Universal photocurrent-voltage characteristics of dye sensitized nanocrystalline TiO 2 photoelectrochemical cells
aa r X i v : . [ c ond - m a t . o t h e r] J a n Universal photocurrent-voltage characteristics ofdye sensitized nanocrystalline TiO photoelectrochemical cells J. S. Agnaldo, J. C. Cressoni, and G. M. Viswanathan
Laborat´orio de Energia Solar, N´ucleo de Tecnologias e Sistemas Complexos,Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o–AL, 57072-970, Brazil (Dated: October 26, 2018)We propose a new linearizable model for the nonlinear photocurrent-voltage characteristics ofnanocrystalline TiO dye sensitized solar cells based on first principles and report predicted valuesfor fill factors. Upon renormalization diverse experimental photocurrent-voltage data collapse ontoa single universal function. These advances allow the estimation of the complete current-voltagecurve and the fill factor from any three experimental data points, e.g., the open circuit voltage,the short circuit current and one intermediate measurement. The theoretical underpinning providesinsight into the physical mechanisms responsible for the remarkably large fill factors as well as theirknown dependence on the open circuit voltage. PACS numbers: 84.60.Jt, 72.80.Le, 05.40.-a, 85.60.-q
I. INTRODUCTION
An important feature of photovoltaic solar cells andof diverse optoelectronic devices studied in semiconduc-tor physics concerns their current-voltage characteris-tics [1, 2, 3, 4]. The pioneering work that led to theinvention of Gr¨atzel or dye sensitized solar cells becamea milestone in the study of photovoltaic devices [5, 6].Previous theoretical and experimental studies have iden-tified the dependence of the photocurrent and photovolt-age on radiant power [7], but not the precise nonlinear de-pendence of the photocurrent on the photovoltage underconditions of constant radiant power. Moreover, vari-ability in the manufacturing process of dye sensitizedsolar cells can lead to differences — e.g., variables in-clude the choice of dye, the sintering temperature, thick-ness of the nanoporous TiO film and choice of chemicaltreatments. This diversity leads to significant qualita-tive and quantitative variation in photocurrent-voltagecharacteristics and of the relevant quantities such as theopen circuit voltage V oc or the fill factor. Such variabilityhas discouraged attempts to identify (possibly “hidden”)dynamical patterns that could yield important insightsinto the regenerative photoelectrochemical mechanismsthat underlie the conversion process. Given the variabil-ity and diversity in the characteristics, which propertiesremain universal and which nonuniversal? More impor-tantly from an experimental point of view, how can wequantitatively model the photocurrent-voltage character-istics, based on fundamental theoretical principles? Herewe answer these questions by deriving from first princi-ples an analytical expression for the photocurrent.The topic of solar energy in general [8, 9, 10, 11] anddye sensitized solar cells in particular [5, 6, 7, 12, 13, 14,15, 16, 17, 18] attracts broad interest from diverse sectorsof society, due to technological, economic, political andenvironmental considerations. The growing scientific in-terest in dye sensitized TiO solar cells stems from theirunusual features and mode of operation that distinguishthem from Si solar cells: (i) efficient charge separation due to ultra-fast injection of electron from the dye onpicosecond and subpicosecond time scales [14, 17]; (ii)conduction consisting only of injected electrons ratherthan electron-hole pairs [5, 6], due to the wide bandgapof the semiconductor TiO ; (ii) high optical density dueto the extremely large surface area of the dye sensitizednanoporous semiconductor [14]; (iv) negligible charge re-combination with the oxidized sensitizer dye [5, 6, 7];and (v) very high quantum yields [14]. One importantfact that will contribute towards derivation and subse-quent interpretation of the current-voltage characteris-tics concerns how the experimentally measured recombi-nation current density vanishes at short circuit [7] — in-dicating that the only significant recombination pathwayproceeds via back electron transfer into the electrolyte.Indeed, charge recombination between redox species (I − ions) in the electrolyte and conduction band electronslocalized at the nanoporous interface result in subopti-mal photovoltage levels — thus limiting the conversionefficiency [19, 20, 21]. II. METHODS
