Electronic structure and optical properties of Si1−xGex alloys

Physica B 322 (2002) 225–235

Electronic structure and optical properties of Si1�xGex alloys

F. Ben Zida, A. Bhouria, H. Mejria, M. Saida,*, N. Bouarissaa,1, J.-L. Lazzarib,F. Arnaud d’Avitayab, J. Derrienb

aUnit!e de Physique des solides, D!epartement de Physique, Facult!e des Sciences de Monastir, 5019 Monastir, TunisiabCentre de Recherche sur les M !ecanismes de la Croissance Cristalline, CRMC2, UPR-CNRS 7251, Laboratoire associ!e aux Universit!es

Aix-Marseille II et III, Campus de Luminy, Case 913, 13288 Marseille cedex 9, France

Received 13 August 2001; accepted 28 February 2002

Abstract

We have investigated the compositional dependence of the electronic structure and optical properties of Si1�xGex

semiconductor alloys with composition x ranging from 0 to 1 by means of two computational methods: the linear-

muffin-tin-orbital method with the atomic-sphere approximation and the empirical pseudopotential method within the

virtual crystal approximation. To make allowance for the compositional disorder effect, this latter has been included as

an effective potential. Moreover, the effect of the deviation from Vegard’s law on the opto-electronic properties has

been discussed. A comparison between the results obtained from the two used methods with respect to available

experimental data has been made. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 71.20.Nr; 71.15�m; 71.20.b

Keywords: Electronic structure; Alloys; SiGe; Disorder; Optical properties

1. Introduction

Semiconductor alloys offer the freedom todesign material properties by choosing appropriatealloy constituents. Group-IV semiconductor alloyshave immense potential for technological applica-tions. Many kinds of electronic and opto-electro-nic devices in strained layers Si1�xGex=Siheterostructures have been realized [1]. Alloys

and ordered compounds of Si–Ge may also haveparticularly unique opto-electronic properties forapplications in quantum-well intersubband tech-nology [2]. Moreover, SiGe alloys suggest manyinteresting possibilities for the future. Severalcompanies offer already SiGe circuits on themarket with considerable performance advantagesover conventional devices [3].Silicon and germanium, both crystallizing in the

diamond lattice, form a continuous series ofSi1�xGex solid solutions with composition x

ranging from 0 to 1. By changing the alloycomposition, physical properties can be controlledat arbitrary values between those of Si and Ge.Crystalline Si1�xGex compounds, for example,

*Corresponding author. Fax: +216-3-500-278.

E-mail addresses: [email protected] (M. Said),

[email protected] (J.-L. Lazzari).1Present address: Physics Department, University of M’sila,

28000 M’sila, Algeria.

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 8 5 - 7

offer the possibility for tuning the electronicenergy band structure with the chemical composi-tion of the alloy in order to adapt the material fordevices utilizing the energy of solar photons at anoptimum [4]. The rapid demonstration of newSiGe-based devices has, in many cases, precededfundamental measurements of materials para-meters which influence device performance [5].Thus, a detailed understanding of the properties ofthe Si–Ge alloys is useful for material design and aprerequisite for successful device applications.Among the many questions concerning the

properties of the Si–Ge alloys is their opto-electronic properties. To this purpose the depen-dence of the opto-electronic properties on thecomposition parameter x assumes particular im-portance. Hence, the composition dependence ofoptical and electronic properties in SiGe semicon-ductor alloys has been investigated theoreticallyand experimentally [4–10]. However, it is wellknown that dealing with alloy properties, thedisorder effect is an important and complicateproblem at least from the theoretical point of view.The application of first principles electronic

band structure methods to the study of disorderedalloys and solid solutions requires some approx-imation for the treatment of the alloy disorder. Adirect approach is to make use of the supercellapproximation, and consider all possible config-urations for a given unit cell, for a particularcomposition x: Such calculations generally requirethe use of very large supercells in order to mimicthe distribution of local chemical environments,and tend to be computationally very demanding.Other possible approach would be to make use ofthe coherent potential approximation (CPA) andthe virtual crystal approximation (VCA). Thereare different ways of deriving the CPA and,depending on the context, each of them maypresent specific advantages [11,12]. Also, theimplementation of the VCA is neither straightfor-ward nor unique [13].Fully self-consistent KKR–CPA, LMTO–CPA

