Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet ${\mathrm{CrI}}_{3}$

Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet ${CrI}_{3}$

Inhee Lee, Franz G. Utermohlen, Daniel Weber, Kyusung Hwang, Chi Zhang, Johan van Tol, Joshua E. Goldberger, Nandini Trivedi, and P. Chris Hammel

Phys. Rev. Lett. 124, 017201 – Published 2 January 2020

Abstract

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality ${CrI}_{3}$ single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with $K \sim - 5.2 meV$ , 25 times larger than the Heisenberg exchange $J \sim - 0.2 meV$ , and responsible for opening the $\sim 5 meV$ gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy $Γ \sim - 67.5 μ eV$ , though small, is crucial for opening a $\sim 0.3 meV$ gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a $μ eV$ -scale quadrupolar contribution to the $S = 3 / 2$ magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher $T_{C}$ as well as magnetically frustrated quantum spin liquids based on Kitaev physics.

Received 29 January 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.017201

Physics Subject Headings (PhySH)

Ferromagnetism Magnetic anisotropy Magnetism Magnetization dynamics Micromagnetism Spin dynamics

Kitaev model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Inhee Lee ^1,*, Franz G. Utermohlen¹, Daniel Weber², Kyusung Hwang^1,3, Chi Zhang¹, Johan van Tol⁴, Joshua E. Goldberger², Nandini Trivedi¹, and P. Chris Hammel^1,†

¹Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
²Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, USA
³School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
⁴National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA

^*lee.2338@osu.edu
^†hammel@physics.osu.edu

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Vol. 124, Iss. 1 — 10 January 2020

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Figure 1
(a) Optical image of the ${CrI}_{3}$ single crystal for the FMR experiment (axis of rotation shown in orange). The internal angles of the cleaved edges are multiples of 30°. The sample thickness is $\sim 35 μ m$ . (b) Schematic of the honeycomb lattice of the ${Cr}^{3 +}$ ions (dark blue) inside the iodine octahedron (upper: violet, lower: pink). Octahedral coordinate axes $x$ , $y$ , $z$ (black), FMR coordinate axes $e_{1}$ , $e_{2}$ , $e_{3}$ , and Kitaev bonds $x$ (red), $y$ (green), $z$ (blue) are indicated. (c) Pair of neighboring edge-sharing octahedra highlighting the local symmetries and the superexchange plane (blue). (d) FMR coordinate system.
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Figure 2
(a) Evolution of the FMR spectrum as $θ_{H}$ is varied, measured at 240 GHz and 5 K. Each spectrum is offset and scaled moderately for clarity. The same offset is applied for $θ_{H}$ and $180 ° - θ_{H}$ . $Δ H_{A}$ and $Δ H_{B}$ are two anisotropy features in $H_{res} . ω / γ_{Cr}$ denotes the corresponding $H_{res}$ for a free ion spin. (b) $H_{res}$ vs $θ_{H}$ obtained from (a). The marker size indicates the signal peak area in the Lorentzian fits of the FMR spectrum. The red (blue) markers and labels indicate the range of angles from 0° to 90° (90° to 180°). The solid and dashed black lines are fits calculated from Landau theory [Eq. (3)] and MFT of our model Hamiltonian [Eq. (1)], respectively. Similarly, (c)–(g) show $H_{res}$ vs $θ_{H}$ for various frequencies and temperatures.
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Figure 3
(a) Dependence of $T_{C}$ on the spin interaction parameters $J$ , $K$ , $Γ$ under the experimental constraint $J + K / 3 \equiv E_{0} = - 1.94 meV$ . (b) Dependence of $T_{C}$ on $J$ and $K$ for fixed $Γ = - 67.5 μ eV$ . In (a) and (b), $(J_{0}, K_{0}, Γ_{0})$ (filled red circles) are the values of $J$ , $K$ , $Γ$ (listed in Table 1) that fit the FMR data and the known $T_{C} = 61 K$ of bulk ${CrI}_{3}$ ; the magenta and green lines are contour lines for $T_{C} = 61$ (bulk) and $T_{C} = 45 K$ (monolayer). (c) Dependence of $T_{C}$ on the anisotropy field $H_{a}$ (and on $Γ$ ) for $Cr X_{3}$ ( $X = Cl, Br, I$ ) bulk crystals [20, 21]. The values of $H_{a}$ used are for temperatures mostly below 5 K. (d) Spin-wave dispersion calculation along the momentum-space path $\tilde{K} - \tilde{Γ} - \tilde{M} - \tilde{K}$ . The blue and red plots correspond to $(J, K, Γ) = (E_{0}, 0, Γ_{0})$ and $(J_{0}, K_{0}, Γ_{0})$ , respectively. Note that the Kitaev interaction is responsible for opening the gap $Δ_{K}$ between the bands at the Dirac point $\tilde{K}$ . We zoom in on the area in the dashed black box to show the gap $Δ_{Γ} = - 3 S Γ$ at the zero-momentum point $\tilde{Γ}$ , where $S = 3 / 2$ is the spin of the ${Cr}^{3 +}$ ions.
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Figure 4
(a) Temperature dependence of the coefficients $K_{p q}$ associated with the basic anisotropy structures shown in (c) for 120 and 240 GHz. (b) Saturation magnetization $M_{s} (T)$ obtained from SQUID magnetometry [out of plane (OP) and in plane (IP)], a MFT analysis of the FMR data, and a zero-field spin-wave theory (SWT) analysis using the values of the spin interaction constants found (listed in Table 1). In (a) and (b), the lines connecting the markers are guides to the eye. (c) Basic anisotropy structure in terms of the direction cosines $α$ , $β$ , $γ$ (the projections of the magnetization onto the $x$ , $y$ , $z$ directions). The sizes are rescaled relative to that for $α β + β γ + γ α$ with the indicated magnifications. Red (blue) denotes positive (negative) values. (d) Total anisotropy FEF $F_{L}$ for 240 GHz and 5 K constructed from Eq. (3). Orange (cyan) represents positive (negative) values. (e) Contribution of $Γ$ to the FEF, $Δ F_{Γ}$ , at 5 K. (c)–(e) are plotted with the coordinate axes $e_{1}$ , $e_{2}$ , $e_{3}$ .
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Physical Review Letters

Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI3

Inhee Lee, Franz G. Utermohlen, Daniel Weber, Kyusung Hwang, Chi Zhang, Johan van Tol, Joshua E. Goldberger, Nandini Trivedi, and P. Chris Hammel

Phys. Rev. Lett. 124, 017201 – Published 2 January 2020

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Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet ${CrI}_{3}$