Classifying superconductivity in ThH-ThD superhydrides/superdeuterides

Satterthwaite and Toepke (1970 Phys. Rev. Lett. 25 741) discovered that Th4H15-Th4D15 superhydrides are superconducting but exhibit no isotope effect. As the isotope effect is a fundamental prediction of electron-phonon mediated superconductivity described by Bardeen, Cooper, and Schrieffer (BCS) its absence alludes to some other mechanism. Soon after this work, Stritzker and Buckel (1972 Zeitschrift f\"ur Physik A Hadrons and nuclei 257 1-8) reported that superconductors in the PdHx-PdDx system exhibit the reverse isotope effect. Yussouff et al (1995 Solid State Communications 94 549) extended this finding in PdHx-PdDx-PdTx systems. Renewed interest in hydrogen- and deuterium-rich superconductors is driven by the discovery of near-room-temperature superconductivity in highly-compressed H3S (Drozdov et al. 2015 Nature 525 73) and LaH10 (Somayazulu et al 2019 Phys. Rev. Lett. 122 027001) as well as two recently discovered high-pressure hydrogen-rich phases of ThH9 and ThH10 (Semenok et al 2019 Materials Today, DOI: 10.1016/j.mattod.2019.10.005). We conclude that all known thorium super-hydrides/deuterides, to date, are unconventional superconductors - along with the heavy fermions, fullerenes, pnictides, cuprates, where we find they have Tc/Tf ratios within a range of of 0.008

conclude that all known thorium super-hydrides/deuterides, to date, are unconventional superconductorsalong with the heavy fermions, fullerenes, pnictides, cuprateswhere we find they have Tc/TF ratios within a range of 0.008 < Tc/TF < 0.120, where Tc is the superconducting transition temperature and TF is the Fermi temperature.

I. Introduction
The discovery of near-room-temperature (NRT) superconductivity in highly-compressed H3S (Tc = 203 K) [1], and the following discovery of superconductivity in LaH10 (Tc = 250 K, P = 150 GPa) [2] (current status of the research in the field can be found elsewhere [3][4][5][6][7]), is widely held [8] as a success of the predictions of Ashcroft [9] and Ginzburg [10]. These predictions were based on electron-phonon interactions of Bardeen, Cooper, and Schrieffer's (BCS) theory of superconductivity [11]. This prediction, of NRT Tc hydrides under pressure, and its subsequent discovery in H3S and LaH10 were taken as affirmations that these systems were indeed conventional (BCS electron-phonon mediated) superconductors. However, as we have shown previously [12,13] the available data for these superhydrides can be successfully interpreted in the phenomenology of unconventional (non-BCS) superconductivitysuggesting that the mechanism is not BCS electron-phonon coupling. To further our analysis, and hopefully reiterate the need for new experimental data on the H3S-D3S system, we revisit the thorium-based hydrides Th4H15-Th4D15, ThH9, and ThH10, to see if a similar conclusion has been overlooked.
The isotope effect in Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity can be expressed in the form: where M is isotope mass, and ≈ 1 2 ⁄ (for weak-coupling limit of BCS theory [11]), is an indispensable feature of electron-phonon mediated superconductivity [1,11]. This effect was observed in several elemental superconductors, but not in all of them [14,15]. Geballe et al [16] were the first to find the absence of the isotope effect in ruthenium (more details can be found elsewhere [14,15]). Later, Satterthwaite and Toepke [17] reported the absence of the isotope effect in Th4H15-Th4D15 super-hydride/deuteride phases. Soon after [17], Stritzker and Buckel [18] experimentally found that the isotope effect in the palladium-hydrogendeuterium (PdHx-PdDx) system has the opposite sign (the reverse isotope effect). Yussouff et al [19] extended this discovery to the full palladium-hydrogen-deuterium-tritium (PdHx-PdDx-PdTx) system. This reverse isotope effect in the PdHx-PdDx-PdTx system is currently the subject of wide discussion [20,21]. As for the thorium hydrides/deuteride systems considered herein, detailed studies by Caton and Satterthwaite [22] reported a reverse isotope effect in Th4H15 Th4D15.
Discovery of NRT superconductivity in H3S-D3S [1] and LaH10 [2] has reinvigorated interest in the isotope effect in the superconducting hydrides/deuterides. It should be stressed that recent experimental results reported by Drozdov et al. [23] show that La-H and La-D NRT phases have different stoichiometry, i.e. LaH10 vs LaD11/LaD12, and, thus, more experimental and theoretical studies are demanded to reveal the effect of the isotope effect on Tc in LaH-LaD system, which should be separated from the effect of different chemical stoichiometry on Tc in these superhydrides/superdeuterides.
These studies will support/disprove our previous proposal that hydrogen-rich compounds (PdHx, H3S, LaH10) are unconventional superconductors [12,13] and, thus, the superconductivity in these compounds is not related to electron-phonon interaction. We should note that, so far, we have not included the following in our analysis or proposals: Here we repeat the analysis described in [6,7], by using the best-known models for upper critical field behaviour we can estimate the ground state coherence length, (0). With this, and the other superconducting parameters, we can calculate the Fermi velocity vF. Then with some knowledge of the effective mass, we can calculate TF and characterise these conductors in the same manner as Uemura et al [28][29][30].

