Dirac magnons in a honeycomb lattice quantum XY magnet CoTiO3

TL;DR

This study reveals Dirac magnons in the 3D honeycomb lattice magnet CoTiO3, demonstrating its potential as a model system for exploring Dirac bosons in quantum XY magnets.

Contribution

First observation of Dirac magnons in a 3D honeycomb lattice quantum XY magnet CoTiO3, highlighting strong easy-plane anisotropy and a gapless Dirac cone.

Findings

Dirac magnon dispersion with a gapless Dirac cone

Strong easy-plane exchange anisotropy observed

CoTiO3 identified as a model for Dirac bosons in 3D

Abstract

The discovery of massless Dirac electrons in graphene and topological Dirac-Weyl materials has prompted a broad search for bosonic analogues of such Dirac particles. Recent experiments have found evidence for Dirac magnons above an Ising-like ferromagnetic ground state in a two-dimensional (2D) kagome lattice magnet and in the van der Waals layered honeycomb crystal CrI$_3$, and in a 3D Heisenberg magnet Cu$_3$TeO$_6$. Here we report on our inelastic neutron scattering investigation on large single crystals of a stacked honeycomb lattice magnet CoTiO$_3$, which is part of a broad family of ilmenite materials. The magnetically ordered ground state of CoTiO$_3$ features ferromagnetic layers of Co$^{2+}$, stacked antiferromagnetically along the $c$-axis. We discover that the magnon dispersion relation exhibits strong easy-plane exchange anisotropy and hosts a clear gapless Dirac cone along the edge of the 3D Brillouin zone. Our results establish CoTiO$_3$ as a model pseudospin-$1/2$ material to study interacting Dirac bosons in a 3D quantum XY magnet.

Full text

The discoveries of graphene and topological insulators have led to significant advances in our understanding of the properties of electron in solids described by a Dirac equation. In particular, the fruitful analogy between fundamental massless Weyl-Dirac fermions in Nature and electrons in graphene or topological semimetals has allowed physicists to simulate theories of particle physics using tabletop experiments Novoselov et al. (2005); Zhang et al. (2005); Katsnelson et al. (2006); Castro Neto et al. (2009); Wehling et al. (2014); Chiu et al. (2016). Remarkably, the concept of Dirac particles is not limited to electrons or other fermionic quasiparticles, prompting a search for analogues in photonic crystals Lu et al. (2014); Ozawa et al. (2019), acoustic metamaterials Jin et al. (2018), and quantum magnets Fransson et al. (2016); Owerre (2016); Pershoguba et al. (2018); Boyko et al. (2018). In particular, Dirac magnons, or more broadly defined topological magnons Zhang et al. (2013); Mook et al. (2014); Chisnell et al. (2015); Chen et al. (2018); Yao et al. (2018); Lu and Lu (2018), have attracted much attention as platforms to investigate the effect of inter-particle interaction or external perturbations on Dirac bosons, and are proposed to be of potential interest in spintronic applications.

In contrast to light and sound, the symmetry broken states and emergent bosonic excitations of quantum magnets depend crucially on dimensionality and spin symmetry, which provides a fertile playground for examining the physics of topological bosons. To date, gapped topological magnons in Ising-like ferromagnets have been reported in a kagome lattice material Cu(1,3-bdc)Chisnell et al. (2015) and in a layered honeycomb magnet CrI3Chen et al. (2018). On the other hand, magnons exhibiting symmetry protected band crossings have been found only in a single material, a three-dimensional (3D) Heisenberg antiferromagnet Cu3TeO6 Yao et al. (2018); Bao et al. (2018). It is thus desirable to explore new test-beds with distinct spin symmetries to expand our understanding of the physics of Dirac magnons.

