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10 "peierls"-- Response of a Partially Filled Electron Band to Electric and Magnetic Fields

"peierls" explores the dynamics of the degenerate electron gas of electrons in electric and magnetic fields for electrons moving in a periodic potential. The principal display shows the momentum (k) space trajectories of individual electrons within the Brillouin zone. A second display may be called which illustrates the hole representation of the same system. The Fermi energy may chosen to fill the band to any desired degree. The exclusion principle is obeyed in the scattering.

In the first example you see the response of the system to crossed electric and magnetic fields. This is a momentum (reciprocal) space representation. The blue square is the boundary of the Brillouin zone and the red line the contour in k-space of the Fermi energy (the Fermi surface). The white dots (displayed for three successive cycles of the simulation) represent electrons which initially occupy only states which lie below the Fermi energy. The trajectories are counter-clockwise circulating circles, slightly distorted by the band structure, with centers displaced downward by an amount proportional to E/B. The electric current can be thought of as the net effect of those states OUTSIDE the Fermi surface which have been filled, plus the states INSIDE the Fermi surface which have been emptied, by the fields. You can see the occasional return, by the scattering process, of electrons outside the Fermi surface to empty states inside. This picture for band electrons, with a small Fermi energy, is like that of the Sommerfeld model for the free electron gas.


The second example differs from the first by having a much larger Fermi energy so that the band is nearly full. One frame of the simulation is shown at the RIGHT below. There is a wide variety of orbits, some (near the center of the zone) which are free electron like, but others near the Fermi energy which are quite complex. It would appear that a description of electron transport in such a material would be quite complicated.

The figure at the LEFT below illustrates the power of the concept of the "hole". Here, instead of displaying the motion of occupied states, we display the motion of the empty states (with a subtle but important inversion in k). Second, we take advantage of the periodicity of E(k) surfaces in reciprocal space to displace three of the quadrants of the Brillouin zone to share a common corner. This corner is then shifted to the center of the display. This "hole" version of the second example has all of the simplicity of the nearly empty band in the first example. Note that the circulation of the "particles" (called "holes") is in the clockwise direction as appropriate for positively charged particles with positive mass.

. . . . . . . . .
Hole representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron representation

If you have time to load an 65K animation , you will see both representations evolving together.


Table of Contents for Chapter 10 of "Simulations for Solid State Physics"
  1. Introduction
  2. Nearly empty band
  3. Half full band
    1. Qualitative
    2. Quantitative**
  4. Nearly full band: holes
    1. Electric field
    2. Magnetic field
  5. Anisotropy*
  6. Summary
  7. Appendix: "peierls" -- the program
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