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Current-voltage characteristics of coupled photodiode-electrocatalyst devices
Matthew R. Shaner, Katherine T. Fountaine, and Hans-Joachim Lewerenz
Citation: Applied Physics Letters
103
, 143905 (2013); doi: 10.1063/1.4822179
View online: http://dx.doi.org/10.1063/1.4822179
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Current-voltage characteristics of coupled photodiode-electrocatalyst
devices
Matthew R. Shaner, Katherine T. Fountaine, and Hans-Joachim Lewerenz
Joint Center for Artificial Photosynthesis, California Institute of Technology, 1200 East California Blvd.,
Pasadena, California 91125, USA
(Received 25 February 2013; accepted 10 September 2013; published online 3 October 2013)
Analytical expressions for the illuminated current-voltage characteristics of coupled photodiode-
electrocatalyst fuel forming devices are derived. The approach is based on combining solid-state
diode behavior with metal electrochemistry via the diode equation and the Butler-Volmer equation
(charge transfer coefficients:
a
A
¼
a
C
¼
a
) or the Tafel equation (
j
g
j
118
mV
n
e
), respectively. The
analytical expression for the current-voltage behavior of the coupled photodiode-electrocatalyst
device (
a
A
¼
a
C
¼
0.5) and an isolated photodiode is plotted, compared, and augmented with band
diagrams at equilibrium, open circuit, short circuit, and the maximum power point to illustrate the
effect of coupling an electrocatalyst to a photodiode. The applicability of the derived equations is
then demonstrated by comparing with a recently reported high efficiency, thin film InP/InO
x
P
y
/Rh
photoelectrosynthetic hydrogen evolution device.
V
C
2013 AIP Publishing LLC
.
[
http://dx.doi.org/10.1063/1.4822179
]
Photoelectrosynthetic fuel formation can provide stor-
able, energy dense fuels, and various approaches are pres-
ently pursued towards this end.
1
5
Focus has been placed on
(i) rectifying semiconductor-liquid junctions and (ii) semi-
conductor buried junctions (photodiodes) both coupled to
electrocatalysts.
6
8
To date, the latter approach, the coupled
photodiode-electrocatalyst system, has achieved superior per-
formance in both efficiency and stability.
6
In coupled
photodiode-electrocatalyst systems, the necessary driving
force for the desired reaction is determined by the thermody-
namic reaction potential and the kinetic overvoltage deter-
mined by the catalyst and operating current; this driving force
is supplied by the built-in voltage of the photodiode junction.
Although impressive devices exist, their (photo)current-
voltage behavior has hitherto only been described experimen-
tally.
2
,
9
,
10
Here, we derive explicit analytical expressions for
the (photo)current-voltage behavior of such coupled systems.
These expressions are generally applicable to photoelectro-
chemical (PEC) systems that utilize rectifying semiconductor
junctions coupled to metal electrocatalysts and can be used to
model, predict and understand device performance.
PEC device behavior consists of generation of excess
electron-hole pairs through semiconductor absorption, sepa-
ration of the excess carriers by the rectifying junction, and
electrochemical reaction at the catalyst/solution interface;
non-ideal processes such as shunt and series resistances are,
in general, present and can impact device performance
similar to conventional photovoltaics. To develop explicit
analytical expressions that describe coupled PEC device
behavior it is useful to begin with an electrical circuit dia-
gram (Figure
1(a)
). This consists of a current and voltage
generating photodiode in series with an electrocatalytic over-
potential to drive the desired chemical reaction and a voltage
loss term that accounts for interfacial, material, solution,
and/or any other resistances that can be described as series
elements; parallel elements, such as shunt resistance, do not
allow for an explicit analytical solution and are thus beyond
the scope of this work. Accordingly, the system voltage is
the linear combination of the voltage generated by the photo-
diode,
V
PV
(
j
), that used by the electrocatalyst,
g
(
j
), and that
used to overcome system series resistances, V
series
(
j
)
(Eq.
(1)
). In contrast, to the convention for the solid-state
(photo)diode relation that describes current density as a func-
tion of voltage, the electrical circuit of coupled PEC devices
necessitates an inverse formulation, voltage as a function of
current density, to obtain an analytical solution
V
ð
j
Þ¼
V
PV
ð
j
Þþ
g
ð
j
Þþ
V
series
ð
j
Þ
:
(1)
The illuminated diode equation is used to described the volt-
age generated by the photodiode, where j
0,PV
denotes the
reverse saturation current, j
L
is the light-induced current
(negative (positive) for a photocathode (photoanode)), and
n
d
is the diode quality factor (Eq.
(2)
). To maintain consis-
tency with the IUPAC convention for (photo)electrochemis-
try, Eq.
(2)
has been written for a photocathode (top signs:
þ
,
þ
) and a photoanode (bottom signs:

,

) where the cur-
rent density,
j
, is negative (positive) for a photocathode
(photoanode)
j
¼
j
L
6
j
0
;
PV
e
6
q
V
PV
n
d
kT

1

:
(2)
Solving for
V
PV
(
j
) gives
V
PV
ð
j
Þ¼
6
n
d
kT
q
ln
6
j
L

j
j
0
;
PV
þ
1

:
(3)
The Butler-Volmer equation (Eq.
(4)
) is used to describe
the current density as a function of overvoltage where j
0, cat
is the exchange current density,
a
A,
and
a
C
are the charge
transfer coefficients for the anodic and cathodic reactions,
respectively, n
e
is the number of electrons transferred and
j
is the current density (negative (positive) for a cathode (an-
ode)).
11
Here, a conformal planar catalyst layer is assumed
such that the current density is identical to that of the
photodiode
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2013 AIP Publishing LLC
103
, 143905-1
APPLIED PHYSICS LETTERS
103
, 143905 (2013)
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j
¼
j
0
;
cat
e
a
A
n
e
F
g
RT

e

a
C
n
e
F
g
RT

;
(4)
2 sinh e
a
n
e
F
g
RT

¼
e
a
n
e
F
g
RT

e

a
n
e
F
g
RT
:
(5)
Two analytical solutions for the overvoltage as a function of
current density are possible by imposing separate assump-
tions. Assuming that the charge transfer coefficients are
equal (
a
A
¼
a
C
¼
a
) affords the use of the identity in Eq.
(5)
,
which results in an analytical solution for the electrocatalytic
overvoltage as a function of current density (Eq.
(6)
)
g
¼
RT
a
n
e
F
sinh

1
j
2j
0
;
cat
!
;
(6)
Inserting Eqs.
(3)
and
(6)
into Eq.
(1)
yields an analytical
expression for the voltage as a function of current density for
a coupled PEC device with
a
A
¼
a
C
¼
a
and V
series
(
j
)
¼
j
R
s
where R
s
is the total area-normalized series resistance
V
ð
j
Þ¼
6
n
d
kT
q
ln
6
j
L

j
j
0
;
PV
þ
1

þ
RT
a
n
e
F
sinh

1
j
2j
0
;
cat
!
þ
j
R
s
:
(7)
For PEC systems where
a
A
¼
a
C
¼
a
is not valid, another solu-
tion is developed. Depending on the operating current density,
electrocatalytic overvolta
ges can develop such that the
Butler-Volmer equation can be approximated by the Tafel
equation (Eq.
(8)
). This approximation is accurate within 1% for
j
g
j
118 mV
n
e
or
j

j
0
;
cat
e

ð
118 mV
Þ
a
C
F
RT
(for a (photo)cathode)
j
¼
j
0
;
cat

e

a
C
n
e
F
g
RT

:
(8)
Under this assumption, the overvoltage as a function of cur-
rent density is given by Eq.
(9)
for a cathode,
g
¼
RT
a
C
n
e
F
ln

j
j
0
;
cat
!
:
(9)
Inserting Eqs.
(3)
and
(9)
into Eq.
(1)
yields another analyti-
cal expression, Eq.
(10)
, for the voltage as a function of
current density for a coupled PEC device operating at
j

