Current-induced oscillation of a magnetic domain wall: Effect of damping enhanced by magnetization dynamics

doi:10.1016/j.cap.2010.06.019

Current Applied Physics

Volume 11, Issue 1, January 2011, Pages 61-64

https://doi.org/10.1016/j.cap.2010.06.019 Get rights and content

Abstract

Based on the Thiele’s approach, we investigate current-induced oscillation of a magnetic domain wall. A special attention is paid to effect of damping enhancement due to magnetization dynamics in the limit of no spin diffusion. Unlike for a translation motion, the enhanced damping due to magnetization dynamics has an important role for a rotational motion of a magnetic domain wall and can significantly reduce its oscillation frequency. The frequency reduction becomes more substantial for a narrower domain wall. This result provides a design strategy of high-frequency devices utilizing domain wall oscillation.

Introduction

Spin-transfer torque [1], [2] can drive a local magnetization into a steady-state precession mode. The current-induced precession mode and consequent microwave generation provide potential applications in telecommunication devices, and have been widely studied in spin-valve structures consisting of two ferromagnets separated by a nonmagnet [3], [4], [5], [6], [7]. In addition to spin-valve structures, a single ferromagnetic nanowire having non-collinear spin texture such as magnetic domain walls [8], [9], [10], [11], [12], [13], [14] or spin-waves [15], [16], [17], [18], [19] also allows generating the spin-transfer torque by applying a dc electrical current into the system. Thus it is also possible that a dc current drives a domain wall into a steady-state precession mode, where the domain wall propagation is localized by an opposing magnetic field [20], where the magnetizations at both ends are pinned by exchange coupling [21], and where the domain wall is confined at a geometric constriction [22], [23]. The frequency of domain wall oscillation was expected to be increased up to tens of GHz [24].

It is known that the oscillation frequency of a domain wall strongly depends on the magnetic damping [24]. Recently, it has been reported that the damping can be enhanced when the magnetization spatially and temporally varies [25], [26], [27], which is related to spin motive force [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. The spin motive force was theoretically suggested as a generalized electromotive force including a Berry phase averaged over the electron spin, and was experimentally measured for a domain wall motion [36], [36](a), [36](b). Temporal and spatial variations of magnetization induce spin-dependent effective electric field which generates both charge and spin currents inside the ferromagnet. The spin current induces spin-transfer torque in the non-collinear spin texture, which works as damping torque. Consequently, the magnetization dynamics generates an additional damping, like the spin-pumping operating at the ferromagnetic–nonmagnetic interface [36], [37].

In this study, we investigate effect of damping enhancement on current-driven domain wall oscillation based on Thiele’s approach [38]. We derive analytical formulae in the limit of no spin diffusion [25], since such analytic form is useful to get the first idea about the underlying physics. We first solve the equations on the translational motion of domain wall and confirm that the analytic solutions well fit to the result from micromagnetic modeling. Next we obtain analytic solutions for the frequency of current-induced domain wall oscillation. We find that the enhanced damping due to the spin motive force is insignificant for the translational motion of a domain wall. In contrast, its effect is significant for the rotational motion and causes a sizable reduction of the oscillation frequency.

Section snippets

Mathematical formulation

We start from the Landau-Lifshitz-Gilbert (LLG) equation including spin transfer torque terms and a generalized damping tensor D [25]; $\partial_{t} m = - γ m \times H_{eff} + m \times (D \cdot \partial_{t} m) + u m \times (m \times \partial_{x} m) + β u m \times \partial_{x} m,$ where m is the unit vector of magnetization, H_eff is the effective field including external field (H_ext = H_ze_z), exchange field, anisotropy field and demagnetizing field, γ is the gyromagnetic ratio, u is the rate of spin angular momentum transfer from the conduction electrons to the local moments (=gμ_BJP/2eM_s, μ_B is the

Results and discussion

We first derive the solutions for the average velocity of the domain wall motion (V_DW) [25], and obtain the difference between the velocities with and without damping enhancement (ΔV_DW = V_DW,η−V_DW,0, V_DW,η for η ≠ 0, and V_DW,0 for η = 0). It is arranged for the field-driven motion above the Walker breakdown field H_w [49] (H_ext > H_w (=α₀K_d/M_s), and u = 0); $Δ V_{D W} = \frac{γ λ}{α_{0}} (\frac{η^{'} \sqrt{H_{z}^{2} - H_{w}^{2}}}{(1 + α_{0}^{2} + η^{'}) (1 + α_{0}^{2})}),$ and for the current-driven motion above the Walker breakdown (H_ext = 0, and u > u_c (= γλH_w/α₀)); $Δ V_{D W} = \frac{1}{α_{0}} ($

