Quanta Magazine

Paradoxical Crystal Baffles Physicists

In a deceptively drab black crystal, physicists have stumbled upon a baffling behavior, one that appearsto blur the line between the properties of metals, in which electrons flow freely, and those of insulators,in which electrons are effectively stuck in place. The crystal exhibits hallmarks of both simultaneously.

“This is a big shock,” said Suchitra Sebastian, a condensed matterphysicist at the University of Cambridge whose findings appeared thismonth in an advance online edition of the journal Science. Insulatorsand metals are essentially opposites, she said. “But somehow, it’s amaterial that’s both. It’s contrary to everything that we know.”

The material, a much-studied compound called samarium hexaborideor SmB6, is an insulator at very low temperatures, meaning it resists

the flow of electricity. Its resistance implies that electrons (the

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Original story reprinted with permissionfrom Quanta Magazine, an editoriallyindependent division ofSimonsFoundation.org whose mission isto enhance public understanding ofscience by covering researchdevelopments and trends inmathematics and the physical and lifesciences.

building blocks of electric currents) cannot move through the crystalmore than an atom’s width in any direction. And yet, Sebastian andher collaborators observed electrons traversing orbits millions ofatoms in diameter inside the crystal in response to a magnetic field—a mobility that is only expected in materials that conduct electricity.Calling to mind the famous wave-particle duality of quantummechanics, the new evidence suggests SmB6 might be neither a

textbook metal nor an insulator, Sebastian said, but “something more complicated that we don’t knowhow to imagine.”

“It is just a magnificent paradox,” said Jan Zaanen, a condensed matter theorist at Leiden University inthe Netherlands. “On the basis of established wisdoms this cannot possibly happen, and henceforthcompletely new physics should be at work.”

It is too soon to tell what, if anything, this “new physics” will be good for, but physicists like VictorGalitski, of the University of Maryland, College Park, say it is well worth the effort to find out.“Oftentimes,” he said, “big discoveries are really puzzling things, like superconductivity.” Thatphenomenon, discovered in 1911, took nearly half a century to understand, and it now generates theworld’s most powerful magnets, such as those that accelerate particles through the 17-mile tunnel ofthe Large Hadron Collider in Switzerland.

Theorists have already begun to venture guesses as to what might be going on inside SmB6. One

promising approach models the material as a higher-dimensional black hole. But no theory yetcaptures the whole story. “I do not think that there is any remotely credible hypothesis proposed at thismoment in time,” Zaanen said.

SmB6 has resisted classification since Soviet scientists first studied its properties in the early 1960s,

followed by better-known experiments at Bell Labs.

Counting up the electrons in the orbital shells that surround its samarium and boron nuclei indicatesthat roughly half an electron should be left over, on average, per samarium nucleus (a fraction, becausethe nuclei have “mixed valence,” or alternating numbers of orbiting electrons). These “conductionelectrons” should flow through the material like water flowing through a pipe, and thus, SmB6 should

be a metal. “That’s the idea people had back when I started working on this problem as a young guy,around 1975,” said Jim Allen, an experimental physicist at the University of Michigan in Ann Arborwho has studied SmB6 on and off since then.

But while samarium hexaboride does conduct electricity at room temperature, things get strange as itcools. The crystal is what physicists call a “strongly correlated” material; its electrons acutely feel oneanother’s effects, causing them to lock together into an emergent, collective behavior. Whereas strong

correlations in certain superconductors cause the electrical resistance to drop to zero at lowtemperatures, in the case of SmB6, the electrons seem to gum up when cooled, and the material

behaves as an insulator.

The effect stems from the 5.5 electrons, on average, that occupy an uncomfortably tight shell encasingeach samarium nucleus. These close-knit electrons mutually repel one another, and “that essentiallytells the electrons, ‘Don’t move around,’” Allen explained. The last half electron trapped in each ofthese shells has a complex relationship with its other, freer, conducting half. Below minus 223 degreesCelsius, the conduction electrons in SmB6 are thought to “hybridize” with these trapped electrons,

forming a new, hybrid orbit around the samarium nuclei. Experts initially believed the crystal turnsinto an insulator because none of the electrons in this hybrid orbit can move.

“The resistivity shows it’s an insulator; photoemission shows it’s a good insulator; optical absorptionshows it’s a good insulator; neutron scattering shows it’s an insulator,” said Lu Li, a condensed matterphysicist at the University of Michigan whose experimental group also studies SmB6.

