8 Microwave-Driven Dynamics of Magnetic Skyrmions …
203
active spin-wave modes of skyrmion crystal on a quasi-two-dimensional thin-plate
magnet under a tilted H ex field in [31]. In this study, micromagnetic simulations
based on the Landau–Lifshitz–Gilbert equation were performed to trace the magnetization dynamics of a skyrmion crystal activated by a microwave magnetic field H
ω .
Using the simulated data of the magnetization dynamics, the spatiotemporal profiles
of the spin-motive force E(r, t) were numerically calculated. For the numerical
calculations, it is convenient to rewrite (8.29) in discretized form as
E μ,i (t) =
2e
m i (t) ·
m i+ ˆ
μ (t) − m i− ˆ
μ (t)
2a
×
m i (t + t) − m i (t − t)
2t
,
(8.30)
where μ = x, y and a(=5 Å) is the lattice constant. Time profiles of the spin voltage were calculated by numerically solving the Poisson equation using the spatial
distribution data of the spin-motive force E(r, t) at each moment t.
The spin-motive force that contains a large DC component can indeed be generated by activation of the spin-wave modes of magnetic skyrmions under a tilted
H ext field. Figure 8.9a–d show time profiles of the spin-motive force simulated for
a 50nm × 50 nm squared system which contains a skyrmion crystal composed of
twelve skyrmions. When the H ex field is perpendicular, the generated spin-motive
force is of pure AC for the counterclockwise rotation mode (Fig. 8.9a) or constantly
zero for the breathing mode (Fig. 8.9b). On the contrary, a spin-motive force with a
large DC component of 0.5-1 μV is generated when the H ex field is tilted by θ = 30
◦
(Fig. 8.9c, d).
The simulated time profiles of spin voltages turn out to be well fitted by an
approximate formula of the forced oscillation with a damping.
V μ = V
DC
μ + V
AC
μ (1 − e
−t/τ
) sin ωt,
(8.31)
with μ = x, y. Here V
DC
μ , V
AC
μ , ω(= 2π f ), and τ are the DC component, the AC
amplitude, the angular frequency, and the decay rate of the induced temporally oscillating spin voltage, respectively. Fig. 8.9e–f show the microwave-frequency dependence of the DC component V
DC
μ (μ = x, y) for different microwave polarizations,
which are evaluated by the fitting. Specifically, Fig. 8.9e shows V
DC
x
for H
ω
x, y,
Fig. 8.9f shows V
DC
y
for H
ω
x, y, Fig.8.9g shows V
DC
x
for H
ω
z, and Fig. 8.9h
shows V
DC
y
for H
ω
z. The results show that the DC component is enhanced significantly when the frequency of the microwave is tuned to an eigenfrequency of
the spin-wave modes, which converts the microwave power to a DC voltage with
high efficiency. The results also show that the sign of the DC voltage depends on the
excited spin-wave mode and the microwave polarization, which indicates that the sign
of the voltage can be switched by tuning the microwave frequency or the microwave
polarization. Note that a large DC voltage is obtained for the counterclockwise rotation mode activated by H
ω
x, y and for the breathing mode activated by H
ω
z,
203
active spin-wave modes of skyrmion crystal on a quasi-two-dimensional thin-plate
magnet under a tilted H ex field in [31]. In this study, micromagnetic simulations
based on the Landau–Lifshitz–Gilbert equation were performed to trace the magnetization dynamics of a skyrmion crystal activated by a microwave magnetic field H
ω .
Using the simulated data of the magnetization dynamics, the spatiotemporal profiles
of the spin-motive force E(r, t) were numerically calculated. For the numerical
calculations, it is convenient to rewrite (8.29) in discretized form as
E μ,i (t) =
2e
m i (t) ·
m i+ ˆ
μ (t) − m i− ˆ
μ (t)
2a
×
m i (t + t) − m i (t − t)
2t
,
(8.30)
where μ = x, y and a(=5 Å) is the lattice constant. Time profiles of the spin voltage were calculated by numerically solving the Poisson equation using the spatial
distribution data of the spin-motive force E(r, t) at each moment t.
The spin-motive force that contains a large DC component can indeed be generated by activation of the spin-wave modes of magnetic skyrmions under a tilted
H ext field. Figure 8.9a–d show time profiles of the spin-motive force simulated for
a 50nm × 50 nm squared system which contains a skyrmion crystal composed of
twelve skyrmions. When the H ex field is perpendicular, the generated spin-motive
force is of pure AC for the counterclockwise rotation mode (Fig. 8.9a) or constantly
zero for the breathing mode (Fig. 8.9b). On the contrary, a spin-motive force with a
large DC component of 0.5-1 μV is generated when the H ex field is tilted by θ = 30
◦
(Fig. 8.9c, d).
The simulated time profiles of spin voltages turn out to be well fitted by an
approximate formula of the forced oscillation with a damping.
V μ = V
DC
μ + V
AC
μ (1 − e
−t/τ
) sin ωt,
(8.31)
with μ = x, y. Here V
DC
μ , V
AC
μ , ω(= 2π f ), and τ are the DC component, the AC
amplitude, the angular frequency, and the decay rate of the induced temporally oscillating spin voltage, respectively. Fig. 8.9e–f show the microwave-frequency dependence of the DC component V
DC
μ (μ = x, y) for different microwave polarizations,
which are evaluated by the fitting. Specifically, Fig. 8.9e shows V
DC
x
for H
ω
x, y,
Fig. 8.9f shows V
DC
y
for H
ω
x, y, Fig.8.9g shows V
DC
x
for H
ω
z, and Fig. 8.9h
shows V
DC
y
for H
ω
z. The results show that the DC component is enhanced significantly when the frequency of the microwave is tuned to an eigenfrequency of
the spin-wave modes, which converts the microwave power to a DC voltage with
high efficiency. The results also show that the sign of the DC voltage depends on the
excited spin-wave mode and the microwave polarization, which indicates that the sign
of the voltage can be switched by tuning the microwave frequency or the microwave
polarization. Note that a large DC voltage is obtained for the counterclockwise rotation mode activated by H
ω
x, y and for the breathing mode activated by H
ω
z,