next program--SSS homepage--program list--postings log

8 "bloch" - band structure of electron states in a periodic potential

"bloch" determines the energy band structure and the wave functions for one electron in a one dimensional periodic potential, with choices of square well, cosine, "atomic", and "molecular". Lattice constant, well depth and width are specified, the dispersion relation E(k) is displayed graphically and the energies at the edges of the bands are given. The wave function within a unit cell is displayed in a three dimensional plot as a complex amplitude versus z. Also plotted are the squared amplitude and the phase of the wave function over several cells.

"bloch" was programmed by Joerg Draeger and Russ Thompson.

The left graph gives in red the band structure for a symmetric square-well potetial of depth 10 eV. In blue is the free electron dispersion relatioin, with the origin of energy taken as the average value of the potential. On the right, the upper half of the graph gives the square of the wave function, in four successive cells of the crystal, for two different energies in the fifth energy band. The lower half of the graph shows the potential. Two puzzles. Why for both energies (and for other energies in the band as well) is the electron more likely to be found in the barrier than in the well? How can it be, that for the blue energy, the solution within the well is a pure traveling wave: i.e., |psi| independent of position?
. . . .

"bloch" can also display wave functions in a 3-dimensional format. The left figure gives a 3D plot with the real axis blue, the imaginary axis green, and the black axis the position within the unit cell. The figure may be rotated to allow visualization in a variety of ways. On the right we see the square of the wave function for a forbidden state, a state with energy in one of the energy gaps.
. . . .
Table of contents for Chapter 8 of Simulations for Solid State Physics
  1. Introduction
  2. Solutions in one dimension
    1. Bloch's theorem
    2. Energy bands and gaps
    3. Finding the Bloch solution**
  3. In-band states
    1. Dispersion relations
    2. Wave functions
    3. Ramsauer effect**
  4. Gap states*
  5. Nearly free electrons (NFE)
  6. Tight binding (TB)
    1. background
    2. Square well potential
  7. `Atomic' and `molecular' potentials
    1. Atoms
    2. Molecules*
  8. Summary
  9. Appendix: "bloch" -- the program
next program--SSS homepage--program list--postings log