Spin and wavefunction of Excitons

Question

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 werent 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 thats 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 $|2,1,-2\rangle$ (n=2,l=1,m=-2)

2) $$|3,2,2\rangle \Rightarrow |1,0,0\rangle + 2 photons $$ is this possible ?
Or is just the following possible: $$|3,2,2\rangle \Rightarrow |2,1,1\rangle + 1 photon \Rightarrow (|1,0,0\rangle + 1 photon) + 1 photon $$

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)

Answer 1

Excitons are quasi-particles made by a bound electron-hole pair. In general one refers to exciton as spin conserving excitations. In case spin is a good quantum number, the spin of the state in which the hole is created in valence is the same as the one in which the electron goes in conduction. As a result the spin of the exciton is zero. Thinking to the exciton in terms of the hydroge atoms, this means that the excitonic wave-function is a singlet. This kind of excitons are the only ones which are optically bright at equilibrium, i.e. they can measured in the absorption spectrum of a material.

More in general one can speak about excitons for any bound particle-hole pair and thus also include pairs which have triplet character. These excitons are optically dark. The overall spin of the state is 1 and it is more correct to call this kind of states magnons.

If spin is not a good quantum number, then in general excitons and magnons can be mixed. For a theory paper you can have a look here: https://arxiv.org/abs/2103.02266

Instead, the nomenclature light hole and heavy hole is more related to the orbital angular momentum. Similarly to the spin case, one can think to the exciton orbital momentum in terms of the relative momentum between the electron and the hole. Similarly to the fact that an excitation must conserve the spin, there are selection rules for the orbital momentum for bright excitons. In general photons can change the orbital momentum of the electrons by one. To jump from l=2 to l=0 you need to photons. Now the intermediate state in the process can be real, but it can also be virtual (as it happens in non linear optics). In the case of a virtual intermediate state one can think in terms of a process where two photons are directly emitted.