Quantifying Exchange Forces of a Non-Collinear Magnetic Structure 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.

Chiral nanoscale spin structures such as magnetic skyrmions [1], as well as individual magnetic adatoms [2], have raised a lot of interest in recent years due to their potential for magnetic storage or logic devices at the atomic scale [3,4]. This has necessitated methods that can quantify and manipulate magnetic interactions between coupled magnetic atoms with atomic-scale precision. Spin-sensitive approaches based on scanning tunneling microscopy (STM) are sensitive to local variations in the exchange interaction at the atomic scale by detecting inelastic tunneling [5][6][7][8], spin resonance [9,10], or magnetization between a spin-sensitive tip and a surface [11,12]. Most notable of these methods is spin-polarized scanning tunneling microscopy (SP-STM), in which the total detected current is sensitive to the relative magnetization between the surface and a magnetized tip [13][14][15]. While SP-STM has been widely used to characterize non-collinear magnetization textures driven by the Dzyaloshinskii-Moriya interaction [16,17], the tunnel current is also sensitive to the tunneling anisotropic magnetoresistance [18], the non-collinear magnetoresistance [19], as well as structural and electronic variations, which are all convoluted in the overall measured signal.
Magnetic exchange force microscopy (MExFM) is a method, based on non-contact atomic force microscopy (nc-AFM), complementary to spin-sensitive tunneling spectroscopy. It directly measures the local exchange force between a magnetic tip and the surface [20,21]. Force-based detection of magnetic interactions presents numerous advantages over current-based detection. Most prominently, MExFM can address magnetic insulators [20,22], and it can be combined with simultaneous spin-polarized current detection (SPEX) [23] to disentangle the geometrical structure from electronic properties of magnetic structures [24,25]. Moreover, magnetic exchange force spectroscopy (MExFS) can directly quantify exchange forces [23,26], which combined with first-principles calculations allows for determining the interplay of various exchange mechanisms [27] as well as the role of chemical functionalization of the tip [28,29]. Nevertheless, this method has been scarcely applied, in contrast to the non-magnetic 3D chemical force field method that has been widely used to quantify chemical forces between the tip and various surfaces [30,31] as well as molecules [32]. However, 3D-MExFS contributes to answer open questions in surface magnetism, for example, what the role of spin-orbit coupling on the total force is, in analogy to the modification of the spinaveraged electronic structure known as the tunneling anisotropic magnetoresistance. A further question is if it is possible to detect and quantify various exchange mechanisms, including Zener-type exchange and anisotropic exchange, of chiral magnetic structures with atomic-scale resolution.
Here, we utilize SPEX to investigate a single atomic layer of Mn on W(110), which exhibits a cycloidal spinspiral ground state driven by the Dzyaloshinskii-Moriya interaction [16]. We demonstrate that the spatially resolved exchange force is much more sensitive to the atomic-scale magnetic exchange force field, including the local atomic sites within the magnetic unit cell, when directly compared to the spin polarization.
Utilizing simultaneous MExFS and I-z spectroscopy (MExFIS), we map out the evolution of the exchange force field and spin polarization between various non-collinear orientations of the tip magnetization relative to the sample. Using first-principles calculations based on density functional theory, taking into account both the tip and the surface, we understand the experimental findings of the exchange force field, and we quantify the influence of chemically different tip terminations on the overall distance dependence and the various possible exchange pathways. The surprisingly weak spin polarization of the tunneling current is explained by the dominant contribution of low spin-polarized pz-states near the Fermi energy, EF, due to the large exchange splitting of d-bands in Mn.
We performed SPEX imaging using a bulk Fe tip with a magnetization out of plane to the surface (see section S1 for more details) in a modified commercial ultra-high vacuum STM/AFM [33] operating at a base temperature of T = 5.8 K. Force detection was done in non-contact frequency-modulation mode, utilizing a tuning fork-based qPlus sensor [34] with the free prong oscillating at its resonance frequency (f0 = 30.8 kHz). The bias voltage Vs was applied to the sample, and we performed both constant-current (cc) and constant-height (ch) imaging. The W(110) surface was cleaned by annealing (T ~ 1800 K) in an oxygen atmosphere (p ~ 3 x 10 -7 mbar) followed by a final flash to T ~ 2800 K. Mn was deposited from a crucible with an electron-beam evaporator onto W(110) kept at room temperature and subsequently annealed (T ~ 800 K). The Mn monolayer has a p(1 x 1) chemical unit cell with respect to the W(110) surface [13]. As shown previously by SP-STM imaging [16], the Mn monolayer exhibits a cycloidal spin spiral magnetic ground state along the [11 ̅ 0] direction with a periodicity of ~ 12 nm. The angle between the magnetic moments of adjacent Mn rows is about 173° resulting in a locally almost antiferromagnetic rectangular c(2 x 2) unit cell (see sketches in Fig. 1a). To characterize our magnetic tips, we utilize the spin-direction dependent appearance of co-adsorbed Co adatoms as a reference [35,36] (see section S1 for further details).
