- Email: [email protected]
First-principles study of Raman intensities in semiconductor systems
Computational Materials Science 20 (2001) 363±370
www.elsevier.com/locate/commatsci
First-principles study of Raman intensities in semiconductor systems Pasquale Pavone *, Bernhard Steininger, Dieter Strauch Institut f ur Theoretische Physik, Universit at Regensburg, 93040 Regensburg, Germany Received 15 February 2000
Abstract We present a general method for the theoretical determination of the o-resonance Raman activity of semiconductor mixed systems such as substitutional alloys or superlattices, with the inclusion of strain eects. As basic tool, the densityfunctional perturbation theory has been used. The calculation of the Raman cross-section requires the knowledge of the lattice dynamics of the complete system. The description of semiconductor alloys or long-period superlattices requires large supercells, this makes the numerical eort of a direct ab initio calculation an unfeasible task. Simple procedures using the virtual-crystal (VCA) and the mass approximation fail in describing the eect of both the polarizability variations and the change in the local environment due to the local strain. We used a perturbative scheme which includes the potential ¯uctuations responsible for chemical disorder and the variations of the atomic Raman tensors. The method is applied to some III±V and Si/Ge semiconductor systems. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Much interest is presently devoted to the theoretical and experimental investigation of semiconductor alloys and superlattices, especially in connection with the electronic and lattice-dynamical properties of the corresponding bulk materials [1]. Due to the small size of the samples, methods of optical spectroscopy such as Raman scattering are found to be much more ecient tools in the characterization of mixed crystals than neutron spectroscopy. A complete theoretical description of the Raman spectra of semiconductor alloys has not been available for a long time. Recently, it has been shown * Corresponding author. Tel.: +49-941-943-2048; fax: +49941-943-4382. E-mail address: [email protected] (P. Pavone).
that well-de®ned phonons for systems with topological disorder due to local strain can be eciently calculated by ab initio methods using the anharmonic force constants of a reference periodic system [2]. However, the calculation of Raman intensities of alloys has long escaped any ab initio approach due to the huge computational eort. Consequently, all the previous work based on ®rst-principles calculations completely neglected the dierence in the Raman tensor of the constituents [2,3]. In this work, we present a method based on ®rst-principles calculation which allows the evaluation of the ®rst-order o-resonance Raman spectra of mixed semiconductor systems. The method has general validity and can be used in all those cases where the deviation from a perfect periodic system can be treated within perturbation theory. We apply this method to the calculation of Raman spectra of various III±V and Si/Ge alloys
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 1 9 5 - 6
364
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
and superlattices. We demonstrate the eciency of the method for mixed strained systems with relative dierences in the lattice parameter of the constituents up to about 3%. 2. Theoretical framework Raman scattering of light by lattice vibrations of a crystal depends on the modulation of the electronic susceptibility Dv by phonons [4]. In fact, the o-resonance Raman cross-section can be expressed to the ®rst order as [5] X d x ÿ xm e Dvm eI 2 ; x / 1 rIF R F xm m where eI (eF ) is the polarization vector of the incoming (scattered) photon, xm the phonon frequency of the mode m at the C point of the Brillouin zone, and the susceptibility variation Dvm is given by ovab X ovab ouc RS Dvmab onm ouc RS onm c;RS
X c;RS
Rabc RS
ouc RS ; onm
2
where nm is the normal coordinate of the mode m, Rabc RS de®nes the ®rst-order Raman tensor, and uc RS is the displacement in c direction of the atom at the equilibrium position RS . To our purposes, it is convenient to de®ne the position vectors RS in the ``super'' structure (alloy or superlattice) in terms of the Bravais (R) and basis sj vectors of a reference periodic system (in our case the zinc-blende structure) as RS R sj wj R;
3
where wj R is the ``local'' deviation from the reference structure. The lattice-dynamical quantities involved in the calculation of the cross-section are the eigenvectors and eigenfrequencies of the dynamical matrix of the system, which is essentially the Fourier transformation of the interatomic force-constants matrix U RS ; R0S ÿ
oF RS : ou R0S
4
The calculation of the interatomic force constants in Eq. (4) for semiconductor alloys or long-period superlattices requires the use of large supercells, this makes a direct ®rst-principles calculation an almost unfeasible task, due to the huge numerical eort. Therefore, alternative procedures must be used. The lowest order approximation to the calculation of the interatomic force constants in large mixed systems is the so-called virtual-crystal approximation (VCA), which assumes the real (disordered or super-periodic) crystal to be substituted by a ®ctitious periodic system whose components are a sort of ``averaged'' (virtual) atoms according to the macroscopic composition of the system. A schematic representation of a virtual crystal is shown in Fig. 1. Two main approximations are included in the VCA: (i) due to the dierent chemical behavior and the dierent bulk lattice constants the position of one atom in the actual system diers from the virtual crystal one by a displacement wj R; (ii) equivalent lattice sites in the zinc-blende structure are occupied by the same type of atom. Therefore, the actual con®guration of a mixed system is exactly determined from that of the reference zinc-blende system when speci®ed the set of local deviation fwg and the type of the atom occupying each site.
