effectively the same potential. More precisely, the potential Vc (r)is supposed to be of Hartree-Fock type, i.e. a selfconsistent mean field (Anderson, 1963; Sutton, 1993). The use of V c ( r ) leads to the concept of the “one-electron’’ approximation: it is taken for granted that all electronic properties of the solid can be described in terms of products of mutually independent one-electron wave functions. Despite the large valence electron density of approxicm-3 in typical semimately ne= 3 x conductors, this concept works surprisingly well and is a good basis for all further treatments of semiconductor properties. Rigorous treatments of the many-electron ground state problem and justifications for the independent-particle description have been systematically developed during the past decades in the framework of densityfunctional theory (Hohenberg and Kohn, 1964; Dreizler and Gross, 1990). Ground state energies and corresponding charge density distributions of all relevant semiconductors have been calculated and found to agree very well with experiment (Payne et al., 1992). Excited states, and hence optical properties, are less well described in the density functional formalism. Though dynamical screening of an excited electronhole pair is difficult to treat, ab-initio calculations of gap energies and optical spectra in good agreement with experiment begin to emerge (Pulci et al., 1997; Aryasetiawan and Gunnarson, 1998; Rohlfing and Louie, 1999).

2.2.1 Band Structure and Electron-Hole Pair Excitations In this paragraph the connection between single-electron band structures En ( k ) and fundamental optical spectra of semiconductors is described on an elementary level.

Excitonic effects and screening effects connected with high e-h pair densities will be discussed further in Ch. 2.3. Filled valence bands, empty conduction bands, and a band gap of width E+kT between them define the ground state of the N-electron system “semiconductor” with energy E,. Since all N electrons may conduct a current, in principle, the terms valence- and conduction band are not meaningful in this context. What counts is the distribution of occupied and empty states in each energy band En( k ) with band index n. An electron in an otherwise empty band, and a hole in a filled band are both charge “carriers”. They are the electronic excitations which constitute the response of the semiconductor to external electric fields, i.e. charge transport and optical properties. An excited state E* of the N-electron system may be created by taking first an electron out of a valence band state, and putting it into an empty conduction band state (see Fig. 2-1). This procedure defines the creation of an electron-hole (e-h) pair. N +N+1

,

electron

e-

\

\ o N

* N-1

hole

Figure2-1. An excited state of the N-electron system (left). The energy-wave vector diagram (right) shows valence band hole and conduction band electron with their kinetic energies EEL and EE!. The total energy E* of the excitation includes the potential energy EG (see text).

2.2 Optical Properties

The total energy change AE connected with this process is the sum of four terms: (i) potential energy of the electron with respect to Vc(r). It is called “gap energy” EG and corresponds to the change in band index n (the analog of the atomic orbital quantum number) ; (ii) kinetic energy EL!: of the initial electron state, resp. the final hole state; (iii) kinetic energy Eg: of the final electron state, resp. the initial hole state; (iv) the interaction energy Vi,, between the excited electron and the remaining hole. At this level of the single-particlescheme, any rearrangement of the valence electron density caused by the excited e-h pair is neglected. We may therefore write (2-1) For wide enough bands and sufficiently small kinetic energies of the carriers the socalled effective-mass-approximation (EMA) applies (Cohen, Chelikowski, 1988; Yu and Cardona, 1996; Chow and Koch, 1999), and the kinetic energies are simply given by + Vim-&

and are counted from the band edges E, and

E,. The coefficients m: and mz represent

the effective masses of electrons and holes (see also Ch.1.14). Electrons and holes of given wave vectors k, and kh are distinguishable particles if their band index n is different. They can be described by fermion creation and annihilation operators, which act on the “Fermi seas” of filled valence and empty conduction band states (Haug and Koch, 1994). If we want to go further than counting numbers, we have to study wavefunctions, and the discussion has to be more specific

of Bulk Crystals

73

(Harrison, 1980; Yu and Cardona, 1996). The prototype semiconductors which we have in mind are crystalline solids with tetrahedral coordination. This class includes the group IV element Ge, the group 111-V compound GaAs, and the 11-VI compound ZnSe. These three prototypes have their chemical homologues, which appear in the columns of the periodic table of elements: the monatomic C, Si, Ge family, the diatomic semiconductors InP, InAs, InSb, and CdS, CdSe, ... etc. It turns out that these materials, which all crystallize in either diamond-, zinkblendeor wurtzite-lattice structures, have many basic features in common: their non-bonding conduction band states at and around E, have the symmetry of atomic L=O or “s-like” orbitals, derived from the symmetric linear combination of sp3 hybrids on neighboring sites, with even parity in the diamond case, and twofold degeneracy (for the electron spin). Their effective mass is usually small compared to mo,i.e. mz/m,< 1. The valence band edge at energy E, has eigen-functions similar to “p-like” orbitals. They are derived from the anti-symmetric linear combination of sp3 hybrids, with odd parity in diamond structure. Its sixfold degeneracy is split by the spin-orbit interaction such that the j=3/2 terms lie usually above the j =1 / 2 split-off level (Kane, 1966). In the diatomic zinkblende and wurtzite lattice structures the potential V, (r) and hence the valence electron density @ (r) is somewhat different on the cation and the anion site. This asymmetry causes characteristic linear terms in the E,(k) relation at k = O (Kane, 1957; Cardona et al., 1986; Yu and Cardona, 1996). Away from the zone center k = O the valence band extremum is split by the k . p interaction into heavy and light hole bands, because of the spatial anisotropy of the sp3 hybrid wave functions which form the

74

2 Optical Properties and Charge Transport

in each lattice cell: “s” Electrons

:,

I=-

m,=*-

1 2

“p,” Heavy Holes 3

m,=+?3

“P,’ i Pvy Light Holes

J”3

2’

___*

m =*-1 I 2

z, k,

Figure 2-2. Cell-periodic wave functions uk(r) in a cubic lattice for wavevector k 11 z. Only the spatial part is shown; the spin parts I o ) are chosen to be aligned in z-direction (schematic).

qn,k ( r )= ~ - 1 ’ 2 eik.r uk ( r )

(2-3)

Common basis functions for the symmetry classification of the cell-periodic part of the wavefunction, uk (r), are either the atomic orbitals “s”, “p”, “d” etc. in Cartesian coordinates, or, including spin, the eigenfunctions q,,, belonging to the total angular momentum operators J =L + S, with j = 3/2 or 1/2. The overlap integrals between wave functions in neighboring cells for a given wave vector direction (i.e. chosen quantization axis) depend on the component mjof J in that direction. This is the physical origin of the difference between heavy and light hole mass; it can be inferred qualitatively from Fig. 2-2. Lucid discussions of crystal symmetry and band degeneracy can be found in Tinkham (1964), Bassani et al. (1975), and Yu and Cardona (1996).

