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Simulation and modelling of charge transport in dye-sensitized solar cells based on carbon

nano-tube electrodes

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2013 Phys. Scr. 87 035703

(http://iopscience.iop.org/1402-4896/87/3/035703)

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IOP PUBLISHING PHYSICASCRIPTA

Phys. Scr.87(2013) 035703 (11pp) doi:10.1088/0031-8949/87/03/035703

Simulation and modelling of charge

transport in dye-sensitized solar cellsbased on carbon nano-tube electrodesYahia Gacemi1, Ali Cheknane1 and Hikmat S Hilal2

1 Laboratoire des Semiconducteurs et Materiaux Fonctionnels, Universite Amar Telidji de Laghouat,

Laghouat Algeria2 SSERL, Chemistry, An-Najah National University, PO Box 7, Nablus, Palestine

E-mail:[email protected]

Received 19 June 2012

Accepted for publication 18 January 2013

Published 13 February 2013

Online atstacks.iop.org/PhysScr/87/035703

Abstract

For a better understanding of the mechanisms of dye-sensitized solar cells (DSSCs), based on

carbon nano-tube (CNT) electrodes, a phenomenological model is proposed. For modelling

purposes, the meso-scopic porous CNT electrode is considered as a homogeneous

nano-crystalline structure with thicknessL. The CNT electrode is covered with light-absorbing

dye molecules, and interpenetrated by the tri-iodide (I/I3 ) redox couple. A simulationplatform, designed to study coupled charge transport in such cells, is presented here. The work

aims at formulating a mathematical model that describes charge transfer and charge transport

within the porous CNT window electrode. The model is based on a pseudo-homogeneous

active layer using driftdiffusion transport equations for free electron and ion transport. Basedon solving the continuity equation for electrons, the model uses the numerical finite difference

method. The numerical solution of the continuity equation produces currentvoltage curves

that fit the diode equation with an ideality factor of unity. The calculated currentvoltage (JV)

characteristics of the illuminated idealized DSSCs (100 mW cm2, AM1.5), and the differentseries resistances of the transparent conductor oxide (TCO) layer were introduced into the

idealized simulated photoJVcharacteristics. The results obtained are presented and

discussed in this paper. Thus, for a series resistance of 4 of the TCO layer, the conversion

efficiency () was 7.49% for the CNT-based cell, compared with 6.11% for the TiO 2-based

cell. Two recombination kinetic models are used, the electron transport kinetics within the

nano-structured CNT film, or the electron transfer rate across the CNTelectrolyte interface.

The simulations indicate that both electron and ion transport properties should be considered

when modelling CNT-based DSSCs and other similar systems. Unlike conventional

polycrystalline solar cells which exhibit carrier recombination, which limits their efficiency,

the CNT matrix (in CNT-based cells) serves as the conductor for majority carriers and

prevents recombination. This is because of special conductivity and visiblenear-infrared

transparency of the CNT. Charge transfer mechanisms within the porous CNT matrix and at

the semiconductordyeelectrolyte interfaces are described in this paper.

PACS numbers: 78.20.Bh, 78.56.a(Some figures may appear in colour only in the online journal)

1. Introduction

The dye-sensitized solar cell (DSSC) is a third-generation

photovoltaic device, with high prospects for use in low-cost

photo-conversion technology in the future. DSSCs are

photo-electrochemical cells involving photoactive electrodes

based on meso-porous nano-structured metal oxide (typicallyTiO2 or ZnO) films[12].

Numerical modelling of solar cells helps us to understand

the principles of the photo-conversion process and to

0031-8949/13/035703+11$33.00 Printed in the UK & the USA 1 2013 The Royal Swedish Academy of Sciences
http://dx.doi.org/10.1088/0031-8949/87/03/035703mailto:[email protected]://stacks.iop.org/PhysScr/87/035703http://stacks.iop.org/PhysScr/87/035703mailto:[email protected]://dx.doi.org/10.1088/0031-8949/87/03/035703


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Phys. Scr.87(2013) 035703 Y Gacemiet al

S S, fall to ground state, (5)

S+ + eCB S, regeneration of dye by injected electrons,(6)

I3+2eCB3I, recombination of injected electrons with I3 ,

(7)

whereS is the dye sensitizer; S is the electronically exciteddye sensitizer; S+ is the oxidized dye sensitizer; and O/R is

the redox couple (e.g. I3 /I).

