ARTICLE pubs.acs.org/JPCC
Theoretical Modeling of Measured Photocurrent Dynamics in Dye-Sensitized Nanostructured TiO2 Solar Cells Liangmin Zhang*,† and Robert Engelken‡ †
Arkansas Center for Laser Applications and Science, Department of Chemistry and Physics, ‡Optoelectronic Materials Laboratory, Department of Electrical Engineering, Arkansas State University, State University, Arkansas 72467, United States
Deliang Cui National Laboratory of Crystal Materials, Shandong University, Jinan, China 250100 ABSTRACT: The general solutions describing the buildup and decay processes of photocurrent in dye-sensitized nanocrystalline TiO2 thin film solar cells have been derived by solving the diffusion equations that include charge transport and recombination parameters of electron lifetime and diffusion coefficient. One analytical solution is obtained under the condition of steady light illumination that may be used to interpret the buildup process of photocurrent. The other solution is derived with the condition that the illumination light is turned off after the photocurrent reaches its steady value. The latter solution can be used to describe the recombination process of charge carriers in the thin films. The two solutions have also been used to simulate experimentally measured results to extract electron lifetime, diffusion coefficient, and diffusion length of electrons for the two conditions under different light intensities. The fitted results show that the lifetime and diffusion coefficient vary with light intensity in opposite senses and the diffusion length only slightly depends on light intensity. In the recombination process, the dark diffusion coefficient is independent of excitation light intensity. These theoretical results confirm the observations reported by other researchers.
’ INTRODUCTION Porous, nanocrystalline semiconducting thin films of wide band gap oxides such as TiO2, ZnO, and SnO2 are an important new class of electronic materials.14 They exhibit extraordinary optical and electronic properties due to a large surface-area-tovolume ratio. When these porous nanocrystalline thin films contact with electrolyte and are sensitized by dye, nanoelectronic junctions with enormous area can be formed inside the film, which are useful in applications such as photovoltaics, photocatalyisis, electroluminescent, and bioanalytical devices.5 Dye-sensitized, mesoporous TiO2 thin film solar cells (DSSCs) are a typical example of these cells. In DSSCs, a layer of transparent mesoporous crystalline TiO2 is used as the electron acceptor, one of ruthenium dyes serves as the sensitizer, iodide/iodine redox liquid is utilized as the electrolyte. These photovoltaic devices rely on injection of electrons from a photoexcited ruthenium dye into the conduction band of TiO2 followed by collection of the contact electrodes. The solar conversion efficiencies of 1015% under standard air mass (AM) 1.5 illumination with good durability69 have been obtained, which makes DSSCs the most ideal alternative to the conventional silicon-based PN junction photovoltaic (PV) devices. Since the particle size in the nanocrystalline phase is small and the electrolyte phase penetrates throughout the porous solid phase, the contribution of internal electric field to carrier collection in the TiO2 phase is expected to be negligible. Electron r 2011 American Chemical Society
transport in the penetrated network of nanoparticles is dominated by a gradient in the chemical potential of the electrons.10,11 Clearly, characterizations of electron transport, trapping, and recombination in these types of nanocrystalline cells have become a major challenge in the investigation of these devices. A few theoretical models and several techniques have been used to extract the important parameters such as electron diffusion coefficient, diffusion length, and lifetime.1016 However, most theoretical and experimental investigations were based on the responses to intensity-modulated light illumination.10,11,1517 Various observations indicate that the photocurrent kinetics dominated by electron transport in the nanocrystalline devices exhibits a slow and multiphasic time constant dependence.4,10,12,18,19 which has not been explained well. In this work, we will first solve the diffusion equation directly under the condition of steady light illumination to investigate the buildup processes of photocurrent, which should be closer to real applications comparing with the results obtained with the intensity-modulated illumination. After the photocurrent reaches its steady value, we assume the light is turned off to derive the solution of the diffusion equation for the second time. The solution should describe the decay process of the photocurrent dominated by the Received: September 28, 2011 Revised: December 12, 2011 Published: December 13, 2011 1293
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x direction. Electron density n varies only in the x direction and electron transport also in the same direction. The current density contains only the diffusion term11,12 J ¼ qD
∂n ∂x
ð1Þ
where J is the current density, q is the electronic charge, and D is the electron diffusion coefficient. Using the electron continuity equation, electron transport in the semiconductor film can be described by11,12 ∂n ∂2 n ¼ D 2 þ GR ∂t ∂x Figure 1. Typical optical absorption curve of (RubiPy2Cl2)Cl2 dyestained nanocrystalline TiO2 thin films.
charge recombination in the solar cell. In order to verify these solutions, we have experimentally measured the buildup and decay processes under different intensities and fit these processes to the solutions. The solutions can be used to explain the multiphasic establishing and decay features of the photocurrent.
