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Charge carrier geminate recombination in one-dimensional polymer structures with traps
Solid State Communications, Printed in Great Britain.
CHARGE
CARRIER
Vol. 76, No. 8, pp. 1035-1040,
GEMINATE
RECOMBINATION STRUCTURES WITH
0038-1098/90 $3.00 + .OO Pergamon Press plc
1990.
IN ONE-DIMENSIONAL TRAPS
POLYMER
Igor V. Zozulenko Institute
for Theoretical
(Received
Physics,
Kiev-130,
252130, USSR
30 March 1990 by A.A. Maradudin)
Recombination kinetics of photogenerated carriers in polymeric crystals with disorders (traps) was investigated. The distribution function and the survival probablility Q,(r) for a correlated pair of quasiparticles diffusing in a segment with absorbing ends are calculated. It is shown that the decay of Q,(r) follows the exponential dependence cxexp (- 7), 7 3 5x* Wt/n’ $ 1 (W is the diffusion rate, n is the segment length), unlike the power-law dependence a (Wr)-‘I’ for an infinite ideal chain. Within the framework of a model of molecular chain with random traps the averaged survival probability for a pair of photogenerated charge carriers in a quasi-one-dimensional polymer crystal (Q,(z)) (which is shown to describe the transient photocurrent at low bias field) is calculated.
1. INTRODUCTION STUDIES of charge carrier dynamics in quasi-onedimensional (1 -d) polymer structures, particularly, in trans-polyacetylene (PA) have been the subject of considerable theoretical and experimental interest over the past ten years. For the interpretation of the novel electrooptical properties of PA the theoretical model based on the soliton transport mechanism predicting the existence of localized quasi-particle states in a midgap is widely adopted [l]. Using the time-resolved technique in the experiments on photoinduced absorption [2-41 or reflectance [5], bleaching [4-61 and photoconductivity [7-91 is one of the most informative tools to study the nature of charge carriers, its transport and spectral properties as well as structure features of a polymeric matrix. Despite of a great amount of experimental data on photoexcitations relaxation kinetics, there is no so far reliable interpretation of a physical picture of charge carrier transport and recombination in PA in the all available interval of observations - from picosecond region to subsecond one. The possibility of an interchain violation of one-dimensional motion, generation, finiteness of quasi-particles recombination rate - this is an incomplete list of factors that complicate the description of a charge carrier transport in real PA crystals. The dynamics of quasi-particles in PA as well as in other l-d systems is especially influenced by the defects - traps or barriers. Both outside impurities,
e.g., oxygen atoms and violation of the ordered struc$-hybridized c-atoms, remaining cisture segments, chain ends may play the role of defects in PA. As the defect concentration is estimated to be 10-3-10-4 per c-atom [9, lo], the trapping of quasiparticles may reveal in kinetics of photoinduced absorption or photocurrent on times t = 12/a2 W loops-1 ns (l( - 4OOA or 300 carbon atoms) is the averaged length of chains of carbon bounds free from defects, W (- lo-” s’) is the diffusion rate, a is the cell unit). At the initial stage of photoinduced excitations relaxation (which follows the thermalization of e-h pair and its conversion to S+ -S- pair on times - 1 ps [1 11) the charge carrier decay kinetics are shown by the numerous experiments [2, 3, 6,9] to describe in the framework of the one-dimensional geminate recombination theory [ 121
(1)
where n(t) - is the probability for a pair of quasiparticles to survive by the moment t (survival probability), n, - is the distance of an initial particle separation. The deviation from the dependence (1) in the subnanosecond region was observed in [5,6,9], where an acceleration of the photocurrent decay [9] and a
1035
CHARGE
1036
CARRIER
GEMINATE
deceleration of the photoinduced absorption [6] and reflectance [5] signal decay have been marked. Such peculiarities of kinetics would be expected to associate with quasi-particle trapping by defects. Indeed, being trapped, the charge carriers do not contribute to the photoconductivity but they do contribute to the photoinduced absorption signal and the probability for a pair with one immobile carrier to recombine is obviously lower than that for a pair in which both carriers are mobile [9]. Since the theoretical description of the geminate recombination kinetics in l-d systems with traps is absent, it is of interest to consider the processes concerned. The paper is organized as follows. In Section 2 we deal with the diffusion of a pair of particles in a segment with traps on its ends, here the distribution function of particles in a segment and the survival probability are obtained. The comparison with the case of geminate recombination in an ideal onedimensional chain (without traps) is carried out. In Section 3 the averaged probability for a pair to survive in an infinite chain with random traps is calculated. The possibilities of experimental verification of the results obtained are discussed.
