Direct evidence for the absence of coupling between shear strain and superconductivity in Sr₂RuO₄

Abstract

The superconducting symmetry of Sr₂RuO₄ has been intensely debated for many years. A crucial controversy recently emerged between shear-mode ultrasound experiments, which suggest a two-component order parameter, and some uniaxial pressure experiments that suggest a one-component order parameter. To resolve this controversy, we use a new approach to directly apply three different kinds of shear strain to single crystals of Sr₂RuO₄ and investigate the coupling to superconductivity. After characterising the strain by optical imaging, we observe variations of the transition temperature T_c smaller than 10 mK%⁻¹ as measured by low-frequency magnetic susceptibility, indicating that shear strain has little to no coupling to superconductivity. Our results are consistent with a one-component order parameter model, but such a model cannot consistently explain other experimental evidence such as time-reversal symmetry breaking, superconducting domains, and horizontal line nodes, thus calling for alternative interpretations.

Introduction

Strains in a crystal have well-defined irreducible representations (irreps) under the symmetry of a lattice. Strains are precisely coupled to the superconducting (SC) order parameter (OP) according to their symmetries, and induce changes in T_c. Thus, the application of strains to a superconductor and the observation of the changes in T_c is a precise method of gaining information on the SC OP. In addition to isotropic hydrostatic pressure, it has recently become clear that uniaxial strains are a powerful tool to investigate superconducting symmetry, since they can profoundly alter crystal symmetry^1,2,3.

The OP of unconventional superconductivity in Sr₂RuO₄⁴ is still unresolved despite 30 years of extensive research efforts^5,6. Spin-singlet-like behaviour was newly established from spin susceptibility measured by nuclear magnetic resonance (NMR)^7,8,9 and polarised neutrons¹⁰. Time-reversal symmetry breaking (TRSB), evidenced by an emerging internal magnetic field below T_c¹¹, has been reported by several groups^12,13,14. Magneto-optic Kerr effect (MOKE) also supports the presence of TRSB in the superconducting state¹⁵. Anomalous switching behaviour in superconducting junctions¹⁶ points to the presence of SC domains characterised by a multi-component OP with one-dimensional (1D) or 2D irreducible representations (irreps).

Uniaxial compression in Sr₂RuO₄ enhances T_c from 1.5 to 3.5 K, and it is associated with changes in the Fermi surface topology (Lifshitz transition)^2,17. A splitting between T_c and appearance of the TRSB phase was observed by muon spin relaxation (μSR) under uniaxial strain¹⁴, pointing to a chiral superconducting OP. However, the splitting is not observed in specific heat and elastocaloric-effect measurements under uniaxial strain^18,19, leaving the issue of TRSB unsettled at present.

Another peculiar behaviour of Sr₂RuO₄ is the jump at T_c of the shear-mode ultrasound velocity related to the elastic modulus c₆₆^20,21. This unusual coupling between SC OP and shear strain implies that the OP has two components: either chiral with 2D irreps (e.g., d_yz ± id_zx), nematic with 2D irreps (e.g., d_yz only, d_zx only, or d_yz ± d_zx), or a combination of 1D irreps (e.g., d + ig, s + id). Results of uniaxial stress and hydrostatic pressure experiments can be combined to compare with results from ultrasound experiments²², but such comparison leads to both qualitative and quantitative discrepancies⁶. Specifically, a V-shaped kink in T_c as a function of uniaxial strain along the [110] direction was not observed, in contrast with the expectations from the observed jump in the ultrasound c₆₆. It is crucial to resolve this discrepancy in order to reach a conclusion about the superconducting state of Sr₂RuO₄.

