of 6
Experimental and
Ab Initio
Ultrafast Carrier Dynamics in Plasmonic Nanoparticles
Ana M. Brown,
1
Ravishankar Sundararaman,
2,3
,*
Prineha Narang,
1,2,4
,
Adam M. Schwartzberg,
5
William A. Goddard III,
2,6
and Harry A. Atwater
1,2
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology,
1200 East California Boulevard, Pasadena, California 91125, USA
2
Joint Center for Artificial Photosynthesis, California Institute of Technology,
1200 East California Boulevard, Pasadena, California 91125, USA
3
Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA
4
NG NEXT, 1 Space Park Drive, Redondo Beach, California 90278, USA
5
The Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
6
Materials and Process Simulation Center, California Institute of Technology, 1200 East California Boulevard,
Pasadena, California 91125, USA
(Received 11 August 2016; published 21 February 2017)
Ultrafast pump-probe measurements of plasmonic nanostructures probe the nonequilibrium behavior
of excited carriers, which involves several competing effects obscured in typical empirical analyses. Here
we present pump-probe measurements of plasmonic nanoparticles along with a complete theoretical
description based on first-principles calculations of carrier dynamics and optical response, free of any
fitting parameters. We account for detailed electronic-structure effects in the density of states, excited
carrier distributions, electron-phonon coupling, and dielectric functions that allow us to avoid effective
electron temperature approximations. Using this calculation method, we obtain excellent quantitative
agreement with spectral and temporal features in transient-absorption measurements. In both our
experiments and calculations, we identify the two major contributions of the initial response with distinct
signatures: short-lived highly nonthermal excited carriers and longer-lived thermalizing carriers.
DOI:
10.1103/PhysRevLett.118.087401
Plasmonic hot carriers provide tremendous opportunities
for combining efficient light capture with energy conversion
[1
5]
and catalysis
[6,7]
at the nanoscale
[8
10]
.The
microscopic mechanisms in plasmon decays across various
energy, length, and time scales are still a subject of consid-
erable debate, as seen in recent experimental
[11,12]
and
theoretical literature
[13
16]
. The decay of surface plasmons
generateshot carriers through several mechanisms, including
direct interband transitions, phonon-assisted intraband tran-
sitions, and geometry-assisted intraband transitions, as we
have shown in previous work
[17,18]
.
Dynamics of hot carriers are typically studied via ultra-
fast pump-probe measurements of plasmonic nanostruc-
tures using a high-intensity laser pulse to excite a large
number of electrons and measure the optical response as a
function of time using a delayed probe pulse
[11,19
25]
.
Various studies have taken advantage of this technique
to investigate electron-electron scattering, electron-phonon
coupling, and electronic transport
[20,22,26
32]
. Figure
1
shows a representative map of the differential extinction
cross section as a function of pump-probe delay time and
probe wavelength. With an increase in electron temper-
ature, the real part of the dielectric function near the
resonant frequency becomes more negative, while the
imaginary part increases
[33]
. This causes the resonance
to broaden and blueshift at short times as the electron
temperature rises rapidly, and then to narrow and shift back
over longer times as electrons cool down, consistent with
previous observations
[34]
. Taking a slice of the map at one
probe wavelength reveals the temporal behavior of the
electron relaxation [Fig.
1(b)
], whereas a slice of the map at
one time gives the spectral response, as shown in Fig.
1(a)
for a set of times relative to the delay time with maximum
signal,
t
max
¼
700
fs.
Conventional analyses of pump-probe measurements
invoke a
two-temperature model
that tracks the time
dependence (optionally the spatial variation) of separate
electron and lattice temperatures,
T
e
and
T
l
, respectively,
which implicitly neglects nonequilibrium effects of the
electrons. Recent literature has focused on the contributions
of thermalized and nonthermalized electrons to the optical
signal in pump-probe measurements using free-electron-
like theoretical models to interpret optical signatures
[20,26,35
38]
. However, these models invariably require
empirical parameters for both the dynamics and response of
the electrons, making unambiguous interpretation of
experiments challenging. This Letter quantitatively identi-
fies nonequilibrium ultrafast dynamics of electrons, com-
bining experimental measurements and parameter-free
ab initio
predictions of the excitation and relaxation
dynamics of hot carriers in plasmonic metals across time
scales ranging from 10 fs to 10 ps. Note that, while metal
thin films or single crystals would provide a
cleaner
experimental system in general, we focus on nanoparticles
here because they enable an important simplification:
electron distributions are constant in space over the length
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=
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=
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=
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scale of these particles, allowing us to treat temporal
dynamics and optical response in greater detail (see
Supplemental Material
[39]
).
