The interfacial Dzyaloshinskii–Moriya interaction (DMI) emerges in ferromagnet/heavy metal thin films with broken z-mirror symmetry, stabilizing Néel domain walls and skyrmions of one chirality as dictated by its Rashba-type field. On the other hand, both Bloch chiralities remain energetically degenerate at static equilibrium for mixed domain walls in films with a moderate DMI. Multiple research groups have nevertheless independently reported significant Bloch chirality asymmetries in the Bz-stabilized domain microstructures of different interfacial DMI systems [1, 2]. In this work, we demonstrate how the asymmetric wall energy landscape in such systems can produce a deterministic Bloch chirality when domain walls are driven away from static equilibrium. For D &lt; Dc, the two ground state domain walls—having Bloch components of opposite chiralities—exhibit different restoring torques and steady state configurations. As dictated by the driving field’s direction and sign of DMI, one wall, which we term the lax wall, becomes labile at a lower Bz prior to the Walker field. At this lability point, the wall lies at an inflection point in its energy landscape and the domain wall susceptibility (χdw ∝ (σΦΦ)-1) diverges. The lax wall therefore reorients at higher fields to the steady state configuration of its counterpart, which we call the locked wall (Fig. 1). Using the Slonczewski equations and the 1D wall energy model, we analytically derived a DMI–Bz phase diagram of steady state domain wall Bloch chirality (Fig. 2), which includes an astroid analogous to the Stoner–Wohlfarth domain switching model. While these equations predict that locked walls precess beyond the Walker field, it was observed that they can instead persist due to DMI-induced asymmetries in vertical Bloch line (VBL) evolution—extending the Bloch monochirality regimes [3]. Here, we generalize these Bloch chirality preferences to arbitrary VBL behavior in these systems, before discussing how the theory of horizontal Bloch line evolution could similarly mediate Bloch chirality preferences [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/c78283a8de7f53f55949028b259b7207.jpg)

Steady state Bloch monochirality via DMI-induced asymmetric evolution of lax & locked walls.

![](https://assets.underline.io/uploads/markdown_image/1/image/e73fbc654808325ad8a75b63241d4af0.jpg)

DMI–Bz phase diagram of steady state domain wall Bloch chirality

Deterministic Bloch Chirality in Interfacial DMI Systems: An Astroid for DW Switching

Typically, stabilizing small, robust skyrmions requires a strong Dzyaloshinskii-Moriya Interaction (DMI). Accurately measuring the strength of DMI in prospective skyrmion host materials is therefore crucial to realizing proposed skyrmion-based devices for racetrack memory and neuromorphic computing.[1–2] In the creep regime, the velocity of a domain wall is governed by an Arrhenius law and is inversely proportional to its elastic energy.[3] Early work on the impact of DMI on domain walls showed that the resting energy of a wall is modulated by the DMI's effective field and that the presence of an in-plane field results in asymmetric domain growth due to the (mis)alignment of the two fields.[4] Subsequent studies have proposed creep models that reflect the intricacies of the relevant velocity and elastic energy scales.[5–6] As such, the sign and strength of the Rashba DMI found in thin film systems has been deduced from the asymmetric creep expansion of domains when an in-plane field is present; wall segments with core magnetizations parallel to the in-plane field have reduced energies, resulting in enhanced velocities—and vice-versa.[5–6] However, Brock et al., recently reported experimental results that deviate remarkedly from the above-mentioned creep models. Observations made in [Co(0.7 nm)/Ni(0.5 nm)/Pt(0.7 nm)]N&gt;3 films include highly asymmetric dendritic domain growth perpendicular to the in-plane field for low fields and sharp variations of the growth direction with the strength of the in-plane field.[7–8] While one study focused on the apparent preference of Bloch walls of a specific handedness and attributed most of the phenomena to interlayer DMI [8], further investigation of the findings in [7] revealed that the trends that clearly defy the symmetries expected from an intrinsically favored wall chirality with Hz.[7] In this work, we address these peculiarities by considering the impact of the driving field on the domain wall magnetization, dispersive stiffness, and creep velocity. We show that even in the creep regime, the core magnetization of a domain wall, as per the Landau-Lifshitz-Gilbert equation, can be driven to a steady-state equilibrium that deviates significantly from the static case. We find that for material parameters comparable to those reported in [7], a bubble domain in the presence of an in-plane field can exhibit sharp discontinuities in its steady-state wall magnetization profile. Similar to the findings reported in [10], such a point represents a sudden reorientation of the magnetizations along the domain wall due to the nuanced competition between the effective DMI field, the domain wall anisotropy field, the in-plane field, the driving field and Gilbert damping. Consequently, the domain wall energy and stiffness landscapes can be equally complex, highly sensitive to the in-plane field and significantly different from the static case. Incorporating a steady-state treatment of domain wall profiles into the dispersive stiffness framework developed in [5], we compute the steady-state stiffness and creep velocity profiles of a an expanding domain for the scenarios reported in [7]. As seen in Figure 2., our modelling of the N=3 film shows remarkable agreement with the observed dendritic growth directionalities as a function of the strength of the in-plane field: first propagating uniformly for very weak fields, then at 90° from the field direction, then gradually reducing to ≈20°, and then abruptly reversing direction to ≈200°. Moreover, the model reproduces the curious observation that that the propagation direction is constant for both driving field directions; switching the domain polarization switches the directions of both the DMI and applied field torques, leaving the Bloch chirality at steady state unchanged. Comparing the magnetization profiles to the propagation directions and stiffness values reveals that segments with the lowest elastic energy (and thus the highest velocity) occur close to reorientation positions in the magnetic profile. The sudden emergence of such points is also responsible for the drastic changes in the growth direction. In light of these insights, we discuss the impact of steady-state profiles on DMI values obtained via bubble expansion experiments. Finally, we consider how our steady-state stiffness model could be used in tandem with these growth experiments to more easily determine the value of Gilbert damping in thin films.

