Topological Spin Excitations in Honeycomb Ferromagnet CrI3

In two-dimensional honeycomb ferromagnets, bosonic magnon quasiparticles (spin waves) may either behave as massless Dirac fermions or form topologically protected edge states. The key ingredient defining their nature is the next-nearest-neighbor Dzyaloshinskii-Moriya interaction that breaks the inversion symmetry of the lattice and discriminates chirality of the associated spin-wave excitations. Using inelastic neutron scattering, we find that spin waves of the insulating honeycomb ferromagnet CrI3 (TC=61K) have two distinctive bands of ferromagnetic excitations separated by a ∼4 meV gap at the Dirac points. These results can only be understood by considering a Heisenberg Hamiltonian with Dzyaloshinskii-Moriya interaction, thus providing experimental evidence that spin waves in CrI3 can have robust topological properties potentially useful for dissipationless spintronic applications.

Two-dimensional (2D) lattices possessing three-fold rotational symmetry display a diverse range of novel electronic band properties [1].For instance, a graphene with honeycomb lattice exhibits linear electronic dispersions near the Fermi surface allowing exotic massless Dirac fermions to appear [2,3].Recently, magnetic versions of Dirac particles have been predicted in 2D honeycomb ferromagnets, which have two magnetic atoms per unit cell [Fig.1(a)] [4][5][6].The magnon (spin-wave) band structure of these ferromagnets are essentially identical to the electronic band structure of graphene with two modes, acoustic and optical spin waves, for each state reflecting two sublattices of the honeycomb lattice.Figures 1(a) and 1(b) show the real and reciprocal space of the honeycomb lattice.If the spins interact only via the Heisenberg exchange couplings, the two spin wave modes cross with each other at K/K points at the corner of the Brillouin Zone (BZ) boundary and form Dirac cones with linear dispersion [4][5][6].The presence of these Dirac points are robust against finite next-nearest neighbor exchanges, which will only shift positions of the Dirac points.Such dispersions have experimentally been observed in 2D ferromagnet CrBr 3 and Cr 2 Si 2 Te 6 [7,8], thus confirming the presence of non-degenerate band-touching (Dirac) points in the magnon excitation spectrum and leading to a massless Dirac Hamiltonian [4][5][6].
In the case of graphene, a finite spin-orbit coupling produces a small bulk semiconducting gap (∼1 µeV), leaving only the topological edge state to be truly conducting at absolute zero temeprature [9,10].A magnetic version of such topological edge states may also be realized if the antisymmetric Dzyaloshinkii-Moriya (DM) interaction opens a gap at the spin-wave crossing Dirac point [6,11].In contrast to electron spin current where dissipation can be large due to Ohmic heating, noninteracting topological magnons, which are quantized spin-1 excitations from an ordered magnetic ground state, are uncharged and can propagate for a long time without dissipation [12].Since the DM interaction will cancel out upon space inversion, a finite DM term may appear only between the next-nearest neighbors on the honeycomb lattices [see Fig. 1(a)].Whereas the symmetry does not restrict the possible orientations of these DM vectors [13], only the term collinear with magnetic moments can induce the Dirac gap.Such DM-induced topological magnons have been predicted and observed in 2D Kagome ferromagnets [14][15][16].Inelastic neutron scattering experiments on the three-dimensional (3D) antiferromagnet Cu 3 TeO 6 also reveal the presence of topological magnon band crossing at Dirac points [17,18].In honeycomb ferromagnets, it is unclear whether such topological edge magnons can exist.In fact, the topology of the next-nearest neighbor bonds on a honeycomb lattice is equivalent to the nearest neighbor bonds of a Kagome lattice.Recent experimental discoveries of intrinsic 2D ferromagnetism in van der Waals materials suggest that the topological spin excitations will probably be more robust in the honeycomb lattices [19][20][21] In this work, we use inelastic neutron scattering to map out energy and wave vector dependence of spin-wave excitations in CrI 3 , one of the honeycomb ferromagnets where topological Dirac magnons are predicted to appear [4,11].The honeycomb lattice, shown in Fig. 1(c), is essentially identical