11institutetext: Tao Yu 22institutetext: Max Planck Institute for the
Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg,
Germany, 22email<EMAIL_ADDRESS>33institutetext: Gerrit E. W. Bauer
44institutetext: Institute for Materials Research & WPI-AIMR & CSRN, Tohoku
University, Sendai 980-8577, Japan,
Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft,
The Netherlands,
44email<EMAIL_ADDRESS>
# Chiral Coupling to Magnetodipolar Radiation

Tao Yu and Gerrit E. W. Bauer

###### Abstract

We review and extend the theory of chiral pumping of spin waves by
magnetodipolar stray fields that generate unidirectional spin currents and
asymmetric magnon densities. We illustrate the physical principles by two
kinds of chiral excitations of magnetic films, i.e., by the evanescent Oersted
field of a narrow metallic stripline with an AC current bias and a magnetic
nanowire under ferromagnetic resonance.

##  0.1 Introduction

“Handedness” or “chirality” of wave propagation is a lively research topic in
optics, acoustics, and condensed matter physics. The “spin” of magnons is
rooted in the time-reversal symmetry breaking of the magnetic order and leads
to chiral coupling with other excitations when locked to the momentum. This
phenomenon is governed by non-universal selection rules. This Chapter
clarifies a specific mechanism, viz. evanescent microwaves that can
efficiently generate chiral dynamics.

Magnonics and magnon spintronics magnonics1 ; magnonics2 ; magnonics3 ;
magnonics4  are emergent fields that hold the promise of a next-generation
low-power and scalable information processing and communication technology.
The generation of coherent and propagating spin waves is a crucial ingredient,
which can be realized by magnetic fields generated by microwave antennas such
as current-biased metallic striplines. In order to generate stray fields with
high-momentum Fourier components these must be small in size and placed close
to the magnetic medium. Not only the amplitude, but also the direction of the
excited spin waves depend on the excitation conditions that obey right-hand
rules and are therefore _chiral_.

In this Chapter, we focus on the chirality of the dipolar coupling between the
magnetization dynamics in ferromagnetic heterostructures chiral_simulation ;
Yu1 ; Yu2 ; nanowire ; accumulation1 ; accumulation2 , while those in optics
chiral_review ; chiral_optics1 ; chiral_optics2 ; chiral_optics3 ;
chiral_optics4 ; chiral_optics5 ; chiral_emitter , plasmonics near_field ;
Petersen , and magnetic structures with Dzyaloshinskii-Moriya interaction are
treated in other chapters. We focus on the favorite material of magnonics,
viz. the ferrimagnetic insulator yttrium iron garnet (YIG) with high Curie
temperature and outstanding magnetic and acoustic quality
low_damping_nanometer . Its magnons can be excited electrically by heavy metal
contacts Ludo , acoustically acoustic , as well as by a large spectrum of
electromagnetic waves from gigahertz (microwaves) to petahertz (light).
Magnonic transducers with spatially separated contact that excite and detect
magnons chiral_simulation ; Ludo ; centimeter ; nanomagnetism ; Dirk_2013 ;
CoFeB_YIG ; Co_YIG ; Haiming_NC ; Haiming_PRL  are sensitive probes to study
magnon transport. We illustrate the chiral physics for thin YIG films with in-
plane magnetizations, but other materials and configurations can be treated by
changing the model parameters.

The spin waves of in-plane magnetized films can be classified by the
interaction that governs their dispersion as a function of wave vector, into
the dipolar, dipolar-exchange and exchange type with energies ranging from a
few gigahertz to many terahertz magnonics1 ; magnonics2 ; magnonics3 ;
magnonics4 ; centimeter . The long-wavelength modes are dipolar, whereas the
short-wavelength ones are exchange. Bulk volume modes and surface (Damon-
Eshbach) modes propagate along or perpendicular to the magnetization direction
with different dispersion relations Walker ; DE ; spin_waves_book ; new_book .
Moreover, the surface modes are chiral: their propagation direction (linear
momentum) is fixed by the outer product of surface normal and magnetization
direction, allowing unidirectional spin current generation by dominantly
exciting one surface of a magnetic film heat_conveyer1 ; heat_conveyer2 ;
heat_conveyer3 ; heat_conveyer4 . However, Damon-Eshbach spin waves are not
well suited for applications — their group velocity tends to be zero when the
linear momentum is larger than the inverse of film thickness, leading to a
small spin conductivity. They are also very sensitive to dephasing by surface
roughness surface_roughness , and do not exist in sufficiently thin films.

An alternative to intrinsically chiral spin waves is the chiral excitation of
non-chiral ones. Micromagnetic simulations chiral_simulation  revealed that
the AC dipolar field emitted by a magnetic nanowire on top of an in-plane
magnetized film with magnetization normal to the wire can excite
unidirectional spin waves. We have been motivated by experiments on an array
of magnetic nanowires on top of an ultrathin YIG film that generated
unidirectional spin waves parallel to the surface and perpendicular to the
nanowires Yu2  to develop a general theory of coherent and incoherent chiral
excitation of magnons Yu1 ; nanowire  by the dipolar interaction between the
dynamics of a magnetic film and a magnetic transducer. The chirality can be
traced to the different stray fields generated by spin waves with opposite
polarization and propagation. By angular momentum conservation electromagnetic
waves with particular polarization emitted by a magnetic transducer couple
only the circularly polarized component of a spin wave with a certain
propagation direction chiral_review . When dipolar or crystal anisotropy mixes
the right and left circularly polarized components, magnons are still excited
preferentially, but not exclusively, in one direction. Finally, a (short-
range) exchange coupling between film and transducer is not sensitive to the
propagation direction, and reduces the chirality.

Recently, two remarkable experiments confirmed our predictions. By NV
magnetometry Bertelli et al. NV_Teono  observed chiral pumping of spin waves
by a stripline antenna. Wang et al. two_wire_Haiming  measured unidirectional
microwave transmission mediated by two magnetic wires on top of a thin
magnetic film, i.e. chiral magnon-magnon coupling.

The chiral coupling to spin waves enables the generation and control of spin
currents Yu1 ; Yu2 ; nanowire  or spin accumulations accumulation1 ;
accumulation2  in ferromagnetic insulators, which is beneficial for spintronic
devices. In this short review, we comprehensively illustrate two kinds of
chiral coupling to the magneto-dipolar radiation, including the evanescent
field of a thin stripline that carries an AC current (Sec. 0.2) and that of a
magnetic wire under resonant excitation (Sec. 0.3).

##  0.2 Chiral excitation of spin waves by metallic stripline

We call a wave “chiral” when it propagates with handedness, i.e. in a certain
direction that is determined by two other control vectors, such as surface
normal and magnetic field. A rotating _electrical_ dipole nano_optics ;
Jackson  excites surface plasmon polaritons in one direction only near_field ;
Petersen ; nano_optics , while a precessing _magnetic_ dipole excite magnons
unidirectionally poineering_1 ; poineering_2 . Here we analyze solutions of
the combined Maxwell and Landau-Lifshitz-Gilbert equations that explain the
available experimental evidence. We analyze the near microwave field from a
normal metal strip line in Sec. 0.2.1 and its effect on a thin magnetic film
in Sec. 0.2.2 (see Fig. 1). We focus for simplicity on a configuration in
which the film normal is along the $x$-direction, $\hat{\mathbf{z}}$ is
parallel to a stripline that is assumed to be very long, and the excited spin
waves propagate in the $y$-direction.

Figure 1: (Color online) Chiral excitation of spin waves in a magnetic thin
film by the near field of a stripline antenna. The ac magnetic field is
axially symmetric with an oscillating modulus and in the film a position-
dependent linear polarization. It excites spin waves with the same frequency
and phase-matched spatial amplitude. The film magnetization direction (here
parallel to the stripline) can be tuned by a static magnetic field.

