Distinct uniaxial stress and pressure fingerprint of superconductivity in the 3D kagome lattice compound CeRu₂

Abstract

The exploration of tunable superconductivity in strongly correlated electron systems is a central pursuit in condensed matter physics, with implications for both fundamental understanding and potential applications. The Laves phase CeRu₂, a pyrochlore compound, exhibits a three-dimensional (3D) kagome lattice type geometry giving rise to flat bands and degenerate Dirac points, where band structure features intertwine with strong multi-orbital interaction effects deriving from its correlated electronic structure. Here, we combine muon spin rotation (μSR), uniaxial in-plane stress, and hydrostatic pressure to probe the superconducting state of CeRu₂. Uniaxial stress up to 0.22 GPa induces a dome-shaped evolution of the critical temperature T_c, with an initial plateau, successively followed by enhancement and suppression without any structural phase transition. Stress is further found to drive a crossover from anisotropic to isotropic s-wave pairing. In contrast, hydrostatic pressure up to 1.9 GPa leaves T_c largely unchanged but alters the superfluid density from exponential to linear behavior at low temperatures, indicative of nodal superconductivity under hydrostatic pressure. These findings identify CeRu₂ as a prime platform for multifold tuning of superconductivity in a 3D correlated material.

The pyrochlore compound CeRu2 hosts a three-dimensional kagome lattice with flat bands and it’s possible to tune superconductivity in the system under various conditions. Here, by combining muon spin rotation with uniaxial stress and hydrostatic pressure, the authors observe distinct and complementary tuning pathways, including a dome-shaped stress-induce devolution of Tc with a crossover from anisotropic to isotropic s-wave pairing, and a pressure-driven change from nodeless to nodal superconductivity.

Introduction

The kagome network, composed of corner-sharing triangles, naturally gives rise to a unique diversity of electronic features such as flat bands, van Hove singularities, and Dirac fermions, producing peculiar interaction profiles resulting from destructive interference for the flat band and the associated sublattice frustration mechanism around the van Hove singularities¹. In kagome metals such as CsV₃Sb₅² where several van Hove singularities are placed near the Fermi level, these effects conspire to reveal a wide range of unconventional electronic ground states including magnetism, nematic order, charge ordering, superconductivity, and more. In the uprising context of geometry-induced quantum effects, flat bands have become another central theme in the study of correlated electron systems, can be provided for in a kagome-type lattice environment, opening up yet another realm of exotic phenomena^1–6.

Although flat bands are typically associated with two-dimensional (2D) kagome systems^3,4,7–9—e.g., CoSn, FeSn, (La,Y)Ru₃Si₂^4,10–15, Ce₂Ru₃Si¹⁶, AV₃Sb₅ and AM₆X₆, recent attention has turned to three-dimensional analogs.

In particular, the pyrochlore lattice which consists of corner-sharing tetrahedra and hosts a 3D-kagome sublattice. They mimic the electronic isolation of 2D layers thanks to its unique crystallographic symmetry (space group Fd$\overline{3}\mathrm{m}$)¹⁷ and the geometrically driven destructive interference that suppresses inter-sublattice coupling¹. Known pyrochlore compounds^18,19 such as CuV₂S₄ and CaNi₂ have also emerged as fertile ground for flat-band physics and topological states. In CaNi₂, flat bands and Dirac cones have been observed¹⁹ by angular resolved photoemission (ARPES) experiments.

In this perspective, the pyrochlore compound CeRu₂²⁰, has recently been re-examined due to its rich band structure arising from the Ru-based 3D kagome network^5,21. Angle-resolved photoemission spectroscopy (ARPES) measurements along the (111) direction have revealed nearly dispersionless flat bands originating from both 3d Ru and 4f Ce states¹⁷, thus demonstrating the effect of the sublattice interference mechanism in preserving the 2D-kagome flat band features onto the third dimension in the pyrochlore system. In addition, this structural framework not only supports flat bands, but also protects topological features, such as 3D Dirac cones at symmetry-enforced points in the Brillouin zone. In particular, CeRu₂ exhibits Dirac crossings constrained to the X point due to its non-symmorphic space group symmetry^22,23, a property shared between pyrochlore materials.

