# Symmetry of the coupling between surface acoustic waves and spin waves in
synthetic antiferromagnets

R. Lopes Seeger  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120, Palaiseau, France  SPEC, CEA, CNRS, Université Paris-
Saclay, 91191 Gif-sur-Yvette, France  L. La Spina  Université de Franche-
Comté, CNRS, Institut FEMTO-ST, 26 rue de l’Epitaphe, 25000 Besançon, France
V. Laude  Université de Franche-Comté, CNRS, Institut FEMTO-ST, 26 rue de
l’Epitaphe, 25000 Besançon, France  F. Millo  Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France  A.
Bartasyte  Université de Franche-Comté, CNRS, Institut FEMTO-ST, 26 rue de
l’Epitaphe, 25000 Besançon, France  S. Margueron  Université de Franche-Comté,
CNRS, Institut FEMTO-ST, 26 rue de l’Epitaphe, 25000 Besançon, France  A.
Solignac  G. de Loubens  SPEC, CEA, CNRS, Université Paris-Saclay, 91191 Gif-
sur-Yvette, France  L. Thevenard  C. Gourdon  Institut des Nanosciences de
Paris, Sorbonne Université, CNRS, UMR 7588, 4 place Jussieu, F-75005 Paris,
France  C. Chappert  T. Devolder  Université Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

###### Abstract

Synthetic antiferromagnets host spin waves that are highly tunable. It is of
practical interest to analyze the symmetry of their coupling to surface
acoustic waves with the design of hybrid devices in view. For this we
calculate the layer-resolved susceptibility tensor of a synthetic
antiferromagnet, the effective magneto-elastic and magneto-rotation fields
associated to a travelling elastic wave, and the power irreversibly
transferred by the elastic wave to the magnetic layers. We consider Rayleigh-
type surface acoustic waves: (a) that travel in an elastically isotropic, non-
piezoelectric substrate, or (b) that propagate along the X direction at the
surface of a Z-cut LiNbO3 substrate, or (c) that are guided in a thin Z-cut
LiNbO3 film grown on a sapphire substrate. In particular, we show that the
complementary angular dependencies of the acoustic and optical spin wave modes
in synthetic antiferromagnets makes it possible to excite spin waves for any
relative orientation of magnetization and acoustic wavevector. In addition, we
discuss the symmetries of the driving fields and of the energy transferred to
the magnetic degree of freedom. We evidence new interaction channels coupling
the magnetization eigenmodes when elastic anisotropy and piezoelectricity of
the substrate are considered.

##  I Introduction

The coupling between surface acoustic waves (SAWs) and spin waves (SWs) has
drawn increasing attention in recent years due to its potential for the
realization of integrated microwave devices benefiting from the best of both
magnonics and microwave acoustics [1, 2, 3, 4, 5, 6]. Particularly, the
coupling of SWs in magnetic thin films with SAWs allows excitation,
manipulation and detection of various magneto-elastic waves. SAWs are excited
on piezoelectric substrates, typically bulk single crystals of LiNbO3. The
SAW-SW coupling is mediated by strains [1] and by the rotational motion of the
lattice [7] within the magnetic material. However the SAW-SW coupling is
generally effective only for certain relative orientations between
magnetization and SAW wavevectors [8]. The coupling often vanishes in the high
symmetry directions, except for specific substrates supporting Rayleigh and
shear horizontal SAW modes simultaneously [9]. Finding ways to couple SWs and
SAWs for arbitrary orientations would enable to design hybrid devices that
fully exploit the anisotropic and non-reciprocal properties of SWs as well as
the high energy-efficiency of SAWs excitation and manipulation. In this paper,
we considered an alternative magnetic material platform: the synthetic
antiferromagnets (SAFs). For the analysis, we developed an extensive modeling
of SAWs propagation and its interaction with SAF SWs.

The SAW-SW coupling mechanism is intrinsically non-reciprocal, because of an
helicity mismatch [10, 5, 6, 11, 12]. In addition, non-reciprocal SAW
transmission can also arise from an asymmetric SW dispersion relation [13, 14,
15, 16]. Hybrid SW/SAW systems made from SAF may thus find applications, such
that the SAW to SW coupling deserves to be studied. SAFs have two families of
precession eigenmodes [17]: In-phase and out-of-phase precessions of the
magnetization in each layer are associated with acoustic and optical SW modes,
respectively. As we will show, the SAW to SW coupling displays different
angular dependencies for each of the SAF eigenmodes.

To ease the modelling of SAW propagation, the substrate is usually assumed to
be uncapped, elastically isotropic and its piezoelectric character is ignored.
It, therefore, hosts an (idealized) Rayleigh SAW [1, 6, 18, 11, 3, 19, 20].
One often postulates that the magnetic layer has the same elastic properties
as that of the substrate or, that the back action of the magnetic films on the
substrate elastodynamics can be neglected. The finite thickness of the
magnetic system is then only translated as a ”filling factor” in the SAW-SW
coupling, save for rare occasions implementing the more complex ”thin film on
substrate approach” [21, 22]. In this article, we reconsider this first
assumption, and consider the impact of the substrate’s elastic anisotropy and
piezoelectricity onto the coupling of SWs to SAWs.

This paper presents a comprehensive symmetry analysis of the coupling between
Rayleigh-type elastic waves traveling in various substrates with the SWs of a
capping CoFeB/Ru/CoFeB SAF film. In particular we compare the case of a
conventional Rayleigh wave in a semi-infinite elastically isotropic substrate
(”iso” model), to the ones of a wave travelling in a real material (Z-cut
LiNbO3: ”Z-cut” model) and of a wave guided in a Z-cut LiNbO3 film grown on an
Al2O3 C-sapphire substrate (”guided” model), see Fig. 1(a). We clarify the
respective roles of the magneto-elastic tickle fields, the magneto-rotation
”rolling” fields and the two SW resonance modes of a SAF when set in the
scissors state by an external magnetic field. Finally, we discuss interaction
channels allowing SAW-SW coupling for all applied field orientations.

The paper is organized as follows. We first assess the magnetics of SAF and
its SW modes (section III). We then compare the strain tensors and the
rotation tensors of Rayleigh surface acoustic waves in the ”iso”, ”Z-cut” and
”guided” models in section IV. The power transmitted to the spin waves excited
by the elastic waves is studied in section V. The amplitude of the different
coupling channels are compared. Section VI focuses then on a model system, the
SAW to SW coupling in the fully isotropic case. The differences with more
realistic materials represented by the other substrate models are finally
discussed in section VII.

##  II Geometry

We consider the geometry described in Fig. 1. It consists in a synthetic
antiferromagnetic film grown on substrate ending at $z=0$ that hosts surface
acoustic waves within its depth ($z<0$). Two coordinate systems are defined.
The coordinate system $\\{XYZ\\}$ is defined with respect to the wavevector
$k_{\textrm{SAW}}\parallel X$ of the surface acoustic wave and the out-of-
plane direction $Z$. This coordinate system will be used to describe the
elastic properties and the surface acoustic waves. Three different substrates
will be used: a semi-infinite elastically isotropic substrate (”iso” model), a
semi-infinite Z-cut LiNbO3 single crystal (”Z-cut” model) and, a 150-nm thick
Z-cut LiNbO3 film grown on a semi-infinite C-sapphire substrate (”guided”
model). It is noteworthy that $Y_{LNO}\perp(0\bar{1}10)_{\textrm{sapphire}}$.
For the two anisotropic substrates, the $X$ direction matches with the $X$
crystalline direction of LiNbO3. The full system is invariant in the direction
transverse to the SAW wavevector (i.e. $\frac{\partial}{\partial Y}=0$).

The other coordinate system, i.e. $\\{xyz\\}$ is defined with respect to the
growth direction $z=Z$ and a uniform static applied field $H_{x}$ oriented in
the film plane at an angle $\varphi$ away from the SAW direction. This
coordinate system will be convenient when describing the magnetics of the
synthetic antiferromagnet. We shall consider that the SAF is thin enough and
elastic enough to replicate perfectly to the strain present at the surface of
the (much thicker) substrate, and not to have a back action on the substrate.
To pass from one coordinate system to the other, it is useful to define the
rotation matrix $\bar{\bar{R}}$ :

$\bar{\bar{R}}=\left(\begin{array}[]{ccc}\cos\varphi&\sin\varphi&0\\\
-\sin\varphi&\cos\varphi&0\\\ 0&0&1\\\ \end{array}\right)$ (1)

##  III Modeling of the Synthetic Antiferromagnet under strain

###  III.1 Magnetic energies

We model the SAF as two layers $\ell=1,2$ of thickness $t_{\ell}$ and
magnetizations $\left(m_{x\ell},m_{y\ell},m_{z\ell}\right)M_{s}$, the
magnetizations being uniform across the thickness of each layer. The in-plane
applied field $H_{x}$ leads to a Zeeman energy per unit surface that is
written as
$E_{\textrm{Zeeman}}=-\mu_{0}M_{s}H_{x}\sum_{\ell=1,2}m_{x\ell}\,t_{\ell}$.
The areal shape anisotropy of the system 111The intralayer exchange energy and
the non-uniform demagnetizating fields arising at finite SW wavectors are
disregarded is
$E_{\textrm{shape}}=+\frac{1}{2}\mu_{0}M_{s}^{2}\sum_{\ell=1,2}m_{z\ell}^{2}\,t_{\ell}$,
and the areal energy due to the interlayer coupling can be written as
$E_{\textrm{J}}=-J\left(m_{x1}m_{x2}+m_{y1}m_{y2}+m_{z1}m_{z2}\right)$. In the
presence of mechanical deformations, there is also a magneto-elastic
contribution written as [8]

$\begin{split}E_{\textrm{mel}}=B_{1}\sum_{\ell=1,2}t_{\ell}\left(m_{x\ell}^{2}{\epsilon_{xx}}+m_{y\ell}^{2}{\epsilon_{yy}}+m_{z\ell}^{2}{\epsilon_{zz}}\right)\\\
+2B_{2}\sum_{\ell=1,2}t_{\ell}\left(m_{x\ell}m_{y\ell}{\epsilon_{xy}}+m_{x\ell}m_{z\ell}{\epsilon_{xz}}+m_{y\ell}m_{z\ell}{\epsilon_{yz}}\right)\\\
\end{split}$ (2)

and a magneto-rotation contribution [6]:

$E_{\textrm{roll}}=+\mu_{0}M_{s}^{2}\sum_{\ell=1,2}t_{\ell}m_{z\ell}(m_{x\ell}{\omega_{xz}}+m_{y\ell}{\omega_{yz}}),$
(3)

where $\bar{\bar{\epsilon}}(X,Z,t)$ is the strain tensor and
$\bar{\bar{\omega}}(X,Z,t)$ is the lattice rotation tensor within the SAF.
Note that since we assume easy plane with isotropic magnetic properties, the
directions within the $xy=XY$ planes are magnetically equivalent. The rotation
$\omega_{xy}$ (i.e. rotation about the $z$ axis) thus does not contribute to
the magnetic energy. The lattice deformations lead to the effective ”tickle”
$h_{\textrm{tick}}^{rf}$ and ”rolling” $h_{\textrm{{roll}}}^{rf}$ driving
fields, as we will discuss further below.

