# Scanning X-ray diffraction microscopy of a 6 GHz surface acoustic wave

M. Hanke<EMAIL_ADDRESS><EMAIL_ADDRESS>Paul-Drude-Institut für
Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V.,
Hausvogteiplatz 5–7, 10117 Berlin, Germany  N. Ashurbekov  Paul-Drude-Institut
für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V.,
Hausvogteiplatz 5–7, 10117 Berlin, Germany  E. Zatterin  ESRF, European
Synchrotron, 71 Ave Martyrs, F-38000 Grenoble, France  M. E. Msall  Department
of Physics and Astronomy, Bowdoin College, Brunswick, Maine 04011, USA  J.
Hellemann  Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im
Forschungsverbund Berlin e.V., Hausvogteiplatz 5–7, 10117 Berlin, Germany  P.
V. Santos  Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im
Forschungsverbund Berlin e.V., Hausvogteiplatz 5–7, 10117 Berlin, Germany  T.
U. Schulli  ESRF, European Synchrotron, 71 Ave Martyrs, F-38000 Grenoble,
France  S. Ludwig<EMAIL_ADDRESS><EMAIL_ADDRESS>Paul-Drude-
Institut für Festkörperelektronik, Leibniz-Institut im Forschungsverbund
Berlin e.V., Hausvogteiplatz 5–7, 10117 Berlin, Germany

###### Abstract

Surface acoustic waves at frequencies beyond a few GHz are promising
components for quantum technology applications. Applying scanning X-ray
diffraction microcopy we directly map the locally resolved components of the
three-dimensional strain field generated by a standing surface acoustic wave
on GaAs with wavelength $\lambda\simeq 500\,$nm corresponding to frequencies
near 6 GHz. We find that the lattice distortions perpendicular to the surface
are phase-shifted compared to those in propagation direction. Model
calculations based on Rayleigh waves confirm our measurements. Our results
represent a break through in providing a full characterization of a radio
frequency surface acoustic wave beyond plain imaging.

##  I Introduction

Synchrotron-based scanning X-ray diffraction microscopy (SXDM) is a non-
destructive tool to directly measure the local strain distribution of
nanoscale devices [1, 2, 3, 4] and to explore nanoscale strain dynamics
operando [5, 6]. In the past, it was applied to visualize and probe the time
resolved dynamics of surface acoustic waves (SAWs) at relatively long
wavelengths [7, 8, 9, 10, 11, 12].

SAWs, already studied in 1885 by Lord Rayleigh [13], are infamous for being
the most destructive seismic waves originated by earthquakes. SAWs of much
smaller wavelengths are important components of integrated circuits serving as
miniature radio-frequency (rf) filters [14, 15], sensors [16, 17], or for
micro controlling fluids [18, 19]. While these devices already in the market
are based on SAWs with wavelengths beyond a few micrometers, SAWs at sub-
micrometer wavelengths are required for new applications in quantum technology
[20], for instance, for high-quality cavities of phonons with energies beyond
$\sim 10\,\mu$eV [21]. Such localized phonons could be used as coherent on-
chip interconnects between quantum bits or as components of hybrid quantum
bits [22, 23].

Methods often used to characterize SAWs at lower frequencies include
electrical rf reflection or transmission measurements [21], laser
interferometry [24] or atomic force microscopy [25]. For high frequency SAWs
with sub-micrometer wavelengths, the electrical measurements suffer from
impedance matching problems and the typically small amplitudes of high
frequency SAWs. While laser interferometry has an insufficient lateral
resolution [24] at wavelengths well below $\lambda<1\,\mu$m, atomic force
microscopy in principle has the necessary lateral and vertical resolution
needed for imaging rf standing SAW (SSAW) modes. However, none of the above
methods can provide a full characterization of a SAW. In contrast, SXDM can
not only resolve rf SSAWs but is sensitive to all components of the three-
dimensional local strain field, the essence of a SAW.

At the ID01 beamline of the European Synchrotron Radiation Facility (ESRF)
[26, 27, 28, 4], we have performed SXDM measurements mapping the local strain
field of an SSAW realized on a GaAs wafer at a phonon wavelength of
$\lambda\simeq 500\,$nm corresponding to a frequency near $f=6\,$GHz.
Comparing our findings with the theory of Rayleigh waves [13, 23] we
demonstrate a remarkable agreement between the measured lattice distortions
and theory predictions, which includes a characteristic phase shift between
the longitudinal versus transversal SAW components.

##  II Setup and Experimental Techniques

Figure 1: (a) Optical micrograph of the SAW cavity consisting of two focusing
IDTs with a pitch of 250 nm (not resolved) and additional guard gates in the
center. The Ti/Al/Ti (10 /30 /10 nm) gates (including the IDTs), which are
fabricated by electron beam lithography, are depicted by a lighter tone on the
GaAs surface with a darker color. (b) The color scale depicts, within the
highlighted rectangular region in (a), the diffusely scattered intensity of
the symmetric GaAs(004) reflection of an SXDM measurement. Dark-blue
corresponds to the intensity of the unstrained crystal, red depicts enhanced
intensity due to strain. The data are assembled of successively measured
individual scans within apparent $5\times 5$ $\mu$m2 squares. The metal gates
including the IDTs are clearly visible. An rf-excitation of IDT2 at 5.690 GHz
causes lattice distortions in the GaAs substrate underneath (red), which
excite an SSAW inside the cavity along $\left<110\right>$ seen as a horizontal
light-blue stripe.

