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Nanophotonics 2016; 5 (1):96–111
Review Article
Open Access
Prineha Narang, Ravishankar Sundararaman, and Harry A. Atwater
Plasmonic hot carrier dynamics in solid-state and
chemical systems for energy conversion
DOI: 10.1515/nanoph-2016-0007
Received October 22, 2015; accepted January 14, 2016
Abstract:
Surface plasmons provide a pathway to effi-
ciently absorb and confine light in metallic nanostruc-
tures, thereby bridging photonics to the nano scale. The
decay of surface plasmons generates energetic ‘hot’ carri-
ers, which can drive chemical reactions or be injected into
semiconductors for nano-scale photochemical or photo-
voltaic energy conversion. Novel plasmonic hot carrier de-
vices and architectures continue to be demonstrated, but
the complexity of the underlying processes make a com-
plete microscopic understanding of all the mechanisms
and design considerations for such devices extremely
challenging. Here, we review the theoretical and computa-
tional efforts to understand and model plasmonic hot car-
rier devices. We split the problem into three steps: hot car-
rier generation, transport and collection, and review the-
oretical approaches with the appropriate level of detail for
each step along with their predictions. We identify the key
advances necessary to complete the microscopic mecha-
nistic picture and facilitate the design of the next genera-
tion of devices and materials for plasmonic energy conver-
sion.
1
Introduction
Surface plasmons are collective oscillations of electrons
in conductors coupled to electromagnetic modes that
are confined to conductor–dielectric interfaces [1–4].
They enable efficient coupling of electromagnetic waves
from free space to nanoscale systems [5–9], and have
therefore found a broad range of applications, includ-
ing spectroscopy, non-linear optics [10, 11], photodetec-
tion [12–15], and solar energy harvesting [16, 17]. Addi-
tionally, novel phenomena continue to be discovered in
the field of plasmonics, such as quantum interference of
Prineha Narang, Ravishankar Sundararaman, Harry A. Atwater:
Joint Center for Artificial Photosynthesis, California Institute of
Technology, Pasadena CA 91125 USA
Prineha Narang, Harry A. Atwater:
Thomas J. Watson Laboratories
of Applied Physics, California Institute of Technology, Pasadena CA
91125 USA
plasmons [18–20], quantum coupling of plasmons to
single-particle excitations, and quantum confinement of
plasmons in single-nm scale plasmonic particles [21–23].
Experimental developments in quantum plasmonics have
shown, remarkably, the ability of surface plasmons to pre-
serve many key quantum mechanical properties of the
photons used to excite them, notably entanglement, inter-
ferometry and sub-Poissonian statistics.
Surface plasmons provide high-mode confinement be-
cause the electric field substantially penetrates into the
metal, but this also increases losses. This trade-off be-
tween localization and loss has hampered widespread ap-
plications of plasmonic waveguides and nano resonators
for applications in integrated photonics. However, losses
in plasmonics also provide unique opportunities. For ex-
ample, the short dephasing times can be used to enhance
the emission of nearby nanoemitters with lower internal
quantum efficiency [24–26]. Additionally, the decay of sur-
face plasmons creates electron–hole pairs with energies
much larger than those in the background thermal distri-
bution, and these ‘hot’ carriers enable processes not pos-
sible with thermalized carriers.
Plasmonic hot carriers provide tremendous opportu-
nities for combining efficient light capture with energy
conversion and catalysis at the nano scale. The hot carri-
ers could be used to directly drive chemical reactions at
the metal surface [27], or they could be transferred to a
semiconductor for use in photovoltaics [28–30] and photo-
electrochemical systems [31–33]. Composite systems allow
combining traditional carrier generation in semiconduc-
tors with enhanced sub-band-gap absorption in metallic
nanoparticles and injection of the generated carriers into
the semiconductor, for additional light harvesting. Plas-
monic systems can potentially also be used to tailor car-
rier energy distributions as well as localize them spatially
and temporally in order to exert tremendous control on
the photocatalytic processes, possibly even for selectivity.
Plasmon-enhanced energy conversion devices, such as for
water splitting, have been demonstrated using compos-
ite metal–semiconductor photocatalysts as well as all-in-
one catalytic noble metal plasmonic nanostructures [34–
43]. The significant experimental effort in plasmonic hot
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
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carrier–driven processes and devices has been the focus
of several recent reviews [44–46]. This review empha-
sizes an overall theoretical understanding of the micro-
scopic mechanisms underlying, and thereby the funda-
mental limits of, plasmon decay to hot carriers and their
utilization for energy conversion.
The microscopic mechanisms in plasmon decays
across various energy, length, and time scales are still
a subject of considerable debate, as seen in recent ex-
perimental and theoretical papers [47–49]. A great ma-
jority of investigations of plasmonic nanostructures fo-
cus on their optical response, usually described ade-
quately within classical electromagnetic theory [3]. How-
ever, quantum effects become important in the single pho-
ton limit since plasmons are fundamentally quantum me-
chanical [21, 50], as well as in small nanostructures due to
carrier confinement, tunneling, and uncertainty principle
effects. The quantum effects on carriers are adequately de-
scribed by time-dependent density functional theory (TD-
DFT) [51, 52] in the jellium approximation (which simpli-
fies the band structure to a free-electron one), for exam-
ple, tunneling effects in nanoparticle dimers [53–56]. First-
principles calculations that retain the full electronic band
structure are necessary in order to capture effects beyond
the free electron limit, for example, anisotropy of surface
plasmon response on noble metal surfaces [57]. More im-
portantly, such calculations are critical for describing all
the mechanisms of plasmon decay and gauging their rela-
tive contributions. Such calculations are currently feasible
for nanoparticles containing up to a few hundred atoms,
that is, 1–2 nm in size, and can explicitly account for the ef-
fects of nanoparticle shape with specific facets and surface
states on the optical response and carrier generation [58–
60].
