## Introduction

The recent breakthrough theoretical calculations1, 2 confirmed by spectacular experimental reports of the superconductivity in hydrogen sulfide, with the record high critical temperature equals to 203 K at pressure closed to 150 GPa3,4,5, open the door to achieving the room-temperature superconductivity in the compressed hydrogen-rich materials6,7,8 or in the pristine metallic hydrogen9,10,11,12. In contrast to the cuprates13, 14, where the mechanism responsible for the superconducting state is still debated15,16,17,18,19,20,21,22, the phonon-mediated pairing scenario is generally accepted in the case of H3S23,24,25,26,27 due to the observed of a large isotope effect which convincingly suggest that hydrogen sulfide is a conventional superconductor3, 4. Therefore, in the theoretical papers28,29,30,31,32, the superconducting properties of hydrogen sulfide are studied in the framework of the mean-field Bardeen-Cooper-Schrieffer (BCS) theory33, 34, or more precisely using the Migdal-Eliashberg (ME) approach35,36,37. For the crystal structures of high-T C hydrides, the high-pressure x-ray diffraction experiments combined with the electrical resistance measurements5, 38 confirm that H3S takes a body-centered cubic $$Im\overline{3}m$$ structure, which is certainly stable above 180 GPa2, 39. The pressure-temperature phase diagram of solid hydrogen sulfide, determined on the base of a sharp falls to zero in resistivity with cooling, shows increase of T C from 95 to 203 K in the range of pressure from 110 to 155 GPa3, 5 and, then, further increasing of compression causes a linear decrease of critical temperature to 170 K at 225 GPa3. There arises the natural question, whether under the influence of the high pressure is it possible to obtain the superconducting state with the value of the critical temperature even higher than 203 K in the case of H3S compound. Below, we present the results of the ab initio calculations, which conclude that T C does not exceed the value of 203 K, while the range of the pressure from 250 to 500 GPa is adopted.

## Computational details

The searches for the stable high pressure structures of H3S system were performed through the evolutionary algorithm implemented in the USPEX code40, 41, which has been applied successfully to a number of compressed systems containing hydrogen42, 43. The computed enthalpy differences relative to the Cccm structure (H − H Cccm ) as a function of pressure for the selected crystal structures are presented in Fig. 1. It can be clearly seen that at low pressure (below 110 GPa) the lowest value of enthalpy corresponds to the orthorhombic Cccm structure and the hexagonal R3m structure has the most stable lattice between 110 and 180 GPa. Then, a cubic $$Im\overline{3}m$$ structure becomes favorable above 180 GPa. This structure is characterized by two S atoms located at a simple body centered cubic lattice and H atom situated midway between the two S atoms (see the inset in Fig. 1). Let us emphasize that for the low pressures our results agree well with the data reported by Duan et al.2. Moreover, for H3S the second-order structural phase transition from R3m to $$Im\overline{3}m$$ is also experimentally observed but for a slightly lower pressure (~150 GPa)5, 44. It should be underlined that over 450 various structures were studied, wherein in any case was obtained enthalpy lower than $${H}_{Im\overline{3}m}$$ in the range of pressures from 250 to 500 GPa. Due to the above fact in this study the critical temperature and the other thermodynamic parameters of the superconducting state of H3S are calculated only for the structure $$Im\overline{3}m$$. Figure 2 presents the curve of the volume-pressure type. This curve can be reproduced with the help of the third-order Birch-Murnaghan equation: $$p(V)=\frac{3}{2}{B}_{0}[{({V}_{0}/V)}^{\mathrm{7/3}}-{({V}_{0}/V)}^{\mathrm{5/3}}]\,\{1+\frac{3}{4}({B}_{0}^{^{\prime} }-4)[{({V}_{0}/V)}^{\mathrm{2/3}}-1]\}$$, where B 0 = 129.8 GPa, $${V}_{0}=158.4\,{{\rm{a}}}_{0}^{3}$$, and $${B}_{0}^{^{\prime} }=3.6$$ 39.

