Origin and Whereabouts of Recombination in Perovskite Solar Cells Supporting Information Lidia Contreras-Bernal a, Manuel Salado a,b, Anna Todinova a, Laura Calio b, Shahzada Ahmad b, Jesús Idígoras a, *, Juan A. Anta a, * a Área de Química Física, Universidad Pablo de Olavide, E-41013, Sevilla, Spain. b Abengoa Research, C/Energía Solar n 1, Campus Palmas Altas, 41014 Sevilla, Spain Table S1. Device configurations, photovoltaic parameters including statistics and best efficiencies of the solar cells studied in this work. Configuration MAI/P3HT MAI/Spiro Thickness (nm) 300 (0.8M) 500 (1.2M) 300 (0.8M) 500 (1.2M) J sc (ma cm -2 ) V oc (V) Fill Factor (%) PCE (%) 14.99±0.51 0.64±0.02 65.48±1.33 6.28±0.13 14.21±2.30 0.75±0.07 66.91±2.16 7.01±2.02 15.89±2.35 0.88±0.06 74.12±8.34 9.31±2.04 16.77±1.70 0.94±0.04 71.75±2.11 10.99±1.71
MIX/Spiro 650 (0.8M) 950 (1.2M) 20.05±1.39 0.98±0.01 68.43±1.22 13.47±1.16 20.68±0.36 0.96±0.01 66.37±2.32 13.27±0.25 Best values: Configuration Thickness (nm) J sc (ma cm -2 ) V oc (V) Fill Factor (%) PCE (%) MAI/P3HT MAI/Spiro MIX/Spiro 300 (0.8M) 14.23 0.687 65.5 6.40 500 (1.2M) 17.46 0.878 67.9 10.41 300 (0.8M) 16.03 0.898 74.0 10.66 500 (1.2M) 19.15 0.969 72.5 13.46 650 (0.8M) 20.48 0.998 69.4 14.23 950 (1.2M) 20.19 0.957 70.7 13.69 Figure S1. SEM cross-sectional images of the studied devices for different device configurations. (A,D) MAI/P3HT, (B,E) MAI/Spiro and (C,F) MIX/Spiro and different concentration of perovskite precursors: (A,B,C) 0.8M and (D,E,F) 1.2M.
Figure S2. UV/Vis absorption spectra of the MHP layers studied in this work. The inset illustrates the estimation of the optical bandgap from the measured spectra. Figure S3. High-frequency resistances as extracted from fittings of the impedance spectra using the two excitation wavelengths of (blue) λ blue = 465 nm and (red) λ red = 635 nm.
Figure S4. Current-voltage curve for backward scans (scan rate = 0.1 V/s) for a concentration of perovskite precursor of 1.4M. Inset shows the SEM cross-sectional image. Figure S5. Low-frequency resistances and associated capacitances as extracted from fittings of the impedance spectra for the configurations indicated using the two excitation wavelengths of (blue) λ blue = 465 nm and (red) λ red = 635 nm. Slopes in units of q/k B T are indicated in the graphs.
Figure S6. Resistance Bode plot at open-circuit under the excitation wavelengths of (blue) λ blue = 465 nm and (red) λ red = 635 nm the configurations described in Table 1 and for a precursor concentration of 1.2 M: (A) MAI/P3HT, (B) MAI/Spiro and (C) MIX/Spiro.
Figure S7. Current-voltage curves for fresh and degraded MIX/Spiro cells (left) and high-frequency recombination resistance as extracted from fittings of the impedance spectra using the two excitation wavelengths of (blue) λ blue = 465 nm and (red) λ red = 635 nm for degraded cells (right). Data at the top corresponds to cells being kept in the dark at ambient humidity during 6 days. In the case shown at the bottom, the cell was kept under ambient illumination during 6 days.
Figure S8. Open-circuit potential as a function of temperature for the aged devices studied in Figure S6 for white light and a light intensity of 14.15 W/m 2 Orange: dark degradation, Green: degradation under illumination. Figure S9. High frequency capacitance as extracted from fittings of the impedance spectra using the two excitation wavelengths of (blue) λ blue = 465 nm and (red) λ red = 635 nm, for the indicated cell configurations.
Figure S10. Total and dark currents versus applied potential for both blue and red illumination as obtained from the numerical solution of Eqs. (3) and (4). Top panels: linear recombination case (γ = 1). Results of the analytical model of Södergren et al. 5 is added for comparison. Middle and bottom panels: non-linear recombination (γ = 0.5) with slow (middle) and rapid (bottom) recombination. Figure S11. Dark currents with both blue and red illumination for the degraded devices of Figure S7. Top: degradation in the dark. Bottom: degradation under illumination.