In this study, we investigate the direct effect of shear strain on the T_c of Sr₂RuO₄. The magnitude of sample strains at low temperatures is measured with optical imaging down to 30 K. Kinks, linear slopes, and quadratic changes in T_c are evaluated by applying three different kinds of shear strain and comparing the results with the estimates of the Ehrenfest relations from the jumps and slope changes of the elastic-moduli at T_c. We find that T_c does not change within the experimental precision of ∂T_c/∂ε_xy = ± 6 mK%⁻¹. This result seriously challenges the interpretation of SC in Sr₂RuO₄ in terms of a two-component OP.

Results

Direct shear-strain quantification at low temperature

To apply shear strain, we construct piezo-sample assemblies by rigidly gluing thinly polished Sr₂RuO₄ crystals directly onto the surface of piezoelectric devices, as shown in Fig. 1a (see “Methods” for construction details). Deformation of a piezo-sample assembly is monitored by an optical microscope, which allows us to calculate the point-by-point in-plane displacements (u_x, u_y) and the in-plane shear strain ε_xy = (∂u_x/∂y + ∂u_y/∂x)/2. We note that this definition is symmetric (ε_xy = ε_yx) and that ε_xy = γ_xy/2 = ε₆/2, where γ_xy is the engineering strain and ε₆ is the xy shear strain in Voigt notation. We show in Fig. 1b a typical optical image taken at room temperature with a voltage applied to the shear piezo V_piezo = + 200 V. By means of digital image correlation^23,24, we extract the point-by-point displacements that are represented by the coloured dots. As indicated by the arrows, the bottom of the piezo-sample assembly is moving to the right (red) while the top is moving to the left (blue) due to the applied shear strain. We note that the piezoelectric device we use predominantly generates a shear displacement component ∂u_x/∂y, while the complementary component ∂u_y/∂x is small due to the device’s geometry and boundary constraints (Fig. S1 in the Supplementary Information). We comment on the importance of measuring both components because relaxation of shear strain could lead to insurgence of a negative ∂u_y/∂x at the sample surface, which corresponds to an overall sample rotation (Fig. S2).

Fig. 1: Optical detection of shear strain.

a Photo of a thin Sr₂RuO₄ crystal (dimensions 0.8 × 0.5 × 0.03 mm³, sample S4) attached onto the active side surface of a shear piezo device (dimensions 2.2 × 0.5 × 2.2 mm³) and corresponding schematics. b Optical microscope image of the Sr₂RuO₄ sample (black) on the shear piezo surface with 200 V applied at room temperature. The point-by-point horizontal displacement u_x is represented by the coloured dots, while the arrows schematically indicate the in-plane shear strain. c Direct optical measurement of shear strain on the piezo device (crosses) and Sr₂RuO₄ (circles) at two selected temperatures. d Magnitude of shear at 100 V and e amount of shear transferred to the Sr₂RuO₄ sample as a function of temperature. The dashed lines are polynomial guides to the eye.

We calculate the average shear strain over the surface of the piezo device and the Sr₂RuO₄ sample as a function of V_piezo in Fig. 1c. At room temperature, the piezo device can apply a maximum of about ε_xy = ± 0.05% at V_piezo = ± 200 V, corresponding to a shear strain transferred to the Sr₂RuO₄ of about ε_xy = ± 0.04%. We emphasize the advantage of this optical technique, which enables non-contact measurement of the effective shear strain directly on the sample surface, allowing us to estimate the shear strain transferred to the bulk of the sample and to neglect the uncertainty due to glue-mediated strain transfer from the piezoelectric device. The shear–voltage characteristics in Fig. 1c are nearly linear, with a small hysteresis that is typical of piezoelectric actuators²⁵. While this hysteresis is generally negligible for the purposes of this study, it introduces an uncertainty in the shear strain that is most significant around zero applied voltage, where the upper bound estimated from the vertical width of the hysteresis is δε_xy ≈ 0.02%. Importantly, this uncertainty is minimal at the largest applied voltages, where the strain response is well defined and reproducible. As shown in Fig. 1d, the shear applied at a certain voltage becomes gradually smaller at lower temperature due to the decreasing capacitance of the piezoelectric device (Fig. S3). We compensate for this effect by applying a larger voltage to the piezo device at low temperature, thus achieving similar values of maximum shear strain at all temperatures.

