Momentum conservation in a direct transition

[hokhani]

When a transition from valence band to conduction band happens (in a direct-gap material), how the momentum is conserved while the group velocity of electron is changed considerably?

 

[Cthugha]

It is important to consider that we are dealing with a quasiparticle model here, where a lot of the microscopic physics goes into some parameters. Away from k=0 there may be a different group velocity at the same k in the conduction and valence band, but at the same time the effective mass will also be different.

 

[hokhani]

Away from k=0 there may be a different group velocity at the same k in the conduction and valence band, but at the same time the effective mass will also be different.

By this you mean that "group velocity times effective mass" is always conserved in the direct transition?

 

[Cthugha]

Yes, but one should take the usual caveats into account like crystal momentum only being conserved up to a reciprocal lattice vector. An electron with crystal momentum $\hbar k$ can be described equivalently to a free particle with momentum $\hbar k$, if we assign it the right effective mass. The effective mass is given by the curvature of the parabola of the band. Therefore, momentum conservation works in that easy way as long as effective mass is a good approximation, that is, when the bands are approximately parabolic at the transition points in k-space. This is usually the case in direct band gap materials around the band gap.

Total momentum is of course also conserved when this is not the case, but one cannot see it that easily.