Solar Cells - American Chemical Society

J. Phys. Chem. B 1997, 101, 8141-8155

8141

Band Edge Movement and Recombination Kinetics in Dye-Sensitized Nanocrystalline TiO2 Solar Cells: A Study by Intensity Modulated Photovoltage Spectroscopy G. Schlichtho1 rl, S. Y. Huang, J. Sprague, and A. J. Frank* National Renewable Energy Laboratory, Golden, Colorado 80401 ReceiVed: April 25, 1997; In Final Form: June 30, 1997X

The charge-recombination kinetics and band edge movement in dye-sensitized nanocrystalline TiO2 solar cells are investigated by intensity modulated photovoltage spectroscopy (IMVS). A theoretical model of IMVS for dye-sensitized nanocrystalline semiconductor electrodes is developed, and analytical expressions for the frequency dependence of the photovoltage response at open circuit are derived. The model considers charge trapping/detrapping and electron transfer from the conduction band and surface states of the semiconductor to redox species at the solid/solution interface. IMVS is shown to be valuable in elucidating the contributions of band edge shift and recombination kinetics to changes of the open-circuit photovoltage (Voc) resulting from surface modifications of the semiconductor. IMVS measurements indicate that surface treatment of [RuL2(NCS)2] (L ) 2,2′-bipyridyl-4,4′-dicarboxylic acid)-sensitized TiO2 electrodes with 4-tertbutylpyridine or ammonia leads to a significant band edge shift concomitant with a more negative Voc. Surfacemodified dye-covered TiO2 electrodes exhibit a much higher photovoltage, for a given concentration of accumulated photogenerated electrons, than the unmodified dye-covered electrode. The accumulated charge in the TiO2 electrode is not sufficient to induce a major potential drop across the Helmholtz layer and cannot thus explain the observed photovoltage. The surface charge density is also not sufficient to support an accumulation layer strong enough to have a major influence on the photovoltage. The movement of the Fermi level of the TiO2 electrode, arising from the accumulation of photogenerated electrons in the conduction band, accounts for the observed Voc. The second-order nature of the recombination reaction with respect to I3- concentration is confirmed. Furthermore, the IMVS study indicates that recombination at the nanocrystallite/ redox electrolyte interface occurs predominantly via trapped electrons in surface states.

Introduction Dye-sensitized nanocrystalline TiO2 solar cells are regarded as a potential low-cost alternative to conventional silicon solar cells. The TiO2 cells show high short-circuit photocurrent densities (Jsc) and good fill factors; but, the open-circuit photovoltage (Voc) is substantially below the theoretically predicted maximum.1-3 A major factor limiting the photovoltage is back electron transfer from TiO2 to the redox electrolyte (I3- + 2e- f 3I-).4,5 A substantial effort has been made in devising surface-treatment schemes to suppress the recombination kinetics as a means of improving the photovoltage. Surface treatment of TiO2 or other semiconductor electrodes with organic molecules, such as pyridines, have lead to a substantial improvement of Voc.4-6 The higher photovoltages have been attributive to the blockage of recombination centers by the adsorbed organic molecules, resulting in 1-4 orders of magnitude decrease in the rate constant for back electron transfer in the TiO2 system.4,5 This explanation neglects, however, the possible contribution of other phenomena, such as band edge displacement.5 The aim of this work is to examine by intensity modulated photovoltage spectroscopy (IMVS) the effect of surface modification of dye-covered TiO2 electrodes on the conduction-band edge potential and recombination kinetics. This is the first reported study of IMVS. The theory of IMVS is developed for nanocrystalline semiconductor electrodes and applied to the study of [RuL2(NCS)2] (L ) 2,2′-bipyridyl-4,4′-dicarboxylic acid)-sensitized TiO2 electrodes. X

Abstract published in AdVance ACS Abstracts, August 15, 1997.

S1089-5647(97)01412-0 CCC: $14.00

Theory of IMVS. IMVS measures the modulation of photovoltage in response to the modulation of incident light intensity. IMVS is closely related to intensity modulated photocurrent spectroscopy, which was developed for single crystalline semiconductor electrodes7-9 and was recently applied to nanostructured TiO2 cells.10 Although IMVS measurements apply generally to constant current conditions, we restrict our discussion to the special case in which no current passes through the external circuit (i.e., at open circuit). We consider dyesensitized nanocrystalline n-type semiconductor electrodes. Optical excitation of sensitizing dye molecules leads to an electron injection current density (Jinj) into the conduction band of the semiconductor. This process is known to be very fast, occurring in the femtosecond-to-picosecond domain.11-12 At open circuit, all of the electrons injected into the conduction band must eventually undergo reaction at the semiconductor/ solution interface. In the following, we refer to this process as recombination. Figure 1 shows the reaction pathways and corresponding rate constants for charge trapping (k1), detrapping (k2), and recombination via the conduction band (k3), and surface states (k4). We assume that recombination is irreversible. At open circuit and constant light intensity (steady state), the recombination current density (Jr) equals Jinj. Neither Jr nor Jinj can be measured directly but is rather inferred from measurements of the short-circuit photocurrent density (Jsc) and the quantum efficiency for the collection of injected carriers (η)

qφIo ) Jinj ) Jsc/η ) -Jr

(1)

where q is the charge of an electron, φ is the ratio of injected electrons to incident photons, and Io is the incident photon flux © 1997 American Chemical Society

8142 J. Phys. Chem. B, Vol. 101, No. 41, 1997

Schlichtrol et al.

Figure 1. Scheme for electron-transfer kinetics. Jinj is the electroninjection current from excited dye molecules into the semiconductor conduction band, k1 and k2 are the respective rate constants for electron capture by surface states and the thermal emission of electrons back into the conduction band, and k3 and k4 are the respective rate constants for back electron transfer from the conduction band and surface states to an electron acceptor at the nanocrystalline semiconductor/redox electrolyte interface. Jinj is assumed to not limit the recombination kinetics. Electron transfer from the semiconductor to the oxidized dye and charge injection from the redox electrolyte to the semiconductor are neglected.

density or incident light intensity. From the kinetic scheme in Figure 1, the accumulated charge in the conduction band Qcb and in surface states Qss can be calculated from the relations

dQcb ) qφIo - Qcb (k1 + k3) + Qss k2 ) 0 dt

(2a)

dQss ) Qcb k1 - Qss (k2 + k4) ) 0 dt

(2b)

(Note: Qcb and Qss have a negative sign.) The solution of the system of differential equations (eqs 2a,b), for a constant incident photon flux Io, is

Qcb )

qφIo(k2 + k4) k1k4 + k2k3 + k3k4

(3a)

Qss )

qφIok1 k1k4 + k2k3 + k3k4

(3b)

Figure 2. Energy-level diagram of a dye-sensitized nanocrystalline semiconductor solar cell. The redox energy of the ground state E(D+/ D) and excited state E(D+/D*) of the dye, the conduction-band edge of the nanocrystalline semiconductor (Ecb) and transparent conducting oxide substrate (Ecb(TCO)) are referenced to the redox electrolyte energy (Eredox). For a highly doped TCO, its Fermi energy (Ef) lies close to Ecb(TCO). Photoinjection of electrons from excited dye molecules may lead to a change of the electric field across the Helmholtz layer and Ecb movement. The difference between Ecb and Ef of the nanocrystalline semiconductor (E(Qcb)) depends on the electron concentration (Qcb) and the density of states (Ncb) in the conduction band of the semiconductor, according to the relation E(Qcb) ) Ecb - Ef ) kT ln(q(1 - P)dNcb/ Qcb), where k is the Boltzmann constant, T is the temperature, and P and d are the respective porosity and thickness of the semiconductor film. For a fixed Qcb, the photovoltage will change with Ecb. To some extent, the energies of the adsorbed dye may also change with Ecb.

h ss are the where ω is the circular frequency and Q h cb and Q respective average charge densities in the conduction band and surface states and are calculated from eq 3. A1, B1, A2, and B2 are given by

A1 )

qφM[(k1 + k3)ω2 + (k2 + k4)S] ω4 + Rω2 + S2

The sum of Qcb and Qss is given by the expression

qφIo(k1 + k2 + k4) Q ) Qcb + Qss ) k1k4 + k2k3 + k3k4

B1 ) -

(5)

where Jph ) 0 mA/cm2 at open circuit. For a small sinusoidally modulated light intensity superimposed on a base (bias) light intensity (I(t) ) Io + M sin(ωt), M , Io), the rate constants (k1-k4) in Figure 1 do not change with intensity modulation, and eq 2 transforms under open-circuit conditions to the following system of differential equations

dQcb ) qφIo + qφM sin(ωt) - Qcb (k1 + k3) + Qss k2 dt dQss ) Qcb k1 - Qss (k2 + k4) dt

ω4 + Rω2 + S2

(8b)

(4)

For modulated light intensities, Jinj is composed of the charging or discharging current density of the semiconductor Jch, the photocurrent density in the external circuit Jph, and Jr

Jinj ) Jch + Jph - Jr

qφMω[ω2 + T]

(8a)

(6a) (6b)

which has the solution

Qcb ) Q h cb + A1 sin(ωt) + B1 cos(ωt)

(7a)

Qss ) Q h ss + A2 sin(ωt) + B2 cos(ωt)

(7b)

A2 )

B2 ) -

qφMk1[-ω2 + S] ω4 + Rω2 + S2

qφMωk1[k1 + k2 + k3 + k4] ω4 + Rω2 + S2

(8c)