Our SSAWs are confined in acoustic microcavities aligned along the $\langle
110\rangle$ direction on a GaAs (001) surface. The cavities, designed for
quantum applications based on the coherent interaction between electrons and
phonons [23], are shaped by two mirror-symmetric focusing interdigital
transducers (IDTs), cf. Fig. 1(a) [24, 20]. To generate an SSAW, we modulate
the right-hand-side IDT gates with an electrical rf-signal, while the finger
gates of the left-hand-side IDT are grounded, such that it functions as a
passive Bragg mirror. In Fig. 1(b) we present a typical large-scale scanning
SXDM image of the cavity including an SSAW mode at
$f_{\text{SAW}}=5.690\,$GHz. The color scale depicts the intensity caused by
diffusive scattering collected near the GaAs(004) reflection but excluding the
intensity maximum. It expresses the local strain distribution perpendicular to
the surface within the GaAs crystal. (Here, the strain includes local tilts of
the surface.) Evident strain patterns below the evaporated metal gates allow
an easy orientation on the surface. The SSAW appears as a horizontal stripe of
enhanced strain (brighter color) centered between the IDTs. The region of
maximal strain (red color) below the driven IDT is the source of the excited
SSAW mode. Various additional speckles and stripes of high strain point to
defects near the surface.

Figure 2: Experimental setup (a) and coplanar scattering geometry (b). X-ray
focusing is achieved by a combination of a 300 $\mu$m gold Fresnel zone plate
(FZP) with an outermost zone width of 20 nm and an order sorting aperture
(OSA). Samples are mounted on a piezo scanning stage for precise three-
dimensional translations, while a hexapod on a rotation axis underneath (not
shown) provides macroscopic sample translations and three orthogonal rotation
axes ($\varphi$, $\chi$ and $\eta$). The scattered intensity is recorded by a
2D MAXIPIX detector. In order to maximize the spatial resolution we decided
for a strongly asymmetric scattering geometry with nearly normal incidence.

The SXDM setup is sketched in Fig. 2(a). It shows the $x$-$y$-$z$
piezoelectric scanning stage, which in turn is fixed to a moveable hexapod
serving for controlling the orientation and rough position of the sample
mounted to the piezo stage. Using a Fresnel zone plate, an X-ray beam with an
energy of 10.185 keV is focused onto the sample at a focal length of 20 mm, a
lateral spot size of $\simeq(30\,\text{nm})^{2}$, and a primary intensity of
$\simeq 5\times 10^{9}$ photons per second.

A Rayleigh wave is expected to cause periodic crystal displacements in its
propagation direction parallel to the surface, in our setup along
$\left<110\right>$ ($x$-axis) and perpendicular to the surface, along
$\left<001\right>$ ($z$-axis). The penetration depth of a Rayleigh wave is
comparable to its wavelength, here $\lambda\simeq 500\,$nm. In order to probe
the related modulations of the diffraction vector $\delta\mathbf{q}$ in these
two directions, we select the asymmetric (335) reflection of GaAs as
$\mathbf{q}_{335}$ is composed of similar components
$q_{x}=4.713\,\text{\AA}^{-1}$ and $q_{z}=5.557\,\text{\AA}^{-1}$. The
resulting diffraction geometry is co-planar, such that the scattering plane
spanned by the incoming and diffracted wave vectors $\textbf{k}_{\text{i,f}}$
is orthogonal to the crystal surface. The incidence of the X-rays is for
$\mathbf{q}_{335}$ almost vertical with $\alpha_{i}=85.2^{\circ}$, cf. Fig.
2(b). This enables a high lateral resolution due to a small beam footprint. It
also leads to a small exit angle of $\alpha_{f}=4.6^{\circ}$, which causes a
noticeable damping of the scattered X-rays due to their path through the
crystal almost parallel to the surface. This damping and the extinction depth
of the incoming X-rays in GaAs of $l_{\text{ext}}\simeq 4.6\,\mu$m, both,
cause a reduction of the contribution to $\delta\mathbf{q}$ as $z$ increases.

The planar screen sketched in Fig. 2(a) shows a detector frame, which displays
a reflection in reciprocal space measuring an individual spot on the surface
of the fixed sample. The reflection is distorted and broadened according to
the imaging function of the focused X-ray beam, including a minor contribution
due to the mapping of a spherical diffraction pattern onto a planar screen.
Importantly, this distortion of the X-ray beam does not affect a relative
shift of the reflection because of the $\delta\mathbf{q}$ caused by an SSAW.

To obtain the complete 3D reflection, we measure a rocking curve by varying
the Bragg angle $\Theta_{\text{B}}$, cf. Fig. 2(b), by $\pm 0.2$°, which
corresponds to $\eta$ in Fig. 2(a). Scanning the sample through the $xy$-plane
then yields a set of 3D detector images mapping the sample surface (in real
space). In summary, a complete surface scan consists of a 5D data set. It is
composed of 3D images of the time-averaged intensity of the GaAs(335)
reflection, while the 2D scan in real space provides its spatial variations.
Applying an SSAW results in time-periodic spatial variations
$\delta\mathbf{q}$ of the scattering vector, modulating the position of the
reflection. Practically, the SSAW induced $\delta\mathbf{q}$ is small compared
to the width of the undisturbed broadened reflection. Moreover, our
measurement averages in time over many SSAW periods and we expect
$\left<\delta\mathbf{q}\right>_{t}\simeq 0$ for moderate SSAW powers within a
linear response regime. To nevertheless analyze the effect of the SSAW, we fit
the scattered intensity around the GaAs(335) reciprocal lattice point (the
reflection peak) with three orthogonal Gaussians and determine their
individual standard deviations $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$.

