Japanese scientists may have found a surprisingly simple way to predict how some unusual materials behave by looking at the size of their crystal structure.

In their latest study, they claim that this structural measurement can predict magnetic ground states across a family of unusual materials more consistently than the electron-counting method scientists have traditionally relied on. 

This could make the search for unusual magnetic materials less of a guessing game. 

Until now, the challenge has been that quasicrystals and related materials have extremely complex atomic arrangements, making it difficult to find a single rule that works across different chemical compositions. 

“Quasicrystals are among the most uniquely structured materials discovered to date and are expected to exhibit novel magnetic states and quantum phenomena not found in ordinary crystals. Until now, there has been no unified guideline for systematically exploring these novel phenomena in quasicrystals and their approximate crystals,” Farid Labib, first study author and assistant professor at Tokyo University of Science, said.

The new study suggests that a crystal’s size could provide that missing common measure when researchers search for unconventional magnetism.

Looking beyond the electron count

The researchers focused on Tsai-type approximant crystals. These materials mimic important structural features of quasicrystals while having a repeating crystal structure that can be studied experimentally. 

Their atoms form nested clusters, with rare-earth elements such as terbium, dysprosium and holmium occupying positions that carry magnetic moments.

Researchers have often used the valence-electron concentration, or the number of valence electrons per atom, to classify magnetic behavior. 

However, the magnetic boundaries associated with this measure shift with the rare-earth element and alloy composition, making it less reliable as a universal guide.

The researchers therefore synthesized Au–Al and Au–Ga-based 1/1 approximant crystals containing terbium, dysprosium and holmium, then compared their structures and magnetic properties. They also combined their measurements with data from previously reported Tsai-type compounds to see whether the pattern held across different compositions.

They found a nearly monotonic inverse relationship between electron concentration and the lattice parameter—the dimensions of the crystal’s repeating unit. This allowed them to reorganize the materials according to lattice size rather than chemical composition.

Three magnetic states separated by crystal size

The distinction turned out to be remarkably clear. Crystals with lattice parameters above about 14.72 Å (14.72 x 10^-10 m) developed a whirling antiferromagnetic state, where magnetic moments form an ordered pattern with opposing contributions. 

Between 14.62 and 14.72 Å, they showed a whirling ferromagnetic state with a net magnetization. Below about 14.62 Å, the materials enter a spin-glass state, where magnetic moments become frozen in a disordered arrangement.

The whirling magnetic states arise in these non-Heisenberg compounds partly because the local atomic environment creates a strong crystal electric field. This field favors particular orientations for the rare-earth magnetic moments, influencing how they ultimately arrange themselves. 

The magnetic moments also interact indirectly through electrons moving through the material, linking the atomic structure to the way those moments arrange themselves.

A structural roadmap for new materials

The important point is not simply that lattice size correlates with magnetism. The study establishes specific structural thresholds that place different compounds on a common magnetic phase diagram.

This makes the lattice parameter an experimentally accessible starting point for deciding which materials may host a desired magnetic ground state.

“The unified magnetic phase diagram constructed in this study can serve as a practical roadmap for systematic exploration of new magnetic quasicrystals and approximant crystals exhibiting novel magnetic orders and quantum phenomena,” Kazuhiro Nawa, study co-author and a researcher at Tohoku University said. 

“It can also provide a guideline for designing new magnetic materials with targeted magnetic ground states, opening new opportunities for discovering unconventional magnetism in quasiperiodic and complex intermetallic systems,” Nawa added.

The finding does not make electron concentration irrelevant. Rather, it suggests that lattice parameter may be a more consistent guide for Tsai-type compounds, while its usefulness in other magnetic materials remains untested.

What remains unclear is exactly why the lattice parameter works so well across these different compounds. The researchers also need to determine whether the same relationship holds in true quasicrystals.

Despite these limitations, the study indeed offers a more systematic route for exploring complex alloys and designing new quantum materials.

The study is published in the Journal of the American Chemical Society.