’ EXPERIMENTAL SECTION The TiO2 thin films were prepared by the following procedures. A total of 7 g of TiO2 power (Degussa P25 with an average diameter of 21 nm), purchased from Degussa Co., were added to 20 mL of diluted nitric acid (pH 4). The solution was then spincoated to conducting indium tin oxide (ITO)-coated glass substrates with spin-coating speeds ranging from 1000 to 1500 rpm to create thin films with a thickness range of 1020 μm. The films were first allowed to airdry and were then sintered in a furnace at 450 °C after which they were slowly cooled to room temperature. The iodine redox mediator electrolyte solution was formed by combining 0.5 M potassium iodide and 0.05 M iodine in water-free ethylene glycol. A ruthenium compound, cis-dichloridebis (2,20 -bypyridy14,40 -dicarboxylic acid) ruthenium(II) complex, abbreviated as (RubiPy2Cl2)Cl2 (purchased from Sigma-Aldrich), was used to sensitize the TiO2 thin films. The ruthenium(II) dye was dissolved in DI water (2 mg/mL) to make the dye solution. The TiO2-coated glass slides were soaked in the solution for about 30 min. The resulted dye-stained films were washed first with DI water, followed by isopropanol. The measured optical absorbance is shown in Figure 1. One can see the absorption peak is located at 520 nm. Then, in order to complete each cell, one drop of iodide/iodine electrolyte solution was placed on the thin film surface and a platinum-coated tin oxide glass counter electrode was press together with the dye-coated thin film. The edges of the cells were sealed with epoxy to avoid moisture intrusion. ’ THEORY Since the nanocrystals are usually small (525 nm in diameter), the electric field may be established spontaneously within the nanocrystals when they come into contact with the electrolyte.12,20 According to the findings reported in refs 12 and 20, however, the electric field is extremely weak and the electric potential is not expected to vary by more than several meV across a nanocrystal. Therefore, in the bulk, potential variations are negligible, in other words, macroscopic electric fields are negligible across the semiconductor phase. Electron drift can be neglected and transport can be modeled with diffusion process alone. We assume that light illuminates the sample in the
ð2Þ
where G and R represent the volume generation and recombination rates, respectively. In the dye-sensitized nanocrystalline thin film, the generation rate can be written as G = αI0 exp(αx),1012 where I0 is the incident photon flux and α is the absorption coefficient of the dye-sensitized film. Recombination is assumed to be proportional to the electron number density and can be written as R = (n n0)/τ, where n0 is the dark electron density and τ is called the electron lifetime determined by back-reaction with I3 in the electrolyte. Equation 2 can now be rewritten1012,21 ∂n ∂2 n n n0 ¼ D 2 þ αI0 expð αxÞ ∂t ∂x τ
ð3Þ
Equation 3 has been widely used to describe electron diffusion processes in semiconductors and dye-sensitized nanocrystalline thin films.1015,21 However, it has been solved analytically only with the condition of intensity-modulated light illumination at a fixed frequency.10,11,15,16 When the steady state of photocurrent has reached, ∂n/∂t = 0. This creates D
∂2 n n n0 þ αI0 expð αxÞ ¼ 0 ∂x2 τ
ð4Þ
Equation 4 has been solved to predict the measured steady-state current density by S€odergren et al.,22 and the solution is Lα cosh J0 ¼ I 0 q
d d þ sinh þ Lα expð dαÞ Lα L L d 2 2 ð1 L α Þ cosh L
ð5Þ where d is the thickness of the film and L is called the diffusion length, L = (Dτ)1/2. One can see eq 5 predicts the steady-state photocurrent is proportional to the incident photon flux I0. After the steady-state photocurrent is established, if the illumination light is turned off, the photocurrent will undergo a decay process due to the recombination of free electrons with holes. In order to understand this recombination process of the charge carriers, we have to let I0 = 0 in eq 3 and seek its solution. When I0 = 0, eq 3 becomes ∂n ∂2 n n n0 ¼ Dd 2 ∂t ∂x τd
ð6Þ
where Dd and τd are now defined as the dark diffusion coefficient and dark lifetime of electrons, respectively. This partial differential equation can be solved by separating variables x and t and using Fourier transformation.23 We have solved this equation with the initial condition of Jt=0 = J0, the solution should describe the decay process of the photocurrent. The obtained 1294
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The Journal of Physical Chemistry C time-dependent photocurrent J(t) ∞ 2qI0 αLd 2 t exp Ck ð2k JðtÞ ¼ τd k ¼ 0 dð1 Ld 2 α2 Þ " # π2 ð2k þ 1Þ2 Dd þ 1Þ exp t 4d2
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∑
ð7Þ
where k is an integer, Ck is constant, and Ld = (Ddτd)1/2 is the dark diffusion length. Using boundary conditions, initial conditions, and Fourier transform,11,15,22,23 Ck can be determined by π ð 1Þk þ 1 α expð αdÞ þ ð2k þ 1Þ 2d Ck ¼ 2 2 π ð2k þ 1Þ L2 þ 2 4d þ
πð2k þ 1Þ 2d 2 2 π ð2k þ 1Þ α2 þ 4d2
ð 1Þk α expð αdÞ
Figure 2. Short-circuit current density versus illumination power. The green circles show measured values, and the solid-line displays the theoretically fitted result using eq 5.
