CHAPTER

A BRIEF REVlEW OF THE SEMICONDUCTING COMPOUNDS AND INTRODUCTION TO THE THEORY

OF ELASTICITY - AND THERMAL EXPANSION

1.1 Introduction

I'echnological and theoretical interest in IT-VI and 111-V

semiconductors and {heir alloys in zinc blende structure has been growing

recently due to their appealing properties in electro-optical and electro-

acoustical devices ( i -4) . 7nS. ZnSe and ZnTe are the prototype 11-VI

semiconductors and their cubic phases, which occur naturally as a mineral,

havc been the subject of intense investigation. Considerable efforts are made

to realite a p-type material by doping or growing with impurities such as Li or

CI. When doped with hfn, the i n based crystals form an interesting group of

tlilutr magnetic senl~conductors (5-7). Moreover, in recent years ZnSe has

proved to he particularly interesting dilute magnetic semiconductor when

tiopcd with Mn (8- 1 0 ) Efforts have also been made to Fdbricate a sustainable

ZnSc hlue laser. Very recent 'l<olccnlar-Bean-Epitaxy (MBE) growth of BCC

Fe and Ni single crystals on ZnSe (1 1,12) has opened up the possibility of

using such materials as micro electronic magnetic switches.

Zinc chalcogenides in the cubic sphalerite structure and direct-gap

semiconductors with energy gap ranging from 2.3eV (ZnTe) to 3.7 eV (ZnS)

exhibit increase in direct-gap energy, which is the behaviour usually observed

in binary tetrahedral semiconductors (13-15). Recent interest in the high

pressure behaviour of zinc chalcogenide has arisen in the context of high

pressure investigations of strain effects in superlattices ( 6 1 9 ) . By using

diamond-anvillcell techniques, the effect of pressure on the direct optical gaps

of ZnSe (20) and ZnTe (21) has been investigated for pressure covering the

full stability range of the tetrahedral phases. Phase transitions at high

pressures (14,20,21,22) occur at about 9.5, 13.7 and 15 GPa in ZnTe, ZnSe

and ZnS (14,22) respectively. In the case of ZnS and ZnSe these phases have

been identified as rocksalt [Bl] type (14,15). The 9.5 GPa phase transition in

ZnTe leads to a lower-symmetry crystal structure (23) followed by phase

transition (21,23) at a higher pressure (-1 1.94 Pa). The high pressure phases

of zinc chalcogenides are found to be metallic (14,22) but for the first high-

pressure phase of ZnTe this is not confirmed (2 1).

Photo emission spectra ( I ) show evidence of cation d bands and recent

investigations (24,25) treating the cation d electron as valance electrons have

indicated that the cation d electrons play a significant role in 11-VI and 111-V

conduction mechanism. However, the ground state structural and cohesive

propertla of these materials have not been widely investigated as has been

done for other semiconductors. Therefore. it is meaninghl to characterise

theoretically tht: phys~cal properties of' cubic ZnS, ZnSe, ZnTe and GaAs

crystals using the anhalmonic effects.

Ihe study of ternperature dependence of acoustic attenuation in

semiconductors in the Akhiezel- regime has stimulated much interest among

researchers in the last three decades (26-3 1 ) . Lambade and Sahasrabudhe (26)

has suggested a methiod of calculation involving the non-linearity constant

which ib a function trf second-and third-order elastic constants. Among the

111-V se~niconducting (:ornpounds. GaAs has been the most widely studied for

its attenuating characteristics (32-36). However, the theoretical investigation

on the temperature ciel~elldence of attenuation in GaAs has not been reported

so tar due to the lack of expcrirnental data on temper;ature dependence of

third-order elastic corrstimt ('l'<)l<C:). Recently, Joharapurkar et al. (37) have

reported experimental determination of temperature dependence of the

complete set of T01:C' for GaAs in the range 77-300K. Lambade and

Sahasrabudhe (26) have obtained the acoustic damping in GaAs. They also

suggcsr new compt~tational approach t o determine the collective phonon-

relaxation time in the range 80-300K. These relaxation times are used to

estimate the ternperatlire depcrrdence of attenuation for the longitudinal waves

along the [ I 101 and [ I I I ] directions and for the shear waves along the [I101

direction.

The method generally used to determine the elastic constant C,, of

alloys usually consists in simple linear combination of those of the constituent

components (38). This method has been employed for ternary halides whose

elastic constants are used as input parameters for studying their hardness (39).

