Interpretation of the Time Constants Measured by Kinetic Techniques in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells

Transcription

1 J. Phys. Chem. B 2004, 108, Interpretation of the Time Constants Measured by Kineti Tehniques in Nanostrutured Semiondutor Eletrodes and Dye-Sensitized Solar Cells Juan Bisquert*, and Vyaheslav S. Vikhrenko Departament de Ciènies Experimentals, UniVersitat Jaume I, Castelló, Spain, and Belarussian State Tehnologial UniVersity, Minsk, Belarus ReeiVed: May 20, 2003; In Final Form: NoVember 18, 2003 The proesses of harge separation, transport, and reombination in dye-sensitized nanorystalline TiO 2 solar ells are haraterized by ertain time onstants. These are measured by small perturbation kineti tehniques, suh as intensity modulated photourrent spetrosopy (IMPS), intensity modulated photovoltage spetrosopy (IMVS), and eletrohemial impedane spetrosopy (EIS). The eletron diffusion oeffiient, D n, and eletron lifetime, τ n, obtained by these tehniques are usually found to depend on steady-state Fermi level or, alternatively, on the arrier onentration. We investigate the physial origin of suh dependene, using a general approah that onsists on reduing the general multiple trapping kineti-transport formalism, to a simpler diffusion formalism, whih is valid in quasi-stati onditions. We desribe in detail a simple kineti model for diffusion, trapping, and interfaial harge transfer of eletrons, and we demonstrate the ompensation of trap-dependent fators when forming steady-state quantities suh as the diffusion length, L n, or the eletron ondutivity, σ n. 1. Introdution Spatially heterogeneous mixtures of nanometer-sale onstituents form new lasses of solar ells. These solar ells, onsisting on a ombination of semiondutor nanopartiles, redox eletrolytes, onduting polymers and photoative organi moleules, are quite appealing due to easy proessability of the materials in a large sale, showing promise for heap and versatile photovoltai devies. 1 The dye-sensitized solar ell (DSSC) is a heterogeneous solar ell where the arriers transferring the hemial energy reated in an exited dye are eletrons in nanorystalline TiO 2 and redox speies in a liquid eletrolyte. Besides being highly effiient for light to eletrial energy onversion, the DSSC is a good model system for the heterogeneous photovoltai onverters beause the different phases in it are ontinuously onneted and physially separated. In general, the heterogeneous onfiguration is widely investigated beause it has the advantage of providing a huge internal area where harge separation an be realized following exitation of the light absorber. It is also essential for the onversion effiieny to maintain the separated arriers in their respetive nanosaled independent hannels until they are olleted at the ontats. Therefore, the determination of quantities suh as the diffusion oeffiient and the lifetime of the different arriers beomes a entral issue in the investigation of these devies. These time onstants are obtained by small perturbation kineti measurements that do not modify the steady state over whih they are measured. Examples of the tehniques are intensity modulated photourrent spetrosopy (IMPS), intensity modulated photovoltage spetrosopy (IMVS), eletrohemial impedane spetrosopy (EIS), and small amplitude time transients. The results of these tehniques in nanostrutured semiondutors and DSSCs indiate the dependene of the eletron * To whom orrespondene should be addressed. bisquert@uji.es. Universitat Jaume I. Belarussian State Tehnologial University. diffusion oeffiient on the eletrohemial potential of eletrons, µj n (also denoted quasi-fermi level, E Fn ). 2-9 For DSSCs, a large variation of the eletron lifetime with inreasing light intensity has been reported as well. 5,10,11 From these observations, it is often inferred that trapping mehanisms mediate transport and reombination in nanostrutured semiondutors permeated with a ondutive phase. 2,5,6,12-14 A major researh effort has been aimed toward measuring the effetive variable diffusion oeffiient, D n, and the effetive variable lifetime, τ n, observed in DSSCs in different onditions. In this paper, we aim at a better physial understanding of these effetive parameters. An interesting observation on the effetive D n and τ n has been indiated by Peter and o-workers. They found that in some ases the produt of these quantities ompensates to a large extent, forming a nearly onstant eletron diffusion length, L n. 5,15 On another hand, reent measurements 16 of the eletroni ondutivity, σ n, of nanostrutured TiO 2 in aqueous eletrolyte show an inrease of nearly 60 mv/deade over a wide range of Fermi level positions with respet to the ondution band potential. This dependene an be explained by an expression of the type σ n ) e2 k B T n D 0 (1) where n is the density of free eletrons in the ondution band, D 0 is the onstant diffusion oeffiient for free eletrons, e is the positive elementary harge and k B T is the thermal energy. Thus, it appears that quantities that an be measured at steadystate, suh as L n and σ n, do not ontain the information on trapping and detrapping effets that is obtained when D n is measured diretly by kineti tehniques. From these and other results, it appears relevant to larify the interpretation of photophysial quantities measured in nanostrutured semiondutors and DSSCs, onsidering both /jp035395y CCC: $ Amerian Chemial Soiety Published on Web 01/23/2004

2 2314 J. Phys. Chem. B, Vol. 108, No. 7, 2004 Bisquert and Vikhrenko 2. Chemial Diffusion Coeffiient 2.1. Kineti-Transport Formalism. We state the kinetitransport equations in the MT model for a single kind of trap, onsisting on the equations of onservation for free and trapped eletrons and Fik s law for the free eletrons: 17 t )- J x - βn [1 - f L ] + ɛn L f L (2) f L t ) β n N L [1 - f L ] - ɛf L (3) Figure 1. Shematis of the steps involved in transport of the photoinjeted eletron and the reombination with the oxidized speies in the eletrolyte in a dye-sensitized solar ell. (E F0) shows the position of the Fermi level in the dark, equilibrated with the redox potential (E redox) of the aeptor speies in solution. (E Fn) is the (quasi)fermi level of eletrons under illumination and E is the ondution band energy. The following steps are indiated: (A) Eletron transport through extended states; (B) eletron apture and thermal release at an exponential distribution of band gap loalized states; (C) eletron transfer through ondution band to the flutuating energy levels of oxidized speies in solution indiated in