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Structural, dynamical and optical properties of the semiconductors Si and Ge and their superlattice
Wei, Siqing, Ph.D.
The Ohio State University, 1991
Copyright ©1991 by Wei, Siqing. All rights reserved.
300 N. Zeeb Rd. Ann Arbor, MI 48106 STRUCTURAL, DYNAMICAL AND OPTICAL PROPERTIES OF THE SEMICONDUCTORS Si AND Ge AND THEIR SUPERLATTICE
DISSERTATION
Presented in Partial Fulfillment of the Requirment for
the Degree of Doctor of Philosophy in the Graduate
School of The Ohio State University
by Siqing Wei, B.S., M.S.
The Ohio State University
1991
Dissertation Committee: Approved by
John W. Wilkins ^ ^ , r \ n
Ken G. Wilson
R. Sooryakumar Adviser
Department of Physics © Copyright by
Siqing Wei
1991 To My Parents
ii ACKNOWLEDGMENTS
To Professor John Wilkins, my thesis advisor, I wish to express my deepest gratitude and appreciation. His everlasting encouragement and his healthy skepti cism combines with his fairness has motivated me to do the best that I can. My special thanks to him for his understanding and kindness towards me who come from a totally different culture background. It is my great fortune to have him as my advisor.
I would like to express heartfelt thanks for Professors John Perdew and Mel
Levy who started me into serious research in physics, and helped me settled in a foreign environment in New Orleans.
Many thanks go to Professor Peter Nordlander with whom I have spent many long working nights when we were at Vanderbilt. It was his suggestion that I finish my graduate research at Ohio State.
I would also like to thank Professors R. Sooryakumar and Ken Wilson for serving on my graduate committee.
During the years at Ohio State University, Dr. Zachary Levine helped me to get deeper understanding of the fields. I owe him many thanks for helpful discussion, advises and suggestions.
My gratitude to Dr. Douglas Allan and Mike Teter at Corning for making their code available to me.
The tough life of a graduate student was eased somewhat by my officemates over the years. They were: Nick Bonesteel, Krishna Balasubramaniana, Ben Hu, Chris
Lin, Jim Reynolds, Dan Sullivan, Mingqiu Sun, Ned Wingreen and Hua Zhong at
Ohio State, and all the others at Vanderbilt and Tulane, with name list so long that I am unable to finish here, with apologies for omissions.
I have fonded the memory of many interactions with the cheerful faculty, vis itors and postdocs at The Ohio State University. These include: Franco Buda,
Eivend Hauge, Myung-Ho Kang, Mark Kvale, Zachary Levine, Pavel Lipavsky,
Ross McKenzie, John Mintmire, Frank Pelzer, Michael Reizer, and Jon Andreas
Stoevneng.
Brett Lowry and Marijan Adam have each lent me able technical assistance with the computers and made computing ‘fun’, Linda Antolik and Karen Costantino have made my life easier in the difficult paper work ‘shuffling’ and provided me a guide to the campus life at Ohio State. Their helps will be remembered.
I would like to express my thanks to my fellow Chinese friends here at Ohio
State who shorten the distance between me and my home in Shanghai. My special thanks to Jingmin Leng with whom I spent undergraduate life together back home,
Zhiqiang Wang with whom I learned a lot, and Shulai Zhao, my long suffering roommate in Columbus.
I would like also to express my gratitude to my friends, H. F. C., X. L., Chie
Soong, Chenjie Wang, and Yuchen Yan. for their friendship supports me through my life at various stages.
My research was supported by grants from the Department of Energy — Basic
Energy Sciences, Division of Materials Research. The computational facilities are provided by National Super-Computer Center at Cornell and Ohio Super-Computer Center.
Finally, my deepest gratitude to my parents who have given me neverending support and encouragement through the years while I am thousands of miles away from home. My deepest admiration for their honesty and fairness. No words can express my thanks to my parents for raising me in a very difficult situation. I owe them everything I have been today. Vitae
August 29, 1963 ...... Born - Shanghai, Shanghai, PR of China
1984 ...... B. S., Fudan University, Shanghai, PR of China 1986-1987 ...... Research Assistant, Department of Physics, Tulane University, New Orleans, LA 1987-1988 ...... Research Assistant, Department of Physics, Vanderbilt University, Nashville, TN 1988 ...... M. S., Department of Physics, Vanderbilt University, Nashville, TN 1988-Presen t ...... Research Assistant, Department of Physics, The Ohio State University
PUBLICATIONS
Mel Levy, Rajeev K. Pathak, John P. Perdew and Siqing Wei, “Indirect-Path Meth ods for Atomic and Molecular Energies, and New Koopmans Theorems,” Phys. Rev. A. 36, 2491 (1987).
John P. Perdew, Mel Levy, G. S. Painter, Siqing Wei and Jolanta B. Lagowski, “Chemical Bond as a Test of Density-Gradient Expansion for Kinetic and Exchange Energies,” Phys. Rev. B. 37, 838 (1988).
Y. Wang, P. Nordlander, Siqing Wei and M. Ronay, “Surface and Diffusion Barriers to Oxygen Penetration on Simple Metals and Their Oxides,” Diffusion and Defect Data 61A, 1 (1988).
FIELD OF STUDY
Major Field: Physics
vi TABLE OF CONTENTS
DEDICATION ...... ii ACKNOWLEDGEMENTS...... iii VITA ...... vi LIST OF TABLES ...... ix LIST OF FIGURES ...... xi
Page Chapter I
Introduction: A Review of Dynamical and Structural Properties of Crystals . 1 1.1. Overview of the T h e s is ...... 1 1.2. Basic Concepts of Harmonic Crystals: A Classical Description .... 3 1.3. Emperical Models for Force Constant and Dynamical Matrix .... 10 1.4. Microscopic Study of Force Constant: Dielectric Matrix Method . . . 16 1.5. Direct (Supercell) Approach of Calculating Frozen Phonon Frequency 26 1.6. Elastic Constants: Direct Calculation of Stress ...... 30 1.7. Green’s Function Approach of Linear Response Calculation ...... 36 1.8. Some Systematic Studies of Local Density Approximation ...... 40 1.9. Summary of Thesis R e s u lts ...... 47 References for Chapter I ...... 50
Chapter II
Elastic Constants of Si/Ge Superlattice and Bulk Si and G e ...... 53 2.1. Introduction ...... 54 2.2. The Molecular Dynamical Scheme and Calculation M ethod ...... 56 2.3. The Systematic Studies ...... 59 2.4. The Calculated Results For Si, Ge and S i/G e ...... 73 vii 2.5. Conclusions ...... 79 References for Chapter I I ...... 81
Chapter III
Strained Silicon: A Dielectric Response Calculation ...... 83 3.1. Introduction ...... 84
3.2. Symmetry Considerations ...... 8 6 3.3. R e s u lts...... 91 3.4. Conclusions ...... 110 References for Chapter III ...... 112
Chapter IV
Open Questions and Future W o rk ...... 115 4.1. Improving the Performance of Local Density Approximation . . . . 115 4.2. Properties of Gen/Sim Superlattices ...... 117 References for Chapter IV ...... 118
Appendix
A: Algebra of Stress, Strain and Elastic Energy ...... 119 B: Total energy and energy cutoff (size of planewave basis) ...... 123 C: Murnaghran Fit for Total Energy and Equilibrium Volume .... 126 References for A ppendix ...... 130
Bibliography
Bibliography . 131 LIST OF TABLES
Table Page Chapter I 1.1: Comparison of calculated phonon frequencies of Si and Ge at T and X with experiment ...... 29 1.2: Calculated values of the macroscopic dielectric constant, £oo, and optical phonon frequencies for Si ...... 39 1.3: Lattice constant and bulk modulus of Si with different energy cutoff . . 43 1.4: Calculated lattice constant and bulk modulus of Si, with different exchange-correlation function forms ...... 47 1.5: Static photoelastic tensor and dielectric constant ...... 50 Chapter II 2 .1 : Special fc-point sets and Ar-point density...... 60 2.2: Example of special fc-points of different symmetries and unit cell size, and its density...... 61 2.3: Energy Cutoff E Cut and number of planewaves, Npw...... 62 2.4: Data Set, the total energies of different linear fc-point density and number of planewaves, E{Ecut, Nf.pt)...... 63 2.5: Fitting parameters for convergence with fc-point densities, fitted by both of power law and logarithm ...... 64 2.6: Fitting parameters for convergence with the size of planewave basis, fitted by both of power law and logarithm ...... 69 2.7: Lattice constants a, bulk modular B , elastic constants Cij of Si, Ge, Si/Ge. Unrelaxed elastic constant C 4 4 , internal relaxation parameters 75 2.8: “Elastic theory” (ET) predictions of lattice constant and bulk modulus for Sii/Gej ...... 79 Chapter III 3.1: Characteristics of special values of the parameter ( ...... 90 3.2: Default parameters for calculation in this paper ...... 91 3.3: The photoelastic coefficient of hydrostatic strain, p \\ 4- 2 p i 2 , calculated from various strains ...... 97 3.4: The effect of lattice constant on the dielectric function and photoelastic tensor at three photon energies ...... 98 3.5: The effect of the self-energy parameter A on the dielectric function and photoelastic tensor at two photon energies. 100 3.6: Direct gap of silicon for various conditions. 101 3.7: Special fc-point sets for various symmetries 102 3.8: The effect of Nf.pi, the number of integration points in the irreducible Brillouin zone, on the dielectric function and photoelastic tensor at three photon energies 103 Appendix A A.l: The relationship between elastic energies, distortions and elastic constants. 121 Appendix B B.l: Total energy of 10 special ^-points for various energy cutoff. .... 124 Appendix C C.l: Total energies of differnt lattice constants and fitted parameters. . . 128 LIST OF FIGURES
Figure Page Chapter I
1.1: Valence force model fit to the measured dispersion relations for diamond along [ 1 0 0 ] and [ 1 1 1 ] direction ...... 1 2
1 .2 : Schematic illustration of short range force constants in shell model. . . 13 1.3: Dispersion relations for diamond along the principle symmetry directions with shell model fit to the data ...... 15 1.4: Calculated phonon frequencies of silicon in A (001), A (110), and S (111) directions...... 24 1.5: Phonon polarization at T and X for the diamond structure ...... 28 1.6: Total energy convergence through self-consistant iterations ...... 42 1.7: Total energy versus lattice constant, various cutoff parameters ...... 46 1.8: Exchange-correlation contribution to : Top, total energy; Bottom, bulk modulus, at electron density of Si (rs ~ 2 )...... 48
Chapter II
2.1: Total energy convergence versus fc-point density, Nj.pi...... 65 2.2: Total energy convergence versus the size of planewave basis, Npw. . . . 70 Chapter III
3.1: The photoelastic coefficient p n + 2 p j 2 as a function of photon energy for silicon under hydrostatic strain, compared to experiment ...... 92 3.2: The photoelastic coefficient p i2 — pn as a function of photon energy for silicon under a strain in the [001] direction ...... 93
3.3: The photoelastic coefficient P 4 4 as a function of photon energy for silicon under a strain in the [111] direction ...... 95
3.4: The photoelastic coefficient —P 4 4 as a function of the internal shift £ for three photon energies ...... 96 3.5: The dielectric function of unstrained silicon in the combination (e(w) — 1 ) 1 as a function of the square of photon energy, for 0 < u>2 < 10 (eV) 2 ...... 104 3.6: The dielectric function of unstrained silicon in the combination (e(u>) — I ) - 1 as a function of the square of photon energy for 0 < w < 1 eV...... 106 3.7: The photoelastic coefficient and dielectric function plotted in the combination 1/(pi 2 --.Pll)£(w) 2 as suggested by the single oscillator model. 109 3.8: The photoelastic coefficient and dielectric function plotted in the combination — l/p 4 4 e(u>)2 as suggested by the single oscillator model. 1 1 0
A p p en d ix B
B.l: Total energy versus energy cutoff, Ecut. 125 CHAPTER I INTRODUCTION: A Review of Dynamical and Structural Properties of Crystals
In this thesis are studies of the structure and the elastic properties of Ge/Si
superlattices together with the systematic studies of the convergence properties of
the underlying calculations. We address some problems which arise in the attempt
to understand the structural and electronic properties semiconductor superlattices.
The Local Density Approximation (LDA) is used in all our calculations, with a
planewave expansions and pseudopotentials for the core-electron interactions. The
Brillouin zone integration is approximated by a summation over a set of special
fc-points.
1.1 Overview of the thesis
This thesis is organized as follows. In Chapter 1 we will first review the theo
ries and models for phonon and elasticity calculations, from earlier formalisms to
current development, and present the theoretic results from various stages of the
development. Emphasis is placed on the recent development of a more accurate and
efficient way for calculating these properties. We will start with the classical dis
cussion of force constant matrix (dynamical matrix) and move onto some empirical
models, such as bond force model and shell model. Further, we will present the mi-
1 croscopic description of the force constant (and response) for the electronic system, with the dielectric matrix approach, the direct (supercell) approach and the most recently, the Green’s function approach. The second part of this chapter is devoted to the convergence studies of the local density approximation and the computa tional scheme. We will investigate the effects on various calculated properties, such as total energies, lattice and elastic constants, due to various approximations made in the calculations, such as the size of planewave basis, different pseudopotentials and exchange-correlation forms. We will also compare the different computational methods on the self-consistent procedures used to find the ground state energy, namely the steepest descent versus conjugated gradient method.
In Chapter 2 we report our calculated elastic properties of Gei/Sii superlattice with comparison to that of bulk Si and Ge, including lattice constant, bulk modulus and other elastic constants. The calculated results for bulk Si and Ge agrees well with experiment. In the chapter, we will discuss extensive convergence tests of the total energy of Si computed within the local density approximation with a planewave basis and pseudopotentials. The convergence study includes the total energy convergence over the size of the planewave basis and Brillouin zone sampling
Appoint density. This work, done with Doug Allan and John Wilkins, has been submitted for publication.
In Chapter 3 we present the calculated optical response for strained Si, where the strain-induced birefringence is calculated with crystalline silicon for pressure applied along the [001] and [111] directions of the crystal. Results for the dielectric function and its change under hydrostatic strain are also given. The results are calculated for photon energies in the range 0 to 3.25 eV, i.e., below the direct gap. 3
The fully self-consistent Kohn-Sham Local Density Approximation calculation uses pseudopotential, a planewave scheme, and a self-energy correction in the form of a rigid shift of the conduction bands of magnitude A = 0.9 eV. Agreement with experiment is very good in the static limit, considering disagreements among the experimental values. This work, done with Zachary Levine, Hua Zhong, Doug Allan and John Wilkins, has been submitted for publication.
Finally, in Chapter 4 we summarize the main results of this thesis and discuss the future work on the strained Ge/Si superlattices, such as structural, electronic and optical (linear and non-lineax) properties.
1.2 Basic Concepts of Harmonic Crystals,
Phonon Frequencies and Elastic Constants:
A Classical Description of The Force
Constant and Dynamical Matrix
We consider a crystal lattice of ions, as discussed in Solid State Physics by
Ashcroft and Mermin* and Introduction to Solid State Physics by Kettel^, and assume that at zero temperature each of the ions sit at the equilibrium position of the Bravais lattice R (here we start with the simple Bravais lattice); and the dis tortion from this equilibrium position, u(R), is small comparing to the interatomic distance. The position of each ion can be expressed as
r(R ) = R + u(R ). (1.1)
We further assume that the ions are interacting with each other through a pair po tential, (f>(R). The total potential energy of these ions is the sum of this interaction over all the lattice sites
V = \ £ «R-R'). (1-2) R.R'^R when all the ions sit in their equilibrium positions. Use Eq. (1.1), we can rewrite the total ion interaction for the distorted crystal as
V = \ £ «r(R) - r(R')) =\ £ - R '+ u(R) - u(R')). (1.3) R,R'#R R.R'^R In classical mechanics, the Hamiltonian consists of both kinetic and potential energy
* = £ ^ + " . a - * ) R where P(R) is the momentum of atom. For harmonic crystals, we can expand the total ion potential of Eq. (1.3) according to Taylor’s theorem as
v = \ £ «R-R') +i £ (u(R)-u(R'))-V,S(R-R') R.RVR R.RVR (1.5) D [u (R)- u (R;) - V ] V ( R - R ;) + 0 ( u 3). R,R'#R
Using the equilibrium condition
£ v * (R - R ') = 0, (1.6) R' we can write the harmonic approximation of total potential energy as
U = Ueq + Uharm, (1.7 ) with
uharm=j £ (u^(R)-up(R'))^(R-R')K(R)-u„(R')), RR’/n/ n Converting Eq. (1.8) into a more familiar form as expressed by the force constant,
D - a three by three matrix, we have
fjharm = I £ u (R )D (R - R')u(R'), (1 .9 ) RR' where u is a three dimensional vector, and the elements of the force constant, Dpi, are defined as
Dpi, = y ] R — R ) — ^pv(R — R ). (1-10) R" Equation (1.10) can be understood as follows: (1) if only one ion moves, R = R;, the energy arises from the pair potential change between this displaced ion and the rest of the ions, hence the summation over Rw; (2) if two ions at different sites are displaced from their equilibrium position, the interaction is through the second order derivative of the pair potential (force constant) between this two sites.
There are several basic symmetries that this force constant matrix has to obey.
These are:
1
D p„(R - R ;) = D „p(R ' - R) (1.11)
which can be obtained straight forwardly from Eq. (1.9) by interchanging R
and R ', physically, this must always hold since the force constant matrix, D, is
the coefficient in a quadratic form;
2
Dpi,(R - R ') = D „p(R - R '), (1.12)
which holds if the Bravais lattice has inversion symmetry. The elastic energy
is the same for the ion at site R to have displacement of u or at site —R and
displacement of —u; 6
3
Y , DpV{ R - R /) = 0, (1.13) R' which comes from the fact that if every ion has the same displacement (the
entire crystal moves), there is no elastic energy involved.
We can write the ion-motion-of-equation, from Eq. (1.4), as
Afu(R) = - ] T D (R - R ;)u(R ;). (1.14) R'
Following the convention by Ashcroft and Mermin 1 and transforming into &-space,
u(R,<) = b(k)ei(k-R" w<), (1.15) we can rewrite Eq. (1.14) as
Mu;2 b(k) = D (k)b(k), (1.16) where D(k) is the dynamical matrix, defined as
D (k) = D (R )e",k R (1.17) R
With the symmetries for D(R) described in Eqs. (1.11,1.12), the dynamical matrix has the following properties: ( 1 ) it is an even function of k, ( 2 ) it is a symmetric matrix, and (3) it is real. From linear algebra, a real symmetric matrix can be diagonalized, hence, we can define the normal modes of the lattice distortion
(phonon) as bs (s=l,2, and 3) with
D (k)bs(k) = A(k)bs(k), (1.18) with the orthogonality condition
bs -bfi/= 5 ss/, (1-19) 7
and the frequency of the normal mode phonons, ws(k)
(1.20) “ - ( k ) = \[W-
For a Bravais lattice with p atoms per there are 3p normal modes, with 3
acoustic and rest 3 (p — 1) optical, if p = 1, zero optical modes. (There will be one
more index on each of the variables discussed in above equations to indicate the
atom within the lattice, but the idea remains the same.) The orthogonality relation
is generalized as
] > > * ■ * « . b-(k)= (i.2 i) t=i
with M i the mass of the z'-th type ion. The frequency of the normal modes can be
obtained from solving the eigenvalue equation
6.et\ix)^ M iSijS^ t, — = 0, (1.22)
where i and j are the index for the atoms within the unit cell, p and v are the
Cartesian directions.