We sense the atomic-scale spatial variation of the magnetic exchange interaction of the cycloidal spin spiral utilizing ch imaging. Fig. 1 shows simultaneously measured maps of (b) the current It ch , and (c) the frequency shift Δf ch and at a constant height z1 that is determined by opening the feedback loop at height z0, at which SP-STM images (Fig. S1) in cc were acquired (Vs = -10 mV; It cc = -2 nA), and moving the tip Δz = -0.29 nm closer to the surface. It ch shows a stripe pattern along the [001] direction ( Fig. 1(b)), illustrating spin contrast consistent with the spin spiral order reported in Ref. [16]. A long-range modulation of the stripe-pattern contrast is observed with a periodicity of Δxm = (5.70 ± 0.03) nm reflecting half the periodicity ( /2) of the spin spiral [16]. We refer to the position with maximum contrast of the stripes as = 0° and = 180° ( Fig. 1(d)), which correspond to positions with reverse alignments of the out-of-plane magnetic moments. We can assign the darker and brighter stripes of It ch to Mn rows exhibiting an antiparallel (ap) alignment at = 0° and a parallel (p) alignment at = 180° with respect to the out-of-plane projection of the tip magnetization (see Fig. 1(d)). The periodicity of Δx1 = (0.43 ± 0.03) nm along the [11 ̅ 0] corresponds to the lattice constants of the magnetic c(2 x 2) unit cell as sketched in Fig. 1(a) [13]. In agreement with previous works [13,16], the lattice constant along the [001] direction (Δx2) is not resolved in It ch . Imaging at constant height (z1) gives spin contrast in the It ch image comparable to SP-STM images at z0 (Fig. S1 and S2), but does reveal more information on the magnetism within the spin spiral.
In contrast, the simultaneously probed Δf ch resolves the atomic lattice for both the [11 ̅ 0] and [001] direction with unprecedented detail. The Δf ch image in Fig. 1(c) reflects the force gradient averaged within z1 ± 50 pm. The total force is composed of chemical and magnetic exchange forces, and results in the observation of well-separated bright protrusions in the Δf ch image, which we assign to the positions of the Mn atoms (see section S4 for more information on the assignment of atomic sites). We note that electrostatic forces due to local charges are not expected to be significant because of the short screening length of the metallic substrate. Our extracted lattice constants (Δx1 = (0.43 ± 0.03) nm and Δx2 = (0.30 ± 0.03) nm agree well with the predicted ones [13]. We refer to the top sites, t, within each c(2 x 2) unit cell, i ( Fig. 1(a)), within the spin spiral as ti, where i = 1 and i = 14 correspond to = 0° and = 180° of the spin-spiral period ( Fig.  1(d)), respectively. An analogous definition is used for the hollow sites hi ( Fig. 1(a)). As the Mn layer is planar due to pseudomorphic growth on W(110) [13], atomic contrast emerging from chemical interaction is expected to result in the same variation of Δf ch at all atomic sites (ti and t'i) along the spin spiral. However, we observe a modulation Δfex,i of the contrast between ti (Δfi ch (z1)) and t'i (Δf'i ch (z1)) along the [11 ̅ 0] direction (cf. solid vs. dashed circles at position i = 13 in Fig. 1(c)). To follow the evolution along the spin spiral (white arrow), we define Δfex,i (z1) = Δfi ch (z1) -Δf'i ch (z1) and obtain Δfex,1 (z1) = -(0.8 ± 0.2) Hz, Δfex,7 (z1) = (0.0 ± 0.2) Hz and Δfex,14 (z1) = (0.8 ± 0.2) Hz, reflecting the reversal of the out-of-plane projection of the surface magnetization within the spin spiral as seen with SP-STM (see also sketch in Fig. 1(d)). We therefore assign Δfex,i to variations of the magnetic exchange force along the spin spiral. We note that for different tips we have also observed an opposite correlation between It ch and Δf ch than that seen in Figs. 1(b) and 1(c). Our measurements show that the exchange force component of the measured force directly correlates with the out-of-plane projection of the surface magnetization. Therefore, this indicates that Δf ch directly probes not only the local antiferromagnetic order, but is also sensitive to the variations of the magnetic exchange interaction of the non-collinear magnetic moments on the surface resulting from the Dzyaloshinskii-Moriya interaction driven spin spiral.
