PHYSICAL REVIEW B VOLUME 20, NUMBER 4 15 AU.GUST 1979

Wave hierarchy interpretation of a multiple-trapping model

W D LakinDepartment of Mathematical and Computing Sciences,

Old Dominion 'Universitv, Norfolk, &irginia 23508

J. NoolandiXerox Research Center of Canada Limited, Mississauga, Ontario, Canada L5L lJ9

(Received 4 October 1978)

We consider a generalized multiple-trapping model for charge transport in a disordered ma-terial. The model is defined in terms of a set of coupled differential equations, in which eachtype of trap is characterized by a capture rate and a release rate. We rescale and recast themultiple-trapping equations, and interpret the motion of charge carriers as a hierarchy of waves.This approach leads to estimates of the mean-transit time which involve appropriate restrictedsums. Using previously determined trap parameters for'a-Se, we show how the superlineardependence of the mean-transit time on material thickness follows naturally from the wavehierarchy interpretation of the multiple-trapping model.

I. INTRODUCTION

In the past several years there has been much ac-tivity in the area of charge transport in disorderedmaterials. Following the early work of Scharfe, ' Pfis-ter and others' have carried out experiments toinvestigate the nature of "anomalous" transit-timedispersion in a variety of systems. As is well kriown,observed photocurrent transients are found to bemonotonically decreasing in time, and relativelyfeatureless. However, in many cases the data on alog-log plot are found to lie on two straight lines, in-tersecting at an apparent transit time ~ . One of thenovel effects involving v, which we will discuss in

this paper, is the observation of a superlinear depen-dence on sample thickness.

Theoretically, the first work in this field was carriedout by Scher and Montroll (SM)7 who recognizedthat the microscopic processes which govern chargetransport in a disordered system must be character-ized by a wide distribution of event times. SM usedthe formalism of continuous-time random walk on alattice to develop a theory of anomalous dispersion,and explained quantitatively a number of novel ex-perimental results.

Recently another theoretical approach to the prob-lem has resulted in the development of a generalizedmultiple-trapping model for anomalous dispersion.The model is similar to conventional multiple-trapping theories, ' but differs in the important as-pect that variations in trap release rates can arisefrom both a distribution of attempt frequencies andtrap energies. The transient photocurrent data for a-Se has been successfully analyzed using this model,and the SM waiting-time distribution function hasbeen calculated for both the disperse (low-

temperature) and nondisperse (high-temperature) re-gimes using the basic equivalence of the continuous-time random-walk and multiple-trapping formalisms.

In the present paper we continue the mathematicalanalysis of the multiple-trapping equations, begunearlier. " Previously, we treated these equations us-ing scaling and perturbation techniques. Thesemethods dealt explicitly with all traps whose scaledcapture times or release times were not order one.For the order one traps, however, a complicatedinversion of the Laplace transform of an exit func-tion was required. The present work avoids this dif-ficulty and includes all traps in a natural manner.An important result of our analysis is the analyticalderivation of the superlinear dependence of the ap-parent transit time v on sample thickness. Previ-ously, numerical work on a-Se indicated that 7

could be calculated from an empirical relation involv-ing a restricted sum over a combination of trapparameters. Here we show that the empirical formulafollows naturally from the wave hierarchy interpreta-tion of multiple trapping. In other words, our workquantifies empirical results already present in theliterature. s'0 "

Consider now the continuum model, whichcorresponds to a homogeneous spacial distribution ofa finite number (n say) of distinct types of trapsEach type of trap is characterized by a lifetime v, anda release time r, ; (i =1,.. . , n). Hence, release proba-bilities per unit time are r = v, , capture probabili-ties per unit time are co = v; ', and co;'To is the ex-pected number of times a carrier will be captured bythe ith type of trap, where To is the transit time offree carriers.