Equation (1.20) shows that the frequency of the normal mode phonons can be
obtained if the dynamical matrix is known. We now further show not surprisingly
that the elastic constants can also be obtained from the dynamical matrix.* Since
elastic constants apply in the long wave length, we use a continuum approximation for the displacement from ions
(u^(R) - U„(R')) = B aU<,(£ ; R')-)(ftr -g where x a is the Cartesian directional vector. Using above relation, then Eq. (1.8) jjharm _ 1 V '' ^ d u ^ R - R/) ^ duv{R - R ') ^ (R-RO/iwr ^ 2 4 ) ( - 5 £<*»■ - O W &- R "X-Rr - R ?). R" Change (R - R') and (R - R") to R in both summations and note that the summations are over all the lattice sites, we can rewrite Eq. (1.24) as V h°™ = \ £ (1.25) R/xi/crr R The elastic energy is defined by jjharmharm j —— = - 2 J flan Ccfuvt Pvt , (1.26) ~ = 2 (TfiVT where Co-pvr is the elastic tensor, 0 is the volume of the unit cell and the strain tensor, /Xcr/i, is defined as Ifdun dua, ci 07\ ^ = 2 (& v + t o ? ' (L27) Compare Eq. (1.25) and (1.26), we can express the elastic constant in terms of the force constant as C(TflVT = — ^ -f* RftD crv(R) R t 4" RaDuri^tyRv 4* R[iD From the symmetries of force constant, £)^j/(R), and Eq. (1.28), we can see that the elastic constant, cailVT is invariant under index transformation (crp <-+ v t ), ( In the case of a cubic symmetry, which is relevant for our calculations of Si and Ge, there are only 3 independent components. Conventionally, they are written in the form of as second rank “tensor” (they do not obey the usual tensor transform actions) as C11 = Cxxxx = tyyyy ~ czzzz C\2 — Cxxyy — Cyyzz — Czzxx (1.29) C44 = Cxyxy = Cyzyz = czxzx- Following this rule, the strain components are further defined as (1.30) with a = 1 for /x = i/, and 2 for \i ^ v. These strain components are further converted into a six-dimensional “vector” via the convention: 1 «-» xx, 2 <-> yy, 3 *-* zz , 4 <-» yz, 5 <-> zx, 6 <-> xy. In a lower symmetry case, such as for Ge/Si superlattices, there will be more independent elastic constants. Under tetragonal symmetry, choosing 2-direction as the symmetry direction, the original elastic constants split as follows: C u —> (C'll>C,33)> £l2 -♦ (Ci2,C13), and C44 -» (C44, Cm ), resulting in 6 independent elastic constant. With even lower symmetry, such as orthorhombic, there are further splitting, resulting in 9 independent ones: C u , C22, G33; C12, C23, C31; and C44, C55, Cqq respectively. For the force constant expressed in Eq. ( 1.10), we consider the potential en ergy change for only the ions. In reality, the total energy change will also involve contribution from the electrons responding to the change of the positive ionic po tential background. As the electron velocity is much faster than the speed of the ions, the adiabatic approximation is usually assumed, namely, that the electrons respond instantaneously to the potential change. The real force constant will be much different from the bare ionic potential change, but the relationship between 10 the force constant (dynamical matrix), phonon frequency [Eq. (1.20)] and elastic constants [Eq. (1.29)] remains the same. In the following sections, we will discuss the efforts and developments in obtaining the force constant (dynamical matrix) using various empirical models and microscopic descriptions. 1.3 Empirical Models For Force Constant and Dynamical Matrix Before the extensive use of large mainframes, supercomputers, et al. , devel opment of the computational techniques, and first principal calculations, several empirical models for the force constants were proposed to fit the experimental data on phonon dispersion curves and elastic constants for the unit cell with two atoms. These models can be divided into two groups: (1) the Valence Force Model , 2 ,4 ,5 which approximates the interactions between ions by forces (“springs”) on bond lengths and bond bending forces, and (2) the Shell (Dipole) Model, 2 ,4 ,5 which ap proximates the ion and its valence electrons as dipoles interacting through Coulomb force. There are many variations in these models as more parameters are included to achieve a better fit to the experimental data. In this section we will briefly introduce the concepts and discuss the validities of these two models. Valence Force Model The idea of the valence force model was first proposed by Born , 5 who assumed that the interaction between a pair of atoms can be approximated by two parameters (spring constants): along and perpendicular to the line between the pair; Born further assumed that only the nearest neighbor interaction (interaction between the two atoms within the same unit cell) needed be included in the force constant matrix for the diamond structure crystals. It has been demonstrated that this simple two-parameter model is unsatisfactory 4 ,7 in reproducing the elastic constants. An improved model, proposed by Keating , 7 expands the ion potential in a variable that automatically satisfies translational and rotational invariant requirement - a scalar product of the ion displacements Ar r ' = MR) • r(R') - R • R')- (1-31) As the condition for the equilibrium, the linear term of the potential expansion in A has to vanish. The harmonic potential can be expressed in terms of this new variable as Uharm _ ^ 7^RR'R"R'"^RR'^R"R"'• (1-32) RR'R"R'" This model for the diamond structure includes only four vectors in the scalar prod uct; three of these four are the vectors from the distorted cell to the adjacent cells (three of the six nearest neighboring cells) and the fourth one is the vector between two species (two subsets) of the atoms in the same distorted cell. Hence there are ten scalar products of A and have ten parameters to fit the experimental re sults. This gives a better fit for the experimental elastic constants than the original nearest neighbor model. This model has later used by McMurry et al. ® to fit the phonon dispersion curves for diamond over the zone; the obtained good agreement with experimental data is shown in Fig. 1 .1 . The solid line is the theoretical fit, and the dots and triangles are the experimental measurement. Despite the simplicity of Keating’s model, it has an obvious problem that the model excludes the long-range interactions. Even though in the long-wavelength 12 AJO) oooo a* > * ft «. *M 04 90 (tool Figure 1.1 Valence force model fit to the measured dispersion relations for diamond along [100] and [111] direction. The fit included 16 valence coordinates: 4 nearest neighbor (C-C) bond lengths (Ar,-); 12 C-C-C bond angles (roA limit, the interactions maybe reduced to a short range one , 7 in order to fit the phonon dispersion through the zone, the long-range interaction should be included.®’® To include this long-range interaction, Martin® assumed an effective charge ±Z* (e. g. for the III-V compounds) on the ion, and these effective charges interact through Coulombic potential. We emphasis here that this model uses the short-range pa rameters (the bond length and angle) from Keating model and adds only one more parameter, the effective charge ±Z*, which is determined by the fitting the exper iment data. A further improvement on the model was to include the electronic 13 F (1 2 ) Figure 1.2 Schematic illustration of short range force constants in shell model. The interactions are shown as: D between cores; S between shells(valence elec trons); k{ between core and shell(valence electron) of the same atom; Fij between core and shell(valence electron) of the neighbor atoms . 4 (From Sinha, CRC Crit. Rev. Sold State Sci., 3, 281, (1973).) polarizations in the ionic potential, but the idea is unchanged: the basic interaction is between bond length and bond angle which are short range, plus some correction to include the long range interaction effects. These modified valence force models improved the quality of the fitting for some m aterials, 9 but not for diamond. This leads to an open question: why the phonon dispersion curve for diamond fitted by McMurry® (Fig. 1.1) agrees so well with the experiment. The fit uses only short-range interaction indicating that the long- range forces are relatively unimportant, a result that can not be explained from a microscopic theory. Shell fDipole) Model 14 The shell model considers the electrons as shells centered at the ion. The ion displacement moves the ion away from the center of the shells, which creates an dipole which interacts with other dipoles similarly produced on other atoms. The idea was originally created for the ionic crystals where the long-range interaction is important, such as alkali halides,** (a schematic illustration of the short-range interaction is shown in Fig. 1.2) but the success of using this model for diamond structure crystal by Cochran,1® makes it useful for studying the phonon frequency for semiconductors, most for III-V compounds. The advantage of the shell model is that it can be used to calculate dielectric function of the crystal2, since the potential change can be induced by the external ekectric field, (f3\E. In this model, there are two dynamical variables: the ion displacement, u, and the dipole momentum, p. The harmonic energy of shell model is expressed as fjharm = + lpSp + uTp> (1 33) where matrices R, S and T are empirical force constants which consist of both a long-range Coulombic interaction and short-range interaction. Using Eq. (1.33) we can write the equation of motion for the displacement, u w2M u = R u + T p , (1.34) and for the dipole, p r W S p = 0. (1-35) Eliminating p in Eq. (1.34) yields « 2M u = (R + TS-'T t)u. (1.36) There are seven independent parameters in this model, two for the matrices, R and T, and three for S. There are further approximations that can be used to reduce .2 ,4 ^ .6 A 10 0 .2 4 6 .8 10 10 i J6 4 0 1 2 3 J, .5 [OOfl PfO] CKO] UK] REDUCED W/WE VECTOR COORDINATE I Figure 1.3 Dispersion relations for diamond along the principal symmetry di rections with shell model fit to the data. The fit has used 11 parameters (including nearest and second nearest neighbors ) . 4 The different symbols (dots, triangles) are the experimental data. (From Sinha, CRC Crit. Rev. Sold State Sci., 3, 295, (1973), originally from Warren et al. Phys. Rev., 158, 805, (1967).) the number of parameter to be fitted (as discussed by Sinha4). The comparison between theory and experiment is shown in Fig. 1.3, where the phonon dispersion for diamond is plotted in the principal symmetry directions. The fit is done with a more sophisticated modification of the simple shell model with 1 1 parameters .4 Though the advantage of the shell (dipole) model is not represented by the fitting of the phonon dispersion of diamond, where the reason of the success of the fitting by short-range-interaction (valence force model) is unexplained, it improves the quality of the fitting for III-V semiconductors4. Moreover, it has the potential to calculate dielectric functions of crystals . 2 16 To conclude this section, we point out that the valence force model is simpler and fits the phonon dispersion curve of diamond better than the dipole model (where the long-range interaction is weak), but the dipole model does better for Si, Ge and III-V compounds.** Since both models are empirical, the quality of their fits vaxies from case to case. 1.4 Microscopic Study of Force Constant, Dielectric M atrix Method The microscopic studies on the force constant are based on the electron response to the external potential disturbance. Before we get into this discussion, we will first introduce the basic ideas of the Density Functional (DF) formalism , 1 1 '* 2 simply because most of the applications of this microscopic theory will employ the Density Functional formalism. The basic theorem of this formalism is that the ground state energy (and hence ground state properties) of a many electron system can be expressed as a functional of the ground state electronic density.12'1** Based on this theorem, we can solve the ground state of a many-electron system, with the Hamiltonian = T + Ve« + (1-37) i by finding the electronic density which minimizes the total energy functional of the form £[n(r)] = J drVext(r)n(r) + F[n] =T[n(r)] + J d rl^ r^ r) -f ^ J + ^zcMr)]. (1.38) 17 T, Vee, and Vext in Eq. (1.37) are the kinetic energy, the interaction of electrons and the external potential (including the ion-electron interaction) respectively. The terms in the second line of Eq. (1.38) are the kinetic energy functional, the external potential, the electrostatic interaction of the electrons and the exchange-correlation energy respectively; The functional F[n] includes everything except the external potential contribution to the total energy of the electronic system. The electronic density is expressed as a sum of the single particle density N »(r) = E l*M IS- > The exchange-correlation energy is further approximated; a local functional form E xc = J drVic(rc(r))n(r), (1.40) is assumed, hence, the name “Local Density Approximation” (LDA). With this approximation we can determine the ground state properties by variational principle to solve the ground state density which corresponding to the minimum of the energy functional expressed in Eq. (1.38). Since this is an approximation, several different functional forms for the exchange-correlation have been developed as discussed by Jones and Gunnarsson 1 3 in their recent review. We will consider the effect of two different functional forms together with other approximations in the calculation later in this chapter and in Chapter 2 . For the moment, we use the energy function expressed by Eq. (1.38) to discuss the lattice dynamics. To start: consider response theory. To analyze the dynamical properties, we have to relate the change in the total energy to the change of the dynamical vari ables, as discussed in Section 1.2. We separate the total change (note we do not yet make a harmonic approximation) of the system as sum of the total energy change 18 of the ionic background due to the distortion and the electron response to this ion distortion, AEtot — A E ion + A E e\ — A E{on + {Ee[[n ] E e[[n]). (1.41) Within linear response, the electronic density response to the change of the ion background potential change SVion can be expressed by the relation fn(r)= /*'x(r,r')«VU(r'), (1.42) where x the linear response function .5,14,1®’16 Restricting ourselves to small perturbations, we use the notation n'(r) = n(°)(r)+ ra^^r) for the perturbed electronic density, and V '(r) = V(6 )(r) + Vr^1 )(r) for the perturbed potential. With this notation, we can calculate energy change of the electron system to second order, using Eq. (1.38) A E ei = ( J drV'(r)n'(r) + Efn^r)]) — ( J drV^(r)n^°)(r) + ir[n^°)(r)]) = JdrV^\r)[n^(r) + n^)(r)] + J d r [V ^ (r ) + ^ n ^ )(r) + — [ drdr1- r^(1 )(r)7^(1 )(r,). 2 J Sn(r)6n( i r1) (1.43) The equilibrium condition is 8F y (°)(r) + = 0. (1.44) £n(r) n(°) We can use the variational principle, 6A E e[/6n (J) = 0, to obtain the relation be tween the the change in potential and the electronic density. This leads to 82F —y^)(r) = n ^ )(r;). (1.45) J dr' 5n(r)5n(r') n(°) Inverting the above equation, we have a linear response of the density to the po tential change. By comparing to Eq. (1.42) rewritten in terms of the relevant variables, nW(r) = J dr'x(r,r')K( 1 )(r')- (1.46) we find c2 p X 1 ( r,r /) = -(^ r^n^ ). (1.47) If we can compute Xi the linear response function, in the above equation, the linear response theory is complete. Unfortunately, Eq. (1.47) only gives the inverse of x> which is not easily invertible. Before discussing the inverting of x-1 j we first derive the formula for force constants in terms of x(r>r/)- We relate this response function x ( r >r 0 to the dielectric function by defining the total (self-consistent) potential, V scf as (not including the exchange-correlation contribution) ysc/=y(l)^+ f ^ ^ 2 ---- 1----- |X (ri,r 2 )U(1 )(r 2 ) J | r i " r 2 1 (1.48) = j dr/e-1 (r,r/)y(1)(r/). We find that e_1(r, r') = % - r') + J drt ^ r ^ x ( r /, ri). (1.49) We can rewrite Eq. (1.49) by operating with Vj! on both side to obtain the linear response function, x 5 as X(r.r') = - f ( r , r')]. (1.50) If we assume the interaction between ions is Coulombic with no overlap between the ions*®, and no external potential except the ionic potential (Coulomb) on the 20 electronic system, we can rewrite the total energy change in Eq. (1.41), using (1.50), as A E tot = J (1.51) In this result the equilibrium conditions j drVW(r)[n(°)(r) + nj^( r)] = 0 (1.52) and ionic potential v V ( r ) = 4TOto,(r) (1.53) have also been used. Using Eq. (1.51) we can calculate the force constant (or dynamical matrix) by writing the potential change due to the distortion of ion positions in terms of the ionic displacements. The ionic potential (Coulomb) in the crystal can be expressed as sum over individual Coulombic interaction between positively charged ion (or core, if the ‘ion’ consists of the ion and the closed core electrons) and the valence electrons Vto,(r) = 5>(r-R), ( 1 -5 4 ) R where v(r) is the Coulomb potential for an individual ion and the summation is over all the ions in the crystal at the sites R. For a small displacement of the ion, the total ion potential change V ^^r) can be expressed by the first-order Taylor expansion, v W (r) = E -ai,'E — UW - t1-55) R 21 Using Eq. (1.51) and (1.55) we can write the total energy change, in the form of force constant, as AJS«* = c E / drdT'8v{r- R)vh -\r, r ') 8 "(^R,R')u(R)u(R') (1.56) RR' In a typical calculation, where the ion (core) interactions are Coulombic while the ion(core)-electron interactions are approximated by a pseudopotential, the total effective interaction is given by V « R ), r'(R')) =ZRZR/t>(r(R), r'(R')) + J drdr'vps( r — H )x(r,rl)vps (r1 — li1), with v(r) is the Coulomb interaction between the ions (cores), and vps(r) is the pseudopotential between ion (core) and electron. So the force constant can be written as IV (R , R') = V p v {R, R') (R ? R#) (1.58) D ^ i R ,R ) = ^ V ) iI/(R ,R /), R' where VW E -H' > - 5 ^ ? - (L59) We convert this force constant to the dynamical matrix, D^^(q, R R r) £V„(q,, RR') =(A/rM r/)5R_ 1 x Y , W * + G Riq + G' R ')-« RR,Y V^ ( G R;G' R"). G G ' R" (1.60) where V^„(q + G R ;q + G ' R;) =exp[z(q + G • R)](q + G)^x V (q + G R; q + G' R')(q + G')„ • exp[i(q + G')R'], (1.61) 22 and V (q + G R; q + G ' R') =ZR ZR ,v(q + G)8g g > +v*>8(q + G; R )X(q + G , q + G V * ( q + G '; R;). (1.62) Note that the Fourier transform of both Coulomb potential of ions (cores) and pseudopotentials can be calculated straight forwardly. From Eq. (1.60) together with (1.61) and (1.62), it is clear that to solve for the force constants D, we have to know the response function, x- So we introduce the polarizability function x, which relates electron density change, 8n, to the induced potential, 8Vind, as 8n = J dr'x(r,r')6Vind(r'). (1.63) Together with the definition of the dielectric function 6V*nd(r) = J dr'e(r,r')v(iy>(r’), (1-64) and Eq. (1.42) and (1.48), we can write the relationship between the linear response function, x> the polarizability function, x, and dielectric function, e in the operator form (not include the exchange-correlation contribution) X = Xe _ 1 e =8 — i>x (1.65) e- 1 =8 + vxe-1 . Using perturbation theory , 5 the momentum-spaceform of polarizability, x(q + G, q + G;) (sometimes written as Xgg'C^)) can written as X(q + G ,q+Q ,)= n -1 E /"'k+<1 ~ ^ X nn'k en'k+ where |nk > is the Bloch state of n-th band of wave vector k, is the eigenenergy and /„k is the occupation number. We can obtain x from the relation shown in Eq. (1.65) as X(q + G, q + G') =x(q + G, q + G;) + ^ x ( q + G ,q +G w)u(q +G ,;)x(q+G7,,q +G'). ^ ^ G" Equation (1.67) for x can be solved iteratively. With Eqs. (1.60), (1.66) and (1.67) we have established a microscopic description of the force constant (dynamical matrix). An application of this approach to predict the phonon dispersion curve for Si is shown in Fig. 1.4.*® Using this iterative method, we can avoid inverting the matrix, as required by Eq. (1*47). There are other ways of solving for the response function, x- Under the Local Density Approximation, an external potential disturbance, 8Vext (corresponding to V ion), will induce the self-consistent potential change, 8Vscf (corresponding to V ind), as 6Vscf = 8Vexi + 8Vcoul + 8VXC, ( 1 .6 8 ) where Vcou\, and Vxc are the Coulomb and exchange-correlation energy, j M O ^Kou/(r) = J ^r' |r _ jjj (1.69) &Vxc{r) = J dr'Kxcirjr^Snft), with c2 j? k ^ = s ^ F ) - X = {l-xV c- xKxc)~lx, (1.71) 24 SILICON 02 00 U 02 03 OCQS Figure 1.4 Calculated phonon frequencies of silicon in A (001), A (110), and S (111) directions (in THz). In the calculation, the test charge dielectric function is used, € = 1 + VcXi which assumed that exchange-correlation does not change during the perturbation. The Slater exchange-correlation, Vxc(n) = — ^(^p )1^, and empirical pseudopotential, V(q) = (cos(V^g) + V^)e^4q, are used in the calculation.1® The dots are the experimental data. (From Devreese et al. Electronic structure, dynamics, and quantum structural properties of condensed matter , edited by J. T. Devreese and P. E. van Camp, (Plenum, New York, 1985), p. 170.) with = j r h r (U 2 ) W ith x known through the perturbation approximation [Eq. (1.67)], we can invert the matrix to solve for x according to Eq. (1.71). Furthermore, we can solve for the dielectric function (microscopic and macro scopic) following the same theme, by combining the first and third equations in Eq. (1.49) (including the exchange-correlation contribution), e-1 = l + (Vc + Kxe)x, (1-73) 25 which includes the exchange-correlation contribution. Alternatively excluding the contribution from exchange-correlation, we get the RPA form eRPA = 1 + M* - xVc)x, (1-74) which is basically Eq. (1.73) with zero K Xcl 7 [compared with Eq. (1.71)]. We relate this microscopic dielectric function to the macroscopic (average) di electric function (which is determined by the diagonal elements of e - 1 , 1 7 as £macro{ci~k' G) = , —-. (1-75) eGGta) W ith x known [Eq. (1.71)], we can solve for the inverse of the microscopic dielectric function, e, and invert its diagonal elements to obtain the macroscopic dielectric function. Because the Coulomb interaction is long range, its Fourier transform must be decomposed into two paxts: G = 0 and G ^ 0, with the contribution from the G = 0 called the local field correction. The calculations by Baroni and Resta1®, Hybertsen and Louis , 1 7 and Levine and Allan 1 9 have shown that this correction is important (the estimate of this local correction contribution ranging from 1 0 % to 30 % by different authors). The advantage of this method is that the force constant (dynamical matrix) depends only on the ground state properties. Consequently, we can use it to cal culate the phonon frequencies over all the Brillouin zone. (This is to be compared to the frozen phonon calculation which calculates the phonon frequencies only for high symmetry points in the zone, which approach we discuss in the next section.) The disadvantage is that only the linear effect is considered, which is not a se rious approximation for the lattice dynamics is concerned. One real difficulty of 26 this method is the huge computational effort in obtaining the dielectric matrix, for which there is a summation over the whole spectrum of the conduction band.1®’2® This difficulty is overcome by a Green Function Approach 2 0 which we will discuss later in this chapter. 