Recently, Krieger and Sigg (40) using Brillouin light scattering in AI,Ga,As

observed that the elastic constant tensor differs slightly from those calculated

by linear interpolation of the binaries C,,. From these results it can be seen

that the method generally used for calculating the ternary alloy's C,, by linear

combination of its binary components could be inexact, because the

requirement of linearity of the elastic constant behaviour is not fulfilled.

The elastic properties of semiconductors ZnS, ZnSe, ZnTe and GaAs

are of interest both for technology and basic research (41-45). Anisotropy in

elasticity is a fundamental property of the semiconducting crystals. We

investigate primarily on 11-VI and 111-V semiconducting compounds. We briefly

review here the status of the experimental and theoretical work (46-52) on the

elastic properties, which give additional insight into the basic microscopic

mechanism underlying the condensed state of these materials.

1.2 Zinc Sulphide (ZnS)

Although less studied than 111-V compound superlattices, wide-gap

semiconductors Zns and its alloys are interesting in connection with opto-

electronic application in the blue-light wavelength regime. They may be tuned

by means of built-in strains produced by epitaxial growth of hetero structures

under controlled condirions. Anal! sis of such effects requires the knowledge

of elastic constants. Casali and (:hristensen (53) study the effects of external

pressurc on the elastic moduli. Such calculations of structural and electronic

properties performed directly for heterojunctions or stained-layer superlattices

under pressure (54-50),

The elastic constants of LnS have been determined by Berlincourt et al.

using a piezo resonance technique (57). Casali and Christensen (53) calculated

the inter strain parametel- as a Iiiriction of pressure which defines the relative

displacement ol' the anion and cation sublattices under a trigonal strain. In

addition. they find the trimsverse optical phonon frequency at the zone center

as well as its pressurc variations. Further, the deformation potentials of the

states ;it the valance-hand maxi~num related to tetragonal and trigonal strains

havc heen estimated as a function of pressure. Benkabou et al. (1) predict the

behaviour of solid Zn!; under temperature and pressure effect. These potential

paralnelers are used to calc~~latc the elastic constants. equilibrium lattice

p;lralnelers, bulk modulus, pressure derivatives and cohesive energies in the

cubic phase of ZnS. They also use these parameters to obtain the thermal

e:ipansion coefficient and specitic. heat.

I,uguev,.~ e l < I / . (58) measure the thermal conductivity of poly crystalline

zinc sulfide under hycln~static pressure upto 0.35 GPa in the temperature range

27:-42OK. I t gives ;111 increaw iii the pressure leading to an increase in the

thc1~111,iI conductivity c'xllicient LnS. 'Thermal atomic vibrations in ZnS, ZnSe

and ZnTe crystals are studied by precise X-ray structure analysis in the

temperature range 300-800K by Rabadanov er a/. (59).

ZnS has an FCC cubic zinc blende structure with 43m point group

(60) and its lattice parameter is a = 0.541nm with the co-ordinates of Zn atoms

are (0,0,0), (0,%,%), (%,0,%), (%,%,0) and those of atom S arc (%,l/q,lA), (]A,??,%),

(%,%,?A), (%,?A,%). ZnS structure can be viewed as two interpenetrating FCC

lattices with one composed entirely of Zn atoms, the other entirely of S atoms.

These FCC lattices are displaced from each other by one-quarter of a body

diagonal. Each atom in this semiconducting crystal has four atoms as first

nearest neighbours and twelve atoms as second nearest neighbours.

1.3 Zinc Selenide (ZnSe)

Large band gap materials including ZnSe have recently gained

increasing interest because they are excellent candidates for the blue emitting

laser diodes (61). In the first ZnSe based laser diode, a ZnSe waveguide with

a ternary Cd, Zn(,.,,Se single quantum well was used for the device emitting

in the blue-green spectral range at 77K (62). Later, considerable improvement

towards higher temperature and shorter wavelength was achieved by use of

quaternary layers on GaAs substrates (63-67). Most of the layers are

considerably stressed due to lattice mismatch and different thermal expansion

coefficients (68). In order to analyse the crystal properties in composite

devices, it is important to understand the elastic and electronic properties of

crystals under biaxial strain (68,69). Sorgel and Scherz (70) performed the

first principle calculi~tor~ based on density functional theory using local

density approximation (71,72) for cubic ZnSe under uniaxial and biaxial

strains. I'hey used st:pa.rable norm conserving ah initio pseudopotentials

(73-75) fi)r the c.alculation with 3d valence states for Zn, they used Troullier-

Martins pseudopotentials (76). Rockwell et al. (77) measured the splitting of

the heavy hole and light hole states in ZnSe epilayers grown on GaAs

substrates by means o f ? h , ~ t o modulated spectroscopy under extemal pressure.

