Structural distortion behind the nematic superconductivity in Sr$_x$Bi$_2$Se$_3$

An archetypical layered topological insulator Bi$_2$Se$_3$ becomes superconductive upon doping with Sr, Nb or Cu. Superconducting properties of these materials in the presence of in-plane magnetic field demonstrate spontaneous symmetry breaking: 180$^\circ$-rotation symmetry of superconductivity versus 120$^\circ$-rotation symmetry of the crystal. Such behavior brilliantly confirms nematic topological superconductivity. To what extent this nematicity is due to superconducting pairing in these materials, rather than due to crystal structure distortions? This question remained unanswered, because so far no visible deviations from the 3-fold crystal symmetry were resolved in these materials. To address this question we grow high quality single crystals of Sr$_x$Bi$_2$Se$_3$, perform detailed X-ray diffraction and magnetotransport studies and reveal that the observed superconducting nematicity direction correlates with the direction of small structural distortions in these samples( $\sim 0.02$\% elongation in one crystallographic direction). Additional anisotropy comes from orientation of the crystallite axes. 2-fold symmetry of magnetoresistance observed in the most uniform crystals well above critical temperature demonstrates that these structural distortions are nevertheless strong enough. Our data in combination with strong sample-to-sample variation of the superconductive anisotropy parameter are indicative for significance of the structural factor in the apparent nematic superconductivity in Sr$_x$Bi$_2$Se$_3$.

We reveal that unusual 2-fold in-plane symmetry of superconductivity, recently explored in Srdoped topological insulator Bi2Se3, and associated with nematic topological superconductivity, is accompanied by co-aligned features in magnetoresistance well above critical temperature, signifying structural anisotropy of the crystal. Our detailed X-ray diffraction studies of the superconducting crystals do reveal small structural distortions (about 0.02% elongation in one in-plane crystallographic direction). Additionally, we reveal that the crystal consists of crystallites with high anisotropy in orientation of the crystallite axes. These observations indicate in favor of structural scenario for the apparent nematic superconductivity in this material. Bismuth selenide is a layered topological insulator material. Being doped with Sr, Nb or Cu it becomes superconductive, with T c around 3K and H c2 about a few Tesla [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Since their discovery it was debated whether these materials are topological superconductors [1] or not [2]. More recently this superconductivity(SC) was found to be nematic, i.e. superconducting properties depend strongly on the in-plane orientation of the magnetic field [3][4][5][6][7][8][9]. Critical magnetic fields, magnetization, resistivity, heat capacitance, and Knight shift have 180 • inplane rotation symmetry, contrary to the trigonal crystal (120 • ) symmetry. An explanation for such nematicity was suggested within the two component topological SC model [16][17][18]. However, extremely high in-plane anisotropy rates in superconductive Sr x Bi 2 Se 3 (in-plane H c2 max /H c2 min ratio ranges from 3 to 8) could not be expected theoretically [17].
An alternative explanation of the apparent nematicity would be self-organized structural stripiness. In this case one would expect both structural and electronic properties above T c to have the same two-fold anisotropy. All experimental efforts so far didn't reveal any structural distortions. In our research we have thoroughly studied Sr x Bi 2 Se 3 single crystals with X-Ray diffraction(XRD) and magnetotransport. We do find that magnetoresistance well above T c has the same two-fold in-plane asymmetry as superconductivity in agreement with slight triclinic distortion of the lattice found in XRD studies. More interestingly, we reveal block structure of the system and apparent anisotropy in orientation of the blocks. Thus our work gives arguments in favor of structural scenario and contrary to the topological one.
A series of Sr x Bi 2 Se 3 samples with nominal Sr content of x = 0.10, 0.15 and 0.20 were prepared using modified Bridgman method [19]. High purity elemental Bi, Se (99.999%) and Sr (99.95%) in the desired molar ratio were loaded in quartz ampoules inside inert atmosphere glove box. Sealed evacuated tubes were heated at 850 • C for 24 hours with periodic stirring followed by a slow cooling to 620 • C at a rate of ∼2 • C per hour. The sam-ples were then annealed at 620 • C for 48 hour and water quenched. The crystals obtained by this method had a mirrorlike surface and were readily cleaved along the basal plane. Their structure was characterized by single crystal XRD (Panalytical XPert Pro MRD Extended).
