SI-GExSI x SUPERLATTICES

AND DEVICE APPLICATIONS

By

JAMES VERNON COLE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1993

This dissertation is dedicated to my parents, whose

support and encouragement made it possible.

ACKNOWLEDGEMENTS

I would like to gratefully acknowledge the guidance and

encouragement provided by Dr. Hong H. Lee through his

assistance in this endeavor. Discussions with my fellow

students David Koopman, Sungmin Cho, and Jay Yoo provided many

new insights and were also a source of encouragement.

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . .

ABSTRACT . . . . . . . . .

CHAPTERS

1 INTRODUCTION . . . . . . .

Historical Remarks . . . . .

Superlattice Modelling Approaches . .

Previous Work with Silicon and Germanium

Introduction to Energy Band Theory

Summary and Goals . . . . .

2 BULK MATERIAL MODEL . . . .

Introduction . . . . . .

k-p Hamiltonians . . . . .

Spin-Orbit Splitting . . . .

Strain Effects . . . . .

Sample Band Structures . . .

Summary . . . . . . .

3 SUPERLATTICE MODELLING AND RESULTS

Introduction . . . . . .

Band Alignment . . . . .

Complex Band Structure . . .

Completeness and Spurious Solutions

Superlattice Model . . .

Superlattice Energy Band Results

Summary and Conclusions . . .

4 OPTICAL PROPERTIES . . . .

Introduction . . . . . .

Optical Matrix Elements . . .

Optical Absorption Coefficient .

Summary and Conclusions . . .

5 DEVICE APPLICATIONS . . . .

iii

S. vi

. 1

. 1

S 3

S 5

S. 6

. 10

. . 28

S. 88

: : : :

.

.

Introduction . . . . . . . . 88

Light-Emitting Diodes . . . . . . . 89

Semiconductor Lasers . . .. . . . 90

Photodiodes . . . . . . . . . 94

Summary . . . . . . . ... . .95

APPENDIX COMPUTER PROGRAMS . . . . . . ... .96

REFERENCE LIST . . . . . . . . .. . 128

BIOGRAPHICAL SKETCH . . . . . . . . . 130

v

Abstract of Dissertation Presented to the Graduate School

of the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Doctor of Philosophy

Si-GexSi1x SUPERLATTICES

AND DEVICE APPLICATIONS

By

James Vernon Cole

May 1993

Chairperson: Dr. Hong H. Lee

Major Department: Department of Chemical Engineering

A k-p method is applied to semiconductor superlattices

containing alternating layers of silicon and silicon-germanium

alloys. The method involves calculation of zone-center Bloch

functions for a reference material with the average

composition of the constituent materials using the

pseudopotential method. The parameters of the k-E

Hamiltonians for each material are calculated from the zone-

center energies and Bloch functions of the reference material,

with the appropriate perturbing potential. The spin-orbit

splitting is added as a first order perturbation in energy of

the Hamiltonians. Effects of strain are included to the first

order in energy using deformation potential theory.

Superlattice wave functions are calculated by applying the

appropriate matching conditions to the bulk material solutions

at the material interfaces, then imposing the Bloch condition

for the superlattice.

Energy band structures are calculated for superlattices

with a (100) growth direction and layer compositions of Si-

Si50.Ge0. or Si-Ge. The effects of the strain configuration

are considered for the Si-Si.5 Ge0.5 system. Direct band gap

superlattices are obtained for systems with total superlattice

periods of 10, 14, and 19 monolayers. The optical matrix

elements for transitions from the valence band minimum to the

conduction band maximum are calculated for the superlattices

with 14 monolayer periods. The values of the matrix elements

are as much as an order of magnitude greater than values

obtained for these materials using the envelope function

approximation. Optical matrix elements are calculated as a

function of the superlattice wave vector for a particular

superlattice configuration, seven monolayers of both Si and

Sio.5Ge0.5 in a period with growth on an Si0.75Ge0.25 substrate.

Substantial band mixing effects are found for transitions

involving the lowest two conduction bands and the highest two

valence bands. The absorption coefficient spectrum is

calculated for this structure, and the magnitude of the

coefficients for photon energies near the band gap is

comparable to that of direct band gap materials. Device

applications with this superlattice as the optically active

layer for several optoelectronic devices are considered. The

use of this superlattice as the optically active layer in a

vii

double heterostructure semiconductor laser with carrier and

optical confinement provided by Si0.75Ge0.25 layers is found to

be worthy of further consideration.

viii

CHAPTER 1

INTRODUCTION

Historical Remarks

The concept of a semiconductor superlattice was

originated by Esaki and Tsu (Esaki and Tsu, 1970). Two types

of superlattices were proposed, compositional and doping. The

compositional superlattices would be obtained by depositing

alternating layers of two different semiconductors during

epitaxial growth. The thicknesses of the layers would be

between several angstroms and hundreds of angstroms, with the

period of the modulation less than the electron mean free

path. Doping superlattices would be produced by periodically

altering the concentration of donor and acceptor impurity

atoms in a single semiconductor. The electric fields

generated by the charged dopants would modulate the electric

potential.

The first attempts to fabricate superlattices involved

chemical vapor deposition of GaAs-GaAsxP,-x (0.1 x 5 0.5)

(Blakeslee and Aliotta, 1970, and Esaki, Chang, and Tsu,

1970). This early work was unsuccessful because of the

relatively large lattice constant mismatch, which is 1.8% for

x = 0.5. The GaAs-GaxAl1_-As material system was the next to

receive attention, since the lattice constant mismatch is very

1

2

small, 0.08% for x = 0.5. The observation of negative

differential resistance in this system was the first

experimental evidence of the quantum mechanical confinement

effects expected in superlattices (Esaki et al., 1972). The

GaAs/GaxAll-xAs superlattices have since become the most

thoroughly characterized because of their relative ease of

deposition.

The improvements in epitaxial growth techniques, such as

molecular beam epitaxy (MBE) and metal-organic chemical vapor

deposition (MOCVD), have allowed many other material systems

to be investigated. The growth of lattice-mismatched

heterostructures is possible if the layers are sufficiently

thin (Frank and van der Merwe, 1949, and Frank, 1963). The

individual layers become strained to accommodate the lattice

mismatch. The resulting coherent strain shifts the energy

band gaps of the constituent materials and also splits the

degeneracy of some of the states near the conduction and

valence band edges.

The developments in Si-Ge molecular beam epitaxy and the

observation of optical transitions in the resulting

superlattices (Pearsall et al., 1987) have renewed interest in

the calculation of superlattice band structures for systems

composed of indirect band gap materials. A superlattice

system composed of such materials would allow integration of

advanced silicon electronic devices with optical devices if

the superlattice had a direct band gap and optical transition

strengths near those of the III-V compound semiconductors.

The ability to accurately calculate superlattice energy bands

and optical transition strengths is necessary for systematic

investigation of these materials.

Superlattice Modelling Approaches

There are two groups of empirical methods for calculating

superlattice band structures: supercell models and boundary-

condition models. The supercell methods are conceptually

simple, but the dimension of the matrices that must be

diagonalized increases rapidly with the superlattice period.

For this reason, supercell calculations have been limited to

single interfaces and superlattices with relatively thin

layers. In the boundary-condition approaches, eigenfunctions

of the constituent materials are matched at the material

interfaces to construct a superlattice eigenfunction. The

boundary-condition approaches take advantage of several

simplifying features of the problem. The calculations can be

confined to the energy region of interest, near the energy

band edges. The empirical parameters are taken from the bulk

material models, which are well developed. These methods can

also be implemented fairly easily because the size of the

perturbing potential is not very large compared to the energy

gap. The result is a matrix eigenvalue problem that is

generally smaller than that of the supercell approaches and

does not depend on the layer thickness.

4

Boundary condition approaches are based upon one of three

methods for modelling bulk materials, empirical

pseudopotential, tight-binding, and k-E methods. The

empirical pseudopotential methods are typically applied to the

single interface problem, but could be extended to

superlattices in principal. The major disadvantage of the

pseudopotential methods is the size of the matrices involved,

since the plane-wave expansion used converges slowly. The

tight-binding calculations can include spin-orbit and strain

effects, are adaptable to optical calculations, and reflect

the translational symmetry of the lattice so that truncation

does not cause spurious solutions for the constituent

materials. The primary disadvantage of this method is the

large number of input parameters involved. The parameters for

the bulk materials are not uniquely determined by the bulk

band structures, and there is no physical basis for choosing

the interface parameters.

The boundary condition approaches based on the k- theory

for bulk semiconductors are widely accepted as providing

reasonably accurate results with a minimum of computational

effort. In these approaches, the superlattice wave function

is expanded as a product of Bloch functions modulated by

slowly varying envelope functions. The kp- theory for the

bulk materials provides a set of coupled equations for the

envelope functions, which are solved in conjunction with

interface matching conditions. The formulation of the wave

5

function matching conditions is a nontrivial problem because

the Bloch functions are not explicitly considered in this

method. The typical approach is to assume that the zone-

center Bloch functions are equivalent in the constituent

materials. This gives the relatively simple matching

conditions of continuity across the interface and Robin's

conditions for flux across the interface for the envelope

functions.

Previous Work with Silicon and Germanium

Approximate calculations of the band structure for Si-

Sio.5Geo.5 superlattices have been performed using the envelope

function approximation with the k-E theory of Cardona and

Pollak (Cardona and Pollak, 1966). The lowest conduction band

of the superlattice has been calculated using this approach.

The spin-orbit coupling to other bands was neglected and the

effects of strain were included to zeroth order in the wave

function and first order in the energy. The resulting

Hamiltonian was reduced from a three-band model to a two-band

model. The calculations demonstrated that the conduction

bands split because the indirect nature of the materials

allows interactions between bulk wave functions at kmin+6k and

kmin-6k (de Sterke and Hall, 1987). A similar analysis of the

heavy-hole valence band and lowest conduction band has also

been performed with a three-band expansion for each

superlattice band (Rajakarunanayake and McGill, 1989a). A

6

direct band structure was obtained for a superlattice with

layer thicknesses of 7.2 monolayers for both Si and Si0.sGe0.5.

The optical matrix element for transitions from the

superlattice valence band to the conduction band was also

determined, and an increase of up to 4 orders of magnitude

over the values for bulk Si and Ge was observed.

Although the results of these simple models are

encouraging, there are fundamental problems with such

treatments. The perturbing potential of the superlattice has

been included to only first order in the energy, which is

inadequate for more than a qualitative description of the

energy bands. The spin-orbit and strain interactions between

bands have been suppressed by the small number of basis states

included in the wave function expansion. The small number of

zone-center functions used for the wave function expansion

does not adequately describe interactions with bulk states far

from the zone center. These interactions must be included for

a superlattice composed of indirect band gap materials.

