High critical field superconductivity at ambient pressure in ${\mathrm{MoB}}_2$ stabilized in the P6/mmm structure via Nb substitution

Recently it was discovered that, under elevated pressures, ${\mathrm{MoB}}_2$ exhibits superconductivity at a critical temperature $T_c$ as high as 32 K. The superconductivity appears to develop following a pressure-induced structural transition from the ambient pressure $\mathrm{R}\overline{3}\mathrm{m}$ structure to an ${\mathrm{MgB}}_2$-like P6/mmm structure. This suggests that remarkably high $T_c$ values among diborides are not restricted to ${\mathrm{MgB}}_2$ as previously appeared to be the case, and that similarly high $T_c$ values may occur in other diborides if they can be coerced into the ${\mathrm{MgB}}_2$ structure. In this paper, we show that density functional theory calculations indicate that phonon free energy stabilizes the P6/mmm structure over the $\mathrm{R}\overline{3}\mathrm{m}$ at high temperatures across the ${\mathrm{Nb}}_{1-x}{\mathrm{Mo}}_x{\mathrm{B}}_2$ series. X-ray diffraction confirms that the synthesized Nb-substituted ${\mathrm{MoB}}_2$ adopts the ${\mathrm{MgB}}_2$ crystal structure. High magnetic field electrical resistivity measurements and specific heat measurements demonstrate that ${\mathrm{Nb}}_x{\mathrm{Mo}}_{1-x}{\mathrm{B}}_2$ exhibits superconductivity with $T_c$ as high as 8 K and critical fields approaching 6 T.

Figure 1

Calculated free energy of ${\mathrm{Nb}}_{1-x}{\mathrm{Mo}}_x{\mathrm{B}}_2$ as a function of composition and temperature. The free energy was calculated with respect to 0 K free energy of stable ${\mathrm{NbB}}_2$ (P6/mmm) and ${\mathrm{MoB}}_2$ ($\mathrm{R}\overline{3}\mathrm{m}$) phases. The blue dots represent the P6/mmm phase and the red dots represent the $\mathrm{R}\overline{3}\mathrm{m}$ phase. Only vibrational entropy contribution to the free energy was considered. The black line is the convex hull line at various temperatures. Electron smearing of 0.5 eV was used for the phonon calculation of P6/mmm ${\mathrm{MoB}}_2$ and ${\mathrm{Nb}}_{0.25}{\mathrm{Mo}}_{0.75}{\mathrm{B}}_2$.

Figure 2

(a) P6/mmm ${\mathrm{MoB}}_2$ phonon dispersion curves and (b) band structure diagram for varying electron smearing values. With increasing electron smearing the P6/mmm phase of ${\mathrm{MoB}}_2$ becomes dynamically stable. Notice the downward shift of the Fermi level with increasing smearing.

Figure 3

X-ray patterns of arc-melted ${\mathrm{MoB}}_2$ (upper trace, green) and arc-melted ${\mathrm{Nb}}_{0.1}{\mathrm{Mo}}_{0.9}{\mathrm{B}}_2$ (third trace from the top, black) compared to calculated patterns for R-3m structure ${\mathrm{MoB}}_2$ (blue, space group 166, second trace) and for P6/mmm structure ${\mathrm{MoB}}_2$ (red, space group 191, bottom trace.) Although there are reflections in the measured patterns that do not agree with the calculated ones (e.g., there are extra lines at $30.0^{\circ }$ and $45.4^{\circ }2\theta$ in the arc-melted ${\mathrm{MoB}}_2$ data which may be from a second phase of the P6/mmm structure), clearly the measured data support the conclusions that arc-melted ${\mathrm{MoB}}_2$ occurs primarily in the R-3m structure while $10\%$ Nb doping (${\mathrm{Nb}}_{0.1}{\mathrm{Mo}}_{0.9}{\mathrm{B}}_2$) causes the arc-melted sample to occur essentially entirely in the P6/mmm structure.

Figure 4

Measured x-ray diffraction patterns for arc-melted ${\mathrm{Nb}}_{1-x}{\mathrm{Mo}}_x{\mathrm{B}}_2, x=$ 0.25, 0.50, and 0.75, and calculated patterns for both ${\mathrm{MoB}}_2$ and ${\mathrm{NbB}}_2$ in the P6/mmm structure, as well as for pure Nb. Again, all of the ${\mathrm{Nb}}_{1-x}{\mathrm{Mo}}_x{\mathrm{B}}_2$ samples, $x=$ 0.25, 0.50, and 0.75, occur in the P6/mmm structure, with no evidence of second phase Nb. Two second phase lines are, however, observed. In ${\mathrm{Nb}}_{0.25}{\mathrm{Mo}}_{0.75}{\mathrm{B}}_2$ there is a second phase line at $23.9^{\circ }2\theta$; in ${\mathrm{Nb}}_{0.5}{\mathrm{Mo}}_{0.5}{\mathrm{B}}_2$ there is a second phase line at $42.{65}^{\circ }2\theta$. As shown, neither of these lines match the Nb x-ray pattern.

Figure 5

Specific heat data of ${\mathrm{Nb}}_{1-x}{\mathrm{Mo}}_x{\mathrm{B}}_2, x=$ 0.25, 0.50, 0.75, 0.9.

Figure 6

Resistivity dependence of ${\mathrm{Nb}}_{0.25}{\mathrm{Mo}}_{0.75}{\mathrm{B}}_2$ (a) on temperature for different magnetic fields, and (b) on magnetic field at constant temperature of 2K. ${\mathrm{T}}_c$ is found to be 8.03 K. (c) Upper critical field values for ${\mathrm{H}}_{c2}$(0) for ${\mathrm{Nb}}_{0.25}{\mathrm{Mo}}_{0.75}{\mathrm{B}}_2$ extracted from (a) and (b). The solid line shows the fitting to Ginzburg-Landau formula given by $H_{c2}=H_{c2}\left(0\right)[(1-t^2)/(1+t^2)]$, where $t$ is the temperature normalized by the $\mathrm{zero}-\mathrm{field}$ transition temperature [9]. This fitting gives $H_{c2}$(0) of 6.71 T.