Dear EPW users and developers,
I'm looking for some help and feedback regarding calculations of optical absorption for p-doped GaAs (5e17 concentration). For most parts I was closely following the "Phonon-assisted optical absorption with EPW" tutorial session from 2024 School, with (hopefully) appropriate adaptations for GaAs.
I'm using QE with PBE and spin orbit coupling for the ground state and phonons with 12x12x12 k-grid and 6x6x6 q-grid, and for the EPW calculations I'm using energies from a PBE0 hybrid run. My motivation is to try to look into IR range absorption where I have some experimental dielectric function available, specifically, I see a broad absorption in the range 0.2-0.6eV with two minor bumps (features (~0.29 and 0.36eV)), see the attached figure. Interestingly, the energy difference of those 2 features is approximately 2 times the energy of TO phonons which is what motivated me to see if I can reproduce something like this with EPW.
What works up to now:
- wannierization looks fine, the wannier band structure seems a good match for the underlying DFT hybrid run.
- I can get the direct optical transitions just fine. The convergence requirements are brutal (over 100x100x100 k-point grid is needed), but still easily doable for the direct transitions. In general the spectra matches well the experimental one, see the attached figure. The discrepancies are shifts related to issues of the hybrid band structure not EPW, specifically the predicted energy difference between the VBM and the split off band bellow is larger in the hybrid than in experiment and I don't see the light-holes heavy-holes transitions experimentally which corresponds to the predicted absorption below 0.3eV (in the experiment the Drude term is already too strong to see the interband transition in that region. However, importantly I don't get any of those weak features that I see in the experiment (~0.29 and 0.36eV), but that could just mean they are not related to direct transitions...
What doesn't work:
- the symetry is broken quite a lot for the direct transitions, the difference between the xx, yy and zz dielectric tensor components can be up to 50%. Quantum espresso runs are using proper symmetry and I'm using also mp_mesh_k = .true. for EPW, so I wonder at which point the symmetry breaks? Is this from the Wannierization? In general the average dielectric function looks fine (even the absolute intensity is a really good match for the experimental dielectric function), so just want to make sure this is not something signalling some deeper issue.
- The indirect transitions are probably impossible to really converge at least based on my current testing and CPU limitations. The best I did was 48x48x48 k-points with 24x24x24 q-points, and that took around 10k core hours. I could probably afford an order of magnitude longer run, but with the k-point * q-point scaling I don't think it would get me much closer to the convergence to be really worth it. I don't see anything resembling the spectral features I'm looking for in the indirect spectra, but as I said its not much converged so who knows. That said, in the theory manuscript [1] the discussion is mostly focusing on the relative intensities of the contributions (there seems to be some features in the indirect "phonon" spectra in FIG. 2 as well, most notably some small peak between 1 and 2 μm in 2(a) also small oscillations in the same region in the 2(b), but they are not discussed AFAICS so I'm assuming they can be also convergence artifacts). Now the tricky part from my current understanding is that the intensity of the indirect transitions depend on the η broadening parameter - a shortcoming which comes directly from the theory. In the tutorial just the ε2|η=0.1eV is shown, while in the [1] 2*ε2|η=0.2eV - ε2|η=0.1eV formula is used in [1] and it references [2]. The referenced ACS Nano paper has a bit of discussion about the reason for the subtraction (cancellation of the diverging term), but for the specific used value of η=0.1eV it just links to the the indirect Si transitions paper [3], but that one has just offhand comment that it was set to a constant value (100 meV) but it doesn't matter in that spectra region (there are no overlapping direct transitions). Now what is a bit problematic is that if I actually use the same 2*ε2|η=0.2eV - ε2|η=0.1eV formula, I'm getting a negative ε2 in some spectral range (~0.4-0.7eV which is most of the range where the relevant direct transitions are).
