Simulating charge transfer in (organic) solar cells

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1 Simulating charge transfer in (organic) solar cells July 24, 2013

2 Organic solar cells 1 c Fraunhofer ISE

3 Organic solar cells Step 1: Microscopic structure

4 Organic solar cells 1 c Fraunhofer ISE

5 Organic solar cells 1 c Fraunhofer ISE

6 Organic solar cells 1 Ruderer, Meier, Porcar, Cubitt, Müller-Buschbaum, JPCL. 3, 683 (2012)

7 Organic solar cells 1 Ruderer, Meier, Porcar, Cubitt, Müller-Buschbaum, JPCL. 3, 683 (2012)

8 Organic solar cells

9 Organic solar cells

10 Loss mechanisms Efficiency of solar electrical energy conversion very low. Most prominent loss mechanisms: Exciton dissociation: electron/hole pair recombines instead of dissociating into individual charges.

11 Loss mechanisms Efficiency of solar electrical energy conversion very low. Most prominent loss mechanisms: Exciton dissociation: electron/hole pair recombines instead of dissociating into individual charges. Charge recombination: charges recombine instead of moving to their respective electrodes.

12 Loss mechanisms Efficiency of solar electrical energy conversion very low. Most prominent loss mechanisms: Exciton dissociation: electron/hole pair recombines instead of dissociating into individual charges. Charge recombination: charges recombine instead of moving to their respective electrodes. Charge mobility: charges only move slowly through the cell.

13 Loss mechanisms Efficiency of solar electrical energy conversion very low. Most prominent loss mechanisms: Exciton dissociation: electron/hole pair recombines instead of dissociating into individual charges. Charge recombination: charges recombine instead of moving to their respective electrodes. Charge mobility: charges only move slowly through the cell.

14 The Aim Predict electron mobility in a macroscopic fullerene crystal.

15 Fullerene Crystals C60 Below 250K, C 60 crystals are stable in FCC configuration and exhibit no special features.

16 Fullerene Crystals C60 Below 250K, C 60 crystals are stable in FCC configuration and exhibit no special features. Over 250K every C 60 molecules pseudo-rotates at its site. Simulations need to consider these different regimes.

17 Fullerene Crystals Phenyl-C61-butyric acid methyl ester (PCBM) At room temperature crystal structure depends on preparation method. Observed PCBM crystals are: SC BCC Hexagonal Monoclinic Triclinic

18 The Approach The mobility of an electron can be defined as the derivative of the drift velocity v with respect to the applied external field E: µ ij = v i(e) E j

19 The Approach The mobility of an electron can be defined as the derivative of the drift velocity v with respect to the applied external field E: µ ij = v i(e) E j need a way to calculate field dependent electron velocities.

20 The Approach The mobility of an electron can be defined as the derivative of the drift velocity v with respect to the applied external field E: µ ij = v i(e) E j need a way to calculate field dependent electron velocities. how do electrons move?

21 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping.

22 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping. A polaron is a quasi-particle composed of a charge and its accompanying polarisation field. In our case: charge polarisation of surroundings stabilises charge

23 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping. A polaron is a quasi-particle composed of a charge and its accompanying polarisation field. In our case: charge polarisation of surroundings stabilises charge Polaron in C60

A polaron is a quasi-particle composed of a charge and its accompanying

24 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping. A polaron is a quasi-particle composed of a charge and its accompanying polarisation field. In our case: charge polarisation of surroundings stabilises charge Polaron in C60 Polaron in water

25 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping. Hypothesis II: Electrons are de-localised in bands and move via band-like conduction.

26 How do Electrons move Hypothesis I: Electrons are localised in the form of polarons and move via hopping. Hypothesis II: Electrons are de-localised in bands and move via band-like conduction. Wenzien, Käckell, Bechstedt, Cappellini, PRB (1995) Semiconductor band structure

27 Hopping Hopping model: Need derivative of average drift velocity v. v E

28 Hopping Hopping model: Need derivative of average drift velocity v. v Within hopping model this is given as the sum of all forward and backward hops in each direction. Determined by rate k l and hopping distance d l E

29 Hopping Hopping model: Need derivative of average drift velocity v. v Within hopping model this is given as the sum of all forward and backward hops in each direction. Determined by rate k l and hopping distance d l E µ ij = v i E j = l k l E j d li Can be solved analytically for crystals.

