# Towards Magnonic Logic and Neuromorphic Computing: Controlling Spin-Waves by
Spin-Polarized Current

Raí M. Menezes  NANOlab Center of Excellence & Department of Physics,
University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium  Jeroen
Mulkers  DyNaMat Lab, Department of Solid State Sciences, Ghent University,
Ghent, Belgium  Clécio C. de Souza Silva  Departamento de Física, Universidade
Federal de Pernambuco, Cidade Universitária, 50670-901, Recife-PE, Brazil
Bartel Van Waeyenberge  DyNaMat Lab, Department of Solid State Sciences, Ghent
University, Ghent, Belgium  Milorad V. Milošević
<EMAIL_ADDRESS>NANOlab Center of Excellence & Department of
Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

###### Abstract

Spin-waves (magnons) are among the prime candidates for building fast yet
energy-efficient platforms for information transport and computing. We here
demonstrate theoretically and in state-of-the-art micromagnetic simulation the
effects that strategically-injected spin-polarized current can have on
controlling magnonic transport. We reveal analytically that the Zhang-Li spin-
transfer-torque induced by applied current is analogous to the Dzyaloshinskii-
Moriya interaction for scattering the magnons in the linear regime, to then
provide a generalized Snell’s law that describes the spin-wave propagation
across regions with different current densities. We validate the latter in
numerical simulations of realistic systems, and exemplify how these findings
may help advance the design of spin-wave logic and neuromorphic computing
devices.

###### keywords:

spin-wave, magnonics, spin-transfer torque, neuromorphic computing,
spintronics

The development of neuromorphic computing hardware has attracted significant
attention in recent years as such platforms are capable of performing complex
information processing tasks, such as classification and pattern recognition
of various types of data, from e-commerce to scientific content 1, 2, 3. A
central challenge of this research is the requirement of highly interconnected
systems, inspired by the biological concepts of the human brain.
Interestingly, wave-based physical systems have been demonstrated to operate
as recurrent neural networks 4, where interference patterns in the propagating
substrate can realize an all-to-all interconnection between points of the
substrate that mimic the action of artificial neurons by scattering and
recombining input waves in order to extract their information.

Similar to other wave phenomena in physics, spin-waves (magnons) travel
through space accompanied by a transfer of energy, which if precisely
controlled can lead to fast information transport and computing applications
in the nanometric to micrometric scale 5, 6. Spin-waves are readily
demonstrated as a promising platform for performing logic operations 7, 8 and
the recent theoretical advances in wave-based computation can pave the way for
spintronic hardware in the field of artificial intelligence 9. However, even
though extensive research has been carried out in recent years, the precise
manipulation of spin waves in nanostructures has not been entirely mastered
and needs to be advanced for the benefit of functional magnonic devices.

One manner of manipulating the spin textures in magnetic materials is by the
application of spin-polarized (SP) currents, having (part of the) spins of the
moving electrons aligned. The interaction of a SP current with the localized
magnetic moments results in a torque on the magnetization, dubbed a Zhang-Li
spin-transfer-torque (STT) 10. Being able to affect the orientation of the
magnetization, STT has become nearly unavoidable in the design of spintronic
nanodevices 11. The effect and applicability of the STT was demonstrated in
racetrack memory concepts, where the position of local magnetic structures,
such as domain walls and skyrmions, is controlled by in-plane currents 12, 13.
Moreover, not only do SP currents modify the equilibrium magnetic state, they
can also significantly influence the propagation of spin waves (SWs). For
instance, in Ref. 14 the authors derived the dispersion relation of SWs in a
ferromagnet subjected to a uniform SP current. The applied current introduces
a term proportional to the wavevector $k$ in the spin-wave dispersion
relation. The nonreciprocity in the dispersion relation for waves with a $k$
vector in the same or the opposite direction as the current flow causes
Doppler shift 15, as first validated experimentally by Vlaminck and Bailleul
16.

A similar nonreciprocal term in the dispersion relation can also be induced by
an electric field 17 or by the antisymmetric exchange interaction, also known
as Dzyaloshinskii-Moriya interaction (DMI) 18. Analogously to the Doppler
shift induced by the currents, a frequency shift has been measured in
ferromagnetic materials possessing DMI 19. For heterochiral ferromagnets
(i.e., with spatially varying DMI), the reciprocal term causes a nontrivial
refraction of spin waves at interfaces between regions with different DMI, as
described by a generalized Snell’s law in Ref. 20. In this regard, the
equivalency in the dispersion relations of SWs in the presence of DMI and SP
current suggests that a spatially varying current density (as illustrated in
Fig. 1) can be used to manipulate the propagation of SWs.

