of 25
1
Supporting Information
:
Determining Quasi
-
Equilibrium Electron and Hole
Distributions
of
Plasmonic Photocatalysts
U
sing
Photomodulated
X
-
ray
Absorption Spectroscopy
Levi D
aniel
Palmer
1
, Wonseok Lee
1
, Chung
-
Li Dong
2
, Ru
-
Shi Liu
3
,
Nianqiang Wu
4
, Scott K
evin
Cushing
1,
*
1
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA.
2
Department of Physics, Tamkang
University, New Taipei City 251301, Taiwan
3
Department of Chemistry, National Taiwan University
and Advanced Research Center for Green Materials Science
and Technology
,
Taipei 10617, Taiwan
4
Department of Chemical Engineering, University of Massachusetts Amherst, Amherst, MA 01003
9303, United
States
*
Corresponding author. Email: scushing@caltech.edu
2
Table of Contents
1.
Interpreting Previous Ultrafast X
-
ray Spectra of Anatase TiO
2
.
................................
.....
3
2.
Quantifying the Fit between Experiment and Theory using Mean Squared Error (MSE)
.
4
3.
Core
-
Shell Nanoparticle UV
Visible Spectra.
................................
................................
5
4.
Photodiode Heterojunction Input Parameters.
................................
.............................
5
5.
Energy
-
Dependent Broadening of Simulated Spectra.
................................
..................
6
6.
Raw, Experimental X
-
ray Spectra and Charging Effects.
................................
...............
7
7.
Comparing Carrier Excitation and Relaxation Rates in Amorphous TiO
2
.
.......................
8
8.
Ground
-
State Calculations.
................................
................................
..........................
8
8A. Ground
-
state DFT (Quantum ESPRESSO): Cutoff energy convergence, variable
-
cell relaxation,
and band structure calculations.
................................
................................
................................
.....
9
8B. Ground
-
state X
-
ray theory (OCEAN): BSE, screening, and scissor shift.
................................
.....
10
9.
Excited
-
State X
-
ray Theory and Statistical Error Calculations: lattice expansion
parameters
(heating), state
-
filling simulations, and mean
-
squared error (MSE) calculations.
11
9A. Heating (Thermal Lattice Expansion)
................................
................................
.......................
11
9B. Hot Electrons Simulated in TiO
2
Anatase
................................
................................
.................
12
9C. Electrons and Holes Simulated in TiO
2
Anatase
................................
................................
........
13
9D. Mean Squared Error (MSE) Calculations
................................
................................
..................
14
9E. Hot Electron X
-
ray Differential Spectra with Temperature Included (Used for the MSE Analysis)
................................
................................
................................
................................
....................
15
10.
References
................................
................................
................................
................
16
Appendix A. Example Quantum ESPRESSO Input Files:
................................
..........................
18
Appendix B. Example OCEAN 2.5.2 Input File:
................................
................................
.......
24
3
1.
Interpreting Previous Ultrafast
X
-
ray
Spectra of Anatase TiO
2
.
The excited
-
state
Bethe
-
Salpeter equation (
BSE
)
approach is tested on previous ultrafast
X
-
ray
measurements of anatase TiO
2
collected in ref.
1
(
Figure
S1
).
1
This test notably benchmarks
the theory’s accuracy in modeling carrier distributions and potential discrepancies introduced
using the TiO
2
anatase phase approximation.
Figure
S1
a
compares the Ti L
2,3
transient spectrum
15 ps before and 1 ps after photoexcitation to the ground
-
state simulation performed in this work.
Femtosecond carrier dynamics in TiO
2
are simpler to model than plasmonic quasi
-
equilibrium hot
carrier distributions because the transient carrier populations relax as a function of time after direct
photoexcitation. Direct photoexcitation also avoids complex photothermal and carrier trappin
g
effects occurring in the steady state.
Figure
S1
b
compares the simulation and
experiment for
ultrafast
X
-
ray
spectra of anatase TiO
2
. Fully thermalized electrons and holes are simulated with
equal contribution. The spectral features are notably similar but the effects at the L
3
t
2g
and e
g
peaks
are under
-
and overapproximated, respectively. This result indicates a relatively strong agreement
between the experiment and theory in the advent of well
-
defined ultrafast carrier dynamics.
