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Recombination kinetics of excess electrons at low temperatures
Radiat. Phys. Chem. Vol. 17, No. 6, pp. 457-464, 1981
0146-5724181/060457--08502.00]0
Printed in Great Britain.
Pergamon Press Ltd.
RECOMBINATION KINETICS OF EXCESS ELECTRONS AT LOW TEMPERATURES Yu. A. BERLIN't" and V. I. GOLDANSKII The Institute of the Chemical Physics, The Academy of Sciences of the U.S.S.R. and D. CECCALDI Laboratoire de Physico-Chemie des Rayonnements, Universit6 de Paris-Sud, Centre d'Orsay, 91405 Orsay, France
(Received 3 April 1981) Abstract--A new theory based on the Eyring type kinetic equation is proposed for describing excess electron recombination at low temperatures. It takes into account two important physical processes governing the recombination kinetics, namely the discrete consecutive stochastic transitions of the excess electron from one localized state to another and its direct quantum-mechanical tunnelling from any localized state to the corresponding unoccupied level of a positive ion. Results obtained in the framework of this theory are compared with experimental data on luminescence of organic glasses at low temperatures.
I. INTRODUCTION THE APPARENT rate of the recombination of excess electrons and positive ions in the condensed medium is determined by three main factors. Namely, it depends on the frequency of collisions between reactants in their random motion, the chemical efficiency of these collisions and on the probability, per unit time, of the direct electron tunnelling from a localized state to a corresponding unoccupied level of the cation. The contribution of enumerated factors to the value of the apparent reaction rate has been calculated by means of the diffusion theory in a number of works. "-6) In addition to well-known conditions of the applicability of such a theory to the kinetic description of chemical reactions <7) the approach used to solve the problem is based on the assumption that a set of positions available for an electron in the process of its migration may be approximated by a continuum independently of the physical mechanism of the transport process (hopping, diffusion, transition sequence from localized state to the quasi-free state and vice versa, etc.). Although such a "continuum" approach proves
tVorobevskoye U.S.S.R.
chausse,
2-b,
117334 Moscow,
to be rather helpful for the theoretical treatment of the recombination in liquids, its applicability to the description of the same reaction in low temperature glassy solids seems to be doubtful, since in the latter case the average number of retrapping events during the migration of excess electrons towards positive ions is too small to consider their transport as a diffusion-type (continuous) random walkfl ~ The alternative "discrete" approach to the recombination kinetics, which, in principle, allows to take into account a trap-to-trap electron transfer without any limitations in the average number of electron jumps, is based on the formalism of differential-difference transport equationsfl > Recently first attempts have been made to use this formalism for treating the behaviour of excess electrons in condensed media. ~7"s"~o-t3~ In the present work we propose a new theory describing the kinetics of excess electron recombination at low temperatures. It takes into account both the discrete consecutive stochastic transitions of the excess electron from one localized state to another and its direct quantum-mechanical tunnelling from any trap being occupied by electron during the life-time to an appropriate energy level of the cation. The general formalism of the theory based on the Eyring type differential-difference equation is considered in Section 2. In order to illustrate its application two simplified models of 457
Yu. A. BERLIN et al.
458
the electron-cation recombination is formulated in Section 3. These models allow to compute decay curves characterizing the evolution of the recombination process in glassy solids with a low and with a high concentration of pre-existing traps. Finally, in Section 4 we discuss the relationship of our approach and the ordinary diffusion theory. The qualitative agreement of obtained theoretical results with experimental data on low temperature luminescence of organic solids is also demonstrated. 2. D I S C R E T E R A N D O M W A L K O F EXCESS E L E C T R O N A N D ITS R E C O M B I N A T I O N AT LOW T E M P E R A T U R E S The major part of charged species produced by low L E T ionizing radiations in "pure" glassy solids without scavengers or formed due to direct photoionization of solute molecules embedded in a glassy matrix may be considered as isolated electron-ion pairs which do not overlap and do not interact with each other. At low temperatures both components of these pairs are initially localized in physical traps (cavities, irregularities of the structure, etc.). The average life-time of charged species in a localized state depends on a number of factors. The most important ones among them are: (a) the temperature of the matrix T; (b) the mobility of solvent molecules and their arrangement around components of isolated pairs; (c) the position of a trapped electron, e~, with respect to a position of a positive ion; (d) electronic states of charged species; (e) the interaction of e~ with vibrational molecular modes of the frozen medium. The action of enumerated factors induces the transition of an excess electron which occupied some localized state i at time t to another localized state j o g J). Physically, a set of states j must contain at least one sink for excess electrons. Denoting such an absorbing state by j = 0 we conclude that in the framework of the "discrete random walk approach" the transition i ~ 0 corresponds to the single-step tunnelling decay of e~ due to the recombination with a positive ion while the transition i ~ ] # 0 represents the elementary step of the stochastic trap-to-trap motion of an excess electron. A fairly general transport equation describing this motion is given by n
(1)
dNidt -
ki,oNi - J='~"(KijNi - KiiNi),,
where N~(t) is the concentration of e~ at the ith state at time t, n is the total number of states available for electron in the frozen system, k~.oand K~.j are transition probabilities, per unit time, characterizing the single-step tunnelling recombination and the elementary step of the electron random walk, respectively.t In the further discussion all k~.o and K~.i factors are implied to be known. Expressions used to evaluate these quantities in some case of interest will be specified in Section 3. Note, however, that by definition Kj.j = 0. The differential-difference equation (1) has been widely used in the past for solving various physical problems including the kinetics of consecutive reactions, "4~ the relaxation of a system of a harmonic oscillator('5~ in a vibrational non-equilibrium distribution, the probable nucleation mechanism in condensed systems "6~ and the permeability of membranes. "7) Some fundamental properties of equation (1) as well as its numerous applications have been considered in Ref. 9. In the present case the quantities of direct physical significance are the total concentration C(t) of e~ at time t and the time dependent rate W(t) of their recombination with cations. In order to find them, it is convenient to introduce the square matrix A of order n with elements constructed of coefficients of equation (l) as follows
(2)
ki.o + ~ K,.i, if i = m, i=1 a,.m = - Km,,,if i ¢ m , m = 1,2 . . . . . n.
If, now, we denote the column matrix having elements N,(t) by N(t), equation (1) can be rewritten as
(3)
d N ( t ) _ AN(t). dt
By using the conventional technique of the matrix calculus it is easy to show that the solution of equation (3) is given by n
(4)
N(t) = ~=, p~P") exp ( - x#),
where pt is a real constant and P") is an eigenvector of A with an eigenvalue x, i.e. A P ")= xtP "~. All components P~) of eigenvectors have been normalized as follows
*For convenience, in the abbreviated form ki.o and Ki.i (5) will be called just transition probabilities.
n
~ p~l) = 1.
