Characterizing an effective magnetic field during asymmetric creep motion of Dzyaloshinskii domain walls

Understanding the role of Landau Lifshitz Gilbert dynamics in describing magnetic domain wall (DW) motion in the creep regime is complicated by the presence of static pinning, but has regained interest due to recently observed directional growth in thin films with significant interfacial Dzyaloshinskii–Moriya interaction. Here, we delve into this directional domain growth behaviour in Pt/Co/Ni-based multi-layers under the influence of combined longitudinal and perpendicular magnetic fields via magneto-optical Kerr effect microscopy. Observations, including the onset field, μ0Honset , where the growth direction reverses by 180 degrees, align with the transient steady-state model predictions. By systematically varying the applied perpendicular magnetic field, we estimate the strength of an effective perpendicular field that acts on the DW during creep movement, which was expectedly found to be much smaller than the applied external field itself. This work further adds to the complexity of asymmetric domain expansion in the creep regime, but also highlights the range of magnetic information that can be extracted from careful analysis of this behavior.


Introduction
In the creep regime, motion of magnetic domain walls (DWs) occurs via thermally activated steps through energy barriers originating from pinning sites, such as defects or inhomogeneities in the magnetic material.The role of Landau Lifshitz Gilbert (LLG) dynamics, which have traditionally been applied to the flow regime where movement is continuous, remain poorly understood in this regime and any effective field acting on the wall to drive it into a non-static configuration is often neglected [1].However, it has been posited that LLG dynamics could describe a transient steadystate of the DW during this creep motion as the wall traverses between pinned locations [2].This is evidenced by systematic changes in directionality of expanding domains under varying magnetic fields in perpendicularly magnetized Co/Ni based multi-layers with a strong interfacial Dzyaloshinskii-Moriya interaction (DMI) at the heavy metal/ferromagnetic interface [1].The interfacial DMI serves to amplify the highly anisotropic elastic energy of the DW when subjected to a longitudinal magnetic field as has been well-established elsewhere [3][4][5][6][7][8][9][10][11][12][13].Indeed, the directional growth seen by Brock et al (and reproduced here) has been explained by augmenting the dispersive stiffness model of Pellegren to include steady-state LLG dynamics [1].It is therefore evident that LLG dynamics and non-static DW configurations play a key role in explaining the abnormal directional growth in some thin films with significant interfacial DMI, under an external longitudinal magnetic field.
Recall that bubbles, skyrmions, or pinned DWs will adopt a static size and magnetic configuration under an externally applied magnetic field such that the effective magnetic field acting on the system is exactly zero in the context of LLG dynamics.Conversely, an unpinned DW in a uniform magnetic medium will experience an effective field exactly equal to the applied field where flow dynamics properly capture its steadystate velocity and magnetic configuration [14].It is reasonable to expect that the case of a DW stochastically moving through a non-uniformly distributed energy landscape as in the creep regime will experience an effective field that falls between these limiting cases.However, there have not previously been attempts to quantify the strength of this effective field based on domain expansion, which is the central focus of this work.Here, we leverage the highly anisotropic DW energy provided by perpendicularly magnetized Pt/Co/Ni multi-layers to perform a systematic study of the impact of perpendicular and longitudinal field on growth direction.Coupled with the steady state dispersive stiffness model noted previously, we are able to estimate the strength of the effective magnetic field acting on the DW during creep motion.

