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Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system


The quantum nature of the proton can crucially affect the structural and physical properties of hydrogen compounds. For example, in the high-pressure phases1,2 of H2O, quantum proton fluctuations lead to symmetrization of the hydrogen bond and reduce the boundary between asymmetric and symmetric structures in the phase diagram by 30 gigapascals (ref. 3). Here we show that an analogous quantum symmetrization occurs in the recently discovered4 sulfur hydride superconductor with a superconducting transition temperature Tc of 203 kelvin at 155 gigapascals—the highest Tc reported for any superconductor so far. Superconductivity occurs via the formation of a compound with chemical formula H3S (sulfur trihydride) with sulfur atoms arranged on a body-centred cubic lattice5,6,7,8,9. If the hydrogen atoms are treated as classical particles, then for pressures greater than about 175 gigapascals they are predicted to sit exactly halfway between two sulfur atoms in a structure with symmetry. At lower pressures, the hydrogen atoms move to an off-centre position, forming a short H–S covalent bond and a longer H···S hydrogen bond in a structure with R3m symmetry5,6,7,8,9. X-ray diffraction experiments confirm the H3S stoichiometry and the sulfur lattice sites, but were unable to discriminate between the two phases10. Ab initio density-functional-theory calculations show that quantum nuclear motion lowers the symmetrization pressure by 72 gigapascals for H3S and by 60 gigapascals for D3S. Consequently, we predict that the phase dominates the pressure range within which the high Tc was measured. The observed pressure dependence of Tc is accurately reproduced in our calculations for the phase, but not for the R3m phase. Therefore, the quantum nature of the proton fundamentally changes the superconducting phase diagram of H3S.

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Figure 1: Crystal structures of the competing phases.
Figure 2: Energetics.
Figure 3: Second-order phase transition.
Figure 4: Phonon spectra and superconducting transition temperature.


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We acknowledge financial support from the Spanish Ministry of Economy and Competitiveness (FIS2013- 48286-C2-2-P), the EPSRC (UK) (grant numbers EP/J017639/1 and EP/K014560/1), the Cambridge Commonwealth Trust, the National Natural Science Foundation of China (grant numbers 11204111, 11404148, 11274136 and 11534003), the 2012 Changjiang Scholars Program of China, and the Natural Science Foundation of Jiangsu province (grant number BK20130223). C.J.P. acknowledges support from the Royal Society through a Wolfson Research Merit award. Work at Carnegie was supported by EFree, an Energy Frontier Research Center funded by the DOE, Office of Science, Basic Energy Sciences under award number DE-SC-0001057. Computer facilities were provided by PRACE and the Donostia International Physics Center (DIPC).

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Authors and Affiliations



I.E., M.C. and F.M. performed the anharmonic and superconducting calculations. All authors contributed to the design of the research project and to the writing of the manuscript.

Corresponding authors

Correspondence to Ion Errea, Matteo Calandra or Francesco Mauri.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Equations of state.

Pressure P as a function of volume V for the (left) and R3m (right) phases, calculated from the static energy EBO that represents the classical nuclei limit (black), and including the vibrational contribution within the SSCHA for both H3S (red) and D3S (green).

Extended Data Figure 2 Bond symmetrization within LDA (left) and BLYP (right).

For each volume V the relative coordinate x = (d2 − a/2)/(a/2), in which d2 is the length of the hydrogen bond and a is the lattice parameter, obtained at the energy minimum is given. When x = 0, the covalent and hydrogen bonds have the same length and the structure is fully symmetric. The energy is calculated at the static level without any vibrational contribution as derived from the BOES (black), and including the quantum anharmonic vibrational contribution both for H3S (green) and D3S (purple). The pressure P below which the cubic structure distorts is given in each case.

Extended Data Figure 3 Vibrational energy.

SSCHA anharmonic vibrational energy () calculated as a function of the relative coordinate x = (d2 − a/2)/(a/2) for different volumes (see legend).

Extended Data Figure 4 Dependence of the equation of state on the density functional.

The equation of state is calculated with different exchange correlation functionals with (right) and without (left) the vibrational contribution to the pressure. At each volume V the pressure P of the structure with minimum energy is given, which depends on whether or not the vibrational contribution is included (see Extended Data Fig. 2). The results are compared with the two curves obtained experimentally10.

Extended Data Figure 5 Phonons of D3S.

Comparison between the harmonic and anharmonic phonons of at two different pressures for D3S: 133 GPa (left) and 155 GPa (right).

Extended Data Figure 6 Anharmonic phonons of R3m H3S.

Anharmonic phonons are shown at two different pressures: 153 GPa (left) and 133 GPa (right). The Eliashberg functions α2F(ω) are also shown.

Extended Data Figure 7 Superconducting properties of H3S.

Anharmonic Eliashberg function α2F(ω) (solid lines) and integrated electron–phonon coupling constant λ(ω) (dashed lines) of the phase as functions of frequency ω at two different pressures: 135 GPa (black) and 157 GPa (red).

Extended Data Table 1 Birch–Murnaghan fit to the equation of state
Extended Data Table 2 Raman and infrared active modes
Extended Data Table 3 Superconducting parameters

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Errea, I., Calandra, M., Pickard, C. et al. Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system. Nature 532, 81–84 (2016).

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