New insight into tuning magnetic phases of RMn₆Sn₆ kagome metals

The study investigates how magnetic order in RMn6Sn6 kagome metals can be tuned via competing interlayer exchange interactions and magnetic anisotropies to control topological electronic states (Dirac fermions, Chern gaps). Kagome lattices host flat bands and Dirac crossings that enable correlated and topological phenomena. In RMn6Sn6 (R = Sc, Y, Gd–Lu), Mn kagome layers are ferromagnetic and interlayer couplings produce either ferrimagnetic (FIM) order (for magnetic R) or spiral order (for nonmagnetic R). The research question centers on identifying the microscopic magnetic interactions and anisotropies in ErMn6Sn6 that determine its magnetic phases and their temperature/field evolution, and whether Er participates in the high-temperature spiral state. The work aims to clarify the mechanism behind the transition between a ferrimagnetic phase and a spiral phase and to map the phase stability near the critical boundary between collinear and noncollinear orders, enabling tuning of spin orientation and spin chirality relevant to topological responses.

Prior studies established RMn6Sn6 as a platform linking magnetism and topology, with strong Mn kagome ferromagnetism and R-dependent interlayer couplings controlling magnetic order. Nonmagnetic R (e.g., Y) exhibit double-spiral order and field-induced noncoplanar textures producing topological Hall and anisotropic magnetoresistance. Magnetic R can impose strong anisotropy that orients Mn spins, enabling Chern insulating behavior and temperature/field-driven spin reorientation (e.g., Tb, Dy, Ho). Earlier reports suggested that in ErMn6Sn6 the Er sublattice decouples above the spiral transition, leaving a Mn-only double spiral. Splitting of magnetic satellite peaks and higher-harmonic satellites have been observed in related materials (YMn6Sn6 and Ga-substituted variants), but their origins remain debated, involving possible domain effects, beating, anisotropy-induced bunching, or amplitude modulation. This study revisits these interpretations with combined neutron diffraction and mean-field modeling for ErMn6Sn6.

• Crystal growth and characterization: Single crystals of ErMn6Sn6 grown from excess Sn flux. Phase purity verified by x-ray diffraction.
• Magnetization: Measured on a Quantum Design MPMS from 1.8 K to 7 T, with field orientations including H ∥ c. Background subtraction using measurements of the mounting disc.
• Single-crystal neutron diffraction: Performed on the Fixed-Incident-Energy Triple-Axis Spectrometer at the High-Flux Isotope Reactor (ORNL). Incident neutrons λ = 2.377 Å selected by double-bounce PG monochromator and PG analyzer (PG 002). Söller collimations 40′-40′-40′-80′; PG filters after each monochromator to suppress higher-order contamination. Samples (109.0 mg and 282.6 mg) aligned with (h,0,l) plane in the scattering plane; temperature control via He closed-cycle refrigerator in exchange gas below 300 K or vacuum above.
• Powder neutron diffraction: Time-of-flight measurements on POWGEN (SNS, ORNL) using 6.874 g of powder obtained by grinding single crystals; high-resolution configuration; sample in 6 mm vanadium can; He closed-cycle refrigerator and PAC used.
• Single-crystal x-ray diffraction: Four-circle diffractometer with Cu Kα1 radiation and Ge(111) monochromator at Ames National Laboratory; sample on Cu mount in He closed-cycle refrigerator; Be domes; small He exchange gas.
• Rietveld refinement: FULLPROF used to refine powder data at T = 200 K and extract spiral parameters and ordered moments.
• Mean-field modeling: Magnetic Hamiltonian includes isotropic exchange between Mn–Mn (intralayer J_MM^⊥ and interlayer J_MM^∥ terms) and AFM Mn–Er coupling J_ME; Zeeman term for Mn (g = 2) and Er (g_J = 6/5, J = 15/2, g_J J = 9); Er crystal-electric-field anisotropy (Stevens operators with B_20, B_22, B_40, B_60) producing weak uniaxial anisotropy; Mn easy-plane anisotropy H_Mn = K^Mn(m_x^2 + m_y^2). Parameter set (meV): B_22 = 0.012, B_20 = −3.69×10^−1, B_40 = 0, B_60 = 1.47×10^−2, K^Mn = 0.17, J_MM^⊥ = −28.8 (FM), J_MM^∥ terms −4.4, −19.2, +2.3, and J_ME = 1.35. Free energy minimized over spin angles θ_Er, θ_Mn, and spiral angles Φ and δ; initial simplification ignores Er planar anisotropy for uniaxial fields. Identification of phases: FIM-ab, FIM-c, vertical-plane-canted (VP-canted), planar spiral, vertical-conical-spiral (VCS), and forced ferromagnetic (FF). Additional analysis of six-fold planar anisotropy via B_60 and an effective planar anisotropy constant K3′ to test for lock-in or fan-like phases.

