surface or interface, while the propagation direction is governed by the outer
product of the surface normal and polarization, i.e., exhibiting chirality, as
explained in more detail below.

Bulk plasmon.—We first recall the concepts of bulk plasmon and its interaction
with electromagnetic waves. Maxwell equations read

$\displaystyle\nabla\cdot{\bf D}$ $\displaystyle=\rho_{f},$
$\displaystyle\nabla\cdot{\bf B}$ $\displaystyle=0,$
$\displaystyle\nabla\times{\bf E}$ $\displaystyle=-{\partial{\bf B}}/{\partial
t},$ $\displaystyle\nabla\times{\bf H}$ $\displaystyle={\bf
J}_{f}+{\partial{\bf D}}/{\partial t},$ (110)

where $\rho_{f}$ is the density of net free charges that contribute to a free
charge current ${\bf J}_{f}$ via the conservation law
$\partial\rho_{f}/\partial t=-\nabla\cdot{\bf J}_{f}$, the electric
displacement field ${\bf D}=\varepsilon_{0}{\bf E}+{\bf P}$, and ${\bf H}={\bf
B}/\mu_{0}-{\bf M}$. On the other hand, the electric polarization ${\bf P}$ is
related to the bounded charge accumulation $\rho_{b}$ according to

$\displaystyle\nabla\cdot{\bf P}=-\rho_{b}.$ (111)

The charge conservation $\nabla\cdot{\bf J}_{b}=-\partial\rho_{b}/\partial t$
defines the associated bounded charge current ${\bf J}_{b}$ that leads to the
displacement current

$\displaystyle{\bf J}_{b}={\partial{\bf P}}/{\partial t}.$ (112)

Here we focus on the electrodynamics of isotropic materials in the linear
response regime. In the frequency and momentum space

$\displaystyle{\bf D}({\bf k},\omega)=\varepsilon_{0}\varepsilon({\bf
k},\omega){\bf E}({\bf k},\omega),$ (113a) $\displaystyle{\bf J}({\bf
k},\omega)=\sigma({\bf k},\omega){\bf E}({\bf k},\omega),$ (113b)

where $\varepsilon_{0}\varepsilon({\bf k},\omega)$ is the dielectric constant
and $\sigma({\bf k},\omega)$ is the dynamic conductivity. The Maxwell
equations (110) with Eq. (112) lead to

$\displaystyle\varepsilon({\bf k},\omega)=1+\frac{i\sigma({\bf
k},\omega)}{\varepsilon_{0}\omega}.$ (114)

The imaginary part of $\sigma({\bf k},\omega)$ contributes to the real part of
$\varepsilon({\bf k},\omega)$, i.e., the ability for a material to be
polarized by the electric field; while the real part of $\sigma({\bf
k},\omega)$ corresponds to the imaginary part of $\varepsilon({\bf
k},\omega)$, i.e., the ability of a material to absorb electromagnetic waves.

At the zeros of the dielectric constant $\varepsilon({\bf k},\omega)={\bf
D}=0$, the polarization ${\bf P}$ exactly cancels the electric field ${\bf
E}$, which implies that the motion of the electrons becomes collective. The
root of $\varepsilon({\bf k},\omega)=0$ defines the dispersion of this
collective mode or “plasmon". The Lindhard formula for the dielectric constant
of free electrons in bulk metals reads [258]

$\displaystyle\varepsilon({\bf q},\omega)=1+\frac{8\pi e^{2}}{q^{2}}\sum_{{\bf
k}}n_{\bf k}\frac{E_{{\bf k}+{\bf q}}-E_{\bf k}}{(E_{{\bf k}+{\bf q}}-E_{\bf
k})^{2}-\omega^{2}},$ (115)

where $n_{\bf k}$ is the Fermi distribution and $E_{\bf k}$ is the electronic
band structure. In the long-wavelength limit with ${\bf q}\rightarrow 0$, the
plasmon dispersion is the root of

$\displaystyle\varepsilon({\bf q},\omega_{q})=1-\frac{\omega_{\rm
P}^{2}}{\omega_{q}^{2}}-\frac{3\omega_{\rm
P}^{2}}{5\omega_{q}^{4}}q^{2}v_{F}^{2}=0,$ (116)

where $\omega_{\rm P}=\sqrt{{4\pi ne^{2}}/{m}}$ is the plasmon frequency from
the Drude model, and $v_{F}$ is the Fermi velocity. The plasmon frequency in
bulk metal is then obtained to be

$\displaystyle\omega_{q}=\omega_{\rm P}\sqrt{1+\frac{3v_{F}^{2}}{5\omega_{\rm
P}^{2}}q^{2}},$ (117)

with a frequency gap $\omega_{\rm P}$.

Plasmon polariton.—The bulk plasmon couples strongly with the optical electric
field to form a hybrid plasmon polariton. The wave equation for the electric
field in metals follows from the Maxwell equations (110) and Eq. (113a):

$\displaystyle{\bf k}({\bf k}\cdot{\bf E})-k^{2}{\bf E}=-\varepsilon({\bf
k},\omega)\frac{\omega^{2}}{c^{2}}{\bf E}.$ (118)

The longitudinal mode with ${\bf k}\cdot{\bf E}=kE$, leads to
$\varepsilon({\bf k},\omega)=0$, i.e., corresponding to the bare plasmon
excitation. The plasmon polariton is therefore associated with the transverse
mode with ${\bf k}\cdot{\bf E}=0$ that satisfies

$\displaystyle k^{2}=\varepsilon({\bf k},\omega)\omega^{2}/c^{2},$ (119)

which leads to its dispersion relation.

In the long-wave length limit of a metal $\varepsilon_{m}(\omega)\rightarrow
1-\omega_{\rm P}^{2}/\omega^{2}$, Eq. (119) leads to the frequency of plasmon
polariton

$\omega_{\rm PP}=\sqrt{\omega_{\rm P}^{2}+k^{2}c^{2}}.$ (120)

The light is then not allowed to propagate below $\omega_{P}$ in the bulk
because ${\bf k}$ thereby becomes imaginary. For an optical field of
$\omega>\omega_{P}$, only the transverse plasmon couples to the optical field
which leads to a hybridization.

Surface plasmon polariton.—Surface plasmon polariton emerges at the interface
between a metal and a material with dielectric constant $\varepsilon_{d}$ (or
surface to the vacuum with $\varepsilon_{d}=1$). For a flat interface between
two semi-infinite spaces of a conductor ($y\leq 0$) and a dielectric ($y>0$),
as in Fig. 22(a), the Maxwell’s equations in frequency space read

$\displaystyle\nabla\times{\bf E}$ $\displaystyle=i\omega\mu_{0}{\bf H},$
(121a) $\displaystyle\nabla\times{\bf H}$
$\displaystyle=-i\omega\varepsilon_{0}\varepsilon_{r}(y){\bf E},$ (121b)

where $\varepsilon_{r}(y\leq 0)=\varepsilon_{m}$ at the metal and
$\varepsilon_{r}(y>0)=\varepsilon_{d}$ at the dielectric side. The solutions
are planes waves that we assume to propagate along the $\hat{\bf
x}$-direction, i.e., $\boldsymbol{\cal H}(y)e^{ik_{x}x}$ and $\boldsymbol{\cal
E}(y)e^{ik_{x}x}$. The TE mode with ${\bf E}=E_{z}\hat{\bf z}$ and ${\bf
H}=H_{x}\hat{\bf x}+H_{y}\hat{\bf y}$ does not solve Eqs. (121b) and (121a)
[257], so the SSP must be a TM mode with ${\bf E}=E_{x}\hat{\bf
x}+E_{y}\hat{\bf y}$ and ${\bf H}=H_{z}\hat{\bf z}$ with wave equation

$\displaystyle\frac{\partial^{2}H_{z}}{\partial
y^{2}}+(\varepsilon_{r}k_{0}^{2}-k_{x}^{2})H_{z}=0,$ (122)

where $k_{0}^{2}=\omega^{2}/c^{2}$. Since the in-plane magnetic field is
continuous across the interface,

$\displaystyle
H_{z}(x,y)=Ae^{ik_{x}x}e^{-\sqrt{k_{x}^{2}-\varepsilon_{r}k_{0}^{2}}y}.$ (123)

The electric field components

$\displaystyle E_{x}$
$\displaystyle=\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\frac{\partial
H_{z}}{\partial
y}=-\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\sqrt{k_{x}^{2}-\varepsilon_{r}k_{0}^{2}}H_{z},$
(124a) $\displaystyle E_{y}$
$\displaystyle=-\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\frac{\partial
H_{z}}{\partial x}=\frac{k_{x}}{\omega\varepsilon_{0}\varepsilon_{r}}H_{z},$
(124b)

are related as $E_{x}=-iE_{y}\sqrt{k_{x}^{2}-\varepsilon_{r}k_{0}^{2}}/k_{x}$.
When $k_{x}\gg k_{0}$, $E_{y}=i{\rm sgn}(k_{x})E_{x}$, i.e., a polarization-
momentum locking of the electric field as illustrated in Fig. 22(a).

Figure 22:  Chirality and dispersion of surface plasmon polaritons. (a)
illustrates the polarization-momentum locking of the electric field of surface
plasmon polaritons: the electric field is right (left) circularly polarized
for the right (left) moving excitations. (b) plots the dispersion of surface
plasmon polaritons. $k_{0}=\omega_{\rm SPP}/c$ is indicated.

We can derive the dispersion relation $\omega_{\rm SPP}(k_{x})$ from the
boundary conditions, i.e., the continuity of the normal components of ${\bf
D}$ (and ${\bf B}$) and that of the tangential components of ${\bf E}$ (and
${\bf H}$). From $E_{x}^{(m)}=E_{x}^{(d)}$,

$\displaystyle\omega_{\rm
SPP}=\sqrt{\frac{\varepsilon_{d}+\varepsilon_{m}(\omega_{\rm
SPP})}{\varepsilon_{d}\varepsilon_{m}(\omega_{\rm SPP})}}c|k_{x}|.$ (125)

Equation (123) is a surface mode if $\varepsilon_{r}k_{0}^{2}<k_{x}^{2}$.
Therefore $\varepsilon_{m}(\omega_{\rm SPP})<0$ and $\omega_{\rm
SPP}<\omega_{\rm P}$. Moreover, $\omega_{\rm SPP}$ must be real and positive,
which requires $\varepsilon(\omega_{\rm SPP})+\varepsilon_{d}<0$ and
$\omega_{\rm SPP}<\omega_{\rm P}/\sqrt{1+\varepsilon_{d}}$. These features are
summarized in Fig. 22(b). The upper branch, if we wonder, is just the
dispersion of plasmon polariton Eq. (120), which is bulk in nature.

SPPs allow sub-wavelength confinement of optical fields. Plasmonic structures
act as antennae of electromagnetic radiation that strongly enhance the
amplitude of incident signals. As reviewed in Sec. 5.2, SPPs can generate
electronic spin currents in normal metals [86, 87, 88, 89, 91].

####  3.3.2 Stray fields by dynamic electric dipoles

Analogous to the magnetic dipoles discussed in the previous Sec. 3.2.2,
electric dipoles emit stray electric fields that become chiral when
evanescent. They are important in the near-field regime of nano-optics [17].
Here we discuss the waves emitted by electric dipoles that oscillate or rotate
in a plane normal to an interface, as formulated by Fortuño et. al. [19].

According to non-relativistic radiation theory [206], Eq. (29), an electric
free or bounded current J generates a vector potential ${\bf A}({\bf
r},t)=({\mu_{0}}/4\pi)\int d{\bf r}^{\prime}{\bf J}({\bf
r}^{\prime},t){e^{ik|{\bf r}-{\bf r}^{\prime}|}}/{|{\bf r}-{\bf
r}^{\prime}|}$. A uniform line of dipoles ${\bf p}=p_{x}\hat{\bf
x}+p_{z}\hat{\bf z}$ along the $\hat{\bf y}$-direction that oscillates with
frequency $\omega$ generates a displacement current

${\bf J}(x^{\prime},z^{\prime})=-i\omega\delta(x^{\prime},z^{\prime}){\bf p}.$
(126)

Its vector potential

$\displaystyle{\bf A}({\bf r},t)=\frac{\mu_{0}}{4\pi}\int
dy^{\prime}\frac{-i\omega{\bf
p}}{\sqrt{x^{2}+(y-y^{\prime})^{2}+z^{2}}}e^{ik\sqrt{x^{2}+(y-y^{\prime})^{2}+z^{2}}}$
(127)

does not depend on $y$ as can be made explicit by using the Weyl identity
(30),

${\bf A}(x,z)=\frac{\mu_{0}\omega}{4\pi}{\bf p}\int
dk_{x}\frac{e^{ik_{x}x+ik_{z}|z|}}{k_{z}}.$ (128)

The vector potential is parallel to ${\bf p}$, and the magnetic field ${\bf
H}({\bf r})=({1}/{\mu_{0}})\nabla\times{\bf A}({\bf r})=(0,H_{y},0)$ becomes
for $z<0$

$H_{y}(x,z)=\int{dk_{x}}e^{ik_{x}x}H_{y}(k_{x},z),$ (129)

where $k_{z}=\sqrt{(\omega/c)^{2}-k_{x}^{2}}$ and

$H_{y}(k_{x},z)=-\frac{i\omega}{4\pi}\Big{(}p_{x}+\frac{k_{x}}{k_{z}}p_{z}\Big{)}e^{-ik_{z}z}.$
(130)

The magnetic field generated by an in-plane dipole is always linearly
polarized. When $k_{x}\gg\omega/c\equiv k_{0}$, it becomes evanescent with

$H_{y}(k_{x},z)=-\frac{i\omega}{4\pi}\Big{(}p_{x}-i{\rm
sgn}({k_{x}})p_{z}\Big{)}e^{-|k_{x}||z|}.$ (131)

When the dipole rotates with $p_{z}=ip_{x}$, this Fourier component

$H_{y}(k_{x},z)=-\frac{i\omega}{4\pi}p_{x}\Big{(}1+{\rm
sgn}({k_{x}})\Big{)}e^{-|k_{x}||z|}$ (132)

is unidirectional, viz., it vanishes when $k_{x}<0$ and
$H_{y}(k_{x},z)e^{ik_{x}x}$ is a linear polarized traveling wave with a
chirality governed by the rotation direction of the dipole.

According to Eq. (121b), the associated electric field in the evanescent limit
($z<0$)

$\displaystyle
E_{x}(k_{x},z)=-\frac{1}{\varepsilon_{0}\varepsilon_{r}}\frac{1}{4\pi}\left(p_{x}-i{\rm
sgn}(k_{x})p_{z}\right)|k_{x}|e^{-|k_{x}||z|},$ (133a) $\displaystyle
E_{z}(k_{x},z)=\frac{i}{\varepsilon_{0}\varepsilon_{r}}\frac{1}{4\pi}\left(p_{x}-i{\rm
sgn}(k_{x})p_{z}\right)k_{x}e^{-|k_{x}||z|},$ (133b)

i.e., $E_{z}(k_{x},z)=i{\rm sgn}(k_{x})E_{x}(k_{x},z)$. A linearly polarized
electric dipole generates a circularly polarized electric field wave that
propagates in both directions, with polarization (spin) locked to the
momentum. On the other hand, a circularly polarized electric dipole emits a
(circularly polarized) electric field in only one direction that is locked to
the rotation direction of the dipole, i.e., being unidirectional.

From the above discussion and Sec. 3.3.1 we understand that the dynamics of an
electric dipole close to a metal surface may chirally excite surface plasmon
polaritons, as reported by Rodríguez-Fortuño et al. [19]. As illustrated in
Fig. 23(a), a line of in-plane dipoles can be created by illuminating a narrow
slit (130 nm) in a thin gold film by circularly or linearly polarized light.
At grazing incidence, the circular polarization of the light’s electric field
is approximately perpendicular to the metal interface. The excited SPPs are
recorded by their leakage radiation on either side of the slit. As shown in
Fig. 23(b), the plasmons escape symmetrically to both sides for a linearly
polarized dipole excitation, but the excitation becomes unidirectional with
increasing polarization.

Figure 23: Directional excitation of surface plasmon polaritons by local
electric-dipole “wire". (a) is the schematics of the experiment, where a plane
wave is an incident at a nearly grazing angle ($70^{\degree}$) onto a 130 nm
slit in a thin gold film. The excitation of the surface plasmon polariton is
recorded by the leakage radiation from either side of the slit. The results in
Fourier space in (b) show the dependence of the excitation direction on the
light polarization. The figures are taken from Ref. [19].

###  3.4 Chiral phonons

The excitations of elastic media such as ultrasound obey wave equations that
are similar, but also characteristically different from those of
electromagnetic radiation or the magnetic order. It should come as no surprise
by now that also sound and lattice vibrations display chiral properties when
evanescent. Continuum mechanics is appropriate to describe pressure and shear
waves at long wavelengths, while at short wavelengths the atomic dynamics
should be taken into account. The quanta of mechanical excitations are
phonons. Their eigenmodes are found by solving the lattice dynamical problem,
i.e., the classical equation of motion of the atomic displacement field. We
consider a lattice periodic system with $r$ atoms per unit cell located at
${\bf R}_{l}+{\bf d}_{s}$, where ${\bf R}_{l}$ is the origin of the $l$-th
unit cell and ${\bf d}_{s}$ is the position of the $s$-th atoms in this unit
cell. Their displacement into the $\alpha$-direction ${u}_{l}^{\alpha}(s,t)$
so obeys the (linearized) equation of motion [259]

$\displaystyle
M_{a}u_{l}^{\alpha}(s)=-\sum_{l^{\prime}s^{\prime}\beta}\Phi_{\alpha\beta}\left(\begin{matrix}l&l^{\prime}\\\
s&s^{\prime}\end{matrix}\right)u_{l^{\prime}}^{\beta}(s^{\prime}),$ (134)

where $M_{a}$ is the atomic mass and the $3r\times 3r$ matrix
$\Phi_{\alpha\beta}\left(\begin{matrix}l&l^{\prime}\\\
s&s^{\prime}\end{matrix}\right)$ contains the force constants between
different atoms. With an ansatz of a plane wave solution with wave vector k,
the $3r$ degrees of freedom in the unit cell lead to the same number of
eigenvalues or Eq. (134) that we label as $\omega_{j=\\{1,2,\cdots,3r\\}}({\bf
k})$. These form $3r$ branches of phonons in the reciprocal space, including 3
acoustic and $(3r-3)$ optical phonon bands, and the $3r$ eigenvectors
${\boldsymbol{\xi}}^{(j)}_{s}({\bf k})$ indicate the phonon polarization. In
second quantization, the bosonic phonon operators $\hat{b}_{\bf k}$ and
$\hat{b}_{\bf k}^{\dagger}$ govern the displacement field for atom $s$ in unit
cell $l$

$\displaystyle\hat{\bf u}_{l}(s)=\frac{1}{\sqrt{NM_{a}}}\sum_{{\bf
k}}\sum_{j=1}^{3r}\sqrt{\frac{\hbar}{2\omega_{j}({\bf k})}}\left(\hat{b}_{{\bf
k}j}+\hat{b}_{-{\bf k}j}^{\dagger}\right)\boldsymbol{\xi}_{s}^{(j)}({\bf
k})e^{i{\bf k}\cdot{\bf R}_{l}},$ (135)

where $N$ is the number of unit cells. The phonon Hamiltonian is a distributed
set of quantum harmonic oscillators $\hat{H}=\sum_{{\bf
k}}\sum_{j}\hbar\omega_{j}({\bf k})(\hat{b}^{\dagger}_{{\bf k}j}\hat{b}_{{\bf
k}j}+1/2)$.

