Different values of lambda and Tc (iso vs aniso?)

jhlee · January 4, 2019, 9:09am

Dear all,

I am calculating lambda using two different methods implemented in EPW.

First one is obtained by activating anisotropic Eliashberg equation related tags.

[code:257u2p1r]
eliashberg = .true.
laniso = .true.
limag = .true.
lpade = .true.
[/code:257u2p1r]

The result is as below:

[code:257u2p1r]
===================================================================
Solve anisotropic Eliashberg equations
===================================================================

 Finish reading &#46;freq file 

              Fermi level (eV) =     1&#46;4490810867E+00
 DOS(states/spin/eV/Unit Cell) =     7&#46;4601791612E-01
        Electron smearing (eV) =     1&#46;0000000000E-01
             Fermi window (eV) =     4&#46;0000000000E-01
 Nr irreducible k-points within the Fermi shell =       134 out of       231
       2 bands within the Fermi window


 Finish reading &#46;egnv file 


 Max nr of q-points =       466


 Finish reading &#46;ikmap files


 Start reading &#46;ephmat files


 Finish reading &#46;ephmat files 

 lambda_max =             1&#46;3814131   lambda_k_max =             0&#46;7463819

 Electron-phonon coupling strength =    0&#46;5890385

 Estimated Allen-Dynes Tc =       1&#46;0488626 K for muc =    0&#46;16000

 Estimated BCS superconducting gap =       0&#46;0001591 eV

[/code:257u2p1r]

and the second one is obtained by activating a2f tags and deactivating anisotropic Eliashberg tags.

[code:257u2p1r]
a2f = .true.
phonselfen = .true.
[/code:257u2p1r]

The result is as below:

[code:257u2p1r]
===================================================================
Eliashberg Spectral Function in the Migdal Approximation
===================================================================

 lambda &#58;    0&#46;5689812
 lambda_tr &#58;    0&#46;7333977

 Estimated Allen-Dynes Tc

 logavg =    0&#46;0006900 l_a2F =    0&#46;5735213
 mu =   0&#46;10 Tc =         2&#46;164145346702 K
 mu =   0&#46;12 Tc =         1&#46;691229225724 K
 mu =   0&#46;14 Tc =         1&#46;276401502717 K
 mu =   0&#46;16 Tc =         0&#46;922975116778 K
 mu =   0&#46;18 Tc =         0&#46;632737937327 K
 mu =   0&#46;20 Tc =         0&#46;405375457882 K

[/code:257u2p1r]

The lambda obtained by the first one is 0.5890385, while the lambda from the second method is 0.5689812. and T_c is also slightly different.

Could it be understood it is because of the effect of anisotropic nature of the system? or some bugs in the code or numerical inaccuracy? I’ve calculated the second part by re-loading previously obtained files that required solving the Eliashberg equation. Any comments would be helpful and appreciate.

Best regards,
Jun-Ho

roxana · January 4, 2019, 2:24pm

Dear Jun-Ho,

The reason that the two values are slightly different is because they are calculated using different formulas.

For the default setting

a2f = .true.
phonselfen = .true.

the e-ph coupling strength is calculated using the imaginary part of the phonon self-energy [Eq. 9 in Computer Physics Communications 209, 116 (2016)]. In this case \lambda= Im[\Pi_{q\nu}] / [\pi N_F \omega^2_{q\nu}].

If you set

delta_approx = .true.

together with

a2f = .true.
phonselfen = .true.

the e-ph coupling strength will be calculated using the double delta approximation [Eq. 11 in Computer Physics Communications 209, 116 (2016)]. This second approach is also used when calculating the e-ph coupling strength with the flag eliashberg=.true.

Just a note, the T_c value are going to be different when solving the Eliashberg equations with laniso=.true. or liso=.true.

Best,
Roxana

jhlee · January 5, 2019, 5:13am

Dear Roxana,

Thank you very much for your reply.

Does the significant difference between the calculated values of lambda obtained by two different methods mean that these are not converged results with respect to fine k- or q-grids?

Best regards,
Jun-Ho

roxana · January 5, 2019, 4:43pm

Hi Jun-Ho,

I think the difference you are seeing is not that large, but you should always check how the value for lambda changes with respect to k- and q-grids regardless of the method you use.

Best,
Roxana

jhlee · January 9, 2019, 12:36am

Thank you very much!

Best,
Jun-Ho

liu_xinbao · March 14, 2019, 3:07am

I calculate phonon selfenergy with and without delta_approx.
And I get different result,I want to ask which one is accurate?
Should I use delta_approx. when I calculate a2F,lambda and Tc?
[img:3fiv6a5g]https://i.imgur.com/zbDPViT.png[/img:3fiv6a5g]

roxana · March 15, 2019, 11:06pm

Hi,

Both expressions for are correct, it depends what you are using them for. The double delta approximation is generally used to estimate a2f, lambda, and Tc for superconductors.

Best,
Roxana

liu_xinbao · March 18, 2019, 6:02am

Thank you very much,roxana