Tunneling recombination

The existence of the (quasi) steady-state in the model of particle accumulation (particle creation corresponds to the reaction reversibility) makes its analogy with dense gases or liquids quite convincing. However, it is also useful to treat the possibility of the pattern formation in the A + B —> 0 reaction without particle source. Indeed, the formation of the domain structure here in the diffusion-controlled regime was also clearly demonstrated [17]. Similar patterns of the spatial distributions were observed for the irreversible reactions between immobile particles - Fig. 1.20 [25] and Fig. 1.21 [26] when the long range (tunnelling) recombination takes place (recombination rate a(r) exponentially depends on the relative distance r and could... 

Fig. 3.2. Two principal mechanisms of defect recombination in solids, (a) Complementary defect annihilation, r is the clear-cut (black sphere) radius, (b) distant tunnelling recombination due to overlap of wave functions of defects. Two principal kinds of hole centres - H and Vk...

Tunnelling recombination of primary F, H pairs can result either in closely spaced v+,i pairs (the so-called a, I centres) which annihilate immediately due to Coulomb interaction and a consequently large instability radius. However some i ions occur in crowdion configurations, and leave vacancy moving away up to 4-5 ao even at 4 K [31]. The distinctive feature of tunnelling recombination is its temperature independence, which makes it one of the major low-temperature secondary processes in insulating solids with defects. 

Fig. 3.4. Processes defending the survival probability of F centres in alkali halide crystals 1 -tunnelling recombination of close F, H defects, 2 - their annihilation, 3 - trapping of mobile H centre at impurity, 4 - formation of immobile dimer centre, 5 - H-centre leaves its geminate partner in random walks on a lattice.

Considering the reaction kinetics in the preceding Sections of Chapter 4, we have restricted ourselves to the simplest case of the recombination rate er(r) corresponding to the black sphere approximation, equation (3.2.16). However, if recombination is long-range, like that described by equations (4.1.44) or (3.1.2), one has to use equations (4.1.23) and (4.1.24), which yield essentially more complicated kinetics, especially for the transient period. Let us discuss briefly the main features of the diffusion-controlled kinetics controlled by tunnelling recombination, equation (3.1.2) (see also [32-34]). 

The intensity of any bimolecular, and in particular, tunnelling recombination, defined as... 

Fig. 4.2, Quasi-steady-state tunnelling recombination profile for Vt centres in KC1 calculated at 180 K (curve 1) and 190 K (curve 2) (equation (4.2.11)). The relevant effective radii, equation (4.2.15), are shown by arrows. Dotted line gives the black sphere approximation, broken line - the transient process (discussed in Section 4.2).

There is a complete analogy here with what has been said about equation (4.1.63). Let us analyze now the effective radius of the diffusion-controlled tunnelling recombination in more detail. From equations (4.2.12) and (4.2.15), one gets... 

The behaviour of the steady-state function y(r) is simplest for weak tunnelling recombination (almost pure contact recombination). In this case y(r) = 0 when r R, and when r > R, equation (4.2.15) holds with Reff = R-... 

A comparison of the time developments of Y(r, t) in two limiting cases (pure contact reaction and strong tunnelling recombination) demonstrates their qualitative difference. In the latter case, the first stage is very short and is finished already at t a(R) x the further change of Y(r,t) is defined here entirely by the non-stationary diffusion. The relevant reaction rate for... 

Another distinctive feature of strong tunnelling recombination could be seen after a step-like (sudden) increase (decrease) of temperature (or diffusion coefficient - see equation (4.2.20)) when the steady-state profile has already been reached. Such mobility stimulation leads to the prolonged transient stage from one steady-state y(r,T ) to another y(r,T2), corresponding to the diffusion coefficients D(T ) and >(72) respectively. This process is shown schematicaly in Fig. 4.2 by a broken curve. It should be stressed that if tunnelling recombination is not involved, there is no transient stage at all since the relevant steady state profile y(r) — 1 - R/r, equation (4.1.62), doesn t depend on >( ). 

In many cases of interest tunnelling recombination of defects is accompanied by their elastic or Coulomb interaction, which is actual, e.g., for F, H and Vk, A0 pairs of the Frenkel defects in alkali halides, respectively. In these cases the equation defining the steady-state recombination profile is... 