RECOMBINATION
when they occupy the same cell g(i,j,
t)
=
0,
i + j
FOR A CORRELATED PAIR IN A FINITE SEGMENT
Let us consider a segment (chain) with absorbing ends consisted of n cells (host molecules). In a segment at the initial moment t = 0 a random pair is located, with particles being separated by the distance of n, cells. The coordinate of one particle (discrete variable i, 1 < i < n) will be counted off from the left end of a segment, the coordinate of another one (discrete variable j, 1 < j < n - i) - from the right end. In the case of a stochastic Marcovian processes of random walks realized by jumps between the nearestneighbour sites with the probability W, the displacement of particles in a segment is determined by the distribution function g,(i, j, t) obeying the Pauli master equation
ag”k
+ W&(&j &(O, j, 0
=
g,(i, 0, t)
l,j, t) + Wg,(i,j -
=
1, t); 0,
n.
(5)
= Wa2&(x, YY f>
YY 4
at
+
Wd2gn(x, Y, 0
ax*
ay*
3
(6) where the variables x, y lie in the triangular region O
= g,(n, y, t)
=
g,(x,
= g,(x,
=
0
and the initial
n, t)
0, t) (7)
condition
g&c, y, t = 0)
=
6(x + y -
n + no>
6(x + y -
n -
no).
(8)
Having been chosen in the form (8) the initial condition satisfies the requirement for the distribution function g,(x, y, t) to equal zero when x + y = 0. Due to the symmetrical choice of the boundary conditions (7) the solution of equations (6)-(8) in the range 0 < x < n, 0 < y < n - x will coincide with the solution of the original problem (3)-(6). The extension of the triangular range of an alternation of variables x, y to the square one 0 < x < n, 0 < y < n makes it possible to apply the Green function method to calculate g,(x, y, t). Note that the analogous way was used in [ 131 to find the distribution function of an uncorrelated pair of annihilating quasiparticles under their random distribution in a segment. The solution of the diffusion equation (6) under the initial condition (8) in the Laplace-transform space 7(‘(s) = 1,” dtf(t) exp (- s Wt) has the form
s>= j dx' { j
&(x, Y, + Wg,(i -
=
We will find the solution of (2)-(4) in the diffusion approximation, changing the discrete variables i, j for continuous ones x, y. Assuming the smoothing of the distribution function (that is justify when n $ 1, t $ W-‘), one obtains instead of equation (2)
2. PROBABILITY TO SURVIVE
Vol. 76, No. 8
0
+ 1, t)
dy'
0
x G,,(x, x',y, y’, s)d(x’ + y' - n + no> (2) (3)
n
r”dy’ 6(x,
r
x’, y, Y',
s>
I’
under the initial condition x 6(x’ + y’ -
g,(i,j,
t = 0)
=
S(i + j -
Let us assume the particles
n + no>.
to recombine
n -
no) ,
(9)
(4) immediately
where G, (x, x’, y, y’, t) = G,,(x, x’, t)G,(y, y’, t) -
is
CHARGE
Vol. 76, No. 8 the Green function boundary condition
CARRIER
of the equation (7) and the initial
G,(x, x’, y, y’, t = 0)
=
6(x -
aG”(z, 0 ~
at
under
(6) under one
x’)60,
functions G,,(x, x’, t), G,,(y, y’, t) the Green functions of the equation
- y’),
(10)
Substituting equation going back to the original bution function
x
(11)
a22
the boundary
the
1037
RECOMBINATION
(17) in equation (9) and space one finds the distri-
are, respectively,
B/P G, (z, r)
=
GEMINATE
x
condition
Gm!?, ,zev { - f$Cm2+ 1’)Wt} xm X
sin
sin
711
n G,,(O, t)
=
G,(n, t)
and the initial G,(z,t=O)
0
x
&z-x’)
=
6(z -
(13)
y’).
The Laplace-transform y’, t) is written as
of the function
=
dq Gn(x, x’, q)c,& -I 27ti ~~iu:
where the functions -
Y’, q - 4,
(14)
G are defined
by the expression
sh&
-
sh&n fi
-
sh@
sh$ sh&n
-
’ < “’
’
’ “”
The normalized factor 1I,/? in equation (19) is defined by the choice of the initial condition in the form R,(t, no) = n - no. Substitution of equation (18) in equation (19) gives us
R,(t,
no)
=
n
1
0
q)
(16)
where the symbol C Res denotes the sum of residues taken at the poles of G,,(x, x’, q)cn(_v, y’, s - q) in the right semi-plane q. Having calculated the residues one has from (16)
=
2 1 x sin nl y’ sin 7~1y_ n n’
hJFT-27.
J)
(17)
Note, that expression (17) does not alter while changing x’ f-f x; y’ c* y; y - x, y’ t, x’; x ++ y’, x’ tf y.
