Modeling of Photovoltage and Photocurrent in Dye-Sensitized

J. Phys. Chem. B 2001, 105, 4895-4903

4895

Modeling of Photovoltage and Photocurrent in Dye-Sensitized Titanium Dioxide Solar Cells Jo1 rg Ferber* and Joachim Luther Fraunhofer Institute for Solar Energy Systems and Freiburg Materials Research Center, Oltmannsstr. 5, D-79100 Freiburg, Germany ReceiVed: August 10, 2000; In Final Form: March 28, 2001

By means of two-dimensional simulation calculations, a detailed analysis of the nanocrystalline TiO2 dyesensitized solar cell (DSC) has been performed. A simplified scheme of the nanoporous structure, which is treated as if the TiO2 film is a continuous medium, is used for modeling. On the basis of material parameters, the model permits the determination of steady-state charge-carrier distributions, the calculation of I-V curves under illumination, dark characteristics, and the spectral response of a DSC. The spatial resolution of the model allows for the answer to the question of the spatial distribution of both the electric and the electrochemical potential in the cell. Thus, a deeper insight into the operation mechanism of a DSC is obtained. Nonnegligible drift currents are found. It is shown quantitatively that the electric potential drops mainly at the TCO/TiO2 interface and not at a Helmholtz layer. The role of the dark interfacial electrical potential difference (built-in potential) for the function of a DSC is discussed. It is shown that, contrary to a conventional p-n junction solar cell, higher photovoltages than those of the dark interfacial electrical potential difference can be obtained.

I. Introduction Models of the nanocrystalline TiO2 dye-sensitized solar cell (DSC, Figure 1) generally treat the spongelike structure of the inner cell as an effective medium.1,2 For charge-carrier transport, in most cases, pure diffusion is assumed.2,3 There is, on the other hand, some evidence of a nonneglectable electric field in the bulk of the cell, giving rise to drift currents also.1 Another question still under discussion is whether a dark interfacial electrical dark potential is necessary for the operation of the cell.1,4-6 In this paper, an extended two-dimensional model of the DSC as well as numerical calculations are presented that address specifically the spatial distribution of the electric and the electrochemical potential in the cell. The model can be regarded as an extension of a model recently published,1 rejecting the effective medium approach; the electric fields within the TiO2 semiconductor and within the redox electrolyte are explicitly distinguished. Applying Poisson’s equation and the equations of continuity and transport to all charge carriers involved allows for the calculation of complete I-V characteristics, chargecarrier densities, and current distributions starting with the material parameters. Furthermore, electric potentials, chemical potentials, and electrochemical potentials may be analyzed. In particular, the transformation of differences in the electrochemical potential of electrons under illumination into the external photovoltage of the cell will be discussed. II. Description of the Model A schematic of a DSC is shown in Figure 1. The inner cell consists of nanosized TiO2 colloids and of the iodide/triiodide redox electrolyte. The TiO2 colloids are sintered together and coated with suitable light-absorbing charge-transfer dye molecules. The electrolyte fills the pores in the nanoporous TiO2 layer. This compound is sandwiched between two glass * To whom correspondence should be addressed. E-mail: ferber@ microlas.de.

Figure 1. Schematic diagram of the DSC. The interconnected TiO2 particles (grey) are covered with light-absorbing dye molecules (small black dots). The free volume between 0 and d is filled with the electrolyte. The coordinates y ) 0 and y ) d indicate the TCO/TiO2 interface or the electrolyte/platinum interface, respectively.

substrates, which are both coated with a transparent conductive oxide (TCO) layer. The contact between the electrolyte and the platinized TCO is modeled as a redox electrode via the Nernst equation and a charge-transfer resistance. The TiO2/TCO contact is modeled as an ohmic metal-semiconductor contact. The electrolyte is also in contact with the unplatinized TCO. However, in the absence of platinum, which acts as a catalyst for the redox reaction, the charge-transfer resistance is very high. Therefore, the current at the electrolyte/unplatinized TCOcontact is very small and is neglected for simplicity. Equations Applying within the TiO2 and within the Electrolyte. The mobile charge carriers involved are the electrons in the TiO2 conduction band and the iodide, the triiodide, and the cations in the electrolyte. It is assumed that the transport of all of these charge carriers can be described by effective diffusion constants D or mobilities b. A simplified geometric structure of the DSC is used for modeling (Figure

10.1021/jp002928j CCC: $20.00 © 2001 American Chemical Society Published on Web 05/08/2001

4896 J. Phys. Chem. B, Vol. 105, No. 21, 2001

Ferber and Luther

3 b ‚Bj i(x,y) ) 0 (i ) I-, I3-, K+)

(3)