The characteristics of the electron structure, the phonon structure, and the electron-phonon interaction was made in the framework of the Quantum-ESPRESSO package45. The calculations were conducted basing on the density-functional methods using the PWSCF code45,46,47. The Vanderbilt-type ultra-soft pseudopotentials for S and H atoms were employed with the kinetic energy cut-off equal to 80 Ry. The phonon calculations were performed for 32 × 32 × 32 Monkhorst-Pack k-mesh with the Gaussian smearing of 0.03 Ry. The electron-phonon coupling matrices were computed using 8 × 8 × 8 q-grid. The superconducting transition temperature (T C ) can be in a simple way estimated using the Allen-Dynes modified McMillan equation48:

$${k}_{B}{T}_{C}={f}_{1}{f}_{2}\frac{{\omega }_{\mathrm{ln}}}{1.2}\,\exp \,[\frac{-1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}],$$
(1)

where k B  = 0.0862 meV/K (the Boltzmann constant), f 1 and f 2 are the correction functions:

$${f}_{1}={[1+{(\frac{\lambda }{2.46(1+3.8{\mu }^{\ast })})}^{\mathrm{3/2}}]}^{\mathrm{1/3}},$$
(2)
$${f}_{2}=1+\frac{(\sqrt{{\omega }_{2}}/{\omega }_{\mathrm{ln}}-1){\lambda }^{2}}{{\lambda }^{2}+{[1.82(1+6.3{\mu }^{\ast })(\sqrt{{\omega }_{2}}/{\omega }_{\mathrm{ln}})]}^{2}}.$$
(3)

The quantity ω 2 represents the second moment of the normalized weight function:

$${\omega }_{2}\equiv \frac{2}{\lambda }{\int }_{0}^{{\omega }_{D}}\,d\omega {\alpha }^{2}F(\omega )\omega$$
(4)

and ω ln is the logarithmic average of the phonon frequencies:

$${\omega }_{\mathrm{ln}}\equiv \exp \,[\frac{2}{\lambda }{\int }_{0}^{{\omega }_{D}}\,d\omega \frac{{\alpha }^{2}F(\omega )}{\omega }\,\mathrm{ln}(\omega )].$$
(5)

More sophisticated calculations can be conducted within the framework of the Eliashberg formalism, which allows a more accurately description of the superconducting state in strong-coupling systems31, 32. The Eliashberg equations for the superconducting order parameter function $${\phi }_{m}\equiv \phi \,(i{\omega }_{m})$$ and the electron mass renormalization function $${Z}_{m}\equiv Z\,(i{\omega }_{m})$$ written in the imaginary-axis formulation take the following form refs 36 and 49:

$${\phi }_{m}=\pi {k}_{B}T\sum _{n=-M}^{M}\,\frac{{\lambda }_{n,m}-{\mu }^{\ast }\theta ({\omega }_{c}-|{\omega }_{n}|)}{\sqrt{{\omega }_{n}^{2}{Z}_{n}^{2}+{\phi }_{n}^{2}}}{\phi }_{n},$$
(6)

and

$${Z}_{m}=1+\frac{\pi {k}_{B}T}{{\omega }_{n}}\sum _{n=-M}^{M}\,\frac{{\lambda }_{n,m}}{\sqrt{{\omega }_{n}^{2}{Z}_{n}^{2}+{\phi }_{n}^{2}}}{\omega }_{n}{Z}_{n},$$
(7)

where the pairing kernel for the electron-phonon interaction is given by:

$${\lambda }_{n,m}=2{\int }_{0}^{{\omega }_{D}}\,d\omega \frac{\omega }{{({\omega }_{n}-{\omega }_{m})}^{2}+{\omega }^{2}}{\alpha }^{2}F(\omega ).$$
(8)

