Semiconductor bandstructure computation

[supermanii]

Homework Statement

To plot the bandstructure of a free electron in silicon using python.

Homework Equations

We have been told that to do this we must use the 15 lowest magnitude reciprocal lattice vectors (so 2pi/A (0,0,0), (1,1,1), (-1,1,1) etc...). We are then told to setup a matrix with kinetic energy terms along the diagonal given by

[itex]\frac{h_bar²\left|k + g\right|²}{2*m*e}[/itex]

where g is the reciprocal lattice vector, k is the wavevector m is the mass of an electron and e is the charge on an electron. For the free electron case all potential terms are set to 0.

The Attempt at a Solution

I used python to loop over the k vectors between (0.5,0.5,0.5) to (0,0,0) to (1,0,0) (L to T to X) and plotted the result. I have attached the plotted bandstructure. This seems wrong particularly the convergence around T (k= 0,0,0). However I am quite sure it is not a programming issue as I have checked the program's results by hand. I think rather that I am misunderstanding the instructions or that the method is incorrect can anyone help shed some light on this? Also if anyone could show me an example plot of free electron silicon bandstructure? Thanks in advance

 

[NateD]

. A:If you want to plot the band structure of free electron in silicon, you should be using the Fermi-Dirac distribution function to calculate the probability of occupation of the energy states. The Fermi-Dirac distribution function is given by:\frac{1}{1+e^{(\epsilon_F-\epsilon)/k_BT}}where εF is the Fermi energy, kB is the Boltzmann constant and T is the temperature. You can then plot the band structure by plotting the probability of occupation of the energy states against the wave vector.