Nickel disulfide, NiS₂, is an unusual material. Its interior behaves as an electrical insulator, blocking the flow of current, while certain parts of its surface conduct electricity freely. This contradiction has puzzled physicists for decades, and a recent study offers a detailed explanation. The conduction, it turns out, is confined to tiny one-dimensional steps on the surface, and these narrow conducting channels are a direct consequence of how electrons are arranged inside the crystal.

Mott insulator

NiS₂ shares its crystal structure with pyrite, the mineral sometimes called fool’s gold. At ordinary conditions, its electrons do not move freely through the lattice: the strong electrical repulsion between them opens an energy gap that keeps the bulk of the material insulating. This behaviour defines what physicists call a Mott, or charge-transfer, insulator, a category of materials that simple band theory cannot explain. Band theory predicts that any material with a partially filled outer electron shell should conduct electricity, yet compounds like nickel oxide obstinately insulate instead. In 1949, the physicist Nevill Mott showed why: if two electrons crowding the same atomic site cost more energy than is available, the electrons stay locked in place and no current flows, regardless of what band theory predicts. NiS₂ behaves in much the same way.

Yet electrical-transport measurements had repeatedly found that crystals of NiS₂ conduct through their surfaces. Earlier scanning-tunnelling experiments helped explain part of the puzzle: the broad, flat surface remains insulating, but the electronic gap narrows sharply near atomic-scale steps, tiny cliffs just one or two atoms high where the surface abruptly changes level. A new study ¹ confirms this picture and examines the steps with much higher spatial resolution than before.

A not that smooth surface

A scanning tunnelling microscope (STM) works by bringing an extremely sharp needle close enough to a surface that electrons can leak across the gap between tip and sample. Measuring that leakage at different energies reveals both the surface’s shape and how easily electrons can enter or leave it. In NiS₂, these measurements uncovered electronic states concentrated tightly along the step edges: the states extend for a long distance parallel to a step but stay confined to roughly one to two nanometres across it. In other words, the electrons are effectively free to move in one direction while remaining trapped in the other two, forming genuine one-dimensional conduction channels.

Two distinct kinds of surface step were observed, depending on whether the outermost atomic layer exposes nickel or sulfur atoms. Both carry electronic states associated with their steps, but they are not equally important. The nickel-terminated steps host states that sit right at the energy where electrical conduction happens, making them the most likely explanation for the surface conductivity that transport experiments had measured for years. The sulfur-terminated steps also host states inside the insulating gap, but their energy is positioned further away, making them less directly responsible for carrying current.

The study also tested what happens when a magnetic field is applied perpendicular to the surface. The step-edge states survived fields as strong as 10 tesla, a field far stronger than any produced by a laboratory magnet used for everyday purposes. Their intensity weakened somewhat, particularly on the sulfur-terminated steps, but the states were never destroyed. This robustness matters because NiS₂ already displays complicated magnetic behaviour of its own, which might easily have disrupted a more fragile conducting state.

The deeper explanation lies in how electrons are distributed inside the crystal. In an ordinary insulator, the occupied electronic states can usually be pictured as clouds centred on the atoms themselves. NiS₂ is different: calculations show that part of its electron density is naturally centred at special positions between atoms, specifically near the midpoints of pairs of sulfur atoms, where no atom actually sits. Physicists call these obstructed charge centres, and materials built around them belong to a class known as obstructed atomic insulators, a concept that grew out of a broader effort over the past decade to classify materials by the geometry, and not just the energy, of their electron states.

Not ordinary metallic defects

This arrangement is not a chemical accident; it is protected by the symmetry of the crystal. When the crystal is cut to form a surface, though, the special positions that hosted this electron density can no longer fit together the way they did in the bulk. At a step edge, this mismatch creates what is known as a filling anomaly: an imbalance in how the available electronic states can be occupied, which leaves behind a state inside the insulating gap, localized right at the edge. Calculations of artificial step structures reproduce the main features seen under the microscope, including this localization, supporting the idea that the conducting channels are not ordinary metallic defects caused by broken chemical bonds, but a direct consequence of the crystal’s underlying electronic geometry.

The study also settles a long-standing question about NiS₂’s magnetism at low temperature. Below about 39 kelvin, the nickel magnetic moments arrange themselves into a non-collinear antiferromagnetic pattern, with neighbouring moments pointing in different directions that largely cancel out. Below about 30 kelvin, they rearrange again into a state that produces a small but measurable net magnetization. New neutron-diffraction measurements support a revised model of this low-temperature arrangement, resolving a debate about the material’s magnetic ground state that had lasted for roughly fifty years.

Taken together, the results give NiS₂ a significance well beyond its peculiar conducting surface. The material shows that strong repulsion between electrons, magnetism, and the geometry of electronic states can coexist within a single crystal. Such geometric protection is normally associated with topological materials, in which certain electronic states are shielded from disorder or defects by the crystal’s symmetry. The new results suggest that this protection can survive even when strong interactions push a material into being an insulator, making NiS₂ a rare setting for studying how the quantum-mechanical arrangement of electrons decides where electricity is allowed to flow within an otherwise insulating crystal.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper/s.

