Electron–phonon coupling constant and BCS ratios in LaH_10−y doped with magnetic rare-earth element

The discovery of a superconducting state with a transition temperature above 200 K in highly compressed H₃S by Drozdov et al [1] with the consequent discovery of high-temperature superconductivity in superhydrides of thorium [2] and near-room-temperature superconductivity (NRTS) in superhydrides of lanthanum [3, 4], yttrium [5, 6] and lanthanum–yttrium [7] represent the most fascinating scientific explorations in the field of superconductivity since the discovery of high-temperature superconductivity in cuprates [8].

While in the majority of theoretical works (a comprehensive review has been published recently [9]) the electron–phonon mechanism is considered to be the primary pairing mechanism in NRTS superhydrides, there are several new and alternative approaches to the nature of charge carrier interaction, and pairing in NRTS is also under development [10–14]. One of the most solid experimental facts that supports the electron–phonon pairing mechanism in superhydrides is the prominent isotope effect with respect to the transition temperature, T_c [1, 3, 15]. However, the effect of hydrogen–deuterium exchange on other fundamental parameters of NRTS superhydrides, for instance, on the lower and upper critical fields, as well as on Bardeen–Cooper–Schrieffer (BCS) ratios (i.e. $\frac{{2{{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}}{T_{\text{c}}}}}$ where ${{\Delta }}\left( 0 \right)$ is the ground state of the superconducting energy gap, ${k_{\text{B}}}$ is the Boltzmann constant), remains to be explored.

Another important question that needs to be answered is the superconducting gap symmetry in NRTS hydrides. To the author's best knowledge, there is general agreement [9] that the superconducting energy gap in superhydrides exhibits s-wave symmetry. However, the first experimental evidence that confirms s-wave gap symmetry was only recently reported by Semenok et al [16]. This research group proposed to verify the s-wave gap symmetry in NRTS by employing one of the conclusions of Abrikosov–Gor'kov [17], Anderson [18] and Openov [19, 20] theories of dirty superconductors. The conclusion is that uniformly distributed (on the atomic level) impurities exhibiting magnetic moment should suppress the superconducting order parameter in s-wave superconductors. However, this kind of impurity should not affect the superconducting order parameter in d-wave superconductors. Nevertheless, non-magnetic impurities should cause the suppression in d-wave superconductors [19, 20], but this kind of doping should not affect the s-wave superconducting state [17–20].

Thus, to reaffirm/disprove s-wave symmetry gap in LaH₁₀, Semenok et al [16] performed gradual doping of lanthanum decahydride by the magnetic rare-earth element, neodymium. Hydrogen and rare-earth element (i.e. H:(La + Nd)) stoichiometry was targeted to be 10:1 in all samples. However, the exact hydrogen content in the synthesized samples of La_1−x Nd_x H_10−y (x = 0.08, 0.09, 0.20, 0.25 and 0.50) [16] should be further established because similar T_c suppression was observed in hydrogen-deficient LaH_10−y samples reported by Drozdov et al [3]). In addition, there is a need for further experimental confirmation that all Nd atoms replace the lanthanum at their sites in the crystal lattice. This is important, because Nd can form its own hydride phases, while the total hydrogen content might remain stoichiometric in the mixture of the two phases.

Semenok et al [16] reported on gradual ${T_{\text{c}}}$ suppression upon the increase in the Nd concentration. One of the interesting consequences associated with this ${T_{\text{c}}}$ suppression is that the upper critical field, ${B_{{\text{c}}2}}\left( T \right)$, is also decreasing. This makes it possible to measure ${B_{{\text{c}}2}}\left( T \right)$ for La_1−x Nd_x H_10-y within a much wider reduced temperature range, $\frac{T}{{{T_{\text{c}}}}}$, in comparison with the range available for undoped stoichiometric NRTS materials H₃S [21] and LaH₁₀ [22], which exhibits a ground state upper critical field well above the value that is measurable in non-destructive experiments.