We show in Fig. 1e that the strain transferred from the piezoelectric device to the Sr₂RuO₄ sample is as large as 75% at lower temperature, where a weak enhancement is possibly caused by a stiffening of the glue used to attach the sample to the piezo surface. This strain transfer is consistent with the value predicted by our finite element simulations (Fig. S2) for samples of 30 μm thickness. A direct correlation between the travel distance of the piezoelectric device and its capacitance (Fig. S3) allows us to calculate a conversion factor ε_xy/V_piezo = 0.007%/100V at 2 K. Because the samples investigated in this study have similar thicknesses, we found this conversion factor to be approximately constant.

Response of T _c to shear strain

We now consider changes in T_c by shear strain. To detect the superconducting transition, we insert the piezo-sample assembly of Fig. 1a into the mutual-inductance coil set shown in Fig. 2a. The mutual-inductance technique allows us to measure the imaginary χ′′ and real ${\chi }^{{\prime} }$ parts of the alternating current (AC) susceptibility extracted from the in-phase and out-of-phase lock-in voltages V_x, V_y, respectively. These are measured as a function of temperature for different applied voltages V_piezo in Fig. 2b, c. We use Sr₂RuO₄ samples with a sharp superconducting transition that appears as a dissipation peak in χ′′, with full width at half maximum of about 20 mK, and as a drop in ${\chi }^{{\prime} }$ that indicates the onset of diamagnetic susceptibility, with a transition width smaller than 40 mK. We compare in Fig. 2d two possible criteria for defining T_c: from the position of the peak in χ′′ (squares), and from the mid-point of the drop in ${\chi }^{{\prime} }$ (dots). We comment that both criteria allow us to overcome time-dependent changes in the electronic signal, which suffers from small voltage offsets over time. The values of T_c resulting from the two criteria show consistent trends, with a minor difference in absolute value. Since the peak in χ′′ is about an order of magnitude smaller than the drop in ${\chi }^{{\prime} }$, the latter provides a more reliable criterion across different samples and will be used to determine T_c throughout this work.

Fig. 2: Effect of shear strain on T_c.

a Assembly of shear piezo and Sr₂RuO₄ sample, attached on a bamboo holder with copper wires, to be inserted into a mutual-inductance coil set for measuring AC susceptibility. (b) Imaginary and (c) real parts of the AC susceptibility measured by the mutual-inductance method (sample S5, shear [110]). The colours represent the chronological order in which the measurements were taken, as indicated by the colourscale in (d). The T_c is detected by two different criteria: fitting a Lorentzian peak (dotted vertical line in (b)) or a sigmoid function (dashed vertical line in (c)). d Resulting changes in T_c as a function of voltage applied to the shear piezo: from the peak maximum in χ′′ (squares) and from the mid-point of the drop in ${\chi }^{{\prime} }$ (circles). We use filled (open) symbols to indicate points taken with increasing (decreasing) voltage. e Changes in T_c with shear strain applied along the [100], f [110], and g [001] crystalline directions of Sr₂RuO₄ using several piezo-sample assemblies. The schematics above each panel indicate the corresponding shear deformation of the Sr₂RuO₄ lattice. A linear regression is performed on each data set to illustrate the variation of the small slopes. The grey shaded regions, marked by the black arrows, indicate confidence intervals chosen to represent the scattering of the data of  ± 0.6,  ± 0.9, and  ± 0.6 mK, respectively.

