ESI Polaron Workshop - Polaron mobility and response functions in the Feynman variational approximation

Jarvist Moore Frost (ICL, UK) Feynman Variational Polarons, ESI, Vienna
10th Dec 2019 Imperial College London @JarvistFrost [email protected] https://jarvist.github.io Polaron mobility and response functions in the Feynman variational approximation Jarvist Moore Frost, Artem Bakulin, Tom Hopper, Lucy Whalley, Aron Walsh.

10th Dec 2019 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!

10th Dec 2019 Dielectric response… Fröhlich Polaron (static picture)

10th Dec 2019 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. )

10th Dec 2019 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

10th Dec 2019 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

10th Dec 2019 'Feynman Polaron' units • Energy in multiples of ħω • Frequency in multiples of ω • Mass in multiples of the bare-band effective mass • Reduced thermodynamic Beta ◦ β(T,ω) = ħω/(kB⋅T) ◦ 'Low temperature' = hard material ◦ Soft material = always high T

10th Dec 2019 "It is typical of modern physicists that they will erect skyscrapers of theory upon the slender foundations of outrageously simplified models." —J.M. Ziman, 1962. "Electrons in metals: a short guide to the Fermi surface"

10th Dec 2019

10th Dec 2019 W

10th Dec 2019 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

10th Dec 2019 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 <exp(S-S1)> • <exp(S-S1)> factors out of the integral ◦ I.e. you only need to path-integrate S1 • By convex nature of exp, this is variational

10th Dec 2019 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 <exp(S-S1)> Practically: Tweak variational parameters (v,w) to lower E. You have thereby specified your polaron quasi-particle

10th Dec 2019 M k → Simple Harmonic Motion (ball and chain) An explicitly quasi-particle theory

10th Dec 2019 FHIP mobility theory • Follows <exp(S1-S)> method for an 'influence functional' • Most often remembered for the asymptotic formula, but first develops the contour integral for (frequency dependent) impedance (as used in Hellwarth & Biaggio 1999)

10th Dec 2019 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.

10th Dec 2019 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.

10th Dec 2019 Effective mass + 40% (Phonon drag) (You could use this in a BTE calculation.) Time scale for scattering. Polaron wavefunction (Gaussian), and scale.

10th Dec 2019 Asymptotically we are all dead • Feynman 1955 - athermal actions ◦ ~asymptotic solutions taken for T~=0 ◦ small-α / large-α limits (often reproduced in textbooks) → ZERO TEMPERATURE; SINGLE OPTICAL MODE

10th Dec 2019 Asymptotically we are all athermal • Osaka finite-temperature (free energy) actions

10th Dec 2019 Mishchenko2019 • How well can the FHIP theory, with direct contour integration of the frequency-dependent impedance, approach these results?

10th Dec 2019 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)

10th Dec 2019 FHIP1962 vs. Mishchenko2019

10th Dec 2019 FHIP mobility theory non-monotonic only for alpha>~=8 α = 8

10th Dec 2019 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

10th Dec 2019 Athermal (Fröhlich) action Mishchenko2019 Fig4: α=6 v=4.7 w=1.9

10th Dec 2019 Osaka ﬁnite-temperature action

10th Dec 2019 Freq-dep mobility (α=1)

10th Dec 2019 Multiple phonon branches cm-1 MAPI Low-frequency dispersion (PCCP, AMA Leguy et al., 2016)

10th Dec 2019 Frohlich model cm-1 • Single phonon mode • No dispersion Require 4 input parameters:

10th Dec 2019 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

10th Dec 2019 Representation of i.r. activity from: Phys. Rev. B 92, 144308 (2015) Soft as wood ∴ frequencies small Ionic ∴ Born-Effective-Charges large

10th Dec 2019 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'

10th Dec 2019 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

10th Dec 2019 → 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

10th Dec 2019 Effective mass + 40% (Phonon drag) (You could use this in a BTE calculation.) Time scale for scattering. Polaron wavefunction (Gaussian), and scale.

10th Dec 2019 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.

10th Dec 2019 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

10th Dec 2019 Fröhlich effective mass polarons

10th Dec 2019 (ionic) dielectric from i.r. activity

10th Dec 2019 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 <S-S1> C(v,w,β) - Enthalpy of model action F(v,w,β,α)

10th Dec 2019 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

10th Dec 2019 Mobility comparison (MAPI params)

10th Dec 2019 'v,w' internal variational parameters

10th Dec 2019 Up to 20% correction to Hellwarth effective mode mobility

10th Dec 2019 Explicit modes • Quite different internal parameters for polaron • … while mobility is only +20% • Probably more accurate for real materials? • Directly using the different phonon modes allows each to be treated correctly in terms of Bose-Einstein statistics. The Hellwarth effective frequency method gets 'distracted' by the high-energy modes. • Naive extension to making the Hellwarth effective mode frequency a variational parameter did not work: the solution collapsed to either 0 or infinite frequency.