A theoretical description of pump-probe measurements of
hot carrier dynamics in plasmonic systems involves two
major ingredients: (i) The optical response of the metal (and
its environment) determines the excitation of carriers by the
pump as well as the subsequent signal measured by the probe
pulse; (ii) the dynamics of the excited carriers, including
electron-electronand electron-phononscattering,determines
the time dependence of the probe signal. We previously
presented
[33]
ab initio
theory and predictions for both the
optical response and the dynamics within a two-temperature
model, where the electrons are assumed to be in internal
equilibrium albeit at a different temperature from the lattice.
Below, we treat the response and relaxation of nonthermal
electron distributions from first-principles calculations, with-
out assuming an effective electron temperature at any stage.
For the optical response, we calculate the imaginary part
of the dielectric function Im
ε
ð
ω
Þ
accounting for direct
interband transitions, phonon-assisted intraband transi-
tions, and the Drude (resistive) response, and calculate
the real part using the Kramers-Kronig relations.
Specifically, we start with density-functional theory calcu-
lations of electron and phonon states as well as electron-
photon and electron-phonon matrix elements using the
JDFTX
code
[40]
, convert them to an
ab initio
tight-binding
model using Wannier functions
[41]
, and use the Fermi
golden rule and linearized Boltzmann equation for the
transitions and Drude contributions, respectively. The
theory and computational details for calculating
ε
ð
ω
Þ
are
presented in detail in Refs.
[17,33]
, and we do not repeat
them here. All these expressions are directly in terms of the
electron occupation function
f
ð
ε
Þ
, and we can straightfor-
wardly incorporate an arbitrary nonthermal electron dis-
tribution instead of Fermi functions. These nonthermal
distributions differ from the thermal Fermi distributions by
sharp distributions of photoexcited electrons and holes that
dissipate with time due to scattering, as shown in Fig.
2
and
discussed below.
We use the
ab initio
metal dielectric function for
calculating the initial carrier distribution as well as the
probed response. The initial carrier distribution following
the pump pulse is given by
f
ð
ε
;t
¼
0
Þ¼
f
0
ð
ε
Þþ
U
P
ð
ε
;
ω
Þ
g
ð
ε
Þ
;
ð
1
Þ
where
f
0
is the Fermi distribution at ambient temperature,
U
is the pump pulse energy absorbed per unit volume,
g
ð
ε
Þ
is the electronic density of states
[33]
, and
P
ð
ε
;
ω
Þ
is the
energy distribution of carriers excited by a photon of energy
ω
[17]
. We then evolve the carrier distributions and lattice
temperature in time to calculate
f
ð
ε
;t
Þ
and
T
l
ð
t
Þ
as
FIG. 1. (a) Map of the differential extinction cross section of
colloidal gold nanoparticles as a function of pump-probe delay
time and probe wavelength for a pump pulse of
68
μ
J
=
cm
2
energy density at 380 nm. At time 0, the pump pulse excites the
sample. As the electrons thermalize internally, extinction near the
absorption peak (533 nm) decreases (negative signal) while
extinction in the wings to either side of the absorption peak
increases (positive signal). After
700
fs, the electrons began to
thermalize with the lattice and the differential extinction decays.
A contour line is drawn in black at zero extinction change.
Differential extinction (b) as a function of probe wavelength at a
set of times relative to the pump-probe delay time with maximum
signal,
t
max
¼
700
fs and (c) as a function of pump-probe delay
time at various probe wavelengths.
FIG. 2. Difference of the predicted time-dependent electron
distribution from the Fermi distribution at 300 K, induced by a
pump pulse at 560 nm with intensity of
110
μ
J
=
cm
2
. Starting
from the carrier distribution excited by plasmon decay at
t
¼
0
,
electron-electron scattering concentrates the distribution near the
Fermi level with the peak optical signal at
700
fs, followed by a
return to the ambient-temperature Fermi distribution and a decay
of the optical signal due to electron-phonon scattering.
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described next. From those, we calculate the variation of
the metal dielectric function
ε
ð
ω
;t
Þ
and, in turn, the
extinction cross section using Mie theory
[42,43]
.To
minimize systematic errors between theory and experiment,
we add the
ab initio
prediction for the change in the
dielectric function from ambient temperature
[33]
to the
experimental dielectric functions from ellipsometry
[44]
.
