Atomic-scale visualization of strain-tailored noncollinear spin textures in an antiferromagnetic ultrathin film

Abstract

Crystalline strain is typically considered as an effective approach to engineer low-dimensional antiferromagnets. However, a direct visualization of strained-tailored noncollinear spin textures in antiferromagnetic atomic layers has so far not been achieved. Here, we uncover a strain-induced transition from a three-dimensional noncollinear spin state in pseudomorphic Mn bilayer to a cycloidal spin spiral with a canted rotation plane in reconstructed Mn bilayer on the Ag(111) surface. These spin states are spatially imaged on the atomic scale by spin-polarized scanning tunneling microscopy revealing the correlation of atomic and magnetic structures. As demonstrated via first-principles electronic structure theory, the three-dimensional noncollinear spin state arises from the superposition of spin spiral and antiferromagnetic order due to higher-order exchange interactions. In reconstructed Mn bilayer, by contrast, the antiferromagnetic order is hindered by interlayer exchange coupling resulting in a pure spin spiral state. Our work highlights the complex interplay of atomic structure, intra- and interlayer exchange, as well as higher-order exchange interactions at antiferromagnetically coupled interfaces.

Antiferromagnets have intrinsic robustness to external perturbations, and while useful for device miniaturization, it makes manipulation of antiferromagnetic spins and ordering challenging. Here, Chen, Drevelow and coauthors successfully image the strain induced transformation of  noncollinear antiferromagnetic order in a manganese bilayer.

Introduction

Antiferromagnets are of significant importance in fundamental research and hold great promises for spintronic applications^1–3. In contrast to a ferromagnet, an antiferromagnet exhibits a vanishing net magnetization and is more robust against magnetic perturbations, enabling a higher packing density for magnetic storage bits due to the absence of interfering cross-talk from magnetic stray fields^4–6. In addition, antiferromagnets preserve a small damping factor allowing for a long decay length of spin current transport, which benefits the design of low-power devices and the development of ultrafast dynamics in THz regime^7–9. While two or more sublattice moments result in a cancellation of total magnetization that provides an immunity to external magnetic fields in antiferromagnetic (AFM) materials, it has been an essential issue as well as a challenging task to control AFM order and spin moments efficiently. Up to date, different approaches have been put forward to manipulate magnetic states in antiferromagnets, including large magnetic field cooling^10–12, ultrashort electric field stimuli and current-induced spin transfer torque^13–15, and optical switching through thermal demagnetization or electronic excitation^16–18.

Given a well-ordered atom arrangement in a crystalline structure, lattice strain naturally offers an effective pathway to engineer AFM order and spin moments in a wealth of low-dimensional AFM systems. For example, spin reorientation and AFM phase transition occur in strained Mn₂Au thin films^19–21, AFM to ferromagnetic (FM) phase transition and enhanced perpendicular magnetic anisotropy in strained FeRh thin layers^22–24, strained-induced Néel vector reorientation in MnPt and γ-FeMn thin films^25–27, and strain manipulation of spin-flop field in NiO and IrMn thin layers^28–30. However, as compared to collinear cases, the influence of strain on noncollinear AFM nanomaterials has been mainly studied on antiperovskite Mn₃XN (X = Ni, Ga, Sn)^31–33 and cubic Mn₃X (X = Sn, Ge, Pt)^34–36 compounds with multiple heterogeneous elements.

Ultrathin films on surfaces represent an important class of model systems to explore the mutual interplay between structural, electronic, and magnetic degrees of freedom. Pseudomorphic growth of AFM Mn atomic layers has been achieved on heavy metal substrates, such as Re, Ir, and W, with strong spin-orbit coupling (SOC)^37–40, leading to the milestone discovery of interfacial Dzyaloshinskii-Moriya interaction (DMI)³⁷ and the emergence of topological noncollinear spin states in atomic thin films^41,42. However, noncollinear magnetism can also arise from the competition of pair-wise Heisenberg exchange interactions in magnetic thin films grown on Ag or Cu substrates with small SOC^43,44. Moreover, as a consequence of a weak film-substrate hybridization, ultrathin magnetic films grown on Ag or Cu substrates approach the limit of ideal two-dimensional (2D) magnets. Despite having several appealing aspects, noncollinear spin textures remain rarely explored in ultrathin AFM films on light metal substrates, let alone atomic-scale spin mapping of relevant strain-induced effects.

Here, we demonstrate atomic-scale visualization of strain-induced tailoring of noncollinear magnetic order in the AFM Mn bilayer on Ag(111) by employing spin-polarized scanning tunneling microscopy (SP-STM). The Néel state of Mn monolayer on Ag(111)⁴⁵ is provided as a reference to characterize the in-plane magnetization directions of magnetic tips. Through three different in-plane tip magnetizations, a cycloidal spin spiral is revealed on the reconstructed Mn bilayer, exhibiting a period of 0.85 ± 0.1 nm and propagating about 60° with respect to the reconstruction lines along high symmetry [$11\overline{2}$] crystalline axis. For the pseudomorphic Mn bilayer, we observe a non-coplanar magnetic superstructure consisting of periods of 0.7 ± 0.05 nm and 0.5 ± 0.05 nm along the [$11\overline{2}$] and [$1\overline{1}0$] crystallographic directions, respectively, indicative of a magnetic superposition state of spin spiral and AFM order.

