These intricate patterns are known as pair density waves, or PDWs. Theoretical physicists first predicted the existence of PDWs about twenty years ago, and previous studies have occasionally observed them coexisting alongside full superconductivity in other exotic metals. However, this new research, which has been published in the prestigious Proceedings of the National Academy of Sciences, goes significantly further than earlier investigations. It provides the very first direct evidence that pair density waves in uranium ditelluride can persist within the material’s "ordinary" phase long after bulk superconductivity has completely disappeared.
"Pair density waves are the Cheshire Cat’s grin of superconductivity," said Eduardo Fradkin, an Illinois Grainger Engineering physics professor and a co-lead of the research project. "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."
The breakthrough was made possible by a combination of cutting-edge experimental hardware and vastly improved material synthesis. "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," said Vidya Madhavan, an Illinois Grainger Engineering physics professor and the other project co-lead. "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."
How Unconventional Superconductors Behave
To understand the significance of this discovery, it helps to examine how superconductors generally behave. Superconductivity is a remarkable quantum mechanical phenomenon that occurs when a metal is able to carry electric current with absolute zero electrical resistance, typically after being cooled below a specific critical temperature. In this pristine state, the material’s free electrons settle into a collective, low-energy quantum state.
This creates a fundamental physical challenge. Electrons belong to a broad group of subatomic particles called fermions. The foundational principles of quantum mechanics explicitly prevent fermions from gathering in the exact same quantum state—a rule known as the Pauli exclusion principle. Superconductivity manages to bypass this strict restriction through a more complex process involving the formation of bound pairs of electrons.
Back in 1957, renowned Illinois physicists John Bardeen, Leon Cooper, and Robert Schrieffer formulated an explanation for this behavior, which quickly became known to the scientific community as BCS theory. According to their landmark model, individual electrons first become correlated through interactions with the metal’s underlying atomic lattice. One electron then effectively binds with another to create what is called a Cooper pair.
Unlike individual electrons, which are fermions, Cooper pairs behave as bosons. This crucial distinction means that a large number of them can occupy the exact same quantum state simultaneously, entirely avoiding the mutual repulsion that normally applies to individual fermions. Once they are successfully paired, these bosons can condense together into a unified macroscopic quantum state, resulting in zero-resistance superconductivity.
BCS theory successfully explained virtually all known superconducting behavior for decades, right up until scientists discovered a new class of "unconventional" superconductors in 1986. These complex materials do not fit comfortably within the foundational assumptions behind BCS theory, even though their constituent electrons still form Cooper pairs and eventually condense. Understanding the exact microscopic mechanisms driving how these unconventional superconductors work remains one of the most active and challenging areas of modern condensed matter physics research.
The Mystery of Pair Density Waves
Unconventional superconductors frequently host other competing or cooperating electronic phases at temperatures well below the superconducting critical temperature. One well-known example of this phenomenon is charge density waves, or CDWs, in which a fraction of the material’s electrons spontaneously form a repeating, nonuniform spatial pattern. The physical result is a periodic variation in electronic charge across the material, with some microscopic regions containing a higher concentration of electric charge and others containing less.
In 2007, Professor Fradkin and his academic colleagues proposed the theoretical existence of yet another possible electronic phase, which they named a pair density wave. In a standard superconductor, Cooper pairs are distributed uniformly throughout the material. However, the researchers suggested that under the right conditions, these pairs might instead organize themselves into distinct, repeating spatial patterns.
Their theoretical model made a particularly unusual and bold prediction: pair density waves could potentially survive at temperatures above the critical temperature of the material. If true, this would mean that Cooper pairs could successfully form even when the bulk material had not yet entered its superconducting phase.
"We were out on a limb when we first suggested it," Fradkin recalled, reflecting on the initial skepticism surrounding the theory. "It’s a very peculiar state, and, although there have been experimental hints over the years, there has been no direct confirmation of the phase’s existence."
The difficulty in confirming the theory stemmed from the nature of the phenomenon itself. "PDWs are tricky to analyze in real materials, because they behave like conventional superconductors in some experiments, and like CDWs in other," explained Julian May-Mann, a former Illinois Grainger Engineering graduate student who worked extensively on the study’s theoretical analysis. "Confirming the existence of a PDW requires both high-quality experimental data as well as careful theoretical analysis."
Why Uranium Ditelluride Stands Out
Until 2019, uranium ditelluride was generally regarded by the scientific community as a relatively ordinary metal. That perception changed dramatically when researchers discovered that it undergoes a phase transition, entering a superconducting state when cooled below 2 kelvins.