• Magnetic phase sequence: With cooling, ErMn6Sn6 transitions from a high-temperature distorted triple-spiral phase to a planar ferrimagnetic (FIM-ab) phase via a first-order transition at T_spiral = 92(1) K. The Néel temperature is T_N = 348(1) K.
• Er participation in spiral: Contrary to previous reports of Mn-only double-spiral above T_spiral, neutron diffraction and mean-field modeling indicate Er moments remain ordered and participate, yielding an ideal triple-spiral reference state distorted in practice.
• Spiral parameters and ordered moments (200 K, powder refinement): τ = 0.1876(6), δ = 14.0(2)°, μ_Er = 3.9(3) μ_B, μ_Mn = 2.0(1) μ_B.
• Higher-harmonic satellites: Observed 2τ and 3τ magnetic satellites around (0,0,2); absence of corresponding peaks in x-ray diffraction confirms magnetic origin. Even-harmonic (2τ) peaks indicate distortion of the ideal triple-spiral order; not attributable to fan-type order or Er planar anisotropy.
• Lineshape splitting: Primary magnetic satellites split into multiple Gaussian components. Around room temperature, three components yield Δτ_⊥ ≈ 0.03; at lower T, two components remain with temperature-dependent splitting, intrinsic to the material. Unequal splittings for τ and 2τ (Δτ_⊥^2 = 1.5 Δτ_⊥^1) support magnetic origin and single-domain beating hypothesis.
• Interactions and anisotropy: Parameter set reproduces temperature evolution of the spiral angle Φ and the small free energy difference between FIM-ab and triple-spiral states, placing ErMn6Sn6 near the critical boundary between spiral and ferrimagnetic orders. Mn exhibits easy-plane anisotropy (K^Mn ≈ 0.17 meV); Er has weak, temperature-dependent uniaxial anisotropy arising from CEF.
• Thermal fluctuations of Er: Increasing temperature induces strong thermal fluctuations of Er moments, which weaken effective Mn–Er coupling and quench Er anisotropy. Planar magnetic anisotropy energy is reduced by ~95% by T_spiral and is negligible above ~150 K.
• Field-driven transitions (H ∥ c): First-order magnetization process (FOMP) rotates spins from FIM-ab to FIM-c with small critical field at low T (μ0H_FOMP ≈ 0.65 T at 2 K), increasing to ~4 T near T_spiral. Above T_spiral, a VCS phase emerges with gradual canting; Er thermal fluctuations raise H_FOMP and suppress the FOMP at higher T.
• High-field predictions (mean-field): At low T, a VP-canted phase appears at μ0H ≈ 23 T (Er spin flop), followed by a transition to forced FM at ≈ 58 T (Er spin flip). The FIM-c to FF crossover occurs at ≈ 47 T where the ordered Er moment is fully quenched due to cancellation of exchange and Zeeman energies.
• Topological implications: Control of FIM-c vs FIM-ab via small fields (FOMP) enables tuning of Chern gaps; triple-spiral phases with tunable periodicity offer routes to explore vector spin chirality effects in transport and optics.

The combined neutron diffraction and mean-field analysis resolves the magnetic structure and phase stability of ErMn6Sn6 near the boundary between spiral and ferrimagnetic orders. Demonstrating Er participation in the high-temperature spiral directly addresses prior assumptions of Er decoupling and explains the temperature-dependent shortening of the spiral period via thermal suppression of Er moment and anisotropy. The observation of even and odd higher-harmonic satellites and lineshape splittings indicates intrinsic distortions of the triple-spiral state not accounted for by Er planar anisotropy, pointing to competition among interlayer exchange interactions and anisotropy that favors non-sinusoidal and long-period modulations. The temperature evolution of the FOMP and emergence of a VCS phase align with the calculated polar magnetic anisotropy: quenching of Er anisotropy with temperature strengthens net easy-plane behavior, altering phase boundaries. The predicted high-field Er-flop and Er-flip transitions clarify how Er’s CEF landscape governs metamagnetism. These insights provide a framework for magnetic tuning to manipulate Dirac fermions, Chern gaps, and real-space spin chirality across RMn6Sn6 by controlling interlayer coupling and anisotropy.

This work maps the temperature- and field-dependent magnetic phases of ErMn6Sn6 and establishes a microscopic model capturing the competition among Mn–Mn and Mn–Er interlayer exchanges and Er/Mn anisotropies. Key contributions include (i) identification of a distorted triple-spiral phase with Er participation above T_spiral, (ii) quantification of interlayer exchange and anisotropy parameters reproducing observed phase transitions and magnetization, and (iii) predictions of high-field Er spin-flop/flip transitions and a complete quenching of Er moment at the FIM-c to FF crossover. The results show that thermal fluctuations of Er moments are central to phase stability, offering a handle to tune collinear, noncollinear, and noncoplanar phases relevant to topological phenomena. Future research should determine the microscopic origin of the spiral distortions and even-harmonic satellites, explore transport/optical signatures of vector spin chirality in triple-spiral states, and extend the parameterized Hamiltonian across the RMn6Sn6 series to engineer desired magnetic-topological phases.

• The origin of the distorted triple-spiral order and the even-harmonic (2τ) satellites remains unresolved; fan-type order and Er planar anisotropy are disfavored but not a complete explanation.
• Lineshape splitting (multi-component satellites) is intrinsic but its precise microscopic mechanism (domains vs single-domain beating) is not fully determined.
• Mean-field treatment may overestimate thermal fluctuations of Er spins and neglects fluctuations beyond mean-field and possible coupling to itinerant electrons.
• Initial modeling simplifies Er planar anisotropy; while later considered via K3′, quantitative effects on distortions near T_spiral remain difficult to assess.
• The study focuses on magnetic structure; direct measurements of electronic topology (e.g., ARPES, magneto-optical spectroscopy) and transport under controlled magnetic textures were not performed here.