to those of another chromium trihalide CrBr 3 , in which spin-wave excitations have long been known [7].The magnetism in CrI 3 is commonly ascribed to Cr 3+ ions surrounded by I 6 octahedra, forming a 2D honeycomb network [Fig.1(c)].The CrI 3 layers are stacked against each other by van der Waals interaction, and have a monoclinic crystal structure at room temperature.Upon cooling, the monoclinic crystal structure transforms to the rhombohedral structure (space group: R 3) over a wide temperature range (100 -220 K) via lateral sliding of the CrI 3 planes with hysteresis [22].At arXiv:1807.11452v2 [cond-mat.str-el] 1 Aug 2018 Curie temperature T C = 61 K, ferromagnetic ordering appears with Cr 3+ spins oriented along the c axis [Fig.1(c)] [23].Since CrI 3 has similar structural and ferromagnetic transitions as that of CrBr 3 albeit at different temperatures, one would expect that spin-wave excitations of CrI 3 should be similar to that of CrBr 3 , which have Dirac points at the acoustic and optical spin wave crossing points [7].Surprisingly, we find that spin waves in CrI 3 exhibit remarkably large gaps at the Dirac points, thus providing direct evidence for the presence of DM interactions and topological edge magnons in CrI 3 [11].
Thin single crystal platelets of CrI 3 were grown by the chemical vapor transport method using I 2 as the transport agent [22].The grown crystals are typically 1 cm Note that the minima appear at l = 3n in the hexagonal setting.The data are normalized to the absolute units of mbarn/meV/Sr/f.u. by a vanadium standard.by 1 cm in area and extremely thin and fragile [23].Our results are reported using a honeycomb structure with inplane Cr-Cr distance of ∼3.96 Å and c-axis layer spacing of 6.62 Å in the low temperature rhombohedral structure [Fig.1(c)] [25].The in-plane momentum transfer q = ha * + kb * is denoted as q = (h, k) in hexagonal reciprocal lattice units (r.l.u.) as shown in Fig. 1(b).The temperature-dependent magnetization and neutron powder diffraction measurements confirmed that the ferromagnetic transition occurs at T C ≈ 61 K with an ordered moment of 3.0 ± 0.2 µ B per Cr 3+ at 4 K, and the magnetic anisotropy has an easy-axis along the c axis [23].To observe spin-wave excitations, we co-aligned and stacked ∼25 pieces of platelets with a total mass of ∼0.3-g.Time-of-flight inelastic neutron scattering experiments were performed using the SEQUOIA spectrometer of the Spallation Neutron Source at the Oak Ridge National Laboratory using three different incident energies of E i = 50, 25, and 8 meV [24].Neutron powder diffraction measurements were carried out using the BT-1 diffractometer of NIST Center for Neutron Research.
Figure 1(d) shows an overview of spin-wave dispersions along high symmetry directions in the (h, k) plane.A nearly isotropic spin-wave mode emerges from the Γ point at the ferromagnetic zone center and moves towards the zone boundary with increasing energy.This Goldstone mode accounts for the in-phase oscillations between the two sublattice Cr spins within an unit cell analogous to an acoustic phonon mode.For this reason, we refer to this low energy mode as the "acoustic" magnon mode.Along the [h, 0, 0] direction towards the M point [Fig.1(b)], the acoustic mode reaches its maximum energy around hω = 10 meV while another mode is visible at high energy between 15 meV and 19 meV.This high energy mode accounts for the out-of-phase oscillations between the two sublattice Cr spins, which we refer to as the "optical" mode.The large separation in energy between the two modes is consistent with the dominant ferromagnetic exchanges.Using a simple Heisenberg Hamiltonian with only the in-plane magnetic exchange couplings and without the DM interaction [11], we can fit to the overall momentum dependence of the spin-wave excitations as the solid lines in Fig. 1(d) [23].While the overall agreement of the Heisenberg Hamiltonian is reasonably good, the calculation apparently fails to explain the observed spin-wave dispersions along the [h, h, 0] direction going through the Dirac K point.As indicated by a thick arrow in Fig. 1(d), the spin-wave intensity exhibits a clear discontinuity where the acoustic and optical modes are expected to cross each other.This observation strongly suggests that magnons at the Dirac points in CrI 3 have a finite effective mass contrary to the Heisenberg-only Hamiltonian [11].
