Fermi distribution function

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] ... 

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) ... 

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... 

We will suppose that the internal relaxation in the leads is fast enough to lead to equilibrium distributions of the electrons. This means that wa R) = f(sa T V/2) (where /(e) is the Fermi distribution function) and V is the applied voltage. 

Approximating the Fermi distribution function so that for electrons, for example,... 

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. 

Fig. V-l.—Fermi distribution function, as function of energy, for several temperatures. Curve a, kT = 0 bf kT = 1 c, kT 2.5...

Let us assume now that the density function of energy levels for electrons in the solid is D (AE). The distribution of electrons is governed by the Fermi distribution function ... 

Fyj ( )= 7 r[rfG (aj)r G (ffl)] is the transmission function, describing electron tunneling ability between lead-/ and lead-a via QDs, and w) is the Fermi distribution function in lead- a In the expression of, the retarded Green functions are written as... 

Here g E) is the distribution of energy states in the metal whereas/( ) is the Fermi distribution function as given by Eq. (1.25), i.e. f(E)p(E) is the number of occupied and (l-f)p(E) the number of empty states in the metal. The exponential terms correspond to the distribution functions of the empty and occupied states of the redox system as illustrated in Fig. 7.5. All terms describing the interaction between electrode and redox system and other factors such as a normalization are summarized in the preexponential factor k which will not be discussed here. 

In view of these uncertainties, it is fortunate that an independent proof of Equation (5.28) exists in solid-state physics.Equation (5.29), the Fermi distribution function, is used to count the number of electrons in the valence band and in the conduction band. The result is that... 

Therefore if we make the potential barrier below Eb or above Eg,., the change of figure of merit 6Z becomes always positive. However the effect of the barrier above 63+ is neglected in following analysis, since the expression of 6Z contains the derivative of Fermi distribution function (eq.(2)) and therefore the contribution of carriers above Eg+ is considered to be small. It is easy to see that eq.(5) gives the generalization of height of the optimal barrier derived earlier[4]... 

While Gurney referred in his treatment of electrochemical charge transfer to the Fermi distribution function for electronic states in the metal, he did not, however, pursue the consequences of using this function in preference to a Boltzmann distribution. In Gerischer s treatment of redox reactions, also referred to by Schultze and Vetter in their treatments of the role of electron tunneling in O2 evolution and other redox process at oxide-covered (Pt) electrodes, the Fermi distribution was explicitly used in the current-potential function which is written (cf. Gurney and Geris-cher ) as... 

Figure 7.11 Fermi distribution function for two different temperatures Ef was assumed to be 1 eV. Note that whereas the distribution shifts to higher energies as the temperature increases, Ef, defined as the energy at which the probability of finding an electron is 0.5, does not change.

When the well representing the atom core is displaced slowly to the right, away from the surface, the atomic levels sharpen again into a single state. As long as the barrier between atom core and metal is small, however, electronic equilibrium can be maintained. The probability of finding an electron at the level Ea is therefore given by the Fermi distribution function... 

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