1. Modelling excitonic solar cells Alison Walker Department of Physics

2. How can modelling help? Materials Patterning, Self-organisation, Fabrication Device Physics Characterization

3. Outline Dynamic Monte Carlo Simulation Energy transport Charge transport

4. Dynamic Monte Carlo Simulation

5. 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)

6. Disordered morphology (a) Interfacial area 3  10 6 nm 2 (b) Interfacial area 1  10 6 nm 2 (c) Interfacial area 0.2  10 6 nm 2

7. 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)

8. Ising Model Ising energy for site i is  i = -½J  [  (s i, s j ) – 1] Summation over 1 st and 2 nd nearest neighbours Spin at site i s i = 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

9.  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

10. External quantum efficiency  EQE For illumination with spectral density S( ) J SC = 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

11. 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

12. 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

13. Challenges Several interacting particle species Many possible interactions: Generation Hopping Recombination Extraction Wide variation in time scales Two site types

14. 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

15. 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

16. Master equation dPdP dtdt =  (W  P  - W  P  )  ,  are configurations P , P  are their probabilities W  are the transition rates

17. 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) Simple derivation of Poisson Distribution R Hockney, J W Eastwood Computer simulation using particles IoP Publishing, Bristol, 1988

18. Integrating dc = dp = P(t) dt gives cumulative probability c(t) =  0 t 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) =  0 t P(t)dt Selecting waiting times

19. eg for a distribution peaked at x 0, most values of r will give values of x close to x 0 x f x0x0 For Poisson distribution, t P t t0t0 r 1 c 0 F x r x0x0 1 0

20. To select times with Poisson distribution from random numbers r i distributed uniformly between 0 and 1, use r 1 =  0 t kexp(-kt)dt Hence t = -1 ln(1-r 1 ) = -1 ln(r 2 ) kk

21. Each reaction i with rate w i 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) wiwi

22. Flow Chart Create 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 queue No

23. 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 E i are all zero for excitons For charge transport, E i include (i) Coulomb interactions (ii) external field due to built-in potential and external voltage

24. Electron(hole) hopping between e(h) sites w ij = w 0 exp[-2  R ij ]exp[-(E j – E i )/(k B T)] if E j > E i w 0 exp[-2  R ij ] if E j < E i w 0 = [6  k B T/(qa 2 )]exp[-2  a]  e =  h = 1.10 -3 cm 2 /(Vs)  = 2 nm -1 Electron(hole) recombination rate w ce = 100 s -1 allows peak IQE to exceed 50% for idealised morphology Electron(hole) extraction w ce =  if electron next to anode/hole next to cathode w ce = 0 otherwise

25. Reaction rates Exciton creation on either e or h site S = 2.4  10 2 nm -2 s -1 Exciton hopping between sites of same type w ij = w e (R 0 /R ij ) 6 w e R 0 6 = 0.3 nm 6 s -1 gives diffusion length of 5nm Exciton dissociation at interface between e and h sites w ed =  if exciton on an interface site w ed = 0 otherwise

26. Disordered morphology (a) Interfacial area 3  10 6 nm 2 (b) Interfacial area 1  10 6 nm 2 (c) Interfacial area 0.2  10 6 nm 2

27. Efficiencies (disordered morphology) a b c

28. 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

29. 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

30. Ordered morphology Achievable with diblock copolymers

31. Efficiencies (ordered morphology)

32. 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.

33. 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

35. Geminate recombination Unexpected difference between rod structures and the others. Recombination 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

36. Angleη gr 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.

37. 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.

38. 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

39. Energy transport Stavros Athanasopoulos, David Beljonne, Evgenia Emilianova University of Mons-Hainaut Luca Muccioli, Claudio Zannoni University of Bologna

40. electronic properties Chemical structure Physical morphology

41. 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

42. SolidSolution PL intensity Indenofluorene chromophores Perylene end-caps  (nm) Spectroscopy on end-capped polymers

43. 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

44. Morphology P3HT- crystalline, high mobility (~0.1 cm 2 /Vs) Disorder could occur parallel to plane of substrate

45. 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

46. Energetic disorder

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

48. 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

49. 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 V da The hopping rate from donor to acceptor is Electronic couplingOverlap factor

50. Trajectories of individual particles are averaged to obtain quantities of interest (note periodic boundary conditions)

51. 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 y x z Mean absolute value = 1.6 nm Mean absolute value = 4.5 nm

52. r F = 3.1 nm N t = 1 nm -3

53. 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

54. Charge transport Jarvist Frost, James Kirkpatrick, Jenny Nelson Imperial College London

55. The waiting time before a hop from site i to a neighbouring site j is  ij = -1 ln(r) w ij where w ij 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

56. Ordered chains

57. Time of flight (ToF) experiment  = d  E

58. 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

59. Field parallel to the chains leads to higher mobility => Intra chain transfer dominates

60. Relaxed Geometry

61. Marcus theory Reorganisation energy intra = intra (A1) + intra (D2) J-L. Brédas et al Chemical Reviews 104 4971 (2004) Donor E QDQD 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 + ) QAQA QAQA ii i intra (A2) intra (A1) 1 2 E D + A + → D + + A

62. Transfer rates k DA = 2  V 2 exp - (  G + ) 2 ħ  (4  k B T) (4 k B T) Electronic coupling potential V from INDO  G is change in free energy from Density Functional Theory (B3LYP)

63. Simulated transient current 

64. Charge transfer in aligned PFO Hole mobility (cm 2 V -1 s -1 ) (Field) 1/2 (V 1/2 m -1/2 )

65. 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

66. Where next? Improved charge and exciton transfer and recombination rates Include triplet excitons Different morphologies Other systems eg display devices

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