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An expression for the
bridge-mediated electron
transfer rate in dye-sensitized
solar cells
rsta.royalsocietypublishing.org
Research
Cite this article: Maggio E, Troisi A. 2014
An expression for the bridge-mediated
electron transfer rate in dye-sensitized solar
cells. Phil. Trans. R. Soc. A 372: 20130011.
http://dx.doi.org/10.1098/rsta.2013.0011
One contribution of 8 to a Theo Murphy
Meeting Issue ‘Theo Murphy International
Scientific Meeting between the UK and China
on the chemistry and physics of functional
materials’.
Subject Areas:
physical chemistry, computational chemistry,
solid state physics
Keywords:
electron transfer, interface, tunnelling,
dye-sensitized solar cell
Author for correspondence:
Alessandro Troisi
e-mail: a.troisi@warwick.ac.uk
Emanuele Maggio and Alessandro Troisi
Department of Chemistry, and Centre for Scientific Computing,
University of Warwick, Coventry CV4 7AL, UK
We have derived an expression for the rate of
electron transfer between a semiconductor and a
redox centre connected to the semiconductor via a
molecular bridge. This model is particularly useful to
study the charge recombination (CR) process in dyesensitized solar cells, where the dye is often connected
to the semiconductor by a conjugated bridge. This
formalism, designed to be coupled with density
functional theory electronic structure calculations, can
be used to explore the effect of changing the bridge on
the rate of interfacial electron transfer. As an example,
we have evaluated the CR rate for a series of systems
that differ in the bridge length.
1. Introduction
Dye-sensitized solar cells (DSSCs) may become an
economically viable alternative to commercial solar cells
based on different architectures if their efficiency can be
increased beyond what is currently achievable [1]. The
innovative element in DSSCs is the photoactive electrode,
made of sintered nanoparticles of semiconducting
material, TiO2 in the original formulation [2]. These
nanoparticles are in contact with an electrolyte solution
and are sensitized to visible light by the adsorption
of a dye, either an inorganic complex or an organic
molecule [3–5]. Upon light irradiation, the molecule is
promoted to an excited state and injects an electron into
the TiO2 conduction band, generating a photocurrent. An
unwanted loss is caused by the charge recombination
(CR) reaction, when the dye cation resulting from the
charge injection (CI) process is neutralized by an electron
in the semiconductor conduction band [6].
2014 The Author(s) Published by the Royal Society. All rights reserved.
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For specificity, we consider the theory of CR rate in DSSCs, i.e. the process of neutralization of the
oxidized dye by electron transfer from the conduction band of the semiconductor. This derivation
follows that proposed in standard textbooks [32], but is required to provide the notation necessary
to describe the charge transfer mediated by a molecular bridge, which is the new equation
derived in this work. The initial state of this electron transfer process is an electronic state
where an electron is present in the semiconductor’s conduction band manifold {l}, and the dye is
positively charged. The manifold {v} collects all vibrational states of the system. These vibrations
are localized on the dye or the solvent and are not affected by the specific state l occupied in the
.........................................................
2. General expression for the charge recombination reaction rate
2
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
Many organic dyes are composed of a light-harvesting unit (a chromophore) connected to
the TiO2 surface via a molecular bridge terminated with an anchoring group [7]. The coupling
between the chromophore and the semiconductor is mediated by the bridge fragment, and these
systems can be considered analogous to the molecular donor–bridge–acceptor (DBA) systems
studied in the photochemistry literature [8]. In this paper, we express the rate of CI and CR
for chromophores connected to the semiconductor through a bridge by combining the theories
developed to study DBA systems with the theories of electrochemical reactions and transport in
molecular junctions.
The optimization of a DSSC is a very challenging task: the modification of one material within
the device is bound to impact upon different microscopic processes taking place [9]. For this
reason, the only systematic pathway towards more efficient cells is the separate optimization of
the devices’ individual components. This approach has been gaining momentum recently, with
studies focused, for example, on the dye’s anchoring group [10,11], where a computational study
has been performed to predict the relative stability of the attachment and the efficiency of charge
transfer across these anchors [12]. The design of new synthetic procedures for TiO2 nanoparticles
exposing different crystallographic faces [13] has added another tool for the engineering of
interfacial electron transfer, with the CI process being noticeably affected by surface morphology
[14–16]. In this respect, being able to express the rate for the interfacial charge-transfer process
solely in terms of the properties of the dye, the bridge, the anchoring group and the interaction
between these with the semiconductor surface is potentially very beneficial.
