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Solar energy conversion via hot electron internal photoemission in metallic
nanostructures: Efficiency estimates
Andrew J. Leenheer, Prineha Narang, Nathan S. Lewis, and Harry A. Atwater
Citation: Journal of Applied Physics
115
, 134301 (2014); doi: 10.1063/1.4870040
View online: http://dx.doi.org/10.1063/1.4870040
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/13?ver=pdfcov
Published by the AIP Publishing
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Solar energy conversion via hot electron internal photoemission in metallic
nanostructures: Efficiency estimates
Andrew J. Leenheer,
1,2
Prineha Narang,
1,2
Nathan S. Lewis,
3,2
and Harry A. Atwater
1,2,
a)
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena,
California 91125, USA
2
Joint Center for Artificial Photosynthesis, Pasadena, California 91125, USA
3
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,
California 91125, USA
(Received 31 October 2013; accepted 15 February 2014; published online 1 April 2014)
Collection of hot electrons generated by the efficient absorption of light in metallic nanostructures,
in contact with semiconductor substrates can provide a basis for the construction of solar
energy-conversion devices. Herein, we evaluate theoretically the energy-conversion efficiency of
systems that rely on internal photoemission processes at metal-semiconductor Schottky-barrier
diodes. In this theory, the current-voltage characteristics are given by the internal photoemission yield
as well as by the thermionic dark current over a varied-energy barrier height. The Fowler model, in
all cases, predicts solar energy-conversion efficiencies of
<
1% for such systems. However, relaxation
of the assumptions regarding constraints on the escape cone and momentum conservation at the
interface yields solar energy-conversion efficiencies as high as 1%–10%, under some assumed (albeit
optimistic) operating conditions. Under these conditions, the energy-conversion efficiency is mainly
limited by the thermionic dark current, the distribution of hot electron energies, and hot-electron
momentum considerations.
V
C
2014 AIP Publishing LLC
.[
http://dx.doi.org/10.1063/1.4870040
]
I. INTRODUCTION
The energy-conversion efficiencies of record-setting
pn
-
junction photovoltaics are rapidly approaching the theoretical
single-bandgap Shockley-Queisser limit of 32% under uncon-
centrated sunlight.
1
Multi-junction solar cells (that still operate
within the Shockley-Queisser limitations for each absorber and
junction) can provide much higher efficiencies partly by reduc-
ing the amount of sub-bandgap light lost, but such devices also
have much higher costs than single-bandgap devices due to the
need to produce multiple high-
purity semiconductor materials
to capture the incident light and convert it into a collected elec-
trical current. Another possibl
e device architecture considered
here consists of a single band g
ap semiconductor homojunction
or heterojunction device used
in combination with a metal-
semiconductor Schottky junct
ion formed from that same light
absorber. In such an approach, in addition to collection of
above band-gap carriers generat
ed in the semiconductor (again
subject to the Shockley-Queisser limit), the metal would addi-
tionally serve to generate “hot”
electron-hole pairs in the metal
which would then be emitted into the semiconductor and col-
lected as an additional photocur
rent. The process of hot carrier
internal photoemission (IPE) from the metal to the semiconduc-
tor over a tuneable Schottky ba
rrier has therefore been pro-
posed as a possible solar energy
conversion device formation
strategy.
2
,
3
This metal-absorber device structure (similar in
some ways to a dye-sensitized solar cell) could therefore pro-
vide an interesting device integration possibility when placed
optically behind a single-junctio
n solar cell, serving to increase
the overall efficiency of the whol
e system by virtue of the pres-
ence of this second capture and conversion system in the
overall device structure. Though referred to as “hot electron” or
“hot hole” emission/capture,
we emphasize that the device
physics are different from “
hot carrier” solar cells.
4
Such
hot carrier metal/semiconductor device structures could,
in principle, be beneficially used in solid-state
2
,
5
10
or
photoelectrochemical
11
16
systems to collect photons having
energies lower than the energy band gap of a semiconductor, in
essence, serving as the second
junction in a tandem structure
but not requiring necessarily a second pure semiconductor light
absorber as in a conventional tandem cell arrangement.
Plasmonic structures have been demonstrated to provide
highly efficient light scattering and trapping elements, in
some cases, providing enhancements in solar energy conver-
sion.
17
In the context of hot-electron devices, the large
extinction cross-section at a surface plasmon resonance ena-
bles very thin films of nanostructures to absorb a significant
fraction of the solar spectrum.
18
The collective plasmon os-
cillation may also play a role in increasing the photoemission
yield,
19
,
20
though the details of the hot-carrier dynamics after
surface plasmon decay are still under study. At the small
dimensions of plasmonic structures, the effects of electron
scattering at surfaces strongly modifies the yield even in the
semiclassical IPE model.
21
,
22
Herein, we present an analysis of the efficiency limits for
energy conversion via IPE, capturing the key optical and elec-
tronic processes in such devices. Section
II
presents the
current-voltage characteristics and energy-conversion effi-
ciency based on simple Fowler theory and thermionic emis-
sion; Section
III
reviews the three-step model of internal
photoemission and describes explicitly the inherent assump-
tions of Fowler theory; Section
IV
refines the yield including
the effect of phonon scattering and thin-film enhancement;
Section
V
presents example calculations of the limiting
a)
E-mail: haa@caltech.edu
0021-8979/2014/115(13)/134301/7/$30.00
V
C
2014 AIP Publishing LLC
115
, 134301-1
JOURNAL OF APPLIED PHYSICS
115
, 134301 (2014)
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efficiency under various assumptions; and Section
VI
dis-
cusses the application of these approaches to plasmonic struc-
tures. Previous estimates of the IPE yield and energy-
conversion efficiency of such systems have used simple
Fowler theory and/or have used a simplified treatment of the
carrier dynamics, with a recent study by White and Catchpole
consequently calculating a maximum best case solar energy-
conversion efficiency of 8% for such systems.
3
In contrast, we
describe the situation in which realistic assumptions are made
and the carrier dynamics are fully treated. Our most generous
efficiency estimates agree with previous “absolute upper-
limit” efficiency values,
2
,
3
which assumed that the momentum
requirements at the interface governing emission (the hot elec-
tron escape cone) can be relaxed for nanostructures. However,
our more in-depth analysis shows that even for nanostructures
of dimensions on the order of 20 nm, the practically obtainable
efficiency is lowered by orders of magnitude due to the limit-
ing effects of the hot electron mean free path in conjunction
with the requirement of a critical momentum normal to the
interface. The lowered efficiency limits calculated herein thus
serve as a more realistic framework for establishing the
expected efficiencies, design parameters, and performance
characteristics, of an actual energy-conversion system based
on metallic hot-carrier internal photoemission.
II. INTERNAL PHOTOEMISSION AND EFFICIENCY
Fowler developed the basic theory of photon-induced
emission of electrons from metals in the early 20th century.
23
Though refinements have been made,
22
,
24
,
25
the simple
Fowler equation has proven to be in accord with experimen-
tal data for the internal photoemission yield
26
in both magni-
tude and spectral behaviour
Y
Fow
ð

