From Atoms to Solar Cells

Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015

Jarvist Moore Frost

Walsh Materials Design Group,University of Bath, UK

[email protected]

From atoms to solar cellsMultiscale physics of Photovoltaics


Jarvist Moore Frost

Walsh Materials Design Group,University of Bath, UK

[email protected]

From solar cells to atomsMultiscale physics of Photovoltaics


Overview● Introduction & why Photovoltaics (PV)● What is a solar cell?● What are the physics?● Thermodynamics, PV, Shockley-Queisser limit, Light trapping● Semiconductor Device Physics● Drift Diffusion, Organic Charge Transport● Theory & Programming

Case studies:-● PCBM-MD● Tight Binding with PCBM-MD● Polarons and Tight Binding● Beta-Phase PFO● Sturm Sequences…● Geometries by statistical mechanics


"I'm afraid I can't put it more clearly," Alice replied very politely, "for I can't understand it myself to begin with."

Lewis Carroll, Alice's Adventure in Wonderland


I really dislike seeing this ~hundreds of times at conferences!- the one utility is to look at the lower right point, and see how recently the

speaker refreshed their slides (NREL republishes this every 3 months)


Not a CO2 Crisis - we have an Energy Crisis

Each person in Britain:-

10'000kg CO2 → ~10MWh pa

10MWh → 36GJ pa

1142W → ~10 Cyclists

The world (2008)

= 18 TW

= 18'000'000'000'000 W

Art Installation, V&A museum, Christmas 2009


What we need is Fusion! [ E=mc2 ]

Opération Canopus, Fangatuafa atoll, 1968. 2.6 MT





Vast Power of the Sun Is Tapped By Battery Using Sand Ingredient; NEW BATTERY TAPS SUN'S VAST POWER

Special to The New York Times.

April 26, 1954, Monday

MURRAY HILL, N. J., April 25 -- A solar battery, the first of its kind, which converts useful amounts of the sun's radiation directly and efficiently into electricity, has been constructed here by the Bell Telephone Laboratories.

[~6% efficiency - PN Silicon Diode]Vanguard 1 (1958 – NASA)

[Still in orbit!]~5cm square silicon cells

[transmitted 7 years]We have the technology!

→ Cost is the key issue - the solar resource is low density (1kW / m^2)


What is a solar cell?...a slab of P and N doped silicon......a bulk heterojunction of fullerene and conjugated polymer…...a convert of individual quanta of light (photons) to movement of quanta of charge (electrons......a thermodynamic engine driven between the heat bath of the sun and the ambient...


Silicon grown by Czochralski process 1956George E. Meyers, Raytheon Corp. semiconductor plant in Newton, Massachusetts, USA

American Radio History


a) Methylammonium lead iodide (CH3NH3I) and PbI2 (precursor solution) dissolves in γ-butyrolactone solution.

b) Spin-coating process ofCH3NH3PbI3perovskite solution on indium tin oxide (ITO)/glass substrate.

8 August 2013, SPIE Newsroom. DOI: 10.1117/2.1201307.005033


(old) mono-crystallinehigh efficiency (~15%) but expensive...


modern mono-crystalline (better anti-reflective → black)highest efficiency (~20%), really quite cheap…

Best lifetimes


SunPower E2020.4% efficient module (!!!!!!!!!)Full back-contacted cells

Commercially available >2013


polycrystalline silicongood efficiency (~14-16%)

(used to be sig. cheaper than mono-crystalline)


CIGS / a-Si - Black MirrorsEfficiencies 7-13%. Were relatively cheap...


CdTeYour flexible,

slightly purple-yfriend

(poisonous, but apparently easiest to make!)

~7-15% efficiencies





What are the phyiscs?● Almost every area of physics outside the nucleus...

● Solid State → crystalline solids (Si, CZTS)● Condensed Matter Theory → amorphous, polymer,

liquid (a-Si, Dye SCs, Perov?)● Quantum Electrodynamics (QED) & Quantum Field

Theory (QFT)→ Matter / Light interaction● Statistical physics (structures, defects)● Thermodynamics (device operation, light

concentration & formation of active layer)● Electromagnetism (classical field theory of light)● Group theory (useful for a lot of the above)


Ziman - extremely clear and intuitive descriptions; a delight to read.