In combination with density functional theory (DFT), we show that for the pseudomorphic Mn bilayer on Ag(111) a magnetic superposition state of spin spiral and AFM order occurs. This intriguing 3D noncollinear magnetic structure is stabilized by frustrated inter- and intralayer AFM exchange and higher-order exchange interactions. Surprisingly, this spin state is energetically favorable and independent of hollow- or bridge-site stacking of surface Mn atoms. The strain-induced effect in the cycloidal spin spiral of reconstructed Mn bilayer can be explained by a lattice shift within the surface reconstruction, preventing the formation of AFM order in the two Mn layers. While SOC is weak, the DMI still couples the spin structures to atomic lattice, resulting in a spin spiral state with cycloidal rotation sense. Unexpectedly, the rotation plane of spin spiral and superposition state is canted by about 45° with respect to the surface normal. Furthermore, SP-STM simulations have been performed for the spin structures obtained from DFT, which are in good agreement with the experimental observations.

Results

Growth of Mn atomic bilayer on Ag(111)

Figure 1a represents the STM topographic overview of 1.0 ± 0.1 atomic layers of Mn deposited onto Ag(111) surface at 200 K. On the contrary to the pseudomorphic growth of monolayer (ML) Mn with fcc stacking⁴⁵, the double layer (DL) Mn appears with periodic reconstruction lines (DL Mn_R) as a relief of uniaxial strain, forming three rotationally symmetric domains. It is noted that surface reconstruction lines from uniaxial strain relief have been reported on the bcc(110)-like structure when bcc materials are epitaxially grown on top of fcc(111) substrates for matching crystalline orientation and reducing lattice misfit^46–50. In addition to DL Mn_R, there is also pseudomorphic strained area of DL Mn (DL Mn_S) as marked by the black arrow in Fig. 1b, c displays the zoom-in image acquired from Fig. 1a (white square frame), and the corresponding tunneling conductance ($\mathrm{d}I/\mathrm{d}U$) maps reveal that the appearance of reconstruction lines depends strongly on bias voltages, e.g., Fig. 1d (−0.6 eV) and Fig. 1e (+0.6 eV), suggesting different electronic contributions as a consequence of distinct local atomic arrangement (see Supplementary Fig. S1).

Fig. 1. Topographic overview of ML and DL Mn on Ag(111).

a STM topographic overview of 1.0 ± 0.1 atomic layer of Mn deposited onto Ag(111) surface, where ML Mn is pseudomorphically grown and DL Mn can be not only reconstructed (DL Mn_R) with three rotationally symmetric domains, but also pseudomorphically strained (DL Mn_S) as indicated by the black arrow in (b) (scan parameters: $U=+1.0$ V, $I=1.0$ nA). c Zoom-in topography from the white square frame in (a) and corresponding $dI/dU$ maps at U = −0.6 and +0.6 V have been shown in (d, e), respectively. f The DL Mn_R structure model in which the DL Mn is compressed about 9 % along [$1\overline{1}0$] direction on top of the pseudomorphic ML Mn, resulting in local bcc(110)-like areas (black rectangles) separated by reconstruction lines (orange, blue) on atomic hollow sites. Black dashed empty circle in (c) marks a surface defect, presumably single atomic vacancy as depicted in (f). g Bridge and hollow site stackings of DL Mn_S.

To mimic the feature of surface reconstruction lines in DL Mn_R, the atomic structure model composed of an uniaxially compressed Mn top layer above the pseudomorphic Mn bottom layer has been constructed in Fig. 1f, where 11 Mn atom distances (top layer) have been placed over 10 Mn atomic separations (bottom layer) along the [$1\overline{1}0$] direction (see Supplementary Fig. S2). This ratio turns out having about 9% uniaxial compression, leading to the bcc(110)-like areas (black rectangles) mirrored symmetrically in between alternating fcc- and hcp-like stacking lines. The deduced period of reconstruction lines is 2.89 nm from Fig. 1f, which is in line with 2.9 ± 0.1 nm from Fig. 1c measured experimentally. Note that both period and direction of reconstruction lines might deviate slightly from the ideal structure model in Fig. 1f, reflecting a manifestation of atomic strain relaxation varied locally. Fig. 1g displays the structure models of DL Mn_S, where bridge- and hollow-site stackings have been illustrated. It is also denoted that atomic resolution images on top surface layer of DL Mn_R and DL Mn_S provide additional support on the structure models of Fig. 1f, g^49,50(see Supplementary Fig. S3).

SP-STM on the reconstructed Mn bilayer

After recognizing the growth of DL Mn_R and DL Mn_S on Ag(111), SP-STM measurements have been performed to map out real-space magnetic spin structures with atomic resolution^37–39,51 (see Supplementary Fig. S4). As shown in Fig. 2a, magnetic periodic stripes are observed on the DL Mn_R by using a bulk Cr tip and they appear with an angle of about 60° relative to the reconstruction lines along the symmetry-equivalent [$11\overline{2}$] direction. Fig. 2b presents the topographic line profile measured from the black dashed line of Fig. 2a and the period of magnetic stripes about 0.85 ± 0.1 nm has been extracted from the DL Mn_R. Note that magnetic stripes on the DL Mn_R exist for three rotationally symmetric domains and they all have a period of about 0.85 ± 0.1 nm (see Supplementary Fig. S5). In addition to the bulk Cr tip, the same period as well as a phase shift of magnetic stripes have been identified by using Fe coated W tip under an external magnetic field. This supports a noncollinear spin spiral (SS) state that also exhibits the stripe-like magnetic contrast as reported on the ML and DL Mn films on refractory W substrates^37–39 (see Supplementary Fig. S6).

