Supplementary Information for the paper entitled “ Phase Diagram and High-Temperature Superconductivity of Compressed Selenium Hydrides ”

Recent discovery of high-temperature superconductivity (Tc = 190 K) in sulfur hydrides at megabar pressures breaks the traditional belief on the Tc limit of 40 K for conventional superconductors, and opens up the doors in searching new high-temperature superconductors in compounds made up of light elements. Selenium is a sister and isoelectronic element of sulfur, with a larger atomic core and a weaker electronegativity. Whether selenium hydrides share similar high-temperature superconductivity remains elusive, but it is a subject of considerable interest. First-principles swarm structure predictions are performed in an effort to seek for energetically stable and metallic selenium hydrides at high pressures. We find the phase diagram of selenium hydrides is rather different from its sulfur analogy, which is indicated by the emergence of new phases and the change of relative stabilities. Three stable and metallic species with stoichiometries of HSe2, HSe and H3Se are identified above ~120 GPa and they all exhibit superconductive behaviors, of which the hydrogen-rich HSe and H3Se phases show high Tc in the range of 40–110 K. Our simulations established the high-temperature superconductive nature of selenium hydrides and provided useful route for experimental verification.

bare electron-ion interaction. Ashcroft firstly proposed that solid hydrogen once being metallic under pressure has the potential to be a high-temperature superconductor 11 . Later on, the idea on metallic superconducting hydrogen was extended into hydrogen-rich compounds 12 , where metallization pressure can be significantly lowered than that in pure hydrogen. A number of hydrogen-rich compounds were subsequently predicted to be good superconductors 13 with estimated T c reaching remarkably high values (e.g. 64 K for GeH 4 at 220 GPa 14 , 107 K for SiH 4 (H 2 ) 2 at 250 GPa 15 , 235 K for CaH 6 at 150 GPa 16 , and 204 K for (H 2 S) 2 H 2 at 200 GPa 17 ). Experimental syntheses of these potential high-temperature superconductors are excitingly ongoing, but challenging.
Sulfur dihydride (H 2 S) has not been considered as the candidate for superconducting hydrides since it was proposed to dissociation into elemental sulfur and hydrogen before the metallization 18,19 . Only recently, first-principles swarm structure searches on high-pressure structures of H 2 S was conducted and H 2 S was excluded from the elemental dissociation, making the prediction of its superconductivity with a high T c ~80 K at 160 GPa possible 20 . Shortly after this report, breakthrough electrical measurement observed high-temperature superconductivity in compressed H 2 S with an unprecedentedly high T c up to 190 K at megabar pressures 21 . The decrease of T c with magnetic field, and the strong isotope shift of T c suggest that H 2 S is a conventional superconductor. There exist two observed superconducting states: the sample prepared at low temperature of 100-150 K has a maximal T c of 150 K at 200 GPa, while the 190 K superconductivity comes from the sample prepared at high temperature of 220-300 K, which likely associates with the dissociation of H 2 S into H 3 S 21-23 . This discovery has stimulated significant interest in studying the underlying superconducting mechanism [23][24][25][26][27] and searching for new high-T c superconductors in other dense hydride systems.
Selenium is a sister and isoelectronic element of sulfur with a larger atomic radius and a weaker electronegativity. By witness of high-T c superconductivity of sulfur hydrides (i.e. H-S system), a natural and immediate thought is to examine whether selenium hydrides (H-Se system) are also high-temperature superconductors at high pressures. To our best knowledge, there is less report on solid phases of H-Se system, except for the existence of three temperature-dependent H 2 Se phases at ambient pressure 28 .
We herein explored the hitherto unknown high-pressure phase diagram of the H-Se system via the first-principles swarm-intelligence based structure search. In the moderate pressure region, different from the H-S system all the H-Se compounds are energetically unstable against the elemental decomposition including the normally expectant H 2 Se stoichiometry. At megabar pressures above 120 GPa, three stable species with stoichiometries of HSe 2 , HSe and H 3 Se were identified. Among them HSe with the high-symmetry P4/nmm structure is the most stable phase and H 3 Se becomes marginally stable at high pressures. They are all metallic and exhibit superconductive behaviors. The latter two H-rich phases, benefiting from fairly strong electron-phonon coupling, are predicted to owe high T c values up to 110 K. Our simulations provide a useful roadmap for discovering high-temperature superconductors in selenium hydrides.