the right; (D) apture by and (F) harge transfer through surfae states. the time onstants obtained from kineti measurements and the steady-state quantities suh as the eletron diffusion length, the eletron ondutivity, and the inident photon-to-urrent onversion effiieny (IPCE). Indeed, both lasses of parameters indiate a different kind of information. The former refer to the time for reovery of equilibrium, either by transport or reombination, i.e., they orrespond to a swithing time, whereas the latter refer to stationary operation and determine the photovoltai effiieny of the devie. It is important, therefore, to examine the relationship of time onstants D n and τ n, obtained from marosopi evolution of arrier densities, to the mirosopi assumptions on eletroni transitions and distribution of states, to see how these onstants, desribing in priniple different phenomena in the solar ell, relate to eah other and how they behave when they are ombined to form other important quantities suh as the IPCE. To analyze these questions, we outline a relatively simple kineti model for diffusion, trapping and interfaial harge transfer of eletron arriers in a nanostrutured semiondutor permeated with a redox eletrolyte. We will use the multiple trapping (MT) model for transport and harge transfer illustrated in Figure 1. This model is adapted to nanostrutured semiondutors from a wide experiene on disordered semiondutors. The MT transport, is summarized by Shmidlin 17 and Tiedje and Rose, 18 and was applied first by Vanmaekelbergh 2 for nanoporous semiondutors, and extended by many workers in the DSSC area. 2,5,6,9,12 The key feature of the MT framework is the restrition that only free eletrons ontribute to the diffusion urrent. 17 Diffusion by diret hopping between loalized states is also possible in materials with a wide distribution of traps, but this mehanism will not be onsidered in this paper. For trapping and reombination, the ideas formulated by Rose 19 an be adapted to DSSC as indiated in ref 11. Here we ombine both aspets, MT transport and reombination, in a single model that may be onsidered a working model that gives an overall view of the more relevant phenomena in the DSSC and shows learly the interonnetion between measured quantities. The meaning of the Fermi-level dependene of D n and τ n will beome transparent in terms of trapping fators and we will show the ompensation of these trapping fators when we form the steady-state quantities L n and IPCE. J )-D 0 x Here N L is the total density of loalized sites (per unit volume), f L is the frational oupany (n L ) N L f L ), and J is the diffusive flux of ondution band eletrons. The rate onstant for eletron apture is determined by the thermal veloity of free eletrons, V, the eletron apture ross setion of the trap, s n, and the density of traps, N L The rate onstant for eletron thermal release from the trap to the ondution band is related to β by Shokley-Read-Hall statistis 20 Here, E is the lower band edge energy, E L is the energy of the loalized state in the band gap, and N is the effetive density of ondution band states. Equations 2-4 an be readily extended to a distribution of loalized levels. We have omitted in these equations the rate of interfaial harge transfer (reombination) whih will be onsidered in setion 3. From eq 3 in steady state, f L / t ) 0, it follows that the eletrons in the free and loalized states maintain an equilibrium with a ommon value of the Fermi level, µj n. The oupanies in the two kinds of states are given expliitly by Note that, in eqs 7 and 8, the Fermi level µj n an be maintained at a different value than the redox potential in solution, E redox (assuming that the exhange of eletrons at the oxide/solution interfae is slow). This is a onstrained equilibrium 21 of the system formed by the eletrons in extended and loalized states in the nanoporous semiondutor in ontat with redox eletrolyte. If the onstraint (bias potential or illumination) is removed, then the system equilibrates all of the eletrons at the same eletrohemial potential, and in this ase, µj n has the unique value E redox. We now define the onditions that enable the redution of the MT framework of eqs 2-4 to the onventional diffusion equations, onsisting in the ordinary onservation equation and Fik s law Quasiequilibrium Condition. In general the quasi-stati ondition applies in thermodynami proesses that are suffiiently slow for the hange to onsist in a suession of (4) β ) N L Vs n (5) ɛ ) N N L β exp[-(e - E L )/k B T] (6) n ) N e (µj n-e )/k B T 1 f L ) 1 + e (E L-µj n )/k B T (7) (8)

3 Interpretation of the Time Constants J. Phys. Chem. B, Vol. 108, No. 7, equilibrium states. In our ase, we are onsidering a system omposed by two lasses of eletroni states, initially at (onstrained) equilibrium indiated by the ommon value of the temperature and the eletrohemial potential in eqs 7 and 8. When equilibrium at µj n is perturbed by some external ause (for example, injeting n eletrons to the ondution band), the subsequent variations ( / t) and ( n L / t) are ruled by the instantaneous oupanies and transition rates desribed in eqs 2-4. We define a partiular kind of evolution as that whih obeys the quasistati ondition n L ) n L t t so that free and trapped eletrons maintain a ommon equilibrium even when the system is displaed away from equilibrium. In pratie, this implies that trap relaxation is muh faster that the frequeny/times of interest in the measured phenomena, for instane, faster than the transit time through the film while measuring diffusion oeffiient. Equation 9 may be written alternatively in terms of kineti fators for trapping and detrapping, n L / ) β/ɛ. However, normally the priniple of detailed balane (that states that for a system in thermal equilibrium, the rates of a proess and of its inverse are equal and balane in detail) is taken as a representation of mirosopi reversibility. 22 Given the rate onstant β, detailed balane gives ɛ in eq 6 through the equilibrium oupanies. 