# What is the momentum of a hole in a semiconductor?

I've been playing around with some ideas of electron-hole pairs in semiconductors lately, have realized that I'm confused about some basic conventions that maybe the physics forum community could help clear up.

Let's imagine that we have a direct gap semiconductor initially at zero temperature. I shine exactly one photon on the system and it excites an electron-hole pair such that the final state of the system now contains an electron with energy E1 in state "k" in the conduction band, and a vacancy at energy -E2 in state "k" in the valence band (energies are relative to the chemical potential and we'll assume that the photon contributes a negligible amount of momentum to the problem).

My question is: according to standard conventions, what is the crystal momentum of the hole that has been created in the valence band? If I were to draw an analogy to electron-positron creation in particle physics, it would seem that the momentum ought to be "-k", but in the textbooks I've read so far, people seem to prefer leaving the hole at positive "k" and negative energy. If the latter convention is the case, wouldn't that seem to rather mangle up the standard equations for conservation of energy and momentum?

## Answers and Replies

DrDu
Science Advisor
Which textbook did you read? Maybe you could provide a quote?

I've mostly been looking at Ascroft & Mermin, pp. 225-229, but also Ziman (Principles of the Theory of Solids), pp. 184-186. There's a bit too much material to quote, but after closer inspection, I guess Ashcroft and Mermin make no statements about the sign of the energy of holes, though their momentum conventions imply that the incoming photon in my thought experiment above would both have momentum "k". Ziman appears to adopt a convention where the signs of hole energies are flipped, but crystal momentum values are not (see Fig. 105 on p. 184), which would again imply that both the electron and its corresponding hole in the thought experiment would have momentum "k".

DrDu
Science Advisor
A&M are not too clear on that point, but after eq. 12.26 they mention the switching of a and k's sign.