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# Unidirectional spin waves measured using propagating spin wave spectroscopy

G. Y. Thiancourt  S. M. Ngom  Université Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120, Palaiseau, France  T. Devolder
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

###### Abstract

The dispersion relation of spin waves can vary monotonously about the center
of the Brillouin zone, allowing zero-momentum wavepackets to flow
unidirectionally, which is of interest for applications. Techniques such as
propagating spin wave spectroscopy are inoperative in such cases because of
the difficulty to identify the spin wave wavevector at a particular frequency
within a spectrum. Here we present a method to analyse this case and apply it
to acoustic spin waves in a synthetic antiferromagnet in the scissors state,
in which we confirm that propagation parallel to the applied fields is
unidirectional. Interestingly, we find that the phase accumulated by the spin
waves propagating between two antenna is not proportional to the antenna
spacing. It is also a function of the two other lengths of the problem: the
antenna width and the spin wave decay length. Accounting for them is required
to avoid wavevector errors in the dispersion relations.

Thin film magnetism finds versatile applications in the radio-frequency
domain, where understanding spin wave (SW) dynamics is crucial. The emergence
of Propagating Spin Wave Spectroscopy (PSWS [1, 2, 3, 4]) has garnered
attention due to its swift, precise, and sensitive capabilities, making it a
useful method for investigating SW propagation on the micron scale [5, 6],
including the assessment of their group velocities [3, 7, 8]. While utilizing
PSWS for determining SW band structure holds promise, it necessitates the
precise disentanglement of each band’s contribution to the overall PSWS
signal[9]. Often, this entails employing semi-empirical models for data
analysis [10, 11, 12, 8], which may yield ambiguous outcomes. Achieving a
robust resolution of dispersion relations for different SW modes typically
mandates either multiple variants of a sample [8] or the external measurement
of ferromagnetic resonance [9]. Notably, the scenarios investigated using PSWS
thus far have predominantly been reciprocal or weakly non-reciprocal; the
development of precise PSWS methodologies for strongly non-reciprocal
situations remains a frontier yet to be explored.

In this study, we implement PSWS on a synthetic antiferromagnet (SAF) whose
dispersion relation $\omega(k)$ has been shown to be highly non-reciprocal
[13, 14] and even monotonic [15] across $k=0$ when the wavevector is parallel
to the applied field. Under these conditions, the group velocity remains
positive regardless of the wavevector’s sign, resulting in a unidirectional
energy flow of spin waves. This paper builds upon the PSWS analysis from ref.
16 by applying it to SAF spin waves. The methodology incorporates field-
differentiation, time-of-flight filtering and precise analysis of PSWS signal
phases. Notably, our study reveals that the existence of several
characteristic lengths –antenna width $w_{\textrm{ant}}$, antenna-to-antenna
distance $r$ and SW attenuation length $L_{\textrm{att}}$– entails that the
phase accumulated by the SW during their journey between antennas is not
directly proportional to $r$, contrary to common assumptions. The correct
analysis of the PSWS signal allows to demonstrate that the unidirectional
energy flow for acoustic SWs in SAF occurs across a broad range of applied
fields.

##  I Propagating Spin Wave Spectroscopy measurement procedure

We studied SAF films with the composition LiNbO3
(substrate)/Ta/CoFeB($t_{\textrm{mag}}~{}=~{}17$ nm)/ Ru(0.7 nm)
/CoFeB($t_{\textrm{mag}}$)/Ru /Ta (cap) (Fig 1.a, growth described in ref.
17). Properties include a CoFeB magnetization of
$\mu_{0}M_{s}=1.7~{}\textrm{T}$, and an interlayer exchange energy
$J_{\text{ex}}=-1.7~{}\textrm{mJ}/\textrm{m}^{2}$, corresponding to an
interlayer exchange field
$\mu_{0}H_{j}=-{2J}/{(M_{s}t_{\textrm{mag})}}=148~{}\textrm{mT}$. Under
applied fields in the 0-100 mT range, the linewidth of the acoustic spin wave
resonance at $k=0$ is almost constant at
$\Delta\omega_{0}/{(2\pi)}~{}=~{}350\pm 50$ MHz, which is consistent with a
Gilbert damping $\alpha=\Delta\omega_{0}/(\gamma\mu_{0}(M_{s}+H_{j}))=0.006\pm
0.001$, where $\gamma$ is the gyromagnetic ratio.

