Unconventional Pressure Dependence of the Superfluid Density in the Nodeless Topological Superconductor $\alpha \text{−}{\mathrm{PdBi}}_2$

We investigated the superconducting properties of the topological superconductor $\alpha \text{−}{\mathrm{PdBi}}_2$ at ambient and external pressures up to 1.77 GPa using muon spin rotation experiments. The ambient pressure measurements evince a fully gapped $s$-wave superconducting state in the bulk of the specimen. Alternating current magnetic susceptibility and muon spin rotation measurements manifest a continuous suppression of $T_c$ with increasing pressure. In parallel, we observed a significant decrease of superfluid density by $\sim 20\%$ upon application of external pressure. Remarkably, the superfluid density follows a linear relation with $T_c$, which was found before in some unconventional topological superconductors and hole-doped cuprates. This finding signals a possible crossover from Bose-Einstein to Bardeen-Cooper-Schrieffer like condensation in $\alpha \text{−}{\mathrm{PdBi}}_2$.

Figure 1

(a) TF $\mu \mathrm{SR}$ asymmetry spectra obtained above and below $T_c$ for $\alpha \text{−}{\mathrm{PdBi}}_2$. The spectra were recorded after field cooling the sample from above $T_c$ in an applied magnetic field of ${\mu }_{0}H=10\mathrm{mT}$. Solid lines correspond to the fitting of the spectra as described in the Supplemental Material (SM) [41]. (b) Normalized Fourier spectra at 0.08 K (green) and 2 K (red) obtained by fast Fourier transformation of the spectra in (a). Inset: Schematic illustration of how muons, as local probes, sense the inhomogeneous magnetic field distribution in the vortex state of a type-II superconductor. (c) Temperature dependence of the superconducting muon spin depolarization rate ${\sigma }_{\mathrm{sc}}$ of $\alpha \text{−}{\mathrm{PdBi}}_2$ measured in an applied magnetic field of ${\mu }_{0}H=10\mathrm{mT}$. The solid lines correspond to different theoretical models as discussed in the text. (d) Field dependence of ${\sigma }_{\mathrm{sc}}$ fitted with the Brandt formula for an isotropic single $s$-wave gap (solid line) as discussed in the text.

Figure 2

Temperature dependence of the real, ${\chi }^{\prime }$, (a) and imaginary, ${\chi }^{\prime \prime }$, (b) components of ACS of $\alpha \text{−}{\mathrm{PdBi}}_2$ (mounted inside a double-wall piston cylinder type pressure cell made of $\mathrm{CuBe}/\mathrm{MP}35\mathrm{N}$ material) at different selected pressures. For better visualization, we vertically shifted the $y$ axis in both (a) and (b).

Figure 3

(a) Pressure($p$)-$T_c$ phase diagram of $\alpha \text{−}{\mathrm{PdBi}}_2$ constructed from ACS (green) and $\mu \mathrm{SR}$ (red) measurements under various applied hydrostatic pressures. Solid lines represent the linear fit through the experimentally obtained data points. Note that $T_c$’s from $\mu \mathrm{SR}$ measurement are lower than those obtained from ACS because we performed the $\mu \mathrm{SR}$ experiments under an applied field of 10 mT. (b) The temperature dependence of ${\sigma }_{\mathrm{sc}}(T)$ (normalized to its value at the zero applied pressure) under different applied pressures. Inset shows the pressure dependence of the superconducting gap ${\mathrm{\Delta }}_0$ normalized to the zero pressure value.

Figure 4

(a) Normalized superfluid density as a function of $T_c(p)/T_c(p=0)$ at different applied pressures shown in a color contour plot. We compared the pressure effect with a few other conventional and unconventional superconductors. Black dashed horizontal line manifests the BCS prediction. Purple dashed line indicates the deviation from the BCS prediction and highlights the linear dependence between superfluid density and $T_c$ in $\alpha \text{−}{\mathrm{PdBi}}_2$. (b) $\mathrm{d}{\sigma }_{\mathrm{nor}}/\mathrm{d}p$ as a function of $\mathrm{d}{T}_{c,\mathrm{nor}}/\mathrm{d}p$ (where $X_{\mathrm{nor}}=[(X_p-X_{p=0})/X_{p=0}]$, with $X$ being $T_c$ or ${\sigma }_{\mathrm{sc}}$ for different superconductors depicted in panel (a).