1.5 Direct (Supercell) Approach of Calculating Phonon Frequency, Frozen Phonons In this section, we turn our attention to the total energy calculation, and relate the difference between the total energy corresponding to different crystal geometry to the phonon frequencies. This approach is called “direct approach” (or “supercell approach”) and the phonon frequencies are those of “Frozen phonons ” . 1 5 ,2 1 The name comes from the frozen phonon approximation, which assumes that each of the distorted crystal structures can be treated as a distinct system unrelated to the undistorted system. The lattice vibration can be ‘frozen’ to form different crystal structures. This method was first used by Wendel and Martin 1 5 to calculate the phonon frequencies for Si. In this direct approach, the undistorted and distorted crystals are treated as two distinct systems. This gives an alternative to ground state perturbation the ory treated by linear response theory we discussed in the previous section. This approach is based on two separate approximations, the adiabatic approximation, which assumes that the electron wavefunctions follows the ion displacement in stantaneously, and the frozen phonon approximation. We can calculate the energy change for the crystal distortion, by calculating the total energies of the distorted and undistorted crystal; if we can relate this energy change to a certain vibrational 27 mode (phonon), we can deduce the phonon frequency. But this idea was purely academic before the Density Functional formalism , 1 1 ,1 2 because of the difficulty of achieving sufficient accuracy in the total energy calculation. As noted by Kune 2 1 the earlier attempt of frozen phonon calculation by tight-bonding method proved to be unsuccessful . 2 2 As pointed out earlier, Wendel and Martin1® were the first to use the local density approximation to obtain a reasonable agreement with the ex periment of the phonon frequency for Si. Later the similar calculations were carried out by Yin and Cohen to extend the calculations to C, Si and Ge ,2 ®’2 '1 and Allan and Teter for Si 0 2 . 2 5 These calculations have proved that the direct (supercell) approach gives an accurate description of the phonon frequency. Having established the basic idea of the method, we will now concentrate on how to relate different phonon modes to the crystal distortions. For the diamond structure crystal, there are 2 atoms per unit cell. Each of them forms a face- centered-cubic lattice of its own; we will call them as A and B species. From the discussion in Section 1.2, we know that there are 6 normal modes: 2 transverse acoustical (TA), 1 longitudinal acoustical (LA), 2 transverse optical (TO), and 1 longitudinal optical (LO). The ionic vibration follows Newton’s laws, which spell out the relationship between the energy change and the velocity (or frequency). The different normal modes corresponding to the different ion vibrations shown in Fig. 1.5. 2 4 Except for the TO(r), for which it is known experimentally that the third order anharmonic term is non-negligible, and the total energy can be written as A Etot{u) = ^{~mSi)u^0u2 + kxyz( - j= f + .... (1.76) 28 (110) P la n e L T O ID — TA(Xj • © TO(X) 1 1 LOA(X) 4 *x Figure 1.5 Phonon polarization at T and X for the diamond structure. Atoms are numbered and denoted by black dots. The solid lines denote an atomic chain in a ( 1 1 0 ) plane, and the dashed lines denote the projection of an atomic chain a distance \/2a/4 away from the plane, where a is the lattice constant. The arrows (or circle with dots or cross) indicate the atom vibration directions. ‘LTO’ stand for LO and TO, and ‘LOA’ for LO and LA. T and X indicate the q vector in the Brillouin zone. 2 4 (From Yin and Cohen, Phys. Rev. B26, 3264, (1982). Each mode has a dominant harmonic term, and the total energy changes can be written as A Etot(u) = ~M u2u2 + ..., (1.77) where in above equations, u is the ion displacement, kXyz is the cubic-anharmonic force constant and M is the effective mass for LO mode). The calculated phonon frequencies for Si and Ge are shown in Table 1.1, with a comparison to the experimental results. * 29 Table 1.1 Comparison of calculated phonon frequencies (in THz) of Si and Ge at T and X with experiment. Calculated with 28 special fc-points (see Chapter II) for diamond structure, and energy cutoff of 10 Ry for Si, and 12 for Ge, under local density approximation, and the Hamann- Schluter-Chiang pseudopotentials. 2 4 (From Yin and Cohen, Phys. Rev. B26, 3265, 3266, (1982). LTO(r) LOA(X) TO(X) TA(X) Si LDA 15.16 12.16 13.48 4.45 EXP 15.53 12.32 13.90 4.49 Ge LDA 8.90 7.01 7.75 2.44 EXP 9.12 7.21 8.26 2.40 We want to emphasize that for all the distortions discussed above, the volume of the unit cell (supercell) is unchanged, we only rearrange the positions of the ions in the cell to form some specific distortions corresponding to the normal modes of the vibrations. One other advantage of this direct (supercell) method is' that we can use it to calculate the elastic constants of the system. The calculation is again straight forward as long as the distortion (strain) of the unit cell is properly chosen (see Chapter 2). Except for the calculation of shear modulus C 4 4 , for which the internal shift is crucial, the ions take positions proportional to the unit cell distortion (strain). (See Chapter II for details of these calculations.) In principle, this approach can be used to calculate the phonon dispersion by calculating each column of the polarizability 1 7 of an arbitrary (but rational) q, by 30 introducing a monochromatic external perturbation A V ext(r) = Vbei(q+GoK (1.78) This perturbation leads to a charge density change for wave vector q + G, which is obtained by direct comparison of the self-consistent calculation of perturbed and unperturbed system, A n(q + G) = V&XGGofa)- (L79) Repeating this for different q and Go generates the whole response matrix, and hence the force constant (or dynamical matrix) as well. The direct (supercell) approach has the advantage of being a clear physical picture; it is not limited to the linear response; and the exchange-correlation is included automatically during each self-consistent calculation. But it has the limi tation that only one frequency for one mode at one single q-value is calculated for each geometry chosen; the size of the supercell increases when frequencies of lower symmetry points (q) are considered. This is a serious limitation in applications because the self-consistent calculation becomes prohibitively large. To combine the advantage of both direct approach and dielectric matrix method, and avoid their disadvantages, Baroni et al. had proposed a new approach: the Green function approach,'*® which we will discuss in Section 7 of this chapter. First we will discuss another method used to calculated the elastic constants. 1 . 6 Elastic Constants: Direct Calculation of Stress In the spirit of the direct approach for calculating the phonon frequency by the energy difference between different ion configurations, a method of calculating the 31 elastic constants through directly calculating the stress was developed by Nielsen and M artin . 2 6 ,2 7 The similar idea for calculating the force directly (instead of total energy) was also used in the phonon frequency calculations carried by Yin and Cohen . 2 4 The idea is to calculate, within the Local Density Approximation, the stress of a cell distorted from its equilibrium structure. The definition of a stress, induced by a strain, (iap, is ( i - 8 o ) where ft is the volume of the cell, a and (3 are the Cartesian directions. The strain (jl scales the positions of the ions as x —> ( 1 + /z)x (with no internal shift). The reciprocal lattice vectors scale as G —► (1 + /i)-4 G under the strain. Hence, we can find the derivative of the reciprocal lattice vector with respect to the strain is BC 5 ^ - - * ^ . ( 1 .8 D There are two invariances under the strain as noted by Nielsen and Martin : 2 6 (1) the structure factor, elG x ; (2 ) the product of ft and charge density |<^|2. The total energy expression in the planewave expansion which was derived by Ihm et al. 2 6 is EtotlSl = £ l« k + G)|2(k + G)2 + 2 ,r£ nkG G + £ £«(G )p (G)* + £ Sr(G)Vr£(G)p(G)* G Gr + Sr(G ~ Q,)Vlr ‘&(k + G, k + G')tf„(k + G)«^(k + G7)* nkGG'ir +(^3 ar)(^-1 Y ^ Zr) + Sl^lEwaldi T T (1.82) 32 where k extends over the first Brillouin zone, G is the reciprocal vector, The total stress can be expressed as separate components contributed by the corresponding components of the total energy: ( 1 ) kinetic energy contribution — X / M k *^)|2(k + G )a(k + G )^, (1.83) nkG (2 ) local pseudopotential contribution ~ E S’-(G K ^ | ^ r 2G‘»G /J + Vr ( G ) W f ( G * ) , (1-84) 33 (3) non-local pseudopotential contribution E Sr(G - G')$(21 + l)[-«„^P|(cos $)FT,{k + G,k + G’) nkGG'/r r/- , r.SFrlik + G.k + O k + Gak + G^ +P,( cos 0)[-(* + G) ------— ------(Jfc + G)2'_ ^ a f r,(* + G,k + G')k -HG-ok + G V - ( * + k + Gak-f- Gg k + G ^ k + G'g k + Gak -J- G'g + k + G/ak + Gg + [c0S«| ■— t + g) 2 ^ + (t + g)i 'l ------ P’,( cos 6)Fr,(k + G,k + G')]M k + G)^„(k + G/)*, (1.85) (4) G = 0 Coulomb contribution - m E ^ ^ ' E ^ ) ’ t 1-86) r r (5) Hartree electron-electron repulsion contribution -E^(^-M . (UK) (6 ) exchange-correlation contribution <«/) E ( £*'(Q) - #‘“ (G)W G)*, (1.88) G where fixc = d(pe)/dp, (7) ion-ion Madelung contribution (Ewald form) 4 r n t E G»/4,+1 ) - M (1.89) + \ E ZrZsH '^D )!^ + 5 ^ ( E Z r)2^ , rr'R, 34 where 77 is a convergence parameter chosen for computational performance, R is the real lattice translational vector, x T is the atomic position, D = xr/ - xr + R ^ 0. And H 1 is the function Hf(x) = (1.90) ox where erfc(x) is the complementary error function.29,2^ The stress contributions, from Eqs. (1.83) - (1.89), add up to the total stress crap. In the calculation, we can use the Local Density Approximation to find the wavefunctions and eigenvalues corresponding to the specific strain pap chosen; the calculated wavefunctions and eigenvalues are then substituted into the above equa tions to calculated the stress. There are many detailed technical aspects (as discussed by Nielsen and Martin2®); we emphasis here: 1. The choice of the pseudopotentials. Since the earlier days of ab-initio calculation using the local density approximation, there have been many efforts to create the “perfect” pseudopotentials. An important step was taken, when Hamann et al. , 2 9 constructed norm-conserving pseudopotentials. Later this semi-local pseu dopotential was made separable by Kleinman and Bylander39. Bachelet et al. 3 1 tabulated the parameters for the pseudopotentials for atoms from H to Pu. Recently, there are many effort at producing better pseudopotentials , 3 2 ,3 3 ,3 4 but the basic requirement remains: the pseudopotential should reproduce the atomic eigenstates; should be transferable from atom to solid; and should con verge with the smallest possible number of planewaves (the convergence study will be discussed in detail in Chapter II). 2. The integration over Brillouin zone. For semiconductors with fully occupied bands the special fc-point scheme is used, where the integration over Brillouin zone is reduced to a summation over those specially chosen fc-points.'^1'^ We note that the special ^-points form a uniform grid in the extended Brillouin zone. We will discussed the convergence over the chosen fc-point density in next Chapter. 3. The cutoff energy, E cut • In the planewave expansion, the number of the planewaves included in the basis function is limited by the cutoff energy which controls the highest kinetic energy of planewaves. The detailed discussion will presented in Chapter II. Here we want to point out that there are two different views on the choice of energy cutoff: ( 1 ) keep it fixed in all calculations for different properties of the crystal (having the highest kinetic energy fixed); ( 2 ) change it to keep the number of the planewave fixed as the volume changes (hav ing the number of momentum-space components fixed). In most calculations is a fixed energy cutoff is accepted. As regard to how large of the energy cutoff should be ( for the calculation to be considered converged), it varies among different groups; we will present our estimates in the next chapter. 4. The matrix diagonalization. Direct matrix diagonalization was used before Car and Parrinello ^ 7 proposed a simulated annealing procedure, which solves for the eigenvalues and eigenvectors iteratively. This iterative method improves the computational performance greatly, reduces an computational effort from NpW to Mf,anciNpW (which improved performance enabled the Green function approach in the linear response calculation by Baroni et al. , 2 0 which will be discussed in the next section). 36 Using the relationship between stress and strain ~ ca0uv flpvi (1-91) ftv we deduct the elastic tensor cap^u. The fourth rank elastic constants are usually converted to the second rank “tensor” see Eq. (1.29). So, by a proper choice of strain, the elastic constant can be calculated straight forwardly. The calculated elastic constants for Si and Ge are shown in Chapter II. 1.7 Green Function Approach to Linear Response Calculation The Green function approach 2 0 combines the advantages of the dielectric matrix and direct approaches and avoids their drawbacks. The unique advantage of this method is related to the recent development of the iterative method,which im proves the computational performance of Local Density Approximation. The Green function approach requires only the ground state properties of the unperturbed sys tem while avoiding the huge summation over the excited spectrum required by the dielectric matrix approach. The approach is not restricted to a local perturbation as in dielectric matrix nor restricted to the short wavelength as in the direct method. First, we will briefly introduce the iterative method to show its advantage over the conventionally direct diagonalization. In the iterative method ^ 7 we introduce a ‘fictitious’ Lagrange with a ‘fictitious’ kinetic energy for the wave function as £ = E 5*1 / ^l^nkl2 + E l M‘Af - Writ}], (1-92) rit 2 Ja I Z 37 where the potential is the local density total energy of the system, and /z is the fictitious mass for the electron wavefunction. Studying the time dependent wave- function discussed by Eq. (1.92) leads to the ground state wavefunction of the system. During the process of solving for the ground state, we use the fast Fourier transformation for which the number of calculations proportional to Npw log Npw, with Npw is the size of the planewave basis. The whole computation has a number of computation step proportional to M\jan(iNpW including the construction of the Hamiltonian matrix, where Afjan(f is the total bands included in the calculation. Comparing to the conventional direct diagonalization, which has the operations proportional to NpW, the save in the computation budget is tremendous when the basis function, Npw, is much much larger than the number of the bands included in the consideration. The main feature of the Green Function approach 2 0 is that it treats the response by a perturbation theory (similar to the dielectric matrix approach), but the total potential is obtained by iterative calculation to self-consistency (not by the inversion of the dielectric matrix). Furthermore, the summation over the conducting bands is avoided by introducing a Green’s function which keeps the numerical effort to ^b a n d ^p w Under the Local Density Approximation, the total self-consistent potential on the electron can be written as V(r) = Ferf(r) + J j ^ T * ' + V«(»(r)), (1.93) where Vext is the pseudopotential for ion-electron interaction when the lattice vibra tion (phonon frequency) is considered, and the local exchange-correlation potential is used for VXc• K the change in the self-consistent potential, 8V, is known, the 38 linear variation in the electronic density can be obtained through the first-order perturbation theory Sn = occ with , Ei E * Converting to momentum-space (using operator form) . p, 4 ^ C uk|e-'( Eq. (1.93) for a small change in charge density, £n(r), due to the external potential change, SVext, is ^ +/ In order to avoid the summation over the whole spectrum of the conducting band in Eq. (1.94), Baroni et al. rewrite the equation as fn(q + G) = £ < »k|e-<(,«+Q)-rA(3b(£:„k)Pc 5n(q + G) = — Y"' < vk|e-,(q+G)‘r|i;k >, (1-97) «k 39 Table 1 . 2 Calculated values of the macroscopic dielectric constant, Coo> and optical phonon frequencies for Si and GaAs. In the calculation, the Bachelet-Hamann-Schluter pseudopotential is used with 28 special fc-points (see Chapter II) and energy cutoff of 14 Ry . 2 0 (From Baroni et al. , Phys. Rev. Lett. 58, 1863, (1987). Coo u to ULO Si LDA 12.7 15.4 15.4 EXP 11.4 15.5 15.5 GaAs LDA 12.3 8.14 8.70 EXP 10.9 8.06 8.75 where (Hscf - > = PcSV\vk > . (1.98) There are algorithms to solve linear systems 3 3 iteratively which requires a number of operations proportional to the square of the size of matrix times the number of terms in the sum in Eq. (1.96) (number of occupied bands times the number of the Appoints). With Eqs. (1.95), (1.97) and (1.98), we can calculate the linear response to external perturbation (non-local) of arbitrary wavelength with a numerical effort growing as MfjanjN pW. This Green function approach has been used to calculate the long wavelength phonon frequencies of Si and GaAs, and phonon dispersion of GeiAli_a;As . 2 0 ,3 9 The agreement with experimental measurement is good as shown in Table 1.2. 40 1.8 Some Systematic Studies of the Local Density Approximation on Various Aspects: Self-consistent Iterations, Total Energy Convergence, Exchange-correlation Functional Forms In this section we will concentrate on some important aspects of our calculations. We will restrict our Hamiltonian to a local exchange-correlation functional form, with the wavefunction expanded in the planewave basis, and use the special fc-point scheme to approximate the Brillouin zone integration. The calculation is performed self-consistently. We seek to determine the effects of each of these approximations on the calculated results: As there are few published systematic studies, we will mainly discuss the results of our studies, and compare with such studies 4 0 ,4 1 as are available. The discussion will be divided into three parts: (1) self-consistency; (2) total energy convergence; and (3) exchange-correlation effects. 1 . Self-consistent Iterations. There are two different ways of solving the Lagrange equation of Eq. (1.92): (i) the second-order-in-time approach of Car and Parrinello , ^ 7 u<}> = 4 . constraint, (1.99) o temperature properties of the system; and (ii) the second approach, first-order- in-time, by Teter et al. p which is used to find the ground state properties of the system. In Eq (1.99) H is a fictitious “mass” of the electron wavefunction; and Eq (1.100) the de- 41 cay rate, chosen to optimal the computational effort, as the time-derivative is approximated by the finite difference, . 6 n+ l -n ( U 0 1 ) The combination of fi/dt controls the change of the wavefunction; if it is too small (the ‘time step’ too big), the wavefunction will be unstable, if it is too big (the ‘time step’ too small), it takes too much steps to get to the convergence. We tested for different values and choose the largest stable ‘time step’ possible in our calculations use Eq (1.100). The constraint in both equations is to guarantee that the new wavefunctions, <^n+1, remains orthonormal. We use the first-order form in all of our calculations since our interests are in the ground state properties. For each time step the wavefunction, and total energy, Et0t, are cal culated; the self-consistency is reached when the total energy change between consecutive time steps is smaller than a preset limit 6E j1 - .£& < <£. (1.102) The study on the self-consistent iterations (time steps) are performed by plot ting the logarithm of the energy difference, log( E ^ 1 — E fot), against the number of iterations, n. We have compared two different ways of finding the energy min imum: (1) the steepest decent, and (2) the conjugated gradient . 2 5 The results are shown in Fig. 1.6. We can see from Fig. 1.6 (dashed line) that using the steepest decent method, the energy change has a neaxly perfect logarithmic rela tionship with the number of the iteratives, though it takes quite a few iterations to reach the final self-consistency (~ 100 iterations). The conjugated gradient 42 -4.0 I -7.5 6 > o - 11.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 n: iterations (time steps) F ig u re 1 . 6 Total energy convergence through self-consistent iterations. The logarithm of the difference of the total energy (in eV) from each iteration (time) step is plotted against the number of iterations. The calculation is for bulk Si with energy cutoff of 10 hartree, 10 special fc-points. Dashed Line: With steepest decent; Solid Line: With conjugated gradient. It is clear that the conjugated gradient takes much less iterations (time steps) to reach self-consistency. takes considerable fewer steps (less than 10) to reach the self-consistency (Fig. 1.6 solid line). In order to compare the relative efficiency of the two procedures we have to take the time consumed at each of the time step into account. For each time step, the conjugated gradient method takes considerably longer (2-3 times) than the steepest decent, but considering the whole process of the self- consistency, the conjugated gradient wins by a factor of 3 (or more). Hence, the conjugated gradient is chosen as the minimum finding procedure in our calculations. 