Using several piezo-sample assemblies with Sr₂RuO₄ glued along different crystalline axes, we apply continuous shear strains along the three directions shown in Fig. 2e to g: a pure ε_xy shear along the [100], a diagonal shear ${\varepsilon }_{{x}^{{\prime} }{y}^{{\prime} }}^{110}={\varepsilon }_{xx}-{\varepsilon }_{yy}$ along the [110], and a c-axis shear ε_zx along the [001]. As shown in the insets above the panels, these correspond to B_2g, B_1g, and E_g strain modes, respectively. We use several samples from at least two different batches of crystals for the measurements. In contrast to ultrasound experiments where the response of Sr₂RuO₄ to shear modes was measured with strains oscillating at high frequencies, our technique allows us to apply static shear strains across the whole dynamic range of the piezoelectric device. Due to the C_4z rotational symmetry of the lattice, a positive and a negative shear are expected to yield identical results [i.e., ΔT_c(ε_xy) = ΔT_c( − ε_xy)], hence a symmetric trend is expected around ε_xy = 0. The data in Fig. 2e to g show no change of T_c within our experimental resolution for all three kinds of applied shear strain, with neither a kink nor a quadratic trend. We estimate upper limits of a purely linear slope of the kind ∂T_c/∂ε by drawing grey shaded regions that correspond to T_c variations smaller than ± 6, ± 9, and ± 6 mK%⁻¹, respectively. Based on the same shaded regions, upper limits for purely quadratic slopes ∂²T_c/∂ε² are estimated to be ± 60, ± 90, and ± 60 mK%⁻². These values will be further discussed in the following and compared with thermodynamic predictions from ultrasound experiments in Table 1.

Table 1 Comparison between strain and ultrasound

We now consider the possible effect of thermally-induced pre-strains in Sr₂RuO₄ due to the difference in thermal contraction with the piezoelectric device. As we show in Fig. S4, an Sr₂RuO₄ sample glued along the ab plane is affected by a biaxial tensile pre-strain up to about + 0.1 to + 0.3%. For a [100] piezo-sample assembly, this pre-strain can be seen as a combination of compressive A_1g and shear B_1g strain modes. In such configuration, the voltage-driven shear-piezo device applies a pure B_2g strain, a shear mode that is not present in the pre-strain. Therefore, the T_c variations that we investigated provide direct information of the SC coupling to a specific shear mode.

Tensile control experiment

To validate our technique of detecting small T_c changes with strain, we perform a similar experiment with a uniaxial tensile piezo. As shown in Fig. 3, tensile strain applied to Sr₂RuO₄ induces a clear T_c change in the range of several mK (see Fig. S5 for a voltage-to-strain conversion and Fig. S6 for extended data). As indicated by the fitted curve (black dashed line), the strain-dependence of T_c follows a quadratic behaviour. This quadratic trend is consistent with that reported by Watson et al.²⁶, although our minimum is not at Δε_xx = 0. A likely cause of this shift is the presence of pre-strains caused by the differential thermal contraction of Sr₂RuO₄ and the piezoelectric device²⁷. We discuss in Fig. S4 that for an assembly of Sr₂RuO₄ glued over the active surface of a tensile piezo there is a biaxial pre-strain up to about + 0.1 to + 0.3%, which is reasonably consistent with the value of + 0.1% extracted from the fit in Fig. 3. Although it is difficult to extract quantitative results from this control experiment with tensile strain, it proves that the technique used in this work is suitable to capture the smallest variations of T_c.

Fig. 3: Tensile strain applied to Sr₂RuO₄.

Variations of T_c measured by mutual inductance with an assembly of a tensile piezoelectric device and an Sr₂RuO₄ sample. A quadratic dependence on uniaxial strain is observed, with a minimum at Δε_xx ~ + 0.1% that indicates presence of pre-strains. To compare the T_c resolution with the shear data, the grey shaded region indicates an interval of  ± 0.6 mK.

Discussion

Our results show that the T_c of Sr₂RuO₄ does not change under the direct application of shear strains ε_xy, ${\varepsilon }_{{x}^{{\prime} }{y}^{{\prime} }}^{110}$, and ε_zx. This is different from the T_c variations that occur under compressive strains. In the following, we compare our results with the prediction of shear-strain response estimated by combining the results of uniaxial compressions and hydrostatic pressure. We also examine the limit of compatibility with the jumps in ultrasound velocities at T_c and with the change in their slope. Finally, we compare with the jumps in thermal expansion at T_c.

