Quantifying Nonadiabaticity in Major Families of Superconductors

The classical Bardeen–Cooper–Schrieffer and Eliashberg theories of the electron–phonon-mediated superconductivity are based on the Migdal theorem, which is an assumption that the energy of charge carriers, kBTF, significantly exceeds the phononic energy, ℏωD, of the crystalline lattice. This assumption, which is also known as adiabatic approximation, implies that the superconductor exhibits fast charge carriers and slow phonons. This picture is valid for pure metals and metallic alloys because these superconductors exhibit ℏωDkBTF<0.01. However, for n-type-doped semiconducting SrTiO3, this adiabatic approximation is not valid, because this material exhibits ℏωDkBTF≅50. There is a growing number of newly discovered superconductors which are also beyond the adiabatic approximation. Here, leaving aside pure theoretical aspects of nonadiabatic superconductors, we classified major classes of superconductors (including, elements, A-15 and Heusler alloys, Laves phases, intermetallics, noncentrosymmetric compounds, cuprates, pnictides, highly-compressed hydrides, and two-dimensional superconductors) by the strength of nonadiabaticity (which we defined by the ratio of the Debye temperature to the Fermi temperature, TθTF). We found that the majority of analyzed superconductors fall into the 0.025≤TθTF≤0.4 band. Based on the analysis, we proposed the classification scheme for the strength of nonadiabatic effects in superconductors and discussed how this classification is linked with other known empirical taxonomies in superconductivity.


Introduction
The majority of experimental works in superconductivity utilize the classical Bardeen-Cooper-Schrieffer (BCS) [1] and Migdal-Eliashberg (ME) [2,3] theories as primary tools to analyze measured data. However, it should be clarified that these theories are valid for superconductors which satisfy the condition designated by the Born-Oppenheimer-Migdal approximation [4]: where is the reduced Planck constant, ω D is the Debye frequency, k B is the Boltzmann constant, T θ is the Debye temperature, T F is the Fermi temperature, and data for lead were reported by Poole [5]. The Born-Oppenheimer-Migdal approximation allows the separation of electronic and ionic motions in metals, because Equation (1) implies that the conductor exhibits fast charge carriers (for which characteristic energy scale is related to the Fermi temperature, T F ) and relatively slow phonons (for which characteristic energy scale is related to the Debye temperature, T θ ).
For experimentalists, it is important to have a simple practical routine to establish the strength of nonantiadiabatic effects in newly discovered superconductors. The most obvious parameter, which serves as an experimentally measured value to quantify the strength of nonantiadiabaticity, is the T θ T F ratio. For practical use of this criterion, there is a need for the taxonomy of possible T θ T F values. To establish the taxonomy, we performed the analysis for a broad a range as possible of superconductors; these range from two-to three-dimensional materials, from elements to compounds of up to five elements, from low-T c (with T c ∼ 0.1 K) to record high-T c (with T c = 240 K) hydrides, and from materials that exhibit a high order of crystalline lattice symmetry to the materials with low symmetry. Namely, we tried to cover all superconductors for which primary characteristic parameters (apart T c , T θ , and T F ), such as the London penetration depth, λ(0), the coherence length, ξ(0), the amplitude of the superconducting energy gap, ∆(0), and the electron-phonon coupling strength constant, λ e−ph , were established. In the results, we presented the analysis of more than 40 superconductors within the families of main superconductors.
Based on our analysis, we proposed the following classification scheme: One of our findings is that for weakly nonadiabatic superconductors (i.e., for materials exhibited 0.025 ≤ T θ T F 0.4), the predicting power of the BCS-ME theories (for instance, the prediction of the superconducting transition temperature) is reasonably accurate. However, all these superconductors are located outside of the BCS corner in the Uemura plot.
We also showed how the proposed classification scheme is linked to other known empirical scaling laws and taxonomies in superconductivity [13,[17][18][19][20][21]; meanwhile, the search for the link of the proposed taxonomy with the recently reported big data [22,23] is under progress.