Fig. 2. Cycloidal SS revealed on DL Mn_R/Ag(111).

a SP-STM topographic image on DL Mn_R, where cycloidal SS, i.e., periodic stripe-like pattern, appears with an angle about 60° to reconstruction lines. b The cycloidal SS period about 0.85 ± 0.1 nm has been extracted from topographic line profile (black dashed line in a). c, d ML Mn Néel state (black square frame from a) together with simulated SP-STM image determine the in-plane tip magnetization direction (black arrow in d). e Magnetic contrast of cycloidal SS on DL Mn_R nearly vanishes after changing to the second in-plane tip magnetization, which can be supported by line profile in (f). The corresponding in-plane tip direction can be deduced from ML Mn Néel state in (g) and SP-STM simulation in (h). i The cycloidal SS appears again on DL Mn_R with the third in-plane tip magnetization, which can be verified by line profile in (j). Through ML Mn Néel state in (k) and SP-STM simulation in (l), the third in-plane tip magnetization can thus be characterized. (scan parameters for a, e, i: U = +10 mV, I = 1.0 nA).

Apart from the SS of DL Mn_R, the 120° AFM Néel spin structure of ML Mn has also been revealed in Fig. 2c, which reproduces the results reported by Gao et al.⁴⁵. Note that Fig. 2c is acquired from the black square frame in Fig. 2a, where a landmark can be referred to the white dashed circle of single atomic defect. Since the atomic spins of 120° Néel state of ML Mn align on the surface in-plane direction⁴⁵, it can serve as a magnetic reference to calibrate the in-plane direction of tip magnetization (see Supplementary Fig. S7). By combining experimental and simulated SP-STM images on the ML Mn of Fig. 2c, d, the in-plane tip magnetization of Fig. 2a has been deduced in the inset of Fig. 2d (black arrow). Aside from the in-plane tip magnetization, one should not rule out the existence of an out-of-plane tip component (see Discussion for details).

Fig. 2e shows the SP-STM image at the same location as in Fig. 2a, but with the second direction of in-plane tip magnetization obtained by applying a small voltage pulse to the bulk Cr tip at a distant sample position. Interestingly, there is an absence of magnetic SS contrast on the DL Mn_R with this second in-plane tip magnetization direction, which can be further supported by the topographic line profile (green dashed line from Fig. 2e) in Fig. 2f. While the magnetic SS disappears on the DL Mn_R in Fig. 2e, the Néel state spin contrast remains observed on the ML Mn (see zoom-in image in Fig. 2g). Through an atom-by-atom comparison between Fig. 2g and the simulated SP-STM image of Fig. 2h, the second direction of in-plane tip magnetization can be identified in the inset of Fig. 2h (green arrow) (see Supplementary Fig. S8).

Moreover, Fig. 2i demonstrates the return of magnetic SS contrast on the DL Mn_R by exploiting the third in-plane tip magnetization prepared through the voltage pulse method described above. On top of that, the topographic line profile in Fig. 2j (blue dashed line from Fig. 2i) restores the periodic modulation of about 0.85 ± 0.1 nm of magnetic SS on the DL Mn_R. According to the observed contrast on the Néel state of ML Mn in Fig. 2k (blue square frame from Fig. 2i) and the resultant SP-STM simulation image of Fig. 2l, the third direction of in-plane tip magnetization can be derived in the inset (blue arrow). Note that the subtraction analyses of two antiparallel magnetic tip configurations^52,53 have been applied to verify the magnetic signals on DL Mn_R (see Supplementary Fig. S9). It is also worth mentioning that the sharp step edge and the atomically-resolved Néel state of ML Mn indicate the absence of considerable tip shape change after a small voltage pulse. As a result of SP-STM measurements on the magnetic SS of DL Mn_R with three different in-plane tip magnetizations, the propagation direction of the SS lies in the rotation plane of atomic spin, leading to a cycloidal SS that is unexpected from the weak interfacial DMI for atomic Mn bilayer on Ag(111).

SP-STM on the pseudomorphic Mn bilayer

Besides uncovering the cycloidal SS state of DL Mn_R, we have also employed SP-STM to atomically resolve the magnetic spin texture on DL Mn_S. Fig. 3a shows the magnetic image on DL Mn_S obtained by SP-STM with a bulk Cr tip, where a rectangular spin lattice consisting of two periodic stripes has been revealed on each of three rotationally symmetric domains. It is worth mentioning that such rectangular spin lattice can not be observed on DL Mn_S with a nonmagnetic tip, insinuating its magnetic origin. Other than that, Fig. 3b represents the Néel state with an AFM domain wall on ML Mn⁴⁵ (black dashed rectangular from Fig. 3a), which can be combined with the simulated SP-STM image in Fig. 3c to determine the in-plane tip magnetization direction as the black arrow sketched in the inset (see Supplementary Fig. S10). Note that the potential contribution of out-of-plane tip component has been examined (see Discussion for details). Fig. 3d represents the topographic line profiles of two periodic stripes from the rectangular spin lattice at the bottom domain of DL Mn_S, i.e., black and gray lines in Fig. 3a, the resultant periods are 0.7 ± 0.05 nm and 0.5 ± 0.05 nm along high symmetry [$11\overline{2}$] and [$1\overline{1}0$] axes, respectively. Note that these two periods are equivalent to those extracted from the other two rotational domains (see Supplementary Fig. S11).