Results
The variable-composition structure searches are performed at a variety of H-Se stoichiometries containing up to 4 formula units per simulation cell at 0, 50, 100, 200 and 300 GPa. At 0 GPa, we find only one stable stoichiometry with respect to the elemental decomposition, H 2 Se (see Supplemental Fig. S1a), which is consistent with available experimental reports 28 . It is in the P3 1 21 symmetry, where the arrangement of Se atoms is nearly the same as that of the Se-I phase 29 and H atoms are accommodated on the line of two adjacent Se to form covalent H-Se bonds and hydrogen bonds simultaneously (Fig. S1c). However, with increasing pressure this phase becomes dramatically unstable (Fig. S1b). The structure search results are summarized in the convex hulls constructed with solid H 2 and Se as the binary variables in Fig. 1 (and Fig. S1a). In the low-pressure region up to 100 GPa, all the stoichiometries are energetically unstable against the elemental decomposition. This is in sharp contrast to the H-S system, where all the investigated stoichiometries are stable with respect to the elemental decomposition 22 . With increasing pressure to 200 GPa, while most of phases still lie above the elemental decomposition line, three stoichiometries (HSe 2 , HSe and H 3 Se) show the tendency of being stabilized and move downward below the line. Eventually HSe 2 and H 3 Se become stable stoichiometries on the hull against any way of decomposition. At 300 GPa, the HSe 2 , HSe and H 3 Se stoichiometries are all clearly located on the hull, and HSe emerges as the most stable phase over other species. The energetic instability of these phases in the low-pressure region and their being restabilized at high pressures may be attributed to the weaker covalent bond of H-Se than that of H-S (resulted from the larger size of Se), which is cumulatively strengthened by volume contraction under compression.
The Se-rich HSe 2 stoichiometry is the most stable phase at 200 GPa. Its lowest-energy structure (Fig. 2a) has a C2/m symmetry, in which the sublattice of Se atoms is isostructural to the Se-IV phase 30 and H atoms passivate alternatively from both sides of the infinite zigzag Se chains. It is stabilized above 124 GPa (as in the inset of Fig. 1). The HSe stoichiometry is stabilized above 249 GPa as a highly symmetric PbO-type structure (space group P4/nmm, Fig. 2b). This compound, which is isostructural to superconductive Fe-based chalchogenides 31 , consists of the stack of two-dimensional layered edge-sharing SeH 4 -tetrahedra networks. Within the layer both Se and H are four-fold coordinated. It is worth mentioning that in addition to the P4/nmm HSe phase, our structure search finds an energetically competitive P2 1 /c structure (though metastable, Fig. 2d). It consists of layered three-fold coordinated Se/H  networks. As to the H 3 Se stoichiometry (stable above 166 GPa), the lowest-energy structure has a high symmetry of Im-3 m, isostructural to that of H 3 S 17 , where Se atoms occupy a body-centered cubic sublattice, each Se being six-fold coordinated by H. This structure is transformed from a molecular R3m phase as the result of pressure-induced hydrogen-bond symmetrization.
We then studied the electronic, phonon and electron-phonon coupling (EPC) properties for the lowest-energy structures of three stable stoichiometries. The band structures of them all exhibit metallic features in their stable pressure regions (see Fig. S3). Phonon calculations indicate their lattice dynamical stabilities evidenced by the absence of any imaginary phonon mode in the whole Brillouin zone.
