The Isotope Effect and Critical Magnetic Fields of Superconducting YH$_{6}$: A Migdal-Eliashberg Theory Approach

TL;DR

This study uses Migdal-Eliashberg theory and density functional calculations to show that YH$_6$ exhibits strong-coupling conventional superconductivity with high critical fields and isotope effects, challenging its perceived anomalous behavior.

Contribution

The paper demonstrates that YH$_6$'s superconducting properties are consistent with strong-coupling conventional theory, extending understanding to related metal hydrides.

Findings

YH$_6$ shows high critical magnetic fields.

Isotope effect aligns with strong-coupling superconductivity.

Strong-coupling corrections modify BCS values for key parameters.

Abstract

The emergence of near-ambient temperature superconductivity under pressure in the metal hydride systems has motivated a desire to further understand such remarkable properties, specifically critical magnetic fields. YH$_6$ is suggested to be a departure from conventional superconductivity, due to apparent anomalous behavior. Using density functional calculations in conjunction with Migdal-Eliashberg theory we show that in YH$_6$ the critical temperature and the isotope effect under pressure, as well as the high critical fields, are consistent with strong-coupling conventional superconductivity; a property anticipated to extend to other related systems. Furthermore, the strong-coupling corrections occur to the expected BCS values for the Ginzburg-Landau parameter ($\kappa_{1}(T)$), London penetration depth ($\lambda_{L}(T)$), electromagnetic coherence length ($\xi(T)$), and the energy gap ($\Delta_{0}$).

Tables

Table 1. Table 1: Zero-temperature calculated parameters of the superconducting state at selected pressures for YH 6 and YD 6 using the solutions of the NLMEEs and LMEEs, in conjunction with Eqs. 1 , 3 and 4 for the upper critical field, London penetration depth ( λ L subscript 𝜆 𝐿 \lambda_{L} ), and coherence length ( ξ 𝜉 \xi ), respectively. The critical fields are in units of Tesla, the Fermi velocity ( v F subscript 𝑣 𝐹 v_{F} ) in × 10 5 ​ m / s absent superscript 10 5 𝑚 𝑠 \times 10^{5}m/s (Eq. S26), λ L subscript 𝜆 𝐿 \lambda_{L} and ξ 𝜉 \xi in nm.

			YH6										YD6
P			$v_F$	$H_c\left(0\right)$	$H_{c1}\left(0\right)$	$H_{c2}\left(0\right)$	${\kappa }_1\left(0\right)$	${\lambda }_L\left(0\right)$	$\xi \left(0\right)$			$v_F$	$H_c\left(0\right)$	$H_{c1}\left(0\right)$	$H_{c2}\left(0\right)$	${\kappa }_1\left(0\right)$	${\lambda }_L\left(0\right)$	$\xi \left(0\right)$
160			8.76	3.79	0.33	141.5	26.37	159.15	1.712			8.75	2.96	0.26	110.8	26.48	158.38	2.18
180			8.68	3.59	0.33	129.0	25.57	157.98	1.884			8.68	2.83	0.25	103.0	25.76	157.38	2.34
200			8.61	3.45	0.31	122.3	25.04	157.40	1.954			8.59	2.74	0.25	97.7	25.27	157.22	2.43
220			8.53	3.35	0.31	116.7	25.64	158.12	2.164			8.51	2.68	0.25	91.9	24.93	157.93	2.54
250			8.39	3.25	0.30	111.6	24.27	159.04	2.110			8.38	2.61	0.24	90.9	24.60	159.28	2.58

Equations

$$H_{c2}(T)=\rho (T)/eD,$$

$$D=l_{tr}{v}_F/3,$$

$${\lambda }_L(T)={\left[\frac{4}{3}\pi N(0)e^2{v}_F^{2}T{\mu }_0\sum _{n=1}^{\infty }\frac{{\tilde{\Delta }}_n^2}{{\left({\tilde{\omega }}_n^2+{\tilde{\Delta }}_n^2\right)}^{3/2}}\right]}^{-1/2},$$

$$\xi (T)=\frac{v_F\hslash }{2}\frac{\left[{\sum }_{n=1}^{\infty }\frac{{\tilde{\Delta }}_n^2}{{\left({\tilde{\omega }}_n^2+{\tilde{\Delta }}_n^2\right)}^{3/2}}\right]}{\left[{\sum }_{n=1}^{\infty }\frac{{\tilde{\Delta }}_n^2}{{\tilde{\omega }}_n^2+{\tilde{\Delta }}_n^2}\right]}.$$