$ (m’ +
I’) Wt)
x (1 - (- I)‘)(1 + (- 1)“) y (m’ - 12)2 x (1 sin 7cm3 + m sin 711t n > c( exp (- 5rc2Wt/n2),
x’, Y, Y’, s)
G”(X, x’, Y, Y’, s)
(19)
0
zc
and the path of integration in equation (14) goes to the right of all the singular points of functions G,,(z, z’, t). Having locked the integration contour to the right, one obtains
c Res G”(x, x’, q)e,,,cV, y’, s -
(18)
f dx ‘rX dy g,,(x, y, t).
x C C exp m=l /=I { (15)
-
.
n>
3
’ z)
&h$
1
=
= i
z’)
GJZ, z’, s) =
G&
n
$0
g+ia
.
lsinnm~+msinxZ~
G,,(x, x’, y,
x’, y, Y’, s) 1
m2 - l2
In real l-d systems the observable characteristics of random walks of quasi-particles are connected with the survival probability of a pair by the moment t Q,,(t, n,)
G&,
I)” - (- l)’
n
(12)
ones =
G,(z, t = 0)
=
Y_(-
5z2 Wt/n2 >>1.
(20) (204
At long times, 57~’Wr/n2 9 1, the decay of Q,(t, no> obeys the exponential dependence cc exp (- 5rr2Wt/n2), as distinct from the power-law dependence a ( Wt)-“2, justified for the case of infinite chain. At small times, 5n2 Wt/n2 < 1, being unable to get the analytical form of the dependence of R, (t, no) from (20) directly, we will estimate R,(t, no) in the following way. It follows from the condition Wt/n’ + 1 that the diffusion length for the time t (= m) is much less than the segment length n, i.e., under n < no, particles do not “feel” the segment ends. Therefore, in the case under consideration the initial problem of diffusion of recombining pair in a finite chain is expected to be equivalent to the problem of geminate recombination in an infinite chain and for the survival probability
CHARGE
CARRIER
GEMINATE
RECOMBINATION
Vol. 76, No. 8
initial distribution of particles in a segment. Averaging equation (20) over all values of n, in the interval 0 < n, < n with the weight of l/n one has 4 n,(t)
=
f
n
0 (1 -
f
i
m=l
/=I
(-
exp
l)‘)(l
- $ (m2 + 12)Wt}
i
+ (-
1)“)
X (m2
-
m2 (22)
7'
P)*
The dependence (22) up to normalized factor coincides with the result obtained in [13] for the survival probability of a pair of diffusing quasi-particles under their uniform initial distribution (the difference in the normalized factors is connected with the difference of the definitions of survival probabilities). Note, that in the case concerned the root-like dependence (la) in decay kinetics of a,(t) is absent, the dependence
Q(t)
= ;
g+4Jzm r XX
1 (
&l
n'n
>
’ (23)
preceding the exponential decrease, i.e., the rate at which an uncorrelated pair vanishes R,(t) = - dQ,,(t)/dt, is equal to the sum of the rates at which each of the particles is trapped R:(t) = l/fi and their annihilation rate R:(t) = $/&; otherwise the particle vanishing on small times (m/n < 1) thorough the trapping and annihilation channels is independent.
57Pwt/n'
Fig. 1. Decay kinetics of survival probability for a pair in a segment (equation (20) solid lines) and in infinite chain (equation (1) dotted lines) in different temporal scales (a), (b); n, = l(top curve), lO(bottom curve), n = 300. R,(t, n,) one can write R,(t, n,)
=
n erf
--!%_ ( 2$iZ
, >
n, 4 n,
E41 n
(21) (214 In Fig. 1 the results the survival probability ness of our assumption for Q,(t, no) presented possible to demonstrate power-law asymptotic one (20a). Now let us consider
of a numerical calculation for (20) confirming the correctare shown. The dependencies in the different scales make it clearly the alternation of the (21a) by the exponential-like the limit case of uncorrelated
3. KINETICS OF GEMINATE RECOMBINATION IN THE CHAIN RANDOM TRAPS
WITH
Action of external sources of pumping, say, exciting radiation, leads to a creation in a solid matrix the correlated (geminate) pairs of charge carriers. Initial distribution of particles in a pair is determined by the function 4(x), which characterizes the probability to find in moment t = 0 a pair whose particles are separated by the distance x. We will restrict ourselves to the case of d-function initial distribution [15, 161 4(x)
=
6(x -
no);
no 6
n.