Utilized Models
Proposed taxonomy is based on the knowledge of three fundamental temperatures of the superconductor, which are T c , T θ , and T F . The superconducting transition temperature, T c , is directly measured in either temperature resistance or in magnetization experiments. It is also important to mention the primary experimental techniques and theoretical models utilized to deduce the Debye temperature, T θ , and the Fermi temperature, T F , in superconductors.
There are two primary techniques to determine the Debye temperature, T θ . One technique is to analyze the measured temperature-dependent normal-state specific heat, C p (T), from which the electronic specific heat coefficient, γ n , and the Debye temperature, T θ , are deduced (see, for instance [24][25][26]): where β is the Debye law lattice heat-capacity contribution, and α is from higher order lattice contributions. The Debye temperature can be calculated: where R is the molar gas constant, and p is the number of atoms per formula unit. Another technique is to fit normal-state temperature dependent resistance, R(T), to the Bloch-Grüneisen (BG) equation [24][25][26][27][28]: where, R sat is the saturated resistance at high temperatures which is temperature independent, R 0 is the residual resistance at T → 0 K , and A is free fitting parameter. Many research groups utilized both techniques (i.e., Equations (4)-(6)) to deduce T θ [24][25][26][27]29]. From the measured T c and the deduced T θ , one can derive the electron-phonon coupling constant, λ e−ph , as a root of either the original McMillan equation [30], or its recently revisited form [27]: (9) where µ * is the Coulomb pseudopotential, 0.10 µ * 0.15 [27,30].
There are several experimental techniques to derive the Fermi temperature, T F , from experimental data. One of these techniques is to measure the temperature dependent Seebeck coefficient, S(T), and fit a measured dataset to the equation [8]: Another approach is to measure the magnetic quantum oscillations [31], from which the magnitude of charge carrier mass, m * = m e 1 + λ e−ph (where m e is bare mass of electron), together with the size of the Fermi wave vector, k F , can be obtained and plugged into [31]: An alternative approach is based on the extraction of the charge carriers mass, m * , and density, n, as two of four parameters from the simultaneous analysis of C p (T), R(T), the muon spin relaxation (µSR), the lower critical field data, B c1 (T), and the upper critical field data, B c2 (T) [32], and plugging these parameters into the equation for an isotropic spherical Fermi surface [32]: where n s is bulk charge curriers density at T → 0 K . For 3D superconductors, n s is given by the equation [33]: where µ 0 is the permeability of free space, l is the charge carrier mean free path, λ(0) is the ground state London penetration depth, and ξ(0) is the ground state coherence length.

Type/Chemical Composition
In Figure 2, we represent the same superconducting materials, but here we display the − ℎ vs.
dataset in a semi-log plot. To our best knowledge, the − ℎ vs. plot was first plotted by Pietronero et al. [13] in linear-linear scales. However, because the ratio for main families of superconductors is varied within four orders of magnitude (Table 1), and 0.4 ≤ − ℎ ≤ 3.0, it is more suitable to use the semi-log plot ( Figure 2). Pietronero et al. [13]. References for original data (Tθ, − ℎ , TF) can be found in Table 1.  Table 1.
In Figure 2, we represent the same superconducting materials, but here we display the λ e−ph vs. T θ T F dataset in a semi-log plot. To our best knowledge, the λ e−ph vs. T θ T F plot was first plotted by Pietronero et al. [13] in linear-linear scales. However, because the T θ T F ratio for main families of superconductors is varied within four orders of magnitude (Table 1), and 0.4 ≤ λ e−ph ≤ 3.0, it is more suitable to use the semi-log plot (Figure 2).  Table 1.
In Figure 2, we represent the same superconducting materials, but here we display the − ℎ vs.
dataset in a semi-log plot. To our best knowledge, the − ℎ vs. plot was first plotted by Pietronero et al. [13] in linear-linear scales. However, because the ratio for main families of superconductors is varied within four orders of magnitude (Table 1), and 0.4 ≤ − ℎ ≤ 3.0, it is more suitable to use the semi-log plot ( Figure 2). Pietronero et al. [13]. References for original data (Tθ, − ℎ , TF) can be found in Table 1.  [13]. References for original data (T θ , λ e−ph , T F ) can be found in Table 1. Figure 3, we represented the same superconducting materials, but here we displayed the T c vs. T θ T F dataset in a log-log plot. This type of plot was chosen because as T c , as T θ T F are varied within several orders of magnitude. Finally, in Figure 3, we represented the same superconducting materials, but here we displayed the vs. dataset in a log-log plot. This type of plot was chosen because as , as are varied within several orders of magnitude. can be found in Table 1.
It should be noted that, in both approaches, the electron-phonon coupling strength constant, − ℎ , was assumed to be − ℎ = 1.76, which is the average value of values calculated by first-principles calculations [55,56], and values extracted from experimental R(T) data [27].
It can be seen in Table 1 and Figure 1 that the calculated values for H3S, by two alternative approaches, are in a very good agreement with each other. To demonstrate the acceptable level of variation in values for the same material, in Table 1 and Figure 1 we present the results of the calculations for pure metals, where was calculated by the two approaches mentioned above and the use of experimental data reported by different research groups.
in HTS cuprates were calculated by the Equations (13) and (15), which do not require the knowledge of the electron-phonon coupling constants, − ℎ . This is despite Ledbetter et al. [7] reporting the so-called effective electron-phonon coupling strength,   Table 1.

