Quantifying exchange forces of a spin spiral on the atomic scale

The large interest in chiral magnetic structures for realization of nanoscale magnetic storage or logic devices has necessitated methods which can quantify magnetic interactions at the atomic scale. To overcome the limitations of the typically used current-based sensing of atomic-scale exchange interactions, a force-based detection scheme is highly advantageous. Here, we quantify the atomic-scale exchange force field between a ferromagnetic tip and a cycloidal spin spiral using our developed combination of current and exchange force detection. Compared to the surprisingly weak spin polarization, the exchange force field is more sensitive to atomic-scale variations in the magnetization. First-principles calculations reveal that the measured atomic-scale variations in the exchange force originate from different contributions of direct and indirect (Zener type) exchange mechanisms, depending on the chemical tip termination. Our work opens the perspective of quantifying different exchange mechanisms of chiral magnetic structures with atomic-scale precision using 3D magnetic exchange force field measurements.


Supplementary Note 1: Sample preparation and magnetic characterization of the tip
Supplementary Fig. 1 shows an overview image of the Mn monolayer on W(110) acquired with a mainly out-of-plane sensitive tip. The Co adatoms adsorb at the hollow sites of the locally nearly antiferromagnetic c(2x2) unit cell (see Fig. 1(a) in the main article) and couple ferromagnetically to the nearest underlying Mn atoms 1 . The orientation of the magnetic moments of Co adatoms adsorbed at hollow sites of neighboring rows within the spin spiral (black box in Supplementary Fig. 1) differs by an angle of about 173°. It was recently shown that the appearance and symmetry of the Co adatoms, as observed in SP-STM, strongly depends on the orientation of their magnetic moments with respect to the magnetization of the tip 2 . For a parallel out-of-plane alignment between the magnetic moments of the tip and the Co adatom, an oval protrusion is observed in SP-STM images, while the Co adatom appears as a dumbbell shape along [11 # 0] for an anti-parallel out-of-plane alignment 1 . For magnetic tips with an in-plane magnetization, the appearance of the Co adatom in SP-STM images is again significantly different 2 . We use of these different appearances of Co adatoms in SP-STM images to characterize the magnetic sensitivity of the tip. We laterally manipulated at least two Co adatoms (manipulation parameters Vs = -2 mV and IT = -30 nA) to neighboring atomic hollow-site rows along [11 # 0] at the location with maximum contrast of the spin-spiral period. We monitored the appearances of the Co adatoms in our SP-STM, and compared them to the expected shapes and symmetries 2 . We continued with gentle dipping of the tip into the Mn layer, as described above, until we observed the expected appearances for an out-of-plane magnetization of the tip. To make sure that we have achieved out-of-plane magnetic sensitivity, we moved the Co adatoms afterward by one lattice constant and monitored if we see the expected change of their appearance in SP-STM images. Using this procedure, we can ensure that all tips used for the SPEX measurements have a dominant out-of-plane magnetic sensitivity. However, we cannot exclude a small misalignment of the magnetization of the tip from the out-of-plane direction that is smaller than the difference between the magnetic moments of Mn atoms in neighboring rows of the spin spiral (~173°). For that, an external magnetic field would be required to provide a better magnetic alignment.

Supplementary Note 2: Experimental details of constant-height imaging and raw data
Prior to the acquisition of constant-height images, an overview constant-current image ( Supplementary   Fig. 1) was always measured in order to ensure that the area is atomically flat and no Co adatoms or other adsorbates as well as defects is present within at least 1 nm around the area of interest of the spin spiral. In case of only Co adatoms present in the area, they were moved away by lateral manipulation.
In order to reduce the creep from the piezo scanner, repeated close-up constant-current images at setpoint Vs = -10 mV and IT cc = -2 nA with zmod = 50 pm were acquired until no lateral creep could be discerned anymore between the images. We used the Δf values at this feedback setpoint to characterize the bluntness of our tips, which were typically between -10 Hz and -17 Hz). Afterwards, the current feedback loop was switched off at z0 at the position of parallel alignment between the magnetization of the tip and the Mn spin spiral (cross in Supplementary Fig. 2(a)). Vs was reduced to about |Vs| = 0.1 mV, and the tip was approached toward the surface by -0.29 nm (corresponding to z1) at which the constant-height imaging was started with a scan speed of 0.35 nm/s. While scanning, we recorded the constant-height frequency shift Δf ch , the constant-height current IT ch , as well as the oscillation amplitude zmod and the excitation voltage Vex. In certain cases, we did not determine the tilt angle of the surface with respect to the tip precisely enough. This misalignment results in a linear change of Δf ch with respect to, e.g., the [11 # 0] direction of a few Hz at z1 between = 0° and = 180° of the spin spiral (see Supplementary Fig. 2(b)). However, on top of this linear offset, we observe variations of the magnitude of Δf ch reflecting the chemical interaction of the tip with the Mn atoms as well as the magnetic exchange interaction, due to the reversal of the out-of-plane projection of the surface magnetization within the spin spiral (see main article Fig. 1(c) and Supplementary Fig. 2(b)).

