378
X. S. Wang and X. R. Wang
The sextic equation does not have a closed-form solution, but their numerical solutions are easily obtained as plotted in dashed lines in Fig. 14.12d. In summary, our
approximate formula agrees very well with the simulations for R w as expected
from our approximation. For smaller skyrmions, the approximation is still not bad,
and qualitatively gives correct parameter dependence. We can also determine the
upper limit of D and lower limits of A, K , and B from the approximate formula.
Since R must be real and finite, we have
16AK > π
2 D
2
,
(14.30)
for B = 0, so that the upper limit of D and lower limits of A and K are obtained
as D <
4
π
√
AK , A >
π
2 D
2
16K
, and K >
π
2 D
2
16A
. The limit is also the critical value separating the uniform state from helical state [89]. These critical values are plotted in
Figs. 14.12a–c as vertical dashed lines that agrees also well with simulations.
It is natural to extend our approach to Bloch skyrmions in the systems with bulk
inversion symmetry broken. The bulk DMI energy E DM = D
m · (∇ × m)d S can
be rewritten as
E DM = 2π Dd sin γ
∞
0
d
dr
+
sin 2
2r
r dr,
(14.31)
where γ = ±π/2 gives minimal energy. Since all other discussions are the same
as those for Néel skyrmions, the formulas of R and w are applicable to the Bloch
skyrmions.
It is worth mentioning that the topological protection does not mean an absolute
stability. The energy barrier between a skyrmion state and the single domain state
is still finite. The skyrmions can be generated from the edges without breaking the
conservation of topology [90], and from the bulk with the conservation of topology
broken [70, 71].
The non-trivial topology of the skyrmions is a double-edged sword. Despite of
the better stability, similar to the vortices, the non-trivial topological number of
skyrmions leads to a finite gyrovector [54] in the Thiele equation (14.12), so there
is a Skyrmion Hall effect which means the existence of a transverse motion in a
longitudinal driving force [52, 68, 79, 91–93]. Thus, there exists a threshold current
density above which skyrmions can annihilate at the film edge [94]. This edge effect
strongly limits the speed of skyrmion propagation which is of vital importance for
real applications.
14.2.3 Hopfions
The 1D and 2D magnetic solitons have been discovered and studied for a long time.
A natural question is whether there are 3D solitons and how they behave. Of course,
there are transient states that can be regarded as 3D solitons, and some of them can be
X. S. Wang and X. R. Wang
The sextic equation does not have a closed-form solution, but their numerical solutions are easily obtained as plotted in dashed lines in Fig. 14.12d. In summary, our
approximate formula agrees very well with the simulations for R w as expected
from our approximation. For smaller skyrmions, the approximation is still not bad,
and qualitatively gives correct parameter dependence. We can also determine the
upper limit of D and lower limits of A, K , and B from the approximate formula.
Since R must be real and finite, we have
16AK > π
2 D
2
,
(14.30)
for B = 0, so that the upper limit of D and lower limits of A and K are obtained
as D <
4
π
√
AK , A >
π
2 D
2
16K
, and K >
π
2 D
2
16A
. The limit is also the critical value separating the uniform state from helical state [89]. These critical values are plotted in
Figs. 14.12a–c as vertical dashed lines that agrees also well with simulations.
It is natural to extend our approach to Bloch skyrmions in the systems with bulk
inversion symmetry broken. The bulk DMI energy E DM = D
m · (∇ × m)d S can
be rewritten as
E DM = 2π Dd sin γ
∞
0
d
dr
+
sin 2
2r
r dr,
(14.31)
where γ = ±π/2 gives minimal energy. Since all other discussions are the same
as those for Néel skyrmions, the formulas of R and w are applicable to the Bloch
skyrmions.
It is worth mentioning that the topological protection does not mean an absolute
stability. The energy barrier between a skyrmion state and the single domain state
is still finite. The skyrmions can be generated from the edges without breaking the
conservation of topology [90], and from the bulk with the conservation of topology
broken [70, 71].
The non-trivial topology of the skyrmions is a double-edged sword. Despite of
the better stability, similar to the vortices, the non-trivial topological number of
skyrmions leads to a finite gyrovector [54] in the Thiele equation (14.12), so there
is a Skyrmion Hall effect which means the existence of a transverse motion in a
longitudinal driving force [52, 68, 79, 91–93]. Thus, there exists a threshold current
density above which skyrmions can annihilate at the film edge [94]. This edge effect
strongly limits the speed of skyrmion propagation which is of vital importance for
real applications.
14.2.3 Hopfions
The 1D and 2D magnetic solitons have been discovered and studied for a long time.
A natural question is whether there are 3D solitons and how they behave. Of course,
there are transient states that can be regarded as 3D solitons, and some of them can be