2.2.2 Dielectric Polarization and Response Function tetrahedrally coordinated bonds (Harrison, 1980; Sutton, 1993). For the same reason, the effective masses at k = O and beyond are anisotropic, in general. With increasing k, the effective mass parameters m $ usually become larger. In the wurtzite structure (with a different bond length along the c-axis), the valence bands are nondegenerate at k = 0. The basic shapes of cell-periodic wave functions and their nearest neighbor overlap are shown qualitatively in Fig. 2-2. A simple 2-dimensional quadratic lattice representing fourfold coordination in a monatomic crystal has been chosen for illustration. The proper eigenstates of both electrons and holes in the periodic crystal potential V, (r)are usually described by Bloch-wavefunctions. They extend over the entire crystal volume, with equal probability density

Optical studies of semiconductors probe their dielectric polarizability. At sufficiently high frequencies the response of any solid is dominated by electronic transitions. When static or low frequency electric fields act on a diatomic, (i.e. polar) semiconductor, there is an additional effect: the two sublattices in zinkblende (or wurtzite) structure will move with respect to each other and give a lattice contribution to the induced polarization. In the following we will restrict ourselves to excitation frequencies that are high in comparison to the vibrational frequencies w,b of the lattice. The natural frequency w, of a simple mass-spring system falls in the infrared part of the spectrum for all common values of lattice force constants D and atomic masses A4 in semiconductors (see, e.g. Ashcroft and Mermin, 1976). D is directly related to

75

2.2 Optical Properties of Bulk Crystals

the inverse bulk modulus and the inter. atomic distance (bond length) d through

E=O, p=O

(2-4)

We find that the oo is in the range 0.32 x 1014 ... 2.4 x 1014 rad/s when we go from the heaviest 11-VI compound CdTe to the lightest group IV crystal diamond (C). In the visible part of the spectrum ( w =3 x 1015rad/s at 2 ev), o , i b is fulfilled and we may safely neglect the contribution of lattice polarizability, because the nuclei cannot follow the driving field. We will now discuss the dielectric properties of semiconductors as a function of incident photon energy Am, or light frequency. The low frequency limit, or “static” case, implies frequencies which are small compared to characteristic electronic excitation energies. Lattice effects will not be considered further. Let us consider as the driving force a spatially uniform electric field Boa exp (-i o t) which interacts with all N electrons in the crystal. The induced polarization p of electronic charge in the directed bonds and (to a much lesser degree) in the inner shells of the atoms will, in general, cause a counteracting and non-uniform electric field in the solid and hence reduce the “effective” field at the atomic sites. The importance of such local field corrections depends on the degree of ionicity and the localization of the bond charges. The reader is referred to the article by Lannoo in this volume for a thorough discussion (see also Resta, 1983; Baldereschi et al., 1993). For simplicity we neglect here the local field effect altogether and keep in mind that the assumption of a uniform electric field implies weak polarizability of electronic wavefunctions (see Fig. 2-3). The electronic polarization density P within the crystal is the sum of all induced

. .

. .

. .

+

. .

E

Figure 2-3. Charge density distribution around nuclei in periodic crystal with no external electric field (upper) and induced dipole momentsp =xkE caused by external electric field (lower).

polarizations pi per volume:

P=

v-’cpi

(2-5)

i

The electric field due to the crystal potential V , (r)close to the nucleus is - 10’ V/cm. Provided that the external E , is much smaller than this, it is reasonable to assume that the induced P is proportional to E,. This basic assumption defines the regime of “linear optics” in solids. For ordinary incoherent light sources, where power densities 1

S = - &2 o

ClE,)*

do not exceed a couple of kilowatts per cm’, the condition is perfectly well fulfilled:

Eo= 19 V/cm

for

S = 1 W/cm2

(2-7)

Only laser light sources offer the possibility of entering the nonlinear regime. Focused and short-pulsed lasers are now routinely used to study the nonlinear optical response of solids (see Sec. 2.6 below). The proportionality constant between P and E is the dielectric polarizability or

x,

76

2 Optical Properties and Charge Transport

phase component to x2 :

“response function”, defined by

P=E,x(o) E

(2-8)

in the frequency domain, with sinusoidally varying fields

E=E,.

(2-9)

e-’W‘

Here it is understood that only the real part of E represents the actual electric field. Summing over frequencies and including appropriate phases in Eq. (2-9) allows to describe any time dependence of E ( t )without loss of generality. x ( w ) will, in general, be a complex function of w

x ( w ) = x, (0) +i

x 2 (0)

(2-10)

with an in-phase “dispersive” term x1 and an out-of-phase “absorptive” term x2. We can extract the dielectric response from the electronic band structure as follows: the driving field E is treated as a small perturbation H ’ to the Hamilton operator Ho. The latter contains V, (r)and stands for the unperturbed crystal. The vector potential is chosen such that

E = - c-’

at

= “w A

c

(2-1 1)

for the plane wave of Eq. (2-9). We add the field momentum e .A / c to the momentum operator p in H,. To lowest order in A the coupling term between a single electron and the light field

H r = - ie. A = - -

eA

E-V, (2-12) mo Here mo is the free electron mass. The next term is proportional to A2 and is neglected here (Bassani and Pastori Paravicini, 1975). The incorporation of Eq. (2-12) into timedependent perturbation theory is straightforward. With the aid of Fermi’s “Golden Rule” one obtains the induced current density in the crystal and relates its out-of-

*

S (E, ( k )- E, ( k )- A W )

(2-13)

V is the volume appropriate for normalization of q9n,k(r),the sum includes all wavevector pairs (c,k; v, k ) which fulfil the condition that the initial state is full and the final state is empty (with a plus sign), or vice versa (with a minus sign). The driving field is chosen along x-direction, and P is taken to be parallel to E . Equations (2-8)(2-13) are the simplest and qualitatively correct semiclassical approach to the linear optical response of semiconductors: (i) absorption of light is mediated by the linear coupling between E and single electron momentap in Eq. (2-12); (ii) the sum in Eq. (2-13) contains transitions from occupied valence to unoccupied conduction band states (“upward transitions”) and their time-reversed counterparts (“downward transitions”). With the appropriate signs, Eq. (2-13) describes both fundamental optical processes of induced absorption and emission.

