Walker's Velocity Research Articles

Magnetic domain wall dynamics in the precessional regime: Influence of the Dzyaloshinskii-Moriya interaction

The domain wall dynamics driven by an out of plane magnetic field was measured for a series of magnetic trilayers with different strengths of the interfacial Dzyaloshinskii-Moriya interaction (DMI). The features of the field-driven domain wall velocity curves strongly depend on the amplitude of the HD field stabilising chiral N\'eel walls. The measured Walker velocity, which in systems with large DMI is maintained after the Walker field giving rise to a velocity plateau up to the Slonczewski field HS, can be easily related to the DMI strength. Yet, when the DMI field HD and the domain wall demagnetising field HDW have comparable values, a careful analysis needs to be done in order to evaluate the impact of the DMI on the domain wall velocity. By means of a one-dimensional model and 2D simulations, we successfully extend this method to explain experimental results to cases where HD and HDW are comparable.

Walker solution for a magnetic domain wall driven by spin-orbit torques

A domain-wall configuration and a current-injection scheme are proposed which ensure the satisfaction of the boundary condition of the dynamic Walker wall driven by the spin-orbit torques (SOTs). The analytic solutions of the wall are obtained in the whole region of current by the traveling-wave method. In the steady regime, the initial velocity mainly depends on the Slonczewski SOT. Meanwhile, the final steady velocity largely relies on the fieldlike one. The critical behavior of velocity is closely linked to the ratio of the fieldlike SOT parameter $\ensuremath{\beta}$ to the Gilbert damping constant $\ensuremath{\alpha}$. Reducing the difference between $\ensuremath{\beta}$ and $\ensuremath{\alpha}$, the Walker breakdown is postponed and the Walker velocity is enhanced. Additionally, by adjusting an applied magnetic field, $\ensuremath{\beta}$ can be measured by halting the wall. In the periodic regime, the wall propagates back and forth with an average velocity increasing almost linearly for high currents.

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. 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.

Solvable continuous-time random walk model of the motion of tracer particles through porous media.

We consider the continuous-time random walk (CTRW) model of tracer motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported by Holzner etal. [M. Holzner et al., Phys. Rev. E 92, 013015 (2015)PLEEE81539-375510.1103/PhysRevE.92.013015]. The particle's passing through one channel is modeled as one step of the walk. The step (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability, mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step duration τ, length l, and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter α. Depending on that parameter, all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds, which was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer diffusion in the porous medium is considered.

Transformation of the Domain-Wall Structure in Permalloy Nanotape Under Electrical Current Pulses

Motion of the magnetic domain walls under the pulses of magnetic field or spin-polarized electric current is studied experimentally by direct visualization of the walls. The experiments are performed in the temperature range from 300 down to 6 K. Coercivity, mobility of the walls, and their variations with temperature are investigated. Wall velocity is observed to increase by orders of magnitude due to cooling from room to helium temperature in the case of current stimulated motion. The velocity becomes as high as the Walker velocity limit, about 1000 m/s. The walls are found to expand and tilt under the current pulses, which leads to their transformation into the domains with transverse magnetization.

Collective dynamics of model microorganisms with chemotactic signaling

Various microorganisms use chemotaxis for signaling among individuals-a common strategy for communication that is responsible for the formation of microcolonies. We model the microorganisms as autochemotactic active random walkers and describe them by an appropriate Langevin dynamics. It consists of rotational diffusion of the walker's velocity direction and a deterministic torque that aligns the velocity direction along the gradient of a self-generated chemical field. To account for finite size, each microorganism is treated as a soft disk. Its velocity is modified when it overlaps with other walkers according to a linear force-velocity relation and a harmonic repulsion force. We analyze two-walker collisions by presenting typical trajectories and by determining a state diagram that distinguishes between free walker, metastable, and bounded cluster states. We mention an analogy to Kramer's escape problem. Finally, we investigate relevant properties of many-walker systems and describe characteristics of cluster formation in unbounded geometry and in confinement.

Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes

We consider a previously devised model describing Levy random walks (Phys. Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that the given model describes Levy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Levy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.

FEEDBACK/FEEDFORWARD SCHEMES FOR MOTION CONTROL OF THE CYBERCARPET

The CyberCarpet is an actuated platform that allows unconstrained locomotion possibilities to a walking user for VR exploration. The platform has two actuating devices (linear and angular) and the motion control problem is dual to that of nonholonomic wheeled mobile robots. The main control objective is to keep the walker close to the platform center. Simple but global kinematic control schemes are presented, addressing in particular the handling of singularity issues. The feedback stabilizing part, which is based only on the user's pose information, is complemented by a feedforward term derived from a walker's velocity observer. Numerical and graphical simulation results are presented.

Perceived velocity of point‐light walkers under complex viewing and background conditions1,2

Abstract: An experiment with a computer‐generated display of nine moving dots representing a human walker (Cutting, 1978) was carried out. The independent variables were the observer's perspective (stabilized or mobile), the background (vertical gratings with spatial frequencies of 0.2, 0.4, and 0.8 cycles/deg), the movement of the background (−1, 0, 1, 2, and 4 cycles/s), and the objective velocity of the walker. The dependent variable was the velocity estimated by the observer. It was predicted that the mobile observer's perspective should lead to a more precise estimate of the walker's velocity, because there were more motion cues available. There were 10 observers and their data were analyzed individually. Five observers perceived the walker's velocity faster under the mobile observer's perspective, three showed the opposite result, and two observed no significant difference. This large variability was attributed to the ambiguity of motion cues when there is no direct information available about the distance of the background from the moving object.

