of 10
Determining
Quasi-Equilibrium
Electron
and
Hole
Distributions
of Plasmonic
Photocatalysts
Using
Photomodulated
X
ray
Absorption
Spectroscopy
Levi Daniel
Palmer,
Wonseok
Lee, Chung-Li
Dong,
Ru-Shi
Liu, Nianqiang
Wu,
and Scott
Kevin
Cushing
*
Cite This:
ACS Nano
2024,
18, 9344−9353
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*
Supporting
Information
ABSTRACT:
Most
photocatalytic
and
photovoltaic
devices
operate
under
broadband,
constant
illumination.
Electron
and
hole
dynamics
in these
devices,
however,
are
usually
measured
by
using
ultrafast
pulsed
lasers
in a narrow
wavelength
range.
In
this
work,
we
use
excited-state
X-ray
theory
originally
developed
for
transient
X-ray
experiments
to
study
steady-state
photo-
modulated
X-ray
spectra.
We
use
this
method
to
attempt
to
extract
electron
and
hole
distributions
from
spectra
collected
at
a nontime-resolved
synchrotron
beamline.
A set
of
plasmonic
metal
core
shell
nanoparticles
is
designed
as
the
control
experiment
because
they
can
systematically
isolate
photo-
thermal,
hot
electron,
and
thermalized
electron
hole
pairs
in
a TiO
2
shell.
Steady-state
changes
in
the
Ti
L
2,3
edge
are
measured
with
and
without
continuous-wave
illumination
of
the
nanoparticle’s
localized
surface
plasmon
resonance.
The
results
suggest
that
within
error
the
quasi-equilibrium
carrier
distribution
can
be
determined
even
from
relatively
noisy
data
with
mixed
excited-state
phenomena.
Just
as importantly,
the
theoretical
analysis
of noisy
data
is used
to provide
guidelines
for
the
beamline
development
of photomodulated
steady-state
spectroscopy.
KEYWORDS:
X-ray
absorption,
X-ray
spectroscopy,
Bethe-Salpeter
equation,
plasmonics,
core
shell
nanoparticles,
hot
carriers,
photocatalysis
A
balance
between
carrier
photoexcitation,
thermal-
ization,
and
recombination
rates
determines
the
quasi-
equilibrium
carrier
distribution
that
controls
photo-
catalytic
and
photovoltaic
device
efficiencies
(Figure
1).
1
3
A
quasi-equilibrium
state
occurs
during
the
thermodynamic
balance
of
the
system’s
photoexcitation
and
relaxation.
Photoexcited
carriers
are
generally
assumed
to
be
fully
thermalized
to
the
band
edges
at
the
device’s
working
conditions.
4
However,
slowed
hot
carrier
cooling
through
phonon
bottlenecks,
5
surface-state
trapping,
6
9
or
dielectric
carrier
Coulomb
screening
10
can
generate
a nonthermal
carrier
quasi-equilibrium.
In
nanoscale
junctions,
photoexcited
carriers
can
transfer
between
active
layers
or
to
surface
catalysts
on
timescales
shorter
than
carrier
thermalization.
11,12
Transferring
the
quasi-equilibrium
hot
carrier
population
into
surface
reactants
then
modifies
a
semiconductor’s
photochemical
redox
potential,
tailoring
resultant
reaction
products.
13
For
example,
plasmonic
metal
semiconductor
junctions
have
been
used
to
increase
semiconductors’
photocatalytic
product
selectivity
14
16
and
solar
power
conversion
efficiency.
17,18
Measuring
the
quasi-equilibrium
carrier
distribution
is
therefore
critical.
However,
few
methods
to
date
can
characterize
the
equilibrium
photoexcited
carrier
population
with
the
same
detail
as
ultrafast
pump
probe
methods
like
two-dimensional,
terahertz,
or
photoemission
spectroscop-
ies.
19
22
While
ultrafast
spectroscopy
is
the
conventional
method
for
measuring
carrier
thermalization
and
recombina-
Received:
August
30,
2023
Revised:
March
9,
2024
Accepted:
March
14,
2024
Published:
March
18,
2024
Article
www.acsnano.org
© 2024
The Authors.
Published
by
American
Chemical
Society
9344
https://doi.org/10.1021/acsnano.3c08181
ACS Nano
2024,
18, 9344
9353
This article is licensed under CC-BY 4.0
tion,
ultrafast
measurements
sum
over
different
relaxation
pathways,
often
use
a high
peak
power
that
exceeds
the
solar
flux,
and
rely
on
laser
sources
that
are
more
narrowband
than
the
solar
spectrum.