Fig. 3. Magnetic superposition state uncovered on DL Mn_S/Ag(111).

a SP-STM topographic image on the DL Mn_S, where rectangular spin lattice on each of three rotationally symmetic domains has been revealed. By comparing the ML-Mn Néel state in b (black rectangle frame from a) with simulated SP-STM image in (c), in-plane tip magnetization direction can be deduced (black arrow in c). The rectangular spin lattice consisting of two magnetic periods of 0.7 ± 0.05 nm and 0.5 ± 0.05 nm can be extracted from topographic line profiles (black and gray lines from a) in (d). e SP-STM topography with the second in-plane tip magnetization that can be derived from ML-Mn Néel state in (f) and SP-STM simulation in (g). By using second in-plane tip magnetization, only magnetic stripe with the period of 0.7 ± 0.05 nm has been resolved in upper right domain, which can be verified by line profiles in (h). i Only magnetic stripe with the period of 0.7 ± 0.05 nm has been observed in upper left domain by the third in-plane tip magnetization that can be identified from ML Mn Néel state in (j) together with SP-STM simulation in (k). Topographic lines profiles in (l) support the presence of 0.7 ± 0.05 nm magnetic stripe in upper left domain. (scan parameters for a, e, i: U = +10 mV, I = 1.0 nA).

By means of a small voltage pulse onto the Cr tip as aforementioned, the magnetic image of Fig. 3e with the second in-plane tip magnetization has been obtained. Based on the change of magnetic spin contrast on ML Mn of Fig. 3f, i.e., acquired from green square frame of Fig. 3e, as well as the comparison with SP-STM simulation in Fig. 3g (see Supplementary Fig. S10), the second in-plane tip magnetization has been deduced as the green arrow depicted in Fig. 3g. Instead of rectangular spin lattice formed by two periodic stripes, interestingly, there is only one stripe contrast left with a period of 0.7 ± 0.05 nm in the upper right domain of Fig. 3e, which is distinct from the other two domains (upper left and bottom). This observation can also be confirmed by the line profiles in Fig. 3h: there is a periodic modulation from the green line, but not the case of light green line. Note that green and light green lines are measured directly from the upper right domain in Fig. 3e. Furthermore, by repeating the same tip treatment again, Fig. 3i displays the magnetic image resolved by the third in-plane tip magnetization, which can be inferred from the zoom-in magnetic image of ML Mn in Fig. 3j together with the corresponding SP-STM simulation in Fig. 3k (see Supplementary Fig. S10). With the third in-plane tip magnetization (blue arrow in the inset of Fig. 3k), one can observe that the upper left domain has only one periodic stripe of 0.7 ± 0.05 nm(Fig. 3i), which can be further corroborated by the line profiles (blue and light blue lines in Fig. 3i) plotted in Fig. 3l. Note that concentric ring-like structures on ML Mn in Fig. 3a, e and i arise from the Ar-ion nanocavities buried underneath Ag(111) after the sample sputtering and annealing processes, leading to the standing wave patterns, i.e., so called Friedel oscillations, on the sample surface^53–55.

Since the identical period has also been observed for the 0.7 ± 0.05 nm magnetic stripe by an out-of-plane magnetized tip, the magnetic spin structure of SS might thus be anticipated (see Supplementary Fig. S6). The magnetic stripe period of 0.5 ± 0.05 nm matches twice the atomic spacing along [$11\overline{2}$] of DL Mn_S/Ag(111), implying a local AFM state locked onto the atomic lattice. Most importantly, one can clearly see the coexistence of these two magnetic stripes from a series of SP-STM measurements with different in-plane tip directions in Fig. 3a, e, i. Taking these observations into account, we could therefore arrive at a 3D spin configuration on the basis of magnetic SS superposed with atomic row-wise AFM order on the DL Mn_S/Ag(111).

First-principles calculations

In order to understand the magnetic properties of Mn bilayers on Ag(111) and to explain the experimental SP-STM results on the pseudomorphic and the reconstructed areas, we have performed DFT calculations (see “Methods”, Supplementary Table 1 and Note 1). Since the unit cell of the reconstructed Mn bilayer (Fig. 1f) is computationally not feasible within DFT, in particular, since noncollinear spin states need to be explored, we have considered a pseudomorphic Mn bilayer in which the Mn atoms of the top layer are in either bridge or fcc hollow site stacking (cf. Fig. 1g). Note that these two types of stackings appear locally within the surface reconstruction (Fig. 1f and Fig. 4). Because of the small influence of the Ag substrate (see Supplementary Note 2), fcc and hcp stacking at the hollow site are very similar (see Supplementary Table 1), which is why only the fcc stacking is presented here. We begin with the SS states because they represent the fundamental solutions of the Heisenberg model on a periodic lattice and thus allow us to scan a large part of the magnetic phase space.

Fig. 4. Local domains of the reconstruction.

Sketch of the RW-AFM state within the model of DL Mn_R. Blue and red circles denote Mn atoms with opposite spin directions. The structural domains of fcc hollow, hcp hollow and brigde site each splits into different magnetic sub-domains, which are named below the sketch. Dashed circles are used to show the nearest-neighbor spin alignment between top and bottom layer Mn atoms.

Bridge site stacking of the Mn bilayer

First, we discuss the energy dispersion of homogeneous SSs obtained via DFT for the DL Mn in bridge site stacking on Ag(111) (Fig. 5a). A SS is characterized by a wave vector q from the 2D Brillouin zone (2D-BZ) along which it propagates. The normalized spin moment at site R_i is given for a flat SS by ${\mathbf{s}}_i\left(\mathbf{q}\right)={\widehat{\mathbf{e}}}_1\cos \left({\mathbf{R}}_i\cdot \mathbf{q}\right)+{\widehat{\mathbf{e}}}_2\sin \left({\mathbf{R}}_i\cdot \mathbf{q}\right)$ where the orthonormal vectors ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$ span the rotation plane of spins. The SS vectors q were chosen along each unique high symmetry path in reciprocal space, which is larger than that of the hexagonal 2D-BZ because the second magnetic layer partially breaks its symmetries (Fig. 5c). At the high symmetry points we obtain collinear magnetic states: the FM state at the $\overline{\mathrm{\Gamma }}$ point, the layered AFM state with FM alignment of the magnetic moments within top and bottom layer and AFM alignment between the two layers at the $\overline{{\mathrm{\Gamma }}^{\prime }}$ point, and the row-wise AFM (RW-AFM) I and II states at the $\overline{{\mathrm{M}}^{\prime }}$ and $\overline{\mathrm{M}}$ points, respectively.