Phonons have been called chiral [260] when i) their polarization is circular,
i.e., the displacement field rotates around the equilibrium position, and ii)
the direction of the polarization rotation (spin) is locked to the linear
momentum under the time-reversal symmetry. We focus here on two types of
chiral phonons, i.e., bulk phonons (Sec. 3.4.1) and surface acoustic waves
(Sec. 3.4.2).

####  3.4.1 Chiral bulk phonons

A chirality of phonon modes in a bulk crystal affects the electron scattering
[261] and the interaction with light [262]. In the reduced zone scheme the
phonon crystal momenta lie in the first Brillouin zone and the phonon
eigenmodes of the $j$-th band are the Bloch waves

$\displaystyle{\boldsymbol{\psi}}^{(j)}_{l}({\bf
k})=\frac{1}{\sqrt{NM_{a}}}\sqrt{\frac{\hbar}{2\omega_{j}({\bf
k})}}\boldsymbol{\xi}^{(j)}({\bf k})e^{i{\bf k}\cdot{\bf R}_{l}}.$ (136)

These basis functions transform according to the point group symmetry of the
lattice spanned by a number of symmetry operations $\hat{O}_{g}$ such that
$\hat{O}_{g}{\boldsymbol{\psi}}^{(i)}_{l}({\bf
k})=R_{ji}(g){\boldsymbol{\psi}}_{l}^{(j)}(g^{-1}{\bf k})$ in terms of the
irreducible representation matrix $R_{ji}(g)$. At high-symmetry points of the
Brillouin zone, eigenmodes tend to have degenerate eigenvalues and higher-
dimensional irreducible representations of the point group. The phonon
polarization is then not well-defined as an arbitrary superposition of the
polarization of the degenerate states. A definite polarization of the phonons
at symmetry points can be generated by breaking a symmetry. Zhang et al.
showed that breaking the inversion symmetry generates a “pseudoangular"
momentum of phonons at special points in the Brillouin zone of a two-
dimensional honeycomb lattice such as monolayer transition-metal
dichalcogenide [260]. The “pseudo-angular" momentum should not be confused
with the “phonon spin", i.e., the rotational motion of a single atom that we
discuss below. The “pseudo-angular" momentum is defined by the transformation
law of the displacement field under the rotation operation, but not that
defined via Noether’s theorem below. Throughout this review, we stick to the
latter description as addressed in Sec. 3.4.3 for the phonon spins and orbital
angular momentum.

We illustrate these abstract notions for a monolayer of a hexagonal lattice
[260]. With $r=2$ atoms, labeled by $A$ and $B$ in the unit cell, we have
$3r=6$ phonon modes of every momentum in total. Limiting attention to the in-
plane motion, 2 acoustic and 2 optical phonon branches remain. At the
Brillouin-zone corners points ${\bf K}$ and ${\bf K}^{\prime}$, the phonon
amplitudes transform according to the $C_{3h}$ symmetry group that contains
six elements including, in particular, the threefold rotation $C_{3}$, as
shown in Table 3 together with their basis functions. All six representations
$\\{A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}\\}$ are one-dimensional and non-
degenerate. The basis of the representations $A_{3}$ and $A_{4}$ are right-
handed and left-handed, respectively, thus corresponding to circular
polarization and pseudo-angular momentum. They are not degenerate because of
the lack of inversion symmetry. The threefold rotation of the basis at ${\bf
K}$, $\hat{O}_{C_{3}}{\boldsymbol{\psi}}_{l}^{(j)}({\bf
K})={\boldsymbol{\psi}}_{l}^{(j)}(C_{3}^{-1}{\bf K})=e^{-2\pi
l_{s}^{(j)}i/3}{\boldsymbol{\psi}}_{l}^{(j)}({\bf K})$ therefore implies a
pseudoangular momentum $l_{s}^{(j)}$of the band with index $j$. For the
representations $A_{3}$ and $A_{4}$, $l_{s}^{(A_{3})}=1$ and
$l_{s}^{(A_{4})}=-1$, while $l_{s}^{(j)}=0$ for all others. These
pseudoangular momenta are governed by the crystal symmetry and generate
selection rules in the matrix elements of electron-phonon scattering [261] and
photon-phonon [262] scattering. Figure 24(a) shows the dispersion of the four
phonon bands of the in-plane motion along the $K^{\prime}$-$\Gamma$-$K$ lines
[260]. The motion of the $A$ and $B$ atoms in each band at $K$ and
$K^{\prime}$ valleys are illustrated in the insets. Here bands 2 and 3 at the
${\bf K}$-valley has pseudoangular momentum $+1$ and $-1$, respectively, with
opposite sign at the ${\bf K}^{\prime}$-valley, as shown in Fig. 24(b). There
are more complicated cases in three dimensions and other symmetry groups
[264].

Table 3: Character table for $C_{3h}$ group [263]. $\omega=e^{2\pi i/3}$. | $e$ | $C_{3}$ | $C^{2}_{3}$ | $\sigma_{h}$ | $S_{3}$ | $\sigma_{h}C^{2}_{3}$ |  basis function
---|---|---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 |  $x^{2}+y^{2}$;$\sigma_{z}$;$(x+iy)^{3}$;$(x-iy)^{3}$
$A_{2}$ | 1 | 1 | 1 | -1 | -1 | -1 |
$A_{3}$ | 1 | $\omega$ | $\omega^{*}$ | 1 | $\omega$ | $\omega^{*}$ |  $x+iy$;$(x-iy)^{2}$
$A_{4}$ | 1 | $\omega^{*}$ | $\omega$ | 1 | $\omega^{*}$ | $\omega$ |  $x-iy$;$(x+iy)^{2}$
$A_{5}$ | 1 | $\omega$ | $\omega^{*}$ | -1 | -$\omega$ | -$\omega^{*}$ | $\sigma_{x}+i\sigma_{y}$
$A_{6}$ | 1 | $\omega^{*}$ | $\omega$ | -1 | -$\omega^{*}$ | -$\omega$ | $\sigma_{x}-i\sigma_{y}$
Figure 24: Dispersion relation [(a)] of phonon modes with in-plane dynamics
and their pseudoangular momentum [(b)] at the $K$ and $K^{\prime}$ valleys of
a two-dimensional honeycomb lattice of monolayer transition-metal
dichalcogenides. The insets in (a) show the motions of the $A$ and $B$ atoms
in every band at $K$ and $K^{\prime}$ valleys. These figures are taken from
Ref. [260].

####  3.4.2 Surface acoustic waves

The phonon modes localized at surfaces and interfaces have a lot in common
with the surface plasmons discussed above (Sec. 3.3.1), but with
idiosyncrasies that induce subtle differences as well. Surface acoustic waves
(SAWs) are evanescent normal to the plane and propagating in the plane, with
chirality as discussed below.

The deformation vector field u(r) in a homogeneous and isotropic elastic
continuum obeys the equation of motion [16, 15]

$\rho\frac{\partial^{2}u_{i}}{\partial
t^{2}}=\sum_{j}\frac{\partial\sigma_{ij}}{\partial x_{j}},$ (137)

where $\rho$ is the mass density and $\sigma_{ij}$ the stress tensor. The
boundary at the surface or interface $S$ to air or vacuum is free, i.e.,

$\displaystyle\sigma_{ij}|_{S}=0.$ (138)

For small deformations Hooke’s Law holds:

$\displaystyle\sigma_{ij}=\sum_{kl}C_{ijkl}\epsilon_{kl},$ (139)

with stiffness coefficient $C_{ijkl}$ and strain tensor
$\epsilon_{ij}=({\partial u_{i}}/{\partial x_{j}}+{\partial u_{j}}/{\partial
x_{i}})/2$ [16]. For the surface acoustic waves propagating along
$\hat{\textbf{x}}$ in Fig. 25, we have to consider the elements $u_{x}(x,z)$
and $u_{z}(x,z)$ that do not depend on $y$.

Figure 25: Rayleigh SAWs with opposite momentum propagate and rotate in
opposite directions, as indicated by the green and black arrows, respectively.

The non-zero strain tensor elements are $\epsilon_{xx}$, $\epsilon_{zz}$ and
$\epsilon_{xz}$, and Hooke’s Law can be simplified to

$\displaystyle\left(\begin{matrix}\sigma_{xx}\\\ \sigma_{zz}\\\
\sigma_{xz}\end{matrix}\right)=\left(\begin{matrix}\lambda+2\mu&\lambda&0\\\
\lambda&\lambda+2\mu&0\\\
0&0&2\mu\end{matrix}\right)\left(\begin{matrix}\epsilon_{xx}\\\
\epsilon_{zz}\\\ \epsilon_{xz}\end{matrix}\right),$ (140)

where $\mu$ and $\lambda$ are the elastic Lamé force constants, The
displacement field can be expressed in terms of the elastic scalar and vector
potentials $V$ and $\mathbf{A}$ as $\boldsymbol{u}=\nabla
V+\nabla\times\mathbf{A}$, with $\mathbf{A}=A\hat{\textbf{y}}$ because the
displacement field of an acoustic wave propagating in the $\hat{\textbf{x}}$
direction does not depend on $y$. With $u_{x}=\partial V/\partial x-\partial
A/\partial z$ and $u_{z}=\partial V/\partial z+\partial A/\partial x$ we
arrive at the wave equations

$\displaystyle\rho\frac{\partial^{2}V}{\partial
t^{2}}=(\lambda+2\mu)\left(\frac{\partial^{2}V}{\partial
x^{2}}+\frac{\partial^{2}V}{\partial z^{2}}\right),$ (141a)
$\displaystyle\rho\frac{\partial^{2}A}{\partial
t^{2}}=\mu\left(\frac{\partial^{2}A}{\partial
x^{2}}+\frac{\partial^{2}A}{\partial z^{2}}\right).$ (141b)

The evanescent plane wave ansatz

$\displaystyle V({\bf r},t)$
$\displaystyle=V_{0}e^{-q|z|}e^{i(kx-\omega_{k}t)},$ (142a) $\displaystyle
A({\bf r},t)$ $\displaystyle=A_{0}e^{-s|z|}e^{i(kx-\omega_{k}t)},$ (142b)

solves the wave equations of a surface at $z=0$, where $\\{V_{0},A_{0}\\}$ are
the wave amplitudes, $q\equiv\sqrt{k^{2}-k_{l}^{2}}$,
$s=\sqrt{k^{2}-k_{t}^{2}}$, and $k_{l}=\omega_{k}\sqrt{\rho/(\lambda+2\mu)}$
and $k_{t}=\omega_{k}\sqrt{\rho/\mu}$ are the wave vectors of the longitudinal
and transverse bulk waves, respectively. The surface acoustic waves exist only
when $k>\\{k_{l},k_{t}\\}$. With boundary condition of vanishing stress along
the surface normal $\hat{\textbf{z}}$, i.e., $\sigma_{zz}|_{z=0}=0$ and
$\sigma_{xz}|_{z=0}=0$,

$\displaystyle\left(\begin{matrix}2\mu k^{2}-(\lambda+2\mu)k_{l}^{2}&-2i\mu
k\sqrt{k^{2}-k_{t}^{2}}\\\
2ik\sqrt{k^{2}-k_{l}^{2}}&-k_{t}^{2}+2k^{2}\end{matrix}\right)\left(\begin{matrix}V_{0}\\\
A_{0}\end{matrix}\right)=0,$ (143)

and hence,

$\displaystyle 4k^{2}qs-(k^{2}+s^{2})^{2}=0.$ (144)

With $\eta=k_{l}/k<1$ and $\xi=k_{l}/k_{t}=\sqrt{\mu/(\lambda+2\mu)}$, Eq.
(144) leads to the characteristic equation [15]

$\displaystyle\eta^{6}-8\eta^{4}+8(3-2\xi^{2})\eta^{2}+16(\xi^{2}-1)=0.$ (145)

Its roots are the eigenfrequencies

$\displaystyle\omega_{k}=\eta\sqrt{\frac{\mu}{\rho}}\left|k\right|=c_{r}|k|,$
(146)

with sound velocity $c_{r}$ that depends only on the Lamé constants. The
amplitudes of these “Rayleigh" surface acoustic waves are entirely transverse

$\begin{split}\psi_{x}&=ik\varphi_{k}\left(e^{qz}-\frac{2qs}{k^{2}+s^{2}}e^{sz}\right)e^{ikx},\\\
\psi_{z}&=q\varphi_{k}\left(e^{qz}-\frac{2k^{2}}{k^{2}+s^{2}}e^{sz}\right)e^{ikx},\end{split}$
(147)

where $\varphi_{k}$ is a normalization constant. The relative phase of the
displacement field ${\mathrm{Arg}}(\psi_{z}/\psi_{x})|_{z=0}=\pm i$ is
opposite for right- and left-propagating waves, which reflects the rotation-
momentum locking [15], as illustrated in Fig. 25. With a proper definition of
phonon spin density (Sec. 3.4.3) below, this implies the spin-momentum locking
of SAWs, which together with the surface normal is well characterized by the
chirality index defined in Eq. (12).

The quantized displacement field $(\hat{u}_{x},\,\hat{u}_{z})$ can be expanded
in terms of these eigenmodes $\boldsymbol{\psi}(k)$ and bosonic phonon
operators $\hat{b}_{k}(t)$ [265]

$\hat{\mathbf{u}}(x,z,t)=\sum_{k}\left[\boldsymbol{\psi}(x,z,k)\hat{b}_{k}(t)+\boldsymbol{\psi}^{\ast}(x,z,k)\hat{b}_{k}^{\dagger}(t)\right].$
(148)

We normalize the mode amplitudes $\boldsymbol{\psi}$

$\int_{-\infty}^{0}dz\left(|\psi_{x}|^{2}+|\psi_{z}|^{2}\right)=\frac{\hbar}{2\rho
L\omega_{k}},$ (149)

to recover the quantum harmonic oscillator form of the Hamiltonian for
Rayleigh SAWs

$\hat{H}_{\mathrm{e}}=\rho\int
d\boldsymbol{r}\,\dot{\hat{\boldsymbol{u}}}^{2}(x,z,t)=\sum_{k}\hbar\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}.$
(150)

The normalization factor

$\varphi_{k}=\frac{1}{2|k|}\sqrt{\frac{2\hbar}{\rho
Lc_{r}}}\left(\frac{1+a^{2}}{2a}+\frac{2a(a-2b)}{b(1+b^{2})}\right)^{-1/2},$
(151)

with dimensionless material constants
$a=q/\left|k\right|=\sqrt{1-(c_{r}/c_{l})^{2}}$ and
$b=s/\left|k\right|=\sqrt{1-\eta^{2}}$. Here
$c_{l}=\sqrt{(\lambda+2\mu)/\rho}$ is the sound velocity of the longitudinal
bulk waves.

####  3.4.3 Phonon orbital and spin angular momentum

The angular momentum of a photon and its separation into a photon spin and
orbital angular momentum is well established [266, 206]. In spite of the
similarities of the respective wave equations, there are differences that show
up when the phonon wavelength approaches either the lattice constant or the
size of the dielectric particle under consideration. The concept of the phonon
spin can be helpful in understanding microscopic processes such as the spin-
transfer processes between magnons, electrons, phonons, and rigid mechanical
rotation. The chirality implies the unidirectional spin current of phonons
(Sec. 4.4.3).

Figure 26: Orbital and spin angular momentum of phonons. (a) and (b)
distinguish the mechanical angular momentum of the rotation of a finite
particle that forms the atomic angular momentum in which the local deformation
field rotates around the equilibrium position. Panels (c) and (d) illustrate
the ambiguity of the phonon angular momentum for large bulk systems. Here the
square lattice sketches a homogeneous elastic continuum, while the red arrows
indicate a deformation field. Process (c) illustrates the invariance with
respect to a global rotation of the whole levitated particle, while the
invariance under rotations of the vector field [(d)] should hold in the
thermodynamic limit of the particle size. The figures are taken from Ref. [44]
[(a) and (b)] and Ref. [272] [(c) and (d)].

Vonsovskii and Svirskii [42] introduced the concept of phonon spin, while
Zhang and Niu [43] brought the concept to recent attention. The total angular
momentum of a homogeneous elastic body is defined by [43, 44]

$\displaystyle{\bf L}=\int d{\bf r}({\bf r}+{\bf u})\times{\bf p},$ (152)

to leading order in a small displacement ${\bf u}({\bf r},t)$ at a position
${\bf r}$ of the body at rest, and ${\bf p}({\bf r},t)=\rho\dot{\bf u}({\bf
r},t)$ is the momentum density. $\bf L$ is the sum of two contributions, as
shown in Fig. 26 (a) and (b) for the motion of media.

$\displaystyle{\bf L}^{(1)}=\int d{\bf r}\rho({\bf r}\times\dot{\bf u})$ (153)

is the angular momentum due to the rotation of the elastic medium relative to
the origin; while

$\displaystyle{\bf L}^{(2)}=\int d{\bf r}\rho({\bf u}\times\dot{\bf u})$ (154)

is the phonon spin, which is finite for circular shear displacements of volume
elements around their equilibrium positions. Accordingly, we can define
$\boldsymbol{l}_{\mathrm{DC}}\left({\bf
r}\right)=\rho\left\langle\mathbf{u}\times\mathbf{\dot{u}}\right\rangle_{t}$
as the time averaged mechanical angular momentum density. This phonon spin
density is constant for the SAWs discussed above [267, 268, 269, 75]. The spin
of water (surface) waves is defined by the water velocity in a similar way
[270]. The equation of motion for the displacement field [Eq. (137)] leads to
the conservation law of total phonon angular momentum ${\dot{\bf L}}={\dot{\bf
L}}^{(1)}+{\dot{\bf L}}^{(2)}=0$ only when including non-linear terms beyond
the harmonic oscillator Hamiltonian [44], which indicates the subtlety of the
concept.

