High pressure electronic properties and elastic stability criteria of AlAs

Materials Science and Engineering B 123 (2005) 87–93

Short communication

High pressure electronic properties and elastic stability criteria of AlAs N.Y. Aouina a , F. Mezrag a , M. Boucenna a , M. El-Farra b , N. Bouarissa b,∗ a b

Physics Department, Faculty of Science and Engineering, University of M’sila, 28000 M’sila, Algeria Department of Physics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia Received 21 April 2004; received in revised form 15 June 2005; accepted 9 July 2005

Abstract We study the electronic properties and elastic stability criteria of AlAs under hydrostatic pressure by the pseudopotential method combined with the Harrison bond-orbital model for determining the elastic constants. The numerically calculated masses and elastic constants are in good agreement with the available experimental data. Computed electronic properties depend non-linearly on the lattice constant. The generalized stability criteria show the material of interest to be mechanically stable at pressures up to 120 kbar. © 2005 Elsevier B.V. All rights reserved. Keywords: Electronic properties; Elastic stability; Bond-orbital model

1. Introduction Semi-conducting III–V compounds are of continuing great interest owing to their vast technical applications, namely high-electron mobility and hetero-structure bipolar transistors, diode lasers, light-emitting diodes, photodetectors, electro-optic modulators and frequency-mixing components, to mention a few [1]. Aluminum arsenide (AlAs) is one of these compounds that is particularly useful in forming epitaxial multilayer structures with GaAs, with which it is closely lattice matched [1–4]. Unlike GaAs, AlAs is an indirect-gap semiconductor whose conductionband minimum lies near the X points of the Brillouin zone and its band structure is similar to those of AlP or GaP [5]. The study of materials at high pressures is experiencing great current activity. High pressure can have a very large effect on the chemical and physical properties of matter and materials often exhibit new crystal phases and novel behavior under pressure. Moreover, the possibility to study structural stability as a function of volume has added a new dimension to the fundamental question concerning the relation between atomic and electronic structure. With the develop∗

Corresponding author. Tel.: +966 72269326; fax: +966 72269326. E-mail address: N [email protected] (N. Bouarissa).

0921-5107/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2005.07.008

ments of the diamond anvill technique and the extension of the range for optical and X-ray measurements under static pressures, a large range of compression can be accessed which has opened a new domain for solid state physics and crystallography [6–8]. In addition to the experimental advances, reliable computational methods for electronic band structure and total energy calculation have made a substantial impact on high pressure physics [9–16], which provides important complementary data to the experimental work. In the specific case studied here, the III–V compounds exhibit tetrahedral coordination under ambient conditions, typically in the zinc blende or wurtzite structure. Applied high pressure probes the physics of interatomic bonding extremely thoroughly and thus changes the physical properties of semiconductors. In general, the metallic behavior is expected for most semiconductors at high pressures since pressureinduced structural phase transformations occur when the atoms become more closely packed [8,9]. Although the high pressure study of the III–V binary compounds has been the subject of numerous experimental and theoretical researches [8–11,14,16,17], the high pressure behavior of the AlAs binary compounds has not been completely understood. Experimentally, AlAs adopts the zinc blende structure at low and moderate pressures. With increasing pressure, AlAs undergoes a transition to a structure which has been

