Abstract
In condensed matter physics, spontaneous symmetry breaking has been a key concept, and discoveries of new types of broken symmetries have greatly increased our understanding of matter^{1,2}. Recently, electronic nematicity, novel spontaneous rotationalsymmetry breaking leading to an emergence of a special direction in electron liquids, has been attracting significant attention^{3,4,5,6}. Here, we show bulk thermodynamic evidence for nematic superconductivity, in which the nematicity emerges in the superconducting gap amplitude, in Cu_{x}Bi_{2}Se_{3}. Based on highresolution calorimetry of singlecrystalline samples under accurate twoaxis control of the magnetic field direction, we discovered clear twofold symmetry in the specific heat and in the upper critical field despite the trigonal symmetry of the lattice. Nematic superconductivity for this material should possess a unique topological nature associated with odd parity^{7,8,9}. Thus, our findings establish a new class of spontaneously symmetrybroken states of matter—namely, oddparity nematic superconductivity.
Main
The features of superconductivity are mostly governed by the superconducting (SC) gap Δ, or equivalently, by the SC wavefunction, which takes complex values with amplitude and phase degrees of freedom. Nematic superconductivity is characterized by spontaneous rotationalsymmetry breaking (RSB) in the amplitude of the SC gap^{8} (Fig. 1a). Thus, once a nematic superconductor is cooled below its SC transition temperature T_{c}, RSB in various bulk properties is expected to emerge, while the RSB is absent in the normal (N) state^{9}. We emphasize that nematic superconductors are distinct from known ‘unconventional’ superconductors such as pwave or dwave superconductors^{2}, in which spontaneous RSB occurs only in the phase factor of Δ, and thus phasesensitive junction techniques are required to detect it^{10}. Nematic superconductivity is also different from nematicity originating from the properties of normalstate conduction electrons recently found in various systems^{3,4,5,6}. Materials realization of nematic superconductivity has never been reported (see Supplementary Note 12 for details).
Among known superconductors, the copperdoped topological insulator Cu_{x}Bi_{2}Se_{3} (ref. 11), consisting of triangularlattice layers of Bi and Se with intercalated Cu between layers (Fig. 1b, c), is rather unique. In addition to the ordinary swave SC state Δ_{1}, possible unconventional oddparity SC states, labelled Δ_{2}, Δ_{3}, Δ_{4x}, and Δ_{4y}, originating from strong spin–orbit interactions and its multiorbital nature have been proposed (Fig. 1d)^{7,12,13}. These oddparity states can be also categorized as topological SC states, which are accompanied by stable surface states originating from the nontrivial topology of the SC wavefunction. Among these states, the Δ_{4x} and Δ_{4y} states are predicted to be nematic SC states with a nonzero nematic order parameter^{8}. Experimentally, the zerobias conductance peak observed in the pointcontact spectroscopy, indicating the existence of unusual surface states, evidences topological superconductivity^{14}. On the other hand, a scanning tunnelling microscopy (STM) experiment on the abplane revealed swavelike tunnelling spectra^{15}, which was later actually found to be inconsistent with the swave (Δ_{1}) scenario^{16}. Rather, it was proposed that the STM spectra may be explained within the topological superconductivity scenario by taking into account the possible quasitwodimensional (Q2D) nature of the Fermi surface^{12,17}. More recently, by means of the nuclearmagnetic resonance (NMR), spinrotational symmetry was revealed to be broken in the SC state^{18}, suggesting realization of the Δ_{4x} or Δ_{4y} states.
In this Letter, based on the highresolution specificheat C measurements of singlecrystalline Cu_{x}Bi_{2}Se_{3} (T_{c} ≈ 3.2 K) under accurate twoaxis fielddirectional control using a vector magnet with a rotating stage^{19}, we report thermodynamic evidence for spontaneous RSB in the SC gap amplitude for the first time among any known superconductors. We further obtained evidence for the Δ_{4y} state, with gap minima (or nodes) along one of the Bi–Bi bonding directions. Our results unambiguously show that Cu_{x}Bi_{2}Se_{3} belongs to a new class of materials with oddparity nematicity and topological superconductivity. We emphasize that the conclusion holds irrespectively of the dimensionality of the actual normalstate electronic structure (see Supplementary Note 9).