In the absence of data in the literature, we shall follow the usual
speculation and assume an isotropic magnetic system which leads to
$B_{1}=B_{2}$. Typical values for Co40Fe40B20 include a magnetization [24]
$M_{s}=1.35$ MA/m, an interlayer exchange coupling [25]
$J=-1.1~{}\textrm{mJ/m}^{2}$ for a $0.7$ nm thick Ru spacer, a first magneto-
elastic constant [26] $B_{1}=-7.6~{}\textrm{MJ/m}^{3}$, which for layer
thicknesses $t_{1}=t_{2}=13~{}\textrm{nm}$ lead to characteristics fields of
$\mu_{0}M_{s}=1.7$ T,
$\mu_{0}H_{\textrm{mel}}=\frac{2B_{1}}{M_{s}}=11.2~{}\textrm{T}$ and
interlayer exchange field
$\mu_{0}H_{J}=-\frac{2J}{M_{s}t_{\ell}}=125~{}\textrm{mT}$.

Figure 1: Schematics of the investigated systems. (a) Studied elastic media.
(b) The SAF is one of the two degenerate scissors state. The coordinate system
$\\{xyz\\}$ is defined with respect to the growth direction $z=Z$ and the
static applied field $H_{x}$ oriented in the film plane. The coordinate system
$\\{XYZ\\}$ is defined with respect to the wavevector of the surface acoustic
wave $k_{\textrm{SAW}}\parallel X$ and the out-of-plane direction $Z$.

###  III.2 Ground state of the synthetic antiferromagnet

In order to allow for analytical calculations we shall simplify the
micromagnetic problem. We consider that the ground state of the SAF is equal
to that of a two-macrospin system. In other words, we neglect any gradient of
the magnetization within the thickness of the magnetic layers [24]. This
assumption will be lifted temporarily in Fig. 2(c) and in the related
discussion (section III.4.2).

We consider that the applied field follows $0\leq H_{x}<H_{J}$, and that the
rf elastic deformations have vanishing time-averages such that at rest the SAF
is in one of the two degenerate scissors states. We choose the one for which
the layers are magnetized as:

$\left(\begin{array}[]{c}m_{x\ell}\\\ m_{y\ell}\\\ m_{z\ell}\\\
\end{array}\right)=\left(\begin{array}[]{c}\frac{H_{x}}{H_{J}}\\\
\delta\sqrt{1-\frac{H_{x}^{2}}{H_{J}^{2}}}\\\ 0\\\ \end{array}\right)$ (4)

where $\delta=1$ for the layer $\ell=1$ and $\delta=-1$ for layer $\ell=2$.

###  III.3 Tickle fields and rolling fields

The description of magnetization dynamics requires to depict the effective
fields acting on the magnetization in the frame of the applied field, i.e.,
$xyz$ [Fig. 1(b)]. The SAW induced lattice deformations lead to the so-called
magneto-elastic ”tickle” fields 222Note that for an in-plane magnetized
system, $\epsilon_{zz}$ does not induce a torque at first order.:

$h_{\textrm{tick}}^{rf}=-\frac{2}{\mu_{0}M_{s}}\left(\begin{array}[]{c}B_{1}\frac{H_{x}}{H_{J}}\,\epsilon_{xx}+B_{2}\delta\sqrt{1-\frac{H_{x}^{2}}{H_{J}^{2}}}\,\epsilon_{xy}\\\
B_{2}\frac{H_{x}}{H_{J}}\,\epsilon_{xy}+B_{1}\delta\sqrt{1-\frac{H_{x}^{2}}{H_{J}^{2}}}\,\epsilon_{yy}\\\
B_{2}\frac{H_{x}}{H_{J}}\,\epsilon_{xz}+B_{2}\delta\sqrt{1-\frac{H_{x}^{2}}{H_{J}^{2}}}\,\epsilon_{yz}\\\
\end{array}\right)$ (5)

as well as to the magneto-rotation ”rolling” fields:

$h_{\textrm{roll}}^{rf}=M_{s}\left(\begin{array}[]{c}0\\\ 0\\\
\omega_{zx}\frac{H_{x}}{H_{J}}~{}+~{}\omega_{zy}\delta\sqrt{1-\frac{H_{x}^{2}}{H_{J}^{2}}}\\\
\end{array}\right)$ (6)

Note that owing to the $\delta$ terms in Eq. 5 and 6, these effective fields
are layer-dependent in a SAF. When a SAW is present, these effective fields
are also space and time dependent and their values can be found by the
substitution $h^{rf}\rightarrow\mathcal{R}e(h^{rf}e^{i(\omega
t-k_{\textrm{SAW}}X)})$.

###  III.4 Spin waves of the synthetic antiferromagnet

####  III.4.1 Uniform resonances

To get the uniform resonances of the SAF, we consider the total energy of the
system and linearize the Landau-Lifschitz-Gilbert equation around the chosen
ground state. We follow the standard procedure and write the dynamical matrix
of the system, whose eigenvalues are the two uniform resonance frequencies of
a SAF [Fig. 2(a,b)]. For $H_{x}<H_{J}$, their real parts read [28]:

$\frac{\omega_{\textrm{ac}}}{\gamma_{0}}=H_{x}\sqrt{\frac{M_{s}+H_{J}}{H_{J}}}\textrm{~{}and~{}}\frac{\omega_{\textrm{op}}}{\gamma_{0}}=\sqrt{\frac{M_{s}}{H_{J}}}\sqrt{H_{J}^{2}-H_{x}^{2}},$
(7)

where the labels recall the acoustic (in-phase) or optical (out-of-phase)
nature of the spin wave mode. The imaginary parts of the eigenvalues lead to
the mode linewidths (full widths at half maximum):

$\frac{\Delta\omega_{\textrm{ac}}}{\alpha\gamma_{0}}=M_{s}+\frac{H_{J}^{2}+H_{x}^{2}}{H_{J}}~{}\textrm{and~{}}\frac{\Delta\omega_{\textrm{op}}}{\alpha\gamma_{0}}=M_{s}+\frac{H_{J}^{2}-H_{x}^{2}}{H_{J}},$
(8)

where $\gamma_{0}$ is the gyromagnetic ratio and $\alpha$ is the damping.

####  III.4.2 Working assumption: spinwaves with vanishing group velocity

We now discuss an important simplifying assumption. We consider that the spin
waves have vanishing group velocities, i.e. we use the definition:

$\omega_{\textrm{ac, op}}(k,\varphi)~{}~{}\triangleq~{}~{}\omega_{\textrm{ac,
op}}(k=0,\varphi)$ (9)

The crude [29, 30, 31] assumption of Eq. 9 is done at the sake of pedagogy but
it has an important consequence [Fig. 2(c)] that deserves a discussion. In
reality the SW mode frequencies $\omega_{\textrm{mode}}$ for $k=0$ and $k\neq
0$ and their angular dependencies with $\varphi$ differ. This is shown in Fig.
2(c), in which $\omega(k=0,\varphi)$ from Eq. 7 is compared with $\omega(k\neq
0,\varphi)$ when calculated in the full micromagnetic framework, as conducted
in Ref. [31]. The non-circular character of $\omega(\varphi,k\neq 0)$ reflects
the anisotropic character of the SWs of a SAF. The non-centro-symetric
character of $\omega(\varphi,k\neq 0)$ recalls their non-reciprocal nature
[30, 29, 31, 12].

In a real experiment, there would be a defined SAW wavevector
$k_{\textrm{SAW}}$ and a defined magnetic field orientation $\varphi$. Within
our working assumption (Eq. 9), if the field amplitude $H_{x}$ is set to have
a given SW mode be at resonance with the SAW, the field can be rotated while
maintaining the resonant character. The angular dependence of the SAW-SW
coupling then only reflects the angular dependence of the driving torque.

In contrast with our working assumption, if the true dispersion of the spin
wave was taken into account, rotating the field would automatically detune the
SW resonances away from the SAW resonance. As a result, the angular dependence
of the SAW-SW coupling would reflect both the angular dependence of the
driving torque (so-called helicity mismatch [12]) and the angular dependence
of the SW frequencies (mismatch between the SAW resonance and the SW
resonance).

We aim to unravel the respective effects of angle-dependent resonance mismatch
on the one hand, and, on the other hand, angle-dependent helicity mismatch.
For that purpose our study will be conducted by assuming spin waves with
vanishing group velocities (Eq. 9) to highlight only the SAW-SW coupling
symmetry from the driving field standpoint. Fully taking into account that
$\omega_{\textrm{ac, op}}(k,\varphi)\neq\omega_{\textrm{ac, op}}(k\neq
0,\varphi)$ is an interesting subject that will be explored in future
investigations.