###  0.2.1 Oersted magnetic fields

We first demonstrate that even though the magnetic field of a stripline is
linearly-polarized in real space (see Fig. 1), it is chiral in momentum space.
Ampere’s Law states that the current density $\mathbf{J}(\mathbf{r})$
generates the vector potential Jackson

$\mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int
d\mathbf{r}^{\prime}dt^{\prime}\frac{\mathbf{J}(\mathbf{r}^{\prime},t^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\delta\left(t^{\prime}+\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}-t\right),$
(1)

where $\mu_{0}$ is the vacuum permeability and the delta-function represents
(non-relativistic) retardation. For a harmonic source
$\mathbf{J}\left(t\right)\sim\mathbf{J}(\omega)e^{-i\omega t}$,

$\mathbf{A}(\mathbf{r},\omega)=\frac{\mu_{0}}{4\pi}\int
d\mathbf{r}^{\prime}\mathbf{J}(\mathbf{r}^{\prime},\omega)\frac{e^{ik|\mathbf{r}-\mathbf{r}^{\prime}|}}{|\mathbf{r}-\mathbf{r}^{\prime}|},$
(2)

where $k=\omega/c$. The current in the stripline is uniform over the cross
section of width $w$ and thickness $t$ as well as length $L\ll c/\omega.$ In
the long wavelength limit and square cross section
$\mathbf{J}(\mathbf{r},\omega)\simeq\delta(x)\delta(y)\mathcal{J}(\omega)\hat{\mathbf{z}},$
where $\mathcal{J}$ is the total electric current, leading to

$\mathbf{A}(\mathbf{r},\omega)=\frac{\mu_{0}}{4\pi}\mathcal{J}(\omega)\hat{\mathbf{z}}\int_{-\infty}^{\infty}dz^{\prime}\frac{e^{ik\sqrt{x^{2}+y^{2}+z^{\prime
2}}}}{\sqrt{x^{2}+y^{2}+z^{\prime 2}}},$ (3)

which does not depend on $z$. Substituting the Weyl identity nano_optics

$\frac{e^{ik\sqrt{x^{2}+y^{2}+z^{2}}}}{\sqrt{x^{2}+y^{2}+z^{2}}}=\frac{i}{2\pi}\int
dk_{y}dk_{z}\frac{e^{ik_{x}|x|+ik_{y}y+ik_{z}z}}{k_{x}},$ (4)

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$ and $k_{x}=|a|+i|b|$ is
complex, into Eq. (3) yields

$\mathbf{A}(x,y,\omega)=\frac{i\mu_{0}}{4\pi}\mathcal{J}(\omega)\hat{\mathbf{z}}\int
dk_{y}\frac{e^{ik_{x}|x|+ik_{y}y}}{k_{x}}.$ (5)

The magnetic field
$\mathbf{H}(\mathbf{r})=\boldsymbol{\nabla}\times\mathbf{A}(\mathbf{r})/\mu_{0}=\left(\partial_{y}A_{z},-\partial_{x}A_{z},0\right)/\mu_{0}$
is transverse to the wire, see Fig. 1. Below the stripline $\left(x<0\right)$,

$\displaystyle H_{x}(x,y,\omega)$ $\displaystyle\equiv\int
dk_{y}e^{ik_{y}y}H_{x}(x,k_{y})=-\int
dk_{y}e^{ik_{y}y}\frac{\mathcal{J}(\omega)}{4\pi}\frac{k_{y}}{k_{x}}e^{-ik_{x}x},$
$\displaystyle H_{y}(x,y,\omega)$ $\displaystyle\equiv\int
dk_{y}e^{ik_{y}y}H_{y}(x,k_{y})=-\int
dk_{y}e^{ik_{y}y}\frac{\mathcal{J}(\omega)}{4\pi}e^{-ik_{x}x},$ (6)

where $k_{x}=\sqrt{(\omega/c)^{2}-k_{y}^{2}}.$ Directly above or below the
wire $H_{x}\left(x,y=0,\omega\right)=0,$ i.e. the magnetic field is linearly-
polarized along $y.$ The polarization rotates as a function of $y$ until
$H_{y}\left(0,y\rightarrow\infty,\omega\right)=0.$ Surprisingly, a circular
polarization emerges in the Fourier components

$\displaystyle H_{x}(x,k_{y},\omega)$
$\displaystyle=-\frac{\mathcal{J}(\omega)}{4\pi}\frac{k_{y}}{k_{x}}e^{-ik_{x}x},$
$\displaystyle H_{y}(x,k_{y},\omega)$
$\displaystyle=-\frac{\mathcal{J}(\omega)}{4\pi}e^{-ik_{x}x}.$ (7)

For an evanescent field with $k_{y}>\omega/c\equiv k$,
$k_{x}=i\sqrt{k_{y}^{2}-k^{2}}$ and $x<0$

$\displaystyle H_{x}(x,k_{y},\omega)$
$\displaystyle=\frac{i\mathcal{J}(\omega)}{4\pi}\frac{k_{y}}{\sqrt{k_{y}^{2}-k^{2}}}e^{\sqrt{k_{y}^{2}-k^{2}}x},$
$\displaystyle H_{y}(x,k_{y},\omega)$
$\displaystyle=-\frac{\mathcal{J}(\omega)}{4\pi}e^{\sqrt{k_{y}^{2}-k^{2}}x}.$
(8)

At microwave frequencies $\omega/(2\pi)\sim 10$ GHz, $k\equiv\omega/c\sim 200$
m-1 and wavelength $\lambda=2\pi/k\sim 3$ cm. The spin wavelength at the same
frequency is much smaller with $\sqrt{k_{y}^{2}+k_{z}^{2}}\gg\omega/c$, so we
are in the near-field limit. The magnetic field component $H_{x}\rightarrow
i\mathrm{sgn}(k_{y})H_{y}$ is then circularly polarized with a sign locked to
its linear momentum.

For a finite rectangular cross section with $0<x<t$ and $-w/2<y<w/2$ the
Fourier components of the magnetic field read

$\displaystyle H_{x}(x,k_{y},\omega)$
$\displaystyle=i\frac{J(\omega)}{4\pi}\mathcal{F}(t,w)\frac{k_{y}}{\sqrt{k_{y}^{2}-k^{2}}}e^{\sqrt{k_{y}^{2}-k^{2}}x},$
$\displaystyle H_{y}(x,k_{y},\omega)$
$\displaystyle=-\frac{J(\omega)}{4\pi}\mathcal{F}(t,w)e^{\sqrt{k_{y}^{2}-k^{2}}x},$
(9)

which differ from the previous results only by the form factor

$\mathcal{F}(t,w)=\frac{4}{k_{x}k_{y}}e^{ik_{x}\frac{t}{2}}\sin\left(k_{x}\frac{t}{2}\right)\sin\left(k_{y}\frac{w}{2}\right).$
(10)

Irrespective to the shape of the stripline, the magnetic field components are
circularly polarized when $\left|k_{y}\right|\gg\omega/c$ but oscillate now as
function of the wave vector.

###  0.2.2 Chiral excitation of spin waves

We focus here on thin YIG films with thickness
$d\sim\mathcal{O}(10~{}\mathrm{nm})$, which allows an analytical treatment of
the dispersion and spin wave amplitudes in the dipolar-exchange regime Yu1 .
An applied magnetic field $H_{\mathrm{app}}\hat{\mathbf{z}}$ parallel to the
stripline corresponds to the Damon-Eshbach configuration, but we stress that
for ultrathin films there are no Damon-Eshbach surface modes. The spin wave
energy dispersion Yu1

$\omega_{\mathbf{k}}=\mu_{0}\gamma
M_{s}\sqrt{\left[\Omega_{H}+\alpha_{\mathrm{ex}}k^{2}+1-f(|k_{y}|)\right]\left[\Omega_{H}+\alpha_{\mathrm{ex}}k^{2}+(k_{y}^{2}/k^{2})f(|k_{y}|)\right]},$
(11)

where $-\gamma$ is the electron gyromagnetic ratio, $M_{s}$ denotes the
saturated magnetization, $\alpha_{\mathrm{ex}}$ is the exchange stiffness,
$\Omega_{H}\equiv H_{\mathrm{app}}/M_{s}$, and

$f(|k_{y}|)=1-\frac{1}{|k_{y}|d}+\frac{1}{|k_{y}|d}\exp(-|k_{y}|d),$ (12)

is highly anisotropic. The spin waves amplitudes across sufficiently thin
films are constant Yu1 :

$m_{x}=\sqrt{\frac{B+1}{4d(B-1)}},~{}~{}~{}~{}~{}m_{y}=i\sqrt{\frac{B-1}{4d(B+1)}},$
(13)

where we chose the normalization Kittel_book ; HP ; surface_roughness

$\int
d\mathbf{r}\left[m_{x}(\mathbf{r})m_{y}^{\ast}(\mathbf{r})-m_{x}^{\ast}(\mathbf{r})m_{y}(\mathbf{r})\right]=-i/2,$
(14)

and

$B=\frac{1/2-(1/2)\left(1+k_{y}^{2}/k^{2}\right)f(|k_{y}|)}{\omega_{\mathbf{k}}/(\mu_{0}\gamma
M_{s})-\left(\Omega_{H}+\alpha_{\mathrm{ex}}k_{y}^{2}+1/2\right)+(1/2)\left(1-k_{y}^{2}/k^{2}\right)f(|k_{y}|)}.$
(15)

When $k_{y}\rightarrow 0$: $f(|{k_{y}}|)=0$, $\lim_{k_{y}\rightarrow
0}\omega_{\mathbf{k}}=\mu_{0}\gamma M_{s}\sqrt{\Omega_{H}(\Omega_{H}+1)}$,
$B\rightarrow-1-2\Omega_{H}-2\sqrt{\Omega_{H}(\Omega_{H}+1)}$. When
$\Omega_{H}\rightarrow 0$ with a small static magnetic field,
$B\rightarrow-1-2\sqrt{\Omega_{H}}$,
$\left|m_{y}\right|\gg\left|m_{x}\right|$, so the Kittel mode is (nearly)
linearly polarized. In the opposite (exchange) limit of$\
\left|k_{y}\right|d\gg 1$ and $\alpha_{\mathrm{ex}}k^{2}\gg 1$,
$f(k_{y})\rightarrow 1$, $\left|B\right|\gg 1$, and the spin waves are right-
circularly polarized with $m_{y}=im_{x}$.

