8 Microwave-Driven Dynamics of Magnetic Skyrmions …
197
8.6 Electrically Driven Spin Torque and Dynamical
Dzyaloshinskii-Moriya Interaction
In the above section, we argued that the application of microwave electric field can
drives translational motion of isolated skyrmions embedded in a ferromagnetic background of magnetic bilayer system through inducing the temporal variation of the
interfacial Dzyaloshinskii-Moriya interaction. In this section, we discuss a theoretical formulation of this electrically induced time-dependent Dzyaloshinskii-Moriya
interaction [35]. From a theoretical perspective, we demonstrate that the spin torques
can be exerted into magnetic bilayer systems via the Rashba spin-orbit interaction
by application of an AC electric voltage. The exerted spin torques turn out to resemble the well-known electric-current-induced torques, i.e., the spin-transfer torque
and the nonadiabatic torque, providing similar controllability of magnetism with
microwave electric fields. The spin torques also turn out to work as an interfacial
Dzyaloshinskii-Moriya interaction, which contains both steady and oscillating components and enables us to create and activate noncollinear magnetism like magnetic
skyrmions by application of a microwave electric field.
In the magnetic bilayer system with broken spatial inversion symmetry, the Rashba
spin-orbit interaction becomes active. This interaction works as an effective magnetic
field acting on the conduction-electron spins, through mediating mutual coupling
between spins and orbital momenta of the electrons [38, 39]. Importantly, strength
and direction of the effective magnetic field are determined by the momentum of
the electron. Therefore, the Rashba spin-orbit interaction can induce nontrivial spin
torques acting on the magnetizations through controlling the spin polarizations of
the conduction electrons which couple to the magnetizations via the exchange interaction. The strength of the Rashba spin-orbit interaction can be tuned by application
of a gate electric voltage normal to the interfacial plane [40], through modulating
the extent of the spatial inversion asymmetry. This suggests that an AC gate voltage
produces nontrivial Rashba-mediated dynamical spin torques.
We consider a magnetic bilayer system with a ferromagnet/heavy-metal interface
(Fig. 8.7), which is fabricated on an insulating substrate. The insulating substrate
prevents the electric-current flow and thus enhances the effects of gate electric voltage
acting on the ferromagnet/heavy-metal interface. The Hamiltonian for this system
has four terms as
H = H K + H R + H ex + H imp
(8.13)
with
H K =
1
2m e
d
2 r
pψ(r, t)
2 − E F
d
2 r ψ
†
(r, t)ψ(r, t),
(8.14)
H R = −
α R (t)
d
2 r ψ
†
(r, t)( p × σ ) z ψ(r, t),
(8.15)
H ex = J ex
d
2 r m(r) · ψ
†
(r, t)σ ψ(r, t),
(8.16)
197
8.6 Electrically Driven Spin Torque and Dynamical
Dzyaloshinskii-Moriya Interaction
In the above section, we argued that the application of microwave electric field can
drives translational motion of isolated skyrmions embedded in a ferromagnetic background of magnetic bilayer system through inducing the temporal variation of the
interfacial Dzyaloshinskii-Moriya interaction. In this section, we discuss a theoretical formulation of this electrically induced time-dependent Dzyaloshinskii-Moriya
interaction [35]. From a theoretical perspective, we demonstrate that the spin torques
can be exerted into magnetic bilayer systems via the Rashba spin-orbit interaction
by application of an AC electric voltage. The exerted spin torques turn out to resemble the well-known electric-current-induced torques, i.e., the spin-transfer torque
and the nonadiabatic torque, providing similar controllability of magnetism with
microwave electric fields. The spin torques also turn out to work as an interfacial
Dzyaloshinskii-Moriya interaction, which contains both steady and oscillating components and enables us to create and activate noncollinear magnetism like magnetic
skyrmions by application of a microwave electric field.
In the magnetic bilayer system with broken spatial inversion symmetry, the Rashba
spin-orbit interaction becomes active. This interaction works as an effective magnetic
field acting on the conduction-electron spins, through mediating mutual coupling
between spins and orbital momenta of the electrons [38, 39]. Importantly, strength
and direction of the effective magnetic field are determined by the momentum of
the electron. Therefore, the Rashba spin-orbit interaction can induce nontrivial spin
torques acting on the magnetizations through controlling the spin polarizations of
the conduction electrons which couple to the magnetizations via the exchange interaction. The strength of the Rashba spin-orbit interaction can be tuned by application
of a gate electric voltage normal to the interfacial plane [40], through modulating
the extent of the spatial inversion asymmetry. This suggests that an AC gate voltage
produces nontrivial Rashba-mediated dynamical spin torques.
We consider a magnetic bilayer system with a ferromagnet/heavy-metal interface
(Fig. 8.7), which is fabricated on an insulating substrate. The insulating substrate
prevents the electric-current flow and thus enhances the effects of gate electric voltage
acting on the ferromagnet/heavy-metal interface. The Hamiltonian for this system
has four terms as
H = H K + H R + H ex + H imp
(8.13)
with
H K =
1
2m e
d
2 r
pψ(r, t)
2 − E F
d
2 r ψ
†
(r, t)ψ(r, t),
(8.14)
H R = −
α R (t)
d
2 r ψ
†
(r, t)( p × σ ) z ψ(r, t),
(8.15)
H ex = J ex
d
2 r m(r) · ψ
†
(r, t)σ ψ(r, t),
(8.16)