h
x
Þ
1
8
E
F
ð

h
x

/
b
Þ
2

h
x
;
(1)
where
É
is the reduced Planck constant,
x
is the incident
light frequency,
u
b
is the barrier height (in units of energy),
and
E
F
is the Fermi energy of the emitter, with the value of
E
F
describing the curvature of the conduction band in mo-
mentum space (Figure
1(a)
) (This treatment assumes a
1-dimensional problem as shown in Figure
1(a)
, though the
results should not differ significantly for the 3-dimensional
case.) The Fowler yield is based on a semiclassical model of
hot electrons emitted over an energetic barrier, with the criti-
cal assumption that the kinetic energy
normal
to the barrier
must be greater than the barrier height. As depicted in Figure
1(b)
, for a spherical Fermi surface, this assumption gives rise
to a limited escape cone for hot electrons, because the mo-
mentum normal to the interface must be larger than a critical
value,
p
crit
¼
[2
m
*(
E
F
þ
/
b
)]
1/2
. The escape cone limitation
results in zero yield at the threshold photon energy as well as
a slow rise with photon energy if the Fermi energy is large
compared to the photon energies of interest. This latter con-
dition is true for visible light incident on noble metal emit-
ters; for instance, both silver and gold have a Fermi energy
near 5.5 eV (which was the value for
E
F
used in our
calculations).
The collector material can be either an insulator or a
semiconductor, and the built-in electric fields of metal-
semiconductor Schottky barriers assist in the collection of the
emitted hot carriers. In principle, a metal-insulator-metal
diode could also be used for energy conversion,
2
butinour
calculations the maximum energy-conversion efficiency was
found to be equivalent to that of a metal-semiconductor diode
(see the supplementary material
27
), so the conceptually and
notationally simpler Schottky barrier case will be discussed
here, in which the metal is the emitter and the semiconductor
is the collector. Considering hot-electron emission, the opti-
mal semiconductor will be a highly doped n-type material,
and the Fermi energy in the semiconductor should be nearly
equal to the conduction-band energy. Equivalent considera-
tions apply to a p-type semiconductor that would collect hot
holes, but here for clarity we consider only the n-type case. To
operate in power-generation mode, the diode must be
forward-biased (by applying a positive voltage to the metal),
in contrast to most internal photoemission detection experi-
ments in which reverse bias aids in extracting the carriers.
The current-voltage characteristics can be determined
by considering the reverse photocurrent density due to inter-
nal photoemission
J
photo
, the dark forward current density
due to thermionic emission from the collector to the emitter
J
dark
, and the properties of the illumination source. The effi-
ciency is given by
% eff
¼
j
J
photo
þ
J
dark
j
V
P
ill

