Transition matrix elements

[sitangshu]

Dear Sir,

While doing the analysis of mono layer h-BN we found that it is an indirect band gap semiconductor but behaves as a direct band gap material because of the high probability of transition at K point in comparison to probability of transition for the indirect case. We need to show this using the "Transition matrix elements" fundamentally necessary to calculate the absorption spectra. We have done the Bethe Salpeter analysis of the mono layer h-BN. Can you please help me to extract the transition matrix elements from this?
Also, Is Bethe Salpeter analysis able to capture indirect transitions? If not then how can we calculate transition probability for indirect transition?

I have also calculated excitonic weights for my highest peak in the absorption spectra attached below. It is showing transitions at k point 19 and k point 16. Can you help me to find out the position of k point 19 and 16 in my band structure?

And at k point 16 it is showing transition from band 8 to band 4. What does this means?

Regards,
Sitangshu
IIIT Allahabad

[Daniele Varsano]

Dear Sitangshu,

While doing the analysis of mono layer h-BN we found that it is an indirect band gap semiconductor

Well, may be you need to check your ground state, here a paper you may have a look for reference (Fig.4) but surely you can find 2D-HBN in literature to compare with:
https://arxiv.org/pdf/1806.06201.pdf

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Also, Is Bethe Salpeter analysis able to capture indirect transitions? If not then how can we calculate transition probability for indirect transition?

Yes, in line of principle it is possible, but the q-finite BSE although implemented in Yambo it has not been yet released in the GPL version, we plan to make it public soon.

Can you help me to find out the position of k point 19 and 16 in my band structure?

In any report file, including the setup, you can find the K vector corresponding to each K index, e.g.

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 *X* K [4] : 0.000000 -0.769211  0.000000 ( cc) * Comp.s 2736 * weight   0.08333
              0.000000 -0.500000  0.000000 (rlu) 

Index 4 that correspond to the wave vector report in cartesian coordinate (cc) and reciprocal lattice unit (rlu).

And at k point 16 it is showing transition from band 8 to band 4. What does this means?

In the BSE you have both the resonant part, transition v-> c and anti resonant c -> v. In the resonant approximation (Tamm Dancoff) the two blocks are decoupled, when introducing coupling the two are mixed.
Are you adding the coupling term in your BSE?

Can you please help me to extract the transition matrix elements from this?

The transition matrix elements are calculated here:
src/pol_function/DIPOLE_transverse.F

you can follow this thread
viewtopic.php?f=13&t=382&p=2621&hilit=dipole#p2621

on how to write them. Please note that if you have the ./SAVE/ndb.dipoles these are not re-calculated.

Other option is here:
src/bse/K_components_folded_in_serial_arrays.F

and write on a txt fie BSS_dipoles
in the loop around line 75 they are assigned, at the beginning of the loop you have the corresponding index of valence, conduction bands and k point, but here I'm afraid that the BZ
referes to the entire BZ and not the irreducible.

Last option (may the easiest) is to use yambo-py to extract them from the database:
https://yambopy.readthedocs.io/en/latest/autodoc.html

Hope it helps,

Best,
Daniele

[sitangshu]

Dear sir,

Actually while doing the DFT analysis, we indeed got a direct gap at K. However, a G0W0 is giving an indirect one. We find this consistent with
https://arxiv.org/abs/1605.09581

Thanks for your response. I am checking the transition matrix element method. Meanwhile, there are few more queries namely:

1. How can I get exciton dispersion along the Brillouin zone.
2. How to plot the excitonic wavefunction distribution over Brillouin zone?
3. In our calculation, we put the hole position above boron atom which lacks electron density. However, I am not sure why the hole in the above mentioned paper is taken above the nitrogen atom.
4. I have used double grid+el-ph coupling+e-h coupling in the BSE. This resulted in a convincing absorption spectra. However, when I projected the excitonic wavefunction on lattice, it is not symmetrical as shown in the attachment below. I found from this paper that one has to sum the excitonic densities to restore the three fold rotational symmetry (as also previously done using yambo in
https://journals.aps.org/prl/pdf/10.110 ... 100.189701 ). Can you let me know this method of summing the exciton densities?

Thanks and Regards.

[Daniele Varsano]

Dear Sitangshu,

1. How can I get exciton dispersion along the Brillouin zone.

As I wrote you above, you cannot get it, as this feature is experimental, it is implemented in the devel version and and not yet released in the GPL version.

2. How to plot the excitonic wavefunction distribution over Brillouin zone?

In the o.weight file obtained using ypp (ypp -e a) beside the weight associated with the electron-hole transition A_{cvk}, on top you have also the weight associated with each k point. essentially
\sum_cv A_{cvk}. By default only the weight higher than 5% are shown, but you can change the threshold using the MinWeight variable in input file (it appears adding -V gen to the ypp command line

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ypp -e a -V gen

), note that this value is multiplied by 5. Essentially this gives you the exciton wave function in k space on a regular grid. In Yambo-py (https://yambopy.readthedocs.io/en/latest/tutorial.html) there should be tools to plot it, unfortunately, I never used it so I can't help on that, but with the Yambo output you can plot them in the flavor you prefer.

3. In our calculation, we put the hole position above boron atom which lacks electron density. However, I am not sure why the hole in the above mentioned paper is taken above the nitrogen atom.

This is something you should ask the authors, in any case, in my opinion, it is not a good idea to put the hole in a point where the valence Bloch states have zero or very small density.

4. I have used double grid+el-ph coupling+e-h coupling in the BSE. This resulted in a convincing absorption spectra. However, when I projected the excitonic wavefunction on lattice, it is not symmetrical as shown in the attachment below. I found from this paper that one has to sum the excitonic densities to restore the three fold rotational symmetry (as also previously done using yambo in
https://journals.aps.org/prl/pdf/10.110 ... 100.189701 ). Can you let me know this method of summing the exciton densities?

A,s far as I know there is not a tool to do that, you can calculate the excitonic wavefunction for the nearly degenerate states and sum the up by hand with a simple script. Other option is to set the Degen_Step variable to a value that the
exciton similar in energy are considered degenerates.

Best,
Daniele