In order to capture as much solar radiation as possible,

the dye must have a wider absorption spectrum. The dye

LUMO must energetically lie above the conduction band

edge of the semiconductor to allow electron injection into

the nano-porous CNT, before it can fall back to its ground

state (reaction (5)) [17, 19]. On the other hand, the dye

HOMO must have a more positive (lower) potential than

Eredox. To prevent recombination of the injected electrons

with the oxidized dye (reaction (6)), hole transfer to theredox couple must be extremely fast. The electrolyte in

turn conducts the holes to the counter-electrode, where the

redox couple itself is regenerated. As VOC corresponds to

the difference between the Eredox and the CNT Fermi level,

the former must be as positive as possible to guarantee

higher photo-voltages. The over-voltage for reduction of

the species in the counter-electrode must be small, since it

represents a loss in the photo-voltage of the cell. To enhance

charge-transfer at the counter-electrode, a charge-transfer

catalyst, such as platinum particles, is commonly used [16].

The over-voltage at the semiconductordye interface must be

high, since the dark current caused by electron back transferto the electrolyte lowers the number of electrons available for

the photocurrent (reaction (7)).

2.1. Electrical model for charge transport

2.1.1. Charge transport in nanocrystalline CNT films

(continuity equation for electrons). The electron diffusion

through the nano-structured CNT film, with variable

generation and recombination terms using appropriate

boundary conditions, is evaluated. By illuminating the

working electrode, electrons are injected in the system,

resulting in an electron density build-up in the film that

is a function of position and time. Electrons are allowedto diffuse towards the working electrode, which in turn

behaves as a collecting interface. The continuity equation is

finally solved by linear geometry, i.e. only the x-coordinate

is considered [20]. The x denotes the location within the

cell, where x= 0 indicates the TCOCNT interface at thefront electrode. Thex= dindicates the interface between theelectrolyte and Pt-TCO where drepresents the thickness of

the simulated cell; andncis the density of free electrons in the

conduction band. Hence, the following differential equation

is solved:

De2ne(x, t)

x2 + G(x)ne(x, t)

neq

e =ne(x, t)

t , (8)

where De is diffusion coefficient for electrons (m2 s1); and

G(x) is the generation rate at distance x, ne(x, t) is the total

density of electrons (cm3) as a function of time (t) anddistance (x) from the working electrode [16], and neq is the

carrier density of electrons at equilibrium in the dark.

Equation (8) is solved by FDM using the lax scheme

[20,21] and with the following two boundary conditions:

x= 0, q De nex

x=0

= Jcell,

x= d, nex

x=d

= 0,(9)

whereq is elementary charge of the electron (1.6 1019 C),and Jcell is short-circuit current density of the cell.

Considering that for t= 0 the cell is in the dark atequilibrium:

t= 0, ne(x, 0) = neq, (10)

wheretis exposure time in seconds.The generations could be simplified in integral

form (equation (11)), where () denotes the absorption

coefficient, ninj is the electron injection efficiency, () the

incident photon flux, and the wavelength:

G(x) = inj max

min

()()e()x d. (11)

Generation rates are integrated in the wavelength range

from 300 (min) to 800 n m (max), where the DSSC is

active[22], while on the other hand, they could also be a result

of the optical simulator. At open circuit only recombinationcurrents flow inside the cell. Electrons recombine with

tri-iodide ions in the electrolyte, or enter the front TCO layer

and recombine from there [23]. The electron transportation

(recombination) to the tri-iodide species follows first-order

kinetics as[16,22]

Re(x) =ne(x, t) neq

e, (12)

where Re(x)is the rate of electron combination as a function

ofx, and eis electron lifetime (s) determined by back reaction

with redox couple I/I3

.