100
;
¼

ð

h
x
max
0
I
ill
Y
ð

h
x
Þ
q
=

h
x
ðÞ
d
ð

h
x
Þþ
J
dark
ð
V
Þ








V
ð

h
x
max
0
I
ill
d
ð

h
x
Þ

100
;
(2)
FIG. 1. (a) Internal photoemission band diagram for hot electrons emitted
from a metal into an n-type semiconductor. (b) Schematic of isotropic distri-
bution of hot electron momentum on a sphere in momentum space with a
limited escape cone. (c) Sketch of a possible energy conversion device lay-
out where light passing through a photovoltaic solar cell and the semicon-
ductor collector is absorbed in the metal emitter.
134301-2 Leenheer
etal.
J. Appl. Phys.
115
, 134301 (2014)
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where
V
is the operating voltage,
P
ill
is the illumination irra-
diance,
I
ill
is the spectral irradiance, and the integration is
performed up to a maximum energy
É
x
max
. Note that here
the yield
Y
is the external quantum yield, but
Y
is assumed to
be equal to the internal quantum yield under the condition of
negligible optical reflection losses. To model the AM1.5 so-
lar spectrum, the spectral irradiance was assumed to be a
5800 K blackbody with a total irradiance of 95 mW cm

2
.
which provides an easily integratable function that generally
matches the shape and irradiance of the AM1.5 spectrum.
The thermionic dark current is given by
J
dark
;
therm
¼
A

T
2
e
q
ð
V

/
b
Þ
kT
;
(3)
where A* is the Richardson’s constant,
T
is the absolute tem-
perature, and
k
is Boltzmann’s constant. Here, we are assum-
ing that the operating voltage is less than the barrier height
but a few times greater than the thermal voltage
kT
. Though
Richardson’s constant is given as 120 A cm

2
K

1
, in our
calculations, we generously assumed the more optimistic
value of A*

50 A cm

2
K

1
which applies for thermionic
emission involving a semiconductor like silicon; however,
this more optimistic value only results in a maximum of 10%
relative efficiency increase relative to the more stringent con-
dition with A*
¼
120 A cm

2
K

1
.
Figure
2(a)
displays the efficiency for hot carrier internal
photoemission assuming the simple Fowler yield based on
Eqs.
(1)
(3)
. Because one application of this concept involves
capture of sub-bandgap illumination below a traditional pho-
tovoltaic cell as shown schematically in Figure
1(c)
,theeffi-
ciency is plotted as a function of maximum photon energy.
Hence, the maximum photon energy would be 1.1 eV for a Si
solar cell, 3.0 eV for a TiO
2
photoelectrochemical device, or
about 4 eV for the entire solar spectrum. The inset shows an
example current-voltage behaviour, which has a shape that is
similar to a standard pn-junction or Schottky solar cell, but at
a much lower operating voltage and current. Figure
2(b)
dis-
plays the barrier height and voltage at the maximum power
point,
V
mpp
for the maximum efficiency values displayed in
Figure
2(a)
. The yield is highest for a small energy barrier, but
avoiding the thermionic dark current requires a larger barrier.
Specifically, for operation at 1 sun and 300 K, a difference of

