Uniaxial-strain control of nematic superconductivity in SrxBi2Se3

Abstract

Nematic states are characterized by rotational symmetry breaking without translational ordering. Recently, nematic superconductivity, in which the superconducting gap spontaneously lifts the rotational symmetry of the lattice, has been discovered. In nematic superconductivity, multiple superconducting domains with different nematic orientations can exist, and these domains can be controlled by a conjugate external stimulus. Domain engineering is quite common in magnets but has not been achieved in superconductors. Here, we report control of the nematic superconductivity and their domains of SrxBi2Se3, through externally-applied uniaxial stress. The suppression of subdomains indicates that it is the Δ4y state that is most favoured under compression along the basal Bi-Bi bonds. This fact allows us to determine the coupling parameter between the nematicity and lattice distortion. These results provide an inevitable step towards microscopic understanding and future utilization of the unique topological nematic superconductivity.

Introduction

In nematic states of liquid crystals, bar-shaped molecules exhibit orientational ordering. Because of the peculiar partial ordering property, the orientation of the molecules and hence the structure around defects are easily controlled by external stimuli, as widely utilized in liquid-crystal displays. Analogous phenomena in electronic systems, nematic electron liquids, where conduction electrons exhibit orientational ordering, have been revealed1,2. Here, orientational properties are also highly controllable, and observations of such tunability have played fundamental roles to clarify driving mechanisms3,4.

A more exotic form of nematicity has been discovered in AxBi2Se3 (A = Cu, Sr, Nb)5,6,7,8,9: nematic superconductivity5, in which the superconducting (SC) gap amplitude spontaneously lifts the rotational symmetry of the lattice as a consequence of Cooper-pair formation. After the initial observations, nematic SC behavior has been consistently observed by using various probes such as transport, thermodynamic, and microscopic techniques10,11,12,13,14,15,16,17,18, as reviewed in ref. 19. We comment here that these doped Bi2Se3 nematic superconductors are quite distinct from iron pnictide superconductors, where nematicity occurs in the normal state as a consequence of orbital ordering and superconductivity eventually occurs at much lower temperatures4. Although control of such normal-state nematicity has been intensively tested in iron pnictides, a direct control of nematicity of Cooper pairs has never been achieved. It is essential to show such control over the nematic SC orientation for further research developments of nematic superconductivity.

In this Article, we report the first control of nematic superconductivity in SrxBi2Se3, through application of in-situ tunable uniaxial stress along the a axis (meaning a Bi−Bi bond direction). We reversibly controlled the nematic domain structure, allowing us to determine the sign of the coupling constant between the nematicity and lattice distortion.

Results

Doped Bi2Se3 nematic superconductors

Our target materials family AxBi2Se3 is derived from the topological insulator Bi2Se320,21, which has a trigonal crystalline symmetry with three equivalent crystalline a axes in the basal plane (Fig. 1a)22. Because the superconductivity induced by A ion intercalation23,24,25 occurs in its topologically non-trivial bands26,27, the resultant superconductivity can also be topologically non-trivial. Indeed, topological SC states have been proposed, among which a pair of SC states in the two-dimensional Eu representation, Δ4x and Δ4y, are nematic SC states5,28,29,30. The SC gap amplitude of the Δ4x and Δ4y states are twofold anisotropic and their maximum amplitude is located along the a and a* axes, respectively (Fig. 1a). Since there are three equivalent basal axes, each state has three degenerate order-parameter orientations, as shown in Fig. 2c. In addition, it has been shown that there is sample-to-sample variation in whether the nematicity aligns along the a or a* axes 11,16. This fact suggests that Δ4x and Δ4y states are nearly degenerate states. Therefore, in total, a AxBi2Se3 sample can contain up to six kinds of nematic SC domains, in which nematicity of each domain is selected by a certain pre-existing local symmetry breaking field such as possible structural distortion or A ion distribution. Here, we probe whether applied uniaxial stress can overcome this pinning and alter the SC domain configuration.

