Ruthenate superconductor
external pageImpurity induced magnetic ordering in Sr2RuO4call_made
Bastian Zinkl and Manfred Sigrist
Ti substituting Ru in Sr2RuO4 in small concentrations induces incommensurate spin density wave order with a wave vector Q≃(2π/3,2π/3) corresponding to the nesting vector of two out of three Fermi surface sheets. We consider a microscopic model for these two bands and analyze the correlation effects leading to magnetic order through non-magnetic Ti-doping. For this purpose we use a position dependent mean field approximation for the microscopic model and a phenomenological Ginzburg-Landau approach, which both deliver consistent results and allow us to examine the inhomogeneous magnetic order. Spin-orbit coupling additionally leads to spin currents around each impurity, which in combination with the magnetic polarization produce a charge current pattern. This is also discussed within a gauge field theory in both charge and spin channel. This spin-orbit coupling effect causes an interesting modification of the magnetic structure, if currents run through the system. Our findings allow a more detailed analysis of the experimental data for Sr2Ru1−xTixO4. In particular, we find that the available measurements are consistent with our theoretical predictions.
Bastian Zinkl and Manfred Sigrist
Non-magnetic impurities can lift the accidental degeneracy of unconventional pairing states, such as the (d+ig)-wave state recently proposed for Sr2RuO4. This type of effect would lead to a superconducting double transition upon impurity doping. In a model calculation it is shown how this behavior depends on material parameters and how it could be detected.
external pageHalf-quantum vortices on c-axis domain walls in chiral p-wave superconductorscall_made
Sarah B. Etter, Wen Huang and Manfred Sigrist
Chiral superconductors are two-fold degenerate and domains of opposite chirality can form, separated by domain walls. There are indications of such domain formation in the quasi two-dimensional putative chiral -wave superconductor Sr2RuO4, yet no experiment has explicitly resolved individual domains in this material. In this work, -axis domain walls lying parallel to the layers in chiral -wave superconductors are explored from a theoretical point of view. First, using both a phenomenological Ginzburg-Landau and a quasiclassical Bogoliubov-deGennes approach, a consistent qualitative description of the domain wall structure is obtained. While these domains are decoupled in the isotropic limit, there is a finite coupling in anisotropic systems and the domain wall can be treated as an effective Josephson junction. In the second part, the formation and structure of half-quantum vortices (HQV) on such -axis domain walls are discussed.
external pageTopological Features in a Chiral p-wave Superconductor under Uniaxial Straincall_made
Yoshiki Imai and Manfred Sigrist
Uniaxial strain effects in the superconducting phase are investigated assuming chiral p-wave pairing for the unconventional superconductor Sr2RuO4. The Fermi surface structure of the γ-band is especially sensitive to this type of strain as it is presumably most important for the superconducting phase and its Fermi surface undergoes a Lifshitz transition. Taking several distinct pairing channels within a lattice model into account, we study the effect of strain on the topological properties of the superconducting phase considering various characteristic quantities such as the thermal Hall conductivity, which gives direct access to the Chern number of the pairing state in the low-temperature limit. While the charge edge current under strain depends on the dominant pairing channel but is essentially insensitive to the Fermi surface topology, we find that the temperature dependence of the thermal Hall conductivity is strongly affected by the Lifshitz transition under strain.
Hirono Kaneyasu, Yuya Enokida, Takuji Nomura, Yasumasa Hasegawa, Toru Sakai and Manfred Sigrist
We use the Ginzburg-Landau theory for a chiral p-wave superconductor to describe the filamentary superconducting phase nucleating at temperatures higher than the bulk transition temperature near the interfaces between Ru metal and Sr2RuO4 in eutectic Sr2RuO4-Ru. The peculiar phase diagram of magnetic field versus temperature shows very distinct properties for different directions of the magnetic field. A clear feature of the magnetic fields in the basal plane of Sr2RuO4, perpendicular to the chiral axis (c-axis), is the occurrence of a second superconducting transition with a new order parameter visible as a zero-bias anomaly in tunneling spectroscopy. For the field along the
c-axis of Sr2RuO4, two transitions merge, indicating that the field drives the second-order parameter through the paramagnetic coupling to a chiral supercurrent. The resulting phenomenology shows that the assumption of a chiral superconducting order parameter in Sr2RuO4
yields an image reproducing the qualitative properties observed in the experiment. This discussion is qualitatively analogous to the corresponding chiral even- and odd-parity channels.