0.7 eV between
/
b
and
V
mpp
is required to keep the thermi-
onic dark current less than the photocurrent. The thermionic
dark current for metal-semiconductor Schottky barrier solar
cells can be reduced by introducing a higher barrier for major-
ity carriers, but internal photoemission is entirely a
majority-carrier process, so any extra barrier will also reduce
the photocurrent.
III. THE THREE-STEP MODEL FOR INTERNAL
PHOTOEMISSION
Because the simple Fowler equation predicts that the
maximum efficiency of an energy-conversion device based
on internal photoemission is

1%, it is useful to analyze the
assumptions and mechanisms involved in derivation of the
Fowler theory to determine the conditions, if any, that could
result in higher efficiencies. The semiclassical model of in-
ternal photoemission involves three steps: hot-electron exci-
tation, hot-electron transport to the interfacial barrier, and
hot-electron emission over the energetic barrier from the
emitter material into the collector material. Although the
actual processes of light absorption and excitation of the col-
lective electron cloud are quantum-mechanical phenomena,
we assume herein that after light absorption, the “hot elec-
tron” behaves as a quasiparticle whose transport can be
described semiclassically within a free-electron-like band
structure.
Light is absorbed in the metal when the photon’s per-
turbing electric field causes electronic transitions.
Consequently, the material response is described macro-
scopically by a frequency-dependent dielectric constant,
e
,
determined empirically for bulk materials. Assuming that
this local, linear permittivity is a valid description for nano-
scale structures such as plasmonic absorbers, Maxwell’s
equations yield the spectral power absorption as
P
abs
¼
1
2
Re
r
S
½
¼
1
2
x
j
E
j
2
Im
e
½
/
g
e
;
(4)
where
S
is the Poynting vector,
E
is the electric field of the
incident electromagnetic wave, and
g
e
is the hot electron
generation rate per length. The spatial distribution of
absorbed power is obtained from Eq.
(4)
, and for antenna-
like structures, the absorbed power is highest near the surfa-
ces around the midpoint where the highest currents flow.
Assuming that the probability is low for an absorbed photon
to couple directly to phonons or multiple electron excitations
(because many-body excitations are not very probable), the
spatial power absorption normalized by the incident power
then directly corresponds to the spatial distribution of hot-
electron generation. Such calculations are readily performed
FIG. 2. (a) Solar conversion efficiency for internal photoemission over a
metal-semiconductor Schottky barrier based upon the simple Fowler equa-
tion. (b) Optimized barrier height and maximum power point voltage (V
mpp
)
used to calculate the curve in (a). Inset: Example current-voltage curve with
maximum power shown as the dotted box.
134301-3 Leenheer
etal.
J. Appl. Phys.
115
, 134301 (2014)
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using, e.g., full-field finite difference time domain simula-
tions, but the generation profile depends significantly on the
geometry of the antenna and system as a whole. Hence, for
simplicity, the generation profile was assumed herein to be
uniform throughout a film of thickness
w
, i.e.,
g
e
¼
1/
w
.
The electron-hole pair excited by light was assumed to
have a total energy equal to the photon energy, so the hot
electron energy,
E
el
, can range from 0 to
É
x
. In the simplest
approximation, the distribution of energies would be uniform
in this range. However, considering the electronic density of
states
g
(
E
) and nondirect transitions in which momentum
can be supplied by surfaces, defects, or phonons, the proba-
bility of excitation to a certain energy
E
¼
E
F
þ
E
el
is just the
multiplied probability of the existing initial and final states,
normalized to the total number of transitions possible
P
0
ð
E
el
Þ
dE
¼
g
ð
E
Þ
g
ð
E


h
x
Þ
dE
ð
E
F
þ

h
x
E
F
g
ð
E
0
Þ
g
ð
E
0


h
x
Þ
dE
0
:
(5)
For a free-electron-like metal with a parabolic band structure
at low temperature, such that the tails of the Fermi distribu-
tion can be ignored, the hot electron energy distribution
becomes
P
0
ð
E
el
Þ
dE
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
F
þ
E
el
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
F
þ
E
el


h
x
p
dE
ð
E
F
þ

h
x
E
F
ffiffiffiffiffi
E
0
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
0


h
x
p
dE
0
(6)
which was used in these calculations. Many metals are free-
electron like near the Fermi energy, e.g., for gold the bands
with d-orbital character lie about 1.6 eV below the Fermi
level, so this approximation is most valid for low photon
energy excitation. The relative distribution of hot electrons
and hot holes varies depending on the material, and low-lying
bands could favor hot holes over hot electrons due to the
increased density of states below the Fermi level; modification
of the “electron distribution joint density of states” could, in
principle, increase (or decrease) the yield and efficiency.
3
After excitation, the hot electron quasiparticle must
move through the material to reach a collecting interface.
Because phonon scattering is a quasielastic process, only
electron-electron scattering is assumed to cause significant
energy loss of the hot electrons. Typically

