Orbital character of the spin-reorientation transition in TbMn6Sn6

Abstract

Ferromagnetic (FM) order in a two-dimensional kagome layer is predicted to generate a topological Chern insulator without an applied magnetic field. The Chern gap is largest when spin moments point perpendicular to the kagome layer, enabling the capability to switch topological transport properties, such as the quantum anomalous Hall effect, by controlling the spin orientation. In TbMn₆Sn₆, the uniaxial magnetic anisotropy of the Tb³⁺ ion is effective at generating the Chern state within the FM Mn kagome layers while a spin-reorientation (SR) transition to easy-plane order above T_SR = 310 K provides a mechanism for switching. Here, we use inelastic neutron scattering to provide key insights into the fundamental nature of the SR transition. The observation of two Tb excitations, which are split by the magnetic anisotropy energy, indicates an effective two-state orbital character for the Tb ion, with a uniaxial ground state and an isotropic excited state. The simultaneous observation of both modes below T_SR confirms that orbital fluctuations are slow on magnetic and electronic time scales < ps and act as a spatially-random orbital alloy. A thermally-driven critical concentration of isotropic Tb ions triggers the SR transition.

Fig. 1. Temperature dependence of the low-energy spin excitations in TbMn₆Sn₆.

The evolution of the low-energy ferrimagnetic spin waves, plotted as χ^″(q, E)/E. Inelastic neutron scattering (INS) data are shown along the a (H, 0, 1) and b (H, 0, 2) directions at T = 100, 150, 200, 240, 300, and 350 K through the spin-reorientation (SR) transition at T_SR = 310 K. Insets at higher energy in a and b correspond to data along (H, 0, 3). At 100 K, the dashed and solid lines correspond to the ground state dispersion as obtained from fits to the Heisenberg model described in Ref. . Calculations of the spin dynamics in the random-phase approximation (RPA) are shown along the c (H, 0, 1) and d (H, 0, 2) directions at the same temperatures as the data. In a and b in the SR-phase at 350 K, the dashed and solid lines correspond to the RPA dispersions obtained using the Heisenberg parameters and a magnetically isotropic Tb ion. Thinner lines at other temperatures are guides to the eye.

Fig. 2. Temperature dependence of the spin gap in TbMn₆Sn₆.

a The evolution of the acoustic mode intralayer spin wave dispersion along q = (H, 0, 2) at T = 100, 150, 200, 240, 300, and 350 K through the spin-reorientation (SR) transition at T_SR = 310 K. The dashed line at T = 100 K corresponds to the ground state dispersion as obtained from fits to the Heisenberg model described in Ref. . b The inelastic neutron scattering spectra at (0,0,2) are plotted as χ^″(q, E)/E for several temperatures up to (solid lines) and above T_SR (dashed lines). c INS spectra at (1,0,2) for several temperatures. d The temperature dependence of the spin gap energy at (0, 0, 2) and (1, 0, 2) as obtained from the maxima in panels b and c. The dashed lines in d correspond to random-phase approximation (RPA) calculations in the uniaxial phase. Observation of dynamical features in the hashed area are obscured by the instrumental elastic resolution. Error bars are statistical in origin and represent one standard deviation.

Fig. 3. Temperature dependence of the zone-boundary spin excitations in TbMn₆Sn₆.

a The evolution of the upper and lower Tb spin wave modes along the (1/2, 0, L) direction at T = 100, 150, 200, 240, 300, and 350 K through the spin-reorientation (SR) transition at T_SR = 310 K. Inset shows data along the (3/2, 0, L) direction. b The inelastic neutron scattering data at (1/2,0,3) as a function of energy and temperature with intensities plotted as χ^″(E). The filled symbols indicate that the energy of the upper mode drops in a similar fashion to the spin gap. c The integrated susceptibility of the upper Tb mode as a function of temperature compared to random-phase approximation (RPA) calculations. d, e The structure factor of the lower mode at 6 meV along (1/2, 0, L) at 300 K and 350 K, respectively. The solid red lines are calculations of the dipole factor times the Tb magnetic form factor for transverse excitations with m_Tb∥[0,0,1] (d) and m_Tb⊥[0, 0, 1] (e) averaged over equivalent domains. Sharp excitations seen at L = ± 6 in panels a, d, and e are phonons. Error bars are statistical in origin and represent one standard deviation.

Fig. 4. Crystal field model for TbMn₆Sn₆.

a Crystal electric field (CEF) energy levels for a J = 6 Tb ion with dominant $B_2^0$ and $B_4^0$ terms. The orange arrow indicates the experimental values with $B_2^0=-0.0347$ meV and $B_4^0=-0.00143$ meV. b The Zeeman splitting of the CEF levels due to the molecular field acting on the Tb ion. The vertical line shows the estimated ground state molecular field strength of $B_{MF}^{\mathrm{Tb}}=22.2$ meV and the blue arrow labels the only dipole-allowed transition out of the ground state. Orbital states are labeled as $∣m_J⟩$. c Tb state populations as a function of temperature, where the molecular field decreases according to the measured Mn sublattice magnetization. T_SR indicates the spin-reorientation transition.

Fig. 5. Quantum alloy model for TbMn₆Sn₆.

a Random-phase approximation (RPA) calculations of the excitations in the spin-reorientation phase for an isotropic Tb ion with $B_2^0=B_4^0=0$. b Instantaneous configuration of a quantum alloy on the Tb triangular lattice. Magnetically uniaxial (Ψ_u) and isotropic (Ψ_i) quantum states of the Tb ion are split by the magnetic anisotropy energy (2K₁/J). c Classical Monte Carlo simulations of a quenched mixed anisotropy alloy with uniaxial fractions f = 1, 1/2, and 0.