High-T_C Superconductivity in Hydrogen Clathrates Mediated by Coulomb Interactions Between Hydrogen and Central-Atom Electrons

Abstract

The uniquely characteristic macrostructures of binary hydrogen-clathrate compounds MH_n formed at high pressure, a cage of hydrogens surrounding a central-atom host, is theoretically predicted in various studies to include structurally stable phonon-mediated superconductors. High superconductive transition temperatures T_C have thus far been measured for syntheses with M = La, Y, and Th. In compressed LaH₁₀, independent studies report T_C of 250 K and over 260 K, a maximum in T_C with pressure P, and normal-state resistance scaling with temperature (suggesting unconventional pairing). According to reported band structure calculations of Fm$\overline{3}$m-phase LaH₁₀, the La is anionic, with the chemical valence electrons appearing evenly split between La and H₁₀. Thus, compressed LaH₁₀ contains the combination of structure, charge separation, and optimal balanced allocation of valence electrons for supporting unconventional high-T_C superconductivity mediated by Coulomb interactions between electronic charges associated with La and H₁₀. A general expression for the optimal superconducting transition temperature for MH_n clathrates is derived as T_C0 = k_B⁻¹Λ[(n + v)/2A]^1/2e²/ζ, where Λ is a universal constant, (n + v) is the chemical valence sum per formula unit, taking unity for H and v for atom M, A is the surface area of the H-polyhedron cage, and ζ is the mean distance between the M site and the centroids of the polyhedron faces. Applied to Fm$\overline{3}$m LaH₁₀, T_C0 values of  249.8(1.3) K and 260.7(2.0) K are found for the two experiments. Associated attributes of charge allocation, structure, effective Coulomb potential, and H-D isotope effect in T_C of Fm$\overline{3}$m LaH₁₀ and Im$\overline{3}$m H₃S are discussed, along with a generalized prospective for Coulomb-mediated superconductivity in MH_n.

Appendix

The following appendix sections present and analyze results for the pressure variation in T_C, normal-state resistance, pressure dependence of the lattice parameter, and accuracy and systematic error of the model, based on this work and published data.

1.1 A1 Pressure Variation in T_C

A comprehensive plot of available experimental data [12, 13, 29] and theoretical results [4,5,6, 21,22,23,24,25,26] pertaining to the variation of T_C with pressure P in LaH₁₀ is presented in Fig. 3. Open circle symbols shown for P ∈ (140, 218) GPa and T_C ∈ (243, 251) K are transcriptions of experimental data from Fig. 1 inset of [12] (error bars omitted for clarity). The arched trend in T_C vs. P, noted in [12], comprises points with highest T_C at 251.5 and 251.3 K (error bars ± 1 K) corresponding to P at 166 and 172 GPa (± 2 GPa), respectively (data for sample 5 in [12]). Taking rounded averages and uncertainties, the values 251(1) K and 169(4) GPa thus determine the experimentally measured optimal T_C and P, respectively, shown in the first row of Table 1.

Fig. 3

Plot of transition temperature T_C vs. pressure P for Fm$\overline{3}$m LaH₁₀ (marked with source references). Larger green symbols denote experimental data from [12, 13, 29]; smaller blue symbols and connecting lines represent theoretical results from [4,5,6, 21,22,23,24,25,26]. Magenta star symbols are at T_C0 and P as given in Table 1

Although experimental measurements of T_C are not specified in [13], full superconducting resistance transition curves for cooling and warming of sample A are presented in Fig. 3 therein. Onset values of T_C are determined herein to about ± 1 K uncertainty from the intersection of tangents drawn through the linear-in-T normal-state region (see Section A2) and the more steeply sloped region just below the transition onset. This method yields T_C of 262(1) and 247(1) K for cooling and warming, respectively, and is shown as the filled square symbols in Fig. 3, where horizontal error bars span the pressures of 188 and 196 GPa associated with the data (i.e., P = 192 ± 4 GPa). The higher T_C for cooling is assumed to be closer to optimal and therefore entered in the second row of Table 1. Coincidentally, Fig. 3 shows by way of comparison that the onset T_C for warming lies in the region of the lower T_C observed in the data from [12] (sample 6 therein). Other resistance-vs.-temperature data in [13] show resistance drops suggesting onsets of superconducting behavior (full transitions not displayed). Figure 4 in [13] shows a resistance drop at 280 K for 0.1 mA applied current and 195 GPa pressure for sample F. Figure S4 in [13] shows drops at 257, 282, and 276 K for pressures of 190, 195, and 202 GPa, respectively, for sample B. Additionally, a small jump at 240 K (at assumed 179 GPa) is described for sample G [13]. The five open square symbols in Fig. 3 represent these other temperatures and pressures drawn from [13]. Cross symbols represent data from Fig. 10 of [29].

