Dynamic Bloch Chirality and Enhanced Velocities from Spin-Orbit Torque Driven Domain Wall Motion in Thick Magnetic Films

Abstract

Spin-orbit torque (SOT) driven domain wall motion has attracted significant attention as the basis for a variety of spintronic devices due to its potential use as a high speed, low power means to manipulate the magnetic state of an object. While most previous attention has focused on ultrathin films wherein the material thickness is significantly less than the magnetic exchange length, recent reports have suggested unique dynamics may be achieved in intermediate and high thickness films. We used micromagnetic modelling to explore the role of the vertically non-uniform spin textures associated with the domain wall in nanowires of varying thickness on SOT driven domain wall motion. We found large velocity asymmetries between Bloch chiralities near the current density required for reversal of the Bloch component of the magnetization and linked these asymmetries to a gradual reorientation of the domain wall structure which drives a non-negligible, chiral Néel component of the domain wall. We further explored the influence of saturation magnetization, film thickness, the Dzyaloshinskii-Moriya interaction, and in-plane fields on domain wall dynamics. These results provide a framework for the development of SOT based devices based on domain wall motion in nanowires beyond the ultrathin film limit.

1. Introduction

The propagation of magnetic domain walls under the influence of spin polarized currents has garnered intense interest due to its potential applications in a variety of spintronic memory, logic, and sensing devices [1,2,3,4,5,6,7,8] as well as the intriguing fundamental physics governing the interaction of spin currents with magnetization [9,10,11,12,13,14]. Traditionally, current induced motion has been explored in magnetic states wherein the static domain wall structure may be described as fully Bloch or Néel-like. In ultrathin films with a strong Dzyaloshinskii-Moriya interaction (DMI) driven by symmetry breaking at interfaces, for example, a Néel domain wall may form [15,16,17]. This Néel structure allows field-free current driven domain wall motion while enhancing domain wall velocities and suppressing the Walker breakdown field [18]. In films without DMI, a Bloch structure is favorable, and, without an in-plane symmetry breaking field, domain wall motion is suppressed [19,20,21,22].

In magnetic films with perpendicular anisotropy, no DMI, and a film thickness greater than the exchange length, however, domain walls cannot be characterized as purely Bloch or Néel, and instead the competition of exchange and demagnetization energies results in a flux closure state, with Néel caps forming at the surfaces with an achiral central Bloch component [23,24,25,26,27], as shown in Figure 1. The sense of rotation of the Néel caps across the domain wall boundary is determined by the relative orientation of the magnetization on each side of the domain wall. For moderate DMI, the Néel cap corresponding to the chirality energetically preferred by the DMI will be enhanced, with the energetically unfavorable Néel cap suppressed. These non-uniform structures play a key role in current driven magnetization dynamics. For instance, this structure modifies the skyrmion Hall angle during current driven skyrmion motion and allows for enhanced or suppressed velocities depending on the specific current injection geometry, even in the absence of DMI [25,26,28,29]. Further, the spin current itself can result in modified domain wall structure, leading to current density dependent dynamics and current driven switching of the Bloch component of the magnetization [30]. However, the role of this structure and its interaction with spin currents on current induced domain wall motion in thick, perpendicularly magnetized nanotracks has been largely unexplored.

In this work, we use micromagnetic simulations to understand the reorientation process in thick magnetic films during application of spin polarized currents and the corresponding domain wall motion as a function of injection geometry, film thickness, current density, and saturation magnetization. We found that, for Bloch chiralities aligned antiparallel to the spin current polarization direction, the domain wall first reorients with a chiral Néel component resulting in an increased propagation velocity near the spin current required for domain wall reorientation, even in the absence of DMI or an applied in-plane field. The maximum velocity and required current density for switching the Bloch chirality may be controlled via application of an in-plane field, either parallel or orthogonal to the wire long axis. We further explored the combined effects of this current driven reorientation and moderate DMI.