,

I

0

0

‘,

J

/A

0

‘LO

m0

Figure 2-4. Light absorption from the crystal ground state, x z > O (left), and light emission from excited state, xz
77

2.2 Optical Properties of Bulk Crystals

The extension towards a full quantum mechanical description is conceptually simple: the vector potential A is quantized and the elementary process of photon absorption (emission) occurs only if the two coupled states (c,k ; v, k ) differ by Aw in energy. Since the photon momentum Ac q - lo5 cm-’ in the visible part of the spectrum is small on the scale k,,, = n/u0- 6 x 1O7 cm-’ of electron quasi-momenta A k, one speaks of “vertical” transitions in k-space. The matrix element of Eq. (2-13) may be rewritten by using the relation

which is valid for localized electron orbitals (Sutton, 1993). Equation (2-14) connects the operator for the dipole moment, e with ~ ( w )and , allows to substantiate the intuitive interpretation of Fig. 2-3: the process of optical absorption may be seen as the time-proportional admixture of “s-like” wave function amplitude to the initial “plike” valence band wave function mediated by the dipole moment operator 2 . The superposition of both eigen-states produces an oscillatory motion of the bond charges with amplitude x out of their equilibrium positions x = O . Inspection of Eqs. (2-5) and (2-14) shows that the expectation value of x corresponding to one optically coupled e-h pair per site in the crystal is ex^,

*I( l&l

2

c v)i Eo cos (w t ) (2-15) mi w 3 It is convenient to introduce the momentum matrix element p (Lawaetz, 197 1) (x) =

(2- 16) According to Eq. (2-15), we find that for a typical semiconductor with @ = 30 eV,

EG = 1.5 eV, the induced dipole moment is 1 x lo-’ nm per V/cm and elementary charge. Equation (2- 13) reveals that all optical transitions are directly connected with momentum matrix elements -just the same quantities which also determine the band gaps and band curvatures near band extrema in the framework of k - p perturbation theory. This connection leads to a number of useful relations (so-called sum rules) which help to determine mass parameters and g-factors from optical spectra, and vice versa (Kane, 1966; Lawaetz, 1971; Harrison, 1980; Yu and Cardona, 1996). Up to this point we have only discussed the dielectric polarization for resonant excitation with light, i.e. Aw2EG. It follows from very general arguments, which are based only on locality, causality and linearity of the system, that x1(0) and x2 ( w ) are closely connected with each other. One relation is (Kronig, 1926; Kramers, 1927; Stem, 1963; Cardona, 1969):

-

It gives an explicit expression for x1( w )

x

(2-18)

The real part of ( w ) describes the response of all electrons weighted with a resonance denominator. The form of the dielectric susceptibility resembles very closely the behavior of an ensemble of massive charges on springs which are characterized by individual eigen-frequencies w ‘= (Eck- Evk)/A and are driven by an external force field with frequency w. This analogy with forced harmonic oscillations has been recognized very early - in fact long before quantum mechanics - in the work of Thomson, Drude

78

2 Optical Properties and Charge Transport

and Lorentz. They interpreted the empirical results of their time, which are now described quantum-mechanically by Eqs. (2-13) and (2- 18), in terms of electro-mechanical oscillators (Jackson, 1975). For a given frequency w one must sum over all possible resonances of Eq. (2-18), i.e. all e-h-pair states with zero total momentum. The occupation of initial and final states is taken into account by summing only over filled valence and empty conduction band states (or vice versa, with opposite sign). For the consistent description of matter in Maxwell’s equations the complex dielectric function is defined E (0) = SO(

1

+x (0))

(2-19)

It describes the dielectric response of a semiconductor and is the quantum mechanical version of the response of a linear system to harmonic time-dependent perturbations (Kubo, 1957). An important extension of Eqs. (2-13)(2-18) can be made in order to include the possibility of partially filled (resp. empty) states k, and k,. Each term of the k-sum is multiplied with scalar distribution functionsf,, fh obeying 0
x

mainly controlled by its real part ten, 1972).

x (Woo-

2.2.3 Correlated Electron-Hole Pairs: Excitons The model of independent electrons moving in a strictly static crystal potential V , (r)works reasonably well in the context It provides of the dielectric response (0). a qualitatively correct description of the important features of optical spectra - band gaps, critical points, density-of-states peaks - for all common semiconductors. However, a closer look at empirical spectroscopic data and their quantitative comparison with the results of one-electron band structure calculations inserted into Eqs. (2- 13) and (2-18) reveals a problem: at and around all critical points of En(k),and especially in the vicinity of band edges, which are of greatest importance for all practical applications (see below), the independent particle scheme is grossly inadequate. The underlying dense electron gas with an average density of n, 3 x cm-3 has correlations, so that the excited electron and the hole interact strongly with each other and also polarize their surrounding (see the term Vint in Eq. (2-1)). In other words, a more realistic picture of interband transitions has at least to include the statically screened Coulomb interaction between electron and hole. In principle we expect three different kinds of correlated motion when “excitons”, i.e. shortform for electron-hole pair excitations, are created in a semiconductor:

x

-

(i) the excited electron (Fig. 2-1) does not leave the atomic site of its origin and stays bound to it. The pair is said to be trapped (Toyozawa, 1980); (ii) both electron and hole are mobile after the excitation, but stay bound together; the pair moves through the crystal but keeps a characteristic separation t-e-h of

2.2 Optical Properties of Bulk Crystals

79

relative motion. The two limiting cases r e - h a. and re.h %a, (a, is the lattice constant) are known as Frenkel resp. Wannier-Mott excitons (Wannier, 1937; Elliott, 1957; Mott, 1961). (iii) electron and hole are in unbound continuum states (“ionization”), the charges are separating after local excitation (Dow, 1976; Stahl and Balslev, 1986).

a frequency corresponding to Ry*IA. In relative coordinates defined by t-.h = I ‘e-rh p represents the reduced effective mass of the two-particle problem “exciton”. The kinetic energy connected with the translational (i.e. center-of-mass) motion of the exciton appears in the third term of Eq. (2-22). Typical values for Ry* are 1- 100 meV, and ug= 1 - 10 nm, in other words

The situation is controlled by several factors: kinetic energies of electrons and holes, the pair density with its influence on screening of the e-h Coulomb interaction, and the strength of electron-phonon coupling. The most common case in group IV, III-V and II-VI semiconductors with band gaps in the range 1-3 eV and moderate electron-phonon coupling is the extended exciton, case (ii). We discuss now briefly its basic features (Elliott, 1963; Dimmock, 1967; Cho, 1979; Rashba and Sturge, 1982; Klingshirn, 1995). The natural length and energy scales of one interacting e-h pair are the exciton Bohr radius ug and the exciton Rydberg energy Ry*:

Ry* 4E ,

-

(2-20) Ry* =

P4 -

2h2 ( E g E*)2

-

e2 2EO E * u;

(2-21)

They describe the discrete levels of the exciton spectrum with orbital quantum number n = 1, 2, ... according to

In this hydrogen-like spectrum the second term describes the binding energy due to relative e-h pair motion in the presence of a long-range attractive Coulomb interaction modified by the dielectric function E ( w ) , where E* is the value of E (w)taken at

1,

-

and

ug 9%. ~

(2-23)

The existence of bound e-h pair states (“discrete excitons”) and the adjacent continuum of unbound, correlated pair states (“continuum excitons”) change the functional form of the dielectric response function E ( w ) appreciably. Instead of the plane wave states, which were assumed in Sec. 2.1 to be the proper eigen-states of the crystal and had been used in the derivation of Eqs. (2-13) and 2-18, we now incorporate the actual envelope wave function for relative e-h motion. The problem is easily tractable for two interacting carriers in the effective-mass-approximation (Wannier, 1937; Cho, 1979; Haug and Koch, 1994).

