RECOMBINATION IN DISORDERED SOLIDS

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Submitted on 1 Jan 1981

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RECOMBINATION IN DISORDERED SOLIDS

H. Scher

To cite this version:

H. Scher. RECOMBINATION IN DISORDERED SOLIDS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-547-C4-550. �10.1051/jphyscol:19814118�. �jpa-00220735�

RECOMBINATION I N DISORDERED SOLIDS

H. Scher

Xerox Webster Research Center, Webster, New York 14580, U.S.A.

Abstract.- The main point of this article is to present a simple physical picture of the recombination process of a carrier hopping in a disordered solid in the presence of molecular recombination centers (rc). First, an exact (asymptotic) calculation of the entire time dependence of the transient photocurrent (TP) and the number of surviving carriers n(t) is carried out for a suitable model system. It is shown that the t'(l-~)-, t - ( l + ~ ) transition does exist in the TP however the transition occurs in over two decades in time (for a=0.5). Further, the time tr corresponding to the intersection of the two slopes in the log(TP)-log(t) plot is the same as the median time for the n(t) decay. The

t,

is next shown to have an alternate simple interpretation. In the low rc concentration limit, tr is determined from a calculation of the number of distinct sites visited by the hopping carrier, as a function of time (t) and electric field (E). The results are compared to recent measurements in a-Si:H.

Introduction.- There has been a considerable experimental [l-61 and theoretical [3,7-91 interest recently in carrier recombination processes in disordered solids. Measurements on non-geminate recombination have been carried out in a-Si:H and a-As7Se3 using the delayed collection field technique [3]. Recombination is observed in the decay ofphitogenerated carriers using transient optical absorption in a-Si:H [4] and a-As7Se3 [l] and in transient photocurrent measurements in a-Si:H [5] and a-As2Se3 [6]. The results-forsurviving carriers n(t) typically show an asymptotic algebraic decay n(t)

-

^{f a (o}

^<

^a

^<

1) [3,4]. Theoretical studies have been carried out with a variety of models and techniques. For hopping transport Shlesinger [7] used a conditional probability for the occurrence of a reaction which was determined from a first passage time distribution to reach a specific set of sinks, while Movaghar [8] solved the Master eqcation, with site specific random loss terms, in a self consistent approximation. Basically they derived the same results, and for dispersive trmsport obtained n(t)- ra.

We shall show that the form of their results can be obtained in a straightforward way using an idea known in the random walk literature [l01 which formed the basis of a model [l11 that succesfully accounted for the unusual field dependent trapping [l21 in a quasi-one- dimensional molecular crystal. In the limit of low recombination center rc (or deep trap) concentration, to an excellent approximation, the recombination time tr can be determined from S(t) the expected number of distinct sites visited in time t. The hopping and/or trapping process can be quite complex but S(t) is calculated from the rc-free transport- First we calculate the complete time dependence of both the transient photocurrent (TP) and n(t) for a carrier undergoing dispersive transport. We discuss the significance of thz results and then interpret the recombination time tr using S(t).

Transient vhotocurrent (TP) and survivors n(ti We first model recombination in a disordered solid by a carrier hopping on a lattice in the presence of a distribution of absorbing sites. The transport is characterized by +(s,t), the probability density that the time interval between successive arrivals is t and the displacement is S. The equation for the continuous time random walk (CTRW) generated by +(s,t) has been shown [l31 to be an

exact

one for the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19814118

C4-548 JOURNAL DE PHYSIQUE

configilration averaged <P(s,t)>, where P(s,t) is the probability of finding the carrier at s at time t if it slnrtcd at S, at t = O in a spccific configuration of randomly distributed sites. Equivalently, the process of configuration averaging defines a self-energy operator

2

for a coherent medium [14]. The descriptions are sinlply related by fs,sl = u$(s-sr,u) [l

-

$(u)]-I and therefore a self- consistent calculation of

5

[8,14] can be used to determine a $(s,t). [$(s,u), ~ ( L I ) are the Laplace transform (LT) of $(s,t) and $(t) r

zs

+(s,t), respectively.] The CTRW approach is particularly fruitful for trapping or recombination problems. Specifically, there have been a .lumber of cases in the RW literature [15,16] which demonstrate that under certain conditions various transport proverties (including diffusion) on lattices depend onlv on the densitv of "defects" and not on thew spatial arrangements. l'he conditions include low defect concentration and spatial dimension d22. We will further demonstrate this fact by the two approaches to the problem taken in this note. An exact solution for <P(s,t)> can be obtained for a general &(s.t) and a periodic distribution of traps [l71 of concentration c = n-3, where n is the distance between traps in units of the lattice constant. The traps are taken to be absorbing sites and the zeroth and the time derivative of the first spatial moment of <P(s,t)> determine n(t) and TP, respectively. The details of this derivation will be given elsewhere. We further specialize the result to the case where $(U)

---

l-bua (O<a<l) (i.e., +(t) is a slowly varying function of time over the region of interest). We also take a = H so that the LT of n(t) and

TP

can be inverted analytically with the result

where f(x) ^E ( n ~ ) ~ e ~ ~ r f c ( x ~ ) , S, is the initial site, 71 E (bg(0)I2

with ^K = 2np/n, pi = 1,2

. . .

n. The prime on the sum in Eq. 2 excludes ~i = 2 ~ .