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One of the most striking discoveries in condensed matter physics over the past few years is that simply rotating one atomically thin crystal on top of an identical layer can create entirely new electronic behavior. This idea became famous in 2018, when physicists found that graphene sheets stacked at a very particular “magic angle” could turn into a superconductor, even though a single sheet of graphene is not. That discovery launched an entire field, sometimes called twistronics, built around moiré patterns: the large-scale interference patterns that appear when two crystal lattices are laid on top of each other with a slight mismatch in angle or spacing. In many materials, these patterns reshape how electrons move, sometimes producing superconductivity or other exotic collective states.

Moiré collapse

Most of this research has focused on materials whose crystal structure looks roughly the same in every direction, such as graphene, where the interesting physics shows up only at very small twist angles, close to perfect alignment. Now, a team of researchers instead looks at a strongly directional material: bilayer black phosphorus, whose atomic structure behaves very differently along two perpendicular directions in the plane, somewhat like a fabric that stretches easily one way but not the other. That built-in directionality turns out to change the whole logic of moiré physics, producing an effect the authors call moiré collapse. ¹

In black phosphorus, the two layers start out rotated 90 degrees relative to each other. As that angle is dialed away from 90 degrees, the moiré pattern does something unusual. Rather than staying a two-dimensional pattern, like a slowly shifting checkerboard, it stretches more and more along one direction until, at a specific angle that depends only on the material’s own geometry, it effectively stops repeating in the other direction altogether. What remains behaves like a one-dimensional crystal, even though the material is still built from ordinary two-dimensional atomic sheets. For twisted bilayer black phosphorus, the calculations place this collapse at a twist of about 71.6 degrees, or equivalently about 18.4 degrees away from the initial 90-degree arrangement.

Pure geometry

This transformation has nothing to do with how electrons repel or attract each other; it is pure geometry. The mathematical description of the moiré pattern, expressed in terms of the directions in which its periodicity repeats, is squeezed from two independent directions down to one. In ordinary space, this means the pattern keeps a well-defined repeat distance along one axis while stretching out indefinitely along the other, with no repeating pattern left in that direction at all.

Such a dramatic structural change reshapes the electrons riding on top of it. At the untwisted 90-degree configuration, the anisotropy of one layer happens to cancel out the anisotropy of the other, so electrons move about equally well in every in-plane direction. As the twist angle approaches the collapse point, that cancellation breaks down. The electron energy bands become sharply directional, and electrons increasingly travel along one preferred line rather than spreading across the plane. Their energy spectrum starts to show the telltale fingerprints of one-dimensional systems, including sharp spikes in the density of available electron states known as one-dimensional van Hove singularities.

Luttinger liquid physics

One-dimensional conductors behave very differently from ordinary metals. In a three-dimensional metal, electrons can mostly be treated as independent particles that only weakly notice one another. Confine them to a single dimension, however, and that picture stops working: electrons cannot slip past each other, so they are forced to move collectively, in a coordinated wave-like fashion. The framework that describes this collective behavior, called Luttinger liquid theory, traces back to work by Sin-Itiro Tomonaga in 1950 and Joaquin Luttinger in 1963, and it has since been confirmed experimentally in systems such as carbon nanotubes. In a Luttinger liquid, the basic excitations are not single electrons at all, but collective ripples of charge and spin, and this leads to electrical transport properties that look qualitatively different from those of a normal metal.

The calculations in this study show that moiré collapse naturally sets up the conditions Luttinger liquid physics requires. As the collapse angle is approached, the one-dimensional electronic channels that form become progressively less coupled to their neighbors, making the correlated, wave-like behavior increasingly favorable. The analysis estimates a Luttinger parameter of about 0.3 for the collapsed black phosphorus system, a value indicating strong electron-electron interactions, and it predicts distinctive collective charge oscillations and unusual, non-metallic patterns in electrical conduction that could, in principle, be measured in the laboratory.

A notable feature of this mechanism is that the electron bands stay quite dispersive even at collapse, with electron speeds exceeding 100 kilometers per second. This sets it apart from most previously studied moiré systems, including twisted graphene, where strong correlations usually arise because the electronic bands become extremely flat and electrons effectively slow to a crawl. Here, the correlated behavior instead comes from the geometric collapse of the moiré lattice itself, with fast-moving electrons confined to one dimension rather than slow ones spread over two.

Moiré collapse is not a peculiarity of one material

To test these ideas, the researchers combined simplified mathematical models with detailed first-principles calculations of the atomic and electronic structure. Beyond black phosphorus, they found the same collapse in twisted bilayer tin selenide, another layered material with strongly directional bonding. The details of the electronic structure differ somewhat because of the specific atomic orbitals involved, but the same geometric collapse into quasi-one-dimensional channels appears, suggesting the phenomenon is not a peculiarity of one material but a general consequence of twisting any sufficiently anisotropic crystal.

