The superconducting transition temperature calculated from the superconducting gap Δ(T) is too high

[fale li]

Dear all，
When I calculated the gap function and the superconducting transition temperature by solving the isotropic Migdal-Eliashberg equation with the example of fcc Pb, the results obtained are as follows：

Finish reading a2f file

Electron-phonon coupling strength = 3.2617374

Estimated Allen-Dynes Tc = 16.932230 K for muc = 0.10000

Estimated w_log in Allen-Dynes Tc = 7.828424 meV

Estimated BCS superconducting gap = 2.568028 meV

Estimated Tc from machine learning model = 26.971873 K

The Allen-Dynes Tc in this result is close to the values calculated by others, but the estimated value of Tc is too large, and the value of the Electron-phonon coupling constant is also too large. Moreover, when I change the size of the Wannier window, fsthick, degaussw, and wscut, the results do not change.

Could you please tell me where I went wrong? Here is my epw.in file

&inputepw
prefix = 'X',
amass(1) = 135.8925
outdir = './'
dvscf_dir = '../ibrav-phonon--3/save'

ep_coupling = .true.
elph = .true.,
! kmaps = .false.
epbwrite = .true.,
epbread = .false.
epwwrite = .true.
epwread = .false.

nbndsub = 6
! nbndskip = 4
bands_skipped = 'exclude_bands = 1:4'
!efermi_read=.true.
!fermi_energy=16.527850

wannierize = .true.
num_iter = 500
! dis_win_max = 30
! dis_win_min = 10
dis_froz_min= 13
dis_froz_max= 18.5
proj(1) = 'X:s,d'
!dis_num_iter=1000

wdata(1) = 'bands_plot = .true.'
wdata(2) = 'begin kpoint_path'
wdata(3) = 'G 0.00 0.00 0.00 H 0.50 -0.50 0.50'
wdata(4) = 'H 0.50 -0.50 0.50 N 0.00 0.00 0.50'
wdata(5) = 'N 0.00 0.00 0.50 P 0.25 0.25 0.25'
wdata(6) = 'P 0.25 0.25 0.25 G 0.00 0.00 0.00'
wdata(7) = 'G 0.00 0.00 0.00 N 0.00 0.00 0.50'
wdata(8) = 'end kpoint_path'
wdata(9) = 'bands_plot_format = gnuplot'

! parallel_k = .true.
! parallel_q = .false.

iverbosity = 0
ephwrite = .true.

elecselfen = .false.
phonselfen = .true.
eliashberg = .true.
liso = .true.
!limag = .true.
!lpade = .true.
!lacon = .true.
wscut = 0.1

nsiter = 500
conv_thr_iaxis = 1.0d-3
conv_thr_racon = 1.0d-3

fsthick = 0.4 ! eV
! temps = 0.075! K
degaussw = 0.1 ! eV
degaussq = 0.5 ! eV

a2f = .true.
muc = 0.1

temps(1) = 2
temps(2) = 3
temps(3) = 4
temps(4) = 5
temps(5) = 6
temps(6) = 7
temps(7) = 8
temps(8) = 9
temps(9) = 10
temps(10) = 11
temps(11) = 12

!dvscf_dir = '../phonon/save-pp-paw'
!mp_mesh_k = .true.
nkf1 = 12
nkf2 = 12
nkf3 = 12

nqf1 = 12
nqf2 = 12
nqf3 = 12

nk1 = 6
nk2 = 6
nk3 = 6

nq1 = 6
nq2 = 6
nq3 = 6
/

[hmori]

Dear Li,

I guess your calculations are collinear and do not account for spin-orbit interaction, which may lead to an overestimation of the Allen-Dynes Tc compared to its expected converged value. With this in mind, I have a few concerns:

1. If the target system is truly fcc Pb, the p orbitals of Pb should be included in the Wannier projections. Your input only includes the s and d orbitals. Have you checked the spread of the Wannier orbitals and the band structure obtained after Wannierization? The Wannierized band structure has to reproduce the original DFT band structure, and the spread of the Wannier orbitals have to be sufficiently minimized.
2. The fine grid size (nkf1, nkf2, ..., nqf1, ...) is too small. It should be at least 24x24x24.
3. If an a2f file exists in the working directory, the message 'a2f file is found and will be used to estimate initial gap' will appear in stdout, and the file will be used without recalculation. To apply new calculation conditions, ensure the a2f file is removed from the directory.

Please address these points.

