1 Chiral Coupling to Magnetodipolar Radiation
17
The excitation of the left nanowire propagates to the right nanowire by the spin waves
in the film. When chiral coupling is perfect, f (ω) vanishes without the back-action.
The microwave output of both left and right nanowires as inductively detected by
coplanar wave guides are denoted ˆ
p
(L)
out (ω) and ˆ
p
(R)
out (ω) with input-output relations
[56, 57]
ˆ
p
(L)
out (ω) = p
(L)
in (ω) +
√
κ p,L ˆ
m L (ω),
ˆ
p
(R)
out (ω) =
√ κ p,R ˆ
m R (ω),
(1.40)
where κ p,R is the additional radiative damping induced by the detector. Therefore,
the elements in the microwave scattering matrix describing reflection (S 11 )and transmission (S 21 ) amplitudes become
S 11 (ω) ≡
ˆ
p
(L)
out
ˆ
p
(L)
in
= 1 −
κ p,L
−i(ω − ω K ) + (κ L + κ p,L )/2 + i
q g 2
q G q (ω) − f (ω)
,
S 21 (ω) ≡
ˆ
p
(R)
out
ˆ
p
(L)
in
= [1 − S 11 (ω)]
κ p,R
κ p,L
i
q g
2
q G q (ω) e
iq(R 2 −R 1 )
−i(ω − ω K ) + κ R /2 + i
q g 2
q G q (ω)
.
(1.41)
The real parts of S 11 and S 12 at different magnetic fields and microwave frequencies
are illustrated in Fig. 1.6 for antiparallel magnetizations. The frequency of the Co
Kittel mode decreases with increasing magnetic field until its direction is reversed
to the magnetic-field direction (here
H app
200 mT). The interference pattern on
the Kittel resonance in Fig. 1.6b reflects the transmission phase delay e
ik(R 1 −R 2 ) in
(1.41). We note that in our model the nanowires do not reflect spin waves, the features
should therefore not be interpreted in terms of standing spin waves.
1.3.4 Incoherent Chiral Pumping
A temperature gradient between the magnetic nanowire and film also injects unidirectional magnon currents, i.e., causes a chiral spin Seebeck effect [52–55] . Here we
consider again two identical transducers, i.e., a magnetic nanowire at r 2 = R 2 ˆ
y that
detects magnons, which are now thermally injected by the nanowire at r 1 = R 1 ˆ
y and
R 1 < R 2 . This is the configuration of the non-local spin Seebeck effect as detected
electrically in many experiments starting with [21]. The magnons in those experiments are believed to be injected by the interface exchange interaction or generated
by a temperature gradient in the bulk and results are interpreted by spin diffusion
models. Here we consider the regime in which the exchange effect is suppressed,
magnon propagation is ballistic and we disregard the bulk spin Seebeck effect due
to possible temperature gradients. We predict a spin non-local spin Seebeck effect
that is caused exclusively by dipolar fields and carried by magnons with long wave
17
The excitation of the left nanowire propagates to the right nanowire by the spin waves
in the film. When chiral coupling is perfect, f (ω) vanishes without the back-action.
The microwave output of both left and right nanowires as inductively detected by
coplanar wave guides are denoted ˆ
p
(L)
out (ω) and ˆ
p
(R)
out (ω) with input-output relations
[56, 57]
ˆ
p
(L)
out (ω) = p
(L)
in (ω) +
√
κ p,L ˆ
m L (ω),
ˆ
p
(R)
out (ω) =
√ κ p,R ˆ
m R (ω),
(1.40)
where κ p,R is the additional radiative damping induced by the detector. Therefore,
the elements in the microwave scattering matrix describing reflection (S 11 )and transmission (S 21 ) amplitudes become
S 11 (ω) ≡
ˆ
p
(L)
out
ˆ
p
(L)
in
= 1 −
κ p,L
−i(ω − ω K ) + (κ L + κ p,L )/2 + i
q g 2
q G q (ω) − f (ω)
,
S 21 (ω) ≡
ˆ
p
(R)
out
ˆ
p
(L)
in
= [1 − S 11 (ω)]
κ p,R
κ p,L
i
q g
2
q G q (ω) e
iq(R 2 −R 1 )
−i(ω − ω K ) + κ R /2 + i
q g 2
q G q (ω)
.
(1.41)
The real parts of S 11 and S 12 at different magnetic fields and microwave frequencies
are illustrated in Fig. 1.6 for antiparallel magnetizations. The frequency of the Co
Kittel mode decreases with increasing magnetic field until its direction is reversed
to the magnetic-field direction (here
H app
200 mT). The interference pattern on
the Kittel resonance in Fig. 1.6b reflects the transmission phase delay e
ik(R 1 −R 2 ) in
(1.41). We note that in our model the nanowires do not reflect spin waves, the features
should therefore not be interpreted in terms of standing spin waves.
1.3.4 Incoherent Chiral Pumping
A temperature gradient between the magnetic nanowire and film also injects unidirectional magnon currents, i.e., causes a chiral spin Seebeck effect [52–55] . Here we
consider again two identical transducers, i.e., a magnetic nanowire at r 2 = R 2 ˆ
y that
detects magnons, which are now thermally injected by the nanowire at r 1 = R 1 ˆ
y and
R 1 < R 2 . This is the configuration of the non-local spin Seebeck effect as detected
electrically in many experiments starting with [21]. The magnons in those experiments are believed to be injected by the interface exchange interaction or generated
by a temperature gradient in the bulk and results are interpreted by spin diffusion
models. Here we consider the regime in which the exchange effect is suppressed,
magnon propagation is ballistic and we disregard the bulk spin Seebeck effect due
to possible temperature gradients. We predict a spin non-local spin Seebeck effect
that is caused exclusively by dipolar fields and carried by magnons with long wave