# Energy-efficient spin injector into semiconductors driven by elastic waves

Andrei V. Azovtsev∗ Andrei I. Nikitchenko  Nikolay A. Pertsev  Ioffe Institute
194021 St. Petersburg Russia
<EMAIL_ADDRESS>
###### Abstract

Generation of significant spin imbalance in nonmagnetic semiconductors is
crucial for the functioning of many spintronic devices, such as magnetic
diodes and transistors, spin-based logic gates, and spin-polarized lasers. An
attractive design of spin injectors into semiconductors is based on a spin
pumping from a precessing ferromagnet, but the classical excitation of
magnetization precession by a microwave magnetic field leads to the high power
consumption of the device. Here we describe theoretically a spin injector with
greatly reduced energy losses, in which the magnetic dynamics is excited by an
elastic wave generated in a ferromagnet-semiconductor heterostructure by an
attached piezoelectric transducer. To demonstrate the efficient functioning of
such an injector, we first perform micromagnetoelastic simulations of the
coupled elastic and magnetic dynamics in $\mathrm{Ni}$ films and
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers traversed by plane longitudinal and
shear waves. For thick $\mathrm{Ni}$ films, it is shown that a monochromatic
acoustic wave generates a spin wave with the same frequency and wavelength,
which propagates together with the driving wave over distances of several
micrometers at the excitation frequencies $\nu\approx 10$ GHz close to the
frequency of ferromagnetic resonance. The simulations of
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers with $\mathrm{Ni}$ thicknesses
comparable to the wavelength of the injected acoustic wave demonstrate the
development of a steady-state magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface. The amplitude of such a precession has
a maximum at $\mathrm{Ni}$ thickness amounting to three quarters of the
wavelength of the elastic wave, which is explained by an analytical model.
Using simulation data obtained for the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, we evaluate the spin current pumped
into $\mathrm{GaAs}$ and calculate the spin accumulation in the semiconducting
layer by solving the spin diffusion equation. Then the electrical signals
resulting from the spin flow and the inverse spin Hall effect are determined
via the numerical solution of the Laplace’s equation. It is shown that
amplitudes of these ac signals near the interface are large enough for
experimental measurement, which indicates an efficient acoustically driven
spin pumping into $\mathrm{GaAs}$ and rather high spin accumulation in this
semiconductor.

##  I Introduction

Semiconductors are attractive for the development of spintronic devices due to
their large spin diffusion lengths in comparison with transition metals Hägele
_et al._ (1998); Kikkawa and Awschalom (1999), long spin relaxation times Bhat
and Kumar (2014), and the possibility of manipulating the electrons’ spin by
polarized light Putikka and Joynt (2004); Kokurin  _et al._ (2013). However,
the application of conventional nonmagnetic semiconductors in spintronics
requires the generation of an internal spin imbalance by an external stimulus
or via an attached magnetic material Hirohata  _et al._ (2020). The simplest
method to create such an imbalance would be the direct injection of a spin-
polarized charge current from a metallic ferromagnet through Ohmic contact,
but the conductance mismatch at the semiconductor-metal interface makes this
method inefficient Schmidt  _et al._ (2000). The presence of a thin insulating
interlayer acting as a tunnel barrier solves the mismatch problem Hanbicki and
Jonker (2002); Jiang  _et al._ (2005); Dash  _et al._ (2009); Kamerbeek  _et
al._ (2014), but requires the fabrication of a high quality interlayer unless
the formation of a natural Schottky barrier with the appropriate parameters
occurs Hanbicki and Jonker (2002). Alternatively, the spin imbalance in the
semiconductor can be created by bringing it into a direct contact with a
precessing ferromagnet Brataas  _et al._ (2002); Tserkovnyak  _et al._ (2005).
The resulting spin pumping into the nonmagnetic semiconductor is due to the
modulation of the interface scattering matrix by the coherent precession of
the magnetization Brataas  _et al._ (2002).

Typically, in spin pumping experiments magnetization dynamics is excited by an
external microwave magnetic field with the frequency matching that of the
ferromagnetic resonance. Efficient generation of spin currents in normal
metals by this technique has been demonstrated experimentally Heinrich  _et
al._ (2003); Saitoh  _et al._ (2006); Bell  _et al._ (2008); Mosendz  _et al._
(2010); Czeschka  _et al._ (2011); Tashiro  _et al._ (2015). The spin pumping
into semiconductors from metallic ferromagnets Ando  _et al._ (2011); Shikoh
_et al._ (2013); Lee  _et al._ (2014); Wang  _et al._ (2017) and ferrimagnetic
insulators Mendes  _et al._ (2018) subjected to microwave radiation has been
revealed as well. However, the power consumption associated with the
generation of microwave magnetic fields appears to be rather high, which
impedes applications of magnetically driven spin injectors in low-power
spintronics. For this reason, alternative spin pumping techniques have been
studied during the past decade, one of which is based on the excitation of
magnetization dynamics in ferromagnets by injected elastic waves Weiler  _et
al._ (2011); Uchida  _et al._ (2011a); Weiler  _et al._ (2012); Kamra  _et
al._ (2015); Polzikova  _et al._ (2016); Azovtsev and Pertsev (2016, 2017);
Polzikova  _et al._ (2018); Azovtsev and Pertsev (2019); Alekseev  _et al._
(2020). Since such waves can be generated by a piezoelectric transducer
coupled to the ferromagnet and subjected to an ac electric field, the power
consumption of elastically driven spin injectors is expected to be
comparatively low Alekseev  _et al._ (2020); Cherepov  _et al._ (2014);
Bhaskar  _et al._ (2020). The experimental and theoretical studies have
demonstrated an efficient generation of spin currents in normal metals by
surface and bulk acoustic waves, but the strain-driven spin pumping into
semiconductors was not investigated so far.

In this paper, we theoretically describe a spin injector into nonmagnetic
semiconductors, which employs the spin pumping generated by a dynamically
strained ferromagnetic film. The injector has the form of a ferromagnet-
semiconductor bilayer coupled to a piezoelectric transducer excited by a
microwave voltage. Such a transducer creates a bulk elastic wave propagating
across the bilayer, which induces a radio-frequency magnetization precession
providing efficient spin pumping into the semiconducting layer. To quantify
the elastically driven magnetic dynamics in the ferromagnetic film, we employ
the state-of-the-art numerical simulations allowing for the two-way coupling
between spins and strains (see Sec. II). The simulations are performed for the
$(001)$-oriented $\mathrm{Ni}$ films and $\mathrm{Ni/GaAs}$ bilayers traversed
by plane longitudinal and transverse acoustic waves. For thick $\mathrm{Ni}$
films, tightly coupled elastic and magnetic dynamics are described (Sec. III),
which involve the generation of a spin wave carried by the propagating elastic
wave. In Sec. IV, we report the results of numerical simulations performed for
$\mathrm{Ni/GaAs}$ bilayers with the $\mathrm{Ni}$ thickness comparable to the
wavelength of the propagating elastic wave and discuss the influence of the
thickness of the ferromagnetic layer and the excitation frequency on the
amplitude of the magnetization precession at the $\mathrm{Ni|GaAs}$ interface
(Sec. IV). Numerical results obtained for the steady-state magnetization
precession at the interface are then used to calculate the spin pumping into
the $\mathrm{GaAs}$ film and to determine the spin accumulation in the
semiconductor by solving the spin diffusion equation (Sec. V). It is shown
that the proposed injector has a high efficiency ensuring significant spin
flux in $\mathrm{GaAs}$, which can be detected experimentally via the inverse
spin Hall effect.

$y$$z$$x$m$\psi$transducer$t_{\text{F}}$$t_{\text{N}}$$w_{\text{N}}$$h_{\text{N}}$k$\mathrm{GaAs}$$\mathrm{Ni}$Fe$V_{\text{s}}$
Figure 1:  Ferromagnet-semiconductor heterostructure comprising $\mathrm{Ni}$
and $\mathrm{GaAs}$ layers. An elastic wave (longitudinal or transverse) with
the wave vector k is injected into the $\mathrm{Ni}$ layer by the attached
piezoelectric transducer. The thicknesses of the $\mathrm{Ni}$ and
$\mathrm{GaAs}$ layers are denoted by $t_{\text{F}}$ and $t_{\text{N}}$,
respectively. Precessing magnetization in Ni creates a spin imbalance in GaAs,
which then produces a measurable voltage $V_{\text{s}}$ between an attached
iron probe and a nonmagnetic contact.

