Effect of Van Hove singularities on high-$T_{\mathrm{c}}$ superconductivity in ${\mathrm{H}}_3\mathrm{S}$

One of the interesting open questions for the high-transition-temperature $\left(T_{\mathrm{c}}\right)$ superconductivity in sulfur hydrides is why high-pressure phases of ${\mathrm{H}}_3\mathrm{S}$ have extremely high $T_{\mathrm{c}}$'s. Recently, it has been pointed out that the presence of the Van Hove singularities (VHS) around the Fermi level is crucial. However, while there have been quantitative estimates of $T_{\mathrm{c}}$ based on the Migdal-Eliashberg theory, the energy dependence of the density of states (DOS) has been neglected to simplify the Eliashberg equation. In this study, we go beyond the constant DOS approximation and explicitly consider the electronic structure over 40 eV around the Fermi level. In contrast with the previous conventional calculations, this approach with a sufficiently large number of Matsubara frequencies enables us to calculate $T_{\mathrm{c}}$ without introducing the empirical pseudo Coulomb potential. We show that while ${\mathrm{H}}_3\mathrm{S}$ has much higher $T_{\mathrm{c}}$ than ${\mathrm{H}}_2\mathrm{S}$ for which the VHS is absent, the constant DOS approximation employed so far seriously overestimates (underestimates) $T_{\mathrm{c}}$ by $\sim 60$ K $(\sim 10$ K) for ${\mathrm{H}}_3\mathrm{S}$ $\left({\mathrm{H}}_2\mathrm{S}\right)$. We then discuss the impact of the strong electron-phonon coupling on the electronic structure with and without the VHS and how it affects the superconductivity. In particular, we focus on (1) the feedback effect in the self-consistent calculation of the self-energy, (2) the effect of the energy shift due to the zero-point motion, and (3) the effect of the changes in the phonon frequencies due to strong anharmonicity. We show that the effect of (1)–(3) on $T_{\mathrm{c}}$ is about 10–30 K for both ${\mathrm{H}}_3\mathrm{S}$ and ${\mathrm{H}}_2\mathrm{S}$. Eventually, $T_{\mathrm{c}}$ is estimated to be 181 K for ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and 34 K for ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa, which explains the pressure dependence of $T_{\mathrm{c}}$ observed in the experiment. In addition, we evaluate the lowest-order vertex correction beyond the Migdal-Eliashberg theory and discuss the validity of the Migdal approximation for sulfur hydrides.

Figure 1

Band structures of (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GP along several high-symmetry lines. Energy is measured from the Fermi level.

Figure 2

Densities of states of (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ and (c) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$. (b) and (d) show the enlarged views of (a) and (c) around the Fermi level, respectively. The unit of the vertical axis is states per an atom. In (b), there is a sharp peak around the Fermi level and the peak width is comparable with the phonon energy scale, while (d) does not show any characteristic structures within the energy range of 1 eV.

Figure 3

Phonon dispersions and Eliashberg functions ${\alpha }^{2}F\left(\omega \right)$ in (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$.

Figure 4

$T_{\mathrm{c}}$ against the number of Matsubara frequencies for (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa within the constant DOS approximation. $T_{\mathrm{c}}$ is calculated for fixed $\mu$ values of (a) $\mu =0.32$ and (b) $\mu =0.16$ evaluated with the RPA. The value of ${\omega }_{\mathrm{el}}$ is fixed at 20 eV for both structures.

Figure 5

Number of Matsubara frequencies dependence of $T_{\mathrm{c}}$ for (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa based on the ME theory with energy-dependent DOS. The screened Coulomb interaction is evaluated with the RPA.

Figure 6

Numerical convergence of $T_{\mathrm{c}}$ in $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa. SC (red line) and one-shot (blue line) denote the self-consistent and one-shot calculations, respectively, for the normal Green's function. The number of Matsubara frequencies is fixed at 1024. $\mathbf{q}$ and $\mathbf{k}$ represent the Brillouin zone sampling employed for the phonon dynamical matrix and the gap equation, respectively.

Figure 7

Renormalization function $Z\left(i{\omega }_n\right)$ as a function of the Matsubara frequency for $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa. Here the Matsubara frequency is defined by ${\omega }_n=(2n-1)\pi /\beta$. We choose the band crossing the Fermi level (the fifth band from the band bottom). SC (red line) and one-shot (blue line) denote the self-consistent and one-shot calculation, respectively, for the normal Green's function. Plotted $Z$ is calculated at 185 K in the SC calculation and 166 K in the one-shot calculation. The temperature dependence of $Z$ is weak and irrelevant.

Figure 8

Band dispersions with (red solid line) and without (blue broken line) ZPR for (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa. Band structures with ZPR are plotted by the Wannier interpolation.

Figure 9

Densities of states with (red solid line) and without (blue broken line) ZPR for (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa near the Fermi level.

Figure 10

Phonon-dispersion relation for (a) $Im\overline{3}m$ ${\mathrm{H}}_3\mathrm{S}$ at 250 GPa and (b) $P\overline{1}$ ${\mathrm{H}}_2\mathrm{S}$ at 140 GPa. The red solid line shows the dispersion considering the anharmonic effect within the SCPH theory, and the blue broken line shows the result within the harmonic approximation.