Universal photocurrent-voltage characteristics of dye sensitized nanocrystalline TiO$_2$ photoelectrochemical cells

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by  J. S. Agnaldo

Universal photocurrent-voltage characteristics of
dye sensitized nanocrystalline TiO photoelectrochemical cells
2
J. S. Agnaldo, J. C. Cressoni, and G. M. Viswanathan
Laborat´orio de Energia Solar, Nu´cleo de Tecnologias e Sistemas Complexos,
Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o–AL, 57072-970, Brazil
(Dated: February 5, 2008)
We propose a new linearizable model for the nonlinear photocurrent-voltage characteristics of
nanocrystalline TiO dyesensitized solar cells based on first principles and report predicted values
2
for fill factors. Upon renormalization diverse experimental photocurrent-voltage data collapse onto
8
a single universal function. These advances allow the estimation of the complete current-voltage
0
curve and the fill factor from any three experimental data points, e.g., the open circuit voltage,
0
theshortcircuitcurrentandoneintermediatemeasurement. Thetheoreticalunderpinningprovides
2
insightintothephysicalmechanismsresponsiblefortheremarkablylargefillfactorsaswellastheir

n known dependenceon the open circuit voltage.
a
J PACSnumbers: 84.60.Jt,72.80.Le,05.40.-a,85.60.-q

8
2
I. INTRODUCTION due to ultra-fast injection of electron from the dye on

] picosecond and subpicosecond time scales [14, 17]; (ii)
r
e An important feature of photovoltaic solar cells and conduction consisting only of injected electrons rather
h than electron-hole pairs [5, 6], due to the wide bandgap
of diverse optoelectronic devices studied in semiconduc-
t
o tor physics concerns their current-voltage characteris- of the semiconductor TiO2; (ii) high optical density due
t. tics [1, 2, 3, 4]. The pioneering work that led to the to the extremely large surface area of the dye sensitized
a nanoporoussemiconductor[14];(iv)negligiblechargere-
invention of Gr¨atzel or dye sensitized solar cells became
m combination with the oxidized sensitizer dye [5, 6, 7];
a milestone in the study of photovoltaic devices [5, 6].
- Previoustheoreticalandexperimentalstudies haveiden- and (v) very high quantum yields [14]. One important
d
fact that will contribute towards derivation and subse-
tifiedthe dependenceofthephotocurrentandphotovolt-
n
quent interpretation of the current-voltage characteris-
o ageonradiantpower[7],butnottheprecisenonlinearde-
tics concerns how the experimentally measured recombi-
c pendence of the photocurrentonthe photovoltageunder
[ conditions of constant radiant power. Moreover, vari- nation currentdensity vanishes at shortcircuit [7] — in-

