Short Notes

phys. stat . sol . (b) as, K121 (1978) Subject classification: 2 and 13.1; 20.1; 22.1.2

Physics Department, The New University of Ulster, Coleraine 1)

K121

Electronic and Optic Properties of Crystalline and Amorphous Si

BY

G. P . SRIVASTAVA

In recent years many theoretical attempts have been made to calculate electron-

ic and optical properties of c- and a-semiconductors ( see for general Joannopou-

10s and Cohen /1/ ). Basically two types of Hamiltonians are used: the empirical

pseudopotential Hamiltonian and the tight-binding Hamiltonian. The method of em-

pirical pseudopotential has been successfully applied in many studies on c- and a-

semiconductors ( see, e .g . Tsay et a l . /2/, Chelikowsky and Cohen /3/, and

Ja rds and Srivastava /4/ ).

In this note we use the recent local form factors of Chelikowsky and Cohen /3/

( hereafter referred t o as CC ) and compute the electronic band structure and the

joint density of states in c-Si. We use a slightly different computational detail than

CC use. We then use the perturbed crystal approach t o model an amorphous dia-

mond solid. The method is closely related to that of Kramer and Treusch /5/ in

that we change a structurally disordered solid into a compositionally disordered

one. A finite cubic structure is considered and the positions of the atoms changed

at random by using a Monte Carlo procedure. Then the empirical pseudopotential

Hamiltonian is modified t o obtain a coherent potential Hamiltonian in the Born ap-

proximation, as done by Kramer and Treusch /5/. In this way we represent a real

a-Si by an effective crystal and are thus able to use suitable modifications of the

pseudopotential scheme to calculate the electronic band structure m d the joint den-

si ty of s ta tes . The results compare well with some other theoretical models.

In the reduced zone scheme the one-electron Schriidinger equation is equivalent

to solving the following secular equation ( in atomic units ) /3/: ,

In the local pseudopotential approximation we write

1) Coleraine, Northern Ireland.

K122 physica status solidi (b) 85

W ( b . - b , l = w ( b ) = s ( b ~ w ~ s ) , i j

w (q ) is the local atomic form factor and S (4 ) is the structure factor given by

?; being the position vectors of N atoms. In the c-diamond solid

where f. a r e the position vectors of N/2 sites in the f . c . c . lattice and i$ J 1 = - 4, = (a/8) (1 .1 , l ) a r e the two atomic positions associated with each E' fact id the c-diamond solid S (8) is symmetric so that

=

In j '

S ( 3 ) = S,(Zf) = cos (4 .Q. (5)

where ?. is chosen as the origin. (2) 3

In modelling a-diamond solid we introduce deviations x!') and A . with the J 3

basis atoms situated at and $ In the Born approximation we still solve the

secular equation (1) with the following modifications: 1 2'

i .e. we replace an atomic form factor by an effective, composition-dependent co-

herent atomic potential, as explained by Kramer and Treusch /5/. We further ex-

press the matrix element Wa( $) as

(7) a 4 = Wa(4 ' ) = s (9) w ( q )

with

when E* i s chosen a s the origin, and (9) is a modified atomic form factor for the

perturbed diamond crystal. % (9) differs from w (9) in that the former is obtained

by taking into account the change in the nearest neighbour distance and the atomic

volume as suggested by Tsay et al . /2 / . However, we only take the symmetric part

1

Short Notes K123

of S (a) , which enables us essentiallly to be along the symmetric local pseudopo-

tential approach.

t o r s 2")and 6") derived from a Gaussian distribution function which has a mean

a

We use a Monte Carlo method to generate on computer two sets of random vec-

j j p and a r. m. s . static displacement u compatible with the experimental results

in a-Si.

The joint density of states (JDOS) is written as

J ( E ) = const &[d3k &(En ( c ) - E n (2) - E l . c' v C V

(9)

Here n represents the conduction bands and n the valence hands. F o r a given pho- C V

ton energy E we evaluate J ( t ) by replacing the integral Over the wave vector k +

by a finite sum over M, say, sampling points in

lows Brust /6/:

%3z 1 J ( c = const - - >,

c s n v A s M

where

the first Brillouin zone which fol-

* @ , -b

k . =1 J

( E - Afd2) 5 ( E n (? t ) - E n (c)) d ( c + A E / ~ ) C V

and 0 is a step function defined by

= o otherwise.

.c$z is the volume of the first Biillouin zone. A € = 0 . 1 eV i s a reasonable

choice in Si.

In the case of c-Si 59 plane waves were used and the determinant in (1) was

solved by using the subroutine F02ABF developed by the Nottingham Algorithm

Group. Spin-orbit interaction was not considered. The density of states (DOS)

curve was obtained by solving the determinant in (I) over the four valence bands

and Six conduction bands for 286 sampling points in the irreducible segment of the

first Brillouin zone of the diamond solid. The JDOS was also calculated by evaluat-

ing the expression in (10) Over the four valence bands and s ix conduction bands for

K124 physica status solidi (b) 85

Fig. 1. Band structures. - c-Si, ---a-Si

286 sampling points in the irreducible segment of the

; Brillouin zone.

Fo r a-Si a cubic solid with dimensions five t imes

that of the conventional cubic structure was considered

f o r which a value of p = 0.04

crease in the nearest neighbour distance rl Over the

value U = 0.06 bl was used. These parameters are com

patible with the experimental facts ( Grigorovici /7/ ).

F o r the amorphous case S (200) is not zero which de-

mands to know

curve for the modified form factors described earlier.

Table 1 gives the form factors. Therefore, we use 65 plane waves in the a-Si and

calculate the energy bands, DOS, and JDOS along the lines presented for the c-Si.

