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D-Wave Superconducting Gap Symmetry as a Model for Nb_{1−x}Mo_{x}B_{2} (x = 0.25; 1.0) and WB_{2} Diborides

^{1}

^{2}

## Abstract

**:**

**2023**, nwad034, 10.1093/nsr/nwad034) reported that ambient pressure $\beta $-MoB

_{2}(space group: $R\overline{3}m$) exhibits a phase transition to $\alpha $-MoB

_{2}(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 $-MoB

_{2}has the same crystalline structure as ambient-pressure MgB

_{2}and the superconducting critical temperatures of $\alpha $-MoB

_{2}and MgB

_{2}are very close, the first-principles calculations show 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 ${\epsilon}_{F}$. 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−x}Mo

_{x}B

_{2}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 WB

_{2}exhibited ${T}_{c}~15\mathrm{K}$ at P~121 GPa. Here, we aimed to reveal primary differences/similarities in superconducting state in MgB

_{2}and in its recently discovered diboride counterparts, Nb

_{1−x}Mo

_{x}B

_{2}and highly-compressed WB

_{2}. By analyzing 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}, 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 $-MoB

_{2}, $\frac{2{\mathsf{\Delta}}_{m}\left(0\right)}{{k}_{B}{T}_{c}}=5.3\pm 0.1$ for Nb

_{0.75}Mo

_{0.25}B

_{2}, and $\frac{2{\mathsf{\Delta}}_{m}\left(0\right)}{{k}_{B}{T}_{c}}=4.9\pm 0.2$ for WB

_{2}. We also found that Nb

_{0.75}Mo

_{0.25}B

_{2}exhibited high strength of nonadiabaticity, which was quantified by the ratio of $\frac{{T}_{\theta}}{{T}_{F}}=3.5$, whereas MgB

_{2}, α-MoB

_{2}, and WB

_{2}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

_{2−x}A

_{x}B

_{5}(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].

_{2}exhibits a phase transition from the $\beta $-MoB

_{2}-phase (space group: $R\overline{3}m$) to the $\alpha $-MoB

_{2}-phase (space group: $P6/mmm$) at a critical pressure of P~70 GPa. This high-pressure phase, $\alpha $-MoB

_{2}, exhibits the same crystalline structure as the ambient-pressure MgB

_{2}. The most intriguing experimental result reported by Pei et al. [44] was that the $\alpha $-MoB

_{2}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 MgB

_{2}[4,45].

_{2}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 $-MoB

_{2}phase exhibits the same crystal structure as MgB

_{2}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 $-MoB

_{2}[44], whereas the in-plane B-B stretching mode in MgB

_{2}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 $-MoB

_{2}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 $-MoB

_{2}phase.

_{2}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 MgB

_{2}.

_{1−x}Mo

_{x}B

_{2}(x = 0.25, 0.50, 0.75, and 0.9) solid solutions. Despite the superconducting transition temperature in Nb

_{1−x}Mo

_{x}B

_{2}(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−x}Mo

_{x}B

_{2}and highly-pressurized $\alpha $-MoB

_{2}.

_{1−x}Mo

_{x}B

_{2}(x = 0.25, 0.50, 0.75, and 0.9) superconductors (in particular, the Debye temperature, ${T}_{\theta}$) were determined.

_{2}(${T}_{c}~15\mathrm{K}$ at P~121 GPa) for which Pei et al. [48] proposed the space group: P6

_{3}/mmc (which is distorted P6/mmm), while Lim et al. [49] concluded that this highly pressurized superconducting phase of WB

_{2}formed by stacking faulted P6

_{3}/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]).

_{2}and in the recently discovered Nb

_{1−x}Mo

_{x}B

_{2}(x = 0.25; 1.0) and WB

_{2}, 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−x}Mo

_{x}B

_{2}(x = 0.25, 1.0) and WB

_{2}(P = 121.3 GPa) exhibit d-wave superconducting gap symmetry. We also found that ambient pressure Nb

_{1−x}Mo

_{x}B

_{2}(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 MgB

_{2}, α-MoB

_{2}, WB

_{2}, pnictides, A15 alloys, Heusler alloys, Laves phase compounds, cuprates, and highly-compressed hydrides by more than one order of magnitude.

