Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T07:00:59.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Systematic derivation of a surface polarisation model for planar perovskite solar cells

Published online by Cambridge University Press:  22 April 2018

N. E. COURTIER ,
J. M. FOSTER ,
S. E. J. O'KANE ,
A. B. WALKER  and
G. RICHARDSON
N. E. COURTIER
Affiliation:
Mathematical Sciences, University of Southampton, SO17 1BJ, UK emails: nc4g14@soton.ac.uk, G.Richardson@soton.ac.uk
J. M. FOSTER
Affiliation:
Department of Mathematics, University of Portsmouth, PO1 3HF, UK email: jamie.michael.foster@gmail.com
S. E. J. O'KANE
Affiliation:
Department of Physics, University of Bath, BA2 7AY, UK emails: A.B.Walker@bath.ac.uk, S.E.J.O'Kane@bath.ac.uk
A. B. WALKER
Affiliation:
Department of Physics, University of Bath, BA2 7AY, UK emails: A.B.Walker@bath.ac.uk, S.E.J.O'Kane@bath.ac.uk
G. RICHARDSON
Affiliation:
Mathematical Sciences, University of Southampton, SO17 1BJ, UK emails: nc4g14@soton.ac.uk, G.Richardson@soton.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Increasing evidence suggests that the presence of mobile ions in perovskite solar cells (PSCs) can cause a current–voltage curve hysteresis. Steady state and transient current–voltage characteristics of a planar metal halide CH3NH3PbI3 PSC are analysed with a drift-diffusion model that accounts for both charge transport and ion vacancy motion. The high ion vacancy density within the perovskite layer gives rise to narrow Debye layers (typical width ~2 nm), adjacent to the interfaces with the transport layers, over which large drops in the electric potential occur and in which significant charge is stored. Large disparities between (I) the width of the Debye layers and that of the perovskite layer (~600 nm) and (II) the ion vacancy density and the charge carrier densities motivate an asymptotic approach to solving the model, while the stiffness of the equations renders standard solution methods unreliable. We derive a simplified surface polarisation model in which the slow ion dynamics are replaced by interfacial (non-linear) capacitances at the perovskite interfaces. Favourable comparison is made between the results of the asymptotic approach and numerical solutions for a realistic cell over a wide range of operating conditions of practical interest.

Keywords

perovskitesolar cellion vacancydrift-diffusionasymptotic analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 3 , June 2019 , pp. 427 - 457
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2018

Footnotes

NEC is supported by an EPSRC funded studentship from the CDT in New and Sustainable Photovoltaics. SEJO'K was supported by EPSRC grant EP/J017361/1. ABW acknowledges funding from the European Union Horizon 2020 research and innovation programme under Grant no. 676629.

