Acta Physica Academiae Scientiarum Hungaricae, Tomus 36 (3), pp. 311--322 (1974)

APPLICATION OF HARRISON'S FIRST-PRINCIPLE PSEUDOPOTENTIAL METHOD FOR SEMICONDUCTORS

II. CALCULATION OF ENERGY BAND STRUCTURES

By

[L. V. KOTOVA, [ E. V. GALAKTIONOV, V. E. KHARTSIEV 1 m

A. F. J O F F E PHYSICO-TECHNICAL INSTITUTE, USSR ACADEMY OF SCIENCES, LENINGRAD, USSR

a n d

T. VSnSs RESEARCH I N S T I T U T E FOR TECHNICAL PHYSICS OF THE HUNGARIAN ACADEMY OF SCIENCES,

BUDAPEST

(Reeeived 14. VI. 1973)

The HARnlso.w first-principle pseudopotential method (FPPM) is combined with thv L~WDrN-BRusT method for the determination of electronic energy levels for Si and Ge. The FP conduetion bands computed by the use of the previously calculated FP forro factors differ from the known data. For this reason the FP forro factors are modified with an empirical correction made. The resulting first three Fourier coefficients and the corresponding band str~ctures agree with the empirical pseudopotential ones. The physical reason of the differenc~ between the FP form factors and the empirical pseudopotential ones is considered in detail.

1. Introduct ion

I t is well known tha t the pseudopotent ia l form factors can be ob ta ined with the aid of several methods; in par t icu lar the empirical pseudopoten t ia l method (EPM), the model pseudopotent ia l me thod (MPM), and the first- principle pseudopotent ia l me thod (FPPM). In the EPM th ey are de termined empirically from crystall ine energy levels [1]. In the MPM th ey are computed from free-atom te rm values [2, 3, 4]. For semiconductors the diffcrence between the form factors of EPM and MPM was explained by PHILLIPS [5] (see also [6, 7, 1]). He pointed out t ha t it arose from the metallic screening of the bare pseudopotent ia l and from the cont r ibut ion of bonding charges.

Other works also emphasize the significance of the inclusion of covalent contr ibut ions. BROVMAN and KAGAN [8] f i t ted the phonon spectrum of white Sn taking into account noncent ra l neares t neighbour in terac t ions (covalent effects), and the pseudopotent ia l was similar to the adequate model potent ia l [7]. MARTIN [9] f i t t ed the phonon spec t rum of Si taking into account ion- ion, bond-ion and bond-bond interact ions; the pseudopotent ia l is comparable to tha t of EPM [7]. WEAInE [10] modif ied the pseudopotent ia l theory (using the Ashcroft forro with the HunnARD--SHAM dielectric funct ion) applied to

zŸ Physica Academiae Scientiarum Hungarir 36, 1974

3]2 L.V. KOTOVA et al.

non-transition metals, particularly Na, Mg, Al, for the calculation of the bind- ing energy and equilibrium atomic volume of semiconductors with the inclu- sion of covalent coutributions.

From the above ir can be seen that : 1. For semiconductors the differ- ence between the form factors of EPM and MPM is understood. 2. When the pseudopotential theory successful for simple metals is applied for semicon- ductors, the theory must be modified by taking the "covalency"into account.

For the FPP theory the situation is different. After HARRISON [11] part I of this work [12] applied the FPP theory of simple metals to semicon- ductors and determined the difference between the forro factors of EPM and FPPM. The aim of the present part of the paper is to understand the sources of this difference. I t happens as follows.

The form factors calculated in [12] make it possible to study the elec- tronic structure. The band structure of semiconductors using the HARRISON FP form factors has not been investigated, although the OPW method is generally used in FP band structure computations for semiconductors. In this paper we do not use the HARRISON perturbat ion theoretical solution of the pseudoequation to calculate conduction band states and energies. Therefore another method must be employed in determining the eigenvalues of the cor- responding pseudoequation. I t is the L6WDIN BUVST method [13 to 17], developed and used in EPM, but the form factors ate computed theoretically, with the aid of FPPM. Thus two different methods are combined: 1. The I-IARRISON FPPM, for the calculation of form factors. 2. The L0WDIN--BRusT method, for studying the electronie structure.