(24)
Now we will calculate the decay kinetics of correlated pair in l-d structures with recombination centers playing the role of deep traps for diffusing particles. Under random distribution of recombination centers in a chain the determination of dependence we are required comes to averaging Q,(t, no) over the Poisson distribution of lengths of segments confined between two traps a (C&,(t)) = c2 dnQ,(t, n,,)e-'"; (25) I "0
CHARGE
Vol. 76, No. 8
CARRIER
GEMINATE
Fig. 2. Charge carriers density decay (n,(t)) in a chain with random traps (equation (26), solid lines). Dotted lines correspond to the dependence (1); n,, = l(top curve), lO(bottom curve), c = lo-‘. c being the trap concentration. Taking into account (20), (21), (23)-(25) we get from (24) (Q”(l)>
=
(Z)X, x
- The author is very grateful to Dr A.I. Onipko for valuable discussions and critical reading of the manuscript. I am also indebted to Dr G. Juselunas who pointed out an alternative way of solving the problem (2)-(5) by the secondary quantatization method. Acknowledgements
z REFERENCES
(1 - t- l)‘)U + (- 1)“) y Cm’- 12)2
I
W.P. Su, J.R. Schrieffer 8z A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
dn n exp
- $ (m’ + 12) Wt - cn
“0
x
1039
current decay kinetics in 1-d polymeric structures with traps. Taking into account the applied electric field E leads to an appearance of the term 2q(ag/+ + ag/dx) (q = eEa/2kT, e is the quasi-particle charge) in the r.h.s. of equation (6). The estimates performed show the contribution of this term in the values of (Cl,(t)) under low field, q 4 c, has no vital importance. Thus, the dependence (26) for (Q,(t)) under the condition r] @ c describes the transient photocurrent in l-d polymeric structures with traps, particularly in polyacetylene crystals. A detailed analysis of the photocurrent kinetics as well as photoinduced absorption kinetics under arbitrary bias field will be given elsewhere.
30
X
RECOMBINATION
I
sin Km 2 + m sin 711z n
PC41 exp (-+(10rc22
Wt)“3),
(26)
(26a)
5.
10x2c2 Wt $ 1. (26b)
6.
Therefore, the decay kinetics of (O,(t)) on times m/c 6 1 is described by the power dependence which is typical for an infinite chain. The effects, connected with the finiteness of the particle motion in segment between two traps are manifest in the decay kinetics of (a,(t)) on times $!% < 1, when the power asymptotic (26a) alters by the nonexponential dependence (26b) (Fig. 2). Note, that the analogous dependence (cc exp (- 3 (2n2 c2 Wt)li3)) describes the decay of the averaged survival probability of single particle, executing random walks in a chain with random traps [17-191. Five-fold difference in the characteristic time scale is caused by the existence of an additional (recombination) channel of particle disappearance in the case of geminate recombination. To conclude, we will discuss briefly the possibility to use the dependence (26) to describe the photo-
7.
C.V. Shank, R. Yen, R.L. Fork, J. Orenstein & G.L. Baker, Phys. Rev. Lett. 49, 1660 (1982). C.V. Shank, R. Yen, J. Orenstein & G.L. Baker Phys. Rev. B28, 6095 (1983). M. Yoshizava, T. Kobayashi, H. Fujimoto 8z J. Tanaka, J. Phys. Sot. Japan 56, 768 (1987); M. Yoshizava, T. Kobayashi, K. Akagi & H. Shirakawa, Phys. Rev. B37, 10301 (1988). D.L. Weidman & D.B. Fitchen, Synthetic Metals 17, 355 (1987). Z. Vardeny, J. Strait, D. Moses, T.-C. Chung & A.J. Heeger, Phys. Rev. Lett. 49, 1657 (1982). M. Sinclair, D. Moses & A.J. Heeger, Solid State Commun. 59, 343 (1986).
10.
H. Blair, S. Roth, Y.Q. Shen, D. Schafer-Siebert & G. Leising, Phys. Rev. B38, 6031 (1988). S.D. Phillips & A.J. Heeger, Phys. Rev. B38 6211 (1988). D. Bloor & B. Movaghar, ZEE Proc. 130, 225
11.
W.P. Su & J.R. Shriefer, Proe. Nat/. Acad. Sci.
12.
A.G. Kofman & A.I. Burshtein, Chem. Phys. 27,
8. 9.
(1983). USA 77. 5626 (1980).
13. 14. 15.
217 (1978). Yu. B. Gaididei, A.I. Onipko & I.V. Zozulenko, Phys. Lett. A132, 329 (1988); A.I. Onipko & I.V. Zozulenko, J. Lumin. 43, 173 (1989). G. Korn & T. Korn, Mathematical Handbook,
Nauka, Moscow (1978) (in Russian). M. Pope & C. Swenberg, Electronic Processes in Organic Crystals, Mir, Moscow (1985) (in Russian).
1040 16. 17. 18.
CHARGE CARRIER
GEMINATE
H. Sher & S. Rackovsky, J. Chem. Phys. 81, 1994 (1984). B. Ya. Balagurov 8z V.G. Vaks, Zh. Esksp. Teor. Fiz. 65, 1939 (1973). [Sov. Phys. - JETP38,968 (1974)]. B. Movaghar, D. Murray, B. Pohlmann & D. Wiirtz, J. Phys. C: Solid State Phys. 17, 1667
19.
RECOMBINATION
Vol. 76, No. 8
(1984); B. Movaghar, B. Pohlmann & D. Wiirtz, Phys. Rev. A29, 1568 (1984). A.I. Onipko & I.V. Zozulenko, J. Phys.: Cond. Mutter 1, 9875 (1989); I.V. Zozulenko & A.I. Onipko, Fiz. Tverd. Tela 32, 1462 (1990) [Sov. Phys. - Solid State, in press (199011.
