Unusual sulfur isotope effect and extremely high critical temperature in H3S superconductor

Recent experiments have set a new record for the transition temperature at which a material (hydrogen sulfide, H3S) becomes superconducting. Moreover, a pronounced isotope shift of TC in D3S is evidence of an existence of phonon-mediated pairing mechanism of superconductivity that is consistent with the well established Bardeen-Cooper-Schrieffer scenario. Herein, we reported a theoretical studies of the influence of the substitution of 32S atoms by the heavier isotopes 33S, 34S and 36S on the electronic properties, lattice dynamics and superconducting critical temperature of H3S. There are two equally fundamental results presented in this paper. The first one is an anomalous sulfur-derived superconducting isotope effect, which, if observed experimentally, will be subsequent argument that proves to the classical electron-phonon interaction. The second one is fact that critical temperature rise to extremely high value of 242 K for H336S at 155 GPa. This result brings us closer to the room temperature superconductivity.

Quantum-Espresso package 33,34 . The ultrasoft Vanderbilt pseudopotentials for S and H atoms were employed with a kinetic energy cut-off equal to 80 Ry. The electronic band structure calculations were performed for 32 × 32 × 32 Monkhorst-Pack k-mesh to sample the Brillouin zone with the Gaussian smearing of 0.03 Ry. The phonon dispersion and electron-phonon coupling matrices were computed within the framework of the linear-response method on the set of 8 × 8 × 8 q-point mesh for the first Brillouin zone integrations. The energy convergence and precision of all presented results are controlled by assuming the sufficiently small (10 −16 Ry) threshold on the change in total energy. The X-ray diffraction experiments 8 confirm that the superconducting phase of H 3 S is in good agreement with the theoretically predicted body-centred cubic Im m 3 structure 35 . The ball-and-stick model of cubic H 3 S structure under compression is shown in the inset of Fig. 1. To evaluate the lattice constant and phase stability of the investigated Im m 3 structure we performed total energy calculations and structural relaxations in a wide range of high pressure. The lattice constant and atomic positions were relaxed according to the atomic forces. This procedure was repeated until the forces on every atom of the unit cell were less than 1 meV/a.u. and the resulting stress less than 1 kbar. In this way, the fully relaxed structural parameters of H 3 S have been obtained. In Fig. 2, the calculated lattice constant as a function of pressure was presented and compared with other theoretical predictions 35,36 . These results coincide with very good accuracy with previous reports, consequently, for the pressure range where the high-T C was measured (155-225 GPa) we assume that Im m 3 structure is stable and we expect that the change of sulfur isotope mass can significantly influences on the superconducting state of H 3 S.