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identified as NiAs [8,11]. The resulting NiAs structure is the only known high pressure phase to 46 Gpa (460 kbar) [8]. The aim of this work is to focus on the theoretical calculations of the pressure dependence of the electronic properties, such as energy band gaps and effective masses. The mechanical stability of AlAs under hydrostatic pressure has been also examined using the generalized stability criteria. The calculations regarding the electronic properties are based on the local empirical pseudopotential method (EPM). The latter has been combined with the bond-orbital model of Harrison so as to determine the elastic constants that are needed for formulating the generalized stability criteria. The approach used is more empirical than the fundamental first principle calculations that are technically involved and computational time consuming. Quasiparticle calculations, which at the present are probably the most fundamental and accurate way of calculating the band gaps, produce energy gaps that are in good agreement with measured values. However, the starting point for such calculations is the results of the self-consistent density functional theory calculations, which are then used to compute complex many-body corrections to ab initio band gaps. This is a complicated and CPU intensive process. More importantly, due to the well known problem of using ground state density functional theory to obtain the electronic band structure of semiconductors, ab initio calculations underestimate the band gaps. This may affect our results regarding the studied properties. The EPM is simple in nature, computationally economical and accurate. Also, it does not underestimate the band gaps. The rest of this paper is organized as follows. In the following section, we briefly describe the method of calculation used in this work. In Section 3, we present and discuss our results regarding the electronic properties and the mechanical stability criteria, comparing with previous theoretical and experimental studies. We make our conclusion in Section 4.

(LS’s) from the experimental ones defined by,
 1/2 m (i,j) }2 (i,j) {E δ= (m − N)1/2

(1) (i,j)

(i,j)

should be minimum. In Eq. (1), E(i,j) = Eexp − Ecal , (i,j) (i,j) where Eexp and Ecal are the observed and calculated LS’s between the ith state at the wave vector k = kI and the jth at k = kJ , respectively, in the m chosen pairs (i, j); N is the number of the empirical pseudopotential (EPP) parameters. The calculated energies given by solving the EPP secular equation depend non-linearly on the EPP parameters. The starting values of the parameters are improved step by step by iterations until δ is minimized. If we denote the parameters by pv (v = 1,2, . . ., N) and write as pν (n + 1) = pν (n) + pν , where pv (n) is the value at the nth iteration. These corrections pv are determined simultaneously by solving a system of linear equations,   N m    (Qiν − Qjν )(Qi  − Qj  ) pv ν ν ν=1

=

(i,j) m 

j

(i,j) (Eexp − Ecal (n))(Qiν − Qν ) (i,j)

(2)

(i,j)

(v = 1,2, . . ., N), where Ecal (n) is the value at the nth iteration, Qiν is given by

 ∗ ∂H(kI ) i i Qν = [Cq (kI )] Cqi  (kI ) (3) ∂pν qq  (i,j)

q,q

H(kI ) is the pseudo-Hamiltonian matrix at k = kI in the planewave representation and the ith pseudo wave function at k = kI is expanded as,  ΨkiI (r) = Cqi (kI ) exp[i(kI + kq )r] (4) q

2. Method of calculation 2.1. Electronic properties The electronic band structure is calculated using the empirical pseudopotential method. The main advantage of using pseudopotentials is that only valence electrons have to be considered. The core electrons are treated as if they are frozen in an atomic-like configuration. As a result, the valence electrons are thought to move in a weak one-electron potential. In EPM, one has to adjust the pseudopotential form factors so as to reproduce measured interband optical transition energies at selected high-symmetry points in the Brillouin zone. In the present work, the optimization model of Kobayasi and Nara [18] has been used for the optimization of the empirical pseudopotential parameters. The non-linear least-squares method used in that model requires that the rootmean-square (rms) deviation of the calculated level spacing

kq being the reciprocal lattice vector. The experimental direct and indirect energy band gaps at zero pressure used in the fitting procedure are given in Table 1. The pseudopotential form factors required in our calculations at different pressures up to 120 kbar have been determined by fitting the calculated pressure coefficients of selected criticalpoint band gaps to data [20,21] in the literature assuming a linear behavior of the energy band gaps versus pressure. The first-order pressure coefficient of the band gap at X highsymmetry point has been fitted to the experimental one of Ref. Table 1 Band-gap energies of AlAs at normal pressure Band-gap energy (eV)

Experimental dataa

Present work

Eg EgX EgL

2.95

2.95

2.16

2.14

2.36

2.36

a

Ref. [19].

