I Taller de Innovación Fotovoltaica y Celdas Solares; March 8 10, 2011, CIE UNAM, Temixco. Numerical simula,on of the performance of the dye sensi,zed solar cell Julio Villanueva Cab 1,*, Elena Guillén 2, Juan Antonio Anta 2 and Gerko Oskam 1 1 Departamento de Física Aplicada, CINVESTAV IPN, Mérida, Yucatán, México 2 Departamento de Sistemas Físicos, Químicos y Naturales, Universidad Pablo de Olavide, 41013 Sevilla, Spain * Current address: Chemical & Materials Science Center, NaQonal Renewable Energy Laboratory, Golden, CO 80401, USA
CB E e - D*/D + h ν recombination D 0 /D + acceptor donor E redox TCO TiO 2 dye electrolyte solution I - photoelectrochemical cell porous, high surface area metal oxide film light absorption by adsorbed sensitizer molecules electron transport in solid and ion transport in solution I - 3 I - + I - I - 3
I Taller de Innovación Fotovoltaica y Celdas Solares; March 8 10, 2011, CIE UNAM, Temixco. Objectives Numerical tool to simulate the current-voltage curve in Dye Sensitized Solar Cells (DSSC) Make a connection with microscopic theories on transport and recombination so the model is as rigorous as possible (but not too complex!!) Make a connection with experimental techniques to obtain the relevant parameters
Dye solar cells: generation, transport and recombination Absorbance x light glass TCO Electron electron density electron acceptors e - e - e - ions Open circuit Short circuit ions Particles are too small for band bending Solution with ions provide shielding Electron transport is impeded by transfer to electron acceptor in the solution x steady-state measurements (Lindquist et al.): photocurrent is dominated by diffusion
Continuity equation n t = 1 e J R + G n is the electron density under illumination J is the current density in the film G and R are generation and recombination rates Current density J = enµ n φ + ed n µ n is the electron mobility φ is the electrical potential D is the electron diffusion coefficient Diffusion transport equation n(x,t) t = D 2 n(x,t) 2 + n(x,t) n 0 τ 0 + Γα exp( αx) = 0 flux recombination electron injection The diffusion coefficient and recombina,on term are dependent to the light intensity transport equa,on is more complex: numerical methods to model electron transport
D is a power law function of the light intensity, i.e, the electron density Conduction Band k 1 k -1 E F,n Valence Band Electron Transport in Porous Nanocrystalline TiO 2 Photoelectrochemical Cells F Cao, G Oskam, G J Meyer, and P C Searson J. Phys. Chem. 1996, 100, 17021-17027. Band diagram showing trap states in the band gap. The rate constants k 1 and k -1 denote trapping and de-trapping of electrons, respectively. The Fermi energy determines which traps dominate the transport kinetics.
Continuity equation for electrons 1-dimensional problem (x is the distance to the substrate) n(x,t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed GENERATION DIFFUSION n(x,t) is the total electron density RECOMBINATION CHARGE TRANSFER FROM TCO SUBSTRATE
n(x, t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed λ max G(x) = φ inj I 0 λ λ min GENERATION ( ) ε Cell (λ) exp[ ε Cell (λ) x] dλ Injection quantum yield (0 < φ inj < 1) Dye absorption coefficient
n(x,t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed DIFFUSION D(n) = D ref f (n) = D ref n n ref 1 α α Density-dependent (Fermi-level dependent) diffusion coefficient α = 0.2-0.5 Multiple trapping mechanism g(e) = αn t k B T αe exp k B T E F (x,t) = k BT α n(x,t) ln N t
n(x,t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed RECOMBINATION from nanostructured film Rate constant is f(e F ): k R = k R ref f (n) = k R ref n n ref β E F (x,t) = k B T α n(x,t) ln N t β = (1-α)/α The same as for diffusion
n(x,t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed RECOMBINATION from nanostructured film Rate is f(v): ref k R = k R f (V ) = k ref R exp bev kt ref k R = k R n(x) n ref b α b 0.5
n(x,t) t = G(x) + D(n) n(x,t) 0 k R (n) n(x,t) n 0 ( ) + J TCO ed TCO CHARGE TRANSFER FROM TCO SUBSTRATE 0 J TCO = J TCO exp (1 b)ev k B T exp bev k B T Butler-Volmer equation b TCO 0.5
n(x) Electron density profile J J SC b n V 0 a V OC n 0 0 0 d x V V = (E E 0 F F ) e = k T B αe ln n 0 V 0 n 0
Practical procedure J Use the experimental short-circuit current to fit either the injection yield or the dye concentration in cell Use the experimental opencircuit voltage to obtain a first estimate of the recombination constant prefactor V
Use the experimental current transient to obtain the trap distribution parameter α Villanueva et al., J. Phys. Chem. C 2009, 113, 19722 19731.
