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Are there ab initio methods to estimate the singlet exciton fraction in light emitting polymers ?

Are there ab initio methods to estimate the singlet exciton fraction in light emitting polymers ?. William Barford. n. R. R. n. Light emitting polymers. Electroluminescence discovered in semiconducting polymers in 1989 at the Cavendish Laboratory. Full colour spectrum. PPV:. PFO:.

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Are there ab initio methods to estimate the singlet exciton fraction in light emitting polymers ?

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  1. Are there ab initio methods to estimate the singlet exciton fraction in lightemitting polymers ? William Barford

  2. n R R n Light emitting polymers • Electroluminescence discovered in semiconducting polymers in 1989 • at the Cavendish Laboratory. Full colour spectrum PPV: PFO:

  3. Al, Ca, Mg polymer layer ITO glass substrate conduction band Device operation: (LUMO) electrons Ca E holes g exciton ITO valence band (HOMO) Light emitting polymer devices

  4. For a random injection of electron-hole pairs and spin independent recombination hs = 25%, as there are three spin triplets to every one spin singlet. Experimentally: £ h £ 20% 80% S Electro-luminescence quantum efficiency,

  5. Inter-conversion: transitions between states with the same spin Inter-system crossing: transitions between states with different spin

  6. What determines the singlet-exciton fraction ? • What are the electron-hole recombination processes ? • What is the rate limiting step in the generation of the lowest triplet • and singlet excitons ? • What are the inter-system crossing mechanisms at this rate limiting step ? • (Spin-orbit coupling or exciton dissociation.)

  7. + + + 1. Unbound electron-hole pair on neighbouring chains: _ _ _ 2. Electron-hole pair is captured to form a weakly bound ‘charge-transfer’ exciton: 3. Inter-conversion to a strongly bound exciton: 4. Singlet exciton decays radiatively: Inter-molecular recombination

  8. Centre-of-mass wavefunction Electron-hole pair wavefunction R 0.6 0.4 0.2 0 n = 1 intra-molecular lowest excitons or inter-molecular charge-transfer excitons n = 2 intra-molecular charge-transfer excitons -5 0 r/d 5 Energy j = 3 j = 2 j = 1 -0.2 -0.4 j = 3 j = 2 j = 1 Effective-particle model of excitons

  9. The model Intermediate, weakly bound, quasi-degenerate “charge-transfer” (SCT and TCT) states. • Efficient inter-system crossing between TCT and SCT. (by spin-orbit coupling or exciton disassociation). The charge-transfer states lie between the particle-hole continuum and the final, strongly bound exciton states, SX and TX. SX and TX are split by a large exchange energy. • Short-lived singlet C-T state (SCT) and long-lived triplet C-T state (TCT).

  10. Energy level diagram electron-hole continuum t t D S / T = “charge-transfer” singlet / triplet exciton (j = 1) S / T = “strongly-bound” singlet / triplet exciton (j = 1) CT CT ground-state X X

  11. Energy level diagram Classical rate equations: electron-hole continuum t t D S / T = “charge-transfer” singlet / triplet exciton (j = 1) S / T = “strongly-bound” singlet / triplet exciton (j = 1) CT CT ground-state X X

  12. a = 4: inter-system crossing via exciton dissociation a = 3: inter-system crossing via spin-orbit coupling 1 0.9 a = 4 0.8 0.7 a = 3 S 0.6 h 0.5 0.4 0.3 b = 0 b = 1 0.2 0 2 4 6 8 10 The singlet exciton fraction

  13. Determined by inter-molecular inter-conversion, which occurs via the electron transfer Hamiltonian, unperturbed Hamiltonian perturbation In the adiabatic approximation the electronic and nuclear degrees of freedom are described by the Born-Oppenheimer states: electronic eigenstate of (parametrized by a configuration coordinate, Q) nuclear (LHO) state Charge-transfer exciton life-times

  14. Transition rates are determined by the Fermi Golden Rule: The matrix elements are: electronic matrix element overlap of the vibrational wavefunctions Adiabatic (Born-Oppenheimer) energy surface n Energy 1 0 Q

  15. + + _ _ chain 1: Final state: chain 2: The electronic states Initial state: chain 1: chain 2:

  16. Assumptions of the model Electron transfer occurs between parallel polymer chains, and between nearest neighbour orbitals on adjacent chains This implies electronic selection rules for inter-molecular inter-conversion

  17. |n' – n| = even Intra-molecular Inter-molecular Energy n = 1 exciton n = 2 charge-transfer exciton j = 1 j = 1 n = 1 strongly bound exciton IC j = 1 Selection rules for inter-molecular inter-conversion • Preserves electron-hole parity, i.e. |n' – n| = even • "Momentum conserving", i.e. j' = j

  18. Chain 1 Chain 2 Energy From the conservation of energy: Vibrational wavefunction overlap: Franck-Condon factors

  19. Chain 1 Chain 2 Energy The polaron and exciton-polaron have similar relaxed geometries

  20. Multi-phonon emission IC VR re-organization energy Huang-Rhys factor Inter-conversion leaves chain 2 in the vibrational level of Subsequent vibrational relaxation with the emission of phonons

  21. Ratio of the rates is: electron-hole continuum t t where, The ratio of the rates is an increasing function of D when The ratio of the rates increases as decreases The inter-conversion rate

  22. Estimate of the singlet exciton ratio

  23. Inter-molecular states Intra-molecular states Energy Chain length dependence • For chain lengths < exciton radius the effective-particle model breaks down. • The "j' = j" selection rule breaks down. • Need to sum the rates for all the transitions.

  24. Conclusions • The singlet exciton fraction exceeds the spin-independent recombination value • of 25% in light-emitting polymers, because: • Intermediate inter-molecular charge-transfer (or polaron-pair) singlets are • short-lived, while charge-transfer triplets are long-lived. • This follows from the inter-conversion selection rules arising from the • exciton model and because the rates are limited by multi-phonon • emission processes. • The inter-system crossing time between the triplet and singlet charge transfer • states is comparable to the life-time of the CT triplet.

  25. The theory predicts that the singlet exciton fraction should increase with • chain length, because the exciton model becomes more valid and the • Huang-Rhys parameters decrease. • The theory suggests strategies for enhancing the singlet exciton fractions: • Well-conjugated, closed-packed, parallel chains. • The theory needs verifying by performing calculations on realistic systems, • i.e. finite length oligomers with arbitrary conformations.

  26. Required Computations • Electronic matrix elements between constrained excited states: • Polaron relaxation energies. • Spin-orbit coupling matrix elements:

  27. Possible ab initio methods ? • Time dependent DFT: doesn’t work for ‘extended’ systems. • DFT-GWA-BSE method: successful, but very expensive. • RPA (HF + S-CI): HOMO-LUMO gaps are too large. • Diffusion Monte Carlo: ?

  28. Estimate of the inter-system crossing rate • Emission occurs from the "triplet" exciton because it acquires singlet character • from the "singlet" exciton induced by spin-orbit coupling. 2. The life-times can be used to estimate the matrix element of the spin-orbit coupling, W: 3. The ISC rate between the charge-transfer states is,

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