A complete theoretical investigation of real plasmonic
hot carrier based energy conversion devices is, how-
ever, extremely challenging. The optical response of these
devices depends on length scales ranging from a few
nanometers to hundreds of nanometers or microns, and
this presents challenges even for classical electromagnetic
simulations. On the other hand, carrier generation re-
quires a quantum mechanical electronic structure treat-
ment where the relevant length scales are in Angstroms,
and the current practical upper limit for such theories is a
few nanometers. Figure 1 illustrates this disparity in length
scales, and also points out the disparate time scales rang-
ing from carrier thermalization by electron–electron scat-
tering tens of femtoseconds after excitation, to equilibra-
tion with lattice by electron–phonon scattering picosec-
onds later.
A single level of theory cannot adequately describe
plasmonic hot carrier devices spanning all the relevant
length and time scales discussed above, and this necessi-
tates a multi-scale, multi-paradigm description. Figure 2
outlines the separation of a plasmonic energy conver-
sion device (either photovoltaic or photocatalytic) into
steps that can each be described at a different appro-
priate level of theory. The coupling of light from free
space to surface plasmons can be treated using classi-
cal electromagnetic theory, and we do not deal with that
well-studied aspect here. Section 2 describes the decay
of surface plasmons to generate carriers in the material
using a combination of jellium and first-principles elec-
tronic structure methods. Section 3 then discusses the
dynamics of the generated carriers, including electron–
electron and electron–phonon scattering described using
electronic structure methods, and their transport in plas-
monic nanostructures. Finally, section 4 deals with injec-
tion of the carriers into molecules at the surface or across
metal–semiconductor Schottky barriers, and describes the
necessary theoretical considerations in each case.
2
Hot carrier generation from
plasmon decay
Surface plasmons can decay either radiatively [14, 61], by
emitting a photon, or nonradiatively, by single-particle
electronic excitations [62] that generate non-thermal elec-
trons and holes, typically referred to as hot carriers. For
plasmons, the collective excitations of electrons in met-
als, this decay to single-particle excitations constitutes
Landau damping. Several microscopic mechanisms con-
tribute to the Landau damping of plasmons, and their rela-
tive importance depends on the material, plasmon energy,
and geometry.
Photons and surface plasmons have negligible mo-
menta compared to the crystal momenta of electrons. In
bulk materials, this implies that direct decay of plasmons
can only create electron–hole pairs with net zero crystal
momentum, and such pairs of electronic states are only
available in the band structure of most metals above a
certain interband threshold energy. Below this interband
threshold, absorption or emission of phonons accompa-
nies the plasmon decay to provide the necessary momen-
tum to excite electrons to states with different crystal mo-
menta. Figure 2 shows the Feynman diagrams correspond-
ing to direct and phonon-assisted decay of surface plas-
mons. Note that we do not distinguish carrier excitations
due to surface plasmons from those due to photons be-
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
Length scales and timescales associated with plasmonic hot carriers
Fig. 1.
Top panel: length scales in plasmonics vary from the atomic to the mesoscale. Shown from left to right: atomic lattice, small gaps
in nanoparticle dimers, nanoparticles and bowtie antennae. Lower panel: typical time scales for the the excitation of hot carriers and their
subsequent relaxation. In the 10 fs regime, carrier distributions do not resemble Fermi distributions at any temperature, but at later times
the dynamics can be described approximately by distinct electron and lattice temperatures,
T
e
and
T
l
.
Plasmonic hot carrier dynamics: Generation, transport and collection
Fig. 2.
Processes involved in the excitation of plasmons, their decay to hot carriers, the transport of hot carriers in plasmonic nanostruc-
tures and their collection either in adsorbed molecules or semiconductors (lower part). The top part of the figure shows the theoretical
methods with a level of detail appropriate for each stage: (a) dielectric functions for plasmon excitation (b) electronic structure theory for
carrier generation and transport, and (c) band / energy-level alignment analysis for collection. Feynman diagrams indicate the relevant
processes at each stage: direct transitions, phonon-assisted transitions, and multiplasmon decay (in the high-intensity range only) for gen-
eration, and electron-electron and electron–phonon scattering for transport. Collection of hot carriers in solid-state systems can be used
for solar energy conversion devices, sensitive photodetectors, and nano-spectrometers. Hot carriers injected into molecules on a surface
can induce photochemical reactions, for example, CO
2
reduction, which is mechanistically very different from solid state collection.
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99
cause carrier generation only depends on the net electric
field distribution in the metal. This field distribution con-
sists of a superposition of photon and surface plasmon
modes for an illuminated metal nanostructure, depending
on the frequency of illumination relative to the plasmon
resonance. We therefore separate the questions of carrier
generation, which focuses on microscopic mechanisms
such as direct and phonon-assisted excitations, from the
question of field distributions, which can be obtained from
classical electromagnetic simulations accounting for pho-
tonic and plasmonic contributions.
