Spin Fluctuations in Sr_{2}RuO_{4} from Polarized Neutron Scattering: Implications for Superconductivity

Distinct spin and orbital dynamics in Sr2RuO4

The unconventional superconductor Sr2RuO4 has long served as a benchmark for theories of correlated-electron materials. The determination of the superconducting pairing mechanism requires detailed experimental information on collective bosonic excitations as potential mediators of Cooper pairing. We have used Ru L3-edge resonant inelastic x-ray scattering to obtain comprehensive maps of the electronic excitations of Sr2RuO4 over the entire Brillouin zone. We observe multiple branches of dispersive spin and orbital excitations associated with distinctly different energy scales. The spin and orbital dynamical response functions calculated within the dynamical mean-field theory are in excellent agreement with the experimental data. Our results highlight the Hund metal nature of Sr2RuO4 and provide key information for the understanding of its unconventional superconductivity.

Topological Superconductors from a Materials Perspective

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

Critical Current Density in d-Wave Hubbard Superconductors

In this work, the Generalized Hubbard Model on a square lattice is applied to evaluate the electrical current density of high critical temperature d-wave superconductors with a set of Hamiltonian parameters allowing them to reach critical temperatures close to 100 K. The appropriate set of Hamiltonian parameters permits us to apply our model to real materials, finding a good quantitative fit with important macroscopic superconducting properties such as the critical superconducting temperature (Tc) and the critical current density (Jc). We propose that much as in a dispersive medium, in which the velocity of electrons can be estimated by the gradient of the dispersion relation ∇ε(k), the electron velocity is proportional to ∇E(k) in the superconducting state (where E(k)=(ε(k)-μ)2+Δ2(k) is the dispersion relation of the quasiparticles, and k is the electron wave vector). This considers the change of ε(k) with respect to the chemical potential (μ) and the formation of pairs that gives rise to an excitation energy gap Δ(k) in the electron density of states across the Fermi level. When ε(k)=μ at the Fermi surface (FS), only the term for the energy gap remains, whose magnitude reflects the strength of the pairing interaction. Under these conditions, we have found that the d-wave symmetry of the pairing interaction leads to a maximum critical current density in the vicinity of the antinodal k-space direction (π,0) of approximately 1.407236×108 A/cm², with a much greater current density along the nodal direction (π2,π2) of 2.214702×109 A/cm². These results allow for the establishment of a maximum limit for the critical current density that could be attained by a d-wave superconductor.

Non-conventional superconductivity in magnetic In and Sn nanoparticles

We report on experimental evidence of non-conversional pairing in In and Sn nanoparticle assemblies. Spontaneous magnetizations are observed, through extremely weak-field magnetization and neutron-diffraction measurements, to develop when the nanoparticles enter the superconducting state. The superconducting transition temperature TC shifts to a noticeably higher temperature when an external magnetic field or magnetic Ni nanoparticles are introduced into the vicinity of the superconducting In or Sn nanoparticles. There is a critical magnetic field and a critical Ni composition that must be reached before the magnetic environment will suppress the superconductivity. The observations may be understood when assuming development of spin-parallel superconducting pairs on the surfaces and spin-antiparallel superconducting pairs in the core of the nanoparticles.

Resonance from antiferromagnetic spin fluctuations for superconductivity in UTe2

Superconductivity originates from the formation of bound (Cooper) pairs of electrons that can move through the lattice without resistance below the superconducting transition temperature T_c (ref. ¹). Electron Cooper pairs in most superconductors form anti-parallel spin singlets with total spin S = 0 (ref. ²), although they can also form parallel spin-triplet Cooper pairs with S = 1 and an odd parity wavefunction³. Spin-triplet pairing is important because it can host topological states and Majorana fermions relevant for quantum computation^4,5. Because spin-triplet pairing is usually mediated by ferromagnetic (FM) spin fluctuations³, uranium-based materials near an FM instability are considered to be ideal candidates for realizing spin-triplet superconductivity⁶. Indeed, UTe₂, which has a T_c ≈ 1.6 K (refs. ^7,8), has been identified as a candidate for a chiral spin-triplet topological superconductor near an FM instability^7-14, although it also has antiferromagnetic (AF) spin fluctuations^15,16. Here we use inelastic neutron scattering (INS) to show that superconductivity in UTe₂ is coupled to a sharp magnetic excitation, termed resonance^17-23, at the Brillouin zone boundary near AF order. Because the resonance has only been found in spin-singlet unconventional superconductors near an AF instability^17-23, its observation in UTe₂ suggests that AF spin fluctuations may also induce spin-triplet pairing²⁴ or that electron pairing in UTe₂ has a spin-singlet component.