Next, we quantify the exchange force field at each atomic site as a function of tip-sample separation. We address both the top and hollow sites to study the influence of the local lattice configuration on the detected magnetic exchange interaction between each site and the tip, which we were not able to resolve in SP-STM mode. First, we focus on out-of-plane magnetic orientation of the atoms in the spin spiral with p and ap alignment ( Fig. 1(d)). We record Δfi (from which we calculate Δfex,i) together with It,i with respect to the tip displacement Δz (Δz = 0 nm corresponds to z0) using the same atomic-scale tip (see Fig. 2(a) and the insets in (b) and (c)). The error margin reflects an uncertainty of ±0.2 Hz for Δfex,i ( Fig. 2(b,c), see supplemental material Fig. S7 for discussion of the influence of a surface tilt). Using the procedure described in Ref. [37], we derive the magnetic exchange forces Fex,i from Δfex,i (Fig. 2(d,e)). As expected, we observe a sign reversal of the distance dependence when comparing Δfex,1 (Δz) and Δfex,14 (Δz), as well as for Fex,1 (Δz) and Fex,14 (Δz). Moreover, we observe a significant difference regarding the distance dependence of the exchange force field between top and hollow sites: for the hollow sites, the curves show a monotonous increase of the magnetic interaction (Fig. 2(c,e)). This is different from the curves acquired at the top sites ( Fig. 2(b,d)) which show a change of the slope. The maximum absolute Δfex values are ~ 3 Hz resulting in magnetic exchange forces up to 40 pN, i.e., smaller by two orders of magnitude compared to the total forces (Fig. S3). The observed exchange forces here are larger by about a factor of three than probed on the non-collinear magnetic nano-skyrmion lattice of a monolayer Fe on Ir(111) with the same bulk tip [23]. The distance-dependent variations of the exchange force field suggest that the measured exchange force results from different exchange mechanisms when comparing data for the top and hollow sites, as a result of the different nearest-neighbor configurations. Moreover, this evidence illustrates that Fex,i is highly sensitive to variations of the magnetic exchange force contributions on different atomic sites.
In order to have a measure for the distance-dependent spin-polarization, we consider the current with It,i (Δz) acquired at ti or hi and I't,i (Δz) acquired at t'i or h'i. The error margins reflect an uncertainty of ±3% for It,i. The different signs of A1 (Δz) and A14 (Δz) (Fig. 2(f, g)) reflect the reverse relative alignments of the magnetic moments of the tip and Mn atoms (see insets in Fig. 2(b,c)).
We observe that A1 (Δz) and A14 (Δz) do not show a significant variation in amplitude regardless of the atomic site (hollow or top) or distance. This is different from previous distance-dependent measurements of the Fe/Ir(111) skyrmion lattice [23], where A(Δz) showed a monotonous increase. In addition, the absolute values of A observed here are one order of magnitude smaller than for one monolayer Fe on Ir(111). Our findings further show that detection of Fex (Δz), is more sensitive to atomic-scale variations, i.e. top and hollow sites, compared to A(Δz), and provides more information about different exchange mechanisms.
So far, we have quantified the magnetic exchange force field at positions of the spin spiral with dominantly p and ap alignment (i.e., for i = 1 and 14). Next, we probe the exchange force field at different sites i along the spin spiral using MExFIS, in order to quantify the variations of the exchange forces for the non-collinear orientations relative to the tip magnetization (Fig. 3a). We determine Fex,i (Δz) and Ai (Δz) (Figs. 3(b,c)) at the hollow sites (hi and h'i) at which we probe larger amplitudes of the exchange forces than on the top sites for the p and ap alignment (Fig. 2(e)). The gradual color change from blue to olive reflects the position within the spin spiral as sketched in Fig. 3(a). We observe a direct correlation between the magnetization of the spin spiral and the measured magnitude of the magnetic exchange force as shown in Fig. 3(b) for ( /4) of the spin spiral. Given our error margins, we observe a gradual decrease of the absolute values of Fex,i (Δz) (as well as of Ai (Δz)) from i = 1 to i = 7. We assign the change of Fex,i (Δz) to the variation of the magnetic exchange interaction due to the different directions of the magnetic moments between = 0° and = 90° (Fig. 3(a)). We note that we also acquired data for i = 7 to i = 14 (see supplemental material Fig.   S7). We carefully acquired and analyzed all data to make sure that the tip has a dominantly magnetic outof-plane sensitivity and that we can rule out measurement artifacts, e.g. piezo creep (for more information on the various types of force curves acquired, see supplemental material Fig. S5). Nonetheless, we cannot exclude a misalignment of the tip magnetization by a few degrees from the out-of-plane direction and small changes of absolute distance due to a slight tilt of the sample (see supplemental material Fig. S8 for more details). Furthermore, we determined a spontaneous change of the tip magnetization during acquisition of the data for i = 7 to i = 14 (see supplemental material Fig. S7 for more information). Therefore, we have excluded this data from our discussion. Nevertheless, our results show that we can probe the variation of the magnetic exchange forces and current asymmetry along the spin spiral.