43 Table 1.3 Lattice constant and bulk modulus of Si with different energy cutoff. The calculation is done with 10 special ^-points and Ceperley-Alder exchange correlation. The equilibrium lattice constant and bulk modulus is fitted by Murnaghan formula (Appendix C). E cut (hartrees) 6 1 0 1 2 16 ao(bohr) 1 0 . 2 0 10.173 10.173 10.174 1.14 0.962 0.975 0.965 2. The total energy convergence. Here we study the effects of two major approxi mations in our calculations: the planewave basis expansion and special fc-point summation. We will briefly discuss the idea here (a detailed description can be found in Chapter II). A similar study was done by Holzschuh 4 1 with a smaller basis (maximum of 411 planewaves and 28 special fc-points) due to the limi tation of computational resources at that time. We emphasis here that it is important to recognize that the special ^-points are no more than the evenly meshed Appoints in the extended Brillouin zone, hence, the fc-point density (ei ther volume or linear density) is a physical variable in studying the total energy convergence. The planewave expansion: the electronic wavefunctions is expanded as a sum of planewaves, which procedure would produce convergence if the planewave basis was infinite. In any real calculation we limit the total number of the planewaves, by limit the kinetic energy of the planewave below a certain value, E Cuti called cutoff energy. We study how the total energy will change as the size of this finite planewave basis increases (detailed discussion in Chapter II). 44 The result of our study is quite surprising, contrary to the common belief , 2 4 ,4 2 it takes a very high energy cutoff (10 hartree or 400 planewaves) to get a poorly converged total energy (absolute energy error per atom of 50 meV), leading to a 9% relative error in calculated energy differences. (This relative error results less than .5% error in lattice constant and 2% in bulk modulus, as the results for bulk silicon is shown in Table 1.3. Note in Table 1.3 the results of bulk modulus calculated with energy cutoff of 6 hartree is off by over 15%.) For a small energy cutoff, 6 hartree, the total energy versus lattice constant is rather unsmooth, as shown in Fig. 1.7, where the total energy per atom of bulk Si is plotted as a function of the lattice constant for few different combinations of energy cutoff and Appoint density: 10 special Appoints, both 6 (A) and 10 (□) hartree; and 6 special Appoints and 10 (o) hartrees. All energies are shown as the relative to the values to the value at lattice constant a = 10.2093 bohr for easy comparison. It is clear that the increasing number of Appoints - curves fitted with o and □ - does not change the lattice constant or bulk modulus; but a different energy cutoff does produce a different lattice constant and bulk modulus (compare o or □ with A, the curvatures are different and the A points do not even make a smooth curve) This is in agreement with Holzschuh’s4* observation, that at a low energy cutoff, the total energy curves of different lattice constants crosses as the number of planewaves increases. It is unfortunate that Holzschuh’s study did not attract much attention to the need of a higher energy cutoff to achieve a converged calculation. Our study indicates that the convergence is too complicated to be modeled by any simple formula (we tried both logarithmic and power law and failed to get a 45 good fit, see Chapter II). One other thing we want to emphasis is that the plot of total energy versus the number of planewave, Npw (or energy cutoff, Ecut) can be quite deceiving. When the low energy cutoff points are included, the range of energy change is so big (a few eV), it obscures all the rough features in convergence at high energy cutoff (larger than 25 hartree region, see Fig. B.l). We suggest the logarithmic plot be used in these kind of convergence study. The special k-points: the Brillouin zone integration is approximated by a sum mation over sets of specially chosen fc-points . ^ ’^ 6 We noted that these special ^-points are no more than the evenly spaced fc-points in the Brillouin zone, so it is sensible to use the fc-point density in our study of the convergence of total energy as the density increases. Though our study shows that the nature of the convergence can be quite complicated (see Chapter II), it is possible to re duce this error contribution to less than 1 % by using a 1 0 special fc-point set,^® which corresponds to a linear density of 4 fc-points per 27T /a for the diamond structure. 3. The effect of different exchange-correlation function forms. We have compared two different exchange-correlation forms: Wigner*'* and Ceperley-Alder^ on calculated lattice constant and bulk modulus, on bulk Si. As shown in Table 1.4 the Wigner form gives a larger lattice constant while the Ceperley-Alder form gives a larger bulk modulus, comparing to the experimental results. The comparison is made with both the experimental measurement and the calcula tion by Holzschuh41. The agreement between our data and Holzschuh’s is good with estimated error by Holzschuh ( 0 . 0 1 A in lattice constant and 5% in bulk modulus) as shown in Table 1.4. We can see that we have to compromise one 46 40.0 □ 10H 10K, o 10H 6K; a 6H 10 30.0 E 2 0 .0 a> 10.0 £ 0.0 10.0 10.0 10.1 10.2 10.3 10.4 10.5 a (Rb) Figure 1.7 Total energy, Et0t(a) per atom (in eV) of Si (per atom) versus lattice constant, a (in bohr). Calculation is done with Ceperley-Alder exchange-correlation and Hamann’s pseudopotentials .'13 The comparison is made with different energy cutoff, Ecut in hartree, and special fc-points. The three data sets are: ( 1 ) □: Ecut = 10 and 10 special fc-points; ( 2 ) o, Ecut = 10 and 6 special fc-points; and (3) A , ECut = 6 and 1 0 special fc-points. The energy zero is adjusted separately for each of the data sets, with Et0t (10.2093) = 0.0. The curves are two separate polynomial fits for □ and o, which are on top of each other. A shows clear disagreement with □ or o, which indicates the poor convergence of total energy against the energy cutoff. Another worse set ( 6 hartree and 6 special fc-points) is not shown here. quantity to get the better agreement for the other one. It has been discussed extensively by Jones 13 that a better exchange-correlation form may be in need. For the moment, we will try to understand where the difference lies between these two different forms. Figure 1.8 show that the exchange-correlation con tribution to total energy and bulk modulus around the electronic density of Si 47 Table 1.4 Calculated lattice constant and bulk modulus of Si, with different exchange-correlation function forms. Our calculated data is for energy cutoff of 10 hartree (~ 400 planewaves) with 10 special ^-points; Holzschuh’s data 4 1 is for 411 planewaves and 10 special fc-points. x c a (A) B (1012 Erg/cm3) Wigner This Work 5.44 0.89 Holzschuh 5.41 0.87 C-A This Work 5.38 0.97 Holzschuh 5.36 0.96 EXP 5.43 0.99 (rs ~ 2). We can see that the exchange-correlation contribution to total energy is negative (attractive) at that electronic density, with Wigner’s less attractive than that of Ceperley-Alder’s; hence, Wigner’s yields a larger lattice constant. Likewise, the contribution to the bulk modulus is also negative, with Wigner’s about 70 kbar more negative than Ceperley-Alder’s; hence, Wigner’s yields a smaller bulk modulus. These observes have been tested in our calculations for both Ge and Si/Ge. Similar trends are found. 1.9 Summary of Thesis Results The next two chapters present the main results of this thesis. • Elastic Properties of Ge/Si Superlattice and Convergence Studies: In Sec. 1.5 the direct approach for calculating the frozen phonon frequency is discussed. In Chapter II, we will use the similar approach, which relates the total 48 - 0.34 -0 .3 4 -0 .3 5 -0 .3 7 -0 .7 Bulk modulus due to Ex-z - 0.8 n 1 -0 .9 - 1.0 1.95 1.982.01 2.04 2.07 2.10 r, = (3/(4Pi*rho))1/3 Figure 1.8 Exchange-correlation contribution to : Top, total energy; Bottom, bulk modulus, at electron density of Si ( rs ~ 2). The Wigner’s form is shown by dashed lines and Ceperley-Alder’s is shown by solid line in this figure. 49 energy change to the elastic constants, to calculate the elastic properties of the “free standing” Ge/Si superlattice. This the the first attempt in theoretically predicting the elastic properties of this yet unsynthesized Gej/Sii. Our calculated results agrees with both the experimental and other calculations for bulk Si and Ge. Also in the chapter, we present the details of the convergence studies discussed in Sec. 1.8. These convergence studies show that to calculate the elastic constants to 9 percent relative error (in the worst situation) requires the use of 400 planewaves; but our calculations shows that the error is less than 2 % in bulk modulus and less than .5% for the lattice constant for a basis of 400 planewaves, though the pressure derivative of the bulk modulus is less converged: we predict that it is between 2 and 3. • O p tical R esponse of S tra in e d Si: In Sec. 1.4 we have shown that the dielec tric function can be calculated from the Local Density Approximation. In Chapter III, we will present the calculated optical response for strained Si, where the strain- induced birefringence is calculated with crystalline silicon. Agreement with experi ment is very good in the static limit, shown in Table 1.5, considering disagreements among the experimental values. Above 2 eV, the calculation predicts less dispersion than seen by the experiments. Thermal effects and electron-hole interactions are estimated to resolve some of the discrepancies with experiment. 50 Table 1.5 Static photoelastic tensor and dielectric constant. All exper iments quoted are at room temperature, except as noted. Distortion LDA EXP photoelastic tensor [0 0 1 ] P ll ~-P 1 2 -0.118 -0.111 ± 0.005 -0.127 ± 0.005 [1 1 1 ] P44 -0.050 -0.051 ± 0.002 -0.051 dh 0.002 hydrostatic PH + 2pi2 -0.067 -0.055 ± 0.006 -0.070 ± 0.008 dielectric constant 10.9 11.7 11.4 (0 K) REFERENCES 1. N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders College, Philadelphia, 1976), p. 421. 2. G. D. Mahan and K. R. Subbaswamy, Local Density Theory of Polarizability, (Plenum, New York, 1990), p. 161. 3. C. Kettle, Introduction to Solid State Physics, (Wiley, New York, 1956). 4. S. K. Sinha, CRC Crit. Rev. Sold State Sci., 3, 273 (1973). 5. L. J. Sham, Dynamical Properties of Solids, edited by G. K. Horton and A. A. Maradudin, (North-Holland, 1974), p. 301. 6 . M. Born, Ann. Physik 44, 605 (1914). 7. P. N. Keating, Phys. Rev., 145, 637 (1966). 8 . H. L. McMurry, A. W. Solbrig, J. K. Boyter and C. Nobel, J. Phys. Chem. Solids, 28, 2359, 1967. 9. R. M. Martin, Phys. Rev., Bl 4005, (1970). 10. W. Cochran, CRC Crit. Rev. Solid State Sci. 2 , 1 (1971). 11. P. Hohenburg and W. Kohn, Phys. Rev. 136, 864 (1964). 12. W. Kohn and L. J. Sham, Phys. Rev. 140, 715 (1965). 13. R. 0. Jones and 0. Gunnarsson, Rev. Mod. Phys. 61 689 (1989). 14. L. J. Sham, Phys. Rev. 188, 1431 (1969). 15. H. Wendel and R. M. Martin, Phys. Rev. B19, 5251 (1979). 16. J. T. Revreese, P. E. van Camp and V. E. van Doren, Electronic structure, dynamics, and quantum structural properties of condensed matter , edited by J. T. Devreese and P. E. van Camp, (Plenum, New York, 1985), p. 157. 17. M. S. Hybertsen and S. G. Louie, Phys. Rev. B35, 5585 (1987). 18. S. Baroni and A. Resta Phys. Rev. B65, 84 (1986). 19. Z. H. Levine, D. C. Allan, Phys. Rev. B 43, 4187 (1991) and Phys. Rev. Lett. 63, 1719 (1989). 20. S. Baroni, P. Giannozzi and A. Testa, Phys. Rev. Lett. 58, 1861 (1987). 21. K. Kune, Electronic structure, dynamics, and quantum structural properties of condensed matter , edited by J. T. Devreese and P. E. van Camp, (Plenum, New York, 1985), p. 227. 22. D. J. Chadi and R. M. Martin, Solid State Commun. 19, 643 (1976). 23. M. T. Yin and M. L. Cohen, Phys. Rev. B24, 6121 (1981). 24. M. T. Yin and M. L. Cohen, Phys. Rev. B26, 3259 (1982). 25. M. P. Teter, M. C. Payne and D. C. Allan, Phys. Rev. B40, 12255 (1989), and D. C. Allan and M. P. Teter, Phys. Rev. Lett. 59, 1136 (1987). 52 26. 0. H. Nieslen and R. M. Martin, Electronic structure, dynamics, and quantum structural properties of condensed matter , edited by J. T. Devreese and P. E. van Camp, (Plenum, New York, 1985), p. 313. 27. 0 . H. Nieslen and R. M. Martin, Phys. Rev. B32, 3780 and 3792 (1985). 28. J. Ihm, A. Zunger and M. L. Cohen, J. Phys. C 1 2 , 4409 (1979). 29. D. R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett. 43,1494 (1979). 30. L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). 31. G. B. Bachelet, D. R. Hamann and M. Schluter, Phys. Rev. B26, 4199 (1982). 32. D. Vanderbilt, Phys. Rev. B41, 7892 (1990). 33. N. Troullier and J. L. Martins, Phys. Rev. B43, 1993 (1991). 34. D. R. Hamann, Phys. Rev. B40, 2980 (1989). 35. D. J. Chadi and M. L. Cohen, Phys. Rev. B 8 , 5747 (1974). 36. H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976). 37. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985) and 60, 204 (1988). 38. J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, Linear Algebra , (Springer-Verlag, Heidelberg, 1971), Vol 2. 39. S. Baroni, S. de Gironcoli and P. Giannozzi, Phys. Rev. Lett., 65, 84 (1990). 40. M. T. Yin and M. L. Cohen, Phys. Rev. B26, 5668 (1982). 41. E. Holzschuh, Phys. Rev. B28, 7346 (1983). 42. P. G. Dacosta, 0 . H. Nielsen and K. Kune, J. Phys. C: Solid State Phys. 19, 3163 (1986). 43. D. R. Hamann, Phys. Rev. B 40, 2980 (1989). 44. E. Wigner, Phys. Rev. 46, 1002 (1934). 45. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). CHAPTER II Elastic constants of Si/Ge superlattice and of bulk Si and Ge In this chapter we report extensive convergence tests of the total energy of Si computed within the local density approximation with a planewave basis and pseu dopotentials. These convergence tests show that to calculate the elastic constants to 9 percent (as the relative energy uncertainty, in the worst case) relative error requires the use of 400 plane waves in the electronic structure calculation and 10 special fc-points to compute the density and energy. Our calculation shows that the error in bulk modulus is less than 3% and less than .5% in lattice constant. Further we find that using the Ceperley-Alder exchange-correlation form in calculating the elastic constants obtains a better agreement with the experimental results than us ing Wigner Form. We report the calculated lattice constants and elastic constants - bulk modulus, C\ \ , C\2 and C44 - of “free standing” Si/Ge ordered superlattice. For comparison, the results for bulk silicon and germanium are in excellent agree ment with existing experiment and other calculations with the exception of C44. The calculation for C44 reported here is the first to allow the “internal” atom in the diamond unit cell to move as the crystal is sheared. The equilibrium position of the sheared crystal can be not be predicted by scaling arguments from the unstrained crystal. An “averaged elastic theory” based on bulk Si and bulk Ge predicts our computed Si/Ge lattice constant and bulk modulus surprisingly well. 53 54 2.1. Introduction The local density approximation has proven to be an effective and useful means for studying of both structural and electronic properties in semiconductors . 1 Al though the method has been in existence for many years, a systematic study of convergence is yet to be done. This is an attempt to rectify this situation in con text of calculating elastic properties. In this chapter, we first report the results of convergence studies on various pa rameters in the method, such as the size of the planewave basis, the density of those special Appoints and the effects of different forms of exchange-correlation functions. Even though we may not fully understand the natures of convergences our studies reveal that, contradictory to the conventional believe, 2’^ even for a simple system like Si, a very large planewave basis (400 planewaves) is needed to ensure an error less than 9% in predicting the lattice and elastic constants. Our study on different exchange-correlation forms, comparison between Wigner and Ceperley-Alder, gives a guideline on whether to compromising the lattice constant or elastic constants depending on the focus of research interests: Wigner potential should be used if the structural properties is desired while Ceperley-Alder should be the choice if elastic constants are to be calculated. Next we present the theoretic predictions for the lattice and elastic constants of the “free standing” Si/Ge superlattice with compar isons to bulk Si and Ge. To ensure the accuracy, we have proposed a scheme to calculate individual elastic constants, detailed in Appendix A. One unique aspect of this calculation is that the shear modulus (C 4 4 ) is obtained while allowing the atoms to relax to their equilibrium positions. Allowing this relaxation is important: it reduces the calculated C44 by 25% to agree with the experiments. The relax 55 ation is made possible with the molecular dynamic scheme developed by Car and Parrinello ,4 and later improved by Teter, Payne and Allan ,5 which scheme will be discussed briefly in this chapter. An outline of this chapter follows. In Sec. 2.2, we will describe the local density approximation with molecular dynamic approach used in our calculation, and the details of our calculations. In Sec. 2.3, we will show the the results of our convergence tests, discuss the importance of each parameter on the results, and give a realistic estimate of the errors in our results. Section 2.4 is devoted to the results and discussion of the calculated lattice and elastic constants, where we will compare our results of bulk Si and Ge with existing data, and present the results for the Si/Ge superlattice. We will demonstrate the consistency of our method by we comparing the elastic constants obtained from different lattice distortions discussed in the appendix. Finally, in Sec. 2.5, we will summarize the main results of this chapter. 56 2.2. The Molecular Dynamics Scheme and Calculation Method The local density formalism ®’7 expresses the total energy of ground state as a functional of total electron charge density, p, as Etot [p] — T[p] + Eion[p] + Eee[p] + Eex[p], (2 .1 ) where the various terms represent the kinetic energy, the core-electron interaction, the electron-electron interaction, and the exchange-correlation energy. The density p is obtained as the sum of the occupied electronic states, occ P(r ) = 2 2 l^nkMI2’ (2>2) nk for materials with gaps in their band structures, such as insulators or semiconduc tors, where the occupied states axe separated from the conduction states, the sum in Eq. (2.2) runs through only those ^-points used to sampling the Brillouin zone, special fc-points as in our calculations,®’® for each valence states. Further, the wavefunctions are expanded in some basis functions. In the case when the planewave basis is used, the wavefunctions VVik’s 3 X 6 represented as the sums of planewaves, Npw M r ) = r ik t J 2 CG,»*eiG*- (2-3) G The sum is cutoff by the size of the basis, Npw, the total number of the planewaves included. In order to avoid the direct matrix diagonalization, where the number of op erations scales as Npw, with Npw is the size of the matrix, Car and Parrinello 4 57 proposed the construction of a Lagrangian with simulated kinetic energy terms for the electronic wavefunctions. The fictitious Lagrangian is written as i = f d \ \ i nVI2 + T \M ,R j - £,„([{£,}, «„k}]. (2-4) t i f Ja I 2 where (j, is pseudo-mass of wavefunction, and M/ and R/ are the masses and coordi nates of ions. Our method is slight different in that instead of solving a second-order derivative-in-time equation as would result from Eq. (2.4), we use a first-order-in time equation.® This method utilizes, except for matrix construction which scales as NpW, the Fast-Fourier-Transform procedure, which has the operations scales as NpW\og(Npw), to solve the lowest eigenvalues and eigenfunctions. As basis func tions increase it is clear that the iterative method is much more effective than the direct matrix diagonalization. This molecular dynamics approach allows the atoms to move at the same time when the electron wavefunctions are converging to the ground stated The ability to move the atoms according to the calculated force gives us a distinct advantage in calculating the shear modulus (C 4 4 ) since in a shear distortion the atoms move to equilibrium positions different from the ones predicted by simple scaling of the distortion. We will discuss this in detail in Sec. 2.4. Throughout our calculations, we employ the generalized norm-conserving pseu dopotentials developed by Hamann . 1 0 with the non-local pseudopotential being made separable using Kleinman-Bylander procedure . 11 In particular the s-wave potential is chosen to be local since that results in direct gap with nearly the same absolute error throughout the Brillouin zone for for both Si and Ge, and in addition that produces a gap in bulk Ge. The Ceperley-Alder functional form , 155 parame terized by Teter , 1 3 is used for the exchange and correlation energy. Except for 58 our systematic studies, we use a 400-planewave basis (corresponding to a kinetic energy cutoff of 10 hartrees) and 10 special Appoints of the FCC irreducible Bril louin Zone .® ’9 These Appoints are generalized as k —► (l+/i)-1k in the distorted lattices, as done by Nielsen and Martin,** with equivalent Appoints added in the distorted lattices of lower symmetry. The equality of these Ar-point sets are verified with calculations performed using both the high and low symmetry sets of same structure. The systematic studies are done for the total energy convergence versus both the size of planewave basis, Npu), and the density of special Appoints; where Npw changes from 60 to 1200, and the number of special Appoints from 2 to 60. 59 2.3. The Systematic Studies The systematic studies are carried out for the diamond structured Si. The studies are designed to test the total energy convergences versus the Appoint sam pling density, (Nkpt), and the size of planewave basis, (N pw or energy cutoff, E cut)- Also we look into the effects of two different exchange-correlation function forms: W igner 1 4 and Ceperley-Alder12. The Ar-point sampling approximates the Brillouin zone integration by a sum over some “special Appoints” 8 ,9 f /(k) Appoints are actually uniformly spaced k-points in the extended zone. With this observation, the Appoint density (whether linear or volume) is the appropriate vari able for a study of the convergence over the number of Appoints. Table 2.1 shows the relationship between those special Appoint sets and and the volume and lin ear densities of Ar-points, measured in units of (27r/agi)8 and (27r/agi), respectively. For distorted cells with lower symmetries, Table 2.1 shows the rapidly increasing numbers of Appoint. The details of how a ‘2 special Appoints’ is represented in dif ferent unit cell choices are show in Table 2.2. In this study, we concentrate on how well the integration leading to the total energy is represented by a summation over successively larger number of Ar-points. The planewave-basis convergence study focus on how well the incomplete basis function (with finite number of planewaves) can represent the electron wavefunc- 60 T able 2 . 