# Non-Contact Spin Pumping by Microwave Evanescent Fields

Tao Yu  Kavli Institute of NanoScience, Delft University of Technology, 2628
CJ Delft, The Netherlands  Gerrit E. W. Bauer  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

###### Abstract

The angular momentum of evanescent light fields has been studied in nano-
optics and plasmonics, but not in the microwave regime. Here we predict non-
contact pumping of electron spin currents in conductors by the evanescent
stray fields of excited magnetic nanostructures. The coherent transfer of the
photon to the electron spin is proportional to the $g$-factor, which is large
in narrow-gap semiconductors and surface states of topological insulators. The
spin pumping current is chiral when the spin susceptibility displays
singularities that indicate collective states. However, 1D systems with linear
dispersion at the Fermi energy such as metallic carbon nanotubes are an
exception since spin pumping is chiral even without interactions.

Introduction.—Efficient transfer of spin information among different entities
is a key objective in spintronics [1]. The electromagnetic field at frequency
$\omega$ carries a spin angular momentum density [2; 3]

${\boldsymbol{\mathcal{D}}}=\frac{1}{4\omega}\operatorname{Im}\left(\varepsilon_{0}\mathbf{E}^{\ast}\times\mathbf{E}+\mu_{0}\mathbf{H}^{\ast}\times\mathbf{H}\right),$
(1)

where $\mu_{0}/\varepsilon_{0}$ are the vacuum permeability/permittivity, and
in the microwave regime the magnetic field component
$\sim\mathrm{Im}(\mathbf{H}^{\ast}\times\mathbf{H})$ dominates the
contribution of the electric field $\mathbf{E}$. The evanescent fields at
boundaries can have local angular momentum even when the (linearly-polarized)
propagating ones have not [4; 5]. A distinguishing feature of such evanescent
fields is the locking between the linear and angular momentum [2; 3; 6; 7].
The chiral electrical near-field of a rotating electrical dipole [4; 5], e.g.,
unidirectionally excites surface plasmon polaritons [6; 8; 5]. Metallic
striplines or coplanar waveguides biased by currents in the GHz regime also
emit chiral magnetic near-fields [9; 10; 11], which is of considerable
interest for magnonics [12; 13; 14; 15], since chiral excitation is a robust
and switchable mechanism to pump a DC unidirectional magnon current by an AC
field [16; 17].

Spin pumping by exchange interaction is established when the magnet and
conductor form a good electric contact, which is difficult to achieve between
metals and semiconductors including graphene because of Schottky barriers and
electronic structure mismatch [18; 19]. Even when a good contact to a magnet
can be established, results may be difficult to interpret due to proximity
effects. Spin pumping at a distance by microwaves solves these issues since it
does not require direct contact between the magnet and the system of interest.
In this Letter, we address the non-contact angular momentum transfer to an
electric conductor by stray magnetic fields emitted by an excited magnet,
thereby generalizing the concept of spin pumping by a contact exchange
interaction [18; 19]. We are motivated by the significant near fields that
couple magnetic nanowires and ultrathin magnetic insulating films, causing
several chiral magnon transport phenomena [21; 22; 23; 20; 17; 24; 25; 26].
Here we demonstrate that a magnetodipolar field pumps electron spins into a
conductor without need of electric contacts. We illustrate the physics for a
simple yet realistic model system of a magnetic nanowire on top of a two-
dimensional electron gas (2DEG) as illustrated in Fig. 1. The latter may be
graphene [30; 27; 28; 29], but the effect is strongly enhanced by spin-orbit
interaction, such as a large $g$-factor in InAs or InSb quantum wells (QWs)
[31; 32] or the surface states of 3D topological insulator [33; 34]. In
contrast to the dipolar spin pumping of magnons, the spin pumping current in
non-interacting conductors is in general not chiral. However, the singular
spin susceptibility in one-dimensional systems with linear dispersion at the
Fermi energy such as metallic carbon nanotubes generates chirality of the spin
injection with and without interactions.

Figure 1: Snapshot of spin pumping by the microwave dipolar field of an
excited magnetic nanowire on top of a 2DEG. A thin tunneling barrier
suppresses any exchange coupling. Orange arrows indicate the direction of
stray field.

