ARTICLE
Received 3 Sep 2014
|
Accepted 7 Nov 2014
|
Published 16 Dec 2014
Theoretical predictions for hot-carrier
generation from surface plasmon decay
Ravishankar Sundararaman
1,
*, Prineha Narang
1,2,
*, Adam S. Jermyn
1,3,
*, William A. Goddard III
1,4
& Harry A. Atwater
1,2
Decay of surface plasmons to hot carriers finds a wide variety of applications in energy
conversion, photocatalysis and photodetection. However, a detailed theoretical description of
plasmonic hot-carrier generation in real materials has remained incomplete. Here we report
predictions for the prompt distributions of excited ‘hot’ electrons and holes generated by
plasmon decay, before inelastic relaxation, using a quantized plasmon model with detailed
electronic structure. We find that carrier energy distributions are sensitive to the electronic
band structure of the metal: gold and copper produce holes hotter than electrons by 1–2 eV,
while silver and aluminium distribute energies more equitably between electrons and holes.
Momentum-direction distributions for hot carriers are anisotropic, dominated by the plasmon
polarization for aluminium and by the crystal orientation for noble metals. We show that in
thin metallic films intraband transitions can alter the carrier distributions, producing hotter
electrons in gold, but interband transitions remain dominant.
DOI: 10.1038/ncomms6788
OPEN
1
Joint Center for Artificial Photosynthesis, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA.
2
Thomas J.
Watson Laboratories of Applied Physics, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA.
3
Division of
Physics, Mathematics and Astronomy, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA.
4
Materials and
Process Simulation Center, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA. * These authors c
ontributed
equally to this work. Correspondence and requests for materials should be addressed to H.A.A. (email: haa@caltech.edu).
NATURE COMMUNICATIONS
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P
lasmons are collective oscillations of electrons that couple
to electromagnetic fields. They exhibit wave-like as well as
particle-like behaviour
1
, support intense electromagnetic
field concentrations
2
and provide a pathway to couple optical
energy from free space in nanoscale systems
3
. Surface plasmons,
electromagnetic modes confined to the surface of a conductor–
dielectric interface, have sparked recent interest because of
their quantum nature
4,5
and their broad range of applications,
including solar energy harvesting
6
, nonlinear optics,
tunable photodetectors
7
and spectroscopy. Decay of plasmons
to hot carriers has recently attracted considerable interest
8
due to applications in energy conversion, photocatalysis and
photodetection
9
.
Surface plasmons can decay either radiatively
10
via emission
of a photon or non-radiatively through the generation of
excited carriers, typically referred to as hot carriers. These
photo-excited hot carriers in metals could be used to directly
drive energetically demanding chemical reactions
11
,or
they could be transferred to a semiconductor for use in
photovoltaics
12,13
and photoelectrochemical systems
6,14
.Ata
metal–semiconductor interface, plasmonic hot-carrier collection
over a tunable Schottky barrier allows the collection of photons
with energies lower than the interband threshold of the
semiconductor, thereby enabling additional energy harvesting.
These excited carriers, both electrons and holes, can be injected
into other materials, for example, graphene
15
and MoS
2
(ref. 16),
thereby enabling plasmonic hot-carrier-induced doping and
phase transitions.
Despite the significant experimental work in this direction, a
complete theoretical understanding of plasmon-driven hot-
carrier generation with electronic structure details has been
evasive. Understanding the initial energy distribution of carriers
generated by plasmon decay, before inelastic relaxation, is the first
key step towards exploiting these phenomena. Theoretical studies
of plasmonic systems have traditionally focused on their optical
response, including quantum jellium models of nanostructured
systems such as nanoparticle dimers
17–20
, and detailed time-
dependent density functional calculations of short-wavelength
surface plasmons on noble metal surfaces
21
. Recently, the initial
hot-carrier distribution generated by plasmon decay has been
estimated within a simple electron gas model for various
geometries
22
and within a jellium model for silver nanoparticles
and nanoshells
23
. These models provide insight into the
mechanisms of plasmonic hot-carrier generation, but do not
capture the material dependence of this process and miss
interband transitions in noble metals since they preclude
transitions involving
d
bands.
In this article, we combine quantized plasmon modes from
experimental dielectric functions with electronic states from first
principles density functional theory (DFT) to calculate the initial
distribution of hot carriers in real materials. We first examine the
direct electron excitations generated by the decay of surface
plasmon polaritons on planar metal–dielectric interfaces, as
shown schematically in Fig. 1a. (Note that the surface plasmon
and the initial photon have the same energy, and a coupling
geometry such as a grating provides the change in momentum
24
.)
This allows us to explore the effects of the electronic structure of
the metal on the generated carrier distributions, independent
from other effects such as geometry. In addition, we focus on
interband transitions that dominate at higher plasmon energies
since these are expected to be more sensitive to the electronic
structure than intraband transitions; the latter dominate at lower
plasmon energies and have been described within simplified
jellium models
23
. Finally, we analyse the effects of geometry on
the generated hot-carrier distribution in real materials by
studying the decay of plasmon modes in thin metallic films of
varying thickness.