Fig. 5. Energy dispersion of SSs for DL Mn in bridge site stacking on Ag(111).

a Energy dispersion of a SS in the bridge site stacking of DL Mn on Ag(111) surface. The SSs are superimposed with the RW-AFM I state with a superposition angle θ (cf. Equation (1)). The dots represent DFT data and the lines are interpolations with the atomistic spin model that is parameterized with the energies of superposition angles of both 45° (yellow) and 90° (blue). For ${\theta }_{\min }\approx$ 14.2° (pink), an energetic minimum between the $\overline{\mathrm{\Gamma }}$ and the $\overline{\mathrm{K}}$ point is predicted. b Zoomed in region marked by the black box in panel a.  Gray lines are interpolations in steps of Δθ = 5°. c Symmetry zone and high symmetry path used for SS calculations. d–f Spin states for θ = 0° and an antiferromagnetic coupling between the layers (RW-AFM I), the flat spiral with θ = 90° and the q-vector of the energetic minimum, and a superposition of both with θ = 45°, respectively. Large (small) cones represent the magnetic moments of the top (bottom) Mn layer. Colors denote in-plane magnetization directions. Black and white cones denote moments pointing down- or upwards along the out-of-plane direction.

In both the RW-AFM I and II state, the top and bottom Mn layers exhibit the RW-AFM state. However, in the RW-AFM I state there is a net AFM coupling between the magnetic moments of the two Mn layers (Fig. 5d). The RW-AFM I state can occur for both local fcc hollow and bridge site stacking (left edge of Fig. 4). In the RW-AFM II state, the net coupling between the Mn layers is FM (c.f. central part of Fig. 4 for fcc hollow and bridge site stackings). Due to the surface reconstruction of the Mn bilayer, there is a shift between the RW-AFM order in the top and bottom Mn layers such that locally RW-AFM I and II states arise (Fig. 4). Note that for the bridge site stacking there is a third possibility, denoted as RW-AFM III state, with a balanced FM and AFM exchange coupling between Mn atoms of the two layers (cf. Fig. 4).

The energy minimum of the dispersion (Fig. 5a) for flat spin spirals (θ = 90°, blue curve), i.e., the energetically lowest SS state, is found for bridge site stacking at the $\overline{{\mathrm{M}}^{\prime }}$-point corresponding to the RW-AFM I state (Fig. 5d). In contrast, the RW-AFM II state ($\overline{\mathrm{M}}$ point) is energetically very unfavorable (Fig. 5a). The RW-AFM III state is also energetically much higher (see Supplementary Table 1). The favoring of the RW-AFM I state can be explained based on an AFM nearest-neighbor exchange coupling. The magnetic moments of Mn atoms in top and bottom layers are 3.2 and 3.0 μ_B, respectively, in the RW-AFM I state and vary little for spin spirals with q. The similar size of the bottom Mn moment shows that the hybridization with the Ag surface is quite weak (see also Supplementary Figs. S12, S13 and Note 2).

In the SP-STM results of DL Mn_S, a rectangular spin lattice has been observed from a longer stripe period of 0.7 ± 0.05 nm superposed with a shorter stripe period of 0.5 ± 0.05 nm (Fig. 3). Therefore, we have studied magnetic superposition states (Fig. 5f) which can be constructed from a flat spin spiral (Fig. 5e) and the RW-AFM I state (Fig. 5d). These superposition states represent conical AFM SSs³⁹ with a spin at site i given by

$${\mathbf{s}}_i^{\mathrm{super}}\left(\mathbf{q},{\theta }_i\right)=\cos \left({\theta }_i\right)\cdot {\mathbf{s}}_i^{\mathrm{AFM}}+\sin \left({\theta }_i\right)\cdot {\mathbf{s}}_i\left(\mathbf{q}\right)$$ 1

where ${\mathbf{s}}_i^{\mathrm{AFM}}$ denotes the spin at site i in the RW-AFM I state and the superposition angle θ_i determines the mixing. Note that the spin orientation of the RW-AFM I state is chosen perpendicular to the rotation plane of the SS (Fig. 5d, e). We assumed a uniform superposition angle θ_i = θ for all lattice sites. For a superposition angle of θ = 45°, the SS energy dispersion was calculated via DFT (yellow symbols in Fig. 5a). Based on the two DFT energy dispersions for θ = 90° and θ = 45°, a parameterization of an atomistic spin model including higher-order exchange interactions has been obtained (see “Methods”, Supplementary Table 2 and Note 3).

This spin model predicts a conical SS minimum for an opening angle of θ ≈ 14.2° (see Fig. 5b). DFT calculations of the conical SS dispersion for θ = 14.2° (magenta symbols in Fig. 5a, b) confirm this minimum. Note that the SS part of this superposition state exhibits a period of λ_b = 0.857 nm that is consistent with the experimental period of 0.85 ± 0.1 nm of the cycloidal SS observed on the DL Mn_R (Fig. 2).