Summary

We derive the modified Thiele’s equations to describe the effect of the damping enhancement on domain wall dynamics. The damping enhancement has a significant effect on the current-driven rotational motion of domain wall, but has a minute effect on the current-driven translational motion. For a typical value of η of conducting ferromagnet, the frequency of current-driven oscillation is largely decreased from the expected value for η = 0. Our results have important implications for applications

Acknowledgements

This study was supported by the Korea University grant.

References (51)

J.C. Slonczewski
J. Magn. Magn. Mater.
(1996)
G. Tatara et al.
Phys. Rep.
(2008)
L. Berger
Phys. Rev. B.
(1996)
S.I. Kiselev et al.
Nature
(2003)
K.J. Lee et al.
J. Appl. Phys.
(2004)
W.H. Rippard et al.
Phys. Rev. Lett.
(2004)
I.N. Krivorotov et al.
Science
(2005)
K.J. Lee et al.
Appl. Phys. Lett.
(2005)
G. Tatara et al.
Phys. Rev. Lett.
(2004)
A. Yamaguchi et al.
Phys. Rev. Lett.
(2004)

Cited by (14)

Self-consistent calculation of spin transport and magnetization dynamics
2013, Physics Reports
Citation Excerpt :
Spin currents cannot be directly measured, but they couple to other processes that can. In ferromagnets, spin currents generate charge currents, which in turn generate electric voltages [125–134], and the generation of spin currents enhances magnetic damping [135–141]. Just as spin transfer torques in multilayers and nanowires are similar processes in different geometries, so are spin pumping and the spin motive force.
A spin-polarized current transfers its spin-angular momentum to a local magnetization, exciting various types of current-induced magnetization dynamics. So far, most studies in this field have focused on the direct effect of spin transport on magnetization dynamics, but ignored the feedback from the magnetization dynamics to the spin transport and back to the magnetization dynamics. Although the feedback is usually weak, there are situations when it can play an important role in the dynamics. In such situations, simultaneous, self-consistent calculations of the magnetization dynamics and the spin transport can accurately describe the feedback. This review describes in detail the feedback mechanisms, and presents recent progress in self-consistent calculations of the coupled dynamics. We pay special attention to three representative examples, where the feedback generates non-local effective interactions for the magnetization after the spin accumulation has been integrated out. Possibly the most dramatic feedback example is the dynamic instability in magnetic nanopillars with a single magnetic layer. This instability does not occur without non-local feedback. We demonstrate that full self-consistent calculations generate simulation results in much better agreement with experiments than previous calculations that addressed the feedback effect approximately. The next example is for more typical spin valve nanopillars. Although the effect of feedback is less dramatic because even without feedback the current can make stationary states unstable and induce magnetization oscillation, the feedback can still have important consequences. For instance, we show that the feedback can reduce the linewidth of oscillations, in agreement with experimental observations. A key aspect of this reduction is the suppression of the excitation of short wavelength spin waves by the non-local feedback. Finally, we consider nonadiabatic electron transport in narrow domain walls. The non-local feedback in these systems leads to a significant renormalization of the effective nonadiabatic spin transfer torque. These examples show that the self-consistent treatment of spin transport and magnetization dynamics is important for understanding the physics of the coupled dynamics and for providing a bridge between the ongoing research fields of current-induced magnetization dynamics and the newly emerging fields of magnetization-dynamics-induced generation of charge and spin currents.
Well-designed rectangular cavity resonator for FMR experiment
2013, Current Applied Physics
Citation Excerpt :
In spintronics society, the Gilbert damping constant (α) of ferromagnetic materials is important because it governs the performances of the high-speed magnetization switching for magnetic random access memory (MRAM) [1,2], spin torque nanooscillators [3], and oscillation of a magnetic domain wall [4], etc.
