Path integral polarons for real semiconductors

TYC 8th Energy Materials workshop: From Electron and Phonon Interactions to Net Zero
1 June 2026 @ 1:30 pm – 3 June 2026 @ 4:30 pm
Venue: The Great Hall, King’s College London, Strand

Path integral polarons for real semiconductors
Jarvist Moore Frost, Imperial College London [contributed]

Most technically relevant behaviours of semiconductors for energy materials are emergent phenomena. Rather than being direct quantum observables, they arise due to the competition between different, similar strength, processes, and are transient and finite-temperature. This poses a considerable challenge for materials modelling.

A particular recent focus of ours has been the charge-carrier mobility in polar semiconductors. The charge-carrier polarises the lattice, and the back-reaction attempts to localise the charge carrier. Trulythis is a finite-temperature quantum field-theory problem (as phonons can be created and destroyed).

Most modelling is limited to semi-classical rate theory-based models (such as Marcus theory;Boltzmann transport equation).Our computational tool of choice is the Feynman path integral variational approximation.

Once the model Hamiltonian parameters are set by electronic structure calculations on the material of interest, we use this finite-temperature quantum-field-theory theory to make direct quantitative predictions of finite temperature charge-carrier response in both inorganic and organic semiconductors.

We extended the Feynman variational method for the solid-state Froehlich Hamiltonian to multiple phonon modes and multiple quasiparticles in the variational solution and developed the numerics to predict finite-temperature frequency-dependent observables (particularly optical absorption and polaron mobility), of more complicated multi-phonon branched semiconductors as are often proposed as future inorganic photovoltaic materials [PRB 107 115203].We developed a Feynman variational method for the Holstein Hamiltonian, allowing us to use the same machinery to simulate ’small’ polarons in organic materials [arXiv: 2207.06846]. Finally, I will describehow this picture of the Feynman dynamic polaron localisation can be used to correct quantumscattering form-factors that otherwise rely on (incorrectly) fully delocalised Bloch waves[unpublished].

[PRB 107 115203] https://doi.org/10.1103/PhysRevB.107.115203
[arXiv: 2207.06846] https://arxiv.org/abs/2207.06846

