A gateway towards non-collinear spin processing using three-atom magnets with strong substrate coupling

A cluster of a few magnetic atoms on the surface of a nonmagnetic substrate is one suitable realization of a bit for spin-based information technology. The prevalent approach to achieve magnetic stability is decoupling the cluster spin from substrate conduction electrons in order to suppress destabilizing spin-flips. However, this route entails less flexibility in tailoring the coupling between the bits needed for spin-processing. Here, we use a spin-resolved scanning tunneling microscope to write, read, and store spin information for hours in clusters of three atoms strongly coupled to a substrate featuring a cloud of non-collinearly polarized host atoms, a so-called non-collinear giant moment cluster. The giant moment cluster can be driven into a Kondo screened state by simply moving one of its atoms to a different site. Using the exceptional atomic tunability of the non-collinear substrate mediated Dzyaloshinskii–Moriya interaction, we propose a logical scheme for a four-state memory.


Supplementary Note 1 | Determination of the giant moment cluster structure
In order to determine the internal structure of the GMCs, we assembled several individual fcc and hcp atoms close to the built GMCs, as shown in the top of Supplementary Fig.1 exemplarily for one of the GMCs. The stacking type (fcc, hcp) of the individual Fe atoms is unambiguously identified by their characteristic ISTS signature 1 . Thereby, we can put a lattice of Pt surface atoms with assigned fcc and hcp adsorption sites on top of the GMCs topography ( Supplementary Fig.1), and conclude that it contains either three fcc or three hcp atoms on nearest neighboring adsorption sites. Moreover, the GMC is not exactly round shaped, but slightly has the form of a downwards pointing triangle. We therefore conclude that this GMC is an fcc top cluster. The same procedure leads to the identification of the internal structure of the three other GMCs, which are shown schematically in the bottom of Supplementary Fig.1. For each of the four configurations, the vertical relaxation of the trimer was calculated using the QUANTUM-ESPRESSO package 5 , imposing a relaxation criterion whereby the vertical force exerted on individual Fe atoms and Pt atoms of the surface layer is <10 −4 Ry a.u. −1 . The computational modeling of the system was done employing the repeated slab approach considering 86 atoms per unit cell, an energy cutoff of 40 Ry, a Gamma-point reciprocal-space mesh and ultrasoft fully relativistic pseudopotentials. In the hcp hollow and fcc hollow configurations, the trimer relaxes vertically 17.5% towards the surface (0% corresponds to the ideal interlayer separation in bulk, a/ √ 3 = 2.26Å), and the height of the Pt surface layer remains approximately constant. In contrast, in the hcp top and fcc top configurations, one of the surface Pt atoms is shifted with respect to the rest of the Pt surface layer, as schematically illustrated in Supplementary Figure 2. Note that this particular Pt atom lies underneath the center of the Fe trimer and is therefore the only substrate atom that is a nearest neighbor of every Fe adatom. Finally, total energy calculations predict that the fcc and hcp hollow configurations are the energetically most stable ones (the calculated energy difference between the two is below our numerical precision), followed by fcc top (35 meV/adatom higher) and hcp top (45 meV/adatom higher).