##  II Modeling of magnetoelastic phenomena in ferromagnetic heterostructures

Owing to the magnetoelastic coupling between spins and strains, the excitation
of an elastic wave in a ferromagnetic material can induce a precessional
motion of the magnetization and the generation of a spin wave Weiler  _et al._
(2011); Uchida  _et al._ (2011b); Weiler  _et al._ (2012); Thevenard  _et al._
(2014); Janušonis  _et al._ (2015); Gowtham  _et al._ (2015); Casals  _et al._
(2020). The backaction of the induced magnetization precession on strain state
of the ferromagnet may significantly affect the propagation of the driving
elastic wave and lead to the appearance of additional “secondary” waves
Azovtsev and Pertsev (2017, 2019). Therefore, the two-way interplay between
elastic and magnetic variables Akhiezer  _et al._ (1958) should be fully taken
into account for an accurate modeling of the magnetoelastic phenomena in
ferromagnets. Such micromagnetoelastic modeling can be realized via the
numerical solution of the system of differential equations comprising the
elastodynamic equation for the mechanical displacement u and the Landau-
Lifshitz-Gilbert (LLG) equation for the magnetization M Chen  _et al._ (2017);
Azovtsev and Pertsev (2017, 2019, 2020). The elastodynamic equation should
allow for the magnetoelastic contribution $\delta\sigma_{ij}^{\text{ME}}$ to
the mechanical stresses $\sigma_{ij}$ in the ferromagnet, which can be
calculated as $\delta\sigma_{ij}^{\text{ME}}=\partial
F_{\text{ME}}/\partial\varepsilon_{ij}$, where $F_{\text{ME}}$ is the
magnetoelastic energy density, and $\varepsilon_{ij}$ are the elastic strains
($i,j=x,y,z$). The influence of strains on the magnetization orientation can
be quantified by adding a magnetoelastic term
$\textbf{H}_{\text{ME}}=-(1/\mu_{0})\partial F_{\text{ME}}/\partial\textbf{M}$
to the effective magnetic field $\textbf{H}_{\text{eff}}$ involved in the LLG
equation ($\mu_{0}$ being the magnetic constant). For cubic ferromagnets such
as nickel, the magnetoelastic contribution to the total energy density F can
be written as

$\displaystyle F_{\text{ME}}$
$\displaystyle=B_{1}[(m_{x}^{2}-\frac{1}{3})\varepsilon_{xx}+(m_{y}^{2}-\frac{1}{3})\varepsilon_{yy}+(m_{z}^{2}-\frac{1}{3})\varepsilon_{zz}]$
(1)
$\displaystyle+2B_{2}\left[m_{x}m_{y}\varepsilon_{xy}+m_{x}m_{z}\varepsilon_{xz}+m_{y}m_{z}\varepsilon_{yz}\right],$

where $\textbf{m}=\textbf{M}/M_{s}$ is the unit vector in the magnetization
direction, $M_{s}$ is the saturation magnetization regarded as a strain-
independent quantity, and $B_{1}$, $B_{2}$ are the magnetoelastic coupling
constants Kittel (1949).

In this work, we performed micromagnetoelastic simulations of $\mathrm{Ni}$
films and $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers subjected to a periodic
displacement $\textbf{u}(x=0,t)=\textbf{u}_{0}(t)$ imposed at the
$\mathrm{Ni}$ surface $x=0$ (Fig. 1). Such a displacement models the action of
a piezoelectric transducer coupled to $\mathrm{Ni}$ film and generates a plane
elastic wave traversing the heterostructure Azovtsev and Pertsev (2017, 2019).
To excite a longitudinal wave characterized by the strain
$\varepsilon_{xx}(x,t)$, we introduced the surface displacement with the
components $u^{0}_{y}=u^{0}_{z}=0$ and $u^{0}_{x}=u_{\text{max}}\sin{(2\pi\nu
t)}$, while a transverse wave with the shear strain $\varepsilon_{xz}(x,t)$
was created by setting $u^{0}_{x}=u^{0}_{y}=0$ and
$u^{0}_{z}=u_{\text{max}}\sin{(2\pi\nu t)}$. The excitation frequency $\nu$
was varied in a wide range spanning the resonance frequency $\nu_{\text{res}}$
of the coherent magnetization precession in the unstrained $\mathrm{Ni}$ film,
which was determined by simulations of the magnetization relaxation to its
equilibrium orientation. To ensure the same maximal strain in the elastic wave
at any excitation frequency $\nu$, the displacement amplitude $u_{\text{max}}$
was taken to be inversely proportional to $\nu$ Azovtsev and Pertsev (2019).
Namely, we used the relations
$u_{\text{max}}=\varepsilon_{xx}^{\text{max}}/k_{L}$ and
$u_{\text{max}}=2\varepsilon_{xz}^{\text{max}}/k_{T}$ for longitudinal and
transverse waves, respectively, where $k_{L}=2\pi\nu/c_{L}$ and
$k_{T}=2\pi\nu/c_{T}$ are the wave numbers of these waves having velocities
$c_{L}$ and $c_{T}$.

The magnetization dynamics in the $\mathrm{Ni}$ film was quantified using the
LLG equation with the effective magnetic field $\textbf{H}_{\text{eff}}$
comprising contributions resulting from the exchange interaction, cubic
magnetocrystalline anisotropy, magnetoelastic coupling, Zeeman energy, and
dipolar interactions between oscillating spins Azovtsev and Pertsev (2016).
For numerical calculations, we characterized $\mathrm{Ni}$ by the saturation
magnetization $M_{s}=4.78\times 10^{5}$ A m-1 Niitsu (2020), exchange constant
$A_{\text{ex}}=0.85\times 10^{-11}$ J m-1 Niitsu (2020), magnetocrystalline
anisotropy constants $K_{1}=-5.7\times 10^{3}$ J m-3, $K_{2}=-2.3\times
10^{3}$ J m-3 Stearns (1986), magnetoelastic constants $B_{1}=9.2\times
10^{6}$ J m-3, $B_{2}=10.7\times 10^{6}$ J m-3 Stearns (1986), and Gilbert
damping parameter of 0.045 Walowski  _et al._ (2008). The elastic stiffnesses
$c_{11}$, $c_{44}$ and mass densities $\rho$ of $\mathrm{Ni}$ and
$\mathrm{GaAs}$, which are involved in their elastodynamic equations of
motion, were taken from Ref. Haynes (2016) and listed in Table 1 together with
the velocities $c_{L}$ and $c_{T}$ of elastic waves in these materials. No
elastic damping was added to the elastodynamic equation in our simulations for
the following reasons. First, we are interested in investigating the purely
magnetic damping of elastic waves in $\mathrm{Ni}$, and the introduction of
the intrinsic elastic damping would obscure simulation results described in
Sec. III. Second, in Sec. IV we consider $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayers comprising $\mathrm{Ni}$ films much thinner than the decay lengths of
longitudinal and transverse elastic waves in $\mathrm{Ni}$, which are measured
to be 5.8 and 29 $\mu$m respectively at the relevant frequency of 9.4 GHz
Homer  _et al._ (1987).

