Electrons in a Periodic Potential

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Electrons in a Periodic Potential

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Energy Bands and Energy Gaps in a Periodic Potential
Metals, Insulators, and Semiconductors
Energy Bands and Fermi Surfaces in 2-D and 3-D Systems
Bloch s Theorem
Using Bloch s Theorem: The Kronig-Penney Model
Empty Lattice Bands and Simple Metals
Density of States for a Periodic Potential
E(k) and N(E) for d-electron Metals
Dynamics of Bloch Electrons in a Periodic Potential
Effective Mass of Electrons
Electrons and Holes

Electron Wavefunctions in a Periodic Potential

Consider the following cases:

Wavefunctions are plane waves and energy bands are parabolic

Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above

Electrons wavelengths approach a, so waves begin to be strongly back-scattered :

Electrons waves are strongly back-scattered (Bragg scattering) so standing waves are formed:

Origin of the Energy Gap

In between the two energies there are no allowed energies; i.e., an energy gap exists. We can sketch these 1-D results schematically:

The periodic potential U(x) splits the free-electron E(k) into energy bands separated by gaps at each BZ boundary.