Fig. 1: Uniaxial-strain control of nematic superconductivity in SrxBi2Se3.
figure1

a Crystal structure of the mother compound Bi2Se3. The right figure shows the definitions of the axes and the field angle ϕab with respect to the crystal structure in the ab plane, with three equivalent a and a* axes. b Photograph of the sample in the uniaxial strain cell with 4-wire terminal configuration. I and V labels next to the gold wires indicate the current and voltage leads, respectively. The large yellow arrows indicate the direction of the external strain, which was applied parallel to the x axis. c Magnetoresistance at specified field directions in the ab plane (ϕab = 0°,  −30°,  −60°,  −90°), with and without Δεxx. The data were obtained at 2.2 K and with 250 μA applied current. A substantial change in the magnetoresistance curves under large Δεxx (green curves) provides evidence for the in-situ uniaxial-strain control of the nematic superconductivity. Source data are provided as a Source Data file.

Fig. 2: Disappearance of nematic superconducting domains in SrxBi2Se3 under compressive strain.
figure2

a b Color polar plot of magnetoresistance for Hab measured at the relative strains of Δεxx = 0 (a) and −1.19% (b) with 250 μA applied current and 2.2 K. The light-green regions extending along ±30 and ±150° in a indicate existence of nematic subdomains, which substantially disappears under applied strain (b). The contours are drawn from 0.5 mΩ to 7.5 mΩ in steps of 1 mΩ. c, Table of the 6 possible nematic superconducting states that can exist in the sample as domains. Xn and Yn (n = 0, 1, 2) domains exhibit Δ4x and Δ4y states with the large Hc2 along one of the a axes (ϕab = (60n)°) and a* axes (ϕab = (90 + 60n)°), respectively, as indicated with the red arrows. The crystal structure in the ab plane of Bi2Se3 is shown with the schematic superconducting wave function in its center. The thickness of the blue crescent depicts the superconducting gap amplitude. Source data are provided as a Source Data file.

Magnetoresistance without and with uniaxial strain

In this work, we measured the magnetoresistance of single-crystalline Sr0.06Bi2Se3 samples (with the critical temperature Tc of 2.8 K; see Supplementary Note 1). The sample was affixed onto a custom-made uniaxial strain cell31, a modified version of the recent invention32, mounted inside a vector magnet. The sample was cut along one of the a axes (Bi-Bi bond direction), which we define as the x axis (Figs. 1a and b). Both the uniaxial force and electric current were applied along this x axis. The angle between the magnetic field and x axis is denoted as ϕab. We comment that twofold behavior in the normal state is practically absent in our sample, and is not altered by the application of compressive strain (Supplementary Note 13). Following the ordinary convention, we defined the sign of the strain so that negative sign corresponds to compressive strain.

In Figs. 1c and 2a, b, we present the magnetoresistance at 2.2 K for various ϕab. First, focus on the data with the relative strain Δεxx of 0%, i.e. zero applied voltage to the piezo stacks (black curves in Fig. 1c), corresponding to the actual strain of around +0.10% (tensile) due to the thermal-contraction difference of the sample and strain cell (“Methods” section). Clearly, superconductivity is more stable for ϕab = −90° (H − y) than 0° (Hx), resulting in a prominent twofold upper critical field Hc2, which is indicative of the nematic superconductivity8,11. This observed anisotropy Hc2 − y > Hc2x is consistent with the Δ4y state with the SC gap larger along y33, which is schematically shown as the Y0 state in Fig. 2c. Interestingly, additional sixfold behavior emerges at the onset of the SC transition between 1 and 2 T, as clearly visible by the green region extending along ϕab = −30° or  +30° in Fig. 2a (see Supplementary Fig. 2 for raw data). This sixfold component indicates that the sample contains minor parts exhibiting large Hc2 along ϕab = ±30°, namely the Y1 and Y2 domains (both belonging to the Δ4y state) in Fig. 2c with their gap maxima along the ±30° directions.

