Question

I am new to reading into the k⃗ ⋅p⃗

approach. Where for a periodic function uk⃗
, which satisfies the Schrodinger equation:
Hk⃗ uk⃗ =Ek⃗ uk⃗
which for a system parabolic-like system the Hamiltonian can be written as:
Hk⃗ =H0+Hk⃗ ′=H0+ℏ2k22m+ℏk⃗ ⋅p⃗ m

If I am particularly looking at a semiconductor system comprised of a single generic valence and conduction band that is non-degenerate and I want to find the dispersion Ek⃗
. The Hamiltonian reads as:
H=⎛⎝Ecℏm0k⃗ ⋅p⃗ vcℏm0k⃗ ⋅p⃗ cvEv⎞⎠
From this expression how would I obtain Ek⃗
, would it be simply the sum of the matrix above when diagonalised?

In addition, to the Hamiltonian above, if we wanted to include a magnetic field to the system where would it be included in this system? Would it be a component in the off-diagonal components?

I have reads that in the k⃗ ⋅p⃗
approach we can include the magnetic field by rewriting p⃗
in terms of the magnetic vector potential

Answer 1

To find the dispersion Ek, the Hamiltonian matrix needs to be diagonalized. Inclusion of a magnetic field can affect the off-diagonal components of the matrix.

Step-by-step explanation:

The dispersion Ek can be obtained by diagonalizing the Hamiltonian matrix. The Hamiltonian matrix given in the question contains off-diagonal terms that represent the coupling between the valence and conduction bands. When diagonalized, these terms will contribute to the dispersion relationship.

If a magnetic field is included in the system, it would generally affect the off-diagonal components of the Hamiltonian matrix. Specifically, the magnetic field can be included by rewriting the momentum vector p in terms of the magnetic vector potential.