Abstract
A nematic topological superconductor has an order parameter symmetry, which spontaneously breaks the crystalline symmetry in its superconducting state. This state can be observed, for example, by thermodynamic or upper critical field experiments in which a magnetic field is rotated with respect to the crystalline axes. The corresponding physical quantity then directly reflects the symmetry of the order parameter. We present a study on the superconducting upper critical field of the Nbdoped topological insulator Nb_{x}Bi_{2}Se_{3} for various magnetic field orientations parallel and perpendicular to the basal plane of the Bi_{2}Se_{3} layers. The data were obtained by two complementary experimental techniques, magnetoresistance and DC magnetization, on three different single crystalline samples of the same batch. Both methods and all samples show with perfect agreement that the inplane upper critical fields clearly demonstrate a twofold symmetry that breaks the threefold crystal symmetry. The twofold symmetry is also found in the absolute value of the magnetization of the initial zerofieldcooled branch of the hysteresis loop and in the value of the thermodynamic contribution above the irreversibility field, but also in the irreversible properties such as the value of the characteristic irreversibility field and in the width of the hysteresis loop. This provides strong experimental evidence that Nbdoped Bi_{2}Se_{3} is a nematic topological superconductor similar to the Cu and Srdoped Bi_{2}Se_{3}.
Introduction
The search for topological superconductors, which support Majorana bound states as low energy excitations, is one of the central topics of current research.^{1,2,3,4,5} This is because of the fact that Majorana bound states are nonAbelian particles that have potential applications in engineering in the form of topologically protected qubits for quantum computation.^{6,7,8} A promising approach to the realization of topological superconductors is to use swave superconductors to induce superconducting pairing on topological insulators^{9} or systems with strong Rashba spin–orbit coupling.^{10,11,12,13,14,15,16,17,18} This results in effective pwave topological superconductors which support Majorana fermions.
Importantly, as shown by many experimental groups recently, topological insulators such as Bi_{2}Se_{3} can become superconducting when doped by metals such as Cu, Sr and Nb.^{19,20,21,22,23,24,25} Due to the nontrivial topology in the normal state band structures, the nature of pairing in these superconducting states has attracted much interest. In particular, it was proposed that Cudoped Bi_{2}Se_{3} could be a topological superconductor with odd parity pairing, which belongs to the A_{u} representation of the D_{3d} point group.^{26} There are Majorana surface states associated with these superconducting phases that should cause zerobias conductance peaks in tunneling experiments.^{21} However, the zerobias conductance peak found in an early experiment on Cu_{x}Bi_{2}Se_{3} is missing in more surface sensitive scanning tunneling experiments.^{27}
On the other hand, recent measurements on Cudoped and Srdoped Bi_{2}Se_{3} observed that the inplane threefold rotational symmetry is spontaneously broken into a twofold rotational symmetry below the superconducting transition temperature.^{28,29,30,31} This spontaneous breaking of rotational symmetry can be explained if the superconducting phase is in the nematic phase, which belongs to the twocomponent E_{u} representation of the point group.^{32,33}
Interestingly, the E_{u} representation, which is a twodimensional representation, allows another topological superconducting phase, which spontaneously breaks the timereversal symmetry. This phase is a nodal Weyl superconductor that supports Majorana arcs on the surfaces.^{33,34,35,36,37} It was proposed by Qiu et al.^{25} that Nbdoped Bi_{2}Se_{3} spontaneously breaks timereversal symmetry below T _{c}. The Nbdoped case may be special because of the finite magnetic moments of the Nb atoms intercalated in the van der Waals gap between the Bi_{2}Se_{3} layers,^{25} which may enhance this Weyl superconducting phase.^{34} On the other hand, other recent experiments on Nbdoped Bi_{2}Se_{3}, including torque magnetometry,^{24} penetration depth measurements,^{38,39} and Andreev reflection spectroscopy,^{40} suggested that the system is in the nematic phase. However, only the torque experiments were directly probing the inplane anisotropy, but the measurement were carried out in the irreversible regime and the results are difficult to interpret. In order to settle this issue we have measured the upper critical fields for magnetic fields applied in different directions in the basal plane, using magnetoresistance and DC magnetization measurements, and found a strong evidence of a twofold rotational symmetry below T _{c}. The twofold symmetry is further reflected in the DC magnetization value of the initial branch of the hysteresis loop after zerofield cooling, in the reversible thermodynamic contribution above the irreversibility field, as well as in the irreversible properties, such as the width of the hysteresis loop at a fixed magnetic field value. Our work thus provides further evidence that Nbdoped Bi_{2}Se_{3} is in the oddparity nematic superconducting phase. Due to the large superconducting volume fraction in Nbdoped Bi_{2}Se_{3} compared to the Cu and Srdoped cases, Nbdoped Bi_{2}Se_{3} can be an ideal material to investigate the topological properties of this nematic phase.
