The role of interband interactions in formation of mobility gap negative-U states in glassy semiconductors

Solid State Communications, Vol. 79, No. 3, pp. 231-233, 1991. Printed in Great Britain.

THE ROLE OF INTERBAND

0038-1098/91$3.00+.00 Pergamon Press plc

INTERACTIONS IN FORMATION OF MOBILITY STATES IN GLASSY SEMICONDUCTORS M.I.Klinger*

GAP NEGATIVE-U

and S.N.Taraskin**

*Ioffe Physico-Technical Institute, 194021 Leningrad **Moscow Engineering Physics Institute, 115409 Moscow

12 May1991

(Received

by V.M.Agranovich)

Interactions between mobility-gap states and non-parent band states were found to be important for formation of negative-U The quantum-mechanical states in glassy semiconductors. theory of the related effect of the gap-state levels repulsion from the non-parent band mobility edge is quantitatively considered. It is shown that the real gap level at the electron (hole) pair self-trapping is effectively stoped near the mobility edge approaching it largely in exponential way.

or mobility) holes) in a

1. It is well-known that interband gap state terms Eq for electrons

approximation

depend on the atomic semiconductor can environment coordinate change (x), this dependence to the related atomic adiabatic contributin x are expressed in potential f the coordinates atomic

length

gap-state

a0

terms

z

Eq(x)

1

1).

Furthermore,

can essentially

El(x)

r Ei + Q”x can be used for

the bare gap term at typical, (z

l),

with

Q” the

not very large

electron-atomic

1x1

interaction

constant (generally, Q” N 3 eV). The interband interactions, includmg those between the gap states ($J~, Es) and the states ($p, EP) of the

the

non-parent

change and

approach even the non-parent band (or mobility-gap) edge when var ‘ng the atomic coordinates. That is important ror electron (hole) trapping in the gap states and particularly for strong self-trapping of singlet electron (hole) pairs, or negative4 centre formation, in atomic soft configurations of glassy semiconductors (GS). The negative-u centre formation was predicted to be associated with quantum mechanical repulsion of the true, renormalized, self-trapped gap state term

should

band

become

ap$roa$es EB(=EV,

(valence essential

the

band, when

repulsing

for electrons).

for

the

band

electrons)

term

E:(x)

mobility

The technique

edge

used below

to account for these interactions is similar to the well-known one given by Haldane and Anderson 5 The result is that for a plausible model of the ’ bare gap density-of-states (DOS) the behaviour

from the non-parent mobility edge ’ . Some attemps have been made to describe the repulsion effect in a more quantitative way 2,3 .

of the gap term E:(x)

does not change

significantly near the gap middle, whereas it is essentially reconstructed to the *true gap term Eq(x)

near

the

mobility

edge EB and

the

gap

In what follows an approach is proposed to a quantitative theory of the repulsion of the state terms, which is due to self-trapping interband interactions, in the mobility gap of the glassy disordered system (GS) containing random

state tiq(x) = c~(x)$~+C cp(x)rJp should contain

fields 4 Important “bare” gap state terms in the original; undisplaced, configuration (x=0 are terms of the respective band tail states, whit h are of small size (N a, the mean interatomic separation) and close, but* not too close, to the for parent mobility edge EA. The situation

;

electrons

is explicitly

the

conduction-band

for

holes being

valenceband

considered, mobility

symmetric

mobility

edge.

with

Ei

edge, the with The

EL

considerable admixture :f the repulsing band states tip, with considerable

I c,(x)Ia=l-l c,(x) I 2N Ic,(x) I

(valence)

a.

The separation

AE(x) of the gap term Eq(x) from

the

edge

mobility

Ei

is

exponentially with increasing .-I earlier result in ‘.

= Ez

a

shown to decrease 1x1, in contrast to a

situation

=

standard

2. The model Hamiltonian

EG the

of the well-known

linear 231

one

we use is a modification 5

232

Vol. 79, No. 3

MOBILITY GAP NEGATIVE-U STATES IN GLASSY SEMICONDUCTORS

0

+

H =aZ-1, 2 E q ( x ) a q a a q a + - p,a ~" eB(P)bp+abpc+

(1)

+ ~ Vpq(X)a + b h.c. pc q a pc + Here eB(p) are the terms of the repulsing band states (p) and Vpq(X) stand for the interband interaction matrix elements. The latter largely are finite due the random fields, even for the nondisplaced atomic configuration (Vpq(O) # 0), and are expected only slightly to dep.end on (q,p) rather close to the mobility edge E B. Therefore,

large

for not very typical Ix] (< 1), the approximations IVpq(X)]-~lVpq(0)l and, close to this mobility edge, IVpq(0) l -~ const =IVI can be used. The contribution of the self---consistent Hubbard interaction for double occupied E q° terms can be neglected here as far as it is generally small compared to the gap width Eg and the fraction of gap states with double occupation is low (see 4 ). The energy spectrum of the Hamiltonian (1) can be obtained by soling the respective Green function equation (E-H)G = I. The diagonal part of G (in the q - states) is ^

o

1

Gqq- < q lG ]q> = fEq(X)-E+l]( E)]- ,

since

generally 2

4IV I / D E g < < I

gB(F,)=gv(E)_~ const - gv at E v > E > E v or gv (E) = 0 otherwise, for electrons (and correspondingly for holes), and then estimate the effect of the band tail states. With this DOS, the eq. (4) is transformed to ** gv [VI 2. In Eq(x)-Ev, , Eq(X)-E v