Results
Interband transition rate formalism
. We describe the surface
plasmon using an explicit quantization of the surface modes
25–27
derived from an experimental dielectric function
28
. The vector
potential operator for plasmons on the surface of a semi-
infinite slab (with normal along the
z
direction) is
^
A
ð
r
;
t
Þ¼
P
k
u
k
ð
r
;
t
Þ
^
a
k
þ
h
:
c
:
, in terms of creation and annihilation
operators,
^
a
y
k
and
^
a
k
, and the normalized mode functions of
wave vector
k
and angular frequency
o
,
u
k
ð
r
;
t
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
‘
o
SL
ð
o
Þ
s
^
k
k
^
z
g
ð
z
Þ
e
i
ð
g
ð
z
Þ
z
þ
k
r
o
t
Þ
:
ð
1
Þ
The modes satisfy the dispersion relation
k
¼
o
=
c
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
ð
o
Þ
=
ð
E
ð
o
Þþ
1
Þ
p
, where
E
(
o
) is the experimental
dielectric function of the metal. The
z
wavenumber satisfies
g
2
ð
z
Þ¼
E
ð
z
Þ
o
2
=
c
2
k
2
, where
E
(
z
)
¼
E
(
o
) for
z
o
0 and 1 and for
z
4
0, and with the sign of Im
g
(
z
) set so that the modes decay
away from the surface. Above,
S
is a test area with periodic
boundary conditions for discretizing the modes and
L
(
o
)isa
normalization length chosen so that each mode has energy
:
o
.
Relative
energy
distributions
Photon
abc
Surface
plasmon
Hot carriers
Au
n
-GaAs
Energy (eV)
p
-GaAs
Au
Energy (eV)
–3
–2
–1
0
1
–3
–2
–1
0
1
E
c
E
f
E
v
E
c
E
f
e
–
e
–
E
v
h
+
h
–
p
E
→
B
→
h
h
+
x
x
Figure 1 | Schematic of plasmonic hot-carrier generation and injection.
(
a
) Schematic for optical excitation of surface plasmons followed by decay to
hot carriers, (
b
) tunnelling of plasmonic hot electrons from gold through a Schottky barrier into
n
-type gallium arsenide using the predicted carrier
distribution from Fig. 3d and typical experimental band offsets
40
and (
c
) barrier-less injection of plasmonic hot holes from gold into
p
-type gallium
arsenide. (
E
f
is the Fermi energy,
E
c
the conduction band minimum and
E
v
the valence band maximum energy.)
ARTICLE
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(See ref. 26 for details.) Although this neglects the possibility of
nonlocal effects in the dielectric matrix, such an approximation is
valid at the wavelengths of interest.
Next, given an approximation of the quasiparticle orbitals
c
s
q
n
ð
r
Þ
and energies
e
q
n
of the metal, the electron field operator is
^
W
s
ð
r
;
t
Þ¼
P
q
n
c
s
q
n
ð
r
Þ
e
i
e
q
n
t
=
‘
^
c
q
n
. Here,
^
c
q
n
and
^
c
y
q
n
are
Fermionic creation and annihilation operators for electrons with
wave vector
q
and band index
n
. We have included the spinor
index
s
in the orbitals to fully treat relativistic effects such as
spin-orbit coupling, when necessary.
Finally, we approximate the plasmon–quasiparticle interaction
Hamiltonian using the lowest order (unrenormalized) vertex,
^
H
int
¼
e
2
m
e
P
s
R
d
r
^
C
y
s
^
A
^
p
^
C
s
, where
^
p
is the electronic momen-
tum operator. Fermi’s golden rule for the decay of a single
plasmon with wave vector
k
and angular frequency
o
to electron–
hole pairs via interband transitions then reduces to
G
¼
p
2
2
o
L
ð
o
Þj
g
ð
z
o
0
Þj
Z
BZ
d
q
ð
2
p
Þ
3
X
n
0
n
ð
1
f
q
n
0
Þ
f
q
n
d
ð
e
q
n
0
e
q
n
‘
o
Þ
^
k
k
^
z
g
ð
z
o
0
Þ
X
s
Z
O
d
r
c
s
q
n
0
ð
r
Þ
e
‘
r
im
e
c
s
q
n
ð
r
Þ
2
;
ð
2
Þ
where
f
q
n
are the occupation factors of the quasiparticles in the
Fermi sea. (See the Methods section for a brief derivation.) The
key approximation above is that the plasmon mode function
varies slowly on the atomic scale (interband approximation).
Electronic structure method selection
. To calculate the carrier
distributions from plasmon decays in real materials using
equation (2), we need a sufficiently accurate prescription for the
quasiparticle orbitals and energies. Figure 2 compares the accu-
racy of different electronic structure methods for the noble metals
copper, silver and gold. The different methods produce identical
results for aluminium since it is a nearly free electron metal; we
omit that comparison for brevity and select the PBEsol general-
ized-gradient approximation
29
within DFT.
