From Atoms to Solar Cells
Transcript of From Atoms to Solar Cells
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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
From atoms to solar cellsMultiscale physics of Photovoltaics
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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
From solar cells to atomsMultiscale physics of Photovoltaics
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
"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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
What we need is Fusion! [ E=mc2 ]
Opération Canopus, Fangatuafa atoll, 1968. 2.6 MT
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Silicon grown by Czochralski process 1956George E. Meyers, Raytheon Corp. semiconductor plant in Newton, Massachusetts, USA
American Radio History
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
(old) mono-crystallinehigh efficiency (~15%) but expensive...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
modern mono-crystalline (better anti-reflective → black)highest efficiency (~20%), really quite cheap…
Best lifetimes
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
SunPower E2020.4% efficient module (!!!!!!!!!)Full back-contacted cells
Commercially available >2013
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
polycrystalline silicongood efficiency (~14-16%)
(used to be sig. cheaper than mono-crystalline)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
CIGS / a-Si - Black MirrorsEfficiencies 7-13%. Were relatively cheap...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
CdTeYour flexible,
slightly purple-yfriend
(poisonous, but apparently easiest to make!)
~7-15% efficiencies
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Thermodynamic LimitSun: 5760 KCell: 2470 KBath: 300 K
ηPCE = 85%
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
1.6 eV
1.0 eV
θ
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Shockley–Queisser limit (@ 1 sun)
Maximum solar conversion efficiency around 33.7%, assuming a single junction with a band gap of 1.34 eV
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Silicon, 2012; 25 Yr life
CIGSCdTeBoth ~killed by Silicon
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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 )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Light Trapping (Ray Limit)
Nothing + Back reflector Inverted Pyramids
Randomised Surface
d= 4 n^2 =~ 50 (Si)
Lambertian emittance
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
PERL Silicon Solar Cell ~23.5% PCE
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Organic Solar Cells!(This one is Rubrene, also the highest mobility organic…)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
1.6 eV
1.0 eV
θ
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Ideal Diode equation
I0
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
PN Junction - From: http://en.wikipedia.org/wiki/P%E2%80%93n_junction
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Nabla electrical potential =charge density / dielectric constant
Actually quite nasty to solve! (self consistency & BCs)
Poisson's Equation
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Full Drift DiffusionElectron Affinity
(variations)
Effective Band Densities of State
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Modelling Charge Transport (in Organics)
Following 5 slides of figures & text reproduced from Joe Kwiatkowski's MPhil thesis (ICL, 2008)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Charge transport - the microscopic view
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Effective potential energy surface
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Marcus theory
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Transfer integral
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
All this scary matrix maths (orbital projection)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Practically… 3 DFT calculations & a matrix inversion
MoleculeA
MoleculeB
MoleculeA + B
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Monte Carlo model of charge transport…
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Mathematician
Theorist
Numerics (Computational)
Experimentalist
The Lay Person
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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 )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
"Computers are bicycles for the mind." - Steve Jobs
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
UK Children of the 80s had a head start...
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Programming languages compared
(Potential) Computational Speed
Algo
rith
mic
Exp
ress
iven
ess
C
Python
Julia
Fortran
Assembly
R
Octavego
parallel
Haskell
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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
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From Christoph Ortner's excellent numerical analysis Julia notebooks.. http://homepages.warwick.ac.uk/staff/C.Ortner/index.php?page=julia
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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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
A simple type of N-state model
Basis of states on monomers… (orthogonal)
Coupled with effective transfer-integrals
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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"
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Psi & E - results of our efforts
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Infinite polymer (10 units)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Motivation
Motivation: Why do organic solar cells work*? (* at all)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Actually quite boring for large polymers...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
100 'polymer', PBCs, slight mid-chain defect
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
2D checkerboard… (without PBCs)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
2D checkerboard… (with PBCs)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
3D Checkerboard
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
3D 10x10x10 simple cubic cell
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
"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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Overview● Tight Binding● PCBM-MD● Tight Binding with PCBM-MD● Polarons and Tight Binding
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Atoms → Smarties, gives you a ~1000x (or more) speedup
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Where to get parameters?
Girifalco 'smeared' C60 for the balls...
And a scaled version for PBM?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Atomistic view...We have an OPLS PCBM model... so let's use it!
2564 PCBMsMD for ~20ps
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Atomistic RDF via COM (Center of Mass) of PBM and C60 fragments
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
1000 Tris PCBM, CG
alter elem c, vdw=5.0alter elem p, vdw=3.0show spheres
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Consider the inter-adduct angles
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
8 Bis isomers; Angles: 36 72 60 90 180 144 120 108
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
45 Unique Tris Isomers... (of 24360)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
C60 (Bucky Balls) Mono PCBM Bis PCBM
Tris PCBM
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Transfer Integrals…Amicable Separation
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Blue = exponential fitPrefactor = 1e6 (?)
