7 Current-Induced Dynamics of Chiral Magnetic Structures
169
δ B between the velocity ˙
X and ˙
ψ to (7.28a) and (7.28b). For the full dynamics,
therefore, both the change of the skyrmion size and the its non-circular distortion
contribute.
In the following sections, we review the dynamics of current-driven instead of
force-driven skyrmions which follow the same basic concepts.
7.5.3.3 Skyrmion Motion Due to Spin-Transfer Torques
A standard example for the application of a Thiele equation is to model the motion
of spin-tranfer torque-driven skyrmions in chiral magnets. The Thiele formalism,
for example, provides a direct mean to explain the skyrmion Hall effect, where the
skyrmions move at an angle relative to the direction of the applied current density j e .
We will discuss this in the following. Analog to the previous chapters we will first
discuss the limit of a pinned helicity and then explain briefly what happens beyond
this regime.
Pinned helicity. Let us consider a frequently used assumption for skyrmions in
chiral magnets, namely that the helicity is pinned by DMI to a fixed value ψ and
does not contribute to the dynamics. Consider, moreover, that the system is translation
invariant, i.e., the position R is a zero mode, and that the skyrmion can be described
by the ansatz in (7.24). The Thiele equation then reads
G × ( ˙
R − v e ) + D s (α ˙
R − βv e ) = 0
(7.29)
where G = 4π Qˆ z is the gyro-vector, and the dissipation matrix D reduces to a scalar
D s as in (7.26). In principle, this equation of motion can be solved for ˙
R which yields
the skyrmion Hall effect. Alternatively, we can interpret the effect of the STTs from
a different perspective. By isolating all terms which originate from the STT on the
right hand side of (7.29), we effectively recover the Thiele equation for a skyrmion
driven by an external force, (7.25), with
F
STT
= G × v e + β D s v e .
(7.30)
The skyrmion Hall angle θ
STT
d
is then the sum of (i) angle θ
STT
F
between the effective
STT-force F
STT and the direction of the current v e and (ii) the deflection angle θ d for
a force-driven skyrmion, see (7.27), and reads
θ
STT
d
= arctan
4π Q
β D s
− arctan
4π Q
α D s
= arctan
4π QD s (α − β)
(4π Q) 2 + αβ D 2
s
. (7.31)
The result reflects the trivial cases θ
STT
d
= 0 for α = β or for Q = 0 where the
magnetic texture just moves along with the current. In contrast to the deflection
angle θ d in the previous section (7.27), the skyrmion Hall angle θ
STT
d
shrinks for
increasing Q and, for typical values of parameters, the maximal θ
STT
d
is at Q =
169
δ B between the velocity ˙
X and ˙
ψ to (7.28a) and (7.28b). For the full dynamics,
therefore, both the change of the skyrmion size and the its non-circular distortion
contribute.
In the following sections, we review the dynamics of current-driven instead of
force-driven skyrmions which follow the same basic concepts.
7.5.3.3 Skyrmion Motion Due to Spin-Transfer Torques
A standard example for the application of a Thiele equation is to model the motion
of spin-tranfer torque-driven skyrmions in chiral magnets. The Thiele formalism,
for example, provides a direct mean to explain the skyrmion Hall effect, where the
skyrmions move at an angle relative to the direction of the applied current density j e .
We will discuss this in the following. Analog to the previous chapters we will first
discuss the limit of a pinned helicity and then explain briefly what happens beyond
this regime.
Pinned helicity. Let us consider a frequently used assumption for skyrmions in
chiral magnets, namely that the helicity is pinned by DMI to a fixed value ψ and
does not contribute to the dynamics. Consider, moreover, that the system is translation
invariant, i.e., the position R is a zero mode, and that the skyrmion can be described
by the ansatz in (7.24). The Thiele equation then reads
G × ( ˙
R − v e ) + D s (α ˙
R − βv e ) = 0
(7.29)
where G = 4π Qˆ z is the gyro-vector, and the dissipation matrix D reduces to a scalar
D s as in (7.26). In principle, this equation of motion can be solved for ˙
R which yields
the skyrmion Hall effect. Alternatively, we can interpret the effect of the STTs from
a different perspective. By isolating all terms which originate from the STT on the
right hand side of (7.29), we effectively recover the Thiele equation for a skyrmion
driven by an external force, (7.25), with
F
STT
= G × v e + β D s v e .
(7.30)
The skyrmion Hall angle θ
STT
d
is then the sum of (i) angle θ
STT
F
between the effective
STT-force F
STT and the direction of the current v e and (ii) the deflection angle θ d for
a force-driven skyrmion, see (7.27), and reads
θ
STT
d
= arctan
4π Q
β D s
− arctan
4π Q
α D s
= arctan
4π QD s (α − β)
(4π Q) 2 + αβ D 2
s
. (7.31)
The result reflects the trivial cases θ
STT
d
= 0 for α = β or for Q = 0 where the
magnetic texture just moves along with the current. In contrast to the deflection
angle θ d in the previous section (7.27), the skyrmion Hall angle θ
STT
d
shrinks for
increasing Q and, for typical values of parameters, the maximal θ
STT
d
is at Q =