To accurately determine the spin-wave gap at Dirac points, we plot in Fig. 2 the constant-energy cuts at different spin-wave energies with an energy integration range of 1.0 meV, obtained by using the E i = 25 meV data.Since CrI 3 is a ferrromagnet, spin-wave excitations stem from the Γ point at low energy transfer about hω = 3 meV [Fig.2(f)].Upon increasing energy to hω = 6 meV, spin waves form an isotropic ring pattern around the Γ point [Fig.2(e)].At hω = 10 meV, the ring breaks into six-folded patterns concentrated around K/K points, revealing the typical Heisenberg gap at M [Fig.2(d)].When the energy transfer is further increased, the six folded pattern becomes invisible at hω ≈ 12 meV and only reappears for hω ≥ 14 meV.Consistent with Fig. 1(d), we find that the spin gap around hω ≈ 12 meV extends to the entire BZ including six equivalent Dirac points.These results suggest that the associated magnon modes obtain finite mass via interactions with each other or with additional degrees of freedom.One candidate may be the spin-wave interacting with lattice excitations (phonons).In general, dynamic spin-lattice coupling can create energy gaps or broadening in the magnon dispersion at the nominal intersections of magnon and phonon modes [26][27][28][29][30].Although CrI 3 has several phonon modes in the vicinity of the spin gap [31], the large magnitude (∼4 meV) and over the entire BZ of the gap suggest that magnon-phonon Constant-hω cuts of spin waves at selected energy transfer values within the (h, k) plane at 5 K.In all plots, data were subtracted by the empty can background, integrated over the energy range of hω ± 0.5 meV as well as integrating over −5 ≤ l ≤ 5.The dashed lines are the BZ boundaries of the 2D reciprocal lattice.In (f), Γ, K and M points are marked along the 1st BZ boundary for clarity.
coupling is unlikely to be the origin of the gap.
To qualitatively understand the observed spin-wave excitations, we fit the spin-wave spectra with the SpinW program [32].By including the DM interaction (A) in the linear spin-wave Heisenberg Hamiltonian as [11] (1) where J ij is magnetic exchange coupling of the spin S i and S j , A ij is the DM interaction between sites i and j, and D z is the DM interaction along the z (c) axis.We fit the data in the following two steps assuming a Cr spin of S = 3/2.First, we integrate the data over l to improve the statistics, and fit the spin-wave dispersions in the (h, k) planes excluding J c .Assuming that the nearest, next-nearest, and next-next nearest neighbor Cr-Cr magnetic exchange couplings are J 1 , J 2 , and J 3 as shown in Fig. 1(c), our best-fit values yield J 1 = 2.01, J 2 = 0.16, and J 3 = −0.08 meV, comparable to the density functional theory calculations [33][34][35].Remarkably, we find |A| = 0.31 meV across the next-nearest neighbors, which is larger than J 2 .We note that the value of D z = 0.49 meV obtained by this method is overestimated because there is a finite spin-wave bandwidth along the l direction (∼1.8 meV) as shown in Fig. 1(e).Both J c and D z were finally obtained by fitting the low energy modes along the l direction while fixing the in-plane exchange constants.The best fit values of these two parameters are J c = 0.59 and D z = 0.22 meV, respectively, which are significantly larger than those in CrBr 3 [7].In particular, the anisotropy term in CrI 3 is an order of magnitude larger than those of CrBr 3 .These results suggest that ferromagnetic order in CrI 3 has much stronger c-axis exchange coupling compared with CrBr 3 , although both materials are van der Waals ferromagnets [36].Since the single layer CrI 3 orders ferromagnetically below about T C ≈ 45 K [20] and not significantly different from the bulk of T C = 61 K, the magnetic ordering temperature of CrI 3 must be mostly controlled by the in-plane magnetic exchange couplings as the c-axis exchange coupling of J c = 0.59 meV in bulk is expected to vanish in the monolayer CrI 3 .