Orbital Isotropy of Magnetic Fluctuations in Correlated Electron Materials Induced by Hund's Exchange Coupling

Characterizing nonlocal magnetic fluctuations in materials with strong electronic Coulomb interactions remains one of the major outstanding challenges of modern condensed matter theory. In this Letter, we address the spatial symmetry and orbital structure of magnetic fluctuations in perovskite materials. To this aim, we develop a consistent multiorbital diagrammatic extension of dynamical mean-field theory, which we apply to an anisotropic three-orbital model of cubic t_{2g} symmetry. We find that the form of spatial spin fluctuations is governed by the local Hund's coupling. For small values of the coupling, magnetic fluctuations are anisotropic in orbital space, which reflects the symmetry of the considered t_{2g} model. Large Hund's coupling enhances collective spin excitations, which mixes orbital and spatial degrees of freedom, and magnetic fluctuations become orbitally isotropic. Remarkably, this effect can be seen only in two-particle quantities; single-particle observables remain anisotropic for any value of the Hund's coupling. Importantly, we find that the orbital isotropy can be induced both at half filling and for the case of four electrons per lattice site, where the magnetic instability is associated with different, antiferromagnetic and ferromagnetic, modes, respectively.

Magnetic-field-controlled spin fluctuations and quantum critically in Sr3Ru2O7

When the transition temperature of a continuous phase transition is tuned to absolute zero, new ordered phases and physical behaviour emerge in the vicinity of the resulting quantum critical point. Sr₃Ru₂O₇ can be tuned through quantum criticality with magnetic field at low temperature. Near its critical field B_c it displays the hallmark T-linear resistivity and a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\,{{{{{{\mathrm{log}}}}}}}\,(1/T)$$\end{document}Tlog(1/T) electronic heat capacity behaviour of strange metals. However, these behaviours have not been related to any critical fluctuations. Here we use inelastic neutron scattering to reveal the presence of collective spin fluctuations whose relaxation time and strength show a nearly singular variation with magnetic field as B_c is approached. The large increase in the electronic heat capacity and entropy near B_c can be understood quantitatively in terms of the scattering of conduction electrons by these spin-fluctuations. On entering the spin-density-wave ordered phase present near B_c, the fluctuations become stronger suggesting that the order is stabilised through an “order-by-disorder” mechanism.

Sr₃Ru₂O₇ exhibits a quantum critical point tunable by magnetic field and has been widely used in the study of criticality. Here, by using inelastic neutron scattering, the authors measure collective magnetic excitations near the quantum critical point and relate them to thermodynamic properties and spin density wave order.

Electronic Structure Correspondence of Singlet-Triplet Scale Separation in Strained Sr2RuO4

At a temperature of roughly 1 K, Sr2RuO4 undergoes a transition from a normal Fermi liquid to a superconducting phase. Even while the former is relatively simple and well understood, the superconducting state has not even been understood after 25 years of study. More recently, it has been found that critical temperatures can be enhanced by the application of uniaxial strain, up to a critical strain, after which it falls off. In this work, we take an “instability” approach and seek divergences in susceptibilities. This provides an unbiased way to distinguish tendencies to competing ground states. We show that in the unstrained compound, the singlet and triplet instabilities of the normal Fermi liquid phase are closely spaced. Under uniaxial strain, electrons residing on all orbitals contributing to the Fermiology become more coherent, while the electrons of the Ru-dxy character become heavier, and the electrons of the Ru-dxz,yz characters become lighter. In the process, Im χ(q,ω) increases rapidly around q = (0.3,0.3,0)2π/a and q = (0.5,0.25,0)2π/a, while it gets suppressed at all other commensurate vectors, in particular at q = 0, which is essential for spin-triplet superconductivity. We observe that the magnetic anisotropy under strain drops smoothly, which is concomitant with the increment in singlet instability. Thus, the triplet superconducting instability remains the lagging instability of the system, and the singlet instability enhances under strain, leading to a large energy-scale separation between these competing instabilities. However, since this happens even without spin-orbit coupling, we believe it is primarily the enhancement in the spin fluctuation glue around quasi-anti-ferromagnetic vectors that drives the Cooper pairing instead of the magnetic anisotropy. At large strain, an instability to a spin density wave overtakes the superconducting one. The analysis relies on a high-fidelity, ab initio description of the one-particle properties and two-particle susceptibilities, based on the quasiparticle self-consistent GW approximation augmented by dynamical mean field theory. This approach is described and its high fidelity confirmed by comparing to observed one- and two-particle properties.

