Fermi level, free-electron theory

In the free-electron theory the Fermi level for nonzero temperature is approximately ... 

The main shortcoming of the molecular dynamics approach discussed in the previous section is that it ignores the fact that an electron transfer at the solution/metal interface occurs between an ion in a well-defined electronic state and a continuum of electronic states in the metal. For example, depending on the ion s orbital energy, the reorganization free energy and the overpotential, the electron could be transferred from, or to, any level around the Fermi level of the metal. Therefore, a sum over all these possibilities must be performed. Analytical theories of electron transfer at the solution/metal interface recognized this issue very early on, and the reader is referred to many excellent expositions on this sub-... 

Note that the exchange term is of the form / y(r,r ) h(r )dr instead of the y (r) (r) type. Equation (1.12), known as the Hartree-Fock equation, is intractable except for the free-electron gas case. Hence the interest in sticking to the conceptually simple free-electron case as the basis for solving the more realistic case of electrons in periodic potentials. The question is how far can this approximation be driven. Landau s approach, known as the Fermi liquid theory, establishes that the electron-electron interactions do not appear to invalidate the one-electron picture, even when such interactions are strong, provided that the levels involved are located within kBT of Ep. For metals, electrons are distributed close to Ep according to the Fermi function f E) ... 

The surface states observed by field-emission spectroscopy have a direct relation to the process in STM. As we have discussed in the Introduction, field emission is a tunneling phenomenon. The Bardeen theory of tunneling (1960) is also applicable (Penn and Plummer, 1974). Because the outgoing wave is a structureless plane wave, as a direct consequence of the Bardeen theory, the tunneling current is proportional to the density of states near the emitter surface. The observed enhancement factor on W(IOO), W(110), and Mo(IOO) over the free-electron Fermi-gas behavior implies that at those surfaces, near the Fermi level, the LDOS at the surface is dominated by surface states. In other words, most of the surface densities of states are from the surface states rather than from the bulk wavefunctions. This point is further verified by photoemission experiments and first-principles calculations of the electronic structure of these surfaces. 

On surfaces of some d band metals, the 4= states dominated the surface Fermi-level LDOS. Therefore, the corrugation of charge density near the Fermi level is much higher than that of free-electron metals. This fact has been verified by helium-beam diffraction experiments and theoretical calculations (Drakova, Doyen, and Trentini, 1985). If the tip state is also a d state, the corrugation amplitude can be two orders of magnitude greater than the predictions of the 4-wave tip theory, Eq. (1.27) (Tersoff and Hamann, 1985). The maximum enhancement factor, when both the surface and the tip have d- states, can be calculated from the last row of Table 6.2. For Pt(lll), the lattice constant is 2.79 A, and b = 2.60 A . The value of the work function is c() w 4 cV, and k 1.02 A . From Eq. (6.54), y 3.31 A . The enhancement factor is... 

It should be appreciated that in contrast to the simple free electron models used in much of our discussion of metals and semiconductors, a treatment of screening necessarily involves taking into account, on some level, the interaction between charge carriers. In the Thomas-Fermi theory this is done by combining a semiclassical approximation for the response of the electron density to an external potential with a mean field approximation on the Hartree level—assuming that each electron is moving in the mean electrostatic potential of the other electrons. 

The quantum mechanical theory leads to a number of conclusions in respect to the kinetics of electrochemical reactions, in particular to the hydrogen evolution reaction. These conclusions are, to a certain degree, opposite to those of the classical approach. Thus, the consistent incorporation of the electronic energy spectrum in the electrode in the theory leads to the conclusion that barrierless and activationless transitions should be observed under certain conditions. In the theories which consider transitions to only one electronic energy level (the Fermi level), the transition probability should increase, reach a maximum, and then decrease with decrease of the reaction free energy. Experiment shows the existence of the barrierless and activationless processes. 

This quasi-chemical modeling will, obviously, preserve the contribution of the two methods of modeling, the chemical and electronic methods. Thus, in this manner, we have to identify the active sites and put them back in the context of the total solid with the possibility of association of stmcture elements and the interaction of free electrons and electron holes with the elements of the solid by the ionization of the defects. Also recall that the Fermi level introduced by the theory of bands has its physicochemical equivalent because it corresponds to the electrochemical potential of the free electrons in quasi-chemical description. 

The simplest of the simple metals, in many respects, are the alkali metals. Not only are the bands of nearly-free-electron form, but since there is only one electron per atom, this means that the first Brillouin zone is only half filled and all of the gaps in the band structure lie above the Fermi level, resulting in a relatively undistorted Fermi surface, as illustrated in Figure 17. Lee has recently reviewed both theory and experiment for the alkali metals (particularly Fermi surface data). He concludes that, with few exceptions, the experimental evidence is consistent with the straightforward NFE band picture, rather than the spin-density-wave or charge-density-wave models which have been advanced to explain supposed discrepancies. 

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