## 2. Utilized Models

_{1−x}Mo

_{x}B

_{2}(x = 0.25; 1.0) and WB

_{2}.

## 3. Results

#### 3.1. P6/mmm $\alpha $-MoB_{2} ($P=109.7\mathit{GPa}$)

_{2}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 $-MoB

_{2}sample).

_{2}($P=109.7\mathrm{GPa}$) superconductor are within the weak-coupling values for d-wave superconductors.

_{2}($P=109.7\mathrm{GPa}$) 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]).

#### 3.2. Ambient Pressure P6/mmm Nd_{0.75}Mo_{0.25}B_{2}

_{1−x}Mo

_{x}B

_{2}($x=0.25$), 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).

_{0.75}Mo

_{0.25}B

_{2}with better accuracy.

_{0.75}Mo

_{0.25}B

_{2}. 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 $\frac{{T}_{c}}{{T}_{F}}=0.042$.

_{0.75}Mo

_{0.25}B

_{2}superconductor exhibits strong nonadiabaticy, because the ratio

#### 3.3. P6_{3}/mmc WB_{2} (P = 121.3 GPa)

_{2}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].

_{2}(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.

_{2}(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 $0.01\le \frac{{T}_{c}}{{T}_{F}}\le 0.05$.

#### 3.4. P6/mmm MgB_{2}

_{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 MgB

_{2}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]:

_{2}by other techniques [83], in particular, by point contact spectroscopy [84].

## 4. Discussion

_{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, we listed calculated ${\lambda}_{e-ph}$ values for all studied dibories (apart MgB

_{2}) in accordance with Equations (2)–(4), with the assumption of ${\mu}^{*}=0.00;0.05;0.10;\mathrm{and}0.13$.

_{0.75}Mo

_{0.25}B

_{2}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 $\frac{{T}_{\theta}}{{T}_{F}}>0.4$.

_{c}in Nb

_{0.75}Mo

_{0.25}B

_{2}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

_{0.75}Mo

_{0.25}B

_{2}(and in all materials in the Nb

_{1−x}Mo

_{x}B

_{2}(x = 0.25; 0.50; 0.75 and 0.9) system) can be interpreted as T

_{c}suppression in d-wave MoB

_{2}superconductors by nonmagnetic impurity—Nb/Mo atoms. However, we need to note that NbB

_{2}and MoB

_{2}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.

_{0.65}Rb

_{0.35}Fe

_{2}As

_{2}(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].

_{2}—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

_{2}and WB

_{2}and ambient pressure superconductors Nd

_{0.75}Mo

_{0.25}B

_{2}. 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

_{0.75}Mo

_{0.25}B

_{2}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 MoB

_{2}superconductors by nonmagnetic impurities (Nb/Mo atoms).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**R(T) data for highly compressed $\alpha $-MoB

_{2}(P = 109.7 GPa) and data fit to Equation (1) (raw data reported by Pei et al. [44]). The green balls indicate the bounds for which R(T) data were used to fit data to Equation (1). (

**a**) Deduced ${T}_{\theta}=301\pm 1\mathrm{K}$, ${T}_{c,0.10}=26.6\mathrm{K}$, ${\lambda}_{e-ph}=1.42$, ${R}_{sat}=0.61\pm 0.02\mathsf{\Omega}$, fit quality is 0.9998. (

**b**) Deduced ${T}_{\theta}=321\pm 1\mathrm{K}$, ${T}_{c,0.10}=28.2\mathrm{K}$, ${\lambda}_{e-ph}=1.41$, ${R}_{sat}=0.50\pm 0.01\mathsf{\Omega}$; fit quality is 0.9998. The 95% confidence bands are indicated by pink shadowed areas.

