Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI$_3$

TL;DR

This study combines experimental and theoretical methods to identify the dominant Kitaev interaction in CrI$_3$, revealing its role in magnetic anisotropy and stability of ferromagnetism in 2D materials.

Contribution

It provides the first quantitative determination of the Kitaev interaction strength and other anisotropies in CrI$_3$ using angle-dependent FMR and symmetry-based modeling.

Findings

Kitaev interaction is the strongest spin interaction in CrI$_3$

Kitaev interaction opens a 5 meV gap at Dirac points

Symmetric off-diagonal anisotropy stabilizes 2D ferromagnetism

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 $\Gamma \sim -67.5$ $\mu$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 $\mu$eV-scale quadrupolar contribution to the $S=3/2$ magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher $T_\text{C}$ as well as magnetically frustrated quantum spin liquids based on Kitaev physics.

Tables

Table 1. Table 1: Values of the spin and quadrupole interaction constants in the Hamiltonian for CrI 3 bulk crystals (Eq. ( 1 )) and the angle dependence of the anisotropies they generate in terms of the direction cosines α , β , γ 𝛼 𝛽 𝛾 \alpha,\beta,\gamma (compare to Fig.  4 (c)). The constants with a subscript Q 𝑄 Q are the quadrupole interaction constants described in the Supplement. The values are determined experimentally (with uncertainties of ∼ similar-to \sim 0.1%) through angle-dependent FMR and the known T C = 61 subscript 𝑇 C 61 T_{\text{C}}=61  K.

Coupling constant	Value ($\mu$eV)	Angle dependence
$J$	-212	1
$K$	-5190	1
$\Gamma$	-67.5	$\alpha \beta +\beta \gamma +\gamma \alpha$
$J_Q+K_Q/3$	2.40	${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$
${\Gamma }_Q$	-2.69	${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$, $\alpha \beta \gamma \left(\alpha +\beta +\gamma \right)$
${\Gamma }_Q^{\prime }$	-0.372	$\alpha \beta +\beta \gamma +\gamma \alpha$, ${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$, $\alpha \beta \gamma \left(\alpha +\beta +\gamma \right)$
$K_Q^{\prime }$	-0.170	${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$

Equations

$$\mathcal{H}={\mathcal{H}}_{\text{S}}+{\mathcal{H}}_{\text{Q}}-g{\mu }_{\text{B}}{\mathbf{H}}_0\cdot \sum _i{\mathbf{S}}_i,$$

$${\mathcal{H}}_{\text{S}}$$

$$+\sum _{⟨ij⟩\in \text{interlayer}}{J}_{\perp }{\mathbf{S}}_i\cdot {\mathbf{S}}_j$$

$$F_{\text{L}}=2\pi M_s^2{\cos }^2\theta +K_{21}(\alpha \beta +\beta \gamma +\gamma \alpha )$$

$$+K_{41}({\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2)+K_{42}\alpha \beta \gamma (\alpha +\beta +\gamma )$$

$$+K_{61}{\alpha }^2{\beta }^2{\gamma }^2+K_{62}({\alpha }^3{\beta }^3+{\beta }^3{\gamma }^3+{\gamma }^3{\alpha }^3)$$

$$+K_{63}\alpha \beta \gamma ({\alpha }^3+{\beta }^3+{\gamma }^3)-{\mathbf{M}}_{\text{s}}\cdot {\mathbf{H}}_0,$$

$$\Delta F_{\Gamma }=F_{\mathcal{H}}(J,K,\Gamma ,J_Q,...)-F_{\mathcal{H}}(J,K,0,J_Q,...),$$

Full text

Two-dimensional (2D) van der Waals (vdW) ferromagnets Huang et al. (2017); Gong et al. (2017) have recently emerged as an exciting platform for the development of 2D spintronic applications Jiang et al. (2018a); Klein et al. (2018) and novel 2D spin order Pershoguba et al. (2018); Liu et al. (2018a). These 2D ferromagnets must have magnetic anisotropy, since the Mermin–Wagner theorem forbids 2D materials with a continuous spin-rotation symmetry from spontaneously magnetizing at finite temperature Mermin and Wagner (1966). Understanding 2D ferromagnets thus requires a thorough knowledge of this anisotropy. However, it remains an open question which fundamental magnetic interactions correctly describe these materials and generate this anisotropy.

In this Letter we answer this question for CrI3, one of the most robust 2D ferromagnets with a $T_{\text{C}}$ of 45 K for the monolayer Huang et al. (2017). We first construct a general Hamiltonian based on its crystal symmetries containing anisotropic Kitaev $K$ and symmetric off-diagonal $\Gamma$ interactions in addition to the Heisenberg $J$ interactions. We determine the strength of these interactions using ferromagnetic resonance (FMR).