Comparison with uniaxial stress and ultrasound experiments

In Table 1 (left columns), we give some of the irreducible representations (irrep) of the D_4h point group that apply to space group I4/mmm of Sr₂RuO₄ crystal structure. The couplings between different strains and superconductivity are dictated by symmetry. Due to the C_4z rotational symmetry, the T_c variation under shear is expected to be symmetric and show either a linear kink for a two-component SC or a quadratic dependence for a one-component SC. For the two-component order parameter η = (η_x, η_y) that belongs to E_g irrep with components {d_xz, d_yz}, the Ginzburg-Landau (GL) free energy density that describes the coupling to lattice strain is given by (more details in the Supplementary Information)

$${f}_{\eta \varepsilon }= {r}_{1}{\varepsilon }_{{A}_{1g,1}}| {{{\boldsymbol{\eta }}}}{| }^{2}+{r}_{2}{\varepsilon }_{{A}_{1g,2}}| {{{\boldsymbol{\eta }}}}{| }^{2} \\ +{r}_{3}{\varepsilon }_{{B}_{2g}}({\eta }_{x}^{*}{\eta }_{y}+{\eta }_{x}{\eta }_{y}^{*})+{r}_{4}{\varepsilon }_{{B}_{1g}}(| {\eta }_{x}{| }^{2}-| {\eta }_{y}{| }^{2}),$$

(1)

up to second order in η_x, η_y and first order in strain, with constant coupling parameters r_i²⁸. The D_4h symmetry does not allow linear coupling to strains ε_yz and ε_zx of irrep E_g and modulus c₄₄, so to lowest order these strains should not affect T_c. For a nematic or chiral OP, we have

$$\frac{\partial {T}_{c}}{\partial {\varepsilon }_{{B}_{2g}}}=\pm | {r}_{3}| \,\,(\pm sign\,for\,{\varepsilon }_{{B}_{2g}}\gtrless 0),\\ \frac{\partial {T}_{c}}{\partial {\varepsilon }_{{B}_{1g}}}=\pm | {r}_{4}| \,\,(\pm sign\,for\,{\varepsilon }_{{B}_{1g}}\gtrless 0),$$

(2)

indicating that application of pure shear strains B_2g or B_1g is expected to lead to a kink in T_c around ε = 0. We show again with a grey confidence interval in Fig. 4 that our experiment proves little to no coupling to ε_xy.

Fig. 4: Comparison with other stress–strain and ultrasound experiments.

Our experimental data from Fig. 2e are shown as circles, while the grey shaded region marked by the black arrows indicates our confidence interval for the lack of a kink or a quadratic strain dependence. The green solid line shows the kink required by the symmetry, with the slope estimated combining stress experiments from refs. ^22,29,30 via Eq. (3). The V-shaped kinks shown by the blue and red dashed lines are estimated from the c₆₆ ultrasound jump at T_c^20,21 via the 1st order Ehrenfest relation. The dotted red line shows the quadratic slope calculated via the 2nd order Ehrenfest relation from the c₆₆ slope change at T_c²¹.

We now compare our direct measurement of shear strain with estimates from other techniques where shear strain was indirectly applied along other strains. By combining the results of other stress–strain experiments (details in the Supplementary Information and Table S1) as

$$\frac{\partial {T}_{c}}{\partial {\varepsilon }_{xy}}=2{c}_{66} \left(2\frac{d{T}_{c}}{d{\sigma }_{{x}^{{\prime} }{x}^{{\prime} }}^{110}}+\frac{d{T}_{c}}{d{p}_{hydro}}+\frac{\partial {T}_{c}}{\partial {\sigma }_{zz}}\right),$$

