Comparison of pressurized sulfur hydride with conventional superconductors

A recent report that sulfur hydride under pressure is an electron-phonon superconductor with a $T_c$ of 190 K has been met with much excitement although it is yet to be confirmed. Based on several electron-phonon spectral density functions already available from density functional theory, we find that the electron-phonon spectrum is near optimum for $T_c$ with a particularly large value of its characteristic phonon energy ${\omega }_{\ln }$ which is due to the small hydrogen mass. We find that the thermodynamic universal BCS ratios are near those for Pb and ${\mathrm{Nb}}_3\mathrm{Sn}$. We suggest that optical measurements could be a useful tool to establish the existence and nature of the superconductivity in this system. Conventional superconductors are in the impurity-dominated dirty limit. By contrast sulfur hydride will be in the clean limit because of its large energy gap scale. The ac optical conductivity will display distinct and separate signatures of the superconducting gap in the low-energy impurity-dominated range of the optical spectrum and additional phonon structures at higher energies where the clean limit applies.

Figure 1

(a) The anharmonic ${\alpha }^{2}F\left(\omega \right)$ spectrum for ${\mathrm{H}}_3\mathrm{S}$ at 200 GPa given in Ref. [3]. The functional derivative $\delta T_c/\delta {\alpha }^{2}F\left(\omega \right)$ vs $\omega$ for this same spectrum is also shown scaled up by a factor of 12. It peaks at $\omega =7k_B{T}_c$ (dashed line) which is very close to the characteristic phonon frequency ${\omega }_{\ln }$ (dash-dotted line) for the ${\alpha }^{2}F\left(\omega \right)$ demonstrating that the spectrum is highly optimized for $T_c$. (b) $T_c/A$ vs ${\mu }^{*}$ where $A$ is the area under the ${\alpha }^{2}F\left(\omega \right)$ spectrum. The black curve with open circles is the theoretical upper limit for $T_c$ with fixed $A$ and the solid circles are data for several superconductors constructed from Table IV of Ref. [9]. The gold star is for the ${\alpha }^{2}F\left(\omega \right)$ spectrum in (a) with $T_c=190$ K and ${\mu }^{*}=0.18$. The blue curve represents how $T_c/A$ would change for varying ${\mu }^{*}$, demonstrating that this spectrum is near optimal.

Figure 2

The gap ratio $2{\mathrm{\Delta }}_0/k_B{T}_c$ vs $T_c/{\omega }_{\ln }$. The solid black points are the data for several conventional superconductors taken from Table IV of Ref. [9]. Some have been labeled. The dashed line is an analytical formula developed for conventional superconductors from strong coupling Eliashberg theory [18]. The gold star is the value calculated for the anharmonic spectrum shown in Fig. 1 and the red stars are the values for the harmonic spectra [10] with $T_c=190$ K.

Figure 3

The real part of the optical conductivity (scaled up by a factor of ${10}^4$) vs $\omega$ at temperature $T=0.1T_c$. Shown are the normal and superconducting state curves (red and blue, respectively). ${\omega }_p$ is the plasma frequency. The ${\alpha }^{2}F\left(\omega \right)$ spectrum from Fig. 1 is shown shifted to the right along the frequency axis by $2{\mathrm{\Delta }}_0$.