CHARGE
CARRIER
Vol. 76, No. 8, pp. 1035-1040,
GEMINATE
RECOMBINATION STRUCTURES WITH
0038-1098/90 $3.00 + .OO Pergamon Press plc
1990.
IN ONE-DIMENSIONAL TRAPS
POLYMER
Igor V. Zozulenko Institute
for Theoretical
(Received
Physics,
Kiev-130,
252130, USSR
30 March 1990 by A.A. Maradudin)
Recombination kinetics of photogenerated carriers in polymeric crystals with disorders (traps) was investigated. The distribution function and the survival probablility Q,(r) for a correlated pair of quasiparticles diffusing in a segment with absorbing ends are calculated. It is shown that the decay of Q,(r) follows the exponential dependence cxexp (- 7), 7 3 5x* Wt/n’ $ 1 (W is the diffusion rate, n is the segment length), unlike the power-law dependence a (Wr)-‘I’ for an infinite ideal chain. Within the framework of a model of molecular chain with random traps the averaged survival probability for a pair of photogenerated charge carriers in a quasi-one-dimensional polymer crystal (Q,(z)) (which is shown to describe the transient photocurrent at low bias field) is calculated.
1. INTRODUCTION STUDIES of charge carrier dynamics in quasi-onedimensional (1 -d) polymer structures, particularly, in trans-polyacetylene (PA) have been the subject of considerable theoretical and experimental interest over the past ten years. For the interpretation of the novel electrooptical properties of PA the theoretical model based on the soliton transport mechanism predicting the existence of localized quasi-particle states in a midgap is widely adopted [l]. Using the time-resolved technique in the experiments on photoinduced absorption [2-41 or reflectance [5], bleaching [4-61 and photoconductivity [7-91 is one of the most informative tools to study the nature of charge carriers, its transport and spectral properties as well as structure features of a polymeric matrix. Despite of a great amount of experimental data on photoexcitations relaxation kinetics, there is no so far reliable interpretation of a physical picture of charge carrier transport and recombination in PA in the all available interval of observations - from picosecond region to subsecond one. The possibility of an interchain violation of one-dimensional motion, generation, finiteness of quasi-particles recombination rate - this is an incomplete list of factors that complicate the description of a charge carrier transport in real PA crystals. The dynamics of quasi-particles in PA as well as in other l-d systems is especially influenced by the defects - traps or barriers. Both outside impurities,
e.g., oxygen atoms and violation of the ordered struc$-hybridized c-atoms, remaining cisture segments, chain ends may play the role of defects in PA. As the defect concentration is estimated to be 10-3-10-4 per c-atom [9, lo], the trapping of quasiparticles may reveal in kinetics of photoinduced absorption or photocurrent on times t = 12/a2 W loops-1 ns (l( - 4OOA or 300 carbon atoms) is the averaged length of chains of carbon bounds free from defects, W (- lo-” s’) is the diffusion rate, a is the cell unit). At the initial stage of photoinduced excitations relaxation (which follows the thermalization of e-h pair and its conversion to S+ -S- pair on times - 1 ps [1 11) the charge carrier decay kinetics are shown by the numerous experiments [2, 3, 6,9] to describe in the framework of the one-dimensional geminate recombination theory [ 121
(1)
where n(t) - is the probability for a pair of quasiparticles to survive by the moment t (survival probability), n, - is the distance of an initial particle separation. The deviation from the dependence (1) in the subnanosecond region was observed in [5,6,9], where an acceleration of the photocurrent decay [9] and a
1035
CHARGE
1036
CARRIER
GEMINATE
deceleration of the photoinduced absorption [6] and reflectance [5] signal decay have been marked. Such peculiarities of kinetics would be expected to associate with quasi-particle trapping by defects. Indeed, being trapped, the charge carriers do not contribute to the photoconductivity but they do contribute to the photoinduced absorption signal and the probability for a pair with one immobile carrier to recombine is obviously lower than that for a pair in which both carriers are mobile [9]. Since the theoretical description of the geminate recombination kinetics in l-d systems with traps is absent, it is of interest to consider the processes concerned. The paper is organized as follows. In Section 2 we deal with the diffusion of a pair of particles in a segment with traps on its ends, here the distribution function of particles in a segment and the survival probability are obtained. The comparison with the case of geminate recombination in an ideal onedimensional chain (without traps) is carried out. In Section 3 the averaged probability for a pair to survive in an infinite chain with random traps is calculated. The possibilities of experimental verification of the results obtained are discussed.