In contrast, for the noble metals, generalized-gradient approx-
imations such as PBEsol predict the
d
band positions to be closer
to the Fermi level than that of experiment because of the self-
interaction error for localized electrons. The GLLB-sc orbital-
dependent functional
30
partially remedies this situation, as shown
by a recent density functional study of plasmon dispersions on
noble metal surfaces
21
. However, Fig. 2 shows that the band
structure predicted by this functional still exhibits significant
deviations from angle-resolved photoemission measurements
31–33
.
These deviations are largest (
B
0.5 eV) near the
L
point in
the Brillouin zone, and this region of the Brillouin zone is
particularly active for interband transitions in these metals.
Many-body perturbation theory methods, such as the quasi-
particle self-consistent
GW
approximation, significantly improve
the agreement with experiment as shown for gold in Fig. 2c (
GW
results from Rangel
et al.
34
), however, the error near the
L
point
remains large (
B
0.5 eV).
We find that the DFT
þ
U
method
35
, which improves the
description of localized electrons with a local semi-empirical
correction on each atom, yields the best agreement (
B
0.1 eV) to
the quasiparticle bandstructure of all three noble metals
(indicated by PBEsol
þ
U
in Fig. 2, since we combine the
U
correction with the PBEsol density functional). We pick the value
of
U
to reproduce the experimental energies at the
G
point, which
results in
U
¼
1.63, 2.45 and 2.04 eV for copper, silver and gold,
respectively.
The calculations presented here account for relativistic effects
including spin-orbit coupling fully self-consistently. The spin-
orbit splitting at the
X
point is particularly relevant, since it
determines the inter-band threshold energy. The magnitude of
the effect is
B
0.5 eV in gold,
B
0.2 eV in silver and
B
0.1 eV in
copper. Therefore, the inclusion of spin-orbit splitting is critical
in calculations of optical transitions in gold, still quite significant
for silver and negligible at the achievable accuracy for copper
(and aluminium).
Hot-carrier distributions from surface plasmon decay
. Sub-
stituting the electronic states and energies from the PBEsol
þ
U
density functional method in equation (2) allows us to predict
accurate carrier distributions. We histogram the contributions to
–6
–4
–2
0
2
4
6
Γ
X
W
L
Γ
K
Energy (eV)
PBEsol+U
GLLBsc
QSGW
Expt
–6
–4
–2
0
2
4
6
Γ
X
W
L
Γ
K
Energy (eV)
PBEsol+U
GLLBsc
Expt
–4
–2
0
2
4
Γ
X
W
L
Γ
K
Energy (eV)
PBEsol+U
GLLBsc
Expt
L
W
Γ
K
X
Cu
Au
Ag
Figure 2 | Comparison of theoretical and experimental band structures.
Theoretical band structures for (
a
) copper, (
b
) silver and (
c
) gold as predicted
by different density functional approximations, compared with angle-resolved ultraviolet photoemission measurements
31–33
and quasiparticle self-
consistent
GW
calculations
34
.(
d
) The high-symmetry paths in the Brillouin zone along which the band structures are plotted. All calculations account
for relativistic effects including spin-orbit coupling. The PBEsol
þ
U
approximation, with
U
fit to the experimental
G
point energies, provides the best
overall agreement with the experimental data for all three noble metals and particularly improves on the accuracy of the other methods near the L point
.
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the decay rate
G
by the electron (final state) and hole (initial state)
energies to generate the distributions shown in Fig. 3 and by the
carrier energies as well as momentum directions to generate
Fig. 4. Energy conservation determines which regions of the
Brillouin zone contribute to the plasmon decay, and the top
panels of Fig. 3 annotate the allowed transitions for each metal at
selected energies.
For aluminium, the band crossing close to the Fermi level near
the W point allows interband transitions that originate from
valence band states with energies ranging continuously from the
Fermi level to
:
o
below it. Consequently, surface plasmon decay
results in both hot electrons and holes with continuous energy
distributions that extend from zero energy to the plasmon energy,
as seen in the bottom panel of Fig. 3a. Additional transitions near
the K point contribute predominantly to hot holes with energies
4
2 eV and lead to the moderate asymmetry between the electron
and hole energy distributions.
For silver, the lowest energy interband transitions, originating
from the
d
bands at the X and L points as well as from the Fermi
level at the L point, all appear at
B
3.6 eV. Consequently, the
decay of a plasmon of that energy produces bimodal energy
distributions for both the electrons and holes as shown in Fig. 3b.
In copper and gold, the allowed interband transitions near the
resonant surface plasmon polariton energies occur near the X and
L points for both metals, and additionally near the K point for
copper, as shown in Fig. 3c,d. Notice that all these transitions
originate in the
d
bands that are approximately
E
t
¼
2 eV below
the Fermi level. Consequently, for these metals, the generated
holes are on average more energetic than the electrons by
E
t
.