Fig. 1. Schematic representation within the VCA (left panel) of a disordered crystal made up of two atom types (right panel).
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
In order to go beyond the VCA, both eects of local displacements and of variation in the site occupation should be included in the description of the system. To de®ne the site occupation, we introduce at each site a ``con®guration'' Ising-like variable u RS . Such variables are chosen to vanish for a given virtual-crystal reference con®guration. As an example, for a zinc-blende mixed 2 system Cat 1 x Cat1ÿx An with con®gurational disorder only at the cationic site the fu RS g can be de®ned, using as reference the virtual atom with symmetric composition, as 8 0 if R sjA is an anionic site > > > > 1 if R sjC is occupied by Cat 1 > < if R sjC is occupied by the uj R 0 1 2 > > Cat virtual atom Cat > 0:5 0:5 > > : ÿ1 if R sjC is occupied by Cat 2 5 Non-integer values of uj R indicate the presence at the site R sj of a virtual atom with a dierent composition than the reference one. Within this description, the main ingredients for the calculation of the Raman cross-section can be explicitly considered as function of the local deviations and the con®guration variables: fwgfug
U RS ; R0S Ujj0 Rabc RS
R; R0 ;
6
365
where j0 means that all derivatives are calculated for {w} 0 and fug 0, i.e., for the reference virtual crystal, and where oFj R 0 9 Njj0 R ÿ R ouj0 R0 0 is the force-constant matrix of the virtual crystal. Furthermore, in Eq. (8) we have explicitly used the formal equivalence between w and u in performing the derivatives of the force. The knowledge of the force constants as well as the mass con®guration allows the calculation of the phonon frequencies and eigenvectors in Eq. (2). Within this context, the use of the virtualcrystal force constants and the correct mass con®guration de®ne the so-called mass approximation. The expansion of the Raman tensor to the ®rst order in fug and {w} reads X 0 fwgfug 0 0 Tjj Rabc;j R Bjabc abc R ÿ R uj0 R
X d;j0 ;R
j0 ;R
2 jj0
Rabcd R ÿ R0 wd;j0 R0 ;
10
where Bjabc is the Raman tensor of the reference virtual crystal and o2 vab jj0 0 11 Tabc R ÿ R : ouc;j R ouj0 R0 0
fwgfug Rabc;j R:
7
Within a perturbation scheme the interatomic force constants and the Raman tensors can be expanded in terms of both the local deviations and the con®guration variables. For the interatomic force constants, we obtain at the ®rst order in {w} and fug [2] fwgfug Ujj R; R0 Njj0 R ÿ R0 0
o2 Fj R 00 ÿ 0 00 wj00 R 0 R ouj00 R ou j 00 j00 R 0 2 X o Fj R 00 00 R ÿ 0 00 uj00 R ; 0 R ou R ou j j00 j00 X
0
8
The last term in the RHS of Eq. (10) depends explicitly on the second-order Raman tensor, 2jj0 Rabcd R, of the virtual system. The contribution of this term vanishes in absence of local strain as, e.g., in the case of GaAs/AlAs mixed systems. In the next, this term will be neglected also for the other studied system, retaining, in practice, localstrain eects only in the calculation of the force constants. The calculation of the ®rst-order terms in Eqs. (8) and (10) requires the knowledge of expansion coecients with the same symmetryselection rules of the perfect periodic virtual crystal. In fact, the number of the independent elements in the linear-order terms of Eqs. (8) and (10) can be strongly reduced by symmetry considerations. However, the presence in the
366
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
expansions of the sum over the atoms makes an explicit calculation of all the independent coecients practically unfeasible. On the other hand, it is reasonable to assume coecients involving pairs of ``distant'' atoms to have a very small in¯uence on the properties in which we are interested. This leads to the possibility of describing our systems within dierent levels of approximation by simply retaining in Eqs. (8) and (10) only the terms which involve atom pairs with distance within a chosen reference value. In practice, we indicate with Mn and Pn the approximation consisting in dropping all the terms involving pairs with distance larger than the nth-nearest-neighbors one in the expansion coecients in Eqs. (8) and (10) for the force constants and the Raman tensor, respectively. In this work, we consider the validity of the approximations up to the second-nearest-neighbors distance. In order to further limit the number of independent parameters we restrict the ``model'' M2 to include only terms referring to triplets of sites that contain at least one ®rst-nearest-neighbor pair. This restriction has been shown to be quite accurate for the description of the interatomic force constants of Si/Ge systems [2]. Furthermore, in polar materials, due to dipolar interactions associated with non vanishing eective 0 charges, the Tjj abc R de®ned in Eq. (11) are not short ranged. However, only dipolar contributions survive at large jRj and the corresponding term in the expression of the Raman tensor can be analytically parametrized. The inclusion of this parametrization is implicitly considered in any model Pn for distances beyond the nth-nearest-neighbors one. The resulting eect of this term is, anyway, very small. 3. Technical ingredients Our calculations are carried out within the density-functional theory together with the localdensity approximation. To describe the electron± ion interaction, we used norm-conserving pseudopotentials. The electronic wave functions are expanded up to a kinetic-energy cuto of 16 and 12 Ry for the calculation of the force constants