In Fig. 2a, we compare the dependence of C/T on the inplane field angle φ for Sample #1 in the SC and N states. Here, as shown in Fig. 1c, we define the xaxis as one of the six equivalent Bi–Bi bond directions within the abplane, the yaxis as the direction perpendicular to x within the plane, and φ as the azimuthal angle of the field with respect to x. In addition, the zaxis is parallel to the caxis and the angle θ is the polar angle with respect to z. Although C/T is independent of φ in the N state, C(φ)/T in the SC state unexpectedly exhibits clear twofold oscillation. For the rhombohedral crystal symmetry of Cu_{x}Bi_{2}Se_{3}, C(φ)/T should exhibit sixfold oscillation. Thus, the observed twofold oscillation in C(φ)/T clearly breaks the rotational symmetry of the underlying lattice. The RSB is more easily recognized in the polar plot of C(φ)/T in Fig. 2b. A possible extrinsic origin for such RSB is field misalignment with respect to the abplane. To examine this possibility, we measured the dependence of C on the polar angle θ for various φ, presented as a surface colour plot in Fig. 2c (also see Supplementary Fig. 3). Evidently, C(θ)/T exhibits minima at θ = 90° for any φ, excluding the possibility of field misalignment. In addition, the twofold oscillation has been reproduced in several samples (Supplementary Fig. 7). Furthermore, one sample (#3) exhibits shifted and smaller oscillations, indicating the existence of multiple ‘nematic domains’, which manifest the spontaneous nature of the RSB (see Supplementary Note 7). Therefore, the rotational symmetry of the lattice is intrinsically and spontaneously broken in the SC state, evidencing nematic superconductivity in Cu_{x}Bi_{2}Se_{3}.
Next, we discuss the inplane anisotropy of H_{c2} presented in Fig. 2d. In Fig. 2e, we present the fieldstrength dependence of C/T at 0.6 K for various inplane field directions. Clearly, C(H)/T curves again do not obey the expected sixfold rotational symmetry: the curves are substantially different between φ = 0° and 60°. We here define H_{c2} as the onset of deviation from the linear field dependence in the N state (see Supplementary Note 10). The obtained H_{c2}(φ) (Fig. 2d) is clearly dominated by twofold oscillation. Indeed, by fitting H_{c2}(φ) with H_{0} + H_{2} cos(2φ) + H_{6} cos(6φ), we obtain μ_{0}H_{0} = 2.37 ± 0.03 T, μ_{0}H_{2} = 0.37 ± 0.04 T, and μ_{0}H_{6} = −0.05 ± 0.04 T. Interestingly, H_{2} is as large as 16% of H_{0}. This striking inplane H_{c2} anisotropy not only supports the nematic SC state of Cu_{x}Bi_{2}Se_{3}, but also indicates the existence of a single nematic domain in this sample (see Supplementary Note 7).
As one can clearly see in Fig. 2, the observed inplane fieldangle dependences of C and H_{c2} are both dominated by the twofold components. The large twofold behaviour is probably due to the dominance of the twofold component originating from the nematic SC order parameter over the ordinary component, such as the one originating from Fermisurface anisotropy, as discussed in Supplementary Note 13.
Among the proposed SC states for Cu_{x}Bi_{2}Se_{3}, only the Δ_{4x} and Δ_{4y} states spontaneously break the inplane rotational symmetry^{8,9}. Thus, the observed bulk nematicity provides strong evidence for the two possible nematic states, the nodal Δ_{4x} state (with nodes along the k_{y}direction, protected by the mirror symmetry^{8}) and the fully gapped Δ_{4y} state (with gap minima along the k_{x}direction; also see Supplementary Note 9) can be distinguished by the position of the gap minima or nodes. With this aim in mind, we investigate C(φ)/T of Sample #1, with a single nematic domain, in more detail. When the SC gap has minima (including nodes), C/T exhibits oscillatory behaviour as a function of field angle, because of the fieldangledependent quasiparticle excitations originating from the gap anisotropy^{20,21}. At lowtemperature and lowfield conditions, C/T exhibits minima when the field is parallel to the Fermi velocity at a gap minimum v_{F}^{min}, as shown in Fig. 3d. Furthermore, it has been predicted and observed that, in addition to the C/T oscillation originating from H_{c2} anisotropy, the C/T oscillations exhibit sign changes depending on the temperature and field conditions: for example, at intermediate temperatures, C/T exhibits maxima for H ∥ v_{F}^{min} (refs 22,23). Thus, detailed experiments as well as comparison with theoretical calculations are required to conclude the gap structure.