Further investigation by physicists soon led them to suspect that uranium ditelluride might be a rare example of a triplet-pair superconductor. Unlike the electron pairs described by conventional BCS theory, which possess opposite spins, triplet electron pairs possess parallel spins and non-zero magnetic moments.
To date, the only definitively confirmed example of a triplet-pair "super-phase" is superfluid helium-3. That exotic quantum system was extensively studied by the late Illinois physicist Anthony Leggett, whose pioneering work ultimately earned him a Nobel Prize in Physics.
"Triplet-pair superconductors with properties analogous to superfluid helium-3 are conjectured to exist, and there are several superconductors believed to be such instances," Fradkin noted. "I would not say that the question is completely settled, but the consensus is that uranium ditelluride is a triplet-pair superconductor."
Uranium ditelluride eventually drew the focused attention of Professor Madhavan’s experimental research group. Using advanced scanning tunneling microscopy, the team identified charge density waves within the material. Yet, something about those observed waves behaved unexpectedly. Crucially, the application of magnetic fields could disrupt and destroy them.
"A charge density wave is just a collective electronic state that is modulated in space, so there is no reason for them to respond to magnetic fields, let alone be destroyed by them," Madhavan said. "We took our data to Professor Fradkin and his students, and we predicted that this could happen if there is also a pair density wave in the system. There aren’t many possible explanations for something like this, and pair density waves are the best one I can think of."
Better Crystals Reveal the Hidden State
To rigorously investigate the hypothesis that pair density waves were present, Madhavan’s team first required much cleaner, higher-purity samples of uranium ditelluride. PDWs are exceptionally delicate quantum states that demand highly regular, defect-free crystal lattices in order to form and remain stable.
Collaborating researchers supplied significantly improved samples produced via a newly developed molten flux growth method. The experimental team then examined these pristine samples using a state-of-the-art vector magnetic field scanning tunneling microscope (STM). This specialized instrument allows scientists to observe with high precision how a material’s atomic surface responds to magnetic fields applied simultaneously from multiple different directions.
"我们 couldn’t see pair density waves in our earlier data because of material impurities that obscured our data," Madhavan explained. "It would have been like trying to spot a light in a cloud of fog. But we succeeded with the better samples because of our new vector magnetic field equipment. The latter was especially important because uranium ditelluride is anisotropic, so we needed the capability to examine the crystals from many directions."
The technical demands of the material made the specialized equipment indispensable. "A key advantage of our experiment was the newly developed vector-magnet equipment, which provides an unusually large magnetic-field range along multiple directions," said Zhen Zhu, an Illinois Grainger Engineering physics postdoctoral research associate who carried out the hands-on measurements. "This capability is particularly well suited to uranium ditelluride, whose superconducting upper critical field is strongly anisotropic. By systematically varying both the magnitude and direction of the field, together with temperature, we could track how these modes evolved and build confidence that the behavior we observed was intrinsic."
Cooper Pairs Persist After Superconductivity Vanishes
The subsequent measurements of the charge density waves revealed distinct modes that changed as a function of both temperature and magnetic field in precisely the manner predicted for pair density waves. Furthermore, the application of magnetic effects successfully destroyed them, aligning neatly with the theoretical framework.
The most striking and significant result, however, emerged when the researchers incrementally raised the temperature of the experimental setup. They observed that certain modes continued to exist even after the material had passed its critical temperature threshold and the bulk superconducting phase had entirely vanished.
"There are foundational principles in condensed matter physics that constrain how different phases can appear and disappear when one changes the temperature or applies a magnetic field," May-Mann observed. "Any explanation of the experimental data that only relies on a CDW is at odds with these principles. The PDW-based explanation, on the other hand, provides a satisfactory and consistent explanation."
The researchers readily acknowledge an important experimental limitation inherent in their methodology: scanning tunneling microscopy imaging only measures electronic effects taking place at a material’s exposed surface. Therefore, the experiment does not directly reveal the microscopic phenomena occurring deep throughout the interior volume of the uranium ditelluride crystals. Even so, the research team remains confident that the findings offer a vital, compelling clue and will open entirely new avenues for studying the mysteries of unconventional superconductivity.
"It is possible for the interior of a material to behave differently than the surface, but these experimental results still give us a very strong hint at what’s happening inside," Fradkin concluded.
"As an experimentalist, one of the most satisfying things is when several independent measurements begin to tell the same story," Zhu added. "Here, the temperature and magnetic-field dependence, together with the improved sample quality, all came together to reveal a remarkably consistent picture of the pair density wave state."
Additional contributions to this research study were made by Yudi Huang, Kaiming Liu, Zheyu Wu, Shanta Saha, Johnpierre Paglione, Alexander Eaton, Andrej Cabala, and Michal Vališka. Financial support for the project was provided by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division.