and of the Raman tensors, respectively. A lower kinetic-energy cuto for the calculation of Raman tensor is allowed because we are more interested in the relative than the absolute intensity of the features in the Raman spectra. The description of the virtual atoms is achieved by mixing the pseudopotentials of the corresponding constituent atoms. The force constants of the reference virtual crystal are obtained using the density-functional perturbation theory [6,7], further details of the calculation of dynamical properties are found in Ref. [7]. We simulated alloys using a cubic 512 atom supercell with randomly distributed atoms at the atomic sites [3]. The coecients in the RHS of Eqs. (8) and (10) are extracted from ®ts of the corresponding equations to a number of independent con®gurations. For the force-constants coecients we have ®tted Eq. (8) to direct ab initio calculations of the force constants of: (i) the virtual crystal with a displaced atom in the unit cell and under the two independent distorsions de®ned by a (0 0 1) and (1 1 1) macroscopic strain; (ii) the two bulk constituents and the virtual crystal at the atomic positions of the virtual crystal. For the Raman-tensor coecients, we have ®tted Eq. (10) to ab initio calculation of the Raman tensor for three short-period superlattices, i.e. Cat 1 An 1 n Cat 2 An 2 n for n 1; 2 in the (0 0 1) direction and for n 2 in the (1 1 0) one. The Raman tensor is obtained by ®nite-dierence calculations of the derivative of the polarizability tensor with respect to an atomic displacement. Dynamical matrices and forces on atoms are calculated from interatomic force constants obtained following Eq. (8). Atomic positions are relaxed iteratively until equilibrium is reached. Phonon frequencies and eigenvectors are then obtained for a given con®guration by standard diagonalization of the dynamical matrix at equilibrium. Finally, the Raman cross-section is calculated following Eqs. (1), (2), and (10) for the equilibrium con®guration. To reduce statistical errors, ®ve random con®gurations are considered for each composition. In order to take into account the experimental resolution, the calculated intensity is convoluted with Gaussians with a full width at half maximum of 2 cmÿ1 .
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
367
4. Results and Discussion
4.2. Gax Al1ÿx As; Gax Al1ÿx Sb, and Ga1ÿx Alx P
4.1. Six Ge1ÿx
In the right-hand panel of Fig. 2, we report the experimental and theoretical Raman spectrum of the alloy Ga0:55 Al0:45 As in the M0; P0 and M0; P2 approximation. Due to the very small lattice mismatch and the chemical similarity between GaAs and AlAs, we use the simple mass approximation (model M0) for the determination of the dynamical matrix of the alloy. Also in this case, the inclusion of the eect of polarizability changes ± as in the model P2 ± is shown to have a very strong eect on the relative intensity of the Raman peaks. The larger polarizability of the Ga than the Al atoms has as direct eect, in comparing with the P0 result, the lowering of the intensity of the AlAs-like peak and a contemporary increase of the intensity of the GaAs-like feature. For this alloy, our calculations extend the results of a previous theoretical work [3], where it is found a general excellent agreement with the experimental position of the features in the spectra, failing, however, in predicting the correct relative intensities of the two main peaks.
The Raman spectra of the Six Ge1ÿx alloy are characterized by the presence of three strong peaks corresponding to the dierent bond vibrations. The picture of Six Ge1ÿx systems as prototype random alloys with no relevant clustering of ordering eects is supported by experimental investigation [8]. In the left panel of Fig. 2 we report the results of our calculation of rxy R x within the M0; P0 and M2; P2 models at the same composition x 0:45 as the experimental work of Ref. [9]. The position of the experimental Si±Ge peak is very well reproduced by both models. However, the position of the remaining features and their relative intensity are better described by the model M2; P2. The comparison with the previous theoretical work of de Gironcoli [2], where the analogue of the model M2; P0 was used, shows, for this alloy, the relevance in the determination of relative intensities of taking into account the polarizability dierence of the constituent atoms.
Fig. 2. Raman cross-section rxy R x calculated within the M0; P0 and M2; P2 approximations for Si0:45 Ge0:55 (left panel) and Ga0:55 Al0:45 As (right panel) in comparison with the experimental data of Refs. [9] and [10], respectively.
