Modelling excitonic solar cells

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Modelling excitonic solar cells

• Alison Walker
• Department of Physics

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How can modelling help?

• Materials
• Patterning, Self-organisation, Fabrication
• Device Physics
• Characterization

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Outline

• Dynamic Monte Carlo Simulation
• Energy transport
• Charge transport

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Dynamic Monte Carlo Simulation
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Excitons generated throughout Electrons confined
to green regions Holes confined to red regions
P K Watkins, A B Walker, G L B Verschoor Nano
Letts 5, 1814 (2005)
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Disordered morphology
(a) Interfacial area 3?106 nm2
(b) Interfacial area 1?106 nm2
(c) Interfacial area 0.2?106 nm2
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Modelled Morphology

• Hopping sites on a cubic latticewith lattice
  parameter a 3 nm
• Sites are either electron transporting polymer
  (e) or hole transporting polymer (h)

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Ising Model

• Ising energy for site i is?i -½J ??(si, sj)
  1
• Summation over 1st and 2nd nearest neighbours
• Spin at site i si 1 for e site, 0 for h site
• Exchange energy J 1
• Chose neighbouring pair of sites l, m and
  findenergy difference ?? ?l - ?m
• Spins swopped with probability

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Internal quantum efficiency ?IQE
?IQE measures exciton harvesting
efficiency Exciton dissociation efficiency ?e
no of dissociated excitons no of absorbed
photons Charge transport efficiency ?c no of
electrons exiting device no of dissociated
excitons Internal quantum efficiency ?IQE no
of electrons exiting device ?e ?c no of
absorbed photons NB Assume all charges reaching
electrodes exit device
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External quantum efficiency ?EQE

• For illumination with spectral density S(?)
• JSC q?d? ?EQE S(?)
• where external quantum efficiency
• ?EQE no of electrons flowing in external
  circuit
• no of photons incident on cell
• ?A?IQE
• photon absorption efficiency
• ?A no of absorbed photons
• no of photons incident on cell
• internal quantum efficiency
• ?IQE no of electrons flowing in external
  circuit
• no of absorbed photons

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Possible reactions

• Exciton creation on either e or h site
• Exciton hopping between sites of same type
• Exciton dissociation at interface between e and h
  sites
• Exciton recombination

• Electron(hole) hopping between e(h) sites
• Electron(hole) extraction
• Charge recombination

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Generation of morphologies with varying
interfacial area

• Start with a fine scale of interpenetration,
  corresponding to a large interfacial area
• As time goes on, free energy from Ising model is
  lowered, favouring sites with neighbours that are
  the same type
• Hence interfacial area decreases
• Systems with different interfacial areas are
  morphologies at varying stages of evolution

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Challenges

• Several interacting particle species
• Many possible interactionsGenerationHoppingRec
  ombinationExtraction
• Wide variation in time scales
• Two site types

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Why use Monte Carlo ?

• Do not have (or want) detailed information about
  particle trajectories on atomic length scales nor
  reaction rates
• Thus can only give probabilities for reaction
  times
• These can be obtained by solving the Master
  equation but this is computationally costly for
  3D systems

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Dynamical Monte Carlo Model

• Many different methods
• These can all be shown to solve the Master
  Equation (Jansen)
• First Reaction Method has been used to simulate
  electrons only in dye-sensitized solar cells

A P J Jansen Phys Rev B 69, 035414 (2004) A P
J Jansen http//ar.Xiv.org/, paper no.
cond-matt/0303028
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Master equation
?
?,? are configurationsP?, P? are their
probabilities W?? are the transition rates
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Simple derivation of Poisson Distribution
Consider a reaction with a transition rate
k. Probability that a reaction occurs in time
intervalt ? t dt dp (Probability reaction
does not occur before t) ?(Probability
reaction occurs in dt) - p(t) k dt Hence
probability distribution P(t) of times at which
reaction occurs normalised such that ?P(t)dt 1
is the Poisson distribution P(t) kexp(-kt)
R Hockney, J W Eastwood Computer simulation using
particles IoP Publishing, Bristol, 1988
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Selecting waiting times

• Integrating dc dp P(t) dt gives
• cumulative probability
• c(t) ?0t P(t?)dt ?
• The reaction has not occurred at t 0 but
• will occur some time, so
• c(0) 0 ? c ? 1 c(?)
• If the value of c is set equal to a random
• number r chosen from a uniform distribution in
• the range 0 ? r ? 1, the probability of selecting
• a value in the range c ? c dc is dc
• Hence
• r c(t) ?0t P(t?)dt ?

