D-Wave Superconducting Gap Symmetry as a Model for Nb_1−xMo_xB₂ (x = 0.25; 1.0) and WB₂ Diborides

Abstract

Recently, Pei et al. (National Science Review 2023, nwad034, 10.1093/nsr/nwad034) reported that ambient pressure $\beta$-MoB₂ (space group: $R\overline{3}m$) exhibits a phase transition to $\alpha$-MoB₂ (space group: $P6/mmm$) at pressure P~70 GPa, which is a high-temperature superconductor exhibiting $T_c=32 \mathrm{K}$ at P~110 GPa. Although $\alpha$-MoB₂ has the same crystalline structure as ambient-pressure MgB₂ and the superconducting critical temperatures of $\alpha$-MoB₂ and MgB₂ are very close, the first-principles calculations show that in $\alpha$-MoB₂, the states near the Fermi level, ${\epsilon }_F$, are dominated by the d-electrons of Mo atoms, while in MgB₂, the p-orbitals of boron atomic sheets dominantly contribute to the states near the ${\epsilon }_F$. Recently, Hire et al. (Phys. Rev. B 2022, 106, 174515) reported that the $P6/mmm$-phase can be stabilized at ambient pressure in Nb_1−xMo_xB₂ solid solutions, and that these ternary alloys exhibit $T_c~8 \mathrm{K}$. Additionally, Pei et al. (Sci. China-Phys. Mech. Astron. 2022, 65, 287412) showed that compressed WB₂ exhibited $T_c~15 \mathrm{K}$ at P~121 GPa. Here, we aimed to reveal primary differences/similarities in superconducting state in MgB₂ and in its recently discovered diboride counterparts, Nb_1−xMo_xB₂ and highly-compressed WB₂. By analyzing experimental data reported for P6/mmm-phases of Nb_1−xMo_xB₂ (x = 0.25; 1.0) and highly compressed WB₂, we showed that these three phases exhibit d-wave superconductivity. We deduced $\frac{2{\mathsf{\Delta }}_m\left(0\right)}{k_B{T}_c}=4.1\pm 0.2$ for $\alpha$-MoB₂, $\frac{2{\mathsf{\Delta }}_m\left(0\right)}{k_B{T}_c}=5.3\pm 0.1$ for Nb_0.75Mo_0.25B₂, and $\frac{2{\mathsf{\Delta }}_m\left(0\right)}{k_B{T}_c}=4.9\pm 0.2$ for WB₂. We also found that Nb_0.75Mo_0.25B₂ exhibited high strength of nonadiabaticity, which was quantified by the ratio of $\frac{T_{\theta }}{T_F}=3.5$, whereas MgB₂, α-MoB₂, and WB₂ exhibited $\frac{T_{\theta }}{T_F}~0.3$, which is similar to the $\frac{T_{\theta }}{T_F}$ in pnictides, A15 alloys, Heusler alloys, Laves phase compounds, cuprates, and highly compressed hydrides.

1. Introduction

There is an experimental quest for high-temperature superconductivity in compounds based on lightweight elements which exhibit high Debye temperature, $T_{\theta }$. Thus, in accordance with the theory of the electron–phonon mediated superconductivity, these compounds can have a high transition temperature, $T_c$. This work started in the 1970s [1,2]. These studies covered hydrides [1] and borides [2]. Surprisingly, Cooper et al. [2] performed detailed studies of Mo-diborides, Nb-diborides, and ternary borides R_2−xA_xB₅ (R = Mo, Nb, A = transition metal), while Fisk [3] reported on discovery of 40 superconducting phases in rare earth and transition metals borides. The diboride of magnesium was first studied on its superconducting properties in 2001 [4].

The discovery of near-room temperature superconductivity in highly compressed sulfur hydride by Drozdov et al. [5] sparked theoretical and experimental studies of a variety of materials that can potentially exhibit high-temperature superconductivity to be compressed at high pressure [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. This research field represents one of the most fascinating scientific explorations in which advanced first-principles calculations are combined with the top world class of experimental studies [30,31,32,33,34,35,36,37,38,39,40,41,42,43].