Sirrcc the suhstl-ate and the epilayer material have different elastic properties,

the nominal lattice mismatch between the constituents varies when pressure is

applied, and consequently tuning varies. Casali and Christensen (53) compared the

results with ab initio th,eoretical calculations of the elastic properties and they

investigated the effect5 of external pressures on the elastic moduli. The elastic

constants of ZnSe have heen determined by Berlincourt et aL (57) using a piezo

resonance technique. and by Lee (68) using ultrasonic echo technique. The

dependence of the crystal lattice constant on isotopic substitution of atomic

masses is investigated in ZnSe using perturbation theory in a density functional

framework by Alberto 1)ebernardi and Cardona (78). Migal et al. (79) report the

first observation of pit:zc~ electric textures formed in poly crystalline ZnSe

during the growth of th.e crystals. LnSe bulk crystals often suffer from severe

electrical compensat1t:n. So~nc theoretical calculations suggest that the

formation energy of rr;itive defect5 would be too high to make them a source

of any significant conipensatioii i i i ZnSe (80.81). Other calculations yielded

loll- ti)rniation cnergic!. lor. vacant?-dopant cc>mplexes in p- and n- type ZnSe

(82,83). Gebauer et al. (84) show that the vacancies in n-ZnSc can directly be

identified to be V, monovacancies forming complexes with the donors, in

accordance with Electron Paramagnetic Resonance (EPR). Dai et al. (85) have

measured the temperature dependence of the energy Eo (T) and broadening

parameter To (T) of the direct gaps of ZnSe using contactless electro

reflectance in the temperature range 2SK<T<400K. Vinogradov et al. (86)

obtained the results of the first measurements of the lattice reflection IR

spectra from ZnSeIZn,., Cd,Se (x = 0.2, 0.4, 0.47) superlattices. Guan el a/.

(87) made the Brillouin scattering spectra for ZnSe-ZnSe,.,S, [001] obtained

superlattices with the composition x = 0.2.

Vodopyanov et al. (88) studied the IR reflectivity and Raman scattering

for the single constituent layers of ZnSeIZnCdSe superlattice grown on GaAs

substrate by Molecular Beam Epitaxy. Vodopyanov has reported the

measurements of lattice vibrations in ZnSelZnCdSelGaAs superlattices using

far infrared reflection spectroscopy. According to Plekhanov (89) the isotopic

composition of a crystal lattice exerts some influence on the thermal, elastic

and vibrational properties of crystals. These effects are large and can be

readily measured by ultrasound, Brillouin, Raman and the neutron scattering.

Mosca et al. (90) have studied chemical and structural evolution of thick

pseudomorphic ZnSe epilayers grown by Molecular Beam Epitaxy (MBE) on

GaAs [001] substrates. ZnSe structure consists of two interpenetrating FCC

lattices displaced from each other by one-quarter of a body diagonal. About

each atom, thert: are four equally distant atoms of opposite kind arranged at

- the comers of' a regular tetrahedron. ZnSe possesses a 43m point group

syrnmetp and thc latticc. parameter is a == 0.567nm.

1.4 Zinc Telluride (ZnTe)

Zir~c telluride at ambient condition is a semiconductor with zinc blende

crystal structure and a fundamental direct energy gap in the visible spectral

range (2.3eV at 300K). S'imilar to other tetrahedral semiconductors (91) the

effect 01' pressure on thc physical properties of ZnTe has been the subject of

nurnerous experimental and theoretical investigations. ZnTe and its solid

solutions are widely ust:d in infrared and laser techniques, in nuclear radiation

detectors. solar cells n d in semiconductor electronics. Sorgel and Scherz

(70) havc calculated tht: second-and third-order elastic constants under

uniaxial strain. The) have used first-principle density-functional theory

applied to super cells with the plane wave method and energy cutoffs at 30-40

Ry. Here. the 3d state ol' Zn is treated as a core state but a non-linear core

correction is used for the excliangc and correlation energy. They determine

internal strain parameter, the density of states and the change of deformation

density t>t'%n'le due to hiaxial strain. Camacho et al. (92) have calculated the

phonon densities of stnts?s. mode (iriincisen parameters and thermal expansion

coefficiie~lts as a funct~on of PI-cssure for ZnTe using the rigid-ion model.

