9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
229
It should be mentioned that the contribution of the oscillating term in total spin
current may seem insignificant. Indeed, in the theory the spin Seebeck effect only the
term given by (9.72) was taken into account in the definition of the spin current [65,
66]. In this case, the second term, which mixes magnons of different helicities, has
vanishing contribution. However, as we discuss below, such processes as the photoexcitation require both terms being considered with equal attention. Moreover, the
contribution of the second term in (9.73) may become dominant in low-dimensional
systems where it may contain geometric phase effects.
9.4.2 Photo-Excitation of Magnon Spin Currents
Let us now turn to a semi-classical theory of photo-excitation of magnon spin currents. For this purpose, we add a magneto-dipole interaction between the magnetic
field component of the electromagnetic wave h(t, r) and the magnetization of the
antiferromagnet, so that the total energy is written as
W t = W −
d
3 r (M 1 + M 2 ) · h(t, r),
(9.74)
where W is determined by (9.40). In this case, (9.43) acquires the additional term
−2γ M s [n × h p (t)], where h p (t) is the Fourier component of the magnetic field. The
system of equations of motion (9.42) and (9.43) can be easily solved by transforming
the ω-domain, which gives
m p (ω) = 2γ M s
ε
(l)
p h p (ω)
ω 2
p − ω 2 ,
(9.75)
l p (ω) = 2iγ M s
ω[n × h p (ω)]
ω 2
p − ω 2 .
(9.76)
The Gilbert damping can be phenomenologically introduced in these equations by
considering complex parameters ε
(α)
p → ε
(α)
p − iηω (α = m, l). Using the definition
of the spin current in (9.67), we find the current excited by the magnetic field vector
(Fig. 9.3)
J
(n)
s = iγ
2 M s
pω
ε
(l)2
p ∂ p ε
(m)
p + ω
2
∂ p ε
(l)
p
ω 2 − ω 2
p
2
h
∗
p (ω) · [n × h p (ω)].
(9.77)
This expression shows that the direct spin current excited by the electromagnetic
wave is the second order effect in h p (ω), and is determined by the asymmetric combination h
∗
p × h p , so that the direction of the current is determined by helicity of the
electromagnetic wave. The effect is resonantly amplified near the antiferromagnetic
resonance ω ≈ ω p .
229
It should be mentioned that the contribution of the oscillating term in total spin
current may seem insignificant. Indeed, in the theory the spin Seebeck effect only the
term given by (9.72) was taken into account in the definition of the spin current [65,
66]. In this case, the second term, which mixes magnons of different helicities, has
vanishing contribution. However, as we discuss below, such processes as the photoexcitation require both terms being considered with equal attention. Moreover, the
contribution of the second term in (9.73) may become dominant in low-dimensional
systems where it may contain geometric phase effects.
9.4.2 Photo-Excitation of Magnon Spin Currents
Let us now turn to a semi-classical theory of photo-excitation of magnon spin currents. For this purpose, we add a magneto-dipole interaction between the magnetic
field component of the electromagnetic wave h(t, r) and the magnetization of the
antiferromagnet, so that the total energy is written as
W t = W −
d
3 r (M 1 + M 2 ) · h(t, r),
(9.74)
where W is determined by (9.40). In this case, (9.43) acquires the additional term
−2γ M s [n × h p (t)], where h p (t) is the Fourier component of the magnetic field. The
system of equations of motion (9.42) and (9.43) can be easily solved by transforming
the ω-domain, which gives
m p (ω) = 2γ M s
ε
(l)
p h p (ω)
ω 2
p − ω 2 ,
(9.75)
l p (ω) = 2iγ M s
ω[n × h p (ω)]
ω 2
p − ω 2 .
(9.76)
The Gilbert damping can be phenomenologically introduced in these equations by
considering complex parameters ε
(α)
p → ε
(α)
p − iηω (α = m, l). Using the definition
of the spin current in (9.67), we find the current excited by the magnetic field vector
(Fig. 9.3)
J
(n)
s = iγ
2 M s
pω
ε
(l)2
p ∂ p ε
(m)
p + ω
2
∂ p ε
(l)
p
ω 2 − ω 2
p
2
h
∗
p (ω) · [n × h p (ω)].
(9.77)
This expression shows that the direct spin current excited by the electromagnetic
wave is the second order effect in h p (ω), and is determined by the asymmetric combination h
∗
p × h p , so that the direction of the current is determined by helicity of the
electromagnetic wave. The effect is resonantly amplified near the antiferromagnetic
resonance ω ≈ ω p .