Elastocaloric determination of the phase diagram of Sr2RuO4.

One of the main developments in unconventional superconductivity in the past two decades has been the discovery that most unconventional superconductors form phase diagrams that also contain other strongly correlated states. Many systems of interest are therefore close to more than one instability, and tuning between the resultant ordered phases is the subject of intense research1. In recent years, uniaxial pressure applied using piezoelectric-based devices has been shown to be a particularly versatile new method of tuning2,3, leading to experiments that have advanced our understanding of the fascinating unconventional superconductor Sr2RuO4 (refs. 4-9). Here we map out its phase diagram using high-precision measurements of the elastocaloric effect in what we believe to be the first such study including both the normal and the superconducting states. We observe a strong entropy quench on entering the superconducting state, in excellent agreement with a model calculation for pairing at the Van Hove point, and obtain a quantitative estimate of the entropy change associated with entry to a magnetic state that is observed in proximity to the superconductivity. The phase diagram is intriguing both for its similarity to those seen in other families of unconventional superconductors and for extra features unique, so far, to Sr2RuO4.

The Lorenz ratio as a guide to scattering contributions to transport in strongly correlated metals.

In many physical situations in which many-body assemblies exist at temperature T, a characteristic quantum-mechanical time scale of approximately [Formula: see text] can be identified in both theory and experiment, leading to speculation that it may be the shortest meaningful time in such circumstances. This behavior can be investigated by probing the scattering rate of electrons in a broad class of materials often referred to as "strongly correlated metals". It is clear that in some cases only electron-electron scattering can be its cause, while in others it arises from high-temperature scattering of electrons from quantized lattice vibrations, i.e., phonons. In metallic oxides, which are among the most studied materials, analysis of electrical transport does not satisfactorily identify the relevant scattering mechanism at "high" temperatures near room temperature. We therefore employ a contactless optical method to measure thermal diffusivity in two Ru-based layered perovskites, Sr3Ru2O7 and Sr2RuO4, and use the measurements to extract the dimensionless Lorenz ratio. By comparing our results to the literature data on both conventional and unconventional metals, we show how the analysis of high-temperature thermal transport can both give important insight into dominant scattering mechanisms and be offered as a stringent test of theories attempting to explain anomalous scattering.

[Formula: see text]-linear resistivity from magneto-elastic scattering: Application to PdCrO2.

An electronic solid with itinerant carriers and localized magnetic moments represents a paradigmatic strongly correlated system. The electrical transport properties associated with the itinerant carriers, as they scatter off these local moments, have been scrutinized across a number of materials. Here, we analyze the transport characteristics associated with ultraclean PdCrO[Formula: see text]-a quasi-two-dimensional material consisting of alternating layers of itinerant Pd-electrons and Mott-insulating CrO[Formula: see text] layers-which shows a pronounced regime of T-linear resistivity over a wide range of intermediate temperatures. By contrasting these observations to the transport properties in a closely related material PdCoO[Formula: see text], where the CoO[Formula: see text] layers are band-insulators, we can rule out the traditional electron-phonon interactions as being responsible for this interesting regime. We propose a previously ignored electron-magneto-elastic interaction between the Pd-electrons, the Cr local moments and an out-of-plane phonon as the main scattering mechanism that leads to the significant enhancement of resistivity and a T-linear regime in PdCrO[Formula: see text] at temperatures far in excess of the magnetic ordering temperature. We suggest a number of future experiments to confirm this picture in PdCrO[Formula: see text] as well as other layered metallic/Mott-insulating materials.

Investigation of Planckian behavior in a high-conductivity oxide: PdCrO2.