Using Noether’s theorem, according to which the conservation of angular
momentum is a consequence of rotational symmetry, Nakane and Kohno derived two
inequivalent expressions for the phonon angular momentum [271]. The invariance
of a (very large or spherical) body under a rotation operator $\cal{R}$, i.e.,
${\bf u}({\bf r})+{\bf r}\rightarrow{\cal R}[{\bf u}({\bf r})+{\bf r}]$ as in
Fig. 26(c), leads to a phonon angular momentum via Eqs. (153) and (154). In
Figs. 26(c) and (d), the square lattice indicates the elastic continuum, while
the red arrow denotes a rotation of a pinned and free vector field. On the
other hand, when the medium is very large, the invariance of the phonon
displacement field under rotation, i.e., ${\bf u}({\bf r})\rightarrow{\cal
R}{\bf u}({\cal R}^{-1}{\bf r})$ as illustrated in Fig. 26(d) appears by the
natural symmetry. The phonon “orbital" and spin angular momentum then read

$\displaystyle\tilde{{\bf L}}^{(1)}$ $\displaystyle=\sum_{\alpha=x,y,z}\int
d{\bf r}\rho\dot{u}_{\alpha}{\bf r}\times(-\nabla)u_{\alpha},$ (155a)
$\displaystyle\tilde{\bf L}^{(2)}$ $\displaystyle=\int d{\bf r}\rho({\bf
u}\times\dot{\bf u}).$ (155b)

Streib [272] referred to the two definitions as the (real) angular momentum
[Eqs. (153) and (154)] and the pseudo-angular momentum [Eqs. (155a) and
(155b)] by the analogy with linear momentum in free space and pseudo-momentum
in crystals. Both definitions lead to the same expression for the phonon spin
[Eqs. (154) and (155b)] but different angular momentum conservation laws. When
considering an embedded spin such as a magnetic moment in an elastic medium
[271], the spin transfer should be invariant to a combined rotation of spin
directions and phonon fields as in Fig. 26(c), i.e., pseudo-angular momentum
should be conserved. On the other hand, when taking into account the angular
momentum transfer of the intrinsic spins of magnons, photons, phonons, and the
global lattice rotation [273], the conserved quantity is the real rather than
the pseudo-angular momentum. The relaxation of magnetization dynamics in a
small particle, therefore, leads to a rigid rotation of the entire particle
[273]. Chaplain et al. demonstrated that the flexural guided elastic waves in
elastic pipes carry a well-defined orbital angular momentum that enables the
transfer of elastic orbital angular momentum to a surrounding fluid in contact
with the pipe [274].

###  3.5 Universal features of chirality of evanescent waves

After reviewing various chiral waves in the condensed matter as summarized in
the left panel of Table 2, we here address some unifying features in terms of
fixed handedness of three vectors as defined in the introduction 1. This
unification is achieved in terms of the transverse spin ${\bf S}_{T}$ (its
direction is denoted by $\hat{\boldsymbol{\sigma}}$), the surface normal ${\bf
n}$ of the propagation plane of interest, and the momentum ${\bf k}$ in this
plane of evanescent waves, which defines the chirality index $Z=\hat{\bf
n}\cdot(\hat{\boldsymbol{\sigma}}\times\hat{\bf k})$, where the hat indicates
unit vectors. We demonstrate the following “theorem":

  * •

Chirality index $Z=\hat{\bf n}\cdot(\hat{\boldsymbol{\sigma}}\times\hat{\bf
k})$ is fixed for the evanescent waves;

  * •

In the presence of time-reversal symmetry, their transverse spin is locked to
the momentum with fixed ${\bf n}$, i.e., the spin-momentum locking, leading to
an opposite spin of opposite propagation;

  * •

The breaking of time-reversal symmetry leads to the non-reciprocal and even
unidirectional propagation of waves with fixed ${\bf n}$, i.e., the spatial
chirality. The wave propagation is reversed with opposite ${\bf n}$.

####  3.5.1 The zoo of chiral evanescent waves

Before showing the unifying features, we first address the diverse description
of the chirality of these waves in the above sections (Sec. 3.1 to 3.4).

  * •

In Sec. 3.1.1, when the time-reversal symmetry is broken by the magnetization,
the propagation direction ${\hat{\bf k}}$ of Damon-Eshbach (DE) modes at the
surface of a ferromagnet is determined by the cross product of the surface
normal ${\bf n}$ and the magnetization ${\bf M}_{s}$ [Fig. 14(a)]. The
chirality of the DE modes in the magnetic films can be understood in terms of
the self-interaction of the magnetic dipoles that interact with the magnetic
charges generated by the spin waves themselves.

  * •

In Sec. 3.2.1, the stripline magnetic field with the surface normal ${\bf n}$
that is perpendicular to the plane of propagation of interest as in Fig. 18,
is polarized with a transverse spin ${\bf S}_{T}$ along the cross product of
its momentum ${\bf k}$ and the surface normal.

  * •

In Sec. 3.3.2, the electric field emitted by the linearly polarized electric
dipoles along a surface normal shares a similar chirality to that of the
stripline magnetic field by being spin-momentum locking at a plane with the
fixed surface normal ${\bf n}$.

  * •

In Sec. 3.3.2, when the time-reversal symmetry is broken with circularly
polarized dipoles, the electric field emitted by it is unidirectional, similar
to that of DE modes in the magnetism.

  * •

In Sec. 3.2.2, the Kittel mode of a magnetic nanowire causes a rotating two-
dimensional magnetic dipole with in general elliptical polarization dynamics.
When being linearly polarized, the spin of its dipolar field shares the same
chirality with the electric counterpart by being spin-momentum locking.

  * •

In Sec. 3.2.2, while when being circularly polarized for the Kittel mode of a
magnetic nanowire, the time-reversal symmetry is broken that renders the
generated dipolar field to be unidirectional in propagation, similar to the
electric stray field by the circularly-polarized electric-dipole wires.

  * •

In Sec. 3.2.2, in sufficiently thin films, spin waves are uniform across the
film and therefore not chiral anymore, but their stray magnetic fields are.
The dipolar field of these spin waves propagating normal to the magnetization
in a direction ${\bf k}$ that is governed by the cross product of the surface
normal ${\bf n}$ and the in-plane magnetization ${\bf M}_{s}$ [Fig. 19(b)];
the spin of the dipolar field is also locked by the momentum and surface
normal.

  * •

In Sec. 3.2.3, for the chirality of a waveguide magnetic field on special
lines or planes, we can also identify a surface normal by which locking
effects between polarization and propagation rotation (spin) direction can be
formulated in terms of vector products, as shown in Fig. 21(a), with similar
physics for the electric field in optical fibers.

  * •

In Sec. 3.3.1, the electric field of the surface plasmon polariton is
circularly polarized, with polarization rotation direction determined by the
chiral relation with the surface normal and the propagation direction [refer
to Fig. 22(a)].

  * •

In Sec. 3.4.2, the surface acoustic waves are evanescent and hold chirality in
that their polarization, momentum, and surface normal direction obey the
handed rule (refer to Fig. 25).

Blow we unify these features with a strict definition of the spin density of
these waves (Sec. 3.5.2) and highlight the role of evanescence for the
chirality (Sec. 3.5.3), with the unifying features summarized in Tables 4 and
5, which will be explained in detail in the following.

Table 4: Universal origin of typical chiral waves exploited in Spintronics. Chiral Waves | Wave Equations | Spin $vs$ Momentum $vs$ Surface
---|---|---
Damon-Eshbach spin waves $\hskip 51.21504pt\boxed{\nabla\cdot{\bf M}\rightarrow 0~{}{\rm for}~{}{\rm large}~{}k}$ |  $\displaystyle\partial{\bf M}/\partial t=-\gamma\mu_{0}{\bf M}\times{\bf H}_{\rm eff}$ $\displaystyle H^{\beta}_{\rm eff}({\bf r})=H_{z}\hat{\bf z}\delta_{\beta z}+\frac{1}{4\pi}\partial_{\beta}\sum_{\alpha}\partial_{\alpha}\int\frac{M_{\alpha}({\bf r}^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}d{\bf r}^{\prime}$ |
Dipolar field of spin waves of film $\hskip 48.36958pt\boxed{\nabla\cdot{\bf h}=0~{}{\rm outside}~{}{\rm the}~{}{\rm magnet}}$ |  $\displaystyle~{}~{}~{}~{}h_{\beta}({\bf r})=\frac{1}{4\pi}\partial_{\beta}\sum_{\alpha}\partial_{\alpha}\int\frac{M_{\alpha}({\bf r}^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}d{\bf r}^{\prime}$ |
Dipolar field of spin waves of nanowire $\hskip 48.36958pt\boxed{\nabla\cdot{\bf h}=0~{}{\rm outside}~{}{\rm the}~{}{\rm magnet}}$ |  $\displaystyle~{}~{}~{}~{}h_{\beta}({\bf r})=\frac{1}{4\pi}\partial_{\beta}\sum_{\alpha}\partial_{\alpha}\int\frac{M_{\alpha}({\bf r}^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}d{\bf r}^{\prime}$ |
Stripline magnetic field $\hskip 48.36958pt\boxed{\nabla\cdot{\bf H}=0~{}{\rm outside}~{}{\rm the}~{}{\rm magnet}}$ |  $\displaystyle{A}_{z}({\bf r},\omega)=\frac{\mu_{0}}{4\pi}\int d{\bf r}^{\prime}{J}_{z}({\bf r}^{\prime},\omega)\frac{e^{ik|{\bf r}-{\bf r}^{\prime}|}}{|{\bf r}-{\bf r}^{\prime}|}$ $\displaystyle{\bf H}=(1/\mu_{0})\nabla\times{\bf A}$ |
Electric field of two-dimensional, linearly polarized electric dipoles $\hskip 48.36958pt\boxed{\nabla\cdot{\bf H}=0~{}{\rm and}~{}\nabla\cdot{\bf E}=0}$ |  $\displaystyle{\bf A}({\bf r},t)=\frac{\mu_{0}}{4\pi}\int d{\bf r}^{\prime}\frac{-i\omega{\bf p}\delta(x^{\prime},z^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}e^{ik|{\bf r}-{\bf r}^{\prime}|}$ $\displaystyle{\bf H}=(1/\mu_{0})\nabla\times{\bf A}$ $\displaystyle{\bf E}=\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\nabla\times{\bf H}$ |
Electric field of two-dimensional, circularly polarized electric dipoles $\hskip 48.36958pt\boxed{\nabla\cdot{\bf H}=0~{}{\rm and}~{}\nabla\cdot{\bf E}=0}$ |  $\displaystyle{\bf A}({\bf r},t)=\frac{\mu_{0}}{4\pi}\int d{\bf r}^{\prime}\frac{-i\omega{\bf p}\delta(x^{\prime},z^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}e^{ik|{\bf r}-{\bf r}^{\prime}|}$ $\displaystyle{\bf H}=(1/\mu_{0})\nabla\times{\bf A}$ $\displaystyle{\bf E}=\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\nabla\times{\bf H}$ |
Table 5: Universal origin of typical chiral waves exploited in Spintronics (continued). Waveguide and cavity microwaves $\hskip 79.6678pt\boxed{\nabla\cdot{\bf H}=0}$ |  $\displaystyle\left(\nabla^{2}-({1}/{c^{2}}){\partial^{2}}/{\partial t^{2}}\right){\bf E}=0$ $\displaystyle{\bf n}\times{\bf E}|_{S}=0$ $\displaystyle{\bf H}=-{i}/({\mu_{0}\omega})\nabla\times{\bf E}$ $\displaystyle{\rm\textit{e.g.},}~{}E_{z}\propto\cos(p\pi z/b)$ |
---|---|---
Surface plasmon polaritons $\hskip 79.6678pt\boxed{\nabla\cdot{\bf E}=0}$ |  $\displaystyle\partial^{2}H_{z}/\partial y^{2}+(\varepsilon_{r}k_{0}^{2}-k_{x}^{2})H_{z}=0$ $\displaystyle H_{z}\propto e^{ik_{x}x}e^{-\sqrt{k_{x}^{2}-\varepsilon_{r}k_{0}^{2}}y}$ $\displaystyle{\bf E}=\frac{i}{\omega\varepsilon_{0}\varepsilon_{r}}\nabla\times{\bf H}$ |
Surface acoustic waves $\hskip 48.36958pt\boxed{\nabla\cdot{\bf u}_{T}=0,~{}{\rm where}~{}{\bf u}_{T}=\nabla\times{\bf A}}$ |  $\displaystyle\rho\frac{\partial^{2}V}{\partial t^{2}}=(\lambda+2\mu)\left(\frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial z^{2}}\right)$ $\displaystyle\rho\frac{\partial^{2}{\bf A}}{\partial t^{2}}=\lambda\left(\frac{\partial^{2}{\bf A}}{\partial x^{2}}+\frac{\partial^{2}{\bf A}}{\partial z^{2}}\right)$ $\displaystyle{\bf u}=\nabla V+\nabla\times{\bf A}$ $\displaystyle{\bf A}\propto e^{-s|z|}e^{i(kx-\omega_{k}t)}$ |

####  3.5.2 Spin density of classical waves

To unify the chirality of these waves, including the magnetization ${\bf M}$,
magnetic field ${\bf H}$, electric field ${\bf E}$, and displacement field
${\bf u}$, we may employ the generalization of spin concept from the
particles, of relativistic origin, to waves [21], of dipolar origin. The
electric polarization ${\bf P}$ may also be contained in this generalization.

The spin operator $\hat{\bf S}$ has elements
$(\hat{S}_{j})_{ab}=-i\epsilon_{abj}$ for the spin-1 representation, which in
matrix form reads

$\displaystyle\hat{S}_{x}=-i\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&-1&0\end{array}\right),~{}~{}~{}~{}~{}\hat{S}_{y}=-i\left(\begin{array}[]{ccc}0&0&-1\\\
0&0&0\\\
1&0&0\end{array}\right),~{}~{}~{}~{}~{}\hat{S}_{z}=-i\left(\begin{array}[]{ccc}0&1&0\\\
-1&0&0\\\ 0&0&0\end{array}\right).$ (165)

The definition of spin density of waves is given for the monochromatic waves,
which for our purpose is sufficient to identify the chirality. This
definition, however, cannot be simply applied to wavepacket. For the
electromagnetic waves propagating in the dispersive medium, the spin density
is defined as the average value between
$\langle\tilde{{\boldsymbol{\psi}}}|=(\tilde{\varepsilon}{\bf
E}^{*},-i\tilde{\mu}{\bf H}^{*})/2$ and $|{\boldsymbol{\psi}}\rangle=({\bf
E},i{\bf H})^{T}/2$, i.e. [21, 40, 41],

$\displaystyle{\bf S}_{\rm EM}=\langle\tilde{{\boldsymbol{\psi}}}|\hat{\bf
S}|{\boldsymbol{\psi}}\rangle=(1/4)\tilde{\varepsilon}{\bf E}^{*}\hat{S}{\bf
E}+(1/4)\tilde{\mu}{\bf H}^{*}\hat{S}{\bf H}=(1/4)\tilde{\varepsilon}{\rm
Im}({\bf E}^{*}\times{\bf E})+(1/4)\tilde{\mu}{\rm Im}({\bf H}^{*}\times{\bf
H}).$ (166)

Here $\tilde{\varepsilon}=\partial[\omega\varepsilon]/\partial\omega$ is the
group permittivity and $\tilde{\mu}=\partial[\omega\mu]/\partial\omega$ is the
group permeability [41]. Because these are defined for density, spatial
variation is allowed which is convenient for addressing the physical process
that happens locally.

Making an analogy to this definition with keeping the same unit, we may define
the spin of magnetization (and electric polarization) of the monochromatic
oscillation as

$\displaystyle{\bf S}_{\rm M}$ $\displaystyle=(1/4)\mu_{r}{\mu}_{0}{\bf
M}^{*}\hat{S}{\bf M}=(1/4)\tilde{\mu}{\rm Im}({\bf M}^{*}\times{\bf M}),$
(167)

a quantity not defined before, which, however, can auxiliarily help to unify
the description of this review article.

Similarly, in terms of $|{\bf u}\rangle=\sqrt{\rho\omega/2}{\bf u}$ and
$\langle{\bf u}|=\sqrt{\rho\omega/2}{\bf u}^{*}$, the spin angular momentum
density is defined for the displacement field of phonon as [267, 268, 269, 75]

$\displaystyle{\bf S}_{\bf u}=\langle{\bf u}|\hat{\bf S}|{\bf
u}\rangle=(\rho\omega/2){\rm Im}({\bf u}^{*}\times{\bf u}).$ (168)

The phonon displacement field of surface acoustic waves contains the
transverse component ${\bf u}_{T}=\nabla\times{\bf A}$ and longitudinal
component ${\bf u}_{L}=\nabla V$, respectively (refer to Sec. 3.4.2).

The spin (density) of waves is longitudinal when it is parallel to the wave
vector. The longitudinal spin ${\bf S}_{L}$ is also known as the helicity of
waves, as discussed in the introduction (Sec. 1.1). While it is transverse
when it is normal to the propagation direction. For a comparison, Figs. 27(a)
and (b) define waves with longitudinal ${\bf S}_{L}$ and transverse ${\bf
S}_{T}$ polarizations, respectively, according to their direction relative to
the momentum (thick red arrow).

Figure 27: Chirality of evanescent waves. (a) and (b) are a comparison between
the longitudinal and transverse spins of evanescent waves. The longitudinal
spin ${\bf S}_{L}$ in (a) is parallel or anti-parallel to the wave propagation
direction, implied by the red thick arrow, while the transverse spin ${\bf
S}_{T}$ in (b) is perpendicular to it. (c) and (d) are two key chiral
relations that lead to the chiral interactions and excitations in spintronics,
where (c) retains the time-reversal symmetry, while (d) breaks the time-
reversal symmetry. (c) describes the spin-momentum locking of waves that often
acts as sources for chiral excitation, while (d) describes the surface-
momentum locking of waves that can both act as a chiral medium and chiral
sources in spintronics devices.

####  3.5.3 Transverse spin and origin of chirality

We recall from Helmholtz’s theorem, also known as the fundamental theorem of
vector calculus, that any sufficiently smooth, rapidly decaying vector field
in three dimensions can be resolved into the sum of a curl-free vector field
and a divergence-free vector field. Here, as listed in the left panel of
Tables 4 and 5, we are allowed to focus on the divergence-free vector field
because the waves we are interested in turn out to be either exactly or
approximately divergence-free (e.g., $\nabla\cdot{\bf M}\approx 0$,
$\nabla\cdot{\bf H}=0$, $\nabla\cdot{\bf E}=0$, and $\nabla\cdot{\bf
u}_{T}=0$).

We now demonstrate that for this kind of elliptically polarized evanescent
waves, the spin density is transverse, viz., the spin direction is
perpendicular to the propagation direction; they contain a locking between the
transverse spin and momentum when fixing a surface normal of the propagation
plane. We consider generally a monochromatic wave of vector field propagating
along the $\hat{\bf y}$-direction with wave number $k_{y}$, decaying along the
surface normal $\hat{\bf x}$-direction by rate $\kappa$, and is homogeneous
along the $\hat{\bf z}$-direction, i.e.,

$\displaystyle{\bf C}({\bf r},t)=\left(C_{x}\hat{\bf x}+C_{y}\hat{\bf
y}+C_{z}\hat{\bf z}\right)e^{ik_{y}y-\kappa x-i\omega t},$ (169)

where $C_{x,y,z}$ are amplitudes and we do not require decaying
$\kappa=|k_{y}|$. The divergence-free condition $\nabla\cdot{\bf C}=0$ renders
that $C_{x}=(ik_{y}/\kappa)C_{y}$ and an arbitrary $C_{z}$. We are interested
in those fields with $C_{z}=0$ and we obtain ${\bf
C}=C_{y}(ik_{y}/\kappa,1,0)^{T}e^{ik_{y}y-\kappa x}$ and the spin angular
momentum density [refer to above Sec. 3.5.2]

$\displaystyle{\bf S}_{T}\propto{\rm Im}({\bf C}^{*}\times{\bf
C})=-2(k_{y}/\kappa)|C_{y}|^{2}e^{-2\kappa x}\hat{\bf z},$ (170)

which is time independent, decays along the surface normal with the rate
$2\kappa$, perpendicular to the propagation direction (transverse), and, more
importantly, is locked to its momentum $k_{y}$. This spin is known as
transverse spin. This analysis demonstrates, quite generally, that an
elliptical, evanescent, and divergence-free vector field contains a transverse
spin that is locked to the momentum when propagating within a particular plane
fixed by a surface normal ${\bf n}$. Reversing the direction of the surface
normal ${\bf n}$ retains this relation. So the chirality is well captured by
the chirality index Eq. (12) defined in this review article.

