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J. Phys. Chem. B 2000, 104, 4292-4294
Effect of the Surface-State Distribution on Electron Transport in Dye-Sensitized TiO2 Solar Cells: Nonlinear Electron-Transport Kinetics J. van de Lagemaat* and A. J. Frank* National Renewable Energy Laboratory, Golden, Colorado 80401 ReceiVed: March 2, 2000
Analytical expressions describing electron transport in dye-sensitized nanocrystalline TiO2 solar cells are derived and verified by intensity-modulated photocurrent spectroscopy measurements. An exponential distribution of surface states can account for the observed power law dependence of the electron-transport rate on the electron population. Use of an effective electron diffusion coefficient to describe the electron transport kinetics is found to be inadequate. A slope of 69 meV is inferred for the surface-state distribution curve.
Introduction The rate of electron transport is a major determinant of the overall efficiency of dye-sensitized nanocrystalline TiO2 solar cells. Following their injection into the conduction band from optically excited dye molecules, electrons can traverse the particle network and be collected at the transparent conducting glass back contact or can react (recombine) with redox species (e.g., I3-) or dye molecules at the particle/electrolyte interface; usually the recapture of an electron by the oxidized dye is prevented by an even faster neutralization of the oxidized dye by the redox electrolyte (e.g., I-). Since the collection of electrons competes with recombination at the nanoparticle/redox electrolyte interface, slow electron transport can lead to a low charge-collection efficiency and hence a low conversion efficiency.1,2 A quantitative description of electron-transport kinetics is thus important not only for improving cell performance but also for enriching our knowledge of the fundamental electron-transport process occurring in nanocrystalline semiconductor thin film based photoelectrochemical solar cells. Electron transport in nanoporous semiconductor films is assumed to proceed by diffusion because of the absence of a significant electrical potential gradient in the film.1,3-10 The dynamics of the electron-transport process in nanocrystalline TiO2 films have been studied by intensity-modulated photocurrent spectroscopy (IMPS)1,3,4,6,7 and time-of-flight (TOF).5,11 The time constants for IMPS (τIMPS) and TOF (τTOF) are found to depend on light intensity.1,3,4,6,7,11 With increasing illumination intensity, τIMPS and τTOF decrease. This decrease has been ascribed to trap filling.3,4,6,11 Electron transport has been modeled in terms of an effective diffusion coefficient Deff,5,7 and τIMPS and τTOF have been correlated with Deff.1,3,7,11 Measured values of Deff range from 10-8 to 10-4 cm2/s, depending on light intensity. These values are lower than the diffusion coefficient calculated3,7,11 from the electron mobility measured for singlecrystal TiO2. The low value of Deff has been attributed to electrons spending a large fraction of their transit time in traps.7 Because of the large surface area of the nanoporous films, these traps are assumed to be located primarily at the particle surface. Evidence for an exponential distribution of surface states has been obtained from intensity-modulated photovoltage spectroscopy12 and TOF measurements.10,11 The IMPS (or TOF) time constant and the short-circuit photocurrent Jsc are important parameters for understanding the
