Regions in the electronic band structure

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Discussion Overview

The discussion revolves around the electronic band structure of semiconductors, specifically focusing on the implications of electrons occupying different valleys (L, X, Gamma) in the band structure. Participants explore concepts related to momentum, current flow, and the statistical behavior of electrons in these materials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the physical meaning of electrons being in the L or X valleys versus the Gamma valley, suggesting that non-zero momentum implies a directional movement of electrons even without an applied electric field.
  • Another participant clarifies that while electrons in a specific valley may move in the same direction, the presence of multiple valleys leads to no net current when averaged across all valleys.
  • A further contribution reflects on the behavior of electrons at thermal energy, proposing that there may be significant numbers of electrons moving in one direction, yet they are balanced by an equal number moving in the opposite direction, raising questions about the interpretation of net current.
  • One participant introduces the concept of Fermi-Dirac statistics, noting that at finite density, most electrons have high momentum, but the average momentum remains zero.

Areas of Agreement / Disagreement

Participants express differing views on the implications of electron movement in different valleys and the interpretation of net current. There is no consensus on whether the cancellation of currents is meaningful or "unnatural."

Contextual Notes

The discussion highlights the complexity of interpreting electron behavior in semiconductors, including assumptions about valley degeneracy and the effects of thermal energy on electron motion. The implications of Fermi-Dirac statistics are also noted but not fully resolved.

erst
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I'm thinking mainly within the context of the useful semiconductors here (Si, GaAs, etc.)...

What does it mean for an electron to be in the L or X valley instead of the Gamma valley? If Gamma is the k = 0 point, then momentum p = 0. Does that mean being in L or X means all the electrons there are moving in some particular direction in real space (since k ≠ 0, so p ≠ 0, giving them some non-zero velocity vector). Does that mean that in the indirect bandgap materials like Si, electrons are moving together in some direction even with no applied E-field?
 
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There are several L (and X) valleys which are energetically degenerate. Hence while all electrons in one valey move in the same direction, on the average over all valleys, there is no net current.
 
Ok, I see. But with just thermal energy, we have large numbers of electrons moving in a direction rather than randomly scattering about. Sure, they're canceled by another set of electrons doing the very same thing in exactly the opposite direction. But this just seems very "unnatural".

Physically in real space (just making up numbers in 1-D), do we have case of 1 A of current being canceled by -1 A, or is the net result of 0 A the only meaningful interpretation?
 
Hm, finally this is a consequence of the Fermi-Dirac statistics. At finite density most electrons move at a high momentum whose absolute value is known as the Fermi-momentum. Only on the average the momentum of the electrons is zero.
 

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