Since pivotal experimental discovery of the first near-room-temperature superconductor (NRTS) H₃S by Drozdov et al.,1 nearly two dozen highly compressed hydrogen-rich superconducting phases have been synthesized in binary and ternary systems.2–17 Experimental studies of NRTS are well supported by first-principles calculations,18–30 but experimental characterizations of NRTS phases are limited by the narrow range of techniques that are available for materials inside diamond anvil cells (DACs).25–27 These comprise x-ray diffraction (XRD) phase analysis, Raman spectroscopy, and magnetoresistance measurements,31–35 although, with the use of some advanced techniques, Hall effect measurements can also be performed.31 Because of this, only two characteristic values of the superconducting state of the NRTS phases are commonly extracted from experimental data, namely, the transition temperature T_c and the extrapolated value for the ground-state upper critical field $B_{c2}\left(0\right)$ or the ground-state superconducting coherence length $\xi \left(0\right)$, which can be derived from the Ginzburg–Landau36 expression$$\xi \left(0\right)=\sqrt{\frac{{\varphi }_0}{2\pi B_{c2}\left(0\right)}},$$(1)where ϕ₀ = h/2e is the superconducting flux quantum, with h being Planck’s constant and e the electric charge of the electron.

Other important parameters of NRTS materials, among which we can mention the Fermi velocity v_F, have not been measured to date, owing to the challenges of performing such measurements on samples inside DACs. However, considering that all NRTS superconductors are hydrides, there is an expectation that these materials will exhibit a universal Fermi velocity v_F,univ, as has been discovered in cuprates, for which $v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\left(2.7\pm 0.5\right)\times 10^5$ m/s, as reported by Zhou et al.37 (see Fig. 1).

The theoretical motivation for the quest for a universal Fermi velocity in NRTS comes, on the one hand, from the recent understanding38 that sulfur in H₃S is analogous to the oxygen in cuprates, and, on the other hand, from the fact that highly compressed hydrides fit nicely with the main global scaling laws for superconductors.39–43 However, it should be noted that an analysis of whether these materials also comply with other scaling laws for superconductors44,45 requires more experimental data on normal-state resistivity $\rho \left(T\right)$31,46–48 and ground-state London penetration depth $\lambda \left(0\right)$.1,49,50

Here, we report the results of our search for a universal Fermi velocity in NRTS materials, based on an analysis of the full inventory of values for the ground-state upper critical field $B_{c2}\left(0\right)$ in these materials. We find that a universal Fermi velocity, v_F,univ, does indeed exist in NRTS materials and obeys the empirical law$$v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\frac{1}{1.3}\times \frac{2\mathrm{\Delta }\left(0\right)}{k_B{T}_c}\times 10^5\mathrm{m}/\mathrm{s},$$(2)where k_B is Boltzmann’s constant and $\mathrm{\Delta }\left(0\right)$ is the ground-state superconducting energy gap.

In the Bardeen–Cooper–Schieffer (BCS) theory of superconductivity,51 the ground-state coherence length $\xi \left(0\right)$ and the amplitude of the ground-state energy gap $\mathrm{\Delta }\left(0\right)$ are linked through the expression$$\xi \left(0\right)=\frac{\hslash v_F}{\pi \mathrm{\Delta }\left(0\right)},$$(3)where ℏ is the reduced Planck’s constant. BCS theory also involves a dimensionless ratio$$\alpha =\frac{2\mathrm{\Delta }\left(0\right)}{k_B{T}_c}.$$(4)

Substitution of Eqs. (3) and (4) into Eq. (1) gives the dependence of the ground-state upper critical field on the transition temperature:$$B_{c2}\left(0\right)=\left[\frac{\pi {\varphi }_0{k}_B^2}{8{\hslash }^2}\right]\frac{{\alpha }^2}{v_F^2}{T}_c^2,$$(5)where the multiplicative prefactor in square brackets is a constant:$$A=\left[\frac{\pi {\varphi }_0{k}_B^2}{8{\hslash }^2}\right]=1.38\times 10^7\mathrm{T}{\mathrm{m}}^2/({\mathrm{s}}^2{\mathrm{K}}^2).$$(6)

