Broken time-reversal symmetry (BTRS) in superconductors is widely regarded as evidence for a multicomponent order parameter, yet the microscopic origin of the associated spontaneous magnetic fields remains unresolved. Sr 2 RuO 4 is a central example, where BTRS has been reported by several probes but its relation to superconductivity remains controversial. Here we use zero-field muon spin relaxation (μSR) to investigate how spontaneous magnetic fields evolve with pair-breaking disorder and crystal inhomogeneity in Sr 2 RuO 4 . We combine measurements on Sr 2−y La y RuO 4 single crystals at ambient pressure with measurements on stoichiometric Sr 2 RuO 4 under hydrostatic pressure, and compare these results with literature data spanning samples with different levels of inhomogeneity and Ru inclusions. We find that the superconductivity-induced enhancement of the exponential muon-spin relaxation rate, ∆Λ, associated with spontaneous magnetic fields decreases monotonically with La substitution and hydrostatic pressure, following approximately ∆Λ ∝ Tc2 when the effective density of field-generating inhomogeneities remains nearly unchanged. In contrast, ∆Λ is enhanced in samples containing stronger structural inhomogeneities, including nonmagnetic random disorder and Ru inclusions. These results support a picture in which spontaneous magnetic fields in Sr 2 RuO 4 are generated by nonmagnetic inhomogeneities within a BTRS superconducting state, with the strength controlled by the superconducting order parameter. More broadly, our findings provide an experimental framework for interpreting local magnetic signatures in multicomponent superconductors.

We investigate the electronic structure at the surface of the correlated oxide Ca_3Ru_2O_7, a low-symmetry ruthenate oxide which hosts an unconventional polar-metal phase. From a combination of angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy measurements, we demonstrate that the surface hosts an insulating phase, a distinct departure from metallicity within the bulk. Utilizing quantitative low-energy electron diffraction in conjunction with electronic structure calculations, we show how this results from a combined surface structure relaxation and the impact of marked electronic correlations in this system. Our findings highlight the proximity of Ca_3Ru_2O_7 to an insulating metallic state, and illustrate how subtle structural distortions can control its emergent electronic phases.

We performed high-resolution magnetostriction measurements on the Pauli-limited superconductor Sr_2RuO_4 using high-quality single crystals. A first-order superconducting transition, accompanied by pronounced hysteresis, was observed under in-plane magnetic fields, where the relative length change of the sample, ΔL/L, was on the order of 10^-8. To ensure the reliability of the measurements, particular attention was paid to minimizing the influence of magnetic torque, which can significantly affect data under in-plane field configurations, via field-angle-resolved magnetostriction. Within the hysteresis regime, slightly below the Pauli-limited upper critical field, a hump-like anomaly in the magnetostriction coefficient was identified. Furthermore, a characteristic double-peak structure in the field-angle derivative of the magnetostriction provides additional support for this anomaly. Although these findings may reflect a lattice response associated with the emergence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in Sr_2RuO_4, the possibility of a broadened first-order transition cannot be excluded. Notably, this magnetostriction anomaly qualitatively deviates from the FFLO phase boundary suggested by previous NMR measurements, highlighting the necessity for further experimental and theoretical investigations to elucidate the nature of the FFLO state in this material.

Uniaxial stress has now been widely used to study correlated electron materials. However, Fermi surface-resolved experimental data on the evolution of the electronic structure under piezoelectrically applied stress are sparse, with no reports of de Haas-van Alphen (dHvA) effects under uniaxial stress. Here we present dHvA measurements under c-axis uniaxial stress on the unconventional superconductor Sr_2RuO_4. This allows us to study the evolution of the electronic structure directly and to gain insight into the contradicting behavior of the predicted enhancement of the electronic density of states and the observed suppression of T_c. We are able to follow all Fermi surfaces for stress up to -1.8 GPa and find that the cross-sectional areas of the hole-like α sheet increase and electron-like β sheet decrease. At the same time, the area of the electron-like γ sheet increases. Therefore, in contrast to in-plane uniaxial stress, charge transfer is the mechanism for approaching the electron-to-hole Lifshitz transition and the associated Van Hove singularity. Additionally, we find that the effective masses on all three Fermi sheets are slightly enhanced as the Lifshitz transition is approached. We compare the dHvA results with quantum oscillations in the magnetostriction and band structure calculations, and find good agreement. At a more general level, our findings show that quantum oscillation measurements under uniaxial stress, combined with band-structure calculations, offer a promising new route for studying quantum materials.