Nanocrystalline TiO2 is weakly n-doped because of a nonperfect crystal structure. By means of conductance measurements on nanocrystalline TiO2 layers (4-point measurement in the absence of any electrolyte), ND+ was estimated to be in the range of 1016 cm-3. This agrees well with published values.7,8 A remark about the meaning of an electron density of 1016 cm-3 for a 20 nm sized particle should be added: Both in conductance measurements and in the laminar structures underlying the presented model, the nanostructure is treated as if the film is a continuous medium. The influence of traps, especially at the nanocrystalline TiO2 surface, on the potential distribution was neglected for simplicity and may be included in a further refinement of the model. The influence of such traps on charge transport by means of trapping and detrapping events9 is, to a certain extent, included by using effective diffusion constants. In principle, when effective diffusion constants are used in eq 2, the electron density terms ne refers to the total (free + trapped) electron density. However, within the model presented, all photoinjected electrons are assumed to be within the TiO2 conduction band. Thus, the density of trapped electrons is zero and the total electron density equals the conduction band electron density. It should be emphasized that, this way, the used electron diffusion constant De is effective both with respect to the nanoporous TiO2 structure and to the influence of traps. Boundary Conditions. For the differential equations introduced above, each valid within one of the two phases, appropriate boundary conditions have to be applied. TiO2/Electrolyte Interface. The net current density across the TiO2/electrolyte phase boundary is determined from the difference of an electron generation current density jG and an electron relaxation current density jR across this interface. This results in the following boundary condition for Bje:

Bj i ) e0((Di3 b ni - bini3 b φ)

(4)

Bj e (x ) 0,y)‚ν bx ) jG(y) - jR(y)

Figure 2. Different models for the DSC. Three steps of simplification are shown. (a) f (b): For symmetry reasons, some TiO2-colloids can be neglected. (b) f (c): The vertical TiO2 structures are approximated by laminar structures. The symmetry element indicated in the laminar structure at the right-hand side was used for the calculations. At the lower right side, coordinates x and y are indicated. For the numerical calculations, aTiO2 ) 10 nm, ael ) 10 nm, dTiO2 ) 10 µm, and d ) 11 µm were used.

2c). If the assumed ordering of the TiO2 colloids into columns (as shown in Figure 2) were real, improved effective diffusion constants or mobilities would be obtained. Apart from this, cell properties should not change. The assumed geometrical structure shown in Figure 2c is essential for the realization of the model calculations. The coordinate system (x, y) used is also shown in Figure 2c. Continuity and transport equations are applied to all of the mobile charge carriers involved:

3 b ‚Bj e(x,y) ) 0

(1)

Bj e ) e0(De3 b ne - bene3 b φ)

(2)

for the electrons within the TiO2

for the iodide, the triiodide, and the cations within the electrolyte. The plus sign corresponds to the negatively charged particles I- and I3-, and the minus sign corresponds to the positive cations K+. n is the density and j the current density of the respective charge carrier, e0 is the elementary charge, and φ is the electric potential. Because within the bulk materials no charge carriers are generated or lost, neither generation nor recombination terms appear in eqs 1 and 3. Charge carrier generation and recombination within the cell result from electron transfer between the electrolyte and the TiO2 (in the case of generation, it is by means of the dye). This is described by appropriate boundary conditions at x ) 0 (see below). The electric potential φ in the cell, and therefore the electric field B E)-3 B φ, is calculated using Poisson’s equation both for the TiO2 semiconductor (TiO2) and for the redox electrolyte (El):

e0 [N + - ne(x,y)] (x, y) ∈ TiO2 ∆φ ) TiO20 D

(5)

e0 [n (x,y) - nI-(x,y) - nI3- (x,y)] ∆φ ) El0 K+ (x, y) ∈ electrolyte (6) The two materials are characterized by the two dielectric constants TiO2 and El. 0 is the permittivity of free space. Apart from the mobile charge carriers involved, the positive charge ND+ of the ionized donor sites of the TiO2 is taken into account.

(7)

b νx is a normalized vector in x-direction, i.e., across the interface. The absorption of each photon in the dye (at x ) 0) is assumed to be coupled with the injection of one electron into the TiO2 conduction band and the subsequent oxidation of the electrolyte. This process is described by the electron generation current density jG across the phase boundary TiO2/electrolyte. jG is calculated from the spectral absorption coefficient of the dye, which is determined by the chosen dye and its concentration within the cell:

dTiO2

jG(y) ) e0

Rf

800nm R(λ) Φ(λ) e-R(λ)y dλ ∫300nm

(8)

The integration limits reflect the characteristics of the absorption coefficient R(λ) of the used ruthenium dyes.10 The ratio of the TiO2 layer thickness dTiO2 and the roughness factor Rf accounts for the correct description of the internal surface of the nanoporous TiO2 layer by the simplified geometrical structure used. Within the cell, only one electron loss mechanism is considered: the capture of conduction band electrons by the oxidized species (triiodide) of the electrolyte. This process is described by an electron relaxation current density jR across the phase boundary TiO2/electrolyte. It is expected that jR depends on both the local electron and the local triiodide density:4 -

jR ) e0ketnIβ3-I3 (nβe e - njβe e)

(9)