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Recombination kinetics Since C(t)= E N~(t), from equations (4) and (5) I=1
we find that the evolution of the recombination process is described by n
C(t) = i~= p~ exp ( - xlt).
(6)
The differentiation of this result with respect to t yields
W(t) =
(7)
dC(t)
dt
=
Otxt exp ( - x~t).
Thus, taking into account the possibility to evaluate p~ for a given initial distribution of e~ in the localized state, N(0), by solving a set of algebraic equations n
(8)
N~(0) = ~
piP~"~,
we conclude that both the total concentration C(t) of e?~ and the recombination rate W(t) are completely determined by eigenvalues and eigenvectors of the matrix A. In the next section components of the matrix A will be specified for two very simple models which describe the behaviour of e?r in systems with a low and with a high concentration of traps, respectively. By computing its eigenvalues and eigenvectors we shall find the evolution of the recombination process in these cases for certain initial distribution of e?~ around positive ions. 3. SOME A P P L I C A T I O N S OF G E N E R A L FORMALISM
Both models have been established with the following assumption. (a) Discrete states For the sake of simplicity we suppose that a discrete state i refers to the position of a trap with respect to the cation only (Fig. 1). In other words, in the case under consideration equation (1) has been used to describe the random walk of an excess electron in the ordinary space, although its stochastic motion in the energetic, configurational or any other abstract physical space may be taken into account by means of the same equation too. (b) Elementary step o[ random walk In order to simplify numerical calculations the displacement by one elementary step, A, of the discrete random walk is assumed to be constant independently of the electron position in the frozen matrix. This implies that we neglect the dispersion of intertrap distances and that only nearest-neighbor transition probabilities are involved throughout subsequent computations. From the latter circumstance it follows that K~j = O f o r a l l ] ¢ i -+ 1.
(ll)
The condition (11) must be considered as a zero approximation. However, it leads to results which are consistent with experimental observations. Generally speaking assumption (b) is not necessary and may be rejected in more precise calculations. "s~ Its use in the present work is necessitated by the aspiration to illustrate the application of the general formalism in the most obvious manner.
1 Preliminary remarks In order to illustrate the application of the general formalism developed in Section 2 the rate of the electron-cation recombination has been computed using two simplified kinetic models. The first model seems to be valid for glassy solids with a low concentration of pre-existing traps, i.e. in the case when the average initial separation distance fo between e~ and the cation and the concentration of traps c,r satisfy the inequality (9)
C,r '~ 3/(4¢rfo3).
The second model describes the evolution of the recombination process in frozen matrices with the relatively high concentration of traps (10)
Ctr ~ 3/(4'n'ro3).
I
/:2
_.gj_,. j.
""
r
,:o
/U,
'j
FIG. 1. Scheme illustrating the simplified models of electron-cation recombination in frozen matrices.
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Yu. A. BERLIN et al.
(c) Initial condition Since due to assumption (a) a state i refers to the distance between some trap and a cation the initial condition N~(0) used to solve equation (1) reduces to the initial population of traps by excess electrons. We have solved equation (1) with two kinds of the initial condition, i,e. either with the rectangular population distribution or with the delta one. The first distribution corresponds to that proposed in Ref. 19 to calculate the rate of e~ tunnelling decay in water-alkaline glasses while the delta distribution is analogous to the deltafunction initial condition often applied to compute the evolution of the neutralization process of ion pairs in a non-polar medium in the framework of the "continuum" approach/2°'2~) Parameters of assumed population distributions will be discussed in subsections 3 and 4. (d) Depth of traps All traps are supposed to have the same depth denoted by Eo. 2 Transition probabilities In order to calculate decay curves describing the kinetics of the electron-cation recombination expressions for transition probabilities ki.o and K~,j are needed. The formula for k~.ocan be found by taking into account that in the condensed medium e~ recombines with a cation both due to the ordinary thermally activated reaction ( T A R ) and due to the long-ranged tunnelling. The latter is believed to be either a temperature independent process ( T I T ) or a thermally assisted one ( T A T ) . (22) Denoting by k fAR, K r,rT, K TAT transition probabilities of enumerated recombination processes we have
(12)
k,.o
=
k TAR + k TIT + k TAT.
In accordance with the assumption (a) quantities in the r.h.s, of equation (12) depend on the distance r~ between e~ and a cation only. This makes possible to use for k TAR, k r~r and k TAT known expressions usually applied to solve the recombination problem by means of the "continuum" approach. Note, however, that by virtue of the assumption (b), rl = it, and hence the electroncation separation distance should be replaced by the discrete variable i,~ in all formulae for transition probabilities derived in the "continuum" approximation. For example, the transition probability of the temperature independent tunnelling is given b y t23) (13)
k StT= 1JTIT exp ( - 2aiM/a),
where uT~r is a frequency factor, a is a characteristic distance for the attenuation of electronic wave-functions of reactants, ai is the coefficient which depends on the barrier shape. In the case of a trap in the Coulomb field ai is defined as (23~ (14)
ai = 1 - I n
fli I/2 + (1 + ill) ~/2 fli(l +ill) jl2
In equation (14) fli=EoEiA/e 2, E is the static dielectric constant of the matrix, e is the charge of an electron. For thermally assisted tunnelling equation (13) should be modified as follows ~22) (15) k f AT = IITAT exp ( - 2ajti/a - ETAT[(kT))n, where E T A T is the activation energy of the thermally assisted tunnelling, k is the Boltzmann constant, T is the temperature. Obviously k T A R = o , if i # l . Otherwise the expression for k fAR coincides with the formula which gives the probability, per unit time, of the thermally activated electron transfer from the trap with i = 1 to the state associated with the cation (j = 0). Following results obtained in Refs. (7 and 12), we get kT AR = UTARexp [ -- (Eo - eA V/2)/(kT)],
if =0,
i= 1 if
i#l,
(16) where A V = e 2 A e ' ( A i + d ) - l ( A ( i + l ) + d ) ~ is the change in the Coulomb potential over the length A and d is the average radius of the trap. We now turn to the discussion of the transition probabilities Ki.j determining the rate of the stochastic motion of charged reactants in glassy solids. Since the random walk of an electron and the electron-cation recombination are due to the same physical processes (i.e. sub-barrier and over-barrier transitions), Ki.i and k~.o should be given by similar expressions. Moreover, formulae for Ki,j contain the same set of parameters ur~r, UrAT, UTAn, a, ETAT and Eo as that used in equations (13)-(16) with exactly the same physical meaning. However, in principle, some parameters characterizing the trap-to-trap migration of an excess electron may differ in their values from those used to describe the recombination. In this case physical quantities which refer to the electron migration will be denoted by this symbol "tilde". Taking into account equation (11) K~.j can be expressed in the general
Recombination kinetics form as follows
exp ( - 2A/ti) + ~TAR exp (
Eo
eA
V/2]
461
non-polar matrices although some exception are known. (22~ Parameters VTAT and ErAr has been found by fitting the computed values of the ratio n
y --if
kr'rN~(t)/~, kfArN~(t) i=1
j=i-+l=0, if
j~i+_l.