In this paper, we analyzed reports by Semenok et al [16] on temperature-dependent resistance, $R\left( T \right)$, and the upper critical field, ${B_{{\text{c}}2}}\left( T \right)$, and deduced the following fundamental parameters for the La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,\text{GPa}$) superconductor:

• (a)  
  Debye temperature, ${T_\theta }$;
• (b)  
  Electron–phonon coupling constant, ${\lambda _{{\text{e}} - {\text{ph}}}}$;
• (c)  
  Ground-state energy gap, Δ(0);
• (d)  
  Relative jump in electronic specific heat at T_c, $\frac{{{{\Delta }}C}}{{\gamma {T_{\text{c}}}}}$ (where $\gamma$ is the Sommerfeld constant);
• (e)  
  Ground-state coherence length, ξ(0);
• (f)  
  Gap-to-transition temperature ratio, $\frac{{2{{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}}{T_{\text{c}}}}}$ (where ${k_{\text{B}}}$ is the Boltzmann constant).
• (g)  
  Fermi temperature, T_F.

2.1. Temperature-dependent resistance of La_1−xNd_xH_10−y (x = 0.09; P = 180 GPa)

Debye temperature can be deduced from the fit of experimentally measured temperature-dependent resistance, $R\left( T \right)$, to the Bloch–Grüneisen equation [23, 24]:

$$\begin{equation}R\left( T \right) = {R_0} + A{\left( {\frac{T}{{{T_\theta }}}} \right)^5}\int \limits_0^{\frac{{{T_\theta }}}{T}} \frac{{{x^5}}}{{\left( {{e^x} - 1} \right)\left( {1 - {e^{ - x}}} \right)}}dx,\end{equation} \tag{ 1 }$$

where ${R_0}$ and $A$ are free-fitting parameters. From the deduced ${T_\theta }$ and measured ${T_{\text{c}}}$, the electron–phonon coupling constant, ${\lambda _{{\text{e}} - {\text{ph}}}}$, can be calculated as a unique root of the advanced McMillan equation [25, 26]:

$$\begin{equation}{T_{\text{c}}} = \left( {\frac{1}{{1.45}}} \right) \times {T_\theta } \times {e^{ - \,\left( {\frac{{1.04\left( {1 + {\lambda _{{\text{e}} - {\text{ph}}}}} \right)}}{{{\lambda _{{\text{e}} - {\text{ph}}}} - {\mu ^{\,*}}\left( {1 + 0.62{\lambda _{{\text{e}} - {\text{ph}}}}} \right)}}} \right)}} \times {f_1} \times f_2^{\,*},\end{equation} \tag{ 2 }$$

where

$$\begin{equation}{f_1} = {\left( {1 + {{\left( {\frac{{{\lambda _{{\text{e}} - {\text{ph}}}}}}{{2.46\left( {1 + 3.8{\mu ^{\,*}}} \right)}}} \right)}^{3/2}}} \right)^{1/3}},\end{equation} \tag{ 3 }$$

$$\begin{equation}f_2^{\,*} = 1 + \left( {0.0241 - 0.0735 \times {\mu ^{\,*}}} \right) \times \lambda _{{\text{e}} - {\text{ph}}}^2,\end{equation} \tag{ 4 }$$

where ${\mu ^{\,*}}$ is the Coulomb pseudopotential parameter, which can be assumed to be ${\mu ^{\,*}} = 0.13$ for all NRST materials.

The fit of $R\left( T \right)$ data set to equation (1) is shown in figure 1. This $R\left( T \right)$ data set reported in figure 2(a) in [16] and the raw data are freely available online by Semenok et al [27]. The deduced parameters are: ${R_0} = \,0.729\left( 5 \right)$, $A = \,5.42 \pm 0.06$ and ${T_\theta } = 1156 \pm 6\,{\text{K}}$. By utilizing the general requirement [26] that ${T_{\text{c}}}$ should be defined at the lowest possible $\frac{{R\left( T \right)}}{{{R_{{\text{norm}}}}\left( T \right)}}$ ratio, and considering that the same criterion should be used to define the ${B_{{\text{c}}2}}\left( T \right)$ data set from the $R\left( {T,B} \right)$ curves (reported in figures 2(b) and (c) in [16]), the ratio of $\frac{{R\left( T \right)}}{{{R_{{\text{norm}}}}\left( T \right)}} = 0.08$ was used. In the result, ${T_{\text{c}}}$ for $R\left( T \right)$ in figure 1 was defined as ${T_{\text{c},0.08}} = \,122\,{\text{K}}$.

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The root of equations (2)–(4) for given ${T_\theta }$, ${T_{\text{c}}}$ and ${\mu ^{\,*}} = 0.13$ is ${\lambda _{{\text{e}} - {\text{ph}}}} = 1.65 \pm 0.01$. The deduced values are in good agreement with ${\lambda _{{\text{e}} - {\text{ph}}}}$ values calculated by first-principles calculations by Semenok et al [27] in their table S5.

