As electrons move through a Bardeen-Cooper-Schrieffer (BCS) superconductor, they attract positive charges in the lattice, and the subsequent deformation leads to an attractive interaction between time-reversed electron states. Below some critical transition temperature Tc, that electron–phonon interaction forms Cooper pairs of electrons with s-wave symmetry, and their collective behavior constitutes a macroscopic quantum state of matter. That is, the electrons stay paired and flow through a superconductor without any resistance. (See the article by Warren Pickett and Mikhail Eremets, Physics Today, May 2019, page 52.)
In 1994 researchers discovered that below a Tc of 1.5 K, strontium ruthenate (Sr2RuO4, or SRO) is superconducting.1 Although its layered perovskite structure is similar to that of high-Tc, cuprate-based unconventional superconductors that don’t strictly follow BCS theory, SRO has several important differences. For example, unlike cuprates, SRO is conducting and superconducting in its stoichiometric form and therefore doesn’t require any dopants to be added to its structure. That clean form provides an ideal framework through which to study the emergence of unconventional superconductivity. (See the article by Yoshiteru Maeno, Maurice Rice, and Manfred Sigrist, Physics Today, January 2001, page 42.)
A year after the discovery, researchers proposed that the similarity between SRO and superfluid helium-3 may favor an electron pairing in SRO with p-wave symmetry.2 Conventional s-wave or unconventional d-wave superconductors are inversion symmetric, and their paired electrons must have opposite spins, known as even parity. But a superconductor with p-wave symmetry would have electron pairs with the same spin, or odd parity. In such spin-triplet pairing, an electron’s spin can exist with one of three values of the quantum spin component. If SRO had such a pairing, it could be manipulated by a magnetic field, which would potentially be useful in spintronics and quantum computing.
Since then, researchers have reported experimental evidence consistent with p-wave symmetry in SRO, including time-reversal symmetry breaking (see Physics Today, December 2006, page 23) and half-quantum vortices (see Physics Today, March 2011, page 17). The findings have been the accepted explanation of SRO’s exotic properties for about 20 years and have tantalized physicists with the promise of revealing new insights in unconventional superconductivity.
But in 2019 a paper by then-postdocs Andrej Pustogow and Yongkang Luo and their adviser Stuart Brown of UCLA questioned the prevailing p-wave explanation.3 They analyzed a superconducting SRO crystal subjected to variable stress, and the results ruled out one of the several possible odd-parity states of SRO. Now a new paper by UCLA graduate student Aaron Chronister, Pustogow, Brown, and their collaborators has provided convincing evidence that excludes all odd-parity superconductivity in SRO. Their results are consistent with an unconventional even-parity state.4
A long experiment
As Brown recounts, he and his collaborators’ investigations were strongly motivated by a 2017 result from Andrew Mackenzie’s team at the Max Planck Institute for Chemical Physics of Solids in Dresden, Germany. That experiment suggested that the uniaxial strain applied to an SRO crystal that raised the Tc from 1.5 to 3.5 K could indicate an odd-parity to even-parity transition of the superconducting state.5
“To us,” says Brown, “this meant that the superconducting state is profoundly affected by this stress, and it opened up an opportunity to revisit the SRO order-parameter question and any possible stress-induced phase change but in a vastly broadened parameter space.”
To measure the electron pairing of SRO, Chronister and Pustogow held a magnetic field in a fixed orientation parallel to the sample’s ruthenium oxide layers and then varied its strength. The spin susceptibility can be inferred from the Knight shift, a measurement of the NMR frequency shift of the oxygen-17 atoms in the superconducting material. Figure 1 shows the crystalline SRO sample with an NMR coil wrapped around it.
Strontium ruthenate is superconducting below a critical transition temperature of 1.5 K. Researchers explored its unconventional superconducting state by wrapping a crystalline sample in an NMR coil and observing its response to a magnetic field. The results showed no detectable magnetic response from the superconducting condensate. That finding rules out previously postulated theories of odd-parity superconductivity in which paired electrons have the same spin. (Courtesy of Andrej Pustogow.)
Strontium ruthenate is superconducting below a critical transition temperature of 1.5 K. Researchers explored its unconventional superconducting state by wrapping a crystalline sample in an NMR coil and observing its response to a magnetic field. The results showed no detectable magnetic response from the superconducting condensate. That finding rules out previously postulated theories of odd-parity superconductivity in which paired electrons have the same spin. (Courtesy of Andrej Pustogow.)
Such an experiment has its challenges. In particular, superconductivity in SRO or any other material is lost at a magnetic field strength higher than the so-called upper critical field Bc2. SRO’s small Bc2 of 1.5 T limits the signal intensity and spectral resolution of the NMR measurements. The signal intensity is further weakened by the need to regulate the RF eddy current that arises from the NMR pulse. Without RF regulation, the SRO crystal could be exposed to significant temperature fluctuations.
