Structural screening of phosphorus sulfur ternary hydride PSH₆ with a high-temperature superconductivity at 130 GPa

Abstract

In our study, we constructed a series of inorganic nonmetallic ternary hydrides PSH₆ by first-principles structural screening under pressure of 200 GPa. The structural stability under lower pressure are examined. Focusing on the structural stability, electronic and phonon properties, as well as the possible superconducting properties within the framework of Bardeen−Cooper−Schrieffer (BCS) theory, we show that PSH₆ with space group \text{textcolor}[RGB]12,108,100Pm\overline{3}m possesses a superconducting transition temperature of 146 K at 130 GPa. In the pressure range of 100−200 GPa, our work suggests that the ternary phosphorus-sulfur-hydrogen would act as a promising compositional and elemental space for achieving high-temperature superconductivity.

Graphical abstract

Keywords

phosphoruses / sulfur, hydrides / high-temperature / superconductivity / low-pressure / structural screening

1 Introduction

Since Ashcroft put forward the concept of hydride superconductivity [1], great progress has been made in superconductivity based on hydride and its derivatives. In the prototype hydride superconductor, solid CaH{}_6 [2] was reported to show the T_{\text{c}} of 263 K at a high pressure of 200 GPa, adopting the same crystalline lattice symmetry of Im\overline{3}m. The highly symmetrical cubic structure makes the hydrogen atoms in this calcium hydrogen compound form a sodalite structure with H{}_{24}. A strong electron-phonon coupling is generated and associated with the d electrons of Ca and the highly localized hydrogen lattice. After that, the SH{}_3 [3] phase was found to achieve a superconducting transition temperature up to 203 K at a high pressure of 200 GPa with space group of Im\overline{3}m and the ClH{}_3[4] has T_{\text{c}} under 150 GPa. In 2018, the discovery of LaH{}_{10} [5] and YH{}_{10} [5] with Fm\overline{3}m showing a record-high T_{\text{c}} of 289 and 300 K at 200 and 250 GPa, respectively triggers ever-increasing interest in the search of novel superconducting structures in complex configurational space.

At the same time, in-depth studies of the derivative structures of these typical hydrides have also been conducted. Compared with the primitive high-pressure compound, a promoted T_{\text{c}} has been demonstrated through doping or substitution of LaH{}_{10} structure, such as TbH{}_{10} [6], AcH{}_{10} [7], LaYH{}_{20} [8], LaH{}_{9.99}B{}_{0.01} [9], KB{}_2H{}_8 [10], LaBH{}_8 [11], LaB{}_2H{}_8 [12], under an ultrahigh pressure above 200 GPa, with the charge transfer governing the superconductivity in these superconductors. For CaH{}_6 families, derivatives such as MgH{}_6 [13], LaKH{}_{12} [14], CaYH{}_{12} [15] and MgCaH{}_{12} [16] were obtained under high pressure (usually 200−300 GPa), and the coupling of electrons from metallic elements and phonon from hydrogen atoms at high vibration frequencies makes the T_{\text{c}} of these superconductors even higher than that of their parent. In addition to above mentioned metal hydrides, high-pressure anionic hydrides based on sulfur [17], selenium [18], phosphorus [19], chloride [20] and their mixtures [21–23] have also been investigated to show a high T_{\text{c}}. In such compounds, the focus is on building supercells large enough to replace the target atoms in them to study the effects of doping on the system, such as H{}_3S{}_{0.925}P{}_{0.075} (280 K, 250 GPa) [24], H{}_3S{}_{0.9375}P{}_{0.0625} (189 K, 200 GPa) [25], H{}_3S{}_{0.5}Se{}_{0.5} (182 K, 200 GPa) [26]. However, most of these compounds can only exist and stable under ultra-high pressure above 200 GPa. Such a high-pressure of achieving the superconducting state acts as the obstacles of promising applications. Most recently, the higher superconducting critical temperature (T_{\text{c}}) up to 287 K has been also observed in carbonaceous sulfur hydride CH{}_4-SH{}_3 [27] at Mbar region, this finding further proves that compressed light-element compounds are potential room temperature superconductors. In addition, theorists have predicted some of these light-element compounds which can attain high-T_{\text{c}} superconductivity when they are compressed to Mbar region. For example, H{}_6SCl [28] exhibits the superconductivity with T_{\text{c}} 155 K at 90 GPa. Realization of any potential high-temperature superconductors under low-pressure or even ambient pressure is of ever-evolving interests and increasingly sought-out field in condensed matter physics.

