Calculated deformation potentials in Si, Ge, and GeSi

~Solid State Communications, Vol. 75, No. i, pp. 39-43, 1990. Printed in Great Britain.

CALCULATED

DEFORMATION

POTENTIALS

0038-1098/9053.00+.00 Pergamon Press plc

I N Si, G e e A N D

GeSi

U. Schmid, N. E. Christensen* and M. Cardona Max-Planck-Institut f~r Festk6rperforschung, HeisenbergstraBe 1, D-7000 Stuttgart 80, Federal Republic of Germany * And: Institute of Physics, Aarhus University, DK-8000 Aarhus C, Denmark (Received 30 April 1990 by T. P. Martin)

We have calculated electronic deformation potentials of Si, Ge, and zinc-blende-like GeSi for both hydrostatic as well as uniaxial strain along the [001] direction. The calculations have been performed with the self-consistent relativistic linear muffin-tin orbital (LMTO) method, including spin-orbit coupling, and show that the uniaxial spin-dependent deformation potential b2 = -(0.05 4- 0.05) eV for both Si, Ge, and GeSi. The volume scaling parameters of the spin-orbit splittings A0 and A1, d lnA0,1/d lnV, were calculated to be ~. -0.6 for all three materials investigated. The error in the Eo gap introduced by the use of the local-density approximation is corrected by the introduction of extra potentials included self-consistently in the calculation, thus leading to pressure coefficients of the direct and indirect gaps which compare well with experiments and other recent calculations.

structures are calculated with the inclusion of these sharply peaked potentials, which are chosen so that gaps at three symmetry points (r, X, L) match the experimental data of unstrained Si and Ge. These extra. potentials are kept invariant under strain [13]. We include "empty spheres" [14], that is, atomic spheres with no net charge, in the empty tetralaedral sites in order to obtain a close-packed structure. Details of this method, and the equilibrium band structures of Si, Ge and GeSi have been published elsewhere [15]. Internal strain parameters of Si for strain along the [III] direction have already been studied with the LMTO method [16]. Most technological applications, however, ale coneerned with samples grown along the [001] direction, although other directions are also of interest [I"/].

I. I N T R O D U C T I O N THE CONCEPT of deformation potentials (DP's) is rather old: it was introduced by Bardeen and Shockley [1] some forty years ago to connect the effects of strain on the electronic band structure with the transport properties of semiconductors. Since then, much theoretical and experimental effort has been spent on the basic understanding of this issue [2,3]. With the recent development of ultrathin Ge/Si strained-layer superlattices [4] strain effects have proven to be of great importance for the electronic and optical properties of these new man-made materials [5-7]. This is due to the large lattice-mismatch of about 4°~ of the two constituting semiconductors, and various possible conditions of pseudomorphic growth and thus strain distribution in the superlattice. Most ab initio calculations of DP's have so far included spin-orbit effects only "a posteriori", although these effects have been studied experimentally [8-11]. The calculations discussed in this communication are performed within the local density approximation (LDA) by means of the self-consistent (relativistic) linear muffin-tin orbital method (LMTO) [12], with the inclusion of spin-orbit coupling as a perturbation to the Hamiltonian. The so-called "band-gap problem" associated with the LDA becomes even more pronounced when relativistic effects are included [13]. This is due to the downshift of the s states. We compensate for the too low gap by adding and hoe" potentials on the atomic sites [13]. The self-consistent band-

II. H Y D R O S T A T I C D E F O R M A T I O N Pressure coefficients of the band gaps of Si have recently been calculated by Zhu et al. from a first-principles quasiparticle approach [18] and - for both Si and Ge - by Ghahramani and Sipe using the semi-ab initio minimal basis orthogonalized Linear Combination of Gaussian Orbitals (LCGO) technique [19]. Chang et al. have also calculated pressure coefficients for various semiconductors with the first-principles pseudopotential method [20]. In Table 1, the calculated linear DP's a = d E / d lnV are listed for various direct gaps, and compared to other theoretical and experimental values. We prefer to give the DP's of the gaps rather than the 39

40

VOI. 75t No. i

CALCULATED DEFORMATION POTENTIALS IN Si, Ge, AND GeSi

Table 1: Comparison of calculated and measured hydrostatic deformation potentials a = dE/d lnV and uniaxial strain deformation potentials, as described in the text, at high symmetry points. Units are in eV. Present

Pseudopotential ( ab initio)

Quasiparticle

LCGO

-0.53 e

-1.38!