368
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
In the case of (Ga/Al)Sb systems, the local strain is expected to have very small in¯uence on the Raman spectra, due to the small relative difference in the lattice parameter of the bulk constituents (about 0.7%). In fact, as it is reported in the left-hand panel of Fig. 3, both the models M0 and M2 yield very similar results in overall good agreement with experiment. For these systems, already the model P1 gives a satisfactory description of the relative intensities, in good agreement with the experimental (unpolarized) results. Notice that for the cross-section in the xy con®guration the selection rules predict the presence of only longitudinal optical phonons. The calculation of the Raman spectra of the Ga1ÿx Alx P alloy presents spurious local-strain effects related to the overestimation of the theoretical mismatch between the bulk lattice parameters of GaP and AlP, 2%, in comparison with the experimental value of about 0.02%. These deviations should be attributed to the overestimation of the strain eects predicted by our density-functional calculations. The calculated spectra are displayed in the left-hand panel of Fig. 4. Here, we report the
dependence of the TO and LO Raman modes of Ga1ÿx Alx P. The results of the M0 calculations deviate to greater extent from those of the M2 model than in the alloys seen above. The M2 frequencies are generally in better agreement with the experimental values than the M0 ones. However, even using the M2 model deviations from experiment of the order of 15 cmÿ1 are found.
4.3. GaAsx P1ÿx and InAsx P1ÿx The lattice mismatch between GaP and GaAs is about 4%. This makes modi®cations in the force constants necessary. Consequently, the M2 model yields better lattice-dynamical results than the M0 one. The larger Raman activity of the arsenic than the phosphorus atoms causes the GaAs-like peak to be much stronger in intensity in the results obtained using the P2 model than in the P0 spectra, as it can be seen in the right-hand panel of Fig. 3. Finally, the right-hand panel of Fig. 4 gives the results of our M2 calculations for InAsx P1ÿx . The
Fig. 3. Raman cross-section rxy R x calculated within the M0; P0 and M2; P1 approximations for Ga0:55 Al0:45 Sb (left panel) and within the M0; P0 and M2; P2 approximations for GaAs0:325 P0:675 (right panel) in comparison with the experimental data of Refs. [11] and [12], respectively.
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
369
Fig. 4. Phonon frequencies corresponding to the position of the Raman peaks in the Ga1ÿx Alx P (left panel) and InAsx P1ÿx (right panel) alloy as a function of the composition x calculated within the M0 and M2 approximations. The experimental data are taken from Refs. [13] and [14], respectively.
agreement with the experiment is in general quite satisfactory. Deviations from the experiments of about 3% are of the same order of magnitude of those obtained for the corresponding bulk constituents. 5. Conclusions Employing DFPT methods, we were able to calculate the ®rst-order o-resonance Raman spectra of various alloys. Both the position and the intensity of the features of the Raman spectra generally well compare with experimental ®ndings. The agreement is less satisfactory for systems which are characterized by a (real or ®ctitious) large mismatch between the lattice parameters of the bulk constituents. This could be due to the neglect of direct local strain eects in the expression of the Raman tensors or to explicit non-linearities to be included in the perturbative expansion we used. The analysis of the validity of such approximations should be the subject of future theoretical investigations on this topic.
Acknowledgements One of us (B.S.) is grateful to the Studienstiftung des Deutschen Volkes for ®nancial support.
References [1] For a recent review see: G. Kanellis, Properties of Phonons in Semiconductor Superlattices, in: T.S. Moss, M. Balkanski (Eds.), Optical Properties of Semiconductors, NorthHolland, Amsterdam, 1994, p. 463. [2] S. de Gironcoli, Phys. Rev. B 46 (1992) 2412. [3] S. Baroni, S. de Gironcoli, P. Giannozzi, Phys. Rev. Lett. 65 (1990) 84. [4] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954. [5] W. Hayes, R. Loudon, Scattering of Light by Crystals, Wiley, New York, 1978. [6] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [7] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [8] M. Hansen, K. Anderko, Constitution of Binary Alloys, second ed., McGraw Hill, New York, 1958, p. 774. [9] M. Alonso, K. Winer, Phys. Rev. B 39 (1989) 10056. [10] A.S. Barker, J.L. Merz, A.C. Gossard, Phys. Rev. B 17 (1977) 3181.
370
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
[11] F. Char®, M. Couaghi, C. Llinares, M. Balkanski, C. Hirlimann, A. Joullie, in: M. Balkanski (Ed.), Lattice Dynamics, 1977, 438. [12] N. Strahm, A. McWorther, in: G. Wright, (Ed.), Light Scattering in Solids, vol. 455, 1969.