**Figure 2.**Temperature-dependent upper critical field, B

_{c2}(T), and data (left Y-axes) (defined by $\frac{R\left(T\right)}{{R}_{norm}\left(T\right)}=0.10$ criterion), calculated by Equation (5). The coherence length $\xi \left(T\right)$ (right Y-axes) for $\alpha $-MoB

_{2}($P=109.7\mathrm{GPa}$) reported by Pei et al. [44] 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 T

_{c}= 28.2 K): (

**a**) s-wave fit, $\mathsf{\xi}\left(0\right)=6.5\left(2\right)\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=2.8\pm 0.1\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.5\pm 0.8$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=2.3\pm 0.2$, the goodness of fit is 0.8267; (

**b**) d-wave fit, $\mathsf{\xi}\left(0\right)=6.2\left(5\right)\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=5.0\pm 0.2\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=0.8\pm 0.1$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=4.1\pm 0.2$, the goodness of fit is 0.9842.

**Figure 4.**Plot of $\frac{{T}_{\theta}}{{T}_{F}}$ vs. ${\lambda}_{e-ph}$ for several superconducting families and diborides. This type of plot was proposed by Pietronero et al. [69]. References to the original data can be found in Refs. [68,69,70,71,72,73,74]. In this plot, we assumed that $\alpha $-MoB

_{2}, WB

_{2}, and the Nb

_{1−x}Mo

_{x}B

_{2}($x=0.25$) exhibit the Coulomb pseudopotential parameter, ${\mu}^{*}=0.13$.

**Figure 6.**Temperature dependent upper critical field, B

_{c2}(T), data (left Y-axes) (defined by $\frac{R\left(T\right)}{{R}_{norm}\left(T\right)}=0.015$ criterion) and calculated by Equation (5). The coherence length $\xi \left(T\right)$ (right Y-axes) for P6/mmm Nb

_{0.75}Mo

_{0.25}B

_{2}reported by Hire et al. [47], and data fit 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 T

_{c}= 7.2 K) (

**a**) s-wave fit, $\mathsf{\xi}\left(0\right)=8.0\left(7\right)\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=0.987\pm 0.038\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.6\pm 0.2$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=3.2\pm 0.1$, the goodness of fit is 0.9534; (

**b**) d-wave fit, $\mathsf{\xi}\left(0\right)=7.5\left(0\right)\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=1.65\pm 0.05\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.13\pm 0.03$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=5.3\pm 0.1$, the goodness of fit is 0.9959.

**Figure 7.**R(T) data for highly compressed WB

_{2}(P = 121.3 GPa) and data fit to Equation (1) (raw data reported by Pei et al. [48]). Green balls indicate the bounds for which R(T) data was used for the fit to Equation (1). Deduced ${T}_{\theta}=440\pm 1\mathrm{K}$, ${T}_{c,0.18}=12.5\mathrm{K}$, ${\lambda}_{e-ph}=0.755$, ${R}_{sat}=\infty $; fit quality is 0.9997. 95% confidence bands are shown by pink shadow areas.

**Figure 8.**Temperature-dependent upper critical field, B

_{c2}(T), data (left Y-axes) (defined by $\frac{R\left(T\right)}{{R}_{norm}\left(T\right)}=0.015$ criterion). Calculated by Equation (5): coherence length $\xi \left(T\right)$ (right Y-axes) for P6

_{3}/mmc WB

_{2}(P = 121.3 GPa) reported by Pei et al. [48] and data fits to s-wave (panel

**a**) and d-wave (panel

**b**) single-band models. Deduced parameters are (

**a**) s-wave fit, T

_{c}= 12.45 K (fixed), $\xi \left(0\right)=13.8\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=1.48\pm 0.06\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.6\pm 0.4$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=2.8\pm 0.1$, the goodness of fit is 0.9019; (

**b**) d-wave fit, ${T}_{c}=12.2\pm 0.2\mathrm{K}$, $\xi \left(0\right)=13.0\mathrm{nm}$, $\mathsf{\Delta}\left(0\right)=2.58\pm 0.02\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.19\pm 0.07$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=4.9\pm 0.1$, the goodness of fit is 0.9986.