Hollow site stacking of the Mn bilayer

We have further studied the hollow site stacking of the top Mn layer, which also occurs locally in the reconstruction (Fig. 4). Three superposition states of the RW-AFM I state (Fig. 6a) and a flat spin spiral (Fig. 6b) are found in our DFT calculations as energetic minima with very similar total energy (see Supplementary Figs. S14, S15, Table 3 and Note 4). The superposition state with a spin spiral direction consistent with the experiments is displayed in Fig. 6c, similar to that found for the bridge-site stacking, but exhibits a spin spiral period of λ_h = 0.686 nm. As shown in Eq. (1), the created superposition state can be described by a superposition angle θ. For a fixed value of q, the energy from the parametrized atomistic spin model becomes

$$E_{\mathbf{q}}\left(\theta \right)=\left(J_{\mathrm{eff}}+B_{\mathrm{eff},1}\right){\cos }^2\left(\theta \right)+B_{\mathrm{eff},2}{\cos }^4\left(\theta \right)+\text{const.},$$ 2

with the effective pairwise (Heisenberg) exchange constant $J_{\mathrm{eff}}$ and the effective higher-order exchange constants $B_{\mathrm{eff},1}$ and $B_{\mathrm{eff},2}$. A superposition state ($0^{\circ }<\theta <90^{\circ }$) can only be stabilized by higher-order interactions since the pairwise exchange favors either the RW-AFM I or the spin spiral state (cf. Fig. 6d and Eq. (2)). The DFT total energies along the path (blue symbols in Fig. 6d) can be well described by Eq. (2). In the superposition state at the energy minimum, both the RW-AFM I and the flat spin spiral state mix evenly with an effective mixing angle of $\theta =45^{\circ }$.

Fig. 6. DFT ground state for DL Mn in hollow-site stacking on Ag(111).

a Schematic of magnetic moments in the row-wise antiferromagnet with antiferromagnetic coupling between the layers (RW-AFM I). b Flat spin spiral. c Superposition of spin spirals and RW-AFM I state, which forms a local energetic minimum. d Energy of a superpositions with superposition angle θ between the RW-AFM I (θ = 0°) and the flat spin spiral (θ = 90°). Dots are DFT total energies and lines are the interpolation of an effective atomistic model. Blue is the total energy, yellow is the contribution of pairwise exchange and green is the contribution of higher order exchange.

Influence of SOC on the Mn bilayer

SOC is a weak effect in Mn and Ag due to their small nuclear charge. Therefore, SOC is not expected to play an essential role for the magnetic ground state in Mn/Ag(111), in contrast to DMI-driven spin spirals observed in Mn monolayers on heavy metal substrates with high SOC constants such as W^37,38. However, SOC determines the orientation of the spin structure with respect to the atomic lattice. To investigate this aspect, we have calculated the energy contribution of SOC within DFT in first-order perturbation theory⁵⁶ for a flat SS with the same q vector as the energetic minimum found for DL Mn on Ag(111) with different tilting angle ϕ of the rotation axis (Fig. 7). Due to symmetry the plane of rotation must always include the q vector. Different examples of such rotation planes are given in Fig. 7a with either clockwise or counter-clockwise rotation sense. Note that spin rotations in a plane perpendicular to q, which are possible in bulk systems with broken inversion symmetry⁵⁷, only gave negligibly small energy contributions in DFT consistent with calculations for an Fe bilayer on Cu(111)⁴⁸.

Fig. 7. Influence of DMI on spin spiral states.

a Projection of the magnetic moments of a SS onto a sphere of unit magnetization. The arrows show the sense of rotation when moving along the direction of the q vector. Different rotation planes are illustrated. b SOC energy of a flat SS dependent on the tilting angle ϕ of the rotation axis. The displayed SSs are the energetic minima for the bridge and hollow sites, i.e., from Fig. 5e (yellow) and Fig. 6b (blue), respectively. Symbols show DFT energies and the line is the interpolation with the DMI. Insets show spin structures.

The calculated energy contributions to SSs due to SOC for both bridge and hollow site stackings are displayed in Fig. 7b. Independent of the atomic stacking, a tilting angle near 45° with a counter-clockwise sense of rotation is energetically preferred by about 0.5 meV/Mn atom. The interpolation (solid lines in Fig. 7b) shows that the SOC energy contribution can be described by the DMI

$$E_{\mathrm{DMI}}=-{\sum }_{ij}{\mathbf{D}}_{ij}\cdot \left({\mathbf{s}}_i\times {\mathbf{s}}_j\right)$$ 3

To check the influence of the magnetocrystalline anisotropy energy, which also results from SOC, we have performed self-consistent calculations of the energetically favorable RW-AFM I state in each stacking (see “Methods” and Supplementary Note 5). All obtained values are very small (see Supplementary Table 4). Therefore, the orientation of the magnetic superposition state is mainly dominated by DMI.

Discussion

Interestingly, for both hollow and bridge site stackings of DL Mn on Ag(111), a superposition state of a flat SS and RW-AFM I state has been found as the magnetic ground state from DFT. Note that their magnetic spin patterns have been further verified by performing SP-STM simulations using the spin-polarized extension^58,59 of the Tersoff-Hamann model⁶⁰ (see Supplementary Fig. S16 and Note 6). Although the rectangular spin lattice is restored in both stackings, their mixing angles are different, i.e., θ ≈ 14° and 45° for bridge and hollow site stacking, respectively. This results in a larger contribution of the RW-AFM state for bridge site stacking. The short period of the magnetic unit cell of the RW-AFM order (0.498 nm) and the long period of the SS, especially λ_h = 0.686 nm of hollow site stacking, agree well with experimental SP-STM measurements on the DL Mn_S/Ag(111) (Fig. 3).

