Effects of defect interaction

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, A pairs of the Frenkel defects in alkali halides, respectively. In these cases the equation defining the steady-state recombination profile is 

As it was said above (Section 3.2), for the elastic interaction this coefficient coincides with the effective radius of recombination, i eff = b, whereas for the Coulomb interaction / eff is defined in equation (3.2.51). Therefore the problem of obtaining the steady-state reaction rate is reduced to the finding the asymptotic coefficient b of the solution of equation (4.2.25). Formally it coincides with the quantum-mechanical scattering length on the potential 

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). 

The steady-state for diffusion-controlled recombination without interaction is given by equation (4.2.15), whereas its analog for the isotropic elastic attraction g 0) without tunnelling reads [59, 60] 

As it follows from equation (4.2.29), at low temperatures decreases with temperature as [61-63] and approaches an annihilation radius at high 

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For any of the several reasons mentioned above, defect interactions cause wall defects to lose translation invariance and, on average, reduce their effective mobility (see Fig. 36a) or to become pinned (Fig. 36b). For the case where forces due to defect interactions are small compared with the field-induced forces, the effect of defect interactions on the effective mobility can be understood at least conceptually in a simple way. The scenario considered here is a single wall defect moving in an environment containing other defects. The velocity of the defect is equal to the sum of the forces imposed on it divided by its intrinsic mobility /i ... 

Within this construct, the case of defect pinning is illustrated in Fig. 36b. The activation barrier around the local minimum is much larger than thermal energy, so the defect cannot escape. An important point is that the effects of defect interactions become stronger with increasing defect density, not only because of the increased number of defect interactions but also because of the reduced size of the average wall defect (and thus reduced electric field-induced driving force). 

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