We begin by assuming that the number of electronsinjected into the conduction band depends only on theincident radiant power — in fact the known very highquantum yields justify this assumption. This assumptionallows us to express the recombination current density J r as a function of the photocurrent density J and the in-jection current density. Since J r vanishes as short circuit,the injection current equals the short circuit current J sc ,so that J r = J sc − J . (1)For a well mixed solution with identical surface and bulkconcentrations (typical for small current densities), theButler-Volmer equation [22, 23] leads to the followingTypeset by REVTEX
Untreated VP TBP
PVP J [ m A / c m ] V [Volts] l n J r Figure 1: Typical photocurrent voltage characteristics undera radiant power of 1.5 AM for dye sensitized solar cells, takenfrom ref. [7]. The four data sets have different characteris-tics due to varied chemical treatments (see text), yet in eachcase the theoretical curves (solid lines) corresponding to Eq. 6can account well for the experimental curves. Inset shows ap-proximate logarithmic relation for the recombination currentdensity J r versus voltage, not inconsistent with Eq. 3. expression for the recombination current density J r : − J r = J (cid:20) exp( − α C uf η ) − exp( α A uf η ) (cid:21) , (2)where J denotes the exchange current density, u thenumber of electrons transferred in the reaction (and con-sequently, the order of the rate of reaction for recom-bination for electrons) α A and α C the anodic and ca-thodic transfer coefficients, η = V the overpotential and f ≡ q/k B T . With some simplification [7], this equationbecomes J r = qk et c m n uα [exp( uαqV /k B T ) − , (3)where α = α A and k et denotes the back electron transferrate constant, c the concentration and m the order of thereaction for the oxidized species, q the electronic chargeand n represents value in dark conditions of the electronpopulation in the semiconductor.How do the prefactors depend on voltage V ? The backelectron transfer rate constant varies with radiant powerand the maximum, i.e. open circuit, photovoltage V oc ,however we do not expect it to depend on the photovolt-age for fixed radiant power. The Nernst Equation forthe potential in terms of the concentrations of the oxi-dized and reduced species holds valid only under equi-librium conditions, yet we know that c varies across theelectrolyte. Indeed, since the reduced species greatly ex-ceeds the oxidized species, we can safely conclude thatthe voltage varies as ∆ V ( x ) ≈ − ( k B T /q ) ln( c /c ( x )),where c ( x ) denotes the concentration at a position x across the cell (i.e., electrolyte), c denotes a reference J (R) V / V OC TBP
PVP
Model
1% Error Untreated VP Y * X* Figure 2: Renormalized photocurrent J ( R ) versus V /V oc forthe curves shown in Fig. 1. The data collapse onto a singleuniversal function stable over a variety of chemical treatmentsand solar cell variability. We have chosen v s = 1 /
40, based onthe value of k B T at room temperature, for illustration, cor-responding to our estimate of an upper bound for fill factors,however we could have renormalized the photocurrent to anyfill factor. Indeed, we can linearize the curves, as shown inthe inset and explained in the text. The dashed line tracesthe upper and lower uncertainties corresponding to an errorof 1% in the short circuit current. Notice how remarkably thedata collapse onto a straight line, within the error tolerance. (or mean) concentration. Nonetheless, we still do notknow, a priori , exactly how it varies with the potentialat the semiconductor-electrolyte interface as the externalload (i.e., impedance) varies, because of the out of equi-librium conditions. In this context, one important cluecomes from the dependence on the photovoltage on theelectron population. Electrons act as charge carriers inthe TiO — just as the redox species do in the electrolyte.In the semiconductor, injected electrons shift the Fermilevel, so that n = n exp( qV /k B T ). This exponentialdependence on potential, together with the exponentialdependence of c on voltage across the electrolyte, hintsat a similar, i.e. exponential, dependence of c on V atthe interface as the external load varies: V = V oc − ( γk B T /q ) ln( c oc /c ) . (4)Here c oc represents the concentration under open cir-cuit conditions and γ represents a free parameter in themodel, quantifying the fraction of the voltage variationthat affects the oxidized species concentration. Since thetriiodide concentration cannot vary very much in the liq-uid, we cannot expect small γ close to equal unity sincethis would imply c ∝ n . For now we only mention thatone would na¨ıvely expect a positive γ since electron in-jection and dye regeneration (associated with larger V )produce oxidized species. -100 -50 0 50 1000.00.20.40.60.81.0 V OC / V S V OPT / V OC =0.55 V OPT / V OC =0.91 FF MIN =0.31 V OC / V S FF MAX =0.88 (a)6 8 10 12 14 16 180.550.600.650.700.750.80 FF (b) Model