Introduction to Energy Band Theory

The energy band theory of solids is based on two major

assumptions. The first is that the atoms of the solid are

stationary and form a crystal lattice with perfect

translational symmetry. The second is that the electrons can

be treated one at a time, with the potential formed by the

other electrons averaged over their wave functions in a self-

7

consistent manner. The crystal lattice is assumed to consist

of an infinite number of identical unit cells. The origin of

each cell is associated with a point, a, in the lattice

defined by

a=3iA1+-N12+a1 (1.1)

where a a2, and a3 are called primitive vectors and 11, 12,

and 13 range over all integers. The origin of a unit cell may

be taken to be anywhere, but it is usually chosen to be an

atom. There are only 14 distinct three-dimensional lattices

for which Eq. (1.1) is satisfied, the Bravais lattices. The

most symmetric of these are the cubic, body-centered cubic,

and face-centered cubic. The diamond lattice, which is of the

most interest for semiconductors, consists of two

interpenetrating face-centered cubic lattices and is a Bravais

lattice with a two point basis.

The symmetry of a Bravais lattice is most apparent in

Fourier space or k space. To expand a function with the

symmetry of the lattice in k space, the only components of k

that are necessary are the reciprocal lattice vectors. The

reciprocal lattice vectors are defined by Gi.a. = 2nij7 for

every aj of the lattice. The reciprocal lattice vectors lie

on a Bravais lattice themselves.

The basis for energy band calculations is Bloch's

Theorem. This theorem states that the eigenfunctions of a

periodic Hamiltonian, such as the one-electron Schr6dinger

equation with a periodic potential, may be written as

8

,E =ei"u-Z (Y) (1.2)

where uk is a function with the periodicity of the potential.

A more fundamental form of Bloch's theorem will be used later

in this work. This form shows explicitly that Bloch functions

are eigenfunctions of translation operators, so that

rj(Z+!) =eilA-y(fZ) (1.3)

where a is any Bravais lattice vector.

A final consequence of Bloch's theorem is that Bloch

states with different k vectors need not be distinct. The set

of states that is necessarily distinct occupies a unit cell in

k-space, with the remaining states distinguished by their

energies. The unit cells in k-space are called the Brillouin

zones. A single Brillouin zone, usually called the first

Brillouin zone, is sufficient for a description of all of the

crystal eigenvalues and eigenfunctions. The states outside

the first Brillouin zone are simply translated to equivalent

states by k = k, + G, where kl is in the first zone and G is

a reciprocal lattice vector.

The k-E representation of the crystal Hamiltonian is a

complete representation of the crystalline Hilbert space, and

as such it can be used to calculate the band structure

throughout the first Brillouin zone (Kane, 1982). Consider

the one-electron Schrodinger equation, in atomic units,

H = [-V2 + V(r) ] =E (1.4)

9

where V is an approximation of the crystal potential and has

the periodicity of the crystal lattice. Substituting the

Bloch form for the electronic wave function, Eq. (1.2),

results in:

[Ho (k-0) p+(k2-k2) ] Un.(fi) =En( ) Unf(f) ((1.5)

where H o is the Hamiltonian at k = ko. Expanding the unk in

terms of the Bloch functions at ko,

Un.( fF) = cum U, (f) (1.6)

multiplying Eq. (1.5) on the left by the conjugate of unkO, and

integrating over the unit cell, one obtains

{( [En (ko) + (k2-k) ] + (- k, (k-o) P~ ) C ) c (1.7)

where the E, are momentum matrix elements defined as

Pnm=i tunka()Pu) cf M dr (1.8)

cell

The k-p representation given by Eq. (1.7) is exact and

may be used to describe the energy bands at any point k. In

practice, the number of Bloch states used as a basis for the

Hamiltonian is reduced by perturbation theory or by simply

neglecting bands that fall outside of the desired energy

range. The Bloch states used for the expansion are typically

the zone-center functions, uno. Inclusion of spin-orbit

coupling adds additional off-diagonal, k-independent terms to

Eq. (1.7) and doubles the dimension of the matrices.

Summary and Goals

The development of advanced epitaxial growth methods has

sparked interest in the use of superlattice structures,

particularly for material systems with lattice constant

mismatches. By selecting the appropriate layer compositions,

layer thicknesses, growth direction, and substrate, it is

possible to design a material with desirable properties. A

variety of approaches are used to predict the properties of a

superlattice design, and each has both advantages and

disadvantages. For the silicon-germanium system considered

here, the model used must be capable of describing the energy

band structures throughout the first Brillouin zone because of

the indirect band gap. Also, the lattice mismatch induces

strains in the materials that must be included in the model.

The goal of this work is to choose a modelling approach

that provides a balance between physical realism and ease of

computation, then screen a variety of superlattice

configurations for direct band gaps. Optical matrix elements,

which provide a measure of suitability to optoelectronic

applications, are then calculated for the direct band gap

configurations. A more detailed analysis of the optical

properties is performed for the most promising superlattice

configuration. The results are then used to evaluate the

suitability of this material for optoelectronic applications.

The model used for this work is based upon the k'g theory

for superlattices (Smith and Mailhiot, 1986). This approach

11

is an extension of the envelope function approximation using

bulk k-E parameters. The problems with the interface matching

are avoided by using the same set of basis functions in each

material. Zone-center Bloch functions are calculated for a

reference material using the pseudopotential method. The k-p

parameters are calculated from these functions, with

perturbing potentials used to recover the desired material

potentials. The superlattice wave functions at a given energy

are then treated as linear combinations of the wave functions

for the bulk materials. This approach satisfies the criteria

mentioned previously for applicability to the silicon-

germanium system, resolves the ambiguity of interface matching

conditions present in the envelope function methods, and

provides the information necessary for optical calculations.

CHAPTER 2

BULK MATERIAL MODEL

Introduction

The k'E method was used to describe the bulk materials,

providing the bulk solutions for interface matching. Instead

of using empirically determined parameters for the k'E

Hamiltonians, the parameters were calculated from zone-center

solutions of a reference Hamiltonian. The spin-orbit

splitting and strain effects were both added as first order

perturbations to the energy after calculation of the material

Hamiltonians.

k-p Hamiltonians

The k-p Hamiltonians for the constituent materials were

calculated as perturbations of a reference Hamiltonian. The

reference Hamiltonian was taken to be the one-electron

Schrodinger equation for a material with a crystal potential

given by the average of the constituent material potentials,

HR = -V2+ [Va() +Vb() ] (2.1)

2

where Vt was the empirical pseudopotential of material 1 and

atomic units were used. The pseudopotential V is the crystal

potential expanded in the reciprocal lattice vectors of the

13

crystal. The expansion is usually divided into symmetric and

antisymmetric parts, resulting in

V(i) = [S(G) VG+iSA(G) VA] e-iD' (2.2)

G

where the S(G) are structure factors and the VG are the

pseudopotential form factors. The diamond lattice structure

of silicon and germanium is actually two interpenetrating fcc

lattices, with two atoms in the primitive unit cell. If the

origin of coordinates is taken as halfway between these atoms,

then their positions are given by r: and r2, with r, = a(1,a,s/)

= r and r2 = --. With this coordinate system, the structure

factors for the diamond lattice are

SS(G) = cos(G') SA(G) = sin(G.) (2.3)

Because of the symmetry of the diamond lattice, the

antisymmetric potentials are zero.

The first five reciprocal lattice vector families have

magnitudes 0, 3, 4, 8, and 11 in units of (2w/a)2. The

symmetric structure factor for G2 = 4 is zero, and the

pseudopotential factor for G2 = 0 only shifts the reference

energy. Therefore, only V3, Vs, and V11 are needed to

approximate the crystal potential. The values used for Si and

Ge are in Table 2.1, and the alloy parameters were calculated

by a unit cell volume weighted average.

The eigenvalues and eigenvectors of HR at k=0 give the

zone-center energies and plane-wave expansions of the

reference Bloch functions as

Table 2.1 Pseudopotential Form Factors, Rydbergs

V3 V8 V11

Si -0.21 0.04 0.08

Ge -0.23 0.01 0.06

H, Rijd = d Rj,d (2.4)

Ud 1 jde (2.5)

where the Bloch functions u were normalized over the unit cell

volume, 2. The matrix form of Eq. (2.4) was obtained by

taking the inner product

= ed<-i < ud> (2.6)

resulting in the eigenvalue problem for energies and expansion

coefficients. The expansion of the zone-center Bloch

functions in terms of reciprocal lattice vectors converges

slowly. The reciprocal lattice vectors used were the families

{000}, (111), {200), {220}, (311), (222}, (400), {420), and

(422). These provide the 113 lowest energy plane wave

solutions that have the appropriate symmetry for the diamond

lattice.

The solution of the reference Hamiltonian provides the

information necessary for transformation to the Bloch function

basis for the wave function description. The wave function

was given by

F- = (Cd () eiR Ud(f) (2.7)

where the Bloch functions ud were given by Eq. (2.5). The

Hamiltonians for the constituent materials were calculated by

perturbing HR to regain the appropriate potentials,

H1 = HR+AV1 (2.8)

where

A V (f) = Vl(r)- [Va (r) +Vb(r)] (2.9)

with the potentials expressed as pseudopotentials. Operating

on the wave function expansion with a material Hamiltonian and

taking the inner product with a Bloch function results in the

k-p Hamiltonian for that material. The elements of the k-E

Hamiltonian matrix are

Hdd' = (d+k2) d, d,-k + (2.10)

where atomic units were used. The expansion coefficients of

Eq. (2.5) were used to calculate the momentum matrix elements,

the terms containing the gradient of the Bloch functions, and

the material potential perturbations.

The k-p Hamiltonian matrix given by Eq. (2.10) has the same

dimension as the pseudopotential Hamiltonian. To take

advantage of the faster convergence of the Bloch function

basis, Lowdin perturbation theory (Lowdin, 1951) was used to

reduce the number of functions explicitly considered to the 15

lowest in energy. The use of only 15 Bloch functions was

16

based upon a successful full-zone k'- model of Si and Ge that

used that level of approximation (Cardona and Pollak, 1966).

The constituent material Bloch functions were then given by

Ud, (F) = Ud(F) + WUp (F) (2.11)

where

U < pl> (2.12)

ed-ep

with the explicitly treated functions labelled by d and the

perturbing functions labelled by P. The resulting wave

function expansions were given by

S(f) -1 e E ud ) (2.13)

7d .j Ud,J(r)

where N was the number of atoms per primitive cell. It was

determined that perturbing the 15 lowest energy functions with

the next 74 solutions gave an accurate description of the bulk

materials.

