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 E. California Blvd, Pasadena CA
2
Joint Center for Artificial Photosynthesis, California Institute of Technology, 1200 E. California Blvd, Pasadena CA
3
Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110 8
th
Street, Troy, NY
4
NG NEXT, 1 Space Park Drive, Redondo Beach CA
5
The Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA
6
Materials and Process Simulation Center, California Institute of Technology, 1200 E. California Blvd, Pasadena CA
(Dated: December 28, 2016)
Ultrafast pump-probe measurements of plasmonic nanostructures probe the non-equilibrium be-
havior of excited carriers, which involves several competing effects obscured in typical empirical anal-
yses. 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 which allow
us to avoid effective electron temperature approximations. Using this calculation method, we ob-
tain 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 non-thermal excited carriers and
longer-lived thermalizing carriers.
Plasmonic hot carriers provide tremendous opportu-
nities for combining efficient light capture with energy
conversion
1–5
and catalysis
6,7
at the nano scale.
8–10
The
microscopic mechanisms in plasmon decays across various
energy, length and time scales are still a subject of con-
siderable debate, as seen in recent experimental
11,12
and
theoretical literature.
13–16
The decay of surface plasmons
generates hot carriers through several mechanisms includ-
ing direct interband transitions, phonon-assisted intra-
band transitions and geometry-assisted intraband transi-
tions, 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
Fig. 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 temperature,
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 blue shift at short times as the electron tempera-
ture rises rapidly, and then to narrow and shift back over
longer times as electrons cool down, consistent with previ-
ous 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
= 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 non-equilibrium effects of the
electrons.
Recent literature has focused on the con-
tributions of thermalized and nonthermalized electrons
to the optical signal in pump-probe measurements us-
ing free-electron-like theoretical models to interpret op-
tical signatures.
20,26,35–38
However, these models invari-
ably require empirical parameters for both the dynamics
and response of the electrons, making unambiguous inter-
pretation of experiments challenging. This
Letter
quan-
titatively identifies non-equilibrium ultrafast dynamics
of electrons, combining experimental measurements and
parameter-free
ab initio
predictions of the excitation and
relaxation dynamics of hot carriers in plasmonic metals
across timescales ranging from 10 fs–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 sim-
plification: electron distributions are constant in space
over the length scale of these particles, allowing us to
treat temporal dynamics and optical response in greater
detail. (See supplementary information.)
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 car-
riers by the pump as well as the subsequent signal mea-
sured by the probe pulse. ii) The dynamics of the excited
carriers, including electron-electron and electron-phonon
scattering, determines the time dependence of the probe
signal. We previously presented
33
ab initio
theory and
predictions for both the optical response and the dynam-
ics within a two temperature model, where the electrons
are assumed to be in internal equilibrium albeit at a dif-
ferent temperature from the lattice. Below, we treat the
response and relaxation of non-thermal electron distribu-
tions from first principles, without assuming an effective
arXiv:1608.03309v2 [physics.optics] 24 Dec 2016
2
FIG. 1. (a) Map of the differential extinction cross section of
colloidal gold nanoparticles as a function of pump-probe de-
lay 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 ex-
tinction decays. A contour line is drawn in black at zero ex-
tinction 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
= 700 fs; and (c) as
a function of pump-probe delay time at various probe wave-
lengths.
electron temperature at any stage.
For the optical response, we calculate the imaginary
part of the dielectric function Im

(
ω
) accounting for
direct interband transitions, phonon-assisted intraband
transitions and the Drude (resistive) response, and cal-
culate the real part using the Kramers-Kronig relations.
Specifically, we start with density-functional theory cal-
culations of electron and phonon states as well as electron-
photon and electron-phonon matrix elements using the
JDFTx code,
39
convert them to an
ab initio
tight-binding
model using Wannier functions,
40
and use Fermi Golden
rule and linearized Boltzmann equation for the transitions
and Drude contributions respectively. The theory and
computational details for calculating

(
ω
) are presented in
detail in Refs. 17 and 33, and we do not repeat them here.
All these expressions are directly in terms of the electron
occupation function
f
(
ε
), and we can straightforwardly
incorporate an arbitrary non-thermal electron distribu-
tion instead of Fermi functions. These non-thermal dis-
tributions differ from the thermal Fermi distributions by
sharp distributions of photo-excited 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 calcu-
lating the initial carrier distribution as well as the probed
response. The initial carrier distribution following the
pump pulse is given by
f
(
ε,t
= 0) =
f
0
(
ε
) +
U
P
(
ε,
̄
)
g
(
ε
)
(1)
where
f
0
is the Fermi distribution at ambient temper-
ature,
U
is the pump pulse energy absorbed per unit
volume,
g
(
ε
) is the electronic density of states,
33
and
P
(
ε,
̄
) is the energy distribution of carriers excited
by a photon of energy ̄
.
