Abbreviations:
---
2TM | Two-temperature model
AFM | Antiferromagnetic
FEL | Free-electron laser
FM | Ferromagnetic
LCM | Linaer chain model
NTE | Negative thermal expansion
PM | Paramagnetic
PSD | Position-sensitive detector
PUX | Picosecond ultrasonics with x-rays
RSM | Reciprocal space map
RSS | Reciprocal space slicing
TDBS | Time-domain Brillouin scattering
TEM | Transmission electron microscopy
UXRD | Ultrafast x-ray diffraction

# Concepts and use cases for picosecond ultrasonics with x-rays

Maximilian Mattern  Alexander von Reppert<EMAIL_ADDRESS>Steffen Peer
Zeuschner  Marc Herzog  Jan-Etienne Pudell  Matias Bargheer  Institut für
Physik & Astronomie, Universität Potsdam, 14476 Potsdam, Germany  Helmholtz
Zentrum Berlin, 12489 Berlin, Germany  European XFEL, 22869 Schenefeld,
Germany

(August 28, 2024)

###### Abstract

This review discusses picosecond ultrasonics experiments using ultrashort hard
x-ray probe pulses to extract the transient strain response of laser-excited
nanoscopic structures from Bragg-peak shifts. This method provides direct,
layer-specific, and quantitative information on the picosecond strain response
for structures down to few-nm thickness. We model the transient strain using
the elastic wave equation and express the driving stress using Grüneisen
parameters stating that the laser-induced stress is proportional to energy
density changes in the microscopic subsystems of the solid, i.e., electrons,
phonons and spins. The laser-driven strain response can thus serve as an
ultrafast proxy for local energy-density and temperature changes, but we
emphasize the importance of the nanoscale morphology for an accurate
interpretation due to the Poisson effect. The presented experimental use cases
encompass ultrathin and opaque metal-heterostructures, continuous and granular
nanolayers as well as negative thermal expansion materials, that each pose a
challenge to established all-optical techniques.

###### keywords:

Picosecond ultrasonics , Ultrafast x-ray diffraction , Ultrafast x-ray
scattering , Ultrafast photoacoustics , Nanoscale heat transfer , Negative
thermal expansion

##  1 Introduction

Picosecond ultrasonics subsumes the generation and detection of laser-induced
nanoscopic strain pulses with GHz or even THz frequency components [1, 2].
Most studies of picosecond ultrasound have been and will be conducted with
all-optical setups, which allow tackling both fundamental and applied research
questions in light-matter interaction, condensed matter and material science
[3, 2, 4, 5, 6]. Picosecond ultrasonics experiments exploit that the energy
deposition of an optical femtosecond pump-pulse creates a stress profile
within an absorbing transducer layer. The subpicosecond rise of the stress
gradient introduces a source term to the elastic wave equation, which is used
to rationalize the occurrence of highly localized strain pulses [7, 6, 8].

Many aspects of the strain propagation can be measured by reflection,
transmission or scattering of optical probe pulses at interfaces and in the
bulk. In the most straight-forward experiments, strain induced changes of the
reflectivity witness the arrival of a strain pulse in the layer and the timing
can be used to infer layer thicknesses or elastic properties [1, 9, 5, 10]. In
transparent materials, the detection process has been interpreted as time-
domain Brillouin scattering (TDBS), which is observed as oscillations of the
optical reflectivity at a frequency that is proportional to the sound velocity
[11, 12, 13]. Although the development of all-optical probing schemes with
ever-growing sensitivity has enabled studies of a multitude of picosecond
ultrasonics effects such as strain pulses in heterostructures [9, 14], shear
waves [15], acoustic solitons [16] or even imaging of elastic properties [17,
18], the reported strain signatures mostly remain on a qualitative level.

In ultrafast x-ray diffraction (UXRD) experiments the optical probe pulses are
replaced by ultrashort hard x-ray pulses that are diffracted from the crystal
structure in motion [19, 20, 21]. Similar to optical reflectivity, x-ray
diffraction can detect propagating strain pulses in two ways: Strain in a
material is heralded by Bragg peak shifts which encode the amplitude, sign and
shape of the elastic wave [22, 23, 24]. Secondly, TDBS of x-rays [12] from the
sum of a reciprocal lattice vector and a phonon wave vector can probe the
phonon population and the elastic properties even for wave vectors close to
the Brillouin zone boundary [25, 26, 27]. For such studies the high degree of
collimation of hard x-rays from synchrotron and free-electron laser (FEL)
sources is highly beneficial. This is also true for the time-resolved
detection of x-rays scattered from incoherent phonons by thermal diffuse
scattering, which yields a time- and wavevector-resolved picture of the phonon
population [28, 29, 30, 31].

X-rays from laser-based plasma sources, in contrast, are sufficient to support
the conceptually simple picosecond ultrasonics experiments that investigate
Bragg peak shifts [32, 33, 34, 35, 36]. These setups are particularly useful
for broad diffraction peaks associated with thin films of imperfect
crystallinity due to the relatively large divergence and bandwidth of the
provided x-rays. For ultrathin layers, crystals consisting of light atoms,
nanoparticles or advanced concepts using resonant diffraction, however, FELs
are ideal as they pair high photon flux and sub-picosecond time resolution
[37, 38, 39]. Synchrotron radiation sources typically lack temporal
resolution, which can be improved via fs-pulse slicing schemes [40, 41], x-ray
streak camera setups [22, 23, 42], special short-pulse filling patterns [43,
44] or picosecond Bragg switches [45, 46], that all come at the cost of a
reduction of the photon flux by multiple orders of magnitude.

In this article, we highlight the conceptual advantages of picosecond
ultrasonics with x-rays (PUX) experiments that observe shifts of Bragg peaks
of layered heterostructures. Each crystalline material scatters at a
characteristic Bragg diffraction angle, and hence, the signal is often a
layer-specific measure of the lattice distortions caused by the strain pulse
propagation and heat flow. In other words, for each pump-probe delay, x-ray
probe pulses can project the Bragg peak of each layer onto the detector with a
peak shift that is proportional to the instantaneous strain averaged over one
layer. The four central advantages of x-ray probing in picosecond ultrasonics
are: (i) Diffraction yields quantitative strain values (ii) with layer
specific information for (iii) strain pulses and quasi static strain. (iv) The
x-ray probe penetrates metals and insulators irrespective of their optical
properties. This permits a very flexible exploitation of dedicated strain-
sensing layers that can be placed within the heterostructure and often
naturally exist as buffer, contact or electrode layers. We elaborate and
illustrate concepts for the analysis of PUX experiments that have emerged to
utilize the quantitative access to the strain response for tracking the energy
flow in laser-excited heterostructures.

We structured the manuscript as follows: In Section 2 we introduce the
measurement principle and observables of PUX experiments based on a
representative example. We show that the layer for optical excitation can be
separated from a dedicated probe layer, which is optimized for x-ray
scattering. This facilitates the separation of the strain pulse traveling at
the speed of sound and quasi-static strain that propagates via heat diffusion.

In Section 3 we reconsider aspects of the elastic wave equation that are
important for a quantitative modeling of the picosecond strain response. It
becomes evident that an in-plane motion introduces a transverse elastic stress
that affects the out-of-plane strain response. Accordingly, constrained in-
plane motion distinguishes the laser-induced thermal expansion response of a
homogeneously excited continuous film from its unconstrained near equilibrium
thermal expansion. We elaborate the direct relation of laser-induced energy
and stress via a Grüneisen parameter and emphasize that the quasi-static
expansion is linearly proportional to the energy density that generates the
stress. The concept of subsystem-specific Grüneisen parameters is introduced
to capture the stress contributions of different quasi-particle excitations to
the strain response.

In Section 4 we present scenarios where ultrashort hard x-ray probe pulses
excel at probing the strain response in laser-excited nanolayered
heterostructures. At first, we illustrate in 4.1 how PUX tracks bipolar and
unipolar picosecond strain pulses launched by an opaque transducer with
transparent capping layers of various thicknesses. The second use case is the
quantitative determination of the picosecond strain response in granular and
continuous FePt thin films, that exhibits a strong dependence on the sample
morphology as shown in 4.2. The third use case demonstrates that the PUX
experiments can distinguish the strain response of nanoscopic heterostructures
that are thinner than the optical penetration depth as demonstrated on Au-Ni
bilayer structures in 4.3. This thin film scenario complements the
introductory example where we access the electronic energy transfer between
layers of an opaque Pt-Cu-Ni heterostructure discussed in Section 2. The last
use case in 4.4 is the detection of ultrafast negative thermal expansion (NTE)
exhibited by a rare-earth transducer, which launches unconventional picosecond
strain pulses towards a buried detection layer. This example furthermore
demonstrates the extraction of subsystem-specific Grüneisen parameters from
equilibrium thermal expansion data.

Section 5 discusses the advantages of large scale sources and provides an
overview on related experimental schemes that utilize x-rays for picosecond
ultrasonics.

The appendix in Section A contains a brief discussion of the diffraction
geometry and the relation between the diffraction angles and the reciprocal
space coordinates. We revisit the concept of reciprocal space slicing (RSS) as
a rapid data acquisition approach that is often sufficient and less time-
consuming compared to the acquisition of reciprocal space maps (RSMs), which
requires scanning the incidence angle of the x-rays on the sample. We explain
the scaling factor that relates the Bragg peak shift in reciprocal space to
the shift measured on an area or line detector in the RSS scheme and discuss
scenarios when time-resolved RSMs are required for a proper strain assessment.

##  2 PUX experiments in a nutshell

A typical picosecond ultrasonics experiment is schematically depicted in Fig.
1, where we illustrate the generic series of events common to laser-excited
heterostructures [6]. The laser-induced strain response in this type of
experiment contains information on all four conceptual steps that occur in
response to the light-matter interaction between the femtosecond laser pulse
and the absorbing transducer layer i.e., the energy deposition profile (1),
the strain generation from a laser-induced stress (2), strain pulse
propagation and reflection (3) and quasi-static strain concomitant with
thermal transport (4).

Figure 1: Generic series of events for picosecond ultrasonics: A femtosecond
laser pulse excites a metallic heterostructure that consists of multiple,
often nanoscopically thin layers. The deposited energy creates a stress that
drives a picosecond strain response consisting of strain pulses propagating at
the speed of sound and a quasi-static thermal expansion which evolves via heat
diffusion. The small skin depth of optical probes limits the probed volume to
the near-surface region whereas hard x-rays often penetrate multiple microns
or more into the bulk.

In applications using optical probe pulses, the main results of such
experiments are often the echo time for thickness determination [5, 10], the
acoustic transmission amplitude through an interface for measuring impedance
mismatches [4, 47] or the oscillations in TDBS to determine the sound velocity
[48, 49, 13]. Here, we demonstrate that PUX provides information on film
thicknesses, strain-pulse reflections and microscopic energy transfer
processes within the heterostructure, that set the space and time-dependent
stress driving the strain response. As indicated in Fig. 1 PUX experiments
benefit from the large extinction length of hard x-rays (e.g.
$8\,\text{keV}$), which is typically on the order of few micrometers
irrespective of the optical properties, e.g. of metals, semiconductors and
insulators. Therefore, the hard x-ray probe pulse can report on all layers of
thicker heterostructures. More importantly, when each layer exhibits a
different lattice constant, i.e., has a characteristic Bragg diffraction
angle, the probe is even layer-specific.

In the remainder of this section, we discuss a representative PUX experiment
on metallic heterostructures composed of Pt, Cu and Ni to exemplify the
measurement principle and data evaluation. This example demonstrates the
advantages of PUX experiments on a sample structure that is frequently used to
study the effect of hot-electron pulses that are launched in a Pt layer and
propagate through an opaque Cu stack towards a buried functional detection
layer [50, 51, 52, 53].

Figure 2: Extraction of layer-specific strain via x-ray diffraction
exemplified for a Pt-Cu-Ni heterostructure: (a) Sketch of the samples and
diffraction geometry using a PSD. Scanning the angles $\alpha_{\text{in}}$ and
$\alpha_{\text{out}}$ yields the RSM (c) around the Bragg peaks of Pt, Cu and
Ni that are separated along the out-of-plane reciprocal coordinate
$q_{\text{z}}$ (b). The yellow and blue lines denote the diffraction intensity
distributions for the respective subset of the reciprocal space that is probed
by the area detector at particular fixed angles as utilized in the time-
resolved experiment. The black line is the integrated intensity distribution
of (c) along $q_{\text{x}}$. Panels (d–f) display the laser-induced shift of
the Bragg peaks of sample 1 that is used to determine the layer-specific
transient strain according to Eq. (2).

Fig. 2(a) introduces the sample structures and the specular i.e., symmetric
and coplanar diffraction geometry. A femtosecond x-ray pulse derived from a
laser-based plasma x-ray source [54] is incident under the angle
$\alpha_{\text{in}}$ with respect to the sample surface. The pixels of a
position-sensitive detector (PSD) record the x-ray intensity distribution
diffracted from the individual layers of the heterostructure. On the PSD we
observe three maxima separated within the diffraction plane which correspond
to the Bragg peaks of the individual metal layers. The respective diffraction
angles $\alpha_{\text{out}}$ are determined by the reciprocal lattice vector
$\bm{G}$ via the Laue condition:

$\displaystyle\bm{q}=\bm{k}_{\text{out}}-\bm{k}_{\text{in}}=k\begin{pmatrix}\cos{(\alpha_{\text{out}})}-\cos{(\alpha_{\text{in}})}\\\
\sin{(\alpha_{\text{in}})}+\sin{(\alpha_{\text{out}})}\end{pmatrix}=\bm{G}\,,$
(1)

where the scattering vector $\bm{q}$ is determined by the wave vector of the
incident ($\bm{k}_{\text{in}}$) and diffracted ($\bm{k}_{\text{out}}$) x-rays,
which have equal magnitude $k=2\pi/\lambda$ according to the wavelength
$\lambda$ of the elastically scattered x-rays. The magnitude of $\bm{G}$
encodes the out-of-plane lattice constant $d_{3}$ and the diffraction order
$n$ via $|\bm{G}|=2\pi n/d_{3}$. Each detector pixel of the PSD probes a small
volume around a specific point in reciprocal space and a symmetric scan of
$\alpha_{\text{in}}$ and $\alpha_{\text{out}}$ maps the reciprocal space along
the out-of-plane direction defining the $q_{\mathrm{z}}$-axis. A
representative RSM, i.e., the intensity scattered along $\bm{q}$, of a Pt-Cu-
Ni heterostructure is depicted in Fig. 2(c) and reveals Bragg peaks of each
(111)-oriented layer. The width of the Bragg peaks along $q_{\text{x}}$ and
$q_{\text{z}}$ is given by the crystalline quality of the layers characterized
by the mosaicity and the in-plane and out-of-plane coherence length of the
crystallites forming the metal layers. A more detailed discussion of the RSM
and the transformation of the angle-dependent intensity to reciprocal space
$q_{\text{x}}$-$q_{\text{z}}$ is provided in appendix A.1 and [54].