The unloaded quality factor of the cavity resonator is the ratio between the stored energy of the cavity resonator to the power loses in the cavity resonator. The homemade rectangular cavity resonator in X-band shows higher unloaded quality factor compare with standard cavity resonator in the TE₁₀₂ mode. Because the inner walls of rectangular cavity resonator are treated through high quality polishing and high purity Au plating. Also the inner walls are made by printed circuit board which has thin Cu foil, two problems such as mechanical vibration and thermal expansion can be solved by minimizing unwanted eddy current. Through the ferromagnetic resonance measurement by using our rectangular cavity resonator, we can be obtained reasonable values of resonance frequency and linewidth by using NiFe thin film. As a result, the Gilbert damping constant from the experimental result is in good agreement with the typical value of damping parameter of the NiFe thin film.
Current-induced domain wall motion in nanoscale ferromagnetic elements
2011, Materials Science and Engineering R: Reports
Citation Excerpt :
For example, Adam et al. [69] in (Ga,Mn)As measured a damping parameter about 1 order of magnitude higher using the dependence of the DW velocity with the magnetic field as compared to the one obtained by ferromagnetic resonance. Actually, several theories predict an enhancement of the damping parameter as a consequence of a fluctuating spin torque due to electronic noise [283] or absorption of the spin current generated by the magnetization dynamics when the DW is moving [284–286]. Enhanced damping is also predicted due to the emission of spin waves during the DW motion [147] or the excitation of internal degree of freedom of the DW in the presence of defects [120].
The manipulation of a magnetic domain wall (DW) by a spin polarized current in ferromagnetic nanowires has attracted tremendous interest during the last years due to fundamental questions in the fields of spin dependent transport phenomena and magnetization dynamics but also due to promising applications, such as DW based magnetic memory concepts and logic devices. We comprehensively review recent developments in the field of geometrically confined domain walls and in particular current induced DW dynamics. We focus on the influence of the magnetic and electronic transport properties of the materials on the spin transfer effect in DWs. After considering the different DW structures in ferromagnetic nanowires, the theory of magnetization dynamics induced by a spin polarized current is presented. We first discuss the different current induced torques and their origin in the light of recent theories based on a simple s-d exchange model and beyond. This leads to a modified Landau-Lifshitz-Gilbert equation of motion where the different spin transfer torques are included and we discuss their influence on the DW dynamics on the basis of simple 1D models and recent micromagnetic simulations studies. Experimental results illustrating the effects of spin transfer in different ferromagnetic materials and geometries constitute the body of the review. The case of soft in-plane magnetized nanowires is described first, as it is the most widely studied class of ferromagnetic materials in this field. By direct imaging we show how confined domain walls in nanowires can be displaced using currents in in-plane soft magnetic materials and that using short pulses, fast velocities can be attained. While a spin polarized current can trigger DW depinning or displacement, it can also lead to a modification of the DW structure, which is described in detail as it allows one to deduce information about the underlying spin torque terms. High perpendicular anisotropy materials characterized by narrow domain walls have also raised considerable interest. These materials with only a few nanometer wide DWs combined several key advantages over soft magnetic materials such as higher non-adiabatic effects leading to lower critical current densities and high domain wall velocities. We review recent experimental results obtained in this class of materials and discuss the important implications they entail for the nature of the spin torque effect acting on DWs.
Micromagnetic modelling on magnetization dynamics in nanopillars driven by spin-transfer torque
2011, Journal of Physics D: Applied Physics
Roles of chiral renormalization on magnetization dynamics in chiral magnets
2018, Physical Review B
Roles of chiral renormalization on magnetization dynamics in chiral magnets
2018, arXiv

View all citing articles on Scopus

View full text

Current-induced oscillation of a magnetic domain wall: Effect of damping enhanced by magnetization dynamics

Abstract

Introduction

Section snippets

Mathematical formulation

Results and discussion

Summary

Acknowledgements

J. Magn. Magn. Mater.

Phys. Rep.

Phys. Rev. B.

Nature

J. Appl. Phys.

Phys. Rev. Lett.

Science

Appl. Phys. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

Nature

Phys. Rev. Lett.

J. Phys. Soc. Jpn.

Science

Phys. Rev.

Phys. Rev. B.

Phys. Rev. B.

Science

Phys. Rev. Lett.

Appl. Phys. Lett.

Phys. Rev. B.

Appl. Phys. Lett.

J. Appl. Phys.

Appl. Phys. Express

Phys. Rev. Lett.