(8d)

where R ) (k1 + k3)2 + (k2 + k4)2 + 2k1k2, S ) k1k4 + k2k3 + k3k4, and T ) (k2 + k4)2 + k1k2. A1 and B1 are called the real (in-phase) and imaginary (out-of-phase) components of the modulation of Qcb, and A2 and B2 are defined as the real and imaginary components of the modulation of Qss, respectively. The influence of Qcb and Qss on the photovoltage can be deduced from Figure 2, which shows an energy-level diagram of a dye-sensitized nanocrystalline semiconductor in contact with a transparent conducting oxide (TCO) and a redox electrolyte. At dark equilibrium, the Fermi levels (Ef) of TCO and the nanocrystalline semiconductor equals the redox electrolyte energy (Eredox). Under illumination, injected electrons from the excited state of the dye accumulate in the conduction band and surface states of the nanocrystalline semiconductor. Qcb determines the difference between Ef and the conduction-band energy of the semiconductor (Ecb). Qss may be of sufficient magnitude to change the potential drop across the Helmholtz layer, resulting in conduction-band edge movement. Because Qcb fixes the position of Ef with respect to Ecb, a shift of Ecb

Dye-Sensititized Nanocrystalline Solar Cells

J. Phys. Chem. B, Vol. 101, No. 41, 1997 8143

will be accompanied by an equal displacement of Ef relative to Eredox. Changing the potential drop across the Helmholtz layer may also alter, to some extent, the energy of the ground and excited states of the adsorbed dye. At open circuit, Ef of TCO equals Ef of the nanocrystalline semiconductor. The bands of highly doped TCO(ND g 1020 cm-3) remain essentially flat, and the potential drop occurs across the Helmholtz layer at the redox electrolyte/TCO interface. The situation at the semiconductor/TCO interface is more complex; but, because of the small contact area, the capacitance of this interface can be neglected. Depending on experimental conditions (e.g., TCO and nanocrystalline semiconductor materials, the thickness and porosity of nanocrystalline semiconductor film, and the redox electrolytesolution composition), the TCO can influence IMVS measurements. If a significant portion of the charge recombination current occurs at the TCO/redox electrolyte interface, the measured time constant will not reflect the charge recombination process at the nanocrystalline semiconductor/redox electrolyte interface. However, for large-surface-area nanocrystalline semiconductor electrodes, recombination at the TCO/redox electrolyte interface can generally be neglected. The charge necessary to raise the Fermi level of TCO to the same level as that of the illuminated nanocrystalline semiconductor may represent a significant fraction of the charge measured by IMVS. Also, the time constant for charging and discharging the TCO/ redox electrolyte interface, which is controlled by electron diffusion within the nanocrystalline semiconductor film, may contribute significantly to the rise and decay time of the photovoltage and may therefore influence IMVS measurements. However, for cells with high collection efficiencies, this time constant is much smaller than the time constant for charge recombination. In this case, the IMVS response is dominated by charge recombination. Modulation of Qcb and Qss leads to modulation of photopotential V with respect to the redox electrolyte potential as expressed by the relations

∫Q0 (t) C

V(Qcb(t)) ) -

V(Qss(t)) ) -

cb

1 dQ (Q cb cb)

∫Q0 (t) C1H dQ ) ss

Qss(t) CH

Voc ) V(Qcb(t)) + V(Qss(t))

(9a)

(9b)

Figure 3. Schematic plot of the frequency response of potential modulation according to the reaction scheme in Figure 1. τ1 and τ2 are time constants, ω is the circular frequency, and S and T depend on the rate constants as defined by eq 8.

In other words, modulation of Qss would not give rise to modulation of the photopotential. In practical photoelectrochemical cells, the surface-state capacitance Css is much smaller than CH. It follows therefore that V(Qss) ) V(Qcb)Css/CH is much less negative than V(Qcb) and that Voc is determined only by Qcb. An important consequence of this relationship is that illumination of such nanocrystalline semiconductor electrodes is not expected to lead to a potential drop across the Helmholtz layer and thus to band edge movement. For this case, the real (re) and imaginary (im) parts of the IMVS response can be expressed, with the aid of eqs 7a, 8a,b, and 9a, as follows:

re(∆Voc) )

qφM[(k1 + k3)ω2 + (k2 + k4)S] (ω4 + Rω2 + S2)Ccb(Q1)

(9c)

where Ccb is the conduction-band capacitance and CH is the Helmholtz capacitance. The potential modulation induced by Qss (eq 9b) is a consequence of a potential drop across the Helmholtz layer; a change of this potential drop gives rise to a shift in the conduction-band edge potential.13-16 The first integral (eq 9a) can be expressed by the Taylor series (around the equilibrium Q h cb), which, for small charge modulations, can be truncated after the linear term. Thus, the modulation of V is linearly proportional to the charge modulation; the proporh cb). Although both Qcb and Qss tionality factor equals 1/Ccb(Q contribute to Voc, one may predominate. If Qss were to contribute significantly to Voc, the modulation of Qss would affect the frequency response of Voc and determine a time constant τ. With knowledge of τ, the accumulated surface-state charge (Qss ) τJinj) and thus the potential drop across the Helmholtz layer can be calculated from the expression: V(Qss) ) τJinj/(rCH), where r is the roughness factor of the nanocrystalline semiconductor electrode. On the other hand, if the term τJinj/(rCH) is much less negative than Voc, for any observed value of τ, then Qss would not contribute significantly to Voc.

im(∆Voc) ) -

qφMω[ω2 + T] (ω4 + Rω2 + S2)Ccb(Q1)

(10a)

(10b)

Mathematical analysis of eqs 10a,b reveals that A1 (eq 8a) increases monotonically with increasing ω, whereas B1 (eq 8b) exhibits one or two maxima, depending on the relative magnitude of k1-k4. The influence of kinetic parameters on charge modulation is illustrated schematically in Figure 3, which shows plots of the modulation of Voc in the complex plane (B1 vs A1). The frequency ω at which the imaginary part reaches a maximum defines a time constant (τ). For two time constants with significantly different values (τ1 , τ2), Figure 3a,b shows two well-separated semicircles, and τ1 and τ2 can be, to a very good approximation (Appendix A), expressed as

τ1 )

1 xR

(11a)

τ2 )

xR S

(11b)


Schlichtrol et al.

S and T (eq 8) determine the relative size (diameter) of the semicircles. If T > S (Figure 3a), the low-frequency semicircle (corresponding to τ2) will be larger than the high-frequency semicircle (corresponding to τ1). Similarly, if T < S (Figure 3b), the low-frequency semicircle will be smaller than the highfrequency semicircle. In the case that T . S or T , S, the smaller semicircle may be too small to be observed. Figure 3c depicts the situation in which τ1 approaches τ2. In this case, the separation of the semicircles becomes less distinct. At some point, one semicircle appears as a shoulder on the other, and only one maximum is present. Evaluation of the IMVS Response. (a) Two Semicircles. If the IMVS response shows two semicircles (cf. Figure 3), the two time constants are well defined by eq 11 and can be determined by fitting the frequency response of the photovoltage with the expressions (see Appendix A):

re(∆Voc) )

im(∆Voc) )

-M1*

-M2* + 2

1 + ω2τ1

M1*ωτ1 2

1 + ω2τ22 M2*ωτ2

1 + ω τ1

2

+

1 + ω2τ22

(12a)

(12b)

where M1* and M2* are scale factors. It can be seen from eq 12a that the quantity -(M1* + M2*) is the low-frequency limit of re(∆Voc). The product of the time constant τ and qφIo (qφIo ) Jinj) gives the charge Q. For the respective time constants τ1 and τ2, Q can be identified with the expressions for Qcb and Qcb + Qss in eq 3 according to the relations (Appendix A):

qφIo (k2 + k4) ) Qcb k1k4 + k2k3 + k3k4

(13a)

qφIo (k1 + k2 + k4) ) Qcb + Qss k1k4 + k2k3 + k3k4

(13b)

qφIoτ1 ≈ qφIoτ2 ≈

qφIok1 qφIo (τ2 - τ1) ) qφIo∆τ ≈ ) Qss (13c) k1k4 + k2k3 + k3k4 A more precise analysis for the case of similar size semicircles is given in Appendix A. (b) One Semicircle. If only one semicircle is observed, and the high-frequency limit of re(∆Voc) is not zero, only the lowfrequency semicircle (corresponding to τ2) is measured. To obtain the second semicircle (corresponding to τ1), the frequency range of measurement must be extended to higher frequencies. On the other hand, if only one semicircle is observed with a high-frequency limit of zero for re(∆Voc), it can be attributed to one of four possibilities: (1) only the high-frequency semicircle (τ1) appears because the measurements do not cover the frequency range of the low-frequency semicircle, (2) only the high-frequency semicircle (τ1) appears because the lowfrequency semicircle is too small (T , S), (3) only the lowfrequency semicircle (τ2) appears because the high-frequency semicircle is too small (T . S), or (4) only one semicircle exists. In case 1, the measured low-frequency limit of the real part of the potential modulation (-M1*) will be less negative than the theoretically predicted value x1/2M dVoc/dIo (Figure 3), and the second semicircle (τ2) will be found by extending the measured frequency range to lower frequencies. (Note: The factor x1/2 reflects the conversion from peak-to-peak to rootmean-square by the lock-in amplifier). However, if -M* ) x1/2 M dVoc/dIo (cases 2-4), only one time constant will be found. A necessary condition for case 4 is that the rate constants

for electron disappearance from the conduction band (k1 + k3) and surface states (k2 + k4) are about the same (Appendix A). Also, it can be concluded that -Q ) -Qcb g -Qss (Appendix A). In practice, this possibility can be excluded when the presence of a single semicircle is independent of the intensity of the bias light (Io). To fit the IMVS response for one semicircle, M1* or M2* in eq 12 is set equal to zero. At depletion, no significant electric field is present in nanosized semiconductor particle films with moderate doping densities (ND e 1018 cm-3).17,18 However, an electric field associated with an accumulation layer may exist as a result of the interaction between photoinjected electrons in the conduction band and compensating positively charged ions at the semiconductor/solution interface. A rough estimation of the upper limit of the photoinduced-potential drop across the accumulation layer (∆Vacc) can be made from an expression for the maximum electric field at the semiconductor particle surface (Eσ ) Qacc/ (os), where o is the permittivity in vacuum, s is the dielectric constant of the semiconductor, and Qacc is the surface charge density). Vacc can be calculated by integrating the electric field from the center of the particle to the surface. Assuming the electric field declines linearly from the surface to 0 V/cm at the particle center, one obtains the relation Vacc ) Qaccφ/(4os), where φ is the diameter of the particle. This relation can be expressed as a capacitance: Cacc ) Qacc/Vacc ) 4os/φ. For s ≈ 10 and particle diameters of 10-100 nm, Cacc ranges from 40-4 µF/cm2, respectively. The magnitude of Cacc is in the range typically found for CH. In the absence of an electric field, Voc depends on Qcb, the potential of the conduction-band edge (Vcb) (vs the redox electrolyte potential), the density of states in the conduction band (Ncb), and the volume of the semiconductor per unit cell area, which is a function of film thickness d and porosity P.19