The integration of the RSM along $q_{\text{x}}$ yields the intensity
distribution along the reciprocal coordinate $q_{\text{z}}$ (Fig. 2(b)) that
encodes the average out-of-plane lattice constant $d_{3}$ of the diffracting
layers by the center-position of their Bragg peaks $q_{\text{z}}=2\pi n/d_{3}$
(see Eq. (1)). Following the laser-induced time-dependent shift of the Bragg
peaks along $q_{\text{z}}$ in a pump-probe experiment yields the layer-
specific transient strain $\eta(t)$ which represents the relative change of
the average out-of-plane lattice constant with respect to its value before
excitation ($t<0$):

$\displaystyle\eta(t)=\frac{d_{3}(t)-d_{3}(t<0)}{d_{3}(t<0)}=\frac{q_{\text{z}}(t<0)-q_{\text{z}}(t)}{q_{\text{z}}(t)}\,.$
(2)

Instead of scanning the incidence angle $\alpha_{\text{in}}$ to create an RSM
at each pump-probe delay, we often follow the transient Bragg peak position by
the time-efficient RSS method [55] that is further discussed in Section A.2.
The PSD simultaneously probes a subset of the reciprocal space as indicated in
Fig. 2(c) by the blue and yellow lines for two different fixed
$\alpha_{\text{in}}$ corresponding to the Bragg angles of Ni and Pt,
respectively. The intensity on the detector exemplarily shown in the inset of
panel (a) for an intermediate $\alpha_{\text{in}}$ probing all three peaks
simultaneously with less intensity is integrated along the $q_{\text{y}}$
coordinate of the detector. For the yellow and blue lines $\alpha_{\text{in}}$
is chosen to efficiently capture slices through the different maxima of the
RSM (Fig. 2(c)), in order to collect RSS (Fig. 2(b)) which closely resemble
the integrated RSM (black line). We relate the transient shifts of the Bragg
peaks on the detector to a transient shifts along $q_{\text{z}}$ depicted in
Fig. 2(d–f) that yield the layer-specific transient out-of-plane strains
according to Eq. (2).

The resulting strain response of the Pt transducer, the Cu propagation layer,
and the Ni detection layer are compared in Fig. 3(a–c) for two different
heterostructures with and without a $5\,\text{nm}$ thin insulating MgO
interlayer in front of the buried Ni detection layer. The near-infrared pump-
pulse mainly deposits energy in the $7\,\text{nm}$ thin Pt transducer as
illustrated by the absorption profile in Fig. 3(e). In absence of an MgO
interlayer, we observe a rapid expansion of both the Pt and the Ni layer upon
laser-excitation and the Cu layer is compressed within the first picoseconds.
The rapid expansion of the Ni layer compressing the adjacent Cu layer
originates from a fast transport of hot electrons from the laser-excited Pt
transducer through Cu to the buried Ni layer where they release their energy
to phonons. This dominant electronic heat transport to Ni causes the
compression of the Cu layer that is heated only on a longer timescale owing to
its weak electron-phonon coupling. This surprising observation was referred to
as "heat transport without heating" in a previous publication [56] that
provides additional information on the role of layer-specific electron-phonon
coupling and the modelling of the strain response.

Here, we highlight the suppression of the crucial electronic energy transport
from Pt to Ni, if the additional MgO interlayer is introduced (blue data in
Fig. 3). Now the Ni layer is rapidly compressed by the expansion of Cu and
compressed even more at $20\,\text{ps}$, when the strain pulse generated in Pt
reaches the Ni layer. The expansion of the Ni detection layer only rises on
the timescale of hundreds of picoseconds after laser-excitation via phononic
heat transport from Cu through the MgO interlayer separating sound and heat in
the time domain. The suppressed electronic energy transport to the buried Ni
detection layer yields a background-free signal of the strain pulse and an
increased expansion of the Cu layer compared to the pure metal
heterostructure.

Figure 3: Comparison of the picosecond strain response in Pt-Cu-(MgO-)Ni
heterostructures: Transient strain in the (a) Pt, (b) Cu and (c) Ni layers of
the heterostructures with and without MgO interlayer as depicted in (d). Blue
lines are for the heterostructure with an insulating MgO barrier between Cu
and Ni. (e) Absorption of the pump-pulses occurs only in the Pt layer and in
the first few nm of Cu. The suppression of the electronic heat transport from
Pt to Ni by the MgO interlayer changes the strain response of the Ni detection
layer from expansion to compression and delays the rise of the quasi-static
expansion.

The experiment nicely illustrates PUX: We obtain the strain response by
tracking the shift of layer specific diffraction peaks that are well separated
due to their material-specific lattice constants. The total strain response is
a superposition of propagating strain pulses and a quasi-static expansion due
to heating that dominates after the strain pulses have propagated into the
substrate. Since PUX measures the strain amplitude, both the strain pulses and
the quasi-static strain from laser-induced temperature changes can be
quantitatively evaluated on equal footing. In addition, the transient strain
accesses the layer thicknesses via the timing of the expansion and compression
pulses.

##  3  Concepts for modelling the strain response

In this section, we discuss the fundamental concepts for quantitatively
modelling the picosecond strain response, driven by a laser-induced spatio-
temporal stress. These dynamics are generally described by the elastic wave
equation, and we elaborate on the special case of a laterally homogeneous
excitation of continuous thin films and heterostructures investigated in
typical picosecond ultrasonic experiments. The spatio-temporal stress is
proportional to the local contributions to the energy density, which changes
in time according to the heat transport within the sample. The different
degrees of freedom such as electrons and phonons contribute differently to the
transport and have to be accounted for in nanoscale metals. We advocate the
Grüneisen concept to model the effect of the energy transfer between these
subsystems on the laser-induced stress. The Grüneisen parameters linearly
relate the energy deposited in different degrees of freedom to the respective
stress contributions. Their superposition yields the total external stress
driving the strain response. Finally, we provide an example of modelling the
strain response of a sample by numerically solving the elastic wave equation
for an educative case of an inhomogeneously excited transducer on a non-
absorbing detection layer.

###  3.1  Poisson stresses in a 3D strain response

In general, the strain response of an elastic solid to a time- and space-
dependent stress is found as a solution of the elastic wave equation, i.e.,
the equation of motion for the displacement field $u_{i}(\bm{x},t)$ at a
specific position $\bm{x}$ in the sample and at a time $t$.111To enhance the
readability of this section the space and time dependencies $(\bm{x},t)$ are
not listed further after the introduction of physical quantities. The index
$i$ enumerates the three spatial dimensions. The mass density
$\rho_{\text{m}}(\bm{x})$ is accelerated by the spatial gradient of a total
stress $\sigma^{\text{tot}}_{ij}(\bm{x},t)$, as described by the elastic wave
equation:

$\displaystyle\begin{split}\rho_{\text{m}}\frac{\partial^{2}u_{i}}{\partial
t^{2}}&=\sum_{j}\frac{\partial}{\partial x_{j}}\sigma_{ij}^{\text{tot}}\\\
&=\sum_{j}\frac{\partial}{\partial
x_{j}}\left(\sum_{k,l}c_{ijkl}\eta_{kl}-\sigma_{ij}^{\text{ext}}\right)\,.\end{split}$
(3)

The deformation of the solid is described by the strain

$\displaystyle\eta_{kl}(\bm{x},t)=\frac{\partial u_{k}(\bm{x},t)}{\partial
x_{l}}\,,$ (4)

which is determined by the displacement $u_{k}$. The proportionality constants
of the elastic stress are the direction-dependent elastic constants
$c_{ijkl}(\bm{x})$222Here, $c_{ijkl}$ does not necessarily denote the elastic
constants along the high symmetry axis of the crystal but along the spatial
directions given by the sample geometry. If the high symmetry axis do not
match the principal axis of the sample a transformation of the elastic tensor
into the coordinate system of the sample is required.. Adding an external
stress $\sigma_{ij}^{\text{ext}}(\bm{x},t)$ drives the atomic motion.

In this publication, we limit our discussion to longitudinal laser-induced
stresses $\sigma^{\text{ext}}_{ii}$ and strains $\eta_{kk}$, i.e., volume
changing elements of the stress-strain relation. Under this limitation the
elastic wave equation (3) simplifies to:

$\displaystyle\begin{split}\rho_{\text{m}}\frac{\partial^{2}u_{i}}{\partial
t^{2}}&=\frac{\partial}{\partial x_{i}}\left(c_{iiii}\eta_{ii}+\sum_{k\neq
i}c_{iikk}\eta_{kk}-\sigma_{ii}^{\text{ext}}\right)\\\
&=\frac{\partial}{\partial
x_{i}}\left(\sigma^{\text{elastic}}_{ii}+\sigma_{ii}^{\text{Poi}}-\sigma_{ii}^{\text{ext}}\right)\,.\end{split}$
(5)

Here, the negative sign in front of the external stress is chosen such that a
positive longitudinal stress $\sigma^{\mathrm{ext}}_{ii}$ that acts from
within the material leads to an expansion $(\eta_{ii}>0)$ along the direction
of the stress. Note that other works [8, 6] use the opposite sign for
$\sigma^{\mathrm{ext}}_{ii}$, which yield intuitive results when the material
is subjected to an external force, for instance, through pressing in a diamond
anvil cell where a positive stress leads to compression $(\eta_{ii}<0)$. If
the gradients in the external stress rise faster than the elastic stress which
is proportional to the strain propagating at sound velocity, a propagating
strain pulse is launched. Its propagation is affected by interfaces between
different layers within the sample structure unless they are acoustically
impedance matched333The acoustic impedance $Z$ of a material is defined by the
product of its mass density $\rho_{\mathrm{m}}$ and the appropriate sound
velocity $v_{\mathrm{s}}=\sqrt{c_{3333}/\rho_{\text{m}}}$ i.e.,
$Z=\rho_{\mathrm{m}}v_{\mathrm{s}}$. The acoustic reflection coefficient $R$
for a strain pulse traversing the interface from material 1 to material 2
considered in the model, is given by $R=\frac{Z_{2}-Z_{1}}{Z_{1}+Z_{2}}$,
which implicitly assumes perfect adhesion [47, 57].. The strain pulses are
partially reflected from the interfaces. When reflection occurs at an
interface to a medium with lower acoustic impedances, e.g. in particular for
reflections at the surface of the sample, the sign of the strain pulse
changes. The strain $\eta_{ii}$ induces an elastic stress
$\sigma^{\text{elastic}}_{ii}(\bm{x},t)$ that partially compensates the
external stress. In addition, the three-dimensional response of the solid
introduces Poisson stress contributions $\sigma_{ii}^{\text{Poi}}$ that
originate from the spatio-temporal strains $\eta_{kk}$ along the perpendicular
directions analogously determined by the elastic wave equation along the
respective directions. In total, the three-dimensional strain response of the
solid to the external stresses $\sigma_{ii}^{\text{ext}}$ requires a solution
of the three coupled differential equations (3).

When the driven strain pulses and their reflections have propagated out of the
volume of interest, the elastic and the Poisson stress contributions fully
compensate the external stress and the vanishing total stress marks a quasi-
static state444The quasi-static state slowly changes due to diffusion of heat
and other slow processes.. The vanishing time derivative in Eq. (5) then
determines the quasi-static strain $\eta_{ii}^{\text{qs}}$ to:

$\displaystyle\begin{split}\eta_{ii}^{\mathrm{qs}}&=\frac{\sigma_{ii}^{\text{ext}}-\sigma_{ii}^{\text{poi}}}{c_{iiii}}\\\
&=\frac{\sigma_{ii}^{\text{ext}}}{c_{iiii}}-\sum_{k\neq{i}}\frac{c_{iikk}}{c_{iiii}}\eta_{kk}^{\text{qs}}=\alpha_{i}\Delta
T\,.\end{split}$ (6)

The anisotropic linear expansion coefficient $\alpha_{i}(\bm{x})$ relates this
quasi-static strain to a temperature increase $\Delta T(\bm{x},t)$ as in
thermal equilibrium555Note, this approach is only true for small $\Delta
T=T_{\text{f}}-T_{\text{i}}$. Otherwise,
$\eta_{ii}^{\mathrm{qs}}=\int_{T_{\text{i}}}^{T_{\text{f}}}\alpha_{i}(T^{\prime})\text{d}T^{\prime}$
holds.. Here, the expansion is not only driven by the externally induced
stress $\sigma_{ii}^{\text{ext}}$ but also reduced by a Poisson contribution
that arises from the expansion along the perpendicular directions
$\eta_{kk}^{\text{qs}}$ driven by the external stresses
$\sigma_{kk}^{\text{ext}}$ (see Fig. 4(a)).

Figure 4: Morphology-dependent strain response of nanoparticles and thin
films: Sketch of the laser-driven quasi-static expansion for an in-plane
nanostructured film (a) and an in-plane homogeneous and continuous thin film
(b) of an isotropic material without an attached substrate. The out-of-plane
expansion of the continuous film is enhanced by the absence of contractive
Poisson stress contributions that would partially compensate the expansive
external stress. The arrows indicate the effective driving stress
$\sigma_{ii}$.

The parametrization of $\eta_{kk}^{\text{qs}}$ in Eq. (6) by their respective
linear thermal expansion coefficient $\alpha_{k}$ yields a relation between
the external stress and the temperature increase:

$\displaystyle\begin{split}\sigma_{ii}^{\text{ext}}(x_{i},t)&=c_{iiii}\left(\alpha_{i}+\sum_{i\neq
k}\frac{c_{iikk}}{c_{iiii}}\alpha_{k}\right)\Delta T\\\
&=\Gamma_{i}\rho_{Q}\,.\end{split}$ (7)

This temperature increase originates from an optically deposited energy
density $\rho_{Q}$: $\Delta T=\rho_{Q}/C_{V}$ determined by the heat capacity
per constant volume $C_{V}$666Note, this approach is only true for small
$\Delta T=T_{\text{f}}-T_{\text{i}}$ where $C_{V}(T)$ only barely changes.
Otherwise it should read
$\rho_{Q}=\int_{T_{\text{i}}}^{T_{\text{f}}}C_{V}(T^{\prime})\text{d}T^{\prime}$.
However, the chosen notation simplifies the introduction of the Grüneisen
parameter.. Equation (7) introduces the direction-dependent Grüneisen
parameter $\Gamma_{i}(\bm{x})$:

$\displaystyle\Gamma_{i}$
$\displaystyle=\frac{c_{iiii}}{C_{V}}\underbrace{\left(\alpha_{i}+\sum_{i\neq
k}\frac{c_{iikk}}{c_{iiii}}\alpha_{k}\right)}_{\alpha_{i}^{\text{uf}}}\,,$ (8)

that linearly relates this deposited energy density to the induced external
stress $\sigma_{ii}^{\text{ext}}$. It describes how efficiently energy density
generates stress and is determined by the direction-dependent
$\alpha_{i}^{\text{uf}}$. This is the expansion coefficient along direction
$i$ assuming that the lattice is clamped along other spatial directions, e.g.
for the ultrafast excitation of a homogeneous thin film (see next section).

