All the magnets you have ever interacted with, such as the tchotchkes stuck to your refrigerator door, are magnetic for the same reason. But what if there were another, stranger way to make a material magnetic?

In 1966, the Japanese physicist Yosuke Nagaoka conceived of a type of magnetism produced by a seemingly unnatural dance of electrons within a hypothetical material. Now, a team of physicists has spotted a version of Nagaoka’s predictions playing out within an engineered material only six atoms thick.

The discovery, recently published in the journal Nature, marks the latest advance in the five-decade hunt for Nagaoka ferromagnetism, in which a material magnetizes as the electrons within it minimize their kinetic energy, in contrast to traditional magnets. “That’s why I’m doing this kind of research: I get to learn things that we didn’t know before, see things that we haven’t seen before,” said study co-author Livio Ciorciaro, who completed the work while a doctoral candidate at the Swiss Federal Institute of Technology Zurich’s Institute for Quantum Electronics.

In 2020, researchers created Nagaoka ferromagnetism in a tiny system containing just three electrons, one of the smallest possible systems in which the phenomenon can occur. In the new study, Ciorciaro and his colleagues made it happen in an extended system — a patterned structure called a moiré lattice that’s formed from two nanometer-thin sheets.

This study “is a really cool use of these moiré lattices, which are relatively new,” said Juan Pablo Dehollain, a co-author of the 2020 study who completed the work at the Delft University of Technology. “It looks at this ferromagnetism in a kind of different way.”

When Your Parallel Spins Cause a Field to Begin

Traditional ferromagnetism arises because electrons don’t like each other very much, so they have no desire to meet.

Imagine two electrons sitting next to each other. They’ll repel each other because they both have negative electrical charges. Their lowest-energy state will find them far apart. And systems, as a rule, settle into their lowest-energy state.

According to quantum mechanics, electrons have a few other critical properties. First, they behave less like individual points and more like probabilistic clouds of mist. Second, they have a quantum property called spin, which is something like an internal magnet that can point up or down. And third, two electrons can’t be in the same quantum state.

As a consequence, electrons that have the same spin will really want to get away from each other — if they’re in the same place, with the same spin, they run the risk of occupying the same quantum state. Overlapping electrons with parallel spins stay slightly farther apart than they would otherwise.

In the presence of an external magnetic field, this phenomenon can be strong enough to cajole electron spins into lining up like little bar magnets, creating a macroscopic magnetic field within the material. In metals such as iron, these electron interactions, which are called exchange interactions, are so potent that the induced magnetization is permanent, as long as the metal isn’t heated too much.

“The very reason that we have magnetism in our everyday lives is because of the strength of electron exchange interactions,” said study co-author Ataç İmamoğlu, a physicist also at the Institute for Quantum Electronics.

However, as Nagaoka theorized in the 1960s, exchange interactions may not be the only way to make a material magnetic. Nagaoka envisioned a square, two-dimensional lattice where every site on the lattice had just one electron. Then he worked out what would happen if you removed one of those electrons under certain conditions. As the lattice’s remaining electrons interacted, the hole where the missing electron had been would skitter around the lattice.

In Nagaoka’s scenario, the lattice’s overall energy would be at its lowest when its electron spins were all aligned. Every electron configuration would look the same — as if the electrons were identical tiles in the world’s most boring sliding tile puzzle. These parallel spins, in turn, would render the material ferromagnetic.

When Two Grids With a Twist Make a Pattern Exist

İmamoğlu and his colleagues had an inkling that they could create Nagaoka magnetism by experimenting with single-layer sheets of atoms that could be stacked together to form an intricate moiré pattern (pronounced mwah-ray). In atomically thin, layered materials, moiré patterns can radically alter how electrons — and thus the materials — behave. For example, in 2018 the physicist Pablo Jarillo-Herrero and his colleagues demonstrated that two-layer stacks of graphene gained the ability to superconduct when they offset the two layers with a twist.

Moiré materials have since emerged as a compelling new system in which to study magnetism, slotted in alongside clouds of supercooled atoms and complex materials such as cuprates. “Moiré materials provide us a playground for, basically, synthesizing and studying many-body states of electrons,” İmamoğlu said.

The researchers started by synthesizing a material from monolayers of the semiconductors molybdenum diselenide and tungsten disulfide, which belong to a class of materials that past simulations had implied could exhibit Nagaoka-style magnetism. They then applied weak magnetic fields of varying strengths to the moiré material while tracking how many of the material’s electron spins aligned with the fields.

The researchers then repeated these measurements while applying different voltages across the material, which changed how many electrons were in the moiré lattice. They found something strange. The material was more prone to aligning with an external magnetic field — that is, to behaving more ferromagnetically — only when it had up to 50% more electrons than there were lattice sites. And when the lattice had fewer electrons than lattice sites, the researchers saw no signs of ferromagnetism. This was the opposite of what they would have expected to see if standard-issue Nagaoka ferromagnetism had been at work.

However the material was magnetizing, exchange interactions didn’t seem to be driving it. But the simplest versions of Nagaoka’s theory didn’t fully explain its magnetic properties either.

When Your Stuff Magnetized and You’re Somewhat Surprised

Ultimately, it came down to movement. Electrons lower their kinetic energy by spreading out in space, which can cause the wave function describing one electron’s quantum state to overlap with those of its neighbors, binding their fates together. In the team’s material, once there were more electrons in the moiré lattice than there were lattice sites, the material’s energy decreased when the extra electrons delocalized like fog pumped across a Broadway stage. They then fleetingly paired up with electrons in the lattice to form two-electron combinations called doublons.

These itinerant extra electrons, and the doublons they kept forming, couldn’t delocalize and spread out within the lattice unless the electrons in the surrounding lattice sites all had aligned spins. As the material relentlessly pursued its lowest-energy state, the end result was that doublons tended to create small, localized ferromagnetic regions. Up to a certain threshold, the more doublons there are coursing through a lattice, the more detectably ferromagnetic the material becomes.

Crucially, Nagaoka theorized that this effect would also work when a lattice had fewer electrons than lattice sites, which wasn’t what the researchers saw. But according to the team’s theoretical work — published in Physical Review Research in June ahead of the experimental results — that difference comes down to the geometric quirks of the triangular lattice that they used versus the square one in Nagaoka’s calculations.

That’s a-Moiré

You won’t be able to affix kinetic ferromagnets to your fridge anytime soon, unless you do your cooking in one of the coldest places in the universe. Researchers evaluated the moiré material for ferromagnetic behavior at a frosty 140 millikelvins.

To İmamoğlu, the substance nonetheless reveals exciting new avenues for probing electrons’ behavior in solids — and in applications that Nagaoka could have only dreamed of. In collaboration with Eugene Demler and Ivan Morera Navarro, theoretical physicists at the Institute for Theoretical Physics, he wants to explore whether kinetic mechanisms like those at play within the moiré material could be used to manipulate charged particles into pairing up, potentially pointing the way toward a new mechanism for superconductivity.

“I’m not saying that this is possible yet,” he said. “That’s where I want to go.”