Quantum muon diffusion and the preservation of time-reversal symmetry in the superconducting state of type-I rhenium

Elemental rhenium exhibiting type-II superconductivity has been previously reported to break time-reversal symmetry in the superconducting state. We have investigated an arc-melted sample of rhenium exhibiting type-I superconductivity. Low-temperature zero-field muon-spin relaxation measurements indicate that time-reversal symmetry is preserved in the superconducting state. Muon diffusion is observed, which is due to quantum mechanical tunneling between interstitial sites. The normal state behavior is characterized by the conduction electrons screening the muons and thermal broadening, and is typical for a metal. Energy asymmetries between muon trapping sites and the superconducting energy gap also characterize the superconducting state behavior.

Figure 1

Normal state resistivity of rhenium as a function of temperature, $\rho \left(T\right)$. Data were fitted using the Bloch-Grüneisen model for a transition metal [51]. Uncertainties in resistivity values arising from the nonuniform shape of the sample are shown by the outer band. The inset shows evidence of type-I superconductivity from magnetization versus field data, where $D=0.29\left(1\right)$ is the demagnetization factor of the sample.

Figure 2

Heat capacity, $C$, divided by temperature, $T$, versus $T$ of rhenium. Normal state data were fitted by including electronic, phonon, and Schottky anomaly contributions. Zero-field (ZF) superconducting state data were also fitted using a conventional, single-gap $s$-wave model. The jump in heat capacity occurs at $T_{\mathrm{c}}^{\mathrm{I}}=1.70\left(1\right)$ K, which is the reported transition temperature of type-I rhenium. Data obtained by Smith and Keesom are also shown [52].

Figure 3

Zero-field muon-spin relaxation spectra for type-I rhenium above and below the superconducting transition temperature $T_{\text{c}}^{\text{I}}=1.7$ K. Fits to each temperature curve were found using Eq. (5) and include static relaxation effects and muon diffusion [56, 57]. Longitudinal field (LF) data in 30 mT are also shown.

Figure 4

Muon hopping rate as a function of temperature, $\nu \left(T\right)$, where $A_{\text{BG}}=0.11$ and $\sigma =0.325\mu {\text{s}}^{-1}$ were fixed in the fitting. Error bars are within the size of the data point markers. Data were fitted in the normal state using $\nu \left(T\right)=a_{\text{NS}}[\mathrm{\Gamma }\left(K\right)/\mathrm{\Gamma }(K+1/2)]T^{2K-1}$, where $a_{\text{NS}}=0.20\left(1\right)\mu {\text{s}}^{-1}{\text{K}}^{1-2K}$, and $K=0.20\left(2\right)$. The three dashed curves below the superconducting state data were simulated using expressions for small (${\varepsilon }_{\text{s}}$) and intermediate (${\varepsilon }_{\text{i}}$) energy asymmetries between muon sites [59], where the energies are normalized by $k_{\text{B}}$.