What is the momentum of a hole in a semiconductor?

• csmallw
In summary: I'm not entirely sure what they're getting at though.In summary, the crystal momentum of the hole created in the valence band in a zero-temperature direct gap semiconductor is not always given by the standard convention of "-k".

csmallw

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?

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".

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

What is the momentum of a hole in a semiconductor?

The momentum of a hole in a semiconductor refers to the motion and direction of the empty space left by an electron in the valence band. It is a fundamental property of the particle and plays a crucial role in understanding the behavior of semiconductors.

How is the momentum of a hole different from that of an electron?

Unlike electrons, which have a negative charge, holes have a positive charge. This means that their direction of motion is opposite to that of electrons. In addition, holes have a lower effective mass compared to electrons, meaning they can move faster in a semiconductor material.

How is the momentum of a hole related to the band structure of a semiconductor?

The band structure of a semiconductor plays a significant role in determining the momentum of a hole. As the electrons move from the valence band to the conduction band, they leave behind holes in the valence band. These holes have a specific momentum and can move freely in the valence band.

Can the momentum of a hole be controlled or manipulated?

Yes, the momentum of a hole can be controlled and manipulated in semiconductors. This is achieved through various techniques such as applying an electric field, which can alter the direction and speed of the holes. Additionally, doping a semiconductor with impurities can also affect the momentum of holes.

How is the momentum of a hole relevant in semiconductor devices?

The momentum of a hole is crucial in understanding the behavior of semiconductor devices such as transistors and diodes. It helps in predicting the movement of charge carriers and the overall performance of the device. In addition, it is also essential in designing and optimizing these devices for specific applications.