1 Special fc-point sets and fc-point density. The relationship between special fc-points and fc-point density (volume or linear) are shown. The special fc-point sets are usually named by the fc-point numbers under FCC symmetry; the lower symmetry cells are those used in calculations of C\\ and C 4 4 , when the cell is chosen tetragnal or orthorhombic, with length scale agi/y/2, agi/y/2, and agi, along ( 1 1 0 ), (1 1 0 ), and ( 0 0 1 ) direction in the equilibrium; The volume density is the total number of Appoints within (2tt/agi)^, and the linear density is the divisions in each of (2ir/agi), the density increases when the unit cell size increases (since the reciprocal cell size decrease which the total number of fc-points remains the same). Number of special fc-point set Full diamond symmetry 0/, 2 6 1 0 28 60 Lower symmetry with double length in (001) tetragnal D 2 d 2 8 1 2 36 80 orthorhombic D 2 h 4 16 24 72 160 Corresponding density of fc-points volume density of fc-points (2ir/agi)3 8 27 64- 216 512 linear density of ^-points (2ttjagi) 2 3 4 6 8 tions of the systems we study. The number of planewaves included is determined by an energy cutoff, Ecut, which limits the highest kinetic energy of the planewaves, \{ k + G ? < E cut. (2.6) We estimate the number of planewaves within a energy cutoff, Ecut, by filling the volume of the sphere of radius kcut = y/2Ecut with cubes with a volume of the Brillouin zone, (27r)3/fl, Npw « (2.7) 61 T able 2 . 2 Example of special fc-points of different symmetries and unit cell size, and its density. The numbers and the coordinates of the 2 special ^-points are shown with different unit cell choices. A 2-fc-point set in full FCC symmetry, O/,, results a 2 fc-points in tetragonal double cell, and 4 ^-points in orthorhombic double cell; with a linear density of 2 fc-points per (27T/a), and 8 fc-points per (27r/a)^. G is the reciprocal vector in (2ir/a) with the direction indicated. Cells G-vectors (2ir/a) Symmetry fc-point coordinates Diamond (1 1 1 ) ( 1 1 1 ) (1 1 1 ) o h (1/4,1/4,1/4) (1/2,1/2,1/4) Tetraganol (IlO) ( 1 1 0 ) (0 0 1 ) D2d (0,1/4,1/4) (1/4,1/2,1/4) Orthorhombic (1 1 0 ) ( 1 1 0 ) (0 0 1 ) £>2 h (0,1/4,1/4) (1/4,1/2,1/4) (1/4,0,1/4) (1/2,1/4,1/4) Extended Zone (1 0 0 ) (0 1 0 ) (0 0 1 ) none (±1/4,±1/4,±1/4) where Q, is the volume of the unit cell, for Si the numbers are shown in Table 2.3. We will study how this incomplete basis function affects the total energy, and in turn the calculated lattice and elastic constants. A. Total energy convergence against special fc-point density A systematic study of the total energy as a function of the linear density of fc-point (special ^-points) requires a well-defined procedure. There is no unique procedure known at this time. Ours is subject to criticism. Our three-part procedure involves (i) assembling the data set, (ii) devising a way of analyzing the data set and (iii) using that analysis to estimate errors in specific properties. After this has been done for several systems, one might hope to be able to generalize the results. As far as we can determine no such systematic 62 Table 2.3 Energy Cutoff E Cut and number of planewaves, Npw- The relationship between numbers of planewaves, Npuh and energy cutoffs, Ecut, are shown for the diamond Si cell. The number of planewaves is related to the energy cutoff by the following approximation Npw ^ where ft is the unit cell volume in bohr®; Ecut in hartree. For Si, ft ~ 263 bohr^. ______Ecu* (hartrees) 3 5 10 14 20 No. of planewaves 6 6 141 397 656 1124 study is in the literature. In this section and the next we undertake that procedure for convergence of the total energy as a function of the linear density of Appoints, Nf.pt, (see Table 2.1) and the numbers of plane waves, Npw, (see Table 2.3). Data set. We compute the total energy for five special Appoint sets, see Table 2.1, which correspond to a linear density of 2, 3, 4, 6 , and 8 fc-points per (2n/a). For each of these sets, the energy is computed for five different cutoff energies, Ecut. We thus accumulate, after 25 IBM3090 hours, twenty-five total energies: E(Ecut, Nf.pt), shown in Table 2.4. To add five more points at the next larger fc-point set would double the CPU time. Analysis of Data. We assume that for any given cutoff energy the total energy will converge as density of fc-points (the number of special fc-points) approaches infinity. Unfortunately there is no mathematical analysis as to what functional form it should have. We have tested two possible convergence forms. The first is 63 Table 2.4 Data Set, the total energies of different linear Appoint density and number of planewaves, E{ECut, Nf.pt). The total energies (per atom) of 2-atom Si cell is shown with various energy cutoffs, Ecut , and Appoints, Nf.pt. Nf.pt is the linear density of Appoints, Ecut is in hartrees, and the total energies are in eV. The total energy is reported here with 10 effective digits as the self-consist ant gives error in the 1 2 th digits, while the finite size of the FFT box gives error in the 11th digits. 2 3 4 6 8 Ecut /E(NpW, Nf.pt) 3 -105.583 134 2 -106.262 820 3 -106.253 6 6 6 3 -106.263 003 5 -106.261 614 4 5 -106.772 855 8 -107.509 512 8 -107.511 826 3 -107.516 217 4 -107.517 320 5 10 -107.231 429 4 -107.982 055 2 -107.990 266 2 -107.991 743 5 -107.991 760 5 14 -107.249 243 5 -108.000 192 7 -108.008 662 9 -108.010 050 7 -108.010 116 1 20 -107.270 395 6 -108.022 052 5 -108.030 458 9 -108.031 870 1 -108.031 914 8 the power law, Nkpt) - E {E cul, E{Ecut, Nf.pt) — E(Ecut,oo) = AE 10 (2.9) Both forms have three parameters which must be fitted to only five data points for each Ecut - a clearly unsatisfactory situation; but this difficulty is true for almost all calculations with a finite computing budget. In Table 2.5, To characterize the fit, we plot the data and E*q. (2.8) in Fig. 2.1, We observe: (1) the “rate of convergence” is set by the power a. The deviation of the total energy from the estimated converged value decreases by an order of magnitude when the Table 2.5 Fitting parameters for convergence with Appoint densities, fit ted by both of power law and logarithm. The two sets of fitting parameters are shown, the two forms are: the power law fit has the form of we record the values of these two sets of parameters. Even though the two E(Ecut, oo) produced by Eqs. ( 2 .8 ) and (2.9) for each Ecut are the same, (differs less them 10 “ 5 meV), neither fit satisfactorily goes through all five data points, which indicates either that the actual convergence form, if one exists, might be more complicated than a simple power law or logarithmic relationship, or that our data is not in the asymptotic region where the form ( 2 .8 ) or (2.9) might apply. E ( E ^ N kpt) - E(Eoa, oo) = Ecut), and the logarithmic fit has the form of Nkpt E{Ecut, Nkpt) ~ E(Ecut,oo) = AE(Ecut) e N°(Ecut). The three parameters, E{Ecut, oo), N0(Ecut ) and a (ECut) or E(Ecuti °o), A E(E cut) and N0{ECUi), are obtained through the least square fit to the data sets shown in Table 2.4. Ecut (hartree) 5 1 0 14 2 0 Power Law E(oo) (eV) -107.517 321 80 -107.991 761 32 -108.010 116 78 -108.031 915 44 No 1.92852 1.94464 1.94285 1.94476 a 8.1564 9.7664 9.4341 9.5597 Logarithm E(oo) (eV) -107.517 320 51 -107.991 760 47 -108.010 116 07 -108.031 914 74 A E (eV) 146.07694 499.31444 380.18680 378.12081 N0 (27r/a) 0.87219 0.70976 0.74063 0.73719 fc-point density increases by a factor of lO1/® (~ 1.3). And the rate of convergence 65 - 1.0 111 -3 .0 -5 .0 965.365 10 a 16.478 14 + -20.234 20 x -63.831 o - 7 . 0 0.0 0.2 0.4 0.6 0.8 1.0 logio(Niq>0» linear linear k-point density (2(2 pi/a) Figure 2.1 Total energy convergence versus fc-point density. The logarithm of the energy differences, log(E(Ecut, Nkpt) — E(Ecut, oo)), are plotted against loga rithm of Nf.pt, the linear density of ^-points. E(Ej<;cut, oo) is obtained from a least square fit of our data, as log(E(Ecut,N kpt) - E(Ecut,oo)) = -a (E cut)log(Nkpt) + \og(A{Ecut)). The o denotes Ecut = 5; A , 10; + , 14; and x, 20 hartree. The slopes are very close for different Ecut as long as Ecut is larger than 1 0 hartree. is not sensitive to Ecut for Ecui greater than 5 hartrees. ( 2 ) The “norm” of the Nkpt, N0(Ecut), is also not sensitive to Ecut for Ecut greater than 5 hartree. (3) The deviations of total energies from the fitted values are less 2.5 meV, for Ecut greater than 5 hartrees. Note here that we have not included the 3-hartree data for the fit since the data does not monotonically decrease as the fc-point density increases-a failure we 66 believe is due to the a very small planewave basis. And the 5-hartree data does not behave as well as higher energy cutoff data due to the same reason. Error Estimate. In the calculations for lattice and elastic constants, we are interested in the energy differences between various geometries, therefore, we are more interested in the relative uncertainty rather than the absolute uncertainty. For example, when we calculate the bulk modulus, we calculate the total energies for unit cells with different lattice constants, and the total energy differences between these unit cells are the elastic energies induced by the distortions. Hence, the error in the bulk modulus results from the error in the energy differences, rather than the absolute error in the calculated energy for each unit cell. We define the relative uncertainty as the total energy change induced by a small change in the linear density of ^-points connected with some physical “measure ment”, e. <7., computing the elastic constants. We use the power law fit (Eq. (2.8)) here, with all the other physical parameters remains the same, SE = E(EmUNkpa) - E(EmUNkpa) = (2.10) Under hydrostatic strain where lattice constant a increases as (for Si), agj — > agi(l + 5), the linear density of fc-points increases as, since the total number of fc-points remains the same while the reciprocal vector length (originally 27r/ag}) decreases, Nf.pt — > -Njfcp*(l -f S). Using Eqs. (2.8,2.10), the relative uncertainty in the energy difference is SE = - (-JL)«(£«i0). (2.11) Nkpt 1 + ° This hydrostatic distortion yields an elastic energy, ^elastic* hence, we can define the relative error as the ratio of relative uncertainty, SE, in the energy difference, 67 to the elastic energy, Elastic 1 SE relative error = —------. (2 .1 2 ) ■^elastic To estimate SE (for Ecut = 10 hartree and 10 special fc-point set) we use the largest volume change in our calculations, 6 %, corresponding to a 2 % change in the lattice constant. (We use a 5 point quadratic fit with ±2%, ±1%, and 0% to obtain the equilibrium lattice constant.) Using the parameters shown in Table 2.5, we find SE « 0.15 meV. In terms of the elastic energy due to a 6 % hydrostatic volume change ~ 75 meV, the relative error is about 0.2%. Even though this estimate is for the hydrostatic distortion, we expect that the relative error should be comparable for uniaxial and shear strains, when the other elastic constants, C\\ and C 4 4 , are calculated. Our evidence for this expectation is that our calculations of 6 and 10 special ^-points in the unit cells with increased number of Appoints associated with the calculations of C\\ and C 4 4 (in cells of lower symmetry) yields same total energies as those with higher symmetry. To compare our error estimate with other works, we consider the relative error for 2 special fc-points. Our estimate yields a relative error about 100%; from Table 2.4 we can see that the total energy changes (decreases) more than 750 meV when the linear Ap-point density increases from 2 to 3. Others have suggested a smaller relative error. For example, Nielsen and Martin® reports 5% for the 2 special Ap points in their calculations for Ge. It is worth pointing out here that a 2 special Ap-point set for the 2-atom unit cell of volume a®/4 has the same Appoint density as a single Appoint (at T) for a 64-atom unit cell of volume ( 2 a)®. The latter is commonly used in molecular dynamics calculations1®, hence we believe has the similar relative errors ( 1 0 0 %). 68 While this is far from a systematic treatment of k-point convergence, we can use 1 0 special fc-point set to ensure that the relative error is much less than 1 % (0 .2 % in hydrostatic case). B. Total energy convergence against the size of the plane wave basis In this study, we use the same data basis established for the convergence study with fc-points density. Again, we assume that for any given linear fc-point density, the total energy will converge as the energy cutoff, Ecut , (or number of planewaves, NpW, as shown in Table 2.3) approaches infinity. We tried both power law and logarithmic form for this fit, as for power law having the form, E(Npw,Nkrt) - E(oo, Nkpt) = (2 .1 3 ) JVpW and logarithmic fit, Npuf E(EPw,Nkpt)-E (oo,N kpt) = AE10 (2.14) Both have three fitting parameters which are shown in Table 2.6. We stress that neither of these two fitting forms represents the data well, and the predicted con verged values from these two form are quite different: the power law gives the converged values about 10 meV lower than that obtained from logarithmic fit (Ta ble 2.6). This big difference in the converged energies affects the reliability of our error estimate. Here we focus on the power law fit shown in Fig. 2.2. We have following observations. ( 1 ) The “rate of convergence” is set by oc(Nj.pt) which is insensitive of Nj.pt; We can characterize the convergence by saying that the deviation from the converged values reduced by an order of magnitude every time the number 69 T able 2 . 6 Fitting parameters for convergence with the size of planewave basis, fitted by both of power law and logarithm. The two sets of fitting parameter are shown, the two fitting forms are: the power law fit has the form of E(Npw,N kpt) - E ^ , N krt) = and the logarithmic fit has the form of Npw E(Npw, Nkpt) - E(oo, Nkpt) = A E e . The three parameters, J5(oo, Nkpt), No{Nf.pt) an(* <*(Nkpt) or E (°°, Nf.pt), AE(Nf.pt) and N0(Nf.pt), are obtained through the least square fit to the data sets shown in Table 2.4. Nkpt ( W a) 2 3 4 6 8 Power Law E(oo) (eV) -107.279 52 -108.031 43 -108.040 39 -108.041 50 -108.041 67 No 92.3519 91.3483 91.1517 91.3469 91.2650 a 1.8777 1.8539 1.8352 1.8458 1.8414 Logarithm E{oo) (eV) -107.270 420 -108.022 072 -108.030 479 -108.031 893 -108.031 934 A E (eV) 3.18219 3.37196 3.37126 3.25131 3.36627 No 231.99 227.90 228.53 231.82 228.54 of planewaves increases by a factor of lO1/4* (~ 3.5). (2) The “norm” of Npw, No(Nf.pt), is likewise insensitive to Nf.pt- The relative error estimate is done the same way as for fc-point density (dis cussed in Section 2.3-A), with the relative uncertainty for two slightly different 70 5 1.0 ui 0.0 ' i z Ui - 1.0 I 1 1440.96 -62.86 - 2.0 10 * -80.77 28 + -82.99 UI 60 x -83.33 o o> o 0.0 1.0 2.0 3.0 4.0 logio(Npw), average number of planewaves Figure 2.2 Total energy convergence versus the size of planewave basis, Npw. The logarithm of the energy differences, log (E(NpW, Nkpt) — E(oo, Nkpt)), are plot ted against logarithm of Npw, the size of the planewave basis. E{oo, Nkpt) is ob tained from a least square fit of our data, as log (E(Npw, Nkpt) - E(oo, N ty)) = ~ a (NkPt) log(Npw) + \og(A(Nkpt)). The □ denotes 2 special Appoint set; o, 6 ; A, 10; +, 28; and x , 60. The slopes are almost the same for all those different Appoint sets considered. numbers of planewaves being SE = (2.15) Npw i NpW2 The relative error (for 10 special Appoint set and 10 hartree energy cutoff) caused by a 6 % volume change with our fitted parameters (shown in Table 2.6) is SE « 6 . 8 meV, compared to the elastic energy of 75 meV. This relative uncertainty in energy differences produces a large error of 9% in our predictions for both lattice constant 71 and bulk modulus. It is a shock that such a high energy cutoff still produces such a big relative error of 9%. (Note this does not indicate that the calculated elastic constant, in this case bulk modulus, should always bear the same error as the relative error in the energy differences; usually, the error in the calculated bulk modulus is much smaller than the error in the energy difference, though in the worst case, it can be as large as the error in the energy difference.) But if we focus on our data shown in Table 2.4, for the 1 0 special fc-point the total energy for Ecut = 2 0 hartree is more than 40 meV lower than that of 10 hartrees. We recognize that we could be overestimating the error since the fit is not satisfactory, see Appendix B. This relative error results from the finite numbers of planewaves which change as the unit cell volume changes with a fixed energy cutoff. We expect that this error estimate for the hydrostatic distortion hold for other uniaxial distortions also. C. Effects of the different exchange-correlation forms, Wigner and Ceperley- Alder Aside from the above convergence studies, we have also considered the effects of two different exchange-correlation forms-Wigner versus Ceperley-Alder-on the bulk modulus and the lattice constants for all three systems we studied: bulk Si, Ge and superlattice Sii/Gei. We find that the Wigner form gives a larger lattice constant (better agreement with experiment) and Ceperley-Alder gives a larger bulk modulus (better agreement with experiment). This is true for all three systems: as summarized by two observations. (1) When we compare the exchange- correlation energies around the electronic density of Si or Ge, ( rs « 2 ), we find that Wigner’s is less attractive than Ceperley-Alder’s this decreased attraction explains why we get larger lattice constant with Wigner’s exchange- 72 correlation potential. (2) When we compare the bulk moduli, we find that the exchange-correlation contribution to bulk modulus is negative around rs = 2 , and it is 70 kbar more negative using Wigner form than using Ceperley-Alder; this larger negative contribution explains the smaller bulk modulus resulted by using Wigner form. Hence, a smaller bulk modulus resulted from the Wigner form. These observations, well known in the electronic structure community1®, point up once again the needs for an improved exchange-correlation form. D. Conclusions (1) The Wigner exchange-correlation potential gives better lattice constants while Ceperley-Alder gives better elastic constants. We choose the Ceperley-Alder potential, since this work is more focused on the elastic constants than on the structural properties. (2) We have introduced the “relative error” as a measure of convergence and error estimate. We find that a 10 special fc-point set will virtually eliminate the error (0.2%) due to finite sampling Appoint in Brillouin zone, but that even a large (10 hartree) energy cutoff could still yields an error of 9%; though we find the error in bulk modulus less than 3% and in lattice constant less than .5%. 73 2.4. The Calculated Results for Si, Ge and Si/Ge The elastic constants are calculated through finite differences, with quadratic fits for 5-7 data points (5 points for bulk modulus, lattice constant and £n; 7 points for £ 4 4 ). According to Nye,1^ the total energy change per volume due to the elastic distortions is expressed as [see Appendix A] — = g ^ ll( el + e 2 + el) + C l2 (el e 2 + e2 e3 + e3 ei) + - £4 4 ( 6 4 + 6 5 + eg), (2.16) where is the supercell volume, and £11,12,44 are the elastic constants, and ei g is the 6 -dimensional strain “vector”. First, we obtain the bulk modulus and the equilibrium lattice constant. This is done by calculating the total energies for different lattice constants, and then simply fitting Eq. (2.16) to find the bulk modulus and minimum energy lattice constant, which is the equilibrium lattice constant we report. Then we proceed to do ground state total energy calculations for different distor tions (geometries) involving other elastic constants, and the strains are calculated for each geometry. With both elastic energies (energy change) and strains known, we can fit Eq. (2.16) to obtain the elastic constants. There are several aspects in this work we want to emphasize: (1) the distortions are chosen to involve only one single elastic constant in the total energy change for a given geometry. With this choice of strain, we can rewrite Eq. (2.16) in the following form, = K C ij e \ (2.17) where the C ij is the elastic constant we wish to calculate, ej. is the strain due to the chosen distortion, and the K is just a proportionality constant, [see Appendix 74 A]. This procedure enables us to reduce the original 3-dimensional fit to a one dimensional fit, and hence to reduce the error in the fitting process. (2) We allow the atoms in the unit cell to relax to their equilibrium positions when unit cell is under a shear distortion. Even though this relaxation is vital in calculating shear modulus, C 4 4 , as discussed by Nielsen and Martin 3 and other earlier experimental and theoretical studies , 1 8 ' 1 9 ,2 0 this is the first attempt to allow this relaxation during the ground state energy calculation. The experimental and theoretic study had found that when the diamond structured Si and Ge (zincblende Si/Ge) are under a strain involving shear distortion, e 4 5 6 , the internal atom will relax by a distance d, in Cartesian coordinates, d = C^(e4>e5 ,e6)> (2-18) where a is the lattice constant, 6 4 ^ 6 are the shear strains, and ( is the internal shift parameter. Our results show that this relaxation reduces the C 4 4 obtained from non-relaxed calculation by about 25% to agree with the experiment value, for both bulk Si and Ge. Si and Ge provide test of method Table 2.7 shows our calculated results together with the experiment 18,19,20,21'22 and other LDA calculation 3 for bulk Si and Ge. The slight differences between ours and Nielsen’s are the result of (1) different exchange-correlation forms used in calculations for Si; they used Wigner form while we use Ceperley-Alder form, (2) different special fc-point sets for Ge; they used 2 special fc-points while we use 10 special fc-points. Comparing our results to the experiment, we can see that, except for £ , 2 3 our results for both Si and Ge agree well (no more than 3% difference). The good Table 2.7. Lattice constants a, bulk modular B, elastic constants Cy of Si, Ge, Si/Ge. Unrelaxed elastic constant C 4 4 , internal relaxation parameters (. a B C n C1 2 Si Ge Si/Ge Si Ge Si/Ge Si Ge Si/Ge Si Ge Si/Ge NMa 1 0 . 