Figure 2: (a) Acoustic and optical SW frequencies in a SAF from Eq. 7 for
$H_{x}<H_{J}$ and extracted from Ref. [28] for $H_{x}>H_{J}$. (b) sketch of
their respective magnetization precessional motions in the scissors state. (c)
Angle-resolved dispersion relations $\omega_{\textrm{mode}}(\varphi)$ for
$\mu_{0}H_{x}=50$ mT, $k=0$ and 7.5 rad/µm within the full micromagnetics
framework.

###  III.5 Susceptibility tensor of a SAF in the reference frame of the
applied field

From the dynamical matrix, a cumbersome algebra leads to the susceptibility of
each layer of the SAF for a zero wavevector. The SAF susceptibility can be
separated into two independent contributions corresponding to the acoustic and
optical eigenmodes, and it is expressed as:

$\bar{\bar{\chi}}_{\ell}=\mathcal{L}^{\textrm{ac}}\left(\begin{array}[]{ccc}0&\lambda\delta&i\eta\frac{\lambda}{\delta}{\color[rgb]{0,0,0}\frac{\omega}{\omega_{\textrm{ac}}}}\\\
0&1&i\eta\frac{\omega}{\omega_{\textrm{ac}}}\\\
0&-i\eta\frac{\omega}{\omega_{\textrm{ac}}}&\eta^{2}\\\
\end{array}\right)+\mathcal{L}^{\textrm{op}}\left(\begin{array}[]{ccc}1&0&0\\\
\frac{\delta}{\lambda}&0&0\\\
-i\frac{\nu}{\delta}{\color[rgb]{0,0,0}\frac{\omega}{\omega_{\textrm{op}}}}&0&0\\\
\end{array}\right)$ (10)

where the dimensionless terms $\eta$, $\lambda$ and $\nu$ describing the
precession ellipticity are: $\eta=\sqrt{\frac{H_{J}}{H_{J}+M_{s}}}$,
$\lambda=\frac{\sqrt{H_{J}^{2}-H_{x}^{2}}}{H_{x}}$ and
$\nu=\sqrt{\frac{H_{J}}{M_{s}}}$. Note the non-hermitian character of the
layer-resolved $\bar{\bar{\chi}}$. The Hermitian character (reflecting the
gyrotropy) is recovered when summing the susceptibilities of the two layers.

The resonant nature of the response of each mode is expressed by the
Lorentzian function:

$\mathcal{L}^{\textrm{mode}}(\omega)=\chi_{\textrm{max}}^{\textrm{mode}}\frac{i\,\omega_{\textrm{mode}}\,\Delta\omega_{\textrm{mode}}\,}{(\omega_{\textrm{mode}}^{2}-\omega^{2})+i\,\omega\,\Delta\omega_{\textrm{mode}}},$
(11)

The partial responses $\mathcal{L}^{\textrm{ac}}$ and
$\mathcal{L}^{\textrm{op}}$ peak at their resonance frequencies (Eq. 7). The
$\mathcal{L}^{\textrm{ac}}$ and $\mathcal{L}^{\textrm{op}}$ are purely
imaginary at their resonances and can be expressed as the dimensionless
numbers:

$\chi_{\textrm{max}}^{\textrm{ac}}=-i\frac{M_{s}}{H_{J}}\frac{\omega_{\textrm{ac}}}{\Delta\omega_{\textrm{ac}}}\textrm{~{}~{}and~{}~{}}\chi_{\textrm{max}}^{\textrm{op}}=-i\frac{M_{s}}{H_{J}}\frac{\omega_{\textrm{op}}}{\Delta\omega_{\textrm{op}}}.$
(12)

Several points are worth noticing in Eq. 10.
(i) As is well known, the optical mode responds to parallel pumping only,
$H_{x}\parallel h^{rf}$ (the non-vanishing susceptibility terms are in the
first column), while the acoustic mode responds only to transverse rf fields,
$H_{x}\perp h^{rf}$ (second and third columns).
(ii) Near resonance, all susceptibility terms are essentially Lorentzian-like
(Eq. 11). However the additional $\omega/\omega_{\textrm{mode}}$ factors in
some of the non-diagonal susceptibility terms (Eq. 10) make them non-
Lorentzian and let them vanish in the static limit.
(iii) The magnitudes of the layer-resolved susceptibilities of the two modes
are very different. At low fields when $H_{x}\ll H_{J}\ll M_{s}$, we have
$\chi^{\textrm{ac}}_{\textrm{max}}(H_{x}\rightarrow
0)\approx-\frac{i}{\alpha}\frac{H_{x}}{H_{J}}\sqrt{\frac{M_{s}}{H_{J}}}$ and
$\chi^{\textrm{op}}_{\textrm{max}}(H_{x}\rightarrow
0)\approx-\frac{i}{\alpha}\sqrt{\frac{M_{s}}{H_{J}}}$. The susceptibility at
low fields is much larger for the optical mode at resonance than for the
acoustic mode at resonance. Conversely, the response of the optical mode
progressively dies as one approaches saturation at $H_{J}$ while the resonant
susceptibility of the acoustic mode attains it maximum with essentially
$\chi^{\textrm{ac}}_{\textrm{max}}(H_{x}=H_{J})\approx-\frac{i}{\alpha}\sqrt{\frac{M_{s}}{H_{J}}}$.

We highlight that since
$\bar{\bar{\chi}}_{\textrm{ac}}h_{\textrm{roll}}^{rf}\neq\vec{0}$, the lattice
rotations are expected to excite the acoustic SW mode. It is the case when the
strain is considered to be uniform within the SAF, which is a good
approximation when it is much thinner than the SAW wavelength. In contrast,
$\bar{\bar{\chi}}_{\textrm{op}}h_{\textrm{roll}}^{rf}=\vec{0}$, meaning that
the lattice rotations do not excite the optical SW mode directly. This does
not mean that the rolling fields could be disregarded for the optical mode, it
would only be the case if
$\bar{\bar{\chi}}_{\textrm{op}}h_{\textrm{tick}}^{rf}\perp
h_{\textrm{roll}}^{rf}$, which is in general not true.

##  IV Strain and rotation tensors of surface acoustic waves in Lithium
niobate systems

The objective of this section is to describe the strain tensor and the lattice
rotation tensor of Rayleigh SAWs travelling in LiNbO3 systems. Targeting SAWs
to operate at higher frequencies, one indeed replaces the traditional bulk
piezoelectric substrate [32, 33] by a piezoelectric thin film, see Fig. 1(a).
In particular, thin films are grown on higher-acoustic-velocity substrates,
typically forming an LiNbO3/sapphire [34] stack [Fig. 1(a)]. In such
structure, the SAW is guided in the thin piezoelectric film, and the clamping
effectively accelerates the acoustic waves while modifying their
polarizations. As a bonus, the high confinement of the acoustic energy is
expected to boost the SAW-SW interaction, with direct application interest.
Therefore, it is of great interest to analyze the symmetry of this coupling in
order to understand to what extent SWs can couple to the guided and non-guided
SAWs. Here we consider the specific case of a SAF with the magneto-rotation
and the magneto-elastic interactions being included on an equal footing.

We first study the case of an elastically isotropic non-piezoelectric medium
mimicking LiNbO3 (”iso” model). We then compare it to the Rayleigh wave
propagating along X${}_{\textrm{LNO}}$ crystalline direction of a semi-
infinite LiNbO3 substrate with a Z${}_{\textrm{LNO}}$-oriented surface
(”Z-cut” model). Finally, we will discuss the additional features that appear
when the SAWs are guided in a thin film of Z-LNO grown on C-sapphire (”guided”
model).

###  IV.1 Rayleigh wave for an elastically isotropic substrate

Following Ref. [35], the velocity $V$ of Rayleigh-type surface acoustic wave
in an elastically isotropic medium obeys:

$\left(1-\frac{\rho
V^{2}}{C_{44}}\right)\left(1-\frac{C_{12}^{2}}{C_{11}^{2}}-\frac{\rho{V}^{2}}{C_{11}}\right)^{2}+\frac{\rho^{2}V^{4}\left(1-\frac{\rho
V^{2}}{C_{11}}\right)}{C_{11}^{2}}=0$ (13)

where $\rho$ is the mass density, $C_{11}^{\textrm{iso}}$ the modulus of axial
compression , $C_{44}^{\textrm{iso}}$ is the shear modulus , and
$C_{12}^{\textrm{iso}}$ is the ratio of longitudinal stress $\sigma_{11}$ to
transverse compression $\epsilon_{22}$, which reduces to
$C_{12}^{\textrm{iso}}=C_{11}^{\textrm{iso}}-2C_{44}^{\textrm{iso}}$ in case
of isotropic elastic properties. The ${}^{\textrm{iso}}$ superscripts were
omitted in the previous equation. The wave velocity is the smallest root of
Eq. 13. [35]

The strict application of Eq. 13 with the material parameters of bulk [36]
LiNbO3 would yield a SAW velocity of $3.37$ km/s. This poorly accounts for
that of a Z-cut orientation for which the reported experimental values are
3.798-3.903 km/s [37, 38]. By definition a Rayleigh wave has finite strains
only in the X and Z directions for an X-propagating wave on a Z-cut substrate.
The strains $\epsilon_{XX}$ and $\epsilon_{ZZ}$ have comparable amplitudes so
that the best approximation for the modulus of axial compression is to take
$C_{11}^{\textrm{iso}}=\left(C_{11}^{\textrm{LNO}}+C_{33}^{\textrm{LNO}}\right)/2$.
Besides, since the active shear component of a Rayleigh wave is
$\epsilon_{XZ}$, the natural choice for the shear modulus is
$C_{44}^{\textrm{iso}}=C_{66}^{\textrm{LNO}}$, and
$C_{12}^{\textrm{iso}}=C_{13}^{\textrm{LNO}}$. Making this substitution and
applying Eq. 13, one gets a Rayleigh mode velocity of 3.629 km/s, now in
better agreement with experimental reports for the case of Z-cut LNO.