30 Hopping Within the hopping model we need to be able to predict electron transfer rates 1 between nearest and next-nearest neighbours. 1 H. Oberhofer and J. Blumberger Angew. Chem. Int. Ed (2010)

31 Hopping Within the hopping model we need to be able to predict electron transfer rates 1 between nearest and next-nearest neighbours. e 1 H. Oberhofer and J. Blumberger Angew. Chem. Int. Ed (2010)

32 Rate equations Rate expressions of the form: k = A e G/k BT 1 Brunschwig, Logan, Newton, Sutin, JACS, 102, 5798 (1980) 2 R. A. Marcus, Rev. Mod. Phys., 65, 599 (1993)

33 Rate equations Rate expressions of the form: k = A e G/k BT semi-classical Landau Zener transition state theory: A = κ el (H ab,λ,ν n ;T)ν n, G = E (H ab,λ, G) 1 Brunschwig, Logan, Newton, Sutin, JACS, 102, 5798 (1980) 2 R. A. Marcus, Rev. Mod. Phys., 65, 599 (1993)

34 Rate equations Rate expressions of the form: k = A e G/k BT semi-classical Landau Zener transition state theory: A = κ el (H ab,λ,ν n ;T)ν n, G = E (H ab,λ, G) non-adiabatic (Marcus) rate: A H ab 2 (λk B T) 1/2, G = E valid for activated processes where λ >> H ab 1 Brunschwig, Logan, Newton, Sutin, JACS, 102, 5798 (1980) 2 R. A. Marcus, Rev. Mod. Phys., 65, 599 (1993)

35 Rate equations Rate expressions of the form: k = A e G/k BT semi-classical Landau Zener transition state theory: A = κ el (H ab,λ,ν n ;T)ν n, G = E (H ab,λ, G) non-adiabatic (Marcus) rate: A H ab 2 (λk B T) 1/2, G = E valid for activated processes where λ >> H ab adiabatic rate: A = ν n, G = E (H ab,λ, G) valid for κ el 1 ( H ab 2 >> hν n λkb T) 1 Brunschwig, Logan, Newton, Sutin, JACS, 102, 5798 (1980) 2 R. A. Marcus, Rev. Mod. Phys., 65, 599 (1993)

36 Example: Marcus Theory Marcus theory 1 of electron transfer gives a configuration dependent rate: k ET (r) = 1 h Hab (r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT

37 Example: Marcus Theory Marcus theory 1 of electron transfer gives a configuration dependent rate: k ET (r) = 1 h Hab (r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT Has to be calculated for every crystallographic direction and depending on the electric field E. 1 R. A. Marcus, J. Chem. Phys (1956)

38 Example: Marcus Theory Marcus theory 1 of electron transfer gives a configuration dependent rate: k ET (r) = 1 h Hab (r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT Has to be calculated for every crystallographic direction and depending on the electric field E. 1 R. A. Marcus, J. Chem. Phys (1956)

39 Example: Marcus Theory k ET (r) = 1 h H ab(r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT 1 R. A. Marcus, J. Chem. Phys (1956)

40 Example: Marcus Theory k ET (r) = 1 h H ab(r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT H ab the electronic transition matrix element accurate calculation of diabatic energies 1 R. A. Marcus, J. Chem. Phys (1956)

41 Example: Marcus Theory k ET (r) = 1 h H ab(r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT H ab the electronic transition matrix element accurate calculation of diabatic energies the reorganisation free energy λ sampling of diabatic states for a given r 1 R. A. Marcus, J. Chem. Phys (1956)

42 Example: Marcus Theory k ET (r) = 1 h H ab(r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT H ab the electronic transition matrix element accurate calculation of diabatic energies the reorganisation free energy λ sampling of diabatic states for a given r Driving force G energy difference of diabatic states 1 R. A. Marcus, J. Chem. Phys (1956)

43 Example: Marcus Theory k ET (r) = 1 h H ab(r) 2 (4πλ(r)k B T) 1/2 e (λ(r) G(r))/4λ(r)k BT Free energy Reaction coordinate 1 R. A. Marcus, J. Chem. Phys (1956)

44 Example: Marcus Theory Free energy Reaction coordinate Diabatic states: Charges localised on either Donor or Acceptor molecules.

45 Concerning the reaction coordinate Charge of Donor or acceptor is not a good reaction coordinate! Actual charge hopping not the slow process.

46 Concerning the reaction coordinate Charge of Donor or acceptor is not a good reaction coordinate! Actual charge hopping not the slow process. Reorganisation of surroundings is slow. Idea: Use energy difference of charge states.

47 Concerning the reaction coordinate Charge of Donor or acceptor is not a good reaction coordinate! Actual charge hopping not the slow process. Reorganisation of surroundings is slow. Idea: Use energy difference of charge states. Electron localised on a site, surroundings polarised but not necessarily in minimum

48 Concerning the reaction coordinate Charge of Donor or acceptor is not a good reaction coordinate! Actual charge hopping not the slow process. Reorganisation of surroundings is slow. Idea: Use energy difference of charge states. Electron jumps de Surroundings still polarised for old charge state

49 How to calculate the parameters Need to construct the charge localised states. DFT suffers from the so called charge delocalisation error.

