Nanometer-Scale Acoustic Wave Packets Generated by Stochastic Core-Level Photoionization Events

Nanometer-Scale Acoustic Wave Packets Generated

by Stochastic Core-Level Photoionization Events

Yijing Huang ,

1,2,3,4

Peihao Sun ,

2,5

†

Samuel W. Teitelbaum ,

2,6

Haoyuan Li ,

Yanwen Sun,

Nan Wang,

Sanghoon Song,

Takahiro Sato,

Matthieu Chollet ,

Taito Osaka ,

Ichiro Inoue,

Ryan A. Duncan,

2,4

Hyun D. Shin ,

Johann Haber,

Jinjian Zhou ,

Marco Bernardi,

Mingqiang Gu ,

James M. Rondinelli ,

Mariano Trigo,

2,4

Makina Yabashi,

Alexei A. Maznev,

Keith A. Nelson ,

Diling Zhu,

and David A. Reis

2,3,4

‡

Department of Physics,

University of Illinois at Urbana-Champaign

, Urbana, Illinois 61801, USA

Stanford PULSE Institute,

SLAC National Accelerator Laboratory

, Menlo Park, California 94025, USA

Department of Applied Physics,

Stanford University

, Stanford, California 94305, USA

Stanford Institute for Materials and Energy Sciences,

SLAC National Accelerator Laboratory

Menlo Park, California 94025, USA

Dipartimento di Fisica e Astronomia

“

Galileo Galilei

”

Universit`

a degli Studi di Padova

Padova 35131, Italy

Department of Physics,

Arizona State University

, Tempe, Arizona 85287, USA

Department of Mechanical Engineering,

Stanford University

440 Escondido Mall, Stanford, California 94305, USA

Linear Coherent Light Source

, SLAC National Accelerator Laboratory,

Menlo Park, California 94025, USA

RIKEN SPring-8 Center

, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan

Department of Chemistry,

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139, USA

Department of Applied Physics and Materials Science,

California Institute of Technology

Pasadena, California 91125, USA

Department of Physics,

Southern University of Science and Technology (SUSTech)

Shenzhen, Guangdong, China 518055

Department of Materials Science and Engineering,

Northwestern University

Evanston, Illinois 60208-3108, USA

(Received 2 January 2024; revised 28 June 2024; accepted 28 August 2024; published 10 October 2024)

We demonstrate that the absorption of femtosecond hard x-ray pulses excites quasispherical, high-

amplitude, and high-wave-vector coherent acoustic phonon wave packets using an all hard-x-ray pump-

probe scattering experiment. The time- and momentum-resolved diffuse scattering signal is consistent with

an ensemble of 3D strain wave packets induced by the rapid electron cascade dynamics following

photoionization at uncorrelated excitation centers. We quantify key parameters of this process, including

the localization size of the stress field and the photon energy conversion efficiency into elastic energy. The

parameters are determined by the photoelectron and Auger electron cascade dynamics, as well as the

electron-phonon interaction. In particular, we obtain the localization size of the observed strain wave packet

to be 1.5 and 2.5 nm for bulk SrTiO

and KTaO

single crystals, respectively. The results provide crucial

information on the mechanism of x-ray energy deposition into matter and shed light on the shortest

collective length scales accessible to coherent acoustic phonon generation using x-ray excitation.

DOI:

10.1103/PhysRevX.14.041010

Subject Areas: Condensed Matter Physics

I. INTRODUCTION

Fundamental x-ray-matter interactions are typically

dominated by photoionization of core electrons creating

highly excited states that initially decay on the femtosecond

timescale through Auger-Meitner decay and characteristic

fluorescence

[1]

. The subsequent secondary electron cas-

cade involves the inelastic scattering of high energy

electrons and reabsorption of the florescence photons.

The cascade creates additional core excited states and a

Contact author: huangyj@stanford.edu

†

Contact author: peihao.sun@unipd.it

‡

Contact author: dreis@stanford.edu

Published by the American Physical Society under the terms of

the

Creative Commons Attribution 4.0 International

license.

Further distribution of this work must maintain attribution to

the author(s) and the published article

’

s title, journal citation,

and DOI.

PHYSICAL REVIEW X

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14(4)

041010(21)

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Published by the American Physical Society

plethora of both single-particle and collective mode exci-

tations including electron-hole pairs, plasmons, polarons,

and phonons in hard condensed matter systems

[2]

. This

exponentially complex process of secondary interactions

can either induce or mitigate radiation damage, depending

on how efficiently it dissipates the high energy density

concentrated around the localized x-ray excitation.

