9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
225
ω
(R)
p = c s | p − p s | + iη(( s − pv s ),
(9.60)
ω
(L)
p = c s | p + p s | + iη(( s + pv s ),
(9.61)
where p s = γ M s v s δ/(2c
2
s ), = γ M s δ/2, and p p s is the wave vector of the spin
waves along n, see Fig. 9.2.
This effect is in contrast to the Doppler shift from a spin polarized current where
both modes are shifted in the same direction so that the degeneracy holds [59]. The
imaginary parts of the frequencies ω
(R)
p and ω
(L)
p also have contributions from the
spin current of the opposite signs for the waves with left and right polarizations. This
can be considered as a spin-current-induced circular dichroims of spin waves, which
occurs at the characteristic length scale CD = c s /(ηv s p).
Interestingly, the effect of spin current on the spin waves in the linear approximation is analogous to the existence of the additional Dzyaloshinskii-Moriya interaction
(DMI) term in the antiferromagnetic energy in (9.40)
W DMI =
v s
2γ M s
d
3 r [m 1 · (∇ n × m 1 ) + m 2 · (∇ n × m 2 )] ,
(9.62)
between the magnetizations on the same sublattices.
9.3.4.2 Asymmetric Energy Absorption
Let us now look at the spin-wave energy absorption. The dissipation rate for the
magnetization dynamics can be expressed through the Rayleigh dissipation function
dW
dt
= −
η
γ
d
3 r
∂ M 1
∂t
2
+
∂ M 2
∂t
2
.
(9.63)
According to the equations of motion (9.57) and (9.58), in the presence of the spin
current we replace ∂ t with ∂ t − v s ∇ n for M 1 and with ∂ t + v s ∇ n for M 2 . The energy
dissipation rate in (9.63) in this case acquires the asymmetric contribution proportional to v s that is written as
dW
dt
χ
=
2ηv s
γ
d
3 r
∇ n m 1 ·
∂ m 1
∂t
− ∇ n m 2 ·
∂ m 2
∂t
.
(9.64)
The expression in parentheses is nothing but the spin-wave chirality density ρ
(m)
χ
written in terms of m 1 and m 2 .
As a result, when a pure spin current is injected into an antiferromagnet, the
asymmetry in the spin-wave energy absorption rate becomes proportional to the
spin-wave chirality, (dW/dt) χ = 2ηv s γ
−1 C
m
χ . This result is a direct analogy with
the result of Tang and Cohen [27] for the electromagnetic energy absorption rate in
chiral metamaterials, see Sect. 9.2.2. In antiferromagnetic materials, the microscopic
225
ω
(R)
p = c s | p − p s | + iη(( s − pv s ),
(9.60)
ω
(L)
p = c s | p + p s | + iη(( s + pv s ),
(9.61)
where p s = γ M s v s δ/(2c
2
s ), = γ M s δ/2, and p p s is the wave vector of the spin
waves along n, see Fig. 9.2.
This effect is in contrast to the Doppler shift from a spin polarized current where
both modes are shifted in the same direction so that the degeneracy holds [59]. The
imaginary parts of the frequencies ω
(R)
p and ω
(L)
p also have contributions from the
spin current of the opposite signs for the waves with left and right polarizations. This
can be considered as a spin-current-induced circular dichroims of spin waves, which
occurs at the characteristic length scale CD = c s /(ηv s p).
Interestingly, the effect of spin current on the spin waves in the linear approximation is analogous to the existence of the additional Dzyaloshinskii-Moriya interaction
(DMI) term in the antiferromagnetic energy in (9.40)
W DMI =
v s
2γ M s
d
3 r [m 1 · (∇ n × m 1 ) + m 2 · (∇ n × m 2 )] ,
(9.62)
between the magnetizations on the same sublattices.
9.3.4.2 Asymmetric Energy Absorption
Let us now look at the spin-wave energy absorption. The dissipation rate for the
magnetization dynamics can be expressed through the Rayleigh dissipation function
dW
dt
= −
η
γ
d
3 r
∂ M 1
∂t
2
+
∂ M 2
∂t
2
.
(9.63)
According to the equations of motion (9.57) and (9.58), in the presence of the spin
current we replace ∂ t with ∂ t − v s ∇ n for M 1 and with ∂ t + v s ∇ n for M 2 . The energy
dissipation rate in (9.63) in this case acquires the asymmetric contribution proportional to v s that is written as
dW
dt
χ
=
2ηv s
γ
d
3 r
∇ n m 1 ·
∂ m 1
∂t
− ∇ n m 2 ·
∂ m 2
∂t
.
(9.64)
The expression in parentheses is nothing but the spin-wave chirality density ρ
(m)
χ
written in terms of m 1 and m 2 .
As a result, when a pure spin current is injected into an antiferromagnet, the
asymmetry in the spin-wave energy absorption rate becomes proportional to the
spin-wave chirality, (dW/dt) χ = 2ηv s γ
−1 C
m
χ . This result is a direct analogy with
the result of Tang and Cohen [27] for the electromagnetic energy absorption rate in
chiral metamaterials, see Sect. 9.2.2. In antiferromagnetic materials, the microscopic