We calculate the time evolution of the carrier distribu-
tions using the nonlinear Boltzmann equation:
d
dt
f
ð
ε
;t
Þ¼
Γ
e
-
e
½
f
ε
Þþ
Γ
e
-ph
½
f;T
l
ε
Þ
;
ð
2
Þ
where
Γ
e
-
e
and
Γ
e
-ph
, respectively, are the contributions due
to electron-electron and electron-phonon interactions to the
collision integral. For simplicity, we assume that the
phonons remain thermal at an effective temperature
T
l
ð
t
Þ
and calculate the time evolution of the lattice temperature
using energy balance,
C
l
ð
T
l
Þð
dT
l
=dt
Þ¼ð
dE=dt
Þj
e
-ph
,
where the term on the right corresponds to the rate of
energy transfer from the lattice to the electrons due to
Γ
e
-ph
,
and
C
l
is the
ab initio
lattice heat capacity
[33]
.
The
ab initio
collision integrals are extremely computa-
tionally expensive to calculate repeatedly to directly solve
Eq.
(2)
. We therefore use simpler models for the collision
integrals parametrized using
ab initio
calculations. For
electron-electron scattering in plasmonic metals, the calcu-
lated electron lifetimes exhibit the inverse quadratic energy
dependence
τ
1
ð
ε
Þ
ð
D
e
=
Þð
ε
ε
F
Þ
2
characteristic of
free-electron models within Fermi liquid theory
[17]
.We
therefore use the free-electron collision integral
[20,26,45]
,
Γ
e
-
e
½
f
ε
Þ¼
2
D
e
Z
d
ε
1
d
ε
2
d
ε
3
g
ð
ε
1
Þ
g
ð
ε
2
Þ
g
ð
ε
3
Þ
g
3
ð
ε
F
Þ
×
δ
ð
ε
þ
ε
1
ε
2
ε
3
Þ
×
f
f
ð
ε
2
Þ
f
ð
ε
3
Þ½
1
f
ð
ε
Þ½
1
f
ð
ε
1
Þ
f
ð
ε
Þ
f
ð
ε
1
Þ½
1
f
ð
ε
2
Þ½
1
f
ð
ε
3
Þg
;
ð
3
Þ
with the constant of proportionality
D
e
extracted from
ab initio
calculations of electron lifetimes
[33]
. In doing so,
we neglect variation of the electron-electron scattering rate
between states with different momenta at the same energy,
which is an excellent approximation for gold where this
variation is
10%
for energies within 5 eV of the Fermi
level
[17]
. For electron-phonon scattering, assuming that
phonon energies are negligible on the electronic energy
scale (an excellent approximation for optical frequency
excitations in metals), we can simplify the electron-phonon
collision integral to
Γ
e
-ph
½
f;T
l
ε
Þ
¼
1
g
ð
ε
Þ
ε

H
ð
ε
Þ

f
ð
ε
Þ½
1
f
ð
ε
Þþ
k
B
T
l
f
ε

;
ð
4
Þ
where
H
ð
ε
Þ
is an energy-resolved electron-phonon cou-
pling strength calculated from
ab initio
electron-phonon
matrix elements
[33]
. (See Supplemental Material
[39]
for
details, derivations, and plots, as well as numerical tabu-
lation of
H
ð
ε
Þ
for four commonly used plasmonic metals:
the noble metals and aluminum).
In our experiments, we use an ultrafast transient absorp-
tion system with a tunable pump and white-light probe to
measure the extinction of Au colloids in solution as a
function of pump-probe delay time and probe wavelength.
The laser system consists of a regeneratively amplified Ti:
sapphire oscillator (Coherent Libra), which delivers 1-mJ
pulse energies centered at 800 nm with a 1-kHz repetition
rate. The pulse duration of the amplified pulse is approx-
imately 50 fs. The laser output is split by an optical wedge
to produce the pump and probe beams and the pump beam
wavelength is tuned using a coherent OperA optical para-
metric amplifier. The probe beam is focused onto a sapphire
plate to generate a white-light continuum probe. The time-
resolved differential extinction spectra are collected with a
commercial Helios absorption spectrometer (Ultrafast
Systems LLC). The temporal behavior is monitored by
increasing the path length of the probe pulse and delaying it
with respect to the pump pulse with a linear translation
stage capable of step sizes as small as 7 fs. Our sample is a
solution of 60-nm-diameter Au colloids in water with a
concentration of
2
.
6
×
10
10
particles per milliliter (BBI
International, EM.GC60, OD1.2) in a quartz cuvette with a
2-mm path length.
The initial excitation by the pump pulse generates an
electron distribution that is far from equilibrium, for which
temperature is not well defined. Our
ab initio
predictions of
the carrier distribution at
t
¼
0
in Fig.