The asymmetry between energy distributions for plasmonic
hot electrons and holes in copper and gold has important
consequences for collection efficiencies across a metal–
semiconductor interface. Consider, for example, the gold to
n
-
type gallium arsenide Schottky junction in Fig. 1b, which includes
the predicted hot-carrier distributions from Fig. 3d. Most of the
hot electrons are not sufficiently energetic to overcome the
Schottky barrier, and would either have to tunnel through the
barrier or would require an additional thermal boost to overcome
it. In contrast, in the corresponding junction to
p
-type gallium
arsenide shown in Fig. 1c, all the holes are sufficiently energetic to
cross into the semiconductor. This barrier-less collection of holes
would exhibit significantly higher efficiency and would require
lower biases than electron collection. Combining the results of
Fig. 3 with this picture, gold and copper are ideally suited for hot
hole injection, whereas silver and aluminium are capable of both.
The nature of transitions accessible by the plasmon also affects
the angular distribution of the excited carriers. For the four
metals considered above, Fig. 4 illustrates the momentum
direction and energy distribution for hot carriers generated by
the decay of a surface plasmon polariton propagating on a [001]
surface along the
x
direction. The radial direction indicates carrier
energy, whereas the orientation indicates the carrier momentum
direction. The maximum energy, illustrated by the grey sphere,
corresponds to the photon energy, and the colour scheme
indicates the relative probability density of carriers at that energy
and momentum direction.
The electric field due to such a plasmon is predominantly along
the
x
direction, and for a free-electron metal we expect a dipole-
antenna-like momentum distribution that peaks along the field
directions. Only the electron and hole distributions in aluminium
exhibit such an orientation dependence; the remaining metals
deviate significantly from that idealized prediction. For the noble
metals, the allowed transitions are on a surface in
k
space
containing the X and L points, which contributes carriers in all
directions, but with a strong anisotropy dominated by the crystal
directions rather than the plasmon field.
The collection efficiency of hot carriers in plasmonic structures
depends on both the initial distribution and transport of the
carriers. In faceted structures smaller than the carrier mean free
path, ballistic transport, which preserves the momentum direc-
tion, is significant compared with diffusive transport. The crystal
orientation-dependent anisotropy in the initial momentum
distribution would therefore become particularly important for
such structures.
–10
–5
0
5
10
Γ
X
W
L
Γ
K
Energy (eV)
Al
abcd
Γ
X
W
L
Γ
K
Ag
Γ
X
W
L
Γ
K
Cu
Γ
X
W
L
Γ
K
Au
–6
–4
–2
0
2
4
6
Relative probability
Ener
g
y (eV)
Electrons
Holes
×
40
–4
–3
–2
–1
0
1
2
3
4
Ener
g
y (eV)
Electrons
Holes
×
150
h
=3.6 eV
–4
–3
–2
–1
0
1
2
Ener
g
y (eV)
Electrons
Holes
×
4
–3
–2
–1
0
1
Ener
g
y (eV)
Electrons
Holes
×
1
h
=2.0 eV
h
=4.0 eV
h
=6.0 eV
h
=2.8 eV
h
=2.6 eV
h
=2.4 eV
h
=2.9 eV
h
=3.4 eV
Figure 3 | Allowed transitions and hot-carrier energy distribution.
PBEsol
þ
U
band structure and predicted plasmonic hot-carrier energy distributions
for (
a
) aluminium, (
b
) silver, (
c
) copper and (
d
) gold. The bottom panels show the energy distribution of hot electrons (positive energies relative to
Fermi level at 0) and hot holes (negative energies) for various photon and plasmon energies,
h
n
. The top panels show the band structure and arrows mark
the allowed transitions for the plasmon energy plotted with a solid line in the corresponding bottom panel. Contrast the almost uniform energy distri
bution
of electron and hole energies in aluminium with the hole-dominant energy distribution in copper and gold and the bimodal hot-hole and hot-electron
distributions in silver due to the position of the
d
bands.
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Geometry effects in decay of thin-film plasmons
. Above we
considered the decay of surface plasmon polaritons on semi-
infinite metal slabs to minimize geometry effects and focus on the
effects of electronic structure. In nano-confined geometries,
crystal momentum or
q
is no longer a good quantum number due
to the uncertainty principle and therefore the transitions excited
by plasmon decay no longer need to be vertical unlike the
situation in Fig. 3. This opens up the possibility of geometry-
assisted intraband transitions without involving phonons. This
mode of localized plasmon decay has been studied within the
context of jellium models with simplified electronic structure
22,23
;
here we analyse this decay mode, including the full electronic
structure of the plasmonic metal.
Direct electronic structure calculations for nanoparticles
require significant computational effort; however, and we there-
fore make two simplifications to enable practical calculations for
systems of experimentally relevant sizes. First, we consider thin
films that are of finite thickness along one dimension; the
electronic structure calculation can then exploit Bloch’s theorem
in the remaining two periodic directions. The resulting predic-
tions would then be a lower bound on the corresponding
geometry effect in nanoparticles, which are confined in all three
dimensions. Second, we adopt an
ab initio
tight-binding
approximation for the electronic structure of the thin film using
the density functional Hamiltonian expressed in the basis of
maximally localized Wannier functions
36,37
. This approach
reproduces the full density functional band structure of the
bulk material exactly by construction, but approximates geometry
effects since changes in orbital shapes within a unit cell are
neglected.