To perform electrical PSWS, the films are patterned into devices (Fig 1.a)
consisting of $20~{}\mu\textrm{m}$-wide SW conduits inductively coupled to two
identical single-wire antennas. The antenna widths are
$w_{\textrm{ant}}=1,~{}2$ and $2~{}\mu\textrm{m}$ and they are respectively
spaced at center-to-center distances $r=3.4,~{}4$ and $6~{}\mu\textrm{m}$ (Fig
1.b).

Figure 1:  Propagating spin waves spectroscopy (PSWS) setup and measurements.
(a) Schematic of the experimental setup with $\mu_{0}\vec{H}_{x}$ the applied
field and $\vec{h}_{rf}$ the exciting field of antenna $1$. (b) Scanning
electron micrograph of a device. Light gray: single-wire antennas with a width
$w_{\textrm{ant}}$. Medium gray: Synthetic antiferromagnet conduit for SWs.
(c) Forward transmission parameter $\tilde{S}_{21}$ for $\mu_{0}H_{x}=93$ mT.
(d) Field derivative of $\text{Re}[\tilde{S}_{ij}]$ versus frequency and
field.

The two antennas are connected to a two-port Vector Network Analyzer (VNA) in
order to measure the forward ($\tilde{S}_{21}$) and backward
($\tilde{S}_{12}$) transmission parameters in the presence of a static field
${H}_{x}$ applied along the SW propagation path (see Fig 1.a). The insertion
loss, i.e., $20\,\textrm{log}_{10}(||\tilde{S}_{21}||)$, is typically -24.7 dB
and -16 dB at 5 and 10 GHz respectively, both for
$\\{w_{\textrm{ant}},r\\}=\\{2~{}\mu\textrm{m},~{}4~{}\mu\textrm{m}\\}$. A
representative transmission spectrum is shown in Fig 1.c. A substantial part
of the transmission signal arises unfortunately from the antenna-to-antenna
cross-talk that is not related to SWs.

Two methods can reveal the SW contributions to the transmission parameters. A
reference signal $\tilde{S}_{ij}^{\textrm{ref}}$ can first be subtracted from
the raw data. Here we use the $H_{x}$-averaged transmission coefficients
$\tilde{S}_{ij}^{\textrm{ref}}=\langle\tilde{S}_{ij}\rangle_{H_{x}}$. The
field-dependent part of the transmission is defined as:

$\overline{S}_{ij}(H_{x},\omega)=\tilde{S}_{ij}(H_{x},\omega)-\tilde{S}_{ij}^{\textrm{ref}}(\omega).$
(1)

Since the reference signal unavoidably includes some SW contribution, Eq. 1 is
an imperfect suppression of the SW-independent background.

Alternatively, if the magnetic susceptibility varies with the applied field,
the contribution of the SWs can also be determined from the field derivative
$\frac{d\tilde{S}_{ij}}{dH_{x}}$ of the transmission signals. This
differentiation effectively enhances the SW-related oscillation in the
transmission signals, however at the expense of increased noise [compare Fig.
1.(a) and (b)]. While the optical spin wave mode of the SAF in the
$[11,15]~{}\textrm{GHz}$ interval was already perceivable in the raw data, the
field differentiation reveals in addition the acoustic SW branch, which is
attributed to the V-shaped signals at lower frequencies (Fig 1.d). The
amplitude of the acoustic branch is typically seven times lower than the
optical one, not exceeding $\textrm{Re}[\frac{d\tilde{S}}{dH_{x}}]\approx
0.002~{}\textrm{mT}^{-1}$.

