14 Topology in Magnetism
377
the area bounded by g(t, x)-curve and t-axis is 1 so that g(t, x) ≈ δ(t − x) resembles
the properties of a delta function. f i under this approximation are,
f 1 (x) ≈ x, f 2 (x) ≈
1
x
,
f 3 (x) ≈
π
2
x, f 4 (x) ≈ 0,
f 5 (x) ≈ x f 6 (x) ≈
x
2
2
+
π
2
24
.
The total energy is then
E(R, w) = 4πd
A
R
w
+
w
R
−
π
2
D R
+ K w R + M s B
R
2
2
+
π
2
24
w
2
. (14.24)
Due to the specific form of the magnetization profile of dw (r ), Rw-term in E Ze
vanishes and E Ze ≈ 4π M s B
R
2
2
+
π
2
24
w
2
. The skyrmion size and wall width are
then the values that make E(R, w) minimal, or
A
1
w
−
w
R 2
−
π
2
D + K w + M s B R = 0,
(14.25)
A
−
R
w 2 +
1
R
+ K R +
π
2
12
M s Bw = 0.
(14.26)
For B = 0, (14.25) and (14.26) can be analytically solved. The results are
R = π D
A
16AK 2 − π 2 D 2 K
, w =
π D
4K
.
(14.27)
The dashed lines in Figs. 14.12a–c are the approximate formulas that compare quite
well with simulation results too. When B = 0, the equation for w is a sextic equation,
432π 2 A 2 D 2 − 3456π A 2 DK w + [−(1728 + 288π 2 + 12π 4 )A 2 (M s B) 2 + 6912 A 2 K 2 ]w 2
+ [3456π ADK 2 − (72 − 6π 2 )π 3 AD(M s B) 2 ]w 3 + [−6912K 2 + 576π 2 (M s B) 2 ]AK w 4
− 6π 5 (M s B) 2 DK w 5 + π 4 (M s B) 2 [12K 2 − π 2 (M s B) 2 ]w 6 = 0,
(14.28)
and R, in terms of w, is
R =
π
2 M s Bw
3
± w
576A 2 − 576AK w 2 + π 4 (M s B) 2 w 4
24(A − K w 2 )
.
(14.29)