Rotational symmetry breaking in the topological superconductor SrxBi2Se3 probed by upper-critical field experiments

Recently it was demonstrated that Sr intercalation provides a new route to induce superconductivity in the topological insulator Bi2Se3. Topological superconductors are predicted to be unconventional with an odd-parity pairing symmetry. An adequate probe to test for unconventional superconductivity is the upper critical field, Bc2. For a standard BCS layered superconductor Bc2 shows an anisotropy when the magnetic field is applied parallel and perpendicular to the layers, but is isotropic when the field is rotated in the plane of the layers. Here we report measurements of the upper critical field of superconducting SrxBi2Se3 crystals (Tc = 3.0 K). Surprisingly, field-angle dependent magnetotransport measurements reveal a large anisotropy of Bc2 when the magnet field is rotated in the basal plane. The large two-fold anisotropy, while six-fold is anticipated, cannot be explained with the Ginzburg-Landau anisotropic effective mass model or flux flow induced by the Lorentz force. The rotational symmetry breaking of Bc2 indicates unconventional superconductivity with odd-parity spin-triplet Cooper pairs (Δ4-pairing) recently proposed for rhombohedral topological superconductors, or might have a structural nature, such as self-organized stripe ordering of Sr atoms.

to a Majorana surface state 20 . However, STS (Scanning Tunneling Spectroscopy) showed that the density of states at the Fermi level is fully gapped without any in-gap states 21 . On the other hand, the superconducting state shows a large inhomogeneity 21 and the superconducting volume fraction depends on quenching conditions 22 . Consequently, the issue of topological superconductivity in Cu x Bi 2 Se 3 has not been settled and further experiments are required, as well as new materials.
Very recently it has been demonstrated that Sr intercalation provides a new route to induce superconductivity in Bi 2 Se 3 23 . Resistivity and magnetization measurements on Sr x Bi 2 Se 3 single crystals with x = 0.06 show T c = 2.5 K. The superconducting volume fraction amounts to 90% which confirms bulk superconductivity. By optimizing the Sr content a maximum T c of 2.9 K was found for x = 0.10 24 . The topological character of Bi 2 Se 3 is preserved upon Sr intercalation. ARPES showed a topological surface state well separated from the bulk conduction band 25,26 . Based on the first measurements of the electronic parameters in the normal and superconducting states, and the close analogy to Cu x Bi 2 Se 3 , it has been advocated that Sr x Bi 2 Se 3 is a new laboratory tool to investigate topological superconductivity 23,24 .
Here we report a study of unusual basal-plane anisotropy effects in the upper critical field, B c2 , of Sr x Bi 2 Se 3 . Bi 2 Se 3 crystallizes in a rhombohedral structure with space group R m 3 . It is a layered material and Sr is intercalated in the Van der Waals gaps between the quintuple Bi 2 Se 3 layers 23  T for x = 0.15 at T/T c = 0.1 (T c = 3.0 K), where a and a * are orthogonal directions in the basal plane. This large effect cannot be explained with the anisotropic effective mass model 27,28 or the variation of B c2 caused by flux flow 29 . The rotational symmetry breaking of B c2 indicates unconventional superconductivity 30,31 , or might have a structural nature, such as preferential ordering of Sr atoms.

Results
The resistivity, ρ(T), of our Sr x Bi 2 Se 3 crystals with x = 0.10 and x = 0.15 shows a metallic temperature variation with superconducting transition temperatures T c of 2.8 K and 3.0 K, respectively, see Fig. S4 in the Supplementary Information 32 . The superconducting volume fractions of the crystals measured by ac-susceptibility amount to 40% and 80%, respectively 32 . In Fig. 1 we show the angular variation of the resistance, R(θ), measured in a fixed field B = 0.4 T directed in the basal plane (aa * -plane), in the temperature range 2-3 K around T c (T c = 2.8 K at B = 0 T), for x = 0.10. Rather than attaining a constant value, the curves show a pronounced angular variation which demonstrates that B c2 (T) (or T c (B)) is field-angle dependent. For instance, at 2.5 K and 0.4 T (violet symbols) the sample is in the normal state at θ = 3° and superconducts (R = 0) at 93°. By raising the temperature from 2 K to 3 K superconductivity is smoothly depressed for all field directions. The data show a striking two-fold symmetry, which is most clearly demonstrated in a polar plot (Fig. 2). We remark, the same two-fold anisotropy is observed in crystals with x = 0.15. In the top panel of Fig. 1 we show R(θ) in the normal state measured in 8 T for x = 0.10. The data have been symmetrized after measuring R(θ) for opposite field polarities to eliminate a small Hall component. R(θ) in the normal state shows the same two-fold symmetry as in Fig. 1a. The variation in R(θ) is small and amounts to 3% in 8 T. The data follow a sin θ dependence, which tells us the variation is due to the classical magnetoresistance related to the Lorentz force F L = BI sin θ, where I is the transport current that flows in the basal plane. R(θ) is minimum in the longitudinal case (B || I) and maximum in the transverse case (B ⊥ I).
In Fig. 3 we report B c2 (T) for two single crystals measured with the B-field along the orthogonal directions in the hexagonal unit cell. The data points are obtained by measuring the superconducting transition in R(T) in fixed fields, where T c is identified by the 50% drop of R with respect to its value in the normal state 32 . In determining the values of B c2 we did not correct for demagnetization effects, since the demagnetization factors calculated for our crystals are small 32

