© Copyright 1999 by the American Chemical Society
VOLUME 103, NUMBER 5, FEBRUARY 4, 1999
LETTERS Driving Force for Electron Transport in Porous Nanostructured Photoelectrodes D. Vanmaekelbergh* and P. E. de Jongh Debye Institute, UniVersity of Utrecht, P.O. Box 80000, 3508 TA Utrecht, The Netherlands ReceiVed: October 16, 1998
Electron transport in a photoelectrode consisting of a porous nanostructured semiconducting or insulating network interpenetrated with an electrolyte solution is considered. Electrons, photogenerated in the solid network by light incident from the electrolyte side, travel through the system and are removed at the substrate/ network boundary. We calculate the driving force for electron diffusion through the network from first principles. It is found that the driving force is in the order of kT/e divided by the thickness of the network and independent of the light intensity. Photoinduced interfacial charging of the network due to electron trapping can enhance the driving force but does not change the transport characteristics.
Porous networks consisting of an insulating or semiconducting solid form a promising basis for several electrical and optoelectrical devices. Such a network is usually prepared by deposition of nanometer-sized particles on a conducting substrate. Electrical contact between the particles and between the particles and the substrate is achieved by thermal treatment. Efficient electrochemical solar cells have been reported; the best known example is the Gra¨tzel-type solar cell based on a porous matrix of electrically connected TiO2 particles sensitized with a dye.1 In this system, a photon is absorbed by a dye molecule, which subsequently injects an electron into the conduction band of the TiO2 matrix. The dye is regenerated by a reduced species in the electrolyte. Alternatively, porous nanostructured systems in which electrons and holes are generated in the solid matrix have also shown high photocurrent quantum yields.2 It is clear that there are fundamental differences in the physics of electron-hole separation and photocurrent flow for porous nanostructured electrodes and bulk single-crystal electrodes which have a macroscopically flat surface.3,4 In bulk single crystals, there is bipolar (electron-hole) diffusion in the bulk phase; eventually electrons and holes are separated by the electrical field in the interfacial depletion layer. In porous matrices, electron-hole separation is due to electrochemical removal of the photogenerated empty state by a reduced species
interpenetrated in the matrix. Recombination occurs predominantly by interfacial charge transfer.5 Under short-circuit conditions and steady-state illumination, there is a net flux of photogenerated electrons through the network which is sustained by the constant photogeneration and the removal of electrons at the interface between the substrate and the porous network. In addition, there is a flow of photooxidized ions in the pores of the network in the direction of the counter electrode. Under steady-state conditions, the number of oxidized species arriving at the counter electrode per time unit is equal to the number of photogenerated electrons arriving at the substrate/network boundary. In contrast with devices based on bulk crystals, there is diffusion of one type of charge carrier (electrons) over macroscopic distances (1-100 µm) in the solid network. There is a great deal of interest in the dispersive electron-transport characteristics in this type of disordered system, in which electron-scattering phenomena and trapping-detrapping phenomena play an important role.3,4 Electron flow in a porous nanostructured network interpreted with an ionic conductor must be due to diffusion since there is, a priori, no macroscopic electrical field. However, the negative charge due to electrons trapped in interfacial band-gap states may shift the band edges to higher energy and induce an electrical field component in
10.1021/jp9840883 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/20/1999
748 J. Phys. Chem. B, Vol. 103, No. 5, 1999
Letters
[
Jn(x,t) ) µnn(x,t) -
]
1 ∂EF,n(x,t) e ∂x
(2)
where µn is the mobility of free electrons in the nanoporous system and (-1/e)(∂EF,n(x,t)/∂x) is the driving force. The electron mobility accounts for scattering of free electrons by the lattice, the solid surface, and grain boundaries. A general expression for the driving force follows from eq 1:
-
Figure 1. Sketch of a photoelectrode consisting of a nanostructured network contacting a conducting substrate at x ) d. Light is incident from the electrolyte side, x ) 0. Photogenerated holes are rapidly removed by electron donation from a reduced species (see insert). The electrons are left in the matrix and diffuse toward the substrate, which acts as a sink. Due to constant photogeneration and removal at the substrate boundary, the electron Fermi level decreases in the direction of the substrate.