calculations have been performed and they are ingood general agreement with the available experi-mental data [12,14]. The interest of such firstprinciples calculations is obvious, but in the casewhere it can reasonably be applied. Unfortunately,

the CPA is generally not well suited for use in first-principles total-energy methods. A much simplerand computationally less expensive approach is toemploy the virtual crystal approximation. There-fore, considerable effort has been exerted towardsthe development of a VCA appropriate to firstprinciples calculations [13,15]. The foremost ques-tion that has been addressed is of the accuracy ofthe VCA approximation. This technique has seenwide use in band structure calculations [13,15,16],and has demonstrated good accuracy for the VCAin some semiconductors. Due to the small chemi-cal differences between Si and Ge atoms, the VCAapproach can be reasonably applied to SiGealloys.Recent studies on several semiconductor alloys

indicate that the VCA breaks down whenever themismatch between the electronic properties ofthe constituent atoms exceeds a certain criticalvalue [17]. To remedy this shortcoming, oneshould go beyond the VCA. In order to helpunderstand the dependence of these properties onalloy composition x; and to gain a deeper under-standing of the relevant physical parameters, wehave embarked upon a theoretical study of thisalloy system using two different types of calcula-tions. The two computational methods are (i) Theempirical pseudopotential method (EPM) withinthe VCA, (ii) The linear muffin-tin orbital(LMTO) method in the atomic sphere approxima-tion (ASA), which is used to test the first method.The LMTO calculations are done for a singleordered compound for a given composition x: Infact considering all possible configurations for agiven unit cell for a particular concentration x; andthen taking the average values for the energies atdifferent symmetry points and for each composi-tion x is for us a huge computational effort.Furthermore, by its nature, the VCA is closely tiedto the pseudopotential approximation. LMTO–ASA–VCA would also be a possibility, but it istrickier to implement, one must select carefullywhich parameters are to be VCA-averaged. Start-ing from these two methods (LMTO–ASA andEPM–VCA), we have treated the effect of compo-sitional disorder as an effective potential on goingbeyond the EPM under VCA. The object is tostudy and to reevaluate fundamental parameters in

F. Ben Zid et al. / Physica B 322 (2002) 225–235226

the Si1�xGex alloys and to examine together theresults of each type of calculation, so that aconsistent overall picture can be obtained.The rest of the paper is arranged as follows. The

computational methods used in the present calcu-lations namely the LMTO and EPM are given inSection 2. The results of the calculated opto-electronic properties of the alloy of interest andtheir interpretation are presented in Section 3.Section 4 sets out the conclusion.

2. Computational details

2.1. LMTO method

We have used the density functional theory(DFT) [18] in the local density approximation(LDA), where exchange and correlation weretreated in the formulation of von Barth and Hedin[19]. The scalar relativistic LMTO method [20] isused in the ASA including the combined correc-tion term. Empty spheres [21] were introduced inorder to have a closed-packed lattices. In thesecalculations the lattice parameter of the alloy ofinterest was obtained using Vegard’s law, wherethe Si and Ge lattice parameters are the experi-mental ones taken from Ref. [10]. The unit cell iscomposed of eight atoms (Si and/or Ge) and eightempty spheres. The radii of the atomic and emptymuffin-tin-spheres were all chosen to be equal.Forty k special points were used in the irreducibleBrillouin zone.It is well known that the band structures, which

formally are obtained from the one-particleenergies of the wave equation in the LDA forsemiconductors and insulators, have gaps that arefar too small [22] when compared to experiments.This suggests an obvious way for introducing

correction to the band structure, which yields s likeconduction bands in better agreement with experi-ment.A simple ad hoc procedure that simultaneously

corrects the gaps and the dispersion proposed byCardona et al. [23] was applied and consists ofadjusting the gaps at three symmetry points, G; Xand L by adding at the atomic positions, as well asin the empty spheres extra sharply peaked

potentials. The same form of the potential usedin Ref. [24] is taken in this work. The parameterswe have used for Si and Ge considered here is alsotaken from Ref. [24].