In our study, we demonstrate the realization of superconductivity in anionic compounds comprising of phosphorus, sulfur, and hydrogen atoms. Through structural exploration of the energetics of phase space, we constructed a series of inorganic nonmetallic ternary hydrides below 200 GPa. We explore a promising superconductor of PSH{}_6 that can work and exist in moderate pressure environment. The structural stability, electronic and phonon properties, as well as the possible superconducting properties of PSH{}_6 were considered within the framework of BCS theory [29]. Our results show that the PSH{}_6 with space group of Pm\overline{3}m can reach the superconducting transition temperature of 146 K under the pressure of 130 GPa. In the pressure range of 100−200 GPa, as far as we know, there are some ternary hydrides with showing superconducting behaviors. For instance, critical superconducting transition of BaReH{}_9 (7 K, 100 GPa) [30], Li{}_5MoH{}_{11} (6.5 K, 160 GPa) [32] and CH{}_4-SH{}_3 (287 K, 267 GPa) [27] were experimentally observed. Theoretical predictions such as MgCH{}_4 (121 K, 105 GPa) [31], MgVH{}_6 (26.7 K, 150 GPa) [33], ScYH{}_6 (52.9 K, 200 GPa) [34], LiPH{}_6 (150 K, 200 GPa) [35], and LiP{}_2H{}_{14}(169 K, 230 GPa) [36] show that ternary hydrides are very promising for achieving high temperature superconductivity.

2 Computational details

In this study, the Perdew−Burke−Ernzerhof (PBE) [37] method in the Generalized Gradient Approximation (GGA) in the density functional theory [38] and the projection enhanced wave pseudopotential (PAW) [39] were adopted for the structural screening and optimization by using the VASP package [40]. A plane wave cut-off energy of 600 eV, and the k-point of the Brillouin zone was 0.01 Å⁻¹ interval distribution of Monkhorst−Pack [41] for the optimization of structures, and the k-point interval of the total energy self-consistent calculation was 0.01 Å⁻¹ or better. The threshold of energy convergence and force convergence were set to {10}^{-5} eV and {10}^{-3} eV/Å, respectively. In addition, we used Car and Perrinello [42] molecular dynamics with a 4 × 4 × 4 supercell containing 512 atoms and NVT ensemble at constant temperature and pressure, and Nosé−Hoover thermostat [43] to ensure that the ambient temperature is controlled at the set value.

To investigate the dynamic stability and possible superconductivity of these cage compounds, the QUANTUM ESPRESSO package (QE) [44, 45] with the approach of density functional perturbation theory (DFPT) and PBE-GGA functional was adopted. The typical value of Coulomb pseudopotential {\mu }^{\star } was set as 0.1 for hydrides. Vanderbilt-type ultrasoft pseudopotentials [46] for P, S, and H were employed in this calculation. A k-mesh of 32\times 32\times 32 was used in the calculation of the electron−phonon interaction matrix element and a q-mesh of 8\times 8\times 8 was used for the phonon spectra calculation. The cut-off energies for the wave function and charge density are 80 Ry and 600 Ry, respectively. At the same time, the forces and stresses of the convergent structure were optimized and controlled within the error range of VASP [40] and QE [44, 45] programs.

We calculated the phonon frequency (\omega) and the Eliashberg electron−phonon spectral function [{\alpha }^{2}F(\omega )]. Based on {\alpha }^{2}F(\omega ), the electron-phonon coupling constant (\lambda, EPC) was calculated, which is defined by integration over the entire frequency domain of {\alpha }^{2}F(\omega ):

(1)\lambda =2{\displaystyle {\int }_0^{\mathrm{\infty }}\frac{{\alpha }^{2}F(\omega )}{\omega }\mathrm{d}\omega .}

For , T_{\text{c}} was estimated by the McMillan equation [47], expressed as

(2)T_{\text{c}}={\displaystyle \frac{{\omega }_{\text{log}}}{1.2}}\exp \left[-{\displaystyle \frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}}\right].

The EPC parameter \lambda is larger than 1.5, which represents very strong electron−phonon coupling for systems. T_{\text{c}} was corrected by Allen−Dynes−corrected McMillan equation [48], expressed as

(3)T_{\text{c}}=f_1{f}_2{\displaystyle \frac{{\omega }_{\text{log}}}{1.2}}\exp \left[-{\displaystyle \frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}}\right],

where

(4)f_1={\left\{1+{\left[{\displaystyle \frac{\lambda }{2.46(1+3.8{\mu }^{\ast })}}\right]}^{3/2}\right\}}^{1/3},

(5)f_2=1+{\displaystyle \frac{{\lambda }^2({\omega }_{/}{\omega }_{\mathrm{l}\mathrm{o}\mathrm{g}}-1)}{{\lambda }^2+[1.82(1+6.3{\mu }^{\ast })(\omega /{\omega }_{\mathrm{l}\mathrm{o}\mathrm{g}}){]}^2}}

are the correction factors. {\omega }_{\text{log}} is the logarithmic average of phonon frequency and is written as

(6){\omega }_{\text{log}}=\exp \left[{\displaystyle \frac{2}{\lambda }}{\displaystyle {\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\alpha }^{2}F(\omega )\log (\omega )}{\omega }}\mathrm{d}\omega }\right].