In this regard, the quest for high-temperature superconductivity in highly compressed borides seems reasonable. One of the interesting results of this conjunctive exploration was reported by Pei et al. [44], who found that the stoichiometric compound MoB₂ exhibits a phase transition from the $\beta$-MoB₂-phase (space group: $R\overline{3}m$) to the $\alpha$-MoB₂-phase (space group: $P6/mmm$) at a critical pressure of P~70 GPa. This high-pressure phase, $\alpha$-MoB₂, exhibits the same crystalline structure as the ambient-pressure MgB₂. The most intriguing experimental result reported by Pei et al. [44] was that the $\alpha$-MoB₂ phase is a high-temperature superconductor with $T_c=32 \mathrm{K}$ (at P = 109.7 GPa); this value is remarkably close to $T_c=39-42 \mathrm{K}$ in MgB₂ [4,45].

First-principles calculations performed by Pei et al. [44] showed that several bands in the $\alpha$-MoB₂ cross the Fermi level, ${\epsilon }_F$, which causes the metallic type of conductivity in this phase. Pei et al. [44] also showed that molybdenum d-orbitals (especially the d_z2 orbital) have larger contributions than boron p-orbitals near the ${\epsilon }_F$. Overall, although the $\alpha$-MoB₂ phase exhibits the same crystal structure as MgB₂ and the superconducting transition temperatures for these compounds are comparable, their electronic structures are different. For instance, the out-of-plane phonon mode of molybdenum ions is strongly coupled with molybdenum d-electrons near the ${\epsilon }_F$ in $\alpha$-MoB₂ [44], whereas the in-plane B-B stretching mode in MgB₂ interacts intensively with the σ-bond in the boron honeycomb lattice near the ${\epsilon }_F$ [44]. Pei et al. [44] also calculated the electron–phonon coupling constant, ${\lambda }_{e-ph}=1.60$, in $\alpha$-MoB₂ at $P=90 \mathrm{GPa}$. Similar findings, including ${\lambda }_{e-ph}=1.60$, were reported by Quan et al. [46], who performed first-principles calculations for a highly pressurized $\alpha$-MoB₂ phase.

These results establish a ground to expect that the $\alpha$-MoB₂ phase can exhibit d-wave superconducting energy gap symmetry (or, at least, s+d-wave gap symmetry with a significant d-wave component), which is different from the two-band s-wave MgB₂.

More recently, Hire et al. [47] showed that the $P6/mmm$-phase can be stabilized at ambient pressure in Nb_1−xMo_xB₂ (x = 0.25, 0.50, 0.75, and 0.9) solid solutions. Despite the superconducting transition temperature in Nb_1−xMo_xB₂ (x = 0.25, 0.50, 0.75 and 0.9) being significantly lower (i.e., $T_c=\left(6.5-8.1\right) \mathrm{K}$ [47]), these values are still high enough to suggest that the same pairing mechanism emerges in ambient pressure superconductors Nb_1−xMo_xB₂ and highly-pressurized $\alpha$-MoB₂.

Hire et al. [47] also performed first-principles calculations and measured the temperature-dependent magnetoresistance, $R\left(T,B\right)$, and specific heat, from which several parameters of Nb_1−xMo_xB₂ (x = 0.25, 0.50, 0.75, and 0.9) superconductors (in particular, the Debye temperature, $T_{\theta }$) were determined.

Pei et al. [48] and Lim et al. [49] extended the family of superconducting diborides by the discovery of the highly compressed phase of WB₂ ($T_c~15 \mathrm{K}$ at P~121 GPa) for which Pei et al. [48] proposed the space group: P6₃/mmc (which is distorted P6/mmm), while Lim et al. [49] concluded that this highly pressurized superconducting phase of WB₂ formed by stacking faulted P6₃/mmc-P6/mmm phases (which can be found to be similar to the stacking faulted 123–124 phases in the Y-Ba-Cu-O system [50,51,52]).

Here, we aimed to determine the difference in the superconducting gap symmetry and other superconducting parameters in MgB₂ and in the recently discovered Nb_1−xMo_xB₂ (x = 0.25; 1.0) and WB₂, which might originate from the difference in the band structure of these materials. To do this we performed a detailed analysis of the magnetoresistance data reported by Pei et al. [44], Hire et al. [47], and Pei et al. [48] and showed that the P6/mmm-phases of Nb_1−xMo_xB₂ (x = 0.25, 1.0) and WB₂ (P = 121.3 GPa) exhibit d-wave superconducting gap symmetry. We also found that ambient pressure Nb_1−xMo_xB₂ (x = 0.25) superconductors characterized by high strength of nonadiabaticity, which can be characterized by the ratio of $\frac{T_{\theta }}{T_F}=3.5$ (where $T_F$ is the Fermi temperature, which exceeds the $\frac{T_{\theta }}{T_F}$ ratio in MgB₂, α-MoB₂, WB₂, pnictides, A15 alloys, Heusler alloys, Laves phase compounds, cuprates, and highly-compressed hydrides by more than one order of magnitude.