1 ability in the manufacturing process of dye sensitized dicatingthatthe onlysignificantrecombinationpathway
proceeds via back electron transfer into the electrolyte.
v solar cells can lead to differences — e.g., variables in-
−
Indeed, charge recombination between redox species (I
4 clude the choiceofdye,the sinteringtemperature,thick- 3
3 ness of the nanoporous TiO film and choice of chemical ions) in the electrolyte and conduction band electrons
2
3 localized at the nanoporous interface result in subopti-
treatments. This diversity leads to significant qualita-
4 mal photovoltage levels — thus limiting the conversion
tive and quantitative variation in photocurrent-voltage
1. characteristics and of the relevant quantities such as the efficiency [19, 20, 21].
0 opencircuitvoltageV orthefillfactor. Suchvariability
oc
8
hasdiscouragedattemptstoidentify(possibly“hidden”)
0
: dynamical patterns that could yield important insights II. METHODS
v into the regenerative photoelectrochemical mechanisms
Xi that underlie the conversionprocess. Given the variabil-
We begin by assuming that the number of electrons
ity and diversity in the characteristics, which properties
ar remain universal and which nonuniversal? More impor- injected into the conduction band depends only on the
incident radiant power — in fact the known very high
tantly from an experimental point of view, how can we
quantitativelymodelthephotocurrent-voltagecharacter- quantumyieldsjustifythisassumption. Thisassumption
allowsustoexpresstherecombinationcurrentdensityJ
istics,basedonfundamentaltheoreticalprinciples? Here r
as a function of the photocurrent density J and the in-
we answer these questions by deriving from first princi-
jectioncurrentdensity. SinceJ vanishesasshortcircuit,
ples an analytical expression for the photocurrent. r
the injection currentequals the shortcircuit currentJ ,
The topic of solar energy in general [8, 9, 10, 11] and sc
so that
dye sensitized solar cells in particular [5, 6, 7, 12, 13, 14,
15,16,17,18]attractsbroadinterestfromdiversesectors
of society, due to technological, economic, political and Jr =Jsc−J . (1)
environmental considerations. The growing scientific in-
terest in dye sensitized TiO solar cells stems from their Forawellmixedsolutionwithidenticalsurfaceandbulk
2
unusual features and mode of operation that distinguish concentrations (typical for small current densities), the
them from Si solar cells: (i) efficient charge separation Butler-Volmer equation [22, 23] leads to the following
Typeset by REVTEX
2

16

1.0
Untreated 1.0 Model
12 VP 0.8 1% Error
0.8
]
2 m TBP
0.6
mA/c8 PVP3 0.48 0.56 0.64 J(R)0.6 Y*0.4
[ 2 0.4 0.2
Untreated TBP
J 4 ln Jr01 0.2 0.0 VP PVP
0.0 0.2 0.4 0.6 0.8 1.0
-1 X*
0.0
0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
V/V
V [Volts] OC
Figure 1: Typical photocurrent voltage characteristics under Figure 2: Renormalized photocurrent J(R) versus V/Voc for
the curves shown in Fig. 1. The data collapse onto a single
aradiantpowerof1.5AMfordyesensitizedsolarcells,taken
universalfunctionstableoveravarietyofchemicaltreatments
from ref. [7]. The four data sets have different characteris-
tics dueto varied chemical treatments (see text), yet in each andsolarcellvariability. Wehavechosenvs =1/40,basedon
casethetheoreticalcurves(solidlines)correspondingtoEq.6 the value of kBT at room temperature, for illustration, cor-
respondingtoourestimateofan upperboundforfillfactors,
canaccountwellfortheexperimentalcurves. Insetshowsap-
howeverwecouldhaverenormalized thephotocurrenttoany
proximate logarithmic relation for therecombination current
fill factor. Indeed, we can linearize the curves, as shown in
density Jr versusvoltage, not inconsistent with Eq. 3.
the inset and explained in the text. The dashed line traces
the upper and lower uncertainties corresponding to an error
of1%intheshortcircuitcurrent. Noticehowremarkablythe
expression for the recombination current density J :
r data collapse onto a straight line, within the error tolerance.
−J =J exp(−α ufη)−exp(α ufη) , (2)
r 0(cid:20) C A (cid:21)
(or mean) concentration. Nonetheless, we still do not
where J denotes the exchange current density, u the
0 know, a priori, exactly how it varies with the potential
numberofelectronstransferredinthereaction(andcon-
atthesemiconductor-electrolyteinterfaceastheexternal
sequently, the order of the rate of reaction for recom-
load (i.e., impedance) varies, because of the out of equi-
bination for electrons) α and α the anodic and ca-
A C librium conditions. In this context, one important clue
thodic transfer coefficients, η =V the overpotential and
comes from the dependence on the photovoltage on the
f ≡ q/k T. With some simplification [7], this equation
B electron population. Electrons act as charge carriers in
becomes
theTiO —justastheredoxspeciesdointheelectrolyte.
2
In the semiconductor, injected electrons shift the Fermi
J =qk cmnuα[exp(uαqV/k T)−1] , (3)
r et 0 B level, so that n = n exp(qV/k T). This exponential
0 B
dependence on potential, together with the exponential
where α=α and k denotes the back electrontransfer
A et
dependence of c on voltage across the electrolyte, hints
rateconstant,ctheconcentrationandmtheorderofthe
at a similar, i.e. exponential, dependence of c on V at
reaction for the oxidized species, q the electronic charge
the interface as the external load varies:
andn representsvalueindarkconditionsoftheelectron
0
population in the semiconductor.
HowdotheprefactorsdependonvoltageV? Theback V =V −(γk T/q)ln(c /c) . (4)
oc B oc
electrontransfer rate constantvaries with radiantpower
and the maximum, i.e. open circuit, photovoltage V ,
oc
howeverwedo notexpectit todepend onthe photovolt- Here c represents the concentration under open cir-
oc
age for fixed radiant power. The Nernst Equation for cuit conditions and γ represents a free parameter in the
the potential in terms of the concentrations of the oxi- model, quantifying the fraction of the voltage variation
dized and reduced species holds valid only under equi- thataffectsthe oxidizedspecies concentration. Since the
librium conditions, yet we know that c varies across the triiodideconcentrationcannotvaryverymuchinthe liq-
electrolyte. Indeed, since the reduced species greatly ex- uid, we cannot expect small γ close to equal unity since
ceeds the oxidized species, we can safely conclude that this would imply c ∝ n. For now we only mention that
the voltage varies as ∆V(x) ≈ −(k T/q)ln(c0/c(x)), one would na¨ıvely expect a positive γ since electron in-
B
where c(x) denotes the concentration at a position x jection and dye regeneration (associated with larger V)
across the cell (i.e., electrolyte), c0 denotes a reference produce oxidized species.
3
validity in the large voltage (V > 80mV) regime [7, 13].
1.0
(a) VOPT/VOC=0.91 On the other hand, below this potential, the recombina-
tion current becomes negligible and uninteresting. For
0.8 VOPT/VOC=0.55 FFMAX=0.88 any useful cell, n≫n by many orders of magnitude, so
0
1.0 that we can reasonably well approximate Eq. 5 with
0.6 0.8 FF exp(Voc/Vs)−exp(V/Vs)
0.6 VOPT/VOC J =Jsc(cid:20) exp(Voc/Vs)−1 (cid:21) , (6)