Micromagnetoelastic simulations were performed with the aid of a homemade
software operating with a finite ensemble of nanoscale computational cells.
Our software solves the elastodynamic equations of $\mathrm{Ni}$ and
$\mathrm{GaAs}$ films by a finite-difference technique with a midpoint
derivative approximation and numerically integrates the LLG equation by the
projective Runge-Kutta algorithm. We employed a fixed integration step $\delta
t=100$ fs and set the size of cubic computation cells to $2$ nm, which is
smaller than the exchange length
$l_{\text{ex}}=\sqrt{2A_{\text{ex}}/(\mu_{0}M_{s}^{2})}\approx 7.7$ nm of
$\mathrm{Ni}$. The system of partial differential equations was appended by
appropriate boundary conditions. At the free surface of the $\mathrm{GaAs}$
layer, the stresses $\sigma_{ix}$ were set to zero, and the “free-surface”
condition $\partial\textbf{m}/\partial x=0$ was imposed at both boundaries of
the $\mathrm{Ni}$ layer. Since a unified ensemble of computational cells
covering the whole $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer was employed in the
simulations, the continuity conditions at the $\mathrm{Ni}|\mathrm{GaAs}$
interface were satisfied automatically. The layers comprising the
heterostructure were considered infinite in $y-z$ plane and the dynamical
quantities were allowed to change only along the $x$ direction.

| $\mathrm{Ni}$ | $\mathrm{GaAs}$
---|---|---
$c_{11}$ ($10^{11}$ J m-3) | 2.481 | 1.188
$c_{12}$ ($10^{11}$ J m-3) | 1.549 | 0.537
$c_{44}$ ($10^{11}$ J m-3) | 1.242 | 0.594
$\rho$ (kg m-3) | 8910 | 5317
$c_{L}$ (m s-1) | 5277 | 4726
$c_{T}$ (m s-1) | 3734 | 3344
Table 1: Elastic stiffnesses and mass densities of $\mathrm{Ni}$ and
$\mathrm{GaAs}$ Haynes (2016) used in numerical calculations. Velocities
$c_{L}=\sqrt{c_{11}/\rho}$ and $c_{T}=\sqrt{c_{44}/\rho}$ of longitudinal and
transverse elastic waves in these materials are also given for information.

##  III Magnetic dynamics excited by longitudinal and transverse elastic waves
in thick Ni films

An elastic wave perturbs the ferromagnetic state when it creates a nonzero
magnetoelastic torque
$\textbf{T}_{\text{ME}}=\textbf{M}\times\textbf{H}_{\text{ME}}$ acting on the
magnetization M. In the case of the longitudinal wave $\varepsilon_{xx}(x,t)$,
the effective field $\textbf{H}_{\text{ME}}$ has the only nonzero component
$H_{x}^{\text{ME}}=-2B_{1}\varepsilon_{xx}m_{x}/M_{s}$. Therefore, an external
magnetic field H creating the direction cosine $m_{x}$ in the in-plane
magnetized $\mathrm{Ni}$ film should be applied to generate the magnetization
dynamics. To additionally stabilize the single-domain state, we introduced the
field along the [111] crystallographic direction (easy axis), taking
$H_{x}=H_{y}=H_{z}=1000$ Oe. At such field, the magnetization in the
unstrained $\mathrm{Ni}$ film has an elevation angle $\psi\approx 46^{\circ}$
($m_{x}=0.137$) and equal projections on [100] and [010] directions
($m_{y}=m_{z}=0.70044$). The same magnetic field was used in the simulations
of magnetic dynamics induced by the shear acoustic wave
$\varepsilon_{xz}(x,t)$, which facilitates the comparison of results obtained
for two types of elastic perturbations. It should be noted that shear waves
impose nonzero magnetoelastic torque $\textbf{T}_{\text{ME}}$ even on the in-
plane magnetized ferromagnetic films. Indeed, the effective field
$\textbf{H}_{\text{ME}}$ created by the strain $\varepsilon_{xz}$ has two
components, $H_{z}^{\text{ME}}=-2B_{2}\varepsilon_{xz}m_{x}/M_{s}$ and
$H_{x}^{\text{ME}}=-2B_{2}\varepsilon_{xz}m_{z}/M_{s}$, and the latter differs
from zero even at $m_{x}=0$ when $m_{z}\neq 0$. However, simulations show that
the amplitude of the magnetization precession increases significantly when
both $H_{x}^{\text{ME}}$ and $H_{z}^{\text{ME}}$ differ from zero.

At the chosen magnetic field H, the resonance frequency $\nu_{\text{res}}$ of
the unstrained $\mathrm{Ni}$ film with in-plane dimensions much larger than
the film thickness $t_{\text{F}}$ was found to be $9.6$ GHz. Accordingly, the
excitation frequency was varied in a wide range around 10 GHz. By selecting
the appropriate amplitudes $u_{\text{max}}(\nu)$ of the surface displacements
$u_{x}^{0}(t)$ or $u_{z}^{0}(t)$, we created the maximal strains
$\varepsilon^{\text{max}}_{xx}=\varepsilon^{\text{max}}_{xz}=10^{-4}$ in the
excited elastic waves near the $\mathrm{Ni}$ surface $x=0$. Simulations were
first performed for $\mathrm{Ni}$ films with thicknesses $t_{\text{F}}$ much
larger than the wave lengths $\lambda_{L}=c_{L}/\nu$ and
$\lambda_{T}=c_{T}/\nu$ of the pure elastic waves, which amount to
$\lambda_{L}=550$ nm and $\lambda_{T}=389$ nm at the resonance excitation
$\nu=\nu_{\text{res}}$. This allows the observation of several wave periods
inside the ferromagnetic film. Since in this section we concentrate on the
propagation of the elastic waves in $\mathrm{Ni}$ and their interaction with
the magnetic subsystem, the simulation time was limited by the period needed
for the wave to reach the opposite boundary of the $\mathrm{Ni}$ film. The
effects of the wave reflection from the $\mathrm{Ni}|\mathrm{GaAs}$ interface
are discussed in Section IV.

Figure 2:  Spatial distributions of strains in the driving longitudinal (a)
and transverse (b) elastic waves and strain-induced variations of the
magnetization direction cosines $m_{i}$ in the 2-$\mu$m-thick $\mathrm{Ni}$
film. Excitation frequency $\nu$ is equal to the resonance frequency
$\nu_{\text{res}}=9.6$ GHz, and snapshots are taken at 0.37 ns (a) and 0.52 ns
(b).

The simulations of the coupled elastic and magnetic dynamics in thick
$\mathrm{Ni}$ films confirmed the creation of periodic, almost sinusoidal
elastic waves at all studied excitation frequencies $\nu$ (see Fig. 2). The
wave emerges at the $\mathrm{Ni}$ surface and propagates away with the
velocity $c_{L}$ or $c_{T}$ characteristic of a pure elastic wave despite an
inhomogeneous magnetization precession excited by the magnetoelastic torque
$\textbf{T}_{\text{ME}}$ (see Fig. 2). However, the strain-induced precession
manifests itself in the generation of the additional elastic waves caused by
the magnetoelastic feedback (see Fig. 3). These secondary strain waves were
already revealed by the micromagnetoelastic simulations performed for Fe81Ga19
and CoFe2O4 films Azovtsev and Pertsev (2017, 2019), but were not detected in
the simulations of the propagation of longitudinal elastic waves in
$\mathrm{Ni}$ Chen  _et al._ (2017). When the driving wave is a longitudinal
one, two transverse secondary waves with the strains $\varepsilon_{xy}(x,t)$
and $\varepsilon_{xz}(x,t)$ having amplitudes $\sim 10^{-7}$ appear in
$\mathrm{Ni}$. Their profiles depend on the position in the film, exhibiting a
peculiar behavior similar to that of the secondary waves arising in CoFe2O4
excited by longitudinal elastic waves Azovtsev and Pertsev (2019). This
behavior is caused by the interference of the two components of each secondary
wave, which have the form of a shear wave with the wavelength $\lambda_{T}$
and velocity $c_{T}$ freely propagating from the $\mathrm{Ni}$ surface and a
forced shear wave with the wavelength $\lambda_{L}$ and velocity $c_{L}$
generated in the whole driving longitudinal wave. When the driving wave is a
transverse one [$\varepsilon_{xz}(x,t)$], a longitudinal secondary wave
$\varepsilon_{xx}(x,t)$ and another shear wave $\varepsilon_{xy}(x,t)$ with
amplitudes $\sim 10^{-6}$ are generated by the magnetization precession.
Similarly to the aforementioned situation, the longitudinal wave appears to be
a superposition of a free wave with the wavelength $\lambda_{L}$ and velocity
$c_{L}$ and a forced wave with the parameters $\lambda_{T}$ and $c_{T}$. In
contrast, the secondary shear wave can be regarded as a single wave because
its wavelength $\lambda_{T}$ and velocity $c_{T}$ match those of the driving
wave.