Fig. 1: The transient photocurrent (TP) and the number of survivors n(7) for carriers under- going dispersive transport

(a = 0.5) and monomolec- ular recombination. The dashed lines have slope -1/2, -3/2.

In Fig. 1 we plot both TP(t), n(t) [Eq.(l)] as a function of [7eW0t]. For

TP

we note that the transition t-(l'a)+t'(l+a) does exist however the transition occurs in over 2 decades of time. This feature of the effect of recombination on TP is in excellent agreement with the data

1 oscur at 7/71 = 0.4 which is very close to the niedian time, 7%/r1 = 0.6, for n(t), i.e. the time for n(t) = 0.5. Thus, ^{7 ,} = ( 0 . 6 ) ~ ~ is a good measure of the recombination time for this dispersive transport which gives rise to an extremely broad distribution of recombination times. In a limited time range (10'~_<~/~,<_1) the TP could appear to have no effects due to recombination while actually a significant fraction of the carriers are recombining. We also note that the 2 decades for the transition from n(7)

--

1.0 to n(7)

-

^7-H is in agreement with this feature of ?he data in Fig.

l a of VORT [4]. This again is in contrast with the H decade transition for n(7) predicted for a TOF experiment [IS], although n(7)

-

^{7-H for T}

^>

² transit times. In the limit of low concentration the sum in Eq. (2) can be replaced by an integral and

( T ~ ) ~ = (0.77)bf1w, W =

f"

^{d 3 ~ [l- A(K, o)]-I (3)}

0

In general rr a exp[ a-lln(~/c)], where K is equal to W aside from a numerical factor 1.0. In a low mobility solid (p$ cm2/v-S) the scattering in the band could be modeled by a RW (i.e., Brownian motion). Hence the +(s,t) could include the frequent fast hops (scattering) as well as the trapping in various localized states. For an exponential distribution in energy levels of the localized states one can show that a = TIT,. Thus ^7, a eAjKT, where A = K T ~ ln(K/c), a rc concentration dependent apparent activation energy not equal to the transit time A. If the process is bimolecular A could be a function of light intensity.

S(t) and Trap~ing. For randomlv positioned rc the probability of encountering a rc with each

new

site sampled is c. Thus the opportunity for recombination increases as cS(t) and hence a recombination time can be defined as CS($.) = l 1101. The LT of S(t) can be expressed terms of the rc free lattice Greens function @,U), s(u) = $(u)[u(l-S(u))B(~,u)]-l where G(0,O) is identical to W in Eq. (3). For dispersive transport we obtain (WoQa = r ( l + a ) ~ C - ~ W and for a = % the value of derived using cS(q)=l is only 15% higher than the value determined from the median time for the n(t) decay in Eq. (1). We see from the above considerations why ta occurs in the combination cta in n(t) in Eq. (1) and in the asymptotic result obtained in refs. [7,8]. In fact in all these cases n(t) const./cS(t) as t-m. This algebraic decay for n(t) resulting from the long tail +(t) is in sharp contrast to the exponential decay n(t)

-

exp(-cS(t)) which results from solving a simple kinetic equation dn/dt = - k(t)n(t), where the trapping rate k(t) is taken to be equal to cdS(t)/dt. The validity of the simple kinetic approach has been discussed in Refs. 7-9. It has also been shown to be a good approximation for an exact solution for a one-dimensional molecular crystal with random traps [19]. When the transport is controlled by a non-exponential +(t) the kinefic equation above is not strictly valid. Since the use of a kinetic equation for a bimolecular recombination process [4] gave the same asymptotic time dependence for n(t) as the monomolecular process considered in this note, one must question this approach to the bimolecular case. The decay of n(t) for the bimolecular problem must be slower than tVa as t-.eo, however the solution obtained in [4] may be a good approximation for the first decade of attenuation of n(t), for low rc concentration and for a spatially uniform initial carrier distribution. A more general approach to the bimolecular process has been initiated in Ref. [20].

Field de~endence of recombination. One can calculate S(t) for any complex transport and hence deternine the recombination time limited by a low concentration of rc. In particular one can include the effect of an applied electric field E on the expected number of sites visited.

In Fig. 1 of Ref. [11] the carrier lifetime is shown as a function of E for an anisotropic molecular

C4-550 JOURNAL DE PHYSIQUE

G

W :

Dispersion parameter a vs e!ectric (Bohr radius) of the localized state. We derive

field. = 0(0)[1- e2]-1, which is in good

agreement with recent simulation data 1211 of crystal. Also shown is the excellent agreement

tge same quantity. Using this relation h e 'can show that ~,.(E)/T,.(o) =(C/K)E~/Q(O) and fields of ~ ~ 0 . 1 are required to reduce T ~ ( E ) to 0.5 ~~(0).

For typical hopping systems e ~ 0 . 1 corresponds to an intersite potential drop KT. This is the same condition derived for ordered, isotropic, 3d systems [ll]!

-

I

0.3 0.2

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0.01 0.1 of #(t). In Fig. 2 we show the calculated E-

C dependence on a, where r = eER0/xT, R. is $4

7

with the mcasured carrier lifetimes [l21 for the quasi-one-dimensional crystal phennnthrene- PMDA. The E-field most dramatically effects S(t) in Id, thus there is a strong interplay between anisotropy, E, and t. We now raise the following

dispersive question: transport in Is 3d, S(t) particularly (and hence sensitive tr) for to E? We have included the effect of an E-field on the hopping transition rates in a calculation