IMPERIAL COLLEGE LONDON

Transient Optoelectronic Characterisation and Simulation of Perovskite Solar Cells

by Philip Calado

A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy

in the Faculty of Natural Sciences Department of Physics

April 2018 2 Declaration of Originality

This thesis is a summary of work undertaken in the Department of Physics at Imperial College London between October 2014 and December 2017 under the supervision of Dr. Piers Barnes and Prof. Jenny Nelson. I declare that the work contained herein is my own except where specific reference is made to the contribution of others.

Philip Calado April 2018

The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.

3 “The best that most of us can hope to achieve in physics is simply to misunderstand at a deeper level.”

Wolfgang Pauli

4 Abstract

Lead halide perovskites are a class of solution-processable semiconductor materials showing great potential for photovoltaic applications. While perovskite solar cell (PSC) efficiencies have escalated rapidly to beyond 22% in recent years, the materials suffer from a number of chemical instabilities and the processes underlying the optoelectronic response of devices are not well understood.

This thesis investigates the device physics of PSCs using novel transient optoelectronic measurements combined with device simulation. A one-dimensional numerical drift-diffusion model capable of solving for electrons, holes and a single ionic charge carrier was developed to simulate perovskite devices. The inclusion of a high density of mobile ionic species in the absorber layer is found to have important consequences on both device performance and the interpretation of established measurements.

Transient optoelectronic measurements are presented showing that mobile ions are present in architectures of PSC that do not exhibit current-voltage hysteresis. Simulations of p-i-n structured devices indicate that a combination of mobile ions and field-dependent interfacial recombination rates are critical to reproducing hysteresis.

Transient ideality factor measurements are used to identify the dominant recombination mechanisms in PSCs. Changes in the perceived ideality factor are correlated to localised recombination, the charge carrier population overlap, and ion density profiles in simulated devices.

Simulations are used to assess the validity of a zero-dimensional model applied to small perturbation transient photovoltage (TPV) measurements on perovskite devices. Two analytical models are proposed to explain the different regimes of behaviour in devices with high rates of interfacial recombination. The TPV decay in perovskites is identified as being predominantly a measure of the transport properties of the absorber layer.

Together these findings demonstrate the combined power of experimental measurements and simulation to improve our physical understanding of new semiconductor technologies.

5 6 Acknowledgements

For their undying faith in the face of certain failure, I would like to thank the following people:

First and foremost my brilliant supervisors Piers and Jenny: thank you for rearing me from an engineering graduate who thought he didn’t understand anything to a postgraduate physicist who knows he doesn’t understand anything.

To Piers ‘Radio Frequency’ Barnes, for being an infinite source of creativity and optimism. For always believing that there was a way to solve a problem no matter what the odds. For having an unquenchable scientific curiosity, even if this did mean a few extended ‘meetings’. For being a genuine human, an individual thinker, and seeing the potential in everyone. In all seriousness though, I should tell you at this point that I really don’t have a clue what you’re talking about most of the time.

To Jenny Nelson for being a brilliant scientist who also cares. For being such an inspiration to everyone in the group and a strong role model to your staff and students. Also for writing such a brilliant and accessible book, on which much of the work here is based.

To Brian O’Regan for being a genuine scientist with real integrity. For being direct yet humble and open with it. For your creativity and for coding the most insanely brilliant transient robot ever.

To Andrew Telford who helped me through the early days when I still had no idea what a negative photovoltage transient meant. Thanks for the your great humour and professionalism.

To Davide Moia for being so utterly clever and such a nice bloke all round with it. I hope one day I will be able to conceptualise things like you, but somehow I doubt it. You will be Prof. Moia in no time.

To Mohammed ‘Moho’ Azzouzi who ‘in that sense’ has an amazing scientific mind! Thank you for such engaging scientific discussions and your great enthusiasm.

To Ilario Gelmetti for always having a big smile on your face and for coming to visit us. I feel like you are one of us and you will be missed. Your work in simulating Photo-induced Impedance Spectroscopy will be long-remembered.

7 To Dr Xingyuan Shi for making me feel better about being incompetent by falsely claiming to be incompetent yourself. I really appreciate it.

To the best Plastic Electronic CDT cohort ever, Cohort 5: To Alex for always wearing a smile, Andika for being strong in adversity, Das for not taking life too seriously, Iain A. for saying it just like it is, Gwen for being a great hippie scientist, Heavy Metal Iain for being hilarious, Jam for being so open, Jason for keeping it ‘Rohr’, Madeleine for sorting us all out, Matt for being a great ski buddy, Nathan for laughing at my bad sense humour, Tony for falling asleep, and Yiren for being a TRPL wizard.

To all the members of the Nelson and Barnes groups past and present: Anna, Aurelien, Beth, Drew, Eli, Jarv, Jizhong, Sachetan, Sam, Scot and Will. Thanks for sharing your scientific observations and great humour.

To Prof James Durrant for directing me towards Jenny and agreeing to examine this work. To Li, Dan, Pabitra, Seb and everyone else in the Durrant group for chemical and lab coat assistance.

To everyone who I shared office H1101 with: Alise, Elysia, Jun, Mike, Michelle, Nicola, and Viktoria. Thanks for creating such a nice working environment.

To Martin Neukom for fighting for a better planet in every way imaginable - from solar research, to being an MP for the Green Party, to shutting down nuclear power stations with direct action. You put the rest of us to shame. Also for the great conversations about how to fix the world. I hope we can do that one day.

To Matt Carney for your amazing humour and humility. We still owe you a publication and I haven’t forgotten.

To Trystan Watson, Joel Troughton and all the staff and students at SPECIFIC.

To Pablo Docampo and Hongi Hu for your incredible efficiency and professionalism.

To Shaun Armstrong and Duncan McLellan for teaching me the fundamentals of mathematics and physics with infinite patience.

And to my sponsors, the UK Engineering and Physical Sciences Research Council, for providing me with this unique opportunity.

8 My personal thanks for their unconditional love and support:

Most importantly to Abigail Mortimer for being my guiding light. For being my ethical compass and my companion in life. And for putting up with an increasingly scruffy mad-scientist-Bubba throughout the thesis writing process. Also for the heroic prof readinfg.

To my soul brother Matt Fairclough for walking side-by-side and sharing so many important experiences with me. It is a great honour to call you my friend.

To George Richardson for seeing me through those earlier lab experiments when I was still just a noob. Big shout out to the Imperial College Hackers Ultimate Distruption (ICCHUD) crew- you know who you are!

To Oliver Levers for making me laugh so much that it hurt my ribs almost everyday during the MRes year and for all the good times since with the General Farmer’s Alliance and ICCHUD.

To General Prospero Taroni Junior Waste for being such a stylish and singular human. Only a mind as creative as yours could’ve invented PEDOT-Spandex my friend.

To Squ Ldr Ian ‘Flash’ Mortimer and Jane for being such supportive and inspirational ‘older people’. The pens were really useful in the end Jane, so thanks.

To Steve Baker for your services to the environment and humanity, and your endless positivity!

To Jamie Innes for showing me that all the answers are in the plants.

To Paul Allen and everyone at the Centre for Alternative Technology for reminding me that the world could be so different.

And finally to my family:

To Maria Bridges for all your support through my various stages of education and life. I promise to stop being a student now mum.

To Paulo, Sophie and the Shrimpmonks for giving me hope in the children of tomorrow.

And finally to my Big-bro Carlos for your strength in the face of adversity. Keep on fighting the good fight and never give up!

9 10 Contents

Declaration of Originality3

Abstract 5

Acknowledgements7

List of Figures 16

List of Tables 19

Physical Constants 20

Abbreviations 21

Symbols 24

1 Introduction 29 1.1 Abstract...... 29 1.2 Motivation...... 29 1.2.1 The climate crisis...... 29 1.2.2 The potential for solar energy...... 30 1.3 Lead halide perovskite solar cells...... 32 1.4 A brief retrospective of the work...... 32 1.5 Scope and outline...... 33 1.6 Publications realised from this work...... 35

2 Background 37 2.1 Abstract...... 37 2.2 Working principles of a solar cell...... 37 2.2.1 Recombination...... 39 2.2.2 Devices...... 40 2.3 The perovskite chemical structure...... 41 2.3.1 Physical properties of CH3NH3PbI3 ...... 42 2.3.1.1 Optical properties...... 42 2.3.1.2 Phases...... 44 2.3.2 Defects and mobile ions...... 46 2.3.3 Inter-bandgap states...... 48 2.3.4 Chemical stability...... 49 2.4 Perovskite devices...... 52

11 Contents 12

2.4.1 Material processing...... 52 2.4.2 Architectures...... 53 2.4.3 Contact selectivity...... 54 2.5 Current-voltage hysteresis...... 55 2.5.1 Possible origins of field screening...... 56 2.6 Transient optoelectronic measurements...... 57 2.7 Device simulation...... 59 2.8 Conclusions...... 60

3 Device simulation 63 3.1 Declaration of contributions...... 63 3.2 Abstract...... 63 3.3 Introduction...... 64 3.3.1 Simulating open and closed circuit conditions...... 64 3.4 Fundamental semiconductor energy levels...... 65 3.4.1 Vacuum energy...... 66 3.4.2 Conduction and valence band energies...... 66 3.4.3 Quasi Fermi levels...... 67 3.4.4 Open circuit voltage...... 68 3.4.5 Energy level diagrams...... 68 3.5 Charge transport (drift-diffusion)...... 69 3.5.1 Transport across heterojunction interfaces...... 70 3.6 The Continuity equations...... 71 3.7 Poisson’s equation...... 72 3.8 Initial conditions...... 73 3.9 Boundary conditions...... 75 3.9.1 Contact selectivity and surface recombination...... 76 3.9.2 Doping for selectivity...... 77 3.9.3 Closed circuit boundary conditions...... 78 3.9.4 Open circuit boundary conditions...... 79 3.10 Generation...... 80 3.11 Recombination...... 81 3.11.1 Band-to-band recombination...... 81 3.11.2 Shockley-Read-Hall (SRH) recombination...... 81 3.12 Spatial and time meshes...... 82 3.13 Comparison of the simulation results with analytical solutions for a p-n junction...... 83 3.13.1 The Depletion Approximation solution to a p-n junction at equilibrium...... 83 3.13.2 The Shockley diode equation...... 84 3.13.3 Current-voltage characteristics...... 84 3.14 Conclusions...... 88

4 Experimental methods 89 4.1 Declaration of contributions...... 89 4.2 Abstract...... 89 4.3 Device fabrication...... 90 Contents 13

4.4 Characterisation...... 90 4.4.1 Current-voltage scans...... 90 4.4.2 Transient optoelectronic measurement experimental set-up.... 91 4.4.3 TELTPV technical specifications...... 92 4.4.4 TRACER technical specifications...... 92 4.5 Conclusions...... 93

5 Evidence for ion migration in perovskite solar cells with minimal hysteresis 95 5.1 Declaration of contributions and publication...... 95 5.2 Abstract...... 95 5.3 Introduction...... 96 5.4 Methods...... 97 5.4.1 Current-voltage scan protocol...... 97 5.4.2 Transients of the transient photovoltage (TROTTR)...... 97 5.4.3 Step-Dwell-Probe transient photocurrent (SDP)...... 98 5.4.4 Simulation...... 99 5.5 Experimental Results...... 99 5.5.1 Current-voltage scans...... 99 5.5.2 Transients of the transient photovoltage...... 101 5.5.3 Step-dwell-probe photocurrent transients...... 102 5.6 Simulation results and discussion...... 104 5.6.1 Reduced ion densities...... 108 5.6.2 Alternative recombination schemes...... 110 5.7 Understanding the J-V scan hysteresis...... 111 5.8 Conclusions...... 114

6 Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 117 6.1 Declaration of contributions...... 117 6.2 Abstract...... 117 6.3 Introduction...... 118 6.4 Background...... 119 6.4.1 Ideality factor...... 119 6.4.2 Zero-dimensional analysis...... 120 6.5 Methods...... 123 6.5.1 Experimental...... 123 6.5.1.1 Devices...... 123 6.5.1.2 Suns-VOC ...... 123 6.5.1.3 Suns-EL...... 124 6.5.2 Simulation...... 124 6.6 Results...... 125 6.6.1 Transient ideality factor simulation...... 125 6.6.2 Experimental results and discussion...... 131 6.7 Conclusions...... 135

7 Reinterpreting small perturbation transient photovoltage measurements in p-i-n devices 137 Contents 14

7.1 Declaration of contributions...... 137 7.2 Abstract...... 137 7.3 Introduction...... 138 7.4 Background...... 139 7.4.1 Derivation of a zero-dimensional kinetic model of recombination. 139 7.4.1.1 Differential capacitance charge extraction (DCCE).... 142 7.5 Methods...... 144 7.5.1 Experimental...... 144 7.5.2 Transient photovoltage simulations...... 144 7.6 Results...... 145 7.6.1 Verification of the simulation tool...... 145 7.6.2 Light intensity dependence in devices dominated by surface recombination...... 146 7.6.3 Inference of reaction orders...... 148 7.7 Discussion...... 150 7.7.1 Thermionic injection limited recombination current model..... 151 7.7.2 Diffusion-limited recombination current model...... 154 7.7.3 Origin of the bi-exponential TPV decay in PSCs...... 156 7.7.4 Mobility dependence in the diffusion limited regime...... 156 7.7.5 Conclusions...... 157

8 New insights 159 8.1 Abstract...... 159 8.2 Bistable short circuit current...... 159 8.2.1 Introduction...... 159 8.2.2 Methods...... 160 8.2.2.1 Current Density Evolution (J-evo)...... 160 8.2.2.2 p-i-n Heterojunction simulation...... 160 8.2.3 Results...... 162 8.2.4 Simulation results and discussion...... 164 8.2.5 Conclusions...... 166 8.3 Simulating time-resolved photoluminescence measurements on perovskite bilayers...... 167 8.3.1 Introduction...... 167 8.3.2 Methods...... 168 8.3.3 Results...... 169 8.3.4 Conclusions...... 172

9 Conclusions 173 9.1 Outlook...... 176 9.1.1 Further development of DRIFTFUSION ...... 176 9.1.2 Large-scale studies...... 176 9.1.3 The effects of contact doping and dielectric properties in full device models...... 177 9.1.4 Defect levels and inter-bandgap states...... 177 9.1.5 Reinterpreting transient measurement techniques...... 177

Bibliography 179 Contents 15

Appendix A Full p-i-n device parameter set 1 210

Appendix B Full p-i-n device parameter set 2 211

Appendix C Full device parameter set 1 212

Appendix D Bilayer device parameter set 213

Appendix E Complementary results for Chapter5 214 E.1 Hysteresis in the Inverted-ZnO device...... 214 E.1.1 Fabrication of the Inverted-ZnO device stack...... 215 E.2 Ion enabled open circuit voltage enhancement...... 215

Appendix F Complementary results for Chapter6 217 F.1 Derivation of the ideality factor...... 217 F.2 Derivation of reaction orders...... 218 F.2.1 Band-to-band recombination...... 218 F.2.2 Shockley Read Hall recombination...... 218 F.2.3 Bulk SRH, Shallow traps...... 219 F.2.4 Bulk SRH, Mid-gap traps...... 219 F.2.5 Surface SRH, Shallow traps...... 219 F.2.6 Surface SRH, Mid-gap traps...... 219 F.3 Recombination scheme parameters...... 220 F.3.1 Band-to-band recombination...... 220 F.3.2 Surface SRH, Shallow traps...... 220 F.3.3 Surface SRH, Mid-gap traps...... 220 F.3.4 Bulk SRH, Shallow traps...... 220 F.3.5 Bulk SRH, Mid-gap traps...... 221 F.4 Complementary experimental data...... 222

Appendix G Complementary results for Chapter7 223 G.1 Thermionic emission numerical model...... 223 G.2 Complementary experimental results...... 225 G.3 Complementary simulation results...... 226

Appendix H Permission documents 227 List of Figures

1.1 Surface area required to power global energy consumption using solar PV 30 1.2 Average selling price of silicon, wafers, cells and modules...... 31

2.1 Conversion of photon to extracted energy of a solar cell...... 38 2.2 Charge carrier recombination mechanisms in semiconductors...... 39 2.3 Mechanisms of charge separation in different solar cell device architectures 40 2.4 Chemical structure of methylammonium lead iodide...... 41 2.5 Absorption coefficients of semiconductor materials commonly employed for photovoltaic applications...... 44 2.6 Temperature dependence of crystal Bravais lattice of methylammonium lead-based perovskites...... 45 2.7 Possible vacancy hopping routes and iodide vacancy activation energy calculation...... 48 2.8 Vacancy-mediated superoxide degradation mechanism...... 49 2.9 Examples of two-dimensional perovskite materials...... 51 2.10 Scanning Electron Microscope images of thermally annealed and solvent annealed CH3NH3PbI3 films...... 52 2.11 Common perovskite devices stacks...... 53 2.12 Energy levels for materials constituting Standard and Inverted PSC architectures...... 54 2.13 Current-voltage scan hysteresis and optical preconditioning independence of short circuit current density in PSCs...... 56 2.14 Long timescale transient measurements on perovskite devices...... 58 2.15 Step-dwell-probe (SDP) measurement on a c-TiO2 device as a function of dwell time...... 59 2.16 Simulated dark and illuminated electric potential profiles and J-V scan simulations...... 60

3.1 Simulation device schematics...... 65 3.2 Semiconductor energy levels...... 66 3.3 Short circuit equilibrium and illuminated p-i-n energy level diagrams and charge densities...... 68 3.4 Electron and hole carrier fluxes...... 69 3.5 Continuity of electronic charge in a one-dimensional system...... 72 3.6 Initial conditions used for p-i-n simulations compared to equilibrium state 75 3.7 Effect of high rates of recombination in a thin layer versus full contact region on open circuit voltage...... 78 3.8 Simulating interfacial recombination...... 79 3.9 Schematic of different recombination mechanisms in a HTL-i-ETL device 81

16 List of Figures 17

3.10 Examples of spatial and time meshes...... 83 3.11 Analytical and numerical solutions to a p-n junction...... 85 3.12 AM 1.5 solar spectrum and maximum theoretical short circuit current.. 86 3.13 Comparison of analytical vs. numerical current-voltage characteristics and open circuit voltage as a function of band gap for idealised p-n junctions 87

4.1 Transient optoelectronic measurement system circuit diagram...... 91

5.1 Experimental timelines for optoelectronic transient measurements..... 98 5.2 Measured and simulated device current-voltage characteristics...... 100 5.3 Transient measurements of the open circuit photovoltage evolution with +1 V precondition...... 102 5.4 Transient measurements of the open circuit photovoltage evolution with 0 V precondition...... 103 5.5 Step Dwell Probe (SDP) transient photocurrent measurements...... 104 5.6 Simulated energy level diagram and charge densities for a device at short circuit and space charge profiles at the device interfaces...... 105 5.7 Energy level diagrams and charge densities for simulated devices at open circuit...... 106 5.8 Simulations of TROTTR measurements on devices with and without surface recombination with 0 V precondition...... 107 5.9 Electrical and chemical contributions to the inverted photovoltage transients, electrostatic potential during J-V scan and simulated Step-Dwell-Probe energy level diagram and charge densities for an Inverted device...... 108 5.10 Simulated effects of varying ion defect density and contact recombination 109 5.11 Voltage transient simulations under different recombination schemes and bi-phasic slow transient VOC experimental measurements...... 110 5.12 Electrostatic potential and electron quasi Fermi levels in simulated devices during the current-voltage scan...... 112 5.13 Schematic of charge carrier distributions and recombination during J-V scan...... 113

6.1 Quasi Fermi level splitting and transient Suns-VOC experimental timeline 121 6.2 Temporal evolution the open circuit voltage and ideality factor in simulated and measured devices...... 126 6.3 Transient ideality and recombination rates in the simulated device as a function of time and pholotoluminescence decay in the mp-Al2O3 device. 128 6.4 Energy level diagrams and charge density profiles for open circuit voltage transient...... 129 6.5 Simulated and experimental transient ideality factor measurements following forward bias preconditioning...... 131 6.6 Example current-voltage scans for the three different architectures of device studied...... 132 6.7 Simulated nid and electron density profile for a device with a static ion distribution...... 134 6.8 Simulated nid and VOC transients for a device with high rates of band-to-band recombination in the contact regions...... 135 7.1 Transient photovoltage experimental timeline and excess charge relaxation after an excitation pulse for an intrinsic semiconductor..... 141 7.2 Example transient photovoltage and photocurrent measurements on a perovskite solar cell...... 143 7.3 Zero-dimensional theoretical prediction versus 1D numerical drift-diffusion model transient photovoltage...... 146 7.4 Current-voltage and transient ideality measurement for the c-TiO2 device used in the study...... 147 7.5 Simulated and experimentally measured transient photovoltage rate constants at steady-state as a function of bias light intensity and associated TPV decays...... 148 7.6 Charge carrier density vs. light intensity and TPV rate coefficient vs. carrier density in measured and simulated devices...... 149 7.7 Excess electron density profiles in a simulated device dominated by surface recombination following an excitation pulse...... 151 7.8 Schottky-like barrier and recombination profile at steady-state at the intrinsic/n-type interface...... 152 7.9 Simulated excess charge decay after a uniform generation pulse and example transient photovoltage measurements...... 155 7.10 Recombination rate coefficient extracted from transient photovoltage as a function of the input rate parameter and mobility in simulated devices dominated by surface recombination with fixed open circuit voltage.... 157

8.1 Short circuit current evolution (J-evo) experimental timeline and example current-voltage characteristics of measured device...... 161 8.2 Repeat short circuit current measurements for a c-TiO2 device...... 162 8.3 Current-voltage scans and short circuit current evolution (J-evo) for c-TiO2 and mp-TiO2 architecture devices...... 163 8.4 Measured and simulated Current Density Evolutions (J-evo)...... 165 8.5 Schematic of the simulated bilayer...... 168 8.6 Time-resolved photoluminescence (TRPL) simulations compared to published results...... 170 8.7 Time-resolved photoluminescence simulation energy level diagram, charge density profiles and excess charge decay profiles...... 171

E.1 Hysteresis in an Inverted cell using a ZnO electron collection layer instead of PCBM...... 214 E.2 Open circuit voltage enhancement in devices with zero built in field... 216

F.1 Measured and simulated evolution of the ideality factor with time following preconditioning at short circuit...... 222

G1 Thermionic emission excess electron charge density decay...... 224 G2 Transient photolvoltage measurements on c-TiO2 device at high light intensities...... 225 G3 Simulated transient photovoltage rate constants at Ref. and excess electron density as a function of bias light intensity for a device with bulk SRH recombination...... 226

18 List of Figures 19

List of Tables

2.1 Band gap, dielectric constant and exciton binding energies for perovskites compared to established photovoltaic materials...... 43 2.2 Diffusion lengths and mobilities for perovskites compared to established photovoltaic materials...... 46 2.3 Formation energies for different Schottky defects...... 47 2.4 Calculated ionic migration activation energies for common perovskite materials...... 47

3.1 Table of key values for analytical and numerical model current-voltage characteristics...... 87

6.1 Reaction order γ, carrier density relationship to QFL splitting β and conceptual ideality factor nid for different recombination mechanisms, trap energies and carrier population overlaps...... 122 6.2 Recombination mechanisms simulated in the study...... 125

A.1 Key device simulation parameters set 1...... 210

B.1 Key device simulation parameters set 2...... 211

C.1 Full device simulation parameters...... 212

D.1 Key bilayer simulation parameters...... 213

G1 Thermionic emission simulation parameters...... 224 Physical Constants

−5 −1 Boltzmann constant kB = 8.62 × 10 eV K Charge of an electron q = 1.60 × 10−19 C −31 Mass of an electron me = 9.11 × 10 kg −12 −3 −1 4 2 Permittivity of free space ε0 = 8.85 × 10 m kg s A Planck’s constant h = 6.63 × 10−34 m2kgs−1 Speed of light c = 3.00 × 108 ms−1

20 Abbreviations

General ASA Advanced Semiconductor Analysis DD Drift-Diffusion DFT Density Functional Theory GHG Greenhouse Gas HPC High Performance Computing PDEPE Partial Differential Equation solver for Parabolic and Elliptic equations SRH Shockley-Read-Hall

Devices CB Conduction Band DSSC Dye Sensitised Solar Cell eDOS Effective density of states EQE External Quantum Efficiency ETL Electron Transport Layer HTL Hole transport Layer OPV Organic Photovoltaic OSC Organic Small Molecule PCE Power Conversion Efficiency PSC Perovskite Solar Cell PV Photovoltaics QFL Quasi Fermi Level VB Valence Band

21 Abbreviations 22

Characterisation DAQ Data Acquisition Module CE Charge Extraction DCCE Differential Capacitance Charge Extraction FF Fill Factor GPIB General Purpose Interface Bus J-V Current-Voltage [Scan] J-evo Transient Current Density Evolution LED Light Emitting Diode NIR Near-Infrared HAXPES Hard X-ray Photoelectron Spectroscopy MOSFET Metal-Oxide-Semiconductor Field Effect Transistor PL Photoluminescence RC Resistor-Capacitor SCLC Space Charge Limited Current SDP Step-Dwell-Probe Photocurrent TROTTR Transients of the Transient Photovoltage TELTPV Time-Evolving Laser Transient Photovoltage [System] TRACER Transient and Charge Extraction Robot TRPL Time Resolved Photoluminescence VJL Voltage-Current-Light [Control Software]

Chemistry CdTe Cadmium telluride c-Si Crystalline silicon

c-TiO2 Compact TiO2 FA Formamidinium FAPI Formamidinium lead iodide FTO Fluorine-doped tin oxide GaAs Gallium arsenide ITO Indium tin oxide mc-Si Multicrystalline Silicon MA Methylammonium Abbreviations 23

MAPI Methylammonium lead iodide MAPBr Methylammonium lead bromide MAPCl Methylammonium lead chloride

mp-TiO2 Mesoporous TiO2 PCBM Phenyl-C61-butyric acid methyl ester SC Single Crystal Spiro-OMeTAD 2,2’,7,7’-Tetrakis[N,N-di(4-methoxyphenyl)amino]-9,9’ -spirobifluorene TF Thin Film Symbols

General

ηEQE External quantum efficiency

ηsep Charge separation quantum efficiency

ηcol Charge collection quantum efficiency

ηPCE Power conversion efficiency ρ Charge density cm−3 λ Photon wavelength nm φ Incident photon flux Sun eq. σ Conductivity S cm

σSD Standard deviation −2 −1 −1 bS Spectral photon flux cm eV s c Constant Variable units d Device thickness cm m Gradient of a slope

nid Diode ideality factor t Time s

vdrift Drift velocity cms-1 x Position cm C Capacitance F

Cgeo Geometric capacitance F

Cchem Chemical capacitance F E Energy eV

JSC Short circuit current density A cm−2

J0 Dark saturation current A cm−2

J0,rad Blackbody recombination current A cm−2

24 Symbols 25

P Power density W cm−2 Q Charge C T Temperature K

TS Blackbody temperature K −3 −1 U0 Blackbody recombination rate cm s

Vth Thermal voltage V

Material properties

αGT F Goldschmidt tolerance factor −1 αabs Absorption coefficient cm µ Mobility cm2V−1s−1

εr Relative permittivity ϕ Concentration cm−3 ρ Charge density C cm−3

ΦEA Electron affinity eV

ΦIP Ionisation potential eV D Diffusion coefficient cm2 s−1

EB Exciton binding energy eV

Modelling −2 −1 φ0 Incident photon flux cm s

τSRH SRH recombination lifetime s

ΦEl Electric potential energy eV

ΦChem Chemical potential energy eV

ΦE−C Electrochemical potential energy eV

Φano Anode workfunction eV

Φcath Cathode workfunction eV a Mobile ionic charge density cm−3 j Flux density cm−2 s−1 −3 −1 kbtb Band-to-band recombination rate coefficient cm s −1 kSRH SRH recombination rate coefficient s −3 n0 Equilibrium electron density cm n Electron carrier density cm−3 Symbols 26

−3 ni Intrinsic carrier density cm −3 nt SRH electron trap parameter cm p Hole carrier density cm−3 −3 pt SRH hole trap parameter cm s Surface recombination rate coefficient cm s−1

wn n-type depletion width cm

wp p-type depletion width cm E Electric field V cm−1

ECB Conduction band energy eV

EVB Valence band energy eV

Ei Intrinsic Fermi energy eV

EF0 Equilibrium Fermi level eV

EFn Electron quasi Fermi level eV

EFp Hole quasi Fermi level eV

Evac Vacuum energy eV

Eph Photon energy eV

Eg Bandgap eV

Et SRH trap energy eV J Current density A cm−2 V Electric potential V

Vapp Applied potential V

Vbi Built-in potential V

VOC Open circuit voltage V G Generation rate cm−3 s−1 S Continuity source term cm−3 s−1 U Recombination rate cm−3 s−1 −3 −1 Ubtb Band-to-band recombination rate cm s −3 −1 USRH SRH recombination rate cm s −3 Na Intrinsic Schottky defect density cm −3 NA Acceptor density cm −3 ND Donor density cm −3 N0 Effective density of states (generic) cm −3 NCB Conduction band effective density of states cm Symbols 27

−3 NVB Valence band effective density of states cm V Electric potential V

Transient methods

β Gradient of charge carrier density to ln(VOC) γ Reaction order τ Characteristic time constant s

τTPV TPV decay time constant s

Φb Schottky barrier energy eV −2 −2 AR Richardson constant A cm K

kα Rate coefficient for reaction order α Variable units

kcalc Calculated recombination rate coefficient Variable units

kinput Input recombination rate coefficient Variable units −1 kTPV TPV decay rate coefficient s k Recombination rate coefficient s−1

nid,FBi Initial ideality factor following forward bias −3 nCE Charge density from charge extraction cm −2 JTE Thermionic emission current A cm

Vpre Preconditioning voltage V

Vprobe Probe voltage V ∆n Excess electron carrier density cm−3 ∆p Excess hole carrier density cm−3

tdwell Dwell time s

tpulse Excitation pulse duration s ∆G Excess generation rate cm−3 s−1 28 Chapter 1

Introduction

1.1 Abstract

Anthropogenic carbon emissions are driving a climate crisis with catastrophic consequences for life on earth. Despite widespread consensus on the need to act, governments around the world appear paralysed by the enormity of the problem and the cultural context within which it exists. Given the dominance of the growth narrative, market competitive renewable energy technologies offer the potential to accelerate the transition to a low carbon economy. Of the existing technologies solar photovoltaics have the greatest potential globally. A new class of high-efficiency, solution-processable lead halide perovskite semiconductor materials are showing great potential for use in tandem photovoltaic architectures. Perovskites suffer from a number of instabilities, however, and the processes underlying the optoelectronic response of devices are not well understood. Transient characterisation techniques and numerical models are key tools in furthering our understanding of these unique materials.

1.2 Motivation

1.2.1 The climate crisis

At the time of writing, anthropogenic carbon emissions are driving a climate crisis with catastrophic consequences for life on earth. The predicted outcomes of maintaining ‘business as usual’ include wide-spread extinction of plant and animal species, global food shortages, the creation of millions of climate refugees and an increased likelihood of conflict.[1] In the most extreme scenarios, positive feedback mechanisms releasing

29 Chapter 1. Introduction 30

Figure 1.1: Surface area required to power global energy consumption using solar PV. 19 areas distributed according to ‘reasonable responsibility’ based on 2009 consumption statistics, with the Saharan dessert powering Europe and Northern African. 20% module efficiency, 2, 000 hours per year, 700 Wm−2, assumed. Adapted from Ref.[11] with kind permission from the Land Art Generator Initiative. trapped methane into the atmosphere could lead to run-away climate change, making the planet all but uninhabitable for humans.[2]

Despite campaigns of misinformation by the fossil fuel industry,[3] at the Paris Climate Accord in December 2015, governments around the world publicly acknowledged that anthropogenic greenhouse gas (GHG) emissions are responsible for the increase in average global temperature and extreme weather events.1 Nevertheless, many leaders still seem reluctant to implement the wide-scale change in infrastructure required to meet the 2 ◦C limit2 set by the Paris Agreement.[6] The combination of fossil fuel-driven economies and the hyper-consumerism of advanced Capitalist nations represents a perfect storm: the requirement for infinite economic growth is exceeding the limits of our finite planet.[5] Politicians appear impotent in the wake of powerful industry forces lobbying to maintain the status quo and the fear of negative public reaction should they pass the necessary legislation. As writer David Roberts puts it, we find ourselves ‘...stuck between the impossible and the unthinkable’.[7]

1.2.2 The potential for solar energy

Given the dominance of the growth narrative one hope is that cleaner renewable energy alternatives might be able to challenge fossil fuels in the market place. Of the currently

1With the notable exception of Donald Trump’s current Republican administration.[4] 2Increase in average global temperatures since pre-industrial times (1850 is taken as the baseline).[5] Chapter 1. Introduction 31

) 2.0 -1 Hyperpure silicon mc-Si wafer 1.5 mc-Si cell mc-Si module 1.0

0.5

0.0 Average selling price (US$ W (US$ price selling Average 2010 2011 2012 2013 2014 2015 Year

Figure 1.2: Average selling price of silicon, wafers, cells and modules. mc-Si indicates multi-crystalline silicon. Reprinted by permission from Macmillan Publishers Ltd: Nature Energy, Ref.[13], copyright, 2016. available renewable generation technologies, solar photovoltaics (PV) have the greatest potential for contributing to meeting global energy demand.[8] The amount of energy that could realistically be harvested from solar radiation has been estimated at 5 × 104 Exajoules (1 EJ = 1018 J), which is around 60 times greater than the total predicted global energy consumption for 2035 (782 EJ).[9] Given this value for consumption, an area approximately the size of Spain covered with 20% efficient PV modules could power the energy needs of the planet.[10] Figure 1.1 shows how this could be distributed globally according to ‘reasonable responsibility’ based on historic emissions.[11]

Notwithstanding, to achieve renewable generation on this scale will require not only a large scale-up in existing manufacturing capacity, but more efficient and larger transmission and storage systems.[12] The economic outlook for solar is promising however: The cost of existing polycrystalline silicon solar modules has dropped rapidly in recent years, from 1.9 US $ W−1 in 2010, to 0.32 US $ W−1 at the time of writing (see Figure 1.2).[13, 14] Much of this cost reduction has been realised by economies of manufacturing scale and local government incentives in China.[15] By the end of 2017, solar was expected to be at grid parity in 80% of countries across the globe.[16] As leading silicon researcher Martin Green announced at the first ever International Conference on Perovskite Solar Cells and Optoelectronics, by 2020 ‘...solar will have won’.[17] Chapter 1. Introduction 32

1.3 Lead halide perovskite solar cells

Lead halide perovskite solar cells (PSCs) have recently emerged as a promising new solution-processable photovoltaic technology. Research activity in the field has accelerated rapidly, due to ease of fabrication and high achievable power conversion efficiencies (PCE).[18] Single junction PCE figures sky-rocketed between 2012 and 2017 from 10.9% to the current record of 22.1%.[19–21] Given market confidence in the incumbent technology, the best opportunity for perovskites appears to be in higher efficiency tandem configurations, either perovskite-perovskite or perovskite-silicon devices.[22] McGehee and co-workers recently realised a 23.6% efficient two-terminal tandem perovskite-silicon device.[23] While this is only marginally higher than the current single junction perovskite record, the fundamental limit for a tandem architecture is significantly higher at 46.1% as compared to 33.7%.[24] Many researchers are now predicting that 30% lab-based devices are imminent.[22, 25]

A number of boundaries to commercialisation of the technology remain, however, including poor stability of the perovskite material in the presence of water vapour, light and oxygen, and the toxicity of lead.[26–29] In addition, a key focus of this work is the dependence of perovskite device performance on prior optical and electronic conditioning. This was first noted as an ‘anomalous hysteresis’ in the characteristic current-density voltage (J-V ) scan.[30] Slow changes in device photocurrent, photoluminescence intensity, and open circuit voltage (VOC) occurring on timescales up to hundreds of seconds have also been measured,[31–34] with magnitudes that tend to increase with ageing/degradation.[26, 35, 36] Understanding this mechanism and its effect on photovoltaic performance is critical for directing future perovskite solar cell research.[37–40]

1.4 A brief retrospective of the work

During my first few months of experimental work measuring perovskite devices, it became apparent that something was seriously wrong. Switching filters on the external quantum efficiency (EQE) set up caused the EQE to jump 10%;3 Crystals of methyl-ammonium lead bromide shined bright green like Kryptonite;4 and two consecutive current-voltage scans on the same pixel had completely different curves, apparently dependent on which way you happened to be scanning and whether

3I later realised that we had switched the light on whilst changing filters and inadvertently biased the device, improving its photovoltaic efficiency. 4In case there is any misunderstanding: I realise that Kryptonite is a fictional compound. Chapter 1. Introduction 33

Punxsutawney Phil had seen his shadow or not that afternoon.5 Logic and causality were in grave deficit. Unlike established solar technologies, perovskites appeared to come with a history and since history implies time, it seemed appropriate to use transient measurements as a way to better understand what exactly was going wrong.

This led to the first challenge: how to capture transient signals simultaneously on vastly different timescales - from nanoseconds to seconds. Slow moves the heavy, cumbersome ion, swift the nimble, dexterous electron. The second big challenge, and the focus of the majority of my PhD, was developing a simulation capable of providing some insight into all this mysterious behaviour. Knowing almost nothing about device simulation meant that the initial learning curve for writing the DRIFTFUSION simulation tool was incredibly steep. Each success and step forward, however, led not only to more accurate simulations but also advanced our research group’s understanding of the fundamental processes in PSCs. Along with the inclusion of mobile ionic charges, a key innovation in the model was the use of a mirrored cell to simulate the device at open circuit. Like a biblical vision, this solution presented itself to Dr. Piers Barnes as he stared at an image of a symmetric biological cell whilst conducting research into the structure of neurons. While he still contends that ‘...there are probably better ways to do it’, this functionality has allowed us to move into new simulation territory- to simulate what no one has simulated before. A recurring theme of this thesis is that zero-dimensional models can lead to incorrect interpretations of the data. I believe this statement extends well beyond the realm of perovskite solar cells: one-dimensional systems must be modelled as such if we are to have any hope of understanding them.

1.5 Scope and outline

The structure of this thesis mirrors that of a scientific journal paper: it has a beginning- Introduction, Background and Methods, a middle- Results and Discussion, and an end- Conclusions and Future Work.

We begin with Chapter2: Background by reviewing the profusion of recent literature and investigating some of the properties that make lead halide perovskites not only good materials for photovoltaics but also behave in unexpected ways electronically. While this chapter is intended to give the reader some general context, more specific background sections are included at the start of individual results chapters to enable for faster referencing of the relevant concepts. Chapter3: Device simulation describes the DRIFTFUSION simulation tool and the various models that were used to simulate PSCs.

5This is a reference to the film Groundhog Day: If Punxsutawney Phil, the groundhog sees his shadow and returns to his hole then that there will be six more weeks of winter.[41] Chapter 1. Introduction 34

Chapter4: Experimental methods gives technical details of the set ups used to make the experimental measurements included in the results chapters. Chapter5: Evidence for ion migration in perovskites solar cells with minimal hysteresis investigates the question of why some perovskite devices show hysteresis in their current-voltage scans while others do not. Particular importance is placed on the the presence or absence of localised recombination centres at interfaces. With the aid of the DRIFTFUSION simulation tool and established ideality theory I devise a way to assess the dominant recombination mechanism in perovskite devices in Chapter6: Identifying recombination mechanisms in perovskite solar cells using transient ideality factors. In Chapter7: Reinterpreting small perturbation transient photovoltage measurements in p-i-n devices, the scope of the simulation is extended from investigating individual cases to parameter spaces. I show that a zero-dimensional interpretation of transient optoelectronic measurements can lead to erroneous conclusions and propose two analytical models for interpreting the results based on the results from DRIFTFUSION. These ideas are further extended Chapter8: New insights where I discuss two new projects involving long timescale electronic measurements and simulating nanosecond time resolved photoluminescence measurements. I conclude the thesis by highlighting the key results and conclusions, and discuss some future possibilities for the DRIFTFUSION simulation tool. Chapter 1. Introduction 35

1.6 Publications realised from this work

Philip Calado, Andrew M Telford, Daniel Bryant, Xiaoe Li, Jenny Nelson, Brian C. O’Regan, and Piers R.F. Barnes. Evidence for ion migration in hybrid perovskite solar cells with minimal hysteresis. Nature Communications, 7, 2016.

