Jarvist Moore Frost, Beth Rice, Jenny Nelson Walsh Materials Design Group, University of Bath, UK [email protected] Tight Binding Polarons MICE-UK, Bath Innovation Centre 2014-07-16 

Motivation Motivation: Why do organic solar cells work*? (* at all) 

Overview ● Tight Binding ● PCBM-MD ● Tight Binding with PCBM-MD ● Polarons and Tight Binding 

A simple type of N-state model Basis of states on monomers… (orthogonal) Coupled with effective transfer-integrals 

1D polymer Tight Binding Hamiltonian "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" 

Psi & E - results of our efforts 

Density of States by Tight Binding Solve Density Matrix Hamiltonian → Eigenvalues (electronic wavefunction expectation energies) … Density of States is the key transport parameter for amorphous / defective devices (effective mass and scattering distance matters little when charges are being energetically trapped) 

Infinite polymer (10 units) 

No content

Actually quite boring for large polymers... 

100 'polymer', PBCs, slight mid-chain defect 

2D checkerboard… (without PBCs) 

2D checkerboard… (with PBCs) 

No content

3D Checkerboard 

3D 10x10x10 simple cubic cell 

"The systematic algebraic technique for the complete exploitation of such symmetry properties is called group theory, and is an essential tool for the theoretical physicist in this field. It can guide us to the form of the solution before we even consider the sordid details of atomic potentials, and enables us to squeeze the last drop out of an actual calculation." - J.M.Ziman, 1962 "Electrons in metals: a short guide to the Fermi surface" Characteristic lattices have particular Van Hove singularities associated with their density of states. These are the DoS at the critical points in the Brillion zone. 

Overview ● Tight Binding ● PCBM-MD ● Tight Binding with PCBM-MD ● Polarons and Tight Binding 

Atoms → Smarties, gives you a ~1000x (or more) speedup 

Where to get parameters? Girifalco 'smeared' C60 for the balls... And a scaled version for PBM? 

Atomistic view... We have an OPLS PCBM model... so let's use it! 2564 PCBMs MD for ~20ps 

Atomistic RDF via COM (Center of Mass) of PBM and C60 fragments 

No content

1000 Tris PCBM, CG alter elem c, vdw=5.0 alter elem p, vdw=3.0 show spheres 

No content

Consider the inter-adduct angles 

8 Bis isomers; Angles: 36 72 60 90 180 144 120 108 

No content

45 (unique) Tris Isomers 36.0 36.0 36.0 36.0 36.0 60.0 36.0 36.0 72.0 36.0 60.0 60.0 36.0 60.0 72.0 36.0 60.0 90.0 36.0 72.0 90.0 36.0 72.0 108.0 36.0 90.0 108.0 36.0 90.0 120.0 36.0 108.0 120.0 36.0 108.0 144.0 36.0 120.0 120.0 36.0 120.0 144.0 36.0 144.0 144.0 36.0 144.0 180.0 #All highly sterically hindered 60.0 60.0 108.0 60.0 60.0 120.0 60.0 72.0 72.0 60.0 72.0 90.0 60.0 72.0 120.0 60.0 90.0 108.0 60.0 90.0 144.0 60.0 108.0 108.0 60.0 108.0 144.0 60.0 120.0 144.0 60.0 120.0 180.0 60.0 144.0 144.0 72.0 72.0 108.0 72.0 72.0 144.0 72.0 90.0 120.0 72.0 90.0 144.0 72.0 108.0 120.0 72.0 108.0 180.0 72.0 120.0 144.0 72.0 144.0 144.0 90.0 90.0 90.0 # EEE 90.0 90.0 180.0 #E/T isomers 90.0 108.0 120.0 90.0 108.0 144.0 90.0 120.0 144.0 108.0 108.0 108.0 108.0 108.0 144.0 #Trans isomers 108.0 120.0 120.0 120.0 120.0 120.0 

45 Unique Tris Isomers... (of 24360) 

C60 (Bucky Balls) Mono PCBM Bis PCBM Tris PCBM 

Transfer Integrals (old AutoJ data) MONO 6.5 meV Bis: 10.5 meV Tris: 4.4 meV Sampling!!! Average of 1000 structures; generated with Packmol 

Transfer Integrals… Amicable Separation 

Degeneracy… :( C60 is 3-fold LUMO degenerate Adducts? DFT KS unoccupied orbitals MONO 30A LUMOs 0 -0.060 -0.284 -1.184 -1.248 (eV) BIS 30A LUMOs 0 -0.235 -0.284 -1.196 -1.443 (eV) TRIS 30A LUMOs 0 -0.005 -0.023 -1.195 -1.215 (eV) Arbitrary (?) 50 meV cutoff for degeneracy gives M non degenerate, B non degenerate, T 3-degenerate. Elephant in the room: does degeneracy influence transport in Tris? How about in M/B/T mixes? 

Blue = exponential fit Prefactor = 1e6 (?) Characteristic length: 0.597 A (MONO) 0.577 A (BIS - E1) 0.580 A (TRIS - EEE) 

Summary... ● We have a CG FF for Mono, Bis, Tris… ● Simulated annealing on 1000 mole samples… ● Transfer integrals - exponential (isotropic) seems to be a good fit to full blown projective method QC 

Overview ● Tight Binding ● PCBM-MD ● Tight Binding with PCBM-MD ● Polarons and Tight Binding 

No content

(Histogram of 1000 eigenvalues) High lying states Low lying states 

No content

Overview ● Tight Binding ● PCBM-MD ● Tight Binding with PCBM-MD ● Polarons and Tight Binding 

What is a polaron? ● bare electron interacts with surrounding medium (Fermi sea) ● becomes dressed in excitation cloud ● interaction tends to self trap particle... (Diagram: A Guide to Feynman Diagrams in the Many-body Problem, R.D. Mattuck) 

Frohlich Polaron ● Consider linear dielectric response - outside- polaron Dielectric response… 

Dielectric response… Site Energy (Polarisation) Self consistent response... 

The SCF loop... Alpha is a 'response parameter'; almost identical to the Frohlich Electron-Phonon coupling (may even be formally identical!) Calculated as ~0.5 eV / e by assuming linear response of dielectric to electron fully localised on 1 nm fullerene molecule 

Simple 1D chain case... S 

Simple cubic lattice Tiny defect to break symm 

No content

No content

Transfer between polarons Polaron transfer integral calculation 

Unperturbed; C60 Simulated Annealing 

Localised Polaron State Perturbed States Polaron; C60 Simulated Annealing 

? 

Acknowledgments WMD Group (Bath) James KP (now Deepmind / Google) - for all the tutorage re: Wave Functions Beth Rice (Imperial) Jenny Nelson (Imperial) 

(405 DFT calculations…) 

No content

Bis PCBM CG-MD with Isomer mix (8 isomers x 125=1000 molecules total)