Kurilo i.1 ui. (93) made an investigation of some mechanical properties of

ZnTe cr>:;~al grown b) the Bricfni;~ri nlerhod, by sublimation and by chemical

. transport reaction'methods. Schall et al. (94) measure the absorption and

dispersion spectra of ZnTe below 3THz, in the temperature range between

10K and 300K to show the importance of interaction between long

wavelength photons and vibrational modes of the crystal lattice in the

interpretation of the spectra.

ZnTe crystallises in zinc blende form. There are eight atoms per unit

cell. The lattice parameter of ZnTe is a = 0.61nm and it possesses a 43m point

group symmetry.

1.5 Gallium Arsenide (GaAs)

Among the 111-V semiconducting compounds, GaAs has been the most

widely studied for its elastic properties (95). As a sensor material, GaAs

possesses many interesting properties. These include properties such as direct

band gap transition and high mobility of electrons. GaAs also has piezo

electric properties comparable with those of quartz, and it exhibits a strong

photoelastic effect leading to birefringing and consequently the opto-

mechanical polarization effect. Moreover, various physical effects give higher

piezo resistive values. GaAs is also considered to be a good material for high

temperature electronics due to its large band gap.

The measured elastic stiffness constants C , , , C , 2 and C44 exhibit

random variations covering as much wider range in the quoted experimental

values. Stefano de Gironcoli et al. (96) calculated the piezo electric constant from

*., j the stress induced by an applied clectric ficld at vanishing ma6mscopic strain. , .A

The linear response t ( , the applied perturbation is obtained by a recently

introduced selt'consistant Green's function method (97). Lambade and

Sahasrabudhe (26) sug,ge:jt a new computational approach to determine the

collective phonon-relax;ition time for GaAs. The approach is based on the work

of 1,lisavskii and Sternin (98). Ihese calculations are performed in the

temperature range 80.- 300K. [!sing these relaxation times, they estimated the

temperature dependence of atte~iuation for longitudinal wave along the [I101 and

[ I 111 directions for shear waves along the [110] direction. Gehrsitz er al. (99)

measured the elastic constants and relaxed lattice parameter of the

technologically important Al,Ga,~,;Zs/GaAs systems by near infrared Brillouin

scattering and high resolution X-ray diffraction. The composition of layers is

specified by inductively coupled plasma atomic emission spectroscopy,

photoluininescence and .liaman spectroscopy. There is a significant and

unexplained drop in the band gap pressure coefficients of 111-V ternary

sen~iconductor alloys grown as strained layers compared with the bulk binary

values.

h~lbacki ef al. 169) introduced the deformation potentials in strained

laycr hetrostrucrure bg measurilig the piezoflectance of biaxially strained

ZnSelCiai\s laycrs undcr ;additiolral application of an external uniaxial stress

along the [I001 ;axis. Sawada er ( 1 1 (100) develop the thermo elastic analysis

for pl-cdictions o f slip defect sc;ic.ration on [001] GaAs wafers by means of

the finite element method by considering the anisotropic structure and slip

system of dislocation. De Caro et al. ( 1 01) introduce a new procedure for the

determination of the Poisson's ratio, the elastic stiffness constant ratios and

the lattice parameters of epitaxial films based on the strain tensor determination

by high-resolution X-ray diffraction using epitaxial layers si~nultaneously grow

on three different substrate orientation and different alloy compositions. A de

Bernabe et al. (38) determine the elastic tensors of thin layers of ln,Ga,.,As and

In,Ga,,P with composition around x = 0.5 using surface acoustic waves

measured by Brillouin light scattering. Constantino et al. (102) report the

stress effects on GaAs at the ZnSe-GaAs hetro interface using the samples

with the structures ZnSe/GaAs/GaAs. Glazov and Pashinkin (103) measure

the thermal expansion of GaAs and InAs in the temperature ranges 396-1 149K

and 441-1206K respectively and calculated the thermal expansion coeflicient,

Debye temperature and root mean square atomic displacements.

Xu et al. (104) reported that a strikingly well defined square two

dimensional lattice of InGaAs dots are formed on the GaAs [3 1 11 R surface.