Thus, it is crucial to experimentally investigate the

energy relaxation processes and the subsequent structural

dynamics following x-ray ionization, across the relevant

length and timescales. For instance, this is especially

critical for experiments utilizing the high flux and short

pulse durations of x-ray free-electron lasers (XFELs) to

create or probe atomic-scale dynamics. Even otherwise

robust materials are susceptible to single-shot radiation

damage within the focused beam of an XFEL, where

intensities can be high enough to saturate the photoioni-

zation cross section

[3,4]

and induce multiphoton K-shell

absorption

–

and Compton scattering

[8]

. In recent

x-ray pump, x-ray probe experiments on diamond, excited

beyond the single-shot damage threshold, atomic motion

appeared to be frozen for the first 20 fs

[9]

, whereas in

proteins, dense-environment effects have been shown to

significantly influence the structural dynamics associated

with local radiation damage

[10]

. Equally important is

understanding the structural dynamics induced by x-ray

absorption below the single-shot damage threshold.

Here we present results of an x-ray pump, x-ray probe

experiment on the oxide perovskites SrTiO

and KTaO

excited at high x-ray pump pulse densities, but below the

multishot damage threshold. We observe high contrast

oscillations in the time- and momentum-resolved x-ray

diffuse scattering corresponding to the fundamental and

second-harmonic frequency of longitudinal acoustic pho-

nons with characteristic wavelengths on the nanometer

scale. The results are consistent with coherent phonon

generation in the form of a three-dimensional strain

comprising a spatially inhomogeneous strained region with

a corresponding outward propagating wave packet. The

characteristic length scale of the strain is significantly

longer than the x-ray wavelength but orders of magnitude

shorter than the pulse dimensions and penetration depth,

indicating considerable spatial heterogeneity in absorption.

Therefore, we model strain generation and propagation as

arising from nanometer-scale stress centers, formed by

localized photoionization events at random, uncorre-

lated sites.

The results have fundamental implications for our under-

standing of x-ray-matter interactions at moderate intensities

below the damage threshold. Specifically, the electron

cascade processand subsequent transportrapidly redistribute

the initial energy deposition from core-level ionization on an

ultrafast timescale, limiting the smallest spatial scales and

largest wave vectors of coherent excitations, such as pho-

nons. These findings also have practical implications for

investigating nanoscale thermal transport in bulk materials,

heterostructures, and devices

[11,12]

, including through

x-ray transient grating spectroscopy

[13]

II. METHODS

The experiment is carried out at the x-ray correlation

spectroscopy (XCS) end station at the Linac Coherent

Light Source (LCLS)

[14]

. The photon energy is set to

9.828 keV, slightly below the Ta L3 edge. A schematic

diagram of the split-delay setup is shown in Fig.

1(a)

. The

hard-x-ray split-delay (HXRSD) unit

[15]

is inserted into

the x-ray beam path, splitting each x-ray pulse into two

branches, a fixed-delay branch (red lines) and a variable-

delay branch (blue lines). The relative delay between the

pulse from the two branches is adjusted by changing the

path length in the variable-delay branch, as indicated by

the blue double-headed arrows; in this work, the delay is

changed between

−

and 10 ps in 0.1 ps steps. After the

crystal C

, the pulses from the two branches, each approx-

imately 30 fs in duration, become nearly collinear and are

focused by a beryllium (Be) lens stack of focal length 3.5 m

to approximately

at the sample position. The

spatial overlap between the two pulses is optimized with

the help of a beam profile monitor consisting of a Ce:YAG

scintillator screen positioned in the same plane as the

sample and a microscope objective.

Because of imperfections in the translation stages, the

angles of crystals C

and C

vary slightly as the delays are

scanned. While the magnitude of the angular deviation is

small compared to the

∼

rad Darwin width (for the

-polarized x rays), this

“

wobble

”

nonetheless results in a

slight variation of the pointing between the two pulses.

Since the wobbling motion is correlated with the motor

positions (which correspond to different delay times), the

variations in the pointing are repeatable and thus are

partially corrected by changing the angles of crystals C

and C

as a function of delay. The remaining variations are

well characterized, and the effect on the signal is accounted

for using an overlap correction factor as a function of the

delay; more details are provided in Appendix

The pulse energies are measured shot to shot at the

120 Hz repetition rate of the FEL by intensity monitors

shown as green dots in Fig.

1(a)

. Specifically, the pulse

energies in the individual branches are measured by the

x-ray diodes

and

placed right before the recombi-

nation of the branches, while the overall pulse intensity is

measured by the intensity monitor

placed between the

Be lens stack and the sample. The conversion from diode

reading to pulse energy is calibrated, as detailed in

Appendix

The experimental geometry is shown in Fig.

