14 Topology in Magnetism
369
freedom. For bosons with the Bose-Einstein distribution, the low-energy states are
always easier to be excited. Thus, the DW has a larger entropy S. The free energy of
the system is F = U − T S where U is the internal energy and T is the temperature.
Therefore, when the DW has higher temperature, the total free energy of the system is
lower. This theory also explains why the DW moves to the hotter side in the presence
of a thermal gradient, which is consistent with the magnon STT theory [51].
14.2.2 Vortices and Skyrmions
In previous section, we discussed vortices in the context of vortex walls and winding
numbers in 2D XY model (d = 2, d
= 1). In this section, we discuss a more general
model on R
2
→ M
2 . Since m is allowed to tilt out-of plane, the spin at vortex
(antivortex) center points perpendicularly out-of-plane rather than be singular as
discussed before. A new topological charge/number called skyrmion number on
R
2
→ M
2 can be defined [52–54], as a generalization of (14.6),
Q = −
1
4π
m ·
∂m
∂x
×
∂m
∂ y
d
2 r.
(14.10)
Geometrically, the integrand is the solid angle spanned by m when one moves all
spins to the Bloch sphere center in the spin space. Thus, Q measures how many times
the vector m wraps the unit Bloch sphere.
For vortices whose m are out-of-plane at the centers and in-plane at the periphery,
the skyrmion numbers are ±1/2 since all m can only wrap a hemisphere. Different
types of vortices are shown in Fig. 14.8, with ansatz profiles
(φ) = cφ + ,(r ) =
π
2
(1 − pe
−r/R
),
(14.11)
where r , φ are radial and angular coordinates of a point in R
2 centered at the vortex
center, and , are polar angle and azimuthal angle of m in M
2 . c takes integer
values, measuring how many turns m goes around the vortex center. p = ±1 denotes
the spin direction at the vortex center ( p = 1 for m z = 1 and p = −1 for m z = −1). c
and p are usually called chirality and polarity of a vortex, respectively [55]. Equation
(14.10) can be written in polar coordinate with the ansatz (14.11) as Q =
1
2
cp.
The topological charge Q can not only characterize the topology of a vortex, but
also directly enters the equation of motion. Thiele derived an equation of motion for
a spin texture, known as Thiele equation [54], under the assumption of a rigid-body
motion. After the discovery of STT [25], the Thiele equation including STT becomes
[37, 53, 56],
γ
M s
T + G × (v − u) −
← →
D · (αv − βu) = 0,
(14.12)
369
freedom. For bosons with the Bose-Einstein distribution, the low-energy states are
always easier to be excited. Thus, the DW has a larger entropy S. The free energy of
the system is F = U − T S where U is the internal energy and T is the temperature.
Therefore, when the DW has higher temperature, the total free energy of the system is
lower. This theory also explains why the DW moves to the hotter side in the presence
of a thermal gradient, which is consistent with the magnon STT theory [51].
14.2.2 Vortices and Skyrmions
In previous section, we discussed vortices in the context of vortex walls and winding
numbers in 2D XY model (d = 2, d
= 1). In this section, we discuss a more general
model on R
2
→ M
2 . Since m is allowed to tilt out-of plane, the spin at vortex
(antivortex) center points perpendicularly out-of-plane rather than be singular as
discussed before. A new topological charge/number called skyrmion number on
R
2
→ M
2 can be defined [52–54], as a generalization of (14.6),
Q = −
1
4π
m ·
∂m
∂x
×
∂m
∂ y
d
2 r.
(14.10)
Geometrically, the integrand is the solid angle spanned by m when one moves all
spins to the Bloch sphere center in the spin space. Thus, Q measures how many times
the vector m wraps the unit Bloch sphere.
For vortices whose m are out-of-plane at the centers and in-plane at the periphery,
the skyrmion numbers are ±1/2 since all m can only wrap a hemisphere. Different
types of vortices are shown in Fig. 14.8, with ansatz profiles
(φ) = cφ + ,(r ) =
π
2
(1 − pe
−r/R
),
(14.11)
where r , φ are radial and angular coordinates of a point in R
2 centered at the vortex
center, and , are polar angle and azimuthal angle of m in M
2 . c takes integer
values, measuring how many turns m goes around the vortex center. p = ±1 denotes
the spin direction at the vortex center ( p = 1 for m z = 1 and p = −1 for m z = −1). c
and p are usually called chirality and polarity of a vortex, respectively [55]. Equation
(14.10) can be written in polar coordinate with the ansatz (14.11) as Q =
1
2
cp.
The topological charge Q can not only characterize the topology of a vortex, but
also directly enters the equation of motion. Thiele derived an equation of motion for
a spin texture, known as Thiele equation [54], under the assumption of a rigid-body
motion. After the discovery of STT [25], the Thiele equation including STT becomes
[37, 53, 56],
γ
M s
T + G × (v − u) −
← →
D · (αv − βu) = 0,
(14.12)