374
X. S. Wang and X. R. Wang
lower left inset of Fig. 14.11 is three typical stable structures. Figure 14.11c is a
skyrmion for D = 3.7 mJm
−2 and B = 0. Figure 14.11d is a single-domain state of
m z = 1 (or m z = −1) for D = 0 and B = 0. Figure 14.11e is a stripe domains state
for D = 5 mJm
−2 and B = 0. Figure 14.11b shows the spatial distribution of m z of
the skyrmion in Fig. 14.11c along three radial directions, φ = 0 (the crosses), 45
◦
(the triangles), and 90
◦ (the circles). All three sets of data are on the same smooth
curve, showing that m z is a function of r , but not φ. The curve can fit perfectly to
(14.22) with R = 25.77 nm, w = 4.94 nm. We plotted also (φ) at randomly picked
spins from the simulated skyrmion. All numerical data (red circles) are perfectly on
the curve of = φ. These results not only confirm the validity of (14.22), but also
suggest that m z (r ) ≡ cos[ dw (r )] follows the 360° DW profile (14.22) although it
is not exact since (14.22) does not satisfy nonlinear partial differential equations for
skyrmions [83].
The energy of a skyrmion can then be obtained from (14.18) by using the 360°
domain wall profile dw (r ). Substituting dw (r ) into (14.15) and (14.18), the total
energy is, in general, a function of R and w [instead of a functional of (r ) and
(φ)] as
E(R, w) = 4πd
A
f 1
R
w
+ f 2
R
w
+ Dw
f 3
R
w
+ f 4
R
w
+ K w
2 f 5
R
w
+ μ 0 M s Bw
2 f 6
R
w
, (14.23)
where f i (x) (i = 1 ∼ 6) are non-elementary functions defined as,
f 1 (x) =
∞
0
2 sinh
2
(x) cosh
2
(t)
sinh
2
(x) + sinh
2
(t)
2 tdt,
f 2 (x) =
∞
0
2 sinh
2
(x) sinh
2
(t)
t
sinh
2
(x) + sinh
2
(t)
2 dt,
f 3 (x) = −
∞
0
t sinh x cosh t
sinh
2 x + sinh
2 t
dt,
f 4 (x) = −
∞
0
sinh x sinh t
sinh
2 x − sinh
2 t
sinh
2 x + sinh
2 t
2
dt,
f 5 (x) =
∞
0
2t sinh
2 x sinh
2 t
sinh
2 x + sinh
2 t
2 dt,
f 6 (x) =
∞
0
sinh
2 x
sinh
2 t + sinh
2 x
tdt.
The skyrmion size R and wall width w are the values that minimize E(R, w). In
[78], w was assumed to be a fixed value such as w =
√
A/K that eventually leads
to a wrong skyrmion size. Figure 14.12 is D− (a), A− (b), K − (c) and B− (d)
X. S. Wang and X. R. Wang
lower left inset of Fig. 14.11 is three typical stable structures. Figure 14.11c is a
skyrmion for D = 3.7 mJm
−2 and B = 0. Figure 14.11d is a single-domain state of
m z = 1 (or m z = −1) for D = 0 and B = 0. Figure 14.11e is a stripe domains state
for D = 5 mJm
−2 and B = 0. Figure 14.11b shows the spatial distribution of m z of
the skyrmion in Fig. 14.11c along three radial directions, φ = 0 (the crosses), 45
◦
(the triangles), and 90
◦ (the circles). All three sets of data are on the same smooth
curve, showing that m z is a function of r , but not φ. The curve can fit perfectly to
(14.22) with R = 25.77 nm, w = 4.94 nm. We plotted also (φ) at randomly picked
spins from the simulated skyrmion. All numerical data (red circles) are perfectly on
the curve of = φ. These results not only confirm the validity of (14.22), but also
suggest that m z (r ) ≡ cos[ dw (r )] follows the 360° DW profile (14.22) although it
is not exact since (14.22) does not satisfy nonlinear partial differential equations for
skyrmions [83].
The energy of a skyrmion can then be obtained from (14.18) by using the 360°
domain wall profile dw (r ). Substituting dw (r ) into (14.15) and (14.18), the total
energy is, in general, a function of R and w [instead of a functional of (r ) and
(φ)] as
E(R, w) = 4πd
A
f 1
R
w
+ f 2
R
w
+ Dw
f 3
R
w
+ f 4
R
w
+ K w
2 f 5
R
w
+ μ 0 M s Bw
2 f 6
R
w
, (14.23)
where f i (x) (i = 1 ∼ 6) are non-elementary functions defined as,
f 1 (x) =
∞
0
2 sinh
2
(x) cosh
2
(t)
sinh
2
(x) + sinh
2
(t)
2 tdt,
f 2 (x) =
∞
0
2 sinh
2
(x) sinh
2
(t)
t
sinh
2
(x) + sinh
2
(t)
2 dt,
f 3 (x) = −
∞
0
t sinh x cosh t
sinh
2 x + sinh
2 t
dt,
f 4 (x) = −
∞
0
sinh x sinh t
sinh
2 x − sinh
2 t
sinh
2 x + sinh
2 t
2
dt,
f 5 (x) =
∞
0
2t sinh
2 x sinh
2 t
sinh
2 x + sinh
2 t
2 dt,
f 6 (x) =
∞
0
sinh
2 x
sinh
2 t + sinh
2 x
tdt.
The skyrmion size R and wall width w are the values that minimize E(R, w). In
[78], w was assumed to be a fixed value such as w =
√
A/K that eventually leads
to a wrong skyrmion size. Figure 14.12 is D− (a), A− (b), K − (c) and B− (d)