The Oersted magnetic fields from the stripline interact with spin waves by the
Zeeman interaction Landau

$\hat{H}_{\mathrm{int}}=-\mu_{0}\int\mathbf{M}(\mathbf{r})\cdot\mathbf{H}(\mathbf{r})dV.$
(16)

The excited magnetization in the film can be expressed by time-dependent
perturbation theory Mahan

$M_{\alpha}(x,\boldsymbol{\rho},t)=-i\int_{-\infty}^{t}dt^{\prime}\left\langle\left[\hat{\mathbf{M}}_{\alpha}(x,\boldsymbol{\rho},t),\hat{H}_{\mathrm{int}}(t^{\prime})\right]\right\rangle.$
(17)

in terms of the retarded spin susceptibility tensor

$\chi_{\alpha\delta}(x,x^{\prime};\boldsymbol{\rho}-\boldsymbol{\rho}^{\prime};t-t^{\prime})=i\Theta(t-t^{\prime})\left\langle\left[\hat{\mathbf{S}}_{\alpha}(x,\boldsymbol{\rho},t),\hat{\mathbf{S}}_{\delta}(x^{\prime},\boldsymbol{\rho}^{\prime},t^{\prime})\right]\right\rangle,$
(18)

where $\hat{\mathbf{S}}_{\alpha}=-\hat{\mathbf{M}}_{\alpha}/(\gamma\hbar)$ is
the spin operator and a sum over repeated indices is implied. Hence Yu1 ; Yu2
; nanowire ,

$\mathbf{M}_{\alpha}(x,k_{y},\omega)=\mu_{0}(\gamma\hbar)^{2}\int_{-d}^{0}dx^{\prime}\chi_{\alpha\beta}(x,x^{\prime},k_{y},\omega)H_{\beta}(x^{\prime},k_{y},\omega),$
(19)

where

$\chi_{\alpha\beta}(x,x^{\prime},\mathbf{k},\omega)=-\frac{2M_{s}}{\gamma\hbar}m_{\alpha}^{\left(\mathbf{k}\right)}(x)m_{\beta}^{\left(\mathbf{k}\right)\ast}(x^{\prime})\frac{1}{\omega-\omega_{\mathbf{k}}+i\Gamma_{\mathbf{k}}}.$
(20)

Here, $\Gamma_{\mathbf{k}}=2\alpha\omega_{\mathbf{k}}$ is the reciprocal
lifetime in terms of the Gilbert damping constant $\alpha$. The excitation
efficiency is determined by
$m_{\beta}^{\left(k_{y}\right)\ast}(x^{\prime})H_{\beta}(x^{\prime},k_{y},\omega)$,
so the excitation of circularly polarized spin waves is chiral (or
unidirectional) by the polarization-momentum locking with the stripline
magnetic field. Since the amplitudes across thin films are constant for $kd\ll
1$, the excited magnetization in time domain and position space is the real
part of the inverse Fourier transform ($q\equiv k_{y}$),

$\displaystyle\mathbf{M}_{\alpha}(x,y,t)$
$\displaystyle=\sum_{q}e^{iqy-i\omega t}\mathbf{M}_{\alpha}(x,q)$
$\displaystyle\approx 2i\mu_{0}\gamma\hbar
dM_{s}m_{\alpha}^{\left(q_{\omega}\right)}m_{\beta}^{\left(q_{\omega}\right)\ast}\frac{1}{v_{q_{\omega}}}e^{-i\omega
t}\left\\{\begin{array}[c]{c}e^{iq_{\omega}y-\delta_{\omega}y}H_{\beta}(q_{\omega},\omega)\\\
e^{-iq_{\omega}y+\delta_{\omega}y}H_{\beta}(-q_{\omega},\omega)\end{array}\text{
for }\begin{array}[c]{c}y>0\\\ y<0\end{array}\right.,$ (25)

where $q_{\omega}+i\delta_{\omega}$ is the positive root of
$\omega_{q}=\omega+i\Gamma_{q}$, and $v_{q_{\omega}}$ is the modulus of the
group velocity $|\partial\omega_{q}/\partial q|_{q_{\omega}}$. The
polarization-momentum locking of the stripline field generates two different
magnetization dynamics. When the excited spin waves are circularly polarized,
they not only propagate in one direction only, but the excitation is also
spatially limited to half of the film, i.e. the chirality is perfect. We can
understand this phenomenon in terms of the interference between the spin waves
and the stripline magnetic field that is constructive and destructive on
opposite sides, as illustrated in Fig. 1.

The dominant excitation direction can be switched with the film magnetization.
For a finite angle $\theta$ between the saturated magnetization and the
stripline, the situation becomes complicated by the reduced symmetry. It is
advantageous to transform Eq. (25) following the Supplements of Refs. Yu2 ;
nanowire :

$m_{x}^{(k_{y})}\rightarrow
m_{x}^{(\mathbf{l)}},~{}~{}~{}~{}m_{y}^{(k_{y})}\rightarrow\cos\theta
m_{y}^{(\mathbf{l})}$

where $\mathbf{l}=(0,q\cos\theta,q\sin\theta)$ and $q$ is determined by
$\omega_{\mathbf{l}}+i2\alpha\omega_{\mathbf{l}}=\omega$. Even for circularly
polarized spin waves, the chirality is not perfect anymore while situation is
complicated for elliptical spin waves since their polarization depends on the
wave vector. For $\theta_{c}=\pi/2$ the chirality always vanishes. Since
mirror symmetry is broken, the two roots
$\left|q_{\omega}^{(+)}+i\delta_{\omega}^{(+)}\right|\neq\left|-q_{\omega}^{(-)}-i\delta_{\omega}^{(-)}\right|$
for $\theta\neq 0,\theta_{c}$. The wavelength and propagation direction of the
excited spin waves may therefore be different on the two sides of the
stripline.

Figure 2 is a plot of the calculated excited magnetization profile for a YIG
magnetic film for constant current density but different excitation
frequencies $\omega/(2\pi)$.

Figure 2: Calculated magnetization amplitude profile
$\left|M_{y}\left(y\right)\right|$ of a YIG film with ground state
magnetization along $z,$ $d=20$ nm, and $\alpha=10^{-4},$ excited by a metal
stripline with $t=100$ nm, $w=1~{}\mathrm{\mu}$m, carrying an AC current with
excitation frequencies $\omega/(2\pi)=3$, 5, and 10 GHz.
$\left|M_{y}\left(y\right)\right|$ is proportional to the current density,
which is here normalized by its maximum value for $\omega/(2\pi)=3$ GHz.

At low frequencies the excitation efficiency is high, but since the dipolar
interaction renders the spin wave precession elliptical, the chirality is
relatively weak. At high frequencies the chirality improves, but the
magnetization amplitude is suppressed by the form factor $\sin(k_{y}w/2)$ that
favors spin waves with wavelengths around $w$. A narrower stripline helps to
excite spin waves with short wavelengths and higher chirality. The spatial
decay on both sides of the stripline is governed by the Gilbert damping.
Chiral spin waves can also be generated by magnetic striplines with high
coercivity that allow efficient excitation and almost perfect chirality at
frequencies $>10$ GHz. The physics is quite different, however, and explained
in the following section.

##  0.3 Chiral spin wave excitation and absorption by a magnetic transducer

Coherent exchange-dipolar spin waves with short wavelengths $\lambda<100$ nm
are attractive information carriers by their long lifetime and high group
velocity. According to the discussion above their excitation is difficult
because striplines cannot be fabricated much finer than this wave length. A
small stripline cross section also increases Joule heating and thereby limits
the maximum applicable currents. A new strategy is to use magnetic nanowires
with high coercivity and resonance frequencies that can be fabricated with the
same feature sizes as normal metal ones. Rather than applying an AC current
directly, magnetic nanowires can be used as “antennas” that are excited by
proximity coplanar wave guides chiral_simulation ; Dirk_2013 ; CoFeB_YIG ;
Co_YIG ; Haiming_NC ; Haiming_PRL . A direct contact between film and
nanowires can suppress chirality by the interface exchange interaction and
associated spin transfer Yu1 , but an insulating spacer of a few atomic
monolayers strongly suppresses exchange without much affecting the dipolar
interaction. Figure3 shows a typical configuration with a Co nanowire on top
of the YIG film.