**References:**

1. R. Tomasello, E. Martinez, R. Zivieri et al., Scientific Reports., Vol. 4, (2014) 2. G. Bourianoff, D. Pinna, M. Sitte and K. Everschor-Sitte, AIP Advances., Vol. 8, p. 055602 (2018) 3. P. J. Metaxas, J. P. Jamet, A. Mougin, Physical Review Letters., Vol. 99, p. 217208 (2007) 4. S. Je, D. Kim, S. Yoo et. al., Physical Review B., Vol. 88, p. 214401 (2013) 5. J.P. Pellegren, D. Lau, and V. Sokalski. Physical Review Letters., Vol. 119, p. 027203 (2017) 6. D. Lau, J. P. Pellegren, H. T. Nembach et al., Physical Review B., Vol. 98, p. 184410 (2018) 7. J. Brock, R. Medapalli and E.E. Fullerton, Bulletin of the American Physical Society. (2020) 8. S.D. Pollard, J. A. Garlow, K. Kim,et al., Physical Review Letters., Vol. 125, p. 227203 (2020) 9. L. Sánchez-Tejerinaa, O. Alejosa, and E. Martínez, Journal of Magnetism and Magnetic Materials., Vol. 423, p. 405–410 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5c391c3ecd610921b55255792a947aef.jpg)
Simulated static (blue/dark) and steady-state (yellow/light) equilibrium wall magnetization profiles for an expanding bubble domain similar to those observed in [7].

![](https://assets.underline.io/uploads/markdown_image/1/image/8a760a928ff1a5a294b48a522d4d7bce.jpg)
(a) Conventions used for defining the growth direction of a domain in the presence of an in-plane field. (b) Experimental (markers) and semi-analytically computed (lines) plots of domain growth direction against in-plane field for the [Co(0.7 nm)/Ni(0.5 nm)/Pt(0.7 nm)]3 sample referenced in [7].

 Impacts of Steady-State Domain Wall Configuration on Domain Wall Stiffness and Directional Domain Propagation in the Creep Regime