Studies of the crystals grown started with the X-ray diffraction selection of the proper samples. The crystals obtained in our growth process consisted of blocks (crystallites) with lateral dimensions 0.05 − 0.3 mm. The blocks had the same structure, slight variation of the caxis lattice parameter and misorientation up to 1 • with respect to each other. For detailed structural investigations and further transport measurement we cleaved samples with lateral dimensions 0.6-3 mm and thickness 50-150 µm. Detailed structural studies (2θ/ω scanning) were performed on the dominating block within the cleaved sample. Single crystals (4-wire resistance 10-100 mOhm) were mounted on the holders and the contact wires (diameter 0.02 mm) were glued with either silver or grahite paint (size of the paint drop was about 0.1 mm and resistance about 20-100Ohm per contact). We aligned the basal plane and plane of rotation by eye with precision ∼ 3 o parallel to each other.
Magnetotransport measurements were performed with four-terminal scheme using lock-in detection at frequencies  Hz and measurement currents up to 500 uA. We have checked that measurement current did not overheat the samples(i.e. did not shift the SC transition temperature). We used Cryogenics 21T/0.3K, dry CFMS 16T, and Quantum Design PPMS 9T systems equipped with platforms, that allowed to rotate the samples in-situ to ∼360 • . Magnetic field sweeps were performed from positive to negative direction and magnetoresistance was obtained by symmetrization of the data. For measurements of in-plane angular dependence of magnetoresistance (AMR), similarly to the previous investigators [3][4][5][6][7][8][9], we applied constant magnetic field at fixed temperature and rotated the sample. To exclude inevitable admixture of the Hall effect we subtracted the lowest harmonic A cos(φ) + B sin(φ) from the AMR. None of studied crystals demonstrates perfect uniformity in SC, i.e. AMR depends on choice of potential probes and never demonstrates a perfect 8-like shape. We studied small samples because they are expected to have less blocks and hence to be more uniform. Fig.  1a shows AMR for sample Sr 0.1 Bi 2 Se 3 #306-2 (1x1x0.05 mm) for two current flow directions, indicated in panel 1c by arrows. One can see that the nematicity is almost insensitive to the current flow direction. This observation is in-line with previous magnetotransport studies [7] where current flow was perpendicular to basal plane as well as with thermodynamic studies (heat capacitance [5] and magnetometry [4]), and it was also recently confirmed in Ref. [9] . Interestingly, SC direction coincides with crystallographical axis in the basal plane. Thus, we have systematically reproduced the results of Refs. [6,7,9] with our small crystals. However, as we have discovered from XRD studies, even this small single crystal Sr 0.1 Bi 2 Se 3 #306-2 consists of at least two blocks. The superconducting properties, including the two-fold axis direction, are stable i.e. don't change after two months exposure at ambient conditions and insensitive to thermal cycling.
AMR for large single crystals (2x3x0.05 mm) shown in Fig.1b suggests the presence of several SC domains. Indeed, for current flow direction from contact 7 to contact 4 AMR is similar to small samples, whereas for perpendicular current flow (from 1 to 4) besides the main SC direction (A), two other directions emerge (B and C). In the particular case of Sample #306-1, the angle between axes A, B and C is 60 o . This observation clearly evidences for presence of domains with various orientation of SC axis located close to contacts 1, and 2, aligned with different crystallographical directions than the main domain. Although this alignment agrees with the previous results [6,7], we show for the first time multidomain character of the single crystalline superconducting Sr x Bi 2 Se 3 samples. We believe that different SC domains correspond to different blocks or groups of blocks. SC multidomain structure of the material was seen by us in numerous samples, as well as it was indirectly seen from Refs. [6,7] (see discussion in Supplementary information).Inplane manifold structure was also observed in Nb x Bi 2 Se 3 with magnetization measurements [4]. The structure was unambiguously interpreted as fingerprint of SC spontaneous symmetry breaking. Our observations pose a question whether this pattern is a consequence of multidomain structure.