j
0
;
cat
e

ð
118 mV
Þ
a
C
F
RT
,
V
ð
j
Þ¼
6
n
d
kT
q
ln
6
j
L

j
j
0
;
PV
þ
1


RT
a
C
n
e
F
ln

j
j
0
;
cat
!
þ
j
R
s
:
(10)
Note that Eq.
(10)
contains the charge transfer coefficient,
allowing analytical calculation for any value of
a
C
.
If neither of these assumptions is met in a particular de-
vice architecture (
a
A
¼
a
C
or
j
g
j
118 mV
n
e
), numerical solu-
tions for the voltage can readily be obtained for the
photodiode and electrocatalyst and applied to Eq.
(1)
,by
selecting voltages at equal operating current densities.
Discussion. To demonstrate the implications of the
derived explicit voltage-current density relations, a cathodic,
coupled PEC device was modeled.
12
Equation
(7)
was cho-
sen in favor of Eq.
(10)
because the latter is not valid
throughout the power-producing region of the device.
Figure
1(a)
shows the modeled structure and an electrical cir-
cuit diagram to demonstrate the series arrangement of a
coupled PEC device. For the case of the cathodic device,
photogenerated electrons travel to the electrocatalyst to drive
a reductive reaction, while photogenerated holes are col-
lected at the back contact. An auxiliary electrode (not
shown) connected to the back contact completes the circuit
by performing an oxidation reaction at a current that matches
the total current flowing through the coupled PEC cathode.
Figure
1(b)
compares the normalized current density (
j
/j
L
)
versus normalized voltage (V/V
OC
) behavior for an isolated
photodiode and a coupled PEC device in the absence of se-
ries resistance (R
s
¼
0).
Figures
2(a)
2(d)
illustrate the effect of an ideal PEC
system (R
s
¼
0
X
-cm
2
) through a series of band diagrams
that span the power-producing region. These band diagrams
are shown for a symmetric p-n junction, but Eq.
(7)
and the
following analysis are generally applicable to ideal photodio-
des of homojunction, heterojunction, liquid-junction, or
Schottky types. Figure
2(a)
shows the device at equilibrium
(in the dark) for reference. Figure
2(b)
illustrates the device
at open circuit, which is identical to that for an isolated pho-
todiode at open circuit because there is no current and thus
FIG. 1. (a) General schematic of
coupled
cathodic
photodiode-
electrocatalyst device. (b) Modeled
normalized current density versus nor-
malized voltage behavior of isolated
photodiode and coupled photodiode-
electrocatalyst devices. Equation
(7)
was used for the coupled curve, with
j
L
¼
35 mA-cm

2
,j
0,PV
¼
5

10

11
mA-cm

2
,j
0,cat
¼
0.25 mA-cm

2
,
n
e
¼
2, n
d
¼
1 and
a
¼
0.5 and R
s
¼
0
X
-cm
2
.
143905-2 Shaner, Fountaine, and Lewerenz
Appl. Phys. Lett.
103
, 143905 (2013)
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no catalytic overvoltage at the electrocatalyst-liquid inter-
face. Thus, the electrocatalyst remains in equilibrium with
the solution redox potential (
E
0
0
(Ox/Red)), while the back
contact of the photodiode is shifted to V
OC
vs
E
0
0
(Ox/Red).
As the potential is varied such that cathodic current is
allowed to flow through the device, the catalytic overvoltage
becomes nonzero and, accordingly, the electrocatalyst is no
longer in equilibrium with
E
o
0
(Ox/Red). The first distinct
feature of the coupled PEC device behavior compared with
the isolated photodiode (Figure
1(b)
) is an exponential
increase in current density near open circuit. This exponen-
tial “turn-on” originates from Butler-Volmer kinetics and is
indicative of an electrocatalytically limited system (Eq.
(4)
and Figure
2(b)
). The degree to which the system is electro-
catalytically limited is dependent on the exchange current
density and Tafel slope for the electrocatalyst, the fill factor
for the photodiode, and the total device current. Namely,
higher efficiency electrocatalysts (large exchange current
density, small Tafel slope) and less efficient photodiodes
(low fill factors) result in the most similar behavior between
the coupled PEC device and isolated photodiode because the
photodiode becomes the limiting factor for the electronic
performance. The second distinct feature is the downward
shift in the maximum power point from the isolated photo-
diode to the coupled PEC device. This shift is caused by the
introduction of a catalytic overvoltage at nonzero currents,
which results in an operating voltage that is the quasi-Fermi
level splitting of the photodiode less the catalytic overvolt-
age in accordance with Eq.
(1)
and depicted in Fig.
2(c)
. For
photodiode devices with high fill factors, like the one shown
in Fig.
1
, the maximum power point maintains a similar cur-
rent density to the isolated photodiode. However, for photo-
diode junctions with poor fill factors the catalytic
overvoltages can significantly lower the current density at
the maximum power point and, thus, further reduce the
efficiency.
Figure
2(d)
shows the device operating at short circuit
(
E
0
0
(Ox/Red)) where, in contrast to an isolated photodiode,
the photodiode in the coupled system maintains a certain
quasi-Fermi level splitting to offset the catalytic overvoltage
required to drive the electrocatalytic reaction. For the case
of a highly active catalyst coupled to an ideal diode
(Figure
1(b)
), the short circuit current value remains virtually
constant despite the overvoltage; however, a considerable
decrease in the quality of either component will result in a
drop in the short circuit current value. A poor catalyst can
have such a slow exponential turn-on to push the light-
limited current point to potentials more negative than short
circuit. Likewise, use of a low fill factor diode, implying
non-unity carrier collection at short circuit, means that the
addition of an overvoltage will further reduce the carrier col-
lection efficiency of the diode, resulting in a reduced short
circuit current. Many experimental results for transition
metal-oxide photoanode materials exhibit this phenomenon,
where the photocurrent continues to increase past
E
0
0
(Ox/Red).
13
,
14
To demonstrate its applicability, Eq.
(7)
has also been
applied to a recently developed high efficiency InP/InO
x
P
y
/
Rh coupled PEC system for hydrogen evolution. The fabri-
cated structure, as determined from TEM and AFM, is
shown in Fig.
3(a)
.
12
The photodiode is a heterojunction of
p-InP and n-InO
x
P
y
, where the photoactive layer is p-type
InP and the voltage is maximized by the use of a thin, highly
doped layer of n-InO
x
P
y
. Figure
3(b)
compares the modeled
and experimental current density versus voltage behavior.
Both curves exhibit the exponential current turn-on charac-
teristic that is demonstrative of an electrocatalytically lim-
ited system. However, the current density quickly becomes
linearly dependent on voltage resulting in a low fill factor
(