FMR provides spectroscopically precise measurements of magnetic anisotropy, magnetization, spin-wave modes, and damping Farle (1998); McMichael et al. (2003); Lee et al. (2010). The structure of the magnetic anisotropy of a given material can be obtained from angle-dependent FMR by measuring the change in the resonance field as the direction of the external field $\mathbf{H}_{0}$ is varied Farle (1998). At 2 K, CrI3 single crystals have a $\sim$3 T anisotropy field $H_{\text{a}}$ oriented normal to the layer plane McGuire et al. (2015); Dillon and Olson (1965). This large $H_{\text{a}}$ results in a resonance frequency of at least $\omega/2\pi\sim 100$ GHz in an out-of-plane field. We performed angle-dependent FMR using a heterodyne quasi-optical electron spin resonance spectrometer van Tol et al. (2005). The measurement was implemented at $\omega/2\pi=120$ and 240 GHz and at $T=5$–80 K. The angle $\theta_{H}$ between $\mathbf{H}_{0}$ and the $e_{3}$-axis normal to the sample plane (see Fig. 1(d)) is varied by rotating the thin CrI3 single crystal plate about the axis indicated by the orange line in Fig. 1(a). A representative example of the FMR spectra for different $\theta_{H}$ at 240 GHz and 5 K is shown in Fig. 2(a).

The resonance field $H_{\text{res}}(\theta_{H},\omega,T)$, plotted in Fig. 2(b)–(g), shows two distinct anisotropy features as $\theta_{H}$ is varied, which we label $\Delta H_{\text{A}}$ and $\Delta H_{\text{B}}$ in Fig. 2(a): $\Delta H_{\text{A}}$ is the shift in $H_{\text{res}}$ from the free ion contribution $\omega/\gamma_{\text{Cr}}$, where $\gamma_{\text{Cr}}$ is the gyromagnetic ratio of Cr3+, and $\Delta H_{\text{B}}$ is the difference in $H_{\text{res}}$ between $\theta_{H}$ and $180^{\circ}-\theta_{H}$. These anisotropy features are crucial to understanding the magnetic behavior of CrI3 and are central to our symmetry-based theoretical analysis.

In order to analyze the anisotropies measured in FMR and determine the microscopic exchange interactions, we begin by writing the most general Hamiltonian allowed by the symmetries of a monolayer with undistorted CrI6 octahedra: the crystal lattice is globally invariant under (i) time reversal, (ii) 120∘ rotations about the $e_{3}$-axis at each Cr3+ ion, (iii) Cr–Cr-bond-centered spatial inversion, (iv) 180∘ rotations about the Cr–Cr bonds, and (v) locally invariant under 180∘ rotations about the axis perpendicular to a Cr–Cr bond’s superexchange plane.

Based on these symmetries, we obtain the general Hamiltonian:

[TABLE]

where

[TABLE]

describes the spin–spin interactions, $\mathcal{H}_{\text{Q}}$ describes the quadrupole–quadrupole interactions (see Supplement), $\mathbf{S}_{i}$ is the spin-3/2 operator for the Cr3+ ion at site $i$, $-g\mu_{\text{B}}\mathbf{H}_{0}\cdot\sum_{i}\mathbf{S}_{i}$ is the Zeeman coupling, $g$ is the g-factor of Cr3+, $\mu_{\text{B}}$ is the Bohr magneton, and $J_{\perp}$ is the interlayer Heisenberg coupling Narath (1965). $\braket{ij}\in\lambda\mu(\nu)$ denotes that the Cr3+ ions at the neighboring sites $i,j$ are interacting via a $\nu$-bond, where $\lambda,\mu,\nu\in\{x,y,z\}$.

We next determine the spin interaction parameters in the Hamiltonian. From the resonance field $H_{\text{res}}(\theta_{H},\omega,T)$ we determine the value of $J+K/3=-1.94$  meV, which appears as a combination in mean field theory (MFT) and determines how quickly $\Delta H_{\text{A}}$ and $\Delta H_{\text{B}}$ shrink with increasing temperature; and $\Gamma=-67.5$ $\mathrm{\SIUnitSymbolMicro}$eV, which determines the size of $\Delta H_{\text{A}}$ at low temperatures. The detailed fitting procedure is described in the Supplement. From the switching field $\sim$0.6 T in bilayer CrI3 Huang et al. (2017); Jiang et al. (2018a, b) we estimate $|J_{\perp}|\sim 0.03$ meV, which is negligible compared to $J+K/3$. Remarkably, the high spectroscopic precision of FMR also enables us to estimate the $\mathrm{\SIUnitSymbolMicro}$eV-scale quadrupole interaction constants (listed in Table 1), which give rise to $\Delta H_{\text{B}}$ in Fig. 2(a). The calculated $H_{\text{res}}$ and $M_{\text{s}}(T)$ are in reasonable agreement with the data at all temperatures and frequencies (Fig. 2(b)–(g)). From the known $T_{\text{C}}=61$ K of bulk CrI3, we then determine the value of $K=-5.2$ meV, which automatically fixes the value of $J=-0.2$ meV (see Fig. 3(a)).