Figure S1.
Simulating carriers in ultrafast Ti L
2,3
edge
X
-
ray
spectra of anatase TiO
2
.
(a) Ground
-
and excited
-
state (
-
15 ps
, blue,
and 1 ps
, red,
before/after photoexcitation) experimental spectra of anatase TiO
2
compared to
the
simulation in this work (black)
. (b) The 1 ps transient differential of anatase TiO
2
modeled with thermalized
electrons and holes using the
ab initio
approach. All experimental spectra were measured in ref.
[1]
.
4
2.
Quantifying
the
Fit
between Experiment and Theory using Mean Squared Error (MSE)
To quantify the agreement between the simulated and experimental spectra, the mean squared error
(MSE) is calculated for each simulated
spectrum. The MSE,
(
y
simulated
y
measured
)
2
x
N
, is
given by the
difference in the spectral intensity for the simulated (y
simulated
) and measured (y
measured
) spectra at each point
in energy normalized to the total number of
analyzed
points in the spectrum (N). The MSE represents the
average squared difference between the experiment and theory across the spectral range.
For Au@SiO
2
@TiO
2
, the differential
X
-
ray
spectrum
for each simulated
lattice temperature (∆T
of 0, 2.5, 5.0, 10, 15, and 20 K)
was calculated, as shown in
Figure
S2a
. The MSE for each temperature
was fit using a quadratic regression model (R
2
= 0.99) to determine the lattice temperature of the
Au@SiO
2
@TiO
2
nanoparticles.
Because the quadratic regression fits the quadratic relationship of the MSE
(mOD
2
)
as a function of temperature (K)
, the vertex reflects the best match between the experiment and
theory, giving an estimate of the photothermal temperature.
The fit minimum at 2.5 K has a MSE of
~
5
mOD
2
, or the average difference between the simulated and measured spectrum is ~2 mOD across the
spectral range, rounding to the first significant digit.
For
A
u@TiO
2
,
the
MSE prediction and quadratic regression model (R
2
= 0.95) were again used to
determine the most likely combination of heating and hot electron occupation. The
~
5 mOD
2
MSE
minimum in the quadratic regression suggests that hot electrons exist up to 0.3 eV above the CBM when
including a 14 K lattice expansion, where both temperature and electron energy are optimized by
minimizing the total MSE (
Figure
S2b
). This MSE again equates to an average difference of ~2 mOD
between the simulated and measured spectra.
For Ag@SiO
2
@TiO
2
, the MSEs for a +5 K
lattice expansion, thermalized electrons, and
thermalized carriers are displayed. There was no determined significant difference between the three
simulations although thermalized electron
-
hole pair reportedly exist in the TiO
2
layer.
Figure S2.
Mean square
d
error (MSE) analysis for the
metal@(SiO
2
)@TiO
2
nanoparticle
X
-
ray
simulations
.
The MSE between the simulated and measured spectra in
Figures
(a)
5, (b) 6, and
(c)
7 within the main text.
The
MSEs
in (a) and
(b)
are fit to a quadratic regression model with
the fit’s vertex
at 2.5 K
and 0.3 eV, respectively
.
The
MSE analysis in (b) contains both the hot electron simulation and a 14 K lattice expansion as shown by the
spectra in
Figure S11
.
5
3.
Core
-
S
hell
N
anoparticle
UV
Visible Spectra
.
Figure S
3
.
UV
visible absorption spectra for the
metal
@
(SiO
2
)
@
TiO
2
nanoparticles
.
The
intensity is scaled
to the TiO
2
absorption
onset
to depict the relative
position between each
localized surface plasmon resonance
(LSPR)
and the TiO
2
absorption
.
4.
Photodiode
H
eterojunction
I
nput
P
arameters
.
Table S1.
Input
Heterojunction Materials Properties
Parameter
Units
Ag
Au
SiO
2
TiO
2
Thickness
[nm]
15
15
10
20
Dielectric
constant
[
]
1200
1200
11.9
11.9
Electron affinity
[eV]
1.0
-
2.0*
4.3
Fermi Level
[eV]
-
5.3
-
5.2
-
4.68 (calc.)
