Superconducting rings show hints of half-quantum vortices

In most known superconductors, electrons pair up to form spin singlets, combinations of spin up and spin down with zero spin angular momentum. But strontium ruthenate (Sr₂RuO₄, or SRO) shows signs of superconductivity based on spin triplets. (See the article by Yoshiteru Maeno, Maurice Rice, and Manfred Sigrist, PHYSICS TODAY, January 2001, page 42 .) There’s also evidence that the triplets in SRO take a particular form, called an equal-spin-pairing (ESP) state, that can be thought of as two weakly interacting condensates, one of spin-up pairs and one of spin-down pairs. Helium-3, another substance that forms a superfluid of pairs of fermions, has a superfluid phase with similar structure.

Magnetic vortices in superconductors are quantized. Conventional superconductors are characterized by a scalar order parameter, whose phase must change by an integer multiple of 2π over a closed loop around the vortex core. In unconventional superconductors, including those with spin-triplet pairing, the order parameter is a tensor, and more complicated topological structures are allowed. In particular, an ESP state allows half-quantum vortices (HQVs), in which one of the condensates has an extra quantum of vorticity relative to the other.

Researchers are interested in HQVs in their own right because of the insight they provide into the nature of the order parameter. And in a superfluid that breaks time-reversal symmetry (as SRO also shows signs of doing; see PHYSICS TODAY, December 2006, page 23 ), the HQVs can harbor Majorana quasiparticles, ¹ which are potentially applicable to quantum computing (see the box on page 20 ). But although it’s been more than 30 years since HQVs were predicted to occur ² in superfluid ³He, they have never been observed in the bulk of any system. Now, Raffi Budakian (University of Illinois at Urbana-Champaign), his student Joonho Jang, and their collaborators have seen the first experimental signs of HQVs in SRO. ³ Their evidence is the magnetic moment of ring-shaped pieces of the superconductor; that magnetization can be made to jump in increments half the size of full-quantum vortices.

Tiny rings

An isolated HQV is not stable in the bulk of a macroscopic sample. Unlike the charge currents of a full-quantum vortex, which fall off exponentially with distance from the vortex core, an HQV’s spin currents are unscreened by the superconductor, so its energy diverges with the sample size. There are two ways around that difficulty: creating HQVs in closely separated pairs and using mesoscopic samples.⁴ Budakian and colleagues took the latter approach. When the dimensions of the sample are similar to the charge currents’ screening length, HQVs are similar in energy to full-quantum vortices.

The researchers used ring-shaped samples, as shown in figure 1, so the vortex cores were excluded. (Technically, that makes them fluxoids, not vortices.) They made that choice for practical reasons: To identify a vortex from its magnetic moment, one needs to know where the vortex is centered. There’s no known reason why an HQV shouldn’t be stable in a micron-sized disk as well as in a ring.

Making a micron-sized ring, though, was a challenge. To do it, they started with a millimeter-sized SRO crystal (supplied by Yoshiteru Maeno, a pioneer of SRO superconductivity) and shattered it. They picked out a fragment of the right size and shape, attached it to a cantilever, and drilled a hole through it with a focused ion beam. Further complicating that time consuming process, SRO’s superconductivity can be destroyed by even small amounts of nonmagnetic impurities, such as the gallium ions in the beam. Coating the SRO in glue before drilling the hole seemed to help.

Magnetic measurements

Once a ring was glued to the cantilever and shown to be superconducting, the next step was to measure the ring’s magnetic moment. The magnetization associated with one quantum of vorticity is tiny. To make sufficiently precise measurements, Jang and Budakian invented a new type of magnetometry.

The oscillation of the cantilever at its resonant frequency can be driven with a piezoelectric transducer. Applying a magnetic field along the cantilever (H_x ) produces a torque proportional in magnitude to the ring’s magnetic moment. If H_x is made to oscillate at the same frequency as the piezo, the torque can change the oscillator’s frequency or its dissipation, depending on the relative phase. Locking the phase of H_x with respect to the phase of the piezo gives the experimenters enough accuracy to deduce the ring’s magnetization from the cantilever’s oscillation dissipation.

Small steps

Scanning H_z , the field normal to the ring, changes the ring’s magnetization in two ways. A smooth, linear response results from the Meissner effect: The superconducting ring sets up edge currents to expel the magnetic field from its bulk. And discrete jumps result from changes in the quantized vorticity.

Figure 2a shows some measurements of the vortex magnetization (with the Meissner response subtracted out). Without an additional static H_x , the magnetization steps are of the right size and position to be full-quantum vortices, as shown in the middle row. When the researchers added a sufficiently strong static H_x (tens or hundreds of oersted versus the 1-Oe modulation used in the cantilever magnetometry), the steps split into half-height increments, as shown in the other rows. Figure 2b shows histograms of the steps’ vertical positions (accumulated over several measurements).

Budakian and colleagues weren’t expecting H_x to stabilize the HQVs, and they’re still not quite sure why it does. Suk Bum Chung, a theorist at Stanford University and a coauthor of the paper, noticed that the experimental observations might be explained by an effect called kinematic spin polarization,⁵ in which the in-plane field couples to the spin degrees of freedom to stabilize the HQVs.

Ruling out alternatives

The researchers are cautious in interpreting their data; they present their observations as being consistent with the presence of HQVs, rather than as conclusive evidence of them, since HQVs are not the only thing that could produce magnetizations in between those of the full-quantum-vortex states. There could, for example, be vortices piercing the bulk of the sample rather than (or in addition to) fluxoids whose cores are excluded by the ring’s hole.

To strengthen the case for HQVs, Budakian worked with theorists David Ferguson, Victor Vakaryuk, and Paul Goldbart to come up with alternative explanations for the data. He and Jang then tested those scenarios in the laboratory. In one such test, they repeated the cantilever experiment with a ring of niobium diselenide (a conventional superconductor similar in some ways to SRO, but not expected to support HQVs) and with an SRO sample too large to allow HQVs. They saw no half-height steps in either case. “Establishing the HQV state with greater certainty requires measuring a quantity unique to HQVs, such as the spin currents themselves,” says Budakian. “But based on the alternatives we’ve considered, the HQV description is the most consistent.”