Let the (dimensional) variables t'and x'denotetime and position through the amorphous material

20 1750 1979 The American Physical Society

20 WAVE HIERARCHY INTERPRETATION OF A. . . 1751

with x"=0 corresponding to the material's surface,and x' = L to the substrate. %e will restrict our-selves here to a small signal flash with constant elec-tric field E. In this case, the photogeneration rate isclosely approximated by a 8 function, and hence thesolution for the flash is a Green's function for thecontiriuous illumination case. The trap-dominatedphotoconduction equations are now

some traps present in the restricted sum for v dropout of the restricted sum for ~™ . %e show later thatthis behavior is due to changes in relative magnitudesof coefficients in the governing equation when thelength scale is decreased.

II. SCALING OF THE MODEL EQUATIONS

O'

Q4

, + p, E P„+$ (p "co;" p;—"r )i 1

+p;"r;'= p"o);" (i -I, ..'. ,n) (1.2)

where p'(x", t") and p;"(x",t") are volume concentra-tions of free carriers and carriers trapped in the ithkind of trap, Np is the exposure in photons per unitarea, and q is the efficiency of conversion into freecarriers. Associated initial conditions are

p'(x", 0) =p;"(x', 0) =0

The transient photocurrent I'per unit area is ob-tained from p' through the relation

(1.3)

I L

I'(t') = ~ pEp" (x', t") dx",L Jp (1.4)

where q is the magnitude of the moving charge, and

p, is the mobility of untrapped carriers.The model equations (1.1)—(1.3) may be solved

exactly using Laplace transforms. In particular, it has-

been shown9'0 that I'(t") involves an n-fold convo-lution of modified Bessel functions of the first kindof order one. Unfortunately, because of its compli-cated structure, information can be easily extractedfrom this exaqt solution only in special cases such aso)~ Tp large.

The present work begins by reexamining the non-dimensionalizing of the model equations. A key ele-ment here is the existence of two intrinsic timescales. The scaled first-order system is then recast asa single higher-order equation involving a hierarchyof wave operators. Analysis of this equation leadsdirectly to an estimate of the mean-transit time ~

for the carriers with trapping. In particular, ~ isfound to be approximately a restricted sum of factors4!lt To/r, "over suitable values of i The final po. rtionof this work concerns the dependence of the mean-transit times v on material thickness L The rela-tion To= p, E/L for the free-—transit time suggests that

should vary linearly with L. However, experi-mental evidence sho~s that this is not the case. Ifthe material thickness is decreased from L toL = L/I(I ) 1), then the new mean-transit time is

=r /I with I ) I. In particular, it appears that

An obvious time scale in this problem is the free-transit time To = p, E/L. As a&; = ru To is the expect-ed number of times a carrier will be captured by theith kind of trap, Tp is clearly an appropriate scalingfactor for ao . However, Tp is not an appropriatescale for the release times r as usually r, 'T«& ~;.Hence, this problem contains a second time scale Tdefined by the condition r; =r T =O(co,).

If o.p is a charge density, an appropriate scale forpt "(x",t") is trp/ti, i.e., a dimensionless volume con-centration for carriers trapped in the ith kind of trap1s

p, (x, t) = q p;"(x",t")Op

(2.1)

As r~'p = aor,p;/tIT, Eqs. (1.2) now show that timeshould be scaled by T and not by Tp. Hence, we de-fine scaled time and length variables t and x by

t = t"/T and x = x"/L (2.2)

If p(x, t) is the scaled volume concentration of freecarriers, the balances in Eqs. (1.2) also show we mustdefine

p(x, t) = ~ p'(x', t")cd Tp

With these scalings, Eqs. (1.2) now become

(2.3)

(2.4)

where

e= To/T and A. =rtNoqT/oo (2.6)

Scaled initial conditions are

p (x, 0) =p;(x, 0) =0 (2.7)

In most cases, Tp && T so ~ will be a small parame-ter. Care must thus be taken in analyzing the scaledmodel equations for small times, e.g. , t =0(e). Thisdoes not cause difficulties in the present work, how-

ever, as we will be interested primarily in order onetimes.

