The formalism developed by Fletcher (1967) to take account of the presence of short range order in the calculation of the electronic energy spectrum of amorphous covalent semiconductors is examined critically and found to have fundamental difficulties

Klima, J.

McGill, T. C.

English

Caltech Authors - Main

J. Phys. C :  Solid St. Phys., 1970, Vol. 3. Printed in Great Britain 
Critical analysis of the 
‘generalized coherent wave approximation’ 
T. C. McG1LL.F and J. KLIMA 
H. H. Wills Physics Laboratory, Bristol University, Bristol BS8 1TL. 
MS.  received 1 l t h  May 1970 
Abstract. The formalism developed by Fletcher to take account of the presence of short 
range order in the calculation of the electronic energy spectrum of amorphous covalent 
semiconductors is examined critically and found to have fundamental difficulties. 
1. Introduction 
Short range order is widely held to be responsible for the presence of an energy gap in 
elemental amorphous covalent semiconductors. Consequently, theoretical interest has 
developed in showing the connection between these two facts. One promising attempt, 
which has received rather widespread attention, is reported in a series of papers by Fletcher 
(1967 a, b, c). Numerical calculations based on this formalism which have been made in 
this laboratory and others, have been largely unsuccessful and have led the authors to 
undertake a critical analysis of this formulation of the problem. As a result of this study, 
the authors have noted certain fundamental difficulties both with the formalism and the 
results. These errors are of sufficient magnitude to call into serious question the validity of 
this approach. 
2. Review 
To put the remarks contained in this paper in proper perspective, the main points of 
Fletcher’s quantative work (Fletcher 1967 b) are reviewed. In this paper a formalism for a 
two dimensional model with strong short range angular correlations but total absence of 
long range order is developed. Each centre is assumed to be surrounded by Z nearest 
neighbours. The formalism is based upon a generalization of the ‘coherent wave’ approxi- 
mation introduced by Phariseau and Ziman (1963) to facilitate the calculation of the 
density of states of a liquid metal. The ‘coherent wave’ approximation assumes that the 
‘average’ wave function about one site is related to that about another site by a trans- 
formation analogous to a Bloch transformation. 
The generalization consists of the introduction of an additional transformation. This 
transformation adapts the wave function about the various sites to the differences in the 
angular orientation of the spatial location of its nearest neighbours. That is 
(1) 
(equation (13), Fletcher 1967 b), where rj/qj(p) is the wave function about site , j  with angular 
orientation of nearest neighbours a j .  xj is the location of site ,j. R(a) is the additional trans- 
formation which is called the generalized rotation operator. In line with an intuitive 
approach to the problem, R(a) is assumed to satisfy the relations 
@ ( p )  = exp {ik. ( x j  - xi)> R(aj - ai) $y(p)  
and 
t NATO Postdoctoral Fellow. 
2240 
Generalized coherent wave approximation 2241 
(equations (14) and (15) Fletcher 1967 b). Equation (2) is the commutativity and additivity 
relationship satisfied by standard rotations in two dimensions, and equation (3) reflects 
the fact that a rotation of 27112 brings the nearest neighbour structure to an orientation 
which is indistinguishable from its original orientation. 
Because of these conditions, R(a) is taken to have the form 
R,,,(a) = Jmmc 1a,,,  exp (inZma) (4) 
n 
(equation (17), Fletcher 1967 by), where m refers to the basis {exp(imq5)/J2n) and an,, are 
coefficients which are believed to be undetermined by the assumed properties of R(cc). 
The development of the formalism proceeds in the manner analogous to the standard 
KKR method (Kohn and Rostoker 1954), adapted for two dimensions and averaged over 
the disorder using techniques similar to those of Phariseau and Ziman (1963). The wave 
function about a site is expanded in the basis {exp(imq5)/J2~} 
$ ( P )  = bm Rmb) ex~(im4) ( 5 )  
m 
where R,(p) is the solution of the radial Schrodinger equation for the single site potential. 
However, the variational step in this calculation differs in two fundamental ways from the 
analogous step in the standard KKR formalism. First, in this formalism the product 
(labeled p,J of the undetermined coefficients in the expansion for the rotation operator, 