Thus, if hydrogen-rich superconductors exhibit a universal Fermi velocity v_F,univ, then a fit of the full inventory of the $B_{c2}\left(0\right)$ vs T_c dataset to the equation$$B_{c2}\left(0\right)=Af{T}_c^{\beta },$$(7)where β and $f={\alpha }^2/v_F^2$ are free fitting parameters, should reveal that$$\beta \cong 2,$$(8)and, if this is the case, then the universal Fermi velocity v_F,univ can be calculated from the deduced free-fitting parameter f as$$v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\frac{\alpha }{\sqrt{f}}=\frac{1}{\sqrt{f}}\frac{2\mathrm{\Delta }\left(0\right)}{k_B{T}_c}.$$(9)

It should be noted that $\alpha =2\mathrm{\Delta }\left(0\right)/k_B{T}_c$ in highly compressed hydrogen-rich superconductors varies within the range4,8,12,27,42,49,52–55$$3.2\le \frac{2\mathrm{\Delta }\left(0\right)}{k_B{T}_c}\le 5,$$(10)where the lower limit is the value deduced from experiment42,50,52 and the upper limit is based on the many results obtained from first-principles calculations, which always predict $4.3\le 2\mathrm{\Delta }\left(0\right)/k_B{T}_c$ in NRTS materials,4,8,12,27,53–55 including very high values of $2\mathrm{\Delta }\left(0\right)/k_B{T}_c\cong 5.0$ for some NRTS phases.4,8,12,27

Equation (7) has the ground-state upper critical field $B_{c2}\left(0\right)$ as dependent variable. However, it is important to note that this value can be determined by the use of several extrapolative models56–59 that utilize experimental $B_{c2}\left(T\right)$ data measured at high reduced temperatures T/T_c. The primary reason why there is a necessity for extrapolative models is that all highly compressed hydrogen-rich superconductors have $B_{c2}\left(T\to 0\mathrm{K}\right)>20\mathrm{T}$, which cannot be measured by the conventional and widely used Physical Property Measurement System (manufactured by Quantum Design), for which the highest measurable magnetic field B_appl = 9–16 T (depending on the specific model). It should be also stressed that $B_{c2}\left(T\to 0\mathrm{K}\right)$ for the NRTS compounds H₃S, LaH₁₀, YH₆/YH₉ and (La,Y)H₁₀ are so high that even measurements at the best available quasi-DC magnetic field facilities worldwide31,48,60 cover only the range of reduced temperatures $\frac{1}{2}\lesssim T/T_c$.

In this paper, from the several extrapolative $B_{c2}\left(T\right)$ models that are available,56–59 we use the following analytical approximate expression from Werthamer–Helfand–Hohenberg (WHH) theory,61,62 which was proposed by Baumgartner et al.:59$$\begin{array}{ll}\hfill B_{c2}\left(T\right)& =\frac{1}{0.693}\frac{{\varphi }_0}{2\pi {\xi }^2\left(0\right)}\left[\left(1-\frac{T}{T_c}\right)-0.153{\left(1-\frac{T}{T_c}\right)}^2\right.\hfill \\ \hfill & -\left.0.152{\left(1-\frac{T}{T_c}\right)}^4\right],\hfill \end{array}$$(11)where ξ(0) and T_c ≡ T_c(B = 0) are two free fitting parameters (this equation is referred to as the B-WHH model hereinafter). Equation (11) was originally proposed for the extrapolation of $B_{c2}\left(T\right)$ data for neutron-irradiated Nb₃Sn alloys,59 and recently several research groups have found that provides good approximations for a variety of superconducting materials.4,63–68 On this basis, in the present study, we used Eq. (11) as a good, robust and simple analytical tool to extrapolate the B_c2(T) curve to the low-temperature high-field region.