Figure 3: Depth profile of the main strain components and lattice rotation
component for a Rayleigh surface acoustic wave traveling on semi-infinite
substrate, elastically isotropic (plain and dashed lines) or along the X
crystalline direction of a Z-LNO substrate (dotted lines). (a) Terms that are
real used as phase reference. (b) Terms that are imaginary (phase in
quadrature). The inset shows the strain tensor in the elastically isotropic
case. Additional non-vanishing terms arise for the Z-LNO case, as plotted in
Fig. 4.

The thickness profile of the Rayleigh wave is defined by dimensionless
penetration coefficients $q_{1}$ and $q_{2}$ [2]:

$q_{1}=-\sqrt{1-\frac{\rho
V_{r}^{2}}{C_{11}}}~{}\textrm{and}~{}q_{2}=-\sqrt{1-\frac{\rho
V_{r}^{2}}{C_{44}}}.$ (14)

The thickness profile of the lattice displacement $\vec{\tilde{u}}(X,Z,t)$ is
the sum of two partial waves:

$\left(\begin{array}[]{c}\tilde{u}_{X}=e^{-kq_{1}z}-\frac{2q_{1}q_{2}}{q_{2}^{2}+1}e^{-kq_{2}z}\\\
\tilde{u}_{Y}=0\\\
\tilde{u}_{Z}=iq_{1}\left(\frac{2}{q_{2}^{2}+1}e^{-kq_{2}z}-e^{-kq_{1}z}\right)\\\
\end{array}\right)e^{i(\text{{\hbox{\omega}}t}-kX)}$ (15)

The lattice rotation tensor
$\bar{\bar{\omega}}=\frac{1}{2}\left(\bar{\bar{\nabla}}\tilde{u}-\bar{\bar{\nabla}}\tilde{u}^{T}\right)$
and the strain tensor
$\bar{\bar{\epsilon}}=\frac{1}{2}\left(\bar{\bar{\nabla}}\tilde{u}+\bar{\bar{\nabla}}\tilde{u}^{T}\right)$
can be straightforwardly obtained as the anti-symmetric part and the symmetric
part of the displacement gradient 333$\bar{\bar{\omega}}$ and
$\bar{\bar{\epsilon}}$ were multiplied by $-i$ to set $\epsilon_{xx}$ as the
phase reference. .

The strain and the lattice rotation tensors are plotted in Fig. 3 for material
parameters meant to best mimic an isotropic equivalent of the Z-cut LiNbO3
substrate. The material parameters of bulk [36] LiNbO3 transform in
$C_{11}^{\textrm{iso}}=213$ GPa, $C_{44}^{\textrm{iso}}=72$ GPa and
$C_{12}^{\textrm{iso}}=65$ GPa, with $\rho=4628~{}\textrm{kg/m}^{3}$. Since
the displacements are strictly restricted to the $XZ$ plane and since for an
X-propagating wave we have $\frac{\partial}{\partial y}=0$, all terms
involving $Y$ cancel within $\bar{\bar{\epsilon}}$ and $\bar{\bar{\omega}}$.

The non-vanishing terms and their phase relations are
$\epsilon_{XX}\in\mathbb{R}$, $\epsilon_{ZZ}\in\mathbb{R}$, $\epsilon_{XZ}\in
i\mathbb{R}^{-}$ and $\omega_{XZ}\in i\mathbb{R}^{+}$. The shear strain
$\epsilon_{XZ}$ exactly vanishes at the free surface. Note also that the
lattice rotation $\omega_{XZ}$ at the surface is greater than all the strain
components, with foreseen implications for the amplitude of the rolling fields
(Eq. 6) and the tickle fields (Eq. 5).

###  IV.2 Rayleigh wave for X-propagation and Z-cut LNO

It is interesting to analyze to what extent the above-studied Rayleigh SAW of
an hypothetically isotropic substrate differs from that of the real
(anisotropic and piezoelectric) material. The latter can be numerically
calculated using the method described in ref. [34], derived from the model of
ref. [40]. This method solves for the resonant lattice displacements and the
electrical potential across the depth of a multilayer, taking into account the
elastodynamic equation, the Poisson equation and the constitutive relations of
piezoelectricity (see [34]), all using the material properties [36] of bulk
LNO.

For X-propagating SAWs on Z-LNO, the lowest frequency mode is calculated to
have a velocity of 3.787 km/s, in agreement with experimental findings [37,
38]. A finite displacement $u_{Y}$ appears as a consequence of the anisotropy
of the rigidity tensor of LNO. However the wave polarization still lies
essentially in the $XZ$ plane, and therefore it will be referred to as a
Rayleigh mode. The depth profiles of its strain tensor and its lattice
rotation tensor are displayed in Figs. 3 (pure Rayleigh terms) and 4 (other
terms). A comparison with the elastically isotropic case (also plotted in Fig.
3) indicates that the large elastic deformations are still the pure Rayleigh
terms $\epsilon_{XX}$, $\epsilon_{ZZ}$ and $\epsilon_{XZ}$, and that they are
still smaller than the lattice rotation main term $\omega_{XZ}$. Actually, if
looking at these sole four terms (Fig. 3), it would be difficult to find major
differences between the Rayleigh waves in real Z-LNO and in its elastically
isotropic imitation. In the real materials $\epsilon_{YZ}$, $\epsilon_{XY}$,
$\omega_{XY}$ and $\omega_{YZ}$ become finite although still remaining small
444A minor difference is that thanks to piezoelectricity, the shear strain
$\epsilon_{XZ}$ does no longer strictly vanish at the free surface. compared
to that of the canonical Rayleigh wave [see the vertical scale in Fig. 4]. We
will discuss further below that these non-vanishing characters actually change
the angular dependence of the rolling fields (Eq. 6) and of the tickle fields
(Eq. 5), with consequences for the energy transfer between the SAW and the
SWs.

Figure 4: Depth profile of the strain components and lattice rotation
components that become non-vanishing when passing from an elastically
isotropic substrate to a Z-LNO substrate for a SAW traveling along the X
crystalline direction.

###  IV.3 Rayleigh SAW guided in Z-LNO/sapphire system

Figure 5: Depth profile of the non-vanishing strain components and lattice
rotation components for the lowest order surface acoustic wave traveling in a
Z-LNO film (150 nm) on a C-sapphire substrate for
$k=7.5~{}\textrm{rad}/\mu\textrm{m}$ (equivalently 5.88 GHz). (a-b) Terms that
would already exist in an elastically isotropic, non-piezoelectric semi-
infinite system like that of Fig. 3. (c) General form of the strain and
lattice rotation tensors, and depth profile of the components that are
specific to this guided SAW.

Additional features emerge when considering guided piezoelectric thin films.
We consider a 150-nm thick Z-LiNbO3 film grown on a C-sapphire (Al2O3)
substrate. The dispersion and elastic tensors of the Rayleigh SAW can again be
numerically calculated using the model of [40, 34] and the elastic properties
of sapphire [42]. For a SAW frequency of 5.88 GHz, with corresponding
wavelength of 833 nm, the guidance within the 150-nm thick LNO increases the
wave phase velocity to 4.899 km/s. In addition to the acceleration effect, the
guidance distorts the depth profiles (Fig. 5) of the elastic deformations and
steers it towards net localization within the LiNbO3 film. In addition, the
mismatch of acoustic impedance between Al2O3 and LiNbO3 induces some waviness
within the depth profile of the elastic deformations at their interface (at
$z=0$ in Fig. 5), in a manner recalling Sezawa waves.

Just as in the pure Z-LNO case (§IV.2), there are non-zero strain/rotation
coefficients related to the finite displacements in the Y direction. These
terms can originate from the anisotropic character of Z-LNO. However a
comparison between Fig. 4 and 5(c) shows that shear strains and lattice
rotations involving the transverse direction (i.e. $Y$) are further increased
by the guidance of the wave, with a strongly enhanced $\epsilon_{YZ}$
recalling Love waves.

###  IV.4 Comparison of elastic tensors at the surface

For magnetic films much thinner than the wavelength of the SAWs, the magneto-
elastic tickle fields and magneto-rotation fields can be anticipated from the
surface strain and surface rotation. The magnetic film can also be assumed not
to exert any back-action on the SAW. These surface deformations are compared
in Table 1 for the ”iso”, ”Z-cut” and ”guided” models. It is noticeable that
the piezo-electricity, the elastic anisotropy and the guidance do not alter
significantly the deformations at the surface for the pure Rayleigh terms. The
signs, amplitude and phase relations between $\epsilon_{XX}$, $\epsilon_{ZZ}$,
$\epsilon_{XZ}$ and $\omega_{XZ}$ are very similar for the 3 material systems.
This similarity leads to conclude that Eq. 15 is a satisfactory description of
$\epsilon_{XX}$, $\epsilon_{ZZ}$ and $\omega_{XZ}$ of the acoustic waves of
Rayleigh-type for the surface of our 3 material systems. While $\epsilon_{XZ}$
stays negligible in all configurations at the surface, the additional terms
$\epsilon_{XY}$, $\epsilon_{YZ}$ and $\omega_{YZ}$ should be considered for
bulk Z-LNO and, even more, for guided configurations 555The lattice rotation
$\omega_{XY}$ is also finite at the surface, but will not lead to a torque for
the isotropic magnetic materials shall we shall consider hereafter..

