Structural transition, metallization, and superconductivity in quasi-two-dimensional layered $\mathrm{Pd}{\mathrm{S}}_2$ under compression

Based on first-principles simulations and calculations, we explore the evolutions of crystal structure, electronic structure, and transport properties of quasi-two-dimensional layered $\mathrm{Pd}{\mathrm{S}}_2$ under compression by uniaxial stress and hydrostatic pressure. An interesting ferroelastic phase transition with lattice reorientation is revealed under uniaxial compressive stress, which originates from the bond reconstructions of the unusual $\mathrm{Pd}{\mathrm{S}}_4$ square-planar coordination. By contrast, the layered structure transforms into a three-dimensional cubic pyrite-type structure under hydrostatic pressure. In contrary to the experimentally proposed coexistence of layered $\mathrm{Pd}{\mathrm{S}}_2$-type structure with cubic pyrite-type structure at intermediate pressure range, we predict that the compression-induced intermediate phase will show the same structure symmetry as the ambient phase, except for sharply shrinking interlayer distances. The coordination of the Pd ions not only plays crucial roles in the structural transition, but also leads to electronic structure and transport property variations, which changes from square planar to distorted octahedron in the intermediate phase, resulting in bandwidth broadening and orbital-selective metallization. In addition, the superconductivity in the cubic pyrite-type structure comes from the strong electron-phonon coupling in the presence of topological nodal-line states. The strong interplay between structural transition, metallization, and superconductivity in $\mathrm{Pd}{\mathrm{S}}_2$ provides a good platform to study the fundamental physics of the interactions between crystal structure and transport behavior, and the competition or cooperation between diverse phases.

Figure 1

Illustration of the pressure-induced structural transition in $\mathrm{Pd}{X}_2$ from (a) the layered orthorhombic $\mathrm{Pd}{\mathrm{S}}_2$-type structure to (b) the 3D cubic pyrite-type structure. The large pink balls represent $X$ atoms and the small gray balls represent Pd atoms, respectively. The structural detail is highlighted by the $X_2$ anion dumbbells (red solid lines), whereas the $\mathrm{Pd}{X}_4$ square-planar and hidden distorted $\mathrm{Pd}{X}_6$ octahedra are plotted in cyan in (a), (b), respectively.

Figure 2

(a) Strain, (b) lattice constants, (c) bond lengths, and (d) energy evolutions in response to the uniaxial compression along the $c$ axis of the orthorhombic phase $\mathrm{Pd}{\mathrm{S}}_2$. The insets in (c) display the Pd-S interatomic distances in the distorted $\mathrm{Pd}{\mathrm{S}}_6$ octahedron. d1 denotes the interlayer distance between Pd and S atoms; d2 and d3 are the Pd-S bond lengths in the square planar ${\left(\mathrm{Pd}{\mathrm{S}}_4\right)}^{2-}$ structural units; d4 is the S-S bond lengths in the ${\left({\mathrm{S}}_2\right)}^{2-}$ anions.

Figure 3

Electron density differences for the three stages of the ferroelastic phase transition: (a) initial phase at 0 GPa, (b) intermediate phase at 0.7 GPa, and (c) final phase at 0.8 GPa. The isosurface value is taken as $0.1e/{\AA }^3$ (light blue and orange indicate the regions of electron depletion and electron enrichment, respectively), and the pink (gray) balls represent S (Pd) atoms, respectively.

Figure 4

The activation energy barrier vs ferroelastic strain in different materials (ML stands for 2D monolayer).

Figure 5

Variation of the lattice parameters of $\mathrm{Pd}{\mathrm{S}}_2$ in response to the hydrostatic pressure: (a) lattice constants and (b) interatomic distances (d1, d2, and d3) in the distorted $\mathrm{Pd}{\mathrm{S}}_6$ octahedra and S-S bond length (d4) in the ${\left({\mathrm{S}}_2\right)}^{2-}$ dumbells. The insets in (b) display the transformation of the coordination polyhedra from square planar to octahedron.

Figure 6

(a) Band structure of orthorhombic $\mathrm{Pd}{\mathrm{S}}_2$ under ambient condition calculated with DFT-D (Grimme) and HSE06 functionals. The color denotes the contributions from Pd-$4d$ (red) and S-$3p$ (blue) states, and strong hybridizations between them. (b) The evolution of band edge and band gap under uniaxial stress calculated within HSE06. The Fermi level and VBM under ambient conditions are set to 0 and as the reference point.

Figure 7

Projected density of states (PDOS) evolutions of the Pd $4d$ state in $\mathrm{Pd}{\mathrm{S}}_2$ for different stages of the phase transitions under compression: (a) initial phase at 0 GPa, (b) intermediate phase at 2 GPa, and (c) high-pressure cubic pyrite-type phase at 4 GPa. The Fermi level is set to zero.

Figure 8

Electronic band structures of orthorhombic $\mathrm{Pd}{\mathrm{S}}_2$ calculated with DFT-D (Grimme) for the (a) ambient pressure phase, and (b) intermediate phase at 2 GPa. The Fermi level is set to zero. The contributions of the $d_{z^2}$ (blue solid balls) and $d_{x^2-y^2}$ (red hollow balls) orbitals are proportional to the size of the colored balls, as indicated in the legend.

Figure 9

Electronic structure of the high-pressure pyrite-type $\mathrm{Pd}{\mathrm{S}}_2$: (a) the band structure and (b) the corresponding Fermi surface. (c) Left: phonon dispersion curves for pyrite-type $\mathrm{Pd}{\mathrm{S}}_2$ at 4 GPa. Right: Eliashberg electron-phonon coupling spectral function ${\alpha }^{2}F\left(\omega \right)$ and the electron-phonon coupling integral $\lambda \left(\omega \right)$.