20 So the fator n L / between equilibrium oupanies in eq 9 appears more fundamental in order to assert that the proportion of the rates of hange of populations of loalized and free eletrons, ( n L / t)/( / t), maintains those populations at the ommon equilibrium values. We remark that the fator ( n L / ) is not the proportion of number of arriers, but rather the relation of variations indued during the small perturbation that leaves invariant the steady state. Inidentally, it is found that n L / n L /n in some ases (as in exponential distribution of traps), but this is not generally true. It is generally possible to establish the time sale of the trapping-detrapping phenomena, and the time onstant of the proess under onsideration must be onsiderably longer to guarantee the possibility of using the quasi-equilibrium relation between n and f L. In the limit of long waves the relaxation time for trapping-detrapping is determined by the expression τ t -1 ) β(1 - f L + n /N L ) + ɛ. Long waves mean that k 2, (D 0 τ) -1, where k is the wave vetor determining the harateristi spatial nonhomogeneities. n and f L orrespond to some quasiequilibrium value of the hemial potential. In frequeny methods, the harateristi patterns of relaxation funtions show the onset of traps relaxation at ω τ t -1, see, for example, refs 23 and 24. There are onditions in whih eq 9 is not satisfied in the frequeny window of the measurement, for instane if the transit time is, τ t ; or else, if harge injetion to the eletrolyte is rather fast Redution of Multiple Trapping to Ordinary Diffusion. Hereafter, we assume, unless otherwise stated, that the measurement operates in quasistati onditions. From eqs 2, 3, and 9, it follows that ( 1 + n L ) t )- J x Equation 10 suggests to form a new partile flux, Ĵ, as (9) (10) where D n is defined as Ĵ )-D n x D n ) (11) n D 0 (12) L and then, provided that D n is approximately independent of position (homogeneous Fermi level), eq 2 an be expressed as t )- Ĵ x (13) The new oeffiient D n obtained in eq 12 will be interpreted more generally in another paper 25 as the hemial diffusion oeffiient of eletrons. Clearly, the redution of MT to fikean diffusion of free arriers is ahieved in eqs The simplifiation involves the removal of some internal degrees of freedom in the system (the oupany of loalized states) that are not expliitly resolved in the quasistati measurement but ontribute to the hemial diffusion oeffiient D n, whih heneforth beomes a funtion of the onentration, D n (n ), or Fermi level, D n (µj n ). Therefore, the experimental results of small perturbation quasistati measurements onsist on an ordinary diffusion proess that takes plae with the hemial diffusion oeffiient, D n.in other words, the transport-kineti equations that desribe the measured transients or frequeny spetra an be onsiderably simplified by heking that quasiequilibrium of trapping is obeyed. In pratie, this type of interpretation has been often adopted, as pointed out in the Introdution, and explained, for instane, in Appendix C of ref 26. Our analysis shows a quite general justifiation for this approah to the analysis of the data. The general signifiane of the result in eq 12 is onfirmed in partiular ases found in the literature, for instane from the omplete solution of the single trap model in EIS 23 and also in IMPS. 27 In these papers, eq 12 is obtained in the low-frequeny limit of the solution of eqs 2-4 for a small perturbation in the frequeny domain. The effet of trapping in the hemial diffusion oeffiient is important only insofar as n L /. 1, as disussed further below. Therefore, normally it is justified to redue eq 12 to the expression D n ) ( n L) D 0 (14) It is worth to emphasize that in general D n (n ) and / x do not ommute, so that in onditions of nonhomogeneous steady-state Fermi level (as in IMPS) eq 13 is not valid. The orret quasiequilibrium transport equation an be formulated using eqs 4 and 10, whih give t - D 0( n L) 2 n x 2 ) 0 (15) 2.4. Chemial Diffusion Coeffiient in Traps Distributions. In the quasistati approximation the fator ( n L / ) -1 an be alulated for any distribution of loalized levels, with abundany g(e) (the density of loalized states, DOLS, at the energy E in the band gap) and oupanies f L (E - µj n ), using the equilibrium distribution of free and trapped arriers indiated

4 2316 J. Phys. Chem. B, Vol. 108, No. 7, 2004 Bisquert and Vikhrenko in eqs 7 and 8. In the approximation of the zero temperature limit of the Fermi funtion, i.e., a step funtion at E ) µj n separating oupied from unoupied states, a hange of Fermi level implies a hange of loalized harge orresponding to the DOLS On another hand, for free eletrons, the Boltzmann statistis of eq 7 gives therefore From eqs 14 and 18 a general expression is found of the hemial diffusion oeffiient: We illustrate this general expression with the derivation of two ases obtained in previous works. 5,8,9,12 For the box distribution of width ɛ L one gets hene whih explains the phenomenologial generalized diffusion equations (with D n n ) used in ref 3 for analyzing transient photourrents. For the exponential distribution with tailing parameter T 0 (with R)T/T 0 ) we obtain from eqs 7 and 18 and it follows that n L µj n ) g(µj n ) (16) µj n ) n k B T (17) n L ) k B T g(µj n n ) (18) D n ) n k B T g(µj n )-1 D 0 (19) g(e) ) N L /ɛ L (20) n L ) N L k B T (21) n ɛ L D n ) n ɛ L N L k B T D 0 (22) g(e) ) N L k B T 0 exp[(e - E )/k B T 0 ] (23) n L )R N L N n R-1 R (24) D n ) N R n 1-R RN D 0 (25) L The result D n I 1-R 0, derived in equation A.11 of ref 26 for the effetive diffusion oeffiient dependene on light intensity, I 0, is similar to eq 25, assuming that n I 0. Experimental observations do show the power-law dependene of the measured, hemial diffusion oeffiient on onentration or light intensity, 5,28 so that MT with the exponential tail of band gap states indiated in eq 23 seems a plausible model for the DSSC. Taking into aount that n L ) Ev E g(e)fl (E - µj n )de ) N L exp[(µj n - E )/kt 0 ] (26) and using eq 7, it follows that n L ) N L N n R R (27) In multiple trapping onditions n L. n, i.e., the total harge n tot n L ; therefore, eq 25 an be written in terms of total eletron density in the following way N D n ) RN n (1-R)/R 1/R tot D 0 (28) L It should be remarked that a ontinuous trap distribution usually auses speifi patterns of anomalous diffusion. An analysis of frequeny features for multiple trapping diffusion in the presene of an exponential DOLS is presented in ref 29. The time domain equations for frational time diffusion in MT are disussed in ref Eletron Lifetime (Response Time) The analysis of the time onstant for reombination in nanostrutured semiondutors requires to onsider two essential fators represented in the sheme of Figure 1. First, the bulk of a nanopartile may ontain a large density of traps, e.g., with the exponential form of eq 23, where trapping and detrapping phenomena take plae, idential to those analyzed before in relation with diffusion (proess B). On another hand, reombination is an interfaial hargetransfer event and ours in the surfae only. Charge transfer may involve a variety of interfaial mehanisms (ondution band and surfae states, as indiated in Figure 1). 