The dark equilibrium carrier density neq corresponds

to the matching between the Fermi level in the CNT

and the Eredox. When the Fermi level is aligned with

electrolyte potential (i.e. kept constant, because electrolyte

concentrations are almost constant and their potential is nearly

constant) this indicates that the carrier density is constant.

Fermi level is affected by majority carrier density.

2.1.2. Charge transport in redox species (continuity equation

for redox species). The oxidation of iodide ions and the

reduction of tri-iodide occur inside the electrolyte species

present in the interstitial space within the CNT nano-particles.Taking into account the stoichiometry of reactions (3) and

(4), the terms of generation and recombination of tri-iodide

and iodide species must be affected by the corresponding

3


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coefficients (1/2) and (3/2), respectively:

DI3

2nI3 (x, t)

x2 +

1

2p

G(x)ne(x, t)neq

e ne(x, t)

t

=nI3 (x, t)

t,

DI 2nI(x, t)

x2 3

2p

G(x)ne(x, t)neq

e ne(x, t)

t

= nI(x, t)t

,

(13)

where DI3 and DI are diffusion coefficients for ions I

3 and

I, respectively;nI3and nIare concentrations for I

and I

ions, respectively (m3); andp is CNT film porosity.In a non-steady state, the accumulated electrons do not

participate in the redox reactions and must thus be discounted

from the ion generation term [16]

ne(x, t)

t electron accumulation

. (14)

At time t= 0, initial concentrations of the electrolytespecies are assumed:

t= 0, nI3 (x, 0) = ninitI3

, nI(x, 0) = ninitI. (15)At the TCOCNT interface only the photo-injected

electrons flow. The net flux of I and I3 ions is zero, thus

x= 0,nI3 (0, t)

x= 0; nI(0, t)

x= 0. (16)

The total amount of the redox species contained in

the CNT nano-pores remains constant during the DSSCoperation. This may be described as [24]

x= d, d

0

nI3 (x) dx= niniI3

d; d

0

nI(x) dx= niniId.(17)

In this study, the approximate equations of

nI3 (x, t), nI3

(x, t) and ne(x, t) are estimated using the

FDM.

3. Numerical solutions of difference equations forvarious types of boundary conditions

3.1. Numerical solutions for electron transport in the CNTelectrode

The platform, the so-called slice here, involves a number

of sub-models. The sub-models describe the structure of the

electrode, the generation of electrons and the contacts. They

also describe the electron and ion mass transport processes,

and the electron transfer reactions between the redox couple

(I/I3 ) and the oxidized dye molecules[25].In addition to transporting charge between respective

interfaces, the electrolyte also influences electron transport

within the nano-porous CNT [2528]. The electrolyte has

both redox and screening properties. Screening is directly

coupled to the ionic ability to charge-compensate electrons inthe nano-structured semiconductor.

The cell is treated as a one-dimensional effective medium

whose thickness,d, is divided into a number of slices, xN. The

-

+

-

-

+

+

+

-

+I-

I-

3

x1x0 x2 xi-1 xi xNxN-1

x

Figure 1. Description of the DSSC by the slice model, andillustration of the multiple-trapping model of transport (left) and thegeometry used for solving the continuity equation (right). The cell isassumed to be a one-dimensional medium whose thickness, d, isdivided into a number of slices,xN, where the length of each slice isxi

=xi

xi

1

=x.

length of an individual slice is expressed as xi= xi xi1 =x. The relation is expressed as

d=xNi=1

x(i ). (18)

A sequence of slices is referred to as a region. A slice

includes a generation region, an electron transport region and

an electrolyte transport region. Figure 1 shows a schematic

slice structure model corresponding to a DSSC. In the

multiple trapping (MT) models, electron transport is assumed

to occur via an extended state Ec. Furthermore, in the presentnumerical model, the recombination processes of electrons

with tri-iodide ions and dye cations may occur in the cell

under operation.