2.2. EPM within VCA

In carrying out the EPM calculations, we haveessentially used the local pseudopotential method(see, for example, Ref. [25]). To ensure theempirical results accurately reflected, what isknown from experiment, the fitting proceduregave a high priority to obtain close agreementfor the highest valence band and lowest conduc-tion band at the high-symmetry points.The method of optimization of the empirical

pseudopotential parameters used in the presentwork is the non-linear least-squares method [26].The dimension of our eigenvalue problem is a(136� 136) matrix. However, 59 plane wavesgenerally give a good convergence.It is straightforward to extend our treatment to

alloys through the use of the VCA. With VCA, thesymmetric and antisymmetric form factors foralloys of interest can be expressed as a function ofthose of the alloy constituents. Since Si and Geboth crystallizing in the diamond structure, theirantisymmetric form factors are zero. Hence, itremains only the symmetric form factors forSi1�xGex alloys, which can be obtained accordingto the following expression:

VSSiGe ¼ ð1� xÞVS

Si þ xVSGe: ð1Þ

The lattice constant of Si1�xGex alloy is given by

a ¼ ð1� xÞaSi þ xaGe: ð2Þ

Using this lattice constant, the correspondingreciprocal lattice vectors and the pseudopotentialform factors VSðGÞ of Eq. (1), we can calculate theband structures of Si1�xGex alloys.It is well known that the VCA does not take into

account the effect of compositional disorder andhence the bowing factors obtained by VCA maydeviate from experiments [17,27,28]. In order toovercome this shortcoming, we have added to theVCA a non-periodic potential due to the composi-tional disorder [29]. By adding this effectivedisorder potential to the virtual crystal potential,

F. Ben Zid et al. / Physica B 322 (2002) 225–235 227

expression (1) becomes,

VSSiGe ¼ ð1� xÞVS

Si þ xVSGe

� p½xð1� xÞ�1=2½VSGe � VS

Si� ð3Þ

p is then treated as an adjustable parameter.

2.3. Optical properties calculations

The two most interesting optical properties of asemiconductor are the absorption edge, or opticalenergy gap Eg; and the refractive index n: It istherefore natural that attempts should have beenmade to find a general relationship between theseparameters, both from the point of view offundamental interest and also as a technologicalaid in estimating the refractive index if onlythe energy gap is known. Many attempts havebeen made to correlate the energy band gap tothe optical refractive index of semiconductors. Thefirst proposal based on an atomic model was madeby Moss [30], on the very general grounds that allenergy levels in a solid are scaled down by a factor1=e2opt; where eopt ¼ n2 is the optical dielectricconstant. The data on the then known materialswere correlated and as a result the so-called ‘‘Mossrelation’’

Egn4 ¼ 95 eV: ð4aÞ

From an analysis of more recent data, Ravindraand Srivastava [30,31] proposed a revised value ofthe constant

(i) Egn4 ¼ 108 eV: ð4bÞ

From a study of a wide range of materials,Ravindra et al. [32] argued that the atomicmodel used by Moss limits the validity of hisequation. Therefore, they proposed a linearrelation between the refractive index n andthe energy gap Eg given by

(ii) n ¼ aþ bEg ð5Þ

with a ¼ 4:084 and b ¼ �0:62 eV�1:The latter relation was examined by Moss

[30], who pointed out that equation isrestricted to energy gaps of less than about4 eV and gives unrealistic results for both lowand high Eg: Herve and Vandamme [33]

proposed an overall relation, which is basedon the classical oscillator theory and givenby

(iii)

n ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

A

Eg þ B

� �2s

ð6Þ

with A ¼ 13:6 eV and B ¼ 3:4 eV.In the following, we will use the three relationsmentioned above that will be called (i) Moss, (ii)Ravindra et al. and (iii) Herve and Vandammerelations, respectively. Eg has been calculatedusing both LMTO and EPM methods.The high-frequency dielectric constant (eN) is

calculated using the relation [34]

eN ¼ n2: ð7Þ

3. Results

The adjusted local symmetric ðVSÞ pseudopo-tential form factors together with lattice constantsfor Si and Ge materials used in our EPMcalculations at 300K are given in Table 1.Table 2 lists the calculated direct and indirect

band gap energies for Si and Ge using the LMTOand EPM, alongside other available results fromthe literature. As can be seen from Table 2, ourEPM results agree well with those of Ref. [35] forboth Si and Ge materials showing therefore thequality of our adjusted pseudopotential formfactors. On the other hand, the agreement betweenour LMTO results and the available experimentaldata cited in Ref. [36] (at low temperatures) isreasonable.Using LMTO method in the ASA and the EPM

within the VCA, we have calculated the direct andindirect energy band gaps, namely EG

g ; EDg ; EX

g and

Table 1

Pseudopotential parameters for Si and Ge

Form factors (Ry)