For the most H-rich H 3 Se stoichiometry, most of conducting states across the Fermi level (E f ) have significant contribution from the H-s orbital, hybridizing strongly with the Se-p states, and there exists the combination of flat bands and steep bands 32 in proximity to the E f (Fig. 3a). These features, resembling those of isostructural H 3 S 17 , are potentially favorable for strong EPC. By consideration of the heavier atomic weight of Se than that of S, one expects H 3 Se will have the generally lower phonon frequencies than those of H 3 S at the same pressure. This is indeed the case for the low-frequency Se-derived vibrations (below 15 THz) and mid-lying H-derived wagging and bending modes (between 15 and 45 THz) as shown in Fig. 3b. However, for the high-lying H-stretching vibrations (above 50 THz), one observes an opposite behavior, i.e. they show the higher frequencies and separate clearly from the mid-lying regime, unlike the situation of H 3 S in which the H-stretching vibrations mix together with the mid-lying phonons. This may be attributed to the stronger H-Se covalent bond as the result of the larger chemical precompression effect 12 induced by Se with the larger atomic radius. The calculated phonon linewidths (Fig. 3b) and EPC spectral function (Fig. 3c) indicate a similar mechanism of EPC to that of H 3 S, where the high-frequency H-stretching modes give the notable contribution (31%) to the integral EPC parameter λ. This mechanism is different from the cases of superconducting CaH 6 16   It is thus highly possible that the calculated λ of H 3 Se here is underestimated in a similar way. Different from the linearly decreased λ with pressure in H 3 S, the λ of H 3 Se shows negligible pressure dependence (Fig. 3d). By substituting λ into the Allen-Dynes modified McMillan equation 36 , we get a weakly pressure-dependent superconducting T c around 110 K for H 3 Se, mildly lower than the values of 160-170 K for H 3 S (see also Table S3).
Turning to the HSe stoichiometry, there is moderate contribution of the H-s state to the conducting states, i.e. only several bands across the E f (e.g. along the A-M, M-Γ and X-Γ lines) are derived from the H-s orbital (Fig. 4a). The Fermi surface ( Fig. 4b and Fig. S8) consists of the relatively small pockets around the M and Z points, and two expanding sheets at the larger wave-vectors. No notable Fermi nesting can be observed. Analysis of bonding feature via the electron localization function (ELF, Fig. 4c) indicates a much weaker H-Se covalent bond (with the maximum ELF magnitude of ~0.5) by comparison with that of H 3 Se (with the maximum ELF of ~0.9). This is further supported by the elongated bond length of HSe (1.69 Å) than that of H 3 Se (1.51 Å) at the same pressure (300 GPa). The weakening of the covalent H-Se bonding results in a remarkably softening of phonon spectrum (solid blue line in the phonon density of states plot of Fig. 4d) compared with the case of H 3 Se (black dash line). The EPC calculations (the upper panel of Fig. 4d) gives a moderately strong EPC parameter λ of ~0.8 (see also  Table S3), where the H-derived vibrations make a ~48% contribution. The smaller magnitude of λ in HSe than that of H 3 Se can be rationalized by the weakened bonding strength in the covalent-bond system 37 . Meanwhile, due to the relatively low logarithmic averaged phonon frequency ω log (800-900 K) as the result of phonon softening, HSe exhibits a moderate T c around 40 K.
For the Se-rich HSe 2 stoichiometry, as expected its electronic structure is predominated by Se rather than H. The calculated rather small λ of 0.45 and low T c of ~5 K (at 300 GPa) is reminiscent of superconducting solid Se at high pressures 38 .

Discussion
After the completion of our work, we were aware of the work by Flores-Livas et al. 35 predicting high-temperature superconductivity (with T c up to 131 K) in selenium hydrides. Our work is different from theirs in several aspects: (i) they only focus on the H 3 Se stoichiometry in analogy to H 3 S, whereas we investigated the entire energy landscape of H-Se system by a more comprehensive global structure search. In addition to H 3 Se, we identified two new energetically stable stoichiometries i.e., HSe 2 and HSe; (ii) we got a qualitatively different picture of energetic stability for selenium hydrides with respect to the elemental decomposition from theirs. Most of their structures are highly stable relative to the elemental decomposition, but in our work the stable stoichiometries against the elemental decomposition can only emerge at quite high pressures (above 100 GPa); (iii) in contrast to their work assuming H 3 Se as the most stable stoichiometry, we find in fact H 3 Se is marginally stable, and HSe 2 and HSe are the most stable species at medium and high pressure region, respectively. The implication of our results is that the synthesis of H 3 Se in experiments may require particular kinetic control process.