a,,,,, with the coefficients in the expansion of the wave function, b,, are varied. This should 
be contrasted with the standard approach in which only the coefficients in the expansion 
for the wave function are varied. 
Further, the set of p,,,, is usually larger than the set of b,. For, if M is the number of m 
values taken in the expansion of the wave function, and N is the number of n values taken 
in the expansion of the rotation operator, then the set of p,,,, is N times as large as the 
set of b,. 
Second, a condition for the attainment of an extremum of the variational functional 
(equation (27), Fletcher 1967 b) which results in only M linearly independent equations is 
used. Since there are N x A4 unknowns in these equations, it is necessary to generate 
N - 1 additional equations from each equation in this set. This is accomplished by taking 
different values for the orientation of the nearest neighbours about the central site In 
contrast, in the standard KKR formalism all of the required number of linearly inde- 
pendent equations are generated by the condition for an extremum of the variational 
functional. 
The results of the variational step when N = 2 and Z is odd, is a set of homogeneous 
equations for the p,,,,. Using a notation in which Po and p1 are column vectors with 
elements Po,, and pl,,, respectively, the set of equations may be written in the form 
( 6) 
where Fo and F1 are M x M matrices with elements 
and 
respectively. 
t Equation (4) differs from his equation (17) by the appearance of a factor m in the exponential. 
The insertion of an extra factor of m in the exponential makes this equation consistent with equation 
(39) of his paper. 
2242 T. C. McGill and J. Klima 
BE;, and Bg;, contain information about the nearest neighbour and more distant 
neighbour radial distributions, respectively. Az measures the degree of angular correlation. 
Am gives the scattering strength of the single site potential. Noting that 
and that 
equation (6) may be written in the form 
This set of equations has two types of solutions. First, a solution is obtained when 
Po # 0 
p' = 0. 
det F' # 0 
Second, a solution is obtained when 
det Fo # 0 
po = 0 
B' # 0. 
det F' = 0 
(The case when both det Fo = 0 and det F' = 0 will not occur in general). 
These conditions for an eigenvalue are compared with those which are obtained from a 
standard KKR calculation for a perfect crystal with two identical atoms per unit cell 
For k = 0 these conditions are 
and 
where Bimt and B;kr are the structure factors for the equivalent sites and nonequivalent 
sites, respectively. The similarity of equation (14) with equation (124 and equation (15) 
with equation (13c) is noted. In particular, the presence of a similar minus sign in equations 
(15) and (13c) leads to the conclusion that the bonding-antibonding character of the 
eigenfunctions which exist in the perfect crystal at k = 0 is also found in the disordered 
case. Since the bonding-antibonding character may lead to a band edge in the ordered 
case, one might expect analogous behaviour in the electronic energy spectrum in the 
disordered case. It should be noted that this analogy is made possible by the increase in 
the number of variational parameters due to the undetermined coefficients in the generalized 
rotation operator. 
Generalized coherent wave approximation 2243 
3. Criticism 
The properties ascribed by Fletcher (1967 b) to the generalized rotation operators allow 
the determination of the set of an,m. As is well known, the additivity and commutativity 
relation given by equation (2) plus the differentiability of the operator with respect to CL 
implies that 
R m m b )  = exp [i{q(m) CI + ym>] hmm, (16) 
(Messiah 1962) where ym is a phase and q(m) is a constant independent of CL. The additional 
condition that R(0) = 1 (equation (3)) implies that ym = 0. Thus, the ‘arbitrary’ coefficients 
in equation (4) must be given by 
a n , m  = d n ,  n ’ ( m ) ,  (17) 
where n’ is an integer which may be a function of m. 
Hence, for a single generalized rotation operator (specified by giving the set of n’ as a 
function of m in equation (17)), only M of the variational parameters p,,,, are different 
from zero. The number of homogeneous equations required is M ,  not N x M ;  and the 
argument which generates N - 1 additional equations is not required. In addition, it is 
impossible to compare the results from a single generalized rotation operator with the 
result from a standard KKR calculation for a perfect crystal with two atoms per unit cell 