Figure 3:  Secondary elastic waves generated by the magnetization precession
induced by the primary longitudinal (a) and transverse (b) waves in the
2-$\mu$m-thick $\mathrm{Ni}$ film. Snapshots are taken at 0.37 ns (a) and 0.35
ns (b).

The magnetization precession induced by the primary elastic wave also affects
its propagation at long distances from the $\mathrm{Ni}$ surface. A careful
evaluation of the local strain amplitudes $\varepsilon_{xx}^{\text{max}}$ and
$\varepsilon_{xz}^{\text{max}}$ in the driving longitudinal and transverse
waves reveals that they decrease with the increasing distance $x$ from the
$\mathrm{Ni}$ surface. This decay is caused by the energy transfer to the
magnetic subsystem, where the strain-driven magnetization precession is
hindered by the Gilbert damping Azovtsev and Pertsev (2019). The analysis of
the simulation results shows that the dependences
$\varepsilon_{xx}^{\text{max}}(x)$ and $\varepsilon_{xz}^{\text{max}}(x)$ can
be fitted by an exponential function $e^{-x/L_{\text{dec}}}$, where the decay
length $L_{\text{dec}}$ depends on the wave frequency $\nu$. At the resonance
excitation $\nu=9.6$ GHz, $L_{\text{dec}}$ amounts to approximately 350 $\mu$m
for the longitudinal wave and about 19 $\mu$m for the shear wave. This finding
explains why no damping of magnetic origin was detected in simulations of the
propagation of longitudinal elastic waves in $\mathrm{Ni}$ through a short
distance of 300 nm Chen  _et al._ (2017). At the same time, it was shown
experimentally that surface acoustic waves (SAWs) can propagate in a
$\mathrm{Ni}$ film over the distance of several millimeters Casals  _et al._
(2020). This absence of significant damping observed for the studied SAWs with
frequencies not exceeding 500 MHz is very different from the results of Homer
et al. Homer  _et al._ (1987), who reported the decay length
$L_{\text{dec}}\approx 5.8\leavevmode\nobreak\ \mu$m for the longitudinal wave
with the frequency of 9.4 GHz in $\mathrm{Ni}$. The reason for such a
difference most probably lies in a drastic reduction of damping, which should
happen when the frequency of the elastic wave changes from about 10 GHz to
several hundreds of MHz. As for the damping of transverse elastic waves in
$\mathrm{Ni}$, our results show that the magnetic damping of elastic waves
($L_{\text{dec}}\approx 19\leavevmode\nobreak\ \mu$m) could be stronger than
the damping of electronic origin (measured $L_{\text{dec}}\approx
29\leavevmode\nobreak\ \mu$m Homer  _et al._ (1987)) at wave frequencies
around 10 GHz.

Most important of all, the magnetoelastic interaction leads to the formation
of a spin wave tightly coupled to the driving elastic wave. The spin waves
predicted by our simulations have sinusoidal time dependences under both types
of elastic excitation, which differs from a non-sinusoidal time dependence
reported in Ref. Chen  _et al._ (2017) for the spin wave generated by the
longitudinal acoustic wave having near-resonance frequency of 10 GHz.
Regarding the spin wave amplitude, the transverse elastic wave appears to be
much more efficient for the generation of spin waves in $\mathrm{Ni}$ than the
longitudinal wave at the chosen equilibrium magnetization orientation [compare
panels (a) and (b) in Fig. 2]. Similarly to elastically generated spin waves
in Fe81Ga19 and CoFe2O4 films Azovtsev and Pertsev (2017, 2019), the spin wave
propagating in a thick $\mathrm{Ni}$ film has the same frequency and
wavelength as the driving strain wave. Since both waves (spin and elastic)
travel with the same velocity $c_{L}$ or $c_{T}$ and obey a purely elastic
dispersion relation $k_{L}=2\pi\nu/c_{L}$ or $k_{T}=2\pi\nu/c_{T}$, the
driving wave acts as a carrier of the spin wave having a forced character.
Furthermore, the decay length of the spin wave carried by the longitudinal
acoustic wave matches the decay length $L_{\text{dec}}\approx
350\leavevmode\nobreak\ \mu$m of the latter in our simulations, which agrees
with the behavior predicted for CoFe2O4 films Azovtsev and Pertsev (2019).
However, in the case of the excitation by a shear wave, the spin-wave decay
length is significantly smaller than that of the driving wave, being roughly 9
$\mu$m instead of 19 $\mu$m. Despite this peculiarity, the acoustically driven
spin waves with frequencies $\nu\approx 10$ GHz can still propagate in Ni over
long distances of several micrometers, which is important for magnon
spintronics.

##  IV Magnetoelastic dynamics in Ni/GaAs bilayers

Now we turn our attention to $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers

Figure 4:  Time dependence of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface excited by longitudinal (a) or
transverse (b) elastic waves with the frequency $\nu=\nu_{\text{res}}=9.6$
GHz. Thickness $t_{\text{F}}$ of $\mathrm{Ni}$ film in each case is equal to a
corresponding wavelength $\lambda_{L}$ or $\lambda_{T}$.

comprising relatively thin $\mathrm{Ni}$ layers with the thickness
$t_{\text{F}}$ comparable to the wavelength of the driving elastic wave with
the frequency $\nu\sim\nu_{\text{res}}$, which are most suitable for
applications in miniature spin injectors (see Sec. V). The magnetoelastic
dynamics in such bilayers is of a more complicated character due to
reflections of the elastic waves from the $\mathrm{Ni}|\mathrm{GaAs}$
interface and the $\mathrm{GaAs}$ free surface. Fortunately, owing to similar
acoustic impedances of $\mathrm{Ni}$ and $\mathrm{GaAs}$, the transmittance of
the driving longitudinal or transverse wave through the
$\mathrm{Ni}|\mathrm{GaAs}$ interface is close to unity (about 0.9 with
respect to energy). In contrast, the driving wave fully reflects from the
$\mathrm{GaAs}$ free surface, and the reflected wave strongly disturbs the
magnetization dynamics when it penetrates back into the $\mathrm{Ni}$ layer.
In order to avoid this complication, we imparted a strong artificial elastic
damping to $\mathrm{GaAs}$, which is sufficient to force any elastic wave to
vanish before it reaches the free surface, but does not change significantly
the strain dynamics at the $\mathrm{Ni}|\mathrm{GaAs}$ interface.

The simulations demonstrated that the elastically driven magnetization
dynamics in $\mathrm{Ni}$ layers with the thicknesses $t_{\text{F}}$ about
$\lambda_{L}$ or $\lambda_{T}$ remains to be highly inhomogeneous at the
resonance excitation $\nu=\nu_{\text{res}}$. Initially the magnetic dynamics
has the form of a spin wave, but it assumes a complex character after several
reflections of the driving elastic wave from the boundaries of the
$\mathrm{Ni}$ layer. However, near the interface the magnetization precesses
with a constant frequency and amplitude in a steady-state regime, which
settles in after a transition period of about 1 ns (Fig. 4). Performing a
series of simulations at different thicknesses of $\mathrm{Ni}$ layers, we
found that the amplitude of the magnetization precession at the interface has
local maxima at Ni thicknesses amounting to 0.25, 0.75, 1.25 and 1.75 of the
wavelength $\lambda_{L}$ or $\lambda_{T}$ (Fig. 5). This result differs from
that obtained for Fe81Ga19/Au and CoFe2O4/Pt bilayers in our previous works
Azovtsev and Pertsev (2017, 2019), where such amplitude maximizes at the
ferromagnet’s thickness equal to one wavelength of the driving elastic wave.