2. Utilized Models

The Debye temperature, $T_{\theta }$, can be deduced from the fit of the experimentally measured temperature-dependent resistance curve, $R\left(T\right)$, to the Bloch–Grüneisen (BG) equation [53,54]. In many reports, the classical BG approach was advanced by introducing the so-called saturation resistance [55,56,57,58,59,60]:

$$R\left(T\right)=\frac{1}{\frac{1}{R_{sat}}+\frac{1}{R_0+A{\left(\frac{T}{T_{\theta }}\right)}^5{{\displaystyle \int }}_0^{\frac{T_{\theta }}{T}}\frac{x^5}{\left(e^x-1\right)\left(1-e^{-x}\right)}dx}},$$

(1)

where $R_{sat}$, $R_0$, $T_{\theta }$, and $A$ are free fitting parameters. From the deduced $T_{\theta }$ and measured $T_c$ (which we defined by as strict as practically possible resistance criterion of $\frac{R\left(T\right)}{R_{norm}}\to 0$, where $R_{norm}$ is the normal state resistance at the onset of the superconducting transition (ee details in [59]), the electron–phonon coupling constant, ${\lambda }_{e-ph}$, can be calculated as the unique root of the advanced McMillan equation [59]:

$$T_c=\left(\frac{1}{1.45}\right)\times T_{\theta }\times e^{-\left(\frac{1.04\left(1+{\lambda }_{e-ph}\right)}{{\lambda }_{e-ph}-{\mu }^{*}\left(1+0.62{\lambda }_{e-ph}\right)}\right)}\times f_1\times f_2^{*},$$

(2)

where

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

(3)

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

(4)

where ${\mu }^{*}$ is the Coulomb pseudopotential parameter, which we assumed (following the approach proposed in [44,47,49]) to be ${\mu }^{*}=0.13$ for Nb_1−xMo_xB₂ (x = 0.25; 1.0) and WB₂.

By following the general logic [59,61,62] that a resistance criterion with the smallest possible value should be in use, we utilized the same criterion of $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.10$, as the one that was used to define the $T_c$ and the $B_{c2}\left(T\right)$. The temperature-dependent upper critical field, $B_{c2}\left(T\right)$, is described by

$$B_{c2}\left(T\right)=\frac{{\varphi }_0}{2·\pi ·{\xi }^2\left(T\right)},$$

(5)

where ${\varphi }_0$ is the superconducting magnetic flux quantum and $\xi \left(T\right)$ is the coherence length. $B_{c2}\left(T\right)$ datasets were fitted to the equation for the temperature-dependent upper critical field for s-wave superconductors [61,62,63]:

$$B_{c2}\left(T\right)=\frac{{\varphi }_0}{2·\pi ·{\xi }^2\left(0\right)}{\left(\frac{1.77-0.43{\left(\frac{T}{T_c}\right)}^2+0.07{\left(\frac{T}{T_c}\right)}^4}{1.77}\right)}^2\times \left[1-\frac{1}{2k_{B}T}{{\displaystyle \int }}_0^{\infty }\frac{d\epsilon }{cos{h}^2\left(\frac{\sqrt{{\epsilon }^2+{\Delta }^2\left(T\right)}}{2k_{B}T}\right)}\right],$$

(6)

where the amplitude of the temperature-dependent superconducting gap, $\mathsf{\Delta }\left(T\right)$, is given by [64,65]

$$\Delta \left(T\right)=\Delta \left(0\right)\times \tan \mathrm{h}\left[\frac{\pi k_B{T}_c}{\Delta \left(0\right)}\sqrt{\eta \frac{\Delta C}{\gamma T_c}\left(\frac{T_c}{T}-1\right)}\right],$$

(7)

where $\eta =2/3$ for s-wave superconductors, $\gamma$ is the Sommerfeld constant, and $k_B$ is Boltzmann’s constant.