It is a necessary to describe the criterion for extracting B_c2(T) datasets from experimentally measured R(T, B_appl) curves. Several criteria are available for the definitions of T_c, B_c2(T), and T_c(B_appl), which have been discussed recently for the case of NRTS in Ref. 69. We have found69,70 that the best match between the electron–phonon coupling constant λ_e–ph extracted from R(T, B_appl = 0) curves and the λ_e–ph computed by first-principles calculation is obtained when T_c is defined at a value of the ratio R(T)/R_norm that is as low as practically possible (where R_norm is the normal-state resistance just above the transition). By analyzing the full inventory of R(T, B_appl) data for NRTS materials herein, we have come to the conclusion that owing to noise and slope issues with real-world R(T, B_appl) curves and the fact that highly compressed superhydrides contain several superconducting phases, the appropriate criterion is$$\frac{R\left(T,B_{\text{appl}}\right)}{R\left(T_c^{\text{onset}},B_{\text{appl}}\right)}=0.05,$$(12)and we use this henceforth in this study.

In the first paper on NRTS superconductors, Drozdov et al.1 reported R(T, B_appl) data for unannealed highly compressed sulfur hydride (P = 155 GPa) in their Fig. 3(a). By using the criterion of Eq. (12) [which is $R{\left(T,B_{\text{appl}}\right)}_{\text{criterion}}=23\mathrm{m}\mathrm{\Omega }$ for the R(T, B_appl) curves shown in bottom insert in Fig. 3(a) in Ref. 1], we extracted the B_c2(T) dataset for this sample, which is shown in Fig. 2. Because this B_c2(T) dataset covers a significant part of the full temperature range 0 K < T ≤ T_c, there was no need to use an extrapolative fit, and instead we fitted this dataset to the model in Ref. 52, which allowed to deduce $\mathrm{\Delta }\left(0\right)$, $2\mathrm{\Delta }\left(0\right)/k_B{T}_c$, and ΔC/γT_c (the last of which is the relative jump in electronic specific heat at T_c, with γ being the so-called Sommerfeld constant):$$\begin{array}{ll}\hfill B_{c2}\left(T\right)& =\frac{{\varphi }_0}{2\pi {\xi }^2\left(0\right)}{\left[\frac{1.77-0.43{\left(\frac{T}{T_c}\right)}^2+0.07{\left(\frac{T}{T_c}\right)}^4}{1.77}\right]}^2\hfill \\ \hfill & \times \left\{1-\frac{1}{2k_{B}T}\underset{0}{\overset{\infty }{\int }}\frac{d\epsilon }{\cos {\mathrm{h}}^2\left[\frac{\sqrt{{\epsilon }^2+{\mathrm{\Delta }}^2\left(T\right)}}{2k_{B}T}\right]}\right\},\hfill \end{array}$$(13)where the temperature-dependent superconducting gap Δ(T) is given by71,72$$\mathrm{\Delta }\left(T\right)=\mathrm{\Delta }\left(0\right)\tanh \left[\frac{\pi k_B{T}_c}{\mathrm{\Delta }\left(0\right)}\sqrt{\eta \frac{\mathrm{\Delta }C}{\gamma T_c}\left(\frac{T_c}{T}-1\right)}\right],$$(14)with η = 2/3 for s-wave superconductors.

We used Eqs. (13) and (14) to extract $\xi \left(0\right)$, $\mathrm{\Delta }\left(0\right)$, T_c, and ΔC/γT_c from B_c2(T) datasets for a variety of superconductors, including two highly compressed hydride phases of H₃S,52 SnH₁₂,42 V₃Si,73 and several iron-based superconductors.73 However, it should be stressed that the approach using Eqs. (13) and (14) is only applicable for B_c2(T) datasets defined by Eq. (12) or by a stricter criterion.