The layered delafossite metal PdCrO[Formula: see text] is a natural heterostructure of highly conductive Pd layers Kondo coupled to localized spins in the adjacent Mott insulating CrO[Formula: see text] layers. At high temperatures, T, it has a T-linear resistivity which is not seen in the isostructural but nonmagnetic PdCoO[Formula: see text]. The strength of the Kondo coupling is known, as-grown crystals are extremely high purity and the Fermi surface is both very simple and experimentally known. It is therefore an ideal material platform in which to investigate "Planckian metal" physics. We do this by means of controlled introduction of point disorder, measurement of the thermal conductivity and Lorenz ratio, and studying the sources of its high-temperature entropy. The T-linear resistivity is seen to be due mainly to elastic scattering and to arise from a sum of several scattering mechanisms. Remarkably, this sum leads to a scattering rate within 10[Formula: see text] of the Planckian value of k[Formula: see text]T/[Formula: see text].

High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure.

A key question regarding the unconventional superconductivity of [Formula: see text] remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat-capacity signature of any second transition of a few percent of that of the primary superconducting transition. The normalized jump in heat capacity, [Formula: see text], grows smoothly as a function of uniaxial pressure, favoring order parameters which are allowed to maximize in the same part of the Brillouin zone as the well-studied van Hove singularity. Thanks to the high precision of our measurements, these findings place stringent constraints on theories of the superconductivity of [Formula: see text].

Evidence for even parity unconventional superconductivity in Sr2RuO4.

Unambiguous identification of the superconducting order parameter symmetry in [Formula: see text] has remained elusive for more than a quarter century. While a chiral p-wave ground state analogue to superfluid 3He-A was ruled out only very recently, other proposed triplet-pairing scenarios are still viable. Establishing the condensate magnetic susceptibility reveals a sharp distinction between even-parity (singlet) and odd-parity (triplet) pairing since the superconducting condensate is magnetically polarizable only in the latter case. Here field-dependent 17O Knight shift measurements, being sensitive to the spin polarization, are compared to previously reported specific heat measurements for the purpose of distinguishing the condensate contribution from that due to quasiparticles. We conclude that the shift results can be accounted for entirely by the expected field-induced quasiparticle response. An upper bound for the condensate magnetic response of <10% of the normal state susceptibility is sufficient to exclude all purely odd-parity candidates.

Low-symmetry nonlocal transport in microstructured squares of delafossite metals.

Intense work studying the ballistic regime of electron transport in two-dimensional systems based on semiconductors and graphene had been thought to have established most of the key experimental facts of the field. In recent years, however, additional forms of ballistic transport have become accessible in the quasi-two-dimensional delafossite metals, whose Fermi wavelength is a factor of 100 shorter than those typically studied in the previous work and whose Fermi surfaces are nearly hexagonal in shape and therefore strongly faceted. This has some profound consequences for results obtained from the classic ballistic transport experiment of studying bend and Hall resistances in mesoscopic squares fabricated from delafossite single crystals. We observe pronounced anisotropies in bend resistances and even a Hall voltage that is strongly asymmetric in magnetic field. Although some of our observations are nonintuitive at first sight, we show that they can be understood within a nonlocal Landauer-Büttiker analysis tailored to the symmetries of the square/hexagonal geometries of our combined device/Fermi surface system. Signatures of nonlocal transport can be resolved for squares of linear dimension of nearly 100 µm, approximately a factor of 15 larger than the bulk mean free path of the crystal from which the device was fabricated.

Probing Momentum-Dependent Scattering in Uniaxially Stressed Sr_{2}RuO_{4} through the Hall Effect.

The largest Fermi surface sheet of the correlated metal Sr_{2}RuO_{4} can be driven through a Lifshitz transition between an electronlike and an open geometry by uniaxial stress applied along the [100] lattice direction. Here, we investigate the effect of this transition on the longitudinal resistivity ρ_{xx} and the Hall coefficient R_{H}. ρ_{xx}(T), when Sr_{2}RuO_{4} is tuned to this transition, is found to have a T^{2}logT form, as expected for a Fermi liquid tuned to a Lifshitz transition. R_{H} is found to become more negative as the Fermi surface transitions from an electronlike to an open geometry, opposite to general expectations from this change in topology. The magnitude of the change in R_{H} implies that scattering changes throughout the Brillouin zone, not just at the point in k space where the transition occurs. In a model of orbital-dependent scattering, the electron-electron scattering rate on sections of Fermi surface with xy orbital weight is found to decrease dramatically.