Band gap
[eV]
0.001
0.001
9.1
3.2
Cond./val. band density
[cm
-
3
]
1E22/1E22
1E22/1E22
2.9E19/2.7E19
1E21/1E21
Electron/hole
mobility
[cm
2
/Vs]
1107/424.6
1107/424.6
0.01/0.001
0.1/0.001
Acceptor/donor
concentration
[cm
-
3
]
0/100*
0/5E14
Electron/hole thermal
velocity
[cm/s]
1E7/1E7
1E7/1E7
1E7/1E7
1E7/1E7
Layer density
[g*cm
-
3
]
10.5
19.3
2.7
3.84
6
*The electron affinity
and dopant concentration
of
SiO
2
should be 0.8 eV
and
0 cm
-
3
, respectively.
However,
different values were used to
avoid
calculation instabilities
from
the conduction band
approach
ing
the vacuum level.
Table S2.
Output
Heterojunction
Results
Result
Units
Au
@
SiO
2
@
TiO
2
Au
@
TiO
2
Ag
@
SiO
2
@
TiO
2
Schottky barrier
[eV]
0.73 (SiO
2
@
TiO
2
interface)
0.94
0.79 (SiO
2
@
TiO
2
interface)
Junction Fermi level
[eV]
-
5.3
-
5.
2
-
5.3
Built
-
in field (in
dielectric)
[V/cm]
1.7E+05
2.6E+05
2.1
E+05
We calculate the
average
built
-
in electric field (
F
) by calculating the difference in the metal and
TiO
2
Fermi levels
before forming a junction
and divid
ing
th
is
difference by the
total
dielectric
and
semiconductor
thickness
(
t
) for
TiO
2
and SiO
2
following
equation S1:
=
(
,
푇푖
2
,
푚푒푡푎푙
)
/
(S1)
5.
Energy
-
Dependent Broadening of Simulated Spectra.
The OCEAN code
uniformly
broaden
s
all simulated
X
-
ray
spectr
a
using
a
set
energy
broadening
input.
T
he calculations in this work include
0.1 eV broadening for
each
output
X
-
ray
absorption
spectrum.
The calculated spectrum
from OCEAN
is then manually broadened in MATLAB to
account for the inherent lifetime broadening of the core
-
level
X
-
ray
transition.
This lifetime
,
or
energy
-
dependent
,
broadening
is a result of
the
energy
-
dependent
loss function
and
the inelastic
mean free path of the core electron and core hole in TiO
2
.
The
spectra
are
manually broadened
by
convoluting
them with
Lorentzian
function
s
with 1.
05
(L
3
t
2g
)
, 1.55
(L
3
e
g
)
,
1.85 (L
2
t
2g
), and 2.25
(L
2
e
g
) eV
bandwidth
s
.
The average
broadening of
2.05
and 1.
3
eV for the L
2
and L
3
edges has a
ratio of 3:1.
9
, comparable to the reported 3:2 lifetime broadening ratio for TiO
2
anatase.
To accurately model the intensity of the differential spectral features, t
he
intensity of the calculated
spectra
was
also
manually modified in MATLAB. The intensity of the convoluted Lorentzian
function during the spectral broadening was manually adjusted for each major
peak in the
experiment
to match the
ground
-
state intensity. These peaks include the t
2g
and e
g
peaks for both
the L
3
and L
2
edges. The relative intensity multiplier
used for each peak
was 1 (L
3
t
2g
), 1.
65
(L
3
e
g
), 1.
3
5 (L
2
t
2g
), and
2.76
(L
2
e
g
)
a.u
.
The post
-
edge region was also manually broadened by 5.0 eV and amplified by 3
a.u
. However,
the differential spectrum in
this region was not interpreted
due to the sample charging in the
7
experimental measurements.
This large (and somewhat arbitrary) manual modification of the post
-
edge region is a testament to the fact that OCEAN
models
core
-
level
features
on
/
before the edge
better
than post
-
edge
features
and extended
X
-
ray
absorption
fine
-
structure.
6.
Raw
,
E
xperimental
X
-
ray
S
pectra
and Charging Effects
.