ð8Þ
one can see that eq 7 predicts multiphasic time constant response of photocurrent. For the establishing process, we have to consider both ∂n/∂t ¼ 6 0 6 0 to find the general solution of eq 3. This general can and I0 ¼ be derived by separation of variables x and t and using Fourier transformation11,23 ∞ 2qI0 αL2 Cm ð2m dð1 L2 α2 Þ m ¼ 0 ( " #) 1 π2 ð2m þ 1Þ2 D þ þ 1Þ exp t τ 4d2
JðtÞ ¼ J0
∑
ð9Þ
J0 is determined by eq 5 and Cm is constant and can be calculated by11,23 π ð 1Þm þ 1 α expð αdÞ þ ð2m þ 1Þ 2d Cm ¼ 2 2 π ð2m þ 1Þ L2 þ 4d2 πð2m þ 1Þ ð 1Þm þ 1 α expð αdÞ 2d ð10Þ 2 2 π ð2m þ 1Þ α2 þ 4d2 m is another integer. The similar solution has also been reported in the index of ref 11. We will use eq 9 to fit the establishing processes of the photocurrent in the following section.
’ MEASUREMENT RESULTS AND DISCUSSION In this section, we will prove the solutions [eqs 7 and 9] of the diffusion eqs 3 and 6 that can be used to describe the decay and buildup processes, respectively. The experimental measurements were conducted using a CW laser beam operating at the wavelength of 532 nm from a Coherent Verdi 5 laser system. The laser beam has a diameter of about 5 mm before the sample. The intensity of the laser beam can be controlled either by the knob of the laser power supply or using a set of neutral density filters. A photodiode was used to monitor the intensity. The illumination period was controlled by a digital shutter (Newport 845-HP)
Figure 3. Measured dynamics of photocurrent density with different illumination intensities. The illumination duration is 10 s. The laser beam of 532 nm line is turned on at t = 0 and turned off at t = 10 s. One can see both the buildup processes after the laser is turned on and the decay processes after laser beam is turned off for each curve.
with a resolution of 1 ms. The short-circuit current was measured by a HP 34401A digital multimeter and the data were collected by a laptop computer connected to the multimeter. Figure 2 shows steady short-circuit current density (J0) as a function of illumination power for a thin film with a thickness of 12.1 μm. The open-green circles indicate the measured result, and the solid line shows the fitted result using eq 5. One can see that the current density approximately exhibits a linear dependence on the laser power. Except when the laser power is high, the discrepancy from the linear dependency becomes larger. Figure 3 shows the photocurrent dynamics under different illumination powers. The flash period for these curves is 10 s. The illumination laser beam was turned on at t = 0 and turned off at t = 10 s by the shutter. One can easily see that the buildup and decay processes on these curves. The powers we used are shown on the figure that varies from 195.9 to 16.1 mW. Apparently, the buildup process takes a shorter time when the laser power is higher. The dark decay process is slow and has a long tail, which were also observed by other colleagues.24,19 Let us concentrate on the dark decay processes first. Since there was no light illumination for these processes (I0 = 0), eq 6 and its solution, eq 7, should be used to describe the processes. It is worthwhile to note that both the exponential term and 1295
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Table 1. Theoretically Fitted Dark Lifetime, Diffusion Length, and Dark Coefficient of Electrons power
dark lifetime,
dark diffusion coefficient,
dark diffusion length,
(mW)
τd (s)
Dd (1011 cm2/s)
Ld (nm)
195. 9 176.9
4.99 5.15
2.06 2.14
32.1 33.2
158.6
5.19
2.19
33.7
138.1
5.48
2.24
35.0
115.3
5.64
2.36
36.5
93.6
5.80
2.49
38.0
73.6
6.06
2.57
39.5
53.2
6.76
2.52
41.3
34.1 16.1
8.25 8.99
2.10 2.12
41.6 43.7
Figure 4. Examples of theoretically fitted decay processes. Circles show the measured results shown in Figure 3. Upper: 195.9 mW; lower: 138.1 mW.
