PHYCS 498CQM: Homework 4

1. Get the files for the calculations from the WWW pages. Carry out a calculation for silicon to see if it matches the one given in the sample output. You may need to change the order of the input file. No need to turn in anything.
2. Change the input files to treat the "empty lattice", i.e., no potential. Do they agree in shape with the bands shown in Thijssen for fcc Al. Are they similar to the real bands in silicon. Turn in the free electron band plot along with short answers to the questions.
3. Carry out a calculation for a 1D Matthieu potential for which the exact solution is known. (See notes with lecture 14 on the WWW pages and passed out in class. Also the Handbook by Abramowitz and Stegun if needed.) Turn in a plot of the 1 dimensional bands from k=0 to k = pi/a and show that they agree with the known results (Abramowitz and Stegun) at the k=0 point.
4. Carry out a calculation for Gallium Arsenide using the input file given. Change the lattice constant (a change of 1% is reasonable) and determine if the band gap increases or decreases with pressure. Experimentally it is found that V(dEgap/dV) is around -10 eV. (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. Give your result for this derivative.
5. Distort the lattice in two ways as listed below. In each case you should find that bands shift and certain degenerate states split. For each case turn in a plot of the bands to turn in.
• 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(exx - eyy - ezz) where exx is the fractional change in the x length, etc., and the value of b is around -2.0 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. Turn in a graph and comment on the change. No need to compare with experiment.
6. Carry out a calculation for metallic hydrogen at a high density which is thought to exist in the heavy planets like Jupiter. Choose an fcc crystal with one atom per primitive cell and a density so that the electron density parameter in atomic units is r_s = 1.0 (This corresponds to a pressure around 10 Mbar.) For the potential choose the Coulomb potential screened by an approximate dielectric function for a homogeneous gas with density rs = 1, i.e., V(q) = (factor/q**2)/epsilon(q), where epsilon(q) = 1. + (ko/q)**2, with ko the Thomas-Fermi wavevector ko = 0.815 kF sqrt(rs). See Ashcroft and Mermin, p. 342. Plot the bands along different directions and estimate the Fermi energy relative to the bottom of the band. You do not need to do this accurately - just an estimate!