The structure factors a r e obtained from (8).

was used for the in-

Oiey crystal value, and a static standard deviation in r of a 1

a x

k'- (200) . We get this by extrapolating the

c-si

Table 1

Lacal pseudopotential form factors for the crystalline and amorphous Si ( in Ryd )

-0.224 0.0 0.055 0.072

I elem&ormfactor I q 2 = 3 q 2 = 4 q 2 = 8 q

I a -Si I -0.246 4.088 0.051 0.071 I

The results of our calculations a r e presented in Fig. 1, 2, and 3. a) c-Si In the case of c-Si our results compare very well with the results of CC who

treated 50 plane waves exactly and used next higher 50 plane waves through the per-

turbation theory used by Brust /6 / . The peaks in the JDOS curve at 4.4 and 5.2 eV

can be associated with the transitions X + Xc and L I + L ~ , respectively. 4 1 3

Short Notes K125

[(el') cfeV)-

Fig. 2 Fig. 3

Fig. 2. Electronic density of states.-c-Si,---a-Si. S , M, and P represent, re- spectively, the s-, mixture of s- and p-, and p- regions

Fig. 3. Joint density of states.-c-Si,--- a-Si

b) a-Si Our calculation showed a thermal gap of 0.37 eV in a-Si when p = 0.04 w and$ = O . 06 1 were used. A number of different combinations of p and d were also

tried yielding that the increase in d rather than in p ( i . e . more loss of long-range

than short-range order) could be responsible for the collapse of the thermal gap in a

modelled a-semiconductor. This i s in contrast t o the finding of /8/ who attributed

the value f o r p corresponding to 11

responsible for the closing of the thermal gap.

increase in the lattice constant in Ge to be

2. In general the valence and the conduction bands come below the crystalline

bands, in contrast to /2/ who observed that the valence bands in a-Si go above those

in c-Si. The conduction bands a r e more sensitive to distortion; the level r2, being

the most sensitive. Various degenerate levels in the crystalline case split when dis-

tortion i s brought in . The removal of degeneracies is a consequence of the non-vanish-

ing of the form factor % (200) and the non-equivalence of some qf the structure factors

Sa for the families (100) , (111) , {220}, and (311) for the reciprocal lattice vectors

considered in the plane wave expansion. This is in agreement with the findings of /5/. "

The splitting of the degeneracies a t XL and X4 i s responsible for the decrease in the gap at

Xinthea-Sicomparedwithc-Si,in line with the suggestion by Phillips /9/. 1

3 . The changes in the DOS observed in Fig. 2 are in general comparable with those of the

works of /2, 5, lo / . However, our simple model does not containmanyodd-number-

ed rings so that the valley between S and M regions is not filled up. Nevertheless we ob-

s e r v e one somewhat broad peak rather than two peaks just above the top of thevalence

K126

band density of states (i . e. in the P region) which is slightly shifted towards the band edge.

Song /11/ regards this coalescence of the two peaks as beingdue to reduction in the s-p

interaction.

physica status solidi (b) 85

4. The JDOS curve has a featureless shape, which is in agreement with /1/ and

/ l a / , The presence of long-range disorder s eems more responsible for this behaviour . We conclude that our calculations in c-Si reproduce CC's results. Based upon

the local pseudopotential approach we have performed electronic and optical calcula-

tions for an effective crystal which represents our modelled a-Si which includes

both short-range and long-range disorder compatible with the experimental facts .The

results of our calculations show the gross features of disorderness in tetrahedrally a-Si . Although we have considered an effective Hamiltonian fo r t h e d i a m o n d solid which can

be justified by using a CPA method, one should really make a self-consistent calcula-

tion a s pointed out in /5/ . Allowance should also be made for the asymmetric part of

the structure factor and corresponding form factors should be determined. Finally the

perturbed crystal approach itself should not be regarded a s very satisfactory a s in a

real a-solid K i s no longer a good quantum number.

Useful discussions with S. Brand and Dr . R .H. Williams a r e thankfully acknowledged

References

/1/ J . D . JOANNOPOULOSandM. L. COHEN, SolidStatePhya . 31, 71 (1976).

/2/ Y . F . TSAY, D.K. PAUL, andS.S. MlTRA. Phys. Rev. B g, 2827 (1973).

/3/ J .R . CHELIKWSKY and M.L. COHEN, Phys. Rev. B 14, 556 (1976).

/4/ M. JAR6kand G . P . SRIVASTAVA, J. Phys. Chem: Solids 38, 1399 (1977).

/5/ B. KRAMER and J. TREUSCH, P roc . 12th Internat. Conf. Phys. Semicond.,

Ed. M.H. PILKUHN, Stuttgart1974; AIPConf. Proc. 20, 133 (1974).

/6/ D. BRUST, Phys. Rev. 134, A1337 (1964).

/7/ R. GRIGOROVICI, in: Electronic and Structural Properties of Amorphous Se-

miconductors, Ed.P.G.LECOhTBER and J . MORT, Academic Press, 1974 (p.191).

/8/ F. HERMAN and J.P. VAN DYKE, Phys. Rev. Letters g, 1575 (1968)

/9/ J .C . PHILLIPS, phys. stat. sol. (b) 9, K1 (1971).

/ l o / W . Y . CHIN and C.C. LIN, Phys. Rev. Letters 34, 1223 (1975).

ill/ K.S. SONG, AIP Proc. Conf. 20, 162 (1974).

/12/ W .Y. CHIN, C. C. LIN, and D. L.HUBER, Phys. Rev. B l 4 , 620 (1976).

(Received December 15, 1977)