Figure 3 compares the calculated spin-wave spectra using these parameters with the experimentally observed dispersions.The left panel in Fig. 3(a) plots the calculated dispersion along the [h, 0] direction, while the right panel shows the data.Similar spin-wave calculations and observed spectra along the [h, h], [h − 1/2, h + 1/2], and [h + 1/3, −2/3] directions are shown in Figs.3(b), 3(c), and 3(d), respectively.In all cases, the spin-wave gap observed at Dirac K points are well reproduced by the calculation.The calculated spectra also reasonably reproduce the strong (weak) intensity of the acoustic (optical) spin-wave modes within the 1st BZ, which becomes weaker (stronger) in the 2nd BZ.At M points, the ferromagnetic nearest neighbor exchange couplings (J 1 > 0) of the Heisenberg Hamiltonian ensures that the acoustic spin-wave mode is always lower in energy than the optical mode.The overall dispersion along the M -Γ-M direction is not significantly affected by the DM interaction A [Fig. 3(e)].On the other hand, we see clear splitting of the acoustic and optical spin-wave modes at K points along the Γ-K-M direction due to the large DM term, which also enhances the magnon density of states at Dirac points.Figure 3(f) shows the calculated overall spin-wave dispersion including the DM interaction.
If we assume that the observed spin gap near Dirac points is due to the presence of DM interactions, spin waves in CrI 3 should have topological edge states in place of massless Dirac magnons [11,16].Such topological magnons emerge from localizes spin-wave modes forming chiral vortices, among which the handedness may be chosen via local magnetic fields.From Γ towards K, the two sublattice spins are displaced from each other along the direction parallel to the wave front.Exactly at K = (1/3, 1/3, 0), the spins of one sublattice will precess by 120 • along the direction of wave propagation.As a result, the excitations of the two sublattice spins will be decoupled from each other since the in-plane Heisenberg exchanges are frustrated.For instance, the two sublattice excitations illustrated in Figs.4(a compatible with each other.Although mutually degenerate via Heisenberg exchanges, their degeneracy is lifted when the next-nearest neighbor DM vectors are introduced [11].Interestingly, the apparent magnitude of the DM interactions in CrI 3 is larger than J 2 along the same next-nearest pair, and as large as 14% of J 1 .Given that no such feature has been observed in CrBr 3 [7], this effect must arise from the larger spin-orbit coupling in CrI 3 when Bromine is replaced by the heavier Iodine. Finally, we discuss temperature dependence of spinwave excitations in CrI 3 .As temperature is increased towards T C , the magnons gradually broaden and soften.At T = 52 K (T /T C = 0.85), the overall spin-wave spectra remain unchanged with a spin gap at Dirac points [Fig.4(b)].In the hydrodynamic limit of long wave-lengths and small q, spin-wave energy hω has the quadratic q dependence via hω = ∆(T )+D(T )q 2 , where D(T ) is temperature dependence of the spin wave stiffness and ∆(T ) is the small dipolar gap arising from the spin anisotropy [37].For a simple 3D Heisenberg ferromagnet, temperature dependence of the spin-wave stiffness D(T ) is expected to renormalize to zero at T C via [(T −T C )/T C ] ν−β , where ν = 0.707 and β = 0.367 are critical exponents [38].The blue solid and dashed lines in Fig. 4(c) show the resulting temperature dependence of D(T ) for 3D Heisenberg and 1D Ising model with ν = 0.631 and β = 0.326, respectively.For CrI 3 , magnetic critical exponent behavior was found to be 3D-like with β = 0.26 [36].While temperature dependence of D(T ) is finite approaching T C clearly different from the 3D Heisenberg or 1D Ising expectation [Fig.4(c)], the results are rather similar to D(T ) in ferromagnetic manganese oxides [39].On the other hand, ∆(T ) vanishes at T C [Fig.4(c)], suggesting that the spin anisotropy fields play an important role in stabilizing the 2D ferromagnetic ordering [35].