Best regards,
Hitoshi

[fale li]

Dear Li,

I guess your calculations are collinear and do not account for spin-orbit interaction, which may lead to an overestimation of the Allen-Dynes Tc compared to its expected converged value. With this in mind, I have a few concerns:

1. If the target system is truly fcc Pb, the p orbitals of Pb should be included in the Wannier projections. Your input only includes the s and d orbitals. Have you checked the spread of the Wannier orbitals and the band structure obtained after Wannierization? The Wannierized band structure has to reproduce the original DFT band structure, and the spread of the Wannier orbitals have to be sufficiently minimized.
2. The fine grid size (nkf1, nkf2, ..., nqf1, ...) is too small. It should be at least 24x24x24.
3. If an a2f file exists in the working directory, the message 'a2f file is found and will be used to estimate initial gap' will appear in stdout, and the file will be used without recalculation. To apply new calculation conditions, ensure the a2f file is removed from the directory.

Please address these points.

Best regards,
Hitoshi

Dear Hitoshi，

Thanks for your help, I deleted the a2f file and then performed the calculation, and the calculation result has changed.

However, the lambda and Estimated Tc from machine learning model obtained by calculation are still too large. I cannot solve this problem by increasing the values of nkf and nqf.
a2f file is not found to estimate initial gap: calculating a2f files


Finish reading a2f file

Electron-phonon coupling strength = 4.0545800

Estimated Allen-Dynes Tc = 15.958319 K for muc = 0.10000

Estimated w_log in Allen-Dynes Tc = 6.823878 meV

Estimated BCS superconducting gap = 2.420319 meV

Estimated Tc from machine learning model = 28.084740 K

About the first two questions you asked：
I used fcc pb as an example to calculate the Tc of MEA TaNbHfZr bcc structure, so I considered the s and d orbitals. Wannier calculates bands that reproduce the original DFT bands

I have another problem: when I calculated the electron-phonon coupling constant alone, the lambda value obtained was different from that obtained by calculating the superconducting gap and superconducting transition temperature，and the calculated lambda value was 5.0733014. Which lambda value was correct? Below is the input file where I calculated the lambda and the ph.in file where I calculated the phonon

epw calculation
&inputepw
prefix = 'X',
amass(1) = 135.8925
outdir = './'

elph = .true.
epbwrite = .true.
epbread = .false.

epwwrite = .true.
epwread = .false.
nbndsub = 6
bands_skipped = 'exclude_bands = 1:4'
!efermi_read=.true.
!fermi_energy=16.527850

wannierize = .true.
num_iter = 10000
! dis_win_max = 40
! dis_win_min = -50
dis_froz_min= 13
dis_froz_max= 18.5
proj(1) = 'X:s,d'
!dis_num_iter=1000

wdata(1) = 'bands_plot = .true.'
wdata(2) = 'begin kpoint_path'
wdata(3) = 'G 0.00 0.00 0.00 H 0.50 -0.50 0.50'
wdata(4) = 'H 0.50 -0.50 0.50 N 0.00 0.00 0.50'
wdata(5) = 'N 0.00 0.00 0.50 P 0.25 0.25 0.25'
wdata(6) = 'P 0.25 0.25 0.25 G 0.00 0.00 0.00'
wdata(7) = 'G 0.00 0.00 0.00 N 0.00 0.00 0.50'
wdata(8) = 'end kpoint_path'
wdata(9) = 'bands_plot_format = gnuplot'

iverbosity = 0

elecselfen = .false.
phonselfen = .true.
!eliashberg = .true.
!liso = .true.
!limag = .true.
wscut = 0.1

fsthick = 0.4 ! eV
temps = 0.075 ! K
degaussw = 0.1 ! eV
degaussq = 0.5 ! eV

a2f = .true.

! dvscf_dir = '../phonon/save-pp-paw'
dvscf_dir = '../ibrav-phonon--3/save'
nkf1 = 24
nkf2 = 24
nkf3 = 24

nqf1 = 24
nqf2 = 24
nqf3 = 24

nk1 = 6
nk2 = 6
nk3 = 6

nq1 = 6
nq2 = 6
nq3 = 6

/

phonon calcuation
&inputph
outdir = './'
prefix = 'X'
fildvscf = 'dvscf'
tr2_ph = 1.0d-18
!epsil = .true.
ldisp = .true.
trans = .true.
!nmix_ph = 10
!alpha_mix = 0.3
amass(1) = 135.8925
nq1 = 6, nq2 = 6, nq3 = 6
fildyn = 'X.dyn'
/

Sincerely,
Li

[hmori]

Dear Li,

Even if the Wannier bands closely reproduce the DFT bands, the Wannier orbitals may not be ideal for EPW. To effectively interpolate the electron-phonon interaction, the Wannier orbitals must be spatially localized. I strongly recommend repeating the calculations using the s and p orbitals of Pb and comparing the results. If these calculations reproduce the electron-phonon coupling constant obtained from ph.x, it would suggest that the issue is caused by using the s and d orbitals.

Best regards,
Hitoshi

[fale li]

Dear Li,

Even if the Wannier bands closely reproduce the DFT bands, the Wannier orbitals may not be ideal for EPW. To effectively interpolate the electron-phonon interaction, the Wannier orbitals must be spatially localized. I strongly recommend repeating the calculations using the s and p orbitals of Pb and comparing the results. If these calculations reproduce the electron-phonon coupling constant obtained from ph.x, it would suggest that the issue is caused by using the s and d orbitals.