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[20], whereas those at Γ and L points have been fitted to those of Ref. [21]. The lattice constant of the material of interest at various pressures has been obtained through the use of the Murnaghan equation of state [22], where the values of the equilibrium bulk modulus (B0 ) and its first pressure derivative (B0 ) are taken to be 74 Gpa and 4 [11], respectively. 2.2. Elastic stability The fundamental basis for studying the mechanical stability of solid state materials lies on the formulation of stability criteria. The usual Born description for the stability criteria is expressed in terms of the elastic constants Cij where for a cubic crystal, it is given by [8,19], C11 + 2C12 0 B= 3

(5)

C44 0

(6)

C =

C11 − C12 0 2

(7)

The quantities B and C are the bulk and tetragonal moduli of a cubic crystal. Recently, Wang et al. [23,24] showed that under external pressure, the conditions (5)–(7) should be modified so as to describe changes in enthalpy rather than energy. Thus, the stability criteria based on the elastic constants are only valid for the special case of zero stress and the generalization of the stability criteria to the non-zero stress case needs the reformulation of the stability criteria in terms of the elastic stiffness coefficients where the elastic stiffness tensor is given by, Bijkl (X) =

∂σij ∂ηkl (X)

(8)

where σ ij is the applied stress and ηkl is the strain defined with respect to finite strain (Lagrange) coordinates. X and x are the coordinates before and after the deformation, respectively. For a cubic crystal, under hydrostatic pressure, the generalized stability criteria are given by, C11 + 2C12 + P M1 = 0 3 M2 = C44 − P0 M3 =

C11 − C12 − P0 2

(9) (10) (11)

where M1 , M2 and M3 are spinodal, shear and Born criteria, respectively. The first criteria is related to the elastic instability associated to the structural transformation. Practically, the structural transition in solids proceed at spinodal. In view of the expressions (9)–(11), the determination of M1 , M2 and M3 needs the knowledge of the elastic constants C11 , C12 and C44 at various pressures. For that, we have adopted the same method used recently by Bouarissa and Bachiri that is based on the combination of the EPM with the Harrison bond-orbital model [25], where the elastic constants

are expressed as, √  h ¯2 3 3/2 C11 = 5 4.37 (5 + λ)( 1 − α2P ) 4d m 2 h ¯ 1/2 − 0.6075(1 − α2P ) m √  h ¯2 3 3/2 C12 = 5 4.37 (3 − λ)(1 − α2P ) 4d m 2 h ¯ 1/2 + 0.6075(1 − α2P ) m

89

(12)

(13)

√ 3 (14) C44 = (α + β) − 0.136 SC0 − Cζ 2 4d √ where d = 3/4a is the nearest-neighbor distance (a is the lattice constant) and λ is the dimensionless parameter with a constant value of 0.738 [26]. For each pressure, the polarity αP is obtained as [27], αP = −

VA (3) VS (3)

(15)

where VS (3) and VA (3) are the symmetric and antisymmetric form factors for a given pressure at G(1 1 1), respectively. G are reciprocal lattice vectors. In fact, the potential term V(r) for an electron in the crystal may be expanded in reciprocal lattice vectors G and expressed as a structure factor S(G) times a pseudopotential form factor V(G). It is convenient, then, to rewrite this in terms of a symmetric and antisymmetric potential, giving  V (r) = (SS (G)VS (G) + iSA (G)VA (G))eiGr (16) G

where the symmetric form factor VS (G) and the antisymmetric form factor VA (G) are given by, 1 [V1 (G) + V2 (G)] 2 1 VA (G) = [V1 (G) − V2 (G)] 2 VS (G) =