slope = 78 mv Use open-circuit voltage versus light intensity and time decay to obtain charge transfer parameters from TCO substrate (J 0 TCO, b)
J Use the current at maximum power point to obtain the total series resistance in the cell V Numerical Method: Forward Time Centered Space (FTCS) with the Lax scheme
TiO 2 /N719/organic electrolyte ZnO/N719/solvent-free electrolyte eff = 6.5% eff = 1.5% Numerical Simulation of the Current-Voltage Curve in Dye-Sensitized Solar Cells Julio Villanueva, Juan A. Anta, Elena Guillén, and Gerko Oskam J. Phys. Chem. C 2009, 113, 19722 19731.
TiO 2 (brookite)/n719/organic electrolyte α = 0.28 eff = 4.0% slope = 33 mv
ZnO/D149/organic solvent electrolyte eff = 2.8% R = 30 Ohm cm 2 α = 0.2 k R 0 = 3.3 10-3 s -1
Parameter Z Cell (ZnO) T Cell (TiO 2 ) Brookite Cell ZnO/D149 C Cell (M) 0.14 0.25 0.24 k 0 R (s 1 ) 9.0 10 7 3.1 10 9 7.3 10 9 3.3 10 3 α 0.18 0.20 0.28 0.2 blocking layer no no yes yes J 0 (TCO) (A cm 2 ) 1.0 10 4 1.1 10 5 1.5 10 9 b TCO 0.50 0.55 0.55 dv oc /dln(int) (mv) 52 78 33 34 R (Ω cm 2 ) 37.5 11.3 15.3 30 L (µm) 10.5 180 117
Transport-limited or transfer-limited recombination Model 1: Transport-limited recombination k R = k R ref n n ref 1 α α Model 2: Transfer-limited recombination k R = k ref R exp bev kt Influence of the recombination mechanism on the IV-curve of dye-sensitized solar cells J Villanueva, G Oskam and J A Anta, Solar Energy Materials & Solar Cells, 94 (2010) 45 50.
D(n) = D ref n n ref 1 α α ref k R = k R n n ref 1 α α k R = k ref R exp bev kt V OC k BT e V OC Ln I k B T (α + b)e Ln I Model 1: Transport-limited recombination Model 2: Transfer-limited recombination
ZnO/D149/organic solvent electrolyte Non-ideality in Voc vs. light intensity curve γ 0.75 for NPs Model 2: Transfer-limited recombination γ = α + b α = 0.2 b = 0.55
Comparison total electron and free electron density models Steady-state conditions 0 = G(x) + D 2 n cb k 2 R ( n cb n 0 ) γ 0 = G(x) + D(n ) n tot tot k R (n tot ) n tot n 0 ( ) For γ = 1, both equations are formally identical Free electron Total electron If γ < 1, we have the case of non-first order recombination light intensity dependence of the electron diffusion length discrepancy results from steady-state & modulation methods Model 2: k R = k ref R exp bev kt Both models are identical with γ = α + b
Conclusions Numerical solution of the continuity equation in DSSC was obtained with explicit consideration of recombination via the oxide and the substrate The model can fit simultaneously current and voltage transients, open-circuit voltage vs. light intensity and the full IV curve The model was tested for several very different kind of cells and different types of recombination kinetics The total electron density model compares well with the free electron model to describe transport & recombination kinetics
Acknowledgements PROYECTO DE EXCELENCIA P06-FQM-01869 CONSOLIDER-INGENIO 2010 CSD2007-00007 FPU fellowships Grant No. 80002-Y Red Temática en Fuentes de Energía