Jan 1, 1981 - We present a theory of dispersive drift transport in amorphous ... ding formalism (renormalized perturbation expansion, EMA, CPA) is ex-.
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. .
HAL Id: jpa-00220871 https://hal.archives-ouvertes.fr/jpa-00220871 Submitted on 1 Jan 1981
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JOURNAL DE 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 Uniuersitat 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 analytical techniques and approximation schemes borrowed from the tight-binding formalism (renormalized perturbation expansion, EMA, CPA) is extremely 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 tt . i(t) can quite generally be expressed in terms of the pulse shape nTx,t) (10,ll)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981424
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 -1 r . -r . I , I ri-ri I ) (barrier-approximation) -1 ., ., and to have the form w ( ~ , E ) = vexp(-26r-€/kBT). In the 2-site effective medium approximation (3,5) theO~ourier-laplacetransform of G (g,t) can be written as
[ u - m(k.u)
G(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
In this time regime the current i (u) is given by (10,ll)
(10)
i (u) = n { (2D(u)n/uL) I-exp(-uL/2D(u)Q ) - 1 (11) 0 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
>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