50 How to calculate the parameters Need to construct the charge localised states. DFT suffers from the so called charge delocalisation error. Instead of this:

51 How to calculate the parameters Need to construct the charge localised states. DFT suffers from the so called charge delocalisation error. DFT will always give this (even with hybrid functionals):

52 Methods To circumvent the delocalisation error we use: Constrained DFT: B. Kaduk, T. Kowalczyk, and T. Van Voorhis, Chem. Rev (2011) H. Oberhofer and J. Blumberger J. Chem. Phys (2009) H. Oberhofer and J. Blumberger J. Chem. Phys (2010)

53 Methods To circumvent the delocalisation error we use: Constrained DFT: B. Kaduk, T. Kowalczyk, and T. Van Voorhis, Chem. Rev (2011) H. Oberhofer and J. Blumberger J. Chem. Phys (2009) H. Oberhofer and J. Blumberger J. Chem. Phys (2010) Fragment orbital DFT H. Oberhofer and J. Blumberger Angew. Chem. Int. Ed (2010) K. Senthilkumar, F. C. Grozema, F. M. Bickelhaupt, L. D. A. Siebbeles, J. Chem. Phys (2003) A. Farazdel, M. Dupuis, E. Clementi, A. Aviram, JACS (1990)

54 Constrained Density Functional Theory w A,D = w D w A = i D ρ i(r R i ) i A ρ i(r R i ) N i=1 ρ i(r R i ) A constraint on charges takes the form 1 w(r)ρ(r) dr N c = 0 1 Q. Wu and T. van Voorhis, Phys. Rev. A (2005)

55 Constrained Density Functional Theory w A,D = w D w A = i D ρ i(r R i ) i A ρ i(r R i ) N i=1 ρ i(r R i ) A constraint on charges takes the form 1 w(r)ρ(r) dr N c = 0 With a new energy functional F[ρ,V] = E[ρ]+V( w(r)ρ(r) dr N c ) 1 Q. Wu and T. van Voorhis, Phys. Rev. A (2005)

56 Constrained Density Functional Theory w A,D = w D w A = i D ρ i(r R i ) i A ρ i(r R i ) N i=1 ρ i(r R i ) A constraint on charges takes the form 1 w(r)ρ(r) dr N c = 0 With a new energy functional F[ρ,V] = E[ρ]+V( w(r)ρ(r) dr N c ) The matrix element is then given by: H ab ψ a H KS ψ b = F B ψ a ψ b V B ψ a w ψ b 1 Q. Wu and T. van Voorhis, Phys. Rev. A (2005)

57 Constrained Density Functional Theory w A,D = w D w A = i D ρ i(r R i ) i A ρ i(r R i ) N i=1 ρ i(r R i ) 1 H. Oberhofer and J. Blumberger J. Chem. Phys (2009)

58 Fragment Orbital DFT Separate donor and Acceptor group and calculate the HOMO s of the charged groups separately.

59 Fragment Orbital DFT Separate donor and Acceptor group and calculate the HOMO s of the charged groups separately.

60 Fragment Orbital DFT Separate donor and Acceptor group and calculate the HOMO s of the charged groups separately. H ab is the off-diagonal Kohn-Sham matrix element of the two HOMO s. H ab Φ HOMO a HKS Φ HOMO b

61 Reorganisation energy Diabatic state a

62 Reorganisation energy Electron jumps Surroundings still polarised for old charge state

63 Reorganisation energy Electron jumps atic state Surroundings reorganise

64 Driving force of the hopping reaction In an ideal bravais crystal all sites are equivalent. Only with an external potential G is non-zero. Given an external Field E the energy difference between two lattice sites A and B is simply: G A,B = ee.(r B r A ) = ee.d AB A B E

65 Some results for hopping in Fullerene crystals H. Oberhofer and J. Blumberger, Phys. Chem. Chem. Phys (2012) F. Gajdos, H. Oberhofer, M. Dupuis, and J. Blumberger J. Phys. Chem. Lett (2013)

66 Results Distribution of (nearest) site-to-site transition matrix elements in crystal Sampled over equidistributed configurations Gaussian distribution arises from different relative orientations of molecules.