##  III Results and Discussion

In Fig. 3(a)

Figure 3: (a) The upper plots show two orthogonal projections of an exemplary
GaAs(335) reflection. The undisturbed widths are
$(\sigma_{0})_{x}=29.1\,\mu$m-1 and $(\sigma_{0})_{z}=5.5\,\mu$m-1. The color
scales of the lower plots depict the standard deviations, $\sigma_{x}$,
$\sigma_{y}$ and $\sigma_{z}$ of Gaussians fitted to the reflection along the
respective directions, while we scanned the center of the cavity inside the
square in Fig. 1(b). In addition to the strain beneath four adjacent gates,
$\sigma_{x}$ and $\sigma_{z}$ are enhanced along vertical stripes with the
period of $\lambda/2=251.6\,$nm, which indicate an SSAW. $\sigma_{y}$ shows no
such periodic signal. (b) Contributions of the SSAW, $\delta\sigma_{x}^{2}(x)$
and $\delta\sigma_{z}^{2}(x)$, to the variances averaged along $y$ within the
regions indicated by horizontal dashed lines in panel (a). The solid lines are
the measurements, the dashed lines (blue) are model predictions for
$C=25\,$pm. The model line of $\sigma_{z}$ is reduced by one order of
magnitude. A vertical solid line facilitates the comparison of the phases
between $\delta\sigma_{x}^{2}(x)$ and $\delta\sigma_{z}^{2}(x)$.

we present the components $\sigma_{i}(x,y)$ with $i=x,y,z$ for a high
resolution SXDM scan of the center of the cavity. The average values of
$\sigma_{x}$ and $\sigma_{y}$ are similar while that of $\sigma_{z}$ is
smaller. This leads to a distortion of the unperturbed reflection which can be
seen in the two inserts at the top of Fig. 3(a). The strong signals below the
Ti/Al-gates indicate distortions of the GaAs lattice, which likely occurred
during cool-down after the evaporation of the metals at an elevated
temperature. This is an interesting observation in itself but not a topic of
this article. In addition, in the central region between the gates
(highlighted by a pair of dashed lines) periodic oscillations of
$\sigma_{x,z}(x)$ appear. They have a period of $\lambda/2=251.6\,$nm and
disappear if no SSAW is generated. As expected for Rayleigh waves no such
oscillations occur for $\sigma_{y}$.

For an analysis we model the SSAW near the center of the cavity as a
superposition of a Rayleigh wave propagating in $x$-direction and its
reflection from a Bragg mirror. We consider the non-vanishing components of a
Rayleigh wave, the longitudinal compression wave in $x$-direction and the
transversal shear wave in $z$-direction. Assuming fixed ends for both relevant
strain directions the SSAW is characterized by the displacement field

$\displaystyle\mathbf{u}(x,z,t)=-2\begin{pmatrix}\chi(z)\\\
\zeta(z)\end{pmatrix}\begin{pmatrix}\cos(\omega t)\\\ \sin(\omega
t)\end{pmatrix}\sin(kx)\,.$ (1)

In Appendix B we discuss alternative reflection conditions. The functions
$\chi(z)=2C\text{e}^{-\Omega_{\text{r}}kz}\cos(\Omega_{\text{i}}kz+\varphi)$
and
$\zeta(z)=2C|\gamma|\text{e}^{-\Omega_{\text{r}}kz}\cos(\Omega_{\text{i}}kz+\varphi+\theta)$
with material-dependent parameters
$\Omega=\Omega_{\text{r}}+\emph{i}\Omega_{\text{i}}$,
$\gamma=|\gamma|\text{e}^{-\emph{i}\theta}$, and $k=2\pi/\lambda$ describe the
penetration of the Rayleigh wave by about one wavelength into GaAs [23]. The
constant $C$ is proportional to the amplitude of the SSAW. A straightforward
calculation based on infinitesimal strain theory, cf. Appendices A and B,
yields the time averaged modulation, $\delta\bm{q}$, of the diffraction vector
caused by an individual scattering event as a function of $x$ and $z$

$\begin{split}\left<\delta
q_{x}^{2}\right>_{t}&=2q_{x}^{2}\left[\chi(z)k\,\cos(kx)+\frac{q_{x}}{q_{z}}\,\frac{\partial\chi(z)}{\partial
z}\sin(kx)\right]\\\ \left<\delta
q_{z}^{2}\right>_{t}&=2q_{z}^{2}\left[\frac{q_{z}}{q_{x}}\,\zeta(z)k\,\cos(kx)+\frac{\partial\zeta(z)}{\partial
z}\sin(kx)\right]\,.\end{split}$ (2)

In Appendix D we have numerically summed up the contributions of all scattered
photons to the actual reflection to simulate $\sigma_{x,z}(x)$. Here we
present an instructive and simplifying approximation, cf. Appendix C which
predicts identical results as our numerical simulation in the phases of
$\sigma_{x,z}$ and the ratio $\sigma_{x}/\sigma_{z}$ while the pre-factor $C$
differs by $<10\,\%$. The SSAW causes an additional local broadening of the
unperturbed reflection by a tiny amount $\sigma_{x,z}(C)/\sigma_{x,z}(0)-1\sim
10^{-5}$. Such a small perturbation motivates us to apply an incoherent model
of statistically independent contributions of individual scattering events to
the broadening of the reflection. Then, the measured variances can be
expressed as $\sigma_{i}^{2}=(\sigma_{0})_{i}^{2}+\delta\sigma_{i}^{2}$, where
$(\sigma_{0})_{i}$ indicates the standard deviations of the unperturbed beam
and $\delta\sigma_{i}$ the small contributions of the SSAW. Because of
$\delta\sigma_{i}^{2}\ll\sigma_{i}^{2}$ we can further express the sum of the
contributions of all scattering events along the path of the incident X-ray
beam as $\delta\sigma_{i}^{2}=\sum\left<\delta q_{i}^{2}\right>_{t}$.

