123
SPRINGER BRIEFS IN PHYSICS
Shinichiro Seki
Masahito Mochizuki
Skyrmions
in Magnetic
Materials
SpringerBriefs in Physics
Editorial Board
Egor Babaev, University of Massachusetts, USA
Malcolm Bremer, University of Bristol, UK
Xavier Calmet, University of Sussex, UK
Francesca Di Lodovico, Queen Mary University of London, UK
Maarten Hoogerland, University of Auckland, New Zealand
Eric Le Ru, Victoria University of Wellington, New Zealand
Hans-Joachim Lewerenz, California Institute of Technology, USA
James Overduin, Towson University, USA
Vesselin Petkov, Concordia University, Canada
Charles H.-T. Wang, University of Aberdeen, UK
Andrew Whitaker, Queen’s University Belfast, UK
More information about this series at http://www.springer.com/series/8902
Shinichiro Seki • Masahito Mochizuki
Skyrmions in Magnetic
Materials
123
Shinichiro Seki
Center for Emergent Matter Science (CEMS)
RIKEN
Wako, Japan
Masahito Mochizuki
Department of Physics and Mathematics
Aoyama Gakuin University
Sagamihara, Japan
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs in Physics
ISBN 978-3-319-24649-9 ISBN 978-3-319-24651-2 (eBook)
DOI 10.1007/978-3-319-24651-2
Library of Congress Control Number: 2015953776
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2016
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Contents
1 Theoretical Model of Magnetic Skyrmions ................................ 1
1.1 What Is a Skyrmion?..................................................... 1
1.2 Stabilisation of Magnetic Skyrmions ................................... 2
1.3 Model and Phase Diagrams.............................................. 4
References ...................................................................... 12
2 Observation of Skyrmions in Magnetic Materials......................... 15
2.1 Skyrmions in Non-centrosymmetric Magnets .......................... 15
2.2 Skyrmions in Centrosymmetric Magnets ............................... 22
2.3 Skyrmions at Interface ................................................... 26
References ...................................................................... 30
3 Skyrmions and Electric Currents in Metallic Materials .................. 33
3.1 Emergent Electromagnetic Fields ....................................... 33
3.2 Electric-Current-Driven Motions of Skyrmions ........................ 35
3.3 Topological Hall Effect .................................................. 41
3.4 Manipulation by Electric Current ....................................... 44
Appendix 1: Landau-Lifshitz-Gilbert-Slonczewski Equation ................ 48
Appendix 2: Derivation of Thiele’s Equation ................................. 50
References ...................................................................... 56
4 Skyrmions and Electric Fields in Insulating Materials ................... 57
4.1 Magnetoelectric Skyrmions and Manipulation by Electric Fields ..... 57
4.2 Magnetoelectric Resonance of Skyrmions .............................. 62
References ...................................................................... 66
5 Summary and Perspective ................................................... 67
References ...................................................................... 69
v
Chapter 1
Theoretical Model of Magnetic Skyrmions
Abstract Skyrmions were originally proposed by Tony Skyrme in the 1960s to
account for the stability of hadrons in particle physics as a topological solution of
the non-linear sigma model. Bogdanov and his collaborators theoretically predicted
their realisation in chiral-lattice ferromagnets with finite Dzyaloshinskii–Moriya
interaction due to the lack of spatial inversion symmetry. In this chapter, an overview
of theoretical aspects of magnetic skyrmions is provided.
1.1 What Is a Skyrmion?
Keen competition among interactions in magnets often gives rise to non-collinear
or non-coplanar spin structures such as vortices, domain walls, bubbles and spirals.