Rebecca A. Belisle, William H. Nguyen, Andrea R. Bowring, Philip Calado, Xiaoe Li, Stuart J. C. Irvine, Michael D McGehee, Piers R. F. Barnes, and Brian C. O’Regan. Interpretation of inverted photocurrent transients in organic lead halide perovskite solar cells: proof of the field screening by mobile ions and determination of the space charge layer widths. Energy & Environmental Science, 2016. 36 Chapter 2

Background

2.1 Abstract

In this chapter the working principles of solar cells are briefly reviewed before the key material properties of CH3NH3PbI3 and its derivatives are discussed in detail. Lead halide perovskites are a class of easily processable semiconductor materials with many properties suited to photovoltaic applications including appropriate band gaps, high absorption coefficients and long carrier diffusion lengths. Perovskites also suffer from a number of material instabilities including reversible hydration upon exposure to water vapour, and irreversible degradation in the presence of light and oxygen. Calculations and experimental evidence indicate that high densities of mobile ionic charge are intrinsic to CH3NH3PbI3. These migrating ionic defects are hypothesised to play a critical role in irreversible chemical degradation via the formation of super oxides. Different architectures of perovskite solar cells (PSCs) and the role of selective contacts in devices are reviewed. Possible mechanisms accounting for the infamous current-voltage (J-V ) hysteresis are explored and mobile ions are found to be the most likely cause. Lastly, transient measurements and drift-diffusion simulations examining some of the anomalous optoelectronic behaviour of PSCs are reviewed.

2.2 Working principles of a solar cell

In this section the working principles of a solar cell will be introduced in brief. The reader is referred to Nelson 2003[42] and Wurfel 2016[43] for richer and more detailed texts.

37 Chapter 2. Background 38

1. Thermal relaxation

2. Charge separation 3. Charge collection ηsep η

Energy col Eg Recombination Extracted energy

Photon Absorption energy e ciency

Figure 2.1: Conversion of photon to extracted energy of a solar cell. Photons with energy greater than the semiconductor bandgap Eg are absorbed, promoting an electron from the valence band to the conduction band, leaving behind a positively-charged hole. Electrons with energy excess to the bandgap thermalise to the conduction band edge. Bound electrons and holes separate with an efficiency ηsep and are collected with an efficiency ηcol. At each stage there is a finite probability of recombination with an opposite carrier. Figure concept taken from Ref.[44].

A solar cell is a device that uses a semiconducting absorber material to convert incident light into electrical energy. In the absorber incoming photons, with energy greater than the material bandgap Eg, excite electrons from valence to conduction band energy states leaving behind positively charged holes.1 Figure 2.1 is a schematic representing the statistical energy losses for an electron-hole pair from generation to extraction.

The probability that an incident photon will be absorbed is defined by the wavelength-dependent extinction coefficient of the absorber material. Once absorbed, the pathway for a photon’s conversion into useful energy involves a number of processes that can reduce the overall quantum efficiency of the device:

1. Electrons and holes with excess energy thermalise to the conduction and valence band edges. Since this process is considerably faster than relaxation of carrier across the band gap, the solar cell is described as being at quasi-equilibrium under optical and electrical bias.

2. Electrons and holes may remain Coulombically bound as an exciton (as in most organic materials) or dissociate quickly to become free charges (as in many crystalline materials). If charges recombine before dissociation, the quantum

efficiency of charge separation ηsep is reduced.

1A hole is a quasi-particle that represents an empty electronic state in the valence band. Since there are generally many more electrons in the valence band than holes, it is far more convenient to track the generation, recombination and transport of holes than electrons in the valence band. Chapter 2. Background 39

3. Free carriers are transported to, and collected at the cell electrodes with a quantum

efficiency ηcol. This may also include energy losses arising from transport between materials with different energy levels.

At each stage of this process, there is a finite probability that electrons and holes will encounter one another and recombine i.e. the electron will transfer energy by some means such that it can occupy the empty lower energy state (hole).

2.2.1 Recombination

As shown in Figure 2.2, recombination processes are broadly divided into three types: 1. Radiative recombination in which a photon is emitted, 2. Auger recombination where the energy from one carrier is imparted to a second carrier via a collision, and 3. Trap-assisted non-radiative recombination2 via localised inter-bandgap states, originating from impurities or vacancies in the material.

In real devices recombination via trapping states is generally dominant and considered to be of most importance since careful material and device design can minimise these losses, whereas radiative and Auger recombination are unavoidable. Trap-assisted recombination is generally further subdivided into categories of bulk and surface recombination. Defect states in the band gap are often found at surfaces or interfaces where broken molecular bonds create a wide distribution of energetic states.

Radiative Auger Trap-assisted

Bulk Surface

Electron Hole Photon Phonon

Figure 2.2: Charge carrier recombination mechanisms in semiconductors.

2Since Auger recombination is generally considered negligible in perovskite devices, the terms ‘non-radiative’ and ‘trap-assisted’ recombination are used interchangeably throughout this work. Chapter 2. Background 40

2.2.2 Devices

In order to drive charge separation, some spatial asymmetry must exist in the energetics of the cell. Figure 2.3a shows the p-n junction architecture used in silicon solar cells. Here, the difference in equilibrium chemical potentials of the n and p-type regions gives rise to an exchange of carriers and associated development of space charge across the junction. Electrons and holes generated in the depletion region are thus swept in opposing directions by the electric field. Figure 2.3b shows an alternative method of charge separation, based on blocking contacts. Here the energetics of the Hole Transport Layer (HTL), absorber layer, and Electron Transport Layer (ETL) are chosen such as to allow current to flow only in a single direction. Thin film devices often incorporate both advantageous fields and blocking layers to maximise the efficiency of charge separation and extraction.

The most important material in a solar cell is that of the absorber3 layer. The optoelectronic properties of the absorber material define both the probability with which photons are converted into electron-hole pairs and the efficiency with which charge carriers are separated and collected. I now turn to focus on to some of the key properties that make lead halide perovskites excellent materials for solar energy conversion. a b

p-type n-type HTL Absorber ETL

Figure 2.3: Mechanisms of charge separation in different solar cell device architectures. (a) The chemical potential difference between the p and n-type layers of a p-n junction generates an electric field which drives charge carrier separation. (b) A ETL-i-HTL heterostructure device: Energetic barriers in the bands ensure that electrons and hole diffuse from the absorber to the Electron Transport Layer (ETL) and Hole Transport Layer (HTL) respectively. Symbol key for electrons and holes as in Figure 2.2.

3Also often referred to as the ‘active’ layer. Chapter 2. Background 41

2.3 The perovskite chemical structure

In 1839 a German mineralist, Gustav Rose discovered calcium titanate (CaTiO3) in the Ural mountains of Russia. Subsequently, the compound was named ‘perovskite’ after the Russian mineralist Count Lev Alekseevich von Perovski.[45] 90 years later in 1926, Victor Goldschmidt identified the perovskite crystal structure, an example of which is given in Figure 2.4. The term perovskite has since become a general name for any of a class of ionically-bonded crystals with the same ABX3 form as CaTiO3, where A and B are cations and X is an anion.[46]

Initial research into the application of perovskites as light absorbing materials for solar energy conversion centred around methylammonium lead iodide (CH3NH3PbI3, abbreviated as MAPI herein, see Figure 2.4).[49–51] More recently, high performing devices commonly substitute iodide with other halides such as chloride and bromide.[52, 53] Substitution of the organic methylammonium cation has also been a focus of research, particularly for tandem top cell applications for which the optimal silicon tandem perovskite band gap is 1.75 eV.[25] Band gap tuning over the range 1.48 - 2.23 eV has been accomplished by multiple groups using formamidinium (FA) and caesium (substitutes for cation A), and varying proportions of iodide to bromide (substitutes for anion X).[25, 54, 55] These substitutions alter the lattice constants of the material, resulting in a different set of electronic energy states.

Altering the atomic spacing of the crystal via ionic substitutions has also been found to influence material stability. The Goldschmidt Tolerance Factor αGT F parametrises the degree of distortion and in perovskite crystals and is defined as:[46]

Iodine (I -) Lead (Pb 2+ ) Carbon

+ Nitrogen Methylammonium (CH 3NH 3 ) Hydrogen

Figure 2.4: Chemical structure of methylammonium lead iodide. Perovskite crystal structure (cubic phase): The crystal is ionically bonded (oxidation states given in brackets). The methylammonium cation sits with the lead-iodide cages and is able to rotate.[47]. Image adapted from Ref.[48], Creative Commons license. Chapter 2. Background 42

rA + rX αGT F = √ 2(rB + rX ) where rA and rB are the ionic radii of the cations and rX is the ionic radius of the anion. CH3NH3PbI3 (MAPI) has αGT F = 0.83, at the lower end of the range of values for stable perovskites of 0.8 < αGT F < 1.0. Including a small amount of Cs (cation A) has been found to increase αGT F and stabilise the material to some degree owing to the smaller size of Cs with respect to MA.[56]

Recently, CsPbBr3-based devices demonstrated improved thermal stability as compared those with MAPbBr3 absorber materials.[57] Likewise, FA1−yCsyPb(I1−xBrx)3 has also shown improved stability over the established MAPI derivatives, albeit with a lower single-junction efficiency.[23] One leading expert recently voiced concern with this ‘dash to alloy’ however, since ‘... history tells us that for every element you add to a compound you add ten years to its research and development’.[58]

In an attempt to deal with the toxicity of lead, tin-based perovskites have been extensively investigated.[59, 60] The atmospheric stability of such materials has generally proven to be very poor, however, as Sn4+ is a more favourable oxidation state than Sn2+.[61] Furthermore, an in-depth toxicity study on tin perovskites found that while the decomposition of SnI resulted in stabilised Sn compounds, the HI formed as a result of the reaction makes ‘...Sn-containing perovskites more acutely toxic than Pb-containing perovskites’.[28]

2.3.1 Physical properties of CH3NH3PbI3

This section briefly reviews some of the key properties that define the optoelectronic response of perovskite solar cells. A more comprehensive analysis with useful references can be found in Leguy 2016.[62].

2.3.1.1 Optical properties

MAPI has a number of physical properties that make it a good material choice for solar energy conversion. MAPI has an optical band gap of around 1.6 eV (see Table 2.1) equating to an absorption edge at 775 nm.[63] This corresponds to a maximum theoretical efficiency of 31% using Detailed Balance theory,[64] close to maximum achievable theoretical efficiency for a single junction of 33 % (optimal band gap ≈ 1.4 eV [42]).[65] Chapter 2. Background 43

Material Band gap Ref. Dielectric Ref. Exciton binding Ref. (eV) constant energy (meV) MAPbI3 1.55 [63] 23 − 100 [66–69] 2 − 55 [70–73] MAPBr3 2.39 [71] 29 [66] 31 − 80 [67, 71, 74] FAPbI3 1.56 [71] 47 (SC) [75] 8 [71] c-Si 1.12 [76] 13 [77] 15† [78] CdTe 1.5 [79] 11 [80] 10 [81] GaAs 1.44 [82] 13 [80] 3 [82] OSC Wide range < 6 [83] > 200 [74, 84]

Table 2.1: Band gap, dielectric constant and exciton binding energies for perovskites compared to established photovoltaic materials. Room temperature values except for †c-Si binding energy recorded at 1.4 K. SC denotes single crystal. Data references obtained in part from Ref.[62].

The absorption coefficient αabs of a material is the exponential decay constant of the photon flux, φ according to the Beer-Lambert law:

φ = φ0 exp(−αabsx) (2.1)

where φ0 is the incident photon flux at the surface of the material and x is the position in the material in the direction parallel to the direction of light propagation. Beyond the sharp absorption edge, the absorption coefficient for MAPI rises from 104 to 105 cm−1 over the range 1.6 - 2.5 eV indicating that a high proportion of incident photons with energy greater than the bandgap would be absorbed in a 400 nm thick MAPI layer with a reflective back contact.[85, 86] This high absorption coefficient makes MAPI an appropriate material for thin film solar cells. Figure 2.5 shows how the absorption coefficient of MAPI compares favourably with other semiconductors commonly employed for photovoltaics. While these absorption properties indicate that MAPI acts like a direct band gap semiconductor, calculations have suggested that the conduction and valence band edges are split by a spin-orbit coupling effect known as Rashba splitting. This effect is thought to suppress radiative electron-hole recombination since direct transitions from conduction to valence band after thermalisation of carriers to the band edges are disallowed.[87] Chapter 2. Background 44

6 ] 10 -1 GaAs

10 5 CdTe

10 4 c-Si MAPI

10 3 P3HT-PCBM 10 2 Absorption coecient [cm Absorption coecient 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Energy [eV]

Figure 2.5: Absorption coefficients of semiconductor materials commonly employed for photovoltaic applications. Data sources: MAPI (CH3NH3PbI3), Ref.[88]; Crystalline silicon (c-Si), Ref.[89]; Gallium arsenide (GaAs), Ref.[90]; Cadmium telluride (CdTe), Ref.[91]; P3HT-PCBM blend, Ref.[92].

2.3.1.2 Phases

At room temperature, CH3NH3PbI3 forms a tetragonal phase, while CH3NH3PbCl3 and CH3NH3PbBr3 form cubic phases. Figure 2.6a summarises the phase transitions for these three materials. As shown in Figure 2.6b, the phase transition from tetragonal to cubic is continuous over the temperature range 161 - 330 K. It has been noted however, that the photovoltaic properties of CH3NH3PbI3 remain largely unaltered in this range.[93, 94]

Molecular dynamics simulations of a MAPI supercell by Frost show that while at 300 K the average structure appears pseudo-cubic, the lattice is incredibly mobile and locally disordered.[95] The fluid nature of the perovskite lattice with its multiple dipoles and degrees of freedom results in a relatively high static dielectric constant, calculations of which range from 5 to 33 (see Table 2.1).[96] In combination with a low effective mass ∗ of m ≈ 0.1me, where me is the mass of an electron,[73] the extent to which excited carriers feel one another is limited and low excitonic binding energies (EB < 80 meV) have been measured in MAPI, MAPBr and FAPI (see Table 2.1). Recent calculations have suggested that long timescale, room temperature binding energies in MAPI are in fact considerably less than the characteristic thermal energy kBT ≈ 26 meV.[96] Table 2.1 shows that the lower estimates are of a similar order as high performance thin film materials GaAs and CdTe, and much lower than solution-processable organic semiconductors. Notwithstanding, even the higher bound experimental values suggest Chapter 2. Background 45

a CH NH PbI Ortho. (< 161 K) Tetra. (161 K - 330 K) Cubic 3 3 3 (> 330 K)

Tetra. (172 K - 177 K)

CH 3NH 3PbCl 3 Ortho. (< 172 K) Cubic (> 177 K)

CH NH PbBr Ortho. (< 149 K) Tetra. Cubic (> 263 K) 3 3 3 (149 K - 236 K) 0 K 100 K 200 K 300 K 400 K

b 350 Cubic 300

250 Tetra. 200

150 Temperature [K] Temperature Ortho. 70 75 80 85 2 Theta [degrees]

Figure 2.6: Temperature dependence of crystal Bravais lattice of methylammonium lead-based perovskites. (a) Phases of CH3NH3PbI3, CH3NH3PbCl3 and CH3NH3PbBr3. Data from Ref.[66]. (b) Neutron diffraction data from powdered CH3NH3PbI3 as a function of temperature between 100 and 352 K. Bright spots indicate peak counts. Adapted from Ref.[93] - Published by and used with kind permission of The Royal Society of Chemistry. The three crystal phases, orthorhombic (Ortho.), tetragonal (Tetra.) and cubic are shown on the right. Adapted with permission from Ref.[69]. Copyright 2017 American Chemical Society. Chapter 2. Background 46 that at room temperature and 1 Sun photoexcitation intensities, a high proportion of electrons and holes exist as free charges.[72]

Mobilities too are unusually high for a solution processable material: long range values of 20 to 66 cm2V−1s−1 have been measured using Hall Effect and photoconductivity techniques in MAPI (see Table 2.2). MAPBr and FAPI have both shown similarly impressive transport properties, comparing favourably with the wide ranging but generally low mobilities found in organic semiconductors (OSC). Importantly, the √ diffusion lengths (LD = Dτ, where D is the diffusion coefficient and τ is the average carrier lifetime) for high quality perovskite single crystals (SC) are significantly longer than the characteristic absorption length (< 500 nm, see Figure 2.5) allowing charges to be extracted in films sufficiently thick to absorb a high proportion of the incident photon flux.

Material Diffusion length Ref. Mobility Ref. (µm) (cm2V−1s−1) MAPbI3 0.1(TF), 2 − 8(SC) [97, 98] 20 − 66 [99, 100] MAPBr3 3 − 17(SC) [98] 20 − 60(SC) [98] FAPbI3 19(SC) [101] 1 − 4(SC) [75] c-Si > 100 [102, 103] > 125 [103, 104] CdTe 2 [105] 600 [106] GaAs 10 [105] 800 [107] OSC < 0.02 [108] 10−5 − 10−1 [108, 109]

Table 2.2: Diffusion lengths and mobilities for perovskites compared to established photovoltaic materials. Perovskite compounds compared to cadmium telluride (CdTe), gallium arsenide (GaAs), crystalline silicon (c-Si), and organic semiconductors (OSC). SC and TF indicate measurements on single crystals and thin films respectively. Data references obtained in part from Ref.[62].

2.3.2 Defects and mobile ions

Prior to the new wave of perovskite research generated by the interest in CH3NH3PbI3, related perovskite materials such as CsPbI3, CsPbBr3, MAGeCl3 and MASnCl3 had already been identified as mixed ionic/electronic conductors.[110] After Snaith et al. published on the current-voltage hysteresis in early 2014,[30] (see Section 2.5) multiple publications followed citing ionic migration as the probable cause.[34, 110–112] More direct evidence was published by Xiao et al., showing that the direction of photocurrent in devices with energetically-symmetric contact materials could be switched by changing the polarity of an applied preconditioning voltage. Furthermore, degradation of the material was apparent after biasing for 110 mins, an effect attributed to the accumulation of mobile ionic vacancies at the electrodes.[37] Further evidence was provided by Yang and co-workers who fabricated a Pb/MAPI/AgI/Ag Coloumetric Chapter 2. Background 47 cell with an ionic electrode made from AgI/Ag. A current flowing through the cell induced a reaction at the electrode indicating that iodide was migrating through the

MAPI layer.[110] Leijtens et al. similarly identified PbI2 accumulation at the electrodes of lateral devices after biasing using Raman spectroscopy. This was interpreted as evidence of methylammonium ions moving through the material.[113].

These experimental findings have been substantiated by formation energies calculations of charge neutral Schottky defects by Walsh et al.[114] Table 2.3 summarises the results from the study.

−3 Label Reaction ∆HS [eV] Nion[cm ] 0 00 ◦ 19 (1) nil→ VMA + VPb + 3VI + MAPbI3 0.14 2 × 10 0 ◦ 20 (2) nil→ VMA + VI + MAI 0.08 2 × 10 00 ◦ 17 (3) nil→ VPb + 2VI + PbI2 0.22 8 × 10

Table 2.3: Formation energies for different Schottky defects. The reactions 0 are described using Kr¨ogerVink notation: V represents a vacancy, signifies a net ◦ negative charge and signifies a net positive charge. ∆HS is the enthalpy of formation, KC is the calculated fraction of vacant lattice sites and Nion is the density of intrinsic defects at room temperature. By way of context the number density of Pb lattice sites, 21 −3 22 NP b = 4.04×10 cm , while the density of MA and I sites is NMA = NI = 1.21×10 cm−3. Data reproduced from Ref.[114].

These intrinsic defect density values of > 1019 cm−3 are far in excess of experimentally measured electronic charge carrier densities of 1014 − 1015 cm−3 under 1 Sun illumination,[31, 115] implying that fields within the perovskite are predominantly determined by ionic charges under photovoltaic operation.

0 00 ◦ Ref. Material VA VB VX [48] MAPbI3 0.88 2.31 0.58 [116] MAPbI3 0.46 0.80 0.08 [117] MAPbI3 0.57 - 0.32 [118] MAPbI3 0.70 - 1.12 1.39 - 1.78 0.28 - 0.45 [119] MAPbI3 0.74 - 0.34 [118] MAPbBr3 0.70 - 1.13 0.16 - 0.29 0.20 - 0.46 [119] MAPbBr3 0.80 - 0.33 [118] CsPbI3 0.59 - 1.16 0.81 - 0.99 0.29 - 0.36 [117] FAPbI3 0.56 - 0.18

Table 2.4: Calculated ionic migration activation energies for common perovskite materials. A is the central cation within the cage e.g. MA−, B is the anion e.g Pb2+, and X is the second cation e.g. I−. See Figure 2.4 for structure. Data format and references reproduced from Ref.[62]. All energies in eV.

Results from numerous computational studies (Table 2.4), indicate that ionic defects are highly mobile in MAPI and its derivatives at room temperature. Figure 2.7 shows lowest energy migration routes for iodide, lead and methylammonium vacancies for one such calculation by Eames et al.[48] Experimental values of activation energies from Chapter 2. Background 48

a c

b

Figure 2.7: Possible vacancy hopping routes and iodide vacancy activation energy calculation. Vacancy hopping routes to neighbouring positions for (a) Iodide and lead and (b) methylammonium. (c) 3-D representation of an iodide vacancy hopping to an adjacent site. Reproduced from Ref.[48], Creative commons license. transient photocurrent and conductivity measurements of between 0.23 and 0.68 eV also correlate well with the range of values calculated for iodide and methylammonium vacancy migration.[27, 48, 120, 121] Using an attempt frequency of 1012 Hz, Eames et al. estimated diffusion coefficients4 for iodide and methylammonium vacancies in MAPI to be on the order of 10−12 and 10−16 cm2s−1 respectively.[48] Based on this estimate, and a typical electric field strength of E ≈ 104 Vcm−1, an iodide vacancy would be expected to drift across a 500 nm layer in around 65 seconds.5 This value correlates well with the timescales on which slow optoelectronic response in perovskite devices have been measured (seconds to tens-of-seconds).[31, 32, 122]

2.3.3 Inter-bandgap states

Typically high defect densities are associated with high rates of non-radiative recombination in semiconductor materials since states within the band gap are easily accessible to electrons via interactions with phonons. First principles density function theory (DFT) calculations on MAPI have shown only shallow states for defects with

4 The diffusion coefficient D can be calculated using D = ν∆x exp(−EA/kBT ), where ν is the migration attempt frequency, ∆x is the hopping distance, EA is the activation energy for a hop, kB is the Boltzmann constant and T is the temperature. 5 The average drift velocity of an electron, vdrift is given by the product of the mobility µe and electric field strength E: vdrift = µeE. Chapter 2. Background 49 low activation energies however.[123, 124] These findings lie in contrast to experimental results by two groups using hard X-ray photo-electron spectroscopy (HAXPES), in which a uniform distribution of mid-gap states in perovskite films and bilayers with

TiO2 have been measured.[100, 125] In addition, the MAPI absorption spectrum Urbach tail, and increase in photoluminescence quantum efficiency with excitation intensity, have also been interpreted as evidence of a band tail.[85, 126] A wide range of trap densities of 108 - 1015 cm−3 have been estimated using space-charge limited current (SCLC) and deep-level transient spectroscopy measurements, although these studies did not take into account how mobile ions might influence the extracted current.[127–129] Ball & Petrozza have suggested that some growth conditions would enable the formation of higher activation energy defects states sufficient to produce ‘... a significant density of non-radiative recombination centres.’.[86] Nonetheless, long carrier diffusion lengths for lead halide perovskites (see Table 2.2) indicate that non-radiative recombination at high carrier densities is relatively low.[97, 101] These long diffusion lengths under illumination are key to enabling perovskite devices to remain efficient at steady-state, even after mobile ions have screened potentially advantageous electric fields in the absorber layer.[32]

2.3.4 Chemical stability

Ionic mobility in perovskites may not just be a problem for stabilised photovoltaic performance (see Chapter5): iodide vacancy migration has also recently been linked to degradation. Haque and co-workers have proposed that the dominant degradation − route in MAPI is via vacancy-mediated superoxide (O2 ) formation.[130] The proposed mechanism is depicted in Figure 2.8 and composed of a number of steps:

1. 2. 3. 4. h+ e- h+ e-

I- 2+ - Pb e O2 - + + I vacancy CH 3NH 3 h H2O

Figure 2.8: Vacancy-mediated superoxide degradation mechanism. Concept for the figure taken from Ref.[130]. Relative atomic size not to scale. Chapter 2. Background 50

1. Oxygen occupies an energetically favourable vacancy site.

2. An electron and hole pair are formed following optical excitation.

− 3. The electron transfers to the oxygen forming superoxide (O2 ) and a CH3NH3PbI3* radical.

4. The superoxide reacts with CH3NH3PbI3* to form PbI2,I2,H2O and CH3NH2.

Assuming that vacancy-mediated superoxide degradation is the dominant pathway in MAPI, migration of iodide vacancies, as opposed to methylammonium ions, could account for the accumulation of iodide at interfaces in lateral devices and electrochemical cells after biasing described in Subsection 2.3.2 above.[110, 113] Certainly, previous studies by the Haque group have established strong links between the presence of oxygen, superoxide formation and degradation of CH3NH3PbI3.[29, 131] The density of photo-generated charges in the material has also been related to the rate of material breakdown, supporting this viewpoint.[132] Furthermore, CH3NH2 has been identified as a degradation product in two separate thermal degradation studies consistent with this process.[133, 134] In other studies however, CH3I and NH3 were found to be the major product gases and alternate thermal degradation pathways were proposed.[135, 136]

Work by the McGehee group has demonstrated that encapsulation may be a route to mitigating thermodynamic instability: they hypothesised that if reaction products are confined to the device then an equilibrium between sublimation and re-inclusion will arise.[94, 137] By sputtering an impermeable ITO top contact they demonstrated that the initial PCE of MAPI devices could be maintained for 250 hours under continuous illumination at maximum power point. Devices tested at 100◦C, however, degraded to 80% PCE in only 150 hours, implying that ITO alone is not an adequate barrier to oxygen diffusion.[137] Encapsulation using a photo-curable polymer has also proven successful in extending device lifetime: Bella et al. found that encapsulated devices showed markedly improved stability compared to their unencapsulated counterparts whilst stored in argon atmosphere over 6 months, supporting the partial pressure hypothesis proposed by McGehee.[138]

Attempts have been made to chemically stabilise MAPI via the application of additives and cross-linkers.[36, 139] ‘Two-dimensional’ perovskites, where ribbons of perovskite are separated by long organic molecules forming self-assembled layers (Figure 2.9) have also shown improved stability compared to the conventional materials, albeit at the expensive of photovoltaic performance.[56, 140–143] Chapter 2. Background 51

Figure 2.9: Examples of two-dimensional perovskite materials. Reprinted by permission from Macmillan Publishers Ltd: Nature Ref.[142], Copyright 2016.

Whilst oxygen and electronic excitation lead to the permanent decomposition of MAPI, Leguy et al. have shown that exposure to water vapour forms reversible hydrated phases.[26] Notwithstanding, device performance was found to be permanently affected after hydration-dehydration cycling, an observation that was interpreted as evidence of recrystallisation of the perovskite layer.[26]

Despite progress being made on many fronts, the research community as a whole appears prone to exaggeration with respect to device stability.[36, 94, 144–146] In one recent example, Grancini et al. claimed ‘One-Year stable perovskite solar cells...’ under ambient conditions using a 2-D/3-D organic junction, yet devices were kept at short circuit between current-voltage scans, reducing the density of photo-generated carriers and associated degradation rates as compared with maximum power point tracking.[132, 146] Other studies demonstrate stability only under inert atmosphere.[138, 147] The community will need to agree upon standardised test protocols under realistic operating conditions to properly monitor advances in device stability.[148]

Martin Green has speculated that the stability of perovskites will need to equal or improve on current silicon module degradation rates of ≈ 0.1% decrease in PCE per year.[17] At the time of writing it is undetermined as to whether these levels can be achieved with the existing MAPI derivatives. New materials systems and improved encapsulation standards may be required before perovskites can become a commercial solar technology. Chapter 2. Background 52

2.4 Perovskite devices

2.4.1 Material processing

Alongside their excellent photovoltaic properties, one of the major appeals of perovskites is the ease with which they can be processed. The precursors for multi-crystalline perovskite films are commonly deposited as a single solution (one-step) or in two stages with the PbI2 solution deposited prior to the addition of an MAI solution (two-step) to produce CH3NH3PbI3 for example.[149–151] Where greater control is required, vacuum co-evaporation of precursor materials can also be used to produce high quality films.[68, 152] The choice of processing method influences both the nano-crystallinity and interfacial roughness of the material, two properties which have been linked to device performance.[68, 153, 154] Nie et al. demonstrated a correlation between crystal grain size and power conversion efficiency (PCE) by depositing the perovskite solution at high temperature, creating millimetre-sized crystal grains.[150] The performance improvements were attributed to reduced recombination via defect states at grain boundaries. Furthermore, solvent annealing has been demonstrated to increase grain size significantly (Figure 2.10),[155] a property that has also been linked to stability: Aristdou and co-workers observed that superoxide formation is slower in films with larger crystallites.[130] a b Thermally Annealed Solvent Annealed

Figure 2.10: Scanning Electron Microscope images of thermally annealed and solvent annealed CH3NH3PbI3 films. Scanning Electron Microscope (SEM) micrographs of (a) thermally annealed and (b) solvent annealed CH3NH3PbI3 films. x50k magnification. Adapted from Ref.[155], Creative Commons license. Chapter 2. Background 53

2.4.2 Architectures

The first PSCs were derived from liquid electrolyte Dye-Sensitised Solar Cells (DSSCs) with the perovskite acting as a sensitiser for the mesoporous TiO2 scaffold.[49, 50] Major breakthroughs in the development of current PSCs included the move from a liquid electrolyte to a solid-state hole transporting material (commonly Spiro-OMeTAD6) and the realisation that a mesoporous TiO2 (mp-TiO2) layer was not necessary for electron transport: an inert Al2O3 scaffold could produce devices with comparable performance.[19, 156, 157] Notwithstanding, architectures including mp-TiO2 remain common in the literature and devices exceeding 20% efficiency have been manufacture both with and without the mesoporous layer.[20, 152, 158, 159] The standard 7 orientation planar, compact TiO2 (c-TiO2) and mesoporous TiO2 (mp-TiO2) device stacks are shown in Figures 2.11a and 2.11b respectively.

a Gold Spiro OMeTAD Planar Perovskite

base-TiO2 FTO

b Gold Spiro OMeTAD Perovskite Mesoporous mp-TiO2 or mp-Al2O3

base-TiO2 FTO

c Gold PCBM Inverted Perovskite

PEDOT:PSS ITO

Figure 2.11: Common perovskite devices stacks.(a) Standard orientation planar FTO/TiO2/CH3NH3PbI3/Spiro-OMeTAD/Ag device stack. (b) Mesoporous FTO/TiO2/mp-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Ag device stack. The mp-TiO2 is sometimes substituted for an inert Al2O3 scaffold in control experiments. (c) Inverted (top cathode) ITO/PEDOT:PSS/CH3NH3PbI3/PCBM/Ag architecture.

62,2’,7,7’-Tetrakis[N,N-di(4-methoxyphenyl)amino]-9,9’-spirobifluorene 7Devices are built from the glass layer upwards. I use the term Standard to distinguish devices with the cathode located at the bottom of the stack, as distinct from Inverted architectures in which the anode is at the bottom of the stack. I note that this is the opposite convention from organic photovoltaics. Chapter 2. Background 54

a -1.9 b

SPIRO- e- e- OMeTAD -3.0 PEDOT: -3.7 -3.7 -3.95 -4.1 PSS -4.7 -4.7 -4.9 MAPI -(4.9) MAPI PCBM -4.7 (-4.6) -4.9 -5.2 Ag Ag -5.5 FTO TiO 2 -5.3 ITO

-5.85 -7.4 h+ h+

Figure 2.12: Energy levels for materials constituting Standard and Inverted PSC architectures. Values for the various materials constituting perovskite device stacks measured in isolation using Kelvin probe and UV photoemission spectroscopy: (a) Standard orientation FTO/TiO2/CH3NH3PbI3/Spiro-OMeTAD/Ag architecture. Values taken from Ref.[163]. Fermi energies (where available) are marked with a dashed line and bracketed values - note the small built in potential of 0.3 eV between electron and hole transport layers. (b) Inverted (top cathode) ITO/PEDOT:PSS/CH3NH3PbI3/PCBM/Ag architecture. Values taken from Ref.[52]. All values in eV.

The precise function of the mp-TiO2 scaffold remains unclear although mesoporous devices may operate in a similar manner to bulk heterojunctions with the perovskite effectively acting as a donor material to the TiO2 acceptor.[160] Indeed, efficient bulk heterojunction devices have been manufactured using a mixed phase of MAPI and PCBM.8[161] Further developments led to the introduction of ‘Inverted’ device architecture, which, using an established organic photovoltaic (OPV) architect, were found to be ‘hysteresis-free’ with respect to their current density-voltage scans(Figure 2.11c).[153, 162] The absence of hysteresis in Inverted architectures is explored in-depth in Chapter5.

2.4.3 Contact selectivity

Figure 2.12 shows energy level diagrams for both Standard ETL-i-HTL and Inverted HTL-i-ETL device architectures, where the ‘i’ layer in an approximately intrinsic absorber material.9

Notable is the apparent low built-in voltage in both architectures (Figure 2.12a). Given the high intrinsic mobile defect densities in the MAPI layer it is expected that the field in the perovskite layer is zero during steady-state operation (see Chapters5-7). In this

8Phenyl-C61-butyric acid methyl ester 9The order is given imagining the transparent electrode to be at the left of the list of layers since this is commonly the order in which the layers are deposited. Chapter 2. Background 55 respect, the built-in field of the device, generated by the difference in Fermi energies of the contact materials, does not appear to be important to efficient charge separation and collection (see Section 2.2.2).[32] Critical, however, are well-passivated blocking contacts that prevent minority carriers from reaching their counter electrodes, where they can easily recombine with majority carriers. Indeed, Zhang et al. showed that cells can only operate efficiently with both electron and hole transporting layers.[112] By fabricating devices with only a single contact layer they found that while a photocurrent could be sustained temporarily, stabilised power outputs were close to zero.[112] The initial photocurrent was hypothesised to result from mobile ions producing a doping gradient. A similar analysis would suggest that the temporary electric field created by the ions could be responsible for the same effect i.e. that the origin of the Fermi level gradient was electrostatic in origin rather than chemical.[122] Transient electric fields in the absorber layer are explored further in Chapter5.

2.5 Current-voltage hysteresis

Current density-voltage (J-V ) scans are the established method for determining the performance characteristics and maximum achievable power conversion efficiency (PCE,

ηPCE) of solar cells. In brief, an applied voltage is scanned across the device at a given rate and the current is measured. ηPCE is then calculated from the ratio of the maximum power point, Pmax on the J-V curve to the incident optical power Pin:

JmaxVmax Pmax ηPCE = = (2.2) Pin Pin where Jmax and Vmax are the current density and voltage at the maximum power point.

An alternative definition of ηPCE uses the concept of the fill factor FF , defined as the ratio of the maximum power point to the product of the open circuit voltage, VOC with the short circuit current density JSC:

J V FF = max max (2.3) JSCVOC

VOCJSCFF ηPCE = (2.4) Pin

Early reports of PSC record efficiencies neglected to note that perovskite J-V characteristics appeared to depend on the direction in which the device was scanned (see Figure 2.13a for example curve).[164, 165] Subsequent investigation revealed Chapter 2. Background 56 a b ) -2 -20

-10

0

10

Currentdensity (mAcm 0 20 40 60 Time (s)

Figure 2.13: Current-voltage scan hysteresis and optical preconditioning independence of short circuit current density in PSCs. (a) The first report of the current voltage scan in PSCs, adapted with permission from Ref.[30]. Copyright 2014 American Chemical Society. (b) Optical preconditioning makes little difference to the short circuit current density following forward biasing at 0.9 V for 40 s. Inset shows the initial 4 s of preconditioning step. Reprinted with permission from Ref.[31] Copyright 2015 American Chemical Society. that the degree of observed hysteresis is dependent on a number of measurement parameters, including device preconditioning, scan protocol and scan rate.[30, 32, 34] The leading model to explain J-V hysteresis (known herein as ‘hysteresis’) is that charge accumulated at the MAPI interfaces reduces or entirely screens the internal electric field, resulting in increased recombination and loss of photocurrent.[32, 37, 48, 110, 118, 166] Kelvin Force Probe Microscopy, scanning the potential across a device cross-section, has revealed that field screening is indeed present in devices.[167] Migration of ionic defects in the perovskite phase, ferroelectric polarisation, and trapping of electrons at the interfaces have all been suggested as origins for the accumulation of this charge.[30]

2.5.1 Possible origins of field screening

In DSSCs, electronic transport is mediated by a tail of trap states in the TiO2 frequently modelled as an exponential distribution.[168] It has been proposed that the hysteresis could be attributed to the slow release of electronic charge from deep trap states.[169] Tress et al. showed, however, that an unrealistically high density of trapped surface charge (≈ 30 mC cm−2) would be required for electronic capacitance alone to account for hysteresis. The severity of the hysteresis was further demonstrated to be independent of bias light intensity implying that if hysteresis could be attributed to a capacitive effect, the capacitance would somehow need to scale linearly with light intensity.[32] Chapter 2. Background 57

Juarez-Perez et al. attributed the ‘giant dielectric constant’10 measured at low frequencies in MAPI to the reorientation of ferroelectric domains.[33, 69] Beilsten-Edmands and co-workers instead argued that the ‘hysteretic charge’ was better described by ionic displacement given that low-frequency conductance of the material was over an order of magnitude greater than that expected for a ferroelectric.[170] Furthermore, established timescales for ferroelectric domain reorientation at room temperature of between 0.1 to 1 ms are far too fast to account for the long electronic relaxation times measured in PSCs.[110, 112]

Numerous authors have noted that light appears to induce hysteresis effects.[34, 171] Contrary to this, O’Regan et al. showed that the photocurrent decay in devices following forward bias preconditioning is largely unaltered by different optical preconditioning (see Figure 2.13b).[31] Conversely, improved performance after being held at short circuit under optical bias has been observed. One plausible explanation is that finite series resistance and an associated charge build-up at the electrodes, mimics an electrically applied bias,[31] an effect explored in Chapter8, Section 8.2.

While all of the processes outlined above may contribute in varying degrees to the anomalous optoelectronic response of PSCs, there is general consensus in the research community that ionic migration in the perovskite phase is primarily responsible for hysteresis.[48, 111, 112, 116, 118, 121, 122] A detailed description of the mechanisms underlying hysteresis is given in Chapter5.

2.6 Transient optoelectronic measurements

Modern optoelectronic transient measurements were primarily developed to characterise DSSCs and have, more recently, been applied to organic photovoltaics.[172, 173] These techniques generally involve perturbing the system with either voltage or light pulses and measuring the electronic response either in the time or frequency domain. A comprehensive review of their theory and application to DSSCs can be found in Barnes 2013.[168]

O’Regan et al. applied a number of DSSC transient techniques to a comprehensive study of PSCs.[31, 168] Using two different charge extraction (CE) techniques combined with transient photovoltage (TPV) measurements, they showed that a larger than reasonable recombination flux could be derived using long timescale CE. The additional charge was attributed to either mobile ions or moving dipoles generating a displacement current.

10Given that the origin of the field screening at low frequencies is likely to be mobile ionic charge, the use of dielectric terminology here is questionable. It also has nothing to do with Giants- it’s just a large value. Chapter 2. Background 58 a b

Figure 2.14: Long timescale transient measurements on perovskite devices. (a) Slow transient photocurrent decay after the voltage was stepped from +1 V to −0.5 V. The delay between the voltage step at time t = 0 s and the illumination switch on time (shown in the legend) was varied showing that the slow process appears to be independent of illumination. (b) Photocurrent transients on a c-TiO2 device showing the development of an overshoot characteristic of diffusion -driven transport after switching from +1 V to −1 V. Reproduced from Ref.[32] with permission from The Royal Society of Chemistry.

The study highlighted the difficulty in decoupling ionic, electronic and dielectric (bound) charge responses, but also demonstrated that time domain measurements could be an effective tool for probing slow transient behaviour in PSCs.

Tress et al. used long timescale photocurrent transients in mp-TiO2 PSCs to show that the slow process responsible for hysteresis is independent of illumination (Figure 2.14a). In the same work, by analysing the shape of photocurrent transients at increasing delay times after switching voltages, they showed compelling evidence that devices slowly shift from being drift to diffusion driven (Figure 2.14b).[32] Photocurrent transients with low frequency voltage modulation were also the subject of an investigation by Shi and co-workers.[174] They interpreted evolving inverted photocurrent transients as evidence of changes in the internal electric field in devices. Two of our own works have used similar techniques to track the evolving field and understand hysteresis phenomena in different device architectures (Figure 2.15).[122, 175]

Many questions remain regarding the appropriate application and interpretation of transient measurements to perovskite devices. Some particular cases are addressed in Chapters7 and8. Chapter 2. Background 59

a b

Figure 2.15: Step-dwell-probe (SDP) measurement on a c-TiO2 device as a function of dwell time. (a) Photocurrent transients show a change in polarity with increasing dwell time following a jump from short circuit to 0.8 V. (b) The magnitude of the photocurrent transient at 6µs (Jtr − 6µs) as a function of dwell time is used to characterise the strength and direction of the electric field in the device. Reproduced from Ref.[122] - Published by The Royal Society of Chemistry. Creative Commons licence.

