7 Current-Induced Dynamics of Chiral Magnetic Structures
171
±1. The properties of θ
STT
d
for different skyrmion-like solitons are schematically
summarized in Fig. 7.6.
The skyrmion Hall angle can change dramatically, for example, when considering
small random defects which additionally reduce the velocity [62, 72]. Extended
defects, such as the DMI-induced twisting at the edge of a sample, in turn, can speed
up the skyrmion motion ∝ |Q|/α when STTs push the skyrmion into nonequilibrium
positions [71]. To accommodate the translationally non-invariant case in the Thiele
formalism one has to take a spatially dependent force in (7.29) into account.
Moreover, the STT-induced torques can distort the skyrmion profile on the level
of the LLGS equation which eventually leads to strong corrections to the skyrmion
Hall effect and, even more importantly, a speed limit above which the STTs destroy
the skyrmion [74]. The latter cannot be derived from a simple Thiele ansatz and
requires more rigorous models or numerical simulations of the LLGS equation.
Unpinned helicity. As discussed above in Sect. 7.5.3.2, for a circular skyrmion, the
coupling between the collective coordinates R and the helicity ψ is absent. Similarly,
because of ∂ X m = −∂ x m, STTs do not directly couple to the helicity. However, STTdriven skyrmions can still show dynamics of the helicity, e.g., in an energy landscape
E(R, ψ) where the position and helicity are coupled. In this case, the helicity of the
moving skyrmion shows dynamics around the local optimum ψ 0 (R), potentially
showing features of an effective helicity mass [51]. Moreover, STTs can deform the
skyrmion which enables the coupling in the Thiele equation, leading to a steady
rotation of the helicity.
7.5.3.4 Skyrmion Motion Due to Spin-Orbit Torques
In contrast to STTs, SOTs couple directly to the magnetization texture and not
derivates thereof. Therefore, SOT induced dynamics are sensitive to the helicity
ψ of the magnetic soliton as we will discuss in more detail in the following. A
skyrmion Hall effect can also be derived for skyrmions with SOTs and was recently
also confirmed experimentally [73, 79].
Pinned helicity. In monolayers on heavy metal substrates or thin films of stacked
heterostructures, skyrmions can be stabilized by a strong DMI with extra stabilizing
support from dipolar interactions. These interactions usually pin the helicity such
that it does not contribute to the dynamics.
For the axially symmetric soliton with m(∞) = ˆ
z and φ = N χ in the ansatz in
(7.24), the gyro-coupling SOT matrix evalutes to zero whereas the dissipative SOT
coupling matrix elements only vanish for |N | = 1. Note that in this convention,
the relation Q = N (1 − m z (0))/2 implies that not only the skyrmion and the antiskyrmion, but also the topologically trivial skyrmionium can be driven by SOTs.
In turn, higher order skyrmions with |Q| > 1 do not react to SOTs within these
approximations. The Thiele equation for these |N | = 1 objects can be written as
G × ˙
R + α D s ˙
R + τ DL σ · D
SOT
R (ψ) = 0 .
(7.32)
171
±1. The properties of θ
STT
d
for different skyrmion-like solitons are schematically
summarized in Fig. 7.6.
The skyrmion Hall angle can change dramatically, for example, when considering
small random defects which additionally reduce the velocity [62, 72]. Extended
defects, such as the DMI-induced twisting at the edge of a sample, in turn, can speed
up the skyrmion motion ∝ |Q|/α when STTs push the skyrmion into nonequilibrium
positions [71]. To accommodate the translationally non-invariant case in the Thiele
formalism one has to take a spatially dependent force in (7.29) into account.
Moreover, the STT-induced torques can distort the skyrmion profile on the level
of the LLGS equation which eventually leads to strong corrections to the skyrmion
Hall effect and, even more importantly, a speed limit above which the STTs destroy
the skyrmion [74]. The latter cannot be derived from a simple Thiele ansatz and
requires more rigorous models or numerical simulations of the LLGS equation.
Unpinned helicity. As discussed above in Sect. 7.5.3.2, for a circular skyrmion, the
coupling between the collective coordinates R and the helicity ψ is absent. Similarly,
because of ∂ X m = −∂ x m, STTs do not directly couple to the helicity. However, STTdriven skyrmions can still show dynamics of the helicity, e.g., in an energy landscape
E(R, ψ) where the position and helicity are coupled. In this case, the helicity of the
moving skyrmion shows dynamics around the local optimum ψ 0 (R), potentially
showing features of an effective helicity mass [51]. Moreover, STTs can deform the
skyrmion which enables the coupling in the Thiele equation, leading to a steady
rotation of the helicity.
7.5.3.4 Skyrmion Motion Due to Spin-Orbit Torques
In contrast to STTs, SOTs couple directly to the magnetization texture and not
derivates thereof. Therefore, SOT induced dynamics are sensitive to the helicity
ψ of the magnetic soliton as we will discuss in more detail in the following. A
skyrmion Hall effect can also be derived for skyrmions with SOTs and was recently
also confirmed experimentally [73, 79].
Pinned helicity. In monolayers on heavy metal substrates or thin films of stacked
heterostructures, skyrmions can be stabilized by a strong DMI with extra stabilizing
support from dipolar interactions. These interactions usually pin the helicity such
that it does not contribute to the dynamics.
For the axially symmetric soliton with m(∞) = ˆ
z and φ = N χ in the ansatz in
(7.24), the gyro-coupling SOT matrix evalutes to zero whereas the dissipative SOT
coupling matrix elements only vanish for |N | = 1. Note that in this convention,
the relation Q = N (1 − m z (0))/2 implies that not only the skyrmion and the antiskyrmion, but also the topologically trivial skyrmionium can be driven by SOTs.
In turn, higher order skyrmions with |Q| > 1 do not react to SOTs within these
approximations. The Thiele equation for these |N | = 1 objects can be written as
G × ˙
R + α D s ˙
R + τ DL σ · D
SOT
R (ψ) = 0 .
(7.32)