Thus, equations (13)-(17) completely define the square matrix A. This result will be used to compute the d e c a y c u r v e W = f ( t ) i n t h e f r a m e w o r k o f m o d e l s presented in the next two subsections.
3 Glassy solids with low concentration of traps (a) Model of electron-cation recombination It is generally accepted that in a glassy solid with a low concentration of traps e?r reacts with a cation by means of the "single-step" tunnelling if T is much less than the temperature of the glass transition, T~. If, however, T ~< T~, excess electrons can migrate through the medium due to diffusion of trapping cavities. "2'24) This motion of an electron may be represented as a sequence of discrete stochastic transitions from one autolocalized state to another. Thus in the temperature range T ~< Tg the recombination results from the competition of "single-step" tunnelling and electron random walk. In order to describe this competition mathematically the general formalism proposed in Section 2 can be used. (b) Choice of parameters Let electrons form cavities with Eo = 0.2 e V (25) and d = 1.5/~ in the frozen non-polar matrix with = 2. We assume that e~ are initially localized within a spatial zone Rz-< r-< R2 around cations and that at t = 0 the population of traps is given by the rectangular distribution. The lower and the upper limit of initial separation distances has been taken as RI = 20/~ and R2 = 50 ./~, respectively. "9) Since tunnelling transitions between any two nearest-neighbor autolocalized states seems to be hardly probable, we use in our computations ~r~r = ~rAr = 0 . The order-of-magnitude estimation of the third frequency factor describing the electron random walk gives
kT ~rAR-- ~ --1-6×1012s -z at T = 7 7 ° K . Two quantities vrlr and a, which characterize the rate of temperature independent tunnelling (see equation (13)), have been chosen as ~'rlr = 1014 s -I and a = 2 A. These values are typical for
i=1
to that obtained experimentally/12) This procedure gives vrAr = 9.6 x 1014 s -J and ETAT = 0.015 eV. The latter value agrees with the activation energy of the isothermal luminescence observed in organic glasses under conditions, when the light emission is caused by the electron-cation recombination. (26) The displacement in one elementary step of the electron random walk, A, has been evaluated from the experimental data on the mobility of e,r in hydrocarbons by means of the following expression (18)
A = (47rD/v) '/2 ~ 2(Dh/(kT)) '/2 exp
(Eol(2kT)) = 2(ph/e) 1/2exp (Eo/(2kT)). Taking / x = 10~Scm2v-ls-I at T = 7 7 ° K cs~ from equation (18) we get A = 2.5 .~. The simplest choice of the total number of autolocalized states involved in the recombination process must take into account that the upper limit of the electron initial distribution can move away from the position of a cation during the time of experiment, tex. Otherwise the rate of the electron-cation recombination may be over-estimated. This displacement does not exceed (4.Dt~x) '/2. As a result the total number of states should be taken as (19)
n = (R2 + (4"Dtex) 1/2- RO/A.
By substituting equation (18) into equation (19) we have (20)
n = (R2 - ROI;t + (vtex/lr) ~/2.
In the typical experimental situation tex~ < 104 s., and therefore for chosen values of parameters n is nearly equal to 20. (c) Computation of recombination kinetics To find the rate of the recombination, W(t) as a function of time, t, we numerically calculate eigenvalues and eigenvectors of the matrix A using the choice of parameters described in the previous subsection. The result has been substituted in equation (7) to obtain decay curves W = f ( t ) .
Yu. A. BERLIN et al.
462
Calculations were p e r f o r m e d on an U N I V A C l l l 0 computer. Figure 2 shows the time d e p e n d e n c e of the r e c o m b i n a t i o n rate under isothermal conditions ( T = 77°K). The decay law may be approximated
W
(t],
a.u.
2o
as
W(t)
(21)
= Bt'-'
where B and s are numerical constants. The effect of the sudden temperature pulse on the r e c o m b i n a t i o n kinetics is illustrated in Fig. 3. At to = 184 s, w h e n T jumps from 77°K up to 81°K, the rate of the e l e c t r o n - c a t i o n recombination increases by a factor of 2 (the segment AB of the kinetic curve). Such a change in the d e c a y trend is due to thermally assisted tunnelling. F o r t > t o we found that (d2W(t)/dt2)
4 Glassy solids with high concentration of traps To describe the evolution of the r e c o m b i n a t i o n process in the f r o z e n matrix with the high concentration of traps both thermally activated transitions and " t r a p - t o - t r a p " tunnelling should be taken into account. T h e r e f o r e , in contrast to the
A
I
0
-
T 81*K 77*K
0
I
1
O0
200 t,
300
$
FIG. 3. The rate of electron-cation recombination, W, vs time, t, computed for glassy solids with a low concentration of pre-existing traps: the effect of a thermal jump. case of glassy solids with the low concentration of traps, w h e n f'rlr = f~rAr = O, we have chosen 1~0 = ~TIT + ~TAT exp ( -
ETAT/(kT)) = lO ts S - 1
at
77°K. In addition we assume ti = a = 2 .A. The displacement in one e l e m e n t a r y step of the random walk, A, has been taken as ;t = 20 ~. This value agrees with the upper limit of the trap concentration in f r o z e n matrices which has been found to be equal to 10~9-102°cm-U 27~ All other physical quantities in equations (13)-(17) were supposed to have just the same values as those used in the previous subsection. The decay curve W = f(t) has been calculated for the delta initial distribution of separation distances: NRO) = ~.3. This implies that at t = 0 all e?r are localized at r -- 60 .~ (i = 3). ~2s~ The total n u m b e r of traps involved in the r e c o m b i n a t i o n has been estimated by means of equation (20) using the following set of parameters
O.U. 20
rO
R~
0 t,
I
1
tO0
200 $
FIG. 2. The rate of the electron-cation recombination, W, vs time, t, computed for glassy solids with a low concentration of pre-existing traps: isothermal conditions (T = 77°K).