Material VS ð3Þ VS ð8Þ VS ð11Þ Lattice constant

( (A)

Si �0.231074 0.05656 0.068254 5.431

Ge �0.294588 0.074802 0.001723 5.6579


ELg for Si1�xGex alloys as a function of Ge

concentration x in the range 0–1. The curves thatfit the calculated LMTO and EPM under VCAdata are plotted in Figs. 1a and b, respectively.The obtained analytical expressions are as follows:

EGg ðxÞ ¼ 2:4� 2:75x þ 1:30x2 ðLMTOÞ: ð8Þ

EGg ðxÞ ¼ 3:41� 1:01x � 1:69x2 ðEPMþ VCAÞ: ð9Þ

EGg ðxÞ ¼ 3:51� 4:35x

þ 1:44x2 ðEPMþ improved VCAÞ: ð10Þ

EDg ðxÞ ¼ 1:01� 0:15x � 0:11x2 ðLMTOÞ: ð11Þ

EDg ðxÞ ¼ 1:07þ 0:20x � 0:29x2 ðEPMþ VCAÞ:

ð12Þ

Table 2

Band-gap energies of Si and Ge

Material Band-gap energy (eV)

EGg ED

g EXg EL

g

Si 3.33a 1.07a 1.19a 2.00a

2.40b 1.01b 1.08b 2.00b

3.4c 1.12c 1.2c 2.0c

1.17d

Ge 0.78a 0.99a 1.10a 0.70a

0.95b 0.76b 0.85b 0.67b

0.8c 1.2c 0.66c

0.0898(1)e 0.744(1)e

aPresent work, using EPM (at 300K).bPresent work, using LMTO.cRef. [35] (at 300K).dRef. [36] (at 0K).eRef. [36] (at 1.5K).

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ene

rgy

gap

(eV

)

(a)

EΓ∆

EΓ

X

EΓ

L

EΓΓ

Composition x

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5(b)

Ene

rgy

gap

(eV

)

EΓ∆

EΓX

EΓ

L

EΓΓ

Composition x

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

EΓ∆

EΓ

X

EΓ

L

EΓ

Γ(c)

Ene

rgy

gap

(eV

)

Composition x

Fig. 1. Direct and indirect band-gap energies in Si1�xGex as a function of alloy composition x (a) using LMTO method (b) using EPM

within VCA (c) using EPM within improved VCA.


EDg ðxÞ ¼ 1:08� 0:34x


EXg ðxÞ ¼ 1:08� 0:28x

þ 0:05x2 ðLMTOÞ: ð14Þ

EXg ðxÞ ¼ 1:19þ 0:23x

� 0:32x2 ðEPMþ VCAÞ: ð15Þ

EXg ðxÞ ¼ 1:20� 0:33x


ELg ðxÞ ¼ 2:06� 1:33x ðLMTOÞ: ð17Þ

ELg ðxÞ ¼ 2:06� 1:32x ðEPMþ VCAÞ: ð18Þ

ELg ðxÞ ¼ 1:83� 1:36x ðEPMþ improved VCAÞ:

ð19Þ

The unit of the energy is eV. The quadratic termsare referred to as band gap bowing factors. FromEqs. (11) and (12), one can notice that the bowingparameters are negative (upward bowings). The fitof experimental data of Weber and Alonso [10] inthe composition range 0pxo0:85 by a least-squares procedure showed that the bowing factorof ED