All experimental data was collected at the
BL20A1 beamline (National Synchrotron Radiation Research
Center in Hsinchu, Taiwan) in total electron yield mode (reflection geometry) depicted in
Figure S
4
.
Figure S
4
.
Photomodulated
X
-
ray
absorption spectroscopy in total electron yield detection geometry.
A
continuous
-
wave (CW) lamp photoexcites the specimen from 1 meter away. The incident
X
-
ray
s probe the
dynamics with the lamp on/off, and a total electron yield detector collects electron scattering to emulate
X
-
ray
absorption spectroscopy.
The raw, experimental data show spectral intensity fluctuations due to charging for both Au@TiO
2
and Ag@SiO
2
@TiO
2
(
Figure
4
a
).
The
influence
of
charging
is apparent by the greater intensity
of the light on spectrum (orange or blue) compared to the light off spectrum (black). The
Au@SiO
2
@TiO
2
nanoparticles do not reflect charging signatures, which validates the control by
suggesting the SiO
2
layer effectively blocks hot electron transfer from Au to TiO
2
. Sample
charging is measurable in the total electron yield geometry because the surface charging causes
the detected electrons to be acquired more or less efficiently, depending on the charge type, due to
Co
ulombic repulsion.
Normalization:
The spectra in
Figure 4
a
were normalized to the
X
-
ray
edge onset maximum near
458 eV to correct for the spectral charging artifacts (
Figure 4
b
). This is indicated by a grey dashed
line to guide the eye. Using the normalized spectra, the differential absorption was calculated by
mOD
=
log
10
(
light
on
light
off
)
10
3
for
each of
the two data sets (
Figure 4
c
)
and averaged (
Figure
S5
)
. The spectra were only interpreted between 458 and 466 eV to avoid regions that may have
8
been affected by the charging normalization.
These spectra are overlaid to have a better direct
comparison of the differential features and intensities (
Figure S5
).
7.
Comparing Carrier Excitation and Relaxation Rates in Amorphous TiO
2
.
With the experiment’s 10 mW/cm
2
power density (2.7*10
16
photons/(s*cm
2
) at 560 nm) and the
reported ~45% injection efficiency of hot electrons from Au into TiO
2
,
roughly 1.2*10
16
hot
electrons/(s*cm
2
) inject into TiO
2
.
2
This corresponds to ~6*10
5
hot electrons/s injected into each
particle’s TiO
2
layer, assuming uniform particle packing. Ultrafast measurements indicate that hot
electrons in crystalline TiO
2
films fully thermalize within 20
50 fs,
3
but other steady
-
state
spectroscopic measurements have reported hot electron trapping in amorphous TiO
2
surface states
that prevents carrier and phonon scattering and extends the carrier cooling time.
4
6
8.
Ground
-
S
tate
C
alculations
.
Figure S
5
.
Photomodulated
X
-
ray
spectra of the Ti L
2,3
edge.
(Top)
Experimental
ground
-
state
and
photomodulated Ti L
2,3
edge spectra and (Bottom)
differential spectra for each core
-
shell nanoparticle system:
Au@SiO
2
@TiO
2
(yellow), Au@TiO
2
(orange), and
Ag@SiO
2
@TiO
2
(blue). The ground
-
state spectra are
depicted in a lighter shade of each color, but are only
distinguishable in the inset. The differential solid lines
depict the average raw differential spectra, and the shading
depicts the standard deviation of each dat
a point across two
averaged spectra. The grey dashed line denotes the point
used for charging (amplitude) norm
alization.
9
8A
.
Ground
-
state DFT (Quantum ESPRESSO)
:
C
utoff energy convergence, variable
-
cell
relaxation, and band structure calculations.
Quantum
ESPRESSO (QE), a
DFT
package, was used to calculate
the
ground
-
state
electronic
structure
inputs for the OCEAN
X
-
ray
calculations. A variable
-
cell crystal structure relaxation was
used to define the simulated atomic coordinates of TiO
2
anatase, and a convergence calculation in
QE was used to define the cutoff energy. QE was also separately used to calculate the projected
density of states (PDOS) and
band structure
of TiO
2
.
Variable
-
Cell Relaxation.
