Modelling excitonic solar cellsModelling excitonic solar cells

Alison Walker

Department of Physics

How can modelling help?

• Materials

• Patterning, Self-organisation, Fabrication

• Device Physics

• Characterization

Outline

• Dynamic Monte Carlo Simulation

• Energy transport

• Charge transport

Dynamic Monte Carlo Simulation

Excitons generated throughoutElectrons confined to green regionsHoles confined to red regions

P K Watkins, A B Walker, G L B Verschoor Nano Letts 5, 1814 (2005)

Disordered morphology

(a) Interfacial area 3106 nm2

(b) Interfacial area 1106 nm2

(c) Interfacial area 0.2106 nm2

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)

Ising Model

• Ising energy for site i isi = -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 )(exp1

)(exp)(

Tk

TkP

B

B

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

Internal quantum efficiency IQE

External quantum efficiency EQE For illumination with spectral density S()

JSC = qd EQE S()where external quantum efficiency

EQE = no of electrons flowing in external circuit no of photons incident on cell

= AIQE

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• Several interacting particle species• Many possible interactions:

GenerationHoppingRecombinationExtraction

• 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. cond-matt/0303028

Master equation

dP

dt= (W P- W P)

, are configurationsP, P are their probabilities

W are the transition rates

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 distributionP(t) = kexp(-kt)

Simple derivation of Poisson Distribution

R Hockney, J W Eastwood Computer simulation using particles

IoP Publishing, Bristol, 1988

Integrating dc = dp = P(t) dt givescumulative probabilityc(t) = 0

t P(t)dt The reaction has not occurred at t = 0 but will occur some time, soc(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 selectinga value in the range c c + dc is dcHencer = c(t) = 0

t P(t)dt

Selecting waiting times

eg for a distribution peaked at x0, most values of r will give values of x close to x0

x

f

x0

For Poisson distribution,

t

P

tt0

r1

c

0

F

x

r

x0

1

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

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

First Reaction Method

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

i = -1 ln(r)wi

Flow ChartCreate a queue of reactions i and associated waiting times i.

Set simulation time t = 0.Select reaction at top of queue

Do top reaction Remove this reaction from queue

Set t = t + top

Set i = i - top

Add enabled reactions

Top reaction enabled?

Yes

Remove from queueNo

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[-2Rij]exp[-(Ej – Ei)/(kBT)] if Ej > Ei

w0exp[-2Rij] if Ej < Ei

w0 = [6kBT/(qa2)]exp[-2a] e = h = 1.10-3 cm2/(Vs) = 2 nm-1

• Electron(hole) recombination ratewce = 100 s-1

allows peak IQE to exceed 50% for idealisedmorphology

• Electron(hole) extractionwce = if electron next to anode/hole next to cathode

wce = 0 otherwise

Reaction rates• Exciton creation on either e or h site

S = 2.4102 nm-2s-1

• Exciton hopping between sites of same typewij = we(R0/Rij)6 weR0

6 = 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 3106 nm2

(b) Interfacial area 1106 nm2

(c) Interfacial area 0.2106 nm2

Efficiencies (disordered morphology)

a

b

c

At large interfacial area ie small scale phaseseparation:

• 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.

• 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?

Gyroids

Geminate recombinationUnexpected difference between rod structures and the others.

Recombination

Bimolecular recombinationNovel structures show little advantage over blends (even at 5 suns). Islands and disconnected pathways not responsible for inefficiency as previously thoughtRod 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 EmilianovaUniversity of Mons-HainautLuca Muccioli, Claudio ZannoniUniversity of Bologna

electronic properties

Chemical structure Physical morphology

• 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

Experimental background

SolidSolutionP

L in

tens

ity

Indenofluorene chromophoresPerylene end-caps

(nm)

Spectroscopy on end-capped polymers

• 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 micrographof PF2/6:Liquid-crystalline state lamellae separatedby disordered regions;molecules inside lamellaeseparate according tolengths

Ordered regions also seen in PIF copolymers

Energetic disorder

Numbers of chromophores per chain, and lengths of individual chromophores are assigned specified distributions:

• 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

Key Features of our Model

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

are averaged to obtain quantities of interest

(note periodic boundary conditions)

• 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

yx

z

Mean absolute value = 1.6 nm

Mean absolute value = 4.5 nm

2

2exp

22tN E

g E

rF = 3.1 nmNt = 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 NelsonImperial College London

• 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

Dynamical Monte Carlo Migration Algorithm

Ordered chains

Time of flight (ToF) experiment

= d E

• 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?

Our Model

Field parallel to the chains leads to higher

mobility

=> Intra chain transfer dominates

Relaxed Geometry

Marcus theory

Reorganisation energy

intra = intra(A1) + intra

(D2)

J-L. Brédas et al Chemical Reviews104 4971 (2004)

Donor

E

QD

Acceptor

intra(D2)

i

ii

1

2

intra(D1)

intra(A1) = E(A1)(A+) – E(A1)(A)

intra(D2) = E(D2)(D) – E(D2)(D+)

QA

QA

ii

i

intra(A2)

intra(A1)

1

2

E

D + A+ → D+ + A

Transfer rates

kDA = 2V2 exp - (G + )2

ħ(4kBT) (4kBT)

Electronic coupling potential V from INDOG is change in free energy

from Density Functional Theory (B3LYP)

Simulated transient current

Charge transfer in aligned PFOH

ole

mob

ility

(cm

2 V-1s

-1)

(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

ToRisto, Martti, Adam,Arkady,Mikko,Teemu

Thanks!!!Thanks!!!