Superconducting phase diagram of H3S under high magnetic fields

The discovery of superconductivity at 260 K in hydrogen-rich compounds like LaH10 re-invigorated the quest for room temperature superconductivity. Here, we report the temperature dependence of the upper critical fields μ0Hc2(T) of superconducting H3S under a record-high combination of applied pressures up to 160 GPa and fields up to 65 T. We find that Hc2(T) displays a linear dependence on temperature over an extended range as found in multigap or in strongly-coupled superconductors, thus deviating from conventional Werthamer, Helfand, and Hohenberg (WHH) formalism. The best fit of Hc2(T) to the WHH formalism yields negligible values for the Maki parameter α and the spin–orbit scattering constant λSO. However, Hc2(T) is well-described by a model based on strong coupling superconductivity with a coupling constant λ ~ 2. We conclude that H3S behaves as a strong-coupled orbital-limited superconductor over the entire range of temperatures and fields used for our measurements.

T he ongoing scientific endeavor to stabilize superconductivity at room temperature led to the discovery of superconductivity, with a very high critical temperature T c = 203 K, in sulfur hydride H 3 S under high pressures of p = 155 GPa 1 . H 3 S along with other hydrides 2,3 seems to be the closest compound, so far, to metallic hydrogen, which is predicted to be a high temperature superconductor [4][5][6] . H 3 S forms as the result of the chemical instability of H 2 S under high pressures, where H 2 S decomposes into elemental sulfur S and H 3 S 7-10 . The structure of H 3 S is believed to be body-centered cubic with Im-3m symmetry above 150 GPa 10 , and characterized by H atoms symmetrically situated midway between two body-centered S atoms. T c shows a sharp increase from 95 K to 203 K in the pressure range of 110-155 GPa, but decreases with further increasing the pressure due to the pressure-induced phase transition from the R-3m to the Im-3m phase of H 3 S 1, 10 .
The superconductivity in H 3 S is believed to be conventional, or whose pairing is mediated by phonons characterized by highfrequency optical modes due to the motion of hydrogen 1,[11][12][13][14] . The high T c arises from both the metallization of the strongly covalent bonds in H 3 S 15 and the high phonon frequencies displayed by its light elements 16 . Band structure calculations indicate that H 3 S is a multiband metal 17 having a large Fermi surface (broad energy dispersive bands) as the result of the hybridization between the H 1s and the 3p orbitals of sulfur 8,9,13,16,18,19 . Band structure calculations also yield small Fermi surface pockets for the high-T c phase of H 3 S 17 .
Despite the very high T c s reported for H 3 S, the studies of H c2 (T) are limited to a narrow range of temperatures close to T c due to the inherent experimental difficulties in performing ultrahigh pressure measurements under very high magnetic fields. The behavior of H c2 (T) provides valuable information such as an estimation of the Cooper pair coherence length, the strength of the electron-phonon coupling, the role of the spin-orbit coupling, and the dominant mechanism breaking the Copper pairs.
In type-II superconductors, the orbital and the spinparamagnetic pair-breaking effects are the two main mechanisms depairing electrons under high magnetic fields. The orbital pair-breaking effect explains the suppression of superconductivity via the formation of Abrikosov vortices 20 in the presence of a field. The superconductivity is suppressed when the kinetic energy associated with vortex currents exceeds the condensation energy of the paired electrons. When the magnetic field approaches a critical value H orb c2 ¼ ϕ 0 =2πξ 2 , referred to as the orbital limiting field, the vortex cores overlap and the system returns to the normal state. Here, ϕ 0 is the flux quantum and ξ is the coherence length. The spin-paramagnetic effect explains the effect of the magnetic field on the spin of the electron based on the work by Clogston 21 and Chandrasekhar 22 . When the magnetic energy exceeds the superconducting gap, i.e. 1=2χ p H 2 p ¼ 1=2NðE F ÞΔ 2 , superconductivity will be suppressed, where χ p , N(E F ), and Δ are the normal state paramagnetic susceptibility, the density of states at the Fermi level, and the superconducting gap, respectively. If we only consider the spinparamagnetic effect, the zero-temperature Pauli limiting field (Chandrasekhar-Clogston limit) for a weakly coupled superconductor is approximately μ 0 H p (0) = 1.86 T c , where 2ΔðT ¼ 0Þ $ 3:52 k B T c (for a conventional Bardeen-Cooper-Schrieffer (BCS) superconductor) and χ p ¼ gμ 2 B NðE F Þ (g is the Landé g factor and μ B is the Bohr magneton). H p can be renormalized by the strength of the electron-phonon coupling or by electronic correlations. In many superconductors H c2 (T) is affected by both pair-breaking mechanisms. The Maki 23 para- is a measure of the relative strength between the orbital and the paramagnetic pair-breaking mechanisms for a given type-II superconductor. The WHH theory 24 includes both pair-breaking effects through the Maki parameter and the spin-orbit constant.