For Eq. (1.1), we have

N

e p + p + X (o),p r;p, ) = XS(x)—g(t), (2.5)Qt 9x

%. D. LAKIN AND J. NOOLANDI 20

III. %AVE HIERARCHY FORMULATION ted, e.g. , H1" =r1+r3. Then, for k =2, ..., n

To obtain the wave hierarchy formulation of themodel equations, we convert the system (2.3) and(2.4) of n + I first-order equations into a single(n + 1)st-order equation for p(x, t) by successivelyeliminating the trapped carrier variablesp;(x, t)(i =1, , n. ..) W. e now have the governingequation

and

a = KHk + ~ ~'Hk-1 akck Hk(i) (3.6)

(3.7)

gn-k$ ak „—+ ck p = Z $ akck5'" "'(t)5(x)

(3.1)

where gt~'(t) is the j th derivataive of 5(t). Eachterm in Eq. (3.1) now involves a wave operator[it/f)t + ck(t)/Bx)]. In particular, solutions of theequations

~here Hk'= T 'Hk, i.e. , Hk'contains the dimen-sional quantities r, rather than the dimensionless r~.

Typical situations involve traps with a wide rangeof values of lifetimes and release times. This, in turnleads to disparate sizes of the coefficients ak in Eq.(3.1), which suggests that asymptotic methods maybe used to approximate p(x, t) and the mean-transittime r . Suppose, for example, that of the n kindsof traps, N traps have "large" values of ~;. Let thesetraps be numbered 1 through N so that if 1 «i ~ Nthen

8 8—+ck /k=0Qt Qx

(3.2) c»t =O(c»;) for I ~j ~ At

co=& and ao=~-1

so aoco =1 and 7o = To. For k =1, we have

(3.3)

with ck & 0 are waves Pk(x, t) = fq(x —c„t) whichpropagate to the right with velocity ck. Each wavewill thus travel a unit distance in time ck, i.e., in di-rnensional variables, the wave will traverse a distanceL in time rk ——ck

' T.' The solution of Eq. (3.1) may

thus be thought of as coming from the interaction ofa collection of n +1 waves with phase speedscp c1 c„. Following Whitham's treatment ofsecond-order equations, '" we will call the setpk(x, t)(k =0, I, . . n) a h. ,ierarchy of waves.

Regardless of the number of traps, the coefficientsao and co in Eq. (3.1) are given by

(u~ (( (u; for N + 1 ~j ~ n (3.8)

By the definition of T, scaled lifetimes and releasetimes are of the sarrie order, so Eqs. (3.8) also holdwith co; and ru& replaced by r; and r&. In most cases,/tt « I. Using Eqs. (3.8) in the expressions (3.3),(3.4), and (3.6) for the coefficients ak in Eq. (3.1)now shows that aN will be large compared to the oth-er coefficients a with m 4 N.

Let p, be a scale for the coefficient aN. Then

=O(1), but =O(p, ')(m W N) . (3.9)p aN.

Similarly, aNcN && a c for m ~ N. Expandingp(x, t) in the form

so

at = g (er; + t»;) and a~et ——g r;i 1

(3.4) p(x, t) =p'0 (x, t) + p, 'p" (x, t) +O(p, '), (3.10)

now implies that the first approximation p (x, t) sa-tisfies the equation

n n

Tt = To+ Q /gt»r; (3.S)

H1 r1 + r2 + I3

H2 = f1f2+ f1f3+ f2f3

To define ak, ck, and ~k for k =2, . . ., n, let H denote

the sum of the combinations of products of the n

quantities r;, .. . , r„ taken m at a time. For example, ifn =3 then

1

gn —N —+ctt p"'(x, t) =) cN8'" n'(t)8(x)gtn —N gt gX

(3.11)

