March 23, 2020 feature
Realizing kagome spin ice in a frustrated intermetallic compound
Exotic phases of matter known as spin ices are defined by frustrated spins that obey local "ice rules"—similar to electric dipoles in water ice. Physicists can define ice rules in two-dimensions for in-plane Ising-like spins arranged on a kagome lattice. The ice rules can lead to diverse orders and excitations. In a new report on Science, Kan Zhao and a team in experimental physics, crystallography, and materials and engineering in Germany, the U.S. and the Czech Republic used experimental and theoretical approaches including magnetometry, thermodynamics, neutron scattering and Monte Carlo simulations to establish the HoAgGe crystal as a crystalline system to realize the exotic kagome spin ice state. The setup featured a variety of partially and fully ordered states as well as field-induced phases at low temperatures consistent with the kagome experimental requisites.
Formation of exotic phases of matter can cause frustrations in spin systems. For example, local constraints in a molecule can lead to a macroscopic number of degenerate ground states or to an extensive ground state in entropy. In two-dimensional setups, ice rules require elaborate arrangements of spins on triangular shaped kagome lattices. Consequently, the kagome spin ices showed multi-stage ordering behavior under changing temperature. Physicists had thus far only experimentally realized kagome spin ices in artificial spin ice systems formed by nanorods of ferromagnets organized into honeycomb networks. In this work. Zhao et al. used multiple experimental and theoretical approaches to demonstrate the intermetallic compound HoAgGe as a naturally existing kagome spin ice with a fully ordered ground state.
The team then conducted structure and magnetometry measurements of HoAgGe. Although neutron diffraction measurements conducted in the past suggested noncollinear magnetic structures of HoAgGe—these experiments were based on powder samples that were insufficient to fully determine the magnetic structure in the presence of frustration. Zhao et al. combined neutron diffraction with thermodynamic measurements in single-crystalline HoAgGe to show its exotic temperature and magnetic-field dependent magnetic structures—consistent with the kagome ice rule. To fully determine magnetic structures from neutron diffraction based on nontrivial spin structures of HoAgGe, Zhao et al. performed single-crystal neutron diffraction experiments, down to 1.8 K. Below a high-temperature transition at 11.6 K, the team observed a magnetic peak.
When they refined the neutron data at 4 K, the team observed a more detailed magnetic structure where the fully ordered ground state indicated alternating clockwise and counter-clockwise hexagonal spins. The resulting √3 x √3 ground state precisely represented the classical kagome spin ice, as theoretically predicted. According to the kagome ice rule, the dominating nearest-neighbor ferromagnetic coupling must occur between co-planar spins with site-dependent Ising-like uniaxial anisotropy. In the present work, Zhao et al. calculated and confirmed Ising-like anisotropy of the crystalline electric field (CEF) for the HoAgGe crystals.
To further confirm the authenticity of HoAgGe as a kagome spin ice, the research team investigated if established ice rules were applicable even outside the fully ordered ground state. Using neutron diffraction under magnetic fields they showed that HoAgGe satisfied these requisites and observed an increasing magnetic field with sudden changes during metamagnetic transitions. For further information, Zhao et al. refined the magnetic structures obtained from neutron scattering and noted magnetic transitions to result from the competition between the external magnetic field and weaker couplings that do not affect the ice rule.
After establishing that the kagome ice rule applied to HoAgGe crystals at low temperature, the team examined thermodynamic behaviors of kagome spin ice by isolating the magnetic contribution to specific heat by deducing contributions from the nuclei, lattice vibrations and itinerant electrons of the crystal. To determine the extent to which Ho ionic spins of the HoAgGe crystal could be viewed approximately as Ising, Zhao et al. next discussed the crystalline electric field (CEF) effects. To directly understand CEF splitting, they conducted inelastic neutron scatting (INS) experiments of HoAgGe crystals using the advanced time-of-flight spectrometer. The results indicated four low-energy CEF modes showing Ising-type anisotropy.
Based on the experimental evidence, they proposed a classical spin model containing Ising-like in-plane spins on a 2-D distorted kagome lattice. Using Monte Carlo simulations of the classical spin model on an 18 x 18 lattice, they reproduced the ground state and partially ordered state to capture the classical spin model and the main characteristics of the HoAgGe magnetism at low temperatures. The model developed in the study differed from both dipolar and short-range kagome ice cases relative to exchange couplings and long-range dipolar interactions, with further investigations requiring a separate study.
In this way, the Monte Carlo simulations of the classical spin model only partially agreed with the experiments. The discrepancy may have resulted from multiple, low-lying CEF levels of the Ho3+ ions. In HoAgGe, the metallicity simultaneously suppressed CEF splitting of Ho3+ ions to enhance exchange coupling between them, making the two energy scales comparable to low-lying CEF levels. The resulting semi-classical model can still be mapped to an Ising model, thereby explaining the validity of the experiment. Compared to other pyrochlore spin ices, the metallic nature of HoAgGe made it a high-temperature kagome ice and may also lead to further exotic phenomena, including interactions between electric currents and magnetic monopoles as well as metallic magnetoelectric effects.
Leon Balents. Spin liquids in frustrated magnets, Nature (2010). DOI: 10.1038/nature08917
Gia-Wei Chern et al. Magnetic charge and ordering in kagome spin ice, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2012). DOI: 10.1098/rsta.2011.0388
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