Mackenzie helped improve the signal intensity by acquiring oxygen that was highly enriched in oxygen-17 from Eric Bauer at Los Alamos National Laboratory. The researchers added it to the SRO samples for the experiments and Knight-shift measurements. Even with that improvement, though, the experiments still took weeks to complete, and meaningful analyses required the team to average the results from thousands of individual NMR spectra.
Then in March 2020, the coronavirus lockdown limited how often the researchers could conduct work in the lab. “We made the best out of the time that we could use,” Pustogow says. He recalls thinking, “If we can only come in twice a week, we’ll do the long experiment that doesn’t need a lot of housekeeping. And it can line up with the coronavirus rules.”
Nail in the coffin
The 2019 results3 contradicted earlier findings and assumptions. To determine the cause of the discrepancies, Chronister and colleagues analyzed the RF-heating effect from the NMR pulses. At the high Tc of cuprate superconductors, RF heating isn’t substantial enough to raise the temperature above Tc. But at the low Tc of SRO, they found that the excess heat was sufficient to nudge SRO from its superconducting state into a normal-conducting one. The UCLA team surmised that the earlier evidence of SRO’s odd-parity state came from measurements that were collected in the normal state, rather than the sought-after superconducting state.
Convincing new evidence that extended beyond the 2019 finding came when the researchers compared two different measurements of the superconducting condensate’s magnetization: their own Knight-shift data and previously acquired observations of the specific heat of a high-quality SRO crystal. Specific heat in that system is sensitive only to quasiparticles that are produced in the superconducting state. As figure 2 illustrates, the researchers found no difference when they subtracted the specific-heat data from their Knight-shift magnetization results.
Electron-spin susceptibility in strontium ruthenate (SRO) can be inferred from the material’s response to a magnetic field, normalized to the upper critical field, and measurable by the NMR Knight shift of oxygen-17 atoms in SRO’s superconducting state. The field-dependent magnetization measured by the Knight shift (red squares and line) doesn’t differ from that induced by nonsuperconducting quasiparticles, as inferred from specific-heat measurements in the system (purple line). The lack of difference means that the electron pairs (blue triangles) in the superconducting SRO condensate don’t contribute detectable magnetization. SRO, therefore, has an even-parity, or spin-singlet, state—rather than the long-thought odd-parity, or spin-triplet, state. (Figure by Andrej Pustogow.)
Electron-spin susceptibility in strontium ruthenate (SRO) can be inferred from the material’s response to a magnetic field, normalized to the upper critical field, and measurable by the NMR Knight shift of oxygen-17 atoms in SRO’s superconducting state. The field-dependent magnetization measured by the Knight shift (red squares and line) doesn’t differ from that induced by nonsuperconducting quasiparticles, as inferred from specific-heat measurements in the system (purple line). The lack of difference means that the electron pairs (blue triangles) in the superconducting SRO condensate don’t contribute detectable magnetization. SRO, therefore, has an even-parity, or spin-singlet, state—rather than the long-thought odd-parity, or spin-triplet, state. (Figure by Andrej Pustogow.)
The agreement between the Knight-shift and specific-heat measurements means that the electrons forming Cooper pairs in the superconducting state of SRO do not contribute detectable spin polarization to the overall magnetization. The pairing state must therefore be even parity.
The road to resolution
Even though the results disagree with the prevailing interpretation of SRO that physicists have had for more than two decades, they are consistent with the emergence of unconventional superconductivity from a well-described Fermi liquid. By starting from perturbative electron–electron collisions, theorists have shown that at low temperatures, resistivity in a Fermi liquid increases as T2, and experimentalists have observed that power-law resistivity in SRO.
Although the odd-parity state has been overturned by the results of Chronister and colleagues, determining how to interpret the spin-singlet state of SRO in conjunction with other recent experimental results remains a challenge. Another method to study the electron pairing in SRO and other superconducting materials is muon-spin spectroscopy. The technique implants spin-polarized muons into a material and then records the interaction of the muon’s magnetic moment with its surroundings.
Vadim Grinenko and Shreenanda Ghosh at Dresden University of Technology in Germany and their collaborators published a paper earlier this year using that method on single-crystalline SRO samples. They reported a small magnetic buildup, which by definition breaks time-reversal symmetry. A uniaxial stress broke the sample symmetry and resulted in a two-component order parameter: a spin-singlet superconducting state followed by one that was interpreted as a chiral state, perhaps with d-wave symmetry. (See “An unconventional superconductor undergoes two transitions,” Physics Today online, 22 March 2021.)
Physicists are also using ultrasound measurements to study SRO. According to Ginzburg–Landau theory, discontinuities in the elastic constant at the superconducting transition reflect electron pairing. Recent results found such a discontinuity in the shear elastic constant of SRO as the temperature was increased across the superconducting transition.6 Like the muon-spin results, the observation is consistent with a two-component order parameter.
Spin states compatible with the recent reports of time-reversal symmetry breaking, ultrasound discontinuities, and spin-singlet pairing are possible. Verifying which particular spin state, however, will require more research. Brown and his colleagues are currently examining the effects of exposing superconducting SRO to especially high stress.