In electron part and electron−phonon parts, we use “Methfessel−Paxton first-order spreading” method [49] and the value of the Gaussian spreading for Brillouin-zone integration was set to 0.01 Ry.

3 Results and discussion

We have screened more than 10000 crystal structures composed of phosphorus, sulfur and hydrogen based on particle swarm optimization (PSO) method using CALYPSO code [50–52], and the results are summarized in Fig. S1 of the Supplemental Material. Among these ternary compounds, three compounds PSH{}_3, PSH{}_6 and PSH{}_8 were found to be thermodynamically stable. Among them, the PSH{}_6 has superconducting properties with superconducting transition T_{\text{c}} of more than 140 K. Although PSH{}_3 and PSH{}_8 also have a good thermal stability, unfortunately, we cannot find low-pressure polymorph of them with promising superconducting phases. Next, we would like to focus on PSH{}_6 and explore its intermediate structures under varying pressure.

By using CALYPSO [50–52], we successfully identified more than 1000 structures of PSH{}_6 under different pressure conditions, suggesting the existence of rich polymorphs of this ternary compound.Amongst the various intermediate structures (Fig.1), we obtained a structure of Pm\overline{3}m showing intriguing superconducting properties with the T_{\text{c}} occurring at relatively low pressure. As shown in Fig.1, in the novel Pm\overline{3}m phase of PSH{}_6 the phosphorus atom is located at the body center of the lattice, and the fractional Cartesian coordinate is (0.5,0.5,0.5), and the sulfur atom occupies lattice vertices, and the Cartesian coordinate is (0,0,0). Six hydrogen atoms occupy the face centers and middle edges, respectively.

As shown in Fig.2, we plotted formation enthalpy of PSH{}_6 at different pressures. Here the enthalpy of formation process of gaseous P + S + H is used as the reference. We found that the C2/m, P{}_4mmc, Cmcm, Pm{}_3 and PH{}_3+SH{}_3 structures are significantly higher than that of P + S + H at the pressure range of 0−200 GPa. This suggests that a thermodynamic stability of these polymorph at the pressure is worse than that of P + S + H, implying a phase decomposition into gaseous elements. Similarly, at the pressure range of 98−200 GPa, as shown in the inset of Fig.2, the Pm\overline{3}m phase is the most thermodynamically stable structure among the existing candidates. Further, we also considered the possibility of the decomposition of PSH{}_6 into other stable binary compounds (as marked in Fig. S1), such as 2PH{}_3 + 2SH{}_2 + H{}_2, P{}_2S{}_7 + 5PH{}_3 + 27/2H{}_2, P{}_2S{}_5 + 3PH{}_3 + 21/2H{}_2, P{}_4S{}_7 + 3PH{}_3 + H{}_2, P{}_4S{}_9 + 5PH{}_3 + 39/2H{}_2 and P{}_4S{}_3 + SH{}_3 + 21/2H{}_2. These results are presented in Fig. S3 of the Supplemental Material, indicating that PSH{}_6 will not decompose into the similar products at above pressures. Therefore, the PSH{}_6 is prone to be thermodynamically stable between 98−200 GPa.

In order to further evaluate the structural stability of Pm\overline{3}m phase of the PSH{}_6, we examine its structural stability at the dual effects of pressure and temperature from the perspective of molecular dynamics. At applying a pressure of 130 GPa, the structure was simulated with ab initio molecular dynamics at a series temperatures of 50,100,200,300 and 500 K. The detailed relaxation process at different temperatures is shown in Fig.3(a). We calculated the root-mean-square displacement (RMSD) of PSH{}_6 at 50,100,300 and 500 K which are found to be 0.135, 0.147, 0.176 and 0.211 Å, respectively. As expected, the actual temperature of the thermostat fluctuates near the set values, as shown in Fig.3(b). The steady evolution around the targeted temperature signifies a robust structure of the PSH{}_6 phase without any abrupt collapse of structure nor bond breaking at disturbances of thermal and stress. Similarly, as shown in Fig.3(c), the in-situ pressure is overall maintained around 130 GPa which also excludes the discontinuity associated with any structural reconstruction. Therefore, at the pressure of 130 GPa and the temperature of 50−500 K, the PSH{}_6 still maintains the stability and integrity of the structure. We also considered higher pressures up to 200 GPa, the atomic RMSD for temperatures between 50−500 K is similar to that in 130 GPa, as shown in Fig. S4 of the Supplemental Material. Therefore, the PSH{}_6 phase still maintains a dynamic stability at broad windows of pressure and temperature.

