First-principles study of superconducting hydrogen sulfide at pressure up to 500 GPa

We investigate the possibility of achieving the room-temperature superconductivity in hydrogen sulfide (H3S) through increasing external pressure, a path previously widely used to reach metallization and superconducting state in novel hydrogen-rich materials. The electronic properties and superconductivity of H3S in the pressure range of 250–500 GPa are determined by the first-principles calculations. The metallic character of a body-centered cubic Im\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\bf{3}}}$$\end{document}3¯m structure is found over the whole studied pressure. Moreover, the absence of imaginary frequency in phonon spectrum implies that this structure is dynamically stable. Furthermore, our calculations conducted within the framework of the Eliashberg formalism indicate that H3S in the range of the extremely high pressures is a conventional strong-coupling superconductor with a high superconducting critical temperature, however, the maximum critical temperature does not exceed the value of 203 K.


Computational details
The searches for the stable high pressure structures of H 3 S system were performed through the evolutionary algorithm implemented in the USPEX code 40,41 , which has been applied successfully to a number of compressed systems containing hydrogen 42,43 . 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 m 3 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 H 3 S the second-order structural phase transition from R3m to Im m 3 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 m 3 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 H 3 S are calculated only for the structure Im m 3 . 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: . The characteristics of the electron structure, the phonon structure, and the electron-phonon interaction was made in the framework of the Quantum-ESPRESSO package 45 . The calculations were conducted basing on the density-functional methods using the PWSCF code [45][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 equation 48 :   The quantity ω 2 represents the second moment of the normalized weight function: and ω ln is the logarithmic average of the phonon frequencies:   where the pairing kernel for the electron-phonon interaction is given by: Symbols µ ⁎ 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 H 3 S system were calculated using the density functional perturbation theory and the plane-wave pseudopotential method, as implemented in the Quantum-Espresso package 45 : where ρ ε ( ) F denotes the density of states at the Fermi energy, ω qν determines the values of the phonon energies, and γ qν represents the phonon linewidth. The electron-phonon coefficients are given by g qν (k, i, j) and ε i k, is the electron band energy.

Results and Discussion
To investigate the electronic properties of H 3 S at ∈ p 250, 500 GPa, we calculate the electronic band structure and density of states (DOS). The Fermi surface of H 3 S at 250 and 500 GPa is shown in Fig. 3. It is formed by five different Fermi surfaces calculated in the bcc Brillouin zone 50 . 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 m 3 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 pressure [3][4][5]52 . The metallic behavior of this system indicates that Im m 3 phase might be superconducting above 250 GPa.
In order to investigate the superconductivity of H 3 S, the phonon band structures, the phonon density of states (PhDOS) and the Eliashberg spectral functions together with the electron-phonon integrals ∫ λ ω ω α ω ω = ω d F ( ) 2 ( )/ 0 2 were carried out. As shown in Fig. 5 there is no imaginary frequency to be found in the whole Brillouin zone, confirming that Im m 3 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 (ω ∈ . 0,76 4 meV) and the high-energy vibrations of hydrogen (ω ∈ . . 95 2,256 7 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  . 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 hydrogen 53 , 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: , where the order parameter is defined as The commonly accepted value of the Coulomb pseudopotential µ = .
⁎ 0 13 was Figure 5. The phonon dispersion relation, the phonon density of states, and the spectral functions for selected values of pressure.
adopted, the exact results of T C are collected in Table 1. The error bars indicate the value range of T C with µ ∈ . . ⁎ 0 11, 0 15 . 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 H 3 S structure loses the dynamical stability.
Then, by using the analytical continuation 37, 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 H 3 S 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 H 3 S 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 H 3 S 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 paper 55 , 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 H 3 S 0.925 P 0.075 at 250 GPa.   Table 1. Density of states at the Fermi level ρ (ε F ) (in units states/eV/f.u.), Fermi energy ε F , electron-phonon coupling constant λ, logarithmic phonon frequency ω ln , critical temperature T C (determined using Eliashberg equations for µ = . ⁎ 0 13), superconducting energy gap Δ(0) and dimensionless ratio 2Δ(0)/k B T C of H 3 S under different pressures. The lattice constant a corresponding with appropriate pressure is also included.