The advantage of using the Grüneisen concept is that the ratios of the
intrinsically temperature-dependent quantities $\alpha_{i}(T)$,
$\alpha_{i}^{\text{uf}}(T)$ and $C_{V}(T)$ in Eq. (8) can be combined to
temperature-independent unitless parameters $\Gamma_{i}$. Equation (7) states
that the energy density $\rho_{Q}$ is proportional to the stresses
$\sigma_{ii}^{\text{ext}}$ which themselves linearly relate to the quasi-
static strain $\eta_{ii}^{\text{qs}}$. Moreover, in materials with several
subsystems hosting the energy density, we find a simple recipe for modeling
the out-of-equilibrium expansion response by adding the stresses for all
subsystems, even if their energy content may correspond to different
equilibrium subsystem temperatures [58, 34, 25, 59] (see Section 3.4 for
further details).

###  3.2  Constraints of the thin film geometry

In most picosecond ultrasound experiments the footprint of the excitation
laser pulse (sub-mm) is much larger than both the thickness of the transducer
(sub-µm) and the footprint of the probe pulse, which results in a laterally
homogeneous excitation of the probed sample volume.

Therefore, on the timescale ($t_{\text{1D}}\ll d_{\text{L}}/v_{\text{s}}$)
given by the size of the pump-laser footprint $d_{\text{L}}$ and the sound
velocity $v_{\text{s}}(\bm{x})$, the in-plane stresses are balanced and only
the spatial derivative $\frac{\partial}{\partial
x_{3}}\sigma_{33}^{\text{ext}}$ remains. Under this condition, the system of
coupled Eqs. (5) simplifies to a one-dimensional equation for the strain along
the out-of-plane direction ($x_{3}$) of the thin film:

$\displaystyle\rho_{\text{m}}\frac{\partial^{2}u_{3}}{\partial t^{2}}$
$\displaystyle=\frac{\partial}{\partial
x_{3}}\left(c_{3333}\eta_{33}-\sigma^{\text{ext}}_{33}\right)\,.$ (9)

Figure 4 compares the three-dimensional response of a thin film with in-plane
nanostructure to the purely one-dimensional expansion of a continuous thin
film. This anisotropic picosecond strain of continuous thin films occurs even
in otherwise isotropic solids. It is driven by gradients in the external
stress along the out-of-plane direction which typically appear at the sample
surface or layer interfaces or at the slopes of the excitation profile that is
slowly changing via heat transport. The arising out-of-plane strain
$\eta_{33}$ upon laser excitation partially compensates the external stress
until the total stress vanishes when the strain pulses have propagated into
the substrate. The absence of in-plane expansion for $t\leq t_{\text{1D}}$
suppresses any Poisson stress contributions. Therefore, the remaining quasi-
static expansion $\eta_{33}^{\text{qs}}$ is directly related to the external
stress $\sigma_{33}^{\text{ext}}$ which simplifies Eq. (6) to:

$\displaystyle\eta_{33}^{\mathrm{qs}}=\frac{\sigma^{\mathrm{ext}}_{33}}{c_{3333}}=\frac{\Gamma_{3}\rho_{Q}}{c_{3333}}=\alpha_{3}^{\text{uf}}\Delta
T\,.$ (10)

Using the concept of a Grüneisen parameter we express the external stress by
$\sigma^{\mathrm{ext}}_{33}=\Gamma_{3}\rho_{Q}$. With the definition of the
Grüneisen parameter in Eq. (8) the quasi-static expansion in the thin film
geometry is related to the ultrafast expansion coefficient
$\alpha_{3}^{\text{uf}}$, which differs from the corresponding equilibrium
expansion coefficient $\alpha_{3}$ that is used for three-dimensional
expansion (Eq. (6)):

$\alpha_{3}^{\text{uf}}=\alpha_{3}\left(1+\sum_{k\neq
3}\frac{c_{33kk}\alpha_{k}}{c_{3333}\alpha_{3}}\right)\,.$ (11)

In case of isotropic solids this expression simplifies due to identical values
of the off-diagonal elements of the elastic tensor and isotropic expansion
coefficients $\alpha_{i}=\alpha$. For a metal with a typical Poisson constant
of $\nu=c_{1133}/(c_{3333}+c_{1133})\approx 1/3$, the ultrafast expansion
coefficient

$\alpha^{\text{uf}}=\alpha\left(1+2\cdot\frac{c_{1133}}{c_{3333}}\right)\approx
2\alpha\,.$ (12)

is approximately twice times larger than the equilibrium constant, because the
Poisson stresses are absent in typical picosecond ultrasonic experiments on
homogeneous thin films. Therefore, the Poisson effect requires considering the
morphology of the thin film (Fig. 4), because it influences the dimensionality
of the strain response and hence the amplitude of the strain response to the
laser-induced temperature increase.

The one-dimensional nature of the picosecond strain response of continuous
thin films was already considered implicitly by the seminal work by Thomsen et
al. [7]. Their formulation of the one-dimensional wave equation (13) can be
transformed to the much simpler Grüneisen formulation (Eq. (15)) by
considering the bulk modulus $B=(c_{3333}+2c_{3311})/3$ and the Poisson factor
$\nu$:

$\displaystyle\rho_{\text{m}}\frac{\partial^{2}u_{3}}{\partial t^{2}}$
$\displaystyle=\frac{\partial}{\partial
x_{3}}\left(3\frac{1-\nu}{1+\nu}B\eta_{33}-3B\alpha\Delta T\right)$ (13)
$\displaystyle=\frac{\partial}{\partial
x_{3}}\left(c_{3333}\eta_{33}-c_{3333}\alpha^{\text{uf}}\Delta T\right)$ (14)
$\displaystyle=\frac{\partial}{\partial
x_{3}}(\underbrace{c_{3333}\eta_{33}}_{\sigma^{\text{elastic}}_{33}}-\underbrace{\Gamma\rho_{Q}}_{\sigma^{\text{ext}}_{33}})\,.$
(15)

Equation (14) illustrates that the modified thermal expansion coefficient
$\alpha^{\text{uf}}$ has to be used to quantify the laser-induced stress in
the absence of an in-plane expansion. This formulation is useful as it
connects $\alpha^{\text{uf}}$ to the quasi-static strain
$\eta_{33}^{\text{qs}}$ (see Eq. (10)) that exists in homogeneous thin films
on timescales where strain pulses have propagated out of the investigated
region of interest, but in plane motion is still negligible due to the large,
homogeneously exited area of the thin film. When $t>t_{\text{1D}}$ the film
laterally relaxes and induces the Poisson stresses that re-establish the
thermal expansion coefficients used in Eq. (6).

Finally, the simple formulation of the wave equation given in Eq. (15)
highlights the direct access to the spatio-temporal energy density by
measuring the transient strain response using the Grüneisen concept. This
perspective is particularly useful in the context of materials, where the
excitation of several degrees of freedom (e.g. electrons, phonons and spins)
simultaneously contribute to the stress with different time-dependencies as
discussed in Section 3.4.

###  3.3  Energy transfer processes and diffusive two temperature models

The elastic wave equation (14) relates the picosecond strain response of a
homogeneous thin film to a laser-induced stress that is characterized by a
temperature increase $\Delta T=T_{\text{f}}-T_{\text{i}}$. This simple
formulation contains the strong assumption that the temperatures of the
different degrees of freedom - in particular electrons and phonons - is the
same.

Under this assumption the shape of both the driven picosecond strain pulses
and the spatio-temporal distribution of the remaining quasi-static expansion
$\eta_{33}^{\text{qs}}$ is determined by the spatio-temporal profile of the
temperature $T(x_{3},t)$ that is a solution to the one-dimensional heat
equation:

$C_{V}(T)\frac{\partial T}{\partial t}=\frac{\partial}{\partial
x_{3}}\left(\kappa(T)\frac{\partial T}{\partial x_{3}}\right)+S\,,$ (16)

with the thermal conductivity $\kappa=\kappa(T(x_{3},t))$ that inherits its
depth-dependence from the temperature profile and differs for different
materials. In addition, the description of heat transport across interfaces
may require considering interface resistances that account for different
dispersion relations of the involved quasi-particles [60, 61, 62]. The
absorption of energy from the incoming photons is treated by the source term
$S(x_{3},t)$ which is in the most simple case described by Lambert-Beer’s law.
Especially in heterostructures and films thinner than the optical skin depth
it is often necessary to consider internal optical reflections, which can be
accounted for using transfer matrix approaches [63, 64]. The one-dimensional
approach is again only valid for thin, lateral homogeneously excited films
where in-plane thermal transport within the probed volume can be neglected.

However, for most of the materials assuming a single temperature, i.e., quasi-
instantaneous equilibration of the electrons, phonons or any other energy
reservoirs in the solid is a strong oversimplification. In typical metals such
as Au, Cu and Pt the laser excitation leads to a sudden increase of the energy
density in the electron system, which is subsequently transferred to phonons
within few picoseconds. This coupling of the subsystems on a timescale
comparable to the relaxation of the lattice may be crucial for the induced
stress and the driven picosecond strain pulses as experimentally demonstrated
for Al [58], Au [34] and Ni [65]. In contrast, the strain response of
materials that exhibit a very strong electron-phonon coupling such as SrRuO3
are in some scenarios sufficiently well described by a single temperature [66,
67].

In addition to the energy distribution among different degrees of freedom, a
one-temperature model also oversimplifies spatial heat transport within metal
heterostructures because it disregards non-equilibrium transport phenomena
like ballistic [68, 69] and super-diffusive [70, 71] electron transport.
Already the modeling of the diffusive transport within the Pt-Cu-Ni metal
stack discussed in Section 2 requires a two-temperature model (2TM) that
captures the electronic thermal conductivity $\kappa^{\text{el}}\propto
T^{\text{el}}/T^{\text{ph}}$ [56, 72] enhanced by a long lasting non-
equilibrium of electrons and phonons ($T^{\text{el}}(x_{3},t)\gg
T^{\text{ph}}(x_{3},t)$) due to weak electron-phonon coupling in Cu. In
general, the propagation of quasiparticle excitations following the non-
equilibrium after optical excitation can be discussed in a quantitative and
state-resolved way by Boltzmann-transport equations [73, 74, 75]. However, in
most cases the conceptually simpler diffusive 2TM suffices:

$\displaystyle\begin{split}C^{\text{el}}(T^{\text{el}})\frac{\partial
T^{\text{el}}}{\partial t}&=\frac{\partial}{\partial
x_{3}}\left(\kappa^{\text{el}}(T^{\text{el}},T^{\text{ph}})\frac{\partial
T^{\text{el}}}{\partial x_{3}}\right)\\\ &\leavevmode\nobreak\ -g^{\text{el-
ph}}\left(T^{\text{el}}-T^{\text{ph}}\right)+S\,,\\\
C^{\text{ph}}(T^{\text{ph}})\frac{\partial T^{\text{ph}}}{\partial
t}&=\frac{\partial}{\partial
x_{3}}\left(\kappa^{\text{ph}}(T^{\text{ph}})\frac{\partial
T^{\text{ph}}}{\partial x_{3}}\right)\\\ &\leavevmode\nobreak\ +g^{\text{el-
ph}}\left(T^{\text{el}}-T^{\text{ph}}\right)\,.\end{split}$ (17)

Such a diffusive 2TM not only includes the coupling between the two subsystems
(here by the electron-phonon coupling constant $g^{\text{el-ph}}$) but also
the individual diffusion of electrons and phonons. Transport across interfaces
can be treated via the spatial dependence of the thermo-physical parameters.
By modeling the thermal conductivity of electrons $\kappa^{\text{el}}$ as a
parameter depending on both the electron and phonon temperature, we can even
rationalize the picosecond ultrasonic response of thin metal nanolayers, where
ballistic or superdiffusive transport occurs. However, the analysis of our PUX
experiments has up to now not depended on finer details of the electron
transport since very rapid processes may be masked by the comparatively slow
rise of the strain limited by the sound velocity. Therefore, studying details
of non-equilibrium electron transport would require ultrathin detection
layers.

In materials with magnetic order, the excitation of the magnetic degrees of
freedom has to be treated in addition to the electron and phonon subsystems.
Studies of the electron-phonon coupling in ferromagnetic (FM) transition metal
elements (Ni, Fe, Co) in the high excitation fluence regime observe a distinct
fluence and temperature dependence of the electron-phonon coupling timescale
[76, 77, 78]. In order to explain this observation the authors explicitly
treated the excitation of magnetic degrees of freedom via electrons by
extending the 2TM. However, the variety of different demagnetization behaviors
[79, 80, 81, 82] is related to different timescales of energy transfer to
magnetic degrees of freedom. Therefore, modeling the spatio-temporal
excitation of quasi-particles requires the explicit treatment of magnetic
excitations in three-temperature, N-temperature or even more complex models
[83, 84, 85].

###  3.4  Subsystem-specific stresses and Grüneisen parameters

The prevailing paradigms in picosecond ultrasonics are thermoelastic stresses
that drive the observed picosecond strain response according to the elastic
wave equation (15). Here, the introduced Grüneisen parameter $\Gamma_{3}$
linearly relates the laser-induced stress $\sigma_{33}^{\text{ext}}(x_{3},t)$
to the spatio-temporal energy density $\rho_{Q}(x_{3},t)$.

This energy density initially deposited to electronic excitations is
subsequently distributed within the sample structure and also locally
transferred to other degrees of freedom as already discussed in Sec. 3.3. This
heat transport and subsystem couplings determine the spatio-temporal energy
densities $\rho_{Q}^{r}(x_{3},t)$ stored in each subsystem $r$ that add up to
the total deposited energy density $\rho_{Q}(x_{3},t)$:

$\displaystyle\rho_{Q}(x_{3},t)=\sum_{r}\rho_{Q}^{r}(x_{3},t)\,.$ (18)

Within the Grüneisen approach the energy density deposited in the respective
degrees of freedom is linearly related to a stress contribution
$\sigma_{33}^{r}=\Gamma_{3}^{r}\rho_{Q}^{r}$ by subsystem-specific Grüneisen
parameters $\Gamma_{3}^{r}(x_{3})$ [34, 8, 25, 86, 87]. In case of different
subsystem-specific Grüneisen parameters $\Gamma_{3}^{r}$ the total stress in
Eq. (15) has to be adapted in order to individually treat the subsystem
contributions to the laser-induced stress $\sigma_{33}^{\text{ext}}$:

$\displaystyle\sigma_{33}^{\text{ext}}=\sum_{r}\sigma_{33}^{r}(x_{3},t)=\sum_{r}\Gamma_{3}^{r}\rho_{Q}^{r}(x_{3},t)\,,$
(19)

i.e., their superposition gives the total laser-induced stress driving the
picosecond strain response.

It depends on the properties of the material under investigation, which
degrees of freedom have to be considered separately to account for the total
spatio-temporal external stress. In general, the individual treatment of
different degrees of freedom is only necessary for describing the picosecond
strain response if the subsystem-specific Grüneisen parameters differ.
Conversely, the strain measurement only provides access to ultrafast
microscopic processes if the involved quasi-particle excitations contribute
differently to the total external stress. The separation of subsystem
contributions to the strain response of the atomic lattice is schematically
visualized in Fig. 5 for the case of a material where electronic excitations,
phonons and magnetic excitations contribute.