(5)

for Ev
Id~/dEI

~

4 • gvlVI 2Eg _ 1

for actual band

(Dv) and gap

(Eg) widths, e.g. IVI < 1 e V, Eg~ 2-3 e V and D ~ 5-10 eV. v Quite different is the situation for Eq(X) close to the mobility-gap edge, since the typical size of the gap states (¢q(X)) generated by the non-parent band is much larger than the size of t h e original gap state Cq (rp>>rq = a). It follows that the 0

QO.

effective interaction constant Q = A Q << So the dependence of the true gap term Eq(x) drastically changes and the term is repulsed from the mobility edge states, whereas the bare term E : ( x ) penetrates the non-parent band to about -£

Veff=4gv IV 12,o~Eq(x)~Veff, o behind obtain solving the eg. (5) that

E v. * One can •

AE(x)=Eq(X) -

o

E ; ~ Dv.eXp { - Ev-Eq(X)}

=

gv Ivl~ o

Dvexp{- 4 6Eel(x)}

(2)

E(E)~_Z]Vpq 12(E---eB(p))-I , (3) P V ~- V = const. Pq so the true, renormalized, gap state term Eq(X) can be found by solving the equation 0 Eq(X) - Eq(X) = Z(Eq(X)). (4) For solving the eq. (4) and determining the repulsion effect of the interband interactions (Vpq(X)) we use first a model step-like DOS for the repulsing, non-parent, * band, with **

Zq(X) - Z°q(X) -~

A = I<¢(~)1¢c>12 is easily seen to be close to 1,

(6)

Veff

•

*

for AE(x)<<{Eg., Dv} and E B = E v (similarly , for holes with E B = Ec, gc and V c substituted for E v , gv and Vv). This relation holds true for sufficiently-large atomic displacements x,1 >lxl_> x*~ ( Eg / Q ) ,o

at x* <~ 1 and Veff<
3. In fact, the model with abrupt band edges considered above should be generalized to account for the influence of the repulsing band tail states with a typical DOS o

*

gt(E) = gt exp {---(IE--EBI/W)#}, # ~ 1"2. The question is whether the tail states can essentially change the term repulsion and the basic relation (6), and we argue that they can not. Let us note in this connection that the typical time of interaction between the gap state (Eq) and a band tail state (Et) is tint ~ ~I~

(Eq-Et)2+IV[ 2I'- ~/~

(7)

,

and near the mobility edge E v. Near the gap middle the interband interactions give rise to a renormalization of the gap term: o Eq(X)-Eq ~ Q x , with Q = AQ° and A =

1 + d~/d

=

Here ~ is a typical energy at the gap middle, and

If the time rcros s during which the term E q crosses at self-trapping the band tail term E t is smaller than ~'int' then the repulsion from the term E t is inefficient. The time rcros s can be estimated as follows: Tcross(X)~IEt--Eq(X) /] dt /" (8) For a "triangular" model of the band tail DOS

Vol. 79, No. 3

MOBILITY GAP NEGATIVE-U STATES IN GLASSY SEMICONDUCTORS

gt (E) and actual band tail size w < < E g ,

233

repulsion effect under consideration largely is related to interactions with the band states around

[dEq/dt [~AE(x(t))Q°xo~/Veff

the mobility edge E B . Thus for actual dependence

gt (E) and actual band tail size w<
Here ~ -- ~D(k/k(°))l/~ ~nd Xo~'(/t/ M~) ~/~ are the typical vibration frequency and atomic displacement in soft configurations with spring constants k, k(U~MWDao2>>-~ k > > 10-2k (°~. It follows that the E q-term repulsion from any band tail term E t is inefficient as far as

WD~(klk(°))tl2q°lglV12w>>1,

i.e. 'Vcross < < Vint. It is easily seen that this inequality holds true for typical soft configurations (k, xo) and tail states (g, V, w). Therefore, the gap state Eq(X) penetrates the band tail states up to the energy range around the mobility edge E B . There are qualitative reasons to believe that this result holds true for realistic band tall DOS as well, so the

of the band D O S g(E) at E z E B , which differs from the model D O S in eq. (5), the behaviour of Eq(X) can hardly differ from the considered in , section 2 near the mobility edge E B , although some difference can occur near the gap middle. Therefore, we expect that the eq. (6) holds true for actual band D O S g(E) as well. 4. To conclude, the quantum mechanical repulsion of the negative-U centre related gap terms from the non-parent band mainly is determined by the band states around the mobility edge. The effect of the band tail is found to be less important.

The relation (6) describing the gap term repulsion is expected to hold true in a rather good appro~mation with IVpq(X)lcIvl= const. The real gap level at the negative-U centre formation approaches the mobihty-gap edge in an exponential way.

REFERENCES

1. 2. 3.

M. I. Klinger, Phys. Reports, 94, 183 (1983) V. G. Karpov, Soviet Phys. JETP, 85, 883 . . Klinger, Doklady Acad. Nauk SSSR, 279,

91 (1984)

4. 5.

M.I. Klinger, Phys. Reports, 165, 275 (1988) F.D. Haldane and P. W. Anderson, Phys. Rev. B, 13, 2553 (1976)