Excimers in light-emitting conjugated polymers

Excimers in light-emitting conjugated polymers

Hsin-Fei Meng

Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan ~Received 19 December 1997!

The electronic structures of the low-lying excited states for two parallel chains of poly

( p-phenylene-vinylene! with arbitrary relative positions are studied. The interchain interaction is treated

mi-croscopically. The energy and charge transfer fraction of the ~light-emitting! lowest singlet state are found

to depend sensitively on the horizontal shift of the two chains, which is in turn determined by the

pack-ing geometry. Our predictions of the excimer-formpack-ing geometry and the relative~dipole-forbidden!

photolu-minescence lifetimes are in excellent agreement with experiments.@S0163-1829~98!07529-8#

The physics of electroluminescent conjugated polymers, especially poly( p-phenylene vinylene! ~PPV!, has been of major interests since the first demonstration of a light-emitting diode ~LED! based on such materials.1Despite the ensuing extensive studies, the photophysical process under-lying both the photoluminescence ~PL! and electrolumines-cence ~EL! in solid state is still not completely settled. Re-cently, it has been observed that the light-emitting species in some PPV derivatives ~CN-PPV and Br-PPV! are found to be quite different from the others, both in terms of spectrum and lifetime.2,3The fact that a major difference in the spectra exists between chemically very similar derivatives is surpris-ing. Since no such spectral difference is observed in the so-lutions, the interchain interaction is believed to be the origin. Excimer formation4,5has been proposed as a possible expla-nation.

Assuming the intrachain electronic structure is not af-fected by the side groups, the question is how does the pack-ing geometry determine the emission process. In order to answer the question, we study in this paper a two-chain model including microscopic interchain interactions. Con-figuration interaction among exciton ~EX! and charge-transfer ~CT! states, common in smaller organic molecules,6,7are found to be indispensable in obtaining the correct excited states for polymer films as well. Our key result is that the electronic structure of the emitting state

~lowest singlet excited state! depends sensitively not only on

the interchain distance, but also the relative horizontal shift of the two chains in a dramatic way. Based on the packing geometry obtained from computer simulation by Conwell

et al.,8a significant CT component is found in the emitting state for CN-PPV, but it is negligible in MEH-PPV and PPV. These results completely agree with the conjecture based on experimental data that excimers are formed in the first but not in the latter two.2,3Furthermore, three experimental sig-natures of excimers, large Stokes shift, broad emission spec-trum, and, most importantly, long radiative lifetime,2 all agree with our calculation. Our results are opposite to the intuitive interpretation8in one crucial aspect. We found that the emitting state is always of even parity, implying the tran-sition to the ground state is dipole forbidden. Vibrational coupling to a higher odd parity state is necessary for the transition to happen, with the transition rate inversely pro-portional to the square of the energy splitting of the two

states due to interchain interaction. This implies that the larger the interchain interaction ~and therefore the CT com-ponent!, the smaller the transition rate is, instead larger as Conwell et al. claimed. This explains the extraordinarily long radiative lifetime of CN-PPV,3 which has a quite large CT component by our calculation.

Consider two chains a and b, such that b is displaced from a by a rigid translation vector dez1Dh. ez is the unit vector normal to the symmetry plane ~approximately the benzene plane! of chain a. Dh is the horizontal shift parallel to the plane. The Hamiltonian for the p electrons contains four pieces:

H5H₀a1H0b1Hi1Hc. ~1!

The first two are intrachain tight-binding Hamiltonians for chains a and b, respectively. Hiis the interchain hopping and

H_C the Coulomb interaction. In second quantized form,

H₀a,b52

(

~i j!,s ti jcˆis

a,b†

cˆa,b_js1H.c. ~2!

Here cˆ_ia,b_s is the annihilation operator of electrons on carbon site i and spin s of chain a and b, respectively. We include only the nearest-neighbor intrachain hopping. ti jis equal to t for benzene bonds, t1 for single bonds, and t2 for double

bonds@see inset of Fig. 1~b!#. For the interchain hopping,

Hi52

(

n51 N

(

l51 8

(

s561i

(

51 m t_li@cˆ_{n f}b†_~l,i!scˆ_na_ls1H.c.#. ~3!