Fig. 4

Distributions of experimental and theoretical results for T_C of Fm$\overline{3}$m LaH₁₀, as presented in Fig. 3 and Table 1. Normalized histograms show distributions in T_C from experiments (solid green) and phonon theories (dashed blue). Vertical pink bars denote T_C0 presented in Table 1 (bar width denotes uncertainty)

Small symbols in Fig. 3 display results of theoretical calculations of T_C for LaH₁₀ from strong electron-phonon coupling based on Migdal-Éliashberg theory, density functional theory, or ab initio methods (labels distinguish the source references [4,5,6, 21,22,23,24,25,26], exclusive of ranges in theoretical stability study of [81]). Results calculated at multiple values of P, connected by broken lines, show that the electron-phonon mechanism reproducibly predicts that T_C has a monotonically decreasing dependence on P with average slope ΔT_C/ΔP ≈ − 0.30 ± 0.08 K/GPa [5, 23,24,25,26], a finding with theoretical basis [26]. The dotted line in Fig. 3 illustrates the linear extrapolation from the calculated point (diamond symbol) to the range of experimental pressure that is used in [22]. Superconducting density functional theory (SCDFT), as applied in [23], predicts T_C = 271 K at P = 200 GPa, whereas SCDFT applied in [25] predicts a much lower T_C = 214 K (interpolated to P = 200 GPa).

The magenta star symbols in Fig. 3 are the results for T_C0 given in Table 1 at the values of P indicated. These two points define a positive change ΔT_C/ΔP = + 0.47 K/GPa, in contrast to negative variations predicted in electron-phonon theory [5, 22, 24,25,26].

The distributions of the T_C values from Fig. 3 are cast in Fig. 4 in the form of histograms corresponding to the experiments (solid green) and phonon theories (dashed blue) normalized to the number of points (30 and 27, respectively), and with bin sizes of 5 K and 10 K, respectively. Statistical averages and ± standard deviations of the distributions are as follows: T_C = 251 ± 10 K (experiments) and 245 ± 26 K (phonon theories); P = 174 ± 27 GPa (experiments) and 232 ± 52 GPa (phonon theories). Theoretical pressure dependence suggests shifting the statistical average for the phonon theories as T_C = 263 ± 29 K at P → 174 GPa. Pre-experimental theories include predictions of 274 and 286 K at 210 GPa in [5] and 288 K at 200 GPa in [6], which may be compared with resistance drops [13] or magnetic onsets [29] at temperatures of 276–283 K and pressures of 185–202 GPa, and represented by the smaller histogram fragment in Fig. 4. The two pink vertical bars (and widths) in Fig. 4 represent the values of T_C0 (and uncertainties) given in Table 1.

Figure 5 is a comprehensive plot of T_C vs. P for YH₆ (circle symbols) and YH₉ (square symbols) where the larger green circles and squares denote experimental data in [14, 15]. The smaller blue circles and filled squares denote theoretical results for YH₆ [6,7,8,9, 14] and YH₉ [6], respectively. Asterisk symbols in the legend express the caveat that phases other than those indicated are determined from X-ray data analysis (e.g., YH₄) [14]. Statistical averages and ± standard deviations of samples identified as YH₆ in [14, 15] are T_C = 219 ± 5 K and P = 188 ± 28 GPa; statistics for the YH₆ phonon theories are T_C = 245 ± 47 K and P = 176 ± 75 GPa. The “dome” trend in T_C vs. P for YH₉ with maximum T_C = 243 K at P = 201 GPa is noted in [14] (filled symbols, before change in P; open symbols, after change in P); phonon theory predicts T_C = 253–287 at P = 150 GPa [6].

