A calculation of the density of electronic states for amorphous semiconductors

Journal of Non-Crystalline Solids 114 (1989) 253-255

253

North-Holland

A CALCULATION OF THE DENSITY OF ELECTRONIC STATES FOR AMORPHOUSSEMICONDUCTORS o

J.L. BEEBY & T.M. HAYES+'% Department of P h y s i c s , L e i c e s t e r U n i v e r s i t y , L e i c e s t e r , LE1 7RH, U.K. Department of Physics, Colorado School of Mines, Golden CO 80401, U.S.A. A novel method for calculating electronic densities of states for tetrahedrally bonded amorphous semiconductors in the tight binding l i m i t is extended to the physically more appropriate situation of weak potentials. Results are presented showing the variation of the density of states as a function of potential strengths for potentials having s-wave phase shifts only. The relationship between short range order and the band gap is discussed. I. INTRODUCTION

include such 'long-hop' propagation, which is

The electronic densities of states of dis-

treated in a self-consistent fashion.

The

ordered systems are usually calculated using

results of calculations for scattering poten-

tight binding models in which the parameters

t i a l s with s-phase shifts and a simple struc-

are determined empirically.

tural model are presented.

This is true

In practice, band

whether the structural disorder is represented

gaps are strongly influenced by short range

through a cluster of fixed size or through an

order even in the weak scattering case. A

extended formal procedure such as a Bethe

qualitative discussion of this is given.

lattice.

The novel procedure proposed e a r l i e r 2. GENERAL FORMULATION

by the authors I and applied to amorphous

The density of electronic states for a dis-

silicon is in this second category. used to describe some of the electronic proper-

ordered system can be written in terms of an average, over the ensemble representing the

ties of semiconductors, f i r s t principles calcul-

structural disorder, of the imaginary part of

Although the tight binding model is often

ations 2 show that the atomic potentials are weak

the total scattering matrix, T, for the system.

and do not lead to deeply bound states.

This, in turn, can be written for a set of non-

This

has the consequence in our procedure, which is

overlapping muffin-tin potentials as a sum over

of a f i r s t principles form using a real, local

a l l possible scattering paths, represented by

potential, that the proper bandwidth and band

sequences of scatterings and propagations4.

gap can not be obtained for any choice of strong

For t i g h t l y bound states with energy E<< O only

binding potential parameters; that i s , the usual empirical parameters do not correspond to such a

propagation to nearest neighbours is important.

physically real, local potential 3. electronic wavefunctions are not confined to

neighbours is equally important. In this work the pair distribution function about a particular atom w i l l be s p l i t into two

near neighbours and so propagation to distant

parts,

I f the scattering potentials are weak then

For weak potentials propagation to distant

gCB) = n(R) + y(R),

neighbours must be included in any multiple scattering formulation. This paper describes

where

the reformulation of the e a r l i e r theory to

y(R) the distant neighbours.

n(R) represents

(I)

the near neighbours and In our e a r l i e r

%Present address: Physics Department, Rensellaer Polytechnic Institute, Troy, NY12180-3590, U.S.A. 0022-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

J.L. Beeby, T.M. Hayes /The density of electronic sta-tes

254

work y(R) was neglected and n(R) represented by

O;

X +

X

X

+

×

X~K

+

..

tetrahedral local order with a d i s t r i b u t i o n of dihedral angles.

In this section a model w i l l

be used in which there is no short range order, i.e.

n(R)=O,and y(R)=l

b)

x/

x -X

.

.

.

~

÷.

outside a 'correlation c)

hole' big enough to hold j u s t one atom.

//

A f i r s t approximation can be obtained by

\

o

x x ~ × x k ~

replacing each propagation within the total scattering matrix by i t s average, , giving = t / ( l - t )

/

i/

--~

\X

(2)

where t is the scattering matrix for each atom. The principal error introduced by this approximation concerns propagation paths in which an

FIGURE l Diagrammatic representation of multiple scattering paths.

electron scatters once from an atom and then returns to scatter from i t again after scat-

and the sequence b) gives the correction in-

tering elsewhere in the system. This second

cluding the factor (3).

Rather than analysing

scattering must occur at exactly the same pos-

this term i t is better to include some further

i t i o n as the f i r s t , whereas the averaged pro-

corrections, as w i l l now be discussed.

pagator summation of equation C2) does not take e a r l i e r scatterings into account.

