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 32S (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 3 Calculated electronic band structure and partial density of states (DOS) for H 3 32S at selected pressures. Full size image

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 \(Im\overline{3}m\) 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 α2F(ω), which weights the phonon density of states with the coupling strengths and appropriately describes the pairing interaction due to phonons:

$${\alpha }^{2}(\omega )F(\omega )=\frac{1}{2\pi N(0)}\sum _{{\bf{q}}{\rm{

u }}}\delta (\omega -{\omega }_{{\bf{q}}{\rm{

u }}})\frac{{\gamma }_{{\bf{q}}{\rm{

u }}}}{\hslash {\omega }_{{\bf{q}}{\rm{

u }}}},$$ (1)

where

$$\begin{array}{l}{\gamma }_{{\bf{q}}

u }=\pi {\omega }_{{\bf{q}}

u }\sum _{ij}\int \frac{{{\rm{d}}}^{{\rm{3}}}k}{{{\rm{\Omega }}}_{BZ}}|{g}_{{\bf{q}}

u }({\bf{k}},i,j{)|}^{2}\delta ({\varepsilon }_{{\bf{q}},i}-{\varepsilon }_{F})\delta ({\varepsilon }_{{\bf{k}}+{\bf{q}},j}-{\varepsilon }_{F}\mathrm{).}\end{array}$$ (2)

Figure 4 Phonon dispersion and projected phonon density of states (PhDOS) for H 3 S with all stable sulfur isotopes. Results for pressure of 155 GPa. Full size image

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.

The Eliashberg spectral function, electron-phonon coupling constant λ and logarithmic average phonon frequency ω ln are investigated to explore the possible record critical temperature of H 3 S. The calculated α2F(ω)/ω functions and integration of λ for H 3 32S, H 3 33S, H 3 34S and H 3 36S at 155 GPa are shown in 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 32S 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 36S. 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 α2(ω) is responsible for complicated shape of the Eliashberg spectral functions. This may leads directly to the non-monotonic changes of magnitudes related to α2F(ω) such as λ, ω ln , and critical temperature with increasing mass of sulfur isotope.

Figure 5 The Eliashberg spectral functions for H 3 S at 155 GPa. Only the stable S isotopes was investigated (upper panel). The electron-phonon coupling parameter integral as a function of frequency (bottom panel). The percent contribution of λ originating from sulfur was marked for all cases. Full size image

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 formalism37,38. It should be noted that in literature T C is usually obtained using the simple approach proposed by McMillan or Allen and Dynes39,40, which represent the weak-coupling limit of the more elaborate Eliashberg approach37. In our previous papers41,42, 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 way43 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 u44.

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 32S, H 3 33S, H 3 34S and H 3 36S at 155 GPa are summarized in 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 36S, λ increases resulting in an enhanced T C to record value of 242 K.

Figure 6 Critical temperature calculated for investigated systems at 155 GPa. Insets present behavior of λ and ω ln . Full size image

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 = 0.4723. 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:

$$\alpha =-\frac{{\rm{l}}{\rm{n}}\,{[{T}_{C}]}_{{{{\rm{H}}}_{3}}^{36}{\rm{S}}}-\,{\rm{l}}{\rm{n}}\,{[{T}_{C}]}_{{{{\rm{H}}}_{3}}^{32}{\rm{S}}}}{{\rm{l}}{\rm{n}}\,{[M]}_{{}^{36}{\rm{S}}}-\,{\rm{l}}{\rm{n}}\,{[M]}_{{}^{32}{\rm{S}}}},$$ (3)

where \({[M]}_{{}^{32}{\rm{S}}}\) and \({[M]}_{{}^{36}{\rm{S}}}\) are the atomic mass of 32S and 32S isotope, respectively. Contrary to most superconducting materials, the calculated isotope coefficient is negative α = −1.5. The inverse and nontrivial 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 pressure47. 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 sulfur12,13,35,48. 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.