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Band Theory
Kronig-Penney Model
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K-P assumed there is a change in periodic potential surrounding the ion cores forming potential wells.
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Insulators
Here the Fermi energy E_F is at the midpoint between the valence band and the conduction band is empty.
At T=0, the valence band is filled and the conduction band is empty.
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Metals
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At T=0, all levels in the conduction band below E_F are filled with electrons while those above E_F are empty
For T>0, some electrons can be thermally excited to energy levels above E_F, but overall there is not much different from the T=0 case.
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Semiconductors
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However for semiconductors the band gap energy is relatively small so appreciable numbers of electrons can thermally excited in to the conduction band.
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Hole
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If multiplied by the electron charge and summed for all electron, rate of current change when applied with an electric field E is given by:
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Effective Mass
Same magnitude of electric field applied to both electrons in vacuum and inside the crystal, the electrons will accelerate a at different rate from each other due to the existences of different potentials inside the crystal.
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Thus, the mass of electron is different in vacuum and in lattice. The altered mass is called as an effective mass.
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