1960 - Electrons and Phonons - Great reference for transport1969 - Elements of Adv. Quantum Theory - Very gentle Intro...1972 - Theory of Solids (2nd Ed.) - Perhaps the best single Solid State Text1980 - Models of Disorder - The only real 'disorder' textbook, slightly dated


My PhD Supervisor's book

Unapologetically physics-based view of solar cells

Clear text, fairly well structured, some interesting 'advanced' topics covered (thin films, light trapping strategies)

Diagrams a little bit coarse





A solar cell...... is a light absorbing material connected to an external circuit in an asymmetric manner. Photogenerated charge carriers are driven towards one or other of the contacts by the built-in spatial asymmetry. - J.Nelson, Physics of Solar cells

A heat engine that silently runs on sunlight and thermodynamics to produce electrical power.


Thermodynamic LimitSun: 5760 KCell: 2470 KBath: 300 K

ηPCE = 85%


1.6 eV

1.0 eV

θ


Shockley–Queisser limit (@ 1 sun)

Maximum solar conversion efficiency around 33.7%, assuming a single junction with a band gap of 1.34 eV



Silicon, 2012; 25 Yr life

CIGSCdTeBoth ~killed by Silicon


Nature Materials 13, 103–104 (2014) doi:10.1038/nmat3837

On the thermodynamics of light trapping in solar cellsUwe Rau & Thomas Kirchartz

Etendue ratio of solar cell system - solid angle of incident radiation vs. outgoing radiation( sort of an light-beam measure of entropy; in a system it can only increase or stay constant )

http://www.nature.com/nmat/journal/v13/n2/full/nmat3837.html#auth-1




Light Trapping (Ray Limit)

Nothing + Back reflector Inverted Pyramids

Randomised Surface

d= 4 n^2 =~ 50 (Si)

Lambertian emittance


PERL Silicon Solar Cell ~23.5% PCE





Organic Solar Cells!(This one is Rubrene, also the highest mobility organic…)





1.6 eV

1.0 eV

θ


Ideal Diode equation

I0


PN Junction - From: http://en.wikipedia.org/wiki/P%E2%80%93n_junction





Nabla electrical potential =charge density / dielectric constant

Actually quite nasty to solve! (self consistency & BCs)

Poisson's Equation


Full Drift DiffusionElectron Affinity

(variations)

Effective Band Densities of State


Drift Diffusion (Assumptions)● e/h form quasi thermal equilibrium (T, Ef)● e/h temperature same as lattice (no hot /

accelerated carriers)● relaxation time approximation (scattering within

band dominates over defects)● e/h are well defined by quantum number k● Boltzmann approximation● Compositional invariance (band edge gradient replaced

with general F)


Drift Diffusion Codes

SCAPS: Solarcell CAPacity SimulatorUsed by the CIGS community.Intuitive + has a new manual.Has built in defects :)

but rather poorly documented :(Good for CV analysis of real cells.



References / Scheme of WorkThomas Kirchartz, Kaori Seino, Jan-Martin Wagner, Uwe Rau, and Friedhelm Bechstedt. “Efficiency Limits of Si/SiO2 Quantum Well Solar Cells from First-Principles Calculations.” Journal of Applied Physics 105, no. 10: 104511. Accessed November 18, 2013. doi:10.1063/1.3132093.

→ Combines first principles calculation with drift diffusionto model performance of novel PV structures

+ Read Jenny's Chapter 3 in The Physics of Solar Cells (Imperial College Press 2003)


Thomas Kirchartz, Kaori Seino, Jan-Martin Wagner, Uwe Rau, and Friedhelm Bechstedt. “Efficiency Limits of Si/SiO2 Quantum Well Solar Cells from First-Principles Calculations.” Journal of Applied Physics 105, no. 10: 104511. Accessed November 18, 2013. doi:10.1063/1.3132093.


