14 Topology in Magnetism
365
respectively the Bohr magneto, the degree of current spin polarization, the electron
g-factor and the electron charge.
To achieve a high DW speed, a large electric current has to be used. Thus, a large
amount of Joule heat is produced [35] that is harmful to device performance. As
explained before, spin waves can take the role of an electric current in the DW motion.
Indeed, both electrons and magnons carry spins. A magnon carrying an angular
momentum of . Similar to the STT from electrons to local magnetic moments, a
STT from magnons to a DW also exists. Since the spins carried by magnons are
always opposite to spins of the domain, magnon STT should drag a DW towards
the opposite direction of magnon flows. The magnonic STT has been predicted [39]
and experimentally verified [40, 41]. The magnons can not only be coherent ones
generated by oscillating electromagnetic fields but also be incoherent ones generated
by nonlinear effects or thermal gradients. The STT from thermal magnons also has
been predicted [42, 43] and experimentally observed [44]. Under a thermal gradient,
the hotter side has a larger magnon population than the colder side. So the magnons
diffuse from the hotter side to the colder side, forming a magnon flow. This flow
drags the DW towards the hotter side.
The d = 1 model is a good approximation for small diameter (much less than
exchange length and DW width) nanowires or very large bulky magnets whose magnetization varies only along one direction. For DWs in nanostrips which are usually
used in experiments and devices, a d = 2 model is necessary. We now discuss a
R
2
→ M
1 model, i.e. the 2D classical XY model, which is applicable to in-plane
magnetized strips [45]. The pioneering work of McMichael and Donahue [46] show
numerically that a HH DW can be a transverse wall or a vortex wall, depending on the
width and the thickness of the strip, as shown in Fig. 14.5. To discuss the topology,
similar to the winding number defined in (14.6), we can define a winding number
along a path C in R
2 ,
W C =
1
2π
C
∇θ · dr,
(14.8)
where C is a closed loop inside a strip denoted as , on which θ is continuum
and smooth. W C in this case is a bulk winding number that must be an integer. If
W C = 0, there is no singularity inside C. If W C = +1 (−1), there is a singularity
inside C, called a vortex (antivortex), as shown in Fig. 14.5 [45]. At the edge of the
strip denoted as ∂, C can be a segment of ∂, and the definition of W C is modified
as
W C =
1
2π
C
∇(θ − θ C ) · dr,
(14.9)
where θ C is the angle of the sample edge. W C of edge defects takes the values ±1/2,
as shown in Fig. 14.5. The sum of all the winding numbers over + ∂ is conserved.
Total winding number of
W C is a topological invariant.
A transverse wall has a pair of edge defects of W C = ±1/2, and a vortex wall
with one vortex of W C = 1 has two edge defects of W C = −1/2. Both walls have
the same total winding numbers of 0. Thus the transverse wall and the vortex wall
365
respectively the Bohr magneto, the degree of current spin polarization, the electron
g-factor and the electron charge.
To achieve a high DW speed, a large electric current has to be used. Thus, a large
amount of Joule heat is produced [35] that is harmful to device performance. As
explained before, spin waves can take the role of an electric current in the DW motion.
Indeed, both electrons and magnons carry spins. A magnon carrying an angular
momentum of . Similar to the STT from electrons to local magnetic moments, a
STT from magnons to a DW also exists. Since the spins carried by magnons are
always opposite to spins of the domain, magnon STT should drag a DW towards
the opposite direction of magnon flows. The magnonic STT has been predicted [39]
and experimentally verified [40, 41]. The magnons can not only be coherent ones
generated by oscillating electromagnetic fields but also be incoherent ones generated
by nonlinear effects or thermal gradients. The STT from thermal magnons also has
been predicted [42, 43] and experimentally observed [44]. Under a thermal gradient,
the hotter side has a larger magnon population than the colder side. So the magnons
diffuse from the hotter side to the colder side, forming a magnon flow. This flow
drags the DW towards the hotter side.
The d = 1 model is a good approximation for small diameter (much less than
exchange length and DW width) nanowires or very large bulky magnets whose magnetization varies only along one direction. For DWs in nanostrips which are usually
used in experiments and devices, a d = 2 model is necessary. We now discuss a
R
2
→ M
1 model, i.e. the 2D classical XY model, which is applicable to in-plane
magnetized strips [45]. The pioneering work of McMichael and Donahue [46] show
numerically that a HH DW can be a transverse wall or a vortex wall, depending on the
width and the thickness of the strip, as shown in Fig. 14.5. To discuss the topology,
similar to the winding number defined in (14.6), we can define a winding number
along a path C in R
2 ,
W C =
1
2π
C
∇θ · dr,
(14.8)
where C is a closed loop inside a strip denoted as , on which θ is continuum
and smooth. W C in this case is a bulk winding number that must be an integer. If
W C = 0, there is no singularity inside C. If W C = +1 (−1), there is a singularity
inside C, called a vortex (antivortex), as shown in Fig. 14.5 [45]. At the edge of the
strip denoted as ∂, C can be a segment of ∂, and the definition of W C is modified
as
W C =
1
2π
C
∇(θ − θ C ) · dr,
(14.9)
where θ C is the angle of the sample edge. W C of edge defects takes the values ±1/2,
as shown in Fig. 14.5. The sum of all the winding numbers over + ∂ is conserved.
Total winding number of
W C is a topological invariant.
A transverse wall has a pair of edge defects of W C = ±1/2, and a vortex wall
with one vortex of W C = 1 has two edge defects of W C = −1/2. Both walls have
the same total winding numbers of 0. Thus the transverse wall and the vortex wall