J . Phys. Chem. 1993, 97,952-951
952
A Study of the Recombination Processes of Photogenerated Charge Carriers in SnO*/Ion Exchange Polymer-ZnTPP/Au Cells Z. Hung,+ A. Ioannidis, and M. F. Lawrence’ Department of Chemistry and Biochemistry, Concordia University, 1455 de Maisonneuve Montreal, Quebec, Canada H3G lM8
W.,
Received: February 26, 1992
Theoretical approaches to the description of charge carrier recombination in insulators and semiconductors containing traps have been used to interpret photoconductive decay curves observed after illumination of SnO2/ ZnTPP-polyXIO/Au cells. Application of the double exponential solution derived from a simplified photoconductivity model accurately reproduces the decays and points to the involvement of both shallow and deep traps in the overall recombination process, which is consistent with experimental results presented in a previous report. From this approach, relevant parameters such as the relative rates of trapping, detrapping, and recombination for photogenerated charge carriers are estimated. However, the relatively large detrapping rate that this model yields for the deep traps involved outlines its limitation in describing recombination processes in systems of this type. A further analysis of these results, basedon a model for bimolecular trapping/annihilation processes where deep traps act solely as recombination centers, helps demonstrate how a single decay can be composed of two distinct recombination mechanisms, in agreement with previously reported data. During about the first 15 s of the decay, bimolecular recombination occurs mainly between charge carriers residing in shallow traps, also referred to as “free” carriers. At longer times, the bimolecular recombination becomes mediated by the deeper traps present in the system. The expression derived from the bimolecular/trapping annihilation model for the decay at longer times, which is arrived at by effectively turning off the free carrier recombination, is equivalent to the expression one gets for a process that is of a unimolecular type. photocurrent decay. To gain more insight on the mechanisms by which photogenerated carriers recombine in these films, this paper In a previous paper dealing with the examination of photoexplores theoretical approaches that generate solutions which fit conductive behavior,’ it was established that the decay of the photocurrent decays observed for the Sn02/polyXI&ZnTPP/ photogenerated charges in Sn02/ionexchange polymer-ZnTPP/ Au system. Au cells is characterized by two distinct recombination processes The first approach used is a simplified photoconductivity model which occur in sequence. A fast process of bimolecular character which usually applies to more conventional semiconductors was shown to operate in the first 15 s or so of photocurrent decay, containing traps. It enables theestimation of lifetimesand related followed by a slower process varying exponentially. parameters of photogenerated charge carriers in the polyXI0The literature on photoconductivity abounds with attempts to ZnTPP films studied. Although approximate in nature, the describe the experimentally observed photocurrent responses for advantages of this simplified model are that it leads to analytical molecular based systems in terms of simple unimolecular and solutions and it also provides a starting point for the proper bimolecular models for the recombination processes in~olved.~-~ interpretation of the observed photocurrent decays. Application These lead respectively to expressions of the following type for of this model gives a first indication that the overall decay involves photocurrent decays the presence of both shallow and deep traps. Drawing on this possibility, a second approach based on bimolecular trapping/ Zph = 1 0 exp(-t/t,,) (1) annihilation reactions is shown to be consistent with the appearand ance of two discernable charge recombination mechanisms following illumination of these polymer4ye films. z p h = Z O / ( ~+ KZot) (2) The experimental data used in this study were taken from the where 10is the steady-state photocurrent, Iph. is the decaying previous article in which the photocurrent decay curves are given photocurrent, t p h is the characteristic response time for the decay for SnO2/PolyXI&ZnTPP/Au cells at various temperatures.l The preparation of these cells and the conditions under which the involving a unimolecular process, and K is the rate constant for a bimolecular process. In general, however, the photoconductive photoconductivity measurements were obtained have also been reported in detail. The best fits to the experimental decay curves response of a specimen is usually more complex, and rationalizing these phenomena in terms of free carriers with recombination obtained with the photoconductivity model and the bimolecular and trapping at localized states requires considerable interplay trapping/annihilation model and thevalues reported in the analysis between experimental results and analytical solutions of the rate of results section have been calculated with the use of an IBM equations of an assumed model. compatible AT-386SX computer. These affirmations are confirmed by our examination of photoconductive behavior in films of the ion exchange polymer 11. The Photoconductivity Model (polyXI0) containing the dye zinc tetraphenylporphyrin (ZnTAs in the previous paper on this subject, many concepts and PP),I where neither of the two simple approaches described by terms that normally apply to materials characterized by longq s 1 and 2, if applied individually, accounts for the overall range order (crystals), where charge transport is mediated by delocalized electronic states forming bands, will be adopted for Address correspondence to this author. Present address: INRS-Energie, C.P. 1020, Varennes. Quebec, Canada. the interpretation of conductivity experiments performed on a
I. Introduction
+
oQ22-3654/58/2097-o952504.00/0
0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 953
dn/dt
=f- ( n / ~ +) emn,- n(N, - n,)Cm
(5a)
In the steady state, we have
no = fr,
Figure 1. Simplified energy level diagramfor the photoconductivity model.