2.2.4 Single-Particle versus Pair Excitations The spatial correlation of electron and hole motion represents the simplest type of collective excitation in a solid: both quasiparticles attract each other by the potential V,,, = e2/(4JC E, E * I re-rh ) and move freely through the crystal. It is plausible to assume that for low e-h pair densities - say npair4ug-3 - the interactions between two different pairs can be ignored. This basic assumption is the concept for the elementary excitation “exciton”. An exciton with given total momentum A K and centerof-mass (CM) coordinate R is then represented by the product of a phase factor exp (i K R ) with the envelope function @

I

-

80

2 Optical Properties and Charge Transport

Ee

T

EPBY

’

EG

a)

b)

Figure 2-5. Single particle versus pair excitations: a) Dispersion relations E, (k,) and Eh (kh)with gap energy EG for non-interacting electrons and holes. b) Pair spectrum with ground state at Epair=Oand Kpai,=O. One class of discrete bound pair states (the “1s exciton“) is shown together with the continuum of unbound, but still correlated e-h-pair states at higher energy in cross-hatched region.

(re-rh). The wavevector K = k e - k , characterizes the CM-motion of the pair in Rspace, the envelope @describes the relative e-h motion in r,-,-space. The new “quasiparticle” exciton is composed of two oppositely charged carriers with half-integral spins; its creation and annihilation operators obey commutator rules which are only approximately, in the limit of npair 4 0, those of bosons (Hopfield, 1969; Tassone et al., 1992). The sum of electron and hole momenta, k, and -kh, is conserved in such an excitonic state. Therefore the pair wavevector, Kpair=ke-khris the natural choice for its translational motion. Figure 2-5 (a) summarizes the essentials of the single-particle description in terms of single electron (resp. hole) momenta and energies, for a noninteracting pair. Fig. 2-5 (b) shows the corresponding energy spectrum with interaction in the so-called exciton (or e-h-pair) representation. The lowest excitation ener-

gy is smaller than the (single-particle) gap energy EG. For K = O we find Els= EG-Ry* because of the attractive e-h Coulomb interaction. Discrete exciton resonances with their energy spectrum ranging from E l , to EG and the continuum of correlated pair states above EG modify the single-particle optical spectrum of Eq. (2-13) in a characteristic manner. This Coulomb enhancement effect has been discussed in great detail in the literature (Dow, 1976; Hanke, 1978; Keldysh, 1997).

2.2.5 Exciton-Photon Coupling: Polaritons We started in Sec. 2.2.1 with a perturbative treatment of optical transitions between uncorrelated initial and final plane wave states, Iki) and lkf), i.e. neglecting the term Vi,, of Eq. (2-1) and taking the external field as a small perturbation which couples the single-particle base states (Eqs. 2-12, 13). In the next step, the unretarded Coulomb interaction was included, v,,,- ( E re-h)-’, and electronic excitations described as correlated two-particle states or “excitons”, with hydrogen-like pair wave functions and modified energies (Eq. 2-22). Pair states withp-like hole wave function components oriented perpendicular to K,, (see Fig. 2-2) are transverse excitons. They have a dipole moment p and can interact with light propagating in the direction of K,, , via their macroscopic polarization. This interaction couples excitons with photons. If it is taken into account, we find the true electromagnetic eigenmodes of the bulk semiconductor (also insulator), the excitonic polaritons. To obtain their dispersion relation, we have, in principle, to quantize also the light field. It turns out that already a semiclassical treatment, based on Maxwell’s equations, gives correct results

2.2 Optical Properties of Bulk Crystals

and leads directly to the wavevector-dependent frequency A w ( q ) of the polariton modes. The discrete bound excitons n = 1, 2, ... below EG and the continuous spectrum of unbound, i.e. ionized excitons with energies E 2 EG are now considered as base states. A closer look at the lowest lying discrete resonance “Is” in a typical 111-V semiconductor reveals more details: there is 8-fold degeneracy of two valence band and one conduction band orbital, each in two spin states. Two of the spin singlet exciton modes are transverse, i.e. dipole-active, a third one is the longitudinal exciton mode LE, see Fig. 2-6 below. The remaining five pure triplet states are lower in energy, do not couple to light and are called “dark states”. Each of the transverse excitons can be seen as a dipole-active resonance. It represents an oscillator of frequency wo(q), whose dielectric function ~ ( wq), , in the spirit of Thomson, Drude and Lorentz, is E

( 0 9 4 )=

Here E, is the background contribution due to all other resonances above and below, and the oscillator strength fos, of the transition is given by (2-25) With the constitutive matter equation D = ~ , E ( oE,) we solve Maxwell’s equations and obtain for longitudinal modes the condition (0, q ) = 0, and for transversal modes the implicit dispersion relation 2

o2E L (w, q ) = E g c q

2

(2-26)

This leads in the simplest case (parabolic exciton dispersion) to an explicit expression of fourth order in q, which is shown in

81

-------- 8

, .

// 0

:

lower polariton branch

q

Figure 2-6. Polariton dispersion E = h w (4)for the eigenmodes of the coupled exciton-photon system (left). Bare exciton o =wT and vacuum photon dispersions are also shown (broken lines).

Fig. 2-6. The key features for small q are the following: (i) two non-crossing branches reveal the mixed mode character of the new eigenstates “polariton”, with strongly o-dependent light-like and excitonlike components, (ii) there is a gap AE,, of frequencies, (“Reststrahlen-Bande”), where propagating wave solutions do not exist, (iii) both branches are twofold degenerate with respect to polarization. The longitudinal-transverse splitting between exciton modes at q = 0 is AEL, =

A e2fosc 2 ~ , ~ o m ~ w ~ V

(2-27)

and lies in the range 0.1 ... 0.5 meV for typical semiconductors (Weisbuch and Ulbrich, 1982). The description in terms of a dielectric function (Eq. 2-24) with quadratic exciton dispersion is equivalent to solving the wave

82

2 Optical Properties and Charge Transport

equation for each of the transverse field components E (r, t ) 1 a2E

2

V, E - - -c 2. -

&2

a2p = k2 at

(2-28)

with d2P/at2 as the local source term, coupled together with the polarization wave equation, where E (r, t ) is the driving term:

-+r--a2p ap at2

at

+ 002

Po

v:

iiwo

M

e2fosc E ( r ,t )

(2-29) m0 V The eigenmodes of the coupled system “light-matter’’ (Eqs. 2-28,29) in a bulk, uniform semiconductor are called excitonic polaritons. They are implicitely contained in the dielectric response function, which appears in Eq. (2-26). They provide the correct description of excitons with quadratic dispersion on a regular lattice with lattice constant u 4 A, including their retarded interaction with the electromagnetic field. Historically, the correct treatment of light interacting with point dipoles in a crystalline solid - though already fully contained in the classical Eq. (2-24) - was first given by Huang and Born in the context of infrared absorption mediated by lattice vibrations (Born and Huang, 1954). For the quantum mechanical description of delocalized Wannier excitons (i.e. extended dipoles) one proceeds as follows: with the z-axis chosen for quantization, the interaction between the transverse excitons and the radiation field is considered. The term which has to be added to the Hamiltonian He, for the exciton problem with wave function Vexciton (re rh) 9

is, this time going beyond the first order approach of Eq. (2-12), the full canonical momentum

H ’ = - L C Ai-$i+-Ce2 mo i 2mo

A?

