7 Current-Induced Dynamics of Chiral Magnetic Structures
159
7.5.1 A Collective Coordinate Approximation: Thiele
Equations of Motion
The main step to obtain the Thiele equations for a given magnetic structure is to
project the LLGS equation onto the corresponding collective coordinates. This said,
the first step is to select suitable collective coordinates for a given magnetic structure.
In principle, these collective coordinates can represent any property of the quasiparticle. To achieve an accurate description of the system with only a few coordinates
it makes sense to choose coordinates which are related to zero modes or low energy
modes as these are most easily activated, and thus most relevant for the low-energy
physics of the system. A suitable choice of coordinates should therefore depend on
the symmetries of the entire setup: the quasi-particle itself, the energy landscape,
and the acting spin-torques.
To give an example for an appropriate collective coordinate, let us consider the
standard assumption of the standard Thiele approach, i.e., a translational invariant
model with a rigid magnetic texture. This means that the magnetic texture does not
change its shape when driven by an electric current. In this situation, the position
of the quasi-particle R(t) is a proper collective coordinate (or more generally, any
position of the rigid magnetic structure) and the magnetization behaves as m(r, t) =
m(r − R(t), 0).
For the derivation of the generalized Thiele equations, suppose that the timedependence of the magnetic texture m(r, t) is described by N collective coordinates
q(t) = {q i (t)} i=1,...,N . We first isolate the expression for the effective magnetic field
B eff by multiplying the LLGS equation (7.2), by m× from the left.
2 Next, we project
the LLGS equation onto the translational mode
dm
dq i
of the i-th collective coordinate q i ,
where the projection P(q i ) is implemented by the scalar product P(q i ) = =
dm
dq i
| . =
dr(
dm
dq i
· . ). Moreover, we explicitly use that all time-dependence is now expressed
in the collective coordinates to replace d t m =
N
j=1 ˙
q j
dm
dq j
where ˙
q j = d t q j . A compact form of the i = 1, ..., N Thiele equations for an arbitrary magnetic texture with
both STTs and SOTs then reads
F i (q) = G i j ˙
q j + αD i j ˙
q j + G
STT
iμ v e,μ + βD
STT
iμ v e,μ + τ FL G
SOT
iμ σ μ + τ DL D
SOT
iμ σ μ
(7.8)
with implicit summation over both, the collective coordinates j = 1, ..., N and the
spatial dimensions μ = x, y, z. The projection of the effective magnetic field B eff =
−δ E[m]/(M s δm) can be interpreted as a force
F i (q) = −
γ
M s
dE[q]
dq i
= −
γ
M s
dm
dq i
·
δ E
δm
dr
(7.9)
2 We exploit that the magnetization is a normalized vector field with |m(r)| = 1. Thus, m ⊥ ∂ i m and
m × (m × ∂ i m) = −∂ i m for all coordinates i = x, y, z, t. Moreover, m ⊥ B eff is always achieved
by adding a term λ(r)(1 − m 2 ) = 0 to the energy functional which does not change the energy but
cancels all components of B eff that are parallel to m(r).
159
7.5.1 A Collective Coordinate Approximation: Thiele
Equations of Motion
The main step to obtain the Thiele equations for a given magnetic structure is to
project the LLGS equation onto the corresponding collective coordinates. This said,
the first step is to select suitable collective coordinates for a given magnetic structure.
In principle, these collective coordinates can represent any property of the quasiparticle. To achieve an accurate description of the system with only a few coordinates
it makes sense to choose coordinates which are related to zero modes or low energy
modes as these are most easily activated, and thus most relevant for the low-energy
physics of the system. A suitable choice of coordinates should therefore depend on
the symmetries of the entire setup: the quasi-particle itself, the energy landscape,
and the acting spin-torques.
To give an example for an appropriate collective coordinate, let us consider the
standard assumption of the standard Thiele approach, i.e., a translational invariant
model with a rigid magnetic texture. This means that the magnetic texture does not
change its shape when driven by an electric current. In this situation, the position
of the quasi-particle R(t) is a proper collective coordinate (or more generally, any
position of the rigid magnetic structure) and the magnetization behaves as m(r, t) =
m(r − R(t), 0).
For the derivation of the generalized Thiele equations, suppose that the timedependence of the magnetic texture m(r, t) is described by N collective coordinates
q(t) = {q i (t)} i=1,...,N . We first isolate the expression for the effective magnetic field
B eff by multiplying the LLGS equation (7.2), by m× from the left.
2 Next, we project
the LLGS equation onto the translational mode
dm
dq i
of the i-th collective coordinate q i ,
where the projection P(q i ) is implemented by the scalar product P(q i ) = =
dm
dq i
| . =
dr(
dm
dq i
· . ). Moreover, we explicitly use that all time-dependence is now expressed
in the collective coordinates to replace d t m =
N
j=1 ˙
q j
dm
dq j
where ˙
q j = d t q j . A compact form of the i = 1, ..., N Thiele equations for an arbitrary magnetic texture with
both STTs and SOTs then reads
F i (q) = G i j ˙
q j + αD i j ˙
q j + G
STT
iμ v e,μ + βD
STT
iμ v e,μ + τ FL G
SOT
iμ σ μ + τ DL D
SOT
iμ σ μ
(7.8)
with implicit summation over both, the collective coordinates j = 1, ..., N and the
spatial dimensions μ = x, y, z. The projection of the effective magnetic field B eff =
−δ E[m]/(M s δm) can be interpreted as a force
F i (q) = −
γ
M s
dE[q]
dq i
= −
γ
M s
dm
dq i
·
δ E
δm
dr
(7.9)
2 We exploit that the magnetization is a normalized vector field with |m(r)| = 1. Thus, m ⊥ ∂ i m and
m × (m × ∂ i m) = −∂ i m for all coordinates i = x, y, z, t. Moreover, m ⊥ B eff is always achieved
by adding a term λ(r)(1 − m 2 ) = 0 to the energy functional which does not change the energy but
cancels all components of B eff that are parallel to m(r).