In this paper, we present a new model 3D quantum XY magnet realizing gapless Dirac magnons, CoTiO3, which has a simple ilmenite crystal structure. The magnetic lattice of Co2+ ions in CoTiO3 is a stacked honeycomb lattice, exactly the same as in ABC stacked graphene. Below $T_{N}\approx 38$ K, this material exhibits magnetic order with ferromagnetic planes stacked antiferromagnetically along the ${\bf c}$-axis. Our inelastic neutron scattering (INS) experiments reveal crystal field excitations and sharp low energy dispersive spin waves. Although Co2+ is a spin $S\!=\!3/2$ ion, resulting in a large magnetic signal, our analysis of the observed low energy crystal field levels provides evidence for strong easy-plane single-ion anisotropy leading to a pseudospin $\tilde{S}=\frac{1}{2}$ doublet ground state. The low energy spin-wave dispersion reveals clear crossings of magnon modes exhibiting Dirac cone dispersion in the 3D Brillouin zone (BZ). A simple model Hamiltonian with dominant nearest neighbor XY ferromagnetic (FM) exchange, and antiferromagnetic interlayer second-neighbor exchange, provides an excellent description of the magnon dispersion. In addition, the observed spin-wave intensities highlight the stark contrast between the two different transverse fluctuation modes, distinguishing this XY magnet from Ising-like or Heisenberg magnets.

Unlike the other materials mentioned above, large, pristine, single crystal samples of this material, suitable for neutron scattering and other experiments, can be grown using the conventional floating zone method. In addition, ilmenites can be grown as epitaxial thin films using conventional oxide film growth methods Varga et al. (2012), permitting future explorations of epitaxial strain and incorporation in spintronic devices.

Crystal structure, crystal field levels: As shown in Fig. 1a, CoTiO3 crystallizes in an ilmenite structure ($R\bar{3}$) that consists of alternating layers of edge-sharing CoO6 or TiO6 octahedra Newnham et al. (1964). Details of the synthesis and other experimental details are provided in the Supplemental Material. The magnetic properties of CoTiO3 are determined by the Co2+ ions which reside in a slightly buckled honeycomb layer; see Fig. 1b. The layers are ABC stacked along the $\mathbf{c}$ direction, with neighbouring honeycomb planes displaced diagonally by a third of the unit cell.

Atomically, each Co2+ in a high spin state is surrounded by a trigonally distorted oxygen environment depicted in the inset of Fig. 2b. Local electronic states in Co2+ are determined by a combination of trigonal distortion $\Delta_{\rm trig}$ and spin-orbit coupling $\lambda$. To elucidate the magnetic ground state of each Co2+ ion, we measured the crystal field excitations using high energy inelastic neutron scattering (INS). The results shown in Fig. 2a and Fig. 2b clearly reveal three transitions at $\sim 30$meV, $\sim$60meV and $\sim$130meV, together with a broad continuum between 60meV-120meV. Their intensities decrease with increasing momentum transfers as shown in Fig. 2a, which confirms their magnetic origin.

We model the crystal field levels using a single-ion Hamiltonian $H_{\rm ion}=\lambda\vec{L}\cdot\vec{S}+\Delta_{\rm trig}L_{z}^{2}$, where Hund’s coupling favors total spin $S\!=\!3/2$ and total orbital angular momentum $L_{\rm eff}\!=\!1$, and $\hat{z}\!\parallel\!{\bf c}$. We find that $\lambda\!\approx\!28$ meV and $\Delta_{\rm trig}\!\approx\!44$ meV provides a good description of the observed crystal field transitions (see Supplemental Material for details), leading to transitions at $(29,58,108,117,129)$ meV. The full electronic level diagram of a single Co2+ ion determined using these parameters is shown in Fig. 2c. The three sharp transitions observed in Fig. 2a and Fig. 2b are indicated by vertical arrows, the other two transitions could not be identified unambiguously, but seem to be buried in the broad continuum intensity.