Supplementary Note 3: Assignments of top and hollow sites
It is not straightforward to determine the top and hollow sites from the contrast in constant-height Δf images that reflect the averaged force gradient at a fixed tip-sample distance. Instead, we compare the total measured F(Δz) at different sites within the c(2x2) unit cell with calculated force-distance curves. . Δd is defined as d-d0, where d0 = 0.5 nm and d is the tip-sample. Separation (cf. Supplementary Fig. 8).

Supplementary Note 4: Details on distance-dependent measurements
In order to quantify the forces between the tip and the spin spiral, distance-dependent measurements on the top (ti and t'i) and hollow (hi and h'i) sites of the c(2x2) unit cell were performed. First, a constant-height image with atomic resolution of the spin spiral was obtained to resolve the top and hollow sites of the c(2x2) unit cell (Supplementary Note 2 and Supplementary Fig. 2). The tip-sample distance was increased to zs (where zs = z0 + 50 pm) and the tip was moved laterally to the measurement position. Then, the tip was brought closer to and subsequently retreated from the surface by Δz ≤ 360 pm in steps of ~ 0.8 pm while recoding the frequency shift Δf, the current IT, as well as the oscillation amplitude zmod and the excitation voltage Vex at each step for 50 to 100 ms. To account for possible lateral piezo creep, we repeatedly acquired data in an alternating pattern at ti /t'i and hi /h'i for three to five times, and averaged the data at equivalent positions. To exclude vertical piezo creep, we compared the frequency shift Δf0 at z0 before and after measurements at the four atomic sites (ti, t'i, hi, h'i). If the variation of Δf0 was smaller than the noise in Δf (i.e. ±0.1 Hz), we considered the vertical creep to be negligible. In addition, we averaged the data from the forward and backward sweeps. The total force F (Δz) was extracted from Δf (Δz) by utilizing the formula in ref. 3 : with zmod = 50 pm, B = 30.8 kHz the resonance frequency and k = 1800 N/m the stiffness of the qPlus sensor 4 . Supplementary Fig. 4 shows the raw data (Δf14, F14, IT,14) for the data shown in the main article Fig. 2 at t14 and h14. Note that the total forces (Supplementary Fig. 4(b)) range up to a few nN. We observe a clear difference in the total forces on the different atomic sites (t14 and h14). We refer to Supplementary Note 3 for the definition of the top and hollow sites. Interestingly, no clear difference in the currents can be resolved ( Supplementary Fig. 4(c)). Our data shows that the forces are more sensitive to the atomic-scale configuration than the current.

Supplementary Note 5: Distance-dependent measurements along half the spin spiral
We perform MExFIS along half a period of the spin spiral as sketched in Supplementary Fig. 5(a).
Prior to the data acquisition, the tip magnetization was prepared to have dominantly out-of-plane magnetization using the procedure described in Supplementary 5(b,c)) as well as Fex,zc(i) and Azc(i) ( Supplementary Fig. 5(d,e)). We observe the expected gradual decrease of the absolute values for Fex,zc(i) and A zc(i) from i = 1 and i = 7 ( Supplementary Fig. 5(d,e)).
In addition, Fex,zc(1) exhibits a different sign than Fex,zc(14), which is expected considering the reversal of the magnetic moments of the spin spiral between = 0° and = 180°. However, we observe indications that the magnetization of the tip has changed during the data acquisition. There is an abrupt change between Fex,zc(12) and Fex,zc(13), which we interpret as a change of the magnetization of the tip due to large magnetic exchange interactions with the surface and the softness of the magnetic tip.
Furthermore, a few curves show a behavior as characterized by the tip type in Supplementary Fig. 7(a), in addition to the tip type Supplementary Fig. 7(d). Therefore, we excluded the data for i = 7 to i = 14 from our discussion. To avoid spontaneous changes of the tip magnetization, magnetic tips with a larger magnetic remanence are required.  Supplementary Fig. 7(a), as well as a small peak in Vex(Δz) at Δz = -320pm in Supplementary Fig. 7(b). We interpret the variations in Vex(Δz) as the presence of inelastic processes.
However, SP-STM images acquired afterwards do no indicate a change of the magnetization nor a change of the geometric structure of the tip. We therefore attribute these findings to small fluctuations of the tip magnetization at small tip-sample distances. For the curves in Supplementary Fig. 7(c), the slope of Δf (Δz) and IT (Δz) changes drastically for Δz < -340 pm, together with a sudden increase in Vex (Δz). We interpret these observations as structural relaxations between the tip and the surface, probably due to an adsorbate at the tip apex. In a few occasions, we observed a short spike in Δf(Δz) which we assign to vibrational/acoustic noise of the cryogenic UHV system ( Supplementary Fig. 7(e)). underneath. In case of the hollow sites, the p and ap alignments were defined with respect to the magnetic moments of the nearest neighbor (nn) surface Mn atoms (cf. Supplementary Fig. 8(b)). The tip was approached toward the surface through a trajectory of discrete points, and for every point we a vacuum of at least 20 Å along the vertical direction between the base of tips in two adjacent unit cells.