2 1 10.57 0.93 0.72 1.59 1.30 0.61 0.45 This Work 10.17 10.51 10.32 0.97 0.77 0 . 8 8 1.646 1.346 1.506 0.64 0.49 0.57 This Work 1.62c 1.32c 1.51c e x p / 10.270 10.680 0.99 0.77 1 . 6 8 1.32 0.65 0.49 a C4 4 C4 4 C Si Ge Si/Ge Si Ge Si/Ge Si Ge Si/Ge Si Ge Si/Ge NMa 1 0 . 2 1 10.57 1 .1 1 0.77 0.85 0 . 6 8 0.53 0.44 This Work(110)d 10.17 10.51 10.32 1 . 1 0 0.89 0.99 0.80 0.69 0.73 0.56 0.57 0.59 This Work(lll)e 0.77 0 . 6 8 0.75 0.53 0.50 0.51 e x p / 10.270 10.680 0.80 0 . 6 8 0.73A 0.72ft exp. 0.64* 0 .6 6 * *. Lattice constant in ao, elastic constants in Mbar(lOOGPa). C 1 2 deduced from B and Cn via C 1 2 = 5 (3B - C n ). a. O.H. Nielsen and Richard Martin, PRB 32, 3797 (1985). b. Volume changed, distortion in (001), [see appendix]. c. Volume uncahnged, depress in (001), expand in both (110) and (110), [see appendix]. d. Expand in (110) and depress in (110), [see appendix]. e. Distortion in (111) direction, [see appendix]. f. H.J. McSkimin, J. Appl. Phys. 24, 988 (1953), 35, 3312 (1964), except otherwise indicated. g. J. Donohue, The Structures of the Elements (Wiley, NY, 1974). h. H. d ' Amour et al, J. Appl. Crystallogr. 15, 148 (1982) and C.S.G. Cousins et al, J. Appl. Crystallogr. 15, 154 (1982). i. A. Segmuller and H.R. Neyer, Phys. Kondens. Materie. 4, 63 (1970). 76 agreement between our results and experimental data prove that our method is accurate, and establishes the solid foundation for predicting the properties of new materials, in our case, the zincblende Sii/Gej superlattice. The large discrepancy between our calculated £ and the experiment values is believed to be experimental error . 2 3 ,2 4 In part, this believe is supported by our estimate of £ using empirical two- and three-body potentials for Si . 2 4 Consider a distortion, 77, in ( 1 1 1 ) direction, the elements of strain tensor (x are Pij = qV i (2.19) Using Eq. (2.18), and the definition of strain tensor, the relaxed atomic position is, r ; = ( 1 + fi)r + d, (2 .2 0 ) where r is chosen as (a/4 )(l, 1,1) in the undistorted cell. W ith both fx and £, we can calculate the changes of the bond lengths and bond angles of the relaxed atom to its four nearest neighbors. The change in bond length, /, along (111) direction dl\ is ( 1 — C)vh while the change of the other bond lengths, d/ 2 , d/3 and d/ 4 , in the directions of (111), (111) and (ill), are same and equal to ( 1 + 3()rjl/9. The three bond angles involving the ( 1 1 1 ) bond are changed, d#io 2 , d#io3 and d$io4 , by the same amount of \/ 8 (l + 3£)/9; while the other three bond angles, dO2 0 3 , dd 2 0 4 and d#3 Q4 , have the change of same magnitude but opposite sign. The numbers in the subscripts indicate the nearest neighbors of the relaxed atom (with 0 being the position of the central atom). The equilibrium bond angle 9 = cos- 1 (—1/3). 77 To estimate the optimal we assume a simple elastic energy form of a quadratic function of bond length (two-body) and bond angle (three-body) £( = , 2 A[(1 - < ) 2 + 1(1 + 302] + + 3C)2]. where A and B are force constants. Minimizing E(dl, dO)/(rj)2, we have the optimal c, _ 2 1 —(2/ In order to evaluate the ratio of (B/A), we use the data from the empirical two- and three-body potential developed by Stillinger and Weber . 2 4 Expand their potential around equilibrium bond length and bond angle, we find B/A = 0.14998. Substituting this ratio into Eq. (2.22), yields £ = 0.5255, which agrees with our calculated result for Si (0.53 to 0.56).25 Results for Sii/G ei superlattice As a first cut at characterizing our results for the Sii/Gei superlattice (Table 2.7), we compare with the corresponding values of bulk Si and Ge. Note that the superlattice results lie intermediate between, as commonsense would suggest . 2 5 For a closer look, we compare our LDA results with “elastic theory” predictions on lattice constant and bulk modulus. What we mean by “elastic theory” is this: we require the compound (Sii/Gei ordered alloy) to have minimum elastic energy with each atomic species retaining its original lattice constant and bulk modulus. In the case of Sii/Gei, the elastic energy per volume is £((°Si/Ge “ °Si)> (°Si/Ge “ °Ge)) 9 „ , (aSi/Ge “ aSi) ,2 , 9 r, , (aSi/Ge “ aGe) ,2 ------n------= i Bsi(------^ ------> +4Ba^— ^ — >' (2.23) where a g i / G e is the lattice constant of Sii/Gei. Minimizing the elastic energy respect to agi/Qe> we have t^ S i i ^Ge \ /f^ S i i ^Ge \ s o o a \ aSi/Ge = (— + —)/ (“ 2“ + T2“ )- i2'24) Si aGe a*Si a Qe By the same token, we can get the “elastic theory” prediction of bulk modulus for Sii/G ei, Ssi/Ge = + »<*>(-f ^ ) 2]- (2-25) 1 1 aSi aGe The LDA results for bulk Si and Ge can then “predict” values for Sii/Gei. See Table 2.8. The difference between the LDA calculated property and the one predicted by ‘elastic theory’ is less than 0 .1 % for the lattice constant and 1 % for the bulk modulus. Such close agreement is unexpected since this mean-field ‘elastic theory’ would be expected to valid at most in the long wavelength limit . 2 5 To sum up, (1) we have developed a scheme to calculate each elastic constant separately, which eliminates the error in the fitting process; ( 2 ) we allow atoms to relax during the calculations of shear modulus, C44, the relaxation reduces the cal culated value by a 25% to agree with the experiment; (3) this is the first prediction for the properties of the as yet unsynthesized Sii/Gei. We argue that the accuracy of these predictions are supported by the agreement between our results for bulk Si and Ge with both previous theory and experimental data. 79 Table 2.8 “Elastic theory” (ET) predictions of lattice constant and bulk modulus for Sii/Gei. The “elastic theory” prediction of Sii/Gei lattice constant and bulk modulus is shown. The lattice constants are in bohr; bulk moduli in Mbar(=100GPa). The predictions are based on our LDA results for bulk Si and Ge (lattice constant and bulk modulus), and the comparison is made with the LDA results of Sii/Gei. The worse agreement seen on bulk modulus is due to the limit digits reported, actual agreement is within 0.4%. Si(LDA) Ge(LDA) Si/Ge(ET) Si/Ge(LDA) a I(n 7 10M 10T33 I(h32 B 0.97 0.77 0.87 0.88 2.5. Conclusions We have reported convergence studies of total energy versus of various aspects in the local density approximation calculations, such as a function of the size of the planewave basis and of the density of the special fc-point grids. The results show that the incompleteness of the planewave basis is the major source of error in these calculation; even with a large energy cutoff of 10 hartrees (400 planewaves), there could be still about 9% relative error in the energy difference, though the error in the calculated bulk modulus (and other elastic constants) is less than 3%, and lattice constant less than .5%. The finite density of fc-points accounts for less error, no more than 0.2% in energy difference, for a 10 special fc-point set. These convergence studies point to a way to realistic error estimation in the future work. But we could stress that we expect the error due to the fc-point set to grow as the energy gap decreases. 80 One unique aspect of our elastic constant calculation is that we actually allow the atoms to relax during our calculation, this relaxation produces agreement be tween theory and experiment for the sheax modulus C 4 4 . As a first step in our effort to understand Si/Ge superlattices, we have presented the results on Sii/Gei; these results are validated by the agreement between our results and experiment on bulk Si and Ge. In conclusion, we have ( 1 ) an efficient and accurate method to determine the ground state properties of Si/Ge systems, and (2) a realistic error estimation, and these methods may be used in more complicated Si/Ge systems. 81 REFERENCES 1. R. 0. Jones and 0. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989), and refer ences therein. 2. M. T. Yin and M. L. Cohen, Phys. Rev. B 24, 6121, (1981), B 26, 3259, (1982). 3. O. H. Nielsen and R. Martin, Phys. Rev. B32, 3797 (1985). 4. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985); 60, 204 2471 (1988). 5. D. C. Allan and M. P. Teter, Phys. Rev. Lett. 59, 1136 (1987). 6 . P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 7. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 8 . D. J. Chadi and M. L. Cohen, Phys. Rev. B 8 , 5747 (1973). 9. H. J. Monkhorst and D. J. Pack, Phys. Rev. B13, 5188 (1976). 10. D. R. Hamann, Phys. Rev. B40, 2980, (1989). 11. L. Kleinman and M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). 12. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566, (1980). 13. Private communication (1988); result is still unpublished. 14. E. Wigner, Phys. Rev. 4 6 ,1002 (1934). 15. G. Galli, R. Martin, R. Car and M. Parrinello, Phys. Rev. B42, 7470, (1990). 16. Ref. 1 page 698. 17. J. F. Nye, Physical Properties of Crystals, (Oxford, London, 1969) p 131. 18. H. dAmour et al, J. Appl. Crystallogr. 15, 148 (1982). 19. C. S. G. Cousins et al, J. Appl. Crystallogr. 1 5 ,154 (1982). 20. A. Segmuller and H. R. Neyer, Phys. Kondens. Materie. 4, 63 (1970). 21. J. Donohue, The Structures of the Elements , (Wiley, NY, 1974). 22. H. J. McSkimin, J. Appl. Phys. 24, 988 (1953); 35, 3312 (1964). 23. Recent comments of C. S. G. Cousins indicate that the earlier experiment results for £ might be over estimated. In eq. (2.22), if we only consider the two-body potential, (B=0), ( = 2/3, this leads to all four bond lengths same; if we only consider the three-body potential, (A=0), ( = —1/3, this leads to all six bond angle same (unchanged). One should expect that £ should be in between -1/3 and 2/3 for normal diamond structured materials. 24. Stillinger and Weber, Phys. Rev. B 32, 5262, (1985). 25. John R. Dutcher, et. al., Phys. Rev. Lett. 65, 1231, (1990). CHAPTER III Strained silicon: a dielectric response calculation Strain-induced birefringence is calculated with crystalline silicon for pressure applied along the [001] and [111] directions of the crystal. Results for the dielectric function and its change under hydrostatic strain are also given. The results are calculated for photon energies in the range 0 to 3.25 eV, i.e., below the direct gap. We have made a fully self-consistent Kohn-Sham Local Density Approximation cal culation, in the pseudopotential, plane wave scheme, with a self-energy correction in the form of a rigid shift of the conduction bands of magnitude A = 0.9 eV. Agree ment with experiment is very good in the static limit, considering disagreements among the experimental values. Values of the photoelastic tensor for [001] strain are P ll ~ P \2 — —0.118 (theory) and —0.111 ±0.005, —0.127 ±0.005 (expt.). For [111] strain, we obtain P 4 4 = —0.050 (theory) and —0.051 ± 0.002, —0.051 ± 0.002 [sic] (expt.); for hydrostatic distortions, p n + 2 p i 2 = —0.067 (theory) and —0.055±0.006, —0.070±0.008 (expt.). For the static dielectric constant, we obtain 10.9, compared to 11.7 and 11.4 (0 K) (expt.). All experiments quoted are at room temperature, except as noted. Above 2 eV, the calculation predicts less dispersion than seen by the experiments. Thermal effects and electron-hole interactions axe estimated to resolve some of the discrepancies with experiment. The experimental data for [001] strains is not consistent with a single oscillator model, and is therefore suspect. 83 84 3.1 Introduction When a cubic crystal is strained, the normally isotropic dielectric tensor be comes anisotropic. For small distortions, the linear relationship between the change in the inverse dielectric function and the strains form the photoelastic tensor. At least two groups have measured these coefficients for strained silicon. Biegelsen has measured the photoelastic coefficients for frequencies below the indirect gap . 1 Cardona and co-workers performed a similar measurement 2 and subsequently ex tended their work to frequencies between the indirect and direct gaps . 3 ,4 On the whole, these measurements are in agreement. For the case of isotropic strain, the agreement among the experiments is somewhat less satisfactory, as reviewed in Ref. 5. Semi-empirical models of the optical properties of strained silicon have been reviewed , 1 but there has never been a calculation using “first principles” techniques to our knowledge in silicon, or any other material; however, a sophisticated calcu lation of the band gap in diamond under [ 0 0 1 ] stress has been performed recently.® The purpose of the this chapter is to present a calculation of silicon under arbitrary anisotropic strains. In the past, two of us have studied other optical properties of solids, specifically the dielectric function of semiconductors®’^’3’® and the nonlin ear susceptibility for second harmonic generation.3’® Our approach (in both these works and the present) is: first, we perform a well-converged Kohn-Sham Local Density Approximation (LDA) calculation using plane waves and pseudopotentials to obtain a ground state potential . 1 3 Second, in calculating optical response, we 85 use a modified Hamiltonian H% = h ! d a + f3-1) where is the LDA Hamiltonian at a point k in the Brillouin zone, Ag is an en- ergy shift, and P j£ is the projection operator onto the conduction bands at k . The term A£ is sometimes called a “scissors” operator. This energy shift represents a self-energy correction which may be obtained through a many-body calculation in the GW approximation.11,12,13,14 In practice, we do not include the ^-dependence of the self-energy correction, which is on the order of 0.1 eV for silicon.13 Generally, self-energy-corrected LDA has been quite successful in predicting optical proper ties of semiconductors, leading e.g., to disagreement with experimental value of the static limit of the dielectric function e Our interest in strained silicon arises from the desire to test the range of appli cability of this approximation. Ultimately, our goal is the reliable prediction of optical properties of semiconductors which have not even been fabricated. Some of the problems we address in strained silicon will arise later in studies of strained semiconductor heterostructures. Our theory has been presented extensively recently,5 so it is omitted here. (How ever, we opted for finding virtual levels through direct diagonalization as in Refs. 8 and 9 rather than the iterative solution of the inhomogeneous Schrodinger equa tion.) Briefly, we are solving for the dielectric function with local field corrections using the scheme first presented by Adler15 and Wiser,15 using the LDA Hamil tonian modified by a self-energy correction in the form of a scissors operator, as shown in Eq. (3.1). 86 3.2 Symmetry Considerations A cubic crystal is optically isotropic. When the crystal is strained anisotropi- cally, its symmetry is lowered, and the crystal becomes birefringent. For a general strain, the resulting dielectric tensor has three independent components, i.e., it is biaxial. Certain particular strains render the crystal optically uniaxial. Compres sion in the [ 0 0 1 ] direction changes the crystal class from cubic to tetragonal; for compression in the [111] direction, it becomes trigonal. Both the tetragonal and trigonal crystal classes are optically uniaxial; the directions are fixed by the crystal geometry and axe independent of the frequency of light passing through them . 1 7 The change in the dielectric tensor due to strain is often described by the photoelastic tensor Pijkl, = PijklPkt (3-2) where the strain is given by the we use the Einstein summation convention in this chapter. The strain is symmetric in its two indices; in the absence of mag netic fields or absorption, the dielectric tensor (and its inverse) axe also symmetric. The tensor Pijkt a fourth rank tensor which is symmetric in both indices. It is conventional to use a compressed notation for such a tensor. Specifically, the index pairs are mapped: 1 1 —► 1 , 22 —> 2, 33 —* 3, 23,32 —► 4, 13,31 —*■ 5, and 12,21 —» 6 . We refer to the resulting set of 6 values as a“6 -vector”; these index pairs may be used for vector and matrix multiplications, but do not have the co-ordinate transformation properties of a vector. We adopt, the conventions of Grimsditch et al* pap = Pijkii where ij and M correspond to a and (3 respectively, with simi lar relations for a , S( 1 /e), and c, but for ir and p. relations such as = ffap, 87 a,/3 = 1,2,3 and 2ttijk£ — nap, a,/? = 4,5,6 hold. The variables resent the stress, elastic moduli, and piezobirefringence tensors, respectively. For silicon’s point group, O^, such a tensor will have three independent components. The tensor takes the form 1 8 ( p \ \ P12 P12 0 0 0 \ P12 Pll P12 0 0 0 P12 P12 Pll 0 0 0 0 0 0 P44 0 0 0 0 0 0 P44 0 \ 0 0 0 0 0 P44/ This matrix may be diagonalized. The eigenvalues (and degeneracies) are p \\ + 2 p i 2 ( 1 ), p ii — p i2 (2), and P 4 4 (3). The eigenvector, i.e., the strain, corresponding to the first eigenvector is a uniform compression. The eigenvectors of strain for the second eigenvalue may be chosen to be a compression along one Cartesian axis with a volume-conserving expansion in the other two directions. For the third eigenvalue, the eigenvectors may be viewed as a contraction in a bonding direction with a volume-conserving expansion in the orthogonal plane. The piezobirefringence tensor 7T relates the changes in the inverse dielectric function to stresses cr(.£ via W e i j ) = vijke°k£- (3-4) Experimentally, a stress is placed along the [001], [111], or [110] direction or applied hydrostatically, leading to a measurement of certain linear combinations of compo nents of the piezobirefringence tensor. In our calculation, we model a strain in the [001] or [111] direction or a hydrostatic distortion. The elastic moduli, defined by the relation ak£ = cijklPiji (3,5) may be used to relate the photoelastic tensor to the piezobirefringence tensor: Pijkt = ^ ijmncmnk£• (3-6) The tensors 7r and c have the same symmetry as p given in Eq. (3.3). In practice, comparison to experiment is facilitated because we are only concerned with those linear combinations of the tensor coefficients which are eigenvalues. Since, viewed as matrices, these are simultaneously diagonalizable, conversion from 7r to p involves a single scalar multiplication. Explicitly, P llll + 2pil22 = (*1111 + 27r1122)(cilll + 2cn22) P llll - P1 1 2 2 = ( * 1 1 1 1 - * ii 2 2 )(c m i - C1 1 2 2 ) (3.7) and P 1 2 1 2 = *1212C1212- The strains we consider are: uniform compression, compression of the [001] direction only, leaving the other directions fixed, and compression of the [111] di rection only, leaving the other directions fixed. (Our [001] and [111] compressions are not eigenvectors of the matrix p in Eq. (3.3) since they result in a change in vol ume.) From uniform compression, we determine the linear combination p \\ + 2pi2- From the [001] compression, we determine both p \\ and P 1 2 , and hence p \\ — P12 and pii -f 2pi2- From [111] compression, we find P 4 4 and p \\ -f 2pi2- The three ways of determining p ii + 2 p i 2 represent an internal consistency check, as discussed below. To determine these photoelastic constants, we calculate the dielectric functions under different strains. Under hydrostatic stain, 89 where 8a is the change in lattice constant and T is the transpose symbol. The symbol denotes a “6-vector”. Using the definition of the photoelastic constant, Sq rn S[ 1/e] = \p][n] = — (pn + 2pi2,pn + 2pi2,pn -f 2pi2,0,0,0) (3.9) u P ll + 2P12 can be determined from the difference between the strained and un strained dielectric constant £[l/e], using Sa ( PH + 2P12 0 0 \ 8[l/e} = - { 0 Pii + 2pi2 0 . (3.10) a \ 0 0 p n -f 2pi2 / Under a strain in [001] direction, M = — (0,0,1,0,0, o f (3.11) a the difference of the dielectric matrix becomes Sa /P12 0 0 \ 8[l/e) = -i 0 p i2 0 . (3.12) ° V 0 0 pn/ The other element in the photoelastic matrix P44 can be determined by changing the length in the [111] direction by 81, i.e., by the strain M = lj(l,l,l,2,2,2)r . (3.13) The factor 2 involved in the formula comes from the definition stated above. Again from the definition of the photoelastic constant, \8 l ( P ll + 2 P12 2P44 2P44 \ ^[1/e] = --T 2p44 pn + 2pi2 2p44 . (3.14) V 2 P44 2 P44 P ll + 2 P12 / The [111] strain permits an internal degree of freedom, often denoted by £. Under a [111] strain, the bond length between two silicon atoms joined in the [111] direction may vary, but the direction will be unchanged. As indicated in Table Table 3.1 Characteristics of special values of the parameter ( which describes the bond length along the [111] direction under strain in the [111] direction. For the unstressed system, the bond lengths along [111], [111], [111], and [111] are all equal, but with stress along [111], there are two distinct values for the bond lengths: [111] and the others. Similarly, in the unstressed case, there is one value for the bond angle, but under [111] strain there are two distinct values for the bond angles. (When the phrase “under compression” appears, under expansion the sign of the indicated phenomenon is reversed.) C Remark —1/3 All bond angles equal > —1/3 [111] [111] angle less than [111] [ill] angle under compression 0 Change in [111] is proportional to overall change in [111] direction <2/3 [111] bond shorter than other bonds under compression 2/3 All bond lengths equal >2/3 [111] bond longer than other bonds under compression 1 New [111] bond length equals uncompressed [111] bond length 3.1, C = 0 indicates the atoms connected along [111] are shifted uniformly with the strain, and £ = 1 indicates this pair of atoms retains its unstressed bond length. These two points suffice to define a scale for the final bond position, but we refer the reader elsewhere for a mathematical definition.*9,20,21,22 91 Table 3.2 Default parameters for calculation in this chapter. Only the changes from these values will be noted in the other tables and the figures. The “Equivalent number of fc-points” refers to the number of integration points (or “special points”23) in the irreducible Brillouin zone of the dia mond structure; see Table 3.7 for more information. Photoelastic tensor components were found using the indicated finite difference scheme. Plane wave energy cut-off 10 hartree Self-energy correction A 0.9 eV Equivalent number of fc-points 60 Lattice constant 10.2646 bohr (300 K expt.) Bands retained hydro. 