Electronic chemical potential levels

Fig. 7.10 Change of energy (for F+, F and F ) as electrons are added to a species. The energies were calculated at the QCISD(T)/6-311+G level. The slope of the curve at any point (first derivative) is the electronic chemical potential, and the negative of the slope the electronegativity, of the species at that point. The curvature at any point (second derivative) is the hardness of the species). See too Table 7.12...

In the internal MEC case, all populational derivatives = 0 for the initial global equilibrium. Therefore, fim = 0 also in the REC description for a given partitioning, Melectron distribution corresponds to the global equilibrium then too, ijf> = dEjdz = (pfp = P Thus, the representative REC chemical potential pf > for the M(k) partitioning, representing the relaxed chemical potential of atom k, is also at the initial global chemical potential level. 

A test of the PMH which is quite different, and more general, has recently been given.It follows earlier work by Gyftopoulos and Hatsopoulos, who used a grand canonical ensemble with a limited number of discrete energy levels, so that the distribution function was known.Those authors then calculated the electronic chemical potential, /z, which was found to be /z = (/-(-y4)/2. The ensemble was a collection of systems containing the three species M , M+, M and with energy levels E°, E° + /) and E° — A). 

On the hypothesis of local equilibrium, a description analogous to that proposed by Kroger can be used to determine the local situation in the crystal. The basic principles and assumptions of this description are briefly recalled in what follows. Here we will use as an independent variable the electron chemical potential Pe. It is the most appropriate parameter to characterize the degree of trap filling. Within a constant, it equals the distance between the Fermi level and the valence band edge. 

A common reference energy level, the so-called crystal zero, can be uniquely defined for both electrons and positrons in perfect solids. Therefore, the energy levels in the calculations are measured relative to this internal quantity. The electron chemical potential p- is defined as the distance of the Fermi levels from the crystal zero (see Figure 4.33). Similarly, the distance of the lowest positron energy level from the crystal zero defines the positron chemical potential p,+ (Figure 4.32). 

SO that Fermi level is located at e, the effective one-electron energy of an atom (pi orbital). The general form of the electron chemical potential for a finite molecule A in the presence of a second molecule P is approximately... 

Fig. 1. The energy levels in a semiconductor. Shown are the valence and conduction bands and the forbidden gap in between where represents an occupied level, ie, electrons are present O, an unoccupied level and -3- an energy level arising from a chemical defect D and occurring within the forbidden gap. The electrons in each band are somewhat independent, (a) A cold semiconductor in pitch darkness where the valence band levels are filled and conduction band levels are empty, (b) The same semiconductor exposed to intense light or some other form of excitation showing the quasi-Fermi level for each band. The energy levels are occupied up to the available voltage for that band. There is a population inversion between conduction and valence bands which can lead to optical gain and possible lasing. Conversely, the chemical potential difference between the quasi-Fermi levels can be connected as the output voltage of a solar cell. Fquilihrium is reestabUshed by stepwise recombination at the defect levels D within the forbidden gap.

Electrons excited into the conduction band tend to stay in the conduction band, returning only slowly to the valence band. The corresponding missing electrons in the valence band are called holes. Holes tend to remain in the valence band. The conduction band electrons can estabUsh an equihbrium at a defined chemical potential, and electrons in the valence band can have an equiUbrium at a second, different chemical potential. Chemical potential can be regarded as a sort of available voltage from that subsystem. Instead of having one single chemical potential, ie, a Fermi level, for all the electrons in the material, the possibiUty exists for two separate quasi-Fermi levels in the same crystal. 

In the quantum level stmctures illustrated in Figure lb, electrons in each band fill up the energy levels, up to the available chemical potential. From Figure lb two critical optoelectronic devices can be explained. 

If we can assume that the electrode material is a good metal, and the electronic gas is fully degenerate, the chemical potential of the electrons is given by the Fermi level, EP, which can be written as... 

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

The Fermi energy, Wp, is reckoned from the energy of the valence-band bottom (zero-point energy) and gives the kinetic energy of the electrons at the highest occupied level of this band. This energy is equal to the chemical potential of the electrons. 

Unlike the values of values of electron work function always refer to the work of electron transfer from the metal to its own point of reference. Hence, in this case, the relation established between these two parameters by Eq. (29.1) is disturbed. The condition for electronic equilibrium between two phases is that of equal electrochemical potentials jl of the electrons in them [Eq. (2.5)]. In Eig. 29.1 the energies of the valence-band bottoms (or negative values of the Fermi energies) are plotted downward relative to this common level, in the direction of decreasing energies, while the values of the electron work functions are plotted upward. The difference in energy fevels of the valence-band bottoms (i.e., the difference in chemical potentials of the... 

At the contact of two electronic conductors (metals or semiconductors— see Fig. 3.3), equilibrium is attained when the Fermi levels (and thus the electrochemical potentials of the electrons) are identical in both phases. The chemical potentials of electrons in metals and semiconductors are constant, as the number of electrons is practically constant (the charge of the phase is the result of a negligible excess of electrons or holes, which is incomparably smaller than the total number of electrons present in the phase). The values of chemical potentials of electrons in various substances are of course different and thus the Galvani potential differences between various metals and semiconductors in contact are non-zero, which follows from Eq. (3.1.6). According to Eq. (3.1.2) the electrochemical potential of an electron in... 

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