7 Current-Induced Dynamics of Chiral Magnetic Structures
167
along a deflected direction that depends on the winding number Q [27]. For both,
skyrmions and magnetic bubbles, the side-drift response can be understood within
the Thiele approach, (7.8). Moreover, this effect occurs not only for field gradients
but for all forces F(q) in the Thiele equation, e.g., due to field gradients or the interaction with defects and other magnetic structures. It is also the source of the unusual
Brownian motion of skyrmions which in two dimensions diffuse less if the Gilbert
damping α is reduced [37, 75, 76]. In the following, analog to the domain wall case,
we will first discuss the limit of a pinned helicity and then consider what happens
beyond this limit.
Pinned helicity. Consider a skyrmion with winding number Q and the position
R = (X, Y ) as the only collective coordinates for the Thiele approach. In a spatially
dependent energy landscape E(R), e.g., due to anisotropy gradients, magnetic field
gradients, defects, or other magnetic textures, R is not a true zero mode but can still
be a good collective coordinate. As the system is dissipative, the skyrmion will at
some point be trapped in a local minimum of E(R). An elegant form of the resulting
Thiele equation then reads
G × ˙
R + αD(R) ˙
R = F(R)
(7.25)
where F(R) = −(γ /M s )∇ R E(R) is the force on the skyrmion, G = 4π Qˆ z is the
gyro-vector, and D(R) is the dissipation matrix. The gyro-vector G couples the
motion of the X and Y coordinates and leads to the side-deflection in the motion of
two-dimensional solitons with a finite topological charge. For a circular skyrmion,
using the notation in (7.24) and the angular dependence φ(χ) = N χ , N ∈ Z, the
dissipation matrix reduces to a scalar with
D(R) = D s (R) =
∞
0
πρ
N
2
ρ 2 sin
2
θ(ρ) +
θ
(ρ)
2
dρ
(7.26)
where ρ is the distance to R, i.e., the center of the skyrmion. Usually, it is assumed
that the texture of the skyrmion does not change much with the position such that
D s (R) ≈ D s is a good approximation.
The Thiele equation, (7.25), can be solved for the skyrmion velocity ˙
R. Its absolute
value | ˙
R| and the direction relative to the force F, parameterized by the deflection
angle θ d , then read
| ˙
R| =
|F(R)|
(4π Q) 2 + α 2 D 2
s
and θ d = − arctan
4π Q
α D s
.
(7.27)
In this formulation, the real-space topological nature of the side-deflection can be
identified as θ d = 0 only for Q = 0. The deflection angle θ d = 0 is schematically
illustrated in Fig. 7.6 for various skyrmion-like textures and Gilbert dampings α.
Moreover, (7.27) reveals that a finite charge Q reduces the (absolute) velocity | ˙
R|. The
dependence on Q should, however, be investigated more thoroughly as the dissipation
167
along a deflected direction that depends on the winding number Q [27]. For both,
skyrmions and magnetic bubbles, the side-drift response can be understood within
the Thiele approach, (7.8). Moreover, this effect occurs not only for field gradients
but for all forces F(q) in the Thiele equation, e.g., due to field gradients or the interaction with defects and other magnetic structures. It is also the source of the unusual
Brownian motion of skyrmions which in two dimensions diffuse less if the Gilbert
damping α is reduced [37, 75, 76]. In the following, analog to the domain wall case,
we will first discuss the limit of a pinned helicity and then consider what happens
beyond this limit.
Pinned helicity. Consider a skyrmion with winding number Q and the position
R = (X, Y ) as the only collective coordinates for the Thiele approach. In a spatially
dependent energy landscape E(R), e.g., due to anisotropy gradients, magnetic field
gradients, defects, or other magnetic textures, R is not a true zero mode but can still
be a good collective coordinate. As the system is dissipative, the skyrmion will at
some point be trapped in a local minimum of E(R). An elegant form of the resulting
Thiele equation then reads
G × ˙
R + αD(R) ˙
R = F(R)
(7.25)
where F(R) = −(γ /M s )∇ R E(R) is the force on the skyrmion, G = 4π Qˆ z is the
gyro-vector, and D(R) is the dissipation matrix. The gyro-vector G couples the
motion of the X and Y coordinates and leads to the side-deflection in the motion of
two-dimensional solitons with a finite topological charge. For a circular skyrmion,
using the notation in (7.24) and the angular dependence φ(χ) = N χ , N ∈ Z, the
dissipation matrix reduces to a scalar with
D(R) = D s (R) =
∞
0
πρ
N
2
ρ 2 sin
2
θ(ρ) +
θ
(ρ)
2
dρ
(7.26)
where ρ is the distance to R, i.e., the center of the skyrmion. Usually, it is assumed
that the texture of the skyrmion does not change much with the position such that
D s (R) ≈ D s is a good approximation.
The Thiele equation, (7.25), can be solved for the skyrmion velocity ˙
R. Its absolute
value | ˙
R| and the direction relative to the force F, parameterized by the deflection
angle θ d , then read
| ˙
R| =
|F(R)|
(4π Q) 2 + α 2 D 2
s
and θ d = − arctan
4π Q
α D s
.
(7.27)
In this formulation, the real-space topological nature of the side-deflection can be
identified as θ d = 0 only for Q = 0. The deflection angle θ d = 0 is schematically
illustrated in Fig. 7.6 for various skyrmion-like textures and Gilbert dampings α.
Moreover, (7.27) reveals that a finite charge Q reduces the (absolute) velocity | ˙
R|. The
dependence on Q should, however, be investigated more thoroughly as the dissipation