Uniaxial pressure is a powerful tuning parameter for quantum materials, but conventional thermodynamic probes such as specific heat are difficult to realize in the constrained geometries of strain apparatus. We develop a quantitative analysis framework for a.c. elastocaloric effect measurements that enable the reconstruction of the absolute entropy and hence specific heat across complex phase diagrams. The absolute accuracy is achieved by combining measurements in the strong coupling regime at low frequencies with high signal-to-noise measurements in the quasi-adiabatic regime at high frequencies. Applying the approach to the correlated superconductor Sr_2RuO_4, we obtain an absolute entropy map across the phase diagram including across phase transitions deep into the superconducting state. We demonstrate that from such data one can derive the absolute specific heat which is currently not possible through other approaches. This data reinforces the finding that the quenching of entropy within the superconductor Sr_2RuO_4 is strongest at the critical strain consistent with the superconducting gap being maximized at the Van Hove singularity (VHs). Furthermore, we demonstrate that, although Δc /(γT) does increase at the VH strain, this increase is much weaker than previously inferred from more indirect caloric experiments.

The nature of the broken time reversal symmetry (BTRS) state in Sr_2RuO_4 remains elusive, and its relation to superconductivity remains controversial. There are various universal predictions for the BTRS state when it is associated with a multicomponent superconducting order parameter. In particular, in the BTRS superconducting state, spontaneous fields appear around crystalline defects, impurities, superconducting domain walls and sample surfaces. However, this phenomenon has not yet been experimentally demonstrated for any BTRS superconductor. Here, we aimed to verify these predictions for Sr_2RuO_4 by performing muon spin relaxation (μSR) measurements on Sr_2-yLa_yRuO_4 single crystals at ambient pressure and stoichiometric Sr_2RuO_4 under hydrostatic pressure. The study allowed us to conclude that spontaneous fields in the BTRS superconducting state of Sr_2RuO_4 appear around non-magnetic inhomogeneities and, at the same time, decrease with the suppression of T_ c. The observed behaviour is consistent with the prediction for multicomponent BTRS superconductivity in Sr_2RuO_4. The results of the work are relevant to understanding BTRS superconductivity in general, as they demonstrate, for the first time, the relationship among the superconducting order parameter, the BTRS transition, and crystal-structure inhomogeneities.

Transport and magnetic properties of a 4 f 2 fcc lattice compound, PrCdNi4, were studied. The magnetic susceptibility chi (T ) follows the Curie-Weiss law from 300 to 20 K, as expected for a free Pr3+ ion. As the temperature decreases below 5 K, chi (T ) approaches a constant, indicating van Vleck paramagnetic behavior. The specific heat C(T ) displays a broad shoulder at around 4 K, which can be reproduced by a doublet-triplet two-level model with an energy gap of 12 K. These results suggest a nonmagnetic P3 doublet ground state of the Pr3+ ion in the cubic crystalline electric field. C(T ) exhibits a peak at TO = 1.0 K and this peak remains robust against magnetic fields up to 5 T. In powder neutron diffraction measurements, no magnetic reflection was observed at 0.32 K < TO. Two anomalies at B = 2.1 and 5.3 T in magnetoresistance rho(B) at 0.05 K likely originate from switching in the order parameter. These results suggest that the phase transition at TO is ascribed to an antiferro-type order of the electric quadrupole or magnetic octupole of the P3 doublet in the 4 f 2 fcc lattice.