A key finding of our analysis is that the Kitaev interaction is the dominant interaction in CrI3, almost 25 times stronger than the Heisenberg interaction. A strong signature of this Kitaev interaction in CrI3 is the $\sim$5 meV Dirac gap ($\Delta_{K}$) at $\tilde{K}$ in the spin-wave dispersion, as shown in Fig. 3(d), which is corroborated by a recent inelastic neutron scattering experiment Chen et al. (2018). Furthermore, in the absence of the Kitaev interaction, $T_{\text{C}}$ is incorrectly estimated to be 100 K (Fig. 3(a)).

It is important to note that Kitaev anisotropic exchange interactions arise naturally for 2D honeycomb networks of edge-sharing octahedrally-coordinated transition metals, as found in CrI3 and discussed previously in $A_{2}$IrO3 ($A=\text{Na},\text{Li}$) Singh et al. (2012); Gretarsson et al. (2013) and $\alpha$-RuCl3 Banerjee et al. (2016). Electrons from a transition metal (TM) cation can hop to a neighboring TM cation via their shared ligands $X$ along two pathways (see Fig. 1(c)) Jackeli and Khaliullin (2009); Rau et al. (2014); Kim et al. (2015). In the presence of strong spin–orbit coupling (SOC) on either the cation, ligand, or both, the destructive interference between competing exchange pathways produce Kitaev interactions and weaken the Heisenberg interaction Stavropoulos et al. (2019). Even though the Kitaev interaction leads to frustration, the spin moments in CrI3 are large ($S=3/2$), so quantum fluctuations are not strong enough to produce a quantum spin liquid state.

We next construct a Landau free energy functional (FEF) to map out the various magnetic anisotropies in CrI3 and further connect the coefficients of the Landau FEF to the exchange interaction constants. The Landau FEF based on the underlying symmetries up to sixth order in the direction cosines $\alpha,\beta,\gamma$ (the components of the saturation magnetization $\mathbf{M}_{\text{s}}$ along the $x,y,z$ directions) (Fig. 1(b)) is given by van Vleck (1937); Zener (1954); Ascher (1966):

[TABLE]

where $2\pi M_{\text{s}}^{2}\cos^{2}\theta$ is the shape anisotropy, $\theta$ is the angle between $\mathbf{M}_{\text{s}}$ and the $e_{3}$-axis (Fig. 1(d)), and $K_{pq}(\omega,T)$ are the coefficients associated with the magnetocrystalline anisotropies plotted in Fig. 4(c). The FEF determines the resonance condition Eq. (S4) of $\omega$ and $H_{\text{res}}(\theta_{H},\omega,T)$ (see Supplement). The values of the $K_{pq}(\omega,T)$ that fit the data are shown in Fig. 4(a), and the corresponding fits are shown in Fig. 2(b)–(g).

We map out the total Landau FEF $F_{\text{L}}$ shown in Fig. 4(d) using the $K_{pq}$ obtained at 5 K for 240 GHz. We find that the uniaxial term $F_{\text{L,21}}=K_{21}(\alpha\beta+\beta\gamma+\gamma\alpha)$ is the dominant anisotropy in CrI3, having $F_{\text{L,21}}(\theta=90^{\circ})-F_{\text{L,21}}(\theta=0^{\circ})\sim 220$ $\mathrm{\SIUnitSymbolMicro}$eV/Cr (corresponding to $H_{\text{a}}\sim 2.5$ T), which primarily accounts for the large $\Delta H_{\text{A}}$ in Fig. 2(a). The higher-order anisotropy terms ($K_{4q}$, $K_{6q}$) in Fig. 4(c) account for the small shift $\Delta H_{\text{B}}$ since they are not symmetric about the film plane.