Superconductivity remains relatively rare in these systems and often competes with other symmetry-breaking orders. However, in CeRu₂, the normal-state electronic structure—dominated by hybridized 3d Ru and itinerant 4f Ce states—appears particularly favorable for the emergence of superconductivity. This is supported by complementary photoemission and de Haas-van Alphen (dHvA) measurements, which highlight the strong hybridization and itinerant character of the Ce-derived bands^24–26. Additionally, anisotropic s-wave superconductivity in CeRu₂ has been reported by μSR, NMR/NQR, angle-resolved specific heat, and photoemission studies^27–30. The superfluid density is consistent with values observed in unconventional superconductors, suggesting an unconventional pairing mechanism. Furthermore, three distinct magnetic anomalies observed up to 8 T²¹ indicate the possible presence of coexisting phases. Taken together, these findings demonstrate that pyrochlore compounds with 3D kagome sublattice represent a promising frontier for the study of flat-band physics, electronic correlation, and emergent superconductivity in three dimensions.

An important question in this study is whether these flat bands can be effectively tuned and, if so, how such tuning influences the microscopic properties of superconductivity. In this context, we present results from a unique combination of uniaxial stress applied along the kagome plane and hydrostatic pressure μSR experiments (Fig. 1a–c), offering valuable insights into the tunable microscopic superconducting properties of CeRu₂. Key findings include (1) a non-monotonic dependence of the superconducting critical temperature on stress, (2) an uniaxial stress-induced crossover from anisotropic to isotropic s-wave superconductivity, and (3) a hydrostatic pressure-induced transition from nodeless to nodal unconventional pairing. Notably, the stress and pressure required to induce these changes differ by an order of magnitude, highlighting the distinct roles of directional and isotropic tuning parameters.

Fig. 1. Schematic overview and transport response.

The CeRu₂ unit cell with marked (111) kagome planes (light gray), schematically showing the effect of uniaxial stress compression (a, stress applied in the (111) plane, although the exact direction within the plane is not known), and b the hydrostatic pressure. c Uniaxial stress device designed for the μSR experiments (1—single-crystalline sample fixed to the detachable holder; 2—coil for simultaneous measurements of AC magnetic susceptibility; 3—force sensors; 4—temperature sensor; 5—printed circuit board). d Temperature dependence of AC susceptibility (left) and the longitudinal resistivity (right). Arrows mark the midpoint of the superconducting transition at T_c ≈ 4.86 K and the normal state anomaly at T^* ≈ 40 K. The inset shows the experimental setup for measurement of magnetotransport properties in CeRu₂ single crystal (1,4—current leads; 2,3—voltage probes for longitudinal resistivity, ρ_xx, measurements; 3,5—voltage probes for Hall resistivity, ρ_xy). e Temperature dependence of magnetoresistance (left) and Hall resistivity (ρ_xy, right) measured under the applied field of 9 T. The vertical gray line marks the T^* anomaly, corresponding to the onset of MR.

Results and discussion

Uniaxial stress

Stress effects on superconducting T_c

The single crystal of CeRu₂ studied here exhibits a superconducting transition onset at T_c ≈ 5 K, consistent with previous reports of high-quality samples. Samples with Ru deficiency typically show^31,32 a higher T_c (6 K). Our sample also shows a clear anomaly in susceptibility around T^* = 40 K (see Fig. 1d). Furthermore, Fig. 1e reveals that the magnetoresistance (MR) at 9 T increases below T^*, reaching 7% MR at 5 K. Superimposed on the plot, we show that the Hall resistivity that saturates under T^* coincides with both the MR increase and the AC broad anomaly (see Fig. 1e). Previous muon spin rotation measurements (μSR)^21,33 revealed a time-reversal symmetry (TRS) breaking in the same temperature regime. This convergence of three independent probes provides compelling evidence for a coupled magnetic and electronic response in CeRu₂ at temperatures well above T_c. Furthermore, the TRS breaking signal was shown to be slightly modified below T_c²¹, indicating an interplay between the TRS-breaking state and superconductivity. While the precise nature of the magnetic transition near 40 K remains to be clarified, the observed TRS breaking is intrinsic and represents the first example of a TRS-breaking normal state in a pyrochlore system.

Figure 2a presents the temperature dependence of magnetic susceptibility as a function of uniaxial stress applied within the kagome (111) direction. The superconducting transition temperature, T_c, is defined as the midpoint of the transition, corresponding to the exclusion of 50% magnetic flux. As shown, T_c remains nearly constant up to 0.07 GPa, and then increases slightly, reaching a maximum at 0.13 GPa, which is then followed by a continuous decrease at higher stress. At 0.22 GPa, T_c is suppressed to 4 K, an approximate 16% reduction compared to ambient pressure. This behavior reflects a dome-shaped stress dependence of T_c. Figure 2c shows that the stress-strain relationship remains linear in the elastic regime of the sample, indicating the absence of any stress-induced structural phase transition in CeRu₂. Thus, the observed modulation of T_c is not related to structural changes but probably originates from stress-induced shifts in the electronic structure, particularly involving flat bands near the Fermi level.