Experiments

Si

a(Eo) a(~o) a(E1) (~a + ½F-.u- av) z~ (Zd + ½F.u -- av) L av b bv

-12.9 -0.43 -5.3 1.2 --4.3 -1.4 a -2.18 1.4

= u~ ~

8.0

D~

5.8

-11.5 b -0.5 b, 1.6b, --3.8 b,

1.3g -2.35 d

"0.48 c 1.7~'d --3.1 d 0.4=i=0.6 h

1.3 ~

1.42/

1.41i, 1.9+0.7 °, -2.14-0.1 ~

1.54-0.3k 1.0p

8.64-0.4k 5.04-1.0q

9.2 d

Ge

a(Eo) a(~o) a(E1) (F.d+½,F,u--av) x (.F,a+ I~F,,~ av) L av b bv

-10.3 -0.~5 -5.8 0.32 -4.2 -0.7 ~

_-a ~tt

-2.30 1.6 8.9

D~

5.5

-9.7 ~, -0.6 b

-9.5 ~

-9.2!

1.1 b, -3.5 b, 1.89

1.3d -2.8 c'd 1.0+0.5 h

-9.5 i,

-9.2 ~

-4.74-0.4" -3.9!

-3.8 m -2.864-0.15"

-2.55 d 9.4 d

5.8q-0.6"

GeSi a(E0) a(~0)

a(E1) (=,, + iV.~ -- aV )A (Ed +

-- av) L

av b bc .-z D~

a DME-Model b Ref. 20 i ReL 21 t Ref. 9

-11.5 -0.51 -5.8 0.86

-4.4

-0.7 a -2.28 1.7 7.8 5.7

e Ref. 27 t Ref. 11

d Ref. 5 m Ref. 35

pressure coefficients, as they are obtained directly from our calculations. If necessary, we used the experimental bulk moduli (B0 = 0.99 Mbar for Si and 0.758 Mbar for Ge) [21] to convert the pressure derivatives from the literature to volume derivatives. T h e D P ' s a(Eo) and a(E~) refer to the transitions from the top of the valence band to the F~ (F1 for GeSi) and to the Fls conduction band, respectively. The DP a(E1) refers to the transition from the L~ valence to the L1 conduction band. In general, the agreement between the various calculations and the experiments is excellent. T h e E~ gaps are relatively insensitive to variations of pressure, compared to the E0 gap. The hydrostatic D P ' s of the zinc-blende like compound GeSi turn out to be approximately the average of the Si and Ge values. All of these hydrostatic D P ' s are insignificantly affected by the presence of the &function potentials. Also, since self energy corrections are proportional to pl/s (p-valence electron density) [22] they should contribute to the gap D P ' s by less than 0.5 eV. The method of adjusting the gaps which we use, i.e. the introduction of the adjusting external potentials, leads to somewhat larger changes (1 eV) of the D P ' s for the indirect gaps whereas those of the direct

e Ref. 18 " Ref. 10

/ Ref. 19 ° Ref. 31

9 Ref. 26 P Ref. 32

h Ref. 28 q Ref. 34

gaps are essentially unaffected. The adjusting potentials of the "empty spheres" [14] need to have a rather long range, and this presumably causes an overestimate of the volume effects on the conduction states near L and in particular X. These states have particularly large densities in the outer parts of the empty spheres. Following the notation of Herring and Vx>gt [23], the energy shift of a conduction band valley along a certain direction with respect to the valence band top under hydrostatic pressure is given by

~E9 = (--~ + ~

- av) T, 7

(1)

Here, ~" denotes the strain tensor, and av the absolute DP of the valence band. Thus, the quantity (=d--+ ~=~1--_ av) refers to relative changes in the band position and is equivalent to the pressure coefficient of the indirect gaps. Along the A direction, we have always chosen the transition with the smallest energy, which occurs for Si and GeSi around 0.85 F - X . For Ge, we have chosen the X-point. The DP's are listed in Table 1. Again, the overall agreement is good.