A metal film of finite thickness supports symmetric and
antisymmetric plasmon modes (with respect to
E
^
k
, the electric
field along the propagation direction) as shown schematically in
the top insets of Fig. 5. We analytically construct the vector
potential for one quantum of each of these modes
38
, analogous to
equation (1), and use Fermi’s golden rule to calculate the
rate of their direct decay to electron–hole pairs with the matrix
elements and electronic energies obtained from the Wannier
representation. See the Methods section for details.
Figure 5 shows the resulting electron and hole energy
distributions generated from the decay of plasmons in gold films.
The results for both the symmetric and antisymmetric plasmon
modes on a film of thickness 40 nm resemble that for the semi-
infinite slab in Fig. 3d: most of the energy is deposited in the hot
holes. However, with decreasing film thickness, the finite-
thickness geometrical effects become stronger and the probability
of generating hot electrons via intraband transitions increases.
The effect is particularly pronounced for the antisymmetric mode
because this lower wave vector mode becomes more light-like
with a smaller fraction of the field in the interior of the metal,
which lowers the contribution of the interband transitions (note
the scale factors in the right hand panels of Fig. 5).
This analysis allows one to weigh the relative importance of the
interband transitions and geometry-assisted intraband transi-
tions. Confining geometries smaller than 10 nm enable finite
probabilities of intraband transitions and allow the generation of
hotter electrons than allowed in the hole-dominant bulk copper
or gold. However, interband transitions are still responsible for a
significant fraction of the generated carriers, and hence an
appropriate choice of material (as discussed in the previous
subsection) is important to maximize the efficiency of hot
electron or hole generation.
Discussion
We have reported first principles calculations that describe
plasmon-mediated hot-carrier generation in aluminium, gold,
silver and copper. These calculations illustrate that the generated
carrier profile is extremely sensitive to the details of the electronic
band structure, especially to the position of the
d
bands in silver,
copper and gold relative to unoccupied states above the Fermi
level. Copper and gold generate hot holes that are much more
energetic than the electrons, silver produces narrow energy
distributions of hot holes as well as hot electrons, while
aluminium generates continuous energy distributions of holes
and electrons. These findings inform material selection for
efficiently collecting carriers of a specific type and energy at
metal–semiconductor interfaces or in surface-adsorbed molecular
species.
Geometry of the plasmonic structure also plays an important
role in determining the efficiency of carrier generation and
collection. Nano-confinement effects allow the generation of
hotter electrons in copper and gold via geometry-induced
intraband transitions. However, interband transitions, which
Electrons
Holes
AI,
h
= 4.0 eV
Ag,
h
= 3.6 eV
Cu,
h
= 3.4 eV
E
=
h
E
p
z
z
1
0
Relative probability (a.u.)
y
x
E
p
x
E
p
y
Au,
h
= 2.8 eV
Figure 4 | Hot-carrier energy and momentum-direction distribution.
Plasmonic hot-carrier energy and momentum-direction distribution in (
a
) aluminium
(
b
) silver (
c
) copper and (
d
) gold. The radial coordinate in each panel is the carrier energy relative to the Fermi level, with the spherical shell indicating the
plasmon (photon) energy,
h
n
, while the angular coordinates correspond to the carrier momentum direction. The asymmetry in electron and hole energies in
the noble metals from Fig. 3 is manifested in the radial extent of the corresponding probability clouds here. None of the metals exhibit the isotropic
orientation distribution assumed in the Fowler theory.
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depend strongly on the electronic band structure, still dominate
the initial energy distribution. The initial momentum distribution
of the carriers depends on both the crystallographic orientation
of the metal and plasmon polarization. The net efficiency of
carrier collection in a specific geometry depends on this initial
distribution as well as the subsequent transport of the carriers to
the surface. Therefore, assessing and optimizing carrier collection
efficiency of plasmonic nano-structures additionally require
models for the transport of hot carriers, a subject for future study.
Methods
Computational details
.
We perform density functional calculations for face-
centred cubic aluminium, silver, copper and gold in the plane-wave electronic
density functional software, JDFTx (ref. 39), using full-relativistic norm-conserving
pseudopotentials at a kinetic energy cut-off of 30 hartrees (816 eV) and at the
experimental lattice geometry. We use the PBEsol
29
exchange-correlation
approximation along with a rotationally invariant DFT
þ
U
correction
35
for the
d
electrons, with
U
¼
1.63 eV for copper, 2.45 eV for silver and 2.04 eV for gold fit to
reproduce experimental photoemission data (no
U
correction for aluminium).
We use Kohn–Sham eigenvalues, Fermi occupations and momentum matrix
elements on a dense 128
3
sampling of the Brillouin zone to calculate the inter-band
transition rate of the plasmon mode given by equation (1) using equation (2). We
use experimental dielectric functions from ref. 28, parametrized as a sum of
Lorentz–Drude responses, to determine the plasmon mode. We replace the energy-
conserving
d
function in equation (2) with a normalized Gaussian of width
B
k
B
T
¼
0.026 eV to accommodate the discrete
k
point sum that replaces the
Brillouin zone integral. We histogram the contributions to the total transition rate
G
in terms of the electron and hole energies and momenta to generate the
distributions plotted in Figs 3 and 4.