Taken together, the findings outline a different strategy for engineering exotic quantum matter. Instead of hunting for a special magic angle that flattens electronic bands, as in twisted graphene, this approach relies on a purely geometric collapse angle at which a two-dimensional moiré material naturally turns one-dimensional. If confirmed in experiments, it would offer a new and versatile platform for studying Luttinger liquids and other correlated electronic phases that only emerge when electrons are squeezed into moving along a single line.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper/s.

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Building electronic devices often requires stacking different crystalline materials on top of one another with atomic precision. This process, called heteroepitaxy, works best when the two crystals share similar atomic patterns. When their symmetries differ, the upper crystal tends to grow in several orientations at once instead of a single one, creating defects that degrade device performance. Engineers have long dealt with this problem by treating the surface of the lower crystal beforehand, for example by etching steps into it, exposing it to plasma, or adding a buffer layer, to help the two materials bond more effectively. Now, a team of researchers explores ¹ a different route: rather than forcing the two crystals to fit together, it takes advantage of a crystal that is naturally flexible at the atomic scale.

Flexible by design

The material at the center of this approach is a layered compound called TaCo₂Te₂. Like graphite, its layers are held together by weak van der Waals forces, which makes it easy to combine with other materials without demanding a perfect chemical match. Unlike most layered crystals, though, its atoms are not arranged in a perfectly regular pattern. At room temperature they undergo a small, periodic shift known as a Peierls distortion, a phenomenon first described in the 1930s by the physicist Rudolf Peierls, who showed that in some materials the electrons and the surrounding lattice can lower their combined energy by nudging atoms slightly out of their ideal positions. When TaCo₂Te₂ is heated above about 523 kelvin, or 250 degrees Celsius, this distortion disappears and the crystal adopts a more symmetric structure, although traces of the original pattern linger as fleeting, temperature-driven fluctuations.

These fluctuations are not random. Calculations of how the crystal vibrates, backed up by Raman spectroscopy and electron diffraction experiments, show that the lattice is especially unstable along one crystallographic direction, meaning atoms can shift more easily there than in the perpendicular direction. This directional softness turns out to be the key ingredient that controls how a second material grows on the surface.

Heating TaCo₂Te₂ also sets its atoms in motion across the surface. Tantalum atoms drift away, while cobalt and tellurium reorganize into a thin crystalline layer whose composition resembles cobalt telluride. Rather than clumping into isolated islands, this new layer spreads sideways first and only then thickens, producing a smooth, continuous interface with the crystal beneath it, a layer-by-layer growth style that electron microscopy confirms is energetically favorable.

Directionally locked

The most striking result is that the new crystal does not settle into just any orientation. Even though its symmetry differs from that of the underlying TaCo₂Te₂, it becomes directionally locked. Along one in-plane direction, the atomic spacings of the two materials match closely, giving a tight, well-ordered registry. Along the perpendicular direction the mismatch is far larger, but this happens to be exactly the direction where TaCo₂Te₂ is most flexible, thanks to its lattice instability. Instead of rotating to relieve that mismatch, as symmetry-mismatched interfaces typically do, the interface absorbs it through a gentle, ongoing structural adjustment. The result is a crystal that stays aligned while only one direction bears most of the strain.

That flexible direction does not settle into a simple repeating pattern either. Instead, it forms what is called a one-dimensional incommensurate superlattice: the repeating atomic spacings of the two crystals never line up exactly, yet they stay locked together over long distances. Physicists have long recognized this kind of arrangement as an efficient way to accommodate mismatched lattice spacings without piling up large numbers of defects, since the strain is spread out through a slowly changing modulation rather than concentrated at sharp boundaries. Diffraction measurements and a geometric analysis of the two lattices both confirm that no perfectly repeating, shared lattice can account for what is observed, so the incommensurate description fits best.

Directional lattice instability is key

To check that the lattice instability is truly responsible for this behavior, the same experiment was repeated with a closely related compound, TaNi₂Te₂, which has nearly the same crystal structure but lacks the structural instability. Under similar heating, a surface layer still forms, but its orientation varies from spot to spot, and rotational disorder sets in. This comparison shows that matching crystal symmetry is not enough on its own to produce a locked, well-ordered interface. The directional lattice instability is the extra ingredient that guides the growth.

The local reconstruction effect

One more effect helps stabilize the interface. Even above the temperature where the bulk crystal should be fully symmetric, the atomic layers right beneath the interface partly slip back into the low-temperature distorted structure. This local reconstruction stays confined to the surface rather than spreading through the whole crystal, and it appears to strengthen the bond between the two materials, keeping the aligned structure stable even at high temperature.

As we have just seen, structural instabilities, usually considered as a liability because they signal that a crystal is on the verge of changing shape, can instead become useful design tools. A substrate whose atoms rearrange more easily in certain directions can guide the growth of otherwise incompatible materials, producing well-aligned interfaces without extensive surface treatment. This idea could widen the range of material combinations available for future electronic, optical, and quantum devices, where the atomic arrangement at an interface often determines how well the whole device performs.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper/s.

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