References

[1] Advanpix (2017) Multiprecision Computing Toolbox for MATLAB version 4.3.2.12144.Google Scholar
[2] Black, J. P., Breward, C. J. & Howell, P. D. (2017) Quantum mechanical effects in continuum charge flow models. IMA J. Appl. Math. 82, 251–279.Google Scholar
[3] Brinkman, D., Fellner, K., Markowich, P. A. & Wolfram, M.-T. (2013) A drift–diffusion–reaction model for excitonic photovoltaic bilayers: Asymptotic analysis and a 2D HDG finite element scheme. Math. Models Methods Appl. Sci. 23, 839–872.Google Scholar
[4] Brivio, F., Butler, K. T., Walsh, A. & van Schilfgaarde, M. (2014) Relativistic quasiparticle self-consistent electronic structure of hybrid halide perovskite photovoltaic absorbers. Phys. Rev. B 89, 155204.Google Scholar
[5] Calado, P., Telford, A. M., Bryant, D., Li, X., Nelson, J., O'Regan, B. C. & Barnes, P. R. (2016) Evidence for ion migration in hybrid perovskite solar cells with minimal hysteresis. Nat. Commun. 7, 13831.Google Scholar
[6] Correa-Baena, J.-P., Abate, A., Saliba, M., Tress, W., Jacobsson, T. J., Grätzel, M. & Hagfeldt, A. (2017) The rapid evolution of highly efficient perovskite solar cells. Energy Environ. Sci. 10 (3), 710–727.Google Scholar
[7] Courtier, N. E., Richardson, G. & Foster, J. M. (2018) A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells. arXiv:1801.05737v1.Google Scholar
[8] de Quilettes, D. W., Vorpahl, S. M., Stranks, S. D., Nagaoka, H., Eperon, G. E., Ziffer, M. E., Snaith, H. J. & Ginger, D. S. (2015) Impact of microstructure on local carrier lifetime in perovskite solar cells. Science 348, 683–686.Google Scholar
[9] Domanski, K., Roose, B., Matsui, T., Saliba, M., Turren-Cruz, S.-H., Correa-Baena, J.-P., Carmona, C. R., Richardson, G., Foster, J. M., Angelis, F. D., Ball, J. M., Petrozza, A., Mine, N., Nazeeruddin, M. K., Tress, W., Grätzel, M., Steiner, U., Hagfeldt, A. & Abate, A. (2017) Migration of cations induces reversible performance losses over day/night cycling in perovskite solar cells. Energy Environ. Sci. 10, 604–613.Google Scholar
[10] Eames, C., Frost, J. M., Barnes, P. R. F., O'Regan, B. C., Walsh, A. & Islam, M. S. (2015) Ionic transport in hybrid lead iodide perovskite solar cells. Nat. Commun. 6, 7497.Google Scholar
[11] Foster, J. M., Kirkpatrick, J. & Richardson, G. (2013) Asymptotic and numerical prediction of current-voltage curves for an organic bilayer solar cell under varying illumination and comparison to the Shockley equivalent circuit. J. Appl. Phys. 114, 104501.Google Scholar
[12] Foster, J. M., Snaith, H. J., Leijtens, T. & Richardson, G. (2014) A model for the operation of perovskite based hybrid solar cells: Formulation, analysis, and comparison to experiment. SIAM J. Appl. Math. 74, 1935–1966.Google Scholar
[13] Gottesman, R., Lopez-Varo, P., Gouda, L., Jimenez-Tejada, J. A., Hu, J., Tirosh, S., Zaban, A. & Bisquert, J. (2016) Dynamic phenomena at perovskite/electron-selective contact interface as interpreted from photovoltage decays. Chem 1, 776–789.Google Scholar
[14] Kim, H.-S., Lee, C.-R., Im, J.-H., Lee, K.-B., Moehl, T., Marchioro, A., Moon, S.-J., Humphry-Baker, R., Yum, J.-H., Moser, J. E., Grätzel, M. & Park, N.-G. (2012) Lead iodide perovskite sensitized all-solid-state submicron thin film mesoscopic solar cell with efficiency exceeding 9%. Sci. Rep. 2, 591.Google Scholar
[15] Kojima, A., Teshima, K., Shirai, Y. & Miyasaka, T. (2009) Organometal halide perovskites as visible-light sensitizers for photovoltaic cells. J. Am. Chem. Soc. 131, 6050–6051.Google Scholar
[16] Koutselas, I. B., Ducasse, L. & Papavassiliou, G. C. (1996) Electronic properties of three- and low-dimensional semiconducting materials with Pb halide and Sn halide units. J. Phys.: Condens. Matter 8, 1217–1227.Google Scholar
[17] Lee, M. M., Teuscher, J., Miyasaka, T., Murakami, T. N. & Snaith, H. J. (2012) Efficient hybrid solar cells based on meso-superstructured organometal halide perovskites. Science 338, 643–647.Google Scholar
[18] Löper, P., Stuckelberger, M., Niesen, B., Werner, J., Filipič, M., Moon, S.-J., Yum, J.-H., Topič, M., Wolf, S. D. & Ballif, C. (2015) Complex refractive index spectra of CH3NH3PbI3 perovskite thin films determined by spectroscopic ellipsometry and spectrophotometry. J. Phys. Chem. Lett. 6 (1), 66–71.Google Scholar
[19] The MathWorks, Inc. (2016) MATLAB version 9.1.0.441655 (R2016b).Google Scholar
[20] Nelson, J. (2003) The Physics of Solar Cells, Imperial College Press, London, UK.Google Scholar
[21] Neukom, M. T., Züfle, S., Knapp, E., Makha, M., Hany, R. & Ruhstaller, B. (2017) Why perovskite solar cells with high efficiency show small IV-curve hysteresis. Sol. Energy Mater. Sol. Cells 169, 159–166.Google Scholar