By combining the microscopic spin interaction and Landau theory approaches, we can provide insight into the magnetic anisotropy produced by each interaction in the Hamiltonian (Eq. (1)). For example, for the $\Gamma$ interaction we look at the free energy difference

[TABLE]

plotted in Fig. 4(e), and compare its angular structure to that of the anisotropies associated with the $K_{pq}$ coefficients in the Landau FEF (plotted in Fig. 4(c)). We find that $\Gamma$ is mainly responsible for the large uniaxial anisotropy in CrI3 associated with $K_{21}$ underlying the $\Delta H_{\text{A}}$. It also plays the crucial role of stabilizing ferromagnetism in a CrI3 monolayer by opening a $\sim$0.3 meV gap ($\Delta_{\Gamma}$) at the zone center $\tilde{\Gamma}$ in the spin-wave spectrum (see Fig. 3(d)). The much smaller quadrupole terms generate the higher-order anisotropy terms associated with $K_{4q}$ and $K_{6q}$ underlying the $\Delta H_{\text{B}}$. Even though $J$ and $K$ generate no magnetic anisotropy, from the MFT estimate $k_{\text{B}}T_{\text{C}}^{\text{MFT}}=-\frac{5}{4}(3J+K+2\Gamma)$ we see that they determine the scale for $T_{\text{C}}$ since they are much larger than $\Gamma$.

Our model also describes the relation between the anisotropy field $H_{\text{a}}$ and $T_{\text{C}}$ for the chromium trihalides ($X=\text{Cl},\text{Br},\text{I}$). By inferring their values of $\Gamma$ using the low-temperature relation $H_{\text{a}}\simeq-3S^{2}\Gamma/(M_{\text{s}}V_{\text{Cr}})$ obtained from MFT, where $V_{\text{Cr}}$ is the volume per Cr3+ ion in CrI3, we can compare the predicted $T_{\text{C}}$ vs. $\Gamma$ relation using the values of $J$ and $K$ obtained for bulk CrI3 to the known values of $T_{\text{C}}$ and $H_{\text{a}}$ for bulk Cr$X_{3}$ (see Fig. 3(c)) McGuire (2017); Kuhlow (1982). We note that although the prediction curve agrees closely with the data for CrCl3 and CrBr3, this does not imply that they have the same $J$ and $K$ as CrI3; in fact, we expect $K$ to be much weaker in CrCl3 and CrBr3 since Cl- and Br- have weaker SOC than I-.

Given that CrI3 has a $T_{\text{C}}$ of 61 K for bulk crystals and 45 K for a monolayer, we can speculate on the changes in the values of the spin interaction constants $J$, $K$, and $\Gamma$ that might occur upon exfoliation. A reduction in the strength of one of these interactions by a factor of 2–3 or of several interactions by a smaller amount, perhaps as a result of crystal distortions, would lower $T_{\text{C}}$ by the appropriate amount (see Fig. 3(a) and (b)). FMR studies on monolayer CrI3 are needed to explore this further.

In conclusion, our symmetry-based theoretical analysis of angle-dependent FMR measurements of single crystal CrI3 has revealed strong Kitaev interactions in honeycomb CrI3, almost 25 times larger than the standard Heisenberg exchange, that open a $\sim$5 meV gap at the Dirac points in the magnon dispersion, our prediction that was recently corroborated by an inelastic neutron scattering study of CrI3 Chen et al. (2018). Such Kitaev interactions arise naturally in edge-sharing octahedra due to SOC and the interference of exchange pathways. We also found a small anisotropic $\Gamma$ exchange that generates the large magnetic anisotropy in CrI3, opens a gap at the zone center, and stabilizes ferromagnetic long-range order in 2D. This is in contrast to previous studies, which have used Ising anisotropy Lado and Fernández-Rossier (2017); Liu et al. (2018a, b); Klein et al. (2018); Zheng et al. (2018); Jin et al. (2018) or single-ion anisotropy Narath (1965); Gong et al. (2017); Xu et al. (2018); Chen et al. (2018) to explain this large magnetic anisotropy; however, the former is not allowed by the crystal symmetries of CrI3, whereas the latter is estimated to be too small Lado and Fernández-Rossier (2017) due to the weak SOC on the Cr3+ ion. Our work also provides insight needed to devise new 2D materials with properties ranging from high-$T_{\text{C}}$ magnetism to quantum spin liquid states.

Angle-dependent FMR and our symmetry-based analysis can readily be applied to other 2D materials in order to correctly characterize their magnetic interactions. In particular, we propose performing these FMR measurements on the $S=1/2$ Kitaev material $\alpha$-RuCl3, which like CrI3 has Kitaev, Heisenberg, and $\Gamma$ interactions, but whose interaction constants are still hotly debated Winter et al. (2017).