Fig. 2. Superconducting critical temperature under uniaxial stress.

a The temperature dependence of AC susceptibility, measured under various uniaxial stress values. The dark dash-dotted line midway through the transition defines the estimation of T_c. b Topographic color plot of normalized susceptibility of CeRu₂ as a function of temperature and applied stress. The light-color border corresponds to the transition from the superconducting to normal state and vice versa. c Stress-strain curve for CeRu₂. The force sensor output is proportional to stress, while the displacement sensor reading is proportional to strain. The dashed lines highlight linear fits to the data across the entire applied stress range, indicating the absence of a structural phase transition within this stress regime.

In CeRu₂, the situation is further complicated by the presence of multiple flat-band features, as revealed by ARPES measurements¹⁷. Three distinct flat-band-like features are observed: (1) the Kondo resonance state ($4f_{5/2}^1$) located at the Fermi level, (2) its spin-orbit sideband ($4f_{7/2}^1$) at approximately −0.25 eV, both characteristic of Ce-based systems, and (3) a disentangled flat band originating from destructive interference in the Ru 3d orbitals of the Kagome pyrochlore lattice, located at −0.11 eV. This proximity implies that a significant portion of the associated density of states (DOS) peak lies near the Fermi level. It is plausible that under the right stress conditions along the kagome plane, this Ru-derived flat band and associated DOS peak could be shifted toward the Fermi level, thereby slightly enhancing the superconducting state through an increase in the density of states (DOS). However, beyond a critical stress threshold, the flat band may overshoot the Fermi level, shifting the DOS center of mass away and thereby suppressing T_c. Whether this 16% reduction in T_c is driven solely by the movement of the kagome flat band, or by collective changes involving both Ce and Ru flat-band features, remains an open question requiring theoretical investigation. A relevant comparison comes from CaNi₂, where a low-lying flat band at −0.4 eV was shifted to the Fermi level through only 2% Ru doping in the isostructural compound¹⁹ Ca(Rh_0.98Ru_0.02)₂. This highlights the sensitivity of the electronic structure in such systems, which will be shown later that even a modest applied stress in CeRu₂ could significantly modify the Fermi surface topology by shifting multiple flat-band features.

In addition to possible stress-induced shifts of the flat-band density of states, one may ask whether stress-induced band broadening plays a role. Order-of-magnitude estimates suggest that the bandwidth of the Ru-4d dominated bands is on the order of several electronvolts, whereas the flat portion of the band hybridized with the Ce-4f states is up to an order of magnitude narrower. Given that the applied uniaxial stress corresponds to an energy scale of only a few millielectronvolts, stress-induced bandwidth renormalization of the Ru-4d bands is unlikely to be the dominant effect and can be considered negligible to leading order. Nevertheless, while the impact of uniaxial stress on the overall band structure may be small, the associated energy scale is comparable to the hybridization strength with the Ce-4f states. As such, uniaxial stress can significantly influence low-energy electronic correlations and superconducting properties, consistent with the pronounced effects observed in our experiments.

Stress effects on superfluid density and gap structure

Next, we examine how the superfluid density and superconducting gap structure evolve under uniaxial stress. This analysis is based on measurements of the superconducting muon spin relaxation rate in the vortex state of CeRu₂, using the transverse field TF-μSR. These time spectra were recorded in both the normal state (10 K) and the superconducting state (0.27 K), with a field of 30 mT applied along the stress direction. The normal-state spectrum shows weak depolarization due to the random local fields from the nuclear moments, and is well described by a single Gaussian function as shown in our previous study²¹. In contrast, the superconducting state exhibits a significantly enhanced relaxation rate due to the formation of the flux-line lattice (FLL) as reported previously²¹. The resulting asymmetric field distribution is modeled using a sum of two Gaussians, from which a single central field value is extracted, as detailed in the “Methods” section. The temperature dependence of the superconducting muon spin depolarization rate, σ_SC, under five different stress conditions (up to 0.18 GPa) is presented in Fig. 3. In order to investigate the symmetry of the SC gap, we underline the fact that the penetration depth λ(T) is related to σ_SC(T) in the presence of an external applied field H_a, which induces a perfect triangular vortex lattice with field strength H_a ≪ H_c2 by the equation³⁴:

$$\frac{{\sigma }_{\mathrm{SC}}}{{\gamma }_{\mu }}=0.06091\frac{{\mathrm{\Phi }}_0}{{\lambda }^2\left(T\right)}$$ 1

where γ_μ is the gyromagnetic ratio of the muon and Φ₀ is the magnetic-flux quantum. The temperature dependence of the superfluid density (related to λ⁻²(T)) was then fitted with nodeless (an)isotropic s-wave and nodal d-wave models to determine the superconducting gap symmetry (see “Methods” section).