1
=
2
of the hot
electron’s energy is lost in an electron-electron scattering
event, and the resulting electron can no longer surmount the
barrier. The mean free path for electron-electron scattering
therefore determines the probability
P
int
that the hot electron
will reach the interface, if starting at a depth
z
at an angle
h
away from normal
P
int
sin
h
d
h
¼
1
2
exp

z
k
e

e
ð
E
el
Þ
cos
h

sin
h
d
h
;
(7)
where the factor of
1
=
2
results from half of the electrons ini-
tially travelling away from the interface. A suitable analyti-
cal model for the electron-electron scattering mean free path
was developed by Quinn and is given as
k
e

e
ð
E
el
Þ¼
24
a
0
ffiffiffiffiffiffiffiffiffiffiffiffiffi
a
e
r
s
=
p
p
3
E
F
=
E
2
el
þ
2
=
E
el

tan

1
ffiffiffiffiffiffiffiffi
p
a
e
r
s
r
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
a
e
r
s
=
p
p
1
þ
a
e
r
s
=
p
;
(8)
where
a
0
is the Bohr radius (0.0529 nm),
a
e
¼
(4/(9
p
))
1
=
2
, and
r
s
is the radius of a sphere equal to the volume of one con-
duction electron in units of the Bohr radius; for gold
r
s
¼
3.
The value of
k
e-e
approximately follows a
E
el

2
behaviour,
with some example values being 100 nm at 1 eV to 10 nm at
3.5 eV (a plot of Eq.
(8)
is included in the supplementary ma-
terial
27
). Thus, the details of the spatial hot electron genera-
tion profile are not critical, because the distances travelled
are relatively long compared to the nanoscale dimensions of
exemplary plasmonic structures. Though the mean free path
can be longer than the characteristic dimension of the metal-
lic nanostructure, the escape cone restriction (vide infra) dic-
tates that, in general, multiple reflections within the metal
will occur before the hot electron can be emitted.
When the hot electron encounters the surface and energy
barrier, Fowler’s theory asserts that the component of kinetic
energy normal to the barrier must equal the barrier energy.
This requirement is illustrated as the limited momentum
escape cone in Figure
1(b)
, with the maximum angle of
approach for which a hot electron can escape given by
cos
h
max
¼
p
crit
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
F
þ
/
b
E
F
þ
E
el
s

1

E
el

/
b
2
E
F
;
(9)
where the approximation holds if
/
b
,
E
el
E
F
.
This angle
defines the maximum angle allowed in Eq.
(6)
. Note that the
fraction of hot electrons reflected by the barrier
R
elec
can be
written as
R
elec
¼
1

T
elec
¼
1

ð
h
max
0
sin
h
d
h
;
R
elec

1

E
el

/
b
2
E
F
;
(10)
where
T
elec
is the transmitted fraction. For large Fermi ener-
gies compared to the excitation energy, the reflected fraction
is nearly unity.
The internal photoemission yield as a function of energy
is obtained by combining the probabilities of absorption,
transport to the barrier, and emission over the barrier,
Y
ð

h
x
Þ¼
ð

h
x
/
b
dE
el
ð
h
max
0
sin
h
d
h
ð
1
0
dzP
0
ð
E
el
Þ
P
int
ð
z
;
h
;
E
el
Þ
g
e
ð
z
Þ
:
(11)
Under the conditions of
p
crit

p
F
so that the escape cone is
small, a small absorption length compared to
k
e-e
, and a con-
stant distribution of hot electron energies, the integrals are eas-
ily evaluated and result in the Fowler yield, Eq.
(1)
,whichisa
good approximation for light incident on a bulk slab of metal.
IV. ENHANCEMENTS DUE TO SCATTERING
For thin metal emitters that have a thickness on the order
of
k
e-e
, the yield can be enhanced significantly due to
134301-4 Leenheer
etal.
J. Appl. Phys.
115
, 134301 (2014)
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Lambertian reflections at the interfaces. Electron-phonon
scattering with a mean free path
k
e-p

