$d$-wave superconductivity as a model for diborides apart MgB$_2$

Recently, Pei et al. (arXiv2105.13250) reported that ambient pressure ${\beta}$-MoB$_2$ exhibits a phase transition to ${\alpha}$-MoB$_2$ (space group: $P6/mmm$) at pressure P~70 GPa and this high-pressure phase is a high-temperature superconductor exhibited $T_c=32 K$ at P~110 GPa. Despite ${\alpha}$-MoB$_2$ has the same crystalline structure as ambient pressure MgB$_2$2 and the $T_c$'s of ${\alpha}$-MoB$_2$ and MgB$_2$ are very close, the first principles calculations showed that in ${\alpha}$-MoB$_2$ the states near the Fermi level, ${\epsilon}_F$, are dominated by the $d$-electrons of Mo atoms, while in MgB$_2$ the $p$-orbitals of boron atomic sheets dominantly contribute to the states near the ${\epsilon}_F$. More recently, Hire et al. (arXiv2212.14869) reported that the $P6/mmm$-phase can be stabilized at ambient pressure in $Nb_{1-x}Mo_{x}B_{2}$ solid solutions, and these ternary alloys exhibit $T_c=8 K$. In addition, Pei et al. (Sci. China-Phys. Mech. Astron. 65, 287412 (2022)) showed that compressed WB$_2$ exhibits $T_c=15 K$ at P~121 GPa. Here, we analyzed experimental data reported for $P6/mmm$-phases of $Nb_{1-x}Mo_{x}B_{2}$ (x = 0.25; 1.0) and highly-compressed WB$_2$, and showed that these three phases exhibit $d$-wave superconductivity. We also deduced the gap-to-transition temperature ratio for these three phases. We found that $Nb_{0.75}Mo_{0.25}B_{2}$ exhibits high strength of nonadiabaticity, which is quantified by the ratio of $T_{\theta}/T_F=3.5$, which is by one order of magnitude exceeds the ratio in MgB$_2$, ${\alpha}$-MoB$_2$, WB$_2$, pnictides, cuprates, and highly-compressed hydrides.


d-wave superconductivity as a model for diborides apart MgB2
I. Introduction.
The discovery of near-room temperature superconductivity in highly compressed sulphur hydride by Drozdov et al [1] sparked theoretical and experimental studies of a variety of materials which potentially can exhibit a high-temperature superconductivity to be compressed at high pressure . This research field represents one of the most fascinating scientific exploration where advanced first principles calculations conjuncts with top world class of experimental studies [26][27][28][29][30][31][32][33][34][35][36][37][38][39].
One of the interesting results in this conjunctive exploration has been reported by Pei et al [40] who found that the stoichiometric compound MoB2 exhibits the phase transition from the -MoB2-phase (space group: 3 ̅ ) to -MoB2-phase (space group: 6/ ) at critical pressure P ~ 70 GPa. This high-pressure phase, -MoB2, exhibits the same crystalline structure as ambient pressure MgB2 and, what is the most intriguing experimental result reported by Pei et al [40], the -MoB2 phase is a high-temperature superconductor with = 32 (at P = 109.7 GPa), which is remarkably close to = 39 − 42 in MgB2 [41,42].
First principles calculations performed by Pei et al [40] showed that several bands in the -MoB2 crossing the Fermi level, , which causes the metallic type of conductivity in this phase. Pei et al [40] also showed the molybdenum d-orbitals (especially the dz2 orbital) have larger contributions than the boron p-orbitals near the . In overall, despite -MoB2 phase exhibits the same crystal structure as MgB2 and the superconducting transition temperature for these compounds are comparable, their electronic structures are different. For instance, the out-of-plane phonon mode of molybdenum ions are strongly coupled with molybdenum d-electrons near the in -MoB2 [40], while the in-plane boron-boron stretching mode in MgB2 interacts intensively with the σ-bond in the boron honeycomb lattice near the [40].
These results give a ground to expect that the -MoB2 phase can exhibit d-wave superconducting energy gap symmetry (or, at least, s+d-wave gap symmetry with significant d-wave component), which is different from the two-band s-wave MgB2.
Hire et al [44] also performed first principles calculation, measurements of the temperature dependent magnetoresistance ( , ), and specific heat measurements from which several parameters of Nb1-xMoxB2 (x = 0.25; 0.50; 0.75 and 0.9) superconductors (and, in particular, the Debye temperature, ) were determined.
Pei et al [45] and Lim et al [46] extended the range of superconducting diborides by the discovery of highly-compressed phase of WB2 (~15 at P~121 GPa) for which Pei et al [45] proposed space group: P63/mmc (which is distorted P6/mmm), while Lim et al [46] concluded that this highly-pressurized superconducting phase of WB2 formed by staking faulted P63/mmc-P6/mmm phase (which can be found to be similar to the stacking faulted 123-124 phases in Y-Ba-Cu-O system [47][48][49]).
Here, we performed detailed analysis of the magnetoresistance data reported by Pei et al [40], Hire et al [44], Pei et al [45], and showed that the P6/mmm-phases of Nb1-xMoxB2 (x = 0.25; 1.0) and WB2 (P=121.3 GPa) exhibit the d-wave superconducting gap symmetry. We also found that ambient pressure Nb1-xMoxB2 (x = 0.25) superconductors characterized by high strength of nonadiabaticity, which can be characterized by the ratio of = 3.5 (where is the Fermi temperature, which is by more than one order of magnitude exceeds the ratio in MgB2, -MoB2, WB2, pnictides, cuprates, and highly-compressed hydrides.