-Voc ) -Vcb -

(

)

1 qNcbd(1 - P) ln ) β Qcb -Vcb - V0 +

1 ln(-Qcb) (14) β

where β ) -q/(kT) ) 1/26 mV at room temperature, and V0 is a film specific parameter that does not change with surface treatment. With increasing accumulation of electrons in the conduction band (Qcb), Voc becomes more negative. Equation 14 can be used to calculate the potential of the conductionband edge. [Note: All potentials (e.g., Voc, Vcb, and the surfacestate potential (Vss)) are referenced to the redox potential of solution. Energies are also defined with respect to the redox potential, according to E(V) ) qV.] It can be shown that for the measured time constant that either eq 13a or eq 13b holds (Appendix A). If eq 13a applies, then from eq 14, the slope dVoc/d ln(-qφIoτ) ) -1/β. Conversely, if the slope is much more negative than -1/β, eq 13b holds, and Qcb + Qss ≈ Qss (Appendix B). The evaluation of IMVS response will therefore yield either Qcb(V) or Qss(V) for one observed time constant or both Qcb(V) and Qss(V) for two observed time constants. With knowledge of Qcb(V), the conduction-band edge position can be calculated with eq 14. Recombination via Conduction band or Surface States. The dependence of the recombination-current density on the open-circuit photovoltage is described by the simplified ButlerVolmer expression5:

( )

Jr ) -qket coxm exp(-βVocuR) ) -qket coxm exp

-Voc ma

(15)



where ket is the rate constant for recombination, cox is the concentration of oxidized species in solution, u and m are the respective order of the rate of reaction for electrons and oxidized species, and R is the electron-transfer coefficient. Equation 15 leads to the well-known linear relation between the logarithm of the incident light intensity and Voc.5,20,21

-Voc ) -ma ln(ketcoxm) + ma ln(-qφIo) ) Va + ma ln(-qφIo) (16) From eqs 15 and 16, we obtain the expression for uR.

uR ) 1/(βma) )

β d ln(Jr) β d ln(-qφIo) ) -dVoc -dVoc

(17)

For metal electrodes, R is explained by a change of transitionstate energy (Et) of a chemical reaction (Figure 4), resulting from a variation of the electric field at the metal surface with potential.22 Usually for nanocrystalline semiconductors, however, the electric field across the Helmholtz layer does not change with photopotential, as discussed above, because the surface-state capacitance is much smaller than the Helmholtz capacitance (-Qss/(rCH) ) -Jinjτ2/(rCH) , -Voc). The conduction-band edge potential does not therefore shift significantly with photopotential, and Voc is just the consequence of an increase of electron concentration in the conduction band. The general expression for the dependence of the rate constant (k′) on the activation energy (Ea) is given by

k′ ) k* exp(-Ea/kT) ) k* exp

( ) βEa q

(18)

where k′ may depend on potential, and k* is a constant. When electron transfer occurs directly from the conduction band, Ea does not change with photopotentialsin the absence of photoinduced-band edge movementsand therefore, the rate constant for this process (k3) does not depend on the photopotential. Because k3 is independent of Voc, Qcb is, from eq 1, proportional to Jr (i.e., Jr ) -k3Qcb), -Jinj, and Io (if φ does not change with light intensity). For recombination via the conduction band, eq 14 yields -1/β for the slope dVoc/d ln(-qφIo). On the other hand, if the slope differs significantly from -1/β, recombination occurs principally via surface states. Influence of Photopotential and Band Edge Shift on Rate Constants k3 and k4. When electron transfer proceeds via surface states, the rate constant for recombination (k4) will increase with photopotential as a consequence of the population of higher energy surface states (cf. Figure 4). From geometric considerations of Figure 4 (Appendix C), the dependence of Ea on the energy of the potential V (E(V)) and the displacement of the conduction-band edge (∆Ecb) induced by surface modification can be obtained as

Ea(E(V)) ) Eao + ∆Ecb

γmD mD - E(V) mD + mA mD + mA

(19)

where mA and mD are the respective slopes of the plots for the dependence of the electron-acceptor-state energy and the electron-donor-state energy on the reaction coordinate, EAo is the energy of the electron-acceptor state in the absence of band edge movement, γ is the ratio of the change of energy of the electron-acceptor state to ∆Ecb, and E(V) is the electron-donorstate energy at a potential V; the parameter γ depends on the rate-determining step for electron transfer and has values

Figure 4. Energy diagrams for an electrochemical reaction on (a) a metal cathode, (b) a nanocrystalline semiconductor cathode without band edge movement, and (c) a nanocrystalline semiconductor cathode before and after band edge movement, resulting from chemical treatment. The straight lines represent linear approximations of the coulomb energy of electron donors and acceptors across the reaction coordinate. The bold lines show the charge-transport pathway. The energies are referenced to the energy of the redox electrolyte Eredox. Varying the potential V applied to the metal cathode will change, in direct proportion, the potential drop across the Helmholtz layer and the energy of the metal electrons E(V). The energy of the transition state (Et) and the electron-acceptor state (EA) may move, to some extent, with applied potential, depending on the location of the transition state and the electron acceptor (within the Helmholtz layer) involved in the rate-limiting step. Varying the potential of the nanocrystalline semiconductor will not change the potential drop across the Helmholtz layer when the surface-state capacitance is small against CH. Therefore, the energies of the electron-acceptor state and electron-donor states E(V) (e.g., surface states or conduction band) do not move with potential. The higher the energy of an electron donating surface state E(V), the lower is the activation energy (Es) for electron transfer. The energetics of electron transfer from the conduction band is not affected by potential. When chemical treatment shifts the energy of the conductionband edge (Ecb), the electron-acceptor state may also move, to some extent, with the band edge. Altering EA will change Ea for electron transfer from surface states and from the conduction band. Mathematical analysis of the activation energy as function of potential and band edge movement is given in Appendix C.

between 0 and unity. When the rate-limiting step is electron transfer from the conduction band or surface states to adsorbed molecules at the semiconductor surface, γ approaches 1. Substituting eq 19 into eq 18 gives the distribution function of the rate constant (k4**) with potential

k4**(V) ) k4* exp

(

βmDγ∆Ecb

) (

q(mD + mA)

exp

-βmDE(V)

q(mD + mA)

)

(20)

were k4* is a constant. k4**(V) is related to the rate constant for recombination via surface states (k4) by the expression

k4(V) )

dQss(V)/dt Qss(V)

-Jr ) ) Qss

∫V0 qNss(V) k4**(V)dV ∫V0qNss(V)dV

(21)

where Nss(V) is the surface-state density. Equation 20 can be written as

k4**(V) ) k4* exp(γ∆Vcb/mb) exp(-V/mb)

(22)

where mb ) (mD + mA)/(βmD), ∆Vcb ) ∆Ecb/q, and V ) E(V)/q. Equation 22 is valid not only for recombination via surface states but also for recombination via the conduction band for V ) Vcb + ∆Vcb.

k4**(Vcb + ∆Vcb) ) k4* exp(γ∆Vcb/mb)exp(-(Vcb + ∆Vcb)/mb) (23)


Schlichtrol et al. and is thus not affected by chemical modification of the semiconductor surface. Combining eqs 26 and 27 gives

[ (

Nss(V) ) Nsso exp ∆Vcb

)] [ (

)]

1 1 1 1 exp -V ma mb ma mb

(28)

The dependence of Voc on s and ∆Vcb is obtained with the aid of eqs 1, 25, and 28.

qφIo )

∫V0

[ ( )] [ ( )] ( ) ( )

oc

exp -V Figure 5. Influence of band edge movement (∆Vcb) on the distribution function of the surface-state density [Nss(V)] relative to the redox potential Vredox. Nss(V) is fixed with respect to the band edge potential. (a) In the absence of band edge shift, Nss(0) ) Nsso; Nss(0) is the surfacestate density at Vredox. (b) After band edge movement, Nss(0) * Nsso (i.e., a band edge shift changes the surface-state density at Vredox).