20 nm (used in the
calculations here) can additionally enhance the yield,
because the hot electron momentum can be redirected into
the escape cone with little loss of energy in the quasi-elastic
collisions. Again considering the case of
p
crit

p
F
, Dalal
22
has derived an enhanced yield expression that takes into
account both phonon and back-surface scattering. In this
model (see Ref.
22
for more details), the angular integral of
P
int
is replaced with a more detailed function
q
(
z
) due to a
sum over multiple reflections at various scattering angles,
producing the following expression for enhanced yield:
Y
enh
ð

h
x
Þ¼
ð

h
x
/
b
dE
el
ð
1
0
dzP
0
ð
E
el
Þ
q
ð
E
el
;
z
Þ
g
e
ð
z
Þ
;
q
ð
E
el
;
z
Þ¼
A
w
e
l
z
þ
B
w
e

l
z
;
A
w
¼
e

2
l
w
B
w
;
B
w
¼
1

R
elec
ð
1

R
elec
Þð
1
þ
e

2
l
w
Þþ
1
þ
k
e

e
k
e

p


1
=
2
ð
1
þ
R
elec
Þð
1

e

2
l
w
Þ
;
l
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
k

1
e

e
þ
k

1
e

p
Þ
2

k

1
e

p
ð
k

1
e

e
þ
k

1
e

p
Þ
q
;
(12)
where
w
is the metal thickness in the
z
direction. Relative to
Fowler’s treatment, phonon scattering effectively boosts the
yield by a factor (
k
e-e
/
k
e-p
)
1/2
for emission from a bulk emit-
ter, and the yield increases strongly with reduced thickness
of the metal; a plot of the thickness effect is included in the
supplementary material.
27
For extremely thin films,
Y
enh
diverges because the
assumption of
R
elec

1 that was used to derive Eq.
(12)
is no
longer valid. Instead, for a best-case yield that may be appro-
priate to describe the behaviour of very thin and rough films
in which the escape cone restriction is relaxed, the condition
R
elec