Table 1: Elastic strain tensors and lattice rotation tensors ($\\{XYZ\\}$ coordinate system) at the surface of the elastically isotropic LiNbO3 (”iso” model §IV.1), of X-prop-Z-cut LNO (”Z-cut” model §IV.2) and, of X-prop-Z-LNO(150nm)/C-sapphire ”guided” model (§IV.3), for a Rayleigh surface acoustic wave. | Surface strain | Surface rotation
---|---|---
| $\bar{\bar{\epsilon}}$ | $\bar{\bar{\omega}}$
iso | $\left(\begin{array}[]{ccc}0.69&0&0i\\\ 0&0&0\\\ -0i&0&-0.23\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0&i\\\ 0&0&0\\\ -i&0&0\\\ \end{array}\right)$
Z-cut | $\left(\begin{array}[]{ccc}0.76&-0.17i&0.04i\\\ -0.17i&0&-0.05\\\ 0.04i&-0.05&-0.18\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0.17i&i\\\ -0.17i&0&-0.05\\\ -i&0.05&0\\\ \end{array}\right)$
guided | $\left(\begin{array}[]{ccc}0.85&-0.22i&0.04i\\\ -0.22i&0&-0.06\\\ 0.04i&-0.06&-0.24\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}0&0.22i&i\\\ -0.22i&0&-0.06\\\ -i&0.06&0\\\ \end{array}\right)$

###  IV.5 Amplitudes of the deformations

So far we described the polarization of the acoustical waves (Table 1). It is
also worth discussing the amplitudes of the elastic deformations that can be
effectively achieved. In practice the SAW is almost always generated/collected
by interdigital transducers (IDTs) forming a delay line. Assuming that the
IDTs are connected to an rf source by lossless cables, at most half of the
electrical power $P_{\textrm{rf}}$ can be transmitted to elastodynamic power
$P_{\textrm{SAW}}$ in the form of a SAW travelling in the propagation
direction imposed by the delay line. Neglecting resistive losses, the
microwave acoustic power below the input IDT reads:

$P_{\textrm{SAW}}=\frac{1}{2}\,P_{\textrm{rf}}\,(1-||S_{11}||^{2}),$ (16)

where $S_{11}$ is the reflection coefficient of the input IDT.

For a Rayleigh wave propagating in an isotropic medium, if the acoustical
power $P_{\textrm{ac}}$ is spread over a wavefront of acoustic aperture
$w_{\textrm{ap}}$ (for width), the so-called total displacement [44] is

$||u_{N}||=\sqrt{2P_{\textrm{SAW}}/(w_{\textrm{ap}}\rho\omega V_{r}^{2})}.$
(17)

The typical strain is $k\,.||u_{N}||$ where $k=\omega/V_{r}$ is the SAW
wavevector.

Experiments typically use an electrical power of $P_{\textrm{rf}}=1$ mW, IDTs
of reflection coefficients $||S_{11}||=0.95$ at a frequency of 600 MHz and an
acoustical aperture of $w_{\textrm{ap}}=35~{}\mu$m. These numbers lead to a
typical strain of $2.3\times 10^{-5}$. Better electrical matching can lead to
a reduced $||S_{11}||$ and therefore much larger elastic deformation. From now
on, we arbitrarily choose $\omega_{xz}=10^{-5}i$ and scale the strain
components accordingly.

##  V Surface acoustic waves dissipation to the SAF

###  V.1 Power supplied to the magnetic material

The lattice deformations create space-time-varying effective fields
$\mathcal{R}e\left((h^{rf}_{\textrm{tick}}+h^{rf}_{\textrm{roll}})e^{i(\omega
t-k_{SAW}X)}\right)$ that in turn excite magnetization precession. The SAW
power absorbed by the magnetic damping is usually calculated using the
effective field approach [8]. The time-averaged power transmitted to the
magnetic layers at position $X$ (per unit surface where the SAW is present)
is:

$\Delta
P(X)=-\frac{\mu_{0}\omega}{2}\textrm{Im}\left(\sum_{layers}t_{\ell}(h^{rf})^{\dagger}\cdot\bar{\bar{\chi}}_{\ell}\cdot
h^{rf}\right)\textrm{~{}~{}~{}in~{}W/m}^{2}$ (18)

where the dagger means transpose-conjugate, and
$h^{rf}=(h_{\textrm{tick}}^{rf}+h_{\textrm{roll}}^{rf})$ is the (complex-
valued) total effective rf field. Being a power, $\Delta P(X)$ is a positive
scalar that is independent from the chosen reference frame. Under the
assumption of Eq. 9 the time-averaged power transmitted to the magnetic layers
is spatially uniform: $\Delta P(X)=\Delta P$ 666Note that if removing the
assumption of Eq. 9 the expression of $\Delta P(X)$ would still hold provided
that the magnetic susceptibility $\bar{\bar{\chi}}$ is expressed for the
wavevector $k_{\textrm{SAW}}$. We shall calculate it in the reference frame
$xyz$ of the magnetic field. For this, we need to express the tickle field and
the rolling field associated to each of the SAWs of our 3 material systems
(Table 1) in the reference frame of the magnetic field. This is done by using
Eq. 5 and 6 and expressing the elastic tensors as:
$\bar{\bar{\epsilon}}_{xyz}=\bar{\bar{R}}\cdot\bar{\bar{\epsilon}}_{XYZ}\cdot\bar{\bar{R}}^{T}$
and
$\bar{\bar{\omega}}_{xyz}=\bar{\bar{R}}\cdot\bar{\bar{\omega}}_{XYZ}\cdot\bar{\bar{R}}^{T}$
where T is the transposition operation.

###  V.2 Magnitudes of the different channels of SAW to SW energy transfer

The time-averaged power transmitted to the magnetic layers $\Delta P$ is a
function of the experimental conditions $H_{x}$, $\omega$, $\varphi$ and of
the magnetic material properties $\gamma_{0}$, $M_{s}$, $H_{J}$, $B_{1,2}$ and
$\alpha$. The two SAF SW eigenmodes (acoustic and optical) are independent,
such that the power supplied to the magnetic material (Eq. 18) can be
expressed as the sum of the contribution of each SW mode as $\Delta P=\Delta
P_{\textrm{ac}}+\Delta P_{\textrm{opt}}$. The contribution of each SW mode
scales with the corresponding susceptibilities, which strongly depend on the
applied field $H_{x}/H_{J}$ through Eq. 12 as was discussed in point (iii) of
section III.5.

Figure 6: Contributions of the pure magneto-elastic term
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
(blue), the pure magneto-rotation term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
(green, magnified by a factor of 100) and the mixed term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
\+
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
(red) to the total power dissipation (black). A Rayleigh wave is assumed in an
elastically isotropic, non-piezoelectric semi-infinite system like that of
Fig. 3. The applied dc field is 25 mT. Panel (a) and (b) are respectively for
applied frequencies that are resonant with the SW acoustic or optical modes.

In contrast, the contributions of the tickle field and of the rolling field
are coupled. Indeed, in addition to the pure magneto-elastic term
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
and the pure magneto-rotation term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
that are unconditionally positive, one should also consider the ”mixed” terms
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
and
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
that can have arbitrary signs. Examples of the expected contributions of these
terms in the ”iso” model are plotted in Fig. 6 and 7 for applied fields
$H_{x}=0.2H_{J}$ and $H_{x}=0.8H_{J}$.

###  V.3 Reciprocity of the different channels of SAW to SW energy transfer

It is also interesting to identify which terms lead to reciprocal or non-
reciprocal transduction, as illustrated for the ”iso” model in Fig. 6 and 7.
Thanks to the assumption of Eq. 9, the non-reciprocity discussed here only
originates from an helicity mismatch. The properties of the rotation operator
ensure that:

$\left\\{\begin{array}[]{ll}\epsilon_{xx,yy,zz,xy}(\varphi)&=~{}~{}\epsilon_{xx,yy,zz,xy}(\varphi+\pi)\\\
\epsilon_{xz,yz}(\varphi)&=-\epsilon_{xz,yz}(\varphi+\pi)\\\
\omega_{xz,yz}(\varphi)&=-\omega_{xz,yz}(\varphi+\pi)\\\ \end{array}\right.$
(19)

In the ”iso” model, we have in addition $\epsilon_{xz}=0$ at the surface.
Besides, in in-plane magnetized isotropic SAF $\epsilon_{zz}$ and
$\omega_{xy}$ do not provide any magnetic torque.

As long as the magnetic layer is much thinner than the SAW peneration depth,
it will be reasonable to neglect the $\epsilon_{xz}$ torque. In the ”iso”
model the tickle field (Eq. 5) lies in the sample plane (no $z$ component) and
obeys $h^{rf}_{\textrm{tick}}(\varphi+\pi)=h^{rf}_{\textrm{tick}}(\varphi)$,
while the rolling field (Eq. 6) is along $z$ and obeys
$h^{rf}_{\textrm{roll}}(\varphi+\pi)=-h^{rf}_{\textrm{roll}}(\varphi)$. As a
result, the pure magneto-elastic term
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
and the pure magneto-rotation term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
are both reciprocal [if taken independently they would yield $\Delta
P(\varphi)=\Delta P(\varphi+\pi)$].

When the material is both subject to magneto-elastic interactions and magneto-
rotation interactions, the ”mixed” terms have to be taken into account. Indeed
the interplay between the tickle field and the rolling field leads to helicity
mismatches that differ for $\varphi$ and $\varphi+\pi$. This induces a non-
reciprocity of the excitation of spin waves, we have $\Delta
P(\varphi)\neq\Delta P(\varphi+\pi)$ even in the simplest ”iso” case. This
mechanism is analogous to that invoked in single layer ferromagnets: the
constructive/destructive interplay between on the one hand the magneto-elastic
torques arising from the longitudinal strain, and on the other hand, the shear
strain components $\epsilon_{XZ}$ [5] and the magneto-rotation torque related
to the lattice rotation tensor $\omega_{XZ}$ [6, 11].

The additional strain components present in the realistic elastic models will
understandably complicate the analysis of non-reciprocity. Let us thus first
discuss the dependence of the SAW to SW coupling on the angle between the SAW
wavevector and the applied magnetic field, first in the ”iso” case.