The quasi-instantaneous electronic stress contribution is captured by an
electronic Grüneisen parameter
$\Gamma^{\text{el}}=\frac{\partial\ln{\gamma_{S}}}{\partial\ln{(V)}}$ that can
be derived as first-order approximation from the electronic density of states
at the Fermi level that determines the Sommerfeld constant $\gamma_{S}$ [88].
The value depends on details of the band structure and for the idealized case
of free electrons in a parabolic band with spherical symmetry in the
Sommerfeld approximation one obtains $\Gamma^{\text{el}}=2/3$ [88]. The
subsequent transfer of energy to phonons gives rise to a phonon stress that
can be parameterized to first order by a macroscopic Grüneisen constant
$\Gamma^{\text{ph}}$ assuming a similar phonon population as in thermal
equilibrium. However, mode-specific electron-phonon coupling can give rise to
long lasting non-equilibria of the phonon system itself [25, 59] which becomes
highly relevant for the modeling of picosecond acoustics in case of strongly
mode-specific Grüneisen parameters
$\Gamma^{\text{ph}}_{l}=\frac{\partial\ln{(\hbar\omega_{l})}}{\partial\ln{(V)}}$.
Here the change of the respective phonon energy $\hbar\omega_{l}$ with volume
$V$ parametrizes the efficiency of a class of phonon modes $l$ to generate
stress. In various semiconducting materials such as Tellurides the transverse
phonon modes exhibit negative Grüneisen parameters in contrast to the
typically positive Grüneisen parameter of longitudinal phonon modes [89, 88].
In thermal equilibrium this results in NTE at low temperatures since their
low-frequency transverse acoustic modes are already excited at lower
temperatures, as opposed to the longitudinal acoustic modes with higher
frequency. In case of strong non-equilibria between phonon modes with strongly
mode-specific Grüneisen parameters upon laser-excitation, the individual
treatment of the different modes is necessary. However, in metals the
application of a macroscopic phononic Grüneisen parameter is typically
sufficient to quantify the transient phonon stress [58, 34, 65, 56, 90].

Figure 5: Grüneisen concept for laser-induced stress on the lattice: This
viewgraph depicts the treatment of magnetic excitations in addition to the
optically excited electrons and phonons for the laser-induced stress using the
Grüneisen concept. The total equilibrium heat capacity is the superposition of
contributions from all subsystems. The energy density stored in their
excitations generates stresses on the lattice according to the subsystem-
specific Grüneisen parameters $\Gamma_{3}^{r}$. Finally, the strain response
is driven by the time- and space-dependent superposition of all subsystem-
stress contributions that exclusively depend on the energy transfer into each
subsystem.

In laser-excited magnetically ordered metals, the laser-induced spin disorder
provides an additional magnetic stress contribution in addition to the
electron and phonon stresses. For a Heisenberg exchange interaction the
respective Grüneisen parameter can be expressed as
$\Gamma^{\text{mag}}=\frac{\partial\ln{(J)}}{\partial\ln{(V)}}$ resulting from
the dependence of the exchange constant $J(V)$ on the volume of the unit cell
[91, 92, 93]. However, so far only few experiments [94, 86, 87, 95, 96] have
investigated the magnetic stress that adds to the electron and phonon stress
contribution as discussed in Sec. 4.4.

Figure 5 graphically represents the main idea of the Grüneisen approach
treating these three stresses $\sigma^{r}_{3}=\Gamma^{r}_{3}\rho_{Q}^{r}$ with
$r=\\{\text{el,ph,mag}\\}$ that all act on the strain $\eta_{33}$. The central
idea is, that by Hooke’s law – and more generally by Eq. (10) – the strain is
a linear measure of each of the stress contributions. In the Grüneisen model
for several subsystems within the same material each of the stresses is
proportional to the respective energy density $\rho_{Q}^{r}$ which can be
expressed via the specific heat contributions $C^{r}$ in thermal equilibrium.
Introducing subsystem-specific Grüneisen parameters is useful as the thermal
expansion coefficients and the heat capacities often share the same
temperature-dependence which originates from the same occupation probability
of the underlying quantum states [97]. In essence, the macroscopic out-of-
plane Grüneisen parameters $\Gamma^{r}_{3}$ encode, how efficiently energy
densities $\rho_{Q}^{r}(x_{3},t)$ in each of the subsystems generate out-of-
plane stress $\sigma^{r}_{33}(x_{3},t)$. Their extraction is exemplarily
demonstrated for the rare-earth element Dy in Section 4.4 and requires the
separation of the specific heat and thermal expansion into the subsystem
contributions [98, 99, 100, 89]

In total, the Grüneisen approach provides a separation of the laser-induced
stress into distinct degrees of freedom, which is linear in the contributing
energy densities, and can be used even if these subsystems are out-of-
equilibrium with each other for many picoseconds [101, 59, 25, 102, 103]. In
such situations, the stress and strain only depend on the energy density
transferred between the subsystems under the boundary condition of energy
conservation. Therefore, the subsystem-specific Grüneisen parameters
determining the transient stress contributions enable the description of
expansion and contraction in thermal equilibrium and after photoexcitation on
an equal footing.

###  3.5  Numerical modelling of the picosecond strain response

Figure 6: Visualization of the contributions to the elastic response of a
laser-excited heterostructure: The transducer layer (thickness $L_{1}$) on top
of a non-absorbing detection layer (thickness $L_{2}>L_{1}$) exhibits an
absorption profile for the optical excitation that is indicated as red area on
top of the sample structure (a). (b) LCM representation of the elastic
response wherein spheres represent a local mass element, springs encode the
elastic coupling and incompressible spacer sticks represent the laser-induced
external stress. The motion of the masses is greatly enlarged, and thermal
fluctuations are omitted for better visibility. The quasi-static strain is
reached when the springs have relaxed to their equilibrium length and a new
inter-atomic distance is attained. (c) Spatio-temporal maps of the external
stress $\sigma^{\text{ext}}_{33}$, the elastic stress
$\sigma^{\text{elastic}}_{33}$, the total stress $\sigma^{\text{tot}}_{33}$,
the atomic displacement $u_{3}$, and its spatial derivative, the strain
$\eta_{3}$. (d) Spatial profiles of the same quantities at selected times. (e)
shows the spatially averaged strain $\eta$ of the individual layers of the
bilayer heterostructure, which can be compared with the center-of-mass
evolution of the Bragg peaks extracted from a PUX experiment.

Modelling the time-dependent strain response allows us to obtain the spatio-
temporal stress profile $\sigma_{33}^{\text{ext}}(x_{3},t)$ that occurs as
source term in the one-dimensional elastic wave equation (9). This provides
insights into energy transfer processes as the energy density distribution
between the subsystems $\rho_{Q}^{r}(x_{3},t)$ determines the external stress
contributions $\sigma^{r}_{33}(x_{3},t)$ as introduced in Section 3.4.
Different approaches for solving Eq. (15) for a given
$\sigma_{33}^{\text{ext}}(x_{3},t)$ exist. Analytical solutions for the strain
field can be constructed for time-independent stress profiles as shown in [7,
104, 6]. Numerical approaches [105, 106] may be easier to implement, when the
time dependence of the stresses by sub-system couplings, thermal diffusion and
interface effects need to be accounted for. A natural spatial grid in a
numerical simulation is provided by the atomic layers, or unit-cells which
also represent the smallest physically meaningful discretization of the strain
response. Linear chain models (LCMs) of masses and springs provide an
intuitive approach for the numerical calculation of the strain field
$\eta_{33}(x_{3},t)$ with unit-cell precision [107]. Publicly available
implementations of a LCM are, for example, provided by the udkm1Dsim Matlab
[105] and Python [106] code-libraries, which include modules for modelling the
pump-laser absorption profile using a transfer matrix model, heat-transport
via diffusive N-temperature models, the strain response and dynamical x-ray
scattering, which can also include magnetic scattering.

In the following, we display and discuss the modeled laser-driven strain
response for a generic sample structure that consists of an opaque transducer
with thickness $L_{1}$ and a transparent detection layer of thickness $L_{2}$
grown on a semi-infinite, transparent substrate. Fig. 6 illustrates the sample
structure and the space- and time-dependent results for the terms in the
elastic wave that is solved numerically using in the udkm1Dsim code [105,
106]. The strain response can be visualized by a LCM as shown in Fig. 6(b),
which serves as a mechanistic analog for the time-dependent elastic response
of the bilayer structure sketched in Fig. 6(a). The time-dependent position of
the masses in the LCM visualizes the displacement $u_{3}$ averaged over a
$5\,\text{nm}$ length-fraction of the sample, where the elongation of the
adjacent spring encodes the corresponding local elastic stress
$\sigma^{\text{elastic}}_{33}$. The thickness $L_{1}$ of the metallic
transducer is chosen to exceed the optical penetration depth leading to an
inhomogeneous energy deposition shown in Fig. 6(a). For simplicity, we assume
an instantaneous rise of the laser-induced external stress
$\sigma^{\text{ext}}_{33}$ that is indicated by the appearance of
incompressible red spacer sticks compressing the springs of the linear chain
directly after excitation at $t=0$ (see Fig. 6(b)). Unbalanced gradients in
the external stress drive the elastic response, which consists of an expansion
wave starting at the sample surface and a compression within the
inhomogeneously excited transducer. The resulting strain pulse propagates at
the speed of sound from the transducer through the detection layer into the
substrate. Reflections at the interfaces are omitted for clarity by choosing
perfect acoustic impedance matching of all constituents. When the strain pulse
has passed, the material reaches its new equilibrium position where the
residual quasi-static expansion is represented by the spacer sticks’ length
that varies due to heat diffusion within the heterostructure. Note that the
LCM representation of the strain response in Fig. 6(b) omits thermal
fluctuations and strongly exaggerates the strain response.

The color maps and selected profiles in Fig. 6(c) and (d) provide a numerical
representation of the elastic stress, the total stress, the driven
displacement $u_{3}$ and the corresponding strain $\eta_{33}$ that appear in
the elastic wave equation (9). Figures 6(c) and (d) illustrate that the finite
total stress $\sigma_{33}^{\text{tot}}$ arising from the unbalanced external
stress $\sigma_{33}^{\text{ext}}$ drives a displacement of the lattice. The
rising strain response $\eta_{33}$ induces an elastic stress
$\sigma_{33}^{\text{elastic}}$ that lowers the total stress by partially
compensating the external stress within the transducer. After the strain pulse
has propagated into the substrate the elastic stress arising from the induced
quasi-static expansion compensates the external stress which results in a
vanishing total stress, which marks the quasi-static state. The resulting
strain is given by the derivative of the displacement and displays both the
quasi-static expansion and the strain pulse propagating through the
heterostructure777The pronounced high frequency oscillations in panel (d) of
Fig. 6 originate from surface vibrations, which occur due to the
discretization of the strain response, which can be relevant for atomically
flat samples [108, 109]. They are not present in analytic solutions of the
continuum elastic wave equation..

Fig. 6(e) displays the average strain of the transducer and the detection
layer that can be inferred from the simulated strain field
$\eta_{33}(x_{3},t)$ and compared to the strain extracted from the Bragg peak
shift in a PUX experiment. While the average strain of the laser-excited
transducer contains contributions from the strain pulse and the quasi-static
expansion, the strain response of the non-absorbing detection layer is
dominated by the propagating strain pulse. The timing of the inflection points
in the average strain response depends on the layer thicknesses, whereas the
shape of the rising and falling edge in the strain response is related to the
stress-profile as discussed in [66, 87]. At the delay
$t_{1}=L_{1}/v_{\text{s}}$ the strain pulse has propagated through the
transducer and the compressive part has fully entered the detection layer.
This causes a maximum expansion of the transducer due to the superposition of
the quasi-static expansion and the expansive part of the strain pulse as well
as a maximum compression of the detection layer. Subsequently, also the
expansive part enters the detection layer, which reduces the expansion of the
transducer until the strain pulse has completely left the transducer at
$2t_{1}$. The entrance of the expansive part and the propagation of the
compressive part from the detection layer into the substrate both result in an
overall expansion of the detection layer. As the compressive part of the
strain pulse has completely entered the substrate at $t_{1}+t_{2}$ the
detection layer reaches its maximum expansion which subsequently decreases as
the expansive part of the strain pulse exits towards the substrate. Finally,
for $t\geq 2t_{1}+t_{2}$ only the quasi-static expansion remains in the
detection layer, which varies due to thermal transport.

Modeling the laser-driven strain response via the elastic wave equation is
independent of experimental probing technique e.g. optical probe pulses or
x-ray diffraction. A direct comparison between experimental data with the
model, however, requires a weighting of the modeled spatio-temporal strain map
$\eta_{33}(x_{3},t)$ by a sensitivity function that is specific for the
detection method. The large penetration depth of hard x-rays on the order of
microns enables the homogeneous probing of strain in nanostructures.
Therefore, the center of mass evolution of the Bragg observed in PUX
experiments is, in many cases, well-approximated by the average strain in the
probed material. This no longer applies in case of variations of the
crystallinity within a layer which modifies the sensitivity and thus require a
weighting the modelled strain map. Furthermore, strongly inhomogeneously
strained layers exhibit peak broadening or even splitting effects [66, 110]
that complicate the relation between Bragg peak shift and average strain and
require explicitly modelling the Bragg peak evolution via kinematical or
dynamical x-ray diffraction [105, 106]. Furthermore the occurrence of laser-
induced structural phase transitions may complicate the extraction of strain
from the diffraction peak evolution [111, 112, 31]. The presented numerical
modelling approach not only helps to rationalize experimental data, but also
aids in the design of the sample structure as the amplitude, shape and timing
of the detected strain signal can be predicted even for complex
heterostructures [56, 36, 113, 87].

##  4 Use cases for PUX

This section is dedicated to the presentation of PUX experiments that
illustrate the theoretical concepts discussed in the previous section by
utilizing the capabilities of hard x-ray diffraction as a quantitative,
material specific probing technique.

In Section 4.1 we exemplify how the large x-ray penetration depth extends the
sensitivity of classical picosecond ultrasonics beyond the near-surface region
of a metallic transducer. A quantitative comparison of the strain response of
a nanogranular and a continuous FePt film in 4.2 highlights the importance of
the Poisson stresses and geometrical constraints discussed in 3.1 and 3.2.
Signatures of the energy transfer processes discussed in 3.3 are demonstrated
in 4.3 using PUX on a nanoscopic Au/Ni heterostructure with and without an
insulating MgO interlayer. Section 4.4, demonstrates the utility of sub-system
specific Grüneisen parameters introduced in 3.4 for a Dy transducer that
exhibits giant magnetic stress contributions which result in a contraction
upon laser-excitation.