0 applies. This condition can be met by modifying Eq.
(6)
to set
h
max
¼
p
/2 and to include one specular reflection off
the back surface. In this situation, phonon scattering is irrele-
vant if
w
<
k
e-p
. In the limiting case of
k
e
-
e
w
, all of the hot
electrons with sufficient energy will be emitted, and in the
best-case scenario
P
int
¼
1. For this hypothetical case, with no
escape-cone limitation, the yield is only limited by the distri-
bution of hot electron energies; the analyses in Refs.
2
and
3
correspond to this extremely thin film assumption.
V. THEORETICAL EFFICIENCIES
Based on the equations for yield outlined in Secs.
III
and
IV
, the efficiency given by Eq.
(1)
can be numerically
evaluated for a variety of conditions and assumptions. Figure
3
shows the optimized efficiency as a function of maximum
photon energy, assuming that the incident light is completely
and uniformly absorbed over the film thickness with no
reflection losses. Figure
3
includes (on a logarithmic scale)
the result from Figure
1(a)
based on the Fowler yield but
also shows results for a 100 nm metal film (
E
F
¼
5.5 eV) at
300 K, a 20 nm film at 300 K, a 20 nm film at 300 K for
which
R
elec
¼
0, and a 20 nm film at 77 K, as well as the best-
case scenario at both 300 and 77 K for which the hot electron
mean free path is much longer than the film thickness. Note
that when the momentum escape cone restriction is included,
with either finite film thickness, the calculated efficiency is
much lower than the best-case scenario in which the escape
cone restriction is not explicitly included in the analysis.
The 100 nm and 20 nm cases at 300 K show lower effi-
ciency than the simple estimate from the Fowler equation,
due to the inclusion of a finite electron mean free path, which
affects light absorbed deeply in the metal. In contrast, the
Fowler case assumed absorption at the surface. Operation of
such devices at lower temperature enhances the efficiency,
because the dark current from thermionic emission is low-
ered significantly as the temperature decreases. Clearly, the
efficiency reaches values significantly in excess of 1% only
if the escape cone restriction is lifted in the best-case sce-
nario and if photon energies above 1.5 eV are included.
VI. DISCUSSION
In this semiclassical model of hot electron internal pho-
toemission, the energy-conversion efficiency is low for two
FIG. 3. Solar conversion efficiency for IPE considering a 5800 K blackbody
spectrum up to a given maximum photon energy fully absorbed in an Au-
like metal considering different film thicknesses (colors), operating tempera-
tures (dashed lines), and escape cone limitations (dotted line). The simple
Fowler case from Fig.
1
is included as the bold black line, and “best case”
represents a very long hot electron mean free path compared to the dimen-
sions of the structure itself.
134301-5 Leenheer
etal.
J. Appl. Phys.
115
, 134301 (2014)
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primary reasons. First, the diode must be operated in forward
bias to capture the reverse photocurrent, so the thermionic
current of electrons flowing from the collector to the emitter
strongly reduces the net current. Second, each photon creates
a hot electron and a hot hole, and the hot electron energy can
easily be less than the photon energy. In contrast to a semi-
conductor photovoltaic device in which the internal energy
of the minority carriers is nearly equal to the bandgap, the in-
ternal energy of the hot electrons involved here is distributed
from zero to the photon energy.
Although the most optimistic assumptions were used in
most cases, the results could be modified by explicit consider-
ation of some other effects. The band structure of real materi-
als can change the distribution of hot electron (and hot hole)
energy, so the transition probabilities linking initial and final
electronic band states should be calculated to determine the
excitation probabilities as a function of both the hot electron/
hole energy and momentum. An optimized band structure or
device layout may allow higher efficiency than that calculated
herein. We have used reasonable numbers for a Au/n-Si junc-
tion, but the efficiency estimates here should be applicable to
a wide range of metal/semiconductor materials because the
hot electron mean free paths do not vary wildly for various
metals. Also, the efficiencies presented here are not true
detailed-balance efficiencies, because no re-radiation of light
was considered. Considering optical reflection and photon
emission would further lower the efficiencies.
Another limitation exists due to the requirement of criti-
cal momentum normal to the interface, which leads to a
small escape cone for hot electrons having an energy just
larger than the barrier energy. For an interface that is rough
on the scale of the electron wavelength, this classical restric-
tion may be relaxed. Indeed, some vacuum photoemission
experiments have seen anomalously high yields from nano-
particles,
20
with a variety of explanations involving geome-
try, escape cone relaxation, and surface chemistry
modification.
28
30
In theory, the quantum mechanical details
of plasmon-mediated hot carrier production may introduce a
momentum-polarization correlation. Qualitatively, this cou-
pling may enhance photocurrent along certain geometry-
dependent directions at specific polarizations. This coupling
tends to relax the escape cone considerations, by allowing
final electron momenta that are usually disallowed by regular
IPE processes. As a result, the yield may be increased up to
the
R
elec
¼
0 case. However, simply relaxing the escape cone
restriction without addressing the dark current and hot elec-
tron energy distribution limitations still results in energy-
conversion efficiencies of a few percent at best (Fig.
3
).
Because conventional photovoltaic cells do not absorb
light below the bandgap energy, exploitation of IPE might be
a potentially interesting method for capturing the otherwise
unutilized part of the solar spectrum by placing the device
behind a solar cell. In this arrangement, use of the metal/se-
miconductor device would be analogous to placing another,
low band gap, semiconductor absorber and associated metal-
lurgical junction in the optical path. The system instead
relies on optical absorption and charge carrier excitation in