When
effects
such
as
Fermi-level
pinning,
defects,
and
surface
states
are
present,
it
can
be
difficult
to
reconstruct
steady-state
carrier
distributions
by
using
ultrafast
measurements
alone.
X-ray
spectroscopy
is
one
potential
method
for
resolving
carrier
distributions
and
dynamics.
Transient
X-ray
spectros-
copy
is
now
routinely
used
to
measure
element-specific
electron
and
hole
energies
in
multielement
catalysts.
23
26
The
same
capabilities
should
also
be
true
for
steady-state,
photomodulated
X-ray
spectroscopy.
27
However,
measuring
and
interpreting
photomodulated
X-ray
spectra
are
challenging
tasks
because
the
decreased
photoinduced
carrier
concen-
tration
and
slower
repetition
rate
make
the
signal-to-noise
ratio
(SNR)
significantly
lower.
Therefore,
accurate
excited-state
X-
ray
theory
is
needed,
even
more
so
than
ultrafast
X-ray
spectroscopy,
to
interpret
the
small
photomodulated
spectral
intensity
within
the
experimental
noise.
Previous
investigations
of
plasmonic
Au@TiO
2
nano-
particles
and
their
photomodulated
X-ray
spectra
suggest
that
quasi-equilibrium
hot
electron
populations
exist
in
TiO
2
.
7,28,29
In
our
prior
work,
the
photoexcited
X-ray
spectra
were
not
modeled
fully
ab
initio
but
were
rather
calculated
using
a
semiquantitative
model
of
phase-space
filling
and
lifetime
effects.
29
Thermal
and
photoexcited
hole
effects
were
not
included.
Here,
we
use
excited-state
density
functional
theory
(DFT)
and
Bethe-Salpeter
equation
(BSE)
calculations
to
evaluate
steady-state
photomodulated
X-ray
spectra.
To
simulate
hot
electron
X-ray
spectra,
we
fill
the
DFT-calculated
conduction
band
states
with
electrons
from
the
conduction
band
minimum
(CBM)
to
a specified
energy
and
calculate
the
corresponding
spectrum
with
the
BSE.
Methods
that
model
transient
charge
configurations
ab
initio
instead
use
density
functional
perturbation
theory
and
the
Boltzmann
transport
equation
to
predict
ultrafast
charge
redistribution
following
electron
phonon
interactions
and
transport.
30
However,
density
functional
perturbation
theory
is
not
used
for
modeling
steady-state
dynamics
due
to
computational
costs.
In
this
work,
we
tested
whether
photomodulated,
steady-
state
X-ray
spectroscopy
can
be
used
to
quantify
quasi-
equilibrium
carrier
distributions.
X-ray
absorption
at
the
Ti
L
2,3
edge
is
measured
for
each
nanoparticle
with
modulated
photoexcitation.
An
adiabatic
approximation
of
the
BSE
is
then
used
to
predict
the
change
in
the
X-ray
spectrum
for
each
possible
photoexcited
configuration.
We
use
a mean
squared
error
(MSE)
analysis
to
compare
the
potential
theoretical
contributions
of
thermal,
hot
electron,
and
dipolar
excitations
to
the
signal
from
plasmonic
core
shell
nanoparticles
designed
Figure
1.
Photoexcited
properties
of
plasmonic
core
shell
nanoparticles.
(a)
Continuous
photoexcitation
of
a metal
nano-
particle’s
LSPR
with
visible
light
results
in
dynamic
carrier
excitation
[ref
2].
Hot
carrier
formation
then
occurs
following
electron
electron
and
electron
phonon
scattering
[ref
3].
The
hot
carriers
transfer
into
TiO
2
and
can
be
probed
with
the
Ti
L
2,3
edge.
(b)
The
hot
carriers
transfer
over
the
Au@TiO
2
Schottky
barrier
(
φ
B
) and
fill
the
TiO
2
CBs
with
an
excess
energy
of 0.3
eV
and
subsequently
(c)
thermalize
in
the
CBs
through
phonon
scattering.
Figure
2. Plasmonic
core
shell
nanoparticle
heterostructure
characterization.
(a
c)
UV
visible
absorption
spectra
for
(a)
Au@TiO
2
, (b)
Ag@SiO
2
@TiO
2
, and
(c)
Au@SiO
2
@TiO
2
core
shell
nanoparticles.