A
variable
-
cell relaxation
(vc
-
relax)
calculation was used to determine
the anatase atomic coordinates
by optimizing the unit cell dimensions
.
The vc
-
relax calculation
was
completed with the cell_dofree = ‘ibrav’ input to maintain consistency in the lattice structure
while relaxing (optimizing) the unit cell axis and angles. See
Appendix A
for input parameters.
Convergence Calculations.
The cutoff energy used for the
X
-
ray
simulations was first checked
for total energy convergence using QE self
-
consistent field
(SCF)
calculations. The
SCF
calculation (
Appendix A
) calculates the total energy of the unit cell lattice, which is used to
determine the calculation’s convergence. The convergence is a representation of the decrease in
the total energy and a higher calculation accuracy following the increase of the wavefu
nction’s
(pseudopotential’s) energy. The purpose is to preserve calculation accuracy while reducing
computational
expense. As shown in
Figure
S6
, the 350 Ry cutoff energy used for all OCEAN
calculations was sufficiently converged.
Figure S
6
. Cutoff energy convergence.
A TiO
2
(anatase) self
-
consistent field calculation was used to calculate
the total
unit cell
lattice energy at various QE cutoff energy input values.
10
The
k
-
point
mesh
convergence was similarly confirmed using the 350 Ry cutoff energy.
However, the
k
-
points largely determine the accuracy of the state
-
filling calculation, so a large
k
-
point
mesh
are desired regardless of the convergence threshold. A 16x16x12
k
-
point mesh was
used for all OCEAN
X
-
ray
calculation (
Appendix B
)
.
Projected Density of States
(PDOS) and Band Structure
.
The band structure simulation
included the following calculation stages: SCF,
non
-
self
-
consistent field (
NSCF
)
, Bands, and
converting/plotting the data.
The SCF calculation calculates the wavefunctions for the unit cell
used for the density of states calculation, which is extrapolated in
k
-
space with the NSCF
calculation using a higher
k
-
point mesh.
The input parameters for each step can be found in
Appendix A
.
Plotted in
Figure
S7B
, the band structure
k
-
path is defined in the Bands calculation,
following previous literature.
7
The calculated Fermi level is located at the valence band edge when
calculated with DFT.
The
P
DOS simulation included the following
calculation stages
:
SCF, NSCF,
and
plotting
the projected state
s for each atomic orbital
.
The PDOS is plotted in
Figure
S7C
for the relevant
valence states, Ti 3
d
(blue) and O 2
p
(orange). Background shading in
Figure
S7C
,D
depicts the
crystal field
-
split t
2g
(blue shading) and e
g
(grey shading) states.
Figure S
7
.
Ab initio
ground
-
state
calculations
of TiO
2
anatase.
(A)
The variable
-
cell relaxed c
rystalline TiO
2
anatase
structure unit cell
with calculations of the
(B) band structure
,
(C) projected density of states (
P
DOS)
, and
(D)
X
-
ray
spectra
. A 1 eV scissor shift
is applied to
extend the
bandgap
to the experimental value
.
The band
structure and PDOS were calculated using Quantum ESPRESSO
, and the calculated
X
-
ray
spectra were calculated
with OCEAN and are also shown in the main text.
8B
.
Ground
-
state
X
-
ray
theory (OCEAN)
: BSE
,
screening
,
and scissor shift.
To simulate the TiO
2
(anatase) L
2,3
edge, the ‘Obtaining Core Excitations from
Ab initio
electronic
structure and the NIST BSE solver (OCEAN)’ code
was implemented
. OCEAN is a DFT and
11
GW/BSE approach to simulat
ing
core
-
level electron excitations.
The
DFT framework is the
Quantum
ESPRESSO
package
,
8,9
specified in the OCEAN input file (see
Appendix B
).
The DFT
stage is first calculated in OCEAN to determine the ground
-
state electronic structure.
Notably, the OCEAN code uses the BSE to simulate the Coulombic effects of the core
-
to
-
valence transition
exciton. OCEAN implements a screening stage to simulate the
screening of the
core
-
hole
by
the valence state
electrons
. The combined BSE and screening
stages alongside the
angular momentum matrix elements of the
X
-
ray
transition Hamiltonian calculate the core
-
level
transition (or
X
-
ray
absorption) spectrum.