Supplementary Note 8: Impact of structural relaxations
The left panel of the Supplementary Fig. 9 shows the relative tip apex displacement (Fe apex tip) compared to the unrelaxed position due to tip-surface interaction. As expected, the displacement is larger at the hollow site as the tip apex atom has much space to move along the z-direction compared to the top position. The right panel of Supplementary Fig. 9 shows the comparison of the exchange energy and the exchange force before and after the geometry relaxation due to tip-sample interaction (Fe apex tip). Our results indicate that the qualitative behavior of the exchange energy and the  Supplementary Fig. 8). exchange force will remain similar even after we consider geometry relaxation due to tip-sample interaction.

Supplementary Note 9: Charge density difference plots
We have calculated the spin-resolved charge density difference Δ at a change of tip-sample distance of Δd = -0.1 nm (see Supplementary Fig. 10, with Δd=d-d0, where d0 = 0.5 nm and d is the tipsample separation) in order to understand the origin of the exchange interaction between the tip and the sample. The charge density differences are calculated using the following formula The plots for the majority and the minority channels are similar to each other with a reversal of charge accumulation and depletion at the tip apex atom and the interacting surface for both channels. This kind of charge distribution is the characteristic for a short-range direct exchange mechanism between the d orbitals due to the formation of spin-dependent covalent bonds 7, 8 which favor antiferromagnetic coupling.
In the parallel magnetic configuration, the characteristic of Δ is different in the two spin channels. In the majority spin channel, we notice a charge depletion at the apex and surface atoms and an accumulation in between them, for both tip terminations. This charge redistribution can be explained based on the indirect Zener double exchange mechanism 9, 10 which leads to ferromagnetic exchange coupling between tip and sample 8 . The origin of this mechanism is due to delocalized sp-conduction electrons which couple with the d states. However, in the minority channel, the charge density is depleted at the apex atom and accumulated at the probed surface which is the characteristic for a direct exchange mechanism between the d orbitals due to the formation of spin-dependent covalent bonds 7 .
For a Mn-terminated tip, the double exchange mechanism is stronger than the direct antiferromagnetic exchange due to the larger s-d coupling. Hence the total exchange interaction is always ferromagnetic for all tip-sample separations in the case of hollow sites and for larger tip-sample separations in the case of top sites. The coupling changes to an antiferromagnetic coupling at close tip-sample separation.
For a Fe-terminated tip, the Zener type exchange mechanism is weaker compared to direct d-d exchange due to a lower s-d coupling 9, 10, 11 , hence we observe an overall antiferromagnetic exchange at the top sites. For the hollow site, this is much weaker at large tip-sample separation and changes to a ferromagnetic exchange at close tip-sample separation.

Supplementary Note 10: Tip-dependent variations of the magnetic exchange force and spin polarization
The out-of-plane magnetic sensitive tips were prepared by gently dipping the bulk Fe tip into Mn monolayer with adsorbed Co-adatoms. Therefore, in addition to different geometric atomic-scale structures of the tip, the apex may also have a different chemical termination (Fe, Mn, Co).
Supplementary Fig. 11 shows the tip-dependent variations of the magnetic contribution to the frequency shift (Δfex), the magnetic exchange force (Fex) and the current asymmetry (A) that we relate to the spin polarization, versus Δz at the positions i = 1 and i = 14 of the spin spiral as defined in the main article Fig. 1(d). Please note that Supplementary Fig. 11 only shows data acquired with tips that have a dominantly out-of-plane magnetic sensitivity according to our characterization in Supplementary Note 1, and belong to the tip types as defined in Supplementary Fig. 7 (b,d). While

Supplementary Note 11: Spin polarization
The spin polarization of the tunneling current found on Mn/W(110) is unexpectedly small (see Fig. 2 in the main article). Our DFT calculation shows a spin polarization value of ~20% of the vacuum LDOS in an energy range of ± 0.2 eV near the Fermi energy for Mn/W(110) (Supplementary Fig. 12).
Supplementary Fig. 13 shows the LDOS above the Mn/W(110) surface in comparison with the orbitally decomposed LDOS at the Mn atom underneath. We find that the peaks in the vacuum LDOS stem from pz states and that there are considerable contributions from spin-up and -down states leading to an overall spin polarization of about 20% in the vicinity of the Fermi energy. The d states of the Mn atoms are pushed far away from the Fermi energy due to the large exchange splitting resulting in a large magnetic moment 5 . This is different from Fe where we have a considerable contribution from the minority d states close to the Fermi energy 6 . In particular, the dz 2 states are energetically relatively far away from the Fermi energy for Mn/W(110). In general, a 3d material with a very large magnetic moment, such as Mn, will have a smaller spin polarization in the vicinity of the Fermi energy while the large exchange interaction with a magnetic tip is not affected. Supplementary Fig. 12 shows a comparison of spin polarization and vacuum LDOS 5 Å above the Mn surface for top (green) and hollow (orange) sites. From the figure, one can clearly observe that the vacuum LDOS does not change significantly between top and hollow sites as found in the experiments (cf. Fig. 2(f,g) in the main article). The calculated spin polarization is ~ 20% for the top site near the Fermi energy. This value is lower by 5% in the case of the hollow site.