Unidirectional tilt of domain walls in equilibrium in biaxial stripes with Dzyaloshinskii-Moriya interaction

The orientation of a chiral magnetic domain wall in a racetrack determines its dynamical properties. In equilibrium, magnetic domain walls are expected to be oriented perpendicular to the stripe axis. We demonstrate the appearance of a unidirectional domain wall tilt in out-of-plane magnetized stripes with biaxial anisotropy and Dzyaloshinskii--Moriya interaction (DMI). The tilt is a result of the interplay between the in-plane easy-axis anisotropy and DMI. We show that the additional anisotropy and DMI prefer different domain wall structure: anisotropy links the magnetization azimuthal angle inside the domain wall with the anisotropy direction in contrast to DMI, which prefers the magnetization perpendicular to the domain wall plane. Their balance with the energy gain due to domain wall extension defines the equilibrium magnetization the domain wall tilting. We demonstrate that the Walker field and the corresponding Walker velocity of the domain wall can be enhanced in the system supporting tilted walls.

The orientation of a chiral magnetic domain wall in a racetrack determines its dynamical properties. In equilibrium, magnetic domain walls are expected to be oriented perpendicular to the stripe axis. We demonstrate the appearance of a unidirectional domain wall tilt in out-of-plane magnetized stripes with biaxial anisotropy and Dzyaloshinskii-Moriya interaction (DMI). The tilt is a result of the interplay between the in-plane easy-axis anisotropy and DMI. We show that the additional anisotropy and DMI prefer different domain wall structure: anisotropy links the magnetization azimuthal angle inside the domain wall with the anisotropy direction in contrast to DMI, which prefers the magnetization perpendicular to the domain wall plane. Their balance with the energy gain due to domain wall extension defines the equilibrium magnetization the domain wall tilting. We demonstrate that the Walker field and the corresponding Walker velocity of the domain wall can be enhanced in the system supporting tilted walls.
Spin orbitronics relies on the manipulation of magnetic textures via spin orbit torques and enables new devices ideas for application in magnetic storage and logics [1][2][3][4][5] . The key component of these devices is a stripe with out of plane easy axis of magnetization and featuring the Dzyaloshinskii-Moriya interaction (DMI). Typically, asymmetrically sandwiched ultrathin films of ferromagnets are used, where the DMI originates from the broken symmetry at the film interfaces 6,7 . There are numerous demonstrations of the energy efficient and fast motion of chiral magnetic solitons including skyrmions [8][9][10] , skyrmion-bubbles 11,12 and domain walls [13][14][15] in stripes.
The orientation of the plane of a domain wall with respect to the stripe main axis has major impact on its dynamics including the maximum velocity 16 and Walker limit 17,18 . It is commonly accepted that in equilibrium the plane of a magnetic domain wall is perpendicular to the stripe main axis. This remains true even if the sample possesses DMI. Domain wall can acquire a tilt yet only if exposed to an external magnetic field [19][20][21][22][23] , driven by a current 16,19,[24][25][26][27] , or pinned on edge roughness during current-induced dynamics 28 . Being exposed to an in-plane magnetic field, domain wall tilts unidirectionaly with the rotation direction determined by the sign of the DMI. The tilt increases linearly with the field strength and the slope of the resulting dependence was proposed to be used for the determination of the DMI constant 19,22 .
Here, we demonstrate that domain walls can acquire a unidirectional tilt even at equilibrium if the out-of-plane maga) Electronic mail: o.pylypovskyi@hzdr.de b) Electronic mail: volodymyr.kravchuk@kit.edu c) Electronic mail: o.volkov@hzdr.de d) Electronic mail: j.fassbender@hzdr.de e) Electronic mail: sheka@knu.ua f) Electronic mail: d.makarov@hzdr.de netized stripe possesses DMI and an additional easy-axis anisotropy in the plane of the stripe. The easy axis direction of the in-plane anisotropy can be given by a crystalline structure of the ferromagnet 29 or induced via exchange bias when the stripe is in proximity to an antiferromagnet 4,30-32 . In contrast to the shape anisotropy 17,33 with an easy axis along the stripe, any misalignment between the in-plane anisotropy and the stripe axes breaks the symmetry of the magnetic texture and tilts (i) the magnetization inside the domain wall as well as (ii) the plane of the domain wall. The motion a domain wall in a biaxial magnetic with DMI has strong impact on the Walker field and the domain wall velocity. The obtained results allow to design stripes with stronger Walker field (i.e. extend range of the linear motion of domain walls) and control the domain wall nucleation process in T-junctions by selecting the initial tilt direction 34,35 .