Modeling of TiO2 DSCs

J. Phys. Chem. B, Vol. 105, No. 21, 2001 4897

Figure 3. Stern’s model of the distribution of charge and of the electric potential φ in the electric double layer at a solid/electrolyte interface.

ket is a rate constant. The exponents βe and βI3- are the reaction orders. βe should be in the range of 0.5 < βe < 2. Values not equal to 1 may result from the involvement of traps in the detailed reaction mechanisms. The determination of the equilibrium electron density nje is described below. As discussed in the literature,1 the back reaction of conduction band electrons with triiodide (which probably occurs via traps and intermediate reactions) is the main loss mechanism. Further reaction paths, e.g., the recombination of conduction band electrons with oxidized dye molecules, could principally be included in a refined model. However, this would lead to additional parameters which are not exactly known. Because the total redox reaction is

I3- + 2e- h 3I-

(10)

the boundary conditions for the iodide and triiodide current densities are

3 bx ) (jG - jR) Bj I-(x ) 0,y)‚ν 2 (ν bx is a normalized vector in the x direction) (11) 1 bx ) (jR - jG) Bj I3-(x ) 0,y)‚ν 2

(12)

The electric double layer at the phase boundary between the electrolyte and the TiO2 or the TCO is modeled by Stern’s modification of the Gouy-Chapman theory,11 which assumes that the ions in the diffuse layer on the electrolyte side of the electric double layer cannot approach the surface any closer than the ionic radius. There is a plane of closest approach for the centers of the ions at some distance xH, which is called the Helmholtz plane (Figure 3). Thus, there is no net charge between the interface and the Helmholtz plane. The conditions of continuity for the electric potential φ and for the dielectric displacement vector (-03 B φ) are applied to the interface. Dye molecules partially block the TiO2/electrolyte interface. The influence of the dye molecules on the potential distribution was neglected. The dye is included in the model only for its content in the cell and its spectral absorption coefficient. TiO2/TCO Interface. Usually fluorine-doped SnO2 is used as the TCO. Because its doping is very high, the large band gap

semiconductor SnO2 becomes degenerate and shows metallic properties, such as resistance increasing with temperature and high conductivity. Therefore, it is treated as a metal, and the TiO2/TCO interface is modeled as a semiconductor/metal contact. This is, of course, an approximation. In fact, when the TCO is in contact with I-/I3-, it is in depletion, and so the electrons are no longer degenerate at the surface. Under illumination, the TCO moves into accumulation and becomes degenerate again.12 However, it was estimated that only 15% of the electrical potential difference at this contact drops within the TCO.13 In principle, there might be a potential barrier at the TiO2/ TCO interface. However, measurements of Gregg and coworkers showed that the work function of the TCO does not influence the photovoltage of the DSC. One possible explanation for this surprising effect is tunneling of the electrons through a very thin barrier.6 Assuming such a contact to be approximately ohmic, boundary conditions similar to those used in modeling silicon solar cells14 are applied: (1) The electric potential φ is continuous:

φ(x, y ) 0) ) φfront TCO

(13)

φfront TCO is the electric potential of the TCO at the front contact, i.e., the TCO/TiO2 contact. (2) Charge neutrality independent of the actual current flow is assumed:

ne(x, y ) 0) ) ND+

(14)

This corresponds to the fact that the electron density and thus the chemical potential of the electrons does not change within a metal. Of course, there might be surface charges. However, at any position within a metal, even very close to the interface, the electron density remains fixed. Thus, also at the interface, a fixed electron density is assumed. Electrolyte/TCO Interface. There are two different electrolyte/ TCO interfaces: at the front side and at the rear side. Contrary to the front side, the TCO at the rear side is covered with some platinum, acting as a catalyst for the redox reaction. Provided the charge-transfer resistance is not too high, it is a good approximation to describe the electrolyte/platinized TCO contact by an ohmic charge-transfer resistance and the Nernst equation. A low charge-transfer resistance corresponds to a high exchange current density, which is essential for a DSC with good performance. For higher charge-transfer resistances, it would be necessary to use the more accurate, generalized currentvoltage equation. A low charge-transfer resistance can simply be added to the ohmic series resistance of the TCO layers. It is included into the model via a simple equivalent circuit, which consists of a series connection of this resistance, and the inner cell, described by the presented model. The Nernst equation has to be fitted into Stern’s model of the electric double layer at the interface. On a microscopic scale, the electric potential is continuous. On a larger scale, there is an electric potential drop between the Helmholtz plane of closest approach of the ions to the interface and the interface itself (Figure 4). This potential drop is described by the Nernst equation, which combines it with the concentrations of the redox ions at the Helmholtz plane. An analysis presented in the Appendix results in the following form of Nernst’s equation, suitable for an integration into the presented model:15


Ferber and Luther Now, fixing an arbitrary reference potential, e.g. φback TCO ) 0.6 V, and varying the cell voltage front U ) φback TCO - φTCO

