Interpolating exciton dispersion

[Guo_BIT]

Dear developers:
I want to calculate the excitonic band structure and further obtain the effective mass of excitons by referring to the websitehttps://www.yambo-code.eu/wiki/index.ph ... Yambo_5.x)
However, due to the limited explanations on the web page, I have a few questions below：
（1）I know that in the original BSE calculation, we set BSEQptR = |1|1|, which means only considering the q = 0 case, for example, the electron can only form excitons with the holes at same K-point.
To calculate the excitonic band structure, we have to consider all the K points. For example, these are 19 K points in my calculation, then I should set BSEQptR = |1|19|. In this case, q is not equal to 0, which means that the electron at the M point can form excitons with the holes at other points like Γ，right?
（2）After the BSE calculation, various output files are generated for each point, such as o-E_BS.alpha_q1_diago_bse. How can I obtain the overall absorption spectrum data？
（3）Most importantly, I conducted tests using 2D AlN (without convergence tests). At BSEQptR = |1|1|, I observed the lowest energy exciton at around 3.6 eV with order -e s 1. However, when I modified it to BSEQptR = |1|19|, I noticed that a 3 eV energy exciton occurs at the M point. I speculate that this may be due to AlN being an indirect bandgap semiconductor, but I cannot confirm if this is caused by my input file.
I have attached both input files to the attachments.

Thank you in advance.

[Daniele Varsano]

Dear Jingda Guo,

1) Right
2) It's not totally clear to me what you mean with "overall absorption spectrum data". The (light) absorption spectrum is the q=0 limit i.e. iq=1, next you will have output for finite momentum excitations, detectable e.g. in electron energy loss experiments.
3) If you are studying an indirect gap semiconductor, most likely the minimum of the exciton dispersion will be for the q connecting the VBM and CBM, you can analyze your exciton composition in terms of band and v_k-->c_k+q using the ypp utility.

Best,
Daniele

[Guo_BIT]

Dear Daniele,
Thank you for your explanation.
(1). My understanding is that when setting BSEQptR = |1|19|, it considers all K points, which means that the hole of the first K point can form an exciton with the electrons of all K points from 1 to 19, and the same to the holes of 2-18, right?
(2). If my understanding is correct, can I calculate the exciton only between the hole of the first K point and the electron of the seventh K point? （only 1 and 7, not 1 to 7）Is this possible to achieve?

[Daniele Varsano]

Dear Jingda,

Actually, in your understanding, there is a misleading between k points and transferred momentum q=k-k'

Using BSEQptR = |1|19|, Yambo perform 19 different calculations and you will have 19 outputs. Each of them is the calculations of excitations having different momentum: electrons and hole are in a different region of the BZ where the momentum of electrons and holes differs by Q. Of course iq=1 is the q=0 limit and electron and hole shares the same momentum.

In other words, for q=0 the excitonic wavefunction of an exciton lambda will be:

psi1.png

where all the k points enters the sum weighted by the Acv, solution of the BSE.

For q different from zero you have:

psi2.png

I hope that now it is more clear,

Best,
Daniele

[Guo_BIT]

Dear Daniele,
Thank you for your patient guidance, it has been very helpful to us!
Setting BSEQptR = |1|19| would generate 19 different Q, just like the figure 3 in this articlehttps://journals.aps.org/prl/abstract/1 ... 124.166401
the holes at each K point will interact with the electrons at K+q to form excitons. so there would be 19 different calculations. Then ypp -e i will consider all possible Q and provide the overall excitonic band structure of the system. Is my understanding correct this time?

After obtaining the excitonic band structure based on the above procedure, I can calculate the effective mass of excitons using a similar method as calculating the effective mass of electrons in electronic band structure, am I correct?

[Guo_BIT]

I hope you don’t mind, but I have another question
Following the procedure mentioned earlier, I calculated all the cases of q by setting BSEQptR = |1|19| and obtained the excitonic band structure using the “ypp -e i” command.

