Incommensurate counterrotating magnetic order stabilized by Kitaev interactions in the layered honeycomb $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$

The layered honeycomb magnet $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ has been theoretically proposed as a candidate to display unconventional magnetic behaviour associated with Kitaev interactions between spin-orbit entangled $j_{\mathrm{eff}}=1/2$ magnetic moments on a honeycomb lattice. Here we report single crystal magnetic resonant x-ray diffraction combined with powder magnetic neutron diffraction to reveal an incommensurate magnetic order in the honeycomb layers with Ir magnetic moments counterrotating on nearest-neighbor sites. This unexpected type of magnetic structure for a honeycomb magnet cannot be explained by a spin Hamiltonian with dominant isotropic (Heisenberg) couplings. The magnetic structure shares many key features with the magnetic order in the structural polytypes $\beta$- and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, understood theoretically to be stabilized by dominant Kitaev interactions between Ir moments located on the vertices of three-dimensional hyperhoneycomb and stripyhoneycomb lattices, respectively. Based on this analogy and a theoretical soft-spin analysis of magnetic ground states for candidate spin Hamiltonians, we propose that Kitaev interactions also dominate in $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, indicative of universal Kitaev physics across all three members of the harmonic honeycomb family of ${\mathrm{Li}}_2{\mathrm{IrO}}_3$ polytypes.

Figure 1

Magnetic peak at $(1,1,6)-\mathit{q}$. (a)–(c) Scans along three different reciprocal space directions (filled/open symbols are at base temperature/above $T_{\mathrm{N}}$). Solid lines are fits to a Lorentzian-squared shape [panel (b) shows a side shoulder attributed to the finite sample mosaic]. (d) Temperature dependence of the integrated peak intensity (solid line is guide to the eye). (e) Energy scan through the magnetic peak (thick blue solid symbols) showing a large resonant enhancement with a maximum at the onset edge of the fluorescence signal from the sample (black solid line, scaled). In contrast, the same energy scan through a structural peak (dotted line) shows minimum intensity near resonance (due to increased x-ray absorption). Data points in all panels are shown with an estimate of the incoherent background subtracted off.

Figure 2

(a) Schematic diagram of the $\left(hk6\right)$ reciprocal plane with filled circles, diamonds, and magenta crosses indicating positions of structural peaks, measured magnetic peaks, and the absence of peaks, respectively. Lattice points are also labeled by the magnetic basis vectors that have finite structure factor for magnetic peaks at satellite $\pm \mathit{q}$ positions. (b) Scan along the $(h,0,6)$ direction observing structural peaks at integer $h=0,2$ (intensity scaled by $3\times {10}^{-4}$ for clarity), and magnetic peaks at satellite positions $h=0\pm q,2\pm q$. Solid (red) line is the calculated magnetic scattering intensity [25] for the magnetic structure model depicted in Fig. 5. Data points are raw counts with an estimate of the incoherent background subtracted off.

Figure 3

Integrated intensity as a function of azimuth for three magnetic Bragg peaks: (a) pure-$F$ and (b) and (c) paired satellites of mixed FA character. Top diagram illustrates the scattering geometry. Data points (filled circles) are integrated peak intensities from rocking curve scans corrected for absorption and Lorentz factor. Thick (red) lines show fits that include all contributions to the MRXD structure factor [25] for the magnetic structure model $(-i{A}_x,F_y,-i{A}_z)$, depicted in Fig. 5. Blue/green curves illustrate that other phase combinations of basis vectors are ruled out.

Figure 4

Neutron powder diffraction at base temperature (5.9  K, red circles) and in the paramagnetic regime (30  K, brown circles) in the lowest-angle detector bank. The four rows of marks below the pattern show (from top to bottom) positions of structural $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ peaks, impurity phase ${\mathrm{IrO}}_2$ peaks, aluminium peaks (from the sample sachet), and magnetic Bragg peaks, respectively, and the blue line underneath represents the difference between data and fit. Insets: zoom into the large $d$-spacing region showing the fundamental magnetic peak indexed as $\left(000\right)\pm \mathit{q}$ (right panel). In all panels the solid black line shows the fit (using FullProf [30]) to the structural and magnetic contributions as discussed in the text.

