Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Imperial College London Email: jarvist.frost@ic.ac.uk Twitter: @JarvistFrost https://jarvist.github.io Sustainable materials modelling for a sustainable future Jarvist Moore Frost. Lucy Whalley, Federico Brivio, Jonathan Skelton, and Aron Walsh. Tom Hopper, and Artem Bakulin.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Carbon cost of High Performance Computing epcc.ed.ac.uk 1.2 MW for 4920 compute nodes (each 24 cores) ⇒ 250 W / node, 6 kWhr in 24 hours Each 24-hour single-node job = 8.6 kAU ⇒ 700 kWhr / MAU But UK as a whole averages about 400 g / kWhr for the year ⇒ 280 kg CO2 / MAU on ARCHER (Flying to Boston and back Economy, for Fall MRS = 800 kg CO2, ~= 3 MAU calculations.) Provisos: ● Electricity is (very quickly) decarbonising ● South Scotland typically has a very low carbon intensity for their electricity (nuclear + wind) ● ARCHER2 will be much more energy efficient Academic Research Computing High End Resource (Archer) Cray XC30

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 UK national supercomputer utilisation … by research area

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 UK national supercomputer utilisation … by language

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Doing what? 'B' term in the 2-electron integrals that make Hartree-Fock theory scale as O(N4)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 "Computers are bicycles for the mind." - Steve Jobs A long way from the hippie ideals of personal computing...

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 J.M. Ziman ©1957 Gonville & Caius College, Cambridge "It is typical of modern physicists that they will erect skyscrapers of theory upon the slender foundations of outrageously simplified models." ~ J.M.Ziman, "Electrons in metals: a short guide to the Fermi surface", 1962 Is almost all by Fermi's golden rule! ● 1st order perturbation theory (Time-dependent Schrodinger equation.) ● Difficult to go beyond this Theorists (>1980) have mainly retreated into ever more involved methods of calculating matrix elements. Link solid state theory ⇔ experiment

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Most solid state (electronic structure) theory based on a fiction of periodicity ● Infinite in all directions ● Perfect registration ● Crystallographic momentum is a good q number Sustainable materials won't be periodic

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Thesis of this talk To model materials one should: ● Build computationally lightweight models, often based on 1950s/1960s solid state theories ● Use standard (DFT) packages to parametrise model ● (preferably) develop in a modern language, that allows natural physical abstractions Why? ● Make technically relevant predictions ○ Many interesting measurements probe response functions ○ Most materials are not single crystals ○ Even single crystals have thermal vibrations ● These predictions are by definition unique ○ You won't get replaced by an exascale computer... ● It's fun!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 A - Molecular Cation - '1+' charge B - {Pb, Sn} - '2+' charge X 3 - Halide {I, Br, Cl*} - '1-' charge Hybrid Halide Perovskites (ABX 3 ) Weber, Dieter. "CH3NH3PbX3, ein Pb (II)-System mit kubischer Perowskitstruktur/CH3NH3PbX3, a Pb (II)-System with Cubic Perovskite Structure." Zeitschrift für Naturforschung B 33.12 (1978): 1443-1445.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Solution processed → defective Soft → thermal disorder ● Slow radiative recombination (for a direct gap material) ● Sufficient mobility to get charges out (But not that high considering effective mass 0.12, ~50 cm2/Vs vs. 1000 cm2/Vs for CdTe) ● Almost absent non-radiative recombination Why can we make efficient solar cells out of solution processed MAPI?

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Methylammonium (CH 3 NH 3 +) ; MA A closed shell (18 e-) molecular cation with a large electric dipole (2.2 D) J. M. Frost et al, Nano Letters 14, 2584 (2014) Deprotonation (pK a ~ 10): CH 3 NH 3 + → CH 3 NH 2 + H+

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 ( Videos on YouTube - search for 'MAPI molecular dynamics' ) https://youtu.be/K_-rsop0n5A Incredibly Soft crystal; large distortions of octahedra ➔ MA ion yaw ➔ ...and roll… ➔ ...CH3 clicks ➔ so does NH3 [2x2x2 Pseudo cubic relaxed supercell, lattice parameters held constant during MD (NVT simulation). PBESol Functional at the Gamma point (forces + energies should converge well). dt = 0.5 fs, T = 300 K ] ~2 ps timescale to MA rotation, And octahedra tilting / distortion Molecular Dynamics

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 100 ps MD; 2x2x2 supercell; Iodine location

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Glazer Tilting - Glazer 1972

