Current Induced Vortex Wall Dynamics in Helical Magnetic Systems
Bahman Roostaei

arXiv:1403.5323v4 [cond-mat.mes-hall] 3 Mar 2015

Department of Physics, Indiana University-Purdue University Indianapolis, Indiana 46202, USA and
Department of Physics, Institute For Research In Fundamental Sciences, Tehran, Iran.
(Dated: March 4, 2015)
Nontrivial topology of interfaces separating phases with opposite chirality in helical magnetic
metals result in new effects as they interact with spin polarized current. These interfaces or vortex
walls consist of a one dimensional array of vortex lines. We predict that adiabatic transfer of angular
momentum between vortex array and spin polarized current will result in topological Hall effect in
multi-domain samples. Also we predict that the motion of the vortex array will result in a new
damping mechanism for magnetic moments based on Lenz’s law. We study the dynamics of these
walls interacting with electric current and use fundamental electromagnetic laws to quantify those
predictions. On the other hand discrete nature of vortex walls affects their pinning and result in
low depinning current density. We predict the value of this current using collective pinnng theory.
PACS numbers: 75.10.-b, 75.60, 75.70, 75.85

Localized magnetic moments in some rare earth metals
such as Ho, Tb, Dy[1, 2] and their alloys, interact through
anisotropic RKKY exchange interaction. This interaction consists of ferromagnetic nearest neighbor and antiferromagnetic next nearest neighbor between planes
while the in-plane interactions are ferromagnetic[1]. As
the result for example in Ho at temperatures below 133 K
a helical order is developed with the helical axis perpendicular to the basal planes and below 19 K the moments
cant away from the planes to form a conical phase[2].
The ground state of this helical system is degenerate
with respect to the sense of rotation of the helices or chirality of the structure. Recently observation of chiral domains in these systems[2] has triggered novel ideas and
predictions about the interface between chiral domains
in a general perspective[3–5]. The most striking one is
the prediction that these interfaces posses vorticity totally unrelated to the boundary effects contrary to what
is seen for example in the case of vortex domain walls
(consisting of a single vortex) in certain ferromagnetic
nanowires[6]. More precisely these interfaces consist of
vortex lines parallel to the basal planes.
Early studies have shown that the peculiar structure of
these walls leads to a different static and dynamic behavior. The interaction between the vortex lines have been
predicted to be weak[5] which implies the elasticity must
originate from a vicinal behavior[4]. On the other hand
the movement of the vortex lines inside the plane of the
interface leads to the mobility enhancement[5].
Here in the present work we focus on the interaction of
the spin polarized current with these walls. The vorticity
of the walls rotates the conduction electron’s spins. This
torque can be seen by electrons as a fictitious magnetic
field generated by the wall. The interplay between this
induced Berry phase and electron motion leads to topological Hall effect as the first result in this letter. This is
an anomaly in resistivity caused by domain walls.
The next result is a dominating mechanism for damp-

ing of the local magnetic moments as a result of vorticity and the domain wall motion. Local fluctuations
and motions of the wall will add time dependent phase
to the wavefunction of the conduction electrons which
can be interpreted as an extra electric field (the so called
spin motive force) which at the end drags the motion
of domain wall. We will show that this dynamical effect will generate an additional damping of the magnetic
moments.
On the other hand the the angular momentum transfer
from conduction electrons to the domain wall results in
a force which can move the domain wall. We predict in
the presence of weak disorder the current necessary to
depinn the wall is smaller than predicted values for walls
between ferromagnetic domains, a promising property for
future technological applications[7].
The discretized Hamiltonian describing the helical
magnetic order in the rare earth metals we are considering is:
X
X
mi,n · mi,n+1 +
mi,n · mj,n − J
H = −J
hiji,n

+ J′

X
i,n

mi,n · mi,n+2

i,n

(1)

which introduces a ferromagnetic(antiferromagnetic) exchange coupling J > 0 (J ′ > 0) between local moments mi,n in basal plane number n. We parameterize the local magnetic moments by spherical coordinates
m = {sin θ cos ϕ, sin θ sin ϕ, cos θ} so we can write the
long range Hamiltonian of the magnetic system as:

Z 
i2 a2

2
a2 h
J
2
2
2
2
+
(∇⊥ ϕ) +
(∂z ϕ) − q
.
∂z ϕ
HS ≈
a r
4
4
(2)
We consider the polar angle θ being almost constant
throughout the system. In the above the helix propa-

2
sumption one can use the well known adiabatic approximate for the electrons by projecting their spins into the
subspace of the local magnetic structure. In this approximation the wavefunction of electron can be written as
|ψi = φ|mi in which φ is the orbital and |mi is the spin
part of the wave function, σ · m|mi = m|mi matching
the local magnetic distribution of the metal.
With this map, the Hamiltonian of a single electron
ˆ = p2 + V (r) + JH σ · m will look like the Hamiltonian
H
2m
of an electron in presence of an external electromagnetic
field:

20

15

ˆ ef f ≈ 1 (p − ea)2 − ea0 + V (r)
H
2m

10

5

z
y

2

4

6

8

10

FIG. 1: Illustration of vorticity. The right and left helix have
opposite chirality. For better visualization spins have been
represented to lie in the y − z plane, parallel to the helical
axis while in reality they lie in x − y plane.

gation wavevector q = Θ/a = a−1 arccos(J/4J ′ ) and Θ
is the angle which moments in successive planes make
with respect to each other. a is the lattice constant of
the crystal and we assume it is the same for all directions. Also we have taken z-axis perpendicular to the
basal planes and ∇⊥ = {∂x , ∂y }.
Obviously the above Hamiltonian is degenerate with
respect to two different chiral ground states ϕ = ±qz.
Li et al.[3] have realized the domain wall between chiral
states must consist of an array of vortex lines.
H This can
be easily seen by performing the line integral ∇ϕ.dl on
a path encircling any part of the interface (in x-z plane in
figure 1). So from now on by vortex wall we mean a one
dimensional array of vortex lines with a rest separation
of π/q (≈ 2.1nm for Ho) which is half the length of one
helix.
Conduction electrons moving inside a magnetic environment interact through Hund’s rule with local magnetic moments at each point. In an adiabatic approximation we can assume the local magnetization acts as a
local magnetic field and aligns the spin of electrons with
its direction. This adiabatic approximation is satisfied
when the typical variation of the local moment direction
is much larger than the electron wavelength, kF ≫ q
which is held in most rare earth metals[1]. With this as-

(3)

h
¯
(1 −
in which the fictitious vector potential is aµ = 2e
cos θ)∂µ ϕ[8]. In this subspace the spin part of the Hamiltonian is now diagonalized and acts as a constant Zeeman
term with a coupling proportional to Hund’s coupling
JH . This term is approximately constant and so ineffective for slowly varying textures in all of our calculations.
In the simplest case of a flat vortex domain wall electrons that are moving transverse to the wall plane with
a velocity v with respect to the wall, feel the fictitious
h
¯
∇ϕ (assuming θ ≈ π/2) which
vector potential a ≈ 2e
is correspondingPto a magnetic field B = ∇ × a =
h
¯
h
x in which x⊥ = {y, z}
n δ[x⊥ − xn (t)]ˆ
2e ∇ × ∇ϕ = 2e
and xn (t) = {−vt, nπ/q} represents a vortex line.
The magnetic field that electrons feel is a direct consequence of the vorticity of the wall and deflects electron’s
H
path resulting in a topological Hall contribution, σyz
to
the conductivity of the metal which can be estimated
using:
H
σyz
/σyy ≈ eBx τ /m

(4)