But this is no garden-variety insulator. Not only does its insulating behavior arise from strongcorrelations between its electrons, but in the past five years, mounting evidence has suggested that it isa “topological insulator” at low temperatures, a material that resists the flow of electricity through itsthree-dimensional bulk, while conducting electricity along its two-dimensional surfaces. Topologicalinsulators have become one of the hottest topics in condensed matter physics since their 2007discovery because of their potential use in quantum computers and other novel devices. And yet, SmB6

does not neatly fit that category either.

Early last year, hoping to add to the evidence that SmB6 is a topological insulator, Sebastian and her

student Beng Tan visited the National High Magnetic Field Laboratory, or MagLab, at Los AlamosNational Laboratory in New Mexico and attempted to measure wavelike undulations called “quantumoscillations” in the electrical resistance of their crystal samples. The rate of quantum oscillations andhow they vary as the sample is rotated can be used to map out the “Fermi surface” of the crystal, asignature property “which is sort of the geometry of how the electrons flow through the material,”Sebastian explained.

Sebastian and Tan didn’t see any quantum oscillations in New Mexico, however. Scrambling to salvageTan’s doctoral project, they measured a less interesting property instead, and, to check these results,booked time at another MagLab location, in Tallahassee, Fla.

In Florida, Sebastian and Tan noticed that their measurement probe had an extra slot with a diving-board-style cantilever on it, which could be used to measure quantum oscillations in the magnetizationof their crystals. After failing to see quantum oscillations in the electrical resistance, they hadn’tplanned on looking for them in a different material property—but why not? “I was thinking, fine, let’s

stick a sample on,” Sebastian said. They cooled down their samples, turned on the magnetic field, andstarted measuring. Suddenly they realized the signal coming from the diving board was oscillating.

“We were like, wait—what?” she said.

In that experiment and subsequent ones at MagLab, they measured quantum oscillations deep in theinterior of their crystal samples. The data translated into a huge, three-dimensional Fermi surface,representing electrons circulating throughout the material in the presence of the magnetic field, asconduction electrons do in a metal. Judging by its Fermi surface, electrons in the interior of SmB6

travel 1 million times farther than its electrical resistance would suggest is possible.

“The Fermi surface is like that in copper; it’s like that in silver; it’s like that in gold,” said Li, whosegroup reported surface-level quantum oscillations in Science in December. “Not just metals… these arevery good metals.”

Somehow, at low temperatures and in the presence of a magnetic field, the strongly correlatedelectrons in SmB6 can move like those in the most conductive metals, even though they cannot conduct

electricity. How can the crystal behave like both a metal and an insulator?

Contamination of the samples might seem likely, if not for another surprising discovery: Not only didSebastian, Tan and their collaborators find quantum oscillations in an insulator, but the form of theoscillations—namely, how quickly they grew in amplitude as the temperature decreased—greatlydiverged from the predictions of a universal formula for conventional metals. Every metal ever testedhas conformed to this Lifshitz-Kosevich formula (named for Arnold Kosevich and Evgeny Lifshitz),suggesting that the quantum oscillations in SmB6 come from an entirely new physical phenomenon. “If

it were coming from something trivial, like inclusions of some other materials, it would have followedthe Lifshitz-Kosevich formula,” Galitski said. “So I think it’s a real effect.”

Amazingly, the observed deviation from the Lifshitz-Kosevich formula was presaged in 2010 by SeanHartnoll and Diego Hofman, both then at Harvard University, in a paper that recast strongly correlatedmaterials as higher-dimensional black holes, those infinitely steep curves in space-time predicted byAlbert Einstein. In their paper, Hartnoll and Hofman investigated the effect of strong correlations inmetals by calculating corresponding properties of their simpler black hole model—specifically, howlong an electron could orbit the black hole before falling in. “I had calculated what would replace thisLifshitz-Kosevich formula in more exotic metals,” said Hartnoll, who is now at Stanford University.“And indeed it seems that the form [Sebastian] has found can be matched with this formula that Iderived.”

This generalized Lifshitz-Kosevich formula holds for a class of metallike states of matter that includesconventional metals, Hartnoll says. But even if SmB6 is another member of this “generalized metal”

class, this still does not explain why it acts as an insulator. Other theorists are attempting to model thematerial with more traditional mathematical machinery. Some say its electrons may be rapidlyvacillating between insulating and conducting states in some novel quantum fashion.

Theorists are busy theorizing, and Li and his collaborators are preparing to try and replicateSebastian’s results with their own samples of SmB6. The chance discovery in Florida was only the first

step. Now to resolve the paradox.

Original story reprinted with permission from Quanta Magazine, an editorially independentpublication of the Simons Foundation whose mission is to enhance public understanding of science bycovering research developments and trends in mathematics and the physical and life sciences.

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