In accordance with the standard methods for obtain-ing Green's functions, for x & 0 and. t & 0, pto'(x, t)will satisfy the homogeneous equation obtained bysetting the right-hand side of Eq. (3.11) to zero.Hence, after an initial organization period, we obtain

H3 = r1r2r3p"'(x, t) = hatt(x, t) = f~(x —cnt) (3.12)

Let H denote H with all terms containing r~ omit-Additional terms in Eq. (3.1) now enter, at earliest,at second order for t = O(1). In particular, terms with

WAVE HIERARCHY INTERPRETATION OF A. . . 1753

y~ = cg T = 7 it/ (3.13)

Let ao be a scale for the ao; with i =1,..., N. Then,by Eqs. (3.6)

' g ~iIIN —l

i 1

Cittt= 6 +

H(3.14)

I

To further analyze the structure of Eq. (3.14), we

note that the dominant term in H~ will be r~r2" rg,i.e.,

coefficients ak with k & N will provide second-orderdispersion while terms with coefficients ak withk & N will provide second-order amplitude decay. Tolowest order, Eq. (3.12) thus shows that for orderone time the signal is a primary wave with phase.speed c~ . A first approximation to the mean-transittime is hence

in Fig. 1 of Ref. 8(b). Although three traps are suffi-cient to represent the experimental data, in thepresent example we have used four traps to better il-

lustrate the behavior with scaling of the sample thick-ness. For the current trace shown we haveTp=4.74 x 10 8 sec, using L =79 p, m and E =10Vlp, m. The mobility of the untrapped carriers is cal-culated assuming a T ' ' law for the temperaturevariation, and using the room-temperature valuep, =0.34 cm'/V sec."

As est, ~2, and co3 must all be considered "large" inthis context, the relatively wide range spanned by r~ ',r2", and r3', somewhat restricts the choice of a valuefor the scale T We m.ust insure that r~ "T= O(0i;)for i = I, 2, 3 [the condition r4 "T= O(i04) is less im-

portant as m4 is "small"] and hence we take T =10.sec. This gives ~=ap=4. 74 x10 && 1. Values ofthe four additional coefficients in the governing equa-tion are now

IIN =r, r2 "rN[I +0(p, )] (3.15) a~ = 17.322, a2 =816.137, a3 =4135.913

Similarly, if pg" ~ denotes IIN'~ excluding all terms in-

volving an r& with j & N, then

and

a4 = 2311.339e iV

X;115'-'i = g,pk" i [I+O(p ')] .

We thus have

(3.16) Hence, the dominant wave in the hierarchy will havedimensionless phase speed c3, and an approximationto the mean-transit time will be the restricted sum

N

civ' a+ X— (3.17)0)g Tp = 1.86 x 10 ~ sec

fg(3.19)

which gives for the (dimensional) mean-transit timethe restricted sum

in good agreement with the observed value shown in

Fig. 4 of Ref. 8(b).

N eTr =T+X

i ) f](3.18)

IV. VARIATION OF MATERIAL THICKNESS

TABLE I. Values of trap parameters for a-Se transientphotocurrent trace, corresponding to T =143 K,L =79@m, E =10 V/pm. As explained in the text, thefree-transit time is Tp=L/p, E =4.74 x10 8 sec.

Trap ao Tp (dimensionless) . r (sec)

i =2I =3i 4

8.5595.3342.'613

0.816

7.43 x 103

1.08 x 103

2.09 x 102

3.04 x 10

As Eq. (3.16) only involves To, this approximation is

independent of our choice of the scale T.To illustrate the application of these results, con-

sider the four trap case given in Table I. The valuesof r;" and cu; shown (i =1,... , 4) were obtained froman analysis of the a-Se photocurrent transient shown

In Sec. III, we found 7 = 7N where ~g involved

Tp but was independent of T. Consider, now themean-transit time v for a material sample with re-duced thickness L =L/I (L ) I). As To=atE/L, thenew wave-transit times Tk(k = I, .., n) are simply.

rk rk/I (4.1)

It might thus appear that r„=r /I However, ex.per-imental results show that this is not the case.Indeed, the reduction in mean-transit time for thethinner sample is found to be much less than the fac-tor I '. We now wish to examine this apparentlynonlinear behavior.