###  4.1 Sensing shape and timing of strain pulses in buried layers

Observing the timing, shape and amplitude of picosecond strain pulses is
central to picosecond ultrasonics experiments. Here, we showcase the ability
of PUX to track the propagation of strain pulses within an opaque
heterostructure consisting of a thick transducer on a thin detection layer
[36]. We illustrate that the presence of a transparent capping layer on top of
the transducer leads to the emission of unipolar strain pulses towards the
sample surface and a pronounced asymmetry of the bipolar strain-pulse that
propagates towards the substrate.

In particular, we discuss the strain response of heterostructures that consist
of a few hundred nm thick TbFe2 transducer on top of a $50\,\text{nm}$ thin Nb
detection layer grown on a Al2O3 substrate, which are capped with an
amorphous, transparent SiO2 layer with a variable thickness ranging from $0$
to $1100\,\text{nm}$ as sketched in Fig. 7(d–f). The samples are excited by
femtosecond laser pulses with an optical penetration depth of $30\,\text{nm}$
in the TbFe2 layer and the resulting strain responses observed via PUX are
depicted in Fig. 7(a–c). Further details on the sample growth and experimental
parameters are given in [36].

Figure 7: Sensing propagating strain waves within an opaque heterostructure
with a variable capping: Transient strain of the metallic TbFe2 transducer
(yellow) and Nb detection layer (blue) for an uncapped structure (a), a
structure with a transparent $550\,\text{nm}$ SiO2 capping (b) and a
$1100\,\text{nm}$ SiO2 capping (c). Solid lines represent the modeled strain
response using the udkm1Dsim toolbox [105] using parameters given in [36].
Panels (d) to (f) sketch the corresponding sample structures that are laser-
excited from the TbFe2 side. A comparison of the strain response shows that
the SiO2 capping layer separates the bipolar strain pulse emitted by the
uncapped transducer (a) into an asymmetric bipolar strain pulse and a unipolar
strain pulse (b, c) whose delay with respect to the bipolar pulse is set by
the thickness of the capping layer and its sound velocity.

The femtosecond laser pump-pulse deposits energy into the near surface region
inducing an expansion of the $450\,\text{nm}$-thick TbFe2 transducer of the
uncapped structure, that launches a bipolar strain pulse with a leading
compression and a trailing expansion propagating through the metal
heterostructure into the substrate as discussed in Sec. 3.5. Fig. 7(a)
displays the corresponding rise of the average strain of the TbFe2 layer and
the bipolar shape of the strain response of the Nb detection layer that are
extracted from the diffraction peak shifts. While the strain pulse travels at
the speed of sound, the deposited energy density causing a quasi-static
expansion of the transducer diffuses into the Nb detection layer on a
nanosecond timescale. The different propagation speeds of sound and heat thus
yield a background-free signature of the strain pulse in the buried Nb
detection layer. The average strain in Nb is determined by the integral over
the part of the strain pulse within the Nb layer. When the compressive part
enters the Nb layer at $100\,\text{ps}$ it exhibits a negative strain until
$125\,\text{ps}$ when the trailing tensile part has entered the layer and the
compressive part has progressed towards the substrate. Subsequently, the
tensile part dominates and the resulting positive strain recedes as the strain
pulse propagates into the substrate. The average strain of the Nb layer
detected via PUX is thus determined by the spatial shape of the bipolar strain
pulse convoluted with the propagation of the strain pulse through the layer.

Adding an additional transparent SiO2 layer on top of a $350\,\text{nm}$ TbFe2
transducer modifies both the shape and timing of the emitted strain pulses as
shown in Fig. 7(b) and (c). As before, the expansion of the TbFe2 launches a
bipolar strain pulse propagating towards the Nb detection layer. In addition,
a unipolar compression pulse is launched into the SiO2 capping. Consequently,
the amplitude of the tensile part of the initial bipolar strain pulse detected
in the buried Nb layer is reduced due to conservation of elastic energy.
Subsequently, we observes a train of unipolar strain pulses with alternating
sign and a period that is given by the propagation time of the strain pulse
back and forth through the capping layer. The presence of the unipolar strain
pulse inside the thick TbFe2 layer is heralded by small plateau-like
signatures in the measured TbFe2 strain, which is rather insensitive to the
shape of the strain pulses in comparison to the short, pronounced peaks that
occur in the strain response of the thin Nb detection layer. It is interesting
to note that if we extrapolated the results in Fig. 7(b) and (c) with a cap
layer thickness of $550\,\text{nm}$ and $1100\,\text{nm}$, respectively, to a
cap layer thickness of zero, the positive unipolar strain pulse would
superimpose onto the asymmetric bipolar pulse in such a way that the symmetric
bipolar pulse from Fig. 7(a) is recovered. One can therefore rationalize the
occurrence of the tensile part of the bipolar strain pulse shape by an
instantaneous and complete reflection of a compression wave driven at the
transducer-air interface.

Figure 8: Modeled strain map: Spatio-temporal strain of the sample structure
without SiO2 capping (a) and with $550\,\text{nm}$ capping (b) simulated using
the modular udkm1Dsim toolbox [105]. The spatio-temporal strain displays the
propagation of the strain pulses through the heterostructure and the
distribution of heat via heat diffusion indicated by the slowly growing
expanded part of the TbFe2 transducer.

We modelled the strain response of the TbFe2 transducer and the Nb detection
layer for all heterostructures using the approach introduced in Section 3.5
with parameters given in [36]. The modeled average layer strains shown as
solid lines in Fig. 7 match the experimental data indicated by symbols.
Moreover, the modelling yields detailed spatio-temporal maps of the strain
inside the heterostructures without and with SiO2 capping layer which are
depicted in Fig. 8(a) and (b), respectively. The modeled strain maps provide a
detailed depiction of the strain profile within the structure at any given
time, which extends the insights from the average layer strain obtained via
PUX experiments. Fig. 8(b) shows that the unipolar strain pulse launched into
the capping layer inverts its sign upon reflection at the sample surface and
that a considerable fraction of the returning wave is reflected at the
SiO2-TbFe2-interface due to an acoustic impedance mismatch between these two
materials. This rationalizes the multiple echoes of unipolar pulse that
consecutively traverse the structure as revealed in Fig. 7(b) and (c).
Modeling the strain allows us to precisely calibrate the layer thicknesses
from the timing of the detected strain pulses. However, the modeled strain map
does not only help to rationalize the strain observed in a PUX experiment but
can furthermore guide and support the interpretation of strain signatures in
all-optical data, e.g. from time-resolved magneto-optics and reflectivity [15,
36, 114].

Overall, we find that the observation of the strain pulses in a thin
crystalline detection layer is advantageous compared to the analysis of the
strain response of a thick transducer layer. The separation of the detection
layer from the laser-excitation region separates the strain signatures of
sound and heat in the time domain which can be exploited for the detection of
strain pulses with an unconventional shape, as shown in Section 4.4. The
current example shows that in principle only the detection layer needs to be
crystalline to observe the strain pulses, which extends the applicability of
PUX to a large class of heterostructures and transducer materials.

###  4.2 Quantifying a morphology-dependent strain response

Here, we compare the qualitative and quantitative picosecond strain response
of thin films for various in-plane expansion constraints that are shown to
affect the out-of-plane strain response. Different sample morphologies change
the nature of the ultrafast strain response from one-dimensional in the case
of a homogeneous film to three-dimensional in the case of nanograins even if
they are attached to a substrate. Describing the ultrafast strain response of
granular morphologies requires a model for the three-dimensional elastic
response in accordance with Eq. (5) which includes Poisson stress
contributions that couple in-plane and out-of-plane motion.

Specifically, we discuss the picosecond strain response of three
$10\,\text{nm}$ thin crystalline $\text{L1}_{0}$-phase FePt layers having
their tetragonal c-axis oriented out-of-plane. Interestingly, the
$\text{L1}_{0}$-phase of FePt exhibits a vanishing expansion along its c-axis
($\alpha_{3}\approx 0$ [115, 95]) under near-equilibrium heating conditions
and a distinct expansion along the in-plane directions
($\alpha_{1}=\alpha_{2}\approx 9\cdot 10^{-6}$) [116, 95]. Fig. 9 contrasts
the laser-induced strain response along the tetragonal c-axis for a continuous
and a granular thin film on an MgO substrate as measured via UXRD [117] and a
quasi-free-standing granular film transferred to a transmission electron
microscopy (TEM) grid measured via ultrafast electron diffraction [94]. The
compiled experiments used comparable laser excitation conditions with a
fluence of $\approx 6\,\mathrm{mJ/cm^{2}}$. Both substrate-supported FePt
samples, i.e., the continuous and the granular thin film, respond on average
by an out-of-plane expansion upon laser-excitation despite the Invar-like
behavior in near-equilibrium heating conditions. However, the free-standing
grains on the TEM grid exhibit an initial out-of-plane contraction while the
lattice expands in-plane (not shown here) [94]. After the coherent strain
pulse oscillations have ceased, one observes a nearly vanishing out-of-plane
expansion, in agreement with the Invar-like behavior that is expected from
quasi-static heating experiments.

Figure 9: Morphology-dependent strain response of FePt $10\,\text{nm}$
specimen: Comparison of the laser-induced out-of-plane strain-response of
$\text{L1}_{0}$-phase FePt for three different in-plane boundary conditions:
(a) continuous thin film (b) FePt nanograins on an MgO substrate (c) free-
standing FePt grains on a TEM grid. The data in panel (a) and (b) are from
UXRD measurements [117] at an incident laser fluence of $\approx
6\,\mathrm{{mJ}/{cm^{2}}}$. The data in panel (c) are reproduced from the
publication by Reid et al. [94] in order to highlight the qualitative
differences in the FePt strain response. Reid et al. determined the strain
response for a comparable laser-fluence of $5\,\mathrm{{mJ}/{cm^{2}}}$ using
time-resolved electron diffraction. Solid lines in in (a–c) are derived from
modeling approaches discussed in the corresponding publications [117, 94] and
schematic insets depict the sample morphologies. Panels (d–f) illustrate the
constrained expansion with respect to the initial condition indicated as black
dashed line.

The schematic depictions (Fig. 9(d–f)) adjacent to the data illustrate the
hypothesis for the data interpretation that rationalizes the observed
behaviors [117, 95]. They sketch the equilibrium film dimensions as a dashed
black rectangle and the change in dimensions upon laser excitation in color.
In case of the free-standing grains the modeling of the picosecond strain
response requires the three-dimensional elastic wave equation Eq. (5) that
includes a coupling of the in-plane and out-of-plane strain response. The non-
vanishing laser-driven in-plane strains introduce a time-dependent out-of-
plane Poisson stress contribution in addition to the laser-induced out-of-
plane stresses from electrons, phonons and spin-excitations [95, 94]. The
rising in-plane strain induces a contractive out-of-plane Poisson stress that
dominates the strain response of the free-standing grains. When the coherent
motion has ceased at $\approx 20\,\text{ps}$, the remaining quasi-static
expansion is given by the near equilibrium thermal expansion coefficient
$\alpha_{3}\approx 0$ according to Eq. (6) as indicated by the out-of-plane
Invar behavior and an in-plane expansion.

In contrast, the strictly one-dimensional picosecond strain response of the
continuous film along the out-of-plane direction can be described by the one-
dimensional elastic wave equation (9) derived in Section 3.2. Because the
homogeneous laser excitation of a thin film lacks in-plane stress gradients,
there is no in-plane motion on ultrashort timescales. The absence of the
Poisson stress contributions modifies the out-of-plane thermal expansion
coefficient on ultrafast timescales according to Eq. (11), which in the case
of FePt simplifies to:

$\displaystyle\alpha_{3}^{\text{uf}}$
$\displaystyle=\alpha_{3}+2\frac{c_{3311}}{c_{3333}}\alpha_{1}=\alpha_{3}+2\frac{\nu}{1-\nu}\alpha_{1}\,.$
(20)

Assuming a Poisson ratio $\nu=1/3$, as is common for metals one finds that
$\alpha_{3}^{\text{uf}}$ is equal to $\alpha_{1}$ and thus positive despite
the vanishing $\alpha_{3}$. According to Eq. (8) this is related to a positive
out-of-plane Grüneisen parameter that translates deposited energy density to a
positive out-of-plane stress $\sigma_{33}^{\text{ext}}$ that is fully
compensated by the Poisson stress in thermal equilibrium.

The nanograins attached to a substrate represent an intermediate case between
the freestanding grains and the continuous film morphology. Accordingly, they
exhibit an intermediate behavior with a short-lived contraction that is
followed by a similar, albeit smaller out-of-plane thermal expansion compared
to the continuous FePt film. The clamping to the substrate partially hinders
the in-plane expansion of the FePt grains as sketched in Fig. 9(e). The nature
of the strain response for this specimen is intrinsically three-dimensional
and further complicated by the additional boundary conditions introduced by
the clamping to the substrate, by a carbon filling in between the FePt grains
and the grain size distribution. Finite-element modeling is however able to
reproduce the observed short-lived contraction followed by an out-of-plane
expansion for these boundary conditions [95]. The partially allowed in-plane
expansion induces a Poisson stress contribution that reduces the observed out-
of-plane expansion in the quasi-static limit for $t\geq 20\,\mathrm{ps}$
compared to the continuous thin film but does not lead to the Invar-like
behavior of the free-standing grains.

This use case illustrates the crucial role of the constraints for the in-plane
expansion on the out-of-plane strain response. Utilizing the picosecond strain
response for a quantitative determination of the laser-induced energy density
or even temperature change therefore requires a careful consideration of the
in-plane boundary conditions in the sample. Application of the near-
equilibrium linear thermal expansion coefficient in the common thin film
geometry overestimates the laser-induced temperature change, due to the
absence of the Poisson stress contributions on ultrafast timescales. Nanoscale
granularity drastically influences the amplitude of the out-of-plane strain,
not only for samples purposely grown as lateral nanostructures but also for
thin granular films or islands that unintentionally form during sample growth.
However, if the Poisson effect is taken into account, time-resolved strain
measurements via PUX may serve for tracking ultrafast energy transfer
processes and enable thermometry applications in nanoscale structures.

###  4.3 Identifying non-equilibrium heat transport between ultrathin layers

Here, we showcase the ability of PUX to observe the non-equilibrium energy
transfer within nanoscale heterostructures down to few-unit-cell thickness.
Time-resolved, quantitative strain measurements extend the applicability of
thermal expansion as a proxy for energy content and temperature changes from
quasi-equilibrium to the picosecond timescale. The shape of the picosecond
strain pulses provides supporting insights into the spatio-temporal stress
profile that is determined by the energy density distribution and heat
transfer within the structure which we quantify by modelling the strain
response.

In the following, we discuss the ultrafast energy transport within a metallic
bilayer of $5\,\text{nm}$ Au and $13\,\text{nm}$ Ni in the framework of a
diffusive 2 TM (Eq. (17)) that we previously applied to capture the energy
transfer and resulting strain response in similar Au-Ni [118, 62] and Au-Fe
bilayers [113] across a large range of layer thicknesses. The picosecond
strain response of both Au and Ni serves as a reference for the heat transfer
that is dominated by electronic transport within the first hundreds of
femtoseconds. In order to experimentally verify this crucial electron energy
transport we compare the strain response of the Au-Ni bilayer to the strain
response of a Au-MgO-Ni trilayer where the $8\,\text{nm}$ thin insulating MgO
interlayer inhibits electronic energy transfer among the metal layers.