The HSe phase in the P4/nmm symmetry represents another interesting high-symmetry structure in addition to the intriguing Im-3m structure discovered originally in H 3 S 17 . It should be pointed out that this structure is completely different from the one of HS (in the low symmetry of C2/m) predicted by Errea et al. 34 . For such a highly symmetric structure, the transition barrier between it and other isomers is usually high, which points to a strong kinetic stability with respect to variations of external conditions, thus favorable to experimental synthesis.
It should be mentioned that since the screened Coulomb repulsion parameter μ* (in the Allen-Dynes modified McMillan equation) cannot be evaluated by any first principle method, our theoretically predicted T c has an intrinsic uncertainty originating from the empirical choice of μ*. Depending on particular materials, μ* usually falls in the range between 0.1 and 0.2 39 . In current study the typical value μ* = 0.1 is used. For completeness, we have calculated the T c values of H 3 Se and HSe (at 250 GPa) by taking a series of μ* = 0.05, 0.10, 0.15 and 0.20, as shown in Fig. S10. As seen, as μ* increases from 0.1 to 0.13, a somehow ubiquitous value for metallic hydrides 12 , T c of H 3 Se and HSe change slightly from ~110 K and ~39 K to ~96 K and ~32 K, respectively.

Conclusion
In summary, with the aim of finding stable and metallic selenium hydrides for potential high-T c superconductors, we explore via a global-minimum structure search method the hitherto unknown energy landscape of H-Se system at high pressures. Despite of similar electronic properties of Se and S, the high-pressure phase diagram of H-Se system is distinct from its H-S analogy. The H 2 Se stoichiometry expected from the normal valence states of H (+ 1) and Se (− 2) is surprisingly unstable against the elemental decomposition. Three energetically stable phases, i.e. HSe 2 , HSe and H 3 Se, are identified above 120 GPa. While the H 3 S stoichiometry dominates as the most stable phase in the H-S system, H 3 Se turns out to be marginally stable and the most stable stoichiometry is the highly symmetric HSe. All of stable phases exhibit metallic features and superconducting activities. The latter two are predicted to have a high T c of 40 K (HSe) and 110 K (H 3 Se). Experimental attempt to synthesize these new phases and verification of their superconductivity are called for.

Methods
The energetic stability of H-Se system is investigated by globally minimizing the potential energy surface at varied stoichiometries via an in-house developed swarm-intelligence based CALYPSO method 40,41 in combination with ab initio density functional theory (DFT) total-energy calculations. Its validity in rapidly finding the stable ground-state structures has been demonstrated by its applications in various material systems ranging from elements to binary and ternary compounds 40,[42][43][44] . The energetic calculations are performed using the plane-wave pseudopotential method within the generalized gradient approximation through the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 45 , as implemented in the VASP code 46 . The electron-ion interaction was described by the projected-augmented-wave potentials with 1s 1 and 4s 2 4p 4 as valence electrons for H and Se, respectively. During the structure search, an economy set of parameters are used to calculate the relative energetics of sampled structures, following which the cutoff energy of 600 eV for the expansion of wave-function and Monkhorst− Pack k-point sampling with grid spacing of 2π × 0.03 Å −1 were chosen to ensure the enthalpy converged to better than 1 meV/atom. The validity of pseudopotentials used at high pressures is carefully examined with the full-potential linearized augmented plane-wave method through the WIEN2k package 47 . The phonon spectrum for evaluating the lattice dynamic stability and electron-phonon coupling for superconducting properties of stable phases are performed within the framework of the linear-response theory via Quantum-ESPRESSO package 48 . The spin-orbit coupling is found to have a negligible effect on the electronic and superconductive properties of the current system ( Fig. S11 and Table S4), and is thus reasonably neglected. See Supplementary information for more details.