Download Chemical capacitance of nanostructured semiconductors: its origin

Departament de Ciències Experimentals, Universitat Jaume I, 12080 Castelló, Spain.
E-mail: [email protected]; Fax: +34 964728066; Tel: +34 964728064
PCCP
Juan Bisquert*
COMMUNICATION
Chemical capacitance of nanostructured semiconductors: its origin
and significance for nanocomposite solar cells
Received 8th September 2003, Accepted 6th November 2003
First published as an Advance Article on the web 14th November 2003
The capacitance measured in dye-sensitised nanocrystalline TiO2
solar cells (DSSC) is interpreted in terms of a chemical capacitance, which is found to be a crucial feature for describing the
dynamic operation of solar cells based on nanoscaled materials.
Interpenetrated networks of nanoscaled materials provide the
opportunity to construct efficient solar cells from low purity
materials and versatile fabrication processes. Instead on relying on long diffusion length in high purity crystals, the nanocomposite solar cells base their ability to convert photon
energy to electricity on fast conversion of the photoexcitation
to energetic carriers spatially separated in distinct nanoscaled
phases, which enables their extraction.
In this paper we discuss the general principles of operation
of these solar cells. We start from the thermodynamic model
of chemical capacitance, and we show its special role in the
dynamic model of the solar cell in terms of equivalent circuits
for small modulated perturbation. We argue that the concept
of chemical capacitance is of critical importance for nanostructured semiconductor solar cells, because it describes the fundamental mechanism whereby photogenerated carriers store
free energy and produce a voltage and current in the external
circuit.
The DSSC1 is so far the most efficient nanocomposite heterogeneous solar cell. In DSSC, the carriers transferring the
chemical energy created in a photoexcited dye are electrons
in nanocrystalline TiO2 , and redox species in a liquid electrolyte. The capacitance of DSSCs can be determined in several
ways: electrochemical impedance spectroscopy (EIS),2–4 cyclic
voltammetry,5 or integrating the current at differential voltage
steps,6 and provides important information on the density of
states (DOS) and the electron density.
Normally, the concept of a capacitance relates to the textbook notion of an electrostatic geometric capacitance, determined by the electrical field between two metal plates with
equal amounts of opposite charges. But this description is
valid only insofar as the excess charge induced by the potential
difference is confined to a very small region of the surface of
the plates. Especially for mesoscopic systems, this is far from
being generally true. In general, the change of electrochemical
potential of electron reservoirs connected through leads to the
mesoscopic plates, causes both an accomodation of the field by
excess charges, and a variation of the Fermi level position with
respect to the conduction band in the plates. Consequently, the
more general electrochemical capacitance has been defined
for small conductors connected to macroscopic reservoirs.7
The basic result is that the DOS contributes a factor to the
capacitance given by
CDOS ¼ e2
5360
dN
dE
ð1Þ
where e is the absolute elementary charge and dN/dE is
the total DOS of the mesoscopic plates.7 When dN/dE is
very large, eqn. (1) can be neglected with respect to the geometric electrostatic capacitance, as in macroscopic metal plate
capacitors.7
This idea can be applied readily to the DSSC configuration,
as indicated in Fig. 1. Here, the two plates forming the capacitor are the whole TiO2 nanostructure connected to transparent
conducting oxide (TCO), and the whole electrolyte connected
to Pt electrode, as indicated in the scheme at the bottom of
Fig. 1. When the electrochemical potential of electrons, m
n ,
is varied at the TCO side (by applying a voltage
dV ¼ d
mn/e at the substrate) the subsequent electrical field
is not located at the contact between the extended plates,
because it is shielded at the TCO/TiO2 contact,8,9 as indicated
by the lower conduction band edge of TiO2 , Ec , being stationary in Fig. 1 (band unpinning is neglected here). Thus the basic
notion of the capacitor does not hold. In fact the modification
of the electrochemical potential at the substrate produces a
change of the chemical potential of electrons in the TiO2
nanostructure, dmn , with associated variation of both free electron density, dnc , and localized electron density in band gap
states, dnL . In this case the electrochemical capacitance is
dominated by a chemical capacitance, Cm , related only to the
Fig. 1 Electron energy diagram illustrating the behaviour of a nanostructured TiO2 electrode (shown in the bottom scheme) when a varian (Fermi level) is
tion dV of the electrochemical potential of electrons m
applied, assuming that conduction band energy (Ec) remains stationary
with respect to the redox level, Eredox . The voltage variation is
absorbed at the TCO/TiO2 electrolyte interface. Inside the TiO2
nanostructure, the Fermi level is displaced towards the conduction
band, i.e., the chemical potential of electrons varies, dmn , what causes
a change of occupancy both of conduction band, dnc , and trapped
electrons in localized levels, dnL (shaded region of the bandgap).