disordered polymeric system to which a dye, ZnTPP, has been added. It is now well accepted that periodicity is not a necessary requirement to electron mobility and that even amorphous materials characterized by a relatively high degree of disorder offer regions where electrons are mobile.5-11 It should be clear, however, that due to the nature of localization in materials like polyXI0, the electronic structure of the polymer is essentially that of an ensemble of molecular subunits and is dictated by the overlap of pendant groups. Charge transport in this case is best described by a modified energy level model in which all available states are necessarily localized and where the term "free carrier" now alludes to charges occupying states separated by the socalled mobility edge from what are considered to be deep traps, in which the carrier mobility drops by several orders of magnitude.12 In what follows, the term "free carrier" relates to charges occupying a 'conduction level" composed of localized energy states characterized by shorter immobilization times (relatively shallow traps allowing better mobility), as opposed to the term "trapped carrier" which designates charges occupying states where immobilization lasts much longer (deep traps). A narrow distribution of electron traps with a width of about 0.1 eV (compared to the energy gap of about 2 eV associated to ZnTPP) and having a density of N,z 3 X 1014cm-3 was shown to be present in the polyXIO-ZnTPP system. It is reasonable to consider this trap distribution as a single set of discrete trapping states located at a certain depth Et (previously determined to be about 0.8 eV) below a conduction level assumed to form from the overlap of the lowest unoccupied molecular orbitals of the dye, in addition to their possible overlap with the r* levels of neighboringpyridineand styrene units present in the polymer.I3-l5 This situation can be approximated by the simple model illustrated in Figure 1. In this case we consider a system in which f carrier pairs are being generated per unit volume per second by absorbed radiation. n / 7 is the rate at which electron-hole pairs are annihilated through recombination. In addition, a set of N, trapping states per unit volume are introduced, of which n, are occupied by electrons. The probability for transition of free electrons (density is n) from the conduction level to the traps is denoted Cm, so that the trapping rate r,,, is given by the product of this probability and the density of states involved in the transition, i.e., rnt
= n(Nt - nt)Cnt
(3)
The thermal generation rate from the traps into the conduction level will be given by the product of the number of trapped electrons, the transition probability to a state in the conduction level, and the density of these final states. The latter two factors are denoted by em, so that the generation rate is given by
emnt (4) With this model, we can write the rate equations for the freeand trapped-electron densities as gm
f4Nt - nto) Cm/e, (6) where no and ntoare the steady-state densities of free and trapped electrons, respectively. In order to solve for the decay of the photocurrent, we make the simplifying assumption that we are working under conditions of low illumination levels and high trap density, N1>> n,. The first of these conditions is clearly satisfied by the experimental conditions under which the photocurrents are generated] and is required to fulfill the second one. Under steady-state illumination, the density of free electrons was determined to be about one-fifth of the deep trap density (3 X 1014~ m - ~and ) , at this point satisfaction of the condition N, >> n, is questionable. However, the condition tends to be satisfied as the decay proceeds to longer times during which the densities of both free and trapped carriers gradually decrease while maintaining a constant ratio (n/n,),l For the photocurrent decay, the light source has been removed and therefore f = 0. In this case eqs 5 become "to
