Superconductivity of ${\mathrm{MoBe}}_{22}$ and ${\mathrm{WBe}}_{22}$ at ambient- and under applied-pressure conditions

${\mathrm{MoBe}}_{22}$ and ${\mathrm{WBe}}_{22}$ compounds belong to the binary $X{\mathrm{Be}}_{22}$ $(X=4d$ or 5$d$ metal) family of superconductors, whose critical temperature depends strongly on $X$. Despite the multiphase nature of these samples, it is possible to investigate the superconducting properties of ${\mathrm{MoBe}}_{22}$ and ${\mathrm{WBe}}_{22}$ at the macro- and microscopic level. A concurrent analysis by means of magnetization and heat-capacity measurements, as well as muon-spin spectroscopy $\left(\mu \mathrm{SR}\right)$ was implemented. At ambient pressure, both compounds enter the superconducting state below $2.6\pm 0.1$ K $\left({\mathrm{MoBe}}_{22}\right)$ and $4.1\pm 0.10$ K $\left({\mathrm{WBe}}_{22}\right)$ and show modest upper critical fields [$({\mu }_0{H}_{\text{c2}}\left(0\right)=48\pm 1$ mT and ${\mu }_0{H}_{\text{c2}}\left(0\right)=58\pm 1$ mT, respectively]. In ${\mathrm{WBe}}_{22}$, the temperature-dependent superfluid density suggests a fully gapped superconducting state, well-described by an $s$-wave model with a single energy gap. Heat-capacity data confirm that such a model applies to both compounds. Finally, ac magnetic susceptibility measurements under applied pressures up to 2.1 GPa reveal a linear suppression of the superconducting temperature, typical of conventional superconducting compounds.

Figure 1

The arrangement of the coordination polyhedra around Mo(W) and Be2 atoms ($16d$ site) in the structures of ${\mathrm{MoBe}}_{22}$ and ${\mathrm{WBe}}_{22}$. For simplicity, only part of the unit cell is shown.

Figure 2

(a), (b) Backscatter electron micrographs of ${\mathrm{MoBe}}_{22}$ sample surface at various magnifications. (c) A cuboid of ${\mathrm{MoBe}}_{22}$, used for the single-crystal diffraction experiments, prepared using focused-ion beam.

Figure 3

(a), (b) Backscatter electron micrographs of ${\mathrm{WBe}}_{22}$ sample surface at various magnifications. (c) A cuboid of ${\mathrm{WBe}}_{22}$, used for the single-crystal diffraction experiments, prepared using focused-ion beam.

Figure 4

Zero-field-cooled (open symbols) and field-cooled (full symbols) temperature-dependent magnetic susceptibility data for ${\mathrm{MoBe}}_{22}$ (a) and ${\mathrm{WBe}}_{22}$ (b) in an applied field $0.5\le {\mu }_{0}H\le 40$ mT.

Figure 5

Total specific heat of ${\mathrm{MoBe}}_{22}$ (a) and ${\mathrm{WBe}}_{22}$ (c) as a function of temperature in an applied field $0\le {\mu }_{0}H\le 50$ mT. The electronic specific heat of ${\mathrm{MoBe}}_{22}$ (b) and ${\mathrm{WBe}}_{22}$ (d) measured in ${\mu }_{0}H=0$ mT. Vertical dashed lines represent equal-entropy constructions. The horizontal dashed line corresponds to the Sommerfeld coefficient ${\gamma }_n$. The solid line is a fit to $C_e/T\propto {\mathrm{e}}^{-\mathrm{\Delta }/T}$ in the superconducting state. Insets of panels (b) and (d) show $C_p/T$ versus $T^2$, from which the values of $\gamma$ and $\beta$ were extracted.

Figure 6

$H$–$T$ phase diagram of ${\mathrm{MoBe}}_{22}$ (orange) and ${\mathrm{WBe}}_{22}$ (blue). The values of the critical field $H_{\text{c2}}\left(T\right)$ are extracted from $M\left(T\right)$ (squares) and $C_{\text{p}}$ (circles) data. The dashed lines correspond to a Ginzburg-Landau fit of the data.

Figure 7

ac susceptibility data versus temperature, measured at various applied pressures for ${\mathrm{MoBe}}_{22}$ (a) and ${\mathrm{WBe}}_{22}$ (b). In both cases, the steep changes in signal reflect the response of the sample and of indium (used here as a pressure gauge). Insets show the almost linear dependence of $T_{\text{c}}$ on pressure. In both cases, error bars are smaller than the symbols.

Figure 8

(a) TF-$\mu \mathrm{SR}$ time-domain spectra of ${\mathrm{WBe}}_{22}$ collected at $T=0.3$ K and $T=5$ K in an applied field of 40 mT. The respective Fourier transforms are shown in (b) and (c). Below $T_{\text{c}}$, three oscillations are needed to reproduce the data. Solid lines are fits to Eq. (8) using three oscillations. Note the absence of a diamagnetic shift below $T_{\text{c}}$ in panel (b)

Figure 9

Superfluid density of ${\mathrm{WBe}}_{22}$ versus temperature, as determined from TF-$\mu \mathrm{SR}$ measurements at ${\mu }_{0}H=15$ and 40 mT. The inset shows the temperature dependence of the three components of the $\mu \mathrm{SR}$ rate ${\sigma }_i\left(T\right)$ at 40 mT (left axis) and the diamagnetic field shift (right axis). Lines represent fits to a fully gapped $s$-wave model with a single superconducting gap (see text for details).