Transverse spin density of microwaves.—We first demonstrate that the
evanescent magnetodipolar field of a magnetic nanowire carries transverse
angular momentum or “spin”. The nanowire with width $w$, thickness $d$ and
equilibrium magnetization $\mathbf{M}_{s}$ along the wire $y$-direction, on
top of an electron gas confined in the $z$-direction on a length scale $s$ as
illustrated in Fig. 1, acts as an antenna for external microwaves with
frequency tuned to the ferromagnetic resonance (FMR) $\omega_{\mathrm{K}}$
[21; 35; 36]. In the following we use a quantum mechanical notation for
convenience, but in the classical limit operators can be simply replaced by
field amplitudes. The quantum formalism may form a starting point to study
quantum effects in the electron gas, such as spin-pumping-induced magnetic
quantum noise [37] at low temperatures or quantum squeezing and entanglement
of the electrons with microwaves via the magnet [38].

The magnetization dynamics expressed by the spin operator
$\hat{\mathbf{S}}(\mathbf{r},t)$ generates a magnetic field by Coulomb’s Law
[39],

$\mathbf{H}_{\beta}(\mathbf{r},t)=-\frac{\gamma\hbar}{4\pi}\partial_{\beta}\partial_{\alpha}\int
d\mathbf{r}^{\prime}\frac{\langle\hat{\mathbf{S}}_{\alpha}(\mathbf{r}^{\prime},t)\rangle}{|\mathbf{r}-\mathbf{r}^{\prime}|},$
(2)

in the summation convention over repeated spatial (or spin) indices
$\\{\alpha,\beta\\}=\\{x,y,z\\}$. $-\gamma$ is the gyromagnetic ratio of the
nanowire. For sufficiently weak excitation the spin operators in the wire can
be expanded into magnon field operators $\hat{\alpha}_{k_{y}}$ and their
amplitudes across the nanowire $m_{x,z}^{k_{y}}(x,z)$:

$\hat{\mathbf{S}}_{x,z}(\mathbf{r})=\sqrt{2S}\sum_{k_{y}}\left(m_{x,z}^{k_{y}}(x,z)e^{ik_{y}y}\hat{\alpha}_{k_{y}}+\mathrm{H.c.}\right),$
(3)

where $S=M_{s}/(\gamma\hbar)$. The static stray field is negligibly small for
sufficiently long nanowires. The dynamic stray field $\mathbf{H}$ is the
response to $\langle\hat{\alpha}_{k_{y}}\rangle$, the coherent amplitude of
magnons with momentum $k_{y}\hat{\mathbf{y}}$ excited by external microwaves.
With Fourier components
$\mathbf{H}_{\beta}(z,\mathbf{k},t)=\tilde{\mathbf{H}}_{\beta}(z,\mathbf{k})e^{-i\omega_{\mathrm{K}}t}+\tilde{\mathbf{H}}_{\beta}^{\ast}(z,-\mathbf{k})e^{i\omega_{\mathrm{K}}t}$
for $\mathbf{k}=(k_{x},k_{y},0)^{T}$, below the nanowire ($z<0$) [22; 23; 17],

$\displaystyle\left(\begin{array}[c]{c}\tilde{\mathbf{H}}_{x}(z,\mathbf{k})\\\
\tilde{\mathbf{H}}_{y}(z,\mathbf{k})\\\
\tilde{\mathbf{H}}_{z}(z,\mathbf{k})\end{array}\right)$
$\displaystyle=F_{\mathbf{k}}\left(m_{z}^{k_{y}}+\frac{ik_{x}}{k}m_{x}^{k_{y}}\right)\left(\begin{array}[c]{c}ik_{x}/k\\\
ik_{y}/k\\\ 1\end{array}\right)$ (10) $\displaystyle\times
e^{kz}\left|\left\langle\hat{\alpha}_{k_{y}}\right\rangle\right|,$ (11)

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.