Results
The electric resistivity of Nb_{0.25}Bi_{2}Se_{3} (Sample 1) in zero magnetic field shows a superconducting transition in form of a sharp drop with onset at 3.2 K (see inset of Fig. 1). In Fig. 1 we show magnetoresistance data taken at fixed temperature of 0.34 K for different directions of inplane magnetic fields varying over more than 180°. A significant angular variation of the fielddriven transition can be seen. The resistive increase occurs at ~ 0.4 T for orientations around 0°, and is shifted to higher fields until the field of the midpoint approximately doubles with a value exceeding 1.4 T peaking at ±90°. To define the upper critical field transition we used different criteria to determine characteristic fields from the data at which for each angle a certain fixed percentage of the normal state resistance is reached: 50, 75 and 90%. The so defined characteristic fields are presented in Fig. 2 as a function of the inplane magnetic field orientation. The inset shows the same data in form of a polar plot. Centered at 0° a broad and rounded minimum is observed, while sharp peaks occur at ±90°. No signature of the threefold crystalline symmetry is obvious. The data is dominated by a pronounced twofold symmetry, which is completely in contradiction with the crystalline symmetry. Note that the normal state resistance above the upper critical field does not show any variation for the different inplane orientations of the magnetic field, thus suggesting an isotropic normal state within the trigonal basal plane. The superconductivity thus appears to be nematic, and the inplane angular dependence of the resistive transition is almost identical to what has been observed for Sr_{x}Bi_{2}Se_{3,} ^{30} although the upper critical field is clearly lower in Nb_{0.25}Bi_{2}Se_{3}.
In Fig. 3 we show a selection of magnetization hysteresis loop data of a second sample (Sample 2), which are taken at T = 1.8 K for different directions of the magnetic field applied parallel to the basal plane. The anisotropy of the upper critical field is evident. The data at the 101° angle have a clearly higher critical field, while the data at 0° show the lowest value as seen in the main figure showing an enlargement of the reversible regime of the magnetization above the irreversibility field H _{irr} below which the hysteresis loops open. Data were collected under identical conditions at 0, 48, 101, 138, 180, 228, 281 and 318 degrees. For clarity, we have only included angles from −43° (318°) to 101°, while larger angles follow exactly the same trend. The anisotropy is also reflected in the irreversible part, where the largest hysteresis occurs at 0° and the smallest at ~ 101°, as shown for the two extreme angles (0° and 101°) in the inset.
In Fig. 4a we show the dependence of the upper critical field H _{c2} on the orientation of the magnetic field vector in the basal plane over the full 360degree range, and in Fig. 4b the irreversibility fields H _{irr} below which flux pinning becomes efficient, thus causing an opening of the hysteresis loops. Both data show a similar twofold anisotropy. Upper critical field data of two different samples (Sample 2 and Sample 3) have been included in Fig. 4a, both of which show a very similar trend. In Fig. 4c, we show the magnetization value for Sample 2 from the initial zerofieldcooled branch of the magnetization loop at 8.5 mT, where the minimum associated with the lower critical field H _{c1} occurs. The dependence on the inplane direction of the magnetic field also reflects the twofold anisotropy of the superconducting phase. The inset shows the corresponding magnetization data at 0° and 101°. The same field angular dependence is observed in the reversible part of the DC magnetization above the irreversibility field (Fig. 4d), which represents the purely thermodynamic contribution to the DC magnetization, and thus provides a bulk thermodynamic proof of nematic superconductivity.