$B_{c2}\left(T\right)$ datasets were also fitted to the equation for the temperature-dependent upper critical field for d-wave superconductors [61,62,63], where the amplitude of the temperature-dependent superconducting gap, $\mathsf{\Delta }\left(T\right)$, is given by [64,65]:

$$\begin{array}{c}{B}_{c2}\left(T\right)=\frac{{\varphi }_0}{2·\pi ·{\xi }^2\left(0\right)}{\left(\frac{1.77-0.43{\left(\frac{T}{T_c}\right)}^2+0.07{\left(\frac{T}{T_c}\right)}^4}{1.77}\right)}^2[1-\frac{1}{2·k_B·T}·\\ {{\displaystyle \int }}_0^{2\pi }co{s}^2\left(\theta \right)·\left({{\displaystyle \int }}_0^{\infty }\frac{d\epsilon }{cos{h}^2\left(\frac{\sqrt{{\epsilon }^2+{\Delta }^2\left(T,\theta \right)}}{2·k_B·T}\right)}\right)·d\theta ],\end{array}$$

(8)

where the superconducting energy gap, $\mathsf{\Delta }\left(T,\theta \right)$, is given by [64,65,66,67]:

$$\Delta \left(T,\theta \right)=\cos \left(2\theta \right)\times {\Delta }_m\left(0\right)\times \tan \mathrm{h}\left[\frac{\pi k_B{T}_c}{\Delta \left(0\right)}\sqrt{\eta \frac{\Delta C}{\gamma T_c}\left(\frac{T_c}{T}-1\right)}\right],$$

(9)

where ${\Delta }_m\left(0\right)$ is the maximum amplitude of the k-dependent d-wave gap, $\eta =7/5$ [68], $\theta$ is the angle around the Fermi surface subtended at $\left(\pi ,\pi \right)$ in the Brillouin zone (details can be found elsewhere [64,65]).

The Fermi temperature, $T_F$, was calculated using the equation [61]:

$$T_F=\frac{{\pi }^2{m}_e}{8·k_B}\times \left(1+{\lambda }_{e-ph}\right)\times {\xi }^2\left(0\right)\times {\left(\frac{2{\Delta }_m\left(0\right)}{\hslash }\right)}^2,$$

(10)

where $m_e$ is the bare electron mass, $\hslash$ is the reduced Planck’s constant, and the other parameters were deduced above.

3. Results

3.1. P6/mmm $\alpha$-MoB₂ ($P=109.7 \mathit{GPa}$)

The fits of $R\left(T\right)$ datasets, measured for the $\alpha$-MoB₂ phase at $P=91.4 \mathrm{and} 109.7 \mathrm{GPa}$ [44] of Equation (1), together with the deduced $R_{sat}$, $T_{\theta }$, and ${\lambda }_{e-ph}$, are shown in Figure 1 (where we utilized $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.10$ criterion to define $T_c$ because the same criterion was used by Pei et al. [44] to define the upper critical field in the same $\alpha$-MoB₂ sample).

The deduced ${\lambda }_{e-ph}\left(91.4 \mathrm{GPa}\right)=1.42$ is in good agreement with the value calculated by first-principles calculations, ${\lambda }_{e-ph}\left(90 \mathrm{GPa}\right)=1.60$ [44,46].

In Figure 2a, the $B_{c2}\left(T\right)$ dataset is fitted to the equation for the temperature-dependent upper critical field for s-wave superconductors (Equations (6) and (7)). However, the deduced $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=2.3\pm 0.1$ (Figure 2a) is too low to be attributed to s-wave superconductivity, for which the weak-coupling limit is $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=3.53$ [66,67]. Additionally, the fit quality was low (coefficient of determination = 0.8267).

Subsequently, we fitted the temperature-dependent upper critical field data to the d-wave gap symmetry model. The fit converged with a better quality (goodness of fit of 0.9842) (Figure 2b). The deduced parameters are $\mathsf{\xi }\left(0\right)=6.2\left(5\right) \mathrm{nm}$, $\mathsf{\Delta }\left(0\right)=5.0\pm 0.2 \mathrm{meV}$, $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=4.1\pm 0.2$, $\frac{\mathsf{\Delta }C}{\gamma T_c}=0.8\pm 0.1$. Considering that the weak coupling limits for d-wave superconductors [64,65,66] are $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=4.28$ and $\frac{\mathsf{\Delta }C}{\gamma T_c}=0.995$, we can conclude that the deduced parameters in $\alpha$-MoB₂ ($P=109.7 \mathrm{GPa}$) superconductor are within the weak-coupling values for d-wave superconductors.