Next, let us focus on the data under applied strain of Δεxx = −1.19% (green curves in Fig. 1c) corresponding to the actual compressive strain of around εxx −1.1%; the largest measured compressive strain in the elastic limit (see Supplementary Note 3). Notably, the magnetoresistance at the SC transition is substantially altered, marking the first in-situ uniaxial-strain control of nematic superconductivity. More specifically, the SC transition becomes sharper with strain except near ϕab = −90° (Hy). Moreover, comparing the color plots in Figs. 2a and b, we can notice that the weak sixfold SC onset due to domains, seen in the Δεxx = 0 data, is substantially reduced by the applied strain. Thus the primary effect of the compressive uniaxial strain is to suppress the minor nematic domains.

Upper critical fields

From the magnetoresistance data, we defined Hc2 as the field where the resistance R(H) divided by the normal-state resistance Rn reaches various criterion values (“Methods” section; Supplementary Note 4). In the strain dependence of Hc2 (Fig. 3), there is a high reproducibility among measurement cycles within the present strain range, manifesting that strain response is repeatable and thus our sample is in the elastic deformation regime. Reproducibility across samples has also been demonstrated (see Supplementary Note 5). Comparing data for various field directions, we can see that Hc2x largely reduces under strain, attributable to the disappearance of minor nematic SC domains. In contrast, Hc2 along the y and z axes (Fig. 3), as well as the zero-field Tc (Supplementary Fig. 7), is only weakly affected by strain, with small decreasing trend under compression.

Fig. 3: Reversible uniaxial-strain control of the nematic superconductivity.
figure3

a Upper critical field (Hc2) at 2.2 K along the a axis (x; ϕab = 0°; circles), a* axis (y; ϕab= −90°; upper triangles), c axis (z; lower triangles), as a function of the relative strain Δεxx induced by an applied voltage to the piezostacks. b, In-plane Hc2 anisotropy Hc2y/Hc2x as a function of strain. At the top of each panel, the estimated actual strain εxxΔεxx + 0.1% is indicated (see “Methods” section) and the gray region illustrates the possible range in the actual zero strain. The numbers in the top corner of each sub-panel indicates the criteria used for determining Hc2 (see “Methods” section). The numbers in the data points indicate the order of the measurements. The blue and red data points indicate the cases that the measurement was performed after a decrease and increase in strain, respectively. Here, the Hc2 anisotropy varies systematically with external strain. Source data are provided as a Source Data file.

The strain control of the nematic subdomains is more evident in the Hc2(ϕab) curves in Fig. 4a. Notice that Hc2 defined with higher values of R/Rn is more sensitive to existence of nematic subdomains. In addition to the prominent twofold anisotropy with maxima at ϕab = ±90° (originating from the Y0 domain) seen in all criteria, Hc2 with the 95% or 80% criteria exhibit additional 4 peaks located at ϕab = ±30° and ±150° for low Δεxx, due to the existence of Y1 and Y2 domains. These peaks are suppressed with increasing Δεxx, indicating the disappearance of the minor Y1/Y2 domains. In contrast, in Hc2 with lower criteria, the additional peaks are absent because the sample resistivity near the zero resistance state is mostly governed by the domain with the highest volume fraction. Nevertheless, even for Hc2 with the lower criteria (e.g. R/Rn = 5%), there is noticeable strain dependence near ϕab = 0°. This dependence is also attributable to the domain change by comparison with a model simulation explained later.

Fig. 4: Evidence for the control of nematic superconducting domains with strain.
figure4

a In-plane field angle ϕab dependence of the upper critical field Hc2 at 2.2 K for various criteria (see “Methods” section), which are indicated with a number in the top-left corner of each sub-panel. The curves colored from black to blue are in the order of increasing compressive strain; the numbers in the legend indicate the value of Δεxx. The results of a simulation are shown in the second column (see “Methods” section) capturing key features of the observation. b Schematic showing spatial configurations of nematic superconducting domains controlled in-situ by the uniaxial strain in our experiment. The yellow regions are the minor domains (Y1 and Y2), which are suppressed by the application of compressive strain, as evidenced by the changes in the Hc2(ϕab) curves. Source data are provided as a Source Data file.