The matrix form of the k-p Hamiltonian was calculated

from the inner product over the unit cell

Hdd' =

Hd = <4ude H e |ifud.> (2.14)

resulting in the eigenvalue problem

[ Hdd' ) -e8dd/ = (2.15)

17

for the energies e allowed at a given k. The energy-dependent

denominators resulting from the perturbation were symmetrized

to evaluate Eq. (2.14). The eigenvalue problem of Eq. (2.15)

can be rewritten as

[ H,(k ) 2+H ,k+H1 ]0,. = (2.16)

where k.t is the component of k.L in the growth direction and

the Hn are the coefficient matrices of k.t to the nth power.

Detailed expressions for the matrices are

Hddi = 1 +^^121 (2.17)

H'2d/: 1 + (2.17)

d (ed+e d) /2-ep

Hdd = 2

+ 2^ __ + __

(E (d+eC,) /2-ep (ed +ed) /2-ep

(2.18)

+ 4. UdIPpX u>

+ (ed+ed) /2-e (Ed+e d) /2-ep

d' = {ed+kf-e}8dd ,+ (uI V1I ud> +2 ,

+[

S(ed+e/) /2 -ep

+ <2 (upAIId (2.19)Ud *

(cd+ed,) /2-e (2.19)

+ 2k.I +

(ed+edI) /2-ep (ed+ed/) /2-ep

where p, is the component of the momentum operator in the

growth direction and p, and k, are the vectors parallel to the

interfaces. After the H2, HiL, and Ho matrices were

calculated, their elements were transformed to correspond to

interactions between zone-center functions of a certain

symmetry. The symmetry groups of the 15 functions explicitly

included in the Hamiltonians were Fr, F25,, F2,, 15 r,, 12,,

r25,, and r2,, in order of increasing energy. The groups

appearing twice will be labelled by an 1 or u corresponding to

the group with the lower or upper zone-center energy. The F25,

symmetry group is triply degenerate, and the Hamiltonian

elements were transformed so that the functions had the

symmetry X=yz, Y=xz, and Z=xy, where x, y, and z are the

coordinates with respect to the crystal axes. The F15 group

is also triply degenerate, and those elements were transformed

19

to correspond to the functions x, y, and z. The doubly

degenerate r12, functions were taken to transform as ./(y2-z2),

labelled A, and 3x2-r2, B.

Spin-Orbit Splitting

The effects of spin-orbit splitting were included to

first order in the energy and zeroth order in the electron

wave functions. The rigorous operator for spin-orbit

splitting leads to a term that is independent of k, analogous

to the atomic spin-orbit splitting, and a second term that is

proportional to k, representing the contribution from the

crystal momentum (Kane, 1956). The k-dependent term was

neglected since its contribution is much smaller than that of

the k-independent term.

The inclusion of spin-orbit effects leads to a doubling

of the number of zone-center Bloch functions used as the basis

set for the k-p Hamiltonian. The Hamiltonian can be expressed

as

H + 0 +H (2.20)

0 H s.o.

where Hkp represents the original k-p Hamiltonian of Eq.

(2.15). The upper left and lower right blocks represent

interactions between states with spin up and spin down,

respectively. The spin-orbit coupling was only included for

interactions between the lower r25, states and between the r15

20

states (Cardona and Pollak, 1966). The spin-orbit interaction

has the form

0 i 0 0 0 -1

-i 0 0 0 0 i

SHs0 0 0 1 -i (2.21)

S 3 0 0 1 0 -i 0

0 0 ii 0

-1 -i 0 0 0 0

for the basis set xt, yt, zt, xi, y}, zI, where 6 is the

magnitude of the splitting.

Strain Effects

Because of the lattice mismatch between Si and Ge, the

effects of strain on the energy bands of the layers must be

considered. The lattice mismatch forces either an expansion

or a compression of the crystal lattice in the plane

perpendicular to the growth direction. This expansion, or

compression, is accompanied by a compression, or expansion, of

the crystal lattice in the growth direction, which minimizes

the elastic energy of the layer.

The effect of the strain upon the energy band structures

of the bulk material is to induce further splitting of

degenerate states and to shift the energy gap between the mean

valence band maxima and the mean conduction band minima. For

the valence band maxima, the sixfold degenerate (X, Y, and Z

symmetries with spin up and down) states behave as atomic p

states. The spin-orbit effect causes a splitting into a

fourfold degenerate P3/2 set of solutions and a twofold

21

degenerate p1/2 set. The fourfold set consists of the light-

and heavy-hole bands, while the p1/2 states are the spin-orbit

split-off states. The strain removes the remaining

degeneracy, so that the fourfold states with J = 3/2 are

separated according to the magnetic quantum number M The M.

1/2 bands move up in energy for compressive stresses along

the [100] crystallographic axis, while the M. 3/2 bands move

down.

The strain Hamiltonian, in the absence of spin, is given

by the deformation potential theory (Pikus and Bir, 1959) as

lex+m (eyy+ezz) nexy nexz

Hs = nex leyy+m (ex+ezz) neyz (2.22)

nexz neyz lezz+m (e+ezz)

for the X, Y, and Z basis set, where the e.j are elements of

the deformation tensor and 1, m, and n are independent

constants. These constants are traditionally expressed in the

forms a=-(l+2m)/3, b=(l-m)/3, and d=n//,. The parameter a

determines the shift of the mean of the valence bands at k=0,

while b and d give the splitting between the bands. As in the

model for spin-orbit splitting, allowing for higher order

terms that couple the spin-orbit and strain effects produces

additional terms in the theoretical expressions for the

constants a, b, and d. These terms produce small changes in

the value of the constants and will be neglected. Such a

treatment is consistent with treating both strain and spin-

22

orbit effects as first order perturbations of the energy and

zeroth order perturbations of the electron wave functions.

As an example, consider the case of [100] growth of Si

and Sio.5Geo.5 layers. The nonzero elements of the strain

tensor are ex and e (=ezz) and are calculated from the

material elastic compliance constants and the lattice

constants. The strains are calculated to minimize the elastic

energy in each layer (Van de Walle and Martin, 1986). For a

configuration in which hi and hSG are the unstrained

thicknesses of the Si and Sio.5Geo.5 layers, the lattice

constants are given by

a, = (asiGsihsi +aGGSGhSG) / (Gsihsi+GsGhsG) (2.23)

al1 = a [l-D1(al/al-1)] (2.24)

where at represents the equilibrium lattice constant for

material 1 and GL is the shear modulus for that material. For

[100] growth the shear modulus is

G, = 2 (cl+2c') (1-ci /c) (2.25)

in terms of the elastic constants. In Eqs. (2.23) and (2.24),

the 1I and subscripts refer to lattice constants parallel and

perpendicular to the growth interface. For [100] growth,

these lattice constants are in the directions of the

crystallographic axes, with denoting the x direction and Ii

referring to the y-z plane. For growth of a superlattice

structure on a substrate, the value of aI may be set as the

lattice constant of the buffer layer.

The values of the strains are given by

el, = (al/al-l) = elyy = el, (2.26)

l. = (a1,1/a-l) = elC (2.27)

The strain Hamiltonian for the basis functions Xt, Yt, and Zt,

or XI, YI, and ZI, reduces to

2b (e-ezz) 0 0

Hs = 0 -b(e-ezz) 0 -la(eX+e,) (2.28)

0 0 -b(ex-ezz)

The term proportional to a in Eq. (2.28) is the shift of the

mean of the valence band maxima. Because that term gives an

absolute shift, instead of a change in the value of the

valence band maxima relative to another band, it cannot be

isolated experimentally. The shift of the mean of the valence

bands will be combined with the shift of the conduction band

mean to yield an average change in the energy gap, which can

be determined from experimental results.

The conduction band minima are also split and shifted by

strain. The deformation potential theory can be used to

describe the strain effects for the conduction bands as for

the valence bands (Hensel and Feher, 1963). Because the

conduction band minima are not at the zone center,

corresponding to the basis functions of the k-p Hamiltonian,

the effects are not given as a perturbing matrix as for the

valence bands. The hydrostatic shift of the average

conduction band minimum, relative to the mean valence band

maximum, is given by

0E 1 -

AE = -(Ed+3-a)1: (2.29)

where Ed and Eu are deformation potential parameters for the

conduction band shift and a is from the valence band shift.

For strain in the [100] direction, the dyadic product of Eq.

(2.29) is the trace of the strain tensor as in Eq. (2.28).

The splitting of the (100) minima relative to the mean

conduction band minimum for strain in the [100] direction is

given by

AEc 2 A( -ez) (2.30)

where 2u is a parameter of the deformation potential theory.

Sample Band Structures

The band structures for Si and Sio.5Geo.5 in the (100)

direction are shown in Figures (2.1) and (2.2), respectively.

The spin-orbit and strain effects have been included, with the

strain corresponding to a substrate composition of Sio.75Ge0.25.

The conduction band minimum shown for Si is one of the global

minima. For pure Ge, the global conduction band minima are at

the L symmetry points, where k = (h,,)2T/a. However, alloys

of Ge and Si have a conduction band structure similar to Si

when the Si concentration exceeds 15% (Bassini and Brust,

1963). Therefore, the conduction band minimum shown in Fig.

(2.2) for the alloy is also the global minimum of the

conduction band when the material is unstrained. For the

strain configuration presented, the conduction band minima for

20

15

10 -

S -

JD

-5

-10 -

-15

0.0 0.5 1.0

k (27t/a)

Figure 2.1. Energy band structure for Si in the (100)

direction with spin-orbit and strain.

20

15-

10

Cm

-5

-10

-15

0.0 0.5 1.0

k (27T/a)

Figure 2.2 Energy bands for Si. 5Geo.5 in the (100) direction

for strain configuration of growth on Sio075Ge0.25.

27

the alloy are at similar points along the (010), (010), (001),

and (001) directions.

Summary

The method used to obtain the energy band structures for

the bulk materials was presented. The materials were modelled

as perturbations of a reference material with an intermediate

composition. Zone-center Bloch functions of the reference

material were used to calculate the parameters of k-p

Hamiltonians for each material. The effects of spin-orbit

splitting and strain were then included to allow description

of the strained layers in the superlattice structures.

CHAPTER 3

SUPERLATTICE MODELLING AND RESULTS

Introduction

The k-E method for superlattice modelling is a boundary

condition method, in which solutions for the allowed

electronic energies and wave vectors in each material are

matched at the interfaces. The procedure involves aligning

the bands at the interfaces, then calculating the bulk

material states and determining which of those should

contribute to the superlattice wave function. The interface

matching for those solutions is performed, and the Bloch

condition is used to determine which of the resulting

superlattice wave functions are valid. This method is applied

to Si-Sio0.Ge.5 superlattices with a strain configuration

corresponding to a freestanding lattice, Si-Si 0.Geo.5

superlattices on an Si.75Ge0.25 substrate, and Si-Ge

freestanding superlattices.

