Quasi-Fermi Distributions

Any additional excitation over the excitations present in equilibrium with background radiation of temperature T leads to an increase in the electron 

In the Boltzmann approximation, the product of the electron and hole concentrations is 

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It is worth remembering that we are still working with the one-electron picture, and that we have applied the Boltzmann relation in order to approximate Fermi and quasi-Fermi distribution functions, assuming the quasi-free electron and hole densities of states in the bands. 

The distributions of excess, or injected, carriers are indicated in band diagrams by so-called quasi-Fermi levels for electrons, Ep or holes, These... 

Before the transfer starts, the energy distribution of electrons takes the form of a Fermi-Dirac distribution function. While the number of electrons is decreasing steadily with time, the distribution of electrons keep the form of a Fermi-Dirac distribution function. This constancy of the distribution is due to the fact that the capture rate of free electrons by the localized states is much faster than the loss of free electrons caused by the transfer when the occupation probability of localized states is not approximately one. Therefore, electrons are considered to be in their quasi-thermal equilibrium condition i.e., the energy distribution of electrons is described by quasi-Fermi energy EF. Then the total density t of electrons captured by the localized states per unit volume can be written as... 

If local traps are distributed in energy (E), they will be filled from bottom to top as electric fields, F, increase. The quasi-Fermi level will scan the distribution shifting towards the transport band, and 0 = rif/rit will become a function of F. A general form of nt = nt(iif) relation can be obtained from a detailed balance equation as [363]... 

Figure 1. Space-charge distribution of the excited current carriers p along z-coordinate within one period of superlattice structures (a) No. 4 and (b) No. Ai as a function of the quasi-Fermi level difference AF at 20 K.

Li Fermi distribution Emax = 20 Mev P> 2/>) Quasi-elastic scattering Chamber-... 

The charge distribution in energy follows a quasi-equilibrium distribution and can be described as the DOS multiplied by a Fermi-Dirac function. 

The density of the 7=3 states determined from the experimental data (Krasznahorkay et al. 1999) is close to a Wigner quasi-probability distribution (Brody et al. 1981) but the mixing-in of some Poisson type distribution is also visible. The density of / = 3 states has also been calculated using the back-shifted Fermi-gas description with parameters determined by Rauscher et al. (1997). The calculated curve had to be shifted by 2.7 MeV to reproduce the experimental values (Krasznahorkay et al. 1999). The energy of the ground state in the third well is obtained to be Em = (3.1 0.4) MeV. 

However, each quasi-photon transition still affects the energy of the quasi-electrons, and each quasi-electron transition still affects the energy of the quasi-photons. We may not simultaneously occupy the quasielectron states according to the Fermi distribution and quasi-photon states according to the Bose distribution. This is permissible only after renormalization of the quasi-particles considered. 

The curves are for different illumination intensities. Each curve is characterized by the quasi Fermi energy E. The good agreement with experiment combined with the conclusion that long-time behavior of the recombination kinetics is determined solely by the trap energy distribution has led to widespread acceptance of the trapping model. 

Note that this occupation probabihty has indeed no longer the same shape as a Fermi-Dirac distribution. Instead of one inflection point (the Fermi level in Fermi-Dirac statistics), there are two inflection points that are sometimes called quasi-Fermi levels for trapped charge [234, 235]. 

The electrical resistivity is determined by the inverse lifetime of the quasi-particles averaged over the Fermi distribution. The calculated p(T) curve, shown in fig. 43, starts from zero at T = 0, rises exponentially until a peak is reached around T = ri/k, and drops smoothly to zero at higher temperatures. Again, variations of /i(0) produces only slightly different results. The low-temperature part is in disagreement with experiments, because the calculation fails to include the mutual scattering effect of quasi-particles. The resistivity peak reflects the breakdown of the band structure, which takes place when the temperature reaches rj/kg. The model does not shed light on the variety of resistivity behaviors shown in fig. 6. 

The next assumption is that the relaxation time approximation is valid. According to Nag, this is equivalent to the assumption that scattering processes are elastic or isotropic and that quasi-Fermi levels can be used to describe carrier distribution [347]. In that case... 

The result of the previous section is only valid close to equilibrium. The electric field is limited such that the difference between donor and acceptor quasi Fermi levels Md - Ma < 2fe7 . Close to equilibrium we can assume that the electron distribution is characterized by the lattice temperature, and we can also linearize the hopping conductivity expression of Eq. (3). 

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