$\tilde{\mathbf{H}}(z,\mathbf{k})$ decays exponentially $\sim e^{-k|z|}$ on a
scale governed by complex momentum
$k_{x}\hat{\mathbf{x}}+k_{y}\hat{\mathbf{y}}-ik\hat{\mathbf{z}}$ [4]. External
microwaves excite the Kittel magnon [40] with $k_{y}=0$, such that
$\tilde{H}_{y}(\mathbf{k})$ 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 [41]. We focus here on circularly polarized spin waves
with $m_{x}\rightarrow im_{z}$ in nanowires with circular/square cross
sections or sufficiently large magnetic field. In this case,
$\tilde{\mathbf{H}}(\mathbf{k})=0$ for $k_{x}>0$, since magnons precess
preferentially in one direction, which distinguishes the chirality found here
from the polarization-momentum locking in optics [2; 3], noting that chirality
would vanish for linearly polarized spin waves. The photon spin density under
the nanowire
${\boldsymbol{\mathcal{D}}}(x,z)=\mu_{0}\mathrm{\mathrm{\operatorname{Im}}}\left[\tilde{\mathbf{H}}^{\ast}(x,z)\times\tilde{\mathbf{H}}(x,z)\right]/(4\omega)$
is purely transverse since ${\boldsymbol{\cal D}}\cdot\mathbf{H}=0$ for
arbitrarily polarized spin waves. As illustrated in Fig. 2 for $w=d=60$ nm,
${\boldsymbol{\mathcal{D}}}$ is symmetric with respect to the center of the
nanowire. At finite distances from the wire the near-singularity at the edges
is smeared out, but the average amplitude remains significant. The photon
magnetic field couples to the electron spins by the Zeeman interaction.
Absorption transfers the photon spin over distances limited by the evanescent
decay length in contrast to conventional spin pumping, which happens directly
at the interface.

Figure 2: Spatial distribution of the microwave photon spin density along the
$y$-direction (normalized by the maximum modulus when $z=0$) generated by a
magnetic nanowire with $d=w=60$ nm under FMR. The distances to the nanowire
are $z=0$ (solid line) and 3 nm (dashed-dotted line).

Formalism.—The photon field derived above can excite spins into any conductor
in its proximity. Here we illustrate the concept by a 2DEG with Hamiltonian
$\hat{H}_{0}$ in which only the lowest subband with envelope wave function
$\phi(z)$ is occupied (see Supplemental Material Sec. III for a 1D quantum
wire [42]). The Zeeman coupling between the conduction electron spin
$\hat{\mathbf{s}}$ and the evanescent (near) field amplitude
$\mathbf{H}(\mathbf{r},t)$ reads [39; 4]

$\hat{H}_{\mathrm{Z}}=\mu_{0}\gamma_{e}\int
d\mathbf{r}|\phi(z)|^{2}\hat{\mathbf{s}}(\boldsymbol{\rho},t)\cdot\mathbf{H}(\mathbf{r},t),$
(12)

where $\mathbf{r}=\boldsymbol{\rho}+z\hat{\mathbf{z}}$ and
$\gamma_{e}=-g_{e}\mu_{B}/\hbar$ with $\mu_{B}$ and $g_{e}$ being the Bohr
magneton and (effective) electron $g$-factor, respectively. In the strictly
2DEG limit $|\phi(z)|^{2}\rightarrow\delta\left(z\right)$ and
$\mathbf{H}(\boldsymbol{\rho},z)\approx\mathbf{H}(\boldsymbol{\rho},z=0)$. The
excited spin density in the linear response reads

$s_{l,\alpha}(\boldsymbol{\rho},t)=-\mu_{0}\gamma_{e}\sum_{\omega,\mathbf{k}}e^{i\mathbf{k}\cdot\boldsymbol{\rho}-i\omega
t}\chi_{\alpha\beta}(\mathbf{k},\omega)H_{\beta}(\mathbf{k},\omega),$ (13)

where $\chi_{\alpha\beta}(\mathbf{k},\omega)$ are the elements of the spin
susceptibility tensor [44; 45; 42]. $\mathbf{s}_{l}$ decays with the dipolar
field on the scale of the wire width. Assuming that the spin diffusion length,
which can be of the order of micrometers in 2DEGs [46] (and even longer in
graphene [47]), exceeds the field decay length (tens of nanometer), we can
compute $\chi_{\alpha\beta}$ by straightforward linear response theory (see
Supplemental Material [42]).