In Fig. 5. we show the temperature dependence of the upper critical field transition as obtained from the magnetoresistance for three different characteristic directions: for inplane fields along −90° (Fig. 5a) and 0° (Fig. 5b), where the maximum and minimum of the critical field is observed, respectively, and for magnetic fields applied perpendicular to the Nb_{0.25}Bi_{2}Se_{3} basal plane (Fig. 5c). The resulting phase diagram (Fig. 5d), compiled using the field at which the magnetoresistance reaches 90% of the normal state value, shows that the twofold inplane anisotropy is present at all temperatures below the superconducting transition temperature, although it becomes weaker with increasing temperature.
Discussion
Both, the magnetoresistive and the magnetic upper critical field transitions of Nb_{x}Bi_{2}Se_{3} show a clear twofold symmetry instead of the expected threefold symmetry, which suggests that the superconducting state of Nb_{x}Bi_{2}Se_{3} is nematic and spontaneously breaks the threefold crystalline symmetry, in perfect agreement with what has been observed for Cu_{x}Bi_{2}Se_{3} ^{28,29} and Sr_{x}Bi_{2}Se_{3}.^{30,31} Since the metal dopants are intercalated in the van der Waals gap between the Bi_{2}Se_{3} layers, it has been argued that 1D clustering in the form of stripe like ion patters could be responsible for the twofold anisotropy.^{30} While we cannot rule this out from our experiments, the fact that the normal state resistance does not show any dependence on the inplane field direction does make this scenario unlikely, and rather suggests a true nematic superconducting state. The superconducting Nbdoped Bi_{2}Se_{3} thus appears to be very similar to Cu_{x}Bi_{2}Se_{3} ^{28,29} and Sr_{x}Bi_{2}Se_{3},^{30} ^{,31} despite it has been shown that the Nb ions exhibit finite magnetic moments.^{25} The latter can be seen in the form of a small paramagnetic contribution in the normal state magnetization in Fig. 6, which follows a Curie Weiss behavior as illustrated by the fit. The paramagnetic behavior is observed down to T _{c} without any sign of magnetic ordering.
The pronounced twofold symmetry observed in inplane upper critical fields can be explained consistently within the GinzburgLandau theory and could be identified as a signature of the nematic superconducting state. It is known that for a nematic superconductor at zero field, the twocomponent order parameter \(\left( {\eta _1,\eta _2} \right)\) is pinned to \(\eta _0\left( {1,0} \right)\) or \(\eta _0\left( {0,1} \right)\) by sixth order terms in the GinzburgLandau free energy that arise from the crystalline anisotropy. Ref. 41 predicts that the inplane upper critical field \(H_{c2}\) of a nematic superconductor would have a sixfold symmetry, assuming that the nematic pinning effect on \(\left( {\eta _1,\eta _2} \right)\) is negligible in the fielddriven phase transition. This assumption, however, only applies if the superconducting phase is sufficiently close to the normal phase. The experimental data of \(H_{c2}\) show a strong twofold symmetry up to the 90% criterion, as shown in Fig. 2, indicating that to the precision of the experiment, the pinning effect on the \(\left( {\eta _1,\eta _2} \right)\) maintains. Based on these facts, we assume that \(\left( {\eta _1,\eta _2} \right)\)is pinned to \(\eta _0\left( {1,0} \right)\) or \(\eta _0\left( {0,1} \right)\) and solve the corresponding GinzburgLandau equation. In this way one obtains the inplane upper critical field as
Here \(\Gamma\) indicates the inplane anisotropy in the superconducting phase and the angle Θ is defined such that Θ = 0 is the normal direction to the mirror plane. The details of the derivation from the GinzburgLandau theory can be found in the Methods section. The above formula well describes the twofold symmetry in the inplane upper critical fields of nematic superconductors. With this formula, we fit the experimentally measured inplane upper critical field of Sample 1 as shown by the lines in Fig. 2, with fitting parameters: Γ = 3.32 and H _{c2}(0) = 1.44 T.