Fig. 3. Superfluid density under uniaxial stress.

a–e The temperature dependence of the superconducting muon spin relaxation rate σ_SC, recorded under various stress conditions up to 0.18 GPa. The dash-dotted and solid lines represent the fits using the anisotropic s-wave and isotropic s-wave models, respectively. f The temperature dependence of the diamagnetic shift, recorded under various uniaxial stress pressures. The error bars represent the s.d. of the fit parameters.

As shown in Table 1, the anisotropic s-wave model, which accounts for variation of the a single gap value from ${\mathrm{\Delta }}_{\min }$ to ${\mathrm{\Delta }}_{\max }$, provides the best fit to the data under zero stress, yielding a gap ratio of ${\mathrm{\Delta }}_{\min }/{\mathrm{\Delta }}_{\max }=0.41$. This result is in very good agreement with previous μSR measurements without a uniaxial stress cell²¹, and is consistent with values reported from NMR (${\mathrm{\Delta }}_{\min }/{\mathrm{\Delta }}_{\max }=0.33$)²⁹ and photoemission experiments (${\mathrm{\Delta }}_{\min }/{\mathrm{\Delta }}_{\max }=0.447$)²⁸. In Fig. 3a we highlight the difference between isotropic(solid line)/anisotropic s-wave(dashed line). Upon increasing the uniaxial stress within the (111)-planes, the superconducting gap evolves toward isotropy, as the anisotropy parameter ${\mathrm{\Delta }}_{\min }/{\mathrm{\Delta }}_{\max }$ vanishes at 0.07 GPa as seen in Table 1. This trend indicates a stress-induced crossover from an anisotropic to an isotropic s-wave superconducting state. We note that the crossover from an anisotropic nodeless state to an isotropic nodeless state is already apparent at a qualitative level, even prior to detailed modeling, by examining the temperature range over which the superfluid density saturates at low temperatures. Specifically, Figs. 3a and 4a present ambient-pressure data measured using two independent μSR instruments, consistent with our previous μSR results²¹. In both cases, the superfluid density saturates only below approximately 1.2 K. In contrast, for measurements performed under 0.07 GPa and higher uniaxial stress values, saturation sets in already below about 2.3 K. This pronounced extension of the saturation regime provides direct, model-independent evidence for a more isotropic superconducting gap under uniaxial stress. Taken together, these qualitative observations, supported by quantitative analysis, provide evidence for a stress-induced crossover from an anisotropic nodeless to an isotropic nodeless superconducting state in CeRu₂.

Table 1.

Uniaxial stress dependence of various parameters

Symmetry	λ0 (nm)	Δmax (meV)	Δmin (meV)
s-wave (0 Gpa)	282(4)	0.82(4)	–
an.s-wave (0 GPa)	278(6)	0.83(4)	0.34(16)
s-wave (0.07 GPa)	294(3)	0.95(5)	–
an.s-wave (0.07 GPa)	294(3)	0.95(5)	–
s-wave (0.10 GPa)	292(4)	0.94(4)	–
an.s-wave (0.10 GPa)	291(5)	0.95(4)	–
s-wave (0.13 GPa)	290(4)	0.93(4)	–
an.s-wave (0.13 GPa)	290(4)	0.93(4)	–
s-wave (0.18 GPa)	272(5)	0.85(5)	–
an.s-wave (0.18 GPa)	272(5)	0.86(5)	–

Fit parameters of gap symmetries extracted from the temperature dependence of superconducting relaxation rate σ_SC under different stress conditions.

Fig. 4. Superfluid density under hydrostatic pressure.

The temperature dependence of the superconducting muon spin relaxation rate σ_SC, recorded under various hydrostatic pressure conditions: a 1.5 GPa and b 1.9 GPa. In a, solid line corresponds to the fit using an anisotropic s-wave model. In b, the dashed and the solid lines represent the fits using the isotropic s-wave and nodal models, respectively. c The temperature dependence of the diamagnetic shift, recorded under various hydrostatic pressures. The error bars represent the s.d. of the fit parameters.

Further insight are gained from the temperature dependence of the diamagnetic shift, shown in Fig. 3f. The shift is defined as ΔB_dia = μ₀(H_int,SC − H_int,NS), where H_int,SC and H_int,NS are the internal magnetic fields in the superconducting and normal states, respectively. A pronounced diamagnetic response of approximately 0.6 mT develops below T_c, reflecting the onset of bulk superconductivity. Importantly, the magnitude and shape of this diamagnetic shift remain unaffected by applied stress.