In order to understand the experimental observations, we have performed first-principles calculations for Mn/W(110) and for the interaction with a magnetic tip (see supplemental material section S8 for computational details). Although an Fe bulk tip was used in the experiments, the tip was prepared by gentle dipping into the Mn layer with co-adsorbed Co adatoms. Therefore, we consider the influence of three different tip terminations (Fe, Co, Mn) of Fe-based tips on the magnetic exchange energy for collinear alignments between the magnetic moments of the tip and both the top and hollow sites of the Mn monolayer (cf. Fig. 1(a)). Mn/W(110) was modelled by a symmetric slab with 5 W layers and 1 layer of Mn on each side. The tip was modelled by a 14-Fe pyramid in a bcc-(001) orientation (see supplemental material   (Fig. 4(c)).
For smaller d, the coupling is antiferromagnetic. In contrast, for a Fe-or Co-terminated tip, only antiferromagnetic coupling is observed. We attribute the change of sign for Mn-terminated tips to a transition from an indirect long-range ferromagnetic Zener-type exchange mechanism between the Mn d-states mediated by s electrons, to a short-range direct antiferromagnetic exchange between d-orbitals of Mn apex and surface atom (as suggested in Ref. [27]). The Zener-type exchange mechanism is much weaker for Fe and Co due to their smaller s-d coupling [38][39][40], and no transition occurs, in agreement with a previous work [26]. The different types of exchange mechanisms can be seen in spin-resolved charge-density difference plots (see supplemental material Fig. S13). For hollow sites, the coupling is always ferromagnetic for the Mn-terminated tip ( Fig. 4(a)). For Fe-and Co-terminated tips, antiferromagnetic coupling at large tipsample separations changes to a ferromagnetic coupling at small tip-sample separations. The relatively small values of Eex and Fex for hollow sites and the change of sign for Fe-and Co-terminated tips can be attributed to competing exchange interactions between the tip apex atom and nearest neighbor and nextnearest neighbor surface Mn atoms (cf. Fig. 1(a)). For Mn-terminated tips the Zener-type exchange dominates. Our calculations reveal that the magnetic exchange force is highly sensitive to competing magnetic exchange interactions between the tip and the Mn layer depending on the chemical tip termination as well as on the atomic site.
The calculated exchange force curves for a Mn-terminated tip agree qualitatively with the experimental data shown in Fig. 2(d,e) while Fe-and Co-terminated tips cannot explain the observations. From a comparison between the onset of magnetic exchange forces in the calculations and in the experimental curves we estimate that d = 0.5 nm corresponds to a tip displacement of about Δz = -0.16 nm. This is also in reasonable agreement, regarding the total measured and calculated forces (see supplemental material Figs. S3 and S4). For the discussion of the calculated and experimental Fex, we focus on the tip-sample separations between d = 0.35 nm and d = 0.5 nm corresponding to Δz = -0.31 nm and Δz = -0.16 nm, respectively. The experimental exchange force Fex (Δz) on top sites first rises up to an absolute maximum value and changes its sign when the tip is brought closer to the surface (Fig. 2(d)). At the hollow sites, Fex (Δz) exhibits a monotonous increase of the absolute values (Fig. 2(e)). This agrees with the change in the calculations from positive exchange force for d > 0.4 nm to negative exchange force for d < 0.4 nm on the top site for the Mn-terminated tip (Fig. 4(d)), while there is a monotonous rise of the exchange force on the hollow sites ( Fig. 4(b)). We point out that we have observed many different types of distance-dependent curves To understand the experimentally observed small and site-independent spin polarization, we analyze the vacuum local density of states (LDOS) at the top and hollow sites. Within the Tersoff-Hamann model of STM [41,42], the differential conductance is proportional to the LDOS a few Å above the surface, and the asymmetry curves in spin-polarized measurements ( Fig. 2(f,g)) can be compared to the spin polarization of the vacuum LDOS [43,44]. As shown in the supplemental material ( Our work provides the first combined experimental and theoretical study to quantify the different exchange force regimes of a chiral magnetic structure on the atomic scale. As a future perspective, SPEX can be used to simultaneously quantify the influence of spin-orbit coupling on both the current and the magnetic interactions in chiral magnetic structures, as well as characterize them with antiferromagnetic probes or functionalized tips. Furthermore, 3D-MExFIS opens a new route to quantify the various direct and indirect exchange forces, with atomic-scale resolution, in addition to extracting dissipation mechanisms, providing direct access to interaction energies in comparison to state-of-the-art methods based solely on tunneling current [6,46].