Full text

superconductivity, isotope effect, Eliashberg theory, superhydride, high pressure

Over the last decade, a new class of materials of stoichiometric to hydrogen-rich metal hydrides under pressure has emerged with theoretical predictions being made about their crystal structures, electronic, dynamic, and coupling properties [1, 2, 3, 4]. As a result of those periodic table spanning predictions [5], several conventional-superconductor candidates have been proposed with critical temperatures ($T_{c}$) approaching room temperature [6, 7, 8]. The experimental breakthrough for these materials came with the discovery of phonon-mediated superconductivity in H3S, with a maximum $T_{c}$ of $203$ K measured at 155 GPa [9, 10, 11]. Subsequently, high-$T_{c}$ superconductivity measurements were reported in other compounds such as LaH10 with a $250$–$260$ K $T_{c}$ at $170$ GPa [12, 13]; YH9 with $T_{c}=262$ K at $182$ GPa [14]; YH6 with $T_{c}=220$ K at approx. $160$ GPa [15, 16]; more recently, CaH6 with $T_{c}=215$ K at $172$ GPa [17]; as well as a reported carbonaceous sulfur hydride with $T_{c}=287$ K at $267$ GPa [18, 19]. Most of the theoretical works on the superconducting state of these novel metal hydrides have shown that a strong electron-phonon coupling and high-energy hydrogen phonon modes play a key role in the high $T_{c}$ calculated values, concluding that they are phonon-mediated strong-coupling superconductors [3, 20, 21, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31, 1, 4, 32].

Beyond the $T_{c}$, there are several other important properties of a superconducting material such as the isotope effect coefficient, the upper, lower, and thermodynamic critical magnetic fields, the penetration depth, and the coherence length. The first has been crucial in elucidating the mechanism responsible for Cooper-pair formation in conventional superconductors. While the last gives us the response of the materials to an external magnetic field. The lower and upper critical magnetic fields ($H_{c1}$ and $H_{c2}$) are specially interesting, since they give a measure of the Meissner effect and the magnitude of the external magnetic field at which superconductivity is totally suppressed. Due to experimental difficulties [33], measurements of the critical magnetic fields of the metal hydrides at high pressures have been done only at temperatures near $T_{c}$. From that data the slope of $H_{c2}(T)$ is fitted and $H_{c2}(0)$ is extrapolated using the Ginzburg-Landau (GL) [34] or the Werthamer–Helfand-Hohenberg (WHH) [35] models, giving rise to different values of $H_{c2}(0)$. For example, the reported $H_{c2}(0)$ values for YH6, the aim of this work, are 107(157) T and 76(102) T at 160 GPa and 200 GPa, respectively, for GL(WHH) [15, 36]. Similar differences are reported in other metal hydrides [9, 12, 33]. Furthermore, these extrapolated values of $H_{c2}(0)$, in conjunction with GL model, Bardeen-Cooper-Schrieffer (BCS) theory [37], and empirical relations [38, 39, 40, 41], are used to get a complete description of other relevant quantities, like the critical magnetic fields, penetration depth ($\lambda$), coherence length ($\xi$), as well as the GL parameter ($\kappa=\lambda/\xi$).

While the $T_{c}$ of the metal hydrides has been widely studied theoretically within a strong-coupling formalism (Migdal-Eliashberg (ME) theory [42]), there are no reports where these other important properties, like the critical magnetic fields and related lengths, are calculated from first principles within a strong-coupling formalism, where corrections to the empirical and weak-coupling values are expected. This, in conjunction with the lack of experimental measurements of the critical fields at low temperatures, has generated confusion and some doubts about the nature of the kind of superconductors they belong to [38, 39, 40, 41]. For YH6, this has even suggested a possible deviation from conventional superconductivity [36]. The aim of this communication is to show that the critical magnetic fields, the London penetration depth, and the electromagnetic coherence length can be calculated from first principles using density functional theory (DFT) in conjunction with ME theory, showing the conventional nature of YH6 and providing a general prescription to describe the superconducting state in the high-$T_{c}$ metal hydrides.

