14 Topology in Magnetism
373
Fig. 14.11 a Schematic diagram of a Néel skyrmion of radius R and wall (between the skyrmion
core and the outer domain) width w in a 2D film. The arrows denote the spin direction. Spin orientations of the skyrmion along x axis are sketched below the main figure. b (Right axis) φ-dependence
of (red circles) and (Left axis) radial (r ) distribution of m z = cos along the diameters of φ = 0
(crosses), 45 ◦ (triangles), and 90 ◦ (circles) for A = 15 pJm −1 , D = 3.7 mJm −2 , K u = 0.8 MJm −3 ,
and B = 0. The green solid line is the fit to dw (r ) = 2 arctan
sinh(R/w)
sinh(r/w)
with fitting parameters
of R = 25.77 nm and w = 4.94 nm. The blue solid line is = φ. c–e Three typical equilibrium
states obtained in the numerical simulations for different parameters. c Isolated skyrmion. d Singledomain state of m z = 1 (or m z = −1). e Stripe domains. The pixel color encodes the m z component
with the color bar shown in the figure. Figure reproduced from [84]
E ex = 2π Ad
∞
0
d
dr
2
+
sin
2
r 2
r dr,
(14.18)
E DM = 2π Dd cos
∞
0
d
dr
+
sin 2
2r
r dr,
(14.19)
E an = 2πK d
∞
0
sin
2
dr,
(14.20)
E Ze = 2π M s Bd
∞
0
(1 − cos dr.
(14.21)
= 0 or π minimizes the DMI term. This kind of skyrmions are called Néel
skyrmions or hedgehog skyrmions. Along a radial direction, the skyrmion profile
looks like a 360° Néel domain wall as illustrated in Fig. 14.11b. This leads us to
model a skyrmion profile by a 360° domain wall originally proposed by Braun [75,
85, 86],
dw (r ) = 2 arctan
sinh(R/w)
sinh(r/w)
.
(14.22)
Whether this is a good approximation subjects to its comparison with micromagnetic
simulations. To test how good ansatz (14.22) is for a skyrmion, we use MuMax3
[87] to simulate various stable spin structures. A = 15 pJm
−1 , M s = 580 kAm
−1 ,
and perpendicular easy-axis anisotropy K u = 0.8 MJm
−3 [68] are used to mimic Co
layer in Pt/Co/MgO system. The initial state is m z = 1 for r > 10 nm and m z = −1
for r ≤ 10 nm. The final stable structures depend on the values of D and B. The
373
Fig. 14.11 a Schematic diagram of a Néel skyrmion of radius R and wall (between the skyrmion
core and the outer domain) width w in a 2D film. The arrows denote the spin direction. Spin orientations of the skyrmion along x axis are sketched below the main figure. b (Right axis) φ-dependence
of (red circles) and (Left axis) radial (r ) distribution of m z = cos along the diameters of φ = 0
(crosses), 45 ◦ (triangles), and 90 ◦ (circles) for A = 15 pJm −1 , D = 3.7 mJm −2 , K u = 0.8 MJm −3 ,
and B = 0. The green solid line is the fit to dw (r ) = 2 arctan
sinh(R/w)
sinh(r/w)
with fitting parameters
of R = 25.77 nm and w = 4.94 nm. The blue solid line is = φ. c–e Three typical equilibrium
states obtained in the numerical simulations for different parameters. c Isolated skyrmion. d Singledomain state of m z = 1 (or m z = −1). e Stripe domains. The pixel color encodes the m z component
with the color bar shown in the figure. Figure reproduced from [84]
E ex = 2π Ad
∞
0
d
dr
2
+
sin
2
r 2
r dr,
(14.18)
E DM = 2π Dd cos
∞
0
d
dr
+
sin 2
2r
r dr,
(14.19)
E an = 2πK d
∞
0
sin
2
dr,
(14.20)
E Ze = 2π M s Bd
∞
0
(1 − cos dr.
(14.21)
= 0 or π minimizes the DMI term. This kind of skyrmions are called Néel
skyrmions or hedgehog skyrmions. Along a radial direction, the skyrmion profile
looks like a 360° Néel domain wall as illustrated in Fig. 14.11b. This leads us to
model a skyrmion profile by a 360° domain wall originally proposed by Braun [75,
85, 86],
dw (r ) = 2 arctan
sinh(R/w)
sinh(r/w)
.
(14.22)
Whether this is a good approximation subjects to its comparison with micromagnetic
simulations. To test how good ansatz (14.22) is for a skyrmion, we use MuMax3
[87] to simulate various stable spin structures. A = 15 pJm
−1 , M s = 580 kAm
−1 ,
and perpendicular easy-axis anisotropy K u = 0.8 MJm
−3 [68] are used to mimic Co
layer in Pt/Co/MgO system. The initial state is m z = 1 for r > 10 nm and m z = −1
for r ≤ 10 nm. The final stable structures depend on the values of D and B. The