In this Letter, we detail the effect of a SP current on controlling the
propagation of spin-waves. We show that the Zhang-Li STT induced by in-plane
current has analogous effect to DMI for confining and controlling the
propagation direction of magnons in the linear regime. We proceed to derive a
Snell’s law to describe the scattering of spin-waves between regions with
different current densities, and validate it by advanced simulations including
solving Poisson’s equation within the micromagnetic framework. Finally we
present selected tailored examples to illustrate how strategically applied
current can be employed to advance logic and neuromorphic computing devices
based on spin-waves.

Figure 1: Schematic illustration of a SW facing a non-uniform distribution of
the SP current. The SW (in red) is induced by the input antenna on the left
side and propagates along the magnetic film. Voltage electrodes induce the
distributed SP current ($j$) which can be tuned to control the SW propagation.

##  1 Results and discussion

###  1.1 Spin-wave dispersion under applied current

Within the micromagnetic framework, we describe the magnetization of a thin
extended ferromagnetic film by considering the vector field $\vec{M}({\bf
r})=M_{\text{s}}\vec{m}({\bf r})$ with constant magnetization modulus
$|\vec{M}|=M_{\text{s}}$ and the normalized magnetization direction
$\vec{m}({\bf r})$ at each point ${\bf r}\in\mathbb{R}^{2}$ of the film 111In
this paper we use an arrow for 3D vector fields such as the magnetization
$\vec{m}$ and the effective magnetic field $\vec{H}_{\text{eff}}$ and bold
variables for 2D dimensional vectors in the plane such as the position ${\bf
r}$, the ${\bf k}$-vector, and the current ${\bf u}$.. The dynamics of the
magnetization is governed by the Landau-Lifshitz-Gilbert (LLG) equation

$\dot{\vec{m}}=-\gamma\vec{m}\times\vec{H}_{\text{eff}}+\alpha\vec{m}\times\dot{\vec{m}}+\vec{\tau}_{\text{STT}},$
(1)

where $\gamma$ is the gyromagnetic ratio, $\alpha$ the dimensionless damping
factor, and $\vec{H}_{\text{eff}}$ the effective field, which can be derived
from the free energy $E[\vec{m}]$ by taking the functional derivative with
respect to the magnetization: $\vec{H}_{\text{eff}}=-\delta E/\delta\vec{M}$.

We extend the LLG equation by adding the torque $\vec{\tau}_{\text{STT}}$
which includes the adiabatic and non-adiabatic STT terms derived by Zhang and
Li 10:

$\vec{\tau}_{\text{STT}}=-\vec{m}\times(\vec{m}\times({\bf
u}\cdot\nabla)\vec{m})+\beta\vec{m}\times({\bf u}\cdot\nabla)\vec{m},$ (2)

where

${\bf u}=-\frac{\mu_{B}P}{eM_{\text{s}}(1+\beta^{2})}{\bf j}.$ (3)

Here, $\nabla$ is the two-dimensional differential operator, $\beta$ is a
dimensionless constant that represents the degree of non-adiabaticity, $e$ the
elementary charge and $\mu_{B}$ the Bohr magneton. The polarization $P$ is a
property of the ferromagnet and does not depend on the magnetization. Although
${\bf u}$ has the units of velocity, we refer to it as _current_ since it is
proportional to the SP current density ${\bf j}$. The most prominent effect of
the STT is best understood by assuming small dissipation terms ($\alpha\approx
0$ and $\beta\approx 0$). In that case it is easy to prove that the solution
of the LLG equation is given by $\vec{m}({\bf r}-{\bf u}t;t)$ if $\vec{m}({\bf
r};t)$ is the solution in absence of the STT. Put differently, adding the STT
shifts the solution with a velocity ${\bf u}$ 21. For instance, due to the
STT, relaxed local structures such as domain walls and skyrmions will move
with a velocity ${\bf v}={\bf u}$ when the current is switched on.
Accordingly, a SW packet traveling in the film will gain an additional
velocity equal to ${\bf u}$. This insight will help to obtain an intuitive
understanding of the results presented in this paper in which we study the
effect of a non-uniform static current ${\bf u}({\bf r})$ on the propagation
of SWs.