Here we briefly discuss another possible interpretation for the change in the Hc2 anisotropy: a strain-induced crossover from an un-pinned single nematic domain state to a strongly-pinned single domain state. Theoretically, sixfold sinusoidal Hc2 has been proposed for a single-domain state in the ideal situation of a complete absence of nematicity pinning, and twofold Hc2 for finite nematicity pinning34, which might have been achieved by the application of uniaxial strain. However, this scenario is less likely compared to the multi-domain scenario, because of the following several reasons. Firstly, if the sixfold Hc2 originates from a single domain, it is difficult to explain the fact that the sixfold behavior is accompanied by a change in the broadness of the superconducting transition. The superconducting transition in magnetic field is rather broad for ϕab = ±90° without the uniaxial strain, whereas the transition becomes sharper under compressive strain (see Fig. 1c); and the sixfold behavior was observed only near the onset of the transition. This is difficult to be explained by a single domain scenario, since the transition width would be insensitive to the field direction or to the applied strain. Secondly, the observed onset Hc2 does not exhibit a perfect sixfold behavior: the onset Hc2 vs ϕab curve exhibits stronger peak at ϕab = ±90° than the peaks at ±30° or ±150°. In contrast, for an un-pinned domain, we expect that Hc2 should be perfectly sixfold symmetric34. Indeed, we demonstrate that the observed Hc2 cannot be fitted with a simple sixfold sinusoidal function even for the onset Hc2 with strongest sixfold component, as shown in Supplementary Fig. 15. This imperfectness of the sixfold behavior in the onset Hc2 is better explained by the multi-domain scenario. Thirdly, we find that, with increasing compressive strain, there is a smooth and anisotropic disappearance of the sixfold pattern. In particular, the pair of satellite peaks at −150° & +30° and −30° & +150° have different strain dependences; as shown in Supplementary Fig. 16, the former pair is still visible even under −1.19% strain whereas the latter pair completely disappears. If we were to assume that the sixfold component of Hc2 is due to a single sixfold domain, then with the application of strain Hc2 would suddenly exhibit a twofold behavior and both satellite peaks should disappear simultaneously. Nevertheless, we cannot completely deny a possibility that some part of the minor domains exhibits the sixfold Hc2 behavior expected for the ideal situation, as a result of accidental cancellation of pre-existing symmetry breaking field and externally applied strain. But this contribution should be, if exists, small and is neglected in the analyses in the following. Lastly, for a realistic situation with certain non-uniformity in the sample, Hc2 defined using higher R/Rn values pick up the most robust superconducting regions and the effective uniformity is enhanced.

Model simulation of the upper critical field

In order to confirm our scenario, we performed a simple model simulation for Hc2. In this simulation, we assume a network consisting of many Y0 domains and one of each Y1/Y2 domain and calculate the net resistance under magnetic fields (“Methods” section). Then Hc2 with various criteria is evaluated from the calculated resistivity curves. Under strain, the minor domains are assumed to change into Y0 domains. We should comment here that our model is free from the choice of the detailed mechanism of appearance of finite resistivity under magnetic fields near Hc2. This is because we model the behavior of each domain by making use of experimental magnetoresistance data under high strain, and also because we used classical equations for circuit resistance calculation (Supplementary Note 10).

As shown in Fig. 4, the simulation reproduces all the observed features described above, even without any change of Hc2 and in-plane Hc2 anisotropy in each domain, although the model is quite simple. We have tried several other circuit configurations and confirmed that the result of the simulations are rather similar, as shown in Supplementary Fig. 17. We find that setting slightly smaller Hc2 (ca. decrease of 10%; the broken curves in Fig. 4a) of the Y0 domain under strain gives a better match with the experimental data. Separately, we also performed fitting of this model to the experimental onset Hc2. As shown in Supplementary Fig. 15, the fitting was reasonably successful. These findings lead one to infer that the circuit model is reasonable and the observed behavior is almost solely explained by the change of the nematic SC subdomains.