(3)

where ${\sigma }_{{x}^{{\prime} }{x}^{{\prime} }}^{110}$ is the uniaxial compression along the [110] direction, σ_zz is uniaxial compression along the z axis, and the hydrostatic pressure p_hydro is positive for compression (opposite sign compared to the σ_i). By using c₆₆ = 65.5 GPa²¹ and values of the three slopes + 71 mk GPa⁻¹²², − 210 mk GPa^−1 29, and + 76 mk GPa^−1 30, respectively, we obtain ∂T_c/∂ε_xy ~ 10.5 mK%⁻¹, which is shown by the green V-shaped line in Fig. 4. This estimate is consistent with the one provided by Jerzembeck et al. $| \partial {T}_{c}/\partial {\varepsilon }_{xy}| < 13\,mK{\%}^{-1}$^[ 22.

Interestingly, we observe no measurable change in T_c under ${\varepsilon }_{{x}^{{\prime} }{y}^{{\prime} }}^{110}$, which corresponds to pure B_1g shear strain (Fig. 2f). This seems in contrast to previous uniaxial stress studies along [100], where an increase in T_c was reported and attributed to the B_1g symmetry channel³¹. In those works, the strain state includes a combination of symmetry components due to the Poisson effect, with variation of both B_1g and A_1g strains. By contrast, our setup applies a pure B_1g shear strain, with constant A_1g strain. Furthermore, the strain magnitude in our experiment is smaller, such that the Fermi surface remains far from the van Hove singularity. At present, the absence of a T_c response under pure B_1g strain remains an open question, but it highlights the complexity of how different strain symmetries couple to the superconducting state.

In addition, we compare our results with estimates from the thermodynamic Ehrenfest relations

$$\left| \frac{\partial {T}_{c}}{\partial {\varepsilon }_{ij}}\right|=\sqrt{-\frac{{T}_{c}\Delta {c}_{ij}}{f({\beta }_{k})\Delta C}},$$

(4)

where Δc_ij is the jump in elastic moduli at T_c and ΔC ~ 570 J m⁻³K^−1 21 is the jump in specific heat, with f(β_k) a function of the 4th order GL free energy coefficients β_k²⁸. Here, the coefficients f(β_k) are parameters that can be estimated in the weak-coupling limit under the isotropic-band approximation²⁸, yielding the equations and estimates presented in the central columns of Table 1. As shown in Fig. 4, the V-shaped kinks estimated from Ehrenfest relations have a much larger slope than our direct measurement. It should be noted that discrepancies are also serious for the A_1g compressional modes (c₁₁ + c₁₂)/2 and c₃₃: the T_c variations under constant stress are both about a factor five smaller than the expectation from the compressional-mode ultrasound jumps⁶. Thus, we find here an inconsistency with the Ehrenfest relations, which has to be resolved.

Finally, we consider the upper limit for quadratic variation in T_c found in this study (grey regions of Fig. 2e to g) and compare it with estimates from second order Ehrenfest relations that depend on the slope jump of elastic moduli at T_c (right columns of Table 1)²⁸. While the quadratic term estimated from c₆₆ is rather small, the one from (c₁₁ − c₁₂)/2 is very large. It is rather puzzling that a second-order coupling is so large in the absence of the first-order coupling. Such discrepancy in both shear and compressional modes suggests that the ultrasound response may contain additional contributions not given by the coupling to the SC OP. In the normal state of Sr₂RuO₄, the elastic moduli c₁₁ and (c₁₁ − c₁₂)/2 are known to exhibit a peculiar temperature dependence compared to the other modes³². These modes involve stretching and shrinking of the in-plane Ru-O bond lengths that may cause additional coupling to the quasiparticles, although it is not clear how such coupling changes below T_c.