Integration along the incident beam finally results in the model curves for
the SSAW contributions, $\delta\sigma_{x,z}^{2}$, plotted as dashed lines in
Fig. 3(b) using the amplitude $C=25\,$pm as the only fit parameter. The
vertical line near $x=-1\,\mu$m helps visualizing a phase shift between
$\delta\sigma_{x}^{2}$ and $\delta\sigma_{z}^{2}$. The corresponding
experimental results are presented as solid lines, where we subtracted
$(\sigma_{0})_{i}^{2}$ using $(\sigma_{0})_{x}=29.1\,\mu\text{m}^{-1}$ and
$(\sigma_{0})_{z}=5.5\,\mu\text{m}^{-1}$ from the undisturbed reflections. It
is remarkable that our model data confirm the measured phase shift. For the
chosen value of $C$ the model agrees with the observed $\delta\sigma_{x}$,
while the measured $\delta\sigma_{z}$ is an order of magnitude smaller
compared to the prediction. We attribute this deviation to limitations in the
quantitative analysis of the amplitude of the measured $\delta\sigma_{z}$,
related with the finite resolution of the X-ray scan, such that only few
points were detected across the narrow width $(\sigma_{0})_{z}$ of the
reflection. Note, that the accuracy of the amplitude of $\sigma_{x}$ as well
as the phases of both components are not affected by these limitations, which
could be avoided in an optimized future SXDM experiment. Given the absence of
fit parameters the agreement between measured and model data regarding phase
shift is remarkable good.

Figure 4: (a) Variance of $\delta\sigma_{z}$ along line scans as shown in Fig.
3 as a function of frequency; $\sigma(\delta\sigma_{z})$ is proportional to
the amplitude of the SAW. The solid line is the sum of Lorentzians
$\sum_{i}A_{i}\beta_{i}f_{\text{SAW}}\,/\,((f_{\text{SAW}}^{2}-f_{i}^{2})^{2}+(\beta_{i}f_{\text{SAW}})^{2})$.
(b) Fit-parameters used for the model curve in (a): quality factors
$Q_{i}=f_{i}\,/\,\beta_{i}$ (circles), amplitudes $A_{i}$ (squares), and
resonance frequencies $f_{i}-f_{1}$ (triangles) as a function of the resonance
index number which we set to $i=1$ at the lowest frequency measured cavity
mode.

So far we focused on the strain field of the SSAW. However, the SXDM
measurements are also suited for characterizing the cavity modes. Larger area
surface scans as the one shown in Fig. 1(b) give an overview of a mode’s
geometry. In Fig. 4(a) we present the frequency dependence,
$\sigma_{z}(f_{\text{SAW}})$, within $5.58\,\text{GHz}\leq f_{\text{SAW}}\leq
5.81\,$GHz determined from high resolution surface scans as in Fig. 3 in the
center of the cavity. The almost equidistant maxima correspond to the cavity
modes. The mode spacing of $\delta f\simeq 25\,$MHz corresponds to an
effective cavity length of $L_{\text{eff}}=c/2\delta f\simeq 56\,\mu$m, which
exceeds the distance between the innermost Bragg mirror fingers by $6\,\mu$m
(where we used $c=\lambda f\simeq 500\,\text{nm}\times 5.68\,\text{GHz}\simeq
2840\,$m/s). The strong decrease of the peak amplitude towards the highest
frequency of $5.81\,$GHz indicates the high-frequency end of the stop-band of
the Bragg mirrors. The solid line is a fit of a sum of Lorentz curves with the
fit-parameters plotted in Fig. 4(b). The $Q$-factor (triangles) of the cavity
reaches 750, corresponding to a phonon dephasing time of $47\,\mu$s, but
decreases towards the edge of the stop-band. The squares indicate the
approximate area below each peak, which is proportional to the energy stored
in the corresponding mode. It has a maximum around $f=5.68\,$GHz. SXDM
provides a direct method for determining the true spectrum of the an rf-SAW
cavity with a striking resolution.

##  IV Summary and Outlook

We present the first direct imaging study of the three-dimensional strain
field of a rf SSAW at a frequency near 6 GHz. Even for studies of SAWs of
comparable frequency, this goes far beyond what alternative methods can
provide, which are at most sensitive to the displacements perpendicular to the
surface. Based on our quantitative extraction of strain through spatially
resolved Bragg diffraction at the GaAs(335) reflection we are able to show a
match of the strain field generated by the SSAW as well as its transversal and
longitudinal surface displacements with the theory of Rayleigh waves. Our
measurements reveal —in agreement with the model— a relative phase shift
between the components of the reflection broadening along the propagation
direction and perpendicular to the surface. Beyond, we demonstrate the
capability of scanning X-ray diffraction microscopy to fully characterize an
SSAW and its generating cavity. Our results pave the way towards new
applications in future quantum technologies, where the complete knowledge of
the three dimensional local strain field of cavity phonons including their
phases will be crucial. An example is the coupling between a coherent phonon
field and a solid state quantum bit, which sensitively depends on the local
amplitude and phase of the strain field components.

## Acknowledgement

The authors thank C. David and F. Koch from the Paul Scherrer Institut for
providing an exceptionally good focusing optics for the X-ray beam, W. Anders
and A. Tahraoui for the fabrication of the SSAW cavities, A. Kuznetsov for
support with preliminary S-parameter measurements, S. Fölsch for an internal
review of the manuscript and C. Bäuerle for providing us with an rf generator
in Grenoble. We highly appreciate beamtime access through project MA-4449 at
the ID01 beamline of the European Synchrotron Radiation Facility. This work
was financially supported by the Deutsche Forschungsgemeinschaft through
grants SA 598/15-1 and LU 819/11-1.

## Contributions of the authors

M. Hanke, S. Ludwig and P. Santos planned the project. M.E. Msall, P. Santos,
J. Hellemann and S. Ludwig designed the rf focusing IDTs and cavities; N.
Ashurbekov and S. Ludwig prepared the samples for the SXDM measurements; N.
Ashurbekov and P. Santos characterized the electrical rf-properties of the
IDTs; E. Zatterin and T.U. Schulli performed the SXDM measurements (locally)
in online sessions in collaboration with N. Ashurbekov, M. Hanke and S. Ludwig
(online). E. Zatterin processed and analyzed the raw data. M. Hanke and S.
Ludwig developed the model, performed numerical calculations as well as
further data analysis and prepared the manuscript.

