Dark Excitons

[WEI Wei]

Dear All,

I am reading a paper published on PRL (Park et al. 96, 126105, 2006), about which I have some questions.
The QP corrected band structures and the BSE absorption spectra (shown as following) were calculated, from which the authors concluded that:
"We also note that, for the (8,0) SWBNNT, there are numerous dark excitons distributed rather uniformly in energy below and among the bright excitons shown in Fig. 2. The energy of the lowest doubly degeberate bound dark exciton (K) is at 4.63 eV. This dark exciton is made up of transitions from the highest valence band to the lowest conduction band (the NFE tubule state) is the quasiparticle band structure, and has a binding energy of 1.94 eV with respect to these interband transition energies."
I was completely confused by these words. So, I want to ask you:
1) How to find a dark exciton from the Band Structure and Absorption Spectra?
2) From the figures of this paper, the energy difference between the highest valence band and the lowest conduction band is apparently larger than 4.63 eV. How the authors determind the binding energy of 1.94 eV?
3) Why the authors said that "there are numerous dark excitons distributed rather uniformly in energy below and among the bright excitons shown in Fig. 2"?
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[Daniele Varsano]

Dear Wei Wei,

1) How to find a dark exciton from the Band Structure and Absorption Spectra?

No way, the dark exciton is a solution of the BSE equation with very small (zero) oscillator strength so you can't see them in the absorption spectra, anyway they are solution of the equation!! in Yambo once you have diagonalized the excitonic matrix you
can see all the excitation by typing ypp -s. Next the dark ones are the excitations with small intensity
(oscillator strength), the intensity is reported in the same file generated by ypp -s.

2) From the figures of this paper, the energy difference between the highest valence band and the lowest conduction band is apparently larger than 4.63 eV. How the authors determind the binding energy of 1.94 eV?

Of course, 4.63 eV is the excitation energy calculated including electron-hole interaction (exciton). The binding energy is the difference between the quasiparticle gap (calculated from the QP band structure) and excitation energy.

Why the authors said that "there are numerous dark excitons distributed rather uniformly in energy below and among the bright excitons shown in Fig. 2"?

Because they find them from the solution of the BSE, of course you cannot see them from the absorption spectrum as they have very small oscillator strength.

Cheers,

Daniele

[WEI Wei]

Dear Daniele,

OK, I see.
Thank you very much!