8 Microwave-Driven Dynamics of Magnetic Skyrmions …
185
All the above-mentioned magnets and magnetic systems have structures with
broken spatial inversion symmetry. Consequently, the Dzyaloshinskii-Moriya interaction becomes active, which favors rotating alignment of magnetizations with a
pitch angle of 90
◦ [25, 26]. This interaction strongly competes with the ferromagnetic exchange interaction, which favors parallel alignment of magnetizations. The
keen competition between these two interactions results in the formation of helical
magnetic order with a moderate pitch angle. When an external magnetic field is
applied to this helical magnetic state, a skyrmion crystal appears on a plane normal
to the applied magnetic field. In each skyrmion constituting the skyrmion crystal,
the magnetizations at periphery are oriented along the external magnetic field so as
to maximize an energy gain of the Zeeman interaction. The magnetizations gradually rotate when we move from the periphery to the center along the radial direction.
Eventually, the magnetizations are oriented antiparallel to the external magnetic field
at the center. In bulk magnets, these two-dimensional skyrmions are stacked along
the three-dimensional direction to form tubular structures. It should also be mentioned that the magnetic skyrmions can be classified into several types according to
the way of the magnetization rotation. Skyrmions in which the magnetizations rotate
within planes perpendicular (parallel) to the radial direction are called Bloch type
(Neel type). The three types of skyrmions, i.e., the hedgehog-type, the Bloch-type,
and the Neel-type, are mutually related via the rotational operation with respect to
the z axis and the stereo projection (see Fig. 8.1).
Skyrmions are characterized by a quantized topological invariant, which represents a sum of the solid angles spanned by three neighboring normalized magnetization vectors m(r),
UC
dxdy
∂ m
∂ x
×
∂ m
∂ y
· m = 4π Q
(Q = ±1).
(8.1)
Here the integration is taken over the area of the unit cell (UC). Because the magnetization vectors constituting a skyrmion point in every direction wrapping a sphere,
the sum of the solid angles is identical to the surface area of a unit sphere (∓4π ).
Consequently, the quantity Q, which is called the topological charge or the skyrmion
number, becomes +1 or −1. The sign is positive (negative) when the magnetization
at the core points upward (downward). This number does not change upon continuous variation of the magnetization alignment. Because the value of Q is zero
for ferromagnetic and spiral magnetic orders, magnetic skyrmions belong to a different topological class from these magnetic structures. This means that magnetic
skyrmions cannot be created or erased in a uniform ferromagnetic state by continuous
variation of the magnetization alignment. Instead, a local flop of the magnetization
is needed to create and erase them, which necessarily requires a large cost of energy,
magnitude of which is comparable to the energy scale of the ferromagnetic exchange
interaction. The topological protection by this large energy barrier makes magnetic
skyrmions robust against external assitations and perturbations.
This significant stability is one of the advantageous properties of magnetic
skyrmions for spintronics application to ubiquitous magnetic devices in the coming
185
All the above-mentioned magnets and magnetic systems have structures with
broken spatial inversion symmetry. Consequently, the Dzyaloshinskii-Moriya interaction becomes active, which favors rotating alignment of magnetizations with a
pitch angle of 90
◦ [25, 26]. This interaction strongly competes with the ferromagnetic exchange interaction, which favors parallel alignment of magnetizations. The
keen competition between these two interactions results in the formation of helical
magnetic order with a moderate pitch angle. When an external magnetic field is
applied to this helical magnetic state, a skyrmion crystal appears on a plane normal
to the applied magnetic field. In each skyrmion constituting the skyrmion crystal,
the magnetizations at periphery are oriented along the external magnetic field so as
to maximize an energy gain of the Zeeman interaction. The magnetizations gradually rotate when we move from the periphery to the center along the radial direction.
Eventually, the magnetizations are oriented antiparallel to the external magnetic field
at the center. In bulk magnets, these two-dimensional skyrmions are stacked along
the three-dimensional direction to form tubular structures. It should also be mentioned that the magnetic skyrmions can be classified into several types according to
the way of the magnetization rotation. Skyrmions in which the magnetizations rotate
within planes perpendicular (parallel) to the radial direction are called Bloch type
(Neel type). The three types of skyrmions, i.e., the hedgehog-type, the Bloch-type,
and the Neel-type, are mutually related via the rotational operation with respect to
the z axis and the stereo projection (see Fig. 8.1).
Skyrmions are characterized by a quantized topological invariant, which represents a sum of the solid angles spanned by three neighboring normalized magnetization vectors m(r),
UC
dxdy
∂ m
∂ x
×
∂ m
∂ y
· m = 4π Q
(Q = ±1).
(8.1)
Here the integration is taken over the area of the unit cell (UC). Because the magnetization vectors constituting a skyrmion point in every direction wrapping a sphere,
the sum of the solid angles is identical to the surface area of a unit sphere (∓4π ).
Consequently, the quantity Q, which is called the topological charge or the skyrmion
number, becomes +1 or −1. The sign is positive (negative) when the magnetization
at the core points upward (downward). This number does not change upon continuous variation of the magnetization alignment. Because the value of Q is zero
for ferromagnetic and spiral magnetic orders, magnetic skyrmions belong to a different topological class from these magnetic structures. This means that magnetic
skyrmions cannot be created or erased in a uniform ferromagnetic state by continuous
variation of the magnetization alignment. Instead, a local flop of the magnetization
is needed to create and erase them, which necessarily requires a large cost of energy,
magnitude of which is comparable to the energy scale of the ferromagnetic exchange
interaction. The topological protection by this large energy barrier makes magnetic
skyrmions robust against external assitations and perturbations.
This significant stability is one of the advantageous properties of magnetic
skyrmions for spintronics application to ubiquitous magnetic devices in the coming