Here, we report measurements of the upper critical field in H 3 S samples under extremely high pressures and high magnetic fields. The ultrahigh pressures can only be obtained through the use of a diamond anvil cell (DAC), however, measurements on samples contained by a DAC are quite challenging, particularly in pulsed fields due to the narrow magnet bore and the large magnetic-field induced currents which heat the cell. The small diameter of the DAC 1 employed in our study allows us to perform transport measurements up to 65 T without heating the sample significantly. Our pulsed field measurements under p > 150 GPa represent a significant pressure increase over a previous pulsed field study under 10 GPa 25 . We find that H 3 S is an orbital-limited superconductor over the entire temperature range and likely a multigap or strongly coupled superconductor.

Results
Temperature dependence of the resistance. Figure 1 displays the resistance as a function of the temperature T for H 3 S samples pressurized up to 155 GPa and 160 GPa. The onset, i.e., the first deviation from the normal state resistivity, of the superconducting transitions are estimated to be T c = 201 K for the sample under 155 GPa, and T c = 174 K for the other sample under 160 GPa. The drop in T c for the higher pressure sample is consistent with the results in ref. 1 . The width of these transitions are ΔT c = 5.5 K and 26 K for the samples under 155 GPa and 160 GPa, respectively, indicating that the latter sample is less homogeneous, or that it is subjected to stronger pressure gradients. The value of T c for the sample under 155 GPa increases slightly over time, indicating an evolution towards a more homogeneous sample or weaker pressure gradients. Notice that the superconducting transition displayed by this sample shows a step around 195 K indicating that its superconducting state indeed is inhomogeneous. For samples synthesized within the confines of the DAC a certain degree of inhomogeneity is inevitable. The multiple SC transition steps merge into a single one above a Notice that the T c for the 155 GPa sample shifts slightly to higher values while its resistance decreases between July and September 2018. Source data are provided as a Source Data file moderate field of 3 T in the 155 GPa sample ( Supplementary  Fig. 1).
We find that R(T) exhibits supralinear behavior in both samples, which is well described by a R ∝ T 2 leading term. A quadratic temperature dependence for the resistivity is usually ascribed to the inelastic electron-electron scattering in Fermi liquid theory. While our estimates point to a predominantly electron-phonon transport scattering at high temperatures ( Supplementary Fig. 2), a linear temperature dependence for the resistivity is typically expected for this regime, hence R ∝ T 2 is quite unusual. Spectroscopic studies on H 3 S attribute the supralinear behavior of R(T) to strong coupling between electrons and a very high energy optical phonon 26 . A similar explanation was given for the supralinear R(T) observed in MgB 2 , which is another BCS superconductor characterized by a strong coupling to an optical phonon mode 27 .
Hall resistance as a function of the temperature. Figure 2a displays the Hall resistance R xy for the sample under p = 155 GPa as a function of the magnetic field μ 0 H near the superconducting transition. The R xy (μ 0 H) do not reveal any non-linearity as a function of μ 0 H all the way up to 35 T. This indicates that near and above T c the electrical transport in H 3 S is dominated by a single type of carrier (electrons) despite H 3 S being predicted to be a multiband metal 17 . Most likely, this indicates that electrons display a considerably higher carrier mobility than holes at these temperatures. The temperature dependence of the Hall coefficient R H is shown in Fig. 2b, obtained by multiplying the slope of R xy (μ 0 H) at high fields by the sample thickness, estimated to be t = (2 ± 1) μm. We observed a noticeable decrease in R H (T) by as much as 20% as the temperature is decreased, corresponding to an increase in the effective carrier density n = 1/eR H . The smaller values of the Hall coefficient at lower temperatures could come from a decrease in electron mobility, a progressive increase in hole mobility, or from an evolution in their relative densities. If one assumes that only one band contributed to the transport of carriers, the Hall coefficient would yield an electron density n = 1/eR H = (8.5 ± 4.3) × 10 22 cm −3 at room temperature, which is relatively close to known values for transition metals such as copper.
