Allen dynes formula for Tc is not giving correct lambda from epw calcs

[simba2828]

Dear EPW Experts,

I am running one anisotropic superconducting calculation where the calculation gives the following output in epw.out file:

Code: Select all

......
    a2f file is found and will be used to estimate initial gap


     Finish reading a2f file

     Electron-phonon coupling strength =    1.6514433

     Estimated Allen-Dynes Tc =     3.105056 K for muc =    0.10000

     Estimated w_log in Allen-Dynes Tc =     2.153136 meV

     Estimated BCS superconducting gap =     0.470929 meV

     Estimated Tc from machine learning model =     4.895778 K
.......

With this given lambda (el-ph strength)=1.6514433, w_log= 2.153136 meV and mu*=0.1, I am not getting Tc= 3.105056 K. Could anyone clarify where I am misinterpreting? Is this lambda (1.6514433), given by Migdal-Eliashberg approach, is different from Allen-dynes lambda? Or w_log is something different?
I am attaching a figure where the Allen-Dynes formula for Tc is given.

Thank you for your response.

Sincerely,
Shubham

[simba2828]

I noticed that w_log is in meV while it should be in Kelvin. Could anyone tell what is the conversion formula for logarithmic average of omega? Is it E = kB.T?

Also, I noticed that from QE calculations I am getting lambda close to 1 while EPW gives 1.65. Why there is a discrepancy?
Additionally, a2F.dos also print lambda at the end of the file which is 0.77, again a discrepancies. Which one should I consider?

Output of the `lambda` file:

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  Electron-phonon coupling constant, lambda

 Broadening   0.0200 lambda       0.9669 dos(Ef) 23.4848 omega_ln [K]     52.2538
 Broadening   0.0400 lambda       0.8246 dos(Ef) 22.8038 omega_ln [K]     57.1286
 Broadening   0.0600 lambda       0.8655 dos(Ef) 20.8352 omega_ln [K]     52.5826
 Broadening   0.0800 lambda       0.9878 dos(Ef) 19.8054 omega_ln [K]     46.4001
 Broadening   0.1000 lambda       1.0742 dos(Ef) 19.6655 omega_ln [K]     43.4068
 Broadening   0.1200 lambda       1.0882 dos(Ef) 20.0924 omega_ln [K]     42.8778
 Broadening   0.1400 lambda       1.0565 dos(Ef) 20.8684 omega_ln [K]     43.8291
 Broadening   0.1600 lambda       1.0120 dos(Ef) 21.8472 omega_ln [K]     45.6556
 Broadening   0.1800 lambda       0.9726 dos(Ef) 22.9001 omega_ln [K]     47.8655
 Broadening   0.2000 lambda       0.9425 dos(Ef) 23.9120 omega_ln [K]     50.0714

[Shashi]

Dear Subham,

The w_log unit in EPW is indeed correct and expressed in meV. When converted to Kelvin, w_log is 24.976 K. The w_log expression depends on electron-phonon coupling strength, which differs between EPW and QE in your case giving differences in w_log values. Using your formula, I estimated the Tc to be 3.1038 K, which matches exactly with the Tc printed in EPW output. Please recheck.

Regarding the discrepancy in electron-phonon coupling strength values between QE and EPW, it could likely be attributed to differences between k and q-grid sizes. Typically, QE uses coarser grids, whereas EPW relies on much finer grid sizes. You can also look at the paper by Hitoshi (Phy. Rev. B 110, 064505 (2024)), where he showed for Pb (Table 2) that the electron-phonon coupling strengths differ with grid sizes.

Regards,
Shashi

[simba2828]

Dear Shashi,

Thank you for your reply and the reference. But isn't lambda is very large? In this paper by Allen and Dynes: https://journals.aps.org/prb/pdf/10.1103/PhysRevB.12.905 , it has been explained whenever lambda > 1.5 there is correction factor used in Allen dynes Tc. Is that factor taken into account for the EPW calculations which can explain lambda 1.65?
Thank you again.

Shubham

[Shashi]

Dear Subham,

The paper you mentioned discusses the scenarios for asymptotic Tc in the case of lambda < 1 and lambda > 1 in Sec. III. To clarify, there is no additional factor applied in EPW for calculating Allen-Dynes' Tc, and the equation used in EPW is the same as the one you shared. If you wish to obtain more accurate Tc values, I would recommend solving the Eliashberg equations, either using the isotropic or anisotropic formalism in EPW. This approach might be better to compare Tc with your semi-empirical estimation.

Regards,
Shashi

[simba2828]

Thank you Shashi for the clarification.