250; [001] 100; [111] 70. Finite differences at lattice constants of ±1% of nominal Pseudopotential Hamann,24 separable23 Exchange-correlation potential Ceperley-Alder23/Teter27 Core corrections None 3.3 Results We have calculated the various independent components of the photoelastic tensor using the method described in the previous sections. Our default parameters are given in Table 3.2; that is, these conditions are used except when we explicitly state they are varied in a sensitivity test. Compared to a previous study by some of us,3 we have increased the plane wave energy cut-off from 9 hartree to 10 hartree, used the experimental lattice constant at 300 K rather than 0 K, and corrected a small, previously reported bug® which shifts the calculated dielectric function by about - 2% upon correction. « 92 0.5 t ------1------r Si, Hydrostatic Strain 0.4 r / ■ Theory j s 0.3 Expt. * (Biegelsen) * (Vetter) j C\2 +H / S 0.2 I 0.1 0 J ______I______I______I------1------L. 0 0.5 1 1.5 2 2.5 3 3.5 Photon Energy (eV) Figure 3.1 The photoelastic coefficient p n +2pi2 as a function of photon energy for silicon under hydrostatic strain, compared to experimental data from Biegelsen1 and Vetter.^® Our results for silicon under hydrostatic strain are plotted in Fig. 3.1. There have been a number of measurements of this quantity, as collected by Ref. 9 for example; here we only present two which we believe to be authoritative. Their static limits are almost in agreement with each other: for p n + 2pi2, Biegelsen gives a value of —0.055 ± 0.006 compared to Vetter’s —0.070 ± 0.008. The present study predicts —0.062, which is in marginal agreement with the both values. The calculated frequency dependence is a reasonable description of the experimental 93 0.15 Si, [001] Strain oo°oo oooo 0.12 T - r — - ^ - 0.09 ■ Theory j ^ & I * Expt. (Biegelsen) f CM & 0.06 o Expt. (Higginbotham) • Expt. (Grimsditch) | 0.03 0 0 0.5 1 1.5 2 2.5 3 3.5 Photon Energy (eV) Figure 3.2 The photoelastic coefficient p\2 — p n as a function of photon energy for silicon under a strain in the [001] direction. Experimental data is from Biegelsen1 and Cardona and co-workers below2 and above4 the indirect band gap. As discussed in the text, by our estimate, thermal effects and electron-hole interactions should account for some of the difference between our calculation and experiment at the higher frequencies, but cannot reproduce the dispersion. data. We are not aware of data in the region above the indirect band gap, so the test of frequency dependence is not particularly stringent in this case. In Fig. 3.2, the calculated photoelastic tensor component p \\ — p\2 is given for strain in the [001] direction. Below the indirect band gap, our calculated values are between the two data sets. While this indicates our theory is reasonable, it gives no support for either set of measurements over the other. Above the indirect 94 band gap, our calculated dispersion is less than half that of the experiment. As discussed below, this may be due to a combination of thermal effects, particle-hole interactions, and errors in the experiments beyond the reported uncertainties. The third independent component of the photoelastic tensor P 4 4 is given for stress in the [111] direction. As discussed in section 3.2 and Table 3.1, the variable £ is a measure of the relaxation of the [111] bond under stress. The value ( = 0.53 is predicted by two independent LDA total energy calculations ;2 1 ,2 2 and a semi- empirical model . 2 2 The value ( — 0.53 is made plausible by noting that C = 2/3 if the bond energy were totally determined by bond lengths alone, and ( = —1/3 if determined by bond angle energies alone. The value suggested by the LDA calculations is about 1/7 of the way from the “bond length only” value to the “bond angle only” value indicating bond lengths dominate the energetics, but bond angles play a significant role. Both measurements 1 ,2 of P 4 4 happen to have the identical value of —0.051 ± 0.002 which is in agreement with our calculated value of —0.050. As seen in Fig. 3.3, our calculation gives an excellent account of the low frequency behavior, but does less well with the dispersion. Although particle-hole interactions and thermal effects may account for this problem, as we proposed for the [ 0 0 1 ] strain, we have no evidence that this is the case. At first glance, it appears that a smaller value of ( might fit the data, but it does not appear possible to obtain good agreement above the band gap without destroying the already good agreement in the static limit. Fig. 3.4 illustrates that P 4 4 is approximately a linear function of £ in the frequency range of interest. 95 0.18 T Si, [111] Strain f 0.15 / Expt. * (Biegelsen) o (Higginbotham) • (Grim^ditch) 0.12 a {= 0 / ' ► £=0.53 $ ft 0.09 ▼ £ =0.8 ^ I 0.06 -— ~-"vr * * _* co o o o ,o pod ____ » ______»— — » — » 0.03 \ 0 j _____ 0 0.5 1 1.5 2 2.5 3 3.5 Photon Energy (eV) Figure 3.3 The photoelastic coefficient P 4 4 as a function of photon energy for silicon under a strain in the [111] direction. The sources for the experimental data are the same as in Fig. 3.2. The internal relaxation parameter £ is defined in section 3.2 and described in Table 3.1. £=0.53 is predicted by LDA total energy calculations ; 2 1 ,2 2 £ = 0 is the value without internal relaxation of the [ 1 1 1 ] bond; £ = 0.8 is an arbitrary third value. As discussed in the text, by our estimate, thermal effects and electron-hole interactions apparently will not account for the difference between our calculation and the experiment at higher frequencies. There may be problems with the measurement. The 1980 work (Ref. 4) was a refinement of work by similar authors two years earlier.5* They undertook the remeasurement to take into account effects of optical absorption above the band gap on their reported values because of an inconsistency of the 1978 measurements and certain Raman scattering data. The 1980 paper gave a downward revision of 96 0.12 Si, No Strain Expt. 100 K (Li) 0.10 Expt. 300 K (Li) Expt. 300 K (Aspnes) 08 3 to 0.06 A=0.9 eV A=0.7 eV 0.04 0 3 6 9 12 Photon Energy Squared [(eV)2] Figure 3.4 The photoelastic coefficient —P 4 4 as a function of the internal shift ( for three photon energies. p44 is seen to be roughly linear in ( at all frequencies. the magnitude of P 4 4 which was about twice the 1978 error estimate, and the new error estimate was increased by about a third. Nevertheless, the authors concluded that even the revised values were not consistent with the Raman scattering data. We may gain a qualitative understanding of the behavior of the piezobirefrin gence coefficients. Imagine a two level model, with an occupied bonding orbital and an unoccupied antibonding orbital. If the bond length is shortened, the energy splitting of the two levels will increase, and hence the polarizability in the direction 97 Table 3.3 The photoelastic coefficient of hydrostatic strain, pn + 2 p i 2 , calculated from various strains. (See section 3.2 for a discussion of how these values are obtained.) The variable £ only applies to [111] strain, with C = 0.53 being our estimate of the experimental value. Agreement at the 2 % level is achieved between all cases, except [ 1 1 1 ] strain with ( = 0 . Photon energy (eV) c 0 2.75 3.25 Hydrostatic strain -0.062 -0.206 -0.404 [001] strain -0.061 -0.203 -0.401 [111] strain 0 -0.236 -0.294 -0.432 0 . 5 3 -0.063 -0.208 -0.410 0.80 -0.063 -0.208 -0.402 of the bond will decrease. Perpendicular to the bond there will be little effect: our calculation indicates that the magnitude of the shift in dielectric function for the component parallel to the strain direction is some 6 times larger for the [ 0 0 1 ] strain and 16 times larger for the [ 1 1 1 ] strain than for the perpendicular compo nent. These simple ideas also account for the £ dependence of the coefficient P 4 4 . The [111] bond length change is less for larger £ (up to 1). To the extent that the [1 1 1 ] bond dominates the changes in the polarizability, a larger value of £ implies less change in the polarizability, hence a smaller P 4 4 than the result given by the purely kinematic distortion, £ = 0. For hydrostatic compression, one expects, and finds, a decrease in polarizability with decreased lattice constant. However, at large pressures, silicon becomes metallic, so this simple picture necessarily breaks down. 98 Table 3.4 The effect of lattice constant on the dielectric function and photoelastic tensor at three photon energies. The LDA lattice constant, obtained by minimizing the total energy in the LDA, is = 10.1733 bohr. The experimental lattice constant is aeXp = 10.2646 bohr. As the band gap is approached, the sensitivity to lattice constant increases; see in particular p n — pi2- Photon energy (eV) 0 2.75 3.25 e aLDA 10.88 16.26 21.66 Qexp 10.94 16.71 23.17 Pll + 2pi2 aLDA -0.041 -0.165 -0.279 o-exp -0.062 -0.206 -0.404 Pll ~ P12 aLDA -0.1193 -0.098 -0.067 aexp -0.1184 -0.087 -0.017 P44 aLDA -0.045 -0.045 -0.045 aexp -0.050 -0.051 -0.051 A. Sensitivities We noted in section 2 that the photoelastic coefficient associated with hydro static strain, p \\ + 2 p i 2 , may be calculated by applying a hydrostatic strain, an [001] strain, or a [111] strain. In principle, these values should be equal. We calcu lated pn + 2 p i 2 using the three different distortions; in the case of the [111] strain, we used three values of (. As seen in Table 3.3, we find the various methods yield agreement at three photon energies within 2%, except for the unphysical case of £ = 0. (Why ( = 0.80, which is also unphysical, remains well behaved remains a 99 minor mystery.) This numerical agreement provides a limit on our overall accuracy, and is a verification of our code. The sensitivity to the lattice constant of the calculation is studied in Table 3.4. For e, p n — P 1 2 , and P44 the lattice constant dependence is moderate, but for PH + 2pi2, the lattice constant dependence is quite large, varying by up to 50% for a change of less than 1% in the lattice constant. Typically, properties become very sensitive just below the direct band gap, which occurs at 3.58 eV in our calculation. In Table 3.4, this enhanced sensitivity occurs for e and pn — P 1 2 for the 3.25 eV photon energy; indeed, the p n —P 1 2 value is so sensitive that we feel it is unreliable. It is unfortunate that the property pn + 2pi2 is so sensitive to the lattice constant — for a novel material, the geometry would not be known in advance; if the LDA total energy method is used to determine the geometries, this might be the largest source of error in the calculation. A similar issue arose in the case of the nonlinear susceptibility for second harmonic generation.®’® The sensitivity to changes in self-energy correction A is given in Table 3.5. On a priori grounds we prefer the value of A = 0.9 eV, indicated by the GW calculation of Zhu and coworkers14 which used a 9-hartree plane-wave energy cut-off, to A = 0.7 eV, which is suggested by two earlier GW calculations which employed energy cut offs of 6.25 hartree13 and 6.5 hartree.12 As illustrated in Table 3.6, A = 0.7 eV happens to make our eigenvalue difference estimate of the direct gap nearly equal to the experimental value. So we chose A = 0.7 eV as a reasonable comparison value; some LDA results, i.e., A = 0, are also given. As illustrated in Table 3.5, in the static limit an adjustment of A by 0.2 eV has a rather modest effect (less then 10%) on the photoelastic tensor components and e. Near the direct band gap, 100 Table 3.5 The effect of the self-energy parameter A on the dielectric function and photoelastic tensor at two photon energies. As the band gap is approached, the sensitivity to A increases. The omitted values are too singular to be evaluated accurately in our integration scheme; u> = 3.25 eV, A = 0.7 eV also leads to unreliable values. A = 0 is the LDA. Photon energy (eV) A 0 2.75 e 0 13.2 0.7 11.4 18.6 0.9 10.9 16.7 Pll + 2pi2 0 -0.085 0.7 -0.067 -0.249 0.9 -0.062 -0.206 P ll -P12 0 -0.105 0.7 -0.115 -0.073 0.9 -0.118 -0.087 P44 0 -0.0459 0.7 -0.0489 -0.0500 0.9 -0.0497 -0.0506 the sensitivity increases in most cases. The fc-dependence of the scissors operator is estimated to be about 0.1 eV for Si.13 Such a variation is certainly negligible for static properties, although may play a minor role in obtaining quantitative agreement near the band edge, i.e., for photon energies above (say) 3 eV. Next, consider the sensitivity to the number of k-points included in our Bril- louin zone quadratures. As detailed in Table 3.7, the strains lower the symmetry 101 Table 3.6 Direct gap of silicon for various conditions. The direct gap occurs at T. Present LDA 2.68 eV Present A = 0.7 eV 3.38 eV Present A = 0.9 eV 3.58 eV Expt.29 300 K 3.45 eV Expt.30 0 K 3.35 eV Expt.31 4 K 3.4 eV Expt.31 shift 4 K to 190 K —50 meV of the crystal, and hence increase the size of the irreducible Brillouin zone. We obtain agreement to no worse than 1 microhartree, and possibly much better, in eigenvalues when the different symmetries are used (i.e., when we allow “accidental degeneracies”) in the calculation of the unstrained crystal. Table 3.8 illustrates the effect of increasing the density of integration points in the full Brillouin zone by a factor of about 2.4, i.e., going from what is usually known as “28 special points”23 to “60 special points”. In the static limit, values change by 1% to 10%; at higher frequencies the quantities become more sensitive to the number of special points. In particular, the variation in p \\ —pi2 at 3.25 eV is so large as to render this value unreliable (and it is not reported in Fig. 3.2). We expect a greater variation in all integrated quantities for photon energies approaching the direct band gap. For this photon energy and above, the integrand becomes singular, and therefore is either slowly convergent or requires sophisticated integration methods. 102 Table 3.7 Special fc-point sets for various symmetries. Each column rep resents an identical set of fc-points for the unstressed crystal. N Sym refers to the number of useful point group operations; as our program includes time reversal symmetry automatically, we present the number of group elements used explicitly by our program: This will be half the number of elements in the point group if T is a subgroup of the point group. All indicated group operations are symmorphic. N at0m refers to the number of atoms in the unit cell, for the cells we have chosen. In the first three lines, Nf.pi refers to the number of integration points in the irreducible Brillouin zone. In the final two lines, Nf.pi represents the linear and volume densities of integration points in the Brillouin zone, respectively. Condition Point Group Nsym Natom Nkpt Undistorted Ofl <£> T/T 24 2 28 60 [001] strain D We did not test the sensitivity of the predictions of this study to the energy cut-off in our plane wave basis. However, earlier work by our group9 indicated that e is converged to a few parts in 103 in the case of GaAs for a 10 hartree cut-off. The convergence in silicon is expected to be somewhat more forgiving, leading to a relative uncertainty of perhaps 10“ 3, or 0.01 in absolute numbers. We regarded this value as small enough not to warrant a detailed test. Table 3.8 The effect of Nkpt, the number of integration points in the irreducible Brillouin zone, on the dielectric function and photoelastic tensor at three photon energies. The lattice constants are given in the caption of Table 3.4. As noted in Table 3.7, the quadratures using 28, 36, and 91 integration points in the irreducible Brillouin zone - the odd lines in the table - represent equivalent integrations in the full Brillouin zone. Similarly the quadratures using 60, 80, and 204 integration points - the even lines in the table - are also equivalent. The sensitivity to the Nf.pt is not expected to vary strongly with the lattice constant. N kpt Photon energy (eV) 0 2.75 3.25 £ aLDA 28 10.95 16.47 22.15 60 10.88 16.26 21.66 P ll + 2pi2 aLDA 28 -0.046 -0.180 -0.337 60 -0.041 -0.165 -0.279 P ll “ P12 aLDA 36 -0.113 -0.083 -0.028 80 -0.119 -0.098 -0.067 P44 aeXp 91 -0.047 -0.046 -0.041 204 -0.050 -0.051 -0.051 Overall, we believe the static values we present are accurate within the model to a few per cent, and those at photon energies of up to 2.75 eV are accurate to 104 0.105 Si, No Strain • - - A=0.9 eV 0.100 A=0.7 eV 0.095 0.090 Expt., 100 K (Li) Expt., 300 K (Li) 0.085 0 0.2 0.4 0.6 0.8 1 Photon Energy Squared [(eV)2] Figure 3.5 The dielectric function of unstrained silicon in the combination (e(u>) — l ) - 1 as a function of the square of photon energy, for 0 < u? < 10 (eV)2. Such a plot is usually nearly linear for semiconductors below their direct band gaps 3 2 which occurs for u >2 « 11.6 in silicon. Experimental values are given for 100 K 3 3 and 300 K . 3 3 ,3 4 The box bounded by the three dotted lines and the ordinate is expanded in Fig. 3.6. B. Thermal Effects and Particle-Hole Interactions To address the discrepancies between our predictions and the measured disper sion of pn — pi 2 and P 4 4 from about 1.5-2.75 eV, we consider thermal effects and particle-hole interactions. Although the photoelastic tensor coefficients have only been measured at room temperature, temperature dependence of e(u>) is available . 3 3 ,3 4 We plot in Fig. 3.5 105 the dielectric function below the gap in a particular combination which is known to give a straight line throughout nearly the entire sub-gap regime.32 (The function is well described by a single oscillator model, as discussed in section C.) An expanded view is given in Fig. 3.6 of the region in which values are known at several temper atures. Extrapolating the two experimental curves to zero temperature will give a value nearly co-incident with our A = 0.7 eV curve both in the static limit (where both experiment at 0 K and A = 0.7 eV lead to e = 11.4) and the slope. A value of A near 0.6 eV would be co-incident with the 300 K line. Hence, the thermal shift from 0 K to 300 K is roughly equivalent to subtracting 0.1 eV from the band gap. Such an observation is consistent with experimental observation of a shift of —50 meV in the direct gap as the temperature is raised from 4 K to 190 K, as presented in Table 3.6. An estimate of the eifect of a —0.1 eV shift in the band gap may be obtained using the data in Table 3.5. For p \2 — P ll, a reduction of about 0.007 is expected at 2.75 eV photon energy, which is much smaller than the 0.03-0.05 reduction required to obtain agreement with experiment. For P 4 4 , there is almost no A dependence, so this mechanism is completely unimportant. For e, our A = 0.9 eV value of 10.9 differs by 0.8 from the 300 K value of 11.7, but only 0.5 from the 0 K value of 11.4. Band theory neglects electron-hole interactions. The GW approximation de termines improved values for the one-electron Green’s function; electron-hole in teractions are outside its scope. As a byproduct of an attempt to understand the optical absorption spectrum of silicon, Hanke and Sham35 calculated the effect of the electron-hole interaction on the static dielectric constant; they predict a shift of +0.55 (from 9.85 to 10.4) for e due to this interaction. Their calculation is per- 106 0.08 Si, [001] Strain 0.07 s, r—nC>J N ^ ■ Theory J ^ s. o Expt. (Higginbotham) 2 0.06 ^ • Expt. (Grimsditch) •■ 4 0.03 0 2 4 6 8 Photon Energy Squared [(eV)2] Figure 3.6 The dielectric function of unstrained silicon in the combination (e(u>) — I ) - 1 as a function of the square of photon energy for 0 < u> < 1 eV. This data is an expanded region of the plot in Fig. 3.5. Experimental values presented at 100 K and 300 K are those suggested by Li . 3 3 The shift due to temperature change of is seen to be roughly equivalent to a shift due to a change in A. formed with a model band structure, so the exact value presented here is not to be taken too seriously. (Hanke and Sham’s 3 5 value of 8.0 for the static dielectric constant in the LDA compares to 13.2 in the present work, and 12.7 and 13.0 in other high quality calculations.36'37) An additional shift of +0.5 in e is just what is required to bring the A = 0.9 eV calculation of e into agreement with the 0 K value of 11.4. In terms of a variation in A, we crudely model the shift due to particle-hole interactions by deducting 0.2 eV from the self-energy correction. Such a shift would 107 translate into a reduction of about 0.015 in p\2 — p \\ at w = 2.75 eV, which, in combination with the 0.007 thermal shift, is nearly sufficient to obtain marginal agreement with the data. Of course, the static limit would move away from the data of Higginbotham et al.2 by about 3% (but towards the data of Biegelsen1) and away from the data of Grimsditch et a/ .4 just above the indirect gap. Hence, it is not likely that particle-hole effects will lead to a match of the measured 4 dispersion of p n — pi 2 above the indirect gap. There is no effect on P 4 4 , since A has very little effect on it, at least for £ = 0.53. The detailed quantitative argument in the above paragraphs is not to be taken literally. The main point is that thermal effects and particle-hole interaction effects are of the correct order of magnitude to remedy some but not all discrepancies with experiment. They are of the same order of magnitude as each other, and their sign is the same. An improved theory should take both effects into account. C. Single Oscillator Model In this section, we compare our calculation and experimental data to the simple “Single Oscillator Model”. Wemple and DiDomenico 5*2 observed that the dielectric function of a large number of semiconductors obeys: e(w) = 1 -I- J o> (3-15) w* — ur for some oscillator strength / and oscillator frequency u>0. Such a relationship suggests that a plot of (e(u>) — l ) - 1 against u >2 will lie on a straight line. This is verified for both theory and experiment in Figs. 3.5 and 3.6. We wish to extend this model to the case of strain-induced birefringence. The simplest model is to imagine that under some strain the dielectric tensor is 108 given by t \ Cl &ijf + fijklPkl (<)-ia\ eij(u>) = S(j + f o t3-16' J (w0+w0iwPktr- ”2 to first order in the strain. Expanding to first order in the strain and neglecting terms of order (w/u;0)4, which is consistent with the single oscillator assumption , ^ 2 we arrive at _ , \ c , Si j f + fijki Pkt ~ 2Sijfuouokt Vkt ,0 eij{u) = Sij + ------~ 2~~~T.T2------(3-17^ U‘-U‘ The constants are not important in this connection — only the 10 dependence. For two directions of polarization with components and ef?