382
X. S. Wang and X. R. Wang
panels are the midplane cross-sections in the x y-plane and xz-plane. The magnetization profile in each x y-midplane cross-sections is Bloch-type (a) or Néel-type (b)
skyrmionium or the target skyrmion [121, 122], while the xz-midplane cross-section
shows a pair of vortices with opposite chirality. The right (x > 0) xz-midplane contains a vortex (antivortex) with chirality +1 (−1) for an H = +1 (H = −1) hopfion.
Outside the hopfions and at the center of the hopfions, the magnetization is along the
z direction, and the donut-shape transition region is chiral (for Bloch-type hopfions)
or hedgehog-like (for Néel-type hopfions). Figure 14.13c, d show the corresponding
preimages (constant-m curves in real space) of Fig. 14.13a, b. The preimages link
with each other once, which is consistent with the Hopf index calculation, justifying
the hopfion nature of the textures in (a) and (b).
Unlike skyrmions, although the topology of a hopfion is nontrivial, the gyrovector G =
FdV of a hopfion vanishes. Consequently, the main drawback of a FM
skyrmion racetrack memory, the skyrmion Hall effect, is absent in the hopfion racetrack memory. In addition to the numerical verification, the vanishing gyrovector of
a hopfion can be understood as follows. Consider a film that is isotropic in the x y
plane. In cylindrical coordinates, F z = n
sin
r
d
dr
. We can rewrite G z =
F z dV as
G z =
V
F z r dr dφdz = −2nπ
d/2
−d/2
cos
r =∞
r =0
dz.
(14.43)
Since in a hopfion the magnetization directions are the same at both the periphery
(r = ∞) and the center (r = 0), G z vanishes. Since the two vortices in any xz (or
yz) midplane cross-section have opposite chirality, as shown in the lower panels
of Fig. 14.13a, b, the integration of F x (or F y ) over the volume gives a vanishing
contribution to G x (or G y ). The components of G are invariant under continuous
deformation [52]; therefore, G = 0 applies to all the hopfions.
According to the Thiele equation (14.12), the direct consequence of the vanishing
gyrovector is the absent of Hall effect. Under spin transfer torque, the hopfions
move straightforward along the current. Under spin Hall torque which emerges in
the heavy metal/ferromagnet heterostructures, Néel-type hopfions move along the
current while the Bloch-type hopfions move transverse to the current.
14.3 Topological Spin Waves
Matter that possesses excitations with nontrivial band topology has attracted enormous attention in recent years because of their interesting and exotic properties
[123–127]. One of such properties is the existence of unidirectional and topologically protected surface/edge states that are robust against internal and external perturbations. The study was initially exclusive for electron systems and was believed
to be a quantum phenomenon. It is now known that the topological states can exist in
classical mechanics [128] and photonics [129]. There are also intensive researches
X. S. Wang and X. R. Wang
panels are the midplane cross-sections in the x y-plane and xz-plane. The magnetization profile in each x y-midplane cross-sections is Bloch-type (a) or Néel-type (b)
skyrmionium or the target skyrmion [121, 122], while the xz-midplane cross-section
shows a pair of vortices with opposite chirality. The right (x > 0) xz-midplane contains a vortex (antivortex) with chirality +1 (−1) for an H = +1 (H = −1) hopfion.
Outside the hopfions and at the center of the hopfions, the magnetization is along the
z direction, and the donut-shape transition region is chiral (for Bloch-type hopfions)
or hedgehog-like (for Néel-type hopfions). Figure 14.13c, d show the corresponding
preimages (constant-m curves in real space) of Fig. 14.13a, b. The preimages link
with each other once, which is consistent with the Hopf index calculation, justifying
the hopfion nature of the textures in (a) and (b).
Unlike skyrmions, although the topology of a hopfion is nontrivial, the gyrovector G =
FdV of a hopfion vanishes. Consequently, the main drawback of a FM
skyrmion racetrack memory, the skyrmion Hall effect, is absent in the hopfion racetrack memory. In addition to the numerical verification, the vanishing gyrovector of
a hopfion can be understood as follows. Consider a film that is isotropic in the x y
plane. In cylindrical coordinates, F z = n
sin
r
d
dr
. We can rewrite G z =
F z dV as
G z =
V
F z r dr dφdz = −2nπ
d/2
−d/2
cos
r =∞
r =0
dz.
(14.43)
Since in a hopfion the magnetization directions are the same at both the periphery
(r = ∞) and the center (r = 0), G z vanishes. Since the two vortices in any xz (or
yz) midplane cross-section have opposite chirality, as shown in the lower panels
of Fig. 14.13a, b, the integration of F x (or F y ) over the volume gives a vanishing
contribution to G x (or G y ). The components of G are invariant under continuous
deformation [52]; therefore, G = 0 applies to all the hopfions.
According to the Thiele equation (14.12), the direct consequence of the vanishing
gyrovector is the absent of Hall effect. Under spin transfer torque, the hopfions
move straightforward along the current. Under spin Hall torque which emerges in
the heavy metal/ferromagnet heterostructures, Néel-type hopfions move along the
current while the Bloch-type hopfions move transverse to the current.
14.3 Topological Spin Waves
Matter that possesses excitations with nontrivial band topology has attracted enormous attention in recent years because of their interesting and exotic properties
[123–127]. One of such properties is the existence of unidirectional and topologically protected surface/edge states that are robust against internal and external perturbations. The study was initially exclusive for electron systems and was believed
to be a quantum phenomenon. It is now known that the topological states can exist in
classical mechanics [128] and photonics [129]. There are also intensive researches