Taken together, these results show that while uniaxial stress does not significantly alter the absolute value of the superfluid density or the diamagnetic response, it plays a crucial role in tuning the superconducting gap symmetry—driving a continuous transition from anisotropic to fully isotropic nodeless behavior.

Hydrostatic pressure

Complementary to stress, hydrostatic pressure μSR^35,36 experiment was conducted on CeRu₂. An external magnetic field of 30 mT was applied, and the sample was field-cooled from above T_c to establish a well-ordered vortex lattice. Measurements were carried out over the temperature range of 0.25–8 K. The Fourier transform of the muon asymmetry spectra reveals an asymmetric internal field distribution f(B), arising from the flux-line lattice (FLL) in the superconducting state as previously reported⁴ and an additional background contribution from the pressure cell. Thus, for temperatures below T_c, the transverse-field (TF) μSR spectra were analyzed using a three-component Gaussian model: two components corresponding to the superconducting vortex state and one accounting for the signal from the pressure cell. From this fit, the superconducting depolarization rate σ_SC was extracted, allowing us to trace the evolution of the superconducting gap structure under hydrostatic pressure (Fig. 4a).

In the analysis, the anisotropy ratio ${\mathrm{\Delta }}_{\min }/{\mathrm{\Delta }}_{\max }$ was fixed at 0.41 for 0 GPa, consistent with stress-dependent measurements and previous results under ambient conditions⁴. Up to 1.5 GPa, the data are well described using an anisotropic s-wave model (Fig. 4b). However, at the highest applied pressure of 1.9 GPa, the best fit is obtained with a nodal gap symmetry (Fig. 4b). At this pressure, σ_SC(T) exhibits a linear temperature dependence at low temperatures, indicating that λ⁻²(T) approaches its zero-temperature value linearly, a hallmark of nodes in the superconducting gap. This pressure-induced nodal behavior suggests that hydrostatic pressure suppresses the minimum gap ${\mathrm{\Delta }}_{\min }$, eventually closing it and allowing for gapless quasiparticle excitations. This, in turn, gives rise to the observed power-law behavior of the relaxation rate at low temperatures. The values of the penetration depth and the superconducting gaps, estimated at various pressures, are summarized in Table 2.

Table 2.

Hydrostatic pressure dependence of various parameters

Symmetry	λ0 (nm)	Δmax (meV)	Δmin (meV)
s-wave (0 Gpa)	300(4)	0.72(2)	–
an.s-wave (0 GPa)	297(4)	0.69(2)	0.29(7)
s-wave (1.5 GPa)	311(3)	0.83(1)	–
an.s-wave (1.5 GPa)	307(4)	0.79(2)	0.32(3)
nodal (1.9 GPa)	303(5)	1.22(4)	0
s-wave (1.9 GPa)	277(4)	0.92(3)	–

Fit parameters of gap symmetries extracted from the temperature dependence of superconducting relaxation rate σ_SC at different pressures.

The diamagnetic response under hydrostatic pressure also differs significantly from that under uniaxial stress. As pressure increases the overall magnitude of the diamagnetic shift nearly doubles at 1.9 GPa. Most notably, a paramagnetic upturn emerges below approximately 1.5 K. Paramagnetic shifts of similar nature have been reported in cuprate and iron-based high-temperature superconductors^37–40 and was interpreted as field induced magnetism. Such anomalous paramagnetic responses have also been discussed as manifestations of unconventional pairing symmetry in the high-temperature superconductors⁴¹. In the absence of magnetism³¹ in CeRu₂ within the investigated pressure range, the observed paramagnetic shift below 1.5 K, together with the emergence of nodal behavior, is likely associated with intrinsic modifications of the superconducting state.

A pressure-induced crossover from a nodeless to a nodal superconducting state has previously been observed⁴² in the iron-based superconductor Ba_1−xRb_xFe₂As₂, where the appearance of accidental nodes was proposed, or as in the case of KFe₂As₂ where the non-monotonic pressure behavior of T_c is proposed⁴³ as an effect of the change of SC pairing symmetry. Additionally, in kagome superconductors such as (K,Rb)V₃Sb₅, a pressure-induced transition from nodal to nodeless gap structure has been reported³ and attributed to the suppression of charge order. In our case, the emergence of nodal behavior under pressure does not necessarily imply a d-wave symmetry, as any nodal gap structure would produce the observed linear temperature dependence of the penetration depth. A detailed microscopic understanding of how pressure influences the superconducting state is still required. Altogether, these findings underscore the powerful role of hydrostatic pressure as a tuning parameter in correlated superconductors. In CeRu₂, this tuning appears to act as a modulator of the superconducting pairing symmetry.

