# Electron-phonon matrix elements and deformation potentials in silicon and germanium in the quasi-ion model.

код для вставкиСкачатьAnn. Physik 1 (1992) 24-33 Annalen der Physik 0 Johann Ambrosius Barth 1992 Electron-phonon matrix elements and deformation potentials in silicon and germanium in the quasi-ion model M. Klenner, C. Falter, and W. Ludwig lnstitut fur Theoretische Physik I1 - Festkorperphysik - der Westfalischen Wilhelms-Universitat Munster, Wilhelni-Klemm-Stral3e 10, W-4400 Munster, Germany Received 1 1 November 1991, accepted 16 December 1991 We have calculated electron-phonon matrix elements relevant for indirect optical absorption, intervalley matrix elements and intravalley deformation potentials as well as splitting parameters for the k = 0 states in silicon and germanium within the rigid quasi-ion model. In contrast to the “atomic” potentials in conventional rigid-ion models the quasi-ion potentials are well-defined. The agreement with experiment and with other theoretical results is satisfactory in most cases. We have also studied the role of distortion corrections to the rigid quasi-ion approximation. We find that the influence of distortions is small, indicating that the rigid quasi-ion model is an adequate description - at least in case of these two materials. Abstract. Keywords: Electron-phonon interaction; Density response; Phonons-lattice dynamics. 1 Introduction Electron-phonon scattering plays an essential role in many optical and transport properties of semiconductors. The calculation of the corresponding electron-phonon matrix elements requires the knowledge of the electron-phonon potential, i. e. the change in the self-consistent crystal potential felt by an electron when an ion is displaced. This quantity is, in general, hard to compute. In linear response theory, e. g., it is obtained from the change in bare potential by screening with the inverse dielectric function. Therefore, the use of models is common practice in this field. The most popular of these models is the so-called “rigid-ion’’ model. It is assumed that the total crystal potential is given as superposition of atomic-like potentials which move rigidly with the ions. The main difficulty in connection with this model is that there seems to be no unique prescription for splitting the total potential into atomic contributions. As is well known, a given lattice periodic function can be decomposed into localized functions centered at the lattice points in an infinite number of different ways. A specific decomposition in direct space corresponds to a specific inter(extra)polation of the discrete potential form factor data in reciprocal space. In general, there is a wide range of curves passing through the few empirically known data (cf. Fig. 3). M. Klenner, C . Falter, W. Ludwig, Electron-phonon matrix elements and . . . 25 2 Theory and method of calculation As stressed above, a unique decomposition of the total crystal potential into atomic contributions is not possible if only the lattice periodic equilibrium configuration is considered. It is, however, possible by consideration of a non-equilibrium configuration with the atoms displaced from their ideal sites. To see this, let us define the vector field describing the change in the crystal potential V(r) due to a unit displacement of an ion of type a in cell m. VF(r) can be decomposed into its gradient and rotational parts by means of the Helmholtz construction [ I ] : The gradient part of VF(r) corresponds to the “rigid” part of the change in potential and hence the quantities T ( r ) in (2) have the meaning of rigid-ion potentials (“quasiion” potentials). It follows from translational invariance that the potentials VF sum up to give the total crystal potential. In connection with the phonon problem we are dealing with the displacement-induced (electron) density redistribution instead of the change in crystal potential. The former quantity is defined quite analogous to (1) by [2] and has a Helmholtz decomposition similar to (2), The quantities p,“ (r) are called “quasi-ion” (or “partial”) densities. They can be calculated, in principle, from P,“(r) which