Derivation of the net recombination rate According to semiconductor theory 1 when the quasi-fermi level lies well below the corresponding band edge (non-degenerate semiconductor), Boltzmann statistics is a good approximation for the Fermi-Dirac formula. Hence, the concentration of carriers is given by n = N $ exp ) - *+),. / 0 p = N V exp E p F +EV k B T (S1) where E C and E V are the energies of the conduction and valence band edges, respectively, and E n F and E p F are the quasi-fermi levels. From Eq. (S1) we get np = N $ N < exp ) =+ ), > +), -. / 0 = N $ N < exp ) =+?<. / 0 (S2) where E g = E C - E V is the band gap and V = (E p F - E n F )/q. At high-injection conditions the photogenerated carrier densities well exceed intrinsic and majority carrier values. 1 In that situation electroneutrality requires that n = p. Therefore n = N $ N < @ A exp ) =+?< A. / 0 (S3) Combining Eqs. (2) and (S3) we can write the recombination current in the device as J CDE = qdu CDE = J II exp γ ) =+?< A. / 0 (S4) where d is the thickness of the active layer and J 00 = qd(n C N V ) 1/2 k B T p 0. Numerical solution of Eqs. (3) and (4) Eqs. (3) and (4) in the main text U CDE LM LN = k 0p I n P (3)
D M L R M LS R U CDE + I I λ α λ exp α λ x = 0 (4) can be solved numerically 2,3 to obtain the current-voltage (J-V) curve and the dark current. The Forward Time Centered Space (FTCS) method was used with the following boundary conditions: 5 n(x= 0,t )=n 0 (V ) n(x,t=0)=n 0 (V ) ( dn(x,t ) dx ) = 0 x=d (S5) (S6) (S7) where d is the thickness of the active layer, V is the applied voltage and n 0 (V )=n 0 0 exp(v / k B T ) (S8) Once the equation is solved the photocurrent density for a given value of the potential V is obtained from the stationary density profile at x = 0 (it is assumed that collection of electrons and holes in external contacts is the same): J (V )= ( dn(x,t) dx ) x=0, t (S9) Solving Eqs. (3)-(4) for different values of V, and using Eq. (S8) and Eq. (S9), the full J-V curve can be obtained. The following parameters were used in the simulations (intended to approximately reproduce the conditions of the MIX/Spiro devices). Absorption coefficient: α (λ = 465 nm) = 5 10 6 m -1 (blue), α (λ
= 635 nm) = 2 10 6 m -1 (red), electron diffusion coefficient: 4 D n = 2 10-6 m 2 s -1, photon flux: I 0 (λ = 465 nm) = 3.5 10 20 m 2 s -1 (blue), I 0 (λ = 635 nm) = 4.8 10 20 m 2 s -1 (red). Two recombination rates where considered: k rec = 2 10 6 m 3(γ-1) s -1, corresponding to a diffusion length of L n = 1 µm 4 and k rec = 2 10 8 m 3(γ-1) s -1, which yields L n = 0.1 µm for linear recombination (γ = 1). The numerical simulation was carried out using 300 points in x-space and 1.2 10 6 points in t-space, which were found to be enough to ensure convergence and the steady-state is reached. Results of the numerical simulation can be found in Figure S9 for both linear (γ = 1), and non linear recombination (γ = 0.5). In the first case, simulated data where compared with the analytical results of Södergren et al. 5 In the second case, both slow and rapid recombination (long and short diffusion lenght) were compared. As explained in the main text, significant differences in the dark currents obtained between the red and the blue light illumination are only found in the non-linear case for a short-diffusion length. References (1) Wiley: Physics of Semiconductor Devices, 3rd Edition - Simon M. Sze, Kwok K. Ng http://eu.wiley.com/wileycda/wileytitle/productcd-0471143235.html (accessed Oct 19, 2016). (2) Anta, J. A.; Idígoras, J.; Guillén, E.; Villanueva-Cab, J.; Mandujano-Ramírez, H. J.; Oskam, G.; Pellejà, L.; Palomares, E. A Continuity Equation for the Simulation of the Current voltage Curve and the Time-Dependent Properties of Dye-Sensitized Solar Cells. Phys. Chem. Chem. Phys. 2012, 14 (29), 10285 10299. (3) Todinova, A.; Idígoras, J.; Salado, M.; Kazim, S.; Anta, J. A. Universal Features of Electron Dynamics in Solar Cells with TiO2 Contact: From Dye Solar Cells to Perovskite Solar Cells. J. Phys. Chem. Lett. 2015, 6 (19), 3923 3930. (4) Stranks, S. D.; Eperon, G. E.; Grancini, G.; Menelaou, C.; Alcocer, M. J. P.; Leijtens, T.; Herz, L. M.; Petrozza, A.; Snaith, H. J. Electron-Hole Diffusion Lengths Exceeding 1 Micrometer in an Organometal Trihalide Perovskite Absorber. Science 2013, 342 (6156), 341 344. (5) Sodergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S. E. Theoretical-Models for the Action Spectrum and the Current-Voltage Characteristics of Microporous Semiconductor-Films in Photoelectrochemical Cells. Journal of Physical Chemistry 1994, 98 (21), 5552 5556.