How Do Ashcroft and Mermin Derive the Cyclotron Effective Mass Expression?

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SUMMARY

Ashcroft and Mermin derive the cyclotron effective mass expression, represented as \(\sqrt{\frac{det M}{M_{zz}}}\), specifically for scenarios near maxima or minima in the band structure, where the dispersion is approximately quadratic. This expression is applicable to certain energy contours, particularly those that exhibit an ellipsoidal constant energy surface, as highlighted on pages 568 and 569 of their text. The assumption of a symmetric effective mass tensor is crucial for solving related problems, such as problem 12.3a.

PREREQUISITES
  • Understanding of band theory in solid-state physics
  • Familiarity with effective mass tensor concepts
  • Knowledge of quadratic dispersion relations
  • Ability to interpret constant energy surfaces in momentum space
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could anyone elucidate on how Ashcroft and Mermin get
\sqrt{\frac{det M}{M_{zz}}}
for the cyclotron effective mass (page 571)?
Is this true for all energy contours or only certain ones? And if that's the case which because they don't seem to specify?
thanks.
 
Last edited:
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It's from problem 12.2 - essentially it's valid near a maximum or minimum in the band (so you get an approximately quadratic dispersion). I haven't solved the problem yet, but I think the comment on page 568/569 about an ellipsoidal constant energy surface is key (in addition to the fact/assumption that the effective mass tensor is symmetric (this assumption is necessary to solve problem 12.3a)).
 

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