Introduction

Following the discovery of topological insulators1,2, the search for a superconducting analogue of a topological insulator has gained considerable interest in the condensed matter physics community. A topological superconductor (TSC)3,4,5 has a bulk superconducting energy gap (nodal or nodeless)6 but has gapless surface states which are of great interest both for fundamental physics, as they can host Majorana quasiparticles7, and also for applied physics, as the non-Abelian statistics of surface-state excitations have important implications for robust quantum computing8,9,10.

The topological nature of the superconducting state is determined by the symmetry of the superconducting order parameter and the shape of the Fermi surface. In a time-reversal and inversion symmetric system, odd-parity pairing, where Δ(−k) = −Δ(k), and a Fermi surface that contains an odd number of time-reversal invariant momenta are necessary requirements5. In materials with weak spin-orbit coupling, odd-parity pairing corresponds to spin-triplet pairing; for certain strong spin-orbit coupling systems, unique unconventional superconducting states are possible11 that may qualify as topological superconductivity. Currently, two paths towards topological superconductivity are being investigated: proximity-induced TSC7,12,13 at the interface between a conventional superconductor and a topological insulator or a strong spin-orbit coupled semiconductor, respectively, and via chemical doping of bulk topological insulators. Of the superconducting doped topological insulators, the M x Bi2Se3 family of materials (M = Cu, Nb, Sr)14,15,16 has generated the most interest as high quality, mm-scale single crystals are available. Topological order observed via ARPES measurements17 and magnetization measurements18,19 is consistent with a spin-triplet pairing state. Calorimetry measurements20 are not in full agreement with conventional BCS theory, and low-temperature penetration depth measurements21 indicate nodes in the superconducting energy gap. The observation of zero-bias conductivity peaks in point-contact spectroscopy measurements22,23,24 has been interpreted as evidence for Majorana surface states.

The M x Bi2Se3 family maintains the trigonal \({R}\bar{3}m\) structure of the parent compound. Following the surprising observation25 of an anomalous Knight shift in CuxBi2Se3 demonstrating spin-triplet pairing and twofold anisotropy in the threefold symmetric basal plane, a nematic superconducting state with a two-component order parameter was proposed26. This model26,27,28 accounted for subsequent observations of a spontaneous twofold symmetry below T c in several quantities29,30,31 including the upper critical field of CuxBi2Se3 as determined by calorimetry32 and of NbxBi2Se3 as determined by magnetization33. This state has E u symmetry and odd-parity pairing, and allows for states with complete, albeit anisotropic, superconducting gap as well as for a gap with point nodes. Despite the unconventional nature, the superconducting state has been shown to be robust against disorder scattering34,35,36. Figure 1(a) shows the \({R}\bar{3}m\) crystal structure of Sr0.1Bi2Se3, the same as that of the parent compound Bi2Se3 with a slightly extended c axis due to intercalation of the Sr atom in the gap between adjacent quintuple layers of Bi2Se337 while Fig. 1(b) shows the threefold symmetric basal plane, with the a (blue) and a* (pink) directions marked by arrows. Figure 1(c) shows the proposed twofold symmetric Δ4 superconducting gap structure, which breaks crystallographic rotational symmetry in the basal plane26.

Figure 1
figure 1

(a) Crystal structure of SrxBi2Se3, with directions a (blue), a* (pink) and c (black) marked. The dopant Sr ion (blue) sits in the van der Waals gap between quintuple layers of Bi (green) and Se (red) ions. (b) The threefold symmetric basal plane. (c) Schematic presentation of the two-dimensional Δ4 superconducting gap with respect to a spherical Fermi surface. The Δ4 gap has basis functions (Δ4x) with nodes on the mirror plane [depicted in (c)] and (Δ4y) with deep minima in the perpendicular direction. This gap breaks the threefold crystal symmetry and gives rise to the nematic state with twofold symmetry.