The dynamics of the magnetization depends strongly on the characteristics of
the magnetic film, incorporated in the free energy functional. In this paper,
we take into account the contribution of exchange interaction and Zeeman
energy due to applied bias magnetic field:

$E[\vec{m}]=\iint\left[A(\nabla\vec{m})^{2}-\vec{H}\cdot\vec{m}\right]\text{d}x\text{d}y,$
(4)

with exchange stiffness $A>0$ and bias field $\vec{H}$.

The ground state magnetization is the ferromagnetic (field-polarized) state in
which the magnetization is aligned with the bias field $\vec{H}$. To study
small deviations from the ground state it is useful to construct the right-
handed coordinate system $(\hat{e}_{a},\hat{e}_{b},\hat{e}_{0})$ with
$\hat{e}_{0}\parallel\vec{H}$. The effective field in this coordinate system
becomes

$\vec{H}_{\text{eff}}({\bf
r},t)=\begin{bmatrix}\frac{2A}{M_{\text{s}}}\nabla^{2}m_{a}({\bf r},t)\\\
\frac{2A}{M_{\text{s}}}\nabla^{2}m_{b}({\bf r},t)\\\
\frac{2A}{M_{\text{s}}}\nabla^{2}m_{0}({\bf r},t)+H_{0}\end{bmatrix},$ (5)

where $\vec{m}=(m_{a},m_{b},m_{0})$ and $H=(0,0,H_{0})$. Here and in the next
section, we focus on first order deviations from the ground state and consider
the SW solution as $m_{0}\approx 1$,
$m_{a}=A_{0}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)-\mu t}$ and
$m_{b}=iA_{0}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)-\mu t}$, where $A_{0}\ll
1$ represents the SW amplitude; $\omega$ is the SW angular frequency; k is the
wave vector, and $\mu$ represents the damping of the SW. Substituting that
into the LLG equation [Eq. (1)], with STT term given by Eq. (2) and effective
field from Eq. (5), yields the dispersion relation (see supplementary material
22)

$\omega=\frac{\gamma
H_{0}}{1+\alpha^{2}}\left(1+\xi^{2}k^{2}\right)+\frac{1+\alpha\beta}{1+\alpha^{2}}{\bf
u}\cdot{\bf k},$ (6)

and damping parameter

$\mu=\alpha\omega-\beta{\bf u}\cdot{\bf k},$ (7)

where we define the length scale $\xi=\sqrt{2A/H_{0}M_{s}}$. Note that the
damping $\mu$ of the SW decreases if the wave vector ${\bf k}$ is parallel to
the electron flow ${\bf u}$. Consequently, the attenuation length of a spin
wave can be increased by applying a current opposite to the propagation
direction, as predicted earlier by Seo _et al._ 23.

Propagation direction.—The dispersion relation consists out of circular
isofrequencies in $k$-space, which are shifted away from the origin due to the
current. The center of the circular isofrequency is given by

${\bf k}_{0}=-\frac{1+\alpha\beta}{2\gamma H_{0}\xi^{2}}{\bf u},$ (8)

and the radius by

$k_{g}=\left|{\bf k}-{\bf k}_{0}\right|=\sqrt{\frac{1+\alpha^{2}}{\gamma
H_{0}\xi^{2}}(\omega-\omega_{0})},$ (9)

with the minimal frequency $\omega_{0}=\gamma H_{0}(1-\xi^{2}k^{2}_{0})$.

The propagation velocity v of a wave packet is given by the gradient of the
frequency in $k$-space:

$\textbf{v}=\nabla_{k}\omega=\frac{2\gamma H_{0}\xi^{2}}{1+\alpha^{2}}{\bf
k}+\frac{1+\alpha\beta}{1+\alpha^{2}}{\bf u}=\frac{2\gamma
H_{0}\xi^{2}}{1+\alpha^{2}}(\textbf{k}-\textbf{k}_{0}),$ (10)

where $\textbf{k}_{g}={\bf k}-{\bf k}_{0}$ defines the propagation direction.
The velocity of the wave packet can be separated as

$v_{\parallel}=\frac{\gamma
H_{0}\xi^{2}}{1+\alpha^{2}}2k+\frac{1+\alpha\beta}{1+\alpha^{2}}u_{\parallel},~{}v_{\perp}=\frac{1+\alpha\beta}{1+\alpha^{2}}u_{\perp},$
(11)

where $\parallel$ and $\perp$ denote the component parallel and perpendicular
to the wave vector respectively. Notice that, in general, the propagation
direction is not parallel to the wave vector k, and the SW can be deflected in
the presence of applied current.

