Solving the anisotropic Migdal-Eliashberg equation with EPW under the anharmonic approximation in combination with SSCHA

[anyi-mxh]

Dear Developer,

Recently I have read many articles (https://doi.org/10.1038/s41467-023-44326-4; https://www.nature.com/articles/s42005-023-01413-8) where the authors used SSCHA to perform free energy minimization to obtain anharmonic_dyn and solve the anisotropic Migdal-Eliashberg equation using EPW under the anharmonic approximation. However, in the tutorials and examples provided by the SSCHA code, there is no case of using EPW under the anharmonic approximation to calculate superconducting properties.

So far, I have successfully completed the first few steps, which are as follows:

Step 1: Calculate the harmonic phonon spectrum using Quantum Espresso (QE) (execute pw.x < pw.scf.in > pw.scf.out; ph.x < ph.in > ph.out). In this process, the k-mesh of scf is 10×10×10 and the q-mesh of ph.in is 5×5×5. Based on these settings, I got 11 harmonic_dyn files.

Step 2: Use SSCHA to calculate the anharmonic phonon spectrum (execute pw.x < pw.scf.in > pw.scf.out; ph.x < ph.in > ph.out; python generate_ensemble.py; bash generate_input_files.sh; mpirun -np NPROC pw.x -i input_file.pwi > output_file.pwo; bash get_energy_forces.sh; python minimize.py). In this step, the k-mesh of scf is still 10×10×10, and the q-mesh of ph.in is 2×2×2. The energy and force are calculated in the 2×2×2 supercell, resulting in 4 anharmonic_dyn files.

However, I encountered a problem in the next step: how to combine the anharmonic_dyn file with EPW. First of all, the number of anharmonic_dyn and harmonic_dyn is not equal. In addition, the series of files such as dvscf required by EPW are not calculated under the anharmonic approximation, which confuses me. Please ask, how can I correctly combine anharmonic_dyn with EPW to solve the anisotropic Migdal-Eliashberg equation under the anharmonic approximation?

Hope to get your answer, thank you for your help!

[DjordjeDangic]

Hi,

The reason you have a different number of dyn files is that you ran a 5x5x5 ph.x calculation for the electron-phonon coupling and you ran a 2x2x2 calculation for SSCHA. If you want the same number of dyn files you have to use the same q-point grid in both cases. This is not important though. With EPW you are never using dyn files anyway.

What you need actually is the force constant file generated by the q2r.x code that comes in the QE package. You ran it for SSCHA dyn files to generate SSCHA force constants and then use them for EPW interpolation. You have instructions on the EPW site on how to do this.

A few notes though:

Check convergence wrt to k-point grid for scf calculation for electron-phonon calculation. Once you generate supercells to run scf calculations for SSCHA minimization you don't need such a large k-point grid. In your case, you can definitely half it, so 5x5x5 for SSCHA calculations.

Make sure your SSCHA calculations converge. That means running the SSCHA process iteratively until your free energy and force constants gradients are significantly below stochastic error.

Make sure that your final structure from SSCHA minimization matches the structure for which you calculate electron-phonon coupling. So I would suggest first running SSCHA minimization (iteratively!) and then taking the last structure and running electron-phonon calculation for it.

Also are you sure you need to solve anisotropic Migdal-Eliashberg equations?

[anyi-mxh]

Thank you for your valuable suggestions! Yes, I need to solve the anisotropic Migdal-Eliashberg equation. Based on your suggestions, I have corrected some parameter settings.

Step 1: Use SSCHA to calculate the anharmonic.dyn and use the q2r.x code in the QE package to read the anharmonic.dyn and generated force constant files

pw.x < pw.vc-relax.in > pw.vc-relax.out (k grid is 10×10×10)
pw.x < pw.scf.in > pw.scf.out (k grid is 10×10×10)
ph.x < ph.in > ph.out (q grid is 2×2×2)
python generate_ensemble.py
bash generate_input_files.sh (generate input_file.pwi related to supercell 2×2×2)
pw.x < input_file.pwi > output_file.pw (perform SCF calculations for SSCHA minimization, k grid is set to 5×5×5)
bash get_energy_forces.sh
python minimal.py
q2r.x < q2r.in > q2r.out (generated force constant files anharmonic.fc)

Steps 2: Use Quantum Espresso (QE) to calculate the harmonic.dyn and dvscf

pw.x < pw.scf.in > pw.scf.out (k-grid is 10×10×10
ph.x < ph.in > ph.out (q-grid is 5×5×5)

In your suggestion, "make sure the final structure of the SSCHA minimization matches the structure for which the electron-phonon coupling is calculated. Run the SSCHA minimization first (iteratively!), then take the last structure and run an electron-phonon calculation for it."

All the minimized scf calculations are done with a 2×2×2 supercell. Can you tell me how the last structure in the SSCHA minimization is determined? The structure I took came from the DFT structure optimization and perform an electron-phonon calculation for it. Is this the wrong approach?

Step 3: Copy anharmonic.fc generated by q2r.x in step 1 to the step 2 folder

cp sscha/anharmonic.fc ../phonon_harmonic/harmonic.fc
python pp.py

I am a little confused in this step. Different q-grids in ph.in will produce different force constant files. Can they be mixed? Specifically, anharmonic.fc is based on a q-grid of 2×2×2, while the overwritten harmonic.fc is based on a q-grid of 5×5×5. Will this lead to inconsistency?

Step 4: Use EPW to calculate the electron-phonon matrix elements and solve the anisotropic Migdal-Eliashberg equation

pw.x < pw.scf.in > pw.scf.out (k-grid is 10×10×10)
pw.x < pw.nscf.in > pw.nscf.out (use kmesh.pl to generate a coarse grid of 5×5×5):
pw.x < pw.scf.in > pw.scf.out (nk: 5×5×5; nq: 5×5×5; nkf: 60×60×60; nqf: 20×20×20）

Thank you very much for your great support!

[DjordjeDangic]

I am not sure what type of SSCHA calculation you are running. Are you relaxing structure as well? SSCHA average forces and stresses are going to be different from the DFT ones unless they are zero by symmetry! If you are relaxing structure, it will change during SSCHA minimization. You will need to calculate electron-phonon coupling and electronic bands for this final SSCHA structure. SSCHA structure will be written in the dyn files generated by SSCHA.

EPW interpolates phonon dynamical matrices on the higher grid. So you can use whatever coarse grid you want, there will be no inconsistencies. Check "lifc" input parameter for EPW calculations to see how to use force constants generated by q2r.x.

Also, when I said to use 5x5x5 k-grid, I meant for the supercell calculations. I hope you are not using a 5x5x5 k-point coarse grid for Wannier interpolation because I said so in the last post.