external pageSuperconducting order parameter of Sr2RuO4: A microscopic perspectivecall_made
Aline Ramires and Manfred Sigrist
The character of the superconducting phase of Sr2RuO4 is the topic of a longstanding discussion. The classification of the symmetry allowed order parameters has relied on the tetragonal symmetry of the lattice and on cylindrical Fermi surfaces, usually taken to be featureless, not including the nontrivial symmetry aspects related to their orbital content. Here we show how the careful account of the orbital degree of freedom and a three-dimensional description lead to a much richer classification of order parameters. We analyze the stability and degeneracy of these new order parameters from the perspective of the concept of superconducting fitness and propose new order parameter candidates which can systematically account for the observed phenomenology in this material.
Bastian Zinkl, Mark H. Fischer, and Manfred Sigrist
Symmetry arguments based on the point group of a system and thermodynamic measurements are often combined to identify the order parameter in unconventional superconductors. However, lattice translations, which can induce additional momenta with vanishing order parameter in the Brillouin zone, are neglected, especially in gap functions otherwise expected to be constant, such as in chiral superconductors. After a general analysis of the symmetry conditions for vanishing gap functions, we study the case of chiral p- and chiral
f-wave pairing on a square lattice, a situation relevant for Sr2RuO4. Specifically, we calculate the impurity-induced density of states, specific heat, superfluid density, and thermal conductivity employing a self-consistent T-matrix calculation and compare our results to the case of a nodal (d-wave) order parameter. While there is a clear distinction between a fully gapped chiral state and a nodal state, the strongly anisotropic case is almost indistinguishable from the nodal case. Our findings illustrate the difficulty of interpreting thermodynamic measurements. In particular, we find that the available measurements are consistent with a chiral (f-wave) order parameter. Our results help to reconcile the thermodynamic measurements with the overall picture of chiral spin-triplet superconductivity in Sr2RuO4.
František Herman, Jonathan Buhmann, Mark H. Fischer, and Manfred Sigrist
Recent experiments revealed non-Fermi-liquid resistivity in the unconventional superconductor Sr2RuO4 when strain pushes one of the Fermi surfaces close to a van Hove singularity. The origin of this behavior and whether it can be understood from a picture of well-defined quasiparticles is unclear. We employ a Boltzmann transport analysis beyond the single relaxation-time approximation based on a single band which undergoes a Lifshitz transition where the Fermi surface crosses a van Hove singularity either due to uniaxial or epitaxial strain. First, analytically investigating impurity scattering, we clarify the role of the diverging density of states together with the locally flat band at the point of the Lifshitz transition. Additionally, including electron-electron scattering numerically, we find good qualitative agreement with resistivity measurements on uniaxially strained Sr2RuO4, including the temperature scaling and the temperature dependence of the resistivity peak. Our results imply that, even close to the Lifshitz transition, a description starting from well-defined quasiparticles holds. To test the validity of Boltzmann transport theory near a van Hove singularity, we provide further experimentally accessible parameters, such as thermal transport, the Seebeck coefficient, and Hall resistivity and compare different strain scenarios.
external pageSpontaneous surface flux pattern in chiral p-wave superconductorscall_made
Sarah B. Etter, Adrien Bouhon, and Manfred Sigrist
In chiral p-wave superconductors, magnetic flux patterns may appear spontaneously when translational symmetry is broken such as at surfaces, domain walls, or impurities. However, in the candidate material Sr2RuO4 no direct signs of such magnetic fields have been detected experimentally. In this paper, the flux pattern at the edge of a disk-shaped sample is examined using the phenomenological Ginzburg-Landau approach. The detailed shape of the flux pattern, including self-screening, is computed numerically for different surface types by systematically scanning a range of boundary conditions. Moreover, specific features of the electronic structure are included qualitatively through the coefficients in the Ginzburg-Landau functional. Both the shape and the magnitude of the flux pattern are found to be highly sensitive to all considered parameters. In conclusion, such spontaneous magnetic flux patterns are not a universal feature of chiral p-wave superconductors.