Substituting k3 for k4**(Vcb + ∆Vcb) and k3* for k4* exp(-Vcb/ mb) into eq 23 yields

k3 ) k3* exp[(γ - 1) ∆Vcb/mb]

(24)

where k3* and k3 are the respective rate constants in the absence and in the presence of band edge displacement. Equation 24 predicts that as the band edge shifts to more negative potentials, the rate constant for recombination via the conduction band increases. However, eq 24 is only valid when the conductionband edge potential lies within the noninverted Marcus region. Influence of Surface Shielding and Band Edge Shift on Voc. In addition to band edge movement, the rate constants for recombination (k3 and k4) can be altered by sterically shielding the semiconductor surface with chemical reagents. This effect can be quantified by introducing a shielding factor (s) which reflects the contribution to the change of the rate constant that is not due to band edge movement. With the introduction of s, the parameters k3* and k4* are, by definition, independent of surface modification. Recombination via Surface States. When recombination proceeds via surface states, the recombination-current density is given by the integral of the product of qNss(V) and k4**(V) (eq 22)

∫V0

Jr ) -

oc

( ) ( )

γ∆Vcb k4* -V exp exp dV s mb mb

qNss(V)

(25)

A comparison of eqs 15 and 25 shows that Nss(V) can be approximated by an exponential function for the photopotential range investigated.

[ (

)]

1 1 ma mb

Nss(V) ) Nss(0) exp -V

(26)

where Nss(0) is the density of surface states at the potential of the redox electrolyte. For a fixed distribution of surface states in relation to the band edges, Nss(V) will change with band edge movement (Figure 5). Nss(0) is related to band edge movement by the expression

[ (

Nss(0) ) Nsso exp ∆Vcb

)]

1 1 ma mb

1 1 × ma mb k4* γ∆Vcb -V exp exp dV (29) s mb mb

qNsso exp ∆Vcb 1 1 ma mb

Integrating eq 29 and solving for Voc gives

-Voc ) ma ln(-qφIo) - ma ln(-qmaNssok4*) + (1 - γ)ma (30) ma ln(s) - ∆Vcb 1 mb

(

Comparing eqs 30 and 16 yields an expression for the experimentally observable parameter Va.

Va ) -ma ln(-qmaNssok4*) + ma ln(s) -

(

∆Vcb 1 -

)

(1 - γ)ma (31) mb

Unless modification of the semiconductor surface alters the recombination mechanism, the parameter mb, which is related to the symmetry factor for the recombination reaction (Figure 4 and eq 22), is not expected to change with surface treatment. Consequently, for a fixed density of surface states with respect to the band edges, it can be concluded from eq 27 that ma is also expected to be independent of surface treatment when the symmetry of the reaction is not altered. The respective independence of ma and mb on the particular surface treatment can be examined experimentally, with the aid of eq 30, from the relation

d(-Voc)

ma )

(32)

d ln(Io)

and eq 37 (see below). From eq 31, we obtain an expression relating the change of Va to s and ∆Vcb, resulting from surface modification.

(

∆Va ) ma ln(s) - ∆Vcb 1 -

)

(1 - γ)ma mb

(33)

For recombination via surface states (Jr ) -Qssk4; eq 21) and qφIoτ2 ) Qcb + Qss ≈ Qss (eq 13b), it can be deduced that τ2 ) 1/k4. The dependence of τ2 on Voc can be obtained by substituting Nss(V) (eq 26 or eq 28) and k4**(V) (eq 22) into eq 21 and integrating the resulting expression to obtain

(

)

( ) ( )

mb - ma k4* γ∆Vcb -Voc 1 exp ) k4 ) exp τ2 mb s mb mb Solving for -Voc in eq 34 yields

(27)

where Nsso is the density of surface states in the absence of band edge movement at the potential of the redox electrolyte

)

-Voc ) -mb ln(τ2) - mb ln

[

]

(34)

k4*(mb - ma) + mb mb ln(s) - γ∆Vcb (35)



For convenience, eq 35 can be rewritten as

-Voc ) -mb ln(τ2) + Vb

(36)

The parameter mb is obtained from eq 36 as

mb ) -

-Voc ) -Vcb - V0 +

d(-Voc)

(37)

d ln(τ2)

From eq 36, the change of Vb, as a consequence of chemical treatment of the semiconductor surface, can be given as a function of s and ∆Vcb.

∆Vb ) mb ln(s) - γ∆Vcb

(38)

Qss can be expressed as qφIoτ2 (eq 13b) and as the integral over the density of occupied surface states (eq 28).

Qss ) qφIoτ2 )

∫V0

oc

[ (

)]

1 1 × ma mb 1 1 exp -V dV (39) ma mb

Nsso exp ∆Vcb

[ (

)]

Integration of eq 39 leads to the expression

[ (

qφIoτ2 ) -Nsso exp ∆Vcb

)]( ) [ ( )]

1 1 ma mb

1 -1 1 × ma mb 1 1 exp -Voc ma mb

(

)[

1 1 ma mb

(40)

-1

ln(-qφIoτ2) - ln(Nsso) +

(

ln

1 1 ma mb

)]

- ∆Vcb (41)

Equation 41 can be represented as

-Voc ) Vc + mc ln(-qφIoτ2)

(42)

where mc ) (1/ma - 1/mb)-1. The experimentally observable parameter Vc relates the density of surface states to the potential and is not affected by shielding of the semiconductor surface. From eqs 41 and 42, we obtain the relation between the change of Vc and ∆Vcb.

∆Vc ) -∆Vcb

(43)

Equations 33, 38, and 43 are interdependent. Therefore, to determine s and the influence of ∆Vcb on Voc and k4, the value of γ must be known. If τ1 is measured (qφIoτ1 ) Qcb; eq 13a), the potential of the band edge can be determined from eq 14.

Vcb ) Voc -

(

)

1 qNcb(1 - P)d ln β qφIoτ1

(44)

The influence of ∆Vcb and s on Voc can be evaluated from eq 33 with knowledge of mb and γ. Recombination via the Conduction Band. Taking into consideration the shielding factor, eq 24 can be rewritten as

k3 )

k3* exp[(γ - 1)∆Vcb/mb] s

(

)

-qφIos 1 ln β k3* exp[(γ - 1)∆Vcb/mb]

(46)

A comparison of eqs 46 and 16 gives the dependence of ∆Va and s and ∆Vcb.

(

∆Va ) -∆Vcb 1 -

)

1-γ 1 + ln(s) βmb β

(47)

When τ1 is known, the potential of the conduction-band edge can be obtained with eq 44. k3 is independent of Voc and is equal to 1/τ1. When τ2 is known, the position of the conductionband edge can be evaluated with eqs 42 and 43. For recombination via conduction band, mb and γ cannot, in general, be determined by IMVS and photovoltage vs light intensity measurements. However, if the potential of the redox electrolyte can be varied, while s is held constant, then the factor [1 - (1 - γ)/(βmb)] in eq 47 or [1 - (1 - γ)ma/mb] in eq 32 can be determined. With this information, the shielding factor can be determined from ∆Va and ∆Vcb. Experimental Section

From eq 40, a relationship between Voc and qφIoτ2 is obtained.

-Voc )

band, -qφIo ) Jr ) -k3Qcb (eq 1 and Figure 1). Substituting eq 45 into eq 14 gives an expression relating Voc to Io.

(45)

When recombination occurs predominantly via the conduction

Materials. cis-Di(thiocyanato)-N,N-bis(2,2′-bipyridyl-4,4′dicarboxylic acid)-ruthenium(II) dihydrate, [RuL2(NCS)2] 2H2O (Solaronix), TiO2 colloidal solution (Solaronix) and ammonium hydroxide (Aldrich; 30% (w/w) NH3 in water) were used as received. 3-Methyl-2-oxazolidinone, NMO (Aldrich), and 4-tert-butylpyridine, TBP (Aldrich), were vacuum-distilled. Acetonitrile (Aldrich, 99.9%) was dried over CaH2 and distilled under N2. Other reagents were obtained from vendors at the highest commercial purity and used as received. The preparation of nanocrystalline TiO2 films was adapted from the literature.20,23 Conducting glass plates (2.5 × 10 cm; Nippon Sheet Glass; F-doped SnO2 overlayer, 81% transmission in the visible, 2-5% haze, 10 Ω/sq) were used as the substrate for depositing TiO2 films. To control the thickness of the TiO2 film and to mask electrical contact strips, 0.5-cm width of the conducting glass plate was covered along the length of each edge with adhesive tape (Scotch), having a nominal thickness of 40 µm. A viscous TiO2 colloidal solution was spread uniformly on the surface of the glass. After removing the adhesive tapes, the assemblage was heated in air for 30 min at 450 °C and then allowed to cool. The TiO2-covered plates were cut into 1.5 × 1.0 cm electrodes. The film was then covered with a layer (0.5 mL/cm2) of 0.3 M TiCl4 in water overnight in a closed chamber filled with air, washed with distilled water, and annealed again at 450 °C for 30 min. The thickness of the resulting film was about 8 µm as measured with a Tencor AlphaStep profiler. The TiO2 electrodes were coated with the dye by soaking them for 2 h in 3 × 10-4 M [RuL2(NCS)2]‚2H2O in absolute ethanol at 80 °C and then drying them in a stream of N2. To minimize rehydration of the TiO2 surface from moisture in ambient air, the electrodes, while still warm (80-100 °C) from annealing, were exposed to the dye solution. Some dye-coated nanocrystalline TiO2 electrodes were soaked in either 4-tertbutylpyridine (TBP) for 15 min and then dried under a N2 stream or exposed to NH3 vapors above an aqueous NH4OH solution for 20 s. These are referred to as TBP- or NH3-treated electrodes.


Schlichtrol et al.

Figure 6. Block diagram of IMVS setup.