To gain insight into the stress and pressure induced changes of the SC gap structure, we next provide a qualitative framework that illustrates how such changes may arise. At first glance, the notion that an anisotropic perturbation—such as uniaxial stress—can enhance the isotropic character of anisotropic s-wave superconductor may appear counterintuitive. However, this behavior can be naturally understood within the context of unconventional s-wave pairing. In such systems, the superconducting gap is not dominated by a simple isotropic constant component but instead arises from a complex superposition of momentum-dependent harmonic contributions. As a consequence, even a moderate stress perturbation can modify the balance among competing electronic ordering tendencies, thereby altering the harmonic composition of the anomalous propagator (i.e., the particle-particle wave function). Depending on how these competing channels are affected, the resulting superconducting state may exhibit either enhanced or reduced gap anisotropy.

For CeRu₂, a plausible scenario consistent with our data is that the system initially hosts a moderately anisotropic unconventional s-wave state. Upon the application of uniaxial stress, the competing subleading ordering tendencies may be suppressed or reshaped such that the preferred harmonic composition of the extended s-wave gap leads to a reduced difference between the maximum and minimum gap values on the stress-distorted Fermi surface, resulting in a more isotropic gap. By contrast, under hydrostatic pressure, starting again from a moderately anisotropic extended s-wave gap, we envision that pressure enhances the gap anisotropy to the extent that accidental nodes emerge. This behavior is reminiscent of phenomena previously observed in several iron pnictide superconductors^42,44. Detailed future theoretical studies, for example using functional renormalization group or random phase approximation approaches, may provide further insight into the microscopic mechanisms underlying this behavior.

Conclusion

We present the first comprehensive study of uniaxial stress and hydrostatic pressure effects on the microscopic superconducting properties of the pyrochlore compound CeRu₂, which hosts a three-dimensional (111) kagome lattice. Our key findings include: (1) a non-monotonic evolution of the superconducting transition temperature under uniaxial stress applied along the kagome plane; in the absence of any structural changes, this behavior suggests an unconventional superconducting state; (2) stress-induced modifications of the superfluid density, consistent with a crossover from anisotropic s-wave to a more isotropic superconducting gap; and (3) a pressure-induced change in the temperature dependence of the superfluid density at low temperatures, indicative of a transition to a nodal pairing state or the emergence of accidental nodes. These results reveal distinct and complementary pathways for tuning superconductivity through mechanical means, underscoring the need for theoretical insight into the interplay between lattice geometry, electronic correlations, and pairing symmetry in CeRu₂. In particular, a systematic ab initio analysis of the impact of point group symmetry breaking from uniaxial stress vs. the point group preserving hydrostatic pressure needs to focus on the itineracy vs. localized nature of electronic degrees of freedom from a heavy fermion perspective, the modification of the correlated character via changing the interaction over bandwidth ratio, and the band structure dynamics of flat bands, van Hove singularities, and eigenstate sublattice distribution both within the Brillouin zone and with respect to the Fermi level. Although similar phenomena are observed across various families of unconventional superconductors, CeRu₂ occupies a unique position at the intersection of heavy-fermion and kagome physics, which are both known for their rich and complex quantum states, combined with an enhanced of electronic order through the material’s 3D character. The mechanical tunability demonstrated here offers a powerful platform to explore unifying mechanisms underlying unconventional superconductivity in broader classes of materials, including cuprates, iron-based superconductors, and nickelates.

Methods

Single crystal growth

The single crystal of CeRu₂ was grown using the Czochralski pulling method with a prepared polycrystalline ingot i5n a tetra-arc furnace. Firstly, the spheroidal ingot of mass 10 g was prepared by arc melting stoichiometric amounts of lumpish cerium and ruthenium, which was then placed on the water-cooled copper hearth of the tetra-arc furnace under argon atmosphere with a titanium ingot used as an oxygen getter. The current of 16A was applied into the ingot to melt it and the velocity of the hearth rotation was 0.3 revolutions per minute. A columnar single crystal was successfully grown after several hours.

Sample characterization

The quantitative elemental composition, atomic ratios and homogeneity of the CeRu₂ crystal were measured using X-ray fluorescence spectroscopy (XRF) on an AMETEK Orbis PC Micro EDXRF spectrometer system. Elemental multiple-points and distribution maps for cerium and ruthenium were collected from a cut surface of the crystal in air by applying a white X-ray beam generated by a rhodium tube set to 35 kV and 500 μA, with a primary beam spot focus of 30 μm. An area of approximately 2 × 2 mm² on the crystal surface was probed using EDX. The Ce:Ru ratio in the crystal was confirmed to be stoichiometric (1:2) within the limits of the measurement accuracy.