It should be noted that the accuracy of the extracted parameters is directly related to the sampling number of the measurement. Thus, further increase in the accuracy of the deduced parameters is possible if more raw $R\left(T,B\right)$ data (especially, measured at low temperatures, down to the milliKelvin level) are available.

From the deduced parameters, one can calculate the Fermi temperature $T_F=1756 \pm 25 \mathrm{K}$. The calculated $T_F$ implies that the P6/mmm $\alpha$-MoB₂ ($P=109.7 \mathrm{GPa}$) phase falls in the unconventional superconductor band in the Uemura plot (Figure 3) because this phase is typical for many unconventional superconductors (for instance, iron-based, cuprates, and hydrogen-rich superconductors) ratio of $\frac{T_c}{T_F}=0.016$. Raw data for this plot were reported by many research groups (Refs. [68,69,70,71,72,73,74,75,76,77,78]).

In addition, we found that the P6/mmm $\alpha$-MoB₂ ($P=109.7 \mathrm{GPa}$) phase exhibits a similar level of nonadiabaticy ($\frac{T_{\theta }}{T_F}=0.18\pm 0.02$) to iron-based, cuprates, and hydrogen-rich superconductors [69,70] (Figure 4 and Figure 5).

3.2. Ambient Pressure P6/mmm Nd_0.75Mo_0.25B₂

In Table 1 of Hire et al.’s work [24], they reported the Debye temperature for P6/mmm Nb_1−xMo_xB₂ ($x=0.25$), which was deduced from low-temperature specific heat measurements, $T_{\theta }=625 \mathrm{K}$. Following the approach implemented in this study, we processed $R\left(T,B=0\right)$ data reported by Hire et al. [24] by utilizing the resistance criterion of $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.015$. We deduced $T_{c,0.015}=7.2 \mathrm{K}$, from which ${\lambda }_{e-ph}=0.573$ was calculated using Equations (2)–(4).

In Figure 6 of Hire et al. [47]’s work, they reported $R\left(T,B\right)$ data, which we processed by utilizing the resistance criterion of $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.015$. We deduced the $B_{c2}\left(T\right)$ dataset. The fits of this dataset to the s-wave (Equations (6) and (7)) and d-wave model (Equations (8) and (9)) are shown in Figure 6.

The deduced parameters for s-wave (Figure 6a) contradict each other, i.e., $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=3.18\pm 0.15$ (which is lower than the s-wave weak-coupling limit is $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=3.53$ [66,67]). The deduced $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.62\pm 0.19$ is larger than the s-wave weak-coupling limit of $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.43$. The fit quality is not high and has a coefficient of determination of 0.9534.

The fit to the d-wave gap symmetry model has a better quality (with a goodness of fit of 0.9959) (Figure 6b). The deduced parameters are $\mathsf{\xi }\left(0\right)=6.2\left(5\right) \mathrm{nm}$, $\mathsf{\Delta }\left(0\right)=1.65\pm 0.05 \mathrm{meV}$, $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=5.3\pm 0.1$, and $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.13\pm 0.03$; these values characterize the material as being a moderately strong coupled d-wave superconductor (considering that the weak coupling limits for d-wave superconductors [64,65,66] are $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=4.28$ and $\frac{\mathsf{\Delta }C}{\gamma T_c}=0.995$). It should be noted that analyzed experimental $B_{c2}\left(T\right)$ dataset has six raw data points which cover the $0.2\le \frac{T}{T_c}\le 1.0$ range. More experimental $B_{c2}\left(T\right)$ data measured at wider temperature ranges can be used to deduce primary superconducting parameters of the Nb_0.75Mo_0.25B₂ with better accuracy.

By the substituting the deduced parameters in Equation (10), the Fermi temperature can be obtained: $T_F=180 \pm 7 \mathrm{K}$ in P6/mmm-phase of Nb_0.75Mo_0.25B₂. The calculated $T_F$ implies that this phase falls in the unconventional superconductors band in the Uemura plot (Figure 3) because this phase is typical for many unconventional superconductor ratios of $\frac{T_c}{T_F}=0.042$.

However, what comes from our analysis and reported by Hire et al. [47] is the Debye temperature: the P6/mmm-phase of Nb_0.75Mo_0.25B₂ superconductor exhibits strong nonadiabaticy, because the ratio

$$0.4\ll \frac{T_{\theta }}{T_F}=3.5\pm 0.3,$$

(11)

is well above the typical range for the moderate level of nonadiabaticity ($0.025\le \frac{T_{\theta }}{T_F}\le 0.4$) observed in the majority of unconventional superconductors, including iron-based, cuprates, and highly compressed hydrides [70] (Figure 4 and Figure 5).