In the atomic model of DL Mn_R on Ag(111) (Fig. 4), both hollow and bridge site stackings occur locally in the bilayer structure. Because the SS does not couple to the crystalline lattice, it can propagate independently, regardless of the local atomic stackings. In contrast to the SS, the RW-AFM order couples to the atomic lattice, it is thus impossible to only obtain the energetically favorable RW-AFM I state, with a net AFM exchange coupling between the magnetic spin moments of top and bottom Mn layers, on the entire surface reconstruction. As shown in Fig. 4, one finds locally different effective magnetic couplings, e.g., either FM or AFM, between top and bottom Mn layers in terms of the RW-AFM state. Hence, the RW-AFM II and III states also turn out becoming energetically unfavorable on the bridge site stacking (cf. Fig. 5a and Supplementary Table 1). Taking these observations into account, we conclude that the RW-AFM state and thereby also the superposition state do not appear in the reconstructed domains of the Mn bilayer, leaving only a flat SS as seen in the experimental results of DL Mn_R/Ag(111) in Fig. 2.

For a direct comparison between experiment and theory, we have carried out SP-STM simulations^58–60 on the canted cyclodial SS and the magnetic superposition state as deduced from DFT for the DL Mn_R and DL Mn_S, respectively. Fig. 8a depicts the magnetic spin texture of cyclodial SS with a canted rotation plane. The top panels from Fig. 8b–d are zoom-in experimental images taken from Fig. 2a, e, i, which have been accompanied by the corresponding SP-STM simulations at the bottom panels. The magnetic stripe patterns have been reproduced in Fig. 8b, d with magnetic tip directions derived from the in-plane Néel state of ML Mn in Fig. 2 and an optimization of out-of-plane tip magnetization. The magnetic periodicity, i.e., 0.857 nm extracted from simulated line profiles in Fig. 8e, g, is consistent with 0.85 ± 0.1 nm obtained from Fig. 2. Note that Fig. 8c (bottom panel) displays only atomic lattice with the corrugation profile in Fig. 8f, validating the absence of magnetic spin contrast in Fig. 2e.

Fig. 8. SP-STM simulations on canted cycloidal SS and magnetic superposition state.

a Magnetic spin texture of canted cycloidal SS, where individual atomic spin is color coded by its out-of-plane spin component. b, c, d Top panels: zoom-in images taken from Fig. 2a, e, i. Bottom panels: magnetic stripe patterns (b, d) and atomic lattice (c) obtained from SP-STM simulations with three magnetic tip directions. The magnetic periodicity of 0.857 nm (e, g) and atomic lattice constant of 2.88 Å (f) are extracted from the simulated line profiles. h Atomic spin model of magnetic superposition state. i, j, k Left panels: magnification of upper right domain acquired from Fig. 3a, e, i. Right panels: simulated SP-STM images with three magnetic tip directions. l, m, n Two magnetic stripe periods in magnetic superposition state are 0.686 nm and 0.498 nm in line with Fig. 3.

Besides the canted cyclodial SS on DL Mn_R, the atomic spin model of magnetic superposition state on DL Mn_S has been constructed in Fig. 8h. Note that SP-STM simulations on three magnetic superposition states of energy minima in hollow site stacking have been examined (see Supplementary Fig. S17). From Fig. 8i–k, the left panels show the magnification of upper right domain acquired from Fig. 3a, e, i, and the right panels are the simulated SP-STM images for side-by-side comparison. The rectangular spin lattice composed of two periodic stripes has been obtained, where three magnetic tip directions are determined by the in-plane Néel state of ML Mn in Fig. 3 combined with an optimized out-of-plane tip magnetization. According to simulated line profiles plotted in Fig. 8l, m, n, two magnetic stripe periods are 0.686 nm and 0.498 nm, reconciling with 0.7 ± 0.05 nm and 0.5 ± 0.05 nm resolved experimentally in Fig. 3. SP-STM simulations on all three rotational domains are shown in Supplementary Fig. S18.

From our combined SP-STM and DFT studies, we have systematically investigated the effect of uniaxial strain relief on the noncollinear spin textures of DL Mn on Ag(111). In addition to the DL Mn, the 120° AFM Néel state has also been resolved on the ML Mn that plays an essential role in characterizing the in-plane magnetization direction of bulk Cr tip. By exploiting distinct in-plane tips, a cycloidal SS has been revealed on the DL Mn_R, with a period of 0.85 ± 0.1 nm and a propagation direction of around 60° relative to high symmetry [$11\overline{2}$] direction. While there is about 9% uniaxial compression of surface reconstruction on the DL Mn_R, neither inter- nor intralayer pair-wise Heisenberg exchange can stabilize the RW-AFM structure, and the cycloidal SS in turn becomes a more favorable ground state.

Regarding the DL Mn_S, we observe a magnetic superposition state of a cycloidal SS with a period of 0.7 ± 0.05 nm and an in-plane RW-AFM order of 0.5 ± 0.05 nm atomic spacing. This intriguing 3D spin structure emerges due to higher-order exchange interactions and is confirmed by the comparison of experimental and simulated SP-STM images with atomic resolution. Due to DMI, the rotation plane of superposition state and cycloidal SS in DL Mn_S and DL Mn_R are both tilted by about 45° with respect to the surface normal. In our work, we have directly visualized strain-induced noncollinear magnetic states on the atomic scale and highlighted how the impact of higher-order interactions on magnetic spin structures can be tuned via strain in AFM thin films.

Methods

Sample preparation

The whole experiment was performed in the ultra-high vacuum (UHV) environment of order ${10}^{-10}$ mbar. The Ag(111) substrate was cleaned by several cycles of Ar⁺ ion sputtering with an ion energy of 500 eV at room temperature and subsequent annealing up to 900 K. High purity manganese (Mn) (99.995%, Goodfellow) was evaporated from a molybdenum (Mo) crucible in an e-beam evaporator (FOCUS) while keeping the substrate at a low temperature (T ≈ 200K). After the sample preparation, it was transferred to the measurement chamber where a low-temperature scanning tunneling microscope (LT-STM) (UNISOKU, USM-1500), with a base temperature of 4.2 K, was employed to characterize the sample.

