Recombination profile

Another issue that can be clarified with the aid of numerical simulations is that of the recombination profile. Mailiaras and Scott [145] have found that recombination takes place closer to the contact that injects the less mobile carrier, regardless of the injection characteristics. In Figure 13-12, the calculated recombination profiles arc shown for an OLED with an ohmic anode and an injection-limited cathode. When the two carriers have equal mobilities, despite the fact that the hole density is substantially larger than the electron density, electrons make it all the way to the anode and the recombination profile is uniform throughout the sample. 

It is shown below (Chapter 4) that its solution is also important for the bimolecular stage of a reaction defining a stationary recombination profile. The second linearly independent solution of this equation could be expressed through y(r) as... 

The formation of the steady-state recombination profile (4.1.62) occurs for the space dimension d = 3 only. For instance, if d = 2, taking into account the change in the diffusion operator A and the expression for the two-dimensional reaction rate... 

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).

Fig. 4.3. The process of steady-state recombination profile formation for the strong tunnelling case parameter x2 = D) x 107 (a) and 2 x 10s (b). Figures in curves show log t.

The effect of an inertial increase in recombination intensity I(t) was observed experimentally in many insulating crystals including alkali halides [51, 52], Ba3(P04)2 [53], a-Al203 [54], Na-salt of DNA (Fig. 4.4) [52], The advantage of this technique which is efficient for the identification of tunnelling stages of reactions in insulators is that the initial and final steady-state recombination profiles are known and could be calculated by means of equation (4.2.11) unlike the initial distribution which is usually believed to be random but in fact is unknown. 

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... 

Figure 4.18 shows the kinetics of the tunnelling luminescence growth after the step-like temperature increase observed for F, Vk centres in pure KBr crystal (for the static decay the kinetics is described well by the Becquerel s law with a 0.8). Unlike the case of rotation, Fig. 4.17, it is obviously delayed with the distinctive r which becomes shorter, as temperature is raised (for more details see [18, 51]). This delay arose directly due to the transient period resulting in the formation of the quasi-steady-state recombination profile. (The same behaviour has been observed for Ag°, centres in KCl-Ag). 

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.

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

Figure 13-12. The recombination profiles for an OLED with an ohmic anode and an injection-limited cathode for various ratios of the electron/hole mobilities. Reproduced with permission from [145],...

Ironic charge injection is facilitated (thus the efficiency increases), while in the bulk of the sample the electronic charge motion is diffusive. Numerical simulations for devices with equal electron and hole mobilities reveal a recombination profile with a maximum at the middle of the sample (contrary to the case of ordinary OLEDs see Fig. 13-12), and a width that depends on the sample thickness. In Figure 13-14, recombination is shown to take place in a thin zone in the middle of the 100 pm thick sample. The electrodynamic model is very successful in the description of the steady-state and transient electrical characteristics of the cells, however, the prediction of a uniformly low electric field in the bulk of the sample is at odds with a previous experimental observation [158]. 

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