3.3. P6₃/mmc WB₂ (P = 121.3 GPa)

Pei et al. [48] measured the $R\left(T\right)$ dataset for the WB₂ phase at $P=121.3 \mathrm{GPa}$, which was fitted to Equation (1) in Figure 7. The fit converged at $T_{\theta }=440 \pm 1 \mathrm{K}$ and $R_{sat}\to \infty$. From the deduced $T_{\theta }$, we found ${\lambda }_{e-ph}=0.755$, for which we utilized the criterion of $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.18$, which is based on the presence of the inflection point in the $R\left(T,B,P=121.3 \mathrm{GPa}\right)$, as shown in Figure 2b,d of Ref. [48].

By utilizing the resistance criterion of $\frac{R\left(T\right)}{R_{norm}\left(T\right)}=0.18$ for $R\left(T,B\right)$ data reported in Figure 2d by Pei et al. [48], we deduced the $B_{c2}\left(T\right)$ dataset for WB₂ (P = 121.3 GPa). The fit of the $B_{c2}\left(T\right)$ dataset to the s-wave (Equations (6) and (7)) and d-wave models (Equations (8) and (9)) are shown in Figure 8.

The deduced parameters for s-wave (Figure 8a) contradict to each other, i.e., $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=2.8\pm 0.1$ (which is lower than the s-wave weak-coupling limit of $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=3.53$ [66,67]), while the deduced $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.6\pm 0.4$ is slightly larger than the s-wave weak-coupling limit of $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.43$. The fit quality is not high and has a coefficient of determination of 0.9019.

The fit to the d-wave gap symmetry model has a better quality (with a goodness of fit of 0.9986) (Figure 8b). The deduced parameters are $\mathsf{\xi }\left(0\right)=13.0 \mathrm{nm}$, $\mathsf{\Delta }\left(0\right)=2.58\pm 0.02 \mathrm{meV}$, $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=4.9\pm 0.1$, $\frac{\mathsf{\Delta }C}{\gamma T_c}=1.19\pm 0.07$. The parameters characterize the material as being a moderately strong coupled d-wave superconductor (considering that the weak coupling limits for d-wave superconductors [64,65,66] are $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=4.28$ and $\frac{\mathsf{\Delta }C}{\gamma T_c}=0.995$).

By substituting the deduced parameters in Equation (10), a Fermi temperature of $T_F=1679 \pm 68 \mathrm{K}$ in WB₂ (P = 121.3 GPa) is calculated. The calculated $T_F$ implies that this phase falls in the nearly conventional superconductors band in the Uemura plot (Figure 3), because this phase exhibits a reasonably low ratio of $\frac{T_c}{T_F}=0.0077\pm 0.0003$, while the typical range for unconventional superconductors is $0.01\le \frac{T_c}{T_F}\le 0.05$.

This superconductor also exhibits a very moderate strength of nonadiabaticy, because the ratio:

$$0.025<\frac{T_{\theta }}{T_F}=0.26\pm 0.01<0.4,$$

(12)

is typical for the majority of high-temperature superconductors, including iron-based, A15 alloys, Heusler alloys, Laves phase compounds, cuprates, and highly compressed hydrides [70] (Figure 4 and Figure 5).

3.4. P6/mmm MgB₂

To demonstrate that the B_c2(T) model (Equations (6)–(9) [61,62,63,74]) can be considered as an alternative model to extract primary superconducting parameters from $R\left(T,B\right)$ datasets (while the B_c2(T) definition criterion is $\frac{R\left(T\right)}{R_{norm}\left(T\right)}\to 0$) in addition to the widely used Werthamer–Helfand–Hohenberg model [79,80], we showed B_c2(T) data in Figure 9. The data were reported by Zehetmayer et al. [81] for single crystal MgB₂ and data fits to the single band s-wave (panel a, Equations (6) and (7)), the single band d-wave (panel b, Equations (8) and (9)), and the so-called two-band α-model [80] under the assumption of s-wave gap symmetry for both bands (panel c) [80,81]:

$$B_{c2,total}\left(T\right)=\alpha \times B_{c2,band1}\left({\xi }_{total}\left(0\right),T\right)+\left(1-\alpha \right)\times B_{c2,band2}\left({\xi }_{total}\left(0\right),T\right),$$