17
We then evolve the carrier
distributions and lattice temperature in time to calcu-
late
f
(
ε,t
) and
T
l
(
t
) as described next. From those, we
calculate the variation of the metal dielectric function

(
ω,t
), and in turn, the extinction cross section using
Mie theory.
41,42
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.
43
We calculate the time evolution of the carrier distribu-
tions using the nonlinear Boltzmann equation
d
dt
f
(
ε,t
) = Γ
e-e
[
f
](
ε
) + Γ
e-ph
[
f,T
l
](
ε
)
,
(2)
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 tempera-
ture
T
l
(
t
) and calculate the time evolution of the lattice
temperature using energy balance,
C
l
(
T
l
)(
dT
l
/dt
) =
(
dE/dt
)
|
e-ph
, where the term on the right corresponds
to the rate of energy transfer from the lattice to the elec-
trons due to Γ
e-ph
, and
C
l
is the
ab initio
lattice heat
capacity.
33
The
ab initio
collision integrals are extremely compu-
tationally expensive to calculate repeatedly to directly
solve (2). We therefore use simpler models for the col-
lision integrals parametrized using
ab initio
calculations.
For electron-electron scattering in plasmonic metals, the
calculated electron lifetimes exhibit the inverse quadratic
energy dependence
τ
1
(
ε
)
(
D
e
/
̄
h
)(
ε
ε
F
)
2
character-
istic of free electron models within Fermi liquid theory.
17
We therefore use the free-electron collision integral,
20,26,44
Γ
e-e
[
f
](
ε
) =
2
D
e
̄
h
1
2
3
g
(
ε
1
)
g
(
ε
2
)
g
(
ε
3
)
g
3
(
ε
F
)
×
δ
(
ε
+
ε
1
ε
2
ε
3
)
[
f
(
ε
2
)
f
(
ε
3
)(1
f
(
ε
))(1
f
(
ε
1
))
f
(
ε
)
f
(
ε
1
)(1
f
(
ε
2
))(1
f
(
ε
3
))
]
(3)
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 scatter-
ing 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
3
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
. Start-
ing from the carrier distribution excited by plasmon decay
at
t
= 0, electron-electron scattering concentrates the dis-
tribution 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.
of the Fermi level.
17
For electron-phonon scattering, as-
suming that phonon energies are negligible on the elec-
tronic energy scale (an excellent approximation for opti-
cal frequency excitations in metals), we can simplify the
electron-phonon collision integral to
Γ
e-ph
[
f,T
l
](
ε
) =
1
g
(
ε
)
∂ε
[
H
(
ε
)
(
f
(
ε
)(1
f
(
ε
)) +
k
B
T
l
∂f
∂ε
)]
,
(4)
where
H
(
ε
) is an energy-resolved electron-phonon cou-
pling strength calculated from
ab initio
electron-phonon
matrix elements.
33
(See Supporting Information for de-
tails, derivations and plots as well as numerical tabula-
tion of
H
(
ε
) for four commonly used plasmonic metals:
the noble metals and aluminum.)
In our experiments, we use an ultrafast transient ab-
sorption system with a tunable pump and white light
probe 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 regener-
atively amplified Ti:sapphire oscillator (Coherent Libra),
which delivers 1mJ pulse energies centered at 800 nm with
a 1 kHz repetition rate. The pulse duration of the am-
plified pulse is approximately 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 OPA. The probe beam is focused onto a
sapphire plate to generate a white-light continuum probe.
The time-resolved differential extinction spectra are col-
lected with a commercial Helios absorption spectrometer
(Ultrafast Systems LLC). The temporal behavior is mon-
itored 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 col-
loids 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
pre-
dictions of the carrier distribution at
t
= 0 in Fig. 2 ex-
hibits high-energy holes in the
d
-bands of gold and lower
energy electrons near the Fermi level. These highly non-
thermal 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 car-
riers 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 re-
sponse additionally includes contributions from short-
lived highly non-thermal carriers excited initially, that
become particularly important at low pump powers when
smaller temperature changes limit the thermal contribu-
tion. Higher energy non-thermal carriers exhibit faster
rise and decay times than the thermal carriers closer to
the Fermi level,
26,35
due to higher electron-electron scat-
tering 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.