Figure 10:  Subsystem-specific results from modelling the strain response of a
thin Au-Ni bilayer excited by 400 nm pump pulses using a diffusive 2TM: (a)
spatio-temporal electron temperature increase $\Delta T^{\text{el}}$, (b)
energy density increase in the electron system $\Delta\rho_{Q}^{\text{el}}$,
(c) phonon temperature increase $\Delta T^{\text{ph}}$ and (d) phonon energy
density increase $\Delta\rho_{Q}^{\text{el}}$. Panel (e) displays the
resulting laser-induced stress that drives the strain response depicted in
panel (f). (g) Layer-selective spatial averaging yields the average strain of
Au and Ni (solid lines) in reasonable qualitative and quantitative agreement
with the experimental strain response (dots). The modelling illustrates the
ultrafast electronic energy transfer, that is evident from the expansion of
the Ni which compresses the Au, that in turn only heats up only within tens of
ps as discussed in [118, 62].

Fig. 10(g) displays the picosecond strain response of the Au (yellow dots) and
the Ni (grey dots) layer of the Au-Ni bilayer to a $100\,\text{fs}$ laser
pulse with a central wavelength of $400\,\text{nm}$ observed via PUX. The
modeling process yields spatio-temporal electron and phonon temperature maps
and the corresponding energy densities that are depicted in Fig. 10(a–d). The
electron temperature map in Fig. 10(a) displays an equilibration of the
electron temperature across the metal stack within the first $200\,\text{fs}$
after laser excitation. The flat 3d-bands of Ni close to the Fermi level
provide a large density of states, which results in a large electronic
specific heat and also a large electron-phonon coupling strength compared to
Au [119]. The thermalization of the Au and Ni electrons thus rapidly transfers
most of the energy to the Ni electrons as visualized by the electronic energy
density map in Fig. 10(b). Subsequently, the energy is transferred to phonons
in Ni via electron-phonon coupling within the first $1\,\text{ps}$ to
$2\,\text{ps}$, which lowers the overall electron temperature and the energy
density stored in electron excitations. The phonon temperatures of both layers
equilibrate only on the timescale of tens of picoseconds as discussed in [118,
62] and illustrated by Fig. 10(c).

The rapidly rising energy density in the electrons and the phonons of Ni
induces a rapidly rising expansive stress via the respective positive
Grüneisen parameters (see Section 3.4) [98, 65] which remains unbalanced at
the Au-Ni interface and drives a rapid expansion of the Ni layer that
compresses the adjacent Au layer. The compression pulse propagates through the
Au layer to the sample surface where it is turned into an expansion pulse upon
reflection. This propagation of the strain pulse through the heterostructure
superimposed with the quasi-static expansion of the layers is displayed in the
strain map in Fig. 10(f) that is obtained by solving the elastic wave equation
(9) for the modelled external stress $\sigma_{33}^{\text{ext}}(x_{3},t)$.
Averaging the spatio-temporal strain over the Au and the Ni layer,
respectively, enables the comparison of the model with the experimental data
for the average strains of the layers shown in Fig. 10(g). The bipolar feature
in the Au strain in combination with the rapidly rising expansion of Ni
indicates the initial compression of the Au layer due to the dominant energy
transfer into Ni. In addition, the slowly rising expansion of Au on tens of
picoseconds indicates the surprisingly slow equilibration of both metal layers
due to a backward transfer of energy from Ni into the Au phonon system via
conduction electrons and weak electron-phonon coupling in Au and via phonon
transport across the Ni-Au interface. For heterostructures with Au thinner
than $\approx 10\,\text{nm}$ one finds that the energy transfer between the
two metals occurs predominantly via phonons [62]. The Au layer reaches its
maximum quasi-static strain and thus temperature at $\approx 100\,\text{ps}$
after which the entire bilayer cools towards the MgO substrate via diffusive
phonon transport, which results in the overall decrease of the strain in Fig.
10(g).

It is instructive to consider further excitation scenarios and sample
structures that support and extend our findings. Fig. 11(a) compares the
strain response of the Au and the Ni layer to an ultrashort laser pulse
excitation with a central wavelength of $400\,\text{nm}$ and $800\,\text{nm}$.
Fig. 11(c) and (d) display the corresponding spatial absorption profiles
calculated from literature values for the optical constants using a transfer
matrix model [120, 121, 118]. While the absorption in Ni is essentially
independent of the excitation wavelength, the Au absorption is heavily
increased at $400\,\text{nm}$. However, the strain response in the bilayer
does not depend on where the optically deposited energy is absorbed except for
an overall scaling factor, because the heat energy in the electron system is
rapidly distributed throughout the metallic heterostructure. The strong
electron-phonon coupling in Ni localizes the energy in Ni phonons that
dominate the overall strain response. The spatial profile of the source term
$S(x_{3},t)$ in the diffusive 2TM (17) is not highly relevant.

Figure 11:  Strain response for wavelength-dependent energy deposition
profiles in bi- and trilayer Au-Ni samples: Comparison between the strain
response of a sample without insulating barrier (a) and with insulating MgO
barrier (b) subjected to a laser pulses with $800\,\text{nm}$ (red lines) and
$400\,\text{nm}$ (blue lines) central wavelength. Solid lines are spline
interpolations to the data that serve as guide to the eye. Panels (c–f)
represent the modeled absorbance per length $a$ in the bi- and trilayer
structure from a transfer matrix model that uses the refractive indices for
similarly thin Au films [122] for $800$ (red areas) and $400\,\text{nm}$ (blue
areas) excitation.

This changes if we insert an insulating MgO layer between Au and Ni that
inhibits the ultrafast electronic energy transfer between the layers (Fig.
11(b)). Now the Au and Ni layers both expand individually due to the optical
excitation of electrons and subsequent local electron-phonon coupling. The
absorbed energy density in the Au and Ni layers (Fig. 11(e) and (f))
determines the stress profile which sets shape and amplitudes of the
picosecond strain pulses and the quasi-static expansion of the individual
layers within the first picoseconds. Phonon heat transport subsequently
equilibrates the temperature in the trilayer within tens of picoseconds and
cools it towards the substrate on a timescale of hundreds of picoseconds. In
the trilayer, both the coherent phonon excitation (picosecond strain pulses)
and the incoherent phonon population (thermal energy) depend drastically on
the excitation wavelength as seen in Fig. 11(b). According to the increased
absorption of the Au layer for 400 nm excitation, Au shows a strongly enhanced
quasi-static expansion around $10\,\text{ps}$ in comparison to
$800\,\text{nm}$ excitation. The strain oscillation signature of the
coherently driven picosecond strain pulse observed at $4\,\text{ps}$ in Au,
which is attributed to the compression pulse launched by the Ni expansion also
heralds the varied energy distribution. In comparison to the bilayer the
compression signature is much less pronounced in the trilayer where the
electron energy transfer from Au is prohibited. Moreover for $400\,\text{nm}$
excitation the laser-induced stress of the Au and Ni layers is almost equal as
indicated by the similar quasi-static strain around $10\,\text{ps}$ to
$20\,\text{ps}$. Accordingly, the amplitudes of the strain pulses launched by
Au and Ni become comparable and the oscillatory signatures in the early Au
strain response essentially cancel.

This use case thus displays various signatures of energy transfer in nanoscale
heterostructures and illustrates that the insertion of an insulating
interlayer suppresses the electronic transport that is key for the
surprisingly slow thermal equilibration in similar structures [118, 113, 62].
Ultrafast electronic energy transport within metallic heterostructures has
been reported and utilized previously in various all-optical experiments [123,
124, 125]. However, the layer dimensions often need to be larger than the
optical skin depth in the involved metals in order to unequivocally attribute
the experimental signals to the layers. The layer-specificity of PUX overcomes
this limitation and thus enables such investigations on few-nm thin films.

###  4.4  Detecting ultrafast negative thermal expansion

In this section, we discuss the strain response of a magnetostrictive
transducer which exhibits NTE caused by a contractive stress originating from
magnetic excitations. Depending on the temperature, i.e., the initial magnetic
state, we observe a laser-induced expansion or contraction of the transducer,
which we identify via Bragg peak shifts. The strain response of a buried
detection layer reveals that parts of the inhomogeneously excited transducer
expand while others contract upon laser excitation. We apply the Grüneisen
concept to individually treat the stress contributions of electrons, phonons
and magnetic excitations via subsystem-specific Grüneisen parameters extracted
from near-equilibrium thermal expansion experiments. Based on the strain
response of the transducer and the detection layer, which encodes the shape of
the strain pulse, we extract the spatio-temporal subsystem-specific stress
contributions. The proportionality of energy and stress in the Grüneisen
concept thus provides insight into energy transfer processes between the
subsystems.

Specifically, we discuss the picosecond strain response of a heterostructure
that consists of a $80\,\text{nm}$ Dy transducer embedded between an Y capping
and buffer layer on top of a $100\,\text{nm}$ buried Nb detection layer (Fig.
12(a)). Below its Néel temperature ($T_{\text{N}}=180\,\text{K}$) the rare-
earth Dy transducer hosts a helical antiferromagnetic (AFM) order of its large
localized 4f-magnetic moments along the c-axis of its hexagonal unit cell
which is oriented along the out-of-plane direction. At its Curie-temperature
($T_{\text{C}}=60\,\text{K}$) the Dy layer undergoes a first-order phase
transition below which the magnetic order becomes FM [126].

Figure 12: Temperature-dependent picosecond strain driven by the laser-excited
Dy transducer: The investigated heterostructure sketched in (a) contains an
opaque Dy transducer and a buried Nb detection layer. Panels (b) and (c)
display the picosecond strain response of Dy and Nb at sample temperatures
above and below $T_{\text{N}}=180\,\text{K}$ for a fixed laser fluence of
$7.2\,\mathrm{mJ/cm^{2}}$. Lowering the temperature below $T_{\text{N}}$
changes the response of Dy from expansive to contractive and modifies the
shape of the emitted picosecond strain pulse that is detected in the Nb layer.
Solid lines depict the modelling results as reported in a previous publication
[87].

Figures 12(b) and (c) display the transient average strain of the Dy
transducer and Nb detection layer, respectively, extracted from the shift of
their material-specific Bragg peaks for a fixed laser fluence of
$7.2\,\mathrm{mJ/cm^{2}}$ [87]. Measurements at selected temperatures above
and below $T_{\text{N}}=180\,\text{K}$ compare the response for initial states
with and without magnetic order, which directly shows that the laser-induced
disorder of the spin system provides a contractive magnetic stress. The Dy
transducer discussed in this section serves as a representative of the class
of heavy rare-earth elements. We obtain similar findings for Gd [127] and Ho
[86] in their respective FM and AFM phases.

In the paramagnetic (PM) phase at $250\,\text{K}$, we observe the conventional
strain response of a laser-excited metallic transducer and a buried detection
layer (see Sec. 3.5). The optical excitation of electrons in Dy within the
optical penetration depth of $\approx 25\,\text{nm}$ and the subsequent energy
transfer to phonons via electron-phonon coupling induces a rapidly rising
expansive electron and phonon stress. This drives a bipolar strain pulse with
a leading compression propagating from the Y-Dy interface through the Dy
transducer into the Nb detection layer. In addition, the laser-induced stress
gives rise to an expansion of the Dy layer reaching its maximum at $\approx
29\,\text{ps}$ at the time when the compressive part has completely left the
Dy layer (Fig. 12(b)). Partial back reflections at interfaces cause the damped
oscillations in the strain response of Dy that superimpose with the quasi-
static expansion, which decays on a nanosecond timescale due to thermal
transport into the buried Nb layer.

The entrance of the leading compression into the unexcited Nb layer leads to a
negative average strain in Nb, while the exit of the compression and the
entrance of the tensile part of the strain pulse cause an expansion (Fig.
12(c)). In total, the bipolar strain response of the Nb layer within the first
picoseconds maps out the spatial stress profile within the inhomogeneously
excited Dy transducer, which determines the spatial profile of the propagating
strain pulse according to Eq. (9). Finally, the quasi-static expansion of Nb
due to heating rises on hundreds of picoseconds via the slow diffusive
transport of thermal energy from Dy into Nb shown in Fig. 12(c).

The excitation of Dy in its AFM or FM state below $T_{\text{N}}=180\,\text{K}$
changes the strain response by the presence of an additional contractive
stress originating from the excitation of the spin system that adds to the
expansive electron-phonon stress. Already slightly below the Néel temperature
at $160\,\text{K}$ we observe a reduced expansion of Dy that continuously
changes to a pronounced contraction at $31\,\text{K}$ upon further decreasing
the initial sample temperature. At the same time, the strain response of Nb
changes from a conventional bipolar shape at $250\,\text{K}$ to a unipolar
expansion at $31\,\text{K}$. At intermediate temperatures we observe an
unconventional composite shape of the strain response consisting of a leading
expansion and a bipolar contribution that indicates a complex space and time
dependence of the total external stress within the inhomogeneously excited Dy
transducer. With decreasing temperature, the leading expansion continuously
becomes more pronounced and the amplitude of the bipolar component decreases.

Figure 13: Separation of the magnetic and phonon stresses in Dy thin films
using the Grüneisen concept: (a) Specific heat of Dy [128] separated into a
magnetic-, phonon- and a very small electron contribution [87, 101]. (b)
Thermal expansion of Dy along the hexagonal c- and a-axis. (c) The associated
Poisson stress $\sigma_{33}^{\text{Poi}}$ (grey dots) and external stress
$\sigma_{33}^{\text{ext}}$ (black dots) along the out-of-plane separated into
the magnetic $\sigma_{33}^{\text{mag}}$ ( solid blue line) and phononic
$\sigma_{33}^{\text{ph}}$ ( solid red line) contribution by the respective
Grüneisen parameters determined from the linear dependence of the stress on
the energy densities $\rho^{Q}_{\text{mag}}$ and $\sigma^{Q}_{\text{ph}}$ in
panels (d) and (e). The zero stress level is arbitrarily chosen at
$T_{\text{N}}=180\,\text{K}$.

The solid lines in Figs. 12(b) and (c) represent the modeled strain response
from a previous work [87] that reproduces the observed temperature dependence
and yields subsystems specific stress contributions. This provides insights
into the energy transfer processes among the different degrees of freedom. The
modeling approach utilizes the concept of subsystem-specific Grüneisen
parameters to individually treat the stress contributions of electrons,
phonons and magnetic excitations as sketched in Figure 5. Fig. 13 exemplifies
the determination of the subsystem-specific Grüneisen parameters along the
out-of-plane direction of the Dy film from heat capacity data and the
anisotropic thermal expansion in equilibrium. In Fig. 13(b) the temperature-
dependent out-of-plane lattice strain of the Dy layer along its hexagonal
c-axis $\eta_{33}^{\text{qs}}$ illustrates an anomalous expansion below
$180\,\text{K}$ that is concomitant to the onset of magnetic order and results
in a NTE during heating. This behavior is universal for the class of the heavy
rare-earth elements from Gd to Er which exhibit a pronounced NTE below their
respective magnetic ordering temperatures as summarized in the works of
Darnell et al. [129, 130, 131].