The bridge between the light-harvesting component of the dye and the semiconductor is one
of the parameters that can potentially be optimized. In DBA systems, extensive studies have been
carried out on the possibility of using different bridges to modulate the charge-transfer rate by
changing the length of the bridge or the positions of its orbital levels [17–19]. Another parameter
that can be controlled is the coupling between the bridge and the dye orbitals. We have recently
shown that, by considering dyes of particular symmetry, it is possible to design a semiconductor–
bridge–dye system with an extremely low rate of CR [20]. In DSSC, the CI rate is the fastest
process, occurring with quantum efficiency close to one. A linear bridge typically controls both
the CI and the CR rates. Being able to systematically reduce the two rates can be beneficial for
the solar cell as long as the quantum efficiency of CI remains very high. With a slower CR, the
efficiency of the cell can increase, because the loss channel of CR is less active [21,22], and a larger
open-circuit voltage can be obtained [23], allowing the use of a redox mediator with more negative
electrochemical potential to regenerate the longer-lived dye cation.
In this paper, we present the theory of CR mediated by a bridge. Several theoretical approaches
have been developed for the related problem of transmission through a molecular bridge
[24,25]; these include spin-boson models [26], quantum dynamics [27,28], scattering formalism
[29,30] and Löwdin partitioning techniques [31]. The latter is similar, in principle, to the approach
presented herein; however, the application to DSSCs has not been previously investigated. The
theory will first be presented for a model Hamiltonian and then be generalized for a system whose
parameters are obtained from an electronic structure calculation with a localized basis set. A few
realistic examples are then considered.
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(a)
(b)
3
l
m
~
V
{w}
m
nuclear displacement
Figure 1. (a) Schematic of the electronic initial states {l}, forming a continuum band, and the final discrete state m.
(b) Representation of the vibronic levels for one of the initial electronic states l and the final state m.
semiconductor. The initial vibronic states can therefore be indicated as {|l, v}. We denote with
m the final electronic state where there is no excess electron in the semiconductor’s conduction
band and a neutral dye is present. When the system is in state m, the equilibrium position of the
vibration on the dye/solvent is changed, and this set of vibrational states after the CR is indicated
with {w}, with the final vibronic states indicated as {|m, w}. The energy of each vibronic state can
be expressed as the sum of electronic and vibrational energy (El,v = εl + Ev , Em,w = εm + Ew ). A
schematic of the energy diagram is shown in figure 1, where, for the sake of simplicity, we show
only one normal vibrational mode.
The total Hamiltonian for the system is given by
H = H0 + Ṽ,
(εl + Ev )|l, vl, v| + (εm + Ew )|m, wm, w|
H0 =
l
and
Ṽ =
Ṽlm v|w|l, vm, w| + h.c.,
(2.1a)
(2.1b)
(2.1c)
l
where H0 represents the Hamiltonian where there is no interaction between initial and final
states. The coupling Ṽ stands for an electronic coupling between states l and m, and the form
used in equation (2.1c) implies that the Condon approximation has been used, i.e. l, v|Ṽ|m, w =
v|wl|Ṽ|v. The coupling Ṽ may indicate a direct coupling between states l and m but also
an effective coupling due to the coupling of states l and m to a bridge that connects them. The
derivation in this section is indifferent to the origin of the coupling Ṽ, which is discussed instead
in §3.
The total rate for the CR, i.e. for all transitions {l, v} → m, {w}, is the sum over the possible initial
and final states weighted by the occupation of the initial states,
Pμ,T (εl )PT (Ev )
kl,v→m,w .
(2.2)
k{l,v}→m,{w} =
l,v
w
Here Pμ,T (εl ) is the probability of occupation of a state in the semiconductor with energy εl , a
function of the chemical potential μ in the semiconductor and the temperature T; and PT (Ev ) is
the probability that the vibrational level with energy Ev is occupied. With the Hamiltonian defined
above,
we have
⎫
2π
⎪
⎪
|l, v|Ṽ|m, w|2 δ(Em,w − El,v )
kl,v→m,w =
⎬
h̄
(2.3)
⎪
2π
⎪
|v|w|2 |Ṽlm |2 δ(εm − εl + Ew − Ev )⎭
=
h̄
.........................................................
