Magnetic field induces spatially varying superconductivity

In 1957 the Bardeen-Cooper-Schrieffer (BCS) theory emerged as the first quantum mechanical model of what would become known as conventional superconductors. Below a critical temperature, the highest-energy electrons in those materials form pairs with antiparallel spins. Pairing up allows the electrons to act like bosons rather than fermions and condense into a collective state that moves without resistance. (See the article by Howard Hart Jr and Roland Schmitt, Physics Today, February 1964, page 31 .)

But other models for superconductivity exist. In 1964, for example, Peter Fulde and Richard Ferrell and, independently, Anatoly Larkin and Yuri Ovchinnikov predicted that a large magnetic field could induce a different type of superconducting state. ¹ Known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity, the state’s parameters would vary periodically in space, unlike the homogeneous BCS state.

Direct evidence of FFLO superconductivity has long been elusive, however, in large part because the predicted state is unstable. A few materials, such as quasi-two-dimensional organics and the heavy-fermion system cerium cobalt indium-5, have shown some signatures of a potential FFLO state. Now Kenji Ishida of Kyoto University in Japan, his graduate student Katsuki Kinjo, and their colleagues have found the most direct evidence to date of the state. ² Their observation of modulations in strontium ruthenate’s spin density, illustrated in figure 1, points to inhomogeneous superconductivity.

Figure 1.

Gaining momentum

The key difference between BCS and FFLO superconductivity is the response to magnetic fields. In the case of BCS, an applied magnetic field, if strong enough, twists the spins apart and destroys the material’s superconductivity—a phenomenon known as Pauli pair breaking. An FFLO superconductor also eventually succumbs to pair breaking under the influence of a sufficiently strong field. But when subjected to slightly weaker fields, the FFLO gains its signature inhomogeneity.

To understand how, consider the simple band structure depicted in figure 2a. In the absence of a magnetic field, spin-up and spin-down electrons zip around with the same magnitude of momentum for a given energy (gray dashed curve). With the addition of a magnetic field, Zeeman splitting shifts the energy band of one spin upward and the other downward. The highest-energy spin-up electrons have different momenta from those of the highest-energy spin-down ones, and the resulting pairs adopt a nonzero net momentum, which creates spatial modulations in the superconducting order parameter and spin density.

Figure 2.

Many superconductors, including most elemental ones, are well-described by BCS theory; finding materials suitable for an FFLO state has been a challenge. For starters, unlike robust conventional superconductivity, even dilute defects in the sample prevent the state’s formation. So a candidate material must be quite pure and nearly perfectly crystalline. Its carriers must also undergo strong Zeeman splitting. Finally, the FFLO state requires high magnetic fields—higher fields than those at which superconductivity disappears in most materials as a result of induced vortex currents. So a suitable material must have Pauli pair breaking, rather than vortex formation, as the limiting factor on superconductivity.

Strontium ruthenate—Sr₂RuO₄ or SRO—is, in many respects, a promising candidate for FFLO superconductivity: It can be fabricated with few defects and possesses charge carriers with large effective mass, which produces large Zeeman splitting. Its layered structure, shown in the bottom left of figure 1, also can hinder vortex formation. Superconducting currents run primarily in the plane of each layer. (See the article by Yoshiteru Maeno, Maurice Rice, and Manfred Sigrist, Physics Today, January 2001, page 42 .) Because electron pairs are effectively stuck in two dimensions, a magnetic field parallel to the plane fails to create the vortices that might otherwise disrupt the superconductivity before the FFLO state has a chance to form.

For many years, however, SRO was thought to be a spin-triplet superconductor, which has electron pairs with parallel spins. In 1998 Ishida’s group was the first to produce NMR data that seemed to confirm that supposition. Spin-triplet superconducting pairs would be manipulable with magnetic fields, which makes them promising for spintronics and quantum computing. But they can’t form an FFLO state. Zeeman splitting would have no effect on the net momentum of spin pairs pointing in the same direction.

After two decades of experiments in support of the spin-triplet interpretation, studies in 2019 and 2021 by Stuart Brown of UCLA and his colleagues reported NMR results that contradicted that picture. ³ (See Physics Today, September 2021, page 14 .) The researchers argued that sample heating from the NMR pulses, negligible in most measurements, was enough to push the material out of the superconducting state in previous experiments. Brown and his colleagues employed low-energy pulses to reduce heating and found results that ruled out spin triplets. After that revelation, Ishida wondered if an FFLO state might be possible in SRO after all.

Seeing double

Ishida and his colleagues first replicated Brown’s results. ⁴ They then used the same technique to examine SRO close to the critical magnetic field, above which the material returns to its normal state. The low-energy NMR pulses may prevent heating, but they also make the signal weak. So each spectrum took an order of magnitude longer than a typical NMR measurement. The researchers tested a range of temperatures from 70 mK to 1.6 K and magnetic fields up to 1.5 T.

The NMR spectra are given in terms of the Knight shift, which quantifies the NMR frequency shift. They indicate the electron-spin susceptibility in the vicinity of the probed nuclei, in this case oxygen-17. In its normal state, SRO has a certain, uniform spin susceptibility, with electrons and their accompanying spins spread out evenly. That state produces one NMR peak, as shown in black in figure 2b. In the homogeneous superconducting state at a low magnetic field, the Knight shift is lower because paired electrons with antiparallel spins reduce the material’s spin susceptibility. Because the electron pairs are equally distributed across the material, the NMR spectrum still has one peak, albeit shifted.

Just below the critical field of 1.4 T, a second NMR peak appears, shown in the red spectra in figure 2b, that can’t be explained by coexisting normal and superconducting phases. The researchers attribute the double peaks to spatial modulations in the spin density—periodic stripes of low and high electron-spin densities, illustrated in figure 1. As the second peak is at a slightly higher Knight shift than the normal state’s peak, it suggests spin-dense regions between ones populated with electron pairs. Although spin-density waves and their accompanying multiple NMR peaks often arise in materials, they haven’t been found before for superconducting electron pairs. The result thus strongly hints at an FFLO state.

Kinjo, Ishida, and their colleagues constructed a full phase diagram for the homogeneous and the FFLO superconducting phases, with the FFLO occupying the high magnetic field, low-temperature region. The magnetic fields in their study were about a tenth of those necessary for previous FFLO candidates, which makes SRO a practical choice for future FFLO investigations.

Although encouraging, the new result isn’t conclusive. The only definitive evidence would be an observation of spatial modulations in the superconducting order parameter through, for example, measurements of the superconducting gap—the small energy gap that opens when electrons pair up. That smoking gun could come in the future from scanning tunneling microscopy measurements.