Indirect chiral magnetic exchange through Dzyaloshinskii–Moriya-enhanced RKKY interactions in manganese oxide chains on Ir(100)

Localized electron spins can couple magnetically via the Ruderman–Kittel–Kasuya–Yosida interaction even if their wave functions lack direct overlap. Theory predicts that spin–orbit scattering leads to a Dzyaloshinskii–Moriya type enhancement of this indirect exchange interaction, giving rise to chiral exchange terms. Here we present a combined spin-polarized scanning tunneling microscopy, angle-resolved photoemission, and density functional theory study of MnO2 chains on Ir(100). Whereas we find antiferromagnetic Mn–Mn coupling along the chain, the inter-chain coupling across the non-magnetic Ir substrate turns out to be chiral with a 120° rotation between adjacent MnO2 chains. Calculations reveal that the Dzyaloshinskii–Moriya interaction results in spin spirals with a periodicity in agreement with experiment. Our findings confirm the existence of indirect chiral magnetic exchange, potentially giving rise to exotic phenomena, such as chiral spin-liquid states in spin ice systems or the emergence of new quasiparticles.


Supplementary Note 1: Temperature-dependent ARPES data
The photoemission spectra of MnO 2 chains on Ir(001) exhibit relatively broad peaks at any given k-value (Fig. 1e). Since the Ir-related bands in the very same spectra are well resolved, this cannot be an effect of insufficient statistics or resolution, but must be related to intrinsic factors, such as the overlap of many Mn-related bands, whose spectral weight changes as a function of k, and their hybridization with the substrate bands, as expected from the DFT calculations (Fig. 1f). In spite of this complexity, the Mn 3d features marked with symbols in Fig. 1e show a dispersion compatible with the 2× AFM ordering along the chains. The link between the electronic structure and the AFM ordering is reinforced in We interpret our observation as due to magnetic disorder at higher temperature, possibly coexisting with AFM short range correlations. In order to determine the easy magnetization axes of the chiral (9 × 2) spin structure of MnO 2 /Ir(001) we have performed experiments on Fe double-layer (DL) films epitaxially grown on W(110), a sample system that is particularly suited as a magnetic reference system.
(i) At high miscut from the ideal (110) orientation the terraces are relatively narrow (typically 5 to 50 nm). As shown in Supplementary Figure 3    There is only a 2 % probability that this finding is accidental, strongly supporting that the spin spirals observed across the transition metal oxide (TMO) chains are indeed chiral.

Supplementary Note 5: Theoretical analysis of the inter-chain spin interaction
At the first glance, the experimentally observed homochiral spin rotation from chain to chain looks similar to what was observed for thin Mn films on W(110) [10] where the Dzyaloshinskii-Moriya interaction (DMI) induced a spin-spiral structure with unique rotational sense. But a closer comparison shows that in the latter case the spins are spiraling in a plane formed by the surface normal and the propagation direction of the spin spiral, q, so the rotational sense is determined by the Dzyaloshinskii vector D lying in the surface plane and perpendicular to q. This mechanism is allowed by symmetry.
In the present case, the structure has a (110) mirror plane (i.e. perpendicular to the MnO 2 chains) and according to Moriya's rules [11] this dictates the absence of a D vector to each other in the chain direction and relaxing the structure again. After relaxation, the symmetry was restored, so that we can exclude this symmetry breaking as source for the observed spin structure. Note, that for all structural relaxations the generalized gradient approximation to the DFT exchange-correlation potential (like in Ref. [12]) was used.
As mentioned in the main text, a small oblique distortion of the (3×1) unit cell is observed in the STM images (also see the following Section VI of this Supplementary Material).
Although this still preserves a two-fold rotational symmetry (with rotation axis along the  also leads to the correct magnetic order of the Mn chains. In some systems, e.g. bulk Gd, such a correlation exists but the effects are also much bigger in that case [13]. Moreover, our calculations with different U parameters did not show significant changes in the results. Another complication that comes with the DFT+U calculations is that the magnetic force theorem, that simplifies the determination of the magnetocrystalline anisotropy and the dispersion of the spin-spirals, cannot be used. Therefore, all calculations had to be performed self-consistently and this limits the cutoff parameters. For the shown results a planewave cutoff of 4.0 (a.u.) −1 and 864 k-points were used. For some cases the convergence was tested with higher values, but again no strong effects on the results were found. direction (hatched black line). Along these stripes a weak corrugation can be recognized.

Supplementary
As indicated by the continuous black line these maxima are not oriented perpendicular to the stripes but significantly rotated. We note that for the same material very similar images have been obtained in Ref. [12] (see Fig. 1c and Supplementary Figure S2 therein).
To our experience this result is not unique for CoO 2 but is also observed for other TMO chains on Ir(001). For example, the STM image presented in Supplementary Figure 9