Figure 3: A magnetic (Co) nanowire transducer separated by a non-magnetic
spacer (optional) from a YIG film. The dipolar coupling is maximized for the
antiparallel magnetization. The direction of the magnon spin currents pumped
into the $\pm\hat{\mathbf{y}}$-directions is indicated by the green arrows,
whose size indicates the magnitude of the magnon currents. The black arrows
indicate the (nearly uniform) microwave input to the magnetic nanowire.

###  0.3.1 Chiral magnetodipolar field

The dipolar field from the magnetic nanowire fundamentally differs from the
Oersted field of the AC current-biased normal metal wire discussed above. The
precessing magnetization is a magnetic dipole and generates a rotating dipolar
field rather than the oscillating axially symmetric field of the normal metal
wire sketched in Fig. 1. The amplitudes of dipolar waves decay faster than
that of (monopolar) current-induced ones, but are still long-ranged compared
to e.g. the exchange interaction. The nanowire and its equilibrium
magnetization are parallel to the $z$-direction as shown in Fig. 3. When
driven with a frequency $\omega$, the macrospin (Kittel) magnetization
dynamics of a wire with thickness $d$ and width $w$ is the real part of

$\tilde{M}_{x,y}(\mathbf{r},t)=\tilde{m}_{x,y}\Theta(x)\Theta(-x+d)\Theta(y+w/2)\Theta(-y+w/2)e^{-i\omega
t},$ (26)

where $\Theta(x)$ is the Heaviside step function and $\tilde{m}_{x,y}$ are
constant amplitudes that depend on the geometry and the excitation power. The
corresponding dipolar magnetic field Landau

$\displaystyle\tilde{h}_{\beta}(\mathbf{r},t)$
$\displaystyle=\frac{1}{4\pi}\partial_{\beta}\partial_{\alpha}\int\frac{\tilde{M}_{\alpha}(\mathbf{r}^{\prime},t)}{|\mathbf{r}-\mathbf{r}^{\prime}|}d\mathbf{r}^{\prime}$
$\displaystyle=\frac{1}{4\pi}\partial_{\beta}\partial_{\alpha}\int
dz^{\prime}\int_{0}^{d}dx^{\prime}\int_{-\frac{w}{2}}^{\frac{w}{2}}dy^{\prime}\frac{\tilde{m}_{\alpha}e^{-i\omega
t}}{\sqrt{z^{\prime 2}+(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}}.$ (27)

We use the Coulomb integral Yu1 ; nano_optics

$\frac{1}{\sqrt{z^{\prime
2}+(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}}=\frac{1}{2\pi}\int
dk_{x}dk_{y}\frac{e^{-|z^{\prime}|\sqrt{k_{x}^{2}+k_{y}^{2}}}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}e^{ik_{x}(x-x^{\prime})+ik_{y}(y-y^{\prime})},$
(28)

a variation of the Weyl identity used in Eq. (4), to express the magnetic
field below the nanowire ($x<0$) with partial Fourier components $k_{y}$

$\displaystyle\tilde{h}_{\beta}(k_{y},x,t)$ $\displaystyle=\int
h_{\beta}(\mathbf{r},t)e^{-ik_{y}y}dy$ $\displaystyle=\frac{1}{\pi}\int
dk_{x}(k_{x}\tilde{m}_{x}+k_{y}\tilde{m}_{y})k_{\beta}e^{ik_{x}x-i\omega
t}\frac{1}{k_{x}^{2}+k_{y}^{2}}\frac{1-e^{-ik_{x}d}}{ik_{x}}\frac{\sin(k_{y}w/2)}{k_{y}}.$
(29)

Closing the contour of the $k_{x}$ integral in the lower half of the complex
plane yields

$\left(\begin{array}[c]{c}\tilde{h}_{x}(k_{y},x,t)\\\
\tilde{h}_{y}(k_{y},x,t)\end{array}\right)=-\frac{i}{4\pi}e^{\left|k_{y}\right|x}(1-e^{-\left|k_{y}\right|d})\frac{2\sin(k_{y}w/2)}{k_{y}\left|k_{y}\right|}\left(\begin{array}[c]{cc}\left|k_{y}\right|&ik_{y}\\\
ik_{y}&-\left|k_{y}\right|\end{array}\right)\left(\begin{array}[c]{c}\tilde{m}_{x}\\\
\tilde{m}_{y}\end{array}\right)e^{-i\omega t}.$ (30)

The perfectly right-circularly polarized wire dynamics of the Kittel mode in
rectangular wires ($\tilde{m}_{y}=i\tilde{m}_{x}$ when $w=d$) implies that the
Fourier components of $\mathbf{\tilde{h}}$ with $k_{y}>0$ vanish. The Fourier
component with $k_{y}<0$ is then perfectly left circularly polarized
$\left(\tilde{h}_{y}=-i\tilde{h}_{x}\right)$. Above the nanowire, the magnetic
field direction and polarization are reversed, as sketched in Fig. 4. The
elliptical polarization of the Kittel mode in rectangular nanowires breaks the
perfect chirality. Analogous expressions can be derived for arbitrarily shaped
magnetic transducers such as discs, but analytical expressions become complex
or may not exist when the symmetry is reduced.

Figure 4: Dipolar magnetic field $\mathbf{\tilde{h}}$ generated by a Kittel
mode excitation of a magnetic nanowire ($\parallel\hat{\mathbf{z}}$). The
thick red and thin blue arrows indicate the propagation and precession
directions of $\mathbf{\tilde{h},}$ respecctively, both above and below the
wire.

Equation (25) can be used also for magnetic fields $\mathbf{\tilde{h}}$
generated by a magnetic transducer, i.e. Eq. (30), a left-circularly polarized
dipolar field that propagates to the left. An ellipticity of the spin waves in
the film does not affect the chirality since the excited magnetization still
propagates to the left and lives only in the left half-space, but it reduces
the excitation efficiency. The same holds when the Kittel mode in a
rectangular nanowire is elliptical and the spin waves in the film are
circularly polarized. We illustrate these conclusion below from different
viewpoint.

Let us compare the dipolar stray fields $\mathbf{\tilde{h}}$ emitted by the
excited magnetic wire and $\mathbf{H}$ generated by a stripline as discussed
in the previous section. The main difference between these “Oersted” vs.
“dipolar” radiation is that the latter has additional chirality that induces a
circularly-polarized magnetic field in real space, in contrast to the
linearly-polarized magnetic field of the former. Equation (30) can be
summarized as
$\tilde{h}_{x}\propto|k_{y}|(\tilde{m}_{x}+i\mathrm{sgn}(k_{y})\tilde{m}_{y})$
and $\tilde{h}_{y}\propto
ik_{y}(\tilde{m}_{x}+i\mathrm{sgn}(k_{y})\tilde{m}_{y}).$
$\tilde{h}_{y}=i\mathrm{sgn}(k_{y})\tilde{h}_{x}$ is the polarization-momentum
locking in reciprocal space, which is the same as that of the evanescent
Oersted field. However, the magnetic chirality affects
$\tilde{m}_{x}+i\mathrm{sgn}(k_{y})\tilde{m}_{y}$: for right circularly-
polarized (when $w=d$) $\tilde{m}_{y}=i\tilde{m}_{x}$, $\mathbf{\tilde{h}}$
simply vanishes for positive $k_{y}$. Thus, the magnetic field is
unidirectional with linear momentum components normal to the wire that are
negative, which is more than just a locking between polarization and momentum.
$\mathbf{\tilde{h}}$ therefore couples chirally to spins with arbitrary
polarizations.