+
dn/dt = -(n/~) e,n, - nN,Cm dn,/dt = -e,,n,
(74
+ nNtCm
(7b)
where N1>> n, has been used. A solution to eq 7a is
n = a, exp(-t/r,)
+ u2 exp(-r/~,)
(8) where 0 1 and a2 are constants of integration, and T~ and ~2 are algebraic functions of the constants in eqs 7. Substituting eq 8 into eq 7b we have
dn,/dt
+ entnt= CmN,a,exp(-t/.r,) + CntNta2 ex~(-t/72) ( 9 )
According to the standard formula dY/dt + W which has the solution
Y = Q(t)
~ ( t =) exp(-JPdt)[JQ(t)(ex~(SP
(10)
dt)) dt + Cl
(1 1)
The solution to eq 9 is therefore nt
= exp(-Jem dt)[JcmNt(a, ex~(-t/r,) +
a, exp(-t/~,)) exp(Se, dt) dt
CmNt u x + c] = -1
em - 71
1
CmNt +exP(-t/72)
-1% + c exp(-e,r) (1 2) en, - 7 2 Now that we know the solutions of n and n, we can substitute eq 8 and eq 12 into eq 7a in order to determine constants T~ and 72. This gives
exp(-t/.r,)
[--en, - 1 / -~N,Cn,+ 1u1 exp(-t/zJ + l / -~N,Cn,+ [+em 1a2exp(-t/r,) 1/~,
TI
72
+
en&exp(-e,t) = 0 (13)
Because exp(-r/T1), exp(-t/T2), and exp(-emt) are linearly independent, we have
954
Huang et al.
The Journal of Physical Chemistry. Vol. 97, No. 4, 1993
c=o
(14)
--
25 5 O C 29 4 O C
where r = l / r l or 1/r2. Rearranging eq 15 we have
r2-(1/r+NtC,+e,,)r+(e,/r) = O
(16) 100
The solutions of eq 16 are
rf = 1/2[(1/r
+ N,C,,, + e,,,) f ((1/r + N,C,, + em)2-4e,,/r)'/2]
300
t (sec)
(17)
Choosing two linearly independent solutions we have r1 = l/r+ and r2 = l/r-
200
Figure 2. The time dependence of photocurrent decay for an Sn02/ polyXIO-ZnTPP/Au cell at three different temperatures (taken from ref 1). Cldcircleeshow thebest fitsobtained withthephotoconductivity model.
(18)
Rearranging eq 28 we have
(19)
1 y = -[Q z em)2- (l/rl - ~ / r ~ ) ~ ] 4em Substituting eq 27 into q 29 we get
+ +
and
n = a, exp(-t/r,)
+ a2exp(-t/r2)
Y
1/(emrIrJ
From eq 27 we have that where r1 = l/r+ and 7 2 = l/r-. The constants, 0 1 and 02, can be found by applying the initial condition for the decay to eqs 7, at t = 0, = f r , and nto = frNtCnt/e,. With this, we have
z=(l/r,
+ 1/r2)-y-e,
Finally we get
r = e,,,rlr2
+
f r = a1 a2
NtC, = (l/rl + 1/r2)- l/r-e, The solutions are summarized in the following:
n = a, exp(-r/T,)
Solving eqs 21 we have
+ a2exp(-t/r,)
If we take a1/a2 = a and rearrange eq 22 we find e,,
where constants a,, a2, 7 , and CmNt are as follows:
l+a =-
+
r2a
71
From eqs 17 and 18 we know that
+ N,C,,, + e,,) + ((1/r + NtC, + e,J2 - 4 e , / ~ ) ' / ~ ] (24)
l / r l = r+ = 1/2[(1/r
N,C,=(l/r,+
1/r2)-1/7-em
III. Analysis of Results ind Disclnrsion Rearranging eqs 24 and 25 gives y
+ z + e, + (0, + z + em)2- 4e2)'I2 = 2/r1
y
+ z + e,, - (0, + z + e,,)'
(26a)
- 4edy)'/, = 2/r2 (26b)
where y = 1/r and z = N,C,, have been used. If we add q 26a and q 26b we have y
+ z + e,, = l / r l + 1/r2
(27)
Similarly, subtracting eq 26b from q 26a we get
+
(0, z + e,,,)'- 4e,y)1/2 = l / r , - 1/r2
(28)
A. The Photoconductivity Model. The photocurrent decay characteristia for the Au/ZnTPP-polyXIO/SnO2 system shown in Figure 2 can be approximately expressed in the following form
+
(37) Iph = A, exp(-t/r,) A, exp(-t/r,) given that the current is proportional to the concentration of free electrons. The concentration of free electrons should have the same characteristic time constants and the same ratios for the coefficients. The response time constants TI and 72, as well as theratioaI/a2= AI/A2,can thereforebefoundfromphotocumnt decay curves. To do so, the best fit to the decay curves were obtained using q 37, as shown in Figure 2 for different temperatures. In the case of photocurrent decay at 23.5 O C , for example, this yields the following parameters
