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Excitonic states of poly(para-phenylene)
ELSEVIER
Synthetic
Metals
101 (1999)
310-311
Excitonic states of poly@ra-phenylene) M. Yu. Lavrentiev’*, W. Barford’, and R. .I. Bursil12 ‘Department
of Physics, The University ofsheffield, Sheffield S3 7RH, United Kingdom
‘School of Physics, University ofNew South Wales, Sydney, NSW 20.52, Aust&ia
Abstract
Usinga recently developedphenomenological Hamiltonianof poly@ara-phenylene)andthe densitymatrix renormalizationgroup approach.we investigatethe transitionfrom boundto unboundstates.Energiesandexciton correlationfunctionsof the lowestexcited statesare calculated.The lowest ‘Br; statessituatedbelow the chargegap are found to be boundexcitons.The 2’Ai state, being unbound,represents a bandthreshold;the exciton bindingenergyisfoundto be0.74 eV. Keywords:Electrondensity,excitationspectracalculations,Poly(phenylene vinylene) andderivatives
1. introduction
In the last decade,after the first light-emittingdevicebased on poly@ara-phenylenevinylene)(PPV) was created [ 11, the phenyl-basedpolymershave attractedmuch attentionowing to their light-emitting and nonlinearoptical properties.Hovewer, evenfor the simplestphenyl-based system,poly@ara-phenylene) (PPP),the electronicstructureandthe natureof the the bluelight emissionremaincontroversial.The mainproblemis the existence of excitonic states,in which the electron and hole are bound together below the conduction band. In order to clarify this questionand study the low-lying electronicexcitationsof PPP, we presenta density matrix renormalizationgroup (DMRG) calculationof the system.
A = 5.26eV; on-siteandnearestneighboursHubbardparameters, U = 2V = 3.67eV; on-site exchangeenergy and pair hopping, X = P = 0.89eV. Theseparameterscorrespondto the caseof intermediatecorrelation. To diagonalizethe Hamiltonian,we usedthe DMRG method which hasproved extremelyusefulfor one-dimensional quantum systems.In our treatment,the systemincreasesby two repeat units at each step (the repeatunits are the phenyl rings). We performeda numberof tests,including comparisonwith-the resultsof exact diagonalizattion, which proved that the method givesresultsexactto aboutfew hundredsof an eV.
3. Results 2. Hamiltonian
The Pariser-Parr-PO+(or extendedHubbard)Hamiltonian, which contains the terms giving both band and localized excitonic states,was adopted.The six atomic n-orbitalsof the phenyl ring are transformed to six molecularorbitals,of which only two areretained,namelythe bondingelp (HOMO) and e,, (LUMO) states.It was shownpreviously [2] that theseorbitals containthe essentialphysicsof the low-lying statesof PPPand PPV. The parametrisationof the Hamiltonianwas performed using the available experimentaldata on the energiesof the lowestexcited statesof benzeneandbiphenyl[3]. Here,we give the values of the Hamiltonian parameters:hybridization, t = 0.895eV: energy difference betweenLUMO and HOMO,
* On leave from: Institute of Inorganic Chemistry, 630090 Novosibirsk.Russia.Supportedby EPSRC(GR/K86343). 0379-6779/99/$ - see front matter 0 1999 Elsevier PII: SO379-6779(98)0 1136-9
Science S.A. All
The energiesof the lowestexcited statesin the ‘Ai and ‘B, symmetries(i.e., the symmetriesof the ground stateand of the stateswhich can be reachedfrom the ground state through a photonexcitation) werecalculated.The resultsare given in Fig. 1. Also, we calculated the charge gap E,, , defined as EcG =E(N t l)+E(N-I)-2E(N), E(hr) being the groundstate energy of the systemwith N electrons.For long systems,the chargegapgivesthe energyof the uncorrelatedelectron-holepair and can thus be used as a boundary between excitonic -and extendedstates. Fig. 1 showsthatthereis a numb-erof statesIv_‘_fh ‘B, symmetrybelow the chargegap, while the lowestexcited ‘Ai state almost coincideswith the charge gap for all the systems studied. This suggests an existenceof excitonic bandof ‘B, states. To check this, we calculatedexciton correlation functions for these states, defined as follows:
rightsreserved.
