2018-03-08_JMFrost_Polarons.pdf

Session S29: Electrons, Phonons, Electron Phonon Scattering and Phononics IV
11:15 AM–2:15 PM, Thursday, March 8, 2018
Sponsoring Units: DCOMP DMP

Abstract: S29.00009 : Polarons in CH3NH3.PBI3: Formation, transport and recombination

Authors:
Jarvist Frost
(University of Bath)

Lucy Whalley
(Materials, Imperial College London)

Jonathan Skelton
(University of Bath)

Pooya Azarhoosh
(Physics, King's College London)

Scott McKechnie
(Physics, King's College London)

Mark Schilfgaarde
(Physics, King's College London)

Aron Walsh
(Materials, Imperial College London)

Hybrid halide perovskites are soft, polar, semiconductors[1]. We propose that low energy (9 meV) optical phonons limit room temperature mobility. We have written open source codes to solve the finite-temperature Feynman polaron state. This provides a temperature-dependent calculation of mobility[2], in good agreement with experiment.

This model suggests a mechanism to explain recent data on slow cooling of photo-excited carriers. The polaron state is stable at high temperature, and has a limited phonon density of states which the hot-electron is in thermal contact with. The low lattice thermal conductivity slows dissipation of this transient hot-spot[3].

We construct a multi-scale model for the formation of the polaron, and its migration through the material. We quantify the beneficial decrease in recombination rate due to segregation of electrons and holes in the 'ferroelectric highways' and relativistic spin-split of the Rashba effect, versus the detrimental decrease in mobility due to disorder. We quantify the contribution of short-range ferroelectric order on carrier stability and electron-hole recombination in this unique class of materials.

[1] JM Frost et al. Acc.Chem.Res. 49 (3) pp 528–535 (2016)
[2] JM Frost. ArXiv:1704.05404
[3] JM Frost et al. ACS Energy Letters (2017)