In nanoscale systems, the electronic states are local-
ized in space and are therefore no longer exact (crys-
tal) momentum eigenstates by the uncertainty principle.
This introduces a finite probability of direct plasmon de-
cay into an electron–hole pair with net crystal momen-
tum, which can occur even for plasmons below the in-
terband threshold energy. For definiteness, we refer to
these as geometry-assisted intraband transitions because
the nano-scale geometry provides the momentum in lieu
of phonons to induce the transitions within the conduc-
tion band of the metal. Some distinguish this process as
plasmon-induced carrier generation in contrast to direct
and phonon-assisted transitions as photoexcited carrier
generation [49], but we emphasize that the key difference
is in the localized electronic states; optical excitation far
from the plasmonic resonance will also induce geometry-
assisted transitions in small nanoparticles.
A complete understanding of plasmonic hot carrier
generation requires accounting for material as well as ge-
ometry effects. The decay mechanisms in bulk materials,
direct and phonon-assisted transitions, are strongly de-
pendent on the electronic band structure of the metal,
whereas the geometry-assisted transitions occur predomi-
nantly in the free-electron like conduction band. Theoreti-
cally, the former require detailed bulk electronic structure
calculations, whereas the latter can be treated using free-
electron-like jellium models but require explicit inclusion
of geometry in the quantum mechanical method. Different
theoretical approaches spanning different levels of detail
and system size have been applied to different aspects of
hot carrier generation, and we need a combination of these
to understand the relative contributions of all mechanisms
as a function of material, energy, and geometry.
2.1
Geometry dependence: intraband
transitions
In nano-scale metallic systems, the free-electron-like con-
duction band splits into several discrete states with non-
zero matrix elements for optical transition between them
(they are zero within the band for the bulk metal). This
is a consequence of the geometry breaking translation in-
variance so that the Hamiltonian no longer commutes with
crystal momentum and the energy eigenstates do not have
definite crystal momentum.
Theoretical calculations can capture this effect by
solving the Schrödinger equation for electronic states
in the actual geometry, and using Fermi’s Golden rule
with optical matrix elements between these states to cal-
culate the transition rate induced by the plasmon. The
level of detail in the electronic states vary from ana-
lytical free-electron solutions [48, 63, 65], jellium time-
dependent density functional theory (TD-DFT), which ne-
glects atomic-scale structure in the nuclear potential and
therefore details in the band structure but approximately
accounts for electron–electron interactions [64], to TD-
DFT calculations with full band-structure for small metal
clusters [59]. The full calculations are currently feasible
only for clusters with very few atoms (2–3 nm across),
whereas the free electron and jellium TD-DFT calculations
can be extended to
∼
20
nm nanoparticles and they yield
qualitatively similar results [64].
Figure 3(a) shows the predicted initial energy distribu-
tion of hot carriers generated due to geometry-assisted in-
traband transitions in gold nano-platelets [63]. These tran-
sitions predominantly generate low-energy electrons near
the Fermi level for larger particles
∼
40
nm, whereas pro-
duce continuous distributions of electrons extending from
the Fermi energy to the plasmon energy above it for very
small particles
.
10
nm. Figure 3(b) shows a similar de-
pendence of the carrier generation in spherical nanopar-
ticles of various sizes [64]. It additionally shows that the
net hot carrier rate decreases with carrier lifetime, by in-
corporating lifetime as a Lorentzian broadening in Fermi’s
golden rule. (We contrast this rudimentary treatment of
transport effects with more complete models below in Sec-
tion 3.)
The free electron / jellium approaches are invaluable
in treating geometry-assisted plasmon decay in experi-
mentally relevant structures, but they are sufficient only
for materials and ranges of plasmon energies that only in-
volve the free-electron-like conduction band, and where
the other decay mechanisms are negligible. This is reason-
ably valid for silver below its interband threshold of 3.6 eV,
which is what most of these studies focus on, but less ap-
plicable to gold and copper, which exhibit direct transi-
tions at lower energies
∼
2
eV, and inapplicable to alu-
minum, which undergoes direct transitions at all energies.
We additionally need to include the material-dependent
direct and phonon-assisted processes, which in turn re-
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
Plasmonic hot carrier generation: intraband transitions in jellium
Fig. 3.
Initial hot carrier energy distributions due to geometry-assisted intraband transitions in (a) gold platelets [63] and (b) silver
spheres [64] using a jellium (free electron) approach. Part (b) additionally incorporates an energy-independent carrier lifetime
τ
as a
Lorentzian broadening in Fermi’s golden rule. The probability of high-energy electrons and holes decreases sharply as
τ
decreases or the
dimension
D
increases. Part (a) adapted from Ref. 63 and part (b) from Ref. 64 (
©
2013 and 2014 respectively, American Chemical Society).
quire a more detailed treatment that fully accounts for the
electronic band structure.
2.2
Material dependence: direct and
phonon-assisted transitions
Electronic structure methods including density-functional
theory (DFT) and many-body perturbation theory within
the GW approximation can predict accurate electronic
band structures and optical matrix elements for the bulk
material. Recently, the decay of surface plasmons via di-
rect transitions in the bulk material has been examined in
detail using Fermi’s golden rule calculations on relativistic
DFT+
U
band structures [66], and confirmed by subsequent
GW calculations [68].
