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Even odder after twenty-three years: the superconducting order parameter puzzle of Sr2RuO4
Andrew P Mackenzie, Thomas Scaffidi, Clifford W Hicks, Yoshiteru Maeno
TL;DR
The review addresses unresolved questions about superconductivity in Sr2RuO4 and frames a new assessment of the evidence. It concludes that the issue remains unsettled, while highlighting discrepancies among key experiments and limitations in interpreting them.
Problem
Significant discrepancies remain among experiments probing superconductivity in Sr2RuO4.
Method
The review reassesses the material’s superconductivity by examining key experiments, including the proximity effect between Sr2RuO4 and metallic magnets.
Results
The issue is not settled, with evidence including good agreement between experiment and theory in magnitude and increased Tc near spatial vicinities.
Takeaways & Limitations
Understanding Sr2RuO4 requires continued attention to the discrepancies and interpretation of key unconventional-superconductivity probes.
Takeaways & Limitations
Time-reversal-symmetry-breaking experiments report substantially different internal-field scales, including a much lower scanning-SQUID limit of ≤ 1 mG than the volume-averaged muon-spin-rotation fields.
Abstract
from arXiv · showhide
In this short review, we aim to provide a topical update on the status of efforts to understand the superconductivity of Sr2RuO4. We concentrate on the quest to identify a superconducting order parameter symmetry that is compatible with all the major pieces of experimental knowledge of the material, and highlight some major discrepancies that have become even clearer in recent years. As the pun in the title suggests, we have tried to start the discussion from scratch, making no assumptions even about fundamental issues such as the parity of the superconducting state. We conclude that no consensus is currently achievable in Sr2RuO4, and that the reasons for this go to the heart of how well some of the key probes of unconventional superconductivity are really understood. This is therefore a puzzle that merits continued in-depth study.
2 Scottish Universities Physics Alliance, School of Physics and Astronomy, University
This section identifies affiliations with the School of Physics and Astronomy at the University of St. Andrews and the Department of Physics at the University of California, Berkeley.
- The School of Physics and Astronomy affiliation is at the University of St. Andrews, St. Andrews KY16 9SS, U.K.
- The Department of Physics affiliation is at the University of California, Berkeley, California 94720, USA.
1. Introduction
The review examines why identifying Sr2RuO4’s superconducting order-parameter symmetry remains unresolved despite unusually favorable experimental and theoretical conditions. It highlights contradictions among key probes and frames the problem as a broader challenge at the theory–experiment interface.
- The review provides a status report on research into Sr2RuO4 superconductivity and its order parameter.
- The review deliberately remains open to all possibilities, including the parity of the superconducting state.
- Sr2RuO4 appears unusually tractable because its normal state is a well-understood Fermi liquid and its clean samples minimize disorder complications.High-quality crystals have enabled precise determination of the Fermi surface and normal-state quasiparticle properties.
- Full understanding remains unattained, motivating continued research into this unresolved quantum-materials problem.
- Its focus is the lack of a fully self-consistent description of key experimentally determined features of the superconducting state.
- The review is selective rather than comprehensive, emphasizing issues judged most important because conflicting experimental techniques produce significant discrepancies.The authors argue that these discrepancies matter beyond Sr2RuO4 because the techniques are widely used across unconventional superconductivity.
2. Summary of the theoretical situation
Theory places Sr2RuO4 near a competition between odd- and even-parity superconductivity, with multiple nearly degenerate states and complex gap structures. These features make the order-parameter symmetry difficult to determine from theory alone.
- Earlier work proposed a spin-triplet state modeled on phases of superfluid 3He, but later analyses emphasize parity as a safer descriptor when spin–orbit coupling is important.The coupling is strongly momentum-dependent, so spin-singlet and spin-triplet labels can be misleading.
- Realistic spin-fluctuation calculations find a small free-energy difference between odd- and even-parity states.
- Either parity can be favored depending on input parameters, while several odd-parity states are also nearly degenerate.
- Near degeneracies and complicated gap structures make determining the order-parameter symmetry nontrivial, although they may permit rich phase diagrams including odd–even parity transitions.
- Sr2RuO4 lies close to the border between odd- and even-parity superconductivity because its susceptibility structure can support different spin-fluctuation-mediated pairing states.