Hierarchy of Lifshitz Transitions in the Surface Electronic Structure of Sr_{2}RuO_{4} under Uniaxial Compression.

We report the evolution of the electronic structure at the surface of the layered perovskite Sr_{2}RuO_{4} under large in-plane uniaxial compression, leading to anisotropic B_{1g} strains of ϵ_{xx}-ϵ_{yy}=-0.9±0.1%. From angle-resolved photoemission, we show how this drives a sequence of Lifshitz transitions, reshaping the low-energy electronic structure and the rich spectrum of van Hove singularities that the surface layer of Sr_{2}RuO_{4} hosts. From comparison to tight-binding modeling, we find that the strain is accommodated predominantly by bond-length changes rather than modifications of octahedral tilt and rotation angles. Our study sheds new light on the nature of structural distortions at oxide surfaces, and how targeted control of these can be used to tune density of state singularities to the Fermi level, in turn paving the way to the possible realization of rich collective states at the Sr_{2}RuO_{4} surface.

Momentum-resolved superconducting energy gaps of Sr2RuO4 from quasiparticle interference imaging.

Sr2RuO4 has long been the focus of intense research interest because of conjectures that it is a correlated topological superconductor. It is the momentum space (k-space) structure of the superconducting energy gap [Formula: see text] on each band i that encodes its unknown superconducting order parameter. However, because the energy scales are so low, it has never been possible to directly measure the [Formula: see text] of Sr2RuO4 Here, we implement Bogoliubov quasiparticle interference (BQPI) imaging, a technique capable of high-precision measurement of multiband [Formula: see text] At T = 90 mK, we visualize a set of Bogoliubov scattering interference wavevectors [Formula: see text] consistent with eight gap nodes/minima that are all closely aligned to the [Formula: see text] crystal lattice directions on both the α and ß bands. Taking these observations in combination with other very recent advances in directional thermal conductivity [E. Hassinger et al., Phys. Rev. X 7, 011032 (2017)], temperature-dependent Knight shift [A. Pustogow et al., Nature 574, 72-75 (2019)], time-reversal symmetry conservation [S. Kashiwaya et al., Phys. Rev B, 100, 094530 (2019)], and theory [A. T. Rømer et al., Phys. Rev. Lett. 123, 247001 (2019); H. S. Roising, T. Scaffidi, F. Flicker, G. F. Lange, S. H. Simon, Phys. Rev. Res. 1, 033108 (2019); and O. Gingras, R. Nourafkan, A. S. Tremblay, M. Côté, Phys. Rev. Lett. 123, 217005 (2019)], the BQPI signature of Sr2RuO4 appears most consistent with [Formula: see text] having [Formula: see text] [Formula: see text] symmetry.

Electronically driven spin-reorientation transition of the correlated polar metal Ca3Ru2O7.

The interplay between spin-orbit coupling and structural inversion symmetry breaking in solids has generated much interest due to the nontrivial spin and magnetic textures which can result. Such studies are typically focused on systems where large atomic number elements lead to strong spin-orbit coupling, in turn rendering electronic correlations weak. In contrast, here we investigate the temperature-dependent electronic structure of [Formula: see text], a [Formula: see text] oxide metal for which both correlations and spin-orbit coupling are pronounced and in which octahedral tilts and rotations combine to mediate both global and local inversion symmetry-breaking polar distortions. Our angle-resolved photoemission measurements reveal the destruction of a large hole-like Fermi surface upon cooling through a coupled structural and spin-reorientation transition at 48 K, accompanied by a sudden onset of quasiparticle coherence. We demonstrate how these result from band hybridization mediated by a hidden Rashba-type spin-orbit coupling. This is enabled by the bulk structural distortions and unlocked when the spin reorients perpendicular to the local symmetry-breaking potential at the Ru sites. We argue that the electronic energy gain associated with the band hybridization is actually the key driver for the phase transition, reflecting a delicate interplay between spin-orbit coupling and strong electronic correlations and revealing a route to control magnetic ordering in solids.