2.7 Device simulation

Numerical drift-diffusion simulations have a long history of application in modelling solar cells, from early analyses of p-n junctions to modelling transients in organic heterojunctions and DSSCs.[176–180] Due to the numerical challenge of solving the drift-diffusion equations for three separate charge carriers (electrons, holes and a mobile ionic species) with vastly different transport rates and high interfacial carrier gradients, initial drift-diffusion simulations of PSCs tended to neglect ionic charge altogether.[150, 181, 182] Progress was made in simulating some of the anomalous photovoltaic behaviour by using fixed surface charge densities at the interfaces between the perovskite and transport layers.[32, 183]

Van Reenen, Kemerink & Snaith were the first to publish simulations from a coupled drift-diffusion model including mobile ionic charge.[184] They found that J-V hysteresis could only be reproduced by including a density of trap states at the perovskite/contact interface facilitating recombination between carriers (Figure 2.16).[184] Given that calculations of a 1.5 nm Debye length11 have been presented,[185] the choice of a 4 nm mesh spacing in the simulations was questionable. Richardson and co-workers used an analytical approximation to the potential drop in the Debye layers to overcome the numerical challenge of solving for the perovskite/contact interfaces.[185, 186] While this approach enabled the reproduction of hysteresis effects using high rates of bulk recombination, the inability to accurately model interfacial recombination between the

11Based on an ion density of 1.6 × 1019 cm−3. Chapter 2. Background 60

a Dark c

b Illuminated

Figure 2.16: Simulated dark and illuminated electric potential profiles and J-V scan simulations. Simulated (a) dark and (b) illuminated electric potential profiles at short circuit following stabilisation at preconditioning at various different voltages. (c) J-V scans for the simulated device following voltage stabilisation at different voltages. In this instance the ions were kept constant during the scan reproducing an experiment by Tress et al.[32] The keys for all figures are the same, with the voltages given in panel (c). Dashed black curves indicate the results for a device without mobile ions. Adapted with permission from Ref.[184]. Copyright 2015 American Chemical Society. perovskite/contact interfaces led to inaccuracies in the simulation.[185] These included the absence of large differences in the open circuit voltage between forward and reverse scans. In a later publication by the same group modelling dark current transients, surface recombination was included, but only at the inner boundary of the Debye layer.[187] Our own initial work simulating devices, discussed in Chapters3 and5, and published in Ref.[175], verified van Reenen et al.’s conclusion that both mobile ions and high rates of interfacial recombination were necessary to observe hysteresis, as well as other slow optoelectronic phenomena in p-i-n solar cells. Neukom et al. recently published a similar study with similar conclusions.[188] They used a commercial package to solve for electronic carriers combined with a separate MATLAB script to solve for ionic densities (personal communication). Collectively, these results indicate that accurate modelling of the interface kinetics is critical to accurate modelling of devices as a whole.

2.8 Conclusions

Lead halide perovskites posses great potential as absorber materials for photovoltaic applications. Appropriate band gaps for light absorption, high absorption coefficients Chapter 2. Background 61 and long carrier diffusion lengths all contribute to making perovskites highly efficient solar materials. A number of material instabilities must be resolved, however, if perovskites are to become a commercially viable photovoltaic technology. These include reversible hydration upon exposure to water vapour and irreversible degradation in the presence of light and oxygen. Superoxide formation at iodide vacancy sites has been hypothesised to facilitate the latter process. Consequently, new materials systems and high standards of encapsulation will be required if perovskites are to become a commercially viable technology. In devices, the high intrinsic mobile ionic charge densities screen built-in electric fields in the absorber layer of devices leading to diffusion-dominated charge transport at steady-state, making selective contacts with passivated interfaces vital to efficient perovskite device operation. Mobile ions can also account for much of the anomalous slow optoelectronic transient behaviour observed in PSCs, including the current-voltage scan hysteresis. Given the system complexity, numerical drift-diffusion simulations provide an appropriate solution for modelling perovskite devices and have led to advances in understanding perovskite device physics. 62 Chapter 3

Device simulation

3.1 Declaration of contributions

The DRIFTFUSION simulation tool is based on an existing MATLAB code developed by Dr Piers Barnes1 to simulate transient measurements in dye sensitised solar cells.

Simulations using the Advanced Semiconductor Analysis (ASA) tool were performed by Mohammed Azzouzi1.

1 Department of Physics, Imperial College London, London, UK

3.2 Abstract

A one-dimensional drift-diffusion simulation capable of solving for electrons, holes, and a single ionic species was developed to simulate perovskite devices. Established semiconductor transport and continuity equations were solved using a MATLAB-based code to obtain charge densities for electronic and ionic carriers and the electric field potential. An analytical solution to the cell equilibrium condition based on the Depletion Approximation was used to calculate the initial conditions for the numerical solver. A mirrored-cell approach was used to solve for open circuit conditions and enable direct read-out of the open circuit voltage. Doping in the n and p-type regions was found to be critical to obtaining a built-in voltage in the mirrored cell and consistent boundary conditions in the standard configuration. Simulation of interfacial recombination at the intrinsic-doped region interfaces was found to model surface recombination at heterojunction interfaces more successfully when recombination was included throughout the n and p-type contact regions. Results from the simulation were compared to those from the Depletion Approximation solution for a p-n junction and 63 Chapter 3. Device simulation 64 an existing commercial drift diffusion package. The numerical solution obtained from the new simulation tool was identical to that obtained from the commercial package and in good agreement with the analytical solution.

3.3 Introduction

In order to unravel some of the subtleties of the device physics of perovskite solar cells (PSCs), I developed a one-dimensional simulation tool (DRIFTFUSION) capable of solving the coupled drift-diffusion equations for three charge carriers: electrons, holes and a mobile ionic species. The one-dimensional approach is justified provided that the materials are homogeneous, isotropic and of uniform layer thickness since all currents in the second and third dimensions cancel. The device physics of DRIFTFUSION are based on established semi-classical transport and continuity equations, which are well described in Nelson, 2003 [42] and Tress, 2011[83]. This chapter is intended as a ‘quick guide’ to the mechanics of the simulation.

DRIFTFUSION is based on a code written to simulate transient measurements in DSSCs by Piers RF Barnes[180] and uses MATLAB’s built-in Partial Differential Equation solver for Parabolic and Elliptic equations (PDEPE). The procedure solves the continuity equations and Poissons equation (see Sections 3.6 and 3.7) for electron density n, hole density p, a positively charged mobile ionic charge density a, and the electrostatic potential V as a function of position x and time t. While the details of the numerical methods employed by the PDEPE solver for discretising the equations go beyond the scope of this work, full details can found in Skeel and Berlizns 1990.[189]

3.3.1 Simulating open and closed circuit conditions

For closed circuit simulations, where the current output of the device is of interest (for example for current-voltage J-V scans) a p-type/intrinsic/n-type (p-i-n) architecture was used. Figure 3.1a shows the typical structure with position labels at the interfaces and boundaries.

At open circuit the cell is disconnected from the external circuit, resulting in zero current for all carriers. In conventional, fixed potential boundary condition simulations,(see subsection 3.9) the open circuit voltage is found by using an iterative convergence method (e.g. Newton-Raphson) to find the potential at which the current is zero. Simulating voltage transients in this way is computationally expensive; In order to accelerate calculation times and enable direct readout of the open circuit voltage VOC, Chapter 3. Device simulation 65

a p-type Intrinsic n-type (200 nm) (400 nm) (200 nm)

x x x x 0 pi in d b p-type Intrinsic n-type n-type Intrinsic p-type (200 nm) (400 nm) (200 nm) (200 nm) (400 nm) (200 nm) x x x x x x x 0 pi in d ni ip 2d

Figure 3.1: Simulation device schematics. Device schematics with position labels and layer thicknesses for (a) the fixed potential boundary condition p-i-n simulation used for closed circuit conditions, and (b) the mirrored p-i-n-n-i-p cell used for open circuit conditions. the method of image charges was used to devise a symmetric p-i-n-n-i-p cell (see figure 3.1b). The technical reasons for this are discussed in section 3.9.4.

3.4 Fundamental semiconductor energy levels

Figure 3.2a shows the energy levels associated with an idealised intrinsic semiconductor. 1 The electron affinity ΦEA and ionisation potential ΦIP are the energies required to remove an electron from the conduction band (CB) and valence band (VB) respectively to the vacuum level Evac. The band gap Eg of the material is defined as:

Eg = ΦEA − ΦIP (3.1)

The equilibrium Fermi energies EFi (intrinsic semiconductor) and EF0 (doped semiconductors) define the energy at which an electronic state has a 50% probability of occupation and the chemical potential of the material. While no states exists in an idealised semiconductor within the band gap, for an intrinsic semiconductor EFi lies close to the middle of the gap. Where the semiconductor is p-type, dopant impurities accept electrons from the bands, shifting the equilibrium Fermi level EF0 towards the VB (Figure 3.2b). Similarly, where the semiconductor is n-type, dopant impurities donate electrons to the bands, shifting EF0 towards the CB (Figure 3.2c). Under electrical and/or optical bias, electrons populate the CB and holes populate the VB leading to the development of a difference between the chemical potentials of the two populations. This is often described as ‘splitting’ of the electron and hole Fermi levels. Since relaxation to the band edges is assumed to be much faster than relaxation

1These energies are chosen to be negative by convention in this work in order to maintain consistency with the semiconductor energy scale. Chapter 3. Device simulation 66 between bands, this results in a quasi-equilibrium state, where separate electron and hole Quasi Fermi levels (QFLs), EFn and EFp (Figure 3.2a, blue and red dashed lines) arise. Below follows a brief description of how these energies are calculated in the simulation. a E b c vac Φ Φ EA IP E CB E E Fn E F0 Energy E g i E E Fp F0 E VB Intrinsic p-type n-type

Figure 3.2: Semiconductor energy levels. (a) An intrinsic semiconductor material showing the vacuum level Evac, electron affinity ΦEA, ionisation potential ΦIP, conduction and valence band energies ECB and EVB, band gap Eg, intrinsic Fermi energy EFi and electron and hole quasi-Fermi levels EFn and EFp.(b) A p-type material: The equilibrium Fermi energy EF0 lies closer to the VB due to acceptor impurities adding holes to the VB. (c) An n-type material: EF0 lies closer to the CB as donor impurities increase the equilibrium electron density.

3.4.1 Vacuum energy

The vacuum energy, Evac is defined as the ‘...the energy to which an electron must be raised to be free of all forces from the solid’.[42] Spatial changes in the electrostatic potential V are therefore reflected in Evac such that:

Evac = −qV (3.2)

3.4.2 Conduction and valence band energies

In the absence of electric fields, the conduction and valence band energies ECB and EVB are simply defined by the chemical potentials ΦEA and ΦIP respectively. Where electric fields are present, the energy required to remove electrons from the bands is modified by qV such that:

ECB = ΦEA − qV (3.3) Chapter 3. Device simulation 67

EVB = ΦIP − qV (3.4) where q is the elementary charge.

3.4.3 Quasi Fermi levels

The QFLs are the sum of electrostatic and chemical potentials at each position. Using the typically applied Boltzmann approximation to the Fermi-Dirac probability distribution function, the QFLs for electrons and holes EFn and EFp can be expressed as:

 n  EFn = ECB + kBT ln (3.5) NCB

 p  EFp = EVB − kBT ln (3.6) NVB where kB is Boltzmann’s constant, T is the temperature of the system and NCB and NVB are the effective density of states (eDOS) in the conduction and valence bands respectively. The eDOS parametrises the curvature of the respective band and is dependent on the carrier’s effective mass. The gradient of the QFLs provides a convenient way to determine the direction of the current since, from the perspective of the electron energy scale, electrons move ‘downhill’, and holes move ‘uphill’ with respect to electrochemical gradients. Furthermore, electron and hole currents, Jn and

Jp, can be calculated using the product of the quasi Fermi level gradient with carrier conductivity:

dE J = σ Fn (3.7) n n dx

dE J = σ Fp (3.8) p p dx where the conductivities can be obtained from the product of the appropriate carrier mobility µ and density with the elementary charge:

σn = qnµn (3.9) Chapter 3. Device simulation 68 a b 0 0

-1 ECB -1 EFp E E Energy [eV] Energy [eV] -2 VB Fn -2 ] ] 20 20 -3 -3 10 10 p 10 10 10 10

0 10 n 0 Carrier density [cm Carrier density [cm 10 0 200 400 600 800 0 200 400 600 800 Position [nm] Position [nm]

Figure 3.3: Short circuit equilibrium and illuminated p-i-n energy level diagrams and charge densities.(a) A simulated p-i-n structure at thermal equilibrium. p and n-type regions are shaded in green and blue respectively. The built-in field predominantly drops across the active layer (white region) while the chemical potential gradient is equal and opposite accounting for the same slope in the n and p profiles plotted on a logarithmic scale. This results in flat quasi-Fermi levels (QFLs) for both electrons and holes indicating that no current is flowing. (b) The same device under optical bias: The QFLs split due to population of the valence and conduction bands with photoexcitated electrons and holes. Here, the Fermi level gradients indicate that net electron and hole currents are flowing in the device.

σp = qpµp (3.10)

3.4.4 Open circuit voltage

The open circuit voltage is the maximum energy per unit charge that can be extracted from the cell and is given by the difference in QFLs between electrons and holes at the two electrodes.

qVOC = EFn(xd) − EFp(x0) (3.11)

3.4.5 Energy level diagrams

Energy level diagrams2 provide a convenient method for graphing the solution of the simulation at a specific time t. The vertical axis graphs the electron energy while the

2The term ‘Band Diagrams’ is also often used interchangeably although I avoid this terminology here to prevent confusion with band structure diagrams. Chapter 3. Device simulation 69 horizontal axis shows the position in the device. The electrochemical potentials (QFLs), and conduction and valence band energies of carriers at each location within the cell are plotted with respect to an electric potential-dependent vacuum energy. This method enables a qualitative (if not quantitative) assessment of the state occupancy of the bands and the electrostatic potential at each position. Since the energy levels in the simulation are all relative, for convenience, the electron affinity of the p-type material is set to zero and all other energies are calculated relative to this reference level in this work. Figure 3.3a and 3.3b shows example energy level diagrams of a p-i-n structure at equilibrium and under illumination respectively.

3.5 Charge transport (drift-diffusion)

Contributions to the electron and hole currents in a semiconductor can be divided into two categories:

1. Drift currents arising from the electrostatic response of charges to an electric field E.

2. Diffusion currents arising from carrier concentration gradients ∂n/∂x, ∂p/∂x, and ∂a/∂x.

In one dimension the expressions for the flux of electrons jn, holes jp, and ions ja, with mobility µ and diffusion coefficients D respectively3 are given by: a b c V n

dV dn p dp j j j p, drift dx n, di! dx p, di! dx

jn, drift Hole density, Hole density, Electron density, Electron density, Electric potential, Electric potential, Position, x Position, x Position, x

Figure 3.4: Electron and hole carrier fluxes. Direction of electron jn and hole jp carrier fluxes in response to positive gradients in (a) the electric field potential (note: dV E = − dx ), and (b) electron and (c) hole carrier densities. The subscripts ‘drift’ and ‘diff’ denote drift and diffusion fluxes respectively. Mobile ionic charge has been assigned a positive charge in the simulation and therefore follows the same direction as holes.

3The corresponding charge carriers are denoted by subscripts. Chapter 3. Device simulation 70

∂n j = −µ nE − D (3.12) n n n ∂x

∂p j = µ pE − D (3.13) p p p ∂x

∂a j = µ aE − D (3.14) a a a ∂x

Figure 3.4 illustrates how the direction of flux is determined from gradients in the electric potential and charge carrier densities. Mobile ionic charge has been assigned a positive charge in the simulation and therefore follows the same direction as holes. The carrier currents are given by the product of the fluxes with the carrier charge, (e.g. Jn = −qjn).

Using the Einstein relation D = µkBT/q to express the diffusion coefficient in terms of mobility, the electron Jn, hole Jp, and ion Ja current densities can be written as:

∂n J = qµ nE + µ k T (3.15) n n n B ∂x

∂p J = qµ pE − µ k T (3.16) p p p B ∂x

∂a J = qµ aE − µ k T (3.17) a a a B ∂x where kB is the Boltzmann constant, T is temperature and q is the electronic charge. This work takes the convention that electrons flowing right and holes flowing left produce a negative current. At steady-state this is also the direction of photocurrent, although, as investigated in Chapter5, it is possible for the direction of photocurrent to become inverted in PSCs.

3.5.1 Transport across heterojunction interfaces

Following the approach by Zeman et al.[190], Equations 3.15 and 3.16 can be modified by including additional gradient terms for changes in ΦEA,ΦIP and effective density of conduction and valence band states at material interfaces. This leads to an adapted set of current equations for electrons and holes: Chapter 3. Device simulation 71

  ∂ΦEA kBT ∂NCB ∂n Jn = µnn qE − − + µnkBT (3.18) ∂x NCB ∂x ∂x

  ∂ΦIP kBT ∂NCB ∂p Jp = µpp qE − + − µpkBT (3.19) ∂x NCB ∂x ∂x

A finite interface thickness must be chosen in order to define discrete gradients for these properties. While the precise nature of transport across device interfaces is not well understood, here I make the assumption that transport is not limited by transfer across interfaces, except for where energetic barriers exists.

3.6 The Continuity equations

The continuity equations are a set of ‘book keeping’ equations that ensure conservation of charge in the system. They describe how the density of charge carriers changes as a function of position and time. In the most general form, for a concentration ϕ with flux jϕ, and source S:

∂ϕ − ∇j − S = 0 (3.20) ∂t ϕ

In solar cells, in the simplest case, the source term S is composed of two parts:

1. Generation G of electronic carriers by photoexcitation.

2. Recombination U of electronic carriers through radiative and non-radiative pathways.

Where chemical reactions take place within the cell, additional generation and recombination of carriers may also contribute to S. In this work, mobile ions are considered inert and as such the source term for ions is zero. Figure 3.5 illustrates how the changes in electron concentration in a thin slab dx are determined by generation and recombination processes, and changes in the flux.

In one dimension the continuity equations for electrons, holes and ions are:

∂n 1 ∂J = n + G − U (3.21) ∂t q ∂x Chapter 3. Device simulation 72

ECB j (x) j (x + dx) n G U n

EVB

Electron Hole

x x + dx

Figure 3.5: Continuity of electronic charge in a one-dimensional system. Schematic illustrating the principle of continuity for electrons in a thin slab dx of material. A difference in the incoming and outgoing flux Jn, and generation G and recombination U processes result in changes in the electron concentration. The conduction and valence bands are denoted ECB and EVB respectively. Figure concept taken from Ref.[191].

∂p 1 ∂J = − p + G − U (3.22) ∂t q ∂x

∂a 1 ∂J = − a (3.23) ∂t q ∂x

Equations 3.21- 3.23 form the first part of the system of equations that must be solved.

3.7 Poisson’s equation

Poisson’s equation relates the electrostatic potential to the space charge density ρ as a function of position. In the simulation the space charge density is defined by the sum of the mobile and static charge densities. Doping is achieved via the inclusion of fixed charge densities terms for donors, ND and acceptors, NA, which effectively generate mobile counter charges. In this work ionic defects are modelled as Schottky defects in which each charged vacancy has an oppositely charged, immobile, counterpart (see subsection 2.3.2). In the simplest approximation, the counter ions are described as being static with uniform charge density of Na. Together these terms yield the following expression: Chapter 3. Device simulation 73

∂V 2 ρ q 2 = − = − (p − n + a + NA − ND − Na) (3.24) ∂x ε0εr ε0εr where ε0 is permittivity of free space and εr is the relative permittivity of the medium. Poisson’s equation completes the set of coupled parabolic and elliptic equations solved by DRIFTFUSION. Upon obtaining the electrostatic potential, the electric field E can be calculated using the negative of the gradient of the potential:

∂V E = − (3.25) ∂x

3.8 Initial conditions

The solver requires a set of initial conditions for charge carrier densities and the electric field potential. For the p-i-n junction used for the majority of this work, true equilibrium conditions were found by starting with an analytical solution derived from the Depletion Approximation. The general approach was to obtain an expression for the electric field potential in the p and n-type depletion regions. The potential was assumed to drop uniformly across the intrinsic layer. Carrier densities were subsequently calculated under the principle that the gradient of the QFLs is zero at equilibrium. The resulting set of initial conditions are as follows:

V = 0 x < (xpi − wp) (3.26) 2 qNA(x − (xpi − wp)) V = (xpi − wp) < x < xpi (3.27) 2εp 2 2 ! 2 (x − xpi) qNAwp qNDwn qNAwp V = Vbi − − + xpi < x < xin (3.28) (xin − xpi) 2εp 2εn 2εp 2 qND(x − (xin + wn)) V = Vbi − xin < x < xin + wn (3.29) 2εn

V = Vbi x > xin + wn (3.30)

n = ND x < (xpi − wp) (3.31)   EFn0,n − (ΦEA − qV ) n = NCB exp (xpi − wp) < x < (xin + wn) (3.32) kBT 2 n = ni − NA x > (xin + wn) (3.33) Chapter 3. Device simulation 74

2 p = ni − ND x < (xpi − wp) (3.34)   (ΦIP − qV ) − EFp0,p p = NVB exp (xpi − wp) < x < (xin + wn) (3.35) kBT p = NA x > (xin + wn) (3.36)

a = Na (3.37)

where εn and εp are the relative permittivities of the n and p-type materials respectively, and ni is the intrinsic carrier density:

  2 Eg ni = NCBNVB exp − (3.38) kBT wn and wp in the above statements are the widths of the depletion region in the n and p-type layers respectively. The analytical solution for the depletion widths described in Section 3.13.1 did not always yield low field results and, as such, the depletion width was adjusted manually to obtain a solution with a low Fermi level gradient after the solving for the correct electric field potential.

The equilibrium Fermi energies in the p-type EFp0,p and n-type EFn0,n are given by:

  kBT ND EFn0,n = ΦEA + ln (3.39) q NCB

  kBT NA EFp0,p = ΦIP − ln (3.40) q NVB where ΦEA and ΦIP are the electron affinity and ionisation potential of the material respectively.

For the initial solution recombination was set to a low rate and mobilities were switched off. The simulation was then run iteratively with the mobilities turned on using increasing time steps until the equilibrium state was reached. The equilibrium state without ions then formed the initial conditions for a second equilibrium state with mobile ions. Following this solutions were ‘symmetricised’ to create initial conditions for the mirrored cell structure for use with the set of open circuit boundary conditions. Chapter 3. Device simulation 75

a b ] 10 18 0 -3 10 16 Numerical ECB [cm -1 E p Fp 10 14 EFn -2 12 Analytical Energy [eV] EVB 10

-3 10 10 0 200 400 600 800 density, Hole 150 200 250 300 Position [nm] Position [nm]

Figure 3.6: Initial conditions used for p-i-n simulations compared to equilibrium state. (a) Energy level diagram generated using the initial conditions (solid curves) compared to the final equilibrium states (dashed curves) for the p-i-n device without mobile ions. (b) Detail of the hole charge density generated from the initial conditions (solid red curve) compared to the final equilibrium state (dashed black curve).

The energy level diagram generated using the analytical approximation (solid curves) is compared to the true equilibrium solution (dashed curves) in Figure 3.6a.

While the agreement is reasonable, as Figure 3.6b shows, the charge densities at the p-type/intrinsic depletion region deviate from the true equilibrium solution by up to a factor of 5. I note that at the time of writing, these conditions have only been rigorously tested in cases where NCB = NVB and NA = ND and with a single dielectric constant in all layers of the cell.

Where heterojunctions were simulated, equilibrium charge carrier densities and Fermi energies were chosen as the initial conditions for each individual layer. The simulation was then run iteratively with increasing time steps until the devices reached equilibrium.

3.9 Boundary conditions

Solving Equations 3.21- 3.24 requires two constants of integration for each variable, which are given by the boundary conditions of the simulation. Two common classes of boundary condition employed in the simulation are:

1. Dirichlet: The value of the variable is specified

2. Neummann: The value of the flux is specified Chapter 3. Device simulation 76

3.9.1 Contact selectivity and surface recombination

Perovskite devices typically employ selective ‘contact’ layers that block minority carriers from being extracted at the wrong electrode via energetic barriers. These are known variously as transport layers, blocking layers, blocking contacts, or selective contacts (see Chapter2, Figure 2.3). In a real solar cell, semiconductor layers are typically sandwiched between two metallic electrodes4 constituted of either pure metals or highly-doped semiconductors e.g. indium tin oxide (ITO). Such materials are numerically challenging to simulate due to the high charge carrier densities and thin depletion widths. Resultantly, a common approach is to simulate only the active layer and use appropriate boundary conditions defined by the properties of the contact or electrode. Neumann boundary conditions defining charge carrier extraction and recombination fluxes are often used for this purpose. A rate coefficient, smaj is used to define the rate of extraction of majority carriers, which for a good contact is expected to be high. For minority carriers, smin effectively defines the ‘surface recombination velocity’: in this instance a low value indicates a highly selective contact, whereas a high value indicates a contact with poor selectivity. The expressions for majority and minority carrier flux densities, jmaj and jmin respectively, are given by:

jmaj = qsmaj(nmaj − n0,maj) (3.41)

jmin = qsmin(nmin − n0,min) (3.42)

where nmaj and nmin are the equilibrium majority and minority carrier densities respectively. In the limiting case, for an Ohmic contact with an infinite rate of extraction (smaj = ∞), Equation 3.41 reduces to the simple Dirichlet condition:

nmaj = n0,maj (3.43)

This assumes Fermi level equilibrium between the electrode and the contact layers.

For self-consistency, the equilibrium majority carrier densities n0,maj and p0,maj can be calculated using Boltzmann statistics such that:

  EVB − Φano p0,maj(x0) = NVB exp (3.44) kBT 4While elsewhere in the literature, the terms ‘contact’ and ‘electrode’ are used interchangeably, here the term ’electrode’ is reserved for metallic layers at the boundary of the device, while the term ‘contact’ is reserved for selective semiconductor transport/blocking layers. Chapter 3. Device simulation 77

  Φcath − ECB n0,maj(xd) = NCB exp (3.45) kBT

5 where Φano and Φcath are the anode and cathode workfunctions respectively. Extraction barriers can also be modelled with this approach by including a term for the barrier energy in the exponent of Equations 3.44 and 3.45.

For a metal the rate of surface recombination is expected to be very high, since no barriers exist to the occupation of states by minority carriers. A similar approach to that used to define an Ohmic contact for majority carriers can be used to model infinite surface recombination for minority carriers using the Dirichlet condition:

nmin = n0,min (3.46)

In this instance, n0,min can be calculated using the law of mass action:

2 n0,majn0,min = ni (3.47)

3.9.2 Doping for selectivity

In the standard p-i-n device model used for the majority of the simulations in this thesis, selectivity is achieved through doping of the n and p-type regions. In doped semiconductors ionised impurities either donate or accept free electrons resulting in a shift in the equilibrium QFLs relative to the bands as can be seen in figure 3.6a. Simulating selectivity in this way leads to some limitations: Firstly, The built-in voltage and p and n-type doping densities are interdependent variables. This potentially prevents intrinsic or lightly doped contact materials from being investigated. Secondly, minority carrier concentrations decrease over a finite depletion width (dependent on the dielectric constant, doping density and band gap) as opposed to a single point at the interface as with a blocking layer. This feature of the p-i-n approximation of full HTL-i-ETL devices can result in unrealistic behaviours, such as large shifts in recombination rates at the perovskite-contact interfaces (positions xpi and xin in Figure 3.1), where interfacial recombination is solely implemented within thin regions at contact/perovskite interfaces (Figure 3.7a). Figures 3.7a and 3.7b show how changes in the ionic charge density profile lead to a spatially-shifting carrier overlap and associated unrealistic open circuit voltage behaviour.

5Here I use the convention that the electron collecting electrode is the cathode and the hole collecting electrode is the anode as well as defining the workfunctions to be negative energies. Chapter 3. Device simulation 78

a b 10 18 0.9 ] Entire contact region -3 0.8 p Experiment 14

10 [V] 0.7 OC

V 0.6 n 10 10 0.5

Open circuit Open 0.4 voltage, voltage, 5nm rec. layer 6 Carrier density [cm 10 180 190 200 210 10 -1 10 0 10 1 10 2 Position [nm] Time [s]

Figure 3.7: Effect of high rates of recombination in a thin layer versus full contact region on open circuit voltage.(a) Charge densities at initial (solid lines) and final (dashed lines) states during migration of ions after the cell has been equilibrated at short circuit and illuminated. The arrow indicates how the position of maximum overlap between the two carriers species shifts into and out of the recombination region (shaded yellow), leading to a sudden drop and then recovery of (b) the open circuit voltage as a function of time (yellow curve). This behaviour is unrealistic as compared to when recombination is implemented throughout the entire contact region (green curve). The black dashed curve shows experimental data from Chapter5 for a planar TiO 2 device for comparison.

In order to circumvent these issues, surface recombination at these interfaces (herein known as ‘interfacial recombination’) was instead implemented by introducing high rates of recombination throughout the contact regions. This approach ensures that changes in the depletion width due to ion migration does not strongly impact the overall rate of recombination at these interfaces. Provided a sufficiently high rate coefficient is chosen, the charge carrier densities become pinned to their equilibrium levels and the state of the cell is effectively the same as if equation 3.46 were satisfied. Figure 3.8b shows how the inclusion of Shockley-Read-Hall recombination with sufficiently low time constants (see Equation 3.57) in the contact regions achieves this in a device at open circuit.

3.9.3 Closed circuit boundary conditions

As discussed in Chapter2, Section 2.2.2, in order for a solar cell device to efficiently extract charges an asymmetry must exist, such that electrons and holes are driven in opposing directions. This can be achieved solely through the use of selective contacts or by choosing contact materials that induce an electric field across the active layer. This built-in field is defined by the difference in equilibrium Fermi energies per unit charge of the n and p-type regions in a p-i-n structure, or alternatively electron and hole transport layers in a device with heterojunctions. Provided that the contacts are in equilibrium with the electrodes (as assumed in DRIFTFUSION), the built in potential Vbi is simply the difference per unit charge between the cathode Φcath and anode Φano workfunctions: Chapter 3. Device simulation 79 a b No Surface Rec. Surface Rec. 0 0 ECB -0.5 E Fn -1 -1 EFp Energy [eV] -1.5 Energy [eV] -2 E

] VB ] -3 10 18 -3 20 p 10 10 16 10 10

10 14 n 10 0

12 -10 Carrier density [cm 10 Carrier density [cm 0 200 400 600 800 10 0 200 400 600 800 Position [nm] Position [nm]

Figure 3.8: Simulating interfacial recombination. Energy level diagrams (top) and charge densities (bottom) for simulated p-i-n structure solar cells at open circuit with a uniform carrier generation profile of G = 2.5 × 1021 cm−3s−1.(a) A device with SRH recombination in the contacts switched off: uniform quasi-Fermi level (QFL) splitting is observed. (b) A device with a high rate of SRH recombination (τnSRH = −15 τpSRH = 2 × 10 s) in the contact regions. Minority carriers are highly depleted in the contact regions leading to convergence of the QFLs.

qVbi = Φcath − Φano (3.48)

As discussed above, in this work the p-type layer/material conduction band is chosen to be the reference level. The electrostatic potential at the left-hand boundary is therefore

fixed at 0 V (Equation 3.49). If an electrical bias, Vapp is applied to the cell then the appropriate complimentary condition at the right-hand boundary, xd for the potential is given by equation 3.50.

V (x0) = 0 (3.49)

V (xd) = Vbi − Vapp (3.50)

3.9.4 Open circuit boundary conditions

The requirement for direct readout of the open circuit voltage required a different approach from that detailed above since the at the right-hand boundary cannot be specified. Accordingly, the method of image charges and the mirrored device architecture Chapter 3. Device simulation 80 shown in Figure 3.1b were implemented to enforce zero flux for all carriers while allowing the potential at xd to float. The fluxes for all carriers and the electric field potential at the true boundaries of the mirrored cell x0 and x2d were also set to zero, leading to the following complete set of boundary conditions for the p-i-n-n-i-p cell:

jn(x0) = jn(x2d) = 0 (3.51)

jp(x0) = jp(x2d) = 0 (3.52)

ja(x0) = ja(x2d) = 0 (3.53)

V (x0) = V (x2d) = 0 (3.54)

While this approach led to reasonably stable solutions, the zero flux conditions (Equations 3.53, 3.52) meant that surface recombination could not be implemented at the boundaries in the usual way. A simple solution, employed in Chapter8, is to introduce high rates of recombination in thin regions close to the real and virtual boundaries (x0, xd and x2d).

3.10 Generation

In order to accelerate solver calculation times a uniform generation profile was used for the the majority of simulations in this work. This is a reasonable approximation in devices in which the rate of transport is significantly greater than that of recombination, such as PSCs. For convenience, I define a total absorbed photon flux of 1017 cm−2s−1 as 1 Sun simeq. This is to an order of magnitude approximation equivalent to the absorbed photon flux under 1 Sun in a 400 nm CH3NH3PbI3 layer.[100]

In Chapter8, Section 8.3, a Beer-Lambert profile is used to model the spectral generation rate G(Eph) for a laser excitation pulse of photon energy (Eph) and incident photon flux

ϕ0, using the common expression:

G(Eph) = αabs(Eph)ϕ0 exp(−α(Eph)x) (3.55)

where αabs(Eph) is the material’s absorption coefficient for photon energy Eph. For an incident spectrum, the integral of G(Eph) can be taken over the range of appropriate photon energies. Chapter 3. Device simulation 81

Band to Bulk Band SRH

SRH interfacial

ElectronHole Photon Phonon

Figure 3.9: Schematic of different recombination mechanisms in a HTL-i-ETL device. Avoidable trap-mediated Shockley Read Hall (SRH) recombination can be localised in the simulation whereas unavoidable band-to-band recombination is implemented through all layers of the device.

3.11 Recombination

The simulation uses two established models for recombination: unavoidable band-to-band recombination and avoidable trap-mediated Shockley-Read-Hall. recombination.

3.11.1 Band-to-band recombination

The rate of band-to-band recombination Ubtb (also commonly described as radiative recombination) at a given location in the cell is proportional to the density of electrons and holes at that position:

2 Ubtb = kbtb(np − ni ) (3.56)

2 where kbtb is the rate coefficient. The ni term provides self-consistency and is equivalent to including an expression for thermal generation: if the charge densities decrease such 2 that their product is less than ni , Ubtb becomes negative and hence generative.

3.11.2 Shockley-Read-Hall (SRH) recombination

Recombination via trap states is modelled using a simplified expression for Shockley-Read-Hall (SRH) recombination[192] in which the capture cross section, mean Chapter 3. Device simulation 82 thermal velocity of carriers, and trap density are collected into SRH time constants,

τn,SRH and τp,SRH for electrons and holes respectively:

2 (np − ni ) USRH = (3.57) τn,SRH(p + pt) + τp,SRH(n + nt) where nt and pt are parameters that define the dependency of the recombination rate to the trap level and are given by the densities of electrons and holes when their respective

QFLs are at the position of the trap energy, Et:

  Ei − Et nt = ni exp (3.58) kBT

  Et − Ei pt = ni exp (3.59) kBT

I note that Equation 3.57 is derived for devices in steady-state. In this work, the rate of trapping and detrapping of carriers is assumed to be fast (ps - ns) compared to the timescale of measurements (µs), such that the Ref. approximation is reasonable. To accurately simulate faster timescale transient measurement, the dynamics of capture and emission of carriers could be included.

3.12 Spatial and time meshes

During the early stage of development of the simulation, the p-i-n devices simulated in Chapters5-6, used a linear spatial mesh with a mesh spacing of 0 .63 nm. For later stage p-i-n simulations (Chapter7) and heterojunction simulations (Chapter8), a piece-wise linear mesh with a high density of points at the interfaces and space charge regions was used. An example is given in Figure 3.10a. This had the twin benefits of both increasing the numerical accuracy of the solution and reducing the total number of points require to obtain accurate solutions.

While in general the time mesh was set to be logarithmic, more complex piece-wise logarithmic meshes, such as the example shown in Figure 3.10b, were used for transient simulations. Chapter 3. Device simulation 83 a b 600 500 p-type n-type Baseline Pulse Decay Space-Charge Regions 400 400 300 200 200 Mesh point Mesh point Absorber 100 Layer 0 0 0 200 400 600 800 10 -8 10 -6 10 -4 Position [nm] Time [s]

Figure 3.10: Examples of spatial and time meshes. (a) Piece-wise linear spatial mesh used for more advanced simulations of p-i-n devices and heterojunctions. Steep gradients are included throughout the space-charge regions and the 4 nm interface between material types. (b) Piece-wise logarithmic time mesh used for the exponential rise and decay of the transient photovoltage signal in Chapter7.

3.13 Comparison of the simulation results with analytical solutions for a p-n junction

To verify the accuracy of the simulation, current-voltage scans and open circuit voltages obtained using analytical and numerical solutions for a 4 µm p-n junction with a band gap of 1.12 eV were compared. Additionally, the numerical solutions from DRIFTFUSION were also compared with those of the Advanced Semiconductor Analysis (ASA) simulation tool, an established commercially available package developed by TU Delft.[190] Since the Depletion Approximation for a p-n junction has been dealt with elsewhere in detail,[42] here the results are used without derivation. In order to simplify the calculation I assume a perfectly absorbing semiconductor (i.e. a step function absorption at the band gap energy).

3.13.1 The Depletion Approximation solution to a p-n junction at equilibrium

The Depletion Approximation enables the continuity equations and Poisson’s Equation (Equations 3.21, 3.22 and 3.24) to be solved analytically.[193] The depletion region at the junction of the device is assumed to have zero free carriers allowing a step function for the space charge density ρ equal to the background doping densities to be used (see figure 3.11a, top panel). Furthermore, since there are no free carriers, transport ∂J ∂J  and recombination in the depletion region are ignored n = p = 0 . Poisson’s ∂x ∂x equation is then solved to give the depletion widths for n and p-type regions, wn and wp respectively: Chapter 3. Device simulation 84

v 1 u 2ε V w = u r bi (3.60) n N u  1 1  A tq + NA ND

v 1 u 2ε V w = u r bi (3.61) p N u  1 1  D tq + NA ND

∂E ρ Using these expressions for wn and wp, Gauss’s law = − can be used to obtain ∂x εr the electric field E from the space charge density ρ. Integrating the negative of the field with respect to position gives the electric potential:

Z xd V = − E dx x0

The solutions for the space charge density, electric field, and electric potential for both analytical and numerical solutions are shown in Figure 3.11a. The simulation is in excellent agreement with the analytical solutions, with the approximate depletion width and electric field gradient and magnitude well reproduced.

3.13.2 The Shockley diode equation

The solution to the Depletion Approximation combined with the superposition of the photo JSC and dark saturation current J0, can be used to derive Shockley’s diode equation describing the total current J for a potential drop across the junction V :

  qV   J = JSC − J0 exp − 1 (3.62) nidkBT

For V >> kBT (approximately 26 meV at 300 K) the −1 from the dark current term is often omitted. Here nid accounts for deviations in the slope of the ln(J) vs. V curve from q/kBT frequently observed in real devices. The ideality factor in perovskite devices is explored more fully in Chapter6.