Experimental setup
Thin film magnetic multi-layers were prepared on oxidized silicon substrates by DC magnetron sputter deposition with a base pressure less than 4 × 10 −8 kPa and a working pressure fixed at 0.333 Pa Argon as reported previously [3,15,16].The film structure examined in this work was Ta(3)/Pt(3.5)/[Co(0.2)/Ni(0.6)]× 4/Ir(0.5)/Pt(3)/Ta(3)(thicknesses in nm).For this film stack, Ta is used as an amorphous adhesion layer, Pt seeds FCC(111) texture for the subsequent Co and Ni layers.Pt and Ir both serve to establish a significant interfacial DMI as previously reported although the contribution made by Ir may be weaker than originally believed [17].The Co/Ni interfaces preserve perpendicular magnetic anisotropy in the system as needed to observe perpendicular domain expansion.
Saturation magnetization and perpendicular magnetic anisotropy were determined via vibrating sample magnetometry at room temperature in both in-plane and out-of-plane geometries (see SI). White light magneto-optic Kerr effect (MOKE) microscopy, configured for polar MOKE sensitivity, was used to image the magnetic domain growth.Out-of-plane magnetic field pulses, µ 0 H z , were generated by a magnetic coil which is powered by a current source, providing up to 20 mT magnetic fields with pulse lengths of at least 100 µs.These field pulses were used to nucleate and expand magnetic domains with velocities and growth direction determined by static images taken before and after each pulse.The static in-plane magnetic fields, µ 0 H x , were varied in magnitude between 0 and 60 mT using an electromagnet with a tapered pole piece to concentrate field.The external fields are aligned with the in-and out-of-plane direction, which was verified by the measured asymmetric domain growth velocity (see SI).The overall growth was confirmed to occur in the creep regime over the range of fields studied as confirmed by fits to the creep scaling law [18][19][20] (see SI): where v 0 is the velocity scaling factor, χ is an energy barrier scaling constant, and ε is the elastic energy of the DW that accounts for changes in wall profile and orientation as it moves.Higher elastic energy results in lower DW motion velocity.

Experimental results
The DW coordinate system used in this work is shown in figure 1. Figure 2 shows an example set of results examining the impact of µ 0 H x (figure 2(b)) and µ 0 H z (figure 2(d)) on the propagation direction of magnetic domains, where the favored domain growth directions were determined from the largest DW movements compared to the nucleation sites (labeled as blue dots).As µ 0 H x increases, there is a gradual change in the growth direction from nearly vertical to nearly horizontal.Upon further increasing µ 0 H x , the growth direction abruptly reverses direction (see figure 2(a) as previously seen by Brock et al and is consistent with the corresponding transient model presented in that work.To avoid numerous factors that can contribute to the absolute value of growth velocities in these systems, we focus our attention on the growth direction, specifically, and use the aforementioned reversal field, henceforth referred to as µ 0 H onset , as a defining parameter of the system's behavior.
As will be discussed later, the transient model overwhelmingly predicts that the DW will be in a state of Walker Breakdown if we input the value of the externally applied perpendicular field as the effective perpendicular field in the LLG equation.Nonetheless, we do see clearly that as the strength of µ 0 H z is increased, there is a marked drop in µ 0 H onset (figure 3).This likely originates from a stronger field torque that drives the DW magnetization further away from its static configuration and ultimately leads to a sweeping change of the angular dependence of the DW's elastic energy.
To quantify the behavior described above; in particular, the strength of the effective perpendicular field resulting from the applied field, we turn our attention to the steady-state model based on the dispersive elastic energy of the DW [1].As noted in prior work, the growth velocity of a DW in the creep regime is directly related to it is elastic energy, which becomes highly anisotropic in the case of thin films with both strong interfacial DMI and a longitudinal magnetic field [4].

