8 Microwave-Driven Dynamics of Magnetic Skyrmions …
201
When the time-dependent Rashba spin-orbit interaction α R (t) is induced by the AC
gate voltage, the vector quantity j s ∝ ∂ t α R (t)z × m gives rise to AC spin torques.
In the clean limit with a long relaxation time (/J ex τ 1), the coefficient β R in
the second term is reduced to /J ex τ . Interestingly, the vector quantity j s defined
here can be regarded as a fictitious spin current because the above expressions of
T 2 and T 3 have equivalent forms with those of the spin-transfer torque and the
nonadiabatic torque in the presence of the real spin current j s , respectively. It should
be noted, however, the present relaxation time τ corresponds to a different time scale.
Specifically, the relaxation time is governed by the coherence of the conductionelectron momenta in the present case, while in the current-induced case, it is governed
by that of the conduction-electron spins.
Assuming the material parameters for the metallic bilayer systems in Table 8.2,
the values of j s = (e/a)D 2 and β R are evaluated as ∼2 A/m and ∼0.07, respectively. These values are large enough to induce the magnetization dynamics. Numerical simulations in [35] indeed demonstrated that not only a skyrmion crystal with
hexagonally packed magnetic skyrmions but also isolated skyrmions embedded in
a ferromagnetic background can be excited resonantly through temporal variation
of the Dzyaloshinskii-Moriya interaction achieved by application of a microwave
electric voltage. This technique provides a means to drive magnetic skyrmions electrically with a low energy consumption. Recently, a lot of ferromagnet/heavy-metal
bilayer systems hosting magnetic skyrmions have been reported. The above theoretical proposals are anticipated to be realized by future experiments on these magnetic
bilayer systems.
8.7 Microwave-Induced DC Spin-Motive Force
We next discuss a proposed method to generate DC electric voltages by exploiting the spin-wave excitations of magnetic skyrmions under a tilted H ext field [31].
It is well known that injection of spin-polarized electric currents can drive noncollinear skyrmion textures in metallic magnets via the spin-transfer torque mechanism, whereas the noncollinear skyrmion magnetizations inversely affect transport
properties of conduction electrons as exemplified by the topological Hall effect. The
spin-driven electromotive force (i.e., an emergent electric field induced by magnetization dynamics) is another important example of the latter kinds of phenomena [43,
44]. It was proposed theoretically that spatially modulated magnetic textures such
as magnetic skyrmions, magnetic helices and ferromagnetic domain walls produce
effective vector potential acting on the conduction electrons via exchange coupling
with the conduction-electron spins, which is called non-Abelian gauge field. When
these magnetic textures are temporally varied by an applied time-periodic field such
as microwave electromagnetic fields, this effective vector potential also changes
temporally. This temporal variation of the vector potential gives rise to an effective
electric field which acts on the conduction electrons. The electric motive force due
to this effective electric field is referred to as the spin-motive force, which is one of
201
When the time-dependent Rashba spin-orbit interaction α R (t) is induced by the AC
gate voltage, the vector quantity j s ∝ ∂ t α R (t)z × m gives rise to AC spin torques.
In the clean limit with a long relaxation time (/J ex τ 1), the coefficient β R in
the second term is reduced to /J ex τ . Interestingly, the vector quantity j s defined
here can be regarded as a fictitious spin current because the above expressions of
T 2 and T 3 have equivalent forms with those of the spin-transfer torque and the
nonadiabatic torque in the presence of the real spin current j s , respectively. It should
be noted, however, the present relaxation time τ corresponds to a different time scale.
Specifically, the relaxation time is governed by the coherence of the conductionelectron momenta in the present case, while in the current-induced case, it is governed
by that of the conduction-electron spins.
Assuming the material parameters for the metallic bilayer systems in Table 8.2,
the values of j s = (e/a)D 2 and β R are evaluated as ∼2 A/m and ∼0.07, respectively. These values are large enough to induce the magnetization dynamics. Numerical simulations in [35] indeed demonstrated that not only a skyrmion crystal with
hexagonally packed magnetic skyrmions but also isolated skyrmions embedded in
a ferromagnetic background can be excited resonantly through temporal variation
of the Dzyaloshinskii-Moriya interaction achieved by application of a microwave
electric voltage. This technique provides a means to drive magnetic skyrmions electrically with a low energy consumption. Recently, a lot of ferromagnet/heavy-metal
bilayer systems hosting magnetic skyrmions have been reported. The above theoretical proposals are anticipated to be realized by future experiments on these magnetic
bilayer systems.
8.7 Microwave-Induced DC Spin-Motive Force
We next discuss a proposed method to generate DC electric voltages by exploiting the spin-wave excitations of magnetic skyrmions under a tilted H ext field [31].
It is well known that injection of spin-polarized electric currents can drive noncollinear skyrmion textures in metallic magnets via the spin-transfer torque mechanism, whereas the noncollinear skyrmion magnetizations inversely affect transport
properties of conduction electrons as exemplified by the topological Hall effect. The
spin-driven electromotive force (i.e., an emergent electric field induced by magnetization dynamics) is another important example of the latter kinds of phenomena [43,
44]. It was proposed theoretically that spatially modulated magnetic textures such
as magnetic skyrmions, magnetic helices and ferromagnetic domain walls produce
effective vector potential acting on the conduction electrons via exchange coupling
with the conduction-electron spins, which is called non-Abelian gauge field. When
these magnetic textures are temporally varied by an applied time-periodic field such
as microwave electromagnetic fields, this effective vector potential also changes
temporally. This temporal variation of the vector potential gives rise to an effective
electric field which acts on the conduction electrons. The electric motive force due
to this effective electric field is referred to as the spin-motive force, which is one of