3.13.3 Current-voltage characteristics

In order to compare numerical and analytical current-voltage (J-V ) characteristics, values for J0 and JSC need to be related to input parameters for the simulation. Under Chapter 3. Device simulation 85 a b 1 0

] Analytical 16 0.5 DRIFTFUSION -0.5 x10 E -3 0 CB EFp

[cm -0.5 -1 ρ E Charge density, Fn -1 EVB ]

4 -1.5 Energy [eV] Energy

x10 0 -1 10 20

-2 ] -3 [Vcm Electric !eld, 15 E 10 p -4

10

[V] 0.6 10 V 0.4 10 5 n 0.2 Electrostatic potential, 0 [cm Carrier density 10 0 1.5 2.0 2.5 0 1 2 3 4 Position [ μm] Position [ μm]

Figure 3.11: Analytical and numerical solutions to a p-n junction.(a) The space charge density in the Depletion Approximation is given by a step function equal to the doping density. The numerical solution, which includes finite charge densities at the junction, shows a smoother transition to the field-free regions. The simulated electric field E and electric potential V show similar features and magnitudes to the analytical solution. (b) Energy level diagram and charge densities for an example device with Eg = 1.12 eV, at equilibrium. Green and white regions indicates the n-type and p-type layers respectively.

the principle of Detailed Balance[64] the efficiency of a solar cell with band gap Eg is always limited by unavoidable radiative recombination. At equilibrium in the dark, the net current is zero implying that the rate of thermal generation equals the rate of radiative recombination. If the cell is surrounded by a black body at temperature TS then the total recombination current J0,rad is given by Planck’s law:[194]

∞ 2 Z 2π ηEQE(Eph)E J = ph dE (3.63) 0,rad h3c2  E  Eg exp ph − 1 kBTS where h is Planck’s constant, and c is the speed of light. In the limiting case, the external quantum efficiency ηEQE is set to unity for photons with energy Eph above the 2 band gap (step function absorption). Using n0p0 = ni in combination with Equations

3.56 and 3.63 the radiative rate of recombination per unit volume U0, and band-to-band rate coefficient kbtb can be obtained for a device of known thickness, d at equilibrium: Chapter 3. Device simulation 86

60 0.2 Power density [mWcm 50 J

] SC, max -2 AM 1.5 G 0.15 40 30 0.1 [mAcm 20

SC, max 0.05 J

10 -2 nm

0 0 -1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ] Photon energy [eV]

Figure 3.12: AM 1.5 solar spectrum and maximum theoretical short circuit current. AM 1.5 power density versus wavelength (right axis, black curve). Integrating the spectral photon flux above the band gap gives the maximum theoretical short circuit current JSC,max (left axis, red curve). The dashed line indicates the position of 1.12 eV.

U0 J0,rad kbtb = 2 = 2 (3.64) ni qdni

For the 4 µm thick device, with a band gap of Eg = 1.12 eV, surrounded by a black −13 3 −1 body at 300 K, kbtb = 2.28 × 10 cm s .

In order to compare illuminated J-V curves and associated open circuit voltages with the simulation, a theoretical maximum current density JSC,max, and corresponding uniform generation rate, G were calculated. Figure 3.12 (right axis, black curve) shows the AM 1.5 Global Tilt6 solar spectrum.

The short circuit photocurrent for a semiconductor of band gap Eg is given by:

Z ∞ JSC(Eg) = q ηEQEbS(Eph)dE (3.65) 0 where bS is the spectral photon flux density. Since ηEQE = 1 for photon energies Eph >=

Eg, and ηEQE = 0 for Eph < Eg, JSC,max is simply the integral of bS from the bandgap energy to infinity. Figure 3.12 (left axis, red curve) shows the maximum achievable current density JSC,max over a range of band gap energies. The dashed line indicates −2 that for a band gap of 1.12 eV, JSC,max = 42.7 mAcm . To calculate a uniform carrier generation rate G, the short circuit flux density (jSC,max = JSC,max/q) is divided by

6’Global Tilt’ is for a south facing surface at an angle of 37 degrees to the horizontal and includes sky diffuse and diffuse reflected light from the ground.[195] Chapter 3. Device simulation 87 a b ] 2 3

-2 10 ] [V]

-2 DF 0 20 10 OC 2.5 ASA Ana V 10 -2 [mAcm 2 J [mAcm

0 J m = q V/k T 10 -4 B Analytical 0.5 0.6 0.7 0.8 0.9 1.5 V [V] DRIFTFUSION -20 1 DF Ana ASA 0.5 -40 Current density, Open circuit voltage, Open circuit voltage, 0 0 0.5 1 0.5 1 1.5 2 2.5 3 V E Voltage, [V] Band gap energy, g [eV]

Figure 3.13: Comparison of analytical vs. numerical current-voltage characteristics and open circuit voltage as a function of band gap for idealised p-n junctions. (a) Current-voltage characteristics for numerical and analytical solutions for a p-n junction with Eg = 1.12 eV. Light blue (AM 1.5) and yellow (dark) curves show the results from DRIFTFUSION (denoted DF), black dashed (AM 1.5) and grey dashed (dark) curves show the results from TU Delft’s Advanced Semiconductor Analysis (ASA) simulation tool.[190]) Purple (AM 1.5) and pink (dark) curves show the analytical solutions using the Depletion Approximation (denoted Ana). (b) Open circuit voltage as a function of band gap for DRIFTFUSION (blue curve, circle markers) using the method of images (see Section 3.3.1) compared to the analytical solution (purple dashed curve).

the device thickness, G = jSC,max/d. Using the given device thickness of 4 µm and −2 20 −3 JSC,max = 42.7 mAcm a value of G = 6.67 × 10 cm is obtained.

Figure 3.13a shows the light and dark current-voltage curves for the analytical solution obtained using equation 3.62, as compared to the numerical solutions. The results from DRIFTFUSION and ASA were indistinguishable from one another. Table 3.1 summarises key metrics from the J-V scans demonstrating that the simulations are also in good agreement with the analytical solution. Since the short circuit current is prescribed, it is the difference in open circuit voltage that accounts for the observed discrepancies in fill factor7 and maximum power density, indicating an increased recombination rate in the simulations. Metric Abbreviation Analytical Numerical % Difference solution solution − Short circuit current JSC [mAcm 2] 42.7 42.7 0.0 Open circuit voltage VOC [mV] 876 850 −3.1 − Maximum power density Pmax [mWcm 2] 32.5 31.5 −3.1 Fill factor FF 0.870 0.867 < 0.1

Table 3.1: Table of key values for analytical and numerical model current-voltage characteristics. cf. figure 3.13

7 The fill factor FF is defined as: FF = VMPJMP/VOCJSC, where VMP and JMP are the maximum power point voltage and current density respectively. Chapter 3. Device simulation 88

In this and other test cases, the open circuit voltage using the mirrored configuration boundary conditions (Section 3.9.4) agreed to beyond 3 decimal places with the solution obtained using conventional boundary conditions and iterating until the applied voltage gave zero current (Section 3.9). As a further verification of the accuracy of the simulation, the theoretical maximum VOC,max under the Depletion Approximation was calculated for a range of band gaps by setting J equal to zero in the ideal diode equation (Equation 3.62):

  kBT JSC,max VOC,max(Eg) = ln + 1 (3.66) q J0, rad

The results are shown in figure 3.13b. Again the agreement is good, with some deviation arising from the approximations made in the derivation of the ideal diode equation.

3.14 Conclusions

Numerical drift-diffusion simulations are powerful tools for modelling solar cells. In order to simulate perovskite devices with electron, hole, and ionic mobile charge carriers in the active layer I developed a new simulation tool (codename: DRIFTFUSION) written in MATLAB. The simulation uses established fundamental semiconductor equations to model transport, generation, and recombination in devices. Initial conditions for p-i-n devices were developed based on analytical solutions to the Depletion Approximation for a p-n junction. Self-consistent boundary conditions were also presented for a p-i-n structured device. In order to solve directly for open circuit, a unique mirrored device approach was used and shown to output the same

VOC as in simulations using fixed potential boundary conditions. Modelling interfacial recombination between the perovskite and n and p-type layers was found to be made more physically meaningful by including recombination throughout the contact regions in p-i-n devices due to shifting of the point of maximum carrier overlap. Unique spatial and time meshes have been developed to account for high carrier concentration gradients at interfaces and non-linear transient device responses. Lastly, the results for a p-n junction from DRIFTFUSION were found to be indistinguishable from those obtained using an existing commercial drift-diffusion package and compare favourably to analytical results using the Depletion Approximation. Chapter 4

Experimental methods

4.1 Declaration of contributions

Xiaoe Li1 refined the device fabrication protocol in Section 4.3 and manufactured devices for the results chapters.

Andrew Telford2 designed, coded and built the TELTPV set-up described in Section 4.4.3.

Brian O’Regan3 designed, coded and built the original TRACER set-up described in Section 4.4.4.

1 Imperial College London, Kensington, London 2 The University of New South Wales, Sydney, Australia 3 Sunlight Scientific, Berkeley, California, USA

4.2 Abstract

Perovskite solar cells (PSCs) exhibit a unique optoelectronic response that spans from nanoseconds to hundreds or even thousands of seconds.[31, 33, 37, 111, 196] The disparate timescales over which electronic and ionic charge transport occurs presents a challenge to making meaningful experimental measurements. In response, and in collaboration with other experimentalists I developed two optoelectronic transient set-ups. This involved coding original and versatile experimental procedures in the Igor programming environment capable of capturing both the fast and slow response of devices. Also included in this Chapter are details on device fabrication and the

89 Chapter 4. Experimental methods 90 standard experimental set-ups that were used for making calibrated current-voltage scans.

4.3 Device fabrication

Devices for this work were obtained from a number of laboratories worldwide including Imperial College London, Stanford University, Swansea University and Ludwig Maximilians University of Munich. Whilst the precise approach followed by Imperial College London is given below, other labs used identical or similar methods to fabricate devices.

The planar bottom cathode devices had the following stack of layers: FTO glass/ compact-TiO2 (≈50 nm)/CH3NH3PbI3 (≈300 nm)/Spiro-OMeTAD (≈200 nm)/Au (80nm) with an active area of 0.08 cm2 (where FTO is fluorine-doped tin oxide; and Spiro-OMeTAD is N2,N2,N20,N20,N7,N7,N70,N70-octakis(4-methoxyphenyl)-9, 90-spirobi[9H-fluorene]-2,20,7,70-tetramine). The devices were prepared as described in Ref.[31]. The planar top cathode stack of layers was ITO glass/PEDOT:PSS

(30nm)/CH3NH3PbI3 (≈300 nm)/PCBM (≈85 nm)/Ca (20 nm)/Al (100 nm) with an active area of 0.1 cm2; the cells were prepared as described in Ref.[154] (where ITO is indium-doped tin oxide; PEDOT:PSS is poly(3, 4-ethylenedioxythiophene) :poly(styrenesulfonate and PCBM is phenyl-C61-butyric acid methyl ester)).

4.4 Characterisation

4.4.1 Current-voltage scans

For the results included in Chapter5, current-voltage ( J-V ) scans were made using a Keithley 236 Source Measure Unit and a xenon lamp solar simulator with AM1.5G filters (Oriel Instruments). The illumination intensity was adjusted to be equivalent to 100 mWcm−2 using a calibrated filtered Si photodiode (Osram BPW21).

As means to compare the degree of hysteresis between devices a hysteresis index, HI can be defined for a given scan rate as the ratio of the maximum power point of the reverse scan, Pmax,r, to that of the forward scan, Pmax,f .

P HI = max,r − 1 (4.1) Pmax,f Chapter 4. Experimental methods 91

VPS2 VPS3 Key to Electronic Symbols

I1 Q3 Photovoltaic Cell D1 D2 Variable Power Supply

Light Emitting Diode D1 Laser Diode V VM1 R1 V DAQ Transistor Switch 10Ω VM2 Voltmeter Q1 V Pulse Generator Q2 DAQ Data Aquisition Module VPS1

Figure 4.1: Transient optoelectronic measurement system circuit diagram. Generalised circuit diagram for the transient data acquisition systems. The yellow shaded area is the primary circuit which allows the cell to be electrically biased and switched between open and closed circuit. The light red shaded area is the laser circuit and the light blue shaded region is the white bias LED circuit.

This differs slightly from the HI introduced by Kim and Park.[197] Elsewhere, the TRACER system (Section 4.4.4) was used for J-V characterisation.

4.4.2 Transient optoelectronic measurement experimental set-up

Two bespoke rigs were used to obtain the transient experimental data presented in Chapters5-8: the Time-Evolving Laser Transient PhotoVoltage (TELTPV) and TRAnsient and Charge Extraction Robot (TRACER) systems. The chief difference between the two systems is that TELTPV uses laser excitation sources and is configured for higher temporal resolution acquisitions (up to 1 ns), whilst TRACER uses Light Emitting Diode (LED) pulse excitations and acquires at lower temporal resolution (max. 0.8 µs). Nonetheless, that switching of the LEDs, an applied bias and open/closed circuit condition transistors can all be directly controlled by the same Data Acquisition hardware module (DAQ) gives TRACER powerful automation capabilities. The circuit design used in the acquisition of experimental data for Chapters5-8 is similar in both cases and is shown in figure 4.1.

In the primary circuit (yellow shaded region) the solar cell is connected in parallel with variable power supply VPS1, transistor Q2 and voltmeter VM1 (1 MΩ input resistance), Chapter 4. Experimental methods 92 which is an input channel of the DAQ. For acquisition rates faster than 1.25 MSs−1, the direct input into the DAQ can be substituted for a digital oscilloscope, which transfers data directly to a PC via an Ethernet connection. When transistor Q11 is switched open the cell is at open circuit and VM1 measures the open circuit voltage. When both Q1 and Q2 are closed the cell is approximately at short circuit (a 10 Ω resistor remains in series) and the voltage across resistor R1 is measured by voltmeter VM2, allowing the current flowing through the resistor to be evaluated. With Q1 closed and Q2 open VPS1 can be used to apply a voltage to the cell, with the dark current measured by R1. The light red shaded region is the excitation pulse circuit. For the TELTPV rig, a laser excitation source is used which is controlled by a pulse generator that can be switched on or off by a transistor operated from a digital output on the DAQ. In the TRACER system the DAQ controls a transistor which directly switches power supply VPS2 in and out to power an array of LEDs. In both systems a ring of white LEDs is used as the bias light (blue shaded circuit). Due to the large spectral mismatch (the white LEDs cut off at λ > 720 nm) these are not accurate 1 Sun measurements. As opposed to calibrating the light intensity for the unique absorption properties of each device, the LED intensity is instead calibrated using a silicon photodiode to obtain the correct short circuit current density as the solar simulator and the illumination intensity is denoted 1 Sun equivalent (1 Sun eq) where applicable.

4.4.3 TELTPV technical specifications

Data acquisition was performed by a Tektronix DPO5104B digital oscilloscope and a National Instruments USB-6361 DAQ. The laser pulse was provided by a digitally modulated Omicron PhoxX+638 nm diode laser with a 100 Hz repetition rate. The laser spot size was expanded to cover the active pixel and the continuous wave intensity over the cell pixel area was approximately 550 mWcm2 during the pulse. The preconditioning bias was applied using the data acquisition card. The system was controlled by a custom Labview code.

4.4.4 TRACER technical specifications

The TRACER (TRAnsient and Charge Extraction Robot) is a flexible light and voltage controller that uses LED lights, MOSFET switches, and custom software running in the IGORTM control and analysis environment. Light was provided by a selection of white, blue, green, red, or NIR (735 nm) LEDs. LED intensity was controlled by two

1In reality these back-to-back transistors allow both forward and reverse biasing of the cell without a leakage current, but are shown as a single transistor on the diagram for convenience. Chapter 4. Experimental methods 93 separate General Purpose Interface Bus (GPIB) controllable power supplies. LEDs were switched on and off by MOSFET switches with <50 ns on/off times. The LEDs had ≤50 ns off times, and rise times of 1 to 2 ms. Switch control and cell voltage was supplied by a National Instruments USB-6251 DAQ board which has 16-bit resolution and 0.8 ms per point analogue-to-digital conversion. The USB-6251 voltage supply to the cell has a nominal slew rate of 20 V ms−1, meaning that switching times of ±1 V were sub-microsecond. Low capacitance MOSFETs (< 10 pF) were used to switch the cell between open and short circuit and to disconnect the voltage supply. J-V s recorded with the TRACER use white LEDs as the bias light source. In order to capture transient phenomena on vastly different timescales a bespoke set of procedures programmed in the Igor were developed. The core procedure VJL (Voltage-Current-Light Control) uses hardware-timed transient switching to allow preconditioning stages on the scale of seconds followed by fast transient acquisitions with microsecond resolution.

4.5 Conclusions

Perovskite devices require a unique approach to fabrication and characterisation. Highly versatile experimental instruments were developed in order to make transient optoelectronic measurements on PSCs. The requirement to acquire data on two vastly different timescales led to a number of innovations in systems control including a new code written in the Igor programming environment capable of synchronised switching of electrical and optical biases. 94 Chapter 5

Evidence for ion migration in perovskite solar cells with minimal hysteresis

5.1 Declaration of contributions and publication

The material for this chapter has been adapted from Ref.[175].

Andrew Telford1 aided with the experimental measurements.

Daniel Bryant2 and Xiaoe Li2 fabricated devices for the study.

Jenny Nelson2 contributed to the interpretation of the measurements.

Piers Barnes2 and Brian O’Regan3 designed the study.

1 The University of New South Wales, Sydney, Australia 2 Imperial College London, Kensington, London 3 Sunlight Scientific, Berkeley, California, USA

5.2 Abstract

The migration of mobile ionic charge has been proposed as a possible cause of photovoltaic current-voltage (J-V ) hysteresis in hybrid perovskite solar cells. A major objection to this hypothesis is that hysteresis can be reduced by changing the interfacial contact materials, which are unlikely to significantly influence the behaviour of mobile ionic charge within the perovskite absorber layer. Here, transient optoelectronic 95 Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 96 measurements combined with device simulation are used to show that electric field screening consistent with ionic migration can be observed both in devices that exhibit hysteresis as well as in those with ‘hysteresis-free’ architectures. p-i-n structure devices are modelled using the DRIFTFUSION simulation tool and hysteresis is only found to be reproducible when both intrinsic mobile ion densities of greater than 1017 cm−3 and high rates of interfacial recombination are present at the n and p-type contact interfaces. Recombination at these locations results in large losses of photocurrent during the forward J-V scan due to the presence of a reverse electric field in the perovskite layer, which drives minority carriers towards these recombination centres. During the reverse scan the field is found to be advantageous for charge extraction leading to improved J-V characteristics. Passivating interfacial recombination enables efficient charge extraction, irrespective of the field direction due to the high selectivity of the contacts, resulting in devices that do not show hysteresis.

5.3 Introduction

Recent simulations performed by Van Reenen et al. have shown that J-V hysteresis (known herein as ‘hysteresis’) in perovskite solar cells (PSCs) could only be accurately reproduced if both ionic migration and recombination via interfacial traps were present in devices.[184] In this chapter I present novel transient optoelectronic measurements that probe the strength and direction of the internal electric field across the perovskite layer in different architectures of PSC. The measurements directly indicate that ionic migration appears in devices both with and without hysteresis. The simulations reproduce the transient device behaviour over all relevant timescales (10−6 s −102 s). The results show that hysteresis is only observed in cases where high rates of recombination exist at the perovskite/contact interfaces of devices. During the forward J-V scan a reverse electric field in the bulk perovskite layer drives electronic charge carriers away from their respective transport layers and high concentrations of minority carriers accumulate at these interfaces. Where these interfaces act as recombination centres, the collection efficiency of the device is adversely affected. Conversely, where recombination at these interfaces is passivated carriers diffuse out of the device without significant recombination losses. Low degrees of hysteresis can, therefore, be explained as an artefact of low interfacial recombination and resultant high photogenerated carrier populations at forward bias, despite the presence of ionic migration. The work presented here experimentally confirms the prediction of van Reenen et al.[184] and indicates that the long timescale transient response of perovskite devices can be determined by the interfacial recombination rate coefficients and the electronic Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 97 properties of the contact materials, without necessarily requiring a change in ion concentration or mobility within the perovskite phase.

5.4 Methods

5.4.1 Current-voltage scan protocol

The J-V measurement protocol was as follows for both the light and the dark measurements: The cell was left at −1 V in the dark for approximately 30 s. For the light measurements the solar simulator shutter was then opened. The applied voltage was then swept from −1 to +1.2 V at a rate of approximately 40 mVs−1 (forward scan) and the current density measured, the optical shutter was then closed. The cells were then held at +1.2 V for a few seconds before the shutter was opened and the voltage swept back to −1 V at 40 mVs−1 (reverse scan).

5.4.2 Transients of the transient photovoltage (TROTTR)

To investigate the processes underlying hysteresis the evolution of open circuit photovoltage (VOC) was examined with steady-state illumination, after preconditioning devices with a fixed bias voltage Vpre. While monitoring the evolution of the VOC, devices were simultaneously excited with a series of short (500 ns) laser pulses to induce small perturbation transient photovoltage (TPV) signals. Figure 5.1a shows a schematic of the experimental timeline for these Transients of the Transient (TROTTR) measurements. Analysis of the transient photovoltage perturbations gives information about changes in the movement and recombination kinetics of photogenerated charges as the background VOC evolves.

PSCs are often preconditioned using an applied forward bias or illuminated open circuit conditions prior to measurement. This procedure changes the polarization of the device to a state in which higher efficiency values can sometimes be inferred from J-V measurements as compared with short circuit or reverse bias preconditioning.[31, 32] To explore this effect two preconditioning voltages were chosen for the TROTTR measurements in this study: 1. Short circuit dark conditions (Vpre = 0 V) where the voltage across the device is defined by the built-in potential between the contacts; 2.

An applied forward bias (Vpre = +1 V or +1.2 V). These two states form the starting conditions for the subsequent transient measurements.

Single exponential small perturbation photovoltage decays (∆V ) were seen in some devices, consistent with organic and dye sensitized solar cells. However, for many devices Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 98 a b Preset Probe Voltage Bias Voltage 0 Light 0 Bias Light 0 0 Light Pulse Photovoltage Transients dV dV Light 0 Pulse t [μs] t [μs] Dark Photocurrent Current Transient Preset Voltage Open Circuit Voltage Current 0 0

-1 0 1 -1 min-20 μs 0 30 μs Time [min] Time

Figure 5.1: Experimental timelines for optoelectronic transient measurements. (a) Transients of the Transient photovoltage (TROTTR) measurement. The device is held in the dark (the bias light state is represented by the yellow line and shaded regions) at a preset voltage (dark blue line) for 1 min before being switched to open circuit with 1 Sun equivalent illumination. During the VOC evolution the cell is pulsed with a 638 nm laser (red line and pink shaded regions) at 1 second intervals and the resulting photovoltage transients acquired. (b) Step-Dwell-Probe (SDP) photocurrent transient measurement: The device is held in the dark for 1 min at a preset applied voltage (Vpre = 0 V in this study), then switched to an applied measurement voltage Vprobe at which it is held for a further 20 µs, allowing the dark current (green dotted line) to stabilize. A 10 µs LED pulse (typically near-infrared) is then used to excite the cell and the transient photocurrent (solid green line) is recorded. the photovoltage transient decays could only be accurately fit using a bi-exponential function as has been reported previously:[31, 33, 198, 199]

−t/τ1 −t/τ2 ∆V = A1e + A2e (5.1)

where A1 and A2 are the amplitudes of the two components and t is time. In contrast to the established interpretation of the TPV decay time constant as a measure of carrier lifetime, in Chapter7 I instead relate the decay to the rate of transport in the absorber layer of PSCs. Accordingly, here a decrease in the TPV lifetime τ is taken as an indication of improved charge transport while an increase in τ is taken as a indication of repressed transport.

5.4.3 Step-Dwell-Probe transient photocurrent (SDP)

In order to probe the electric field in the dark, a second novel measurement protocol was implemented. The experimental timeline for the Step-Dwell-Probe transient photocurrent (SDP) is given in Figure 5.1b. As in the TROTTR protocol, devices were preconditioned Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 99

at a preset voltage Vpre = 0 V for at least 1 minute. The voltage applied to the cell was then switched to a probe value Vprobe = 0.4 V. After a dwell time tdwell of 100 µs to allow the dark current to stabilise, a 10 µs square wave LED light pulse was applied. This technique is similar to a method presented by Shi and co-workers in Ref.[174] where modulated voltage pulses were applied to hole transport layer-free perovskite devices in order to monitor the evolution of the internal electric field.

5.4.4 Simulation

Simulations were performed using the DRIFTFUSION tool described in Chapter3. The focus in this study was not to realistically simulate all aspects of the devices but to explore principal device behaviour. For simplicity the devices were simulated as p-i-n structures in which a 400 nm intrinsic perovskite layer was sandwiched by contacts approximated by 200 nm p-type and n-type regions with identical band gaps (see Chapter 3, Figure 3.1 for an example device schematic). The p-i-n approach can be justified for perovskite devices since the high mobile ionic charge densities are capable of fully screening the built-in field. Fields within the active layer are therefore variable and determined by the ionic charge distribution, making the precise choice of Vbi of lesser importance. The perovskite layer was set to contain a uniform density of 1019 cm−3 positively charged mobile ionic carriers (corresponding to I− vacancies for example) with a corresponding uniform density of negative static ionic species. As a first approximation, ions were confined to the intrinsic region of the device. The Inverted architecture was assumed to have no Shockley-Read-Hall (SRH) recombination in the contact regions, while the Standard c-TiO2 device was assumed to have high recombination rates in the contact regions, simulating interfacial recombination as discussed in Chapter3, Section 3.9.2. For simplicity the same SRH time constants (see Chapter3, Section 3.11, Equation 3.57) were used for electrons and holes in both n and p-type regions. A full list of the key device parameters used in this chapter is given in Appendix A.1.

5.5 Experimental Results

5.5.1 Current-voltage scans

Two common device architectures of CH3NH3PbI3 solar cells were examined: Inverted devices showing limited J-V hysteresis and Standard compact TiO2 cells showing significant hysteresis in the photovoltaic performance at room temperature. The devices architectures are reproduces in Figures 5.2a and 5.2b. As shown in Figure 5.2c, with a scan rate of 40 mVs−1, the current-voltage scan of the Inverted ITO/PEDOT:PSS/ Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 100

a Al b Au PCBM Spiro OMeTAD Rec. CH NH PbI 3 3 3 CH 3NH 3PbI 3 Rec. TiO PEDOT:PSS 2 ITO FTO

5 c Measured 5 d Measured ] ] 0 0 -2 -2 -5 -5

-10 -10 [mA cm [mA cm J J -15 -15

5 e Simulated 5 f Simulated 0 0

-5 -5 Current density, density, Current Current density, density, Current -10 -10

-15 -15

0.0 0.4 0.8 0.0 0.4 0.8 Voltage, V [V] Voltage, V [V]

Figure 5.2: Measured and simulated device current-voltage characteristics. (a) Inverted and (b) c-TiO2 perovskite solar cell device architecture stacks - blue arrows indicate locations of interfacial recombination (labelled Rec.). Measured J-V curves in the dark and under 1 Sun equivalent illumination scanned at approximately 40 mVs−1 in the forward (reverse-to-forward bias) and reverse (forward-to-reverse bias) directions for the (c) Inverted cell and (d) the c-TiO2 cell. Dashed orange and light blue curves show the reverse dark scan for top and c-TiO2 devices, respectively. Black curves show the forward 1 Sun scans, while dashed grey curves show the forward dark scan. The corresponding simulated J-V scans in each scan direction are shown for a p-i-n device structure with mobile ions, without (e) and with (f) recombination in the p and n-type contact layers. The simulated scan protocol was similar to the experimental measurement at 40 mVs−1 with a ion mobility of 10−12 cm2V−1s−1.

CH3NH3PbI3/PCBM/Al device had a power conversion efficiency of 9.3% and a low Hysteresis Index (HI, see Chapter4, Section 4.4.1) of 0 .05.

By contrast the FTO/c-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Au device exhibited a significant J-V hysteresis, with a reverse scan power conversion efficiency of 7.7% and an HI of 1.71 (Figure 5.2d). The simulated devices showed remarkably similar HIs for the same scan speed and similar scan protocol (see Chapter4 Section 4.4.1 for full details of experimental and simulated scan protocols), with HI’s of 0.00 and 1.81 for devices without and with high rates of recombination in the contact regions Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 101 respectively. The discrepancy in fill factor between experiment and simulation can be accounted for by the absence of both high series and low shunt resistances in the model.

5.5.2 Transients of the transient photovoltage

Figure 5.3a shows the photovoltage evolution of an Inverted PSC, preconditioned at Vpre = +1 V in the dark for 60 s prior to switching the cell to open circuit and simultaneously turning the bias light on. This preconditioning step is analogous to the forward bias, or equivalent ‘light soaking’ at open circuit, often applied to perovskite solar cells prior to measuring a J-V curve. After the initial development of the VOC to approximately 990 mV in less than 50 µs, a small increase in VOC (≈ 20 mV) follows during the subsequent 30 seconds of the measurement. Throughout the measurement there was no significant change in the shape (Figure 5.3b) and time constants (Figure 5.3a right-hand axis) of the transient photovoltage decays. This might be expected given that there is only a small change in the background VOC over the course of the measurement. Similarly stable results were also observed with this cell when it was preconditioned at short circuit (Vpre = 0 V) in the dark (Figure 5.4a). These observations are consistent with the absence of significant hysteresis seen in the J-V curves in Figure 5.2a.

In contrast, the c-TiO2 devices exhibited very different behaviour upon preconditioning at different bias voltages. Figures 5.3c and 5.3d show the VOC TROTTR measurement performed on a c-TiO2 architecture PSC, which showed significant hysteresis. Following a forward bias (Vpre = +1 V) preconditioning step in the dark, the VOC declines steadily from around 850 mV to 720 mV over the course of 40 s under continuous illumination. Although the magnitudes of the small perturbation transient photovoltage decays decreased, the time constants increased by a factor of approximately 2 during the course of the measurement. I interpret this as an indication of improved transport during the initial stages of the measurement as compared to the later stages. Figure 5.4c shows the evolution of the VOC for the c-TiO2 device following preconditioning at short circuit

(Vpre = 0 V) in the dark. In this case the photovoltage rises by around 300 mV from 400 to 700 mV over a period of 42 seconds. The simultaneous transient photovoltage signals in Figure 5.4b exhibited anomalous behaviour: the initial photovoltage transients during the excitation pulse showed a negative deflection. Following the end of the pulse the photovoltage recovered to positive values before decaying.

The initial negative deflection of the transient photovoltage measurements indicates the existence of a positive internal current within the device (note that the sign of normal photocurrent is defined to be negative, cf. Figure 5.2). This positive current and associated negative displacement voltage at open circuit during the light pulse Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 102 a b 1.1 15 V = +1 V V Inverted preset OC 0.8 τ 10 t = 0 s 1.0 [s x10 0.6 [V] [mV] 5 t = 40 s V OC τ -6

V 0.9 Pulse Δ 0.4 ] 0 0.2 0.8 0 10 20 30 40 0 1 2 3 4 5 Time [s] Time [ µs] c 0.9 10 -5 d 40

c-TiO 2 V = +1 V 30 preset

τ τ 1 [s] V 20 [mV] [V] 0.8 OC τ

2 V OC 10 Δ V -6 10 0 0.7 0 10 20 30 40 0 1 2 3 4 5 Time [s] Time [ µs]

Figure 5.3: Transient measurements of the open circuit photovoltage evolution with +1 V precondition. Slow timescale evolution of the open circuit voltage (left column, blue lines with circle markers) and corresponding small perturbation photovoltage transients (right column) for (a, b) Inverted and (c, d) c-TiO2 devices after preconditioning at Vpre = +1 V. The shaded pink region represents the duration of the laser pulse. The measurements were made using the protocol shown in Figure 5.1a. The time constants τ1 (gold curve, square markers) and τ2 (red curve, diamond markers) from single or double exponential function fits to the tails of the photovoltage decay transients are plotted on the right-hand axis in the left column. is apparent for the first 20 seconds of the bias light exposure at open circuit. Given the selective nature of the contacts an inverted diffusion current is an implausible explanation. Instead, I interpret the negative signal as evidence of an inverted electric field in the perovskite region resulting from the accumulation of ionic space charge at the contact interfaces during the preconditioning stage of the measurement. Further evidence for this hypothesis is given by the decrease in the TPV time constants, suggesting that transport towards the correct contacts is improving throughout the course of the measurement.

5.5.3 Step-dwell-probe photocurrent transients

Given this interpretation, the results in Figure 5.3 might suggest that mobile ions are only present in c-TiO2 architecture devices since the Inverted device did not Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 103 a b 1.0 Inverted V = 0 V 1.0 0.8 15 preset t = 0 s τ

VOC [ 10 0.6 µ s [V] ] 0.9 [mV] t = 40 s OC 5 0.4 V V τ Pulse Δ 0 0.2 0.8 0 10 20 30 40 0 1 2 3 4 5 Time [s] Time [ µs] c d 0.7 10 -4 20 Vpreset = 0 V c-TiO 2 10 0.6 VOC

τ τ [s] [V] 1 10 -5 0 [mV] OC 0.5 V V τ Δ -10 0.4 2 -20 10 -6 0 10 20 30 40 0 1 2 3 4 5 Time [s] Time [ µs]

Figure 5.4: Transient measurements of the open circuit photovoltage evolution with 0 V precondition. Slow timescale evolution of the open circuit voltage (left column, blue lines with circle markers) and corresponding small perturbation photovoltage transients (right column) for (a, b) an Inverted architecture device and (c, d) a c-TiO2 device following pre-biasing at Vpre = 0 V. Labels as described in the caption for Figure 5.3. exhibit a negative TPV deflection. To further probe the electric field in devices, SDP measurements were performed on both device types after stepping from short circuit

(Vpre = 0 V) to an applied forward bias (Vprobe = +0.4 V) in the dark. Based on previous estimates for ion diffusion coefficients in CH3NH3PbI3,[48] the dwell time was set to be sufficiently short (tdwell = 100 µs) to allow the dark current to stabilise without significant redistribution of ionic charge. By using an applied electrical bias, instead of an optical bias to generate a photovoltage, the charge carrier transport direction can be probed without flooding the device with photogenerated charges. The sign of the transient photocurrent reflects the direction of the dominant electric field in the cells. The results of the SDP measurements on the two architectures are given in Figure 5.5.

Both devices showed remarkably similar behaviour: The control measurements (Vpre

= Vprobe = 0 V) produced negative transient photocurrents, whilst switching to Vprobe = 0.4 V produced positive signals in both cases. This result is consistent with the negative photovoltage transients observed for the c-TiO2 device in Figure 5.4, which Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 104

a 1.5 b 1.5 V = 0.4 V V = 0.4 V ] 1.0 probe ] 1.0 probe

-2 -2 0.5 0.5 0.0 0.0 -0.5 -0.5

[mA cm [mA [mA cm [mA

J J -1.0 Vprobe = 0 V -1.0 Vprobe = 0 V -1.5 -1.5 0 5 10 15 20 0 5 10 15 20 Time [ µs] Time [ µs]

Figure 5.5: Step Dwell Probe (SDP) transient photocurrent measurements. Measurements were taken using a 10 µs, 735 nm, pulse after a 100 µs dwell time following switching from Vpre = 0 V to an applied forward bias in the dark (Vprobe = 0.4 V) following the protocol shown in Figure 5.1b. The solid red curve (a) and blue (a) curves show the positive photocurrent transients for Inverted and c-TiO2 cells respectively. In both cases, control photocurrent transients at short circuit are also presented (solid grey lines). The dashed lines show the corresponding simulated photocurrent transients at (Vprobe = 0.8 V). Note that the rise and fall times of the experimentally measured transients are limited by the resistor-capacitor (RC) time constant of the device and the switch-on time of the LEDs used to create the light pulse (≈ 2 to 3µs). result from a positive internal photocurrent in the device. This is strong experimental evidence that an accumulation of slow moving space charge is also present in the Inverted architecture devices, despite the absence of significant hysteresis in the J-V scan at room temperature.

5.6 Simulation results and discussion

As discussed in Chapter2, Section 2.3.2, there is compelling theoretical and experimental evidence that high concentrations (> 1019 cm−3) of mobile ionic defects are intrinsic to

CH3NH3PbI3.[37, 48, 110, 112, 118] The accumulation of these charged defects at the contacts of devices, owing to the internal electric field within the perovskite layer, has been used as a model to understand some hysteresis behaviour.[32, 184, 185].

Figure 5.6a shows the simulated energy level profiles and charge carrier density distributions of a device without interfacial recombination at short circuit equilibrium in the dark. The simulated data for the device including interfacial recombination under these conditions is virtually identical (data not shown). At equilibrium mobile ionic charges (Figure 5.6b, black curve) screen the cell’s built-in potential, originating from the difference in Fermi energies of the p and n-type regions (green and blue shaded regions respectively), resulting in a field-free region in the bulk of the device. Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 105 a 1.5 b 0 ] Initial -3 1.0

E cm -1 EFp CB 19 0.5 E -2 EVB Fn [x10 Energy [eV] ρ Final 0 -3 190 200 210 220 ] -3 15 p n 10 1.5 c Initial 10 10 1.0

10 5 0.5

Space charge density, charge density, Space Final 10 0 0 Carrier density [cm 0 200 400 600 800 190 200 210 220 Position [nm] Position [nm]

Figure 5.6: Simulated energy level diagram and charge densities for a device at short circuit and space charge profiles at the device interfaces. (a) Device without interfacial recombination at short circuit in the dark. Mobile ionic charge has drifted to completely screen the built-in potential between the p-type (green shaded region) and n-type (blue shaded region) contact regions. ECB (gold) EVB (purple) EFn (blue) and EFp (red) refer to the energies of the conduction band, valence band, electron quasi-Fermi level and hole quasi-Fermi level respectively. n (blue) and p (red) refer to the densities of electrons and holes, respectively. Space charge densities at the p-type/intrinsic interface for simulated devices (b) without and (c) with interfacial recombination. Ionic charge accumulates at the interface in the intrinsic layer creating regions of high density space charge. The black line indicates the charge density at equilibrium and shortly (< 100 µs) after illumination at open circuit. The dashed red and blue curve shows the space charge after ion migration to Ref. at open circuit under illumination.

Illuminating the device with interfacial recombination switched off with open circuit boundary conditions (simulating the experimental conditions in Figure 5.1a), results in quasi-Fermi level splitting and the development of an open circuit voltage within 100 µs of the illumination being switched on (Figure 5.7a). Initially the distribution of ionic defects remains very similar to that of the equilibrium state since they have not had time to move in response to the new potential across the intrinsic layer generated by the VOC. This ionic charge accumulation at the interfaces results in electrostatic potential minima, which are described herein as band valleys. As shown in Figure 5.7, high concentrations of photogenerated electrons and holes rapidly redistribute to fill these valleys, resulting in screening of the initial E-field generated by ion accumulation (Figure 5.6b).

Over a period of tens of seconds ions migrate away from the p-type region (Figures 5.6b, red dashed curve). This migration is driven by the high ion concentration gradient at the interfaces and the inversion of the field direction, which serve to drive ions back into the bulk by diffusion and drift respectively. There is an accompanying redistribution of Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 106

a b 0 ECB 0 E No Interface Rec. Fn -1 -1 E Fp -2 Energy [eV] Energy [eV] E Interface Rec. VB 18 ] -2 ] 10 -3 -3 17 10 p n 10 16 10 16 10 14 10 15 10 12 10 14 10 10 10 13 Carrier density [cm Carrier density 0 200 400 600 800 [cm Carrier density 0 200 400 600 800 Position [nm] Position [nm]

Figure 5.7: Energy level diagrams and charge densities for simulated devices at open circuit. Simulated devices (a) without and (b) with interfacial recombination at open circuit under illumination following short circuit in the dark. a, n (blue) and p (red) refer to the densities of mobile ionic charge, electrons and holes, respectively. Dashed and solid curves indicate initial and final states respectively. the electrons and holes throughout this time. The key observation is that despite this ionic and electronic charge rearrangement, the measured change in VOC is very small (≈ 1 mV, see Figure 5.8a), in agreement with the magnitude of the change seen in Figure 5.4a.

It is this same process of electronic screening of the mobile ionic charge that results in the absence of a VOC hysteresis in the simulated J-V curve shown in Figure 5.2e, despite ionic charge being in quite different configurations during forward and reverse scans (Figure 5.12 shows the resulting potential profiles).

When recombination in the contact layers was included, the simulation results replicated the slow evolution of the VOC and TPV time constants following dark short circuit preconditioning (compare Figure 5.4a with Figure 5.8c). From the simulated energy and charge distribution diagrams, shown in Figures 5.6c and 5.7b, it is apparent that band valleys are formed immediately following illumination. In this case photogenerated electrons and holes collected in the valleys rapidly recombine due to high rates of recombination in the n and p-type contact regions, and the electric field across the intrinsic layer remains unscreened. Since the presence of the photovoltage somewhat negates the built-in potential, the concentration of ionic defects at the contacts decreases as the defects migrate away (Figure 5.6c, dashed blue curve) until the E-field in the bulk of the cell is zero as in the previous example. At this point, the VOC reaches a plateau.