{v}
{l}
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
E
e
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and the total rate is
4
The second summation in equation (2.4) is known as the thermally and Franck–Condon averaged
density of vibrational states:
F(εm − εl ) =
v,w
PT (Ev )|v|w|2 δ(εm − εl + Ew − Ev ).
(2.5)
The expression for the total rate can then be rewritten as
⎫
2π ⎪
⎪
Pμ,T (εl )|Ṽlm |2 F(εm − εl )
⎪
⎪
⎪
h̄
⎪
l
⎪
⎪
⎪
⎪
2π ⎪
2
⎪
=
dE fμ (E)|Ṽlm | F(εm − E)δ(E − εl ) ⎪
⎪
⎬
h̄
l
⎪
⎪
2π ⎪
= dE fμ (E)
|Ṽlm |2 δ(E − εl ) F(εm − E)⎪
⎪
⎪
⎪
h̄
⎪
⎪
l
⎪
⎪
⎪
⎪
⎪
˜
⎭
= dE fμ (E)Γ (E)F(εm − E).
k{l,v}→m,{w} =
(2.6)
The probability of an electronic level being occupied is given by the product of the density of
states at that energy with the Fermi–Dirac distribution fμ (E) (where the temperature dependence
is understood). The spectral density Γ˜ (E) has been defined as
Γ˜ (E) =
2π |Ṽlm |2 δ(E − εl ).
h̄
(2.7)
l
This quantity can be considered a measure of the facility of electron transfer between state l and
state m at energy E and in the absence of nuclear modes. It would correspond exactly to the
lifetime of a state prepared initially in m and degenerate with the levels {l}. This latter situation
was investigated in reference [33].
The Franck–Condon term F can be evaluated analytically if the vibrational wave functions
are assumed to be displaced harmonic oscillators. In the limit of high temperature, when
these oscillators can be treated classically, the Franck–Condon term takes a particularly simple
form [34]:
F(x) = √
(x + λ)2
1
exp −
,
4λkB T
4π λkB T
(2.8)
where λ represents the reorganization energy (a measure of the geometry change between states
l and m) and kB T is the thermal energy.
3. Bridge-mediated semiconductor–dye electronic coupling
In §2, we have considered the states {l} and m as directly coupled by an effective coupling Ṽ. Here,
we consider more specifically the situation of semiconductor states coupled directly with the dye
or through states localized on the bridge. The total electronic Hamiltonian can therefore be written
as the sum of the electronic Hamiltonians on the semiconductor (HSel ), the bridge (HBel ) and the
.........................................................
(2.4)
l
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
2π Pμ,T (εl )|Ṽlm |2
PT (Ev )|v|w|2 δ(εm − εl + Ew − Ev ).
k{l,v}→m,{w} =
h̄
v,w
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(a)
e
5
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rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
{l}
{b}
tlb
kbm
m
(b)
Figure 2. (a) Schematic of the energy levels of the semiconductor, the bridge and the molecule and their coupling. (b) Physical
partition of the systems into semiconductor, bridge and adsorbed molecule, which approximately corresponds to the partition
of the Hamiltonian. (Online version in colour.)
el ) plus the interactions between these three subsystems (V el = V el + V el + V el ),
molecule (HM
SB
SM
BM
i.e.
el
el
el
el
+ VSB
+ VSM
+ VBM
,
Hel = HSel + HBel + HM
el
HSel =
εl |ll|, HBel =
εb |bb| +
Vbb |bb |, HM
= εm |mm|
l∈L
and
el
VSM
=
b∈B
l∈L
γlm |lm|,
el
VSB
=
b ∈B
b =b
τlb |lb|,
l,b
el
VBM
=
κbm |bm|.
(3.1a)
(3.1b)
(3.1c)
b∈B
A graphical representation of this system is given in figure 2.