26,33,45
Combining
ab initio
predictions and ex-
perimental measurements of 60-nm colloidal gold solu-
tions, we quantitatively identify these signatures of ther-
mal and non-thermal electrons, first as a function of pump
power and then as a function of probe wavelength.
Fig. 3(a) first shows that our
ab initio
predictions of
electron dynamics and optical response quantitatively
capture the
absolute
extinction cross section as a func-
tion of time for various pump pulse energies. Note that
the agreement is uniformly within 10%, which is the level
of accuracy that can be expected for parameter-free DFT
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 local-
ized
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 trans-
fer from electrons to the lattice via electron-phonon scat-
tering. At higher pump pulse energies, the electrons ther-
malize to a higher temperature. For
T
e
<
2000 K, the
electron heat capacity increases linearly with tempera-
ture, 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 Fig. 3(b,c). Again, we find
quantitative agreement between the measurements and
ab initio
predictions with no empirical parameters.
Rise of the measured signal arises from electron-
electron scattering which transfers the energy from few
excited non-thermal electrons to several thermalizing
4
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. Part (a) compares absolute measurements
(circles) and calculated values (solid lines) of the differential
cross-section, while the remaining parts 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.
electrons closer to the Fermi level. Higher power pump
pulses generate a greater number of initial non-thermal
carriers, requiring fewer electron-electron collisions to
raise the temperature of the background of thermal car-
riers. 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 mea-
surements 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 ther-
mal and non-thermal electron contributions with pump
power. Fig. 4 shows the sub-picosecond variation of mea-
sured response for two different pump powers, but now
with a pump wavelength of 380 nm with a higher energy
photon that excites non-thermal 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 re-
sponse. The response has a slow rise and decay time for
the higher pump power, as observed previously in cases
-1
-0.5
0
0.5
1
-0.2
0
0.2
0.4
0.6
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 elec-
trons dominate at lower pump power, resulting in a fast sig-
nal rise and decay. (Correspondingly smaller signals cause the
higher relative noise in the measurements shown in (a).)
-0.2
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
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 sec-
tions for 560 nm pump pulse with 110
μ
J
/
cm
2
intensity, nor-
malized 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 non-thermal
effects are relatively more important.
where thermal electrons dominate. However for the lower
pump power, the thermal contribution is smaller making
the non-thermal contribution relatively more important,
resulting in a faster rise and decay time. Once again, the
measurements and
ab initio
calculations, which include
all these effects implicitly, are in quantitative agreement.
Finally, we examine the variation of the temporal sig-
natures with probe wavelength. Thermalized electrons in
noble metals predominantly contribute near the resonant
d
s
transitions, and therefore non-thermal signatures
become relatively more important at probe wavelengths
far from these resonances. Fig. 5(a) indeed shows a faster
rise and decay due to non-thermal 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 in-
volves 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
Let-
5
ter
presents a complete theoretical description of pump-
probe measurements, free of any fitting parameters that
are typical in previous analyses.
35,46–48
The theory here
accounts for detailed energy distributions of excited car-
riers (Fig. 2) instead of assuming flat distributions,
36,37,44
and accounts for electronic-structure effects in the density
of states, electron-phonon coupling and dielectric func-
tions beyond the empirical free-electron or parabolic band
models previously employed.
20,26,37,44,46–51
This frame-
work, by leveraging Wannier interpolation of electron-
phonon matrix elements, enables quantitative predic-
tions, while avoiding the empiricism that could hide can-
cellation 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 thermal-
izing carriers near the Fermi level in plasmonic nanopar-
ticles. By demonstrating the predictive capabilities of our
theory for metal nanoparticles, we open up the field for
similar studies in other materials
52
where fits are not nec-
essarily possible or even reliable eg. semiconductor plas-
monics, and where
ab initio
theory of ultrafast dynamics
will be indispensable.
Acknowledgements
: This material is based upon
work performed by the Joint Center for Artificial Pho-
tosynthesis, a DOE Energy Innovation Hub, supported
through the Office of Science of the U.S. Department
of Energy under Award Number 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 Gradu-
ate Research Fellowship and by the Resnick Sustainability
Institute. A. B. is supported by a National Science Foun-
dation Graduate Research Fellowship, a Link Foundation
Energy Fellowship, and the DOE ‘Light-Material Inter-
actions in Energy Conversion’ Energy Frontier Research
Center (DE-SC0001293).
sundar@rpi.edu
prineha@caltech.edu
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