1(b)

. The

samples are placed in reflection geometry at room temper-

ature, with the beam incident angle on the sample fixed to

5° grazing. The incident x-ray fluence is kept below the

multiple pulse damage threshold of the sample. The x rays

YIJING HUANG

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PHYS. REV. X

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scattered by the sample are collected by an area detector

(Jungfrau-1M, pixel size

)

[16]

placed around

130 mm away from the sample. In the elastic scattering

limit, each pixel on the detector maps to a

out

−

where

and

out

are the incoming and outgoing wave

vectors, respectively, with amplitudes

j¼j

out

j¼

where

is the x-ray wavelength 1.26 Å. The sample

is rotated around its surface normal

ˆ

until the Bragg

condition for a low-order Bragg peak is found, and then

rotated by at most 1° to tune off the Bragg peak to access

the diffuse scattering about the peak. For the cubic

perovskite samples SrTiO

and KTaO

with surface normal

(001), the targeted Bragg peak was

, which is an

asymmetric reflection (i.e., nonspecular with respect to the

surface).

III. RESULTS

A. Extraction of the pump-probe signal

We begin by examining the general features of the pump-

probe signal, taking SrTiO

as an example. The detector

measures x rays from both pulses, such that the scattered

intensity detected

;

Þ¼

Þþ

;

where

is the time delay.

and

denote the pulse

energies in the variable-delay and fixed-delay branches,

respectively, which are measured separately as shown in

Fig.

1(a)

. Here and in the rest of the text,

denotes the

scattering wave vector,

the nearest reciprocal lattice

vector (i.e., the Bragg peak), and

≡

−

the reduced

wave vector (i.e., the deviation from the Bragg peak). The

first two terms on the right-hand side of Eq.

(1)

represent

the intensities of diffuse scattering in thermal equilibrium,

which are proportional to the pulse energies, and

the diffuse scattering structure factor independent of the

pulse energies. The last term,

;

, represents

the pump-probe signal which depends on both the pump

and probe pulse energies and the relative delay between the

two pulses.

To extract the pump-probe signal

;

,we

first note that the x-ray pulse intensity delivered onto the

sample varies shot to shot due to the fluctuating overlap

between the x-ray spectrum coming into the split-and-delay

system and the bandpass of the crystals in the system

[17]

The ratio between the intensities in the two branches,

, also fluctuates due to jitter in the beam position at

the splitting crystal C

. Therefore, throughout the mea-

surement, we collect a large set of images with a wide

distribution of pulse energies

and

. As an example, a

histogram of the distribution of

;

at delay

is shown in Fig.

2(a)

. The distributions at other time

delays are similar. This wide distribution of

;

helps

isolate the pump-probe signal

;

: From all

shots at time delay

, we select

“

low-intensity

”

shots

(

;

J) where the pump-probe signal

is expected to be small, and

“

high-intensity

”

shots

(

;

J) where the pump-probe signal

should be large. These ranges are indicated by the solid

and dashed boxes in the histogram in Fig.

2(a)

. We then

calculate the normalized image for each category by

dividing the summed image by the summed pulse

intensities.

(a)(b)

FIG. 1. The split-delay setup and experimental geometry. (a) Schematic diagram of the split-delay setup. After the FEL x-ray pulse

passes through an upstream double-crystal, diamond (111) monochromator, it arrives at the HRXSD unit and is split into two branches

by a silicon crystal with a polished edge (C

): the fixed-delay branch (red lines) consisting of two channel-cut crystals (CC

and CC

and the variable-delay branch (blue lines) consisting of four crystals (C

−

). Silicon (220) reflections were used for all crystals of the

HRXSD unit. X rays from the two branches are combined after crystal C

and are focused by a Be lens stack onto the sample. The

relative delay between the two branches is adjusted by changing the path length in the variable-delay branch, specifically by changing

the positions of C

and C

using linear translation stages aligned along the blue double-headed arrows. The black circles around the

crystals denote the rotation motor stages. Green dots indicate x-ray intensity monitors. (b) Schematic diagram of the experimental

geometry. X rays from the two branches, denoted with red and blue pulses, separated by

in time, are focused onto the same position on

the sample at an incidence angle of 5°. The sample is rotated around its surface normal

ˆ

to go on and off the Bragg condition. The

scattered x rays are collected by an area detector.

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The normalized low- and high-intensity images for

SrTiO

ps are shown in Figs.

2(b)

and

2(c)

Since the original detector images are in analog-to-digital

units (ADUs), the normalized images are in units of

ADU

J. Note that the long white streaks are due to

scattering from the tails of the Bragg peak (from the surface

truncation rod). Comparing these two images, one can see

modulations away from the central region appear in the

high-intensity image, which becomes clearer when dividing

the high-intensity image by the low-intensity image as

shown in Fig.