To this end, we start by showing that the behavior of the critical temperature under pressure can be reproduced within the ME formalism, using as input the Eliashgberg function ($\alpha^{2}F\left(\omega\right)$) calculated from first principles in the optimized $lm\bar{3}m$ crystal structure (see Supplemental Material (SM) [43]). As a first step, the Linearized Migdal-Eliashberg Equations (LMEE, Eq. S8) are solved at a fixed pressure, where $T_{c}$ is known from experiment. Here, the experimental $T_{c}\,=\,$220 K at 160 GPa [15] for YH6 and $T_{c}\,=\,$165 K at 200 GPa [36] for YD6 are employed as the initial data. From this solution, the functional derivative of $T_{c}$ with respect to $\alpha^{2}F\left(\omega\right)$ is calculated within the Bergmann and Rainer [44, 45] formalism. These functional derivatives allow a link between the observed changes in $\alpha^{2}F\left(\omega\right)$, by a pressure variation (from $P_{0}$ to $P_{i}$), to changes in the critical temperature $\Delta T_{c}(P_{0},P_{i})$ (Eq. S17). Fig. 1a shows the behavior of $T_{c}$ under pressure for both, YH6 and YD6. It can be observed that our calculated $T_{c}$ is in excellent agreement with the experimental values in the whole pressure interval (160 – 330 GPa), and even better than the calculations reported previously in the literature, showing that the behavior of $T_{c}(P)$ is driven mainly by the changes in the electron-phonon interaction (Fig. S4). The temperature isotope effect coefficient, $\alpha$, is calculated taking into account the changes in the electron-electron and the electron-phonon interaction (Eq. S19), coming from phonon changes due to the isotope mass substitution, within the Rainer and Culetto [46, 47, 48] formalism. Fig. 1b presents $\alpha$ as a function of pressure, also in excellent agreement with experiment. This first step confirms the conventional nature of superconductivity in YH6 and show us that the solutions of the gap function, $\tilde{\Delta}_{n}$, correctly reproduce the superconducting state using the $\alpha^{2}F\left(\omega\right)$ determined from first principles calculations as an input for the LMEEs.

We now focus on the critical magnetic fields. In the dirty limit for a strong-coupling superconductor, $H_{c2}(T)$ has to be evaluated from the LMEEs (Eqs. S13 and S14) in the presence of a homogeneous magnetic field [49] where the pair-breaking parameter $\rho(T)$ and $H_{c2}(T)$ are related by

[TABLE]

[TABLE]

where $D$ is the diffusion constant, $v_{F}$ the Fermi velocity, $l_{tr}$ the mean free path, and $e$ is the electron charge. Figure 2 shows the temperature dependence of $H_{c2}(T)$ calculated within the ME (Eq. 1, solid lines), GL (dashed lines), and WHH (doted lines) formalism for YH6 at 160 and 200 GPa, and YD6 at 173 GPa. These pressures were selected to make a direct comparison with available experimental data (symbols) [15, 36]. $v_{F}$ was calculated from the dispersion of the electronic band structure (see Table 1), and $l_{tr}=1.655(1.645)$ Å for YH6(YD6) was fitted using Eq. 2 to get the experimental $H_{c2}$ data close to $T_{c}$. Near $T_{c}$, the calculated values of $H_{c2}(T)$ are similar for the three models, and in very good agreement with experiment. As the temperature starts to decrease, the difference between the models’ results increases, with the ME results being in between the higher WHH and lower GL values. The largest differences between ME and the GL and WHH models ($\Delta{H_{c2}}$) are at $T=0$ K, with values that go from \Delta{H_{c2}}(\mbox{YH{}{6}})=7(18) T at 200 GPa to as high as \Delta{H_{c2}}(\mbox{YH{}{6}})=17(33) T at 160 GPa, in respect to WHH(GL). Although the GL theory is applicable to practically all superconductors, it is a phenomenological theory restricted to temperatures close to $T_{c}$. Therefore, GL theory is not expected to give accurate results at lower temperatures, giving rise to the huge differences at T=0 K between itself and the other theories. The WHH formalism is based on weak-coupling BCS theory, and thus should be valid for the whole temperature range; whereas the ME formalism covers weak- and strong-coupling. Thus, the ME results show a distinct strong-coupling correction to the WHH results.