When $\Delta_{\rm trig}\gg\lambda$, the orbital angular momentum is quenched, $L_{z}=0$, leading to a pure spin $S=3/2$ moment. SOC splits this spin-quartet via an effective single-ion term $H^{\rm eff}_{\rm ion}=\Delta S^{2}_{z}$ with $\Delta>0$, leading to a ground doublet $S_{z}\!=\!\pm 1/2$. In the opposite regime, $\Delta_{\rm trig}\ll\lambda$, SOC will lead a ground $j_{\rm eff}=1/2$ doublet and excited $j_{\rm eff}=3/2,5/2$ levels Liu and Khaliullin (2018); Sano et al. (2018). Weak $\Delta_{\rm trig}$ splits the excited levels, and mixes different $j_{\rm eff}$ wavefunctions Ross et al. (2017). With increasing $\Delta_{\rm trig}$, the ground $m_{j}\!=\!\pm 1/2$ doublet smoothly connects with the $S_{z}\!=\!\pm 1/2$ doublet in the first scenario. For the fitted values of ($\lambda$, $\Delta_{\rm trig}$), the ratio between $g$-factor parallel ($g_{\parallel}$) and perpendicular ($g_{\perp}$) to the honeycomb plane is $g_{\parallel}/g_{\perp}\!\approx\!1.6$, leading to a large anisotropy $(g_{\parallel}/g_{\perp})^{2}$ in the magnetic susceptibility, qualitatively consistent with experiment Watanabe et al. (1980). Upon projection to the ground state doublet, even a simple Heisenberg model for the $S\!=\!3/2$ spins leads to a strong easy-plane exchange anisotropy between $\tilde{S}\!=\!1/2$ pseudospins.

Ordered state and magnon dispersion: Below the Neel temperature $T_{N}\approx 38K$, Co2+ magnetic moments confined within the $\mathbf{ab}$ plane are ordered ferromagnetically within each honeycomb layer, and antiferromagnetically along the $\mathbf{c}$ direction, giving rise to the ordered structure shown in Fig. 1bNewnham et al. (1964). Energy and momentum resolved magnon spectra of CoTiO3 were obtained by inelastic neutron scattering with an incident energy, Ei=50meV. In Fig. 3, we show magnon spectra for momentum transfers along (H,-H) (Fig. 3a,b) and (H,H) (Fig. 3c) which lie within the honeycomb plane at fixed L=0.5 as well as along L at (1, 1, L) (Fig. 3d). The directions of these momentum transfers have been denoted by thick blue lines in the 2D reciprocal space map in Fig. 3j. Strongly dispersive magnon modes extending up to $\sim$12 meV are observed in all directions, which indicates the presence of significant intra-plane and inter-plane couplings in CoTiO3. Magnetic Bragg peaks are located at (1,0,0.5) in Fig. 3a and (1,1,$\pm 1.5$) in Fig, 3d consistent with earlier neutron diffraction results Newnham et al. (1964). Acoustic magnon modes are found to emanate from these Bragg peaks. As shown in Fig. 1c, the magnetic unit cell is $\frac{2}{3}$ of the conventional structural unit cell along $\mathbf{c}$; the magnon dispersion should therefore exhibit a periodicity of 1.5 in L. This is inconsistent with the L=3 periodicity apparent in Fig. 2d. As we show below, this reflects the strong XY-nature of magnetic interactions in CoTiO3.

Our data clearly show linear crossings of magnon bands at Dirac points whose positions in reciprocal space are denoted by $\mathbf{K}$ and marked with red circles in Fig. 3j. These crossings occur at $\hbar\omega^{\star}\sim$8.5meV as highlighted by the red arrows in Fig. 3b and Fig. 3c. The linear dispersions of magnon modes away from these points along (H,-H) and (H,H) are well resolved in our data. These observations unambiguously demonstrate the existence of Dirac magnons in CoTiO3. To visualize Dirac magnons in momentum space, we show slices of magnon dispersions in the (H,K) plane at the energy transfers of $\hbar\omega^{\star}$ and $\hbar\omega^{\star}\pm 1$ meV in Fig. 4. The constant energy slice at $\hbar\omega^{\star}$ in Fig. 4c shows discrete points at $\mathbf{K}$ that come from crossings of magnon modes. Due to conical dispersions near the Dirac points, a slice of magnon dispersions at an energy below $\hbar\omega^{\star}$ should show closed loops in momentum space as illustrated in Fig. 3b. This is not clear from our data due to finite energy resolution and the fact that the loop traverses three different BZs, each with a very different dynamical structure factor. However, this is clearly captured in our calculation as described below. Remarkably, the Dirac points at different L appear to merge into a nodal line in in the 3D BZ (Fig. 5a), with no discernible dispersion along $L$ in our data (Fig. 5b).