- Predicted gap structures are complex, with variation between and within Fermi-surface sheets, deep minima in some nodeless odd-parity states, and richer nodal structures for even-parity states.
3. Identification of key experiments
The review argues that experiments must distinguish among complex candidate gap structures and directly probe symmetry, because theory alone cannot select a definitive order parameter. Existing probes provide suggestive but not conclusive evidence, while proximity studies offer another promising route.
- Theory cannot provide a definitive answer because candidate odd- and even-parity states have insufficiently separated free energies.
- Thermodynamic measurements show signatures beyond a single gap, and heat-capacity data support two or more gaps with different magnitudes.
- Measurements near gap nodes require temperatures of 50 mK or below, the highest-purity samples, and discrimination between accidental nodes and symmetry-imposed nodes.
- Parity-sensitive tunneling generally favors odd-parity states, but sample-to-sample reproducibility is insufficient for conclusive interpretation.
- Half-flux-quantum vortices have been observed under special conditions and interpreted as evidence for an odd-parity two-component state, but they do not prove it conclusively.
- Proximity effects between Sr2RuO4 and metallic magnets are presented as an early but promising approach for obtaining symmetry-sensitive information.
Experiments probing time reversal symmetry breaking
μSR and Kerr rotation observations support spontaneous time-reversal-symmetry breaking associated with superconductivity, but quantitative discrepancies and unresolved mechanisms remain. Two-component order parameters are favored by prevailing inference, while null edge-current searches do not exclude TRS breaking.
- Experimental evidence for TRS breaking: μSR measurements indicate spontaneous magnetism near implanted muons in zero external field, later confirmed by polar Kerr rotation.Both datasets are presented as evidence associated with the superconducting state.
- Experimental evidence for TRS breaking: The origin of the TRS breaking is not firmly established, and further experimental work is considered desirable.The review also notes difficulty in quantitatively interpreting the relevant signals.
- Interpretation and order-parameter structure: The prevailing inference is that μSR and Kerr observations arise from an order parameter with two degenerate orbital components.The review notes that the possible role of the spin degree of freedom has been less widely investigated, especially with strong spin-orbit coupling.
- Experimental evidence for TRS breaking: A quantitative discrepancy separates the 0.5 G volume-averaged μSR fields from the scanning-SQUID limit of ≤ 1 mG.These measurements probe substantially different internal-field scales.
- Interpretation and order-parameter structure: In a tetragonal crystal without spin-orbit coupling, dxz and dyz provide the non-accidental degenerate d-wave pair, whereas dxz ± idyz implies horizontal line nodes and interplane pairing.Interplane pairing is described as exotic and intuitively unlikely for Sr2RuO4’s strongly two-dimensional Fermi surface.
- Interpretation and order-parameter structure: The px ± ipy state remains extensively discussed because tetragonal symmetry preserves degenerate p-wave components with in-plane pairing.This state is connected to expected edge currents, although more sophisticated calculations allow much smaller currents than naïve estimates.
- Interpretation and order-parameter structure: Extensive edge-field searches are mostly null, but the absence of observed edge currents does not currently rule out a TRS-breaking superconducting order parameter.The issue remains unsettled because predicted edge currents may be substantially reduced by realistic effects.
- Interpretation and order-parameter structure: Qualitative observations are consistent with superconducting domains, although estimates of their characteristic sizes vary widely.The passage frames domain formation as another possible consequence of a two-component order parameter.
Cooper pair formation and spin susceptibility in the superconducting state
Knight-shift, NMR, and neutron-scattering measurements probe spin susceptibility to constrain Cooper-pair parity, but spin-orbit coupling complicates interpretation. Reported data show no susceptibility drop and have been interpreted as favoring odd-parity order parameters.
- Measurement and interpretation: Spin susceptibility is expected to drop for a simple even-parity superconductor with weak spin-orbit coupling upon entering the superconducting state.Non-magnetic singlet pairs are removed from the conduction-electron reservoir available for field-induced spin polarization.
- Measurement and interpretation: Knight-shift measurements and inelastic neutron scattering are used to isolate or analyze the spin contribution to magnetic susceptibility despite orbital diamagnetism.Spin-orbit coupling complicates this analysis.