The Zeeman interaction $\sim\mathbf{M}\cdot\mathbf{\tilde{H}}$ between the
wire and film is governed as used above is completely equivalent to the
interaction $\sim\mathbf{\tilde{M}}\cdot\mathbf{h}$, where
$\mathbf{\tilde{M}}$ is the wire magnetization and $\mathbf{h}$ the dipolar
field generated by the spin waves in the film. It is instructive to discuss
the physics from this second viewpoint. We assume again that the equilibrium
wire magnetization is fixed by the form anisotropy to the $z$-direction. A
sufficiently soft film magnetization can be rotated in the $x$-$z$ plane by an
applied magnetic field, but we address here only (anti)parallel magnetizations
but general wave propagation direction Yu2 ; nanowire . We allow for the
elliptical spin wave polarization in the magnetostatic regime. At frequency
$\omega$ and in the coordinate system defined in Fig. 3 with in-plane wave
vector $\mathbf{k}=k_{y}\hat{\mathbf{y}}+k_{z}\hat{\mathbf{z}},$ we define
$M_{x}(\mathbf{r},t)=m_{R}^{\left(\mathbf{k}\right)}(x)\cos(\mathbf{k}\cdot\boldsymbol{\rho}-\omega
t)$ and $M_{y}(\mathbf{r},t)\equiv-
m_{R}^{\left(\mathbf{k}\right)}(x)\sin(\mathbf{k}\cdot\boldsymbol{\rho}-\omega
t),$ where $m_{R}^{\left(\mathbf{k}\right)}(x)$ is the time-independent
amplitude into the film and
$\boldsymbol{\rho}=y\hat{\mathbf{y}}+z\hat{\mathbf{z}}.$ The dipolar field
outside the film with $\alpha,\beta=\\{x,y,z\\}$ Landau ,

$h_{\beta}(\mathbf{r},t)=\frac{1}{4\pi}\partial_{\beta}\partial_{\alpha}\int
d\mathbf{r}^{\prime}\frac{M_{\alpha}(\mathbf{r}^{\prime},t)}{|\mathbf{r}-\mathbf{r}^{\prime}|},$
(31)

then reads

$\left(\begin{array}[c]{c}h_{x}(\mathbf{r},t)\\\ h_{y}(\mathbf{r},t)\\\
h_{z}(\mathbf{r},t)\end{array}\right)=\left(\begin{array}[c]{c}\left(k+\eta
k_{y}\right)\cos\left(\mathbf{k}\cdot\boldsymbol{\rho}-\omega t\right)\\\
\left(\frac{k_{y}^{2}}{k}+\eta
k_{y}\right)\sin\left(\mathbf{k}\cdot\boldsymbol{\rho}-\omega t\right)\\\
k_{z}\left(\frac{k_{y}}{k}+\eta\right)\sin\left(\mathbf{k}\cdot\boldsymbol{\rho}-\omega
t\right)\end{array}\right)\frac{1}{2}e^{-\eta kx}\int
dx^{\prime}m_{R}^{\mathbf{k}}\left(x^{\prime}\right)e^{\eta kx^{\prime}},$
(32)

where $x>0$ ($x<-s$) indicates the dipolar field above (below) the film,
$\eta=1$ ($-1$) when $x>0$ ($x<-s$), $k=\left|\mathbf{k}\right|$, and the
spatial integral is over the film thickness.

When $k_{z}=0$, $k_{y}\neq 0$ spin waves propagate normal to the wire and
$h_{z}=0$. The distribution of the dipolar field above and below the film then
strongly depends on the sign of $k_{y}$: the dipolar field generated by the
right (left) moving spin waves only appears above (beneath) the film Yu1 ; Yu2
; nanowire  and precesses in the opposite direction of the magnetization.
These features provide an alternative explanation of the chiral coupling
between these spin waves and _any_ magnet close to the film surface Yu1 ; Yu2
. The chiral dipolar coupling is most pronounced when the magnetizations of
the film and wire are antiparallel Yu1 ; Yu2 ; nanowire .

When the film magnetization is rotated by 90 degrees in perpendicular to the
wire, the wire magnetization excites spin waves that propagate parallel to the
magnetization ($k_{y}=0$, $h_{y}=0$), which for thick films correspond to the
backward moving bulk modes. Surprisingly, these also couple chirally to the
wire dynamics, but by a different mechanism. According to Eq. (4),
$h_{x}\propto\left|k_{z}\right|\cos\left(k_{z}z-\omega t\right)$ and
$h_{z}\propto\eta k_{z}\sin\left(k_{z}z-\omega t\right)$. The dipolar fields
generated by spin waves with positive (negative) $k_{z}$ are left (right)
circularly polarized, respectively, while below the film, the polarizations
are reversed. These spin waves chirally interact with the transducer magnet
since the polarization of the transverse magnetization dynamics of the latter
has to match that of the stray field $\mathbf{h}$ chiral_simulation .

Therefore, two mechanisms contribute to the chiral excitation, depending on
the magnetic configuration. When spin waves propagate perpendicular to the
magnetization with opposite momenta, their dipolar fields vanish on opposite
sides of the film; when propagating parallel to the magnetization, their
dipolar field is chiral, i.e., polarization-momentum locked. Purely chiral
coupling between magnons can be achieved in the former case without
constraints on the polarization of the local magnet, but in the latter case
elliptical polarization of the wire leads to partial chirality.

The resonance frequency of a magnetic nanowire can be tuned by an applied
magnetic field and excites spin waves in a frequency window that is governed
by the wire form factor. The magnetodipolar field emitted by a coherently
excited magnetic nanowire array can also be chiral Yu1 ; Yu2 . However, such a
nanowire grating with period $a$ and translational symmetry
$na\hat{\mathbf{y}}$ excites discrete spin waves with momenta
$(m\pi/a)\hat{\mathbf{y}}$, where $\\{m,n\\}\in Z_{0}$ that are observable as
sharp and intense feature in the microwave transmission (more details are
shown below).

###  0.3.2 Non-local detection

Here we illustrate the principle of non-local excitation and detection of
magnons by a device consisting of two magnetic nanowires on top of a YIG film.
The generation of DC currents by AC forces in the absence of a DC bias is
generally referred to as “pumping” pumping_Buttiker . Spin pumping is the
injection of a spin current by the magnetization dynamics of a magnet into a
normal metal contact by the interface exchange interaction
spin_pumping_electron ; non_local . C _hiral spin pumping_ is the generation
of unidirectional spin waves by the dynamics of a proximity magnetic wire as
discussed above. Its inverse is the _chiral spin absorption,_ i.e. the wire
dynamics induced by the stray fields caused by spin waves in the film. We
develop below a semi-analytic theory of chiral spin pumping/absorption for
antiparallel magnetic configurations and describe two effects — non-reciprocal
microwave transmission and chiral spin Seebeck effect. Whereas the former is
due to coherent pumping by applied microwaves, the latter represents the
incoherent (thermal) pumping by a temperature difference Spin_seebeck_exp ;
Spin_seebeck_theory1 ; Spin_seebeck_theory2 ; Spin_caloritronics . Both
effects can be observed in terms of the magnon population or temperature in
the detector, e.g., inductively or by light scattering.

We switch from a purely classical picture of previous sections to a quantum
description of the chiral coupling in terms of Hamiltonian matrix elements
between generalized harmonic oscillators. This does not introduce new physics
since we can simply replace operators by classical amplitudes, but it provides
a compact formalism used in many other fields such as nanomechanical systems
and optics, and prepares the stage for the treatment of real quantum problems.
For simplicity, we focus on the antiparallel magnetic configuration with
maximized dipolar coupling (for arbitrary magnetization directions see
nanowire ). The dipolar coupling of the wire magnetization
$\mathbf{\tilde{M}}$ with that of a film $\mathbf{M}$ is governed by the
Zeeman interaction with the respective stray magnetic fields $\mathbf{h}$ and
$\mathbf{\tilde{h}}$ Landau

$\hat{H}_{\mathrm{int}}/\mu_{0}=-\int\mathbf{\tilde{M}}(\mathbf{r},t)\cdot\mathbf{h}(\mathbf{r},t)d\mathbf{r}=-\int\mathbf{M}(\mathbf{r},t)\cdot\mathbf{\tilde{h}}(\mathbf{r},t)d\mathbf{r},$
(33)

where $\mathbf{h}$ and $\mathbf{\tilde{h}}$ have been introduced in Eqs. (32)
and (30). The magnetization dynamics of film ($\mathbf{\hat{M}}$) and nanowire
($\mathbf{\hat{\tilde{M}}}$) are now interpreted as operators with Cartesian
components $\beta\in\left\\{x,y\right\\}$. To leading order of the expansion
in magnon creation and annihilation operators Kittel_book ; HP ;
surface_roughness ,