The magnetic flux density here corresponds to one vortex
line Bx ≈ φ0 /(π/q)2 and τ is the typical scattering time
of the metal.
This topological Hall conductivity is the result of the
exerted torque on the spin of electron by the vortex domain wall and in principle will cause anomaly in magnetotransport in multi-domain samples for example in
Ho below 133 K for which with typical values of scatH
tering time σyz
/σyy ≈ 0.1. This effect has been already
realized in skyrmion lattices in some noncentrosymmetric magnetic systems such as MnSi and also in magnetic
nanowires and disks[10, 11].
Wall equation of motion: We parameterize each vortex line in the vortex domain wall by the collective
coordinates u(x, n, t) = {uy (x, n, t), nπ/q + uz (x, n, t)}
with respect to the wall’s rest position. We assume
the vortices are rigid and consider long wavelength distortions of the domain wall |u(x, n, t) − u(x′ , m, t)|2 ≪
(n − m)2 π 2 /q 2 + (x − x′ )2 and build a long wavelength
Lagrangian for this texture.
The kinetic energy part of the Lagrangian reflects the

3
spin dynamics or the Berry phase of each spin:
Z
hS
¯
K= 3
d3 r(cos θ − 1)ϕ˙
a

motion[13]:
(5)

This energy
R 2in terms of the vortex coordinates will be
q
˙
K ≈ 2πa
d xk G·[u×u].
Here u(xk , t) is the coordinate
of each vortex line in continuum limit and xk = {x, z}.
The average Gyrovector of the vortex line is introduced
x where p is the average dimensionalso as G = h¯aS3 (π/q)pˆ
less value of local moment component out of the basal
plane direction. This value is nonzero in conical phase of
Holmium.
Gyrovector is always perpendicular to the plane of
the vortex. In most of the conventional systems the
moments forming the vortex lie inside the plane of the
vortex however in our case the moments lie inside the
x − z plane instead of x − y plane. This is why the
Gyrovector is not perpendicular to the plane of the vortex:
R 2 Indeed Gyrovector is a second rank tensor, Gij =
d rm · ∂i m × ∂j m. In our system the out of plane component (cos θ(y)) of moments vary slowly as a function
of y as one approaches from y → ∞ toward the center of the wall y = 0. Assuming full symmetry in the
x-direction the only nonzero components of this tensor
are Gyz = −Gzy . This leads to a pseudovector in the
x-direction.
The interaction of the spin current with the domain
wall is also obtained by summing over contribution from
all vortex lines:
Z
Z
q
d2 xk G · [vs × u]
(6)
Hint = − d3 rj · a ≈
πa
Here we have defined electron’s velocity vs = a3 /(2eS)j
considering perfect polarization of the current. Finally
the energy cost for distortions of the domain wall can be
approximated by a local elastic term[3, 4]:
Z
σ
Hel =
d2 xk |∇u|2 .
(7)
2
Damping of the local moments can be incorporated into
the equation of motion by adding the dissipation function
to the Lagrange equation[12]:
Z
Z
qαd
˙2=
D d2 xk u˙ 2
(8)
W =h
¯ Sαd /2a3 d3 m
2πa
Here αRd is the Gilbert damping constant and D =
h
¯S
2
a3 π/q dxdy|∇m0 | is the profile function for one vortex which behaves as logarithm of the ratio of the system
size to the vortex core size.
We can now construct the total Lagrangian of the domain wall L = K − Hint − Hel and obtain its equation of

u˙ =

αd D
πaσ
G × ∇2 u + vs + 2 u˙ × G
qG2
G

(9)

Note that the dissipation function enters the Lagrange
˙
equation by derivative of W with respect to u.
The first conclusion is that a flat vortex wall will start
moving with the same velocity as electrons in the absence
of damping. On the other hand the boundaries of the
sample acts as an effective potential to keep the vortex
line array inside the system. Damping torque will distort
the vortex lines in transverse and longitudinal directions
since uz and uy are conjugate of each other[14]. However
the average longitudinal movements uz will be confined
by the boundary.
Damping: As the result of the vortex wall movement
the conduction electrons feel a time dependent magnetic
flux density. This magnetic flux density vector, assuming
symmetry in the ˆz direction is:
B(r, t) = φ0 ρv (x⊥ , t)ˆ
x
(10)
P 2
in which ρv (x⊥ , t) =
n δ [x⊥ − un (t)] is the vortex
wall vortex density and un (t) represents an undistorted
vortex line. Here we ignore the distortions of the vortex
wall.
The resulting change in the magnetic flux according
to Lenz’s law creates an electric field which in the conduction electron’s rest frame will induce another current
depending on the conductivity of the medium. Let’s calculate this electric field. The time derivative of the magnetic flux density is:
∇×E = −