Symbols $${\mu }^{\ast }$$ and θ denote the Coulomb pseudopotential and the Heaviside function with cut-off frequency ω c equal to three times the maximum phonon frequency (ω D ). The α 2 F(ω) functions, called the Eliashberg functions, for H3S system were calculated using the density functional perturbation theory and the plane-wave pseudopotential method, as implemented in the Quantum-Espresso package45:

$${\alpha }^{2}F(\omega )=\frac{1}{2\pi \rho ({\varepsilon }_{F})}\sum _{{\bf{q}}\nu }\,\delta (\omega -{\omega }_{{\bf{q}}\nu })\frac{{\gamma }_{{\bf{q}}\nu }}{{\omega }_{{\bf{q}}\nu }},$$
(9)

with

$${\gamma }_{{\bf{q}}\nu }=2\pi {\omega }_{{\bf{q}}\nu }\sum _{ij}\,\int \frac{{d}^{3}k}{{{\rm{\Omega }}}_{BZ}}{|{g}_{{\bf{q}}\nu }({\bf{k}},i,j)|}^{2}\delta ({\varepsilon }_{{\bf{q}},i}-{\varepsilon }_{F})\delta ({\varepsilon }_{{\bf{k}}+{\bf{q}},j}-{\varepsilon }_{F}),$$
(10)

where $$\rho \,({\varepsilon }_{F})$$ denotes the density of states at the Fermi energy, ω q ν determines the values of the phonon energies, and γ q ν represents the phonon linewidth. The electron-phonon coefficients are given by g q ν  (k, i, j) and $${\varepsilon }_{{\bf{k}},i}$$ is the electron band energy.

## Results and Discussion

To investigate the electronic properties of H3S at $$p\in \langle 250,500\rangle \,{\rm{GPa}}$$, we calculate the electronic band structure and density of states (DOS). The Fermi surface of H3S at 250 and 500 GPa is shown in Fig. 3. It is formed by five different Fermi surfaces calculated in the bcc Brillouin zone50. As has been previously reported by Bianconi and Jarlborg, the red small Fermi surface centered at the Γ-point and covering the surfaces #1 and #2, appears above 95 GPa with the change of the Fermi surface topology. This change of the Fermi surface topology is called a L1 Lifshitz transition for a new appearing Fermi surface spot and occurs where the bands at the Γ-point cross the chemical potential. The L2 Lifshitz transition for neck disrupting occurs around 180–200 GPa and is connected with appearing of the small tubular necks in the Fermi surface (in particular in the surface #4)50, 51.

The results presented in Fig. 4 clearly show that $$Im\overline{3}m$$ structure is a good metal with a large DOS at the Fermi level (0.418–0.511 states/eV/f.u.). This is in a good agreement with the previous theoretical and experimental results obtained for the lower pressure3,4,5, 52. The metallic behavior of this system indicates that $$Im\overline{3}m$$ phase might be superconducting above 250 GPa.

In order to investigate the superconductivity of H3S, the phonon band structures, the phonon density of states (PhDOS) and the Eliashberg spectral functions together with the electron-phonon integrals $$\lambda (\omega )=2{\int }_{0}^{\omega }\,d\omega {\alpha }^{2}F(\omega )/\omega$$ were carried out. As shown in Fig. 5 there is no imaginary frequency to be found in the whole Brillouin zone, confirming that $$Im\overline{3}m$$ is a dynamically stable structure. In the case of the pressure of 250 GPa, the clearly separated lines respectively associated with the low-energy vibrations of sulfur ($$\omega \in \langle \mathrm{0,76.4}\rangle \,{\rm{meV}}$$) and the high-energy vibrations of hydrogen ($$\omega \in \langle \mathrm{95.2,256.7}\rangle \,{\rm{meV}}$$) can be noticed in the phonon dispersive relation. Such fact directly translates into the shape of the function of the phonon density of states, which consists of two parts separated by the gap of the energy (about 19 meV). On the basis of the diagram related to the spectral function it can be seen that the contribution to the electron-phonon coupling constant comes mainly from hydrogen, and is equal approximately to 79%. At the pressure of 350 GPa, the maximum energy of the hydrogen vibrations becomes larger and equals to 308.7 meV. Still visible is the division of functions of the phonon density of states on the part related to sulfur and hydrogen. However, the energy gap width decreases, and is approximately 14 meV. The contribution of hydrogen to the electron-phonon coupling constant is still dominating (around 66%). The increase of the pressure by the further 150 GPa causes the renewed increase in the maximum vibration energy of hydrogen $${[{\omega }_{D}]}_{500{\rm{GPa}}}/{[{\omega }_{D}]}_{350{\rm{GPa}}}=1.179$$. However, the division of the phonon density of states on the portion derived from sulfur and hydrogen is galling. The disappearance of the sulfur-hydrogen separation results also in the slight increase of the value of the electron-phonon coupling constant in the range of the lower frequencies $${\lambda }_{500{\rm{GPa}}}(\omega =120\,{\rm{meV}})/{\lambda }_{350{\rm{GPa}}}(\omega =120\,{\rm{meV}})=1.29$$. Above the pressure of 500 GPa we found the imaginary (negative) phonon frequencies which is an indication of the structural instability. This is one of the reason why we have limited our calculations to this range of pressures, the second one is that higher pressures are far beyond the ability of the experiment. In light of the latest results on the metallization of hydrogen53, compression up to 500 GPa is possible to achieve in laboratory.