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##  Appendix A Modulation of the diffraction vector by a displacement field

In the following we derive the modulation $\delta\mathbf{q}(x,z)$ caused by an
SSAW of the scattering vector $\mathbf{q}$ of the GaAs(335) reflection used in
our SXDM experiment. As in the main article, we define the $x$-axis to be
parallel to the propagation direction [110] and the $z$-axis to be directed in
[00$\overline{1}$], which points inside the crystal perpendicular to the
(001)-surface. In this notation, the diffraction vector is

$\mathbf{q}=\begin{pmatrix}q_{x}\\\
q_{z}\end{pmatrix}=2\pi\begin{pmatrix}{1}/{a_{x}}\\\
{1}/{a_{z}}\end{pmatrix}=\frac{2\pi}{a}\begin{pmatrix}3\sqrt{2}\\\
-5\end{pmatrix}$ (3)

with the lattice constant of GaAs $a=5.6533\,$Å, while $q_{y}=0$. The
reciprocal of $\mathbf{q}$,
$\mathbf{a}=(a_{x},a_{z})^{\text{T}}=a(\sqrt{2}/6,-1/5)^{\text{T}}$, is normal
to the Bragg planes and its absolute value of $\simeq 1.7\,$Å is twice the
distance between adjacent Bragg planes.

To determine the modulation of the diffraction vector caused by the
displacement field of the SSAW we apply the concept of infinitesimal strain
theory. We are interested in the variation
$\delta\mathbf{q}=\mathbf{q^{\prime}}-\mathbf{q}$ corresponding to a variation
$\delta\mathbf{a}=\mathbf{a^{\prime}}-\mathbf{a}$. In Fig. 5

Figure 5: Vectors $\mathbf{a}$, $\mathbf{a^{\prime}}$ connecting adjacent
Bragg planes, and displacements $\mathbf{u_{1}}$ and $\mathbf{u_{2}}$ of
adjacent Bragg planes. The Bragg planes, which without strain are orthogonal
to $\mathbf{a}$, are shifted by $\mathbf{u_{1}}$ and $\mathbf{u_{2}}$ and
tilted, such that they become orthogonal to $\mathbf{a^{\prime}}$.

we sketch the connection between the displacement vector field $\mathbf{u}$ of
the SSAW and the vector $\mathbf{a}$. The displacement field shifts and tilts
the Bragg planes such that $\mathbf{a}\rightarrow\mathbf{a^{\prime}}$. From
the sketch we can immediately see that
$\delta\mathbf{u}=\mathbf{u_{2}}-\mathbf{u_{1}}=\mathbf{a^{\prime}}-\mathbf{a}=\delta\mathbf{a}$.

To determine the variation $\delta\mathbf{q}$ of the diffraction vector it is
convenient to start from its total differential

$\text{d}\mathbf{q}=\frac{\partial\mathbf{q}}{\partial
x}\text{d}x+\frac{\partial\mathbf{q}}{\partial z}\text{d}z\,.$ (4)

Assuming constant strain on the lengthscale of $|\mathbf{a}|$ (infinitesimal
strain theory), we can replace $dx\rightarrow a_{x}$ and $dz\rightarrow a_{z}$
and write

$\delta\mathbf{q}=\frac{\partial\mathbf{q}}{\partial
x}a_{x}+\frac{\partial\mathbf{q}}{\partial z}a_{z}\,.$ (5)

Inserting the components $q_{x,z}$ from Eq. (3) while using $\partial
u_{i}/\partial j=\partial a_{i}/\partial j$ (because
$\delta\mathbf{u}=\delta\mathbf{a}$) we find

$\displaystyle\delta q_{x}$
$\displaystyle=-q_{x}\left(u_{xx}+\frac{q_{x}}{q_{z}}u_{xz}\right)$
$\displaystyle\delta q_{z}$
$\displaystyle=-q_{z}\left(u_{zz}+\frac{q_{z}}{q_{x}}u_{zx}\right)\,,$ (6)

where we introduced the notation $u_{ij}=\partial u_{i}/\partial j$.

##  Appendix B Standing Rayleigh waves on a GaAs wafer

SAWs on a homogeneous crystal are Rayleigh waves which can be described as a
combination of a compression wave in the propagation direction and a shear
wave perpendicular to the surface. The corresponding displacements of a
Rayleigh wave can be written as

$\mathbf{u}=\begin{pmatrix}u_{x}\\\
u_{z}\end{pmatrix}=\begin{pmatrix}\phantom{-}\chi(z)\sin(\omega t-kx)\\\
-\zeta(z)\cos(\omega t-kx)\end{pmatrix}\,.$ (7)

For GaAs the functions $\chi(z)$ and $\zeta(z)$, which describe the
penetration of the SAW into the crystal, are [23]

$\begin{split}\chi&=2C\text{e}^{-\Omega_{\text{r}}kz}\,\cos\left(\Omega_{\text{i}}kz+\varphi\right)\\\
\zeta&=2C|\gamma|\text{e}^{-\Omega_{\text{r}}kz}\,\cos\left(\Omega_{\text{i}}kz+\varphi+\theta\right)\,\end{split}$
(8)

with $\Omega=\Omega_{\text{r}}+i\Omega_{\text{i}}\simeq 0.5+0.48i$,
$\gamma=|\gamma|\text{e}^{-i\theta}\simeq 1.345\text{e}^{-4.182i}=-0.68+1.16i$
and $\varphi\simeq 1.05$. Directly at the surface $\chi(z)$ and $\zeta(z)$ are
both positive, which results in a retrograde elliptical motion of surface
elements for a propagating Rayleigh wave.