(17)

here, V1 (G) and V2 (G) are the pseudopotential form factors of the individual atoms in the unit cell. For the zinc blende lattice, which is common for III–V semiconductors, the two elements involved each occupy one of the two interpenetrating fcc lattices. If the origin is taken half way between the two fcc lattices, omitting the G = 0 term which gives a constant average potential, the symmetric structure factor is only non-zero for G’s in the zinc blende lattice, the first three of which are given by G2 = 3,8,11, in units of (2π/a)2 , whereas the antisymmetric structure factor is non-zero for G2 = 3,4,11,. . . in units of (2π/a)2 . Hence, we require VS (G2 = 3,8,11) and VA (G2 = 3,4,11) to specify the pseudopotential for zinc blende structure of the material of interest. In Table 2, we present our computed final adjusted

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Table 2 ˚ of AlAs at various pressures Symmetric and antisymmetric form factors (in Ryd) and lattice constant (in A) Form factors

P = 0 kbar

P = 30 kbar

P = 60 kbar

P = 90 kbar

P = 120 kbar

VS (3) VS (8) VS (11) VA (3) VA (4) VA (11) Lattice constant

−0.212694 0.00 0.092750 0.068833 0.05 −0.0075 5.6611

−0.212931 0.00 0.099816 0.068074 0.05 −0.0075 5.6012

−0.212725 0.00 0.106973 0.066888 0.05 −0.0075 5.5406

−0.211967 0.00 0.114209 0.065781 0.05 −0.0075 5.4875

−0.210679 0.00 0.121637 0.064803 0.05 −0.0075 5.4403

local symmetric and antisymmetric pseudopotential form factors along with the lattice constants at different pressures ranging from 0 to 120 kbar. α and β in Eq. (14) are the short-range force parameters. They represent the bond-stretching and bond-bending force constants, respectively and are expressed as [25], (18)

d β = √ [(C11 − C12 ) − 0.053 SC0 ] 3

(19)

whereas ζ is the internal-strain parameter. The quantities S and C0 are obtained using the following expressions, z∗ ε0

2

(20)

and C0 =

e2 d4

(21)

S is an effective charge parameter, C0 has the dimensions of an elastic constant and z* is the effective charge. More details about the determination of the quantities C, ζ and z* are given in Ref. [25]. ε0 is the static dielectric constant. At ambient pressure, ε0 is taken to be 10.06 [5]. However, for a given pressure rather than zero, we have assumed a linear behavior of ε0 under pressure. On this basis, ∂ε0 /∂P has been calculated as follows [28]. Using the relation which holds between the static (ε0 ) and high-frequency (ε∞ ) dielectric constants within the Harrison model [28], ε0 − 1 =1+υ ε∞ − 1

∂ε∞ ε∞ − 1 = −η ∂P 3B

(26)

3. Results and discussion 3.1. The band structure and applied pressure Fig. 1 shows the electronic energy band structure calculated for zinc blende AlAs at the high-symmetry points and along the high-symmetry direction in the Brillouin zone at ambient and 120 kbar pressure. The top of the valence band is taken as zero energy. At normal pressure, the material of interest is an indirect band-gap semiconductor with the lowest minimum conduction band at the X valley, as is well known. The overall features of the band structure are similar to those of III–V zinc blende compound semiconductors [14,16]. The full valence band width is estimated to be about 11.6 eV. Applied pressure affects the electronic band structure of AlAs (Fig. 1, dashed lines). All the relative

(22)

ε∞ at zero pressure is taken to be 8.16 [5], whereas υ is given by [28], υ=

α2P (1 + 2α2C ) 2α4C

(23)

αC is the covalency of the compound defined as, α2C = 1 − α2P

(24)

(25)

where η = 2(1 − 3α2P )

d d α = √ (C11 + 3C12 ) + √ (1.473 SC0 ) 3 3 3

S=

The function ∂ε∞ /∂P has been obtained according to the following expression derived in Ref. [28], using the definition of the bulk modulus B [29],

Fig. 1. Band structure for AlAs.