Vol. 75, No. 1

CALCULATED DEFORMATION POTENTIALS IN Si, Ge, AND GeSi

The pressure dependence of the spin-orbit splitting Ao at the top of the valence bands in Ge is of particular theoretical interest. In first-order perturbation theory, the spin-orbit splitting is proportional to the momentum operator ~ o¢ V - i and to the valence charge density Pc of the core [24]. The exclusion principle prevents pc from simply scaling o¢ V -1, so that the measured and calculated volume scaling parameter p( Ao) = d InAo/ d lnV

(2)

has an absolute v~lue smaller than that of the value -1.33 estimated above. The value for Ge recently measured by Gofii et al., p(Ao) = -0.7(4), is in excellent agreement with our result (--0.63). It is listed together with various other spin-orbit splitting volume scaling parameters in Table 2. All calculated p(A0)'s are around -0.6, which means that the effect of the exclusion principle to "freeze~ the core is significant and constant for group IV materials. This also holds for the pressure dependence of the valence band spinorbit splitting A1 at the L point. Similar results have been reported for II-VI compounds by Cerdeira et al. [p(A0,,) ~ -0.5] [25]. The situation is different for the conduction bands. Our results clearly show that the spin-orbit splitting of the lowest p-like conduction bands, A[, is affected much less by a change of volume. This has to do with the fact that it is much easier to compress the antibonding 1~1s wave functions, for which the electron density is small in the center of the bond, than those of the antibonding states, (I'~s), which have a maximum of the electron density between the bonds. In the case of GeSi, our calculations even indicate a positive value for p(A~). This is consistent with the large magnitude of the value we found for the valence bands, P(~o)- We have, however, no detailed explanation of why p(A~) is positive. Absolute hydrostatic DP's account for the absolute shift of the top of the valence band at the F-point under pressure. While this quantity cannot be uniquely defined for a crystal extending to infinity, this is not true for the related DP's which represent the coupling of the electrons to acoustic phonons. In this case, longrange electrostatic contribution to the electron-phonon interaction have to be included. These terms can either be evaluated in large scale ab initio calculations in a supercell geometry [26-28], or in model theories, based on ab initio LMTO deformation potentials, for which screening effects have been added analytically with the Table 2: Calculated volume scaling parameters of the various spin-orbit splittings as defined in Eq. (2).

p(Ao) p(A1)

Si -0.56 -0.65

Ge -0.63 -0.61

GeSi -0.75 -0.65

p(/%)

-0.28

-0.25

+0.17

41

dielectric midgap energy (DME) model [29]. In the latter, the evaluation of the screening with the dielectric function e(w, q) contains two approximations: a) for the evaluation of e only transitions from the top of the valence bands to the bottom of the conduction bands are considered, and b) the screening is estimated by using the midpoint of an average dielectric gap (DME) evaluated at the first Baldereschi special point. To elucidate these approximations, we repeated the calculations as described in Ref. 29 with the inclusion of the b-like potentials, but estimated the DME with 10 special points [30]. The increase in the number of sampling points results in an increase of the absolute DP av by 0.5 eV. This reduces the discrepency between our values and the values given by Van de Walle and Martin [26] slightly, as can be studied in Table 1. Although in both cases the abolute values of the DP's are small, their signs are different. Recent experiments [31,32] indicate for S i a small positive value of 1 - 2 eV for av. It still remains unclear whether a fullscale integration over all bands for the evaluation of e (i. e. the "cure" of approximation b) would yield positive absolute DP's. We should point out that the corresponding electronacoustical phonon interaction depends on ([q for ( ~ 0 [33]. The value estimated here should correspond to its isotropic component.

III. U N I A X I A L S T R A I N In the absence of strain, the spin-orbit interaction splits the threefold degenerate (not counting_ Kramer's degeneracy) top of the valence band F[s at k = 0 into 3 3 1 1 the I~, ~) (v2) and the ]~, ~> (vl) doublet and the ]~, ~) (v3) singlet. A uniaxial strain [8,10] splits the vl, v2 3 1 doublet, and introduces a coupling between the [~, ~) and the 1½, ~) state, causing a non-linear stress dependence of vl and v3. The v2 band is not coupled to the others for [001] and [111] strains and therefore shows a linear dependence on strain (see Fig. 1). In the case that the induced strain-splittings are much smaller than the spin-orbit splitting A0, the change of energy of the various bands can be written as [8] 1 5E,,2 = --~6 Eool ,

(3a)

6E~= +.g~E~ + (~E;.~)UAo+...,

(3b)

6E,z= --Ao L- 2(6E~o,)2/Ao + . . . .