These spin structures endow hosting materials with interesting physical properties
and useful device functions, which have attracted intense research interest from
viewpoints of fundamental science and technical applications. For example, domain
walls and vortices in metallic ferromagnets can be driven by spin-polarised electric
currents [1–3], and their application to magnetic storage devices such as race-track
memory is anticipated [4]. Magnetic spirals in insulating magnets often exhibit
rich magnetoelectric cross-correlation phenomena due to the coupling between
magnetism and electricity through the generation of ferroelectric polarisation via a
relativistic spin–orbit interaction [5–7]. In addition to these spin structures, magnetic
skyrmions, vortex-like swirling spin structures characterised by a quantised topo-
logical number, are attracting considerable research attention because it has turned
out that their peculiar response dynamics to external fields hold highly promising
properties with applications to spintronic device functions [8–10].
Skyrmions were originally proposed by Tony Skyrme in the 1960s to account for
the stability of hadrons as quantised topological defects in the three-dimensional
(3D) non-linear sigma model [11, 12]. They have now turned out to be highly
relevant to a spin structure in condensed-matter systems. A magnetic skyrmion
comprises spins pointing in all directions wrapping a sphere similar to a hedgehog,
as shown in Fig. 1.1a. The number of such wrappings corresponds to a topological
invariant, and thus, the skyrmion has topologically protected stability. It has been
found that skyrmions are indeed realised in quantum Hall ferromagnets [13, 14],
© Springer International Publishing Switzerland 2016
S. Seki, M. Mochizuki, Skyrmions in Magnetic Materials, SpringerBriefs
in Physics, DOI 10.1007/978-3-319-24651-2_1
1
2 1 Theoretical Model of Magnetic Skyrmions
Fig. 1.1 (a) Schematic of the original hedgehog-type skyrmion proposed by Tony Skyrme in
the 1960s, whose magnetisations point in all directions wrapping a sphere. (b) Schematic of the
helical state realised in chiral-lattice magnets as a consequence of the competition between the
Dzyaloshinskii–Moriya and ferromagnetic exchange interactions. (c) Schematic of a skyrmion
recently discovered in chiral-lattice magnets, which corresponds to a projection of the hedgehog-
type skyrmion on a two-dimensional (2D) plane. Its magnetisations also point in all directions
wrapping a sphere. (d) Schematic of the skyrmion crystal realised in chiral-lattice magnets under
an external magnetic field in which skyrmions are hexagonally packed to form a triangular lattice
ferromagnetic monolayers [15], doped layered antiferromagnets [16], liquid crys-
tals [17] and Bose–Einstein condensates [18]. Recently, the realisation of magnetic
skyrmions in chiral-lattice magnets was theoretically predicted [19–21] and later
experimentally discovered [6, 22].
1.2 Stabilisation of Magnetic Skyrmions
There are several mechanisms for skyrmion formation in magnets. One major
mechanism is the competition between the Dzyaloshinskii–Moriya and ferromag-
netic exchange interactions [19–21]. In chiral-lattice ferromagnets without spatial
inversion symmetry, such as B20 compounds (MnSi, FeGe, Fe
1x
Co
x
Si) and copper
oxoselenite Cu
2
OSeO
3
, the Dzyaloshinskii–Moriya interaction, which originates
from the relativistic spin–orbit coupling, becomes finite [23, 24]. In a continuum
spin model, the Dzyaloshinskii–Moriya interaction is expressed by
1.2 Stabilisation of Magnetic Skyrmions 3
H
DM
/
Z
drM .r M/; (1.1)
where M is the classical magnetisation vector. This interaction alone favours a
rotating magnetisation alignment with a turn angle of 90
ı
and competes with
the ferromagnetic exchange interaction that favours a collinear ferromagnetic spin
alignment. As a result of their competition, a helical spin order with a uniform
turn angle shown in Fig. 1.1b is realised in the absence of an external magnetic
field [25–28]. On the application of a weak magnetic field, skyrmions appear as
vortex-like topological spin textures shown in Fig. 1.1c in a plane normal to the
field irrespective of field direction. In a skyrmion, magnetisations are parallel to
an applied magnetic field at its periphery but antiparallel at its centre. This spin
structure corresponds to a projection of the original hedgehog-type skyrmion on a
2D plane. The topological nature of this projected skyrmion is characterised by the
topological invariant
G D
Z
d
2
r
@
O
n
@x
@
O
n
@y
O
n; (1.2)
where
O
n D M=jMj is the unit vector pointing in the direction of magnetisation. This
quantity is a sum of solid angles spanned by three neighbouring magnetisations; for
a single skyrmion, its value is given by 4Q, with Q.D˙1/ being the skyrmion
number. The sign of Q corresponds to that of magnetisation at the skyrmion core,
i.e. Q DC1 (Q D1) for up (down) magnetisation at the core. Skyrmions
often form a hexagonal lattice, the so-called skyrmion crystal shown in Fig. 1.1d.