Comparison with thermal expansion

Another thermodynamic comparison can be made with jumps in thermal expansion coefficient α_i = ∂ε_ii/∂T using the different Ehrenfest relation

$$\frac{\partial {T}_{c}}{\partial {\sigma }_{ii}}=-{T}_{c}\frac{\Delta {\alpha }_{i}}{\Delta C},$$

(5)

where C is the specific heat, and Δ refers to the jumps at T_c. Note that, unlike Eq. (4) for elastic moduli, which involves elastic couplings to the superconducting order-parameter in the free energy, this Ehrenfest relation is a more direct thermodynamic requirement. From the negative jumps Δα_a = − 0.42 × 10⁻⁷ K⁻¹ and Δα_c = − 0.52 × 10⁻⁷ K⁻¹ observed by Grube et al.³³, we obtain ∂T_c/∂p_hydro = − 360 mk GPa⁻¹ and ∂T_c/∂σ_zz = + 140 mk GPa⁻¹. These estimates are within a factor two in agreement with the results of hydrostatic pressure − 210 mk GPa^−1 29 and c-axis uniaxial-stress + 76 mk GPa^−1 30. Thus, the strong disagreements of ultrasound jumps in both shear and compression modes suggest that the ultrasound jumps may contain additional contributions which are not properly included in the Ehrenfest relation of Eq. (4).

To conclude, we investigated the coupling of Sr₂RuO₄ superconductivity to shear strain. For this purpose, we developed a technique where thin sample crystals were glued on shear-strain piezoelectric devices that allowed us to directly apply static shear strains along three symmetry channels. The magnitudes of strains were evaluated from optical imaging down to 30 K. We found no detectable change of T_c, with a resolution limit smaller than 10 mK%⁻¹. This poses strict limits on the possible nature of the superconducting order parameter, pointing at ruling out a two-component order parameter. We note, however, that there remains an unresolved discrepancy between the results of ultrasound and static deformation (stress/strain) experiments, which requires further investigation. Our analysis indicates that Sr₂RuO₄ has a one-component OP, or it is a host of more exotic OPs that are required to fully explain its superconducting state. The new method developed in this study can be readily applied to other superconductors for which a two-component OP is suggested, such as UPt₃, or to other materials showing different kinds of phase transitions where a two-component order parameter may emerge.

Methods

Piezo-sample assemblies and sample crystals

In order to apply various shear strains to Sr₂RuO₄ samples, we used a shear piezoelectric device (Thorlabs, PL5FB) consisting of a single piezoelectric ceramic layer of PbZr_xTi_1−xO₃ (PZT) with gold-plated electrodes on the top and bottom surfaces. The device provides a free-stroke displacement of 1.3 μm with the application of ± 200 V at room temperature. We cut the commercial 5 × 5 × 0.5 mm³ shear devices into four pieces of about 2 × 2 × 0.5 mm³ with a diamond-disc cutter, and polished the cut planes using a polishing machine (Musashino Denshi, MA-200) with a diamond slurry of 3 μm. We left a suitable surface roughness to secure strong adhesion to the sample crystal and to provide a clear optical contrast for measuring actual strain. For the control experiment, we used a tensile strain device (Thorlabs, PA3CE, dimension 2 × 2 × 2 mm³) consisting of piezoelectric layers stacked in series. The device provides a free-stroke expansion displacement of 2  μm with the application of + 100 V at room temperature.

The Sr₂RuO₄ single crystals used in this study were grown by a floating-zone method using an image furnace (Canon Machinery, SC-E15HD)³⁴. The cystal orientation was determined by X-ray Laue patterns (Rigaku, RASCO BL-II with a CCD camera). We selected Sr₂RuO₄ crystals with a sharp superconducting transition at T_c = 1.5 K, ensuring the absence of metallic-Ru inclusions that would induce broad enhancement of T_c up to 3.5 K³⁵. The Sr₂RuO₄ crystals were cut into a typical size of 0.4 × 1 mm² and polished down to a thickness of 30 ± 5 μm with a diamond slurry of 3 μm. Each thin crystalline sample was glued to the active side surface of a piezo device with epoxy (EPO-TEK, 353ND) cured at 110–120 °C for 20 min. The lap shear of EPO-TEK 353ND (13.5 MPa) is much larger than the typical lap shear of other low-temperature epoxy, such as Stycast 2850FT (3 MPa, with catalyst 24LV), ensuring good adhesion between the Sr₂RuO₄ and the piezo element.