Temperature dependent upper critical field
Pei et al [40] in their Figure 2,d utilized several resistance criteria ( ) ( ) = 0.10, 0.50, 0.90 to derive the upper critical field, 2 ( ), from measured ( , , = 109.7 ) curves. By following general logic [56,58,59] that as low as possible resistance criterion should be in use, here we utilized the same criterion of ( ) ( ) = 0.10, as the one which was used to define the in Fig. 1 and by the lowest criterion to defined 2 ( ) by Pei et al [40].
where the superconducting energy gap, (T,), is given by [61,62,65]: where m(T) is the is the maximum amplitude of the k-dependent d-wave gap,  = 7/5 [65],  is the angle around the Fermi surface subtended at (, ) in the Brillouin zone (details can be found elsewhere [61,62]). ) reported by Pei et al [40] and data fits to s-wave (panel a) and d-wave (panel b) single-band models. Deduced parameters are (for both panels the critical temperature was fixed to the observed value of Tc = 28. The fit converged with a better quality (with the goodness of fit is 0.9842) (Fig. 2,b).
Deduced parameters are: ξ(0) = 6.2 (5) , It should be noted, that the accuracy of the extracted parameters is directly related to the sampling number of the measurement, and, thus, further increase in the accuracy in the deduced parameters, can be possible if more raw ( , ) data (especially, measured at low temperature, down to miliKelvin level) will be available.

The Fermi temperature and the strength of the nonadiabaticity
The Fermi temperature can be calculated by the equation [58]:

The electron-phonon coupling constant
Hire et al [44] in their Table 1 reported the Debye temperature for P6/mmm Nb1-xMoxB2 ( = 0.25) which was deduced from low-temperature specific heat measurements, = 625 . Follow the approach implemented in this study, we processed ( , = 0) data reported by Hire et al [44] by utilizing the resistance criterion of

The Fermi temperature and the strength of the nonadiabaticity
The substitution of deduced parameters in Eq. 9 returns the Fermi temperature = 180 ± 7 in P6/mmm-phase of Nb0.75Mo0.25B2. Calculated implies that this phase falls in unconventional superconductors band in the Uemura plot (Fig. 3), because this phase exhibits typical for many unconventional superconductors ratio of = 0.042.
However, what comes from our analysis and reported by Hire et al [44] the Debye temperature, that P6/mmm-phase of Nb0.75Mo0.25B2 superconductor exhibits strong nonadiabaticy, because the ratio: is well above typical range for moderate level of nonadiabaticity (0.025 ≤ ≤ 0.4) observed in majority of unconventional superconductors, including iron-based, cuprates and highly compressed hydrides [67] (Figs. 4,5).

The Debye temperature and the electron-phonon coupling constant
Pei et al [45] measured ( ) datasets for WB2 phase at = 121.3 which we fitted to Eq. 1 in Fig. 7. The fit converged at = 440 ± 1 and → ∞. From deduced we found − ℎ = 0.755, for which we utilized the criterion of ( ) ( ) = 0.18, which is based on the presence of the inflection of the transition ( , , = 121.3 ) which can be seen in Fig. 2(b,d)
And the fit quality R =0.9019 is not high. The fit to the d-wave gap symmetry model has a better quality (with the goodness of fit is 0.9986) (Fig. 8,b)

The Fermi temperature and the strength of the nonadiabaticity
The substitution of deduced parameters in Eq. 9 returns the Fermi temperature = 1679 ± 68 in WB2 (P = 121.3 GPa). Calculated implies that this phase falls in nearly conventional superconductors band in the Uemura plot (Fig. 3), because this phase exhibits reasonably low ratio of = 0.0077 ± 0.0003, while typical range for unconventional superconductors is 0.01 ≤ ≤ 0.05.
Also, this superconductor exhibits very moderate strength of nonadiabaticy, because the ratio: is typical for majority of high-temperature superconductors, including iron-based, cuprates and highly compressed hydrides [67] (Figs. 4,5).

Temperature dependent upper critical field
To show that our Bc2(T) model (Eqs where to reduce the number of free-fitting parameters, we implemented the restriction [80]: The deduced parameters for single band s-wave model (Fig. 9,a) contradict to each other, i.e. 2Δ(0) = 3.3 ± 0.1 (which is lower than the s-wave weak-coupling limit), while Δ = 2.3 ± 0.3 is much larger than the s-wave weak-coupling limit. The deduced ratio of 2Δ (0) = 7.1 ± 0.3 for d-wave model is nearly twice larger the d-wave weak-coupling limit of 2Δ (0) = 4.28, which is too large to be realistic value.

IV. Conclusions
In this work, we deduced primary superconducting parameters in three diborides, i.e.
P6/mmm phases of Nb1-xMoxB2 (x = 0.25; 1.0) and WB2. It was shown that these phases exhibit d-wave superconducting gap symmetry. We proposed that many fold suppression of the superconducting transition temperature (down to = 8 ) in Nb0.75Mo0.25B2, can be related to either strong nonadiabaticity in this phase (which exhibits the ratio = 3.5), either to the effect of the suppression in d-wave MoB2 superconductors by nonmagnetic impurity (which is Nb atoms).

Acknowledgement
The author thanks

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

Declaration of interests
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.