Houben-Weyl Methoden der organischen Chemie vol.E6a by Kreher R., et al. (eds.)

By Kreher R., et al. (eds.)

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Example text

The conductivity band width Ew and the value dE (see formula (23) in the text) are shown. e. electrons in the conductivity band and holes in the valence band, are epsilon squared. In another case, electrons from the conductivity band will be captured by the acceptor, or holes from the valence band will get annihilated by the electron on the donor. Then, the tunneling transition becomes impossible because either the acceptor level becomes populated or the donor level is emptied. The Green’s function of electron in a crystal has the following spectral representation [9,10]: Gð~ r;~ r 0 ; EÞ ¼ X cs~ð~ r Þcsà ð~ r 0 Þ X ctà ð~ r Þctk~ð~ r 0Þ k k~ k~ À ~ ~ E À s ðkÞ E À t ðkÞ ~ k;s (21) ~ k;t r Þ is the wave function of the electron state with quasi-momentum Here cnk~ð~ ~ in nth band.

Therefore, the asymptotics CaD ð~ substituted in the matrix element (9) instead of the wave function CD ð~ r; EÞ. The value of the matrix element Vif is proportional to exp{ÀkrDA} due to the exponential decay of CaD ð~ r; EÞ. The evaluation of the correction to the wave function CA ð~ r À~ rA ; EÞ due to the interaction with the donor obviously shows that it is also proportional to expfÀkrDA g oo1. Therefore, to calculate vif to the first order of the value exp{ÀkrDA}, it should substitute the wave function CA ð~ r À~ rA ; EÞ in the expression (9) defined on the potential U A ð~ r À~ rA Þ only, and the same designation is kept for this function.

In the general case, the different multipole potentials in addition to Coulomb’s potentials of the donor and the acceptor may be included in the Hamiltonian (2) to the definition of Green’s function. These multipole potentials are non-zero in the asymptotic region only, but in MREL they are a part of the short-range potentials U SD;A ð~ r Þ. In the asymptotic region the values jV D j; jV A j oojE j, and their inequalities permit us to calculate the Green’s function by the quasi-classic method [4,8].

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