(19)

the corresponding net current through the cell and thus the whole I-V characteristic can be calculated. Calculations can be performed for different illumination levels and different material parameters. The simulations are performed numerically, approximating the differential equations by finite-difference equations, which are solved iteratively by means of the Newton method.16 The mathematical procedure closely follows the route given by Selberherr14 for the simulation of crystalline silicon solar cells, where differential equations with similar structure have to be solved. Dark Cell in Equilibrium. To evaluate eq 9, the equilibrium electron density nje at the TiO2/electrolyte interface is needed. Therefore, the Poisson-Boltzmann equation has to be solved. The Poisson-Boltzmann equation for the electrolyte is

∑i zin0i e(-z e (φ-φ

-El0∆φ ) e0

i 0

ref El )/kT)

(20)

(i ) iodide, triiodide, and cation. Charge number zi ) -1 for iodide and triiodide, and zi ) +1 for the cations). The Poisson-Boltzmann equation for the TiO2 semiconductor is Figure 4. Principal sketch of the electric potential φ at the electrolyte/ platinized TCO contact (A) in equilibrium (electrode potential U0) and (B) when a current is flowing (electrode potential U). The platinized TCO is assumed to be metallic. Dark interfacial electrical potential difference φTCO/El , overvoltage η, thickness of the rigid Helmholtz bi layer xH.

φback TCO - φEl(d - xH) ) φTCO/El bi

(

)

nI3-(d - xH)/nI03kT + ln 2e0 {n -(d - x )/n0-}3 I H I

(y ) d) (15)

by + 3Bj I3- (x, y ) d)‚ν by ) 0 φ bI-(x, y ) d)‚ν (ν by is a normalized vector in the y direction) (16) Equations of ConserVation. Two additional equations are necessary in order to obtain a complete system of equations. These result from the fact that neither cations nor iodine atoms are generated or lost, i.e., their integrals over the volume VEl of the electrolyte (VEl depends on ael, aTiO2, and d) remain constant: +

El

∫V (nI El

-

3

dx dy ) n0K+VEl

(17)

1 1 + nI- dx dy ) n0I3- + n0I- VEl 3 3

)

(

)

(18)

(21)

In both equations, the electric potential φ has to base on a reference potential characterizing the separated, undisturbed material. The difference of the two electric reference potentials involved equals the difference in the chemical potentials of the two materials (see the Appendix), usually called diffusion or build-in potential: bi ref φref TiO2 - φEl ) φ

The upper index “0” denotes initial concentrations, and d - xH is the position of the Helmholtz layer. The dark interfacial is set up at such an electrical potential difference φTCO/El bi interface, formed by two different materials with different initial electrochemical potentials. φTCO/El would be the standard bi Galvani potential difference, if standard concentrations would be used instead of the initial concentrations n0. Because of the total redox reaction (9), the iodide current density jI- at the interface is linked to the triiodide current density jI3-:

∫V nK

ref

-eTiO20∆φ ) e0{ND+ - n0e e(e0(φ-φTiO2)/kT)}

(22)

It should be pointed out that an electric potential of this order is not necessarily built up directly at a nanocrystalline TiO2/ electrolyte contact (see below). Therefore, in this work, to avoid confusion, φbi will be called the dark interfacial electrical potential difference (see the Appendix for details). Because differences in electrical potentials only have physical meaning, one of these reference potentials may be arbitrarily chosen, e.g. φref El :) 0 for the dark equilibrium condition. Using the conditions of continuity for the electric potential φ and for the dielectric displacement vector (-03 B φ) as described above, the Poisson-Boltzmann eqs 20 and 21 have to be solved simultaneously, resulting in both the electron distribution and the distribution of the electric potential within the DSC in equilibrium. III. Material Parameters The material parameters used for the simulations are compiled in Table 1. Numerical values or at least orders of magnitude for all of these parameters were taken from literature. The rate constant for the relaxation current density ket corresponds to a mean electron lifetime τe ) 20 ms approximately. The mobilities b of the charge carriers are connected to the diffusion constants D by means of the Einstein relation b ) e0D/kT. The geometrical parameters are indicated in Figure 2. For the I-V curves


Figure 5. Electric potential distribution at the electrolyte/TiO2 interface in the x direction (1 µm e y e 9 µm, see Figure 2 for coordinates and dimensions). TiO2 colloid radius aTiO2 ) 10 nm. Two-dimensional modeling of the DSC in thermal equilibrium (nonilluminated cell).