Band Alignment

The appropriate alignment of the energy bands for the two

materials must be determined before the interface matching can

be performed. The reference energy for each individual

material is arbitrary, but an energy scale with a common

28

29

reference must be imposed in order to describe the interface

between the materials. For this work, the reference energy

was taken to be the valence band maximum for Si in the absence

of the spin-orbit and strain effects.

The valence band alignment was calculated by adding an

average valence band offset, 0.265 eV for Sio.5Ge0.5, to the

material containing Ge (Van de Walle and Martin, 1986). The

spin-orbit and strain portions of the Hamiltonians then shift

and split the zone-center valence band energies to provide the

appropriate alignment. The result for a configuration with

equal numbers of layers of Si and Sio0.Geo.5 on an Si 0.7Ge0.25

substrate was an alignment with the alloy valence band maximum

at 0.290 eV above the Si maximum, which compares well with

experimental values and other theories (Rajakarunanayake and

McGill, 1989b).

Determining the conduction band offset was slightly more

involved because the strain changed the bandgap. The diagonal

elements of the Ho matrices, which give the zone-center

energies, were shifted so that the conduction band minima were

equivalent to the bandgap for the unstrained materials without

spin-orbit splitting. The value of AEvav was added to the

SixGe1-x material as for the valence band alignment, then the

hydrostatic change in the energy gap due to strain, AEgo from

Eq. (2.29), was added to each material (Van de Walle and

Martin, 1986). Because the hydrostatic shift of Eq. (2.29)

represents the change in the average conduction band minimum

30

relative to the average valence band maximum, as opposed to an

absolute shift in either or both of the extrema, this

procedure ensured that the conduction and valence bands had

the same reference. The splitting of the conduction bands by

strain given in Eq. (2.30) was then included. For the sample

case of equal numbers of layers of Si and Si0.sGe0.5 mentioned

previously, this procedure results in a conduction band offset

of 0.28 eV between the minima corresponding to the growth

direction. The band alignment for this case is presented

schematically in Fig. (3.1).

Complex Band Structure

The k-p Hamiltonians of Eq. (2.16) provide the allowed

energies for an electronic state with a specified wave vector

k. For the superlattice calculation, the energy of interest,

E, and the component of k parallel to the material interfaces,

k,, will be specified and the allowed values of the component

of k in the growth direction, k, must be calculated. This

procedure was simplified by transforming Eq. (2.16) into

0 I C

[ I= k [kC (3.1)

-(H2)-IH1 -(H2)-1H k kC

so that the values of k were the eigenvalues (Chang and

Schulman, 1982).

The transformed Hamiltonian of Eq. (3.1) is non-

Hermitian, so that complex values of k may result. The

complex and purely imaginary values of k correspond to

1.38

Eg Eg St

St

1.17 1.10 1.305

0.36

|5 S-O, St

I 0.265

0.07

S-0, St

0.0

Si Si Ge

0.5 0.5

Figure 3.1 Schematic diagram of band alignment.

32

evanescent states which can contribute to the superlattice

wave function if their decay length is sufficiently short.

The complex solutions of k must occur as conjugate pairs since

they also satisfy Eq. (2.16), which is Hermitian. It can also

be shown from symmetry arguments that if k is an eigenvalue

then -k is an eigenvalue (Smith and Mailhiot, 1986).

The complex band structures of Si and Si0.5Ge0.5 are shown

in Figures (3.2) and (3.3). The left, or negative k, portion

of the figures presents imaginary values while the right, or

positive k, portion presents real values. For each of these

band structures, the strain was not included and the value of

k, was 0. It should be noted that the periodicity of the

energy bands in k space was not present for the bands outside

of the first Brillouin zone. The loss of periodicity results

from truncating the pseudopotential and k-p Hamiltonians to a

finite number of basis functions. The non-periodic, or

spurious, solutions do not have a physical significance and

should not be included in the superlattice solutions.

The orthogonality conditions for the eigenvectors of Eq.

(3.1) may be obtained in the usual manner (Smith and Mailhiot,

1986). Consider the eigenvalue problem of Eq. (2.16) for

eigenvalues kj and ki,, where ki, represents the conjugate of

ki. Taking the adjoint of the equation for k,, multiplying

each equation on the left by the adjoint of the other's

eigenvector, and subtracting the results gives

(3.2)

(kj -ki) ( i"' [ H2 (kj +ki) +H1 I -Ci ) =

10.0 -

9 5.0

>.

r--

0.0

-5.0 :

-2.0

-1.0 0.0 1.0

k (27T/a)

Figure 3.2 Complex band structure of Si. The solid curves

represent real and purely imaginary values of k. The dashed

curves represent the complex solutions.

2.0

3.0

10.0

5.0

L,

0.0 -/ I

/ II

/ / \ II I

-5.0

-2.0 -1.0 0.0 1.0 2.0 3.0

k (27/ra)

Figure 3.3 Complex band structure for Si,.Ge05. The solid

curves represent the real and purely imaginary values of k.

The dashed curves represent the real and imaginary parts of

the complex solutions.

so that either k. and ki are equivalent or the vector product

vanishes. The product may be interpreted as the x component

of the current density for state 7j averaged over a unit cell,

so that

C+. [H2 (kj+ki) +HI] (3

.-0= J 6j .jj (3.3)

This condition was used to orthogonalize the eigenvectors of

degenerate solutions whenever they occurred.

Completeness and Spurious Solutions

The eigenvectors of Eq. (3.1) are the expansion

coefficients for the electron wave functions of Eq. (2.13).

Completeness relations for the set of eigenvectors may be

derived by expanding an arbitrary 2N vector, where N is the

dimension of the Hamiltonians, in terms of the eigenvectors.

The orthogonality relation, Eq. (3.2), is then used to

evaluate the components of the arbitrary vector. By

considering various possibilities for the components of the

arbitrary vector, one finds that

=; E (Cdj.),d Cd2

o = (Cdl d Cdj (3.4)

0 di HQ d J jj (3.5)

0o = ) H1dk + Hidkj] C (6)

j di QJj.j

and

dd' d (3.7)

are satisfied for all d and d' (Smith and Mailhiot, 1986).

These relations imply that all eigenvalues kj of Eq. (3.1),

including the spurious solutions, will be needed to form a

complete basis set for the superlattice wave functions.

Completeness relations equivalent to Eqs. (3.4) through

(3.7) can also be derived for the solutions of pseudopotential

or k- Hamiltonians with an infinite number of basis functions

(Smith and Mailhiot, 1990). These Hamiltonians use the

infinite number of allowable reciprocal lattice vectors or

zone-center Bloch functions as basis functions for the

expansion of the material wave functions. The resulting

Hamiltonians have the correct periodicity in k, so that

changing the value of k by a reciprocal lattice vector does

not alter the energies obtained as eigenvalues.

The effect of translations in k are more easily described

for the pseudopotential Hamiltonian. For this Hamiltonian,

the wave functions are

Z = Cj, exp[i (+G) -f] (3.8)

G

and the Hamiltonian satisfies

Z H,,dco. = Ce(E) Cd,,W (3.9)

where Ec(k) is the energy at k for the band labelled by G. The

periodicity conditions are

37

Hy,,(k+Gi ) = Hy,+GiB G ( k) (3.10)

and

C k- = CG,~l (3.11)

so that translating k by a reciprocal lattice vector reorders

the rows and columns of the Hamiltonian and reorders the

components of the eigenvectors. These conditions are only

valid for the infinite basis set. For a finite basis set, the

reordering requires information from reciprocal lattice

vectors outside the set and the periodicity of the energies in

k is lost. Because the kjE and pseudopotential Hamiltonians

are related by a linear transformation, the periodicity

condition for the eigenvectors of the infinite basis set k-p

Hamiltonian is

C-d,+ d R-/ ,d) + Rd', d CdI', (3.12)

where R is the transformation matrix of Eq. (2.4).

The completeness relations of Eqs.(3.4) to (3.7) may now

be viewed in two ways. The sum over j could include the

spurious solutions, so that these physically meaningless

states must be included in the superlattice wave function.

Alternatively, the sum over j can be thought of as a sum over

both the calculated values in the first Brillouin zone and

repeated-zone solutions related to the calculated solutions by

Eq. (3.12). The use of such constructed repeated-zone

solutions as replacements for the spurious solutions appears

to be an attractive approach, and has been advocated for

38

indirect band gap systems such as Si-Ge (Smith and Mailhiot,

(1990). A third alternative is to modify the form of the k-p

Hamiltonian so that the spurious solutions do not arise. This

has been accomplished by dropping terms that represent higher-

order interactions between the explicitly included states (de

Sterke and Hall, 1987 and Smith and Mailhiot, 1986). Such an

approach would be impractical for the full-zone model desired

here.

The use of constructed repeated-zone solutions was

considered, but proved to be impractical. The eigenvectors

calculated from the k-E Hamiltonian for the (100) conduction

band minima of Si with spin-orbit splitting neglected were

used as a test case. Neglecting spin interactions allows a

block diagonalization of the 15x15 Hamiltonians mentioned

previously, so that only three zone-center Bloch functions

contribute to the lowest conduction band. The full

transforming matrix R, order 113, was used to shift solutions

near the conduction band minima by G=(200)7/a and G=-(200)r/a,

giving the nearest repeated-zone solutions. For k=kmin, kmin-

(200)r/a is the nearest repeated-zone solution to k=-kmi in

the (100) direction. The eigenvectors Cd,k that resulted from

application of Eq. (3.12) were physically meaningless,

primarily because the information from only 80 of the original

113 reciprocal lattice vectors was used by the mapping. The

elements of the constructed eigenvectors were two orders of

magnitude smaller than the elements of the original

39

eigenvectors. The length 15 eigenvectors for repeated-zone

solutions were also constructed, with similar results. The

largest elements were two orders of magnitude smaller than

those of the calculated C ,k and the elements did not reflect

the block diagonalizability of the Hamiltonian. Therefore,

the use of this approach to satisfy the completeness

conditions was eliminated from further consideration.

The completeness relations may be used to derive the

identities

1 jab 1 rba

j..-i.. ,= 6j (3.13)

,a -1 7 b 23

J3.j 2i.i

and

1 Tjba 1 ab

b i' J J'it = 6ii (3.14)

J J7

where Jab and Jba represent current densities across an

interface from material a to material b and from b to a,

respectively (Smith and Mailhiot, 1986). ja and jb represent

the current densities within materials a and b defined in Eq.