in turn has to be determined, e.g., via the density response function. In practice, the quasi-ion densities are determined from calculated lattice-periodic quasi-ion densities [3] together with empirical data consisting of total valence electron densities and phonon frequencies. For silicon the quasi-ion density is shown in Fig. 1. The most remarkable features are the maxima located at approximately 1/3 of the bond length. For a further discussion we refer to the literature [2]. The quasi-ion density in germanium looks similar. For shortness, also the remaining figures refer to silicon only. Taking into account in Eq. (4)the gradient part only, P,“(r) = -Vp,“(r) (5) we obtain the rigid quasi-ion model for P,“(r). Figure 2 shows the phonon dispersion curves in the main symmetry directions as calculated within the rigid quasi-ion model. 26 Ann. Physik 1 (1992) Fig. 1 Quasi-ion density of silicon. Units are electrons per cell. The plot refers to a (Oil) plane. r h X 1 r h L Fig. 2 Phonon dispersion curves of silicon within the rigid quasi-ion model. Experimental data are indicated. Now, the central point is that the electron-phonon potential V E(r) can be expressed by the displacement-induced density redistribution PE (r) via the relation where Vt", is the ion (pseudo-)potential and V is the effective electron-electron interaction including exchange and correlation [2]. Hence, P,"(r) can be considered as the basic quantity also in connection with the electron-phonon interaction. M. Klenner, C. Falter, W. Ludwig, Electron-phonon matrix elements and . . . 27 Specific models for PF lead to corresponding models for VF. In particular, the rigid quasi-ion model (5) yields a rigid-ion model for Vg, when additionally a homogeneous form of fi is assumed. In this case the quasi-ion potentials T ( r ) are given by V F . ~ =) VE(r) + j d3 J fi(r - r’) p:(r’) (8) In Fig. 3 we have displayed the potential form factor corresponding to the quasi-ion potential of Eq. (8). For comparison we have also included the empirical form factors from Cohen and Bergstresser [4]. We used an ion pseudo-potential of AppelbaumHamann type [5] with parameters adjusted to yield accurate energy bands (Fig. 4). In case of the effective electron-electron interaction 6 we employed a simple homogeneous model suggested by Falter et al. [6]. Note that there is some anisotropy in the form factor which is, however, small as compared to the anisotropy of the quasi-ion densities. 0.2 I 1 I 1 I -I 0.0. I -z -0 2 -a: - 0 . 4 - -- ------ 0 > - 0 . e y Fig. 3 Potential form factor of silicon as calculated from the quasiion density of Fig. 1. -1 .o 0 , I 3 q I 0 , , , 4 5 6 ~ 2 1 <100>-direction <llO>-direction Cohen-Bergstresser 1 7 12n/a) 1 r (3, L a, C W Fig. 4 Energy bands of silicon as calculated from the form factor of Fig. 3. Experimental data are indicated. 1 -121 -14 L A r B X y z r 28 Ann. Physik 1 (1992) Of particular interest are the regions of small and large wave vector q, respectively, where an extrapolation of the empirical form factors is required within conventional rigid-ion models. Our potentials are similar to those of Glembocki and Pollak [7,81 with respect to their curvature at small q. In contrast, the potential suggested by Bednarek and Rossler [9] has V(0) = 0. In the region defined by q > 5 * (2n/a) most extrapolations choose V ( q ) = 0. From the figure, however, it can be seen that our potentials have substantial values even in this region. In addition to the rigid contribution (Eq. (5)) to P," models for distortions can be included in Eq. (6). Finally, it should be stressed that besides yielding a unique decomposition of the crystal potential a further advantage of the quasi-ion model lies in the fact that the calculation of the electron-phonon potential (including the band structure potential) and of the phonon dispersion are intimately related. This is in contrast to conventional rigid-ion models were the phonon frequencies and eigenvectors are usually taken from phenomenological lattice dynamical models. In solving the secular equation for the electronic states we used a plane wave basis taking into account