14 Topology in Magnetism
379
topologically non-trivial [95]. However, the existence of stable 3D solitons in magnetic systems is a highly non-trivial problem. The Derrick theorem [96] indicates that
there is no stable solitons in three or higher dimension for Heisenberg ferromagnets.
However, the theorem does not forbid the existence of stable 3D solitons in the presence of DMI or higher-order exchange interactions. In addition, the Derrick theorem
is only applicable for infinite systems. In finite systems with boundary confinement,
stable 3D solitons are also possible.
The existence of 3D string-like topological solitons has been proposed by Ludvig
D. Faddeev [97] as a limit of the Skyrme model [64]. These 3D topological solitons
are known as Faddeev-Hopf knots [98] or hopfions, which are classified by a topological charge called the Hopf index [99]. Hopfions have been discussed in many physical systems, such as gauge theories [97, 100], cosmic strings [101], ferromagnets
[95] (as a special case of dynamical vortex rings), low-temperature bosonic systems
[102–104], fluids [105], and liquid crystals [106–108]. Recently, stable magnetic
hopfions were numerically predicted in finite-size noncentrosymmetric FM systems
with Dzyaloshinskii-Moriya interaction (DMI) [109, 110] and interfacial perpendicular magnetic anisotropy (PMA) [111–113] or higher-order exchange interaction
[114].
To study the statics and dynamics hopfions, we consider a film of thickness d t ,
and start with the energy functional [56],
F =
V
A ex |∇m|
2
+ E DM
m,
∂m
∂x i
+ K b (1 − m
2
z )
+ B M s (1 − m z )
dV +
z=±d/2
K s (1 − m
2
z )dS + E d , (14.32)
where A ex is the exchange constant; E DM is the aforementioned DMI energy density
functional, which depends on the symmetry of the system; K b and K s are the bulk
PMA and the interfacial PMA, respectively; B is a perpendicular magnetic field; M s
is the saturation magnetization; and E d is the demagnetization energy.
Figures 14.13a, b show the typical magnetization profiles of Bloch-type (corresponding to Bulk DMI) and Néel-type hopfions (corresponding to interfacial
DMI), respectively, obtained by numerical relaxation of the total energy using
conjugate gradient method [115]. The simulations are performed using MuMax3
[87] at zero temperature with mesh size 0.5 nm×0.5 nm×0.5 nm. We consider a
128 nm×128 nm×16 nm film with A ex = 0.16 pJ m
−1 and M s = 1.51 × 10
5 A m
−1 ,
representing MnSi parameters [111]. No external field is applied. The Bloch-type
(Néel-type) hopfions are favorable in bulk (interfacial) DMI systems. In Fig. 14.13a,
we use K s = 0.5 mJ m
−2 , K b = 41 kJ m
−3 , and D b = 0.115 mJ m
−2 , while in
Fig. 14.13b, we use K s = 0.5 mJ m
−2 , K b = 20 kJ m
−3 , and D i = 0.115 mJ m
−2 .
The surface PMA is modelled by imposing a very strong PMA K = 10
6 J m
−3 on
two additional layers attached to the top and bottom surfaces. This corresponds to a
surface anisotropy K s = 0.5 mJ m
−2 by multiplying the mesh size.
379
topologically non-trivial [95]. However, the existence of stable 3D solitons in magnetic systems is a highly non-trivial problem. The Derrick theorem [96] indicates that
there is no stable solitons in three or higher dimension for Heisenberg ferromagnets.
However, the theorem does not forbid the existence of stable 3D solitons in the presence of DMI or higher-order exchange interactions. In addition, the Derrick theorem
is only applicable for infinite systems. In finite systems with boundary confinement,
stable 3D solitons are also possible.
The existence of 3D string-like topological solitons has been proposed by Ludvig
D. Faddeev [97] as a limit of the Skyrme model [64]. These 3D topological solitons
are known as Faddeev-Hopf knots [98] or hopfions, which are classified by a topological charge called the Hopf index [99]. Hopfions have been discussed in many physical systems, such as gauge theories [97, 100], cosmic strings [101], ferromagnets
[95] (as a special case of dynamical vortex rings), low-temperature bosonic systems
[102–104], fluids [105], and liquid crystals [106–108]. Recently, stable magnetic
hopfions were numerically predicted in finite-size noncentrosymmetric FM systems
with Dzyaloshinskii-Moriya interaction (DMI) [109, 110] and interfacial perpendicular magnetic anisotropy (PMA) [111–113] or higher-order exchange interaction
[114].
To study the statics and dynamics hopfions, we consider a film of thickness d t ,
and start with the energy functional [56],
F =
V
A ex |∇m|
2
+ E DM
m,
∂m
∂x i
+ K b (1 − m
2
z )
+ B M s (1 − m z )
dV +
z=±d/2
K s (1 − m
2
z )dS + E d , (14.32)
where A ex is the exchange constant; E DM is the aforementioned DMI energy density
functional, which depends on the symmetry of the system; K b and K s are the bulk
PMA and the interfacial PMA, respectively; B is a perpendicular magnetic field; M s
is the saturation magnetization; and E d is the demagnetization energy.
Figures 14.13a, b show the typical magnetization profiles of Bloch-type (corresponding to Bulk DMI) and Néel-type hopfions (corresponding to interfacial
DMI), respectively, obtained by numerical relaxation of the total energy using
conjugate gradient method [115]. The simulations are performed using MuMax3
[87] at zero temperature with mesh size 0.5 nm×0.5 nm×0.5 nm. We consider a
128 nm×128 nm×16 nm film with A ex = 0.16 pJ m
−1 and M s = 1.51 × 10
5 A m
−1 ,
representing MnSi parameters [111]. No external field is applied. The Bloch-type
(Néel-type) hopfions are favorable in bulk (interfacial) DMI systems. In Fig. 14.13a,
we use K s = 0.5 mJ m
−2 , K b = 41 kJ m
−3 , and D b = 0.115 mJ m
−2 , while in
Fig. 14.13b, we use K s = 0.5 mJ m
−2 , K b = 20 kJ m
−3 , and D i = 0.115 mJ m
−2 .
The surface PMA is modelled by imposing a very strong PMA K = 10
6 J m
−3 on
two additional layers attached to the top and bottom surfaces. This corresponds to a
surface anisotropy K s = 0.5 mJ m
−2 by multiplying the mesh size.