(3.3). The explicit form of Jab will be given later, when the

interface matching conditions are presented. The identities

of Eqs. (3.13) and (3.14) imply flux conservation through a

superlattice period. For the superlattice calculations to be

presented later, the spurious solutions were not included in

the interface matching. For the higher energy conduction

bands, some solutions outside the first Brillouin zone were

needed. These solutions were obtained from the k-p

40

Hamiltonian, instead of being constructed from Eq. (3.12).

The criteria used to exclude spurious solutions while

retaining the needed second-zone solutions was that Iki was

less than or equal to 1.5(27/a). The completeness of the set

of eigenvalues k used for the interface matching was then

evaluated by checking for flux conservation. The deviation of

the flux conservation calculations from the Kronecker delta

was no larger than 10-3, and typically several orders of

magnitude smaller than that.

Superlattice Model

The two principal features of superlattices, interfaces

between the constituent materials and the periodicity of the

layer thicknesses in the growth direction, will be considered

next. The wave functions for superlattice solutions will be

constructed from linear combinations of the solutions

calculated for each of the constituent materials. These

solutions must be matched, using appropriate boundary

conditions, at the interfaces. This gives sets of

coefficients for the linear combination, but not all of these

sets of coefficients generate superlattice wave functions with

the appropriate periodicity. The Bloch condition for the

superlattice will be used to determine which combinations

produce allowable solutions and to calculate the corresponding

superlattice wave vector.

Interface Matching

The solutions for propagating and evanescent states in

the constituent materials were matched at the interfaces with

boundary conditions derived from the current density operator.

The interfaces were assumed to be abrupt. To derive the

matching conditions, consider an interface defined by the

plane x=xo between layers of materials a and b. The solutions

in the neighborhood of the interface were expressed as

T = a8 (-X+Xo) +*bO(X-X) (3.15)

where

a = A = Aei''f C, jUd j(f) (3.16)

and *b was expressed as the analogous linear combination of

eigenstates for material b. The solutions which were included

in the sum over j of Eq. (3.16) were those for which the

magnitude of kj was less than 1.5(2r/a), as described

previously.

The boundary conditions for wave functions that solve

Hamiltonians of the form of Eq. (2.15) are continuity of the

wave function and its first derivative. Continuity of the

current density is an equivalent condition, and is often more

convenient for multiband models such as this (Altarelli,

1983). The boundary conditions at the interface were taken to

be

Jx(Xoy, z) 4a = Jx(Xo y, z) *b (3.17)

where the current density operator is defined as

42

Jx(R) = 6(r-R)p+px8 (i-R) (3.18)

The boundary condition of Eq. (3.17) was converted into

a system of equations relating the Aj and Bi coefficients.

This was accomplished by substituting Eq. (3.16) and its

analogue for 7b into Eq. (3.17), overlapping with j.a, and

then integrating over the x=xo plane. The integrals were

divided by the area of the plane to give an average value of

the fluxes, resulting in

EIdydz A =

(3.19)

Slfdydz < l (xoyz) 11>B

The integrals on the left-hand side correspond to the current

densities within material a, Jjja, as given by Eq. (3.3).

Those on the right-hand side represent the fluxes across the

interface from an eigenstate of material a to each eigenstate

of material b. The vector product representation of these

integrals is

b i(k-k)xo = ei(k-k)o (C ) [H2 (k +k ) +Hl ] (3.20)

where A is a matrix arising from the perturbation of the Bloch

functions. The A matrix is first order in the AV

perturbations, and is non-Hermitian. A more rigorous

derivation of Eq. (3.20), which uses an interface state to

justify the integral on the right-hand side of Eq. (3.19), is

available elsewhere (Smith and Mailhiot, 1986). The relations

43

between Aj and Bi may also be derived using a variational

argument (Schlosser and Marcus, 1963).

The relation that results from Eqs. (3.19) and (3.20),

after applying the orthogonality condition, is

Aeik-xo -ikxoBe (3.21)

J-j

and the analogous equation for the Bi is

Beikfxo 1 Oeikx (3.22)

i'l*

The combination of Eqs. (3.21) and (3.22) to eliminate either

the Aj or the B, leads to the flux conservation criteria.

Superlattice Bloch Condition

The interface matching conditions are not sufficient for

determining the allowed superlattice wave functions. The

additional restriction needed comes from the periodicity of

the superlattice in the growth direction. The application of

Bloch's Theorem to the superlattice wave functions results in

Y(F+B) = eidQ-W(f) (3.23)

where Q is a wave vector analogous to k for the constituent

materials and D is a superlattice translation vector. If a is

defined as the primitive superlattice translation vector in

the growth direction, then Eq. (3.23) for the superlattice

wave function in material a gives

, (f+n~) = e in-" ,o(r)

(3.24)

44

where n labels the superlattice period and n is the

superlattice quantum number. From Eq. (3.24) and its analogue

for material b, the relationship between expansion

coefficients of the different superlattice cycles is found to

be

A = ei(4l,-- a)-niAj (3.25)

with an analogous expression for the Bi. These expressions

may be combined with the interface matching condition for the

interface between material b in cycle n and material a in

cycle n+l to give a relationship between the expansion

coefficients, superlattice wave vectors, and layer

thicknesses. Using the matching condition of Eq. (3.22) to

eliminate the B, results in the eigenvalue problem

j Mjj,A, = ei(a+b)Aj (3.26)

where Q is the (100) component of the superlattice wave vector

and a and b are the layer thicknesses of materials a and b,

respectively. The matrix Mjj, is given by

M = eik a .eikfb 1 (3.27)

"-j i'i

Solution of the eigenvalue problem gives the superlattice wave

vectors Q and the expansion coefficients for the wave

function in material a (Smith and Mailhiot, 1986). The

matching condition of Eq. (3.22) was used to calculate the

coefficients for material b when necessary.

45

Alternative forms of the eigenvalue problem for the

superlattice wave vector are useful if the inverse of M may be

computed efficiently. The flux conservation identities may be

used to show that

M-1 = J-1 J (3.28)

where ~ is an adjoint operator given by

Mji = (M(,).j.) (3.29)

and J is the diagonal matrix

Jj, = J = 6j, (3.30)

(Smith and Mailhiot, 1986). The eigenvalue problem for

exp[iQ(a+b)] can then be expressed in the equivalent forms

M + J-1MJ] = cos[1Q(a+b)]A (3.31)

and

M- j-1MJa, = sin[Q (a+b)]A (3.32)

2i -9

These forms were used to obtain the results presented here.

Superlattice Energy Band Results

The k-' theory described above was used to calculate the

energy band structures for three Si-Ge systems. The

calculation procedure followed the presentation of the theory.

The elements of the k-p Hamiltonians were calculated from the

solutions of the pseudopotential reference Hamiltonian at k=0.

The spin-orbit splitting was added using values obtained from

the literature (Van de Walle and Martin, 1986).

46

The superlattice calculations were begun by calculating

the band offsets and strain interactions for the desired layer

thicknesses. The energy was chosen, then the values of k. and

Cdj were calculated from the transformed Hamiltonians of Eq.

(3.1). After discarding the spurious solutions, the current

densities within each material and across the interfaces were

calculated and the eigenvalue problems for the superlattice

wave vector, Eqs. (3.31) and (3.32), were assembled. The

solution of those eigenvalue problems gave the allowed

superlattice wave vectors, those for which cos(Qd) and sin(Qd)

were real with magnitudes less than or equal to unity. Unless

otherwise noted, the results presented here are for [100]

growth direction with the component of k parallel to the

interfaces, k,, set equal to zero. The energy reference is

the maximum of the valence bands of Si with spin-orbit and

strain effects neglected.

Si-Si0.5Geo.5 Freestanding Superlattices

The first system considered was superlattices with

alternating layers of Si and Si0.5Ge0.5. The strains were

calculated for a freestanding superlattice, in which the

macroscopic elastic energy of the superlattice is minimized

(Van de Walle and Martin, 1986). This configuration is

equivalent to growth on an Si0.75Ge0.25 substrate when the same

number of monolayers are used for each material.

47

The contour plots of Figs. (3.4) through (3.7)

demonstrate the trends of important electronic properties as

the material thicknesses in the superlattice are altered.

Although the use of contour plots may imply otherwise, it

should be stressed that only integer numbers of monolayers

were considered.

The values of the superlattice wave vector at which the

superlattice conduction band minimum occurs are presented in

Fig. (3.4). Over the range of layer thicknesses considered,

the minimum appears to depend only on the total thickness of

the superlattice period. Such a dependence was expected,

since the value of k at the (100) conduction band minimum was

approximately the same in each material. The configurations

of interest are the direct, or near-direct, ones with

superlattice periods of 10, 14, and 19 monolayers.

The energies at which the superlattice conduction band

minimum occurred are in Fig. (3.5). The minima are primarily

a function of the Si layer thickness because the band

alignment leaves the Si minimum below the alloy minimum. The

valence band maximum energies of Fig. (3.6) depend primarily

on the alloy layer thickness because the Si layer is the

barrier for valence bands. The variation in the energies of

the valence band maxima, 0.1 eV, is twice that of the

conduction band minima. Consequently, the energy gaps of Fig.

(3.7) are nearly independent of the thickness of the Si

layers.

Q of Conduction Band Minimum (rn/d)

10

a>

0

c-

0

a,

0.

4 5 6

7 8 9 10

Si Monolayers

Figure 3.4 Value of the superlattice wave vector at the

conduction band minimum as a function of the layer thicknesses

of Si and Si 0.Ge05 within a superlattice period. The strains

were calculated for freestanding superlattices.

Conduction Band Minimum Energy (eV)

10

>1

C,

0

ca

UC

0;

0,

d~

4 5 6 7 8 9 10

Si Monolayers

Figure 3.5 Values of the energy for the conduction band

minimum as a function of the layer thicknesses of Si and

Sio.5Geo.5 within a superlattice period. The strains were

calculated for freestanding superlattices.

Valence Band Maximum Energy (eV)

10

LO

C,

a

o

a,

5 6

7 8 9 10

Si Monolayers

Figure 3.6 Values of the energy for the valence band maximum

as a function of the layer thicknesses of Si and Si0 Ge05

within a superlattice period. The strains were calculated for

freestanding superlattices.

Energy Gap (eV)

10

C,

0

0

L

a,

(D

LO

6

C/

4 5 6 7 8 9 10

Si Monolayers

Figure 3.7 Values of the energy gap as a function of the

layer thicknesses of Si and Sio.5Ge0.5 within a single

superlattice period. The strains were calculated for

freestanding superlattices.