Summary of results

Summarizing our findings, we succeeded in repeatable in-situ uniaxial-strain control of nematic SC domains in SrxBi2Se3, covering pre-existing tensile regime to the compressive regime. The primary effect of the increasing compressive strain is the suppression of minor Y1/Y2 domains, well reproduced by a simple model simulation. Other properties are rather insensitive to the strain, but there are decreasing trends in Tc as well as Hc2 of the main domain under compression. We comment here that, in most superconductors, domains with different but degenerate SC order parameters are absent. The only exception has been chiral SC domains with opposite Cooper-pair angular momenta suggested for the chiral-superconductor candidate Sr2RuO4 but their existence and control are not yet confirmed35. The present study provides the first demonstration of SC-domain engineering, a superconductor counterpart of domain engineering common in magnets.

Discussion

The coupling between the nematic superconductivity and uniaxial strain has been proposed using the Ginzburg–Landau (GL) formalism34,36. The strain couples to the nematic superconductivity through the free energy \({F}_{\varepsilon }=g[({\varepsilon }_{xx}-{\varepsilon }_{yy})(| {\eta }_{x}{| }^{2}-| {\eta }_{y}{| }^{2})+2{\varepsilon }_{xy}({\eta }_{x}{\eta }_{y}^{* }+{\eta }_{y}{\eta }_{x}^{* })]\), where g is the coupling constant, ηx and ηy are the amplitudes of the Δ4x and Δ4y components. Here we considered only the lowest-order coupling terms for simplicity. However, this treatment should be valid when considering the situation near Tc. This relation indicates that uniaxial εxx strain prefers one of the Δ4x or Δ4y states, depending on the sign of g. If a pre-existing symmetry breaking field exists, the nematic SC order parameter is initially fixed to the pre-existing field direction but eventually the state most favored by the external strain direction will be chosen with increasing strain. This is true even when the pre-existing field and external strain have a finite angle, as in the case for the Y1 or Y2 domains: the nematicity gradually rotates toward the external strain (see Supplementary Fig. 8). However, in these phenomenological theories, the sign of g remains arbitrary and should be determined based on experiments. Our result, a multi-domain sample driven to a mono-domain Δ4y state by εxx < 0 as shown in Fig. 3c, indicates that g is negative, an important step toward modeling of the nematic SC phenomenon. Moreover, this negative g provides a crucial constraint to realistic microscopic theories on the pairing mechanism. Such model should also explain the observed weak sensitivity of Tc on εxx. For example, a proposed odd-parity fluctuation model making use of phonons dispersing along the kz direction37,38,39,40 can be compatible with our observation, since such kz phonons should be less sensitive to the in-plane distortions.

Our results also provide information about the magnitude of g. The present work shows that compressive strain of around  −1% is required to alter the nematic SC domains. This is unexpectedly much larger than the strain required to homogenize the normal-state nematicity in iron pnictides (typically around 0.01–0.1%3). Thus the nematicity-lattice coupling is much weaker in doped Bi2Se3 than in iron-based superconductors. This difference may be due to the difference in the size of the nematic ordering elements. In nematic superconductors, the nematicity is carried by Cooper pairs, which are non-local objects with sizes much larger than inter-atomic distances. In contrast, in iron pnictides, the nematicity occurs due to orbital ordering, which occurs within one atomic site. We speculate that large strain is required to control a large and non-local object.

Coming back to the GL theories, they predict that Tc linearly increases with increasing strain in either tensile or compressive directions, accompanied by a kink in Tc(εxx) at the strain where the nematic state changes between Δ4x and Δ4y34,36. This prediction, at first glance, seems to be inconsistent with our decreasing trend of Tc with increasing εxx. However, we should note that Tc of doped Bi2Se3 decreases under hydrostatic pressure, i.e. under isotropic strain10. This effect is not taken into account in the above mentioned GL free energy, which couples only to the anisotropic strains. In the actual experiments, a combination of the increasing and decreasing trends in Tc due to anisotropic and isotropic strains, respectively, is observed. If the latter is relatively stronger, the observed small decrease of Tc by compressive strain is explained. Moreover, the existence of multiple domains weakens the predicted kink in Tc(εxx), because each domain’s Tc(εxx) curve convolves. This will result in a rounded kink, further obfuscating the linear behavior predicted from a mono-domain model.