all plane waves 1 k + G ) with I k + G I * I 16. (Zn/a)*(or 65 plane waves at k = 0) exactly and the plane waves satisfying 16 (27r/a)* < 1 k + G 1 5 27 (2n/a)* (or 104 additional plane waves at k = 0) within the Brust-Lowdin perturbation scheme. The expressions for calculating the electron-phonon matrix elements within the plane wave basis are easily worked out and will not be given here. In the case of scattering at long-wavelength acoustic phonons the electron-phonon matrix elements can be expressed in terms of the so-called (acoustic) deformation potentials [lo]. These are independent of the direction of the wave vector only if the electron-phonon potential is short-ranged in the sence of containing no quadrupole and octupole contributions [ 1-11. The quadrupole and octupole contributions have recently been calculated by Resta et al. [ I21 in case of the valence band edge deformation potential in silicon. Whereas the octupole contribution is small the quadrupole contribution is significant, according to their findings. One easily verifies that the electron-phonon potential within a rigid-ion model in a crystal with diamond or zinc-blende symmetry is free of quadrupole and octupole contributions. In this case the deformation potential tensor for a Bloch state I k v) with wave vector k and band index v is given by ' - 1 €3 p + C [(R," - r) €3 V,"(r) + V,"(r).ra] I k v ma where m e is the electron mass, p is the momentum operator, r, is the internal strain tensor for sublattice a, and €3 denotes a dyadic product. Note that the potential form factor is calculated in our approach from the corresponding quasi-ion densities (using certain models for the ion pseudopotential and the effective electron-electron interaction) instead of being obtained by inter(extra)polation of empirical form factor data as in conventional rigid-ion models. Especially, the limiting behaviour as q --t 0 is essential in connection with the absolute deformation potentials. By the deformation potential theorem [ 101 the tensor Z(k v) describes also the shift of band energies due to strain. Note, however, that the absolute enery zero is an ill-defined quantity in an infinite crystal [13]. Hence, only relative energy shifts are described meaningfully by the deformation potentials. The absolute meaning of the deformation potentials is, however, established by their connection with the electron-phonon matrix elements for long-wavelength acoustic phonons. M . Klenner, C. Falter, W. Ludwig, Electron-phonon matrix elements and . . . 29 3 Results As a first application we have calculated the matrix elements involved in the indirect optical absorption process [14] in silicon and germanium. The results are summarized in Table 1 for silicon. Since the phases of the matrix elements depend on the phases of the corresponding wave functions and phonon eigenvectors, only the absolute values are given. The theoretical results of Glembocki and Pollak [7] have been divided by 2 since the authors seem to have used an improper normalization of the potential form factor [15]. The experimental values which are available only in case of the TO matrix elements are significantly larger than the theoretical results. One should bear in mind, however, that there have been several assumptions and approximations in deriving the “experimental” values from the measured data. Table 1 Electron-(hole-)phonon matrix elements (mRy) involved in the indirect optical absorption process in silicon. RPDM: Rigid partial density model; PDM, Rot.: Partial density model including model distortion corrections (“rotations” [2]); DR: Density redistribution from the model density response function of Ref. [16]; DR, Rig.: Same, but only “rigid” contribution taken into account; GP: Theoretical values of Glembocki and Pollak [7], divided by 2; Expt.: Experiment [14]. iL-rr-T electron-phonon hole-phonon RPDM PDM, Rot. DR DR, Rig. GP Expt . 