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 955
Charge Carriers in Sn02/ZnTPP-polyXIO/Au Cells
rl = 131 S,
72
= 10.8 s
TABLE I: Photocurrent Decry Data and Trap Related Parameters for tbe SnOJZnTPP-polvXIO/Au Cell ~
a1/a2= A l / A , = 2.0
From eqs 36 we have e,,, = 0.02 s-I, T = 26 s, and C,Nt = 0.05 s-1. The ratio of free to trapped electrons can be obtained if the first few seconds of the decay curve are omitted. The second terms for both n and n, (eqs 34) become negligible after the first few seconds because T ] >> 72. So for most of the decay, the ratio, n/nt, can be taken as the ratio of the first terms. We then have
The results thus obtained are summarized in Table I. By rewriting eq Sa as follows
~~
~~
23.5
25.5
29.4
response time T I (s) response time 1 2 (s)
131 11 2.0 0.02 26 0.2 0.05
126 9 1.9 0.02 23 0.2 0.06
91 7 1.8 0.03 17 0.2 0.07
ado2 e,,t (s-9
lifetime nlnt CntNt
T
(s)
(s-')
TABLE Ik Relative Recombmtion, Trapping, md Detnpping Rates Calculated from the Photoconductivity Model for SnOJZnTPP-polyXIO/Au cell 23.5 25.5 29.4
0.038 0.043 0.059
The present analysis has shown that photocurrent decay in polyXIO-ZnTPP films can be described by the sum of two exponentials; one representing a fast component in the beginning and the other a slower component at longer times. Based on the mathematical analysis of Blakney and Grunwald for smallcurrent transients in insulators,'6 the number of such exponential terms appearing in the sum is equal to the number of distinct trapping species present in the material. For a material possessing two sets of trap states (shallow and deep, for example), the expression for the transient is that given by eq 37, and the response time constants thus obtained in fitting the decays (Table I) are in good agreement with the time scales over which the two separate recombination processes wereshown tooperate.' Ifoneconsiders that in polyXIO-ZnTPP the states allowing electron and hole migration are effectively shallow traps (disordered system's analogue of a conduction "band"), which are in the presence of a deep set of electron traps, the two essentially distinct but sequential charge recombination mechanisms at work in this system can now be focused on and identified. This approach is developed in the following section.
B. Bimoleculu Trapping/Annihil.tion Processes. The treatment given in this section stems from an analysis made by Rasaiah et al. consisting of computer simulations of diffusion controlled bimolecular recombination in a two-dimensional square lattice where trapping can 0 ~ c u r . ITheir ~ analysis makes use of a mean field theory in which the position of each particle present in the lattice is randomized after each time step. For our purposes we have used the three dimensional version of this approach, and have applied the resulting kinetic rate equations for bimolecular trapping/annihilation procespes todescribe the photocurrent decay in polyXIO-ZnTPP films. As indicated in the simulations performed by Rasaiah et aI.,l7 the densities appearing in the following equations are actually normalized with respect to a certain lattice site density. These normalized densities, noted with primes to avoid confusion, are
0.10 0.10 0.15
0.05 0.06 0.07
the values used in the calculations. The reactions on which the treatment is based are k
(39) we can make a comparison between the relative rates of recombination ( l / ~ )detrapping , (e,,,n,/n), and trapping (NtC,,,). These relative rates were calculated from Table I and are listed in Table 11. The results show that the rates for recombination and trapping areapproximately the same. The thermal excitation rates are 2 times larger but not large enough to consider the other two rates as negligible.