Magnetisations align ferromagnetically in a stacking direction to form rod-like
or tube-like structures. Typically, skyrmions in chiral-lattice ferromagnets are
3–100nm in size, which is determined by the ratio of the Dzyaloshinskii–Moriya
interaction D to the ferromagnetic exchange interaction J.
Another major mechanism of skyrmion formation is the competition between
magnetic dipole interaction and easy-axis anisotropy [29–32]. In thin-film spec-
imens of ferromagnets with perpendicular easy-axis anisotropy, the anisotropy
favours out-of-plane magnetisations, whereas a long-range magnetic dipole interac-
tion favours in-plane magnetisations. Their competition results in a periodic stripe
with spins rotating in a thin-film plane. An application of a magnetic field normal
to the thin-film plane turns the stripe into a periodic arrangement of magnetic
bubbles or skyrmions. Skyrmions or bubbles of this origin tend to be large, typically
3–100m in size, which is orders of magnitude larger than skyrmions in chiral-
lattice ferromagnets. In addition to these two mechanisms, frustrated exchange
interactions [33] and four-ring exchange interactions [15] have been theoretically
proposed as origins of skyrmion formation. Skyrmions of these origins tend to be
atomically small.
4 1 Theoretical Model of Magnetic Skyrmions
1.3 Model and Phase Diagrams
To describe the magnetism in MnSi as a prototypical chiral-lattice ferromagnet, the
continuum spin model that was proposed by Bak and Jensen in 1980 [34]isas
follows:
H D
Z
d
3
r
J
2a
.rM/
2
C
D
a
2
M .rM/
1
a
3
B M
C
A
1
a
3
.M
4
x
C M
4
y
C M
4
z
/
A
2
2a
Œ.r
x
M
x
/
2
C .r
y
M
y
/
2
C .r
z
M
z
/
2
: (1.3)
The first and second terms represent the ferromagnetic exchange interaction (J >0)
and the Dzyaloshinskii–Moriya interaction, respectively. The third term denotes the
Zeeman coupling to an external magnetic field B. The fourth and fifth terms are
magnetic anisotropies allowed by a cubic crystal symmetry, but they turn out to
play a minor role as far as realistically small values of A
1
and A
2
are considered.
Here, a is the lattice constant.
In Ref. [35], the stability of a skyrmion-crystal phase was theoretically studied
based on this model by writing the Ginzburg–Landau free energy functional near
T
c
as
FŒM D
Z
d
3
r
r
0
M
2
C J.rM/
2
C 2DM .rM/
CUM
4
B M
: (1.4)
When a uniform component of magnetisation M
uniform
is induced by a magnetic
field, we obtain the term
X
q
1
;q
2
;q
3
.M
uniform
m
q
1
/.m
q
2
m
q
3
/ı.q
1
C q
2
C q
3
/ (1.5)
from the quartic term in Eq. (1.4), where m
q
is the Fourier component of M.r/.
Wave vectors q
1
, q
2
and q
3
should have a fixed modulus determined by two
competing gradient terms, i.e. the ferromagnetic exchange term and Dzyaloshinskii–
Moriya term. In addition, the energy change should be proportional to M
uniform
On
by symmetry, where On is a vector normal to the plane spanned by the three wave
vectors. Therefore, one can gain energy from this term for the skyrmion-crystal