Pt counter electrodes with a mirror finish were prepared by electron beam, depositing a 60 nm layer of Pt on top of a 40nm layer of Ti on a glass plate. The Pt electrode and the dyecoated TiO2 electrode were sealed together with 0.5-mm wide strips of Surlyn (Dupont, grade 1601; Pierson Industries), sandwiched along the length of each edge. Sealing was accomplished by pressing the two electrodes together at a pressure of 900 psi and a temperature of 100 °C for 15 s. A small quantity of redox electrolyte solution was introduced into the cell through one of two notches cut on the edge of the Pt counter electrode. The notches were then sealed with an epoxy resin. The dye-coated TiO2 film was illuminated through the transparent conducting glass support. Apparatus. The samples were illuminated with 680-nmwavelength light from a laser diode (SDL Model 7421 H1; 250mW maximum output power). The laser diode was the source of both bias and modulated light. At 680 nm, the attenuation of light by the dye-covered TiO2 films was very small, implying homogeneous illumination throughout the TiO2 film. This results essentially in a constant charge-injection rate within the TiO2 layer and minimizes diffusion processes at short circuit. A SDL Model 800 laser diode driver was used to obtain a welldefined output power and to modulate the intensity of the light beam. A Stanford Research Systems Model 830 lock-in amplifier was used to control light modulation and to measure the modulation of photopotential. The amplitude of the sinusoidal modulation of the light intensity M (eq 6) had an average value of 0.04Io and ranged from 0.01Io to 0.1Io over some 40 measurements. The dc coupled input of the lock-in amplifier was used because of the low modulation frequency (g0.01 Hz) of the light intensity. A block diagram of the IMVS setup is shown in Figure 6. Jsc and Voc were measured with a Keithley Model 177 multimeter. Description of Sample Sets and Data Evaluation. Measurements were performed on three sets of samples, denoted as A, B, and C. All electrodes within a set were made from the same TiO2 film, covered with [RuL2(NCS)2]. Set A consisted of untreated electrode and a TBP- and a NH3-treated electrode. Set B included an untreated and a TBP-treated electrode. Set C contained two untreated electrodes. The redox electrolyte contained CH3CN/NMO (50:50 wt %), 0.3 M LiI and 0.03 M I2 for all sets except for one sample in set C, denoted as ∆{[I3-],[I-]}, for which the LiI (1.5 M) and I2 (0.15 M) concentrations were 5-fold higher. The parameters ma (eq 16), mb (eq 36), and mc (eq 42) were obtained respectively from linear fits of Voc as a function of ln(Jsc), ln(τ), and ln(Jscτ); the slope (ma, mb, or mc) and the offset (Va, Vb, or Vc) were used as fit parameters. In determining ∆Va, ∆Vb, and ∆Vc, the curves were refitted; however, in this case, only the offset (Va, Vb, and Vc) was used as the fit parameter. For ma, mb, or mc, the slope of the untreated (reference) sample of each set was used in the curve-fitting procedure for each sample set. For calculating the accumulated charge (eq 13), η (eq 1) was taken as unity.24-26

Figure 7. Typical IMVS response and theoretical fit according to eq 11. (a) The real and imaginary response of a NH3-treated electrode vs the modulation frequency and (b) the same curves in the complex plane are shown.

Results and Discussion Because recombination takes place predominantly at the TiO2/ redox electrolyte interface27 and the nanocrystalline TiO2 solar cells exhibit high charge-collection efficiencies,24-26 we can neglect the influence of the SnO2 conducting glass in evaluating the IMVS response. Figure 7 illustrates a typical IMVS response curve with the theoretical fit. The response shows only one semicircle in the complex plane for all measurements. The theoretical fit for one time constant (eq 12) is in excellent agreement with experimental data. In all measurements, the low-frequency limit of the real part of the potential modulation (-M*) corresponds closely with the predicted value of x1/2M dVoc/dIo obtained from the change of dc photovoltage with light intensity and the amplitude of the modulated light intensity. The accumulated charge (Qcb or Qcb + Qss), calculated as Jscτ (eqs 1 and 13) vs the photopotential, is shown in Figure 8. The charge in Figure 8 is normalized to the real surface area of the TiO2 electrode, using a roughness factor of 800. The parameter mc (eq 42) is about 100 mV for all samples, implying that the calculated charge is not Qcb (eq 14) but is instead Qss + Qcb ≈ Qss. The photocapacitance of the cells (Cp) was determined from Figure 8. In terms of the geometric cell area, values of Cp range typically from 80 µF/cm2 at the lowest light intensity to 1.7 mF/cm2 at the highest light intensities. The photocapacitance of the cells is always much larger than that of the SnO2 conducting glass, which is determined to be about 5 µF/cm2 and can therefore be neglected. Assuming that TiO2 and SnO2 have the same Helmholtz capacitance and that the nanocrystalline TiO2 films have a roughness factor of 800, we estimate the Helmholtz capacitance of TiO2 to be about 4 mF/cm2, which is not substantially larger than values of Cp obtained at the


Figure 8. Dependence of calculated charge (Q ) Jscτ/r) on Voc for untreated (reference) and (TBP or NH3)-treated [RuL2(NCS)2]-coated nanocrystalline TiO2 electrodes in CH3CN/NMO (50:50 wt %) containing I2 (30 mM) and LiI (0.3 M) except for ∆{[I3-],[I-]} in (c) in which 0.15 M I2 and 1.5 M LiI were used with an untreated electrode. Jsc equals the charge-injection current into TiO2, r ) 800 is the roughness factor of the TiO2 electrode (surface area/cell area), and τ is the measured time constant. For comparing Vc values (eq 42) within a given data set, mc obtained for the untreated (reference) dye-sensitized TiO2 electrode was used for curve fitting of other samples within a set. Furthermore, to avoid overweighting larger values of -Q, linear fits of Voc vs ln(-Q) plots were made.

highest light intensities. The photoinduced band edge movement can be estimated from the accumulated charge and the Helmholtz capacitance (∆Vcb ) Q/CH) to be about 40 mV at the highest light intensities. Because the accumulated charge depends exponentially on Voc (eq 42 with mc ≈ 100 mV, Figure 8), about half of the band edge shift (∆Vcb ≈ 20 mV) occurs during the last 70 mV increase of -Voc. In other words, as the ratio of Cp to CH of TiO2 become larger with increasing light intensity, the contribution of band edge movement to the change of Voc becomes more important. Thus, although photoinduced band edge movement does not contribute substantially to Voc (∆Vcb/Voc , 1), it can make a significant contribution to the change of Voc (d∆Vcb/dVoc not much less than 1) at high light intensities. Calculations show that for highly insulating TiO2 particles (doping densities < 1018 cm-3), no significant space-charge layer is present, in the case of depleted particles, because of their small size.17,18 In contrast to the depletion case, an accumulation layer can be very thin, even compared to nanosized TiO2 particles. At high photopotentials, an accumulation layer may form when the accumulated charge in the conduction band


Figure 9. Relation of Voc to the light intensity represented by the shortcircuit current (Jsc) for untreated (reference) and (TBP or NH3)-treated [RuL2(NCS)2]-coated nanocrystalline TiO2 electrodes in CH3CN/NMO (50:50 wt %) containing I2 (30 mM) and LiI (0.3 M) except for ∆{[I3],[I-]} in (c) in which 0.15 M I2 and 1.5 M LiI were used with an untreated electrode. For comparing Va values (eq 16) within a given data set, ma obtained for the untreated (reference) dye-sensitized TiO2 electrode was used for curve fitting of other samples within a set. Furthermore, to avoid overweighting larger values of -Jsc, linear fits of Voc vs ln(-Jsc) plots were made.

exceeds the doping density. An upper limit for the potential drop across the accumulation layer is determined by the strength of the electric field at the TiO2 surface induced by cations from solution, which compensate the charge of photoinjected electrons. For a surface-charge density of 200 nC/cm2 (Figure 8), the electric field Eσ ) Q/(os); s(TiO2) ) 31 28) at the TiO2 surface is about 105 V/cm, decreasing toward the bulk of the particle. This restricts the potential drop within the accumulation layer to less than 30 mV for 15-nm-sized particles. The actual potential drop within the accumulation layer may be much smaller. It will depend not only on the particle size and the surface-charge density but also on the photoinduced free-carrier concentration in the conduction band and the doping density of the particle. The photoaccumulated charge in the TiO2 film can be converted to an electron concentration (n) by the expression

n)

qφIoτ q(1 - P)d

(48)

Assuming a film porosity P of 0.329 and a thickness d of 8 µm, the photoinduced electron concentration in the TiO2 film was


Schlichtrol et al. from the equation (cf. eq 14):

Vcb ) -

Figure 10. Dependence of the time constant (τ) on Voc for untreated (reference) and (TBP or NH3)-treated [RuL2(NCS)2]-coated nanocrystalline TiO2 electrodes in CH3CN/NMO (50:50 wt %) containing I2 (30 mM) and LiI (0.3 M) except for ∆{[I3-],[I-]} in (c) in which 0.15 M I2 and 1.5 M LiI were used with an untreated electrode. For comparing Vb values (eq 36) within a given data set, mb obtained for the untreated (reference) dye-sensitized TiO2 electrode was used for curve fitting of other samples within a set. Furthermore, to avoid overweighting higher values of τ, linear fits of Voc vs ln(τ) plots were made.

