A topological gauge field in nanomagnets: spin wave excitations over a slowly moving magnetization background

Abstract

We introduce a topological gauge vector potential which influences spin wave excitations over arbitrary non-uniform, slowly moving magnetization distribution. The time-component of the gauge potential plays a principal role in magnetization dynamics, whereas its spatial components can be often neglected for typical magnetic nanostructures. As an example, we consider spin modes excited in the vortex state magnetic dots. It is shown that the vortex/ spin wave interaction can be described as a consequence of the gauge field arising due to non uniform moving vortex magnetization distribution. The coupled equations of motion of the vortex and spin waves are solved within small excitation amplitude approximation. The model yields a giant frequency splitting of the spin wave modes having non-zero overlapping with the vortex mode as well as a finite vortex mass of dynamical origin. Comment: 14 pages, 2 figures

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A topological gauge field in nanomagnets: spin wave excitations over a slowly
moving magnetization background



Konstantin Y. Guslienko1,2*, Gloria R. Aranda1, and Julian M. Gonzalez1

1Dpto. Fisica de Materiales, Universidad del Pais Vasco, 20018 Donostia-San Sebastian, Spain
2IKERBASQUE, the Basque Foundation for Science, 48011 Bilbao, Spain

We introduce a topological gauge vector potential which influences spin wave excitations over
arbitrary non-uniform, slowly moving magnetization distribution. The time-component of the
gauge potential plays a principal role in magnetization dynamics, whereas its spatial components
can be often neglected for typical magnetic nanostructures. As an example, we consider spin
modes excited in the vortex state magnetic dots. It is shown that the vortex – spin wave interaction
can be described as a consequence of the gauge field arising due to non uniform moving vortex
magnetization distribution. The coupled equations of motion of the vortex and spin waves are
solved within small excitation amplitude approximation. The model yields a giant frequency
splitting of the spin wave modes having non-zero overlapping with the vortex mode as well as a
finite vortex mass of dynamical origin.

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In the recent years a lot of attention was paid to influence of the gauge fields on semi-classical
equations of motion of quantum particles such as electrons in solids [1]. It was shown within the
local models of itinerant magnetism [2, 3] and s-d exchange interaction [4] that a topological
gauge field induced by a non-uniform magnetization acts on electron motion as a real external
magnetic or electric field [5]. From the other side, spin-polarized electric current can essentially
contribute to the equation of motion of magnetization [6] transferring angular momentum. The
conception of such gauge field has led to prediction and explanation of various observable
magneto-electric effects in patterned nanostructures related to spin angular momentum transfer
(spin Hall effect, magnetoresistance, spin pumping, domain wall dynamics etc.), for recent review
see Ref. 1 and references therein.
Precise knowledge of the dynamic processes in patterned nanomagnetic materials is important
due to their applications in magnetic storage media, magnetic sensors, spintronic devices including
arrays of microwave nano-oscillators [7] etc. Understanding dynamic response of nanomagnets to
external magnetic field [8-9] or spin-polarized current [10-11] involves rich underlying physics
and is also an interesting fundamental issue. Among the different patterned magnetic structures
considered so far, flat ferromagnetic particles (dots) with submicron lateral sizes occupy a special
place due to their unique, non-uniform vortex ground state [12]. Regular arrays of such dots are
being considered as a new high-density non-volatile recording media [13] characterized by two
Boolean variables: chirality and polarity of magnetic vortex core [12]. Special interest in the
vortex core, small area where magnetization deviates from the dot plane (~10-15 nm in size [14]),
is inspired by the possibility of easy and fast switching its magnetization direction [15,16]. The
vortex-state dots as well as domain walls in magnetic nanostripes [5, 11] represent the simplest
patterned nanosystems with topologically non-trivial magnetization distribution. However, there is
still lack of understanding of the spin wave dynamics in such strongly non-uniform ground states.
In this Letter we present a gauge potential approach to description of the spin wave modes
excited over arbitrary non-uniform slowly moving background magnetization distribution
υ
M . It

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is shown that a coordinate transformation to the local quantization axis directed along
υ
M allows
introducing a topological gauge potential, which leads to an inter-mode interaction even in the
linear regime of small amplitude spin excitations.
Let us consider a ferromagnetic body with a time dependent magnetization field
() t , rM
defined
in the point r. Assuming conservation of magnetization vector length
s
M
=
M
it is convenient to
introduce the reduced magnetization
s
M/Mm =
, 1
2=
m . To describe magnetization dynamics
we use the Lagrangian corresponding to the Landau-Lifshitz equation of motion of m:


() t,d
3
rrλ
∫
=Λ
,
( )
m
()
mmm
?
D
α
λ∂−⋅=
,w
, (1)
where ( ) (
D
)() []
mnnmm
×⋅+=
−1
1/γ
s
M
is the vector potential related to the Dirac string in the
direction of an arbitrary unit vector n [17], the dot over symbol means derivative with respect to
time (t), w is the magnetic energy density
()
Hm
wwAw
++∂=
2
m
α
(α= x, y, z),
2/
msm
MwHm⋅−=
is the magnetostatic energy density,
Hm⋅−=
sH
Mw
is the Zeeman energy
density, A is the exchange stiffness,
m
H and Hare the magnetostatic field and external fields.
We distinguish two subsystems in the body: slowly moving magnetization (e.g., a vortex or a
domain wall, characteristic time of motion is ~10 ns) + fast magnetization oscillations described as
spin waves (SW, with period about of 0.1 ns) and express magnetization as a sum
s
mmm
+=
υ
of
the slow magnetization (υ) and spin wave (s) contributions. Recalling the restriction
1
2=
m the
components of
s
m are the simplest in a moving coordinate frame x’y’z’, where the axis Oz’
(quantization axis) is directed along the local direction of
υ
m . We assume that the fast s-
subsystem can instantly follow slow υ-subsystem magnetization
υ
m (adiabatic approximation),
but the spin wave length can be comparable with a characteristic size of
υ
m non-uniformity. Then,
we perform a rotation of the initial xyz coordinate system to the direction of
υ
m . The