2
exhibit high-energy
holes in the
d
bands of gold and lower-energy electrons near
the Fermi level. These highly nonthermal carriers rapidly
decay within 100 fs, resulting in carriers closer to the Fermi
level which thermalize via electron-electron scattering in
several 100 fs, reaching a peak higher-temperature thermal
distribution at
700
fs in the example shown in Fig.
2
.These
thermalized carriers then lose energy to the lattice via
electron-phonon scattering over several picoseconds.
The conventional two-temperature analysis is only valid in
that last phase of signal decay (beyond 1 ps) once the
electrons have thermalized. The initial response additionally
includes contributions from short-lived highly nonthermal
carriers excited initially, which become particularly impor-
tant at low pump powers when smaller temperature changes
limit the thermal contribution. Higher-energy nonthermal
carriers exhibit faster rise and decay times than the thermal
carriers closer to the Fermi level
[26,35]
, due to higher
electron-electron scattering rates. Their response also spans a
greater range in probe wavelength compared to thermal
electrons, which primarily affect only the resonant
d
band to
Fermi level transition
[26,33,46]
, Combining
ab initio
pre-
dictions and experimental measurements of 60-nm colloidal
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gold solutions, we quantitatively identify these signatures of
thermal and nonthermal electrons, first as a function of pump
power and then as a function of probe wavelength.
Figure
3(a)
first shows that our
ab initio
predictions of
electron dynamics and optical response quantitatively cap-
ture the
absolute
extinction cross section as a function of
time for various pump pulse energies. Note that the agree-
ment is uniformly within 10%, which is the level of accuracy
that can be expected for parameter-free Density functional
theory predictions, given that the first-principles band
structures are accurate to 0.1
0.2 eV and optical matrix
elements are accurate to 10%
20%, with the larger errors for
localized
d
electrons
[18]
. We then examine the cross section
time dependence normalized by peak values to more clearly
observe the changes in rise and decay time scales.
Decay of the measured signal is because of energy
transfer from electrons to the lattice via electron-phonon
scattering. At higher pump pulse energies, the electrons
thermalize to a higher temperature. For
T
e
<
2000
K, the
electron heat capacity increases linearly with temperature,
whereas the electron-phonon coupling strength does not
appreciably change with electron temperature
[20,33]
.
Therefore, the electron temperature, and, correspondingly,
the measured probe signal, decays more slowly at higher
pump powers, as shown in Figs.
3(b)
and
3(c)
. Again, we
find quantitative agreement between the measurements and
ab initio
predictions with no empirical parameters.
The rise of the measured signal arises from electron-
electron scattering which transfers the energy from few
excited nonthermal electrons to several thermalizing elec-
trons closer to the Fermi level. Higher power pump pulses
generate a greater number of initial nonthermal carriers,
requiring fewer electron-electron collisions to raise the
temperature of the background of thermal carriers.
Additionally, the electron-electron collision rate increases
with temperature because of increased phase space for
scattering
[20]
. Both these effects lead to a faster rise time
at higher pump powers, as seen in the measurements shown
in Fig.
3(d)
, as well as in the
ab initio
predictions shown in
Fig.
3(e)
, once again in quantitative agreement.
Next, we examine the variation of the ratio of thermal
and nonthermal electron contributions with pump power.
Figure
4
shows the subpicosecond variation of measured
response for two different pump powers, but now with a
pump wavelength of 380 nm with a higher-energy photon
that excites nonthermal carriers further from the Fermi
level. Additionally, the probe wavelength of 560 nm is far
from the interband resonance at
520
nm, so that the
thermal electrons contribute less to the measured response.
The response has a slow rise and decay time for the higher
pump power, as observed previously in cases where
thermal electrons dominate. However, for the lower pump
power, the thermal contribution is smaller, making the
nonthermal contribution relatively more important, result-
ing in a faster rise and decay time. Once again, the
(a)
(b)
(c)
(d)
(e)
FIG. 3. Comparison of measured and predicted differential
cross sections at 530-nm probe wavelength for pump excitation at
560 nm with intensities of 21, 34, 68, and
110
μ
J
=
cm
2
as a
function of time. Panel (a) compares absolute measurements
(circles) and calculated values (solid lines) of the differential
cross section, while the remaining panels are normalized by the
peak value: (b) and (c) show measurements and predictions,
respectively, over the full time range, while (d) and (e) focus on
the initial rise period. Increased pump power generates more
initial carriers, which equilibrate faster (shorter rise time) to a
higher electron temperature (larger signal amplitude), which
subsequently relaxes more slowly due to increased electron heat
capacity. The
ab initio
predictions quantitatively match all these
features of the measurements.