Although many excellent t reatments of the L0WDIN BRUST method are known [1, 13 to 17], those questions as essential for understanding the present investigations will be shortly outlined. So the FP band structure of Si and Ge will be determined. With the FP forro factors and band structure known the FP form factors will be modified and the origin of contributions of the above-mentioned correction considered.

2. Theory and results

The pseudoequation for an electron in a crystal is

{ 'q g~" ) , . H*',,k = -- -2m- + ~ q',,i = Enk q 'n , , (1)

where W is a local pseudopotential, @n.k is the pseudo-wave-function, and En, k is the true eigenvalue for the state of reduced zone rec tor k and of band index n. The pseudo-wave-function ~.~.k is expanded in terms of plane waves:

{~rl,k : ~ a..k(Qj) lk + Q]>, (2) Qr

Acta Physica Academiae Scientiarum Hungaricae 36, 1974

APPLICATION OF HARRISON'S F IRST-PRINCIPLE PSEUDOPOTENTIAL METHOD II. 313

where thc Qj terms ate the reciprocal lat t ice vectors. Subs t i tu t ing the expan- sion (2) into (1), mult iplying on the left by <k + Q~] and integrat ing over a unit cell of the crystal , the pseudoequat ion (1) is reduced to a set of linear algebraic equat ions for the coefficients an.k(Qj ). From the consistency condi- tion ah infinite de terminant secular equat ion is given for the determinat ion of eigenvalues

IHi, J(k) E(k)6ijI = 0 , (3) where

¡ H/,J(k) ---- <k § Q;[H[k § Qj> - Ik§ 2 6,.j § 2 4 7 2 4 7 >. (4)

2m

I t is shown in [11] tha t

<k + QiIWlk § Qj> = S(Qi - Qj)<Q;IwIQj> = S(Q)V(Q), (5)

where Q = Q; Qy is the difference of reciprocal lat t ice vectors, S(Q) is the s t ructure factor, and V(Q) the forro factor, where Q is the magni tude of the reciprocal lat t ice veetor. As usual the s t ructure factor is determined in such a way tha t the origin of coordinates is hal fway between the two atoms in the unir cell, the coordinates of which ate:

( 1 1 1) ~ : b = • , , ,

8 8 8

where a is the lat t ice constant . So

S(Q)V(Q) = cos(Q .b) 2-�91 w(r)e-iQ"dr, (6) - ( d i

where ~Q is the volume of the unir eell, and w(r) is the pseudopotent ia l due to single atoms.

The first f ive reciprocal lat t ice vectors in units of 2:t/a ate: (0, 0, 0), (1 ,1 ,1 ) , ( 2 ,0 ,0 ) , (2, 2 ,0) , (3 ,1 ,1 ) . In consequence of S ( 0 , 0 , 0 ) # 0 the diagonal eontains the Fourier coefficient V(0) of the potent ial . I t is assumed to be zero. Fur the r on S(2, 0, 0) = 0. So three form factors V(Q1), V(Q2), V(Q3) remain, where Q1, Q2, Q3 are the magni tudes of the reciprocal latt ice vectors (1, 1, 1), (2, 2, 0), (3, 1, 1) in units of 2zc/a. The form faetors corresponding to the fur ther reciproeal latt ice vectors are neglected.

To get a good convergence of eigenvalues of (3) (within 0.1 eV) one should operate wi th matrices of large size (~v 50 • 50). The LiiWDIN--BaUST scheme based on the per turba t ion theory offers a possibil i ty of determining eigenvalues for such large matrices by wa y of reducing the size of matrix. Following BRVST the secular equat ion is wri t ten

!U n,m -- E6 ",mI = O, (3a)

Acta Physica Academiae Sr ltuntgarir.a8 36, 1974

3 1 4 L . V . KOTOVA et al.

where r Hn.r Hv,m

U'~, '= H"," § , ~ , (7) ~=N+I E - Hr,"

where n, m ~ N and 7 " ~ 7 > N. N and 7' are integers. The indices n and m label the plane waves treated rigorously, ~ refers to the highest plane waves treated through the perturbation theory. So some lower E N and upper E r cutoff energies correspond to the plane waves N and 7", and the plane waves

tŸ of energy-~m [k§ < EN ate taken into account rigorously, and the plane

?/2 waves of energy EN < 2-~ lk§ <E/ , are treated through the perturbation

scheme (see also the discussion in [1]). The valueschosen in this paper are:

EN=-8 40 eV, and therefore the average secular equation is 2m

~-~24X24. Around 90 instead of ~-, 70 [14] plane waves wcre taken into account through the perturbation theory; however, the upper cutoff E r was

2~r somewhat higher, because the set of plane waves t k + (4, 2, 0)> was left

a

out. The increase of lower and upper cutoff improved somewhat the conver- gence of eigenvalues, which was within 0.1 eV. The eigenvalue dependence of the U n'm elements [Eq. (7)] was eliminated so that the eigenvalues were replaced by a reasonable average of the eigenvalues in the case of off-diagonal elements, and by the kinetic energy in the case of diagonal ones. In this

w ~ ~2 { ? ) z ~ 10 e v 2 m (i.e., the average of the first eight energy

levels) was used in all cases for these matrix elements. With the aid of the FP form factors of Si and Ge version (i) (i.e. Har t ree- -

Fock wave functions, the Slater exchange [12]) on the basis of this method the band structures along the A (10 points -4-/ ' point) a n d ~ (20 points) axis and along the X - IV (5 points), I V - K (2 points), K - 7" (8 points) lines werc computed in the optical, near- and far-ultraviolet regions without taking the spin-orbit intcraction into aceount. So the energy levels were computed at 46 points of the reduced zone. At first, fixed values of N and F were used (N-~ 25, 7" = 120). After it the dependence of N and /~ on k, E~v and E r was taken into consideration and the results were controlled at seven points

(in units of 2zr/a) of reduced zone: 7" (O, O, O), L l -} , l ) , A {-}, 0, 01 , A~in

(0"85' O' 0)' X (l ' 0' 0)' Z" {-2 - ' 1 } ( 1 ' ) -~ , 0 and IV 1, ~- 0 .The energylevels for Si

and Ge (i) are shown in Table I in eV at the points 7" (0, 0, 0), L 2 2 2

Acta Physir Academiae Scientiarum Hungarir 36, 1974

APPLICATION OF HARRISON'S F IRST-PRINCIPL E PSEUDOPOTENTIAL METHOD II. 315

a n d X (1, 0, 0) of reduced zone. At each point the list gives 4 valence levels

(below), and 7 conduct ion levels for point /,, 4 for point L and 2 for point X (above). These are the lowest conduct ion levels for these points and it is so

for the FP energy levels, too, exeept point L of Ge, where some other

conduet ion levels ate below the level L r The zero of energy is at F25, the top of the valence band. 7 conduct ion levels are given for point /,, because

for Ge /'2' lies above the levels /'12"-

Table I FP and FP ~- C eleetronic energy levels for Si and Ge in eV

Si Ge

Level FP FP A- C FP version (i) FP -~ C

F 1 2 ,

F1 -Y'x5

F z ,

/~25"

F1

L 2 ,

La

L1

La,

L1

L z ,

X 1

X4

X1

8.07

3.86

2.42 --2.41

0.00 -- 14.46

8.10

7.47

3.42

4.00

0.00

7.13

7.44

3.44

8.42

0.00

- 12.74 --11.13

8.09 6.16

3.26

0.89

0.00

--12.18

--9.05 3.09

5.83

--1.42

--1.75

7.88 3.33

4.01 3.82 1.89 13.54

--1.23 --1.04 -- 7.40 -- 6.35

--12.28

--0.98

--3.41

--10.41

--10.31

0.90

--2.57

--8.47

--8.88

1.35

--2.67

--6.871 - - 7 .09]

7.42

4.02

0.62

--1.11

--7.15

--10.13

0.82

-- 2.68

-8.39

F rom the results the following conclusions can be drawn: The Si F band s t rueture seems to be ah exaggerated version of the GROVES--PAU

modcl for semimetal ~-Sn [18]. The si tuat ion is very similar to the band

s t ructure shown for dilated Ge crystal in [19], except t ha t on the A axis the E - - k curve of F + (in double group notat ion) is concave towards the k-

axis, while on the A axis, as in [19], it is convex towards the k-axis, and the

/ , point is its inflection. On the eont rary , the FP Ge is a semiconductor . As seen above, only the first three Fourier coefficients were taken into

account. KA~E [20] (see also [1]) found for Si t ha t enlarging the number of nonzero V(Q)'s in addit ion to the first three did not improve the fit to the

energy band s t ructure .