[13] G. Lucovsky, R. Burnharm, A.S. Alimonda, Phys. Rev. B 14 (1976) 2503. [14] R. Carles, N. Saint-Cricq, J.B. Renucci, R.J. Nicholas, Solid State Phys. 13 (1979) 899.
www.elsevier.com/locate/commatsci
First-principles study of Raman intensities in semiconductor systems Pasquale Pavone *, Bernhard Steininger, Dieter Strauch Institut f ur Theoretische Physik, Universit at Regensburg, 93040 Regensburg, Germany Received 15 February 2000
Abstract We present a general method for the theoretical determination of the o-resonance Raman activity of semiconductor mixed systems such as substitutional alloys or superlattices, with the inclusion of strain eects. As basic tool, the densityfunctional perturbation theory has been used. The calculation of the Raman cross-section requires the knowledge of the lattice dynamics of the complete system. The description of semiconductor alloys or long-period superlattices requires large supercells, this makes the numerical eort of a direct ab initio calculation an unfeasible task. Simple procedures using the virtual-crystal (VCA) and the mass approximation fail in describing the eect of both the polarizability variations and the change in the local environment due to the local strain. We used a perturbative scheme which includes the potential ¯uctuations responsible for chemical disorder and the variations of the atomic Raman tensors. The method is applied to some III±V and Si/Ge semiconductor systems. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Much interest is presently devoted to the theoretical and experimental investigation of semiconductor alloys and superlattices, especially in connection with the electronic and lattice-dynamical properties of the corresponding bulk materials [1]. Due to the small size of the samples, methods of optical spectroscopy such as Raman scattering are found to be much more ecient tools in the characterization of mixed crystals than neutron spectroscopy. A complete theoretical description of the Raman spectra of semiconductor alloys has not been available for a long time. Recently, it has been shown * Corresponding author. Tel.: +49-941-943-2048; fax: +49941-943-4382. E-mail address: [email protected] (P. Pavone).
that well-de®ned phonons for systems with topological disorder due to local strain can be eciently calculated by ab initio methods using the anharmonic force constants of a reference periodic system [2]. However, the calculation of Raman intensities of alloys has long escaped any ab initio approach due to the huge computational eort. Consequently, all the previous work based on ®rst-principles calculations completely neglected the dierence in the Raman tensor of the constituents [2,3]. In this work, we present a method based on ®rst-principles calculation which allows the evaluation of the ®rst-order o-resonance Raman spectra of mixed semiconductor systems. The method has general validity and can be used in all those cases where the deviation from a perfect periodic system can be treated within perturbation theory. We apply this method to the calculation of Raman spectra of various III±V and Si/Ge alloys
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 1 9 5 - 6
364
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
and superlattices. We demonstrate the eciency of the method for mixed strained systems with relative dierences in the lattice parameter of the constituents up to about 3%. 2. Theoretical framework Raman scattering of light by lattice vibrations of a crystal depends on the modulation of the electronic susceptibility Dv by phonons [4]. In fact, the o-resonance Raman cross-section can be expressed to the ®rst order as [5] X d x ÿ xm e Dvm eI 2 ; x / 1 rIF R F xm m where eI (eF ) is the polarization vector of the incoming (scattered) photon, xm the phonon frequency of the mode m at the C point of the Brillouin zone, and the susceptibility variation Dvm is given by ovab X ovab ouc RS Dvmab onm ouc RS onm c;RS
X c;RS
Rabc RS
ouc RS ; onm
2
where nm is the normal coordinate of the mode m, Rabc RS de®nes the ®rst-order Raman tensor, and uc RS is the displacement in c direction of the atom at the equilibrium position RS . To our purposes, it is convenient to de®ne the position vectors RS in the ``super'' structure (alloy or superlattice) in terms of the Bravais (R) and basis sj vectors of a reference periodic system (in our case the zinc-blende structure) as RS R sj wj R;
3
where wj R is the ``local'' deviation from the reference structure. The lattice-dynamical quantities involved in the calculation of the cross-section are the eigenvectors and eigenfrequencies of the dynamical matrix of the system, which is essentially the Fourier transformation of the interatomic force-constants matrix U RS ; R0S ÿ
oF RS : ou R0S
4
The calculation of the interatomic force constants in Eq. (4) for semiconductor alloys or long-period superlattices requires the use of large supercells, this makes a direct ®rst-principles calculation an almost unfeasible task, due to the huge numerical eort. Therefore, alternative procedures must be used. The lowest order approximation to the calculation of the interatomic force constants in large mixed systems is the so-called virtual-crystal approximation (VCA), which assumes the real (disordered or super-periodic) crystal to be substituted by a ®ctitious periodic system whose components are a sort of ``averaged'' (virtual) atoms according to the macroscopic composition of the system. A schematic representation of a virtual crystal is shown in Fig. 1. Two main approximations are included in the VCA: (i) due to the dierent chemical behavior and the dierent bulk lattice constants the position of one atom in the actual system diers from the virtual crystal one by a displacement wj R; (ii) equivalent lattice sites in the zinc-blende structure are occupied by the same type of atom. Therefore, the actual con®guration of a mixed system is exactly determined from that of the reference zinc-blende system when speci®ed the set of local deviation fwg and the type of the atom occupying each site.
