The interplay between surface and bulk properties in materials significantly impacts their overall behavior. While the surface-bulk correspondence is well-established in topological phases, its understanding in Landau-like phases remains limited. Theoretical work dating back to the 1970s predicted the possibility of surface order emerging at a higher temperature than the bulk order – an 'extraordinary' transition – but experimental evidence has been lacking. This counterintuitive phenomenon significantly enriches the complexity of phase transitions. This research addresses this gap by investigating CrSBr, a van der Waals (vdW) layered antiferromagnet, known for its unique magnetic properties and layered structure, making it an ideal candidate to explore surface-dominated phase transitions. The study leverages the surface sensitivity of electric dipole SHG and the bulk sensitivity of electric quadrupole SHG to probe both surface and bulk magnetic ordering simultaneously. The researchers hypothesized that the strong intralayer interactions and weak interlayer interactions in vdW materials could lead to the observed surface-dominated phase transition, as surface modifications might outweigh the reduced interactions at the surface compared to the bulk. This study is crucial because surfaces are ubiquitous in real-world materials, and understanding their influence on phase transitions is essential for controlling and manipulating material properties for various technological applications, particularly in the burgeoning field of two-dimensional (2D) magnetism. The ability to enhance surface magnetism in 2D materials, as suggested by this research, could have significant implications for developing novel spintronic devices.

Previous theoretical studies on semi-infinite systems predicted three distinct phase transitions: 'ordinary' (simultaneous surface and bulk ordering), 'surface' (surface ordering without bulk ordering), and 'extraordinary' (bulk ordering after surface ordering). The conditions for observing 'surface' and 'extraordinary' transitions require enhanced surface interactions compared to bulk interactions. In three-dimensional (3D) materials, where intralayer and interlayer interactions are comparable, such a scenario is unlikely. However, in quasi-2D materials like vdW magnets, where intralayer interactions dominate, minor surface modifications could lead to the observed split transitions. Despite extensive research on vdW and 2D materials, experimental evidence for these split transitions has remained scarce, largely due to the lack of suitable experimental tools capable of probing both surface and bulk transitions simultaneously. Recent advancements in second harmonic generation (SHG), particularly the ability to detect electric quadrupole (EQ) contributions in addition to electric dipole (ED) contributions, have provided a promising avenue to address this challenge. ED-SHG is highly surface sensitive, while EQ-SHG can probe bulk properties. Combining both techniques allows for a comprehensive investigation of surface and bulk phase transitions.

The researchers used high-quality CrSBr single crystals grown via a direct solid-vapor method to minimize defects that could confound magnetic measurements. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) confirmed the high crystalline quality of the samples, revealing a scarcity of atomic and stacking defects. Temperature-dependent heat capacity measurements reproduced the three characteristic temperatures reported in the literature (Tγ = 185 K, Tγ* = 155 K, TN = 132 K) and additional magnetization measurements identified a possible ferromagnetic (FM) state at a lower temperature (TF = 30-40 K). The core of the experimental methodology involved temperature-dependent oblique incidence SHG rotation anisotropy (RA) measurements. This technique allows for probing different SHG contributions (ED and EQ) and their interference, thereby differentiating surface and bulk magnetic ordering. In the SHG RA measurement, the intensity of reflected SHG light is recorded as a function of the azimuthal angle between the crystallographic a-axis and the scattering plane, in different polarization channels. The data was collected during warming cycles with a slow heating rate (0.5 K/min) and a 5-minute waiting period to ensure thermal equilibrium. The SHG RA data were fitted by the coherent superposition of surface ED and bulk EQ contributions. The model used to fit the data involved identifying the relevant SHG radiation sources and their corresponding point groups. The EQ contribution, based on the point group mmm, was expected at all temperatures, while the ED contribution from the surface, related to the magnetic point group m'm2', was included at low temperatures. The fitting procedure yielded temperature-dependent coefficients for both ED and EQ contributions, revealing the behavior of surface and bulk magnetic order. The researchers also performed first-principles density functional theory (DFT) calculations to understand the microscopic origins of the observed surface magnetism enhancement. These calculations involved computing the Curie-Weiss temperature (TcW) based on calculated exchange couplings (J1-J7) from an isotropic Heisenberg spin Hamiltonian using the VASP and FPLO codes. Four structural configurations were considered: bulk CrSBr, a rigid monolayer, a fixed ab monolayer, and a free monolayer. The Hubbard U parameter was varied to check the robustness of the results. The goal was to understand the change in exchange couplings from bulk to surface, providing insights into the mechanism for the enhanced surface magnetism.