One of the most important deduced parameters, $\alpha =2\mathrm{\Delta }\left(0\right)/k_B{T}_c=3.2\pm 0.3$, is in remarkable agreement with the corresponding values deduced for highly compressed annealed H₃S (P = 155–160 GPa), α = 3.20 ± 0.0249 and 3.55 ± 0.31,52 and for highly compressed annealed SnH₁₂ (P = 190 GPa), α = 3.28 ± 0.18.42 The deduced ΔC/γT_c = 0.7 ± 0.1 is also below the weak-coupling limit of BCS theory ΔC/γT_c = 1.43, as is the corresponding value for the annealed H₃S material, ΔC/γT_c = 1.2 ± 0.3.52 It should be mentioned that to deduce ΔC/γT_c with higher accuracy requires more B_c2(T) data points, especially at T ∼ T_c. The deduced $B_{c2}\left(0\right)$ and T_c are given in Table I.

We processed reported R(T, B_appl) datasets for several annealed highly compressed hydrides by using Eq. (12) to extract B_c2(T) datasets. The obtained datasets were fitted to Eq. (11), and the deduced values are given in Table I. These materials are as follows:1.Sulfur superhydride H₃S (P = 155 and 160 GPa), for which the raw data were reported by Mozaffari et al.³¹2.Cerium superhydride CeH_n (P = 88, 137, and 139 GPa), for which the raw data were reported by Chen et al.¹²3.Lanthanum superhydride LaH₁₀ (P = 120, 136 GPa), for which the raw data were reported by Sun et al.⁶⁰4.Yttrium superhydride/superdeuteride YH₆/YD₆ (P = 172 and 200 GPa), for which the raw data were reported by Troyan et al.⁴5.Lanthanum–yttrium superhydride (La,Y)H₁₀ (P = 182, 183, and 186 GPa), for which the raw data were reported by Semenok et al.⁸6.Tin superhydride SnH₁₂ (P = 190 GPa), for which the raw data were reported by Hong et al.¹¹7.Thorium superhydrides ThH₉ and ThH₁₀ (P = 170 GPa), for which the raw data were reported by Semenok et al.¹⁶

The respective fits are shown in Figs. S1–S7 in the supplementary material.

All deduced $B_{c2}\left(0\right)$ and T_c values for superhydride phases are collected in Table I, where we have also added data for the Th₄H₁₅ phase reported by Satterthwaite and Toepke.74

The full dataset from Table I is shown in Fig. 3, together with the fit to Eq. (7). Although this dataset has a large scatter, it can be seen in Fig. 3(a) that the free-fitting power-law exponent β = 2.07 ± 0.14 is practically undistinguishable from the expected value β ≡ 2 [Eq. (5)]. When β is the free-fitting parameter [Fig. 3(a)], the deduced $f=\left(1.19\pm 0.90\right)\times 10^{-10}$ s²/m² has a large uncertainty. However, when β is fixed to 2 [Fig. 3(b)], the free-fitting parameter f can be deduced with high accuracy as$$f=\frac{{\alpha }^2}{v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}^2}=\left(1.68\pm 0.08\right)\times 10^{-10}{\mathrm{s}}^2/{\mathrm{m}}^2,$$(15)from which we can obtain$$v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\frac{\alpha }{1.30\pm 0.03}\times 10^5\mathrm{m}/\mathrm{s}\cong \frac{1}{1.3}\times \frac{2\mathrm{\Delta }\left(0\right)}{k_B{T}_c}\times 10^5\mathrm{m}/\mathrm{s}.$$(16)

By substituting the lower $[2\mathrm{\Delta }\left(0\right)/k_B{T}_c=3.2$] and upper [$2\mathrm{\Delta }\left(0\right)/k_B{T}_c=5.0$] limits of the ratio $2\mathrm{\Delta }\left(0\right)/k_B{T}_c$ [Eq. (10)] into Eq. (16), we can establish the lower and upper limits of v_F,univ in superhydrides:$$2.5\times 10^5\mathrm{m}/\mathrm{s}\lesssim v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}\lesssim 3.8\times 10^5\mathrm{m}/\mathrm{s}.$$(17)

The deduced v_F,univ for hydrogen-rich superconductors [Eq. (16)] is similar to the corresponding value for high-T_c cuprates, $v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\left(2.7\pm 0.5\right)\times 10^5$ m/s,37 if Eq. (10) is taken into account.