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Lead Iodine Pb: Lone pair / 2nd order Jahn-Teller distortion Carbon (Methylammonium) "It's as soft as jelly!" (Bulk modulus ~wood.)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Experimental validation... 2D infrared spectroscopy ~ 3 ps Bakulin et al. J. Phys. Chem. Lett., 2015, 6 (18), pp 3663–3669 Quasi-Elastic Neutron Scattering (QENS) ~14 ps ; Leguy et al., Nature Communications 2015, 6, 7124 ~5 ps (higher SNR); Chen et al. Phys. Chem. Chem. Phys., 2015,17, 31278-31286 (2015) DFT Molecular Dynamics → 2x2x2 unit cell ~2.5 ps ; Bakulin et al. ~2 ps (FAPI) ; Weller et al. J. Phys. Chem. Lett., 2015, 6 (16), pp 3209–3212

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 A total of 58 ps (2319 frames) of data was used for analysis, after an equilibration run of 5 ps. This generated 18547 unique MA alignment vectors.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Reduce to first octant… abs(r), r=[x,y,z]

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Apply reflection symmetry… sorted(abs(r)) , r={x,y,z} = 48 fold increase in SNR (Tom Ruen, Wikimedia Commons)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 FACE (X) DIAGONAL (R) EDGE (M)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 FACE (X) DIAGONAL (R) EDGE (M) FACE: 42% EDGE: 31% DIAG.: 26% (weighted by MC integration of random sphere points)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Lattice Dynamics (Phonons)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Quantum (an)Harmonic Oscillators & e-ph coupling Motivation: How to treat soft phonon modes? What is the repercussion for the electronic structure (and electron-phonon coupling) for such large tilting modes?

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Adiabatic electron-phonon coupling Born-Oppenheimer approximation (full wavefunction is product of electronic and nuclear wavefunctions) Adiabatic approximation: treat Nuclear and Electronic degrees of freedom separately. Mean-field expectation. Solve Sch. Eqn. for nuclear degree of freedom. Solve electronic Sch. Eqn. varying nuclear degree of freedom (i.e. deformation potential). Combine in mean-field manner.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 The 1D Schrodinger equation is easy to solve!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 ~15 meV well persists in structure to > 600 K BE Distribution 600 K 1 K

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Iodine 300K Iodine 200K

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 (Calculations: Lucy Whalley) Band-gap as a function of Q (Deformation Potential)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 (R acoustic mode at Brillouin-Zone boundary [tilt])

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 R M Phonon anharmonicity, lifetimes, and thermal transport in CH 3 NH 3 PbI 3 from many-body perturbation theory LD Whalley, JM Skelton, JM Frost, A Walsh - Physical Review B, 2016

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 What is a Polaron? ➔ bare electron interacts with polar modes of lattice → polaron (the i.r. active lattice vibrations) ➔ becomes dressed in a cloud of excitations ➔ interactions energetically trap particle… ➔ And shield interaction between particles... (A Guide to Feynman Diagrams in the Many-body Problem, R.D. Mattuck) e + + + + + + A Quasiparticle!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Dielectric response… Fröhlich Polaron (static picture)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Fröhlich effective mass polarons α GaAs: 0.068 CdTe: 0.29 AgCl: 1.84 SrTiO3: 3.77 (Devreese 2005) We need: ➔ Difference of dielectric constants ➔ Characteristic frequency of 'Linear Optical' mode ➔ Effective mass of electron This is the long-range dielectric electron-phonon interaction (the 1/q divergence that causes issues in ab-initio calculations). (Original form Landau (1933); this follows Jones & March (1985), "Theoretical Solid State Physics Vol 2". See also Devreese (2016), arXiv:1611.06122. )

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 W

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 Infinite quantum field of phonon excitations Path Integrals for Pedestrians (2016) https://doi.org/10.1142/9183

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 M k → Simple Harmonic Motion (ball and chain) An explicitly quasi-particle theory