Figure 4 shows the theoretical predictions for the car-
rier distributions generated by direct transitions in several
plasmonic metals, and annotates the dominant transitions
on the band structures of the metals [66]. Direct transitions
are very sensitive to the band structure because of the se-
lection rule that requires initial and final electronic states
with equal crystal momentum. In copper and gold, these
transitions are predominantly from the
d
-bands deep be-
low the Fermi level to the conduction band, which results
in high-energy holes and lower energy electrons, whereas
in silver, transitions from the conduction band can addi-
tionally produce high-energy electrons with lower energy
holes. Aluminum produces a continuous energy distribu-
tions of electrons as well as holes ranging from zero to
the plasmon energy. Additionally, this selection rule also
results in strongly anisotropic initial momentum distribu-
tions dominated by the crystal directions (rather than the
polarization direction) for noble metals, as shown in the
bottom panels of Figure 4. The polarization direction con-
trols the carrier momentum distribution only in aluminum
because of the band crossing near the Fermi level [66].
Ab initio
electronic structure calculations have re-
cently also become feasible for phonon-assisted transi-
tions and have been used to study the optical proper-
ties of indirect bandgap semiconductors [69, 70]. Subse-
quently, this approach has been generalized to metals,
which requires additional careful treatment of singular
contributions from sequential processes, and applied to
the study of plasmon decay [67]. These calculations show
that phonon-assisted transitions also generate uniform
electron and hole energy distributions ranging from zero
to the plasmon energy, similar to geometry-assisted tran-
sitions.
Most importantly, accounting for direct and phonon-
assisted transitions, along with resistive losses at lower
frequencies,
quantitatively
accounts for the experimen-
tal absorption in metals from mid-IR to UV frequencies
as shown for gold in Figure 5(a) [67]. Combined with jel-
lium estimates for geometry-assisted transitions in spher-
ical nanoparticles, this also enables estimating the rel-
ative contributions of bulk and geometry-dependent de-
cay for particles of various sizes, as shown in Figure 5(b-
c) [67]. Direct transitions dominate above the interband
threshold even for very small particles
∼
10
nm, whereas
phonon-assisted and geometry-assisted intraband transi-
tions compete below threshold. These two intraband tran-
sitions contribute approximately equally for 40 nm parti-
cles, and the relative importance of the geometry-assisted
transitions varies inversely with particle diameter.
This combination of
ab initio
electronic structure
methods for detailed material properties with free electron
/ jellium calculations for nano-scale geometry enables un-
derstanding the relative contributions of different plas-
mon decay mechanisms and predicting the initial energy
distribution of hot carriers thus generated. Such calcula-
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Plasmonic hot carrier generation: interband transitions (
ab initio
)
Fig. 4.
Initial hot carrier energy distributions due to interband transitions in (a) aluminum, (b) silver, (c) copper, and (d) gold as predicted
using detailed electronic structure methods [66]. The top panels annotate the allowed interband transitions on the electronic band struc-
ture of each metal. The position of the
d
bands relative to the Fermi level produces hotter holes than electrons in copper and gold, com-
pared to hot holes and electrons in silver, and nearly uniform energy distributions of electrons and holes in aluminum. The bottom panels
show that the energy–momentum distribution of hot carriers, plotted with energy on radial axis and direction set by momentum, is highly
anisotropic and, for the noble metals, is determined by crystal directions rather than polarization. Adapted with permission from Ref. 66
(
©
2014 Nature Publishing Group).
Absolute plasmon linewidths and contribution of different mechanisms (
ab initio
)
Fig. 5.
Absolute linewidths estimated from measured complex dielectric functions compared to theoretical predictions [67] including con-
tributions from resistive losses, direct, geometry-assisted and phonon-assisted transitions for (a) surface plasmon polaritons, and relative
contributions of these mechanisms in (b) 40 nm and (c) 20 nm Au spheres. The geometry-assisted contributions are negligible for the sur-
face plasmon, comparable to the phonon-assisted and resistive contributions below threshold for the 40 nm sphere and increase with de-
creasing particle size, but direct transitions always dominate above the interband threshold energy. Adapted with permission from Ref. 67
(
©
2015 American Chemical Society).
tions can therefore guide material and nanostructure de-
sign for applications that require carriers of specific polar-
ity and energy.
3
Transport of plasmonic hot
carriers
The decay of surface plasmons generates hot carri-
ers through several mechanisms including direct inter-
band transitions, phonon-assisted intraband transitions,
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
and geometry-assisted intraband transitions, as discussed
above. These carriers can be generated with finite prob-
ability anywhere inside the plasmonic metal with a dis-
tribution of initial momenta, and travel through the ma-
terial scattering against electrons, phonons, and defects
in the metal. These scattering events thermalize the car-
riers and bring their energies closer to the Fermi level
of the metal, on average. Plasmonic hot carrier applica-
tions, on the other hand, require carriers far from the
Fermi level to more efficiently drive reactions in molecules
or cross-Schottky barriers into semiconductor bands. The
challenge in designing efficient plasmonic hot carrier de-
vices is, therefore, to collect the hot carriers at the surface
of the plasmonic nanostructure with minimal scattering.
