Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Imperial College London Email: [email protected] Twitter: @JarvistFrost https://jarvist.github.io Mobility, Defect Scattering, Vibrational Response of Feynman Polarons in Halide Perovskites Jarvist Moore Frost.

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 Lattice Dynamics (Phonons) cm-1 MAPI Low-frequency dispersion (PCCP, AMA Leguy et al., 2016)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Representation of i.r. activity from: Phys. Rev. B 92, 144308 (2015) Soft as wood ∴ frequencies small Ionic ∴ Born-Effective-Charges large

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 What is a Polaron? ● an electron polarises the lattice → polaron (via the polar, i.r. active, lattice vibrations) ➔ Back reaction attempt to 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) NanoGe Computational Perovskites 8th September 2020 Dielectric response… Fröhlich Polaron (small-polaron / static picture)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 The most simple polaron theory ~ Hamiltonian from the 1950s - Fröhlich, Landau. → Single effective-mass electron (bare band effective mass) → Interacts with harmonic lattice vibrations (boson), Via a dielectric (long-range) coupling

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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 phonon frequency ➔ Effective mass of electron This is the long-range dielectric electron-phonon interaction that dominates for polar materials. (Original form Landau (1933); this follows Jones & March (1985), "Theoretical Solid State Physics Vol 2". See also Devreese (2016), arXiv:1611.06122. ) MAPI: 2.4

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 W

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 M k → Simple Harmonic Motion (ball and chain) An explicitly quasi-particle theory

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Free energy of polaron, by path integration. Optimisation by automatic-differentiation. Explicit contour integration of polaron self-energy on complex plane github.com/jarvist/PolaronMobility.jl

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Drude model (1900): (Simon, The Oxford solid state basics)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Sommerfield model (1927): Why this actually works... (Ziman, Principles of the theory of solids, 2nd edition)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 'Ab initio' mobility calculation... A lot of work calculating matrix elements… (But for a plane-wave Ansatz) ● To put into a semi-classical Drude-like transport theory ○ Fermi's Golden Rule (1st order time-dep perturbation theory) ○ (normally) Relaxation time approximation

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 FHIP mobility theory ● Extends method for an 'influence functional' ● Fully quantum-mechanical theory of mobility ○ Includes all phonon scattering processes up to all orders

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Mishchenko2019 ● How well can the FHIP theory, with direct contour integration of the frequency-dependent impedance, approach these results?

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 FHIP1962 vs. Mishchenko2019

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 FHIP mobility theory non-monotonic only for alpha>~=8 α = 8

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Athermal (Fröhlich) action Mishchenko2019 Fig4: α=6 v=4.7 w=1.9

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 n= -0.46 ~= -0.5 n= -1.33 n= -0.95 T-dependence can suggest nature of scattering; polaron optical phonon scattering has a lower exponent than the textbook value.

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 w = −0.53 "Impact of the Organic Cation on the Optoelectronic Properties of Formamidinium Lead Triiodide" Christopher L. Davies et al. J. Phys. Chem. Lett., 2018, 9 (16), pp 4502–4511 Figure 4. Effective charge-carrier mobility ϕμ as a function of temperature for a thin film of FAPbI3. Here, μ is the charge-carrier mobility and ϕ the photon-to-free-charge branching ratio, which is expected to decrease from a high-temperature value of 1 when the temperature is lowered below the value of EX/kb ≈ 60 K and excitons become thermally stable. The solid line shows a fit of μ ∝ Tw to the data for temperatures 60 K and above. A power-law behavior with an exponent of w = −0.53 is found, in agreement with predictions34 based on charge-carrier interactions with polar optical phonons.20

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 How do polarons 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) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 Polaron-renormalised phonon modes + ?

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Polaron-renormalised phonon modes F. Brivio et al, Physical Review B 89, 155204 (2014) Dresselhaus Splitting (SOC)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Conclusions ● Halide perovskites are soft and polar ⇒ strongly interacting large polarons ● Feynman polaron theories seem a good model for the correct physics of charge-carriers in this material ● We can use them to predict:- ○ Temperature-dependent mobilities with no empirical parameters ○ Reduction in defect absorption cross-section / scattering power ○ Renormalisation of phonon energies ○ Slow carrier cooling:- ■ https://doi.org/10.1021/acsenergylett.7b00862 ■ https://doi.org/10.1021/acsenergylett.8b01227

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 The End

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Multiple phonon branches cm-1 MAPI Low-frequency dispersion (PCCP, AMA Leguy et al., 2016)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Nb: Log scale! Dynamic disorder, phonon lifetimes, and the assignment of modes to the vibrational spectra of methylammonium lead halide perovskites AMA Leguy, et al. Physical Chemistry Chemical Physics 18 (39), 27051-27066 (2016) → You'd like to see something like this (semi-classical scattering rates from Ridley's book) → As electron energy (and therefore temperature) increases, scattering pathways 'turn on'

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Fröhlich model cm-1 ● Single phonon mode ● No dispersion Require 4 input parameters:

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Lack of dispersion not so terrible... cm-1 ● I.r. activity only couples around Gamma ● Polar phonon modes are flat here (due to the i.r. coupling!) ● To get the dielectric constant, one integrates over these modes