~
temperature ("C)
n'+ p'+ annihilation
(40)
ki
n'+ T'+ n,'
(41)
kt
n,' + p ' +
T'
(42)
wheren'andp'are the freeelectronand holedensities, respectively, n( is the trapped electron density, and T'is the empty trap density. Free carriers are annihilated with a rate constant k (eq 40). The rate, k,,at which free electrons become trapped (eq 41), is also the rate at which trapped electrons are annihilated through encounter with an oppositely charged free carrier, with the end result being regeneration of a free trap site (eq 42). As will be made clear in the followingdiscussion, these rate constantsactually represent the probability that a specific event will occur, in accordance with the formalism previously developed.17 If one assumes that n' 2: p', as is the case for the polyXIOZnTPP system after illumination once the steady-state photocurrent is reached,] we may write dn'ldt = -kn5
- k,nT'-
k,nh,'
(43)
and since the total trap density, N;, is given by
N,' = T'+ n,'
(44)
the kinetic equation for the free electron density becomes dn'ldt = -n'(kn'+
(45)
k,N,')
A comparison of this expression with eq 39 derived from the photoconductivity model (Figure l), exposes a fundamental difference between the two approaches, namely, in reactions 4042, the possibilityof having trapped carriers becomingfree again isnonexistent. Integrationofeq45 gives theratiooffreeelectrons at time r (during the decay) to free electrons at time 0 (when light is removed) n w -n'(0)
1
+
exp(-k,N,'t) K( 1 - exp(-k,N,'t))
10
(46)
where Zph is the photocurrent decaying with time and ZO is the steady-state photocurrent. The coefficient K is given by
K = CY/@
(47)
956
Huang et al.
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 1.0
0.8
0.6
decay, in an SnO&AyXIO-ZnTPP/Au cell at 23.5 OC. These f ’ 1 points were calculated taking n’(0) = 0.007, N,‘ = 0.02, and a
A
value of 1.2 for 8, which yields a value of K = 0.29. The fit is excellent up to approximately 15 s of decay, in good agreement with the fact that beyond this time a definite transition is found to occur in the decay rate.’ Similar fits are obtained for early decay times measured at higher temperatures by taking into account the corresponding increase in n’(0) which results in a proportional increase of the coefficient K. At the limit when the free carrier annihilation represented by eq 40 is turned off, meaning that k equals 0, we have K = 0 and eq 46 becomes
i.1. I
\
n’(r)/n’(O) = exp(-k,N(t)
which corresponds to the previous interpretation proposed for the exponentialphotocurrent decay at longer times (> 15S ) , ~expressed as
0 0
= fph/fo
100
200
n = no exp(-t/t,,)
0 300
t(sec)
Figure 3. Variation of I p h /with l ~ time for the photocurrent decay at 23.5 O C (from Figure 2). Open circles and closed circles show the best fits obtained with eqs 46 and 51, respectively, derived from the bimolecular trapping/annihilation model.
where a = n’(O)/N(
and (49)
Equation 46 was used to fit the initial photocurrent decays shown in Figure 2. To do so the rate constant for free carrier annihilation was taken to be k = 5 / t , (as prescribed by the threedimensional treatment where a site has six nearest neighbors, as opposed toonly four in the two dimensionalcase), and theconstant for trapped carrier annihilation is kt= 1.’’ According to eqs 41 and 42, a kt of 1 implies that every time a free electron encounters an empty deep trap or a free hole encounters an occupied deep trap, thecorresponding event (trapping or annihilation) doesoccur. The value of k is derived from a combinatorial analysis which assumes an equal probability of occupation for all the lattice sites. In determining k,it is considered that the charge carriers participating in the bimolecular recombination process at early times (