Based on above analysis from thermodynamics and structural dynamics, we can safely conclude that the novel Pm\overline{3}m phase of PSH{}_6 will not be decomposed at the condition of 98−200 GPa, and can still maintain the stability of the structure at the condition of 98−200 GPa. This sets a solid foundation of exploring its superconducting properties.

After confirming that the Pm\overline{3}m PSH{}_6 can stably exist at certain pressures, we next focus on the electronic structure of this phosphorus-sulfur-hydrogen compound. The ternary compound can be regarded as a cubic structure formed by the nesting of PH{}_3 and SH{}_3 in space, which is also confirmed by the electronic local function (ELF) as shown in Fig.4(a). Owing to a stronger electronegativity, the sulfur atoms tend to attract more electrons from hydrogens, which is proved by bader charge analysis. Each S and P atom get 0.189 and 0.177 electrons respectively, while the H atom around the S (P) atom loses 0.119 (0.004) electron. Therefore more electrons are localized around the S atom. From the ELF, it is found that there is stronger polar bond between S and H than between P and H, which is because S has stronger electronegativity than P. Compared with S atom, there are fewer electrons localized around P atom, indicating that the electrons of H tend to be distributed at the joint region between H and P atoms.

Band structure around the Fermi level is plotted in Fig.4(b). A good metallicity is found for the Pm\overline{3}m PSH{}_6 phase as reflected by the partially occupied bands crossing the Fermi level. Another feature of the electronic band structure is the rich distribution of valleys, for instance, the electron pockets along the X−M−Γ path. Here different colors are used to differentiate each band crossing the Fermi energy, as shown in Fig.4(b). The Fermi surface corresponding to each band is plotted in the inset, from which the pocket states exist around the zone center and high symmetrical sites at face centers and corners. The projected electronic density of states (DOS) of PSH{}_6 are shown in Fig.4(b). At the Fermi level, the P, S, and H contribute roughly equally and lead to a robust metallicity. This is quite surprising as H only having one valence electron and the evenly distributed H states spread over the whole energy range. In real space, those H atoms at edge and face centers form hybridized states with S and P in form of conducting channels along the cubic edge connecting the vertices (S atom) and cubic center (P atom).

Finally, we would like to show the possible aspect of achieving superconduction and its related mechanism of PSH{}_6. As shown in Fig.5(a), the frequency range of phonon vibration is 0−2000 cm⁻¹. Among them, the high- and middle-frequency vibrational modes (more than 600 cm⁻¹) are mainly associated with the vibration of hydrogen atoms, while the low-frequency vibration at 0−600 cm⁻¹ mainly involves with the vibration of phosphorus and sulfur atoms with a minor contribution of hydrogen. By integrating the Eliashberg function, we can obtain the electron-phonon coupling constant \lambda. In the low frequency part, mainly contributed by the heavier phosphorus and sulfur atoms, the growth rate of \lambda is slow but steady. In the high-frequency region with vibrations dominated by hydrogen, the \lambda grows quickly and finally saturates to 3.02, comparably contributed by the hydrogen and phosphorus and sulfur, and which is larger than 2.19 of H{}_3S at 200 GPa.

The variation of superconducting transition temperature T_{\text{c}} with pressure is shown in Fig.5(b). At 130 GPa, when {\mu }^{\star } is 0.1 and 0.13 the T_{\text{c}} is 146.5 K and 139.9 K, respectively. In a similar pressure range, as far as we know, there is no inorganic nonmetallic ternary hydrides with T_{\text{c}} higher than PSH{}_6. We found that the T_{\text{c}} decreases with the increase of pressure. When the pressure increases to 200 GPa, T_{\text{c}} is reduced to 112.13−102.44 K. The \lambda has a similar trend with the increase of pressure, while the {\omega }_{\mathrm{l}\mathrm{o}\mathrm{g}} shows the opposite trend [Fig.5(b) right].

Unfortunately, although other structures can be metallized at pressure, it is less likely to be promising for superconducting due to the unstable phonon structures.

4 Conclusion

In summary, via screening the energy landscapes of the P−S−H ternary phase space at pressure, we have identified and explored a metastable phase of PSH{}_6 with respect to its structural stability (thermodynamic stability and dynamic stability), electronic structure, phonon structure and superconducting properties. We found that the Pm\overline{3}m structure reveals a good metallicity, stable phonon dispersions, and a relatively high superconducting transition temperatures of 146 K at a moderate pressure of 130 GPa. The discovery of PSH{}_6 superconductivity suggests that the P−S−H systems are promising for the exploration of relatively low-pressure and high-temperature superconductors.

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