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In this paper, we study the likelihood of chaos appearance during domain wall motion induced by electronic transfer. Considering a time-varying current density theory, we proceed to a numerical investigation of the dynamics. Using the dissipation parameter, amplitude and frequency of current density as control parameters; we show how periodic regime as well as chaotic regime can be exhibited in nanomagnetic systems. Numerical results allow setting up the periodicity and quasi-periodicity of system and chaotic phenomena occurring during magnetization switching process in nanomagnet through electronic transfer.

Since the appearance of pioneering work done by Berger [1,2] and Slonczewski [

Nowadays, it is well known that the concept and methods for the theory of nonlinear processes have substantially enriched many fields of physics, such as from elementary particle physics to biophysics, to mention just a few. An important field in nonlinear physics is the study of chaos in various dynamical systems [

The aim of this paper is to investigate the phenomenon of chaos in a nanomagnetic (spin) system under electronic transfer. It is important to understand the behavior of domain-wall (DW) whose dynamic, may depend not only on an induced-current through electron motion or either on an applied field but also on the fundamental characteristics of metal. Such chaos investigation may lead to dynamical characterization of such system that are of great importance for determining the suitable parameters for magnetic storage of nanomagnets. Probing the influence of each parameter on the DW motion may help in determining the spin switches behavior which is one of the key processes in reading and writing for magnetic memories devices.

Before pursuing, one should keep in mind that the damping parameter doesn’t only influence the nature of motion; it is rather a key parameter that is responsible for chaos appearance as current density amplitude and current density frequency in the system as shown in this paper.

The first experimental evidence of low-current operation due to the resonant DW motion induced by oscillating currents was carried out by Saitoh et al. [

In the present work, time-varying electrical density current in the form is applied along with j_{0 }being the amplitude of the current’s density and w_{0}, is the frequency of oscillation.

We consider a ferromagnet consisting of localized spins and conduction electrons. The spins are assumed to display an easy axis and a hard axis. In the continuum approximation, considering harmonic pinning potential, while assuming to work in the framework of a one dimensional spin lattice, the equation of motion of the wall we consider is given by [

where

and

has the dimension of velocity. The equations of motion in terms of dimensionless parameters are given by [15,16]:

where; t is time parameter,; X position of the domain wall center and λ the wall width parameter, where p is the spin current polarization and S spin magnitude,; j_{0 }is current density amplitude is where V_{0}

pinning potential magnitude and ξ is the width of pinning potential,.

By eliminating from the previous Equations (2) and (3) we obtain the following dynamical equation in function of the angle ϕ shown in

where β is the non-adiabatic parameter, P is the spin current polarization. Considering a Taylor development around; we obtain the magnitude of transfer function given by:

The parameters ω_{n}_{ }and ζ are the angular corner frequency of DW and damping factor, respectively defined by:

The resonance angular frequency is the minimum of the square of denominator transfer function denoted by η(ω) as:

In this framework, we determine the resonance angular frequency.

Since the resonance frequency corresponds to the highest displacement of the domain wall, the frequency of the current density has to be chosen taking into account the resonance frequency to keep the wall safe. In this respect, to avoid the highest vibration of the wall, the frequency of the current density has to be less than the resonance frequency.

The numerical investigation has been done while using a Matlab software in addition Fortran 90 codes. The outcomes were obtained by solving equation (7) with help of the standard fourth-order Runge-Kutta algorithm (Matlab’s ode45). For more detail concerning DW dynamics we have proceed to a meticulous bifurcation analysis.

Domain Wall BifurcationA Bifurcation analysis is always useful and it is a widely studied subﬁeld of dynamical systems. The observation of the bifurcation scenario allows one to draw qualitative and quantitative conclusions about the structure and dynamics of a given system. Several problems have been investigated using such a theoretical approach.

In the numerical results that follow, we investigate the dependence of the DW behaviour on the dissipation parameters α. Here, the amplitude of the current density and the frequency of the current’s density are used as the control parameters.

While looking at

for. In this specific case (time varying current density), the threshold is lower as shown in

It is worth mentioning the fact that the bifurcation diagram presented here consists of a projection of the attractors in the phase space into one of the DW coordinates versus α. To gain further insight on the dynamics of the equation under investigation, we compute the phase space and a Lyapunov spectrum (Lya). These results are obtained solving equation (7) with the help of the standard fourth-order Runge-Kutta algorithm.

ures it is realized that for the parameter region ranged from α = 0.017 to α = 0.0185, the Feigenbaun scenario is exhibited by the period doubling bifurcation. This is almost known as one of the routes toward chaos in this system [

a chaotic strange attractor. The Poincare section extends our investigations. This technique is particularly suited to the study of periodic system since it clearly distinguishes the quasi-periodic regimes of chaotic system. The poincare section is defined by the set of points of intersection of a plane with the trajectory of the state vector in space.

While looking at

The Poincaré section of the phase trajectory is shown in

From the results of the study of chaos in this ferromagnetic system, we realized that it is of capital interest and in this respect it deserves a peculiar attention. The above outcome shows the periodicity and unpredictability of DW motion for some values of control parameters where the system falls into periodic or chaotic, regime. We consider the physical value given by [

and λ = 70 nm. By considering these parameters the resonance frequency is f_{c} = 2.5 GHz. When the wall moves periodically, in a chosen phase subspace such a motion corresponds to the point attractor. In this case the Bloch surface of the wall is uniform and therefore the spin switches can be proceed successively. Chaotic motion corresponds to a strange attractor in the phase space. In this case, the strong oscillations of the wall occur.

The amplitude of these oscillations is of order of π. This means that the vertical Bloch lines are generated in the wall and therefore the switching process occurs randomly [

The knowledge of this unpredictable behavior of the DW motion is very important in technology based on the switching processes. It is clear that such a study would

help in determining suitable range of parameter for the safe and sustainable switching process, which is of outmost importance for magnetic recording technology.

In summary, we have introduced a model of periodically time-varying current density. We have shown that the

model admits a very complex dynamical structure that strongly depends on the control parameters. Using a phase space diagram; periodic, quasiperiodic and chaotic attractors of the DW motion were presented, setting then up three different regimes of spin switches when a nano-magnet undergoes an electronic transfer. Furthermore, the structures found in this model can be perfectly traced by the Lyapunov spectrum. This spectrum shows spikes at the bifurcation point reﬂecting a sudden change in the system. The outcomes of this work are element of an important step toward determining chaos in magnetic nanowire. However, further studies are necessary for controlling possible chaos in the system. Hence, further numerical and experimental investigations are encouraged.