the driving force. To our knowledge, the effect of a light-induced electrical field was not taken into account in the existing models. In this letter, the driving force (the gradient of the electron Fermi level) leading to the net flow of photogenerated electrons through a porous nanostructured semiconducting electrode will be calculated for steady-state conditions. Diffusion will be considered first. Then, it will be shown that the effect of nonuniform photoinduced charging of the semiconducting network due to electron trapping can be taken into account, by using a generalized boundary condition. Finally, the implications of our results for the electron-transport characteristics and for the experimental methods using a modulation of the incident light intensity will be discussed. We consider a porous nanostructured semiconductor electrode illuminated from the electrolyte side with monochromatic light of intensity Φ(0) at x ) 0 (see Figure 1). The generation rate of electrons and holes is denoted as RΦ(x,t), R being the effective absorption coefficient taking into account light scattering in the porous matrix. We assume that valence-band holes, photogenerated in a structural unit of the semiconductor matrix, are scavenged effectively by a reduced species present in the interpenetrating solution. In the case of a dye-sensitized electrode, the dye is regenerated by electron donation from the reduced species. The photooxidized ions diffuse through the pores of the solid network and are reduced at the counter electrode. Photogenerated electrons travel through the network over macroscopic distances; eventually, they can be temporarily localized in bandgap states or lost by recombination.5 The concentration of free electrons (in the conduction band) of the solid matrix at a given distance (d - x) from the substrate will be denoted as n(x,t). The Fermi level of electrons in the nanoporous network can be written as ref EF,n(x,t) - Eref F,n ) -e[φ(x,t) - φ ] + kT ln
n(x,t) nref
(1)
where Eref F,n stands for a reference Fermi level, characterized by a concentration nref of free electrons having a potential energy -eφref and e and k are the positive elementary charge and the Boltzmann constant, respectively. The net particle flux of electrons traveling through the network in the direction of the substrate is denoted as Jn(x,t) (cm-2 s-1) and given by
1 ∂EF,n(x,t) ∂φ(x,t) kT 1 ∂n(x,t) ) e ∂x ∂x e n(x,t) ∂x
(3)
Under steady-state conditions, the driving force consists of an electric field component, dφ(x)/dx and a component [(-kT/e)(1/n(x))(dn(x)/dx)] due to the spatially nonuniform concentration of free electrons in the matrix which arises from constant photogeneration in the network and electron removal at the matrix/substrate boundary. It is useful to refer to conditions of spatial equilibrium in the electron system, i.e. dEF,n(x)/dx ≡ 0. This is, for instance, the case in the depletion layer of a semiconductor in thermodynamic equilibrium with an adjacent phase; the electrical field component dφ(x)/dx is then equal to the chemical component [(-kT/e)(1/n(x))(dn(x)/dx)] but opposite in sign. We will show that for a nanoporous photoelectrode under short-circuit conditions, the electrical field component and the “chemical” component are also related but operate in the same direction. The rate equation for the (excess) concentration of free electrons is given by:
∂Jn(x,t) ∂n(x,t) ) RΦ(x,t) ∂t ∂x
Ec
∂f(E,x,t) dE - R(x,t) ∂t (4)
∫X(E) Ev
Here, RΦ(x,t) is the photogeneration rate due to the photon flux Φ(x,t) and Jn(x,t) is the net electron particle flux in the direction of the substrate (increasing x). The third term on the right-hand side accounts for temporary localization of electrons in (interfacial) band-gap states, with a volume density X(E) (cm-3 eV-1) and an electron occupation factor f(E,x,t). Here, scattering phenomena are accounted for by the mobility, while trappingdetrapping is accounted for separately by the third term in eq 4. The term R(x,t) stands for the recombination rate of photogenerated electrons. Several loss mechanisms have been discussed in the literature.3,5,6 For the sake of simplicity, it will be assumed here that R(x,t) can be neglected with respect to RΦ(x,t) - (∂Jn(x,t)/∂x); this means that the photogenerated holes are scavenged rapidly by a reduced species and that photogenerated electrons are not transferred to the oxidized species in the electrolyte. Under short-circuit conditions, real systems can, indeed, show a photocurrent quantum yield close to unity.1-3 We consider steady-state conditions. It is assumed that the boundary x ) 0 acts as an ideal reflector for electrons.7 In models which only account for diffusion, the boundary condition is