Discussion
Having conclusively established the two-fold anisotropy of B c2 in the basal plane, we now turn to possible explanations. A first explanation could be a lowering of the symmetry caused by a crystallographic phase transition below room temperature. However, the powder X-ray diffraction patterns measured at room temperature and T = 10 K are identical (see Fig. S2 in ref. 32). Moreover, the resistivity traces (T = 2-300 K, Fig. S4) and the specific heat (T = 2-200 K, Fig. S8) all show a smooth variation with temperature and do not show any sign of a structural phase transition 32 . We therefore argue our crystals keep the R m 3 space group at low temperatures. A second explanation for breaking the symmetry in the basal plane could be the measuring current itself. Since the current flows in the basal plane it naturally breaks the symmetry when we rotate the field in the basal-plane. Indeed B c2 is largest for B || I and smallest for B ⊥ I. In the latter geometry, and for large current densities, the Lorentz force may cause flux lines to detach from the pinning centers, which will lead to a finite resistance, a broadened R(B)-curve and a lower value of B c2 29 . This effect has been observed for instance in the hexagonal superconductor MgB 2 by rotating B with respect to I in the basal plane 33 . For a current density 30 A/cm 2 , the two-fold anisotropy obtained just below T c = 36 K is small, ~8% 33 . In our transport experiments the current densities are ≤ 0.4 A/cm 2 and we did not detect a significant effect on the resistance when the current density was varied close to T c (see Fig. S9 32 ). Also, when flux flow has a significant contribution, one expects the R(B)-curves for B ⊥ I to be broader than the curves for B || I. However, we observe the reverse (see Fig. 3a,b). Moreover, the anisotropy is still present at T/T c = 0.1 and is much larger (of the order of 300%, see Fig. 4) than can be expected on the basis of flux flow. In order to further rule out the influence of the current direction we have investigated B c2 (θ) in the basal plane with the transport current perpendicular to the layers (I || c) and thus keeping B ⊥ I (see Fig. S11, ref. 32). The angular variation of the resistance, measured in this geometry using a two-probe method, is similar to that reported in Fig. 1. Thus the two-fold anisotropy in B c2 is also present for the B-field in the aa * -plane and the current along the c-axis.
Next we address whether the variation of B c2 in the basal plane can be attributed to the anisotropy of the effective mass. Within the Ginzburg-Landau model 27 The Fermi surface of n-doped Bi 2 Se 3 , with a typical carrier concentration n ~ 2 × 10 19 cm −3 representative for the superconducting Sr x Bi 2 Se 3 crystals 23,24 , has been investigated by the Shubnikov -de Haas effect 23,35,36 . It can be approximated by an ellipsoid of revolution with the longer axis along the k c -axis. A trigonal warping of the Fermi surface due to the rhombohedral symmetry has been detected, but the effect is small: the variation of the effective mass in the basal plane amounts to a few % only 35 . This also explains why R(θ) in the normal state (Fig. 1a), does not show a 2π/3 periodicity superimposed on the two-fold symmetry induced by the current. Clearly, the two-fold symmetry (Fig. 4), while three fold is expected, and the calculated large ratio ⁎ m m / a a using the Ginzburg-Landau model are at variance with the experimental Fermi-surface determination 35 and we discard this scenario. In the experiments for x = 0.15 the crystal was not mounted on the rotator but oriented by eye, which adds some inaccuracy as regards field alignment. The current direction was always along the a-axis, with a precision of several degrees.
Having excluded these conventional explanations for the rotational symmetry breaking we now proceed to a more exciting scenario. Nagai (ref. 30) and Fu (ref. 31) recently proposed a model for odd parity spin-triplet superconductivity developed in the context of Cu x Bi 2 Se 3 , and investigated the experimental consequences of Δ 4 pairing in the two-orbital model 18 . Here, superconductivity is described by an odd-parity two-dimensional representation, E u , where the attractive potential pairs two electrons in the unit cell to form a spin triplet, i.e. a vectorial combination of c 1↑ c 2↑ and c 1↓ c 2↓ . The indices 1, 2 refer to the two orbitals and the arrows to the spin. The Δ 4 state has zero-total spin along an in-plane direction n = (n x , n y ) that is regarded as a nematic director and breaks rotational symmetry. By taking into account the full crystalline anisotropy in the Ginzburg-Landau model, it can be shown that n is pinned to a direction in the basal plane. For =n x, point nodes in the superconducting gap are found along ŷ, whereas for =n y two gap minima occur at ±k x Interpreting ξ as the Cooper-pair size, this implies that the pairing interaction is strongest along the a * and c-axis, and weakest along the a-axis. The observation that ξ ξ ξ > ≈ ⁎ a a c can naively be translated to the gap structure consistent with the one predicted for =n y. More recent calculations show that B c2 for the two-dimensional E u representation retains the hexagonal symmetry of the crystal lattice, but its symmetry can be lowered to two-fold in the presence of a symmetry breaking field 37,38 . As regards Sr x Bi 2 Se 3 the origin of the symmetry breaking is not clear yet. Possible candidates are sample shape, residual strain and local ordering of Sr atoms. We remark that rotational symmetry breaking in the spin system has been observed by Nuclear Magnetic Resonance (NMR) in the related superconductor Cu x Bi 2 Se 3 , which is considered to provide solid evidence for a spin-triplet state 39 .
Yet another interesting possibility is a self-organized structural stripiness in the optimum for superconductivity due to ordering of Sr atoms in the Van der Waals gaps. This could naturally lead to an anisotropy of B c2 when measured for a current in the basal plane, because of an effective reduced dimensionality. The higher B c2 -values will then be found for B || I along the stripes. On the other hand, for I perpendicular to the layers the basal-plane anisotropy of B c2 is found as well 32 . This calls for a detailed compositional and structural characterization of Sr x Bi 2 Se 3 by techniques such as Electron Probe Microprobe Analysis (EPMA) or Transmission Electron Microscopy (TEM). Notice that in Cu x Bi 2 Se 3 crystals EPMA has revealed that the Cu concentration shows variations on the sub-mm scale, which gives rise to superconducting islands 40 . Moreover, a STM study reports an oscillatory behaviour of the Cu pair distribution function due to screened Coulomb repulsion of the intercalant atoms 41 .
In conclusion, we have investigated the angular variation of the upper critical field of superconducting crystals of Sr x Bi 2 Se 3 . The measurements reveal a striking two-fold anisotropy of the basal-plane B c2 . The large anisotropy cannot be explained with the anisotropic effective mass model or the variation of B c2 caused by flux flow. We have addressed two alternative explanations: (i) unconventional superconductivity, with an odd-parity triplet Cooper-pair state (Δ 4 pairing), and (ii) self-organized striped superconductivity due to preferential ordering of Sr atoms. The present experiments and results provide an important benchmark for further unraveling the superconducting properties of the new candidate topological superconductor Sr x Bi 2 Se 3 .
After completion of this work we learned that rotational symmetry breaking has been observed in two related superconductors, namely in Cu x Bi 2 Se 3 by means of specific heat experiments 42 and in Nb x Bi 2 Se 3 by means of torque magnetometry 43 .

