Physics 573: Problem set 4

1. A two-dimensional electron gas has a density of n electrons per area. Obtain the density of levels for the gas (you will find that it is a constant).
   Now obtain the chemical potential for the gas.

2. For free electrons in an FCC lattice, plot the energy bands along the [111] direction. The lowest band will reach the corner of the BZ with K = (π/a)(1,1,1) and will set an energy scale E₀ at that point. Plot all bands for energies up to about 6E₀. Notice that the bands cross in places that are not on the zone boundary These crossings will become gaps, band minima, etc once interactions are included, so it is perfectly possible to have gaps in the "bulk" of the BZ.

3. Calculate the crystal field U_K for a diamond lattice when K = 2b₁ (referring to the conventional cubic cell).

4. A square lattice has a crystal potential field
   
   U(x,y) = - 4U₀ cos{2πx/a} cos{2πy/a}.
   
   What is the gap at the corner of the BZ (π/a)(1,1)?

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