2.2. Temperature-dependent upper critical field of La_1−xNd_xH_10−y (x = 0.09; P = 180 GPa)

By utilizing the same criterion of $\frac{{R\left( T \right)}}{{{R_{{\text{norm}}}}\left( T \right)}} = 0.08$, the ${B_{{\text{c}}2}}\left( T \right)$ dataset was derived from the $R\left( {T,B} \right)$ curves shown in figures 2(b) and (c) of [16]. In figure 2, the ${B_{{\text{c}}2}}\left( T \right)$ data set is fitted to the equation for temperature-dependent upper critical field for s-wave superconductors [28, 29]:

$$\begin{align}{B_{{\text{c}}2}}\left( T \right) &= \,\frac{{{\phi _0}}}{{2 \cdot \pi \cdot {\xi ^2}\left( 0 \right)}}{\left( {\frac{{1.77 - 0.43{{\left( {\frac{T}{{{T_{\text{c}}}}}} \right)}^2} + 0.07{{\left( {\frac{T}{{{T_{\text{c}}}}}} \right)}^4}}}{{1.77}}} \right)^2}\nonumber\\ &\quad \times \left[ {1 - \frac{1}{{2{k_{\text{B}}}T}}\int\limits_0^\infty \frac{{d\varepsilon }}{{{\text{cos}}{{\text{h}}^2}\left( {\frac{{\sqrt {{\varepsilon ^2} + {{{\Delta }}^2}\left( T \right)} }}{{2{k_{\text{B}}}T}}} \right)}}} \right],\end{align} \tag{ 5 }$$

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where the amplitude of temperature-dependent superconducting gap, Δ(T), is given by [30, 31]:

$$\begin{equation}\Delta \left( T \right) = \Delta \left( 0 \right) \times {\text{tanh}}\left[ {\frac{{\pi {k_{\text{B}}}{T_{\text{c}}}}}{{\Delta \left( 0 \right)}}\sqrt {\eta \frac{{\Delta C}}{{\gamma {T_{\text{c}}}}}\left( {\frac{{{T_{\text{c}}}}}{T} - 1} \right)} } \right]{} ,\end{equation} \tag{ 6 }$$

where η = two-thirds for s-wave superconductors.

The fit converges with a high quality (with goodness-of-fit R = 0.9976) (figure 2). The deduced parameters are: $\xi\left( 0 \right) = 2.33\, \pm 0.02\,\,{\text{nm}}$, ${{\Delta }}\left( 0 \right) = 20.2 \pm 1.3\,{\text{meV}}$ and $\frac{{2{{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}}{T_{\text{c}}}}} =$ $4.0 \pm 0.2$, $\frac{{{{\Delta }}C}}{{\gamma {T_{\text{c}}}}} = 1.68 \pm 0.15$. Considering that the weak coupling limits of the BCS theory [30–32] are $\frac{{2 \cdot {{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}} \cdot {T_{\text{c}}}}} = 3.53$ and $\frac{{{{\Delta }}C}}{{\gamma {T_{\text{c}}}}} = 1.43$, and the upper limits for low-T_c electron–phonon mediated superconductors are $\frac{{2 \cdot {{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}} \cdot {T_{\text{c}}}}} = 5.2$ (for Pb_0.50Bi_0.50 alloy [33]) and $\frac{{{{\Delta }}C}}{{\gamma {T_{\text{c}}}}} = 3.0$ (for Pb_0.70Bi_0.30 alloy [33]), one can conclude that La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,{\text{GPa}}$) is moderately strongly coupled superconductor.

2.3. La_1−xNd_xH_10−y (x = 0.09; P = 180 GPa) in the Uemura plot

Final characterization of the superconducting properties of La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,{\text{GPa}}$) was to position this hydride in the empirical Uemura plot [34, 35], where heavy fermions, fullerenes, cuprates, pnictides and hydrogen-rich superconductors [28, 36, 37] form a narrow band that exhibits the ratio of the superconducting transition temperature, T_c, to the Fermi temperature, T_F, within a range:

$$\begin{equation}0.01 \lesssim \frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}} \lesssim 0.05,\end{equation} \tag{ 7 }$$

while all low-T_c conventional superconductors have much smaller $\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}}$ ratio:

$$\begin{equation}\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}} \lesssim 0.001.\end{equation} \tag{ 8 }$$