Figure 5

(a) Magnetic structure in a honeycomb layer of $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ viewed along the monoclinic $\mathit{c}$ axis. Three unit cells are shown horizontally (along the propagation direction) and two vertically, with unit cell edges indicated by thin gray rectangles. The global phase of the moment rotation was chosen such as to have the magnetic moments at the origin pointing straight up along the $b$ axis. Left curly arrows illustrate counterrotation of the magnetic moments between consecutive sites along $\mathit{b}$. In unit cell 2 the light shaded ellipses show the envelopes of the moment rotation. In unit cell 3 the color coding of the bonds indicates the anisotropy axes of Kitaev exchange (black, green, red for $\mathsf{x},\mathsf{y},\mathsf{z}$, respectively). (b) Projection of the magnetic structure onto the $\mathit{a}\mathit{c}$ plane showing ferromagnetic order between adjacent layers stacked along $\mathit{c}$. The magnetic propagation vector is along the horizontal direction (${\mathit{a}}^{*}$, dashed arrow). Thin gray lines at each site give the projection of the elliptical envelopes of moment rotation. The Cartesian axes $\left(x,y,z\right)$ used to describe the magnetic moment components are shown in blue at the bottom of the figure.

Figure 6

(a) Honeycomb lattice showing the $a\times b$ unit cell (dashed rectangle). (b) Reciprocal space diagram of the honeycomb lattice showing the position of the magnetic Bragg peaks (blue stars) corresponding to the incommensurate magnetic order in $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. Labels $F$ and $A$ next to zone center positions $\mathbf{\tau }$ indicate the character of the magnetic basis vectors that can contribute to the intensity of the corresponding magnetic Bragg peaks at $\mathbf{\tau }\pm \mathit{q}$. The inner solid line hexagon is the first Brillouin zone of the honeycomb lattice, and the other symbols are magnetic Bragg peak positions for other types of magnetic structures such as Néel, “zigzag” with spins ferromagnetically aligned on the zigzag bonds and antialigned along the vertical bonds, and “stripy” with spins ferromagnetically aligned along the vertical bonds and antialigned along the zigzag bonds.

Figure 7

(a) Magnetic structure in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ projected onto the orthorhombic ${\mathit{a}}_o{\mathit{c}}_o$ plane (from Ref. [16]). (b) For comparison, the magnetic structure in one honeycomb layer of $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ is plotted for a scaled propagation vector, $f\mathit{q}$, with $f=0.89$, such that it shows the same periodicity of moment rotation as in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. For both structures left curly arrows indicate counterrotation of moments between consecutive sites along ${\mathit{c}}_o$ and both magnetic structures are plotted for three orthorhombic cells along the horizontal direction. In (a) light/dark shaded elliptical envelopes in unit cell 2 illustrate the alternation of the orientation of the plane of rotation between adjacent vertically stacked zigzag chains, whereas no such alternation occurs in the $\alpha$ magnetic structure [panel (b)]. In unit cell 3 the color of bonds indicates the anisotropy axis of Kitaev exchange with black/green/red for $\mathsf{x},\mathsf{y},\mathsf{z}$. The Kitaev axes are normal to the Ir-${\mathrm{O}}_2$-Ir bond planes and are shown by the unit vectors $\widehat{\mathsf{x}},\widehat{\mathsf{y}}$ ($\widehat{\mathsf{z}}$ into the page), as defined in Eq. (B2). In (b) the axes labels ${\mathit{a}}_m,{\mathit{b}}_m$, and ${\mathit{c}}_m$ indicate the “symmetrized” monoclinic axes defined in Eq. (B1).

Figure 8

Phase diagram of model B described in the text, computed in a soft spin approximation. We find that the observed counterrotating magnetic order ($q=0.32$, light blue line), with a uniformly tilted plane of rotation, can be captured by adding “truncated-dipole” superexchange interactions, $I_d$ and $I_c$, within the regime $|I_c|>|I_d|,I_d/K>0$. These interactions couple spins to the spatial honeycomb plane, which combines with the counterrotation due to Kitaev exchange to produce the experimentally observed magnetic order pattern. The observed spiral phase is shown in a color gradient corresponding to the magnitude of the propagation vector $q$, from 0 to 0.5 in units of $2\pi /a$. Comparing to Fig. 6, the wave vector is along the horizontal direction, with units such that $q=1$ would correspond to the white diamond symbol. Labels stripy-XY and stripy-Z denote antiferromagnetic patterns where spins are aligned with one of their three nearest neighbors, across a diagonal/vertical bond for stripy-XY/Z, respectively, and antialigned with the other two. Label C-spiral denotes an incommensurate counterrotating order with propagation vector along the $\mathrm{\Gamma }\text{−}M$ line, e.g., the vertical $({\widehat{\mathit{c}}}_o={\widehat{\mathit{b}}}_m$) direction in Figs. 6 and 6.