Non-collinear spin states in bottom-up fabricated atomic chains

Non-collinear spin states with unique rotational sense, such as chiral spin-spirals, are recently heavily investigated because of advantages for future applications in spintronics and information technology and as potential hosts for Majorana Fermions when coupled to a superconductor. Tuning the properties of such spin states, e.g., the rotational period and sense, is a highly desirable yet difficult task. Here, we experimentally demonstrate the bottom-up assembly of a spin-spiral derived from a chain of iron atoms on a platinum substrate using the magnetic tip of a scanning tunneling microscope as a tool. We show that the spin-spiral is induced by the interplay of the Heisenberg and Dzyaloshinskii-Moriya components of the Ruderman-Kittel-Kasuya-Yosida interaction between the iron atoms. The relative strengths and signs of these two components can be adjusted by the interatomic iron distance, which enables tailoring of the rotational period and sense of the spin-spiral.

The chains were built with several nanometer separation (as indicated, the white scale bar has a length of 2 nm) in order to minimize mutual interactions between the chains (V s = −6 mV, I stab = 0.5 nA). Nevertheless, the two end atoms in each chain are not completely equivalent considering their distances to the atoms of the neighboring chains: the left end atom in each chain has a slightly larger distance to the atoms of the neighboring chains than the right end atom. Taking into account the residual magnetic interactions between neighboring chains, this leads to a tiny residual shift of the excitation energy which is different for the two end atoms, as seen in Figure 1a of the main manuscript. This effect is more pronounced for the Fe 4 (atoms 1-4) and Fe 3 (atoms 5-7) chains as compared to the Fe 2 (atoms 8,9) pair, as they have a smaller separation. However, note, that these residual couplings are negligible as compared to the couplings within each chain, and will therefore not affect our conclusions on the non-collinear spin states.

Supplementary Note 1 | Labeling and coordinate system
In order to clarify the coordinate system used within the main manuscript text, a sketch of all directions and vector components is shown in Supplementary Figure 1. Here, a chain built from three Fe atoms is placed on top of the Pt(111) surface with a mutual distance of three lattice sites (8.31Å). The gray coordinate system on the top right defines the labeling of the components of all vectors mentioned. Figure 2 | Sketch of directions. The schematic ball model shows a chain consisting of three Fe atoms on a Pt(111) surface to explain the used nomenclature. The different directions are defined by the gray coordinate system and are adapted from a previous publication 1 . The components of the magnetic field B, the spin S and the Dzyaloshinskii-Moriya vector D are labeled accordingly.

Supplementary Note 2 | Additional ISTS measurements of 4a chains
In addition to the measurements of the Fe 4 chain shown in Figure 2 of the main manuscript, the spectra of the remaining 4a chains (atoms 5 -10) are shown in Supplementary Figure 3. The complementary colorplot representation of the spectra is given in Supplementary Figure 4 and the corresponding calculations using third-order perturbation theory 2 are plotted in Supplementary Figure 5. The different effects of the Heisenberg and Dzyaloshinskii-Moriya interaction on the spectra, and that we can indeed distinguish them, is shown in Supplementary Figure 6.
norm. dI/dV (a.u.)    norm. dI/dV (a.u.)  b Calculated ISTS from the perturbation theory model using the same parameters as in Figure 2 and Table 1

Supplementary Note 3 | Additional SPSTM & -ISTS measurements of 3a chains
An STM topograph taken at 3 T of all investigated 3a chains is shown in Supplementary Figure 7. It was recorded with a magnetically sensitive tip resulting in the alternating apparent heights of the atoms, i.e. atoms 2, 4, 5, 7 and 9 are higher than atoms 1, 3, 6 and 8. This height contrast is consistent with the calculated Ŝ -component of the atoms' magnetization (Supplementary Figure 12 and 13) assuming a sensitivity of the used tip to the ||-component of the atom magnetization.