Charge carrier transport is of critical importance in
several fields, including traditional semiconductors [71]
and low-dimensional materials [72]. The transport is usu-
ally analyzed in one of two regimes: ballistic, where the
carriers rarely scatter within the characteristic structure
dimensions, and diffusive, where the carriers scatter sev-
eral times on length scales much smaller than the struc-
ture. Ballistic transport in low-dimensional systems is of-
ten treated as a quantum mechanical transmission prob-
lem, such as within Landauer theory [73, 74]. Diffusive
transport often involves small perturbations from equilib-
rium that can be described using spatially varying quasi-
Fermi-levels, such as the drift-diffusion model for semi-
conductors [71].
The general case – structures that are neither much
smaller nor much larger than the mean free path between
carrier scattering events – falls between the ballistic and
diffusive regimes. This intermediate regime can be treated
using the Boltzmann equation that evolves phase space
(momentum and spatial) distribution functions [75] or us-
ing Monte Carlo methods [76], but these are computation-
ally feasible only in special (high-symmetry) limits or with
approximate simplifications tailored for the specific prob-
lem. Plasmonic hot carrier devices often belong to this in-
termediate regime, as we discuss below, and the goals ex-
plicitly require carriers far from equilibrium, which makes
analyzing the carrier transport particularly challenging.
The full description of non-equilibrium carrier trans-
port involves several variables: the spatial and temporal
evolution of energy and momentum distributions of the
hot carriers. We examine plasmonic hot carrier dynam-
ics from two orthogonal simpler perspectives: ultrafast dy-
namics in pump–probe measurements, where spatial de-
pendence is usually unimportant [77], and the steady-state
transport in nanostructures at continuous-wave illumina-
tion, where time dependence is not important. The former
is more convenient for (indirectly) probing the carrier dy-
namics experimentally, whereas the latter is relevant for
plasmonic energy conversion devices.
3.1
Hot carrier dynamics: ultrafast and fast
timescales
Ultrafast pump–probe measurements of plasmonic nanos-
tructures use 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 [77, 78]. The
typical signal observed in these experiments is an initial
fast rise (10–100 fs) attributed to electron–electron scat-
tering that converts fewer high-energy excited carriers into
several more lower-energy carriers, followed by a slower
decay (100 fs to 1 ps) attributed to electron–phonon scat-
tering.
Figure 1 schematically shows the typical electron dis-
tributions as a function of time in such experiments. This
is phenomenologically described by a “two-temperature
model” that tracks the time dependence (and optionally
also spatial variation) of separate electron and lattice tem-
peratures,
T
e
and
T
l
respectively. The initial excitation
generates an electron distribution that is far from equilib-
rium for which temperature is not well-defined.
Electron–electron scattering equilibrates the elec-
trons with each other in
∼
100
fs, with a typical mean-free-
time between collisions for each electron ranging from
∼
10
fs to
∼
100
fs depending on the energy of the elec-
tron relative to the Fermi level. Figure 6(a) shows
ab ini-
tio
calculations of the electron–electron scattering lifetime
of electrons in gold [79]. As expected from Fermi liquid
theory, the phase space for collisions increases quadrat-
ically with energy difference from the Fermi level, and
the lifetime drops approximately as
(
E
−
E
f
)
−
2
. Electron-
defect scattering also contributes in polycrystalline metal
structures, but electron–electron scattering dominates for
high-energy carriers due to the quadratically increasing
phase space. At times
&
100
fs, the electrons can be de-
scribed approximately by an elevated electron tempera-
ture
T
e
, and the subsequent dynamics can be adequately
described within the two-temperature model.
Electron–phonon scattering in the noble metals and
other plasmonic metals such as aluminum occurs with
a mean-free-time
∼
10–30 fs, but the energy transferred
in each collision is limited by their small Debye energies
∼
0.02–0.04 eV compared to the typical excitation ener-
gies of
∼
1–3 eV. Therefore, in spite of electron–phonon
collisions occurring at approximately the same rate as
electron–electron collisions, the net energy transfer oc-
curs
∼
100
times slower resulting in the observed slower
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
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103
equilibration of electrons with the lattice at picosecond
timescales.
The two-temperature model can be used to extract
phenomenological estimates of electron and phonon cou-
pling strengths and relaxation times from experiment, or
these parameters can be calculated
ab initio
[80, 81]. De-
viations due to overlap of electron thermalization and
electron–phonon equilibration time scales can also be
dealt with phenomenologically [82] or calculated using
spatially independent jellium Boltzmann equation ap-
proaches [83] However, most of the detailed information
currently available from these approaches deals with par-
tially or fully thermalized electrons, which is “too late”
for the efficient use of plasmonic hot carriers for energy
conversion. Extensions of the two-temperature model that
account for non-thermal electrons as an additional en-
ergy source for the thermalized electrons and phonons [84]
make it possible to analyze the effect of these non-thermal
electrons on photocatalysis at metal surfaces [85]. We fo-
cus next on using scattering rates derived from theory, and
corroborated by experiment in certain regimes, to under-
stand the spatial dependence of hot carrier transport and
its implications for device design.
3.2
Spatial dependence of hot carrier
distributions
The effect of carrier scattering on the efficiency of plas-
monic hot carrier devices has been estimated approxi-
mately using different approaches. The simplest approach
is to incorporate the lifetime
τ
of the carriers directly in the
calculation of the carrier generation as a Lorentzian broad-
ening
~
τ
−
1
of the energy conservation in Fermi’s golden
rule [64], which results in the predictions shown in Fig-
ure 3(b) for
τ
ranging from 50 fs to 1 ps in nanoparti-
cles of different sizes. However, this approach ignores the
strong energy dependence of the electron lifetime due to
electron–electron scattering (Figure 6(a)), and the spatial
dependence of the carrier distributions due to transport.
