The Fermi-Dirac Distribution Function

Now we find the change of A, when the iV7s are varied, keeping temperature and the fixed. We find at once 

Equation (3.5) expresses the Fermi distribution law, which we shall now proceed to discuss, 

let us see that the Fermi-Dirac distribution law reduces to the Maxwell-Boltzinann law in the limit when the Nt s are small. In that case, it must be that the denominator in Eq. (3.5) is large compared to 

The quantity co is to be determined by the condition that the total number of particles is N. Thus we have 

---

Examination of the Fermi-Dirac distribution function Eq. (2.41) shows that the condition for applicability of the ideal-gas distribution to electron velocities is... 

Overhauser s original derivation of the effect employed the Fermi-Dirac distribution functions for electrons and was an involved calculation. Kittel ISl), Slichter 1S2), and others supplied simple derivations for this effect and Abragam 133) extended it to nonmetallic systems. 

The probability distribution, F(E), is the Fermi-Dirac distribution function... 

Extension to finite temperature T can be made by using the Fermi-Dirac distribution function for fk in Eq. (82)... 

Modification of this model to get the potential function is obtained considering the Fermi-Dirac distribution function for the electron density and the Boltzmann distribution for the ionic density. This was done by Stewart and Pyatt [58] to get the energy levels and the spectroscopic properties of several atoms under various plasma conditions. Here the electron density was given by... 

The important point is, that the leads are actually in the equilibrium mixed state, the single electron states are populated with probabilities, given by the Fermi-Dirac distribution function. Taking into account all possible single-electron tunneling processes, we obtain the incoming tunneling rate... 

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) ... 

Now the Fermi-Dirac distribution function of Eq. (8.2.2) will be proved. For a system of N particles in equilibrium at a finite temperature T (where N is very large, of the order of Avogadro s number), statistical mechanics suggests that the statistical weight PN(E) for the energy state E is given by... 

Fermi energy — The Fermi energy of a system is the energy at which the Fermi-Dirac distribution function equals one half. In metals the Fermi energy is the boundary between occupied and empty electronic states at absolute temperature T = 0. In the Fermi-Dirac statistics the so-called Fermi function, which describes the occupation fraction as a function of energy, is given by f(E) = —pjrj—> where E is the energy, ft is the - chem-... 

When r > 0 K, the population of the electron energy levels is described by the Fermi-Dirac distribution function (See Section 4.3.3, Eq. 22). At T > 0 K, electrons from the valence band can be thermally excited into the conduction band. As a result, the bottom of the CB becomes partly populated and the top of the VB partly depopulated [Figure 7(b)]. An empty electron level at the top of the valence band is called a (valence-band) hole. The concentration of holes, p, and of electrons, n, can be expressed as a function of the electrochemical potential with Eq. 22. We denote the density of electron levels within IcbT from the top of the VB and the bottom of the CB as the effective density of valence band levels, Avb, and conduction band levels, Nqb, respectively. The electron occupancy of the electron levels at the bottom of the CB is... 

For metal electrodes, the Fermi level is embedded within a broad distribution of closely spaced electronic levels. The Fermi level describes the occupancy of energy levels of a system at equilibrium and can simply be thought of as the chemical potential of electrons in the solid [10]. When employed in the Fermi-Dirac distribution function, Eq. 3 results ... 

The distribution of the electrons among the allowed energy states in the semiconductor crystal at thermal equilibrium is described by the Fermi-Dirac distribution function." It is denoted by ME), which has the form... 

In the case of electrochemical ET, the relevant overlap was between the gaussian density-of-states function of the reacting species in solution and the Fermi-Dirac distribution function of the charge carriers in the electrode (Fig. 4.21). Figure 4B. 1 shows the analogous density-of-states functions for a homogeneous ET reaction. The rate... 

At temperature above T = 0 K the population (F) of the orbitals has a Boltzmann-type distribution which is described by the Fermi-Dirac distribution function ... 

Here d is the molecule length, F(E) is the Fermi-Dirac distribution function, andDff, g(E) is the tunneling probability, which is described elsewhere [5]. 

Gas adsorption is a chemical interaction between the gas molecules and the semiconductor surface. This interaction is accompanied by charge exchange creating acceptor or donor like band gap level, whose occupation probability is given by the Fermi-Dirac distribution function (Kireev 1978). Its conduction behavior as acceptor or donor will depend on the type of the adsorbed molecule. 

The energy states of the surface, denoted by Es, are located with respect to the Fermi energy value Ep, which corresponds to the reference state of the solid. In a similar way to the bulk states, each surface state can be empty or occupied by an electron. The ratio of surface states Es occupied by electrons, denoted by ft (Es), is given by the Fermi-Dirac distribution function ... 

These quantities are illustrated in Figure 11. Also shown is the distribution of occupied levels in a metal electrode in equilibrium with a redox couple (equimolar oxidized and reduced forms) of = 0.6V calculated from the Fermi-Dirac distribution function... 

---