8 Microwave-Driven Dynamics of Magnetic Skyrmions …
189
d m i
dt
= −γ m i × H
eff
i +
α G
m
m i ×
d m i
dt
.
(8.5)
The first term of the right-hand side is the gyrotropic term where γ = gμ B / is the
gyromagnetic ratio. This term describes the precessional motion of magnetizations
m i around the effective local magnetic field H
eff
i . The second term is the Gilbertdamping term introduced phenomenologically to depicts the dissipation of gyration
energy. The effective magnetic field H
eff
i , which acts on the local magnetization m i ,
is calculated from the Hamiltonian H = H 0 + H
(t) as,
H
eff
i = −
1
γ
∂H
∂ m i
,
(8.6)
where H 0 is the model Hamiltonian given by (8.3). Here an additional term of the
Hamiltonian H
(t) is introduced, which describes the dynamical coupling between
the skyrmion magnetizations and a time-dependent magnetic field or a microwave
magnetic field in the form,
H
(t) = −H(t) ·
i
m i .
(8.7)
The initial magnetic configuration of a skyrmion crystal is prepared by the MonteCarlo thermalization at low temperatures and by further relaxing it in the micromagnetic simulation without applying a microwave magnetic field. The microwaveinduced dynamics are simulated by applying a microwave field to thus obtained
sufficiently converged skyrmion-crystal configuration.
8.3 Microwave-Active Spin-Wave Modes
A skyrmion crystal confined in a quasi-two-dimensional magnet exhibits characteristic microwave-active spin-wave modes (Fig. 8.3a–c) [27]. An in-plane polarized
microwave magnetic field H
ω
x, y activates a pair of rotation modes, in which all
the skyrmions constituting the skyrmion crystal uniformly circulate in counterclockwise and clockwise ways. The counterclockwise rotation mode has a lower resonance
frequency and a larger intensity than the clockwise rotation mode. On the other hand,
an out-of-plane polarized microwave magnetic field H
ω
z activates the breathing
mode in which all the skyrmions in the skyrmion crystal uniformly expand and shrink
in an oscillatory manner. Then how are these spin-wave modes modulated when the
H ex field is tilted from the perpendicular direction?
The spin-wave modes and their resonance frequencies can be identified by calculating the dynamical magnetic susceptibility,
189
d m i
dt
= −γ m i × H
eff
i +
α G
m
m i ×
d m i
dt
.
(8.5)
The first term of the right-hand side is the gyrotropic term where γ = gμ B / is the
gyromagnetic ratio. This term describes the precessional motion of magnetizations
m i around the effective local magnetic field H
eff
i . The second term is the Gilbertdamping term introduced phenomenologically to depicts the dissipation of gyration
energy. The effective magnetic field H
eff
i , which acts on the local magnetization m i ,
is calculated from the Hamiltonian H = H 0 + H
(t) as,
H
eff
i = −
1
γ
∂H
∂ m i
,
(8.6)
where H 0 is the model Hamiltonian given by (8.3). Here an additional term of the
Hamiltonian H
(t) is introduced, which describes the dynamical coupling between
the skyrmion magnetizations and a time-dependent magnetic field or a microwave
magnetic field in the form,
H
(t) = −H(t) ·
i
m i .
(8.7)
The initial magnetic configuration of a skyrmion crystal is prepared by the MonteCarlo thermalization at low temperatures and by further relaxing it in the micromagnetic simulation without applying a microwave magnetic field. The microwaveinduced dynamics are simulated by applying a microwave field to thus obtained
sufficiently converged skyrmion-crystal configuration.
8.3 Microwave-Active Spin-Wave Modes
A skyrmion crystal confined in a quasi-two-dimensional magnet exhibits characteristic microwave-active spin-wave modes (Fig. 8.3a–c) [27]. An in-plane polarized
microwave magnetic field H
ω
x, y activates a pair of rotation modes, in which all
the skyrmions constituting the skyrmion crystal uniformly circulate in counterclockwise and clockwise ways. The counterclockwise rotation mode has a lower resonance
frequency and a larger intensity than the clockwise rotation mode. On the other hand,
an out-of-plane polarized microwave magnetic field H
ω
z activates the breathing
mode in which all the skyrmions in the skyrmion crystal uniformly expand and shrink
in an oscillatory manner. Then how are these spin-wave modes modulated when the
H ex field is tilted from the perpendicular direction?
The spin-wave modes and their resonance frequencies can be identified by calculating the dynamical magnetic susceptibility,