Figure 6. Five typical buildup processes. Circles show the measured results and the solid-line displayed theoretically fitted curves. The five buildup processes correspond to 195.9, 158.6, 115.3, 73.6, and 34.1 mW in Figure 3. Figure 5. Other examples of theoretically fitted decay processes. Circles show the measured results taken from Figure 3. Upper: 176.9 mW; lower: 53.2 mW.
coefficient Ck drop quickly with increasing k, and we assume that the finite terms in eq 7 can be used to fit the decay processes. Figures 4 and 5 show the fitting results when we take the first seven terms in eq 7. In these figures, the solid lines display the theoretically fitted results using eq 7 and the circles show the measured data. One can see the discrepancies between the measured results and fitted curves are acceptable. The two decay processes shown in Figure 4 are taken from the two curves with flash powers of 195.9 (upper) and 138.1 mW (lower) displayed in Figure 3. In the two curves, the several points before t = 0 are just used to show the steady values. The two curves shown in Figure 5 are taken from the curves of 176.9 (upper) and 53.2 mW (lower) in Figure 3. We have also theoretically fitted all the decay processes displayed in Figure 3 using eq 7. The fitted values of the dark diffusion coefficient Dd and time constant τd can be found in Table 1. From the fitted results, it is very interesting to note that the dark diffusion coefficient does not depend on excitation power significantly; this is because the electron diffusion process should rely on the gradient of free electron density more considerably and the free electrons cannot be continued increasing after the light is turned off. Meanwhile, the dark lifetime strongly
depends on excitation power, and the dark diffusion length weakly depends on excitation power. In order to examine eq 9, the photocurrent buildup processes shown in Figure 3 have also been theoretically fitted to eq 9. Five of these curves are shown in Figure 6. The pink solid curves show the fitted results and the symbols display the measured data. One can see that the fit is nearly perfect. The obtained parameters of the diffusion constant D, electron lifetime τ, and diffusion length L for the ten curves shown in Figure 3 are listed in Table 2. One can see that the diffusion length L changes only slightly with increasing the illumination because the diffusion coefficient D and electron lifetime τ vary with the illumination power in opposite senses, which should be an important finding and has been confirmed by other observations as well.2527 The computation simulations of the photocurrent buildup and decay processes show that the first seven terms of the two solutions [eqs 7 and 9] can be used to essentially model electron diffusion and recombination in the thin films. This may be interpreted that the diffusion and recombination processes involve electron injection, electron transfer and electron recombination with holes through several chemical reactions and each of them has a different time constant.2528 For example, refs 25 and 28 have reported more than five chemical reactions may occur in the cell during the electron transport processes. Each of them exhibits a different time constant. In addition to the 1296
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Table 2. Theoretically Fitted Lifetime, Diffusion Length, and Diffusion Coefficient of Electrons under CW Light Illumination diffusion
diffusion
power
lifetime,
coefficient,
length,
(mW)
τ (s)
D (1011 cm2/s)
L (nm)
195. 9
1.27
7.70
98.9
176.9
1.47
7.29
103.5
158.6 138.1
1.54 1.76
6.56 6.42
100.5 106.3
115.3
2.03
5.86
109.1
93.6
2.38
5.49
114.3
73.6
2.59
5.26
116.7
53.2
2.95
4.79
118.8
34.1
3.24
4.62
122.4
16.1
3.72
3.89
120.3
co-occurrence of several chemical reactions, there may be also several types of traps inside the thin film, which can exhibit different time constants when electrons recombine with these traps as well. In general, to simulate experimental observations correctly, one has to decide how many terms should be used in eqs 7 and 9 based on how many chemical reactions happen in the film. Since the value of each term decreases quickly with increasing k and m, the first several terms usually have considerable effect on the simulated result.
’ CONCLUSIONS Electron diffusion processes in porous nanocrystalline thin films with and without light illumination are investigated. The diffusion equation that has been widely used to describe charge diffusion processes within semiconductor materials has been solved under different conditions. The solutions may be used to describe both the buildup process of photocurrent under light excitation and the decay process after the light is turned off. The experimentally measured steady-state values and dynamics of photocurrent observed in ruthenium dye-sensitized mesoporous TiO2 thin films are used to examine the validity of the solutions. We have shown that the solutions of the diffusion equation under different initial and boundary conditions can be used to simulate the buildup and decay processes to derive the electron diffusion coefficient, lift-time, and diffusion length. Since the improvement of the light-to-electricity conversion efficiency significantly relies on our understanding of the charge transport processes, the solutions are useful to optimize DSSCs systems to increase the conversion efficiency.
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