Al Critical Temperature Calculation FAILED

[Farhad Jalali_Asadabadi]

Dear ELK Developers and Users Community,

I hope this message finds you well. I am currently working on a project where I'm attempting to calculate the critical temperature of a simple single-atom compound, Aluminum (Al), using Elk.

In my approach, I've varied the ngridk values up to 42 with a fixed ngridq of 6. Despite these modifications, the computed critical temperature does not seem to align more closely with the experimental value of 1.2 K. Instead, my results fluctuate drastically around this value without converging to the precise figure. I am hoping to understand why this might be happening and would appreciate any insights the community could provide.

One of the principal advantages of using Elk, specifically the SCDFT method, is its predictive power. This tool enables us to predict the properties of even a new superconductor based solely on computational analytics, without relying on empirical parameters like mu* in Eliashberg theory to conform our models to experimental data.

However, the reliability of these predictive capabilities becomes questionable if we can't obtain consistent results for the critical temperature (Tc) for a seemingly straightforward model like Aluminum.

Here are some of my calculated results for Tc, all obtained with ngridq=6 and varying ngridk values:

With ngridk=6, Tc=1.8663
With ngridk=12, Tc=7.1790
With ngridk=18, Tc=10.2652
With ngridk=24, Tc=2.3046
With ngridk=30, Tc=0.4717
With ngridk=36, Tc=1.04326
With ngridk=42, Tc=0.8451
I have experimented with varying values for q, including 7, 8, and even 18. However, these alterations seem to affect nothing beyond increasing the computational cost. All my calculations have been performed using the Density Functional Perturbation Theory (DFPT) method.

In addition, I've employed the Supercell method along with Generalized Gradient Approximation (GGA) and optimized lattice parameters for ngridk=24 and q=6. The result from this method was a critical temperature (Tc) of 0.15.
I would greatly appreciate any guidance from the community on how to improve the consistency of these results and better align them with experimental data.
I have also found the optimization process for the k and q parameters to be rather empirical and randomly tedious. I believe there should be a mechanism in place allowing the code to comprehend the level of accuracy I require for my results, while also providing me with feedback regarding the grid values for q and k.

My ultimate goal is to utilize Elk effectively to accurately reproduce the critical temperature of various materials, starting with this simple Aluminum model. Any insights or suggestions that could assist me in this endeavor would be truly valuable.

Thank you in advance for your time and support.

Best regards,
Farhad

[J. K. Dewhurst]

Hi Farhad,

The electron-phonon coupling constant can be one of the most difficult properties to converge in electron structure calculations. This is because it has to be sampled with a double delta function over the Fermi surface.

Even something as simple as aluminium can be difficult because the coupling constant is small and the Fermi surface is nearly spherical.

It is important to converge both the q-point and k-point sets. I used an 8x8x8 q-point set with a 16x16x16 k-point set for the DFPT phonon calculation.

I then ran a series of electron-phonon coupling calculations from 16³ to 48³ k-point sets. It is important to use a smearing width which is larger than the default (0.001). We find that swidth=0.01 allows for relatively fast convergence of λ with respect to the k-point set. The choice of smearing function can also improve convergence. I used stype=1, which is Methfessel-Paxton order 1.

I've attached my input file as well as λ vs. k-point grid for swidth=0.005, 0.01 and 0.02. The converged value of λ is 0.42, which is in good agreement with previous results.

Best wishes,
Kay.

[Farhad Jalali_Asadabadi]

Dear Kay,

I am sincerely grateful for your detailed response and the invaluable information you've shared with me. It has provided me with substantial insights that I'm confident will help steer my project towards a successful direction.

I would like to mention a couple of additional observations I've made during my studies. I found that using the LSDA functional, it is challenging to accurately meet the experimental lattice parameter for Aluminium. In contrast, the GGA method seems to yield considerably improved results. I noticed in your elk.in file that the lattice parameter you used is 7.5306, whereas the experimental value is closer to 7.65. This is in line with my findings where the converged value using GGA was 7.66, which is much closer to the experimental value.

However, a limitation I encountered was that the DFPT method doesn't appear to currently be compatible with the GGA functional. To work around this, I switched to the supercell method, which does support GGA. I found that the supercell method provided more accurate values for Tc even with smaller k and q point grids.

Your advice and insights have been invaluable, and I wish to extend my deepest appreciation for the time and effort you took to assist me. I plan to apply your recommendations and share the results of my new calculations in due course.

Furthermore, if these calculations and results have been published, I would be very interested in reading the entire article. It would greatly help me in deepening my understanding of the processes involved.

Best regards,

Farhad

[TOP GUN]

https://forum.epw-code.org/viewtopic.php?f=3&t=1388&p=3713&hilit=topgun#p3713

[J. K. Dewhurst]

Hi Farhad,

We still have to add the exchange-correlation kernel f_xc for GGA. The code is written but needs checking.

Regards,
Kay.