|
dn(x) )0 dx x)0 Since we consider both the electric field and the “chemical” component of the driving force, we will use
|
dEF,n(x) ≡0 dx x)0
(5)
Letters
J. Phys. Chem. B, Vol. 103, No. 5, 1999 749
as a general boundary condition. The net electron flux in the direction of increasing x under steady-state conditions is calculated from eq 4 and the boundary condition
Jn(x) ) Φ(0)[1 - e-Rx]
(6)
The steady-state photocurrent density is determined by the flux of electrons arriving at the substrate (x ) d) and is hence given by eJn(d) ) eΦ(0)[1 - e-Rd]. First, electron transport due only to diffusion will be discussed; in other words, we assume that the electrical field term dφ/dx in the driving force (eq 3) is zero. From eqs 2 and 3 it follows that
Jn(x) ) -
dn(x) kT dn(x) µ ) -Dn e n dx dx
(7)
Hence, Fick’s first law is found from the more general equations if the electric field component in the driving force is neglected. The excess concentration of free electrons at a distance d-x from the substrate with respect to the concentration at the boundary between the porous matrix and the substrate (x ) d) is readily calculated from eqs 6 and 7
n(x) - n(d) )
Φ(0) [R(d - x) + e-Rd - e-Rx] RDn
(8)
The concentration of free electrons at the boundary between the porous matrix and the substrate (electron sink), n(d), cannot be calculated from our model. This would require additional assumptions concerning the electron exchange between the substrate and the semiconducting matrix.6 We will consider n(d) as an external parameter. The driving force for electron flow by diffusion follows from eq 3 (with dφ/dx ) 0) and eq 8
-
1 dEF,n(x) ) e dx [Φ(0)/Dn](1 - e-Rx) (kT/e) (9) n(d) + [Φ(0)/RDn][R(d - x) + e-Rd - e-Rx]
The driving force is independent of the incident light intensity Φ(0) and the diffusion constant Dn if n(d) can be neglected with respect to n(x)- n(d) equal to Φ(0)/(RDn)[R(d - x) + e-Rd - e-Rx] in eq 9. Then, the net velocity of the electrons in the direction of the substrate can indeed be regarded as the product of two independent physical quantities, i.e., the drift mobility µn and the driving force [(-1/e)(dEF,n(x)/dx)] equal to [(kT/e)(1 - e-Rx)]/[(d - x) + (1/R)(e-Rd - e-Rx)]. It is clear that the driving force for electron diffusion in the direction of the substrate depends on the absorption depth of the light with respect to the thickness of the porous network and on the position x in the network. For relatively strong light absorption (Rd > 1), the driving force is a fraction x/(1/R) of (kT/e)/d nearby the electrolyte contact, while it becomes ((kT/e)/d)(d/(d - x)) in the region deeper than the absorption depth 1/R. The remark that the condition n(x) - n(d) > n(d) must hold; this means that from our model we cannot predict the driving force near the boundary between the porous matrix and the substrate. In Figure 2a, a steady-state concentration profile of free electrons in the solid network is shown for Rd ) 1 and 10. In Figure 2b, the driving force, normalized with respect to (kT/e)/d, is presented; the normalization is used because it is often assumed that the aVerage driVing force in the system is (kT/e)/d.3 We
Figure 2. (a) Concentration profile of free electrons in the solid matrix, n(x) - n(d), for two absorption depths (Rd ) 1 and Rd ) 10) and two values of the incident light flux Φ(0) ) 1015 and 1016 cm-2 s-1. The thickness of the porous network is 1 µm, and the diffusion constant of the electrons is 10-5 cm2 s-1. (b) Profile of the driving force (-1/e)(dEF,n(x)/dx) (eq 9) for net electron diffusion in the solid network (thickness d) in units of (kT/e)/d. The profiles are shown for two different absorption depths and two light intensities (see Figure 2a).
