Postings for `peierls': transport in a partially full electron band

• 10.2 -- 3/19/99 -- Bug in NEW version --but see update 1.11
  With the revision of the software which allows use with lower resolution screens we have also managed to add a few bugs into the programs. You may receive an error message when you try to switch between "show holes" and "show electrons" with the graph display open. To fix this problem download:
  • for Macintosh: peierls.tcl;
  • for Windows: peierls.tcl;
  • or for Unix: peierls.tcl;
  and replace the peierls.tcl file in your sss/lib/debye folder with this corrected file. (This bug is on the CD with the revised program and on earlier versions of our updaters which replace the original (high resolution only) version of the software with the newer version. It is NOT on the original SSS CD.)
  
  postings log.
• 10.1 -- 12/1/97 -- k conventions for holes
  You chose to follow Kittel's description of holes, and not to mention Ashcroft and Mermin (A&M) at all at that point.

  As far as I can guess, the reason is that there is, for once, an important difference between the two descriptions. A&M assign the hole exactly the same characteristics and behaviour as the possibly missing electron. Kittel and you, on the other hand, change the sign of k, which I suppose is associated with the change in convention for the energy of the hole, measured downwards from the top of the band in this description. Is this correct, or did I miss a point?

  -----MS

  I find it very difficult to argue with sufficient care about holes. They are VERY slippery. Fortunately, once we agree that the concept is a valid one, we can be quite careless in using it, without getting into trouble.

  I think that, in fact, A&M and Kittel are not all that far apart: it's just that Kittel carries the arguments a little more deeply than A&M do. A&M talk always about the behavior of electronic states in the applied field, and do not fully define a complete hole description. The telling argument, for me, for the negative k business is the following. If the electron bands are all full, the total wave vector of the crystal (the sum of the wave vectors of all the occupied states) is zero. If I remove an electron from the state (k-sub-e), then the total wave vector of the crystal is -(k-sub-e). If I wish to represent that state of the full crystal in terms of occupying a single "hole state", I must assign to the hole state the wave vector (k-sub-h) = -(k-sub-e).

  The need for this description is also seen in the simple optical absorption experiment. Consider the absorption of a photon (wave vector approximately zero), which excites an electron from the state k in the valence band to the state k in the conduction band, in a crystal that initially has all bands either full or empty. The wave vector of the crystal is essentially unchanged (we take k-photon = 0). In the pure electron language, there is an electron missing from k in the valence band and one occupying the state k in the conduction band, so the total k of the crystal remains zero. If we use the hole language for the valence band, we MUST associate the wave vector -k with the hole in order to get the sum of the wave vectors of hole and electron to be zero. (I am confident that A&M would support this argument.)

  I hope these remarks help rather than confuse.

  -----RHS

  postings log.