To calculate the thermodynamic critical field, $H_{c}$, of an isotropic strong-coupling superconductor, the Non-linear MEEs (NLMEE) have to be solved (Eqs. S5 and S6). With the knowledge of $\tilde{\Delta}_{n}$ and $\tilde{\omega}_{n}$, the difference in free energy between the normal and the superconducting states of the metal, $\Delta F(T)=F_{n}-F_{s}$, can be calculated directly. By definition, $H_{c}$ is given by the relation $H_{c}(T)=[8\pi(\Delta F(T))]^{1/2}$. Then, from the calculated fields, the GL parameter and the lower critical field can be evaluated from the relations $\kappa_{1}=(1/\sqrt{2})H_{c2}/H_{c}$ and $H_{c1}H_{c2}=H_{c}^{2}\text{ln}(\kappa)$, respectively. Expanding Figure 2 to higher pressures to span the whole pressure range studied, Figure 3 shows the calculated behavior of $H_{c}(T)$, $H_{c1}(T)$, and $H_{c2}(T)$ for YH6 (solid lines) and YD6 (dashed lines) at 160, 200, and 250 GPa. As a function of pressure, there is a steady shift of the critical fields to lower values as the pressure is incremented. In particular, at $T$=0 K there is a considerable suppression in $H_{c2}$ from 141.5(110.8) T to 111.8(91.0) T for YH6(YD6), at 160 and 250 GPa respectively (see Table 1). As it can be seen, the three fields show an isotopic shift to lower values due to the replacement of hydrogen by deuterium.

The strong-coupling behavior of the calculated $H_{c2}(T)$ and $H_{c}(T)$ by the ME-model is confirmed by the deviation function, $D(t)$ (Eqs. S27 and S28), which shows the standard behavior (positive values) for strong-coupling materials, in contrast to the intermediate-coupling behavior (change in sign) that is observed in the WHH model (Fig. S6). Fig. 3d shows the parameter $\kappa_{1}(T)$, which has values between approximately 24 and 27 for both compounds at $T$=0 K (Table 1), while at $T_{c}$ (where $\kappa_{1}$ is similar to the GL $\kappa$ [50]) this parameter has its minimum values and vary from 17.5 to 17.7. The ratio $\kappa_{1}(0)/\kappa_{1}(T_{c})$ give us an estimation of the strong-coupling correction to BCS values. While in the weak-coupling formalism there is a universal ratio $\kappa_{1}(0)/\kappa_{1}(T_{c})$=1.12 [49], the values for YH6(YD6) vary from 1.49(1.5) to 1.38(1.4) at 160 and 250 GPa, respectively. Such enhanced values clearly show that these systems are within the strong coupling regime across this noted pressure range, and the slight decrease with respect to pressure shows the tendency towards a less strong-coupling (intermediate) regime, as expected from the general trend under pressure of the coupling parameter (Fig. S4).

Despite expressions which are valid within Migdal-Eliashberg theory being derived several years ago [51, 52], there are very few strong-coupling superconductor systems with reported numerical results of their electromagnetic properties. Here we are interested in the magnetic field penetration depth (in the London limit) $\lambda_{L}(T)$, and the electromagnetic coherence length $\xi(T)$. Both quantities are studied in the clean limit, due to the very large energy scale associated with the phonon frequencies and superconducting gap for the metal hydrides, as was previously suggested for H3S by Nicol and Carbotte [20]. The London-limit penetration depth, which applies when $\lambda_{L}(T)\gg\xi(0)$, is given by

[TABLE]

where $\mu_{0}$ is the permeability and $N(0)$ is the single spin density of electronic states at the Fermi energy. The electromagnetic coherence length, which describes the nonlocality in the electromagnetic response of a superconductor, is given by

[TABLE]