Figure 6 ilustrates the pressure dependence of the superconducting critical temperature. Close and open circles corresponds to the experimental results presented by Drozdov et al.3 and Einaga et al.5, respectively. The red dashed lines drawn by eye represents the trend of the experimental data above 150 GPa and great combine together with the theoretical range of T C calculated for the high pressures. These theoretical results were obtained using the Eliashberg equations and the following relation: $${{\rm{\Delta }}}_{m=1}(T={T}_{C})=0$$, where the order parameter is defined as $${{\rm{\Delta }}}_{m=1}={\phi }_{m=1}/{Z}_{m=1}$$. The commonly accepted value of the Coulomb pseudopotential $${\mu }^{\ast }=0.13$$ was adopted, the exact results of T C are collected in Table 1. The error bars indicate the value range of T C with $${\mu }^{\ast }\in \langle 0.11,0.15\rangle$$. The obtained results show that T C decreases from 164 to 129 K in the range of pressure from 250 to 350 GPa. Then above 350 GPa the superconducting critical temperature starts increasing. This is a promising result, however in the range of the pressure from 350 to 500 GPa the critical temperature does not exceed the value of 203 K. This can be explained by the unfavorable, and simultaneously weak, variation of the electron-phonon coupling constant and the logarithmic phonon frequency (the insert in Fig. 6). From the conducted ab initio calculations comes the conclusion that this is caused by the small or unfavorable influence of the pressure on the electron density of states and the electron-phonon matrix elements. It can be, however, noticed that the value of T C in the range of the very high pressures is relatively high and does not drop below 120 K. We did not study the critical temperature under extreme pressures because beyond 500 GPa the H3S structure loses the dynamical stability.

Then, by using the analytical continuation37, 49, we determine the superconducting energy gap Δ(0) and the dimensionless ratio 2Δ(0)/T C which in the BCS theory takes the constant value 3.53. As we can see in Table 1, the obtained results significantly exceed the value of BCS predictions. This is connected with the strong-coupling and retardation effects, which in the framework of the Eliashberg formalism are not neglected. During the preparation of this manuscript, a superconducting energy gap of H3S compressed to 150 GPa was experimentally found (2Δ = 73 meV)54. Taking into account this result and the previously determined value of T C at this same pressure (203 K)3 we can evidence that 2Δ/k B T C  = 4.17 is surprisingly close to our predictions for higher pressures. The above fact proves the correctness of our calculations.

## Conclusions

In this paper we showed that H3S exhibits the superconducting properties in the range of the very high pressures (250–500 GPa), however, the critical temperature does not exceed the value of 203 K. The obtained result is related to the weak and unfavorable volatility of the electron-phonon coupling constant and the logarithmic phonon frequency. From the microscopic point of view, this results from the small or unfavorable influence of the pressure on the value of the electron density of states at the Fermi surface and the electron-phonon matrix elements. According to the above, it can be seen that the increase of the pressure alone is not sufficient to obtain the superconducting state in H3S at the room temperature. It is possible that the better method to achieve this goal is the appropriate partial atomic substitution of S atoms by other atoms. Interesting theoretical results were obtained by Ge et al. in the paper55, where the increase of the P-substitution rate causes the increase of the DOS, the phonon linewidths and the electron-phonon coupling constant. Finally, T C  = 280 K for H3S0.925P0.075 at 250 GPa.