II. The upper critical field models
The ground state upper critical field, Bc2(0), in the Ginzburg-Landau theory [31] is given by: where 0 = 2.068 • 10 −15 Wb is magnetic flux quantum, and (0) is the ground state coherence length. For real world experiments only a part of full Bc2(T) temperature dependence can be measured; although there are several models were proposed to deduce extrapolated values for (0) from raw Bc2(T) data measured at high reduced temperatures.
One such model, proposed by Werthamer, Helfand, and Hohenberg [32,33], is an extrapolative expression: which we designate as the WHH model.
Another model, which is based on the primary idea of the WHH model [32,33], but accurately extrapolates the full Bc2(T) curve from experimental data measured at high reduced temperatures, T/Tc, was proposed by Baumgartner et al [34]: we will designate this as the B-WHH model.

III. Th4H15-Th4D15 superconductors in Uemura plot
We start our consideration with the first discovered superhydride/superdeuteride superconductors i.e. Th4H15 and Th4D15 [17]. From the author's knowledge, experimental data available to date for the upper critical field, Bc2(T), for Th4H15 and Th4D15 is limited by values reported by Satterthwaite and Toepke [17]. Both Th4H15 and Th4D15 compounds have ground state upper critical fields of: From these values and Eq. (1), the ground state coherence length, (0), for Th4H15 and Th4D15 phases, must be: (0) = 11.0 ± 0.5 .
Miller et al [37] for both phases reported the BCS ratio within a range: By using the superconducting transition temperature for Th4H15 and Th4D15 phases [17]: one can deduce ground state superconducting energy gap: and by using well-known BCS expression [10]: where ℏ = h/2 is reduced Planck constant, one can calculate the Fermi velocity, vF, in Th4H15 and Th4D15 phases: To classify Th4H15 and Th4D15 in the Uemura plot [28][29][30] we need to make assumption about the effective charge carrier mass, * , to calculate the Fermi temperature, TF: which leads to the Fermi temperature: and upper bound to the Tc/Tf ratio: = 0.12 ± 0.01.
For an upper bound on * we use the highest value reported for a highly compressed hydrides, * = 3.0 • [39]. The corresponding lower bound for the Tc/TF value is then: = 0.020 ± 0.002.

IV. ThH9 (P = 170 GPa) in Uemura plot
Semenok et al [44] reported the discovery of a high-temperature superconducting phase of ThH9 at P = 170 GPa which exhibits P63/mmc crystallographic symmetry and superconducting transition temperature of Tc = 146 K. They also performed first principles calculations and deduced the effective mass in this superconductor: which is remarkably close to the effective mass of * = 2.76 • in compressed H3S [45].
Despite the orthodox view, several new, alternative, approaches were developed to explain NRT superconductivity in compressed hydrides: Hirsch and Marsiglio [53], Souza and Marsiglio [54], Harshman and Fiory [55], as well as Kaplan and Imry [56]. For instance, Kaplan and Imry [56] showed that in the case of highly compressed H3S their model gives an  within the weak-coupling BCS limit: This  value is in a good agreement with ones deduced from experimental Bc2(T) [12] and the self-field critical current density, Jc(sf,T), data [46,57]. Assuming all hydrogen-rich superconductors have the same primary mechanism for the superconductivity, the value of  = 3.53 was taken as the lower bound for our calculations.

V. ThH10 (P = 174 GPa) in Uemura plot
Semenok et al [44] also reported on the discovery of another high-temperature superconducting phase of ThH10 at P = 174 GPa, which exhibits 3 ̅ crystallographic symmetry and superconducting transition temperature of Tc = 159 K. In Fig. 4 we show raw upper critical field, Bc2(T), data for this phase [44] and data fit to Eqs. 2-5.  As expected, highly-compressed ThH10 superconductor is located within unconventional superconductors band of the Uemura plot, see Fig. 2.

VI. Conclusions
Recent interest in the near-room-temperature superconductivity has revived interest in the hydride superconductors. While the latest generation of hydride superconductors, H3S-D3S and LaH10-LaD11/LaD12, are widely considered to be conventional BCS conductors, we point out that this is not supported in other hydrides such as the Th4H15-Th4D15, and PdHx-PdDx-PdTx. Critically, these previously discovered hydride systems exhibit the reverse isotope effect, which cannot be explained in BCS theory. In addition, we stress that the isotope effect in LaH-LaD system should be further studied, as available experimental data show that at high-pressure conditions La-H and La-D NRT superconducting phases have different stoichiometry [23].
To further this analysis, we have classified (conventional vs unconventional) the superconductivity in the thorium hydrides. We analyse experimental Bc2(T) data for several thorium based superhydrides and Th4D15 superdeuteride. This analysis was completed for thorium hydrides where fundamental superconducting parameters beyond Tc were available i.e., Th4H15, Th4D15, ThH9 and ThH10. For all these materialsall thorium hydrides where analysis is possiblewe find that they fall into the band of unconventional superconductors, as seen in an Uemura plot. This along with similar analysis of other hydrides, previously done, further necessitates understanding the hydrides outside of conventional BCS theory.