(13)

To reduce the number of free-fitting parameters, we implemented the restriction [82]:

$$T_{c1}=T_{c2},$$

(14)

$$\frac{\Delta C_1}{{\gamma }_1{T}_{c1}}=\frac{\Delta C_2}{{\gamma }_2{T}_{c2}}.$$

(15)

The deduced parameters for the single band s-wave model (Figure 9a) contradict each other, that is, $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}=3.3\pm 0.1$ (which is lower than the s-wave weak-coupling limit). $\frac{\mathsf{\Delta }C}{\gamma T_c}=2.3\pm 0.3$ is much larger than the s-wave weak-coupling limit. The deduced ratio of $\frac{2{\mathsf{\Delta }}_m\left(0\right)}{k_B{T}_c}=7.1\pm 0.3$ for the d-wave model is nearly two times as large as the d-wave weak-coupling limit of $\frac{2{\mathsf{\Delta }}_m\left(0\right)}{k_B{T}_c}=4.28$, which is too large to be a realistic value.

However, the parameters deduced for the two-band α-model, $\alpha =0.77\pm 0.06$, $\frac{2{\mathsf{\Delta }}_1\left(0\right)}{k_B{T}_c}=4.1\pm 0.3$, and $\frac{2{\mathsf{\Delta }}_2\left(0\right)}{k_B{T}_c}=1.7\pm 0.2$, are in good agreement with the values deduced for MgB₂ by other techniques [83], in particular, by point contact spectroscopy [84].

4. Discussion

In the consideration above, we calculated the electron–phonon coupling constant, ${\lambda }_{e-ph}$, with the assumption that diborides exhibit the Coulomb pseudopotential parameter, ${\mu }^{*}=0.13$. The latter value is typical value for s-wave superconductors [67]. While our analysis of the upper critical field, B_c2(T), showed that the materials exhibited d-wave gap symmetry, it is useful to show the variation in ${\lambda }_{e-ph}$ calculated in the assumption of d-wave superconductivity. Santi et al. [85] reported that d-wave superconductors exhibit much lower ${\mu }^{*}$ values in comparison with s-wave superconductors. In

Table 1. Calculated the electron–phonon coupling constant, ${\lambda }_{e-ph}$, for assumed ${\mu }^{*}=0.00;0.05;0.10;\mathrm{and} 0.13$ for studied diboride compounds α-MoB₂, Nb_0.75Mo_0.25B₂, and WB₂.

Compound	Tθ (K)	Tc (K)	Assumed μ*	λe−ph
α-MoB2	321	28.2	0.00	0.935
(109.7 GPa)			0.05	1.10
			0.10	1.29
			0.13	1.41
Nb1−xMoxB2	625	7.2	0.00	0.337
(x = 0.25)			0.05	0.422
			0.10	0.514
			0.13	0.573
WB2	440	12.5	0.00	0.475
(121.3 GPa)			0.05	0.575
			0.10	0.685
			0.13	0.755

, we listed calculated ${\lambda }_{e-ph}$ values for all studied dibories (apart MgB₂) in accordance with Equations (2)–(4), with the assumption of ${\mu }^{*}=0.00;0.05;0.10;\mathrm{and} 0.13$.

The P6/mmm-phase of Nb_0.75Mo_0.25B₂ exhibits pronounced nonadiabaticity, $\frac{T_{\theta }}{T_F}=3.5$. This value is well above an empirical border, $\frac{T_{\theta }}{T_F}\cong 0.4$. The majority of conventional and unconventional superconductors are located below this value (Figure 4 and Figure 5). We can propose that the strength of the nonadiabaticity is a primary reason for the relatively low T_c in this material in comparison with other diboride counterparts. A good support for this hypothesis can be seen in Figure 5, where the T_c suppression within four dibories is linked to the increase in the strength of the nonadiabaticity. It can also be seen in Figure 5 that no materials simultaneously exhibit $T_c>10 \mathrm{K}$ and $\frac{T_{\theta }}{T_F}>0.4$.