31 Let us assume for the moment that reombination ours preferentially through the ondution band (proess C), at a rate U 0 )- n τ n0 (29) so that τ n0 is the onstant free arrier lifetime; that is, the lifetime with respet to injetion to the eletrolyte, in the absene of trapping. The measurement of the eletron lifetime onsists of determining the time for the system to reover equilibrium under a small perturbation of the steady state, by removal of the exess arriers by reombination. 11 In DSSC, the lifetime an be determined by monitoring diretly the variation of the position of the Fermi level with time (open-iruit photovoltage deay tehnique, OCVD). 11 To desribe this evolution, one an solve the kineti eqs 2-4, where eq 29 is an additional term in eq 2. However, this is not neessary provided that ertain onditions are satisfied. The effet of trapping and detrapping in the bulk is simplified by the quasi-equilibrium ondition of eq 9, provided that the rate onstants for trapping and detrapping are muh faster than τ n0. Therefore, the displaement of the Fermi level involves the reombination by interfaial harge transfer of both the trapped and free harge, and the observed time onstant is onsiderably longer than τ n0. 19 Furthermore, the proess of

5 Interpretation of the Time Constants J. Phys. Chem. B, Vol. 108, No. 7, ondution band transport (A) is assumed to be also fast so that no relevant inhomogeneities of harge exist (this point is further disussed in the Appendix). Then the time onstant for the deay takes the form τ n ) ( 1 + n L ) τ n0 (30) as shown in detail in a reent paper. 11 Equation 30 was formulated for amorphous semiondutors by Rose, 19 and the observed reombination time, τ n, is denoted the response time. In the ase n L /. 1, in whih trapping and detrapping governs the response time, eq 30 an be simplified τ n ) ( n L ) τ n0 (31) Equation 31 implies a Fermi-level dependene of τ n as we have seen above for D n. Using the exponential distribution of eq 23 in eq 31, we obtain with eq 24 the power law form on free eletron onentration (exponential dependene on illumination intensity) that is usually found in measurements 11 τ n ) RN N n R-1 R τ n0 (32) Besides injetion from the ondution band, another important reombination hannel is a two-step proess involving trapping at band-gap surfae states (D) and subsequent isoenergeti transfer to eletrolyte levels (F). In addition, there may exist a distribution of surfae states that partiipate in harge transfer. 10 The different possibilities have been analyzed theoretially in ref 31, but experimentally, the details about the relevant levels for harge transfer have not yet been larified, and the subjet lies outside the sope of the present work. However, it is important to realize that the result of eq 31 is valid generally in the quasiequilibrium onditions stated above. 11 Indeed, the previous argumentation is the same when the quantity τ n0 of eq 31 involves a ombination of interfaial mehanisms. In this ase, however, τ n0 may aquire its own dependene on the steady state, as disussed in ref 11. In this paper, it was shown that eq 31 desribes orretly the major features of the response time in DSSC, but more work is neessary in order to establish the details of τ n dependene on Fermi level. The analysis of transient deays in ref 32 in terms of mirosopi models for eletrons transitions provides more detailed insight into the temporal evolution of reombination mehanisms. Nelson et al. 32 pointed out that reombination in the multiple trapping regime is governed by arriers redistribution in the energy levels, and ref 33 onfirmed this idea, whih supports our simpler, quasiequilibrium approah. However, these papers did not alulate the steady-state time onstants onsidered here. 4. Eletron Diffusion Length We now turn our attention to the measurements that are realized in steady-state onditions, for instane by measuring the photourrent in the solar ell at a onstant inident illumination. In those ases, imposing / t ) 0 and n L / t ) 0, eqs 2-4 redue to D 0 2 n x 2 - n τ n0 + G ) 0 (33) Figure 2. Shematis of the apaitive ontributions in a dye-sensitized solar ell: (A) Chemial apaitane due to inreasing hemial potential (onentration) of eletrons in the TiO 2 phase, obtained when the eletrode potential, V, displaes the eletron Fermi level, E Fn, with respet to the lower edge of the ondution band, E, in the semiondutor nanopartiles. (B) Eletrostati apaitane of the Helmholtz layer (and semiondutor bandbending) at the interfae between the exposed surfae of the transparent onduting oxide substrate and the eletrolyte. (C) Eletrostati apaitane at Helmholtz layer at the oxide/eletrolyte interfae. The remaining transport equation is equivalent to the standard diffusion model for DSSC 34 and ontains no information on trapping. This is beause in the steady state the traps simply adjust their oupany to the Fermi level. Aordingly the diffusion length is given by a onstant, L n ) (D 0 τ n0 ). The IPCE, whih depends only on L n and geometrial fators, 34 is also onstant. Both L n and IPCE an be measured from the steady-state photourrent. On the other hand, determinations of the diffusion oeffiient and lifetime by kineti measurements provide D n and τ n,as disussed in the previous setions. However, from eqs 12 and 31, we realize that the fators ( n L / )ind n and τ n ompensate when forming the diffusion length from measured quantities. The result is a onstant onsistent with eq 33 L n ) D n τ n ) D 0 τ n0 (34) The meaning of the ompensation is lear when we note that the origin of the fator ( n L / ) lies in arrier equilibration in the energy spae (proess B in Figure 1), both for hemial diffusion oeffiient in MT (D n ), and for response time (τ n ). Peter and o-workers 5,15 and also Nakade et al. 28 have reported for DSSCs the ompensating behavior indiated in eq Chemial Capaitane In the previous setions, the DOLS of eletrons was seen to exert a onsiderable influene over the measured time onstants. Fortunately, the DOLS an be determined quite diretly in nanostrutured semiondutors, by measurements of apaitane. It is onvenient to emphasize the diret relationship between the measured apaitane and thermodynami funtion (hemial potential) of eletrons, otherwise one may loose valuable information by attempting to desribe the apaitane in terms of onventional ideas of dieletri onstant and spae-harge regions. Therefore, we try to larify the point in the following disussion Eletrostati and Chemial Capaitors. The apaitane of nanoporous semiondutor films an be determined in several ways: EIS, 35,36 yli voltammetry, 37 or integrating the urrent at differential voltage steps. 38 There are several physial effets ontributing to the measured apaitane, as indiated in Figure 2. The proess (B) indiates polarization at the interfae between the transparent onduting substrate (TCS) and the eletrolyte, and (C) indiates the Helmholtz layer at the oxide/eletrolyte interfae. The former effet is important when the eletron density is low in the semiondutor 39 and