Additionally, the assumed constant lifetimes are much
longer than the practical values for the hot electrons with
energies
∼
1–3 eV.
Exponential attenuation models
exp(
−
L
/
λ
)
based on
a fixed mean free path
λ
may also be used to estimate car-
rier collection [86]. These approaches also usually neglect
the tremendous variation in carrier mean free path with
energy. Further, all these approaches neglect the carriers
that are generated from the scattering of the initially ex-
cited carriers.
Ab initio
calculations of electron–electron
scattering [79] indicate that the energy transferred by a
hot electron is uniformly distributed between zero and its
energy relative to the Fermi level (Figure 6(b)), in agree-
ment with phase-space arguments from Fermi liquid the-
ory. Therefore, secondary electrons (after scattering once)
can also carry energy comparable to the primary electrons,
and it is important to account for the evolution of carrier
energy distributions through the first few scattering events
at least.
A full analysis of the transport of plasmonic hot car-
riers that correctly accounts for all of these effects re-
quires treatment using the Boltzmann equation with spa-
tial dependence. This analysis would be highly desirable
for understanding the efficiency of plasmonic energy con-
version, but it would be computationally challenging and
does not seem to have been performed as yet.
To gain some insight into the complex strongly energy-
dependent transport of plasmonic hot carriers, we present
a simplified version of the Boltzmann analysis that cap-
tures the essential features. We solve the time-dependent
spatially–homogeneous Boltzmann equation, and infer
the dependence on the transport distance
L
from the time
dependence using
v
d
f
d
L
=
Γ
[
f
]
,
(1)
where
v
is the carrier velocity,
f
is the energy and momen-
tum probability distribution, and
Γ
is the collision integral
calculated using Fermi’s golden rule for electron–electron
and electron–phonon scattering that implicitly includes
the strong energy dependence of carrier lifetimes.
Figure 6(c) shows the evolution of energy and mo-
mentum distribution of carriers in gold with transport
distance, starting from plasmonic excitation at
~
ω
=
2
.
6
eV. The initial carrier distribution is dominated by di-
rect transitions, consists of high-energy holes and lower
energy electrons, and is strongly anisotropic constrained
to crystal directions [66]. The high energy holes are ex-
tremely short-lived and scatter significantly even at the
small transport distance of
L
= 3
nm, where the lower en-
ergy electrons are still mostly unscattered. By
L
= 10
nm,
the holes have a continuous energy distribution from zero
to the plasmon energy, and by
L
= 30
nm, both electrons
and holes are predominantly within 1 eV of the Fermi level.
The electron and hole distributions thermalize with each
other and become symmetric by
L
= 100
nm. The initial
anisotropy in the momentum distribution also disappears
within this transport distance.
Hot carrier transport establishes the link between the
electromagnetic field profiles, which determine hot carrier
generation and the spatial dependence of carrier collec-
tion. The strong dependence of electron–electron scatter-
ing rates on energy implies vastly different length scales
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
Ab initio
lifetimes and transport of plasmonic hot carriers
Fig. 6.
(a)
Ab initio
calculations of electron–electron scattering show that the carrier scattering rates depend strongly on carrier energy –
approximately quadratically with energy relative to Fermi level, and (b) that each scattering event transfers energy with an almost uniform
distribution [79]. Note that the maximum lifetime of carriers
>
1
eV is less than the smallest constant lifetime
τ
= 0
.
05
ps assumed for
the jellium estimates in Fig. 3(b). (c) The energy–momentum distribution of hot carriers evolves with transport distance
L
due to electron–
electron and electron–phonon scattering, starting from a strongly anisotropic distribution with hotter holes than electrons (as in Fig. 4(d)).
The holes relax to energies
∼
1
eV within
L
= 10
nm, and the distribution becomes isotropic and symmetric between electrons and holes
within
L
= 100
nm. Parts (a, b) adapted with permission from Ref. 79 (
©
2004 American Physical Society).
depending on the carrier energy: 1 eV electrons in gold
may be collected efficiently 100 nm away, whereas 2 eV
holes must be collected within 10 nm from where they are
generated. Detailed computational modeling of hot carrier
transport that fully accounts for these material-dependent
effects simultaneously with the geometry of plasmonic
nanostructures, the electromagnetic field, and carrier gen-
eration profiles is therefore highly desirable.
4
Collection and injection of
plasmonic hot carriers in devices
The carriers generated by plasmon decay impinge upon
the surface of a plasmonic nanostructure, either directly
or after scattering against other carriers and phonons in
the metal. Combining theoretical calculations for carrier
generation with a transport simulation should be able to
predict the energy and spatial distribution of this carrier
flux. The final step, as illustrated in Fig. 2, is the injection
of the carriers into molecules to drive chemical reactions or
their collection in a semiconductor. In this section, we re-
view the considerations for efficient molecular and solid-
state collection, as well as the promising alternative of di-
rectly generating carriers across a metal–semiconductor
interface.
We discuss the collection efficiency of electrons and
holes separately for simplicity, but both carriers must be
collected in a practical device. For example, if electrons are
collected more efficiently than holes, the plasmonic struc-
ture will develop a charge imbalance, building up a po-
tential that opposes the electron collection until a steady
state is reached where both carriers are collected equally.