Incommensurate Spin Fluctuations in the Spin-Triplet Superconductor Candidate UTe_{2}

Spin-triplet superconductors are of extensive current interest because they can host topological state and Majorana fermions important for quantum computation. The uranium-based heavy-fermion superconductor UTe_{2} has been argued as a spin-triplet superconductor similar to UGe_{2}, URhGe, and UCoGe, where the superconducting phase is near (or coexists with) a ferromagnetic (FM) instability and spin-triplet electron pairing is driven by FM spin fluctuations. Here we use neutron scattering to show that, although UTe_{2} exhibits no static magnetic order down to 0.3 K, its magnetism in the [0,K,L] plane is dominated by incommensurate spin fluctuations near an antiferromagnetic ordering wave vector and extends to at least 2.6 meV. We are able to understand the dominant incommensurate spin fluctuations of UTe_{2} in terms of its electronic structure calculated using a combined density-functional and dynamic mean-field theory.

Magnetic order and fluctuations in the quasi-two-dimensional planar magnet Sr(Co1-xNix)2As2

We use neutron scattering to investigate spin excitations in Sr ( Co 1 - x Ni x ) 2 As 2 , which has a c -axis incommensurate helical structure of the two-dimensional (2D) in-plane ferromagnetic (FM) ordered layers for 0.013 ⩽ x ⩽ 0.25 . By comparing the wave vector and energy dependent spin excitations in helical ordered Sr ( Co 0.9 Ni 0.1 ) 2 As 2 and paramagnetic SrCO 2 As 2 , we find that Ni doping, while increasing lattice disorder in Sr ( Co 1 - x Ni x ) 2 As 2 , enhances quasi-2D FM spin fluctuations. However, our band structure calculations within the combined density functional theory and dynamic mean field theory (DFT+DMFT) failed to generate a correct incommensurate wave vector for the observed helical order from nested Fermi surfaces. Since transport measurements reveal increased in-plane and c -axis electrical resistivity with increasing Ni doping and associated lattice disorder, we conclude that the helical magnetic order in Sr ( Co 1 - x Ni x ) 2 As 2 may arise from a quantum order-by-disorder mechanism through the itinerant electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions.

Knight Shift and Leading Superconducting Instability from Spin Fluctuations in Sr_{2}RuO_{4}

Recent nuclear magnetic resonance studies [A. Pustogow et al., Nature 574, 72 (2019)] have challenged the prevalent chiral triplet pairing scenario proposed for Sr_{2}RuO_{4}. To provide guidance from microscopic theory as to which other pair states might be compatible with the new data, we perform a detailed theoretical study of spin fluctuation mediated pairing for this compound. We map out the phase diagram as a function of spin-orbit coupling, interaction parameters, and band structure properties over physically reasonable ranges, comparing when possible with photoemission and inelastic neutron scattering data information. We find that even-parity pseudospin singlet solutions dominate large regions of the phase diagram, but in certain regimes spin-orbit coupling favors a near-nodal odd-parity triplet superconducting state, which is either helical or chiral depending on the proximity of the γ band to the van Hove points. A surprising near degeneracy of the nodal s^{'} and d_{x^{2}-y^{2}} wave solutions leads to the possibility of a near-nodal time-reversal symmetry broken s^{'}+id_{x^{2}-y^{2}} pair state. Predictions for the temperature dependence of the Knight shift for fields in and out of plane are presented for all states.

Interplay of Electronic and Spin Degrees in Ferromagnetic SrRuO_{3}: Anomalous Softening of the Magnon Gap and Stiffness

The magnon dispersion of ferromagnetic SrRuO_{3} was studied by inelastic neutron scattering experiments on single crystals as a function of temperature. Even at low temperature the magnon modes exhibit substantial broadening pointing to strong interaction with charge carriers. We find an anomalous temperature dependence of both the magnon gap and the magnon stiffness, which soften upon cooling in the ferromagnetic phase. Both effects trace the temperature dependence of the anomalous Hall effect and can be attributed to the impact of Weyl points, which results in the same relative renormalization in the spin stiffness and magnon gap.

Time Reversal Symmetry Breaking Superconductors: Sr2RuO4 and Beyond

Recent work done on the time reversal symmetry (TRS) breaking superconductors is reviewed in this paper. The special attention is paid to Sr 2 RuO 4 believed to be spin triplet chiral p-wave superconductor which break TRS and is expected to posses non-trivial topological properties. The family of TRS breaking superconductors is growing relatively fast, with many of its newly discovered members being non-centrosymmetric. However not only Sr 2 RuO 4 but also many other superconductors which possess center of inversion also break TRS. The TRS is often identified by means of the muon spin relaxation ( μ SR) and the Kerr effect. Both methods effectively measure the appearance of the spontaneous bulk magnetic field below superconducting transition temperature. This compound provides an example of the material whose many band, multi-condensate modeling has enjoyed a number of successes, but the full understanding has not been achieved yet. We discuss in some details the properties of the material. Among them is the Kerr effect and by understanding has resulted in the discovery of the novel mechanism of the phenomenon. The mechanism is universal and thus applicable to all systems with multi-orbital character of states at the Fermi energy.