2016-11-29_JMFROST_MRS-Boston-Organics

## Density of states of conjugated polymers by tight binding
### Jarvist Moore Frost, Beth Rice, Jenny Nelson

Organic electronic materials are highly spatially disordered. Resultant fluctuations in wavefunction overlap leads to band tailing in the electronic density of states. In turn this limits the charge carrier mobility. Methods to understand these relationships will enable the design of higher performance organic semiconductors.

We will discuss the multi-scale simulation methods required to solve the electronic density of states of a conjugated polymer, focusing on amorphous P3HT[1]. Here we undertook atomistic molecular dynamics, calculated transfer integrals from frozen snapshots with the molecular orbital overlap method, and then solved a tight binding model, to have an entirely ab-initio prediction of the Urbach tail of charges. Our chief results were: inter-monomer torsional disorder dominates intra-chain disorder; and that the Urbach tail was composed of extremal configurations, and so required very large calculations to converge on a value.

Our latest work calculates the electronic density of states for conjugated polymer chains, with the off-diagonal disorder simulated by statistical mechanics based on an ab-initio torsional potential[1,2]. The method uses the linear-scaling Sturm sequences to solve the tight binding Hamiltonian and construct the densities of states. This enables an extremely high signal to noise ratio, for minimal computational effort. This enables us to make quantitative predictions on how varying the backbone of a conjugated polymer can directly influence the resulting charge transport characteristics of both holes and electrons, and so indicate routes in chemical synthesis to materials of superior performance.

Finally we will discuss how organic materials can offer unique performance benefits over crystalline, with the use of Wavefunction Engineering to engender unique excited state properties.

[1] Parameter free calculation of the subgap density of states in poly (3-hexylthiophene), JM Frost et al, Faraday discussions 174, 255-266
[2] https://github.com/jarvist/Teclo

EM4:
## Density of states of conjugated polymers by tight binding
### Jarvist Moore Frost, Beth Rice, Jenny Nelson

Organic electronic materials are highly spatially disordered. Resultant fluctuations in wavefunction overlap leads to band tailing in the electronic density of states. In turn this limits the charge carrier mobility. Methods to understand these relationships will enable the design of higher performance organic semiconductors.

We will discuss the multi-scale simulation methods required to solve the electronic density of states of a conjugated polymer, focusing on amorphous P3HT[1]. Here we undertook atomistic molecular dynamics, calculated transfer integrals from frozen snapshots with the molecular orbital overlap method, and then solved a tight binding model, to have an entirely ab-initio prediction of the Urbach tail of charges. Our chief results were: inter-monomer torsional disorder dominates intra-chain disorder; and that the Urbach tail was composed of extremal configurations, and so required very large calculations to converge on a value.

Our latest work calculates the electronic density of states for conjugated polymer chains, with the off-diagonal disorder simulated by statistical mechanics based on an ab-initio torsional potential[1,2]. The method uses the linear-scaling Sturm sequences to solve the tight binding Hamiltonian and construct the densities of states. This enables an extremely high signal to noise ratio, for minimal computational effort. This enables us to make quantitative predictions on how varying the backbone of a conjugated polymer can directly influence the resulting charge transport characteristics of both holes and electrons, and so indicate routes in chemical synthesis to materials of superior performance.

Finally we will discuss how organic materials can offer unique performance benefits over crystalline, with the use of Wavefunction Engineering to engender unique excited state properties.

[1] Parameter free calculation of the subgap density of states in poly (3-hexylthiophene), JM Frost et al, Faraday discussions 174, 255-266
[2] https://github.com/jarvist/Teclo

