408
Z. Li et al.
center of the nanodisk and K is the spring constant. By neglecting the viscous force
term, we can derive the gyration frequency of an isolated vortex with ω 0 = K /|G|.
However, it is well known that magnetic bubbles and skyrmions in particular
manifest an inertia in their gyration motion [63, 92]. The mass effect thus should
be taken into account for describing the skyrmion (or bubble) oscillation. Therefore,
the Thiele’s equation can be generalized as:
− M
d
2 U j
dt 2 + G ˆ
z ×
dU j
dt
− α D
dU j
dt
+ F j = 0,
(15.3)
with M being the inertial mass of magnetic soliton. Similarly, we can calculate the
gyration frequency of skyrmion (or bubble) with:
ω ± = −G/2M ±
(G/2M) 2 + K /M.
(15.4)
The positive and negative values of the ω in (15.4) indicate that there are two kinds
of gyration modes with clockwise and counterclockwise direction, respectively.
Magnetic skyrmions are typical magnetic solitons stabilized by Dzyaloshinskii
Moriya interactions (DMIs) [66, 93, 94]. They can be manipulated by spin-polarized
electrical current with extremely low density [73, 74]. Skyrmion also can be drived
by other external force, such as spin wave [95, 96], microwave field [97, 98], and
temperature gradient [99–101]. Interestingly, very recently, it has been proposed
that twisted photons [102] and magnons [103] carrying orbital angular momentum
(OAM) can act as “optical tweezers” and “magnetic tweezers” to drive the rotation of
skyrmion, as shown in Fig. 15.2 and Fig. 15.3, respectively. Theoretical calculation
and numerical simulation show that the topological charge of twisted photons (or
magnons) can determine both the magnitude and the handedness of the rotation
velocity of skyrmions.
Skyrmions are the ideal information carrier in spintronic devices. However, a
skyrmion can not move in a straight line along the driving current direction because
of the Magnus force which leads to the shift of its motion trajectory, such behavior is
called skyrmion Hall effect (SHE) [104, 105]. In practical applications, the skyrmion
may be destroyed when touching the device boundaries. To overcome this issue, a lot
of methods have been proposed. For example, X. Zhang et al. [106] showed that the
antiferromagnetically exchange-coupled bilayer system containing two skyrmions
with different polarities can suppress the SHE, leading to a perfectly straight trajectory for skyrmion driven by a spin-polarized current. Moreover, the antiferromagnetic
(AFM) skyrmion can also avoid SHE [107, 108]. Interestingly, recent research shows
that the AFM skyrmion can emerge in a ferromagnet with gain (negative α) [109].
Figure 15.4 shows the dynamical process of the formation of AFM skyrmion in ferromagnet (here α is set to be −0.01), with the initial magnetization profile being
random [see Fig. 15.4a]. At t = 0.035 ns, local magnetic moments quickly evolve to
an antiparallelly aligned state, as shown in Fig. 15.4b. Then, all spins inside a circle
of radius 5 nm in the film center are randomized. At t = 0.14 ns, an AFM skyrmion
is stabilized (see Fig. 15.4d). When an in-plane spin-polarized electric current was
Z. Li et al.
center of the nanodisk and K is the spring constant. By neglecting the viscous force
term, we can derive the gyration frequency of an isolated vortex with ω 0 = K /|G|.
However, it is well known that magnetic bubbles and skyrmions in particular
manifest an inertia in their gyration motion [63, 92]. The mass effect thus should
be taken into account for describing the skyrmion (or bubble) oscillation. Therefore,
the Thiele’s equation can be generalized as:
− M
d
2 U j
dt 2 + G ˆ
z ×
dU j
dt
− α D
dU j
dt
+ F j = 0,
(15.3)
with M being the inertial mass of magnetic soliton. Similarly, we can calculate the
gyration frequency of skyrmion (or bubble) with:
ω ± = −G/2M ±
(G/2M) 2 + K /M.
(15.4)
The positive and negative values of the ω in (15.4) indicate that there are two kinds
of gyration modes with clockwise and counterclockwise direction, respectively.
Magnetic skyrmions are typical magnetic solitons stabilized by Dzyaloshinskii
Moriya interactions (DMIs) [66, 93, 94]. They can be manipulated by spin-polarized
electrical current with extremely low density [73, 74]. Skyrmion also can be drived
by other external force, such as spin wave [95, 96], microwave field [97, 98], and
temperature gradient [99–101]. Interestingly, very recently, it has been proposed
that twisted photons [102] and magnons [103] carrying orbital angular momentum
(OAM) can act as “optical tweezers” and “magnetic tweezers” to drive the rotation of
skyrmion, as shown in Fig. 15.2 and Fig. 15.3, respectively. Theoretical calculation
and numerical simulation show that the topological charge of twisted photons (or
magnons) can determine both the magnitude and the handedness of the rotation
velocity of skyrmions.
Skyrmions are the ideal information carrier in spintronic devices. However, a
skyrmion can not move in a straight line along the driving current direction because
of the Magnus force which leads to the shift of its motion trajectory, such behavior is
called skyrmion Hall effect (SHE) [104, 105]. In practical applications, the skyrmion
may be destroyed when touching the device boundaries. To overcome this issue, a lot
of methods have been proposed. For example, X. Zhang et al. [106] showed that the
antiferromagnetically exchange-coupled bilayer system containing two skyrmions
with different polarities can suppress the SHE, leading to a perfectly straight trajectory for skyrmion driven by a spin-polarized current. Moreover, the antiferromagnetic
(AFM) skyrmion can also avoid SHE [107, 108]. Interestingly, recent research shows
that the AFM skyrmion can emerge in a ferromagnet with gain (negative α) [109].
Figure 15.4 shows the dynamical process of the formation of AFM skyrmion in ferromagnet (here α is set to be −0.01), with the initial magnetization profile being
random [see Fig. 15.4a]. At t = 0.035 ns, local magnetic moments quickly evolve to
an antiparallelly aligned state, as shown in Fig. 15.4b. Then, all spins inside a circle
of radius 5 nm in the film center are randomized. At t = 0.14 ns, an AFM skyrmion
is stabilized (see Fig. 15.4d). When an in-plane spin-polarized electric current was