william barford are there ab initio methods to estimate the singlet exciton fraction in light...
TRANSCRIPT
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