A standing wave can be described as the superposition of a left-moving and a
right-moving wave of equal amplitudes and frequencies, where their phase
differences depend on the reflection conditions at the ends of the cavity.
Since the SAW penetrates into the Bragg mirror, it is convenient to introduce
an effective cavity length $L_{\text{eff}}$, which exceeds the distance
between the innermost mirror gates $L$. In the main article, we determined
$L_{\text{eff}}=56\,\mu$m from the frequency separation of the modes, which
corresponds to a penetration depth of $\delta
L\equiv(L_{\text{eff}}-L)/2\simeq 6\lambda$.

Our cavity is mirror symmetric, cf. Fig. 1(a). As a consequence, the standing
wave modes of the cavity alternate in being mirror versus point symmetric
(corresponding to even versus odd parity) in respect to the center of the
cavity. Further, we can describe the reflection conditions in terms of open or
fixed ends. For $\lambda\ll L_{\text{eff}}$, as is the case for our cavity,
the mode number $2L_{\text{eff}}/\lambda$ is large and, hence, the choice of
open versus fixed ends is irrelevant for the observed standing wave modes.
However, the phase difference between the two SSAW induced strain components,
e.g., $\delta\sigma_{x}$ versus $\delta\sigma_{z}$ depends on whether the two
components $u_{x}$ and $u_{z}$ have the same or opposite parity. (In respect
to the center of a mirror symmetric cavity, $u_{x}$ and $u_{z}$ have identical
parity for equal reflection conditions or opposite parity, if $u_{x}$ and
$u_{z}$ have opposite reflection conditions.) This leaves us with two choices,
which result in qualitatively different standing Rayleigh waves: Either
$u_{x}$ and $u_{z}$ have equal reflection conditions (both ends fixed or both
ends open) or $u_{x}$ and $u_{z}$ have opposite reflection conditions ($u_{x}$
has fixed ends and $u_{z}$ open ends or vice versa).

###  B.1 SSAW with fixed ends for $u_{x}$ and for $u_{z}$

In the main article, we considered the case of equal reflection conditions,
say fixed ends for both SAW components $u_{x}$ and $u_{z}$. The corresponding
standing wave has the form

$\displaystyle\mathbf{u}=-2\begin{pmatrix}\chi(z)\\\
\zeta(z)\end{pmatrix}\begin{pmatrix}\cos(\omega t)\\\ \sin(\omega
t)\end{pmatrix}\sin(kx)\,.$ (9)

The strain field of the SSAW for fixed ends only described by the partial
derivatives of the displacements is

$\displaystyle u_{xx}$ $\displaystyle=-2\chi$ $\displaystyle k\cos(\omega
t)\cos(kx)$ $\displaystyle u_{xz}$
$\displaystyle=-2\frac{\partial\chi}{\partial z}$ $\displaystyle\cos(\omega
t)\sin(kx)$ $\displaystyle u_{zx}$ $\displaystyle=-2\zeta$ $\displaystyle
k\sin(\omega t)\cos(kx)$ $\displaystyle u_{zz}$
$\displaystyle=-2\frac{\partial\zeta}{\partial z}$ $\displaystyle\sin(\omega
t)\sin(kx).$ (10)

Inserting Eq. (B.1) into Eq. (A), we find the local variations of the
diffraction vector caused by the SSAW

$\begin{split}\delta
q_{x}&=2q_{x}\left[\chi(z)k\,\cos(kx)+\frac{q_{x}}{q_{z}}\frac{\partial\chi(z)}{\partial
z}\sin(kx)\right]\cos(\omega t)\\\ \delta
q_{z}&=2q_{z}\left[\frac{q_{z}}{q_{x}}\zeta(z)k\,\cos(kx)+\frac{\partial\zeta(z)}{\partial
z}\sin(kx)\right]\sin(\omega t)\,.\end{split}$ (11)

Taking the time average of the square of Eq. (11) yields Eq. (2) in the main
article. By inserting Eq. (8) these variations can be written in the explicit
form

$\begin{split}\delta
q_{x}&=2q_{x}k\chi(z)\left[\phantom{\frac{q_{z}}{q_{x}}}\cos(kx)-\,\frac{q_{x}}{q_{z}}\,\Big{(}\Omega_{\text{r}}+\Omega_{\text{i}}\tan(\Omega_{\text{i}}kz+\phi)\Big{)}\;\sin(kx)\right]\cos(\omega
t)\\\ \delta
q_{z}&=2q_{z}k\zeta(z)\left[\frac{q_{z}}{q_{x}}\cos(kx)-\Big{(}\Omega_{\text{r}}+\Omega_{\text{i}}\tan(\Omega_{\text{i}}kz+\phi+\theta)\Big{)}\sin(kx)\right]\sin(\omega
t)\,.\end{split}$ (12)

###  B.2 SSAW with fixed ends for $u_{x}$ and open ends for $u_{z}$

Next we perform the same calculation as in Sec. B.1 but assuming a standing
Rayleigh wave with fixed ends for $u_{x}$ and open ends for $u_{z}$. It has
the form

$\displaystyle\mathbf{u}(\mathbf{r},t)=-2\begin{pmatrix}\chi(z)\\\
\zeta(z)\end{pmatrix}\begin{pmatrix}\sin(kx)\\\
\cos(kx)\end{pmatrix}\cos(\omega t)\,.$ (13)

The strain field of the SSAW is described by the partial derivatives of the
displacements

$\displaystyle u_{xx}$ $\displaystyle=-2\chi$ $\displaystyle k\cos(\omega
t)\cos(kx)$ $\displaystyle u_{xz}$
$\displaystyle=-2\frac{\partial\chi}{\partial z}$ $\displaystyle\cos(\omega
t)\sin(kx)$ $\displaystyle u_{zx}$ $\displaystyle=\phantom{-}2\zeta$
$\displaystyle k\cos(\omega t)\sin(kx)$ $\displaystyle u_{zz}$
$\displaystyle=-2\frac{\partial\zeta}{\partial z}$ $\displaystyle\cos(\omega
t)\cos(kx).$ (14)