The evolution of VOC during this time is a complex combination of 1. Redistribution of Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 107 a b 1.1 No Interface Rec. 1.6 12 τ Vpreset = 0 V VOC [s x10 8 t = 0 s 1.2 [V] τ 1.0 [mV] -6 OC V 4 t = 40 s ] V τ 0.8 Δ Pulse 0 0 0.9 0 10 20 30 40 50 0 1 2 3 4 5 Time [s] Time [ µs] c d 0.7 1.2 20 Interface Rec. τ V = 0 V [s 10 preset 0.6 V 0.8

OC x10 [mV] [V]

0.5 V 0 -6 OC Δ ] V 0.4 τ 0.4 -10 Nion = 0 0.0 0 10 20 30 40 50 0 1 2 3 4 5 Time [s] Time [ µs]

Figure 5.8: Simulations of TROTTR measurements on devices with and without surface recombination with 0 V precondition. (a) Simulated open circuit voltage and TPV time constants τ, and (b) photovoltage transients for a device 19 −3 with mobile ions (Na = 10 cm ) and SRH recombination in the p and n-type contact layers switched off (τn,SRH = τp,SRH = ∞ s, labelled No Interface Rec.) and preconditioned with Vpre = 0 V. (c) and (d) Simulations using the same protocol for a device with high rates of SRH recombination in the contact regions (τn,SRH = τp,SRH = 2 × 10−15 s, labelled Interface Rec.). Dashed lines indicate time constants and TPV traces for a device without mobile ions (Na = 0, labelled τ0). ionic space charge; 2. Electronic charge rearrangement in response to ionic migration; and 3. An increase in the concentration of photogenerated charge carriers due to their displacement from fast SRH recombination centres in the contacts.

The simulation also reproduces the anomalous transient photovoltage behaviour observed at early times during the VOC evolution (Figures 5.8c and 5.8d). The negative transients are explained by increasing rates of recombination at the interfaces during the pulse.[200] Minority carriers are driven towards the ‘wrong’ contact by the inverted field where they recombine with majority carriers, temporarily reducing the chemical potential. Figure 5.9a shows the contributions from the chemical, ΦChem and electrostatic potentials ΦEl to the electrochemical potential ΦE−C, read-out from the contact interfaces (ΦE−C = EFn(xin) − EFp(xpi)) during the inverted transient. The chemical potential drop between the two interfaces indicates a change in the space charge in the associated depletion regions, which translates into a change in the electric Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 108

20 a c Φ Interfaces 10 El 0 ECB 0 E Φ -1 Fn -10 E-C EFp

-20 Energy [eV] Φ -2 Chem EVB

20 ] b -3 Boundaries p n 10 10 15

Potential [meV] 0 -10 10 10 -20 Carrier density [cm 0 200 400 600 800 0 2 4 Time [ µs] Position [nm]

Figure 5.9: Electrical and chemical contributions to the inverted photovoltage transients, electrostatic potential during J-V scan and simulated Step-Dwell-Probe energy level diagram and charge densities for an Inverted device. Electrical and chemical contributions to the inverted photovoltage transients read-out from (a) the interface locations, xpi and xin and (b) the simulation boundaries, x0 and xd. See Figure 3.1a for position labels. (c) Energy level diagram and charge densities after stepping to an applied forward bias of 0.8 V from short circuit in the dark for the Inverted device. Initial states (shown after 50 ms) are indicated by solid lines while dashed lines designate final states after 50 s of ion migration.

potential at the boundaries of the simulation (see Figure 5.9b), from where the VOC is read-out.

Simulation of the jump-to-voltage transient photocurrent measurements are shown in Figures 5.5 (dashed lines) and energy level diagrams and charge densities for the Inverted device are given in Figure 5.9. Changing the SRH rate coefficient in the contact layers makes little difference to the simulated current transients since there are no photogenerated charge carriers prior to the pulse. Due to the ion charge distribution, positive photocurrent transients following the step to a forward bias measurement voltage are obtained in the simulation for both devices with and without interfacial recombination, consistent with the measured results in both Inverted and c-TiO2 architectures.

5.6.1 Reduced ion densities

For completeness, simulations with zero and lower concentrations of mobile ions, with and without interfacial recombination, were tested. The results showed that mobile ion Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 109

17 −3 concentrations of Na > 10 cm are required to reproduce the observed behaviour. Figure 5.10a shows that, while an initial uniform density of 1017 cm−3 gave rise to screening of the field between approximately x = 210 nm and x = 440 nm at equilibrium (dashed lines), the asymmetry caused by a single ionic carrier type and associated ‘Giant’ Debye length at the perovskite/n-type interface resulted in the formation of a band valley further from the contact. Although a high recombination coefficient was used in the contact region, minority carriers were concentrated further from the contact 18 −3 19 −3 than in the Na = 10 cm and Na = 10 cm cases. This led to a lower overall recombination rate and resultant higher VOC. It was noted that decreasing the SRH time constant beyond a certain limit did not further reduce the VOC with saturation occurring at approximately 720 mV, 320 mV higher than the experimentally observed value. Furthermore, the negative TPV deflection could not be reproduced. These results suggest that, consistent with predictions and observations in Chapter2, Section 2.3.2, in the tested devices, > 1017 cm−3 ionic carriers are present in the perovskite phase. Figures 5.10c and 5.10d show a simulated device with 1018 cm−3 mobile ionic defects equilibrated at short circuit (dashed lines) and at open circuit under illumination (solid ] a b 20 Ion density 0 -3 10 10 17 p a n 15 -1 10 10 15 10 13 10 -2 10 11 10 [cm

Energy [eV] 17 -3 N = 10 cm -3

ion ]

-3 Carrier density [cm 0 200 400 600 800 0 200 400 600 800 Position [nm] Position [nm]

19 Ion density c d ] 10

0 -3 10 17 10 18 -1 10 15 10 13 10 17 -2

11 [cm 10 10 16 Energy [eV] 18 -3 -3 N = 10 cm 9 -3 ion 10 ] 0 200 400 600 800 Carrier density [cm 0 200 400 600 800 Position [nm] Position [nm]

Figure 5.10: Simulated effects of varying ion defect density and contact recombination. Energy level diagrams (left) and charge densities (right) for simulated p-i-n structure solar cells at open circuit with a uniform carrier generation profile of 21 −3 −1 G0 = 2.5 × 10 cm s . The recombination rates in the contacts were set to high values and intrinsic mobile ion carrier densities were chosen to be (a) 1017 cm−3, and (b) 1018 cm−3. Dashed lines indicate the equilibrium state at short circuit and solid lines indicate the initial state directly after the cell has been switched to open circuit under illumination. Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 110

lines). In order to reach the experimentally observed VOC, an increased SRH time −16 constant of τn,SRH = τn,SRH = 3 × 10 s was required. All other results in this case, 19 −3 however, remained similar to the case for Na = 10 cm due to shorter Debye lengths and effective screening of the built-in field.

5.6.2 Alternative recombination schemes

Other possible recombination schemes in the presence of ionic migration were also −17 investigated. Implementing SRH recombination (τn,SRH = 10 s) in a single contact (n-type) allowed the negative TPV transient to be reproduced (Figure 5.11b blue trace) using TROTTR protocol. a 0.8 c 5 SRH single contact BTB Contacts 0 BTB Contacts BTB Bulk 0.6 BTB All Layers BTB All Layers -5 [mV]

V -10

0.4 Δ BTB Bulk -15 SRH Single Contact -20 0.2 Open circuit voltage [V] 0 10 20 30 40 50 -1 0 1 2 3 4 5 Time [s] Time [ µs] c 0.9 d 1.0 Single Exp. 0.8 Measured Bi Exp. 0.7

voltage [V] voltage voltage [V] voltage 0.9 0.6

0.5

Open circuit Open circuit 0.4 0.8 0 10 20 30 40 0 2 4 6 8 10 Time [s] Time [s]

Figure 5.11: Voltage transient simulations under different recombination schemes and bi-phasic slow transient VOC experimental measurements. (a) Slow open circuit voltage transient simulations after the cell had reached equilibrium at short circuit (Vpre = 0 V) for four different recombination schemes with band-to-band (BTB) and trap assisted Shockley-Read-Hall (SRH) recombination mechanisms. (b) Initial photovoltage transients (t = 0 s) for the same recombination schemes trialled in the study. (c) a c-TiO2 device after preconditioning at short circuit for 60 seconds before switching to open circuit and (d) an Inverted device. The characteristic hump in this case is suggestive of a higher rate of recombination in a single contact. Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 111

The simulated VOC at time t ≈ 0 was 200 mV higher than that observed experimentally (Figure 5.11a, blue trace) and converging with respect to increasing SRH recombination rate coefficient. A number of factors could lead to this discrepancy between simulated and measured VOC such as lower-than-real rates of band-to-band recombination, a higher-than-real built-in voltage (which would alter the doping density in the contacts), and the uniform generation profile implemented in the simulation. The slow VOC transient also exhibited a characteristic hump occasionally observed in Inverted VOC transients (see Figure 5.11c and 5.11d for examples of experimental data). Implementing −2 3 −1 high rates of band-to-band recombination (kbtb = 10 cm s ) in all layers of the device also enabled reproduction of the negative TPV at early times in the TROTTR

(Figure 5.11b, purple trace). In this instance the slow VOC transient exhibited an initial negative deflection of around 120 mV after which it plateaued at approximately 100 mV above its initial value (Figure 5.11a, purple trace). Setting the bulk perovskite band-to-band recombination rate coefficient to 10−1 cm3s−1 allowed the experimental

VOC of the c-TiO2 device to be obtained but neither the negative TPV nor the slow

VOC transient were reproduced (Figures 5.11a and 5.11b, red traces). Finally, switching from high rates of SRH to band-to-band recombination in the contacts produced very similar results to when SRH was implemented in the main findings (Figures 5.11a and 5.11b yellow traces).

5.7 Understanding the J-V scan hysteresis

The absence of hysteresis in the simulated device J-V scan cannot be solely attributed to electrostatic screening, since a reverse field exists in the device during the forward scan as shown in Figure 5.12a. In this case the absence of interfacial recombination centres allows charge carriers to be extracted before significant recombination takes place: selective contacts and the low rate of band-to-band recombination of the perovskite phase enable high extraction efficiencies despite the presence of a reverse field. Figures 5.12a and 5.12b show that the simulated potential is similar both including and excluding interfacial recombination at low forward bias. When high rates of recombination are switched on at the perovskite/contact interfaces, charge carriers rapidly recombine after minority carriers are driven to these locations by the reverse field. The large negative gradients in the electron quasi Fermi levels at the p-type/intrinsic interface in Figure 5.12b compared with the flat gradients at the same location in Figure 5.12a are indicative of a strong drive for recombination. This process leads a reduced collection efficiency and associated low photocurrent during the forward scan. During the reverse scan the field drives charge carriers to the correct electrode for extraction and the collection efficiency is enhanced. Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 112 a b 0 No Interfacial Rec. 0 [eV] [eV] V -0.5 V -0.5

0 V fwd 1.1 V rvs -1 0.4 V fwd 0.8 V rvs -1 Electric !eld 0.8 V fwd 0.4 V rvs Electric !eld potential, 1.1 V fwd 0 V rvs potential, Interfacial Rec. 0 200 400 600 800 0 200 400 600 800 Position [nm] Position [nm] c d -0.4 -0.4 Interfacial Rec. -0.6 -0.6 -0.8 -0.8 [eV] [eV] Fn Fn

E -1 E -1 -1.2 -1.2

level, -1.4 No Interfacial Rec. level, -1.4 Electron Fermi Electron Fermi Electron 0 200 400 600 800 0 200 400 600 800 Position [nm] Position [nm]

Figure 5.12: Electrostatic potential and electron quasi Fermi levels in simulated devices during the current-voltage scan. Electrostatic potential during J-V scans (see Figure 5.2). (a) The device without surface recombination (τn,SRH = τp,SRH = ∞ s, labelled No Interface Rec.) and (b) the device with high −15 rates of interfacial recombination (τn,SRH = τp,SRH = 2 × 10 s, labelled Interface Rec.). Corresponding electron quasi-Fermi levels for devices (c) without (labelled No Interfacial Rec.) and (d) with interfacial recombination (labelled Interfacial Rec.). The simulation outputs are shown for different applied voltage points (0, 0.4, 0.8 and 1.1 V) during the J-V scans at 40 mV s−1 from reverse to forward bias (fwd) and from forward to reverse bias (rvs).

These processes are summarised in Figure 5.13, using two ionic charge carriers for greater clarity.

This analysis indicates that diffusion-driven devices will always suffer greater recombination losses in devices with high interfacial recombination rates since large minority charge carrier concentrations must be present at the contact interfaces for a diffusive current to flow. In devices with an advantageous electric field, charge accumulation at interfaces is not required for efficient current collection in this way.

The results shown here demonstrate that J-V hysteresis requires the combination of at least one mobile ionic species and localised recombination centres such that the changing ion distribution during the scan significantly alters the overall rate of recombination. To further confirm this hypothesis Inverted architecture devices in which the phenyl-C61-butyric acid methyl ester (PCBM) was replaced by defective ZnO nanoparticles were fabricated. The ZnO devices showed significant J-V hysteresis compared to their PCBM counterparts (see Figure E.1). Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 113 a Pre-fabrication b Short Circuit + - + - + - - + - + - + - + - + - + - + - + - - + - + + - + - + - + - + + - - + - + - + - + - + - + - + + - + - + - - + - + c Forward Scan c Reverse Scan - + - + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - + - + + - - + - + - + - + - + - - + + - + - + - + - + + - + - + Positive ionic charge + Hole Surface trapping state - Negative ionic charge - Electron Photon

Figure 5.13: Schematic of charge carrier distributions and recombination during J-V scan. (a) Prior to fabrication ionic charge is uniformly distributed in the perovskite layer. (b) Mobile ionic charges move to screen out the built-in potential arising from the difference in Fermi energies of the electron and hole transport layers, shaded blue and green respectively. Note: in the simulations, only a single ionic charge carrier is present. (c) During the forward scan minority carriers are driven towards the wrong contacts and recombine with majority carriers. (d) During the reverse scan, carriers are driven to the correct contacts and interfacial recombination is reduced.

While the discussion here has generally focussed on the effects of interfacial recombination, another possibility is that the electric field in Inverted architectures of device falls predominantly across the PCBM layer. PCBM is both nominally intrinsic and has a low dielectric constant implying that electric fields are not well screened by the material.[201] In this situation, the potential across the device would have less influence on the position of ions within the perovskite layer and hence the rate of recombination via the mechanism discussed above. Furthermore, this explanation can account for why PCBM devices generally have lower open circuit voltages than their Standard architecture counterparts: Interfacial recombination may in fact be similarly high during both the forward and reverse J-V scans. An alternative explanation is that recombination may occur directly between contact layers via pinholes, a process which also would not be expected to be significantly influenced by the ion distribution. Further work will be required to model full device stacks with PCBM transport layers to ascertain where hysteresis-free devices can be simulated with high rates of Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 114 interfacial recombination. Notwithstanding, the work presented here shows that for high performance PSC devices in which the transport layers are highly doped, the absence of interfacial recombination corresponds to an absence of J-V hysteresis.

5.8 Conclusions

In summary, this study confirms that CH3NH3PbI3 shows behaviour consistent with a mixed ionic/electronic conductor at room temperature. Transient photovoltage measurements on c-TiO2 devices, taken directly after preconditioning at short circuit, showed an anomalous inverted deflection, not observed in ‘hysteresis-free devices. This was taken as an indication of an inverted electric field in the active layer of the c-TiO2 architecture under illumination. Step-Dwell-Probe photocurrent transient measurements demonstrated that the effects of ionic migration can, in fact, be observed in both Standard and Inverted architectures of device in the dark where electron and hole concentrations are low.

Simulation of the measurements showed that J-V hysteresis, slow timescale changes in

VOC, and negative TPV behaviour can only be reproduced when both high intrinsic mobile ion densities of greater than 1017 cm−3 in the perovskite phase and high rates of interfacial recombination are present. Intrinsic mobile ion densities of this magnitude were found to be capable of fully screening the electric field in the active layer with space charge widths of less than 10 nm in the perovskite phase. This screening leads to a reverse electric field during forward the J-V scan which generates high minority carrier densities at the contact interfaces and accelerates recombination in devices with high interfacial recombination rate coefficients. During the reverse scan, the electric field is favourable for charge extraction, reducing interfacial recombination and accounting for the higher collection efficiency. In simulated devices that include high mobile ion densities but low rates of surface recombination electronic carriers diffuse out of the cell before recombining resulting in a high collection efficiency for both forward and reverse scan directions. This hypothesis is consistent with lower hysteresis in higher performing

Standard architecture c-TiO2 and mp-TiO2 devices. An alternative explanation for the absence of hysteresis in Inverted architectures considering the potential drop across the PCBM electron transport layer was proposed. While this can account for the lower general open circuit voltages in Inverted devices, further simulation work will be required to ascertain whether this is indeed the case.

The results presented here provide experimental confirmation of the predictions from simulations by van Reenen et al.[184]: both mobile ionic charge and interfacial recombination are required for hysteresis to be observed. In addition to demonstrating Chapter 5. Evidence for ion migration in PSCs with minimal hysteresis 115 the role of localised recombination in the presence of mobile ions in a semiconductor, this study demonstrates the viability of controlling the measurable consequences of this ionic migration. This suggests the interesting possibility of exploiting these effects for use in other electronic applications where a memory of previous operating conditions would influence device behaviour.[38, 39] 116 Chapter 6

Identifying recombination mechanisms in perovskite solar cells using transient ideality factors

6.1 Declaration of contributions

Daniel Burkitt1 initiated the study and fabricated devices.

Joel Troughton1, Daniel Bryant2, Jasper Law2 and Trystan Watson1 fabricated the devices.

The experimental data in this section was recorded by Jizhong Yao3.

1 SPECIFIC, Swansea University, Swansea, UK 2 Imperial College London, Kensington, London, UK 3 Hangzhou Microquanta Semiconductor Co., Ltd.

6.2 Abstract

The ideality factor, derived from the open-circuit voltage-dependence on light intensity, has classically been used to identify the dominant recombination mechanism in solar cells. Here, different architectures of perovskite solar cell are cyclically preconditioned and illuminated at increasing intensity to record the evolution of the ideality factor

117 Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 118 with time. Drift-diffusion simulations on p-i-n devices including mobile ions are used to interpret these transient ideality measurements. The simulation results indicate that the ideality factor can evolve in devices without a change in the dominant recombination mechanism. Established ideality theory is found to apply to the initial ideality value following a forward bias preconditioning step, owing to separation of the carrier concentrations at the contact interfaces. Furthermore, the evolution of the transient ideality curve resulting from ion migration is found to provide a signature for different recombination types enabling an accurate assignment of dominant recombination processes to perovskite devices. Five separate recombination schemes are explored and the results are correlated to experimental data acquired using open circuit voltage vs. light intensity (Suns-VOC), electroluminescence and photoluminescence measurements on three different architectures of perovskite device. Surface recombination is found to be the dominant recombination mechanism in a mp-Al2O3 architecture device, whereas a mp-TiO2 device shows transient idealities consistent with mixed bulk-surface recombination processes. The transient ideality curve in an Inverted architecture device is interpreted as indicating surface recombination via intermediate energy states. Finally, the results from a batch of less stable devices, showing a range of behaviours, are discussed. Idealities of less than one are interpreted as a sign of device degradation during the measurements. This work provides a new measurement tool for identifying the recombination mechanisms in devices in which fields in the active material are dominated by mobile ionic charge.

6.3 Introduction

Understanding the dominant recombination mechanisms in perovskite solar cells (PSCs) is critical to improving both device performance and stability, and advancing the field of research. In Chapter5 I showed evidence that both mobile ions and high rates of surface recombination are present in PSCs that exhibit high degrees of current-voltage (J-V ) hysteresis and observable long-timescale open circuit voltage 1 (VOC) transients.[175, 184, 200] The ‘light’ ideality factor , nid determined from the

VOC- dependence on light intensity, has classically been employed as a means of identifying the dominant recombination mechanism in silicon and organic absorber solar cells.[202–204] Applying this technique to PSCs with hysteresis is challenging owing to precondition-dependent slow transient changes in open circuit voltage. Accordingly, ideality factor measurements on this class of device also exhibit a time-dependence. Tress et al. circumvented this issue by conducting light intensity scanning sweeps in both increasing and decreasing directions and taking the VOC after 3 seconds

1Known variously as the ‘quality factor’. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 119 as a compromise between stabilisation of the VOC and irreversible degradation of devices.[205] As an alternative, Pockett et al. observed anomalously high ideality factors (nid > 5) by using the VOC value after the cell has reached steady-state.[206] To date, however, a detailed understanding of the time-dependent ideality factor in PSCs remains elusive.

In this chapter the DRIFTFUSION simulation tool described in Chapter3 is used to investigate the ideality factor in idealised p-i-n architecture solar cells with mobile ions in the active layer. The results show that both changes in the overlap of electron and hole populations, and the light intensity-dependence of the mobile ionic charge distribution, can alter the measured ideality without requiring a change in the dominant recombination mechanism of the device. Following this analysis, I show that the signature evolution of the ideality factor in a cell that has been preconditioned with a sufficient forward bias can be correlated to theoretical predictions for different recombination types. This framework is then applied to experimental measurements made on various device architectures and active layer compositions. In less stable devices, where irreversible losses occur during the measurement, the variance in initial-value ideality indicates that recombination processes are highly sensitive to processing conditions. In more stable devices the signature of the transient ideality correlates well to theoretical predictions from the model, suggesting that surface recombination via deep surface states is the dominant recombination mechanism in devices with mp-Al2O3 Standard orientation architectures. In Inverted devices the relatively stable idealities are taken as an indication of the field partially dropping across the PCBM layer. Correspondingly, surface recombination via intermediate (i.e. between shallow and mid-gap) energy states are inferred to be the primary recombination pathway in this architecture of device.

6.4 Background

6.4.1 Ideality factor

The ideality factor nid is an additional parameter included in the exponent of the ideal diode equation (Chapter3, Equation 3.62) to describe deviations in the J-V characteristic of a solar cell from ideal behaviour. The adapted expression is known as the Shockley diode equation or non-ideal diode equation:

  qV   J = JSC − J0 exp − 1 (6.1) nidkBT Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 120

Conventionally, the ideality factor of a solar cell is measured either from the dark J-V or, in order to avoid the effects of series resistance, by measuring the open circuit voltage as function of photon flux φ (known as a ‘Suns-VOC’ measurement). Using Equation 6.1, and assuming φ ∝ JSC, the ideality factor can be obtained from the slope of the VOC vs. ln(φ) curve:

qVOC ln(φ) = ln(J0) − ln(qcEQE) + (6.2) nidkBT where the intercept is dependent on both J0 and cEQE, a constant of proportionality describing the external quantum efficiency of the device. At open circuit the generation rate G is equal to the recombination rate U. By assuming that G ∝ φ, nid can also be expressed in terms of recombination, U and VOC:

q dVOC nid = (6.3) kBT d ln(U)

6.4.2 Zero-dimensional analysis

Ideality factors are typically related to recombination mechanisms and their associated reaction orders using a zero-dimensional theoretical framework. Firstly, a relationship 2 between the quasi Fermi-level (QFL) splitting, ∆EF and charge carrier density and n is defined such that:

   β  kBT np kBT n ∆EF = ln 2 = ln 2 (6.4) q ni q ni where kBT/q is the thermal voltage and ni is the equilibrium charge carrier density. β is thus a parameter defining the charge carrier density relationship to the perturbation of 2 the QFLs from equilibrium ∆EF. If n ≈ p, then np → n and β → 2. In this case ∆EF is split equally between the individual electron and hole QFLs (see Figure 6.1a). Where a majority carrier type exists, β → 1. This can be shown by expressing n and p in terms of a change in the electron ∆n and hole ∆p concentration from their quasi-equilibrium values n0 and po respectively:

n = (n0 + ∆n)

p = (p0 + ∆p)

2 In a zero-dimensional approximation, VOC = ∆EF. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 121 a b Intensity n = p p >> n E 0 0 0 0 Bias Light CB 0 EFn

∆EF ∆EF Preset EF0 Voltage Open Circuit Voltage 0 EFp E 0 VB Time [s]

Figure 6.1: Quasi Fermi level splitting and transient Suns-VOC experimental timeline. (a) Schematic of idealised quasi Fermi level splitting ∆EF in a zero-dimensional model. Where n = p, the Fermi levels split with equal magnitude from equilibrium EF0 under bias. Where one carrier is in excess (p >> n shown) the shift in the minority carrier Fermi energy dominates the change in ∆EF.(b) Transient Suns-VOC experimental timeline: the cell is preconditioned in the dark at a preset voltage Vpre for > 100 s. At t = 0 the bias light is switched on and the cell is simultaneously switched to open circuit. The VOC is subsequently measured for > 100s. The protocol is cyclically repeated at increasing bias light intensity and the ideality factor as a function of time is calculated.

If n0 << p0, and ∆n = ∆p >> n0:

np (n0 + ∆n)(p0 + ∆p) np0 2 = 2 ≈ 2 ni ni ni

Here, ∆EF is almost entirely influenced by the change in QFL position of the minority carrier (electrons in this example). A critical assumption is that for p-i-n structured devices charge carrier densities are assumed to be equal in the bulk and separated at the interfaces due to respective low and high field conditions. In a p-n junction, the same assumption is valid owing to high doping densities in the p and n-type regions.

Under the assumption that a single recombination order γ dominates the device, such γ that U = kγn , by substituting Equation 6.4 into Equation 6.3 the relationship between nid and γ can be described as:[207]

βk T    d B ln(n) − ln n(2−m) q q i n = id k T    (1−γ) B d γ ln(n) + ln kγ

It can be shown that this expression reduces to: Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 122

β n = (6.5) id γ

A full derivation of this result is given in AppendixF, Section F.1.

Table 6.1 summarises the reaction order γ, carrier density relationship to QFL splitting

β, and conceptual ideality factor nid, for different recombination mechanisms, trap energies, and carrier population overlaps. In the limiting cases, γ and β often cancel, resulting in an ideality of 1. SRH recombination with mid-gap trap energies is the only mechanism that results in a value of 2.

† Recombination Trap energy Overlap β γ nid mechanism Band-to-band N/A n >> p 1 1 1 Band-to-band N/A n = p 2 2 1 SRH Shallow n >> p 1 1∗ 1 SRH Shallow n = p 2 2 1 SRH Mid-gap n >> p 1 1 1 SRH Mid-gap n = p 2 1 2

Table 6.1: Reaction order γ, carrier density relationship to QFL splitting β and conceptual ideality factor nid for different recombination mechanisms, trap energies and carrier population overlaps. Note that in the case of shallow traps, the values of β, γ and nid tend towards those for band-to-band recombination. †Please refer to AppendixF for full derivations of the reaction orders given here. ∗γ = 1 (and thus nid = 1) in this situation only if n is held approximately constant.

A change in ideality as a function of time thus implies either a shift in the reaction order, for example due to a change from SRH dominated to band-to-band dominated recombination, or a change in the charge density dependence on internal voltage as defined by parameter β.

In working solar cells, non-radiative recombination via defect states in the bandgap is consdered the most important recombination pathway since it can be mitigated via passivation of defects, particularly at grain boundaries and interfaces.[42, 43, 208] Furthermore, while multiple inter-band defect levels may exist, those closest to the centre of the gap contribute to the largest recombination fluxes: in this instance the nt and pt terms in the denominator of the the SRH expression (Equation 3.57) are minimum. In devices with a built-in field, the population overlap at interfaces is likely to be small, hence idealities close to 1 are often correlated with surface recombination while idealities closer to 2 are identified with bulk recombination. In organic solar cells, experimental values of between 1 and 2 have frequently been observed suggesting a range of behaviours from surface to bulk recombination.[209–211] Idealities of > 2 have previously been attributed to a tail of states in the bandgap or non-linear shunt resistance, although Kirchartz et al. has shown that these processes were unnecessary Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 123 to reproduce unusually high idealities in thin devices where carrier population overlap is highly dependent on carrier densities.[212, 213]

6.5 Methods

6.5.1 Experimental

6.5.1.1 Devices

Measurements were performed on three separate device architectures:

1. Mesoporous Al2O3 (mp-Al2O3)

FTO/compact-TiO2/mp-Al2O3/CH3NH3PbI3/Spiro-OMeTAD/Au

2. Mesoporous TiO2 (mp-TiO2)

FTO/compact-TiO2/mp-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Au

3. Inverted

ITO-glass/PEDOT:PSS/CH3NH3PbI3/PCBM/LiF/Ag

The device stacks are illustrated in Chapter2 Figure 2.11. The aluminium oxide scaffold is inert and used for comparison with the n-type mesoporous titanium dioxide. Devices were fabricated according to the methods described in Chapter4, Section 4.3.

6.5.1.2 Suns-VOC

The light ideality factor of a solar cell is obtained by measuring the open circuit voltage as a function of light intensity φ and taking the slope of the VOC vs. ln(φ) plot (see Equation 6.2). The history-dependent nature of the open circuit voltage in perovskite devices required an adaptation of the conventional method. Figure 6.1b shows the experimental timeline for the transient ideality measurements presented herein. Cells were preconditioned at a preset voltage Vpre for greater than 30 s. At time t = 0 devices were switched to open circuit and simultaneously illuminated with a white LED bias light. The VOC was then measured for at least 30 s. The same protocol was repeated at different light intensities and the average slope of VOC vs. ln(φ) was used to calculate nid as a function of time. The TRACER system, described in Chapter4, was used to perform these measurements. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 124

6.5.1.3 Suns-EL

An alternative method to determine nid was also checked using electroluminescence (EL) vs current density measurements. EL was measured using a Shamrock 303 spectrograph ◦ combined with an iDUS InGaAs array detector cooled to −90 C. As with the Suns-VOC measurement, prior to acquiring the EL intensity the device was preconditioned at a voltage, Vpre. After this polling step, the device was set to a constant injection current level ranging between J = 1.25 and 1250 mA cm−2. A pair of optical lenses were used to focus the resulting EL signal allowing sampling at time intervals of between 0.02−0.1 s. The EL signals were normalised relative to the value at t = 1 s. The relative emission flux for each injection current was subsequently determined by integration over the whole spectrum. The EL emission flux φEL can be described as:

  J0,rad qV φEL = exp (6.6) q kBT which is proportional to the radiative component of the total recombination. Here, J0,rad is the dark saturation current for the radiative recombination only. Since the radiative component of recombination comes from band-to-band transitions, Equation 6.6 has an effective ideality factor of unity (see Table 6.1). We can combine Equation 6.6 with the diode Equation 6.1 (where, with no illumination, JSC = 0) to obtain an expression for ideality factor nid of the device:

dφ (t) n (t) = EL (6.7) id d ln(J)

6.5.2 Simulation

The DRIFTFUSION simulation tool, described in Chapter3, was used to simulate p-i-n structure devices with a 1.6 eV bandgap. Here, the mirrored-cell model and open circuit boundary conditions, described in Subsection 3.3.1, were used to solve directly for the open circuit voltage as a function of time. Generation was implemented as a uniform profile with the rate set to 2.5 × 1021 cm−3s−1 throughout the length of the 400 nm intrinsic layer. VOC transients were simulated for 9 different light intensities from 0.1 to 25.6 Sun simeq. The range was wider than in the experimental measurements to account for uncertainty in the values of the density of conduction and valence band states. As elsewhere, a single mobile ionic species and a uniform static distribution of counter-ions were included in the active layer. Low rates of second order band-to-band recombination were implemented in all layers, whereas first order Shockley-Read-Hall Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 125

(SRH) recombination was used in different locations of the device dependent on the recombination scheme. The five different recombination schemes studied are summarised in Table 6.2. Scheme no Dominant recombination Trap energy mechanism 1 Band-to-band only N/A 2 Surface* SRH Shallow 3 Surface* SRH Mid-gap 4 Bulk SRH Shallow 5 Bulk SRH Mid-gap

Table 6.2: Recombination mechanisms simulated in the study. Shallow traps were set to be in the bandgap and 0.2 eV away from the conduction and valence bands in the n and p-type layers respectively. *Surface recombination is defined as recombination at the intrinsic/contact layer interfaces as discussed in Chapter3, Section 3.9.1.

Auger recombination processes were neglected for the purposes of this study. As discussed in Chapter3, Section 3.9.1, surface recombination was simulated by selecting low SRH time constants throughout the contact regions. The coefficients were adjusted dependent on the trap energy to simulated approximate experimental VOC values (approximately 0.8 V - 1.1 V at 1 Sun). Where trap levels are described as ‘shallow’ the energies were set to be in the bandgap, 0.2 eV away from the conduction and valence bands in the n and p-type layers respectively. While symmetrical trap distributions such as this are highly improbable in reality, this approached enabled the study of location-dependent recombination in isolation of asymmetric electron and hole recombination rates. The parameter sets used for the different recombination schemes can be found in AppendixF, Section F.3.

6.6 Results

6.6.1 Transient ideality factor simulation

Example VOC transients for increasing light intensity simulated using for Scheme 3 (surface SRH with mid-gap trap energies) are given in Figures 6.2a and 6.2b. It is apparent that, following Vpre = 1.1 V, the VOC gradually decreases over the course of 100 s from an initially higher value towards a steady-state value for each light intensity.

In contrast, following Vpre = 0 V the VOC rises towards a steady-state value. As described in Chapter5, the redistribution of mobile ions leads to a change in both the electric field and charge carrier density profiles on the timescale of seconds. This influences the rate of recombination and distribution of charge in the space charge region, leading to changes in the open circuit voltage. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 126

Simulated Measured 1.0 1.0 0.8 0.8 0.6 a 0.6 d φ 0.4 0.4

ge [V] 0.2 V 0.2 V = +1.2 V pre = +1.1 V pre 0.0 0.0 b e 0.8 0.8

ircuit Volta ircuit 0.6 0.6 0.4 0.4

Open C 0.2 0.2 V V = 0 V pre = 0 V pre 0.0 0.0 0 20 40 60 80 100 0 5 10 15 20 25 30 Time [s] Time [s] 1.1 c 1.0 f n =1.0 1.0 id n 0.9 id =1.0 0.9 n id =1.7 n =1.6 0.8 0.8 id n =2.0

ltage [V] ltage id en Circuit t = 0.01 s t = 0 s t = 0.1 s Vo 0.7 n 0.7 Op =1.9 id t = 25 s n = 2.3 t = 1 s t = 90 s id t = 28 s 0.6 -3 -2 -1 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 ln( φ) ln( φ)

Figure 6.2: Temporal evolution the open circuit voltage and ideality factor in simulated and measured devices. (a) The simulated evolution of the VOC for different incident photon fluxes (φ = 0.1 − 25.6 Sun sim eq.) from a Vpre = 1.1 V dark equilibrium solution and (b) from a Vpre = 0 V dark equilibrium solution. In this example Recombination Scheme 3 was used in the simulation (Table 6.2, band-to-band in bulk and interfacial SRH recombination via mid-gap states). (c) Simulated VOC vs incident photon flux, φ, at different delay times following illumination for Vpre = +1.1 V (panel (a)) shows an increase in ideality factor with time. Examples of the evolution of measured VOC for different incident photon fluxes (φ = 0.01 − 2 Sun eq.) with time following dark preconditioning with (d) Vpre = +1.2 V and (e) Vpre = 0 V for a mesoporous Al2O3 cell. (f) The measured VOC plotted against incident photon flux, φ, at different delay times following illumination for the Vpre = 0 V case shown in (a) shows a similar increase in ideality from 1 to 2 as in the simulation. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 127

Vales for the open circuit voltage for the Vpre = +1.1 V simulation (Figure 6.2a) are plotted against the logarithm of the light intensity in Figure 6.2c for three separate times (0 s, 25 s and 90 s) to illustrate how the ideality factor is changing in the device.

Qualitatively similar results were observed in the mp-Al2O3 device as shown in Figures 6.2d - 6.2f, as discussed in Section 6.6.2 below.

Figure 6.3a shows the evolution of the ideality factor with time for the two preconditioning voltages. For Vpre = +1.1 V nid shows a gradual increase from around

1 to 1.9 over 100 s. With the device preconditioned at short circuit (Vpre = 0 V), the nid starts at a relatively high value (≈ 1.8) and decreases to a minima at around t = 8 s before recovering to the same final value as for the forward bias precondition. It was noted that, in many of the simulations for Vpre = 0 V, the open circuit voltage did not show a purely linear relationship to ln(φ) and, as such, the mean gradient was used for the ideality value. The shaded regions in Figure 6.3a indicate the large range of nid values over the light intensities studied in the Vpre = 0 V case and Figure 6.3b shows example data taken at t = 0.26 s, with a range of nid value from approximately 1.1 to

2.4. In contrast, the VOC vs. ln(φ) curves following the Vpre = +1.1 V precondition were highly linear.

Classically, these shifts in nid might be interpreted as either a change from shallow to deep trap dominated SRH recombination (see Table 6.1) or else as a shift in the reaction order from second to first order (e.g. a shift in the balance of radiative to surface recombination). Since this recombination scheme uses fixed mid-gap trap energies the first explanation can be disregarded, although in real devices where a change in the ion defect density could influence local trap distributions and energies, this possibility must also be considered. Owing to high rate coefficients, SRH recombination at the contact interfaces dominates recombination in the simulation at all times for both pre-bias conditions (Figure 6.3c, dashed curves). It follows that a shift in the reaction order also cannot account for the changes in nid observed in the simulations. I note that there is significant net reduction in the fraction of recombination arising from band-to-band recombination during the simulation (Figure 6.3c, solid curves), which is consistent with the reduction in photoluminescence (PL) signal with illumination time observed over a timescale of seconds in the mp-Al2O3 device (Figure 6.3d). This decrease in PL suggests that non-radiative recombination is accelerating throughout the measurement. I now consider how the changing charge distribution within the device could affect such a change.

Energy level diagrams, and electron n, hole p, and mobile ionic a charge density profiles, for initial and final states of the slow VOC transient with φ = 1.6 Sun simeq. are shown in Figure 6.4. In the case of Vpre = +1.1 V (Figure 6.4a), a low internal Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 128

a b Idea

id 2.5

n t 0.6 = 0.26 s 2.4

V factor, lity 2.0 pre = 0 V [V]

OC 2.0

V factor, factor, 1.5 0.5 1.6

en circuit

1.0 age, n

V Op 0.4 n = +1.1 V 1.2 id

Ideality pre volt = 1.8 0.5 id 10 -1 10 0 10 1 10 2 -2 -1 0 1 2 3 4 Time [s] ln( φ) c 400 d ] 14 -1 300 mp-Al O

s Interfacial SRH 2 3 200 ]

6 -2 12 100 10 φexcite cm 0.8 15 φ 8 0.6 bias 6

untsx10 0.4 Intensity 4

BTB, Radiative PL

0.2 [Co

Recombination Recombination 2 Rate [x10 0.0 0 0 1 2 3 0 2 4 6 8 Time [arb] Time [s]

Figure 6.3: Transient ideality and recombination rates in the simulated device as a function of time and pholotoluminescence decay in the mp-Al2O3 device. (a) Average Simulated ideality factor nid as a function of time for Vpre = +1.1 V (red) and Vpre = 0 V (blue) pre-bias conditions. Shaded areas indicate the range of values due to (b) the light intensity-dependence of the ideality curve. The example curve was taken at t = 0.26 s.(c) Simulated integrated band-to-band (BTB, radiative, solid curves) and interfacial SRH recombination (dashed curves) fluxes as a function of time for a cell with recombination scheme 3 for Vpre = 1.3 V. (d) The integrated photoluminescent flux density vs time for different excitation flux densities following the application of Vpre = 1.2 V to a mp-Al2O3 cell shows a qualitatively similar decay.

field during the pre-biasing step (due to pre-biasing close to the built-in voltage, Vbi = 1.3 V) led to a relatively small accumulation and associated depletion of mobile ionic charge at the p-type/intrinsic and n-type/intrinsic interfaces (herein known as the contact interfaces) respectively. The electronic charge carrier profile, in this instance, was similar in character to a simulated device under identical conditions without mobile ions as shown in Figure 6.4b. Notable is the separation of n and p populations at the contact interfaces (x = 200 nm and x = 600 nm) as compared with the final state of the device (Figure 6.4d) where screening of the internal field results in n and p being similar values.