55%) that suggests the presence of a series resistance.
Thus, R
s
was floated in the model to obtain the best fit to the
experimental behavior, which resulted in R
s
¼
4
X
-cm
2
. The
close agreement between experiment and model throughout
the power-producing region demonstrates this analytical
models ability to accurately describe experimental data and
to extract experimental parameters that would otherwise not
be known without further experiments.
Analytical expressions were developed for a coupled
PEC device using the illuminated diode equation and the
Butler-Volmer equation, respectively. Specifically, equations
were presented for two limiting cases: (i) transfer coefficient,
a
, equal to 0.5 and (ii) use of the Tafel approximation to the
FIG. 2. Band diagrams for a coupled
device at (a) equilibrium, (b) open cir-
cuit, (c) maximum power point, and
(d) short circuit.
143905-3 Shaner, Fountaine, and Lewerenz
Appl. Phys. Lett.
103
, 143905 (2013)
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Butler-Volmer equation. The coupling of an electrocatalyst
to a photodiode was shown to have distinctly different char-
acter and performance than an isolated photodiode due to the
electrocatalytic overvoltage associated with driving a chemi-
cal reaction. This was shown through a series of band dia-
grams spanning the power-producing region. Finally, these
equations were applied to a recently reported coupled PEC
device structure to demonstrate the applicability.
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. K.F. is supported by the National Science
Foundation Graduate Research Fellowship under Grant No.
DGE-1144469. M.S. acknowledges the support of the Resnick
Sustainability Institute. The authors would like to thank Dr.
Nathan S. Lewis and Dr. Harry A. Atwater for advising during
this project and Dr. Shane Ardo for helpful discussions.
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FIG. 3. (a) Schematic of the p-InP/n-InO
x
P
y
film/Rh catalyst/electrolyte junction;
X
denotes the ohmic contact, the arrows indicate the origin and transport
of the excess minority carriers. (b) Experimental and modeled current density versus voltage behavior of hybrid InP/InOx/Rh photodiode-electroca
talyst de-
vice. Note that the photocurrent variations of the half cell result from hydrogen evolution. Equation
(7)
was used to model the coupled PEC device with
j
L
¼
35 mA/cm
2
,j
0,PV
¼
3

10

11
mA/cm
2
,j
0,cat
¼
0.25 mA/cm
2
,n
e
¼
2, n
d
¼
1, and
a
¼
0.5.
143905-4 Shaner, Fountaine, and Lewerenz
Appl. Phys. Lett.
103
, 143905 (2013)
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