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arXiv:cond-mat/0605125v2 [cond-mat.mtrl-sci] 8 May 2006

Current-driven vortex domain wall dynamics

J. He, Z. Li and S. Zhang

Department of Physics and Astronomy,

University of Missouri-Columbia, Columbia, MO 65211

(Dated: February 6, 2008)

Abstract

Current-driven vortex wall dynamics is studied by means of a 2-d analytical model and micro-

magnetic simulation. By constructing a trial function for the vortex wall in the magnetic wire, we

analytically solve for domain wall velocity and deformation in the presence of the current-induced

spin torque. A critical current for the domain wall transformation from the vortex wall to the

transverse wall is calculated. A comparison between the field- and current-driven wall dynamics is

carried out. Micromagnetic simulations are performed to verify our analytical results.

PACS numbers: 75.60.Ch, 75.75.+a, 75.70.Kw

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I. INTRODUCTION

Magnetic domain walls in magnetic films have various structural forms which are de-

termined by geometrical and material parameters. For a magnetic wire with an “infinite”

length and a finite width, there are commonly two types of domain walls: transverse wall

(TW) and vortex wall (VW). Both of them are stable. Depending on the wire thickness

and width, one of the walls is usually more stable [1, 2]. However, in a certain range of the

parameters, the static energies of these two walls are comparable and thus one can produce

both types of walls in the same wire . By using different initialization methods, one can

create either wall [1]. When a magnetic field or an electric current is applied, both TW and

VW are able to move along the wire. The dynamics of the walls is generally very complex

and micromagnetic simulations are required in order to describe the details of the domain

wall motion. For the TW, a simplified and yet very insightful analytical treatment was

developed by Walker [3]. A 1-d wall profile, i.e., the magnetization direction in the wall

depends only on the coordinate along the wire, has been used to approximate the TW pro-

file. With this approach, one can analytically calculate the wall velocity and wall distortion

in the presence of magnetic field and electrical current [4]. For the VW, however, the 1-d

wall profile fails to capture the wall structure and one needs at least to use a 2-d model to

approximately characterize the vortex structure. In this paper, we propose such a 2-d model

for the VW. Our focus will be on the analytical calculation of the dynamic behavior of the

VW. Within our model, we are able to describe the vortex wall motion, including the wall

distortion, wall velocity and wall structure transformation, in terms of material parameters

and external magnetic field or electric current. In particularly, we show how the vortex core

moves toward the edge of the wire when a current or a field is applied along the wire. With

a sufficiently large current or field, the vortex core may vanish at the boundaries of the wire

edges and the transformation from the VW to the TW occurs. This paper is organized as

follows. In Sec.II, we develop the analytical 2-d model for the VW. An equation of motion

for the domain wall is established. The steady state motions driven by the current and by

the field are investigated. We also compare the dynamics between the TW and the VW.

In Sec.III, the micromagnetic simulations are performed. We compare the simulated results

with the analytical ones. Finally, we summarize the different features of dynamics for TW

and VW in Sec.IV.

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II. ANALYTICAL MODEL

A. Equation of motion

We start with the generalized Landau-Lifshitz-Gilbert equation including the spin transfer

torque terms [5]:

∂M

∂t

= −γM × Heff+

−bJ

M2

s

α

MsM ×∂M

M ×∂M

∂x

∂t

−cJ

(1)

M ×

??

MsM ×∂M

∂x

where γ is the gyromagnetic ratio, and Heff= −δW

is the total energy density which could be written as W = A∇2m+(K/M2

M−(1/2)Hd·M, where A is exchange constant, K is anisotropy, Heis external field, and Hd

is magnetostatic field. α is the Gilbert damping parameter, and bJ= PjeµB/eMs(1+ξ2) and

δMis the effective magnetic field, and the W

s)(M × ex)2−He·

cJ= ξbJ, where P is the spin polarization of the current; jeis the current density along the

length direction of magnetic wire; µBis Bohr magneton, Msis saturation magnetization and

ξ (small, ∼ α) is a dimensionless constant which describes the degree of the nonadiabaticity

between the spin of the nonequilibrium conduction electrons and local magnetization.