The
LSPR
and
UV
bandgap
for
amorphous
TiO
2
(gray
dashed
line
at
3.34
eV,
370
nm)
are
marked
[refs
43,
44].
(d
f)
Schematic
representations
of
each
nanoparticle’s
band
alignment
and
hot
carrier
distribution.
(d)
Hot
electrons
up
to
0.3
eV
above
the
CBM,
relative
to the
Schottky
barrier
(
φ
B
), have
sufficient
energy
to transfer
into
TiO
2
directly
[ref
3].
(e)
The
Schottky
barrier
and
SiO
2
layer
prevent
hot
electron
transfer.
Instead,
the
localized
electromagnetic
field
from
the
plasmon
couples
with
TiO
2
electron
hole
excitation,
and
carriers
are
created
at both
the
CBM
and
the
VBM.
(f)
The
SiO
2
layer
prevents
electron
transfer
into
TiO
2
.
ACS Nano
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Article
https://doi.org/10.1021/acsnano.3c08181
ACS Nano
2024,
18, 9344
9353
9345
to
isolate
each
one
of
these
effects.
Even
in
the
case
of
relatively
noisy
spectra,
quasi-equilibrium
hot
carrier
distribu-
tions
are
differentiated
from
photothermal
heat.
Separating
electron
from
hole
effects
on
a
photothermal
background
is
more
difficult
because
of
the
hole’s
smaller
perturbation
to
the
Ti
L
2,3
edge.
Within
experimental
noise,
electron
versus
hole
populations
were
not
separable
by
statistical
significance.
However,
our
calculations
demonstrate
that
electrons
and
holes
have
distinct
spectral
features
and
could
be
differentiated
with
improved
X-ray
measurements,
and
we
discuss
the
required
experimental
conditions
for
such
measurements.
Our
findings
suggest
that
photomodulated
X-ray
spectroscopy
at
nontime-resolved
beamlines
can
be
used
to
separate
electron,
hole,
and
thermal
excitations,
but
continued
improvement
in
experimental
SNRs
is
needed
when
multiple
photoexcited
processes
simultaneously
exist.
RESULTS
AND
DISCUSSION
A
nanoparticle’s
localized
surface
plasmon
resonance
(LSPR)
can
be
used
to
transfer
energy
to
a
semiconductor
through
multiple
mechanisms.
Here,
a
SiO
2
layer
is
used
to
system-
atically
control
three
such
mechanisms
between
a
Au
or
Ag
nanoparticle
core
and
a
TiO
2
shell.
For
Au@TiO
2
nano-
particles,
plasmonic
hot
electrons
in
Au
can
overcome
the
interfacial
Schottky
junction
to
inject
into
TiO
2
(Figure
2a,d).
3,31,32
Ag@SiO
2
@TiO
2
nanoparticles
use
the
plasmon’s
dipole
moment
to
increase
the
light
absorption
rate
in
the
tail
of
a semiconductor’s
absorption
edge,
creating
electron
hole
pairs
(Figure
2b,e).
20,28
In
Au@SiO
2
@TiO
2
nanoparticles,
a
SiO
2
layer
prevents
carrier
transfer
from
Au
into
TiO
2
, so
the
TiO
2
shell
only
experiences
heating
from
the
Au
core
to
provide
a
control
experiment
(Figure
2c,f).
28
Past
work
has
verified
that
these
core
shell
nanoparticles
isolate
these
excited-state
effects.
28
This
paper
theoretically
interprets
previously
measured
X-
ray
spectra
of
the
nanoparticles
synthesized
and
characterized
in
ref
28.
In
these
previous
measurements,
the
nanoparticles
were
reported
to
have
a 15
nm
radius
Au
or
Ag
core,
a 10
nm
SiO
2
insulating
layer
(not
present
in
Au@TiO
2
),
and
a 10
20
nm
amorphous
TiO
2
outer
shell.
UV
visible
absorption
spectroscopy
was
used
to
measure
the
LSPR
center
wavelength
at
420
nm
for
Ag
and
560
nm
for
Au
(Figure
2a
c).
The
approximate
interfacial
band
bending
of
each
heterojunction
is
calculated
using
a 1D
drift-diffusion
model
implemented
in
the
Automat
FOR
Simulation
of
HETero-
structures
(AFORS-HET)
(Figure
2d
f).
33
This
approach
does
not
consider
nanoscale
near-field
or
photoexcited
effects.
The
approximate
Schottky
barriers
are
0.9
eV
for
Au@TiO
2
,
0.8
eV
for
Ag@SiO
2
@TiO
2
, and
0.7
eV
for
Au@SiO
2
@TiO
2
.
