154
J. Masell and K. Everschor-Sitte
with the vector field F defined above in (7.5) and A being an appropriate vector
potential ∇ × A = F. They have been predicted to occur in magnetic systems [13,
14] and recent progress in the development of new experimental techniques led to
the first experimental signatures of hopfions in magnetic multilayers [114].
Other topological magnetic textures apart from the above mentioned ones, are
predicted including those which have a more complex order parameter than just
the normalized magnetization. Several of them have not yet been observed experimentally. However, the vast progress in recent years, allowing to engineer coupling
strengths and image magnetizations in more and more detail, might reveal more
exotic states in the future.
7.4 Creation of Magnetic Solitons
In this section we discuss, from a theoretical point of view, how to create magnetic solitons in different dimensions. These solitons comprise domain walls and
skyrmions, see Sect. 7.3, and can be introduced into a given magnetic background,
such as the ferromagnetic or helical state. Before discussing specific properties of
different creation mechanisms, we first comment on a few very generic principles.
A soliton is stable and does not decay into magnons if it is protected by a (free)
energy barrier. Vice versa, the creation of a soliton is also associated with an energy
barrier, otherwise the solitons would just spontaneously proliferate and trigger a
phase transition. As an example, Fig. 7.2 shows two possible mechanisms for the
creation/annihilation of a skyrmion in a finite-size two-dimensional system. On the
blue path, the magnetization twists at the edge and a skyrmion enters from outside
Fig. 7.2 Energetics of skyrmion creation in a finite size system with interfacial DMI. The energy
barrier depends on details of the interactions but also on the creation process which can, for example,
involve a continuous change of the winding number via the edge of the system (blue path) or a
discontinuous change via the creation of a skyrmion in the bulk (red path)
J. Masell and K. Everschor-Sitte
with the vector field F defined above in (7.5) and A being an appropriate vector
potential ∇ × A = F. They have been predicted to occur in magnetic systems [13,
14] and recent progress in the development of new experimental techniques led to
the first experimental signatures of hopfions in magnetic multilayers [114].
Other topological magnetic textures apart from the above mentioned ones, are
predicted including those which have a more complex order parameter than just
the normalized magnetization. Several of them have not yet been observed experimentally. However, the vast progress in recent years, allowing to engineer coupling
strengths and image magnetizations in more and more detail, might reveal more
exotic states in the future.
7.4 Creation of Magnetic Solitons
In this section we discuss, from a theoretical point of view, how to create magnetic solitons in different dimensions. These solitons comprise domain walls and
skyrmions, see Sect. 7.3, and can be introduced into a given magnetic background,
such as the ferromagnetic or helical state. Before discussing specific properties of
different creation mechanisms, we first comment on a few very generic principles.
A soliton is stable and does not decay into magnons if it is protected by a (free)
energy barrier. Vice versa, the creation of a soliton is also associated with an energy
barrier, otherwise the solitons would just spontaneously proliferate and trigger a
phase transition. As an example, Fig. 7.2 shows two possible mechanisms for the
creation/annihilation of a skyrmion in a finite-size two-dimensional system. On the
blue path, the magnetization twists at the edge and a skyrmion enters from outside
Fig. 7.2 Energetics of skyrmion creation in a finite size system with interfacial DMI. The energy
barrier depends on details of the interactions but also on the creation process which can, for example,
involve a continuous change of the winding number via the edge of the system (blue path) or a
discontinuous change via the creation of a skyrmion in the bulk (red path)