Excersise 1.2 semiconductors

In excersise 1.2 of semiconductors, the solutions say that the group velocity of the holes should be negative. But i dont understand why this is the case. Should it not be positive? since the derrivative of the hole energy E_h is upwards (negative of E)?

kind regards,
thim

Hi Thim,

I think it’s still negative as you’re looking at a hole in the valance band. Even though you’re talking about holes, it’s still in the valence band which goes to minus infinity for k>0. So I think you’re confusing holes in the valance band with holes in the conduction band where the group velocity is positive.

I’m not a TA but I hope this helps!

To expand on this: the group velocity of holes may be both positive and negative, and it depends on which hole we’re looking at. In the exercise we’re looking at a hole with k>0 and in the valence band. This may mean two things: a hole that is created after removing an electron with k>0. This hole has k<0. Alternatively, the question may refer to a hole that has k>0. The question was ambiguously formulated and referred to the first case.

The dispersion relation of holes in the valence band is

E_h = -E_v + \hbar^2 k^2/2m_h,

which means that the group velocity is v_g = \hbar k / m_h, so it is negative in the second case and positive in the first case.

I have clarified the exercise to refer to the second case (hole with k>0).