Of the three single crystals of different shapes we measured (see Methods section for details), no apparent correlation was observed with the orientation of the macroscopic crystal shape. In fact, the demagnetization factors for our samples with the parallel field orientation are hardly expected to have any significant influence on the inplane variation of the magnetic properties. Indeed, the initial ZFC branch of the magnetization for the different inplane field orientations all fall on top of each other (see inset of Fig. 4c). In any case, the upper critical field is not affected by the demagnetization factor and provides a solid proof of the nematic superconducting state. The theoretical model fits out data very well and further supports the existence of nematic superconductivity in Nb_{x}Bi_{2}Se_{3}. One open question is what determines the direction of the twofold superconducting gap function within the threefold symmetry of the basal plane. The gap function appears pinned along one of three identical crystalline directions and remains there during the entire experiment, even if the sample is brought into the normal state in between. Which direction the anisotropic superconducting gap function chooses likely only depends on microscopic details of the single crystal, such as surface roughness, defects, microcracks or local stress. However, the inplane anisotropy for Sample 1 (64% at 1.8 K) is much greater than that for Sample 2 and 3 (18% and 21% at 1.8 K, respectively). This is not an artifact from the different methods used since the upper critical field values for Sample 3 from magnetization and magnetoresistance agree very well (see Fig. 4a). This difference can only be understood if there are different domains in these samples in which the 2fold gap function is pinned along different crystalline directions and partially cancel the macroscopic anisotropy. Sample 1 is probably almost a monodomain sample with large anisotropy factor Γ = 11, which causes the particularly sharp maxima in Fig. 2. Otherwise, additional smaller peaks should occur at ±30°. In fact, there is actually a small bump at +30°, which could indicate an ~ 10% volume fraction of a minority domain rotated by 60° with respect to the majority domain. This is illustrated by the dashed line in Fig. 2, which represents a fit that takes into account such an admixture of a minority phase. However, this feature is very weak and at the resolution limit. For samples 2 and 3, the maxima are much broader, which is taken into account in the fitting curve in Fig. 4a by considering a smaller anisotropy factor Γ = 1.44. For a small Γ value, the cosine term dominates the angular inplane H _{c2} dependence. In the presence of minority domains, the individual sinusoidal terms would basically merge together without causing any side maxima in the angular H _{c2} dependence, but reducing the overall variation. What causes the strong variation in Γ is unclear, but Sample 1 is likely to be of higher crystalline quality, as evidenced by its shiny flat surfaces. Sample 2 and 3 have rather rough surfaces, and surface roughness, along with internal crystal irregularities, could lead to a broadening of the 2fold gap structure, in addition to the occurrence of minority domains.
To summarize, we have carefully determined the fieldangular dependence of the magnetoresistive and the magnetic upper critical field transitions of Nbdoped Bi_{2}Se_{3} with 0.25 Nb atoms per formula unit. The inplane angular dependence when the field is applied strictly parallel to the Nb_{0.25}Bi_{2}Se_{3} basal plane shows a pronounced twofold symmetry very similar to Cu_{x}Bi_{2}Se_{3} ^{28,29} and Sr_{x}Bi_{2}Se_{3},^{30,31} and thus provides further experimental evidence from two different experimental methods that nematic superconductivity also exists in Nb_{x}Bi_{2}Se_{3}. The twofold symmetry is also reflected in the absolute value of the magnetization (for example in the initial curve of the hysteresis after zerofield cooling, as well as in the thermodynamic reversible regime above the irreversibility field) and in the anisotropic characteristics, such as the width of the hysteresis loop and the irreversibility field. The inplane anisotropy of the upper critical field can be perfectly fitted with a theoretical model^{32,33,41} for nematic superconductivity. The existence of magnetic moments without macroscopic magnetic order could provide a way of tuning the superconducting properties, e.g., by varying the Nb concentration or by introducing different magnetic ions with stronger moments to see whether the unconventional pairing symmetry of the superconducting state could be dramatically altered by the internal magnetic fields.
