THEORY OF DISPERSIVE TRANSPORT IN AMORPHOUS SEMICONDUCTORS

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THEORY OF DISPERSIVE TRANSPORT IN AMORPHOUS SEMICONDUCTORS

K. Godzik, W. Schirmacher

To cite this version:

K. Godzik, W. Schirmacher. THEORY OF DISPERSIVE TRANSPORT IN AMORPHOUS SEMICONDUCTORS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-127-C4-131.

�10.1051/jphyscol:1981424�. �jpa-00220871�

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JOURNAL D E PHYSIQUE

CoZZoque C4, suppzdment au nO1O, Il'orne 42, octobre 1981 page C4-127

THEORY OF DISPERSIVE TRANSPORT IN AMORPHOUS SEMICONDUCTORS

K. Godzik and W. Schirmacher

Technische U n i u e r s i t a t Miinchen, D 8046 k r c h i n g , F . H . G .

Abstract.- We present a theory of dispersive drift transport in amorphous photoconductors. Expressions for the transient current i(t) are derived from the two-site effective medium approximation (Elm) which is equivalent to a generalized master equation approach. The occurrance of Gaussian or dispersive transport is shown to depend on an interplay of three characteristic time constants. Explicit expressions for these time constants are given in terms of microscopic parameters. We show that dispersive transport in hopping systems can only exist for very small times and densities. Experimental findings can be much easier explained within a trapping model which is solved by means of the CPA.

1. Introduction.- The master equation approach combined with analyti- cal techniques and approximation schemes borrowed from the tight-bin- ding formalism (renormalized perturbation expansion, EMA, CPA) is ex- tremely useful in describing the transport properties of disordered semiconductors as shown by Movaghar at this conference (1). Ee and his coworkers have demonstrated that an overwhelming amount of features of amorphous Systems (e.g. Mott's TUL' law ( 2 , 3 ) i can be qualitatively and quantitatively explained within this framework.

It is the aim of this contribution to show that by means of this

generalized master equation approach (GMEA) also the problem of dispersive transport (4) can be solved in a transparent way yielding reliable expressions for the transient photocurrent in terms of microscopic quantities. As pointed out in a recent paper (5) (from here referred to as I) the present theory can be considered to be a generalization of previous approaches (6-11). All the former results can immediately be re-derived from our theory under certain model assumptions.

2. Formulation of the problem.- The problem of dispersive transport arises from the fact that in time-of flight experiments (4) the observed transient current traces decrease monotonically with time according to a power law

.

-

( 1 - a )

where tT is roughly the transit time of the fastest carrier. Such a behaviour contradicts conventional Gaussian statistics which predicts the current to be constant for t<tT and zero for t>t

.

i(t) can quite generally be expressed in terms of the pulse shape nTx,t) (1 0,ll)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981424

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JOURNAL DE PHYSIQUE

where no is the initial charge density and G(r,t) is the averaged propagator of the carriers in the continuum representation.

3. Hopping transport.- We now assume the transport to be characterized by a set of master equations (1) with transfer rates W . that depend explicitly on the applied external field E=2qkBT/e

,

^'j (q>0) via

where the w . . are the transfer rates without an applied field which we assume to bk3symmetric: wii=w (

I

r

.

^-r^.

I , I

ri-ri

I

⁾ (barrier-approximation)

-1 -1

., .,

and to have the form w ( ~ , E ) = v exp(-26r-€/kBT). In the 2-site effective medium approximation (3,5) theO~ourier-laplace transform of G (g, t) can be written as

G(k.u) =

[

^u^-^m(k.u) ⁺ ^m(g,u) ^{( 5 )}

with the memory kernel m(k,u) given by

m m

with m(u)=a m(0,u)

.

n is the site density, a =exp(-1) is a factor which corrects for double counting, p(r) is th& radial pair distribution of sites and p(r) is a normalized barrier height distribution function. Equation (5) represents G(k,u) as the solution of a generalized master equation with memory kernel m(r,t).

In the hydrodynanic limit, i.e. expanding m(&,u) up to terms linear in the field and up to second order ill

l f ,

we obtain

where the frequency dependent diffusion coefficient D(u) is defined in terms of the zero field memory kernel mo(k,u):

a, m

( m (u) is obtained selfconsistently from equation(6) in the limBt

k=g

and with h(x)=l). As shown by Movaghar et al. (1-3) D(u) calculated according to equation ( 9 ) quite accurately describes the d.c. and a.c. transport in disordered hopping systems. In the d.c.

limit it gives the same results as percolation theory whereas the a.c.

behaviour almost perfectly agrees to numerical network calculations(2).

