7 Current-Induced Dynamics of Chiral Magnetic Structures
161
The magnetic field-driven dynamics of one-dimensional magnetic domain walls
have been extensively studied over many decades and can be well described in the
Thiele framework. Also magnetic domain walls in higher dimensions can be well
described by this simple technique. Here, additionally to the one-dimensional case,
the position of the domain wall is not only a one-dimensional parameter, but characterized by an extended line or surface. The additional degrees of freedom that then
typically become relevant is the tilting [39] or bending of the hyperplanes of the
domain walls.
To demonstrate the Thiele approach, let us consider a domain wall in an effectively
one-dimensional system. This means we assume that a domain wall is located in a
nanowire which is narrow compared to the length scale of the variations of the
magnetic texture. In such a system, a simple ansatz for the domain wall profile can
be written as
m (x − X, ψ) = (cos ψ sin θ (x − X ) , sin ψ sin θ (x − X ) , cos θ (x − X )) .
(7.11)
where X and ψ are the position and the helicity of the domain wall, respectively,
where ψ = ±π/2 describes a Bloch type wall and ψ = 0 or π describes a Néel type
wall. θ(x) is the azimuthal angle of the magnetization. Here we assumed that the
nanowire is along the ˆ
x-direction and that the helicity is not spatially dependent.
Consider now the standard thin film setup as introduced in Sect. 7.2, where the
DMI is the relevant source of the twisting of the magnetization and magnetostatic
interactions only enters on the level of a modified uniaxial anisotropy. In its simplified
form, the only parameters that enter the energy functional (7.1) for a low energy
description are the uniform exchange A, interfacial DMI D > 0, and the easy-axis
anisotropy K > 0. In one spatial dimension, the energy functional then explicitly
takes the form
E[m] =
A
dm
dx
2
− D
m x
dm z
dx
− m z
dm x
dx
− K m
2
z dx .
(7.12)
A domain wall which connects two polarized phases m(−∞) = −ˆ z and m(∞) = ˆ
z
minimizes this energy functional for the profile
θ(x) = −2 arctan
e
−
√
K /A x
and ψ = π .
(7.13)
Here, the DMI term fixes the helicity ψ = π while the other terms are independent of
ψ. In the following, we will use the ansatz, (7.11), and the profile, (7.13), to discuss
the current-driven motion of domain walls on the Thiele level.
Note that, in broader nanowires, the additional spatial dimension can allow for
more complex domain wall profiles and also dynamics. In particular, domain walls in
finite-width systems with DMI show a tilting of the domain wall normal [64] which
can be explained by the interaction with the edges of the system [65]. The dynamics
161
The magnetic field-driven dynamics of one-dimensional magnetic domain walls
have been extensively studied over many decades and can be well described in the
Thiele framework. Also magnetic domain walls in higher dimensions can be well
described by this simple technique. Here, additionally to the one-dimensional case,
the position of the domain wall is not only a one-dimensional parameter, but characterized by an extended line or surface. The additional degrees of freedom that then
typically become relevant is the tilting [39] or bending of the hyperplanes of the
domain walls.
To demonstrate the Thiele approach, let us consider a domain wall in an effectively
one-dimensional system. This means we assume that a domain wall is located in a
nanowire which is narrow compared to the length scale of the variations of the
magnetic texture. In such a system, a simple ansatz for the domain wall profile can
be written as
m (x − X, ψ) = (cos ψ sin θ (x − X ) , sin ψ sin θ (x − X ) , cos θ (x − X )) .
(7.11)
where X and ψ are the position and the helicity of the domain wall, respectively,
where ψ = ±π/2 describes a Bloch type wall and ψ = 0 or π describes a Néel type
wall. θ(x) is the azimuthal angle of the magnetization. Here we assumed that the
nanowire is along the ˆ
x-direction and that the helicity is not spatially dependent.
Consider now the standard thin film setup as introduced in Sect. 7.2, where the
DMI is the relevant source of the twisting of the magnetization and magnetostatic
interactions only enters on the level of a modified uniaxial anisotropy. In its simplified
form, the only parameters that enter the energy functional (7.1) for a low energy
description are the uniform exchange A, interfacial DMI D > 0, and the easy-axis
anisotropy K > 0. In one spatial dimension, the energy functional then explicitly
takes the form
E[m] =
A
dm
dx
2
− D
m x
dm z
dx
− m z
dm x
dx
− K m
2
z dx .
(7.12)
A domain wall which connects two polarized phases m(−∞) = −ˆ z and m(∞) = ˆ
z
minimizes this energy functional for the profile
θ(x) = −2 arctan
e
−
√
K /A x
and ψ = π .
(7.13)
Here, the DMI term fixes the helicity ψ = π while the other terms are independent of
ψ. In the following, we will use the ansatz, (7.11), and the profile, (7.13), to discuss
the current-driven motion of domain walls on the Thiele level.
Note that, in broader nanowires, the additional spatial dimension can allow for
more complex domain wall profiles and also dynamics. In particular, domain walls in
finite-width systems with DMI show a tilting of the domain wall normal [64] which
can be explained by the interaction with the edges of the system [65]. The dynamics