since the determinental conditions which are obtained in the two cases have different 
orders. 
However, the solutions obtained by Fletcher are consistent with the above noted require- 
ments on the an,m. For, when the solution is given by equations (12), then equation (124 is 
satisfied by taking all the 
a 1 , m  = 0 (184 
ac)m = 1. (18b) 
When the solution is given by equations (13), then equation (13b) is satisfied by taking all the 
a0,m = 0 (194 
a 1 , m  = 1 (1%) 
The authors believe that this consistency is not due to the original assumptions about 
R(a) but is directly related to the manner in which the set of N x M homogeneous equa- 
tions were obtained from M homogeneous equations. 
The question then arises as to whether one might view Fletcher’s results as coming from 
two separate generalized rotation operators with the an,m given by equations (18) and 
equations (19), respectively. Then perhaps the solution and analogy would be valid. The 
answer to this question seems to be no. For, when the solution and analogy are studied 
carefully, other difficulties appear which seem to invalidate the analogy. 
Comparison of the definitions of the structure factors in equations (7) and (8) with those 
in equations (14) and (15) indicates that even though they are labeled in a similar manner 
they are somewhat different. Neither B;,, nor BE;, which contain information about the 
location of atoms throughout the whole crystal, are directly analogous to BgA,, which only 
contains information about the average nearest neighbour locations. This dependence of 
B z i  leads to the rather unusual result that eigenvalue5 produced by a solution to equations 
(13) will depend only on the average nearest neighbour structure in contrast with the case 
of a perfect solid where the eigenvalues are dependent upon the location of all the atoms in 
the crystal. 
and equation (12b) is consistent with taking all the 
and equation (136) is consistent with taking all the 
2244 T. C .  McGill and J. Klima 
The weakness of the analogy is further demonstrated by looking at the variation of the 
wave function from site to site using transformation (1) and the solutions for the an,,, given 
in equations (18) and (19). The wave functions at k = 0 are found to exhibit a character 
which is different from the standard bonding-antibonding character found in the case of a 
perfect crystal. To illustrate this point, consider a wave function with only a m = U com- 
ponent. In the case where a,,, = 1 and u1,,  = 0, the generalized rotation operator reduces 
to the unit operator. Thus, 
IC/,@) = $ I @ )  (20) 
and the wave function is of the bonding type. However, in the case where U,,, = 0 and 
= 1 the generalized rotation operator is just the ordinary rotation operator. Since 
the m = 0 component is invariant under rotation, 
= $,(PI (21) 
and again one has a bonding solution. Thus, the two solutions are both bonding like. If the 
analogy were valid, one would expect one of these to be of antibonding character. The 
addition of higher m values does not resolve this point, and results in a wave function 
which in the case of equations (19) may not be identifiable as either bonding or antibonding. 
Hence, the analogy hardly hardly seems valid. It should be noted that for a perfect crystal 
the antibonding s-wave functions about various sites are not related by a simple rotation. 
4. Conclusion 
The above-stated difficulties force the authors to conclude that the formalism and 
analogy used by Fletcher suffers from some rather fundamental inconsistencies which are 
of such magnitude as to question seriously the validity of this approach to the discussion 
of the energy spectrum of amorphous covalent semiconductors. 
Acknowledgment 
vestigation and Professor N. H. Fletcher for his correspondence with us. 
We would like to acknowledge Professor J. M. Ziman for suggesting this area of in- 
References 
FLETCHER, N. H., 1967a, Adv. Phys., 16,7034.  
~ 1967 b, Proc. Phys. Soc., 91, 724-35. 
- 1967c, Proc. Phys. Soc., 92, 265-7. 
KOHN, W., and ROSTOKER, N., 1954, Phys. Rev.,  94, 11 11-20. 
MESSIAH, A., 1962, Quantum Mechanics (Amsterdam: North Holland) Pp 653-4. 
PHARISEAU, P. and ZIMAN, J. M., 1963, Phil. Mag., 8,1487-50. 


Critical analysis of the 'generalized coherent wave approximation'

Quantum Mechanics

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Critical analysis of the 'generalized coherent wave approximation'

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