Figure 7: Contributions of the pure magneto-elastic term
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
(blue), the pure magneto-rotation term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
(green, magnified by a factor of 100) and the mixed term
$(h_{\textrm{roll}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{tick}}^{rf})$
\+
$(h_{\textrm{tick}}^{rf})^{\dagger}\cdot\bar{\bar{\chi}}\cdot(h_{\textrm{roll}}^{rf})$
(red) to the total power dissipation (black). A Rayleigh wave is assumed in an
elastically isotropic, non-piezoelectric semi-infinite system like that of
Fig. 3. The applied dc field is 100 mT. Panel (a) and (b) are respectively for
applied frequencies that are resonant with the SW acoustic or optical modes.

Note that in these examples, but also for the ”Z-cut” and ”guided” models (not
shown), the response due to the tickle field is by far the largest
contribution. This is related to the fact that we have chosen a strongly
magnetostrictive material, therefore $\frac{2B_{1}}{M_{s}}\gg\mu_{0}M_{s}$
while the elastic rotations $\omega_{xz}$ and the elastic strain components
($\epsilon_{xx}$, etc.) are of the same order.

##  VI SAW-SW coupling in the elastically isotropic case

Fig. 6 and 7 evidenced that the $\varphi$-dependence of $\Delta P$ at the
acoustic and optical SW resonances depends strikingly on the applied field
strength. It can feature 2-fold or 4-fold behaviors, with in addition a
variable non-reciprocity amplitude, as shown in Fig. 8 for a few
representative frequencies and for all values of the applied field leading to
non-collinear (”scissors”) states. The behavior can be cast in three regimes.

Figure 8: Dependence within the plane of the applied field of the power
absorbed (Eq. 18) by a CoFeB-based SAF in the scissors state. The energy is
supplied by the elastic deformations and lattice rotations at the surface of
an elastically isotropic equivalent of Z-LiNbO3 carrying a Rayleigh surface
acoustic wave (Eq. 15, ”iso” model). The magnetic parameters are
$\mu_{0}H_{J}=125$ mT , $\mu_{0}M_{s}=1.7$ T, $\alpha=0.01$,
$B_{1}=B_{2}=-7.6$ MJ/m3. The ”A” and ”O” symbol label the contributions of
the acoustic and optical SWs. (a) Schematics of the static magnetizations of
the SAF. Power dependances (b) for a low frequency SAW, (c) for a SAW at 6
GHz, (d) for SAW at 9.3 GHz, (e) for a SAW at 10 GHz and (f) for a SAW at 12.5
GHz. The numbers in the upper left corners of each panel indicate the maximum
absorption. (g) Angular dependence of $\Delta P$ when
$\omega_{\textrm{ac}}=\omega_{\textrm{op}}$ that highlights the
omnidirectional coupling in a SAF.

###  VI.1 Low applied frequencies: weak coupling

At low SAW frequency [Fig. 8(b)], $\Delta P$ is bound to be very low because
of the $\omega$ prefactor in Eq. 18. It is therefore a situation of little
practical interest. The resonant condition for the SW acoustic (or optical)
mode requires a low (or high) applied field, and the SAF is in a quasi-
antiparallel (q-AP) (or quasi-parallel [q-P]) state with magnetizations almost
perpendicular (or parallel) to the applied field. The angular dependence of
$\Delta P_{\textrm{op}}(\varphi)$ recalls that of a single layer [1]. The
coupling vanishes for $\varphi=n\pi/2,~{}n\in\mathbb{Z}$ and is maximal for
$\varphi=\pi/4+n\pi/2,~{}n\in\mathbb{Z}$. In contrast, $\Delta
P_{\textrm{ac}}(\varphi)$ vanishes only for $\varphi=n\pi,~{}n\in\mathbb{Z}$,
though it remains also very weak.

###  VI.2 Intermediate frequencies with resonances in scissors configurations:
inductive-like behavior

Increasing the frequency yields a much higher coupling [Fig. 8(c-e)], and the
resonances are met for ”archetypal” scissors states (i.e. very distinct from
the collinear states). A coupling to the SW acoustic mode happens whenever
$\varphi\neq n\pi,~{}n\in\mathbb{Z}$. A coupling to the SW optical mode occurs
whenever $\varphi\neq\pi/2+n\pi,~{}n\in\mathbb{Z}$. In this frequency range,
the magneto-elastic interaction acts very much as if the SAF excitation was
done using a linearly polarized induction field parallel to the SAW
wavevector. Indeed the parallel (perpendicular) pumping of a SAF using an rf
magnetic field excites only the optical (acoustic) SW mode [17].

Interestingly, there exists a peculiar combination of field (87 mT) and
frequency (9.3 GHz) for which the SW modes are both resonant [Figs. 2(a) and
8(d)], leading to no extinction of the SAW-SW coupling. Fig. 8(g) shows a cut
at 87 mT that illustrates this behavior. At this specific applied field, the
efficiencies of the excitation of the two eigenmodes have complementary
angular variations, which prohibits absorption vanishing. In addition, this
$\Delta P$ angular dependence is independent of the symmetry of the tickle
field associated to the magneto-elastic interaction, and of the rolling field
associated with the magneto-rotation interaction.

###  VI.3 High frequencies with resonances for quasi-collinear states: single-
layer-like behavior

Finally at very high SAW frequency [Fig. 8(f)], the resonant condition for the
acoustic mode requires a high applied field, so that the SAF is in a quasi
parallel state. As a result, the acoustic mode now responds in a manner
recalling that of a single layer, i.e., maximal for
$\varphi=\pi/4+n\pi/2,~{}n\in\mathbb{Z}$ and vanishing for $\varphi=0+n\pi/2$.
Note that some amplitude non-reciprocity at the acoustic SW frequency is still
visible as resulting from the mixed terms. See, for example, Fig. 8(f) where
the power absorption is weaker in the $[0,\pi]$ interval than in the
$[\pi,2\pi]$, as in the single layer case.

Conversely, the resonance with the optical SW mode requires now a low field,
hence a quasi antiparallel state. The maximum couplings are also obtained for
$\varphi=\pi/4+n\pi/2,~{}n\in\mathbb{Z}$ and the efficiency map resembles the
four-leaf clover pattern that is classically [8] encountered in single layers
subject to a pure longitudinal strain ($\epsilon_{xx}$).

##  VII Comparison of the three substrate models

We now study to what extent the anisotropy of the elastic properties and the
piezoelectric character of the substrate may change the angular dependence of
power absorption by the spin waves, and also whether this is more prominent in
the guided configuration. The angular dependence of the absorption within the
3 models of substrates are compared in Fig. 9 and 10 for two different
magnitudes of the applied field. The data were normalized to ease the
comparison.

In the case of the elastically isotropic substrate (”iso” model, black curves
in Fig. 9 and 10), the absorption by the acoustic SW mode strictly
extinguishes at $\varphi=0$. That of the optical SW mode strictly extinguishes
at $\varphi=\pi/2$. As a result, the total absorptions at both resonances were
always almost 777The non-resonant absorption due to the other SW mode,
although being very weak, does never vanish. vanishing at some angles during
angular scans of the applied field, except for the very specific applied field
leading to a doubly resonant situation with $\omega_{ac}\approx\omega_{opt}$
like in Fig. 8(d).

In stark contrast, this extinction of the absorption at some angles no longer
occurs when taking into account the elastic anisotropy and the piezolectricity
of the substrate (”Z-cut” and ”guided” models), indicated by the blue and red
curves in Fig. 9 and 10. The resulting new interaction channels couple the
magnetization eigenmodes to the elastic waves for all applied field
directions. This suppression of the extinction can be understood from the
single layer case where the field orientation dependence of the magneto-
elastic coupling is well-known [8]. Indeed in single layer films the (here
dominating) longitudinal strain $\epsilon_{xx}$ leads to a 4-fold coupling
that extinguishes at $\varphi=0+\mathbb{~{}}n\pi/2$,$~{}n\in\mathbb{Z}$.
However the anisotropic character of the substrates results in a finite in-
plane shear strain $\epsilon_{xy}$ (see Table 1) known to lead to a 4-fold
efficiency that is maximum at $n\pi/2$, precisely when the longitudinal strain
looses efficiency, thereby suppressing any possibility of angular extinction,
as discussed in Ref. [9]. Besides, the piezoelectricity present in the ”Z-cut”
and the ”guided” models allows for finite out-of plane shear strains
$\epsilon_{xz}$ and $\epsilon_{yz}$ at the free surface. In single layer
films, these out-of-plane shear strains lead to 2-fold maximum efficiencies at
$n\pi$ and $\pi/2+n\pi$, respectively.

Figure 9: Dependence of the power absorbed with the angle between the magnetic
field and SAW wavevector for the three studied substrates (elastically
isotropic equivalent of Z-LiNbO3 [”iso” model], X-propagating Z-cut LiNbO3
[”Z-cut” model] and guided LiNbO3/sapphire [”guided” model]. The CoFeB-based
SAF is in the scissors state. The applied dc field is 25 mT. Panel (a) and (b)
are respectively for applied frequencies that are resonant with the SW
acoustic or optical mode. The data have been normalized by the maximal
absorption.  Figure 10: Dependence of the power absorbed with the angle
between the magnetic field and SAW wavevector for the three studied substrates
(elastically isotropic equivalent of Z-LiNbO3 [”iso” model], X-propagating
Z-cut LiNbO3 [”Z-cut” model] and guided LiNbO3/sapphire [”guided” model]. The
CoFeB-based SAF is in the scissors state. The applied dc field is 100 mT.
Panel (a) and (b) are respectively for applied frequencies that are resonant
with the SW acoustic or optical mode. The data have been normalized by the
maximal absorption.