∂B
∂ρv
ˆz
= −φ0
∂t
∂t

(11)

Since the total charge of the system is zero we have
∇ · E = 0. The solution is of the form E = E⊥ × x
ˆ
v
.
We
integrate
this
equation
on
satisfying ∇ · E⊥ = φ0 ∂ρ
∂t
a cylinder with radius ≈ π/q around each vortex line to
find E⊥ . The induced rotating electric field, E will push
the electrons and create a current jd = σ0 E in which σ0
is the conductivity of the medium. The resulting contribution in conduction electron velocity will be:
vd = α′d u˙ × x
ˆ

(12)

where we have defined the new damping constant:
α′d = φ0

a3 σ0
4πeS(π/q)2

(13)

This will in turn induce a drag force on the domain
wall and contribute to the damping of its motion. The
ratio of the damping terms in the equation of motion (9)

4
of the domain wall will be then:
rd = φ0

σ0
1
2
(π/q) f αd ns

(14)

3

in
R which2 ns = 2eS/a the electron spin density, f =
|∇m0 | is the vortex profile function introduced ber
fore. To estimate this ratio we use the typical values for
Holmium[1, 15], J ≈ 0.7meV , Θ ≈ π/6, σ0−1 ≈ 60µΩ.cm,
αd ≈ 0.005 and sample size of a micron which result in
a value rd ≈ 6.7. Thus this effect can be dominating in
principle.
The emergent damping mechanism introduced above
is originated from the time dependence of the vector potential a[m]
˙ which has lead to generation of an electric
current. This current affects the dynamics of moments
through the term jd · ∇m in Landau-Lifshitz-Gilbert
equation. This in turn has lead to appearance of the
damping term in the equation of motion of the domain
wall thus maybe distinguished in FMR linewidth measurements as domains are removed. Also note that the
origin of Gilbert damping (αd ) is the spin-orbit coupling
which eventually relaxes the magnetic moments and violates the conservation of total spin. On the other hand
this new damping term in equation of motion of the domain wall vanishes as the system approaches uniform
(as opposed to non-collinear) magnetization (α′d → 0 as
q → 0).

Depinning from Disorder : There are two types of disorder that can pin the vortex domain wall. The one we
discuss is the vacancies or non-magnetic impurities. The
vortex lines in the wall could gain exchange energy by
moving into those points. The second type is the charged
impurities which cause some local variation in electron
density hence variation in exchange energy J. This will
create a random potential that can collectively pin the
wall. Here in this study we consider only vacancy disorder.
We assume vacancies or non-magnetic impurities are
spheres occupying lattice sites in random with radius a of
the order of lattice constant. Then the total wall disorder
energy will be:
Hni =

Z

d3 r

Np
X
X
i=1

n

a3 εv [x⊥ − u(x, n)] δa3 (r − ri ) (15)

in which εv (x⊥ ) is the energy per unit volume of a
straight vortex line, ri is the vacancy position and
δa (x) is a smooth delta function on a scale a. This
energy
can be written in continuum limit
R 2
P as Hni ≈
d xk VD (xk , uy )ρ(z − uz ) where ρ(z) = n δ(z − nπ/q).
The average of random potential energy density VD (r)
vanishes and its correlation is:
Z
6
∆(r) = a ni δa (x) d2 x′⊥ εv (x⊥ − x′⊥ )εv (x′⊥ )
(16)

in which ni is the impurity density. By average we mean
averaging over all equally probable sets of {ri }.
The disorder potential exerts torque on the vortex
lines. This torque will be added to the equation of
πa
motion (9) of the vortex wall via the term − qG
2G ×