0.7 1.1 1.1 1.2 3.5 - 15.2 15.2 12.9 12.4 12.5 - 17.1 17.1 15.5 14.9 18 - 14.3 14.2 13.7 13.9 12 18.7 0.8 I .o 0.8 1.1 1 - 12.7 12.7 10.5 9.7 11.5 - 4.0 4.0 2.4 2.4 6.5 - 10.9 10.8 10.5 10.5 9.5 15.3 The possibly most important conclusion that can be drawn from the results of Table 1 is that the effect of distortions seems to be not dramatic. This is true in the case of the model distortion corrections (“rotations”) as well as in the case of the full density redistribution calculated from the model density response function [ 161 when compared to its rigid counterpart. The fact that the “rotations” do not influence the matrix elements for longitudinal phonons at all is a consequence of the crystal symmetry. In Table 2 we list the results for the T-L electron-(hole-)phonon matrix elements relevant for indirect optical absorption in germanium. The initial states for the two LA e-ph matrix elements are r2,,c and r15,c,, respectively. The first of the two TO h-ph matrix elements refers to the component of r25,,v which transforms according to A 3 while the second refers to the component transforming according to A , . The results of Glembocki and Pollak [8] have been divided by 2 again. The agreement with our results is nearly perfect in case of most of the matrix elements. Only for the first LA e-ph matrix element (r2,c - L , , c )and the LA h-ph matrix element the deviations are somewhat larger. Intervalley scattering, i. e. scattering between two of the equivalent conduction band minima, is another subject of special interest. The corresponding electron-phonon matrix elements are usually expressed in terms of “deformation potentials”, Ann. Physik 1 (1992) 30 where S is the electron-phonon matrix element, D the related deformation potential, M is the mass of the elementary cell and w is the phonon frequency. In silicon, there are two different possibilities for intervalley scattering: The two valleys involved may lie on the same cubic axis, in which case the scattering process is called “gprocess”, or they may lie on axes perpendicular to each other, in which case it is called Table 2 f-Lelectron-(hole-)phonon matrix elements (mRy) in germanium. GP: Theoretical results of Glembocki and Pollak [8],divided by 2. e-ph LA RPDM GP 6.0 10.9 4.5 10.5 11-ph TO LA TO I .8 13.4 10.5 - 13.5 5.3 13.0 5.0 - 2.0 - “f-process”. The corresponding deformation potentials are given in Table 3. The theoretical results of Palomo and VergCs [17] were obtained within a generalized Wannier function method. The agreement of the different theoretical and experimental values is satisfactory, especially in case of the f ( A )-matrix element. The largest discrepancy in comparison with experiment occurs for g-scattering. Again, distortions only play a minor role. The influence of the “rotations” vanishes by symmetry in case of the g-process. Table 3 Intervalley deformation potentials (eV . A -’) in silicon. P V Theoretical results of Palorno and Verges [17]. Expt.: Experimental values from Ref. [ 181. See Table 1 for the remaining abbreviations. RPDM PDM, Rot. DR DR, Rig. PV Expt . 5.3 2.3 4.6 5.3 2.5 4.5 5.8 1.9 3.3 5.6 2.1 3.7 4.3 2.1 5.3 3 2.5 4 In germanium, there is only one possibility for intervalley scattering. The phonon wave vector has X-point symmetry and only the two degenerated LA and LO phonons, respectively, lead to non-vanishing matrix elements. Because of the degeneracy, one defines D = ( D t A + Dio)1’2.The rigid quasi-ion model yields D = 0.37 eV A while Zollner et al. [15] give D = 0 . 3 eV A and D = 0.7 eV- A - I , respectively. The latter value refers to an extrapolation of the potential form factor with V(0) = 0. In case of intravalley scattering the electron remains in the same conduction band valley. The results for the corresponding acoustic deformation potentials in silicon and germanium are compiled in Table 4. Most interesting are the volume deformation potentials where quite controversial values have been reported in the past [17, 19, 201. Recently, Van de Walle and Martin [21] have determined absolute deformation potentials - -’ - -’ M. Klenner, C. Falter, W. Ludwig, Electron-phonon matrix elements and . . . 