Transport and magnetic properties of a 4f^2 fcc lattice compound, PrCdNi_4, were studied. The magnetic susceptibility, χ(T), follows the Curie–Weiss law from 300 K to 20 K, as expected for a free Pr^3+ ion. As the temperature decreases below 5 K, χ(T) approaches a constant, indicating van-Vleck paramagnetic behavior. The specific heat, C(T), displays a broad shoulder at around 4 K, which can be reproduced by a doublet triplet two-level model with an energy gap of 12 K. These results suggest a non-magnetic Γ_3 doublet ground state of the Pr^3+ ion in the cubic crystalline electric field. C(T) exhibits a peak at T_ O = 1.0 K and this peak remains robust against magnetic fields up to 5 T. In powder neutron diffraction measurements, no magnetic reflection was observed at 0.32 K < T_ O. Two anomalies at B = 2.1 and 5.3 T in magnetoresistance ρ(B) at 0.05 K likely originate from switching in the order parameter. These results suggest that the phase transition at T_ O is ascribed to an antiferro-type order of the electric quadrupole or magnetic octupole of the Γ_3 doublet in the 4f^2 fcc lattice.

Transport and magnetic properties of a 4$f^{2}$ fcc lattice compound, PrCdNi$_4$, were studied. The magnetic susceptibility, $\chi(T)$, follows the Curie--Weiss law from 300 K to 20 K, as expected for a free Pr$^{3+}$ ion. As the temperature decreases below 5 K, $\chi(T)$ approaches a constant, indicating van-Vleck paramagnetic behavior. The specific heat, $C(T)$, displays a broad shoulder at around 4 K, which can be reproduced by a doublet triplet two-level model with an energy gap of 12 K. These results suggest a non-magnetic $\Gamma_3$ doublet ground state of the Pr$^{3+}$ ion in the cubic crystalline electric field. $C(T)$ exhibits a peak at $T_{\rm O}$ = 1.0 K and this peak remains robust against magnetic fields up to 5 T. In powder neutron diffraction measurements, no magnetic reflection was observed at 0.32 K $&lt;$ $T_{\rm O}$. Two anomalies at $B$ = 2.1 and 5.3 T in magnetoresistance $\rho(B)$ at 0.05 K likely originate from switching in the order parameter. These results suggest that the phase transition at $T_{\rm O}$ is ascribed to an antiferro-type order of the electric quadrupole or magnetic octupole of the $\Gamma_3$ doublet in the 4$f^2$ fcc lattice.

The perfect linear temperature dependence of the electrical resistivity in a variety of “strange” metals is a real puzzle in condensed matter physics. For these materials also other non-Fermi liquid properties are predicted or detected. In particular we mention the results derived from holographic theories which conclude that plasmons should be overdamped due to a low energy continuum in the electronic susceptibility. These predictions were supported by electron energy-loss spectroscopy in reflection on cuprates and ruthenates. Here we use electron energy-loss spectroscopy in transmission to study collective charge excitations in the layer metal Sr 2 RuO 4 . This metal has a transition from a perfect Fermi liquid below T ≈ 30 K into a “strange” metal phase above T ≈ 800 K. In this compound we cover a complete range between in-phase and out-of-phase oscillations. Outside the classical range of electron-hole excitations, leading to a Landau damping, we observe well-defined plasmons. The optical (acoustic) plasmon due to an in-phase (out-of-phase) charge oscillation of neighbouring layers exhibits a quadratic (linear) positive dispersion. Using a model for the Coulomb interaction of the charges in a layered system, it is possible to describe the range of optical plasmon excitations at high energies in a mean-field random phase approximation without taking correlation effects into account. In contrast, resonant inelastic X-ray scattering data show at low energies an enhancement of the acoustic plasmon velocity due to correlation effects. This difference can be explained by an energy dependent effective mass which changes from ≈ 3.5 at low energy to 1 at high energy near the optical plasmon energy. There are no signs of over-damped plasmons predicted by holographic theories.