Superconducting phase diagram. To determine the temperature dependence of H c2 , we measured the isothermal resistance as a function of the magnetic field μ 0 H at selected temperatures ranging from T = 55 K to 200 K. Figure 3a, b show the magnetic field dependence of the resistance for the samples under 155 GPa and 160 GPa at fixed temperatures. At each temperature the resistance changes from zero to a finite value as μ 0 H increases due to the field-induced suppression of the superconductivity. H c2 is defined as the intersection between an extrapolation of the resistance in the normal state and a line having the slope of the resistive transition at its middle point 28 . The same criterion is used across DC and pulsed field traces. The field value at the intersect with R = 0 axis, H * , marks the onset of dissipation associated with melting of the vortex lattice. In contrast to the unconventional high-temperature superconductors (HTS), vortex   The resistive onset has been found to match the thermodynamic H c2 obtained through other experimental probes [29][30][31] . The resistive transition shifts to higher fields as the temperature is lowered while broadening slightly. The H c2 (T) values obtained from pulsed field measurements agree with the curvature and the values extracted under DC fields. At very high pulsed magnetic fields the self-heating of the metallic DACs by the magnetic-field induced currents becomes noticeable at temperatures below 55 K. Figure 4 shows the extracted H c2 as a function of T for both samples under p = 155 GPa and 160 GPa as a function of the reduced temperature t = T/T c . To avoid the influence of possible superconducting vortex phases, we plot just the onset of their resistive transition. These phase boundaries were obtained from both isothermal field scans and from temperature scans under fixed magnetic fields. For both samples H c2 (T) increases almost linearly upon decreasing the temperature under fields all the way up to 60 T yielding the slopes jdH c2 =dTj T c ¼ 0:62 and 0.50 T K −1 for the samples under 155 GPa and 160 GPa, respectively.

Discussion
HTS hydrides are considered to be conventional BCS-type superconductors. At the same time, the linear H c2 (T) dependence reported here is quite unusual for a weakly coupled BCS superconductor. The linear dependence of H c2 (T) over an extended range of temperatures was also observed in the two-band superconductor MgB 2 32,33 and in the multigap Fe-pnictide superconductors for fields along certain, but not for all crystallographic orientations, and claimed to result from the orbital limiting effect [34][35][36] . Multigap effects might explain our phase diagram for H 3 S in analogy with those systems. However, this multiband scenario contrasts with the linear in field Hall data in Fig. 2a, which indicates conduction dominated by a single type of carrier. Measurements of, for example, the penetration depth as a function of T should help clarify the nature of the superconducting state in this system. The estimated weak coupling Pauli limiting fields for the samples under 155 GPa and 160 GPa are H p (0) ≃ 374 T and 324 T, respectively. In the dirty limit, the H c2 (0) values derived from the slope of the H-T phase boundary at T c are, H orb c2 ð0Þ ¼ 0:69dH c2 =dTj T c T c ¼ 86 T for the sample under 155 GPa and 60 T for the one under 160 GPa. These latter values are much closer to the extrapolation of our experimental H c2 (T) towards zero-temperature, when compared to the Pauli limiting values which differ by a factor of~5, thus indicating that H 3 S is an orbital-limited superconductor.
Subsequently, we fit our experimentally determined phase boundary to two different expressions in order to estimate the value of H c2 at T = 0 K (see Methods). At temperatures near T c , our experimental H c2 values seem to follow the GL-expression, which is used to estimate H c2 (0) in hydrides from the low field data 1,2 , but at low temperatures they deviate from it considerably. This is particularly apparent for the sample under 160 GPa, for which we were able to reach smaller t values, given that the GL theory is formulated for ð1 À tÞ ( 1. Fitting the upper critical fields to the WHH expression yields very small values for α which points again to rather small orbital limiting fields relative to the Pauli limiting ones, with no apparent saturation in H c2 (t) at low ts. We obtained the best fits to the WHH formula for α values ranging between 0.0 and 0.3 for the sample under 155 GPa and α = 0-0.2 for the sample under 160 GPa. For such small values of α, the WHH formula is almost insensitive to the strength of the spin-orbit interaction. We obtained the superconducting coherence length by extrapolating the WHH fit to zero-temperature. This extrapolation of the H c2 (T)s yields H c2 (T = 0) = 88 T and 70 T for the H 3 S samples pressurized up to 155 GPa and 160 GPa, respectively. Thus, we find coherence lengths of ξ = 1.84 nm and 2.12 nm for the samples under 155 GPa and 160 GPa, respectively, which are smaller than the initial estimate ξ ≃ 4 nm from first-principles calculations 15 . As for the effect of pressure on the H 3 S samples, our experimental upper critical fields do not reveal any change in the relative strength of both pair-breaking mechanisms between both pressures. As seen in the inset of Fig. 4, the superconducting phase diagrams for both samples fall into a single curve when plotted in reduced units. What is remarkable is the linearity of the phase boundary as a function of t all the way down to t~0.3. Again, this resembles behavior reported, for example, for the Fe based superconductors 36 .
