Computational Molecular Science (2019) Life and Times of an Electron in a Perovskite Solar Cell Prof. Aron Walsh Imperial College London, UK Yonsei University, Korea Materials Design Group: https://wmd-group.github.io @lonepair

Life of a Photoexcited Electron* *In a semiconductor host, where the terms electron and (charge) carrier are used

“Total” Crystal Hamiltonian Crystals are not frozen in space and time. Should consider vibrational and electronic excitations Source: D. C. Wallace – Statistical Physics of Crystals and Liquids (2002) Crystal Potential static model Electronic Excitations Harmonic Phonons vibrations Anharmonic Interactions Electron-Phonon Coupling

Exciting Perovskites A. (Rapid) Theory of Crystal Dynamics B. Vibrations of CH3 NH3 PbI3 C. What Size is an Electron? D. Research Challenges

Why Phonons? Collective vibrational excitations of crystals: N atoms vibrate as 3N phonon modes, ⍵(q) Essential for: • Free energy of crystals • Vibrational spectra • Thermal expansion • Phase transformations • Heat flow • Electrical conductivity Crystal momentum

Theory of Crystal Dynamics Anharmonicity All higher-order terms Harmonic Phonons Ionic Forces = 0 at equilibrium Crystal potential expanded with ion displacements (r) Crystal Potential static model Born and Huang, Dynamical Theory of Crystal Lattices (1958) Collective vibrational excitations of crystals: N atoms vibrate as 3N phonon modes, ⍵(q)

Harmonic Approximation (HA) U(Q) Q Real PES HA ! " = 1 2 &"' ( " = − *! " *" = −&" • Analytic solutions • 3N normal modes w/ frequency ω i • ": normal mode coordinate Energy Force Schematic courtesy of Dr Jonathan Skelton (now University of Manchester) Weak anharmonicity

“Jacob’s Ladder” of Crystal Dynamics Frequency shifts temperature & pressure Variety of anharmonic techniques available – alternative is (large-scale) molecular dynamics Mode lifetimes linewidths & conductivity Thermal expansion Grüneisen parameters Phonon frequencies thermodynamics & spectra Phonopy, Phon, and built into most DFT packages Phono3py ShengBTE AlmaBTE Alamode, D3q, SCALID TDEP, Phono4py… machine learning assisted Many other codes available

Organic-Inorganic Phonon Challenges J. K. Bristow et al, PCCP 18, 29316 (2016) Complex structures with large vibrational range

Issue Cause Solution Large unit cells 100–1000s of atoms Large computers / interaction cutoff radii Missing H No information on H location from XRD Electron counting and chemical knowledge Defective structures Missing ligands or clusters (up to 25%) Explicit simulation of defect processes Symmetry breaking Average structure from X- ray diffraction (with no H) Check phonons for imaginary modes Unphysical dynamics Large phonon range (0– 3500 cm-1) can result in slow MD equilibration Avoid initialisation from random displacements Organic-Inorganic Phonon Challenges

Exciting Perovskites A. (Rapid) Theory of Crystal Dynamics B. Vibrations of CH3 NH3 PbI3 C. What Size is an Electron? D. Research Challenges

Hybrid Organic–Inorganic Perovskites Brief History (1958) – Photoconductivity in CsPbI3 (Møller) (1978) – Synthesis of CH3 NH3 PbI3 (Weber) (1994) – Metallic transition in CH3 NH3 SnI3 (Mitzi) (2009) – Perovskite dye cell (Miyasaka) (2012) – Planar thin-film solar cell (Snaith) Inorganic CsPbI3 Hybrid CH3 NH3 PbI3 or MAPI

Phonon Modes of CH3 NH3 PbI3 High-frequency: confined CH3 NH3 + modes

Phonon Modes of CH3 NH3 PbI3 Low-frequency: PbI3 - and coupled modes Unit: cm-1

Harmonic Phonon Dispersion, ω(#) Gold-Parker et al, PNAS 115, 11905 (2018) Calculated acoustic modes of orthorhombic CH3 NH3 PbI3 (Phonopy; PBEsol) Measured inelastic neutron scattering on single crystals (led by Mike Toney, SLAC) Excellent agreement for energies and dispersion Calculations can be compared to IR/Raman spectra (q~0) or X-ray/neutron scattering 3 meV = 0.7 THz = 24 cm-1

Anharmonic Phonon Lifetimes, ω(#, τ) Phonons are quasiparticles with individual lifetimes determined by their scattering rates Gold-Parker et al, PNAS 115, 11905 (2018) Lifetime and linewidth Theory (Phono3py; PBEsol) gives upper limit of lifetime as only 3-phonon scattering is included (assumed to be dominant) Acoustic phonon linewidth: From Γ to X point

Ultra-Low Thermal Conductivity Heat carrying modes in CH3 NH3 PbI3 involve Pb-I octahedral network with very short lifetimes Gold-Parker et al, PNAS 115, 11905 (2018) Calculated lattice thermal conductivity (Phono3py; PBEsol) At T = 300 K GaAs: 48 W/m/K CH3 NH3 PbI3 : 0.1 W/m/K Integrated thermal conductivity 20 meV = 4.8 THz = 160 cm-1

Low-Frequency Dielectric Response High-frequency from QSGW+SOC Low-frequency from harmonic phonons (DFT/PBEsol) Low-frequency optic phonon modes Sum over phonon modes λ with Born charges Z