Utilizing the anisotropic expansion of Dy $\eta_{kk}^{\text{qs}}(T)$
($k=\\{1,2,3\\}$) shown in Fig. 13(b), we extract the external stress
$\sigma_{33}^{\text{ext}}(T)$ along the out-of-plane direction taking the
Poisson stress contribution $\sigma_{33}^{\text{Poi}}(T)$ into account that
arises from the three-dimensional expansion in thermal equilibrium according
to Eq. (6). The hexagonal symmetry of Dy above $T_{\text{C}}$ simplifies Eq.
(6) by $c_{1133}=c_{2233}$ and
$\eta_{11}^{\text{qs}}(T)=\eta_{22}^{\text{qs}}(T)$ to:

$\displaystyle\begin{split}\sigma_{33}^{\text{ext}}(T)&=c_{3333}\eta_{33}^{\text{qs}}(T)+\sigma_{33}^{\text{Poi}}(T)\\\
&=c_{3333}\eta_{33}^{\text{qs}}(T)+2c_{1133}\eta_{11}^{\text{qs}}(T)\,,\end{split}$
(21)

with the Poisson stress contribution arising from the expansion within the
hexagonal plane $\eta_{11}^{\text{qs}}(T)$ 888We assume the same in-plane
expansion as for bulk Dy taken from literature [132, 133]. However, we shift
the strain signature of the first-order phase transition in temperature to
account for the lower Curie-temperature of our thin film sample in comparison
to bulk Dy due to the Y layers, which stabilize the spin helix i.e., the AFM
order [126].. Figure 13(c) displays the resulting out-of-plane Poisson (grey
dots) and external stress that is dominated by phonon excitations well above
$T_{\text{N}}=180\,\text{K}$ due to a vanishing magnetic and a negligible
electron heat capacity contribution shown in Fig. 13(a). The linear dependence
of the external stress on the energy density in phonon excitations is given by
the integral of the heat capacity contribution $\rho_{Q}^{\text{ph}}=\int
C^{\text{ph}}\text{d}T$. This yields the phononic Grüneisen parameter
$\Gamma_{3}^{\text{ph}}=1.1$ (Fig. 13(d)) that determines the phonon stress
contribution over the entire temperature range. The difference of the total
external stress and the phonon contribution is given by the magnetic stress
contribution that vanishes above $T_{\text{N}}$ since no additional energy
density can be deposited in magnetic excitations, as indicated by the finite
integral of the magnetic heat capacity contribution. Although
$\sigma_{33}^{\text{mag}}$ is a strongly nonlinear function of the
temperature, it depends linearly on the energy density in magnetic excitations
$\rho_{Q}^{\text{mag}}=\int C^{\text{mag}}\text{d}T$ (Fig. 13(e)). The
resulting large negative magnetic Grüneisen parameter
$\Gamma_{3}^{\text{mag}}=-2.9$ rationalizes that the contractive magnetic
stress dominates the smaller phonon driven expansion below $T_{\text{N}}$. In
Fig. 13(c) the superposition of the temperature-dependent magnetic (blue solid
line) and phononic stress contribution (red solid line) modelled within the
Grüneisen approach (solid black line) matches the temperature-dependent total
external stress and thus describes the expansion of Dy along the out-of-plane
direction in thermal equilibrium.

Figure 14: Model of the external stress separated into subsystem
contributions: Extracted spatio-temporal electron-phonon stress contribution
(a–c), magnetic stress contribution (d–f) and total external stress (g–i) for
three representative sample temperatures. The laser-induced magnetic stress
contributions at $130\,\text{K}$ and $31\,\text{K}$ display a saturation of
the magnetic stress in the strongly excited near-surface region and an
increase of the maximum magnetic stress for lower temperatures. The spatio-
temporal strain (j–l) driven by the total external stress shows that at
$130\,\text{K}$ the saturated magnetic stress results in a non-monotonous
total stress that launches a composite strain pulse into Nb due to an
expansion of Dy at the front and a contraction at its backside.

Finally, the extracted Grüneisen parameters are used to model the laser-
induced strain response of Dy and Nb shown in Fig. 12(b) and (c) [87]. The
modelled strain response (solid lines) excellently reproduces the experimental
data and thus reveals the spatio-temporal laser-induced stresses separated
into an expansive electron-phonon and a contractive magnetic stress
contribution shown in Fig. 14. Within the Grüneisen approach introduced in
Section 3.4 the total laser-induced stress is given by the superposition of
the subsystem contributions that are determined by their Grüneisen parameters
and the deposited energy densities as stated in Eq. (19). This approach
fulfills energy conservation as the local sum of the transient subsystem
energy density contributions corresponds to the total laser-deposited energy
density (Eq. (18)) which we calibrate in the PM phase in the absence of
magnetic excitations. In addition, we determine the finite rise time of the
electron-phonon stress from the amplitude and shape of the strain pulse
detected by the Nb strain response, which we model by assuming a smaller
Grüneisen parameter for the electrons than for the phonons and introduce an
electron-phonon coupling constant that corresponds to a timescale of
$2\,\text{ps}$ for the used excitation. However, we limit our discussion in
the following to the combined expansive electron-phonon stress in contrast to
the contractive magnetic stress since the phonon contribution dominates by far
after electron-phonon equilibration.

The magnetic degrees of freedom are locally excited by energy transfer from
the electron and phonon subsystem on two timescales with a sub-picosecond
contribution and a slower $15\,\text{ps}$ timescale resulting in a reduced
expansive phonon stress contribution below $T_{\text{N}}$ as shown in Fig.
14(b) and (c). The maximum amount of energy density transferred to magnetic
excitations is given by the finite integral of the magnetic heat capacity
(Fig. 13(a)) starting at the initial sample temperature $T_{\text{i}}$. The
Grüneisen concept linearly relates this energy density to a maximum magnetic
stress that is reached in the strongly excited near-surface region of the Dy
transducer (see Fig. 14(e) and (f)). In contrast, the phonon system can take
up very large amounts of energy up to the melting point. For large pump
fluences, the expansive electron-phonon stress in the near-surface region
therefore dominates the saturated magnetic stress (red color in Fig. 14(h)).
In contrast, the contractive magnetic stress (blue) dominates in the Nb-near
region of the transducer, because the weak excitation does not saturate the
spin excitation. Hence the energy is shared among the spin- and phonon system
almost equally according to Fig. 13(a) and the contractive spin stress
prevails because of its large negative Grüneisen constant.

This complex interplay of a saturable magnetic stress and a non-saturable
phonon stress results in a spatially non-monotonic total stress within the
inhomogeneously excited Dy transducer at $130\,\text{K}$ and $31\,\text{K}$ as
displayed in Figs. 14(h) and (i), respectively. The corresponding strain maps,
which are generated by solving the elastic wave equation (15), are depicted in
Fig. 14(k) and (l). They display an expansion of Dy at the frontside launching
a bipolar strain pulse into Nb. The weaker and more slowly rising contraction
at the backside drives a unipolar expansion pulse into the adjacent Nb layer.
Their superposition yields the composite shape of the average strain response
of the Nb detection layer as revealed by PUX in Fig. 12(c) for intermediate
temperatures. With decreasing temperature and increasing maximum magnetic
stress the expansive electron-phonon stress and the amplitude of the bipolar
strain pulse are less pronounced, while the amplitude of the unipolar
expansive strain pulse driven by the contraction in Dy is enhanced.
Furthermore, the total stress averaged over the Dy transducer becomes more
contractive, which results in the increasing contraction of Dy that is even
enhanced within tens of picoseconds due to the second slowly rising magnetic
stress contribution (cf. Fig. 12(b)). In summary, the Grüneisen model reveals
that the saturation of the magnetic stress due to the finite amount of energy
that can be deposited in magnetic excitations is crucial to explain the
temperature dependence of the strain response of the Dy and the Nb layer.
Utilizing the sensitivity of the Nb strain to the stress profile within Dy we
were also able to extract the spatial-profile of the magnetic stress and thus
gain insights into the spatio-temporal excitations of the AFM order in Dy from
our PUX experiment.

In summary, this use case demonstrates the unambiguous detection of NTE on
picosecond timescales via PUX and exemplifies the extraction of the subsystem-
specific stress contributions utilizing subsystem-specific Grüneisen constants
extracted from thermal expansion in equilibrium. In heavy rare-earth elements
NTE arises from magnetostrictive stresses. However, several other mechanisms
for NTE and Invar behavior exist e.g. dominant transverse phonon contributions
at low temperature and repulsive deformation potential contributions [100,
134]. For each of these mechanisms it is interesting to investigate, if the
NTE is also active in non-equilibrium states after ultrashort laser pulse
excitation. An ultrafast NTE or Invar behavior may lead to novel transducer
applications that may allow for tunable waveforms for picosecond ultrasonics
as discussed in [135].

##  5 Conclusion and Outlook

We have presented an overview of the capabilities and concepts of PUX, where
hard x-ray probe pulses detect laser-induced transient material strain via
Bragg peak shifts. Layered heterostructures provide excellent use cases,
because the strain propagation and reflections connect the signals from
different layers, which – thanks to the large penetration depth of x-rays –
are accessible even beneath thick metal layers. PUX experiments often provide
an intuitive picture of both the strain pulses and the quasistatic strain from
the local increase in the energy density as these effects can be separated in
the time-domain due to their different propagation speeds.

Motivated by the quantitative access to the strain response, we have revisited
continuum elasticity theory, to connect the established formulation of
picosecond ultrasonics in homogeneously excited continuous thin films via the
thermal expansion coefficient to a formulation that incorporates NTE and is
able to reveal the Poisson effect on ultrafast timescales.

We have shown that PUX experiments may serve as an ultrafast probe of the
local energy content in nanoscopic layers. The observed strain is driven by a
stress, which is in many scenarios directly proportional to the energy
densities in different subsystems. The proportionality constants are the sub-
system specific Grüneisen parameters for electrons, phonons and magnetic
excitations which enable a simplified expression for the driving stress in the
inhomogeneous elastic wave equation.

The selected examples of PUX experiments illustrate scenarios where x-ray
probing is advantageous compared to established and versatile all-optical
experiments. Strain pulse detection in buried layers that allows us to
separate propagating strain pulses and quasi-static strain in the time-domain
is one of the key advantages of PUX experiments as demonstrated for a thin Nb
layer buried below an opaque TbFe${}_{\text{2}}$ transducer. We discussed the
case of granular vs. continuous FePt films to illustrate the morphology
dependence of the strain response and expansion amplitude. For Au-Ni
heterostructures that are thinner than the optical penetration depth of
visible probe pulses we showcased that the material-specific Bragg peaks in
PUX experiments enable the detection of energy transfer, where all-optical
techniques would convolute the contributions from both layers. Finally, the
NTE on ultrafast timescales was exemplified by a Dy transducer layer whose
ultrafast giant magnetostriction is capable of launching unconventional
picosecond strain pulses into a buried Nb layer.

The presented experiments work in the symmetric diffraction of rather dim and
poorly collimated x-rays from a laser-driven plasma source [54]. Already this
technologically simple approach has many more applications beyond the current
manuscript. Moreover, PUX can be extended to asymmetric diffraction geometries
that are sensitive to in-plane strain and to shear waves [136]. Optical
phonons, i.e., coherent atomic motions within the crystallographic unit cell,
are accessible via the modulations of the x-ray intensity through the
structure factor [137, 138, 139]. The excitation can be tailored by changing
the wavelength and duration of the pump-pulse, by multipulse-excitation
sequences [104, 95] or by combining optical excitation with externally applied
magnetic or electric fields [135]. Time-resolved studies of phase transitions
that are accompanied by lattice strain [111, 140, 141] or non-thermal stresses
that arise in laser-excited piezo- and ferroelectric materials [142, 143]
represent another intriguing field for the application of PUX.

Large scale facilities from synchrotron radiation sources to FELs provide many
additional opportunities beyond the basic PUX experiments discussed here. The
excellent collimation and energy resolution of the x-ray beam supports
grazing-incidence measurements [111, 144], which allow for tuning the
penetration depth of x-rays in the bulk and provide a higher resolution of the
transient shape changes of the diffraction peaks in reciprocal space. The
enormous photon flux at FELs enables the investigation of even thinner layers,
pushing the spatial strain-pulse dimensions further towards the single layer
limit. Inelastic scattering provides alternative access to phonon dynamics via
thermal diffuse scattering [28], Fourier-Transform inelastic scattering [29],
TDBS in the x-ray range [12, 25, 27] and time-resolved x-ray reflectivity on
non-crystalline specimen [145, 146]. The coherence of 4 generation
synchrotrons and FELs even allows for advanced strain-field imaging techniques
[147, 148, 149].

##  A Appendix: Detecting Bragg peak shifts in reciprocal space

Here, we outline details of the x-ray diffraction process that is central to
PUX experiments. We discuss both RSM and RSS schemes that utilize a PSD for
the detection of Bragg peak shifts that encode the laser-induced strain
response of the sample.

In this article, we limit the discussion to the symmetric and coplanar
scattering geometry, where the sample normal is parallel to the scattering
plane, generated by the incident and the diffracted x-ray beam. This specular
Bragg diffraction geometry specifically accesses the out-of-plane lattice
constant. A quantitative analysis of the three-dimensional strain response as
discussed in Section 4.2 requires access to asymmetric Bragg reflections. This
is also true for the detection of shear waves via PUX [136, 26]. More details
on the theory of x-ray diffraction can be found in dedicated books [150, 151,
152], and treatises that specialize on x-ray diffraction from thin films [153,
154] or time-resolved x-ray diffraction [155, 54, 156].

###  A.1 Reciprocal space mapping

As typical for picosecond ultrasonics experiments, we employ a pump-probe
geometry where the probe pulses are focused sufficiently well that they are
scattered from a region of the sample that is homogeneously laser-excited
within the sample plane. The laser-induced strain response then exclusively
develops along the out-of-plane direction of the sample as explained in
Section 3.2. This results in a change of the out-of-plane lattice constant
$d_{3}$, i.e., a change of the spacing of the lattice planes parallel to the
sample surface that can be probed via symmetric x-ray diffraction.