RECOMBINATION
when they occupy the same cell g(i,j,
t)
=
0,
i + j
FOR A CORRELATED PAIR IN A FINITE SEGMENT
Let us consider a segment (chain) with absorbing ends consisted of n cells (host molecules). In a segment at the initial moment t = 0 a random pair is located, with particles being separated by the distance of n, cells. The coordinate of one particle (discrete variable i, 1 < i < n) will be counted off from the left end of a segment, the coordinate of another one (discrete variable j, 1 < j < n - i) - from the right end. In the case of a stochastic Marcovian processes of random walks realized by jumps between the nearestneighbour sites with the probability W, the displacement of particles in a segment is determined by the distribution function g,(i, j, t) obeying the Pauli master equation
ag”k
+ W&(&j &(O, j, 0
=
g,(i, 0, t)
l,j, t) + Wg,(i,j -
=
1, t); 0,
n.
(5)
= Wa2&(x, YY f>
YY 4
at
+
Wd2gn(x, Y, 0
ax*
ay*
3
(6) where the variables x, y lie in the triangular region O
= g,(n, y, t)
=
g,(x,
= g,(x,
=
0
and the initial
n, t)
0, t) (7)
condition
g&c, y, t = 0)
=
6(x + y -
n + no>
6(x + y -
n -
no).
(8)
Having been chosen in the form (8) the initial condition satisfies the requirement for the distribution function g,(x, y, t) to equal zero when x + y = 0. Due to the symmetrical choice of the boundary conditions (7) the solution of equations (6)-(8) in the range 0 < x < n, 0 < y < n - x will coincide with the solution of the original problem (3)-(6). The extension of the triangular range of an alternation of variables x, y to the square one 0 < x < n, 0 < y < n makes it possible to apply the Green function method to calculate g,(x, y, t). Note that the analogous way was used in [ 131 to find the distribution function of an uncorrelated pair of annihilating quasiparticles under their random distribution in a segment. The solution of the diffusion equation (6) under the initial condition (8) in the Laplace-transform space 7(‘(s) = 1,” dtf(t) exp (- s Wt) has the form
s>= j dx' { j
&(x, Y, + Wg,(i -
=
We will find the solution of (2)-(4) in the diffusion approximation, changing the discrete variables i, j for continuous ones x, y. Assuming the smoothing of the distribution function (that is justify when n $ 1, t $ W-‘), one obtains instead of equation (2)
2. PROBABILITY TO SURVIVE
Vol. 76, No. 8
0
+ 1, t)
dy'
0
x G,,(x, x',y, y’, s)d(x’ + y' - n + no> (2) (3)
n
r”dy’ 6(x,
r
x’, y, Y',
s>
I’
under the initial condition x 6(x’ + y’ -
g,(i,j,
t = 0)
=
S(i + j -
Let us assume the particles
n + no>.
to recombine
n -
no) ,
(9)
(4) immediately
where G, (x, x’, y, y’, t) = G,,(x, x’, t)G,(y, y’, t) -
is
CHARGE
Vol. 76, No. 8 the Green function boundary condition
CARRIER
of the equation (7) and the initial
G,(x, x’, y, y’, t = 0)
=
6(x -
aG”(z, 0 ~
at
under
(6) under one
x’)60,
functions G,,(x, x’, t), G,,(y, y’, t) the Green functions of the equation
- y’),
(10)
Substituting equation going back to the original bution function
x
(11)
a22
the boundary
the
1037
RECOMBINATION
(17) in equation (9) and space one finds the distri-
are, respectively,
B/P G, (z, r)
=
GEMINATE
x
condition
Gm!?, ,zev { - f$Cm2+ 1’)Wt} xm X
sin
sin
711
n G,,(O, t)
=
G,(n, t)
and the initial G,(z,t=O)
0
x
&z-x’)
=
6(z -
(13)
y’).
The Laplace-transform y’, t) is written as
of the function
=
dq Gn(x, x’, q)c,& -I 27ti ~~iu:
where the functions -
Y’, q - 4,
(14)
G are defined
by the expression
sh&
-
sh&n fi
-
sh@
sh$ sh&n
-
’ < “’
’
’ “”
The normalized factor 1I,/? in equation (19) is defined by the choice of the initial condition in the form R,(t, no) = n - no. Substitution of equation (18) in equation (19) gives us
R,(t,
no)
=
n
1
0
q)
(16)
where the symbol C Res denotes the sum of residues taken at the poles of G,,(x, x’, q)cn(_v, y’, s - q) in the right semi-plane q. Having calculated the residues one has from (16)
=
2 1 x sin nl y’ sin 7~1y_ n n’
hJFT-27.
J)
(17)
Note, that expression (17) does not alter while changing x’ f-f x; y’ c* y; y - x, y’ t, x’; x ++ y’, x’ tf y.
