9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
223
Similar expression exists for the Lipkin’s zilch written in terms of the polarized
photon modes [30]. For a monochromatic spin wave, C
(m)
χ becomes proportional to
the spin angular momentum component along n, which in terms of magnon number
operators is given by S
(n)
=
p (N
(L)
p − N
(R)
p ) [30].
9.3.4 Spin-Wave Chirality in Dissipative Media
By now, we have established that spin waves in antiferromagnets can be characterized
by the chiral invariant C
(m)
χ , which is analogous to the Lipkin’s zilch in optics. Similar
to the optical case, we may ask a question: how can we make this chirality of the spin
waves visible? To answer this question, we should look at the symmetries. Since C
(m)
χ
is a pseudoscalar that is odd under P and even under T , we have to break the same
symmetries inside the antiferromagnet following the idea discussed in Sect. 9.2.2 for
the light-matter interactions in chiral metamaterials.
Since our model in (9.40) is not chiral, we should provide some symmetry breaking mechanism. One interesting possibility of such mechanism that is relevant for
spintronic applications is based on electron spin current [38]. The flow of spin angular momentum is odd under the spatial inversion and even under the time reversal
transformation, therefore, its interaction with antiferromagnetic spin waves is able
to provide the necessary symmetry breaking.
The microscopic mechanism beyond this symmetry breaking is as follows. Let us
consider an electron spin current flowing along the magnetic ordering direction n,
which can be injected into an antiferromagnetic insulator film by a proximity effect
or can be created in bulk metallic antiferromagnets. A pure spin current consists of
a number of spin majority electrons (↑) polarized along n flowing with the velocity
v s parallel to n balanced by the same amount of spin minority electrons (↓) moving
with the velocity −v s , so that the net electric charge transport is zero. Since the
spin-wave dynamics is slow with respect to that of the electrons, the latter are able
to exert a spin transfer torque on the magnetization dynamics via the Zhang-Li
mechanism [58]. If the local s-d interactions between the electrons and sublattice
magnetizations are in the exchange dominant regime [59], which means that we can
neglect the intersublattice electron scattering, the spin majority (minority) electrons
couple mostly to M 1 (M 2 ) sublattice magnetization. In this situation, the spin-↑
electrons create the spin transfer torque acting mostly on the magnetization M 1
T 1 = −
1
M 2
s
M 1 × (M 1 × (v s · ∇)M 1 ) −
ξ
M s
M 1 × (v s · ∇)M 1 ,
(9.56)
where the first (second) term is the adiabatic (non-adiabatic) torque component, and
ξ 1 is the dimensionless parameter [58, 59]. At the same time, spin-↓ electron
flow produce the spin transfer torque T 2 = −T 1 applied to M 2 . Therefore, a pure
spin current in the exchange dominant regime of the electron-spin interaction is able
223
Similar expression exists for the Lipkin’s zilch written in terms of the polarized
photon modes [30]. For a monochromatic spin wave, C
(m)
χ becomes proportional to
the spin angular momentum component along n, which in terms of magnon number
operators is given by S
(n)
=
p (N
(L)
p − N
(R)
p ) [30].
9.3.4 Spin-Wave Chirality in Dissipative Media
By now, we have established that spin waves in antiferromagnets can be characterized
by the chiral invariant C
(m)
χ , which is analogous to the Lipkin’s zilch in optics. Similar
to the optical case, we may ask a question: how can we make this chirality of the spin
waves visible? To answer this question, we should look at the symmetries. Since C
(m)
χ
is a pseudoscalar that is odd under P and even under T , we have to break the same
symmetries inside the antiferromagnet following the idea discussed in Sect. 9.2.2 for
the light-matter interactions in chiral metamaterials.
Since our model in (9.40) is not chiral, we should provide some symmetry breaking mechanism. One interesting possibility of such mechanism that is relevant for
spintronic applications is based on electron spin current [38]. The flow of spin angular momentum is odd under the spatial inversion and even under the time reversal
transformation, therefore, its interaction with antiferromagnetic spin waves is able
to provide the necessary symmetry breaking.
The microscopic mechanism beyond this symmetry breaking is as follows. Let us
consider an electron spin current flowing along the magnetic ordering direction n,
which can be injected into an antiferromagnetic insulator film by a proximity effect
or can be created in bulk metallic antiferromagnets. A pure spin current consists of
a number of spin majority electrons (↑) polarized along n flowing with the velocity
v s parallel to n balanced by the same amount of spin minority electrons (↓) moving
with the velocity −v s , so that the net electric charge transport is zero. Since the
spin-wave dynamics is slow with respect to that of the electrons, the latter are able
to exert a spin transfer torque on the magnetization dynamics via the Zhang-Li
mechanism [58]. If the local s-d interactions between the electrons and sublattice
magnetizations are in the exchange dominant regime [59], which means that we can
neglect the intersublattice electron scattering, the spin majority (minority) electrons
couple mostly to M 1 (M 2 ) sublattice magnetization. In this situation, the spin-↑
electrons create the spin transfer torque acting mostly on the magnetization M 1
T 1 = −
1
M 2
s
M 1 × (M 1 × (v s · ∇)M 1 ) −
ξ
M s
M 1 × (v s · ∇)M 1 ,
(9.56)
where the first (second) term is the adiabatic (non-adiabatic) torque component, and
ξ 1 is the dimensionless parameter [58, 59]. At the same time, spin-↓ electron
flow produce the spin transfer torque T 2 = −T 1 applied to M 2 . Therefore, a pure
spin current in the exchange dominant regime of the electron-spin interaction is able