Magnetic moments, orbital moments and exchange parameters The magnetic moments obtained using the KKR-GF approach are summarized in Supplementary Table 1. In all cases, we find that the magnetic moments of individual Fe adatoms are close to 3.5 µ B , while the surrounding Pt atoms have a substantial total contribution that ranges between 1.4 and 2.2 µ B depending on the cluster type. It is particularly noteworthy that in the hcp top and fcc top configurations, the Pt atom with 10% vertical distance height (see previous subsection and Supplementary Figure 2) becomes strongly spin-polarized by 0.23 µ B , a factor 3 larger than its closest Pt neighbors. Finally, our calculations show a substantial orbital magnetic moment of both Fe adatoms and the Pt cluster. The calculated magnetic exchange interaction parameters of the Fe trimer are summarized in Supplementary Table 2. For completeness, we have included calculations performed in the hcp top and fcc top structures where the single Pt surface atom underneath the center of the trimer was not shifted with respect to the rest of surface Pt atoms; we add a label 'c. h.' (constant height) to denote this structures. The MAE, denoted by K, has been calculated ab initio following the magnetic force theorem 6 . The rest of the parameters have been evaluated based on the generalized Lichtenstein formula [7][8][9] . Due to the C 3v symmetry of the system the three Dzyaloshinskii-Moriya (DM) vectors are fully specified by two parameters, D and D ⊥ (see Suppl. Figs. 3 and 4): In addition, we have employed a scheme 8  find that all the calculated J's shown in Supplementary Table 2 are fairly large and negative, thus favouring a ferromagnetic coupling between the spins of the Fe trimer. Moreover, the renormalization induced by the Pt substrate has the net effect of increasing the magnitude of the J's by approximately 20-30%, thus favouring even more the ferromagnetic coupling. In comparison, the DM interaction is nearly one order of magnitude smaller, except for the case of the hcp top structure, both in the normal and c. h. case. It is noteworthy that the J/D ratio varies by nearly 30%, thus revealing a large effect induced by the single shifted Pt atom.
Non-collinear spin structure Our DFT calculations have found a close competition between two almost ferromagnetic, slightly non-collinear spin configurations for the four trimers. In one of them, the magnetic moments point mainly along the out-of-plane direction with a non-collinearity polar angle θ as illustrated in Supplementary  Table 2. This figure shows that the nearly out-of-plane configuration is the most stable one for hcp top, fcc hollow and fcc top structures. In the particular case of hcp top, we note that it is necessary to consider the structure with the single shifted Pt atom in order to obtain the correct orientation of the magnetic moments that is in accordance with experimental measurements. Furthermore, we emphasize the importance of the energy minimization induced by the non-collinearity for the hcp top cluster; even though the ferromagnetic alignment favors the in-plane direction by nearly 1.5 meV/adatom, the energy gained by the non-collinearity reverses the trend and favours instead a nearly out-ofplane orientation with large canted angles of approximately θ = 17 • by ∼ 0.1 meV/adatom. In comparison, the fcc top and fcc hollow configurations show a small non-collinearity angle θ < 4 • . On the opposite side, the favoured configuration for the hcp hollow cluster type is the one with  The opening of the non-collinearity angles can be related to the exchange parameters by making use of a classical Heisenberg model: wherem i = (cos ϕ i sin θ i , sin ϕ i sin θ, cos θ i ), i = 1, 2, 3 are the magnetization unit vectors of the adatoms arranged as in Supplementary Figures 3 and 4. Note that within the present convention J < 0 (J > 0) gives rise to a ferromagnetic (antiferromagnetic) coupling, and K < 0 (K > 0) favors out-of-plane (in-plane) orientation of the magnetic moments.
We now compute an analytic solution for the direction of the spin moments illustrated in Supplementary Figures 3 and 4. Let us begin with the nearly in-plane configuration and thus set the azimuthal angle of all adatoms to ϕ i = 0, i.e. the direction of the spin-moments arê m i = (sin θ i , 0, cos θ i ), i = 1, 2, 3. The three contributions to the Hamiltonian of Eq. 4 are then given by: H K = K cos 2 θ 1 + cos 2 θ 2 + cos 2 θ 3 .
We seek next for solutions near θ i = π/2 by using sin θ i = sin(π/2 + x i ) = cos x i ∼ 1 − x 2 i /2, cos θ i = cos(π/2 + x i ) = − sin x i ∼ −x i , allowing to write the full Hamiltonian as The x i that minimize this Hamiltonian are obtained from the solution of three coupled equations ∂H/∂x i = 0, yielding The corresponding total energy for the nearly in-plane configuration is then given by Given that J is negative (ferromagnetic coupling), this energy is also negative.