2(d)

. These modulations appear like ripples

emanating from the center, which corresponds to the closest

point to the

Bragg peak on the detector (i.e., on the

Ewald sphere), reflecting the acoustic phonon excitation in

the sample. Note that the relative signal level is rather high:

The modulations reach more than 100% of the diffuse

scattering background approximated by the low-intensity

image in Fig.

2(b)

. In comparison, the pump-probe signal

appears negligible around zero time delay: Figure

2(e)

shows the results for delay

ps, which does not

contain any modulation like in Fig.

2(d)

. Therefore, we use

the data at time zero as the background diffuse scattering, as

will be further detailed below.

Having observed the general features of the pump-probe

signal, we next demonstrate that it is bilinear in the pump

and probe pulse energies. Because the pump-probe signal,

;

, should be proportional to both the probe

pulse energy and the amount of lattice distortion created by

the pump pulse, the bilinearity is expected if the latter is

proportional to the number of photons in the pump. In this

case, we may write

;

Þ¼

, where

is the pump-probe response coefficient indepen-

dent of the pulse energies. In this case, the normalized

scattered intensity

;

Þþ

We now test the validity of Eq.

(2)

. Using the extracted

pump-probe signal in Fig.

2(d)

, we select a region of

interest (ROI) with a clear signal, as indicated by the

red dashed line. Figure

2(f)

shows the summed intensity

within this region

ROI

normalized by the total pulse energy

, plotted as a function of

. The

results for delay

and 0.0 ps are shown as blue and

green circles, respectively, where each circle corresponds to

the average over a bin in the histogram in Fig.

2(a)

with at

least 5 shots. These data are consistent with a linear trend

with the same intercept at

, which supports the

validity of Eq.

(2)

and hence the bilinearity of the pump-

probe signal. Therefore, the results verify our expectation

that the total lattice distortion is proportional to the pump

pulse energy. Moreover, while the data for

ps show

a clear slope, the data for

appear independent from

the pulse energies, confirming the absence of pump-probe

signal at zero time delay.

(a)

(b)

(d)(e)

(c)

(f)

FIG. 2. General features of the pump-probe signal. (a) 2D histogram of the distribution of

;

ps; the solid (dashed) box

indicates the range corresponding to the low-intensity (high-intensity) image. Panels (b) and (c) show the normalized low-intensity and

high-intensity images at

ps, whose ratio is shown in (d). Panel (e) shows the ratio at

ps following the same procedure,

which does not exhibit the ripplelike feature in (d). Panel (f) shows the sum over the ROI indicated by the dashed red line in (d), plotted

against the value of

at the two delays. Only bins with at least 5 counts are considered. Red lines show linear fits fixing

the intercept to be the average value for

ps.

YIJING HUANG

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PHYS. REV. X

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Since we have demonstrated that the pump-probe signal

is negligible around zero delay, to increase the signal-to-

noise ratio we use the normalized intensity including all

valid shots at

ps,

norm

, as the equilib-

rium diffuse scattering structure factor

, in the

absence of the effect of the pump. Using Eq.

(2)

, the

pump-probe coefficient at delay

is thus obtained from

the experimental dataset as

norm

−

;

wherethe sumisoverallshots

atdelay

.Here,

denotes

the correction factor of order unity which accounts for

changes in the overlap between the two beams on the sample

during the delay scan due to the aforementioned wobbling

motion of the delay scan stages (see Appendix

An example of the pump-probe signal, obtained using

Eq.

(3)

for

ps, is shown in Fig.

3(a)

. The green line

shows the direction where the reduced wave vector is

approximately parallel to the reciprocal lattice vector

(

). Along this line, we take several

points (indicated

by the colored dots) and plot the time dependence of

the pump-probe signal in Fig.

3(b)

, where the labels

indicate the magnitude

≡

for each trace. These

curves exhibit damped oscillations, whose frequency

increases with increasing

. The curves do not resemble

a perfect sinusoidal function but feature flat minima,

suggesting the existence of even-order frequency over-

tones. With a Fourier transformation, we obtain the

spectral weights along this

direction, which are shown

in Fig.

3(c)

The results indicate that the excited modes are predomi-

nantly LA phonons, for the following reasons. Firstly,

the direction of the strongest modulation [green line in

Fig.

3(a)

] coincides with the direction

, while the

modulation vanishes in the perpendicular direction, con-

sistent with the

dependence in the scattering

intensity where

is the phonon polarization vector.

