Time-reversal symmetry breaking in topological superconductor ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_2{\mathrm{Se}}_3$

The single helical Fermi surface on the surface state of three-dimensional topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ is constrained by the time-reversal invariant bulk topology to possess a spin-singlet superconducting pairing symmetry. In fact, the Cu-doped and pressure-tuned superconducting ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ show no evidence of the time-reversal symmetry (TRS) breaking. We report on the detection of the TRS breaking in the topological superconductor ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_2{\mathrm{Se}}_3$, probed by zero-field $\mu \mathrm{SR}$ measurements. The TRS breaking provides strong evidence for the existence of a spin-triplet pairing state. The existence of TRS breaking is also verified by longitudinal-field $\mu \mathrm{SR}$ measurements, which negates the possibility of magnetic impurities as the source of TRS breaking. The temperature-dependent superfluid density deduced from transverse-field $\mu \mathrm{SR}$ measurements yields nodeless superconductivity with low superconducting carrier density and penetration depth $\lambda =1622\left(134\right)\mathrm{nm}$. From the microscopic theory of unconventional pairing, we find that such a fully gapped spin-triplet pairing channel is promoted by the complex interplay between the structural hexagonal warping and higher order Dresselhaus spin-orbit-coupling terms. Based on Ginzburg-Landau analysis, we delineate the mixing of singlet- to triplet-pairing symmetry as the chemical potential is tuned far above from the Dirac cone. Our observation of such spontaneous TRS breaking chiral superconductivity on a helical surface state, protected by the TRS invariant bulk topology, can open avenues for interesting research and applications.

Figure 1

(a) ZF-$\mu \mathrm{SR}$ spectra collected at 0.1 and 3.7 K and LF-$\mu \mathrm{SR}$ spectra taken at 1.4 K under an LF field of 5 mT. (b), (c) Temperature variation of the relaxation rates ${\sigma }_{\mathrm{ZF}}$ and ${\lambda }_{\mathrm{ZF}}$, indicating spontaneous fields appearing in the superconducting state of ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 2

TF-$\mu \mathrm{SR}$ spectra taken at (a) 0.09 and (b) 3.6 K under 10 mT applied magnetic field. (c) Variation of depolarization rate with temperature. (d) Variation of internal field in the material with temperature.

Figure 3

Variation of ${\lambda }^{-2}$ as a function of temperature for ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_2{\mathrm{Se}}_3$. Solid lines are the fit to the data using $s$-wave model.

Figure 4

(a) Single helical surface state of a 3D topological insulator in the $k_x-k_y$-plane. (b), (c) Constant energy cuts (b) far from the Dirac cone and (c) close to the Dirac cone. Red arrows show the corresponding spin alignments across the corresponding constant energy contours. Dashed arrows dictated the $k$ and $-k$ points in which spin is antiparallel, while the solid arrows show similar points where antiparallel alignment of the spin is weakened due to higher order SOC and hexagonal warping effects.

Figure 5

(a) Two superconducting transition temperatures for the singlet $\left(T_{c,s}\right)$ and triplet $\left(T_{c,t}\right)$ pairings, assuming same coupling constant $u_s=u_t$ but different density of states. The cross terms in the free energy $(\gamma ,\delta )$ are neglected. (b) As the cross terms are introduced, both phases become complementary, with a small region of their coexistence near the tricritical point ${\mu }^{*}$.