=
0, R2
=
60 A, v ~ vo exp ( - 2M6)
+ ~TAR exp ( - Eo/(kT)). Since for chosen values of Vo, A, a, kTan, Eo T = 77°K v = 2 x 10 6 s - I , equation (20) yields 8, if tex does not e x c e e d 4 × 10 -5 s. The time d e p e n d e n c e of the recombination c o m p u t e d for T = 77°K is shown in Fig. 4. As
and n = rate can
Recombination kinetics
w
463
the first order and replacing A V by the product ~(OV/Or), the formula for K~.j can be rewritten as
(t). Io ~
(23)
[ , _ eA OV'~ K,.i= v, + v2~, + ~-~--~r } ( ] # i - l ) ,
2 --
where l,] = 17rlr exp ( - 2A/ti) p --
and
I jO-8
rO-r
10-6 t t
i0-5
$
FIG. 4. The rate of the electron-cation recombination, W, vs time, t, computed for glassy solids with a high concentration of pre-existing traps: isothermal conditions (T = 77°K). be seen, the decay curve obtained has a maximum at t = tm = 10-6 s. The position of the maximum shifts towards shorter times as T increases (for example, t,, = 5 x 10-s s. at T = 140°K). 4. D I S C U S S I O N The description of the electron-cation recombination at low temperatures by means of the "discrete" approach proposed in the present work has two main advantages over the "continuum" approximation. First, it makes possible to consider accurately the behaviour of e~ in terms of transition probabilities involved in the transport process. Second, the limitations in the distance and time involved in transport phenomena are small in the "discrete" approach compared with "continuum" one. Moreover, it is possible to show that under certain conditions two simplified models proposed in Section 3 reduce to the Smoluchowski equation with an exponential sink term. Such an e q u a t i o n h a s been applied to investigate the influence of tunnelling on the rate of diffusion-controlled reactions. (3-5) Thus, the relation between the "discrete" approach and the "continuum" description of the electron-cation recombination can be established. To prove this, one should express the transport flux of electrons in the following form (22)
v2 = ~'rAR exp (--~-~) + ~rAr exp ( - - ( 2 M a + ~ T r ) ) .
tO-4
In the limit of very small A, when (24)
A ~ ~o
so that (25)
A grad N~ ~ N .
the quantity Ni+l may be represented by the first two terms of a Taylor series. This yields (26) [ J l = - A 2 ( v l +v2)
-t k T v l + v z - ~
Obviously the total number of electrons crossing a sphere of the radius r per unit time is given by the surface integral §J dr. On the other hand, by virtue of the continuity condition the value of the integral must be equal to the rate of the electron concentration decrease inside the sphere. Since this decrease results from the change in the position of etr due to random walk and from its singlestep tunnelling decay, we have (27)
~sJdr=f~rdivJdv=-f
(ONikot+ki'°Ni) dv"
where symbols Sr and v, denote integration over surface and volume, respectively. Substitution of equation (26) into equation (27) gives after discarding the indices the following kinetic equation
ON D 0 [ r 2 ( O N + y _ ~ _ ~ (28) ~ = r - ~ - ~ [ k 0 r
N
)] - k ( r ) N
IJI = A[K~.~+~Nj(t)- K~+,.iN~+I(t)] (i# 0). where
For a small change in the Coulomb potential over the length ,~ we expand exp ( + eA V[(kT)) in equation (17) in a power series. Neglecting terms above
ij.
V = - e/(er), k(r) = (Prrr + ~rAr exp ( - ErAT/(kT)) exp ( - 2adt~),
464
Yu. A. BERLIN et al. 3' = ~2](vl + v2), D = A2(vl + v2)/(4"rr).
If v2>>vl, equation (28) exactly coincides with the conventional diffusion equation which takes into account the additional long-range interaction between reactants. ~3-5) Note, however, that in the general case equation (28) differs from the ordinary one by the factor y in the second term in the parentheses because resonant tunnelling has been assumed to be almost independent of the Coulomb field. As can be seen from the derivation of equation (28) the "discrete" approach reduces to "continuum" approximation if inequalities (24) and (25) are satisfied. In this sense the proposed models of the electron-cation recombination seems to be more general than those based on the diffusion theory. Usually the behaviour of e~ at low temperatures has been interpreted either in terms of single-step tunnelling (see, for example, reviews "9'22) and references therein) or by means of the trap-to-trap mode of the electron migration in frozen media. ~29-3" This has led to a controversy concerning whether in glassy solids the rate of e?, decay is determined by the transport of excess electrons or by their direct sub-barrier transitions. ~32'33~ The important feature of the present theory is the possibility to take into account both processes mentioned above. Our calculations of decay curves should be considered only as order-of-magnitude estimations since values of physical parameters used in computations do not refer to any specific system. Nevertheless most of them is typical for non-polar glasses. Although the lack of data concerning parameter ,values do not allow to carry out a quantitative comparison of our theoretical results with the experiments, we have found a good qualitative agreement of the decay curve computed for solids with a low concentration of traps (Fig. 3) and the shape of the luminescence curve observed after the photoionization of 3-methylpentane glass dopped by TMPD. "2~ In particular, the isothermal section of the experimental curve can be described by the kinetic law given by equation (21). The intensity of the isothermal luminescence following the photoionization of the sample at 77°K sharply increases at time t = to, when a thermal jump is applied, passes through the maximum and then decreases. This trend of the experimental curve is also consistent with predictions of our model.