M.Y. Lavrentiev
et al. / Synthetic
Metals
101
(1999)
7 3
11
19
27
35
Oligomer size Fig. 1. Energies of the lowest excited states of PPP as function of oligomer size (number of phenyl(ene) units). Dashed line: the charge gap, empty squares: 2’Ai , empty diamonds: n’B, : states
310-311
311
11 15 Oligomer size
19
Fig. 2. Exciton sizesof the lowest excited statesof PPP as function of oligomersize. Solid lines correspondto the states shownin Fig. 1; dottedlineswith symbols:resultsin the absence of electron-electron interaction( I’B, assolidsquaresand 2’116 as empty squares);dotted lines without symbols: hi / 2 and N / 4, N beingthe oligomersize.
below the charge gap belong to the ‘B, exciton band.
4. Conclusion
where SI$ is a singlet exciton creation operator, which removes a particle from the HOMO on molecular site j and places it into the LUMO on site i . Then, the quantity e, = Ci / CC: ‘I servesasthe distributionfunction for the particle-holeseparation. The plot showingmeanexciton-holedistancefor somelow-lying statesis shownin Fig. 2. The ‘B,; statesbelongingto the band belowthe CG are thosewith the smallestelectron-holedistance, which approachesa finite limit as the systemsize increases. Conversely,the spatialextentsof the 2’Ad and n’B,i statesare proportional to the oligomer size, as are those of the lowest excited (unbound)statesof the non-interactingmodel.Thus, we found a ‘B,; exciton (excitonic band)in PPP,with the binding energyfor long oligomersapproaching0.74eV. The lowestband stateisfound to have ’ “i symmetry.
This study wasdevotedto clarifying the nature of the lowlying ‘Ai and ‘B, electronicexcitationsof PPP.It was shown that the lowestexcited stateis a ‘B, exciton (excitonic band), situatedbelowthe chargegap,which marksthe onsetof extended states(the conductionband). The exciton binding energy was found to be 0.74 eV. The chargegap separates bound excitonic andunboundstates.
References
[l] J.H. Burroughes,D.D.C. Bradley, A.R. Brown et al., Nature347(1990)539. [2] W. Barford, R.J. Bursill, Chem.Phys. Lett. 268 (1997) 535. [3] R.J. Bursill, C. Castleton,W. Barford, Chem.Phys. Lett.. in press.
Synthetic
Metals
101 (1999)
310-311
Excitonic states of poly@ra-phenylene) M. Yu. Lavrentiev’*, W. Barford’, and R. .I. Bursil12 ‘Department
of Physics, The University ofsheffield, Sheffield S3 7RH, United Kingdom
‘School of Physics, University ofNew South Wales, Sydney, NSW 20.52, Aust&ia
Abstract
Usinga recently developedphenomenological Hamiltonianof poly@ara-phenylene)andthe densitymatrix renormalizationgroup approach.we investigatethe transitionfrom boundto unboundstates.Energiesandexciton correlationfunctionsof the lowestexcited statesare calculated.The lowest ‘Br; statessituatedbelow the chargegap are found to be boundexcitons.The 2’Ai state, being unbound,represents a bandthreshold;the exciton bindingenergyisfoundto be0.74 eV. Keywords:Electrondensity,excitationspectracalculations,Poly(phenylene vinylene) andderivatives
1. introduction
In the last decade,after the first light-emittingdevicebased on poly@ara-phenylenevinylene)(PPV) was created [ 11, the phenyl-basedpolymershave attractedmuch attentionowing to their light-emitting and nonlinearoptical properties.Hovewer, evenfor the simplestphenyl-based system,poly@ara-phenylene) (PPP),the electronicstructureandthe natureof the the bluelight emissionremaincontroversial.The mainproblemis the existence of excitonic states,in which the electron and hole are bound together below the conduction band. In order to clarify this questionand study the low-lying electronicexcitationsof PPP, we presenta density matrix renormalizationgroup (DMRG) calculationof the system.
A = 5.26eV; on-siteandnearestneighboursHubbardparameters, U = 2V = 3.67eV; on-site exchangeenergy and pair hopping, X = P = 0.89eV. Theseparameterscorrespondto the caseof intermediatecorrelation. To diagonalizethe Hamiltonian,we usedthe DMRG method which hasproved extremelyusefulfor one-dimensional quantum systems.In our treatment,the systemincreasesby two repeat units at each step (the repeatunits are the phenyl rings). We performeda numberof tests,including comparisonwith-the resultsof exact diagonalizattion, which proved that the method givesresultsexactto aboutfew hundredsof an eV.