Analytical modeling
The transient steady-state model is built on four governing equations, which can be used to analytically predict the unusual DW growth directions recently reported [1,5,21,22].In the presence of only in-plane magnetic field, the 1D linear DW energy density σ is expressed as [4]: In equation ( 2), the θ and φ are azimuthal orientations and DW core magnetization direction of the DW section, respectively (shown in figure 1).The first term is the Bloch wall energy, σ 0 = 4 √ AK eff .The second term is the DW anisotropy energy, which arises from the Néel wall demagnetizing field and degenerately favors Bloch type DWs of either chirality, where t f is film thickness, µ 0 is vacuum permeability, and M s is saturation magnetization.The third term is the energy contribution stemming from the interfacial DMI which drives the DW towards a chiral Néel configuration, where D is the DMI constant.The last term is the Zeeman energy which is associated with the misalignment between DW magnetization and the direction of applied in-plane field, where λ 0 represents the DW width.Therefore, the static state magnetization profile of the DW can be determined from minimized energy states with respect to φ at all azimuthal positions θ.
When a perpendicular magnetic field µ 0 H z is applied and the magnetic domain expands, the internal magnetization of DW will experience a torque which rotates the magnetization angle away from the static state.As noted previously, pinning effects from the sample will counter the applied perpendicular field leading to an effective field that is appreciably smaller.The role of such an effective field is captured by the LLG dynamics of equation ( 3) [14,23,24] where γ is the gyromagnetic ratio, α is the Gilbert damping constant and Ω A = σφ 2λ0Ms .The subscripts to σ denote partial derivatives.In a steady state, φ = 0. Consequently, the torque exerted on the internal magnetization of the DW can be represented as σ φ = 2λ0µ0MsHz α .In the creep regime, the thermally activated DW motion follows an Arrhenius scaling law, given in equation ( 1) [18][19][20].As highlighted by Pellegren et al, when DW energies are anisotropic in θ, the elasticity energy density ε is given by the dispersive stiffness σ instead of DW energy σ [4].For DWs of arbitrary length L, the generalized dispersive stiffness is given by equation ( 4) [4].
where all partial derivatives are evaluated at the steady state φ and θ, Λ = λ 0 √ σ0 σφ φ , and The first term is the DW energy density from equation ( 2), the second term involves the curvature of the DW, capturing how the spatial variation in curvature contributes to the overall elastic behavior, and the third term establishes a relation between the length of the DW section L, and the vertical Bloch line length Λ.A similar degree of agreement between the experimental data and the analytical modeling was found over a wide range of L values in a short-wavelength limiting case of the stiffness model (L → 0).
In the presence of in-plane field H x , the velocity expression 1 can be reformulated as equation ( 5) [4] Therefore, by computing the DW dispersive stiffness σ with respect to azimuthal positions θ, we can deduce the relative DW velocity v v0 as well as the preferred growth direction.The parameters used for calculation were as follows: D = −0.0738mJ m −2 (see SI), L = 70 nm, A ex = 5.4 pJ m −1 [16], K eff = 7.5 × 10 4 J m −3 , and the experimentally determined M s = 600 kA m −1 [16].A range of values were used for α   In the case of the µ 0 H x dependent calculation in figure 3(a), for lower µ 0 H x the favored domain growth is nearly perpendicular to the in-plane field.As µ 0 H x is increased, the favored domain growth direction undergoes a clockwise rotation, and grows opposite to µ 0 H x .Further increasing µ 0 H x , a strong directionality flipping occurs.We note that there is not a single field where µ 0 H onset occurs either experimentally or in the model.Instead this occurs over a narrow range where the preference for either direction is modest.Therefore, the average of starting and ending fields of the transitional regions are used to quantify a single value for µ 0 H onset .In calculation, a clear trend can be observed from figure 3(b) that µ 0 H onset decreases with increasing perpendicular field µ 0 H z which matches with the experiments (figure 4).At even higher field in figure 3(a), the domain growth to the right becomes dominant.
As anticipated, there is an offset between modeled effective µ 0 H z and experimentally applied µ 0 H z in figure 2(c).This discrepancy can be described by a propagation field µ 0 H propagation required for the DW to begin moving, where µ 0 H z,effective = µ 0 H z,applied − µ 0 H propagation [25].In creep regime, when the DW jumps between pinning sites with the aid of thermal activation and external fields, the DW section bounded by pinning sites experiences a restoring field from the curved DW, resulting in µ 0 H propagation as the threshold of DW expansion.By extracting the µ 0 H onset and corresponding effective µ 0 H z from analytical data, µ 0 H propagation can be visualized from onset curves plotted in figure 4. Therefore, µ 0 H propagation can be deduced from the gap between experimental onset curve (plotted in larger blue dots and line on right) and computational onset curve (plotted in larger blue empty circles and line on left) that share the same slope, which explained the gap labeled in figure 2(c).The slopes of onset curves are directly related to the damping constant, where lower damping constant results in steeper onset curve, following a power function relationship (shown in inset figure in figure 4).Thus, µ 0 H propagation is approximately 4.7 mT based on gap measuring, and the damping constant of our sample is 0.033 based on the slope matching with calculation, which is comparable to the value of 0.038 for a similar [Co/Ni] × 4 sample according to Rai et al [16] Therefore, by using extracted value α = 0.033, the computational results were plotted in figures 2(a) and (c), showing a good match with experiments.In a multi-layer thin film system, as the heavy metal layer (such as Pt here) increases in thickness or repetition, the damping constant α will increase and results in a smaller dependence of onset fields on µ 0 H z (more gradual onset curve slope).While within the thin film limit, the iDMI and increased thin film thickness will lead to a reduced equilibrium domain size [26], and these dendritic domains, compared to circular domains in thinner films, have more curved domain tips which may act against domain expansion and contribute to larger propagation field.However, how the film thickness affects effective field still remains unclear.