Inserting Eq. (B.2) into Eq. (A) we find

$\begin{split}\delta
q_{x}&=2q_{x}\left[\chi(z)k\,\cos(kx)+\frac{q_{x}}{q_{z}}\frac{\partial\chi(z)}{\partial
z}\sin(kx)\right]\cos(\omega t)\\\ \delta
q_{z}&=2q_{z}\left[\frac{\partial\zeta(z)}{\partial
z}\cos(kx)-\frac{q_{z}}{q_{x}}\zeta(z)k\,\sin(kx)\right]\cos(\omega
t)\,.\end{split}$ (15)

The component $\delta q_{x}$ is identical as for two fixed ends, while $\delta
q_{z}$ is shifted in phase, both along the $x$-axis and in time. By inserting
Eq. (8) these variations can be written in the explicit form

$\begin{split}\delta
q_{x}&=\phantom{-}2q_{x}k\chi(z)\left[\quad\,\cos(kx)-\;\frac{q_{x}}{q_{z}}\;\Big{(}\Omega_{\text{r}}+\Omega_{\text{i}}\tan(\Omega_{\text{i}}kz+\phi)\Big{)}\;\sin(kx)\right]\cos(\omega
t)\\\ \delta
q_{z}&=-2q_{z}k\zeta(z)\,\left[\frac{q_{z}}{q_{x}}\sin(kx)+\Big{(}\Omega_{\text{r}}+\Omega_{\text{i}}\tan(\Omega_{\text{i}}kz+\phi+\theta)\Big{)}\cos(kx)\right]\cos(\omega
t)\,.\end{split}$ (16)

##  Appendix C Model used in the main article

In our SXDM measurements we probe the standard deviations $\sigma_{x,z}$ of
the broadened reflection $\mathbf{q}$ in reciprocal space. Following the model
outlined in the main article, the contribution of the strain field of the SSAW
can be expressed as $\sigma_{x,z}^{2}=\left<\delta q_{x,z}^{2}\right>_{t}$,
which can be easily calculated analytically starting from Eq. (12) for equal
reflection conditions or from Eq. (16) for opposite reflection conditions of
$u_{x}$ versus $u_{z}$. Our last step is adding up all contributions to the
actually measured reflection by numerically integrating along the incident
beam of photons.

In GaAs, the absorption length of X-rays at the energy of 10.185 keV is
$l_{\text{abs}}\simeq 55\,\mu$m. In comparison, the extinction depth of the
X-ray beam for the (335)-reflection is much smaller because of the Bragg
reflection, $l_{\text{ext}}\simeq 6.6\,\mu$m. As sketched in Fig. 2(b) of the
main article, a photon of the X-ray beam penetrates into the crystal until it
is randomly scattered at a penetration length
$l_{\text{i}}=z/\sin\alpha_{\text{i}}\simeq 1.004z$. Then the photon travels
for the distance $l_{\text{o}}=z/\sin\alpha_{\text{f}}\simeq 12.5z$ before it
leaves the crystal. For such a diffraction process at a given depth
$z(l_{\text{i}})$ below the surface the effects of extinction and absorption
can be combined in a damping factor

$\beta(z)=\text{e}^{-[l_{\text{i}}/l_{\text{ext}}+(l_{\text{i}}+l_{\text{o}})/l_{\text{abs}}]}\,.$
(17)

For our $\lambda\simeq 500\,$nm we find
$\beta(z)\simeq\text{e}^{-0.2z/\lambda}$. The contribution of a diffraction at
$z=500\,$nm to the reflection is therefore reduced by approximately 18%. The
SSAW itself decays much faster, namely within one wavelength below the
surface, according to $\chi(\lambda)/\chi(0)\simeq-0.05$ and
$\zeta(\lambda)/\zeta(0)\simeq 0.1$. To simulate our experiment we integrate
$\left<\delta q_{x,z}^{2}(x_{z},z)\right>_{t}$ along the incident beam
trajectory. This integration is particularly relevant because of the
oscillating behavior of $\chi(z)$ and $\zeta(z)$ for $z\leq\lambda$. Using
$x_{z}=x(z=0)+z/\tan\alpha_{\text{i}}\simeq x(z=0)+0.084z$ and including the
depth dependent attenuation we arrive at our prediction for the contribution
of the SSAW to the shape of the reflection

${\delta\sigma_{x,z}^{2}}=\frac{\int_{0}^{\infty}\text{d}z\left<\delta
q_{x,z}^{2}(x_{z},z)\right>_{t}\beta(z)}{\int_{0}^{\infty}\text{d}z\beta(z)}\,.$
(18)

To determine the model curves in Fig. 3(d) we computed this integral
numerically inserting $\delta q_{x,z}$ from Eq. (12), which applies for equal
reflection conditions of both components of the SSAW.

By applying Eq. (18) we incoherently add up the contributions to the Bragg
reflections of infinitesimal volume elements along the incident X-ray beam.
This incoherent approach might be questioned, as the penetration depth of the
SSAW and the nominal longitudinal X-ray coherence length of about 800 nm [4]
are similar. However, the photon coherence is much more reduced by the lateral
focusing of the X-ray beam, which by orders of magnitude dominates the
broadening of the reflections and, thus, justifies our incoherent model.