Fig. 5

Plot of transition temperature T_C vs. pressure P for Im$\overline{3}$m YH₆ and P6₃/mmc YH₉ (marked with source references), denoted by circular and square symbols, respectively. Larger green symbols denote experimental data [14, 15]; smaller blue symbols and connecting lines represent phonon theories [6,7,8,9, 15]

1.2 A2 Normal-State Resistance

Unique features signifying unconventional high-T_C superconductivity are a linear temperature dependence in the normal-state resistivity, ρ_N(T) ∝ T, and the absence of saturation at high temperature, such that the negative curvature of ρ_N(T) for conventional strong electron-phonon coupled metals is absent [43]. Such unconventional behavior in ρ_N(T), observed in optimally and overdoped high-T_C cuprates, is ascribed generally to “Planckian” dissipation phenomena [44, 45]. Superconducting LaH₁₀ and YH₉ are exemplary of this behavior for T > T_C, as can be seen in Fig. 6 showing resistance vs. temperature for (a) Fm$\overline{3}$m LaH₁₀ sample 3 from Fig. 1 of [12] (blue curve, P = 150 GPa, T_C ~ 249 K) and (b) P6₃/mmc YH₉ sample 2 from Fig. 1a of [14] (red curve, P = 201 GPa, T_C = 243 K). Dotted lines illustrate linear dependence for T > T_C extrapolating to the origin. Resistance of LaH₁₀ sample A in [13] (Fig. 3 therein, cooling curve) and Im$\overline{3}$m YH₆ sample M1 in [15] are also consistent with normal-state linearity in T. Table 3 summarizes normalized normal-state slopes obtained from the temperature dependence of the resistance R_N(T) reported for various samples of La, Y, and Th superhydrides and H₃S, defined as S = [T_C/R_N(T_C)] (ΔR_N/ΔT) in terms of slope ΔR_N/ΔT taken asymptotically above T_C and R_N(T_C) as extrapolated to T_C. Resistance in applied magnetic fields, which depress the superconducting transition, shows linear-in-T extending to T < T_C; S = 0.91 at 9 Tesla is obtained for sample 3 from Fig. 2(a) in [12], and extension to above 60 Tesla is reported in [30]. Extrapolated residual resistance ratio is given by R_N(0)/R_N(T_C) = 1 – S. The statistical distribution in S with bin size 0.25 is shown in Fig. 7 (from Table 3 and near unity S assumed from [30]), indicating that highest probability occurs at S ≈ 1. Ideally, S approaches unity for (unconventional) optimal high-T_C superconductors.

Fig. 6

Resistance vs. temperature for (a) Fm$\overline{3}$m LaH₁₀ sample 3 from Fig. 1 of [12] (blue curve) and (b) P6₃/mmc YH₉ sample 2 from Fig. 1(a) of [14] (red curve). Dotted lines show extrapolations of normal-state resistance to the origin

Fig. 7

Statistical distribution of data for normalized normal-state resistance temperature slope parameter S from Table 3

Dissipation from inelastic electron scattering in the normal state is determined from the scattering rate formula τ_N⁻¹ = ρ_Ne²n/m, written in terms of density n and mass m of the carriers. An estimated ρ_N ~ 10⁻⁵ Ω cm at T ≈ 285 K is calculated for sample F in [13] from measured resistance ≈ 0.11 mΩ at 0.1 mA indicated in Fig. 4, thickness ~ 2 μm mentioned in the supplemental material of [13], and an assumed Van der Pauw factor of 4.53. Carrier density n is obtained from the charge fraction σ = 6.5 associated with H₁₀ (6.5e per formula unit volume); m is derived from theoretical superconducting parameters as m = ħk_F/v_F = ħ²k_F/πΔξ with theoretical gap Δ = 2.3 k_BT_C [22] and experimental coherence distance ξ = 1.7 nm (average determined in [12]); Fermi wave vector is determined as k_F = (3π²n/g)^1/3 with g = 2 (approximating the theoretical Fermi surface in [22] as two equal-area spheres). The Planckian dissipation energy evaluates to ħτ_N⁻¹ ~ 4k_BT from the above data for LaH₁₀. Acknowledging that this analysis provides only a single order of magnitude estimation, the dissipation may be considered as similar to (1.3–2.3)k_BT determined for optimal cuprate superconductors [46,47,48]. While conventional theory of strong electron-phonon scattering might mimic linearity for T between T_C and 300 K, if designed with judicious choices of residual and saturation resistivities, it appears rather implausible in view of the preponderance of S ≈ 1 listed in Table 3 and displayed in Fig. 7.