Transcript

1. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Department of Chemistry / Department of Physics Imperial College London Email: jarvist.frost@ic.ac.uk https://frost-group.github.io/ Path integral polarons for real semiconductors Bradley A. Martin, Jarvist Moore Frost.
2. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Black box Electronic structure a la carte • 😊 Bulk modulus • 😊 Refined lattice size • 😊 Bulk modulus • 😊 Lattice dynamics (phonons) • 😊 Molecular dynamics • 😊 Static contribution to dielectric constant • 😐 Band gap • 😐 Band structure • 😐 Effective mass • 😑 Optical properties • 😑 Optical contrib to dielectric • 😑 Magnetism Restaurants VASP, Gaussian, xtb, ABINIT, Questaal, Turbomole, FHI-AIMS, DFTB+, DFTK, GAMESS(UK)... https://en.wikipedia.org/wiki/List_of_quantum_chemistry_a nd_solid-state_physics_software Cost DFT O(N^3) Hybrid-DFTO(N^4) QSGW O(N^4) ++ CCSD O(N^6) / O(N^4) ++ Full-CI O(N!) Unit cell / Molecular Structure
3. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Drude model (1900): (Simon, The Oxford solid state basics)
4. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Sommerfield model (1927): Why this actually works... (Ziman, Principles of the theory of solids, 2nd edition, 1972)
5. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 'Ab initio' mobility calculation... Incredible amounts of work calculating matrix elements… Across two Brillouin zones (q and k). • But assumes a plane-wave Ansatz ?! • Then you just square the matrix element ?!
6. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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 (TDSE on Landau-Zener Hamiltonian.) Theorists (>1980) have mainly put effort into ever more involved methods of calculating matrix elements. Solid state theory ⇔ experiment
7. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Drude model (1900): (Simon, The Oxford solid state basics)
8. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 What is a Polaron? • an electron polar ises 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!
9. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Dielectric response… Fröhlich Polaron (small-polaron / static picture)
10. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Large polarons (continuum) The large-polaron Fröhlich Hamiltonian → Single effective-mass electron (bare band effective mass) → Interacts with harmonic lattice vibrations (boson), Via a dielectric (long-range) electron-phonon coupling (Fröhlich 1950, Landau 1933)
11. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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
12. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Units: m/F (inverse of vacuum permittivity) • The coupling between the electron and the phonon field Units: F • The capacitance of the phonon field Units: m^-1 • Treat harmonic oscillator to make it dimensionless (scale by KE and phonon E) Fröhlich effective mass polarons
13. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 W
14. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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
15. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 → You want to solve this 'model' path integral, but you can't. • Coulomb interaction with lattice disturbance • Which dies out exponentially → … instead, factor 'trial' 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
16. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Slow Electrons in a Polar Crystal, Phys. Rev. 97, Feynman 1955 → … Feynman proposed this (soluble) 'trial' action (S1). • quadratic (harmonic) interaction (C) • But with tunable dampening (w) • Not just a substitution of the mode Action S → trial Action S1 • Includes <exp(S-S1)> Practically: Tweak variational parameters (v,w) to lower E.
17. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 M k → Simple Harmonic Motion (ball and chain) Trial action: An explicitly quasi-particle theory
18. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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
19. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Effective mass + 40% (Phonon drag) (You could use this in a BTE calculation.) Time scale for scattering, energy loss (thermalisation). Polaron wavefunction (Gaussian), and scale. Device physics?
20. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Asymptotically we are all dead • Textbook method: Feynman 1955 - asymptotic solution ◦ athermal (T~=0) ◦ small-α / large-α limiting behaviour → ZERO TEMPERATURE; SINGLE OPTICAL MODE Halide Perovskites
21. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Asymptotically we are all athermal • Osaka finite-temperature (free energy) actions Any soft semiconductor ⇒ 'high temperature' regime
22. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 1/ 'Large' polarons Fröhlich model as a 'fruit fly' 2/ Hybrid lead halide perovskites (MAPI + friends) a/ Temperature dependent mobility b/ Frequency dependent mobility 3/ 'Small' polarons Holstein-Peierls coupling, Rubrene 4/ Polaron device physics a/ How do polarons scatter? Application of the FVA for polarons
23. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Mishchenko2019 • How well can the FHIP theory approach these results?
24. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 FHIP mobility theory • Extends <exp(S1-S)> method for an 'influence functional' • Fully quantum-mechanical theory of mobility ◦ Includes all phonon scattering processes up to all orders • But not variational! Only an approximation.
25. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Numeric integration • Doubly oscillatory integral (in the complex plane) Exp. decay as exp(-Beta) Solution: Don't change the contour! → Finite temperature frequency-dependent numerics are easier if you directly integrate Eqn. 35 in FHIP1962. (Bradley Martin)
26. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Mott-Ioffe-Regel (MIR) critereon Independent scattering approximation For any soft*, moderate mobility, material, this criterion is violated. ⇒ we shouldn't use any perturbation theory based mobility theories. (Scattering is too strong; Drudge model incorrect.) *soft = material where the relevant energy (el-ph coupling OR phonon vibrational mode energy) is comparable to kB T i.e. most semiconductors which are not silicon or diamond!
27. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 FHIP1962 vs. Mishchenko2019 FHIP theory (numerical) α = 2.5 MIR Limit
28. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 FHIP mobility theory non-monotonic only for alpha>~=8 α = 4.0 MIR Limit
29. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Non monotonic mobility α = 6.0 MIR Limit (We only see this pronounced 'ski jump' structure with a larger alpha ~8.)
30. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Frequency dependent mobility ω (1+v)ω (1+2v)ω (1+3v)ω Vibronic / Franck-Condon series - washed out by temperature
31. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Frequency dependent mobility α = 6 However, not as washed out as the DiagMC results! Assumption of perfect harmonic bath, integrated out degree of freedom?
32. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 1/ 'Large' polarons Fröhlich model as a 'fruit fly' 2/ Hybrid lead halide perovskites (MAPI + friends) a/ Temperature dependent mobility b/ Frequency dependent mobility 3/ 'Small' polarons Holstein-Peierls coupling, Rubrene 4/ Polaron device physics a/ How do polarons scatter? Application of the FVA for polarons
33. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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. "Soft as Jelly!"
34. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 → 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 (JM Frost - PRB, 2017)
35. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 1/ 'Large' polarons Fröhlich model as a 'fruit fly' 2/ Hybrid lead halide perovskites (MAPI + friends) a/ Temperature dependent mobility b/ Frequency dependent mobility 3/ 'Small' polarons Holstein-Peierls coupling, Rubrene 4/ Polaron device physics a/ How do polarons scatter? Application of the FVA for polarons
36. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Polaron trial action, 'n' parameters See also: • Sels, D., 2016. Dynamic polaron response from variational imaginary time evolution. arXiv:1605.04998 [cond-mat]. • Poulter, J., Sa-yakanit, V., 1992. A complete expression for the propagator corresponding to a model quadratic action. J. Phys. A: Math. Gen. 25, 1539. https://doi.org/10.1088/0305-4470/25/6/015 (Diagram - Dris Sels, 2016)
37. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Small (~20% effect) on DC mobility MAPI (Per-phonon-branch el-ph coupling)
38. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Single effective mode; single scattering pathway Full multicomponent solution: quantum resonances in energy loss rate Much more structure in freq response!
39. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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) Semi-classical rates from Ridley's book
40. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Almost perfect agreement with (Bakulin lab) THz pump-probe measurements… Theory - no empirical or fitted parameters. Experiment Zheng , X., Hopper, T.R., Gorodetsky, A., Maimaris, M., Xu, W., Martin, B.A.A., Frost, J.M., Bakulin, A.A., 2021. Multipulse Terahertz Spectroscopy Unveils Hot Polaron Photoconductivity Dynamics in Metal-Halide Perovskites. J. Phys. Chem. Lett. 12, 8732–8739. https://doi.org/10.1021/acs.jpclett.1c02102
41. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 1/ 'Large' polarons Fröhlich model as a 'fruit fly' 2/ Hybrid lead halide perovskites (MAPI + friends) a/ Temperature dependent mobility b/ Frequency dependent mobility 3/ 'Small' polarons Holstein-Peierls coupling, Rubrene 4/ Polaron device physics a/ How do polarons scatter? Application of the FVA for polarons
42. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 (5,6,11,12-Tetraphenyltetracene) Ordejón, P., et al. 2017. Phys. Rev. B 96, 035202. X'tal organic crystals by FHA
43. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Rubrene: Holstein-Peierls Similar to alpha Relate to effective mass Ordejón, P., et al. 2017. Phys. Rev. B 96, 035202.
44. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Holstein/Peierls → Fröhlich form g ~= α = 2.20 - middling polaronic, quite similar in fact to hybrid halide perovskite Feynman quasi-particle solution (with Frohlich Hamiltonian) v = 3.27796 w = 2.69318
45. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Tight-binding Hamiltonian + local el-ph: Electronic: • Transfer integral • Nearest neighbours • Lattice site n, n’ El-ph coupling: • Kronecker deltas • Electron on site n • Ion of site m 3 Variational lattice polarons Ions: • Einstein oscillators • Mass M • Lattice site m
46. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Holstein Fröhlich El-ph coupling El-ph coupling strength (“alpha”) Effective self-interacti on
47. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 [Ultraviolet catastrophe → momentum cut-off due to the discrete lattice] [Kronecker deltas!] Holstein self-interaction functional
48. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 The variational method for Holstein polarons Variational inequality for the free energy: Momentum cut-off function: Ultraviolet momentum cut-off due to the discrete lattice
49. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Parabolic band Holstein model Some pathologies (tight-binding nature of Holstein Hamiltonian super important at strong couplings!); but still exhibits a lot of 'small polaron' physics.
50. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Perfect agreement with (effective mass dispersion) DiagMC Thanks to Stefano Ragni for the DiagMC data (+ parabolic).
51. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Quantum random walk Recast polaron as a (reconnecting) random walk on a lattice. Holstein-interaction becomes an energetic benefit ('non-local in time self-interaction') if you retrace your steps. (Makes sense in terms of thinking about what the reorganisation energy is…'walking on sand') (Feynman) Variational Approximation ⇒ minimise Kullback-Leibler divergence between Poisson processes. Currently ⇒ map to one parameter trial pure diffusion (missing phonon drag term)
52. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 OLD DATA - PARABOLIC MODEL Rubrene: Temp. dep. mobility ~ 12 cm2V-1s-1 @ 300K ~ 409 cm2V-1s-1 @ 300K Very good agreement with expt.
53. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 54.4 cm2V-1s-1 @ 300K 409 cm2V-1s-1 @ 300K
54. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 !?
55. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026
56. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026
57. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026
58. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 1/ 'Large' polarons Fröhlich model as a 'fruit fly' 2/ Hybrid lead halide perovskites (MAPI + friends) a/ Temperature dependent mobility b/ Frequency dependent mobility 3/ 'Small' polarons Holstein-Peierls coupling, Rubrene 4/ Polaron device physics a/ How do polarons scatter? Application of the FVA for polarons
59. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 (Quantum processes in Semiconductors, Ridley, 2013)
60. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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.)
61. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Polarons are not a plane wave! M k ~15 nm (Athermal solution.)
62. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 "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.
63. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 (Coulomb) Scattering of Gaussian wavepackets (polarons) Polaron scattering / capture cross section attenuated by: • Classical contribution from localising the electron • Quantum contribution from incoherency of Gaussian wavepacket ◦ Small total scattering / capture cross section ◦ Scattering moved to lower angles (important for transport!) Scattering cross section
64. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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.
65. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Collaborators:- Piers Barnes Mark van Schilfgaarde Pooya Azarhoosh Aron Walsh Federico Brivio Artem Bakulin WMD Group, Bath/ICL Acknowledgment:- Royal Society - URF/R1/191292 Michael Toney Jonathan Skelton Tom Hopper Jenny Nelson Lucy Whalley Beth Rice
66. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Diagrammatic Monte Carlo (DiagMC) Daryl Lee, Xiaoyi Yang https://github.com/Frost-group/Tethys.jl Conclusions & future work • Can we make the Feynman VA more ab-initio? ◦ Multiple phonons (in the model action) ✔ ◦ Multiple parameters (in the variational solution) ✔ ◦ Temperature and frequency dependent mobility ✔ ◦ Extend to small-polaron regime ✔ • Future work ◦ Improve on the variational approximation? ◦ Extend 1950s device physics from Bloch waves to polarons? ◦ Directly use DFT based k-space el-ph couplings. Path Integral Monte Carlo (PIMC) Logan Filipovich, YC Wong, George Su https://github.com/Frost-group/PolaronQMC.jl
67. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 BELOW HERE - EXTRA SLIDES
68. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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? Whalley, Skelton, Frost, Walsh. PRB 94, 220301(R), 2016.
69. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Born-Oppenheimer approx. Adiabatic: treat Nuclear and Electronic degrees of freedom separately. • Solve Sch. Eqn. for nuclear degree of freedom. • Solve electronic Sch. Eqn. varying nuclear degree of freedom (i.e. deformation potential). • Combine piecewise (adiabatically) manner. Build adiabatic el-ph expectation value, by sandwiching Q-dependent operator between nuclear wavefunctions.
70. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 The 1D Schrodinger equation is easy to solve!
71. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Textbook double-well behaviour!
72. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 (R acoustic mode at Brillouin-Zone boundary [tilt]) (From soft-phonon mode following, MAPI.)
73. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 ~15 meV double-well persists in structure to > 600 K BE Distribution 600 K 1 K
74. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 (Calculations: Lucy Whalley) Band-gap as a function of Q (Deformation Potential)
75. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 R M
76. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 Papers • Multiple Phonons
77. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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. (JM Frost - PRB, 2017)
78. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 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
79. Jarvist Moore Frost (ICL, UK) TYC el-ph conference, KLC 3 June 2026 What limits solar cell efficiency? Sunlight is ~ Blackbody 50% of energy is IR With single bandgap: most energy lost to either transmission or thermalisation Energy Environ. Sci., 2015, 8, 103–125