The
metal
semiconductor
junction
produces
band
bending
and
built-in
electric
fields
in
the
TiO
2
and
SiO
2
layers.
The
average
built-in
field
estimated
for
the
semiconducting
layers
is
10
5
V/cm.
The
SiO
2
insulator
acts
as
a
carrier
tunnelling
barrier
between
the
metal
and
TiO
2
, and
the
Schottky
barrier
in
such
cases
refers
to
the
energetic
barrier
for
electron
transfer
at
the
SiO
2
TiO
2
interface.
Considering
the
4.4
eV
Au-SiO
2
Schottky
barrier,
which
exceeds
the
maximum
hot
electron
energy
by
3.2
eV,
Au
hot
carriers
would
need
to
tunnel
through
SiO
2
to
reach
TiO
2
. Similar
junctions
with
a 4.8
nm
SiO
2
oxide
were
previously
measured
to
have
a <10
10
A/cm
2
tunneling
current
at
a 10
5
V/cm
applied
bias.
34
Therefore,
as
experimentally
observed,
photoexcited
electrons
would
not
transfer
to
TiO
2
for
the
Au@SiO
2
@TiO
2
system.
See
the
Supporting
Information
for
numerical
input
parameters
and
field
calculation.
The
ground-state
electronic
structure
and
X-ray
absorption
of
TiO
2
are
first
calculated
as
shown
in
Figure
3.
The
Ti
L
2,3
X-
ray
absorption
edge
(456
468
eV)
was
measured,
which
corresponds
to
a core
electron
transition
from
Ti
2p
to
Ti
3d
states
(Figure
1a,
right).
The
DFT
calculated
projected
density
of
states
(PDOS)
for
anatase
TiO
2
is
given
in
Figure
3a.
A
1 eV
scissor
shift
is
applied
to
the
bandgap.
In
the
PDOS,
the
O
2p
orbitals
dominate
the
valence
band
and
the
Ti
3d
orbitals
compose
the
conduction
band.
The
crystal
field
characteristi-
cally
splits
the
Ti
3d
conduction
band
into
the
t
2g
(blue
shading)
and
e
g
(gray
shading)
orbitals
in
the
electronic
structure
and
ground-state
X-ray
absorption
(Figure
3a,b).
35
Figure
3b
compares
the
BSE
simulated
Ti
L
2,3
edge
to
the
measured
ground-state
Au@SiO
2
@TiO
2
spectrum.
An
energy-
dependent
broadening
method
of
the
predicted
spectrum
was
used
to
replicate
the
experimental
core-hole
lifetime
broad-
ening,
which
has
a 3:2
broadening
ratio
(L
2
:L
3
) for
the
TiO
2
Ti
L
2,3
edge
(see
Supporting
Information).
36
According
to
the
PDOS
in
Figure
3a,
the
photomodulated
Ti
L
2,3
edge
predominantly
probes
photoexcited
electrons
over
holes
through
the
Ti
3d
states;
however,
because
of
the
screening
and
angular
momentum
coupling
matrix
elements
in
the
X-ray
transition
Hamiltonian,
holes
will
still
perturb
the
core-to-
valence
transition
excitons.
37,38
Figure
3.
Calculation
of
TiO
2
electronic
structure
and
X-ray
absorption.
(a)
DFT-calculated,
projected
density
of
states
for
anatase
TiO
2
. The
Fermi
level
(E
F
) represents
the
valence
band
edge.
(b)
Calculated
(black)
and
measured
(gray)
Ti
L
2,3
X-ray
spectra
for
TiO
2
. The
theory
spectrum
is broadened
to match
the
experiment
with
the
bottom
spectrum
being
the
unbroadened
output.
The
blue
and
gray
shading
depict
the
Ti
t
2g
and
e
g
states,
respectively.
(c)
The
difference
between
theory
and
experiment.
ACS Nano
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ACS Nano
2024,
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9353
9346
This
work
approximates
the
nanoparticle’s
amorphous
TiO
2
as
purely
anatase
phase.
Although
10
20
nm
TiO
2
nano-
particles
typically
consist
of
a mixture
of
anatase
and
brookite,
anatase
is
a slightly
more
stable
phase,
and
this
approximation
reduces
the
otherwise
insurmountable
computational
costs
of
excited-state
X-ray
BSE
calculations
for
hundreds
of
atoms.
39
42
We
find
this
to
be
a valid
approximation
due
to
the
excellent
match
between
the
ground-state
experiment
and
theory
(Figure
3b).
