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I want to calculate a superconducting critical temperature.

I see in papers like this one: link that they calculate $T_c$ in Quantum Espresso. They use the formula (Eq. 2 in that paper): $$T_c=\frac{\omega_{ln}}{1.2}\exp\left(-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right)$$

From this and this papers, I get that

$$\lambda=2\int\limits_0^\infty\,d\omega\,\alpha^2F(\omega)/\omega$$ $$\omega_{ln}=\frac{2}{\lambda}\int\limits_0^\infty\,d\omega\,\alpha^2F(\omega)$$ $$\frac{1}{\mu^{*}}=\frac{1}{\mu}+\ln\left(\frac{\omega_{el}}{\omega_{ph}}\right)$$ $$\mu=N(0)|V_c|$$ where $N(0)$ is the electronic density of states at the Fermi surface, and I don't quite understand which value should I take for $V_c$. Also, $\omega_{el}$ is a plasma frequency or a high-frequency peak in $Im[1/\epsilon(\omega)]$, where $\epsilon(\omega)$ is the dielectric function, and $\omega_{ph}$ is the high-frequency cutoff in the Eliashberg spectral function $\alpha^2F(\omega)$.

How to calculate this in VASP? Namely, how to calculate the Eliashberg spectral function $\alpha^2F(\omega)$ as other quantities are calculated from it?

Are there any tutorials or papers where these calculations are done in VASP?

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    $\begingroup$ We try to limit posts here to just one question. This can encourage answerers who could otherwise only address one of multiple questions asked. It also makes things more than easily searchable in the future if each post corresponds to a single question. $\endgroup$
    – Tyberius
    Commented Jul 19, 2023 at 12:34
  • $\begingroup$ @Tyberius Edited. Restricted my question to the Eliashberg spectral function as it is the most important step $\endgroup$ Commented Jul 19, 2023 at 12:47

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Calculating the Eliashberg spectral function α^2F(ω) in VASP requires some post-processing steps as VASP itself does not provide direct tools for this specific analysis. The Eliashberg spectral function is typically obtained from electron-phonon coupling calculations, which are not directly supported by VASP.

To calculate α^2F(ω) in VASP, you would need to perform the following steps:

Perform Electron-Phonon (e-ph) coupling calculation: Start by performing an electron-phonon coupling calculation using a separate software package that supports this analysis. For example, Quantum ESPRESSO, ABINIT, or similar packages are commonly used for such calculations. In this step, you calculate the electron-phonon coupling matrix elements and phonon frequencies.

Extract relevant data: Extract the electron-phonon coupling matrix elements (g_kq), phonon frequencies (ω_q), and the electronic band structure from your e-ph calculation. The electron-phonon coupling matrix elements are responsible for the interaction between electrons and phonons.

Determine the Eliashberg function: The Eliashberg spectral function α^2F(ω) can be calculated using the following formula:

α^2F(ω) = ∑_q ∑_k |g_kq|^2 δ(ω - ω_q)

Where:

∑_q: Sum over phonon wave vectors in the Brillouin zone.

∑_k: Sum over electronic states in the Brillouin zone.

g_kq: Electron-phonon coupling matrix element between electronic state k and phonon mode q.

ω_q: Phonon frequency of mode q.

δ(ω - ω_q): Delta function that takes into account the energy conservation.

Perform the sum: Implement the summation over phonon wave vectors (q) and electronic states (k) numerically in your preferred programming language or analysis tool.

Visualize the Eliashberg spectral function: Once you have computed the Eliashberg spectral function α^2F(ω), you can visualize it using plotting tools like Gnuplot, Matplotlib, or any other suitable visualization software.

Remember that calculating the Eliashberg spectral function involves advanced computations and might require some familiarity with electronic structure theory and condensed matter physics. Make sure to follow the proper theoretical background and understand the underlying concepts before proceeding with the calculations. Additionally, the exact procedure might vary depending on the specific software packages and approaches you are using for electron-phonon coupling calculations.

[Collected and summarized]

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    $\begingroup$ What does the [Collected and summarized] mean at the bottom of your answer? $\endgroup$
    – Tyberius
    Commented Jul 20, 2023 at 3:30

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