Methods
The detailed growth method and characterization of Nb_{0.25}Bi_{2}Se_{3} in the single crystalline form can be found in refs. 24,25. Measurements have been done on three different samples of dimensions and geometry as illustrated in Fig. 7. The magnetic field directions of 0 and 90 degrees are marked by arrows.
Electrical transport
We carried out standard fourprobe resistance measurements in a 15 T magnet cryostat with ^{3}He variable temperature insert. The single crystal (Sample 1) of approximate dimensions 1.7 × 0.8 × 0.2 mm^{3} and demagnetization factor of ~ 0.7 was mounted on an Attocube ANR51/RES precise nano rotary stepper with resistive encoder, which provides a millidegree precision with integrated angle measurement. The current was injected along the long direction (Fig. 7) and coincided with the magnetic field direction for an inplane field orientation of 225°. The Nb_{0.25}Bi_{2}Se_{3} basal plane was aligned precisely parallel to the magnetic field and the rotator allowed us to rotate the sample so that the magnetic field direction could be varied within the basal plane. In a subsequent separate measurement, we oriented the same device with the basal plane perpendicular to the field to derive the complete temperature dependence of upper critical fields along different crystalline directions. The measurements were performed with both an AC and a DC method to check for consistency. All resistance data shown in this article were measured with the AC technique with an alternating current of 0.1 mA amplitude and a frequency of 17.7 Hz using a Keithley 6221 AC/DC current source. The signal was sent through a bandpass filter and measured by a SRS830 digital lockin amplifier.
DC magnetization
The DC magnetization was measured on two singlecrystalline samples of 847 μg (Sample 2) and 4.5 mg (Sample 3) mass with a Quantum Design Vibrating Sample SQUID magnetometer (VSMSQUID). The measurement was deliberately performed on samples of different shapes and sizes to eliminate any effects of the sample shape and quality. The shape of Sample 2 can be approximately described as a rectangular prism of 1.5 × 0.8 × 0.1 mm^{3} with demagnetization factor of ~ 0.8. Sample 3 yields an approximately cylindrical shape of ~ 1 mm diameter and 0.3 mm height with demagnetization factor of ~ 0.6. The samples were first cleaved to obtain a shiny and very flat bottom plane of the crystal, which was then fixed with vacuum grease to the flat surface of a quartz sample holder so that the basal plane of Nb_{0.25}Bi_{2}Se_{3} was oriented parallel to the applied magnetic field in all experiments. The vacuum grease allowed us to rotate the sample at room temperature by carefully pushing the sample with a toothpick into the desired angular orientation characterized by the inplane angle θ, carefully avoiding a contact with the grease. The sample holder was then inserted into the magnetometer. The sample was cooled to 1.8 K in zero field and a full hysteresis loop with maximum field of ±1 T was measured. Subsequently, the sample was heated above T _{c} and again cooled in zero field to repeat the measurement, but with reverse field direction to obtain data at an angle of θ + 180°. After this measurement cycle, the sample holder was removed, the sample was rotated by ~ 45°, and the procedure was repeated until the full 360° angle range was obtained. The exact angular orientations were determined with an accuracy of 0.5° from a photograph by measuring the angle between a linear edge of the sample with respect to the sample holder using a protractor. The absolute values of the inplane field direction θ are chosen so that θ = 0 is the normal direction to the mirror plane within the trigonal basal plane.
