1. Are there ab initio methods to estimate the singlet exciton fraction in light emitting polymers ?
William Barford
The title of my talk is “What determines the electroluminesence efficiency of light-emitting polymers”. The sub-title is “What is the singlet exciton fraction”, because, as I will explain, this is a key factor in determining the overall efficiency.
In fact, the singlet exciton fraction has been an important topic in the physics of light-emitting polymers for the last few years, because it’s value is highly controversial both from an experimental and theoretical point of view.
So, in this talk I’m going to describe a theory to predict it’s value. First, however, I want to give a brief introduction to LEPs and the operation of LEP devices.

2. Light emitting polymers
Electroluminescence discovered in semiconducting polymers in 1989
at the Cavendish Laboratory.
Full colour spectrum n
PPV:
Electroluminescence was discovered (accidentally) in phenyl-based organic semiconductors in 1989 in the Cavendish Lab. Most LEPs are characterized by being conjugated polymers containing phenyl rings.
By tailoring the chemical structure it is now possible to have the full colour spectrum of R, G & B. n R
PFO:

3. Light emitting polymer devices
glass substrate
ITO
polymer layer
Al, Ca, Mg
exciton
holes
electrons E g
conduction band
(LUMO)
valence band
(HOMO) Ca
ITO
Device operation:
In this talk I’m going to concentrate on the performance of light-emitting devices.
A typical device consists of a polymer thin film sandwiched between 2 electrodes and mounted on a glass substrate. ITO is transparent.
The electrodes have different workfunctions and under a suitable bias electrons are injected into the conduction band and holes are injected into the valence band.
Under the influence of the electric field the electrons and holes drift across the polymer film. When sufficiently close an electron-hole pair will be captured to form a bound state, or exciton, which fairly rapidly recombines to the ground state. However, only spin 0 (or spin singlet) excitons decay radiatively. Spin 1 (or spin triplet) excitons decay non-radiatively, by emitting phonons to the environment, which eventually is wasted as heat.

4. Electro-luminescence quantum efficiency,
The figure of merit that defines the efficiency of LEP devices is the electroluminescence quantum efficiency.
Only singlet excitons recombine radiatively.
The lowest singlet exciton is usually either dipole active (as in LEPs) or not (as in trans-PA). So this factor is either 1 or 0.
A random injection of electron-hole pairs, followed by spin-independent recombination => a singlet exciton fraction of 25%.
The light-outcoupling is < 20% because of losses due to internal reflection. Energy is lost by the creation of surface plasmon modes at the electrodes.
Note, that there is no direct way of measuring the singlet fraction.
Notice that if the \eta_S is 25%, \eta_{EL} < 5% - which although useful for LEDs, is far too low an efficiency LEPs are to be used as efficient light sources to replace incandescent light sources.
We need to increase both \eta_S and the light-outcoupling.
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:
80%
20% S h

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. Inter-molecular recombination
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: + _
To understand why the singlet might exceed the statistical limit we need to understand the electron-hole recombination mechanisms. I’m going to propose that the e-h recombination is a 4-stage process.
First, consider inter-molecular e-h recombination.
First the e-h will be polarons on neighbouring polymer chains, and so far apart that their thermal energy exceeds their Coulomb attraction.
Very rapidly they will be captured when their separation is such that their Coulomb attraction exceeds their thermal energy. This weakly bound captured state is called a charge-transfer state (or a bound P+ - P- pair).
This state then undergoes a transition to the lowest, strongly bound exciton on the same polymer chain,
which subsequently decays to the ground state, emitting a photon if it’s a singlet, and phonons if it’s a triplet.
Now, the crucial step is the inter-conversion from the C-T state to the exciton. If this is fast for the singlet and slow for the triplet and if there is reasonably efficient conversion of triplet to singlet CT states (ISC), then it’s possible to increase the singlet fraction. Fairly rapid ISC is possible because the C-T states are quasi-degenerate.
I’m going to show why these conditions are satisfied.
Note that ISC cannot take place between the strongly bound (lowest energy) singlet and triplet excitons because of the large exchange energy.
4. Singlet exciton decays radiatively:

8. Effective-particle model of excitons
-0.4
-0.2
0.2
0.4
0.6 -5 5
r/d
Electron-hole pair wavefunction
Centre-of-mass wavefunction R
j = 3
j = 2
j = 1
n = 2
intra-molecular
charge-transfer excitons
n = 1
lowest excitons or
inter-molecular
charge-transfer
excitons
Energy
I’d now like to explain more quantitatively what I mean by strongly bound and charge-transfer excitons by briefly reviewing the effective-particle theory of excitons in conjugated polymers.
The overall exciton wavefunction is a product of the wavefunction of the e-h pair (labelled by n) and the centre-of-mass wavefunction (labelled by j).
There are families of excitons with the same principal quantum number, n, each branch labelled by j, the pseudo-momentum quantum number).
Bands of exciton states.
Intra-molecular: n = odd => even parity (singlet and triplets split by an exchange interaction)
n = even => odd parity (singlet and triplets are degenerate), because the exchange interaction is local.
Inter-molecular: n = odd => even parity (singlet and triplets degenerate)

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. electron-hole continuum
Energy level diagram
S / T = “charge-transfer” singlet / triplet exciton (j = 1)
S / T = “strongly-bound” singlet / triplet exciton (j = 1) X CT
electron-hole continuum
ground-state D t
The energy level diagram shows the energy levels and life-times of the respective states for PPV-DOO. Note that there is a large exchange energy between the lowest singlet and triplet states (ca. 0.7 eV), whereas the charge-transfer states are quasi-degenerate. This large exchange energy is the reason for the different life-times of the charge-transfer states, as their inter-conversion to the excitons is determined by multi-phonon emission (as I'll show presently).
The classical rate equations can be solved under steady state conditions to give the singlet fraction.

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

12. The singlet exciton fraction
a = 4: inter-system crossing via exciton dissociation
a = 3: inter-system crossing via spin-orbit coupling
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 2 4 6 8 10 h S
b = 0
b = 1
a = 4
a = 3
Definition of the singlet fraction. It is determined by the ratio of the life-times of the charge-transfer states and the ISC time.
Beta = 0
The graph shows that as gamma -> 0 the ratio -> 1 or Conversely, as gamma -> infinity the ratio -> ¼.
We need to be able to predict the life-times of the charge-transfer states.

13. Charge-transfer exciton life-times
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:
nuclear (LHO) state
electronic eigenstate of
(parametrized by a configuration coordinate, Q)

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

15. The electronic states Initial state: + _ Final state: _ + chain 1:
Initial state: n = 1, j = 1 inter-molecular charge-transfer state
Final state: n and j undefined intra-molecular state.

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. Selection rules for inter-molecular inter-conversion
|n' – n| = even
Preserves electron-hole parity, i.e. |n' – n| = even
"Momentum conserving", i.e. j' = j
j = 1
n = 1
exciton
strongly bound exciton
Energy
n = 2
charge-transfer exciton
Inter-molecular
Intra-molecular IC
IC from inter-molecular n= 1, j = 1 to intra-molecular n = 1, j = 1 (and not to higher-lying momentum states).

18. Vibrational wavefunction overlap: Franck-Condon factors
Energy
Chain 1
From the conservation of energy:
Chain 2
Now we need to consider the overlap of the vibrational wavefunctions for each chain.

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

20. Multi-phonon emissionVR IC
Huang-Rhys factor
re-organization
energy
F is a Poisson distribution, maximized when \nu_{GS} = S_p. Thus, if S_p < 1, F decreases as \nu increases > 1.
Inter-conversion leaves chain 2 in the vibrational level of
Subsequent vibrational relaxation with the emission of phonons

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

22. Estimate of the singlet exciton ratio

23. 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.
Inter-molecular states
Energy
Intra-molecular states

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.
The life-times can be used to estimate the matrix element of the spin-orbit
coupling, W:
Using perturbation theory to calculate the mixing of the singlet and triplet states and the Einstein equations for spontaneous emission thhe fluorescent lifetime of S1 and the phosophorescent lifetime of T1 can be used to predict W.
3. The ISC rate between the charge-transfer states is,