# How to understand the spin of exciton?

In very recent years, exciton physics has been intensively explored as the emergence of 2D materials. Exciton is a bound electron-hole pair and is bosonic, namely hosting an integer spin.

• My first question is what the integer spin is? We all know the spin of an electron is $$\dfrac{1}{2}$$ and the hole will hold a positive charge compared to the kicked electron. Can we consider that the spin of the hole will be $$-\dfrac{1}{2}$$ and the exciton holds an integer spin $$0$$.

• My second question is how the formation of the exciton is affected by the spin of electrons? For example, the following figure from this paper indicates that the spin is closely related to the formation of the exciton.

• My third question is how can we demonstrate these spin-related exciton landscapes with first-principles calculations?

• Interesting question. Ch 9 of Cohen and Louie's Fundamentals of Condensed Matter Physics has a brief explanation on the spin: basically it can be either a singlet ($S=0$) or triplet ($S=1$) state. Perhaps that helps address the first two questions.
– tmph
Feb 9 '21 at 2:59
• +1 but you got lucky with Xivi76 answering all three questions right away. Otherwise I woukd have asked you to follow the "1 question per post" rule. Feb 9 '21 at 3:40
• @NikeDattani Questions (1) and (2) are definitely inter-related. I'm not completely conversant with GW-BSE, so I wouldn't be able to do justice to question (3) - If OP wants to know how GW-BSE works, it should be a separate question for sure. Feb 9 '21 at 4:43
• @Xivi76 I would have liked question (3) to be asked as a separate question, and your paragraph about it in your answer, could have been a comment or slightly extended answer. On some SE sites like the quantum computing stack exchange, the question would have been closed right away as being "too broad" because they are much more strict about multi-question posts. Here I would still like all posts to ask one question (within reason) but don't believe we need to close-vote to get there: I trust the users to do this one their own. The reasons why we want one question per post are large in number! Feb 9 '21 at 5:26
• @NikeDattani I will consider your suggestion.
– Jack
Feb 11 '21 at 7:56

As you mentioned in the question, excitons are indeed bound electron-hole pairs. They are often considered to be the signature of optical spectra in insulating solids. Adding onto the comment by tmph, there are two types of excitons: opposite spins of electrons will lead to a dark exciton with S=0 (since it doesn't allow for momentum conservation). Same spin of electrons will lead to a bright exciton with S=1 (since it allows for recombination). Therefore the electronic band-structure can often serve as an indicator for optical selection rules of excitons. The dipole strength (that dictates the relative strength of 'bright' or 'dark') is a term that is basically a superposition of dipoles from DFT (read: Fermi's golden rule). The dark exciton is not trivial though, it has been observed in experiments like this one.

Regarding the third part of your question, the method of choice to model excitons seems to be the GW-BSE method, since the exciton itself is inherently an excited state. But the mean-field starting point is Kohn-Sham DFT in most GW-BSE calculations.

Edit: Thanks to Anyon for pointing out a class of excitons with S=2 occurring in certain semiconductors, this should potentially be considered as well.

• +1 for answering all three questions! But why is GW-BSE the only option, what about TD-TDFT or excited-states using Monte Carlo or plane-wave coupled cluster? Feb 9 '21 at 3:38
• Sometimes one also encounters $S=2$ dark excitons. This can occur in e.g. semiconductor band structures with "heavy holes" that have spin-$3/2$. Feb 9 '21 at 3:39
• @NikeDattani Maybe I should the remove the word 'only', what I have noticed is GW-BSE is the method of choice for studying excitonic spectra. Anyon, thanks for pointing that out, I will edit it into the answer. Feb 9 '21 at 4:38
• I think you may have the spin counting the other way around? I agree that the electron doesn't change the spin as it is excited, but this means that you go from a, say, spin up valence electron to a spin up conduction electron, leaving a hole behind with the opposite spin. So in fact, spin singlet is the bright exciton and spin triplet the dark one. This is also why, for example, singlet excitations undergo fluorescence whereas triplet excitations undergo phosphorescence. Feb 10 '21 at 10:32
• @ProfM Yes, that sounds right. Parallel electron spins say in VB and CB would be equivalent to Anti-parallel electron and hole spins (between electron in CB and hole in VB). I felt this was self-explanatory, and also, I've not seen many texts that mention the 'spin of a hole'. Therefore, I just explained with electrons. Feb 10 '21 at 17:25