SP-STM measurements

The SP-STM measurements were performed by using either chemically etched bulk Cr tip or Fe coated W tip in the LT-STM (UNISOKU, USM-1500) setup. For topographic images, STM was operated in the constant-current mode with the bias voltage U applied to the sample. For tunneling spectroscopy (STS) measurements, a small bias voltage modulation was added to U (frequency ν = 3991 Hz), such that tunneling differential conductance $\mathrm{d}I/\mathrm{d}U$ spectra and mappings can be acquired by detecting the first harmonic signal from a lock-in amplifier. External magnetic fields up to 8 T were available to be applied to the sample along the surface normal direction.

DFT calculations

Structural relaxations of Mn bilayers on Ag(111) were carried out in DFT using the projector augmented wave (PAW) method as implemented in the Vienna ab initio simulation package (VASP)^61–63 for the p(1 × 1) and p(2 × 1) supercells of the fcc surface. The system was modeled by a symmetric slab consisting of 13 Ag layers with two Mn layers on both sides. The theoretical bulk Ag lattice constant of 0.404 nm was used. The Mn layers and the interfacial Ag layers were relaxed into the z-direction perpendicular to the surface until all forces were smaller than 10 meV/Å. All other layers were kept fixed. The calculations were done within the generalized gradient approximation using the exchange correlation functional of Perdew, Burke and Ernzerhof⁶⁴, with a plane wave energy cutoff parameter of 400 meV. 21 × 21 and 13 × 26 k-point grids were used for p(1 × 1) and p(2 × 1) supercells, respectively.

Non-collinear magnetic calculations for Mn bilayers on Ag(111) were performed using the full-potential linearized augmented plane-wave method as implemented in the FLEUR code^44,56,65. The DFT calculation of incommensurate spin spirals is possible using the generalized Bloch theorem⁶⁶ which is implemented in the FLEUR code⁴⁴. An asymmetric film with 5 layers of Ag and 2 Mn layers on one side of the film was used with the interlayer distances of the top three layers taken from the structural relaxations discussed above. We have carried out convergence with the number of Ag layers which showed that the total energy differences between magnetic states changed only on the scale of a few meV. The spin spiral energy dispersions were obtained in the local density approximation using the exchange-correlation potential of Vosko, Wilk, and Nusair⁶⁷. The basis function cutoff was set to (k_max = 4.1 a.u.⁻¹), and the radii of the muffin-tin spheres were set to 2.3 a.u. for both Ag and Mn atoms. Depending on the size of the supercell used in each calculation, 48 × 48 k-points were used for the p(1 × 1) and 24 × 48 k-points were used for the p(2 × 1) supercell. The magnetocrystalline anisotropy energy was calculated via self-consistent total energy calculations for a magnetization along different crystallographic directions including SOC⁶⁸. To determine the DMI, SOC was included in first order perturbation theory⁵⁶.

Atomistic spin model

To interpolate between the total energies of DFT and to investigate which magnetic interactions cause them, an extension of the classical Heisenberg model was used. The Hamiltonian of the spin model is given by

$$H=-{\sum }_{ij}{J}_{ij}\left({\mathbf{s}}_i\cdot {\mathbf{s}}_j\right)-{\sum }_{ijkl}{B}_{ijkl}\left({\mathbf{s}}_i\cdot {\mathbf{s}}_j\right)\left({\mathbf{s}}_k\cdot {\mathbf{s}}_l\right)$$ 4

with normalized spins ${\mathbf{s}}_i$ localized at lattices sites ${\mathbf{r}}_i$. The first summation describes the bilinear Heisenberg exchange interactions with the exchange constants $J_{ij}$, while the latter models higher-order exchange interactions of fourth order with the interaction constants $B_{ijkl}$. This model was mapped to total DFT energies from scalar-relativistic calculations, i.e., neglecting SOC. To parameterize the spin model, interaction constants were fitted to best reproduce the total DFT energies.

To efficiently sample the space of magnetic states, spin spirals with spin moments

$${\mathbf{s}}_i\left(\mathbf{q}\right)={\widehat{\mathbf{e}}}_1\cos \left({\mathbf{r}}_i\cdot \mathbf{q}\right)+{\widehat{\mathbf{e}}}_2\sin \left({\mathbf{r}}_i\cdot \mathbf{q}\right)$$ 5

that rotate homogeneously in the direction of a spin spiral vector q in the plane spanned by the orthonormal vectors ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$ were used. Since without SOC the orientation of the rotational plane does not matter, the energy dispersion $E\left(\mathbf{q}\right)$ of the spin spirals has been used to parameterize the atomistic spin model.

Author contributions

C.J.C., Y.T.L., Y.P.C., T.Y.C., Y.H.L., and P.J.H. carried out the SP-STM experiments and analyzed the data. C.J.C., Y.T.L., Y.P.C., T.Y.C., and T.D. performed the SP-STM simulations. T.D. developed the spin model and conducted the DFT calculations. T.D. and S.H. analyzed the DFT results, the SP-STM simulations and the spin model calculations. All authors discussed the results and contributed to the manuscript.

Peer review

Peer review information

Nature Communications thanks Yingshuang Fu, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data presented in this paper are available from the authors upon reasonable request.

Code availability

The code for the spin model calculations is available from the authors upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-62465-8.