the metal portion of the metal/semiconductor system (with a
large band gap semiconductor, in principle), to produce the
additional current and thus augment the device efficiency.
As displayed in Figure
3
, however, the yield and efficiency
increase strongly as the photon energy is increased, and IPE
is particularly inefficient for a spectrum that only includes
energies below 1–2 eV. Hence, such an approach would be
more appropriate for larger-bandgap devices that normally
only absorb ultraviolet light, such as a TiO
2
-based photoelec-
trochemical system than for an additional absorber approach
to a conventional solar cell arrangement.
The metallic emitter was implicitly assumed to be a nano-
structure that possessed plasmonic resonances so as to provide
high absorption in a very thin structure
18
and additionally to
take advantage of the enhancements in scattering. The spatial
distribution of hot electron generation may vary for such
nanostructures, likely depending on the position of absorption
based on the Poynting vector (Eq.
(4)
). For example, a dipole
antenna has the highest current and dissipation of energy near
its center. The hot electrons may possibly instead be generated
near areas of high field enhancement.
12
Regardless, for these
relatively low-energy hot electrons, the electron-electron scat-
tering mean free path is on the same order as a plasmonic
nanoparticle’s dimensions, so the specific location of hot elec-
tron generation is of minor importance. The hot electron’s ini-
tial momentum is, however, very important for the yield, as
particles with a momentum vector inside the escape cone have
a much higher probability of escape than those with momen-
tum vectors outside the escape cone.
The efficiencies presented here are somewhat lower than
some previous published calculations. Wang and Melosh
2
considered power conversion using Kretschmann coupling to
surface plasmon polaritons in a symmetric metal-insulator-
metal geometry and obtained a calculated maximum effi-
ciency of 2.7%. Their calculation assumed no escape cone
restriction, no carrier reflections, a uniform energy distribu-
tion of excited carriers, and an energy-independent
k
e-e
¼
56 nm. The result is on the same order as that calcu-
lated here including, however, the assumption (for which the
justification is unclear) of no escape cone restriction; simi-
larly, White and Catchpole calculated a maximum efficiency
of 8% by assuming that all hot electrons with sufficient
energy in a perfect absorber were emitted.
3
Although it is
tempting to assume that for nanostructured metallic absorb-
ers the hot electron mean free path will be sufficiently longer
than the device dimension and thus that the momentum
escape cone restriction can be neglected,
2
,
3
we have shown
herein that even a small non-zero thickness (of 20 nm) of
metal lowers the efficiency from 8% to 0.25% (c.f. Figure
3
blue solid line). Schmidt
et al.
postulated energy-conversion
efficiencies up to 10% even with the escape cone restric-
tion;
31
however, the thermionic dark current was neglected
in their approach and their treatment additionally incorrectly
used the barrier height as the operating voltage in Eq.
(2)
.
Finally, some deleterious effects were not included in
our model. Scattering of hot electrons back into the emitter
from the collector will reduce the yield, especially for diodes
operated in forward bias with a weak electric field in the col-
lector. Similarly, internal photoemission from the nominal
collector to the emitter reduces the net photocurrent.
Energetic losses due to phonon scattering also could
134301-6 Leenheer
etal.
J. Appl. Phys.
115
, 134301 (2014)
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somewhat reduce the yield. Emission of hot holes into the
same material could take place if the collector is a low-
bandgap semiconductor, reducing the yield. Last, interfacial
and bulk defects present in real materials will lower the hot
electron mean free paths and collection efficiency, decreas-
ing the device efficiency. In principle, this metal-emitter
junction could be placed at the back of a photovoltaic device,
but the details of device integration and effects on the photo-
voltaic efficiency are beyond the scope of this work.
VII. CONCLUSIONS
The process of internal photoemission in which the
absorbing material is a metal rather than a semiconductor
was evaluated as a candidate for utilization in solar energy-
conversion devices. The semiclassical three-step model of
internal photoemission for hot electrons over an energetic
Schottky barrier was reviewed, and the energy-conversion
efficiency was calculated considering the IPE photocurrent
produced by complete absorption of a 5800 K blackbody
spectrum in a nanoscale metal and the thermionic emission
dark current as a function of voltage. The optimum effi-
ciency values were found to be

1% for room-temperature
operation with a metal similar to Au or Ag. The efficiency
could approach 10% if the escape cone restriction is
removed, the mean free path of hot electrons is very long
compared to the metal dimensions, and the illumination
spectrum includes visible and ultraviolet light, in which case
the efficiency is still limited by the thermionic dark current
as well as by the distribution of hot electron energies (with-
out modifying the metal’s joint density of states). We have
shown herein that considering the momentum escape cone
imposes a significant limit on efficiency even for nanostruc-
tures. Additional work to determine the applicability of this
admittedly semiclassical model would be useful because the
normal momentum requirement might be relaxed when con-
sidering quantum effects or surface chemistry. Alternatively,
a device geometry in which light capture is decoupled from
hot electron-hole generation in a metal bi-layer could possi-
bly reduce the emitter thickness to the nm-size thickness
required to justify neglecting the escape cone restriction.
ACKNOWLEDGMENTS
This material is based upon work performed by the Joint
Center for Artificial Photosynthesis, a DOE Energy
Innovation Hub, supported through the Office of Science of
the U.S. Department of Energy under Award No. DE-
SC0004993. P.N. is supported by a National Science
Foundation Graduate Research Fellowship and by the
Resnick Sustainability Institute.
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