To describe the motion of an entire domain wall, it is useful to introduce a total force

acting on the wall. Following Thiele [6], we define the total force

F{θ,φ} ≡

?

dV ∇W =

? ?δW

δθ(∇θ) +δW

δφ(∇φ)

?

dV

(2)

where θ, φ are the angular components of M in the spherical coordinate. For the steady-

state motion of a domain wall, we may write θ = θ(r − vt),φ = φ(r − vt), where v is the

steady velocity, then we have

˙θ = −v · ∇θ,

˙φ = −v · ∇φ. (3)

The above steady-state condition immediately reduces the temporal-spatial differential equa-

tion, Eq. (1), to a differential equation with spatial variables only. By writing Eq. (1) in the

angular components and by placing them into Eq. (2), we obtain the equation of motion for

the domain wall [7]:

F + G × (v + bJˆ x) + D · (αv + cJˆ x) = 0

(4)

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where G is the domain gyrocoupling vector

G = −Msγ−1

?

dVsinθ(∇θ × ∇φ) (5)

and D is the domain dissipation dyadic (tensor)

D = −Msγ−1

?

dV(∇θ∇θ + sin2θ∇φ∇φ) (6)

The domain wall force defined in Eq. (2) can be simplified when the domain wall undergoes

uniform motion. Let us separate the force in terms of the internal force Finand the external

force Fext,

F = Fin+ Fext. (7)

Fincontains all the forces from the internal energy including anisotropy energy, exchange

energy, and the magnetostatic energy. When one sums over the internal energy contribution,

the total internal force vanishes due to Newton’s third law. Therefore, one may simply

consider the external energy contribution to the force in Eq. (4) and (2), i.e.,

F = Fext=

?

dV

?

(∇θ)∂

∂θ+ (∇φ)∂

∂φ

?

(−H · M). (8)

where we have assumed that the external energy is solely from the external field, W =

−H · M. We will show later that we must consider other external forces on the vortex wall

when the wall reaches the boundary of the wire. Note that if the profile of domain structures

(i.e. M(x,y,z) in the moving frame of the steady motion) is determined, the gyrocoupling

vector, dissipation dyadic and static force can be calculated from Eqs. (5), (6) and (8), and

the steady velocity will be then readily derived from Eq. (4).

In this Section, we shall apply the equation of motion, Eq. (4), to study the domain wall

velocity of the vortex wall, driven by an external magnetic field and by spin transfer torques.

Let us first consider a simplified head to head transverse wall as shown in Fig. (1a). For the

transverse wall, we assume the wall profile can be modeled by the Walker’s trial function

[3],

φ(x) = 2tan−1exp

?x

∆

?

,θ(x) =π

2. (9)

where φ(x) and θ(x) are the angles between the direction of the magnetization and the wire

length direction (+x-axis) and wire plane normal (+z-axis) respectively. ∆ is the domain

wall width. By placing the above wall profile into Eq. (5)-(8), we find that the gyrocou-

pling vector is zero G = 0 and the dissipation dyadic has only one non-zero component

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Dxx= −2Ms/γ∆. The external force is F = 2HMsˆ x. By inserting them into Eq. (4), we

immediately find the velocity:

vx=γH∆

α

This result had been obtained previously [3, 4, 7].

−cJ

α.

(10)

It is difficult to analytically model the profile of the vortex wall depicted in Fig. (1b) by a

single elementary function as we did for the transverse wall. From previous simulation results

[1, 2], the VW structure might be understood as two symmetrical transverse walls diagonally

crossing the wire and a central vortex core connecting the two TWs. It is noted that the two

TWs have opposite polarities, namely, the magnetization of the centers of these two TWs

orient in opposite directions. The transitional region between these two TWs is sometimes

called as the Bloch line [11], which characterizes the wall-polarity reversal in analogy to the

Bloch wall. For the VW we consider here, this transitional region contains a vortex core. The

magnetization of the inner vortex core has a significant out-of-plane component and thus

the inner radius of the vortex core must be very small since the out-of-plane magnetization

enhances demagnetization energy [14]. Outside the inner vortex core the magnetization lies

in the plane and the outer radius of vortex core is limited by the transverse wall width and

the wire width. To characterize the entire VW profile, we separate the wall into three parts:

two transverse walls and a vortex core (schematically shown in Fig. (1c) and (1e)), and for

the model in Fig. (1e), they will be assigned to different trial functions given below . For

the vortex part [13, 14, 15, 16],

θ =

2tan−1

?√

x2+y2

rcore

?

, (0 ≤ x2+ y2< r2

core),

π

2,(r2

core≤ x2+ y2< R2),

φ = q · arg(x + iy) + cπ

2,(0 ≤ x2+ y2≤ R2).

(11)

where rcore and R are the inner-core and outer radius of the vortex respectively, q(=

±1,±2...) is the vorticity of the vortex, c(= ±1) is the chirality of the vortex, and i =√−1.

Here we just use the arg function of complex variable x+iy as a convenient way to express

φ [15]. In this paper, we only consider a single vortex (q = 1). And we note that the

vortex profile we introduced in Eq. (11) does not include its image profile [15]. For the wire

structure, the image profile would consists of a series of terms formed by multiple reflections

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