Figure 5:  Amplitude of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface as a function of the $\mathrm{Ni}$
thickness $t_{\text{F}}$ normalized by the wavelength $\lambda$ of the driving
longitudinal or transverse elastic wave. The plots show the maximal change
$\Delta m_{x}$ of the out-of-plane direction cosine $m_{x}(x=t_{\text{F}},t)$
normalized by the largest value of $\Delta m_{x}$ in the studied thickness
range. The excitation frequency equals $\nu_{\text{res}}=9.6$ GHz.

To understand the revealed behavior of ferromagnetic-nonmagnetic (F/N)
bilayers, we investigated the dependence of the strain amplitude at the
interface on the thickness $t_{\text{F}}$ of the F layer. The analysis of the
results of simulations showed that in all studied bilayers the precession
amplitude in the steady-state regime becomes maximal whenever the strain
amplitude maximizes. Therefore, we considered a general elasticity problem of
finding the strain distribution in an elastic F/N bilayer subjected to a
periodic surface displacement
$u_{i}^{\text{F}}(x=0,t)=u_{\text{max}}e^{-i\omega t}$. Despite multiple
reflections of the elastic waves from the boundaries of the F layer, the
steady-state solution for the elastic displacement $u_{i}^{\text{F}}(x,t)$
inside this layer can be written as a superposition of two waves with the same
frequency $\omega=2\pi\nu$. Indeed, due to the principle of superposition in
linear elasticity any number of interfering sinusoidal waves with the same
frequency but different amplitudes and phases produce another sinusoidal wave
of the same frequency with its own amplitude and phase Sadd (2005). Hence, we
can write

$u_{i}^{\text{F}}(x,t)=A_{i}^{\text{F}}e^{i(k_{i}^{\text{F}}x-\omega
t)}+B_{i}^{\text{F}}e^{-i(k_{i}^{\text{F}}x+\omega t)},$ (2)

where the first term corresponds to the waves propagating towards the F$|$N
interface, while the second term describes the waves reflected from the F$|$N
interface; $A_{i}^{\text{F}}$ and $B_{i}^{\text{F}}$ are the unknown
amplitudes of these waves, and $k_{i}^{\text{F}}$ is the wavenumber of the
longitudinal ($i=x$) or transverse ($i=y$ or $z$) wave in the F layer.

Figure 6:  Expected thickness dependences of the strain amplitudes at the
interface between $\mathrm{Ni}$ and $\mathrm{GaAs}$ layers and CoFe2O4 and Pt
layers. The amplitude of each strain is normalized by its maximal value. The
thickness $t_{\text{F}}$ of the ferromagnetic layer is normalized by the
wavelength $\lambda$ of the excited longitudinal or transverse elastic wave.
The excitation frequency equals 9.6 GHz ($\mathrm{Ni}$/$\mathrm{GaAs}$) and 11
GHz (CoFe2O4/Pt).

Since we neglect the reflections from the free surface of the N layer, only
the transmitted elastic wave exists in it, and the displacement
$u_{i}^{\text{N}}(x,t)$ has the form

$u_{i}^{\text{N}}(x,t)=A_{i}^{\text{N}}e^{i(k_{i}^{\text{N}}x-\omega t)},$ (3)

where $A_{i}^{\text{N}}$ and $k_{i}^{\text{N}}$ are the amplitude and
wavenumber of the transmitted wave. The mechanical boundary conditions at the
F$|$N interface $x=t_{\text{F}}$ yield the displacement continuity
$u_{i}^{\text{F}}(x=t_{\text{F}},t)=u_{i}^{\text{N}}(x=t_{\text{F}},t)$ and
the stress continuity
$\sigma_{ix}^{\text{F}}(x=t_{\text{F}},t)=\sigma_{ix}^{\text{N}}(x=t_{\text{F}},t)$.
In our model case, the stresses are given by the relations
$\sigma_{ix}^{\text{F}}(x,t)=c_{\alpha\alpha}^{\text{F}}(1/\sqrt{\alpha})\partial/\partial
x[u_{i}^{\text{F}}(x,t)]$ and
$\sigma_{ix}^{\text{N}}(x,t)=c_{\alpha\alpha}^{\text{N}}(1/\sqrt{\alpha})\partial/\partial
x[u_{i}^{\text{N}}(x,t)]$, where $c_{\alpha\alpha}^{\text{F}}$ and
$c_{\alpha\alpha}^{\text{N}}$ are the elastic stiffnesses of the F and N
layers, respectively ($\alpha=1$ at $i=x$ and $\alpha=4$ at $i=y$ or $z$).
Combining the boundary conditions at the F$|$N interface and the F surface
$x=0$ and using Eqs. (2) and (3), one can derive analytic relations for the
unknown amplitudes $A_{i}^{\text{F}}$, $B_{i}^{\text{F}}$, and
$A_{i}^{\text{N}}$. The substitution of these relations back to Eqs. (2) and
(3) yields the formulae for the displacements $u_{i}^{\text{F}}(x,t)$ and
$u_{i}^{\text{N}}(x,t)$, which render possible to calculate the strains
$\varepsilon_{ix}^{\text{F}}=(1/\sqrt{\alpha})\partial
u_{i}^{\text{F}}/\partial x$ and
$\varepsilon_{ix}^{\text{N}}=(1/\sqrt{\alpha})\partial
u_{i}^{\text{N}}/\partial x$ in the F and N layers. For the strains
$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)$ at the F$|$N interface after
some mathematical manipulations we obtain

$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)=\frac{2i\varepsilon_{ix}^{\text{max}}Z_{\alpha}e^{-i\omega
t}}{(1+Z_{\alpha})F^{-}+(1-Z_{\alpha})F^{+}},$ (4)

where

$Z_{\alpha}=\sqrt{\frac{c_{\alpha\alpha}^{\text{N}}\rho_{\text{N}}}{c_{\alpha\alpha}^{\text{F}}\rho_{\text{F}}}},F^{+}=e^{ik_{i}^{\text{F}}t_{\text{F}}},F^{-}=e^{-ik_{i}^{\text{F}}t_{\text{F}}}.$

Equation (4) shows that the amplitude of
$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)$ depends on the input strain
$\varepsilon_{ix}^{\text{max}}=(1/\sqrt{\alpha})u_{\text{max}}k_{i}^{\text{F}}$,
the relative thickness $t_{\text{F}}/\lambda_{L}$ or
$t_{\text{F}}/\lambda_{T}$ of the F layer, and the dimensionless parameter
$Z_{\alpha}$ of the F/N bilayer, which is governed by the elastic stiffnesses
and densities of the involved materials. Using Eq. (4), we calculated the
dependences of the discussed strain amplitudes on the relative thickness of
the F layer for $\mathrm{Ni}$/$\mathrm{GaAs}$ and CoFe2O4/Pt bilayers
subjected to the resonance excitation $\nu=\nu_{\text{res}}$. The results
presented in Fig. 6 show that, for a given bilayer, the amplitudes of
$\varepsilon_{xx}^{\text{F}}(x=t_{\text{F}},t)$ and
$\varepsilon_{zx}^{\text{F}}(x=t_{\text{F}},t)$ normalized by their maximal
values follow similar curves (almost identical in case of
$\mathrm{Ni}$/$\mathrm{GaAs}$) when plotted as a function of
$t_{\text{F}}/\lambda_{L}$ and $t_{\text{F}}/\lambda_{T}$, respectively.
However, the maximal strain amplitude is reached at the thicknesses
$t_{\text{F}}=(0.25+0.5n)\lambda$ in $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers
and at $t_{\text{F}}=(0.5+0.5n)\lambda$ in CoFe2O4/Pt ones
($\lambda=\lambda_{L}$ or $\lambda_{T}$, $n=0,1,2,3$…). These conditions
explain dissimilar results of our micromagnetoelastic simulations performed
for $\mathrm{Ni}$/$\mathrm{GaAs}$ and CoFe2O4/Pt bilayers, which showed that
the precession amplitude at the interface has a maximum at
$t_{\text{F}}=0.75\lambda$ ($\mathrm{Ni}$/$\mathrm{GaAs}$) and
$t_{\text{F}}=\lambda$ (CoFe2O4/Pt). Furthermore, the analysis of Eq. (4)
reveals that the character of the strain-amplitude thickness dependence is
governed by the magnitude of the parameter $Z_{\alpha}$. Namely, when
$Z_{\alpha}<1$, the strain amplitude maximizes at
$t_{\text{F}}=(0.25+0.5n)\lambda$ as it happens in the
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers ($Z_{1}\approx Z_{4}\approx 0.53$),
whereas at $Z_{\alpha}>1$ the optimal thicknesses satisfy the condition
$t_{\text{F}}=(0.5+0.5n)\lambda$ holding for the CoFe2O4/Pt bilayers
($Z_{1}\approx 2.34$, $Z_{4}\approx 1.91$). The derived simple criteria open
the possibility to predict the optimal thickness of the ferromagnetic layer
that maximizes the strain and precession amplitudes at the interface for any
F/N bilayer.

