20
T. Yu and G. E. W. Bauer
ρ L ≡ ≡ ˆ
m
†
L (t) ˆ
m L (t) = n L +
dω
2π
κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2 (n q ∗ − n L ),
(1.46)
ρ R ≡ ≡ ˆ
m
†
R (t) ˆ
m R (t) = n R +
dω
2π
2
1 κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2
2 (n L − n q ∗ ),
(1.47)
where the damping in the high-quality film has been disregarded
κ q → 0
. In the
linear regime the non-local thermal injection of magnons into the right transducer
by the left one then reads
δρ R =
S CSSE (T L − T R )
0
when
T L > T R
T L ≤ T R
,
S CSSE =
dω
2π
2
1 κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2
2
dn L
dT
T =(T L +T R )/2
.
(1.48)
where we defined the chiral (or dipolar) spin Seebeck coefficient S CSSE .
The device therefore operates as a heat diode, apparently acting as a “Maxwell
demon” that rectifies the thermal fluctuations at equilibrium. However, in thermal
equilibrium all right and left moving magnons are eventually connected by reflection
of spin waves at the edges and absorption and re-emission by connected heat baths.
The Second Law of thermodynamics is therefore safe, but it might be interesting to
search for chirality-induced transient effects.
1.4 Conclusion and Outlook
Handedness or chirality of wave propagation is a popular research topic in optics,
acoustics, and condensed matter physics. Here we contribute by a theory for the coherent and incoherent chiral pumping of spin waves into thin magnetic films through the
chiral magnetodipolar radiation generated by the Oersted field of metallic striplines
and dipolar field of magnetic nanostructures. Spin waves excited coherently in the
film under magnetic resonance of the nanowire are unidirectional, generating a nonequilibrium magnetization in only half of the film. A temperature gradient between a
local magnet and a film leads to the unidirectional excitation of incoherent magnons,
i.e., a chiral spin Seebeck effect.
PT symmetry has been predicted to amplify unidirectional response [62–64].
Even though our system is dissipative and therefore not PT symmetric, the nonreciprocal coupling of the two wires still allows directional amplification [9, 10]. It
would be interesting to introduce PT symmetry into our system via gain in one wire
that compensates the damping in the other, possibly leading to enhanced effects.
T. Yu and G. E. W. Bauer
ρ L ≡ ≡ ˆ
m
†
L (t) ˆ
m L (t) = n L +
dω
2π
κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2 (n q ∗ − n L ),
(1.46)
ρ R ≡ ≡ ˆ
m
†
R (t) ˆ
m R (t) = n R +
dω
2π
2
1 κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2
2 (n L − n q ∗ ),
(1.47)
where the damping in the high-quality film has been disregarded
κ q → 0
. In the
linear regime the non-local thermal injection of magnons into the right transducer
by the left one then reads
δρ R =
S CSSE (T L − T R )
0
when
T L > T R
T L ≤ T R
,
S CSSE =
dω
2π
2
1 κ
(ω − ω K ) 2 + (κ/2 + 1 /2) 2
2
dn L
dT
T =(T L +T R )/2
.
(1.48)
where we defined the chiral (or dipolar) spin Seebeck coefficient S CSSE .
The device therefore operates as a heat diode, apparently acting as a “Maxwell
demon” that rectifies the thermal fluctuations at equilibrium. However, in thermal
equilibrium all right and left moving magnons are eventually connected by reflection
of spin waves at the edges and absorption and re-emission by connected heat baths.
The Second Law of thermodynamics is therefore safe, but it might be interesting to
search for chirality-induced transient effects.
1.4 Conclusion and Outlook
Handedness or chirality of wave propagation is a popular research topic in optics,
acoustics, and condensed matter physics. Here we contribute by a theory for the coherent and incoherent chiral pumping of spin waves into thin magnetic films through the
chiral magnetodipolar radiation generated by the Oersted field of metallic striplines
and dipolar field of magnetic nanostructures. Spin waves excited coherently in the
film under magnetic resonance of the nanowire are unidirectional, generating a nonequilibrium magnetization in only half of the film. A temperature gradient between a
local magnet and a film leads to the unidirectional excitation of incoherent magnons,
i.e., a chiral spin Seebeck effect.
PT symmetry has been predicted to amplify unidirectional response [62–64].
Even though our system is dissipative and therefore not PT symmetric, the nonreciprocal coupling of the two wires still allows directional amplification [9, 10]. It
would be interesting to introduce PT symmetry into our system via gain in one wire
that compensates the damping in the other, possibly leading to enhanced effects.