$\displaystyle\hat{M}_{\beta}(\mathbf{r})$
$\displaystyle=-\sqrt{2M_{s}\gamma\hbar}\sum_{\mathbf{k}}\left[m_{\beta}^{\left(\mathbf{k}\right)}(x)e^{i{\mathbf{k}}\cdot\boldsymbol{\rho}}\hat{\alpha}_{\mathbf{k}}+\mathrm{H.c.}\right],$
$\displaystyle\hat{\tilde{M}}_{\beta}(\mathbf{r})$
$\displaystyle=-\sqrt{2\tilde{M}_{s}\gamma\hbar}\sum_{k_{z}}\left[\tilde{m}_{\beta}^{(k_{z})}(x,y)e^{ik_{z}z}\hat{\beta}_{k_{z}}+\mathrm{H.c.}\right],$
(34)

where $M_{s}$ and $\tilde{M}_{s}$ are the respective saturation
magnetizations, $m_{\beta}^{\left(\mathbf{k}\right)}(x)$ and
$\tilde{m}_{\beta}^{(k_{z})}(x,y)$ are the spin wave amplitudes across the
film and nanowire, and $\hat{\alpha}_{\mathbf{k}}$ and $\hat{\beta}_{k_{z}}$
denote the magnon (annihilation) operator in the film and nanowire,
respectively. The total system Hamiltonian then reads

$\displaystyle\hat{H}/\hbar$
$\displaystyle=\sum_{\mathbf{k}}\omega_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}^{\dagger}\hat{\alpha}_{\mathbf{k}}+\sum_{k_{z}}\tilde{\omega}_{k_{z}}\hat{\beta}_{k_{z}}^{\dagger}\hat{\beta}_{k_{z}}$
$\displaystyle+\sum_{\mathbf{k}}\left(g_{\mathbf{k}}e^{-ik_{y}y_{0}}\hat{\alpha}_{\mathbf{k}}^{\dagger}\hat{\beta}_{k_{z}}+g_{\mathbf{k}}^{\ast}e^{ik_{y}y_{0}}\hat{\beta}_{k_{z}}^{\dagger}\hat{\alpha}_{\mathbf{k}}\right),$
(35)

where $\omega_{\mathbf{k}}$ and $\tilde{\omega}_{k_{z}}$ are the frequencies
of spin waves in the film and nanowire and the coupling

$g_{\mathbf{k}}=F(\mathbf{k})\left(m_{x}^{(\mathbf{k})\ast},m_{y}^{(\mathbf{k})\ast}\right)\left(\begin{array}[c]{cc}|\mathbf{k}|&ik_{y}\\\
ik_{y}&-k_{y}^{2}/|\mathbf{k}|\end{array}\right)\left(\begin{array}[c]{c}\tilde{m}_{x}^{(k_{z})}\\\
\tilde{m}_{y}^{(k_{z})}\end{array}\right),$ (36)

with
$F(\mathbf{k})=-\mu_{0}\gamma\sqrt{M_{s}\tilde{M}_{s}/L}\phi\left(\mathbf{k}\right)$.
The form factor
$\phi\left(\mathbf{k}\right)=2\sin(k_{y}w/2)(1-e^{-kd})(1-e^{-ks})/(k_{y}{k^{2}})$
couples spin waves with wavelengths of the order of the nanowire width (mode
selection) and $\lim_{\mathbf{\ k}\rightarrow
0}\phi\left(\mathbf{k}\right)=wsd$. Pure exchange waves are right-circularly
polarized with $m_{y}^{(k_{y})}=im_{x}^{(k_{y})}$ and their coupling is
perfectly chiral since $g_{-|k_{y}|}=0$ and $g_{|k_{y}|}\neq 0$.

Eqs. (11) and (13) give the spin-wave dispersion and amplitudes in the thin
film. The spin waves propagate in the nanowire along $\hat{\mathbf{z}}$ with
amplitudes Yu1 ; nanowire

$\tilde{m}_{x}^{k_{z}}=\sqrt{\frac{1}{4\mathcal{D}(k_{z})wd}},~{}~{}~{}~{}\tilde{m}_{y}^{k_{z}}=i\sqrt{\frac{\mathcal{D}(k_{z})}{4wd}},$
(37)

where

$\mathcal{D}(k_{z})=\sqrt{\frac{H_{\mathrm{app}}+N_{xx}\tilde{M}_{s}+\tilde{\lambda}_{\mathrm{ex}}k_{z}^{2}\tilde{M}_{s}}{H_{\mathrm{app}}+N_{yy}\tilde{M}_{s}+\tilde{\lambda}_{\mathrm{ex}}k_{z}^{2}\tilde{M}_{s}}}.$
(38)

$H_{\mathrm{app}}$ and $\tilde{\lambda}_{\mathrm{ex}}$ are the applied
magnetic field and the exchange stiffness of the nanowire, respectively. The
demagnetization factors $N_{xx}\simeq w/(d+w)$ and $N_{yy}=d/(d+w)$ Yu1  also
govern the spin waves frequency

$\tilde{\omega}_{k_{z}}=\mu_{0}\gamma\sqrt{(H_{\mathrm{app}}+N_{yy}\tilde{M}_{s}+\tilde{\lambda}_{\mathrm{ex}}k_{z}^{2}\tilde{M}_{s})(H_{\mathrm{app}}+N_{xx}\tilde{M}_{s}+\tilde{\lambda}_{\mathrm{ex}}k_{z}^{2}\tilde{M}_{s})}.$
(39)

When the magnetic field is antiparallel to the nanowire magnetization we
require
$\left|H_{\mathrm{app}}\right|<\min\\{N_{yy}\tilde{M}_{s},N_{xx}\tilde{M}_{s}\\}$.
The ellipticity of the Kittel mode with $k_{z}=0$ is strongly affected by the
shape anisotropy when the applied field is sufficiently small and the aspect
ratio large: when $d\ll w$, $N_{xx}\rightarrow 1$, $N_{yy}\rightarrow 0$,
$\mathcal{D}$ is large and the mode is nearly linearly-polarized. On the other
hand, when $d\approx w$, $\mathcal{D}\rightarrow 1$, and the Kittel mode is
circularly polarized. When $d\lesssim w$, and the Kittel mode traces an
elliptical orbit. Figure 5 illustrates the chirality of the coupling parameter
$g_{\mathbf{k}}$ of the $k_{z}$-Kittel mode in a nanowire of dimensions $w=70$
nm and $d=20$ nm and magnons in a film of thickness $s=20$ nm with wave vector
$\mathbf{k}=\left(0,k_{y},k_{z}\right)$ nanowire . The coupling maximum can be
shifted to larger momenta by a smaller feature size of the wire. The
excitation of such short-wavelength spin waves is possible with a magnetically
hard transducer that has a high ferromagnetic resonance frequency
chiral_simulation ; Dirk_2013 ; CoFeB_YIG ; Co_YIG ; Haiming_NC ; Haiming_PRL
.

Figure 5: Momentum dependence of the dipolar coupling strength
$|g_{\mathbf{k}}|$ of a magnetic nanowire and film (parameters in the text)
nanowire .

###  0.3.3 Coherent chiral spin wave transmission

The quantum description leads to expressions that are fully equivalent with
Eq. (25) obtained from the classical description input_output1 ; input_output2
. The excitation of magnons saps nanowire energy and angular momentum, thereby
contributing to the magnetization damping, which can be observed as an
increased linewidth of the ferromagnetic resonance spectrum. In the quantum
description, this broadening is determined by the imaginary part of the
magnetic self-energy, which in the first Born approximation or the Fermi-
golden rule reads

$\delta\tilde{\kappa}_{k_{z}}=2\pi\sum_{k_{y}}|g_{\mathbf{k}}|^{2}\delta(\tilde{\omega}_{k_{z}}-\omega_{\mathbf{k}}).$
(40)

We predict a very significant additional damping for a Co nanowire with width
$w=70$ nm, thickness $d=20$ nm, magnetization $\mu_{0}\tilde{M}_{s}=1.1$ T Yu2
; Haiming_PRL , and exchange stiffness
$\tilde{\lambda}_{\mathrm{ex}}=3.1\times 10^{-13}$ cm2 Co_exchange . We adopt
a YIG film $s=20$ nm with magnetization $\mu_{0}M_{s}=0.177$ T and exchange
stiffness $\lambda_{\mathrm{ex}}=3.0\times 10^{-12}$ cm2 Yu2 ; Haiming_PRL ;
surface_roughness . A magnetic field $\mu_{0}H_{\mathrm{app}}=0.05$ T is
sufficient to switch the film magnetizations antiparallel to that of the wire
to maximize the effect Haiming_NC ; Haiming_PRL . The calculated additional
damping of nanowire Kittel dynamics is then
$\delta\alpha_{\mathrm{Co}}=\delta\tilde{\kappa}_{k_{z}=0}/(2\tilde{\omega}_{k_{z}=0})=3.1\times
10^{-2}$, which is one order of magnitude larger than the intrinsic Gilbert
damping coefficient $\alpha_{\mathrm{Co}}=2.4\times 10^{-3}$ Co_damping .