We next turn to the nearly out-of-plane configuration illustrated in Supplementary Figure 3 by fixing the azimuthal angles ϕ 1 = 30 • , ϕ 2 = 150 • and ϕ 3 = 270 • and setting the same polar angle for the three adatoms:m Then, the three parts of the Hamiltonian can be expressed as (after straightforward algebra in the J and DM parts): We next assume small deviations sin θ ∼ θ, cos θ ∼ 1−θ 2 /2, allowing to write the full Hamiltonian as The solution of ∂H/∂θ = 0 is In all cluster types studied in this work the hierarchy |J| |D |, |D ⊥ |, |K| is fulfilled. Therefore, the angles predicted by Eqs. 10 and 19 reveal a larger impact of non-collinearity on the nearly out-of-plane configuration than in the nearly in-plane configuration by a factor of approximately 2/ √ 3 ∼ 1.15 , which is in reasonable accordance with the ab initio band calculations shown in Supplementary Figure 5.
Inserting Eq. 19 into Eq. 18 one obtains the total minimum energy of the nearly out-of-plane configuration, Thus, the energy difference between Eq. 11 and Eq. 20 gives the relative energy gain induced by non-collinearity between the nearly in-plane and out-of-plane configurations: The above equation shows that, in absence of MAE (K ∼ 0), E out − E in < 0 as long as the coupling is ferromagnetic, i.e. the nearly out-of-plane direction is favoured by non-collinearity. Interestingly, even if K > 0 (MAE favours in-plane orientation), the overal total energy may still favour the nearly out-of-plane configuration provided 9D 2 /4J > |3K| is fulfilled.

Supplementary Note 3 | Fano functions
In order to extract the magnetic field dependent shift and temperature induced broadening of the Kondo resonance (Fig.3 of the main manuscript), we fitted the measured voltage dependent spectra dI dV (V ) to the following sum of two Fano functions 10 including a linear background: Here, q is the so-called form factor, Γ is the full width at half maximum and E i are the energetic positions of the Kondo resonances. The parameters extracted by fitting such functions to the experimental data are given in Supplementary Tables 3 and 4, and the two fitted Fano functions for the case of the magnetic field dependence are shown in Supplementary Fig.6. Note, that the resulting q-factors are negative, indicating strong phase shifts between the two tunneling paths 11 .
Supplementary Table 3 | Parameters for magnetic field dependent Fano functions. Parameters used for the fits of the B dependent spectra of the hcp-hollow cluster (Fig.3a of main manuscript). Parameters used for the fits of the T dependent spectra measured on the hcp-hollow cluster (Fig.3b of main manuscript). In order to estimate the Kondo temperature T K , the resulting temperature dependence of Γ was fitted to the power law 12 Γ (T ) = 1 2 (αk B T ) 2 + (2k B T K ) 2 resulting in T K = 4.19 K (α = 11.8) which is illustrated as a grey line in Fig.3d together with the data. Additionally, we compare to published numerical renormalization group calculations for a spin-1/2 impurity in the strong coupling regime 13 , given by the grey dots in Fig.3d, resulting in a similar value of T K = 4.64 K. The associated d 2 I/dV 2 curves (right panels) have peaks and dips at the corresponding bias voltages. This behavior is the fingerprint of a spin excitation of an out-of-plane easy axis system 1 . Note, however, that the shape of the features in the spin excitation spectra partly deviates from the usual step-like appearance typically found for magnetic atoms which are more strongly decoupled from a metallic substrate 14,15 . This might indicate deviations from the simple effective-spin model where the magnetic impurity is artificially separated into an interior part (effective spin) that is excited by the tunneling electrons, and an exterior part that interacts with the effective spin leading to damping of the spin excitations. In the present system, this border is not well defined due to the GMC character of the magnetic impurity, which can lead to excitation spectra that strongly deviate from the simple step shape 16 . We extract the corresponding excitation energies ∆ 01 by searching the center of the symmetric peaks/dips in the d 2 I/dV 2 curves, which are marked by crosses, both in the dI/dV as well as in the d 2 I/dV 2 curves. The extracted excitation energies ∆ 01 are plotted in Supplementary Fig.8 as a function of B. Indeed, there is a linear behavior, as expected from the out-of-plane system. By fitting a linear function to these plots, we extracted the g-factors via g = 1/µ B · d∆ 01 (B)/dB. The accordingly determined parameters ∆ 01 and g of the three GMCs are given in Supplementary Table 5. Here, we distinguish between the values determined from the excitation on the negative (neg.) and positive (pos.) bias voltage sides, which allows us to estimate the error, which is given in the Table of