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Spin-orbit Rashba effect applies a torque on the magnetization of a ferromagnetic nanostrip in the case of structural inversion asymmetry, also affecting the steady domain wall motion induced by a spin-polarized current. This influence is here analytically studied in the framework of the extended Landau-Lifshitz-Gilbert equation, including the Rashba effect as an additive term of the effective field. Results of previous micromagnetic simulations and experiments have shown that this field yields an increased value of the Walker breakdown current together with an enlargement of the domain wall width. In order to analytically describe these results, the standard travelling wave ansatz for the steady domain wall motion is here adopted. Results of our investigations reveal the impossibility to reproduce, at the same time, the previous features and suggest the need of a more sophisticated model whose development requires, in turn, additional information to be extracted from ad hoc micromagnetic simulations.

Magnetization dynamics in nanodevices has been intensely investigated in the last decades as it provides a wide variety of technological applications in the area of storage and logic devices. Formerly, the manipulation of the magnetic configuration was typically achieved by means of external magnetic fields but the contemporary demand of miniaturizing storage devices and of increasing their capacity would have required higher and higher fields. An alternative method, realized by using spin-polarized currents, was outlined by the discovery of spin-transfer torque effect [

Among the different geometries used for spintronic devices, a more recent attention is directed to magnetic nanowires and strips [

DW motion can be also strongly modified by spin-orbit interaction [

From the theoretical viewpoint, DW motion in ferromagnetic thin layers is ruled by the extended Landau-Lifshitz-Gilbert (ELLG) equation including the current-driven spin-torque effects [

In this work, we propose to analytically study, by using a one-dimensional mathematical model, the steady DW motion in ferromagnetic nanostrips subject to the action of spin-polarized currents and Rashba fields. In particular, we investigate on the appropriateness of using a standard travelling wave ansatz describing a Bloch DW structure which rigidly shifts under the action of the external source. In order to validate the developed model, these analytical results are then compared, at qualitative level, with those arising from recent numerical and experimental investigations. The model also accounts for the nonlinear dry-friction dissipation function in order to evaluate how such a contribution affects the current-driven steady DW motion in the presence of a Rashba field.

As depicted in Figure

Schematics of a ferromagnetic nanostrip exhibiting a Bloch DW.

Current-driven DW dynamics in such a thin layer is described by the ELLG equation [

The effective magnetic field

We also assume that the strip is made by a material exhibiting a high perpendicular magnetic anisotropy, so that we can express

Finally, the Rashba field is given by [

The dissipative torque

The spin transfer torque

Taking into account (_{1} in the following form:
_{1} the parameter _{2}, let us remind that, in the absence of the Rashba field and nonlinear dissipations, we recover the classical definition of DW width _{2}, after some algebraic steps, leads to

As pointed out in some previous works [

If we exclude the additive dry friction term in the dissipation function, (

In order to determine the Walker breakdown, from the definition (_{1}, we can write

By comparing the expression of the DW velocity (

Let us discuss, now, the solutions obtained as a function of the strength of the quantity

For

For this reason, we can hypothesize that the correct solution has to be searched for

In order to estimate the orders of magnitude of the quantities involved in the model and to validate our assumption, we carry out a numerical evaluation of the travelling wave profile. For this reason, we take into account the parameter setup proposed in [

Figure

Comparison among the travelling wave profiles with and without Rashba field (

Starting from these results, we evaluate the new demagnetizing factors corresponding to the modified situation. Because of the DW width is reduced, the difference

Current-driven DW velocity in the steady dynamic regime with and without Rashba field (

Current-driven DW velocities in the steady dynamic regime with and without the inclusion of the dry-friction dissipation function in the mathematical model (

Summarizing, the case

We can state, therefore, that the classical travelling wave ansatz for the ELLG equation (

The usage of a dry-friction dissipation function already turned to be useful to model the effects of crystallographic defects, structural disorder, including surface roughness, on the DW motion [

In this work, we have analyzed the bias-field-free current-driven DW motion in a ferromagnetic nanostrip subject to the Rashba field and dry-friction dissipation. The study has been mathematically carried out by modifying the extended Landau-Lifshitz-Gilbert equation with the inclusion of the Rashba contribution into the effective field. The standard travelling wave ansatz generally used for the equation of motion, within the steady regime, does not succeed in confirming simultaneously both the key features revealed in recent numerical and experimental observations : increase of the Walker breakdown value and enlargement of the DW width.

This result suggests that the system (

However, solving this system without simplifying assumptions is not trivial at all. For example, the missing information could be argued from ad hoc micromagnetic simulations which should provide the accurate profile of the travelling wave variable and their dependence on the strength of the Rashba field. Therefore, we strongly encourage numerical investigations in this direction to overcome this issue.

G. Consolo gratefully acknowledges support from GNFM-INdAM.