found to be about 1018 cm-3 at the highest light intensity. Taking the volume of a TiO2 particle to be 10-18 cm-3 (φ ≈ 15 nm), we find that there is at most about one electron per particle, which is located predominately in a surface state. From eq 14, the conduction-band electron concentration in the dark ncbo can be expressed by the relations

ncbo ) ncb exp(βVoc) ,

ncb Qss exp(βVoc) ) n exp(βVoc) Qcb (49)

where ncb is the conduction-band electron concentration, which is much smaller than the electron concentration in the TiO2 film (n ≈ 1018 cm-3, eq 48). The values of n exp(βVoc) at the highest light intensity (cf. Figure 8) are in the range of 104-106 cm-3 for TBP- and NH3-treated samples and 106-108 cm-3 for the reference samples. Deviations of the n exp(βVoc) values among treated and reference samples may be explained by variations of Qss/Qcb. These upper limits for ncbo are much less then any realistic doping density of TiO2, which implies that the TiO2 particles are depleted of electrons in the dark. An upper limit (i.e., the least negative value) for the conduction-band potential vs the redox potential (which is close to SCE) can be calculated

( )

(

Ncb Ncb 1 1 ln < - ln β ncbo β n expβVoc

)

(50)

Taking ncb0 < 106 cm-3 for the reference samples, ncbo < 104 cm-3 for the TBP- and NH3-treated electrodes, and Ncb ≈ 1021 cm-3 30, we find that Vcb < -900 mV for the reference samples and Vcb < -1020 mV for the treated samples. Figure 8 shows that Q vs Voc plots of untreated (reference) and treated electrodes in each sample set have almost the same slope (less than 10% deviation), indicating that over the investigated potential range the distribution of surface states within the band gap is not changed by surface treatment. The shift of the curves (∆Vc) for the TBP- and NH3-treated samples can thus be directly correlated with the displacement of the band edge (-∆Vcb) induced by the surface treatment of the TiO2 electrode (eq 43). Presumably, TBP and NH3 deprotonate the TiO2 surface, which is partially protonated during adsorption of the acidic dye, resulting in the observed band edge shift to negative potentials. For all samples, mc lies between 88 and 104 mV. The shift of potential ∆Vc (i.e., -∆Vcb) is given in Table 1. A large band edge shift was observed for TBP-treated (129 and 102 mV) and NH3-treated (171 mV) samples. As expected, increasing the I3- and I- concentrations by fivefold, while keeping the redox potential constant, did not alter the potential of the band edges. Figure 9 displays a semilogarithmic plot of Jsc vs Voc, fitted according to eq 16. Values of ma (eq 32) ranged from 40 to 55 mV for all samples and deviated less than 10% within each sample set. The observation that ma is significantly larger than 1/β ) 26 mV implies, as discussed in the theory section, that recombination occurs principally via surface states rather than via the conduction band. The TBP- and NH3-treated samples show a more negative Voc at a given light intensity than the untreated (reference) samples. The change of Voc (i.e., ∆Va) of the TBP-treated samples is close to the observed band edge movement (Table 1). In contrast, the change of Voc of the NH3treated sample is much less than the band edge displacement. Figure 10 shows the dependence of τ on Voc for various samples. It is seen that the average time interval between the photoinjection of electrons into TiO2 and their transfer to the redox electrolyte ranges from about 10-1000 ms. This time domain corresponds to respective light intensities equivalent to about 1-10-3 suns. The decrease of τ with more negative Voc reflects the increase of the average energy of occupied surface states and thus the decrease of the average activation energy for electron transfer. The values of mb, which is related to the symmetry factor for the recombination reaction (eq 22 and Figure 4), obtained from independent fits (eq 36), were 100 mV (references), 109 mV (TBP) and 137 mV (NH3) in sample set A (Table 1). For other sample sets, the deviation of mb for all samples was less than 5 mV from that of the untreated (reference) electrode. The relatively large change of mb for the NH3-treated electrode suggests that the recombination reaction is affected by NH3. Because of the presence of one dominant semicircle in the IMVS response (Figure 7) and -Qss . -Qcb (this corresponds to case (ii) of MB1 , MB2 in Appendix A), we conclude that k1 . k2 and k3 and that k2 . k4, which means that charge trapping/ detrapping is much faster than charge transfer to the redox electrolyte (Figure 1). Furthermore, inasmuch as -Qss . -Qcb and recombination occurs via surface states, it follows (as discussed under theory) that τ ) 1/k4. For τ ) 1/k4, the ratio of the rate constants for two different samples is calculated from



TABLE 1: Effect of Chemical Treatment and Redox Electrolyte Composition on Parameters Describing Voc Dependence on Jsc, τ, and Q for [RuL2(NCS)2]-Sensitized Nanocrystalline TiO2 Electrodesa set A Va (mV) Vb (mV) Vc (mV) ma (mV) mb (mV) mc (mV) ∆Va (mV) ∆Vb (mV) ∆Vc (mV)

set B

set C

ref

TBP

NH3

ref

TBP

ref

∆{[I3-],[I-]}

617 1000 215 50 100 99

736 1109 344 53 109 104 119 109 129

730 1054 386 54 137 90 113 54 171

528 778 163 41 70 95

629 881 265 40 73 88 101 103 102

527 910 38 44 78 95

384 657 28 40 74 89 143 253 10

a Sample preparation and curve-fitting procedure for data evaluation are described in the Experimental Section. Within a given sample set, the respective parameters ma, mb, and mc of the reference sample are used as fixed quantities in the curve-fitting procedure for determining Va, Vb, and Vc.

eq 36 as k4(1)/k4(2) ) exp(∆Vb/mb). From ∆Vb and mb values in Table 1, it can be shown that the rate constant of TBP- and NH3-treated electrodes for back electron transfer is a factor of 1.7-4.4 smaller than that of the untreated reference. This decrease is substantially less than reported values,4,5 which were determined from Jsc vs Voc data as exp(∆Va/ma). The determination of the rate constant from ∆Va/ma does not take into account the effect of band edge movement on Voc and does not distinguish between charge accumulated in surface states and that in the conduction band.5 From sample set C, it is seen that increasing the I3- and I- concentrations by 5-fold leads to a factor of 25.6 increase of the rate constant for recombination, implying that the reaction is second order with respect to the I3- concentration, as recently reported.5 The second-order nature of the recombination reaction can be explained by the following mechanism.5

I3- h I- + I2

(51)

e- + I2 f I2•-

(52)

slow

2I2•- 98 I3- + Inet: 2e- + I3- f 3I-

(53) (54)

Equation 52 indicates that I2 is the electron acceptor in the back transfer from TiO2. Two necessary conditions for the proposed recombination mechanism to yield a second-order rate constant are that the adsorbed I2 concentration be proportional to the I3concentration in solution and that the reduction of I2 (eq 52) be reversible. Reversibility implies that the I2- radical anions remain associated with the TiO2 surface and therefore that the rate-limiting dismutation reaction (eq 53) takes place on the TiO2 surface. The influence of surface shielding and band edge movement on the rate constant can be calculated from eq 34. The effect of band edge movement on the rate constant depends on the parameter γ, which describes how much the energy of the electron-acceptor state moves with the band edge potential. When a surface reaction is rate limiting, γ ≈ 1. On the other hand, when charge transport across the Helmholtz layer is rate limiting, γ ≈ 0. From eq 33 and a given data set (∆Va, ma, and mb), one can determine the shielding factor for a certain value of γ. Figure 11a shows calculated values of s for different γ for TBP- and NH3-treated electrodes. It should be emphasized that changing γ will alter ∆Va (i.e., -∆Voc) but will not necessarily affect the shielding factor. Furthermore, from eq 33 one can calculate the contribution of band edge movement and shielding factor to the change of Voc as function of γ.

Figure 11. Calculations of (a) the shielding factor and the effect of (b) band edge movement and (c) surface shielding on Voc for assumed values of γ (eq 33) for TBP- and NH3-treated [RuL2(NCS)2]-coated nanocrystalline TiO2 electrodes in CH3CN/NMO (50:50 wt %) containing I2 (30 mM) and LiI (0.15 M). ∆Vcb was obtained from ∆Vc.

Taking the dismutation reaction (2I2•- f I3- + I-) on the TiO2 surface as rate limiting,5 one expects the value of γ to be close to unity. Figure 11 shows that when γ ) 1, there is very little surface shielding, and all change in Voc for TBP-treated electrodes can be attributed to effects of ∆Vcb. Even in the unlikely event that electron transfer is limited by transport across the Helmholtz layer and therefore γ is close to zero, a comparison of Figure 11b and 11c shows that at least half of the change of Voc is due to band edge movement. In other words, for any reasonable value of γ, the principal cause of the increase of photovoltage, resulting from treating the TiO2 surface with TBP, is the consequence of band edge displacement. Figure 11 shows that for γ ≈ 1, NH3 appears to promote recombination (s < 1). This possibility is supported by τ vs Voc data, which shows that NH3 treatment significantly affects


Schlichtrol et al.

mb, as discussed above. However, the effect of s on Voc is smaller than the increase of Voc, arising from the NH3-induced band edge movement. Further study is necessary to obtain a more detailed understanding of the role of NH3 in the recombination process.