High-voltage electrodes for the piezo elements were provided by copper wires (diameter 0.14 mm) attached using silver paste (EPO-TEK, H20E) cured at 120 °C for 2 h. For SC measurements, each piezo-sample assembly was mounted on a bamboo holder on which ten insulated copper wires (diameter 0.1 mm) were used for thermal anchoring, glued with  varnish (GE, 8530) as shown in Fig. 2a.

Image correlation to optically measure the strains

We employed an optical-imaging technique to quantify the shear and tensile strain applied by the piezoelectric devices to Sr₂RuO₄ samples. Digital photographs were taken with different V_piezo using an optical microscope (MicroSupport, SS306) with 10–40 × zoom. Digital image correlation (DIC) with DICe software was employed to compute in-plane displacements in the x and y directions^23,24. The average strain over the sample and piezo surfaces was calculated line-by-line averaging the displacement data, while the error was estimated from the standard deviation of the data.

Calibration of strain

The conversion factor from V_piezo to shear ε_xy and uniaxial ε_xx was measured at different temperatures between 30 and 295 K in an optical cryostat (Thermalblock, custom design). Since the piezo displacement per unit volt depends on temperature, the capacitance of the piezo was used to bridge between the measurements at 30 K, lowest temperature accessible by our optical cryostat, and the measurements of T_c, which were performed at and below 2 K. The strain transfer ratio discussed in Fig. 1e was estimated on different samples and found to be rather constant among the several samples due to their comparable thicknesses.

Determination of T _c

Evaluation of T_c with exceptionally high precision and accuracy was crucial in this study. We adopted alternating-current (AC) susceptibility measurements using a mutual-inductance coil set³⁶ and a lock-in amplifier (Stanford Research Systems, SR830) to probe both the imaginary χ′′ and real ${\chi }^{{\prime} }$ parts of the AC susceptibility defined as $\chi=\partial M/\partial H=-i{\chi }^{{\prime\prime} }+{\chi }^{{\prime} }$. By driving an alternating current of the form ${I}_{AC}(t)={I}_{0}\cos (2\pi ft)$ in the excitation coil, the resulting voltage in the pickup coil is ${V}_{AC}(t)=(1/\alpha ){V}_{sample}{I}_{0}f(-{\chi }^{{\prime\prime} }\cos (2\pi ft)+{\chi }^{{\prime} }\sin (2\pi ft))$, with α a constant related to the geometry of the coils, V_sample the sample volume, I₀ the amplitude of the excitation current, and f its frequency³⁷. The coil set consisted of a pair of in-series pick-up/reference coils (400 turns each, internal diameter 5 mm, length 5.5 mm, copper wires diameter 50 μm) wound in opposite directions, and an excitation coil (2400 turns, internal diameter 6.0 mm, length 36 mm, copper wires diameter 100 μm), producing a magnetic field of 84 μT mA⁻¹. An alternating current (frequency 17.7 kHz, root mean squared amplitude of 1 mA) was sourced to the excitation coil using the voltage output of the lock-in amplifier and a series resistance of 10 kΩ. In-phase (V_x) and out-of-phase (V_y) lock-in components were calibrated at 2 K by considering the phase delay of the circuit involving only the excitation coil. Several values of current were tested to ensure that the effect of heating was below 1 mK near T_c. The sample temperature was measured with a thermometer (Lake Shore, Cernox CX-1030-SD-HT) mounted on the copper plate to which each piezo-sample assembly and mutual-inductance coil set were thermally anchored.