Calculating Electron Mass at Band Bottom & Brillouin Corner

[Cheetox]

Homework Statement

What is the effective mass of the electron at the bottom of the band and at the corner of the Brillouin Zone? What is the significance of these results? Atoms lie on a square lattice of side a.

Homework Equations

E(k_x,k_y) = C{1 - cos(k_xa) - cos(k_y)}

The Attempt at a Solution

I don't really understand how to use the effective mass tensor to get the result I need, can I somehow use the fact that the areas where I need to know the effective mass have high symmetry, e.g. the contours of constant energy are circular...

 

[Jhero]

Thank you for your question. The effective mass of an electron at the bottom of the band and at the corner of the Brillouin Zone can be calculated using the effective mass tensor. This tensor is a mathematical representation of the electron's mass and its behavior in a particular crystal lattice. In this case, we are dealing with a square lattice of atoms, so the effective mass tensor can be simplified to a 2x2 matrix.

To calculate the effective mass at the bottom of the band, we can use the following equation:

m* = ħ^2 / (∂^2E/∂k^2)

Where m* is the effective mass, ħ is the reduced Planck's constant, and ∂^2E/∂k^2 is the second derivative of the energy with respect to the wave vector k. This can be calculated using the energy dispersion relation given in the homework statement.

Similarly, for the corner of the Brillouin Zone, we can use the same equation but with the wave vector k corresponding to the corner point of the Brillouin Zone.

The significance of these results is that the effective mass of an electron in a crystal lattice is not a constant value, but rather depends on the crystal structure and its band structure. It can have different values at different points in the Brillouin Zone, as we have seen in this case. This has important implications in understanding the electronic properties of materials and their behavior in different conditions.

I hope this helps clarify your understanding of the effective mass tensor and its application in this problem. If you have any further questions, please do not hesitate to ask.