Symmetry in Indirect exciton in presence of phonons

[sitangshu]

Dear developers,

I have a 2D hexagonal crystal with C3v (3m) point group in which I am studying the PL emission. I realised that the Q=0 exciton has E-type symmetry.
I noticed that the indirect (lowest) exciton is located between K-\Gamma of the exciton dispersion. The phonon transferred momenta (q) is between \Gamma-M, where the symmetry is C_s(m). From the character table of C3v(m) and C_s(m), I could find the C3 axis of C3v (3m) is equivalent to mirror axis (\sigma) in C_s(m). However, I am not able to understand the "Wonderful Orthogonality Theorem" in this case from Dresselhaus book, which can tell me about the information about the dipole operation as discussed in Phonon assisted spectra of hBN: https://journals.aps.org/prb/abstract/1 ... .99.081109
Note that, in the phonon assisted DOS (https://www.yambo-code.eu/wiki/index.ph ... emperature), I am getting phonon replicas with finite contribution from out of plane phonon modes (both ZA and ZO).
For D_6h case ZO is zero, but is it true for all?
Can anyone suggest this issue?

Regards,
Sitangshu

[claudio]

Dear Sitangshu

I'm not an expert on symmetries of phonon-assisted excitons

try to have a look in the Supp Info of
https://www.nature.com/articles/nphoton.2015.277#Sec4

and also Chapter 7 of the book
https://academic.oup.com/book/11564

best
Claudio

[palful]

Dear Sitangshu,

Let me add something. Indeed, the main issue is to map which symmetries from the larger point group of the crystal correspond to those in the reduced point of the wave vector. We are working on an automatic assessment of the exciton representations at all momenta, but it will take some time for it to be released.

One can do it manually by carefully analysing the exciton wave functions, the phonon representations and considering all possible constraints on the symmetries. For example, in the case of AA' stacking hBN (D6h symmetry), it turns out that at the relevant exciton momenta (middle of GK), you can show that the important exciton states split as E_2g (D6h) -> A1+B1 (C2v).

I also want to say that what you find looks absolutely reasonable.
In boron nitride, the AA' stacking has D6h symmetry and this causes all the Z-phonons to be forbidden (because they have A2+B2 symmetry and cannot couple with the above giving another E-type state).
However, the rhombohedral BN stacking has C3v symmetry, like in your system, and this causes the couplings with the Z-phonons to become allowed! This is because in that case you pass from E (C3v) to the identity A+A as along GK there are no symmetries, so every coupling is allowed. You can have a look at this reference: https://journals.aps.org/prl/abstract/1 ... 131.206902

Cheers,
Fulvio