Fig. 1. Schematic representation within the VCA (left panel) of a disordered crystal made up of two atom types (right panel).
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
In order to go beyond the VCA, both eects of local displacements and of variation in the site occupation should be included in the description of the system. To de®ne the site occupation, we introduce at each site a ``con®guration'' Ising-like variable u RS . Such variables are chosen to vanish for a given virtual-crystal reference con®guration. As an example, for a zinc-blende mixed 2 system Cat 1 x Cat1ÿx An with con®gurational disorder only at the cationic site the fu RS g can be de®ned, using as reference the virtual atom with symmetric composition, as 8 0 if R sjA is an anionic site > > > > 1 if R sjC is occupied by Cat 1 > < if R sjC is occupied by the uj R 0 1 2 > > Cat virtual atom Cat > 0:5 0:5 > > : ÿ1 if R sjC is occupied by Cat 2 5 Non-integer values of uj R indicate the presence at the site R sj of a virtual atom with a dierent composition than the reference one. Within this description, the main ingredients for the calculation of the Raman cross-section can be explicitly considered as function of the local deviations and the con®guration variables: fwgfug
U RS ; R0S Ujj0 Rabc RS
R; R0 ;
6
365
where j0 means that all derivatives are calculated for {w} 0 and fug 0, i.e., for the reference virtual crystal, and where oFj R 0 9 Njj0 R ÿ R ouj0 R0 0 is the force-constant matrix of the virtual crystal. Furthermore, in Eq. (8) we have explicitly used the formal equivalence between w and u in performing the derivatives of the force. The knowledge of the force constants as well as the mass con®guration allows the calculation of the phonon frequencies and eigenvectors in Eq. (2). Within this context, the use of the virtualcrystal force constants and the correct mass con®guration de®ne the so-called mass approximation. The expansion of the Raman tensor to the ®rst order in fug and {w} reads X 0 fwgfug 0 0 Tjj Rabc;j R Bjabc abc R ÿ R uj0 R
X d;j0 ;R
j0 ;R
2 jj0
Rabcd R ÿ R0 wd;j0 R0 ;
10
where Bjabc is the Raman tensor of the reference virtual crystal and o2 vab jj0 0 11 Tabc R ÿ R : ouc;j R ouj0 R0 0
fwgfug Rabc;j R:
7
Within a perturbation scheme the interatomic force constants and the Raman tensors can be expanded in terms of both the local deviations and the con®guration variables. For the interatomic force constants, we obtain at the ®rst order in {w} and fug [2] fwgfug Ujj R; R0 Njj0 R ÿ R0 0
o2 Fj R 00 ÿ 0 00 wj00 R 0 R ouj00 R ou j 00 j00 R 0 2 X o Fj R 00 00 R ÿ 0 00 uj00 R ; 0 R ou R ou j j00 j00 X
0
8
The last term in the RHS of Eq. (10) depends explicitly on the second-order Raman tensor, 2jj0 Rabcd R, of the virtual system. The contribution of this term vanishes in absence of local strain as, e.g., in the case of GaAs/AlAs mixed systems. In the next, this term will be neglected also for the other studied system, retaining, in practice, localstrain eects only in the calculation of the force constants. The calculation of the ®rst-order terms in Eqs. (8) and (10) requires the knowledge of expansion coecients with the same symmetryselection rules of the perfect periodic virtual crystal. In fact, the number of the independent elements in the linear-order terms of Eqs. (8) and (10) can be strongly reduced by symmetry considerations. However, the presence in the
366
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
expansions of the sum over the atoms makes an explicit calculation of all the independent coecients practically unfeasible. On the other hand, it is reasonable to assume coecients involving pairs of ``distant'' atoms to have a very small in¯uence on the properties in which we are interested. This leads to the possibility of describing our systems within dierent levels of approximation by simply retaining in Eqs. (8) and (10) only the terms which involve atom pairs with distance within a chosen reference value. In practice, we indicate with Mn and Pn the approximation consisting in dropping all the terms involving pairs with distance larger than the nth-nearest-neighbors one in the expansion coecients in Eqs. (8) and (10) for the force constants and the Raman tensor, respectively. In this work, we consider the validity of the approximations up to the second-nearest-neighbors distance. In order to further limit the number of independent parameters we restrict the ``model'' M2 to include only terms referring to triplets of sites that contain at least one ®rst-nearest-neighbor pair. This restriction has been shown to be quite accurate for the description of the interatomic force constants of Si/Ge systems [2]. Furthermore, in polar materials, due to dipolar interactions associated with non vanishing eective 0 charges, the Tjj abc R de®ned in Eq. (11) are not short ranged. However, only dipolar contributions survive at large jRj and the corresponding term in the expression of the Raman tensor can be analytically parametrized. The inclusion of this parametrization is implicitly considered in any model Pn for distances beyond the nth-nearest-neighbors one. The resulting eect of this term is, anyway, very small. 3. Technical ingredients Our calculations are carried out within the density-functional theory together with the localdensity approximation. To describe the electron± ion interaction, we used norm-conserving pseudopotentials. The electronic wave functions are expanded up to a kinetic-energy cuto of 16 and 12 Ry for the calculation of the force constants