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Here, we present the first thermodynamic determination of the anisotropy of the upper critical field of Sr0.1Bi2Se3 single crystals through measurements of the temperature dependence of the reversible magnetization, in addition to angular-dependent magnetotransport measurements. Both quantities yield a large twofold in-plane anisotropy of Hc2 in which the high-Hc2 direction is aligned with the a axis of the crystal structure. We find that the normal state resistivity of Sr0.1Bi2Se3 is isotropic in pairs of samples cut at 90° from the same starting crystal, which excludes conventional mass anisotropy from being the cause of the anisotropy in Hc2. Furthermore, temperature dependent measurements of the normal-state magnetization show that Sr0.1Bi2Se3 is diamagnetic with an isotropic susceptibility of \(\sim \,-\,2\cdot {10}^{-6}\) (CGS) which largely originates from the core diamagnetism. X-ray diffraction studies indicate that the extinction rule for the \({R}\bar{3}m\) crystal structure is fulfilled to a level of 10−6 implying that deviations from the ideal \({R}\bar{3}m\) structure are exceedingly small. We thus conclude that the origin of the twofold anisotropy of the superconducting properties is likely caused by an anisotropic gap structure consistent with the nematic E u state.

Results

We present results on a series of SrxBi2Se3 crystals. Bar-shaped crystals #1a and #1b were cut from the same starting piece oriented at 90° with respect to each other. Field-angle dependent resistivity measurements (Fig. 2) reveal that the nematic state is not tied to the current flow direction. Detailed resistivity measurements as function of applied magnetic field, field orientation and temperature (Figs 3 and 4) on crystal #2 yield the anisotropic phase diagram. On crystal #3 we performed magnetic measurements of the nematic state (Fig. 5) and of the normal state susceptibility (Fig. 6). Bar-shaped crystals #4a and #4b, used for resistivity measurements (Fig. 6), were cut from the same starting piece such that they are oriented parallel and perpendicular to the nematic axis, respectively. High-resolution x-ray diffraction on crystal #2 (Fig. 7) reveals high crystalline quality. Finally, we determined the in-plane superconducting anisotropy using magnetization and resistivity measurements on crystal #5, shown in Supplemental Information.

Figure 2
figure 2

R(θ) in an applied magnetic field of |H ab | = 10 kG in two crystals of Sr0.1Bi2Se3 cut at 90° relative to each other from a single larger crystal in the temperature range 1.7 K (black) to 2.9 K (purple) in 0.1 K steps; I = 0.1 mA. 0° marks the crystallographic a-axis; the red arrow indicates the direction of current. As temperature increases, the twofold nematic symmetry becomes rotationally isotropic. Rotating the direction of current 90° relative to the crystalline axes results in no change of the nematic axis, demonstrating that vortex motion is not the cause of the observed twofold asymmetry.

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Figure 3
figure 3

ρ(T) of Sr0.1Bi2Se3 crystal #2 in increasing magnetic field with the field vector in different orientations. (a) Field vector H//a. (b) Field vector H//a*. (c) Field vector H//c. (d) Magnetic phase diagram. There is a large anisotropy of ~4 between the two in-plane orientations which are 90° apart. The inset shows the in-plane \({\rm{\Gamma }}={H}_{c2}^{a}/{H}_{c2}^{{a}^{\ast }}\).

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Figure 4
figure 4

(a) R(θ) for Sr0.1Bi2Se3 crystal #2 in an applied magnetic field of |H ab | = 10 kG in 0.1 K increments, from 1.7 K (cyan) to 3 K (green). 0° marks the a-axis; the red arrow, the direction of current. (b) T c (θ) extrapolated from R(θ) data for the same crystal, taken as where R(θ) is half the normal-state value, represented by the blue circle in (a). Several additional small θ windows were measured at multiple temperatures to increase data density. The red line is a fit to the data following the Ginzburg-Landau effective mass formula (see text) yielding Γ ≈ 3.8.

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Figure 5
figure 5

(a) M(T) curves as measured by dc SQUID magnetometry on Sr0.1Bi2Se3 crystal #3 in 4 kG for two different orientations of field 90° apart in the basal plane, with linear fits (dashed lines) below T c . There is a clear difference in T c taken as where a linear fit of the response (dashed lines) crosses zero. The inset shows the slope of the linear fit vs angle for multiple measurements. A twofold symmetry in T c is evident. (b) T c (θ) with θ the orientation of magnetic field in the basal plane as measured by dc SQUID magnetometry on a single crystal of Sr0.1Bi2Se3 in fields of 4 kG (blue) and 6 kG (red). In both fields, T c is again twofold symmetric.