charge-transport kinetics in dye-sensitized solar cells. The time constant τIMPS represents the average time for the collection of injected electrons, and Jsc is a measure of the electron-transport rate. The product of τIMPS and Jsc is the number of electrons Q in the film in the absence of recombination. Although the underlying cause has not been established, a power law dependence of Jsc on Q (Jsc ∝ Qm), where m ≈ 2.64,1 is observed. Similarly, it can be shown that τIMPS ∝ J(1/m)-1 ; a sc double logarithmic plot of τIMPS vs Jsc yields a slope of about -0.62. Similar values of the slope of this plot have been observed by others.1,3,4,6,7,13 The light intensity I (I ∝ Jsc) dependence of Q and τIMPS implies that the electron-transport rate in nanocrystalline films depends nonlinearly on the steadystate population of electrons in the film.1 To model the electronconcentration profile of the cell using the continuity equation, it has been necessary to include this nonlinear (power law) kinetic behavior.1 However, the origin of this nonlinear kinetic behavior is not well understood. In this paper, we derive analytical expressions that suggest a cause for the nonlinear electron transport kinetics in dyesensitized solar cells and apply them to IMPS measurements. Results and Discussion The continuity equation for conduction-band electrons is given by
dncb(x) 1 dJn(x) ) RηinjI(x) + - Ut(x) + Ue(x) dt q dx
(1)
where at x, ncb is the density of electrons in the conduction band, R is the effective absorption coefficient, ηinj is the chargeinjection efficiency, I(x) is the photon flux or light intensity, q is the unit of charge, and Jn is the electron current density. The first term on the right in eq 1 (Rηinj I(x)) is the rate at which electrons are injected into TiO2 by the excited dye at x, the second term is the transport component, corresponding to the rate at which electrons, via the transport process, are supplied to x, Ut(x) is the rate at which electrons are captured by traps, and Ue(x) is the rate at which electrons are thermally emitted back to the conduction band from traps. Recombination of electrons from surface states to the redox electrolyte is implicitly contained in eq 1 as the difference between the terms for electron
10.1021/jp000836o CCC: $19.00 © 2000 American Chemical Society Published on Web 04/18/2000
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J. Phys. Chem. B, Vol. 104, No. 18, 2000 4293
trapping and emission (detrapping) (Ut - Ue). The electron current density Jn is given by the relation14
Jn ) qµnncb
( ) 1 dEF q dx
(2)
where µn is the mobility of free electrons and EF is the Fermi level; µn is limited only by electron scattering at grain boundaries and impurity sites. When electric field effects due to charging of the particles are neglected,10,12,14 eq 2 simplifies to
dncb dx
Jn ) Dnq
(3)
where Dn is the diffusion constant of free (untrapped) electrons and is calculated from the Einstein relation15 Dn ) (kBT/q) µn, where kB is the Boltzmann constant and T is the temperature. Since µn for nanocrystalline anatase is about 40 cm2 V-1 s-1,16,17 Dn is estimated to be about 1 cm2 s-1. At short-circuit, in the absence of recombination (Ut(x) ) Ue(x)), the steady-state freeelectron concentration ncb at x can be obtained from eqs 1 and 3
ncb(x) ) ncb(0) +
ηinjI(0) [1 - Rxe-Rd - e-Rx] RDn
(4)
where the boundary condition of no electron current entering or leaving the TiO2 film at the outermost TiO2/redox electrolyte interface (Jn(d) ) 0) is used; only a negligible fraction of the TiO2 film makes electrical contact with the counter electrode. With knowledge of ncb, the difference between the Fermi level in the dark EF0 and in the light EF is calculated from the expression
EF - EF0 ) kT ln
ncb(x)
(5)
ncb(0)
Intensity-modulated photovoltage spectroscopy12 and TOF studies10,11 indicate that the energy distribution of surface states Nss(E) is described by a single exponential
NSS(E) ) NSS0 e(E-EF0)/mc
(6)
where Nss0 is the surface-state density at EF0, and mc corresponds to the slope of the surface-state distribution curve. Because recombination is much slower than trapping and detrapping,12 the quasi-Fermi level for electrons in traps is the same as EF. The electron density in traps nt(x) can thus be obtained by integrating eq 6 from EF0 to EF.