For Vpre = 0 V, the preconditioning step resulted in a comparatively large accumulation of ionic space charge at the contact interfaces (ρ > 1019 cm−3). Switching the device to open circuit under illumination resulted in an inverted electric field and an associated build-up of minority carriers at the contact interfaces, accelerating recombination as previously described in Chapter5. As shown in Figure 6.4c, the initial charge carrier Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 129

0 E a CB 0 b E -1 Fn -1 EFp E Energy [eV] -2 VB -2 ]

-3 p n 10 16 10 16 10 14 10 14 12 10 12

Carrier 10 10 10 10 10 density [cm

V = +1.1 V No ions

] 2 pre 2 -3 1.5 t = 1 ms 1.5

cm 1 1 19 0.5 0.5 [x10 Ion density Ion 0 0 200 400 600 200 400 600 Position [nm] Position [nm]

0 0 c ECB d

-1 E -1 Fn EFp -2 -2 Energy [eV] EVB ] -3 10 16 10 16 10 14 10 14 12 12

Carrier 10 10 10 p n 10

density [cm density 10 10

V = 0 V t ] 2 pre 2 = 100 s -3 t = 1 ms 1.5 1.5 cm

19 1 1 0.5 0.5 [x10 Ion density Ion 0 0 200 400 600 200 400 600 Position [nm] Position [nm]

Figure 6.4: Energy level diagrams and charge density profiles for open circuit voltage transient. Energy level diagrams, electron n and hole p and mobile ionic a charge densities after switching to open circuit with φ = 1.6 Sun simeq., for (a) a device preconditioned at Vpre = +1.1 V at t = 1 ms following illumination, (b) a device without mobile ions at steady-state, (c) a device preconditioned at Vpre = 0 V at t = 1 ms, and (d) a device with mobile ions at t = 100 s, following ion migration and the establishment of a new quasi-equilibrium state. (Note Vpre= 0 V and Vpre = +1.1 V are in identical states after 100 s). Green and blue shaded areas designate p and n-type regions respectively. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 130 profile in the active layer is an inversion of that seen in the Vpre = +1.1 V case. The situation in the contact regions is more complex in this instance. At the interfaces the populations are separated whereas within approximately 20 nm of the contact regions there is a point of intersection due to the selectivity of the contact. As discussed in Chapter3, Subsection 3.9.2, in a heterojunction device the position at which n = p would be the interface itself. The final quasi-equilibrium state, following ion migration and the establishment of a new quasi-equilibrium state, was identical to that for the Vpre = +1.1 V (Figure 6.4d).

The similarity between the initial state of the device with mobile ions following a forward bias (Figure 6.4a) and the device without mobile ions (Figure 6.4b) is highly significant since it implies that the established theory, discussed in Section 6.4.2, can be applied to the former. The initial value of the transient ideality following forward bias nid,FBi is close to 1 (Figure 6.2b), which is the expected value for ‘surface recombination’ in the conventional analysis. The nid,FBi value does not, however, provide information regarding the energy level of the trap (see Table 6.1). This additional information is encoded in the evolution of the transient ideality curve. As ions migrate to screen out the internal field, the carrier overlap increases in the contact regions and the ideality tends towards 2 as would be expected for mid-gap traps with n = p. More formally, following a sufficient forward bias, the parameter β, describing the charge carrier relationship to Fermi level splitting, always increases from 1 to 2 at the interfaces. In some cases, such as where SRH surface recombination via mid-gap traps is dominant, the ideality is sensitive to this change. In other cases, for example where bulk recombination dominates, the ideality factor will remain largely unaffected. For surface recombination with shallow trap energies the reaction order also changes from 1 to 2 (see AppendixF for details) cancelling the change in β and leading to no overall change in ideality (nid = 1 throughout). The transient ideality curve obtained after forward biasing, therefore, provides a signature for different recombination mechanisms in the device.

Figure 6.5a shows simulations for nid vs. time after preconditioning, at Vpre = +1.3 V for the recombination schemes described in Table 6.2. Full details of the parameters used in each scheme are given in AppendixF, Section F.3. Significantly, each signature is distinct from the others enabling the unique assignment of different recombination mechanisms to devices. The initial and final nid values can be correlated to those given in Table 6.1 for changes in β from 1 to 2 for surface recombination processes and for β = 2 for bulk recombination processes as described above. Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 131

id 2.0 a V b

n = 1.3 V pre 2.5 1.8 mp-Al O mp-TiO 2 3 1.6 Bulk SRH Surf SRH 2.0 2 mid shallow 1.4 1.5 ent Ideality, 1.2 Band- Inverted Surf SRH mid to-band V = 1 - 1.2 V 1.0 1.0 pre Bulk SRH shallow

Transi 10 -1 10 0 10 1 10 2 10 -3 10 -2 10 -1 10 0 10 1 Time [s] Time [s]

Figure 6.5: Simulated and experimental transient ideality factor measurements following forward bias preconditioning. (a) Simulated nid as a function of time after pre-biasing at Vpre = +1.3 V for different recombination schemes. The initial value after forward biasing correlates to the nid expected in a conventional solar cell without mobile ions. The evolution of the nid provides a signature for different recombination schemes in perovskite devices. (b) Experimental apparent idealities after pre-biasing at Vpre = 1 − 1.2 V for three different architectures of perovskite device. Experimental data used with kind permission of Jizhong Yao.

6.6.2 Experimental results and discussion

Figure 6.5b shows transient ideality measurements for the three architectures of device analysed in the study using the protocol described in Figure 6.1b following preconditioning at forward bias. Transient idealities curves following Vpre = 0 V are presented in AppendixF, Figure F.1 for reference although their interpretation questionable owing to the light intensity dependence seen in Figure 6.3 and the anomalous high experimental initial values (nid > 3). For this reason the following analysis focusses on the forward bias case.

The current-voltage scans for the same devices are given in Figures 6.6a - 6.6c. The initial nid values obtained using the Suns-VOC measurements were cross-checked with electroluminescence as a function of injection current (Suns-EL), an example of which is given in Figure 6.6d. While the Suns-EL results for the mp-Al2O3 and mp-TiO2 devices were consistent with those of the Suns-VOC, the Inverted device showed a higher nid than expected (nid = 1.7 as compared to 1.1). This is likely due to the longer capture window

(0.1 s) making the measurement more representative of the steady-state condition (nid ≈ 1.6).

After forward biasing, the mp-Al2O3 device exhibited a transient ideality signature (Figure 6.5b, green curve, circle markers) similar to that of a surface recombination dominated device with deep traps (Figure 6.5a, pink curve, circle markers) with a transition from nid ≈ 1 to nid ≈ 2. This recombination scheme also reproduced key features of the VOC transient measurements for the mp-Al2O3 device (see Figure 6.2 Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 132

] -2 a b c m 0

mp-Al 2O3 mp-TiO 2 Inverted

[mAc -5

J

-10

-15

-20

Current Density, 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Voltage, V [V]

d 30 e 7 mp-Al 2O3

] 25 mp-Al O 6 6 n 2 3 id = 1.0 20 mp-TiO J 5 2 injection 15 Initial EL n = 1.7 4 id 10 Inverted

EL intensity EL 3 [Counts x10 5 n = 1.7

log10(EL Intensity) id 0 0 0.04 0.08 0.12 0.16 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Time [s] log10(Injection Current)

Figure 6.6: Example current-voltage scans for the three different architectures of device studied. Current-voltage scans under AM1.5G conditions for (a) mp-Al2O3,(b) mp-TiO2 and (c) Inverted devices. The measurement begins at forward bias and scans back to reverse bias (red curve). The forward scan (blue curve) is then performed. The black curves with square symbols are steady-state current curves. The scanning rate was 0.15 Vs−1.(d) Integrated electroluminescent emission flux density vs time for different injection current densities, J = 107, 213, 426, 533 −2 and 853 mA cm following the application of Vpre = 1.1 V to a mesoporous Al2O3 cell.(e) The initial values measured using EL after forward biasing for the mp-Al2O3 and mp-TiO2 correlate well with those measured using Suns-VOC cf. Figure 6.5. Data reproduced with the kind permission of Jizhong Yao. for comparison). As discussed in Chapter5, the large hysteresis in the device’s J-V scan (Figure 6.6a) is consistent with high rates of surface recombination at the contact interfaces. While computational studies on defective CH3NH3PbI3 crystals have not shown deep trapping states with easily accessible activation energies[123, 124], the

MAPI interface with the highly defective TiO2 compact layer is likely to contain a high density of inter-gap states as commonly observed in dye sensitised solar cells.[173] Furthermore, recent studies have shown that recombination rates at the hole transport layer interface are significantly influenced by doping.[214] As such, the MAPI interface with the lithium-doped Spiro-OMeTAD may also be contributing strongly to the recombination kinetics.

The transient ideality factor profile of the mp-TiO2 (Figure 6.5b, light blue curve, Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 133 triangle markers) following Vpre = 1 V shows an intermediate behaviour between surface (Figure 6.5a, pink curve, circle markers) and bulk SRH recombination (Figure 6.5a, dark blue curve, triangle markers) via mid-gap traps. One possible explanation for this is that at the Spiro-OMeTAD interface the charge populations are separated (β ≈ 1), while within the mesoporous region and at the c-TiO2 interface, the charge populations are approximately equal (β ≈ 2) following a forward bias precondition. The scaffold is likely to introduce deep trapping states in the region of this extended interface, leading to a reaction order of γ ≈ 1. If both interfaces have similar SRH time constants then a steady-state ideality of 1 < nid < 2 is expected: While the device is still dominated by surface recombination, the overlapping electron and hole populations (n ≈ p) throughout the mesoporous region leads to an ideality factor greater than 1. The reduced hysteresis in the cells J-V curve by comparison with the mp-Al2O3 device (Figure 6.6b) also suggests that recombination is dominated by a mix of surface and bulk effects. Further work incorporating the scaffold region into models will be required to test the above hypotheses.

Finally the low hysteresis (Figure 6.6c), Inverted device showed an initial ideality of nid

≈ 1, increasing to a final value of nid ≈ 1.6, consistent with surface recombination via intermediate energy trap states (Figure 6.5, grey and pink curves). This is somewhat contradictory to the findings in Chapter5 although the device exhibited poor J-V characteristics and may not be wholly representative of this architecture of device. As discussed in Chapter5, whilst hysteresis appears to require surface recombination, the converse is not necessarily true i.e. surface recombination may be dominant in low hysteresis devices yet recombination may not be strongly influence by the distribution of ions. Further modelling on full devices with distinct contact energetics will be required to investigate whether a surface recombination device can also exhibit low hysteresis J-V characteristics.

The origin of initial idealities greater than 2 following forward biasing is unknown although simulations of devices dominated by bulk SRH with deep traps did yield idealities up to nid = 3 follow preconditioning at Vpre = 0 V (Figure 6.5a, red curve). Idealities of greater than 2 were also observed in a simulated device dominated by bulk SRH recombination via mid gap energy states at high light intensities with the same pre-bias step where the ion mobility was switched off prior to illumination (Figure 6.7a).

This can be explained by the capacitive effect of the reverse electric field: At low light intensities charges are driven into electrostatic potential minimums which act as charge reservoirs. As the light intensity is increased photgenerated carriers begin to flood the device reducing the strength of the field and the proportion of charge driven to recombination centres (Figure 6.7b). This generation rate-dependent charge density Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 134

a b ] 0.65 18 -3

I 10 n 2.4 deali id 25.6 Sun 10 16 0.55 2.0 ty fa

10 14 ctor 0.45 V 1.6 OC 12 , , 10 0.1 Sun n n id = 1.58 1.2 id φ 0.35 10 10

-2 0 2 4 [cm density Electron 0 200 400 600 800

Open circuit voltagecircuit [V] Open ln(φ) Position [nm]

Figure 6.7: Simulated nid and electron density profile for a device with a static ion distribution. (a) Suns-VOC (black) and nid (gold) curves for a cell following preconditioning at short circuit (Vpre = 0 V). Prior to illuminating the device the ion mobility was switched off. While the average ideality factor is 1.58, the curve exhibits a non-linear relationship between VOC and ln(φ), with nid > 2 at high light intensities. (b) Electron charge density profiles for different light intensities (0.1 to 25.6 Sun simeq.) for simulated device dominated by bulk SRH recombination via mid gap trap energies. profile enables the ideality factor to exceed zero-dimensional theoretical predictions. The screening effect at high light intensity also reduces the ionic drift current and results in the lower VOC transients time constants observed in Figure 6.2a as compared to lower intensities. The negative idealities observed in the simulated devices with shallow traps (Figure 6.5a, yellow curve) and those with extremely high rate coefficients in the contact regions (Figure 6.8) are extreme examples of this distortion effect.

Although not included in our simplistic model, the effects of chemical reactions at interfaces and degradation in real devices must also be considered. These could include a reduction in the quantity of ionic charge or the creation and/or annihilation of trapping states resulting in a change in the effective doping at the interfaces (provided that these do not act as recombination centres).

I note that, while these results appear consistent with those from Chapter5, less stable devices showed wide-ranging and inconsistent behaviour. For a batch of 6 planar compact TiO2 cells manufactured in a different laboratory to those described above, initial idealities after forward biasing were 0.97 < nid,FBi < 4.22 (¯nid,FBi = 1.78, σSD =

1.16), while for mp-TiO2 architectures the values were 0.04 < nid,FBi < 1.12 (¯nid,FBI =

0.75, σSD = 0.62, for 6 devices). Mesoporous TiO2 devices with triple cation active layers showed greater consistency with initial idealities after forward biasing of 0.40 < nid,FBi < 1.04 (¯nid,FBI = 0.77, σSD = 0.32, for 3 devices). The values close to 1 here might suggest that recombination at the compact TiO2 layer interface is more significant than at the mp-TiO2/CH3NH3PbI3 interface in these devices. However, the unexpectedly high nid and standard deviation values make questionable the application of the theoretical framework presented herein. Degradation during measurements could Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 135 a b ] id 2.0 0.8 n V = 0 V 1.5 0.7 pre 1.0

eality, V pre = +1.1 V 0.6 0.5 φ φ

uit voltage [V 0.0 0.5 -0.5 V = 0 V pre 0.4

Transientid -2 -1 0 1 2

10 10 10 10 10 Open circ 0 5 10 15 20 25 30 Time [s] Time [s]

Figure 6.8: Simulated nid and VOC transients for a device with high rates of band-to-band recombination in the contact regions. (a) Simulated average ideality factor, nid for Vpre = +1.1 V (yellow) and Vpre = 0 V (green) pre-bias conditions as a function of time for a device with high rates of band-to-band recombination in the −2 3 −1 contact regions (kbtb = 1 × 10 cm s ). The shaded regions indicate the range of values. (b) Transient VOC curves at different light intensities for Vpre = 0 V illustrating the switch from negative to positive ideality.

account for particularly low ideality factors since the VOC transients were measured in order of intensity from lowest to highest; If degradation was continuous throughout the measurement VOCs would be lower than expected at higher light intensity, reducing the measured nid. Further work on high performing, stable devices will be required to ascertain whether or not the signature transient idealities in Figure 6.5 can be used to accurately identify the dominant recombination mechanism in perovskite devices more generally.

6.7 Conclusions

I have shown, using drift-diffusion simulations on p-i-n structured devices, that changes in the measured ideality factor as a function of time can be attributed to the movement of mobile ions within the intrinsic layer. The rearrangement of ionic space charge both alters the overlap of electron and hole populations and changes the open circuit voltage as a function of time at differing rates dependent on light intensity. While the results presented here do not preclude the possibility that trap energy and distribution in the device may be dependent on the ion distribution, anomalous idealities, including nid

< 0 and nid > 2, were observed in simulated devices where the reaction order is in the range 1 - 2. In cases where devices have been preconditioned with a sufficient forward bias, the initial ideality factor can be interpreted using an established zero-dimensional framework. Furthermore, the transient ideality following forward bias preconditioning provides a signature curve for different recombination mechanisms. These signature ideality curves can be used to infer the dominant recombination mechanisms in real-world Chapter 6. Identifying recombination mechanisms in perovskite solar cells using transient ideality factors 136 devices. The results indicate that the measured mp-Al2O3 and mp-TiO2 devices were dominated by surface recombination via deep trapping states. The measured Inverted device showed results consistent with surface recombination via intermediate energy level trapping states. Finally, less stable devices exhibit a range of behaviours implying that particularly low idealities could be a strong indication of degradation during the measurement. Together these findings provide new insight on the correct interpretation of ideality measurements in PSCs and a valuable new tool for verifying the dominant recombination mechanism in devices. Chapter 7

Reinterpreting small perturbation transient photovoltage measurements in p-i-n devices

7.1 Declaration of contributions

Piers Barnes1 provided conceptual insight for the study. Mohammed Azzouzi1 assisted in measuring the high light intensity TPV data.

1 Imperial College London, Kensington, London, UK

7.2 Abstract

Transient photovoltage (TPV) and charge extraction (CE) methods have gained research attention as methods to quantify both charge carrier lifetimes and the power law relation between carrier density and the rate of recombination in solar cells. The established analysis for these techniques is based on a zero-dimensional model. In this chapter the DRIFTFUSION simulation tool is used to assess the validity of this model in p-i-n devices with mixed electronic/ionic conducting intrinsic layers at steady-state. The

TPV rate coefficient, kTPV, relationship to bulk carrier density is investigated in a simulated device dominated by first order interfacial recombination. The results are compared and contrasted with experimental data collected from a c-TiO2 architecture perovskite solar cell (PSC). At open circuit voltages (VOC) below approximately 0.9 137 Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 138

V, kTPV in the simulated device exhibits a linear dependency on charge density that, under the conventional analysis, would be misinterpreted as evidence of a second-order recombination process. Recombination in this regime is found to be limited by the thermionic emission of majority carriers from the contact layers to the absorber/contact interfaces. The high reaction order recorded for the measured device suggests that a similar mechanism may be partially accountable for the result. At higher VOCs charge is confined to the intrinsic layer and recombination is limited by diffusive transport to the contacts. An analytical model is proposed based on a solution to the diffusion equation, which also offers a plausible explanation for the bi-exponential TPV decay character in poorer performing devices. Lastly, mobilities are investigated using fixed open circuit voltage simulations in devices with interfacial recombination. The results suggest that the TPV decay in PSCs at high open circuit voltages may predominantly be a measure of transport.

7.3 Introduction

The power conversion efficiency of a solar cell is limited by recombination processes that reduce the fraction of photoexcited charges extracted from the device. Consequently, methods to quantify both average charge carrier lifetimes and the power law relation between the average carrier density and the rate of recombination (the reaction order) are essential to improving device performance and progressing solar cell research. Transient optoelectronic measurements for this purpose, have gained research attention in recent years owing to their relative accessibility and versatility. These methods, which include transient photovoltage (TPV) and differential capacitance charge collection (DCCE) were originally developed to measure charge carrier lifetimes and carrier densities in Dye Sensitised Solar Cells (DSSCs).[173] The optoelectronic response at open circuit in DSSCs is well modelled using zero-dimensional kinetic theory since the electric field in the mesoporous TiO2 phase is considered to be zero.[168] Transient techniques have also been applied to bulk heterojunction organic photovoltaic (OPV) and organic small molecule (OSM) devices, where the built-in electric field plays a critical role in exciton dissociation and charge extraction.[160, 215–218] More recently, these techniques have been used to measure carrier lifetimes and reaction orders in perovskite solar cells (PSCs).[31, 115, 155, 198, 199, 219–222] Typical TPV lifetimes in PSCs are on the order of microseconds in contrast to time-resolved photoluminescence measurements where lifetimes of 10s - 100s of nanoseconds have been recorded.[75, 97, 98, 222] The discrepancy between the order of magnitude of the lifetime measured using the two techniques remains unexplained. While zero-dimensional and equivalent circuit models[31][223] have been proposed to explain recombination processes in PSCs, a systematic theoretical Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 139 study of TPV using a one-dimensional drift-diffusion simulation has not been performed to date. As shown in Chapter5, PSCs can exhibit reverse currents and inverted photovoltage transients when stepped between bias conditions, producing a reverse field across the active layer (see Chapter5, Figure 5.4 for example measurements). Such states cannot be accurately modelled with zero-dimensional theory. To a first approximation, devices at steady-state with sufficiently high mobile ion densities would be expected to be field-free in the ion-conducting absorber layer. Given that charge carrier densities are uniform in field-free regions, the question arises as to whether a zero-dimensional model is appropriate for PSCs at steady-state.

7.4 Background

TPV measurements have frequently been used in combination with charge extraction (CE) techniques to determine rate coefficients and reaction orders in perovskite devices.[31, 115, 155, 199, 219–222] The theoretical framework used to interpret these measurements is based on a zero-dimensional (0-D) kinetic model in which electron and hole populations are assumed to act like dimensionless gases:[224] Charge transport, capacitive effects, drift currents and localised regions of recombination are neglected. In this section, the established 0-D theory is reviewed and expressions for idealised small perturbation open circuit voltage transients are derived.

7.4.1 Derivation of a zero-dimensional kinetic model of recombination

Under the assumption that the parabolic band and Boltzmann approximations are valid,1 the charge densities of electrons n and holes p for an intrinsic semiconductor with intrinsic carrier density ni and Fermi energy EFi are given by:[42]

  EFn − EFi n = ni exp (7.1) kBT

  EFi − EFp p = ni exp (7.2) kBT

Using Equations 7.1 and 7.2, the open circuit voltage VOC in a field free, zero-dimensional material, where electron and hole charge carrier concentrations are equal (n = p), can be expressed as:

1Criteria for validity: the band edges can be modelled as parabolic functions and the quasi Fermi levels should be > 3kBT from their respective bands.[42] Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 140

  2kBT n VOC = ln (7.3) q ni

Here, n is dependent on the generation rate G and the recombination model. For example, in an idealised device with band-to-band recombination only, the recombination rate U is given by:

2 2 U = kbtb(n − ni )

where kbtb is the band-to-band recombination coefficient. At open circuit steady-state, generation is equal to recombination (U = G) leading to:

 1/2  1/2 G 2 G n = + ni ≈ (7.4) kbtb kbtb

During a small perturbation measurement a small additional charge density ∆n << n is injected into the device such that the state of the system is not significantly altered. In a transient photovoltage measurement this charge is generated by an excitation light pulse imposed onto a background bias light (Figure 7.1a).

Figure 7.1b is a schematic showing how the relaxation of the excess electron carrier density after a pulse results in a change in the electron quasi Fermi level (QFL). The recombination of excess charge with rate constant krec, results in an associated decay of the QFLs. In this 0-D representation, the open circuit voltage is defined by the difference in electron and hole QFLs (qVOC = EFn − EFp) and decays with the same rate constant as the charge. In the most general case, following a small perturbation ending at t = 0, d∆n the rate of change of addition charge dt can be expressed using a small perturbation rate constant, kTPV (see Ref.[225], pp. 101 − 103 for full derivation):

d∆n ≈ −k ∆n (7.5) dt TPV

   X αi αj k = k nαi pαj + (7.6) TPV ij n p ij where i and j are summation indices and the exponents αi and αj can take any value

(including non-integers). kij is the associated rate coefficient for the reaction order. The solution to Equation 7.5 is an exponential: Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 141

a Bias Light 0 Excitation 0 Pulse V k t [mV] ∆ = Aexp(- rec ) V

Open Circuit ∆ Voltage 0 0 0 Time Time

b g(E) E CB E E Δn f(E) F(n+Δn) Fn k rec Energy E V n f(E) Fn OC g(E) E Fp E Density of states g(E), Probability of occupation f(E) , i Charge density n

Figure 7.1: Transient photovoltage experimental timeline and excess charge relaxation after an excitation pulse for an intrinsic semiconductor. (a) Experimental timeline and (b) Illustration of the delta open circuit voltage (∆V ) obtained from the small perturbation. The exponential decay rate constant of ∆V equates to krec in the 0-D model: (c) The electron carrier density n in the conduction band is the integrated product of the density of states function (g(E)) and the R ∞ probability of occupation function (f(E)): n = 0 f(E)g(E)dE. An excitation pulse adds an additional electron density ∆n. After the pulse ends, the excess charge recombines with rate coefficient krec, which can be read out as a change in the electron quasi Fermi level, from EF(n+∆n) to EFn. A similar process happens with holes in the valence band. EFi indicates the position of the equilibrium Fermi energy.

∆n = exp(−kTPVt) for t > 0 (7.7)

Using Equation 7.3, the change in open circuit voltage ∆VOC produced by a light pulse introducing an additional charge ∆n can be expressed as:

2k T  ∆n 2k T ∆n ∆V = B ln 1 + ≈ B OC q n n

Substituting for ∆n using Equation 7.7 leads to: Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 142

2k T ∆V = B exp(−k t) for t > 0 (7.8) OC qn TPV

A similar method can be used to find the change in charge carrier density and voltage rise during a pulse of duration tpulse:

∆G ∆n = (1 − exp(−kTPV(t + tpulse))) for − tpulse < t <= 0 (7.9) kTPV

2kBT ∆G ∆V = (1 − exp(−kTPV(t + tpulse))) for − tpulse < t <= 0 (7.10) qn kTPV where ∆G is the additional generation rate from the pulse light.

In experimental measurements, a single empirical reaction order γ, with the corresponding rate constant kγ, is typically assumed to dominate recombination such that:

γ U = kγn (7.11)

Accordingly, the slope of the log(kTPV) vs. log(n) plot can be used to determine γ:

log(kTPV) = (γ − 1) log(n) + log(γkγ) (7.12)

7.4.1.1 Differential capacitance charge extraction (DCCE)

TPV measurements are often coupled with differential capacitance charge extraction (DCCE) measurements to obtain information about the carrier density relationship to light intensity. This in turn enables the reaction order to be extracted from the slope of the log(kTPV) vs log(n) plot. DCCE uses the rate of change of voltage and charge obtained from small perturbation transient measurements to obtain the capacitance as a function of VOC.[226, 227]

dQ dQ dt C(VOC) = = (7.13) dV dt dVOC

The initial rise of a photovoltage transient generated by the same pulse is used to calculate dt/dV (Figure 7.2a, inset). dQ/dt is simply the current generated by the excitation pulse and can be obtained from a transient photocurrent peak with the device Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 143

a b ] 30 -2 20 25 15 dV/dt 15 dQ/Ad t 20 0.24 Suns 10

[mV]

V

V] 5 15 ∆ 10 0.53 Suns 0 [m 10 -10.0 -9.0 -8.0 V 5

1 Sun time [μs] [mAdensity cm ∆ 5 τ 0 0 Pulse Pulse -5 -10 0 10 20 Current 0 20 40 60 80 Time [ μs] Time [ μs]

Figure 7.2: Example transient photovoltage and photocurrent measurements on a perovskite solar cell. (a) Example transient photovoltage measurements at increasing bias light intensity. The decay time constant is the reciprocal of the rate coefficient (kTPV = 1/τTPV). The gradient of the initial TPV dV rise dt is used to calculate the differential capacitance as a function of open circuit voltage. (b) Example transient photocurrent measurement made at short circuit. Figure adapted from Ref.[228]. at short circuit or reverse bias (see Figure 7.2b). In the experimental measurements presented herein, the photocurrent showed a non-linear relation to the pulse intensity implying that recombination was non-trivial at short circuit. This is likely due to field-screening int he active layer by mobile ions at steady-state: diffusive currents require charge accumulation at the contact interfaces, leading to losses where interfaces are not well passivated as discussed in Chapter5. An alternative method was used to estimate dQ/dt under the assumption that the excess generation rate increased linearly with pulse intensity (G ∝ φpulse). The rate of injected charge was then calculated using a JSC value of 14 mAcm−2, measured using the reverse J-V scan (see Figure 7.4) under 1 Sun eq. intensity. This method can be justified since the electric field across the active layer of the perovskite at steady-state is assumed to be zero due to field screening by the high intrinsic mobile ion densities. The possibility of field-dependent generation in the active layer is thereby inconsequential except for within the Debye layer regions (< 2 nm)[185] at the interfaces. Given that the red LED excitation source is weakly absorbed by the 400 nm perovskite layer, generation in these interfacial regions can be neglected.

The capacitance of the device is conceptualised as consisting of two components:

A constant geometric capacitance Cgeo and a chemical capacitance Cchem that is the charge stored per unit electrochemical potential2 in the density of states of the semiconductor.[173] Cgeo can be estimated using a parallel plate capacitance model:

  A C = 0 r (7.14) geo d

2In the zero-dimensional case this is a chemical potential Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 144

where 0 is the permittivity of free space, r is the dielectric constant (or relative permittivity) of the dielectric material, A is the area of the plates, and d is thickness of the dielectric. Cgeo is subtracted from the measured total capacitance to give Cchem.

Empirically, the Cchem vs VOC is often fitted with an exponential function, which is subsequently integrated with respect to VOC to obtain the charge Q:[226]

Z Q(V ) = Cchem(V )dV (7.15)

Q can then be divided by the active layer volume to yield the average carrier density n. In the ideal case, the slope of the VOC vs. ln(n) slope should equal 2kBT/q (see Equation 7.3). Deviations from this slope in DSSCs have typically been interpreted as an exponential tail of states in the band gap.[226] In this work, a trapezoidal integration is used to calculate n rather than an exponential fit since the density of states function of the device is unknown.

7.5 Methods

7.5.1 Experimental

Current-voltage (J-V ) scans, transient ideality, and TPV measurements were obtained using the experimental set-ups described in Chapter4, Sections 4.4.2- 4.4.4. The measured devices were Standard FTO/c-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Au

(c-TiO2) planar architecture (see Chapter2, Section 2.11) and fabricated according to the protocol described in Chapter4, Sections 4.3.

7.5.2 Transient photovoltage simulations

The DRIFTFUSION simulation tool described in Chapter3 was used to simulate transient photovoltage measurements on a p-i-n structure dominated by recombination at the intrinsic/contact interfaces. Unless otherwise stated, the parameter set described in AppendixB was used. For simplicity, a mid-gap SRH trap energy was chosen with electron and hole time constants set equal to one another (kSRH = 1/τn,SRH = 1/τp,SRH). 12 −1 A relatively high rate constant of kSRH = 10 s was chosen for the n and p-type layers in the simulation to model surface recombination (see Chapter3, Subsection 3.3.1, for rationale) and the mobility of both electrons and holes was fixed at 1 cm2V−1s−1. For simulations including mobile ionic vacancies in the intrinsic layer, the initial conditions of the simulation use a uniform mobile ionic carrier density of 1019 cm−3 following Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 145 an estimation of 2 × 1019 cm−3 Schottky defects at room temperature by Walsh et al.[114] A equal and uniform charge density was used to simulate static counter-ions as in Chapters5 and6. Prior to recording the simulated transients solutions were run until the open circuit voltage was stable. To achieve this state, the ionic charge was accelerated by using an ion mobility of 10−8 cm2V−1s−1. As in Chapter6, here a total absorbed photon flux of 1017 cm−2s−1 is defined as 1 Sun simeq., with a uniform generation profile used throughout. The excitation pulse generation profile was similarly set to be uniform and adjusted using a Newton-Raphson iteration to obtain a voltage perturbation of between 3−4 mV.3 A polynomial function was used to fit the ln(∆V ) vs. t curve obtained from the TPV simulation and a decay rate constant kTPV extracted. During the excitation pulse the ion mobility was switched off to reduce computational complexity and accelerate solving times.4

As discussed in the introduction, an established experimental methodology uses the slope of the log(kTPV) vs. log(n) plot to determine the power law relationship between kTPV and n. In such cases, n is typically determined using a charge extraction experiment. A reaction order γ is then inferred, under the assumption that a single reaction order γ−1 is dominant, kTPV ∝ n (see Equation 7.6). In the simulation the reaction order is defined by the recombination scheme. In Sections 7.6.1 and 7.7.4, the quantity kcalc is used to describe the rate coefficient calculated using the appropriate reaction order from the simulated transient photovoltage decay constant kTPV. This is the equivalent to asking the question: If the reaction order were known, what is the implied value of the rate coefficient? In the case of first order SRH recombination kcalc = kTPV. For second order band-to-band recombination, kcalc = kTPV/2nCE. Here, nCE is the average carrier density in the absorber (intrinsic) layer obtained by subtracting the equilibrium carrier density and averaging the value of n in the intrinsic layer. If 0-D kinetic theory is an appropriate model for TPV then kcalc should be equal to kinput.

7.6 Results

7.6.1 Verification of the simulation tool

To verify the simulation tool with the 0-D model described in Subsection 7.4.1, transient photovoltage simulations were conducted on a single 100 nm slab of intrinsic material

3It was noted that for perturbations below this magnitude, numerical errors from relatively small changes in charge densities reduced the resolution of the transients. 4I note that in unpublished simulation work, at very low light intensities resulting in low TPV decays 3 −1 −12 2 −1 −1 (kTPV< 10 s , with µa = 10 cm V s ), the movement of ions can influence the rate of the TPV decay in counter-intuitive ways. Given that the simulated and measured decay rates contained herein are outside of this range, these effects are ignored. Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 146 a b E 3 0 CB )

V -3 E 2 -0.5 Fn log10(

[mV] -4 -1 V 1 0 1 2 3 EFp ∆ Time [µs] -1.5 0 Energy [eV] E Drift di"usion 0D model VB -2 0 5 10 15 0 20 40 60 80 100 Time [ µs] Position [nm]

Figure 7.3: Zero-dimensional theoretical prediction versus 1D numerical drift-diffusion model transient photovoltage. (a) Transient photovoltage traces using the zero-dimensional model described in Section 7.4.1 (black dashed line) and the DRIFTFUSION simulation (blue solid line). (b) Steady-state energy level diagram for the spatially uniform simulated device. with a bandgap of 1.6 eV. The current at the boundaries was set to zero for all carriers representing perfect blocking contacts and the generation rate was set to G = 1.89×1021 cm−3s−1 based on the AM 1.5G solar spectrum and a step function absorption. The −10 second order band-to-band recombination coefficient was arbitrarily set to kbtb = 10 cm3s−1 and SRH recombination was switched off. The results of the comparison between theory and simulation are given in Figure 7.3.

The steady-state charge carrier densities and open circuit voltage obtained using DRIFTFUSION agreed to within 10 decimal places with the analytical values calculated 15 −3 using Equations 7.4 and 7.3( n = 4.35 × 10 cm , VOC = 1.08 V (3 s.f.)). The transient photovoltage perturbation obtained using an additional generation rate of ∆G = 3.79 × 1020 cm−3s−1 also behaved as predicted, with the value of the rate constant extracted from fitting the TPV decay correct to within 3 significant figures 5 −1 of that calculated using the band-to-band rate constant (kcalc = 8.70 × 10 s , see Equation 7.6).

7.6.2 Light intensity dependence in devices dominated by surface recombination

I have previously shown strong evidence that both mobile ions and interfacial (surface) recombination are present in CH3NH3PbI3 solar cells with Spiro-OMeTAD and

TiO2 transport layers (see Chapters5 and6).[175] Figure 7.4a shows an example current-voltage scan for the planar device used in the study; the J-V shows a large hysteresis, typical of compact-TiO2 (c-TiO2) devices and indicative of high rates of interfacial recombination. As further evidence that surface recombination is the dominant in this cell, the transient ideality factor nid was measured following Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 147 a b

] 2.0 -2 5 Simulated id

Dark n 1.8 Surface Recombination 0 1.6 -5 1.4 Experimental

-10 1 Sun eq. 1.2 Transiet ideality, 1.0 Current density [mAcm density Current -15 0.0 0.4 0.8 10 -3 10 -2 10 -1 1 10 10 2 Voltage [V] Time [s]

Figure 7.4: Current-voltage and transient ideality measurement for the −1 c-TiO2 device used in the study.(a) J-V scan at 0.2 mVs for the c-TiO2 device used for the kcalc vs. φ study cf. Figure 7.5. The large hysteresis is indicative of high rates of surface recombination (see Chapter5). ( b) Transient ideality factor nid measurement on the same device after 50 s at Vpre= +1.1 V for each light intensity (gold solid curve with cross markers). The transition from an ideality of nid= 1 to around 1.8 confirms that interfacial recombination via deep trap states is predominant in the device (see Chapter6). The grey dashed curve shows the simulated data for a p-i-n device with high rates of surface recombination. Time units for the simulated data have been scaled accordingly to account for the accelerated mobility of the ionic defect population. preconditioning the cell at 1.1 V in the dark (Figure 7.4b). The ideality factor evolved a value of nid ≈ 1 to an end value of nid ≈ 1.8, consistent with the surface recombination signature described in Chapter6, Section 6.6.2.

Figures 7.5a and 7.5b show the results for the light intensity dependence of kTPV in simulated and real-world devices respectively. In the experimental measurement, kTPV showed an approximately linear relationship with light intensity (Figure 7.5b). While the experiment was limited to a maximum light intensity of φ ≈ 14 Sun eq., the simulation enabled rate constants to be obtained over a much wider range of intensities, from φ = 10−3 to 105 Sun simeq. The curve shown in Figure 7.5a has two distinct phases: at low light intensities, as with the experiment, kTPV shows a linear relationship with light intensity up to around 10 Sun simeq. (m = 0.98, Regime I5). Above φ ≈ 10

Sun simeq. the curve plateaus and the rate coefficient tend to a constant value of kTPV = 1.53×108 s−1 (Regime II). Similar modelling experiments were carried out on a device with bulk SRH recombination and a similar light intensity dependence was observed (see AppendixG, Figure G3) albeit with an inflection point at high light intensities rather than a plateaux.

5In the literature, the character m is often used to characterised the slope of the charge carrier density verses VOC exponential relation. Here I use m as a general symbol for the gradient of a linear polynomial, i.e. y = mx+c, and the character β is reserved for the relationship n = n0 exp(qVOC/βkBT ). Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 148 a b 12 1.2 Simulated 7 Measured 10 1.0 ) ) V TPV TPV OC k k 8 0.8 6 [V] 6 0.6 log10( log10( 5 4 m = 0.98 0.4 m = 0.94 2 0.2 4 -2 0 2 4 -2 -1 0 1 log10( φ) log10(φ) 4 e c d 10 -2 3 ] Regime II

[V]

[mV 2 φ

V

V

Regime I ∆

∆ 10 -3 kTPV = 1 0 0.5 1 0 2 4 -20 0 20 40 60 80 100 Time [ µs] Time [ns] Time [µs]

Figure 7.5: Simulated and experimentally measured transient photovoltage rate constants at steady-state as a function of bias light intensity and associated TPV decays. (a) TPV rate constants kTPV (blue curve, cross markers) and open circuit voltage (red curve) extracted from the simulated p-i-n device with mobile ions, and high rates of interfacial recombination (full parameters are given in AppendixB) as a function of light intensity φ. The kTPV curve shows two regimes of behaviour: Regime I, where kTPV is dependent on light intensity (transients shown in panel (c)) and Regime II, where the rate constant tends to the diffusion limited analytical solution described in Subsection 7.7.2 (transients shown in panel (d)). (b) kTPV rate constants extracted from transient photovoltage decays on a FTO/c-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Au device as a function of light intensity between approx. 0.017 to 14 Sun eq. The pulse light intensity was adjusted to produce perturbations of less than 12 mV. The green curve with crosses shows measurements taken using the TRACER system up to 1 Sun eq. intensity (raw data shown in panel (e)), while the dark yellow curve with crosses shows higher light measurements (2 to 14 Sun eq.) using the TELTPV system (see AppendixG, Figure G2 for raw data). The black dashed curves in are fits, with associated gradients m.

7.6.3 Inference of reaction orders

Figures 7.6a and 7.6b show how the charge carrier density n varied with light intensity φ in simulated and measured devices respectively. While in the simulation n showed a linear dependence on φ, in the measured device an approximately n ∝ φ1/2 relationship was observed. The origin of the non-linearity in this instance is unclear but such effects in DSSC and OPV devices have historically been attributed to a tail of states in the bandgap.[215, 226, 229] The absence of traps in the DRIFTFUSION model may Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 149 a b 18 16.6 Simulated 16.4 Measured 16 ) )

n 16.2 n 14 m = 1 16.0 m = 0.52 log10( log10( 12 15.8 15.6 10 15.4 8 15.2 -2 0 2 4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 log10( φ) log10( φ) c 10 d 6.5 Simulated Measured 6.0

) 8

DD )

TPV 5.5 TPV k

TE k 6 m = 1 5.0 log10( log10( 4.5 4 m = 1.64 Regime I Regime II 4.0 10 12 14 16 18 15.2 15.6 16.0 16.4 log10(n) log10(n)

Figure 7.6: Charge carrier density vs. light intensity and TPV rate coefficient vs. carrier density in measured and simulated devices. Charge carrier density n as a function of light intensity φ for (a) the simulated and (b) measured devices. The differential capacitance charge extraction method described in Section 7.4.1.1 was used to obtain the experimental carrier densities whilst the average value of the bulk electron density was used in the simulated device plot. (c) Simulated device TPV decay rate kTPV as a function of charge carrier density n. Blue crosses are solutions using DRIFTFUSION (DD), while the orange curve denotes solutions using the thermionic emission (TE) approximation described in Section 7.7.1.(d) kTPV vs. n for the measured device. m in all cases denotes the gradient of the fits. account for the discrepancy between the experimental and simulated results and further development of the model will be required to properly investigate this possibility.

