Electronic band structure¶

Attention

This tutorial is under construction.

Single-band model¶

2-band model¶

3-band model¶

8-band model¶

Output of effective mass in the multiband case¶

Rescaling of \(\mathbf{k} \cdot \mathbf{p}\) parameters (for multiband)¶

When diagonalizing the \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian for a given wave vector, if the coefficient \(S(L+1)\) of \(k^4\) in the secular equation is positive, two different \(k\) may correspond to the same eigenenergy. One is the expected correct solution, but the other is an oscillatory solution with a large \(k\), and a smooth wave function may not be obtained. To prevent this, the Material{ RescaleS } option rescales \(S\) to 0 (per default) while maintaining the effective mass of the conduction band.

The effect of rescaling on \(S\) and \(E_P\) is the following:

(4.2.1)¶\[S \to S' E_P \to E_P'\]

while the effective mass at bandedge is conserved.

(4.2.2)¶\[S + \frac{E_P}{E_g} = S' + \frac{E_P'}{E_g}\]

while for 3 bands it corresponds to:

(4.2.3)¶\[S + \frac{E_P(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})} = S' + \frac{E_P'(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})}\]