$ (m’ +
I’) Wt)
x (1 - (- I)‘)(1 + (- 1)“) y (m’ - 12)2 x (1 sin 7cm3 + m sin 711t n > c( exp (- 5rc2Wt/n2),
x’, Y, Y’, s)
G”(X, x’, Y, Y’, s)
(19)
0
zc
and the path of integration in equation (14) goes to the right of all the singular points of functions G,,(z, z’, t). Having locked the integration contour to the right, one obtains
c Res G”(x, x’, q)e,,,cV, y’, s -
(18)
f dx ‘rX dy g,,(x, y, t).
x C C exp m=l /=I { (15)
-
.
n>
3
’ z)
&h$
1
=
= i
z’)
GJZ, z’, s) =
G&
n
$0
g+ia
.
lsinnm~+msinxZ~
G,,(x, x’, y,
x’, y, Y’, s) 1
m2 - l2
In real l-d systems the observable characteristics of random walks of quasi-particles are connected with the survival probability of a pair by the moment t Q,,(t, n,)
G&,
I)” - (- l)’
n
(12)
ones =
G,(z, t = 0)
=
Y_(-
5z2 Wt/n2 >>1.
(20) (204
At long times, 57~’Wr/n2 9 1, the decay of Q,(t, no> obeys the exponential dependence cc exp (- 5rr2Wt/n2), as distinct from the power-law dependence a ( Wt)-“2, justified for the case of infinite chain. At small times, 5n2 Wt/n2 < 1, being unable to get the analytical form of the dependence of R, (t, no) from (20) directly, we will estimate R,(t, no) in the following way. It follows from the condition Wt/n’ + 1 that the diffusion length for the time t (= m) is much less than the segment length n, i.e., under n < no, particles do not “feel” the segment ends. Therefore, in the case under consideration the initial problem of diffusion of recombining pair in a finite chain is expected to be equivalent to the problem of geminate recombination in an infinite chain and for the survival probability
CHARGE
CARRIER
GEMINATE
RECOMBINATION
Vol. 76, No. 8
initial distribution of particles in a segment. Averaging equation (20) over all values of n, in the interval 0 < n, < n with the weight of l/n one has 4 n,(t)
=
f
n
0 (1 -
f
i
m=l
/=I
(-
exp
l)‘)(l
- $ (m2 + 12)Wt}
i
+ (-
1)“)
X (m2
-
m2 (22)
7'
P)*
The dependence (22) up to normalized factor coincides with the result obtained in [13] for the survival probability of a pair of diffusing quasi-particles under their uniform initial distribution (the difference in the normalized factors is connected with the difference of the definitions of survival probabilities). Note, that in the case concerned the root-like dependence (la) in decay kinetics of a,(t) is absent, the dependence
Q(t)
= ;
g+4Jzm r XX
1 (
&l
n'n
>
’ (23)
preceding the exponential decrease, i.e., the rate at which an uncorrelated pair vanishes R,(t) = - dQ,,(t)/dt, is equal to the sum of the rates at which each of the particles is trapped R:(t) = l/fi and their annihilation rate R:(t) = $/&; otherwise the particle vanishing on small times (m/n < 1) thorough the trapping and annihilation channels is independent.
57Pwt/n'
Fig. 1. Decay kinetics of survival probability for a pair in a segment (equation (20) solid lines) and in infinite chain (equation (1) dotted lines) in different temporal scales (a), (b); n, = l(top curve), lO(bottom curve), n = 300. R,(t, n,) one can write R,(t, n,)
=
n erf
--!%_ ( 2$iZ
, >
n, 4 n,
E41 n
(21) (214 In Fig. 1 the results the survival probability ness of our assumption for Q,(t, no) presented possible to demonstrate power-law asymptotic one (20a). Now let us consider
of a numerical calculation for (20) confirming the correctare shown. The dependencies in the different scales make it clearly the alternation of the (21a) by the exponential-like the limit case of uncorrelated
3. KINETICS OF GEMINATE RECOMBINATION IN THE CHAIN RANDOM TRAPS
WITH
Action of external sources of pumping, say, exciting radiation, leads to a creation in a solid matrix the correlated (geminate) pairs of charge carriers. Initial distribution of particles in a pair is determined by the function 4(x), which characterizes the probability to find in moment t = 0 a pair whose particles are separated by the distance x. We will restrict ourselves to the case of d-function initial distribution [15, 161 4(x)
=
6(x -
no);
no 6
n.