REFERENCES 1. M. J. PILLING and S. A. RICE, J.C.S. Faraday H 1975, 71, 1333. 2. Yu. A. BERLIN, Dokl. Phys. Chem. Akad. Sci. U.S.S.R. 1975, 223, 625. 3. Yu. A. BERLIN, Dokl. Phys. Chem. Akad. Sci. U.S.S.R. 1976, 228, 628. 4. P. R. BUTLERand M. J. PILLING, Discuss. Faraday Soc. 1977, 63, 38. 5. Yu. A. BERLINand R. F. KHAIRUTDINOV,Khim. Vys. Energij 1978, 12, 3. 6. Yu. A. BERLINand V. I. GOLDANSKn,Radiat. Phys. Chem. To be published. 7. Yu. A. BERLIN,Radiat. Phys. Chem. 1976, 8, 305. 8. G. F. NOVlKOV,Ph.D. Thesis, Institute of Chemical Physics, Academy of Science of the U.S.S.R., Moscow, 1974. 9. F. H. REE, T. S. REE, T. REE and H. EYRING, Advances in Chem. Phys. (Edited by J. Prigogine), Vol. 4, p. 1. Interscience, New York, 1962. 10. B. S. YAKOVLEVand G. F. NOVIKOV,Radiat. Phys. Chem. 1975, 7, 679. ! 1. B. S. YAKOVLEV,K. K. AMETOVand G. F. NOVtKOV, Radiat. Phys. Chem. 1978, 11,219. 12. D. CECCALDI,Radiat. Phys. Chem. 1976, 8, 539. 13. D. CECCALDI,Can. J. Chem. 1977, 55, 2517. 14. H. DOSTAL,Monatshe.fte (Vienna) 1941, 70, 324. 15. E. W. MONTROLLand K. E. SHULER,J. Chem. Phys. 1957, 26, 454. 16. D. TURNBULL,Am. Inst. Min. Metall. Engrs Tech. Pub., No. 2365. 17. B. J. ZWOLINSKI,H. EYRING and C. E. REESE, J. Phys. & Coll. Chem. 1949, 53, 1426. 18. Yu. A. BERLIN,N. CHEKUNAEVand V. N. FLEROV,J. Phys. D. To be published. 19. K. I. ZAMARAEVand R. F. KHAIRUTDINOV,Chem. Phys. 1974, 4, 181. 20. K. M. HONGand J. NOOLANDI,J. Chem. Phys. 1978, 68, 5163. 21. E. CROC, Mathematics Thesis, Universit6e Provence, June, 1979. 22. I. V. ALEXANDROV,R. F. KHAIRUTDINOVand K. I. ZAMARAEV,Chem. Phys. 1978, 32, 123. 23. A. I. MICKHAILOV, Dokl. Phys. Chem. Akad. Sci. U.S.S.R. 1971 197, 123. 24. M. MAGAT,Ber. Bunsenges. physik. Chem. 1971, 75, 666. 25. M. BURTON,M. DILLONand R. REIN, J. Chem. Phys. 1964, 41, 2228. 26. M. WYPYCH and R. LERZCZYNSKI, Radioanal. Radiochem. Lett. 1978, 36, 309. 27. M SHIROM and J. E. WtLLARD,J. Am. Chem. Soc. 1968, 90, 2184. 28. C. LAPERSONNE-MEYER and M. SCHOTT, Chem. Phys. (in press). 29. G. V. BUXTONand K. G. KEMSLEY,J.C.S. Faraday Transactions I, 1975, 71,568. 30. K. FUNARASHIand W. H. HAMlEt., Phys. Rev. B, 1977, 16, 5523; Phys. Rev. 'Lett. 1978, 56, 175; J. Phys. Chem. 1978, 82, 2073. 31. A. PLONKA, W. LEFIK and J. KROH, Chem. Phys. Lett. 1979, 62, 271. 32. Cz. STRADOWSKIand J. KROH, Radial. Phys. Chem. 1978, 12, 175: 33. A. A. RICE and G. A. KENNEY--WALLACE,Phys. Rev. B, 1980, 21, 3748.
0146-5724181/060457--08502.00]0
Printed in Great Britain.
Pergamon Press Ltd.
RECOMBINATION KINETICS OF EXCESS ELECTRONS AT LOW TEMPERATURES Yu. A. BERLIN't" and V. I. GOLDANSKII The Institute of the Chemical Physics, The Academy of Sciences of the U.S.S.R. and D. CECCALDI Laboratoire de Physico-Chemie des Rayonnements, Universit6 de Paris-Sud, Centre d'Orsay, 91405 Orsay, France
(Received 3 April 1981) Abstract--A new theory based on the Eyring type kinetic equation is proposed for describing excess electron recombination at low temperatures. It takes into account two important physical processes governing the recombination kinetics, namely the discrete consecutive stochastic transitions of the excess electron from one localized state to another and its direct quantum-mechanical tunnelling from any localized state to the corresponding unoccupied level of a positive ion. Results obtained in the framework of this theory are compared with experimental data on luminescence of organic glasses at low temperatures.
I. INTRODUCTION THE APPARENT rate of the recombination of excess electrons and positive ions in the condensed medium is determined by three main factors. Namely, it depends on the frequency of collisions between reactants in their random motion, the chemical efficiency of these collisions and on the probability, per unit time, of the direct electron tunnelling from a localized state to a corresponding unoccupied level of the cation. The contribution of enumerated factors to the value of the apparent reaction rate has been calculated by means of the diffusion theory in a number of works. "-6) In addition to well-known conditions of the applicability of such a theory to the kinetic description of chemical reactions <7) the approach used to solve the problem is based on the assumption that a set of positions available for an electron in the process of its migration may be approximated by a continuum independently of the physical mechanism of the transport process (hopping, diffusion, transition sequence from localized state to the quasi-free state and vice versa, etc.). Although such a "continuum" approach proves
tVorobevskoye U.S.S.R.
chausse,
2-b,
117334 Moscow,
to be rather helpful for the theoretical treatment of the recombination in liquids, its applicability to the description of the same reaction in low temperature glassy solids seems to be doubtful, since in the latter case the average number of retrapping events during the migration of excess electrons towards positive ions is too small to consider their transport as a diffusion-type (continuous) random walkfl ~ The alternative "discrete" approach to the recombination kinetics, which, in principle, allows to take into account a trap-to-trap electron transfer without any limitations in the average number of electron jumps, is based on the formalism of differential-difference transport equationsfl > Recently first attempts have been made to use this formalism for treating the behaviour of excess electrons in condensed media. ~7"s"~o-t3~ In the present work we propose a new theory describing the kinetics of excess electron recombination at low temperatures. It takes into account both the discrete consecutive stochastic transitions of the excess electron from one localized state to another and its direct quantum-mechanical tunnelling from any trap being occupied by electron during the life-time to an appropriate energy level of the cation. The general formalism of the theory based on the Eyring type differential-difference equation is considered in Section 2. In order to illustrate its application two simplified models of 457
Yu. A. BERLIN et al.
458
the electron-cation recombination is formulated in Section 3. These models allow to compute decay curves characterizing the evolution of the recombination process in glassy solids with a low and with a high concentration of pre-existing traps. Finally, in Section 4 we discuss the relationship of our approach and the ordinary diffusion theory. The qualitative agreement of obtained theoretical results with experimental data on low temperature luminescence of organic solids is also demonstrated. 2. D I S C R E T E R A N D O M W A L K O F EXCESS E L E C T R O N A N D ITS R E C O M B I N A T I O N AT LOW T E M P E R A T U R E S The major part of charged species produced by low L E T ionizing radiations in "pure" glassy solids without scavengers or formed due to direct photoionization of solute molecules embedded in a glassy matrix may be considered as isolated electron-ion pairs which do not overlap and do not interact with each other. At low temperatures both components of these pairs are initially localized in physical traps (cavities, irregularities of the structure, etc.). The average life-time of charged species in a localized state depends on a number of factors. The most important ones among them are: (a) the temperature of the matrix T; (b) the mobility of solvent molecules and their arrangement around components of isolated pairs; (c) the position of a trapped electron, e~, with respect to a position of a positive ion; (d) electronic states of charged species; (e) the interaction of e~ with vibrational molecular modes of the frozen medium. The action of enumerated factors induces the transition of an excess electron which occupied some localized state i at time t to another localized state j o g J). Physically, a set of states j must contain at least one sink for excess electrons. Denoting such an absorbing state by j = 0 we conclude that in the framework of the "discrete random walk approach" the transition i ~ 0 corresponds to the single-step tunnelling decay of e~ due to the recombination with a positive ion while the transition i ~ ] # 0 represents the elementary step of the stochastic trap-to-trap motion of an excess electron. A fairly general transport equation describing this motion is given by n
(1)
dNidt -
ki,oNi - J='~"(KijNi - KiiNi),,
where N~(t) is the concentration of e~ at the ith state at time t, n is the total number of states available for electron in the frozen system, k~.oand K~.j are transition probabilities, per unit time, characterizing the single-step tunnelling recombination and the elementary step of the electron random walk, respectively.t In the further discussion all k~.o and K~.i factors are implied to be known. Expressions used to evaluate these quantities in some case of interest will be specified in Section 3. Note, however, that by definition Kj.j = 0. The differential-difference equation (1) has been widely used in the past for solving various physical problems including the kinetics of consecutive reactions, "4~ the relaxation of a system of a harmonic oscillator('5~ in a vibrational non-equilibrium distribution, the probable nucleation mechanism in condensed systems "6~ and the permeability of membranes. "7) Some fundamental properties of equation (1) as well as its numerous applications have been considered in Ref. 9. In the present case the quantities of direct physical significance are the total concentration C(t) of e~ at time t and the time dependent rate W(t) of their recombination with cations. In order to find them, it is convenient to introduce the square matrix A of order n with elements constructed of coefficients of equation (l) as follows
(2)
ki.o + ~ K,.i, if i = m, i=1 a,.m = - Km,,,if i ¢ m , m = 1,2 . . . . . n.
