Pressure-induced superconductivity up to 13.1 K in the pyrite phase of palladium diselenide $\mathrm{PdS}{\mathrm{e}}_2$

The evolution of electrical transport properties, the electronic band structure, and lattice dynamics of $\mathrm{PdS}{\mathrm{e}}_2$ is studied under high pressure. The emergence of superconductivity is reported in the high-pressure pyrite-type phase of $\mathrm{PdS}{\mathrm{e}}_2$. In this transition-metal dichalcogenide, the critical temperature of superconductivity rapidly increases with pressure up to 13.1 K. Ab initio electronic band structure calculations indicate the presence of Dirac and nodal-line fermions in the vicinity of the Fermi energy protected by the pyrite structure symmetry, which can lead to interesting superconducting states. Raman spectroscopy shows a direct correlation between critical temperature and bonding strength of Se-Se dumbbells in $\mathrm{PdS}{\mathrm{e}}_2$, underlining the crucial role of bonding for tuning the superconductivity.

Figure 1

Crystal structures of $\mathrm{PdS}{\mathrm{e}}_2$. The ambient pressure $\mathrm{Pd}{\mathrm{S}}_2\text{−}\mathrm{type}$ structure (a) can be described as an elongated pyrite structure, which transforms under pressure ($>6\mathrm{GPa}$) to the pyrite-type structure [19] (b). The change in the Raman spectra between 5 and 9.2 GPa (c) indicates the structural transition to a pyrite-type structure. The mode assignment within the pyrite structure is shown on the spectrum at 9.2 GPa. Blue arrows schematically show the unusual evolution of the $E_{\mathrm{g}}$ libration and the $A_{\mathrm{g}}$ stretching vibrations of the $\mathrm{S}{\mathrm{e}}_2$ dumbbells with pressure. The pyrite phase of $\mathrm{PdS}{\mathrm{e}}_2$ is stable up to 37 GPa, above which the Raman spectrum changes (the green lines for 38.5 GPa show the deconvolution of the observed spectrum into two broad peaks), indicating a structural transition to a next high-pressure phase of $\mathrm{PdS}{\mathrm{e}}_2$ referred to as the γ phase.

Figure 2

Pressure evolution of the electronic transport properties of $\mathrm{PdS}{\mathrm{e}}_2$. (a) Application of pressure tunes the electronic properties of the $\mathrm{Pd}{\mathrm{S}}_2\text{−}\mathrm{type}$ phase of $\mathrm{PdS}{\mathrm{e}}_2$ from a semiconductor to a metal at pressures above 3 GPa. (b) With the transition to the pyrite phase, superconductivity emerges with $T_{\mathrm{c}}$ showing a dome-shaped dependence on pressure with a maximum $T_{\mathrm{c}}$ of 13.1 K at $P\sim 23\mathrm{GPa}$. (c) Demonstration of the correlation between the pressure evolutions of $T_{\mathrm{c}}$ and frequencies of internal stretching vibration ($A_{\mathrm{g}}$ mode) and librational ($E_{\mathrm{g}}$ mode) modes of the $\mathrm{S}{\mathrm{e}}_2$ dumbbells. The sudden increase of the room-temperature resistivity at $P=37\mathrm{GPa}$ and the change of the temperature dependence ρ($T$) in the normal state [a very weak temperature dependence, as illustrated by the curve at 40 GPa in (a)] indicates a structural transition to the γ phase. The γ phase remains superconducting [curve at 40 GPa in (a)] with $T_{\mathrm{c}}$ weakly depending on pressure.

Figure 3

Calculated electronic structure of $\mathrm{PdS}{\mathrm{e}}_2$. (a) The electronic band structure of the semiconducting $\mathrm{Pd}{\mathrm{S}}_2\text{−}\mathrm{type}$ phase at ambient pressure. The valence band maximum at the Γ point has Pd $d_{z^2}$ character hybridized with the $p_{\mathrm{z}}$ state of Se. The conduction band minimum is between the $S$ and $Y$ high-symmetry points and consists of Pd $d_{x^2-y^2}$ and Se $p$ states. (b) At about 2 GPa the band structure shows the metallization of $\mathrm{PdS}{\mathrm{e}}_2$ with the $\mathrm{Pd}{\mathrm{S}}_2\text{−}\mathrm{type}$ structure. (c) The electronic structure of $\mathrm{PdS}{\mathrm{e}}_2$ in the pyrite phase with bands crossing the Fermi energy is very different from that of the $\mathrm{Pd}{\mathrm{S}}_2\text{−}\mathrm{type}$ structure. (d) A further increase of the pressure pushes the bands’ extrema away from the Fermi energy. (e) Dirac crossing along the $\mathrm{\Gamma }M$ direction and two fourfold degeneracies separated by $\sim 0.01\mathrm{eV}$ at the $M$ point near $E_{\mathrm{F}}$ (f) obtained including SOC. (g) The Fermi surface of the pyrite phase for the band structure (c). The network along the edges of the Brillouin zone (lower panel) is formed by small electron and hole pockets. At high pressure these bands move away from the $E_{\mathrm{F}}$ (d) and the Fermi surface will only contain the cubelike pockets.