8 " Acta Physica .4cademiae Sr Htmgaricae 36, 1974

316 L . V . KOTOVA et al .

2.1 Modification of the F P form factors

N o w we sha l l e o n s i d e r t h e m o d i f i c a t i o n o f t h e F P P fo rm f a c t o r s . L e t us

d e n o t e t h e m o d i f i e d F P P f o r m f ac to r s b y VFp+c, t h e n the r e s p e c t i v e b a n d

s t r u c t u r e s a te t h e F P a n d F P + C ones. T h e F P f o r m f a c to r s a n d b a n d s t r u c -

t u r e a t e k n o w n f rom [12] a n d t h e p r e c e d i n g c a l c u l a t i o n a n d i t is a s s u m e d t h a t

t h e e x p e r i m e n t a l v a l u e s o f some e n e r g y gaps a re a lso k n o w n . T h e d i f f e r e nc e

b e t w e e n t h e m e a s u r e d v a l u e o f a gap EFp+c a n d t h e c o r r e s p o n d i n g c o m p u t e d

one E F p is dE i -- (E~p)i - - (E~p+c) i , i = 1 . . . . . n, w h e r e t h e i n d e x i l a b e l s

t h e d i f f e r e n t gaps . T h e r e s p e c t i v e d i f f e rence o f f o r m f a c t o r s is

dV(Qj) = V p , ( Q j ) Vp,+c(Qj) , j = 1 . . . . . m,

Table II

VFp+c forro factors for Si and Ge in Ryd

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Si Ge

--0.6074 --0.5677

--0.5920 --0.5611

--0.5668 --0.5447

--0.5331 --0.5192

--0.4928 --0.4862

--0.4477 --0.4474

--0.3997 --0.4043

--0.3503 --0.3587

--0.3009 --0.3123

--0.2524 --0.2663

--0.2059 --0.2220

--0.1619 --0.1801

--0.1207 --0.1413

--0.0827 --0.1057

--0.0477 --0.0733

--0.0159 --0.0439

0.0128 --0.0171

0.0386 0.0077

0.0617 0.0307

0.0823 0.0547

(8)

w h e r e t h e i n d e x j shows t h e n u m b e r of for ro f a c t o r s t a k e n in c o n s i d e r a t i o n

fo r t h e b a n d - s t r u c t u r e c o m p u t a t i o n . I n [21] i r is n o t e d t h a t f o r a f i x e d k t h e

e n e r g y o f a gap E t is a f u n c t i o n o f t h e F o u r i e r coe f f i c i en t s V(Qj) ( i f t h e l a t t i e e

c o n s t a n t is f i xed ) , i .e . ~:~. = ~ : , . ( v ( Q j ) ) ,

Aeaa Physice Acad,~nir Sr H u n g a r ~ ~6, 1974

APPLICATION OF HARRI$ON'S F IRST-PRINCIP L E PSEUDOPOTENTIAL METHOD II. 317

SO

d E i = . , ~ __3Ei dV(Qj) i = l , . . . , n; j ---- 1, . . . , m, (9) Q,~Q, 8 V(Qj)

where b e y o n d Q0 the form fac tors b e l o n g i n g to the rec ip roca l l a t t i ce vec to r s

~Ei ate n e g l e c t e d . - is the p a r t i a l d e r i v a t i v e of the E i gap accord ing to 8V(Qj) the form fac tors V(Q]), which can be d e t e r m i n e d in the F P scheme. F u r t h e r

on i t is a s s u m e d t h a t i = j = 1 . . . . , n. D e n o t i n g the d e t e r m i n a n t of the set

of e q u a t i o n s b y D, a n d the cofac to r of the e l e m e n t

be w r i t t e n

. ~ Di'QJ d V(Qj) . . . . . d E i .

i=~ D

OE i b y Di,Q~ , i t can OV(Qj)

(10)

E a c h form fac to r d i f ference dV(Qj) can be w r i t t e n in a p o l y n o m i a l f o r m:

I Qj q dV(O/) = -- .~i=, al ( V J 00a)

Table III

The VFp(Q ) Fourier coefficients, the contributions of the dV(Q) correction and the resulting Fourier coefficients V~p+c(Q ) for Si and Ge version (i), (ii) and (iii) in Ryd

VFe(QI) [121

Screening [61

Bonding [61

Wanted correction

Si Gr (i) Ge (ii) Ge (iii)

--0.2204

--0.1078

0.0310

0.0904

--0.1546

--0.0769

0.0220

--0.0125

--0.2663

--0.1326

0.0220

0.1549

VFp+c(Q1) -- 0.2068 -- 0.2220 -- 0.2220

VFp(Q 2) [12] 0.0122 0.0771 --0.0524 --0.0160

Screening [27, 28] --0.0024 0.0142 --0.0097 --0.0030

Bonding [5, 29] . . . .

Wanted correction 0.0532 --0.0836 0.0698 0.0267

VFP+C(Q 2) 0.0386 0.0077 0.0077 0.0077

VFp(Q3) [12] 0.0171 0.1188 --0.0164 0.0170

Screening [27, 281 0.0025 0.0168 --0.0023 0.0024

Bonding [5, 29] . . . .

Wanted correction 0.0627 --0.0809 0.0734 0.0353

VFp+c(Qa) 0.0823 0.0547 0.0547 0.0547

--0.2202

--0.1097

0.0220

0.0859

--0.2220

Acta Ph.ysica Academiae Scienliaru,n H!t.t~qricae 36, 1974

318 L . v . KOTOVA et al.

where k F is the Fermi wave n u m b e r of a free electron gas wi th dens i ty equal to t h a t of the valence electrons. The coefficients of po lynomia l a i can be de t e rmined from the equa t ion

-- ~ a t-Fiel - - , = , O

Since the po lynomia l is val id for all rec iprocal- la t t ice vectors t a k e n in con- s idera t ion when comput ing the band s t ruc ture , i t can be generalized for the cont inuous form fac tor curve. The modif ied pseudopoten t i a l fo rm fac to r is

Vpp+c(q) -- VFp(q) - aV(q) = Vpp(q) 4- . ~ ai i=1

where q denotes the eont inuous var iable . When apply ing this m e t h o d first of all the degree of po lynomia l is to

be f ixed. ] t is evident t h a t n can have different values for different mater ia ls . F r o m the EPM works [1, 14 to 17] it is known t h a t for Si and Ge the poly- nomia l is of th i rd degree. Assuming tha t the th i rd degree t e rm eont r ibu tes a small value, it can be negleeted. The coeffieients aa, a 2 can be ea leula ted f rom the following equat ions :

s ; Q2

g l =

D (Q1 ke

. , ~ [ k p D - - -

a 2

Q1 Q2 Di'Qz} dEi

kp

kF Q2 Di'Q~)

v(o2 011 kF kp )

(12)

The coeflicients a l , a 2 can be considered as ad jus tab lc p a r a m e t e r s to be deter- mined f rom the expe r imen ta l da ta . They are i n R y d for Si: a I 0.0100, a 2 = 0.0214, for Ge version (i): a 1 0.0962, a 2 0.0323, and they are used fu r the r on. They were de te rmined in such a w a y t h a t for Ge the expe r imen ta l

values of energy gaps F2~, ~ 12' (0.90 eu f rom [22, 23, 24]) and -~25' ----* FI5 (2.98 eV f rom [24], see also [25]) were adjusted. For Si the cons tan ts of quadra - t ic equa t ion giving the difference be tween the Four ier coefficients of E P M and F P P M were used, wi th the aid of which the va lues of energy gap F25, --~

F 2, publ ished in [23] can be adjus ted wi th in 0.1 eV. The F P + C forro fac tors for Si and Ge are shown in Table I I . So the Four ier coefficients for

Si in rydbergs are: VFp+c(Qx) = - 0 . 2 0 6 8 , VFp+c(Q2) = 0.0386, VFp+r =

- - 0.0823, for Ge VFP+c(Q1) = --0.2220, VFP+c(Q2) = 0.0077, VFP+C ( Q s ) :

Acta Physica .4vademiae Sr Hungarir 36, 1974

APPLICATION OF HARRI$ON'S FIRST-PRINCIPLE PSEUDOPOTENTIAL METHOD II. 319

--0.0547. Wi th the aid of the difference VFp(Qj) VFp+c(Qj) the coefficients a i calculated f rom the above equat ions ver i fy the values al, a2, a n d a s ir was assumed, the th i rd degree t e rm gives a small contr ibut ion. However , when there is a s ignif icant difference be tween the EFp-Value and the required exper imenta l one in relat ion (9), then the equat ions (9), (10), (11), (12) eannot be applied in one process to the exper imenta l values.