1.3 A3 Pressure Dependence of the Lattice Parameter

X-ray diffraction is employed to determine volumes of LaH₁₀ referenced to the fcc cubic cell for samples 2 and 3 in [12] and referenced per La for superconducting samples A–C and E–G in [13]. Samples 2 and 3 are reported to contain minority phase material [12]. Samples F and G are described as mixed phase [13]. Pressure dependence in the fcc lattice parameter a are plotted in Fig. 8 as filled circles (data from [12]) and filled squares ([13]). Square symbols inscribed with a white cross correspond to samples C and E for which no information on transition temperature is provided in [13]. The straight line is a linear fit (exclusive of the two white-crossed squares) to the relation a = a₀ − sP with intercept a₀ = 5.78 ± 0.15 Å, slope s = 0.00459 ± 0.000091 Å/GPa, and root-mean-square deviation of 0.043 Å between the data and the line. The fitted function is used to extract values of a, and therefrom d, A, ζ, ℓ, and T_C0, at experimental optimal pressures (Table 1).

Fig. 8

Lattice parameter (face centered cubic) vs. pressure for Fm$\overline{3}$m LaH₁₀ from X-ray data in [12] (circles) and [13] (squares). The solid line shows the linear dependence obtained by fitting the data exclusive of the two white-crossed squares (see text)

1.4 A4 Accuracy and Systematic Error of the Model

The accuracy of Eq. (1) rests on that of the constant Λ, which was originally obtained by fitting Λ in the right-hand side expression to experimental data for T_C0 taken for the left-hand side. The fitted data included 31 phase-pure, optimal high-T_C compounds, with superconducting transitions in the range 10.5 K ≤ T_C ≤ 145 K [40]. Subsequently, 24 other compounds (see [41, 49,50,51,52,53,54,55]), optimized according to the developed criteria and inclusive of the two LaH₁₀ samples, have been added to the list of optimal superconductors with T_C values demonstrated to be in congruence with Eq. (1), bringing the total number to 55 from eleven disparate superconducting families (with T_C ranging from ~ 2 to 260 K). Comparisons of T_C0 calculated for LaH₁₀ at 169(4) GPa [12] and at 192(4) GPa [13] with the measured average T_C values (from Table 1) yield differential accuracies of 0.48% and 0.50%, respectively. The optimal transition temperatures are predicted with an overall statistical accuracy of ± 4.3% over a range in T_C from ~ 2 to 260 K, with an average fractional deviation of 0.7%. These results are graphically represented in Fig. 9, which plots the measured optimal transition temperature T_C vs. (ℓζ)⁻¹, and compared with the theoretical expression for T_C0 given in Eq. (1) represented by the solid line. Deviations of T_C from the theoretical line have a root-mean-square average of 1.3 K. The statistical distribution of fractional deviation between calculated T_C0 and measured T_C is given in the inset of Fig. 9, where the dashed curve shows the fitted Gaussian form.

Fig. 9

Optimal measured transition temperatures T_C vs. (ℓζ)⁻¹ for 55 superconductors (grouped as per legend) are shown in comparison with calculated T_C0 (solid line). Inset: statistical distribution of fractional deviation between calculated T_C0 and measured T_C; dashed curve shows fitted Gaussian form

Nearly abrupt superconducting transitions are also predicted theoretically for the hydrides LaH₁₀ and H₃S, by virtue of their being three-dimensional bulk crystals. One might assume that improved synthesis of the hydride samples would lead to sharper transitions, reduced uncertainty in determining T_C and, in turn, the value of Λ.