Phys. Chem. Chem. Phys., 2003, 5, 5360–5364
This journal is # The Owner Societies 2003
DOI: 10.1039/b310907k
variation of the electron chemical potential in the TiO2
‘‘ plate ’’ of the capacitor. In conclusion the sytem of Fig. 1 is
a macroscopic chemical capacitor constituted by mesoscopic
units.
Another, independent but equivalent formulation of the chemical capacitance is obtained from general thermodynamic
considerations that define the capacitor as an ideal element
that stores energy by virtue of a generalized displacement.10,11
If the volume element stores chemical energy due to a thermodynamic displacement, the chemical capacitance per unit
volume is
Cm ¼ e2
@Ni
@mi
ð2Þ
This is indeed another version of eqn. (1), but here the chemical capacitance is defined locally, in a small volume element,
and in this interpretation the capacitance reflects the capability
of a system to accept or release additional carriers with density
Ni due to a change in their chemical potential, mi .10,11 From
eqn. (2), in the case of an ideal mixture for which the chemical
potential of the ith component is mi ¼ m0i + kBT ln Ni , where
kB is Boltzmann ’s constant and T the temperature, we get
CmðiÞ ¼ e2
Ni
kB T
ð3Þ
In this case energy is stored reversibly in the chemical capacitor
as a change in entropy.
In the same way, using the Boltzmann distribution function
for conduction band electrons,
nc ¼ N c eðmmn E c Þ=kB T
ð4Þ
decade1 have been interpreted in terms of the trapped electrons in the exponential distribution described in eqn. (8),3,5
which greatly exceeds the number density of free electrons in
the conduction band.
As an example, in Fig. 2(a) we show the measurement of the
capacitance of nanostructured TiO2 electrodes in aqueous
solution at pH 3, in conditions in which charge transfer to
the electrolyte is very low. By using a model fitting,5 the ohmic
potential drop can be removed in order to extract the variation
of capacitance with the Fermi level position with respect to the
conduction band. The results, shown in Fig. 2(b), reveal a
chemical capacitance dominated by the exponential distribution
of bandgap states as in eqn. (9), with a value of a ¼ 0.25. This
interpretation of Cm(V) is confirmed by other techniques such
as a stepping charge-extraction method.12
The expressions in eqns. (3) and (5) for a dilute species are
quite familiar in solid-state electrochemistry.11,13 For semiconductors, the meaning of the capacitance in eqn. (5) as a charge
storage element is clearly recognized by Sah who denoted it the
electron charge storage capacitance.14
On another hand, many semiconductor device books15 call
eqn. (5) the diffusion capacitance (we have also used eventually
this denomination in the past16). The reason for this is indicated in the scheme of Fig. 3. When two adjacent volume
elements are at a different chemical potential value, a diffusion
process follows; this is an irreversible loss that attempts to
equalize the charge in the chemical capacitors. Therefore, the
diffusion process always requires the chemical capacitors,
and these indeed appear in the well known equivalent circuit
for diffusion11,16,17 (the transmission line composed of the
repetition of the arrangement in Fig. 3). But the converse is
not true. While the extended chemical capacitor does indeed
where Nc is the effective density of conduction band states, the
expression
nc
ð5Þ
CmðcbÞ ¼ e2
kB T
is obtained from eqn. (2). The total chemical capacitance of the
system in Fig. 1, is given by the integral of eqn. (5) throughout
the volume of the nanostructured film.