0.4
0.4 where the potential
0.2 0.2 V ≡(k T/q) 1 ≈ 1 Volts , (7)
s B uα+m/γ 40(0.7+2/γ)
FFMIN=0.31 0.-0100 -50 0 50 100
0.0 representsacharacteristicscaleoftheexponentialdecay.
0 10 20 30 40 Specifically, V quantifies the photovoltage drop corre-
s
VOC/VS sponding to a decrease in recombination current density
by a factor of 1/e where e here denotes Euler’s number.

Eqs. 5 and 6 represent the first out of three new results
0.80
(b) of this article.
Fig. 1 compares the model with photocurrent-voltage
0.75
curves taken from ref. [7], of untreated and pyridine
derivative-treated [RuL (NCL) ]-coated nanocrystalline
2 2
0.70 Model TiO electrodes in CH CN/MNO (50:50 wt %) con-
2 3
FF Untreated taining Li(0.3M) and I (30mM), for a radiant power of
2
0.65 VP 100mW/cm2 (AM 1.5). The electrodes had treatment
TBP with following substances: 3-vinylpyridine (VP), 4-tert-
0.60 PVP butylpyridine (TBP), and poly(2-vinylpyridine) (PVP).
The good agreement with the data validates the model
represented by Eqs. 5 and 6.
0.55
6 8 10 12 14 16 18 The largest possible power output divided by J V
sc oc