∇u V (xk , uy )ρ(z − uz ) . Let’s estimate the value of the
current density necessary to depinn the vortex wall in
the weak pinning regime. The typical variations of the
disorder energy for a set of N vortex lines (N ≫ 1) each
with a length L is:
˜

 2
∆(0)
q
Hni ≈ LN
π
ξd

(17)

where tilde sign mean Fourier transform. We are assuming the correlations of the disorder distribution is of short
range (ξd ). The average elastic energy of this part of the
array assuming they have distorted by π/q is σπ 2 /q 2 .
Thus the total energy cost per vortex line per unit of
length will be:
√
1/2
˜
E ≈ σπ 2 /N Lq 2 − [q ∆(0)/πξ
/ N L.
(18)
d]
Minimizing this eergy will give the so called Larkin
˜
length[16] L = N L ≈ 4π 5 σ 2 ξd /q 5 ∆(0).
Depinning threshold is when the external torque is
compensated by the disorder torque at Larkin length
scale:
jc ≈



2eS πa 

G × ∇u V (xk , uy )ρ(z − uz )  L (19)
3
2
a qG

The disorder energy density at the Larkin length scale per
4
˜
vortex line per unit length is hV ρiL ≈ q 4 ∆(0)/2π
ξd σ.
From this energy density we can find out the typical disorder torque at Larkin length assuming the domain wall distortion is of the order of π/q in this scale:
5
˜
h|∇u (V ρ)|i ≈ q 5 ∆(0)/2π
ξd σ. This estimation gives us
an estimate for the threshold current density:
jc ≈

˜
eS/a2 ∆(0)
4
(π/q) |G| σξd

(20)

Because of strong screening of the vortices by helical
background (the vortex effect on local magnetic moments
vanishes very fast away from the wall into the helical
phase)[5] the vortex energy density vanishes at distances
longer than π/q consequently we can use an approximate
exponential model for the energy density of vortex lines
and find the above correlation function:
∆(r) ≈

σ2 6
2
(x⊥ )
a ni δa (x)δπ/q
q2

(21)

where we have used the fact that the total energy of one
˜
vortex line is ≈ σLπ/q. From this we will have ∆(0)
≈

5
σ 2 ni a6 π 2 /q 2 and:
jc ≈

(aq)3
1 e
(ni a3 )
(σa2 )
4
2π ¯
h
p



1
aξd



partment of Physics at Indiana University - Purdue University Indianapolis for supporting this work.
(22)

The energy of one vortex line has been calculated
easlwhere[3] using variational calculation which results
in σ ≈ Jq/a. Also we approximate ξd ≈ π/q. Again for
Holmium and typical disorder concentration ni a3 ≈ 10−5
and p ≈ 0.1 (conical phase) we obtain jc ≈ 5×105 A.m−2
which is much lower than the value observed for more
conventional domain walls[17].
In conclusion, we have presented a microscopic theory
for the interaction of spin polarized current with a new
type of magnetic domain wall. This theory predicts the
existence of a topological Hall effect because of the topological nature of the wall. Also the collected Berry phase
by the electron motion interacting with a moving vortex
wall affects their motion which in turn damps the wall
motion. One should note that besides the Gilbert damping term there is another higher order term also associated with the relaxation of spins, the so called β-term[8].
This term is almost comparable (if not smaller)[18] in
magnetic metals to the Gilbert damping term. That is
why it is plausible to assume the drag force found in this
work to be an effective force needed to be considered in
vortex domain wall dynamics under spin polarized current.
Finally this current can depinn the wall from nonmagnetic impurity sites which have collectively pinned
the wall. Present theory predicts a very low depinning
current which together with the lack of intrinsic pinning
for vortices makes the vortex walls in helical magnets
attractive for technological applications.
I would like to thank Dr. Yogesh N. Joglekar and De-

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