Figure 7:  Dependence of the amplitude of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface on the excitation frequency $\nu$. The
points show the maximal deviation $\Delta m_{x}(\nu)$ of the out-of-plane
magnetization direction cosine $m_{x}$ from its equilibrium value. The
thickness of the $\mathrm{Ni}$ layer equals three quarters of the wavelength
of the driving longitudinal or transverse elastic wave.

In conclusion of this section, we discuss the dependence of the amplitude of
the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface on
the excitation frequency $\nu$. For the optimal $\mathrm{Ni}$ thickness
$t_{\text{F}}=0.75\lambda$ and the driving waves with the initial strain
amplitudes
$\varepsilon_{xx}^{\text{max}}=\varepsilon_{xz}^{\text{max}}=10^{-4}$, the
simulations predict that the maximal deviation $\Delta m_{x}(\nu)$ of the
magnetization direction cosine $m_{x}$ from the equilibrium value varies with
the frequency as shown in Fig. 7. It can be seen that $\Delta m_{x}(\nu)$
reaches a peak at a frequency $\nu_{\text{max}}$ slightly higher than the
resonance frequency $\nu_{\text{res}}=9.6$ GHz. Namely, $\nu_{\text{max}}$
amounts to 9.9 GHz for the precession excited by the longitudinal elastic
waves and to 10 GHz for that induced by the transverse waves. In agreement
with the results described in Sec. III, the shear waves appear to be much more
efficient for the excitation of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface (see Fig. 7).

##  V Spin pumping into GaAs layer

The magnetization precession occurring near the interface between the
ferromagnet and a nonmagnetic conductor generates the spin pumping into the
latter Tserkovnyak  _et al._ (2005). Using the results obtained for the
magnetization dynamics $\mathbf{m}(x=t_{\mathrm{F}},t)$ induced by the elastic
waves at the $\mathrm{Ni}|\mathrm{GaAs}$ interface, we can calculate the spin
current flowing in the $\mathrm{GaAs}$ layer. The spin-current density
$\mathbf{J}_{s}(x,t)$ is a second-rank tensor characterizing the direction of
spin flow and the orientation and magnitude of the carried spin polarization
per unit volume Dyakonov and Perel (1971). In the vicinity of the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, the density
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$ of the spin current pumped into
$\mathrm{GaAs}$ can be evaluated via the approximate relation
$\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SP}}\simeq(\hbar/4\pi)\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\mathbf{m}\times\dot{\mathbf{m}}$,
where $\mathbf{e}_{n}$ is the unit vector normal to the interface and pointing
into $\mathrm{GaAs}$, $\hbar$ is the reduced Planck constant,
$g^{r}_{\uparrow\downarrow}$ is the reflection spin-mixing conductance per
unit area, and a small contribution caused by the imaginary part of
$g^{r}_{\uparrow\downarrow}$ is neglected Zwierzycki  _et al._ (2005). Since
$\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}$ may be set equal to
$1.5\times 10^{17}$ m-2 for the $\mathrm{Ni}|\mathrm{GaAs}$ interface Ando
_et al._ (2011), the above relation and the simulation data on the temporal
variation of $\mathbf{m}(x=t_{\mathrm{F}},t)$ enable us to evaluate the spin-
current density $\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$.

Figure 8 shows time dependences of three nonzero components
$J^{\mathrm{SP}}_{xj}$ of the tensor $\mathbf{J}_{\mathrm{SP}}$, which settle
in the steady-state regime of the magnetization precession at the excitation
frequency $\nu=\nu_{\mathrm{max}}$. It can be seen that the transverse wave
creates much stronger spin pumping into $\mathrm{GaAs}$ than the longitudinal
one. For both types of elastic excitations, the amplitude of
$J^{\mathrm{SP}}_{xx}$ is about two times larger than almost equal amplitudes
of $J^{\mathrm{SP}}_{xy}$ and $J^{\mathrm{SP}}_{xz}$.

Figure 8:  Time dependences of the spin-current densities
$J_{xj}^{\mathrm{SP}}$ pumped into $\mathrm{GaAs}$ by the longitudinal (a) and
transverse (b) elastic waves with the frequencies 9.9 GHz and 10 GHz,
respectively. The time period shown in the figure corresponds to the steady-
state regime of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface. The thickness of the $\mathrm{Ni}$
layer equals three quarters of the wavelength of the driving longitudinal or
transverse elastic wave.

The averaging over the period $1/\nu$ of (almost sinusoidal) spin-current
variations shows that $\langle J^{\mathrm{SP}}_{xx}\rangle_{t}$ is negligible,
whereas there are very small nonzero dc components $\langle
J^{\mathrm{SP}}_{xy}\rangle_{t}=\langle J^{\mathrm{SP}}_{xz}\rangle_{t}$ of
the pumped spin current. It should be noted that, owing to relatively small
reflection spin mixing conductance of the $\mathrm{Ni}|\mathrm{GaAs}$
interface, the spin pumping into $\mathrm{GaAs}$ does not significantly
increase the effective damping of the magnetization precession in
$\mathrm{Ni}$ Nikitchenko and Pertsev (2020).

The pumped spin current generates non-equilibrium spin accumulation
$\bm{\mu}_{s}(x,t)$, which gives rise to a spin backflow at the interface with
the density $\mathbf{J}_{\mathrm{SB}}(x=t_{\mathrm{F}},t)$ amounting to
$\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SB}}\approx-\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\bm{\mu}_{s}/4\pi$
Tserkovnyak  _et al._ (2005). The overall spin-current density
$\mathbf{J}_{s}(x,t)$ decays inside the $\mathrm{GaAs}$ layer due to spin
relaxation and diffusion. The spatial distribution of the density
$\mathbf{J}_{s}$ depends on that of the spin accumulation $\bm{\mu_{s}}$
Tserkovnyak and Brataas (2002), being defined in our one-dimensional model by
the relation
$\mathbf{e}_{n}\cdot\mathbf{J}_{s}(x,t)=-[\sigma\hbar/(4e^{2})]\partial\bm{\mu}_{s}(x,t)/\partial
x$, where $e$ is the elementary positive charge, and $\sigma$ is the
electrical conductivity, which amounts to $3.68\times 10^{4}$ S m-1 for
n+-$\mathrm{GaAs}$ Kikkawa and Awschalom (1998); Nikitchenko and Pertsev
(2020). We find the spin accumulation $\bm{\mu}_{s}(x,t)$ by solving the
diffusion equation Tserkovnyak and Brataas (2002) appended by the boundary
conditions for the spin currents at the $\mathrm{Ni}|\mathrm{GaAs}$ interface
$x=t_{\mathrm{F}}$ and the $\mathrm{GaAs}$ free surface
$x=t_{\mathrm{F}}+t_{\mathrm{N}}$, which read
$\mathbf{J}_{s}(x=t_{\mathrm{F}})=\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}})+\mathbf{J}_{\mathrm{SB}}(x=t_{\mathrm{F}})$
and $\mathbf{J}_{s}(x=t_{\mathrm{F}}+t_{\mathrm{N}})=0$. The calculation
yields