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Hellwarth & Biaggio approach ● Show (analytically) that collective effect of multiple modes → single effective action f W 4.02 0.082 3.89 0.006 3.53 0.054 2.76 0.021 2.44 0.232 2.25 0.262 2.08 0.234 2.03 0.062 1.57 0.037 1.02 0.013 1.00 0.007 1.00 0.010 0.92 0.011 0.80 0.002 0.57 0.006 ↦ ω effective Used as ω LO

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 w = −0.53 "Impact of the Organic Cation on the Optoelectronic Properties of Formamidinium Lead Triiodide" Christopher L. Davies et al. J. Phys. Chem. Lett., 2018, 9 (16), pp 4502–4511 Figure 4. Effective charge-carrier mobility ϕμ as a function of temperature for a thin film of FAPbI3. Here, μ is the charge-carrier mobility and ϕ the photon-to-free-charge branching ratio, which is expected to decrease from a high-temperature value of 1 when the temperature is lowered below the value of EX/kb ≈ 60 K and excitons become thermally stable. The solid line shows a fit of μ ∝ Tw to the data for temperatures 60 K and above. A power-law behavior with an exponent of w = −0.53 is found, in agreement with predictions34 based on charge-carrier interactions with polar optical phonons.20

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 (ionic) dielectric from i.r. activity

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Ōsaka, adapted by Hellwarth1999 A(v,w,β) - log(Z), partition function of model action B(v,w,β,α) - Electron-phonon coupling terms, Frohlich action with C(v,w,β) - Enthalpy of model action F(v,w,β,α)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Extend variational method to explicit sum over multiple phonons β(T,ω) = ħω/(kB⋅T) F(v,w,T,ω e ,α) = A(v,w,β(T,ω e )) + B(v,w,β(T,ω e ),α) + C(v,w,β(T,ω e )) F(v,w,T,ω e ,{ω i },{α i }) = A + ∑ i B(v,w,β(T,ω i ),α i ) + Explicit modes: solve variational problem with inner loop over individual phonon modes (and thus phonon frequency + electron-phonon coupling) Still need an effective frequency for model Action

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Mobility comparison (MAPI params)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 julialang.org ● Quite easy to get within 2x of C/Fortran ● High level of abstraction possible (due to multiple dispatch) ● High level of code reuse ● Small (compared to Python) collection of high quality numeric computation libraries ● Powerful Forward / Backwards / Source-to-source automatic differentiation ● Code-generation can target X86 / ARM / Cuda / TPUs etc. ● Built in tools enable iterative optimisation improvements

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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) NanoGe Computational Perovskites 8th September 2020 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 → You want to solve this path integral, but you can't. ● Coulomb interaction with lattice disturbance ● Which dies out exponentially → … instead, factor model Action S1 out... → … replace (S-S1) with its average ● factors out of the integral ○ I.e. you only need to path-integrate S1 ● By convex nature of exp, this is variational

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 → … Feynman proposed this (soluble) model action (S1). ● quadratic (harmonic) interaction (C) ● But with tunable dampening (w) ● Not just a substitution actual Action S→ S1 mode Action ● Includes Practically: Tweak variational parameters (v,w) to lower E. You have thereby specified your polaron quasi-particle

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Diagrammatic Monte-Carlo ● Diagrammatic Monte-Carlo to directly sample the Green's function for the polaron propagator. ● A fairly simple method; developing a new open-source code. ● Aim is to use it as reference values for new improved variational path-integral techniques. ● Calculates the zero-temperature fully quantum ground-state 1) Hahn, T.; Klimin, S.; Tempere, J.; Devreese, J. T.; Franchini, C. Diagrammatic Monte Carlo Study of Fröhlich Polaron Dispersion in Two and Three Dimensions. Phys. Rev. B 2018, 97 (13), 134305. https://doi.org/10.1103/PhysRevB.97.134305 . 2) Mishchenko, A. S.; Prokof’ev, N. V.; Sakamoto, A.; Svistunov, B. V. Diagrammatic Quantum Monte Carlo Study of the Fröhlich Polaron. Phys. Rev. B 2000, 62 (10), 6317–6336. https://doi.org/10.1103/PhysRevB.62.6317 . Tethys.jl

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Path-integral Monte-Carlo ● Describe a N-dimension quantum theory, as a N+1-dimension classical theory ● Often applied to quantum-nuclear effect (i-PI) code ● A natural way to include finite temperature effects in a quantum theory ○ (In the limit as T->0, it becomes full quantum & the effort diverges.) https://courses.physics.illinois.edu/phys466/sp201 3/projects/2007/team_pimc/pimc.html

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 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 Fröhlich effective mass polarons

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Technical challenge: ● Quadrature! ● Oscillatory integral ● Slow exponential decay (in Beta) ● Oscillation builds in (nu) Partial solution: Adapt Fourier-integral method to cos(vu) term Exp. decay as exp(-Beta)

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 2020 Non-monotonic mob vs. T, as a function of alpha α=4 α=2.4 α=6 α=8 α=10 α=12 α=14

Jarvist Moore Frost (ICL, UK) NanoGe Computational Perovskites 8th September 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 put effort into ever more involved methods of calculating matrix elements. Link solid state theory ⇔ experiment