Modelling excitonic solar cells
Alison Walker
Department of Physics
How can modelling help?
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Materials
Patterning, Self-organisation, Fabrication
Device Physics
Characterization
Outline
• Dynamic Monte Carlo Simulation
• Energy transport
• Charge transport
Dynamic Monte Carlo
Simulation
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)
Disordered morphology
(a) Interfacial area
3106 nm2
(b) Interfacial area
1106 nm2
(c) Interfacial area
0.2106 nm2
Modelled Morphology
• Hopping sites on a cubic lattice
with lattice parameter a = 3 nm
• Sites are either
electron transporting polymer (e)
or hole transporting polymer (h)
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 find
energy difference  = l - m
• Spins swopped with probability
exp  (k BT )
P( ) 
1  exp  (k BT )
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
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
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
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
Challenges
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Several interacting particle species
Many possible interactions:
Generation
Hopping
Recombination
Extraction
Wide variation in time scales
Two site types
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
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. condmatt/0303028
Master equation
dP
dt
=
(W P- W P)

, are configurations
P, P are their probabilities
W are the transition rates
Simple derivation of Poisson Distribution
Consider a reaction with a transition rate k.
Probability that a reaction occurs in time interval
t  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
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 
eg for a distribution peaked at x0, most values of r will
give values of x close to x0
F1
f
r
x0
0
x
x0
x
t0
t
For Poisson distribution,
c
1
P
r
t
0
To select times with Poisson distribution
from random numbers ri distributed
uniformly between 0 and 1, use
r1 = 0t kexp(-kt)dt 
Hence
t = -1 ln(1-r1) = -1 ln(r2)
k
k
First Reaction Method
• Each reaction i with rate wi has a waiting time
from a uniformly distributed random number r
i = -1 ln(r)
wi
• List of reactions created in order of increasing i
• First reaction in list takes place if enabled
• List then updated
Create a queue of reactions i and
associated waiting times i.
Set simulation time t = 0.
Select reaction at top of queue
Top reaction enabled?
Yes
Do top reaction
Remove this reaction from queue
Set t = t + top
Set i = i - top
Add enabled reactions
No
Flow Chart
Remove from
queue
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
• Electron(hole) hopping between e(h) sites
wij = w0exp[-2Rij]exp[-(Ej – Ei)/(kBT)] if Ej > Ei
w0exp[-2Rij]
if Ej < Ei
w0 = [6kBT/(qa2)]exp[-2a]
e = h = 1.10-3 cm2/(Vs)
 = 2 nm-1
• Electron(hole) recombination rate
wce = 100 s-1
allows peak IQE to exceed 50% for idealised
morphology
• Electron(hole) extraction
wce =  if electron next to anode/hole next to cathode
wce = 0 otherwise
Reaction rates
• Exciton creation on either e or h site
S = 2.4102 nm-2s-1
• Exciton hopping between sites of same type
wij = 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
Disordered morphology
(a) Interfacial area
3106 nm2
(b) Interfacial area
1106 nm2
(c) Interfacial area
0.2106 nm2
Efficiencies (disordered morphology)
b
c
a
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
Sensitivity of IQE to input parameters
a) As the exciton generation rate increases,
IQE decreases at all interfacial areas due
to enhanced charge recombination
b) For larger external biases, the peak IQE
increases and shifts to larger interfacial
areas
c) Similar changes to (b) seen for larger
charge mobilities and if charge mobilities
differ
Ordered morphology
Achievable with diblock copolymers
Efficiencies (ordered morphology)
• 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.
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?
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
Angle
ηgr
0°
~22%
90°
~26%
180°
~83%
E
• Most time is spent tracking at the interface.
• A polymer with a range of interface angles is far less
efficient than a vertical structure.
• 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.
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
Energy transport
Stavros Athanasopoulos, David Beljonne,
Evgenia Emilianova
University of Mons-Hainaut
Luca Muccioli, Claudio Zannoni
University of Bologna
Chemical structure
Physical morphology
electronic properties
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
Spectroscopy on end-capped polymers
Solid
PL intensity
Solution
 (nm)
Indenofluorene chromophores
Perylene end-caps
• 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
Morphology
P3HT- crystalline, high mobility
(~0.1 cm2/Vs)
Disorder could occur parallel to
plane of substrate
Electron micrograph
of PF2/6:
Liquid-crystalline state
lamellae separated
by disordered regions;
molecules inside lamellae
separate according to
lengths
Ordered regions also
seen in PIF copolymers
Energetic disorder
Numbers of chromophores per chain, and
lengths of individual chromophores are
assigned specified distributions:
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
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
Trajectories of individual particles
(note periodic
boundary
conditions)
are averaged to obtain quantities of interest
• 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
Mean
absolute value
= 4.5 nm
– Also, the more numerous interchain
hops can involve a non-negligible
z
z component
x
y
Mean
absolute value
= 1.6 nm
 E2 
g E 
exp   2 
 2
 2 
Nt
rF = 3.1 nm
Nt = 1 nm-3
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
Charge transport
Jarvist Frost, James Kirkpatrick, Jenny Nelson
Imperial College London
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
Ordered chains
Time of flight (ToF) experiment
= d
E
Our Model
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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:
1. Chemical structure?
2. Molecular packing?
Field parallel to the chains leads to higher
mobility
=> Intra chain transfer dominates
Relaxed Geometry
Marcus theory
D + A+ → D+ + A
Donor
Acceptor
2
E
intra
2
ii
(D2)
i
E
intra(A2)
QA i
1
ii
intra(D1)
1
intra(A1)
QA
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)
Transfer rates
Electronic coupling potential V from INDO
G is change in free energy
kDA = 2V2
exp - (G + )2
ħ(4kBT)
(4kBT)
 from Density Functional Theory (B3LYP)
Simulated transient current

Hole mobility (cm2V-1s -1)
Charge transfer in aligned PFO
(Field)1/2 (V1/2 m-1/2)
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
Where next?
• Improved charge and exciton transfer and
recombination rates
• Include triplet excitons
• Different morphologies
• Other systems eg display devices
Thanks!!!
To
Risto,
Martti,
Adam,
Arkady,
Mikko,
Teemu