An additional
input, core_offset = .true., was included in the OCEAN calculations here to
calculate the core
-
level shifts. The core
-
level shifts are important because they produce the Kohn
-
Sham potentials at each atomic site, which improves the calculation’s accuracy by accounting for
the unique core
-
level shifts at each atom and screening radi
us even if the atoms are equivalent
sites.
A 1 eV scissor shift was also used to adjust the simulated band gap to the 3.2 eV experimental
value, but this additional s
tep did not appear to significantly affect the output
X
-
ray
spectrum.
9.
Excited
-
S
tate
X
-
ray
T
heory and
S
tatistical
E
rror
C
alculations
: lattice expansion
parameters
(heating), state
-
filling simulations, and mean
-
squared error (MSE)
calculations.
An adiabatic approximation was taken to simulate the photoexcited and quasi
-
equilibrium
dynamics. In other terms, the excited
-
state dynamics are longer lived than the initial electron field
excitation from the photon field.
9A
.
Heating (
T
hermal
L
attice
E
xpansion
)
P
hotoexcited thermal effects
are accounted for
using the thermal expansion
coefficient
of
TiO
2
anatase.
10
The lattice expansion is anisotropic with the two expansion coefficients being
4.469E
-
06 K
-
1
(a and b directions) and 8.4283E
-
06 K
-
1
(c direction). Equation S2 describes the
calculation of
an
expanded lattice parameter (
d
) at an elevated temperature (
T
)
using the lattice
expansion coefficient (
)
with an
assumed ground
-
state/room temperature at 300 K.
The initial
lattice constants at 300 K
(
d
0
)
were 7.052 Bohr (a and b directions) and 17.81 Bohr (c direction).
=
(
0
(
300
)
)
+
0
(S2)
The lattice expansion was calculated and simulated for
302.5 K,
305 K, 310 K,
315 K,
and
320 K
.
The
percent expansion
for each temperature
is shown in
Table S3
.
The spectra calculated
with these OCEAN input lattice constants are found in the main text
Figure
5
a
.
12
Table S3:
TiO
2
Anatase
Lattice Expansion Calculation Results
Temperature (K)
a
(Bohr)
b (Bohr)
c (Bohr)
a % change
c % change
300
7.05288
7.05289
17.81326
302.5
7.05296
7.05297
17.81363
0.0011
0.0021
305
7.05304
7.05305
17.81401
0.0022
0.0042
310
7.05320
7.05321
17.81476
0.0044
0.0084
315
7.05336
7.05337
17.81551
0.0067
0.0126
320
7.05352
7.05352
17.81626
0.0089
0.0168
9B
.
Hot
E
lectrons
Simulated in TiO
2
Anatase
The OCEAN v2.5.2 source code was previously modified to include photoexcited carriers
using state
-
filling simulations
.
11
Following these modifications, the OCEAN code automatically
outputs an array file after the CNBSE stage that contains unsorted valence states in k
-
space defined
as occupied (contain
s
an electron
= 1) or unoccupied (no electrons
= 0). These states are the valence
band and conduction band, respectively, and the number of states is set using the
k
-
point mesh
defined in the OCEAN input file (1
6
x1
6
x12 =
3072
states in this work).
Figure
S8A
,B
depict the
array file output from the OCEAN code where the
k
-
points are not sorted along a specific
k
-
path
following high
-
symmetry
k
-
points.
Each
k
-
point has an associated energy as shown in
Figure
S8A
.
Then, the
k
-
points can be manually sorted into
the same high
-
symmetry path used for the band
structure (
Figure
S8C
).
Sorting the
k
-
points into the band structure
is an essential step for later
visualizing the simulated carrier distributions.
Figure S
8
.
Unsorted and sorted
k
-
points for the array file and TiO
2
anatase band structure.
(a)
The
k
-
point
grid depicts the number of
k
-
points simulated
in
reciprocal space
. (b) The unsorted
k
-
points plotted by energy. The
1 eV scissor shift
of the band gap
is not included, and the Fermi level (0 eV) is defined at the valence band edge.