Before concluding, we compare our findings with recent and related works on doped Bi2Se3 superconductors. Kuntsevich et al. found that their SrxBi2Se3 samples grown using the Bridgman method exhibits small lattice distortions (0.02% in-plane elongation and 0.005° c-axis inclination) at room temperature and weak twofold angular magnetoresistance (1–4%) even in the normal state16. In contrast, SrxBi2Se3 samples grown using melt-growth technique, as the samples we used in this work, does not seem to have detectable lattice distortions or normal-state nematicity in resistivity11,13 (Also see Supplementary Note 13). Regarding the normal-state nematicity, we should mention here that twofold behavior in the magnetic-field-angle dependence of specific heat of SrxBi2Se3 has been recently reported using samples grown with a self-flux method (equivalent to the melt growth method)17. Nevertheless, the origin of large normal-state specific-heat oscillation requires further consideration.

More recently, Kuntsevich’s group examined the dependence of superconducting nematicity on the as-grown lattice distortion using their Bridgman-grown samples18. They find that Hc2 is larger along the x axis for a sample with x-axis compressive strain, and it is larger along y under x-axis elongation. This tendency is, interestingly, opposite to the findings of our present work. Given that the strain range investigated in their work (±0.04%) is much smaller than that in our work (+0.1% to −1.2%), the origin of the nematicity in the samples with as-grown strain may be attributed to another underlying symmetry breaking field (e.g. alignment of intercalated Sr), rather than a simple uniaxial lattice distortion.

In NbxBi2Se3, a spontaneous lattice distortion occurring at a temperature slightly above Tc has been experimentally reported41, apparently consistent with the theoretical proposal of vestigial nematic order induced by superconducting fluctuation42. In this experiment, Hc2 along x is found to become larger (corresponding to the Δ4x state) after the lattice spontaneously shrinks along the y (a*) axis, inferring a negative coupling constant g. This observation, obtained with passive lattice response, is consistent with and complementary to the observation of the present work using active lattice control.

To conclude, we provide the first experimental demonstration of uniaxial-strain control of nematic superconductivity in doped Bi2Se3. Firstly, the x-axis compression suppresses minor domains while stabilizing the Δ4y state. Secondly, we determined the sign of the nematic coupling constant. These findings should provide bases toward resolving open issues of this highly attractive superconductor. Additionally, this work points to possible engineering of topological nematic superconductivity by uniaxial strain.

Methods

Sample preparation and characterization

Single crystals of SrxBi2Se3 (nominal x = 0.06) were grown from high-purity elemental of Sr chunk (99.99%), Bi shot (99.9999%), and Se shots (99.9999%) by a conventional melt-growth method. The raw materials were mixed with a total weight of 5.0 g and sealed in an evacuated quartz tube. The tube was heated to 1223 K and kept for 48 h with intermittent shaking to ensure the homogeneity of the melt. Then it was cooled slowly to 873 K at a rate of 4 K h−1 and finally quenched into ice water. It is worth pointing out that quench is essential for obtaining superconducting samples with high shielding fraction. The sample used here was cut from a large shiny crystal by wire saw, and the size is 4 mm (length) × 0.53 mm (width) × 0.5 mm (thickness; along the c axis) with the longest dimension along the a axis.

Strain cell and sample mounting

We constructed a custom-made piezoelectric-based uniaxial strain cell (ref. 31), based on the design of ref. 32. The bar-shaped sample was mounted between two anvils by a strong epoxy (Stycast 2850FTJ, Henkel Ablestik Japan Ltd.). The anvils can apply compressive or tensile strain on the sample by applying a positive voltage on the inner or outer piezo stacks, respectively. Thus the strain was applied parallel to the a axis, as shown in Fig. 1. The maximum applied voltage range for each piezo stack was −400–600 V corresponding roughly to −13 μm to 20 μm length changes of the piezo stacks used (P-885.11, PI) at cryogenic temperatures. A parallel-plate capacitor was mounted on the anvils to track the distance between the two plates by measuring the capacitance using a capacitance bridge (2500A, Andeen-Hagerling). The strain was then determined by the displacement divided by the exposed sample length, which was 1.14 ± 0.05 mm in this study.