Theory
In the GinzburgLandau theory for a superconductor, the phenomenological free energy density f _{tot} is the sum of a homogeneous term f _{hom} and a gradient term f _{D} so that f _{tot} = f _{hom} + f _{D}. The superconducting Nb_{ x }Bi_{2}Se_{3} has a crystal structure belonging to the D_{3d} point group. In the E_{u} representation, the homogeneous part of the free energy density up to the sixth order has the form^{33}
where A, B _{1, 2} and C _{1, 2, 3} are the GinzburgLandau coefficients and \(\eta _ \pm = \eta _1 \pm i\eta _2\). Here A ∝ (T−T _{c}), B _{2} > 0 for the nematic phase, and C _{1} is responsible for pinning the nematic state from \(\left( {\eta _1,\eta _2} \right)\) to \(\eta _0\left( {1,0} \right)\) or \(\eta _0\left( {0,1} \right)\). Defining the covariant derivative \(D_i =  i\partial _i  qA_i\), where \(A_i\) is the electromagnetic vector potential and q = 2e, the gradient term, f _{D} can be written as^{41}
where we set i = x, y, and a = 1, 2, and J _{1}, J _{2}, J _{3}, J _{4}, J _{5} are the phenomenological GinzburgLandau coefficients. Here \(\varepsilon _{\left( {i,j} \right),\left( {a,b} \right)}\)is the antisymmetric tensor and \(\tau _{\left( {ij} \right),\left( {ab} \right)}^{\left( x \right),\left( z \right)}\) is the Pauli matrix acting on the index (ij) and (ab). In the presence of the inplane magnetic field \(\vec H = H\left( {\cos \Theta ,\sin \Theta ,0} \right)\) with the gauge \(\vec A = Hz\left( {\sin \Theta ,  \cos \Theta ,0} \right)\), the covariant derivative is defined as \(D_x =  i\partial _x + 2eHz\sin \Theta\), \(D_y =  i\partial _y  2eHz\cos \Theta\) and \(D_z =  i\partial _z\). The linearized GinzburgLandau equation can be derived from the functional variation with respect to the pairing field \(\frac{{\delta f_{{\mathrm{tot}}}}}{{\delta \eta _a^ * }} = 0\) so that
where \(\vec \eta = \left( {\eta _1,\eta _2} \right)^T\) and \(D_ \bot = D_x\sin \Theta  D_y\cos \Theta\). The parallel covariant derivative terms\(\left( {D_x\sin \Theta  D_y\cos \Theta } \right)\eta _a\) are dropped. We set \(\left( {\eta _1,\eta _2} \right)\) to be \(\eta _0\left( {0,1} \right)\) (which is a fully gapped state^{31,32}) and the coupled GinzburgLandau equations reduce to one equation
The righthand side of Eq. 5 can be treated as the Hamiltonian for a simple Harmonic oscillator, and the inplane upper critical field can be obtained from its ground state energy level
Here the dimensionless parameter \(\Gamma ^2 = \frac{{J_1J_3  J_3J_4}}{{J_1J_3 + J_3J_4  J_5^2}}\) indicates the inplane anisotropy and \(H_{c2}\left( 0 \right) =  \frac{A}{{2e\sqrt {J_1J_3 + J_3J_4  J_5^2} }}\). The inplane upper critical field has the same form for the nematic phase \(\eta _0\left( {1,0} \right)\) while the definitions of related parameters are different.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
R.L. thanks U. Lampe for technical support. This work was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (SBI15SC10, HKUST3/CRF/13G and C602616W). Y.S.H. acknowledges support from National Science Foundation DMR1255607. K.T.L. acknowledges the financial support from the Croucher Foundation and the Dr. TaiChin Lo Foundation.
Author information
Affiliations
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
 Junying Shen
 , WenYu He
 , Noah Fan Qi Yuan
 , Zengle Huang
 , Changwoo Cho
 , Kam Tuen Law
 & Rolf Lortz
Department of Physics, Missouri University of Science and Technology, Rolla, MO, 65409, USA
 Seng Huat Lee
 & Yew San Hor
Authors
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Contributions
J.S. performed the electrical transport experiments with the help of C.W.C. R.L. performed the magnetization measurements. S.H.L. and Y.S.H. did the sample preparation and characterization. J.S., R.L., C.W.C and H.Z. analyzed the data. W.Y.H., N.F.Q.Y. and K.T.L. did the theoretical interpretation and fitting of the data. R.L. and K.T.L. contributed to the original idea and supervised the project. J.S., R.L., K.T.L., W.Y.H. and N.F.Q.Y. wrote the manuscript. All authors contributed to the discussion and data interpretation and have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing financial interests.
Corresponding author
Correspondence to Rolf Lortz.
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Further reading

Superconducting and normalstate anisotropy of the doped topological insulator Sr0.1Bi2Se3
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