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The Electron-Phonon Coupling Constant and the Debye Temperature In The electron-phonon coupling constant and the Debye temperature in superconducting polyhydrides of thorium Evgeny F. Talantsev1,2 1M.N. Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, 18, S. Kovalevskoy St., Ekaterinburg, 620108, Russia 2NANOTECH Centre, Ural Federal University, 19 Mira St., Ekaterinburg, 620002, Russia Abstract Milestone experimental discovery of superconductivity with transition temperature above 200 K in highly-compressed sulphur hydride by Drozdov et al (Nature 525, 73 (2015)) sparked experimental and theoretical investigations in metallic hydrides. Since then, a dozen of binary and ternary polyhydrides of metallic and non-metallic elements with superconducting transition temperature above 100 K have been discovered. One of these elements, thorium, forms three discovered to date superconducting polyhydride phases: Th4H15, ThH9, and ThH10. By following a widely accepted assumption that the electron-phonon pairing is the mechanism for the emergence of superconductivity in polyhydrides, here we analysed the temperature dependent resistance, R(T), of I-43d-Th4H15 and of P63/mmc-ThH9 phases and deduced the electron-phonon coupling constant, e-ph, and Debye temperature, T, in these superhydrides. In the result, we found that I-43d-Th4H15 phase exhibits e-ph = 0.82-0.99 which is in a very good agreement with experimental value of e-ph = 0.84 deduced from heat capacities measurements (Miller et al, Phys. Rev. B 14, 2795 (1976)). However, both values are twice higher than e-ph = 0.38 reported by the first principles calculations (Shein et al, Physica B 389, 296 (2007)). For P63/mmc-ThH9 phase subjected to pressure of P = 170 GPa we deduced e-ph(170 GPa) = 1.39 ± 0.07, which is in a reasonable agreement with e-ph(150 GPa) = 1.73 reported by Semenok et al (Materials Today 33, 36 (2020)), who computed this value by first principles calculations. 1 The electron-phonon coupling constant and the Debye temperature in polyhydrides of thorium I. Introduction There are three polyhydride superconducting phases of thorium, 퐼43푑-Th4H15 (푇푐 ≅ 8 퐾) [1], P63/mmc-ThH9 (푇푐 ≅ 145 퐾) [2], and 퐹푚3푚-ThH10 (푇푐 ≅ 160 퐾) [2]. The low- Tc compound, 퐼43푑-Th4H15, was discovered by Satterthwaite and Toepke [1] based on an idea [1]: “…There has been theoretical speculation [3] that metallic hydrogen might be a high-temperature superconductor, in part because of the very high Debye frequency of the proton lattice. With high concentrations of hydrogen in the metal hydrides one would expect lattice modes of high frequency and if there exists an attractive pairing interaction one might expect to find high-temperature superconductivity in these systems also.” The discovery of the superconductivity in 퐼43푑-Th4H15 phase (for which the stoichiometric composition can be written in form of ThH3.75) [1,4] was a part of a wider search of superconducting polyhydrides [5]. For instance, Satterthwaite and Peterson [4] reported on the absence of the superconductivity above T= 1.2 K in ThH2, (Th1/3Zr2/3)H3.5, VH2, NbH2, TaH. It should be noted that elemental thorium exhibits the superconducting transition temperature 푇푐 = 1.37 퐾 [5]. More than four decades later, in 2015, Drozdov et al [6] reported on the discovery of the near-room temperature superconductivity in polyhydride of sulphur, H3S, which is widely acknowledged [7,8] to be a confirmation for the electron-phonon pairing in hydrogen-rich materials. Further experimental and first-principles calculations studies leaded to the discovery of about a dozen of superconducting superhydrides with the superconducting transition temperature above 100 K [2,9-16]. 2 Here, based on general consensus [7,8] that the electron-phonon pairing is primary mechanism governs the superconductivity in polyhydrides, we deduced the electron-phonon coupling constant, e-ph, and the Debye temperature, T, for 퐼43푑-Th4H15 and P63/mmc-ThH9 phases from the analysis of temperature dependent resistance data, R(T). In the result, deduced value of e-ph = 0.8-0.9 for 퐼43푑-Th4H15 phase is in excellent agreement with e-ph = 0.83 reported by Miller et al [17] who obtained it from the analysis of the specific heat measurements. For high-compressed (P = 170 GPa) P63/mmc-ThH9 phase we deduce e-ph = 1.57 ± 0.06, which is in a good agreement with e-ph(150 GPa) = 1.73 computed by first principles calculations (at P = 150 GPa) by Semenok et al [2]. 