The excited spin density is a source term for the kinetic equations, from
which we can calculate spin transport, and when the spin-orbit coupling is
sufficiently weak, the spin current. For the spin dynamics in the 2DEG we need
not only $\mathbf{s}_{l}(\boldsymbol{\rho},t)$ from Eq. (13), but also its
time derivative (“$I$” for interaction representation) [49; 18; 48; 1; 50],
which is derived in the Supplemental Material [42]:

$\displaystyle\left\langle\frac{\partial\mathbf{\hat{s}}_{I}\left(\boldsymbol{\rho},t\right)}{\partial
t}\right\rangle\equiv{\boldsymbol{\mathcal{R}}}(t)=\frac{\partial\mathbf{s}_{l}(\boldsymbol{\rho},t)}{\partial
t}+\mu_{0}\gamma_{e}\mathbf{s}_{l}(\boldsymbol{\rho},t)\times\mathbf{H}(\boldsymbol{\rho},t).$
(14)

This relation recovers that in the conventional spin pumping [18; 19] when
replacing microwave field $\mathbf{H}$ by the magnetization $\mathbf{m}$ at
the interface in Eqs. (13) and (14) [49], in which case
${\boldsymbol{\mathcal{R}}}(t)$ can be interpreted as the spin injection rate
across the interface or spin-current gradient.

When the spin-orbit coupling is negligible, the spin current operator is
defined through the commutator
$-\nabla_{\boldsymbol{\rho}}\cdot\hat{{\boldsymbol{\cal
J}}}_{I}=(i/\hbar)\left[\hat{H}_{0}^{I},\mathbf{\hat{s}}_{I}(\boldsymbol{\rho},t)\right]$,
leading to [49; 48]

$-\nabla_{\boldsymbol{\rho}}\cdot{\boldsymbol{\cal
J}}(\boldsymbol{\rho})={\partial\mathbf{s}_{l}(\boldsymbol{\rho},t)}/{\partial
t}+\mu_{0}\gamma_{e}\mathbf{s}_{l}(\boldsymbol{\rho},t)\times\mathbf{H}(\boldsymbol{\rho},t),$
(15)

where ${\boldsymbol{\mathcal{J}}}(\boldsymbol{\rho})$ is the spin current
tensor with elements $\mathcal{J}_{\alpha}^{\delta}(\boldsymbol{\rho},z)$
($\alpha$ and $\delta$ are the spatial and spin indexes). Substituting Eq.
(13) into Eq. (15) leads to DC and AC spin currents. When the susceptibility
is well behaved at low frequencies and assuming that the Fermi energy
$E_{F}\gg\hbar\omega,$ we may use the adiabatic approximation for the excited
spin density [51; 48; 44; 51; 52],

$\displaystyle
s_{l,\alpha}(\boldsymbol{\rho},t)=-\mu_{0}\gamma_{e}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\mathrm{\mathrm{\operatorname{Re}}}\chi_{\alpha\beta}(\mathbf{k},\omega\rightarrow
0)H_{\beta}(\mathbf{k},t)$
$\displaystyle+\mu_{0}\gamma_{e}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\left.\frac{\partial\mathrm{\operatorname{Im}}\chi_{\alpha\beta}(\mathbf{k},\omega)}{\partial\omega}\right|_{\omega\rightarrow
0}\frac{d{H}_{\beta}(\mathbf{k},t)}{dt}.$ (16)

In the long wavelength limit in which
$\mathrm{\mathrm{\operatorname{Re}}}\chi_{\alpha\beta}(\mathbf{k}\rightarrow
0,\omega\rightarrow 0)$ is constant the first, reactive term on the r.h.s. of
Eq. (16) causes a pure AC contribution $\sim\mathbf{\dot{H}}$ to the spin
current through the first term $\dot{\mathbf{s}}$ in the r.h.s. of Eq. (15).
The second, dissipative term contributes to a DC spin current. We disregard
the nearly homogeneous external microwaves that excites the wire FMR since it
does not contribute to the DC response and assume that a small static magnetic
field that align the wire magnetization has a negligible effect on the 2DEG
spins. The spin current injected under the nanowire can be used as a boundary
condition for a spin transport theory [53].

The DC spin current is evaluated below for two model systems with large
$g$-factors, viz. the 2DEGs in narrow-gap semiconductor heterostructures and
topological surface states and for the 1DEG in the Supplemental Material [42].