A number of techniques (perturbation theory [35], scattering theory [29], partitioning methods
[36], Green’s function methods [37–39]) have been used to express the effective coupling between
subsystems S and M in an effective Hamiltonian that does not contain the bridge states
explicitly, i.e.
el
el
+ ṼSM
.
Hel = HSel + HM
(3.2)
Here, we consider the effective coupling operator expressed from scattering theory as
Ṽ = V el + V el GV el ,
(3.3)
where the retarded Green’s function operator can be defined as G = ((E + iη)I − Hel )−1 with I
being the identity operator, E an independent variable and η a real positive infinitesimal. The
effective coupling between states l and m is
l|V el |bb|G|b b |V el |m.
(3.4a)
l|Ṽ|m = l|V el |m +
b,b ∈B
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where the indices run over the bridge states, and the Green’s function weighted by the overlap
matrices has been indicated with the symbol gbb to lighten the notation. The next step is to
compute the spectral density Γ˜ (E) using the effective coupling expression above.
In the absence of a bridge (or if we assume all states of the bridge as part of the dye), we have
Ṽlm = γlm and the spectral density Γ˜ (E) can be expressed as [43]
Γ˜ (E) =
2π |γml |2 δ(E − εl ).
h̄
(3.5)
l
In the presence of the bridge, the direct coupling is very small (because it decreases exponentially
with distance) and so it is reasonable to neglect the direct coupling γlm , i.e.
τlb gbb (E)κb m .
(3.6)
Ṽlm =
b,b ∈B
The modulus squared of the coupling is given by
†
=
τlb gbb (E)κb m
κa† m g†a a (E) τal†
Ṽlm Ṽml
(3.7)
a,a ∈B
b,b ∈B
and the spectral density reads
Γ˜ (E) =
⎫
2π †
†
†
⎪
τlb gbb (E)κb m
κma
ga a (E) τal δ(E − εl )⎪
⎪
⎪
h̄
⎬
a,a ∈B
l b,b ∈B
=
a,a ∈B b,b ∈B
Γab gbb (E)κb m κa† m g†a a (E),
⎪
⎪
⎪
⎪
⎭
(3.8)
where, for the last equality, we have introduced the semiconductor’s spectral density
Γab (E) =
2π †
τal τlb δ(E − εl ).
h̄
(3.9)
l
By further defining Ka b = κb m κa† m we obtain
Γ˜ (E) =
Γab gbb (E)Kb a g†a a (E) = tr{Γ gKg† },
(3.10)
a,a ∈B b,b ∈B
where tr{·} denotes the trace operator over the bridge states and the bold type has been used for
the matrices involved (whose size is the total number of bridge states present).
Equation (3.10) is one of the main results of this work. This is the form of the spectral density
that should be used in the expression for the rate equation (2.6) when the coupling between
.........................................................
b,b
a,b,c,d
6
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The first term on the right-hand side of equation (3.4a) takes care of the direct coupling
between the initial and final electronic states. The second term accounts for the bridge-mediated
contribution to the electron transfer process, with the tunnelling probability across the bridge
being given by the Green’s function matrix elements.
The Green’s function operator for a bridge interacting with the semiconductor slab and
the molecular fragment attached to it can be recast in terms of the Green’s function for the
non-interacting bridge plus a self-energy contribution that collects the perturbative effects on
the bridge subsystem due to the interaction [40]; an expression for the self-energy has been
reported elsewhere [41]. Because the introduction of the self-energy in the calculation of the
bridge’s Green’s function would further complicate the theory without impacting on the results
that will be presented (consistently with the weak semiconductor–bridge coupling regime), we
approximate it as GB (E) ≈ G0B (E) where GB is the matrix whose elements appear in equation (3.4a).
For the case of a non-orthogonal basis set, the Green’s function for the non-interacting bridge can
−1 el −1 −1
[42], where SB is the overlap matrix over the
be computed as G0B = ((E + iη)S−1
B − SB HB SB )
bridge states. We can then compute the effective coupling by
0 −1
τla S−1
τlb gbb κb m ,
(3.4b)
Ṽlm = γlm +
ab Gbc Scd κdm ≡ γlm +
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The bridge not only mediates the coupling between chromophore and semiconductor, but also
produces effects of quantum interference. If more than one orbital localized on the bridge is
coupled to the electrode, then different tunnelling pathways are possible and they may interfere
constructively or destructively [45]. Destructive interference gives rise to very low dips in the
function Γ˜ (E) at special energies that are known as antiresonances. Energies where Γ˜ (E) is
maximum are known as resonances and occur when the energy is degenerate with one of the
bridge orbital levels [46].