The temperature-dependent SHG measurements revealed a clear order-parameter-like upturn at 140 K, which is higher than the bulk Néel temperature (TN = 132 K). This observation strongly suggests the presence of a surface magnetic ordering transition (Ts = 140 K) at a higher temperature than the bulk ordering. The SHG RA measurements at low temperatures (T < TN) revealed two distinct SHG RA patterns, attributed to two degenerate magnetic domain states. These patterns demonstrated interference between the bulk EQ and surface ED contributions. Temperature-dependent SHG RA data, combined with magnetization measurements, confirmed the presence of three distinct temperature scales: T** = 155 K, Ts = 140 K, and TN = 132 K. The surface ED coefficient (CED), proportional to the Néel vector (N), displayed an order-parameter-like onset at Ts = 140 K and a kink at TN = 132 K. This confirmed surface antiferromagnetic ordering above the bulk ordering. The EQ coefficient (DEQ), scaling with N·N, showed a peak at Ts and a kink at TN, indicating sensitivity to spin correlations both at the surface and in the bulk. The DFT calculations revealed that two factors contribute to the enhanced surface magnetism: the absence of neighboring layers and the intra-unit cell lattice relaxation. Removing neighboring layers leads to an increase in the strongest intralayer exchange couplings (J1 and J2), and lattice relaxation further enhances these couplings (J2 and J3), thus raising the Curie-Weiss temperature. The 8 K difference between Ts and TN represents a clear experimental demonstration of a split surface and extraordinary phase transition in CrSBr.

The findings provide compelling experimental evidence for the long-predicted extraordinary phase transition, where surface magnetism sets in at a higher temperature than bulk magnetism. This counterintuitive behavior challenges conventional understanding of magnetic phase transitions and highlights the significant role of surfaces in shaping the overall magnetic properties of 3D materials. The 8 K temperature difference between the surface and bulk transitions in CrSBr is significant, demonstrating a clear separation. This highlights the importance of considering surface effects in understanding the magnetic behavior of materials, particularly in thin films or nanostructures. The DFT calculations provide a plausible mechanism explaining the enhanced surface magnetism, attributing it to the suppression of AFM-FM competition and changes in exchange interactions due to surface effects and lattice relaxation. The success of the combined SHG and DFT approach opens new avenues for investigating similar phenomena in other vdW magnets.

This study provides definitive experimental and theoretical evidence of surface and extraordinary phase transitions in the vdW antiferromagnet CrSBr. The observed 8 K temperature difference between surface and bulk transitions highlights the importance of surface effects in determining magnetic ordering. DFT calculations elucidate the mechanism behind the enhanced surface magnetism, attributed to changes in exchange interactions. Future research directions include investigating other vdW magnets with similar properties, exploring strain engineering and nonlinear phononics to tune magnetism, and examining moiré superlattices of CrSBr to explore novel magnetic properties.

The study focused on high-quality CrSBr single crystals. The presence of defects or impurities could potentially influence the observed phase transitions. The DFT calculations, while providing a plausible explanation, are not fully quantitative and might not perfectly match the experimental values. The analysis focused on a specific temperature range; investigations across a broader temperature range could provide a more complete picture.