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22 Band Theory - Coggle Diagram
22 Band Theory
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Bloch’s Waves
the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as Ψ(𝑥) = 𝑒𝑖𝑘𝑥𝑢(𝑥) where 𝑢(𝑥) is a periodic function which satisfies 𝑢(𝑎 + b) = 𝑢(𝑥).
only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function 𝑢(𝑥) is also continuous and smooth
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Metals
At T=0, all levels in the conduction band below EF are filled with electrons while those above EF are empty
T>0, some electrons can be thermally excited to energy levels above EF, but overall there is not much difference from the T=0 case
However electrons can be easily excited to levels above EF by applying a (small) electric field to the metal
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Insulators
At T=0, the valence band is filled and the conduction band is empty
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Semiconductors
At T=0, the valence band is filled and the conduction band is empty
However for semiconductors the band gap energy is relatively small (1-2 eV) so appreciable numbers of electrons can be thermally excited into the conduction band
Hence the electrical conductivity of semiconductors is poor at low T but increases rapidly with temperature
Effective Mass
the electrons will accelerate at a different rate from each other due to the existence of different potentials inside the crystal
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Holes in semiconductor
particularly in dealing with semiconductors when there are a few unoccupied electron states at the top of the valence band
much easier to consider the few unoccupied states than the entire electron band minus these few states
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