Periodic Crystals: Solution of Schrodinger Eq. for independent particles in periodic potential
The lecture follows the Postscript file for Classnotes on Periodic Crystals

Outline

1. Periodic systems
   • Translation vectors: a(i,j) = a_i(j); j = vector components, i lables translation vectors
   • Bravais Lattice of translations T
   • Density and potential as examples of periodic functions
   • Fourier Analysis of periodic functions
   • Reciprocal lattice G - primitive translation vectors: b(j,i) = b_i(j)
2. Crystals: Periodic systems
   • A crystal is a periodic array of atoms
   • Types, positions of atoms specified by basis
   • The crystal is specified by the lattice of translations and the basis
3. Excitations in periodic systems
   • The Brillouin Zone, conservation of crystal momentum k
   • The Bloch theorem
   • Single particle solutions of Schrodinger Eq. in periodic system
   • Discrete eigenstates for each k - bands - e_n(k); n = 1,...
   • Solution of matrix (determinant) equations for each k separately
   • Independent particle solutions for electrons in crystals
   • Metals with a Fermi surface in k space
   • Insulators with filed bands and an energy gap at the Fermi energy
4. Solution of Schrodinger-like equations for given effective potential
   • Setting up problem in plane wave basis
   • Hamiltonian matrix
   • Kinetic energy diagonal (1/2)(k+G)²
   • Potential off-diagonal V(G-G')
   • Solution by diagonalization
5. Examples of types of solutions expected
   • One dimensional examples
   • Nearly free electron metals
   • Insulators like He
   • Semiconductors like silicon, diamond
6. Constructing programs: (start discussion in this lecture)
   Input needed:
   • translation vectors, a(i,j)
   • positions, types of atoms in unit cell
   • form of potential
   • number of electrons
7. Self-consistent solutions of local density equations
   • The density as a sum over squares of band eigenstates
   • The effective potential in the Kohn-Sham equations
   • Self-consistent solutions
   • Flow chart for full Self-consistent Kohn-Sham solution
   • Output: Density, total energy, eigenvalues
8. Simpler version
   Assume form of potential (or matrix elements of hamiltonian instead of doing full self-consistent calculation
   • Illustrates key points
   • Empirical pseudopotential
   • Tight binding