4. Charge recombination rate calculations for realistic dyes
To exemplify the methodology, we compute for example the CR rate for the series of dyes reported
in figure 3 with bridges of different length. In these dyes, the coupling between bridge and
chromophore is particularly weak, as described in detail in [20], and the partitioning method
proposed above should be particularly suitable. The length of the bridge will allow us to disregard
the direct coupling contribution.
The electronic structure calculation for the evaluation of the spectral density in equation (3.10)
proceeds in two steps: using the SIESTA code [47], we have simulated independently (i) the
isolated TiO2 slab, and (ii) the whole system semiconductor plus dye. The anatase (101) surface
was modelled using a 3 × 3 surface supercell containing 108 atoms and four atomic layers.
Geometry relaxation was performed while keeping the coordinates of the bottom atomic layer
fixed at the values of the bulk phase. The Brillouin zone was then sampled in 48 k-points (this
density of the k-point grid was obtained by setting the cut-off parameter in SIESTA to 30 Å). The
electronic structure was computed with a GGA–PBE exchange and correlation functional with
double-zeta basis set and Troullier–Martins pseudo-potential. A similar geometry optimization
was also carried out in the presence of formic acid in order to establish the adsorption geometry
of the dye’s anchoring group. The rest of the dye molecule was attached to the carboxylic group
preserving the relative orientation. Single point energy calculations were then performed at the
same level of theory but including only the Γ -point on the slab plus adsorbate system shown
in figure 2 in order to obtain the matrix elements of the full Hamiltonian operator equation
(3.1a). This last calculation was performed at the Γ -point only, because we are not considering a
periodic array of dyes interacting with each other, but rather only one dye adsorbed on a periodic
semiconductor surface.
.........................................................
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7
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
semiconductor and dye is mediated by a bridge. The spectral density Γ˜ (E) includes a matrix
Γ (that quantifies the coupling between semiconductor and bridge), a matrix K (that quantifies
the coupling between bridge and dye) and the Green’s function matrix g (that expresses the
propagation across the bridge). As we mentioned in §2, Γ˜ (E) would also be the rate of CI
(tunnelling) of a state initially prepared in m and degenerate with the manifold {l}. It is therefore
particularly appealing that the form obtained for Γ˜ (E) is very similar to the standard Landauer
formula for the transmittance across a molecular system connected to two electrodes [44]. In
the Landauer formula, instead of the matrix K, there is another term such as Γ , describing the
coupling with a second electrode.
The aim of this paper is to produce a formalism that can be applied in conjunction with
electronic structure computations of the system of interest. The semiconductor–molecule interface
can be routinely simulated, and, for typical chemical applications, the Hamiltonian and overlap
matrices are expressed in terms of a localized, non-orthogonal basis set, such as a linear
combination of atomic orbitals. Therefore, if we consider the chromophore unit, we can identify
the state |m with the isolated molecule’s highest occupied molecular orbital (HOMO), which can
be expanded in terms of the localized basis set as: |m = cm
j |φj . The coupling between bridge
states and state |m can be expressed in terms of these localized states:
el
el
|m =
cm
(3.11)
κbm = b|VBM
j b|VBM |φj .
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8
CN
n = 0,1,2,3,4
Figure 3. Chemical structure of the dyes considered.
1017
~
G (s–1)
1016
1015
n=0
n=1
n=2
n=3
n=4
bridge length
increase
1014
1013
1012
–4.2
–4.1
–4.0
–3.9
energy (eV)
–3.8
–3.7
Figure 4. Spectral density for the dyes considered. The energy range that contributes most to the rate is highlighted assuming
a sharp limit for the conduction band at −4.06 eV (shaded). (Online version in colour.)
The semiconductor’s Hamiltonian matrix HSel was obtained from the isolated slab calculation
and it has been used to evaluate the semiconductor’s eigenvalues required in equation (3.9).