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positions of the valence bands are shifted downwards. The shift of the first conduction band is downwards at the X point but, upwards at the Γ point. This indicates that the material AlAs remains an indirect band gap (Γ → X) semiconductor even at 120 kbar pressure. In addition, the full valence width increases from ∼11.6 to ∼12.3 eV as the pressure increases from 0 to 120 kbar. In general, as the pressure increases, the lattice constant decreases and hence, the valence band width increases. This trend can be ascribed to a reduction of hybridization with the increase of separation between atomic constituents. It should be noted as well that the antisymmetric gap (the gap between the first and the second valence bands at the X point) is affected by pressure which leads to a reduction of this gap. Since ionic materials presumably have larger gaps than the covalent ones [30], our result suggests an enhancement of the covalent character of the material of interest at 120 kbar pressure. Such behavior is consistent with that of the ionicity in semiconductors under pressure. The change in the energy band gaps is often presented against the ratio a/a0 (where a = aP − a0 and aP and a0 are the lattice constants at pressure P and at zero pressure, respectively). The dependencies of the direct and indirect band gaps, namely Eg , EgX and EgL on a/a0 is plotted in Fig. 2. The solid lines represent the results of leastsquares fits to the data obtained using the following quadratic relation, ∗

E =a +b

∗



a a0

+c

∗



a a0

2 (27)

The resulting pressure coefficients b* and c* from the quadratic fits are presented in Table 3. The behavior of energy band gaps against a/a0 is clearly non-linear. This behavior is, qualitatively similar to that reported in an earlier work [16] on other III–V semiconductor compounds.

Fig. 2. Direct and indirect band gaps in AlAs as a function of a/a0 .

91

Table 3 Calculated pressure coefficients of Eg , EgX and EgL energy band gaps for AlAs Band gap

b* (eV)

c* (eV)

Eg EgX EgL

−19.28

174.84

3.50

−22.21

−6.43

58.28

3.2. The effective masses and applied pressure The effective mass, which is strongly connected to the mobility of carriers, is known to be one of the most important spectral parameters. Its value at principal band extrema provides useful information about the properties of semiconductors. In the case of AIII BV compounds, the shape of the conduction-band relationship E(k) is nearly parabolic close to Γ and X extremum. This is also apparent in Fig. 1. So, one scalar electron effective mass m∗e may be used in the present calculations. This effective mass is defined as, 1 1 ∂2 E(k) = 2 ∗ me ∂k2 h ¯

(28)

The effective mass has been calculated from the band curvature at different pressure values. Calculations were carried out at the Γ and X valleys of the conduction-band minimum and at the Γ valley of the valence band maximum. The values of m∗e obtained for AlAs at normal pressure are 0.15 m0 at the Γ valley and 0.25 m0 at the X valley (m0 being the free electron mass). The value of the electron effective mass found at the Γ point of the Brillouin zone is in excellent agreement with most experimental and theoretical values [1,31,32]. The longitudinal and transverse effective electron masses at the X point of the Brillouin zone in AlAs reported in Ref. [33] at 300 K are 1.1 and 0.19 m0 , respectively. The value of the transverse electron effective mass is in reasonable accord with the calculated one of 0.25 m0 obtained in the present work at the X valley. The heavy-hole effective mass found in this work as 0.48 m0 agrees also with the value of 0.5 m0 determined by optical absorption measurements at 6 K [34]. The effect of applied pressure up to 120 kbar on the effective masses was examined. Calculations in this respect have shown that both the electron effective mass at the X valley and the heavy-hole effective mass are not so sensitive to the pressure applied over the range 0–120 kbar. The values found for these masses at ambient pressure have not shown any significant change with increasing pressure. However, the electron effective mass at the Γ valley has been found to depend non-linearly on the ratio a/a0 as can be seen in Fig. 3. This behavior can be fitted with the quadratic polynomial,