(C) The sorted
k
-
points along the
k
-
path of the labelled high
-
symmetry
k
-
points
.
13
The state
-
filling model takes the valence states defined as ‘0’ or ‘1’ and redefines their
values to simulate photoexcited electrons and/or holes. F
or the
hot
electron state
-
filling algorithm
in this work, the electronic states up to a specified value are all
fully
filled.
Hot electrons up to
0.0
(fully thermalized),
0.1, 0.3, 0.45, and 0.6 eV above the conduction band minimum were simulated
in this work, shown in the main text. The input hot electron state
-
filling simulations used for all
calculations are shown in
Figure
S9
. The electron distributions were interpolated onto the band
structure because OCEAN uses the unsorted
k
-
points as an input
(
Figure S8B
)
. Note that this state
-
filling method would not typically be
k
-
point selective for other band structures; however,
all
conduction band
states at or below 0.6 eV
for
TiO
2
anatase
have occupations at
or near
the gamma
point. Therefore,
the
band structure
allows for
the
simulation
of hot electrons
at the gamma point
alone as compared to other, more complicated band structures that would distribute the hot
electrons in
k
-
space.
Figure S
9
.
Hot electron
state
-
filling
simulation
s
.
Electronic states simulated with hot electrons (colored in yellow
to
orange
gradient
) filling from the conduction band minimum up to (A)
0.0 eV
thermalized/one conduction state
filled, (B)
0.1 eV, (
C
) 0.3 eV, (
D
) 0.45 eV, and (
E
) 0.6 eV.
Each state
-
filling occupation was separately input into
OCEAN.
A
1 eV scissor shift is included to accurately depict the simulation input. The bands were interpolated in
k
-
space to project the simulated state
-
filling onto the band structure.
9C
.
Electrons and
H
oles
Simulated in TiO
2
Anatase
Similar to
the procedure for
hot electrons alone, fully thermalized electrons and holes were
simulated with OCEAN state
-
filling at the conduction band minimum and valence band maximum.
This approach was taken to simulate the effect of plasmon
-
induced dipole coupling on the
photomodula
ted
X
-
ray
spectra. The energies of the thermalized electrons and holes were
at the
14
conduction and valence band edges (only one state filled)
.
Figure
S10
depicts the state
-
filling
simulation with the electrons and holes in each respective band. The M
indirect transition is
assumed due to the indirect band gap and lowest transition energy
for
anatase
TiO
2
.
Figure S
10
.
Fully thermalized electron and hole
state
-
filling simulations
.
Thermalized electrons and holes were
input into the OCEAN
simulations
as
the state
-
filled band structure
shown
.
A
1 eV scissor shift is included to
accurately depict the simulation input. The bands were interpolated in
k
-
space to project the simulated state
-
filling
onto the band structure
.
9D
. Mean
Squared Error
(MSE)
Calculations
The MATLAB mean
squared error
(MSE)
goodnessOfFit
function
was used to
quantify
the
difference between the simulated differential spectra from OCEAN and the
raw,
photomodulated
experimental data.
Specifically, both the simulated and experimental differential spectra were
compared in the 45
8
460.6 eV and 462.4
466 eV energy range
s
of interest
using the ‘MSE’ fit
function
.
These energy windows were selected because of the accuracy of the TiO
2
anatase
approximation for the experimentally measured amorphous TiO
2
(
Figure 3
).
For the lattice heating
simulations, the intensity of the dif
ferential spectrum was unmodified and determined solely by
the intensity of the simulated
X
-
ray
spectrum output. For the state
-
filling simulations, the intensity
of the spectrum was first chosen to minimize the
MSE
between the simulated and experimental
X
-
ray
differential spectra. This is essential because the OCEAN code cannot accurately predict the
exact differential intensity (∆OD) of carriers as partial state occupations
,
and a low carrier density
exist
s
experimentally. The state
-
filling simulation’s differential intensity was then normalized to
the number of simulated carriers to account for intensity fluctuations caused by the increased
carrier density.
The values selected for normalizing the hot electron state
-
filling spectral intensity
are shown in
Table S4
, and the total intensity of the spectrum per carrier simulated is 4,500,000.