In the $R$Al(Si,Ge) ($R$: lanthanides) family, both spatial inversion and time-reversal symmetries are broken. This may offer opportunities to study Weyl-fermion physics in nontrivial spin structures emerging from a noncentrosymmetric crystal structure. In this study, we investigated the anomalous Hall effect (AHE) in NdAlGe via magnetotransport, magnetization, and magnetic torque measurements down to 40 mK (0.4 K for magnetization). The single crystals grown by a laser-heated floating-zone method exhibit a single magnetic phase transition at $T_{\rm M}$ = 13.5 K, where the $T_{\rm M}$ is the transition temperature. With the magnetic field parallel to the easy $\lbrack$001$\rbrack$ axis, the AHE gradually evolves as the temperature decreases below $T_{\rm M}$. The anomalous Hall conductivity (AHC) reaches $\sim$320 $\Omega^{-1}$cm$^{-1}$ at 40 mK in the magnetically saturated state. Except in low-temperature low-field plateau phases, the AHC and magnetization are proportional, and their ratio agrees with the ratios for conventional ferromagnets, suggesting that the intrinsic AHE occurs by the Karplus-Luttinger mechanism. Below $\sim$0.6 K, the curves of Hall resistivity against the field exhibit plateaus at low fields below $\sim$0.5 T, correlating with the plateaus in the magnetization curve. For the first plateau, the magnetization is one order of magnitude smaller than the magnetically saturated state, whereas the AHE is more than half that in the saturated state. This finding under well below $T_{\rm M}$ suggests that the AHE at the first plateau is not governed by the magnetization and may be interpreted based on a multipole or spin chirality.

Magnetic skyrmions were initially identified fi ed in non-centrosymmetric materials. However, recent theoretical advancements have suggested their potential emergence in centrosymmetric compounds, which has been experimentally observed in cases such as GdRu2Si2.. In this study, we comprehensively explored the magnetic properties of GdOs2Si2 , a centrosymmetric counterpart to GdRu2Si2,, with a focused investigation into the crystal direction-dependent behavior of key parameters. By combining magnetic, thermal, and transport measurements, we constructed a detailed magnetic phase diagram along two crystal directions, revealing intriguing similarities and differences ff erences compared to GdRu2Si2. . Notably, we observed a distinct suppression of the Hall resistivity in GdOs2Si2 - a phenomenon absent in its counterpart compound.

Superconductivity in quasicrystals poses a new challenge in condensed matter physics. We measured the resistance and ac magnetic susceptibility of a Ta1.6Te dodecagonal quasicrystal, which is superconducting below Tc ~ 1 K. We show that the upper critical field increases linearly with a large slope of − 4.4 T/K with decreasing temperature down to 0.04 K, with no tendency to level off. The extrapolated zero-temperature critical field exceeds the Pauli limit by a factor of 2.3. We also observed flux-flow resistance with thermally activated behavior and an irreversibility field that is distinct from the upper critical field. We discuss these peculiarities in terms of the nonuniform superconducting gap and spin-orbit interaction in quasicrystal structures.

In bulk Sr_2RuO_4, the strong sensitivity of the superconducting transition temperature T_c to nonmagnetic impurities provides robust evidence for a superconducting order parameter that changes sign around the Fermi surface. In superconducting epitaxial thin-film Sr_2RuO_4, the relationship between T_c and the residual resistivity ρ_0, which in bulk samples is taken to be a proxy for the low-temperature elastic scattering rate, is far less clear. Using high-energy electron irradiation to controllably introduce point disorder into bulk single-crystal and thin-film Sr_2RuO_4, we show that T_c is suppressed in both systems at nearly identical rates. This suggests that part of ρ_0 in films comes from defects that do not contribute to superconducting pairbreaking, and establishes a quantitative link between the superconductivity of bulk and thin-film samples.

There is considerable evidence that the superconductivity of Sr2RuO4 has two components. Among this evidence is a jump in the shear elastic modulus c66 at the critical temperature Tc, observed in ultrasound measurements. Such a jump is forbidden for homogeneous single-component order parameters, and it implies that Tc should develop as a cusp under the application of shear strain with (110) principal axes. This shear strain should split the onset temperatures of the two components, if they coexist, or select one component if they do not. Here, we report measurements of Tc and the elastocaloric effect of Sr2RuO4 under uniaxial stress applied along the [110] lattice direction. Within experimental resolution, we resolve neither a cusp in the stress dependence of Tc, nor any second transition in the elastocaloric effect data. We show that reconciling these null results with the observed jumps in c66 requires extraordinarily fine tuning to a triple point of the Ginzburg-Landau parameter space. In addition, our results are inconsistent with homogeneous time-reversal symmetry breaking at a temperature T2 Tc as identified in muon spin relaxation experiments.