Optimisation of Thickness - a Goldilocks situationThick enough / dark enough to absorb...Mobility sufficient to extract charges through this thickness... avoiding Langevin recombination

Sub-gap states will eat your lunchAbove-gap states will never hurt youDispersion / traps are fairly bad



Organic materials are soft: 'effective mass' so high that KE negligible→ Small Polaron / hopping regime

Transport occurs at 'resonance'Marcus theory offers a useful description

Key transport parameter is the 'Electron Transfer Integral' or J (sometimes 't')

→ calculate by projective method (DFT) ~ hrs / calculationOR Molecular Orbital Overlap (MOO) ~ ms / calculation


Modelling Charge Transport (in Organics)

Following 5 slides of figures & text reproduced from Joe Kwiatkowski's MPhil thesis (ICL, 2008)


Charge transport - the microscopic view


Effective potential energy surface


Marcus theory


Transfer integral


All this scary matrix maths (orbital projection)


Practically… 3 DFT calculations & a matrix inversion

MoleculeA

MoleculeB

MoleculeA + B


Monte Carlo model of charge transport…





Mathematician

Theorist

Numerics (Computational)

Experimentalist

The Lay Person


Mathematician

Theorist

Numerics (Computational)

ExperimentalistPURI

TYU

TILITY

After taking a course in mathematical physics, I wanted to know the real difference between Mathematics and Physicists. A professor friend told me "A Physicist is someone who averages the first 3 terms of a divergent series". - Benjamin Jones

An engineer thinks that equations are an approximation to reality.

A physicist thinks reality is an approximation to equations.

A mathematician doesn't care.

--- (from CANONICAL LIST OF MATH JOKES )

http://www.cnonline.net/~TheCookieJar/math_jokes_01.html




The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. - Paul Dirac, 1929

We don't need any new physics (as in, extra forces, processes, etc.) - we need computationally tractable models, that are sufficiently accurate for what we are experimentally probing.


"Computers are bicycles for the mind." - Steve Jobs


UK Children of the 80s had a head start...

Programming languages compared

(Potential) Computational Speed

Algo

rith

mic

Exp

ress

iven

ess

C

Python

Julia

Fortran

Assembly

R

Octavego

parallel

Haskell

Static or Dynamically typedCompiled or Interpreted (JIT'ed)Procedural or FunctionalHomoiconic (or not)Mathematically powerful, or not

Fortran/C: Static, Compiled, Procedural, laboriousPython: Dynamic, Interpreted, Procedural (+ fn), Matrices 'bolted on'Julia: Dynamic and Static, JIT'ed, Proc./fn., homoiconic, extremely strong maths support

Programming languages discussed

From Christoph Ortner's excellent numerical analysis Julia notebooks.. http://homepages.warwick.ac.uk/staff/C.Ortner/index.php?page=julia

http://homepages.warwick.ac.uk/staff/C.Ortner/index.php?page=julia




My recommendations...Everyone (probably especially experimentalists) should use a iPython or iJulia web notebook; plot data, manipulate, back of the envelope calculations etc.Julia is a lot of fun → responsive, quick, less frustrating (than other languages)

From a professional skills development point of view, programming skills are probably the most valuable thing you can acquire during a PhD…

"“Well, Mr. Frankel, who started this program, began to suffer from the computer disease that anybody who works with computers now knows about. It's a very serious disease and it interferes completely with the work. The trouble with computers is you *play* with them. They are so wonderful. You have these switches - if it's an even number you do this, if it's an odd number you do that - and pretty soon you can do more and more elaborate things if you are clever enough, on one machine. " - Richard Feynman





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)


Motivation

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



Actually quite boring for large polymers...


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


2D checkerboard… (without PBCs)


2D checkerboard… (with PBCs)



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 PCBMsMD for ~20ps


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



1000 Tris PCBM, CG

alter elem c, vdw=5.0alter elem p, vdw=3.0show spheres



Consider the inter-adduct angles


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



45 (unique) Tris Isomers36.0 36.0 36.036.0 36.0 60.036.0 36.0 72.036.0 60.0 60.036.0 60.0 72.036.0 60.0 90.036.0 72.0 90.036.0 72.0 108.036.0 90.0 108.036.0 90.0 120.036.0 108.0 120.036.0 108.0 144.036.0 120.0 120.036.0 120.0 144.036.0 144.0 144.036.0 144.0 180.0