The correc-

tion for this error contains a factor

3. SELF-CONSISTENCY During the scattering and propagation steps

~(zBi)+y(l~Bil)-I where a is the Dirac delta function and

(3)

Ri is

the displacement in the i ' t h step of the scat-

between leaving an atom and returning to i t s neighbourhood, i t is possible for an electron to return to an intermediate atom. This has

tering sequence. The three terms correspond

the effect of renormalising the scattering and

respectively to ( i ) precise return to the orig-

propagation sequence and can be treated self-

inal atom ( i i ) return to a neighbour of the

consistently by including the set of diagrams

original atom with probability given by the pair

of the form i l l u s t r a t e d in c) of figure I.

d i s t r i b u t i o n function and ( i i i )

This familiar procedure leads to a self-con-

subtraction of

the incorrect contribution implied by the simple approximation of equation C2).

T = t/(l-ts)

The f u l l formal development w i l l be presented elsewhere, but can be represented simply here using a diagrammatic procedure.

sistent result in which equation (2) becomes

Let a

where the effective propagator ~ has an imaginary part inside the band. Direct iteration of the defining equation

scattering be represented by a cross, an aver-

for z, which is a function of momentum and en-

aged propagation by a straight line and the

ergy, allows the density of states to be com-

correction (3) for returns to the same atom by a

puted within this model. The result for a

dashed line joining the relevant crosses.

square well muffin t i n potential with radius

These

diagrams are of a type discussed e a r l i e r by

1.5au and depth 2.53Ry is shown by the f u l l

Lloyd and Oglesby 5 and are i l l u s t r a t e d in f i g -

line in figure 2.

ure I.

radius 3au. The lack of structure is to be ex.

In this figure the sequence a) sums to

give the simple approximation of equation (2)

The correlation hole is of

pected for such a simple potential and pair

J.L. Beeby, T.M. Hayes / T h e density of electronic states

1-0

2.0

I

n(E)

255

hops from the given atom to a nearest neighbour. A l l the summations can be completed formally

[

and the computational needs are r e a l i s t i c . The band gap in a - S i , unlike that of f i g u r e \

2, arises from a s p l i t w i t h i n the bands of f i g ure 2 caused by the presence of both s- and pphase s h i f t s .

Within the theory described

above i t can be shown that without short range order there can be no true gap f o r energies -0.5-

above the zero of energy.

I t has not been d i s -

covered whether the r e s u l t is also true in the presence of short range order. ENERGY Ry

This is d i f f i -

c u l t to t r e a t numerically because the muffin t i n model does not e a s i l y give a good d e s c r i p t ion of the Si band s t r u c t u r e . Below the muffin t i n zero gaps do arise and depend on the potential strengths and local

-IO

structure. FIGURE 2 Density of states for a square well potential.

This enables a r e a l i s t i c density of

states to be predicted by a s u i t a b l e choice of potential parameters.

However, a f u l l

first

p r i n c i p l e s c a l c u l a t i o n w i l l not be presented distribution function.

The behaviour of the

here.

density of states as a function of potential strength is i l l u s t r a t e d by the dashed curve in

5. CONCLUSIONS

f i g u r e 2, which is the density of states f o r a

The extension of an e a r l i e r novel method for

s l i g h t l y weaker p o t e n t i a l , radius 1.5au, depth

c a l c u l a t i n g e l e c t r o n i c densities of states to a

2.5Ry.

s e l f - c o n s i s t e n t form has been shown to be t r a c -

These curves are f u l l y consistent with

q u a l i t a t i v e expectations.

table and p h y s i c a l l y sensible.

4. LOCAL ORDER AND THE BAND GAP

ACKNOWLEDGEMENTS

In materials such as amorphous s i l i c o n , both short range and long range c o r r e l a t i o n s must play a part in the density of states.

We have benefitted from the support of a NATO Collaborative Research Grant.

The f o r -

mer can be treated by the method published

REFERENCES

e a r l i e r , w h i l s t the l a t t e r can be treated by the

I . J.L. Beeby and T.M. Hayes, Phys. Rev.B32 (1985) 6464.

method described above.

In t h i s case, however,

the f u n c t i o n y(R) must describe a pair d i s t r i bution with a hole at the centre big enough to

2. A. Zunger and M.L. Cohen, Phys. Rev.B20 (1979) 4082.

hold both the central atom and the near neigh-

3. J.L. Beeby and T.M. Hayes, to be published.

bours included w i t h i n n(R).

4. J.L. Beeby, Proc.Roy.Soc. A279 C1964) 82.

The converse of

t h i s is that there are c o n t r i b u t i o n s to coming from the nearest neighbours and i n c l u d ing propagation paths which involve only long

5. P. Lloyd and J. Oglesby, J.Physo C9 (1976) 4383.