6 2318 J. Phys. Chem. B, Vol. 108, No. 7, 2004 Bisquert and Vikhrenko the latter when the density is very high and the semiondutor enters the state of band unpinning. Both these ontributions an be thought of as ordinary eletrostati apaitors, where the harges in two highly onduting plates sustain an eletrial field in between. In the intermediate range of Fermi level variation, a different kind of apaitive effet is found (A). The semiondutor bands are pinned, and the harge aumulation is related to the displaement of the Fermi level position with respet to the ondution band edge, i.e., to a variation of hemial potential of eletrons, µj n ) µ n. Hene, the inrement of harge (eletroni and ioni) ours in the volume of the nanostrutured eletrode with no onomitant eletrial field variation in the volume, beause the eletrial field is shielded near the TCS. 40 Therefore, (free) energy storage in the apaitor is by hemial, not eletrostati, energy. As a onsequene, it is a hemial apaitor, and not an eletrostati apaitor. To appreiate the physial basis for this new onept, we remind that the impedane funtion an be defined generally for any thermodynami system, and haraterizes the linear response of the system to an applied fore. 41 Indeed, the flutuation-dissipation theorem 41 shows that the admittane is related diretly to the equilibrium flutuations of the system, and although the ondutivity haraterizes the irreversible response of the system, a lossless element that indiates the reversible response onstitutes a apaitane. 42 In partiular, for a volume element that stores hemial energy due to a thermodynami displaement, the hemial apaitane per unit volume is defined as 43 C ) e 2 N i µ i (35) So the hemial apaitane reflets the apability of a system to aept or release additional arriers with density N i due to a hange in their hemial potential, µ i. 43 The general physial meaning of eq 35 has been explained reently using another route. 44 In mesosopi apaitors, 44 the eletrial field related to the eletrohemial potential differene between the leads is partially shielded and it annot propagate toward the surfae of the plates of the apaitor, whih auses a displaement of the Fermi level with respet to the ondution band. Büttiker et al. 44 have shown that this effet introdues a fator proportional to dn/dµ n in the eletrohemial apaitane, in agreement with eq Chemial Capaitane in Nanostrutured Semiondutors. The hemial apaitane (A in Figure 2) is a major feature in TiO 2 nanostrutured eletrodes. For instane, in measurements of these eletrodes in aqueous solution, this effet auses an exponential inrease of the apaitane by 3 orders of magnitude in an interval of potentials of 0.8 V. 37 Similar results are obtained in DSSC. 35 Besides, the hemial apaitane is a onept of ruial signifiane for solar ell appliations, beause it desribes properly the splitting of Fermi levels aused by exitation of arriers in the light absorber material. 45 Considering the variation of the eletron density upon a hange of the loal hemial potential in a DSSC, we obtain for the total hemial apaitane C (tot) h ) e 2 (n + n L ) (36) µ n One may distinguish the two omponents in eq The first is related to the free ondution band eletrons. Using eq 17, we find C (b) h ) e 2 ) e n 2 µ n k B T (37) The seond omponent of eq 36 is related to loalized states in the band gap. From eq 16, it is seen readily that this omponent is just proportional to the DOLS at the Fermi level C (trap) h ) e 2 n L ) e 2 g(µj µ n ) (38) n In the ase of the exponential distribution given in eq 23, eq 38 provides the form C (trap) h ) e 2RN L k B T e(µj n-e )/k B T 0 (39) We may also express eq 39 in terms of the free eletron density, in whih ase we obtain C (trap) h ) e RN 2 L k B TN n R R (40) The two hemial apaitors in eq 36 are onneted in parallel. (trap) If it is C h. C (b) h, as required in the trapping models, resolving C (b) h is not possible by simply measuring the low (tot) (trap) (b) frequeny apaitane, whih is C h ) C h + C h C (trap) h. 47 Note that both eq 37 and 39 show an exponential dependene with the bias, although with different slopes. The ideal statistis of eq 37 give a slope (d log C/dV) ) -e/ (2.30k B T), i.e., 60 mv/deade at room temperature. This is not normally found in nanostrutured TiO 2. The exponential apaitane is observed with a muh less steep rise, of about 300 mv/deade. This has been interpreted in terms of eq 39, i.e., the manifestation of the exponential distribution of band gap states whih gives (d log C/dV) )-e/(2.30k B T 0 ), 35,37 with T K. For instane, measurements of apaitane of nanostrutured TiO 2 eletrodes in aqueous solution at ph 3 yield a value of R)0.25 and a total trap density of N L m The exponential DOLS and mentioned (d log C/dV) values are also supported by the results of a stepping harge-extration method. 48 It must be remarked, however, that the exponential distribution that desribes well different piees of experimental data (suh as the hemial diffusion oeffiient and lifetime) is so far a phenomenologial formula the origin of whih is not well understood. So we hoose to use in Figure 1 the simplest approah to desribe this feature, whih is a stationary distribution of eletron sites related to intrinsi disorder, usually found in amorphous semiondutors. However, in doped rystalline semiondutors, the spread of ondution band states is aused by loal distortion of energies near the dopant atoms. In rystalline nanopartiles surrounded by eletrolyte, interation effets between the eletrons and ioni speies that modify the eletrons energy levels annot be disarded to aount partially for the tailing distribution. As mentioned before, surfae harging hanges the potential differene in the Helmholtz layer, produing an upward shift of the semiondutor energy levels, V H ) φ H. The ombined effet of eletron aumulation and partial band unpinning implies that the Helmholtz apaitane, C H, is onneted in series 37 to the hemial apaitane, C h, so that the position of the bands will remain pinned insofar as C h, C H.