67 Results Where do these variations come from?

68 Results Temperature dependence of mobilities for different rate equations. Experimental value µ(t = 300K) = 0.5cm 2 /s/v 1 1 Frankevich, Maruyamaa, Ogataa, CPL , (1993)

69 Rate equations Rate expressions of the form: k = A e G/k BT semi-classical Landau Zener transition state theory: A = κ el (H ab,λ,ν n ;T)ν n, G = E (H ab,λ, G) non-adiabatic (Marcus) rate: A H ab 2 (λk B T) 1/2, G = E valid for activated processes where λ >> H ab adiabatic rate: A = ν n, G = E (H ab,λ, G) valid for κ el 1 ( H ab 2 >> hν n λkb T) 1 Brunschwig, Logan, Newton, Sutin, JACS, 102, 5798 (1980) 2 R. A. Marcus, Rev. Mod. Phys., 65, 599 (1993)

70 Some words on the hopping model adiabatic or non-adiabatic? Adiabaticity is controlled by ratio: 2πγ 1 : adiabatic regime 2πγ 1 : non-adiabatic regime here 2πγ = 0.65! 2πγ = π3/2 H ab 2 hν n kb Tλ

71 Some words on the hopping model adiabatic or non-adiabatic? neither Are there even charge localised states?

72 Some words on the hopping model adiabatic or non-adiabatic? neither Are there even charge localised states? Diabatic states exist if there is a finite barrier separating the sites: In our picture that means H ab 3λ/8

73 Some words on the hopping model adiabatic or non-adiabatic? neither Are there even charge localised states? Diabatic states exist if there is a finite barrier separating the sites: In our picture that means H ab 3λ/8 For 5% of all configurations there is no barrier. Considering nuclear quantum effects (zero point energy) barrier even lower.

74 Some words on the hopping model adiabatic or non-adiabatic? neither Are there even charge localised states? Diabatic states exist if there is a finite barrier separating the sites: In our picture that means H ab 3λ/8 large barrier localised states small barrier slightly delocalised states no barrier no localised states

75 Some words on the hopping model adiabatic or non-adiabatic? neither Are there even charge localised states? No, due to low reorganisation energy and high H ab Hopping models based on localised site-to-site rates only suitable as first approximation. Can yield a starting point for future investigations.

76 A better way to simulate charge transport Direct propagation of a model Hamiltonian parametrised from ab-initio calculations A. Troisi J. Chem. Phys (2011) F. Gajdos, M. Dupuis, and J. Blumberger in preparation

77 Simple Example Assume a time independent Hamiltonian. no coupling between electron and nuclear motion (not really correct, see above)

78 Simple Example Assume a time independent Hamiltonian. no coupling between electron and nuclear motion (not really correct, see above) Write Hamiltonian in basis of site localised states (these correspond to diabatic states)

79 Simple Example Assume a time independent Hamiltonian. no coupling between electron and nuclear motion (not really correct, see above) Write Hamiltonian in basis of site localised states (these correspond to diabatic states)

80 Simple Example Assume a time independent Hamiltonian. no coupling between electron and nuclear motion (not really correct, see above) Write Hamiltonian in basis of site localised states (these correspond to diabatic states) Model Hamiltonian: off diagonal elements = H ab diagonal elements = Eigenenergies of diabatic states

81 Simple Example Solution to i Ψ(r,t) = HΨ(r,t) t Choose initial state Ψ I (r,t 0 )

82 Simple Example Solution to i Ψ(r,t) = HΨ(r,t) t Choose initial state Ψ I (r,t 0 ) SolveHΦ i (r) = ε i Φ i (r)fortimeindependenteigenvaluesε i andeigenfunctions Φ i (r)

83 Simple Example Solution to i Ψ(r,t) = HΨ(r,t) t Choose initial state Ψ I (r,t 0 ) SolveHΦ i (r) = ε i Φ i (r)fortimeindependenteigenvaluesε i andeigenfunctions Φ i (r) Numerically propagate Ψ I (r,t 0 ) for a time t: Ψ(r,t) = i Φ i(r) Ψ I (r,t 0 ) e iε i(t t 0 )/ Φ i (r)

84 Simple Example Solution to i Ψ(r,t) = HΨ(r,t) t Choose initial state Ψ I (r,t 0 ) SolveHΦ i (r) = ε i Φ i (r)fortimeindependenteigenvaluesε i andeigenfunctions Φ i (r) Numerically propagate Ψ I (r,t 0 ) for a time t: Ψ(r,t) = i Φ i(r) Ψ I (r,t 0 ) e iε i(t t 0 )/ Φ i (r) Work in progress for solar cells!

85 Simple Example 1D Example: Propagation of an excited core electron in LiCN 1 M. Ludwig, internship report (2013)

86 Simple Example 1D Example: Propagation of an excited core electron in LiCN Time dependent states time series of atomic charges can follow the movement of the electron through the molecule 1 M. Ludwig, internship report (2013)

87 Simple Example What is missing (and some ideas how to proceed): Electric field modify site energies Absorbing boundary conditions imaginary site energies Movement of the atoms force field or Born Oppenheimer DFT Coupling of electronic and nuclear motion need non-adiabatic coupling element ψ j (r,r) R ψ i (r,r)

Supporting Information for Electron transport in crystalline PCBM-like fullerene derivatives: a comparative computational study

Electronic Supplementary Material (ESI) for Journal of Materials Chemistry C. This journal is The Royal Society of Chemistry 2014 Supporting Information for Electron transport in crystalline PCBM-like