52

Examples of the calculated band structures are presented

in Fig. (3.8). The most significant feature of these

structures is the splitting of the lowest conduction band into

a doublet. This splitting occurs because of the indirect band

structure of the bulk materials. The superlattice conduction

band states are dominated by the bulk wave functions with wave

vectors near kmin, and interference effects between the states

with wave vectors kmin+Ak and kmin-Ak produce the splitting.

This splitting is least for the near-direct structures, in

which the interference should be the least.

The effects of the wave vectors having a nonzero

component parallel to the interfaces was also considered. The

band structure for the 7x7 configuration with k, varying from

0 to 7/d in the (001) direction is shown in Fig. (3.9). The

right-hand side of the figure is the superlattice band

structure for k set to zero, while the left-hand side depicts

the energies at which Q is zero for the given value of k It

should also be noted that there was not another reduction of

degeneracy for nonzero k.. For a superlattice with layers of

Si and Ge, the degeneracy would be appropriate because the

inversion symmetry is retained. The Si0.5Ge0.5 alloy was

modeled with a weighted average of the pseudopotential

parameters of the pure materials, which may result in higher

symmetry than is present in an actual alloy.

The band structures for nonzero k are important for the

calculation of transport and optical properties of thick

7x6 7x7 7x8

1.3

1.1

S0.3

0.1

-0.1

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0

Q (T/d) Q (T/d) Q (/rd)

Figure 3.8 Superlattice band structures for the Si-Si0.5Ge0.5

freestanding system. The notation nxm represents the number

of Si monolayers, n, and Si0.5Ge0.5 monolayers, m, in a

superlattice period.

2.0

1.5 -

1.0

0.5

0.0

-0.5

-1.0

1.0 0.5 0.0

0.5 1.0

kz (r/d) Q (n/d)

Figure 3.9 Band structure for the 7x7 Si-Si0 Ge0,5

freestanding superlattice. The left-hand side of the figure

depicts the band structure for nonzero k .

a)

a)

C

wL

55

superlattices because the band structures have little

dispersion, or curvature. As an example, local maxima or

minima near Q=0 in the kl directions can produce significant

peaks in the optical absorption coefficient spectrum (Chang

and Schulman, 1985). For the relatively thin period

superlattices presented here, these effects are much less

important. The primary reason for performing this calculation

was to provide a sterner test for the method used to handle

the spurious solutions. The flux check agreed well for the

smaller values of kz, but elements that should have been zero

were of order 10"1 for kz=7/d in the conduction band

calculation.

Si-Sio.sGe0.5 Superlattices on Si0.75Ge.25 Substrate

The effects of the strain upon the superlattice

electronic properties were considered by calculating band

structures of Si-Si0.5Ge0.5 superlattices on a Si0.75Ge0.25

substrate. The values of the superlattice wave vector at the

conduction band minima, presented in Fig. (3.10), were nearly

unchanged. This result supports the earlier argument for the

location of the minima depending on the total thickness of a

superlattice period.

The trends in the extrema of the conduction and valence

bands were also unchanged. The differences between the

results for the freestanding superlattices and those for the

superlattices on a single substrate can be explained by

Q of Conduction Band Minimum (n/d)

10

CD

a,

>-

0

d

LO

0

LO

d

4 5 6

7 8 9 10

Si Monolayers

Figure 3.10 Values of the superlattice wave vector at the

conduction band minimum as a function of the layer thicknesses

of Si and Si0,5Ge0.5 within a superlattice period. The

substrate was Si0 7Ge0.25

0.75 0.25'

Conduction Band Minimum Energy (eV)

5 6

7 8 9 10

Si Monolayers

Figure 3.11 Values of the energy for the conduction band

minimum as a function of the layer thicknesses of Si and

Si0.5Ge0.5 within a superlattice period. The substrate was

Sio175Ge0o.25

10

(I,

0

c:

0

LO

6

a,

-51

(C

o;

c3

44

58

considering the behavior of a particle in a potential well.

The ground state energy for the particle decreases as the well

is widened or deepened. The well depths do not change for the

superlattices grown on the Si0.75Ge.25 substrate, but do change

with the relative thickness of the layers for the freestanding

case. Note that for the valence band solutions, holes, the

highest available energy is the ground state. As a result,

the potential well is in the material with the higher valence

band maximum, the alloy.

The variation with layer thickness in the minimum energy

of the conduction bands, Fig. (3.11), was approximately twice

that for the freestanding configurations. The greater

variation of the energy was not caused by the conduction band

offsets, as these remained constant to within one meV. For

the freestanding case, the net effect of the hydrostatic shift

in the band gaps and the strain splitting of the conduction

bands was to lower the energy of the Si conduction band

minimum as the thickness of the Si layers decreased. This

lowering of the minimum energy countered the effect of thinner

wells raising the minimum energy of the superlattice solutions

for decreasing Si layer thicknesses. For a single substrate

composition, all layer configurations have the bulk conduction

band minima at the same energies, removing the damping effect.

The energies of the valence band maxima, Fig. (3.12),

varied over nearly the same range as for the freestanding

case. The dependence of the energy maxima on the Si0 .Ge0.5

Valence Band Maximum Energy (eV)

10

C,,

0

C:

0

2:

LO,

O

a,

4 5 6 7 8 9 10

Si Monolayers

Figure 3.12 Values of the energy for the valence band maximum

as a function of the layer thicknesses of Si and Si .5Geo.5

within a superlattice period. The substrate was Sio075Geo025*

Energy Gap (eV)

5 6

7 8 9 10

Si Monolayers

Figure 3.13 Values of the energy gap as a function of the

layer thicknesses of Si and Sio.5Geo.5 within a superlattice

period. The substrate was Sio.75Geo.25'

10

CO

C,

a

0

c5

(O

4

61

layer thickness was slightly enhanced. This occurred because

the valence band offsets were the same for all layer

configurations. For the freestanding case, the offsets

decreased as the alloy thickness increased, raising the bottom

of the potential energy wells for the holes and countering the

trend of increasing energy for the maxima.

The energy gaps as a function of the layer thicknesses,

Fig. (3.13), reflect the changes in the variation of the

conduction and valence band extrema. Because the variation of

the conduction band extrema increased more than that of the

valence bands, the energy gaps have a greater dependence on

the thickness of the Si layers than for the freestanding case.

The overall variation of the energy gaps also decreased.

Si-Ge Freestanding Superlattices

The effects of the layer composition upon the

superlattice electronic properties were considered by

calculating the band structures for superlattices with

alternating layers of Si and Ge. The calculations were

performed for the freestanding superlattice configurations.

The results for the value of the superlattice wave vector at

the conduction band minima, Fig. (3.14), were nearly

equivalent to the Si-Si0.5Ge0.5 results. This was expected, as

the primary difference between the Ge and Si.5 Ge0.5 band

structures is that the conduction band minima are at the

Brillouin zone edges in the (111) directions for Ge. The

Q of Conduction Band Minimum (T/d)

10

9

a>

o

0

0

0

4 5 6

7 8 9 10

Si Monolayers

Figure 3.14 Values of the superlattice wave vector at the

conduction band minimum as a function of the layer thicknesses

of Si and Ge within a superlattice period. The strains were

calculated for freestanding superlattices.

Conduction Band Minimum Energy (eV)

10

9

CD

ac

0

a>

0

4 5 6 7 8 9 10

Si Monolayers

Figure 3.15 Values of the energy for the conduction band

minimum as a function of the layer thicknesses of Si and Ge

within a superlattice period. The strains were calculated for

freestanding superlattices.

Valence Band Maximum Energy (eV)

10

9

0

o

0

C,

(5

4 5 6 7 8 9 10

Si Monolayers

Figure 3.16 Values of the energy for the valence band maximum

as a function of the layer thicknesses of Si and Ge within a

superlattice period. The strain was calculated for

freestanding superlattices.

Energy Gap (eV)

10

9

C,

0

C:

0

0

(5

5

4

4

5 6 7 8 9 10

Si Monolayers

Figure 3.17 Values of the energy gap as a function of the

layer thicknesses of Si and Ge within a single superlattice

period. The strains were calculated for freestanding

superlattices.

66

bands in the (100) direction have a local minimum at

approximately 0.8(27/a) with curvature similar to that of the

corresponding global minima of Si.

The values of the energy at the conduction band minimum,

Fig. (3.15), and the valence band maximum, Fig. (3.16), have

much greater variation with layer thickness than those of the

Si-Si0.5Ge0.5 system. This can be attributed to two factors.

The strains, and the corresponding splitting and shifting of

the bulk energy bands, were approximately doubled by the

change from Si0.5Ge0.5 to Ge. The energy offsets between the

materials were also approximately doubled. The increased

offsets allowed for greater variation in the energy extrema

and also strengthened the trend for the thickness of the well

material to control the value of the extrema. The energy gaps

of Fig. (3.17) exhibit a much greater variation with the layer

thicknesses than those of the Si-Sio.5Ge0.5 system. The energy

gaps were relatively small, with a maximum value of 0.85 eV.

The larger variation of the valence band maxima with layer

thickness results in the energy gaps depending primarily on

the thickness of the Ge layers.

Summary and Conclusions

The method used to calculate the superlattice band

structures was presented. The solutions for the superlattice

wave function were obtained by matching the bulk material wave

67

functions at the interfaces to determine the appropriate

linear combination of bulk solutions. Imposing the Bloch

condition for the superlattice gave the values of the

superlattice wave vector Q. The band structures were

calculated for superlattices with layer compositions Si-

Si05Ge0.5 and Si-Ge and individual layer thicknesses ranging

from 4 to 10 monolayers. The effect of the strain

configuration upon the band structures was also investigated.

The most important trend for the superlattice systems

studied was that the directness of the energy bands depended

exclusively on the superlattice period. Because of this, it

should be possible to design quasi-direct energy gap materials

based on silicon with broad ranges of band gaps and other

electronic properties. The energy extrema of the valence and

conduction bands depend primarily upon the thickness of the

well materials, Si and Si1lxGex respectively. The valence band

energy maxima vary much more with layer thickness than the

conduction band minima, probably because of the smaller

effective mass of the holes. As a result, the maximum

attainable band gaps correspond approximately to the band gap

of the germanium containing material.

CHAPTER 4

OPTICAL PROPERTIES

Introduction

A direct band structure is not adequate to ensure that a

particular material will be useful for optical devices. The

efficiency of the optical transitions, measured by the optical

absorption coefficient, is also important. The optical

absorption coefficient for a given energy is proportional to

the sum of the squared optical matrix elements for all

transitions between states separated by that energy. The

values of the optical matrix elements at the zone center

provide a useful measure of the optical efficiency of a given

superlattice configuration. In this chapter, the optical

matrix elements for the direct superlattice configurations

will be presented. The optical absorption spectrum for a

specific configuration will then be given.