Absence of the structural indications of the threefold symmetry breaking in previous works [3][4][5][6][7][8][9] motivated us to perform detailed studies of the system above T c , namely magnetoresistance. Magnetoresistance in this system appeared to be non-monotonic and quite complex (See Fig. 2c,d ), nevertheless it allowed us to find some symmetries. Fig.2a shows angular dependence of SC suppression (red curve) at T = 2.3K and magnetoresistance (black curve) of the same sample at T = 5K B = 10T. The direction of the strongest SC coincides with the direction of maximal magnetoresistance. Apparently, current flow direction should generate asymmetry in the system and if it was the only case, magnetoresistance would depend on angle between current flow and magnetic field. However, experimentally applying current flow in perpendicular direction doesn't merely rotate AMR, but rather changes it in complex manner: A · cos(2φ) component vanishes. Such change is an indicator of the crystalline anisotropy, it was observed systematically in various samples. In some of our samples Fig.2b direction of maximal magnetoresistance coincides with direction of minimal H c2 . The magnetoresistance slowly weakens with temperature (Fig. 2c,d), and preserves its 2-fold symmetry up to 200K. In other words, the SC in-plane axis can be anticipated from AMR not only close to T c , as recently predicted in Ref. [20], but also well above T c , i.e, this effect is of structural nature.
In the most of previous structural studies of doped SC Bi 2 Se 3 [6,7,9,10] either Laue or the powder diffraction was used, that didn't allow to resolve fine structural imperfections. Strong (0 0 n) symmetrical reflections, explored previously for Sr x Bi 2 Se 3 by single crystal XRD [10,14], are sensitive only to the lattice parameter value along the c-axis and do not allow to study in-plane lattice anisotropy. We use reflection indices in the hexagonal lattice notations with omitted triple index in the basal plane, which is equal to minus sum first two indices. In order to find the lattice distortion in the basal plane, we used intensive (2 0 5) and (1 1 15) asymmetrical reflections. We used the grazing diffraction geometry for our studies (Fig. 3a)  . φ-scanning curves of the (2 0 5) reflection apparently demonstrate three-peak 120 • rotation symmetry of the studied crystalline structure. This reflection is already sensitive mainly to lattice parameter in the basal plane. We used a detector with the triple crystal-analyzer 3×Ge(220) for high resolution (2θ − ω)-scanning curves for each of these 3-fold (2 0 5) and 6-fold (1 1 15) peaks.
It is evident that when lattice distortions are absent, the maxima of (2 0 5) and (1 1 15) reflections in all azimuthal positions must be unchanged. However, as we have systematically observed in all our samples, the positions of these peaks change upon in-plane rotation. Fig.3b shows an example for the most anisotropic sample Sr 0.2 Bi 2 Se 3 #318-1 (anisotropy factor H c2 max /H c2 min = 8, see Supplementary Information). Variation of the (2 0 5) reflection peak position for φ = 114.7 • and φ = 354.3 • is about 0.02 • in Fig. 3b, and corresponds to 0.02% lattice parameter elongation along a-axis, as shown schematically in Fig.3d.
However, the same diffraction patterns might be caused also by a small deviation of the c-axis from the perpendicular to the basal plane (Fig.3e). In order to evaluate the role of these two types of distortions we used the (1 1 15) reflection that nominally has six-fold rotational symmetry. Were the c-axis not inclined, the positions of (1 1 15) and (-1 -1 15) reflection maxima obtained after 180 • rotation around the φ-axis would be the same. However, as Fig.3c shows, these positions of (1 1 15) (φ = 145.8 • ) and (-1 -1 15) (φ = 323.5 • ) differ by 0.01 • that corresponds to the 0.005 • inclination of the caxis towards a-axis. Inclinations towards the other axes in the basal plane for the same sample are negligible.