\ /«(1)-(1) »(2) a (2) v , - f ’f ««(») - (6f ^ X V " <3-18^ o By the definition of the photoelastic tensor in Eq. (3.2), the combination pe(w ) 2 is proportional to the left side of Eq. (3.18): The single oscillator model predicts a linear plot of l/(pe(u;)2) against u>2. We show such plots for the [001] strain in Fig. 3.7 and [111] strain in Fig. 3.8. The single oscillator model is obeyed by the theory in both plots. For the [001] strain, the low frequency measurements 2 are in accord with the single oscillator model, but the values above the indirect gap, the data 4 obeys a substantial devi ation. For the [111] direction, the low and high frequency measurements show a marginal disagreement with the single oscillator model, primarily in the form of a misalignment of the low and high frequency data. A decrease in the high frequency value of P 4 4 would improve the agreement with the single oscillator model, as well as agreement with our calculation. Noise in the slope of the low frequency data for the [ 1 1 1 ] direction make it a less reliable predictor of the high frequency regime than for the [001] case. The indirect gap plays no role in our theory, but the absorptions 109 0.20 Si, [111] Strain ^ ■ Theory 0.15 tM ^ ^ o Expt. (Higginbotham) ■o^ ^ ^ • Expt. (Grimsditch) ^ 0.10 a \ T 0.05 0.00 0 2 4 6 8 Photon Energy Squared [(eV)2] Figure 3.7 The photoelastic coefficient and dielectric function plotted in the combination 1 /(pi2 — P llM ^)2 35 suggested by the single oscillator model. The single oscillator model suggests that this data should fall on a straight line. This condition obtains for the calculation and the data of Higginbotham et al? but not that of Grimsditch et alA considerably complicate the measurements. The failure of the data, particularly for [001] strain, to obey the single oscillator model suggests that these data may not be valid. Wemple and DiDomenico have considered an alternate extension, leading to the conclusion that a plot of p /( 1 — 1/e)2 against u>2 will lie on a straight line.^® We do not present these plots; however, their analysis leads to substantially the same con clusion: a well-defined inconsistency for the low and high frequency measurements 110 0.18 Si, [111] Strain 0.15 0.12 ft 0.09 0.06 0.03 0.00 0.0 0.2 0.4 0.6 0.8 Figure 3.8 The photoelastic coefficient and dielectric function plotted in the combination — l/p 4 4 £(w)2 as suggested by the single oscillator model. The single oscillator model suggests that this data should fall on a straight line. The low frequency data of Higginbotham et al? and the higher frequency data of Grimsditch et a/.4 axe in marginal agreement with a single oscillator model. of p ii — p\2 and an ambiguous result for the P44. Our calculated values obey the 1 single oscillator model (i.e., fall on a straight line) for this plot as well. 3.4 Conclusions We have calculated all independent components of the photoelastic tensor of sil icon using the Kohn-Sham Local Density Approximation (LDA) with a self-energy I l l correction in the form of a “scissors operator”, as well as the ordinary dielectric function. This work is an extension of previous successful calculations of the di electric function and nonlinear susceptibility for second harmonic generation of semiconductors. We achieve good agreement with all components for low frequencies. It is necessary to use the experimental lattice constant (rather than the one predicted by minimizing the LDA total energy) to achieve this agreement in the case of the components associated with hydrostatic compression. We omit thermal effects and electron-hole interactions in the calculation. While we estimate that these effects are probably sufficient to bring the calculated dielec tric function into agreement with experiment, the agreement for the piezobirefrin gence coefficients may be improved only to a limited extent. Consideration of a single oscillator model indicates that the measured values below and above the in direct gap may not be consistent with each other, particularly for strains applied in the [001] direction. 112 REFERENCES 1. D. K. Biegelsen, Phys. Rev. Lett. 32, 1196 (1974); Phys. Rev. B 12, 2427 (1975). 2. C. W. Higginbotham, M. Cardona, and F. H. Poliak, Phys. Rev. 184, 821 (1969). 3. M. Chandrasekhar, M. H. Grimsditch, and M. Cardona, Phys. Rev. B 18, 4301 (1978). 4. M. H. Grimsditch, E. Kisela, and M. Cardona, Phys. Stat. Sol. (a) 60, 135 (1980). 5. Z. H. Levine and D. C. Allan, Phys. Rev. B 43, 4187 (1991). 6. M. P. Surh, S. G. Louie, and M. L. Cohen, to be published. 7. Z. H. Levine and D. C. Allan, Phys. Rev. Lett. 63, 1719 (1989). 8. Z. H. Levine and D. C. Allan, Phys. Rev. Lett. 66, 41 (1991). 9. Z. H. Levine and D. C. Allan, to be published. 10. M. P. Teter, M. C. Payne, and D. C. Allan, Phys. Rev. B 40 12255 (1989). 11. L. Hedin, Phys. Rev. 139, A796 (1965); L. Hedin and S. Lundqvist in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1969), Vol. 23, p. 1. 12. M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). 13. R. W. Godby, M. Schliiter, and L. J. Sham, Phys. Rev. B 37, 10159 (1988). 14. X. Zhu, S. Fahy, and S. G. Louie, Phys. Rev. B 39, 7840 (1989); erratum 40, 5821 (1989). 15. S. L. Adler, Phys. Rev. 126, 413 (1962). 16. N. Wiser, Phys. Rev. 129, 62 (1963). 113 17. M. Born and E. Wolf, Principles of Optics (Permagon, New York, 1975), p. 679. 18. J. F. Nye, Physical Properties of Crystals, second edition, (Oxford, New York, 1985), Chapters 8 and 13. 19. L. Kleinman, Phys. Rev. 128, 2614 (1962). 20. A. Segmuller and H. R. Neyer, Phys. Kondens. Materie 4, 63 (1965). 21. 0 . H. Nielsen and R. M. M artin, Phys. Rev. B 32, 3792 (1985). 22. S. Wei, J. W. Wilkins, and D. C. Allan, to be published. This paper gives additional references regarding the internal relaxation parameter (. 23. H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). 24. D. R. Hamann, Phys. Rev. B 40, 2980 (1989). 25. L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). 26. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45 566 (1981). 27. M. P. Teter, unpublished. Teter has provided a new parameterization of the electron gas data of Ceperley-Alder which is differentiable at all electron den sities. 28. R. Vetter, Phys. Stat. Solidi (a) 8, 443 (1971). 29. M. Welkowsky and A. Braunstein, Phys. Rev. B 5, 497 (1972). 30. D. R. Masovic, F. R. Vukajlovic, S. Zekovic, J. Phys. C 16, 6731 (1983). 31. D. E. Aspnes and A. A. Studna, Solid State Comm. 11, 1375 (1972). 32. S. H. Wemple and M. DiDomenico, Jr., Phys. Rev. Lett. 23, 1156 (1969); Phys. Rev. B 3, 1338 (1971). 33. H. H. Li, J. Chem. Phys. Ref. Data 9, 561 (1980). 34. D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983). 35. W. Hanke and L. J. Sham, Phys. Rev. B 21, 4656 (1980). 114 36. S. Baroni and R. Resta, Phys. Rev. B 33, 7017 (1986). 37. M. S. Hybertsen and S. G. Louie, Phys. Rev. B 35, 5585 (1987). 38. S. H. Wemple and M. DiDomenico, Jr., Phys. Rev. B 1, 193 (1970). CHAPTER IV Open Questions and Future Work In this thesis, we have addressed some of the questions which arises in the attempt to understanding the properties of the covalent semiconductors, such as Si, Ge and Ge/Si superlattices. What we have done is to: (1) establish an accurate and effective method for calculating the elastic constants by the “direct approach”; (2) propose a method for studying the total energy convergence with respect to various aspects of the Local Density Approximation, and in the process, to produce serious error estimates; (3) calculate the strain-induced birefringence of crystalline Si. In this chapter we briefly discuss the open questions and possible future work. 4.1 Improving the Performance of Local Density Approximation As discussed in Sec. 2.3, the convergence of the total energy in the Local Density Approximation calculation remains a problem. . We start with the less troubled special fc-point scheme. Even though our study has found that the total energy is well converged for a 10 special fc-points set1 (within 1-2 meV), the actual convergence form is far from clear, except that the convergence is faster than the power law.2 Neither of two fitting forms, power law and logarithmic, represent the data well enough to make a prediction, nor are there any theoretic predictions on 115 116 how well this finite sum should present the Brillouin zone integration. One of the defects in our study is the limited data points. It would certainly be our hope that this study might motivate future effort. Our deeper concern in the convergence study is with the planewave expansion; we have discussed this study in both Chapter II and Appendix B. As shown in Fig. B.l, to reach sub meV convergence in total energy (for Si), we have to use a planewave basis of over 3000 planewaves (corresponding to an energy cutoff 40 hartrees). Even at that level, the convergence is not smooth (shown in the insert of Fig. B.l). At this moment it is hard to determine which is the underlying factor causing this convergence difficulty. Two of the most likely candidates are: (1) the poor quality of the pseudopotentials; (2) the Kleimann-Bylander procedure used for producing separable pseudopotentials. There have been recent active efforts3,4'® to develop better pseudopotentials. These efforts should be extended with a serious study of the convergence, and a more realistic error estimate. Here are two suggested approaches to pinpoint the source of error concerning this difficult convergence: (1) a full non-local pseudopotential calculation, which could be used to check whether the Kleimann-Bylander procedure introduces any problem; (2) an all electron cal culation, which can be used as a measure if the pseudopotentials can represent the core-valence electron interaction properly. These axe certainly not easy tasks, but one might be informative. With increasing computational power, we hope that the nature of the convergence will not remain a puzzle in the future. A final note on our systematic study concerns the different exchange-correlation functional forms, as discussed in Sec. 1.9. It has been noted by Jones et al. ® that a better functional form is needed. This could lead to the question on the validity of 117 Local’ Density Approximation. There is no proof that exchange-correlation can be approximated by a local form, it is more of a limitation on computational resource than a choice. It is possible that the future work on the theory7 will produce better functional form for the Density Functional formalism. 4.2 Properties of Gen/Sim Superlattices Aside from our concerns discussed in the previous section, we wish we could have devoted more time on the strained Gen/Sim superlattices. There are many questions to be answered and much interesting physics to be studied. To name a few, we would like to know the exact structures of these superlattice, the size of the unit cell, the exact location of the atoms within the cell. On the electronic structure side, we would like to know the band structures, and how the strains will affect the band structures, whether the direct gap superlattices can be made from these two originally non-direct materials, and whether the internal atomic relaxation will enhance the dipole transition momentum of those originally forbidden transitions. On the optical response side, we would like to know if the broken inversion symmetry will lead to a significant second-harmonic-generation in the odd period superlattices. 118 REFERENCES 1. H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976). 2. David Gottlieb and Steven A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, 26, (SIAM, Philadelphia, 1986), p. 26. 3. D. Vanderbilt, Phys. Rev. B41, 7892 (1990). 4. N. Troullier and J. L. Martins, Phys. Rev. B43, 1993 (1991). 5. A. M. Rappe, K. M. Rabe, E. Kaxiras and J. D. Joannopoulos, Phys. Rev. B 4 1 ,1227 (1990). 6. R. 0. Jones and 0. Gunnarsson, Rev. Mod. Phys. 61 689 (1989). 7. A. Gorling and M. Levy To be published. A ppendix A Algebra of Stress, Strain and Elastic Energy First we define the translational matrix, which is constructed by the transla tional vectors of a unit cell, expressed as follows, R,Xx R2x R3x \ Rly R2y R$y I > (A.l) (R lz R 2 z RZ z / A where aj and are the lengths and directions of each ‘primitive’ translation vector for the chosen unit cell. With this matrix, we then define a reduced coordinate, where the vector t has the relationship to the real Cartesian space vector X as, t = [ R ^ X . (A.2) When the unit cell is distorted, the atom, original at position X , is now moved to I X, so the displacement u equals, u = X ' - X . (A.3) We can rewrite Eq. (A.3) in reduced coordinate, u = [#]**7 - [R]t. (A.4) In the case of no internal shift, which means t 1 = t, Eq. (A.4) can be further expressed as, S=([ie']-[i!])[iJJ-1X. (A.5) 119 120 We can obtain the strain tensor as it is defined as following, 1 duj duj 2(9Xj + dXi’ (A.6) = [M W rtj - h i + - hi< where i and j run through x ,y , and z. Conventionally, this 3x3 tensor is transformed to a six dimensional vector, e^..^6, as (el> e2> e3, e4, e5, e6) = (pxx,Pyy, fizz, (Pyz + Pzy), (Pzx + Pxz), (Pxy + Pyx)), (A.7) The elastic energy due to the strain is found to be E J = 2^1l(el + e2 + el) + C,12(ele2 + e2e3 + e3el) + 2*-'44(e4 + e5 + ei)> (A-8) analysis the relationship between energy and lattice distortion described in Eq. (A.8), we can rewrite the energy as a function of single elastic constant with properly chosen lattice distortion. First we consider the usual cubic diamond cell, where translational matrix, [R] is chosen as, ( 0 f f \ % 0 % , (A.9) where a,- is the lattice constant. Under hydrostatic strain, cti = a0(l + 5), using Eqs. (A.6)-(A.9) we get energy change as shown in the top of Table A.l, as we can see that the total energy change is only related to the bulk modulus, hence we can calculate the total energies for different a, and fit the energy form to get both bulk modular, B , and equilibrium lattice constant, oq. 121 Table A .l The relationship between elastic energies, distortions and elastic constants. The relationship between energy change (elastic energy) and distortion is shown. The choice of our distortion is to involve only one elastic constant in the energy change. The distortion is made in such a way that only the length scale of each translational vector changes. 8{ is the fractional change of the length scale in one of the three vectors with the direction indicated in the table. Elastic Constants unit cells distortions energy E/Cl Si 82 h 2-atom cell(diamond) (Oil) (101) (110) g _ (C11+ 2 C12) 6 8 6 4-atom cell (001) (110) (110) (001) C u 0 0 6 O u 26 26 -6 § c n s2 C44 8 - 8 0 2C44S2 2-atom cell (111) ( 1 1 0 ) (Oil) (HI) C44 6 6 -2 8 6C44S2 For other elastic constants, one of the unit cell choices is to include two primitive cell alone (001) direction, where the translational m atrix, [R], is °\ * = l o f y ( A ' 1 0 ) where a,- are the length scales for each translational vector, with equilibrium value ao, <*i = ao(l + £»)> (A.ll) using Eqs. (A.6)-(A.9), the distortions and elastic energies axe shown in Table A.I. We can see that there are two different ways to calculated C\\ with different choices of distortion: ( 1 ) distortion in ( 0 0 1 ) only, ( 2 ) combined distortion in both ( 1 1 0 ), (IlO) and (001) with the ratio of -2. The C\\ obtained from both distortion is shown in Table 2.7. Also, with the distortion in (110) and (110) with ratio -1 will give us C44 directly. As for the sheax distortion chosen above, fig will be the only non-zero strain ele ment, hence, the internal relaxation will alone the ( 0 0 1 ) (z-) direction, as expressed in Eq. (2.18), this observation reduces a three-dimensional atomic relaxation to a one-dimensional one and reduces the computational time. Another way to obtain an independent C 4 4 is to compress is in (111) and expand in both ( 0 1 1 ) and ( 1 1 0 ) directions, this leads a translation matrix as, / 0 - f R = [ % % % , (A.1 2 ) V - * 0 f / where aj, the length scale, equal to ao in the equilibrium structure. The relationship between energy and distortion is shown in the bottom of Table A.I. Again, observe carefully, we find that under this distortion, the internal relaxation will only allowed in the ( 1 1 1 ) direction, since fX4=fi5=fig’ One final note, there is no way to get C\2 independently with any distortion, so it is deduced from (3 B - C u ) /A1_, Appendix B Total energy and energy cutoff (size of planewave basis) Since, as pointed out in Section 2.3-B, neither a power law nor a logarithmic fit are satisfactory, we decided to look more carefully on how the total energy changes as the energy cutoff increases. Table B.l shows the total energies of 10 special k- points with E cut from 10 to 40 hartree with interval of 2 hartree. The total energies is plotted against energy cutoff in Fig. B.l, we can see that in the full scale plot, the total energy decreases as E cut goes up. But in the insert for E cut between 25 and 40, we see that the convergence is not smooth: the rate of converge slows, and then speeds up and slows down again. This change in convergence pace makes it impossible to fit the data set with any simple form and results in a poor fit as discussed in Section 2.3-B. 123 TABLE B.l. Total energy of 10 special ^-points for various energy cutoff. The calculated total energy for 2-atom Si cell is shown here. The total energy is reported with 10 effective digits as the error in our self-consistant shows in the 12th digits and finite FFT box shows in the 11th digits. Ecut is in hartrees, total energies are in eV, and Npw is the average numbers of the planewaves for 10 special fc-points used in the calculations. Ecut (hartrees) 3 5 10 14 16 18 total energies(eV) -106.253 666 3 -107.511 826 3 -107.990 266 2 -108.008 662 9 -108.018 612 7 -108.026 144 5 Npuj 65.06 140.16 397.75 658.0 804.09 960.5 Ecut (hartrees) 20 22 24 26 28 30 total energies(eV) -108.030 458 9 -108.032 325 5 -108.032 886 0 -108.032 990 6 -108.032 999 7 -108.033 032 2 Npuj 1125.0 1298.9 1479.0 1666.8 1864.1 2065.19 ECut (hartrees) 32 34 36 38 40 total energies(eV) -108.033 098 4 -108.033 176 8 -108.033 244 9 -108.033 294 7 -108.033 327 6 Npxv 2274.75 2491.44 2716.56 2944.12 3181.59 125 5 E 25.0 -3 2 .5 co* © detailed curve T— E CM co + o .o -3 3 .0 N + I - 2 5 .0 - 3 3 .5 lli J u f Full set of data points - 5 0 .0 10.0 20.0 30.0 40.0 Ecut hartrees Figure B .l Total energy versus energy cutoff, E cut. The total energy, in eV, is plotted against the energy cutoff, E Cut in hartrees, the insert shows a “step” for E cut between 25 and 40. This step in the convergence makes it impossible to fit the data with any simple form. The solid curve is the fit of the total energy by power law, and the dashed curve is by logarithm, the dots are the data points used for the fit (see Table 2.6); note that the two lowest data points, at E cut = 3 and 5, are not shown. Appendix C Murnaghran Fit for Total Energy and Equilibrium Volume There are different ways to fit the calculated total energies of different vol ume (lattice constants); besides from the simplest polynormia! fit, there is another widely1,2 used method: Murnaghan fit.*1 The idea of this fit is assume that the bulk modulus is a function of pressure, p B(p) = Bq + B^p. (C.l) By definition, the bulk mudulus, B is B = - v ^ w )W°' ( c -2) together with the definition of pressure, p, dE (r p = w , < ) the Murnaghan expression for the total energy, E, as a function of volume, V, can be written as W = + + (C.4) -°o -°o - 1 where E q is a constant. Using the non-linear least-square fit, we can obtain the bulk modulus, Bq , equilibrium volume, Vo, and Bq with a set of calculated energies for different volumes. 126 127 For small changes in volume from the equilibrium volume, Vq, as V = V q+ 8V = Vo(l + 8 ), we can expand Eq. (C.4) into a polynormial of 8 , as E(V0(1 + {)) =(E0 + + i i W 2 _ ® Q + iSoy0^ + (B& + 1)(-Bi + 2)W 4 (C.5) 6 24 J^+l)(^ +2)^ + 3)^ + 0({6) Instead of doing a difficult non-linear least-square fit, it is possible to use a poly normial fit, for small volume distortion, to obtain the same information. From the experimental knowledge we know that J Bq is positive and of order one (around 3 for Si), it is sufficient to fit the data to the fourth order as long as the volume change is smaller than 10%. Table C.l shows energies of different lattice constants calculated with diferent energy cutof and the fitted bulk modulus, B q, a3 lattice constants, ao (M) = ~£)i an(^ o• From Table C.l, we can see that at energy cutoff of 6 hartree, the quality of the data set, E (a), is poor (not smooth). Our effort to fit the parameters, Bq, ao and B q, showed that it is impossible to reach a stable B q. We believe that the pressure derivative of the bulk modulus, B q, is more sensitive to E cut than the bulk modulus, Bq. This seems reasonable since it is measure of anharmonic effects which probe the hard core potential. So any evaluation of B’ made at 6 H is very likely incorrect. With our limited data, we predict that for bulk Si, Bq is between 2 and 3. One final note on the Murnaghan fit, in order to show the quality of the fit, it is necessary to calculate the deviation of the fit from the original data, as - E i f , (C.6) 128 Table C .l Total energies of differnt lattice constants and fitted parameters. The total energies, E(a), (in meV) are calculated with different energy cutoff, Ecut. The bulk modulus, B , (in Mbar), equilibrium lattice constant, ao, (in bohr) and Bq are obtained from fitting E(a) with Eq. (C.5) to the fourth order. (Comparied with the third-order fit, the coefficient for the third-order term is stable.) The calculation is for bulk Si with Ceperley-Alder exchange-correlation function and 10 special fc-points. Ecut 6 10 12 16 Npw 185 397 522 804 a E(a) 9.9733 43.6673 43.4786 10.0593 31.0151 13.4629 10.0733 10.4457 10.5523 10.1093 5.53105 4.16151 10.1233 2.55742 10.1433 0.787437 10.1593 5.54616 .163084 10.1733 0.0 0.0 0.0 10.2033 1.05843 10.2093 0.0 1.09188 10.2233 2.37785 10.2593 7.29107 7.30503 10.2733 9.90739 9.52287 10.3593 23.1666 33.3594 10.3733 38.3992 37.7618 10.4592 58.3724 76.7597 ao 10.2 10.173 10.173 10.174 So 1.136 .963 .975 .965 s j - 1 2.6 2.0 with the energy at the fitted equilibrium set to be zero. This deviation should be an useful indicator of how good the fitted curve reflects the original data set, compared 129 with the energy differences between different volumes. We realize that it could be quite deceptive to report the x-square as v (E(V,j) - E j f Y ’ with E(Vi) in hundred of eV, while the real meanful quantity is the difference of the energy E(V{) — E(Vj) (in tens of meV). 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