We report on a new technique for measuring the dynamic Young's modulus, $E$, of quantum materials at low temperatures as a function of static tuning strain, $\epsilon$, in piezoactuator-driven pressure cells. In addition to a static tuning of stress and strain, we apply a small-amplitude, finite-frequency a.c. (1 Hz$ \lesssim \omega \lesssim $1000 Hz) uniaxial stress, $\sigma_{ac}$, to the sample and measure the resulting a.c. strain, $\epsilon_{ac}$, using a capacitive sensor to obtain the associated modulus $E$. We demonstrate the performance of the new technique through proof-of-principle experiments on the unconventional superconductor Sr$_2$RuO$_4$, which is known for its rich temperature-strain phase diagram. In particular, we show that the magnitude of $E$, measured using this a.c. technique at low frequencies, exhibits a pronounced nonlinear elasticity, which is in very good agreement with previous Young's modulus measurements on Sr$_2$RuO$_4$ under [100] strain using a d.c. method (Noad et al., Science 382, 447-450 (2023)). By combining the new a.c. Young's modulus measurements with a.c. elastocaloric measurements in a single measurement, we demonstrate that these a.c. techniques are powerful in detecting small anomalies in the elastic properties of quantum materials. Finally, using the case of Sr$_2$RuO$_4$ as an example, we demonstrate how the imaginary component of the modulus can provide additional information about the nature of ordered phases.

This review describes the recent progress of floating-zone techniques for bulk single-crystal growth. The most crucial point of the crucible-free technique is to keep the molten zone stable. It has been investigated and reported to yield a steeper temperature gradient at the liquid–solid interface along the growth direction and a homogeneous molten liquid along the rotation axis. This article overviews several recent achievements starting from the conventional setup, particularly for lamps equipped in horizontal or vertical configurations, tilting mirrors, shielding the irradiation, and filament sizes for the optical-lamp floating-zone furnaces. Also, the recently advancing laser-heated floating-zone furnaces are described. Throughout the article, the author emphasizes that the floating-zone technique has been a powerful tool for crystal growth since the 1950s with its roots in the zone-melting method, and it has still been advancing for further materials’ growth such as quantum materials with modern scientific concepts.

There is considerable evidence that the superconductivity of Sr2RuO4 has two components. Among this evidence is a jump in the shear elastic modulus c66 at the critical temperature Tc, observed in ultrasound measurements. Such a jump is forbidden for homogeneous single-component order parameters, and it implies that Tc should develop as a cusp under the application of shear strain with 〈110〉 principal axes. This shear strain should split the onset temperatures of the two components, if they coexist, or select one component if they do not. Here, we report measurements of Tc and the elastocaloric effect of Sr2RuO4 under uniaxial stress applied along the [110] lattice direction. Within experimental resolution, we resolve neither a cusp in the stress dependence of Tc, nor any second transition in the elastocaloric effect data. We show that reconciling these null results with the observed jumps in c66 requires extraordinarily fine tuning to a triple point of the Ginzburg-Landau parameter space. In addition, our results are inconsistent with homogeneous time-reversal symmetry breaking at a temperature T2≤Tc as identified in muon spin relaxation experiments. Published by the American Physical Society 2024

By determining the superconducting lower and upper critical fields H-c1(T) and H-c2(T), respectively, in a high-purity spherical Sr2RuO4 sample via ac-susceptibility measurements, we obtain the temperature dependence of the coherence length xi and the penetration depth lambda down to 0.04T(c). Given the high sample quality, the observed T-2 dependence of lambda at low temperatures cannot be explained in terms of impurity effects. Instead, we argue that the weak type-II superconductor Sr2RuO4 has to be treated in the nonlocal limit. By comparing our data with existing theory in that limit, the penetration depth in Sr2RuO4 agrees with a gap structure having vertical line nodes, while horizontal line nodes cannot account for the observation. The work highlights the potential benefits of purifying other unconventional superconductors in order to access the fascinating nonlocal regime in more materials and to determine their Cooper pair wave functions.