(24)
Now we will calculate the decay kinetics of correlated pair in l-d structures with recombination centers playing the role of deep traps for diffusing particles. Under random distribution of recombination centers in a chain the determination of dependence we are required comes to averaging Q,(t, no) over the Poisson distribution of lengths of segments confined between two traps a (C&,(t)) = c2 dnQ,(t, n,,)e-'"; (25) I "0
CHARGE
Vol. 76, No. 8
CARRIER
GEMINATE
Fig. 2. Charge carriers density decay (n,(t)) in a chain with random traps (equation (26), solid lines). Dotted lines correspond to the dependence (1); n,, = l(top curve), lO(bottom curve), c = lo-‘. c being the trap concentration. Taking into account (20), (21), (23)-(25) we get from (24) (Q”(l)>
=
(Z)X, x
- The author is very grateful to Dr A.I. Onipko for valuable discussions and critical reading of the manuscript. I am also indebted to Dr G. Juselunas who pointed out an alternative way of solving the problem (2)-(5) by the secondary quantatization method. Acknowledgements
z REFERENCES
(1 - t- l)‘)U + (- 1)“) y Cm’- 12)2
I
W.P. Su, J.R. Schrieffer 8z A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
dn n exp
- $ (m’ + 12) Wt - cn
“0
x
1039
current decay kinetics in 1-d polymeric structures with traps. Taking into account the applied electric field E leads to an appearance of the term 2q(ag/+ + ag/dx) (q = eEa/2kT, e is the quasi-particle charge) in the r.h.s. of equation (6). The estimates performed show the contribution of this term in the values of (Cl,(t)) under low field, q 4 c, has no vital importance. Thus, the dependence (26) for (Q,(t)) under the condition r] @ c describes the transient photocurrent in l-d polymeric structures with traps, particularly in polyacetylene crystals. A detailed analysis of the photocurrent kinetics as well as photoinduced absorption kinetics under arbitrary bias field will be given elsewhere.
30
X
RECOMBINATION
I
sin Km 2 + m sin 711z n
PC41 exp (-+(10rc22
Wt)“3),
(26)
(26a)
5.
10x2c2 Wt $ 1. (26b)
6.
Therefore, the decay kinetics of (O,(t)) on times m/c 6 1 is described by the power dependence which is typical for an infinite chain. The effects, connected with the finiteness of the particle motion in segment between two traps are manifest in the decay kinetics of (a,(t)) on times $!% < 1, when the power asymptotic (26a) alters by the nonexponential dependence (26b) (Fig. 2). Note, that the analogous dependence (cc exp (- 3 (2n2 c2 Wt)li3)) describes the decay of the averaged survival probability of single particle, executing random walks in a chain with random traps [17-191. Five-fold difference in the characteristic time scale is caused by the existence of an additional (recombination) channel of particle disappearance in the case of geminate recombination. To conclude, we will discuss briefly the possibility to use the dependence (26) to describe the photo-
7.
C.V. Shank, R. Yen, R.L. Fork, J. Orenstein & G.L. Baker, Phys. Rev. Lett. 49, 1660 (1982). C.V. Shank, R. Yen, J. Orenstein & G.L. Baker Phys. Rev. B28, 6095 (1983). M. Yoshizava, T. Kobayashi, H. Fujimoto 8z J. Tanaka, J. Phys. Sot. Japan 56, 768 (1987); M. Yoshizava, T. Kobayashi, K. Akagi & H. Shirakawa, Phys. Rev. B37, 10301 (1988). D.L. Weidman & D.B. Fitchen, Synthetic Metals 17, 355 (1987). Z. Vardeny, J. Strait, D. Moses, T.-C. Chung & A.J. Heeger, Phys. Rev. Lett. 49, 1657 (1982). M. Sinclair, D. Moses & A.J. Heeger, Solid State Commun. 59, 343 (1986).
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H. Blair, S. Roth, Y.Q. Shen, D. Schafer-Siebert & G. Leising, Phys. Rev. B38, 6031 (1988). S.D. Phillips & A.J. Heeger, Phys. Rev. B38 6211 (1988). D. Bloor & B. Movaghar, ZEE Proc. 130, 225
11.
W.P. Su & J.R. Shriefer, Proe. Nat/. Acad. Sci.
12.
A.G. Kofman & A.I. Burshtein, Chem. Phys. 27,
8. 9.
(1983). USA 77. 5626 (1980).
13. 14. 15.
217 (1978). Yu. B. Gaididei, A.I. Onipko & I.V. Zozulenko, Phys. Lett. A132, 329 (1988); A.I. Onipko & I.V. Zozulenko, J. Lumin. 43, 173 (1989). G. Korn & T. Korn, Mathematical Handbook,
Nauka, Moscow (1978) (in Russian). M. Pope & C. Swenberg, Electronic Processes in Organic Crystals, Mir, Moscow (1985) (in Russian).
1040 16. 17. 18.
CHARGE CARRIER
GEMINATE
H. Sher & S. Rackovsky, J. Chem. Phys. 81, 1994 (1984). B. Ya. Balagurov 8z V.G. Vaks, Zh. Esksp. Teor. Fiz. 65, 1939 (1973). [Sov. Phys. - JETP38,968 (1974)]. B. Movaghar, D. Murray, B. Pohlmann & D. Wiirtz, J. Phys. C: Solid State Phys. 17, 1667
19.
RECOMBINATION
Vol. 76, No. 8
(1984); B. Movaghar, B. Pohlmann & D. Wiirtz, Phys. Rev. A29, 1568 (1984). A.I. Onipko & I.V. Zozulenko, J. Phys.: Cond. Mutter 1, 9875 (1989); I.V. Zozulenko & A.I. Onipko, Fiz. Tverd. Tela 32, 1462 (1990) [Sov. Phys. - Solid State, in press (199011.