Exciting Perovskites A. (Rapid) Theory of Crystal Dynamics B. Vibrations of CH3 NH3 PbI3 C. What Size is an Electron? D. Research Challenges

Carrier Generation and Cooling Long-lived hot carriers upon photoexcitation Effective mass and heat diffusion models: Frost, Whalley, Walsh, ACS Energy Letters 2, 2647 (2017)

Nature of Electron and Hole Carriers Charge carriers in crystals are quasiparticles defined by electron-lattice interaction: polarons Effective mass (Bohr) radius r ~ 10 nm

Nature of Electron and Hole Carriers Charge carriers in crystals are quasiparticles defined by electron-lattice interaction: polarons Large polaron radius r ~ 2.5 nm

Nature of Electron and Hole Carriers Fröhlich electron-lattice interaction ! = 1 2 1 %& − 1 %( ) ℏ+ 2,+ ℏ - . GaAs = 0.1 CdTe = 0.3 CH3 NH3 PbI3 = 2.4 SrTiO3 = 3.8 Intermediate coupling regime: Large polaron Variational solution for Feynman polaron model rP = 4 unit cells mP * = 0.2 me (+30%) µP < 200 cm2V-1s-1 APL Materials 2, 081506 (2014); ACS Energy Lett. 2, 2647 (2017)

Transport of Electron and Hole Carriers Reality: non-parabolic band structure, anharmonic vibrations, multi-mode dielectric response Lucy D. Whalley et al, Phys. Rev. B 99, 085207 (2019) Electronic band structure Predicted carrier mobility Mobility limited by Fröhlich scattering https://github.com/jarvist/PolaronMobility.jl T = 300 K Parabolic m* regime

Exciting Perovskites A. (Rapid) Theory of Crystal Dynamics B. Vibrations of CH3 NH3 PbI3 C. Life of an Electron D. Research Challenges

Semiconductors with a Twist Current-voltage hysteresis Snaith et al, JPCL (2014); Unger et al, EES (2014) Rapid chemical conversion between halides Pellet et al, CoM (2015); Eperon et al, MH (2015) Photoinduced phase separation Hoke et al, CS (2015); Yoon et al, ACS-EL (2016) Electric-field induced phase separation Xiao et al, NatM (2015); Yuan et al, AEM (2016) Photo-stimulated ionic conductivity Yang et al, AChemie (2015); Kim et al, NatM (2018)

Electron-Hole Recombination Conversion efficiencies of solar cells are limited by non-radiative trap-mediated recombination Band (large polaron) electron

Electron-Hole Recombination Conversion efficiencies of solar cells are limited by non-radiative trap-mediated recombination Electron trapped at point defect

Shockley & Read, Phys. Rev. 87, 835 (1952); Hall, Phys. Rev. 87, 387 (1952) Electron-Hole Recombination Conversion efficiencies of solar cells are limited by non-radiative trap-mediated recombination Beyond SRH: defects levels are not fixed, but vary with the charge state. Recombination is a multi-level phonon-emission process Valence band Conduction band Trap level Shockley-Read- Hall (SRH) process 1st order kinetics k SRH ∝ [e]

Structural relaxation (electron-phonon coupling) is a critical component of carrier capture Q = configuration coordinate [change in local structure with charge state] Huang & Rhys, Proc. RS 204, 406 (1950); Henry & Lang, Phys. Rev. 15, 989 (1977) Radiative (weak coupling) Defect luminescence Defect in charge states E1 and E2 Non-Radiative (strong coupling) Phonon emission Electron-Hole Recombination

Requires: (a) defect concentrations, trap levels, configurational coordinates of each charge First-Principles Carrier Capture Rates [Point defect review] Ji-Sang Park et al, Nat. Rev. Mater. 3, 195 (2018) Chemical Potential Limits github.com/jbuckeridge/cplap Self-consistent Fermi Level github.com/jbuckeridge/sc-fermi Equilibrium defect populations

First-Principles Carrier Capture Rates Group developers: Dr Sunghyun Kim Dr Samantha Hood Defectq=0 + eCB - Defectq=-1 Requires: (b) vibrational wavefunctions and electron-phonon interaction strength github.com/WMD-group/CarrierCapture.jl Anharmonicity Included Solve Schrödinger equation for each potential energy surface Following harmonic approach of Alkauskas et al, Phys. Rev. B 90, 075202 (2014) Static coupling approximation

Redox Processes in Metal Halides 2X# + h& → X( # V centre X# + X) # + h& → X( # H centre Whalley, Crespo-Otero, and Walsh, ACS Energy Letters 2, 2713 (2017) Hole trapping in V and H centres studied in metal halides since the 1950s Predicted excited-states TDDFT (PBE0 with SOC) in DALTON2016 On-going work: Carrier trapping rates by Lucy Whalley (PV-CDT)

Conclusion Electrons in semiconductors are under the influence of lattice vibrations. It is often necessary to go beyond the harmonic approx. especially for “soft” halide perovskites. Next step: ab initio prediction of solar cell efficiency limits Project Collaborators: Sunghyun Kim, Jonathan Skelton, Jarvist Frost, Lucy Whalley, Samantha Hood; Ji-Sang Park; Mark van Schilfgaarde (Kings); Sam Stranks (Cambridge); Mike Toney (SLAC) Slides: https://speakerdeck.com/aronwalsh