The Fermi temperature can be calculated by following equation [37]:

$$\begin{equation}{T_{\text{F}}} = \frac{{{\pi ^2}}}{{8 \cdot {k_{\text{B}}}}} \times {} \left( {1 + {\lambda _{{\text{e}} - {\text{ph}}}}} \right) \times {\xi ^2}\left( 0 \right) \times {\left( {\frac{{2\Delta \left( 0 \right)}}{\hbar }} \right)^2},\end{equation} \tag{ 9 }$$

where all parameters were deduced above. In the result, calculated Fermi temperature is ${T_{\text{F}}} = \,4430 \pm 50\,{\text{K}}$ and, thus, $\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}} = 0.027$ for this superhydride. As a result, La_1−x Nd_x H₁₀ (x = 0.09; $P = 180\,\,{\text{GPa}}$) falls into the unconventional superconductor band in the Uemura plot (figure 3), and is located in close proximity to YBa₂Cu₃O_7−δ and Bi₂Sr₂Ca₂Cu₃O₁₁ cuprates and other NRTS counterparts.

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To address the possible issue of the electron–phonon mediated materials being located in the unconventional superconductor band, we should point out that superhydrides are not the only known electron–phonon mediated superconductors that are located in the unconventional superconductor band in the Uemura plot (for instance, we can mention bulk s-wave A₃C₆₀ (A = K, Rb) superconductors).

On the other hand, the Uemura plot is not intended to reveal the pairing mechanism, but rather indicates the geometrical ratio of the characteristic size of the Cooper pair (which is proportional to ${{\xi }}\left( T \right)$) with average spatial distance between the centers of the Cooper pairs. This implies that materials with low superfluid density tend to locate closer to the Bose–Einstein condensate (BEC) line, $\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}} = 0.22$, while materials with high volume concentration of Cooper pairs tend to be located closer to pure metals, such as aluminium, for which $\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}}\sim {10^{ - 5}}$. This limit is also known as the BCS limit. Detailed studies of the transition of the same material from BEC to BCS pairing state (under high pressure) can be found elsewhere [38].

We summarize our findings and mention that while the detection of superconductivity in elemental highly compressed hydrogen is an ongoing task [39–41], the near-room-temperature superconductivity has been observed in several superhydires [1–7]. One of these superhydrides, LaH₁₀ [3, 4], represents a versatile platform to study the physics of NRTS because T_c and other superconducting parameters in this compound can be modified by:

• (a)  
  Hydrogen stoichiometry [3];
• (b)  
  Pressure, for which similar to cuprates [42] (where the dome-like T_c versus charge carrier concentration was observed), the dome-like T_c dependence versus pressure was observed [3];
• (c)  
  Lanthanum substitution by yttrium [7], neodymium [16] and cerium [43]. It should be noted that there is a possibility that other rare-earth elements can also be utilized for the substitution.

Here, we analyzed experimental magnetoresistance data, R(T,B), for highly compressed La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,{\text{GPa}}$) superconductor in which the superconducting order parameter was suppressed by magnetic rare-earth element (neodymium) impurity. Raw experimental R(T,B) data sets for La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,{\text{GPa}}$) were recently reported by Semenok et al [16, 27]. Deduced parameters, such as the gap-to-transition temperature ratio, $\frac{{2{{\Delta }}\left( 0 \right)}}{{{k_{\text{B}}}{T_{\text{c}}}}} = 4.0 \pm 0.2$, and the relative jump in specific heat at the transition temperature, $\frac{{{{\Delta }}C}}{{\gamma {T_{\text{c}}}}} = 1.7 \pm 0.1$, indicate that La_1−x Nd_x H_10−y (x = 0.09; $P = 180\,\,{\text{GPa}}$) is moderately strongly coupled superconductor. This hydride exhibits a ratio of the superconducting transition temperature to the Fermi temperature of $\frac{{{T_{\text{c}}}}}{{{T_{\text{F}}}}} = 0.027$, and it falls into the unconventional superconductor band in the Uemura plot.

The author thanks Dmitrii V Semenok (Skolkovo Institute of Science and Technology) and co-workers of [16, 27] for making raw experimental data is freely available prior the peer-review publication of their paper.

The research funding from the Ministry of Science and Higher Education of the Russian Federation (theme 'Pressure' No. АААА-А18-118020190104-3 and Ural Federal University Program of Development within the Priority-2030 Program) is gratefully acknowledged.

The data that support the findings of this study are available upon reasonable request from the author.

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.