Model Hamiltonian: To understand the magnetic excitations in CoTiO3, we carried out linear spin wave calculations using a minimal model containing only nearest neighbour (NN) $\it{intra}$-plane and next nearest neighbour (NNN) $\it{inter}$-plane coupling labelled as 1, $2^{\prime}$ and $2^{\prime\prime}$ in Fig. 1b. The magnitudes of interactions along $2^{\prime}$ and $2^{\prime\prime}$ are set to be the same due to similar exchange path geometry. Using $\Delta_{\rm trig}$ and $\lambda$ determined for CoTiO3, the magnetic interactions between the $\tilde{S}$ pseudospins are expected to have strong easy-plane anisotropy (see Supplemental Material). Interactions between the pseudospins are therefore taken to be of the XXZ-type, given by $J_{\parallel}(\tilde{S}_{x,i}\tilde{S}_{x,j}+\tilde{S}_{y,i}\tilde{S}_{y,j})+J_{\perp}\tilde{S}_{z,i}\tilde{S}_{z,j}$.

The neutron data shown in Figs.  3 and Fig. 4 are described very well by this simple model Hamiltonian with just two parameters: a ferromagnetic XY exchange coupling along bond 1 shown in Fig. 1c ($J_{\parallel,1}=-4.4(9)$ meV and $J_{\perp,1}$=0 meV), and a Heisenberg antiferromagnetic exhange interaction on bonds $2^{\prime}$ and $2^{\prime\prime}$ ($J_{\parallel,2^{\prime}=2^{\prime\prime}}=J_{\perp,2^{\prime}=2^{\prime\prime}}=0.6(1)$ meV). We note that the fitting results are not changed by adding a small NN inter-plane coupling ($\lesssim$1 meV), which connects two Co2+ directly on top of each other along the $\mathbf{c}$ axis. The effect of such a NN inter-plane coupling on the magnon dispersion is likely to be negligible compared to $2^{\prime}$ and $2^{\prime\prime}$ interactions because each spin has only one out-of-plane NN but 9 NNN’s. To compare our calculation directly with the data, we calculate the magnetic inelastic intensity using the expression $I(\mathbf{Q},\omega)\propto\frac{1}{2}g_{\parallel}^{2}\mathcal{S}_{\parallel}+g_{\perp}^{2}\mathcal{S}_{\perp}$ where $\mathcal{S}_{\parallel}$ ($\mathcal{S}_{\perp}$) denotes the in-plane (out-of-plane) fluctuations of pseudospins. (See Supplemental Material for derivation of this expression and further fitting details.) Magnon spectra calculated within the linear spin wave theory is shown in Fig. 3e-h. Our calculation gives four magnon modes due to the presence of four spins in a primitive unit cell. The scattering intensity in CoTiO3 mostly comes from the two modes contributing to $\mathcal{S}_{\parallel}$ because of the larger $g_{\parallel}$ as well as stronger in-plane spin fluctuations. This provides a natural explanation for the apparent $L=3$ periodicity in Fig. 3d. The prominent magnon mode emanating from $L=\pm 1.5$ is due to the $\mathcal{S}_{\parallel}$ contribution, while the much weaker $\mathcal{S}_{\perp}$ contribution is shifted by 1.5 along $L$ with respect to $\mathcal{S}_{\parallel}$. The overall periodicity in $L$ is therefore 1.5, but the dominant intensity of $\mathcal{S}_{\parallel}$ makes it difficult to see the true periodicity. The weak magnon intensity due to $\mathcal{S}_{\perp}$ can be identified in the data in Fig. 3a and Fig. 3d, in support of our effective spin model.

The calculated magnon spectra in Fig. 3f and Fig. 3g clearly show crossings of magnon modes at the $\mathbf{K}$ points, which is consistent with results shown in Fig. 3b and c. By taking into account finite energy resolution, constant energy slices in the vicinity of $\hbar\omega^{\star}$ determined from our model (Fig. 4b,d,f) are in good agreement with Fig. 4a,c,e. In particular, the triangular loops in the constant energy slice below $\hbar\omega^{\star}$ broaden into two lines connecting bigger rings (Fig. 4b), entirely consistent with our data in Fig. 4a.

