172
J. Masell and K. Everschor-Sitte
Here, we have introduced the 3 × 2 dissipation tensor (D
SOT
R ) μi , μ = x, y, z, and
i = X, Y , for the SOT-induces torque which reads
D
SOT
R (ψ) =
ˆ
z × ψ, −N ψ
πδ |N |,1
∞
0
cos θ(ρ) sin θ(ρ) + ρθ
(ρ) dρ
(7.33)
with ψ = (cos ψ, sin ψ, 0). The Kronecker delta δ |N |,1 indicates that only solitons
with N = ±1 give finite contributions. The asymmetric N -dependence in only the
second column of D
SOT
R
is an artefact of the ansatz (7.24), which flips m y → −m y
for N → −N .
Similar to the discussion in Sect. 7.5.3.3, we can interpret the SOT-induced terms
as an external force F
SOT and derive the skyrmion Hall angle from the direction of
this effective force. Assuming again the standard spin Hall setup for the SOT with
j e = j e ˆ
x and σ = ˆ
z × j e = j e ˆ
y, the skyrmion Hall angle θ
SOT
d
becomes
θ
SOT
d
(ψ) = −N
ψ + π − arctan
4π Q
α D s
with |N | = 1 ,
(7.34)
which is only well-defined for |N | = 1 as otherwise the soliton does not move. Note
that the skyrmion Hall angle θ
SOT
d
is a function of the helicity ψ and can result in
a motion in arbitrary directions, including parallel to the current, by fine-tuning the
DMI [80]. This angular dependence is also schematically summarized in Fig. 7.6 for
various soliton configurations and parameters.
Unpinned helicity. For circular two-dimensional solitons driven by SOTs, the same
physics arises as for the other driving mechanisms, namely that neither the collective
coordinates R couple directly to ψ nor do the Thiele matrices G
SOT
ψi and D
SOT
ψi yield
a finite coupling between σ and ψ (except for D
SOT
ψ z , which is usually not relevant
as σ z = 0).
A distinguished feature of SOTs is that they tilt the background magnetization.
This naturally leads to deformations of the soliton, breaking the axial symmetry and
enabling a finite coupling of ψ and ˙
R, see Sect. 7.5.3.2. Thus, while moving with
velocity ˙
R at a skyrmion Hall angle θ
SOT
d
(ψ), see (7.34), the helicity ψ changes
which feeds back on θ
SOT
d
(ψ). Consequently, the skyrmion with an activated helicity
degree of freedom can end up orbiting around a fixed point [81] or, for sufficiently
asymmetric energy landscapes E(ψ), perform a trochoidal motion [82] which is a
combination of translation and orbiting. Moreover, once the helicity becomes dynamical, it can also lead to an effective mass in the Thiele equation [51, 81].
J. Masell and K. Everschor-Sitte
Here, we have introduced the 3 × 2 dissipation tensor (D
SOT
R ) μi , μ = x, y, z, and
i = X, Y , for the SOT-induces torque which reads
D
SOT
R (ψ) =
ˆ
z × ψ, −N ψ
πδ |N |,1
∞
0
cos θ(ρ) sin θ(ρ) + ρθ
(ρ) dρ
(7.33)
with ψ = (cos ψ, sin ψ, 0). The Kronecker delta δ |N |,1 indicates that only solitons
with N = ±1 give finite contributions. The asymmetric N -dependence in only the
second column of D
SOT
R
is an artefact of the ansatz (7.24), which flips m y → −m y
for N → −N .
Similar to the discussion in Sect. 7.5.3.3, we can interpret the SOT-induced terms
as an external force F
SOT and derive the skyrmion Hall angle from the direction of
this effective force. Assuming again the standard spin Hall setup for the SOT with
j e = j e ˆ
x and σ = ˆ
z × j e = j e ˆ
y, the skyrmion Hall angle θ
SOT
d
becomes
θ
SOT
d
(ψ) = −N
ψ + π − arctan
4π Q
α D s
with |N | = 1 ,
(7.34)
which is only well-defined for |N | = 1 as otherwise the soliton does not move. Note
that the skyrmion Hall angle θ
SOT
d
is a function of the helicity ψ and can result in
a motion in arbitrary directions, including parallel to the current, by fine-tuning the
DMI [80]. This angular dependence is also schematically summarized in Fig. 7.6 for
various soliton configurations and parameters.
Unpinned helicity. For circular two-dimensional solitons driven by SOTs, the same
physics arises as for the other driving mechanisms, namely that neither the collective
coordinates R couple directly to ψ nor do the Thiele matrices G
SOT
ψi and D
SOT
ψi yield
a finite coupling between σ and ψ (except for D
SOT
ψ z , which is usually not relevant
as σ z = 0).
A distinguished feature of SOTs is that they tilt the background magnetization.
This naturally leads to deformations of the soliton, breaking the axial symmetry and
enabling a finite coupling of ψ and ˙
R, see Sect. 7.5.3.2. Thus, while moving with
velocity ˙
R at a skyrmion Hall angle θ
SOT
d
(ψ), see (7.34), the helicity ψ changes
which feeds back on θ
SOT
d
(ψ). Consequently, the skyrmion with an activated helicity
degree of freedom can end up orbiting around a fixed point [81] or, for sufficiently
asymmetric energy landscapes E(ψ), perform a trochoidal motion [82] which is a
combination of translation and orbiting. Moreover, once the helicity becomes dynamical, it can also lead to an effective mass in the Thiele equation [51, 81].