If, now, we denote the column matrix having elements N,(t) by N(t), equation (1) can be rewritten as
(3)
d N ( t ) _ AN(t). dt
By using the conventional technique of the matrix calculus it is easy to show that the solution of equation (3) is given by n
(4)
N(t) = ~=, p~P") exp ( - x#),
where pt is a real constant and P") is an eigenvector of A with an eigenvalue x, i.e. A P ")= xtP "~. All components P~) of eigenvectors have been normalized as follows
*For convenience, in the abbreviated form ki.o and Ki.i (5) will be called just transition probabilities.
n
~ p~l) = 1.
459
Recombination kinetics Since C(t)= E N~(t), from equations (4) and (5) I=1
we find that the evolution of the recombination process is described by n
C(t) = i~= p~ exp ( - xlt).
(6)
The differentiation of this result with respect to t yields
W(t) =
(7)
dC(t)
dt
=
Otxt exp ( - x~t).
Thus, taking into account the possibility to evaluate p~ for a given initial distribution of e~ in the localized state, N(0), by solving a set of algebraic equations n
(8)
N~(0) = ~
piP~"~,
we conclude that both the total concentration C(t) of e?~ and the recombination rate W(t) are completely determined by eigenvalues and eigenvectors of the matrix A. In the next section components of the matrix A will be specified for two very simple models which describe the behaviour of e?r in systems with a low and with a high concentration of traps, respectively. By computing its eigenvalues and eigenvectors we shall find the evolution of the recombination process in these cases for certain initial distribution of e?~ around positive ions. 3. SOME A P P L I C A T I O N S OF G E N E R A L FORMALISM
Both models have been established with the following assumption. (a) Discrete states For the sake of simplicity we suppose that a discrete state i refers to the position of a trap with respect to the cation only (Fig. 1). In other words, in the case under consideration equation (1) has been used to describe the random walk of an excess electron in the ordinary space, although its stochastic motion in the energetic, configurational or any other abstract physical space may be taken into account by means of the same equation too. (b) Elementary step o[ random walk In order to simplify numerical calculations the displacement by one elementary step, A, of the discrete random walk is assumed to be constant independently of the electron position in the frozen matrix. This implies that we neglect the dispersion of intertrap distances and that only nearest-neighbor transition probabilities are involved throughout subsequent computations. From the latter circumstance it follows that K~j = O f o r a l l ] ¢ i -+ 1.
(ll)
The condition (11) must be considered as a zero approximation. However, it leads to results which are consistent with experimental observations. Generally speaking assumption (b) is not necessary and may be rejected in more precise calculations. "s~ Its use in the present work is necessitated by the aspiration to illustrate the application of the general formalism in the most obvious manner.
1 Preliminary remarks In order to illustrate the application of the general formalism developed in Section 2 the rate of the electron-cation recombination has been computed using two simplified kinetic models. The first model seems to be valid for glassy solids with a low concentration of pre-existing traps, i.e. in the case when the average initial separation distance fo between e~ and the cation and the concentration of traps c,r satisfy the inequality (9)
C,r '~ 3/(4¢rfo3).
The second model describes the evolution of the recombination process in frozen matrices with the relatively high concentration of traps (10)
Ctr ~ 3/(4'n'ro3).
I
/:2
_.gj_,. j.
""
r
,:o
/U,
'j
FIG. 1. Scheme illustrating the simplified models of electron-cation recombination in frozen matrices.
460
Yu. A. BERLIN et al.
(c) Initial condition Since due to assumption (a) a state i refers to the distance between some trap and a cation the initial condition N~(0) used to solve equation (1) reduces to the initial population of traps by excess electrons. We have solved equation (1) with two kinds of the initial condition, i,e. either with the rectangular population distribution or with the delta one. The first distribution corresponds to that proposed in Ref. 19 to calculate the rate of e~ tunnelling decay in water-alkaline glasses while the delta distribution is analogous to the deltafunction initial condition often applied to compute the evolution of the neutralization process of ion pairs in a non-polar medium in the framework of the "continuum" approach/2°'2~) Parameters of assumed population distributions will be discussed in subsections 3 and 4. (d) Depth of traps All traps are supposed to have the same depth denoted by Eo. 2 Transition probabilities In order to calculate decay curves describing the kinetics of the electron-cation recombination expressions for transition probabilities ki.o and K~,j are needed. The formula for k~.ocan be found by taking into account that in the condensed medium e~ recombines with a cation both due to the ordinary thermally activated reaction ( T A R ) and due to the long-ranged tunnelling. The latter is believed to be either a temperature independent process ( T I T ) or a thermally assisted one ( T A T ) . (22) Denoting by k fAR, K r,rT, K TAT transition probabilities of enumerated recombination processes we have
(12)
k,.o
=
k TAR + k TIT + k TAT.