This remarkable self-organization behaviour is explained by the special

ordering mechanism originating from the high elastic anisotropy of the GaAs.

According to Dowens et al. (105) there is a significant drop in the band gap

pressure coefficients of 111-V ternary semiconductor alloys grown as strained

layers compared with the bulk binary values. Miyazaki et al. (106) develop a

computer code for simulation of dislocation density in a bulk single crystal.

In the computcl code they have taken into account the effects of crystal

anisotropy such as elastic constants and slip direction by averaging the

Young's modulus, the P'oisson's ratio and resolved shear stress along the

azimuthal direction. lising high contrast Brillouin spectroscopy Zuk et a[. (107)

investigated thy acoustic waves in supported layers formed on GaAs

substrates. Miluaki e2t 01. (108) develop a computer code for dislocation

density evaluation ot' a single crystal ingot during annealing process. In this

the temperature in ;I single crystal ingot is used as input data, which are

obtained from a transient heat conduction analysis. Flannery et al. (109) have

investigated both experimentally and th~:oretically, the Surface Acoustic Wave

(SAW) velocity behaviour on 10011, [ I l l ] , and [311] surfaces for GaAs.

Several different acoustic modes can bc measured on each surface. Fil (1 10)

has considered the piezo electric mechanism for the orientation of stripes in

tw'o di111~nsioniil elec~ron systems in GaAs-AIGaAs hetro:;tructures. When the

anisotropy ot' the elaslic constants and the influence of the boundary of the

sample are taken into account. the theory gives orientations of the stripes

along I I 101 direction. l ~ h i c h is in agreement with the experimental data.

.4 theoretical study of variation in exciton binding energy with

quantunl dot 51,cc a11cl dut in plane and perpendicular separation for periodic

arrays 01 ' sell clrgani.<eti In CiaAs/(~ia24s quantum dots was conducted by

Andrev and ()'reill! ( 1 11) . I'letal rt ui. (1 12) present a complete first-

pr i~~ci l~le siuci! o f trcnds in thc latticc dynamics of tetrahedral semiconductors

in the zinc blende or diamond structure, using the plane-wave pseudopotential

method within density-functional perturbation theory. They also present

phonon-dispersion curves, eigen vector phases and the results of the

calculation of some linear and non-linear elastic constants.

Sorokin et al. (1 13) modified the simplified phenomenological theory

based on a conception of finite strains, which appear in a solid during thermal

expansion and the propagation of small amplitude bulk acoustic waves under

such conditions. Thus the temperature dependence of the second-order elastic

constants is calculated for a series of cubic crystals with various type of

predominant chemical bonding. To make a deeper understanding of the physio-

chemical nature of melts of these compounds, Glazov and Shchelikov (114)

conducted a detailed study of the temperature dependence of the specific volume

of their melts. The composition dependence of the elastic constants and the

related properties for quaternary semiconductor alloy AI,Ga,.,As,Sbl.,/InAs

are reported by Bouarissa and Bachiri (1 15). Applying lithographic and

Focused Ion Beam (FIB) techniques for InAsI(3aAs hetrostructures on [ I l l ]

surface, Yamaguchi et al. (1 16) have successfully fabricated and characterized

several Free Standing (FS) structure with InAs thickness ranging from 50 to

300nm.

Sorgel and Scherz (70) have made ab initio calculations for the higher

order elastic constants, internal strain parameters and relative deformation

potentials of GaAs. Detlef Conrad and Kurt Scheerschmidt (1 17j proposed a

parameter set tor ( iahs, yicldrng elastic constants and surface defect

properties. < i d s exists in zinc blende structure, which can be considered as a

special lorm of diamold structure. Unlike in diamond structure, the two

interpenetrating 1:CC sublattices arc of different atoms-one composed entirely

of <j;l atoms. the other elltirely of As atoms. The four nearest neighbours of an

atom are of the opposite kind and disposed tetrahedrally.