Optical Matrix Elements

The optical matrix elements, PM, are essentially the

expectation values of the momentum operator for interactions

between superlattice states m and n. For transitions within

a superlattice, there is a selection rule analogous to that

for bulk materials so that the matrix elements are zero unless

68

69

both states have the same value of Q (Rajakarunanayake and

McGill, 1989a). For a transition between superlattice states,

the optical matrix element was

Pmn= (Yj|T (4.1)

where p is the momentum operator, -iV, and the subscripts m

and n denote the states. From the expressions for the

superlattice solutions, Eqs. (3.15) and (3.16), the matrix

elements may be expressed as a sum of contributions from

derivatives of the Bloch functions and contributions from the

derivative of the plane-wave portions.

Selection rules for the allowed transitions can be found

by applying simplifying assumptions to the superlattice wave

function model. If the values of k are sufficiently small,

the expansion for a superlattice solution may be approximated

as

S= EFdd (4.2)

where

Fd = AjeiC-fCd, (4.3)

and the k dependent perturbations of the Bloch functions were

neglected. This is the envelope function approximation for

the superlattice wave functions (Bastard, 1981). Because the

envelope functions, Fd, must vary much more slowly than the

Bloch functions for the approximation to be acceptable, the

spatial dependence of the envelope functions within a unit

70

cell is neglected during calculation of the matrix elements.

The resulting expression is

P.= [( II U F + ( lu)f F ,)i] (4.4)

where the inner products involving the Bloch functions are

only integrated over a unit cell. The symmetry of the zone-

center Bloch functions can be used to infer selection rules

for the two terms of Eq. (4.4) (Karunasiri, Park, and Wang,

1990). For transitions between different types of bands, such

as light-hole to heavy-hole, the second term should vanish

because of orthogonality of the Bloch functions. Because of

orthogonality of the envelope functions, the first term should

vanish for interactions between bands of the same type with

different quantum numbers Since the assumption of slowly-

varying envelope functions was not valid for the superlattice

conduction bands of silicon-germanium systems, and because of

band mixing for the valence bands, these selection rules were

not expected to be rigorously valid for this work.

The selection rules were useful in evaluating the level

of approximation that could be used for calculating the matrix

elements. It was discovered that for transitions between

superlattice conduction bands, the two contributions were of

the same order of magnitude when the slowly-varying

approximations were used. The next level of approximation,

with the integration performed exactly for the first order

perturbation of the Bloch functions, gave the expected

71

results. The contribution to the matrix elements in material

a was calculated as

P= E (Aj)A, (Cd, j) Cd', j

3 d,d (4.5)

X [d, d/(j. j') + ( jd, d' ( j)]

where S and a represent the integrals for Bloch functions d

and d' with wave vectors k1 and k1,, and : is the unit vector

in the growth direction. To first order in the perturbation

theory for the Bloch functions, these quantities are

dld (J'J,) = Ad d-' (Rd,) "Rd, jG'

a+MbaacG, 6 I (RI d

S(4.6)

1 -e --- .-

k [ -k+G- -G

and

Od,'d(J'/) = a- Mb G, (Rd, Rdi~'

(4.7)

1 -ei kJ-kj+G -G )a (4.7)

k, -k +G -G,

where a is the layer thickness of material a, Mb is the number

of monolayers of material b in a superlattice period, and aa

is the lattice constant of material a. The expressions for

the contribution from material b were analogous to Eqs. (4.5)

through (4.7), with sign changes because the range of

integration was from 0 to b instead of -a to 0. The expansion

coefficients for the superlattice, Aj and Bi, were normalized

over a single superlattice period.

72

The optical matrix elements for transitions from the

valence band maximum to the conduction band minimum were

calculated for the direct superlattice systems. The

calculations were performed by choosing Q, using the energies

from the band structure calculations as initial guesses, and

then iterating over energy to refine the estimate and obtain

the wave vectors and expansion coefficients. The search

routine was relatively fast, primarily because of the quality

of the initial values. The normalization and optical matrix

element calculations were very slow, although these were

essentially matrix multiplications. The need for exact

representation of the zone-center Bloch functions as

expansions in terms of 113 plane waves resulted in large

dimensional matrices that slowed the computation. As a

result, there was no attempt made to replicate results or to

identify any erroneous results caused by insufficient

convergence of the search routine.

The squared optical matrix elements for transitions from

the conduction band minima to the valence band maxima for some

freestanding Si-Sio.5Ge0.5 superlattices are presented in Fig.

(4.1). These results are for direct configurations with a

period of 14 total monolayers. The lower curve, labelled x,

is for the transitions caused by light polarized in the growth

direction. The curve labelled y,z represents the sum of the

squared matrix elements for light polarized in the plane

perpendicular to the growth direction.

0.020

0.015

Cli

C.'_

0.010

0.005

x

0.000

4 5 6 7 8 9 10

Si Monolayers

Figure 4.1 Squared optical matrix elements for direct

Si .5Geo.5 superlattices with a total period of 14 monolayers.

The strains correspond to freestanding superlattices.

The results for Si-Si0.5Geo.5 superlattices on a Si0.75Ge0.25

substrate are given in Fig. (4.2). Only results for the

direct configurations with 14 monolayers in the superlattice

period are presented. The principal trend is that the values

of the matrix elements decrease as the thickness of the alloy

decreases. The contributions of each material were considered

to determine the cause of the trends. A sharp increase in the

contribution to the matrix elements from the silicon layers

causes the relatively high values for the 7x7 configurations.

Because the conduction band to heavy-hole band transitions are

favored, the contributions from the alloy layers dictated the

trend for the results with the Si0.75Ge0.25 substrate. For both

the silicon and alloy, the energy bands at the conduction band

minimum have the same symmetry. As a result, one would expect

that the nature of the superlattice states at the conduction

band minimum was relatively independent of the thicknesses of

the individual layers. Because the strain splits the light

and heavy-hole valence bands, the nature of the superlattice

states at the valence band maximum should vary with the layer

thicknesses. The silicon layers were in compression in the

growth direction, so that the light-hole valence band was

above the heavy-hole band. The alloy layers were under

tension in the growth direction, resulting in the heavy-hole

band being above the light-hole band.

For the calculations with the superlattices grown on the

Sio075Ge025 substrate, Fig. (4.2), the strain was constant and

0.030

0.025

0.020

S0.015

n \y,z

0.010

0.005

0.000 x

4 5 6 7 8 9 10

Si Monolayers

Figure 4.2 Squared optical matrix elements for Si-Sio SGe, 5

superlattices with direct band structures and total thickness

of 14 monolayers. The substrate was Si0.75Ge.25 for each

configuration.

76

the mixture of light and heavy-hole bands in the superlattice

solution should remain fairly constant. As the thickness of

the silicon layers increases, the thickness of the alloy

layers drops and the contribution of the alloy to the matrix

element also drops. For the freestanding superlattices of

Fig. (4.1), the tension increases in the alloy layers as the

number of silicon layers increases. As a result, the

separation between the heavy-hole and light-hole bands

increases and the superlattice states would be expected to

have more heavy-hole character. The thinning of the alloy

layers is countered by the changing nature of the valence

bands. Therefore, the alloy contribution to the matrix

element remains fairly constant and the matrix elements only

reflect the higher contribution of the silicon layers for the

7x7 configuration.

The optical matrix elements calculated for the Si-Ge

freestanding superlattices with direct band gaps are presented

in Fig. (4.3). The results were similar to those for the

freestanding Si-Si0 5Ge0.5 superlattices, although the values

were reduced by a factor of two. The value for either the 5x9

or the 6x8 configuration may be incorrect, but additional

calculations to resolve the question were not performed

because of the resource limitations mentioned previously. The

contributions of the germanium layers were responsible for the

relatively large values of the 5x9 configuration and for the

small values of the 6x10 configuration.

0.010

0.008

0.006

0U

CL

0.004

0.002 x

0.000 ,- ,

4 5 6 7 8 9 10

Si Monolayers

Figure 4.3 Squared optical matrix elements for freestanding

Si-Ge superlattices. The total thickness of a superlattice

period was 14 monolayers.

78

The freestanding superlattice structure with seven layers

of silicon and seven layers of the silicon-germanium alloy in

a period was considered the most promising for optical

applications. This decision was based on the values of the

optical matrix elements for transitions from the conduction

band minimum to the valence band maximum presented above. The

optical matrix elements for nonzero values of k and Q also

contribute to the optical absorption coefficient and should be

considered. Because of the large dispersion of the conduction

bands for nonzero k', the contributions of those superlattice

states was expected to be negligible. The superlattice

subbands, however, were relatively flat so that transitions

between states away from the zone center should be considered.

The optical matrix elements were calculated for

transitions from the lowest two conduction bands to the

highest two valence bands for the 7x7 Si-Si 0.Geo.5 freestanding

superlattice. These bands will be labelled as Cl, C2, V1, and

V2, with C and V denoting conduction and valence bands,

respectively. The bands are numbered from the energy gap out,

so that Cl and V1 are the lowest energy conduction band and

the highest energy valence band, respectively. The squared

optical matrix elements for transitions involving Cl are

presented in Fig. (4.4). The results were fairly scattered,

so the values were simply posted. The higher values for

transitions to V2 for large values of the superlattice wave

vector, particularly at 0.8(r/d) and 0.9(7/d), were probably

0.025

0.020

0.015

c\J

0.010

0.005 /

0.000

0.0

Figure 4.4 Optical matrix

Si-Siq- Ge0.5 freestanding

transitions to Vl, dashed

polarization, circles the

0.5 1.0

Q (7T/d)

elements for the Cl band of the 7x7

superlattice. Solid lines are

lines to V2. Squares denote the x

y,z polarization.

0.010

I -o

0.008 ,

i I

>, 0.006

0 0.004 ,

0.002 Jo

-0 -0-

0.000 J

0.0 0.5 1.0

Q (7T/d)

Figure 4.5 Squared optical matrix elements for the C2 band of

the freestanding 7x7 Si-Si05Ge0 5 superlattice. Solid lines

are transitions to Vl, dashed lines to V2. Squares are for x

polarization, circles are y,z polarization.

81

a result of the band mixing mentioned previously. The

probability of transitions to the first valence band were

stronger, as expected from the selection rules. The squared

matrix elements for transitions involving C2, given in Fig.

