Recombination kinetics

One of the characteristic features of the luminescence in a-Si H is the broad distribution of recombination times. Fig. 8.14 shows the luminescence decay extending from 10 s to 10 s (Tsang and Street 1979). The data are inverted in the lower part of the figure to give the distribution of lifetimes, which has its peak at 10 -10 s at low temperature and is 2-3 orders of magnitude wide. The shape of the distribution is sensitive to the excitation intensity for reasons discussed shortly and the time constants are even longer at very low intensity. 

Radiative tunneling is the only recombination mechanism which can explain the decay data satisfactorily. The electron and hole are localized at different sites separated by a distance R, and the recombination time is, from Eq. (8.5) 

Light induced ESR measures the density of band tail electrons and holes, and provides a different method of measuring the recombination 

15 for illumination conditions typical of the luminescence. The decay continues out to 10 s and even then a substantial density of unrecombined carriers remain. At first sight this result seems to conflict with the luminescence decay which does not extend beyond 10 s. The difference is a consequence of the range of electron-hole pair separations. 

At time t, all pairs for which R R. have recombined, while the more distant pairs remain. From the shape of the distribution in Fig. 8.16, about 20 % of the pairs remain after 1 s, reducing to about 5 % after 1000 s, and this is consistent with the LESR data. The measurable range of the recombination over which the radiative tunneling mechanism applies is therefore more than 10 orders of magnitude, from 10 to 10 s. 

The charge carriers formed upon absorption of light (reaction (7.1)) can recombine in a radiative or non-radiative way according to reactions (7.12) to (7.15). This is clearly seen from the rather rapid depletion of the transient absorption spectra recorded during laser flash photolysis studies (see Fig. 7.4). 

The recombination kinetics of the charge carriers have been studied in detail by the groups of Gratzel, Serpone and Colombo [4e, 5, 6]. Since recombina-tion of electrons and holes is monitored by transient absorption techniques most of the observed decay is due to reaction (7.15). 

Serpone et al. have examined colloidal titanium dioxide sols (prepared by hydrolysis of TiCl4) with mean particle diameters of 2.1, 13.3, and 26.7 nm by picosecond transient absorption and emission spectroscopy [5]. Absorption decay for the 2.1 nm sols was found to be a simple first-order process, and electron/hole recombination was 100% complete by 10 ns. For the 13.3 and 26.7 nm sols absorption decay follows distinct second-order biphasic kinetics the decay times of the fast components decrease with increase in particle size. 10 ns after the excitation pulse, about 90% or more of the photogenerated electron/hole pairs have recombined such that the quantum yield of photooxidations must be 10% or less. The faster components are due to the recombination of shallow-trapped charge carriers, whereas the slower components (x 20 ns) reflect recombination of deep-trapped electrons and holes. 

Bowman and coworkers characterized the subpicosecond dynamics of titanium dioxide sols employing particle sizes of about 2 nm prepared by hydrolysis of titanium tetraisopropoxide [6]. From their spectral results the authors inferred that the average lifetime of an electron/hole pair is 23 5 ps, and substantial electron/hole recombination occurs within the first 30 ps. A second-order recombination rate constant of (1.8 0.7) x 1CT10 cm3 s 1 for trapped electrons with holes has been obtained [6a]. 

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Bibo A M and Peterson I R 1989 Disclination recombination kinetics in water-surface monolayers of 22-tricosenoic acid Thin Solid Films 178 81-92... 

Low-temperature, photoaggregation techniques employing ultraviolet-visible absorption spectroscopy have also been used to evaluate extinction coefficients relative to silver atoms for diatomic and triatomic silver in Ar and Kr matrices at 10-12 K 149). Such data are of fundamental importance in quantitative studies of the chemistry and photochemistry of metal-atom clusters and in the analysis of metal-atom recombination-kinetics. In essence, simple, mass-balance considerations in a photoaggregation experiment lead to the following expression, which relates the decrease in an atomic absorption to increases in diatomic and triatomic absorptions in terms of the appropriate extinction coefficients. 