Supplementary Note 4 | Calculated magnetization curves
To estimate the states of all individual spins within the investigated chains, magnetization curves were calculated using exact diagonalization (ED) (see methods section in main manuscript). Supplementary Figure 11 shows the results for the Fe 4 chain with d NN = 4a. The pseudo-vectors given in Figure 2a of the main text were calculated from the spin-expectation values at B z = 3 T of these magnetization curves. For the 3a chains, the resulting magnetization curves of the Fe 3 and the Fe 4 chain are plotted in Supplementary Figure 12 and 13. Driven by the magnetic field pointing in z-direction, the expectation values Ŝ z of all atoms behave almost the same. Therefore, this component of the spin expectation values cannot be responsible for the alternating magnetic contrast which is observed in the SP-ISTS data in Supplementary Figure 8 and 9 as well as in Figure 3 of the main manuscript. In contrast, Ŝ indeed reveals alternating signs, which can explain the experimentally observed magnetic contrasts. For example, at a positive field of about 5 T, Ŝ of atom 5 is positive, zero for atom 6 and negative for atom 7. This behavior can explain the different conductance levels at positive bias and large magnetic field in Supplementary Figure 9d, e and f if we assume that tip has a sensitivity to the ||-component of the magnetization. Similarly, in the case of the Fe 4 chain, the sign of Ŝ alternates from atom to atom, which can explain the alternating conductance levels in Figure 3c and d of the main manuscript. We conclude, that the used magnetic tip shows a certain amount of sensitivity to the ||-component of the magnetization, which enables to image the non-collinear states of the spin chains.

Supplementary Note 5 | SPSTS images of the Co stripes
In order to prove that the magnetic tip used for the measurements of Figure 4 of the main manuscript has a sensitivity to the out-of-plane component of the sample magnetization, and that the magnetic orientation of the Co stripe the Fe 16 chain is coupled to can be reversed, spin-polarized dI /dV images recorded at B z = 0 T are shown in Supplementary Figure 14. The imaged area contains two Co stripes (left and right) and the top part of a Co island (middle), which are known to have a remanent magnetization perpendicular to the surface 3  In the image on the top, the left Co stripe appears bright, while the right stripe as well as the island appear darker. The contrast difference is ascribed to an opposite out-of-plane magnetization of the left and right stripe imaged by the tip with a sensitivity to the out-of-plane component of the sample magnetization. In the following, we define the magnetization state of the bright stripe as state 0, and that of the darker stripe as state 1. Next, the magnetic field is ramped to B z = −1.7 T and back to 0 T. The spin-resolved image recorded afterwards of the same area (bottom image) reveals a dark contrast on both Co stripes and the island. The contrast change from bright to dark of the left stripe is consequently assigned to a reversal of its magnetization to state 1. Note, that we cannot fully exclude a magnetization reversal of the tip and all magnetic structures but the left stripe, as this would lead to the same behavior of the magnetic contrasts in the images. However, this possibility is rather unlikely. Most importantly, even in this case, the relative orientations of the tip and Co stripe magnetizations between the two images are still reversed. Therefore, the analysis of the data and all conclusions made in the main manuscript remain valid.

Supplementary Note 6 | System-size dependence of the DMRG calculation
Supplementary Figure 15 shows the system-size dependence of the spin structure factor Ŝ z (q) , calculated with DMRG and normalized to the area. One observes that a chain length of N = 16 used in the experiment is already sufficient for a well-defined peak in Ŝ z (q) that does not change much with increasing size. With N = 128, one already obtains the converged result, such that considering larger systems is not necessary. This is due to the expectation values Ŝ z,i and Ŝ ,i approaching zero for sites i far away from the local B z -field at the first site.  Figure 16 shows the spin-structure factors Ŝ z (q) calculated for chains with the first spin pinned along z and the different coupling parameters which are experimentally accessible using hcp Fe atoms with the mutual distances d investigated in our previous publication 1 . The coupling parameters, corresponding wavevectors extracted from the maxima of the curves, and the resulting spin-spiral wavelengths λ = 2π /q are given in Table 1. For the chains with antiferromagnetic Heisenberg exchange (J NN < 0) q varies only weakly in the range of 3π /4d NN < q < π /d NN . Nevertheless, due to the variation of the NN distance, λ can be tuned over a rather large range. Even larger wavelengths are expected for the chains with zero and positive (ferromagnetic) Heisenberg exchange. For the purely DM driven spin-spiral (J NN = 0) q is exactly π /2d NN corresponding to a 90 • rotation of neighboring spins. The largest wavelength is achieved for the NN distance with ferromagnetic Heisenberg exchange (J NN > 0).  Figure 16 | DMRG calculated spin-structure factor Ŝ z (q) for chains with different couplings. The spin-structure factors have been calculated assuming the experimentally accessible Heisenberg and DM couplings in chains with NN distances d NN given in the legend. The corresponding coupling parameters and the resulting q and λ are given in Supplementary Table 1.