These monochromatic waves can be real evanescent waves of particular momentum
or the Fourier components of decaying electric and magnetic dipolar fields
that we decompose with Weyl or Coulomb identities [Eqs. (30) and (82)]. So
there exists not a few but many such waves that are governed by various wave
equations, such as those listed in the middle panel of Tables 4 and 5. The
locking between the transverse spin, momentum and the surface normal of the
propagation plane of various chiral waves are depicted in the right panel of
Tables 4 and 5. This locking may be treated as a generalization of the locking
of spin and momentum of electrons where the relativistic effect overrules, or
generalized spin-orbit interaction, a new resource that inspires spintronics
functionality and devices. The chiralities in the propagation as summarized
above all hold for near-field evanescent wave fields with polarization
perpendicular to the wave vector and surfaces or interfaces. We do not find
chirality in waves that are longitudinal in spins or fully propagating. For
Rashba-like spin-orbit interaction of surface states of a topological
insulator [26, 27], the direction of the spin is also perpendicular to the
momentum, so maybe also a transverse spin that is well characterized by the
chirality index. They propagate in one direction at a given boundary and are
immune to (weak) backscattering.

With a fixed maximal chirality index $Z=+1$ (or $-1$), there are two
categories for the wave propagation as classified in Figs. 27(c) and (d),
respectively. When there exists time-reversal symmetry, the transverse spin is
allowed to be reversed. So with a fixed surface normal ${\bf n}$ for the wave
propagation, the propagation is bidirectional, but the transverse spin is
reversed with the opposite wave vector, as shown in Fig. 27(c). Many examples
in Tables 4 and 5 belong to this category. On the other hand, additional
features arise when the sources of waves break the time-reversal symmetry.
Particularly, interesting phenomena appear when the sources become circularly
polarized such as magnetic and electric dipoles, in which the transverse spin
is fixed by the source. Accordingly, the direction of the wave vector is fixed
as well by the restriction of the chirality index, so being unidirectional, as
shown in Fig. 27(d). Sometimes, the sources hold both spin components but of
different amplitudes, in which case the propagation of their evanescent waves
is non-reciprocal with different amplitudes of opposite directions. Examples
vary from the stray fields of elliptical magnetic dipoles to that of the
elliptical electron dipoles as listed in Tables 4 and 5.

The chirality index thereby provides an intuitive explanation of the
unidirectionality of well-known examples. There are unidirectional waves
propagating within one plane such as the Damon-Eshbach spin waves, magnetic
stray fields generated by circularly polarized spin waves, and electric
dipolar fields generated by the circularly polarized electric dipoles. The
broken time-reversal symmetry forces that when close to one surface normal
there is no counter-propagating modes that otherwise break the transverse-
spin–momentum locking effect. The counter-propagating modes are near the
opposite surface or plane that renders a momentum-surface locking for waves.
The transverse spins of these waves close to the opposite surface are the same
since fixed by the equilibrium magnetic or electric dipoles. Figure 27(d)
depicts that for the same transverse spin at opposite surfaces, the
propagation direction must be opposite.

####  3.5.4 An example

We may intuitively associate the “spin density" of classical waves with a
vector field that locally rotates around a fixed axis. This can be the
magnetization vector of spin waves that precess around the saturation
magnetization in a ferromagnet or circularly polarized electric, magnetic, or
displacement fields. When divergence free and evanescent, the precession axis,
or spin, of polarization vectors of these waves are transverse to their
propagation direction. In the previous sections we reviewed this concept for
several types of classical waves and its appeal for spintronics. Here we
illustrate nature of the classical spin density by explicitly calculating the
transverse spin of a particular chiral wave, i.e., the magnetic stray field
emitted by the ferromagnetic magnetic resonance (Kittel) mode of a magnetic
nanowire [85], as sketched in Fig. 28. We also refer to the review of the
theoretical formulation and experimental evidence of chirality in Chapter 5.

Figure 28: The transverse spin of the magnetic stray field emitted by a
magnetization (Kittel mode) of a magnetic nanowire that precesses in frequency
$\omega$ and an anti-clockwise direction around its equilibrium $\hat{\bf
y}$-direction. The stray field precesses at the same frequency $\omega$ but in
opposite direction, as illustrated by the black arrow for the magnetization
precession and the green arrow for the stray-field rotation.

Here we analyze in detail the transverse angular momentum or “spin” [85] of
the evanescent magnetodipolar field emitted by an excited magnetic nanowire
(Sec. 3.2.2) with thickness $d$, width $w$, and equilibrium magnetization
$\mathbf{M}_{s}$ along the wire $\hat{\bf y}$-direction (see Fig. 28). The
magnetization and magnetic stray field of the fundamental mode precess at the
same frequency $\omega$ but in opposite direction. For later convenience (Sec.
5.1), we adopt a quantum mechanical notation, but in the classical limit
operators can be simply replaced by the field amplitudes.

The magnetization dynamics of the wire as expressed by the spin operator
$\hat{\mathbf{S}}(\mathbf{r},t)$ generates a magnetic stray field that obeys
Coulomb’s Law [208, 206],

$\mathbf{h}_{\beta}(\mathbf{r},t)=-\frac{\gamma\hbar}{4\pi}\partial_{\beta}\partial_{\alpha}\int
d\mathbf{r}^{\prime}\frac{\langle\hat{\mathbf{S}}_{\alpha}(\mathbf{r}^{\prime},t)\rangle}{|\mathbf{r}-\mathbf{r}^{\prime}|},$
(171)

where we adopted the summation convention over repeated indices
$\\{\alpha,\beta\\}=\\{x,y,z\\}$ and $-\gamma$ is the gyromagnetic ratio. The
magnon modes in the wire move with wave number $k_{y}$ along the wire
$\hat{\bf y}$-direction. For sufficiently weak excitation the spin operator
can be expanded into magnon field operators $\hat{\alpha}_{k_{y}}$ with
amplitudes $m_{x,z}^{k_{y}}(x,z)$ [refer to Eq. (22a) for magnetic films]:

$\hat{\mathbf{S}}_{x,z}(\mathbf{r})=\sqrt{2S}\sum_{k_{y}}\left(m_{x,z}^{k_{y}}(x,z)e^{ik_{y}y}\hat{\alpha}_{k_{y}}+\mathrm{H.c.}\right),$
(172)

where $S=M_{s}/(\gamma\hbar)$ in terms of the saturation magnetization
$M_{s}$. The static stray field vanishes for long wires. The dynamic stray
field $\mathbf{h}$ is a response to $\langle\hat{\alpha}_{k_{y}}\rangle$, the
coherent amplitude of magnons with momentum $k_{y}\hat{\mathbf{y}}$ that are
excited by external microwaves with frequency $\omega=\omega_{k_{y}}$. We are
interested in the Fourier components $\mathbf{h}_{\beta}(z,\mathbf{k},t)$ of
the magnetic stray field Eq. (171) with momenta $\mathbf{k}=k_{x}\hat{\bf
x}+k_{y}\hat{\bf y}$ that represent chiral waves propagating in the $x$-$y$
plane. At frequency $\omega$

$\displaystyle\mathbf{h}_{\beta}(z,\mathbf{k},t)=\tilde{\mathbf{h}}_{\beta}(z,\mathbf{k})e^{-i\omega
t}+\tilde{\mathbf{h}}_{\beta}^{\ast}(z,-\mathbf{k})e^{i\omega t}$ (173)

is evanescent, i.e. decays exponentially with distance from the $x$-$y$ plane.
Below the nanowire ($z<0$) [50, 48] Eq. (171) leads to

$\displaystyle\left(\begin{array}[c]{c}\tilde{\mathbf{h}}_{x}(z,\mathbf{k})\\\
\tilde{\mathbf{h}}_{y}(z,\mathbf{k})\\\
\tilde{\mathbf{h}}_{z}(z,\mathbf{k})\end{array}\right)$
$\displaystyle=F_{\mathbf{k}}\left(m_{z}^{k_{y}}+\frac{ik_{x}}{k}m_{x}^{k_{y}}\right)\left(\begin{array}[c]{c}ik_{x}/k\\\
ik_{y}/k\\\
1\end{array}\right)e^{kz}\left|\left\langle\hat{\alpha}_{k_{y}}\right\rangle\right|,$
(180)

where $F_{\mathbf{k}}=-\gamma\hbar\sqrt{2S}(1-e^{-kd})\sin(k_{x}w/2)/k_{x}$ is
the form factor of the rectangular wire, which decays exponentially $\sim
e^{-k|z|}$ on a scale governed by the complex momentum
$k_{x}\hat{\mathbf{x}}+k_{y}\hat{\mathbf{y}}-ik\hat{\mathbf{z}}$ [206].

Spatially homogeneous microwaves excite the Kittel magnon [275] with $k_{y}=0$
by ferromagnetic resonance (FMR). Then, according to Eq. (180),
$\tilde{h}_{y}(\mathbf{k})$ vanishes and
$\tilde{h}_{x}(\mathbf{k})=i\mathrm{sgn}(k_{x})\tilde{h}_{z}(\mathbf{k})$. The
polarization of the Kittel mode is governed by the shape anisotropy and
applied magnetic field. In nanowires with circular/square cross sections or
sufficiently large magnetic field the spin waves are circularly polarized with
$m_{x}\rightarrow im_{z}$ (refer to Sec. 4.2.1 below for details). The Fourier
components $\tilde{\mathbf{h}}(\mathbf{k})=0$ for $k_{x}>0$, i.e. the stray
field is unidirectional.

Electromagnetic fields in vacuum are divergence-free, so when $k_{y}=0$ Eq.
(180) tells us that
$ik_{x}\tilde{h}_{x}(z,k_{x})+|k_{x}|\tilde{h}_{z}(z,k_{x})=0$. The photon
spin density is at GHz frequency strongly dominated by the magnetic component.
Under the nanowire in vacuum Eq. (166) leads to [21, 40, 41]:

$\displaystyle{\boldsymbol{\mathcal{D}}}(z,k_{x})$
$\displaystyle=\mu_{0}\mathrm{\mathrm{\operatorname{Im}}}\left[\tilde{\mathbf{h}}^{\ast}(z,k_{x})\times\tilde{\mathbf{h}}(z,k_{x})\right]/(4\omega)$
$\displaystyle=\mu_{0}{\rm
Im}\left[\tilde{h}_{x}(z,k_{x})\tilde{h}_{z}^{*}(z,k_{x})\right]/(2\omega)\hat{\bf
y}$ (181)

is purely transverse since ${\boldsymbol{\cal D}}\cdot\mathbf{h}=0$ and normal
to the propagation $x$-$y$ plane ${\boldsymbol{\cal D}}\cdot\hat{\bf z}=0$.
The vector $\hat{\boldsymbol{\sigma}}$ in the chirality index $Z=\hat{\bf
n}\cdot(\hat{\boldsymbol{\sigma}}\times\hat{\bf k})$ defined in the beginning
of this review is here the unit vector in the direction of the spin density
${\boldsymbol{\cal D}}$.

The photon magnetic field may couple to the electron spin by the Zeeman
interaction non-locally over distances limited by the evanescent decay length
[85] (refer to Sec. 5.1). This can be compared with conventional spin pumping
by interface exchange, which is a contact interaction.

Spintronics is concerned with the information carried by the spin degree of
freedom either by electron particle transport or by spin waves. Here we showed
that spin dynamics generates chiral magnetic stray fields that transfer and
share this information with other waves or quasi-particles in a chiral manner.
Section 4 reviews the associated dynamical chiral interaction (Fig. 13).

##  4 Dynamic chiral interactions: unidirectional transport with transverse
spins

As reviewed in Sec. 2 and 3, the physical origin of the chirality of given
states either comes from the ground-state Hamiltonian that contains strong
chiral interactions, such as the dipolar and relativistic spin-orbit coupling
or is captured by the chirality of the excitations (Fig. 13). For the latter,
the interaction between the excitations or quasiparticles is usually weak and
can be treated by perturbation theory. Nevertheless, the interaction can be
chiral that is governed by the generalized spin-orbit interaction that renders
the dynamics in nanostructures to be totally determined by the chirality but
not the trivial ground state that itself does not have any chirality. In this
Chapter, we first address a universal quantum description of chiral
interaction between excitations (Sec. 4.1) and then address their numerous
realizations in magnetism, spintronics, and magnonics (Sec. 4.2, 4.3, and
4.4).

###  4.1 Chiral interaction between excitations

Much of condensed matter physics is based on the concept of a quasiparticle,
i.e., a collective excitation with a long lifetime due to weak interaction
with other degrees of freedom. The quasiparticle picture for electrons works
well because charges in metals are very efficiently screened and fields can be
mapped on a quadratic Hamiltonian when amplitudes are small. Similarly, the
excitations of ground states such as photons, phonons, and magnons, are
usually weakly interacting (Bosonic) quasiparticles. Nevertheless,
interactions are essential for many phenomena, such as superconductivity and
magnetism. While electron-electron, phonon-phonon, and magnon-magnon
interactions are of course important and interesting, we focus here on
interactions between different quasiparticles, say $\hat{\alpha}$ and
$\hat{\beta}$, that can be Fermions or Bosons. The Hamiltonian can be written

$\hat{H}=\hat{H}_{\alpha}+\hat{H}_{\beta}+\hat{H}_{\alpha\beta}.$ (182)

When the system is translational invariant in a spatial direction the
conjugate linear momentum $k$ is a good quantum number. The non-interacting
quasiparticle Hamiltonian then read in the second quantization

$\displaystyle\hat{H}_{\alpha}$
$\displaystyle=\sum_{k}\epsilon_{k}\hat{\alpha}_{k}^{\dagger}\hat{\alpha}_{k},$
(183a) $\displaystyle\hat{H}_{\beta}$
$\displaystyle=\sum_{n}\hbar\omega_{n}\hat{\beta}_{n}^{\dagger}\hat{\beta}_{n},$
(183b)

where $\epsilon_{k}=\hbar\omega_{k}$ is the energy of a quasiparticle
$\hat{\alpha}$ with continuous momentum variable $k$ created by field operator
$\hat{\alpha}_{k}^{\dagger}$, while $n$ labels the modes of quasiparticle
$\hat{\beta}$ that we assume here to be discrete. The leading term in the
interaction between $\hat{\alpha}$ and $\hat{\beta}$ then reads

$\hat{H}_{\alpha\beta}=\sum_{k,n}g_{k,n}\hat{\alpha}_{-k}^{\dagger}\hat{\beta}_{n}+\mathrm{H.c.}$
(184)

We note that we do not address interactions involving the electron charge that
cannot be expressed in a bilinear form of field operators. We call the
interaction “chiral” when the interaction vanishes for one sign of $k$ but is
finite for the other sign. It is “non-reciprocal” when
$|g_{k,n}|\neq|g_{-{k,n}}|$.

The coupling between quasiparticles or excitations such as electron spins,
phonons, photons, and magnons often leads to undesired effects such as
dephasing and reduced lifetimes. However, when coherently controlled, it can
also be a boon, e.g., in transducers for telecommunication or resulting in an
attractive effective interaction that causes superconductivity. Chiral
interaction between different excitations may help to build key components for
future communication technologies.

Here we focus on the phenomena in spintronics, which is rich in chiral and
non-reciprocal effects. Table 6 summarizes a few of its chiral interactions,
illustrating the theories and typical experimental observations. We here
define some key functionalities of chirality that are addressed in different
circumstances in this review.

  * •

“Diodes" are two-terminal (electronic) components that conduct current
primarily in one direction with an asymmetric conductance.

  * •

“Isolators" are wave devices that are based on chirality. They are two-port
diodes that transmit a coherent wave signal only in one direction, stabilizing
device performance by suppressing back reflections.

  * •

“Circulators" are three-port devices that route a signal between ports in a
clockwise, or counter-clockwise, direction.

This Chapter focuses on the (linear) chiral interaction between quasi-
particles. Table 6 summarizes the key effects, approaches, and experimental
evidence of chiral interactions in spintronics as discussed in detail in the
sections below.

Table 6: Chiral interactions in spintronics.  Systems | Theory |  Samples and phenomena
---|---|---
Propagating spin wave spectroscopy |  • Chiral spin pumping (input-output [67, 50, 48], Green function [50], micromagnetic simulation [47, 276]) • Chiral spin Seebeck effect [50, 67] • Additional damping due to spin and magnon pumping [50, 51] • Magnon trapping [51] and unidirectional blocking [243] |  • Magnetic nanowire arrays on thin YIG films [49] • Antiparallel spin valves [54, 55] • Two magnetic nanowires on thin YIG films [52] • Enhanced damping and magnon trap [53]
Magnons and photons |  • Chiral magnon pumping by microwave striplines [67] • Chiral magnon polaritons (master equation [61], input-output [51]) • Spin skin effect [60, 61]) • Cavity microwave circulators (input-output theory [63, 62], finite element simulations [63]) • Spin-wave Doppler shift [59] |  • Radiative damping in waveguides [381] • Chiral pumping of propagating magnons [64, 65, 66] • Chiral damping of spin waves [363] • Broadband non-reciprocity of magnon polaritons [68]
Magnons and surface phonons |  • Phonon diodes by magnetoelasticity [77] (input-output [75] and scattering [76] theory) • Chiral pumping of surface phonons (input-output theory [75], magnetoelasticity [82], simulation [82], and Landauer-Büttiker [76]) • Spin Seebeck effect by chiral bulk phonon (Boltzmann equations [277]) |  • Chiral attenuation of surface phonon • AC spin current generation in copper • Chiral phonon transmission [77, 81, 79, 83, 84] • Non-reciprocal magnetoelastic waves
Chiral photons and electron spin |  • Transverse spin of magnetic near-field (Maxwell equations [21, 85]) • Contactless spin pumping
(nonlinear response theory [85]) • Chiral spin pumping in Luttinger
liquid (linear response theory [85]) |  • No conclusive experiment available
Electron spin and SPP |  • Transverse spin of electric near-field (Maxwell equations [21, 89, 88]) • Spin-current generation in metals (spin-diffusion equation [88, 89]) |  • Enhanced spin current signals by surface plasmon resonance [87, 86, 91]

###  4.2 Ferromagnets

The physical properties and weak excitations of ferromagnets are affected by
the dipolar self-interaction, i.e., the Zeeman energy of local spins in the
stray field of all other spins, which introduces strong anisotropies in the
magnon dispersion that depend on the shape of the magnets such as small
particles, thin or thick films, or wires (Sec. 3.1). The dipolar interaction
causes the intrinsic chiralities in the Damon-Eshbach (DE) surface modes of
ferromagnetic spheres and films (Sec. 3.1). This chirality ceases to exist
when the film thickness $s$ is much smaller than the exchange length
$\lambda_{\rm ex}=2\pi\sqrt{\alpha_{\rm ex}}$, which is of the order of 100 nm
in YIG. The bulk and surface modes then merge into perpendicular standing spin
waves (PSSW) with $\omega_{l\bf k}=\omega_{-{l\bf k}}$, $l$ being the band
index. However, a chiral interaction emerges in the form of the dipolar
coupling to microwaves as generated by excited nanoscale magnetic wires and
striplines. Here we set the stage by discussing the magnon modes in thin films
and nanowires, without chirality themselves, in Sec. 4.2.1 before introducing
the chiral coupling in Sec. 4.2.2, below.