#All highly sterically hindered

60.0 60.0 108.060.0 60.0 120.060.0 72.0 72.060.0 72.0 90.060.0 72.0 120.060.0 90.0 108.060.0 90.0 144.060.0 108.0 108.060.0 108.0 144.060.0 120.0 144.060.0 120.0 180.060.0 144.0 144.0

72.0 72.0 108.072.0 72.0 144.072.0 90.0 120.072.0 90.0 144.0

72.0 108.0 120.072.0 108.0 180.072.0 120.0 144.072.0 144.0 144.0

90.0 90.0 90.0 # EEE

90.0 90.0 180.0 #E/T isomers90.0 108.0 120.0 90.0 108.0 144.090.0 120.0 144.0

108.0 108.0 108.0108.0 108.0 144.0 #Trans isomers108.0 120.0 120.0120.0 120.0 120.0


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 fitPrefactor = 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





(Histogram of 1000 eigenvalues)

High lying states

Low lying states





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 E

nerg

y

(Pol

arisa

tion)

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 latticeTiny defect to break symm




Transfer between polarons

Polaron transfer integral calculation


Unperturbed; C60 Simulated Annealing


Localised Polaron State

Perturbed States

Polaron; C60 Simulated Annealing


?





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"


What is β-Phase PFO?Poyfluorene-di-octyl (PF8) [but not the other polyfluorenes] sometimes exhibits noticeable green fluorescence.

"Formation of the β-phase effectively corresponds to crystallization in onedimension, a remarkably uncommon phenomenon in nature." Nano Lett., 2007, 7 (10), pp 2993–2998


5K PL-spectra, Octamers


Normal 'Wiggly' Polyfluorene...


HOMO Spin Density - centre crimped flat

Hypothesis:-

Is Beta Phase a hole trap?

Are 'bubbles' of beta phase responsible for reduced mobility in processing?


Auto-generated varying 'Beta phase' with ModRedundant

Me Gusta

[11:43:04]jmf02@login-0:> cat magicnumbers.txt65 64 93 94 =180.0 F86 85 114 115 =180.0 F44 43 72 73 =180.0 F107 106 135 136 =180.0 F23 22 51 52 =180.0 F128 127 156 157 =180.0 F2 1 30 31 =180.0 F

[11:43:07]jmf02@login-0:> cat daughters.shfor i in ` seq 7 `do cat mother.com > "${i}.com" head -n "${i}" magicnumbers.txt >> "${i}.com" echo >> "${i}.com"done

[11:43:12]jmf02@login-0:> cat mother.com%chk=tmp.chk%mem=8GB%nprocshared=8#p opt=ModRedundant am1

01010101.pdb

0 1...

Jarv, evidently pleased with himself circa. March 2012


01234567


Energy of Increasing Beta Phase

eV

Enthalpically very uphill - must be driven by a strongly entropic / enthalpic process external to the backbone. Or perhaps chemistry?


TD-DFT Singlets (Absorption)

0.log: Excited State 1: Singlet-A 3.1101 eV 398.65 nm f=6.0596 1.log: Excited State 1: Singlet-A 3.0180 eV 410.81 nm f=5.6641 2.log: Excited State 1: Singlet-A 2.9314 eV 422.95 nm f=5.4683 3.log: Excited State 1: Singlet-A 2.8662 eV 432.57 nm f=5.5560 4.log: Excited State 1: Singlet-A 2.8262 eV 438.69 nm f=5.7538 5.log: Excited State 1: Singlet-A 2.7974 eV 443.22 nm f=6.0612 6.log: Excited State 1: Singlet-A 2.7840 eV 445.35 nm f=6.2768 7.log: Excited State 1: Singlet-A 2.7727 eV 447.15 nm f=6.5119

Octamer model of flattened segments correctly predicts green band formation...


TD-DFT Singlets (Absorption)

0

7


HOMO / LUMO Energy

eV


(Holes float... )

0 HOMO: -5.066416 eV +0 meV1 HOMO: -5.028321 eV +38 meV2 HOMO: -4.991586 eV +75 meV3 HOMO: -4.964103 eV +102 meV4 HOMO: -4.946688 eV +120 meV5 HOMO: -4.934171 eV +132 meV6 HOMO: -4.928456 eV +138 meV7 HOMO: -4.923558 eV +143 meV


Inner Sphere Reorganisation Energy

eV

--> Losing a degree of freedom... -->


Ran into the limits of what we can do with QC...

