Fermi degeneracy

A symmetry-imposed band degeneracy occurs for the [(/i-3/2)k bands at the Fermi level. 

Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

K). T is the measurement temperature and Tq is the "degeneracy temperature," equal to kEo, where k is the Boltzmann constant. According to a two-dimensional electron gas model for graphitic carbons (see ref. 2a), is the energy "shift" from the Fermi level (Ep), to the top of the valence band. Small values of To ( <344 K) and consequently of Eq signify a more perfect graphite... 

Degeneracy can be introduced not only by heavy doping, but also by high density of surface states in a semiconductor electrode (pinning of the Fermi level by surface states) or by polarizing a semiconductor electrode to extreme potentials, when the bands are bent into the Fermi level region. 

We see that at the leading order in I //j, expansion, the energy is independent of the residual momentum, l , perpendicular to the Fermi velocity. In HDET, therefore, the perpendicular momentum labels the degeneracy and should satisfy a normalization condition... 

The Hamiltonians of the previous sections describe realistic vibrational spectra of linear triatomic molecules except when accidental degeneracies (resonances, cf. Section 3.3) occur. A particularly important case is that in which the bending overtone (02°0) is nearly degenerate with the stretching fundamental (10°0) of the same symmetry Fermi, 1929, resonance). This situation occurs when the coefficient in Eq. (4.67) is nearly equal to -A (Figure 4.13). The Majorana... 

Figure 4.13 Schematic representation of the effects of the Majorana operator Mn, which removes the degeneracy of the local modes and of the Fermi operator hi, which splits the degenerate (when Xn = - 1) normal multiplets.

For high density electron ensembles such as free valence electrons in solid metals where electrons are in the state of degeneracy, the distribution of electron energy follows the Fermi function of Eqn. 1-1. According to quantum statistical dynamics [Davidson, 1962], the electrochemical potential, P., of electrons is represented by the Fermi level, ep, as shown in Eqn. 1-10 ... 

In cases in which the surface state density is high Nc/i,Nm, Ny/i,Nm - 1), electron distribution in the siuface state conforms to the Fermi function (the state of degeneracy) and the Fermi level is pinned at the surface state level. This is what is called the Fermi level pinning at the surface state. 

Fig. 2-81. Surface degeneracy caused by Fermi level pinning at a surface state of high state density (a) in flat band state (Ep ep), G>) in electron equilibrium (cp = cp). cp = surface Fermi level = surface ccmduction band edge level.

The semiconductor surface where the Fermi level is pinned at a surface state of high density (Fig. 2-31) is in the state of degeneracy of electron levels, because of the high electron state density at the surface Fermi level. Similarly, the surface degeneracy is also established when the band bending becomes so great that the Fermi level is pinned either in the conduction band or in the valence band as shown in Fig. 2-32. 

In the state of Fermi level pinning, the Fermi level at the interface is at the surface state level both where the level density is high and where the electron level is in the state of degeneracy similar to an allowed band level for electrons in metals. The Fermi level pinning is thus regarded as quasi-metallization of the interface of semiconductor electrodes, making semiconductor electrodes behave like metal electrodes at which all the change of electrode potential occurs in the compact layer. 

Such an interfacial degeneracy of electron energy levels (quasi-metallization) at semiconductor electrodes also takes place when the Fermi level at the interface is polarized into either the conduction band or the valence band as shown in Fig. 5-42 (Refer to Sec. 2.7.3.) namely, quasi-metallization of the electrode interface results when semiconductor electrodes are polarized to a great extent in either the anodic or the cathodic direction. This quasi-metallization of electrode interfaces is important in dealing with semiconductor electrode kinetics, as is discussed in Chap. 8. It is worth noting that the interfacial quasi-metallization requires the electron transfer to be in the state of equilibrimn between the interface and the interior of semiconductors this may not be realized with wide band gap semiconductors. 

As the potential Ai )sc of an inversion layer increases and as the Fermi level at the electrode interface coincides with the band edge level, the electrode interface is in the state of degeneracy (Fermi level pinning) and both the capacity Csc and the potential A4>sc are maintained constant. Figure 5-48 shows schematically the capacity of a space charge layer as a function of electrode potential. As the electrode potential shifts in the anodic (positive) direction from a cathodic (negative) potential, an accumulation, a depletion, and an inversion layer are successively formed here, the capacity of the space charge layer first decreases to a minimum and then increases to a steady value. 

The density of states function given here follows from the assumption of a parabolic energy band and includes spin degeneracy (Bube, 1974, p. 172). (Note that the Fermi function effectively cuts off the integrand at a few kT above eF, so that for kT the upper limit may be extended to infinity.) Also we have set c n = 0. 

This band often appears as a doublet probably resulting from the accidental degeneracy of an overtone of another low-lying vibration mode (Fermi resonance). 7a, 6 8) The C—H stretching vibrations also undergo a high-frequency shift and appear at 3050 cm-1, a position generally characteristic of the methylene stretch in cyclopropyl systems.67)... 

As seen earlier, some fundamental vibrations are relatively weak. Furthermore, some overtone and combination bands become unusually strong when Fermi resonance (accidental degeneracy) occurs. A typical example is given by C02, where the frequency of the first overtone of the v2(667 cm-1) is very close to that of the vj fundamental (1,337 cm-1). Since vj and 2v2 belong to the same symmetry species ( +), they interact with each other to give rise to two strong Raman bands at 1,388 and 1,286 cm-1. Finally, it should be noted that the point group symmetry in the crystalline state is not necessarily the same as that in the isolated state. Thus, this method must be applied with caution. 

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