I t is appa ren t t ha t the FP + C Fourier coefficients approx imate ly agree with the EPM ones de termined with an accuracy of 0.01 Ry d [16]. The dif- ference between the EPM [26] and F P + C Fourier coefficients VEpM(Q) -- -- I/'FP+c(Q) is for Si 0.0042, 0.0017, and --0.0018 Ryd , for Ge 0.0100, 0.0031, and 0.0057 Ryd for Q1, Q2 and Q3-

According to [1, 7] the form factors are character ised by the values of qo/kz, where the form factor equals zero. For the F P -+- C form factor these values are 1.65 for Si and 1.77 for Ge.

In the case of Ge the version (i) was the s tar t ing p o in t to de te rmine the

V E P M ( Q ) empirical forro faetors. The version (i) is not the best eompared with Ge [12], nevertheless the empirieal eorreet ion has given eorrect forro faetors. I t verifies t ha t for the de te rmina t ion of the empirical forro faetors one need not s tar t f rom the FP form factors eoming nearest to the EPM ones. Each version equally leads to the eorrect EPM Fourier eoefficients, of course, by

using different coefficients al, a 2.

2.2 Meaning of correction dV(Q)

As a rcsult of the invest igat ion ment ioned in Sect ion 1 the difference between the VFp(Q ) and VFp+c(Q ) Four ier coefficients is shown to originate from the following sources: the applicat ion of metallic screening funct ion, the cont r ibut ion of bonding charges, and the diffcrencc bctween the MPM and F P P M form factors of a bare ion.

The difference between the MPM and EPM form factors is given from the scrcening and bonding contr ibut ions [5]. They should be taken into account for the F P P M form factors, too. Screening was discussed in [12] and the screening correct ion was secn to involve two contributim~s: 1. the dif- ference between the Har t ree screening and ~s(q) [6], and 2. the exchange among conduct ion electrons.

The necessi ty of a fur ther cont r ibut ion is shown numerica l ly in Table I I I for Q1, Q2, Q3, which confirms the existence of correction. The necessary cor- rect ion was de te rmined on the basis of the difference VFp+c(Q ) -- [VFp(Q ) +

+ Screening " Bonding] , where Scrcening = - VFp VFp, and E is the

Har t ree dielcctric funct ion for free electrons, E* is the screening funct ion [6] including the exchange correct ion among conduct ion electrons [12]. In the

Acta Physica Acctdemiae Scientiarum Hungaricc*e 36, 1974

3 2 0 L .V. KOTOVA et al.

case of Q1 the Cs-data needed for the calculation of Screening are taken from [6], for Q2 and Q3 the corresponding ones are from [27, 28]. For Q1 the Bonding contribution is taken from [6], for Q2 and Qs it is neglected, as the Q2 component contributes nothing to the bonding charge [5, 29].

Denoting by VMpd(Q) the MPM and by vvpd(Q) the FP forro factors of bare ion the statement tha t the required correction C(Q) is given from their difference will be verified. Taking the C(Q) correction into account the FP -f- C form factor is written

VFp+c(Q) = vvp,i(Q) + C(Q) + Sb(Q) Vb(Q), (13) ~~(q) So(Q)

where V~(Q) is the bonding forro factor, S~(Q)/Sa(Q) is the ratio of the structure factors of the bonding charges to those of the atoms. From this C(Q) is expres- sed and used that

VFp+c(Q) Sb(Q) Vo(Q ) ~ VMp(Q) _ v~pa(Q) ' (14) Sa(Q) ~s(Q)

because

VFP+c(Q) ~ V~pM(Q) ~ W(Q) = VMp(Q) + Sb(Q) Vb(Q ) , Sa(Q)

(15)

where Vp(Q) 1 is the complete pseudopotential form factor. It follows thus, that

C(Q) = VMp.~(Q) --VFpd(Q).