In solving equation (8), for charge transport, the FDM

will be used. For the sake of simplicity, it is assumed

that the mesh with nodes in points 1, . . ., i, . . ., N is

homogeneous. Thus, xi= xi xi1 = x= constant. Thenode with subscript 1 is located at the electrode surface, and

the node with subscript Nis located at the outer boundary of

the diffusion layer.

Dene|ki +1 2n e|ki +n e|ki1

x2 +

f

k

i =ne|ki ne|k1i

tfori= 1, . . . ,N 1, (19a)

withfk

i= G|ki

ne|ki neqe

. (19b)

In the inner nodes of the mesh, for the spatial

discretization of equations (19a)and(19b) of CNT electrode

transfer, the central differences are used. Therefore

Dex2

ne

k

1

+

2

De

x2+

1

t

ne

k

2

=1

tne

k1

1+ f

k

1 +

De

x2 ne(0, tk

)

Dex2

ne

k

1

+

2

De

x2+

1

t

ne

k

2

Dex2

ne

k1

3

4


6/12


= 1t

ne

k1

2

+ fk

2

. . . = . . .

Dex2

ne

ki

+

2 De

x2+

1

t

ne

ki +1

Dex2

ne

k1i

= 1t

ne

k1

i

+ fk

i

. . . = . . .

Dex2

ne

k

N2+

2

De

x2+

1

t

n e|kN1

= 1t

ne

k

1

N1+ fk

N1+ De

x2ne(d, tk), (20)

k designates the point order of the node in the mesh, andshould not be confused with the Boltzmann constant whichis denoted askB elsewhere.

3.2. Numerical solutions for ion transport in the electrolyte

In the uni-dimensional case, for the approximation of thetheory of equations (13), ion transfer in dimensionless formcan be written as

DI3

nI3

k

i +1 2n

I3 k

i

+nI3

k

i1x2

+afki

=nI3

ki nI3

k1i

t+a ne|

ki ne|k1i

t,

DInI

ki +1 2nI

ki

+nIk

i1x2

+b fk

i

=nI

ki nI

k1i

t+b

ne

ki ne

k1i

t,

De

ne

ki +1 2n e

ki

+n e

ki1

x2 + f

k

i

=nek

i nek1

i

t,

DI3

nI3

ki +1 2nI3

ki

+nI3

ki1

x2 +af

ki

=nI3

ki nI3

k1i

t+a

ne

ki ne

k1i

t,

DInI

ki +1 2nI

ki

+nIk

i1x2

+b f

k

i

= nI k

i nI k1

i

t+b ne

k

i nek1

i

t.

(21)

In the inner nodes of the mesh, for the spatialdiscretization of equations of electrolyte species transfer (21),

the central differences are used. Thus

2

DI3

x2+

1

t

nI3

k1

+at

ne

k1

DI3

x2 nI3

k2

= 1t

nI3

k1

1+an e

k1

1

+af

k

1+

DI3x2

nI3

0, tk

,

2

DI

x2+

1

t

nI

k1

+b

tne

k1 DI

x2 nI

k21

1

t

nI

k11

+bnek1

1

+b. f

k1

+DI3x2

nI

0, tk

,

. . . = . . .

DI3x2

nI3

k

i1+

2

DI3x2

+ 1

t

nI3

k

i+at

ne

k

i

DI3

x2 nI3

ki +1

= 1t

nI3

k1i

+an ek1

i

+af

ki

,

DIx2

nIk

i1+

2

DI

x2+

1

t

nI

ki

+b

tne

ki

DIx2

nIk

i +1= 1

t

nI

k1i

+bnek1

i

+bf

ki

,

. . . = . . .