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Figure 6
figure 6

(a) Zero-field superconducting resistive transition (T c  ≈ 3.0 K) of Sr0.1Bi2Se3 crystals #4a and #4b, cut at right angles to each other out of a single larger crystal. The inset shows a typical ~1 mm2 crystal before and after cutting two transport samples out of it at right angles to each other. The anisotropy in ρ is within the uncertainty of the dimensions of the electrical contacts. (b) Magnetization vs temperature of Sr0.1Bi2Se3 crystal #3 with the field along the a axis (yellow), the a* axis (blue), and at 45° to either in the aa* plane (red). The material is diamagnetic, with a Curie-Weiss component (dashed line) possibly due to impurity contamination. The magnetization is essentially isotropic in-plane.

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Figure 7
figure 7

(a) Scans centered at (h, 0, l) for multiple values of l on Sr0.1Bi2Se3 crystal #2 used for transport measurements. Multiple values of h are shown; h = 0 (green circles), h = 1 (blue squares), h = 2 (pink triangles). The trigonal structure enforces an extinction rule unless 2h + k + l = 3n, where n is an integer. The allowed peaks show approximately 5 orders of magnitude more intensity than at l values that are not allowed, showing the high quality of the crystal. Any distortions away from a perfect crystal structure would appear as violations of the extinction rule; none are seen. An additional crystal examined shows similar results. (b) Rocking curve centered at (h, k, l) = (0, 0, 21), showing three closely aligned major grains with a narrow mosaic spread of \(\sim 0.04\)°.

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The resistivity as a function of in-plane angle in an applied magnetic field of magnitude 10 kG is shown in Fig. 2 for Sr0.1Bi2Se3 crystals #1a and #1b at temperatures ranging from 1.7 K (black) to 2.9 K (purple) in increments of 0.1 K. The two crystals were cut from the same starting material at 90° with respect to each other (see inset of Fig. 6). Here, the crystals were intentionally cut such that the long axes did not lie along or perpendicular to the nematic axis. 0° in Fig. 2 marks the crystallographic a axis. The red arrow indicates the direction of current with respect to the crystal axes. The angular dependence of the resistivity thus reflects the angular dependence of the upper critical field Hc2, as directions with higher Hc2 will remain superconducting whereas directions with lower Hc2 will be resistive at a fixed temperature. Twofold anisotropy is evident in both Fig. 2(a,b). As temperature is increased from base temperature through the superconducting transition, the twofold anisotropy is eventually lifted, reaching an angle independent normal state. Our observation that the nematic axis does not depend on the direction of the applied current demonstrates that the observed twofold anisotropy is tied to the crystal structure, and is not an effect due to current flow, such as Lorentz force driven vortex motion.

To further investigate the angular anisotropy of Hc2, a series of ρ(T) curves were measured on crystal #2 with T c  ≈ 2.9 K in different applied magnetic fields with the field vector along the directions of maximum and minimum in-plane Hc2 as well as along the c axis of the crystal [Fig. 3(a,b,c)]. X-ray diffraction on this sample (see Fig. 7) reveals that the directions of high (low) in-plane Hc2 correspond to the crystallographic a and a* directions, respectively (see Fig. 1), consistent with previous reports29,38. Figure 3(a,b,c) show that on increasing field the transitions stay sharp and shift uniformly to lower temperatures. A weak normal-state magnetoresistance is observed only for H//c. Figure 3(d) shows the magnetic phase diagram along the principal axes with T c taken as the midpoint of the resistive transitions. The in-plane anisotropy Γ is ~4.5. Reported values for the anisotropy (also determined from the resistive midpoints) range from 6.8 for nominal 10% doping to 2.7 for nominal 15% doping, both at 1.9 K29, whereas on samples with unspecified doping levels an in-plane anisotropy value of ~2.8 was obtained38. In the temperature range covered here, Γ is approximately temperature-independent.

The nematic E u state that has been proposed as a possible explanation of the twofold anisotropy of superconducting properties is characterized by a two-component order parameter11,26, which can be expressed as a linear superposition of the two basis gap functions Δ4x and Δ4y (see Fig. 1). An in-depth analysis of the upper critical field of a superconductor with trigonal symmetry and two-component order parameter has been presented in Ref.39. Three nematic domains related by rotations of 120° should arise in the sample, giving rise to overall threefold symmetry. Instead, the vast majority of reported data including those presented here reveal a simple twofold anisotropy indicative of a single nematic domain. The theoretical analysis39 reveals that, in contrast to a single component order parameter, a two component order parameter couples linearly to strain fields as parameterized by a coefficient δ, and that such strain fields may serve to pin the nematic vector into a single domain. In particular, for sufficiently strong pinning δ and near Tc0, the two component order parameter is effectively reduced to a single component33 which for δ > 0 is approximately Δ4x and for δ < 0 it is approximately Δ4y.

