William Barford

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

emitting polymers ?

• Electroluminescence discovered in semiconducting polymers in 1989

at the Cavendish Laboratory.

n

n

RR

PPV:

PFO:

Full colour spectrum

Light emitting polymers

Light emitting polymer devices

exciton

holes

electrons

Eg

conduction band

(LUMO)

valence band(HOMO)

Ca

ITO

Device operation:

glass substrateITO

polymer layer

Al, Ca, Mg

Electro-luminescence quantum efficiency, EL

For a random injection of electron-hole pairs and spin independentrecombination s = 25%, as there are three spin triplets to every one spin singlet.

Experimentally: 80%20% S

photons emitted #

photons detected #

%201~

pairs hole

-electron injected #

excitons #

fraction

exciton singlet S

excitons #

excitonssinglet #

1~

excitonssinglet #

photons emitted #

pairs hole-electron injected #

photons detected #EL

Inter-conversion: transitions between states with the same spin

Inter-system crossing: transitions between states with different spin

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.)

Inter-molecular recombination

4. Singlet exciton decays radiatively:

h

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:

+ _

j = 3j = 2j = 1

j = 3j = 2j = 1

n = 2intra-molecularcharge-transfer excitons

n = 1intra-molecular lowest excitons orinter-molecularcharge-transferexcitons

Ene

rgy

)()(),( RrRr jnnj

Effective-particle model of excitons

)(1 rn)(2 rn

-0.4

-0.2

0

0.2

0.4

0.6

-5 0 5r/d

Electron-hole pair wavefunctionhe rrr

Centre-of-mass wavefunction

R

)(1 Rj

)(2 Rj)(3 Rj

2he rr

R

The model

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.

Intermediate, weakly bound, quasi-degenerate “charge-transfer” (SCT and TCT) states.

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).

Energy level diagram

S / T = “charge-transfer” singlet / triplet exciton (j = 1)

S / T = “strongly-bound” singlet / triplet exciton (j = 1)X

CT

X

CT

electron-hole continuum

ground-state

CTS

XS

CTT

XT

CTS

XS

CTT

XT

ISC

CTCT

CTCT

SISCS

ISC

TS NNN

dt

dN

11

4

CTCT

CTCT

TISCT

ISC

ST NNN

dt

dN

11

4

3

XCT

CTX

S

SX

S

SS NN

dt

dN

X

X

CT

CTX

T

T

T

TT NN

dt

dN

Classical rate equations:Energy level diagram

S / T = “charge-transfer” singlet / triplet exciton (j = 1)

S / T = “strongly-bound” singlet / triplet exciton (j = 1)X

CT

X

CT

electron-hole continuum

ground-state

CTS

XS

CTT

XT

CTS

XS

CTT

XT

ISC

The singlet exciton fraction

)1(4//

/

XXXX

XX

TTSS

SSS NN

N

CT

CT

CT

T

ISC

T

S

0, = 4: inter-system crossing via exciton dissociation

= 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

0 2 4 6 8 10

S

= 0= 1

CTTISC /

Charge-transfer exciton life-times Determined by inter-molecular inter-conversion, which occurs via the electron transfer Hamiltonian, H

HHH 0

unperturbed Hamiltonian perturbation

In the adiabatic approximation the electronic and nuclear degrees of freedomare described by the Born-Oppenheimer states:

nuclear (LHO) state electronic eigenstate of (parametrized by a configuration coordinate, Q)

0H

aQa ;A

Transition rates are determined by the Fermi Golden Rule:

)(2 2

FIFI EEFHIk

overlap of the vibrational wavefunctions

The matrix elements are:

fifHiFHI

electronic matrix elementE

nerg

y

Q

0

1

Adiabatic (Born-Oppenheimer) energy surface

)(QEi

fQiQ

)(QE f

+

_

chain 1:

chain 2:

The electronic states

11~,

jnPPi Initial state:

chain 1:

chain 2:

+ _

Final state:)2()1(

'')2()1( GSGSEXf jn

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

Selection rules for inter-molecular inter-conversion

• Preserves electron-hole parity, i.e. |n' – n| = even• "Momentum conserving", i.e. j' = j

dRRRdrrrtfHi jjnn )()()()(~'1'1

|n' – n| = even jj '

drrrt nn )()(~1'1

j = 1

j = 1

n = 1exciton

n = 1strongly bound exciton

Ene

rgy

j = 1

n = 2charge-transfer exciton

Inter-molecular Intra-molecular

IC

Vibrational wavefunction overlap: Franck-Condon factors

2)1(0 0 EXPEX

F E

nerg

y

01E

EXE1

1Q

)(1 EXE

1Q

)(1PE EX

Chain 1

P0

From the conservation of energy: GSEX EE 21

GSE202E

2Q

)(2PE

GS

)(2 GSE

2)2(0 0 GSPGS

F

Chain 2

P0

Chain 1

Ene

rgy

01EEXE1

1Q

)(1 EXE

01 Q

)(1PE

The polaron and exciton-polaron have similar relaxed geometries

012 EE GS

01 Q

Chain 2

GSE2

2Q

)(2 GSE

)(2PE

GS

EXEXEXEXFQ 00 0

)1(01

Inter-conversion leaves chain 2 in the vibrational level of

GS

GS

Subsequent vibrational relaxation with the emission of phonons

GS

!

)exp()2(0

GS

ppGS

GS

SSF

Huang-Rhys factor re-organization energy

/EXCTGS EE

Multi-phonon emission

)1( / dp ES

2Q

)(2 GSE

)(2PE

GSVR

dE

IC

The inter-conversion rate )(

!

)exp()()(~2 2

1'1 ESS

drrrtk fGS

ppnnFI

GS

!

!)(

S

Tp

TT

SSST

XCT

XCT Sk

k

Ratio of the rates is:

where,

electron-hole continuum

CTS

XS

CTT

XT

CTS

XS

CTT

XT

ISCS

T

)( ST

/XCT SSS EE

/XCT TTT EE

The ratio of the rates is an increasing function of when

The ratio of the rates increases as decreases

1pS

pS

Estimate of the singlet exciton ratio

4/))()(( XCTS SESE

5.7/))()(( XCTT TETE

1 eV, 2.0 eV, 1.0 ,4 pStAd

fs 500~CTS

ns 3~CTT

ns 3.01.0~ ISC

%70 S

Inter-molecular states

Ene

rgy

Intra-molecular states

• 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.

Chain length dependence

Conclusions

The singlet exciton fraction exceeds the spin-independent recombination valueof 25% in light-emitting polymers, because:

1. 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 theexciton model and because the rates are limited by multi-phonon

emission processes.

2. The inter-system crossing time between the triplet and singlet charge transfer states is comparable to the life-time of the CT triplet.

• 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.

• 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.

Required Computations

1. Electronic matrix elements between constrained excited states:

2. Polaron relaxation energies.

3. Spin-orbit coupling matrix elements:

PPHEXGS ,

CTSOCT THS

Possible ab initio methods ?

1. Time dependent DFT: doesn’t work for ‘extended’ systems.

2. DFT-GWA-BSE method: successful, but very expensive.

3. RPA (HF + S-CI): HOMO-LUMO gaps are too large.

4. Diffusion Monte Carlo: ?

Estimate of the inter-system crossing rate

1. 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,

2

23

W

E

E

X

X

X

X

S

T

T

S

110

2

10

)(||2

s

EWk fISCCT