####  4.2.1 Magnons in thin films and nanowires

In thin films, the exchange interaction dominates the dipolar one in the
magnon energy when $ks\gg 1$, where $k$ is the wave number and $s$ the
magnetic film thickness. So in that limit, the dispersion relation is
$\omega_{k}\sim k^{2}$ and the magnetization undergoes circular precession.
When the dipolar interaction kicks in at longer wavelengths, the precession
becomes elliptical, as shown below.

Ultrathin magnetic films.—The magnetic mode amplitude $\mathbf{M}$ is the
solution of the Landau-Lifshitz equation [208]

${d}\mathbf{M}/{dt}=-\mu_{0}\gamma\mathbf{M}\times\left(\mathbf{H}_{\mathrm{app}}+\mathbf{H}_{d}+\mathbf{H}_{\mathrm{ex}}\right),$
(185)

where $\mathbf{H}_{\mathrm{app}}=H_{\mathrm{app}}\mathbf{\hat{z}}$ is the
applied magnetic field that saturates the film magnetization $M_{s}\hat{\bf
z}$, $\mathbf{{H}}_{d}$ is the dipolar field [Eq. (2)] [278], and the exchange
field
$\mathbf{H}_{\mathrm{ex}}=\alpha_{\mathrm{ex}}\left(\partial_{x}^{2}\mathbf{M}+\partial_{y}^{2}\mathbf{M}+\partial_{z}^{2}\mathbf{M}\right)$.
By translational symmetry in the film ($\hat{\mathbf{y}}$-$\hat{\mathbf{z}}$)
plane,
${M}_{x,y}(\mathbf{r})={m}_{x,y}^{\mathbf{k}}(x)e^{ik_{y}y+ik_{z}z}e^{-i\omega_{\bf
k}t}$ with $\mathbf{k}\equiv k_{y}\hat{\mathbf{y}}+k_{z}\hat{\mathbf{z}}$. We
choose free boundary conditions $d\mathbf{M}(\mathbf{r})/dx|_{x=0,-d}=0$ [279,
280, 281], since the lowest mode in sufficiently thin films is not affected by
dipolar pinning [282, 283, 284, 285]. Thus, by writing ${\bf m}^{{\bf
k}}_{\pm}(x)={\bf m}^{{\bf k}}_{x}(x)\pm i{\bf m}^{{\bf k}}_{{y}}(x)$, we have
the Fourier series [278, 286],

${\bf m}^{{\bf
k}}_{\pm}(x)=\sum_{l=0}^{\infty}\frac{\sqrt{2}}{\sqrt{1+\delta_{l0}}}\cos\big{(}l\pi
x/s\big{)}{\bf m}^{{\bf k}}_{l,\pm}.$ (186)

When the film is sufficiently thin such that
$k_{y}^{2}\ll\left({l\pi}/{s}\right)^{2}$ for $l\geq 1$, we may confine our
attention to the spin waves in the lowest branch ${l=0}$ with amplitude
governed by [278, 286, 48]

$\displaystyle\omega_{\bf k}\left(\begin{array}[]{cc}{\bf m}^{{\bf
k}}_{0,+}\\\ {\bf m}^{{\bf k}}_{0,-}\\\
\end{array}\right)=\mu_{0}\gamma{M}_{s}\left(\begin{array}[]{cc}-\Omega_{H}-\alpha_{\rm
ex}{k}^{2}-\frac{1}{2}+\frac{1}{2}(1-\frac{{k}^{2}_{y}}{{k}^{2}})f({k})&-\frac{1}{2}+\frac{1}{2}(1+\frac{{k}^{2}_{y}}{{k}^{2}})f({k})\\\
\frac{1}{2}-\frac{1}{2}(1+\frac{{k}^{2}_{y}}{{k}^{2}})f({k})&\Omega_{H}+\alpha_{\rm
ex}{k}^{2}+\frac{1}{2}-\frac{1}{2}(1-\frac{{k}^{2}_{y}}{{k}^{2}})f({k})\end{array}\right)\left(\begin{array}[]{cc}{\bf
m}^{{\bf k}}_{0,+}\\\ {\bf m}^{{\bf k}}_{0,-}\\\ \end{array}\right),$ (193)

where $f({k})\equiv 1-{1}/{(s|k_{y}|)}+{1}/{(s|k_{y}|)}\exp({-s|k_{y}|})$ and
$\Omega_{H}\equiv H_{\mathrm{app}}/{M}_{s}$. The dispersion [278, 279, 280,
281]

$\omega_{\bf k}=\mu_{0}\gamma{M}_{s}\sqrt{\left(\Omega_{H}+\alpha_{\rm
ex}{k}^{2}+1-f({k})\right)\left(\Omega_{H}+\alpha_{\rm
ex}{k}^{2}+\frac{{k}_{y}^{2}}{{k}^{2}}f({k})\right)}$ (194)

is reciprocal with $\omega_{{\bf k}}=\omega_{-{\bf k}}$, but is not isotropic.

The amplitudes $m_{x}$ and $m_{y}$ are constant and when normalized by Eq.
(23) read

$\displaystyle m_{x}({\bf k})=\sqrt{\frac{F({\bf k})+1}{4d\left(F({\bf
k})-1\right)}},~{}~{}~{}~{}~{}m_{y}({\bf k})=i\sqrt{\frac{F({\bf
k})-1}{4d\left(F({\bf k})+1\right)}},$ (195)

in which

$\displaystyle F({\bf
k})=\frac{1/2-(1/2)\left(1+k_{y}^{2}/k^{2}\right)f(|k_{y}|)}{\omega_{\bf
k}/(\mu_{0}\gamma M_{s})-\left(\Omega_{H}+\alpha_{\rm
ex}k_{y}^{2}+1/2\right)+(1/2)\left(1-k_{y}^{2}/k^{2}\right)f(|k_{y}|)}.$ (196)

The local magnetization precesses in an orbit with ${\bf k}$-dependent
ellipticity. When the wavelengths are shorter than the exchange length,
$|F|\gg 1$, and the precession becomes circular with ${m}_{y}^{{\bf
k}}=i{m}_{x}^{{\bf k}}=i\sqrt{1/(4s)}$. When ${\bf k}\rightarrow 0$,
$f(|{k_{y}}|)\rightarrow 0$, $\omega_{\bf k}\rightarrow\mu_{0}\gamma
M_{s}\sqrt{\Omega_{H}(\Omega_{H}+1)}$, and

$F\rightarrow\frac{1}{2}\left(\sqrt{\Omega_{H}^{2}+\Omega_{H}}-2\Omega_{H}-1\right)^{-1}=-1-2\Omega_{H}-2\sqrt{\Omega_{H}(\Omega_{H}+1)}.$

When $\Omega_{H}\rightarrow 0$ with a small static magnetic field,
$F\rightarrow-1-2\sqrt{\Omega_{H}}$, $|m_{y}|\gg|m_{x}|$, leading to a
(nearly) linearly polarized Kittel modes.

Magnetic nanowires.—The stray fields of excited magnetic nanowires are
excellent pumps of exchange spin waves in magnetic films. For an equilibrium
magnetization, $\tilde{M}_{s}\hat{\bf z}$ parallel to the wire and applied
magnetic field the dipolar field Eq. (185) becomes

$\displaystyle{\bf H}_{d}=-N_{xx}\tilde{M}_{x}\hat{\bf
x}-N_{yy}\tilde{M}_{y}\hat{\bf y},$ (197)

where the $N_{xx}\simeq w/(d+w)$ and $N_{yy}\simeq d/(d+w)$ [48, 51] are
demagnetization factors for thickness $d$ and width $w$. With ansatz
$\tilde{M}_{x,y}\propto e^{i\kappa z}e^{-i\omega_{\kappa}t}$, where $\kappa$
and $\omega_{\kappa}$ are the momentum and frequency of the spin waves, the
equation of motion reads

$\displaystyle i\omega_{\kappa}\tilde{M}_{x}=\left(\mu_{0}\gamma H_{\rm
app}+N_{yy}\mu_{0}\gamma\tilde{M}_{s}+\lambda_{\rm
ex}\kappa^{2}\mu_{0}\gamma\tilde{M}_{s}\right)\tilde{M}_{y},$ (198a)
$\displaystyle i\omega_{\kappa}\tilde{M}_{y}=-\left(\mu_{0}\gamma H_{\rm
app}+N_{xx}\mu_{0}\gamma\tilde{M}_{s}+\lambda_{\rm
ex}\kappa^{2}\mu_{0}\gamma\tilde{M}_{s}\right)\tilde{M}_{x}.$ (198b)

Here $\lambda_{\mathrm{ex}}$ is the exchange length in the nanowire. The
dispersion of the lowest mode

$\omega_{\kappa}=\mu_{0}\gamma\sqrt{\left(H_{\mathrm{app}}+N_{xx}\tilde{M}_{s}+\lambda_{\rm
ex}\kappa^{2}\tilde{M}_{s}\right)\left(H_{\mathrm{app}}+N_{yy}\tilde{M}_{s}+\alpha_{\rm
ex}\kappa^{2}\tilde{M}_{s}\right)},$ (199)

reduces to the Kittel formula [275] when $\kappa=0$. The normalized amplitudes
of the dipolar-exchange spin waves

$\tilde{m}_{x}^{\kappa}=\sqrt{\frac{1}{4\mathcal{D}(\kappa)wd}},~{}~{}~{}~{}\tilde{m}_{y}^{\kappa}=i\sqrt{\frac{\mathcal{D}(\kappa)}{4wd}},$
(200)

where

$\mathcal{D}(\kappa)=\sqrt{\frac{H_{\mathrm{app}}+N_{xx}\tilde{M}_{s}+{\lambda}_{\mathrm{ex}}\kappa^{2}\tilde{M}_{s}}{H_{\mathrm{app}}+N_{yy}\tilde{M}_{s}+{\lambda}_{\mathrm{ex}}\kappa^{2}\tilde{M}_{s}}}.$
(201)

precess elliptically unless the wire cross section is rectangular or
cylindrical with $w=d$ and $\mathcal{D}=1$.

An analytic treatment of the anisotropic magnon dispersion of finite systems
in the dipolar-exchange regime is complicated. However, when the films are
much thinner than the magnon wave length, the splitting between the
perpendicular modes becomes large and the intrinsic pinning is suppressed
[232]. The amplitude of the lowest mode is then constant over the thickness,
which drastically simplifies the problem. This regime is relevant for high-
quality ultra-thin YIG films that can be grown down to nanometer thickness
[242, 287, 288].

####  4.2.2 Chiral pumping of spin waves

Polushkin et al. [289] proposed an all-metallic magnetic microwave-to-spin-
wave transducer. They pointed out that the evanescent dipolar stray field of a
magnetic wire antenna enhances the incident microwave field by an order of
magnitude. Au et al. [47] studied the coupling between an excited magnetic
wire and a magnetic ribbon mediated by the local dynamical dipolar field.
Figures 29(a) and (b) show the resonantly excited spin waves in the waveguide.
Micromagnetic simulations reveal that a wire can unidirectionally pump
exchange spin waves, see Fig. 29(c) with the direction that can be switched by
the wire or film magnetization. These authors only address spin wave vectors
normal to the equilibrium magnetizations, i.e., the Damon-Eshbach
configuration. In the thick film limit, this chirality is therefore expected,
as explained in Sec. 3.1.1, but it was surprising for thin films.

Figure 29: Micromagnetic simulations of the unidirectional excitation of spin
waves in a Py magnetic ribbon by the dynamical dipolar field of a nearby
cylindrical Py wire resonantly excited by a microwave stripline (not shown).
Here $L$ is the thickness of the ribbon, $R$ is the radius of the wire, and
$d$ is the distance between the bottom of the wire and the upper surface of
the ribbon. (a) and (b) are sketches of the configuration and the dipolar
field of the wire, respectively. (c) shows calculated results for 100 nm wide
and 10 nm thick ribbon, crossed by 50 nm wide, 150 nm long, and 30 nm thick
cylindrical wires with the shortest distance of 10 nm. Py’s saturation
magnetization is 800 kA/m and the Gilbert damping coefficient $\alpha=0.005$.
The stray field of the wire excites the magnetization component $m_{y}$ of
half of the waveguide that can be chosen by the direction of the wire
magnetization. In the last panel of (c), the excited magnetization appears at
the opposite half space compared with the other panels with reversed saturated
magnetization direction. The figures are taken from Ref. [47].

Yu et al. [50, 48] addressed the dipolar fields emitted by the Kittel mode of
a magnetic wire with rectangular cross section. The analytic form Eq. (84)
clearly reveals the spin-momentum locking of the stray magnetic field. The
Kittel mode’s orbit is elliptic, i.e., a linear combination of right and left
circularly polarized components that strongly depends on the shape of the wire
cross section [refer to Eq. (200)]. Pure exchange spin waves in an underlying
film magnetized parallel to the wire as in Fig. 29(a) are circularly polarized
and couple only to the matching component of the wire dipolar field, which
fixes the propagation direction of the excitation. Equivalently, we may start
from the dipolar fields of spin waves magnetic films that possess the spatial
chirality illustrated by Fig. 19: the dipolar field of right (left) moving
spin waves vanishes below (above) the film, i.e., being unidirectional (also
refer to Sec. 3.5). This implies that spin waves propagating along a
particular direction normal to the magnetization can couple only to
nanomagnets on one side of the film.

In the following, we review the properties of the magnetic wire$\&$film device
for non-collinear magnetic configurations and elliptic precessions. When the
(exchange and/or dipolar) coupling between the wire and film exceeds the
lifetime broadening caused by magnetization and radiation damping, we are in
the strong coupling regime and dynamics that can be accessed by solving the
coupled Maxwell and LLG equations. Here we focus on the limit of weak dipolar
and contact exchange interactions that are relevant for most experiments. We
may then use perturbation/linear-response theory to capture the physics by
transparent analytic results.

Linear response theory.—Here we address the excitation of magnetization in a
thin film by a proximity wire that acts as a microwave antenna, taking into
account arbitrary magnetization direction [50, 48]. We use perturbation theory
[258, 291, 290, 292] combined with the input-output theory of the microwave
scattering matrix (see below). Linear-response/perturbation theory holds in
the “weak coupling" regime, disregarding the back-action of the film dynamics
on the wire. We consider an infinite long magnetic wire of rectangular cross
section with thickness $d$ and width $w$ above an in-plane magnetized magnetic
film of thickness $s$, as illustrated in Fig. 30(a) for a Co wire on top of
YIG film. The Zeeman interaction Hamiltonian

$\hat{H}_{\mathrm{int}}=-\mu_{0}\int\mathbf{\tilde{M}}(\mathbf{r},t)\cdot\mathbf{h}(\mathbf{r},t)d\mathbf{r}=-\mu_{0}\int\mathbf{M}(\mathbf{r},t)\cdot\mathbf{\tilde{h}}(\mathbf{r},t)d\mathbf{r},$
(202)

between the magnetization $\mathbf{\tilde{M}}$ of the wire and that of the
film $\mathbf{M}$ via the stray dipolar magnetic fields $\mathbf{h}$ and
$\mathbf{\tilde{h}}$ [208], respectively. Both expressions are equivalent but
lead to quite different physical pictures, as briefly mentioned above. We
introduce the $3\times 3$ Cartesian Green function tensor

$\mathcal{G}(\boldsymbol{k},x-x^{\prime})=e^{-|x-x^{\prime}||\mathbf{k}|}\left(\begin{array}[]{ccc}\frac{|\mathbf{k}|}{2}&-\frac{ik_{y}}{2}\mathrm{sgn}\left(x-x^{\prime}\right)&-\frac{ik_{z}}{2}\mathrm{sgn}\left(x-x^{\prime}\right)\\\
-\frac{ik_{y}}{2}\mathrm{sgn}\left(x-x^{\prime}\right)&-\frac{k_{y}^{2}}{2|\mathbf{k}|}&-\frac{k_{y}k_{z}}{2|\mathbf{k}|}\\\
-\frac{ik_{z}}{2}\mathrm{sgn}\left(x-x^{\prime}\right)&-\frac{k_{y}k_{z}}{2|\mathbf{k}|}&-\frac{k_{z}^{2}}{2|\mathbf{k}|}\end{array}\right)-\delta_{\mathrm{in}}(x-x^{\prime})\mathcal{I},$
(203)

where $\mathcal{I}$ is the $3\times 3$ unity tensor and
$\delta_{\mathrm{in}}(x)$ vanishes when $x$ lies outside the magnetic film, as
a convenient instrument to formulate the dipolar fields. A spin wave in the
film of the form ${\bf
m}_{\alpha}(\mathbf{k},x)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$, for example,
emits the magnetic field

$\left(\begin{array}[c]{c}{{h}}_{x}(\mathbf{r})\\\ {{h}}_{y}(\mathbf{r})\\\
{{h}}_{z}(\mathbf{r})\end{array}\right)=e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\int_{-s}^{0}dx^{\prime}\mathcal{G}(\mathbf{k},x-x^{\prime})\left(\begin{array}[c]{c}{{m}}_{x}(\mathbf{k},x^{\prime})\\\
{{m}}_{y}(\mathbf{k},x^{\prime})\\\
{{m}}_{z}(\mathbf{k},x^{\prime})\end{array}\right).$ (204)

${\mathbf{h}}$ is a demagnetization field that naturally satisfies the
electromagnetic boundary condition, i.e., continuity of the electromagnetic
fields and currents at the surfaces of the film [278].

Figure 30: Chiral pumping of spin waves by a magnetic wire under the
ferromagnetic resonance. (a) depicts a magnetic (Co) nanowire transducer
separated by a non-magnetic spacer (optional) from a YIG film. The
$\pm\hat{\bf y}$-directions of the magnon spin currents are indicated by the
green arrows, whose size indicates their magnitude. (b) illustrates the top
view of “spatial chirality” of the excited magnetization, viz., the injection
of magnetization, energy and momentum into half-space with $y>\tilde{y}$. The
figure [(a)] is taken from Ref. [67].