Untreated VP TBP PVP FF V OPT / V OC Figure 3: (a) Theoretically predicted fill factors FF and ratio V OPT /V oc of optimal operating voltage V OPT to open circuitvoltage V oc versus V oc /V s , where V s represents a characteristicvoltage (Eq. 7). We estimate a worst-case lower bound on thefill factor using V oc /V s = 1, and an idealized upper boundusing V oc /V s = 40. Inset shows a more complete picture. (b)Comparison of theoretical and experimental values for FF forthe data shown in Fig 1. Useful real cells will likely have FFwithin the range shown. III. RESULTS
These considerations immediately lead to an analyticalexpression for the photocurrent J as a function of thevoltage V across the cell: J = J sc (cid:20) − exp( mq ( V − V oc ) /γk B T )exp( uαqV oc /k B T ) − × (cid:18) exp( uαqV /k B T ) − (cid:19)(cid:21) . (5)Here m ≈ uα ≈ . V > n ≫ n by many orders of magnitude, sothat we can reasonably well approximate Eq. 5 with J = J sc (cid:20) exp( V oc /V s ) − exp( V /V s )exp( V oc /V s ) − (cid:21) , (6)where the potential V s ≡ ( k B T /q ) 1 uα + m/γ ≈ . /γ ) Volts , (7)represents a characteristic scale of the exponential decay.Specifically, V s quantifies the photovoltage drop corre-sponding to a decrease in recombination current densityby a factor of 1 /e where e here denotes Euler’s number.Eqs. 5 and 6 represent the first out of three new resultsof this article.Fig. 1 compares the model with photocurrent-voltagecurves taken from ref. [7], of untreated and pyridinederivative-treated [RuL (NCL) ]-coated nanocrystallineTiO electrodes in CH CN/MNO (50:50 wt %) con-taining Li(0.3M) and I (30mM), for a radiant power of100mW/cm (AM 1.5). The electrodes had treatmentwith following substances: 3-vinylpyridine (VP), 4- tert -butylpyridine (TBP), and poly(2-vinylpyridine) (PVP).The good agreement with the data validates the modelrepresented by Eqs. 5 and 6.The largest possible power output divided by J sc V oc defines the fill factor FF. Notice that V s changes the fillfactor (via γ ). A value V s → ∞ (corresponding to purelyresistive or Ohmic behavior) leads to FF=1/4, whereas V s → T = 0 K. Most dye sensitizedsolar cells have FF=0.6–0.7 (Fig. 1).We now turn our attention to the question of whethera single universal current-voltage relation can describeall TiO solar cells. According to the theory presentedabove, all dye sensitized solar cells must satisfy Eq. 5 ifnot Eq. 6. If we renormalize the photovoltage to obtainan adimensional measure V ∗ ≡ V /V oc , then every singledye sensitized solar cell must satisfy the following relationfor an idealized renormalized photocurrent: J ( R ) ≡ − (cid:20) − ( J/J sc ) (cid:18) − exp( − V oc /V s ) (cid:19)(cid:21) V s /V oc v s − exp( − /v s ) . (8)Here v s fixes the shape or fill factor of the renormalizedphotocurrent. Fig. 2 shows the predicted data collapse.We have chosen a value v s = 1 /
40, due to its significancefor an idealized solar cell with maximum FF (see below)at room temperature. However, we can obtain data col-lapse for any v s (not shown). This is perhaps more clearif we linearize the curves. We define coordinates X ∗ ≡ − V /V oc (9) Y ∗ ≡ − V s /V oc ln (cid:20) − JJ sc (cid:18) − exp( − V oc /V s ) (cid:19)(cid:21) . (10)The inset of Fig. 2 shows how the data collapse onto astraight line. All dye sensitized solar cells thus follow thesame universal pattern of photocurrent-voltage behavior.We next consider the problem from the point of viewof scale invariance symmetry. The fill factor cannotdepend on J sc , since it cancels in the power ratio.It also remains invariant under a scale transformation V oc → λV oc , V s → λV s . In fact, no dilation can alter aratio of geometric areas. The invariance of FF for arbi-trary λ implies that FF can depend on