2. Utilized model Debye temperature, T, of the metallic conductor can be deduced as a free parameter from the fit of normal part of temperature dependent resistance, R(T) (or resistivity, (T)), to the Bloch-Grüneisen (BG) equation [18,19]: 푇 푇 5 휃 푥5 푇 푅(푇) = 푅0 + 퐴 ∙ ( ) ∙ ∫ 푥 −푥 ∙ 푑푥 (1) 푇휃 0 (푒 −1)∙(1−푒 ) where R0, A and T are free-fitting parameters, and the former term is residual resistance appeared due to the conduction electrons scattering on the static defects of the lattice. In considered case of superconductors, Eq. 1 is only valid to fit the normal part of R(T) and recently [20] we proposed to split full R(T) curve in two part: the normal part (which is fitted 표푛푠푒푡 표푛푠푒푡 by Eq. 1, i.e. for 푇푐 < 푇, where 푇푐 is the onset of the superconducting transition), and the transition and zero resistance part is approximated by simplified equation proposed by Tihkham [21]: 표푛푠푒푡 푅(푇푐 ) 푅(푇) = 2 (2) 3⁄2 푇 (퐼0(퐹⋅(1− 표푛푠푒푡) )) 푇푐 3 where F is free-fitting parameter and I0(x) is the zero-order modified Bessel function of the 표푛푠푒푡 first kind. These two parts (Eqs. 1,2) are stitched at 푇 = 푇푐 using the Heaviside function, 휃(푥), and full fitting equation is [20]: 푅(푇표푛푠푒푡) 푅(푇) = 푅 + 휃(푇표푛푠푒푡 − 푇) ⋅ 푐 + 휃(푇 − 푇표푛푠푒푡) ⋅ (푅(푇표푛푠푒푡) + 퐴 ∙ 0 푐 2 푐 푐 3⁄2 푇 (퐼0(퐹⋅(1− 표푛푠푒푡) )) ( 푇푐 ) 푇 푇 5 휃 푇 5 휃 푥5 푇표푛푠푒푡 표푛푠푒푡 푥5 푇 푐 푇푐 (( ) ∙ ∫ 푥 −푥 ∙ 푑푥 − ( ) ∙ ∫ 푥 −푥 ∙ 푑푥)) (3) 푇휃 0 (푒 −1)∙(1−푒 ) 푇휃 0 (푒 −1)∙(1−푒 ) The Debye temperature, T, and the transition temperature, Tc, are linked through the McMillan equation [22], which was recently [23] simplified by excluding one parameter which cannot be deduced from experiment (see details Ref. 23): 1.04∙(1+휆푒−푝ℎ) −( ) 1 ∗ 푇 = ( ) ∙ 푇 ∙ 푒 휆푒−푝ℎ−휇 ∙(1+0.62∙휆푒−푝ℎ) ∙ 푓 ∙ 푓∗ (4) 푐 1.45 휃 1 2 1⁄3 휆 3⁄2 푓 = (1 + ( 푒−푝ℎ ) ) (5) 1 2.46∙(1+3.8∙휇∗) ∗ ∗ 2 푓2 = 1 + (0.0241 − 0.0735 ∙ 휇 ) ∙ 휆푒−푝ℎ. (6) where * is the Coulomb pseudopotential parameter (ranging from * = 0.10-0.16 [24-36]). Eqs. 4-6 have the single solution in respect of 휆푒−푝ℎ, for given T, Tc and *. Due to * is varying within a range of 0.10-0.16 [24-36], in this work we used the mean value of * = 0.13 in all calculations. There is a need to clarify, that we define Tc in Eqs. 4-6 based on the approach to be as closed as possible to the criterion [23]: 푅(푇) 표푛푠푒푡 → 0.0 (7) 푅(푇푐 ) which is in given paper implemented as: 푅(푇) 표푛푠푒푡 = 0.03 (8) 푅(푇푐 ) It should stress that opposite definitions of Tc: 4 푅(푇) 표푛푠푒푡 → 1.0 (9) 푅(푇푐 ) including widely used criteria of: 푅(푇) 표푛푠푒푡 = 0.5 (10) 푅(푇푐 ) 푅(푇) 표푛푠푒푡 = 1.0 (11) 푅(푇푐 ) are not reliable, because Eqs. 10,11 cannot be used to distinguish the difference between the change in the resistance originated from any phase transition in the sample (for instance, the atomic ordering, the ferromagnetic ordering, structural phase transition, etc., see, for instance, Refs. 37-41) and the superconducting transition. The latter has one essential property which distinguishes the superconducting transition from all other phase transition, and this is zero resistance. Extraordinary example of the mismatch based on the definition of Tc by Eq. 11 can be found in recent literature, where Ashcroft’s [42] group claimed “hot” superconductivity with Tc = 550 C. For instance, in Fig. 1(a) we showed R(T) curve [42] and data fit to the equation [20]: 표푛푠푒푡 푅(푇푐 ) 푅(푇) = 푅 + 푘 ⋅ 푇 + 휃(푇표푛푠푒푡 − 푇) ⋅ + 0 푐 2 푇 3⁄2 (퐼0 (퐹 ⋅ (1 − 표푛푠푒푡) )) ( 푇푐 ) 표푛푠푒푡 표푛푠푒푡 표푛푠푒푡 휃(푇 − 푇푐 ) ⋅ (푅(푇푐 ) + (푘 − 푘1) ⋅ 푇푐 + 푘1 ∙ 푇) (9) where k1 and k1 are free-fitting parameter describing linear slopes of R(T) curve. Formally 표푛푠푒푡 ∗표푛푠푒푡 deduced 푇푐 = 556.7 ± 0.1 퐾, which designated as 푇"푐" in Fig. 1(a), because there is no superconducting transition in any of R(T) curves reported in Ref. 42. 푅(푇) It should be stressed, that by using the same criterion of 표푛푠푒푡 = 1.0 (Eq. 11), one 푅(푇푐 ) ∗표푛푠푒푡 would claim the superconducting transition temperature 푇"푐" = 865.6 ± 0.4 퐾 in 5 Cu0.6Pd0.4 alloy (raw temperature dependent resistance was reported by Antonova and Volkov [37]), which would be characterized as a “refractory” superconductivity. 13 a "Hot" superconductivity 12 in La-based superhydride 11 10 9 resistance (Ohm) raw R(T) 8 fit to Eq. 9 *onset T"c" = 556.7 ± 0.1 K 7 520 530 540 550 560 570 580 590 temperature (K) 50 b "Refractory" superconductivity 40 in Cu0.6Pd0.4 alloy 30 raw R(T) Ohm·cm) fit to Eq. 9 T *onset = 865.6 ± 0.4 K 20 "c" resistivity ( 10 0 300 450 600 750 900 1050 1200 temperature (K) Figure 1. a - Resistance data, R(T), and data fit to Eq.
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