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1. Bands for Si and Gallium Arsenide. See lecture notes.
2. Changes in GaAs bands with lattice constant.
   Answer: Experimentally it is found that V(dEgap/dV) is around -10 eV for GaAs. (Reference: Yu and Cardona, Fundamentals of Semiconductors, Springer Verlag, 1996, p. 118.) Here V is the volume per cell which is simply related to the lattice constant. If you are using Hartree units, note that 1 Hartree = 27.2 eV. In the graph below the energies are given in eV.

   

   Our calculations give the gap at k=0 to be:
   
   a = 10.6769569, Egap = 9.13899 - 7.95793  =  1.181 eV 
   
   a = 10.5769569, Egap = 9.69436 - 8.42144  = 1.273 eV 
   
   Thus V(dEgap/dV) = -.09/.03 = -3.  compared to the experimental value of -10.

3. Distort the lattice in two ways as listed below. In each case you should find that bands shift and certain degenerate states split.
• Distort the lattice with no change in volume by compressing along the x axis by 1% and expanding along y and z by 0.5%. Experimentally it is found that the highest occupied bands split by delta E = 3b(e_xx - e_yy - e_zz) where the strain e_xx is the fractional change in the x length (e_xx = -0.01 in this case), etc., and the value of b is around -2.0 eV.

Answer:
My calculation gives delta E = 7.970 - 7.862 = 0.108 eV or b = - 1.97 eV.

• Displace the two atoms in the unit cell along the 111 axis by changing the value of the internal parameter from .125 to a different value. Do the answers make sense? Do states split as expected?
  

  Answer:
  See postscript figure of bands for GaAs with atoms displaced to +- .12,.12,.12
  The results in the figure for .125 changed to .12 (i.e. .125 - delta/2) show that the bands at the top of the valence bands split. This is the splitting due to the fact that the crystal is not cubic. The one band that goes down is has weight centered on the bond that gets stronger (shorter); the two that go up are linear combinations of the three bonds that get weaker (longer). The 2-fold degeneracy is due to trigonal symmetry around the 111 axis when the atoms are displaced. The splitting is around delta E ~ 0.7eV or delta E/delta ~ 7eV.

4. An example of a calculation is the free electron or "empty lattice" case. This is easily done by editing a Si input file an replace the species label "Si" with the label "El" which denotes an "Empty lattice" with zero potential.
   See the postscript figure of free electron bands at the density of Si. .
5. Discussion of the conjugate gradient minimization using an FFT is given in the book by Martin and in the material in the ABINIT tutorials.