$\bm{\mu}_{s}^{\omega}=\frac{4\pi
e^{2}\cosh{[\kappa(t_{\mathrm{N}}+t_{\mathrm{F}}-x)]}}{e^{2}\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\cosh{(\kappa
t_{\mathrm{N}})}+\pi\sigma\hbar\kappa\sinh{(\kappa
t_{\mathrm{N}})}}\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SP}}^{\omega},$ (5)

where $\bm{\mu}_{s}^{\omega}$ and $\mathbf{J}_{\mathrm{SP}}^{\omega}$ denote
the complex amplitudes of the harmonics having the angular frequency $\omega$,
which represent the Fourier components of the spin accumulation
$\bm{\mu}_{s}(x,t)$ and spin pumping density
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$, and the parameter
$\kappa=\lambda_{\mathrm{sd}}^{-1}\sqrt{1+i\omega\tau_{\mathrm{sf}}}$ depends
on spin-diffusion length $\lambda_{\mathrm{sd}}$ and spin-flip relaxation time
$\tau_{\mathrm{sf}}$. Equation (5) differs from a similar relation derived in
Ref. Tserkovnyak and Brataas (2002) by the account of the spin backflow. In
the case of $\mathrm{GaAs}$, the spin backflow cannot be neglected because it
appears to be rather strong at $\lambda_{\mathrm{sd}}=2.32$ $\mu$m Kikkawa and
Awschalom (1998) and $\tau_{\mathrm{sf}}=0.9$ ns Bhat and Kumar (2014). Since
the elastically driven spin pumping is almost monochromatic in our setting,
Eq. (5) is valid for the sought relation between $\bm{\mu}_{s}(x,t)$ and
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$ as well, which enables us to
calculate the overall spin-current density $\mathbf{J}_{s}(x,t)$.

Owing to the inverse spin Hall effect (ISHE), the spin current in the
$\mathrm{GaAs}$ layer generates a charge current with the density
$\mathbf{J}_{c}^{\mathrm{ISHE}}$ defined by the formula Mosendz  _et al._
(2010)

$\mathbf{J}_{c}^{\mathrm{ISHE}}(x,t)=\alpha_{\mathrm{SH}}(2e/\hbar)\mathbf{e}_{n}\times[\mathbf{e}_{n}\cdot\mathbf{J}_{s}(x,t)],$
(6)

where $\alpha_{\mathrm{SH}}=0.007$ is the spin Hall angle of $\mathrm{GaAs}$
Ando  _et al._ (2011). Under considered open-circuit electrical boundary
conditions, the transverse charge current $\mathbf{J}_{c}^{\mathrm{ISHE}}$
flowing along the interface should create a charge accumulation at the lateral
boundaries of the $\mathrm{GaAs}$ film. Such an accumulation induces an
electric field $\mathbf{E}$ in $\mathrm{GaAs}$, which causes a drift current
with the density $\mathbf{J}_{c}^{\mathrm{drift}}=\sigma\mathbf{E}$. To
calculate the spatial distribution of the electric potential $\varphi$ in the
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer and the total charge current density
$\mathbf{J}_{c}=\mathbf{J}_{c}^{\mathrm{ISHE}}+\mathbf{J}_{c}^{\mathrm{drift}}$,
we numerically solve the Laplace’s equation $\nabla^{2}\varphi=0$ with the
appropriate boundary conditions. The latter follow from the absence of charge
current across the outer surfaces of the bilayer, and the absence of
$\mathbf{J}_{c}^{\mathrm{ISHE}}$ inside $\mathrm{Ni}$. It should be noted that
the potential $\varphi$ should be regarded as a complex quantity since the
parameter $\kappa$ affecting the spin-current density $\mathbf{J}_{s}$
involved in Eq. (6) has a substantial complex part at
$\omega\tau_{\mathrm{sf}}>>1$.

In the numerical calculations, we consider only the component $J^{s}_{xy}$ of
the elastically generated spin current $\mathbf{J}_{s}$, because $J^{s}_{xx}$
does not create any charge flow, and the components $J^{s}_{xy}$ and
$J^{s}_{xz}$ have almost equal magnitudes and can be probed independently via
transverse voltages
$V_{z}^{\mathrm{ISHE}}=\varphi(z=w_{\mathrm{N}}/2)-\varphi(z=-w_{\mathrm{N}}/2)$
and $V_{y}^{\mathrm{ISHE}}=\varphi(y=0)-\varphi(y=-h_{\mathrm{N}})$,
respectively (Fig. 1). Figure 9 shows the amplitude $\delta
V_{z}^{\mathrm{ISHE}}(x)$ of the oscillating voltage
$V_{z}^{\mathrm{ISHE}}(x,t)$ calculated at the excitation frequency
$\nu=\nu_{\mathrm{max}}$ for the $\mathrm{GaAs}$ films with the thickness
$t_{\mathrm{N}}=5$ $\mu$m.

Figure 9:  Amplitude $\delta V_{z}^{\mathrm{ISHE}}(x)$ of the ac voltage
between the lateral sides of the $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer excited
by longitudinal (a) and transverse (b) elastic waves with the frequencies 9.9
GHz and 10 GHz, respectively. The thickness of the $\mathrm{GaAs}$ layer
equals 5 $\mu$m, and its width $w_{\mathrm{N}}$ is indicated in the figure.

It can be seen that $\delta V_{z}^{\mathrm{ISHE}}$ varies nonmonotonically
with the distance $x-t_{\mathrm{F}}$ from the $\mathrm{Ni}|\mathrm{GaAs}$
interface, reaching its maximum inside the semiconductor at
$x-t_{\mathrm{F}}=100-200$ nm. The voltage amplitude $\delta
V_{z}^{\mathrm{ISHE}}(x)$ grows with the increasing width $w_{\mathrm{N}}$ of
the $\mathrm{GaAs}$ film, and the voltage peak becomes much higher at the
excitation of magnetization dynamics by the shear elastic wave (see Fig. 9).
Importantly, the transverse ac voltage $V_{z}^{\mathrm{ISHE}}$ characterizing
the spin pumping induced by either type of elastic waves is high enough for
the experimental measurement near the $\mathrm{Ni}|\mathrm{GaAs}$ interface.

Another method to evaluate the spin pumping into a normal metal or
semiconductor experimentally is known as a nonlocal spin detection scheme
Johnson and Silsbee (1985); Lou  _et al._ (2007). This scheme measures a
voltage $V_{s}$ between a ferromagnetic probe and a nonmagnetic electrode
brought into contact with the semiconductor. Since the voltage $V_{s}$ is
directly proportional to the product
$\bm{\mu}_{s}\cdot\mathbf{M}_{\mathrm{probe}}$, where
$\mathbf{M}_{\mathrm{probe}}$ is the probe magnetization, it is possible to
detect all three components of the vector $\bm{\mu}_{s}$ by using differently
magnetized ferromagnetic contacts. As a representative example, we consider an
iron probe magnetized along the $x$ axis, which is placed on the lateral side
of the $\mathrm{GaAs}$ layer (Fig. 1), and a normal-metal electrode deposited
on the free surface $x=t_{\mathrm{F}}+t_{\mathrm{N}}$ of the 5-$\mu$m-thick
$\mathrm{GaAs}$ film. In this case, the spin voltage $V_{s}(x,t)$ is defined
by the relation
$V_{s}(x,t)=\eta_{\mathrm{IE}}p_{\mathrm{Fe}}\mu_{x}^{s}(x,t)/(2e)$, where
$\eta_{\mathrm{IE}}$ is the spin transmission efficiency of the
$\mathrm{GaAs}|\mathrm{Fe}$ interface, $p_{\mathrm{Fe}}$ is the spin
polarization of $\mathrm{Fe}$ at the Fermi level, and the spin accumulation
$\mu_{x}^{s}$ beneath the probe with nanoscale dimensions is assumed uniform.
Figure 10 shows the amplitude $\delta V_{s}(x)$ and phase $\phi_{s}(x)$ of the
ac spin voltage calculated using the parameters $\eta_{\mathrm{IE}}\approx
0.5$ and $p_{\mathrm{Fe}}\approx 0.42$ characteristic of the Schottky tunnel
barrier between Fe probe and n+-$\mathrm{GaAs}$ Lou  _et al._ (2007).