The goal here is to accurately determine the SW dispersion relations
$\omega(k)$ of our SAF from PSWS data. This requires a detailed analysis of
the phase of the transmission coefficients [3, 16]; noise and background
should thus be minimized. Time-of-flight spectroscopy using time-gating of the
PSWS data [9] can be used as an supplemental processing step for that purpose.

Figure 2:  Spin wave spectra after various processing steps for
$\mu_{0}H_{x}=-50~{}\textrm{mT}$ and $r=4~{}\mu\textrm{m}$. (a) Field
derivative of $\tilde{S}_{21}$.(b) Field dependent part of $\tilde{S}_{21}$.
(c) Spinwave wavepacket in time domain. The rectangular window of the time
gating is defined at $t_{\textrm{start}}=0.025~{}$ns and ending at
$t_{\textrm{end}}=6~{}$ns. (d-e) Time gated versions of panels (a) and (b).
(f-g) Zooms of the contribution of the acoustic spin wave mode.

Fig. 2 details the time-gating procedure for a representative transmission
spectrum and its field derivative. For qualitative understanding of the group
velocities of the optical and acoustic SW branches, it is interesting to
display separately the impulse responses corresponding to the sole high
frequency part of the spectrum, and to the sole low frequency part of the
spectrum. The wavepacket of the optical SWs (blue curve in Fig. 2c) arrives at
a group delay $t_{\textrm{g}}$ of approximately 1 ns. The wavepacket of the
slower acoustic SWs (cyan curve in Fig. 2c) arrives later at a longer group
delay of circa 2 ns. A rough estimate of the group velocities is $v_{g}\approx
r/t_{\textrm{g}}$, which amount to $\approx 4~{}\textrm{km.s}^{-1}$ for the
optical SWs and to $\approx 2~{}\textrm{km.s}^{-1}$ for the acoustic SW.

A time gate from $t_{\textrm{start}}=0.25$ ns to $t_{\textrm{end}}=6$ ns is
suitable to retain the two SW wavepackets while excluding both the long-delay
part of the white noise, and the short-delay antenna-to-antenna
electromagnetic cross-talk [see Fig. 2(b)]. The back transformation to
frequency domain yields field-derivative spectra that are free of background
and less noisy [compare Fig. 2(a) and (f)]. The phase of such signals can be
accurately defined even on the low signal of the acoustic SW branch. The
background suppression is more efficient when using the field derivative
instead of Eq. 1 [compare Fig. 2(f) and (g)] which gives artifacts such as
phase distortions, as well as an oscillatory modulus that is unphysical for
single-wire antennas [12]. In what follows we combine field-differentiation
and time-gating when analysing the smallest SW signals.

##  II Model of propagating spin wave spectroscopy

The general formalism relating PSWS data to SW dispersion relations is
reviewed in ref. [16]. We use the same notations. For an ultrathin antenna in
close vicinity with the magnetic film, if the dispersion relation is
monotonous with positive slope [labeled as ”/” to mimic a linear increase of
$\omega(k)$], the experimental transmission parameter $\tilde{S}_{21}$ for
$|r|>w_{\textrm{ant}}$ has a spectral content identical to the term (Eq. 22 of
ref. [16]):

$\tilde{S}_{21f}^{yy}(\omega,r>w_{\textrm{ant}},/)~{}~{}\propto~{}\mathcal{L}(\omega)P_{1}(r,\omega)U(r,\omega)~{},$
(2)

where
$\mathcal{L}(\omega)\propto{\left({{}(\omega-{\omega_{0}})}-i{v_{g}}/{L_{\textrm{att}}}\right)^{-2}}$
is a Lorentzian centered at $\omega_{0}=\omega(k=0)$ with a full width at half
maximum $\Delta\omega_{0}={2v_{g}}/{L_{\textrm{att}}}$, which is the ratio of
the group velocity to the attenuation length111We shall neglect any variation
of the attenuation length with the wavevector..
$P_{1}(r,\omega)=e^{i\frac{r(\omega-{\omega_{0}})}{v_{g}}}e^{-\frac{r}{L_{\textrm{att}}}}$
is the propagating factor with a phase rotation $\propto r$ and an exponential
decay in space. $U(r,\omega)$ is a unidirectional term that also accounts for
the finite antenna width. For $r>w_{\textrm{ant}}$, it reads[16]:

$U(r,\omega)=-4+2\left(e^{\frac{w}{L_{\textrm{att}}}+\frac{iw(\omega-{\omega_{0}})}{v_{g}}}+e^{\frac{-iw(\omega-{\omega_{0}})}{v_{g}}-\frac{w}{L_{\textrm{att}}}}\right).$

The experimental datasets that are the most adequate for reliable analysis are
the time-gated versions of the $\frac{\partial\tilde{S}_{ij}}{\partial
H_{x}}(\omega)$ spectra. We can deduce the dispersion relation from these
spectra by relying on the following three observations.

Firstly, $\frac{\partial\tilde{S}_{21}}{\partial
H_{x}}(\omega)=\frac{d\omega}{dH_{x}}\frac{\partial\tilde{S}_{21}}{\partial\omega}$
and since $\frac{d\omega}{dH_{x}}$ is a real number, the phase of field-
derivatives and frequency-derivatives are equivalent for
$\frac{d\omega}{dH_{x}}>0$. Moreover, $\frac{d\omega}{dH_{x}}$ is almost
constant for the acoustic SW of a SAF [19]; so the modulus of field-
derivatives and of frequency-derivatives are equivalent for practical
purposes.

Second, the modulus of derivative-spectra obey 222This holds up to fourth
order in $\Delta\omega_{0}/\omega_{0}$, hence in all practical situations of
PSWS.:

$\Big{\lvert}\Big{\lvert}\frac{\partial\tilde{S}_{21f}^{yy}}{\partial
H_{x}}(\omega)\Big{\rvert}\Big{\rvert}\textrm{~{}has~{}a~{}maximum~{}at~{}}\omega=\omega_{0}$
(3)

This property is useful to determine the frequency $\omega_{0}/(2\pi)$ of the
acoustic SW at $k=0$.

Third, one can show from Eq. 2 that

$\begin{split}&\textrm{Arg}\left(\frac{\partial\tilde{S}_{21f}^{yy}}{\partial
H_{x}}(\omega)\right)+\frac{\pi}{2}+2\pi\mathbb{Z}=\\\
&k\left(r+\frac{w_{\textrm{ant}}^{2}}{6L_{\textrm{att}}}+\frac{w_{\textrm{ant}}^{2}}{6r}\right)+O(w_{\textrm{ant}}^{4}).\end{split}$
(4)

The $2\pi\mathbb{Z}$ uncertainty within Eq. 4 can be resolved by substituting
in any known point along the dispersion relation. This known point can be
simply $k=0$ and the frequency $\omega_{0}$ determined from Eq. 3.

It is worth emphasizing the Eq. 4 entails that the phase of a spectrum rotates
at a pace that is faster than simply $kr$, in contrast to the usual assumption
that the variation of is exactly equal to $kr$ [10, 11, 8, 9]). Eq. 4 means
that the effective propagation distance is not $r$ but

$r_{\textrm{eff}}=\left(r+\frac{w_{\textrm{ant}}^{2}}{6L_{\textrm{att}}}+\frac{w_{\textrm{ant}}^{2}}{6r}\right)\in[r,r+w_{\textrm{ant}}].$
(5)

In situations of interest for PSWS, we have narrow antennas satisfying
$w_{\textrm{ant}}\ll 6L_{\textrm{att}}$ such that
$r_{\textrm{eff}}\in[r,r+w_{\textrm{ant}}]$. This recalls that while the
center-to-center distance $r$ is the intuitive (hence often chosen)
propagation distance, SWs can in fact also be emitted and collected at the
outer edges of the antennas, which are separated by the distance
$r+w_{\textrm{ant}}$.