g is 0.206 eV. This value is in disagreementwith those of LMTO and EPM within VCAcalculations. This discrepancy is essentially attrib-uted to the neglect of the alloying disorder effect.On the other hand, the LMTO calculations showthat the absorption at the fundamental optical gapis indirect ðG-DÞ in the range 0–0.98 and indirectðG-LÞ in the range 0.98–1.0 indicating thereforethat the crossover point occurs at x ¼ 0:98 (for0K) which is in disagreement with the experiment[10]. This shows that neither the ab-initio methodnor the semi-empirical one used in our calculationsfor SiGe has given good agreement with theexperiment in the absence of the alloying disordereffect in the calculations. Krishnamurthy et al. [7]have shown that the incorporation of bothchemical and structural disorder into the calcula-tion of the CPA band structure of Si1�xGex alloysleads to an L–X(D) crossover taking place nearxE0:87; which agrees very well with the experi-mental data. In our case, the treatment of the

compositional disorder effect needs methods thatgo beyond the LMTO–ASA calculations done fora single order compound. This would be possible,but it is somewhat more demanding computation-ally and tricky to implement. Therefore, we haveemployed the EPM within the VCA and added anon-periodic potential due to the compositionaldisorder according to relation (3). This hasimproved significantly the value of the bowingparameter of ED

g with respect to experiment. Thisvalue is reached for p ¼ �0:46: Unfortunately, thecrossover from the Si-like D conduction-bandminimum to the Ge-like L conduction-bandminimum occurs at x ¼ 0:65 (Fig. 1c) which islower than the experimental one. This discrepancylies in the fact that our adjustment of the p

parameter was only done for EDg : The deviation

from Vegard’s law for the lattice constant couldalso affect our results. We have checked this effectby using relation [10]

aðxÞ ¼ 5:431� 0:1992x þ 0:02733x2 ð (AÞ: ð20Þ

We found that there is no significant effectregarding the crossing point.In Figs. 2a–c, we show the electron energy-band

structure of Si0:50Ge0:50 using LMTO, EPM withinVCA alone and EPM within VCA where thedisorder effect is included, respectively. Accord-ingly, the main difference with the band structureof Si or Ge [6,7] is, of course, the increased ordecreased fundamental gap, generally however, theband structure remains topologically similar.Moreover, the valence bands are qualitativelysimilar to those of Ref. [7]. The top of the valenceband consists of the triply degenerate G15: The pstates of Si and Ge form the upper-valence band.The s states of Si and Ge form a band at about12.29 eV below the top of the valence band(according to the EPM within VCA calculations).The lowest conduction band stems from theinteraction of the Si conduction band states withthe Ge-derived s states. The lowest gap betweenthe top of the valence band to the bottom of theconduction band in Si0:50Ge0:50 is predicted to beindirect associated with the G� D transition,whatever the method used in our calculations. Itshould be noted that the conduction bands aremore dispersive and band crossing than the


valence ones. The reason lies in the fact that theconduction bands are more delocalized and more‘free-electron like’ than valence states. A compar-ison between Figs. 2b and c shows that the relativepositions of conduction bands seems to be moreaffected by the inclusion of the compositionaldisorder than the valence ones. Thus, the majorcompositional disorder lies in the conductionband, whereas the topmost valence band, with itsrich p content is least affected. This agrees wellwith the results of Krishnamurthy et al. [7].According to the authors of Ref. [7], the fact thatthe relative positions of conduction bands aremore affected by the chemical disorder effect ascompared to the valence ones in SiGe semicon-ductor alloys is due to the s scattering which isdominant in these alloys. On the other hand, it isto be noted here that the general shape of theelectronic band structures of the alloy of interestcalculated using different methods (Figs. 2a–c) issimilar except that the bands calculated using

LMTO method (Fig. 2a) are more dispersive thanthose obtained using EPM within VCA andimproved VCA (Figs. 2b and c). The reason forthat lies in the fact that the alloy Si0.5Ge0.5 whichassumed to have a diamond structure in the EPM–VCA calculations has in fact lower symmetrystructure that was considered in the LMTOcalculation. The same reason could also beattributed to the noticed difference betweenFigs. 2a–c, where a gap between the first and thesecond valence bands at the high-symmetry pointX appears when the LMTO method is used(Fig. 2a), whereas it is absent in both cases ofEPM calculations (Figs. 2b and c). This ‘antisym-metric’ gap has been proposed as a measure of thecrystal ionicity [37]. Therefore, the results ofLMTO method indicate that the crystal ionicityof the equimolar alloy Si0:50Ge0:50 is not purelycovalent, whereas those of EPM within VCA andimproved VCA show that Si0:50Ge0:50 are purelycovalent. Our EPM results are consistent with