2. Methods

Micromagnetic modelling of domain wall velocities in a nanowire was carried out using MuMax3 [31,32]. The nanowire geometry was chosen with dimensions of $1024 \mathrm{nm}\times 128 \mathrm{nm}\times t$, where $t$ is the film thickness in nanometers with a mesh of $512\times 128\times N$ cells and $N$ varied from 1 to 10. Unless stated otherwise, $t=10 \mathrm{nm}$ with $N=5$. Material parameters were chosen to be consistent with standard Co-based multilayers and alloys [1,33,34,35]. The saturation magnetization, $M_s$, was set to $400 \mathrm{kA}/\mathrm{m}$ unless otherwise stated. For all simulations, we used a damping $\alpha =0.1$, an exchange stiffness $A_{ex}=10 \mathrm{pJ}/\mathrm{m}$, and a uniaxial anisotropy $K_u=4.0\times {10}^5 \mathrm{J}/{\mathrm{m}}^3$. For $M_s=400 \mathrm{kA}/\mathrm{m}$, this corresponds to an exchange length $l_{ex}$ of nominally $10 \mathrm{nm}$, where $l_{ex}=\sqrt{2A_{ex}/{\mu }_0{M}_s^2 }$ and ${\mu }_0=4\pi \times {10}^{-7} \mathrm{H}/\mathrm{m}$ is the vacuum permeability. We note that, for realistic films, the anisotropy may vary significantly with the film thickness, and more work needs to be carried out in order to better understand the potential effect this may have in determining the specific domain velocity scaling of the geometries studied in this work. The nanowire was initialized and relaxed in a two-domain state, with the magnetization pointing in the positive z-direction to the left of the domain wall and in the negative z-direction to the right of the domain wall. The central Bloch component of the domain wall was chosen to lie along the positive y-direction or negative y-direction, wherein wires with central Bloch magnetization along the positive and negative y-directions are denoted as $M_Y=1$ and $M_Y=-1$ respectively. The system was then allowed to relax at zero magnetic field and without any applied currents to obtain the initial magnetization state. The current density, $\overrightarrow{j}$, was modeled as a vertical current fully spin polarized along the y-axis. Note that only damping-like torques, which can originate from the spin Hall effect and current flowing along the x-direction, were considered in this work. The spin current was modeled as uniform across the film thickness, as is the case for bulk spin orbit torques (SOTs) recently found in a variety of ferro- and ferrimagnetic multilayers and alloys [14,35,36]. To exclude edge effects during domain wall motion, the simulation window was dynamically shifted such that the domain wall remained in the center, approximating an infinitely long wire. An example simulation file is included as a Supplementary File S1. As the damping-like term acts to align the magnetization along the spin polarization direction [30], large current densities will align the Bloch component of the magnetization along the spin polarization direction. As such, near the reorientation current density, a dynamic step size in current density was used with measuring domain wall velocities as a function of current density, with a step size of $2\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$ immediately prior to reversal. This allows for a better capture of the maximum domain wall velocities near the critical current density. For the simulations carried in this work, a step size on the order of $1\times {10}^{-13} \mathrm{s}$ was used, corresponding to roughly 500,000 total steps per velocity measurement. Results are also compared to thin films with a weak DMI of $\left|D\right|=50 \mathsf{\mu }\mathrm{J}/{\mathrm{m}}^2$ throughout the film thickness, comparable to recently discovered bulk DMI in a variety of ferro- and ferrimagnetic alloys [33,35,37]. All simulations were carried out at $0 \mathrm{K}$, unless otherwise noted. In simulations at non-zero temperatures, a temperature of $300 \mathrm{K}$ is implemented by modelling the temperature as an effective fluctuating thermal field [31,38].

3. Results and Discussion

As shown in Figure 2, at large current densities, only a single Bloch chirality is stable due to realignment of the domain wall along the spin polarization direction. The chirality is determined by the current spin polarization direction, and the velocity for both chiralities is negative with a speed of a few m/s. However, at low current densities, both Bloch chiralities are stable. In the case wherein the Bloch component of the domain wall is aligned antiparallel to the current polarization direction, a sharp increase of the domain wall velocity is observed up to the current required for chirality reversal. For the 10 nm thick film with $M_s=400 \mathrm{kA}/\mathrm{m}$, the velocity reaches 25.6 m/s at a current density of $58\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$ prior to Bloch reversal at $60\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, when the central Bloch component is aligned antiparallel to the spin polarization, compared to a velocity of −1.7 m/s observed for the Bloch magnetization parallel to the spin polarization direction at equal current densities. This significant enhancement of velocity is achieved at moderate current densities, in line with experimentally accessible injection schemes. For example, assuming a spin Hall angle of 0.1, typical of heavy metals such as Pt, this would correspond to a true injection current density of $5.8\times {10}^{11} \mathrm{A}/{\mathrm{m}}^2$, which has been previously achieved in a variety of switching measurements.

For thinner films, this asymmetry is weaker, and disappears entirely in the ultrathin limit. For a 2 nm thick film, no velocity was observed for either chirality, even at large current densities. This is in line with previous predictions in ultrathin films, wherein an in-plane field is necessary in order to induce motion. For thicker films, greater maximum velocities were observed, as shown in Figure 3a. The motion for low thicknesses is in line with previous reports showing that, without DMI, an in-plane field is required to induce domain wall motion [34,35,39]. The enhanced domain wall velocity further scales with saturation magnetization, which necessitates larger current densities for high saturation magnetization and lower current densities for lower saturation magnetizations, as shown in Figure 3b.