(m∗e ) a 2 a + 47.86 = 0.15 + 0.42 (29) m0 a0 a0 The ratio a/a0 in this polynomial is negative due to the fact that the applied positive pressure reduces the value of a. Consequently, the electron effective mass at the Γ valley

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N.Y. Aouina et al. / Materials Science and Engineering B 123 (2005) 87–93

Fig. 3. Effective mass of the electron (in units of the free electron mass m0 ) at the Γ point of the Brillouin zone in AlAs as a function of a/a0 .

increases with the increase of applied hydrostatic pressure up to 120 kbar. This means that a high applied pressure can reduce the electron mobility in the material of interest. As for the dielectric constant ε0 , the numerically calculated values at various pressures up to 120 kbar are given in Table 4. It can be noticed that ε0 decreases monotonically with increasing pressure. The calculated value for ∂␧0 /∂P is found to be −7.64 Mbar−1 . 3.3. The elastic constants, stability criteria and the applied pressure The elastic constants C11 , C12 and C44 have been calculated at various pressure values using the relations (12)–(14). The values of these calculations at zero pressure are 12.35, 5.35 and 5.75 (1011 dyn/cm2 ), respectively. Krieger et al. [35] reported experimental values of 11.99 ± 0.12, 5.75 ± 0.13 and 5.66 ± 0.07 (1011 dyn/cm2 ), respectively, while recent experimental results of Gehrsitz et al. [36] were 11.93 ± 0.07, 5.72 ± 0.1 and 5.72 ± 0.04 (1011 dyn/cm2 ), respectively. It can be seen that a very good agreement exists between calculated elastic constants and published experimental results. Table 4 Calculated static dielectric constant (ε0 ) of AlAs at various pressures Pressure (kbar)

ε0

0 30 60 90 120

10.06 9.83 9.60 9.37 9.14

Fig. 4. Generalized stability criteria M1 , M2 and M3 as a function of pressure for AlAs.

This makes it possible to proceed with the mechanical stability criteria. Fig. 4 shows the variation of the generalized stability criteria M1 , M2 and M3 with applied pressure up to 12 Gpa (120 kbar) for the zinc blende structure of AlAs. This figure demonstrates that both, the spinodal stability criterion M1 and the generalized shear stability criterion M2 , increase with increasing pressure. The Born generalized criterion M3 , however, decreases with increasing pressure. RodriguezHernandez et al. [13] have reported a qualitatively similar behavior of these criteria in the zinc blende BeSe and BeTe employing first principles calculations. Although M3 decreases with applied pressure (Fig. 4), it does not vanish at any pressure in the range studied. It may be reasonable to assume, therefore, that the material under consideration is mechanically stable against small deformations due to a hydrostatic pressure lower than 120 kbar. The elastic criteria prescribe a particular homogeneous deformation. However, in other materials, these criteria have been shown to correspond exactly to the transition [37]. The metallization transition for the material of interest has been observed at 124 kbar in bulk-like thin films by Raman spectroscopy [38], visible microscopy [39] and X-ray diffraction [40]. The transition is reversible, with a hysteresis of about 100 kbar [8]. These experimental observations are consistent with our prediction of mechanical stability of AlAs due to a hydrostatic pressure lower than 120 kbar. 4. Conclusion The effect of hydrostatic pressure on electronic properties of semi-conducting AlAs has been computed over a pressure

N.Y. Aouina et al. / Materials Science and Engineering B 123 (2005) 87–93

range of 0–120 kbar. An empirical pseudopotential method combined with the Harrison bond-orbital model have been employed to determine the elastic constants. These constants have been used to formulate the generalized stability criteria to analyze the mechanical instability of the material of interest. The calculated quantities, namely the effective masses and elastic constants agreed very well with available experimental data. The computed electronic properties are shown to exhibit a non-linear behavior with the ratio a/a0 . Generalization of the elastic stability criteria proved to be useful. The results obtained has indicated that small deformations are not expected to take place in AlAs due to pressure values lower than 120 kbar.

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