DI3x2

nI3

kN2

+

2DI3

x2+ 1

t

nI3

kN1

+ at

nekN1

= 1t

nI3

k1i

+an ek1

i

+af

kN1

+DI3x2

nI3

d, tk

,

DIx2

nIkN2

+

2

DI

x2+

1

t

nI

kN1

+b

tne

kN1

= 1t

nI

k1i

+b nek1

i

+bf

kN1

+DI3x2

nI

d, tk

(22)

witha = 1

2p, (23a)

b = 32p

. (23b)

4. Numerical model for electron conductivity innano-structured CNT semiconductors

The measured capacitance, which is believed to be a chemical

type in nature as shown below, is affected by a number

of parameters, as shown in figure 1 [29, 30]. Process (B)

indicates polarization at the transparent conducting substrate(TCS)electrolyte interface, and (C) indicates the Helmholtz

layer at the oxideelectrolyte interface. The former effect is

important when the electron density in the semiconductor

5


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Figure 2. Schematics of the capacitive contributions in a DSSC:(A) Chemical capacitance due to increasing chemical potential(concentration) of electrons in the CNT phase, obtained when theelectrode potential,V, displaces the electron Fermi level, EFn , withrespect to the lower edge of the conduction band, Ec, in thesemiconductor nanoparticles. (B) Electrostatic capacitance of theHelmholtz layer (and semiconductor band-bending) at the interfacebetween the exposed surface of the transparent conducting oxidesubstrate and the electrolyte. (C) Electrostatic capacitance at theHelmholtz layer at the oxideelectrolyte interface.

is low [31] and the latter when the density is very high

with semiconductor band unpinning. Both contributions can

be thought of as ordinary electrostatic capacitors, where the

charges in two highly conducting plates sustain an electrical

field in between.

In the intermediate range of Fermi level variation, a

different kind of capacitive effect is found, as shown in section

(A) of figure 2. The semiconductor bands are pinned, and

the charge accumulation is related to the displacement of the

Fermi level position with respect to the conduction band edge.

This means variation of the chemical potential of electrons as

n =

n

. Hence, the increment of charge (both electronic

and ionic) occurs inside the nano-structured electrode with no

concomitant electrical field variation therein. This is because

the electrical field is shielded near the TCS [31]. Therefore,

freeenergy storage in the capacitor is due to chemical rather

than electrostatic energy. Therefore, it is a chemical capacitor

rather than an electrostatic one. The chemical capacitance

((A) in figure 2) is a major feature in CNT nano-structured

electrodes. Besides, the chemical capacitance is a concept

of crucial significance for solar cell applications, because it

properly describes the splitting of Fermi levels caused by

excitation of carriers in the light absorber material [30, 32,

33]. Considering the variation of the electron density upon a

change in local chemical potential in a DSSC, the chemicalcapacitance occurs. Two components in equation (24) may be

distinguished[34]. The first is related to the free conduction

band electrons.

C(cb)ch = q2

nc

kBT= q 2 nc

n, (24)

where C(cb)ch is the chemical capacitance in the conduction

band; nc is the conduction band electron density; kB is the

Boltzmann constant andTis the temperature (K).

To analyse these questions a relatively simple kinetic

model is outlined for diffusion, trapping and interfacial

charge transfer of electron carriers in a nano-structuredsemiconductor permeated with a redox electrolyte. We will

use the MT model for transport and charge transfer illustrated

earlier [1618]. This model is adapted to nano-structured

semiconductors from a wide experience on disordered

semiconductors.

In the context of the MT model, the electron transport is

carried out by a single kind of state, the extended states of the

conduction band. Carriers trapped in localized states do not

contribute to the dc conductivity until they are released again.

The kinetic transport equations are stated in the MT model fora single kind of trap, involving the equations of conservation

for free and trapped electrons [30]:

nc

t=Je

xnc[1 fL] + fL, (25)

whereJeis the electron flux and fLis the fractional occupancy

(nL = NLfL, where NL is the density of traps); is thesymmetry coefficient; and is the rate constant for electron

thermal release from the trap to the conduction band.

As explained before, diffusive transport of electrons

within the CNT film towards the TCO is considered. Equation

(26) relates the electron flux, Je, at any position x to thegradient of electron concentration across the CNT layer, ne,

by means of the electron diffusion coefficient, De in Ficks

law [22,30]

Je =Denc

x. (26)

The rate constant for electron capture is determined by the

thermal velocity of free electrons, , the electron capture cross

section of the trap,sn , and the overall density of traps, NL, as

= NLsn , (27)

where NL is the total density of localized sites.