presented in Figures 9 and 10, additional series resistances due to the TCO (6 Ω) and to the charge-transfer at the platinum electrode (0.26 Ω corresponding to an exchange current density of 0.1 A/cm2) were included. According to the assumption of an ohmic TCO/TiO2 contact, the initial electrochemical potentials of the TCO and the TiO2 were assumed to be equal, i.e., 2 φTCO/TiO ) 0. From this hypothesis, it follows directly that φbi bi TCO/El ) φbi . IV. Results DSC in Equilibrium. The calculated electric potential distribution at the electrolyte/TiO2 interface within the cell in equilibrium, i.e., in the dark and without any external applied voltage, is shown in Figure 5. The dark interfacial electrical potential difference was estimated to be 0.6 eV.17 When connecting two materials with different electrochemical potentials, charge carriers are exchanged until a constant electrochemical potential is adjusted. This usually results in the setup of an electric potential difference, called build-in or dark interfacial electrical potential difference. The magnitude of this dark interfacial electrical potential difference multiplied by e0 equals the initial difference in the electrochemical potentials of the separate phases. It is important to note that because of the small size of the TiO2 colloids, the setup of an electric potential difference of 0.6 V cannot be realized inside the colloids. There is only a very small electric potential drop of 0.1 mV, and band bending within a colloid is negligible.18 The electric potential cannot drop in the Helmholtz layer of the electrolyte either. This follows directly from the Poisson-Boltzmann eqs 20 and

J. Phys. Chem. B, Vol. 105, No. 21, 2001 4899 21 and the continuity equations for the electric potential and the dielectric displacement vector. Instead of the build-up of an electric potential difference, the density of the conductionband electrons and thus the chemical potential of the TiO2 changes by means of charge exchange with the electrolyte. However, the electric potential has to drop somewhere. The model calculations show, as proposed by Schwarzburg and Willig,5 that the dark interfacial electrical potential difference builds up at the TCO/TiO2 interface (Figure 6). A principle sketch of the electric potential distribution within the nonilluminated DSC is shown in Figure 7, which clarifies the points discussed. As was discussed already, in fact, a small part of the electric potential difference at the TCO/TiO2 interface drops at the TCO side. For simplicity, this was neglected in the model. However, the TCO contains a large amount of electrons. Thus, Poisson’s equation allows a large curvature (second derivative) of its electric potential distribution if the TCO is depleted at the interface. Instead of this large curvature and the corresponding potential drop within the TCO, a discontinuity in the first derivative of the electric potential (i.e., the electric field) was allowed at the boundary (y ) 0). Thus, near the TCO/TiO2 contact, large electrical potential differences can occur on length scales in the order of the TiO2 colloid size. On the contrary, within the bulk of the nanoporous TiO2 layer, Poisson’s equation allows only for very small electrical potential drops, because of the small colloidal size (respectively the size of the laminar structure in the x direction), the low space-charge density available, and the condition of continuity of the electric field within the TiO2. DSC under Operation. The electric potential distribution at the electrolyte/TiO2 interface within the illuminated cell (photon flux equivalent to the AM1.5 spectrum) is shown in Figure 8. Under short-circuit conditions, the potential distribution does not differ significantly from the equilibrium state. Because of the electron injection from the dye, a higher electron concentration in the TiO2 is found. Thus, a somewhat higher electric potential difference of about 1-10 mV between the electrolyte and the nanocrystalline TiO2 colloids is found. However, the overall properties remain unchanged, because there is no photovoltage under both short-circuit and equilibrium conditions. The large electric field which corresponds with the electric potential drop at the TCO/TiO2 contact fits into the picture of Schwarzburg and Willig:5 within the cell, injected electrons are short-range screened by cations. These neutral quasiparticles move to the front contact, where they are separated in the described strong electric field -3 B φ.

TABLE 1: Material Parameters Used for Modeling of the DSC parameter

symbol

rate constant reaction orders electron diffusion constant iodide, triiodide diffusion constant initial concentration of iodide initial concentration of triiodide TiO2 donor concentration TiO2 dielectric constant electrolyte dielectric constant dark interfacial electrical potential difference incident spectral photon flux density

ket βe, βI3De DI-, DI3n0In0I3ND+ TiO2 El φbi Φ(λ)

optical loss from front TCO glass substrate dye content dye absorptivity thickness of rigid Helmholtz layer

R(λ) xH

numerical value 10-24

cm4/s 1 5‚10-5 cm2/s 8.5‚10-6 cm2/s 0.45 M 0.05 M 1016 cm-3 114 36 0.6 V ) ˆ AM1.5 global solar spectrum, 100 mW/cm2 16% ) ˆ 1000 monolayers of dye measured spectrum 0.15 nm


Ferber and Luther

Figure 6. Electric potential distribution in the y direction within the electrolyte (x ) ael ) 10 nm), within the TiO2 (x ) aTiO2 ) 10 nm), and at the electrolyte/TiO2 interface (x ) 0). The electric potential within the TCO (y < 0 nm) and within the platinized TCO (y > 11 000 nm) is also indicated. Two-dimensional modeling of the DSC in equilibrium (nonilluminated) under the assumption of the TCO/TiO2 contact to be an ohmic metal/semiconductor contact. The dark interfacial electrical potential difference drops within the first TiO2 colloids.

Figure 7. Principle sketch of the electric potential distribution in the y direction within the DSC in electrochemical equilibrium (nonilluminated).