The primary assumption of the Migdal–Eliashberg (ME) theory of electron–phonon-mediated superconductivity75–77 is that the ratio of characteristic phonon energy ℏω_D (where ω_D is the Debye frequency) to the Fermi energy E_F is very small, ℏω_D/E_F ≪ 1. In normal metals ℏω_D/E_F ≲ 10⁻²,78–82 and this is why ME theory is quantitatively accurate. However, for many high-temperature superconductors, the application of ME theory cannot be justified. In fact, in our previous studies,69,70 we deduced the Debye temperature T_θ ≅ 1500 K, in highly compressed H₃S from a fit of experimentally measured temperature-dependent resistance R(T) to the Bloch–Grüneisen equation.83,84 This temperature can be converted into the Debye energy k_BT_θ = ℏω_D ≅ 0.13 eV, and, considering that the Fermi energy was deduced in our previous study52 as E_F = 0.5–1.0 eV, we can conclude that 0.13 ≤ ℏω_D/E_F ≤ 0.26, and thus ME theory75,76 does not provide an exact description of highly compressed H₃S.

The first concern that nonadiabatic effects (i.e., effects beyond ME theory75,76) are important in highly compressed H₃S was expressed by Pietronero et al.,82 who also pointed out that: “… The fingerprints of non-adiabatic effects are: – position of the material in the Uemura diagram;…” Although the traditional way to position a material in the Uemura plot requires knowledge of the ground-state London penetration depth $\lambda \left(0\right)$ (which has only recently been reported for H₃S46), the present author utilized the ground-state coherence length $\xi \left(0\right)$ [deduced from the B_c2(T) data] and found52 that H₃S falls into the unconventional superconductors band in the Uemura plot.41 In later work,42,85–88 it was established that LaH₁₀, Th₄H₁₅, ThH₉, ThH₁₀, YH₆, SH₁₂, and H₃(S,C) also fall into the unconventional superconductors band in the Uemura plot. This is direct evidence that the ratio ℏω_D/E_F in superhydrides has values well above those typical of conventional superconductors, ℏω_D/E_F ≲ 10⁻².

On the basis of the derived universal Fermi velocity in superhydrides, v_F,univ [Eq. (16)], we can conclude that the strength of the nonadiabatic effects in a superhydride (as quantified by the ratio ℏω_D/E_F) can be revealed if the Debye temperature T_θ of the compound can be deduced from the normal part of the temperature-dependent resistance R(T).69,70,86 It should be noted that the Debye temperature in superhydrides varies from T_θ ≅ 870 K (D₃S, P = 173 GPa69) up to T_θ ≅ 1700 K (R3m-phase of H₃S, P = 133 GPa69).

An analysis of the first experimental R(T) data89 measured for metallic hydrogen phase III (compressed at P = 402 GPa) revealed that T_θ ≅ 730 K.70 If we assume that metallic hydrogen phase III complies with the established v_F,univ [Eq. (16)] and that it has $2\mathrm{\Delta }\left(0\right)/k_B{T}_c=3.53$ and exhibits no effective mass enhancement, then the ratio ℏω_D/E_F can be estimated as ℏω_D/E_F = 0.3. This implies that metallic hydrogen should exhibit pronounced nonadiabatic effects,78–82 which could prevent the emergence of a superconducting state in this metal at high temperature.

It should be also mentioned that first principles calculations (FPC) are an essential part of current NRTS phase searches.27 The accuracy and powerful capabilities of FPC became obvious after the pivotal prediction of the $\mathrm{I}\mathrm{m}\bar{3}m-{\mathrm{H}}_3\mathrm{S}$ phase,90,91 which was later discovered experimentally.1 Another achievement of the FPC approach was demonstrated recently when Li et al.92 and Ma et al.93 reported the discovery of a calcium superhydride phase with transition temperature T_c = 200–215 K (at a pressure P = 160–190 GPa), which was predicted by Wang et al.94 in 2012. However, it should also be mentioned that the superconductivity predicted by FPC in some binary systems (e.g., AlH₃95,96), has never been observed experimentally. This implies that further development of FPC techniques to take account of non-adiabatic effects is highly desirable.