In standard single-band Ginzburg-Landau (GL) theory the anisotropy of Hc2 is given by the anisotropy of the effective masses40

$$\frac{{H}_{c2}^{(i)}}{{H}_{c2}^{(j)}}=\sqrt{\frac{{m}_{j}}{{m}_{i}}}={\rm{\Gamma }} > 1$$
(1)

where m i and m j are the effective masses along the principal crystal directions i and j. The unit vectors i, j, k define a Cartesian coordinate system; here, i = a, j = a*, k = c. The angular variation of Hc2 in the ij-plane is then given by40

$${H}_{c2}(T,\theta )=\frac{{H}_{c2}^{(j)}}{\sqrt{co{s}^{2}(\theta )+{{\rm{\Gamma }}}^{-2}si{n}^{2}(\theta )}}$$
(2)

where θ is measured from the j-direction (low Hc2). Approximating the Hc2-line as linear, the angular dependence of T c (H,θ) in a given field H can be obtained from Eq. 2 as

$${T}_{c}(H,\theta )={T}_{c0}+\frac{H}{\partial {H}_{c2}^{(j)}(T)/\partial T}\sqrt{co{s}^{2}(\theta )+{{\rm{\Gamma }}}^{-2}si{n}^{2}(\theta )}.$$
(3)

Data in a field of H = 10 kG are obtained from the polar diagram of ρ(T,θ) [(Fig. 4(a)] by tracing for which values of T and θ the resistivity crosses the 50% value. The results for T c (θ) are shown in Fig. 4(b) together with a fit to Eq. 3. The fit yields an in-plane anisotropy of Γ~3.8, in reasonable agreement with the data in Fig. 3(d). The small difference in anisotropy may arise from deviations from linearity of the phase boundaries.

We obtain the first thermodynamic measurement of the in-plane anisotropy of the upper critical field of Sr0.1Bi2Se3 from the temperature dependence of the magnetization of crystal #3 with T c  ~ 3 K. Figure 5 shows data taken in a field of 4 kG applied along the high- and low-Hc2 directions, respectively. A shift in T c , defined as the intersection of a linear fit to the M(T)-data with the M = 0 line, and a change in the slope dM/dT with field angle are clearly seen. The inset of Fig. 5 displays the twofold symmetric angular variation of the slope dM/dT in which a low value of the slope corresponds to a high value of T c . Such behavior is expected in conventional single-band GL theory of anisotropic superconductors, for which the slope in field direction i is given as

$$\frac{\partial {M}^{(i)}}{\partial T}=-\,\frac{1}{8\pi {\beta }_{A}{({\kappa }^{(i)})}^{2}}\frac{\partial {H}_{c2}^{(i)}}{\partial T}$$
(4)

where \({H}_{c2}^{(i)}={\varphi }_{0}/\mathrm{(2}\pi {\xi }_{j}{\xi }_{k})\) and \({\kappa }^{(i)}=\sqrt{({{\rm{\lambda }}}_{j}{{\rm{\lambda }}}_{k})/({\xi }_{j}{\xi }_{k})}\gg 1\) are the upper critical field and Ginzburg-Landau parameter in direction i, respectively, and β A  = 1.16 is the Abrikosov number. With the T-linear variation of Hc2 near T c one finds that the ratio of the slopes for the high-Hc2 and low-Hc2 directions is given by the inverse anisotropy

$$\frac{\partial {M}^{(i)}}{\partial T}/\frac{\partial {M}^{(j)}}{\partial T}=\frac{{{\rm{\lambda }}}_{i}}{{{\rm{\lambda }}}_{j}}=\frac{{\xi }_{j}}{{\xi }_{i}}=\frac{1}{{\rm{\Gamma }}} < 1.$$
(5)

Thus, the data shown in the inset of Fig. 5 indicate an anisotropy of \({\rm{\Gamma }}\sim \,2\) which is smaller than the value deduced from the resistivity measurements (Fig. 3). Data such as shown in Fig. 5(a) taken over the entire angular range in fields of 4 kG and 6 kG yield the angular dependence of T c as shown in Fig. 5(b) for 4 kG (blue) and 6 kG (red). Although there is sizable scatter in the data [the error bars in Fig. 5(b) reflect the scatter in T c obtained on repeated runs], a twofold angular symmetry in this thermodynamic determination of T c is clearly seen consistent with the twofold symmetry observed in magnetotransport measurements. The data shown in Fig. 5(a) also demonstrate that the superconductivity observed in our Sr0.1Bi2Se3 crystals is a bulk phenomenon and not filamentary.