Figure A.1(a) introduces the corresponding experimental geometry: The incident
angles of the x-ray beam with respect to the sample surface and to the lattice
planes are $\alpha_{\text{in}}$ and $\omega$, respectively. The diffracted
intensity is recorded on the PSD under the angles $\alpha_{\text{out}}$ and
$\theta$ relative to the sample surface and the lattice planes, respectively.
In case of symmetric diffraction from lattice planes parallel to the sample
surface, i.e., $\alpha_{\text{in}}=\omega$ and $\alpha_{\text{out}}=\theta$,
Bragg’s law

$\displaystyle n\lambda=2d_{3}\sin(\theta_{\text{B}})$ (A.1)

relates the out-of-plane lattice constant $d_{3}$ to the Bragg angle
$\theta_{\text{B}}$ for a fixed wavelength of the x-rays $\lambda$ and a given
diffraction order $n$. The Bragg angle $\theta_{\text{B}}$ represents the
center and the maximum of the Bragg peak intensity reflected by a set of
parallel lattice planes that are exposed to a monochromatic x-ray probe beam
when scanning the angle $\theta$ relative to the lattice planes. Note that
irrespective of the detector position, Bragg diffraction only occurs if the
incoming x-rays are incident at $\theta_{\text{B}}$ with respect to the
lattice planes, i.e., $\omega=\theta_{\text{B}}$. Within this simplified
picture it is required to carry out $\theta-2\theta$ scans that symmetrically
vary both $\alpha_{\text{in}}$ and $\alpha_{\text{out}}$ to identify the
changing lattice constant $d_{3}$ from the corresponding change of the Bragg
angle $\theta_{\text{B}}$.

Figure A.1: Sketch of the diffraction geometry: (a) symmetric diffraction
geometry where the angle of incidence relative to the sample surface plane
$\alpha_{\text{in}}$ is equal to the reflection angle $\alpha_{\text{out}}$
which in turn are equal to the angles in respect to the lattice planes
$\omega$ and $\theta$, respectively. In this case the scattering vector is
aligned along the out-of plane reciprocal space direction $q_{\text{z}}$. (b)
Sketch of an asymmetric scattering event where reflection occurs from a
scattering vector that has a component along the in-plane reciprocal space
coordinate $q_{\text{x}}$. The asymmetry can equally well result from the
tilting of nanoscale crystallites in respect to the surface and from the
associated nanoscale inhomogeneity within the film plane (finite coherence
length). The diffraction processes in both panels are elastic so that
$|\bm{k}_{\text{in}}|=|\bm{k}_{\text{out}}|$, but the scattered photons are
detected on different pixels of the PSD. Sketch adapted from the book of Holý
et al [154].

This basic description of x-ray diffraction via Bragg’s law has to convoluted
with the finite divergence of the x-ray sources. However, investigating thin
films with mosaicity or finite crystalline coherence length requires more
work. Figure A.1(b) illustrates the diffraction from a mosaic film that
consists of small crystallites tilted with respect to the sample surface in
comparison to the idealized situation depicted in Fig. A.1(a). In case b) the
diffraction under the condition of $\alpha_{\text{in}}\neq\omega$ and
$\alpha_{\text{out}}\neq\theta$ the description by the Laue condition

$\bm{k}_{\text{out}}-\bm{k}_{\text{in}}=\bm{G}\,$ (A.2)

is better suited. Here, $\bm{k}_{\text{in}}$ and $\bm{k}_{\text{out}}$ are the
wave-vectors of the incident and elastically scattered x-rays
($|\bm{k}_{\text{in}}|=|\bm{k}_{\text{out}}|=k={2\pi}/{\lambda}$) which define
the scattering vector $\bm{q}=\bm{k}_{\text{out}}-\bm{k}_{\text{in}}$. In
reciprocal space, Bragg peaks occur at scattering vectors $\bm{q}$ being equal
to a reciprocal lattice vector $\bm{G}$ as stated by the Laue condition Eq.
(A.2). As the reciprocal lattice vectors $|\bm{G}|={2\pi n}/{d_{3}}$ are
inversely related to the lattice plane spacing $d_{3}$, the determination of
the position of a Bragg peak in reciprocal space yields the lattice strain.

Figure A.1(b) displays the application of this description to diffraction from
a tilted crystallite with lattice planes slightly tilted with respect to the
surface of the sample. Even under this condition the diffraction from the
lattice planes only occurs at Bragg condition when $\omega=\theta=\theta_{B}$.
However, the corresponding reciprocal lattice vector $\bm{G}$ does not point
along the out-of-plane direction anymore but is tilted relative to the
reciprocal space coordinates $q_{\text{x}}$ and $q_{\text{z}}$ that are
defined with respect to the sample surface, where $q_{\text{z}}$ denotes the
out-of-plane direction. Under this condition the maximum of the Bragg peak
occurs at $\alpha_{\text{in}}\neq\alpha_{\text{out}}$. In addition, a finite
nanoscopic in-plane size of the crystallites relaxes the $q_{\text{x}}=0$
condition by broadening of the diffraction condition along $q_{\text{x}}$,
even in case of lattice planes parallel to the surface. This leads to
diffraction intensity even if the Bragg condition is not fulfilled as
illustrated in Fig. A.1(b) by the diffuse spread of the reciprocal lattice
vector $\bm{G}$. In total, mosaicity leads to a broadening of the Bragg peak
along the in-plane direction $q_{\text{x}}$ and similarly along
$q_{\text{y}}$. Hence, diffraction can even occur if both the incoming angle
$\alpha_{\text{in}}$ and the scattering angle $\alpha_{\text{out}}$ deviate
from $\theta_{B}$. The PSD simultaneously captures a range of
$\alpha_{\text{out}}$ and additionally probes diffraction intensity
perpendicular to the diffraction plane, i.e., along $q_{\text{y}}$. However,
we restrict our discussion to the diffraction plane characterized by
$q_{\text{x}}$ and $q_{\text{z}}$ and integrate the recorded intensity along
$q_{\text{y}}$. This assumes equivalent in-plane directions of the film and
allows us to treat the PSD effectively as line detector. The transformation
from angular $\alpha_{\text{in}}$-$\alpha_{\text{out}}$-space to reciprocal
$q_{\text{x}}$-$q_{\text{z}}$-space by geometric considerations is then given
by [154, 54]:

$\bm{q}=\begin{pmatrix}q_{\text{x}}\\\
q_{\text{z}}\end{pmatrix}=k\begin{pmatrix}\cos{(\alpha_{\text{out}})}-\cos{(\alpha_{\text{in}})}\\\
\sin{(\alpha_{\text{in}})}+\sin{(\alpha_{\text{out}})}\end{pmatrix}\,.$ (A.3)

According to this transformation the PSD captures a range of
$\alpha_{\text{out}}$ and therefore probes a subset of the reciprocal
$q_{\text{x}}$-$q_{\text{z}}$-space for a fixed $\alpha_{\text{in}}$. The
individual treatment of $\alpha_{\text{in}}$ and $\alpha_{\text{out}}$ is a
generalization of Bragg’s law (A.1) that is recovered from the Laue condition
(A.2) for $\alpha_{\text{in}}=\alpha_{\text{out}}=\theta$ implying
$q_{\text{x}}=0$ as in the symmetric case in Fig. A.1(a). The variation of
$\alpha_{\text{in}}$ during a symmetric $\theta-2\theta$ scan with a line
detector yields the diffraction intensity along several lines within the
$q_{\text{x}}$-$q_{\text{z}}$-space with an offset along $q_{\text{z}}$. This
set of reciprocal space slices yields the RSM in the vicinity of the Bragg
peak in the form of diffraction intensity versus $q_{\text{x}}$ and
$q_{\text{z}}$ as exemplified by Fig. 2(c).

While the width of the Bragg peak along $q_{\text{x}}$ is a measure of the in-
plane coherence length and mosaicity of the layer, its width along
$q_{\text{z}}$ encodes the layer thickness or the out-of-plane coherence
length. The central observable of PUX is the $q_{\text{z}}$-position of the
Bragg peak which encodes the average out-of-plane lattice constant of the
corresponding crystalline layer. Therefore, a heterostructure consisting of
different materials with distinct out-of-plane lattice constants results in
layer-specific Bragg peaks separated in reciprocal space along $q_{\text{z}}$
which enables probing the layer-specific laser-induced strain response as
demonstrated in Section 2. The angular divergence and the spectral width of
the x-rays impinging on the sample, i.e., the instrument function, broaden the
recorded diffraction peaks [55].

Time-resolved RSM measurements follow the shift of the material-specific Bragg
peaks along $q_{\text{z}}$ by recording the diffraction intensity in
reciprocal space via $\omega$ scans or $\theta-2\theta$ scans for each pump-
probe delay [54]. The transient position of the center-of-mass of the Bragg
peak yields the laser-induced out-of-plane strain by the relative change of
the $q_{\text{z}}$ coordinate according to Eq. (2).

Most PUX experiments focus on the average strain of each layer within the
heterostructure. However, strain gradients within a material can be observed
as peak broadening. Especially the case of inhomogeneous energy deposition
within the transducer region or localized, high amplitude strain pulses can
result in a Bragg peak splitting for the inhomogeneously strained layer where
one maximum corresponds to the compressed part and one maximum for the
expanded part of the layer [66]. In extreme cases dynamical x-ray diffraction
analysis is required to reconstruct the underlying strain profile from the
observed Bragg peak deformations. Such capabilities are provided by modules of
the udkm1Dsim Matlab [105] and Python [106] code libraries.

###  A.2 Reciprocal space slicing

Recording RSMs around multiple Bragg peaks for each time delay between laser
pump pulse and x-ray probe pulse contains the full information about changes
of the crystalline structure. To date, most PUX experiments only measure the
out-of-plane expansion encoded by Bragg peak shifts along $q_{\text{z}}$,
because this is the natural strain response on a picosecond timescale in the
standard experimental setting utilizing a thin film excited by a pump laser
profile with much larger lateral extent than the film thickness (see Section
3.2). In these situations we can often apply the RSS technique, which
considerably speeds up the measurement, because only one or very few slices of
reciprocal space and hence angular positions $\alpha_{\text{in}}$ are
required. This means the measurement is done at an angle that offers a large
number of diffracted x-ray photons, whereas RSMs contain many positions in the
reciprocal space with a low number of diffracted photons. Details of this
technique and its application to different types of x-ray sources have been
reported previously [55].

Fig. A.2 displays the central idea of the RSS technique. Due to the finite
coherence length of nanoscopic thin films both along $q_{\mathrm{z}}$
(thickness) and $q_{\mathrm{x}}$ (mosaicity) asymmetrically scattered x-rays
contribute to the scattered intensity distribution around the reciprocal
lattice vector $\bm{G}$. Eventually, this results in a characteristic extent
of the Bragg peak as indicated by the yellow contour lines around the
equilibrium reciprocal lattice vector $\bm{G}$ in Fig. A.2(a). In the case of
a purely one-dimensional strain response of the film along the out-of-plane
direction, as in typical picosecond ultrasonic experiments, the Bragg peak
exclusively shifts along $q_{\text{z}}$ as represented by the grey contour
lines around $\bm{G^{\prime}}$. The pixels of the PSD simultaneously record
the x-ray intensities at various diffraction angles $\alpha_{\text{out}}$ for
a fixed $\alpha_{\text{in}}$. The PSD positioned at a fixed angle thus records
a specific lineout of the Bragg peak’s intensity profile as indicated by the
line with ticks. The PSD position is initially chosen to slice the reciprocal
space through the center of the Bragg peak which maximizes the recorded
diffraction intensity profile (orange profile). This exactly corresponds to
the symmetrical Bragg condition. After the shift of the Bragg peak, the fixed
PSD intersects the Bragg peak’s intensity distribution in its wings. Hence,
the recorded intensity profile shifts on the detector and is reduced in
intensity (grey peak). The red arrow denotes the shift of the Bragg peak
projection observed on the detector which is closely related to the actual
shift of the Bragg peak (contour lines) along $q_{\text{z}}$ in reciprocal
space.

Figure A.2: Sketch of the RSS scheme: (a) RSS utilizes a PSD that probes
multiple diffraction angles $\alpha_{\mathrm{out}}$ without any mechanical
movements of the goniometer. The detector is approximated as a straight line
in reciprocal space with $\alpha_{\mathrm{in}}$ and $\alpha_{\mathrm{out}}$
chosen such that the maximum intensity occurs on the center pixel of the
detector. The peak shifts by $\Delta q_{\mathrm{z,RSM}}$ as a result of out-
of-plane strain. The projection $\Delta q_{\mathrm{z,RSS}}$ of this shift in
the reciprocal space slice defined by the PSD yields the measured strain.
Panels (b–d) display the relation of both shifts for different shapes of the
Bragg peak in reciprocal space, where both shifts are nearly identical for a
Bragg peak that is elongated along $q_{\text{x}}$.

Figures A.2(b–d) show a schematic zoom into the intersection of the detector
slice (black line) and shifting Bragg peaks with different widths along
$q_{\text{x}}$ and $q_{\text{z}}$. The ratio of the shift of the recorded
intensity profile on the detector $\Delta q_{\text{z,RSS}}$ and the shift of
the Bragg peak in reciprocal space $\Delta q_{\text{z,RSM}}$ depend on the
shape of the Bragg peak in reciprocal space. These shifts are related by
geometric arguments. The corresponding strain $\eta_{\mathrm{RSM}}$ is given
by the projection of the RSM:

$\displaystyle\begin{split}\eta_{\mathrm{RSM}}&=-\frac{\Delta
q_{\text{z,RSM}}}{q_{\text{z,0}}}\\\ &=-\frac{\Delta
q_{\text{z,RSS}}}{q_{\text{z,0}}}\underbrace{\left(1+\left(\frac{w_{\text{z}}}{w_{\text{x}}}\right)^{2}\tan^{2}{(\theta_{\text{B}})}\right)}_{:=s}\,,\end{split}$
(A.4)

and can be related to the shift $-(\Delta q_{\text{z,RSM}})/q_{\text{z,0}}$ of
the RSS on the PSD via a scaling factor $s$. Here, $w_{\text{z}}$ and
$w_{\text{x}}$ denote the width of the Bragg peak in reciprocal space along
the $q_{\text{z}}$ and $q_{\text{x}}$ coordinate, respectively, which can be
characterized for a given sample by standard RSM techniques. Under symmetrical
scattering conditions the angle $\theta_{\text{B}}$ in Fig. A.2(b) corresponds
to the chosen fixed incident angle of the x-ray beam
$\alpha_{\text{in}}=\theta_{\text{B}}$ that corresponds to the Bragg angle of
the film yielding maximum intensity on the detector by slicing the Bragg peak
through its center. Figures A.2(b–d) illustrate the dependence of the
correction factor $s$ on the width of the Bragg peak in reciprocal space. The
observed shift on the detector and in reciprocal space nearly coincide for a
peak that is broad along $q_{\text{x}}$ (Fig. A.2(d)), resulting in a
correction factor $s\approx 1$. However, $s$ can become rather large for a
Bragg peak that is narrow along $q_{\text{x}}$ and broad along $q_{\text{z}}$
(Fig. A.2(c)). This situation is met e.g. for very thin films with excellent
in-plane epitaxy, i.e., with negligible mosaicity. Although the strain may
considerably shift the Bragg peak along $q_{\text{z}}$, the shift of the peak
on the detector can become very small in such cases, thus limiting the
sensitivity and accuracy of the RSS technique.

In general, samples exhibiting Bragg peaks elongated along $q_{\text{x}}$ and
small Bragg angles $\theta_{\text{B}}$ are well suited for the RSS probing
scheme that is able to reduce the measurement time by an order of magnitude by
avoiding $\theta-2\theta$ scans or full RSMs at each pump probe delay. The