This deviation from the conventional WHH formalism could be explained by the fact that it was formulated for the weak limit of electron-phonon coupling or for a coupling constant λ ≪ 1 . Recent ab initio calculations report substantial enhancement of λ in H 3 S due to the proximity to a structural instability between the Im-3m and the R3m crystal symmetries 37 . As the phonon frequencies ω p soften near this instability, the Cooper pairing becomes weaker at elevated temperatures by the thermal phonons, while it remains robust in the T → 0 limit, leading to a relative enhancement in H c2 (T) at low temperatures. The calculated λ~2 9,37 would correspond to a nearly linear H c2 (T) for t > 0.3 within the framework of the strong coupling model 40,41 , which was also found to be applicable to the unusually strong electron-phonon coupled case of Bi-III 38 . Coincidentally, the strong coupling model also predicts a very high T c~3 00 K-400 K for a lattice containing light atoms like H 40 . The fact is that our experimental H c2 (T) values display a much better agreement with the temperature dependence predicted by the strong coupling model 40 with an electron-phonon coupling constant λ = 2.
In summary, we have investigated the temperature dependence of the upper critical fields of H 3 S under magnetic fields up to 65 T. At lower fields the phase boundary separating normal and superconducting states is relatively well described by the Werthamer, Helfand, and Hohenberg formula. Pronounced deviations from the WHH formula are observed at lower temperatures due to the linearity of H c2 (T). Overall, the phase boundary between superconducting and metallic states indicates that the orbital-effect suppresses superconductivity over the entire temperature range. The linearity of H c2 (T) observed over an extended range of reduced temperatures suggests that this system might indeed be a multigap superconductor as predicted theoretically and in analogy with similar results for the Fe-based superconductors. Alternatively, enhanced electron-phonon coupling and softening of hydrogen vibrational modes in the vicinity of a structural instability could also explain the deviation from the WHH predictions. We extract values for the coherence length ξ ranging between ξ = 1.84 nm and 2.12 nm. Above the transition temperature the Hall-effect remains linear up to high magnetic fields indicating that H 3 S is a very good metal and that at high temperatures carrier conduction is dominated by electrons.

Methods
Sample preparation. Two samples of H 3 S were synthesized in-situ inside the DACs using two different techniques. The first sample was prepared from condensed liquid H 2 S via a disproportionation reaction as described in ref. 1 , the final pressure inside the sample was as high as 160 GPa. We refer to this sample as the 160 GPa sample. The second sample was synthesized directly from elemental sulfur and hydrogen at high pressures. For this sample a small piece of elemental sulfur (purity of 99.98%) with a lateral dimension of about 20 μm and a thickness of~3 μm-5 μm was placed in a DAC having diamonds with culets of~60 μm-70 μm. Excess hydrogen was introduced in the DAC at a gas pressure of~130 MPa-150 MPa. Subsequently, the DAC was pressurized up to 150 GPa and then heated up to 1000 K with a YAG laser at room temperature to initiate the chemical reaction. The pressure increased slightly up to 155 GPa after the synthesis. The vibrational properties of the pressurized initial reactants, and of the synthesized products, were probed using a triple grating Raman spectrometer equipped with a HeNe laser having a wavelength of 633 nm. Sputtered gold electrodes were thoroughly isolated from the metal gasket by a layer made from magnesium oxide, calcium fluorite, and an epoxy glue mixture. This layer also prevented hydrogen penetration into the rhenium gasket. The pressure was estimated using the Raman shift of stressed diamond 39 and the H 2 vibron's wavenumber, previously calibrated in a separate experiment 42 . The uncertainty in these pressure values is estimated to be ±10%. Both scales indicated a pressure of 155 GPa after the laser heating. Both DACs with H 3 S samples remain under pressure after initial synthesis throughout the duration of the study. The sample under 155 GPa experienced slight decrease in resistance and increase in T c between July and September 2018 (Fig. 1), but no further change was noted.
Magnetotransport measurements. The magnetoresistance and the Hall signal was recorded using a conventional 4-probe configuration. Under continuous fields up to μ 0 H = 35 T the resistance of the sample is measured using a commercial lowfrequency (~13 Hz AC resistance bridge). A high-frequency (345 kHz) lock-in amplifier technique is used under pulsed fields up μ 0 H = 65 T as follows. An AC current of 5 × 10 −4 (RMS) is applied to one pair of sample leads through an isolating radio-frequency transformer. The voltage drop across the second pair of leads was amplified using instrumentation amplifier and recorded at 30 MSPS rate with a high speed digitizer, which is synchronized with the AC current source. The corresponding in-phase and quadrature components of the AC signal are extracted from the recorded waveform by a software lock-in.
Theoretical models for the upper critical fields. First, the H c2 (T) were fit to a conventional Ginzburg-Landau (GL) expression: where ϕ 0 is the quantum of flux and ξð0Þ is an average Ginzburg-Landau coherence length at T = 0.
Subsequently to the WHH formalism where the temperature dependence of H c2 (T) defined by orbital and spin-paramagnetic effects in the dirty limit is given by 24 : where, h ¼ ð4=π 2 Þ½H c2 ðTÞ=ðÀdH c2 =dTÞ T c , α is the Maki parameter, and λ SO is the spin-orbit constant. The results of the fits are summarized in Table 1.