Equation A.3 can be split into the sum of two partial fractions, separating the time constants τ1 and τ2:

A1 )

-UMA1 ω2 + U

Conclusions Detailed information on the recombination kinetics and band edge movement in [RuL2(NCS)2]-sensitized nanocrystalline TiO2 solar cells was obtained by intensity modulated photovoltage spectroscopy. IMVS reveals that the transfer of photoinjected electrons from TiO2 to the redox electrolyte takes place predominantly via surface states and that electron trapping and detrapping are much faster than recombination. The contribution of electric fields (photoinduced-potential drop across the Helmholtz layer and accumulation layer) to the photovoltage is negligible. Thus the photopotential is the result of an increase of the electron concentration in the TiO2 conduction band. The time constant between the photoinjection and recombination of electrons varies from about 10 ms at 1 sun to 1000 ms at 10-3 suns. The recombination reaction was confirmed to be second order with respect to the I3- concentration. The photoaccumulated charge density in surface states is determined to be about 200 nC per cm2 of TiO2 at 1 sun, corresponding to about one electron per TiO2 particle. Treatment of the dye-covered TiO2 surface with 4-tert-butylpyridine and NH3 is shown to produce significant band edge movement. This band edge movement is found to be the major determinant of the more negative photovoltage observed for the treated electrodes. An upper limit of the conduction-band edge potential was determined. Acknowledgment. We are grateful to Dr. M. Gra¨tzel at the Ecole Polytechnique Fe´de´rale de Lausanne for the suggestion of treating the TiO2 surface with ammonia. This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences (G.S., J.S., and A.J.F.), and the Office of Utility Technologies, Division of Photovoltaics (S.Y.H.), U.S. Department of Energy, under Contract DE-AC36-83CH10093. Appendix A The real and imaginary components of charge modulation in the conduction band can be written as

A1 ) -F

B1 ) F

Xω2 + YS ω + Rω2 + S2 ω(ω2 + T)

(A.1b)

ω4 + Rω2 + S2

ω4 + Rω2 + S2 ) (ω2 + U)(ω2 + V)

B1 ) F

Xω2 + YS 2 (ω + U)(ω2 + V) ω(ω2 + T) 2

2

(ω + U)(ω + V)

ω2 + U

+

-MA2 + 2

1 + ω2τ1

ωxVMB2

ωxUMB1

B1 )

ω2 + V

1 + ω2τ22

ωτ2MB2

ωτ1MB1 )

(A.4a)

+ 2

1 + ω2τ1

1 + ω2τ22 (A.4b)

where τ1 ) 1/xU, τ2 ) 1/xV, MA1 ) F(XU - YS)/(U(U - V)), MA2 ) F(YS - VX)/(V(U - V)), MB1 ) F(U - T)/(xU(U V)), and MB2 ) F(T - V)/(xV(U - V)). In the following, we consider the possible cases: (1) τ1 ≈ τ2 and (2) τ1 , τ2. (1) Only One Time Constant (τ1 ≈ τ2). This condition is equivalent to X ≈ Y . xZ (cf. eqs A.2 and A.4). Thus it follows from eq A.2 that

U≈V≈

R ≈ X2 2

(A.5)

Substituting eq A.5 into eq A.3 yields

A1 ≈ -F B1 ≈ F

X(ω2 + X2) 2

2 2

)

(ω + X )

ω(ω2 + X2) (ω2 + X2)(ω2 + X2)

)F

-M* 1 + ω2τ2

(A.6a)

ωτM* ω ) ω2 + X2 1 + ω2τ2 (A.6b)

where M* ) F/X and τ ) 1/X. Setting the derivative of the imaginary component B1 to zero in eq A.6b yields the maximum for B1 at ω ) 1/τ, confirming that τ is a time constant. A comparison of the charge (Q ) qφIoτ) with Qcb and Qss in eq 3 shows that

-Q ) -qφIoτ ≈

-qφIo -qφIoY ≈ ) X XY - Z -qφIok1 ) -Qss (A.7) -Qcb g XY - Z

(2) Two Time Constants (τ1 , τ2). This condition is equivalent to U . V S R . S (cf. eqs A.2 and A.4). It follows from eq A.2 that

U)

where F ) -qφM (scale factor), X ) k1 + k3, Y ) k2 + k4, Z ) k1k2, S ) XY - Z, R ) X2 + Y2 + 2Z, and T ) Y2 + Z. The denominator of eq A.1 can be rewritten as

A1 ) -F

ω2 + V

-MA1 )

(A.1a)

4

where U ) R/2 + xR2 / 4 - S2 and V ) R/2 Substituting eq A.2 into eq A.1 yields

-VMA2 +

V) (A.2)

R 2

x

R + 2

x

R2 - S2 ≈ R w τ1 ≈ 1/xR 4

[ x ]

R R2 - S2 ) 14 2

1-

xR2 / 4 - S2 (A.3a)

4S2 S2 w ≈ R R2 xR (A.9) τ2 ≈ S

In the following, we investigate the conditions determining the relative values of MB1 and MB2. From the definition of MB1 and MB2 in eq A.4, one obtains the expression

MB1 (A.3b)

(A.8)

M B2

F(U - T)xV(U - V) )

xU(U - V)F(T - V)

(U - T)xV )

(T - V)xU

(A.10)



(i) For U . T . V, eq A.10 becomes

MB1 MB2

≈

. X or (ii) Z . XY - Z. (i) Because XY - Z > 0, condition (i) leads to Y2 . Z, and eq A.18 simplifies to

xUV S ) T T

(A.11)

(ii) for U . T J V w xUV . T, and eq A.10 becomes

MB1 M B2

≈

xUV xUV > w MB1 . MB2 T-V T

τ2 ≈

Y XY - Z

(A.19)

With eqs A.19 and 3a, one finds for the accumulated charge

Q ) qφIoτ2 ≈

(A.12)

qφIoY ) Qcb XY - Z

(A.20)

(iii) For U J T . V w xUV , T, and eq A.10 becomes

Furthermore, from the definition of MB2 and MA2 in eq A.4, one obtains the relation

MB1

MB2

M B2

≈

(U - T)xV TxU

not much greater than

xV w xU MB1 , MB2 (A.13)

From the analysis of eqs A.11-A.13, the following deductions can be made:

MB1 . MB2 (i or ii) w xUV ) S . T and U . T

(A.14a)

MB1 , MB2 (i or iii) w xUV ) S , T and T . V

(A.14b)

MB1 ≈ MB2 (i) w xUV ) S ≈ T and U . T . V

(A.14c)

In the case of two time constants in the IMVS response, the analysis is divided into three parts: (a) the high frequency semicircle dominates (MB1 . MB2); (b) the low frequency semicircle dominates (MB1 , MB2); and (c) MB1 ≈ MB2 indicates that MB1 and MB2 have roughly the same values and includes all situations that are not covered by eqs A.14a and A.14b. If one semicircle is dominant, we restrict our analysis to this semicircle. (a) MB1 . MB2. From eq A.14a, it follows that XY - Z . Y2 + Z, which implies that X . Y, XY . Z, and R ≈ X2 . S. Under these conditions and eq A.8, one obtains

τ1 ) 1/xR ≈ 1/X

(A.15)

From the definition of MB1 and MA1 in eq A.4, one obtains the equation

MB1 M A1

F(U - T)U(U - V) )

xU(U - V)F(XU - YS)

)

(U - T)xU X3 ≈ )1 (XU - YS) X3 (A.16)

From eq A.16, one predicts an almost ideal semicircle in the complex plane. With the aid of eq A.15 and eq 3a, the accumulated charge (Q ) qφIoτ1) is found to equal Qcb.

Q ) qφIoτ1 )

qφIoY qφIo ≈ ) Qcb X XY - Z

(A.17)

(b) MB1 , MB2. Using the definitions of R and S given in eq A.1, eq A.9 can be rewritten as

xR xX2 + Y2 + 2Z τ2 ) 1/xV ) ) S XY - Z

M A2

F(T - V) V(U - V) )

xV(U - V)F(YS - VX)

)

(T - V)xV ≈ (YS - VX) Y(XY - Z) ) 1 (A.21) Y(XY - Z)

Eq A.21 is consistent with an almost ideal semicircle. (ii) Z . XY - Z is equivalent to Z ≈ XY, and eq A.18 simplifies to

τ2 ≈

X+Y XY - Z

(A.22)

From the definition in eq A.1, it follows that XY ≈ Z S (k1 + k3)(k2 + k4) ≈ k1k2, implying that k1 . k3 and k2 . k4. These relations show that charge trapping/detrapping is much faster than charge transfer to the electron acceptor. For this case, the accumulated charge can be identified as the sum of Qcb and Qss (cf. eq 4).

Q ) qφIoτ2 ≈

qφIo (X + Y) ≈ Qcb ) Qss XY - Z

(A.23)

Furthermore, from eqs A.1, A.4 and A.9, it can be shown that

M B2 M A2

)

(T - V)xV Y(XY - Z) ≈ )1 (YS - VX) Y(XY - Z)

(A.24)

(c) MB1 ≈ MB2. For a more precise analysis of the case in which both semicircles are of similar size, we introduce κ as the ratio of the size of the semicircles and rewrite the condition MB1 ≈ MB2 as MB1 ) κMB2, where κ is roughly one. From eq A.11, it follows that XY - Z ) κ(Y2 + Z). The inequality U . T (eq A.14c) implies that X2 . Y2 and X2 . Z. With these constraints and eq A.8, it follows that

τ1 )

1

xX

2

≈

2

+ Y + 2Z

1 X

(A.25)

For X . Y, we find for XY - Z ) κ(Y2 + Z) that Z ≈ XY/(1 + κ). Using eqs A.25 and 3a, the accumulated charge can be calculated as

Z qφIo qφIo Y - X κ Q ) qφIoτ1 ≈ ) ) Q X XY - Z 1 + κ cb

(

)

(A.26)

Furthermore, from eqs A.1, A.4, and A.8, it can be shown that

(A.18)

The condition XY - Z , Y2 + Z (eq A.14b) requires that (i) Y

M B1 M A1

)

(U - T)xU X3 ≈ )1 (XU - YS) X3

(A.27)


Schlichtrol et al.