and of the Raman tensors, respectively. A lower kinetic-energy cuto for the calculation of Raman tensor is allowed because we are more interested in the relative than the absolute intensity of the features in the Raman spectra. The description of the virtual atoms is achieved by mixing the pseudopotentials of the corresponding constituent atoms. The force constants of the reference virtual crystal are obtained using the density-functional perturbation theory [6,7], further details of the calculation of dynamical properties are found in Ref. [7]. We simulated alloys using a cubic 512 atom supercell with randomly distributed atoms at the atomic sites [3]. The coecients in the RHS of Eqs. (8) and (10) are extracted from ®ts of the corresponding equations to a number of independent con®gurations. For the force-constants coecients we have ®tted Eq. (8) to direct ab initio calculations of the force constants of: (i) the virtual crystal with a displaced atom in the unit cell and under the two independent distorsions de®ned by a (0 0 1) and (1 1 1) macroscopic strain; (ii) the two bulk constituents and the virtual crystal at the atomic positions of the virtual crystal. For the Raman-tensor coecients, we have ®tted Eq. (10) to ab initio calculation of the Raman tensor for three short-period superlattices, i.e. Cat 1 An 1 n Cat 2 An 2 n for n 1; 2 in the (0 0 1) direction and for n 2 in the (1 1 0) one. The Raman tensor is obtained by ®nite-dierence calculations of the derivative of the polarizability tensor with respect to an atomic displacement. Dynamical matrices and forces on atoms are calculated from interatomic force constants obtained following Eq. (8). Atomic positions are relaxed iteratively until equilibrium is reached. Phonon frequencies and eigenvectors are then obtained for a given con®guration by standard diagonalization of the dynamical matrix at equilibrium. Finally, the Raman cross-section is calculated following Eqs. (1), (2), and (10) for the equilibrium con®guration. To reduce statistical errors, ®ve random con®gurations are considered for each composition. In order to take into account the experimental resolution, the calculated intensity is convoluted with Gaussians with a full width at half maximum of 2 cmÿ1 .
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
367
4. Results and Discussion
4.2. Gax Al1ÿx As; Gax Al1ÿx Sb, and Ga1ÿx Alx P
4.1. Six Ge1ÿx
In the right-hand panel of Fig. 2, we report the experimental and theoretical Raman spectrum of the alloy Ga0:55 Al0:45 As in the M0; P0 and M0; P2 approximation. Due to the very small lattice mismatch and the chemical similarity between GaAs and AlAs, we use the simple mass approximation (model M0) for the determination of the dynamical matrix of the alloy. Also in this case, the inclusion of the eect of polarizability changes ± as in the model P2 ± is shown to have a very strong eect on the relative intensity of the Raman peaks. The larger polarizability of the Ga than the Al atoms has as direct eect, in comparing with the P0 result, the lowering of the intensity of the AlAs-like peak and a contemporary increase of the intensity of the GaAs-like feature. For this alloy, our calculations extend the results of a previous theoretical work [3], where it is found a general excellent agreement with the experimental position of the features in the spectra, failing, however, in predicting the correct relative intensities of the two main peaks.
The Raman spectra of the Six Ge1ÿx alloy are characterized by the presence of three strong peaks corresponding to the dierent bond vibrations. The picture of Six Ge1ÿx systems as prototype random alloys with no relevant clustering of ordering eects is supported by experimental investigation [8]. In the left panel of Fig. 2 we report the results of our calculation of rxy R x within the M0; P0 and M2; P2 models at the same composition x 0:45 as the experimental work of Ref. [9]. The position of the experimental Si±Ge peak is very well reproduced by both models. However, the position of the remaining features and their relative intensity are better described by the model M2; P2. The comparison with the previous theoretical work of de Gironcoli [2], where the analogue of the model M2; P0 was used, shows, for this alloy, the relevance in the determination of relative intensities of taking into account the polarizability dierence of the constituent atoms.
Fig. 2. Raman cross-section rxy R x calculated within the M0; P0 and M2; P2 approximations for Si0:45 Ge0:55 (left panel) and Ga0:55 Al0:45 As (right panel) in comparison with the experimental data of Refs. [9] and [10], respectively.