1.6 Finite Strain Theory of EXasticity

('onsider an elastic mediuni where the co-ordinates of any point can be

denoted as (a, , a,. a,). (3-loose a set of orthonormal vectors el, el, e, as the

basis vectors for the io-onlinate system and denote the kth component of the

stress acting on the plane e, = O hy o,, where i and k are the component

indices. Consider the cquilibrium of a small element centered at the point ai

and bounded by the plane a, + %da,. Let u, denote the elastic displacement of

the point a, of the body and I, rhc density of this point. The equation of

volunre elcment can bc derived lbq considering the total fbrce acting on the

volume clement. If we isnore the body forces, the equations of motion for an

elastic solid can he writleri as (thc: convention that repeated indices indicate

surnnia~ion over the indices will he tbllowed here).

pu, = i3nIb/(3ak (1.1)

where thc stress tensor n , ~ i,i given hq

where 41 is the crystal po~ential and i : , ~ are the components of the strain tensor

given hy

o , ~ and E , ~ are symmetric tensors of second rank. According to Hook's law

The constants Clklm form a fourth rank tensor with 8 1 components

From equations (1.2) and (l.4), we have

Hence the elastic constants C,hI, are multiple strain derivatives of the

state fimctions and since the strains EI, are symmetric, the elastic constants

possess complete Voigt symmetry. Thus,

C k l m = Ch$lrn = Ctkml = ClmA (I,6)

These quantities are symmetric with respect to interchange of the

subscripts. It will be convenient to abbreviate the double subscript notation to

the single subscript Voigt notation running from 1 to 6, according to the

following scheme:

1 1 + 1 ; 2242; 33+3; 23+4; 13+5 and 12+6.

Hence the matrix of elastic constants Ciklm would contain a 6 x 6 array of

36 independent quantities in the most general case. This number is, however,

reduced to 21 by the requirements that the matrices be symmetric on

interchange of double indices. The number of independent elastic constants

will be further reduced by the symmetry operations of the respective crystal

classes. The cubic compounds have three independent elastic constants (1 18).

The elastic constant matrix for this class of compounds is given by

In the equation of motion for an elastic medium, the forces on an

element of volume are given by the divergence of the stress field.

Using equations (1.3) and (l.4), the equation (1 .l) can be written as

For an elastic plane wave, we have

u, : A, exp i(mt - k.a) (1.9)

where .4k are the components of the amplitude of vibrat.ion, o is the angular

frequency and k is the wave vector corresponding to the wavelength 1 = 2dk.

The resulting equatior~s of motion from equation (1.8) are

( P ~ ' ~ , ~ , ~ C,,I,,, k, k ~ ) u,, 0 (1.10)

Substituting k = kI?. where I? is the unit vector, we get

2 (T,,I, n,n, - v &,) u, = 0 (1.11)

where T,,,,, = C,,l,,/p are the reduced elastic constants and v is the phase

velocity given by v = o/k. The components of second rank tensor A are given

by

A,I = Tllkl njn~ (1.12)

Hence equation (I. 1 1) can be written as

(A - v2)u = 0 (1.13)

This shows that u is the eigen vector of tensor A where eigen value is v2.

Hence v2 is the root of the equation

( A - v ~ ~ = 0 (1.14)

The theory of elastic waves generally reduces to find u and v for all

plane waves propagating in an arbitrary direction for crystals possessing

different symmetries. In this situation, all terms in equation (1.1 1) that

involves differentiation with respect to co-ordinates other than that along the

propagation direction drop out.

A more fundamental significance to the elastic constants is implied by

their appearance as the second derivatives of elastic energy with respect to

strains. It should be noted here that the stored elastic energy is only a part of

the complete thermodynamic potential of the crystal, since it depends on many

other variables. Also, one can introduce elastic constants as a constitutive,

local relation between stress and strain for materials in which long-range

atomic forces are unimportant.

Let the position co-ordinates of a material particle in the unstrained

state be a, ( i = 1, 2, 3). Let the co-ordinates of the material particle in the

strained state be x,. Consider two material particles located at a, and a, + da,.

Let their co-ordinates in the deformed state be x, and x, + dx,. The elements

dxi are related to cia, by the equation

'I he conventio~~ that repeated indices indicate summation over the

indices will be followed here. a,, is the Kronecker delta and E , are the

defi~rmation parameters The Jacohian of'the transformation

is taken iv be positive for all real transformations. I f dV, is a volume element

in the natural state and clV:, its volutne after deformation

where p,, and p are the densities i l l the natural and strained states respectively.