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Free energy of polaron, by path integration. Optimisation by automatic-differentiation. Explicit contour integration of polaron self-energy on complex plane Mobility, polaron mass, spring constant, absorption profile etc.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 julia> import Pkg; Pkg.add("PolaronMobility") julia> using PolaronMobility julia> MAPIe=polaronmobility(300, 4.5, 24.1, 2.25E12, 0.12) ... T: 300.000000 β: 2.41e+20 βred: 0.36 ħω = 9.31 meV Converged? : true VariationalParams v= 19.86 w= 16.96 || M=0.371407 k=106.835753 POLARON SIZE (rf), following Schultz1959. (s.d. of Gaussian polaron ψ ) Schultz1959(2.4): rf= 0.528075 (int units) = 2.68001e-09 m [SI] Polaron Free Energy: A= -6.448815 B= 7.355626 C= 2.911977 F= -3.818788 = -35.534786 meV Polaron Mobility theories: μ(FHIP)= 0.082049 m^2/Vs = 820.49 cm^2/Vs Eqm. Phonon. pop. Nbar: 2.308150 μ(Kadanoff1963 [Eqn. 25]) = 0.019689 m^2/Vs = 196.89 cm^2/Vs Tau=1/Gamma0 = 1.15751e-13 = 0.115751 ps μ(Hellwarth1999)= 0.013642 m^2/Vs = 136.42 cm^2/Vs ... https://github.com/jarvist/PolaronMobility.jl Open source! Please use / adapt.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 → Semonin et al.: The Journal of Physical Chemistry Letters 7, 3510 (2016) → Saidaminov et al.: Nature Communications 6, 7586 (2015) → Milot et al.: Advanced Functional Materials 25, 6218 (2015) μ(electron) = 136 cm^2/Vs μ(hole) = 94 cm^2/Vs μ(Saidaminov) = 67.2 cm^2/Vs μ(Milot/Herz) = 35 cm^2/Vs μ(Semonin) = 115 cm^2/Vs

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Effective mass + 40% (Phonon drag) (You could use this in a BTE calculation.) Time scale for scattering. Polaron wavefunction (Gaussian), and scale.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Where does the hot-electron E go? 78 meV / ps Phonon band structure cm-1 ~15 nm

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 ACS Energy Lett., 2017, 2, pp 2647–2652. October 23, 2017. DOI: 10.1021/acsenergylett.7b00862

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 By bulk thermal conductivity... MAPI has extremely low thermal conductivity: a phonon glass. MAPI: κ = 0.05 W m−1K−1 CsPbI3: κ = 0.5 W m−1K−1 CdTe: κ = 9 W m−1K−1 GaAs: κ = 38 W m−1K−1 ( Phono3py RTA; PBESol VASP DFT ) MAPI: 41,544 displacements!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 (Final) experimental data

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Experiment : Theory ● Computer environments are really built for hypothesis testing and model ● Hybrid perovskite have ~twice the heat capacity of inorganic (count the modes! 16 vs. 9 in thermal window) Polaron scattering rate Heat capacity + polaron size

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 How do electrons scatter? Born approximation assumes: 1) Weak scattering (perturbation theory) 2) Input and output states of the charge-carrier are plane waves (Bloch states) These rates underly almost all device physics models (impurity scattering, non-radiative recombination, defect capture cross section etc.)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 "Scattering of wave packets on atoms in the Born approximation" D.V. Karlovets, G.L. Kotkin, and V.G. Serbo PRA 92, 052703 (2015) A very similar problem explored recently in accelerator physics. (Airy beams - electron accelerators can focus to < 1nm.) Standard Born Approximation: Fourier-Transform of potential Karlovets2015: Multiply with transverse wavefunction before Fourier transform.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Scattering of Gaussian wavepackets (polarons) Polaron scattering attenuated by: ● Classical contribution from localising the electron ● Quantum contribution from incoherency of Gaussian wavepacket Derivation follows: "Scattering of wave packets on atoms in the Born approximation" D. V. Karlovets, G. L. Kotkin, and V. G. Serbo Phys. Rev. A 92, 052703 (2015)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Weighted for transport Effect further strengthened for transport-relevant scattering. Q) Why did the polaron cross the defective semiconductor? A) Because it was too incoherent to scatter.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Examples of this approach 1/ Hybrid lead halide perovskites (MAPI + friends) a/ Molecular dynamics interpretation b/ Exceeding the Born-Oppenheimer approximation with an 'undergraduate' 1D QM model 2/ Polarons a/ 1950s theories b/ beyond Fermi's golden rule mobilities 3/ Polaron devices physics a/ Explaining slow-cooling in hybrid halide perovskites b/ Scattering beyond the Born approximation c/ Polarons + large scale Monte-Carlo structural models

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Real space disorder: STARRYNIGHT codes Classical Metropolis algorithm simulation of cage:cage dipole interactions. Analytic Hamiltonian, interaction strength parameterised by DFT. ( Apl Materials 2 (8), 081506, 2014. Open source on GitHub https://github.com/WMD-Group/Starrynight )