Almost perfect chiral pumping by a nanowire _array_ has been observed by the
microwave transmission and Brillouin light scattering in Ref. Yu2 . We here
focus on the new features in the broadband non-local excitation-detection by
_two nanowires_. The magnetic order in two nanowires located at
$\mathbf{r}_{1}=R_{1}\hat{\mathbf{y}}$ and
$\mathbf{r}_{2}=R_{2}\hat{\mathbf{y}}$ act as transducers for microwaves that
are emitted or absorbed by local microwave (normal metal) antennas such as
coplanar wave guides. The observable is the scattering matrix of the
microwaves with excitation (input) at $R_{1}$ and the detection (output) at
$R_{2}$, which can be formulated by the input-output theory input_output1 ;
input_output2 . The equation of motion of magnons localized at $R_{1}$ and
$R_{2}$ with operators $\hat{m}_{L}$ and $\hat{m}_{R}$ and coupled by the film
magnons with operators $\hat{\alpha}_{q}$ (not to be confused with the Gilbert
damping constant) read

$\displaystyle\frac{d\hat{m}_{L}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{L}(t)-i\sum_{q}g_{q}e^{iqR_{1}}\hat{\alpha}_{q}(t)-\left(\frac{\kappa_{L}}{2}+\frac{\kappa_{p,L}}{2}\right)\hat{m}_{L}(t)-\sqrt{\kappa_{p,L}}\hat{p}_{\mathrm{in}}^{(L)}(t),$
$\displaystyle\frac{d\hat{m}_{R}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{R}(t)-i\sum_{q}g_{q}e^{iqR_{2}}\hat{\alpha}_{q}(t)-\frac{\kappa_{R}}{2}\hat{m}_{R}(t),$
$\displaystyle\frac{d\hat{\alpha}_{q}}{dt}$
$\displaystyle=-i\omega_{q}\hat{\alpha}_{q}(t)-ig_{q}e^{-iqR_{1}}\hat{m}_{L}(t)-ig_{q}e^{-iqR_{2}}\hat{m}_{R}(t)-\frac{\kappa_{q}}{2}\hat{\alpha}_{q}(t).$
(41)

Here, $\kappa_{L}$ and $\kappa_{R}$ are the intrinsic damping of the Kittel
modes in the left and right nanowires, respectively, $\kappa_{p,L}$ is the
additional radiative damping induced by the microwave photons
$\hat{p}_{\mathrm{in}}^{(L)}$, i.e. the coupling of the left nanowire with the
microwave source, and $\kappa_{q}$ denotes the intrinsic (Gilbert) damping of
magnons in the films. In frequency space:

$\displaystyle\hat{\alpha}_{q}(\omega)$
$\displaystyle=g_{q}G_{q}\left(\omega\right)\left[e^{-iqR_{1}}\hat{m}_{L}(\omega)+e^{-iqR_{2}}\hat{m}_{R}(\omega)\right],$
$\displaystyle\hat{m}_{R}(\omega)$
$\displaystyle=\frac{-i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}\hat{m}_{L}(\omega),$
$\displaystyle\hat{m}_{L}(\omega)$
$\displaystyle=\frac{-\sqrt{\kappa_{p,L}}}{-i(\omega-\omega_{\mathrm{K}})+(\kappa_{L}+\kappa_{p,L})/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)-f(\omega)}\hat{p}_{\mathrm{in}}^{(L)}(\omega),$
(42)

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

$f(\omega)\equiv-\frac{\left(\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{1}-R_{2})}\right)\left(\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}\right)}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}.$
(43)

The excitation of the left nanowire propagates to the right nanowire by the
spin waves in the film. When chiral coupling is perfect, $f(\omega)$ vanishes
without the back-action. The microwave output of both left and right nanowires
as inductively detected by coplanar wave guides are denoted
$\hat{p}_{\mathrm{out}}^{(L)}(\omega)$ and
$\hat{p}_{\mathrm{out}}^{(R)}(\omega)$ with input-output relations
input_output1 ; input_output2

$\displaystyle\hat{p}_{\mathrm{out}}^{(L)}(\omega)$
$\displaystyle=p_{\mathrm{in}}^{(L)}(\omega)+\sqrt{\kappa_{p,L}}\hat{m}_{L}(\omega),$
$\displaystyle\hat{p}_{\mathrm{out}}^{(R)}(\omega)$
$\displaystyle=\sqrt{\kappa_{p,R}}\hat{m}_{R}(\omega),$ (44)

where $\kappa_{p,R}$ is the additional radiative damping induced by the
detector. Therefore, the elements in the microwave scattering matrix
describing reflection $\left(S_{11}\right)$ and transmission
$\left(S_{21}\right)$ amplitudes become

$\displaystyle S_{11}(\omega)$
$\displaystyle\equiv\frac{\hat{p}_{\mathrm{out}}^{(L)}}{\hat{p}_{\mathrm{in}}^{(L)}}=1-\frac{\kappa_{p,L}}{-i(\omega-\omega_{\mathrm{K}})+(\kappa_{L}+\kappa_{p,L})/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)-f(\omega)},$
$\displaystyle S_{21}(\omega)$
$\displaystyle\equiv\frac{\hat{p}_{\mathrm{out}}^{(R)}}{\hat{p}_{\mathrm{in}}^{(L)}}=\left[1-S_{11}(\omega)\right]\sqrt{\frac{\kappa_{p,R}}{\kappa_{p,L}}}\frac{i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)e^{iq(R_{2}-R_{1})}}{-i(\omega-\omega_{\mathrm{K}})+\kappa_{R}/2+i\sum_{q}g_{q}^{2}G_{q}\left(\omega\right)}.$
(45)

The real parts of $S_{11}$ and $S_{12}$ at different magnetic fields and
microwave frequencies are illustrated in Fig. 6 for antiparallel
magnetizations.

Figure 6: Microwave reflection $\mathrm{Re}S_{11}$ [(a)] and transmission
$\mathrm{Re}S_{12}$ [(b)] amplitudes, Eqs. (45), for a system of two Co
nanowires on a YIG film as a function of frequency $\omega_{\mathrm{in}}$. The
radiative damping of both nanowires is $\kappa_{p}/(2\pi)=10$ MHz and other
parameters are given in the text.

The frequency of the Co Kittel mode decreases with increasing magnetic field
until its direction is reversed to the magnetic-field direction (here
$\left|H_{\mathrm{app}}\right|\lesssim 200$ mT). The interference pattern on
the Kittel resonance in Fig. 6(b) reflects the transmission phase delay
$e^{ik(R_{1}-R_{2})}$ in Eq. (45). We note that in our model the nanowires do
not reflect spin waves, the features should therefore not be interpreted in
terms of standing spin waves.

###  0.3.4 Incoherent chiral pumping

A temperature gradient between the magnetic nanowire and film also injects
unidirectional magnon currents, i.e., causes a chiral spin Seebeck effect
Spin_seebeck_exp ; Spin_seebeck_theory1 ; Spin_seebeck_theory2 ;
Spin_caloritronics . Here we consider again two identical transducers, i.e., a
magnetic nanowire at $\mathbf{r}_{2}=R_{2}\hat{\mathbf{y}}$ that detects
magnons, which are now thermally injected by the nanowire at
$\mathbf{r}_{1}=R_{1}\hat{\mathbf{y}}$ and $R_{1}<R_{2}$. This is the
configuration of the non-local spin Seebeck effect as detected electrically in
many experiments starting with Ref. Ludo . The magnons in those experiments
are believed to be injected by the interface exchange interaction or generated
by a temperature gradient in the bulk and results are interpreted by spin
diffusion models. Here we consider the regime in which the exchange effect is
suppressed, magnon propagation is ballistic and we disregard the bulk spin
Seebeck effect due to possible temperature gradients. We predict a spin non-
local spin Seebeck effect that is caused exclusively by dipolar fields and
carried by magnons with long wave lengths and lifetimes. We focus on the
Kittel magnons in the wires since the dipolar coupling between the film and
higher bands in the nanowire is very small. The coupling strength
$|g_{\mathbf{k}}|$ in Fig. 5 illustrates that magnons with wavelength around
half of the nanowire width (here $\pi/w=0.045$ nm-1) dominate the coupling.
Pumping from other than the those modes can therefore be disregarded even at
elevated temperatures. Furthermore, the spin current in the film is dominated
by spin waves with small momentum and long mean-free paths, so the effects of
magnon-magnon and magnon-phonon interactions that otherwise render magnon
transport phenomena diffuse Ludo  should be negligibly small. The narrow-band
thermal injection requires an inductive (or optical) detection of the magnons
accumulated in the detector contact, since the inverse spin Hall effect with
heavy metal contacts is very inefficient.