In Fig. 6

Figure 6: Identical measured data as in Fig. 3(d) of the main article (solid
lines, black). Dashed lines (blue) are model curves for fixed ends for both
SSAW components, as in Fig. 3(d) of the main article. The dot-dashed line
(red) is the corresponding model curve but using open ends for the
$z$-component of the SSAW. Its phase is shifted by $\pi$ compared to the case
of fixed ends. Model lines of $\sigma_{z}$ are reduced by a factor 10.

we compare the two cases of equal versus different reflection conditions of
the $x$\- and $z$-components of the SSAW, by insertion of Eq. (12) versus Eq.
(16) into Eq. (18). Because we varied the reflection condition for the
$z$-component of the SAW while keeping it unchanged for the $x$-component, we
find two model curves for $\delta\sigma_{z}^{2}$ with opposite phase but only
one model curve for $\delta\sigma_{x}^{2}$. Note, that the phase relation
between the components is fixed while we have chosen the overall phase for
best agreement with the measured data (solid lines). Our measured data are
clearly better described by the combination of equal reflection conditions,
e.g., both ends fixed.

##  Appendix D Numerical simulation of the SXDM experiment

We tested our model described above by comparing its results to numerical
simulations of the SXDM experiment. Experimentally we find, that the
projections of the unperturbed three-dimensional reflection to the $q_{x}$-,
$q_{y}$\- and $q_{z}$-axes of reciprocal space are well described by Gaussian
distributions
$\propto\text{e}^{-\left({q_{i}}/{2(\sigma_{0})_{i}}\right)^{2}}$. The
displacement field of the SSAW causes variations of $\mathbf{q}(x,z)$. Hence,
the reflections of individual scattering events correspond to Gaussian
reflections of the same shape but slightly shifted according to

$A_{i}(x,z,t)\propto\beta(z)\text{e}^{-\left(\frac{q_{i}-\delta
q_{i}(x,z,t)}{2(\sigma_{0})_{i}}\right)^{2}}\,,$ (19)

where $\delta q_{i}(x,z,t)$ has to be inserted from Eq. (12) or Eq. (16),
depending on the reflection conditions and $\beta(z)$ is the damping factor
introduced in Eq. (17) above.

Following the actual experiment, we next determine the shape of the actual
reflection by numerically integrating over $z$ (along the incident beam) and
then over time

$\alpha_{i}(z)\propto\int_{t=0}^{2\pi/\omega}\text{d}t\int_{z=0}^{\infty}\text{d}z\,A_{i}(x,z,t)\,.$
(20)

Finally, we numerically determine the variances $\sigma_{i}^{2}$ and, for a
direct comparison with our approximate model above,
$\delta\sigma_{i}^{2}=\sigma_{i}^{2}-(\sigma_{0})_{i}^{2}$. The numerical
results resemble our approximate model results one-to-one in the phase
difference between $\sigma_{x}^{2}$ and $\sigma_{z}^{2}$ and the ratio
$\sigma_{x}/\sigma_{z}$, while the prefactor $C$ differs by $<10\,\%$. This
good agreement between the direct numerical calculation and our analytical
approximation is expected, because of the small modifications of the
undisturbed reflection by the displacement field of the SSAW.

##  Appendix E Displacement amplitudes of the SSAW — classical regime

The displacement amplitudes of the SSAW at the surface are
$U_{x}=2C\chi(0)\simeq 0.8C$ and $U_{z}=2C|\gamma|\zeta(0)\simeq 1.75C$, cf.
Eq. (1), where we used $\chi(0)\simeq 0.4$, $\zeta(0)\simeq 0.65$ and
$|\gamma|\simeq 1.345$. Fitting our model described above to our measurements
presented in Fig. 3 we find $C\simeq 25\,$pm, corresponding to displacement
amplitudes at the surface of $U_{x}\simeq 20\,$pm and $U_{z}\simeq 44\,$pm.

From our measurements presented in Fig. 1(b) and Fig. 3 we can estimate the
mode area as $A\simeq 56\,\mu\text{m}^{2}$. This allows us to further estimate
the displacement that a single phonon captured inside the cavity would cause,
$C_{0}=\sqrt{\hbar f\lambda/HA}$, where the material constant $H=28.2\times
10^{10}\,$N/m2 for GaAs [23]. Using $\lambda=503.2\,$nm and $f\simeq
5.675\,$GHz we find $C_{0}\simeq 0.138\,$fm. With
$C=\sqrt{N_{\text{ph}}}C_{0}$ we can finally estimate the number of phonons
contributing to our SSAW, which is $N_{\text{ph}}\sim 10^{10}$. This large
number of phonons clarifies that our measurements are probing the classical
regime of SSAWs.

The resolution limit of our method makes such a large amplitude necessary.
Another limiting factor is the small energy of our resonant phonons, which is
about three orders of magnitude smaller compared to that of the dominant room
temperature thermal phonons. Nevertheless, as long as the measured SSAW is not
distorted by non-linear effects, we expect that the SSAW of a single phonon
corresponds to our measurements but scaled down to the amplitude $C_{0}$.


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|  Jing Liu is currently pursuing the Ph.D. degree with the Institute of
Computing Technology, Chinese Academy of Sciences. Her research interests
include graph neural networks, heterogeneous graph representation learning,
and knowledge distillation.
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|  Tongya Zheng is currently pursuing the Ph.D. degree with the College of
Computer Science, Zhejiang University. He received his B.Eng. Degree from
Nanjing University of Science and Technology. His research interests include
graph neural networks, temporal graphs, and explanation for artificial
intelligence. He has authored and co-authored many scientific articles at top
venues including IEEE TNNLS and AAAI. He has served with international
conferences including AAAI and ECML-PKDD, and international journals including
Information Sciences.
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|  Guanzheng Zhang is currently pursuing the M.S. degree with the College of
Information Engineering, China University of Geosciences. His research
interests include machine learning and graph mining.
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|  Qinfen Hao received the Ph.D. degree in computer system architecture from
the Institute of Computing Technology, Chinese Academy of Sciences, China, in
2006. He is currently a Researcher at the Institute of Computing, Chinese
Academy of Sciences. His research interests include computer architecture and
graph computing.
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