166
J. Masell and K. Everschor-Sitte
m(r − R, ψ) = (cos(φ + ψ) sin θ, sin(φ + ψ) sin θ, cos θ ) ,
(7.24)
where φ = φ(r − R) sets the inplane magnetic profile and θ = θ(r − R) determines
the m z profile. R is the position of the skyrmion and ψ is the helicity. For circular
skyrmions R is usually their center position, the profile only depends on the radial
coordinate ρ = |r − R| and φ depends only the axial coordinate χ of the cylindrical
coordinate system centered at R. In this convention, the Bloch-type skyrmion shown
in Fig. 7.1b is described by φ = χ , ψ = −π/2, and θ = θ(ρ) with θ(0) = π and
θ(∞) = 0. The antiskyrmion in Fig. 7.1c is described by φ = −χ , ψ = π/3. Other
skyrmion-like structures, e.g., higher order skyrmions with Q = −N can be described
by setting φ = N χ , and the topologically trivial skyrmionium is characterized by
θ(0) = 2π and θ(∞) = 0.
7.5.3.1 Pinning and Deformation
At ultra low current densities all magnetic solitons are pinned by material defects. For
skyrmion lattices, it has been shown experimentally that the critical current density for
depinning is very low [69, 70]. Theoretically, the influence of disorder on skyrmion
lattices was studied in various micromagnetic simulations [71]. The micromagnetic
results agree well with particle model simulations which are based on the Thiele
equation of motion of skyrmions which interact with each other and with random
pinning sites [72]. Also for isolated skyrmions in the presence of defects, the pinning,
depinning, and motion has been studied experimentally [73] and can be described in
a generalized Thiele equation when taking deformations due to defects into account
[62]. For simplicity, however, we will neglect pinning effects in the following.
Deformations of moving solitons also occur in the absence of impurities, for
example, due to internal dynamics, as has been shown already in early studies in
magnetic bubble dynamics with Bloch lines [27]. For skyrmions, which do not have
Bloch lines, deformations can also arise due to spin-torques. In this case, the matrix
elements of the Thiele equation (7.10), become dependent on the current strength
which leads to non-trivial corrections of the particle-like motion [74]. These effects,
as well as deformations due to thermal fluctuations, interactions with defects or other
magnetic textures can induce an effective mass for two-dimensional solitons which
might potentially be described by the broader term automotion [27]. In the limit
where skyrmions can be treated as rigid objects, i.e., when the bound state excitation
gap of the skyrmion is large, deformation effects can be neglected, as we will assume
in the following.
7.5.3.2 Skyrmion Motion Due to External Forces
Historically, before spin-torques became an active research field, the motion of magnetic bubbles was studied intensively, for example, with pulsed field gradients. It
was found that the bubbles do not move along the direction of the external force, but
J. Masell and K. Everschor-Sitte
m(r − R, ψ) = (cos(φ + ψ) sin θ, sin(φ + ψ) sin θ, cos θ ) ,
(7.24)
where φ = φ(r − R) sets the inplane magnetic profile and θ = θ(r − R) determines
the m z profile. R is the position of the skyrmion and ψ is the helicity. For circular
skyrmions R is usually their center position, the profile only depends on the radial
coordinate ρ = |r − R| and φ depends only the axial coordinate χ of the cylindrical
coordinate system centered at R. In this convention, the Bloch-type skyrmion shown
in Fig. 7.1b is described by φ = χ , ψ = −π/2, and θ = θ(ρ) with θ(0) = π and
θ(∞) = 0. The antiskyrmion in Fig. 7.1c is described by φ = −χ , ψ = π/3. Other
skyrmion-like structures, e.g., higher order skyrmions with Q = −N can be described
by setting φ = N χ , and the topologically trivial skyrmionium is characterized by
θ(0) = 2π and θ(∞) = 0.
7.5.3.1 Pinning and Deformation
At ultra low current densities all magnetic solitons are pinned by material defects. For
skyrmion lattices, it has been shown experimentally that the critical current density for
depinning is very low [69, 70]. Theoretically, the influence of disorder on skyrmion
lattices was studied in various micromagnetic simulations [71]. The micromagnetic
results agree well with particle model simulations which are based on the Thiele
equation of motion of skyrmions which interact with each other and with random
pinning sites [72]. Also for isolated skyrmions in the presence of defects, the pinning,
depinning, and motion has been studied experimentally [73] and can be described in
a generalized Thiele equation when taking deformations due to defects into account
[62]. For simplicity, however, we will neglect pinning effects in the following.
Deformations of moving solitons also occur in the absence of impurities, for
example, due to internal dynamics, as has been shown already in early studies in
magnetic bubble dynamics with Bloch lines [27]. For skyrmions, which do not have
Bloch lines, deformations can also arise due to spin-torques. In this case, the matrix
elements of the Thiele equation (7.10), become dependent on the current strength
which leads to non-trivial corrections of the particle-like motion [74]. These effects,
as well as deformations due to thermal fluctuations, interactions with defects or other
magnetic textures can induce an effective mass for two-dimensional solitons which
might potentially be described by the broader term automotion [27]. In the limit
where skyrmions can be treated as rigid objects, i.e., when the bound state excitation
gap of the skyrmion is large, deformation effects can be neglected, as we will assume
in the following.
7.5.3.2 Skyrmion Motion Due to External Forces
Historically, before spin-torques became an active research field, the motion of magnetic bubbles was studied intensively, for example, with pulsed field gradients. It
was found that the bubbles do not move along the direction of the external force, but