-15

-10

-5

0

5

10

15

ΓxL Γ

(a)

Ene

rgy

(eV

)

-15

-10

-5

0

5

10

15

∆

(b)

LxΓ

Ene

rgy

(eV

)

-15

-10

-5

0

5

10

15

Lx∆Γ

(c)

Ene

rgy

(eV

)

Fig. 2. Band structure for Si0:50Ge0:50 (a) using LMTO method (b) using EPM within VCA (c) using EPM within improved VCA.


those of Refs. [6,7] which showed that the value ofthe antisymmetric gap is equal to zero. Due tothe diamond structure of Si0:50Ge0:50 assumed inthe EPM–VCA calculation, no conclusion could beachieved about the disorder effect on the antisym-metric gap and hence on the crystal ionicity. Thisis not the case for zinc-blende semiconductoralloys, such as Ga0:50In0:50N; where it has beenfound that the antisymmetric gap is slightlyaffected by the presence of the compositionaldisorder [38].Using LMTO, EPM within VCA and EPM

within improved VCA, the refractive index (n) hasbeen calculated and plotted against Ge concentra-tion x in Figs. 3a–c, respectively, for the threedifferent models described in Section 2. Ourcalculated data for n are fitted by a least-squaresprocedure. The analytical expressions are as

follows:

(i) Moss relation:nðxÞ ¼ 3:22þ 0:07x þ 0:18x2 ðLMTOÞ: ð21Þ

nðxÞ ¼ 3:20� 0:49x

þ 0:72x2 ðEPMþ VCAÞ: ð22Þ

nðxÞ ¼ 3:23� 0:57x þ 1:19x2

ðEPMþ improved VCAÞ: ð23Þ

(ii) Ravindra et al. expression:

nðxÞ ¼ 3:46þ 0:08x þ 0:08x2 ðLMTOÞ: ð24Þ

nðxÞ ¼ 3:44� 0:34x


nðxÞ ¼ 3:45� 0:17x þ 0:51x2


0.0 0.2 0.4 0.6 0.8 1.03.0

3.2

3.4

3.6

3.8

4.0 Moss Ravindra et al. Herve & Vandamme

(a)

Ref

ract

ive

inde

x

Composition x

0.0 0.2 0.4 0.6 0.8 1.03.0

3.2

3.4

3.6

3.8

4.0

(b) Moss Ravindra et al. Herve & Vandamme

Ref

ract

ive

inde

x

Composition x

0.0 0.2 0.4 0.6 0.8 1.03.0

3.2

3.4

3.6

3.8

4.0 Moss Ravindra et al. Herve & Vandamme

(c)

Ref

ract

ive

inde

x

Composition x

Fig. 3. Refractive index as a function of alloy composition x in Si1�xGex calculated from different relations (a) using LMTO method

(b) using EPM within VCA (c) using EPM within improved VCA.


(iii) Herve and Vandamme relation:

nðxÞ ¼ 3:24þ 0:08x þ 0:11x2 ðLMTOÞ: ð27Þ

nðxÞ ¼ 3:22� 0:38x


nðxÞ ¼ 3:24� 0:26x þ 0:66x2


Through Figs. 3a–c and Eqs. (21)–(29), we notethat the refractive index increases non-linearlywith increasing Ge concentration x from 0 to 1independently from the used methods and models.However, it seems that the refractive index bowingparameters (quadratic terms) calculated accordingto the LMTO and EPM within VCA methods areless important than those obtained using the EPMwithin improved VCA. This is an expected resultsince the last method includes the alloying disordereffect. On the other hand, we point out that theincrease of Ge content in Si1�xGex has a moreimportant effect on n when Moss relation based onan atomic model is used. As the fundamental bandgap energy decreases on going from Si to Ge(Figs. 1a–c), we note that in the alloy Si1�xGex thesmaller the fundamental band gap energy materialhas a large value of the refractive index. Such abehavior has been also observed for III–Vcompound semiconductor alloys [39]. The fittedexperimental data of the refractive index whichare independent of the photon energy for the SiGealloys reported by Humlicek in Ref. [10] wasgiven as