The interfacial Dzyaloshinskii-Moriya Interaction has had profound consequences on the possible magnetic configurations that can be stabilized in multi-layer thin films, most notably chiral Neel skyrmions and domain walls.[1] More recently, there have been theoretical calculations and experimental work confirming the existence of an appreciable anisotropic DMI in epitaxial thin films with further reduced symmetry – namely the C2v (or mm2) point group.[2] However, so far the systems under consideration have been based on cubic crystals where the magnetocrystalline anisotropy is nearly isotropic. Here, we identify a new epitaxial relationship with C2v symmetry based on HCP Co(10.0)/FCC Pt(110). Although it is quite common to grow FCC Co(111)/Pt(111), for the case of a Pt(110) underlayer, the lattice mismatch is actually smaller for the HCP Co(10.0) orientation [as opposed to FCC Co(110)] where the c-axis lies along a single direction in the plane of the film (figure 1). Such an orientation is expected to offer some degree of anisotropic DMI based on the heavy metal Pt and reduced symmetry as well as the added feature of significant uniaxial in-plane magnetocrystalline anisotropy. Samples were prepared by DC magnetron sputtering on HF-etched Si(110) single crystal substrates. Following the work of Yang et al., we use a Ag (10nm) buffer layer, which is known to grow with (110) orientation on Si(110) at room temperature, thereby avoiding further unwanted interface diffusion.[3] This is followed by deposition of the Pt (10nm) /Co (10-50nm) bilayers, which are the focus of this work. All depositions were done in an Ar atmosphere with the working pressure fixed at 2.5 mTorr. The films were characterized structurally by standard θ/2θ x-ray diffraction (XRD) measurements and phi-scans to characterize the epitaxial relationship. The XRD scan shows primarily Ag(220), Pt(220), and Co(10.0) peaks with no identifiable Co(220) peak suggesting the film is predominantly HCP. Phi scans (figure 2) on Pt(111), Ag(111), and Co(10.1) confirm the epitaxial relationship shown in figure 1. Moreover, reciprocal space q-scans on the HCP Co spots were leveraged to characterize directional peak broadening due to stacking faults. As it is known that stacking faults cause broadening only of certain peaks (h-k ≠ 3m) along the c* direction (with breadth also determined by the parity of l), it is possible to extract the density of growth and deformation faults in the Co layer, which are found to be approximately 10% and 3% respectively in 50nm Co films. Details on how these stacking fault densities are calculated will presented in detail. Alternating gradient field magnetometry was used to measure M-H loops along different crystallographic directions in the plane of the film. A clear in-plane easy axis is identified along Si[001] (parallel to Co[0001]). The hard axis saturation field of 1.5 kOe suggests a notably smaller magnetocrystalline anisotropy than would be expected for bulk cobalt, likely due to the aforementioned stacking faults. Although the relatively thick Co layer in these films makes any interface induced DMI unlikely, the epitaxial relationship is expected to be preserved in thinner samples along with the reported uniaxial anisotropy. Together with the expected interfacial anisotropic DMI, this in-plane anisotropy makes the proposed material system an interesting playground for exploring new chiral magnetic ground states.

**References:**

[1] Thiaville, A., Rohart, S., Jué, É., Cros, V. and Fert, A., 2012. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. EPL (Europhysics Letters), 100(5), p.57002. [2] Hoffmann, M., Zimmermann, B., Müller, G.P., Schürhoff, D., Kiselev, N.S., Melcher, C. and Blügel, S., 2017. Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions. Nature communications, 8(1), pp.1-9. [3] Yang, W., Lambeth, D.N. and Laughlin, D.E., 1999. Unicrystal Co–alloy media on Si (110). Journal of applied physics, 85(8), pp.4723-4725.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cf10786ba9d1e164cc2eadffc57f6d46.jpg)
Fig 1. Schematic of the lattice matching for Ag(110)/Pt(110) with bottom left) FCC Co(110) or bottom right) HCP Co(10.0). The HCP (10.0) orientation has a better match along the Pt [001] direction.

![](https://assets.underline.io/uploads/markdown_image/1/image/78b7907db3cc501cd82fd177111f404e.jpg)
Fig 2. Phi scans for Pt (111), Ag (111), and Co (10.1) confirming the epitaxial relationship shown in figure 1.

 Structural and Magnetic Characterization of Epitaxial Co(10.0)/Pt(110) Multi-layers for Future Anisotropic DMI Systems Based on C2v Symmetry