Therefore, the net collection efficiency of a hot carrier de-
vice will be limited by the smaller of the electron and hole
collection efficiencies.
4.1
Solid-state collection
The collection of photo-excited hot carriers over a metal–
semiconductor Schottky junction, has been studied exten-
sively and is well-described by the Fowler theory for in-
ternal photoemission [90] and its refinements [91]. The
Fowler yield estimate results from a semiclassical model
of electrons overcoming an energy barrier, as shown in
Fig. 7(a, b) [87]. Carriers with energy less than the bar-
rier height are reflected. For carriers that cross the barrier,
only the normal component of the momentum changes at
the interface, which implies that the tangential momen-
tum must be small enough that its kinetic energy contri-
bution in the semiconductor must be less than the excess
energy over the barrier; carriers with greater tangential
momentum will be reflected (analogous to total internal
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105
Injection of hot carriers into semiconductors or molecules
Fig. 7.
(a) Energy and (b) momentum-matching considerations for carrier injection through a metal–semiconductor Schottky junction: car-
rier energy must exceed the Schottky barrier and its incidence angle must lie within an “escape cone” [87]. (c) Carrier collection in semi-
conductors can occur via plasmon decay in the metal followed by injection or directly by exciting interfacial charge-transfer transitions that
generate carriers in the semiconductor [88]. (d) Injection of plasmonic hot carriers can push a molecule into a charged anti-bonding state to
induce bond dissociation, in contrast to dissociation via thermal activation [89]. Parts (a, b) adapted with permission from Ref. [87] (
©
2014
AIP Publishing), (c) from Ref. 88 (
©
2015 AAAS), and (d) from Ref. 89 (
©
2015 Nature Publishing Group).
reflection of light). Therefore, only carriers in an “escape
cone” around normal incidence cross the barrier, and the
angle of this cone increases with carrier energy starting
from zero for carriers with the threshold energy to cross
the barrier. This results in a quadratic dependence of the
injection rate with energy above threshold.
The collection of plasmonic hot carriers has been
demonstrated in insulators as well as semiconductors, and
the built-in electric fields in the Schottky junction assist in
the collection of the emitted hot carriers [28, 92, 93]. Fowler
theory remains qualitatively valid regardless of the band
structure of the semiconductor or insulator [94, 95], but the
magnitudes can differ substantially for materials with high
density-of-states near the band edges, such as transition
metal oxides. In this case, the shallow electron dispersion
relation (localized states) that causes the high density of
states makes the escape cones narrower reducing the ef-
ficiency of injection. Experimentally, roughened surfaces
can overcome these restrictions by providing tangential
momentum for the carrier injection [86]. Another opportu-
nity for plasmonic hot carriers in context of solid state sys-
tems is to modify the photoluminescence from wide band
gap semiconductors or for up-conversion in semiconduc-
tor quantum wells.
4.2
Molecular injection: plasmon-enhanced
catalysis and femtochemistry
Hot carriers generated by plasmon decay can also inject di-
rectly into molecules adsorbed at the surface, directly driv-
ing chemical reactions for photochemical energy conver-
sion. Surface photochemistry, chemical reactions driven
by photoexcited carriers at metal surfaces, has been well
studied in many contexts including solar energy conver-
sion and atmospheric chemistry [37, 96–98]. The possibil-
ity of efficiently absorbing light and generating hot car-
riers in metal nanostructures using plasmon resonances
has driven a lot of research in plasmon-driven photocatal-
ysis [33, 45, 99–103]. In particular, “semiconductor-free”
water-splitting devices using only plasmonic hot carriers
have been demonstrated recently [45].
Understanding mechanisms of chemical reactions
driven by hot carriers is an important direction for further
advancing plasmonic photochemistry. The most detailed
experimental probes of such mechanisms are pump–
probe measurements, where two laser pulses with a vari-
able delay are used to excite electronic transitions in a
molecule and then probe the progress of the chemical re-
action that ensues on femtosecond timescales. In the field
of femtochemistry, such techniques have been invaluable
for tracking reaction mechanisms or even for changing the
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106
Ë
Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
branching ratios of different processes [104–115]. These
techniques are now becoming useful to probe the highly
non-equilibrium processes driven by plasmonic hot carri-
ers [36, 39, 116].
The basic proposed mechanism for plasmon-driven
chemistry involves the injection of an electron from the
metal into an anti-bonding state of an adsorbed molecule,
causing either desorption or the dissociation of a bond in
the adsorbate [117, 118]. Variants of this mechanism can
explain the observed chemical activity of various metal
nanostructures, as discussed in detail in a recent review
article [89]. The adsorbate forms a transient ion strongly
coupled to the plasmonic particle, and the relevant po-
tential energy surface is that of an excited state of the
adsorbate–plasmonic particle complex, which must be
treated together. Theoretical calculations of the excited-
state energy landscape using first-principles beyond-
ground-state electronic structure methods are therefore
essential for predicting such mechanisms and for inter-
preting femtochemistry experiments [119–121].