Considering the variation of both free and localized
electrons indicated in Fig. 1 we obtain for the chemical
capacitance of electrons in the DSSC
Cm ¼ e2
@ðnc þ nL Þ
¼ CmðcbÞ þ CmðtrapÞ
@mn
ð6Þ
where
CmðtrapÞ ¼ e2
@nL
¼ e2 gðm
n Þ
@mn
ð7Þ
Here g(E) is the density of localized states in the bandgap, at
the energy E, as in eqn. (1). For the exponential distribution
with tailing parameter Tc (with a ¼ T/Tc)
gðE Þ ¼
NL
exp½ðE Ec Þ=kB Tc kB Tc
ð8Þ
the chemical capacitance in eqn. (7) can be expressed as
CmðtrapÞ ¼ e2
aNL
n Ec Þ=kB T
exp½aðm
kB T
ð9Þ
In conclusion, eqns. (5) and (9) show that the chemical capacitance in nanostructured TiO2 is characterized by an exponential dependence on the bias potential. The electrochemical
capacitance has been determined experimentally many times
in nanostructured semiconductors and DSSCs, see e.g. refs.
3,5, and the exponential dependence is normally found. However the ideal statistics of eqn. (5), with a slope (e/
kBT )1 ¼ 0.026 V decade1 for (d ln C/dV)1, is not observed.
The experimental results of about (d ln C/dV)1 ¼ 0.100 V
Fig. 2 Chemical capacitance of a 2.6 mm thick film of anatase TiO2
nanoparticles (12 nm radius) in aqueous solution (pH 3) determined
from cyclic voltammetry at scan rate s ¼ 0.2 V s1. (a) The capacitance
as obtained from the measured current (Cm ¼ I/(dV/dt) ¼ I/s), as a
function of potential (vs. Ag/AgCl reference electrode), and model fitting.5 The curved region at less cathodic potentials is due to a deep
monoenergetic surface state that is neglected in the model. (b) Chemical capacitance vs. electron Fermi level, after subtraction of the ohmic
drop IR in the potential axis in (a).
Phys. Chem. Chem. Phys., 2003, 5, 5360–5364
5361
We linearize eqn. (4) and obtain the relationship
en
rn ¼ j
kB T n
ð12Þ
Multiplying eqn. (11) by e and substituting eqn. (12), we obtain
Fig. 3 Representation of the diffusion process: two adjacent chemical
capacitors cm at different chemical potentials jn(x) transfer carriers by
diffusion current i(x) through the transport (diffusion) resistance rt .
require some degree of conduction for the charges to equilibrate into it, the chemical capacitance itself is a reversible
quantity not related to the transport process. Therefore, the
denomination of diffusion capacitance, adopted recently in
some papers in the DSSC area,18,19 is not very fortunate,
and the concept of the chemical capacitance7,11 should be
prefereable, especially for solar cell theory.
To emphasize the significance of the chemical capacitance in
the process of photon energy conversion in a solar cell, let us
consider a simple model for the absorber material that forms
the heart of any solar cell. In general the light absorber can
be a single molecule, a semiconductor crystal network, or an
organic polymer. For convenience we consider the process of
excitation of a homogeneous slab of p-doped silicon crystal
as indicated in Fig. 4. We choose this example because it contains no internal interfaces, so there are no possible electrostatic capacitors in the bulk of this system (indeed, there are
no electrical field variations, except if global space charge is
formed, which can be ignored in this model), and the effect
of the chemical capacitance becomes unmistakeable.
The absorption of sunlight, Fig. 4(b), has the effect of
increasing the concentration of electrons in the conduction
band with respect to the thermal equilibrium situation, Fig.
4(a). The concentration of holes in the valence band is not
changed significantly. Therefore sunlight irradiation, with generation rate G, causes the separation of the quasi-Fermi levels
for electrons and holes, fn and fp , that are indicated in units
of electrical potential (fn ¼ mn/e), Fig. 4(b). The electron
density n is governed by the conservation equation,
dn
n
¼G
dt
t
ð10Þ
where t is the lifetime.
We analyse the behavior of this system under a small perturbation of some physical variable. Let n, fn and G denote the
steady state values, and rn , jn and g the corresponding small
perturbation values, so that the concentration is n + rn(t), and
the quasi-Fermi level is fn + jn(t). Since the steady-state
values are time-independent, it follows from eqn. (10) that
drn
r
¼g n
dt
t
Phys. Chem. Chem. Phys., 2003, 5, 5360–5364
ð13Þ
Eqn. (13) takes the form of current conservation equation (per
unit volume). So eqn. (13) is identical with Kirchhoff ’s rule and
can be expressed as an electrical circuit that describes the relationship of small perturbation voltages, jn , and currents, i.
