Polaron approximation

A number of recent calculations have compared the classical result with quantum mechanical calculations. In many cases, the results from the latter techniques confirm those from classical calculations with a gratifying accuracy. However, one topic on which there is continuing controversy is the nature of the polarons in transition metal oxides. Since the classical method subsumes all the quantum mechanics of the problem into the potential function, it can only tackle problems of electronic structure in a few specific cases, the most common example of which is in non-stoichiometric oxides. Here the question is the location of the electronic hole when the system is metal deficient. The only way such a problem can be tackled by classical methods is to use the small polaron approximation and assume that the hole resides on an ion to produce a new (in effect substitutional) ion with an extra positive charge. This can be successful and the use of the small polaron approximation in crystals is discussed in detail by Shluger and Stoneham (1993). However, all calculations on the first-row transition metal oxides have assumed that the extra charge resides on the metal ion. Recent quantum calculations (Towler et al., 1994) have thrown doubt on this assumption, suggesting that the hole is on the oxide ion. Moreover, the question of whether the hole is a small polaron for all these oxides is, at present, quite uncertain. Further discussion is given in Chapter 8. 

This approximation has a philosophical and mathematical resemblance to the linked-cluster expansion that has been applied successfully to the small polaron problem. The linked-cluster expansion is an exponential resummation of... 

We use these equations first to study formation of the polaron in the absence of an electric field, i.e., with A=0 [65]. Because only small motions of the sites are involved in the formation, the results should be approximately correct even for DNA in solution. To begin with we find the static undistorted solution (y =0) for a stack of AT base pairs with 2N n electrons. Using this solution as the initial condition for t=0, we integrate Eqs. 15 and 16 numerically for the case of 2N-1 n electrons on the stack. The parameters used were fo=0.3 eV, a=0.6 eV/A, and fC=0.85 eV/A. The result for the sequence given at the bottom of the figure is shown in Fig. 5. It is seen that the polaron is fully formed at 4 ps. The time of formation is much longer than was found for polarons in polyacetylene. The calculations of Su and Schrieffer... 

The motion described in the last paragraph, although it might be approximately correct for a polaron on DNA in air or vacuum, is not applicable for... 

In order to demonstrate important properties of displacements (p, and fl dependence) it is sufficient, except for the region of importance of the reflection (Fig. 3) inside the heavy region apparent for y2. to calculate them in the limit 17 = 0. We shall use equation (22) approximated for the cases of both heavy and light polaron and obtain implicit expressions for y,-, which, however, are good to visualize their behaviour in both regions ... 

For our formal treatment of the fermionic 2-level system we assume that we may describe the behaviour of electrons in the one-electron approximation. Then each electron is represented by a wave function that is independent of the wave functions of other electrons, and the individual wave functions may be linearly superimposed. This picture often proves useful in the context of inorganic semiconductors [2,3]. However, it may be highly questionable in organic and molecular matter, where excitonic [4] and polaronic effects are often predominant [5]. 

If the exciton binding energy relative to the separate charge pair is larger than that of the polarons relative to their free particle bands, the polaronic optical transitions will be hidden under the excitonic ones, and a one-electron approximation of the polaron certainly fails. This is yet another reason why optical signatures of the polaron may give dubious results (see Section IV.C.4). Such a situation can have practical consequences, to which we return in the discussion of electroluminescence in Chapter 12, Section V.C. 

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