Figure 10:  Amplitude $\delta V_{s}$ and phase $\phi_{s}$ (inset) of the ac
spin voltage between the lateral Fe probe and a normal-metal contact at the
$\mathrm{GaAs}$ free surface plotted as a function of the distance
$x-t_{\mathrm{F}}$ from the $\mathrm{Ni}|\mathrm{GaAs}$ interface. The
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer is excited either by the longitudinal
wave with frequency 9.9 GHz or by the transverse wave with frequency 10 GHz.
The thickness of the $\mathrm{GaAs}$ layer equals 5 $\mu$m.

Importantly, the voltage amplitude $\delta V_{s}$ appears to be rather large
near the $\mathrm{Ni}|\mathrm{GaAs}$ interface, exceeding 650 nV under the
excitation by the transverse elastic wave and 80 nV in the bilayer excited by
the longitudinal wave (see Fig. 10). Although $\delta V_{s}(x)$ gradually
decreases with the increasing distance $x-t_{\mathrm{F}}$ from the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, it remains to be measurable
experimentally even at the distances over 0.5 $\mu$m. The phase $\phi_{s}(x)$
of the spin voltage varies linearly inside the $\mathrm{GaAs}$ layer and
changes strongly already at $x-t_{\mathrm{F}}\sim 0.5$ $\mu$m (Fig. 10). This
result demonstrates that the phase difference between the spin accumulation
inside $\mathrm{GaAs}$ and the spin pumping at the $\mathrm{Ni}|\mathrm{GaAs}$
interface may be large owing to the condition $\omega\tau_{\mathrm{sf}}>>1$.

The input electric power $W$ necessary for the functioning of the proposed
spin injector can be estimated from the generated acoustic power using the
relation $W=\frac{1}{K^{2}}\frac{1}{2}A\rho c\omega^{2}u_{\text{max}}^{2}$,
where $K^{2}$ is the electromechanical transduction efficiency of the
piezoelectric transducer Bhaskar  _et al._ (2020), $c$ is the velocity of the
generated longitudinal or transverse acoustic wave, and $A$ denotes the
dynamically strained area of the ferromagnetic film having the mass density
$\rho$. Expressing $u_{\text{max}}$ via the maximal strain
$\varepsilon_{\text{max}}$ in the acoustic wave, we obtain
$W=\alpha\frac{1}{K^{2}}\frac{1}{2}A\rho c^{3}\varepsilon_{\text{max}}^{2}$.
For the device with $A<25\leavevmode\nobreak\ \mu$m2 and $K^{2}=12\%$ Bhaskar
_et al._ (2020), which is driven by a transverse ($\alpha=4$) or longitudinal
($\alpha=1$) wave with $\varepsilon_{\text{max}}=10^{-4}$, the calculation
yields $W<2$ mW. This value is much smaller than the lowest power consumption
$W\approx 25$ mW of the spin injector driven by the microwave magnetic field
Ando  _et al._ (2011).

##  VI Conclusions

In this work, we theoretically studied the coupled elastic and spin dynamics
induced in $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers by longitudinal and
transverse acoustic waves generated by the attached piezoelectric transducer
(Fig. 1). Using advanced micromagnetoelastic simulations, we first modeled the
elastically driven magnetization dynamics in thick $\mathrm{Ni}$ films at the
wave frequencies around the resonance frequency $\nu_{\mathrm{res}}$ of the
coherent magnetization precession in unstrained $\mathrm{Ni}$ film. The
simulations showed that this dynamics has the form of a forced spin wave
having the frequency and wavelength of the monochromatic driving wave.
Remarkably, the transverse elastic wave creates much stronger spin wave than
the longitudinal one at the considered external magnetic field (Fig. 2). The
backaction of travelling spin wave on the elastic dynamics manifests itself in
the generation of weak secondary elastic waves created by the magnetization
precession (see Fig. 3). These waves are characterized by oscillating strains
$\varepsilon_{ij}(x,t)$ different from the strain $\varepsilon_{xx}(x,t)$ or
$\varepsilon_{xz}(x,t)$ in the primary driving wave, and they were not
reported in the previous work on the modeling of magnetization dynamics
induced by longitudinal elastic waves in $\mathrm{Ni}$ Chen  _et al._ (2017).
The magnetoelastic feedback also influences the driving elastic wave, leading
to a gradual reduction of its amplitude during the propagation in
$\mathrm{Ni}$, which adds to the “acoustic” decay caused by the wave
attenuation of electronic origin Homer  _et al._ (1987). At the considered
wave frequencies $\nu\approx 10$ GHz, the decay resulting from the energy
transfer to the magnetic subsystem is stronger than the acoustic decay for the
transverse waves but small for longitudinal ones. Importantly, both types of
elastic waves are expected to carry spin signals over significant distances of
several micrometers in $\mathrm{Ni}$.

We also modeled the magnetoelastic dynamics of $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayers at the excitation frequencies $\nu\sim\nu_{\mathrm{res}}$, focusing
on the $\mathrm{Ni}$ thicknesses comparable to the wavelength of the injected
acoustic wave. The simulations allowed for the reflections of the elastic
waves from the boundaries of the $\mathrm{Ni}$ layer and demonstrated the
excitation of a nonhomogeneous magnetization dynamics in it. Importantly, a
steady-state magnetization precession with frequency equal to the excitation
frequency and constant amplitude was revealed at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface after a short transition period of about
1 ns (Fig. 4). The simulations performed for $\mathrm{Ni}$ layers of different
thicknesses showed that the amplitude of stationary precession has a maximum
at $\mathrm{Ni}$ thickness amounting to three quarters of the driving elastic
wave wavelength. This finding, which differs from the results of simulations
carried out for $\mathrm{Fe}_{81}\mathrm{Ga}_{19}$/$\mathrm{Au}$ and
$\mathrm{CoFe}_{2}\mathrm{O}_{4}$/$\mathrm{Pt}$ bilayers Azovtsev and Pertsev
(2017, 2019), was explained by an analytical model giving simple criteria for
the optimal geometry of an elastic bilayer that maximizes the strain amplitude
at the interface.

Numerical results obtained for the steady-state magnetization precession at
the $\mathrm{Ni}|\mathrm{GaAs}$ interface were used to evaluate the spin-
current densities pumped into $\mathrm{GaAs}$ by the dynamically strained
$\mathrm{Ni}$ film (Fig. 8). The spin accumulation in the semiconductor was
then calculated by solving numerically the spin diffusion equation with the
account of the spin pumping into $\mathrm{GaAs}$ and the spin backflow into
$\mathrm{Ni}$. Since the spin current creates a charge current owing to the
ISHE, the spin generation in $\mathrm{GaAs}$ can be detected via electrical
measurements. Therefore, we also determined the distribution of the electric
potential in the $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer with open-circuit
electrical boundary conditions by numerically solving the Laplace’s equation.
This enabled us to evaluate the transverse voltage appearing between the
lateral sides of the dynamically strained $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayer. It was shown that the amplitude of this ac voltage is large enough
for the experimental detection near the $\mathrm{Ni}|\mathrm{GaAs}$ interface
(Fig. 9). Furthermore, spin accumulation manifests itself in the voltage
between a ferromagnetic probe and a nonmagnetic electrode brought into contact
with the semiconductor (Fig. 1). Performing calculations of this ac spin
voltage, we found that it retains measurable amplitude even at the distances
over 0.5 $\mu$m from the $\mathrm{Ni}|\mathrm{GaAs}$ interface (Fig. 10).

Thus, our theoretical study of the $\mathrm{Ni}$/$\mathrm{GaAs}$
heterostructure demonstrated that the spin injector employing elastic waves is
promising for the spin generation in semiconductors. Since the proposed device
can be driven electrically via the strain-mediated magnetoelectric effect, it
has much lower power consumption than the spin injector excited by a microwave
magnetic field Ando  _et al._ (2011).

##  VII Acknowledgements

The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics ”BASIS”.

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