###  1.2 Generalized Snell’s law

Let us now examine the propagation of SWs when experiencing nonuniform current
distributions. For simplicity, we start with the example of a SW propagating
between two regions with different current densities j, i.e. $\textbf{j}(x\leq
0)=0$ and $\textbf{j}(x>0)=j_{0}\hat{y}$. It is well known that SWs reflect at
material boundaries, where the momentum parallel to the interface should be
conserved. In the case of SP current, the change in current density is
equivalent to an interface, and the momentum perpendicular to $\nabla j$,
i.e., $\textbf{k}\cdot\hat{\tau}\equiv|\textbf{k}-(\textbf{k}\cdot\hat{\nabla
j})\hat{\nabla j}|$, with $\hat{\tau}$ the vector tangent to the interface,
should be conserved. In our simple example that corresponds to
$k^{(1)}_{y}=k^{(2)}_{y}$, where the indices 1 and 2 refer to the incident and
refracted waves respectively. If the propagation direction is parallel to the
k vector, the well-known Snell’s law applies:
$k^{(1)}\sin(\phi_{1})=k^{(2)}\sin(\phi_{2})$, with $\phi_{1}$ and $\phi_{2}$
the incident and refracted angles respectively. However, since in our case the
dispersion relation is asymmetric, the Snell’s law has to be adjusted as
follows

$k^{(1)}_{g}\sin(\phi_{1})+\textbf{k}^{(1)}_{0}\cdot\hat{\tau}=k^{(2)}_{g}\sin(\phi_{2})+\textbf{k}^{(2)}_{0}\cdot\hat{\tau},$
(12)

where the angles $\phi_{i}$ are taken with respect to $\hat{\nabla}j$ (i.e.,
the direction normal to the interface). Similarly generalized Snell’s laws for
the refraction of SWs at domain walls and heterochiral interfaces were derived
in Refs. 24 and 20.

###  1.3 Micromagnetic simulations

Figure 2: Generalized Snell’s law for SW refraction. (a-d) Snapshots of the
simulated spin-wave propagation across a sharp interface (at $x=0$) where the
current density changes from 0 to $\textbf{j}=j_{0}\hat{y}$, for different
incident angles $\phi_{1}$. White arrows indicate the propagation direction
following the generalized Snell’s law [Eq. (12)]. In these simulations we
considered SW frequency $f=20$ GHz, bias field $H_{0}=0.5$ T applied along
$+\hat{x}$ direction, $\textbf{j}=2\times 10^{12}$ Am${}^{-2}\hat{y}$,
$\alpha=0.001$ and $\beta=0.002$. Figure 3: SWs under nonuniform current
distribution. (a) Snapshots of the SW propagating across a nonuniform current
distribution induced between the voltage contacts (gray regions). White arrows
indicate the SW trajectory calculated by iterating Eq. (12) locally along the
current gradient in the propagation direction. Parameters are the same as in
Fig. 2. (b) Contour plot of the SP current density considered in (a). The
shown isolines should be seen as interfaces where the SW is sequentially
refracted during propagation. $\Delta\textbf{k}_{\perp}$ (black arrows) shows
the accompanying change in the propagation direction (see text).

To realistically simulate a SW system in the presence of the SP current, we
employ the micromagnetic framework and specifically its mumax3 implementation
25, 26 to calculate the SW refraction when propagating between two regions
with different current densities.

The SW beams are created by a sinusoidal oscillating field
$\textbf{h}=h_{0}\sin(\omega t)\hat{z}$ applied in a narrow rectangular region
(input antenna, see Fig. 2 (a)), where the field amplitude $h_{0}$ has a
Gaussian profile in the transverse direction and $f=\omega/2\pi$ is the
oscillation frequency. For all simulations we consider $h_{0}=0.01H_{0}$,
where $H_{0}=0.5$ T is the bias field applied along $+\hat{x}$ direction [see
Methods section]. Fig. 2 (a-d) shows snapshots of the simulated SW propagation
across the interface where the current density sharply changes, for different
incident angles $\phi_{1}$. White arrows denote the SW trajectories predicted
by our Eq. (12), which are in excellent agreement with the micromagnetic
simulations. Notice that the STT induced by the SP current can either deflect
or confine the SWs. The critical incident angle $\phi^{\ast}$ above which the
SW can not propagate for a given applied current, i.e., the SW undergoes total
internal reflection, is obtained from our Snell’s law by imposing that the
refracted wave is parallel to the interface ($\phi_{2}=\pm\pi/2$), as

$\phi^{\ast}=\pm\arcsin\left(\frac{\pm
k^{(2)}_{g}+(\textbf{k}^{(2)}_{0}-\textbf{k}^{(1)}_{0})\cdot\hat{\tau}}{k^{(1)}_{g}}\right).$
(13)

Note that due to the asymmetric dispersion relation, the refraction is not
symmetric for positive and negative incident angles. This is seen in the
results for $\phi_{1}=60^{\circ}$ and $\phi_{1}=-60^{\circ}$ in Fig. 2 (a,d),
where only for the second case the total reflection of the SW is achieved.