The equation of motions of the Kittel modes in the nanowire and film spin
waves with momentum $q$ in the coupled system read

$\displaystyle\frac{d\hat{m}_{L}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{L}-\sum_{q}ig_{q}^{\ast}e^{iqR_{1}}\hat{\alpha}_{q}-\frac{\kappa}{2}\hat{m}_{L}-\sqrt{\kappa}\hat{N}_{L},$
$\displaystyle\frac{d\hat{m}_{R}}{dt}$
$\displaystyle=-i\omega_{\mathrm{K}}\hat{m}_{R}-\sum_{q}ig_{q}^{\ast}e^{iqR_{2}}\hat{\alpha}_{q}-\frac{\kappa}{2}\hat{m}_{R}-\sqrt{\kappa}\hat{N}_{R},$
$\displaystyle\frac{d\hat{\alpha}_{q}}{dt}$
$\displaystyle=-i\omega_{q}\hat{\alpha}_{q}-ig_{q}e^{-iqR_{1}}\hat{m}_{L}-ig_{q}e^{-iqR_{2}}\hat{m}_{R}-\frac{\kappa_{q}}{2}\hat{\alpha}_{q}-\sqrt{\kappa_{q}}\hat{N}_{q},$
(46)

where $\kappa$ is caused by the same Gilbert damping in both nanowires, and
$\hat{N}_{L}$ and $\hat{N}_{R}$ represent the thermal noise in the left and
right nanowires, with
$\langle{\hat{N}_{\eta}^{\dagger}(t)\hat{N}_{\eta^{\prime}}(t^{\prime})}\rangle=n_{\eta}\delta(t-t^{\prime})\delta_{\eta\eta^{\prime}}$.
Here, $\eta\in\\{L,R\\}$ and
$n_{\eta}=1/\left\\{\exp\left[\hbar\tilde{\omega}_{\mathrm{K}}/(k_{B}T_{\eta})\right]-1\right\\}$
and $T_{R}$ is also the film temperature. Integrating out the spin-wave modes
in the film, we obtain equations for dissipatively coupled Yao_Yu ;
Magnon_radiation  nanowires. In frequency space,

$\displaystyle\left(-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}+\Gamma_{2}}{2}\right)\hat{m}_{L}(\omega)+\Gamma_{2}e^{iq_{\ast}|R_{2}-R_{1}|}\hat{m}_{R}(\omega)$
(47)
$\displaystyle=\sum_{q}ig_{q}^{\ast}e^{iqR_{1}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{L}(\omega),$
$\displaystyle\left(-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}+\Gamma_{2}}{2}\right)\hat{m}_{R}(\omega)+\Gamma_{1}e^{iq_{\ast}|R_{2}-R_{1}|}\hat{m}_{L}(\omega)$
$\displaystyle=\sum_{q}ig_{q}^{\ast}e^{iqR_{2}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{R}(\omega),$
(48)

where $\Gamma_{1}=|g_{q_{\ast}}|^{2}/v_{q_{\ast}}$ and
$\Gamma_{2}=|g_{-q_{\ast}}|^{2}/v_{q_{\ast}}$ are assumed constant (for the
Kittel mode). Here, $q_{\ast}$ is the positive root of
$\omega_{q_{\ast}}=\tilde{\omega}_{\mathrm{K}}$.

For perfectly chiral coupling with $\Gamma_{2}=0$ the solutions of Eqs. (48)
read

$\displaystyle\hat{m}_{L}(\omega)$
$\displaystyle=\frac{\sum_{q}ig_{q}^{\ast}e^{iqR_{1}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{L}(\omega)}{-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}}{2}},$
$\displaystyle\hat{m}_{R}(\omega)$
$\displaystyle=\frac{\sum_{q}ig_{q}^{\ast}e^{iqR_{2}}\sqrt{\kappa_{q}}G_{q}(\omega)\hat{N}_{q}(\omega)-\sqrt{\kappa}\hat{N}_{R}(\omega)-\Gamma_{1}e^{q_{\ast}(R_{2}-R_{1})}\hat{m}_{L}(\omega)}{-i(\omega-\omega_{\mathrm{K}})+\frac{\kappa}{2}+\frac{\Gamma_{1}}{2}}.$
(49)

With $\hat{m}_{L,R}(t)=\int e^{-i\omega
t}\hat{m}_{L,R}(\omega)d\omega/(2\pi)$, the Kittel modes are occupied
according to

$\displaystyle\rho_{L}$
$\displaystyle\equiv\langle\hat{m}_{L}^{\dagger}(t)\hat{m}_{L}(t)\rangle=n_{L}+\int\frac{d\omega}{2\pi}\frac{\kappa}{(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}}(n_{q_{\ast}}-n_{L}),$
(50) $\displaystyle\rho_{R}$
$\displaystyle\equiv\langle\hat{m}_{R}^{\dagger}(t)\hat{m}_{R}(t)\rangle=n_{R}+\int\frac{d\omega}{2\pi}\frac{\Gamma_{1}^{2}\kappa}{\left[(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}\right]^{2}}(n_{L}-n_{q_{\ast}}),$
(51)

where the damping in the high-quality film has been disregarded
$\left(\kappa_{q}\rightarrow 0\right)$. In the linear regime the non-local
thermal injection of magnons into the right transducer by the left one then
reads

$\displaystyle\delta\rho_{R}$
$\displaystyle=\left\\{\begin{array}[c]{c}\mathcal{S}_{\mathrm{CSSE}}(T_{L}-T_{R})\\\
0\end{array}\text{ when }\begin{array}[c]{c}T_{L}>T_{R}\\\ T_{L}\leq
T_{R}\end{array}\right.,$ (56) $\displaystyle\mathcal{S}_{{}_{\mathrm{CSSE}}}$
$\displaystyle=\int\frac{d\omega}{2\pi}\frac{\Gamma_{1}^{2}\kappa}{\left[(\omega-\omega_{\mathrm{K}})^{2}+(\kappa/2+\Gamma_{1}/2)^{2}\right]^{2}}\left.\frac{dn_{L}}{dT}\right|_{T=\left(T_{L}+T_{R}\right)/2}.$
(57)

where we defined the chiral (or dipolar) spin Seebeck coefficient
$\mathcal{S}_{{}_{\mathrm{CSSE}}}.$

The device therefore operates as a heat diode, apparently acting as a “Maxwell
demon” that rectifies the thermal fluctuations at equilibrium. However, in
thermal equilibrium all right and left moving magnons are eventually connected
by reflection of spin waves at the edges and absorption and re-emission by
connected heat baths. The Second Law of thermodynamics is therefore safe, but
it might be interesting to search for chirality-induced transient effects.

##  0.4 Conclusion and outlook

Handedness or chirality of wave propagation is a popular research topic in
optics, acoustics, and condensed matter physics. Here we contribute by a
theory for the coherent and incoherent chiral pumping of spin waves into thin
magnetic films through the chiral magnetodipolar radiation generated by the
Oersted field of metallic striplines and dipolar field of magnetic
nanostructures. Spin waves excited coherently in the film under magnetic
resonance of the nanowire are unidirectional, generating a non-equilibrium
magnetization in only half of the film. A temperature gradient between a local
magnet and a film leads to the unidirectional excitation of incoherent
magnons, i.e., a chiral spin Seebeck effect.

$\mathcal{PT}$ symmetry has been predicted to amplify unidirectional response
PT_symmetry1 ; PT_symmetry2 ; PT_symmetry3 . Even though our system is
dissipative and therefore not PT symmetric, the nonreciprocal coupling of the
two wires still allows directional amplification accumulation1 ; accumulation2
. It would be interesting to introduce $\mathcal{PT}$ symmetry into our system
via gain in one wire that compensates the damping in the other, possibly
leading to enhanced effects.

Magnons can interact remotely by their chiral dipolar magnetic fields with
other quasiparticles including other magnons, photons, phonons, and conduction
electron spins. Strong chiral coupling between magnons and photons exist,
e.g., in microwave waveguides or cavities that contain chains of small magnets
on special lines accumulation1 ; accumulation2 . Large magnon numbers
accumulate at one edge of a chain of magnets when excited by local antennas
accumulation1 ; accumulation2 . Spin currents by electrons or phonons may be
generated by the chiral magnetodipolar radiation as well. Chirality is a
functionality that has not yet been employed much in spintronics, but could be
the basis for a new generation of spin-based devices made from conventional
materials.

###### Acknowledgements.

This work is financially supported by the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO) as well as JSPS KAKENHI Grant No. 19H006450.
We thank Yaroslav M. Blanter, Haiming Yu, Bi-Mu Yao, Toeno van der Sar,
Sanchar Sharma, Yu-Xiang Zhang, Weichao Yu, and Xiang Zhang for helpful
discussions.

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