The solution of the equation (4.2.26) cannot be found in an analytical form and thus some approximations have to be used, e.g., variational principle. Its formalism is described in detail [33, 57, 58] for both lower bound estimates and upper bound estimates. Note here only that there are two extreme cases when a(r)/D term is small compared to the drift term, reaction is controlled by defect interaction, in the opposite case it is controlled by tunnelling recombination. The first case takes place, e.g., at high temperatures (or small solution viscosities if solvated electron is considered). 

One can argue that for the strong tunnelling, x 1> equation (4.2.10), when the diffusion-controlled tunnelling recombination is characterized by the effective radius... 

Fig. 4.5. Effective radii of the diffusion-controlled tunnelling recombination for the Coulomb attraction (a) and repulsion (b) (after [65]). Curve 1 and 2 are results of computer calculations with parameters cr = 107s l, r = 20 A, Rd = 4 A and cr = 10l4s l, r = 2 A, Rd = 4 A respectively. L - the Onsager radius, equation (3.2.55), Ro - radius of strong tunnelling recombination, equation (4.3.7).

Fig. 4.6. Schematic pattern of the tunnelling recombination of positively changed Vk centre with neutral electron centre (e.g., F centre) (a) and with oppositely charged activator atom (e.g., Tl°) (b). In the second case the Coulomb field traps Vk at long distance R n, but electron transfer itself occurs at much shorter distance.

It is convenient to consider a model of an anisotropic recombination region the reflecting recombination sphere (white sphere) with black reaction spots on its surface [77, 78], The measure of the reaction anisotropy here is the geometrical steric factor Q which is a ratio of a black spot square to a total surface square. Such a model could be actual for reactions of complex biologically active molecules and tunnelling recombination when the donor electron has an asymmetric (e.g., p-like) wavefunction. Note the non-trivial result that at small Q, due to the partial averaging of the reaction anisotropy by rotational motion arising due to numerous repeated contacts of reactants before the reaction, the reaction rate is K() oc J 1/2 rather than the intuitive estimate Kq oc Q. 

The quasi-steady-state hopping recombination rate K(oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this Ifu is defined by the asymptotics of the solution, Y(r,oo) = y(r), as r —> oo. It is important, however, that R ff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination... 

Note that the steady-state y(r) obtained from equation (4.3.9) as t — oo does not completely agree with the asymptotic behaviour of the quantity established earlier due to absence here of the term R ff/r. This term comes from the diffusion character of distant tunnelling recombination. The term Reff/r arises from the next after equation (4.3.8) term in the expansion of the integral (4.3.5) in parameter (R/X)2 deriving equation (4.3.10) the asymptotic value of y(r) is only used rather than its behaviour at large r. A comparison of hopping recombination with equation (4.2.18) yields... 

To give a transparent physical interpretation to both hopping and diffusion kinetics controlled by strong tunnelling recombination, the reaction rate could be presented in the universal form as... 

Kinetics of the tunnelling recombination depends greatly upon the defect mobility (whether a static tunnelling luminescence regime at low temperatures or the diffusion-controlled regime arising at higher temperatures when defects become mobile) and their spatial distribution. 

Fig. 4.20. The influence of the tunnelling recombination anisotropy upon the non-steady-state luminescence kinetics. Curves 1 and 2 correspond to the defect reorientation switching-on and -off, respectively at the moments given by broken lines [106],...

Fig. 4.22. The time-development of the recombination profile of dissimilar defects [102]. Curves 1, 2, 3 show the effect of static tunneling recombination 4 - a short time after the diffusion has started broken curve - the profile as t —> oo.

As the temperature stimulation is switched off, the static kinetics is governed by equation (4.1.40) with the initial distribution function y(r) from equation (4.2.11). However, all attempts [102] to describe in such a way the experimental tunnelling luminescence decay for F and in KBr (Fig. 4.18) were unsuccessful. Both this observation and the absence of the plateau of 7(f) during the temperature stimulation, characteristic for the quasi-steady states, argue that the tunnelling recombination takes place in correlated pairs. This is in line with the conclusion [107] that for ordinary defect concentrations 1016 cm-3 (X-ray sample excitation for minutes) and the time 105 s the slope is dose-independent but 7(f) oc dose [95]. 

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... 

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