(2-31)

i

The first term is the exciton-photon coupling of Eq. (2-12), the second represents the electromagnetic field energy. Because of translational symmetry, the interaction couples only e and h states with the wavevector difference k,-k,= q . Hopfield and Pekar have shown (Hopfield, 1958; Pekar, 1962) that this problem can be diagonalized by a suitable transformation of states and gives eigenvalues w (q) which, in the limit of vanishing exciton density, coincide with the eigen-energies of the semiclassical treatment leading to Eq. (2-26). Only exciton base states with wavevectors K,, comparable to the wavevector of light Klight= o (Kex)/cat the exciton frequency interact strongly with the light field, as is evident from Fig. 2-6. The term “polariton” was coined later (Hopfield, 1958) to indicate that the quantum mechanical treatment leads to bosonlike elementary excitations which correspond to quanta of “mixed” electromagnetic/excitonic modes in a dielectric with E, B and nonvanishing P, just as the term “photon” stands for the quantum of a field mode - with only E, B - in vacuum. The dispersion relation o(K,,) can be seen from the standpoint of pulse propagation through a dispersive medium: a planeparallel slab with an excitonic or vibronic resonance described by Eq. (2-24), and excited from outside with a light pulse, will propagate polariton modes within the slab. They are true eigen-modes of the bulk semiconductor, so that in principle no damping or decay into other modes occurs. Consistent descriptions of all details of linear optical spectra, like reflection, transmis-

2.3 Interband Absorption Spectra

83

Figure 2-7. Polariton pulse propagation through a thin slab of GaAs, with driving frequencies covering the 1s exciton resonance. The pronounced beat phenomena indicate full coherence and are due to interference between modes with different group velocities on upper and lower polariton branch.

0

2000

4000

6000

sion, elastic and inelastic light scattering can be given in terms of polariton modes, provided the boundary conditions are treated correctly (see e.g. Stahl and Balslev, 1982; Artoni and Loudon, 1997; Henneberger, 1998). As an example of coherent spectroscopy on the exciton-polariton resonance we show in Fig. 2-7 the time-resolved transmission signal of an ultrashort light pulse having propagated through a thin semiconductor slab. Interference between upper and lower polariton-branch modes in the spirit of Eq.

(2-24,2-27)) the polariton description coincides with the common phenomenological expressions in classical optics, e.g. Snellius’ and Beer’s law, and Fresnel’s equations. The polariton approach (Eqs. 2-28, 29) handles the phenomenon of radiative damping of spatially confined systems (see Sec. 4 below) without conceptual difficulty. The transition from superradiant behav-

aooo

ior (i.e. excitations coherently delocalized over many lattice sites) to the bulk polariton regime has been discussed for the case of a slab with thickness t comparable to wavelength A (Knoester, 1992; Citrin, 1993). The solution of both the wave equation and the polarization oscillator equations with appropriate boundary conditions has been treated extensively for local and non-local response functions (Andreani, 1995).

2.3 Interband Absorption Spectra

2.3.1 Direct versus Indirect Transitions The conservation of crystal momenta k , and k , in Eq. (2-13) requires that only “vertical” transitions occur in the fundamental absorption and emission process. It is natural to classify band structures accordingly into “direct” and “indirect” with respect to their smallest (or any larger) gap.

84

2 Optical Properties and Charge Transplort

Figure 2-8. Dispersion relations (left) and full wavefunctions in periodic crystal potential (right). In a “direct-gap’’ semiconductor the interband polarizations on neighboring sites are in phase, giving nonvanishing macroscopic polarization P (top). The “indirect-gap” case has alternating phase factors which lead to destructive interference and vanishing rates of interband transitions.

+I A At ./ There is a qualitative difference between both cases, due to the symmetry of the wavefunctions at the band extrema. The upper valence band edge of all common semiconductors belongs to p-like wavefunctions, and is centered around k=O. Figure 2-8 shows schematically the full wavefunctions uk(r)with phase factors exp (i k . r ) for a direct gap material like, e.g. InP, and an indirect semiconductor like, e.g. Ge in reciprocal space (left) and direct space (right). In dipole approximation the small wavevector of light is neglected and the matrix element of Eq. (2-13) gives zero upon integration over all crystal unit cells in the indirect case: the transition is called “dipole-forbidden”. Only second-order processes, e.g. the additional emission or absorption of a phonon (or the participation of an impurity) may cause light absorption. For intrinsic Ge and Si it has been shown that both variants of electron-phonon and hole-phonon coupling which are depicted in Fig. 2-8 can occur. They involve phonon modes of different symmetry, so that in high-resolution absorption and emission spectra both paths can be distinguished. Any perturbation of the symmetry of the crystal lattice, e.g. through the presence

+-

&

of impurities, interfaces or a free surface (like, e.g. in porous Si), or an electric field, will increase the probability of such forbidden transitions (Abstreiter, 1984; Abstreiter et al., 1984; Jung et a]., 1993; Linnros, 1999).

2.3.2 Radiative Recombination Rate It is of basic interest to calculate the induced transition probability Wcv of one conduction band state (say at k = O ) with the matching valence band state under the influence of a resonant electric field Ere,. We obtain directly from Eqs. (2-13), (2-14) and (2- 18) that (2-33) If we insert for Eresthe amplitude of the electric field in one photon mode filling the crystal volume at frequency A wgap with occupation number N = 1/2, we may identify Eq. (2-33) with the inverse spontaneous radiative lifetime trad of the coupled e-h pair at k=O: (2-34)

2.3 Interband Absorption Spectra

We find with the parameters @ = 30 eV, Eg= 1.5 eV and E (agap) = 12 a value of trad" 1.05 ns. This is in good agreement with empirical values of the interband radiative recombination time in bulk semiconductors, which were determined from low temperature bound-exciton decay times (Henry, Nassau, 1970) and also from direct band edge luminescence decay in connection with absolute quantum efficiency measurements (Hwang, 1972). The derivation of Eq. (2-34) is based on 3-d bulk electronic and photonic density-ofstates. In confined structures, especially micro-cavities (see Sec. 2.4.4 below), both quantities may be qualitatively different from the 3-d value, and Eq. (2-34) is not meaningful. The e-h recombination rate of indirect gap materials, like Si or Ge, is orders of magnitude smaller and reflects the strength of coupling between phonons and (i) zonecenter valence band and (ii) zone-edge conduction band electronic states. Values of Tife- 35 ms for Si and 20 ps for Ge have been found experimentally in high-purity intrinsic crystals (Yablonovitch et al., 1986; Thomas et al., 1979). Electron-phonon coupling depends sensitively on details of the wavefunction uk(r),and it is not surprising that there is only qualitative agreement between the experimental data and ab initio calculations.