7 Interpretation of the Time Constants J. Phys. Chem. B, Vol. 108, No. 7, Resolution of Free Carrier Time Constants. We have ommented that the hemial apaitane provides diret information on the density of states in the nanostrutured semiondutor. It is important therefore to emphasize that the measurements of apaitane indiate an exponential DOLS with muh higher apaitane than the ondution band omponent, C (trap) h. C (b) b, i.e., n L /. 1, as this is a neessary requisite for the interpretation of time onstants in terms of trapping models that we have exposed in the previous setions. This point an be expressed in quantitative form, leading to interesting onsequenes. Indeed note that the hemial diffusion oeffiient of eq 14 for MT model may be written, alternatively, in terms of the hemial apaitanes of eqs 37 and 38, as 12 D n ) (b) C h C (b) h + C D (trap) 0 (41) h So MT transport ours when C (trap) h. C (b) h, giving the result D n ) C (b) h C D (trap) 0 (42) h whih is idential to eq 14. Also eq 25 for the exponential DOLS an be obtained from eqs 37, 40, and 42. Similar identifiations an be made with the response time of eq 31, i.e. τ n ) C (trap) h C τ (b) n0 (43) h In the previous setions, we have remarked that the time onstants measured in kineti tehniques are Fermi-level dependent, D n (µj n ) and τ n (µj n ), as indiated in eqs 42 and 43. However, this is when the trapping degrees of freedom are obsured in the quasi-equilibrium measurement, as explained in setion 2. Now eqs 42 and 43 show that it is really possible to measure the free eletrons diffusion oeffiient and lifetime, D 0 and τ n0, whih are determining the diffusion length, eq 30, but for this, it is neessary to resolve separately the free and trapped harge through the orrespondent hemial apaitanes. As mentioned before, the low frequeny apaitane gives only C (tot) h C (trap) h. The EIS tehnique, for example, permits the observation of relaxation of the harge in extended states at high frequenies (f. Figures 1 and 4 in ref. 23), where traps do not respond anymore (i.e., breaking quasi-equilibrium at short (b) times). In this way, C h would be determined from highfrequeny data. Note that this proedure requires resorting to the omplete impedane model inluding the traps relaxation. 23,47,49 In a similar way, it should be possible to separate free and trap omponents of the hemial apaitane by light absorption tehniques, and this is disussed in ref Relationship of Chemial Capaitane to Diffusion. Although the manifestation of traps in the hemial apaitane is a neessary ondition for MT diffusion, as already remarked, it must also be pointed out that the measurements of apaitane do not give detailed information on the diffusion proess itself. Chemial apaitane indiates the equilibrium distribution of eletrons in the available states of the system or, more generally, the hemial potential of eletrons, but not the proess of transport between those states. The apaitane does not indiate whether the loalized states belong to the surfae or interior of nanopartiles, whih is an important issue for reombination models. Traditionally, the hemial apaitane assoiated to the storage of ondution band eletrons, eq 37, was observed in solid-state pn juntions at a forward bias and was termed a diffusion apaitane. This denomination, adopted reently in some papers in the DSSC area 50,51 (and also used by us sometimes), is not very fortunate beause diffusion is an irreversible energy loss proess, whereas the apaitane is a reversible energy storage element. It is well-known that diffusion is aused by a loal differene of hemial potentials. The hemial apaitane, distributed in spae, is the element that provides a hemial potential that depends on the position. So the hemial apaitane is a prerequisite for diffusion (and this is the reason the hemial apaitane is always a omponent of diffusion impedane, 43 either in DSSCs or solid-state pn juntions 52,53 ). However, the onverse is not true, the apaitane of eq 37 is not diffusional in origin Eletron Condutivity Another important quantity for many appliations is the eletron ondutivity, whih an be measured in the steady state as reported elsewhere. 16 In the ontext of the MT model, the eletron transport is arried by a single kind of state, the extended states of the ondution band. Carriers trapped in loalized states do not ontribute to the d ondutivity until they are released again. The ondutivity related to the eletron diffusion proess an be obtained from the generalized Einstein relation σ n ) e 2 µ n D 0 (44) where µ n is the hemial potential of eletrons. From eq 37, we an write eq 44 as σ n ) C (b) h D 0 (45) and the standard expression of eq 1 is obtained with the seond equality of eq 37. Let us analyze the onditions required for determining the free arrier difusion oeffiient from the ondutivity. From eq 42, we an write the ondutivity also as σ n ) C (trap) h D n (µj n ) ) e 2 g(µj n )D n (µj n ) (46) The first equality of eq 46 shows that the quotient of quantities σ n /C h, whih an be measured at low frequeny, gives the hemial diffusion oeffiient D n, so again we need C (b) h for obtaining D 0. From eq 1, we an obtain D 0 from σ n if n (E - µj n ) is known, but this also requires to resolve the free eletrons omponent of the hemial apaitane. 7. Final Remarks and Conlusion In this paper, we have disussed the interpretation of photoeletrohemial tehniques in nanoporous semiondutor eletrodes in terms of the model for eletron diffusion, trapping in the bulk and reombination indiated in Figure 1. We argued that the effets of trapping appear in transient and kineti quantities but not in steady-state quantities. Time onstants suh as the hemial diffusion oeffiient, D n, and eletron response time, τ n, are measured by means of small perturbation of the steady state. These time onstants aquire dependenies on the steady-state Fermi level due the presene of internal degrees of freedom orresponding to trapping and detrapping of eletrons that are not observed separately. In ontrast, the free arrier