For the thin-film calculations, we generate the Wannier functions, Hamiltonian
and matrix elements using a coarser 10
3
sampling of the Brillouin zone, which
results in an
ab initio
tight-binding-like Hamiltonian with a range of
B
5 unit cells
in each direction. We then calculate the electronic states of the thin film in the
Wannier basis and calculate the transition rate of the plasmon modes given by
equations (8)–(13) using equation (14), histogrammed by electron and hole
energies to generate the distributions shown in equation (4).
Surface plasmon polariton decay rate
.
Here we sketch the derivation of
equation (2), the decay rate of a single surface plasmon to electron–hole pairs via
interband transitions. The initial state for this decay is the Fermi sea of
quasiparticles and a single surface plasmon with wave vector
k
, which we can denote
by
j
k
i¼
^
a
y
k
j
0
i
, if we define |0
i
to consist of the vacuum of surface plasmons
and the Fermi sea. The possible final states each consist of a single electron–hole
pair on the Fermi sea and no plasmon,
j
q
n
;
q
0
n
0
i¼
^
c
y
q
0
n
0
^
c
q
n
j
0
i
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
f
q
0
n
0
Þ
f
q
n
p
,and
are therefore labelled by the electron and hole wave vectors
q
,
q
0
and band indices
n
,
n
0
. (The occupation factors normalize the final states, since
f
^
c
q
n
;
^
c
y
q
0
n
0
g¼
d
qq
0
d
nn
0
and
f
q
n
h
0
j
^
c
y
q
n
^
c
q
n
j
0
i
.)
Fermi’s golden rule for this decay process is therefore
G
¼
2
p
‘
X
q
0
q
n
0
n
d
ð
e
q
0
n
0
e
q
n
‘
o
Þ
M
k
q
n
;
q
0
n
0
2
;
ð
3
Þ
with the transition matrix element
M
k
q
n
;
q
0
n
0
:
¼h
q
n
;
q
0
n
0
j
^
H
int
j
k
i
¼
h
0
j
^
c
y
q
n
^
c
q
0
n
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
f
q
0
n
0
Þ
f
q
n
p
e
2
m
e
X
s
Z
d
r
^
C
y
s
^
A
^
p
^
C
s
"#
^
a
y
k
j
0
i
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
f
q
0
n
0
Þ
f
q
n
q
O
L
z
S
X
s
Z
L
z
S
d
r
c
s
q
0
n
0
ð
r
Þ
e
2
m
e
u
k
ð
r
;
0
Þ
‘
r
i
c
s
q
n
ð
r
Þ
2
6
4
3
7
5
:
ð
4
Þ
The expectation value of the plasmon and quasiparticle creation and annihilation
operators in the second line above against the vacuum and Fermi sea reduces to
(1
f
q
0
n
0
)
f
q
n
. In the final line above,
S
is the surface area for plasmon
quantization
26
, and we quantize the electrons on a box of area
S
on the surface that
extends a depth
L
z
into the surface, with
L
z
4
4
1/|
g
(
z
o
0)|, the decay length of the
plasmon mode in the metal. The factor
O
/(
L
z
S
) above accounts for the fact that the
orbitals are normalized on the unit cell of volume
O
instead of on the quantization
volume.
Substituting the plasmon mode function given by equation (1), noting that it
varies slowly on the length scale of the orbitals (both
k
and |
g
(
z
o
0)| are small in
atomic units) and splitting
q
¼
q
z
^
z
þ
q
k
, the normal- and surface-plane
components, the matrix element reduces to
M
k
q
n
;
q
0
n
0
:
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
f
q
0
n
0
f
q
n
q
d
z
2
L
z
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
‘
o
SL
o
ðÞ
s
^
k
k
^
z
g
z
o
0
ðÞ
d
q
0
k
q
k
1
1
e
d
z
j
g
ð
z
o
0
Þj þ
iq
z
q
z
0
ðÞ
ðÞ
X
s
Z
O
d
r
c
s
q
0
n
0
r
ðÞ
e
‘
r
im
e
c
s
q
n
r
ðÞ
;
ð
5
Þ
×
1
Relative probability
Holes electrons
×
2
Holes electrons
×
1
×
5
×
1
×
20
–2
–1
0
1
2
×
1
–2
–1
0
1
2
×
80
Carrier energy,
E
–
E
f
(eV)
Carrier energy,
E
–
E
f
(eV)
L
z
=5 nm
L
z
=10 nm
L
z
=20 nm
L
z
=40 nm
L
z
=20 nm
L
z
=10 nm
L
z
= 5 nm
L
z
=40 nm
Symmetric mode (higher
k
)
Anti-symmetric mode (lower
k
)
z
k
L
z
E.k
z
L
z
k
E.k
Figure 5 | Energy distribution of hot carriers from thin-film plasmon decay.