Characteristic length:0.597 A (MONO)0.577 A (BIS - E1)0.580 A (TRIS - EEE)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Overview● Tight Binding● PCBM-MD● Tight Binding with PCBM-MD● Polarons and Tight Binding
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
(Histogram of 1000 eigenvalues)
High lying states
Low lying states
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Overview● Tight Binding● PCBM-MD● Tight Binding with PCBM-MD● Polarons and Tight Binding
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Frohlich Polaron● Consider linear dielectric response -
outside- polaron
Dielectric response…
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Dielectric response…
Site E
nerg
y
(Pol
arisa
tion)
Self consistent response...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Simple 1D chain case...
S
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Simple cubic latticeTiny defect to break symm
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Transfer between polarons
Polaron transfer integral calculation
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Unperturbed; C60 Simulated Annealing
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Localised Polaron State
Perturbed States
Polaron; C60 Simulated Annealing
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
A simple type of N-state model
Basis of states on monomers… (orthogonal)
Coupled with effective transfer-integrals
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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"
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
5K PL-spectra, Octamers
![Page 130: From Atoms to Solar Cells](https://reader034.fdocuments.net/reader034/viewer/2022042716/55a6768e1a28abc2758b4579/html5/thumbnails/130.jpg)
Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Normal 'Wiggly' Polyfluorene...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
HOMO Spin Density - centre crimped flat
Hypothesis:-
Is Beta Phase a hole trap?
Are 'bubbles' of beta phase responsible for reduced mobility in processing?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
01234567
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
TD-DFT Singlets (Absorption)
0
7
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
HOMO / LUMO Energy
eV
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
(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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Inner Sphere Reorganisation Energy
eV
--> Losing a degree of freedom... -->
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 201550meV Gaussian Site Energy Disorder
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Beta Phase Energetic Trap(5 sites)
DFT Trap Depth
Reproduced!
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Beta Phase Energetic Trap(10 sites)
For Wide Enough trap ~10 sites (no confinement),Trap depth = site Energy change
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Theta ~ 45 degrees for segmentTheta = 0 degrees for beta phase bit--> 0.18eV change in J--> 310 meV trap depth
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Disordered (torsionally)TWISTED (~PFO) polymer
596meV
PFO MODEL
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Disordered (torsionally)But FLAT (minima) polymer
70meV
P3HT MODEL
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
No site disorder...
No J disorder...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
PFO trap states from J variation are persistent
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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.
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Random Matrices...
Random Matrix Theory...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Tridiagonal Matrices?
( Reading maths paper on the ArXiv isn't always a complete waste of time. )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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? )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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 )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
~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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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
![Page 161: From Atoms to Solar Cells](https://reader034.fdocuments.net/reader034/viewer/2022042716/55a6768e1a28abc2758b4579/html5/thumbnails/161.jpg)
Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Snakes Under Pressure
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Generating GeometriesAb-initio MDEmpirical MD
Coarse grain MDStat Phy
Make Stuff up'optimised geom'
Phys
ical
Ac
cura
cy?
Tim
e (e
xp)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
MD is quite a pain...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
∇U = FF = ma
At the core of MD...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Statistical mechanics viewAt thermodynamic equilibrium,difference in population:
(We don't need no trajectory (history, kinetics) - just ΔE )
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Polyfluorene - a 9 yr love / hate relationship
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Early PhD work… design a PFO MD forcefield
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Polymers are freaky...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Polyfluorene (by MD)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Given a U, how to populate DoS?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
"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...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
How to Z?Sum over configurations
orSum over energy (caring for degeneracy)Continuous variable U(theta) → simple integral
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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!
![Page 174: From Atoms to Solar Cells](https://reader034.fdocuments.net/reader034/viewer/2022042716/55a6768e1a28abc2758b4579/html5/thumbnails/174.jpg)
Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
I love Julia.
Nb: Z is Z(T , U). Therefore need to reevaluate if they change...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
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)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Zero pressure potential (U from QC, MP2 PFO dimer)
eV
Red = potentialGreen = distribution @ 300K
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Increasing (holding flat) sin potential [[ Pressure ]]
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Populate Density of States Hamiltonian...
Distribution of thetas from stat mech...
Model for transfer integral...
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Feed to TB DoS machinery
Pressure
STURM!10'000 sites
< 0.1 s
[ quite fast ]
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
U is not a Hamiltonian
Assume no correlation Theta1-Theta2-Theta3
Sidechains & cohesive action weird (U not a simple function)
Polymers are hard...
Problems?
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
Acknowledgments
WMD Group (Bath)
James KP (Imperial, now Deepmind / Google) - for all the tutorage re: Wave Functions
Beth Rice (Imperial)
Jenny Nelson (Imperial)
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Jarvist Moore Frost (University of Bath, UK) From atoms to solar cells: Multiscale physics of photovoltaics 5th March 2015
“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