Overall, remarkable progress has been achieved in this field from the first theoretical predictions of high-temperature superconductivity in metallic hydrogen97,98 and hydrogen-dominated alloys74,99 five decades ago to the remarkable experimental and FPC results1,27 reported recently. It should be stressed that all the primary discoveries in the field (e.g., the direct searches for and syntheses of the H₃S, LaH₁₀, and YH₆ phases) have come from a perfect conjunction of theory and experiment. An excellent example of this is the story of the discovery of near-room-temperature superconductivity in highly compressed sulfur hydride.1 In February 2014, Li et al.100 reported results of FPC calculations for highly compressed sulfur hydride. These calculations showed that at P = 160 GPa, the sulfur hydride retains the composition of H₂S and that this compound exhibits a superconducting transition temperature of T_c ∼ 80 K. In November 2014, an alternative theoretical result was reported by Duan et al.,91 who performed thorough FPC using USPEX software101–103 and predicted that sulfur hydride would decompose into a mixture of elemental sulfur and an (H₂S)₂H₂ phase at high pressure. This result was in a good accord with an earlier report by Strobel et al.,104 who showed experimentally that at P > 3.2 GPa, the H₂S–H₂ mixture exhibited structural ordering with the formation of the (H₂S)₂H₂ phase (with four formula units per unit cell). The predicted transition temperature for the (H₂S)₂H₂ phase was T_c = 191–204 K at P = 200 GPa.91 On December 1, 2014, Drozdov et al.105 reported a milestone experimental result on the observation of T_c ≈ 190 K in sulfur hydride compressed at P > 150 GPa.

Another remarkable story should also be mentioned here, namely, the discovery of the $Fm\bar{3}m-\mathrm{L}\mathrm{a}{\mathrm{H}}_{10}$ phase, for which T_c ≅ 274–286 K at P = 210 GPa was theoretically predicted by Liu et al.106 in June 2017. This NRTS phase was experimentally discovered by Drozdov et al.107 on August 21, 2018, and, two days later, Somayazulu et al.108 confirmed this discovery.

In this study, we have proposed that hydrogen-rich superconductors exhibit a universal Fermi velocity v_F, which is given by empirical expression $v_{F,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}=\left(1/1.3\right)\times \left[2\mathrm{\Delta }\left(0\right)/k_B{T}_c\right]\times 10^5$ m/s. Considering that the gap-to-transition temperature ratio $2\mathrm{\Delta }\left(0\right)/k_B{T}_c$ in hydrogen-rich superconductors varies within the range $3.2\le 2\mathrm{\Delta }\left(0\right)/k_B{T}_c\le 5.0$, we conclude that v_F,univ varies within the range 2.5 × 10⁵ m/s ≤ v_F,univ ≤ 3.8 × 10⁵ m/s.

The Debye temperature T_θ can be deduced from the temperature-dependent resistance R(T) of the conductor,69,83,84 and so this universal Fermi velocity v_F in superhydrides [Eq. (16)] can be used to calculate the ratio ℏω_D/E_F, which determines the strength of nonadiabatic effects in the superconductor. Calculations for metallic hydrogen phase III (compressed at P = 402 GPa) have shown that ℏω_D/E_F = 0.3, which implies strong nonadiabatic effects in this metal.

Acknowledgment. The author thanks Luciano Pietronero (Universita’ di Roma) for comments about the limitations of the applicability of the Migdal–Eliashberg (ME) theory of electron–phonon-mediated superconductivity to high-temperature superconductors.

The author is grateful for financial support provided by the Ministry of Science and Higher Education of the Russian Federation through the theme “Pressure” Grant No. AAAA-A18-118020190104-3 and through a Ural Federal University project within the Priority-2030 Program.