Superconductivity is supposed to be an all-or-nothing quantum state. Cool a material below a critical temperature, and electrons pair up and enter a state that lets electricity flow without resistance. Warm it back up, and that collective state disappears. 

However, experiments on uranium ditelluride suggest that part of the superconducting behavior can survive after the main phase is gone.

Physicists at the University of Illinois Urbana-Champaign (UIUC) have found evidence that Cooper pairs—the paired electrons behind superconductivity—can arrange themselves into a repeating pattern called a pair density wave (PDW). 

What’s even more interesting is that the pattern persists above the temperature where uranium ditelluride stops being superconducting. These findings provide the first direct evidence that a PDW can exist independently of the main superconducting phase.

“We were out on a limb when we first suggested it. It’s a very peculiar state, and, although there have been experimental hints, there has been no direct confirmation of the phase’s existence,” Eduardo Fradikin, one of the researchers and a physics professor at UIUC, said.

Finding the hidden state

In conventional superconductors, electrons form Cooper pairs and those pairs condense into one collective quantum state. Unconventional superconductors can behave differently and host additional electronic phases, including charge density waves (CDWs), where electric charge varies periodically across the material.

PDWs are similar in that they form a spatial pattern, but the quantity being modulated is the density of Cooper pairs. This idea was proposed in 2007, including the striking possibility that these paired electrons could exist above the superconducting transition. 

However, “PDWs are tricky to analyze in real materials, because they behave like conventional superconductors in some experiments, and like CDWs in other,” Julian May-Mann, one of the study authors and a former graduate student from UIUC, said.

Uranium ditelluride, or UTe₂, offered an unusual place to look. Once considered an ordinary metal, it was found in 2019 to become superconducting below about 2 kelvins. It is also widely considered a candidate for an unconventional triplet-pair superconductor, in which the paired electrons can carry magnetic moments.

The researchers had previously detected CDWs on the material’s surface using scanning tunneling microscopy, but something did not fit, as magnetic fields could destroy the waves. A normal CDW should not respond so strongly to a magnetic field, prompting the researchers to consider whether a PDW was hiding alongside it.

Better crystals reveal the signal

The researchers first needed cleaner UTe₂ crystals because PDWs are extremely delicate and impurities can obscure their signatures. So they produced higher-quality samples using a new molten-flux growth method.

“Thanks to new methods for growing higher-quality samples, we were able to observe spectral signatures that respond to temperature and magnetic fields exactly as pair density waves should,” Vidya Madhavan, one of the lead researchers and a physics professor at UIUC, said.

They examined the crystals with a vector magnetic-field scanning tunneling microscope. Unlike a conventional setup, the instrument could vary both the strength and direction of the magnetic field—important because UTe₂ is anisotropic, meaning its properties depend strongly on direction.

The measurements revealed electronic modes whose responses to temperature and magnetic fields matched theoretical expectations for a PDW, including modes that survived above the superconducting critical temperature.

“We even showed that the modes persist above the temperature at which superconductivity disappears, a telltale theoretical prediction that has not been convincingly observed before now,” Madhavan added.

The researchers also found that a CDW-only explanation conflicted with established principles governing how electronic phases should respond to temperature and magnetic fields. The PDW interpretation provided a more consistent explanation of the observations.

A clue, but not the final answer

The result could reshape how physicists think about unconventional superconductors by suggesting that Cooper pairing can begin before the material develops its full superconducting state. 

This could help researchers understand why some unconventional materials develop complex phases and potentially guide the search for new superconducting states.

“Pair density waves are the Cheshire Cat’s grin of superconductivity. They are the vestige that remains once the phase itself has disappeared. In conventional superconductors, Cooper pairs form when the full phase transition occurs, but, in this system, their observation in PDWs above the transition point shows that they are formed beforehand in a different state,” Fradikin explained.

However, there is an important limitation as well. Scanning tunneling microscopy probes the material’s surface, so the researchers cannot yet prove that the same PDW exists throughout the interior.

The next challenge is therefore to determine whether the hidden state extends into the bulk of UTe₂ and to establish its nature more firmly. 

The study is published in the journal PNAS.