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eg for a distribution peaked at x0, most values
of r will give values of x close to x0
F
x
For Poisson distribution,
P
t
t
t0
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To select times with Poisson distribution from
random numbers ri distributed uniformly between 0
and 1, use r1 ?0t kexp(-kt?)dt ? Hence
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First Reaction Method

• Each reaction i with rate wi has a waiting time
  from a uniformly distributed random number r

• List of reactions created in order of increasing
  ?i
• First reaction in list takes place if enabled
• List then updated

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Flow Chart
Create a queue of reactions i and associated
waiting times ?i. Set simulation time t
0. Select reaction at top of queue
Remove from queue
No
Yes
Do top reaction Remove this reaction from queue
Set t t ?top Set ?i ?i - ?top Add enabled
reactions
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Simulation details

• Hops allowed to the 122 neighbours within 9 nm
  cutoff distance
• Exclusion principle applies ie hops disallowed to
  occupied sites
• Periodic boundary conditions in x and y
• Site energies Ei are all zero for excitons
• For charge transport, Ei include(i) Coulomb
  interactions(ii) external field due to built-in
  potential and external voltage

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• Electron(hole) hopping between e(h) sites wij
  w0exp-2?Rijexp-(Ej Ei)/(kBT) if Ej gt Ei
  w0exp-2?Rij if Ej lt Eiw0
  6?kBT/(qa2)exp-2?a ?e ?h 1.10-3
  cm2/(Vs) ? 2 nm-1
• Electron(hole) recombination ratewce 100
  s-1allows peak IQE to exceed 50 for
  idealisedmorphology
• Electron(hole) extractionwce ? if electron
  next to anode/hole next to cathode wce 0
  otherwise

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Reaction rates

• Exciton creation on either e or h siteS
  2.4?102 nm-2s-1
• Exciton hopping between sites of same typewij
  we(R0/Rij)6 weR06 0.3 nm6s-1 gives
  diffusion length of 5nm
• Exciton dissociation at interface between e and h
  sites wed ? if exciton on an interface site
  wed 0 otherwise

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Disordered morphology
(a) Interfacial area 3?106 nm2
(b) Interfacial area 1?106 nm2
(c) Interfacial area 0.2?106 nm2
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Efficiencies (disordered morphology)
b
c
a
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• At large interfacial area ie small scale phase
• separation

• excitons more likely to find an interface
  before recombining
• thus exciton dissociation efficiency increases
• charges follow more tortuous routes to get to
  electrodes
• charge densities are higher
• charge recombination greater
• thus charge transport efficiency decreases
• Net effect is a peak in the internal quantum
  efficiency

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Sensitivity of ?IQE to input parameters

• As the exciton generation rate increases, ?IQE
  decreases at all interfacial areas due to
  enhanced charge recombination
• For larger external biases, the peak ?IQE
  increases and shifts to larger interfacial areas
• Similar changes to (b) seen for larger charge
  mobilities and if charge mobilities differ

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Ordered morphology
Achievable with diblock copolymers
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Efficiencies (ordered morphology)
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• As for disordered morphologies, see a peak in
  ?IQE, here at a width of 15 nm
• Maximum ?IQE is larger by a factor of 1.5 than
  for disordered morphologies
• Peak is sharper since at large interfacial areas,
  excitons less likely to find an interface and the
  charges are confined to narrow regions so there
  is a large recombination probability.

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Gyroids

• Continuous charge transport pathways, no
• disconnected or cul-de-sac features
• Free from islands
• A practical way of achieving a similar
• efficiency to the rods?

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Comparison with other morphologies
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Recombination
Geminate recombination Unexpected difference
between rod structures and the others.

• Bimolecular recombination
• Novel structures show little advantage over
  blends (even at 5 suns). Islands and disconnected
  pathways not responsible for inefficiency as
  previously thought
• Rod structures significantly better, even at
  small feature sizes
• Short, direct pathways to electrodes
• Can keep charges entirely isolated

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Why?
Angle ?gr
0 22
90 26
180 83

• Most time is spent tracking at the interface.
• A polymer with a range of interface angles is
  far less
• efficient than a vertical structure.

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• Feature size dependence of fill factor, shift in
  optimum feature
• size when examining complete J-V performance.
• Islands shift the perceived optimum feature
  size.
• New morphologies not as efficient as hoped,
  despite absence
• of islands and disconnected pathways.
• Morphology can still inhibit geminate separation
  at large
• feature sizes.
• Rods have noticeably lower geminate and
  bimolecular
• recombination, but for different reasons.
• Angle of interface is critical, morphologies
  with a
• range of angles less efficient than vertical
• structures.