Estimation of the thermally-induced strain

The effect of thermal contraction of the sample and the strain cell should be taken into consideration. Because the materials used in the strain cell are placed symmetrically between the compressive and tensile arms, the thermal strain on the sample originates only from the asymmetric part31; on the compressive arm, the sample with the length Lsample of 1.14 mm is placed, but on the tensile arms there are Ti blocks. This 1.14-mm length Ti part shrinks less than the sample, resulting in a tensile strain to the sample after cooling down from the epoxy curing temperature (around 350 K). The shrinkage of the sample ΔLsample/Lsample is evaluated as [a(4 K) − a(350 K)]/a(350 K) = −0.36%. Here, we used the lattice constants of Bi2Se3 reported in ref. 43. We note that a(4 K) and a(350 K) is estimated by a linear extrapolation because ref. 43 reports a values only between 10 K and 270 K. For Ti, the shrinkage ΔLTi/LTi is evaluated to be −0.23% by integrating the linear thermal expansion coefficient between 4 K and 350 K reported in ref. 44. The thermal expansion coefficient at 4 K and 350 K were obtained after linearly extrapolated. Thus, the thermally-induced strain to the sample is tensile and (ΔLTi − ΔLsample)/Lsample = +0.13% considering LTi = LSample. In addition, because of the stiffness of the component materials, in particular the epoxy, the actual strain transmitted to the sample may be reduced by roughly 56%31. Thus, the value +0.13% should be considered as the upper bound, and the lower bound should be 0.13% × 0.56 = 0.07%. To conclude, by taking the average, the thermally-induced strain is evaluated to be 0.10 ± 0.03%: the actual strain εxx is given as εxx Δεxx + 0.1%, where Δεxx is the strain applied relative to the situation of zero applied voltage to the piezo stacks.

Resistivity measurement

Sample resistivity was measured by four-terminal sensing: we applied a DC current using a current source (6221, Keithley Instruments) to the two outer wires and measure the resultant voltage by a nanovoltmeter (2182A, Keithley Instruments) on the inner two wires. To subtract the voltage offset, we use the Delta Mode of the combined operation of these instruments: the polarity of the current was periodically changed to acquire only the voltage component that is dependent on the current. We mostly used 250-μA current for resistivity measurements. We have checked that Joule heating or any other current-induced suppression of superconductivity is absent with 250 μA, by comparing resistivity behavior under various current (Supplementary Note 12). Au wire (20 μm diameter) were directly connected to the ac surface of the sample by Ag paint (4929N, Du Pont). To improve the mechanical stability of the wires, the Au wires were anchored onto the ab surface by Ag epoxy (H20E, EPOTEK), which has been confirmed to be electrically insulating to the sample. The four contacts were equispaced by about 0.2 mm. The contact resistance was on the order of 100 Ω at room temperature.

Temperature and magnetic-field control

We used a 3He/4He dilution refrigerator (Kelvinox 25, Oxford Instruments) to cool down the sample. It was inserted into the vector magnet described below. The lowest temperature achievable is roughly 80 mK, well below the superconducting transition temperature of  ~2.8 K. The temperature was measured using a resistive thermometer (Cernox, Lakeshore) and a resistance bridge (AVS-47, Picowatt). A 350-Ω strain gauge (KFG-1-350-C1-16, KYOWA), that was used as a heater for temperature control, was mounted close to the strain cell.

We applied the magnetic field using a vector-magnet system45, which consists of two orthogonal superconducting magnets (pointing in the vertical and horizontal directions in the laboratory frame) inside a dewar that rests on a horizontal rotation stage. This system allows us to direct the magnetic field accurately in any direction in space while the refrigerator, as well as the sample, is fixed. The superconducting magnets can apply fields up to 3 T (vertical) and 5 T (horizontal). The magnetic field can be controlled with a resolution of 0.1 mT. The precision of the horizontal rotation of the helium dewar is 0.001°, with negligible backlash. The strain cell was fixed with a sample mounted, so that the a axis is roughly along the vertical direction in the laboratory frame. The precise directions of the crystalline axes with respect to the laboratory frame are determined by making use of the anisotropy in Hc2. Once the directions of the crystalline axes are determined, we can rotate the magnetic field within the sample frame. All field angle values presented in this Article are defined in the sample frame. Refer to Supplementary Note 8 and Supplementary Fig. 9 for the detailed mathematical explanation for the vector transformations and a demonstration of the field alignment. In addition, see Supplementary Note 9 for the rationale behind the choice of temperature and magnetic field value for the alignment.