When length is decreased from L to L, the expect-ed number of times a'carrier is trapped by the jthkind of trap is reduced from aiz to mq/I. This has theeffect of changing some traps from "large" co traps to"not large" co traps. Consequently, the time scale Tfor the release times r may be chosen with morefreedom as we need insure that r T = O(oi;) forfewer values of i. This suggests that an appropriate

%. D. LAKIN AND J, NOOLANDI

scale for the new release times is not T/I but

T=T/I, with I ) I (4.2)

If the systems (1.1) and (1.2) are now rescaled usingL and T, On reduction to wave hierarchy form wc ob-tain Eq. (3.1), with ak and ck replaced by ak and ek

where

Qk CkQk= ~ 1

and ck=II

(4.3)

2 O'T

=1.52 x10 3 seclfJ

(4.4)

Comparing Eqs. (4.4) and (3.19) shows that the thirdkind of trap is dropped from the restricted sum whenthe sample thickness is reduced to

4L. The value of

obtained is in excellent agreement with the a-Se

The factor l ' in ak is now capable of changing therelative magnitudes of the coefficients in the govern-ing differential equation. The primary wave in thenew hierarchy for the reduced thickness L (andhence the new mean-transit time r ) may thus bedifferent than the primary wave for thickness L, i.e.,it may be associated with an index N ( N. Consider,for example, the previous four trap case with samplethickness reduced to L =

4L. Scaling by 1=4 gives

co1 =2.139, co2 = 1.335, ao3 =0.688, and eo4 =0.204.Unlike the previous case with thickness L, ao3 is not"large" here. %e may thus choose T so as to makethe implicit constant in the order relationr~ = r~ "T= O(ao~) more nearly unity, e.g. , T =10This choice gives I = 10, r1 = 7.432, and r 2

= 1.077.The coefficients ak are now a0=4.74.10, a1=4.33,a2=20.4, a3=10.35, and a4=0.58. In particular, thedominant coefficient now has index. W = 2 whereaspreviously we had N =3. The dominant wave now-1 -1has phase speedi2, not c, . Hence, r =~2 ~2/I

rather than r3/I, i.e.,

data for L =20p, m ', and with earlier theoretical cal-culations shown in Fig. 4 of Ref. 8(b). It should alsobe noted that the above result does not depend criti-cally on the choice of Tor l. Any reasonable scalingsuch that F

~

= ~~ "T=0(~~) is nearly unity will leadto the dominance of the a2 coefficient and the resultfor r given by Eq. (4.4).

The association of different primary waves withdifferent length scales is a situation that is not un-common in fluid mechanics. A particularly strikingexample of this situation is the relationship betweenthe length of a ship and the bow wave excited by theship's motion. '6

V. DISCUSSION

The multiple-trapping equations have been nondi-mensionalized using a time scale associated with traprelease times and the resulting system of equationshas been recast as a single higher-order equation in-volving a hierarchy of wave. operators. This approachallows an approximate description of the signal as aprimary wave with second-order dispersion and de-cay. The approximate mean-transit time is found toinvolve a restricted sum over traps which capture thefree carriers a relatively large number of times. Fi-nally, we have examined the apparently nonlinearbehavior of the mean-transit time when the thicknessof the material sample is reduced. This behavior hasbeen shown to come from changes in the relativemagnitudes of coefficients in the wave hierarchyequation which lead to a different primary wave forthe thinner sample. As an illustration we have calcu-lated the expected transit time for a 20-p, rn sample ofa-Se, using parameters obtained from a numericalanalysis of the photocurrent transient for a 79-p, msample, and found excellent agreement with experi-ment.

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