Poisson solver.—In practice, the current applied into the magnetic film will
not exhibit a step-like distribution as considered in our previous example,
but will spread continuously in the material obeying Poisson’s equation 27. To
precisely calculate the interaction of SWs with such nonuniform current
distributions, we implemented a Poisson solver in the micromagnetic simulation
package mumax3 25. Fig. 3 (a) shows a snapshot of the simulated SW propagation
across the nonuniform SP current density created between shown finite voltage
contacts at the sample edges. Notice that the SW is pertinently deflected
while crossing the region where current is applied, to finally reach a shifted
propagation direction upon leaving the area pierced by current. Although both
the direction and magnitude of the current continuously change as the SW
propagates, our generalized Snell’s law still applies locally - the SW facing
a local gradient in current density $\nabla j$ is effectively analogous to the
earlier interface example. White arrows in Fig. 3 (a) indicate the SW
trajectory calculated by iterating Eq. (12) as the SW propagates in the
current distribution, where in each step the angle $\phi_{i}$ is taken with
respect to the local $\hat{\nabla}j$. The predicted trajectory is in very good
agreement with the simulation, demonstrating the general applicability of Eq.
(12) even for nonuniform current distributions. Fig. 3 (b) shows the contour
plot of the applied current density, where isolines can be seen as interfaces
where the SW is refracted. $\Delta
k_{\perp}=(\textbf{k}^{(2)}_{0}-\textbf{k}^{(1)}_{0})\cdot\hat{\tau}$
quantifies the change in the propagation direction of the SW, shown as black
arrows in Fig. 3 (b).

###  1.4 Multichannel SW selector

Performing logic operations with SWs generally requires combining different
input waves that interfere with each other to generate a desired logic output
state 5. The ability to guide SWs through nanochannels is therefore vital to
the development of more complex SW-based circuitry. In this regard, SP
currents can be used to precisely guide the SW in such devices, for example,
to selectively “write” SWs in one of multiple nanotracks or logic gates in a
larger microprocessor. We exemplify here such an application by simulating a
multichannel SW selector, illustrated in Fig. 4 (a). The input SWs are
generated in a channel on the left-hand side of the sample and propagate
across a region where SP current is applied. By tuning the magnitude or
direction of the applied current one can precisely deflect the SWs towards one
of the output channels on the right side of the sample, as shown in Figs. 4
(b,c). Likewise, a frequency selector can be implemented considering the fact
that SWs with different frequencies experience different deflections under the
same applied current and can therefore be isolated into separate output
channels.

Figure 4: Guiding SWs for logic and neuromorphic computing. (a) A multichannel
SW selector demonstrated based on one input and two output channels. Voltage
leads induce the SP current in the central region of the sample. (b-c)
Snapshots of SW propagation simulated for the setup shown in (a). By changing
magnitude or direction of the applied current one can guide the SW towards the
desired output channel with minimal losses. (d) Scheme of the envisioned
neural network hardware. The (18) input SWs are created on the left and
propagate across the matrix of 80 voltage contacts (grey dots) of 100 nm in
diameter each. (e) SP-current profile induced by the applied voltage that best
performs the desired operation. (f-g) Snapshots of SW simulations after
training the neural network. The voltage (current) pattern was trained to
focus the waves of two different frequencies to the desired outputs. The bar
charts show the normalized intensities at the output locations integrated over
$10$ ns. In these simulations we considered SW frequency $f=15$ GHz (b, c, f)
and $f=14.5$ GHz (g), and other parameters same as previously.