see that in a major part of the system, the driving force is indeed in the order of (kT/e)/d. We next consider whether charging of the matrix due to trapping of photogenerated electrons in interfacial band-gap states can influence the driving force. Since we have assumed that photogenerated holes are scavenged rapidly by a reduced species in the solution, the interfacial states are predominantly occupied by electrons. As a consequence, the surface charge density, qs, is negative and compensated by the charge of positive ions in the (outer) Helmholtz plane. This leads to an upward shift of the band edges equal to -eqs(x)/CH, CH being the capacitance of the Helmholtz double layer per unit of physical surface area. The spatial derivative of the potential energy is
-e
dφ(x) e dqs(x) )dx CH dx
(10)
We will assume, for reasons of simplicity, that the density of interfacial states per unit of energy, s(E), does not depend strongly on the energy in the band gap. The states are occupied with electrons from the valence-band edge to the electron Fermi level; the surface charge density in the solid network can hence be written as -es[EF,n(x) - EV(x)]. Since EF,n(x) ) Ec(x) + kT ln[n(x)/Nc], it follows from eq 10 that
dφ(x) 1 dqs(x) e2s kT 1 dn(x) ) )dx CH dx CH e n(x) dx
(11)
From eqs 3 and 11, it can be seen that the electrical contribution to the driving force is proportional to (-1/n(x))(dn(x)/dx) in a
750 J. Phys. Chem. B, Vol. 103, No. 5, 1999 similar way as the “chemical” contribution. Moreover, both contributions have the same sign and hence operate in the same direction. From eqs 2, 3, and 11, n(x) can be calculated in the same way as above, and subsequently the driving force can be expressed as a function of the parameters R, x, d. The generalized driving force is given by an expression similar to that described by eq 9, however, multiplied by [1 + (e2s)/CH]. The magnitude of the term (e2s)/CH with respect to 1 describes the extent to which surface charging may increase the driving force for electron flow. For instance, with reasonable values of 10-5 F/cm2 for CH and 1013/eV cm2 for s, a value of 0.16 is obtained for (e2s)/CH. It can be concluded that self-induced electric field effects due to charging of the internal surface of the semiconducting network by trapped electrons do not alter the electron-transport characteristics in a fundamental way. An increase of the driving force and for the directed flow of photogenerated electrons through the semiconductor matrix might be expected. Electron loss due to recombination may partly reduce this effect. We found that although electron transport in a porous electrode is characterized by various dispersive phenomena, the driving force for the net electron flow can be calculated in a relative simple way. The driving force depends on the absorption depth of the light with respect to the thickness of the network and is not uniform, i.e., depends on the distance from the macroscopic boundary with the electrolyte. This causes a nontrivial dispersion in the transit times of photogenerated electrons through the solid network, which will be considered in more detail elsewhere. For reasonable conditions, the driving force in a large region of the nanoporous matrix is in the order of (kT/e)/d, independent of the incident light intensity. This has important implications for the interpretation of the measurements in which electron transport is probed by a modulation of the light intensity; we showed that this modulation does not lead to a change in the driving force for the electron flow. Hence, the modulation method consists essentially of the generation
Letters of a small excess of free electrons which travel through the system under the steady-state constraints imposed by the background light intensity. This assumption was made by several research groups, either explicitly or implicitly.3,4,6 With nanoporous particulate TiO2 photoelectrodes, it was found that the transit time for photogenerated electrons through the TiO2 network decreases with increasing background light intensity. This effect was accounted for by assuming that electrons, traveling through the network, can be temporarily localized in band-gap states of energy close to the Fermi level.3,4,6 This explanation is given a more solid foundation here, since we found that the driving force for electron flow toward the substrate does not depend on the background light intensity in a large region of the porous film. References and Notes (1) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (2) Hodes, G.; Howell, I. D.; Peter, L. M. J. Electrochem. Soc. 1992, 139, 3136. Erne´, B. H.; Vanmaekelbergh, D.; Kelly, J. J. AdV. Mater. 1995, 7, 739. (3) So¨dergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S.-E. J. Phys. Chem. 1998, 98, 5552. Peter, L. M.; Vanmaekelbergh, D. Time and frequency resolved studies of photoelectrochemical kinetics. In AdVances in Electrochemical Science and Engineering; Alkire, R., Kolb, D. M., Eds.; 1998; Vol. 6. (4) Schwarzburg, K.; Willig, F. Appl. Phys. Lett. 1991, 58, 2520. Hoyer, P.; Weller, H. J. Phys. Chem. 1995, 99, 14096. Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. Solbrand, A.; Lindstro¨m, H.; Rensmo, H.; Hagfeldt, A.; Lindquist, St.-E.; Sodergren, S. J. Phys. Chem. B 1997, 101, 2514. Vanmaekelbergh, D.; Iranzo Marı´n, F.; van de Lagemaat, J. Ber. Bunsenges. Phys. Chem. 1996, 100, 616. de Jongh, P. E.; Vanmaekelbergh, D. Phys. ReV. Lett. 1996, 77, 3427. de Jongh, P. E.; Vanmaekelbergh, D. J. Phys. Chem. B 1997, 101, 2716. (5) Huang, S. Y.; Schlichtho¨rl, G.; Nozik, A. J.; Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. B 1997, 101, 2576. Schlichtho¨rl, G.; Huang, S. Y.; Sprague, J.; Frank, A. J. J. Phys. Chem. B 1997, 101, 8141. (6) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, J. J. Phys. Chem. B 1997, 101, 10281. (7) Crank, J. The mathematics of diffusion, 2nd ed.; Oxford University Press, 1975.
