Abstract
Recently it was demonstrated that Sr intercalation provides a new route to induce superconductivity in the topological insulator Bi_{2}Se_{3}. Topological superconductors are predicted to be unconventional with an oddparity pairing symmetry. An adequate probe to test for unconventional superconductivity is the upper critical field, B_{c2}. For a standard BCS layered superconductor B_{c2} 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 Sr_{x}Bi_{2}Se_{3} crystals (T_{c} = 3.0 K). Surprisingly, fieldangle dependent magnetotransport measurements reveal a large anisotropy of B_{c2} when the magnet field is rotated in the basal plane. The large twofold anisotropy, while sixfold is anticipated, cannot be explained with the GinzburgLandau anisotropic effective mass model or flux flow induced by the Lorentz force. The rotational symmetry breaking of B_{c2} indicates unconventional superconductivity with oddparity spintriplet Cooper pairs (Δ_{4}pairing) recently proposed for rhombohedral topological superconductors, or might have a structural nature, such as selforganized stripe ordering of Sr atoms.
Introduction
Currently, topological insulators (TIs) are at the focus of condensed matter research, because they offer unprecedented possibilities to study novel quantum states^{1,2,3}. 3D TIs are bulk insulators with a nontrivial topology of the electron bands that gives rise to surface states at the edge of the material. The gapless surface states have a Diractype energy dispersion with the spin locked to the momentum and are protected by symmetry. This makes TIs promising materials for applications in fields like spintronics and magnetoelectrics^{1,2}. The concept of a TI can also be applied to superconductors, where the superconducting gap corresponds to the gap of the band insulator^{4,5}. Topological superconductors are predicted to be unconventional with an oddparity pairing symmetry^{6,7}. Much research efforts are devoted to 1D and 2D superconductors, where Majorana zero modes exist as protected states at the edge of the superconductor^{8,9}. Majorana zero modes with their nonAbelian statistics offer a unique platform for future topological quantum computation devices^{10}. Prominent candidates for 3D topological superconductivity are the Cu intercalated TI Bi_{2}Se_{3}^{11,12}, the doped topological crystalline insulator Sn_{1−x}In_{x}Te^{13} and selected topological halfHeusler compounds^{14,15,16}.
Among the 3D topological superconductors, Cu_{x}Bi_{2}Se_{3}, which has a superconducting transition temperature T_{c} = 3 K for x = 0.3^{11,12}, is the most intensively studied material. ARPES (Angle Resolved PhotoEmission Spectroscopy) experiments conducted to study the bulk and surface states reveal that the topological character is preserved when Bi_{2}Se_{3} is intercalated with Cu^{17}. By evaluating the topological invariants of the Fermi surface, Cu_{x}Bi_{2}Se_{3} is expected to be a timereversal invariant fullygapped oddparity topological superconductor^{6,7}. This was put on a firmer footing by a twoorbital pairing potential model where oddparity superconductivity is favoured by strong spinorbit coupling^{18}. Several experiments have been interpreted in line with topological superconductivity. The specific heat shows a full superconducting gap^{12}. The upper critical field exceeds the Pauli limit and has a temperature variation that points to spintriplet superconductivity^{19}. Much excitement was generated by the observation of a zerobias conductance peak in point contact spectroscopy, that was attributed 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 ingap 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 basalplane 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 . It is a layered material and Sr is intercalated in the Van der Waals gaps between the quintuple Bi_{2}Se_{3} layers^{23}. For a standard BCS (Bardeen, Cooper, Schrieffer) layered superconductor the anisotropy of B_{c2} is expressed by the parameter , where and are measured with the Bfield parallel and perpendicular to the layers, respectively^{27}. Whereas is normally isotropic, Sr_{x}Bi_{2}Se_{3} presents a unique exception. Fieldangledependent magnetotransport experiments demonstrate a large twofold basalplane anisotropy of B_{c2}, with T and 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 acsusceptibility 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 fieldangle 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 twofold symmetry, which is most clearly demonstrated in a polar plot (Fig. 2). We remark, the same twofold 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 twofold 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 Bfield 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}. As expected from the data in Fig. 1, we observe a large difference between and , with an inplane anisotropy parameter of 6.8 (at 1.9 K) and 2.6 (at 0.3 K) for x = 0.10 and x = 0.15, respectively. For both crystals . Obviously, the B_{c2} ratio γ for the field  and ⊥ to the layers now depends on the field angle and ranges from 1.2 to 3.2 for x = 0.15. In ref. 24 a value for γ of 1.5 is reported, whereas from the data in ref. 23 we infer a value of 1. In the top panels of Fig. 3 we show ρ(B) measured along the a, a^{*} and c axis at T = 2.0 K and T = 0.3 K for x = 0.10 and x = 0.15, respectively. The B_{c2}(T) values are determined by the midpoints of the transitions to the normal state and are indicated by open symbols in the lower panels. The agreement between both methods (field sweeps and temperature sweeps) is excellent. For the x = 0.15 sample we see a remarkable broadening for B  a. The initial small increase of ρ(B) between 4 and 6 T is most likely related to a sample inhomogeneity, because a similar tail is also observed in the R(T) data^{32}.
In Fig. 4 we show the angular variation of the upper critical field, B_{c2}(θ). For this experiment the crystals are placed on the rotator and the field is oriented in the basal plane. The data points are obtained as the midpoints of the transitions to the normal state of the R(B) curves measured at temperatures of 2 K for x = 0.10 and of 0.3 K and 2 K for x = 0.15 (see Fig. S7^{32}). All data sets show the pronounced twofold basalplane anisotropy of B_{c2}, already inferred from Figs 1 and 2.