VOC/VS defines the fill factor FF. Notice that Vs changes the fill
factor(via γ). Avalue V →∞(correspondingto purely
Figure3: (a)TheoreticallypredictedfillfactorsFFandratio s
resistive or Ohmic behavior) leads to FF=1/4, whereas
VOPT/Voc of optimal operating voltage VOPT to open circuit V → 0 leads to unity fill factor — perfect but theoreti-
voltageVocversusVoc/Vs,whereVsrepresentsacharacteristic s
voltage(Eq.7). Weestimateaworst-caselowerboundonthe cally impossible except at T = 0 K. Most dye sensitized
fill factor using Voc/Vs = 1, and an idealized upper bound solar cells have FF=0.6–0.7 (Fig. 1).
usingVoc/Vs =40. Insetshows a morecomplete picture. (b) We now turn our attention to the question of whether
ComparisonoftheoreticalandexperimentalvaluesforFFfor a single universal current-voltage relation can describe
the data shown in Fig 1. Useful real cells will likely haveFF all TiO solar cells. According to the theory presented
2
within therange shown. above, all dye sensitized solar cells must satisfy Eq. 5 if
not Eq. 6. If we renormalize the photovoltageto obtain
an adimensional measure V∗ ≡ V/V , then every single
III. RESULTS oc
dyesensitizedsolarcellmustsatisfythefollowingrelation
for an idealized renormalized photocurrent:
Theseconsiderationsimmediatelyleadtoananalytical
Vs/Vocvs
expression for the photocurrent J as a function of the
1− 1−(J/J ) 1−exp(−V /V )
voltage V across the cell: (cid:20) sc (cid:18) oc s (cid:19)(cid:21)
J(R) ≡ .
1−exp(−1/v )
exp(mq(V −V )/γk T) s
J =J 1− oc B × (8)
sc(cid:20) exp(uαqV /k T)−1
oc B Here v fixes the shape or fill factor of the renormalized
s
photocurrent. Fig. 2 shows the predicted data collapse.
exp(uαqV/k T)−1 . (5)
(cid:18) B (cid:19)(cid:21) We havechosena value vs =1/40,due toits significance
for an idealized solar cell with maximum FF (see below)
Here m ≈ 2 because of the second order reaction and at room temperature. However, we can obtain data col-
uα ≈ 0.7 [7, 13]. This analytical expression must hold lapse for any v (not shown). This is perhaps more clear
s
trueforalldyesensitizedsolarcells,yetinthecontextof if we linearize the curves. We define coordinates
providinggreaterclarityandinsightwecanrenderitsim-
X∗ ≡ 1−V/V (9)
plerbutmoreuseful. Moreover,fromapracticalpointof oc
view,wecansimplifyitfurtherbymakingadditionalyet Y∗ ≡ −V /V ln 1− J 1−exp(−V /V ) (1.0)
realistic assumptions. Eq. 3 for the recombination has s oc (cid:20) J (cid:18) oc s (cid:19)(cid:21)
sc
4
The inset of Fig. 2 shows how the data collapse onto a and will thus find practical application. We estimated
straightline. Alldyesensitizedsolarcellsthusfollowthe the error bars for the experimental points from the re-
sameuniversalpatternofphotocurrent-voltagebehavior. gression fits used to arrive at the values of V . Devices
s
We next consider the problem from the point of view thatweconstructedlocallyhadvaluesofFFwithinthese
of scale invariance symmetry. The fill factor cannot bounds.
depend on Jsc, since it cancels in the power ratio. Finally, our findings explain the very large fill factors
It also remains invariant under a scale transformation of dye solar cells. The recombination current becomes
Voc →λVoc, Vs →λVs. In fact, no dilation can alter a insignificantassoonasthevoltagedropstoV =Voc−Vs
ratio of geometric areas. The invariance of FF for arbi- (Eqs. 3, 6). If V ≪ V (as in fact happens), then the
s oc
trary λ implies that FF can depend on Voc and Vs only photocurrentjumpsfromzerotoclosetoitsshortcircuit
via their ratio: value even if the voltage only drops slightly (i.e., by V ).
s
Notice from Fig. 3 that increases in V — e.g., due to
oc