A second class of possible mechanisms involves plas-
monic enhancement of transitions between states local-
ized on the adsorbate [37, 96]. Here, the involved excited
states are primarily those of the adsorbate, and the metal
nanostructure serves to enhance the matrix element for
the transition by plasmonic enhancement of the surface
electric field. Both types of mechanisms require low-lying
molecular orbitals on the adsorbate, but the second class
does not rely on alignment of adsorbate levels with those
of the metal or on hot carrier transport in the metal. These
neutral-adsorbate mechanisms will therefore play an im-
portant role in future applications of high-efficiency plas-
monic catalysis.
Finally, efficient light capture in plasmonic nanos-
tructures can also cause significant local heating and en-
hance the rates of thermal reactions (not driven by photo-
carriers) [34]. Photo-thermal reaction enhancement has
been demonstrated experimentally though explicit calcu-
lations of these processes has also been elusive [34, 122–
124].
4.3
Interface science for plasmonic hot
carriers
Traditionally, semiconductor–semiconductor heterostruc-
tures, hot carriers and photosensitizers (dye sensitizers
for example) have been used to expand the range over
which absorption can occur but band alignment and inter-
facial charge-transfer issues limit the realizable enhance-
ment. This has also been a challenge for plasmonic hot car-
rier injection into semiconductors with unfavorable band
alignment. Recent work has demonstrated a direct, in-
stantaneous transfer of plasmon-derived electrons into
interfacial semiconductors called plasmon-induced inter-
facial charge-transfer transition. The mechanism here is
via the decay of a plasmon that directly excites an elec-
tron from the metal to a strongly coupled acceptor. Such
an interfacial charge transfer could have very high quan-
tum efficiencies (highest demonstrated so far has been
24 percent) [88]. Similarly in molecular systems, chemi-
cal damping at the interface can involve a direct charge-
transfer mechanism where excited carriers are directly in-
jected into an adsorbate acceptor state without first occu-
pying available states in the metal [101, 125–128]. Interface
damping, solid-state and molecular, present unique op-
portunities for high-efficiency plasmonic hot carrier injec-
tion and collection and require further experimental and
theoretical investigation for a complete understanding of
their underlying mechanisms and fundamental limits.
5
Outlook
Splitting plasmonic hot carrier driven processes into steps:
plasmon excitation, hot carrier generation, carrier trans-
port and collection, allows building a complete micro-
scopic picture with different levels of theory appropriate
for each step. The vast majority of theoretical investiga-
tions focus on plasmon excitation (optical response) and
hot carrier generation, while fewer studies deal with car-
rier collection and transport. Here, we conclude with a
short discussion of open directions and unanswered ques-
tions that future work in plasmonic hot carrier devices
should address.
Hot carriers generated by plasmon decay are limited
by the photon energy. A given energy-conversion process is
usually associated with a characteristic energy, for exam-
ple, semiconductor band gap in photovoltaics, such that
lower-energy photons are incapable of driving the pro-
cess, while the excess energy of higher-energy photons is
wasted. The efficiency of solar energy conversion could
be improved by harvesting the energy of these higher and
lower energy photons, by producing multiple electron–
hole pairs (down-conversion) or single electron–hole pairs
from multiple photons (up-conversion). Sub-band-gap in-
jection from plasmonic metals to semiconductors can be
used to achieve up-conversion, whereas careful design in
geometry to optimize for a single electron–electron scat-
tering event could, in principle achieve, the generation of
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Prineha Narang et al., Plasmonic hot carrier dynamics in solid-state
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107
two electron–hole pairs, analogous in effect to multiple ex-
citon generation in semiconductors.
The design of hot carrier devices that carefully op-
timize against or even exploit carrier scattering require
developments in theoretical and computational meth-
ods that account for nonequilibrium transport in greater
detail. As we discussed in Section 3, these methods
should be able to account for a few scattering events,
at least, while retaining energy, momentum, and spatial
dependence, perhaps using an appropriate yet practica-
ble limit of the Boltzmann equation. Extension to nonlin-
ear perturbations additionally including time dependence
will be necessary to provide a link between femtosec-
ond pump–probe measurements and the typically low-
intensity steady-state regime of plasmonic energy con-
version. Detailed analysis of scattering mechanisms and
transport can also provide insight into designing new
materials, which are suitable for carrier transport, es-
pecially with doped-semiconductor plasmonic materials
where electronic band structures, and hence phase-space
for generation as well as scattering, could be controlled by
altering composition.
The final link between hot carrier generation and
the energy conversion process is the collection or injec-
tion mechanism. Understanding mechanisms and kinet-
ics of chemical reactions driven by hot carriers that are
far from equilibrium requires further development of the-
oretical methods [89]. Direct plasmonic hot carrier gener-
ation in semiconductors via interfacial charge-transfer ex-
citations [88] is a particularly exciting direction that could
eliminate bottlenecks due to carrier transport, but requires
further analysis of its contribution relative to generation in
the metal followed by injection in order to guide the design
of optimum interfaces.
As the fields of quantum plasmonics and quantum op-
tics merge with electronic structure theory, there are many
questions about the fundamental nature of plasmons to
be answered, including a many-body understanding of
plasmons in the dispersive regime. While there have been
demonstrations of the quantum behavior of plasmons, be-
ing close to light-line has limited the insight we obtain of
the quantum nature of “lossy” plasmons.
Acknowledgment:
This material is based upon work per-
formed 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
Number DE-SC0004993. P. N. is supported by a National
Science Foundation Graduate Research Fellowship and by
the Resnick Sustainability Institute.
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