##  III Dispersion relations of acoustic spin waves

Figure 3:  (a) Phase (green) and modulus (black) of
$\frac{\partial\tilde{S}_{21}}{\partial H_{\textrm{x}}}(f)$ for
$\\{r=3.4~{}\mu\textrm{m},~{}w_{\textrm{ant}}=1~{}\mu\textrm{m}\\}$ at 50 mT.
The SNR in the shaded areas prevent a reliable extraction of the phase. (b)
Field-dependence of the frequency leading to a maximum of
$\Big{\lvert}\Big{\lvert}\frac{\partial\tilde{S}_{21}}{\partial
H_{x}}\Big{\rvert}\Big{|}$ as function of the field for different propagation
distances ($r$). For the acoustic (optical) mode, this corresponds exactly
(approximately) to the uniform resonance frequency. The dashed lines are the
prediction of the two-macrospin model [19].

Equations 3 and 4 show that it is essential to analyse both the modulus and
the phase of field-derivative spectra in order to correctly determine the
dispersion relation. Fig. 3(a) displays a representative spectrum taken at 50
mT: The frequency of the acoustic mode at $k=0$ is easily found to be
$f_{0}=6.7$ GHz where the signal-to-noise ratio is $\textrm{SNR}\approx 10$.
In the frequency region where the modulus is above the noise, the phase is
reliably found to evolve rather monotonically. When the $\textrm{SNR}\leq 1$,
the measurement noise prevents the estimate of the phase whose variations
exhibit large fluctuations [shaded areas in Fig. 3(a)]. The phase is reliable
in a typically 2 GHz-wide frequency window.

The field dependence of the frequency at $k=0$, obtained with Eq. 3, is
displayed in Fig. 3 (b). The different propagation distances yield consistent
estimates of ${\omega_{0}}/{(2\pi)}$ with a maximum disagreement of $220$ MHz.
The frequency of the uniform acoustic SW is essentially proportional to the
field, in line with the behavior predicted by a 2-macropin model [19] (i.e.,
$\omega_{0}=\gamma\mu_{0}H_{x}\sqrt{(M_{s}+H_{j})/H_{j}}$, dashed line in Fig.
3), with an additional curvature linked to the finite exchange stiffness
within the CoFeB layers [17] and a finite gap related to their magnetic
anisotropy [21].

Quantitative estimates of $k(\omega)$ requires an estimate of the attenuation
length. We use the estimated group velocity from the impulse response of Fig.
2(b) and the definition $L_{\textrm{att}}={2v_{g}}/{\Delta\omega_{0}}$. This
yields a starting value of $L_{\textrm{att}}=11.5~{}\pm 1.5~{}\mu$m, which is
subsequently refined with further analysis. This $L_{\textrm{att}}$ results in
effective propagation distances $r_{\textrm{eff}}$ that are slightly longer
than the antenna-center-to-distances $r$ by $\\{0.06,0.2,0.17\\}~{}\mu$m for
$r=\\{3.4,4,6\\}~{}\mu$m.

The dispersion relation is found by subtracting the phase at
$f_{0}={\omega_{0}}/{(2\pi)}$ and then dividing by $r_{\textrm{eff}}$ in the
region of reliable phase extraction [e.g. in the non-shaded areas in Fig.
3(a)],

$k~{}=~{}\frac{1}{r_{\textrm{eff}}}\Big{[}\textrm{Arg}\left(\frac{\partial\tilde{S}_{21}(f)}{\partial
H_{x}}\right)-\textrm{Arg}\left(\frac{\partial\tilde{S}_{21}(f_{0})}{\partial
H_{x}}\right)\Big{]}.$ (6)

The result of the procedure is displayed in Fig. 4 for a representative
device. The unidirectional character of the spin waves is clearly apparent,
where we find a positive group velocity irrespective of the sign of the
wavevectors.

Figure 4:  Experimental dispersion relations obtained from Eq. 6 for a device
with $\\{r,w_{\textrm{ant}}\\}=\\{4~{}\mu\textrm{m},2~{}\mu\textrm{m}\\}$. The
applied field spans from 32 to 136 mT. The given velocities are obtained from
the slopes of linear fits across the whole range of wavevectors.

