Thomas–Fermi Approximation and Basics of the Density Functional Theory As stated at the beginning of section 2.7 the total energy is a key function describ-ing the basic physical and chemical properties of materials: the ground state. It consists of both kinetic (describing motion) and potential energy parts.
The topics discussed are the nature of interatomic forces, theoretical models of the atom, interatomicc potentials based on Thomas- Fermi theory, empirical interatomic potentials, pseudopotential theory, pair potentials based on pseudopotential theory, atomic collision theory and interatomic potentials, experiments on the scattering of atoms and ions, liquid metal pair interaction potentials, the application of interatomic potentials. Also the Thomas- Fermi screening function, Hartree dielectric screening of the pseudopotential, the cohesion of ionic crystals, interatomic potentials derived from planar channeling data are covered.
We have developed a semimicroscopic theory for the electrostatic potential due to an isolated charge near a semiconductor surface whose surface states do not contribute free carriers. It employs the linearized version of the Debye-Hueckel (or, equivalently, the Thomas-Fermi) approximation. This includes the screening effects both of the plasma of free carriers due to bulk donors or acceptors, and of the bound polarizable charge associated with the bulk dielectric, but does not include free charge from intrinsic or extrinsic surface states. Results are obtained for a source charge above the semiconductor, within the semiconductor, and on the semiconductor surface, but we emphasize the last case. Although there is a dipole moment associated with source charge on the surface, the surface potential at long distances is quadrupolar; at intermediate distances greater than a characteristic atomic dimension it is a screened exponential, with screening length equal to the bulk value. The case of intermediate distance provides a rigorous basis for the exponentially screened surface potential commonly employed to analyze scanning tunneling microscopy images of the depletion or accumulation regions surrounding isolated charges on III-V(110) cleavage surfaces. Certain of these results also apply to colloids.
(c) 2000 The American Physical Society.
The plasma frequency in the study of solids arises in many different contexts. One of the most illuminating ways to look at the plasma frequency is as a measure of the screening response time in solids., but I think it is worth repeating and expanding upon.What I mean by “screening response time” is that in any solid, when one applies a perturbing electric field, the electrons take a certain amount of time to screen this field. This time can usually be estimated by using the relation:Now, suppose I introduce a time-varying electric field perturbation into the solid that has angular frequency. The question then arises, will the electrons in the solid be able to respond fast enough to be able to screen this field? Well, for frequencies the corresponding perturbation variation time is. This means the the perturbation variation time is longer than the time it takes for the electrons in the solid to screen the perturbation. So the electrons have no problem screening this field.
However, if and, the electronic plasma in the solid will not have enough time to screen out the time-varying electric field.This screening time interpretation of the plasma frequency is what leads to what is called the plasma edge in the reflectivity spectra in solids. Seen below is the reflectivity spectrum for aluminum (taken from Mark Fox’s book Optical Properties of Solids):One can see that below the plasma edge at 15eV, the reflectivity is almost perfect, resulting in the shiny and reflective quality of aluminum metal in the visible range. However, above =15eV, the reflectivity suddenly drops and light is able to pass through the solid virtually unimpeded as the electrons can no longer respond to the quickly varying electric field.Now that one can see the effect of the screening time on an external electric field such as light, the question naturally arises as to how the electrons screen the electric field generated from other electrons in the solid. It turns out that much of what I have discussed above also works for the electrons in the solid self-consistently. Therefore, it turns out that the electrons near the Fermi energy also have their electric fields, by and large, screened out in a similar manner. The distance over which the electric field falls by is usually called the Thomas-Fermi screening length, which for most metals is about half a Bohr radius.
That the Thomas-Fermi approximation works well is because one effectively assumes that, which is not a bad approximation for the low-energy effects in solids considering that the plasma frequency is often 10s of eV.Ultimately, the fact that the low-energy electrons near the Fermi energy are well-screened by other electrons self-consistently permits one to use the independent electron approximation — the foundation upon which band theory is built. Therefore, in many instances that the independent electron approximation is used to describe physical phenomena in solids, it should be kept in mind the hidden role the plasmon actually plays in allowing these ideas to work.Naively, from my discussion above, it would seem like the independent electron approximation would then break down in a band insulator.
However, this is not necessarily so. There are two things to note in this regard: (i) there exists an “interband plasmon” at high energies that plays essentially the same role that a free-carrier plasmon does in a metal for energies and (ii) whether the kinetic or Coulomb energy dominates will determine the low energy phenomenology.