Jia-Long Zhang, Wen Huang, Manfred Sigrist, and Dao-Xin Yao
Chiral superconductors support chiral edge modes and potentially spontaneous edge currents at their boundaries. Motivated by the putative multiband chiral p-wave superconductor Sr2RuO4, we study the influence of the interference between different bands at the edges, which may appear in the presence of moderate edge disorder or in edge tunneling measurements. We show that interband interference can strongly modify the measurable quantities at the edges when the order parameter exhibits phase difference between the bands. This is illustrated by investigating the edge dispersion and the edge current distribution in the presence of interband mixing, as well as the conductance at a tunneling junction. The results are discussed in connection with the putative chiral p-wave superconductor Sr2RuO4. In passing, we also discuss similar interference effects in multiband models with other pairing symmetries.
external pageA note on the upper critical field of Sr2RuO4 under straincall_made
Aline Ramires and Manfred Sigrist
In the light of the recently discussed mechanism for suppression of superconductivity in multi-orbital systems called inter-orbital effect, here we extend our analysis of the upper critical field in Sr2RuO4 now under strain. We show that the presence of the standard orbital effect and the new mechanism can consistently account for the qualitative changes observed in the upper critical field for the strained system, in particular, combining the overall enhancement of the critical field with the reduction in the anisotropy. The proposed picture holds for a triplet superconducting state, showing that a singlet state is not the only possibility to account for the observations. We suggest further experiments in order to clarify our understanding about the superconducting phase of Sr2RuO4.
Yoshiki Imai, Katsunori Wakabayashi, and Manfred Sigrist
Motivated by the spin-triplet superconductor Sr2RuO4 p-wave phase with a fully opened quasiparticle excitation gap, the temperature dependence of the thermal Hall conductivity has a temperature linear term associated with the topological property directly and an exponential term, which shows a drastic change around the Lifshitz transition. Examining f-wave states as alternative candidates with d=Δ0z(kx2−ky2)kx±iky) and Δ0zkxky(kx±iky) with gapless quasiparticle excitations, we study the temperature dependence of the thermal Hall conductivity, where for the former state the thermal Hall conductivity has a quadratic dependence on temperature, originating from the linear dispersions, in addition to linear and exponential behavior. The obtained result may enable us to distinguish between the chiral p-wave and f-wave states in Sr2RuO4.
external pageCurrent inversion at the edges of a chiral p-wave superconductorcall_made
Adrien Bouhon and Manfred Sigrist
Motivated by Sr2RuO4, edge quasiparticle states are analyzed based on the self-consistent solution of the Bogolyubov–de Gennes equations for a topological chiral p-wave superconductor. Using a tight-binding model of a square lattice for the dominant γ-band, we explore the nontrivial geometry and band structure dependence of the edge states and currents. As a peculiar finding, we show that, for high band fillings, currents flow in a reversed direction when comparing straight and zigzag edges. We give a simple explanation in terms of the positions of the zero-energy bound states using a quasiclassical picture. We also show that a Ginzburg-Landau approach can reproduce these results. Moreover, the band filling dependence of the most stable domain wall structure is discussed.
external pageLeggett modes and multiband superconductivity in Sr2RuO4call_made
Wen Huang, Thomas Scaffidi, Manfred Sigrist, and Catherine Kallin
Sr2RuO4 is a prototypical multiband superconductor with three bands crossing the Fermi level. These bands exhibit distinct dimensional characteristics, with one quasi-2D γ-band and two quasi-1D α- and β-bands. This leads to the expectation that the superconductivity on the γ-band may be only weakly Josephson-coupled to that on the other two bands. Based on an explicit microscopic weak coupling calculation appropriate for Sr2RuO4, we study the collective Leggett modes associated with the relative phase oscillations between the bands and show that a relatively soft Leggett mode exists due to the comparatively weaker interband Josephson coupling. These calculations also provide insight into why the superconducting gap magnitudes may be comparable on all three bands, despite the noticeable differences between the γ- and α/β-bands. The analyses can be readily applied to other multiband superconductors.