-

2.3.3 Angular Dependence of Interband Transitions In full analogy with atomic spectroscopy, the orbital and spin parts of the initial and final electron wave functions qn,k, Jr) determine the angular characteristics of interband optical transitions. We can therefore anticipate selection rules with a pronounced angular dependence of the matrix element in Eq. (2-13). The wavefunctions at

a5

the valence band extremum in zincblende structure are such that at k = O an optical transition to the conduction band creates both heavy- and light holes, and mixed spin states are produced in the process. In the lower (hexagonal) symmetry of the wurtzite lattice the lightheavy hole degeneracy is lifted and we find two different transition energies with a strict selection rule at k = 0. An electric field vector E 11 c couples the heavy hole (Mj=3/2), and E I c the light hole band (Mj = 112) with the conduction band states. Away from k = 0 the symmetry is lower, in general, and mixing of orbital p and s-components occurs (Bassani and Parravicini, 1975). Electron spins in zincblende crystals may be partially aligned in the process of optical absorption by proper selection of levels and suitable choice of incident photon angular momentum (Lampel, 1974; Zakharchenya and Meier, 1984). The method is analog to optical pumping in atomic spectroscopy and has been applied to transitions at the r-point of the Brillouin zone (Lampel, 1974) and also at the L-point (Plane1 et al., 1980). Recent work in doped semiconductors and quasi 2-d structures has led to a renaissance of spin-dependent effects in semiconductors, and new concepts in spindependent transport have emerged: injection of spin-polarized electrons, and use of the zeeman-splitting for control of transport (see Sec. 2.5.5 below and Oestreich, 1996; Kikkawa and Awshalom, 1998,1999; Awshalom, 1999).

2.3.4 Fundamental Gap Spectra at Low Excitation Under conditions of low temperature and low excitation power, the direct band edge spectrum of intrinsic semiconductors is always dominated by sharp excitonic line spectra.

86

2 Optical Properties and Charge Transport

In bulk crystals, where the sample dimensions are much larger than the exciton Bohr radius a:, the natural unit for e-hpair density is the reciprocal exciton volume. The pair density for complete volume filling with 1s excitons is n,- 2 x 1017cmP3 in a typical semiconductor like GaAs with a; 10 nm. Taking the radiative recombination time of Eq. (2-34), and converting it to a generation rate, we find that at least 2 x lo4 W/cm2 of resonant cw pump power are needed to maintain such a density in a layer of 1 prn thickness, which is roughly the absorption length a-l at A wgap. At densities much lower than n, ,discrete bound e-h pair resonances below the gap energy Eg and a smooth continuum above give clear experimental evidence for the long-range attractive Coulomb interaction between electrons and holes. Figure 2-9a shows the absorption spectrum of a 4.2 prn thick high-purity GaAs crystal at low temperature, T = 1.2 K. The material was grown by vapor-phase epitaxy and had residual shallow acceptor and donor impurity concentrations NA
-

-

regime of effective-mass-approximation (Elliott, 1963; Dow, 1976; Cho, 1979). This is remarkable, because of the valence band degeneracy and the strong mixing of light and heavy hole wavefunctions in the exciton problem. Nevertheless, a simple twoband description with average masses rnz = 0.067 rno and rn; = 0.32 inogives a reasonable idea of the wavefunctions involved. In Fig. 2-9b we have plotted quantitative exciton envelope wave functions for four different energies below and above EG (see the optical spectrum shown in Fig. 2-9a). Energies are given in units the exciton Rydberg Ry*, only states with angular momentum L=O are shown, and normalization is carried out for a fixed excitonic volume (Ulbrich, 1988). The strong spatial correlation between electron and hole at small relative distances r = I r e - t h I in Fig. 2-8 leads to the sharp excitonic peaks below A ugapand the enhancement above, which are both observed in all empirical spectra. As a rule, the true optical absorption spectrum around A ugaPevolves from the single-particle of Eq. (2-13) by a red-shift spectrum x2 (u) of Ry* and multiplication by the square of the exciton envelope wave function at r = 0. At the gap energy, and even several Ry* above, the persistent peak of qj (r) at r = 0 enhances the transition probability appreciably. The magnitude of this “Coulomb enhancement” effect is demonstrated in Fig. 2-9a. The dashed curve is the calculated absorption spectrum for a fictitious system with no Coulomb interaction between electron and hole: here the interband matrix elements are taken between plane wave states according to the single-particle description which led to Eq. (2-13).

2.3 lnterband Absorption Spectra

-

::::2;b'lr

I

t ' l

"; E

-

m

0 u $

w

10

u

I& L

w

.

0 U

= 50 -

r a U 0

-

m

.

V)

4

0

Figure 2-9. a Band edge absorption spectrum of high-purity gallium arsenide at low temperature. The spectrum shows a Rydberg series of discrete bound pair states n = 1 , 2, 3, ... and the smooth continuum above E,. Additional sharp lines below n = 1 and n = 2 are due to residual impurities and their localized bound pair states. A fictitious plane-wave (independent-particle) absorption spectrum is shown for comparison (dashed line). b Wave functions for the relative motion of a single electron-hole pair. The parameter k indicates the pair energy relative to ____-.---__-*--EG. in units of the exciton Rydberg _-* ' ' R y * . Note that there is strong corre~~

n=l

c

I

87

"

"

"

"

~

~

"

2 302

'

e-h pair in the presence of n, and nh free carriers:

I

E(j

+

a) both e and h repel or attract, i.e. polarize the bound valence charges and the nuclei; b) the e-h pair interacts with the bipolar plasma of mobile extra electrons and

k = + l

Ry ,

distances.