Evidence for a vestigial nematic state in the cuprate pseudogap phase.

The CuO2 antiferromagnetic insulator is transformed by hole-doping into an exotic quantum fluid usually referred to as the pseudogap (PG) phase. Its defining characteristic is a strong suppression of the electronic density-of-states D(E) for energies |E| < [Formula: see text], where [Formula: see text] is the PG energy. Unanticipated broken-symmetry phases have been detected by a wide variety of techniques in the PG regime, most significantly a finite-Q density-wave (DW) state and a Q = 0 nematic (NE) state. Sublattice-phase-resolved imaging of electronic structure allows the doping and energy dependence of these distinct broken-symmetry states to be visualized simultaneously. Using this approach, we show that even though their reported ordering temperatures T DW and T NE are unrelated to each other, both the DW and NE states always exhibit their maximum spectral intensity at the same energy, and using independent measurements that this is the PG energy [Formula: see text] Moreover, no new energy-gap opening coincides with the appearance of the DW state (which should theoretically open an energy gap on the Fermi surface), while the observed PG opening coincides with the appearance of the NE state (which should theoretically be incapable of opening a Fermi-surface gap). We demonstrate how this perplexing phenomenology of thermal transitions and energy-gap opening at the breaking of two highly distinct symmetries may be understood as the natural consequence of a vestigial nematic state within the pseudogap phase of Bi2Sr2CaCu2O8.

Itinerant ferromagnetism of the Pd-terminated polar surface of PdCoO2.

The ability to modulate the collective properties of correlated electron systems at their interfaces and surfaces underpins the burgeoning field of "designer" quantum materials. Here, we show how an electronic reconstruction driven by surface polarity mediates a Stoner-like magnetic instability to itinerant ferromagnetism at the Pd-terminated surface of the nonmagnetic delafossite oxide metal PdCoO2 Combining angle-resolved photoemission spectroscopy and density-functional theory calculations, we show how this leads to a rich multiband surface electronic structure. We find similar surface state dispersions in PdCrO2, suggesting surface ferromagnetism persists in this sister compound despite its bulk antiferromagnetic order.

Multicritical Fermi Surface Topological Transitions.

A wide variety of complex phases in quantum materials are driven by electron-electron interactions, which are enhanced through density of states peaks. A well-known example occurs at van Hove singularities where the Fermi surface undergoes a topological transition. Here we show that higher order singularities, where multiple disconnected leaves of Fermi surface touch all at once, naturally occur at points of high symmetry in the Brillouin zone. Such multicritical singularities can lead to stronger divergences in the density of states than canonical van Hove singularities, and critically boost the formation of complex quantum phases via interactions. As a concrete example of the power of these Fermi surface topological transitions, we demonstrate how they can be used in the analysis of experimental data on Sr_{3}Ru_{2}O_{7}. Understanding the related mechanisms opens up new avenues in material design of complex quantum phases.

Hydrodynamic Electron Flow and Hall Viscosity.

In metallic samples of small enough size and sufficiently strong momentum-conserving scattering, the viscosity of the electron gas can become the dominant process governing transport. In this regime, momentum is a long-lived quantity whose evolution is described by an emergent hydrodynamical theory. Furthermore, breaking time-reversal symmetry leads to the appearance of an odd component to the viscosity called the Hall viscosity, which has attracted considerable attention recently due to its quantized nature in gapped systems but still eludes experimental confirmation. Based on microscopic calculations, we discuss how to measure the effects of both the even and odd components of the viscosity using hydrodynamic electronic transport in mesoscopic samples under applied magnetic fields.

Determination of the dynamic Young's modulus of quantum materials in piezoactuator-driven uniaxial pressure cells using a low-frequency AC method.