The discussion so far has been empirical in the sense that Laplace transform method has been utilized in conjunction with an experimentally determined scavenging function without a theoretical model for the recombination kinetics. A theoretical model will be attempted in the following subsections. 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

Equation (7.30) shows that the fundamental information on recombination kinetics is contained in the solution of the scavenger-free case, from which the recombination kinetics with a scavenger may be obtained via an exponential transformation. The scavenger reaction probability is now given by... 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

In the ME model of recombination kinetics in a multiple ion-pair spur, the probability PN that N ion pairs will remain extant at time t is given by (Green and Pimblott, 1990)... 

Bartczak et al. (1991 Bartczak and Hummel, 1986, 1987,1993, 1997) have used random flight MC simulation of ion recombination kinetics for an isolated pair, groups of ion-pairs, and entire electron tracks. The methodology is similar... 

With r = 28.45 nm, r = 3.0 nm, and r = 8.39 nm, Bartczak and Hummel (1986) compute the escape probability Pesc = 0.0336, 0.0261, and 0.0230 respectively for N = 1, 2, and 3. While the first is comparable to the Onsager value, the latter are new results. The kinetics of recombination for the isolated pair, found by Bartczak and Hummel (1987) using MC, is very similar to that obtained by Abell e al. (1972). For N > 1, these authors found the recombination kinetics to be faster than that for the isolated pair. For two pairs, the calculated escape probability increased with the external field, but not as strongly as for N = 1. 

Figure 4 of Mozumder (1971) compares the kinetics of neutralization in an isolated ion-pair, a spherical blob, and a short track. For the isolated ion-pair there is vey little recombination until 5 x 10 u s, but most recombination is over by -Ins. The main difference in the recombination kinetics is between the isolated and multiple ion-pair cases. There is not such a great difference among the different multiple ion-pair blobs or short tracks. In the multiple-ion-pair cases, the neutralization is gradual and much faster than that in the isolated ion-pair case at short times. However, this could be an artifact of the model predicated by close proximity of the positive ions having essentially zero mobility (vide infra). 

The first serious attempt to treat the recombination kinetics of radiolysis was made by Jaffe in 1913 [39] following Wilson s experimental demonstration of the columnar nature of a... 

An important result of the theoretical studies of the multipair effects is that the recombination kinetics in a cluster of ions, in which the initial separation between neighboring cations is 1 nm, is faster than the corresponding decay kinetics of a single ion pair [18]. Furthermore, the escape probability is lower than the Onsager value [Eq. (15)], and decreases with increasing number of ion pairs in the cluster (a relative decrease of about 30% for two ion pairs, and about 50% for five ion pairs). The average electron escape probability in radiation tracks obviously depends on the distribution of ionization events in the tracks, which is determined by the type of radiations and their energy. 

A study of electron scavenging in multipair clusters [21] has shown that the total scavenging probability decreases with increasing number of ion pairs in the cluster. However, the Laplace transform relationship [Eq. (28)] between the scavenging probability and the recombination kinetics was found to work reasonably well also in the multipair case. 

One variant of ESR method should be mentioned in this connection, namely, the recombination-kinetic method widely used by our laboratory in the Institute of Chemical Physics (Academy of Sciences, Moscow, U.S.S.R). This method opened the possibilities of observation of extremely slow diffusion (D 5 1 O 17 to 10 18 cm2 s linear velocities of paramagnetic centers 10 7 cms-1) and of the studies of both intra-globular (e.g., caused by segmental motion) and interglobular recombination of paramagnetic labels in proteins and other biopolymers (see Ref. 1 for general description of the method). 

It is clear that a complete treatment of the recombination kinetics requires careful incorporation of mutual particle distribution, including fluctuations in their local concentrations due to diffusion and reaction. At present the role of fluctuations in reactant concentrations in chemical kinetics is well known... 