3. Results 2. Hamiltonian
The Pariser-Parr-PO+(or extendedHubbard)Hamiltonian, which contains the terms giving both band and localized excitonic states,was adopted.The six atomic n-orbitalsof the phenyl ring are transformed to six molecularorbitals,of which only two areretained,namelythe bondingelp (HOMO) and e,, (LUMO) states.It was shownpreviously [2] that theseorbitals containthe essentialphysicsof the low-lying statesof PPPand PPV. The parametrisationof the Hamiltonianwas performed using the available experimentaldata on the energiesof the lowestexcited statesof benzeneandbiphenyl[3]. Here,we give the values of the Hamiltonian parameters:hybridization, t = 0.895eV: energy difference betweenLUMO and HOMO,
* On leave from: Institute of Inorganic Chemistry, 630090 Novosibirsk.Russia.Supportedby EPSRC(GR/K86343). 0379-6779/99/$ - see front matter 0 1999 Elsevier PII: SO379-6779(98)0 1136-9
Science S.A. All
The energiesof the lowestexcited statesin the ‘Ai and ‘B, symmetries(i.e., the symmetriesof the ground stateand of the stateswhich can be reachedfrom the ground state through a photonexcitation) werecalculated.The resultsare given in Fig. 1. Also, we calculated the charge gap E,, , defined as EcG =E(N t l)+E(N-I)-2E(N), E(hr) being the groundstate energy of the systemwith N electrons.For long systems,the chargegapgivesthe energyof the uncorrelatedelectron-holepair and can thus be used as a boundary between excitonic -and extendedstates. Fig. 1 showsthatthereis a numb-erof statesIv_‘_fh ‘B, symmetrybelow the chargegap, while the lowestexcited ‘Ai state almost coincideswith the charge gap for all the systems studied. This suggests an existenceof excitonic bandof ‘B, states. To check this, we calculatedexciton correlation functions for these states, defined as follows:
rightsreserved.
M.Y. Lavrentiev
et al. / Synthetic
Metals
101
(1999)
7 3
11
19
27
35
Oligomer size Fig. 1. Energies of the lowest excited states of PPP as function of oligomer size (number of phenyl(ene) units). Dashed line: the charge gap, empty squares: 2’Ai , empty diamonds: n’B, : states
310-311
311
11 15 Oligomer size
19
Fig. 2. Exciton sizesof the lowest excited statesof PPP as function of oligomersize. Solid lines correspondto the states shownin Fig. 1; dottedlineswith symbols:resultsin the absence of electron-electron interaction( I’B, assolidsquaresand 2’116 as empty squares);dotted lines without symbols: hi / 2 and N / 4, N beingthe oligomersize.
below the charge gap belong to the ‘B, exciton band.
4. Conclusion
where SI$ is a singlet exciton creation operator, which removes a particle from the HOMO on molecular site j and places it into the LUMO on site i . Then, the quantity e, = Ci / CC: ‘I servesasthe distributionfunction for the particle-holeseparation. The plot showingmeanexciton-holedistancefor somelow-lying statesis shownin Fig. 2. The ‘B,; statesbelongingto the band belowthe CG are thosewith the smallestelectron-holedistance, which approachesa finite limit as the systemsize increases. Conversely,the spatialextentsof the 2’Ad and n’B,i statesare proportional to the oligomer size, as are those of the lowest excited (unbound)statesof the non-interactingmodel.Thus, we found a ‘B,; exciton (excitonic band)in PPP,with the binding energyfor long oligomersapproaching0.74eV. The lowestband stateisfound to have ’ “i symmetry.
This study wasdevotedto clarifying the nature of the lowlying ‘Ai and ‘B, electronicexcitationsof PPP.It was shown that the lowestexcited stateis a ‘B, exciton (excitonic band), situatedbelowthe chargegap,which marksthe onsetof extended states(the conductionband). The exciton binding energy was found to be 0.74 eV. The chargegap separates bound excitonic andunboundstates.
References
[l] J.H. Burroughes,D.D.C. Bradley, A.R. Brown et al., Nature347(1990)539. [2] W. Barford, R.J. Bursill, Chem.Phys. Lett. 268 (1997) 535. [3] R.J. Bursill, C. Castleton,W. Barford, Chem.Phys. Lett.. in press.