Discussion
The family of near-room temperature superconductors (NRTS) is represented in Table 1 and Figure 1 by H 3 S (P = 155 GPa), SnH 12 (P = 190 Gpa), and La 1-x Nd x H 10 (x = 0.09, P = 180 Gpa). Two independent approaches were used to perform calculations in H 3 S:
It should be noted that, in both approaches, the electron-phonon coupling strength constant, λ e−ph , was assumed to be λ e−ph = 1.76, which is the average value of values calculated by first-principles calculations [55,56], and values extracted from experimental R(T) data [27].
It can be seen in Table 1 and Figure 1 that the calculated T F values for H 3 S, by two alternative approaches, are in a very good agreement with each other. To demonstrate the acceptable level of variation in T F values for the same material, in Table 1 and Figure 1 we present the results of the calculations for pure metals, where T F was calculated by the two approaches mentioned above and the use of experimental data reported by different research groups.
T F in HTS cuprates were calculated by the Equations (13) and (15), which do not require the knowledge of the electron-phonon coupling constants, λ e−ph . This is despite Ledbetter et al. [7] reporting the so-called effective electron-phonon coupling strength, λ e−ph,e f f , from which the effective mass can be deduced, m * = 1 + λ e−ph,e f f × m e .
In addition, it should be noted that for YBa 2 Cu 3 O 7 , Uemura [83] reported the relation [83]: m * m e = 2.5 (17) from which λ e−ph = 1.5 can be derived. Calculated values are in a reasonable agreement with experimental m * m e values reported by several research groups [108][109][110] in YBa 2 Cu 3 O 7-x . However, because the phenomenology of the electron-phonon mediated superconductivity cannot describe the superconducting state in cuprates, and the T θ for cuprates were taken as experimental values (see, for instance, report by Ledbetter et al. [7,88]), all cuprate superconductors are shown in Figures 1 and 3 and are not shown in Figure 2.
It should be mentioned that the result of the T F calculation in MATBG (Table 1), T F = 16.5 K, which was primarily based on the London penetration depth, λ(0) = 1860 nm, was deduced in Ref. [96] from the self-field critical current density, J c (s f , T), by the approach proposed by us [84]: The remarkable agreement of the deduced value, T F = 16.5 K, and the value reported in the original work on MATBG by Cao et al. [95], T F = 17 K, which was calculated based on normal state charge carriers density in MATBG, independently validates our primary idea [84] about the fundamental nature of the self-field critical current in weaklinks samples [84,85,111]. This concept was recently proven by Paturi and Huhtinen [112], who utilized the fact that the London penetration depth, λ(0), in real samples, depends on the mean free-path of charge carriers, l: where λ(0) is the effective penetration depth, and λ clean limit (0) is the penetration depth in samples, exhibiting a very long mean free-path, l ξ(0). Paturi and Huhtinen [112] varied l in YBa 2 Cu 3 O 7-x films and showed that the change in J c (st, T) satisfies Equations (18) and (19).
Materials, in which λ(0) was deduced by the mean temperature dependent self-field critical current density, J c (s f , T) (Equation (18)), have designation "J c (s f , T)" in Figures 1-3.
The MATBG does not show in Figure 2, because the derivation of λ e−ph cannot be performed by the used phenomenology: m * = 1 + λ e−ph × m e , because m * m e = 0.2 [96]; however, this material is shown in Figures 1 and 3, because λ e−ph is not required for these plots.
Returning back to hydrides, we need to note that Durajski [56] performed firstprinciples and studied the strength of the nonadiabatic effects in highly-compressed sulfur hydride and phosphorus hydride. Calculations show that the strength of the nonadiabatic effects can be quantified as moderately weak in comparison with the classical nonadiabatic superconductor SrTiO 3 . This is in a good agreement with our result (see Figure 3 and Table 1), that all deduced T θ T F values for NRTS are within the range of: Moreover, the classical nonadiabatic superconductor SrTiO 3 falls into the intermediate zone between unconventional and BCS superconductors; this is because this material exhibits T c T F = 0.0066, and by this criterion, SrTiO 3 is similar to the Laves phase materials, intermetallics, A-15 alloys, and Heusler alloys, which cannot be considered to be a correct manifestation of primary uniqueness for this nonadiabatic material.
More unexpectedly, a two dimensional LiC 6 (which is a lithium-doped graphene) superconductor falls into the BCS metals zone in the Uemura plot (Figure 1), despite the fact that this material exhibits reasonable strength in the nonadiabatic effects, T c T F = 0.15 [99]. However, in Figures 2 and 3, the outstanding separations of all nonadiabatic superconductors from their adiabatic and moderate nonadiabatic counterparts are clearly manifested.
By looking at the data in Figures 2 and 3, it is easy to recognize that 3 /4 (32 of 42) of the analyzed superconductors fall into a reasonably narrow band: Based on this, we proposed that the values in Equation (21) were used as empirical limits for the adiabatic superconductors ( T θ T F ≤ 0.025), moderate nonadiabatic superconductors (0.025 ≤ T θ T F ≤ 0.4), and strong nonadiabatic superconductors ( T θ T F ≥ 0.4). It also follows from our analysis that all strong nonadiabatic superconductors exhibit low superconducting transition temperatures, T c ≤ 1.2 K (Figure 3).

Conclusions
In this work, we proposed a new classification scheme to quantify the effects of nonadiabaticity in superconductors. By performing the analysis of experimental data for more than 40 superconductors, which represent the primary families of superconductors, we found that 3 4 of all analyzed superconductors fall into a narrow 0.025 ≤ T θ T F ≤ 0.4 band. Based on this, we proposed the taxonomy for the strength of the nonadiabatic effects in superconductors.