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Display direction of Dipole by point on HSV sphere p (Nb: Simulation linear scaling + very fast; here I present 2D slices of ~20x20, as any larger and you can't see what's going on!)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Parameters via DFT 25-75 meV (nearest neighbour) 25 meV (nearest neighbour) 1-5 meV at solar cell fields

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 https://github.com/WMD-group/StarryNight Metropolis (local spin move) Monte Carlo code written in C99. Efficient & on lattice → millions of moves per second. Analysis code built in, and additional Julia post processing tools. Open source!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 T= 0 K (Ground State - but a bit out of eqm, due to MC) CageStrain = 0 ---> Anti-Ferroelectric (The potential at a site from the dipole on the nearest neighbour (= 1 in the internet units of Starrynight) is simply 0.165 V.)

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 T= 0 K (Ground State - but a bit out of eqm, due to MC) CageStrain = 50 meV / neighbour ---> Ferroelectric Ferroelectric order parameter tricked by disorder...

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Cagestrain=25 meV → Semi-ordered Ferroelectric ground state; Intermediate long range order (dynamic) at finite T 0K 128K 64K 256K 384K

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 0K 128K 256K 384K Cagestrain=25 meV → Semi-ordered Ferroelectric ground state; Intermediate long range order (dynamic) at finite T

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 POLARON POLARON NORALOP Polarons ~6 lattice units (Frost2014), by Asymptotic Feynman solution Polarons ~4 lattice units (Frost2017) by finite-temperatur e numeric solution (Both numbers are the s.d. of the Gaussian wavefunction.) Real space potential fluctuations

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Real space recombination model:

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Statistical mechanic argument (thermalised population) V h+ h+ h+ e- e- Recombination...

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Boltzmann / mid-gap Fermi Dirac Fermi level

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Fermi-Dirac e- quasi Fermi level h+ quasi Fermi level Fermi Level

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 POLARON POLARON NORALOP How to model polarons in a static potential? POLARON

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Gaussian blur!

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Recombination Rate Polaron size (lattice units) ( Relaxor ferroelectric @ 300 K )

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Recombination Reduction

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Simple thermal de-trapping model (Boltzmann distribution of electrons, at 300 K) Mobility Reduction, from disorder?

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Conclusion ★ You can do a lot with a modest computation ★ Some of this may even have been useful! ★ Basing these calculations on the outputs of electronic structure theory, gives you chemical specificity Everyone in this room can program… Everyone in this room has a physic-y / chemistry-y background… So why not combine them?

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Collaborators:- Piers Barnes Aron Walsh Artem Bakulin WMD Group, Bath/ICL Acknowledgments:- EPSRC - EP/K016288/1 Royal Society - URF/R1/191292 Tom Hopper

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 The End

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 2012-2019 What has theory told us? The fundamental question from 2012: ⇒ Why is a solution-processed material such a good photovoltaic? Does not yet have a definitive answer. But there are lots of suggestions! ● Soft crystal structure. Dynamic local disorder, even for Cs. ● Spin-split indirect-gap reduces recombination rate ● Relativistic band structure could potentially support an IBSC ● Polaron theories of mobility predict predict experimental values ● Slow cooling can be explained by bulk thermal conductivity and polaron model ● Relaxor ferroelectric structure generates local fluctuations in electric potential - affecting recombination and transport. Simple theories sometimes have the most to say. Theory tells us what is possible; experiment tells us what is present.

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Solution processed → defective Soft → thermal disorder ● Slow radiative recombination (for a direct gap material) ■ Slightly-indirect gap due to Rashba splitting (350x) ■ Electrostatic potential fluctuations reduce recombination (10x-100x) ● Sufficient mobility to get charges out (But not that high considering effective mass 0.12, ~50 cm2/Vs vs. 1000 cm2/Vs for CdTe) ⇒ Polarons! ● Almost absent non-radiative recombination ■ ? Few mid gap defects ■ Lower (polaron) cross-section for recombination Why can we make efficient solar cells out of solution processed MAPI?

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 Fröhlich effective mass polarons Units: m/F (inverse of vacuum permittivity) ● Usually viewed as some 'bulk' phenomenological quantity Units: F ● Can view this as the capacitance of the phonon field Units: m^-1 ● Scale the matrix element to make it dimensionless

Jarvist Moore Frost (ICL, UK) New horizons in materials modelling; Sustainable models 10th Jan 2020 General heading Section heading Orange goodness heading Close to dark-grey green thingy Style guide 36 Montserrat - SemiBold 22 Questrial // 22 Montserrat - SemiBold 14 Questrial