The film magnetization excited by the dynamical stray field follows from the
time-dependent perturbation theory [258, 293] as:

$\displaystyle{M}_{\alpha}(x,\boldsymbol{\rho},t)$
$\displaystyle=-i\int_{-\infty}^{t}dt^{\prime}\left\langle\left[\hat{M}_{\alpha}(x,\boldsymbol{\rho},t),\hat{H}_{\mathrm{int}}(t^{\prime})\right]\right\rangle$
$\displaystyle=\mu_{0}(\gamma\hbar)^{2}\sum_{\mathbf{k}}\int_{-\infty}^{\infty}dt^{\prime}\int_{0}^{d}d\tilde{x}d\tilde{\boldsymbol{\rho}}\int_{-s}^{0}dx^{\prime}\tilde{M}_{\beta}(\tilde{x},\tilde{\boldsymbol{\rho}},t^{\prime})\mathcal{G}_{\beta\xi}(-\mathbf{k},\tilde{x}-x^{\prime})\chi_{\alpha\xi}\left(x,x^{\prime};\mathbf{k};t-t^{\prime}\right)e^{i\mathbf{k}\cdot(\boldsymbol{\rho}-\tilde{\boldsymbol{\rho}})},$
(205)

where the spin operator $\hat{S}_{\alpha}=-\hat{M}_{\alpha}/(\gamma\hbar)$ and
we expanded the retarded spin susceptibility
$\chi_{\alpha\delta}(x,x^{\prime};\boldsymbol{\rho}-\boldsymbol{\rho}^{\prime};t-t^{\prime})=i\Theta(t-t^{\prime})\left\langle\left[\hat{S}_{\alpha}(x,\boldsymbol{\rho},t),\hat{S}_{\delta}(x^{\prime},\boldsymbol{\rho}^{\prime},t^{\prime})\right]\right\rangle$,
in the spin wave modes
$m_{\alpha}^{\mathbf{k}}(x)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ with
frequency $\omega_{\mathbf{k}}$. The coefficients in the momentum-frequency
space read

$\chi_{\alpha\xi}(x,x^{\prime};\mathbf{k};\omega)=-\frac{2M_{s}}{\gamma\hbar}\frac{m_{\alpha}^{\mathbf{k}}(x){m_{\xi}^{\mathbf{k}*}(x^{\prime})}}{\omega-\omega_{\mathbf{k}}+i0_{+}}.$
(206)

In the presence of Gilbert damping, the infinitesimal $0_{+}$ should be
replaced by $2\alpha_{G}|\omega|$. Under FMR of the Kittel frequency
$\omega_{\rm K}$ the wire magnetization precesses uniformly (Kittel mode). In
the perturbation regime, it is safe to employ the rotating wave approximation

$\tilde{M}_{\beta}(\tilde{x},\tilde{\boldsymbol{\rho}},t^{\prime})\approx\tilde{M}_{\beta}(\tilde{x},\tilde{\boldsymbol{\rho}},t)e^{i\omega_{\mathrm{K}}(t-t^{\prime})},$
(207)

where $\mathbf{}{M}(t)$ varies slowly on the $1/\omega_{K}$ time scale. We
address the effects of finite damping and back-action by the film in the
input-output theory section below. With Eqs. (206) and (207), the film
magnetization (205) becomes

$\displaystyle M_{\alpha}(\mathbf{r},t)$
$\displaystyle=-2\mu_{0}M_{s}\gamma\hbar\sum_{\mathbf{k}}\int_{0}^{d}d\tilde{x}d\tilde{\boldsymbol{\rho}}\int_{-s}^{0}dx^{\prime}e^{i\mathbf{k}\cdot(\boldsymbol{\rho}-\tilde{\boldsymbol{\rho}})}\frac{m_{\alpha}(\mathbf{k},x)\tilde{M}_{\beta}(\tilde{x},\tilde{\boldsymbol{\rho}},t)}{\omega_{\mathrm{K}}-\omega_{\mathbf{k}}+i0_{+}}$
$\displaystyle\times\mathcal{G}_{\beta\xi}(-\mathbf{k},\tilde{x}-x^{\prime}){m_{\xi}^{*}(\mathbf{k},x)}.$
(208)

When the nanowire and its equilibrium magnetizations are parallel to
$\mathbf{\hat{z}}$, the excited magnetization does not depend on $z$ and the
momentum integral in

$\displaystyle M_{\alpha}(x,y,t)$
$\displaystyle=-2\mu_{0}M_{s}\gamma\hbar\int\frac{dk_{y}}{2\pi}\int_{0}^{d}d\tilde{x}d\tilde{y}\int_{-s}^{0}dx^{\prime}e^{ik_{y}(y-\tilde{y})}\frac{1}{\omega_{\mathrm{K}}-\omega_{k_{y}}+i0_{+}}$
$\displaystyle\times
m_{\alpha}(k_{y},x)M_{\beta}(\tilde{x},\tilde{y},t)\mathcal{G}_{\beta\xi}(-k_{y},\tilde{x}-x^{\prime}){m_{\xi}^{\ast}(k_{y},x)}$
(209)

can be evaluated by contours in the complex plane. The zeros of the
denominator $\omega_{\mathrm{K}}-\omega_{k_{y}}+i0_{+}$ generate two
singularities at $k_{\pm}=\pm(\kappa+i0_{+})$ where $\kappa$ is the positive
root of $\omega_{k_{y}}=\omega_{\rm K}$, so $k_{+}$ and $k_{-}$ lie in the
upper and lower half planes, respectively. When $y>\tilde{y}$ the contour
should be closed in the upper half plane and

$\displaystyle
M_{\alpha}^{>}(x,y,t)=2i\mu_{0}M_{s}\gamma\hbar\frac{1}{v_{\kappa}}\int_{0}^{d}d\tilde{x}d\tilde{y}\int_{-s}^{0}dx^{\prime}e^{i\kappa(y-\tilde{y})}m_{\alpha}(\kappa,x)M_{\beta}(\tilde{x},\tilde{y},t)\mathcal{G}_{\beta\xi}(-\kappa,\tilde{x}-x^{\prime}){m_{\xi}^{\ast}(\kappa,x)},$
(210)

A small spin-wave group velocity $v_{\kappa}=\left.\partial\omega_{k}/\partial
k\right|_{k=\kappa}$ corresponds to a large density of states and excitation
efficiency. When $y<\tilde{y}$,

$M_{\alpha}^{<}(x,y,t)=2i\mu_{0}M_{s}\gamma\hbar\frac{1}{v_{\kappa}}\int_{0}^{d}d\tilde{x}d\tilde{y}\int_{-s}^{0}dx^{\prime}e^{-i\kappa(y-\tilde{y})}m_{\alpha}(-\kappa,x)M_{\beta}(\tilde{x},\tilde{y},t){\cal
G}_{\beta\xi}(\kappa,\tilde{x}-x^{\prime}){m_{\xi}^{\ast}(-\kappa,x)}.$ (211)

When the spin waves in the film are circularly polarized with $m_{y}=im_{x}$,

${\cal
G}_{\beta\xi}(-\kappa,\tilde{x}-x^{\prime}){m_{\xi}^{\ast}(\kappa,x)}\longrightarrow\frac{e^{-|\tilde{x}-x^{\prime}||\kappa|}}{2}\left(\begin{array}[c]{cc}\kappa&i\kappa\\\
i\kappa&-\kappa\end{array}\right)\left(\begin{array}[c]{c}m_{x}\\\
-im_{x}\end{array}\right)=\left(\begin{array}[c]{c}0\\\ 0\end{array}\right),$
(212)

leading to zero $M_{\alpha}^{<}(x,y,t)$, but finite $M_{\alpha}^{>}(x,y,t)$,
i.e., perfect chirality. When the magnetic nanowire excites only spin waves
with positive momentum, energy and momentum are injected into the half-space
$y>\tilde{y}$, as sketched in Fig. 30(b). This “spatial chirality” is a
consequence of the causality or retardation in electrodynamics and persists in
the presence of magnetic damping.

Input-output theory.—References [295, 294] addressed the scattering matrix of
waves in finite systems such as cavities that includes (quantum) noise and
dissipation. Here we employ this formalism to calculate the transmission and
reflection amplitudes of microwave photons between emitters and detectors
formed by striplines or co-planar wave guides, which are modulated by the
interaction with the spin waves [50, 51]. As above, we adopt the configuration
of a weakly excited magnetic nanowire on a thin magnetic film and use the
language of second quantization. The Cartesian components
$\beta\in\left\\{x,y\right\\}$ of the magnetization dynamics of film
($\mathbf{\hat{M}}$) and nanowire ($\mathbf{\hat{\tilde{M}}}$) can be expanded
into magnon creation and annihilation operators [210, 209],

$\displaystyle\hat{M}_{\beta}(\mathbf{r})$
$\displaystyle=-\sqrt{2M_{s}\gamma\hbar}\sum_{\mathbf{k}}\left[m_{\beta}(\mathbf{k},x)e^{i{\mathbf{k}}\cdot\boldsymbol{\rho}}\hat{\alpha}_{\mathbf{k}}+\mathrm{h.c.}\right],$
(213a) $\displaystyle\hat{\tilde{M}}_{\beta}(\mathbf{r})$
$\displaystyle=-\sqrt{2\tilde{M}_{s}\gamma\hbar}\sum_{k_{z}}\left[\tilde{m}_{\beta}(k_{z},x,y)e^{ik_{z}z}\hat{\beta}_{k_{z}}+\mathrm{h.c.}\right],$
(213b)

where $M_{s}$ and $\tilde{M}_{s}$ are the saturation magnetizations,
$m_{\beta}(\mathbf{k},x)$ and $\tilde{m}_{\beta}(k_{z},x,y)$ are the
normalized spin wave mode amplitudes, while $\hat{\alpha}_{\mathbf{k}}$ and
$\hat{\beta}_{k_{z}}$denote the magnon (annihilation) operators in the film
and nanowire, respectively.

We focus, from now on, on ultrathin films with typical dimensions
$s,d=\mathcal{O}\left(10~{}\mathrm{nm}\right)$ and nanowires of width
$w=\mathcal{O}\left(50~{}\mathrm{nm}\right)$. These structures can be
fabricated with high quality only recently [287, 288, 49, 296]. The weakly
excited magnetizations of film and nanowire (centered at
$y_{0}\hat{\mathbf{y}}$) are then nearly constant over a cross section of film
and wire (Sec. 4.2.1) [50, 48], viz.:

$\displaystyle m_{\beta}^{\left(\mathbf{k}\right)}(x)\approx
m_{\beta}^{\left(\mathbf{k}\right)}\Theta(-x)\Theta(x+s),$ (214a)
$\displaystyle\tilde{m}_{\beta}^{\left(k_{z}\right)}(x,y)\approx\tilde{m}_{\beta}^{\left(k_{z}\right)}\Theta(x)\Theta(-x+d)\Theta(y-y_{0}+w/2)\Theta(-y+y_{0}+w/2),$
(214b)

where $\Theta(x)$ is the Heaviside step function. The mode selectivity of the
dipolar coupling allows us to disregard higher perpendicular standing spin
waves, even when they are thermally occupied and important for diffuse magnon
transport. The system Hamiltonian then simplifies to

$\displaystyle\hat{H}/\hbar=\sum_{\mathbf{k}}\omega_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}^{\dagger}\hat{\alpha}_{\mathbf{k}}+\sum_{k_{z}}\tilde{\omega}_{k_{z}}\hat{\beta}_{k_{z}}^{\dagger}\hat{\beta}_{k_{z}}+\sum_{\mathbf{k}}\left(g_{\mathbf{k}}e^{-ik_{y}y_{0}}\hat{\alpha}_{\mathbf{k}}^{\dagger}\hat{\beta}_{k_{z}}+g_{\mathbf{k}}^{\ast}e^{ik_{y}y_{0}}\hat{\beta}_{k_{z}}^{\dagger}\hat{\alpha}_{\mathbf{k}}\right),$
(215)

where $\omega_{\mathbf{k}}$ and $\tilde{\omega}_{k_{z}}$ are the frequencies
of spin waves in the film and nanowire and the Zeeman coupling constant
follows from the results of the previous Sec. 4.2.1

$g_{\mathbf{k}}=F(\mathbf{k})\left(m_{x}^{\ast}(\mathbf{k}),m_{y}^{\ast}(\mathbf{k})\right)\left(\begin{array}[c]{cc}|\mathbf{k}|&ik_{y}\\\
ik_{y}&-k_{y}^{2}/|\mathbf{k}|\end{array}\right)\left(\begin{array}[c]{c}\tilde{m}_{x}(k_{z})\\\
\tilde{m}_{y}(k_{z})\end{array}\right),$ (216)

where
$F(\mathbf{k})=-\mu_{0}\gamma\phi\left(\mathbf{k}\sqrt{M_{s}\tilde{M}_{s}/L}\right)$.
The form factor

$\phi\left(\mathbf{k}\right)=2\sin(k_{y}w/2)\frac{(1-e^{-kd})(1-e^{-ks})}{k_{y}{k^{2}}}$
(217)

is significant when the wavelengths are of the order of the nanowire width.
Right-circularly polarized exchange waves with
$m_{y}^{(k_{y})}=im_{x}^{(k_{y})}$ couple with perfect chirality
$g_{-|k_{y}|}=0$.

The microwave photon operator
$\hat{p}_{k_{z}}(t)=\int\hat{p}_{k_{z}}(\omega)e^{-i\omega t}d\omega/(2\pi)$
represents a coherent input centered at the frequency $\tilde{\omega}_{k_{z}}$
with expectation value $\langle\hat{p}_{k_{z}}(\omega)\rangle\rightarrow
2\pi\mathit{\mathcal{D}}\delta(\omega-\tilde{\omega}_{k_{z}})$ and amplitude
$\mathcal{D}$. The Heisenberg-like operator equations of motion of the input-
output formulation are then

$\displaystyle\frac{d\hat{\beta}_{k_{z}}}{dt}$
$\displaystyle=-i\tilde{\omega}_{k_{z}}\hat{\beta}_{k_{z}}(t)-\sum_{k_{y}}ig_{\mathbf{k}}^{\ast}e^{ik_{y}y_{0}}\hat{\alpha}_{\mathbf{k}}(t)-\left(\frac{\tilde{\kappa}_{k_{z}}}{2}+\frac{\zeta_{k_{z}}}{2}\right)\hat{\beta}_{k_{z}}(t)-\sqrt{\tilde{\kappa}_{k_{z}}}\hat{\tilde{N}}_{k_{z}}(t)-\sqrt{\zeta_{k_{z}}}\hat{p}_{k_{z}}(t),$
(218a) $\displaystyle\frac{d\hat{\alpha}_{\mathbf{k}}}{dt}$
$\displaystyle=-i\omega_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}(t)-ig_{\mathbf{k}}e^{-ik_{y}y_{0}}\hat{\beta}_{k_{z}}(t)-\frac{\kappa_{\mathbf{k}}}{2}\hat{\alpha}_{\mathbf{k}}(t)-\sqrt{\kappa_{\mathbf{k}}}\hat{N}_{\mathbf{k}}(t),$
(218b)

where $\tilde{\kappa}_{k_{z}}\equiv 2\tilde{\alpha}_{G}\tilde{\omega}_{k_{z}}\
$($\kappa_{\mathbf{k}}\equiv 2\alpha_{G}\omega_{\mathbf{k}}$) is the magnon
dissipation rate in terms of the Gilbert damping constant $\tilde{\alpha}_{G}$
$(\alpha_{G})$ in the nanowire (film) and $\zeta_{k_{z}}$ is the photon
dissipation rate. The thermal environment causes random fluctuations in the
magnon amplitudes $\hat{N}_{\mathbf{k}}$ [295, 294]. Their correlation
functions can be calculated assuming Markovian statistics that obeys the
(quantum) fluctuation-dissipation theorem with
$\langle{\hat{N}_{\mathbf{k}}}\rangle=0$ and
$\langle{\hat{N}_{\mathbf{k}}^{\dagger}(t)\hat{N}_{\mathbf{k}^{\prime}}(t^{\prime})}\rangle=n_{\mathbf{k}}\delta(t-t^{\prime})\delta_{\mathbf{k}\mathbf{k}^{\prime}}$.
$n_{\mathbf{k}}=1/\left\\{\exp\left[{\hbar\omega_{\mathbf{k}}}/({k_{B}T_{2}})\right]-1\right\\}$
is the magnon population at temperature $T_{2}$ of the film, and $k_{B}T_{2}$
should be larger than the spectral width of the mode $\mathbf{k}$ (narrow band
white noise). The thermal fluctuations $\hat{\tilde{N}}_{k_{z}}$ in the
nanowire are governed by a different temperature $T_{1}$ and distribution
$\tilde{n}_{k_{z}}=1/\left\\{\exp\left[{\hbar\tilde{\omega}_{k_{z}}}/({k_{B}T_{1}})\right]-1\right\\}$.
The solutions to the equations of motion are then

$\displaystyle\hat{\beta}_{k_{z}}(\omega)$
$\displaystyle=\frac{i\sum\limits_{k_{y}}\gamma_{\mathbf{k}}G_{\mathbf{k}}\hat{N}_{\mathbf{k}}(\omega)-\sqrt{\tilde{\kappa}_{k_{z}}}\hat{\tilde{N}}_{k_{z}}(\omega)-\sqrt{\zeta_{k_{z}}}\hat{p}_{k_{z}}(\omega)}{-i(\omega-\tilde{\omega}_{k_{z}})+\frac{\tilde{\kappa}_{k_{z}}}{2}+\frac{\zeta_{k_{z}}}{2}+i\sum_{k_{y}}|g_{\mathbf{k}}|^{2}G_{\mathbf{k}}\left(\omega\right)},$
(219a) $\displaystyle\hat{\alpha}_{\mathbf{k}}(\omega)$
$\displaystyle=G_{\mathbf{k}}\left(\omega\right)\left(g_{\mathbf{k}}e^{-ik_{y}y_{0}}\hat{\beta}_{k_{z}}(\omega)-i\sqrt{\kappa_{\mathbf{k}}}\hat{N}_{\mathbf{k}}(\omega)\right),$
(219b)

where the Green function
$G_{\mathbf{k}}\left(\omega\right)=\left((\omega-\omega_{\mathbf{k}})+i\kappa_{\mathbf{k}}/2\right)^{-1}$
and
$\gamma_{\mathbf{k}}=ig_{\mathbf{k}}^{\ast}e^{ik_{y}y_{0}}\sqrt{\kappa_{\mathbf{k}}}$
reveal the effect of thermal fluctuations in both wire
($\hat{\tilde{N}}_{k_{z}}$) and film ($\hat{N}_{\mathbf{k}}$) on
$\hat{\beta}_{k_{z}}(\omega)$. Moreover, the magnon pumping enhances the
damping in the nanowire by

$\displaystyle\delta\tilde{\kappa}_{k_{z}}=2\pi\sum_{k_{y}}|g_{\mathbf{k}}|^{2}\delta(\tilde{\omega}_{k_{z}}-\omega_{\mathbf{k}})$
(220)

to $\tilde{\kappa}_{k_{z}}^{\prime}$and the frequency shift to
$\tilde{\omega}_{k_{z}}^{\prime}$ follows from the real part of
$\sum_{k_{y}}|g_{\mathbf{k}}|^{2}G_{\mathbf{k}}$ . The (plasmonically
enhanced) uniform microwave field drives the Kittel mode ($k_{z}=0$) at the
ferromagnetic resonance of the nanowire, which can be pushed to much higher
frequencies than the Kittel mode of the film, e.g., by the high form
anisotropy associated with a large magnetization. Spin waves with finite
$k_{y}\equiv q$ in the film are excited directly only by the inhomogeneous
components of the microwave field, which is dominated by the stray fields
emitted by the magnetization dynamics of the wire.