[22] Niu, G., Guo, X. & Wang, L. (2015) Review of recent progress in chemical stability of perovskite solar cells. J. Mater. Chem. A 3, 8970–8980.Google Scholar
[23] O'Kane, S. E. J., Richardson, G., Pockett, A., Niemann, R. G., Cave, J. M., Sakai, N., Eperon, G. E., Snaith, H. J., Foster, J. M., Cameron, P. J. & Walker, A. B. (2017) Measurement and modelling of dark current decay transients in perovskite solar cells. J. Mater. Chem. C 5, 452–462.Google Scholar
[24] Please, C. (1982) An analysis of semiconductor P-N junctions. IMA J. Appl. Math. 28, 301–318.Google Scholar
[25] Pockett, A., Eperon, G. E., Peltola, T., Snaith, H. J., Walker, A., Peter, L. M. & Cameron, P. J. (2015) Characterization of planar lead halide perovskite solar cells by impedance spectroscopy, open-circuit photovoltage decay, and intensity-modulated photovoltage/photocurrent spectroscopy. J. Phys. Chem. C 119, 3456–3465.Google Scholar
[26] Ravishankar, S., Almora, O., Echeverría-Arrondo, C., Ghahremanirad, E., Aranda, C., Guerrero, A., Fabregat-Santiago, F., Zaban, A., Garcia-Belmonte, G. & Bisquert, J. (2017) Surface polarization model for the dynamic hysteresis of perovskite solar cells. J. Phys. Chem. Lett. 8, 915–921.Google Scholar
[27] Richardson, G. (2009) A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials. Math. Med. Biol. 26, 201–224.Google Scholar
[28] Richardson, G., O'Kane, S. E. J., Niemann, R. G., Peltola, T. A., Foster, J. M., Cameron, P. J. & Walker, A. B. (2016) Can slow-moving ions explain hysteresis in the current-voltage curves of perovskite solar cells? Energy Environ. Sci. 9, 1476–1485.Google Scholar
[29] Richardson, G., Please, C. & Styles, V. (2017) Derivation and solution of effective medium equations for bulk heterojunction organic solar cells. Eur. J. Appl. Math. 28, 973–1014.Google Scholar
[30] Richardson, G. & Walker, A. B. (2016) Drift diffusion modelling of charge transport in photovoltaic devices. In: Da Como, E., De Angelis, F., Snaith, H. & Walker, A. (editors), Unconventional Thin Film Photovoltaics, Royal Society of Chemistry, Cambridge, UK, pp. 297–331.Google Scholar
[31] Schmeiser, C. (1992) Free boundaries in semiconductor devices. Proc. Free Boundary Problems: Theory and Applications. In: Chadham, J. & Rasmussen, H. (editors), Pitman Research Notes Mathematics Series, vol. 3, Longman, Harlow, pp. 268–268.Google Scholar
[32] Schmeiser, C. & Unterreiter, A. (1994) The derivation of analytic device models by asymptotic methods. In: Coughran, W. M. Jr., Cole, J., Lloyd, P. & White, J. K. (editors), vol 59, Semiconductors. The IMA Volumes in Mathematics and its Applications, Springer, New York, NY, pp. 343–363.Google Scholar
[33] Schulz, P., Edri, E., Kirmayer, S., Hodes, G., Cahen, D. & Kahn, A. (2014) Interface energetics in organo-metal halide perovskite-based photovoltaic cells. Energy Environ. Sci. 7, 1377–1381.Google Scholar
[34] Shen, H., Jacobs, D. A., Wu, Y., Duong, T., Peng, J., Wen, X., Fu, X., Karuturi, S. K., White, T. P., Weber, K. & Catchpole, K. R. (2017) Inverted hysteresis in CH3NH3PbI3 solar cells: Role of stoichiometry and band alignment. J. Phys. Chem. Lett. 8, 2672–2680.Google Scholar
[35] Snaith, H. J., Abate, A., Ball, J. M., Eperon, G. E., Leijtens, T., Noel, N. K., Stranks, S. D., Wang, J. T.-W., Wojciechowski, K. & Zhang, W. (2014) Anomalous hysteresis in perovskite solar cells. J. Phys. Chem. Lett. 5, 1511–1515.Google Scholar
[36] Stoumpos, C. C., Malliakas, C. D. & Kanatzidis, M. G. (2013) Semiconducting tin and lead iodide perovskites with organic cations: Phase transitions, high mobilities, and near-infrared photoluminescent properties. Inorg. Chem. 52, 9019–9038.Google Scholar
[37] Stranks, S. D., Burlakov, V. M., Leijtens, T., Ball, J. M., Goriely, A. & Snaith, H. J. (2014) Recombination kinetics in organic-inorganic perovskites: Excitons, free charge, and subgap states. Phys. Rev. Appl. 2, 034007.Google Scholar
[38] Stranks, S. D. & Snaith, H. J. (2015) Metal-halide perovskites for photovoltaic and light-emitting devices. Nat. Nanotechnol. 10, 391–402.Google Scholar
[39] Tan, H., Jain, A., Voznyy, O., Lan, X., de Arquer, F. P. G., Fan, J. Z., Quintero-Bermudez, R., Yuan, M., Zhang, B., Zhao, Y., Fan, F., Li, P., Quan, L. N., Zhao, Y., Lu, Z.-H., Yang, Z., Hoogland, S. & Sargent, E. H. (2017) Efficient and stable solution-processed planar perovskite solar cells via contact passivation. Science 355, 722–726.Google Scholar
[40] van Reenen, S., Kemerink, M. & Snaith, H. J. (2015) Modeling anomalous hysteresis in perovskite solar cells. J. Phys. Chem. Lett. 6, 3808–3814.Google Scholar
[41] Walsh, A., Scanlon, D. O., Chen, S., Gong, X. G. & Wei, S.-H. (2015) Self-regulation mechanism for charged point defects in hybrid halide perovskites. Angew. Chem. 127, 1811–1814.Google Scholar