368
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
In the case of (Ga/Al)Sb systems, the local strain is expected to have very small in¯uence on the Raman spectra, due to the small relative difference in the lattice parameter of the bulk constituents (about 0.7%). In fact, as it is reported in the left-hand panel of Fig. 3, both the models M0 and M2 yield very similar results in overall good agreement with experiment. For these systems, already the model P1 gives a satisfactory description of the relative intensities, in good agreement with the experimental (unpolarized) results. Notice that for the cross-section in the xy con®guration the selection rules predict the presence of only longitudinal optical phonons. The calculation of the Raman spectra of the Ga1ÿx Alx P alloy presents spurious local-strain effects related to the overestimation of the theoretical mismatch between the bulk lattice parameters of GaP and AlP, 2%, in comparison with the experimental value of about 0.02%. These deviations should be attributed to the overestimation of the strain eects predicted by our density-functional calculations. The calculated spectra are displayed in the left-hand panel of Fig. 4. Here, we report the
dependence of the TO and LO Raman modes of Ga1ÿx Alx P. The results of the M0 calculations deviate to greater extent from those of the M2 model than in the alloys seen above. The M2 frequencies are generally in better agreement with the experimental values than the M0 ones. However, even using the M2 model deviations from experiment of the order of 15 cmÿ1 are found.
4.3. GaAsx P1ÿx and InAsx P1ÿx The lattice mismatch between GaP and GaAs is about 4%. This makes modi®cations in the force constants necessary. Consequently, the M2 model yields better lattice-dynamical results than the M0 one. The larger Raman activity of the arsenic than the phosphorus atoms causes the GaAs-like peak to be much stronger in intensity in the results obtained using the P2 model than in the P0 spectra, as it can be seen in the right-hand panel of Fig. 3. Finally, the right-hand panel of Fig. 4 gives the results of our M2 calculations for InAsx P1ÿx . The
Fig. 3. Raman cross-section rxy R x calculated within the M0; P0 and M2; P1 approximations for Ga0:55 Al0:45 Sb (left panel) and within the M0; P0 and M2; P2 approximations for GaAs0:325 P0:675 (right panel) in comparison with the experimental data of Refs. [11] and [12], respectively.
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
369
Fig. 4. Phonon frequencies corresponding to the position of the Raman peaks in the Ga1ÿx Alx P (left panel) and InAsx P1ÿx (right panel) alloy as a function of the composition x calculated within the M0 and M2 approximations. The experimental data are taken from Refs. [13] and [14], respectively.
agreement with the experiment is in general quite satisfactory. Deviations from the experiments of about 3% are of the same order of magnitude of those obtained for the corresponding bulk constituents. 5. Conclusions Employing DFPT methods, we were able to calculate the ®rst-order o-resonance Raman spectra of various alloys. Both the position and the intensity of the features of the Raman spectra generally well compare with experimental ®ndings. The agreement is less satisfactory for systems which are characterized by a (real or ®ctitious) large mismatch between the lattice parameters of the bulk constituents. This could be due to the neglect of direct local strain eects in the expression of the Raman tensors or to explicit non-linearities to be included in the perturbative expansion we used. The analysis of the validity of such approximations should be the subject of future theoretical investigations on this topic.
Acknowledgements One of us (B.S.) is grateful to the Studienstiftung des Deutschen Volkes for ®nancial support.
References [1] For a recent review see: G. Kanellis, Properties of Phonons in Semiconductor Superlattices, in: T.S. Moss, M. Balkanski (Eds.), Optical Properties of Semiconductors, NorthHolland, Amsterdam, 1994, p. 463. [2] S. de Gironcoli, Phys. Rev. B 46 (1992) 2412. [3] S. Baroni, S. de Gironcoli, P. Giannozzi, Phys. Rev. Lett. 65 (1990) 84. [4] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954. [5] W. Hayes, R. Loudon, Scattering of Light by Crystals, Wiley, New York, 1978. [6] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [7] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [8] M. Hansen, K. Anderko, Constitution of Binary Alloys, second ed., McGraw Hill, New York, 1958, p. 774. [9] M. Alonso, K. Winer, Phys. Rev. B 39 (1989) 10056. [10] A.S. Barker, J.L. Merz, A.C. Gossard, Phys. Rev. B 17 (1977) 3181.
370
P. Pavone et al. / Computational Materials Science 20 (2001) 363±370
[11] F. Char®, M. Couaghi, C. Llinares, M. Balkanski, C. Hirlimann, A. Joullie, in: M. Balkanski (Ed.), Lattice Dynamics, 1977, 438. [12] N. Strahm, A. McWorther, in: G. Wright, (Ed.), Light Scattering in Solids, vol. 455, 1969.
[13] G. Lucovsky, R. Burnharm, A.S. Alimonda, Phys. Rev. B 14 (1976) 2503. [14] R. Carles, N. Saint-Cricq, J.B. Renucci, R.J. Nicholas, Solid State Phys. 13 (1979) 899.