Thus, XRD clearly signifies reduced symmetry of the system. It should be noted, that for such fine structural measurements, high crystalline quality (large sizes of uniform blocks, absence of bending) is needed and special precautions were taken in order to avoid systematic shifts of reflection maximum. Before recording each (2θ − ω) curve the sample was (i) rotated around the ψ axis to achieve vertical position for the diffraction plane and (ii) shifted along the goniometer X-and Y-directions in order to achieve illumination of the whole sample by X-Ray. The precision of the lattice parameter measurements from the (2θ − ω) curves in our case was limited by crystal quality and was better than 0.0001Å.
In all studied single crystals distortion of both types was revealed. Although parameters of the deformation vary from sample to sample, the value of deformation itself was about the same. The direction of the SC axis was either parallel or perpendicular to in-plane direction of the maximal crystalline deformation, similarly to Ref. [7].
We also systematically detect correlation between the direction of the longest (or shortest, depending on sample) axis and superconductivity (see Figs.1a,b,and 1a,b) [TO ADD]. Apparently this preferable direction emerges during the crystal growth, because the latter occurs in a fixed direction (vertically) and generally coincides with one of the in-plane crystallographical axes. Real puzzle is a scale of H c2 -anisotropy: how can 0.02% lattice deformation cause up to a factor of eight large and sample dependent H c2 ratio?
To answer this question we studied φ-dependence of the rocking curve. We have chosen 6-fold reflection (1 1 15) to probe anisotropy with higher resolution on φ (each 60 • ). Fig.4a shows that rocking curve measured at φ = 40 • has smaller width than the ±60 • neighbors of φ = 40 • . As broadening of rocking curve is determined by misalignment of the blocks in the corresponding direction, the most realistic interpretation of these XRD data is a block structure shown in Fig.4b and Fig.4c. The direction of a-axis is the same for all blocks (as the red curve is the narrowest). At the same time there is much spread of the c-axis direction from block to block (Fig.4c) that causes wide rocking curve in the direction φ = -20 • .
Interestingly, the direction of blocks reasonably coincides with the direction of maximal elongation and maximal H c2 in the same sample, as shown in Fig.4. The blocks are evidently separated by some transition regions (grain boundaries), indistinguishable by XRD. To obtain a complete picture of the grain/boundary structure by real-space imaging of individual blocks additional studies by electron backscatter diffraction (EBSD) [21], scanning X-ray nano-beam diffraction microscopy (SXRM) [22] and transmission electron microscopy(TEM) are needed.
Another indication of grain boundaries is straightforwardly seen from the magnetotransport in perpendicular magnetic field. Figure 4d  #308-3. Hall mobility µ Hall ≈ 400 cm 2 /Vs is clearly seen from the Hall slope to resistivity ratio. At the same time, Shubnikov-de Haas mobility is found from the field of magnetooscillations onset µ SdH ∼ 1/B ons ≈ 1000cm 2 /Vs. In single-component uniform system both mobilities are governed by the same scattering processes and µ SdH /µ Hall is less than 2 [23]. µ SdH /µ Hall ∼ 2.5 ratio, observed in our case, as well as in Refs. [10,15] implies that for some reason resistivity is too high. Indeed, if grain boundaries are responsible for this high resistivity, whereas low-disorder crystallites provide intensive magnetooscillations starting from relatively low fields, this high µ SdH /µ Hall ratio is naturally explained. Anisotropic splitting of the sample into blocks also explains a complex character of the AMR above T c : transport current might flow either along grain boundaries or across them, therefore AMR depends not only on current-to-field angle, but also on current-to-grain boundary angle. Non-monotonicity of magnetoresistance (Fig.2c,d) also reflects various conductivity mechanisms in the grains and across the grain boundaries.