The solutions of the NLMEEs (Eqs. S5 and S6), $\tilde{\Delta}$ and $\tilde{\omega}$, are required to get both $\lambda_{L}(T)$ and $\xi(T)$. Numerical results for the temperature variation of $\xi(T)$ are given in Fig. 3e. In superconductors, $\xi(T)$ is the range of the perturbation of the current density caused by an applied electromagnetic field, which is different from the GL coherence length $\xi$ that describes the perturbation of the superconducting pair density. In the weak-coupling theory, both quantities are related at $T=0$ K through the relation $\xi=0.739\xi(0)$. From these strong-coupling formalism calculations, $\xi(0)$ is found to be 1.71(2.18) nm and 2.11(2.58) nm for YH6(YD6) at 160 and 250 GPa, respectively. Using the GL relation, $\xi=\sqrt{\frac{\phi_{0}}{2\pi H_{c2}}}$, the GL coherence length $\xi$ is found to be 1.53(1.72) nm and 1.73(1.94) nm for YH6(YD6), at the same applied pressures which are in agreement with GL experimental estimations of 1.4–1.8 nm at 160 GPa for YH6 [15, 36]. These values yield ratios $\xi/\xi(0)$ 0.895(0.787) and 0.82(0.75) for YH6(YD6) at 160 and 250 GPa, respectively, which clearly deviate from the weak-coupling limit of 0.739. As the temperature increases, $\xi(T)$ falls quickly near $T_{c}$, where the ratio $\xi(T_{c})/\xi(0)$ drops to a value of about 0.82(0.83) for YH6(YD6) at 160 GPa, which is larger than the BCS value of 0.752 [53].

Fig. 3f shows the temperature dependence of $\lambda_{L}(T)$. $\lambda_{L}(0)$ is found to be approximately 159 nm for both, YH6 and YD6 at 160 GPa, which is close to the reported BCS values of 164 and 147 nm for H3S and LaH10, respectively [41]. The strong-coupling deviation function of $[\lambda_{L}(0)/\lambda_{L}(T)]^{1/2}$ with respect to the two fluid model for YH6 is shown in Fig. S6. The deviation function (Eq. S29), gives a minimum value of -0.05 GPa, which is in contrast to the -0.22 GPa result from BCS weak-coupling theory [54]. By means of Padé approximants [55], the energy gap, $\Delta_{0}$, can be found from an analytic continuation to the real axis of $\tilde{\Delta}_{n}$ (computed from the NLMEEs). Here, $\Delta_{0}$ is 46.68(37.6) meV and 37.7(30.9) meV for YH6(YD6) at 160 and 250 GPa, respectively. From these values, the BCS ratios $2\Delta_{0}/k_{B}T_{c}$ are 4.94(5.13) and 4.50(4.74) which shows a strong-coupling correction to the 3.52 BCS value.

To summarize, the superconducting properties and electromagnetic field response of the yttrium hydride YH6 are computed here from first principles, using DFT in conjunction with ME theory. From these, the experimental behavior of $T_{c}$ as a function of applied pressure and the isotopic effect can be perfectly reproduced within the harmonic approximation. Similarly, the calculated $H_{c2}(T)$ with the ME formalism shows excellent agreement with the available experimental data at temperatures near $T_{c}$. As $T$ goes to zero, it shows intermediate values between the GL and WHH models, providing an important strong-coupling correction to these currently used phenomenological models. The implemented formalism even allows a full description of the $H-T$ phase diagram by calculating $H_{c}(T)$, $H_{c1}(T)$, and $\kappa_{1}(T)$. Finally, the description of the YH(D)6 superconducting state is completed by calculating $\xi(T)$ and $\lambda_{L}(T)$ within the clean limit, as well as the energy gap $\Delta_{0}$. From our results, we found that any deviation from BCS behavior is well explained as a strong-coupling correction, and in this basis we are able to discard any possible anomalous behavior or departure from conventional superconductivity as was previously suggested  [36]. Even more, we consider that ME theory, in conjunction with Rainer-Bergmann [49] and Nam’s [51, 52] formalism, can provide a general prescription to describe the superconducting state in the high-$T_{c}$ metal hydrides.