Discussion
Having conclusively established the twofold 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 Xray 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 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 basalplane. 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 twofold 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 twoprobe method, is similar to that reported in Fig. 1. Thus the twofold anisotropy in B_{c2} is also present for the Bfield in the aa^{*}plane and the current along the caxis.
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 GinzburgLandau model^{27,34} the anisotropy of B_{c2} is attributed to the anisotropy of the superconducting coherence length, ξ, which in turn relates to the anisotropy of the effective mass. For a layered superconductor the anisotropy ratio ^{28}. Here m and M are the effective masses  and ⊥ to the layers. In the rhombohedral structure and M = m_{c}, where the subscripts a, a^{*} and c refer to the effective masses for the energy dispersion along the main orthogonal crystal axes (i.e. in the hexagonal unit cell). For a field rotation in the aa^{*}plane is in general isotropic, since . For a 3D anisotropic superconductor the angular variation B_{c2}(θ) in a principal crystal plane can be expressed as , where Γ = B_{c2}(90°)/B_{c2}(0°). To provide an estimate of Γ for Sr_{0.15}Bi_{2}Se_{3}, we compare in Fig. 4b the measured B_{c2}(θ) with the angular variation in the anisotropic effective mass model (solid line). We obtain B_{c2}(0°) = 2.3 T, B_{c2}(90°) = 7.4 T and Γ = 3.2. The effective mass ratio ^{34} would then attain the large value of 10.2. As we show below, this is not compatible with the experimental Fermisurface determination.
The Fermi surface of ndoped 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 twofold symmetry induced by the current. Clearly, the twofold symmetry (Fig. 4), while three fold is expected and the calculated large ratio using the GinzburgLandau model are at variance with the experimental Fermisurface determination^{35} and we discard this scenario.
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 spintriplet superconductivity developed in the context of Cu_{x}Bi_{2}Se_{3} and investigated the experimental consequences of Δ_{4} pairing in the twoorbital model^{18}. Here, superconductivity is described by an oddparity twodimensional 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 zerototal spin along an inplane 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 GinzburgLandau model, it can be shown that n is pinned to a direction in the basal plane. For , point nodes in the superconducting gap are found along , whereas for two gap minima occur at ^{31}. Our B_{c2}data can be interpreted as reflecting a strongly anisotropic superconducting gap function. The superconducting coherence length, ξ, along the main axes can be evaluated from the GinzburgLandau relations , and . Here Φ_{0} is the flux quantum. With the experimental B_{c2}values, taken at T/T_{c} = 0.1 in Fig. 3d for x = 0.15, we calculate ξ_{a} = 19.6 nm, nm and ξ_{c} = 5.4 nm. Interpreting ξ as the Cooperpair size, this implies that the pairing interaction is strongest along the a^{*} and caxis and weakest along the aaxis. The observation that can naively be translated to the gap structure consistent with the one predicted for . More recent calculations show that B_{c2} for the twodimensional E_{u} representation retains the hexagonal symmetry of the crystal lattice, but its symmetry can be lowered to twofold 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 spintriplet state^{39}.
Yet another interesting possibility is a selforganized 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 basalplane 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 submm 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 twofold anisotropy of the basalplane 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 oddparity triplet Cooperpair state (Δ_{4} pairing) and (ii) selforganized 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 highpurity elements at 850 °C in sealed evacuated quartz tubes, followed by slowly cooling till 650 °C at the rate of 3 °C/hour. Powder Xray diffraction confirms the space group (see Supplementary Information ^{32}). Laue backscattering diffraction confirmed the singlecrystallinity and served to identify the crystal axes a and a^{*}. Thin barlike 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 PPMSDynacool (Quantum Design) in the temperature range from 2 K to 300 K and magnetic fields up to 9 T and in a 3Helium cryostat (Heliox, Oxford Instruments) down to 0.3 K and fields up to 12 T. The resistance was measured with a lowfrequency actechnique in a 4point 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 insitu measurements of the angular magnetoresistance the crystals were mounted on a mechanical rotator in the PPMS and a piezocrystalbased rotator (Attocube) in the Heliox. The samples were mounted such that the rotation angle corresponds to B ⊥ I. Care was taken to align the aaxis with the current direction, but a misorientation of several degrees can not be excluded.
Additional Information
How to cite this article: Pan, Y. et al. Rotational symmetry breaking in the topological superconductor Sr_{x}Bi_{2}Se_{3} probed by uppercritical field experiments. Sci. Rep. 6, 28632; doi: 10.1038/srep28632 (2016).
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Acknowledgements
The authors acknowledge discussions with A. Brinkman, U. Zeitler, R.J. Wijngaarden and Liang Fu. This work was part of the research program on Topological Insulators funded by FOM (Dutch Foundation for Fundamental Research of Matter).
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Y.P. magnetotransport and acsusceptibility in the PPMS, data analysis; A.M.N. and G.K.A. magnetotransport in the Heliox. Y.K.H. crystal synthesis and Laue singlecrystal diffraction; Y.M. and T.N. temperature dependent Xray measurements. A.d.V. experiment design, supervision measurements, manuscript writing with contributions of Y.P.
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Pan, Y., Nikitin, A., Araizi, G. et al. Rotational symmetry breaking in the topological superconductor Sr_{x}Bi_{2}Se_{3} probed by uppercritical field experiments. Sci Rep 6, 28632 (2016). https://doi.org/10.1038/srep28632
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