**Figure 9.**Temperature-dependent upper critical field, B

_{c2}(T), data (left Y-axes). Calculated by Equation (5): coherence length $\xi \left(T\right)$ (right Y-axes) for P6/mmm MgB

_{2}reported by Zehetmayer et al. [81] and data fits to single band s-wave (panel

**a**, Equations (6) and (7)), single band d-wave (panel

**b**, Equations (8) and (9)), and two-band s-wave [82,83] (panel

**c**, Equations (6) and (7), Equations (13)–(15)) models. Deduced parameters are: (

**a**) s-wave fit, ${T}_{c}=36.7\pm 0.4\mathrm{K}$, $\xi \left(0\right)=10.4\mathrm{nm}$ $\mathsf{\Delta}\left(0\right)=5.22\pm 0.09\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=2.3\pm 0.3$, $\frac{2\mathsf{\Delta}\left(0\right)}{{k}_{B}{T}_{c}}=3.3\pm 0.1$, the goodness of fit is 0.9887; (

**b**) d-wave fit, ${T}_{c}=37.8\pm 0.3\mathrm{K}$, $\xi \left(0\right)=10.0\mathrm{nm}$, ${\mathsf{\Delta}}_{m}\left(0\right)=11.6\pm 0.5\mathrm{meV}$, $\mathsf{\Delta}C/\gamma {T}_{c}=1.15\pm 0.07$, $\frac{2{\mathsf{\Delta}}_{m}\left(0\right)}{{k}_{B}{T}_{c}}=7.1\pm 0.3$, the goodness of fit is 0.9975. (

**c**) two conditions where used: ${T}_{c1}={T}_{c2}=37.2\pm 0.2\mathrm{K}$ and $\frac{\mathsf{\Delta}{C}_{1}}{{\gamma}_{1}{T}_{c1}}=\frac{\mathsf{\Delta}{C}_{2}}{{\gamma}_{2}{T}_{c2}}=1.8\pm 0.1$, and other free-fitting parameters are: ${\xi}_{total}\left(0\right)=10.3\mathrm{nm}$, $\alpha =0.77\pm 0.06$, ${\mathsf{\Delta}}_{1}\left(0\right)=6.5\pm 0.4\mathrm{meV}$, $\frac{2{\mathsf{\Delta}}_{1}\left(0\right)}{{k}_{B}{T}_{c}}=4.1\pm 0.3$, ${\mathsf{\Delta}}_{2}\left(0\right)=2.7\pm 0.4\mathrm{meV}$, $\frac{2{\mathsf{\Delta}}_{2}\left(0\right)}{{k}_{B}{T}_{c}}=1.7\pm 0.2$, the goodness of fit is 0.9984.

**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-MoB

_{2}, Nb

_{0.75}Mo

_{0.25}B

_{2}, and WB

_{2}.

Compound | T_{θ} (K) | T_{c} (K) | Assumed μ* | λ_{e−ph} |
---|---|---|---|---|

α-MoB_{2} | 321 | 28.2 | 0.00 | 0.935 |

(109.7 GPa) | 0.05 | 1.10 | ||

0.10 | 1.29 | |||

0.13 | 1.41 | |||

Nb_{1−x}MoxB_{2} | 625 | 7.2 | 0.00 | 0.337 |

(x = 0.25) | 0.05 | 0.422 | ||

0.10 | 0.514 | |||

0.13 | 0.573 | |||

WB_{2} | 440 | 12.5 | 0.00 | 0.475 |

(121.3 GPa) | 0.05 | 0.575 | ||

0.10 | 0.685 | |||

0.13 | 0.755 |

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## Share and Cite

**MDPI and ACS Style**

Talantsev, E.F. *D*-Wave Superconducting Gap Symmetry as a Model for Nb_{1−x}Mo_{x}B_{2} (x = 0.25; 1.0) and WB_{2} Diborides. *Symmetry* **2023**, *15*, 812.
https://doi.org/10.3390/sym15040812

**AMA Style**

Talantsev EF. *D*-Wave Superconducting Gap Symmetry as a Model for Nb_{1−x}Mo_{x}B_{2} (x = 0.25; 1.0) and WB_{2} Diborides. *Symmetry*. 2023; 15(4):812.
https://doi.org/10.3390/sym15040812

**Chicago/Turabian Style**

Talantsev, Evgeny F. 2023. "*D*-Wave Superconducting Gap Symmetry as a Model for Nb_{1−x}Mo_{x}B_{2} (x = 0.25; 1.0) and WB_{2} Diborides" *Symmetry* 15, no. 4: 812.
https://doi.org/10.3390/sym15040812