0.5b7j In the static limit o=O,assuming temperature T= 0, neutrality n, = nh= n , and for simplicity just inZ=mz and hence one Fermi energy EF,the model of Thomas and Fermi (Platzman and Wolff, 1973) gives aqualitative idea of the screened potential

- 0.25

OO

b

5

10

15

YaB

20

2.3.5 Exciton Screening What is the influence of other free or mutually bound carriers on the e-h pair correlation? One has to inspect the screening of the e-h interaction more closely to answer this question. Consider first the mutual Coulomb interaction V(lr,-rh I) of a given

V (r ) = . -. exp(--) e (2-35) 4 ~ E*r d ~ ~ ATF with the Thomas-Fermi screening length

4

EF

=

Eg E *

6 n n e2

(2-36)

Taking n = 2 x 1015~ m - m: ~ , = 0.067 mo we find that EF=l meV and ATF= 13.5 nm. More realistic treatments of bipolar screening of an e-h pair by electrons and holes are

aa

2 Optical Properties and Charge Transport

difficult. They have to take into account exchange and correlation effects in the carrier plasma (Appel and Overhauser, 1982; Zimmermann, 1990). Authoritative reviews on high density e-h plasma effects were written by Rice (1977), Haug (1985) and Haug and Schmitt-Rink (1985). In any case, the potential V ( r )in Eq. (2-35) contains the two distinct screening mechanisms mentioned above. On the one hand, E* stands for the static screening via polarizable valence bond charges. This contribution can be traced back to the interband terms in Eq. (2-18) and may be referred to as “virtual” e-h pair excitations driven at low frequency w -0. On the other hand, the exponential function in Eq. (2-35) is the result of a balance between (i) attraction of mobile screening carriers by the screened electron (resp. hole) and (ii) the electrostatic repulsion in the screening cloud, both considered at frequency cr) 0. In the model of Thomas and Fermi the radial dependence of the extra charge density around the screened point charge is calculated from an approximate, i.e. smoothed solution of the Schrodinger wave equation with the inclusion of the electrostatic energy term (Platzman and Wolff, 1973). If we deal with the same situation at finite temperature T, the thermal motion of the electrons adds to the repulsive action of the screening charges. Here the approach of Huckel and Debye leads in the limiting case of kT % EFto the earlier “classical” version (Debye and Huckel, 1923) of the screening length A, which has to be used in Eq. (2-35):

-

(2-37)

Both expressions were derived for the static case, that is the screened charge is at rest, w=O. If the screened point charge moves through the crystal, one has to

account for the dynamics of the screening carriers. We will return to the concept of dynamical screening below. The length scale a: which was introduced for the extended exciton states (Eq. 2-20) separates two excitation density regimes: if the mean interparticle distances, i.e. re, and r h h exceed a:, the picture of isolated bound pairs is adequate and we may think of a dilute gas of excitons. If the condition is not met, the situation is more complicated. Screening of the long range part of the Coulomb interaction between e and h will become increasingly important, leading eventually to a screening length shorter than u;, so that individual pair states (eqs. 2-20, 2-21) are no longer a good basis for description. The term e-h “plasma” is used for such a dense system of electrons and holes in a semiconductor (Rice, 1977; Comte and Noziere, 1982). Two essential characteristics of the bipolar plasma are the existence of two distinct quasi-Fermi-levfor both types of carriers, els, EF’ and a reduced (renormalized) gap energy E& and acoustic plasmons (Appel and Overhauser, 1982). A second qualitative change is expected for still higher densities n, and nh. It marks the transition toward true metallic behavior, and is accompanied by the closing of the band gap EA (Combescot and Bok, 1983). In the dilute case, n 4 ll(~;)~ one tries to describe the relative motion of e and h in terms of plane wave states originating from the unperturbed conduction and valence bands. Solutions of the wave equation for this problem of screened exciton motion have been extensively discussed (Zimmermann et al., 1978; Haug, 1985; Haug, 1988). We skip here a thorough discussion of details and sketch only plausible aspects of the problem. Suppose the relative e-h motion of a single exciton is described by the envelope

2.4 Systems with Restricted Dimensionality

function

a,s= V-+exp[-E)

(2-38)

The expansions of this wavefunction into plane wave states I k,) and 1 kh)have Lorentzian shapes, are centered around k, = k, = 0 and extend to roughly a;-' in k-space (Landau, Lifschitz, 1966). If we consider the available density of states in this wavevector range, we can estimate that approxiexcitons can be mately a density of n, created in a given crystal volume before single particle phase space occupations reach f,-fh 1. A further increase in e-h pair density requires more kinetic energy, because of the Pauli principle, and increases the Fermi energies E$-) and ELh' . The interplay between the increase of kinetic energy and lowering of the screened potential of the e-h pair motion controls the behavior of the system as a function of density. There is a relatively smooth transition from individual excitons at low density into a plasma of highly correlated e-h pairs around n,, and finally into a dense metallic plasma with density much higher than n,, and less pronounced long range correlation effects (Rice 1977; Haug, 1985).

-

-

2.4. Systems with Restricted Dimensionality In the past two decades, much effort in engineering of semiconductor materials has been directed towards controlled carrier confinement in one ore more dimensions (Esaki, 1993). Growth techniques such as molecular beam epitaxy have made it possible to fabricate semiconductor structures down to nm scales. The almost arbitrary indepth control of material composition and impurity doping, and the utilization of selforganized growth have been key elements.

89

Not only fundamental gap energies and band-offsets, but also band structures in general have been purposely designed for electrical transport and optical properties. The ability to synthesize nanoscale structures, such as 2-d wells and superlattices, 1-d wires, and O-d dots has in turn spawned new classes of semiconductor devices. Two classes of mesoscalic structures can be distinguished: down-scaled conventional device geometries (like e.g. a field-effect transistor) and structures made to exploit quantum effects. The former class is usually operated in regimes of temperature and electric field strengths, for which the electron coherence length is smaller than the smallest device dimension: charge carriers are still treated as classical particles. In quantum confined structures, on the other hand, the de-Broglie wavelengths AB of electrons close to EF are fundamental to the phenomena in focus. Here, one or more structural dimensions are usually made comparable to AB, and are typically much smaller than the phase-breaking length &. It is a measure for the mean distance over which the phase spread of the propagating electron wave due to fluctuating scatterers is of the order one. The inelastic mean-free path L,, which is the distance electrons move on the average without loosing energy, is determined by the electron-phonon interaction (see Sec. 2.5 below) and is of the order of 1 ...lo0 nm at room temperature, but can reach tens of pm at low T. Mesoscalic devices exhibit properties directly connected with the wavemechanical properties of electrons, even though the overall structure may still be many orders of magnitude larger than the underlying atomic scale of the solid, characterized by its lattice parameter ao. A semiconductor structure with restricted dimensions resembles a low-loss electron waveguide more than a bulk conductor.

90

2 Optical Properties and Charge Transport

2d - growth

2d - MQW

-

Od dot

I d wire

-

Figure 2-10. Semiconductor structures of restricted dimensionality: starting with planar crystal growth techniques (liquid phase, gas phase, or molecular resp. atomic deposition), followed by lateral structuring with lithographic (wet etch, dry etch, ion beam) techniques, devices with typical length scales L, and quasi 2-d, 1-d and 0-d electronic behavior can be made.

