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J. Masell and K. Everschor-Sitte
material with Bloch DMI, the antiskyrmion will annihilate and only a Bloch skyrmion
will remain. Similarly, skyrmions can be created by SOTs [52–54].
An alternative is to create skyrmions via exploiting the tailored geometry of the
material, see Fig. 7.4 for different options. For example, one can convert a domain
wall pair into a skyrmion [55] or one can generate skyrmions through what has
become known as “blowing bubble” technique [56] where a worm domain is sent
through constrictions and “chopped” into pieces, i.e. skyrmions, by means of the
diverging current upon leaving the constriction. Or skyrmions can be produced at
a notch [57]. The latter technique leads over to the another principal option, i.e. to
create skyrmions at the boundaries of a sample. Here the effect of the chiral surface
states are helpful in pre-twisting the magnetic configurations. By means of a properly
chosen protocol of an applied field strength one can even generate a whole train of
skyrmions at the boundary [58].
While in a ferromagnetic background single magnetic skyrmions are (meta-)stable
states, magnetic skyrmions can be the ground state of a chiral magnet in the form of
skyrmion lattices under certain conditions [29]. To switch from the competing stripe
domain phase into the skyrmion phase several methods exist, including triggering
the magnetic material by means of AC field excitations [59].
7.5 Motion of Magnetic Solitons
The micromagnetic dynamics of the magnetization are mainly governed by the LLGS
equation (7.2). This equation describes the local precession and relaxation of the
magnetization, formulated in terms of a local effective magnetic field which accounts
for the interaction of the magnetization with itself and its environment and, moreover,
additional torques due to current-induced STTs and/or SOTs. In general, these nonlinear dynamics lead to a complicated dynamical behavior which can even trigger
the creation of magnetic solitons as described in Sect. 7.4 and can usually only be
solved numerically.
However, once the solitons are created, they are influenced by the applied spintorques and other external forces, e.g., due to field gradients. The reaction of the
magnetization is most strongly expressed in the low energy degrees of freedom. An
effective and potentially more efficient description of the soliton dynamics can therefore be formulated by taking only a few collective coordinates into account. We will
review the derivation of these effective equations of motion, known as (generalized)
Thiele equations,
1 in the following. We then show examples for their application
when we use them as the starting point for the discussion of the dynamics of currentdriven magnetic solitons.
1 Note that the original equation that Thiele derived in his seminal works [60, 61] refers to the
steady-state motion of domain walls. By now the concept that Thiele used to obtain his equation
of motion for the domain wall has been generalized for any structure described by a finite set of
collective coordinates.
J. Masell and K. Everschor-Sitte
material with Bloch DMI, the antiskyrmion will annihilate and only a Bloch skyrmion
will remain. Similarly, skyrmions can be created by SOTs [52–54].
An alternative is to create skyrmions via exploiting the tailored geometry of the
material, see Fig. 7.4 for different options. For example, one can convert a domain
wall pair into a skyrmion [55] or one can generate skyrmions through what has
become known as “blowing bubble” technique [56] where a worm domain is sent
through constrictions and “chopped” into pieces, i.e. skyrmions, by means of the
diverging current upon leaving the constriction. Or skyrmions can be produced at
a notch [57]. The latter technique leads over to the another principal option, i.e. to
create skyrmions at the boundaries of a sample. Here the effect of the chiral surface
states are helpful in pre-twisting the magnetic configurations. By means of a properly
chosen protocol of an applied field strength one can even generate a whole train of
skyrmions at the boundary [58].
While in a ferromagnetic background single magnetic skyrmions are (meta-)stable
states, magnetic skyrmions can be the ground state of a chiral magnet in the form of
skyrmion lattices under certain conditions [29]. To switch from the competing stripe
domain phase into the skyrmion phase several methods exist, including triggering
the magnetic material by means of AC field excitations [59].
7.5 Motion of Magnetic Solitons
The micromagnetic dynamics of the magnetization are mainly governed by the LLGS
equation (7.2). This equation describes the local precession and relaxation of the
magnetization, formulated in terms of a local effective magnetic field which accounts
for the interaction of the magnetization with itself and its environment and, moreover,
additional torques due to current-induced STTs and/or SOTs. In general, these nonlinear dynamics lead to a complicated dynamical behavior which can even trigger
the creation of magnetic solitons as described in Sect. 7.4 and can usually only be
solved numerically.
However, once the solitons are created, they are influenced by the applied spintorques and other external forces, e.g., due to field gradients. The reaction of the
magnetization is most strongly expressed in the low energy degrees of freedom. An
effective and potentially more efficient description of the soliton dynamics can therefore be formulated by taking only a few collective coordinates into account. We will
review the derivation of these effective equations of motion, known as (generalized)
Thiele equations,
1 in the following. We then show examples for their application
when we use them as the starting point for the discussion of the dynamics of currentdriven magnetic solitons.
1 Note that the original equation that Thiele derived in his seminal works [60, 61] refers to the
steady-state motion of domain walls. By now the concept that Thiele used to obtain his equation
of motion for the domain wall has been generalized for any structure described by a finite set of
collective coordinates.