PFO minima at theta = 45 deg

P3HT (+ most other polymers)minima attheta= 0 deg


Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 201550meV Gaussian Site Energy Disorder


Beta Phase Energetic Trap(5 sites)

DFT Trap Depth

Reproduced!


Beta Phase Energetic Trap(10 sites)

For Wide Enough trap ~10 sites (no confinement),Trap depth = site Energy change




Theta ~ 45 degrees for segmentTheta = 0 degrees for beta phase bit--> 0.18eV change in J--> 310 meV trap depth


Disordered (torsionally)TWISTED (~PFO) polymer

596meV

PFO MODEL


Disordered (torsionally)But FLAT (minima) polymer

70meV

P3HT MODEL


No site disorder...

No J disorder...


PFO trap states from J variation are persistent


Summary of TB-BetaPhaseModel of flattened Beta phase segments produces horrific hole traps in TB model

We don't see these being so deep in QC - our system is too small, our model for the distribution of thetas too primitive (athermal).

We're using highly devoid from reality distributions for Theta and Sigma. (Because we don't know any better.)

PFO-type polymers (or other non flat minima systems) have a fundamental propensity for generating polaron traps at finite temperature / disorder.





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


Random Matrices...

Random Matrix Theory...


Tridiagonal Matrices?

( Reading maths paper on the ArXiv isn't always a complete waste of time. )


A 'Mathematical Trick'...

( 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? )


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 )


~BOOM~ ( ~1'000'000 times faster)

N=10'000;elapsed time: 9.4411e-5 seconds ( 86'912 bytes allocated)elapsed time: 112.3307 seconds (803'281'176 bytes allocated)





Snakes Under Pressure


Generating GeometriesAb-initio MDEmpirical MD

Coarse grain MDStat Phy

Make Stuff up'optimised geom'

Phys

ical

Ac

cura

cy?

Tim

e (e

xp)


MD is quite a pain...


∇U = FF = ma

At the core of MD...


Statistical mechanics viewAt thermodynamic equilibrium,difference in population:

(We don't need no trajectory (history, kinetics) - just ΔE )


Polyfluorene - a 9 yr love / hate relationship


Early PhD work… design a PFO MD forcefield


Polymers are freaky...


Polyfluorene (by MD)


Given a U, how to populate DoS?


"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...


How to Z?Sum over configurations

orSum over energy (caring for degeneracy)Continuous variable U(theta) → simple integral


How to Z?Sum over configurations

orSum over energy (caring for degeneracy)Continuous variable U(theta) → simple integral

Transcendental function - an absolute pain to analytically integrate!


I love Julia.

Nb: Z is Z(T , U). Therefore need to reevaluate if they change...


Codes to take arbitrary potential function→ integrate to get Z (=partition function)→ populate configurational density of states→ parameterise electron transfer Js from result→ if tridiagonal, extremely fast Sturm sequence else, standard eigenvalue + histogram → DoS for band of interest

Partition Functions are cool(and not scary, honest)


Zero pressure potential (U from QC, MP2 PFO dimer)

eV

Red = potentialGreen = distribution @ 300K


Increasing (holding flat) sin potential [[ Pressure ]]


Populate Density of States Hamiltonian...

Distribution of thetas from stat mech...

Model for transfer integral...


Feed to TB DoS machinery

Pressure

STURM!10'000 sites

< 0.1 s

[ quite fast ]


U is not a Hamiltonian

Assume no correlation Theta1-Theta2-Theta3

Sidechains & cohesive action weird (U not a simple function)

Polymers are hard...

Problems?


Acknowledgments

WMD Group (Bath)

James KP (Imperial, now Deepmind / Google) - for all the tutorage re: Wave Functions

Beth Rice (Imperial)

Jenny Nelson (Imperial)


“It is simply this: do not tire, never lose interest, never grow indifferent—lose your invaluable curiosity and you let yourself die. It's as simple as that.” ― Tove Jansson, Fair Play

Science

From Atoms to Solar Cells