Methods
Sample preparation. Single crystals Sr x Bi 2 Se 3 with x = 0.10 and x = 0.15 were prepared by melting high-purity elements at 850 °C in sealed evacuated quartz tubes, followed by slowly cooling till 650 °C at the rate of 3 °C/hour. Powder X-ray diffraction confirms the R m 3 space group (see Supplementary Information 32 ). Laue back-scattering diffraction confirmed the single-crystallinity and served to identify the crystal axes a and a * . Thin bar-like samples with typical dimensions 0.3 × 1.5 × 3 mm 3 were cut from the bulk crystal for the transport measurements.

Magnetotransport experiment.
Magnetotransport experiments were carried out in a PPMS-Dynacool (Quantum Design) in the temperature range from 2 K to 300 K and magnetic fields up to 9 T and in a 3-Helium cryostat (Heliox, Oxford Instruments) down to 0.3 K and fields up to 12 T. The resistance was measured with a low-frequency ac-technique in a 4-point configuration with small excitation currents, I, to prevent Joule heating (I = 0.5-1 mA in the PPMS and 100 μA in the Heliox experiments). The current was applied in the basal plane along the long direction of the sample. For in-situ measurements of the angular magnetoresistance the crystals were mounted on a mechanical rotator in the PPMS and a piezocrystal-based rotator (Attocube) in the Heliox. The samples were mounted such that the rotation angle θ 0 corresponds to B ⊥ I. Care was taken to align the a-axis with the current direction, but a misorientation of several degrees can not be excluded.