Results and Discussion
To analyze the electronic properties of H 3 S the electronic band structure and the partial density of states (DOS) were calculated. In Fig. 3 we can see the results for investigated pressures 155, 175, 200 and 225 GPa and for one of stable sulfur isotope 32 S (94.99% natural abundance). The existence of electrons in the Fermi level indicates the  metallic character of all cases. The van Hove singularity near the Fermi level can enhance the electron-phonon coupling strength and hence can be responsible for high-temperature superconductivity. Furthermore, very similar shape of electronic band structure and DOS are found in whole range of pressure. Also the change of sulfur isotope in elemental cell has no effect on the electronic properties of studied system. On this basis, we can suppose that phonons properties in hydrogen sulfide systems are actually responsible for change in their thermodynamic properties. Figure 4 shows the calculated phonon band structure and projected phonon density of states (PhDOS). Phonon calculations did not give any imaginary frequency vibration mode in the whole Brillouin zone, indicating the dynamic stability of Im m 3 structure. Based on the PhDOS, we found that the vibration frequency is divided into two parts as a result of the different atomic masses of S and H atoms. The low-frequency bands mainly result from the vibrations of the S atoms, whereas the H atoms are mostly related to vibrations with higher frequency modes. Note that the contribution derived from sulfur is shifted towards the lower frequencies together with increasing S isotope mass. This should be reflected in the shape of the Eliashberg electron-phonon spectral function α 2 F(ω), which weights the phonon density of states with the coupling strengths and appropriately describes the pairing interaction due to phonons: Symbols N(0), γ qν , and g qν (k, i, j) denote the density of states at the Fermi energy, the phonon linewidth, and electron-phonon matrix elements, respectively.  Fig. 5. The main contribution to the electron-phonon coupling constant derived from hydrogen and it should be highlights that the H atoms play a significant role in the superconductivity of hydrogen sulfide. For H 3 32 S nearly 22% of λ originates from sulfur. With increasing mass of sulfur isotope, it is very interesting to note that, the part coming from S changes and finally decreases to 7% for H 3 36 S. The comparison of Eliashberg functions with phonon density of states shows that the square of the matrix element of the electron-phonon interaction averaged over the Fermi surface α 2 (ω) is responsible for complicated shape of the Eliashberg spectral functions. This may leads directly to the non-monotonic changes of magnitudes related to α 2 F(ω) such as λ, ω ln , and critical temperature with increasing mass of sulfur isotope.
The high vibrational phonon frequency and the strong electron-phonon coupling constant lead directly to a high superconducting critical temperature which was calculated using the Eliashberg formalism 37,38 . It should be noted that in literature T C is usually obtained using the simple approach proposed by McMillan or Allen and Dynes 39,40 , which represent the weak-coupling limit of the more elaborate Eliashberg approach 37 . In our previous papers 41, 42 , we proved that the McMillan or Allen-Dynes-modified McMillan formulas and Eliashberg equations lead to similar results for small λ and Coulomb pseudopotential μ ★ . For larger λ and μ ★ , however, the analytical formulas predicts underestimated T C values. In the case of the hydrogen sulfide the electron-phonon interaction is strong, hence the analytical formulas are inappropriate. The isotropic Migdal-Eliashberg equations were solved in a numerical way 43 using 2201 Matsubara frequencies ω n = (π/β)(2n−1), where n = 0, ±1, ±2, …, ±1100, and a Coulomb pseudopotential which was chosen to match the measured value of T C for standard S atomic weight of 32.06 u 44 .
Such an assumption ensures the stability of the numerical solutions for T ≥ 1 K. The superconducting transition temperature was estimated to be in the range of 202-242 K at 155 GPa. The calculated T C , λ and ω ln for H 3 Fig. 6. It is very interesting to note that T C is strongly correlated with λ and despite decrease in ω ln for H 3 36 S, λ increases resulting in an enhanced T C to record value of 242 K.
The isotope effect of superconducting critical temperature is best described in terms of the isotope effect coefficient α. For experimental results of hydrogen and deuterium sulfide at p = 155 GPa we have α = 0.47 23 . This value is very close to the theoretical value of 0.5 predicted within the framework of the BCS scenario. In this paper, for the most extreme case of sulfur isotopes at 155 GPa we have the following relation:  behavior can be also observed for other isotopes and higher pressures, as shown in Fig. 7. On the other hand, some other systems also display values that are smaller than zero. For example the inverse superconducting isotope coefficient has been observed in uranium (α = −2) 45 , metal hydride PdH (α = −0.25) 46 or lithium where α sign changes with increasing pressure 47 . Let us strongly emphasize, however, that the isotope effect in superconductivity is taken as evidence for phonon mediation. Coming back to the Fig. 7, we can additionally observed that with increasing pressure the critical temperature decreasing which is in a general agreement with the trend established by the experimental results. Moreover, it should be emphasized that correctness of our methods and numerical calculations was confirmed by comparison the obtained results with the previous ones for the natural isotope concentration of sulfur 12,13,35,48 . To benchmark the validity of the results obtained in the present work, in next section we have shown the calculated electronic structure and phonon dispersions together with the results previously reported by Duan et al. 35 . Moreover, we examined the isotope effect for H 3 S and D 3 S.

Proving correctness of the presented results
A benchmark study on correctness of our results was done to electronic structure and phonon dispersions which were collate with the results previously reported by Duan et al. 35 . This comparison for the natural isotope concentration of sulfur in H 3 S at 200 GPa was shown in Fig. 8. On this base we can found that here is almost exact coincidence which proves the correctness of the results reported in the present work.
Moreover, we compared the isotope coefficient resulting from the experimental critical temperature of H 3 S and D 3 S (see Fig. 1) with our estimations conducted within the framework of the Eliashberg formalism. In both cases the isotope coefficient decreases with pressure which is connected with decreasing difference between  critical temperature for H 3 S and D 3 S. As shown in Fig. 9, the very high level of consistency was achieved. This is another argument in favor of correctness and high-value of presented herein results.

Conclusion
We reported the influence of the substitution of 32 S atoms by the heavier isotopes 33 S, 34 S and 36 S on the electronic properties, lattice dynamics and superconducting critical temperature in H 3 S. We observe that for a pressure of 155 GPa this substitution causes a strong (20%) enhancement of T C from 202 to 242 K. This unexpectedly high T C far exceeds the previous record of 203 K and bring us closer to achieving room-temperature superconductivity in hydrogen-rich materials at high pressure. The second very important and interesting result, reported in this paper, is uncommon sulfur isotope effect. We noted the strong negative isotope coefficient (α =−1.5) between H 3 32 S and H 3 36 S, and variation of the isotope effect with the increasing pressure. We expected that our significant findings can stimulate future high-pressure experiments and that suggested pathway to increase T C can be appropriate to reach near-room-temperature superconductivity.  The isotope coefficient as a function of pressure calculated from the averaged experimental critical temperature of H 3 S and D 3 S at the same pressure (blue square) and our results (red triangles) obtained using Eliashberg formalism with Coulomb pseudopotential estimated for standard S atomic weight.