Semiconductor crystals are composed of a large number of atoms with nuclei and electrons. A solution of the Schrödinger equation of this manybody problem would yield the electron states and their characteristics for a given arrangement of the atoms. The equilibrium arrangement would be found by varying it until the minimum energy configuration is located. Semiconductor band theory simplifies this many-body problem to a single electron problem. The Born–Oppenheimer adiabatic approximation takes account of the fact that electron masses are small compared to those of nuclei and therefore electrons respond more rapidly than nuclei to changes in local potentials. The electrons are divided into two classes, core electrons that are localized on their nucleons to form ions, and valence electrons that move through the solid lattice. The adiabatic approximation is the basis of a theory in which the movements of the valence electrons can be decoupled from those of ions. In this way, the many-body problem is reduced to a many-electron problem with the electrons moving in the localized potentials of the ions and the other valence electrons. Then the Hartree-Fock approximation, in which each electron is assumed to be in the fixed potential field of all the lattice ions, and in the Coulomb potential of an average distribution of all other electrons, reduces the many electron problem to a single electron problem. Because the potential distribution from both the lattice ions and the averaged Coulomb interaction from all other electrons is periodic, the problem is reduced to the movement of an electron in a periodic potential.
KeywordsBand Structure Brillouin Zone Heavy Hole Secular Equation Bravais Lattice
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