(a)

(c)

(b)

FIG. 3. Measured x-ray pump, x-ray probe signal

in SrTiO

. (a)

ps. The green line shows

the direction

, which coincides with the direction of the largest intensity modulation. (b) The time dependence of

selected wave vectors

along the red line in (a). The corresponding locations on the detector are indicated as colored dots in (a). An

offset is added between traces of different

values for clarity;

is zero at

. The black lines are fit results, to be

discussed in Sec.

III B

. (c) Fourier transform spectral amplitudes along the direction of the red line in (a). The red and blue lines show the

dispersion of the LA phonon and the LA second harmonic obtained from DFT calculations.

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Secondly, in Fig.

3(c)

, the red line, which is obtained

through fit to data (to be elaborated in Sec.

III B

below),

connects the maxima of spectral amplitudes and has a slope

s, which is well captured by the LA phonon

sound velocity

s according to density functional

theory (DFT) results along the selected direction

.We

see a second-harmonic overtone [blue line in Fig.

3(c)

]

because the scattering picks up the spatial Fourier compo-

nents of the mean-square displacements [see Eq.

(6)

Sec.

III B

below].

The observation of oscillations at the fundamental

frequency of the LA is indicative of coherent phonon

generation

, i.e., the LA phonons are produced with a finite

expectation value of the amplitude, as opposed to an

increase in the phonon population which would not lead

to an oscillating signal, or squeezed phonons which would

oscillate only at the second harmonic

[18,19]

. Importantly,

the coherent phonons can only give rise to oscillations in

the diffuse scattering

probe

signal at the fundamental

phonon frequency if there is a corresponding

static

Fourier component of the displacement, for example, as

generated via static spatially inhomogeneous stress. Thus,

the emission of phonons from a uniform hot electron gas

would

not

produce the signal that we see, even if they

are emitted on a timescale that is short compared to the

phonon period.

B. Model

We present a model below that is consistent with our

experimental observations. In particular, the model pro-

vides a mechanism of translational invariance breaking,

which allows the generation of coherent phonons and hence

the temporal oscillations in the mean-square displacements

with the corresponding frequencies of the phonons. Thus,

we construct the model based on the following physical

picture. The stochastic absorption of x-ray photons from

the pump pulse causes the creation of a large number of

uncorrelated photoelectrons and core holes, leading to

individual cascades that eventually become low-energy

electron clouds at uncorrelated centers. This process is

expected to be mostly complete well within 100 fs

[20,21]

much faster than the period of the acoustic phonons that we

detect. For SrTiO

at 9.828 keV, photoabsorption is

dominated by the Sr sites, with a mean distance between

absorption events on the order of 30 nm (considering a

J x-ray pulse shined on a

spot, and the

photoionization cross section). 30 nm is an order of

magnitude larger than the inverse of the maximum

Fig.

3(a)

with observable

“

ripple

”

feature, which is around

−

≈

nm. Thus, since the distribution

of photoabsorption is random and sparse, we measure the

incoherent sum of scattering amplitudes contributed by

individual stochastic sites, and thus the response of the

lattice to an ensemble average of absorption events (more

details below and in Appendix

We build a model under the following assumptions and

approximations. (1) The pump pulse creates many excita-

tion centers that are stochastically and sparsely distributed

within the illuminated volume, and the number of these

centers is proportional to the pump fluence. (2) Around

each excitation center, a step-function-like (in time) stress

field causes a sudden change in the equilibrium lattice

constant and therefore a sudden strain. We approximate the

excitation as instantaneous compared to the phonon periods

which are on the order of picoseconds [see Fig.

3(c)

], and

the atomic displacements are zero. (3) The strain

field is approximated as isotropic but inhomogeneous with

a Gaussian spatial profile in the ensemble average limit.

This Gaussian form is justified by the combination of the

stochastic nature of x-ray photoabsorption and the fact

that the spatial distribution of excitation centers across the

illuminated volume is given by a binomial probability

distribution. See derivations in Appendix

leading to

Eq.

(A8)

. (4) The strain field can be treated in the

continuum limit since the smallest length scales considered

(several nanometers, corresponding to the inverse of the

maximum

range of visible ripples) are still significantly

larger than the size of the unit cell. Furthermore, for

simplicity, we approximate the material as elastically

isotropic, which is not inconsistent with the fact that the

anisotropy of secondary electron distribution becomes

small by 100 fs

[20]

and that the distribution is isotropic

in the ensemble average limit

[22]

. Note that in the model

assumptions, we do not consider either the microscopic

mechanism of energy relaxation channels of photoionized

electrons or the diffusion and transport of these electrons.

These subjects are the implications of the results obtained

from the model.

