Periodic potential electronic levels

The band structure that appears as a consequence of the periodic potential provides a logical explanation of the different conductivities of electrons in solids. It is a simple case of how the energy bands are structured and arranged with respect to the Fermi level. In general, for any solid there is a set of energy bands, each separated from the next by an energy gap. The top of this set of bands (the valence band) intersects the Fermi level and will be either full of electrons, partially filled, or empty. 

If many atoms are bound together, for example in a crystal, their atomic orbitals overlap and form energy bands with a high density of states. Different bands may be separated by gaps of forbidden energy for electrons. The calculation of electron levels in the periodic potential of a crystal is a many-electron problem and requires several approximations for a successful solution. 

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 current control at the one-by-one electron accuracy level is feasible in mesoscopic devices due to quantum interference. Though the electric charge is quantized in units of e, the current is not quantized, but behaves as a continuous fluid according to the jellium electron model of metals. The prediction of the current quantization dates back to 1983 when D. Thouless [Thouless 1983] found a direct current induced by slowly-traveling periodic potential in a ID gas model of non-interacting electrons. The adiabatic current is the charge... 

As a consequence, Anderson chose two levels of sophistication for treating the electrons belonging to the magnetic ions and the ligand, respectively. The first level consists in treating the extra electrons in terms of the simple one-electron Hartree-Fock functions while the second is treating them as excitations of a many-body system. Thus, Anderson considers the extra electrons in the periodic potential of the nuclei and the core electrons as... 

Appendix 6.D Electronic Levels in Periodic Potentials and Bloch s Theorem.160... 

The analysis of the electronic levels in a periodic potential can be first treated in 1-dimension. The advantage is obtaining an exact solution to the problem and not an approximated one as in the generalization. Suppose that we have a periodic crystal lattice of a periodic potential energy of the form... 

APPENDIX 6.D ELECTRONIC LEVELS IN PERIODIC POTENTIALS AND BLOCH S THEOREM... 

The development of correlation schemes at the highest levels of theory (the CCSD(T) technique) allowed for very accurate DCB predictions of atomic properties for the heaviest elements up to Z=122 (see Chapter 2 in this book). Reliable electronic configurations were obtained assuring the position of the superheavy elements in the Periodic Table. Accurate ionization potentials, electron affinities and energies of electronic transitions (with the accuracy of below 0.01 eV) are presently available and can be used to assess the similarity between the heaviest elements and their lighter homologs in the Periodic Table. 

Thus, the periodicity of the density of probabilities (3.80) was lowered at the level of eigen-function such as Eq. (3.88), regaining the celebrated Bloch theorem of Eq. (3.34), here in a generalized form the eigen-function of an electron in a periodic potential can be written as a product of a function carrying the potential periodicity and a basic exponential factor exp(/lx) . 

College, Philadelphia, 1976, Chapter 8, p. 135. Electron Levels in a Periodic Potential. 

We will next employ a slightly more elaborate model to argue that surface states are localized near the surface and have energies close to the Fermi level. We consider again a ID model but this time with a weak periodic potential, that is, a nearly-free-electron model. The weak periodic potential in the sample will be taken to have oidy two non-vanishing components, the constant term Fq and a term which... 

Fig. 2.8 EJlectrons in the periodic potential of the nuclei ( ), schematic (e.g. Na). bmer electrons (in the case of Na Is, 2s, 2p) may be compared to electrons at an isolated atom, outer electrons (3s) are quasi-free. The sharpness of the levels of the inner electrons has been exaggerated for the purpose of illustration.

---