###  1.5 Neuromorphic computing

Finally, we demonstrate the use of SP current for the design of neural-network
hardware, based on SW propagation, where weights and interconnections of the
network are realized by a pattern of the SP currents applied to the
propagating substrate. Fig. 4 (d) illustrates the envisioned device. The input
signal is created on the left and propagates across a region with a matrix of
80 voltage contacts [gray dots in Fig. 4 (d)] of 100 nm in diameter each. The
read-out is taken from the two output antennas on the right side. An
arbitrarily powered voltage matrix induces a distribution of the SP currents
in the substrate that interacts with the input SWs. Training the neural
network is equivalent to finding the current pattern that realizes the desired
input-output mapping, for example, to classify different input signals by
focusing them in different outputs. As suggested in Refs. 4 and 9, a back-
propagation machine learning algorithm can be used for training a similar SW-
based network, which can perform tasks such as vowel recognition and frequency
classification. Here, we demonstrate that a simple Monte Carlo (MC) algorithm
can perform the same task of training the neural network for simple
classification problems. In our example, we perform a frequency-recognition
operation, where we consider input SWs with frequencies $f=15$ and $14.5$ GHz.
The neural network is trained to focus the SWs with 15 GHz to the output O$1$
and SWs with 14.5 GHz to the output O$2$ [see Figs. 4 (f,g)]. The voltage at
each contact is randomly initialized in one of the three values:
$U_{i}=-u_{0}$, $0$ or $+u_{0}$, with $u_{0}=0.15$ V, and a Metropolis
algorithm is implemented by changing the voltage of one of the contacts at
every MC step [see supplemental material 22]. The finally trained
configuration [resulting in the current distribution shown in Fig. 4 (e)] is
able to focus SWs of each frequency to the desired outputs as shown by the bar
charts in Figs. 4 (f,g), and can therefore perform the classification
operation. In the supplementary material, we show that the effects of Oersted
and demagnetizing fields are negligible in our example 22. One should note
however that the proposed neuromorphic application can be generalized for
arbitrary scenarios, as long as all the relevant magnetic interactions are
considered during the training process of the neural network. The here
proposed fully electronic control enables facilitated reconfiguration of the
neural network to perform different tasks within the same integrated circuit.

To identify the role of different material parameters in the current-induced
SW scattering, and consequently in the proposed applications, let us consider
the case of an SW propagating across two regions with different current
densities, as in Fig. 2. In this scenario, for an incident angle $\phi_{1}=0$
and making use of Eqs. (12) and (3), the refraction angle of the SW is given
by
$\phi_{2}=\sin^{-1}\left(\frac{1+\alpha\beta}{\sqrt{(1+\alpha\beta)^{2}+\eta^{2}}}\right)$,
where $\eta=\frac{4e\gamma}{\mu_{B}}\frac{Ak}{Pj}$. Therefore, the current-
induced SW scattering is maximized when $\eta$ is minimized. In other words,
the SW scattering is maximized in materials with small exchange stiffness $A$
and for SWs with small wavenumber $k$, as well as by increasing the SP
current, represented by the polarization $P$ and applied current density $j$.
Moreover, it is worth mentioning here that applied current $j$ may locally
increase the temperature of the system, and consequently affect the magnetic
parameters such as the saturation magnetization $M_{s}$, hence additional
scattering may arise from such non-uniformity of the magnetic landscape in
which SWs propagate. In Table 1 we show material properties of representative
low-damping, metallic magnetic materials that can host spin waves and are
therefore convenient candidates for the proposed control of SWs by SP current.
As an additional example, we reproduced the neuromorphic application for the
case of a Ni80Fe20 thin film in the supplementary material 22.

|  $M_{s}$ (MA/m) |  $A$ (pJ/m) |  $\alpha$ ($\times 10^{-3}$) |  $P$ $(\Omega m)^{-1}$ | References
---|---|---|---|---|---
Ni80Fe20 | 0.7 | 10 | 7 | 0.5 | [16, 5]
CoFeB | 1.3 | 15 | 4 | 0.65 | [5, 28]
CoFeAlB | 1.0 | 9 | 3 |  | [29]
Table 1: Material properties of representative low-damping, metallic magnetic
materials that can host spin waves and are therefore candidates for the
proposed control of SWs by SP current.

##  2 Conclusions

We have demonstrated the use of non-uniform spin-polarized current for the
manipulation of spin-waves. We showed that the spin-transfer torque induced by
the applied current has an effect analogous to DMI for confining spin-waves
and controlling their propagation direction in the linear regime, and derived
a generalized Snell’s law that describes the scattering of spin-waves between
regions with different current densities. Finally, we implemented the
calculation of the current distribution in micromagnetic simulations by
solving the Poisson’s equation within the simulation package mumax3 (to be
made available in the upcoming release of mumax4), in order to (i) validate
the derived Snell’s law and (ii) demonstrate how strategically applied current
distributions can be employed in magnonic logic and neuromorphic computing
devices, thereby advancing the prospects of low-power spintronic hardware and
artificial intelligence.

