Van Hove singularities and spectral smearing in high-temperature superconducting ${\mathrm{H}}_3\mathrm{S}$

The superconducting phase of hydrogen sulfide at $T_c=200$ K observed by Drozdov and collaborators at pressures around 200 GPa is simple bcc $Im\overline{3}m {\mathrm{H}}_3\mathrm{S}$ from a combination of theoretical and experimental confirmation. The various “extremes” that are involved—high pressure implying extreme reduction of volume, extremely high H phonon energy scale around 1400 K, extremely high temperature for a superconductor—necessitates a close look at new issues raised by these characteristics in relation to high $T_c$ itself. First principles methods are applied to analyze the ${\mathrm{H}}_3\mathrm{S}$ electronic structure, beginning with the effect of sulfur and then focusing on the origin and implications of the two van Hove singularities (vHs) providing an impressive peak in the density of states near the Fermi energy. Implications arising from strong coupling Migdal-Eliashberg theory are studied. It becomes evident that electron spectral density smearing due to virtual phonon emission and absorption must be accounted for in a correct understanding of this unusual material and to obtain accurate theoretical predictions. Means for increasing $T_c$ in ${\mathrm{H}}_3\mathrm{S}$-like materials are noted.

Figure 1

Crystal structure of $Im\overline{3}m {\mathrm{H}}_3\mathrm{S}$. Nearest neighbor S-H bonds are shown. At this nearest neighbor level, the structure consists of two interleaved ${\mathrm{ReO}}_3$ sublattices, displaced relative to one another by the body-centering vector (1,1,1)$a$/2.

Figure 2

DFT band structure of ${\mathrm{H}}_3\mathrm{S}$ at $a=5.6$ a.u., with the Fermi level set to zero. The darker bands in the $-10$ eV to +4 eV region have more H $1s$ character than the others. The position of the van Hove singularity discussed in the text is not visible along these symmetry lines.

Figure 3

Density of states of ${\mathrm{H}}_3\mathrm{S}$ compared to that of ${\mathrm{H}}_3\mathrm{H}$ at the same volume, from Wien2k calculation. The bandwidth is extremely large due to the broadening caused by short interatomic distances. The peak in $N\left(E\right)$ arises from two van Hove singularities within 300 meV of each other and occurring in the region of the Fermi energy. $N\left(E_F\right)=0.63$ states/eV for ${\mathrm{H}}_3\mathrm{S}$. The impact of H bonding with the S $3p$ orbitals is evident.

Figure 4

Density of states (per eV-f.u.) of the sequence ${\mathrm{H}}_{3}X,X=\text{P}$, S, Cl, aligned at the band filling of ${\mathrm{H}}_3\mathrm{S}$. The Fermi levels of ${\mathrm{H}}_3\mathrm{P}$ and ${\mathrm{H}}_3\mathrm{Cl}$ (using the ${\mathrm{H}}_3\mathrm{S}$ structure, hence hypothetical) are shown by the vertical lines at appropriate band fillings. The similarity is striking; a rigid band (or virtual crystal) picture of alloys near ${\mathrm{H}}_3\mathrm{S}$ holds qualitatively.

Figure 5

Left: sulfur $3p_z$ Wannier function, showing strong mixing with H $1s$ contributions above and below. Center: sulfur $3s$ Wannier function, with minor hybridization to neighboring $1s$ orbitals. Right: hydrogen centered Wannier function, revealing strong hybridization of H $1s$ with S $3p\sigma$ above and below,, as well as mixing with neighboring H $1s$ orbitals along the other two axes.

Figure 6

Isosurface of the charge density obtained from states in the energy range ${\mathrm{E}}_F\pm 1$ eV. Sulfur atoms (yellow spheres) lie at the corners and at the body center; H atoms are denoted by small blue spheres.

Figure 7

Top: isoenergy contour from FPLO for the vHs at $E=-0.43$ eV. Bottom: similarly for the vHs at $E=-0.11$ eV. The red circles pinpoint two of the symmetry related vHs in each case. The red ellipse outlines the region joined by the two van Hove singularities; two energy surfaces are “unzipped' as the energy increases between the two vHs. Color denotes the velocity, which ranges from zero (darkest tan, at the vHs) to $2.5\times {10}^8$ cm/s (deep blue).

Figure 8

${\mathrm{H}}_3\mathrm{S}$ density of states without broadening (black line, Wien2k DOS), with thermal broadening at 200 K (green line, which is hard to distinguish from the unbroadened one), and the virtual phonon broadened effective DOS $\mathrm{N}\left(E\right)$ using a Lorentzian halfwidth of 0.38 eV (red line). Note the large drop in the value at the Fermi energy (dashed vertical line).