In accordance with the assumption (a) quantities in the r.h.s, of equation (12) depend on the distance r~ between e~ and a cation only. This makes possible to use for k TAR, k r~r and k TAT known expressions usually applied to solve the recombination problem by means of the "continuum" approach. Note, however, that by virtue of the assumption (b), rl = it, and hence the electroncation separation distance should be replaced by the discrete variable i,~ in all formulae for transition probabilities derived in the "continuum" approximation. For example, the transition probability of the temperature independent tunnelling is given b y t23) (13)
k StT= 1JTIT exp ( - 2aiM/a),
where uT~r is a frequency factor, a is a characteristic distance for the attenuation of electronic wave-functions of reactants, ai is the coefficient which depends on the barrier shape. In the case of a trap in the Coulomb field ai is defined as (23~ (14)
ai = 1 - I n
fli I/2 + (1 + ill) ~/2 fli(l +ill) jl2
In equation (14) fli=EoEiA/e 2, E is the static dielectric constant of the matrix, e is the charge of an electron. For thermally assisted tunnelling equation (13) should be modified as follows ~22) (15) k f AT = IITAT exp ( - 2ajti/a - ETAT[(kT))n, where E T A T is the activation energy of the thermally assisted tunnelling, k is the Boltzmann constant, T is the temperature. Obviously k T A R = o , if i # l . Otherwise the expression for k fAR coincides with the formula which gives the probability, per unit time, of the thermally activated electron transfer from the trap with i = 1 to the state associated with the cation (j = 0). Following results obtained in Refs. (7 and 12), we get kT AR = UTARexp [ -- (Eo - eA V/2)/(kT)],
if =0,
i= 1 if
i#l,
(16) where A V = e 2 A e ' ( A i + d ) - l ( A ( i + l ) + d ) ~ is the change in the Coulomb potential over the length A and d is the average radius of the trap. We now turn to the discussion of the transition probabilities Ki.j determining the rate of the stochastic motion of charged reactants in glassy solids. Since the random walk of an electron and the electron-cation recombination are due to the same physical processes (i.e. sub-barrier and over-barrier transitions), Ki.i and k~.o should be given by similar expressions. Moreover, formulae for Ki,j contain the same set of parameters ur~r, UrAT, UTAn, a, ETAT and Eo as that used in equations (13)-(16) with exactly the same physical meaning. However, in principle, some parameters characterizing the trap-to-trap migration of an excess electron may differ in their values from those used to describe the recombination. In this case physical quantities which refer to the electron migration will be denoted by this symbol "tilde". Taking into account equation (11) K~.j can be expressed in the general
Recombination kinetics form as follows
exp ( - 2A/ti) + ~TAR exp (
Eo
eA
V/2]
461
non-polar matrices although some exception are known. (22~ Parameters VTAT and ErAr has been found by fitting the computed values of the ratio n
y --if
kr'rN~(t)/~, kfArN~(t) i=1
j=i-+l=0, if
j~i+_l.
Thus, equations (13)-(17) completely define the square matrix A. This result will be used to compute the d e c a y c u r v e W = f ( t ) i n t h e f r a m e w o r k o f m o d e l s presented in the next two subsections.
3 Glassy solids with low concentration of traps (a) Model of electron-cation recombination It is generally accepted that in a glassy solid with a low concentration of traps e?r reacts with a cation by means of the "single-step" tunnelling if T is much less than the temperature of the glass transition, T~. If, however, T ~< T~, excess electrons can migrate through the medium due to diffusion of trapping cavities. "2'24) This motion of an electron may be represented as a sequence of discrete stochastic transitions from one autolocalized state to another. Thus in the temperature range T ~< Tg the recombination results from the competition of "single-step" tunnelling and electron random walk. In order to describe this competition mathematically the general formalism proposed in Section 2 can be used. (b) Choice of parameters Let electrons form cavities with Eo = 0.2 e V (25) and d = 1.5/~ in the frozen non-polar matrix with = 2. We assume that e~ are initially localized within a spatial zone Rz-< r-< R2 around cations and that at t = 0 the population of traps is given by the rectangular distribution. The lower and the upper limit of initial separation distances has been taken as RI = 20/~ and R2 = 50 ./~, respectively. "9) Since tunnelling transitions between any two nearest-neighbor autolocalized states seems to be hardly probable, we use in our computations ~r~r = ~rAr = 0 . The order-of-magnitude estimation of the third frequency factor describing the electron random walk gives
kT ~rAR-- ~ --1-6×1012s -z at T = 7 7 ° K . Two quantities vrlr and a, which characterize the rate of temperature independent tunnelling (see equation (13)), have been chosen as ~'rlr = 1014 s -I and a = 2 A. These values are typical for
i=1
to that obtained experimentally/12) This procedure gives vrAr = 9.6 x 1014 s -J and ETAT = 0.015 eV. The latter value agrees with the activation energy of the isothermal luminescence observed in organic glasses under conditions, when the light emission is caused by the electron-cation recombination. (26) The displacement in one elementary step of the electron random walk, A, has been evaluated from the experimental data on the mobility of e,r in hydrocarbons by means of the following expression (18)
A = (47rD/v) '/2 ~ 2(Dh/(kT)) '/2 exp
(Eol(2kT)) = 2(ph/e) 1/2exp (Eo/(2kT)). Taking / x = 10~Scm2v-ls-I at T = 7 7 ° K cs~ from equation (18) we get A = 2.5 .~. The simplest choice of the total number of autolocalized states involved in the recombination process must take into account that the upper limit of the electron initial distribution can move away from the position of a cation during the time of experiment, tex. Otherwise the rate of the electron-cation recombination may be over-estimated. This displacement does not exceed (4.Dt~x) '/2. As a result the total number of states should be taken as (19)
n = (R2 + (4"Dtex) 1/2- RO/A.
By substituting equation (18) into equation (19) we have (20)
n = (R2 - ROI;t + (vtex/lr) ~/2.
In the typical experimental situation tex~ < 104 s., and therefore for chosen values of parameters n is nearly equal to 20. (c) Computation of recombination kinetics To find the rate of the recombination, W(t) as a function of time, t, we numerically calculate eigenvalues and eigenvectors of the matrix A using the choice of parameters described in the previous subsection. The result has been substituted in equation (7) to obtain decay curves W = f ( t ) .
Yu. A. BERLIN et al.
462
Calculations were p e r f o r m e d on an U N I V A C l l l 0 computer. Figure 2 shows the time d e p e n d e n c e of the r e c o m b i n a t i o n rate under isothermal conditions ( T = 77°K). The decay law may be approximated
W
(t],
a.u.