##  VIII Summary and conclusion

We have analysed how surface acoustic waves travelling in a substrate transfer
energy irreversibly to a SAF cap. Our study was conducted under several
simplifying assumptions. (i) We considered that the SAF cap does not perturb
the elastic wave traveling in the underlying material. In other words, we
disregarded any elastic back-action and any magneto-elastic back-action. (ii)
In addition, we have considered that the spin waves have a vanishing group
velocity and that their susceptibility at finite wavevector does not differ
from the uniform one.

We obtained a compact form for the layer-resolved susceptibility tensor. The
expression separates the contributions of the SW acoustic eigenmode and of the
SW optical eigenmode. The eigenmodes react to the tickle field associated to
the magneto-elastic interaction, and to the rolling field associated with the
magneto-rotation interaction.

The model of a Rayleigh surface acoustic wave traveling in a elastically
isotropic substrate is first recalled. Under these assumptions, the surface
strain has only two non-vanishing components, the longitudinal strain and the
(here magnetically inactive) out-of-plane strain. They are smaller but of the
same order as the lattice rotations at the surface. We then analysed the
Rayleigh surface acoustic wave traveling in a realistic material
(X-propagating, Z-cut LiNbO3) as well as in a similar material but in the form
of a film clamped by a sapphire substrate (guided geometry). Additional strain
components (in-plane and out-of-plane shear strains) emerge, allowing for
additional SAW-SW coupling channels. Besides, we confirm that the clamping of
the guided wave substantially accelerates the elastic wave, and changes the
thickness profile of the elastic deformations though it marginally affects the
largest strain components at the surface.

The SAW-SW coupling were evaluated by the energy supplied by the elastic waves
to the magnetic degree of freedom. The magneto-elastic effects, especially the
longitudinal compression term, dominate for our strongly magnetostrictive
CoFeB-based SAF. However the rolling fields must still be considered to
correctly account for the energy transfer and the non-reciprocity thereof.
Indeed the interplay between the rolling field and the tickle field induces a
substantial ”mixed” term in the energy supplied by the SAW to the SW, and this
term can be the main source of the amplitude non-reciprocity of the SAW-SW
transduction. We also emphasize that the non-reciprocity originated from the
SAF SW dispersion relation, as discussed in Ref. [31], is not taken into
account in the present work.

The dependence of the energy transfer versus the angle between the applied
magnetic field and the wavevector of the SAW was finally studied for an
elastically isotropic substrate. It appears that at low and high applied
fields, when the SAF is in close-to-collinear magnetic configurations, the
angular dependence of the energy transfer can be understood by analogy with
the Dreher-Weiler model [8] of a single magnetic layer subjected to a purely
longitudinal strain. Conversely in scissors state, the behavior bears some
similarity with the one that would be induced by an rf magnetic field oriented
along the propagation direction of the SAW.

An important outcome of this study is the symmetry analysis of the SAW-SW
coupling. The symmetries of the two SW modes of SAF are complementary, giving
rise to a large coupling whatever the field direction. This is specially
interesting for applications as it lifts geometry limitations. In addition, a
slightly more complex behavior is obtained when allowing for elastic
anisotropy and by taking into account the piezoelectricity of the substrate,
in contrast to the isotropic case. It results in additional tensor components.
The main feature is that the resulting symmetry of the magneto-elastic and
magneto-rotation driving fields leads to new interaction channels allowing
omnidirectional coupling.

###### Acknowledgements.

This work was supported by public grants overseen by the French National
Research Agency (ANR) as part of the “Investissements d’Avenir” and France
2030 programs (Labex NanoSaclay, reference: ANR-10-LABX-0035, project SPICY.
Also: PEPR SPIN, references: ANR 22 EXSP 0008 and ANR 22 EXSP 0004). R. L. S,
L. L. S, and F. M acknowledge the French National Research Agency (ANR) under
Contract No. ANR-20-CE24-0025 (MAXSAW).

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# Semantic-WER: A Unified Metric for the Evaluation of ASR Transcript for End
Usability

###### Abstract

Recent advances in supervised, semi-supervised and self-supervised deep
learning algorithms have shown significant improvement in the performance of
automatic speech recognition (ASR) systems. The state-of-the-art systems have
achieved a word error rate (WER) less than 5%. However, in the past,
researchers have argued the non-suitability of the WER metric for the
evaluation of ASR systems for downstream tasks such as spoken language
understanding (SLU) and information retrieval. The reason is that the WER
works at the surface level and does not include any syntactic and semantic
knowledge. The current work proposes Semantic-WER, a metric to evaluate the
ASR transcripts for downstream applications in general. The Semantic-WER can
be easily customized for any downstream task.

Index Terms—  speech recognition, word error rate, semantic-wer

##  1 Introduction

Speech recognition systems back in early 2000 were mainly HMM-based models
[1]. In the last two decades, the entire landscape has changed for machine
learning and deep learning in general and speech recognition in particular. We
now have access to thousands of hours of annotated speech databases in
multiple languages [2]. There are many open-source toolkits available for
developing an ASR system [3, 4, 5, 6, 7, 8]. The popular architectures for
getting good performance are DNN-HMM [9], LSTM-RNN [10], time-delay neural
network [11] and CNN [12]. Nowadays, most of the end-to-end ASR systems use
the popular architecture called transformer [13, 8]. Despite such a giant
leap, the means of evaluating the quality of a speech recognition system has
remained mostly unchanged. WER is still the de facto standard metric for ASR
system assessment. It is calculated by the total error count normalized by the
reference length (Nr), as shown in equation 1. The total error count (i.e.,
the sum of substitutions(S), insertions(I), and deletions(D)) is computed by
performing the Levenstein alignment for reference and hypothesis word
sequences.

$WER=\frac{S+D+I}{N_{r}}$ (1)

WER is a direct and objective measure for evaluating the quality of ASR
transcripts. However, there are certain limitations of WER and its application
to end usability [14, 15, 16, 17, 18]. The main two limitations are stated
below.

  * •

The numerator in WER is not bounded by the length of reference because of the
inclusion of insertions in the total number of edits. Therefore, the
normalization by Nr is not bounded to [0, 1].

  * •

Both content words and function words are equally important, not ideal for
most downstream tasks.

The applicability of WER for downstream uses may not be straightforward. The
following examples further demonstrate the fuzziness of WER for its relevance
to the downstream tasks.

  * •

Ref: f*ck you

  * •

Hyp1: thank you

  * •

Hyp2: okay

Hyp1 and Hyp2 are hypotheses from two ASR models in the above example, and Ref
denotes the reference transcript. The meaning of Hyp1 and Hyp2 utterly
opposite to the meaning of the Ref. However, the WER fails to capture these
semantic differences and assigns a score of 0.5 and 1.0 to Hyp1 and Hyp2,
respectively. It implies that low WER may not be a good indicator for better
sentiment analysis accuracy.

  * •

Ref: My name is harvey spelled as h. a. r. v. e. y.

  * •

Hyp1: My name is hurdy spelled as empty

  * •

Hyp2: My name is hurdy spelled as age a. r. v. e. y.

The ’empty’ word in Hyp1 denotes the blank or no output. Both the hypotheses
cannot capture the name accurately. However, the Hyp2 is better in capturing
the spelled out for the proper noun. Considering each spelled letter as a word
results in a WER of 0.58 and 0.16 for Hyp1 and Hyp2, respectively. However,
one can argue that the spelled letters should combine to form a single word
before WER. In that case, the WER for Hyp1 and Hyp2 would be 0.28 and 0.37.
Such inconsistent scoring is far from reliable for spoken language
understanding [16].

The motivation for the present work is stated below.

  * •

SlotWER is computed alongside the WER for the evaluation of SLU systems [19].
However, we need a unified evaluation framework that is direct and objective
and can be used for accessing the quality of the ASR transcript for end
usability.

  * •

In the conversational domain, a single word or phrase may determine the
overall sentiment of a transcript; therefore, such words should have higher
weight while scoring. The present work decides weight by the semantic distance
and the effect of a miss of an entity on the entire utterance.

  * •

The weight can also be configured for particular words or phrases where
semantic distance is not effective in general.

  * •

Many organizations using ASR transcripts have access to customer relationship
management (CRM) data for lookup or post-processing. In such cases, we should
allow a certain level of discounting for spelled-out entities. In other words,
if the ASR transcript misses one or two-character, e.g., ”h a r v e y” is
transcribed as ”h r v e y”, then it should not be penalized much.

  * •

Finally, the metric must be bounded in the range of [0, 1].

In this work, we propose an alternative evaluation metric called Semantic-WER,
which leverages the benefits of WER and augments it with the semantic weight
of a word according to its importance to the end usability as described above.
The user can easily customize the proposed system according to the downstream
tasks at hand.

The remainder of this paper is organized as follows. Section 2 describes the
related works. Section 3 describes the Semantic-WER, and its effectiveness for
the downstream tasks in general. In Section 4 we validate the proposed metric
by finding it’s correlation with the human errors. The conclusion and
limitation are described in Section 5.

##  2 Related Works

A weighted WER is proposed in [20, 21] to avoid the bias in calculating the
total error due to dynamic programming alignment. The bias is avoided by
reducing the weight of insertion and deletion by a factor of 2. A similar
approach is followed in [22, 17] for avoiding the alignment bias. An
alternative metric called Word information lost (WIL) proposed in [16]
effectively bounds the error in the range of [0, 1] as shown in equation (2)
and equation (3). However, the normalization by both the reference length (Nr)
and hypothesis length (Nh) as shown in equation (2) can only be effective for
the cases having hypothesis length longer than the length of a reference.
Moreover, WIL does not use any syntactic or semantic knowledge-based weight to
penalize the miss of essential words. On a similar information theoretic
standpoint [17] proposes that precision and recall can be a better alternative
to WER. However, precision and recall also have the same limitations and do
not include syntactic or semantic knowledge. A work that closes the proposed
work is [23]. It presents three additional metrics for information retrieval
end usability. These metrics are named entity word error rate (ne-wer),
general IR-based, and query-word word-error-rate(qw-wer). The general IR based
metric has three components- i) stop-word-filtered word error rate (swf-wer),
ii) stemmed stop-word-filtered word error rate (sswf-wer), and iii) IR-
weighted stemmed stop-word-filtered word error rate (IRW-WER).