When applied to C2V ferromagnetic/heavy metal interfaces, Moriya’s rules do not deterministically predict the orientation of interatomic DM vectors. Low-symmetry magnetic systems can exhibit both Rashba (DR) and Dresselhaus (DD) forms of the Dzyaloshinskii-Moryia interaction (DMI), as both spin-orbit couplings are generally present [1]. We assert that the spatial variation of the DM vectors—and thus that of the strength and direction of the effective DMI field—is determined by the DMI’s character (i.e., ||DD||/||DR||), as well as the unit cell’s aspect ratio [2]. Overall, the DMI is thus nominally anisotropic and hybridized—potentially stabilizing antiskyrmions and Bloch-Néel (i.e. mixed) skyrmions [2-3]. Ascertaining the existence and tunability of [anti]skyrmions in such systems is of technological interest and would elucidate the nature of DMI [2]. Employing the Wulff construction [3], ξ-vector formalism [4], and an augmented wall energy that accounts for DMI character, this work models the equilibrium shape and wall magnetization profile of bubble domains and skyrmions in thin films with C2v symmetry. We show that for sufficiently large DMI, square-shaped antiskyrmions are stable for certain DM vector orientations while lens- and oval-shaped mixed skyrmions are favored otherwise. A symmetry-breaking applied in-plane field stabilizes domain shapes resembling ovals and Reuleaux triangles. The impact of the unit cell’s aspect ratio, the DM vector’s orientation, and other material parameters on the shape of bubble domains and skyrmions are discussed. Moreover, our results reveal unusual relationships between faceting behavior and wall stiffness under rather reasonable conditions. Elucidating the interplay between the different effective fields present, we explore the validity of these results based on pedal plots, discontinuities in the magnetization profiles, and local extrema in the wall energy.References: U. Güngördü et al., Physical Review B., 93, 064428 (2016) M. Hoffman et al., Nature Communications., 8, 308 (2017) J. P. Pellegren, D. Lau, and V. Sokalski, Physical Review Letters., 119, 027203 (2017) A. A. Wheeler, Journal of Statistical Physics., 95, 1245–1280 (1999)

Energetics, equilibrium shape, and vorticity of skyrmions and antiskyrmions stabilized by the anisotropic Dzyaloshinskii-Moriya interaction in thin films with C2v symmetry

Antiskyrmions and other unconventional topological excitations can potentially be supported by low-symmetry ferromagnet--heavy metal (FM-HM) thin films by the Dresselhaus-type anisotropic Dzyaloshinskii-Moriya interaction (aDMI) arising at the FM-HM interfaces. This work presents the potential of C2V Ag/Pt/Co-based films to exhibit aDMI and stabilize new chiral magnetic textures. X-ray diffraction 2θ-ω scans of Ag/Pt/Co/Pt films grown epitaxially on Si(110) single crystal substrates show that Co grows predominantly with (10.0) hexagonal closed-packed (hcp) orientation, deviating from the (110)-oriented face-centered cubic (fcc) structure of Ag and Pt. These findings were corroborated by selected area electron diffraction patterns and further supported by a strong in-plane uniaxial anisotropy measured along the hcp c-axis by alternating gradient field magnetometry. Crosstie domain walls in these films were directly observed by Lorentz transmission electron microscopy, which are also predicted from micromagnetic simulations. The materials system introduced above creates the previously unexplored situation where an in-plane easy axis is coupled with a strong (potentially anisotropic) DMI, which may lead to interesting spin configurations. However, to break the z-mirror plane [v1] and create a non-zero DMI, we fabricate thinner Ag/Pt/[Co/Ni]x/Pt films. X-ray diffraction analysis suggest that the hcp structure is preserved throughout the Ni layers. Structural and magnetic observations of this lower symmetry film stack are compared to micromagnetic simulations and discussed in the context of C2V films potentially exhibiting Dresselhaus-type aDMI and hosting antiskyrmions.References: 1. U. Güngördü et al., Physical Review B., 93, 064428 (2016) 2. M. Hoffman et al., Nature Communications., 8, 308 (2017)

In Pursuit of the Anisotropic Dzyaloshinskii-Moriya Interaction: Structural and Magnetic Characterization of Uniaxial Pt/Co-based C2V Thin Films

**Session chair:** Sadamichi Maekawa, RIKEN


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Michael Kitcher

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SHORT BIO

Presentations

Deterministic Bloch Chirality in Interfacial DMI Systems: An Astroid for DW Switching

Impacts of Steady-State Domain Wall Configuration on Domain Wall Stiffness and Directional Domain Propagation in the Creep Regime

Structural and Magnetic Characterization of Epitaxial Co(10.0)/Pt(110) Multi-layers for Future Anisotropic DMI Systems Based on C2v Symmetry

Energetics, equilibrium shape, and vorticity of skyrmions and antiskyrmions stabilized by the anisotropic Dzyaloshinskii-Moriya interaction in thin films with C2v symmetry

Energetics, equilibrium shape, and vorticity of skyrmions and antiskyrmions stabilized by the anisotropic Dzyaloshinskii-Moriya interaction in thin films with C2v symmetry

In Pursuit of the Anisotropic Dzyaloshinskii-Moriya Interaction: Structural and Magnetic Characterization of Uniaxial Pt/Co-based C2V Thin Films

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