(4.5), also exhibit band mixing effects for large values of

the superlattice wave vector. The transitions to the second

valence band were more favorable for the y,z polarization, and

were nearly equivalent for the x polarization. Except for the

peak in the values for the y,z polarization at 0.4(r/d), the

transitions to the first valence band were slightly more

likely. This implies that the second conduction band has

similar symmetry to the first band.

The results for the optical matrix elements compare very

favorably to previous work for the Si-Si 0.Ge0.5 system. The

trends for variation of the matrix elements with layer

thickness agree with results obtained using the envelope

function approximation (Rajakarunanayake and McGill, 1989a),

but the values obtained here were approximately an order of

magnitude greater. It appears that the more rigorous

evaluation of the matrix elements resulted in larger values,

although the band structures obtained in this work were not

significantly different from those obtained with the envelope

function approximation.

Optical Absorption Coefficient

The absorption coefficient of a superlattice is given by

2= pdch w W" (4.8)

Vnrf2Ao m,n kl,Q

where W. is the rate of transitions from state m to state n,

V is the volume of the superlattice, o is the frequency of the

incident light, Ao is the amplitude of the vector potential

for the light, and nr is the refractive index of the

superlattice (Ahn and Chuang, 1987). The transition rate is

given by

= 2 JnlHoJI12 (m-fn) 8 (EnE-_h) (4.9)

where fm and fn are the Fermi distribution functions and Em and

En are the energies of states m and n. The Hamiltonian for

the optical interaction is

Hopt = -- p (4.10)

m0

where e and m0 are the charge and rest mass of an electron, A

is the vector potential of the incident light, and p is the

momentum operator. The matrix elements may be approximated as

n nloptl -e Ao0 ng )' (4.11)

2m0

where e is the polarization vector of the light.

The form used to calculate the absorption coefficients

was obtained by substituting the relations for Wmn and

83

replacing the summations over k and Q with integration,

resulting in

a [n 11

m,n n2M n__2 PI2

mlnn 2o +e kT (4.12)

X r/2 do

[En-E-h]2 + (r/2) 2

since the dependence of P on k was neglected. The delta

function of Eq. (4.9) was replaced by the normalized

Lorentzian function in order to represent broadening of the

spectrum by scattering.

The absorption coefficients were calculated for the 7x7

freestanding Si-Sio0.Ge0.5 superlattice. The dependence of the

optical matrix elements and energies on the superlattice wave

vector was fit with cubic spline functions and the integration

was performed numerically. The broadening parameter, r, was

taken to be 8 meV. The effective mass, m*, of Eq. (4.12) was

actually the sum of effective masses for the valence and

conduction bands. The hole effective mass was taken to be the

conductivity effective mass, while the transverse effective

mass was used for the conduction bands (Chang, Chiou, and

Khoshnevissan, 1992). A weighted average of these quantities

for silicon and germanium was used, so that the value of m

was 0.5mo.

The absorption coefficients were calculated at 300 K with

Fermi levels at 0.3, 0.7, and 1.1 eV, corresponding to hole

concentrations of approximately 1017, 1010, and 103 cm"3,

IE

0

1.2

1.0

0.8

0.6

0.4

0.2 !1

0.0 4-

1.0

1.1 1.2 1.3

1.4

Energy (eV)

Figure 4.6 Absorption coefficient spectrum of the 7x7

freestanding Si-Sio0.Ge0.5 (100) superlattice for x polarization

and Fermi levels of 0.3, 0.7, and 1.1 eV.

5.0

4.0

E

0

0

C

3.0

2.0

1.0

0.3

0.0

1.0 1.1 1.2 1.3 1.4

Energy (eV)

Figure 4.7 Absorption coefficient spectrum of the 7x7

freestanding Si-Sio0.Ge0.5 (100) superlattice for y or z

polarization and Fermi levels of 0.3, 0.7, and 1.1 eV.

86

respectively. The results for incident light with x

polarization, Fig. (4.6), and y or z polarization, Fig. (4.7),

were dominated by the transitions to the first conduction

band. The peaks at 1.01 eV correspond to C1-V1 transitions,

and the peaks at 1.05 eV arise from the C1-V2 transitions for

both polarizations. The structure at 1.15 and 1.29 eV was

produced by the band mixing effects for the valence bands. The

absorption coefficients dropped off rapidly beyond 1.3 eV,

since only the transitions to C2 contributed. The maximum

value for the x polarization, 104 cm'1, is approximately the

same as the absorption coefficient of GaAs for photon energies

near the bandgap. The decay seen for larger photon energies

occurs because only the lowest conduction bands and highest

valence bands were used in the calculations. Inclusion of

other bands would significantly increase the high energy

absorption coefficient.

Summary and Conclusions

The applicability of the direct-gap superlattice

configurations was evaluated by calculation of the optical

matrix elements for transitions from the valence band maximum

to the conduction band minimum. The matrix elements were

significantly larger than those for bulk silicon or germanium,

and compare favorably to previous calculations for this

superlattice system. The absorption coefficient spectrum was

calculated for the freestanding Si-Si0. Ge0.5 superlattice with

87

seven monolayers of each material in a period. Values of the

absorption coefficient for photon energies near the bandgap

were comparable to those of typical optoelectronic materials.

CHAPTER 5

DEVICE APPLICATIONS

Introduction

The use of the freestanding 7x7 Si-Sio SGe0.5 superlattice

as the optically active material in optoelectronic devices is

considered in this chapter. The discussion is predominately

qualitative, since the material is not characterized well

enough for meaningful calculations of device performance.

Although the energy levels and optical transition strengths

are available from the previous results, parameters such as

carrier diffusivities and non-radiative recombination rates

would have to be estimated. Details of the energy

distribution of the density of states would also need to be

estimated. For the highly non-equilibrium conditions

prevailing in some optical devices, the density of states can

not be fit to the usual parabolic dependence upon energy.

Instead, the density of states extends into the forbidden gap

with an exponential decrease as the difference from the

equilibrium conduction or valence band edges increases.

Theoretical models for this phenomenon, band tailing, involve

parameters which must be fit to experimental data.

Interpolation of the properties for bulk silicon and germanium

would be possible for some of these quantities, but the

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uncertainty of any measures of device performance would be

unacceptable.

The calculated properties of the superlattice will be

used as a basis for discussion of this material in light-

emitting diodes, semiconductor lasers, and photodiodes. The

superlattice will be considered to have been grown on an

Si 0.Ge0.2s buffer layer, which would provide the same strain

configuration as the freestanding superlattice. The band

alignment of the entire superlattice and buffer system can be

determined from the theory used in the superlattice

calculations (Van de Walle and Martin, 1986). The buffer

layer would have an energy gap of 1.035 eV, with the valence

band maximum at 0.17 eV and the conduction band minimum at

1.205 eV. With this alignment, the entire superlattice and

buffer layer system has a direct band gap, since the global

conduction band minimum and valence band maximum would be in

the superlattice.

Light-Emitting Diodes

Light-emitting diodes, LEDs, are semiconductor p-n diodes

that, under the proper conditions, spontaneously emit light

through radiative recombination when forward biased. The

superlattice system considered here would be of some interest

as a near-infrared light source for optical communication, an

application for which LEDs are employed. The strong optical

transitions at photon energies of 1.01 eV correspond to an

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emitted wavelength of 1.2 jm, for which the optical loss in

optical fibers is low, 0.6 dB/km (Sze, 1981).

The quantum efficiency of a diode constructed with the

superlattice should be comparable to that of the GaxIn1-xAs1. P

materials currently used for the near-infrared region.

However, the use of the superlattice removes one of the main

advantages of LEDs over semiconductor lasers as light sources

for optical fiber communication, ease of fabrication. Since

the LED does not require sophisticated driving or control

circuitry, the ease of integration with silicon based

electronic devices would not be sufficient to make the

superlattice a viable alternative to current materials for LED

fabrication.

Semiconductor Lasers

A semiconductor laser is similar to conventional lasers

in that it produces a highly monochromatic, highly directional

beam of light. The fundamental differences are that in the

semiconductor laser the transitions are not between discrete

states, and that the divergence of the beam is greater because

of the small optically active region. In both conventional

and semiconductor lasers, the emitted light is produced by

stimulated emission. The population of occupied states is

inverted, so that a significant fraction of excited states are

filled with a corresponding number of empty ground states. A

photon impinging upon an excited electron causes it to make

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the transition to a ground state, emitting a photon with the

same phase and energy as the incident photon. Within a

semiconductor, the Fermi distribution function gives the

fraction of occupied states with a given energy. For non-

equilibrium conditions, quasi-Fermi levels for the valence and

conduction bands, Fv and F., are used to determine the fraction

of states which may be occupied. If the population inversion

occurs at a temperature of OK, all conduction band states at

energies below Fc and all valence band states at energies

above Fv are filled.

The buffer and superlattice system considered here would

be suitable for use in a semiconductor laser. Consider the

configuration in which the superlattice is between two

Si.75Ge0.25 layers, a double heterostrucure laser. The larger

energy gap of the alloy layers would provide electrical

confinement of the carriers injected into the superlattice, so

that population inversion could be achieved. Optical

confinement of the generated light would also be provided by

the alloy layers. The index of refraction for the alloy

should be approximately 3% less than that of the superlattice,

based upon the general rule that the index of refraction

decreases with increasing energy gap.

For a semiconductor material to lase, there must be a net

amplification of the generated light. The rate of stimulated

emission is the difference between the rates of spontaneous

emission and absorption. The condition necessary for

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stimulated emission can be found by considering this net rate,

and is given by

F,-F, > he (5.1)

where Fc and Fv are the quasi-Fermi levels and o is the

frequency of the emitted light (Bernard and Duraffourg, 1961).

A gain coefficient can be calculated in the same manner

as the absorption coefficient with the substitution of the

quasi-Fermi levels into the distribution functions. The

absorption spectrum calculated for Fv = 0.17 eV and Fc = 1.205

eV, corresponding to the band edges of the Si.75Ge0.25 buffer

layers, is presented in Fig. (5.1). The parabolic energy

dependence used for the densities of states in this

calculation would not be accurate for the conditions

prevailing in the active region of a laser, but the spectrum

should be qualitatively similar to that obtained from a more

accurate model. The region of negative absorption, or gain,

at photon energies of 1.01 eV corresponds to stimulated

emission from the lowest conduction band to the highest

valence band. The average gain for randomly polarized light

is approximately 200 cm-', which is large enough for one to

expect that gain would occur for an actual device despite the

approximations involved in the calculation.

The Si-Si0.5Ge0.5 superlattice is a promising material for

use as the active layer in a semiconductor laser. The

possibility for integration of a laser with silicon based

electronics makes this device particularly attractive, since