Evaluation of the upper critical field

The upper critical field Hc2 was evaluated by the value of the magnetic field at which the sample resistivity reaches a certain percentage of the normal-state resistivity. If the resistivity value falls in between two data points, then Hc2 is determined by using linear interpolation. For a more detailed methodology see Supplementary Note 4. Temperature dependence of Hc2 and its anisotropy is given in Supplementary Note 11.

Model simulation

The experimental Hc2(ϕab) curve shows three peaks indicating three nematic domains. Thus, we simulated the Hc2 behavior of multi and single-domain samples by considering an electrical circuit consisting of a network of resistive elements representing the three possible nematic SC domains. For the simulation shown in the main text, the circuit is assumed to be a 3D network (see Supplementary Fig. 13) of 12 elements to model the situation that current passes from end to end through a 3D distribution of domains. For the multi-domain simulation corresponding to the non-strained sample, the 12 elements are divided into 10 Y0 nematic domains, and one of each Y1 and Y2 domains. The exact positions are described in Supplementary Note 10. For the single-domain simulation corresponding to the highly compressed sample, all the 12 elements are assumed to be Y0 domains. The calculation of Hc2 is done as follows: firstly, for a fixed H and ϕab, resistivity of each circuit element is calculated from an empirical relationship among resistance, applied magnetic field H and field direction ϕab, by taking into account Hc2 anisotropy of each domain (see Supplementary Note 10 for details). Secondly, the total circuit resistance of the network Rtotal is calculated. The first and second step is iterated while varying H and ϕab, to obtain the H dependence of Rtotal for each ϕab. Lastly, Hc2 at ϕab is determined from the Rtotal(H) curve at ϕab using the same method as that used for the experimental data analysis (see Supplementary Note 4).

Data availability

The source data underlying Figs. 1c, 2a-c, 3a-b, 4a, and Supplementary Figs. 17, 9, 10, 12, 1820 are provided as a Source Data file. The other data that support the findings of this study are available from the corresponding author upon request.

Code availability

The codes used for the simulation are available from the corresponding author upon request.

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Acknowledgements

The authors acknowledge H.-H. Wen, J. Schmalian, P. T. How, S.-K. Yip, V. Kozii, and J. W. F. Venderbos, H.-S. Xu, and G. Mattoni for valuable discussions. We also acknowledge C. W. Hicks, M. E. Barber, A. Steppke, F. Jerzembeck, and A. P. Mackenzie for sharing their knowledge into the construction of a strain cell. This work was supported by JSPS Grant-in-Aids for Scientific Research on Innovative Areas on Topological Materials Science (KAKENHI JP15H05851, JP15H05852, JP15H05853, JP15K21717), on Quantum Liquid Crystals (KAKENHI 20H05158), by JSPS Grant-in-Aid KAKENHI 17H04848, and by the JSPS Core-to-Core Program. The work at Kyoto was also supported by ISHIZUE 2020 of Kyoto University Research Development Program. The work at Cologne was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 277146847—CRC 1238 (Subproject A04).

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This study was designed by I.K., S.Y., and Y.M.; I.K. performed resistivity measurements and analyses, with the assistance of S.Y. and the guidance of Y.M.; Z.W. and Y.A. grew single crystalline samples and characterized them. The uniaxial strain cell was designed and constructed by I.K.; The manuscript was prepared mainly by I.K. and S.Y., based on discussion among all authors.

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Correspondence to Ivan Kostylev or Shingo Yonezawa.

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Kostylev, I., Yonezawa, S., Wang, Z. et al. Uniaxial-strain control of nematic superconductivity in SrxBi2Se3. Nat Commun 11, 4152 (2020). https://doi.org/10.1038/s41467-020-17913-y

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