So the statement for the correction dV(Q) is confirmed:

(16)

dV(Q) = v~P'i(Q~) - VFp,i(Q)_ _ Sb(Q__~) Vb(Q ) _ VMz.i(Q)-- VFp,~(Q) (17) E(Q) ~s(Q) Sa(Q) ~s*(Q), '

which includes the screening, bonding and C(Q) correction of FP form factors of bare ion. I t should be noted here that Eq. (17) cannot be generalized for q --~ 0.

Eq. (16) also explains the differences of C(Q) corrections between the versions given for Ge. The values of vFp,i(Q ) depend on the selection of the exchange potential and core eigenvalues and it causes the dependenc e of C(Q) on version.

So on the basis of (17) the difference dV(Q) for Si and Ge (iii) can be interpreted as follows: for Q 1 - dV(Q1)~ O, for Q2, Q 3 - dV(Q)> o and increases, because for Q1 the negative screening, positive bonding and positive C(Q) approximately compensate each other, for Q2, Q3 the positive C(Q) domi- nates, the screening and bonding give a small contribution.

Acta Physir Academiae Scientiarttm Hungaricae 36, 1974

APPLICATION OF HARRI5ON'S FIRST-PRINCIPLE PSEUDOPOTENTIAL METHOD II. 32I

3. Discussion

With the aid of FP + C form factors the FP § band structures were computed. The electronic energy levels are shown in Table I. For Si the com-

parison with the EPM band structures does not show any peculiar difference,

for Ge it shows tha t the values of gaps/"25" ~ / ~ 2 , and/'25" ~ I"15 were adjusted in this paper. The comparison with the FP electronic energy levels shows the significance of correction dV(Q).

Fur ther , the following invest igations were made: the Fourier coefficients

of SixGel_ • alloy were determined from a linear in terpolat ion between those

of pure Si and Ge and the band s t ructures were computed as the funct ion of Si mole percent for 5 different values. The results are compat ible with those

of [31]. The Fourier coeffieients of compressed Si and Ge were also determined

and the band structures calculated a s a funct ion of lat t ice cons tant for 4 dif- ferent values in each case. The pressure coefficients of form factors and energy

gaps were also computed and the results were found to be compatible with

those of [32, 33].

These facts suggest tha t if the HARmSON F P P M is to be applied to semi- conductors, or semiconductors ate to be approached from the a priori theory

of simple metals, then some essential differences between the simple me-

tals and semiconductors must be taken in consideration, namely :

1. The screening potent ial of semiconductors is different from tha t of

simple metals. So instead of the Har t ree screening the semiconductor sereen- ing funct ion tak ing the exchange between the conduct ion eleetrons into ac-

eount mus t be used.

2. The bonding forro factors mus t be taken in consideration. 3. The C(Q) correction of the F P bare ion forro factor mus t be taken in

consideration.

The analyses made in these works indicate t ha t the general f ramework

of the first-principle pseudopotent ia l me thod can be const ructed for semi- conductors, similarly to the HARRISON's a priori theory of simple metals.

Acknowledgement

The authors are grateful to Prof. A. A. KAPLYANSKII, Prof. I. ABARENKOV, Prof. W. A, HARRtSON, Prof. Z. BOD£ Dr. F. BELEZNAY, and Dr. T. GESZTI for the discussions. The sugges- tion of the topic by Prof. G. SZIO~TI is gratefu]]y acknowledged. We would like to thank Dr. N. N. VASSIL'EVA fol" scrutinizing the style of the manuscript. One of the authors (T. V.) wishes to express his gratitude to the A. F. 3offc Physico-Technical Institute of the USSR Acadcmy of Sciences and to Prof. V. M. TUCHKEVICH and Prof. E. F. GRoss for the kind hospitality. He great]y appreciates the support of the l~esearch Institute for Technica] Physics of the Hungarian Academy of Sciences.

Acta Physica Academiae Scientiarum Hungaricae 36, ~1974

322 L . V . K O T O V A et al.

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Acta Physica Academiae Scientiarum Hungaricae 36, 1974