Dipolar spin pumping.—For the free electron gas, the spin susceptibility is
isotropic [44; 45],

$\chi(\mathbf{k},\omega)=\frac{\hbar^{2}}{2}\sum_{\mathbf{q}}\frac{f(\xi_{\mathbf{q}})-f(\xi_{\mathbf{k}+\mathbf{q}})}{\hbar\omega+i0_{+}+\xi_{\mathbf{q}}-\xi_{\mathbf{k}+\mathbf{q}}},$
(17)

where $\xi_{\mathbf{k}}=\hbar^{2}k^{2}/(2m^{\ast})-\mu$ is the electron energy
with effective mass $m^{\ast},$ relative to the chemical potential $\mu$, and
$f(\xi_{\mathbf{k}})=\\{\exp[\xi_{\mathbf{k}}/(k_{B}T)]+1\\}^{-1}$ is the
Fermi-Dirac distribution at temperature $T$. In the microwave regime for the
nanowire, $\left|k_{x}\right|<2k_{F},$ where $k_{F}$ is the 2DEG Fermi vector
[in semiconductors $k_{F}=O\left(\mathrm{nm}^{-1}\right)$] and
$\mathrm{Re}\chi(|k_{x}|,\omega\rightarrow
0)=m^{\ast}/\left(\pi\hbar^{2}\right)$ [45] and the reactive first term
vanishes. The DC spin current then reduces to

$\displaystyle{\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(x)$
$\displaystyle=(\mu_{0}\gamma_{e})^{2}\int_{0}^{x}dx\sum_{k_{x}}e^{ik_{x}x}\left.\partial_{\omega}\mathrm{\operatorname{Im}}\chi(|k_{x}|,\omega)\right|_{\omega=0}$
$\displaystyle\times\left\langle\dot{\mathbf{H}}(k_{x},t)\times\mathbf{H}(x,t)\right\rangle_{\mathrm{DC}},$
(18)

where we used the symmetry relations ${\boldsymbol{\cal
J}}_{x}^{\mathrm{D}C}(k_{x})=-{\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(-k_{x})$
and ${\boldsymbol{\cal J}}_{x}^{\mathrm{DC}}(x=0)=0$ [42]. Assuming for the
moment that $\chi(|k_{x}|,\omega)\approx\chi(k_{\mathrm{ave}},\omega)$ with
$k_{\mathrm{ave}}\sim\pi/(2w)$, we obtain the simplified expression at FMR

$\displaystyle{\boldsymbol{\cal J}_{x}^{\mathrm{DC}}(x)}$
$\displaystyle\approx\left.-2(\mu_{0}\gamma_{e})^{2}\omega_{\mathrm{K}}\partial_{\omega}\mathrm{\operatorname{Im}}\chi\left(|k_{x}|\rightarrow
k_{\mathrm{ave}},\omega\right)\right|_{\omega=0}$
$\displaystyle\times\int_{0}^{x}dx^{\prime}\mathrm{\operatorname{Im}}\left[\tilde{\mathbf{H}}^{\ast}(x^{\prime})\times\tilde{\mathbf{H}}(x^{\prime})\right].$
(19)

Hence, the DC spin-current below the nanowire is (approximately) proportional
to the transverse spin of the magnetic field, implying transfer of the photon
spin angular momentum to the electron spin with an efficiency governed by
$\left.\partial_{\omega}\mathrm{\operatorname{Im}}\chi\left(|k_{x}|\rightarrow
k_{\mathrm{ave}},\omega\right)\right|_{\omega=0}$. The spin current is
polarized in the $-y$-direction, i.e., opposite to the magnetization direction
of the nanowire.

Since ${\boldsymbol{\mathcal{J}}}_{x}(x)\propto\mathrm{sgn}(x)$, the excited
spin is not chiral, but flows into both directions on both sides of the
nanowire as indicated by the blue arrow in Fig. 1, just as in conventional
spin pumping [18; 48; 51; 52]. Although excited by the same field, this result
is in stark contrast to the magnon spin current [22; 23; 17; 11] or the chiral
energy currents of surface plasmon polaritons excited by a rotating electric
dipole [6; 8; 5], which are both unidirectional and flow in half space. We can
trace the different physics to the collective nature of magnons/plasmons with
a well-defined dispersion relation that in the present geometry are symmetric
in $k$-space, but of which a chiral dipolar field selects only one. The
susceptibility of the non-interacting electron gas, on the other hand, is made
up by a broad spectrum of electron-hole pair excitations at the Fermi energy,
and chirality vanishes in the integral over wave vectors at a given frequency.
Interestingly, chirality emerges for metallic carbon nanotubes or interacting
electrons in a quantum wire that cross the magnetic wire at right angles [42],
because the spin susceptibility in the 1DEG with a linear dispersion at the
Fermi energy such as metallic carbon nanotubes with (Tomonaga-Luttinger
liquid) or without interactions is singular [44; 42]