Figure 3a–c represents the observed C(φ)/T curves in various conditions. Twofold oscillation with a minimum at φ = 0° (H ∥ x) is observed at 0.6 K in the SC state. However, at higher temperatures, the oscillation inverts sign above ∼1.5–2.0 T at 1.0 K and ∼1.0 T at 1.5 K, exhibiting a maximum at φ = 0°. The temperature and field dependence of the oscillation prefactor A_{2} is summarized in Fig. 3e as a colour plot. A boundary between positive and negative A_{2} exists within the SC phase. These observations are compared with theoretical calculations based on the Kramer–Pesch approximation in Fig. 3f, g. Here, we assume a gap structure with gap minima or point nodes along an inplane direction φ = φ_{min} on a spherical Fermi surface. The calculated C(φ)/T curves for both cases of gap minima and nodes resemble the observed ones fairly well: they exhibit twofold oscillation with a minimum for φ = φ_{min} at low temperatures and low fields, and reversed oscillation with a maximum for φ = φ_{min} at elevated temperatures. This oscillation inversion is due to contributions from the density of states at finite energies, as well as of quasiparticle scattering by vortices^{22}. As shown in Fig. 3h, the phaseinversion line passes T/T_{c} ∼ 0.35 near H/H_{c2} = 0 and T/T_{c} ∼ 0.15 near H/H_{c2} = 0.25, again qualitatively similar to the observation (Fig. 3e).
From these agreements between experiment and theory, we conclude that the SC gap of Cu_{x}Bi_{2}Se_{3} is Δ_{4y}, possessing gap minima or nodes lying along the k_{x}direction. Although it is not straightforward to distinguish gap minima or nodes from our data alone, it is more natural to expect that the Δ_{4y} state is fully gapped to have gap minima, due to symmetry and energetic reasons^{8} (see Supplementary Note 9).
The oddparity nematic SC state in Cu_{x}Bi_{2}Se_{3} unveiled here is accompanied by a nematicity in the macroscopically coherent oddparity wavefunction. In this respect, it is clearly distinct from any known nematic states realized in liquid crystals or in nonSC electrons. In the future, it would be interesting to investigate the unusual consequences of oddparity nematic SC ordering, such as unusual quantization phenomena associated with topological defects or new collective modes arising from the nematic order parameter.
Note added in proof: We notice that a report on twofold symmetric behaviour in resistivity of a related compound, Sr_{x}Bi_{2}Se_{3}, appeared on the arXiv server (arXiv:1603.04197; 14 March 2016) shortly after ours (arXiv:1602.08941; 29 February 2016). This work has been recently published^{24}.
Methods
Sample preparation and characterization.
Single crystals of Bi_{2}Se_{3} were grown by a conventional meltgrowth method. Singlecrystal samples of Cu_{x}Bi_{2}Se_{3} were then obtained by intercalating Cu to single crystals of Bi_{2}Se_{3} by means of an electrochemical technique^{25}. The value of the Cu content x was determined by the total charge flow during the intercalation process, as well as by the mass change between before and after the intercalation. The crystal axis directions were determined from Laue photos before the intercalation. The onset T_{c} was checked by using a superconducting quantum interference device (SQUID) magnetometer (MPMS, Quantum Design). After this process, the samples were stored in vacuum until they were mounted to the calorimeter. In the present study, we used three samples, which are labelled Samples #1, #2 and #3. All samples are single crystals of rectangular shapes with x ≃ 0.3. The characteristics of each sample are listed in Supplementary Table 1.
Specificheat measurement.
We used a ^{3}He^{4}He dilution refrigerator (Kelvinox 25, Oxford Instruments) to cool down the samples. We performed specificheat measurement in the temperature range 0.09 K ≤ T ≤ 4 K. We constructed a custommade highresolution calorimeter^{26}, shown in Supplementary Fig. 1a. In our calorimeter, a sample was sandwiched between a thermometer and a heater, both of which were made using RuO_{2} chip resistors. We mounted a sample inside a glove box with an Ar atmosphere. We measured the specific heat by using the a.c. method^{27}: we applied an a.c. current to the heater using a current source (6221, Keithley Instruments), and measured the resultant temperature modulation amplitude T_{ac} and the phase shift φ_{ac} using lockin amplifiers (SR830, Stanford Research Systems). The offset sample temperature T was also recorded by another lockin amplifier. An excitation current to the thermometer was applied using another current source. For most of the data, the raw heat capacity C_{raw} was then obtained as C_{raw} = [P/(2ω_{H}T_{ac})]sinφ_{ac}, where P is the a.c. heat flow produced by the heater and ω_{H} is the frequency of the heater current. Multiplication by the factor sinφ_{ac} allows for more accurate evaluation of the heat capacity, even for smaller frequencies^{28}. Notice that the frequency of the temperature oscillation is two times higher than ω_{H}. We typically used ω_{H} = 1.3 Hz and T_{ac}/T ∼ 1.5–2.0% for measurements. For the temperature dependence of C/T below 0.6 K and at zero field (Supplementary Fig. 2), we evaluated C_{raw} by another method to improve the accuracy: we measured the ω_{H} dependence of T_{ac} and fitted T_{ac}(ω_{H}) with the function T_{ac}(ω_{H}) = [P/(4ω_{H}C_{raw})][1 + (2ω_{H}τ_{1})^{2} + (2ω_{H}τ_{2})^{−2}]^{−1/2} to obtain C_{raw}, where τ_{1} and τ_{2} are the external and internal relaxation times, respectively. The heat capacity of the sample stage (addenda) was measured separately by using a piece of pure silver as a reference sample, and subtracted from the total heat capacity to extract the sample contribution. The addenda contribution is typically 10–20% of the total heat capacity. We confirmed that the addenda does not produce detectable changes in either the θ or φ dependences.