The scaling of velocity in thick magnetic films may be qualitatively understood via modifications to the domain wall structure due to the spin polarized current, as shown in Figure 4 and Figure 5. For a static domain wall, the thickness averaged component of the magnetization along the wire long axis is $M_x=0$ due to the symmetric nature of the Néel caps at top and bottom surfaces, as previously shown in Figure 1. As the domain driving current increases, the transition from one Bloch chirality to another begins with the gradual reorientation of the magnetization through a mixed Bloch/Néel state. This results in an increase in domain wall velocity due to the interaction of the spin current with the increasing Néel component of the magnetization. The direction of the rotation depends on the direction of the spin current and the Bloch chirality of the domain wall. This rotation is most pronounced at the wire edge as compared to the central portion of the wire and leads to switching propagating from the wire edge inward.

Simulations using an effective thermal field with a temperature of 300 K were carried out to ensure the asymmetric motion was preserved at realistic temperatures. Thermal fluctuations will result in local variations in the domain wall structure, which will modify the domain wall velocity. This leads to variations in the domain wall velocity as a function of time, as seen in Figure 6. However, over time scales of 100 ns, the drastic asymmetries are preserved, with a sharp increase in the domain wall displacement with a current density of $-50\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$ relative to current densities further from the critical current density for switching of the Bloch component of the domain wall.

To understand the influence of external fields on domain wall motion, external magnetic fields were applied either parallel (longitudinal) or orthogonal (transverse) to the wire’s long-axis. The longitudinal field lies along the x-axis while the transverse field lies along the y-axis, as described in Figure 1b. A transverse magnetic field aligned parallel to the domain wall can be used to either enhance the required current density for switching of the Bloch component of the domain wall when the field is applied parallel to the Bloch component or decrease the required current density when applied antiparallel to the Bloch component, i.e., when the field is applied 180 degrees from the Bloch component of the domain wall. As shown in Figure 7a, the field parallel to the Bloch component results in a larger velocity achieved prior to switching, at the tradeoff of a larger required current density for the onset of velocity enhancements due to the suppressed Néel component of the wall at lower current densities. For a field of 40 mT applied antiparallel to the Bloch magnetization, a velocity of 2.6 m/s is achieved immediately below the switching current density at $12\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$. For the field applied parallel to the Bloch component, but opposite the spin current polarization, the necessary current density for switching increases to $102\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$ with a maximum velocity of 61.7 m/s. This suggests potential use of transverse fields to either reduce the required currents necessary to achieve enhanced velocities, or to increase maximal velocities at the expense of higher required current densities.

For a field applied along the nanowire, shown in Figure 7b, the field results in a rotation of the domain wall magnetization along the applied field direction, which results in a sign change in velocity for negative applied fields [40]. The asymmetric change in velocities is a result of this rotation either adding to or subtracting from the intrinsic dynamic Néel chirality determined by the initial Bloch chirality, as shown in Figure 4. The in-plane field also results in a decrease of the current density required to switch the Bloch chirality. This is most pronounced for fields applied parallel to the intrinsically favored dynamic Néel chirality.

The influence of DMI on domain wall velocity is shown in Figure 8. As expected, the velocities at large current densities are significantly enhanced compared to the case without DMI. This is a result of the partial alignment of the domain wall towards a Néel wall due to the DMI effective field. The magnitude of the velocities for positive and negative current densities are not equal due to either enhancement or suppression of the Néel component of the domain wall, as determined by the sign and magnitude of the DMI [16]. For low current densities, an achiral Bloch component persists provided the DMI is not too strong. This Bloch component behaves similarly to that without DMI, however with a smaller extent due to the rotation of the domain wall towards Néel-type [25,26,28]. Of note, the current required to switch the Bloch component along the polarization direction of the current is lower than that obtained for no DMI. Further, the velocity difference between opposite Bloch chiralities is lower compared to the asymmetries at low fields observed in Figure 2, and no sign change in the direction of domain wall propagation upon reversal of the Bloch component was observed, as previously seen for domain walls in ultrathin films with DMI [40].

4. Conclusions

We investigated the SOT driven dynamics of domain walls in thick ferromagnetic films, wherein competition between demagnetization and exchange energies result in a mixed Bloch-Néel domain wall structure. We found the Bloch component of the domain wall can strongly influence domain wall dynamics, leading to enhanced velocities even without an applied in-plane field or DMI at low current densities, and unidirectional motion at high current densities. This is explained via the rotation of the domain wall magnetization towards a Néel structure, before switching to align with the applied spin current. The velocities obtained in this work near the transition current density, even without the presence of DMI, are on the same order of those found for magnetic skyrmions in thick multilayers [25,41] and in TbCoFe wires [42] with DMI. These effects can be modified through application of in-plane magnetic fields and persist with the presence of weak DMI. These findings deviate from standard models of ultrathin films and provide a framework for the development of spintronic devices utilizing ferromagnetic films with thicknesses greater than the exchange length and generating bulk spin-orbit torques.