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