Therefore my main question is, can I find somewhere some more extensive explanation of the η=0.1eV value or is this mostly just empirical (as it now seems to be from my current reading of the manuscripts)? Additionally, if I disregard the negative ε2, in the other parts of the IR spectra the indirect transitions are maybe 10x weaker than the direct ones, would it still be reasonable to make the claim that the visible features are not phonon-assisted transitions, based just on the predicted magnitude of the (non-completelly converged) indirect absorption with the η=0.1eV value? That however still leaves the question where the features comes from. Could he issue of the hybrid (I could probably try GW) or could excitons still be a thing in this spectral range and doping concentrations? But thats obviously beyond the scope of this forum...
Figure with the dielectric function, QE and epw inputs and some relevant outputs are attached.
Best regards
Pavel Ondračka
Department of Plasma Physics and Technology
Masaryk University, Brno, Czechia
P.S.: I was not sure about the right forum section for this post, there are some technical questions as well as some more theoretical, so hopefully "running the code" is the correct one.
[1] "Ab initio theory of free-carrier absorption in semiconductors" PHYSICAL REVIEW B 106, 205203 (2022)
[2] "Nonradiative Plasmon Decay and Hot Carrier Dynamics: Effects of Phonons, Surfaces, and Geometry" ACS Nano 2016, 10, 957−966
[3] "Phonon-Assisted Optical Absorption in Silicon from First Principles" PRL 108, 167402 (2012)
IR optical absorption in doped GaAs
Moderator: stiwari
IR optical absorption in doped GaAs
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Re: IR optical absorption in doped GaAs
Dear Pavel,
Thank you for your question! A few thoughts related to your question and discussion:
For the ones that worked:
-Looking at the spectra, the bump that you observed in the region between 0.4-0.5 eV seem to be caused by spin-orbit splitting. In fact, as you mentioned, if the predicted energy difference between VBM and the split off band are larger in hybrid, it seem quite possible that the weak feature you are seeing in experiment around 0.3 eV is exactly the effect of spin-orbit interaction. As for the double bump, I suspect it can be related to the fact off-Gamma, the first and second VB are not degenerate anymore, and transitions can happen from the split off band to either the first or the second VB. Therefore a very small energy difference might be seen in these two types of transitions. I would recommend further decreasing degaussw, as a broadening of 0.05 eV, which is what you are using now, is close to the range where you see the difference between the two bumps, therefore if there was a difference, it might have been smeared out.
For the ones that are problematic:
-Symmetry breaking is indeed expected. For a cubic system and with mp_mesh_k, particularly when one considers e.g. free carrier absorption which involves a very small region, one should average the dielectric function across different directions. This is normal and not signaling deeper issue. Though for this case, it would always be helpful to zoom into the band maximum to make sure, for example, Wannierzation is not causing the band to curl up near the Gamma point (i.e. instead of Gamma being the maximum, you see a bump near Gamma.)
-Regarding the broadening parameter, it is indeed an intrinsic limit of the theory. The correction scheme you saw in e.g. ACS nano can be very sensitive to parameters such as k-point sampling, smearing, which two eta you choose, etc, so indeed the negative epsilon_2 in some regions can be expected(particular in the strongly diverging region, e.g. when direct absorption is allowed.)
The most rigid way that one should consider for this case is to use the qdabs module, with the process described in the following paper: Phys. Rev. B 109, 195127, where quasidegenerate perturbation theory is used to consider the changes in the wavefunction in the degenerate region. See also the presentation and tutorial by Tiwari on Saturday at: https://epw2024.oden.utexas.edu/74-schedule. As of now, the qdabs module does not yet support inputing ncarrier and carrier directly, but I believe what you can do is to manually input the Fermi level obtained from the indirect optics module (i.e. in your case 8.18465 eV), and it should allow you to examine the combined direct and indirect contribution without worrying about the divergence.
