Credit:
The programs were written primily by W. Mattson, and the CG programs were written by D. Das.

Solutions
Solutions to problems are posted at
http://w3.physics.uiuc.edu/~rmartin/css/lab/lab-sol.html
These are available for you to compare with your results.

1. Follow instructions in the lab to access the programs.
2. Carry out a calculation for GaAs using the input file gaas.dat. At the prompt type:
   >pw
   >gaas.dat
   (The file si.dat is in the directory provided and also included below in these instructions.)

   To plot the bands type at the command prompt:
   >gnuplot gnuplot.dat

   Check to see if your result matches the one given in the sample output given in the lecture.

   Repeat the calculations for Si using the data file si.dat

   Note that the lowest state at the k=0 (Gamma or Ga) point is non-degenerate (it is the bonding state formed from s-like orbitals of each Si atom) and the next higher state at Gamma is 3-fold-degenerate (the bonding state formed from p-like orbitals of each Si atom).
   Compare the bands of Si and GaAs. Note similarities and differences. What do you think is the nature of the lowest band in GaAs?
   The band gap for both Si and GaAs is the smallest energy difference between the lowest empty band (#5) and the highest filled band (#4).

   The next two problems require editing the input file. Save the edited file with a different name in your directory.

3. 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 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. Note that energies are given in Hartrees = 27.2 eV.
4. 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(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. Do the answers make sense? Do states split as expected?
5. Carry out a calculation for metallic hydrogen at a high density which is thought to exist in the heavy planets like Jupiter. (Think about the problem yourself, but an input file input file h1.dat is provided that will run the calculation for the following case.)
   Choose an fcc crystal with one atom per primitive cell and a density so that the electron density parameter in atomic units is 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) = (4pi/vol)(1/q**2)/epsilon(q), where epsilon(q) = 1. + (k₀/q)**2, with k₀ the Thomas-Fermi wavevector k₀ = 0.815 k_Fermi sqrt(rs) = 1.56/sqrt(rs) (in atomic units: m=hbar=e=1). See Ashcroft and Mermin, p. 342. Plot the bands just as for Si or GaAs since this is also an fcc crystal. Is hydrogen nearly-free-electron-like at this density? Qualitatively, where is the Fermi energy? Can you estimate it from your calculations?
6. Carry out a calculation for Si bands using the conjugate gradients method. The input file that contains the commands is si-cg.dat. The answers should agree. This is a "worst case" test case where the CG iterations start from random vectors.
7. To show the effects of preconditioning do calculations for the k=0 point for Si using the two files given below. The preconditioning is the form suggested by Teter, et al.Phys. Rev. B 40,12255 (1989):
   K(x) = (27+18x+12x²+8x³)/ (27+18x+12x²+8x³ +16x⁴),
   where x = (1/2) |k + G|²/ T_iⁿ,
   with T_iⁿ = kinetic energy of state i at iteration n. In each case after doing the calculation the convergence of the eigenvalues versus iteration number can be graphed by typing:
   >gnuplot gnuplot.dat
   • Preconditioned: file si-cg-1-pre.dat
   • No preconditioning: file si-cg-1-no.dat
8. You can explore many possiblities with the empirical potentials given: "Si", "Ga", "As", "El" (no potential). This could include Si in the fcc structure that exists at high pressure, elemental As, etc.
9. Other potentials with anlaytic form can be added by editing the function atomPotential in Module AtomPotentialMod in the PW directory. If the files are edited they must be recompiled. This can be done only by access to the source code. See R. Martin or D. Das if you are interested.
10. Tight-binding calculations for Si are implemented in the data file. This can be done only by creating the executable file. See R. Martin or D. Das if you are interested. (Note that now the program ignores the iformation for the planewave calculation and looks only for the information relevant to tight binding identified by the keywords.)

Sample input files:
The following file works for Si (plane waves or tight binding) with ordinary matrix diagonalization. GaAs plane data given at the bottom is ignored. It will be read if it is first in the file.

***  Information for PW or TB calculation of Si

***  Information for lattice
NumberOfDimensions 3
LatticeConstant 10.2612170
LatticeVectors
0.0 0.5 0.5
0.5 0.0 0.5
0.5 0.5 0.0

*** Information for atomic positions
NumberOfAtoms 2
NumberOfSpecies 1
ChemicalSpeciesLabel
1 14 Si
(Chemical Species label (first number) assumed to be in sequential order)

AtomicCoordinatesFormat ScaledByLatticeVectors
AtomicCoordinatesAndAtomicSpecies
-0.125 -0.125 -0.125 1
 0.125  0.125  0.125 1

*** Information for k-points and number of bands to calculate
NumberOfBands 8
NumberOfLines 5
NumberOfDivisions 15
KPointsScale ReciprocalLatticeVectors
KPointsAndLabels
0.0   0.0   0.0      Ga
0.375 0.375 0.75     K
0.5   0.5   0.5      L
0.0   0.0   0.0      Ga    
0.0   0.5   0.5      X
0.25  0.625 0.625    U


*** Information for Plane Wave Calculation (atomic units assumed)
EnergyCutOff 6.0

*** Information for number of bands to include in density plot
NumberOfOccupiedBands 4

*** Information for Tight Binding Calculation
MaximumDistance 5.5
EnergiesInEV
TightBindingModelType  1   
OrbitsAndEnergies
4
0  0  -13.55  
1  1  -6.52  
1  2  -6.52 
1  3  -6.52


***  Information for PW calculation of GaAs

***  Information for lattice
NumberOfDimensions 3
LatticeConstant 10.6769569
LatticeVectors
0.0 0.5 0.5
0.5 0.0 0.5
0.5 0.5 0.0

*** Information for atomic positions
NumberOfAtoms 2
NumberOfSpecies 2
ChemicalSpeciesLabel
1 31 Ga
2 33 As
(Chemical Species label (first number) assumed to be in sequential order)

AtomicCoordinatesFormat ScaledByLatticeVectors
AtomicCoordinatesAndAtomicSpecies
-0.125 -0.125 -0.125 1
 0.125  0.125  0.125 2

*** Information for k-points and number of bands to plot
NumberOfBands 8
NumberOfLines 5
NumberOfDivisions 15
KPointsScale ReciprocalLatticeVectors
KPointsAndLabels
0.0   0.0   0.0      Ga
0.375 0.375 0.75     K
0.5   0.5   0.5      L
0.0   0.0   0.0      Ga    
0.0   0.5   0.5      X
0.25  0.625 0.625    U

*** Information for Plane Wave Calculation (atomic units assumed)
EnergyCutOff 6.0