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, ϵ, in piezoactuator-driven pressure cells. In addition to a static tuning of stress and strain, we apply a small-amplitude, finite-frequency AC (1 Hz ≲ ω ≲ 1000 Hz) uniaxial stress, σac, to the sample and measure the resulting AC strain, ϵac, 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 Sr2RuO4, which is known for its rich temperature-strain phase diagram. In particular, we show that the magnitude of E, measured using this AC technique at low frequencies, exhibits a pronounced nonlinear elasticity, which is in very good agreement with previous Young's modulus measurements on Sr2RuO4 under [1 0 0] strain using a DC method [Noad et al., Science 382, 447-450 (2023)]. By combining the new AC Young's modulus measurements with AC elastocaloric measurements in a single measurement, we demonstrate that these AC techniques are powerful in detecting small anomalies in the elastic properties of quantum materials. Finally, using the case of Sr2RuO4 as an example, we demonstrate how the imaginary component of the modulus can provide additional information about the nature of ordered phases.

Observation of the Anomalous Hall Effect in a Layered Polar Semiconductor.

Progress in magnetoelectric materials is hindered by apparently contradictory requirements for time-reversal symmetry broken and polar ferroelectric electronic structure in common ferromagnets and antiferromagnets. Alternative routes can be provided by recent discoveries of a time-reversal symmetry breaking anomalous Hall effect (AHE) in noncollinear magnets and altermagnets, but hitherto reported bulk materials are not polar. Here, the authors report the observation of a spontaneous AHE in doped AgCrSe2 , a layered polar semiconductor with an antiferromagnetic coupling between Cr spins in adjacent layers. The anomalous Hall resistivity 3 µ Ω c m $\mu \Omega \, \textnormal {cm}$ is comparable to the largest observed in compensated magnetic systems to date, and is rapidly switched off when the angle of an applied magnetic field is rotated to ≈80° from the crystalline c-axis. The ionic gating experiments show that the anomalous Hall conductivity magnitude can be enhanced by modulating the p-type carrier density. They also present theoretical results that suggest the AHE is driven by Berry curvature due to noncollinear antiferromagnetic correlations among Cr spins, which are consistent with the previously suggested magnetic ordering in AgCrSe2 . The results open the possibility to study the interplay of magnetic and ferroelectric-like responses in this fascinating class of materials.

Avoided metallicity in a hole-doped Mott insulator on a triangular lattice.

Doping of a Mott insulator gives rise to a wide variety of exotic emergent states, from high-temperature superconductivity to charge, spin, and orbital orders. The physics underpinning their evolution is, however, poorly understood. A major challenge is the chemical complexity associated with traditional routes to doping. Here, we study the Mott insulating CrO2 layer of the delafossite PdCrO2, where an intrinsic polar catastrophe provides a clean route to doping of the surface. From scanning tunnelling microscopy and angle-resolved photoemission, we find that the surface stays insulating accompanied by a short-range ordered state. From density functional theory, we demonstrate how the formation of charge disproportionation results in an insulating ground state of the surface that is disparate from the hidden Mott insulator in the bulk. We demonstrate that voltage pulses induce local modifications to this state which relax over tens of minutes, pointing to a glassy nature of the charge order.

Layered metals as polarized transparent conductors.

The quest to improve transparent conductors balances two key goals: increasing electrical conductivity and increasing optical transparency. To improve both simultaneously is hindered by the physical limitation that good metals with high electrical conductivity have large carrier densities that push the plasma edge into the ultra-violet range. Technological solutions reflect this trade-off, achieving the desired transparencies only by reducing the conductor thickness or carrier density at the expense of a lower conductance. Here we demonstrate that highly anisotropic crystalline conductors offer an alternative solution, avoiding this compromise by separating the directions of conduction and transmission. We demonstrate that slabs of the layered oxides Sr2RuO4 and Tl2Ba2CuO6+δ are optically transparent even at macroscopic thicknesses >2 µm for c-axis polarized light. Underlying this observation is the fabrication of out-of-plane slabs by focused ion beam milling. This work provides a glimpse into future technologies, such as highly polarized and addressable optical screens.