10th Dec 2019 Polaron device physics What can we try to explain in device physics, with variational polarons? 1) Real-space disorder & its influence on radiative recombination. → Gaussian wavefunction connects Bloch/Band structure to a more mechanistic picture of individual electrons wandering. 2) Slow carrier cooling in hybrid pervoskites, and trends in polaron thermalisation → More limited Kadanoff1963 'Boltzmann equation for polarons' transport theory gives you scattering rates. 3) Much reduced polaron Coulomb-potential (charged defect) scattering rates → An additional (quantum) contribution due to incoherency of Gaussian wavepacket, beyond dielectric screening + localisation.

10th Dec 2019 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 )

10th Dec 2019 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!)

10th Dec 2019 Parameters via DFT 25-75 meV (nearest neighbour) 25 meV (nearest neighbour) 1-5 meV at solar cell ﬁelds

10th Dec 2019 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!

10th Dec 2019 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.)

10th Dec 2019 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...

10th Dec 2019 Cagestrain=25 meV → Semi-ordered Ferroelectric ground state; Intermediate long range order (dynamic) at ﬁnite T 0K 128K 64K 256K 384K

10th Dec 2019 0K 128K 256K 384K Cagestrain=25 meV → Semi-ordered Ferroelectric ground state; Intermediate long range order (dynamic) at ﬁnite T

10th Dec 2019 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

10th Dec 2019 Real space recombination model:

10th Dec 2019 Statistical mechanic argument (thermalised population) V h+ h+ h+ e- e- Recombination...

10th Dec 2019 Boltzmann / mid-gap Fermi Dirac Fermi level

10th Dec 2019 Fermi-Dirac e- quasi Fermi level h+ quasi Fermi level Fermi Level

10th Dec 2019 POLARON POLARON NORALOP How to model polarons in a static potential? POLARON

10th Dec 2019 Gaussian blur!

10th Dec 2019 Recombination Rate Polaron size (lattice units) ( Relaxor ferroelectric @ 300 K )

10th Dec 2019 Recombination Reduction

10th Dec 2019 Simple thermal de-trapping model (Boltzmann distribution of electrons, at 300 K) Mobility Reduction, from disorder?

10th Dec 2019 Where does the hot-electron E go? 78 meV / ps Phonon band structure cm-1 ~15 nm

10th Dec 2019 ACS Energy Lett., 2017, 2, pp 2647–2652. October 23, 2017. DOI: 10.1021/acsenergylett.7b00862

10th Dec 2019 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!

10th Dec 2019

10th Dec 2019 (Final) experimental data

10th Dec 2019 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

10th Dec 2019 Polaron recombination Why are lead-halide perovskite solar cells so insensitive to defects? Is there something special about the polaron state?

10th Dec 2019 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.)

10th Dec 2019 "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.

10th Dec 2019 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)

10th Dec 2019 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.

10th Dec 2019 Collaborators:- Piers Barnes Aron Walsh Artem Bakulin WMD Group, Bath/ICL Acknowledgments:- EPSRC - EP/K016288/1 Royal Society - URF/R1/191292 Tom Hopper

10th Dec 2019 The End

10th Dec 2019 Ship wakes: Kelvin or Mach angle? M. Rabaud and F. Moisy, Phys. Rev. Lett. 110, 214503 (2013). Editor's suggestion

10th Dec 2019 Polaron response functions → J. Devreese, J. De Sitter and M. Goovaerts, "Optical Absorption of Polarons in the Feynman-Hellwarth-Iddings-Platzman Approximation". PRB 5,6,2367--2381 (1972). → A. S. Mishchenko, N. Nagaosa, N. V. Prokof'ev, A. Sakamoto, B. V. Svistunov, "Optical conductivity of the Frohlich polaron". PRL 91,23 (2013). M k How are materials characterised? By perturbing them! Often 'ground state' features such as effective masses and mobilities are generated by fitting a (often simplistic) model to experimental data. → simulate actual polaron response.

10th Dec 2019 Frequency dependent mobility (Feynman et al. 1962)

10th Dec 2019 Calculated Im(χ) ( No published data to compare to. But not Drude-like. ) Massive additional loss when optical phonons are generated (2 THz)) Quadrature becomes numerically unstable (highly oscillatory function)

10th Dec 2019 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?

10th Dec 2019 Slow cooling of photo excited states Gifted a lot of computer time, we did 40'000+ DFT calculations, to get 3rd order force constants. The resulting thermal conductivity is orders of magnitude lower than traditional semiconductors. Phonon anharmonicity, lifetimes, and thermal transport in CH3NH3PbI3 from many-body perturbation theory Lucy D. Whalley, Jonathan M. Skelton, Jarvist M. Frost, and Aron Walsh Phys. Rev. B 94, 220301(R) – Published 8 December 2016

10th Dec 2019 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.

10th Dec 2019 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+

10th Dec 2019 ( 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

10th Dec 2019 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

10th Dec 2019 Iodine location, MAPI, ~100 ps MD

10th Dec 2019 Glazer Tilting - Glazer 1972

10th Dec 2019 Lead Iodine Pb: Lone pair / 2nd order Jahn-Teller distortion Carbon (Methylammonium) "It's as soft as jelly!" (Bulk modulus ~wood.)

10th Dec 2019 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

ESI Polaron Workshop - Polaron mobility and res...

ESI Polaron Workshop - Polaron mobility and response functions in the Feynman variational approximation

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