-1
-0.5
0
0.5
1
-0.2
0
0.2
0.4
0.6
Δ
C
Ext
(normalized)
Delay Time [ps]
(a)
34
μ
J/cm
2
110
μ
J/cm
2
0
0.2
0.4
0.6
Delay Time [ps]
(b)
34
μ
J/cm
2
110
μ
J/cm
2
FIG. 4. (a) Measured and (b) calculated differential cross
sections normalized by peak value for 380-nm pump pulse with
34- and
110
-
μ
J
=
cm
2
intensities, monitored at 560-nm probe
wavelength. Contributions from the nonthermal electrons domi-
nate at lower pump power, resulting in a fast signal rise and decay.
[Correspondingly, smaller signals cause the higher relative noise
in the measurements shown in (a).]
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measurements and
ab initio
calculations, which include all
these effects implicitly, are in quantitative agreement.
Finally, we examine the variation of the temporal
signatures with probe wavelength. Thermalized electrons
in noble metals predominantly contribute near the resonant
d
s
transitions, and, therefore, nonthermal signatures
become relatively more important at probe wavelengths far
from these resonances. Figure
5(a)
indeed shows a faster
rise and decay due to nonthermal electrons for a probe
wavelength of 620 nm, compared to that at 510 nm, which
is near the interband resonance (530 nm). Capturing the
wavelength dependence of the dielectric function in simple
theoretical models
[26]
is challenging because it involves
simultaneous contributions from a continuum of electronic
transitions with varying matrix elements. Our
ab initio
calculations [Fig.
5(b)
] implicitly account for all these
transitions and are therefore able to match both the spectral
and temporal features of the measurements, with no
empirical parameters.
To conclude, by combining the first-principles calcula-
tions of carrier dynamics and optical response, this Letter
presents a complete theoretical description of pump-probe
measurements, free of any fitting parameters that are typical
in previous analyses
[35,47
49]
. The theory here accounts
for detailed energy distributions of excited carriers (Fig.
2
)
instead of assuming flat distributions
[36,37,45]
, and
accounts for electronic-structure effects in the density of
states, electron-phonon coupling, and dielectric functions
beyond the empirical free-electron or parabolic band
models previously employed
[20,26,37,45,47
52]
. This
framework, by leveraging Wannier interpolation of
electron-phonon matrix elements, enables quantitative pre-
dictions, while avoiding the empiricism that could hide
cancellation of errors or obscure physical interpretation of
experimental data. For example, we clearly identified the
temporal and spectral signatures of short-lived highly
nonthermal initial carriers and the longer-lived thermalizing
carriers near the Fermi level in plasmonic nanoparticles. By
demonstrating the predictive capabilities of our theory for
metal nanoparticles, we open up the field for similar studies
in other materials
[53]
where fits are not necessarily
possible or even reliable, e.g., semiconductor plasmonics,
and where
ab initio
theory of ultrafast dynamics will be
indispensable.
This material is based upon work performed by the Joint
Center for Artificial Photosynthesis, a DOE Energy
Innovation Hub, supported through the Office of Science
of the U.S. Department of Energy under Award No. DE-
SC0004993. Work at the Molecular Foundry was supported
by the Office of Science, Office of Basic Energy Sciences,
of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. The authors acknowledge support
from NG NEXT at Northrop Grumman Corporation.
Calculations in this work used the National Energy
Research Scientific Computing Center, a DOE Office of
Science User Facility supported by the Office of Science of
the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. P. N. is supported by a National
Science Foundation Graduate Research Fellowship and
by the Resnick Sustainability Institute. A. M. B. is sup-
ported by a National Science Foundation Graduate
Research Fellowship, a Link Foundation Energy
Fellowship, and the DOE
Light-Material Interactions in
Energy Conversion
Energy Frontier Research Center (DE-
SC0001293).
*
sundar@rpi.edu
prineha@caltech.edu
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.
-0.2
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
Δ
C
Ext
(normalized)
Delay Time [ps]
(a)
480nm
510nm
620nm
0
1
2
3
4
5
Delay Time [ps]
(b)
480nm
510nm
620nm
FIG. 5. (a) Measured and (b) calculated differential cross
sections for 560-nm pump pulse with
110
-
μ
J
=
cm
2
intensity,
normalized by peak value, for probe wavelengths of 480, 510,
and 620 nm. Rise and decay are faster for probe wavelengths far
from the interband resonance at 530 nm, where nonthermal
effects are relatively more important.
PRL
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