Band Filling Control of the Dzyaloshinskii-Moriya Interaction in Weakly Ferromagnetic Insulators

We observe and explain theoretically a dramatic evolution of the Dzyaloshinskii-Moriya interaction (DMI) in the series of isostructural weak ferromagnets, ${\mathrm{MnCO}}_3$, ${\mathrm{FeBO}}_3$, ${\mathrm{CoCO}}_3$, and ${\mathrm{NiCO}}_3$. The sign of the interaction is encoded in the phase of the $x$-ray magnetic diffraction amplitude, observed through interference with resonant quadrupole scattering. We find very good quantitative agreement with first-principles electronic structure calculations, reproducing both sign and magnitude through the series, and propose a simplified “toy model” to explain the change in sign with $3d$ shell filling. The model gives insight into the evolution of the DMI in Mott and charge transfer insulators.

Figure 1

Local atomic and magnetic orders in the weak ferromagnets of this work. The ions of the two magnetic sublattices are represented by blue (site 1) and red (site 2) spheres, with black arrows denoting the direction of their spins. Oxygen atoms between the two adjacent transition metal layers are represented as yellow spheres. The dotted circles highlight the twist of the oxygen layer. The bottom panel shows the occupation of the 3d level of a magnetic ion. The left and right panels show the two possible magnetic configurations which stabilize depending on the $3d$ occupation and, therefore, the sign of the DMI, for a net ferromagnetic moment pointing along the magnetic field $\mathbf{H}$. ${\mathbf{S}}_{\mathrm{A}\mathrm{F}\mathrm{M}}$ denotes the direction of the antiferromagnetic spin structure.

Figure 2

$X$-ray diffraction experiment: schematic view and main results. Normalized experimental values of the diffraction intensity versus magnet angle $\eta$, for the series of weak ferromagnets. The blue curves are measured below the resonance energy and show the pure magnetic scattering intensity, which is symmetric and insensitive to the scattering phase. The red curves are on resonance and include a strong interference term that breaks the symmetry and gives the phase of the magnetic scattering, revealing the sign of the DMI. Experimental data (symbols) are shown with their fits (plain lines) against Eq. (2). For the off-resonance data, $Q(E)=0$ is enforced. The deviation from the 90°–270° symmetry axis, particularly strong in the case of ${\mathrm{CoCO}}_3$ and not explained by Eq. (2), is discussed in Supplemental Material [25].

Figure 3

Visualization of the toy tight-binding model proposed for explaining the change of DMI sign in the weak ferromagnets of this work. The filling of the energy levels in the ground state electronic configuration for $N=2$ [(a), left] and $N=6$ [(b), center] is shown. Arrows denote hopping (interatomic) and spin-orbit coupling (intra-atomic) excitations (between orbital states $n$, $n^{\prime }$, $m$, $m^{\prime }$) corresponding to the Dzyaloshinskii-Moriya interaction. [(c), right] Comparison of the (a) and (b) excitation processes reveals the difference in hoppings between excited and ground states.