Let the square of the length of' arc from a, to a, + dai be dl6 in the unstrained

state and dl' in the strained state. 'l'hen

1 dl: = tix,dx, da,da,

where 1 1 , ~ are the Lagrangian strain components which are symmetric with

respect to the interchange of the indices j and k. In terms of &,,,

1 1 ~ ~ I,:! c I1 1 L,,t,, 1 (1.19)

The internal energy function U (S,q,,) for the material is a function of

the entropy S and Lagrangian strain components. U can be expanded in

powers of the strain parameters about the unstrained state as

azu a'u t

The linear term in strain is absent because the unstrained state is one

where U is minimum. We shall define the elastic constants of different orders

referred to the unstrained state as

and

Here the derivatives are to be evaluated at equilibrium configuration and

constant entropy. C ~ , , , a n d ~ ~ , , , , ~ , are the adiabatic elastic constants of

second- and third-orders respectively. They are tensors of fourth and sixth

ranks. The number of independent second-and third-order elastic constants

for different crystal classes are tabulated by Bhagavantam (1 18).

1.7 Quasi-harmonic Theory of Thermal Expansion

In the harmonic approximation, the atoms in a solid are assumed to

oscillate symmetrically about their equilibrium positions, which remain

unaltered irrespective of the temperature. The thermal expansion of a solid,

therefore, is a property arising strictly due to the anharmonicity of the lattice.

In the quasi-harmonic approximation, the oscillations are still assumed to be

harmonic in nature but the frequencies are taken to be functions of the strain

components in the lattice. The strained state of the lattice is specified fully by

the six strain components q, (r, s -- 1,2,3; q, = qsr).

A normal mode with frequency w(qj) makes a contribution F(qj) to

the total vibrational free energy I:,,,,, given by

F(qj ) = K~T[F,X + log(] - e-')I (1.23)

where X = f i o( q j ) , t;,% T, h = W2n, h being the Plank's constant, KB is the

Bolt~rnann's constant, II I S the absolute temperature and q is the wave vector

of the j"' ,icoust~c modt: The total vibrational free energy is, therefore

F.,s - CF(q3) '4 1

= K*T 1 [:>x + log (1 -e-')] 9 1

The thermal coefficient., u,, of the crystal are obtained as

D,, arc the cotnponcnts of the stress tensor, St,,, are the compliance

coefficients relating q,, ,~nd 01, and

Here. y,, itre the generalised Griincisen parameters (Gps) of the normal mode

frequencies. In cquatior~ ( 1.25), thc subscript 0' means that all other o,, are to

be held constant whilc dfferentiating with respect to o,,,, and subscript o

2 -x X 2 . means that all o,,,, are held consrant. tr(w(q,j)T) = X e l(1-e- ) IS the

Einstein specific heat function. In the quasi-harmonic approximation, the Gps

are assumed to be constants independent of temperature.

It is more advantageous to choose such strains that do not alter the

crystal symmetry, instead of choosing any arbitrary strain, while defining Gps.

For ZnS, ZnSe, ZnTe and GaAs, there is only one principal thermal expansion

coefficient namely a Here, i t is convenient to use the following strains for the

determination of the thermal expansion.

(i) A uniform areal strain E in the basal plane

(ii) Then q l l = q22 = q,3 = E/ : . All other q,, vanish.

From equation (1.25), we now obtain

va = C [2~,2y(q,.i) + S I I y(q.i)l K ~ o ( ~ ( q j ) ) 4.1

The effective Griineisen parameters are defined as

Y,.(T) = [ (c~~+~cA)J ] V / C P (1.29)

The C: are the adiabatic elastic constants, Cp is the specific heat at

constant pressure and V is the volume of the crystal.

Comparing equations (1.29) with (1.28), we get

The expression (1.30) gives the temperature dependence of effective

Griineisen parameter.

In the lo\\ ternpcrature lin11t. only the low frequency acoustic modes

make a contribution to the specific heat. The number of such normal modes

in the j"' acoustic branch is proportional to v:(0,I$), wliere v,(0,I$) is the

velocity of the j lh acoustic mode ravelling in the direction (0,$). The Gp

y(q,.j) depends only on tt-~e branch index j and the direction (0,O). It is

independent of the magnitude of the wave vector q. The effective lattice

Gruneisen parameters y, approach the limits defined below, at low

temperatures.

where y(t1,Q) are the Gps for the acoustic modes propagating in the direction

(0,I$). Caliulatiorr of thy low temperature limit of ?(T) is possible knowing

the pressure derivatives of the second-order elastic constants (SOEC) or the

third-order elast~c conhtants ( T O H ) of the crystal. The evaluation of the low

temperature limits y (0) for ZnS, LnSe, ZnTe and GaAs is given in Chapter 6

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