Figure 8. Electric potential distribution in the y direction within the electrolyte (x ) ael ) 10 nm), within the TiO2 (x ) aTiO2 ) 10 nm), and at the electrolyte/TiO2 interface (x ) 0). Two-dimensional modeling of the illuminated DSC under open-circuit (OC) and short-circuit (SC) conditions, respectively.

Contrary to p-n junction solar cells,19 the open-circuit voltage is not limited to a value that compensates the dark interfacial electrical potential difference (Figure 9). This points out the

principal difference between a conventional p-n junction solar cell and a DSC: In a p-n junction cell, charge separation occurs within one material, both charge-carriers being subject to the same electric potential. In a DSC, the separated charges are transported in two different materials (TiO2 and electrolyte) under principally different electric potentials. The dark interfacial electrical potential difference depends on both the TiO2 conduction band edge ECB and the doping density ND+. A variation of ECB directly influences the photovoltage, as expected (Figure 9). However, the photovoltage is not affected by changing the TiO2 doping density. Also, the overall I-V characteristics of the DSC are not nearly influenced by the doping density (Figure 10). Under operation, significant electric fields can appear within the TiO2 semiconductor, which can be seen from the gradient of the electric potential of the illuminated DSC under shortcircuit conditions (Figure 11). Thus, beside diffusion, there are also significant drift currents (Figure 12). Nevertheless, the current-voltage behavior of the one-dimensional modeled DSC1 could be confirmed qualitatively. Only slight changes in the effective electron diffusion constant De had to be made in order to obtain the same photocurrent for the different models. In the two-dimensional modeling, higher drift currents are largely compensated by lower diffusion currents. Thus, the modeled overall currents do not change significantly. (Therefore, several types of experiments, such as intensity modulated photocurrent spectroscopy,3 can be described well under the assumption of a purely diffusive electron transport). The magnitude of the current density is determined by the gradient in the quasi-Fermi

Figure 9. I-V characteristics of the modeled DSC for different dark interfacial electrical potential differences φbi. Contrary to a conventional p-n junction cell, the open-circuit voltage is not limited by the dark interfacial electrical potential difference φbi established in the dark (left). Here, φbi was changed by varying the position of the TiO2 conduction band (right). The doping density and therefore the relative position between the conduction band edge ECB and the initial Fermi level EF0 remain constant.



Figure 10. I-V characteristics of the modeled DSC for different dark interfacial electrical potential differences φbi (left). In these simulations, φbi was changed by varying the doping density ND+ of the TiO2 (right). The position of the TiO2 conduction band edge was held fixed.

Figure 11. Electric potential distribution in the y direction within the electrolyte (x ) ael ) 10 nm) and within the TiO2 (x ) aTiO2 ) 10 nm), see Figure 2 for the coordinates. Two-dimensional modeling of an illuminated DSC under open-circuit (OC) and short-circuit (SC) conditions. At the TCO/TiO2 contact at y ) 0, large potential drops occur, which can be seen completely in Figure 8.

Figure 12. Drift and diffusion components of the electron current density within the TiO2 semiconductor of the modeled DSC. DSC under illumination and maximum-power-point conditions.

level, i.e., the gradient in the electrochemical potential of the electrons in the TiO2 conduction band (Figure 13). V. Experimental Support for the Model and Discussion The underlying concepts of the model presented are proven concepts of semiconductor physics and electrochemistry. Among others, the question remains whether the results found for the simplified geometrical structure shown in Figure 2c also describe the much more complicated situation within a real DSC. Qualitatively, experimental support for the operation mechanisms and potential distribution found was given by Schwarzburg and Willig.5 Briefly, from the analogy in the qualitative behavior

of a DSC and a p-n junction cell, an analogue photovoltaic functioning principle is derived, emphasizing the role of the dark interfacial electrical potential difference. A capacitance of about 10 µF/cm2 was also measured with the DSCs produced at our institute. This is within the expected order for the capacitance of the space-charge region at the front TCO contact. The question remains whether the TiO2/TCO contact is really an ohmic one. This contact is not yet fully understood and needs further experimental investigations, especially because the presence of the electrolyte may influence the behavior of the TiO2/TCO contact.20,21 The existence of an additional potential barrier at this contact was suggested by Peter et al.3 The exponential term for the current through this interface


Ferber and Luther

Figure 13. Distribution of the electrochemical potential in the y direction within the TiO2 (x ) aTiO2 ) 10 nm). Two-dimensional modeling of the DSC under open-circuit (OC), maximum-power-point (MPP) and short-circuit (SC) conditions. EF ) 0 corresponds to the redox potential of the electrolyte; thus, EFn(0) corresponds to the cell voltage.

the back-reaction rate constant and the electron diffusion constant have a strong influence on the performance of the modeled DSC. VI. Summary

Figure 14. Measured I-V curve of a DSC build at Freiburg Materials Research Center. Cell area 4 cm2, irradiance 100 mW/cm2 of sun simulator, and efficiency 5.2%.

