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Outline

1	Electron Energy Bands The interaction between the electrons and the periodic potential due to the ion cores gives rise to gaps in the energy of the electrons.
2	Felix Bloch
3	Energy Bands
4	One dimensional model: energy gaps
5	Periodic square well potential: Kronig-Penney Model
6	Energy gaps: Kronig-Penney Model
7	Energy eigenvalue vs wavenumbers for Kronig-Penney
8	Low-lying free electron bands: scc lattice [100] direction
9	Low-lying freee elecron energy bands of empty sc lattice
10	Eigenvalue spectrum in periodic zone scheme
11	Ratio of expansion coefficients
12	How many orbitals in a given band? Consider a linear chain (linear crystal) with an even number N of primitive cells. We know that with periodic boundary conditions the allowed values of the electron wavenumber k in the first Brillouin zone are integer multiples of
13	2N orbitals in each band
14	Metals and Insulators If the valence electrons completely fill one or more bands, leaving the others empty, the crystal will be an insulator The reason for this: if a filled band is separated from the next higher band by an energy gap, there is no continuous way to change the total momentum of the electrons by applying an external electric field.
15	Necessary (but not sufficient) condition to be insulator A crystal can be an insulator only if the number of valence electrons in its primitive cell is an even integer. If the primitive cell does have an even number of valence electrons, then one must next consider whether there is any overlap in energy bands. If there is, then there can be two partially filled bands, producing a metal
16	a)Insulator, b) Metal or semi-metal, c)Metal
17	Examples The alkali crystals and the noble element crystals have one valence electron per primitive cell. Therefore they be metals The alkaline earth metals have two valence electrons per primitive cell. They could be insulators. But the bands overlap, so they are (not very good) metals.
18	Examples Diamond, silicon and germanium each have two atoms of valence four There are eight valence electrons per primitive cell. The bands do not overlap and the pure crystals are insulators at absolute zero temperature.