What are the reasons for the observed structural anisotropy? Our XRD studies reveal that one of crystallographical axes in the basal plane (let's call it a-axis) is typically co-aligned with the ampoule axis, i.e. with the vertical temperature gradient and, correspondingly, the growth direction. It means that among all nucleated crystallites survive only those with a-axis orientation along the temperature gradient. They grow in huge blocks with different orientations of c-axis. During the subsequent cooldown the inevitable stress occur. In order to relax the stress, each huge block is split into smaller blocks with small c-axis misorientation.
Possible scenario of the anisotropic SC would be if the grain boundaries host surface states with SC properties different from that of bulk carriers. In this case anisotropy would be caused by grain boundary. Scanning tunneling microscopy (STM) studies of the Sr x Bi 2 Se 3 crystals with bulk T c about 3K revealed for the surface state in Sr x Bi 2 Se 3 SC gap closing at T c ∼ 5 K [13], thus supporting the idea of surface SC. Interestingly, these STM studies revealed no in-plane anisotropy of SC properties in the basal plane. Thus, superconductivity along defects and grain boundaries could be a fruitful idea for explanation of the phenomenology of this system. This scenario is however in contradiction with the large SC fraction (up to 90% of the bulk), detected in Sr x Bi 2 Se 3 [7,10,14]. It is also inconsistent with sharp (∆T ∼ 0.1K) transitions from normal resistance to zero, because interfaces of different blocks should be spread over properties and lead to smooth SC transition.
The most promising scenario is recently suggested 2component order parameter with p-wave pairing, belonging to D 3 -symmetry class [16]. In this case the role of structural anisotropy is to favor the direction of nematic TSC orientation [17]. This scenario seems to explain a bunch of experimental facts: anisotropies of heat capacitance [5], magnetization [4], Knight shift [3], anisotropy survival under hydrostatic pressure [11], and proton bombardment [12]. The stability of the system originates from spin-orbit interaction [24]. However it remains unclear why the value and the spread of anisotropy factor in Sr x Bi 2 Se 3 is so high? Indeed, within nematic topological theories, degree of anisotropy is governed by material-specific parameters and structural distortions only establish the direction of the nematicity.
There is also a possibility , that small 0.02% lattice elongation and c-axis incline might cause much stronger modification of electronic spectrum and/or deformation of the ordinary s-wave SC pairing. In this case stability of the SC is protected by the Anderson theorem. However, we are not aware of any theoretical or experimental arguments for this possibility.
We consider the following scenario. The H c2 value is determined by coherence length, that has a small inplane anisotropy, because it reflects the thermodynamics of the system and should be weakly sensitive to the defects and grain boundaries. However, resistive measurements in the vicinity of H c2 probe essentially vortex phase, very sensitive to the pinning of the vortices. Ap-parently, preferable defect direction and grain boundaries make the pinning along the defect direction stronger (in line with our observations). Crucial experiment to elucidate the role of boundaries would be an observation of SC anisotropy (or its absence) in µm scale and sub-µm scale samples. If superconductivity anisotropy in defect-free micro-scale samples is absent, the overall idea of nematic SC in bismuth chalcogenides becomes doubtful. Presence of such anisotropy favors bulk topological nature of the superconductivity, however anisotropic s-wave with anisotropic pinning should also be examined.
Our results point to necessity of revision of a bunch of data to understand whether similar structural features correlate with anisotropy in Cu x Bi 2 Se 3 and Nb x Bi 2 Se 3 . In respect of pinning, vortex phase for in-plane magnetic field configuration should be carefully examined in these materials. To completely detune from vortices and yet have a chance to detect anisotropy, of crucial importance would be experiments on in-plane H c1 anisotropy.
Thus our experimental research reveals the block structure and intrinsic anisotropy of superconducting Sr x Bi 2 Se 3 . It now should be clarified to what extent these properties are inherent to the other superconducting bismuth chalcogenides and whether should the TSC scenarios in these materials be revisited.