The equations of motion can now be solved for the magnon operators. With
$T_{1}=T_{2}\equiv T_{0}$

$\hat{\alpha}_{q}(t)=\int\frac{d\omega}{2\pi}\frac{e^{-i\omega
t}ig_{q}e^{-iqy_{0}}}{-i(\omega-\omega_{q})+\frac{\kappa_{q}}{2}}\frac{\sqrt{\zeta_{0}}\hat{p}_{0}(\omega)}{-i(\omega-\tilde{\omega}_{0}^{\prime})+\frac{\tilde{\kappa}_{0}^{\prime}}{2}+\frac{\zeta_{0}}{2}}.$
(221)

Under coherent excitation of the wire
$\langle\hat{\alpha}_{q}^{\dagger}(t)\hat{\alpha}_{q}(t)\rangle=\langle\hat{\alpha}_{q}^{\dagger}(t)\rangle\langle\hat{\alpha}_{q}(t)\rangle$
and in the absence of damping, $\kappa_{q}$ is a positive infinitesimal that
safeguards causality. The magnetization in position and time domain reads

$\displaystyle\delta{M}_{\beta}(\mathbf{r},t)=\sqrt{2M_{s}\gamma\hbar}\mathcal{D}\frac{1}{v_{q_{\ast}}}\frac{e^{-i\tilde{\omega}_{0}t}\sqrt{\zeta_{0}}}{(\tilde{\omega}_{0}-\tilde{\omega}_{0}^{\prime})+i(\tilde{\kappa}_{0}^{\prime}+\zeta_{0})/2}\left\\{\begin{array}[c]{c}m_{\beta}^{(q_{\ast})}(x)g_{q_{\ast}}e^{iq_{+}(y-y_{0})}+\mathrm{h.c.}\\\
m_{\beta}^{(-q_{\ast})}(x)g_{-q_{\ast}}e^{iq_{-}(y-y_{0})}+\mathrm{h.c.}\end{array}\text{
for }\begin{array}[c]{c}y>y_{0}\\\ y<y_{0}\end{array}\right..$ (226)

$\tilde{\omega}_{0}-\omega_{q}+i\kappa_{q}/2$ vanishes for
$q_{\pm}=\pm(q_{\ast}+i\delta_{\Gamma})$ where $q_{\ast}>0$, $\delta_{\Gamma}$
is the inverse propagation length, and
$v_{q_{\ast}}=\left.\partial\omega_{q}/\partial q\right|_{q_{\ast}}$ is the
magnon group velocity. Perfect chiral coupling $g_{-q_{\ast}}=0$ implies
actuation of the magnetization only in the half space $y>y_{0}$.

Microwave transmission spectra.—Coherent chiral pumping can be observed in
microwave transmission spectra [57, 58, 296] with two nanowires at
$\mathbf{r}_{1}=R_{1}\hat{\mathbf{y}}$ and
$\mathbf{r}_{2}=R_{2}\hat{\mathbf{y}}$ that act as inductive actuators and
detectors. When representing the magnons in the wires $R_{1}$ and $R_{2}$ by
the operators $\hat{m}_{L}$ and $\hat{m}_{R}$, respectively, the equations of
motion in the time domain become

$\displaystyle\frac{d\hat{m}_{L}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{L}(t)-i\sum_{q}g_{q}e^{iqR_{1}}\hat{\alpha}_{q}(t)-\left(\frac{\kappa_{L}}{2}+\frac{\kappa_{p,L}}{2}\right)\hat{m}_{L}(t)-\sqrt{\kappa_{p,L}}\hat{p}_{\mathrm{in}}^{(L)}(t),$
(227a) $\displaystyle\frac{d\hat{m}_{R}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{R}(t)-i\sum_{q}g_{q}e^{iqR_{2}}\hat{\alpha}_{q}(t)-\frac{\kappa_{R}}{2}\hat{m}_{R}(t),$
(227b) $\displaystyle\frac{d\hat{\alpha}_{q}}{dt}$
$\displaystyle=-i\omega_{q}\hat{\alpha}_{q}(t)-ig_{q}e^{-iqR_{1}}\hat{m}_{L}(t)-ig_{q}e^{-iqR_{2}}\hat{m}_{R}(t)-\frac{\kappa_{q}}{2}\hat{\alpha}_{q}(t),$
(227c)

where $\kappa_{L}$ and $\kappa_{R}$ are the intrinsic magnetization damping
rates of the Kittel modes in the left and right nanowires, respectively,
$\kappa_{p,L}$ represents the radiative damping of the microwave photons
$\hat{p}_{\mathrm{in}}^{(L)}$ by the coupling of the left nanowire with the
microwave source, and $\kappa_{q}$ denotes the intrinsic damping of the spin
waves in the film. In frequency space:

$\displaystyle\hat{\alpha}_{q}(\omega)$
$\displaystyle=g_{q}G_{q}\left(\omega\right)\left[e^{-iqR_{1}}\hat{m}_{L}(\omega)+e^{-iqR_{2}}\hat{m}_{R}(\omega)\right],$
(228a) $\displaystyle\hat{m}_{R}(\omega)$
$\displaystyle=\frac{-i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}\hat{m}_{L}(\omega),$
(228b) $\displaystyle\hat{m}_{L}(\omega)$
$\displaystyle=\frac{-\sqrt{\kappa_{p,L}}}{-i(\omega-\omega_{\mathrm{K}})+(\kappa_{L}+\kappa_{p,L})/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)-f(\omega)}\hat{p}_{\mathrm{in}}^{(L)}(\omega),$
(228c)

where
$G_{q}\left(\omega\right)=\left[(\omega-\omega_{q})+i\kappa_{q}/2\right]^{-1}$
and

$f(\omega)\equiv-\frac{\left(\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{1}-R_{2})}\right)\left(\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}\right)}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}.$
(229)

When the chiral coupling is perfect $f(\omega)$ vanishes without the back-
action. The spin waves excited by the left nanowire in the film propagate to
the right nanowire, where microwave striplines or wave guides detect their
stray fields inductively. The relation between the output
$\hat{p}_{\mathrm{out}}^{(L)}(\omega)$ and
$\hat{p}_{\mathrm{out}}^{(R)}(\omega)$ and the input
$\hat{p}_{\mathrm{in}}^{(L)}(\omega)$ amplitudes is [295, 294]

$\displaystyle\hat{p}_{\mathrm{out}}^{(L)}(\omega)$
$\displaystyle=p_{\mathrm{in}}^{(L)}(\omega)+\sqrt{\kappa_{p,L}}\hat{m}_{L}(\omega),$
(230a) $\displaystyle\hat{p}_{\mathrm{out}}^{(R)}(\omega)$
$\displaystyle=\sqrt{\kappa_{p,R}}\hat{m}_{R}(\omega),$ (230b)

where $\kappa_{p,R}$ is the radiative damping rate induced by the detector.
The elements in the microwave scattering matrix, i.e., microwave reflection
$\left(S_{11}\right)$ and transmission $\left(S_{21}\right)$ amplitudes are
then

$\displaystyle S_{11}(\omega)$
$\displaystyle\equiv\frac{\hat{p}_{\mathrm{out}}^{(L)}}{\hat{p}_{\mathrm{in}}^{(L)}}=1-\frac{\kappa_{p,L}}{-i(\omega-\omega_{\mathrm{K}})+(\kappa_{L}+\kappa_{p,L})/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)-f(\omega)},$
(231a) $\displaystyle S_{21}(\omega)$
$\displaystyle\equiv\frac{\hat{p}_{\mathrm{out}}^{(R)}}{\hat{p}_{\mathrm{in}}^{(L)}}=\left[1-S_{11}(\omega)\right]\sqrt{\frac{\kappa_{p,R}}{\kappa_{p,L}}}\frac{i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}.$
(231b)

Figure 31 shows the real components of $S_{11}$ and $S_{12}$ when the
magnetizations of Co nanowire and YIG film are anti-parallel, which can be
realized in experiments because the magnetic form anisotropy of the wire is
much larger than that of the film. The absorption of the microwaves at the FMR
of the Co wire causes the pronounced dip in the reflection amplitude in Fig.
31(a). The fringes in the transmission spectra in Fig. 31(b) reflect the spin
wave propagation in the film from the left to the right nanowire, caused by
the propagation phase factor $e^{iq(R_{1}-R_{2})}$ in Eq. (231b) that
oscillates with the momentum $q$ selected by the frequency $\omega_{in}$. This
is not an interference pattern since spin waves are not reflected at the right
nanowire. The chirality is implicit: When $g_{-|q|}=0$ and $R_{2}>R_{1}$, spin
and energy are transmitted from 1 to 2 and $S_{21}\neq 0$, while no waves can
travel in the opposite direction, i.e., $S_{21}=0$.

Figure 31: Reflection $\operatorname{Re}(S_{11})$ [(a)] and transmission
$\operatorname{Re}(S_{12})$ [(b)] amplitudes of microwaves, Eq. (231b),
between two parallel magnetic nanowires on a magnetic film.
$\omega_{\mathrm{Co}}$ is the Kittel mode frequency Eq. (199) of the cobalt
nanowire at a fixed applied magnetic field ($H^{0}_{\mathrm{app}}=0.05$ T)
that forces the magnetization of the soft YIG film to be antiparallel to that
of the wires. $\omega_{\mathrm{in}}$ is the microwave frequency and
$\omega_{\mathrm{K}}$ is the Co Kittel mode frequency as a function of
$H_{\mathrm{app}}$ close to $H^{0}_{\mathrm{app}}$. The radiative damping rate
for both nanowires is chosen as $\kappa_{p}/(2\pi)=10$ MHz. The Co nanowire
with magnetization $\mu_{0}\tilde{M}_{s}=1.1$ T is $w=70$ nm wide and $d=20$
nm thick. We adopt an exchange stiffness $\alpha_{\rm ex}^{({\rm
Co})}=3.1\times 10^{-13}$ cm2, and the Gilbert damping coefficient
$\alpha_{G}^{\mathrm{(Co)}}=2.4\times 10^{-3}$. The YIG film thickness $s=20$
nm with magnetization $\mu_{0}M_{s}=0.177$ T and exchange stiffness
$\alpha^{(\rm YIG)}_{\rm ex}=3.0\times 10^{-12}$ cm2. The figures are adopted
from Ref. [67].

Experimental evidence.—Chiral microwave transmission was reported in two
different systems. Chen et al. [49] first reported unidirectional pumping of
dipolar-exchange spin waves in a Co nanowire array on top of ultrathin ($s=$20
nm) YIG films, as illustrated in Fig. 32(a). Subsequently, they simplified the
sample to only two magnetic nanowires [52], as shown in Fig. 32(e), which is
the configuration discussed above. In the earlier experiments, a co-planar
waveguide on top of the device generates nearly uniform microwaves that hardly
excite propagation spin waves in the YIG film at low resonant frequencies.
When reaching the resonance of the Co nanowire array at much higher
frequencies, the dipolar or interface exchange interaction with the precessing
Co magnetization excites spin waves in the YIG film with momenta up to the
order of the reciprocal wire width. The spin waves travel to and inductively
excite another coplanar waveguide. Figure 32(b) shows that the device
functions indeed as an almost perfect and switchable microwave diode. For the
negative applied field (anti-parallel magnetizations) the transmission
$S_{21}$ from waveguide 1 to 2 is finite, but it vanishes for the opposite
direction $S_{12}$. Brillouin light scattering detects an imbalance in the
magnon numbers in the Co wires on both sides of a waveguide, as illustrated in
Figs. 32(c) and (d) [53]. Since magnetizations are anti-parallel, excess
magnons are detected on the right side. The effects observed for the nanowire
array can be interpreted in terms of the chiral pumping mechanism via the
dipolar interaction as explained above with minor modifications for the array
configuration [48]. The unidirectionality of the spin-wave excitation switches
with the YIG magnetization from the parallel to anti-parallel configurations
as shown in Fig. 31(b) with symmetry $S_{12}(\textbf{M})=S_{21}(-\textbf{M})$
because of the chirality of dipolar field of the wire that is locked to its
momentum of the spin wave in the film (refer to Fig. 19) [50, 67, 48]. The
spin wave polarization switches sign with the film magnetization and couples
to the dipolar field of the wire with opposite momentum components. The theory
[50] also predicts a critical angle $\arccos(\sqrt{d/w})$ (for $w>d$) between
the magnetizations of film and wire at which the chirality vanishes.

Figure 32: Observation of chiral pumping of spin waves with magnetic wire
array [(a)] or two magnetic wires [(b)] on top of thin YIG films. The Co wire
magnetizations are either parallel or anti-parallel to that of YIG film in
these observations. (b) and (f) are the microwave transmission amplitudes from
port 1 to 2 ($S_{21}$) and vice versa ($S_{12}$). (c) illustrates the setup of
the Brillouin light scattering experiments of the magnon distribution in the
magnetic wires with results shown in (d). The figures are taken from Ref. [49,
52].

The transmission of the spin waves between the microwave source and detector
is complicated by the nanowire configuration. Subsequent experiments [52]
retain only two Co nanowires on the YIG films, as illustrated in Fig. 32(e).
They are capped by metallic striplines that act as source and detector of wire
magnons (not shown). The microwave transmission in Fig. 32(f) is also non-
reciprocal, but now the phase coherence of the magnons is resolved in terms of
fringes that reflect the propagation phase in Fig. 31(b) [50, 67] as explained
above. Recently, Temdie et al. further downscaled the chiral coupling
technique implementing permalloy nanowires arrays, with which they observed
the non-reciprocal transmission of exchange spin waves with shorter
wavelengths down to $\lambda\approx 80$ nm [297].

DMI.—The chirality from the dipolar interaction may coexist with that caused
by DMI. Szulc et al. [56] demonstrated by micromagnetic simulations that the
interfacial DMI can induce a non-reciprocity of the magnon propagation in the
Damon-Eshbach configuration in metallic magnetic spin valves. The DMI induced
by the interface to heavy metal on one side shifts the spin-wave dispersion in
one layer such that its stray fields couple in-phase with spin waves in the
other layer that propagate in one direction, but out-of-phase with those
moving into another direction. The authors suggested that spin-wave diode and
circulator functionalities may be useful in magnonic logic circuits.

####  4.2.3 Chiral spin Seebeck effect

The chirality of the dipolar spin-photon interaction is ubiquitous and can
significantly affect observables other than the microwave scattering matrix
for ferromagnets.

Chiral spin Seebeck effect.—The generation of a spin current by a temperature
gradient is called spin Seebeck effect [298, 299, 300, 301]. It can be
detected by the perturbation of the equilibrium balance that cancels spin
pumping and torque at the interface of the magnet to a normal metal. “Thermal
spin pumping” can be generated by the Joule heating caused by an electric
current in a metallic contact on top of a magnetic film, while the magnon
accumulation can be detected electrically in a second contact via the inverse
spin Hall effect [302]. In contrast to the single-mode propagating microwave
spectroscopy discussed above, the magnon transport is diffuse. In principle,
all thermal magnons contribute, at room temperature up to THz [303].

The chiral spin Seebeck effect refers to the directional generation of a spin
current in a ferromagnet under a temperature gradient. Here we illustrate its
input-output theory by focusing on the Kittel modes of two magnetic nanowires
at $\mathbf{r}_{2}=R_{2}\hat{\mathbf{y}}$ (the detectors) and at
$\mathbf{r}_{1}=R_{1}\hat{\mathbf{y}}$ with $R_{1}<R_{2}$ on top a magnetic
insulator. We focus on the configuration in which the magnetization of the
wires is opposite to that of the film. We disregard thermal occupation of
higher standing magnon modes in the wires and the film, which is allowed at
low temperatures, fine wires, and thin films. We assume that the magnon mean-
free path is much larger than the distance between the wires. The propagating
magnons in the long nanowires then couple coherently only to the magnons in
the film that propagate perpendicular to the wires. We assume that the non-
local dipolar interaction between the wires and film dominates the contact
exchange interaction and therefore the conventional spin Seebeck effect.