Conclusions
In conclusion, asymmetric and highly directional domain growth induced by both in-plane and out-of-plane field was evaluated in a Pt/Co/Ni-based thin film with interfacial DMI.An analytical model, based on the DW dispersive stiffness and dynamic symmetry breaking of DW magnetization configurations, was used to analyze the abnormal DW growth directionality including the abrupt reversal in direction as described in Brock et al [1].We show that the onset field where this reversal occurs is inversely related to the magnitude of the applied perpendicular field.Moreover, we are able to use this data to extract both the damping constant of the thin film and the effective perpendicular field acting on the DW.We conclude that the latter is non-zero despite common assumptions to the contrary.This analysis further highlights the complexity of creep growth of magnetic DWs, but also adds to myriad magnetic properties that can be elucidated from such experiments.

Figure 1 .
Figure 1.Domain wall coordinate system, where θ is the azimuth of the DW normal, and φ is that of the DW core magnetization direction.The positive horizontal field goes from left to right, and the positive perpendicular field aligns with the out-of-plane direction.

Figure 2 .
Figure 2. (a) Domain wall favored growth direction with respect to µ 0 Hx at µ 0 Hz = 6.17 mT.(b) Kerr images of domain expansion with µ 0 Hz = 6.17 mT and varying µ 0 Hx, where the blue dots indicate the initial nucleation sites of the dendritic strip domains, and blue arrows indicate the favored domain growth directions.(c) Domain wall favored growth direction with respect to µ 0 Hz at µ 0 Hx = 40 mT.(d) Kerr images of domain expansion with µ 0 Hx = 40 mT and varying µ 0 Hz (α = 0.033 was used for calculation fitting in (a) and (c), which was extracted from onset curve fitting in figure 4).

Figure 3 .
Figure 3. (a) Predicted DW magnetization profiles under the assumption of static equilibrium (red arrows) and steady-state dynamic reorientation (blue arrows) under applied out-of plane field of µ 0 Hz = 0.9 mT and various in-plane fields.The polar profile shown in the center of each diagram represents the relative domain wall growth velocity v/vmax as a function of azimuthal position.(b) Modeling results of favored domain growth direction for µ 0 Hz = 0.7 mT, 0.8 mT, and 0.9 mT, where µ 0 Honset are labeled by dashed lines, which qualitatively match with experimental observations (A range of values were used for α in the calculation, from 0.02 to 1.0, and here shows an example of α = 0.04)

Figure 4 .
Figure 4. µ 0 Honset vs. µ 0 Hz of experimental result and calculation with different damping constants.The experimental results are plotted in larger blue dots.The onset curve slopes follow a power function relationship shown in the inset figure, where the calculated damping constant is labeled by a red dot.