Time-reversal invariant and fully gapped unconventional superconducting state in the bulk of the topological compound ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$

Recently, the niobium (Nb) doped topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, in which the finite magnetic moments of the Nb atoms are intercalated in the van der Waals gap between the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ layers, has been shown to exhibit both superconductivity with $T_c\simeq 3$ K and topological surface states. Here we report on muon spin rotation experiments probing the temperature and field dependence of effective magnetic penetration depth ${\lambda }_{\mathrm{eff}}\left(T\right)$ in the layered topological superconductor candidate ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. The exponential temperature dependence of ${\lambda }_{\mathrm{eff}}^{-2}\left(T\right)$ at low temperatures suggests a fully gapped superconducting state in the bulk with the superconducting transition temperature $T_c=2.9\mathrm{K}$ and the gap to $T_c$ ratio $2\mathrm{\Delta }/k_B{T}_c=3.95\left(19\right)$. We also reveal that the ratio $T_c/{\lambda }_{\mathrm{eff}}^{-2}$ is comparable to those of unconventional superconductors, which hints at an unconventional pairing mechanism. Furthermore, time-reversal symmetry breaking was excluded in the superconducting state with sensitive zero-field $\mu \mathrm{SR}$ experiments. We hope the present results will stimulate theoretical investigations to obtain a microscopic understanding of the relation between superconductivity and the topologically nontrivial electronic structure of ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 1

Crystal structure and bulk superconducting transition for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Crystal structure of Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ emphasizing the position of Nb intercalation between ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ layers (highlighted as quintuple layer). (b) Temperature dependence of DC magnetic susceptibility (left axis) in $H(\parallel c)=1$ mT and zero-field superconducting part of the specific heat divided by temperature (right axis) for the single-crystalline sample of ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. The susceptibility and specific-heat measurements show similar transition temperature ($\sim 2.9$ K).

Figure 2

Zero-field (ZF) and transverse-field (TF) $\mu \mathrm{SR}$ time spectra for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. Zero-field $\mu \mathrm{SR}$ (a) and transverse-field (b) $\mu \mathrm{SR}$ time spectra obtained above and below $T_c$ for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ (after field-cooling the sample from above $T_c$). Inset of (a) shows the temperature dependence of the electronic relaxation rate measured in zero magnetic field of ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ with $T_c\sim 3$ K. Error bars are the standard error of the mean in about ${10}^6$ events. The error of each bin count $n$ is given by the standard deviation of $n$. The errors of each bin in $A\left(t\right)$ are then calculated by standard-error propagation.

Figure 3

Superconducting muon spin depolarization rate ${\sigma }_{\mathrm{sc}}$ and the field shift for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Temperature dependence of the superconducting muon spin depolarization rate ${\sigma }_{\mathrm{sc}}$ measured in an applied magnetic field of ${\mu }_{0}H=0.01\mathrm{T}$ for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. The error bars represent the standard deviation of the fit parameters. (b) Temperature dependence of the difference between the internal field ${\mu }_0{H}_{\mathrm{SC}}$ measured in the SC state and the one measured in the normal state ${\mu }_0{H}_{\mathrm{NS}}$ at $T=5\mathrm{K}$ for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 4

Temperature and field evolution of ${\lambda }^{-2}$ for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) The temperature dependence of ${\lambda }^{-2}$ for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$, measured in an applied field of ${\mu }_{0}H=0.01\mathrm{T}$. The solid black line corresponds to a single-gap BCS $s$-wave model. The dashed purple line corresponds to anisotropic $s$-wave model. The dashed red line corresponds to clean-limit $d$-wave model. The dashed blue line corresponds to a dirty-limit $d$-wave model [a power law $1-{(T/T_c)}^n$ with $n=2]$. The error bars are calculated as the standard error of the mean. (b) Field dependence of the superconducting muon spin depolarization rate ${\sigma }_{\mathrm{sc}}$.

Figure 5

Temperature evolution of specific heat for ${\mathrm{Nb}}_{0.25}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. Temperature evolution of the electronic part of specific heat fitted with single $s$-wave gap model (solid line).