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9 "ziman" - One electron dynamics of a band electron

"ziman" displays single electron orbits, in both k-space and real space, for electrons moving in a two dimensional periodic potential. An electric field may be applied in any direction in the plane, and a magnetic field perpendicular to the plane. The E(k) dispersion relation is illustrated by contour lines in the k-space plots. The crystal lattice is taken to be square, but the band structure can be altered to give an x-y anisotropy which leads to the possibility of open orbits. In the following examples single frames show portions of one electron trajectories. The green lines are constant energy contours of the energy band in which the electron moves. A click on any of the figures will give a full size version of the same figure.
. . . . . . 
             r-space                             k-space

The simple example of the one electron trajectory in real (on the left) and k-space (on the right) for an electric field illustrates a number of ideas. Analysis of the real space orbit reveals first the relation between the electron group velocity and the gradient (with respect to k) of the energy E(k). A more careful examination reveals a real space acceleration describable in terms of positive and negative effective masses in different portions of the trajectory. Perhaps the most important feature is that, whatever the complexity of the real-space orbit, the electron moves in k-space at a constant rate, though the "Bragg reflections" at the zone boundaries are an essential complication. An appropriate choice of field direction allows a simple demonstration of Bloch oscillations.


The next example is a series of pairs of orbits (real and k-space) in the presence of a magnetic field perpendicular to the plane. Clicking on the frame buttons gives orbits at successively higher energy. The number pair labels are the initial (kx, ky) of the electron in units of pi/(lattice constant)

start (30K)

Not surprisingly, these follow the constant energy contours in k-space. In real space we see the qualitative change in character from closed to open orbits, in animation the change from counter-clockwise to clockwise circulation is evident on moving from the bottom (positive effective mass) to the top (negative effective mass) of the band, and the demonstration that the real and k-space orbits are images of one another, with a rotation of 90 degrees.


Next is an animation of the k-space trajectory of an electron in crossed electric and magnetic fields to show the complexity of the orbits which arise in this situation. Surprisingly, it is possible to analyze some features of this example quantitatively.

movie (220K)....here is the final frame from the movie:


The final pair of frames shows the real and k-space trajectories in the crossed fields for about 4.5 cycles of the periodic motion.

. . . . . . 

             r-space                             k-space
Table of contents for Chapter 9 of "Simulations for Solid State Physics"
  1. Introduction
  2. E(k) and the contour plot
  3. Group velocities
  4. Acceleration in electric fields
    1. Acceleration theorem
    2. Effective mass
    3. Band anisotropy*
  5. Magnetic field
    1. Orbits of positive and negative mass
    2. Anisotropy and real space orbits*
  6. Crossed fields**
  7. Summary
  8. Appendix: "ziman" -- the program
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