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From Wikipedia, superconductivity is

the set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor.

One of the conventional explanations is the BCS theory (Bardeen–Cooper–Schrieffer theory) that dictates that the superconductivity is due to the condensation of electron pairs called Cooper pairs.

Let's suppose I have a given material: is there a theoretical method or modeling tool that I could use to test if the material will behave as a superconductor?

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    $\begingroup$ looking for an answer as well. $\endgroup$ – Shahid Sattar Jul 27 '20 at 12:18
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It is possible to calculate the superconducting critical temperature $T_{\mathrm{c}}$ of phonon-mediated superconductors using first principles modelling methods. However, the calculations are not trivial.

Theory. The basic quantity that goes into the calculation is the electron-phonon matrix element:

$$ g(\mathbf{k},n;\mathbf{k}',n';\mathbf{q},\nu)=\left\langle\mathbf{k}n\left|\frac{\partial V}{\partial u_{\mathbf{q}\nu}}\right|\mathbf{k}'n'\right\rangle. $$

This represents the scattering from an initial electron state $(\mathbf{k}',n')$ to a final electron state $(\mathbf{k},n)$ mediated by a phonon $(\mathbf{q},\nu)$, where the electron-phonon interaction is the change in the potential $\delta V$ due to the presence of a phonon of amplitude $\delta u_{\mathbf{q}\nu}$. Once you have this matrix element, everything follows:

  1. First, you realise that the electrons that contribute to superconductivity are those around the Fermi energy $\varepsilon_{\mathrm{F}}$, so that you calculate the average of the electron-phonon coupling matrix element for a phonon $(\mathbf{q},\nu)$ over the Fermi surface:

    $$ \langle\langle|g_{\mathbf{q}\nu}|^2\rangle\rangle=\frac{\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k},n}\frac{1}{N_{\mathbf{k}'}}\sum_{\mathbf{k}',n'}\delta(\epsilon_{\mathbf{k}n}-\epsilon_{\mathrm{F}})\delta(\epsilon_{\mathbf{k}'n'}-\epsilon_{\mathrm{F}})|g(\mathbf{k},n;\mathbf{k}',n';\mathbf{q},\nu)|^2}{\left[\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k},n}\delta(\epsilon_{\mathbf{k}n}-\epsilon_{\mathrm{F}})\right]^2}. $$

    The sums run over a grid of $N_{\mathbf{k}}$ $\mathbf{k}$-points and the delta functions select only those electrons whose energies are near the Fermi energy. In this expression I have directly written the sums over a discrete set of $\mathbf{k}$-points (rather than the integrals that one gets from the analytical theory) to prepare for the discussion of the numerics below.

  2. One then usually defines the so-called electron-phonon coupling constant of phonon mode $(\mathbf{q},\nu)$ as

    $$ \lambda_{\mathbf{q}\nu}=\frac{2N(\varepsilon_{\mathrm{F}})}{\omega_{\mathbf{q}\nu}}\langle\langle|g_{\mathbf{q}\nu}|^2\rangle\rangle, $$

    where $N(\varepsilon_{\mathrm{F}})$ is the density of states at the Fermi level and $\omega_{\mathbf{q}\nu}$ is the phonon frequency. The total electron-phonon coupling constant is then obtained by summing (integrating) over the phonon Brillouin zone:

    $$ \lambda=\frac{1}{N_{\mathbf{q}}}\sum_{\mathbf{q},\nu}\lambda_{\mathbf{q}\nu}. $$

  3. You can then calculate the superconducting critical temperature, with methods ranging from the semiempirical McMillan formula to the Green's function based Migdal-Eliashberg formalism. In any case, the basic quantity is still the electron-phonon matrix element above.