2o
as
W(t)
(21)
= Bt'-'
where B and s are numerical constants. The effect of the sudden temperature pulse on the r e c o m b i n a t i o n kinetics is illustrated in Fig. 3. At to = 184 s, w h e n T jumps from 77°K up to 81°K, the rate of the e l e c t r o n - c a t i o n recombination increases by a factor of 2 (the segment AB of the kinetic curve). Such a change in the d e c a y trend is due to thermally assisted tunnelling. F o r t > t o we found that (d2W(t)/dt2)
4 Glassy solids with high concentration of traps To describe the evolution of the r e c o m b i n a t i o n process in the f r o z e n matrix with the high concentration of traps both thermally activated transitions and " t r a p - t o - t r a p " tunnelling should be taken into account. T h e r e f o r e , in contrast to the
A
I
0
-
T 81*K 77*K
0
I
1
O0
200 t,
300
$
FIG. 3. The rate of electron-cation recombination, W, vs time, t, computed for glassy solids with a low concentration of pre-existing traps: the effect of a thermal jump. case of glassy solids with the low concentration of traps, w h e n f'rlr = f~rAr = O, we have chosen 1~0 = ~TIT + ~TAT exp ( -
ETAT/(kT)) = lO ts S - 1
at
77°K. In addition we assume ti = a = 2 .A. The displacement in one e l e m e n t a r y step of the random walk, A, has been taken as ;t = 20 ~. This value agrees with the upper limit of the trap concentration in f r o z e n matrices which has been found to be equal to 10~9-102°cm-U 27~ All other physical quantities in equations (13)-(17) were supposed to have just the same values as those used in the previous subsection. The decay curve W = f(t) has been calculated for the delta initial distribution of separation distances: NRO) = ~.3. This implies that at t = 0 all e?r are localized at r -- 60 .~ (i = 3). ~2s~ The total n u m b e r of traps involved in the r e c o m b i n a t i o n has been estimated by means of equation (20) using the following set of parameters
O.U. 20
rO
R~
0 t,
I
1
tO0
200 $
FIG. 2. The rate of the electron-cation recombination, W, vs time, t, computed for glassy solids with a low concentration of pre-existing traps: isothermal conditions (T = 77°K).
=
0, R2
=
60 A, v ~ vo exp ( - 2M6)
+ ~TAR exp ( - Eo/(kT)). Since for chosen values of Vo, A, a, kTan, Eo T = 77°K v = 2 x 10 6 s - I , equation (20) yields 8, if tex does not e x c e e d 4 × 10 -5 s. The time d e p e n d e n c e of the recombination c o m p u t e d for T = 77°K is shown in Fig. 4. As
and n = rate can
Recombination kinetics
w
463
the first order and replacing A V by the product ~(OV/Or), the formula for K~.j can be rewritten as
(t). Io ~
(23)
[ , _ eA OV'~ K,.i= v, + v2~, + ~-~--~r } ( ] # i - l ) ,
2 --
where l,] = 17rlr exp ( - 2A/ti) p --
and
I jO-8
rO-r
10-6 t t
i0-5
$
FIG. 4. The rate of the electron-cation recombination, W, vs time, t, computed for glassy solids with a high concentration of pre-existing traps: isothermal conditions (T = 77°K). be seen, the decay curve obtained has a maximum at t = tm = 10-6 s. The position of the maximum shifts towards shorter times as T increases (for example, t,, = 5 x 10-s s. at T = 140°K). 4. D I S C U S S I O N The description of the electron-cation recombination at low temperatures by means of the "discrete" approach proposed in the present work has two main advantages over the "continuum" approximation. First, it makes possible to consider accurately the behaviour of e~ in terms of transition probabilities involved in the transport process. Second, the limitations in the distance and time involved in transport phenomena are small in the "discrete" approach compared with "continuum" one. Moreover, it is possible to show that under certain conditions two simplified models proposed in Section 3 reduce to the Smoluchowski equation with an exponential sink term. Such an e q u a t i o n h a s been applied to investigate the influence of tunnelling on the rate of diffusion-controlled reactions. (3-5) Thus, the relation between the "discrete" approach and the "continuum" description of the electron-cation recombination can be established. To prove this, one should express the transport flux of electrons in the following form (22)
v2 = ~'rAR exp (--~-~) + ~rAr exp ( - - ( 2 M a + ~ T r ) ) .
tO-4
In the limit of very small A, when (24)
A ~ ~o
so that (25)
A grad N~ ~ N .
the quantity Ni+l may be represented by the first two terms of a Taylor series. This yields (26) [ J l = - A 2 ( v l +v2)
-t k T v l + v z - ~
Obviously the total number of electrons crossing a sphere of the radius r per unit time is given by the surface integral §J dr. On the other hand, by virtue of the continuity condition the value of the integral must be equal to the rate of the electron concentration decrease inside the sphere. Since this decrease results from the change in the position of etr due to random walk and from its singlestep tunnelling decay, we have (27)
~sJdr=f~rdivJdv=-f
(ONikot+ki'°Ni) dv"
where symbols Sr and v, denote integration over surface and volume, respectively. Substitution of equation (26) into equation (27) gives after discarding the indices the following kinetic equation
ON D 0 [ r 2 ( O N + y _ ~ _ ~ (28) ~ = r - ~ - ~ [ k 0 r
N
)] - k ( r ) N
IJI = A[K~.~+~Nj(t)- K~+,.iN~+I(t)] (i# 0). where
For a small change in the Coulomb potential over the length ,~ we expand exp ( + eA V[(kT)) in equation (17) in a power series. Neglecting terms above
ij.
V = - e/(er), k(r) = (Prrr + ~rAr exp ( - ErAT/(kT)) exp ( - 2adt~),
464
Yu. A. BERLIN et al. 3' = ~2](vl + v2), D = A2(vl + v2)/(4"rr).
If v2>>vl, equation (28) exactly coincides with the conventional diffusion equation which takes into account the additional long-range interaction between reactants. ~3-5) Note, however, that in the general case equation (28) differs from the ordinary one by the factor y in the second term in the parentheses because resonant tunnelling has been assumed to be almost independent of the Coulomb field. As can be seen from the derivation of equation (28) the "discrete" approach reduces to "continuum" approximation if inequalities (24) and (25) are satisfied. In this sense the proposed models of the electron-cation recombination seems to be more general than those based on the diffusion theory. Usually the behaviour of e~ at low temperatures has been interpreted either in terms of single-step tunnelling (see, for example, reviews "9'22) and references therein) or by means of the trap-to-trap mode of the electron migration in frozen media. ~29-3" This has led to a controversy concerning whether in glassy solids the rate of e?, decay is determined by the transport of excess electrons or by their direct sub-barrier transitions. ~32'33~ The important feature of the present theory is the possibility to take into account both processes mentioned above. Our calculations of decay curves should be considered only as order-of-magnitude estimations since values of physical parameters used in computations do not refer to any specific system. Nevertheless most of them is typical for non-polar glasses. Although the lack of data concerning parameter ,values do not allow to carry out a quantitative comparison of our theoretical results with the experiments, we have found a good qualitative agreement of the decay curve computed for solids with a low concentration of traps (Fig. 3) and the shape of the luminescence curve observed after the photoionization of 3-methylpentane glass dopped by TMPD. "2~ In particular, the isothermal section of the experimental curve can be described by the kinetic law given by equation (21). The intensity of the isothermal luminescence following the photoionization of the sample at 77°K sharply increases at time t = to, when a thermal jump is applied, passes through the maximum and then decreases. This trend of the experimental curve is also consistent with predictions of our model.
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