However,
aspects
of
the
amorphous
phase
electronic
structure
are
not
considered.
First,
there
is
a
discrepancy
between
anatase
and
amorphous
TiO
2
for
the
Ti
L
3
e
g
states
at
461
eV
(Figure
3c).
43
The
core-hole
exciton
effects
calculated
by
BSE
are
therefore
not
accurately
modeled
at
these
energies
and
are
not
considered
during
the
MSE
analysis.
Further,
defect-induced
midgap
states
(depicted
in
Figure
2d
f)
are
not
modeled.
Midgap
states
show
little
effect
on
the
ground-state
TiO
2
spectra
but
may
appear
as
a shoulder
of
the
L
2
t
2g
peak
at
463
eV
(Figure
3b,c).
The
previously
measured
X-ray
spectra
were
collected
using
a light-on,
light-off
sequence
with
a 1 min
collection
time
per
spectrum.
A
continuous-wave
lamp,
filtered
below
the
3.34
eV
amorphous
TiO
2
bandgap,
was
used
to
photoexcite
the
nanoparticles’
LSPR.
44,45
Surface
charging
from
photoexcita-
tion
creates
a baseline
drift
in
photomodulated
spectra
for
the
total
electron
yield
detection
method.
46
To
account
for
charging,
each
spectrum
is
normalized
to
the
edge
onset
maximum
near
458
eV
(gray
dashed
line
in
Figure
4)
after
the
baseline
background
subtraction.
The
charging
normalization
creates
artifacts
directly
below
and
above
the
X-ray
absorption
edge,
so
only
the
458
466
eV
range
is
compared
to
theory
herein
(see
average
spectra
for
all
samples
overlaid
in
Figure
S5).
Charging
is
not
measured
for
the
Au@SiO
2
@TiO
2
sample,
which
confirms
that
only
heat
results
from
photo-
excitation.
However,
the
insets
of
Figure
4a
reflect
that
even
the
light-off
spectra
change
after
modulating
photoexcitation,
meaning
residual
charges
may
perturb
the
total
electron
yield
acquisition
even
after
sample
relaxation.
The
measured
differential
absorption,
calculated
as
the
log
of
spectra
collected
with
the
lamp
on
divided
by
those
with
the
lamp
off
and
averaged
across
two
data
sets,
is
used
to
identify
photoexcited
carrier
and
structural
effects
on
the
X-ray
spectra
(Figure
4).
Because
the
total
electron
yield
detection
only
probes
the
first
4
nm
of
TiO
2
, only
carriers
in
surface
trap
states
in
the
amorphous
TiO
2
are
probabilistically
measured.
47
The
spectra
in
Figure
4
are
relatively
noisy
due
to
the
lower
power
of
the
excitation
source
and
slower
modulation
time
of
the
steady-state
measurement.
Therefore,
we
first
test
the
accuracy
of
our
ab initio
approach
by
comparing
to
a previous
ultrafast
X-ray
absorption
spectrum
of
anatase
TiO
2
(Figure
S1).
27
The
measured
transient
spectrum
is
analyzed
using
an
adiabatic
approximation
to
excited-state
effects
in
the
BSE.
This
approach
has
been
verified
previously
for
other
transient
X-ray
data
sets
and
is
described
in
the
Methods
section.
26,37,38
The
ultrafast
time
slice
is
after
carrier
thermalization
(1
ps
after
photoexcitation).
The
proposed
ab
initio
method
accurately
reproduces
the
transient
X-ray
spectrum
at
all
energies
besides
458
460
eV.
The
discrepancy
is
likely
due
to
the
reported
onset
of
carrier
transfer
to
midgap
states
(see
Supporting
Information).
The
electron,
hole,
and
thermal
signals
are
reproduced
in
this
case,
giving
a baseline
for
the
accuracy
of
the
photomodulated
data
presented
here.
Given
this
verification,
we
proceed
to
analyze
the
photo-
modulated
spectra.
Three
differences
are
observed
between
each
plasmonic
excitation
mechanism,
although
it
must
be
noted
that,
since
only
two
averages
are
used,
the
statistical
significance
must
be
interpreted
with
caution.
First,
all
nanoparticle’s
spectra
have
different
amplitudes
just
after
the
L
3
edge
at
458
eV.
Second,
the
Au@TiO
2
has
decreased
absorption
centered
at
462
eV.
Lastly,
the
Ag@SiO
2
@TiO
2
nanoparticles
have
decreased
absorption
at
465.5
eV.