Energy distributions of hot carriers (electrons in blue, holes in red) generated
by the decay of symmetric and antisymmetric plasmon modes of energy 2.8 eV on gold thin films of various thicknesses. As the film thickness decreases,
the relative probability for generating hot electrons via geometry-assisted intraband transitions (compared with hot holes generated via interba
nd
transitions) increases.
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6788
6
NATURE COMMUNICATIONS
| 5:5788 | DOI: 10.1038/ncomms6788 | www.nature.com/naturecommunications
&
2014
Macmillan Publishers Limited. All rights reserved.
where
d
z
is the separation between lattice planes. The term with the exponential in
the denominator arises from the sum of a geometric series over lattice planes and is
a sharply peaked function of
q
z
q
z
0
with width
B
|
g
(
z
o
0)|. We can therefore
approximate it by
d
q
z
q
0
z
in the total transition rate with weight equal to
X
q
0
z
1
1
e
d
z
j
g
z
o
0
ðÞjþ
iq
z
q
0
z
ðÞ
ðÞ
2
N
z
2
d
z
j
g
z
o
0
ðÞ
j
;
ð
6
Þ
where
N
z
¼
L
z
/
d
z
is the number of lattice planes in the quantization volume.
Reducing the double sum over
q
using
d
q
0
k
q
k
and
d
q
z
q
0
z
, the total transition rate
simplifies to
G
¼
p
2
2
o
L
o
ðÞj
g
z
o
0
ðÞj
1
L
z
S
X
q
n
0
n
1
f
q
n
0
f
q
n
de
q
n
0
e
q
n
‘
o
^
k
k
^
z
g
ð
z
o
0
Þ
X
s
Z
O
d
r
c
s
q
n
0
ð
r
Þ
e
‘
r
im
e
c
s
q
n
ð
r
Þ
2
ð
7
Þ
Finally, replacing the discrete average over wave vectors
1
L
z
s
q
by its continuum
limit
R
d
q
2
p
ðÞ
3
gets rid of all dependence on the fictitious plasmon quantization area
and electron quantization volume, and results in equation (2) in the main text.
Thin-film plasmon decay rate
.
Here we briefly sketch the construction of the
plasmon mode functions, the Wannier basis approximation to the electronic states
and the Fermi golden rule calculation for the surface plasmon decay in thin
metallic films (as shown in Fig. 5).
Given a thin metal film of thickness
L
z
¼
2
H
centred at
z
¼
0 described by a
local dielectric function
E
(
o
), we can solve Maxwell’s equations analytically to
obtain the symmetric mode
u
sym
k
ð
r
;
t
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
‘
o
L
sym
o
ðÞ
S
s
e
i
ð
k
r
o
t
Þ
e
g
out
j
z
j
H
ðÞ
^
k
þ
^
z
ik
g
out
sign
z
ðÞ
;
j
z
j
4
H
1
cosh
g
in
H
ðÞ
^
k
cosh
g
in
z
ðÞ
^
z
ik
g
in
sinh
g
in
z
ðÞ
;
j
z
j
H
8
<
:
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
u
sym0
k
ð
z
Þ
;
ð
8
Þ
and the antisymmetric mode
u
asym
k
ð
r
;
t
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
‘
o
L
asym
o
ðÞ
S
s
e
i
ð
k
r
o
t
Þ
e
g
out
j
z
j
H
ðÞ
^
k
sign
ð
z
Þþ
^
z
ik
g
out
;
j
z
j
4
H
1
sinh
g
in
H
ðÞ
^
k
sinh
g
in
z
ðÞ
^
z
ik
g
in
cosh
g
in
z
ðÞ
;
j
z
j
H
8
<
:
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
u
asym0
k
z
ðÞ
:
ð
9
Þ
Here,
g
out
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
2
o
2
=
c
2
p
and
g
in
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
2
E
ð
o
Þ
o
2
=
c
2
p
, where
k
satisfies the
dispersion relation
k
¼
o
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
ð
o
Þð
E
ð
o
Þ
tanh
2
ð
g
in
H
Þ
1
Þ
E
2
ð
o
Þ
tanh
2
ð
g
in
H
Þ
1
s
ð
10
Þ
for the symmetric mode and
k
¼
o
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
ð
o
Þð
e
ð
o
Þ
coth
2
ð
g
in
H
Þ
1
Þ
E
2
ð
o
Þ
coth
2
ð
g
in
H
Þ
1
s
ð
11
Þ
for the antisymmetric mode (obtained by imposing continuity of
D
^
z
across the
interface).
