Negative value of superconducting gap

[simba2828]

Dear EPW developers and Users,

After successfully finishing the phonon, scf and nscf calculations I am trying to calculate superconducting properties for my 2D system.
I notice that the Min. value of superconducting Gap is negative after all the cycles with different temperatures (8K, 12K, 17K and 20K I tried) with Estimated Tc = 14.91 K. The following is from my epw.out file:

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Estimated Allen-Dynes Tc =    14.910578 K for muc =    0.05000

     Estimated w_log in Allen-Dynes Tc =    12.156787 meV

     Estimated BCS superconducting gap =     2.261414 meV

     Estimated Tc from machine learning model =    17.992366 K


     WARNING WARNING WARNING

     The code may crash since tempsmax =   20.000 K is larger than Allen-Dynes Tc =    14.911 K

     temp(  1) =      8.00000 K

     Solve anisotropic Eliashberg equations on imaginary-axis

     Total number of frequency points nsiw(     1) =     23
     Cutoff frequency wscut =     0.1018

The followings are the Min/Max values of superconducting gaps with different temps:
temp=8K

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Convergence was reached in nsiter =     25

     Chemical potential (itemp =   1) =    -1.8116930967E+00 eV

     Temp (itemp =   1) =    8.000 K  Free energy =    -0.009477 meV

     Min. / Max. values of superconducting gap =   [color=#FF0000] -0.340667[/color]    4.740647 meV
     iaxis_imag   :     10.70s CPU     10.78s WALL (       1 calls)


     Pade approximant of anisotropic Eliashberg equations from imaginary-axis to real-axis
     Cutoff frequency wscut =     0.1000

        pade    Re[znorm]   Re[delta] [meV]
         20   1.999132E+00   3.304228E+00

     Convergence was reached for N =     20 Pade approximants

     Min. / Max. values of superconducting gap =[color=#FF0000]    -0.353636 [/color]   5.027398 meV
     raxis_pade   :      1.13s CPU      3.37s WALL (       1 calls)

temp = 12K

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 Convergence was reached in nsiter =     15

     Chemical potential (itemp =   2) =    -1.8116930967E+00 eV

     Temp (itemp =   2) =   12.000 K  Free energy =    -0.006623 meV

     Min. / Max. values of superconducting gap =  [color=#FF0000]  -0.335653 [/color]   4.569238 meV
     iaxis_imag   :     13.65s CPU     13.76s WALL (       2 calls)


     Pade approximant of anisotropic Eliashberg equations from imaginary-axis to real-axis
     Cutoff frequency wscut =     0.1000

        pade    Re[znorm]   Re[delta] [meV]
         14   2.068469E+00   3.184121E+00

     Convergence was reached for N =     14 Pade approximants

     Min. / Max. values of superconducting gap =   [color=#FF0000] -0.333999[/color]    4.926744 meV
     raxis_pade   :      2.31s CPU      6.50s WALL (       2 calls)

temp = 17K

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Convergence was reached in nsiter =     29

     Chemical potential (itemp =   3) =    -1.8116930967E+00 eV

     Temp (itemp =   3) =   17.000 K  Free energy =    -0.002380 meV

     Min. / Max. values of superconducting gap =    [color=#FF0000]-0.290275 [/color]   3.723432 meV
     iaxis_imag   :     16.98s CPU     17.13s WALL (       3 calls)


     Pade approximant of anisotropic Eliashberg equations from imaginary-axis to real-axis
     Cutoff frequency wscut =     0.1000

        pade    Re[znorm]   Re[delta] [meV]
         10   2.152384E+00   2.402631E+00

     Convergence was reached for N =     10 Pade approximants

     Min. / Max. values of superconducting gap =   [color=#FF0000] -0.332317 [/color]   5.359331 meV
     raxis_pade   :      3.47s CPU      9.73s WALL (       3 calls)

temp = 20K

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Convergence was reached in nsiter =     47

     Chemical potential (itemp =   4) =    -1.8116930967E+00 eV

     Temp (itemp =   4) =   20.000 K  Free energy =    -0.000623 meV

     Min. / Max. values of superconducting gap =    [color=#FF0000]-0.224141[/color]    2.739538 meV
     iaxis_imag   :     20.75s CPU     20.94s WALL (       4 calls)


     Pade approximant of anisotropic Eliashberg equations from imaginary-axis to real-axis
     Cutoff frequency wscut =     0.1000

        pade    Re[znorm]   Re[delta] [meV]
          8   2.308737E+00   1.123975E+00

     Convergence was reached for N =      8 Pade approximants

     Min. / Max. values of superconducting gap =    [color=#FF0000]0.224891[/color]  3.098743 meV

The following is my epw input file:

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&inputepw
  prefix      = 'nsms',
  outdir      = './out/'

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

  epwwrite = .true.
  epwread  = .false.