Extreme events as foundation of Lévy walks with varying velocity

Abstract In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.W. Montrell (Eds.), Fluctuation Phenomena, SSM, vol. VII, North-Holland, Amsterdam, 1979, p. 63; J.-P. Bouchaud, M. Potters, Theory of Financial Risks from Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2001; D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer, Berlin, 2000] in determining the scaling properties of Levy walks with varying velocity. This model is an extension of the well-known Levy walks one [J. Klafter, G. Zumofen, M.F. Shlesinger, in M.F. Shlesinger, G.M. Zaslavsky, U. Frisch (Eds.), Levy Flights and Related Topics ion Physics, Lecture Notes in Physics, vol. 450, Springer, Berlin, 1995, p. 196; G. Zumofen, J. Klafter, M.F. Shlesinger, in: R. Kutner, A. Pȩkalski, K. Sznajd-Weron (Eds.), Anomalous Diffusion. From Basics to Applications, Lecture Note in Physics, vol. 519, Springer, Berlin, 1999, p. 15] introduced in the context of chaotic dynamics where a fixed value of the walker velocity is assumed for simplicity. Such an extension seems to be necessary when the open and/or complex system is studied. The model of Levy walks with varying velocity is spanned on two coupled velocity–temporal hierarchies: the first one consisting of velocities and the second of corresponding time intervals which the walker spends between the successive turning points. Both these hierarchical structures are characterized by their own self-similar dimensions. The extreme event, which can appear within a given time interval, is defined as a single random step of the walker having largest length. By finding power-laws which describe the time-dependence of this displacement and its statistics we obtained two independent diffusion exponents, which are related to the above-mentioned dimensions and which characterize the extreme event kinetics. In this work we show the principal influence of extreme events on the basic quantities (one-step distributions and moments as well as two-step correlation functions) of the continuous-time random walk formalism. Besides, we construct both the waiting-time distribution and sojourn probability density directly in a real space and time in the scaling form by proper component analysis which takes into account all possible fluctuations of the walker steps in contrast to the extreme event analysis. In this work we pay our attention to the basic quantities, since the summarized multi-step ones were already discussed earlier [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565]. Moreover, we study not only the scaling phenomena but also, assuming a finite number of hierarchy levels, the breaking of scaling and its dependence on control parameters. This seems to be important for studying empirical systems the more so as there are still no closed formulae describing this phenomenon except the one for truncated Levy flights [Phys. Rev. Lett. 73 (1994) 2946]. Our formulation of the model made possible to develop an efficient Monte Carlo algorithm [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565] where no MC step is lost.

Bloch wall dynamics in ultrathin ferromagnetic films

The domain wall dynamics in `easy axis' anisotropy films without magnetic loss is considered. Similar to the Walker model of the infinite one-dimensional domain wall, the dipole–dipole interaction is responsible for the presence of a critical velocity in this case. However, its experimentally measured value is small compared to Walker's velocity for bulk materials. This fact is connected with the effective demagnetization factor reduction when decreasing d – the film thickness. Comparison with the experimental results obtained for ultrathin cobalt films with `easy axis' anisotropy shows excellent agreement of theory and experiment. The influence of the relatively weak planar (perpendicular to the anisotropy axis) magnetic field on the dynamics of the domain wall and its critical velocity is also analyzed.

Magnetodipole mechanism of flexural instability of a moving domain boundary

The steady motion of a one-dimensional domain boundary in a single-axis ferromagnet has been studied. It has been shown that the flat domain boundary becomes unstable with respect to flexural deformation of its structure at a uniform translation velocity that is less than the Walker velocity. Instability of the domain boundary flat state is associated with the nonlocal nature of the magnetodipole interaction. An expression for the limit velocity is determined as a function of the relationship between the nonuniform exchange constants.

Applications of the gyrocoupling vector and dissipation dyadic in the dynamics of magnetic domains

This paper extends the theory of magnetic domains with emphasis on recent developments in ``hard bubbles''. A spin configuration of a planar Bloch wall containing periodic Bloch lines is presented which minimizes the magnetostatic energy to first order in the parameter 2πMs2/Ku for arbitrary period. The form of this solution is found to suggest the form of the dynamic breakdown of this spin configuration. The remainder of the paper consists of applications of the gyrocoupling force and vector, fg = g × v and g = − (Ms/|γ|)sinθ(▿θ) × (▿φ), respectively, and the dissipation force and dyadic, fa = d · v and d = − α(Ms/|γ|)[(▿θ)(▿θ) + sin2θ(▿φ)(▿φ)]. The use of fg and fa produces results with fewer assumptions and with less calculation than with previous methods. The magnitude of g is found to be an invariant local measure of the ``hardness'' of the domains. Integrating fg and fα produces a general planar wall response function from which the hard bubble dynamic equation is obtained. It is found that the difference between the hard bubble and normal bubble damping parameters can be accounted for by examination of the hard bubble spin-wave spectrum. An estimate of the velocity required for the production of horizontal Bloch lines is made using fg. This velocity is a substantial fraction of the Walker velocity. The vector g is used as an aid in the visualization of the mechanism by which ion implantation suppresses hard bubbles. From the point of view of both mobility and hard bubble suppression, materials having a large in-plane anisotropy are found to be desirable.