$WIP=\frac{H}{N_{r}}\frac{H}{N_{h}}=\frac{I(X,Y)}{H(Y)}$ (2)

$WIL=1-WIP$ (3)

##  3 Semantic-WER

Semantic-WER (SWER) is a direct and objective measure similar to WER. Unlike
WER, SWER has specific weights for substitution, deletion, and insertion. The
substitution weight (Wsub), as shown below in equation (4), has four cases.

$W_{sub}=\begin{cases}1,\text{if
r${}_{w}$}\in\text{${NE}$}\cup\text{${SENT}$}\\\ cer(r_{w},h_{w}),\text{if
r${}_{w}$}\in\text{${SE}$}\\\ 1,\text{if similarity(r${}_{w}$, h${}_{w}$) $<$
0.6 and r${}_{w}$}\notin\text{${NE}$}\cup\text{${SENT}$}\\\ 0,\text{if
similarity(r${}_{w}$, h${}_{w}$) $>$ 0.6 and
r${}_{w}$}\notin\text{${NE}$}\cup\text{${SENT}$}\\\ \end{cases}$ (4)

${NE}$ $\cup$ ${SENT}$ represents a set having named entities and sentiment
words. rw and hw denote the reference word and hypothesis word, respectively.
The first case does not differentiate between short and long utterances having
words belonging to ${NE}$ $\cup$ ${SENT}$. In other words, both long and short
utterances are penalized equally for error. $SE$ represents the spelled out
entities. It is ubiquitous in telephonic conversation to spell out the
entities to the listener. Therefore, the second case uses character error rate
(cer) for spelled out entities. Most of the organizations which analyze
telephonic conversations have access to CRM data. Therefore, one can have a
threshold to relax the number of characters’ substitution for the spelled-out
entities. Current work sets the number of character substitution thresholds to
zero, i.e., even a single character substitution counts. Unlike [23], the last
two instances use the cosine similarity score between the embedding of rw and
hw for obtaining the substitution penalty. Such similarity-based scores have
the following unique advantages.

  * •

It yields a high score for semantically similar words and therefore assigns
zero substitution penalty. e.g., rw = ”go” and hw = ”goes”.

  * •

It yields low score semantically dissimilar words and therefore assigned a
substitution penalty of one, e.g., rw = ”tortoise” and hw = ” rise”.

  * •

The above two benefits allow us to use all words without any pre-processing
(i.e., lemmatization and stop word removal) as proposed in [23].

Similar to Wsub, the deletion weight (Wdel) has three cases, as shown in
equation (5). The first two cases are the same as of Wsub.

$W_{del}=\begin{cases}1,\text{if
r${}_{w}$}\in\text{${NE}$}\cup\text{${SENT}$}\\\ cer(r_{w},h_{w}),\text{if
r${}_{w}$}\in\text{${SE}$}\\\ \frac{1}{N_{r}},\text{otherwise}\end{cases}$ (5)

Unlike [17, 20], the third case assigns a weight of $\frac{1}{N_{r}}$ to
deletion for rest of the words. Finally, the insertion weight Wins is equal to
the probability mass distributed over the hypothesis, i.e., $\frac{1}{N_{h}}$
as shown in equation (6).

$W_{ins}=\frac{1}{N_{h}}$ (6)

The normalization by hypothesis length (Nh) rather than Nr is because the
insertions take place in the hypothesis. Following are the reasons for
assigning such a small weight to the insertion penalty.

  * •

The objective of an ASR system is to transcribe the input speech. However, in
most practical cases, it has been observed that the insertions are due to not
having a better voice activity detection (VAD) or speech activity detection
(SAD) system. Therefore, it is better not to penalize the ASR system for the
VAD errors.

  * •

Some overlapped or non-comprehensible speech part which is not transcribed in
the reference; however, the ASR system transcribes it.

  * •

It is also likely that the human transcribers have missed some words in the
reference which ASR systems are correctly capturing.

  * •

Given that the acoustic models are trained on the thousands of hours of data,
the insertions may be due to not fusing the in-domain text in the language
model.

An intermediate score called scorea aggregates the substitution, deletion, and
insertion errors as shown below in equation (7).

$score_{a}=\sum\limits_{S,D\notin NE\cup SENT}W_{sub}*S+W_{del}*D+W_{ins}*I\\\
$ (7)

###  3.1 Importance Weight and Distributed Weight

Entities and sentiment words can be pretty important as well as sensitive in a
transcription. The importance weight (IW) is based on the assumption that a
wrong entity can maximally affect all other entities in a sentence. In
contrast, a wrong sentiment word can change the semantics of the entire
sentence or a phrase. Therefore, the IW lies between 1 to the maximum number
of entities present in an utterance for a wrong entity. And the IW of a wrong
sentiment word lies between 1 to the maximum number of words in a phrase. The
following equations (9) and (10) compute the distributed weight (DW) and the
SWER respectively. #E(NE∪SENT) denotes the number of wrong entities and
sentiment words in an utterance.

$accuracy=1-score_{a}$ (8) $DW=\frac{accuracy}{N_{r}-\\#E_{(NE\cup SENT)}}$
(9)

$SWER=score_{a}+DW\times IW$ (10)

Three example sentences having the same WER and different SWER demonstrate the
effectiveness of SWER, as shown below in Table 1.

Ref and Hyp | WER | SWER
---|---|---
what did you do in paris |  |
what did u do in phariz | 0.33 | 0.46
i love switzerland |  |
i love switjerlan | 0.33 | 0.66
ram loves sita |  |
ram love sita | 0.33 | 0.0
Table 1: Example sentences for comparison of WER and SWER. The value of SWER
is computed using the default importance weight i.e., IW = 1.

##  4 Validation

The CoNLL-2003 NER dataset marks the begining of a new topic by ”-DOCSTRART”.
It is used for extracting sentences from each domain to capture a better
distribution in terms of unique named entities. Only two sentences are
extracted from each domain. A total of 946 sentences extracted for the
validation purpose. These sentences are labelled for categories like person
(PER), organization (ORG), location (LOC) and miscellaneous (MISC) at word
level. A sample sentence with it’s label is shown below in Table 2.

EU | rejects | German | call | to | boycott | British | lamb | .
---|---|---|---|---|---|---|---|---
I-ORG | O | I-MISC | O | O | O | I-MISC | O | O
Table 2: Sample CoNLL-2003 sentence labeled for all named entities.

The detailed description of the dataset-NER can be found in the Table 3. All
946 sentences are synthesized at a sampling rate of 16KHz using an end-to-end
Text-to-Speech Synthesizer [5].

# Unique Words | 2500
---|---
# I-ORG | 529
# I-LOC | 221
# I-PER | 456
# I-MISC | 637
Table 3: The distribution of category wise named entities in the dataset-NER

###  4.1 SUBJECTIVE SCORING

The sentences from the dataset-NER are categorized into three categories,
namely Cat-I, Cat-II, and Cat-III, based on the number of entities. Cat-I,
Cat-II, and Cat-III consist of sentences having one, two, and three entities
respectively. Cat-I has 354, Cat-II consists of 344, and Cat-III contains 146
sentences. A total of sixty sentences are selected by randomly extracting
twenty sentences from each category. A perceptual experiment was designed
using the Praat MFC experiment. Each audio stimuli has several text responses.
The number of responses equals two times the number of entities (i.e., one for
substitution and one for deletion) present in the reference text as shown
above in Fig. 1. Since it is pretty unusual to consider the insertion of an
entity or sentiment word, we deliberately kept the insertion error out of the
experiment. A total of 10 participants rated each response on a scale of 1
(poor) to 5 (excellent). The subjective accuracy of each response is
calculated by averaging the responses for each stimulus and then dividing it
by the highest score i.e., 5. Finally, the human assigned WER (HWER) is
calculated as one minus the subjective accuracy.

Fig. 1: Praat MFC Experiment for Subjective Scoring for reference ”ram goes to
paris”. The top two response stumli are hypotheses post deletion and bottom
two are for substitution error. ”empty” word is used to show the deletion of a
word. Fig. 2: The distribution of errors (WER, SWER and HWER) per category.
Cat-I, Cat-II and Cat-III denote the sentences in which one, two and three
named entities are substituted/deleted respectively.

###  4.2 Correlation of WER, SWER and HWER

The difference between WER and HWER is significantly higher than between SWER
and HWER, as shown in Fig 2. The SWER_IW=1, SWER_IW=2, SWER_IW=3 represent the
SWER with importance weight of 1, 2, and 3, respectively. The correlation of
WER and HWER is less than 0.7 across categories. However, the correlation of
WER and SWER is mainly in the range of 0.7 to .85. The SWER_IW=1 has the
highest correlation with HWER followed by SWER_IW2. It implies that the
default importance weight is good for sentences with one entity and the
importance weight of two for the rest of sentences. The distribution of
correlation of errors with HWER across categories is shown in Fig 3.

Fig. 3: Correlation of WER, SWER_IW=1, SWER_IW=2 and SWER_IW=3 with HWER for
all three categories.

##  5 CONCLUSION and LIMITATIONS

Entities, in general, are challenging to recognize compared to non-entity
words. Also, some entities are harder to recognize compared to other entities.
One can find a helpful benchmark describing the accuracy of different ASR
systems for entities in [24]. Librispeech and Switchboard corpora are old
corpora and not good enough to further track the progress in the field of
speech recognition [25]. On a similar note, we advocate the urgent need for a
new metric capable of penalizing more for wrong entities than other common
non-entity words. The SWER metric is a proposal for the same and correlates
better with the subjective score (HWER) than WER. One limitation of SWER i.e.,
the reference must be tagged for the entities to be evaluated.

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