The coupled equations of motion of the Kittel modes in the nanowires
$\hat{m}_{L}$, $\hat{m}_{R}$ and spin waves in the film $\hat{\alpha}_{q}$
then read

$\displaystyle\frac{d\hat{m}_{L}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{L}-\sum_{q}ig_{q}^{\ast}e^{iqR_{1}}\hat{\alpha}_{q}-\frac{\kappa}{2}\hat{m}_{L}-\sqrt{\kappa}\hat{N}_{L},$
(232a) $\displaystyle\frac{d\hat{m}_{R}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{R}-\sum_{q}ig_{q}^{\ast}e^{iqR_{2}}\hat{\alpha}_{q}-\frac{\kappa}{2}\hat{m}_{R}-\sqrt{\kappa}\hat{N}_{R},$
(232b) $\displaystyle\frac{d\hat{\alpha}_{q}}{dt}$
$\displaystyle=-i\omega_{q}\hat{\alpha}_{q}-ig_{q}e^{-iqR_{1}}\hat{m}_{L}-ig_{q}e^{-iqR_{2}}\hat{m}_{R}-\frac{\kappa_{q}}{2}\hat{\alpha}_{q}-\sqrt{\kappa_{q}}\hat{N}_{q},$
(232c)

where $\kappa=\alpha_{G}\omega_{\mathrm{K}}$ represents the Gilbert damping in
the contacts and $\kappa_{q}$ that of the film. We model the equilibrium
thermal fluctuations $\hat{N}_{L}$ and $\hat{N}_{R}$ by the white noise time-
correlation functions
$\langle{\hat{N}_{\eta}^{\dagger}(t)\hat{N}_{\eta^{\prime}}(t^{\prime})}\rangle=n_{\eta}\delta(t-t^{\prime})\delta_{\eta\eta^{\prime}}$,
where $\eta\in\\{L,R\\}$,
$n_{\eta}=1/\left\\{\exp\left[\hbar\omega_{\mathrm{K}}/(k_{B}T_{\eta})\right]-1\right\\}$.
By integrating out the spin-wave modes in the film, we obtain equations for
the dynamics of the two wires that are dissipatively coupled by the incoherent
thermal fluctuations in the film. The wires couple differently to right
$\left(\Gamma_{1}=|g_{q_{\ast}}|^{2}/v_{q_{\ast}}\right)$ and left-moving
$\left(\Gamma_{2}=|g_{-q_{\ast}}|^{2}/v_{q_{\ast}}\right)$ modes, where
$q_{\ast}$ is the positive root of $\omega_{q_{\ast}}=\omega_{\mathrm{K}}$.
The wire-film interaction is fully chiral when $\Gamma_{2}=0$. With
$\hat{m}_{L,R}(t)=\int e^{-i\omega t}\hat{m}_{L,R}(\omega)d\omega/(2\pi)$ and
$G_{q}(\omega)=1/[(\omega-\omega_{q})+i\kappa_{q}/2]$,

$\displaystyle\hat{m}_{L}(\omega)$
$\displaystyle=\frac{\sum_{q}ig_{q}^{\ast}e^{iqR_{1}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{L}(\omega)}{-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}}{2}},$
(233a) $\displaystyle\hat{m}_{R}(\omega)$
$\displaystyle=\frac{\sum_{q}ig_{q}^{\ast}e^{iqR_{2}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{R}(\omega)-\Gamma_{1}e^{q_{\ast}(R_{2}-R_{1})}\hat{m}_{L}(\omega)}{-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}}{2}}.$
(233b)

Disregarding the damping in the film $\left(\kappa_{q}\rightarrow 0\right)$ is
an excellent approximation when the contacts distance is much smaller than the
magnon propagation length. The Kittel mode occupation number then becomes

$\displaystyle\rho_{L}$
$\displaystyle\equiv\langle\hat{m}_{L}^{\dagger}(t)\hat{m}_{L}(t)\rangle=n_{L}+\int\frac{d\omega}{2\pi}\frac{\kappa}{(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}}(n_{q_{\ast}}-n_{L}),$
(234a) $\displaystyle\rho_{R}$
$\displaystyle\equiv\langle\hat{m}_{R}^{\dagger}(t)\hat{m}_{R}(t)\rangle=n_{R}+\int\frac{d\omega}{2\pi}\frac{\Gamma_{1}^{2}\kappa}{\left[(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}\right]^{2}}(n_{L}-n_{q_{\ast}}).$
(234b)

We can now compute the effect of a temperature change $T_{L}$ in $n_{L}$ in
the left wire with $n_{R}=n_{q_{*}}$ is at the same thermal equilibrium
temperature $T_{0}$. To leading order in the temperature difference, the non-
local thermal injection of magnons into the passive right nanowire by the
active left one then reads from Eq. (234b)

$\displaystyle\delta\rho_{R}$
$\displaystyle=\mathcal{S}_{\mathrm{CSSE}}(T_{L}-T_{0}),$ (235a)
$\displaystyle\mathcal{S}_{{}_{\mathrm{CSSE}}}$
$\displaystyle=\int\frac{d\omega}{2\pi}\frac{\Gamma_{1}^{2}\kappa}{\left[(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}\right]^{2}}\left.\frac{dn_{L}}{dT}\right|_{T_{0}}.$
(235b)

On the other hand, when we only change the temperature of the right wire
$\delta\rho_{L}=0$. This defines the chiral (or dipolar) spin Seebeck effect
with coefficient $\mathcal{S}_{{}_{\mathrm{CSSE}}}.$

We note that the chirality does not cause a rectification of the heat current
that in the linear response would require a “Maxwell Demon". When
$T_{L}>T_{0}$, magnons are injected from the left wire to the right one, while
when $T_{L}<T_{0}$ the current is induced by “holes" in the thermal
distribution, which is equivalent to a spin and heat current in the opposite
direction. We hereby resolve an issue raised in the Supplemental Material of
Ref. [50]. Instead, the chirality is exposed in the form of a non-local heat
current switch.

Experiments in diffuse regime.— As sketched above, chiral coupling can affect
magnon transport in the diffuse regime even without any spin-orbit coupling.
Han et al. [304] studied the non-local magnon transport in a YIG film
(thickness $s=$50 nm, Gilbert damping $\alpha_{G}=10^{-4}$) under a gate of
the magnetic metal alloy NiFe (40 nm thick) as sketched in Fig. 33(a). They
reported tunable non-reciprocal propagation of magnons between Pt wires at
room temperature. Here a large number of thermally excited magnons up to
several THz contribute to the signal. The NiFe layer has a strong in-plane
uniaxial form anisotropy with an easy axis along the $\hat{\bf y}$ direction,
while the soft YIG magnetization direction can be rotated freely in the film
plane by an applied field $\mathbf{H}$, as illustrated in Fig. 33(a). In the
bare YIG film the non-local spin-Hall voltage depends as $\sin^{2}\varphi$ on
the field angle $\varphi$ [the left panel of Fig. 33(b)] [302]. The NiFe gate
distorts the signal line shape and introduces a dependence on the magnon
current direction, i.e., when the role of source and detector are exchanged
[the right panel of Fig. 33(b)]. Figure 33(c) shows the magnon diffusion
lengths extracted from experiments with different gate widths. The difference
on current direction is the largest when $\varphi$ is close to $90^{\degree}$
and $270^{\degree}$, but tends to vanish when $\varphi=0^{\degree}$ and
$180^{\degree}$. The sudden jump of the signal when $\varphi\sim
100^{\degree}$ and $\sim 280^{\degree}$ is attributed to an abrupt switch of
the $y$-component of NiFe with a form anisotropy of the order of the applied
magnetic field.

Han et al. [304] attributed this non-reciprocity in terms of a chiral dipolar
interaction as sketched in Fig. 33(d), arguing that the dipolar interaction
modifies the damping of right and left moving magnons in the film differently
[304]. Whether the additional damping by chiral interactions [67] can really
explain these experiments requires additional theoretical efforts.

Figure 33: Observation of asymmetric diffusion of thermal magnons in YIG
capped by a film of the magnetic alloy NiFe. (a) is the configuration with two
Pt wires that excite and detect the magnon transport. The dependence of the
detected voltage in one Pt wire on the magnetization direction is plotted in
(b) without and with the NiFe overlayer. The magnon diffusion length depends
on the width of the NiFe strip as shown in (c) for different magnetization
configurations. (d) illustrates the chiral interaction mechanism. The figures
are taken from Ref. [304].

####  4.2.4 Magnon trap

In a field-effect transistor, an electronic gate controls the flow of
electrons, amplifying a gate signal or switching the source-drain current on
and off. “Spin transistors” are three-terminal spintronic devices that
manipulate a source-drain spin current by a gate that couples to the spin
degree of freedom. Manipulation of magnon currents in an on-chip circuit is a
key functionality in magnonics [309, 46, 305, 306, 307, 308]. Electric gates
easily control electrons, but manipulating magnons on a small length scale is
challenging. Chumak et al. [310] etched a magnonic crystal of 20 parallel
grooves into the surface of a YIG strip. The interaction with magnons injected
into this region suppresses the source-drain magnon currents by three orders
of magnitude. Wu et al. reported a 32$\%$ modulation of magnon current in a
YIG|Au|YIG perpendicular spin valve when comparing a parallel and anti-
parallel alignment [311], while Cramer et al. [312] reported a 290$\%$
difference in a cobalt|cobalt oxide|YIG trilayer. By injecting magnons by a Pt
modulator into the source-drain channel of YIG films, the magnon conductivity
can be strongly enhanced [313, 314, 315].

In spite of the recent progress, magnon transistors still manage only small
on-off ratios of the source-drain current. We believe that chiral spin pumping
in multi-terminal magnonic devices may offer more efficient control
mechanisms. An example is the trapping and release of spin wave excitations
[51] as realized between long magnetic nanowires on top of a thin film of YIG
as in Fig. 34(a). We can treat this case by extending the theory of coherent
chiral excitation by a single wire (Sec. 4.2.2) to two or more wires. Here we
address the effect of passive wires that act as gates on the coherent spin
waves underneath. A Kittel magnon mode $\hat{m}_{l}$ with frequency
$\omega_{\mathrm{K},l}$ in the $l$-th magnetic wire at $R_{l}\hat{\mathbf{y}}$
interacts with the propagating spin waves in the magnetic film
$\hat{\alpha}_{k}$ of frequency $\omega_{k}$ [47, 50]. By the chiral coupling,
the precession in the wires interacts preferentially with the traveling waves
propagating in one direction when either one of the coupling constants
$g_{|k|}$ or $g_{-|k|}$ vanishes. Even more than that in the diffuse situation
discussed in the previous Sec. 4.2.3, the passive wire exerts strong effects
in the coherent case. The spin waves that connect magnetic wires can be
integrated out of the problem mediates an effective interaction that couples
their dynamics.

Let us put the wires on $\mathbf{r}_{1}=R_{1}\hat{\mathbf{y}}$ and
$\mathbf{r}_{2}=R_{2}\hat{\mathbf{y}}$ and assume couplings
$g_{k,1}=g_{k,2}=g_{k}$ and frequencies
$\omega_{\mathrm{K},1}=\omega_{\mathrm{K},2}=\omega_{\mathrm{K}}$, the
equation of motion becomes

$\displaystyle\frac{d}{dt}\left(\begin{matrix}\hat{m}_{L}\\\
\hat{m}_{R}\end{matrix}\right)+i\left(\begin{matrix}\tilde{\omega}_{\mathrm{K}}-i\Gamma(\omega)&-i\Gamma_{12}(\omega)\\\
-i\Gamma_{21}(\omega)&\tilde{\omega}_{\mathrm{K}}-i\Gamma(\omega)\end{matrix}\right)\left(\begin{matrix}\hat{m}_{L}\\\
\hat{m}_{R}\end{matrix}\right)=\left(\begin{matrix}\hat{P}_{1}\\\
\hat{P}_{2}\end{matrix}\right),$ (236)

where $\tilde{\omega}_{\mathrm{K}}=\omega_{\mathrm{K}}-i\kappa/2$ and
$\kappa=2\alpha_{\mathrm{G}}\omega_{\mathrm{K}}$ is the material damping rate
of the Kittel modes in terms of the Gilbert constant $\alpha_{\mathrm{G}}$,
while $\hat{P}_{l}$ is the input power of the local antennas. The self-
interaction

$\Gamma(\omega)=\frac{1}{2v(k_{\omega})}\left(|g_{k_{\omega}}|^{2}+|g_{-k_{\omega}}|^{2}\right)$
(237)

is a pumping-induced loss rate for a single nanowire [50, 67]. With
$R_{2}>R_{1}$, the directional coupling rate reads

$\displaystyle\Gamma_{12}(\omega)$
$\displaystyle=\frac{1}{v(k_{\omega})}|g_{-k_{\omega}}|^{2}e^{ik_{\omega}(R_{2}-R_{1})},$
(238a) $\displaystyle\Gamma_{21}(\omega)$
$\displaystyle=\frac{1}{v(k_{\omega})}|g_{k_{\omega}}|^{2}e^{ik_{\omega}(R_{2}-R_{1})},$
(238b)

where $v(k)$ is the group velocity of the traveling spin waves and
$k_{\omega}$ is the positive root of $\omega_{k}=\omega=\omega_{\mathrm{K}}$
at the ferromagnetic resonance (FMR). When $g_{-k}=0$,
$|\Gamma_{21}(\omega)|=2\Gamma(\omega)$, i.e., twice as large as the magnon
broadening Eq. (237) induced by the local spin pumping. In the perfect chiral
limit, one magnet affects the other without any back action. The input-output
theory introduced in Sec. 4.2.2 leads to expressions for the transmission
amplitude of microwaves that excite one of the magnetic wires and inductively
escape from the other [51, 75]. The two magnetic amplitudes are related as

$\hat{m}_{R}(\omega_{\mathrm{K}})=\eta(\omega_{\mathrm{K}})e^{i\pi+ik_{\omega}(R_{2}-R_{1})}\hat{m}_{L}(\omega_{\mathrm{K}}),$
(239)

where
$\eta(\omega_{\mathrm{K}})={2\Gamma(\omega_{\mathrm{K}})}/({\kappa/2+\Gamma(\omega_{\mathrm{K}})})$
is the spin wave transmission amplitude. We observe that the magnon
propagation from $R_{1}$ to $R_{2}$ contributes a phase factor

$\displaystyle\Delta\phi=\pi+k_{r}(R_{2}-R_{1}),$ (240)

where $k_{r}(R_{2}-R_{1})$ is the conventional plane wave phase delay, while
$\pi$ is twice the dissipative phase shift $\pi/2$ between the Kittel modes
and the propagating waves. Remarkably, when $\kappa/2\ll\Gamma$,
$\eta\rightarrow 2$, it corresponds to an enhancement of the magnon amplitude
by a factor of 2. The damping $\Gamma$ and coupling $\eta$ can be tuned by the
external magnetic field as shown in Fig. 34(b). When $\kappa/2=\Gamma$,
$\eta=1$ and we observe a perfect “trapping” of magnons in and under the
passive magnet, as illustrated in Fig. 34(c).

Figure 34: Chirality-induced trapping of magnons. (a) The device consists of
two parallel magnetic wires on top of a thin YIG film: microwaves excite only
the left one. (b) compares the intrinsic (Gilbert) and pumping-induced damping
($\Gamma$) and the transmission amplitude ($\eta$) as a function of magnetic
field. (c) shows the localized magnetization amplitude in the film around the
second contact under “trapping” conditions. (d) illustrates a partial trapping
in which some spin wave energy escapes to the right. The figures are taken
from Ref. [51].

We can express the excited magnetization in the real space by the Green
function method introduced in Sec. 4.2.2. For $y<R_{1}<R_{2}$,

$\displaystyle\hat{M}^{L}_{\alpha}(x)=\frac{2i}{v_{k_{\omega}}}\sqrt{2M_{s}\gamma\hbar}\hat{m}_{L}(\omega_{\mathrm{K}})m_{\alpha}^{(k_{\omega})}(x)g_{-k_{\omega}}\left(e^{ik_{\omega}(R_{1}-y)}-\eta(\omega_{\mathrm{K}})e^{ik_{\omega}(2R_{2}-R_{1}-y)}\right)$
(241)

vanishes for chiral coupling $g_{-k}=0$. The magnetization amplitude to the
right of both wires

$\displaystyle\hat{M}_{\alpha}^{R}(x)=\frac{2i}{v_{k_{\omega}}}\sqrt{2M_{s}\gamma\hbar}\hat{m}_{L}(\omega_{\mathrm{K}})m_{\alpha}^{(k_{\omega})}(x)g_{k_{\omega}}e^{-ik_{+}^{\ast}(R_{1}-y)}\left(1-\eta(\omega_{\mathrm{K}})\right)$
(242)

vanishes for $\eta(\omega_{\mathrm{K}})\rightarrow 1$ as in Fig. 34(c). Here,
$k_{+}^{\ast}=(k_{\omega}+i\epsilon)$ in terms of the inverse extinction
length $\epsilon$. Between the wires ($R_{1}<y<R_{2}$),

$\hat{M}_{\alpha}^{M}(x)=\frac{2i}{v_{k_{\omega}}}\sqrt{2M_{s}\gamma\hbar}\hat{m}_{L}(\omega_{\mathrm{K}})m_{\alpha}^{(k_{\omega})}(x)g_{k_{\omega}}e^{-ik_{+}^{\ast}(R_{1}-y)}$
(243)

is a right-propagating wave for $g_{-k}=0$. The excited magnetization decays
on the length scale of the inverse of $1/$Im$k_{+}^{\ast}$, which is not
significant in the region plotted here. The trapped magnetization is not a
standing wave that requires the interference of back and forth reflections.
The magnetic energy that is injected from the left accumulates in a small
region under the contact and for weak damping reaches a large steady-state
density. Tuning $\eta$ allows a controlled escape of magnons to the right as
illustrated by Fig. 34(d).

Experiment.—The experiments carried out by the Wang et al. on the chiral
magnon pumping by an array of Co wires Ref. [53] into YIG thin films provides
additional evidence on the magnon trapping. They measured microwave
transmission, see Fig. 35(a,b), and Brillouin light scattering (BLS), see Fig.
35(c,d) when the wire and YIG film magnetizations are parallel and anti-
parallel. The broadening of the FMR of Co wires in Fig. 35(b) is larger in the
anti-parallel configuration. The BLS signals in Fig. 35(c) image the magnon
density in the Co wires that decays with distance from the coplanar wave guide
on a scale much shorter than the spin wave decay length in YIG, see Fig.
35(d). Larger coupling strengths $g_{k}$ for the parallel alignment [50, 49],
i.e., Eq. (237) explains the faster decay in that configuration. The authors
interpreted the strong enhancement of the BLS signal in the third wire in Fig.
35(d) in terms of the magnon trapping effect. Wang et al. [52] observed the
phase shift Eq. (240) with two magnetic nanowires on top of the magnetic film
by microwave spectroscopy.

Figure 35: Tunable damping of magnons in an array of Cobalt nanowires on top
of a thin YIG film. (a) shows the microwave transmission in the Damon-Eshbach
configuration with parallel and anti-parallel magnetization. The extracted
line width in the parallel (“P”) and anti-parallel (“AP”) configurations of Co
and YIG magnetizations are shown in (b). The Brillouin light scattering
microscopy sketched in (c) measures the spatial distribution of the excitation
over the individual cobalt wires as shown in (d). The figures are taken from
Ref. [53].

Miscellaneous.— A not yet solved issue is the relative importance of the
dipolar coupling compared to the interface exchange interaction between a
magnetic insulator film and a metallic magnetic wire. This is not universal,
but depends on the material and interface quality. The short-range exchange
interaction can be suppressed by atomically thin insulating spacers. Reference
[316] reported a large antiferromagnetic interface exchange effect in the FMR
of Co|YIG bilayers with direct contact [316]. Model calculations pointed to an
important exchange contribution to the strong coupling observed between
propagating magnons in YIG films and Ni wires [48] that agree with
observations [296]. Reference [53, 49] found evidence for a dominant dipolar
interaction over the exchange ones for Co wires. First principle calculations
[317] can help in assessing the importance of the interface exchange for
different interfaces.

Equation (237) in Refs. [50, 51, 67] holds for a single magnetic wire on top
of magnetic film [50, 51] and weak spin wave damping. The magnon trapping
effect was originally formulated for two-wire configuration [51]. On the other
hand, the dynamics of magnonics crystal formed by a closely spaced array of
nanomagnets [53, 49, 296] is collective [76, 60, 61, 318, 71], which deserves
more attention in the future.

####  4.2.5 Synthetic antiferromagnets

The chiral dipolar field emitted by a magnon can drive other magnets. Here we
address a spin valve of two magnets that are separated by a thin normal metal
spacer that non-locally couples into an anti-parallel configuration, i.e., a
“synthetic antiferromagnet” (SAFM). References [54, 55, 320, 319] reported
non-reciprocal spin wave propagation in SAFMs. We argue now that the dipolar
coupling discovered in the nanowire on film geometry [53, 49, 296] is also at
work here, which may come as a surprise since the dipolar coupling does not
affect the equilibrium configuration in extended thin films. To this end, we
proceed from the quantum formalism introduced above [50, 320, 319] and discuss
the relevant publications.

Bilayer.—Figure 36(a) shows a bilayer of two magnetic films with thicknesses
$h$ and $s$. The magnetodipolar interlayer coupling reads

$\hat{H}_{d}=-\mu_{0}\gamma^{2}\frac{1}{4\pi}\int d{\bf r}\hat{\tilde{\bf