Best regards,
Hitoshi

Dear Hitoshi

I used the s, p orbitals to calculate Tc and lambda for TaNbHfZr, but the results were still too large.These calculations did not reproduce the electron-phonon coupling constant obtained from ph.x. The problem should have nothing to do with the orbit

a2f file is not found to estimate initial gap: calculating a2f files


Finish reading a2f file

Electron-phonon coupling strength = 3.7393846

Estimated Allen-Dynes Tc = 15.277166 K for muc = 0.10000

Estimated w_log in Allen-Dynes Tc = 6.711158 meV

Estimated BCS superconducting gap = 2.317012 meV

Estimated Tc from machine learning model = 26.009001 K

Sincerely,
Li

[hmori]

Dear Li,

I initially misunderstood that you were using EPW for Pb, but you are actually working with TaNbHfZr, correct?

To rule out the possibility that the issue originates from the orbitals, it would be prudent to re-examine them, especially when dealing with a new system in EPW. Have you plotted the spatial decay of the Wannier orbitals? If not, please check the decays by following the tutorial "Interpolation of electron-phonon matrix elements", which you can find at:

https://docs.epw-code.org/doc/School2024.html

Additionally, I noticed you are running EPW with 24x24x24 k- and q-point grids, but in ph.x, you are using a 6x6x6 q-point grid. Could you confirm the k-point grid size used in ph.x? If a coarse k-point grid was used for calculating the electron-phonon coupling in ph.x, it might be necessary to revisit and adjust the input parameters for ph.x. Keep in mind that if you use different grid sizes between ph.x and epw.x, the results are likely to differ.

Best,
Hitoshi

[fale li]

Dear Li,

I initially misunderstood that you were using EPW for Pb, but you are actually working with TaNbHfZr, correct?

To rule out the possibility that the issue originates from the orbitals, it would be prudent to re-examine them, especially when dealing with a new system in EPW. Have you plotted the spatial decay of the Wannier orbitals? If not, please check the decays by following the tutorial "Interpolation of electron-phonon matrix elements", which you can find at:

https://docs.epw-code.org/doc/School2024.html

Additionally, I noticed you are running EPW with 24x24x24 k- and q-point grids, but in ph.x, you are using a 6x6x6 q-point grid. Could you confirm the k-point grid size used in ph.x? If a coarse k-point grid was used for calculating the electron-phonon coupling in ph.x, it might be necessary to revisit and adjust the input parameters for ph.x. Keep in mind that if you use different grid sizes between ph.x and epw.x, the results are likely to differ.

Best,
Hitoshi

Dear Hitoshi，

Yes, I calculated TaNbHfZr. And I just drew the space decay, the picture is attached.

Thanks for your reminder, I have checked the k points I set. Fortunately, before doing the ph.x calculation, I set the k point of the scf.in file to 24*24*24. I also set the k point of the scf.in file to 24*24*24 before performing the epw calculation.

Sincerely,
Li

[hmori]

Dear Li,

Thank you for sharing the spatial decays. They appear to be well localized with respect to the lattice vectors. While it might be possible to slightly improve the spatial localization by increasing the number of Wannier orbitals, it is unlikely that this is the main cause of the discrepancy at this stage.

Since you are using the same k-grid size for both ph.x and EPW, the discrepancy may stem from the difference in q-grid sizes: a 6x6x6 q-grid in ph.x versus a 24x24x24 qf-grid in EPW. You could try using a 6x6x6 qf grid in the EPW calculation to see if this resolves the discrepancy.

Best regards,
Hitoshi

[fale li]

Dear Li,

Thank you for sharing the spatial decays. They appear to be well localized with respect to the lattice vectors. While it might be possible to slightly improve the spatial localization by increasing the number of Wannier orbitals, it is unlikely that this is the main cause of the discrepancy at this stage.

Since you are using the same k-grid size for both ph.x and EPW, the discrepancy may stem from the difference in q-grid sizes: a 6x6x6 q-grid in ph.x versus a 24x24x24 qf-grid in EPW. You could try using a 6x6x6 qf grid in the EPW calculation to see if this resolves the discrepancy.

Best regards,
Hitoshi

Dear Hitoshi,

Thank you for your suggestions. I tried to set nkf to 24*24*24 and nqf to 6*6*6 in epw calculation, but the results are still too large

a2f file is not found to estimate initial gap: calculating a2f files


Finish reading a2f file

Electron-phonon coupling strength = 4.7684881

Estimated Allen-Dynes Tc = 16.074877 K for muc = 0.10000

Estimated w_log in Allen-Dynes Tc = 6.554248 meV

Estimated BCS superconducting gap = 2.437997 meV

Estimated Tc from machine learning model = 29.847448 K

Sincerely,
Li