With these said, in general since direct absorption is a first order process, and indirect is second order, the former indeed tend to dominate the resonant region. So in your case, I still suspect that the spectra feature you are seeing in experiment is likely related to spin-orbit splitting the direct transitions associated with it. The discrepancy likely come from the difference between the value of the splitting from theory and experiment - a difference of 0.1 eV, e.g. the bump you see in experiment compared to the bump you see in the direct spectra is getting close to the error bar of the theory. Excitons may still matter - usually in metallic system/heavily doped semiconductors, metallic screening renders excitons not impactful anymore in the region you are looking at, but in your case, doping concentration is not as high (as you can tell the spectra is not yet blowing up in the region where you see the bump). Indeed this would be beyond the capability of EPW now, but again, if it is indeed spin-orbit coupling, the energy difference (0.3 eV vs 0.4 eV) is close to the error bar of the theory so if one just want to match the energy, it is hard to say if further adding exciton may help or not.
This is very interesting result - please see if you find the above helpful, and I'd be happy to discuss more!
Thanks!
Xiao
Thank you for your question! A few thoughts related to your question and discussion:
For the ones that worked:
-Looking at the spectra, the bump that you observed in the region between 0.4-0.5 eV seem to be caused by spin-orbit splitting. In fact, as you mentioned, if the predicted energy difference between VBM and the split off band are larger in hybrid, it seem quite possible that the weak feature you are seeing in experiment around 0.3 eV is exactly the effect of spin-orbit interaction. As for the double bump, I suspect it can be related to the fact off-Gamma, the first and second VB are not degenerate anymore, and transitions can happen from the split off band to either the first or the second VB. Therefore a very small energy difference might be seen in these two types of transitions. I would recommend further decreasing degaussw, as a broadening of 0.05 eV, which is what you are using now, is close to the range where you see the difference between the two bumps, therefore if there was a difference, it might have been smeared out.
For the ones that are problematic:
-Symmetry breaking is indeed expected. For a cubic system and with mp_mesh_k, particularly when one considers e.g. free carrier absorption which involves a very small region, one should average the dielectric function across different directions. This is normal and not signaling deeper issue. Though for this case, it would always be helpful to zoom into the band maximum to make sure, for example, Wannierzation is not causing the band to curl up near the Gamma point (i.e. instead of Gamma being the maximum, you see a bump near Gamma.)
-Regarding the broadening parameter, it is indeed an intrinsic limit of the theory. The correction scheme you saw in e.g. ACS nano can be very sensitive to parameters such as k-point sampling, smearing, which two eta you choose, etc, so indeed the negative epsilon_2 in some regions can be expected(particular in the strongly diverging region, e.g. when direct absorption is allowed.)
The most rigid way that one should consider for this case is to use the qdabs module, with the process described in the following paper: Phys. Rev. B 109, 195127, where quasidegenerate perturbation theory is used to consider the changes in the wavefunction in the degenerate region. See also the presentation and tutorial by Tiwari on Saturday at: https://epw2024.oden.utexas.edu/74-schedule. As of now, the qdabs module does not yet support inputing ncarrier and carrier directly, but I believe what you can do is to manually input the Fermi level obtained from the indirect optics module (i.e. in your case 8.18465 eV), and it should allow you to examine the combined direct and indirect contribution without worrying about the divergence.
With these said, in general since direct absorption is a first order process, and indirect is second order, the former indeed tend to dominate the resonant region. So in your case, I still suspect that the spectra feature you are seeing in experiment is likely related to spin-orbit splitting the direct transitions associated with it. The discrepancy likely come from the difference between the value of the splitting from theory and experiment - a difference of 0.1 eV, e.g. the bump you see in experiment compared to the bump you see in the direct spectra is getting close to the error bar of the theory. Excitons may still matter - usually in metallic system/heavily doped semiconductors, metallic screening renders excitons not impactful anymore in the region you are looking at, but in your case, doping concentration is not as high (as you can tell the spectra is not yet blowing up in the region where you see the bump). Indeed this would be beyond the capability of EPW now, but again, if it is indeed spin-orbit coupling, the energy difference (0.3 eV vs 0.4 eV) is close to the error bar of the theory so if one just want to match the energy, it is hard to say if further adding exciton may help or not.
This is very interesting result - please see if you find the above helpful, and I'd be happy to discuss more!
Thanks!
Xiao