The simulation of the semiconductor plus dye system allowed us to extract from the total
Hamiltonian matrix the semiconductor–bridge and the bridge–chromophore coupling matrix
elements (indicated with τ and κ, respectively, in §3), together with the bridge’s Hamiltonian
matrix HBel . The use of a localized basis set makes it possible to partition the dye molecule
(and its Hamiltonian) into the bridge and chromophore fragments as required by our theoretical
scheme. The chromophore fragment has been identified as the aromatic core in figure 3, including
the diphenylamino substituent; the bridge has been defined as the polyene chain, including the
carboxylic anchor and the cyano group attached to it. The coefficients of the molecular orbital on
the chromophore fragment, indicated by cm
j in equation (3.11), have been extracted from a similar
single point energy calculation on the isolated dye.
In figure 4, we report the spectral densities for the sensitizers considered; Γ˜ (E) is shown
as a function of the independent energy variable introduced in the definition of the Green’s
operator. To improve the agreement between the position of the conduction band edge of the
simulated system (in this case, the isolated slab) with the values attained experimentally in a
DSSC (reported in [48]) we have acted on the semiconductor’s states with a scissors operator,
leading to a shift of 0.6 eV in the conduction band states. We resort to this strategy because
it is well known how problematic it can be to predict the correct energy alignment in these
systems [49]. Furthermore, the conduction band edge position in DSSCs is susceptible to the
presence of additives [50] or electrolytes [51] co-adsorbed on the semiconductor surface. By
aligning the conduction band minimum with its experimental counterpart, we implicitly account
for these experimental variables in our model [52]. The energy range relevant for the integral in
equation (2.6) is highlighted in figure 4 by a shaded marker. A shift of this integration interval
of 0.1 eV relative to the chosen energy scale does not change our conclusions. The spectral
.........................................................
n
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COOH
N
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9
10–6.3
tCR (s)
10–6.6
10–6.7
10–6.8
4
6
8
10
bridge length (Å)
12
14
Figure 5. Charge recombination lifetime as a function of the bridge length. (Online version in colour.)
density is a complicated function of the bridge’s electronic Hamiltonian (which also reflects the
chemical connectivity between its atoms) and the coupling matrices with the other subsystems.
The occurrence of peaks, as mentioned before, is related to the bridge’s eigenvalues, whereas
the antiresonances are associated with signatures of quantum interference. If a model system
is studied, with the interaction restrained to the bridge’s terminal sites, then antiresonances do
not appear, whereas, if the electron is allowed to hop on different bridge orbitals and then to
propagate across, then the molecule interference is generated [45].
The position of the peaks in figure 4 relates well with the bridge’s eigenenergies: by changing
the length of the bridge, we are effectively modifying the energy spectrum of that subsystem,
hence shifting the energies at which the peaks are located. The presence of antiresonances
is associated with the anchoring group used and with its coupling to the semiconductor.
Interestingly, a sharp interference feature appears close to the conduction band edge, and it is
not affected by changing the length of the bridge. The presence of this antiresonance is associated
with the anchoring group used and with its coupling to the semiconductor. In particular, this
feature is associated with the different symmetry of the states occurring in that energy range
and it disappears when the calculation is done on a hypothetical model molecule without
an anchoring group and coupled only through the terminal atom to the electrode. This is
possibly an explanation for the success of the carboxylic acid anchoring group in DSSCs, because
having an antiresonance located in this energy range is going to slow down the CR to the
oxidized dye.
Experimental investigations are particularly suited to obtain the lifetimes of the processes
taking place in DSSCs. We report our results in terms of the recombination lifetime, τCR , defined
as the reciprocal of the reaction rate in equation (2.6). Besides the spectral density, other quantities
are required to calculate τCR : these are the chemical potential μ in the semiconductor, the
reorganization energy λ associated with the nuclear and dielectric rearrangements involved with
the charge transfer and the free-energy variation εm of the molecular moiety. The value of the
chemical potential is chosen to be 0.2 eV below the conduction band edge to mimic an out-ofequilibrium electron population within the conduction band states. This value lies within the
experimental range attained for typical DSSC architectures [53]. The quantities εm and λ have
been evaluated following the computational procedure in [52] for the molecules in figure 3; their
values vary only marginally along the series (by 0.1 and 0.05 eV, respectively). This suggests that,
to simplify the analysis of the results and to focus on the impact of the spectral density on the
rate, we can fix these two quantities to the values λ = 0.4 eV and εm = −5.0 eV, i.e. the average for
the molecules considered.