$\chi(k,\omega)=\frac{\hbar
kL}{2\pi}\left(\frac{1}{\omega+i0_{+}-v_{F}k}+\frac{1}{\omega+i0_{+}+v_{F}k}\right),$
(20)

where $L$ is the system length and $v_{F}$ is the Fermi velocity. By contour
integration and the chiral dipolar field with
$\mathbf{H}(k_{x}>0,\omega_{\mathrm{K}})=0$ for the right-circularly polarized
spin waves,

$\mathbf{s}_{l}(x,t)=\mathrm{Im}\left(-\frac{2\mu_{0}\gamma_{e}\omega_{\mathrm{K}}}{v_{F}^{2}}\mathbf{H}\left(k_{-},\omega_{\mathrm{K}}\right)e^{ik_{-}x-i\omega_{\mathrm{K}}t}\right),$
(21)

when $x<0$ but vanishes when $x>0$, with $k_{-}=-{\omega_{\mathrm{K}}}/v_{F}$,
implying that the excited spin density lives only in half of the nanowire. The
DC spin current vanishes when $x>0$, but flows in the same half space $x<0$
with

${\boldsymbol{\mathcal{J}}}_{x}(x,t)=-\mu_{0}\gamma_{e}\int_{0}^{x}dx^{\prime}\left.\mathbf{s}_{l}(x^{\prime},t)\times\mathbf{H}(x^{\prime},t)\right|_{\mathrm{DC}},$
(22)

recovering the chiral excitation of a spin-density current [42] found earlier
in magnetic films. This example proves that quite generally chiral excitation
by dipolar radiation is not caused by a hidden symmetry, but requires poles in
the spin-susceptibility generated by degenerate electron-hole pairs, or the
plasmon, magnon, phonon excitations of a rigid ground state.

We now estimate the magnitude of the DC spin current and/or spin-injection
rate by the dipolar field from an excited magnetic nanowire. We choose a
symmetric QW with $s=20$ nm of a semiconductor with small effective mass such
as InSb with $m^{\ast}=0.015m_{e}$ [31] and electron density $n_{e}=3\times
10^{11}~{}\mathrm{cm}^{-2}$ (corresponding to a Fermi energy $E_{F}\sim 50$
meV and Fermi temperature 560 K), such that only the lowest band is populated
even at room temperature. The Dresselhaus-type spin-orbit coupling with
coefficient $\gamma_{D}=220$ $\mathrm{eV\mathring{A}}^{3}$ [31; 54] causes a
small correction $\gamma_{D}(\pi/s)^{2}k_{F}\sim 0.7$ $\mathrm{meV}\ll E_{F}$
that we disregard. The $g$-factor of electron is $g_{e}=-36$ [31; 32], but the
sign is not important here. At temperature $T=100$ K the system is degenerate
with subband splitting $\hbar^{2}(\pi/s)^{2}/(2m^{\ast})=63$ $\mathrm{meV}\gg
k_{B}T$. For a Co or CoFeB nanowire with $w=d=60$ nm and $\mu_{0}M_{s}=1.2$ T
[23], we assume a coherent magnon density
$\rho_{m}=|\langle\hat{\alpha}_{k_{y}=0}\rangle|^{2}=10^{9}$ in Eq. (3) that
corresponds to a transverse magnetization amplitude $M_{x,y}\sim
2\sqrt{2\gamma\hbar M_{s}\rho_{m}}m_{x,y}^{k_{y}=0}$, i.e. a small precession
cone angle $\sim 3.2\times 10^{-3}$ degrees that is easily excited by FMR.

We plot the DC spin current under the nanowire from Eqs. (18) and (19) in Fig.
3(a), in which the bidirectional spin current is indicated by the black
arrows, and the spin-injection rate in Fig. 3(b). The simplified Eq. (19)
describes the pumped current by Eq. (18) well. With the same conditions, the
pumped spin current is four times in magnitude smaller in InAs QWs
($m^{\ast}=0.023m_{e}$ and $|g_{e}|=14.3$ [31]). The spin current is of the
same order as the spin Hall current generated by an electric field of
$0.1~{}\mathrm{kV}/\mathrm{cm}$ and a spin Hall conductivity
$\sigma_{x}^{y}=10^{6}~{}(\Omega\cdot\mathrm{m})^{-1},$ which should be easily
measurable [55]. Under the same conditions, the spin current pumped by the
dipolar interaction is comparable with that from interfacial exchange
interaction with an exchange splitting $JM_{s}\hbar\sim 10$ meV [56], but does
not require good electric contact between magnet and semiconductor.