If D(u) does not depend on u G(k,u) reduces to the Gaussian propagator as can be seen from equations (7-8). If the relation lu/D(u)rl2l < < I is valid the diffusion term in equation(7),i.e. the term ~ ( u ) k ' can be neglected (5). This defines a relaxation time tR=l/uR after which the diffusion term is no more important:

uR/D (uR)

"

⁼ ¹ ⁽¹⁰⁾

In this time regime the current i (u) is given by (1 0,ll)

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i (u) = n { (2D (u) n/uL) I-exp (-uL/2D (u) Q )

-

1

0 (1 1)

As already shown (10,ll) this expression leads to dispersive behaviour according to equation (1 ) if D (u) has the form D(u) =Dl U1-a with a transit time tT=!L/2nD1)1/a. This agrees with experimental data as well as with the predictions of the Scher and Montroll theory (6). On the other hand, for t<<tR the field dependent drift term can be neglected against the diffusion term. In this limit one obtains, as shown by Butcher and Clark (Il), an initial current i(t) tl-a/2

In the E M A we are able to calculate D (u) explicitly for any distribution functions, site density and temperature. To be specific we would like to discuss two hopping models that are relevant for the discussion of computer simulations (12) and experimental data (4,13,14) For u<<vo

,

or equivalently for times t">l/vo

,

the numerical results for both models can be represented as

where the parameters R, mo(0) and a are given by:

"r-hopping": p (r) = l

,

p ( E ) = 6 ( E - C ~ )

m (0) = v exp(-c /kBT) .exp(-il =Goexpl-<) ; a n

=

1.73 n -1 /3G

0 0 0

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"E-hopping": (fixed range hopping with exponential barrier height distribution) p (r) = (ZR/2) 6 (r-R)

p,exp(c/kBT) E < E

P ( E ) = ( ~ E > E ~ m with p i 1 = kgT{exp(~m/kBT)

-

1 1

ino ( 0 ) =&{exp (cm/kBTc) -1 _;G exp (-rm/kgT) (141

with &=voexp (-26R) (Zapan/sin (an) ) c

IT

; R=(Znn) - I 3 ; a = T/Tc The expressions (13) and (14) are only valid for a<l. If the temperature (or site density, res?.) becomes so large thata would be 11 D(u) becomes independent of u, i.e. a transition from dispersive to Gaussian transport occurs.Before we proceed in the further discussion we would like to investigate an alternative model to study dispersive transport.

4. Multiple trapping.- We now assume that at each site i the carrier can be trapped with a rate w = w and released from this trap with a rate r . = w exp ( - r . /kBT), where :E ts the trap depth. As shown in I this leads to a diagoXal perturbation In the set of rate equations of the 0

form

Aif ⁼ U W ~ / ( U + W ₀exp(-c. ₁/k T) _B 1 (15) In the coherent potential approximation (CPA) (1) the total propagator G (k,u) of the combined system, transport states and traps, can be w$(lEt~n as

[

h t m ( g , r ( u ) +u)-m(k.: (ul +ul

I

Gtot (klu) = U +

1-1

⁽¹⁶⁾

with m(k,u) being the memory kernel of the trap free system, as given by equation(5). The self energy (u) is determined from the CPA-condi-

N(c) is a normalized distribution function of trap depths and

Go={u+m(O,u)

I-'

is the averaged trap free single site Green function

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C4- 130 JOURNAL DE PHYSIQUE

in the EMA. For small lul equation (17) can be replaced by

Z(U) = J ~ E N(E)A(u,E) (1 8) We note that Gtot(~,t) again obeys a generalized master equation with a kernel that can be explicitly calculated in terms of microscopic distribution functions. If we perform a hydrodynamic expansion we get

G ( ~ , u ) ={U-D,£~ (u) (2nikx- k 2 )

I-'

(19) with

D,,, (u) =UD(X (u)+u) 1 1 2 ( u ) + u ~ = ~ ( u ) D(X (u) +u) (20) where D(u) is the trap-free diffusion coefficient as given by equ.(9).

As pointed out in I equation (16) reduces to the model of Noolandi (8) and Schmidlin ( 9 ) if equation ( 1 8) is taken for L (u) and D (u) is

assumed to be independent of u (band transport). If we take an expo- nential trapdepth distribution, i.e. N(&)=(l/kBTc)exp(-€/kBTC) for

E < E ~ and zero otherwise, this model yields for u<<wo with

010) = { e x p ( ~ ~ / k B T c )-1 )-Tc/T= exp(-ct/kBT) (22) in agreement with earlier treatments (7-9,151. As pointed out in I we can map the result (21) onto the €-hopping result by making the identi- fications D =DE(-), ct=cm and o

=w .

The more general formula(20) corresponds ?o what has been callgd ( 4 ) "trap controlled hopping".

5. Discussion.- The main result of our theory is that the problem of Gaussian vs. dispersive transport can be discussed in terms of an interplay of several time constants, the relaxation time tR, the transit time tT and to=l/mo(0). For a frequency dependent diffusion coefficient D(u) as given by equation (12) (or equ. (2l),resp.) the transient current in the time interval tR<t<tT takes the form

Obviously t is the time at which the dispersive part becomes smaller than the d.g. part, i.e. i (t) becomes constant. The other two time constants tR and tT scale with t

.