Magneticfield control.
We applied the magnetic field using a vectormagnet system^{19}, which consists of two orthogonal superconducting solenoids (pointing in the vertical and horizontal directions, respectively) and a rotation stage, as schematically shown in Supplementary Fig. 1c. Within the laboratory frame, we can rotate the magnetic field both vertically, by changing the relative field strengths of the two orthogonal solenoids, and horizontally, by using the horizontal rotation stage. For the vertical rotation, the magnetic field of the solenoids can be controlled with a resolution of 0.1 mT, which results in an angular resolution of 0.006° at 1 T and 0.06° at 0.1 T. The horizontal rotator has an angular resolution of 0.001°, with negligible backlash. Thus, this system allows for a precise twoaxis control of the field direction. The directions of the crystalline axes with respect to the laboratory frame are determined by making use of the anisotropy in H_{c2}. Once the directions of the crystalline axes are determined, we can rotate the magnetic field within the sample frame with the aid of our automation software. All field angle values presented in this Letter are defined in the sample frame. The precision and accuracy of the field alignment are approximately 1°. Notice that this is worse than those achieved for more anisotropic superconductors such as Sr_{2}RuO_{4} (ref. 26), because of the relatively small outofplane H_{c2} anisotropy of Cu_{x}Bi_{2}Se_{3} (see Supplementary Note 11). Nevertheless, this small anisotropy of Cu_{x}Bi_{2}Se_{3} in turn makes the misalignment effect rather small, as explained in Supplementary Note 3.
Theoretical calculation.
The fieldangledependent heat capacity is calculated on the basis of the Kramer–Pesch approximation, which is appropriate in a lowmagneticfield region^{29}. By using the quasiclassical framework, a Dirac Bogoliubovde Gennes (BdG) Hamiltonian which describes a topological superconductivity with point nodes derived from the firstprinciple calculations is reduced to a BdG Hamiltonian for spintriplet pwave superconductivity. The corresponding dvector is d(k_{F}) = (v_{Fz} sinφ_{N}, −v_{Fz} cosφ_{N}, v_{Fy} cosφ_{N} − v_{Fx} sinφ_{N}). For this state, point nodes are located in the φ_{N}direction on the abplane^{30}. Here, we consider a threedimensional spherical Fermi surface. In the case of a fully gapped order parameter with gap minima, we consider the gap Δ(k) = d(k_{F}) (1 − r) + d_{max}r, where d_{max} is the maximal value of d(k_{F}) and r (0 < r < 1) corresponds to the ratio between the minimal and maximal values of Δ(k_{F}). We also performed calculations for a Q2D Fermi surface. We used a Q2D tightbinding model for the normalstate electronic band^{17} and assumed a pointnodal gap^{31}, as schematically shown in the inset of Supplementary Fig. 6c.
Data availability.
Data supporting the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge T. Watashige, S. Kasahara and Y. Kasahara for their technical assistance and L. Fu, M. Ueda, Y. Yanase, J. Yamamoto, Y. Matsuda, A. Yamakage, Y. Tanaka and T. Mizushima, for fruitful discussions. This work was supported by JSPS GrantinAids for Scientific Research on Innovative Areas on ‘Topological Quantum Phenomena’ (KAKENHI JP22103002, JP22103004) and ‘Topological Materials Science’ (KAKENHI JP15H05852, JP15H05853, JP16H00995), JSPS GrantinAids KAKENHI JP26287078, JP26800197, and DFG CRC1238 ‘Control and Dynamics of Quantum Materials’, Project A04.
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This study was designed by S.Y., Y.A. and Y.M.; K.T. and S.Y. performed specificheat measurements and analyses, with assistance from S.N. and guidance from Y.M.; Z.W., K.S. and Y.A. grew singlecrystalline samples and characterized them. Y.N. performed theoretical calculations. The manuscript was prepared mainly by S.Y. and K.T., based on discussions among all authors.
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Yonezawa, S., Tajiri, K., Nakata, S. et al. Thermodynamic evidence for nematic superconductivity in Cu_{x}Bi_{2}Se_{3}. Nature Phys 13, 123–126 (2017). https://doi.org/10.1038/nphys3907
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DOI: https://doi.org/10.1038/nphys3907
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