Under these assumptions, the Fourier transform of the

average displacement field for a single excitation center is

(see Appendix

for detailed derivations)

̃

Þ¼

−

cos

qvt

−

ˆ

;

where

describes the amplitude of the displacement field,

is the root mean square (rms) extent of the distortion field,

is the longitudinal sound velocity,

−

is a phenom-

enological decay factor that accounts for the observed

decay of the oscillations [see Fig.

3(b)

], and

ˆ

is the unit

vector in the direction of

. Here,

̃

has the unit of

length. One can reasonably interpret the displacement

field in Eq.

(4)

as a strain wave packet, and sudden local

stress can only produce longitudinal acoustic coherent

phonons of periods longer than the time it takes for sound

to propagate across the typical length of a wave packet. The

−

cos

qvt

term is typical of displacivelike excitation,

where the equilibrium position of the lattice suddenly shifts

and atoms oscillate around the new equilibrium

[23]

.For

simplicity, we take a common decay time

, for both the

YIJING HUANG

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decay of the oscillation amplitude and the decay of the new

equilibrium back to the original equilibrium. Although the

Gaussian profile in Eq.

(4)

takes a similar form as a Debye-

Waller factor, we note that the former is a function of

while the latter is a function of

, due to their totally

different physical origins.

Since we observe that the modulations of the diffuse

scattering (see Fig.

) happen at the regime of relatively

small

≡

≪

, and we assume that the spatial

distribution of excitation centers is sparse and random,

the change in diffuse scattering intensity due to the

distortions is derived as for the Huang diffuse scattering

due to static defects

[24]

. Hence the intensity modulation,

∝

̃

Þj

;

where

is the probe pulse energy;

≪

is the concen-

tration (number per unit cell) of excitation centers that are

expected to be proportional to the pump pulse energy

Thus,

is proportional to

as expected. The full

expression for

considering all geometric factors is

provided in Appendix

.NotethatinEq.

(5)

, the term

̃

Þj

gives rise to the angular dependence

∝

ˆ

[even for an isotropic

̃

Þ¼

̃

ðj

], in agree-

ment with the highly anisotropic scattering intensity pattern

in Fig.

3(a)

. The thermal-equilibrium diffuse scattering

, on the other hand, is presumed to be dominated

by thermal phonons, for simplicity. The expression for

thermal diffuse scattering is given in Eq.

(A37)

Based on this model, the pump-probe signal is (see

Appendix

for detailed derivations)

−

cos

qvt

−

ˆ

;

where the prefactor

takes into account [see Eq.

(A53)

for

the full expression] geometric factors (e.g., the beam size),

Debye-Waller factor, the x-ray linear absorption coefficient,

thermal diffuse scattering background assuming phonon

frequencies and eigenvectors as obtained from DFT, as well

as other known constants (e.g., x-ray atomic scattering form

factors at the given

and photon energy), all of which are

independent of parameters of the model. Thus, the prefactor

can be calculated for any given

. We explicitly write

out in Eq.

(6)

only the following terms: the time depen-

dence

−

cos

qvt

−

, the angular dependence

ˆ

[which determines the intensity anisotropy of the

ripples in Fig.

3(a)

], the size of the distortion field

, and

the energy conversion coefficient

. Here

is the

absorbed energy density and

is the energy density of the

launched acoustic phonons, both defined in the bulk

(a)

(b)

FIG. 4. Time dependence of the pump-probe signal in SrTiO

, and its model predictions using fit parameters. (a) The pump-probe

signal

along the

line cut in Fig.

3(a)

, at selected delay times. Colored lines show the experimental data, while black

lines are predictions by the model. (b)

on the area detector compared between the experiment (top row) and the model

predictions (bottom row) at

, 7, 10 ps.

NANOMETER-SCALE ACOUSTIC WAVE PACKETS GENERATED

...

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average limit. Note that the

−

cos

qvt

term in Eq.

(6)

gives both the fundamental phonon frequency and its

second-harmonic overtone. As detailed in Appendix

the physical meaning of the fundamental frequency is an

interference term in the scattering from the static, non-

propagating longitudinal strain field and the propagating

wave

[25]

Using Eq.

(6)

, we fit our model to the experimentally

measured

to extract the main physical

quantities of interest: the size of the distortion field

and the energy conversion efficiency

. The longi-

tudinal acoustic phonon velocity

is also obtained and is

used to draw the red line in Fig.

3(c)

. The fit is done in the

following way. First, we estimate the decay constant

with

the time-dependent

, the colorful traces in

Fig.

3(b)

, assuming that

is independent of

[i.e., a global

estimate to all traces in Fig.

3(b)

ps for SrTiO

Then, we vary the parameters

, and

to fit the

model to the data in the

range of

–

−

along the direction

[i.e., the green line cut in Fig.