The procedure to extract the dispersion relations from experimental data
should obviously not depend on the propagation distance, nor on the antennas
used. Fig. 5.a, displays a comparison of $\omega(k)$ deduced from three
different devices at selected applied fields, as well as
$\omega(k)=\omega_{0}+v_{g}k$ computed from the prediction of ref. 15 which
uses a sign-dependent group velocity,

$v_{g}=\frac{\gamma_{0}M_{s}t}{2}\left(\sqrt{1-\left(\frac{H_{x}}{H_{j}}\right)^{2}}+\frac{|H_{x}|\textrm{sgn}(k)}{\sqrt{H_{j}(M_{s}+H_{j})}}\right).$
(7)

The experimental dispersion relations are found to be almost identical with
slopes differing by at most $\pm$6 %. The experimental dispersion relations
are probed in a range of wavevectors that is naturally larger for narrower
antennas (here: $w_{\textrm{ant}}=1~{}\mu\textrm{m}$). To a lesser extent, the
k-range is also affected by the signal strength: shorter propagation distances
yield transmission signals that are usable up to slightly larger wavevectors.

Figure 5:  Comparison of the dispersion relations deduced from the
measurements of devices with different antenna widths and separations. The red
line is the prediction from Eq. 7.

As pointed in ref. 15, Eq. 7 accounts quantitatively for the dispersion
relation only when the SAF ground state is well-described by two layers that
are uniformly magnetized across their thickness. For thick SAFs like the films
we studied, this translates in the condition $H_{x}\ll H_{j}$ [17], which is
satisfied only for the lowest applied fields investigated. For instance at 32
mT, Eq. 7 would predict 2.58 km/s, while a linear fit through the experimental
$\omega(k)$ yields 2.61 km/s.

The disagreement between Eq. 7 and the experimental results becomes notably
more pronounced as the applied field is increased. For the fields
$\\{50,70,90,111,133\\}$ mT highlighted in Fig. 5, the average slope of the
experimental dispersion relations are $\\{2.1,2.1,2.1,1.9,1.7\\}$ km/s, while
Eq. 7 would predict $\\{2.5,2.3,2.1,1.7,1.2\\}$ km/s, i.e., a more pronounced
reduction in the SW velocity with the applied field.

##  IV Conclusion

We have employed propagating spin wave spectroscopy to study the acoustic spin
waves of a SAF when the spin wave wavevector is parallel to the applied field.
The SW dispersion relation is found to be non-reciprocal and monotonic near
$k=0$. We show that this monotonic character holds up to large applied fields.
The standard semi-empirical procedure of analysing propagating spin wave
spectroscopy experiments cannot be applied to analyse this situation. This
stems from three difficulties: several spin wave modes coexist within each
propagation spectrum, the $k=0$ spin wave frequencies are difficult to extract
from the spectra, and the effective length of propagation of the spin waves is
ill-defined.

We developed an exact analysis technique by expanding the work reported in
ref. 16. The method relies on field-differentiation, time-of-flight filtering,
and detailed analysis of the phase of the PSWS signal. We show in particular
(Eq. 5) that since there are several characteristic lengths in the PSWS
problem (antenna width $w_{\textrm{ant}}$, spacing $r$ between the antennas
and spin attenuation length) the phase accumulated by the spin wave
propagating between the two antennas is not strictly proportional to $r$,
which contrasts with what is usually assumed. Not accounting for the
difference between the effective propagation distance and the antenna-to-
antenna spacing can yield errors in estimating the wavevectors within the
dispersion relations. The error is minimized by using as narrow as possible
antennas, which also broadens the range of accessible wavevectors. The error
is also minimized when using materials with long SW decay lengths, and devices
with long propagation distances.

###### Acknowledgements.

This work was supported by the French National Research Agency (ANR) as part
of the “Investissements d’Avenir” and France 2030 programs. This includes the
SPICY project of the Labex NanoSaclay: ANR-10-LABX-0035, the MAXSAW project
ANR-20-CE24-0025 and the PEPR SPIN projects ANR 22 EXSP 0008 and ANR 22 EXSP
0004. We acknowledge Aurélie Solignac for material growth and Joo-Von Kim and
Maryam Massouras for helpful comments.

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