Due to the existence of two quite different distinctive distances (scale factors) - lo and l - the recombination kinetics also reveals two stages called monomolecular and bimolecular respectively. The defects survived in their geminate pairs go away, separate and start to mix and recombine with dissimilar components from other pairs. It is clear that the problem of kinetics of the monomolecular process is reduced to the time development of the probability w(f) to find any single geminate pair AB as a function of the initial spatial distribution of the pair components f(r), recombination law cr(r) and interaction Uab (r). The smaller the initial concentration of defects, n(0) —> 0, as lo —> oo, the more correct is the separation of the kinetics into two substages, whereas the treatment of the case of semi-mixed geminate pairs is a very difficult problem discussed below. 

Until the geminate pairs start to mix, i.e., at relatively short times r relative diffusion coefficient, the monomolecular kinetics reads n(t) = n(0)u>(t), with n(0) = nA(0) = ne(0) being initial particle concentration. The distinctive feature of this stage is the linearity of the recombination kinetics n(t) with respect to the irradiation dose n(0). 

Since in the isotropic medium the recombination kinetics is governed by the relative distance r — rA — fB, it is convenient to introduce new coordinates f, R. The latter,... 

They have calculated the continuous diffusion equation (3.2.30) with U(r) = -a/r3 for several kinds of nn F, H centres in the crystalline lattice. Figure 3.9 demonstrates well that both defect initial separation and an elastic interaction are of primary importance for geminate pair recombination kinetics. The 3nn defects are only expected to have noticeable survival probability. Its magnitude agrees well with equation (3.2.60). 

To treat the problem of recombination kinetics for arbitrary hopping lengths 0 < A < oo, it is convenient to restrict ourselves to the exponential hopping length distribution known as the Torrey model [82]. In this particular case the integral kernel [Pg.211]

Computer simulations of bimolecular reactions for a system of immobile particles (incorporating their production) has a long history see, e.g., [18-22]. For the first time computer simulation as a test of analytical methods in the reaction kinetics was carried out by Zhdanov [23, 24] for d, = 3. Despite the fact that his simulations were performed up to rather small reaction depths, To < 1, it was established that of all empirical equations presented for the tunnelling recombination kinetics (those of linear approximation - (4.1.42) or (4.1.43)) turned out to be mostly correct (note that equations (5.1.14) to (5.1.16) of the complete superposition approximation were not considered.) On the other hand, irrespective of the initial reactant densities and space dimension d for reaction depths T To his theoretical curves deviate from those computer simulated by 10%. Accuracy of the superposition approximation in d = 3 case was first questioned by Kuzovkov [25], it was also... 

In its turn Fig. 6.2 illustrates the effect of the initial concentration on the static tunnelling recombination kinetics. The latter is defined by a competition of three distinctive scales - the tunnelling recombination radius ro, mean distance between particles Iq = n(0) 1/d and lastly, the time-dependent correlation radius . At long time curves corresponding to different initial concentrations could be coincided by their displacements along ordinate axis, which confirms existence of the universal asymptotic decay law. 

Therefore, in the symmetric situation, D = D, the recombination kinetics may be also separated into two subsequent stages of dynamic and statistical aggregation. At long times the particle density in these aggregates (domains), characterized by the maximum values of the correlation functions Xu(r —> 0,t), is not very high. The reaction rate is governed by the ratio of two distinctive spatial scales — and re. 

Recombination of the surface electron Fs+ -centres and of the bulk hole V -centres in y-irradiated highly dispersed oxide CaO has been studied [69]. The recombination kinetics is weakly dependent on temperature in the range 4.2-77 K. The formal activation energy has a value of only 30 cal mol l. At small irradiation doses (less than 2 x 1019eVcm-3) the recombination appears to be of geminate character, i.e. it occurs only in the parent donor-acceptor pairs, the process kinetics being well described by the linear dependence of the concentration of centres on the logarithm of observation... 

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