Although transport and magnetization measurements yield similar qualitative features of the superconducting phase diagram of SrxBi2Se3, i.e. a sizable in-plane anisotropy, there are clear quantitative differences in the value of the anisotropy deduced from both techniques. Generally, such differences may arise since magnetization and resistivity represent different quantities, the expectation value of the magnitude-squared of the superconducting order parameter and the onset of phase coherence across the sample, respectively. Furthermore, the resistively determined phase boundaries depend on the resistivity criterion used; here we use the 50% criterion. Nevertheless, considering that the resistive transitions shown in Fig. 3 appear ‘well-behaved’, a difference in anisotropy by a factor of ~2 is surprising. In order to rule out the doping-dependence as a cause of the difference in anisotropy seen in magnetization and magnetoresistance measurements, we performed detailed magnetization and resistivity measurements on a large single crystal, sample #5, shown in Supplemental Information, and reproduce the result that the magnetically determined in-plane superconducting anisotropy is smaller than the resistive result: Γ ~ 2.6 versus 5. The reasons for this unexpected behavior are not understood at present, and may be related to the unusual positive curvature observed in Hc2 in all samples as determined by magnetotransport, or to the existence of surface states which may have different superconducting properties30 than the bulk.

Figure 6(a) shows the temperature dependence of the resistivity of cross-cut crystals with very sharp superconducting transitions at an onset temperature of ~3.0 K oriented such that the current in crystal #4a flows along the a direction and in crystal #4b along a*, respectively. The anisotropy in the normal-state resistivity is small, <10%. We note that absolute values of the resistivity have an uncertainty of \(\sim \,\pm 15{\rm{ \% }}\) due to uncertainties in the dimensions of the samples and contact geometry. At the same time, the upper critical field displays a sizable in-plane anisotropy as expressed by the ratio of the effective masses (Eq. 1). For superconductors with essentially isotropic gaps, these effective masses are the same as those entering the normal state conductivity. Our observed sizable in-plane Hc2-anisotropy would imply an in-plane resistivity anisotropy of more than 4, which is clearly not consistent with the data shown in Fig. 6. Furthermore, quantum oscillation measurements41,42,43 on the Nb and Cu homologues suggest that the planar cross-section of the Fermi surface shows little warping, indicating that effective mass anisotropy cannot be the sole cause of the anisotropy in Hc2.

However, for the more general case of anisotropic gaps the GL effective masses are given as44

$$\frac{1}{{m}_{i}}=\frac{1}{4{\pi }^{3}\hslash N}\oint dS{\varphi }^{2}({\bf{k}})\frac{{v}_{i}^{2}}{{v}_{F}}$$
(6)

Here, dS denotes an integral over the Fermi surface, N is the electron density, v i is the i-component of the Fermi velocity, and v F is the magnitude of the Fermi velocity, both in general k-dependent. ϕ(k) describes the anisotropy of the gap over the Fermi surface, normalized such that its Fermi surface average is unity. For instance, for a spherical Fermi surface (isotropic normal state electronic structure) and a model gap anisotropy of ϕ(k) = sin(θ) (corresponding to two point nodes on the c-axis) the Hc2-anisotropy for fields applied along the c-axis and for fields applied transverse is \(\mathrm{1/}\sqrt{2}\). We expect that nodal gap structures with different forms of ϕ(k) and gaps with deep minima will show similar qualitative behavior, namely, that Hc2 measured along the line connecting the nodes (minima) is lower than in a transverse direction. This anisotropy can be expressed in terms of the original two-component model39 as \({H}_{c2}^{(a)}/{H}_{c2}^{({a}^{\ast })}=\sqrt{({J}_{1}+{J}_{4})/({J}_{1}-{J}_{4})}\) for δ > 0 and \({H}_{c2}^{(a)}/{H}_{c2}^{({a}^{\ast })}=\sqrt{({J}_{1}-{J}_{4})/({J}_{1}+{J}_{4})}\) for δ < 0 with an angular dependence that is given by the conventional form (Eq. 2). Here, J1 and J4 are coefficients of the gradient terms in the two-component GL free energy in the notation of Ref.39. Thus, depending on the values of these coefficients, a sizable temperature independent in-plane anisotropy of Hc2 can arise even when the electronic structure is essentially isotropic. In particular, our observation that \({H}_{c2}^{(a)} > {H}_{c2}^{({a}^{\ast })}\) implies that the nodal Δ4x state is realized.

