Fermi—Dirac distribution probability, energy

Intrinsic Semiconductors. For semiconductors in thermal equilibrium, (iV(E)), the average number of electrons occupying a state with energy E is governed by the Fermi-Dirac distribution. Because, by the Pauli exclusion principle, at most one electron (fermion) can occupy a state, this average number is also the probability, P(E), that this state is occupied (see Fig. 2c). In equation 2, K... 

The significance of the electrochemical potential is apparent when related to the concepts of the usual stati.stical model of free electrons in a body where there are a large number of quantum states e populated by noninteracting electrons. If the electronic energy is measured from zero for electrons at rest at infinity, the Fermi-Dirac distribution determines the probability P(e) that an electron occupies a state of energy e given by... 

Before considering the expressions for the rate constants, it is convenient to introduce a refinement [37] that has been ignored so far. It has been assumed that the electron transfer only involves the electronic state corresponding to the Fermi level of the electrode. However, the continuum of electronic levels (e) must be considered such that energy levels around the Fermi one can participate and the overall rate of electron transfer is a sum of the rates for each electronic state, weighted by the probability of occupancy/vacancy according to the Fermi-Dirac distribution ... 

Figure 5.4.5-1 Probability functions as a function of energy-photon wavelength. Lower curves show the probability profiles of two individual energy bands. Normally, the n-band is full and the 7i -band is empty. The probability of an electron transferring between these bands when excited by a photon is a function of the energy of the photon. The upper curve illustrates this probability as a function of photon energy and wavelength for the long wavelength photoreceptor of vision. This form is frequently described as the Fermi-Dirac distribution or function.

In this regard, the probability of finding an electron in a state with energy E is given by the Fermi-Dirac distribution function, fiE), which is expressed as follows (Figure 1.10) ... 

In this regard, if the probability of occupancy of a state at an energy E is fm(E), in agreement with the Fermi-Dirac distribution, we are dealing with electrons, which are fermions. Then, the product ffI)(E)g(E) is the number of electrons per unit energy per unit volume. Consequently, the area under the curve with the energy axis gives... 

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... 

More exactly, it is the energy where the occupation probability is 0.5 in the distribution of electrons among the various energy levels (the Fermi-Dirac distribution). See Sections 3.6.3 and 18.2.2 for more discussion of Ep. 

The probability/( ) that an energetic level of a solid is occupied by electrons can be determined by the Fermi-Dirac distribution function (Dekker, 1957). ft applies to fermions (particles with half-integer spin, including electrons, photons, neutrons, which must obey the PauU exclusion principle) and states that a given allowed level of energy E is function of temperature (T) and of the Fermi level,, according to the following equation ... 

At absolute zero, the highest occupied energy level is referred to as the Fermi level (in 3-D Fermi surface), derived from Fermi-Dirac statistics.The Fermi-Dirac distribution function, f(E), describes the probability that a given available energy state will be occupied at a given temperature ... 

In (124)/(e) is the Fermi-Dirac distribution function for the probability that a state k in the electrode with an energy e(k) is occupied, and l//(e)l is an electron wave number k-weighted interaction coupling element [186, 187],... 

The probability that an electron occupies an energy state is given by the Fermi-Dirac distribution function ... 

The probability that a given energy level is occupied by an electron is provided by the Fermi-Dirac distribution function/= exp[(e eiTj/feBT] + 1 derived in Section 10.1. Near T=0, f is essentially a rectangular distribution function, with /= 1 for e < ep and/= 0 for e > ep. 

In a scheme of available energy states, a population of electrons distributes according the Fermi-Dirac statistics The probability f(E) of having an electron in a state of energy E, is, at temperature T... 

As electrons obey Fermi-Dirac statistics, the distribution function giving the occupation probability f(E) of a state of energy E at equilibrium is... 

As will be seen further, modifications of the free carriers concentration will probably involve new catalytic properties. But one has to stress that other states of electronic excitation should be taken into account, for instance, excitons. Furthermore, under irradiation, the distribution of the electrons among all the characteristic energy levels of the solid do not correspond to the thermal distribution given by the Fermi-Dirac statistics. Let us also indicate that the electronic imperfections, transient by nature, may sometimes possess a quasi-permanent character with regard to the trapping phenomenon. 

Since the exponential can be written identically as exp(-AG V ) exp(j8VF/RT), this implies that the electrical energy quantity is also Boltzmann distributed. However, it could be argued that since it is electrons in the metal at the Fermi level that are involved, the term expipVF/RT) should in some way be written as a Fermi-Dirac factor rather than a Boltzmann factor. The forms of such a term would be P E) = 1/ 1 + exp[(E - Ep VF)//cT]. Introduction of such a probability factor into the equation for electron transfer rate is to be found in Gerischer s treatment (see below). The effect of this is a broad-... 

We now describe the behavior of charge carriers in an intrinsic semiconductor (i.e., pure) at equilibrium. The electrical properties of any extended solid depend on the position of the Fermi level, defined as the highest occupied state at T = 0 K. An alternative definition, stemming from the Fermi-Dirac statistics that govern the distribution of electrons, the Fermi level is the energy at which the probability of finding an electron is If the Fermi level falls within a band, the band is partially filled and the material behaves as a conductor. As shown in Fig. 3, the valence and conduction band edges of an intrinsic semiconductor straddle the Fermi level. At T = 0 K, no conduction is possible since all of the states in the valence band are completely filled with electrons while aU of the states in the conduction band are empty. 

Fig. 5 The steady-state occupation probability as a function of site energy, for sites with a Gaussian energy distribution. The curves show the fitted Fermi-Dirac functions. The inset shows the same data on a logarithmic scale...

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