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corresponding 3x3 real rotation matrix
()()
()
yz
JiJiR
υυυυ
ΘΦ=ΦΘ
expexp,
is defined by the
spherical angles of
()
υυυ
ΦΘ ,
m
, and
s
m components in new coordinate system are
ss
′
Rmm =
,
() 1 , 0 , 0
=
′υ
m
. Here
α
J are the angular momentum components for J=1 in the Cartesian basis
representation (pure imaginary). To preserve the Lagrangian (1) in the same form after
transformation
mmmR
=
′
→
we need to introduce to Eq. (1) covariant derivatives ()
μμ
A
−∂

instead
μ
∂ , where
μ
A is a gauge vector potential (the index μ = 0, 1, 2, 3 denotes the time and
space coordinates xμ = t, x, y, z, and
μμ
x
∂∂=∂
/
). The
μ
A components are transformed as
11
−−
⋅∂+=
′
μ
→
RRR RAAA
μμμ
. The term
1
−
⋅∂
RR
μ
has sense of a topological contribution to the
vector potential related to the “inertial” moving frame x’y’z’. We denote it as
1
ˆ
A
−
⋅∂=
RR
μμ
and
put
0
=
μ
A
in the laboratory coordinate system xyz.
μ
Aˆ acts on the SW magnetization and can be
represented in the explicit form by the time- and spatial derivatives of the angles ()
υυΦΘ , as


⎟
⎠
⎟
⎟
⎞
⎜
⎝
⎜
⎜
⎛
Θ∂ΦΘ∂Φ
Θ∂Φ−Φ∂−
Θ∂Φ−Φ
0
∂
=
0 sincos
sin
cos0
ˆ
A
υμυυμυ
υμυυμ
υμυυμ
μ
. (2)
The skew-symmetric matrix of the operator
μ
Aˆ can be represented as
α
α
μμ
G
ˆ
AA
ˆ =
via the
generators
α
Gˆof SO(3) group of 3-dimensional rotations, which are expressed in the Cartesian
basis as ()
αβγ
ε
βγ
α
=
Gˆ
, ε ˆ is the unit antisymmetric tensor, α, β, γ= x, y, z. There is a simple
equation
mAm
×=
μμ
Aˆ
for arbitrary vector m due to definition of
μ
Aˆ, where
()
υμυμυυμυμ
Φ∂−Θ∂Φ−Θ∂Φ=
,cos,sin
A
is a dual vector. The gauge vector potential
μ
Aˆ
defined by Eq. (2) introduces a “minimal” interaction between the υ- and s-subsystems and can be
applied to a wide class of problems related to excitation of the spin waves in the non-uniform
magnetization ground state. The first, kinetic, term of the Lagrangian (1) can be written in the

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frame x’y’z’ as
()
s
′
s
′
kin
A mm
?
D
− ⋅ ′=
0ˆ
λ
, where the term
s
′
A m
0
D
⋅ ′−=
int
ˆ
λ
corresponds to the
dynamic υ – SW interaction. The component
0ˆA is always important for spin dynamics, but it was
neglected in Refs. 18,19 because the authors considered only static
υ
m configurations, where
α
Aˆ
appear in the exchange energy terms. Moreover, in the case of body sizes >>
e L
(
se
MAL
/2
=
~10 nm is the exchange length),
0ˆA dominates over other components of
μ
Aˆ
because all the exchange energy terms including spatial derivatives can be neglected, whereas
υυΦΘ∝
?
?,
ˆ0 A
determined by the magnetostatic fields is not, in general, small. The Lagrangian (1)
then can be re-written in the form
int
Λ+Λ+Λ=Λ
sw
υ
, where
int
3
int
λ
r
∫
=Λ
d
is the interaction
term between the slowly moving non-uniform magnetization
υ
m described by the Lagrangian
υ
Λ
and spin waves, which are described by the Lagrangian density
()
s
′
s
′
s
′
sw
wmmm
?
D
∂−⋅ ′=
α
λ
,
.
To demonstrate how useful and powerful the approach based on the potential
μ
Aˆ is, we apply
this formalism to the problem of coupled vortex – SW motion in submicron size cylindrical
particles (dots) in the vortex state. Magnetic dot with the vortex core possesses two qualitatively
different kinds of spin excitation modes: the lowest in frequency gyrotropic mode (typically
several 100 MHz) localized near the dot center [9, 20] and high frequency (several GHz) spin
waves [9, 10, 21]. The excited mainly outside the vortex core radially and azimuthally symmetric
SW modes are described by integers (n, m), which indicate number of nodes in the dynamic
magnetization along radial (n) and azimuthal (m) directions. For in-plane magnetic driving field
the azimuthal SW and gyrotropic mode (
1
±=
m
) with non-zero dipolar moments can be only
excited [8, 9, 22]. The azimuthal SW with m=±1 having the same symmetry as the vortex
gyrotropic mode are especially important because they are responsible for the vortex core
distortion resulting in the core switching.