From eq A.1, eq A.9, and the inequalities X2 . Y2 and X2 . Z, τ2 can be expressed as

τ2 )

xX2 + Y2 + 2Z XY - Z

≈

X 1+κ ≈ XY - Z κY

(A.28)

For this case, the relationship between the accumulated charge qφIoτ2 and Qss (eq 3b) is given by

k1 + k3 qφIoX ) Qss Q ) qφIoτ2 ≈ XY - Z k1

(A.29)

where the factor (k1 + k3)/k1 is estimated as 1 e (k1 + k3)/k1 e 1 + κ, using Z ) XY/(1 + κ) S k1k2(1 + κ) ) (k1 + k3)(k2 + k4) S κ ) (k1k4 + k2k3 + k3k4)/(k1k2) > k3/k1. Furthermore, from eqs A.1, A.4, A.9, A.14c, and the conditions X2 . Y2, X2 . Z and Z ≈ XY/(1 + κ), it can be shown that

MB2

(T - V)xV ≈ ) MA2 (YS - VX)

(Y + 1XY+ κ)XS ≈ 1 YS+ κ ) 1 XY YS X(XY 1 + κ) 1 + κ YS - S 2

X2

Appendix B Taking the difference of -(Qcb + Qss) ) a exp(-V/mc) (cf. eqs 13b and 42) and -Qcb ) b exp(-V/β) (cf. eq 14) yields

-Qss ) a exp(-V/mc) - b exp(-βV)

(A.31)

The derivative of -Qss in eq A.31 gives

dQss a )exp(-V/mc) + bβ exp(-βV) g 0 dV mc

(A.32)

where we recall that - dQss/dV ) qNss(V) g 0. Solving for a in eq A.32 gives

a g mcbβ exp[-βVmax(1 - 1/β(mc))]

(A.33)

where Vmax is the most negative photovoltage. With the aid of eq A.33 and the expressions for Qcb + Qss and Qcb, one obtains the inequality

[

(

Qcb + Qss 1 g βmc exp β(-Vmax + V) 1 Qcb βmc

Appendix C The linear approximations of the coulomb energy for the donor states (LD) and the acceptor states (LA) in Figure 4 can be expressed by the relation

)]

(A.34)

LD ) E(V) + mDx

(A.35)

LA ) EA0 + mA + γ∆Ecb - mAx

(A.36)

where γ is the ratio of the movement of the energy of the acceptor state to the band edge shift ((EA1 - EA0)/∆Ecb). The intersection of these lines has the coordinates:

x)

(A.30)

Summary. If only one large semicircle is measured (the lowfrequency limit of the semicircle equals the predicted value x1/2 M dV/dIo, as discussed in the introduction), the product qφIoτ1 equals either Qcb or Qss + Qcb. If two distinct semicircles of similar size are measured with the ratio MB1/MB2 ) κ, the smaller time constant (τ1) allows the calculation of Qcb ≈ qφIoτ1 (1 + κ)/κ and the larger time contant (τ2) yields an estimation of Qss ) κ*qφIoτ2 with 1/(1 + κ) e κ* e 1. Mathematically, MA1 * MB1 and MA2 * MB2. When both semicircles have similar size (i.e., MB1 and MB2 have roughly the same value), we have shown that MA1 ≈ MB1 (eq A.27) and MA2 ≈ MB2 (eq A.30). In this case, eq 12 can be used to fit the IMVS response. Furthermore, it has been shown that when MB1 . MB2, MA1 ≈ MB1 (eq A.16) and when MB1 , MB2, MA2 ≈ MB2 (eqs A.21 and A.24). In these cases, eq A.6 can be used to fit the IMVS response (i.e., the dominant semicircle). (Note: eq A.6 is a special case of eq 12 when M1* or M2* in eq 12 is set to zero.)

-

When mc > 1/β ) 26 mV, the ratio (Qcb + Qss)/Qcb (eq A.34) increases exponentially with increasing difference between V and Vmax. For sufficiently large values of V - Vmax, the ratio (Qcb + Qss)/Qcb becomes much larger than one, and Qss . Qcb. For example, when mc ) 95 mV, Qss/Qcb is at least 10 at V Vmax ) 40 mV.

Et ) E(V) +

EA0 + mA + γ∆Ecb - E(V) mA + mD

mD (E + mA + γEcb - E(V)) mD + mA A0

(A.37)

Using eq A.37, the activation energy Ea(V) from donor level E(V) is given by

Ea(E(V)) ) Et(V) - E(V) )

mD (E + mA + mD + mA A0 γ∆Ecb - E(V)) (A.38)

Inserting Ea0 ) (mD/(mD + mA)) (EA0 + mA) into eq A.38 (Ea0 is the activation energy in the absence of band edge movement when E(V) ) 0) gives

Ea(E(V)) ) Ea0 +

mD (γ∆Ecb + E(V)) (A.39) mD + mA

References and Notes (1) Matthews, D.; Infelta, P.; Gra¨tzel, M. Sol. Energy Mater. Sol. Cells 1996, 44, 119. (2) Smestad, G. Sol. Energy Mater. Sol. Cells 1994, 32, 273. (3) Liska, P. Ph.D. Thesis, Swiss Federal Institute of Technology, Lausanne, 1994 (No. 1264). (4) Nazeeruddin, M. K.; Kay, A.; Rodicio, I.; Humphry-Baker, R.; Mu¨ller, E.; Liska, P.; Vlachopoulos, N.; Gra¨tzel, M. J. Am. Chem. Soc. 1993, 115, 6382. (5) Huang, S. Y.; Schlichtho¨rl, G.; Nozik, A. J.; Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. B 1997, 101, 2576. (6) Parkinson, B. A.; Furtak, T. E.; Canfield, D.; Kam, K.-K.; Kline, G. Discuss. Faraday Soc. 1980, 70, 233. (7) (a) Peak, R.; Peter, L. M. Ber. Bunsen-Ges. Phys. Chem. 1987, 91, 381. (b) Schlichtho¨rl, G.; Ponomarev, E. A.; Peter, L. M. Electrochem. Soc. 1995, 142, 3062. (c) Ponomarev, E. A.; Peter, L. M. J. Electroanal. Chem. 1995, 396, 219. (8) Vanmaeckelbergh, D.; Iranza Marin, F.; van de Lagemaat, J. Ber. Bunsen-Ges. Physik. Chem. 1996, 100, 616. (9) Schefold, J. J. Electroanal. Chem. 1992, 341, 111. (10) Oskam, G.; Frei, C.; Searson, P. C. Proceedings of the Symposium on Nanostructured Materials in Electrochemistry; Searson, P. C., Meyer, G. J., Eds.; The Electrochemical Society: Pennington, NJ, 1995; p 98. (11) (a) Eichberger, R.; Willig, F. Chem. Phys. 1990, 141 159. (b) Burfeindt, B.; Hannappel, T.; Storck, W.; Willig, F. Abstracts of the Third International Meeting on New Trends in Photoelectrochemistry, Estes Park, Colorado, 1997. (12) Moser, J.-E.; Gra¨tzel, M.; Durrant, J. R.; Klug, D. R. In Femtochemistry: Ultrafast Chemical and Physical Processes in Molecular Systems; World Scientific: River Edge, NJ, 1996.

Dye-Sensititized Nanocrystalline Solar Cells (13) (a) Kelly, J. J.; Memming, R. J. Electrochem. Soc. 1982, 129, 730. (b) McEvoy, A. J.; Etman, M.; Memming, R. J. Electroanal. Chem. 1985, 190, 225. (c) Tubbesing, K.; Meissner, D.; Memming, R.; Kastening, B. J. Electroanal. Chem. 1986, 214, 685. (14) Frese, K. W., Jr. in Photoelectrochemistry: Fundamental Processes and Experimental Techniques; Wallace, W. L., Nozik, A. J., Deb, S., Wilson, R. H., Eds.; The Electrochemical Society Proceedings Series PV 82-3; Pennington, NJ, 1982; pp 157-171. (15) (a) Allongue, P.; Cachet, H. J. Electrochem. Soc. 1984, 131, 2861. (b) Allongue, P.; Cachet, H.; Horowitz, G. J. Electrochem. Soc. 1983, 130, 2352. (16) Mao, D.; Kim, K.-J.; Frank, A. J. J. Electrochem. Soc. 1995, 142, 1869. (17) Hagfeldt, A.; Gra¨tzel, M. Chem. ReV. 1995, 95, 45. (18) Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. 1982, 86, 2964. (19) Sze, S. M. Physics of Semiconductor DeVices, 2nd ed.; Wiley-Interscience: New York, 1981; p 27.

J. Phys. Chem. B, Vol. 101, No. 41, 1997 8155 (20) Gra¨tzel, M.; Kalyanasundaram, K. Curr. Sci. 1994, 66, 706. (21) Kumar, A.; Sanatangelo, P. G.; Lewis, N. S. J. Phys. Chem. 1993, 96, 835. (22) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley and Sons: New York, 1980, p 92. (23) Smestad, G.; Bignozzi, C.; Argazzi, R. Sol. Energy Mater. Sol. Cells 1994, 32, 259. (24) Gra¨tzel, M. Renewable Energy 1994, 5, 118. (25) Gra¨tzel, M. J. Sol-Gel Sci. Technol. 1994, 2, 673. (26) Eichberger, R.; Willig, F. Chem. Phys. 1990, 141, 159. (27) Stanley, A.; Matthews, D. Aust. J. Chem. 1995, 48, 1294. (28) Roberts, S. Phys. ReV. 1949, 76, 1215. (29) Papageorgiou, N.; Gra¨tzel, M.; Infelta, P. P. Sol. Energy Mater. Sol. Cells 1996, 44, 405. (30) Redmond, G.; O’Keefe, A.; Burgess, C.; MacHale, C.; Fitzmaurice, D. J. Phys. Chem. 1993, 97, 11081.

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