3(a)

]

and in the available delay range from 0 to 10 ps. Data at

−

are excluded because of low signal

levels, while data at

−

are excluded

because of their sensitivity to inaccuracies in

-space

calibration and in the modeling of the diffuse scattering,

which may contain a background from static disorder

besides the thermal diffuse scattering considered above.

TheresultsarepresentedinFig.

,whichshowsthemeasured

pump-probe signal

(colored lines) and fit

results (black lines) as a function of

at different delays. The

extracted fit parameters are

−

nm,

¼ð

−

for SrTiO

. Additionally,

−

sisin

reasonable agreement with picosecond ultrasound spectros-

copy measurement of

[26]

and the DFT calcu-

lations of

s. The error bars represent the variations

in fit parameters that increase the total fit error by a factor

of 2. The fits are shown as black lines in Fig.

3(a)

Figure

4(b)

shows

data on the selected

area of the detector (top row) and model predictions using

the fit parameters (bottom row), at delays of 4, 7, and

10 ps. Based on the general agreement between the model

predictions and the experimental data, we consider our

few-parameter model to be robust. Note, however, that

is model dependent, and it may change if one assumes a

different form of the source profile other than a Gaussian

one (e.g., an exponential decay profile in real space). In

KTaO

, the extracted fit parameters are

−

and

¼ð

−

; more detailed results

are shown in Appendix

IV. DISCUSSION

The results are in stark contrast to ultrafast optical

excitation in opaque materials where the energy absorbed

is distributed uniformly over the illuminated area and

exponentially along a depth (on the order of the absorption

length) much shorter than the optical wavelength and beam

size, which typically leads to an effective 1D strain wave

propagating into the bulk with characteristic wavelength

given by the optical penetration depth. In the all-x-ray

experiment reported here, coherent acoustic phonons propa-

gate in 3D. Even though the absorption of the x rays on

average leads to an exponentially decaying density profile

into the bulk, the typical wavelength of the generated

coherent acoustic phonons is many orders of magnitude

shorter than both the x-ray spot size and the penetration

depth. This demonstrates that x-ray excitation induces much

more localized electron distributions than optical pump,

likely because the optical pump excites the delocalized

valence electrons whereas x rays excite localized core holes,

potentially involving dramatically different electron-phonon

coupling mechanisms. The generation and detection of

coherent high-wave-vector acoustic waves as reported here

do not involve engineered interfaces or inhomogeneities,

such as a transducer layer

[27

–

29]

or a superlattice structure

[30

–

32]

, which would normally be required for generation

and detection of high-wave-vector acoustic waves using

optical pulses.

It is remarkable that our simple model, which assumes

only that spherical strain waves are launched from random,

uncorrelated, and three-dimensionally localized sources of

electrons, reproduces our experimental data and allows for

the quantification of key parameters of this process,

including the localization size

of strain wave packets

and the photon energy conversion efficiency

into

the elastic waves. In the case of SrTiO

, we find

nm and

−

. From

one can

obtain the product of the strain amplitude and concentration

of localized excitation centers,

, though the number

density of strain wave packets

and their amplitude

cannot be determined separately (see Appendix

) because

the noninterfering sum of scattering from uncorrelated sites

reflects only their average behavior. If we assume the

concentration of excitation sites is equal to the initial

density of photoexcited atoms (

∼

−

), the ampli-

tude

nm corresponding to a dilation at the

excitation center of

∼

10%

.Wenotethatthevalueof

when estimated solely from the material

’

s photoabsorp-

tion cross section, represents a lower bound, as the

estimation does not consider the contributions of

Auger electrons from multiple elements (e.g., both Ti

and Sr atoms in SrTiO

), the reabsorption of fluorescence

photons or the ionization by secondary electrons.

Therefore, the estimated value of

provides an upper

bound.

The fraction of x-ray energy deposited in acoustic waves,

, is on the order of

∼

−

(

−

for SrTiO

and

−

for KTaO

). Most of the energy transferred to

the lattice is in the form of incoherent phonons, which is

not unexpected, as heating is always a major effect in

YIJING HUANG

et al.

PHYS. REV. X

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041010 (2024)

041010-8

pump-probe experiments. The incoherent phonons span the

entire Brillouin zone and eventually become thermally

populated. In contrast, the coherent phonons are produced

over a much smaller fraction of the Brillouin zone and

dominate the mean-square displacements for modes in the

corresponding region. Additionally, the efficiency of opti-

cal pump energy deposition into coherent phonons would

be on a similar or smaller scale, given that the phonons are

coupled through higher-order lattice effects beyond har-

monic approximations, including deformation potentials

and thermoelastic mechanisms.