We consider an infinitely long magnetic thin stripe of thickness h and width w. The total magnetic energy of the stripe is where the integration is performed over the sample's area in xy plane (x axis is along the stripe). The first energy term is the exchange energy density E x = A ∑ i=x,y,z (∂ i m) 2 with A being the exchange stiffness, m = M /M S being the unit magnetization vector and M S being the saturation magnetization. The second term is the anisotropy energy density of a biaxial magnet, The easy axis of the in-plane anisotropy e 2 lies in the stripe's plane at an angle α to thex direction. The third energy term is the energy density of the DMI 36,37 The last energy term is the Zeeman energy density E Z = −M S Bm z with B being an external magnetic field intensity. We assume, that the magnetostatic interaction can be reduced to a local anisotropy and results in the renormalization of the first anisotropy constant strength of the magnetic anisotropy with an out-of-plane easy axis.
To describe the structure of the domain wall, we apply the following ansatz 19 : where the magnetization vector is parametrized as m = {sin θ cos φ , sin θ sin φ , cos θ } in the local spherical reference frame with θ and φ being polar and azimuthal angles, respectively, p = ±1 is the topological charge of the domain wall (kink or anti-kink), = A/K 1 is the magnetic length, ∆ and q are domain wall width and position of its center, respectively, measured in units of . The origin is placed in the center of the stripe. The angle ψ ∈ (−180 • , 180 • ] describes the tilt of the magnetization inside the domain wall with respect toŷ axis and the angle χ ∈ (−90 • , 90 • ) characterizes the mechanical tilt of the plane of the domain wall with respect toŷ, see Fig. 1(a). In this notation, ψ = 0 or 180 • and ψ = 90 • or −90 • with χ = 0 corresponds to Néel and Bloch domain walls, respectively. The plane of the domain wall is perpendicular to the stripe axis when χ = 0.
The total energy, normalized by E 0 = 2K 1 hw , reads where is the dimensionless parameter characterizing the DMI strength and b = M S B/K 1 is the normalized magnetic field B alongẑ. Note, that the DMI parameter d 0 incorporates the topological charge of the domain wall p. The static domain wall configuration (b = 0, and q = 0 without loss of generality) is given by the minimum of the energy (2) with respect to ∆, χ and ψ. The equilibrium domain wall width is ∆ 0 (ψ) = 1/ 1 − k 2 sin 2 (ψ − α). The relation between values of the angles χ and ψ in equilibrium reads After the substitution of (3) in (2), we obtain the expression for the energy as a function of the angle ψ, characterizing the orientation of the magnetization in the wall: There are several limiting cases related to the absence of the in-plane anisotropy (k 2 = 0) or DMI (d 0 = 0). If k 2 = 0 and d 0 = 0, we obtain a classical case when a magnetic stripe with perpendicular anisotropy can support Bloch domain walls (ψ = 0, 180 • as a consequence of minimization of magnetostatic energy), with a plane of the domain wall being perpendicular to the stripe axis (χ = 0). For any finite k 2 (still when d 0 = 0), the magnetization in the domain wall is tilted to its equilibrium value of ψ = α ± 90 • . This corresponds to the two equivalent minima in the energy (4), see red line Fig. 1(b). However, the plane of the domain wall remains perpendicular to the stripe axis (χ = 0). This result is expected from the analysis of (3)  In an extended film, the domain wall is always oriented perpendicularly to e 2 and the DMI energy favors ψ = χ or ψ = χ + 180 • (magnetization rotates perpendicularly to the domain wall plane). In a stripe of a finite width, the balance between the domain wall tension energy (proportional to its length and increasing with χ) and the DMI energy results in a certain value of χ, which is different from ψ. Note that the model (1) is applicable for relatively narrow stripes, where the domain wall shape can be approximated by a straight line. For wide stripes the curvilinear distortion of the domain wall shape must be taken into account. The domain wall tilt angle χ rapidly increases when d 0 approaches its critical value. The domain wall structure is shown in Fig. 1(d), where tilt angles χ and ψ as well as the orientation of the easy axis of the in-plane anisotropy are depicted. The size of the bistability region in terms of the DMI parameter d bis 0 is shown in Fig. 1(e). Note, that the state α = 0 is degenerated with the domain wall tilt χ ≡ 0.
The dependencies of the domain wall (χ) and magnetization (ψ) tilt angles on the orientation of the easy axis of the in-plane anisotropy (α) is summarized in Fig. 2(b,c). The sign of χ is given by the direction of the anisotropy axis α, while the sign of ψ is opposite to the sign of d 0 . The domain wall tilt angle monotonically increases with the increase of d 0 and k 2 , while the magnetization tilt angle is mainly determined by the k 2 for the case of strong DMI.