Practical calculations. The basic matrix element $g(\mathbf{k},n;\mathbf{k}',n';\mathbf{q},\nu)$ can be calculated relatively easily within density functional theory, either using finite differences or linear response, and codes that implement this include Quantum Espresso and Abinit. The major challenge of these calculations comes from the double sum over the electronic Brillouin zone (sums over $\mathbf{k}$ and $\mathbf{k}'$) and the sum over the phonon Brillouin zone (sum over $\mathbf{q}$). These sums converge very slowly, so many terms need to be included. It is often the case that the number of terms needed is impossibly large for a direct treatment, so what is typically done is to calculate the electron-phonon matrix elements on a coarse grid of $\mathbf{k}$ and $\mathbf{q}$ points, and then some interpolation method is used to obtain these terms on finer grids. Perhaps the most-used approach to this is Wannier interpolation, as implemented in the EPW code.

Other comments. (i) The approach described above is possibly the most extensively used approach to calculate $T_{\mathrm{c}}$ using first principles methods, and it leads to reasonable values for most phonon-mediated superconductors. There are alternative approaches to perform these calcualtions, such as so-called density functional theory for superconductors (SCDFT), but I don't know enough about it to write an answer. Hopefully someone more knowledgeable will. (ii) I don't think it is possible to study superconductors that are not phonon-mediated using first principles methods, but I would be happy to learn more if someone knows better.

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  • $\begingroup$ +1. I was surprised to see no mention of "BCS" here but it is indeed mentioned in the question at least. $\endgroup$ – Nike Dattani Jul 27 '20 at 16:36
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    $\begingroup$ @NikeDattani However, the answer mentions Migdal-Eliashberg theory, which reduces to BCS at weak electron-phonon coupling. $\endgroup$ – Anyon Jul 27 '20 at 16:52
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    $\begingroup$ @Anyon It's a great contribution to the site. This is why we all love ProfM! $\endgroup$ – Nike Dattani Jul 27 '20 at 16:53
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As mentioned in ProfM's answer, there is an extension to DFT known as SCDFT, that aims to account for the symmetry breaking that occurs in a superconductor. Lecture notes on SCDFT from Antonio Sanna can be found here.

However, like ProfM, I don't know enough about this approach to provide any real details. Instead, the reason I write this answer is to mention a very recent development that was published yesterday: A. Sanna, C. Pellegrini, and E. K. U. Gross, "Combining Eliashberg Theory with Density Functional Theory for the Accurate Prediction of Superconducting Transition Temperatures and Gap Functions," Physical Review Letters 125, 057001 (2020)

In short, it seems that Sanna and collaborators have managed to create a functional that approximates Eliashberg theory much better than the standard SCDFT functional, LM2005. Their method is computationally cheaper than solving the full Eliashberg equations, and appears to be rather accurate for the systems they've tested so far, which may open up a new door to predicting new materials with phonon-mediated superconductivity. Below is shown a figure comparing the Eliashberg result, the new SCDFT functional and LM2005 for a simple model as a function of the electron-phonon coupling $\lambda$. Below that is their figure comparing theory and experiment. $T_c$ is the critical temperature, and $\Delta$ is the gap.

Figure 2: Comparison of critical temperatures as a function of electron-phonon coupling

Figure 3: Comparison between theoretical and experimental critical temperatures and gaps

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  • $\begingroup$ +1. I have only 4 votes left today! Is SCDFT something that could be summarized in 2-3 paragraphs here? mattermodeling.stackexchange.com/q/1513/5 $\endgroup$ – Nike Dattani Jul 28 '20 at 18:34
  • $\begingroup$ @NikeDattani I guess it's your push for new questions that's led to much activity on the site recently! I think SCDFT probably could be summarized well by someone who understands it properly. $\endgroup$ – Anyon Jul 28 '20 at 18:40
  • $\begingroup$ Thank you. I pushed hard for it, to get us within the Area51 guidelines by Day 90, and we finally made it to 10 questions/day! In addition to posting on Meta, I campaigned hard with members of the Facebook Group and Followers on our Twitter page. Our % answered dropped due to the huge number of new questions, but once it gets back up to "Excellent" on A51, I'll archive the page on WayBackMachine as a memory of our success :) $\endgroup$ – Nike Dattani Jul 28 '20 at 18:47

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