Figure 3: Excited spin current ${\boldsymbol{\mathcal{J}}}_{x}/h$, with
$h=2\pi\hbar$ [(a)] and spin-injection rate [(b)] under an excited Co
nanowire. In (a), the blue and red curves are calculated from Eqs. (18) and
(19), respectively. In (b), the blue and red curves are the spin injection
rates for a semiconductor 2DEG and the surface state of a topological
insulator, normalized by the maximal magnitude $R_{0}/\hbar=2.7\times 10^{18}$
m-2 of the blue curve.

The excited spin current under the transducer drives diffusive spin transport
over the spin diffusion length scale [53]. The spin signal can be converted to
a transverse voltage by the inverse spin Hall effect in the 2DEG itself or by
heavy metal contacts [55] or the inverse Edelstein effect [57]. The cyclotron
resonance excited by evanescent microwave magnetic fields in the quantum Hall
regime could be an interesting extension of the present work.

Finally, we estimate the efficiency of the dipolar spin pumping for surface
states of the $n$-doped topological insulator Bi2Se3 [58] at a low temperature
$T=30$ K for which a good exchange interaction with magnetic contacts is
difficult to achieve [34] and perhaps not desired because of an associated
proximity effect. Since the spin current is not conserved, we focus on the DC
spin injection rate ${\boldsymbol{\mathcal{R}}}$ defined in Eq. (14) and
compared with the semiconductor case in Fig. 3(b). Only the diagonal terms of
the susceptibility tensor [59]

$\chi(|k_{x}|,\omega)=\frac{\hbar^{2}}{8}\sum_{\mathbf{q}}(1-\cos\phi_{\mathbf{q}})\frac{n_{F}(\xi_{\mathbf{q}-\mathbf{k}})-n_{F}(\xi_{\mathbf{q}})}{\hbar\omega+i0_{+}+\xi_{\mathbf{q}-\mathbf{k}}-\xi_{\mathbf{q}}},$
(23)

where $\xi_{\mathbf{k}}=\hbar v_{F}k-\mu$, contribute to the DC spin injection
(see Supplemental Material [42]). With $n_{e}=10^{11}$ cm-2, $v_{F}=10^{5}$
m/s, and $\left|g_{e}\right|=20$ [33], the spin injection rate is of the same
order as that of InSb semiconductor 2DEG.

Discussion.—Spin pumping by evanescent microwaves is a coherent mechanism for
the generation of pure spin currents in conventional spintronic systems and
devices that requires nanomagnets rather than extended films. The effect is
not small: spin currents generated by the stray fields of Co nanowires on top
of a thin yttrium iron garnet film exceed those by the exchange coupling [23].
Here we focus on transverse spin pumping into a conductor, which is most
efficient for low-dimensional electron systems. We predict here a transverse
spin current density $10^{-13}~{}\mathrm{J/m}$ pumped into an InAs 2DEG, which
is almost two orders of magnitude larger than what has been observed for the
spin-pumping by a Py slab into graphene [29], whose signal should indeed be
much smaller with small $g$-factor. We therefore cannot exclude that the
observations are caused by dipolar fields at the edge of Py, and not exchange
interactions at interface. In general, however, spin pumping into 2DEGs by
extended magnetic films [60; 61; 62; 63] (and in basically all planar
structures used in conventional spin pumping experiments) should be dominated
by the exchange mechanism.

The photon angular momentum is inherent to the evanescent stray fields of a
precessing magnetization, but it also exists in microwave cavities or
waveguides. The dipolar spin pumping is contactless and avoids possible
artifacts by the magnetic proximity effect. The excited spin current is not
chiral for 2DEGs, but chirality re-emerges in the 1DEG. The spin pumping by a
magnetic transducer into a 2DEG and a surface state of a topological insulator
are estimated large enough to be observable. Our study bridges the concepts
and understandings in different fields including spintronics [1; 12; 13; 64;
19], nano-optics [5] and plasmonics [2; 3].

###### 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 Tserkovnyak, Alejandro O. Leon and Jin Lan for useful
discussions.

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