3.41 Superconductivity

The present implementation obtains phonon linewidths from the friction/EPC machinery and then:

1. 1.

   constructs ${\alpha }^{2}F(\omega )$ on a user-defined frequency grid,

2. 2.

   evaluates $\lambda =2\int 𝑑\omega {\alpha }^{2}F(\omega )/\omega$,

3. 3.

   evaluates ${\omega }_{\log }$ and $T_{\mathrm{c}}$ via Allen-Dynes.

For periodic systems, the current friction-based EPC workflow corresponds to the $\mathrm{\Gamma }$-point / $𝐪=0$ setup. The superconductivity driver can be combined with scans in friction broadening and electronic temperature (see keywords below).

The Eliashberg function is evaluated as

$${\alpha }^{2}F(\omega )=\frac{1}{2\pi N_{{\epsilon }_{𝐅}}}\sum _{𝐪,\nu }\delta \left(\omega -{\omega }_{𝐪\nu }\right)\frac{{\gamma }_{𝐪\nu }}{{\omega }_{𝐪\nu }}.$$

(3.185)

The electron-phonon coupling parameter is

$$\lambda =2\int 𝑑\omega \frac{{\alpha }^{2}F(\omega )}{\omega }.$$

(3.186)

In the present implementation, the linewidths ${\gamma }_{𝐪\nu }$ passed from the friction module are treated as full widths at half maximum (FWHM).

The superconducting transition temperature is computed with the Allen-Dynes form

$$T_{\mathrm{c}}=\frac{f_1{f}_2{\omega }_{\log }}{1.20}\exp \left(-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}\right),$$

(3.187)

with

	$f_1$	$={\left(1+{\left[{\displaystyle \frac{\lambda }{2.46(1+3.8{\mu }^{\ast })}}\right]}^{3/2}\right)}^{1/3},$		(3.188)
	$f_2$	$=\left(1+{\displaystyle \frac{{\lambda }^2\left({\overline{\omega }}_2/{\omega }_{\log }-1\right)}{{\lambda }^2+{\left[1.82(1+6.3{\mu }^{\ast })({\overline{\omega }}_2/{\omega }_{\log })\right]}^2}}\right).$

The superconductivity driver reuses friction settings for linewidth generation:

• •

  friction_broadening_width: one value, or a scan start end n_steps.

• •

  friction_temperature: one value, or a scan T_start T_end n_steps.

• •

  friction_print_tensors: defaults to .false. for superconductivity runs to reduce verbose tensor output; set explicitly to .true. if tensor printing is desired.

If both are scans, all combinations are evaluated (Cartesian product of smearing and temperature grids).

Two files are written by the superconductivity module:

• •

  eliashberg_function.out: frequency grid (eV) and ${\alpha }^{2}F$ columns for each parameter point.

• •

  superconductivity_tc.out: one row per parameter point with smearing (eV), electronic temperature (K), ${\mu }^{\ast }$, $\lambda$, ${\omega }_{\log }$ (meV), DOS at $E_F$ (1/eV/spin), Allen-Dynes prefactors $f_1$, $f_2$, and $T_c$ (K).

Additionally, use empty_states large enough (or calculate_all_eigenstates) to ensure sufficient unoccupied states are available for the EPC/frequency-resolved analysis.