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

[BeauGeste]

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.

 

[drummerguy]

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)).

 

[charng]

The cyclotron effective mass is a measure of the effective mass of an electron in a magnetic field. It is used in solid state physics to study the behavior of electrons in a crystal lattice under the influence of a magnetic field. Ashcroft and Mermin derived the expression \sqrt{\frac{det M}{M_{zz}}} for the cyclotron effective mass on page 571 of their book "Solid State Physics". This expression is derived from the cyclotron frequency, which is related to the energy contours of the material. The authors do not specify which energy contours this expression is valid for, but it is generally applicable to all energy contours, as long as the material has a well-defined Fermi surface.

To understand how Ashcroft and Mermin arrived at this expression, we must first understand the concept of effective mass. In a crystal lattice, electrons behave as if they have a mass that is different from their free electron mass. This is due to the interactions between the electrons and the lattice ions. The effective mass is a measure of this mass.

In a magnetic field, the energy levels of electrons are quantized into Landau levels. The energy contours, or Fermi surfaces, are the boundaries between these Landau levels. The cyclotron frequency is related to the curvature of the energy contours. The effective mass can be calculated using the curvature of these energy contours and the cyclotron frequency.

Ashcroft and Mermin use the concept of the tensor effective mass, which takes into account the anisotropy of the material. The expression for the cyclotron effective mass is derived by taking the square root of the determinant of the tensor effective mass divided by the component of the tensor in the direction of the magnetic field (M_{zz}). This expression is valid for all energy contours because it takes into account the anisotropy of the material.

In summary, the expression \sqrt{\frac{det M}{M_{zz}}} for the cyclotron effective mass is a general expression that is valid for all energy contours in a material with a well-defined Fermi surface. It takes into account the anisotropy of the material and is derived from the curvature of the energy contours and the cyclotron frequency.