Electronic band structure

The theory of electron energy levels in solids is an application of the principles of quantum mechanics. In principle, the motions of electrons can be predicted by solution of Schrodinger's equation for the potential field of a particular arrangement of atoms in a crystal. Since a general solution is quite difficult, various simplifying assumptions are used to represent the actual system. A fundamental observation leading to the development of quantum mechanics is that the energy levels of an electron around an atom does not vary continuously, but instead occurs in discrete quantum states called "orbitals", each associated with an amount of energy. Another observation, stated as the Pauli exclusion principle, is that no two electrons can occupy exactly the same quantum state; so, not all the electrons of the atom fall into the lowest state, but occupy increasingly energetic "shells" around the atom. Putting two atoms together leads to delocalized orbitals across two atoms, yielding a partially covalent bond. Additional quantum states are possible, in this molecular orbital, with different energy levels. In a crystal, many atoms are adjacent and many energy levels are possible for electrons. Since there are so many (on the order of 1022) atoms in a macroscopic crystal, the resulting energy states available for electrons are very closely spaced. Since the Heisenberg principle limits the precision of any measurement of the co bination of an electron's momentum (related to energy) and its position, in a crystal effectively the available energy levels form a continuous band of allowed energy levels. The mathematical solution of the Schrodinger equation gives two kinds of solutions depending on the energy of the electrons. One type of solution represents an electron moving indefinitely through the crystal as a plane wave; the particular solutions for a periodic regular crystal lattice are called Bloch functions. A second type of solution occurs for energy levels in the so-called "forbidden" gaps between "allowed" states - in this case, the electron cannot travel indefinitely through the crystal with that energy and will either be reflected at the edges of the region, or possibly must pass through the region in a phenomenon called "quantum tunnelling". For semiconductor materials, one band of "allowed" electron energies is called the "valence band" - these can be thought of electrons bound to a particular atom. A higher-energy band is called the "conduction band", where electrons may travel through the crystal.[6] The energy of an electron may be increased by increasing its temperature or by applying an electric field to it. If a band of allowable energies is completely filled by electrons, it cannot carry any electrical current, because that would require the electron's energy to be increased. Conduction can only occur with partially filled bands.