It has been reported that magnetic effects may play an important role in the formation of the superconducting state in Bi2Se3-derived superconductors, i.e., NbxBi2Se316,45. We therefore explored the temperature dependence of the normal state magnetization of Sr0.1Bi2Se3. Figure 6(b) shows data for crystal #3 measured in a field of 10 kG applied along various in-plane directions. Within the experimental uncertainties, the normal state magnetization is isotropic in the basal plane ruling out a magnetic origin of the observed in-plane anisotropy of the superconducting state. Furthermore, in its normal state, Sr0.1Bi2Se3 is diamagnetic, approaching a volume susceptibility of −2⋅10−6 (CGS) at high temperature. The measured magnetic susceptibility, χ, contains several contributions46, χ = χ core  + χ P  + χ L  + χ VV  + χ CW . Here, χ core represents the core diamagnetism, χ P and χ L the Pauli paramagnetism and Landau diamagnetism of the conduction electrons, respectively, χ VV the van Vleck paramagnetism and χ CW a Curie-Weiss contribution, possibly due to magnetic impurities. χ core is temperature independent and isotropic, whereas χ P , χ L , and χ VV are temperature independent but in general anisotropic, depending on the band structure and orbital structure. Since the charge count of Sr0.1Bi2Se3 is much lower than that of typical metals we neglect χ P and χ L . With the help of tabulated values47, χ core of Sr0.1Bi2Se3 can be estimated as −2.3⋅10−6 (CGS). Thus, the observed isotropic diamagnetic response of Sr0.1Bi2Se3 is in large part caused by its core diamagnetism, which mainly stems from the Se2− ions. The van Vleck contribution may account for the difference between the measured and expected diamagnetic signals, \({\chi }_{VV} \sim 0.3\cdot {10}^{-6}\) (CGS). In addition, superimposed onto the diamagnetic signal is a paramagnetic contribution, which approximately follows a Curie-Weiss dependence [Fig. 6(b)]. This contribution is also isotropic, and we attribute it to residual magnetic impurities.

Deviations from the ideal \({R}\bar{3}m\) crystal symmetry have been proposed as possible causes of the twofold anisotropy itself or as mechanism of pinning the nematic vector into one domain. We performed x-ray diffraction studies on the crystals used here in order to search for these effects. These measurements revealed a high-degree of structural coherence and phase purity. The determined room-temperature lattice parameters are a = 4.146 Å and c = 28.664 Å, consistent with a rhombohedral \({R}\bar{3}m\) crystal symmetry derived from the Sr-intercalated Bi2Sr3 structure15,37. Figure 7 shows l scans centered on various (h, 0, l) zones performed on the same crystal whose transport measurements are shown in Figs 3 and 4. Multiple h values are shown; h = 0 (green circles), h = 1 (blue squares), h = 2 (pink triangles). At all (h, 0, l) zones examined, only Bragg peaks for which 2h + k + l = 3n is satisfied are observed. This is the extinction rule for the \({R}\bar{3}m\) structure. The data shown in Fig. 7 reveal that this extinction rule is satisfied to a level of 10−6 implying that deviations from the ideal \({R}\bar{3}m\) structure are exceedingly small. Over the large illuminated area of the order of 0.4 × 0.4 mm2, comparable to the sample size, there are three closely aligned grains with a mosaic of \(\sim 0.04\)° each [see Fig. 7(b)], which is remarkable for a crystal formed from intercalating atoms between stacks of weakly coupled “quintuple layers”. These measurements do not reveal, at room temperature, any long-range lattice modulations or compositional variations of Sr that could account for the large twofold anisotropy seen in the superconducting properties. It is unlikely, based on the smooth behavior observed in transport and magnetization data (Fig. 6) and calorimetry data29, that there is any structural change at low temperature.