We
then
investigate
how
these
trends
compare
to
the
theoretically
predicted
photomodulated
spectra,
which
is
found
to
successfully
model
ultrafast
dynamics
(Figure
S1).
Figure
4. X-ray
absorption
spectra
of core
shell
nanoparticles
with
and
without
photoexcitation.
(a)
Raw
light
on
and
light
off
Ti
L
2,3
edge
X-ray
spectra
of the
amorphous
TiO
2
outer
shell.
The
spectral
intensity
increased
throughout
the
data
collection
of all
four
spectra.
The
spectra
are
background-subtracted.
Each
inset
magnifies
the
edge
maxima’s
intensity
differences
caused
by
charging.
Each
inset
window
size
is 0.15
eV
width
but
a variable
amplitude.
(b)
Spectra
from
(a)
normalized
to the
edge
maximum
near
458
eV
(gray
dashed
line)
to correct
for
charging
that
broadly
increases
the
spectral
amplitude.
(c)
Photoexcited
differential
spectra
of the
normalized
data
in (b)
to highlight
photomodulated
energetic
shifts
in
the
L
2,3
edge.
The
lighter
spectra
are
from
the
first
light
on,
light
off
collection.
The
ground-state
experimental
spectrum
is shown
above
for
reference.
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9347
The
steady-state
spectral
signatures
of
photothermal
effects
are
first
compared
to
theory
by
using
the
Au@SiO
2
@TiO
2
experimental
control
(Figure
5).
Photothermal
heating
arises
from
the
heat
produced
by
carrier
thermalization
in
the
metal
nanoparticle
after
LSPR
photoexcitation
and
relaxation.
48
The
photomodulated
Au@SiO
2
@TiO
2
nanoparticles’
experimental
spectra
lack
the
surface
charging
artifact
that
results
from
photoexcited
carriers
in
the
other
two
nanoparticle
systems
(Figure
4a),
confirming
that
photoexcited
carriers
are
not
excited
in
TiO
2
.
Heating
is
modeled
through
DFT
and
BSE
calculations
by
an
expansion
of
the
TiO
2
lattice.
Calculations
are
performed
for
lattice
expansions
equivalent
to
2.5,
5.0,
10,
15,
and
20
K
temperature
increases
above
300
K
(Figure
5a).
49
The
spectrum’s
peak
positions
linearly
red-shift
with
increasing
lattice
temperature,
increasing
and
decreasing
the
differential
intensity
at
pre-
and
postedge
regions,
respectively.
The
spectrum
mainly
red-shifts
because
of
the
Ti
atoms’
reduced
crystal
field.
The
simulated
X-ray
differential
absorption
for
a
2.5
K
lattice-expanded
TiO
2
crystal
is
compared
to
the
measured
differential
absorption
in
Figure
5b.
The
Au@SiO
2
@TiO
2
nanoparticle
temperature
after
photo-
excitation
was
predicted
to
be
+2.5
K
through
a MSE
fit
of
all
simulations
in
Figure
S2.
Subtracting
the
2.5
K
heating
differential
from
the
experiment
(Figure
5c)
reflects
that
the
general
spectral
changes
are
reproduced.
The
heating
simulation’s
largest
disagreement
results
from
the
anatase
TiO
2
approximation
at
460
462
eV.
The
differential
between
theory
and
experiment
shows
bias,
as
in
it
is
not
centered
around
zero,
so
a statistical
conclusion
is
difficult
even
if trends
are
similar
by
eye.
The
MSE
fit
predicts
an
2.5
K
rise
in
the
TiO
2
layer
and
is
consistent,
within
an
order
of
magnitude,
with
other
studies
reports
of
7.7
K
(theoretical)
50
and
2.6
±
2.3
K
(experimental)
51
heating
of
aqueous
Au
nanoparticles
when
using
similar
excitation
densities.
The
thermal
dissipation
will,
of
course,
differ
in
the
vacuum
environment
for
the
experimental
X-ray
measurements.
Next,
the
Au@TiO
2
nanoparticle
sample
that
has
both
photothermal
and
hot
electron
effects
is
examined.
To
simulate
hot
electron
transfer
in
the
Au@TiO
2
nanoparticles,
electrons
up
to
a specific
energy
above
the
TiO
2
CBM
(0.0
eV
for
fully
thermalized
electrons,
0.1,
0.3,
0.45,
and
0.6
eV)
are
included
in
the
BSE
calculation
(Figure
6a).