In the following, we address the dynamics of domain walls driven by an external magnetic field applied alongẑ. We apply a collective variables approach 39,40 , considering the wall position q(t), the magnetization tilt ψ(t), domain wall tilt χ(t) and the domain wall width ∆(t) as time-dependent quantities. The solutions of the corresponding Euler-Lagrange-Rayleigh equations are found numerically, see Supplementary Materials for details and compared with micromagnetic simulations 41 , see Fig. 3(a). The analysis is performed in the fields, which are smaller than the Walker field (b < b W ) 42 . In this case, the tilt angles ψ and χ and the domain wall width quickly relax to their equilibrium values ψ ∞ and χ ∞ , respectively (see also Eqns. (S-11)-(S-13) in Supplementary Materials). The domain wall velocity with equilibrium values of its width and angles reads v = pb  where the dimensionless velocity v is measured in units of 2γ √ AK 1 /M S with γ being gyromagnetic ratio and η being Gilbert damping. Note, that the maximum of the Walker field and, hence, the largest velocity is reached at the angle α 0 ≈ 60 • for the given parameters, which does not coincide with a shape anisotropy along the stripe main axis 17,33 . Asymptotic analysis shows a good coincidence with numerical solution of equations of motion even for large enough fields and material parameters, see Fig. 3

(d) and Supplementary Materials for details.
Upon the motion of the domain wall, its internal structure changes dependent on the direction of the easy axis of the in-plane anisotropy α and on the strength of the applied magnetic field b (Fig. 3). There exists a symmetry break with a favorable magnetization tilt direction (indicated as fav states in Fig. 3), in a small angular range about the orientation of e 2 (along or opposite to it). It results in a higher velocity at a given field. The domain wall with unfavorable tilt angle (indicated as unf states in Fig. 3) will switch the internal magnetization in the wall to the e 2 direction at the beginning of motion. The red dashed line in Fig. 3(b) indicates the smallest field, which is needed for switching of the magnetization angle in the wall. The positive b results in the appearance of "unf" state for negative α and vice versa. Sign of χ coincides with sign of b for fields, close to b W . We note a certain similarity with the texture-induced chirality breaking for moving magnetic domain walls in nanotubes due to non-local magnetostatics 43,44 .
To summarize, we investigate the internal domain wall structure and its orientation in an out-of-plane mangetized stripe with biaxial (in-plane and out-of-plane) anisotropy and Dzyaloshinskii-Moriya interaction. The cooperative effect of the DMI and the additional anisotropy with an in-plane easy axis results in a unidirectional tilt of domain wall in equilibrium and in a symmetry break of a domain wall static state with respect to the stripe axis. The domain wall dynamics in an applied out-of-plane magnetic field exhibits slow and fast motion similarly to vortex domain walls in tubes 43,44 . We demonstrate that the Walker field but also the associated Walker velocity strongly depend on the orientation of the easy axis of the in-plane anisotropy. There appears an optimal angle of the orientation of the in-plane easy axis to maximize the Walker field and the Walker velocity. This optimal angle does not coincide with the direction of the easy axis of the shape anisotropy. These results are relevant for the optimization of the domain wall dynamics in data storage and logic devices, relying on spintronic and spin-orbitronic concepts. See

I. ASYMPTOTIC ANALYSIS OF STATIC DOMAIN WALL
Equilibrium values of domain wall and magnetization tilt angles χ and ψ can be found analytically considering corrections ∆χ and ∆ψ to the case of d 0 = 0 and k 2 > 0 to be small and by expanding energy (2) to ∆χ 2 and ∆ψ 2 , see main text. In the case of small d 0 and k 2 , the tilt angles read: with respect to the stripe axisx. The system considered can be described by Lagrangian 3 and dissipative function 4 where τ = ω 0 t is the dimensionless time with ω 0 = 2K 1 γ/M S and dot means derivative with respect to τ. The normalized Lagrange and dissipative functions read where E 0 = 2K 1 hw , and tan χ ∆χ∆ (S-6) withw = w/ being the dimensionless stripe width. Then, (S-4) read . (S-12) The value of ψ ∞ can be found numerically from the following equation: Comparison of numerically found χ ∞ and ψ ∞ from Eqns. (S-12) and (S-13) with simulations for different material parameters is shown in Fig. S-4. Considering case of small deviations of values of χ lin static and ψ lin static during the domain wall motion, one can found the following asymptotic expressions for the stable branch: where values for the stable branch of χ and ψ (+ and − sign, respectively) are used. Comparison of asymptotics (S-14) with numerical solution of (S-7)-(S-10) for different material parameters is shown in Fig. S-5.