The rate constant for electron thermal release from

the trap to the conduction band, , is described by the

ShockleyReadHall statistics

= Ncexp[(Ec EL)/kBT] , (28)

where Ec is the lower band edge energy, EL is the energy

of the localized state in the band gap and Nc is the effective

density of conduction band states. The occupancies, fL, in the

two kinds of states are given explicitly by

fL = 11 + e(ELn )/kB T

. (29)

Note that, in equation (29), the Fermi level n can be

maintained at a value different from the redox potential in

solution,Eredox [30].

In the one-dimensional case, considering the

approximation of the theory of equations, electron

conductivity transfer in dimensionless form can be written as

follows:

nc|ki nc|k1it

= Denc|ki +1 2n c|ki +n c|ki1

x2

[1 fL]n c|ki + fori= 1, . . . ,M 1. (30)

In the inner nodes of the mesh, for the spatial discretization

of equations(30) of electronic conductivity CNT transfer, the

6


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Table 1.Simulator input parameters.

Parameter Value Reference

AC 0.158cm2 [16]

e 23.6 ms [16]De (electrode TiO2) 1.10 104 cm2 s1 [16]e(electrode CNT) 10

8 cm2 V1 s1 [14]

De (electrode CNT) k.T

e e [14]DI 8.5 106 cm2 s1 [23]DI

38.5 106 cm2 s1 [23]

NCB 1.0 1021 cm3 [16]inj 0.90 [16]() 1000 cm1 [16]p 5% [37]() 1.0 1017 cm2 s1 [38]me 5.6me [23]Resistivity (CNT) (300 K) 0.1 cm [14]T 298 K ECB Eredox 0.93 eV [16]n0

I3

2.71 020 cm3 [23]n0

I3

3.01 1019 cm3 [23]n0e 10 1010 cm3 [23]

central differences are used. Therefore

r r 0 . 0 0r r r . 0 00 r r r . 0. . . . . .

0 0 0 . r rr r

nc|k1nc|k2nc|k3

.

nc|kM2nc|kM1

=

nc|k11 + nc 0, tk

+ t

nc|k

1

2 + tnc|k13 + t

.

nc|k1M2+ tnc|k1M1+ nc

M, tk

+ t

. (31)

Another important quantity for many applications is the

electron conductivity, which can be simulated in the steady

state as reported elsewhere. Using equation (24), the

conductivity related to the electron diffusion process, n, can

be obtained from the generalized Einstein relation where nis the chemical potential of electrons [30,34]as

n = q2 ncn

D0 = C(cb)ch D0, (32)

where D0 is the constant diffusion coefficient for free

electrons.

5. Results and discussion

In order to validate the present numerical model, the

simulation results are compared with the numerical results

in the literature, particularly with TiO2-based DSSC

performance results reported by Andradeet al[16]. All input

parameters used in the simulation are shown in table1.

Figure3compares the evolution of the transient electrondensity profile obtained from the present simulation with the

numerical results of the FDM. The plot shows that as the

electron density distribution increases, the diffusion of the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

0

1

2

3

4

5x 10

17

x [cm]

ne-[cm

-3]

CNT-based so lar cell (our approach)

TiO2-based solar cell Ref [16]

Figure 3. The distribution of the electrons within the simulated1 104 cm thick active layer in CNT- and TiO2-based DSSCs.

tri-iodide towards the electrode becomes more limiting. Thisis primarily due to their low initial concentration, with the

limited density of 4.6 1017 cm2 obtained for this case. Theplot shows very good agreement with the numerical results

obtained from FDM which forms the basis of the present

model. Parametric studies based on the validated model are

discussed here. Figure3 shows the evolution of the electron

density profiles under short-circuit conditions before reaching

the steady-state condition.