Transcript

1. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 King's College London / Imperial College London, UK @JarvistFrost jarvist.frost@imperial.ac.uk https://jarvist.github.io Polarons in CH3NH3.PBI3—Formation, Transport and Recombination Jarvist Moore Frost, Lucy Whalley, Jonathan Skelton, Pooya Azarhoosh, Scott McKechnie, Mark Schilfgaarde, and Aron Walsh JM Frost, PRB 96 (19), 195202 (2017). 10.1103/PhysRevB.96.195202
2. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Design of semiconductors: μ Electronic structure is pretty predictive! • (DFT: Structures, lattice response. QSGW: Band-gaps, band dispersion relation.) Charge carrier mobility is not so predictive. • Phenomenological quantity • Most methods are semi-empirical ◦ Kinetic-gas inspired Boltzmann equation (definite scattering events) ◦ Linearised Boltzmann Transport Equation (small field) ◦ Often in the relaxation time approximation ◦ Perturbative (Fermi golden-rule) treatment of el-ph interaction ◦ Often, assume a relaxation / scattering time Many new technological materials are polar (oxides, halides, chalcogenides) ⇒ Polar electron-phonon coupling (Frohlich) domaintes This is often large! Perturbation theory breaks down. ⇒ Polaron mobility theories
3. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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! REAL SPACE
4. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Dielectric response… Fröhlich Polaron (static picture)
5. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 (Every electron in a band structure is delocalised between ALL unit cells.) • Effective mass renormalisation (phonon drag) • Localisation of wavefunction → chops up + blurs perfect band structures
6. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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
7. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 "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"
8. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018
9. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018
10. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 W
11. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 Infinite quantum field of phonon excitations
12. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 (Coulomb interaction with exponentially relaxation polarisation field, left in the lattice by the motion of the electron.) (Proposed quadratic action, with fitted dampening.)
13. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 M k → Simple Harmonic Motion (ball and chain) Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955
14. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Free energy of polaron, by path integration. Optimisation by auto-differentiation. Explicit contour integration of polaron self-energy on complex plane Mobility, polaron mass, spring constant, absorption profile etc.
15. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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.
16. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Iodine location, MAPI, ~100 ps MD "Plastic crystals" - extremely soft @ RT
17. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Fröhlich effective mass polarons α GaAs: 0.068 CdTe: 0.29 AgCl: 1.84 SrTiO3: 3.77 (Devreese 2005) We can calculate this Fröhlich parameter from: ➔ 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. )
18. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 (Florian Marquardt, Wikipedia, CC) Lattice Dynamics (Phonons) MAPI Low-frequency dispersion (PCCP, AMA Leguy et al., 2016) MAPI is as soft as wood!
19. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Dielectric response (i.r. activity) from Gamma-point modes
20. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Figure: Beau Lambert, Kenneth A. Mauritz. 33 24 4.5 Dielectric response function (not a constant)!
21. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 LO-like modes @ 2 THz TO-like modes @ 1 THz Representation of i.r. activity from: Phys. Rev. B 92, 144308 (2015) Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory Xavier Gonze and Changyol Lee Phys. Rev. B 55, 10355 – Published 15 April 1997
22. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Representation of i.r. activity from: Phys. Rev. B 92, 144308 (2015) Soft as wood ∴ Frequencies small Ionic ∴ Born-Effective-Charges large
23. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Representation of i.r. activity from: Phys. Rev. B 92, 144308 (2015) Remap to a single effective mode. Hellwarth et al. PRB 1999, = 2.25 THz
24. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 MAPI Parameters MAPI is soft - the phonons are very low energy; finite population. Debye temperature ~ 100 K.
25. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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
26. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Asymptotically we are all athermal • Osaka finite-temperature (free energy) actions
27. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Effective mass + 40% (Phonon drag) (You could use this in a BTE calculation.) Time scale for scattering. Polaron wavefunction (Gaussian), and scale.
28. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Mobility theory • "Mobility of Slow Electrons in a Polar Crystal", Feynman, Hellwarth, Iddings, and Platzman. Phys. Rev. 1962 ◦ FHIP mobility (T~=0); and direct integrals to eval. • "Boltzmann Equation for Polarons" Leo P. Kadanoff, Phys. Rev. 130, 1364 (1963). • Assumes a Boltzmann equation process (independent scattering events), and then solves for a scattering time. • Hellwarth et al. 1999 PRB Gives prescription for multiple phonon branches Uses the 1950s Osaka free energies (finite T) Shows how the 1962 paper can be revisited with numerical evaluation • MAPI needs numeric evaluation.
29. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 → 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
30. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018
31. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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.
32. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 BaSnO3: phonon drag magnifies effective mass difference [1] High-mobility BaSnO3 grown by oxide molecular beam epitaxy Santosh Raghavana APL Materials 4, 016106 (2016); doi: http://dx.doi.org/10.1063/1.4939657 [2] High Mobility in a Stable Transparent Perovskite Oxide Published 22 May 2012 • ©2012 The Japan Society of Applied Physics Applied Physics Express, Volume 5, Number 6 https://doi.org/10.1143/APEX.5.061102 [3] Wide bandgap BaSnO3 films with room temperature conductivity exceeding 104 S cm 1 Nature Communications 8, Article number: 15167 (2017) https://doi.org/10.1038/ncomms15167 m_e = 0.19 m_h = 0.6 Phonon drag = +77% Phonon drag = +33%
33. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 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.
34. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Frequency dependent mobility (Feynman et al. 1962)
35. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Calculated Im(χ) ( No published data to compared to. But not Drude-like. Massive additional loss when optical phonons are generated (2 THz)) Quadrature becomes numerically unstable (highly oscillatory function)
36. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Polarons → Formation, Dielectric (Frohlich coupling) has essential physics Exciton → Polaron phonon sub pop, explains high carrier T Slow thermal transport in Perov., explains slow cooling → Transport, Feynman theories provide ab-initio mobilities Experimental agreement is very good More temperature-dependent data would be useful! → and Recombination Polarons (finite-size charge carriers) provide a real-space view Simple theories have lots to say!
37. Jarvist Moore Frost (King's College London, UK) APS March - Los Angeles 2018 5th March 2018 PolaronMobility.jl: • Really easy to apply these methods to new materials! • Standard parameters needed: ◦ Effective mass ◦ Gamma-point phonons + BEC. • Outputs ◦ T-dep Mobilities with three different methods ◦ Phonon-drag effective mass; radius ◦ Polaron response functions (oscillation, optical absorption) • Further work ◦ Try more complicated actions (directly use all the Frohlich matrix elements; each phonon response) PAPER: JM Frost, PRB 96 (19), 195202 (2017). CODES: github.com/jarvist/PolaronMobility.jl
38. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Collaborators:- Piers Barnes; Jenny Nelson + Groups - Imperial College London Mark van Schilfgaarde, Pooya Azarhoosh, Scott McKechnie - King's College London Piers Barnes Jenny Nelson Mark van Schilfgaarde Pooya Azarhoosh WMD Group, ICL/Bath Acknowledgments:- EPSRC - EP/K016288/1 EPSRC Archer - EP/L000202 University of Bath HPC Imperial College London HPC https://wmd-group.github.io
39. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 INPUT: Dielectric constants, effective mass, phonon frequencies (+ IR activity). OUTPUT: • Renormalised effective mass, • T-dependent mobilities ◦ (including a relaxation time), • Polaron-character ◦ (vibrational structure, size, response).
40. Jarvist Moore Frost (King's College London, UK) APS Los Angeles 2018 March 2018 Polaron mobility: • Fully ab-initio temperature-dependent prediction of mobility ◦ (No fitted / empirical / free parameters.) ◦ Good agreement to (single-crystal TRMC) data for MAPI and CsSnI3. ◦ Suggests optical phonon scattering limits mobility • Theory is based on effective-mass approximation, and only considers dielectric electron-phonon coupling • BUT, within those limits ◦ Coupled electron-phonon system is treated fully (correlation integrated out with path integration) ◦ Strong coupling regime + transition to weak ◦ In a polar material, the long range dielectric electron-phonon coupling is expected to be large (and perhaps dominate)
41. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 CH 3 NH 3 PbI 3 (MAPI for short) Configuration: PbII [5d106s26p0]; I-I [5p6] F. Brivio et al, Physical Review B 89, 155204 (2014) Relativistic QSGW theory with Mark van Schilfgaarde (KCL) Conduction Band Valence Band Dresselhaus Splitting (SOC) [Molecule breaks centrosymmetry]
42. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 F. Brivio et al, Physical Review B 89, 155204 (2014) Bands are not parabolic, but… m h */m ~ 0.12 (light holes) m e */m ~ 0.15 (light electrons) [sampled within k B T of band edges] Optical Absorption Hole Effective Mass [110] [112] [111] (Nb: requires extremely large k-space grid for sufficient points for fit near extrema!)
43. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 Free Charges or large Polarons or small Polarons? (Arguments for these follow Landau (1933); from Jones & March (1985), "Theoretical Solid State Physics Vol 2" ) MAPI: α GaAs: 0.068 CdTe: 0.29 AgCl: 1.84 SrTiO3: 3.77 (Devreese 2005) m h */m ~ 0.12 (holes)
44. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 Free Charges or large Polarons or small Polarons? (Arguments for these follow Landau (1933); from Jones & March (1985), "Theoretical Solid State Physics Vol 2" ) (Perturbation theory, small alpha limit, R.Feynman, 1955)
45. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 Free Charges or large Polarons or small Polarons? (Polaron Binding Energy) (Arguments for these follow Landau (1933); from Jones & March (1985), "Theoretical Solid State Physics Vol 2" ) (Jones & March, 1985)
46. Jarvist Moore Frost (Imperial College London, UK) WMD Research Day 2017 Thurs 27th July 2017 Exciton binding from effective mass theory: Carrier mass & dielectric screening favour free carrier generation (t→infinity). But initial exciton is commensurate with polaron size! Where does the energy go? Radius = 36 Å = 5.75 Lattice Spacings (Epsilon infinity) Free charge generation or excitons? (MAPI; Iodine material)
47. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 (β>1 ∴ T<100 K, for halide perovskites)
48. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 Variational codes validated against Hellwarth 1999 https://github.com/jarvist/PolaronMobility-FeynmanKadanoffOsakaHellwarth (Hellwarth 1999 PRB)
49. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 [1.61] from Devreese 2016, based on: Boltzmann Equation for Polarons Leo P. Kadanoff Phys. Rev. 130, 1364 – Published 15 May 1963 Kadanoff 1963 - adds a term for the emission of phonons. BUT - is still meant for low-T. Feynman, Hellwarth et al.1962 "Dissipation at low temperatures" T=300 , β=0.36 : v= 19.82 w= 16.92; M=0.372 k=107 μ(Kadanoff) = 112 cm^2/Vs
50. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 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)
51. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 Ship wakes: Kelvin or Mach angle? M. Rabaud and F. Moisy, Phys. Rev. Lett. 110, 214503 (2013). Editor's suggestion
52. Jarvist Moore Frost (Imperial College London, UK) MRS Boston Fall 2017 November 2017 Ship wakes: Kelvin or Mach angle? M. Rabaud and F. Moisy, Phys. Rev. Lett. 110, 214503 (2013). Editor's suggestion