Superconductivity in metastable phases of phosphorus-hydride compounds under high pressure

Hydrogen-rich compounds have been extensively studied both theoretically and experimentally in the quest for novel high-temperature superconductors. Reports on sulfur hydride attaining metallicity under pressure and exhibiting superconductivity at temperatures as high as 200 K have spurred an intense search for room-temperature superconductors in hydride materials. Recently, compressed phosphine was reported to metallize at pressures above 45 GPa, reaching a superconducting transition temperature $\left({\mathrm{T}}_{\mathrm{C}}\right)$ of 100 K at 200 GPa. However, neither the exact composition nor the crystal structure of the superconducting phase have been conclusively determined. In this work, the phase diagram of ${\mathrm{PH}}_n (n=1,2,3,4,5,6)$ was extensively explored by means of ab initio crystal structure predictions using the minima hopping method (MHM). The results do not support the existence of thermodynamically stable ${\mathrm{PH}}_n$ compounds, which exhibit a tendency for elemental decomposition at high pressure even when vibrational contributions to the free energies are taken into account. Although the lowest energy phases of ${\mathrm{PH}}_{1,2,3}$ display ${\mathrm{T}}_{\mathrm{C}}$'s comparable to experiments, it remains uncertain if the measured values of ${\mathrm{T}}_{\mathrm{C}}$ can be fully attributed to a phase-pure compound of ${\mathrm{PH}}_n$.

Figure 1

Calculated enthalpies for ${\mathrm{PH}}_n (n=1,2,3,4,5,6)$ for the low-lying structures found in this work. Values are given with respect to the elemental decomposition (P + $1/2{\mathrm{H}}_2$). The reference structures for hydrogen are $P{6}_{3}m$ (0–120 GPa) and $C2/c$ (120–300 GPa) from Ref. [37]. The reference phases for phosphorus are $Pm\overline{3}m$ (0–10 GPa), $R3m$ (20–110 GPa), $P6mm$ (120–240 GPa), $Im\overline{3}m$ (250–270 GPa), and $I\overline{4}3d$ (280–300 GPa). The change in slope at around 120 GPa is due to the phase transition in elemental hydrogen.

Figure 2

Low-lying enthalpy structures found for different compositions under pressure at 120 GPa. The large and small spheres denote the P and H atoms, respectively. The ELF at a fixed value of 0.8 is shown in the upper part of each structure.

Figure 3

Predicted formation enthalpies of ${\mathrm{PH}}_n$ with respect to decomposition into P and H at 120 GPa. The solid red line denotes the convex hull of stability. Black triangles show PBE and green squares hybrid functional values (HSE06). The color-gradient scale indicate the free-energy within the harmonic approximation at temperatures up to 400 K (color bar in Kelvin on the right) as computed on top of PBE energies.

Figure 4

(Left) SCDFT calculated critical temperatures ${\mathrm{T}}_{\mathrm{C}}$ for PH (blue), ${\mathrm{PH}}_2$ (green), and ${\mathrm{PH}}_3$ (red) as a function of pressure. Experimental ${\mathrm{T}}_{\mathrm{C}}$ by resistivity measurements from Drozdov et al. [36] are shown in black squares. Overall a good agreement is found between the experimental values and those for PH, ${\mathrm{PH}}_2$, and ${\mathrm{PH}}_3$. Although the ${\mathrm{PH}}_2$ composition shows significantly better agreement. (Right) Trend in pressure of the (BCS-like) electron phonon coupling coefficient $\lambda$ (top panel) and of the phononic characteristic frequency ${\omega }_{ln}$ (bottom panel) as a function of pressure.

Figure 5

Eliashberg spectral function (solid line) and frequency-dependent $ep$ coupling parameters $\lambda \left(\omega \right)$ (dashed line)—top panels—and phonon density of states (DOS)—bottom panels—for the three ${\mathrm{PH}}_n$ compounds considered in the present study and ${\mathrm{SH}}_3$ at 200 GPa. The shaded area (red) in the DOS panels is the P/S partial phonon DOS.