8 2320 J. Phys. Chem. B, Vol. 108, No. 7, 2004 Bisquert and Vikhrenko diffusion oeffiient, D 0, and the free arrier lifetime, τ n0, annot be measured separately using tehniques at quasi-equilibrium onditions. However, the free arrier time onstants an be inferred from D n, τ n, and total harge relaxation C (tot) h, all of whih an be measured at low frequenies, provided that additional information on free arrier density is available. In addition, quantities suh as L n, σ n, and IPCE an be measured diretly in the steady state. Kineti effets of the multiple trapping disappear in L n, σ n, and IPCE, beause in the steady state the trap oupany remains stationary. Illustrations of our interpretation of measured time onstants with impedane 23 and optoeletrial tehniques (IMPS) 27 have been mentioned. Another example of this is found in the model of Vanmaekelbergh et al. 4 that onsiders a ombination of proesses in nanoporous TiO 2 eletrodes in aqueous solution. It an be seen form their results of the optoeletrial transfer funtion (IMPS) (see eqs 15 and 20 in ref 4) that the low-frequeny limit, i n /eφ(0), orresponding to steady-state photourrent quantum yield, is independent of internal traps parameters. On the ontrary, the IMPS frequeny ω min, related to the transit time as τ d ) 2.5/ω min, 4 is mainly determined by trapping fators. Having introdued different diffusion oeffiients, i.e., the hemial and free arrier diffusion oeffiients, we may ask whih is their relative signifiane. D n provides the time for restoring equilibrium by transport when an exess of arriers is injeted, whereas D 0 determines (with the arrier density) the arrier flux at the steady state. So one may be more interested in one or the other depending on the partiular devie and appliation. For instane in photoopiers the transient behavior of exess arriers generated by a flash of light, indiated by D n, is ruial, and this led H. Sher and others to identify the anomalous transient-time dispersion. 54 In ontrast, for solar ells, the main issue is the olletion effiieny at steady state. Aording to the model illustrated in Figure 1, the ompensation of the density-dependene of both hemial diffusion oeffiient and response time (lifetime) is absolute, giving a stritly onstant diffusion length, hene, D 0, and the free arrier lifetime, τ n0, appear to be the entral physial parameters determining the solar ell operation. Nonetheless, we remark that frequeny or time transient methods remain essential for the haraterization of heterogeneous solar ells suh as DSSC. Clearly, information on quantities suh as D n and τ n is neessary in order to obtain a piture of the dynami behavior of the solar ell, and to larify the transport and reombination phenomena that are relevant for steady-state operation. The model outlined in Figure 1, based on the ontributions of many workers, provides a desription of disorder in nanostrutured TiO 2, through the traps distribution, and shows good agreement with the main features of the measured hemial diffusion oeffiient of eletrons, ranging from 10-4 to 10-8 m 2 /s depending on light intensity 8,27 and also with the eletron lifetime dependene on open-iruit photovoltage. 11 To inquire further whih is the degree of reality of this simple model, and how it should be improved, let us emphasize the main physial assumption behind the model: it is that there are many eletron traps that do not at as reombination enters. For nanostrutured TiO 2 the obvious realization of this feature is that there are both internal traps and surfae states in the nanopartiles, as suggested in the sheme of Figure 1. Further evidene for the distintion between internal traps and surfae states remains important for establishing this piture. One way to approah this question is to hange the size of partiles in the eletrodes, thus modifying the surfae-to-volume ratio. Reent results of Nakade et al. 55 using this method show that modifiation of the partiles surfae (by dye adsorption) enhanes the hemial diffusion oeffiient signifiantly, while maintaining the light-intensity dependene of D n. The authors 55 remarked that these results indiate the presene of eletron traps loated inside the nanopartiles. Indeed, in terms of Figure 1, when traps near the surfae are removed, the total density N L dereases, and / n L in eq 14 inreases by a onstant value while the dependene of D n on E Fn persists. In relation to this question, we omment in the Appendix on the very interesting results of Kopidakis et al. 56 that were published when this paper was nearly ompleted. We have emphasized in this report that the fator ( n L / ) imparts a Fermi level-dependene to the time onstants measured in quasistati onditions suh as D(µj n ) and τ(µj n ). Considering for instane D n (µj n ) in eq 14, it is appreiated that the kineti harateristis of transport appear through the ondution band diffusion oeffiient, D 0, whih is a onstant, whereas the variable fator ( n L / ) is related to loal redistribution of harge in the energy axis when the Fermi level is modified. Similar remarks an be made about τ(µj n ) in eq 27. So the time onstants D n (µj n ) and τ n (µj n ) ontain different omponents that are either kineti or thermodynami in origin. It is interesting to arry out this distintion preisely, as one may be able to extrat relevant onsequenes from models without having to solve them ompletely. This question is investigated speifially in a separate report for the hemial diffusion oeffiient. 25 Aknowledgment. This work was supported by Fundaió Caixa Castelló under Projet P11B Appendix: Diffusion-Limited Reombination The authors of ref 56 have measured the time onstants of DSSCs with areful onsideration to maintaining a homogeneous steady state and applying small perturbation, so the results reported are the hemial diffusion oeffiient, D n, and lifetime (response time), τ n, that we have disussed above. We first omment the many ommon aspets of their explanation and ours, based on their observation of the features of time onstants under modifiation of the thermodynami funtion of eletrons by lithium interalation. Thereafter, we onsider a point of ontrast onerning the interpretation of the measured lifetime, τ n. In ref 56, lithium ions were interalated into TiO 2 in DSSCs to substantial levels, either potentiostatially or illuminating the solar ells for a long time. This modified the shape of the exponential distribution for eletrons, indiated by a large hange of the tailing parameter R that is determined from D n (n tot ) in eq 28, whih is similar to eq 4 of ref 56. Furthermore, a linear model for reombination is formulated in ref 56 to obtain τ n (n tot ). In ommon with previous reports, 5,15,28 Kopidakis et al. find the onjugate tendenies in D n and τ n dependene on eletron onentration that leads to their ompensation in L n, as we have also disussed, and this is maintained even under variation of R. The authors also onfirm that these huge hanges in the distribution of eletron traps have only a small effet on the olletion effiieny of the solar ell, and they remark on the signifiane of this point, whih we have disussed also. So the results of this report, 56 and the explanation suggested by its authors, are muh in agreement and provide strong support for the general approah to the interpretation of time onstant presented here. We also wish to omment on a point of ontrast between the interpretation of the reombination mehanism in ref 56 and

