Fermi: distribution

According to Fermi statistics, the probability of finding an electron in a state in which it has energy 2 is 

The probability of finding a state at 1 which is not occupied by an electron, i.e., the probability of finding a hole, is 

The total concentrations ne and n , of electrons and holes, respectively, follow from these relations by multiplying by the appropriate densities of states Dx and D2 at energies 1 and 2  

If the numbers of electrons and holes are much smaller than the corresponding densities of states, the Fermi functions in (4.4) and (4.5) may be approximated by Boltzmann distributions by neglecting the +1 in the denominator. 

An interesting consequence of the Boltzmann approximation is that the product of the electron and hole densities, viz., 

The previous section demonstrated in extenso how the quantum description of electrons in crystals requires a discretization of the eigen-eneigies of crystalline orbitals in energetic bands the present discussion follows (Putz, 2006). 

Therefore, the eleetrons showcase discontinuous eneigetic spectrum of bands in crystals but how are they disposed in between these bands, if any  

An idea was already previously introduced, regarding the necessity of the existence of a maximum level of occupancy, i.e., the Fermi level. 

The position of this level in the entire energetie speetrum of bands in crystals is essential, as long as it establishes the maximum limit of the possibilities of electronie oeeupaney. 

But how are disposed the electrons and what is the probability with which the successive energetie bands are occupied imtil the maximum limit of Fermi level  

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Fig. 6. Schematic energy levels of a soHd as a function of interatomic distance where the vertical line represents the equiUbrium spacing (68). A band of states obeying Fermi distribution is required by the PauH principle. High electron velocities and equivalent temperatures exist in conductors even when the...

In perfect semiconductors, there are no mobile charges at low temperatures. Temperatures or photon energies high enough to excite electrons across the band gap, leaving mobile holes in the Fermi distribution, produce plasmas in semiconductors. Thermal or photoexcitation produces equal... 

This result means that p(q)is constant in the range -qT < q < qF. At finite temperature, however, p(q) has a finite width of kBT at qF due to the Fermi distribution... 

Anti-protonic atoms. Recently neutron density distributions in a series of nuclei were deduced from anti-protonic atoms [30], The basic method determines the ratio of neutron and proton distributions at large differences by means of a measurement of the annihilation products which indicates whether the antiproton was captured on a neutron or a proton. In the analysis two assumptions are made. First a best fit value for the ratio I / of the imaginary parts of the free space pp and pn scattering lengths equal to unity is adopted. Secondly in order to reduce the density ratio at the annihilation side to a a ratio of rms radii a two-parameter Fermi distribution is assumed. The model dependence introduced by these assumptions is difficult to judge. Since a large number of nuclei have been measured one may argue that the value of Rj is fixed empirically. 

Fe Oj coatings, 40 105 FejOj/y-AljOj, MSssbauer spectra, 37 30 FCjOj-I catalyst, 37 181-183 FcjOj superacid, 37 199-201 Fermi distribution, 34 228 Fermi energy, 27 217 Fermi golden rule, 34 243 Fermi level, 27 4, 5 Fermi s Golden Rule, 35 19-20 Ferric aluminate as catalyst, 20 109-112 chemical structure and catalytic activity of, 20 111, 112... 

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

Fig. 6-46. Differential capacity observed and computed for an n-type semiconductor electrode of zinc oxide (conductivity 0. 59 S cm in an aqueous solution of 1 M KCl at pH 8.5 as a function of electrode potential solid curve s calculated capacity on Fermi distribution fimction dashed curve = calculated capacity on Boltzmann distribution function. [From Dewald, I960.]...

The tunneling current can be evaluated by summing over all the relevant states. At any finite temperature, the electrons in both electrodes follow the Fermi distribution. With a bias voltage V, the total tunneling current is... 

If ksT is smaller than the energy resolution required in the measurement, then the Fermi distribution function can be approximated by a step function. In this case, the tunneling current is (see Fig. 1.20) ... 

The Fermi-distribution factor in Eq. (14.2), imposes another limit on spectroscopic resolution. At room temperature, ksT. Ol eV. The spread of the energy distribution of the sample is IkeT O.OSl eV. The spread of the energy distribution of the tip is also IksT O.OSl eV. The total deviation is LE AkeT OA eV. 

Figure 6.3 The Fermi distribution function (a) at absolute zero and (b) at a finite temperature, (c) The population density of electrons in a metal as a function of energy. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

Another way of obtaining the same result is the following. The Fermi distribution function / in the presence of a field F along the x-axis is given by... 

The electrons in the Fermi level in a metal—those that undergo the Fermi distribution law—are mobile and that is where the difference comes from electrons in solution which are, in fact, in the bound levels of ions. Such electrons are not mobile and the statement that they have a Fermi energy may therefore be misleading, for they do not obey the same distribution law as the electrons with which they are in equilibrium.4... 

The Fermi distribution law deals with the probability of occupancy by electrons in metals of states of a given energy. The density of states represents a number of states per unit volume having a given energy, (e) What, then, is an expression for the number of electrons per cubic centimeter having an energy between E and E + dE ... 

A degenerate electron gas is an electron gas that is far below its Fermi temperature, thai is. which must be described by die Fermi distribution. The essential characteristic or this state is that a very large proportion of the electrons completely fill the lower energy levels, and are unable to lake pan in any physical processes until excited out of these levels. 

A somewhat more sophisticated approach to the problem of defining the nuclear size and density is to assume the nuclear density distribution, p(r), assumes the form of a Fermi distribution, that is,... 

Choosing for the electrons a Fermi distribution with values of p from 0 to pmax, we obtain for the total energy of interaction the following cumbersome expression ... 

Comparing this expression with the actual one, we can say that, because of the Pauli principle, only a thin layer of electrons in the Fermi distribution (Pm x P ) > h /d, rather than all of the electrons from 0 to pmax, takes part in the interaction or works. The feasibility of this formulation is related to the form given above for I(p, p") as a function of (p —p"). 

The Pauli principle and the Fermi distribution for electrons in the metal lead to the fact that in the interaction corresponding to this value of... 

Insertion of equation 3 into equation 1, approximation of the Fermi distribution by a classical Maxwell-Boltzmann distribution, and integration of equation 1 yield the expression for the total number of electrons in the conduction band ... 

At these temperatures the distribution of occupied levels in the conduction bands ( the Fermi distributions ) in the two metal electrodes ( Fig.l ) are quite sharp, with a boundary between filled and empty states ( the Fermi level ) of characteristic width k T ( k =0.08617 meV/K=0.69503 cm Vk ). An applied bias voltage V between the two electrodes separates the Fermi levels by an energy eV. If the barrier oxide is sufficiently thin electrons can tunnel from one electrode to the other. This process is called tunneling since the electrons go through a potential barrier, rather than being excited over it. The barrier must be thin for an appreciable barrier to flow. For a typical 2 eV barrier the junction resistance is proportional to, where s is the barrier width in Angstroms (17). The... 

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