- Experimental findings: No drop in extracted spin susceptibility has been observed in NMR or neutron-scattering measurements of Sr2RuO4.The measurements collectively report no experimental inconsistency in their signals.
- Experimental findings: A small rise in spin susceptibility was reported in the most precise measurements to date.This result is part of the broader set of data interpreted in terms of odd-parity order parameters.
- Experimental findings: The available susceptibility data have been interpreted as supporting odd-parity order parameters, although their lack of field-orientation dependence creates an interpretive concern.One proposed resolution requires rotating the vector order parameter in applied fields of order 20 mT.
Apparently contradictory results
Sr2RuO4 exhibits critical-field behavior consistent with Pauli limiting for in-plane fields, yet Knight-shift results show no corresponding susceptibility reduction. These observations leave the order-parameter symmetry unresolved, especially because spin-orbit coupling may allow odd-parity states to show limiting as well.
- Critical-field discrepancy: For H//ab, bulk diamagnetic orbital limiting predicts approximately 4.5 T, whereas the measured upper critical field is 1.5 T.The predicted value follows from the superconducting anisotropy parameter of 60.
- Critical-field discrepancy: The low-temperature transition at H//ab is first-order, as expected for Pauli limiting rather than diamagnetic orbital limiting.This contrasts with the second-order transition expected for orbital limiting.
- Apparently contradictory probes: The critical-field observations conflict with Knight-shift measurements showing no evidence for a spin-related magnetic energy competing with condensation energy.The review identifies reconciling critical-field limiting with the absence of a susceptibility reduction in NMR as an urgent theoretical problem.
- Pressure response: Uniaxial pressure raises Tc to 3.5 K and increases Hc2 for H//c twentyfold, from 0.075 T to 1.5 T, but Hc2 for H//ab rises only to 4.5 T.The modest in-plane increase contrasts with the enormous value expected from the anisotropy factor of 60.
- Interpretation: Critical-field limiting is quantitatively consistent with simple Pauli-limiting predictions for an even-parity order parameter, but this does not establish that symmetry.Odd-parity superconductors may also experience critical-field limiting when spin-orbit coupling is present.
- Interpretation: Microscopic spin and orbital susceptibilities are difficult to separate in the presence of spin-orbit coupling, so realistic electronic-structure calculations are needed.The strength of this effect in Sr2RuO4 remains a matter for precise calculation.
4. Summary and future work
The review finds no consensus on Sr2RuO4’s superconducting order parameter because major experimental results remain qualitatively contradictory, despite high-quality, multiply verified data. It identifies continued experimental and theoretical work aimed at resolving these puzzles.
- Summary: Major experimental discrepancies, rather than poor data quality, continue to prevent consensus about Sr2RuO4’s superconducting order parameter.The review emphasizes multiply verified experiments on high-quality samples and argues that key probes of unconventional superconductivity are not yet fully understood.
- Future work: Future priorities include parity-sensitive tunneling, improved sample purity and strain control, low-temperature gap measurements, and theories incorporating kz-dependent spin-orbit coupling.The review also highlights proximity studies, collective modes, and further investigation of junction critical-current anomalies.
- Future work: Uniaxial pressure increases Tc by a factor of 2.3, from 1.5 K to 3.5 K, while lowering crystal symmetry and potentially enabling a richer phase diagram.The review suggests pressure-tuned samples could support experiments on new regimes and possible order-parameter transitions.
- Summary: Many different order parameters are close to degenerate, while the free-energy difference between odd- and even-parity states is small.This near-degeneracy makes parity-sensitive measurements particularly valuable for distinguishing competing candidate states.
- Summary: Time-reversal-symmetry-breaking signals coexist with a Knight-shift result that shows no decrease below Tc.Muon relaxation and polar Kerr rotation indicate spontaneous fields or broken time-reversal symmetry, while Knight-shift and neutron measurements resolve no corresponding decrease.
- Summary: Critical field limiting is clearly observed, yet it is not accompanied by the expected decrease of the Knight shift below Tc.The review identifies this combination as a major discrepancy in the interpretation of spin polarization and superconducting limiting mechanisms.