The values obtained for the CR lifetime τCR are reported in figure 5. The data show an
exponential increase with the length of the bridge fragment, which is defined as the distance
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10–6.4
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In this paper, we have derived an expression for the rates of the CR process taking place at the
semiconductor–molecule interface in a DSSC. In particular, we have developed a formalism that
allows the study of CR mediated by a molecular bridge that connects the chromophore with
the semiconductor. The main result is a combination of Marcus theory of electron transfer at the
interface and the theory of electron transport in molecular junctions, with several interesting
similarities found between the Landauer formula and the expression of the rate. From the
practical point of view, this formalism can be used to explore the effect of changing the bridge
on the rate of interfacial electron transfer. The formalism was designed to be coupled with
density functional theory (DFT) electronic structure calculation and was exemplified with the
calculation of the CR rate for a series of systems that differ in the length of the bridge. The choice
of this particular level of theory, regarded as a good compromise in terms of computational
constraints, is subjected to well-known shortcomings, stemming from limitations in the DFT
implementation used [59] and the basis set used [60], which have been amply addressed in the
literature on molecular electronics. Besides computational limitations, other uncertainties, when
comparing the results presented with experimental realizations, might stem from the breakdown
of the Condon approximation [61], thermal fluctuations of the energy levels [62], inaccuracies in
the computational scheme for the reorganization energy [63], and geometrical distortions [64]
from the ideal case analysed here. As mentioned beforehand, modifications in the electrolyte
composition may also impact on the energy alignment across the interface: for the dye families
considered experimentally and to benchmark our model, however, the range of values typically
attained is generally quite modest [52].
.........................................................
5. Conclusion
10
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130011
between the carbon atom of the anchoring group and that in the pyrene core linked to the bridge.
If the data are fitted to the exponential τCR = τ0 exp(−βR), where R is the bridge length, then we
obtain β = 0.1 Å−1 in complete agreement with the value reported for similar systems [54]. This
dependence on the distance is not unexpected: it was first derived by Beratan for tight binding
model Hamiltonians [55], and it has been related to the imaginary part of the band structure
in Green’s function-oriented studies of metal–organic chain interfaces [56]. The underpinning
assumption in the latter approach is that the molecular environment in the bridge portion can be
approximated by that of an infinite chain, disregarding the modifications due to the interaction
with the interface. This type of approximation is not required in the present scheme, as we
explicitly account for the bridge portion, which does not have to be made up by periodically
repeated units. The impact on the CR lifetime is rather weak: increasing the bridge length by
10 Å has slowed down the CR only by a factor 3 for the range of molecules considered. Therefore,
increasing the length of the bridge fragment is going to have a modest impact on the performances
of the DSSCs, as confirmed by experiment [57]. Moreover, by increasing the length of the bridge,
the coupling between the semiconductor and the lowest unoccupied molecular orbital (LUMO) of
the neutral molecule, responsible for the CI process, is most likely to get smaller, hence reducing
the quantum efficiency for the CI. A strategy to slow down the CR without affecting the injection
process has been developed recently [20] without imposing any constraints on the bridge’s
chemical structure: only the attachment to the chromophore unit plays a role. On the other hand,
the theory presented in this study will enable modifications to the chemical connectivity in the
bridge unit to be considered explicitly.
The electron transfer processes taking place in DSSCs depend critically on the characteristics
of the semiconductor surface: for instance, the presence of trap states is known to speed up
the CR reaction by creating a localized state below the conduction band edge that has faster
recombination kinetics in the Marcus inverted region [58]. This extra electronic state may well
affect the term in equation (3.10) responsible for the coupling between the semiconductor and the
bridge’s anchor. However, it is reasonable to assume that, for a fixed defect position, this term
will be equally modified for all the bridges considered here, hence the CR rate will show the same
dependence on the bridge length.
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Héctor Vázquez for helpful comments.
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