Moreover, our model reproduces the nodal line of magnon modes crossings along L which intersects the 2D BZ at $\mathbf{K}$. The nodal line observed in CoTiO3 is a magnon analogue of the Dirac Nodal Line (DNL) in 3D semimetals Chiu et al. (2016), or the symmetry protected line degeneracy of 3D electronic bands. Similar to Dirac nodal lines in 3D electronic materials without SOC, the XXZ spin model for CoTiO3 features a topological Berry phase winding which ensures the stability of the DNL observed in our data even when such perturbations due to weak further neighbour interactions are included (see Supplemental Material for further discussion). In the presence of significant nearest-neighbour interplane XXZ exchange (denoted by $J_{3}$), the DNL is no longer pinned at $\mathbf{K}$ for an arbitrary L , but starts to spiral around it. Within our momentum resolution, we could not resolve any deviation of the DNL away from the $\mathbf{K}$ points, which justifies the assumption of ignoring further-neighbour interactions in our model and linear spin wave calculations. Furthermore, given the in-plane magnetic order, the impact of the SOC induced Dzyaloshinskii-Moriya interaction also vanishes at this high symmetry point by virtue of $C_{3}$ symmetry. Since relative magnitudes between magnetic interactions can be tuned by changing the lattice structure, one might be able to drive a systematic movement of the DNL in CoTiO3 by an applied pressure. Such DNLs may be also found in other ilmenite magnets, e.g., FeTiO3 Kato et al. (1986); however, the location of DNL might vary depending on their anisotropies Goodenough and Stickler (1967).

Beyond the XXZ model: One important observation which is not captured by our XXZ model is the existence of a small magnetic anisotropy within the honeycomb plane. This is inferred from the highly non-linear magnetization at $T\!=\!5$ K for in-plane fields $\sim\!1$-$4$ Tesla, with a peak susceptibility at $\sim\!2$ Tesla, which is likely a result of rotation of magnetic domains Balbashov et al. (2017). An in-plane anisotropy also implies the existence of a small gap at the magnetic zone center in the magnon dispersion. Although our experimental resolution does not allow us to determine the gap size directly, extrapolation of the magnon dispersion in Fig. 3a suggests a gap of order $\sim\!1$ meV (See Supplemental Material). Such a gap to the Goldstone mode can arise from bond-anisotropic exchange couplings, like the Kitaev interaction, due to quantum order by disorder which pins the order parameter to the crystal axes. A phenomenological way to account for this is via a pinning field $(-1)^{z}g_{\parallel}\mu_{B}h\tilde{S}_{x}$, staggered from layer to layer, deep in the ordered phase. We find that incorporating a pinning field $\sim\!2$ Tesla, based on the magnetization data, leads to a $\sim\!1$ meV zone center gap in the magnon dispersion consistent with the above INS estimate. The Hamiltonian including such a pinning field continues to support a DNL near the $\mathbf{K}$ points. A complete theoretical study of such weak bond-anisotropic exchanges will be discussed in a separate publication Kha .

Conclusion: In conclusion, we show that the ilmenite antiferromagnet CoTiO3 is a magnon analogue of ABC stacked graphene. Our identification of the Dirac magnon phase, and possibly a magnon analogue of a Dirac Nodal Line, in CoTiO3 serves as a starting point to study transitions into other topological phases such as 3D Weyl magnons and magnon topological insulators. The ease of chemical substitution on both Co and Ti sites and high mechanical strength of CoTiO3 make it an ideal material for future studies of the impact of doping, hydrostatic pressure, strain, and magnetic fields, on gapless Dirac magnons. The close resemblance between CoTiO3 and graphene allows a direct comparison between bosonic and fermionic responses to external perturbations. For example, the Dirac magnons in CoTiO3 are an ideal model system to study emergent gauge fields through strain engineering Guinea et al. (2009), or renormalization of the magnon bands resulting from the inter-particle interaction between Dirac bosons Yelon and Silberglitt (1971); Pershoguba et al. (2018). Finally, unlike a simple Heisenberg ferromagnet, for which the electronic analogue is a simple tight-binding hopping Hamiltonian, the full model Hamiltonian for CoTiO3 contains pairing terms analogous to the Bogoliubov de-Gennes Hamiltonian Chiu et al. (2016) of a superconductor. Such unconventional superconducting phases on the layered honeycomb lattice have begun to garner great attention due to the recent discovery of superconductivity in ‘magic angle” twisted bilayer graphene Cao et al. (2018), providing further impetus to explore such remarkable analogies between electronic quasiparticles and bosonic magnons.