Transcript

1. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Jarvist Moore Frost (a,b), Beth Rice (b), Jenny Nelson (b) a) University of Bath, UK b) Imperial College London, UK Walsh Materials Design Group, Imperial College London, UK jarvist.frost@imperial.ac.uk Density of states of conjugated polymers by tight binding
2. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 'Exponential' band tailing from disorder / thermal fluctuations Discrete states merging at band edge Distinct states within the gap E Why care about density of states?
3. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 A simple type of N-state model Basis of states on monomers… (orthogonal) Coupled with effective transfer-integrals
4. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 "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" Tight Binding Hamiltonian No correlation!
5. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Psi & E - results of our efforts
6. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Perfectly ordered (+ with trap states) PFO polymer No site disorder... No J disorder...
7. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Disorder doesn't remove trap states
8. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016
9. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Parameter free calculation of the subgap density of states in poly(3-hexylthiophene) Jarvist M. Frost,ab James Kirkpatrick,ac Thomas Kirchartzad and Jenny Nelson*a Show Affiliations Faraday Discuss., 2014,174, 255-266 DOI: 10.1039/C4FD00153B
10. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Still not fast enough… (only ~1000 DoS points / s) Mostly empty Hamiltonian… yet spending a lot of time solving it Sparse matrix routines? Break into sub problems & combine? Direct mathematical analysis - random matrices? Mathematical interlude
11. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 In 1D polymers; a tridiagonal Hamiltonian ( Reading maths paper on the ArXiv isn't always a complete waste of time. )
12. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 ( For any tridiagonal matrix; technique of Sturm sequences is universal but it is slower for full matrix than traditional solvers. What about the intermediate regime w/ offdiagonals? ) A 'mathematical trick'
13. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Sturm Sequences (in Julia) O(n) time complexity, for m bins (m<<n) ( vs O(n*n) time complexity, O(m) time complexity to bin eigenvalues & much much more memory use )
14. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 ~BOOM~ ( ~1'000'000 times faster) N=10'000; Tri-diagonal matrix Sturm sequence method: elapsed time: 9.4411e-5 seconds ( 86'912 bytes allocated) Standard dense eigensolver: elapsed time: 112.3307 seconds (803'281'176 bytes allocated)
15. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Polyfluorene (by MD)
16. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 ∇U = F F = ma Where to get structures? At the core of MD... HEAT BATH Rough potential energy landscape
17. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Statistical mechanics view At thermodynamic equilibrium, difference in population: (We don't need no trajectory (history, kinetics) - just ΔE )
18. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 PFO potential energy landscape
19. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Really need a free energy (including entropy)
20. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 "This fundamental law is the summit of statistical mechanics, and the entire subject is either the slide-down from this summit, as the principle is applied to various cases, or the climb-up to where the fundamental law is derived and the concepts of thermal equilibrium and temperature T clarified." Feynman says... Statistical Mechanics: A Set Of Lectures (Advanced Book Classics) by Richard Feynman
21. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 How to Z? Sum over configurations or Sum over energy (caring for degeneracy) Continuous variable U(theta) → simple integral
22. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 How to Z? Sum over configurations or Sum over energy (caring for degeneracy) Continuous variable U(theta) → simple integral Transcendental function - an absolute pain to analytically integrate!
23. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Nb: Z is Z(T , U). Therefore need to reevaluate if T or U change...
24. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 (PFO)
25. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Populate Density of States Hamiltonian... Distribution of thetas from stat mech... Model for transfer integral...
26. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Molecular Dynamics; 2 Yrs of PhD (PFO)
27. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 (P3HT)
28. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 (PFO) (P3HT)
29. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 (PFO) (P3HT)
30. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Rigidity → improved mobility ? Hugo Bronstein, UCL ( Fit via Chebyshev approximation + simplification -> ApproxFun )
31. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016
32. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016
33. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Density of states of conjugated polymers: 1D • Effective torsional potential → integrate to partition function • Take these generate assemblies & with a model for the electron transfer (hopping) integral → Hamiltonian • Solve this Tridiagonal H via Sturm sequences • generates disordered DoS extremely efficiently • (everything is linear in time + space, O(n)) 'P3HT'-like potential energy surfaces probably most important for exp. DoS tails; i.e. symmetric around Theta=0. https://github.com/jarvist/Teclo
34. Jarvist Moore Frost (Imperial College London, UK) MRS Fall - Boston 2016 Tues 29th Nov 2016 Acknowledgments Beth Rice (Imperial) Jenny Nelson (Imperial) James KP (Imperial, now Deepmind / Google)