V oc and V s onlyvia their ratio: F F = F F ( V oc /V s ) . (11)The exact functional dependence appears to involve atranscendental equation. We are still attempting an an-alytical solution using the Lambert W function. Never-theless, it is susceptible to numerical solution. Fig. 3(a)shows FF as a function of V oc /V s .We next comment on the values typically found for V s and their physical significance. The values found cor-respond to negative γ and thus suggest that the con-centration of redox species (I − ions) decreases with thephotovoltage. This may at first seem counter-intuitive.Indeed, higher voltage suggests larger electron popula-tion and more injection. Moreover, the regeneration ofthe dye creates I − species, in the proportion of one ionfor every two electrons injected.So do we face an apparent inconsistency? The impor-tant fact, mentioned earlier, of zero recombination cur-rent density J r = 0 under short circuit conditions, hintsat the correct explanation: the rate of regeneration ofthe oxidized dye depends not on the photovoltage (zerounder short circuit) but rather on the rate of electron in-jection — thus only on the open circuit photovoltage, oralternatively, on the radiant power. The finding agreeswith the expectation of a smaller depletion layer for moreexternal current drain.The above findings allow us to estimate lower and up-per limits for FF. Purely Ohmic behavior correspondsto V s → ∞ and FF=1/4, however we cannot imaginethis scenario. For any useful device, the largest conceiv-able value of V s should not exceed V oc , which gives usa lower bound for FF of FF=0.31 and an optimal op-erational voltage of V OPT = 0 . V oc . By considering V s = 1 /
40 Volts, i.e. idealizing uα = 1 , m/γ = 0, and V oc = 1, we arrive at an upper bound of FF=0.88 and V OPT = 0 . V oc , as shown in Fig. 3(a). For uα = 0 . V s , V oc and F F from the other two and will thus find practical application. We estimatedthe error bars for the experimental points from the re-gression fits used to arrive at the values of V s . Devicesthat we constructed locally had values of FF within thesebounds.Finally, our findings explain the very large fill factorsof dye solar cells. The recombination current becomesinsignificant as soon as the voltage drops to V = V oc − V s (Eqs. 3, 6). If V s ≪ V oc (as in fact happens), then thephotocurrent jumps from zero to close to its short circuitvalue even if the voltage only drops slightly (i.e., by V s ).Notice from Fig. 3 that increases in V oc — e.g., due togreater radiant power — should indeed lead to higher FFif V s varies much less than V oc , as in fact occurs. IV. DISCUSSION AND CONCLUSION
We briefly comment on conversion efficiency. Since wecannot expect to change V s significantly, further increasesin efficiency will depend mainly on increasing the opencircuit photovoltage, which in turn also requires reducinglosses due to charge recombination at the nanoporousinterface — the main challenge indeed. Moreover, weknow that c ∼ n /γ , where γ has the negative value γ ≈−
6. In other words, we find evidence of a fractional powerlaw or scaling exponent, indicating self-affine behavior.We hypothesize that the negative value arises due to thefact that larger n leads to greater recombination, whichconsumes the oxidized species. Localization effects andand transport properties play an important role in thiscontext [18].In summary, our theoretical results appear to accountwell for the observed behavior of real dye sensitized solarcells and seem to provide new insights into their function-ing. Among the important results reported here, we notethat Fig. 3 allows one to calculate FF knowing V oc /V s orvice versa. Moreover, knowing either one or the other,one can readily obtain the entire photocurrent-voltagecurve, via Eq. 6. From just 3 points in the photocurrent-voltage curve, one can reconstruct the entire curve. Thefindings reported here may thus allow further advancesand eventually lead to technological innovations.We thank BNB, CAPES and CNPq for research fund-ing and I. M. Gl´eria, A. S. Gon¸calves, M. L. Lyra, M. R.Meneghetti, S. M. P. Meneghetti, F. A. B. F. de Moura,A. F. Nogueira and L. S. Roman, E. C. Silva for discus-sion. GMV thanks S. B. Manamohanan for discussionsin 1989. [1] S. M. Sze, Physics of Semiconductor Devices (J. Wiley& Sons, New York, 1981).[2] D. A. Fraser,
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