For our specific models they are

given by ⁰

2 2 1 /at

- { 6

n

1 tR<'t0

( 2 4 ) ; tT={ I 31/n2n 1

'

^lato

t~ - 2 2 2 t~"to (25)

(6/R rl ) to tR>>to (3L/R n) to tT>>to It is obvious that "pure" dispersive transport in the sense of Scher and Montroll, i.e. a behaviour

i (t) /n0=r (a) t ~ ~ t - ( ' - ~ ) titT ; i(t)/no=7(l-a)a -1 tT a t-(l+a) t>tT

will only occur under the conditions (26)

a < 1 and tR<<t <<t

T 0 (27)

A third trivial condition is that t must be greater than the experimental time scale. If a<l the syste$ is "in principle" capable of exhi- biting dispersive transport. The second condition which can be cast into the form

1 < < Ln/2 < < R 2 2 q /6 (28)

determines the time window in which pure dispersion can be observed.

If we want to discuss the multiple trapping model in terms of these conditions we have to take

-

¹

to= (o,Q(O)) and R = (6Do/oO) 1 /2 ( 2 9 )

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If Do is assumed to stem from a Gaussian hopping process:

Do = Rhopvhop/6 2 ^t we have R = Rhop(vhop/wo) 1 /2

Let us now discuss the computer simulation of Marshall (12), which corresponds to our r-hopping model. Since he uses a site density n = ~ - ~ = l , a field ~ < 1 and a sample "thickness" of L=50 it follows immediately from equations (24-25) that in his case t <tR<<tT applies which leads to "Gaussian" transport with relaxation ?ffects for t<tR irrespective of the choice of the other parameters. The present con- siderations also explain why his data are insensitive with respect to variations in 6. 6 only enters the expression for t but not the condition whether or not Gaussian transport occurs. ~ i g result for t : t =ex? ( 2 6 ) agrees quite well with our formula (1 3)

.

To meet condigion

(28) he would have to take R greater than 10. We explicitly remark that R does

not

scale with l/d.

We turn now to the discussion of experimental data. In the Ohmic region, q/6 < < 1, n must be of the order of 107 cm-l or smaller. This estimate corresponds to the linear region in the field dependence of the mobility shown by Pfister and Scher (4). For samples that are 1 ~ - ~ c m thick the first part of conditicn (28)

,

nL>>2 is always fulfiled To neet the second part values of R larger than 10-5cm are necessary.

For our hopping models this means that the site density must be smaller than 1 ~ ' ~ c m - ~

.

For trapping we see that the effective hopping distance R contains an enhancement factor (uho /wO)

'

/2. Therefore condition (28) can be met in a much easier wayby

B

trapping model with

w < < v hop. This confirms the more qualitative arguments of other

authors (7-9) in favour of the trapping model. A much more severe argument against hopping ( in the sense of oursimple models ) as a possible origin cf dispersive transport is that the limiting time to in hopping systems is of the order of 10-i2 (3r much smaller) which is far below the range of experimental observation.

We conclude that for a simple explanation of the observed data the trapping model seems to be more favourable. This agrees with the interpretation that Hvam and Brodsky (14) gave recently to explain their i(t)-data in a-Si-H:P. They observed an a that linearly in- creased with T and attributed it to trapping with an exponential trap depth distribution.

On the other hand we have to emphasize that on the base of our theory much more detailed calculations using more sophisticated rnicro- scopic models such as trap controlled hopping can be done.

References

1. Movaghar B.,Ninth 1nt.Conf.on Amorph-and Liquid Semiconductors(l981) 2. Movaghar B.

,

Pohlmann B. ,Sauer G.W. ,phys. stat. sol. (b)X(1980) 553 3. Movaghar B. ,Schirmacher W., J.Phys.= (1981) 859

4. Pfister G.,Scher H., Adv.Phys.

7

(1978) 747 5. Schirmacher W., Sol.St.Com. in press

6. Scher H.,Montroll E.W., Phys-Rev. B12 (1975) 2455 7. Pollak M., Phil.Mag. 36 (1977) 1 7 5 7

8. Noolandi J., Phys.~ev.~ (1977) 4466 9. Schmidlin F.W., Phys-Rev.

B16

(1977) 2362 10.Leal Ferreira G.F., Phys-Rev.

B16

(1977) 4719 11.Butcher P.N.,Clark J.D., Phil.Marj.

B42

(1980) 191 1Z.Marshall J.M.,Phil.Mag. B38 (1978) 335,

B43

(1981) 401

13.Fuhs W.,Milleville M.,~tuke J., phys.stat.sol.(b)

89

(1978) 459 14 .Hvam J.M., Brodsky M. H., Phys. Rev. Lett.

46

⁽¹⁹⁸¹⁾ ³⁷¹

15.Tiedje T.,Rose A., Sol.St.Com.

37

(1980) 49