##  3 Methods

###  3.1 Micromagnetic simulations

For the simulations, we employ the micromagnetic framework MUMAX3 25, 26,
where we consider $10$ nm thick magnetic films with saturation magnetization
$M_{\text{s}}=0.14$ MAm-1, exchange stiffness $A=3.65$ pJm-1 30 and damping
constant $\alpha=0.001$. The considered free energy density is given by Eq.
(4), and the dynamics of the magnetization is governed by the LLG equation
[Eq. (1)] with STT terms [Eq. (2)] accounting for in-plane applied currents.
The in-plane bias field is set to $\textbf{H}=H_{0}\hat{x}$, with $H_{0}=0.5$
T. For all simulations, we consider a system discretized into cells of size
$10\times 10\times 10$ nm3. The polarization rate of the SP current is fixed
at $P=0.4$. The calculation of the current distribution is made by solving the
Poisson’s equation for the considered geometries, where we assume the
electrical conductivity as $\sigma=1\times 10^{6}$ ($\Omega$m)-1.

This work was supported by the Research Foundation - Flanders (FWO-Vlaanderen,
under Grant No. K226322N and 12A9223N), EoS ShapeME project, and Brazilian
Agencies FACEPE, CAPES and CNPq.

Calculation of spin-wave dispersion relation under spin-polarized current;
Monte Carlo training of neural network; Effect of Oersted and demagnetizing
fields; Example of a spin-wave neural network in a Ni80Fe20 thin film.