Another explanation for the relatively low T_c in Nb_0.75Mo_0.25B₂ is the Abrikosov–Gor’kov [86], Anderson [87], and Openov [88,89] theory of dirty superconductors. The theory established that impurities with magnetic moments suppress the superconducting transition temperature, if the material exhibits s-wave superconductivity. However, magnetic impurities not affect the superconducting transition temperature in d-wave superconductors. From other hand, non-magnetic impurities cause the suppression of transition temperature in d-wave superconductors, and these impurities not affect the s-wave superconductors transition temperature.. Considering that the (Nb,Mo)-(0001) planes in P6/mmm-phase have chemical atomic disorder, because Hire et al. [47] did not report any evidence for the atomic ordering within Nb-Mo atoms in the (0001) planes, it appears that the T_c suppression in Nb_0.75Mo_0.25B₂ (and in all materials in the Nb_1−xMo_xB₂ (x = 0.25; 0.50; 0.75 and 0.9) system) can be interpreted as T_c suppression in d-wave MoB₂ superconductors by nonmagnetic impurity—Nb/Mo atoms. However, we need to note that NbB₂ and MoB₂ are non-superconductors and these compounds exhibit different crystalline structures ($P6/mmm$ and $R\overline{3}m$, respectively). Thus, the influence of the Nb/Mo atoms composition in (0001) planes on band structure and phonon spectra required more detailed experimental and first-principles calculation studies.

To consider the problem of the superconducting gap symmetry in diborides in a more general context, we should mention that there are several theoretical possibilities for the interplay between s- and d-wave symmetries for different compounds within the same group of superconductors, and even for the same compound at different condition. For instance, we can mention the infinite layer nickelates, where both gap symmetries were observed in experiment [90,91,92]. Wang et al. [93] reported the theoretical consideration that a very delicate boundary between a realization for one of two symmetries has been established.

Considering that all diborides considered here exhibit a layered structure with alternative atomic layers along the c-axis, one can expect a similarity between considered diborides and other anisotropic superconductors. For instance, in one of the seminal papers regarding cuprates (Uemura et al. [94]), the muon spin relaxation ($\mu SR$) data for Tl-based cuprates (showed in their Figure 1 [94]) can be interpreted exclusively as data supporting s-wave superconducting energy gap symmetry in cuprates. However, four years later, Uemura et al. [95] reported more extended $\mu SR$ data for Tl-based cuprates, where temperature-dependent superfluid data for samples with low doping states can be still interpreted within s-wave symmetry, while low temperature data for overdoped samples are typical for d-wave linear dependence on temperature.

In addition, while our consideration is mainly focused on highly pressurized materials, we can mention another class of layered superconductors: iron-based superconductors. In these superconductors, the interplay between s- and d-wave gap symmetries was observed for the same compound in experiment. For instance, Guguchia et al. [96] reported that, at a pressure of several gigapascals, Ba_0.65Rb_0.35Fe₂As₂ (a two-band s-wave superconductor) exhibits a transition into a d-wave superconductor: “…hydrostatic pressure promotes the appearance of nodes in the superconducting gap…” [96].

Another purely theoretical possibility exists for a crossover between the electron–phonon and the electron–plasmon mediated pairing in layered superconductors. This possibility was recently proposed by in ’t Veld et al. [97]. This is another possibility that shows how high-pressure (which, as a rule, changes the screening length in the compound) can induce the change in the pairing symmetry from s-wave (which is widely considered to be a consequence of the electron–phonon interaction) to nodal symmetry (which is widely accepted to be attributed to non-electron-phonon mediated pairing).

In overall, our analysis of experimental data on dibories—apart from MgB₂—showed that d-wave gap symmetry can explain experimental data with much better consistency. However, theoretical understanding of this result is still ongoing.

5. Conclusions

The field of experimental and theoretical studies of materials with strongly correlated charge carriers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,46,47,48,49,57,59,60,61,74,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116], including diborides [44,46,47,48,49,107,108], is experiencing a boom. In this work, we deduced the primary superconducting parameters of highly compressed diborides: the P6/mmm phases of MoB₂ and WB₂ and ambient pressure superconductors Nd_0.75Mo_0.25B₂. It was shown that these the compounds exhibit d-wave superconducting gap symmetry. We proposed that the suppression of the superconducting transition temperature (down to $T_c=8 \mathrm{K}$) in Nb_0.75Mo_0.25B₂ can be either related to strong nonadiabaticity in this phase (which exhibits the ratio $\frac{T_{\theta }}{T_F}=3.5$) or to the effect of the $T_c$ suppression in d-wave MoB₂ superconductors by nonmagnetic impurities (Nb/Mo atoms).