close

Вход

Забыли?

вход по аккаунту

?

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

код для вставкиСкачать
Ann. 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].
Financial support by the Deutsche Forschungsgerneinschaft Proj. No. Fa 170/1-3 and 17012- I is
gratefully acknowledged.
M. Klenner, C. Falter, W. Ludwig, Electron-phonon matrix elements a n d . . .
33
References
[I]
[2]
[3]
[4]
[5]
[6]
171
[8]
191
[ 101
[ 1 11
[I21
[I31
[I41
[I51
[I61
[ 171
[IS]
[I91
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
I311
[32]
[33]
W. E. Pickett, J. Phys. C 12 (1979) 1491
C. Falter, Phys. Rep. 164, Nos. L and 2 (1988)
C. Falter, M. Selmke, W. Ludwig, K. Kunc, Phys. Rev. B 32 (1985) 6518
M. L. Cohen, T. K. Bergstresser, Phys. Rev. 141 (1966) 789
J. A. Appelbaum, D. R. Hamann, Phys. Rev. B 8 (1973) 1777
C. Falter, H. Rakel, M. Klenner, W. Ludwig, Phys. Rev. B 40 (1989) 7727
0. J. Glembocki, F. H. Pollak, Phys. Rev. Lett. 48 (1982) 413
0. J. Glembocki, F. H. Pollak, Phys. Rev. B 25 (1982) 7863
S . Bednarek, U. Rossler, Phys. Rev. Lett. 48 (1982) 1296
W. Brauer, H. Streitwolf, Theoretische Grundlagen der Halbleiterphysik (Vieweg, Braunschweig
1977)
P. Lawaetz, Phys. Rev. 183 ( 1 969) 730
R. Resta, L. Colombo, S. Baroni, in: PHONONS 89, Proc. of the Third Int. Conf. on Phonon
Physics, Eds. S. Hunklinger, W. Ludwig, G. Weiss, Vol. 1, p. 208 (World Scientific, Singapore,
1990)
L. Kleinman, Phys. Rev. B 24 (1981) 7412
0. J. Glembocki, F. H. Pollak, Phys. Rev. B 25 (1982) 1193
S. Zollner, S. Gopalan, M. Cardona, J. Appl. Phys. 68 (1990) 1682
C. Falter, M. Selmke, W. Ludwig, W. Zierau, J. Phys. C 17 (1984) 21
A. Palomo, J. A. Verges, Phys. Rev. B 30 ( I 984) 21 04
C. Jacoboni, L. Reggiani, Adv. Phys. 28 (1979) 493
J. A. Verges, D. Glotzel, M. Cardona, 0. K. Andersen, phys. stat. sol. (b) 113 (1982) 519
M. Cardona, N. E. Christensen, Phys. Rev. B 35 (1987) 6182; 36 (1987) 2906 (E)
C. G. Van de Walle, R. M. Martin, Phys. Rev. Lett. 62 (1989) 2028
D. D. Nolte, W. Walukiewicz, E. E. Haller, Phys. Rev. Lett. 59 (1987) 501; Phys. Rev. B 36 (1987)
9392
G. S. Cargill, J. Angilello, K. L. Kavanagh, Phys. Rev. Lett. 61 (1988) 1748
Landolt-Bornstein, Vol. 17a (Springer, Berlin, 1982) I I
A. Blacha, H. Presting, M. Cardona, phys. stat. sol. (b) 126 ( 1 984) 11
0. H. Nielsen, R. M. Martin, Phys. Rev. B 32 (1985) 3792
C. G. Van de Walle, Phys. Rev. B 39 ( 1 989) 1871
C. S. G. Cousins, L. Gerward, J. S. Olsen, B. Selsmark, B. J. Sheldon, J. Phys. C: Solid State Phys.
20 (1987) 29
R. D. King-Smith, R. J. Needs, J. Phys.: Condens. Matt. 2 (1990) 343 I
L. D. Laude, F. H. Pollak, M. Cardona, Phys. Rev. B 3 (1971) 2623
C. Jacoboni, G. Gagliani, L. Reggiani, 0. Turci, Solid State Electron. 21 (1978) 315
I. Balslev, Phys. Rev. 143 (1966) 636
W. E. Pickett, Rev. Mod. Phys. 61 (1989) 433
Документ
Категория
Без категории
Просмотров
0
Размер файла
515 Кб
Теги
deformation, potential, matrix, elements, phonons, mode, ion, silicon, germanium, electro, quasi
1/--страниц
Пожаловаться на содержимое документа