We remark that these ‘‘ voltages ’’ are the local values of electrochemical potential. Further, in the model of Fig. 4 variations of jn correspond exclusivelly to modification of the
chemical potential. Note also that the ‘‘ currents ’’ iph , irec do
not represent transport currents, but the rates of creation
and destruction of conduction band electrons.
The first term in eqn. (13) can be recognized as a capacitive
current (I ¼ CdV/dt). The capacitance per unit volume is
obviously the chemical capacitance, Cm , given in eqn. (5).
The second term in eqn. (13) is a current generator iph ¼ eg
that stands for the excitation process. The third one, when
written jn/Rrec , defines the recombination resistance,
Rrec ¼ kBTt/ne2. More exactly, the steady state value jnss ¼ 0
0 is taken as the baseline for potentials jn , so that the small
perturbation value of the recombination current is irec ¼
Rrec1(jn jnss). In summary, the circuit in Fig. 4(c) is an
exact representation of the processes described in eqn. (13).
More general equivalent circuits for electrons and holes and
deffect-assisted recombination have been derived by Sah.14
Let us discuss the process of photon energy conversion in
terms of this circuit. Switching on the illumination immediately
charges the chemical capacitor, so that the photon energy
has been converted to chemical energy. In this way free energy
(corresponding to the splitting of Fermi levels) is available in
the form of energetic carriers by virtue of the excitation process.
For the solar cell operation it must also be possible to convert the chemical energy to electrical power. This process of
‘‘ charge separation ’’ has been discussed elsewhere in terms
of selective contacts to the absorber.20 This is indicated in
Fig. 5 for DSSC. The local circuit obtained in Fig. 4(c) for
the absorber material, is combined with the appropriate contacts to form a solar cell. Note that each wire unites one side
of the absorber (i.e. either of the carriers with separated Fermi
levels) to one and only one metal electrode. The overall asymmetric configuration of the circuit, with the different lines of
ð11Þ
Fig. 4 Light absorber material (p-doped semiconductor) in dark
equilibrium (a) and under photon irradiation (b). The increase of excitation rate in (b) promotes an increase of the population of electrons in
the conduction band (Ec) and the separation of Fermi levels Df for
electrons (fn) and holes (fp). (c) The equivalent circuit describing
the response of the system to a small transient or periodic perturbation
of the light intensity.
5362
ne2 djn
ne2
þ eg þ
j ¼0
kB T dt
tkB T n
Fig. 5 Scheme of the structure of a DSSC. The lower part shows the
equivalent circuit for a small transient or periodic perturbation of the
light intensity. The transverse elements represent the three main processes occurring in the solar cell: A photocurrent generator iph stands
for excitation of the dye from ground to high energy level; the chemical
capacitance cm accounts for the energy storage by virtue of carrier
injection; and the resistance rct indicates the charge transfer events
from the phase of high to the phase of low Fermi level. The upper line
shows a diffusion resistance rt along the nanoporous TiO2 , and the
lower line represents the resistanceless contact from the low energy
state of the absorber to the Pt (redox electrolyte).
selective contacts, converts the excess, photo-induced charge in
the chemical capacitor to an output (photo)voltage and
(photo)current.
With the insight obtained from the equivalent circuit model,
we can appreciate that the macroscopic chemical capacitor of
Fig. 1 incorporates also the selective contacts, so that it
converts excess chemical energy of electrons in the TiO2 nanoparticles (with respect to electrons in redox ions), to the
electrochemical potential difference of carriers in outer leads.
One can conclude that the capacitor shown in Fig. 1 is indeed
the central structure of the solar cell.
Returning to Fig. 4(c) we can discuss the different processes
involved in photon energy conversion. The complete structure
of the solar cell is an electrochemical capacitor shorted by two
elements: The current generator (excitation process) transfers
charge from one plate to the other increasing the electrochemical potential difference. In DSSC this is realized by the dye
molecules absorbed in the TiO2 surface. And a leakage,21
recombination resistance, tends to reduce the electrochemical
potential difference. Both the current generator and the leakage resistance for radiative recombination correspond to the
same physical process occurring in inverse directions in time.
