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.

Answer 2

Classical consistent treatment of excitons can be found in Theory of excitons by Knox.

A typical treatment of excitons starts with assuming non-interacting electrons in a valence band, one of which is excited to the conduction band. The Coulomb interaction is then imposed to calculate the binding energy and other properties. One can work further adding various details of (non-interacting/one-particle) band structure, interactions between the pairs of excitons, etc., but this doesn't cure the fact that the initial premise is wrong.

All the electrons in the valence band interact via the Coulomb interaction, and the exciton is an excitation of this interacting multi-electron system. Know's approach is to take a step back and redo the band structure derivation in terms of Bloch waves, taking into account the Coulomb interaction. He then shows that, under some conditions, one can reduce the energy calculations to that of a hydrogen-like atom - the result no less striking then, e.g., Landau Fermi liquid description of an interacting electron gas in terms of non-interacting quasi-particles(see, e.g., Why do Drude/Sommerfeld models even work?.)

Looked from this perspective, the spin of an exciton is not a combination of spins of two constituting particles plus the orbital momentum (as would be the case for a true hydrogen atom), but rather a spin of the whole interacting electron system. It is therefore in principle is undefined, since we assume the crystal to contain an infinite number of electrons (even for periodic boundary conditions, the number of electrons is uncertain - we can't even say whether it is even or odd.)

In problems where the exciton is tied to an optical excitation, we can compare its spin to that of the band before the excitation - knowing that the photon carries spin 1. In the naïve theory starting with non-interacting electrons and holes, this is then collapsed onto the optical selection rules for the heavy and the light holes or something similar. The absorption of the following photons, is then supposedly governed by the selection rules of the hydrogen atom.

Related:
Can an electron have a spin opposite to the hole?
Parallels between Tight-binding wavefunctions and Bloch states
Excitons - Increase in absorption