  !etf_mem     =  1 

  nbndsub     =  17,
  !nbndskip = 0
  bands_skipped = 'exclude_bands = 1:12,30,31'

  wannierize  = .false.
  num_iter    = 500
  dis_froz_max= 0.9
  dis_froz_min= -7.5

  !wdata(1)    = 'exclude_bands=30,31'
  wdata(1)   = 'dis_mix_ratio   = 0.5'
  wdata(2)   = 'dis_num_iter    = 2000'
  wdata(3)    = 'bands_plot : true'
  wdata(4)    = 'bands_num_points : 300'
  wdata(5)    = 'bands_plot_format : xmgrace gnuplot'
  wdata(6)    = 'begin kpoint_path'
  wdata(7)    = 'M 0.5 0.0 0.0 K 0.3333 0.3333 0.0'
  wdata(8)    = 'K 0.3333 0.3333 0.0 G 0.0 0.0 0.0'
  wdata(9)    = 'G 0.0 0.0 0.0 M 0.5 0.0 0.0'
  wdata(10)    = 'end kpoint_path'
  

  
  iverbosity  = 2

  eps_acustic = 0.1    ! Lowest boundary for the phonon frequency 
  ephwrite    = .true. ! Writes .ephmat files used when Eliasberg = .true.

  nsmear      = 1
  delta_smear = 0.01 ! eV 0.04

  degaussq     = 0.5 ! meV
  nqstep       = 500

  eliashberg  = .true.

  laniso = .true.
  limag = .true.
  lpade = .true.

  !lifc = .true.

  fermi_plot = .true.

  conv_thr_iaxis = 1.0d-4


  !nstemp   = 1     ! Nr. of temps
  !temps    = 15.00 ! K  provide list of temperetures OR (nstemp and temps = tempsmin  tempsmax for even space mode)

  temps(1)  = 8
  temps(2)  = 12
  temps(3)  = 17
  temps(4) = 20
  !temps(5) = 6

  !max_memlt = 5  !GB
  !vme=.false. 

  nsiter   = 500
  [color=#FF0000]degaussw    = 0.04 ! eV ~  1/4 of fsthick
  wscut = 0.5   ! eV 10 times of  Upper limit over frequency integration/summation in the Elisashberg eq(1 cm-1 ~ 1/8000 eV)
  fsthick     = 0.18  ! eV ~ 4 times the maximum phonon frequency 
  muc     = 0.05[/color]

  dvscf_dir   = '../phonon_angs/save'
  
  !system_2d = .true.

  nk1         = 20
  nk2         = 20
  nk3         = 1

  nq1         = 4
  nq2         = 4
  nq3         = 1

  mp_mesh_k = .true.
  nkf1 = 60
  nkf2 = 60
  nkf3 = 1

  nqf1 = 60
  nqf2 = 60
  nqf3 = 1
 /

The Wannier projected bands are in good agreement with the DFT band structure. And I selected the values according to a previous post (viewtopic.php?f=3&t=1682) which I also mentioned in the epw input file. The maximum frequency in the phonon dispersion is around 380 cm-1. So accordingly I set fsthick and wscut.

Also When I plot Superconducting gap along the imaginary axis from the file "nsms.imag aniso 008.00", I see that the gap value goes in the negative region. I am attaching the figure here.


Could someone please help me out why I am getting the negative Min. superconducting gap and this negative region delta?

Sincerely,
Shubham
IIT Kharagpur, India

[hpaudya1]

Hi Shubham,

The negative value of superconducting gap is not physical. However, we do not restrict the calculation based on the -/+ value of the superconducting gap, any numerical value is possible in the calculation. If the negative value is too small, that can the a numerical error.

I recommend you to do some convergence tests based on your fsthick, fine k/q meshes, and wscut and make sure that the calculation is converged wrt these parameters.

Happy EPWing,
Hari

[simba2828]

Thank You Hari,

Could you please tell me which quantities I check for the convergence for (1) fsthick (2) k/q point and (3) wscut? Say, in scf calculation we can check the ground state energy but what should I check for the above flags? Though I checked several tests with respect to fstick/k/q/wscut but the negative value of SC gap persists.
I checked for (1) k/q 32 32 1/32 32 1 upto 240 240 1 / 120 120 1
(2) fsthick: 2 to 0.4
(3) wscut = 0.4 to 1.0 ! eV 10 times of Upper limit over frequency

What I found that with very fine k/q mesh the lower gap in "prefix.imag_aniso_012.00" file shifts below the 0.0 value. Does it mean that the system does not show a two-gap nature?

Thank you