In practice, recombination occurs through parallel radiationless pathways of lower resistance. Preferentially the leakage
between the plates should be reduced, but it cannot be
removed beyond the radiative recombination limit. A major
strategy for improving the efficiency of nanocomposite solar
cells is the treatment of the interface between nanoscaled components for increasing the recombination resistance, while
maintaining fast separation of the excitation created by light.
We remark also that the leakage element provides the electronic connectivity between plates that is required for achieving the thermodynamic equilibrium in the dark, with equal
Fermi levels in the two plates (at Voc ¼ 0), regardless of
kinetics.
It is often argued that a diffusion process of the separate carriers is responsible for the photovoltaic effect. However, it is
clear that diffusion is a purely irreversible process that cannot
bring to bear a free energy gain. This paradox is resolved when
we note that diffusion is important only for charge equalization into one plate of the macroscopic chemical capacitor. So
diffusion is not a component of the fundamental structure of
the solar cell shown in Fig. 4(c), but it allows to make this
structure spatially extended, as indicated in Fig. 5. This is
an important point for harvesting a substantial amount of
incident photons.
We note also that the diode characteristic of solar cells is
usually attributed to a diffusion process, following the model
of Shockley for the p-n junction. However, the crucial effect
governing the diode characteristic is the recombination of
the injected carriers, and not their diffusion. Indeed, the diode
characteristic is obtained directly from the leaky capacitor
model of Fig. 4. The leakage current increases exponentially
as the Fermi level of electrons approaches the conduction band
edge, as follows from the expression Rrec ¼ kBTt/ne2 and eqn.
(4). In reverse, Rrec tends to the equilibrium excitation value.
In addition, very often it is thought that electron-hole
separation at an electrical field is the primary act of photovoltaic energy conversion. However, the electrostatic capacitor
does not describe directly chemical energy (excess carriers) storage. Furthermore, in electrostatic capacitor change of electrical field and change of carrier density occur at the same place.
The chemical capacitance concept establishes that the locus of
energy storage by charge accumulation is not generally linked
to the locus of electrical field variation.
Why has the significance of the chemical capacitance not
been generally recognized so far in solar cell theory? This is
probably because device modelling in the dominant silicon
solar cells usually relies on steady-state measurements
(current-voltage curves) where the chemical capacitor cannot
possibly appear. In contrast, in the DSSC area small perturbation light-modulated tecniques became popular rather early. In
consequence the chemical capacitance of eqn. (5) was identified
as a conduction band capacitance22,23 and the global circuit of
Fig. 5 (or at least the analytic version of it) was derived.24 In
fact the circuits in Figs. 4(c) and 5 provide the transfer function
for different modulated techniques (IMVS, IMPS,22,24
impedance spectroscopy).
Besides the chemical capacitance, the system of Fig. 1 contains normal geometric capacitance components. The substrate
capacitances, at the TCO/TiO2 and TCO/electrolyte interface,
have been studied in previous work;25 these in fact are predominant when the electron density in TiO2 is low, but can be
neglected thereafter.4 The capacitance at the TiO2/electrolyte
interface, which is associated to band unpinning, is also geometric-electrostatic. In DSSC the redox process at the
adsorbed dye molecules contributes another component to
the chemical capacitance. The chemical capacitances of nanostructured TiO2 and photoactive absorbed (viologen) molecules can be discriminated experimentally using a combination
of electrochemical and electro-optical techniques.26
In conclusion, we argued that the fundamental structure of a
solar cell is a leaky electrochemical capacitor. The solar cell is
composed of two plates that maintain carriers with different
values of the electrochemical potentials (Figs. 1 and 5). Excitation is a charging of the electrochemical capacitor. Charge
separation relies on a good connectivity into each of the
macroscopic plates. Recombination is a leakage between the
plates that should be minimized but cannot be suppressed.
The electrochemical capacitor automatically describes the fundamental feature of selectivity of contacts in a solar cell.20,27
These conclusions for DSSCs can be probably extended to
other nanocomposite photovoltaic converters such as the
organic solar cells consisting on a blend of conjugated
polymers and fullerene molecules. Indeed, the concept of the
chemical capacitor enables to formulate quantitative models
for solar cell characterization by small perturbation techniques, and this will be shown more extensively in subsequent
publications.
Acknowledgements
This work was supported by Fundació Caixa Castelló under
project P11B2002-39.
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