Transcript The singlet exciton fraction in conjugated polymers

Are there ab initio methods to estimate
the singlet exciton fraction in light
emitting polymers ?
William Barford
Light emitting polymers
• Electroluminescence discovered in semiconducting polymers in 1989
at the Cavendish Laboratory.
Full colour spectrum
PPV:
n
PFO:
n
R
R
Light emitting polymer devices
Al, Ca, Mg
polymer layer
ITO
glass substrate
conduction band
(LUMO)
Device operation:
holes
electrons
Ca
Eg
exciton
ITO
valence band
(HOMO)
Electro-luminescence quantum efficiency,
 EL


# detected photons

#
injected
electron
hole
pairs


 EL  




# excitons

  # singlet excitons   # emitted photons   # detected photons 





# excitons
# injected electron -  
  # singlet excitons   # emitted photons 


hole pairs


~1
S  singletexciton
~1
 20%
fraction
For a random injection of electron-hole pairs and spin independent
recombination s = 25%, as there are three spin triplets to every one spin singlet.
Experimentally: 20%   S  80%
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
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:
h
Effective-particle model of excitons
 nj (r , R)   n (r ) j ( R)
Electron-hole pair wavefunction
r  re  rh
Centre-of-mass wavefunction
R
0.6
re  rh
2
 j1 ( R)
 n1 (r )
0.4  n 2 ( r )
0.2
0
-5
0
r/d
5
-0.2
R
 j3 ( R)
Energy
-0.4
n=2
intra-molecular
charge-transfer
excitons
 j2 ( R)
j=3
j=2
j=1
j=3
j=2
j=1
n=1
intra-molecular
lowest excitons
or
inter-molecular
charge-transfer
excitons
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).
Energy level diagram
electron-hole continuum
SCT
t
t SCT
t
t ISC
TCT
t TCT
SX
D
TX
t SX
t TX
ground-state
SCT / TCT= “charge-transfer” singlet / triplet exciton (j = 1)
SX / TX = “strongly-bound” singlet / triplet exciton (j = 1)
Energy level diagram
Classical rate equations:
electron-hole continuum
SCT
t
t SCT
t
t ISC
dN SCT
TCT
dNTCT
t TCT
SX
dt
D
TX
t SX
dt
 1
N NTCT
1


 N SCT 

 t ISC t S
4t t ISC

CT




 1
3N N SCT
1


 NTCT 

 t ISC t T
4t t ISC

CT




dNTX
dt

NTCT

NTX

N SCT

N SX
t TCT
t TX
t TX
dN S X
dt
ground-state
SCT / TCT= “charge-transfer” singlet / triplet exciton (j = 1)
SX / TX = “strongly-bound” singlet / triplet exciton (j = 1)
t SCT
t SX
The singlet exciton fraction
N S X /t S X

 
4(1     )
 = 4: inter-system crossing via exciton dissociation

t SCT
 0,
t TCT

t ISC
t TCT
S 
N S X / t S X  NTX / t TX
 = 3: inter-system crossing via spin-orbit coupling
1
0.9
4
0.8

0.7
3
0.6
0.5
0.4
0.3
0.2
0
=1
2
=0
4
6
  t ISC / t TCT
8
10
Charge-transfer exciton life-times
Determined by inter-molecular inter-conversion, which occurs via the
electron transfer Hamiltonian, H 
H  H0  H
unperturbed Hamiltonian
perturbation
In the adiabatic approximation the electronic and nuclear degrees of freedom
are described by the Born-Oppenheimer states:
A  a; Q  a
electronic eigenstate of H 0
nuclear (LHO) state
(parametrized by a configuration coordinate, Q)
Transition rates are determined by the Fermi Golden Rule:
k I F 
2
2
I H  F  ( EI  EF )

The matrix elements are:
I H F  i H f  i  f
electronic matrix element
overlap of the vibrational wavefunctions
E f (Q)
Adiabatic
(Born-Oppenheimer)
energy surface
Ei (Q )

1
0
Qi
Qf
Q
The electronic states


Initial state: i  P , P  ~n1 j 1
chain 1:
+
chain 2:
_
Final state:
chain 1:
chain 2:
f  EX
(1)
GS
( 2)
  n' j '
+
(1)
_
GS
( 2)
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
i H  f  t ~n1 (r ) n' (r )dr   j 1( R) j ' ( R)dR
 t ~n 1 (r ) n'1 (r )dr
 j' j
|n' – n| = even
• Preserves electron-hole parity, i.e. |n' – n| = even
• "Momentum conserving", i.e. j' = j
Intra-molecular
Energy
Inter-molecular
n=1
exciton
n=2
charge-transfer exciton
j=1
j=1
IC
j=1
n=1
strongly bound exciton
Vibrational wavefunction overlap: Franck-Condon factors
Chain 1
Chain 2
F0(1)  0 P   EX
2
F0(2)  0 P   GS
EX
2
GS
E1 ( EX )
 EX
E1 ( P  )
E2 ( P  )
DE1 EX
DE10
0P
 GS
 DE2
DQ1
0P
 GS
 DE20
Q1
E2 (GS )
Q2
From the conservation of energy: DE1 EX  DE2 GS
The polaron and exciton-polaron have similar relaxed geometries  DQ1  0
Chain 1
Chain 2
E1 ( EX )
E2 ( P  )

E1 ( P )
DE1 EX  DE10
E2 (GS )
 DE2 GS
DQ1  0
Q1
 GS
Q2
DQ1  0  F0(1)   0 EX   EX  0 EX
EX
 DE2 GS  DE10
Multi-phonon emission
E2 ( P  )
F0(2)
GS
E2 (GS )
IC
VR
Q2
 GS
Ed

exp(  S p ) SpGS
 GS !
 GS  DECT  DE EX  / 
S p  Ed /  ( 1)
Huang-Rhys factor
re-organization
energy
Inter-conversion leaves chain 2 in the  GS vibrational level of GS
Subsequent vibrational relaxation with the emission of  GS phonons
The inter-conversion rate
kI F

2 exp(  S p ) S pGS
2
~

t  n 1 (r ) n'1 (r )dr
 f (E)

 GS !
Ratio of the rates is:
electron-hole continuum
k S CT  S X
kTCT TX

 !
S p ( T  S ) T
 S!
t
t
SCT
t SCT
 S
T
SX
t TCT
D  ( T  S )
where,
 S  DESCT  DES X / 
TCT
t ISC
TX
t SX
 T  DETCT  DETX / 
The ratio of the rates is an increasing function of D when S p  1
The ratio of the rates increases as S p decreases
t TX
Estimate of the singlet exciton ratio
d  4 A, t  0.1 eV,   0.2 eV, S p  1
 S  ( E ( SCT )  E ( S X )) /   4
 T  ( E (TCT )  E (TX )) /   7.5
t S CT ~ 500 fs
t TCT ~ 3 ns
t ISC ~ 0.1  0.3 ns
  S  70%
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.
Energy
Inter-molecular states
Intra-molecular states
Conclusions
The singlet exciton fraction exceeds the spin-independent recombination value
of 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 the
exciton 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 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.
Required Computations
1.
Electronic matrix elements between constrained excited states:
GS EX H  P  , P 
2.
Polaron relaxation energies.
3.
Spin-orbit coupling matrix elements:
SCT H SO TCT
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:
t S X  DETX

t TX  DES X
3
 W2

 D2

3. The ISC rate between the charge-transfer states is,
ISC
kCT

2
| W |2  f ( E )

 1010 s 1