Electron density Fermi-Dirac statistics

Electrons obey Fermi-Dirac statistics, but Photons obey Bose-Einstein statistics. Kittel and Kroemer (1980) describe this phenomenon with the phrase It has been said that bosons travel in flocks . Imagine the variation in customer density at a fastfood restaurant if most customers arrived in a few tour buses instead of walking in off the street. 

In fact it is not the necessity of applying Fermi-Dirac statistics per se which distinguishes the ionic problem from the electronic one (Fermi-Dirac statistics have also been applied for the ionic defects, cf. Eq. (5.18)), it is the different state densities. In the Boltzmann approximation, this difference is reflected by a different standard state. 

In this derivation, it was assumed that the charge-transfer complex is formed between the p- or n- primary dopant and the gas acts as a secondary dopant. In fact, the interaction of the secondary dopant with any energy state in the matrix is possible that would lead to the same result, as long as the exchanged electron density becomes part of the electron population governed by the Fermi-Dirac statistics. The analytical utility of this relationship has been shown for several inorganic gases (Janata and Josowicz, 2003). 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

The density of states is useful to our discussion because it allows the calculation of an extremely important numerical quantity, the effective density of states in the conduction band of the semiconductor, N. As a first approximation, can be considered to be the number of electronic states in the conduction band of the semiconductor that are available to accept electrons from a donor. is more formally defined as the number of electronic states, N E), for aU energy E within 3kT of the conduction band edge [ (i cb — E) < 3kT]. Given the density of states per unit energy (equation 1), and using Fermi-Dirac statistics to describe the distribution of electrons as a function of energy, it can be shown that ... 

It would be of interest to apply the method of March and Murray [12] to convert C, the electron density for non-degenerate electrons, into results applicable to intermediate degeneracy governed by Fermi-Dirac statistics. Unfortunately, without switching on the model potential F(r), this is already difficult to handle by purely analytically methods, as can be seen from the case of complete degeneracy for the harmonic oscillator alone. No doubt, numerical procedures will eventually enable present results to be transformed according to the route established in [12]. 

Electrons escape from the material by tunnelling through a potential barrier at the surface which has been reduced in thickness to about 1.5 nm by the applied field, Figure 2. If the solid is assumed to contain free electrons which obey Fermi-Dirac statistics, the current density J of field emitted electrons is simply related to the applied field Fand work function cf) by the Fowler-Nordheim (FN) equation... 

To determine more accurate values for the number of holes and electrons present in an intrinsic semiconductor it is appropriate to use Fermi-Dirac statistics and the density of states at the bottom of the conduction band (see Section S4.8). To a good approximation, it is found that the number of electrons in the conduction band per unit volume,... 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

Figure 2.9 Electron relaxation dynamics for GaAs (100). (a) Compares the hot electron lifetimes as a function of excess energy (above the valence band) of a pristine surface prepared using MBE methods with device-grade GaAs under the same conditions. The higher surface defect density of the device-grade material increases the relaxation rate by a factor of 4 to 5. (b) The electron distribution as a function of excess energy for various time delays between the two-pulse correlation for MBE GaAs. The dotted lines indicate a statistical distribution corresponding to an elevated electronic temperature. The distribution does not correspond to a Fermi-Dirac distribution until approximately 400 fs. The deviation from a statistical distribution is shown in (c) where the size of the error bars on the effective electron temperature quantifies this deviation.

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

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