234
I. Proskurin and R. L. Stamps
In the case when both v k and K k are odd under the transformation k → −k, the
spin current is determined by the asymmetric part of h
(−)
k (ω)h
(+)
−k (−ω), which is
proportional to i[h
∗
k (ω) × h k (ω)] z . In the limiting case → 0 and φ k = 0, we can
combine both kinds of terms in (9.91), which eventually gives
ˆ
J s (t) =
i
2
ωk
q
2
k ∂ k p k − ω
2
∂ k q k
(ε
2
k − ω 2 ) 2
[h
∗
k (ω) × h k (ω)] z ,
(9.94)
where p k = A k + |B k | and q k = A k − |B k |, which coincides with (9.77) obtained
from the semi-classical equations of motion [44].
9.4.4 Magnon Spin Photocurrents in Antiferromagnetic
Insulators and Low Dimensional Materials
We have demonstrated that in antiferromagnetic materials magnon spin currents contain intraband terms, proportional to the group velocity of magnons, and interband
terms, which by analogy to the relativistic mechanics can be associated with the
Zitterbewegung effect of magnons. The latter is proportional to the fast-oscillating
factors, which makes these terms irrelevant as far as response to a static perturbation
is concerned. For the thermal excitation of spin currents, for example, the antiferromagnetic spin current can be taken in the form of (9.72) [65, 69, 70].
The response to a dynamic perturbation is different. Since spin photocurrent is
the second-order effect, the interband terms that oscillate at the double frequency
should be taken into account together with the intraband contributions, so that the
resulting response current is given by (9.94).
For practical applications, the most interesting frequency region is near the antiferromagnetic resonance, ω ≈ ε k . In this area, the response current is resonantly
amplified and determined by the damping of the material. In the case of ballistic magnon transport, when ε k , we can replace ω − ε k ± i → ±i and
ω + ε k ± i → 2ω r near the resonance ω r . In this limit, the dominant contribution
in (9.91) comes from the first term proportional to v k
ˆ
J s res ≈
iq k
4ω r
v k
2 [h
∗
(ω r ) × h(ω r )] z ,
(9.95)
where we used monochromatic microwave field with h k (ω) [44]. This expression
allows to estimate the order of magnitude for the spin photocurrent excited with
circularly polarized light as ˆ
J s res ≈ χg
2
μ
2
B J 1 S
2 c s I B /(2a 0 c
2
η
2
ω r ), where we take
= ηω r , χ = ± denotes helicity of the wave, I B = |B(ω r )|
2 is intensity, and
linear magnon energy disperison is implied, |v k | = c s . For a typical material with
c s = 3 × 10
4 m/s, J s = 200 K, ω r = 3 × 10
13 s
−1 , η = 10
−4 , and a 0 = 0.5 nm, we
I. Proskurin and R. L. Stamps
In the case when both v k and K k are odd under the transformation k → −k, the
spin current is determined by the asymmetric part of h
(−)
k (ω)h
(+)
−k (−ω), which is
proportional to i[h
∗
k (ω) × h k (ω)] z . In the limiting case → 0 and φ k = 0, we can
combine both kinds of terms in (9.91), which eventually gives
ˆ
J s (t) =
i
2
ωk
q
2
k ∂ k p k − ω
2
∂ k q k
(ε
2
k − ω 2 ) 2
[h
∗
k (ω) × h k (ω)] z ,
(9.94)
where p k = A k + |B k | and q k = A k − |B k |, which coincides with (9.77) obtained
from the semi-classical equations of motion [44].
9.4.4 Magnon Spin Photocurrents in Antiferromagnetic
Insulators and Low Dimensional Materials
We have demonstrated that in antiferromagnetic materials magnon spin currents contain intraband terms, proportional to the group velocity of magnons, and interband
terms, which by analogy to the relativistic mechanics can be associated with the
Zitterbewegung effect of magnons. The latter is proportional to the fast-oscillating
factors, which makes these terms irrelevant as far as response to a static perturbation
is concerned. For the thermal excitation of spin currents, for example, the antiferromagnetic spin current can be taken in the form of (9.72) [65, 69, 70].
The response to a dynamic perturbation is different. Since spin photocurrent is
the second-order effect, the interband terms that oscillate at the double frequency
should be taken into account together with the intraband contributions, so that the
resulting response current is given by (9.94).
For practical applications, the most interesting frequency region is near the antiferromagnetic resonance, ω ≈ ε k . In this area, the response current is resonantly
amplified and determined by the damping of the material. In the case of ballistic magnon transport, when ε k , we can replace ω − ε k ± i → ±i and
ω + ε k ± i → 2ω r near the resonance ω r . In this limit, the dominant contribution
in (9.91) comes from the first term proportional to v k
ˆ
J s res ≈
iq k
4ω r
v k
2 [h
∗
(ω r ) × h(ω r )] z ,
(9.95)
where we used monochromatic microwave field with h k (ω) [44]. This expression
allows to estimate the order of magnitude for the spin photocurrent excited with
circularly polarized light as ˆ
J s res ≈ χg
2
μ
2
B J 1 S
2 c s I B /(2a 0 c
2
η
2
ω r ), where we take
= ηω r , χ = ± denotes helicity of the wave, I B = |B(ω r )|
2 is intensity, and
linear magnon energy disperison is implied, |v k | = c s . For a typical material with
c s = 3 × 10
4 m/s, J s = 200 K, ω r = 3 × 10
13 s
−1 , η = 10
−4 , and a 0 = 0.5 nm, we