# Spin and wavefunction of excitons

• I
• dabafsdf

#### dabafsdf

TL;DR Summary
understanding heavy holes and the wavefunction of those exciton who obtain a heavy hole
My Bachelor thesis is all around Excitons (specifially transitions between excitons of different energies). During my work I often had trouble with the spin and the wavefunction of them. Is there maybe some good (free) literature about the theory of excitons ? I found some books in the internet but they weren't for free. And my university hasnone.

1)I read about "heavy holes" that can have spin of +-3/2. How can these be created ? And for example a 3/2 hole bonds with a 1/2 electron. Then the Exciton has spin 2. How would the wave function of this look like ? I know that in 3D the wavefunctions of hydrogen can be used for the wavefunctions of the exciton. And that's why I think if there is a spin 2 Exciton, it can only exist at higher Energies E(n) (since a hydrogen atom only can have spin 2 when it's n is 3 or higher). But on the other side, my professor now wrote me that there is also a |2,1,-2> (n=2,l=1,m=-2) state of an exciton

2) |3,2,2> => |1,0,0> + 2 photons is this possible ?
Or is just the following possible: |3,2,2> => |2,1,1> + 1 photon => (|1,0,0> + 1 photon) + 1 photon

Or in Words:
If now there is an Exciton with spin 2 and n=3. Can it for example jump directly to n=1 with the emission of 2 photons (afterwards it will be an exciton with spin 0)? Or can it only First jump to n=1 with the emission of 1 photon (then it will be an Exciton with spin 1) and then to n=0 with the emission of another Photon (then it will be an Exciton with spin 0)

Last edited:

My Bachelor thesis is all around Excitons (specifially transitions between excitons of different energies). During my work I often had trouble with the spin and the wavefunction of them. Is there maybe some good (free) literature about the theory of excitons ? I found some books in the internet but they weren't for free. And my university hasnone.

The standard work would be Theory of Excitons by R.S. Knox, but it is from 1963 and therefore somewhat old-fashioned and not exactly easy to read. Are you doing theoretical or experimental work? Is your emphasis on excitons or spin physics?
Your further questions make it look like some basic bokk on semiconductor physics would be more helpful. Almost any book that covers semiconductor optics will treat excitons at some point.

1)I read about "heavy holes" that can have spin of +-3/2. How can these be created ?

Heavy holes with spin 3/2 are what you typically get in III-V semiconductors. It is not a general rule that holes need to have that spin, but as III-V semiconductors are very common, you encounter them quite frequently. What do you mean by "how can these be created"? You lift an electron from the valence band to the conduction band and the residual "empty" state in the valence band is the hole. In order to know what spin it has, you need to consider spin and orbit contributions. In III-V semiconductors, the valence band will usually have P-like character. So you get a contribution of 1/2 from the electron and a contribution of 1 from the orbital angular momentum of the valence band. These may add up to 3/2 (heavy hole and light hole) or 1/2 (split-off band). If the valence band had s-type character instead, the hole spin would be 1/2 instead.

And for example a 3/2 hole bonds with a 1/2 electron. Then the Exciton has spin 2.

No! Standard QM texts should treat addition of angular momenta. You may add them up to end up with states of spin 2 or spin 1. The former states are dark excitons, the latter ones are bright excitons.

How would the wave function of this look like ?
That depends. The exciton wavefunction consists of terms corresponding to the wavefunctions of the bands (usually p-like valence band and s-like conduction band for III-V materials) and the envelope wavefunction. The envelope wavefunction contains the orbital angular momentum in the relative motion of electron and hole with respect to their center of mass. This is analogous to the orbitals of the electrons in hydrogen. However, electrons and holes have almost the same mass in excitons, while the electron is much lighter than the proton in hydrogen, so the problem is a bit more difficult for excitons.

I know that in 3D the wavefunctions of hydrogen can be used for the wavefunctions of the exciton. And that's why I think if there is a spin 2 Exciton, it can only exist at higher Energies E(n) (since a hydrogen atom only can have spin 2 when it's n is 3 or higher). But on the other side, my professor now wrote me that there is also a |2,1,-2> (n=2,l=1,m=-2) state of an exciton

Actually, the spin 2 exciton is usually at lower energy compared to the spin 1 exciton.
It seems to me that you are mixing up spin, orbital angular momentum and total angular momentum. For example, the highest orbital angular momentum state you can get for hydrogen states with principal quantum number n is n-1, which means that you need n=3 to get l=2. However this is not the same as spin. Maybe it would be helpful to revisit this topic. And yes: |2,1,-2> states of excitons are possible. However, there are not that many materials where you see exciton states of principal quantum numbers larger than 1.

Or in Words:
If now there is an Exciton with spin 2 and n=3. Can it for example jump directly to n=1 with the emission of 2 photons (afterwards it will be an exciton with spin 0)? Or can it only First jump to n=1 with the emission of 1 photon (then it will be an Exciton with spin 1) and then to n=0 with the emission of another Photon (then it will be an Exciton with spin 0)

For the heavy holes discussed above, it s impossible to end up with spin 0 as such a state does not exist. Maybe it really helps to consult a good book. Spin Physics in Semiconductors by Dyakonov has a good very brief introduction in the first chapter. Besides that, it is helpful to understand k.p perturbation theory for calculating band structures beforehand as this enhances the understanding of the band properties. This is covered in many books, e.g. the standard semiconductor physics book by Cardona and Yu.

• • dchafatinos and berkeman
Thanks a lot for your detailed answer. I will have a look at the books you mentioned