## References

  * Christensen et al. 2022 Christensen, D. V.; Dittmann, R.; Linares-Barranco, B.; Sebastian, A.; Le Gallo, M.; Redaelli, A.; Slesazeck, S.; Mikolajick, T.; Spiga, S.; Menzel, S., et al. 2022 roadmap on neuromorphic computing and engineering. _Neuromorphic Computing and Engineering_ 2022,
  * Romera et al. 2018 Romera, M.; Talatchian, P.; Tsunegi, S.; Abreu A., F.; Cros, V.; Bortolotti, P.; Trastoy, J.; Yakushiji, K.; Fukushima, A.; Kubota, H., et al. Vowel recognition with four coupled spin-torque nano-oscillators. _Nature_ 2018, _563_ , 230–234
  * Shainline et al. 2017 Shainline, J. M.; Buckley, S. M.; Mirin, R. P.; Nam, S. W. Superconducting optoelectronic circuits for neuromorphic computing. _Physical Review Applied_ 2017, _7_ , 034013
  * Hughes et al. 2019 Hughes, T. W.; Williamson, I. A. D.; Minkov, M.; Fan, S. Wave physics as an analog recurrent neural network. _Science advances_ 2019, _5_ , eaay6946
  * Mahmoud et al. 2020 Mahmoud, A.; Ciubotaru, F.; Vanderveken, A. V., F.and Chumak; Hamdioui, S.; Adelmann, C.; Cotofana, S. Introduction to spin wave computing. _Journal of Applied Physics_ 2020, _128_ , 161101
  * Barman et al. 2021 Barman, A.; Gubbiotti, G.; Ladak, S.; Adeyeye, A. O.; Krawczyk, M.; Gräfe, J.; Adelmann, C.; Cotofana, S.; Naeemi, A.; Vasyuchka, V. I., et al. The 2021 magnonics roadmap. _Journal of Physics: Condensed Matter_ 2021, _33_ , 413001
  * Chumak et al. 2014 Chumak, A. V.; Serga, A. A.; Hillebrands, B. Magnon transistor for all-magnon data processing. _Nature Communications_ 2014, _5_ , 1–8
  * Chumak et al. 2015 Chumak, A. V.; Vasyuchka, V. I.; Serga, A. A.; Hillebrands, B. Magnon spintronics. _Nature Physics_ 2015, _11_ , 453–461
  * Papp et al. 2021 Papp, A.; Porod, W.; Csaba, G. Nanoscale neural network using non-linear spin-wave interference. _Nature Communications_ 2021, _12_ , 1–8
  * Zhang and Li 2004 Zhang, S.; Li, Z. Roles of Nonequilibrium Conduction Electrons on the Magnetization Dynamics of Ferromagnets. _Physical Review Letters_ 2004, _93_ , 127204
  * Ralph and Stiles 2008 Ralph, D. C.; Stiles, M. D. Spin transfer torques. _Journal of Magnetism and Magnetic Materials_ 2008, _320_ , 1190–1216
  * Wang et al. 2022 Wang, W.; Song, D.; Wei, W.; Nan, P.; Zhang, S.; Ge, B.; Tian, M.; Zang, J.; Du, H. Electrical manipulation of skyrmions in a chiral magnet. _Nature Communications_ 2022, _13_ , 1–7
  * Peng et al. 2021 Peng, L.; Karube, K.; Taguchi, Y.; Nagaosa, N.; Tokura, Y.; Yu, X. Dynamic transition of current-driven single-skyrmion motion in a room-temperature chiral-lattice magnet. _Nature Communications_ 2021, _12_ , 1–7
  * Bazaliy et al. 1997 Bazaliy, Y. B.; Jones, B. A.; Zhang, S.-C. Modification of the Landau-Lifshitz Equation in the Presence of a Spin-Polarized Current in CMR and GMR Materials. 1997, _57_ , 3213–3216
  * Fernández-Rossier et al. 2004 Fernández-Rossier, J.; Braun, M.; Núñez, A. S.; MacDonald, A. H. Influence of a uniform current on collective magnetization dynamics in a ferromagnetic metal. _Physical Review B_ 2004, _69_ , 174412
  * Vlaminck and Bailleul 2008 Vlaminck, V.; Bailleul, M. Current-induced spin-wave doppler shift. _Science_ 2008, _322_ , 410–413
  * Mills and Dzyaloshinskii 2008 Mills, D. L.; Dzyaloshinskii, I. E. Influence of electric fields on spin waves in simple ferromagnets: Role of the flexoelectric interaction. _Physical Review B - Condensed Matter and Materials Physics_ 2008, _78_ , 1–4
  * Moon et al. 2013 Moon, S.-M., J.-H.and Seo; Lee, K.-J.; Kim, K.-W.; Ryu, J.; Lee, H.-W.; McMichael, R. D.; Stiles, M. D. Spin-wave propagation in the presence of interfacial Dzyaloshinskii-Moriya interaction. _Physical Review B_ 2013, _88_ , 184404
  * Cortés-Ortuño and Landeros 2013 Cortés-Ortuño, D.; Landeros, P. Influence of the Dzyaloshinskii–Moriya interaction on the spin-wave spectra of thin films. _Journal of Physics: Condensed Matter_ 2013, _25_ , 156001
  * Mulkers et al. 2018 Mulkers, J.; Van Waeyenberge, B.; Milošević, M. V. Tunable Snell’s law for spin waves in heterochiral magnetic films. _Physical Review B_ 2018, _97_ , 104422
  * Thiaville et al. 2005 Thiaville, A.; Nakatani, Y.; Miltat, J.; Suzuki, Y. Micromagnetic understanding of current-driven domain wall motion in patterned nanowires. _Europhysics Letters_ 2005, _69_ , 990–996
  * 22 See supplementary material at [url].
  * Seo et al. 2009 Seo, S.-M.; Lee, K.-J.; Yang, H.; Ono, T. Current-induced control of spin-wave attenuation. _Physical review letters_ 2009, _102_ , 147202
  * Yu et al. 2016 Yu, W.; Lan, J.; Wu, R.; Xiao, J. Magnetic Snell’s law and spin-wave fiber with Dzyaloshinskii-Moriya interaction. _Physical Review B_ 2016, _94_ , 140410
  * Vansteenkiste et al. 2014 Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Van Waeyenberge, B. The design and verification of MuMax3. _AIP Advances_ 2014, _4_ , 107133
  * Leliaert et al. 2018 Leliaert, J.; Dvornik, M.; Mulkers, J.; De Clercq, J.; Milošević, M. V.; Van Waeyenberge, B. Fast micromagnetic simulations on GPU—recent advances made with. _Journal of Physics D: Applied Physics_ 2018, _51_ , 123002
  * Griffiths 2005 Griffiths, D. J. Introduction to electrodynamics. 2005
  * Huang et al. 2008 Huang, S. X.; Chen, T. Y.; Chien, C. L. Spin polarization of amorphous CoFeB determined by point-contact Andreev reflection. _Applied Physics Letters_ 2008, _92_ , 242509
  * Conca et al. 2017 Conca, A.; Nakano, T.; Meyer, T.; Ando, Y.; Hillebrands, B. CoFeAlB alloy with low damping and low magnetization as a candidate for spin transfer torque switching. _Journal of Applied Physics_ 2017, _122_ , 073902
  * Papp et al. 2017 Papp, A.; Porod, W.; Csurgay, A. I.; Csaba, G. Nanoscale spectrum analyzer based on spin-wave interference. _Scientific Reports_ 2017, _7_ , 1–9