The orientation of the single-crystal sample was determined using a Siemens KRISTALLOFLEX 760 Laue X-ray diffractometer equipped with a Mo point-focus X-ray source and a Coolview Fast Digital Imagers (FDI) Photonic Science Laue dual camera. The Laue backscattering pattern confirms that the [111] crystallographic direction is orthogonal to the exposed surface of the aligned CeRu₂ crystal, demonstrating that the applied uniaxial stress is oriented along the [111] direction.

Hydrostatic pressure cell

A double-wall piston-cylinder cell, constructed from CuBe and MP35N materials and specifically designed for μSR experiments under pressure, was utilized to generate pressures up to 1.9 GPa^35,36. Daphne 7373 served as the pressure-transmitting medium. The pressure calibration was established by measuring the superconducting transition of a small indium plate inside the sample volume from AC susceptibility. The pressure cell was optimized for maximum filling factor. Approximately 40% of the muons stopped within the sample during the experiments.

Uniaxial stress cell

For the μSR and AC susceptibility measurements, we used a cryo-compatible, piezoelectric-driven uniaxial pressure cell^45–47, compatible with μSR experiments and designed to fit the He cryostat of the Dolly instrument at PSI’s πE1 beamline (see Fig. 1c). The piezo actuators enable continuous in situ pressure tuning. The CeRu₂ single crystal (7 × 4.4 × 0.6 mm³, oriented along [111]) was mounted on a detachable grade-2 titanium holder using Stycast-2850 FT epoxy. The holder, detailed in our previous work⁴⁶, includes flexures to minimize torques and is designed to maximize muon acceptance by allowing missed muons to reach the veto counter. Hematite foils masked the sample plates to absorb muons outside the target area, resulting in 40% muon stopping efficiency. A pair of 100-turn coils was positioned near the sample for in situ AC susceptibility measurements to track under various stress conditions. The muon-facing area was 4  × 4.5 mm².

By using the force-sensor calibration described in detail elsewhere^45,48, we estimated the force applied to CeRu₂. We find that 1 μV (change in the force sensor reading from the zero force position) corresponds to  ~ 31 N and the maximum applied force is  ~ 560 N. Our sample had a cross section area of 2.6 mm², which results in a maximum applied stress of  ~ 0.22 GPa.

Magnetotransport measurements

Magnetotransport measurements were carried out in a standard four-probe method using the Physical Property Measurement System (PPMS, Quantum Design). The diagonal arrangement of voltage contacts was used and the magnetoresistance and Hall resistance were obtained by symmetrization and anti-symmetrization of the measured data, respectively.

Muon spin rotation experiments

Transverse-field (TF) μSR experiments with uniaxial stress experiment and high pressure were performed on the VMS instrument and the GPD instrument, equipped with an Oxford Sorption Pumped ³He Cryostat type Heliox/Variox, at the Swiss Muon Source (SμS) at the Paul Scherrer Institut, in Villigen, Switzerland.

In a μSR experiment nearly 100 % spin-polarized muons μ⁺ are implanted into the sample one at a time. The positively charged μ⁺ thermalize at interstitial lattice sites, where they act as magnetic microprobes. In a magnetic material the muon spin precesses in the local field B_μ at the muon site with the Larmor frequency ν_μ = γ_μ/(2π) B_μ (muon gyromagnetic ratio γ_μ/(2π) = 135.5 MHz T⁻¹). Using the μSR technique, important length scales of superconductors can be measured, namely the magnetic penetration depth λ and the coherence length ξ. If a type-II superconductor is cooled below T_c in an applied magnetic field ranged between the lower (H_c1) and the upper (H_c2) critical fields, a vortex lattice is formed which in general is incommensurate with the crystal lattice with vortex cores separated by much larger distances than those of the unit cell. Because the implanted muons stop at given crystallographic sites, they will randomly probe the field distribution of the vortex lattice. Such measurements need to be performed in a field applied perpendicular to the initial muon spin polarization (so called TF configuration).

Zero-field and high transverse-field experiments were carried out to probe normal state properties on the HAL instrument in a field range of 0.01 to 8 T. The sample in the form of compressed pellet (diameter 8 mm) was placed on the silver sample holder and mounted in the cryostat. The TF- and ZF-µSR data were analyzed by using MUSRFIT program⁴⁹.