nðxÞ ¼ 3:42þ 0:37x þ 0:22x2: ð30Þ

This relation agrees better with our fitted expres-sions when using the Ravindra et al. relationwhatever the method used. This can be explainedas the moss relation, which is based on an atomicmodel, uses only one parameter for all materials,while the Ravindra et al. and Herve and Van-damme relations which are based on a morereasonable model, use a set of two parameters. Onthe other hand, Herve and Vandamme formula isderived for a wide range of materials, while theRavindra et al. formula is more specific forsemiconductors.Another interesting optical property is the high-

frequency dielectric constant (eN). This latterhas been obtained for Si1�xGex according torelation (7) given in Section 2, using LMTOand EPM calculations. Our results for eN atvarious compositions x (0pxp1) for Si1�xGex

alloys are presented in Tables 3 and 4. In view ofthese tables, one can notice once again that theRavindra et al. expression gives better resultswith respect to the experiment as compared toMoss and Herve and Vandamme relations forboth used methods. The agreement betweenour results and the experimental ones is reasonablefor Si, however, for Ge it is not as good as forSi. Interestingly as well, we note through Table 4that eN is sensitive to the effect of compositionaldisorder in Si1�xGex since our EPM results withinVCA differ from those of EPM within theimproved VCA. Moreover the inclusion of thecompositional disorder has increased the valuesof eN: According to the experimental values of Siand Ge reported in Table 4, we do think that these

Table 3

The calculated high-frequency dielectric constants (eN) for Si1�xGex alloys using LMTO method compared with available

experimental data

Material eN

Moss relation Ravindra et al. relation Herve and Vandamme relation Experiment

Si 10.34 11.96 10.51 12.1a

Si0:7Ge0:3 10.63 12.19 10.75 —

Si0:50Ge0:50 10.91 12.40 10.97 —

Si0:3Ge0:7 11.26 12.65 11.23 —

Ge 12.70 13.46 12.17 16.5a

16.0a

aRef. [36] (at 4.2K).


values are better than those obtained by VCAalone.

4. Conclusion

In conclusion, we have presented a theoreticalstudy of the electronic structure and opticalproperties in Si1�xGex semiconductor alloys withcomposition x ranging from 0 to 1 using twodifferent theoretical approaches. Our results showgenerally reasonable agreement with the availableexperimental data. The variation trend of theopto-electronic properties with composition x isgenerally found to be similar for both LMTO andEPM within improved VCA methods. Before thecrossover, the minimum gap is found to be the Dgap. After the crossover, the minimum gap is the Lgap. While the LMTO method gives a weakðG-DÞ band gap bowing parameter as comparedto experiment, the value of the crossover pointobtained by this method is found to be over-estimated. However the EPM under the improvedVCA that gives the experimental bowing factorunderestimates the crossing point. The computa-tion of the electronic band structure of Si0:50Ge0:50using LMTO and EPM methods shows generally asimilar shape of the band structures except that theLMTO calculated bands are more dispersive and

show an antisymmetric gap at X point. Thecalculated high-frequency optical dielectric con-stant for Si is in reasonable agreement withexperiment which is not the case for Ge. It isfound that Ravindra et al. expression gives betterresults with respect to experiment as compared toother used relations. Furthermore, the high-frequency optical dielectric constant in Si1�xGex

alloys is found to be slightly sensitive to the effectof compositional disorder.

Acknowledgements

One of us (N.B) would like to thank the Facultyof Science of Monastir, Monastir, Tunisia, inparticular the U.R. of Solid State Physics, for theirfinancial support and kind hospitality during hisstay at Monastir. Part of this work has been doneat the International Center for Theoretical Physics(ICTP) Trieste, Italy. N.B and M.S gratefullyacknowledge the ICTP support during their stay atthe Center.

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Table 4

The calculated high-frequency dielectric constants (eN) for Si1�xGex alloys using EPM compared with available experimental data

Material eN

Moss relation Ravindra et al. relation Herve and Vandamme relation Experiment

Si 9.53 11.20 9.78 11.7c

11.9d

Si0:7Ge0:3 9.89a 11.56a 10.12a —

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Electronic structure and optical properties of Si1−xGex alloys