The simulated and measured kTPV vs. n log-log plots are given in Figures 7.6c and 7.6d respectively. Regime I in the simulation has a slope of m = 1 due to the linear dependence of both the kTPV and n on light intensity. In a zero-dimensional approximation, with purely first order recombination, the small perturbation rate constant kTPV would not be expected to show a dependence on the background carrier density. Instead, under the conventional analysis, the results shown in Regime I of Figure 7.6c would be interpreted as an indication of a second order recombination Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 150

process (γ = 2) since kTPV ∝ n (see Equation 7.12). This result is inconsistent with the first order recombination scheme used in the simulation. The slope of m = 1.64 in Figure 7.6d for the measured device reflects the non-linearity of the charge carrier light intensity-dependence. This gradient would indicate a high reaction order of γ = 2.64 in the 0-D analysis, a value at the lower limit of those measured elsewhere for PSCs using the same techniques (γ = 2.5 to 4.5).[31, 115] While the possibility of tail states in the bandgap may be partially accountable for these high empirical reaction orders, it is entirely plausible that the same phenomenon leading to the anomalous result in the simulated device also contributes to these high values. Given the relative simplicity of the simulations presented here, however, it would be unwise to generalise the modelling results to devices and analyses that include trap distributions. Consequently, I now turn to focus solely on examining the results from the simulation and leave a complete interpretation of the experimental results open to future investigation.

7.7 Discussion

To obtain further insight into the physical processes behind the linear carrier dependence of kTPV in Regime I in the simulated device, the excess electron carrier density ∆n profiles were analysed for TPV decays at bias light intensities of 10−2, 102 and 104 Sun simeq. (Figure 7.7). Figures 7.7a and 7.7c are indicative of Regimes I and II respectively. The electric field potentials are plotted on the right-hand axes (grey curves). In each case the device was at steady-state leading to electric field screening in the intrinsic layer by the high density of mobile ionic charges. Figure 7.7a shows that for low VOC excess charge carriers move into the doped contact regions. As illustrated in Figure 7.7d, recombination in this regime is limited by the thermionic emission of majority carriers from the contacts into the absorber layer where a reservoir of minority carriers is accumulated at the interface. At high VOC the excess charge is predominantly confined to the intrinsic layer. As illustrated in Figure 7.7e, in this regime transport to the interfaces is the limiting process for recombination. Figure 7.7b shows the transitional state between these two regimes, where the excess charge is distributed both in the absorber layer and contact. It should be noted that the rate coefficient at the interface is not the limiting factor in determining the recombination rate in either case. Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 151 a 3 0.2 d ]

-3 0 µs 0 Regime I: 2 Thermionic

cm -0.2

14 injection -0.4 1 limited

[x10 -0.6 n Electric #eld potential, Electric ∆ 0 -0.8 φ = 10 -2 Sun simeq. 320 µs b 8 -1 ] 0 µs -3 0.0 6 Increasing cm -0.2 V 14 4 OC -0.4

[x10 2 -0.6 n

∆ 0 1.6 µs -0.8 φ = 10 2 Sun simeq. -1 V 8 [V] c 0 ns ] e -3 6 0 -0.2 cm

16 4 Regime II: -0.4 Transport 2 -0.6 limited [x10 n

∆ 0 -0.8 φ = 10 4 Sun simeq. 32 ns -1 0 200 400 600 800 Electron Hole Position [nm] Recombination zone

Figure 7.7: Excess electron density profiles in a simulated device dominated by surface recombination following an excitation pulse. Change in electron density ∆n for a simulated device dominated by surface recombination with (a) 10−2 Sun simeq., (b) 102 Sun simeq., and (c) 104 Sun simeq. bias intensity. The pulse intensity was adjusted in each case to produce a 3-4 mV voltage perturbation. The grey curves (right-hand axis) in each case show the electric field potential. (d)&(e) Schematics showing the processes limiting recombination in low and high VOC regimes.

7.7.1 Thermionic injection limited recombination current model

In the following section, the recombination of electrons as majority carriers will be discussed in relation to the intrinsic/n-type interface. In the model symmetric system6 presented here an equivalent process occurs with holes at the p-type/intrinsic interface.

In Regime I of the simulated kTPV vs. n curve (Figure 7.6c) contact selectivity causes a diffusion gradient which drives excess electrons into the n-type region (Figure 7.7a). While SRH recombination is implemented throughout the contact region, minority charge carrier densities in the bulk region of the contact are very low compared to

6In this case I refer to symmetry in the n and p-type region properties i.e. the doping densities are equal and opposite, dielectric constants and mobilities are equal etc., rather than the mirrored cell model described in Chapter3 Section 3.3. Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 152 a 0 Jrec

-0.5 b ECB Φ b ] 2.5 -1 s -1 -3 2 0 µs EFn cm 1.5

VOC ≈ 2kT/q ln( nbulk/ni) 22 -1.5 1 E

Energy [eV] Fp 0.5 10 - 160 µs Recombination

rate [x10 rate 0 -2 600 605 610 615 E VB Position [nm] -2.5 550 600 650 700 750 Position [nm]

Figure 7.8: Schottky-like barrier and recombination profile at steady-state at the intrinsic/n-type interface. (a) The simulated device at open circuit, steady-state in Regime I (φ = 10−2 Sun simeq.) forms a Schottky-like barrier at the intrinsic/n-type interface. The barrier height Φb is given by the difference in intrinsic conduction band level and the electron QFL in the n-type region. As the open circuit voltage increases Φb is reduced accordingly. (b) Recombination rate U as a function of position at the intrinsic/n-type interface during the decay. The profile remains relatively constant from 10 µs to 160 µs after an initial fast decay. at the interface. Consequently, as shown in Figure 7.8b, SRH recombination is confined to an approximately 5 nm layer at the interface.

At quasi-equilibrium the state at the interface is similar to that of a Schottky barrier between a metal and a semiconductor (Figure 7.8a).[191] In this instance, mobile ionic charges are responsible for electric field screening in the intrinsic region rather than the sea of electrons residing in the metal’s conduction band. The doping concentration of the n-type material creates the difference in equilibrium Fermi energies between the two regions, resulting in a depletion region in the n-type layer and associated band-bending.7 Since the device is at open circuit, no net current flows across the junction. As shown in Figure 7.8a, the barrier height Φb is defined by the difference between the intrinsic region conduction band energy and electron Fermi energy in the n-type region, EFn.

The high minority carrier concentration and recombination rate coefficient at the interface ensures that excess electrons moving over the barrier recombine. The interface, therefore, acts as a current collector for majority charges. An approximation to the current across a Schottky junction, ignoring the band-bending profile, is given by the thermionic emission current density JTE:[191]

7The term ‘band-bending’ is often used to describe the curvature of the electric field potential due to charge accumulation in the depletion region. Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 153

      2 −Φb qVSD JTE = ART exp exp − 1 (7.16) kBT kBT where AR is the Richardson constant:

4πqm k2 A = e B = 120 A cm−2K−2 (7.17) R h3

Here, q and me are the charge and mass of an electron respectively, h is Planck’s constant, kB is the Boltzmann constant, and T is temperature. The introduction of excess charge in the contact region generates an additional potential that acts in a similar way to an applied forward bias across a Schottky diode VSD. The potential difference across the junction is defined by the excess charge carrier at the intrinsic/n-type boundary ∆n(xip):

    kBT ∆n(xip) + nn−type(xip) nn−type(xip) VSD = EF(n+∆n)(xip)−EF(n)(xip) = ln −ln q ni ni

  kBT ∆n(xip) VSD ≈ ln 1 + (7.18) q ND

Key to understanding the kTPV dependence on carrier density in this regime is the relationship between the barrier height and the open circuit voltage. As noted by Richardson et al., where a single ionic species is present in sufficiently high concentrations the built-in voltage of the device Vbi drops by roughly an equal amount at both interfaces.[185] As such, the built-in voltage of the junction can be approximated as:

    Vbi − VOC ND Φb = q + kBT ln (7.19) 2 NCB where ND is the contact donor density and NCB is the conduction band density of states. In order to describe heterojunctions between absorber and contact regions additional terms for band offsets and the appropriate density of states for the ETL would be required. In less doped semiconductors the contribution from free charges to the n-type quasi-Fermi level (QFL) would also require consideration.

Provided that the carrier density in the bulk is proportional to light intensity (Figure 7.6a), the open circuit voltage can be approximated using Equation 7.3. Substituting Equations 7.18 and 7.19 into Equation 7.16 yields: Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 154

  2 nbulk −qVbi JTE = ART ∆n(xip) exp (7.20) niNCB 2kBT

The thermionic emission current is, then, directly proportional to the charge carrier density in the bulk nbulk. Since the measurement is operating in the small perturbation ∂∆n regime, the rate of change of excess charge ∂t is also proportional to JTE:

∂∆n ∝ J ∝ ∆n ∂t TE

The solution of which is an exponential with a rate constant proportional to the constant in Equation 7.20. Consequently, the TPV rate constant scales linearly with the bulk charge carrier density. Where nbulk also scales linearly with light intensity, there will be a direct proportionality between light intensity and kTPV as seen in the simulation (Figure 7.5a). Where n does not scale linearly with φ, for example where a significant density of inter-bandgap trapping states exist, this will also be reflected in the kTPV vs. n slope and the hence the measurement of γ.

As a means to verify this model, a simplified MATLAB script, employing the same partial differential equation solver (PDEPE) as DRIFTFUSION, was used to solve the diffusion equation:

∂∆n ∂2∆n = −µk T (7.21) ∂t B ∂x2

Zero flux was used for the right-hand boundary condition, whereas Equation 7.20 was used at the left-hand boundary. The details of the numerical implementation are given in AppendixG and the results are shown in Figure 7.6c (orange curve). The TPV decay rates extracted from the diffusion only model are in good agreement with those from the full drift-diffusion model, deviating by less than a factor of 4 in the linear regime, indicating that the thermionic emission model is indeed a good approximation.

7.7.2 Diffusion-limited recombination current model

As shown in Figure 7.7c, and discussed in Section 7.6.2, a defining feature of Regime II in Figure 7.5b is that excess charge remains in the intrinsic layer before recombining. Given that the field is fully screened by mobile ions in this region, it follows that transport is purely diffusive. Furthermore, where high rates of interfacial recombination exist, the boundary conditions for ∆n in the absorber layer effectively become the same as those for a single slab of material with and infinite extraction coefficient i.e. ∆n(xin) = 0, where Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 155

a b 1 1

c-TiO 2 t [norm] n 0.5 mp-TiO 2 [Norm] 0.1 V

∆ Inverted Excess electron Excess

density, ∆ density, 0 400 450 500 550 600 0 2 4 6 8 10 12 14 Position [nm] Time [ µs]

Figure 7.9: Simulated excess charge decay after a uniform generation pulse and example transient photovoltage measurements. (a) Analytical solution to the excess charge carrier density ∆n (normalised) following a pulse excitation with a uniform generation profile (Equation 7.22).[230](b) TPV decays for c-TiO2, mp-TiO2, and Inverted architectures of perovskite device.

xin is the position of the intrinsic/n-type interface (see Figure 3.1). In this limiting case, the excess charge ∆n profile as a function of time t, generated by a uniform generation profile (approximating a weakly absorbed excitation pulse), can be described by an analytical model applied by Duffy et al. to describe the transient photocurrent response of DSSCs:[230]

∞     2 2  4∆nini(x) X 1 πx(2g + 1) π (2g + 1) D ∆n(x, t) = sin exp − t (7.22) π 2g + 1 2d 4d2 g=0

Here, ∆nini is the initial excess charge density created by a uniform generation profile, d is the thickness of the layer, and D is the diffusion coefficient of the material. Figure 7.9a shows how the ∆n profile evolves with time. The x-axis has been translated (x0 =

(xip − x)) to enable easy comparison with Figure 7.7c.

At sufficiently long times, terms with g > 0 become negligible and the recombination current ∆Jrec created by the excess charge tends to:[168]

2qD∆n  π2D  ∆J ≈ ini exp − t (7.23) rec d 4d2

In this diffusion limited case the TPV decay tail rate constant is purely a function of the diffusion coefficient and the absorber layer thickness. This result suggests the possibility of using TPV at high light intensities to measure the diffusion coefficient of the absorber layer in devices with sufficiently high ion concentrations and rates of surface recombination. Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 156

7.7.3 Origin of the bi-exponential TPV decay in PSCs

Figure 7.9b shows examples of transient photovoltage measurements on three different architectures of PSC: c-TiO2, mp-TiO2, and Inverted (see Chapter2, Section 2.4.2, Figure 2.11, for full description of the architectures). All of the curves show some degree of non-linearity, with the mp-TiO2 device being the most clear example.

The ‘bi-exponential’ TPV decay character observed here is typically seen in lower performing devices[219] and has previously been explained as resulting either from transport limitations[31] or multiple recombination pathways with distinct rate coefficients.[220]

Ultimately, any such multi-phase decay to a small perturbation measurement must originate from some kind of transport and/or redistribution of charge since any superposition of recombination rates and/or reaction orders can only result in a single exponential decay to the excess carrier density (Equations 7.5- 7.7). I have shown here that where high rates of surface recombination and/or low mobilities exist, bi-exponential decay characteristics are expected in devices with mobile ions. This is consistent with the observation that multi-phase decays are seen in poorer performing devices whilst high-efficiency devices tend to show purely mono-exponential decays.[175, 200, 220] Further evidence for this position is given by the strong multi-phase character often present in TPV decay of mp-TiO2 devices (example shown in Figure 7.9b, blue curve) as compared to c-TiO2 devices: the mobility of CH3NH3PbI3 in the mesoporous phase of this architecture is reported to be an order of magnitude lower than that of the pure perovskite phase.[100]

7.7.4 Mobility dependence in the diffusion limited regime

In Figure 7.5b it can be seen that the simulated device entered Regime II when the open circuit voltage of the device was approximately 0.9 V. In order to distinguish between the effects of transport and injection limited recombination the variation of kcalc with kinput for devices with and without mobile ions, over a range of mobilities and with the open circuit voltage fixed at 1.0 V, was investigated (Figure 7.10).

Where mobile ions are present, and where recombination is not transport limited, kcalc follows a broadly linear relationship with kinput, offset by around 2 orders of magnitude (Figure 7.10a, dark blue curve with diamond markers). Strikingly, where recombination becomes transport limited, the TPV decay rate coefficient tends to the analytical solution described by Equation 7.23 (indicated by the dashed lines). I note that the SRH recombination rate constants used in the simulations for Chapters5 and6 Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 157

a b 10 12 10 12 Ions µ = 10 3 No ions µ = 10 3 10 10 10 VOC = 1.0 V 10 VOC = 1.0 V µ = 10 ] 8 ] 8 -1 10 µ = 10 -1 10 -1 [s -1 [s µ = 10 6 µ = 10 6 calc 10 calc 10 k k µ = 10 -3 10 4 µ = 10 -3 10 4 Perovskite device regime 10 2 10 2 10 6 10 8 10 10 10 12 10 14 10 6 10 8 10 10 10 12 10 14 -1 -1 kinput [s ] kinput [s ]

Figure 7.10: Recombination rate coefficient extracted from transient photovoltage as a function of the input rate parameter and mobility in simulated devices dominated by surface recombination with fixed open circuit voltage. Recombination rate coefficients kcalc extracted from simulated transient photovoltage measurements (perturbation kept to below 4 mV) as a function of the input rate coefficient (kinput = 1/τSRH) in a p-i-n device (a) with mobile ions 19 −3 −3 (Na = 10 cm ) and (b) without mobile ions (Na = 0 cm ). The generation rate was adjusted to produce and open circuit voltage of 1 V. The dashed lines are the rate constants calculated using Equation 7.23. range between 1010 and 1014 s−1, and typical mobilities for perovskites are considered to be (to an order of magnitude approximation) 10 cm2V−1s−1.[100, 231] This regime, in which recombination is partially transport limited, is indicated by the grey box in Figure 7.10a. From this I conclude that for perovskite devices with high surface recombination under high intensity optical bias TPV is predominantly a measure of transport of charges to the contacts.

Figure 7.10b shows that, for devices dominated by surface recombination in which mobile ions are not present, kcalc tends to a constant value with a fixed offset from the diffusion limited analytical solution. This result suggests a further analytical solution exists with an additional, field-dependent, constant term. Such a solution would be particularly appropriate to organic photovoltaics where active material mobilities are typically less than 0.1 cm2V−1s−1.[109] Further work will be required to develop an analytical approximation that includes the effects of an electric field.

7.7.5 Conclusions

The results presented here show that, for p-i-n structured devices dominated by surface recombination, a zero-dimensional kinetic model is not sufficient to correctly interpret photovoltage transients: Capacitive charging effects, charge transport, and localised recombination centres must also be accounted for. These effects can only Chapter 7. Reinterpreting small perturbation TPV measurements in p-i-n devices 158 be accurately modelled using a one-dimensional system including distinct absorber, electron, and hole transport layers. In particular, I have shown that in devices with ions the extracted TPV rate coefficient shows a linear dependence on charge density for low open circuit voltages. This dependence would incorrectly be interpreted as evidence of second order recombination in the conventional analysis of the transient data. The investigation has presented two analytical models that explain different regimes of VOC-dependent TPV behaviour in the simulated device. At low open circuit voltages excess charge is transported to the contact regions and recombination is limited by the thermionic emission of carriers over a Schottky-like barrier created at the contact. The recombination current across the barrier was found to be proportional to the bulk carrier density, accounting for the observed kTPV dependence on bulk carrier density. At higher open circuit voltages, where excess charge is confined to the intrinsic layer, recombination is limited by diffusion-driven transport to the contacts. In this instance the photovoltage decay can be described by an analytical model in which the characteristic time constant is dependent only on the intrinsic layer diffusion coefficient and thickness. This diffusive transport model further provides a plausible explanation for the bi-exponential TPV decay character observed in poorer performing

PSCs. Finally, analysis of the kcalc dependence on both the defined rate constant kinput and the active layer mobility revealed that perovskite devices are close to the diffusion limited regime under fixed VOC = 1 V simulations. In devices without mobile ions kcalc tended to a value with a fixed offset from the diffusion limited analytical solution, implying that TPV could be a useful tool in measuring transport in low mobility materials such as organic photovoltaics. Chapter 8

New insights

8.1 Abstract

The world of perovskite research moves fast. Here, the preliminary results from two ongoing studies are presented: Long timescale bistability short circuit current density

(JSC) measurements and simulations of time-resolved photoluminescence measurements on perovskite bilayers. Standard compact-TiO2 and mesoporous TiO2 architectures of device are found to exhibit partially reversible bistable short circuit currents after pre-biasing for longer than 200 s. Unexpectedly, a lower final JSC value is consistently recorded for devices that have been forward biased as opposed to those preconditioned at short circuit and the effect is persistent for longer than 10, 000 s. Cycling of the experiment reveals that progressive losses in JSC after forward biasing are partially recoverable. Simulation of the results shows that changes to the material properties at the interfaces due to the accumulation of ionic vacancies can account for the experimental result. Time-resolved photoluminescence (TRPL) measurements on bilayers are also simulated, justifying the application of a diffusion-driven transport model. The TRPL simulations show the power of one-dimensional drift-diffusion simulations to reproduce and properly interpret experimental results.

8.2 Bistable short circuit current

8.2.1 Introduction

In Chapters5 and6 I showed that PSCs can exhibit open circuit voltage transients on a timescale of seconds. Short circuit current (JSC) transients similarly show remarkably slow changes over time periods of 10s to 1, 000s of seconds.[31, 196] 159 Chapter 8. New insights 160

Here, using long preconditioning times of greater than 100 seconds, the long timescale stabilised photocurrent of both compact TiO2 (c-TiO2) and mesoporous TiO2 (mp-TiO2) architectures of PSC is shown to depend on the preconditioning voltage. While at early times forward biasing leads to an increased photocurrent as would be expected from a favourable electric field generated by the ion distribution, at long times the JSC is in fact lower than when devices have instead been preconditioned at short circuit. This effect appears repeatable after multiple preconditioning cycles with alternating or sequential patterns. I speculate that the observed bistability is the result of a large accumulation of defects at the perovskite/ETL interface during the forward biasing stage which degrades the material and reduces its transport properties. This degradation can partially be recovered by returning the device to short circuit and reducing the accumulation of mobile defects at the interface. The implication of this work is that large scale defect accumulation accelerates degradation processes in the material, supporting previous observations.[113, 130]

8.2.2 Methods

8.2.2.1 Current Density Evolution (J-evo)

Figure 8.1a shows the experimental timeline for a Current Density Evolution (J-evo) measurement. In a similar method to the VOC transients discussed in Chapter5, Section

5.1, devices were preconditioned in the dark at a preset voltage Vpre for > 100 s. The TRACER system (see Chapter4) was used to synchronise switching of the optical bias

LEDs, and the electrical bias to a probe voltage Vprobe, with millisecond accuracy. For the results discussed in this section Vprobe = 0 V (short circuit) although, in principle, any voltage could be used. The evolution of the current density was subsequently monitored.

Vpre was alternated between an applied forward bias and short circuit conditions (i.e.

Vpre= 0 V), typically using an ABBA or other non-cyclic patterns.

8.2.2.2 p-i-n Heterojunction simulation

A full p-i-n heterojunction device was simulated to model the photocurrent transients with symmetric 0.5 eV band offsets for the electron and hole transport layers (ETL and HTL respectively), using the methods described in Chapter3. The band energy levels of the contact layers are not related to those of real-world materials but nevertheless provide a high level of carrier extraction selectivity. The full parameter set is given in AppendixC. Literature values for mobility in the ETL and HTL of 10 −3 and 4 × 10−4 cm2V−1s−1 respectively were used to simulate charge transport in Spiro-OMeTAD[232] Chapter 8. New insights 161

a b ] Preset -2 2 Probe Voltage Voltage After J-evo 0 0

Bias Light [mAcm 0 J -2

Dark -4 Current 0 -6 Before J-evo Photocurrent

Current density, -8 Evolution Time -0.4 0.0 0.4 0.8 secs secs Voltage, V [V]

Figure 8.1: Short circuit current evolution (J-evo) experimental timeline and example current-voltage characteristics of measured device. (a) J-evo experimental timeline: The device is biased in the dark at a preset voltage Vpre for > 100 s. The cell is then switched to a probe voltage Vprobe under illumination and the circuit current density is measured over a period of > 100 s. Measurements typically follow an ABBA or more complex, non-cyclic pattern. (b) J-V scans of the device before (blue curve) and after (red curve) the 16-step J-evo measurement shown in Figure 8.2. The scan rate was 50 mVs−1. Arrows indicate the scan direction.

and TiO2[233]. Lower contact region mobilities µ contribute to a larger layer series resistance RS according to:

d R = (8.1) S qnµA where, d is the layer thickness, q is the elementary charge, n is the carrier density and A is the device area.

As previously noted, the long diffusion lengths in PSCs enable devices to operate efficiently without a built-in electric field, provided that interfacial recombination is sufficiently passivated.[32, 112] Furthermore, recent Kelvin probe measurements[163] have indicated that the built-in potential in devices with TiO2 and Spiro-OMeTAD as electron and hole transporters may be negligible. Consequently, the built-in potential of the simulated device here was chosen to be zero. To ensure that the charge carrier boundary conditions were consistent with those of the electric field potential, 100 nm regions of the ETL and HTL were used to simulate electrodes by using a high mobility, low dielectric constant1 and high SRH rate coefficient.

1While metals are often considered to have an infinite dielectric constant, if the polarizability of a material is defined in terms of bound charge,[234] then the dielectric constant of a metal is close to zero. By setting a low dielectric constant in these regions, free charges are not well screened and the √ depletion width is reduced (wn ∝ r, see Equation 3.60). This ensures that the field goes to zero at the boundaries leading to consistent boundary conditions. Chapter 8. New insights 162

0 a ]

-2 -1 -2 V pre = +1 V -3 [mAcm SC J -4 -5 -6 b -1 V = 0 V -2 pre -3 -4 -5 V Short circuit current density, pre = +1 V -6 0 1 2 3 4 5 Time [s x 10 3]

Figure 8.2: Repeat short circuit current measurements for a c-TiO2 device. The FTO/compact-TiO2/CH3NH3PbI3/Spiro-OMeTAD/Au device was first preconditioned at Vpre = 1 V for 5000 s, then switched to short circuit under illumination (1 Sun eq.) for 8 cycles. Following this, a further 8 cycles were recorded with the device preconditioned at Vpre = 0 V for 5000 s.

8.2.3 Results

Figure 8.1b shows the J-V curve for a FTO/c-TiO2/CH3NH3PbI3/Spiro-OMeTAD/

Au (c-TiO2) device before and after the J-evo measurement shown in Figure 8.2. The cell was already somewhat degraded and showed both a low JSC and VOC. The J-evo measurement in Figure 8.2 consisted of a 5000 s pre-biasing stage with an initial 8 cycles with the preconditioning voltage Vpre set to = +1 V, followed by a further 8 cycles with

Vpre = 0 V.

After forward biasing, the JSC for each measurement shows a initial value of ≈ 6.5 mAcm−2 followed by a large decay of 40 − 60% in the photocurrent. Consecutive measurements tended to progressively lower final current densities after the 5000 s illumination stage (Figure 8.2a). When the precondition voltage was switched to Vpre = 0 V (Figure 8.2b) the trend was reversed: for each cycle the initially low photocurrent of ≈ 1 mAcm−2 increased by between 250 − 350%. Consecutive measurements tended to progressively higher final values with the JSC at t = 5000 s of the final cycle Chapter 8. New insights 163 approximately 0.3 mAcm−2 lower than that of the initial cycle. I interpret this as a partial reversal of the degradation caused by the forward biasing cycles. These results support recent results from Domanski et al. demonstrating that short-term losses (occurring in less than 5 hrs) in the efficiency of high performance devices (PCE ≈ 20 %) at maximum power point could mostly be recovered by switching devices to open circuit in the dark.[196]

Figures 8.3a and 8.3c show J-V characteristics and J-evo measurements on a mp-TiO2 cell that similarly exhibited JSC bistability under an optical bias of 2 Suns eq. after only 200 s of pre-biasing. The measurement protocol followed an ABAB pattern, with the final current density following the illumination step approximately 2 mAcm−2 less for the Vpre = +1.0 V as compared to the Vpre = 0 V precondition. Figure 8.3d shows

] a ] b

-2 -2 4 c-TiO 2 4 mp-TiO 2

cm cm

0 0

-4 -4

t density [mA t density [mA

-8 -8

Curren Curren -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.2 0.0 0.2 0.4 0.6 0.8 Voltage [V] Voltage [V]

]

] -14 -13 c -2

-2 V d pre = -0.5 V V = +1 V -16 pre 4 -14

[mAcm

2 [mAcm 2 -18 -15 3 3 -20 1 -16 1 4

rrent density rrent density -22 -17

Cu

Cu 0 1 2 3 4 5 6 0 2 4 6 8 10 Time [s x 102] Time [s x 10 3]

Figure 8.3: Current-voltage scans and short circuit current evolution (J-evo) for c-TiO2 and mp-TiO2 architecture devices. Current density versus voltage scans for (a) planar (c-TiO2) and (b) mesoporous (mp-TiO2) architecture perovskite solar cells in the dark and under 1 Sun eq. illumination at a scan rate −1 of 0.2 Vs .(c) J-evo measurement on a mp-TiO2 device after pre-biasing for 200 s. (d) J-evo measurement on a c-TiO2 device with a 2000 s pre-bias stage. Dashed and solid curves indicate Vpre = −0.5 V and Vpre = +1 V respectively. Colours indicate pairs of measurements and bold numbering indicates the order of the measurements. J-evo measurements made at 2 Suns eq. optical bias. Chapter 8. New insights 164

that the effect in another c-TiO2 device (J-V shown in Figure 8.3b) can be sustained for longer than 2.7 hrs (10, 000 s). The final current densities in this case were different by approximately 1 mAcm−2. While these preliminary results do not prove that bistability is present in all perovskite devices, they show that different architectures can be equally affected.

8.2.4 Simulation results and discussion

Figure 8.4a shows the 8th and 9th J-evo cycles of the measurement presented in Figure

8.2. After preconditioning the cell at Vpre = 0 V the current density gradually increased from 1 to 2.6 mAcm−2 during 5000 s of illumination. When the precondition voltage −2 was switched to Vpre = +1.2 V the JSC decreased from approximately 6.5 mAcm to a −2 −2 final value of 2.1 mAcm , 0.5 mAcm less than following the Vpre = 0 V precondition. The results are simulated in Figure 8.4b using the full device model described in Section 8.2.2.2. Figures 8.4c and 8.4d are the initial (t = 0) energy level diagrams, electronic charge densities (n and p), and mobile ion defect densities (a) for Vpre = 0 V and Vpre

+1.2 V respectively. Following preconditioning with Vpre = 0 V, the low mobilities in ETL and HTL induced a series resistance in the cell sufficient to develop a 180 mV potential difference across the active layer (Figure 8.4c). This resulted in a reverse field across the device, similar to that produced after stepping to a forward bias.[122] The subsequent gradual increase in photocurrent is attributable to ionic charge screening of this field. The reduction of the field results in a decrease in the minority carrier density at the interfaces and an associated decrease in recombination. Equilibrating the simulated device instead at Vpre = 1.2 V before illuminating at short circuit resulted in an advantageous field for charge extraction (Figure 8.4d). In this instance, surface recombination was initially suppressed due to lower minority carrier concentrations at the interfaces. As time moves forward in the measurement and ionic charge screens the field, minority carrier concentrations at the contacts increase and the extracted photocurrent is reduced. These are the same processes responsible for the J-V hysteresis described in Chapter5.

Unaltered, the simulation was unable to reproduce bistable photocurrent observed in the experiment (Figure 8.4b, black dashed line) since the drift-diffusion equations have a unique solution for a given set of parameters and boundary conditions. In order to reproduce the bistable behaviour two strategies were tested: 1. A decrease in the surface −13 −14 recombination SRH time constant from τSRH = 10 to 5×10 s in a 2 nm layer at the perovskite/ETL interface (Figure 8.4b, blue curve) and, 2. A reduction of the mobility 2 −1 −1 by an order of magnitude (from µinter = 20 to 2 cm V s ) in the same region (Figure 8.4b, gold curve). Both adjustments resulted in a reduction of photocurrent consistent Chapter 8. New insights 165

0 a Measuredb Simulated -1 -8 ] V ] V = 0V -2 = 0V -2 pre -2 pre High Rec. -10 -3 Low Mob. [mAcm -4 [mAcm -12 SC SC J -5 J Vpre = +1.2 V Vpre = +1.2 V -6 -14 0 1 2 3 4 5 0 1 2 3 4 Time [s x 10 3] Time [arb] c d -3 -3

ECB -4 -4

E EFn -5 Fp -5 Energy [eV] Energy [eV]

-6 EVB -6 10 20 10 20 ] ] -3 -3 10 15 10 15

10 10 p n 10 10 Carrier Carrier

5 5 density [cm density density [cm density 10 10

10 0 10 0 3 3 Vpre = 0 V Vpre = +1.2 V ] ] -3 -3 2 2 cm cm 19 19 1 1 [x10 [x10 Ion density Ion density 0 0 0 200 400 600 800 0 200 400 600 800 Position [nm] Position [nm]

Figure 8.4: Measured and simulated Current Density Evolutions (J-evo) (a) J-evo measurements from Figure 8.2 preconditioning at Vpre = 0 V and Vpre= +1.2 V for 5000 s.(b) The J-evo from a simulated device with zero built-in potential initially shows similar behaviour before stabilising at the same value for both forward bias (black dashed curve) and short circuit preconditions (green curve). The gold curve (labelled ‘Low Mob.’) shows the J-evo with Vpre = 0 V for the same device in which the mobility had been decreased by an order of magnitude from 20 to 2 cm2V−1s−1 in a 2 nm region at the perovskite/ETL interface. The blue curve (labelled ‘High Rec.’) shows the results from a simulation in which the SRH rate coefficient had been decreased from 10−13 to 5 × 10−4 s in the same region. Energy level diagrams, and electron n, hole p, and mobile ion a charge densities for (c) Vpre = 0 V and (d) Vpre = 1.2 V. Green and blue shaded areas indicate hole and electron transport layers respectively, while the grey areas denote electrodes. Nominal device parameters are given in AppendixC. Chapter 8. New insights 166 with the experimental results. Degradation of the perovskite could be expected to cause one or both of these effects simultaneously. As can be seen in Figure 8.4d, in a device with a low built-in voltage, forward biasing leads to a large accumulation of positive mobile ionic charges at the perovskite/ETL interface. As discussed in Chapter2, the most likely candidate for mobile ionic charge are positively charged iodide vacancies. A − large accumulation of I vacancies at the MAPI/TiO2 interface could result in a higher density of inter-bandgap surface states that are able to act as recombination centres. Vacancy accumulation could also cause domains of degradation products to build-up at the interface as the stoichiometry of the material is being drastically altered. This process would reduce the effective surface area of perovskite in contact with the TiO2, increasing the series resistance of the interface (refer to Equation 8.1). The simulations show that an increase in contact resistance leads to an accumulation of carriers at the interface and associated increase in recombination.

8.2.5 Conclusions

I have shown that PSCs can exhibit bistable short circuit current densities that are determined by the device preconditioning. Counter-intuitively, devices tend to lower final photocurrent densities after forward biasing as compared to devices that have been preconditioned at short circuit. Sustained forward biasing of PSCs has been shown to cause partially reversible degradation. Bi-stable behaviour was observed in both c-TiO2 and mp-TiO2 devices and shown to persist for 10, 000 s. Simulations of full device stacks have shown that either an increase in inter-bandgap states and/or a decrease in the effective mobility of the absorber material at the perovskite/ETL interface, both of which lead to higher rates of recombination, could account for the observed bistable behaviour. Furthermore, these changes in the perovskite material properties could be explained by a localised accumulation of ionic defects at interfaces.

The proposed mechanism presents an obstacle for sustained device operation with present perovskite architectures since a load will act to forward bias the device leading to an accumulation of ionic defects at one of the contacts. In principle, a careful choice of contact workfunctions could allow devices to be designed such that vacancy accumulation at interfaces is minimised at maximum power point. In practice, however, the choice of contact materials is limited. This work highlights the imperative to find materials without intrinsic mobile defect concentrations or alternatively without degradation mechanisms mediated by such defects. Chapter 8. New insights 167

8.3 Simulating time-resolved photoluminescence measurements on perovskite bilayers

8.3.1 Introduction

In Chapter7 transient photovoltage (TPV) measurements were analysed using the DRIFTFUSION simulation tool with respect to the commonly employed zero-dimensional kinetic theory. I showed that in devices dominated by surface recombination, at high open circuit voltages the TPV decay can be modelled by an analytical solution to the diffusion equation that depends only on the diffusion coefficient and the absorber layer thickness. For convenience I reproduce the expression for the excess carrier density ∆n as a function of position and time here:

∞     2 2  4∆nini X 1 πx(2g + 1) π (2g + 1) D ∆n(x, t) = sin exp − t (8.2) π 2g + 1 2d 4d2 g=0

I noted in Chapter7 Subsection 7.7.3 that the biexponential character observed in poorer devices could be the result of the solution to this equation. Here I explore the possibility that the biexponential decay observed in time-resolve photoluminescence (TRPL) quenching measurements on bilayers (absorber/quencher)[53, 97, 198, 235, 236] has the same origin as that seen in the high VOC TPV simulations. Solutions to the diffusion equation similar to Equation 7.22 have already been applied to time-resolved photoluminescence measurements on bilayers to extract carrier diffusion lengths LD in organic materials and perovskites.[97, 235, 237] Stranks and co-workers employed a numerical solver to fit the TRPL decay using a time-of-flight model on CH3NH3PbI3 and CH3NH3PbI3−xClx,[97, 238], while Xing et al.[235] used a similar expression to

Equation 8.2 to calculate LD. Notably absent in these works was any discussion of the role of ions in the perovskite phase and a reasonable justification for ignoring field effects (drift terms) in the active layer.

Here I present simulations using DRIFTFUSION that clearly demonstrate that field screening due to ionic charge migration in the perovskite phase can account for the results from existing TRPL measurements on bilayers published by Bi et al.[53] The simulations reproduce the experimental data with remarkable accuracy, including the effect of illumination from different sides of the device. The results indicate that TRPL is a good measure of diffusive transport and that the extraction of a diffusion coefficient Equation 8.2 is well justified. The extracted mobility from fitted experimental decays indicates a low mobility of between 0.04 and 0.17 cm2V−1s−1 in the measured Chapter 8. New insights 168 solution processed films. The results presented here show the power of one-dimensional drift-diffusion simulations to reproduce and properly interpret experimental results in perovskite devices.

8.3.2 Methods

In order to simulate TRPL quenching measurements a heterojunction bilayer was created in DRIFTFUSION. A schematic of the bilayer is given in Figure 8.5. The full device parameters are given in AppendixD. The bilayer was composed of a 400 nm intrinsic perovskite absorber layer in contact with an electron transport layer (ETL) with conduction and valence bands offsets of 0.5 eV. The built-in voltage between the perovskite and ETL was set at 0.25 eV, following Kelvin probe measurements by Harwell et al.[163]. The ETL is split into two 200 nm regions; In the second region (yellow striped region on grey background) high rates of recombination are included to simulate the surface recombination effect of a metal interface. The mirrored cell approach and boundary conditions described in Chapter3, Section 3.3.1 were used to simulate the device at open circuit.

The radiative bulk recombination rate coefficient was set to a value defined by the −20 3 −1 Shockley-Queisser limit (krad = 3.97×10 cm s ) for the given active layer thickness

Absorber 0.5 eV

1.6 eV

ETL Electrode

400 nm 400 nm Electron Hole Recombination zone

Figure 8.5: Schematic of the simulated bilayer. Electrons are extracted to the electron transport layer (ETL), whilst holes are blocked by the valence band offset. The rate of extraction defines the rate at which radiative recombination is quenched. Chapter 8. New insights 169 and material bandgap. In this instance, radiative recombination is not expected to be significant compared to the rate of charge extraction to the electron transport layer (ETL). To simulate the perovskite/TiO2 interface, Shockley Read Hall (SRH) −12 −1 recombination with a high rate coefficient (kSRH = 10 s ) was used and confined to a 2 nm region in the absorber layer at the material interface (yellow striped region- not to scale). For generation, a Beer-Lambert excitation profile (see Chapter3, Section 3.10 was used to model the 406 nm excitation pulse used to obtain the experimental −1 data in Ref.[53]. An absorption coefficient of αabs = 258, 353 cm was used following optical constant measurements from Ref.[63]. The photoluminescence (PL) intensity

φPL was assumed to be proportional to the integrated product of the electron and hole densities:

Z xd φPL(t) ∝ n(x, t)p(x, t) dx (8.3) 0

8.3.3 Results

Figure 8.6a and 8.6b show results from experimental TRPL measurements on 2 FAMAPIBr /TiO2 bilayers by Bi et al.[53], and associated simulations using DRIFTFUSION, respectively. In the experimental measurements the device was pulsed with a short wavelength laser from both the FAMAPIBr surface (Figure 8.6a, dark blue curve) and the FTO/TiO2 side (Figure 8.6a, red curve). Distinct from the transport measurements discussed in Subsection 8.3.1, in this instance the fast and slow decay components were attributed to non-radiative (first order) and radiative (second order) recombination pathways respectively.[53]

The simulations reproduce the experimental data with remarkable accuracy. The solid blue and red curves in Figure 8.6b show the simulated TRPL decays for the perovskite

‘Surface’ and FTO/TiO2 illumination sides respectively, in a device where the electron and hole mobilities were both set to 0.1 cm2V−1s−1. The slow decay component slope of simulated PL transient from the TiO2 side is in good agreement with the prediction from the solution to the diffusion equation (Equation 7.23 and Figure 8.6b, black dashed curve). This agreement was further improved by switching off surface recombination processes (Figure 8.6b, green curve). In this situation p effectively becomes constant at d(np) dn long times resulting in dt ∝ dt . The simulated TRPL curve for illumination through the perovskite layer (blue solid curve) had both a lower magnitude initial decay, and faster long-time decay, likewise reproducing the experimental result. The increased long-time decay rate in this instance is not currently well understood and will require

2 (CHNH3)0.85(CH3NH3)0.15Pb(I0.85Br0.15)3. Chapter 8. New insights 170 a b 10 0 10 0 Surface 10 -1 10 -1 Analytical 10 -2 10 -2 No SRH 10 -3 TiO -3 2 10 μ = 0.1 cm 2V-1 s-1

Normalised PL Int -4 Normalised PL Int 10 μ = 1 cm 2V-1 s-1 10 -4 0 200 400 600 0 200 400 600 800 Time(ns) Time(ns)

Figure 8.6: Time-resolved photoluminescence (TRPL) simulations compared to published results (a) Experimental TRPL data for (CHNH3)0.85(CH3NH3)0.15Pb(I0.85Br0.15)3 (FAMAPIBr) films deposited on FTO/c-TiO2 pulsed from different directions through the FAMAPIBr layer (labelled ‘Surface’) and the FTO/TiO2 layers using a 406 nm laser pulse. (b) Simulated TRPL decays using a MAPI/ETL bilayer with high rates of surface recombination (dark blue curve - Surface illumination, red curve - FTO/TiO2 illumination) and a mobility of 0.1 cm2V−1s−1 in the perovskite layer. The green curve indicates the same simulation with surface recombination switched off (FTO/TiO2 illumination), and the black dashed curve is the analytical solution described by Equation 7.23 (uniform generation). Dashed orange and light blue curves show FTO/TiO2 and Surface illumination sides respectively for a device in which the mobility was increased to 1 cm2V−1s−1. Experimental data reproduced from Ref.[53]; c The Authors; exclusive license American Association for the Advancement of Science. Distributed under a Creative Commons Attribution Non-commercial License 4.0 (CC BY-NC). http://advances.sciencemag.org/content/2/1/e1501170 further investigation. In order to confirm that the PL decay rate is indeed dependent on the carrier mobility, the mobility was increased to 1 cm2V−1s−1 (dashed orange and blue curves). Work in progress appears to confirm that the rate constant obtained under a uniform generation profile corresponds solely to the mobility of the extracted carrier and active layer thickness, in agreement with the findings from Refs.[235] and [97].