Analysis of TF-μSR data

In order to model the asymmetric field distribution (P(B)) in the SC state, the TF-μSR time spectra measured below T_c are analyzed by using the following two-component functional form:

$$A_{\mathrm{TF}}\left(t\right)={\sum }_{i=0}^2{A}_{s,i}\exp \left[-\frac{{\sigma }_i^2{t}^2}{2}\right]\cos \left({\gamma }_{\mu }{B}_{\mathrm{int},s,i}t+\phi \right)$$ 2

Here A_s,i, σ_i and B_int,s,i is the initial asymmetry, relaxation rate, and local internal magnetic field of the i-th component. φ is the initial phase of the muon-spin ensemble. γ_μ/(2π) ≃ 135.5 MHz/T is the muon gyromagnetic ratio. The first and second moments of the local magnetic field distribution are given by ref. ⁵⁰

$$⟨B⟩={\sum }_{i=0}^2\frac{A_{s,i}{B}_{\mathrm{int},s,i}}{A_{s,1}+A_{s,2}}$$ 3

and

$${⟨\mathrm{\Delta }B⟩}^2=\frac{{\sigma }^2}{{\gamma }_{\mu }^2}={\sum }_{i=0}^2\frac{A_{s,i}}{A_{s,1}+A_{s,2}}\left[{\sigma }_i^2/{\gamma }_{\mu }^2+{\left(B_{\mathrm{int},s,i}-⟨B⟩\right)}^2\right].$$ 4

Above T_c, in the normal state, the symmetric field distribution could be nicely modeled by only one component. The obtained relaxation rate and internal magnetic field are denoted by σ_ns and B_int,s,ns. σ_ns is found to be small and temperature independent (dominated by nuclear magnetic moments) above T_c and assumed to be constant in the whole temperature range. Below T_c, in the SC state, the relaxation rate and internal magnetic field are indicated by σ_SC and B_int,s,sc. σ_SC is extracted by using ${\sigma }_{\mathrm{SC}}=\sqrt{{\sigma }^2-{\sigma }_{\mathrm{ns}}^2}$. B_int,s,sc is evaluated from $⟨B⟩$ using Eq. (3).

Analysis of λ(T)

λ(T) was calculated within the local (London) approximation (λ  ≫  ξ) by the following expression^49–51:

$$\frac{{\sigma }_{\mathrm{SC}}\left(T,{\mathrm{\Delta }}_{0,i}\right)}{{\sigma }_{\mathrm{SC}}\left(0,{\mathrm{\Delta }}_{0,i}\right)}=1+\frac{1}{\pi }{\int }_0^{2\pi }{\int }_{\mathrm{\Delta }\left(T,\phi \right)}^{\infty }\left(\frac{\partial f}{\partial E}\right)\frac{EdEd\phi }{\sqrt{E^2-{\mathrm{\Delta }}_i{\left(T,\phi \right)}^2}},$$ 5

where $f={\left[1+\exp \left(E/k_{\mathrm{B}}T\right)\right]}^{-1}$ is the Fermi function, φ is the angle along the Fermi surface, and Δ_i(T, φ) = Δ_0,iΓ(T/T_c)g(φ) (Δ_0,i is the maximum gap value at T = 0). The temperature dependence of the gap is approximated by the expression $\mathrm{\Gamma }\left(T/T_{\mathrm{c}}\right)=\tanh \left\{1.82{\left[1.018\left(T_{\mathrm{c}}/T-1\right)\right]}^{0.51}\right\}$⁵², while g(φ) describes the angular dependence of the gap and it is replaced by 1 for both an s-wave and an s + s-wave gap, and $\mid \cos \left(2\phi \right)\mid$ for a d-wave gap⁴².

Author contributions

Z.G. conceived, designed, and supervised the project. Crystal growth: H.X.L. and Y.S. Detailed macroscopic characterization of the crystal: D.G. and Z.G. Magnetotransport experiments under ambient pressure: V.S., O.G., and Z.G. Preparation for uniaxial stress experiments: Z.G., D.D., and C.M.III. μSR and AC susceptibility experiments under uniaxial stress and corresponding discussions: Z.G., D.D., O.G., V.S., C.M.III., P.K., S.S.I., J.N.G., V.G., R.S., H.H.K., T.S., J.C., R.T., R.K., and H.L. μSR data analysis: O.G. and Z.G. Figure development and writing of the paper: Z.G. and O.G. with contributions from all authors. All authors discussed the results, interpretation, and conclusion.

Peer review

Peer review information

Nature Communications thanks Subhajit Roychowdhury and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All the results are available from the authors upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s42005-026-02553-3.