introduced by them might describe a tunnel contact. Nevertheless, tunnel barriers are also ohmic contacts.22 Recently, Gregg and co-workers investigated the role of the work function of the TCO electrode. The result was that the TCO work function does not influence the photovoltage.6 However, in these experiments, the chemical potential of the TiO2 was not changed. Thus, no conclusions on the role of the difference in the initial electrochemical potentials of the TiO2 semiconductor and the electrolyte can be drawn from these experiments. However, in accordance with the results presented here, the results of Gregg and co-workers proved that the photovoltage is not limited by a built-in potential. This was also observed in plastic solar cells made from interpenetrating networks of fullerenes and polymers.23 The photovoltage is built up from the difference of two electrochemical potentials. In a DSC, these two electrochemical potentials are the redox potential of the electrolyte and the quasi-Fermi level of the electrons in the TiO2 conduction band. As suggested by Schwarzburg and Willig,5 the dark interfacial electrical potential difference should be measured directly. Also, other material parameters are known only approximately, e.g., the back-reaction rate constant or the electron mobility and diffusion constant. Additionally, they might differ strongly between different cells. It is possible to fit measured I-V curves with results from the model, but the same fit quality can be obtained with different combinations of material parameters. Thus, a determination of material parameters by means of fitting calculations to measured results is not possible. A measured I-V curve of a 1 × 4 cm2 DSC is shown in Figure 14. Higher efficiencies have been reported for smaller cells.24 Beside for the dark interfacial electrical potential difference, particularly,

A simplified electric model of the DSC is presented. The model allows us to calculate cell characteristics such as I-V curves from material parameters. Analogies and differences to conventional p-n junction solar cells were analyzed. The distribution and the role of the electric field within a DSC could be clarified. Nonnegligible drift currents were found. It was shown quantitatively that the electric potential drops mainly at the TCO/TiO2 interface and not at a Helmholtz layer between the TiO2 and the electrolyte. The role of the dark interfacial electrical potential difference for the performance of a DSC was discussed. It was shown that contrary to a conventional p-n junction solar cell the photovoltage of a DSC is not limited by the dark interfacial electrical potential difference. Acknowledgment. This work was partially financed by the BMBF (Bundesministerium fu¨r Bildung und Forschung, Germany) and by the European Commission (Contract No. JOR3CT98-0261). Appendix The electrochemical potential of charged species (ionic charge ze0) is defined as

µ˜ i ) µi + ze0φ

(A1)

where µi is the chemical potential of the charged species and φ is the electrical (Galvani) potential of the phase. The chemical potential is defined as

µi ) µsti + kT ln

() ni

nsti

(A2)

Especially for the electrons, µ˜ e ) µe - e0φ. If two different phases are in equilibrium, the electrochemical potential of the ˜ (2) electrons is equal in both of the phases: µ˜ (1) e ) µ e . To adjust the electrochemical equilibrium, generally an electric (Galvani) potential difference is built up:

φbi :) φ(1) - φ(2) )

1 (1) (µ - µ(2) e ) e0 e

(A3)



Redox Electrolyte. The generalized redox reaction within a redox electrolyte is

νOxOx+zOx + ne0 h νRR+zR

(A4)

where νOx and νR are the stoichiometric coefficients and zOx and zR are the ionic charge numbers of the oxidized and the reduced species, respectively. n is the number of transferred electrons. In equilibrium, the electrochemical potentials have to be equal, taking into consideration the stoichiometric coefficients:

νOxµ˜ Ox + nµ˜ e ) νRµ˜ R

(A5)

Using eqs A1 and A2 for µ˜ i, it follows from eq A5

(

)

(

)

nOx nR νOx µstOx + kT ln st + nµe ) νR µstR + kT ln st nOx nR

(A6)

nstOx and nstR are the standard concentrations, and µstOx and µstR are the standard chemical potentials of the oxidized and the reduced species, respectively. The electrical potential φ cancels out, because of charge conservation, νOxzOx - n ) νRzR. Equation A6 can be solved for the electron chemical potential µe:

µe )

(

)

νRµstR - νOxµstOx kT (nR/nstR)νR + ln n n (nOx/nstOx)νOx

(A7)

Redox Electrolyte near an Electrode. The electrical potential near the electrode (e.g., TCO) may differ from its bulk value within the undisturbed redox electrolyte, φEl(xH) * φ0El, compare Figure 4. However, in equilibrium, the electrochemical potential within the electrolyte is constant, especially at the position xH of the Helmholtz layer, µ˜ El(xH) ≈ µ˜ 0El. Thus

φ0El - φEl(xH) )

1 0 (µ - µe(xH)) e0 e

(A8)

For the simulations, there is no position known a priori within the DSC with φEl ) φ0El. φ has to be referred to a fixed potential, in this case the potential of the TCO back contact back (φback TCO). Adding φTCO to eq A8, according to the definition TCO/El back :) φTCO - φ0El and inserting eq A7, one immediately φbi obtains eq 15. Nanocrystalline TiO2/Electrolyte Contact. Because of the small size of the TiO2 structures, the build-up of a large electrical potential difference is not possible at the nanocrystalline TiO2/ electrolyte contact. However, the right-hand side of eq A3 may be used for the definition of φbi:

φbi :)

1 (1) (µ - µ(2) e ) e0 e

(A9)

Because φbi is not an electrical potential build up at the nanocrystalline TiO2/electrolyte contact, it is called dark interfacial electrical potential difference. Notation D: diffusion constant (m2/s). EFn: quasi-Fermi energy (eV). ERedox: redox energy (eV). ND+: density of ionized donors (m-3). Rf: roughness factor (dimensionless). T: temperature (K). U: voltage (V).

aEl, aTiO2: geometric dimensions, cf. Figure 2 (m). b: mobility (m2/Vs). d: thickness of the inner cell (m). e0: elementary charge (As). ji: current density (A/m2), i ) e, I-, I3-, and K+. j0: exchange current density at Pt electrode (A/m2). jG: generation current density (A/m2). jR: relaxation current density (A/m2). ket: second-order rate constant (m4s-1). k: Boltzmann’s constant (eV/K). n: particle density (m-3). x, y: coordinates (m). xH: thickness of Helmholtz layer. Φ: spectral incident photon flux density (m-3s-1). 3 B : differential operator (m-1). ∆: Laplace operator (m-2). R: absorptivity (m-1). β: symmetry parameter (dimensionless). : dielectric constant (dimensionless). 0: permittivity of free space (As/Vm). λ: wavelength (m). µ: chemical potential (eV). µ˜ : electrochemical potential (eV). φ: electric potential (V). φbi: dark interfacial electrical potential difference (V). b νx: normalized vector in the x direction. b νy: normalized vector in the y direction. References and Notes (1) Ferber, J.; Stangl, R.; Luther, J. Sol. Energy Mater. 1998, 53, 29. (2) So¨dergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S. E. J. Phys. Chem. 1994, 98, 5552. (3) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1998, 102, 1745. (4) Huang, S. Y.; Schlichtho¨rl, G.; Nozik, A. J.; Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. B 1997, 101, 2576. (5) Schwarzburg, K.; Willig, F. J. Phys. Chem. B 1999, 103, 5743. (6) Pichot, F.; Gregg, B. A. J. Phys. Chem. B 2000, 104, 6. (7) Rothenberger, G.; Fitzmaurice, D.; Gra¨tzel, M. J. Phys. Chem. 1992, 96, 5983. (8) Zaban, A.; Meier, A.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 7985. (9) Jongh, P. E.; Vanmaekelbergh, D. J. Phys. Chem. B 1997, 101, 2716. (10) Ferber, J.; Luther, J. Sol. Energy Mater. 1998, 54, 265. (11) Bard, A. L.; Faulkner, L. R. Electrochemical methods; John Wiley & Sons: New York, 1980. (12) Peter, L. Personal communication, 2000. (13) Bisquert, J.; Garcia-Belmonte, G.; Fabregat-Santiago, F. J. Solid State Electrochem. 1999, 3, 337. (14) Selberherr, S. Analysis and Simulation of Semiconductor DeVices; Springer-Verlag: New York, 1984. (15) Ferber, J. PhD Thesis, Universita¨t Freiburg, 1999. (16) Stoer, J.; Burlisch, R. Introduction to Numerical Analysis; SpringerVerlag: New York, 1980. (17) The difference between the energetic position of the TiO2 conduction band edge (ECB) and the redox potential of the electrolyte should be in the order of 0.8-1 eV, depending on the electrolyte composition and especially the radius of the cations used. With a TiO2 dopant concentration of 1016 cm-3 and an effective electron mass me* ) 5.6 me,25 the position of the initial Fermi level is 0.27 eV below ECB. Thus, φbi should be on the order of 0.53-0.73 V. (18) Albery, W. J.; Bartlett, Ph. N. J. Electrochem. Soc. 1984, 131, 315. (19) Wu¨rfel, P. Physik der Solarzellen; Spektrum Akademischer Verlag: Berlin, Germany, 1995. (20) Shiga, A.; Tsujiko, A.; Ide, T.; Yae, Sh.; Nakato, Y. J. Phys. Chem. B 1998, 102, 6049. (21) Knebel, O.; Lauermann, I.; Pohl, J. P.; Uhlendorf, I. Poster presented at IPS-12 conference, Berlin 1998. (22) Sze, S. M. Physics of Semiconductor DeVices; John Wiley & Sons: New York, 1981. (23) Sariciftci, N. S. Personal communication, 2000. (24) Gra¨tzel, M.; McEvoy, A. J. Proc. 14th Eur. Solar Energy Conf.; Barcelona, 1997. (25) Fitzmaurice, D. Sol. Energy Mater. 1994, 32, 289.

Modeling of Photovoltage and Photocurrent in Dye-Sensitized

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