Figure 8.6a shows the energy level diagram, electron n, hole p, and mobile ionic a carrier densities for the simulated bilayer without interfacial recombination (cf. Figure

8.6, green curve) directly following an excitation pulse from the FTO/TiO2 side. Prior to the pulse the device was in equilibrium with mobile ionic charges accumulated at the interfaces, screening the built-in potential in the active layer. Consequently, the built-in potential of the junction drops predominantly in the ETL, as seen in Chapter7, Section 7.7.1. Directly following the excitation pulse, the initial excess electron density ∆n is localised at the contact interface due to the Beer-Lambert generation profile. As electrons move into the ETL contact, the profile collapses to a sinusoidal function similar to that seen in the analytical solution (cf. Figure 8.7b and 7.9). The valence band offset of the two materials prevents holes generated by the pulse from entering Chapter 8. New insights 171

a ] 2

0 -3 b ECB

E cm Fn 1.5 -1 13 EFp t

[x10 1 EVB -2 n Energy [eV] 0.5 20

10 electron Excess ]

-3 n 0 10 15 ∆ density, 0 100 200 300 400

10 10 Position [nm] Carrier

10 5 ] -3 2 density [cm density p c t cm

1.4 14 1.5 ]

-3 1.3 ] cm -3 1 1.2 [x10

19 1 p 1.1 cm 19 a [x10 1 0.5 0 5 10 hole Excess 0.5 Position [nm] [x10 Ion density Ion

0 ∆ density, 0 0 200 400 600 800 0 100 200 300 400 Position [nm] Position [nm]

Figure 8.7: Time-resolved photoluminescence simulation energy level diagram, charge density profiles and excess charge decay profiles. (a) Energy level diagram, electron n, hole p, and ionic a charge density profiles for the simulated device without surface recombination following an exponentially decaying excitation pulse (ETL-side illumination). Inset shows the ionic charge density in the first 10 nm of the perovskite layer. Blue and grey shaded regions are the n-type contact and electrode respectively. Excess (b) electron density (∆n) and (c) hole density (∆p) profiles directly following the excitation pulse. See Figure 8.6, green curve for the TRPL decay. the ETL and confines them to the perovskite layer. Here, in contrast to ∆n, the excess hole profile, ∆p collapses to an approximately uniform function (Figure 8.6c). The longer initial decay in the simulation compared to the analytical model results from the initial localisation of charge at the contact interface and associated higher currents generated by the high carrier gradients. The development of positive space charge within the active layer, due to the depletion of electrons with respect to holes, increases the potential across the active layer by less than 5 mV. While this was not sufficient to significantly influence electron transport to the ETL using the given parameter set, a more extensive investigation will be required to ascertain the conditions for which diffusion only transport is a valid assumption in this respect.

Using the decay rate constant given in Figure 8.6a (k = 4.54 × 106 s−1) and an approximate device thickness of 500 nm obtained from Ref.[53], a mobility of 0.17 cm2V−1s−1 can be inferred using Equation 8.2. This value is less than one order Chapter 8. New insights 172 of magnitude lower than that obtained from space charge limited current mobility measurements on CHNH3PbI3 single crystals.[75] Given that the experimental data in Figure 8.6a was collected on solution-processed films, the extracted mobility value appears consistent with the conclusion that the decay can be modelled by Equation 8.2.3

8.3.4 Conclusions

The results shown here provide further evidence that,∆n for perovskite devices at steady-state, transport in the absorber layer is purely diffusion driven. In the absence of perfect Fermi level alignment between absorber and quencher layers, mobile ionic charges migrate to screen out the field in the perovskite layer. The absence of strong electric fields in perovskite enables the interpretation of experimental results using analytical solutions to the diffusion equation. TRPL on quenching layers is one example where this methodology can be employed to accurately extract the mobility of the perovskite absorber layer. Further work will be required to ascertain how general these solutions may be, and whether adapted solutions can be applied to systems where the electric field plays an important role in transport.

3I note that analogues to Equation 8.2, accounting for arbitrary generation profiles exist and could be applied to model the initial response to short wavelength pulses more accurately. Chapter 9

Conclusions

Perovskite solar cells (PSCs) have generated huge research interest in recent years owing to the relative ease with which devices can be fabricated and high measured power conversion efficiencies.[20, 21, 149, 152–154] To date, however, all of the perovskite compounds that have been investigated for photovoltaic use show a number of reversible and irreversible instability types that affect the optoelectronic response of devices.

Prior to this work, the precise mechanisms underlying the device physics of PSCs were still being hotly debated. While there remain many questions to answer, particularly relating to permanent changes in the material with time, this work presents a number of advances in our understanding of these unique devices.

1. PSCs can only be properly understood using (at least) a one-dimensional model. One of the key achievements of this work was the development of a versatile one-dimensional numerical drift-diffusion model (DRIFTFUSION) capable of accurately simulating devices and reproducing results from complex transient optoelectronic experiments. Key innovations were the inclusion of mobile ions, localised recombination centres, and the ability to solve directly for open circuit conditions. In Chapters5-7 I showed how established zero-dimensional models often fail to explain experimental observations or else lead to misinterpretations of the data. Through the ability to track charge transport and recombination in devices, DRIFTFUSION has provided great insight into the internal kinetic and dynamic processes at work in PSCs.

2. Mobile ions are present in perovskite devices that do not show current-voltage hysteresis. Prior to this work, a debate existed over the origin of the J-V hysteresis in PSCs. 173 Chapter 9. Conclusions 174

A proportion of the research community believed that hysteresis was related only to mobile ionic charge and that the absence of hysteresis could be correlated to the passivation of mobile ions. In Chapter5 evidence was presented, using novel transient optoelectronic techniques, demonstrating that mobile ions and associated electric field screening are apparent in PSCs that do not show J-V hysteresis. These techniques provide a fast, non-destructive probe for assessing the prevalence of mobile ions in devices and will be of value for investigating new perovskite compounds. In agreement with previous work,[184] I have also shown that hysteresis can only be accurately reproduced when both mobile ions and trap-assisted recombination at the perovskite/contact interfaces are present.

3. Transient ideality factor measurements can be used to identify the dominant recombination mechanisms in PSCs. In Chapter6, simulations of p-i-n devices with a range of recombination schemes and mobile ions were used to show how changes in the carrier population overlap and ion density profiles could be correlated to changes in the perceived ideality factor. The instantaneous ideality factor after forward biasing was identified as the value corresponding to the established model. Furthermore, different recombination schemes were found to exhibit unique transient ideality signatures that can be used to identify dominant recombination mechanisms in real devices.

4. The transient photovoltage response in p-i-n devices is determined by a complex interplay of carrier transport dynamics, recombination kinetics and contact energy offsets. Transient photovoltage (TPV) measurements are commonly interpreted using a zero-dimensional kinetic model to extract recombination rate coefficients and reaction orders. In Chapter7 I showed that for idealised p-i-n junctions this interpretation can be erroneous and identified two distinct regimes of behaviour in devices with high rates of interfacial recombination and mobile ions. At lower open circuit voltages the TPV decay rate is limited by the thermionic emission of electrons and holes from the contact regions into the intrinsic layer. At higher open circuit voltages the TPV decay is a measure of the rate of charge transport to the interfaces and can be modelled analytically using a solution to the diffusion equation.

5. Transport in the perovskite layer of PSCs at steady-state is purely diffusive making passivated interfaces critical to efficient device operation. The presence of high concentrations of mobile ionic defects in perovskite materials leads to screening of the electric field in the absorber layer at steady-state in Chapter 9. Conclusions 175

PSCs. Consequently, transport is diffusion-driven under stabilised operation, making highly selective contacts essential for charge extraction. Furthermore, owing to the accumulation of minority carriers at contact interfaces required to drive diffusive currents, passivated interfaces are critical to efficient device performance. This lies in contrast to device types where an electric field is present to drive charge separation and extraction.

6. Charge transport in perovskites can be measured using time-resolved photoluminescence (TRPL) on bilayers. As with full devices, for perovskite bilayers at steady-state, transport is expected to be diffusion-driven owing to electric field screening by the high concentration of mobile ionic charge. The TRPL decay in bilayers is limited by the rate at which the charge is transported into the contact material. TRPL can, therefore, be used to measure the diffusion coefficient of the perovskite material by applying the same analytical solution to the diffusion equation described in Chapter7. In Chapter8, Section 8.3 I showed initial simulations that justified this hypothesis and reproduced published TRPL results on perovskite bilayers with great accuracy.

7. Perovskite devices produce bistable short circuit currents after electronic preconditioning. I have presented multiple examples of devices for which the stabilised short circuit current was found to be dependent on electronic preconditioning (Chapter 8, Section 8.2). Counter-intuitively, a long forward bias step was found to lead to progressively lower short circuit currents than when the device was preconditioned at ≤ 0 V. Cyclic forward biasing appeared to degrade cells, an effect which was partially reversed by cyclic reverse biasing. I hypothesised that the accumulation of ionic defects at the perovskite/contact interface during the forward biasing stage leads to degradation of the material. I showed simulations of full devices without a built-in potential that could accurately reproduce the observed experimental behaviour using modifications to the transport and recombination properties at the interfaces. Chapter 9. Conclusions 176

9.1 Outlook

The work presented in this thesis forms a foundation on which future work can be built. In this final section, I suggest a number of future directions for the DRIFTFUSION simulation tool.

9.1.1 Further development of DRIFTFUSION

The versatility of DRIFTFUSION presents many opportunities for future development. The present focus will be on improving both accessibility and stability of the code in order to enable new users to obtain results with relative ease. Current users have already contributed to the development of DRIFTFUSION and these contributions need to be collated and incorporated into the current version of the software. Likewise, the multitude of tools that have been developed for simulating transient measurements and analysing data need to be properly catalogued and updated for new users. This will include management of a GitHub repository for version control. In the longer-term, I aim to port the code to an open-source software platform, such as Python, although the success of any such project will rely on finding an appropriate open-source solver.

9.1.2 Large-scale studies

DRIFTFUSION enables complex experimental transient procedures to be easily simulated. In Chapter7, recombination rate coefficient parameter spaces were explored with respect to transient photovoltage measurements. One aim of repackaging the code will be to simplify large-scale explorations of solar cell parameter spaces. In combination with high performance computing (HPC) facilities, relatively large-scale studies can be conducted in a matter of minutes. This was recently evidenced by the calculation of 500 transient photovoltage simulations in less than 1 hour by Mohammed Azzouzi, using Imperial College London’s HPC facilities. As the quantity of data extracted from the simulation increases, new tools for data analysis will be required. Furthermore, procedures for automated error checking will be required to ensure the accuracy of numerical solutions. Chapter 9. Conclusions 177

9.1.3 The effects of contact doping and dielectric properties in full device models

The results from Chapter5 implied that Inverted architecture devices do not exhibit J-V hysteresis because surface recombination is passivated as compared to architectures where TiO2 is used as an electron transport material. This hypothesis is not, however, compatible with the lower open circuit voltages generally measured in Inverted devices as compared to high-performing Standard architectures devices. Given the presence of mobile ions and absence of hysteresis in these devices, recombination must be independent of the ion configuration. At least two possibilities can be considered: 1. The potential drops predominantly across the PCBM layer due to the absence of doping and the material’s low dielectric constant. The influence of ions on surface recombination in this instance would likely be reduced. 2. Pin holes in the material enable direct recombination of electrons and holes between contacts. The full heterojunction device model provides an opportunity to explore the first of these options by allowing contacts with the accurate energetics and doping densities to be modelled.

9.1.4 Defect levels and inter-bandgap states

Inter-band gap states (as described in Chapter2, Section 2.3.3) are not included in the current version of DRIFTFUSION and may have important implications for charge transport and recombination kinetics in perovskite devices. Carrier densities in trapped states can be included as additional input variables to the solver, with capture and escape of carriers included in the source terms of the continuity model. If the application of DRIFTFUSION is to be extended to simulate organic photovoltaics and dye sensitised solar cells, trap distribution models will be essential.

9.1.5 Reinterpreting transient measurement techniques

In Chapter7 I showed how the zero-dimensional analysis applied to transient photovoltage measurements can lead to erroneous conclusions. There are a host of transient techniques, including Impedance Spectroscopy, Intensity Modulated Photovoltage Spectroscopy, Intensity Modulated Photocurrent Spectroscopy, and Charge Extraction by Linearly Increasing Voltage that were originally developed for characterising dye cells and that rely on zero-dimensional or diffusive transport-only models.[168, 206, 239, 240] In some cases, it may be perfectly reasonable to apply these same models to organic semiconductor and perovskite-based thin film devices. In Chapter 9. Conclusions 178 other cases it seems plausible that a more sophisticated model may be required. The correct interpretation of measurements is vital to further the field of solar research, and DRIFTFUSION offers an opportunity to assess, and where necessary reinterpret, these measurements. Bibliography

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[242] Jingbi You, Chun-chao Chen, Letian Dou, Seiichiro Murase, Hsin-sheng Duan, Steven A Hawks, Tao Xu, Hae Jung Son, Luping Yu, Gang Li, and Yang Yang. Metal Oxide Nanoparticles as an Electron-Transport Layer in High-Performance and Stable Inverted Polymer Solar Cells. Advanced Materials, 24:5267–5272, 2012. doi: 10.1002/adma.201201958. Appendix A

Full p-i-n device parameter set 1

Parameter name Symbol Value Unit Band gap[63] Eg 1.6 eV 20 −3 Effective density of states N0 10 cm Built-in voltage[241] Vbi 1.3 V Relative dielectric constant[66] r 20 19 −3 Mobile ion defect density[48] Na 10 cm −12 2 −1 −1 Ion mobility µa 10 cm V s 2 −1 −1 Electron mobility[100] µn 20 cm V s 2 −1 −1 Hole mobility[100] µp 20 cm V s −10 3 −1 Band-to-band recombination coefficient kbtb 10 cm s 17 −3 n-type donor density ND 3.0 × 10 cm 17 −3 p-type acceptor density NA 3.0 × 10 cm −15 p-type & n-type SRH time constants* τn,SRH, τp,SRH 2.0 × 10 s SRH trap energy, n-type Et,n−type -0.2 eV** SRH trap energy, p-type Et,p−type -1.4 eV** 25 −3 −1 Incident photon flux (uniform generation) G0 6.25 × 10 cm s

Table A.1: Key device simulation parameters set 1. All values for the perovskite phase are approximately based on literature values. The Shockley-Read-Hall (SRH) recombination rate coefficient and trap energies were chosen to reproduce the approximate open circuit voltage for the bottom cathode device. The generation rate was chosen to yield an absorbed photocurrent of ≈ 16 mA cm−2. The ionic mobility was adjusted from our previous literature estimate of about 4 × 10−11 cm2 V−1 s−1 (ref. [48]) to a lower value of 10−12 cm2 V−1 s−1 to correctly approximate the evolution time of the transient. *SRH coefficient used in bottom cathode device only. **In the band gap, below conduction band (0 V)

210 Appendix B

Full p-i-n device parameter set 2

Parameter name Symbol Value Unit Band gap[63] Eg 1.6 eV 20 −3 Effective density of states N0 10 cm Built-in voltage[241] Vbi 1.3 V Relative dielectric constant[66] r 20 19 −3 Mobile ion defect density[48] Na 10 cm −12 2 −1 −1 Ion mobility µa 10 cm V s 2 −1 −1 Electron mobility[100] µn 1 cm V s 2 −1 −1 Hole mobility[100] µp 1 cm V s −12 3 −1 Band-to-band recombination coefficient kbtb 10 cm s 17 −3 n-type donor density ND 3.0 × 10 cm 17 −3 p-type acceptor density NA 3.0 × 10 cm −12 p-type & n-type SRH time constants* τn,SRH , τp,SRH 10 s SRH trap energy, n-type Et,n−type -0.8 eV** SRH trap energy, p-type Et,p−type -0.8 eV** 25 −3 −1 Incident photon flux (uniform generation) G0 6.25 × 10 cm s

Table B.1: Key device simulation parameters set 2. All values for the perovskite phase are approximately based on literature values. The Shockley-Read-Hall (SRH) recombination rate coefficient and trap energies were chosen to reproduce the approximate open circuit voltage for the bottom cathode device. The generation rate was chosen to yield an absorbed photocurrent of ≈ 16 mA cm−2. The ionic mobility was adjusted from our previous literature estimate of about 4 × 10−11 cm2 V−1 s−1 (ref. [48]) to a lower value of 10−12 cm2 V−1 s−1 to correctly approximate the evolution time of the transient. *SRH coefficient used in bottom cathode device only. **In the band gap, below conduction band (0 V)

211 Appendix C

Full device parameter set 1

Parameter name Symbol p-type Intrinsic n-type Unit Electron affinity ΦEA 0 −0.5 −1.0 eV Ionisation potential ΦIP −1.6 −2.1 −2.6 eV Band gap[63] Eg 1.6 1.6 1.6 eV 20 20 20 −3 Effective density of states N0 10 10 10 cm Built-in voltage Vbi 0 0 0 V Relative dielectric constant r 2 12 80 19 −3 Mobile ion defect density Na - 10 - cm −12 2 −1 −1 Ion mobility µa - 10 - cm V s −4 −3 2 −1 −1 Electron mobility[232, 233] µn 4 × 10 20* 10 cm V s −4 −3 2 −1 −1 Hole mobility[232, 233] µp 4 × 10 20* 10 cm V s −42 −40 −47 3 −1 Band-to-band rec. coefficient kbtb 1.6 × 10 1.0 × 10 1.8 × 10 cm s 16 −3 Donor density ND - - 4.36 × 10 cm 14 −3 Acceptor density NA 9.12 × 10 - - cm −12 −12 SRH time constants τSRH 10 - 10 s SRH trap energy Et −0.8 - −1.8 eV 25 −3 −1 Absorbed photon flux G0 - 6.25 × 10 - cm s

Table C.1: Full device simulation parameters. All values for the perovskite phase are approximately based on literature values. The band energy, doping density and effective density of the states values for n and p-type layers are not based on real material properties- large energetic barriers lead to numerical errors due to very low charge densities of minority carriers. Hence, the barrier heights were changed to 0.5 eV. SRH recombination was implemented in a 2 nm layer at the perovskite/contact interfaces. *To simulate an increase in contact resistance the mobility in this layer was reduced to 2 cm2V−1s−1. Where references are absent, order of magnitude approximations were used based on values from Chapter2

212 Appendix D

Bilayer device parameter set

Parameter name Symbol Intrinsic layer n-type Unit Electron affinity EEA 0 −0.5 eV Ionisation potential EIP −1.6 −2.1 eV Band gap[63] Eg 1.6 1.6 eV 20 20 −3 Effective density of states N0 10 10 cm Built-in voltage Vbi 0.25 0.25 V Relative dielectric constant r 12 12 19 −3 Mobile ion defect density[48] Nion 10 0 cm −12 2 −1 −1 Ion mobility µa 10 0 cm V s 2 −1 −1 Electron mobility µn 1 1 cm V s 2 −1 −1 Hole mobility µp 1 1 cm V s −20 −20 3 −1 Band-to-band rec. coefficient kbtb 4.0 × 10 4.0 × 10 cm s 17 −3 Donor density ND 0 3.0 × 10 cm −12 SRH time constants τSRH - 10 s SRH trap energy Et - -1.3 eV 25 −3 −1 Absorbed photon flux G0 6.25 × 10 0 cm s (uniform generation)

Table D.1: Key bilayer simulation parameters.

213 Appendix E

Complementary results for Chapter5

E.1 Hysteresis in the Inverted-ZnO device

a b ] -2 5 Al 0 ZnO -5

CH 3NH 3PbI 3 -10 PEDOT:PSS ITO -15 -20

Current density cm [mA Current -0.2 0 0.2 0.4 0.6 0.8 1.0 Voltage [V]

Figure E.1: Hysteresis in an Inverted cell using a ZnO electron collection layer instead of PCBM. (a) Device architecture for a ITO/PEDOT:PSS/ CH3NH3PbI3/ZnO/Al perovskite solar cell (b) Forward (black curve) and reverse (red curve) current-voltage scan (125 mV/s) exhibiting large J-V hysteresis (ITO/ PEDOT:PSS/CH3NH3PbI3/PCBM/Al devices showed minimal hysteresis cf. Figure 5.2c). The current-voltage sweeps for this control device were hysteresis free, similar to those shown in Figure 5.2c, and indicate that the IPA solvent and heating steps are not responsible for the presence of hysteresis in the device containing ZnO.

214 Appendix E. Complementary results for Chapter5 215

E.1.1 Fabrication of the Inverted-ZnO device stack

The ITO/PEDOT:PSS/CH3NH3PbI3 stack of layers was fabricated as described in reference [154]. The CH3NH3PbI3 layer was then contacted with a layer of ZnO nanoparticles. The 20 mg/ml solution of ZnO nanoparticles was prepared as described in reference [242], only in this instance the solvent was replaced completely with IPA. The nanoparticle solution was spin cast on top of the perovskite layer at 2000 rpm for 45 seconds followed by heating at 120 ◦C for 5 minutes. An evaporated Al layer was then deposited in the same way as for the PCBM containing device. A control ITO/

PEDOT:PSS/CH3NH3PbI3/PCBM/Al device was also prepared in which pure IPA was spun onto the PCBM layer and then heated prior to the evaporation of Al.

E.2 Ion enabled open circuit voltage enhancement

As shown in Chapter2, Section 2.4.2, Figure 2.12, the built-in potential in perovskite solar cells (PSCs) is expected to be very low (< 0.3 eV). Provided that devices include highly selective and well passivated contacts and active material diffusion lengths are sufficiently long, the absence of a built-in field does not necessarily lead to a low efficiency device.[32, 37] However, in devices with a low built-in field and poorly passivated contact interfaces, the VOC and hence efficiency of devices without mobile ionic species becomes limited by the development of a reverse electric field induced by the photovoltage. In devices with mobile ions however, screening of the field can lead to an enhancement of the VOC. Figure E.2 compares open circuit conditions for simulated devices with and without mobile ions. By redistributing minority carriers away from the interfaces, and corresponding high recombination rate coefficients, the presence of a mobile ionic species leads to a significant enhancement in VOC of over 120 mV. Appendix E. Complementary results for Chapter5 216

a E b 0 CB 0

EFn -1 EFp -1 Energy [eV] Energy [eV] E -2 VOC = 0.66 V VB -2 VOC = 0.78 V

17 ] ] 10 15 10 -3 -3 p 10 15 10 13 10 13 Carrier Carrier n 11 11 10 density [cm density density [cm density 10 0 200 400 600 800 1.05 ]

Position [nm] -3

cm 1 19 Ion density Ion [x10 0.95

0 200 400 600 800 Position [nm]

Figure E.2: Open circuit voltage enhancement in devices with zero built in field. (a) Without mobile ions, a reverse field arises in the cell, limiting the open circuit voltage. (b) The presence of mobile ions in the cells leads to screening of the reverse field an enhancement of the open circuit voltage. The generation rate was set to 1 Sun simeq. Appendix F

Complementary results for Chapter6

F.1 Derivation of the ideality factor

From Equation 6.3, the ideality factor nid expressed in terms of the open circuit voltage

VOC and the recombination rate U:

q dVOC nid = kBT d ln(U)

α Substituting the expression in Equation 6.4 and U = kαn yields:

  β  kBT n d ln 2 q q ni nid = α kBT d ln(kαn )

βk T    d B ln(n) − ln n(2−β) q q i = k T    (1−α) B d α ln(n) + ln kα

  (2−β) β d ln(n) − ln ni = α   (1−α) d ln(n) + ln kα

217 Appendix F. Complementary results for Chapter6 218

Let: x = ln(n)

 (2−β) b = − ln ni

 (1−α) c = ln kα

β d(x + b) n = id α d(x + c)

Using the variable substitution: x = x0 − c β d(x0 + b − c) β n = = id α dx0 α

F.2 Derivation of reaction orders

F.2.1 Band-to-band recombination

For n = p: 2 Ubtb = kbtbnp = kbtbn

Hence α = 1.

For n >> p, the majority carrier density can be considered a constant, leading to:

Ubtb ∝ p

Hence α = 1.

F.2.2 Shockley Read Hall recombination

The compact form of the SRH recombination expression is:

2 (np − ni ) USRH = τn(p + pt) + τp(n + nt)

nt and pt are the electron and hole densities when the Fermi level is at the trap level

Et, for example for electrons:   Et − ECB nt = ni exp kBT Appendix F. Complementary results for Chapter6 219

where ECB is the conduction band energy.

F.2.3 Bulk SRH, Shallow traps

For shallow traps close to the conduction band, in most cases, EFn < Et, and n << nt, p >> pt, and p << nt

Where n = p the rate of recombination is determined by both carrier densities:

np n2 USRH ≈ ≈ τnp + τpnt τpnt

Hence α = 2.

F.2.4 Bulk SRH, Mid-gap traps

For mid gap traps, under bias, EFn > Et, hence n >> nt and p >> pt. Where n = p:

np n2 n USRH ≈ = = τnp + τpn n(τn + τp) (τn + τp)

Hence α = 1.

F.2.5 Surface SRH, Shallow traps

For shallow traps close to the conduction band, in most cases, EFn < Et, and n << nt, p >> pt, and p << nt. Where n >> p, recombination is limited by the availability of holes: np p USRH ≈ ≈ τnp + τpnt τpnt Hence α = 1.

F.2.6 Surface SRH, Mid-gap traps

For mid gap traps, under bias, EFn > Et, hence n >> nt and p >> pt. Where n >> p, recombination is again limited by the availability of holes:

np np p USRH ≈ ≈ ≈ τnp + τpn τpn τp

Hence α = 1. Appendix F. Complementary results for Chapter6 220

F.3 Recombination scheme parameters

Trap energies are referenced to the electron affinity of the material.

F.3.1 Band-to-band recombination

Parameter p-type Intrinsic layer n-type Unit −10 −10 −10 3 −1 kbtb 10 10 10 cm s τn - - - s τp - - - s Et - - - eV

F.3.2 Surface SRH, Shallow traps

Parameter p-type Intrinsic layer n-type Unit −10 −10 −10 3 −1 kbtb 10 10 10 cm s −15 −15 τn 10 - 10 s −15 −15 τp 10 - 10 s Et -1.4 - -1.4 eV

F.3.3 Surface SRH, Mid-gap traps

Parameter p-type Intrinsic layer n-type Unit −10 −10 −10 3 −1 kbtb 10 10 10 cm s −12 −12 τn 10 - 10 s −10 −10 τp 10 - 10 s Et -0.8 - -0.8 eV

F.3.4 Bulk SRH, Shallow traps

Parameter p-type Intrinsic layer n-type Unit −10 −10 −10 3 −1 kbtb 10 10 10 cm s −15 τn - 10 - s −15 τp - 10 - s Et - -0.2 - eV Appendix F. Complementary results for Chapter6 221

F.3.5 Bulk SRH, Mid-gap traps

Parameter p-type Intrinsic layer n-type Unit −10 −10 −10 3 −1 kbtb 10 10 10 cm s −10 τn - 10 - s −10 τp - 10 - s Et,n−type - -0.8 - eV

While the nominal band-to-band rate coefficients were set reasonably high, it can be shown that SRH recombination would still dominate in all cases provided that τSRH << kbtbn. We start by expressing recombination U as a sum of SRH and band-to-band recombination:

1 2 U = n + kbtbn τSRH

If SRH dominates then: 1 2 n >> kbtbn τSRH which leads to the condition: 1 n << τSRHkbtb

−10 20 −3 Hence where τSRH = kbtb = 10 as in Scheme 5, provided n < 10 cm , which is the effective density of states, SRH will dominate. Appendix F. Complementary results for Chapter6 222

F.4 Complementary experimental data

id a Bulk SRH b

n V V = 0 V = 0 V 3.0 mid pre 3.5 pre Surf Bulk SRH SRH mid 3.0 mp-TiO 2 2.0 shallow Band- 2.5 to-band mp-Al O 1.0 2.0 2 3

0.0 Surf SRH 1.5 Inverted shallow

Transient Ideality, 1.0 10 -1 10 0 10 1 10 2 10 -3 10 -2 10 -1 10 0 10 1 Time [s] Time [s]

Figure F.1: Measured and simulated evolution of the ideality factor with time following preconditioning at short circuit. (a) Simulated transient ideality factor, nid, derived from simulations of VOC with Vpre = 0 V, with the following recombination mechanisms: bulk band-to-band recombination (black curve, diamond markers), interfacial SRH recombination via shallow traps (grey curve, square markers), interfacial SRH recombination via deep traps (pink curve, circle markers), bulk SRH recombination via shallow traps (yellow curve, inverted triangle markers), and bulk SRH recombination via deep traps (dark blue curve, triangle markers). The distribution of ions immediately following illumination leads to a reverse internal electric field which results in an accumulation of free holes at the n-type interface such that p > n and an accumulation of electrons at the p-type interface, n > p (Figure 6.4c). Although deeper into the n and p-type contact there is a point where n = p, which will also contribute to recombination. This is very different from what would be expected if no mobile ionic charge were present. A consequence is that an instantaneous ideality factor with an intermediate value of nid(t ≈ 0) ≈ 1.75 is observed. During the subsequent redistribution of ions the cell tends towards an ideality factor of 2 since n ≈ p at the interfaces in the final state (β = 2 and γ = 1). However, due to the different rates of ion redistribution which are slower with increasing φ the net result is a dip in the transient ideality factor during this process resulting in the observed signature nid.(b) Measured nid for mesoporous TiO2 (blue curve, triangle markers), mesoporous Al2O3 (green curve, circle markers) cells following dark Vpre = 0 V for the mp- Al2O3 and mp-TiO2 devices). Experimental data reproduced with the kind permission of Jizhong Yao. Appendix G

Complementary results for Chapter7

G.1 Thermionic emission numerical model

The same MATLAB partial differential equation solver for parabolic and elliptic equations (pdepe) was used to solve the diffusion equation for a single carrier ∆n:

∂∆n(x, t) ∂2∆n(x, t) = −µk T ∂t B ∂x2

The boundary conditions were:

2   ∂∆n(x0, t) ART nbulk −qVbi = −jTE = ∆n(x0, t) exp ∂x q niNCB 2kBT

∂∆n(x , t) d = 0 ∂x

A uniform charge density ∆nini was used for the initial condition:

∆n(x, 0) = ∆nini

Table G1 lists the parameters used in the model.

223 Appendix G. Complementary results for Chapter7 224

Parameter Value Unit −2 −2 AR 120 A cm K T 300 K 6 −3 ni 3.6 × 10 cm 20 −3 NCB 10 cm Vbi 1.3 V 2 −1 −1 µn 1 cm V s 17 −3 ND 3.0 × 10 cm 14 −3 ∆nini 1 × 10 cm d 1 × 10−5 cm

Table G1: Thermionic emission simulation parameters.

The layer thickness was 100 nm and the spatial mesh consisted of 200 linearly-spaced points while the time mesh was logarithmic with 1000 points. Figure G1 shows an 10 −3 example of ∆n as a function of position for nbulk = 4.3 × 10 cm . ∆n was integrated with respect to position and exponential fitting functions were used to obtain rate constants for the decay. 10 10 -12 s

] 8 -3

cm 6 13 4 [x10

n 2 ∆ 10 -5 s 0 0 20 40 60 80 100 Position [nm]

Figure G1: Thermionic emission excess electron charge density decay. 10 −3 Solution for nbulk = 4.3 × 10 cm . Appendix G. Complementary results for Chapter7 225

G.2 Complementary experimental results

10-2

φ 1.2 Sun [V] 10-3

V

∆ 15.6 Sun

10-4 -0.5 0 0.5 1 1.5 2 Time [ µs]

Figure G2: Transient photolvoltage measurements on c-TiO2 device at high light intensities. The bias light was comprised of a white ring of LEDs in combination with a constant wave 473 nm laser. Appendix G. Complementary results for Chapter7 226

G.3 Complementary simulation results

a 9 k = 10 8 s -1 8 SRH 3 0.2 t ) d 0 µs 40 µs 7 ] 10 µs 80 µs -3 TPV k m = 0.98 2 20 µs 160 µs 0 6 #eld potential [V] Electric cm 5 14 1 -0.2 log10( Bulk SRH 4 recombination [x10 n 0 -0.4

3 ∆ -3 -2 -1 0 1 2 3 φ= 10 -3 Sun sim eq. log10(φ) -0.6 0 s 8 µs e 1 µs 32 µs 0 ] 2 µs 64 µs 10 -3 b c 6 0 10 0 t cm

17 4 -0.2 -1 10 3 [x10 φ = 10 [Norm] 2 n

V -0.4 -3 3 ∆ ∆ φ = 10 φ = 10 Sun sim eq. Sun sim eq. 10 -1 Sun sim eq. 0 -2 -0.6 10 0 200 400 600 800 0 0.2 0.4 0 2 4 6 8 Position [nm] Time [ms] Time [ µs]

Figure G3: Simulated transient photovoltage rate constants at Ref. and excess electron density as a function of bias light intensity for a device with bulk SRH recombination (a) Simulated TPV rate constants in a p-i-n device with mobile ions, dominated by bulk SRH recombination. The curve shows a linearly light-intensity dependent regime and an inflection point close to the expected value 8 −1 (kSRH = 10 s ). (b) and (c) The TPV traces show predominantly mono-exponential decay characters independent of light intensity. (d) At low light intensities excess charge drifts into the contact region as with surface recombination dominated devices. (e) At high light intensities, the excess electron density is distributed approximately uniformly in the device. Appendix H

Permission documents

Reference [175], Philip Calado, Andrew M Telford, Daniel Bryant, Xiaoe Li, Jenny Nelson, Brian C. ORegan, and Piers R.F. Barnes. Evidence for ion migration in hybrid perovskite solar cells with minimal hysteresis. Nature Communications, 7, dec 2016.

227 Appendix H. Permission documents 228

Reference [122], Rebecca A. Belisle, William H. Nguyen, Andrea R. Bowring, Philip Calado, Xiaoe Li, Stuart J. C. Irvine, Michael D McGehee, Piers R. F. Barnes, and Brian C. ORegan. Interpretation of inverted photocurrent transients in organic lead halide perovskite solar cells: proof of the field screening by mobile ions and determination of the space charge layer widths. Energy & Environmental Science, 2016. Appendix H. Permission documents 229

Reference [11], Surface area required to power the world with zero carbon emissions and with solar alone, Land Art Generator, 2009.

Friday, 15 December 2017 at 12:07:21 Greenwich Mean Time

Subject: Re: Copyright request Date: Thursday, 12 October 2017 at 12:11:37 Brish Summer Time From: Elizabeth and Robert To: Calado, Phil Hi Phil,

Thanks for your email.

Yes, it is perfectly fine to reproduce that graphic. Please provide credit to the Land Art Generator Iniave and a URL to our website if possible. If you need a higher resoluon version let us know.

Best of luck with your thesis. Send us a copy too when it is finished!

Cheers, Elizabeth and Robert

The Land Art Generator Iniave Robert Ferry & Elizabeth Monoian Founding Directors, LAGI landartgenerator.org windnest.org artenergycamp.org +1 509 961 6237

LAGI is a project of Society for Cultural Exchange, a 501(c)(3) nonprofit organizaon and is supported in part by The J.M.K. Innovaon Prize, a program of the J.M. Kaplan Fund.

On Oct 10, 2017, at 4:16 AM, Calado, Phil wrote:

Dear Elizabeth and Robert,

I am a PhD student at Imperial College London, currently in the process of wring up my thesis. I would like to request permission to use one of your graphics from this page on the area of land required to power the world using solar energy:

hp://landartgenerator.org/blagi/archives/127

I would be really grateful if you could let me know if this would be ok?

All the Best,

Phil

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Reference [13], Commercial progress and challenges for photovoltaics. Nature Energy, 15:15015, 2016.

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Reference [48], Christopher Eames, Jarvist M. Frost, Piers R. F. Barnes, Brian C. O’Regan, Aron Walsh, and M. Saiful Islam. Ionic transport in hybrid lead iodide perovskite solar cells. Nature Communications, 6:7497, 2015.

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Reference [93], Mark T. Weller, Oliver J. Weber, Paul F. Henry, Antonietta M. Di Pumpo, and Thomas C. Hansen. Complete structure and cation orientation in the perovskite photovoltaic methylammonium lead iodide between 100 and 352 K. Chem. Commun., 51(20):41804183, 2015. Appendix H. Permission documents 233

Reference [69], Emilio J Juarez-Perez, Rafael S Sanchez, Laura Badia, Germa Garcia, Yong Soo Kang, Ivan Mora-sero, and Juan Bisquert. Supporting Information -Photoinduced Giant Dielectric Constant in Lead Halide Perovskite Solar Cells. Journal of Physical Chemistry Letters, pages 1-10, 2014

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Reference [142], Hsinhan Tsai, Wanyi Nie, Jean-Christophe Blancon, Constantinos C. Stoumpos, Reza Asadpour, Boris Harutyunyan, Amanda J. Neukirch, Rafael Verduzco, Jared J. Crochet, Sergei Tretiak, Laurent Pedesseau, Jacky Even, Muhammad A. Alam, Gautam Gupta, Jun Lou, Pulickel M. Ajayan, Michael J. Bedzyk, Mercouri G. Kanatzidis, and Aditya D. Mohite. High-efficiency two-dimensional Ruddlesden-Popper perovskite solar cells. Nature, 536(7616):312316, 2016.

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Reference [155], Zhengguo Xiao, Qingfeng Dong, Cheng Bi, Yuchuan Shao, Yongbo Yuan, and Jinsong Huang. Solvent Annealing of Perovskite-Induced Crystal Growth for Photovoltaic-Device Efficiency Enhancement. Advanced Materials, 26:6503 6509, 2014.

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Reference [30], Henry J. Snaith, Antonio Abate, James M. Ball, Giles E. Eperon, Tomas Leijtens, Nakita Kimberly Noel, Samuel David Stranks, Jacob Tse-Wei Wang, Konrad Wo- jciechowski, and Wei Zhang. Anomalous Hysteresis in Perovskite Solar Cells. The Journal of Physical Chemistry Letters, 5:1511-1515, 2014.

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Reference [31], Brian C ORegan, Piers R.F. Barnes, Xiaoe Li, Chunhung Law, Emilio Palomares, and Jose Manuel Marin-Beloqui. Opto-electronic studies of methylammonium lead iodide perovskite solar cells with mesoporous TiO2; separation of electronic and chemical charge storage, understanding two recombination lifetimes, and the evolution of band offsets during JV hysteresis. Journal of the American Chemical Society, 137:50875099, 2015.

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Reference [32], W. Tress, N. Marinova, T. Moehl, S. M. Zakeeruddin, Mohammad Khaja Nazeeruddin, and M. Gr atzel. Understanding the rate-dependent JV hysteresis, slow time component, and aging in CH3NH3PbI3 perovskite solar cells: the role of a compensated electric field. Energy Environ. Sci., pages 9951004, 2015.

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Reference [184], Stephan van Reenen, Martijn Kemerink, and Henry J. Snaith. Modeling Anomalous Hysteresis in Perovskite Solar Cells. The Journal of Physical Chemistry Letters, 6:3808-3814, 2015.

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Reference [158], Dongqin Bi, Wolfgang Tress, M Ibrahim Dar, Peng Gao, Jingshan Luo, Clementine Renevier, Kurt Schenk, Antonio Abate, Fabrizio Giordano, Juan-pablo Correa Baena, Jean-david Decoppet, Shaik Mohammed Zakeeruddin, Mohammad Khaja Nazeeruddin, Michael Gratzel, and Anders Hagfeldt. Efficient luminescent solar cells based on tailored mixed-cation perovskites. Science Advances, (January), 2016.