We
simulate
spectra
by
approximating
an
average
electron
energy
in
the
CBs
and
not
a
distribution.
The
added
electrons
change
the
core-hole
screening
and
prevent
X-ray
transitions
into
the
newly
blocked
states,
leading
to
complex
differential
absorption,
as
shown
in
Figure
6b.
In
Figure
6b,
the
intensity
of
the
hot
electrons’
differential
absorption
is
normalized
to
the
total
number
of
simulated
hot
electrons
to
better
evaluate
excited-state
trends
with
increasing
hot
electron
energy.
Unlike
photothermal
heating,
changes
in
TiO
2
’s
simulated
differential
absorption
are
not
perfectly
linear
with
increasing
electron
energy
(Figure
6b).
Instead,
spectral
intensity
increases
with
the
hot
electron
energy,
and
there
are
differential
peaks
from
changes
in
the
screening
of
the
core
valence
exciton
and
X-ray
transitions
blocked
by
hot
electrons.
The
differential
peaks
are
mainly
a
result
of
hot
electrons
affecting
the
screening
and
angular
momentum
components
of
the
core
valence
exciton
when
measured
energies
are
above
the
hot
carriers
at
the
bottom
of
the
t
2g
bands
(Figure
6a).
The
differential
features
blue-shift
as
more
hot
electrons
screen
the
exciton
in
the
BSE.
State-filling
effects
of
hot
electrons
blocking
X-ray
transitions
begin
to
appear
at
463
eV
when
the
simulated
hot
electrons
fully
occupy
states
above
0.45
eV.
The
measured
Au@TiO
2
nanoparticle
differential
absorp-
tion
spectrum
is
compared
to
a simulated
differential
spectrum
with
0.3
eV
hot
electrons
and
14
K
lattice
expansion
in
Figure
6c.
Compared
to
Au@SiO
2
@TiO
2
,
the
simulated
X-ray
spectrum
with
hot
electrons
has
a new
minimum
around
462
eV,
consistent
with
the
measured
spectral
differences
between
samples
with
and
without
hot
electrons
(Figure
4).
An
MSE
analysis
was
used
to
determine
the
most
likely
temperature
and
hot
electron
energy
based
on
the
simulated
spectra.
Each
modeled
hot
electron
distribution
is
shown
separately
in
Figure
S9,
and
the
differential
X-ray
spectra
with
both
hot
electrons
and
temperature
simulated
are
in
Figure
S11.
Here,
the
differential
between
the
experiment
and
theory
is
centered
around
zero,
but
again,
the
experimental
results
must
be
interpreted
with
some
caution
given
the
low
number
of
averages.
Using
the
MSE
fit,
the
TiO
2
lattice
temperature
is
found
to
be
hotter
than
for
the
Au@SiO
2
@TiO
2
nanoparticles,
attributed
to
the
larger
plasmon
intensity
that
leads
to
a greater
level
of
hot
electron
thermalization
in
TiO
2
(Figure
S3).
Further,
the
0.3
eV
hot
electron
quasi-equilibrium
distribution
is
consistent
with
recent
studies,
as
steady-state
and
ultrafast
Raman
measurements
estimate
that
hot
electrons
exceeding
0.32
and
0.34
eV,
respectively,
transfer
from
Au
nanoparticles
Figure
5. Photothermal
effects
on
the
Ti
L
2,3
edge
in Au@SiO
2
@
TiO
2
. (Top)
Raw
spectra
for
reference.
The
spectral
range
analyzed
by
the
MSE
is shaded.
(a)
Simulated
differential
X-ray
spectra
for
a 0
20
K
anatase
lattice
expansion.
The
arrows
highlight
the
differential
amplitude
change
with
an
increasing
lattice
parameter.
(b)
The
lattice-expanded
differential
spectra
predicted
for
2.5
and
20
K heating
are
compared
to
the
Au@
SiO
2
@TiO
2
differential
spectrum.
The
yellow
shaded
region
depicts
the
experiment’s
standard
deviation.
(c)
The
measured
differential
subtracted
from
the
optimized
simulation
differential
at
2.5
K,
selected
using
the
MSE
analysis.
The
experiment’s
standard
deviation
(yellow
shading)
is included
for
reference.
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9353
9348
to
nearby
molecules.
52,53
An
approximate
calculation
compar-
ing
the
electron
excitation
and
relaxation
rates
in
the
amorphous
TiO
2
is
given
in
the
Supporting
Information,
but
the
main
conclusions
of
Figure
6
are
that
photothermal
and
hot
electron
effects
can
be
differentiated
within
a
relatively
noisy
spectrum.