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Dynamical Monte Carlo Summary

• Dynamical Monte Carlo methods are a useful way
  of modelling polymer blend organic solar cells
  because (i) they are easy to implement, (ii)
  they can handle interacting particles (iii) they
  can be used with widely varying time scales

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Energy transport
Stavros Athanasopoulos, David Beljonne, Evgenia
Emilianova University of Mons-Hainaut Luca
Muccioli, Claudio Zannoni University of Bologna
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electronic properties
Chemical structure Physical morphology
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Experimental background

• Polyphenylenes eg PFO used for blue emissive
  layers in blue OLEDs but emission maxima close to
  violet
• Polyindenofluorenes intermediate between PFO and
  LPPP show purer blue emission
• The solid state luminescence output has been
  related to the microscopic morphology

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Spectroscopy on end-capped polymers
Solid
Solution
PL intensity
l (nm)
Indenofluorene chromophores
Perylene end-caps
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• Transfer rates from chromophore to perylene are
  much faster than those between chromophores
• Different spectra are observed for the polymer in
  solution, and as a film

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Morphology
P3HT- crystalline, high mobility (0.1 cm2/Vs)
Disorder could occur parallel to plane of
substrate
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Electron micrograph of PF2/6 Liquid-crystalline
state lamellae separated by disordered
regions molecules inside lamellaeseparate
according tolengths
Ordered regions also seen in PIF copolymers
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Energetic disorder
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• Numbers of chromophores per chain, and lengths
  of individual chromophores are assigned specified
  distributions

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Key Features of our Model

• Exciton diffusion takes place within a realistic
  morphology consisting of a 3D array of PIF chains
• Excitons hop between chromophores
• Averaging over many exciton trajectories,
  properties such as diffusion length, ratio of
  numbers of intrachain to interchain hops, spectra
  etc are explored

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Quantum Chemical Calculation of Hopping Rates

• Mons provide rates of exciton transfer between
  chromophores
• They use quantum chemical calculations employing
  the distributed monopole method
• This takes into account the shape of donor and
  acceptor chromophores in calculating the
  electronic coupling Vda
• The hopping rate from donor to acceptor is

Electronic coupling
Overlap factor
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• Trajectories of individual particles

(note periodic boundary conditions)
are averaged to obtain quantities of interest
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• Intrachain hops are less common
• (No. interchain hops) / (No. intrachain hops) ? 7
• Yet motion parallel to the chain axes is more
  prevalent why?
• Intrachain hops involve
• long distances
• Also, the more numerous interchain
• hops can involve a non-negligible
• z component

z
y
x
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rF 3.1 nm Nt 1 nm-3
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Summary for exciton transport

• A physically valid method of simulating transport
  in conjugated polymers (towards a multiscale
  approach)
• Advantages over cubic-lattice approaches
• Energetic disorder is crucial

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Charge transport
Jarvist Frost, James Kirkpatrick, Jenny
Nelson Imperial College London
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Dynamical Monte Carlo Migration Algorithm

• The waiting time before a hop from site i to a
  neighbouring site j is
• ?ij -1 ln(r)
• wij where wij is the hopping rate between
  sites i and j, and r is a random number uniformly
  distributed between 0 and 1.
• When the exciton hops, we always choose the hop
  with the shortest waiting time ?ij

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Ordered chains
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Time of flight (ToF) experiment

• d
• ?E

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Our Model

• Localized polarons on single conjugated segments
• Alternative is Gaussian disorder model which
  involves hopping between sites on a cubic lattice
  subject to some disorder
• Questions
• Chemical structure?
• Molecular packing?

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• Field parallel to the chains leads to higher
• mobility
• gt Intra chain transfer dominates

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Relaxed Geometry
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Marcus theory
D A ? D A
Acceptor
Donor
E
QD

• Reorganisation energy
• intra ?intra(A1) ?intra (D2)

?intra(A1) E(A1)(A) E(A1)(A) ?intra(D2)
E(D2)(D) E(D2)(D)
J-L. Brédas et al Chemical Reviews 104 4971
(2004)
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Transfer rates
Electronic coupling potential V from INDO ?G is
change in free energy
kDA 2?V2 exp - (?G ?)2
h?(4??kBT) (4?kBT)
? from Density Functional Theory (B3LYP)
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Simulated transient current
?
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Charge transfer in aligned PFO
Hole mobility (cm2V-1s -1)
(Field)1/2 (V1/2 m-1/2)
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Summary for charge transport

• We can relate charge transport to chemical
  structure up to a point
• The fact that intrachain transport is much faster
  than interchain transport is crucial to
  understand charge mobilities in polymer films
• Good agreement with experimental ToF hole
  mobility data for aligned films

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Where next?

• Improved charge and exciton transfer and
  recombination rates
• Include triplet excitons
• Different morphologies
• Other systems eg display devices

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Thanks!!!
To Risto, Martti, Adam, Arkady, Mikko, Teemu