But the guiding is defined by a potential Vconf(r),unlike the conducting walls of an ordinary electromagnetic waveguide. The essential point is that the long-range Coulomb part of Vconf(r) can be easily controlled from outside, e.g. an external charge reservoir. Such devices offer the unique possibility of exploiting quantum-mechanical interference for information handling and storing purposes. At ambient temperature T, an electron in any specific quantum state may pick up the energy AE of order kT from thermal fluctuations. If AE is less than the energy difference between adjacent discrete states (confined in three, two, or just one spatial dimension) one speaks of a quasi-0, 1, or 2-dimensional system. To characterize the system further, one defines three length scales L,, L,, L, (see Fig. 2-10). If one, two, or three of these quantities tend towards zero, the electron’s spatial degrees

of freedom are said to be fully confined to 2, 1, or 0 dimensions. If the lengths are reduced such that only the weaker AE criterion is met, i.e. AEconf>kT, then the prefix “quasi”-n-dimensional is more appropriate. Lithographically defined patterns on the 10 nm to 100 nm scale have produced a large variety of such electron wave-guides and resonators (Wilson et al., 1993). The terms quantum-“well”, quantum“-wire” and “-dot” are now established shortforms for structures with typical lengths L, between the lattice constant a. of the atomic constituents, and the inverse of a relevant electron (hole) wavenumber k:

aoc Lsck-’

(2-39)

Here k represents the magnitude of the thermal wavevector of a carrier distribution, or the Fermi wavevector kF of a denser ensemble of camers which is confined in the structure (Capasso, 1993). Fig. 2-10 shows the geometries of quasi 2-d, 1-d and 0-d structures made on planar substrates by atomic deposition with indepth control of material composition and subsequent lithographic or cleavage techniques to laterally build the structure. Buried structures based on Stranskii-Krastanov growth have been developed (Woggon, 1997). In an effort to obtain ideal, i.e. atomically smooth interfaces on larger scales, Tshaped wire structures were fabricated by overgrowing in-situ freshly cleaved surfaces (Wegscheider et al., 1994, 1997). However, substitutional disorder still affects the confinement potential. The assembly of any crystalline heterostructure, via doping in a pn-junction, or via heteroepitaxy in the advanced systems, has always to cope with a principal problem: different materials have, in general, different lattice constants. In practice, the use of ternary alloys allows to “lattice-match”

2.4 Systems with Restricted Dimensionality

materials with different electronic gap energies and valence band edge energies. But the strain fields which are induced in composite structures can as well be exploited for specific applications (Bauer and Richter, 1996; Pryor et al., 1997)

I r

91

b

2.4.1 Electronic Confinement The fundamental effect of confining a particle within a potential well has been known since the early days of quantum mechanics. Any deviation of the crystal potential Vc (r)(see Sec. 2.2.1) from translational symmetry, caused by chemical composition and/or structural disorder, will modify the spectrum of electron energies. There are two basic types of interaction (see Fig. 2-1 1): (i) electron and hole change their energies in covariant form, (ii) electrons and holes change their energies individually because of local chemical bonding. The first mechanism is connected with the long-range Coulombic attraction (repulsion) of the carriers by a donor (acceptor) impurity. More general, any localized charge distribution, like an impurity point charge, a line charge, a charged layer, or a 3-d space charge cloud, causes this kind of covariant change of E, and E,. To transcribe the concept of extended states k, with global eigen-energies E (k,) (see Sec.2.2) to a local desription in r-space, we consider an electron at rest, define for the ideal periodic case (i.e. L, =) its total energy E at and around r as E, , and add to this the confinement energy AEconf for the actual L,. An infinitely deep well which confines just one degree of freedom, gives

-

(2-40)

Figure 2-11. The two types of carrier confinement: (i) covariant change of electron and hole energy caused by long-range Coulomb attraction (like e.g. donor impurity) and (ii) individual energy change due to local variation in chemical binding.

For rn; = 0.1 m, the confinement energy is AE= 10 meV for L,= 5 nm. The hole case is more complicated because of the 6-fold band degeneracy at kh=O. Strong mixing of j = 312 and j = 1/2 states occurs there and leads, in general, to a manifold of valence bands with highly anisotropic and non-parabolic E ( k )dispersion relations (Weisbuch and Vinter, 1990). The second type of interaction is the local chemical bond effect: the cell-to-cell variation of Vc(r) in H , leads to states which cannot, in principle, be classified as Bloch states (see Fig. 2-1 1). There are changes of Ec and Ev with chemical composition, that can be either covariant or contravariant, and hence cause edge variations on relatively small length scales: the gap energy EG changes “locally” with chemical composition. Ab-initio calculations for ideal planar interfaces have been made and showed that band offsets are not uniquely determined by the bulk properties of the constituents, but that interface-specific phenomena also exist, especially for heterovalent IV-I11 -V systems (Baldereschi et al., 1993).

92

2 Optical Properties and Charge Transport

In randomly substituted III-V alloys an approximate k-representation can be justified (Wang et al., 1998). Pseudopotential calculations of single-particle energies and wavefunctions for edge states in disordered III-V alloys have been made (Mader and Zunger, 1994). In the more phenomenological effective mass approximation, the breaking of translational symmetry at, e.g. a heterojunction, is simply described by the spatial variation of the effective mass parameters, rnz(r)and mE(r), over short distances. The effective mass operator is incorporated into the kinetic energy according to (Ridley, 1997; Foreman, 1998)

In the following we discuss the various manifestations of carrier confinement in structures of 2, 1 and O-d. 2.4.2 Quantum Wells, -Wires and -Dots

In addition to the more or less “abrupt” discontinuities of band structures at an hetero-interface, the transfer of charge between the constituents causes so-called bandbending. This effect plays a central role for the overall performance of a heterostructure, a superlattice or a quantum well (see Fig. 2-12). To describe this effect, one starts with a single-particle trial wave-function, calculates the charge distribution Q (r) throughout the system, solves the Poisson equation to find the potential distribution, and adds the latter as an external quantity (2-42) to the crystal potential V,(r).This is repeated until self-consistency is reached. In planar geometry it is straightforward and

I

t

well

substrate

Figure 2-12. Potential landscape of quasi-2d structure (quantum well) and l-d quantum wire. Band bending caused by charge transfer from the constituents (bulk, substrate) controls the equilibrium situation in a self-consistent way.

gives global potentials Vtot(r)as shown in Fig. 2-12 (Bastard, 1988; Grahn, 1995). With suitable doping of the bulk (substrate) material and/or application of an external electrical field, a metallic quasi-2d electron gas can be established in the region of quantum confinement. The most important characteristic of such a 2-d electron gas is that the mobile carriers are supplied by the remote bulk material, so that there are no corresponding ionized charges to scatter electrons, as would be the case in a doped bulk semiconductor. The key characteristics of quantum wells made from III-V compounds - almost ideal interfaces, and small effective electron mass -have resulted in electron mobilities more than two orders of magnitude larger than those of bulk Si. Nanostructures made of GaAsGaAlAs are at present the main tool for studying 2-d, 1-d and O-d