Further
light-intensity-dependent
control
experiments
would
be
necessary
to
quantify
the
hot
electron
concentration
and
would
also
be
useful
to
clarify
the
nonlinear
change
with
hot
carrier
concentration
versus
the
temperature-induced
shift.
We
approximate
the
measured
hot
electron
density
using
the
relative
occupation
of
the
state
filling
in
the
band
structure.
The
state-filling
simulation
for
states
up
to
0.3
eV
above
the
CBM
best
matches
the
experiment
(Figures
6c
and
S2b).
The
integrated
total
DOS
in
this
energy
range
is
150
states
of
the
10
4
total
possible
calculated
conduction
states.
Using
this
number,
and
the
measured
DOS
for
nanocrystalline
TiO
2
(8
×
10
18
cm
3
),
one
can
approximate
an
electron
concentration
of
10
16
cm
3
.
54
Lastly,
the
Ag@SiO
2
@TiO
2
nanoparticle
system
is
examined
and
is
expected
to
have
thermalized
electron
and
hole
pairs
(Figure
7).
This
is
the
most
challenging
example
to
model
as
electrons,
holes,
and
photothermal
effects
are
present
within
the
noisy
experimental
spectrum,
and
the
hole
more
weakly
perturbs
the
spectrum
than
the
electron
since
the
probed
Ti
2p
states
predominantly
compose
the
conduction
band
(Figure
3a).
The
holes
only
change
the
screening
and
angular
momentum
components
of
the
core
valence
excitons
in
the
X-ray
spectrum
without
adding
or
blocking
new
transitions,
usually
the
larger
signal.
The
theoretical
differential
absorption
in
Figure
7a
as
compared
to
thermal
and
hot
electron
changes
does
demonstrate
that
adding
holes
to
the
calculation
should
have
a measurable
effect
but
experimentally
requires
a better
SNR.
Namely,
introducing
holes
to
a photoexcited
electrons-
only
model
produces
an
increase
in
the
differential
absorption
largely
at
462
eV
and
across
the
spectrum.
The
Ag@SiO
2
@TiO
2
nanoparticles’
measured
differential
absorption
is
compared
to
the
simulated
differential
for
thermalized
carriers
in
Figure
7b.
Calculating
the
differential
absorption
spectra
and
corresponding
MSE
for
thermalized
electrons,
thermalized
electron
hole
pairs,
and
photothermal
Figure
6. Hot
electron
effects
on
the
Ti
L
2,3
edge
in Au@TiO
2
. (a)
Simulated
hot
electrons
0.0,
0.1,
0.3,
0.45,
and
0.6
eV
above
the
TiO
2
CBM.
(b,
Top)
Raw
spectra
for
reference.
The
spectral
range
analyzed
by
the
MSE
is shaded.
(b)
Simulated
differential
X-ray
spectra
for
each
hot
electron
occupation
in
TiO
2
. The
spectral
intensity
is normalized
to the
number
of electrons
simulated
in all
but
the
thermalized
(0.0
eV)
case.
(c)
The
optimized
simulation
with
hot
electrons
0.3
eV
above
the
CBM
with
a 14
K lattice
expansion
compared
to
the
Au@TiO
2
measured
differential
absorption.
The
orange
shaded
region
depicts
the
experiment’s
standard
deviation.
(d)
The
measured
differential
subtracted
from
the
optimized
simulation
differential
shown
in
(c)
with
the
standard
deviation.
Figure
7. Thermalized
electron
plus
hole
effects
on
the
Ti
L
2,3
edge
in Ag@SiO
2
@TiO
2
. (Top)
Raw
spectra
for
reference.
The
spectral
range
analyzed
by
the
MSE
is shaded.
(a)
Simulated
differential
spectra
for
a +5
K
lattice
expansion
(light
gray),
thermalized
electrons
(gray),
and
thermalized
carriers
(black).
The
thermalized
carriers
are
simulated
at
each
respective
band
edge.
(b)
Thermalized
carriers
compared
to
the
Ag@SiO
2
@TiO
2
measured
differential
spectrum.
The
blue
shaded
region
depicts
the
experiment’s
standard
deviation.
(c)
The
measured
differential
subtracted
from
the
optimized
simulation
differential
shown
in (b)
with
the
standard
deviation.
ACS Nano
www.acsnano.org
Article
https://doi.org/10.1021/acsnano.3c08181
ACS Nano
2024,
18, 9344
9353
9349