9 Interpretation of the Time Constants J. Phys. Chem. B, Vol. 108, No. 7, our approah presented in setion 3. The other report 56 bases the interpretation of the response time τ n on a diffusion-limited (or transport-limited) reombination proess. The Fermi leveldependene of the response time is obtained in their eq 12 from the measured diffusion oeffiient in the form τ n 1/D n (our notation). This follows also from our eqs 14 and 31; however, it should be noted that our expression for τ n in eq obtains the fator ( n L / ) diretly from arguments of quasiequilibrium of free and trapped eletron density, so that eq 31 does not ontain diffusion parameters, in ontrast to eq 12 of ref 56. Therefore, our model explains all the experimental results of ref 56 without assuming a diffusion-limited reombination mehanism. This differene of interpretation raises an interesting point. The meaning of marosopi time onstants beomes a ritial issue for diserning mirosopi mehanisms, as the observed dependenies an be understood in different ways. So we should like to make a preision on the model of ref 56, and by the way, we larify also one of the aspets of our model of setion 3. The relationship τ n 1/D n indiated in ref 56 ould be misleading, beause the measured D n is the hemial diffusion oeffiient that desribes diffusion under a marosopi gradient of onentration. 25 That is, D n governs the flux that is measured in the transients of photourrent of Figure 1a of ref 56. However, during the open-iruit photovoltage deays of Figure 1b of ref 56 for measuring τ n, there are no suh marosopi fluxes, beause the eletron distribution is basially homogeneous at eah time. The only option for gradients to our seems to be from the enter to the surfae of individual partiles. For the measured hemial diffusion oeffiient of eletrons on the order of D n ) 10-5 m 2 s -1 and partiles of radius a ) 10 nm, the time of equilibration of onentration gradients into TiO 2 nanopartiles is τ dif a 2 /D n ) 10-7 s, whereas the OCVD takes a muh larger time, on the order of τ n ) 10-1 s. This is why the eletron density an be assumed homogeneous during the measurement of τ n, as argued in setion 3, so that the diffusion oeffiient does not appear in our eq 31. In omparison, the irumstanes are very different for the lithium interalation proess that is onsidered by Kopidakis et al., beause the hemial diffusion oeffiient of lithium ions in metal oxides an be as low as D h(li) ) m 2 s -1, 57 so that the time for equilibration of gradients is τ dif a 2 /D h(li) ) 1 s, and may dominate the interalation phenomena. 58 Indeed, the former point is lear to Kopidakis et al., as their argument does not involve any onentration gradients of eletrons during OCVD but a sarity of aeptor speies that obliges the eletron to effet a long random displaement over thousands of nanopartiles before it an reombine. So the relationship assumed in ref 56 for diffusion-limited reombination should be τ n 1/D J, where D J is the jump (or traer) diffusion oeffiient that desribes the random walk of eletrons. 25 This omes to no importane for their argument, beause in the ase of the exponential distribution of traps the thermodynami fator, χ T, that relates both diffusion oeffiients, D n ) χ T D J, is a onstant, χ T ) 1/R, as we show in another report, 25 so indeed the random walk of an eletron, governed by D J,is affeted by the total eletron onentration in the same way as by the measured D n. That is, D J n /n L beomes larger when the Fermi level is higher, so that diffusion-limited reombination beomes faster. In DSSC, the I - /I 3- ouple provides two eletron aeptors: I - 3 and I - 2. Reently it was shown that the reombination path depends on the illumination intensity. 59 The eletron reation with I - 2 beomes kinetially favorable only at high light intensities. It is believed that under normal solar onditions the reombination with I 3 - dominates whih makes it the only relevant proess from the pratial point of view. 59,60 Aording to our understanding, Kopidakis et al. 56 base the harateristis of the response time on the relationship τ n 1/D J, and this leads them to selet the muh rarer I 2 as the dominant aeptor, beause for this low-onentration speies the time for the eletron to find a target for reombination would govern the τ n. However, the initial assumption, τ n 1/D J, is not neessary to explain the variations of τ n. In our model, τ n0 is the rate onstant for harge transfer, for any kind or onentration of the aeptor speies. However the measured τ n ) ( n L / )τ n0 is muh longer than τ n0, beause the Fermi level annot deay but with equilibration of free and trapped eletron density. The results of refs 32 and 33 also suggest that reombination is not governed by diffusion in onfigurational spae but rather in energy spae (energy redistribution). Here, we reahed this onlusion on the assumption of the existene of a large density of traps in the bulk of partiles. The results of Kopidakis et al. would also seem to support this idea of internal traps, beause the interalation of lithium into nanopartiles is affeting markedly the tailing parameter R observed in the transport parameter, D n. However, as remarked in setion 5.2, the exat effet that produes the marked departure from Boltzmann statistis is not lear yet. In summary, the results of ref 56 do not prove the diffusionlimited reombination mehanism, that requires a sarity of aeptor speies. The experimental results an be explained more simply on the basis of the normal eletron harge-transfer mehanisms in DSSC and a ommon origin of the Fermi-level dependene of both measured D n and τ n, whih originates in an exponential distribution of traps in the bulk of partiles. This last idea desribes major features, bot not so far the details, of reombination in DSSC, and it is likely that it should be improved with a more elaborated mirosopi piture. In this sense, the model suggested in ref 56 is a rather interesting idea that shows the need for determining the relationship of marosopi, steady-state time onstants, to mirosopi models. In partiular, the onnetion between long-range eletron transport, energy redistribution, and interfaial harge transfer requires further studies. Referenes and Notes (1) Grätzel, M. Nature 2001, 414, 338. (2) de Jongh, P. E.; Vanmaekelbergh, D. Phys. ReV. Lett. 1996, 77, (3) Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. 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