The electron density profiles during the first micrometers

of the TiO2 and CNT film steadily increase, becoming

more flat afterwards. The active layer, presented by

the finite difference method, is electrically modelled

as a one-dimensional pseudo-homogeneous layer. Theelectro-active charge carriers included in the model are

photo-generated free electrons (injected in nano-porous

CNT), iodide and tri-iodide ions in the electrolyte. The

structure of the DSSC is first divided into two different

regions: the active layer (a mixture of electrolyte and

nano-pores covered with dye which is treated as the

pseudo-homogeneous layer) and the bulk electrolyte layer.

Figures 35 show the distribution of the electrons, iodide

and tri-iodide ions within the simulated 1 104 cm-thickactive layer of DSSC under the short-circuit operation

conditions. Figure 3 shows that the electron concentration

distribution is much smaller under short-circuit conditions,since most of electrons reach the front contact. The first few

micrometres of the CNT contribute most to the total JSC,

while electrons generated deeper in the active layer partly

recombine with tri-iodide ions. Therefore under short-circuit

conditions, the electron concentration almost equals zero

at the front contact and monotonically increases along the

positive x-direction. I ions are normally generated at thecounter-electrode. Therefore, the concentration of I ionsmonotonically decreases with decreasingx, figure4.The role

of iodide is to restore the positively charged photo-excited

dye into its ground state. Consequently, under short-circuit

conditions, the concentration of I ions is smallest near theworking electrode.

On the other hand, iodide could also be formed

by a recombination process, especially under open-circuit

conditions. Therefore, the concentration of I ions becomes

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are

n(0) = n0 =

Ec

fL(E)N(E) dE (33)

and dne

dx

x=d

= 0. (34)

The short-circuit current density JSCcan thus be obtainedas

JSC =

qL()[L() cosh(d/L) + sinh(d/L)

+L()exp (dx)]

(1L2()2) cosh(d/L) , (35)

whereqis the charge of an electron;Lis the electron diffusion

length that equals

De; and d is the thin film thickness

[35,36].

Cells with an external resistive load can be modelled if

an appropriate functionality for the electron density at the

working electrode is taken into account. In the presence of

an applied voltage or under higher illumination intensity, theelectron density in the film increases. Here, we compute the

electron density as a function of voltage at x= 0, n e(V), asfollows:

ne = n0exp (eV/kBT), (36)where is a parameter that reflects the average depth of

the distribution of the trap state energy below the conduction

band.

For each defined external current density the cell

voltage is calculated using equation (37), where Vdenotes

the cells voltage and ER(d) the redox potential at the

counter-electrode. The EnF (0) is the Fermi level of the CNT

at the TCOCNT interface (x= 0), which is determined fromthe concentration of free electrons at the TCOCNT interface,

ne(0), equation (38). The NCBis the effective density of states

in the CNT conduction band (equation(39)), whereme is the

effective electron mass andh is Plancks constant [22,23]:

V= 1e

EnF (0) ER(d)

, (37)

EnF (0) = ECB+ kBTlnn e(0)

NCB, (38)

NCB = 22 me kBT

h2

3/2

. (39)

The presented model, which was implemented in a

thin film solar cell simulator [1, 23], permits the steady-

state simulation of the complete JV characteristics,

depth-dependent concentration of electro-active spices and

their current densities in a DSSC. The numerical solution

of the continuity equation (equation (8)), in combination

with equations (36) and (37), and the boundary conditions,

produces currentvoltage curves that fit the diode equation

with an ideality factor of unity. Thus, the solar cell

behaves as an ideal diode at different series resistances. The

calculatedJVcharacteristics of illuminated TiO2-DSSC[16]

and CNT-DSSC-based solar cells (100 mW cm2, AM1.5) fordifferent series resistances of TCO layer were introduced into

the idealized simulated photoJVcharacteristics. The results

are presented in figures8and9.

0 100 200 300 400 500 600 700 8000

2

4

6

8

10

12

V [mV]

J[mA.cm-2]

currentvoltage characteristicRs = 0 ohm

Rs= 4 ohm

Rs= 8 ohm

Figure 8. Calculated photoJVcharacteristics forTiO2-DSSC-based solar cells with different series resistances of the

TCO layer using the input parameters of [16].

