8 Microwave-Driven Dynamics of Magnetic Skyrmions …
191
χ μ (ω) =
M μ (ω)
μ 0 H μ (ω)
(μ = x, y, z),
(8.8)
where H μ (ω) and M μ (ω) are the Fourier transforms of the time-dependent magnetic field H(t) and the simulated time-profile of the net magnetization M(t) =
M(t) − M(0) with M(t) =
1
N
N
i=1 m i (t). In the calculations, a short rectangular
pulse is used for H(t) whose component is given by,
H μ (t) =
H pulse
0 ≤ t ≤ 1
0
others
(8.9)
where t = (J/)τ is the dimensionless time with τ being the real time. An advantage
of using the short pulse is that the Fourier component H μ (ω) becomes constant being
independent of ω up to first order in ωωt for a sufficiently short duration t with
ωωt 1. The Fourier component is
H μ (ω) =
t
0
H pulse e
iωt dt =
H pulse
iω
e
iωωt
− 1
∼ H pulse t.
(8.10)
Consequently, we obtain the relationship χ μ (ω) ∝ M μ (ω). The imaginary part of
thus calculated χ μ (ω) corresponds to the microwave absorption spectrum. Figure 8.3d
shows the spectra for the in-plane polarized microwave field H
ω
x, y under perpendicular (θ = 0
◦ ) and tilted (θ = 30
◦ ) H ext fields, whereas Fig. 8.3e shows the
spectra for the out-of-plane polarized microwave field H
ω
z [29]. When the H ext
field is perpendicular (θ = 0
◦ ), two spectral peaks appear in Fig. 8.3d originating
from the two rotation modes, whereas a single peak appears in Fig. 8.3e originating
from the breathing mode, indicating that the in-plane (out-of-plane) microwave field
can activate the rotation modes (the breathing mode) only when the H ext field is
perpendicular. On the other hand, when the H ext field is tilted (θ = 30
◦ ), three spectral peaks appear in both Fig. 8.3d, e, indicating that all the three spin-wave modes
become active to both microwave polarization under the tilted H ext field.
8.4 Microwave-Magnetic-Field-Driven Translational
Motion of Skyrmion Crystal
The continuous spin-wave excitation by microwave irradiation under a tilted H ext
field can induce translational motion of a skyrmion crystal [28, 29]. Figure 8.4 shows
simulated snapshots of a skyrmion crystal driven by an in-plane microwave field
H
ω
x (right upper panel) and the same skyrmion crystal driven by an out-of-plane
microwave field H
ω
z (right lower panel) at t = 400 ns after the microwave irradiation commences. The figure also shows the initial configuration of the skyrmion
crystal at t = 0 (left panel) under application of the tilted magnetic field H ex =
191
χ μ (ω) =
M μ (ω)
μ 0 H μ (ω)
(μ = x, y, z),
(8.8)
where H μ (ω) and M μ (ω) are the Fourier transforms of the time-dependent magnetic field H(t) and the simulated time-profile of the net magnetization M(t) =
M(t) − M(0) with M(t) =
1
N
N
i=1 m i (t). In the calculations, a short rectangular
pulse is used for H(t) whose component is given by,
H μ (t) =
H pulse
0 ≤ t ≤ 1
0
others
(8.9)
where t = (J/)τ is the dimensionless time with τ being the real time. An advantage
of using the short pulse is that the Fourier component H μ (ω) becomes constant being
independent of ω up to first order in ωωt for a sufficiently short duration t with
ωωt 1. The Fourier component is
H μ (ω) =
t
0
H pulse e
iωt dt =
H pulse
iω
e
iωωt
− 1
∼ H pulse t.
(8.10)
Consequently, we obtain the relationship χ μ (ω) ∝ M μ (ω). The imaginary part of
thus calculated χ μ (ω) corresponds to the microwave absorption spectrum. Figure 8.3d
shows the spectra for the in-plane polarized microwave field H
ω
x, y under perpendicular (θ = 0
◦ ) and tilted (θ = 30
◦ ) H ext fields, whereas Fig. 8.3e shows the
spectra for the out-of-plane polarized microwave field H
ω
z [29]. When the H ext
field is perpendicular (θ = 0
◦ ), two spectral peaks appear in Fig. 8.3d originating
from the two rotation modes, whereas a single peak appears in Fig. 8.3e originating
from the breathing mode, indicating that the in-plane (out-of-plane) microwave field
can activate the rotation modes (the breathing mode) only when the H ext field is
perpendicular. On the other hand, when the H ext field is tilted (θ = 30
◦ ), three spectral peaks appear in both Fig. 8.3d, e, indicating that all the three spin-wave modes
become active to both microwave polarization under the tilted H ext field.
8.4 Microwave-Magnetic-Field-Driven Translational
Motion of Skyrmion Crystal
The continuous spin-wave excitation by microwave irradiation under a tilted H ext
field can induce translational motion of a skyrmion crystal [28, 29]. Figure 8.4 shows
simulated snapshots of a skyrmion crystal driven by an in-plane microwave field
H
ω
x (right upper panel) and the same skyrmion crystal driven by an out-of-plane
microwave field H
ω
z (right lower panel) at t = 400 ns after the microwave irradiation commences. The figure also shows the initial configuration of the skyrmion
crystal at t = 0 (left panel) under application of the tilted magnetic field H ex =