The size of the coherent strain wave packet is presum-

ably closely related to the distribution of secondary

electrons and their coupling to the lattice. This depends

on the details of the energy and momentum relaxation

channels of the photoelectrons and Auger electrons

[21,22,33

–

35]

. Multi-keV

photoelectrons are expected

to yield an electron distribution over the range of tens

of nanometers

[21,22,36,37]

, comparable with the mean

distance between ionization events. However, lower-

energy secondary electrons are expected to initiate

cascades that extend to a smaller, few-nanometer length

scale comparable to

obtained from the experiment

[20

–

22,35,38]

Whileweobservescatteringintensitymodulationsbeyond

100% in the two oxide perovskites, SrTiO

and KTaO

,we

do not detect an observable signal for the tetrahedral semi-

conductors GaAs and GaP over a similar

range at similar

excitation densities. We estimate that the strain amplitude is

at least

≥

timessmaller inGaAsthanforSrTiO

, giventhat

the electron cascade dynamics and distributions are not

significantly different

[21]

and the strain wave packets are

of similar size. However, differences in the stress generated

by either thermoelastic or low-energy deformation potential

coupling are not sufficient to explain the

≥

times differ-

ence between the two types of materials (see Appendix

Of course, the detailed spatiotemporal profile of the stress

depends strongly on the electron cooling rate and whether

there is significant transport across the initial excitation

region during the sound propagation time

[28]

.Eveninthe

optical regime, this can reshape the coherent acoustic

phonon pulses, as seen, for example, in x-ray diffraction

experiments from photoexcited Ge

[39,40]

.Moreover,we

note that in oxide perovskites, the electron-phonon cou-

pling can be highly spatia

lly nonuniform resulting in

polaron formation

[41

–

43]

as a source of localized strain.

Additional measurements with higher sensitivity will help

resolve the differences between these materials.

As the first all-x-ray pump-probe experiment of its kind,

our study reveals fundamental x-ray-matter interaction

processes on the subpicosecond timescale and nanometer

length scale, at modest intensities below the damage

threshold. We demonstrate a method for exciting and

probing high-wave-vector coherent phonons in bulk dielec-

tric crystals without the need to fabricate layered structures,

which is typically required in picosecond ultrasonics

[27,28,30,31]

. Our findings show that significant coherent

phonon generation can occur at nanometer length scales

before spatial homogenization due to the ultrafast elec-

tronic transport and cascade. This has implications for the

limits of selective excitation of short-wavelength coherent

phonons using x-ray transient grating methods

[44

–

52]

.It

also impacts the study of nanoscale transport phenomena

[13,50]

, particularly by enabling measurements of micro-

scopic anharmonic coupling channels of selectively excited

phonons

—

extending beyond what was recently demon-

strated for optically excited zone-center phonons

[53]

Additionally, the element specificity of the photoabsorption

cross sections enable selective pumping and probing of

different layers in heterostructures, offering a powerful tool

for studying transport across these structures.

The data that support the findings of this study are

openly available in Research Data Unipd at

[54]

ACKNOWLEDGMENTS

The authors thank J. B. Hastings for useful discussions.

This work was supported by the U.S. Department of Energy

(DOE), Office of Science, Office of Basic Energy Sciences

(BES) through the Division of Materials Sciences and

Engineering under Contract No. DE-AC02-76SF00515.

Measurements were carried out at the Linac Coherent

Light Source, a national user facility operated by Stanford

University on behalf ofthe U.S.Department of Energy,Office

of Basic Energy Sciences under Contract No. DE AC02-

76SF00515. Preliminary experiments were performed at

SACLA with the approval of the Japan Synchrotron Radia-

tion Research Institute (JASRI) (Proposal No. 2017B8046).

P. S. acknowledges funding from the European Union

’

Horizon 2020 research and innovation programme

under the Marie Sk

ł

odowska-Curie Grant Agreement

No. 101023787. The work of H. L. was supported by the

U.S. DOE, Office of Science, BES under Award No. DE-

SC0022222.TheparticipantsfromM. I. T.weresupportedby

the U.S. DOE, Office of Science, BES under Award No. DE-

SC0019126. M. G. and J. M. R. were supported by the U.S.

DOE under Grant No. DE-SC0012375.

Y. H., P. S., and S. W. T. contributed equally to this work.

APPENDIX A: DERIVATIONS FOR THE MODEL

1. Spherical wave solution

In this appendix, we present the derivation of the

spherical strain wave model which is used in the main

text. Two main assumptions are made. Firstly, we take the

continuum limit, which is appropriate given that we are

considering lengths scales of tens of nanometers and above,

which is large compared with the size of the unit cell.

Secondly, we assume that the material is isotropic, which

greatly simplifies the mathematical form of the results.

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