Hirono Kaneyasu, Sarah B. Etter, Toru Sakai, and Manfred Sigrist
The evolution of the filamentary 3-Kelvin (3-K) superconducting phase at the interface between Sr2RuO4 and Ru-metal inclusions is discussed for Pb-Ru-Sr2RuO4 contacts. Using the Ginzburg-Landau model, the influence of proximity-induced superconductivity in Ru on the topology of the 3-K phase is analyzed. Because the s-wave order parameter in Ru favors a 3-K state of trivial topology, the onset temperature of the phase with a nontrivial topology, which is compatible with the bulk phase of Sr2RuO4, is essentially reduced to the bulk transition temperature. Because the topology of the superconducting state in Sr2RuO4is crucial for the Josephson effect through Pb-Ru-Sr2RuO4 contacts, this model qualitatively reproduces the experimental observation of the anomalous temperature dependence on the critical current.
external pageQuantum transport signatures of chiral edge states in Sr2RuO4call_made
Rakesh P. Tiwari, W. Belzig, Manfred Sigrist, and C. Bruder
We investigate transport properties of a double quantum dot based a Cooper pair splitter, where the superconducting lead consists of Sr2RuO4. The proposed device can be used to explore the symmetry of the superconducting order parameter in Sr2RuO4 by testing the presence of gapless chiral edge states, which are predicted to exist if the bulk superconductor is described by a chiral p-wave state. The odd orbital symmetry of the bulk order parameter ensures that we can realize a regime where the electrons tunneling into the double-dot system come from the chiral edge states and thereby leave their signature in the conductance. The proposed Cooper pair splitter has the potential to probe the chirality of the order parameters in topological superconductors.
external pageTopological and edge state properties of a three-band model for Sr2RuO4call_made
Yoshiki Imai, Katsunori Wakabayashi, and Manfred Sigrist
Modeling the spin-triplet superconductor Sr2RuO4 through a three-orbital tight-binding model, we investigate topological properties and edge states assuming chiral p-wave pairing. In concordance with experiments, the three Fermi surfaces consist of two electronlike and one hole like surface, corresponding to the α-, β-, and γ-band on the level of a two-dimensional system. The quasiparticle spectra and other physical quantities of the superconducting phase are calculated by means of a self-consistent Bogoliubov-de Gennes approach for a ribbon-shaped system. While a full quasiparticle excitation gap is realized in the bulk system, at the edges gapless states appear, some of which have linear and others of which have a nearly flat dispersion around zero energy. This study shows the interplay between spin-orbit-coupling-induced spin currents, chiral edge currents, and correlation-driven surface magnetism. The topological nature of the chiral p-wave state manifests itself in the γ-band characterized by an integer Chern number. As the γ-band is close to a Lifshitz transition in Sr2RuO4, changing the sign of the Chern number, the topological nature may be rather fragile.
external pageInfluence of the domain walls on the Josephson effect in Sr2RuO4call_made
Adrien Bouhon and Manfred Sigrist
A detailed theoretical interpretation of the Josephson interference experiment between Sr2RuO4 and Pb reported by Kidwingira et al (2006 Science 314 1267) is given. Assuming chiral p-wave pairing symmetry, a Ginzburg–Landau theory is derived in order to investigate the structure of domain walls between chiral domains. It turns out that anisotropy effects of the Fermi surface and the orientation of the domain walls are essential for their internal structure. Introducing a simple model for a Josephson junction, the effect of domain walls intersecting the interface between Sr2RuO4 and Pb is discussed. It is shown that characteristic deviations of the Fraunhofer interference pattern for the critical Josephson current as a function of the magnetic field occurs in qualitative agreement with the experimental finding. Moreover, the model is also able to account for peculiar hysteresis effects observed in the experiment.