31 from the potential lineup at interfaces between two differently strained regions in a crystal. The result for silicon given in the table is an average with respect ,to different orientations of the interface. In case of silicon the theoretical and experimental data now agree about a positive value of the volume deformation potential. In case of germanium there is still a discrepancy in sign when comparing the result within the quasi-ion model with that obtained by Van de Walle and Martin. The tendency to lower values, as compared to silicon, is, however, reproduced within the quasi-ion model. Table 4 Volume deformation potentials and uniaxial deformation potentials (eV) for the conduction band minima in silicon and germanium. RPDM - Si 4.9 8.5 2.3 12.1 a -U - Ce a -U a [17] [21] Calc. ' [22] 8.2a, 3.3b 9.16e - 1.Ob 15.13e 10.2=, [23] [27] Expt. 2.4', 3.3d 8.7' -5.6g 16.3* [I91 [24] In Tables 5 and 6 we list the results for the volume deformation potentials and splitting parameters for the states r25t, and rj5 , in silicon and germanium, respectively. We use the notation of Ref. 25. For comparison, we have included results of several calculations. Blacha et al. [25] employed a rigid-ion model with interpolated potential form factors. (It should be mentioned that the same authors have obtained corresponding results [not shown in the tables] within a tight-binding approach.) Nielsen and Martin [26]performed self-consistent density functional calculations. Van de Walle (271 combined selfconsistent band structure calculations with a model-solid theory in order to obtain absolute deformation potentials. Note that the volume deformation potentials of Blacha et al. are strongly negative. This is related to the assumption that the avarage potential does not change with volume, corresponding to V(0) = 0. The sign of the d' (rZ5,,") deformation potential is reversed in our calculation as compared to the other theoretical results. In case of the remaining deformation potentials the agreement is satisfactory. The constant d involves the internal strain parameter <.The rigid quasi-ion model yields the values = 0.68 and = 0.71 in silicon and germanium, respectively. This agrees well < < Table 5 Deformation potentials (eV) for the states rZy, and TlS, in silicon. b a d d' ~~~~ r2s3 v r n c a RPDM Blacha et al. [25] Nielsen and Martin [26] Van de Walle [27] Expt. RPDM Blacha et al. Van de Walle 3.9 -10.2 2.46 1.8a I 3.1 - 10.0 1.98 -3.1 3.0 -2.28 -2.35 -2.1b 0.9 From [22] and a gap deformation potential of 1.5 eV [30]. [30] [25] [31] -5.3 -8.7 -5.41 ~ 1.2 -2.3 -1.52 38.2 35 29.83 40' 26.6d -11.0 -19.8 - 16.9 -5.32 -4.85b -7.6 d0 Ann. Physik 1 (1992) 32 Table 6 Deformation potentials (eV) for the states f z s . , v and f r sin , cgermanium. l a r25,". fl5.c RPDM Blacha et al. [25] Van de Walle [27] Expt . RPDM Blacha et al. 4.9 -12.4 1.24 4.2 -8.5 b d -3.0 -3.1 -2.55 -2.3a -5.3 -7.2 -5.50 -S.Oa 0.9 d' -6.7 1.7 -0.8 d0 39.4 34.7 34a -10.1 -19.1 - 15.4 with the early experimental data. Most recent measurements [28] as well as self-consistent density functional calculations [26, 291 converge, however, around [ = 0.53. The results for d given by Blacha et al. are based on = 0.73 and ( = 0.74, respectively. Finally, Table 7 contains the volume deformation potentials for the lowest direct and indirect gaps in silicon and germanium. Since the change in the average potential cancels in this case, the agreement should be better here. As can be seen from the table, this is true in fact. Table 7 Volume deformation potentials (eV) of the lowest direct and indirect gaps in silicon and germanium. I RPDM Van de Walle [27] Expt. 4 Silicon adir aind -0.8 0.48 - I .0 1.72 Isa - Germanium adir - 8.5 -9.48 - 12.7b alnd -2.6 -2.78 - 2.0C Conclusion We have shown that within the quasi-ion approach it is possible to define "atomic" potentials uniquely summing up to the total crystal potential. When these are used in a rigid-ion model, reasonable results are obtained for electron-phonon matrix elements and deformation potentials. Distortion corrections to the rigid quasi-ion approximation were found to play only a minor role. In the new high-temperature superconductors, however, potential changes of distortion type seem to be essential in producing large values of the coupling constant A [33]. 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