In summary, our inelastic neutron scattering experiments reveal a large gap in the spin-wave excitations of CrI 3 at Dirac points.The acoustic and optical spinwave bands are separated from each other by ∼4 meV, most likely arising from the next-nearest neighbor DM interaction that breaks inversion symmetry of the lattice.This leads to a nontrivial topological magnon insulator with magnon edge states, analogous to topological insulators in electronic systems but without electric Ohmic heating.These properties make CrI 3 appealing for highefficiency and dissipationless spintronic applications [12].Our analysis of the observed spin waves suggests that the DM interaction is stronger than the Heisenberg exchange coupling between the next-nearest neighbor pairs in the 2D honeycomb lattice of CrI 3 .
The neutron scattering work at Rice is supported by the US NSF Grant No.
The CrI 3 single-crystal synthesis work was supported by the Robert A. Welch Foundation Grant No.
C-1839 (P.D.).The work of J.H.C. was supported by the National Research Founda-

FIG. 1 .
FIG. 1.(a) Top view of the honeycomb lattice and the outof-plane components of the antisymmetric DM vectors.The triangular arrows mark the second nearest neighbor bond orientations that share the common sign of DM vectors.(b) The BZ zone boundaries and the high symmetry points on the (h, k) plane.(c) Crystal and magnetic structures of CrI3 with the Heisenberg exchange paths.Iodine sites are not displayed for simplicity.The dotted lines denote the hexagonal unit cell boundary.(d) Inelastic neutron scattering results of spin waves with Ei = 50 meV along the high symmetry directions in the (h, k) plane, which are observed at 5 K and integrated over −5 ≤ l ≤ 5.The superimposed solid lines are the calculations by the Heisenberg-only model.(e) Lowenergy spin-wave mode along the l direction (Ei = 8 meV).Note that the minima appear at l = 3n in the hexagonal setting.The data are normalized to the absolute units of mbarn/meV/Sr/f.u. by a vanadium standard.

FIG. 3 .
FIG. 3. (a-d) The neutron scattering intensities experimentally observed at 5 K are directly compared with the calculated intensities and dispersions discussed in the text.The experimental data on the right panels were integrated over −5 ≤ l ≤ 5.The calculations on the left panels used the best fitting parameters with Jc = 0 as discussed in the text, and convoluted by the effective energy resolution of δ(hω) = 1.8 meV considering the band with along l.The displayed cuts are described in (e) within the (h, k) plane.(f) 3D view of the 2D spin-wave excitations of CrI3.
FIG. 4. (a) The spin displacements of the two sublattice spins at the right-handed chiral wave at K points.The long arrow at the center denotes the direction of spin-wave propagation.The thin triangular arrows mark the direction of rotation for the right-handed chiral within a hexagon.(b) Spin-wave excitations at T = 52 K.The dashed lines are the calculations using the bet-fit parameters from the data at T = 5 K but with Dz set to be 0. (c) Temperature dependencies of the stiffness of the acoustic mode [D(T ), circle] and the anisotropy gap (Eg, square) obtained by fitting the data integrated over −5 ≤ l ≤ 5.For this reason, the absolute values of Eg are overestimated by the factor of ∼2.The open and closed symbols are the results obtained using Ei = 8 meV and 25 meV, respectively.The blue solid and dashed lines are expected D(T ) from 3D Heisenberg and Ising models, respectively.