152
J. Masell and K. Everschor-Sitte
7.3 Magnetic Solitons
In this part we review the most common magnetic structures focusing on chiral
solitons, shown in Fig. 7.1.
Magnetic domain walls are rather ubiquitous one-dimensional textures that connect
two distinctly polarized phases. The reason for this is that they do not require any
particular stabilization mechanisms; the two distinct ferromagnetic ordered phases
can be fixed by the boundary conditions. Therefore, magnetic domain walls have
been observed and studied already long ago and can be found in many different samples with different properties. By continuation in further dimensions, domain walls
can also be hosted in two-dimensional or three-dimensional systems. For example,
in symmetric thin films Bloch- or Néel-type domain walls can be stabilized. Their
helicity is determined by magnetostatic interactions and therefore depends on the
film geometry. For very thin films, Néel type domain walls are formed, where the
magnetization winds from one out-of-plane polarized state to the oppositely polarized state in the plane spanned by the out-of-plane state and the direction of rotation,
as shown in Fig. 7.1. For thicker films mainly Bloch type domain walls are realized. In
two-dimensional systems, domain walls can be effectively described as strings [27,
28] and closing this string leads to structures that are called magnetic bubbles. Furthermore, domain walls can obey localized defects for example in the version of
Bloch lines, i.e., localized windings in the domain wall where the helicity switches
from one Bloch handedness to the other handedness.
Magnetic skyrmions are localized whirls in two dimensions which can be viewed
as a closed magnetic domain wall, embedded as defects in a surrounding background
phase or they can be ordered in a lattice. In three-dimensional systems, skyrmions
form extended strings. Skyrmions received lots of attention in particular due to their
non-trivial real-space topology. The two-dimensional winding number for skyrmions
(located in the x y-plane)
Q =
1
4π
dr m · (∂ x m × ∂ y m) =
1
4π
dr F z ∈ Z
(7.4)
evaluates to Q = −1 for the skyrmion and to Q = +1 for the antiskyrmion shown
in Fig. 7.1, when integrating over the open area of the skyrmion. Note that Q only
evaluates to an integer if is a closed surface, i.e. ∂∂ = 0, which can, however, be
mapped to an open area with a topologically trivial boundary ∂∂. In the second
equality we have introduced the solenoidal gyro-vector field F as
F α =
1
2
αβγ m ·
m
∂r β
×
m
∂r γ
.
(7.5)
While for the skyrmion only one component of this vector field is important, the
topological index in 3D—the Hopf invariant—involves all components, see below.
J. Masell and K. Everschor-Sitte
7.3 Magnetic Solitons
In this part we review the most common magnetic structures focusing on chiral
solitons, shown in Fig. 7.1.
Magnetic domain walls are rather ubiquitous one-dimensional textures that connect
two distinctly polarized phases. The reason for this is that they do not require any
particular stabilization mechanisms; the two distinct ferromagnetic ordered phases
can be fixed by the boundary conditions. Therefore, magnetic domain walls have
been observed and studied already long ago and can be found in many different samples with different properties. By continuation in further dimensions, domain walls
can also be hosted in two-dimensional or three-dimensional systems. For example,
in symmetric thin films Bloch- or Néel-type domain walls can be stabilized. Their
helicity is determined by magnetostatic interactions and therefore depends on the
film geometry. For very thin films, Néel type domain walls are formed, where the
magnetization winds from one out-of-plane polarized state to the oppositely polarized state in the plane spanned by the out-of-plane state and the direction of rotation,
as shown in Fig. 7.1. For thicker films mainly Bloch type domain walls are realized. In
two-dimensional systems, domain walls can be effectively described as strings [27,
28] and closing this string leads to structures that are called magnetic bubbles. Furthermore, domain walls can obey localized defects for example in the version of
Bloch lines, i.e., localized windings in the domain wall where the helicity switches
from one Bloch handedness to the other handedness.
Magnetic skyrmions are localized whirls in two dimensions which can be viewed
as a closed magnetic domain wall, embedded as defects in a surrounding background
phase or they can be ordered in a lattice. In three-dimensional systems, skyrmions
form extended strings. Skyrmions received lots of attention in particular due to their
non-trivial real-space topology. The two-dimensional winding number for skyrmions
(located in the x y-plane)
Q =
1
4π
dr m · (∂ x m × ∂ y m) =
1
4π
dr F z ∈ Z
(7.4)
evaluates to Q = −1 for the skyrmion and to Q = +1 for the antiskyrmion shown
in Fig. 7.1, when integrating over the open area of the skyrmion. Note that Q only
evaluates to an integer if is a closed surface, i.e. ∂∂ = 0, which can, however, be
mapped to an open area with a topologically trivial boundary ∂∂. In the second
equality we have introduced the solenoidal gyro-vector field F as
F α =
1
2
αβγ m ·
m
∂r β
×
m
∂r γ
.
(7.5)
While for the skyrmion only one component of this vector field is important, the
topological index in 3D—the Hopf invariant—involves all components, see below.