(3c)

For ~ 11[0011 ()~ denotes the strain), the energy terms 6Eoox and 6E~ol are given by 6E0o, = 2(/,1 + 2b2)(Sn - S n ) X = 2b(ez, - er,), (4a) 6E~0, = 2(b~ - 2~)(Sn - S n ) X = 2b'(e,z - e=~), (45) thus introducing the uniaxial DP's b and b'. The strain dependence of the spin-orbit coupling is characterized by the DP ~. We have determined the DP b from the

42

CALCULATED

DEFORMATION

-0.5

IN Si, Ge, AND GeSi

VOI. 75, NO. 1

for all semiconductors mentioned here. Consequently, we list in Table 1 only the values for b ~ b'. They are in good accordance with experiments, and - for the case of Ge - even in better agreement with other ab initio calculations. The deformation potential b of the Fls conduction states, by, were calculated to be about -0.7b. The change of sign is consistent with the pseudopotential calculations of Blacha et al. [3].

-0.7

~-0.9

Uniaxial strain does not only split the p-like bands at the F point, but also affects the band-structure along lines of high symmetry, and introduces electronic anisotropy. The 6 degenerate conduction band minima along the A direction split under uniaxial strain along the [001] direction into two equivalent [001] and four [010], [100] minima by the amount

K_

(D

c -1.1 i,i

13i¸

--1.5 0.00

I 0.01

0.02

stroin Fig. 1

POTENTIALS

I 0.03

(-y)

, 0.04

~Ex

= -~ ( ,, -

~=).

0.05

(5)

While the valleys along the A direction remain degenerate under [001] strain (no interband splitting), the spin-orbit splitting AI of the A3 valence bands is increased (intraband splitting) [8]. For small strains, this effect is quadratic in the stress:

Splitting of the top of the valence band of Ge due to uuiaxial strain 7 = ~(ezz e=). The dashed lines indicate the linear portion of the splitting.

6EA = A 1 "~ 2~2/A1,

splitting of the vl and v2 states for extremely small strains [7 ---- 32-(ezz -- e==) ~ - - 1 0 - 4 ] , for which the quadratic terms in Eq. (3) can be neglected. Furthermore, we determined b' and b2 in the region where 6Eool >> Ao (7 ~ --0.05) from the vl - v3 splitting, using the full expressions for vl and v3 [8], and b2 from the quadratic downshift of the v3 state for various strains as demonstrated in Fig. 1 for Ge. The linear regime of the strain splitting is indicated with dashed lines in this figure. Within the accuracy of our calculation, we thus determined the spin dependent DP for all of the three materials investigated to be b2 = -(0.05 4- 0.05) eV. This is not surprising for Si due to the small magnitude of Ao, and also in accordance with experimental data. Laude et al. [9] found from the indirect exciton spectrum of Si b2 = -(0.1 4- 0.15) eV. In the case of Ge, however, one would expect a somewhat larger magnitude. Chandrasekhar et al. determined b2 from electroreflectance measurments to be -(0.14=i=0.08) eV, in agreement with our calculations. Christensen [13] calculated b2 = 0.0 for GaAs, for which the reported experimental value is even larger than for Ge [b2 : (+0.32 4-0.1) eV] [10]. In view of these results, we believe that b2 is basically zero

,f =

(2) 1'2

D~(e,,,, - e=).

(6a) (6b)

Again, the calculated values for the DP's introduced in Eqs. (5) and (6), ~ and D~, are given in Table 1. The calculated values of D3a for both Ge (5.5 eV) and Si (5.8 eV) are within the experimental error bars of (5.8 4- 0.6) eV [10] and (5 4- 1) eV [34], respectively. IV. C O N C L U S I O N S We have calculated various strain coefficients for Si, Ge and GeSi and compared them to other recent ab initic calulations and experiments. The overall'agreement is excellent, and promises good results for the strainedlayer superlattice calculations which we are performing [7]. We have shown that the volume scaling parameter of Ao defined in Eq. (2) is essentially the same for Si, Ge and GeSi, and agrees with recent experiments by Gofii et al. [11]. The spin-dependent uniaxial DP b2 has been shown to be close to zero. A C K N O W L E D G E M E N T S - We are grateful to S. Zollner for fruitful discussions. The calulations have been performed at "Hhchstleistungsrechenzentrum ffir Wissenschaft und Forschung', Jfilich.

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B a4, 5621 (1986)

VOl. 75, No. 1

CALCULATED DEFORMATION POTENTIALS IN Si, Ge, AND GeSi

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43

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