Above, the prefactors in equations (8) and (9) normalize the energy of the
plasmon modes on an area
S
to that of a single quantum,
:
o
. (See ref. 38 for more
details about quantization of surface plasmons on thin films.) Integrating the field
intensities and enforcing this normalization then results in
L
sym
ð
o
Þ¼
ð
g
2
in
þ
k
2
Þ
sinh
ð
g
in
L
z
Þþð
g
2
in
k
2
Þ
g
in
L
z
d
ð
oE
Þ
d
o
þ
oE
c
2
sinh
ð
g
in
L
z
Þ
g
in
L
z
½
2
g
3
in
cosh
ð
g
in
L
z
Þþ
1
ðÞ
þ
k
2
2
g
3
out
ð
12
Þ
for the symmetric mode and
L
asym
ð
o
Þ¼
ð
g
2
in
þ
k
2
Þ
sinh
ð
g
in
L
z
Þð
g
2
in
k
2
Þ
g
in
L
z
d
ð
oE
Þ
d
o
þ
oE
c
2
sinh
ð
g
in
L
z
Þþ
g
in
L
z
½
2
g
3
in
cosh
ð
g
in
L
z
Þ
1
ðÞ
þ
k
2
2
g
3
out
ð
13
Þ
for the antisymmetric mode.
As for the electronic states, we start by computing the maximally localized
Wannier functions
36,37
w
s
a
ð
r
R
Þ¼
1
N
q
P
q
b
U
q
ab
e
i
q
ð
r
R
Þ
c
s
q
b
ð
r
Þ
for the bulk
metal, which involves finding the unitary rotations
U
q
ab
that minimize the spatial
variance of
w
s
a
ð
r
R
Þ
. Here,
c
s
q
b
are the eigen orbitals of the bulk metal and
let
E
q
b
be the corresponding eigen energies. Now, we can use the unitary
transformations to construct the Hamiltonian in the basis of Wannier functions,
H
R
a
;
R
0
a
0
¼
1
N
q
P
q
b
U
q
a
0
b
U
q
ab
E
q
b
e
i
q
ð
R
R
0
Þ
E
q
b
. The fact that the Wannier functions are
localized then implies that the elements of
H
R
a
,
R
0
a
0
decrease rapidly with increasing
|
R
R
0
| and can be truncated after a finite number of sites. This is therefore a
multi-orbital tight-binding-like model (with many neighbours) that exactly
reproduces the original eigenfunctions and eigenvalues since it is obtained from a
unitary transformation of the original Hamiltonian. See ref. 37 for a detailed
exposition.
Next we approximate the Hamiltonian for the thin film by starting with the
above Wannier basis Hamiltonian for the bulk system and setting any matrix
elements that involve a site outside the film to zero. Diagonalizing this discrete
Hamiltonian then gives us the eigenvalues
E
q
n
and eigenfunctions
c
s
q
n
¼
P
a
R
C
q
naZ
e
i
q
R
w
s
a
ð
r
R
Þ
as a linear combination of the localized Wannier
functions on various atom sites
R
. Here we have used Bloch’s theorem in the two
periodic directions to obtain the diagonalizing factor
e
i
q
R
is a wave vector in the
two-dimensional (2D) Brillouin zone. The coefficients
C
q
naZ
(where
Z
¼
^
z
R
) are
obtained from numerical diagonalization of the discrete Hamiltonian matrix
constructed above.
Finally, we apply Fermi’s golden rule to calculate the transition rates using the
vector potentials for the thin-film plasmon modes and the eigenfunctions in the
Wannier basis. Assuming that
u
sym
=
asym
k
vary slowly on the atomic scale in the two
periodic directions, we can show analogously to the previous section that the
transition rates for the symmetric/antisymmetric modes are
G
sym
=
asym
p
2
‘
2
e
2
O
k
m
2
e
o
L
sym
=
asym
ð
o
Þ
X
n
0
n
Z
d
q
ð
2
p
Þ
2
ð
1
f
q
n
0
Þ
f
q
n
d
ð
e
q
n
0
e
q
n
‘
o
Þ
X
a
0
aZ
0
Z
C
q
n
0
a
0
Z
0
C
q
naZ
u
sym0
=
asym0
k
ð
z
Z
0
Z
a
0
a
Þ
X
R
k
e
i
q
R
k
h
i
ri
R
k
a
0
a
2
:
ð
14
Þ
Here,
O
||
is the area of the 2D surface unit cell,
z
Z
0
Z
a
0
a
is the
z
centre of the
product density of the Wannier basis functions indexed by
aZ
and
a
0
Z
0
and
h
i
ri
R
k
a
0
a
is the momentum matrix element between Wannier functions
a
and
a
0
at two sites
separated by
R
||
, a lattice vector parallel to the surface. As before, we generate
the carrier energy distributions by histogramming contributions to the integral in
the above expression in terms of the initial and final electronic state energies.
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Acknowledgements
We thank Marco Bernardi for detailed feedback on and suggestions towards improving
this manuscript; Tonatiuh Rangel, Jamal Mustafa and Marco Bernardi for private
communications of noble metal GW band structures; and Yuan Ping for useful discus-
sions. This material is based on the 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 Number DE-SC0004993. P.N. is sup-
ported by a National Science Foundation Graduate Research Fellowship and by the
Resnick Sustainability Institute.
Author contributions
All authors contributed to all aspects of this work.
Additional information
Competing financial interests:
The authors declare no competing financial interests.
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How to cite this article:
Sundararaman, R.
et al.
Theoretical predictions for
hot-carrier generation from surface plasmon decay.
Nat. Commun.
5:5788
doi: 10.1038/ncomms6788 (2014).
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