Physicists in the US and Taiwan have performed new experiments that verify long-standing theoretical predictions of how long-range magnetic order can emerge in atomically thin materials. Led by Edoardo Baldini at the University of Texas at Austin, the researchers showed how the transformation occurs through two distinct phase transitions – possibly paving the way for new generations of ultracompact magnetic materials.
Atomically thin two-dimensional (2D) materials are widely studied for their diverse electrical, optical, mechanical and thermal properties. So far, however, their magnetic properties have generally remained far more elusive. Underlying the problem are inevitable thermal fluctuations, which make it extremely difficult to sustain magnetic order over distances larger than atomic scales.
For decades, theorists have investigated a possible exception to this rule in “2D XY” systems: featuring flat arrays of spins that can rotate continuously within the plane and interact with neighbouring spins. One particularly interesting extension of this model describes how a phase transition can occur when these spins become locked into one of six preferred directions, corresponding to the symmetry of the crystal lattice.
“In the 1970s, theoretical work showed that 2D XY magnetic systems with this six-fold anisotropy could exhibit an unusual sequence of phase transitions described by the six-state ‘clock model’, including an intermediate Berezinskii–Kosterlitz–Thouless (BKT) phase,” Baldini explains. “These ideas became central to the theory of low-dimensional magnetism.”
Since these theories emerged, however, such effects have proven far more challenging to observe in real 2D materials.
Verifying the predictions
To tackle this challenge, Baldini’s team turned to a technique involving nonlinear optical microscopy, based on second-harmonic generation: where a material probed by intense light at one frequency emits secondary light at twice that frequency. Crucially, the polarization of this secondary light is highly sensitive to magnetic behaviour. This allowed the researchers to examine magnetic order in the atomically thin antiferromagnet nickel phosphorus trisulphide (NiPS3) without disrupting the system with invasive electrical contacts.
“By tracking how the optical response evolves with temperature, we were able to directly follow successive magnetic phase transitions and determine the universality class of the emergent magnetic phases,” Baldini explains. “In addition, polarization-resolved measurements allowed us to reconstruct the symmetry of the magnetic order parameter.”
As the researchers cooled the material, their measurements revealed two key phase transitions – each occurring suddenly below a distinct critical temperature. “The first transition marks the onset of a BKT phase, an unusual state in which magnetic correlations extend over long distances without forming conventional long-range order,” Baldini says.
In this phase, the material forms bound pairs of vortices and antivortices: topological defects in the spin field triggered by thermal fluctuations. Within these swirling patterns, spins collectively curl around single points, either in clockwise or anticlockwise directions.
At higher temperatures, these swirling patterns are isolated and can roam freely through the material, disrupting the emergence of long-range magnetic order. But when vortices and antivortices are bound together, their disruptive influences largely cancel each other out: allowing spin correlations to persist over longer distances, while still remaining sensitive to thermal fluctuations.
As the researchers cooled the NiPS3 further they observed a second phase transition, in which vortices and antivortices are suppressed and a six-state clock phase emerges. But this symmetry was constrained even further: across the whole system the six possible spin orientations could themselves be arranged in just two distinct ways. This interplay between six- and two-fold anisotropy ultimately gives rise to stable long-range magnetic order, just as earlier theories had predicted.
Diamond microscope probes magnetism in 2D materials
Through their experimental validation, the team’s results shed new light on the rich and unexpected magnetic phenomena that can emerge in 2D materials. Revealing two distinct phases, the work highlights how magnetism can arise in fundamentally different ways to that seen in more familiar three-dimensional materials.
“More broadly, these results establish atomically thin magnets as a powerful platform for exploring topological phase transitions and may inspire new approaches to controlling magnetism at the nanoscale for future ultracompact technologies,” Baldini says.
The findings are reported in Nature Materials.