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#     |q| (a.u.)         e1                 e2                q_x (rlu)          q_y (rlu)          q_z (rlu)
#
       0.00000000        0.447456911E-1     0.456438325E-1      0.00000000         0.00000000         0.00000000
      0.310200434E-1     0.834466144E-1     0.101891957        0.277777780E-1      0.00000000         0.00000000
      0.620400868E-1     0.135892168        0.167817146        0.555555560E-1      0.00000000         0.00000000
      0.930601284E-1     0.177934423        0.197894797        0.833333284E-1      0.00000000         0.00000000

(1). According to the output, the energy of e1 is 0.0447 when q=0, which corresponds to the lowest energy exciton appearing at the M point (direct band gap) when BSEQptR=|1|1|. Then, e2 corresponds to the overall second lowest exciton when q=0, for example, the lowest energy exciton appearing at the Γ point.
(2). When qx = 0.5, qy = qz = 0, e1 and e2 correspond to the lowest excitons at, for example, the electron K point (0.1, 0, 0) and hole K point (0.6, 0, 0), or the transition from (0, 0.1, 0) to (0.5, 0.1, 0).
Is my understanding correct?

[Daniele Varsano]

Dear Jingda,

to understand the transitions participating in each exciton, you need to analyze them using ypp (ypp -a -b inq). In the output, you will find in which part of the BZ the electrons and holes are located. Please note that besides some particular cases, the weight is distributed over a region of the BZ and not just one k point.

To perform this analysis, it is needed that the variable WRbsWF in the diagonalization input file is uncommented.

Best,
Daniele

[Guo_BIT]

Dear Daniele,
Thank you for your reply. Is there a tutorial webpage for this command? I only found this command in the outline of YPP instructions,
https://www.yambo-code.eu/wiki/index.ph ... sing_(ypp)

By using ypp -a -e a -b 91, I tried to analyze the excitons with q=91, which is (0.5, 0.5, 0), as is shown in output

Code: Select all

[K] Q-point                                     : -0.499999970    -0.499999970      0.00000000

and the output shows

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# Electron-Hole pairs that contribute to Excitonic State 1 for iq=91 more than  5.000000%
#                       K-point [iku]                         Weight
# :    0.00000000       -0.500000000         0.00000000         1.00000000   
#    
#
#    Band_V             Band_C             Kv-q ibz           Symm_kv            Kc q ibz           Symm_kc            Weight             Energy [eV]
#
         36                 37                13                 3                  13                 1                  0.974979           0.073244
         36                 37                13                 1                  13                 3                  0.974975           0.073244

Where Kc-q ibz = Kv-q ibz = 13 (M)
Here is my input file

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# .-Input file  ypp_AMPL.in
# | BSiq                             # [R] Excitonic State Momentum
# | excitons                         # [R] Excitonic properties
# | avehole                          # [R] Average hole/electron wavefunction
# | amplitude                        # [R] Amplitude
# | States= "1 - 2"                  # Index of the BS state(s)
# | BSQindex=  91                    # Q-Index of the BS state(s)
# | Degen_Step= 0.010000       eV    # Maximum energy separation of two degenerate states

Appreciate your reply in advance

[Daniele Varsano]

Dear Jingda,

>ypp -e a generates the input to analyze your exciton and the output provide in which part of the BZ electrons and hole are (it was mispelled in previous post).

The BZ point is given by the k index in the irreducible BZ and the symmetry index (the symmetry operation can be found in the report file) that reads:
you have the electron at kc point index 13, and the hole in kv obtained applying the symmetry with index 3 to k=kc-q=13.

By default, only transition weighting more than 5% are reported, you can change this threshold in input by changing the EXCITON_E_treshold which by default is set to 0.05.

Please note that it seems you have to degenerate (or nearly degenerate) excitation, and in this case ypp will merge them. This is seen by the fact that the sum of the weight sum to 2.

If you want to analyze just one of the excitations, you need to set to zero the Degen_Step value in input.

Best,
Daniele

[Guo_BIT]

Dear Daniele,
Thank you very much for your patient guidance This is a great help to me

In this case, the horizontal axis for the excitonic bands is not K but Q. Therefore, the effective mass of excitons cannot be obtained through the band’s E-K dispersion relationship. So, how should we determine the effective mass of excitons with excitonic band strucures?

Appreciate your reply in advance