Discussion

In addition to angular-dependent magnetotransport measurements we present the first thermodynamic determination of the anisotropy of the upper critical field of Sr0.1Bi2Se3 crystals through measurements of the temperature dependence of the reversible magnetization. Both quantities yield a large twofold in-plane anisotropy of Hc2 in which the high-Hc2 direction is aligned with the a-axis of the crystal structure. Transport measurements on pairs of samples cut at 90° from the same starting crystal demonstrate that the in-plane anisotropy of Hc2 is tied to the crystal structure and is not induced by the current flow, consistent with the thermodynamic observations. These measurements also show that the normal state resistivity of Sr0.1Bi2Se3 is isotropic in the plane, thereby excluding conventional effective mass anisotropy as a cause of the Hc2-anisotropy. Furthermore, temperature dependent measurements of the normal-state magnetization reveal that Sr0.1Bi2Se3 is diamagnetic with an isotropic susceptibility of \( \sim \,-\,2\cdot {10}^{-6}\) (CGS) which largely originates from the core diamagnetism. These results rule out a possible magnetic origin of the superconducting anisotropy. In addition, x-ray diffraction studies reveal a high degree of structural coherence and phase purity without any detectable deviations from the \({R}\bar{3}m\) crystal structure that could cause the twofold anisotropy. We thus conclude that the origin of the twofold anisotropy of the superconducting properties is likely caused by an anisotropic gap structure as realized in the nematic E u state. In fact, by specializing the general form of the GL free energy applicable to the two-component E u order parameter to the Δ4x and Δ4y basis functions, we retrieve an anisotropic single-component GL expression that can account for the experimental observations, and indicates the Δ4x state is selected.

Methods

Large high quality single crystals of Sr0.1Bi2Se3 were grown by the melt-growth technique described in Ref.38. All crystals regularly showed high volume fraction of superconductivity via magnetic susceptibility measurements with small variation in T c ranging from 2.9 K to 3.05 K. Thin crystals were cut from as-grown bulk crystals. The material cleaves easily in the basal plane yielding naturally flat surfaces parallel to aa* in the lattice. Gold contact pads were evaporated, and gold wires were then attached to the crystals using silver epoxy in a conventional 4-point measurement configuration. We typically cut several samples from the same starting piece as shown for example in the inset of Fig. 6(a). After measuring the first sample the overall orientation of the starting piece is known and we can cut all subsequent samples with approximately known orientation such as mutually perpendicular or at arbitrary angles. Some pairs were aligned parallel and perpendicular to the high-Hc2 direction [such as in Fig. 6(a)] whereas others were intentionally misaligned (such as in Fig. 2). The crystals were mounted with their long axes parallel to each other such that the angle between current and applied in-plane magnetic field were always the same for both. An AMI 10 kG superconducting 3-axis vector magnet was used to apply magnetic field in arbitrary directions without having to physically rotate the sample, and currents smaller than or equal to 1 mA were used for the measurements. Slow rotation of the field direction in 2° increments ensured thermal equilibrium was maintained. The field was swept clockwise from 0° to 400° to eliminate any magnetic hysteresis effects. By recording the magnetoresistance during theta scans (a field with fixed magnitude turning from the z-axis of the magnet into the horizontal plane and beyond) for two azimuthal angles, the orientation of the a-a* plane of the sample with respect to the magnet can be determined with a precision of less than a degree. Such a small level of planar misalignment cannot account for the large observed in-plane anisotropy of Hc2. Magnetization measurements were performed in a 70 kG Quantum Design MPMS with samples mounted on an approximately 25 cm long quartz glass fiber with GE varnish to minimize the background signal. The fiber is suspended from the standard SQUID sample rod which is centered inside the sample chamber with a 7.6 mm diameter spacer ring. The fiber is kept straight with a 7.6 mm diameter plastic weight at the bottom. We estimate that the possible misalignment of this arrangement with respect to the SQUID axis is less than 2°. For angular dependent measurements, the sample is remounted onto the fiber for each angle, which is determined from microscopy photos with a precision of better than 1°. X-ray measurements were performed at the 6-ID-B beamline at the Advanced Photon Source. A vertically focused x-ray beam of 8.979 keV was delivered to the sample. The sample was oriented such that measurements using a reflection geometry from a naturally cleaved surface normal to the c-axis can be carried out.