nt(x) ) mcNSS0[e(EF(x)-EF0)/mc - 1] ) mcNSS0
[( ) ncb(x)
ncb(0)
kT/mc
]
- 1 (7)
Figure 1 shows the concentration profiles of free electrons (eq 4) and n(x) (i.e., n(x) ) ncb(x) + nt(x) ≈ nt(x)) (eqs 4 and 7) across a 10-µm thick film. It can be seen that n(x) and ncb(x) depend differently on x. The plot for n(x) displays more curvature than does that of ncb(x). The continuity equation (eq 1) can also be expressed in terms of dn(x)/dt instead of dncb(x)/dt.1,3 With this approach, the electron current density (eq 3) must be expressed as a function of n(x): Jn(x) ) qDn(dncb/dn)(dn/dx).1 IMPS studies1 reveal that dncb/dn ∝ nmx (see eq 8), where mx ≈ 1.6. When this power law relation is used to calculate n(x) from the continuity equation by a numerical method, the resulting plot has the same shape as n(x) in Figure 1.1 On the other hand, the approximation5,7 that Dn(dncb/dn) ≈ Deff yields a plot of n(x) vs x that mirrors the shape of the ncb profile in Figure 1. The use of a single effective diffusion coefficient thus underestimates the curvature of the n(x) vs x plot. Since n(x) ≈ nt(x), the total number of electrons in the film Q ≈ Qt (where Qt denotes the number of trapped electrons) can be calculated by integrating eq 7 over the film thickness. By inserting eq 4 into eq 7 and assuming that ncb(x) is generally much larger than ncb(0) at short circuit, one obtains
Q ) mcNSS0ncb(0)mc/kT
( ) ηinjI(0) RDn
kT/mc
∫0d[1 - Rx e-Rd - e-Rx]kT/m dx c
(8)
Validation of the power law dependence of ncb on n as expressed by eq 8 has been obtained by IMPS measurements.1
(9)
From eq 9 and the relation Jsc ∝ I(0), which can be inferred from eqs 3 and 4, it follows that Q ∝ I(0)kT/mc or
Jsc ∝ Qmc/kT
In practice, ncb(x)/ncb(0) . 1, which implies that ncb(x) is proportional to nt(x) by the relation ncb(x) ∝ nt(x)mc/kT. Because the density of trapped electrons is much larger than that of free electrons,1,12 the electron concentration n(x) ) ncb(x) + nt(x) is essentially equal to nt(x) (eq 7). Thus ncb(x) is related to n(x) by the expression
ncb(x) ∝ n(x)mc/kT
Figure 1. Concentration profiles of free electrons ncb (eq 4) and n(x) (i.e., n(x) ) ncb(x) + nt(x); eqs 4 and 7) across a 10-µm thick film. I(0) ) 1016 cm-2 s-1, R ) 100 cm-1, Dn ) 1 cm2 s-1, ncb(0) ) 107 cm-3, T ) 300 K, NSS0 ) 1016 cm-3 eV-1, and mc ) 70 meV.
(10)
The derivation of eq 10 implies that an exponential dependence of trap (surface) states on energy gives rise to the observed power-law dependence of the short-circuit current on the number of electrons in the film. Figure 2 shows a double logarithmic plot of Jsc vs Q ) JscτIMPS (see Introduction) for a dye-sensitized nanocrystalline TiO2 solar cell. (Note: τIMPS is related to the short-circuit current density according to the relation τIMPS ∝ c)-1 J(kT/m .) From the slope of this plot, it can be shown that sc mc/kT ) 2.64 (eq 10). Thus the slope of the surface-state distribution curve mc ) 69 meV, which is in good agreement
4294 J. Phys. Chem. B, Vol. 104, No. 18, 2000
Letters surface-state distribution curve is found to have a slope of 69 meV. Acknowledgment. This work was supported by the Office of Science, Division of Chemical Sciences and the Office of Utility Technologies, Division of Photovoltaics, U.S. Department of Energy, under contract DE-AC36-99GO10337. References and Notes
Figure 2. Relation of the short-circuit photocurrent Jsc to the product of Jsc and the IMPS time constant τIMPS measured at short-circuit. The solar cell consisted of a [RuL2(NCS)2] (L ) 2,2′-bipyridyl-4,4′dicarboxylic acid)-coated, 8-µm-thick nanocrystalline TiO2 electrode in CH3CN containing I2 (50 mM) and LiI (0.6 M).
with values based on time-of-flight data10,11 and electrical impedance spectroscopy2 in the light intensity range (