FF =FF(Voc/Vs) . (11) greaterradiantpower—shouldindeedleadtohigherFF
if V varies much less than V , as in fact occurs.
s oc
The exact functional dependence appears to involve a
transcendental equation. We are still attempting an an-
alytical solution using the Lambert W function. Never-
IV. DISCUSSION AND CONCLUSION
theless, it is susceptible to numerical solution. Fig. 3(a)
shows FF as a function of V /V .
oc s
We next comment on the values typically found for V We briefly commenton conversionefficiency. Since we
s
and their physical significance. The values found cor- cannotexpecttochangeVssignificantly,furtherincreases
respond to negative γ and thus suggest that the con- in efficiency will depend mainly on increasing the open
centration of redox species (I− ions) decreases with the circuitphotovoltage,whichinturnalsorequiresreducing
3
photovoltage. This may at first seem counter-intuitive. losses due to charge recombination at the nanoporous
Indeed, higher voltage suggests larger electron popula- interface — the main challenge indeed. Moreover, we
tion and more injection. Moreover, the regeneration of know that c∼n1/γ, where γ has the negative value γ ≈
the dye creates I− species, in the proportion of one ion −6. Inotherwords,wefindevidenceofafractionalpower
3
for every two electrons injected. law or scaling exponent, indicating self-affine behavior.
So do we face an apparent inconsistency? The impor- We hypothesize that the negativevalue arisesdue to the
tant fact, mentioned earlier, of zero recombination cur- fact that larger n leads to greater recombination, which
rent density J = 0 under short circuit conditions, hints consumes the oxidized species. Localization effects and
r
at the correct explanation: the rate of regeneration of and transport properties play an important role in this
the oxidized dye depends not on the photovoltage (zero context [18].
undershortcircuit)butratheronthe rateofelectronin- In summary, our theoretical results appear to account
jection — thus only on the open circuit photovoltage,or wellforthe observedbehaviorofrealdyesensitizedsolar
alternatively, on the radiant power. The finding agrees cellsandseemtoprovidenewinsightsintotheirfunction-
withtheexpectationofasmallerdepletionlayerformore ing. Amongtheimportantresultsreportedhere,wenote
external current drain. thatFig.3 allowsone to calculateFF knowingVoc/Vs or
The above findings allow us to estimate lower and up- vice versa. Moreover, knowing either one or the other,
per limits for FF. Purely Ohmic behavior corresponds one can readily obtain the entire photocurrent-voltage
to V → ∞ and FF=1/4, however we cannot imagine curve,viaEq. 6. Fromjust3pointsinthephotocurrent-
s
this scenario. For any useful device, the largest conceiv- voltage curve, one can reconstruct the entire curve. The
able value of V should not exceed V , which gives us findings reported here may thus allow further advances
s oc
a lower bound for FF of FF=0.31 and an optimal op- and eventually lead to technologicalinnovations.
erational voltage of V = 0.55 V . By considering We thank BNB, CAPES and CNPq for researchfund-
OPT oc
V = 1/40 Volts, i.e. idealizing uα = 1 , m/γ = 0, and ing andI. M.Gl´eria,A.S. Gonc¸alves,M. L. Lyra,M. R.
s
V = 1, we arrive at an upper bound of FF=0.88 and Meneghetti, S. M. P. Meneghetti, F. A. B. F. de Moura,
oc
V = 0.91 V , as shown in Fig. 3(a). For uα = 0.7 A. F. Nogueira and L. S. Roman, E. C. Silva for discus-
OPT oc
we obtainslightly smallerFF. Fig.3(b) allowsone to es- sion. GMV thanks S. B. Manamohanan for discussions
timate one among V , V and FF from the other two in 1989.
s oc
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