0 100 200 300 400 500 600 7000

2

4

6

8

10

12

14

16

18

V [mV]

J[mA.cm-2]

currentvoltage characterist icRs = 0 ohm

Rs= 4 ohm

Rs = 8 ohm

Figure 9. Calculated photoJVcharacteristics forCNT-DSSC-based solar cell having different series resistances (ourapproach).

Based on simulation results in a simple electrical

model the calculated short-circuit current density (JSC) is

18.8mAcm2 for CNT-based DSSC, compared with the values

for TiO2-based based DSSC which are generally about

12.2mAcm2. The calculated open-circuit voltage (VOC) is

700 mV for CNT-based DSSC, which is less than the VOCfor TiO2-based DSSC. The FF was 69.30% for CNT-based

DSSC, when the idealized cell without TCO resistance

was simulated (RS = 0 ). The conversion efficiency ofsimulation results in a CNT-based solar cell without additional

series resistance was 8.29%, compared with 7.02% for

the TiO2-based counterpart. The difference is in agreement

with expectations since the material parameters used in

simulation are not optimized yet. Performance parameters

for the two cell types are summarized in table 2 (entry 1).

Unlike polycrystalline-based solar cells which exhibit carrierelectronhole recombination, which limits their efficiency, the

CNT matrix (in CNT-based cells) serves as the conductor

for majority carriers and prevents such a recombination. That

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Table 2.Performance parameters of simulated DSSC having different series resistances for CNT- and TiO2-based systems.

JSC (mA cm2) VOC(mV) FF (%) (%)

CNT-based TiO2-based CNT-based TiO2-based CNT-based TiO2-based CNT-based TiO2-basedDSSC DSSC DSSC DSSC DSSC DSC DSSC DSSC

RS () (our approach) ([16]) (our approach) ([16]) (our approach) ([16]) (our approach) ([16])

0 18.8 12.2 700 740 69.30 68.91 8.29 7.02

4 18.8 12.2 700 740 62.68 62.95 7.49 6.118 18.8 12.2 700 740 54.25 60.16 6.49 5.70

is because of special conductivity and visible-near infrared

transparency of the CNT.

Different series resistances were then introduced into the

idealized simulated photo JVcharacteristics for CNT-based

DSSC. The results are again shown in figures 8 and 9 and

are summarized in table 2 (entries 23). As expected, the

JSC and VOC values did not change for either cell. In the

CNT-based cell, the FF decreased from 69.30 to 62.68%

when RS

=4 was added, and to 54.25% when RS

=8

was added. Consequently, the initial conversion efficiency wasreduced from 8.29% to 7.49 and 6.49% when RS = 4 and 8 were added, respectively. The TiO2-based cell also exhibited

lowering in values of FF and conversion efficiency by adding

series resistances, as shown in table2.

6. Conclusion

A mathematical model for a new type of photovoltaic solar

cell based on dye-sensitized CT electrodes is discussed. In

contrast to conventional solar cells, where light absorption

is due to band gap excitation of the semiconductor itself,

the CNT with its wide band gap is transparent in the visiblespectrum. While the efficiency of conventional polycrystalline

solar cells is limited by charge carrier recombination at grain

boundaries, bulk defects and impurities, the CNT serves only

as the conductor for majority carriers (injected electrons),

so that electronhole recombination in the semiconductor is

absent. The unique characteristics of carbon films as window

electrodes are due to their excellent conductivity and good

transparency in both visible and near-infrared regions.

The model describes charge transfer and charge transport

within the porous CNT electrode of the DSSC.

Charge transfer at the semiconductordyeelectrolyte

interface within a DSSC is also described. The model is basedon solving the continuity equation for electrons, using the

numerical finite difference method. The equations were used

to describe the charge transport within the electrolyte filled

pores and the porous semiconductor.

Acknowledgments

YG and AC acknowledge Universite Amar Telidji de

Laghouat, Algerie, for financial support. HSH acknowledges

the Al-Maqdisi Project for help.

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