Here n is the index for the unit cells and N is the total number of cells in each chain.l is the index for the carbon atoms within each unit cell@see inset of Fig. 1~b!#. One-to-many interchain electron hopping is allowed from carbon site l on chain a to site f (l,i) on chain b with hopping integral t_li. m is the maximal number of i. Hopping from b to a is in the H.c. part of Hi. Note that the p electrons are assumed to be superpositions of carbon p_z orbitals only. Both intrachain and interchain carbon-hydrogen coup-lings are ignored. The eigenstates of Ha and Hb are the Bloch states, with the annihilation operators Aˆkas

5(nleiknaz_la(k)cˆnls a

, Bˆkas5(nleiknaz_la(k)cˆnls b

, where a is the lattice constant. k lies in the first Brillouin zone PRB 58

@2p/a,p/a#. a51•••8 is the band index. In terms of Aˆkas and Bˆkas, Ha,b is in diagonized form: Ha

5(ksaE_a

0

(k)Aˆ_k†_asAˆkas, Hb5(ksaE_a

0

(k)Bˆ_k†_asBˆkas. The band structure E_a0(k) is well known.9The lower four bands are filled in the ground state. In terms of the Bloch states, Hi becomes Hi52

(

kaa8s T_aa8~k!Aˆ_k†_asBˆka8s1H.c., T_aa8~k!5N

(

l51 8

(

i t_liz_la~k!za_f~l,i!8* _{~k!. ~4!}

The hopping integral is assumed to be the form t_li

5A exp(2mr_li), where r_li is the distance between site l

on chain a and site f (l,i) on chain b, which depends on, besides l and i, both the vertical distance d and horizontal shift Dh between the two chains.m is the size of the carbon

pz orbital and A is a fitting parameter.10In order to simplify our problem, we include only the middle two bands, the conduction (a5c) and the valence band (a5v), and rewrite the Hamiltonian in terms of the electron operators

ak,s[Aˆkcs,gk,s[Bˆkcs, and hole operatorsb2k,2s[Aˆk†vs,

d2k,2s[Bˆkvs

† _{. Within such a two-band approximation, the}

total Hamiltonian becomes H5H˜a1H˜b1H˜C1H˜i. Here

H˜a5(k_sEe(k)aks

† _a

k_s1Eh(k)bks

† _b

k_s. H˜b can be obtained by making the substitutiona→g andb→d. The renormal-ized single-particle spectra for electrons and holes are

Ee(k)5Ec(k)1DEc, Eh(k)52Ev(k)1DEv. The band edge renormalizations DE_c andDE_v are integrals involving the Bloch wave functions and the Coulomb potential V(r). The Coulomb part H˜C splits further into five pieces: H˜C

5HC1 a _1H C2 a _1H C1 b _1H C2 b _1H C ab , with H_C1a 51 2 _kk

(

_8qÞ0V1~q!~ak1qs † _a k82qs8 † ak8s8aks 1bk1qs † _b k82qs8 † bk8s8bks 22ak1qs † _b k82qs8 † bk8s8aks! ~5! and H_C2a 5

(

kk8q V2~q!a_k†_1qsb_2k2s† b_2k81q2s8ak8s8. ~6! As before, H_C1,2b can be obtained by the substitution a→g andb→d. The interchain Coulomb interaction H_Ca,bis

H_Cab51 2 _kk

(

_8q Vab~q!~ak1qs † _g k82qs8 † gk8s8aks 1bk1qs † _d k82qs8 † dk8s8bks2ak1qs † _d k82qs8 † dk8s8aks 2gk_1qs † _b k82qs8 † bk8s8gks!. ~7! In order to get the eigenstates of the Hamiltonian, we first break the total Hamiltonian into ‘‘free’’ and ‘‘interaction’’ parts: H5H01H˜i, with H05H˜a1H˜b1H˜C. There are four low-lying singlet excited states for the ‘‘free’’ Hamiltonian

H0. They are the exciton in chain a, the exciton in chain b,

and two CT states for which the electron and hole reside on different chains. H˜icauses mixing~configuration interaction! among those four states. The resulting 434 eigenvalue prob-lem can be further reduced to two 232 problems, for even and odd parity subspaces, respectively. Diagonization of the two 232 matrices gives the physical eigenstates.

The basis states for singlet elementary excitations in the two-band approximation are uk

&

A, uk

&

B, uk

&

C, and

uk

&

D, which are defined as

11 _uk

_&

A[(1/

A

2)(ak↑ † _b 2k↓ † 1ak↓ † _b 2k↑ † _)u0

_&

_,uk

_&

B[(1/

A

2)(gk↑ † _d 2k↓ † _1g k↓ † _d 2k↑ † _)u0

_&

_{, uk}

_&

C

[(1/

A

2)(g_k†_↑b_2k↓† 1g†_k↓b_2k↑† )u0

&

, and uk

&

D[(1/

A

2)

[~1/2!~gk↑†b2k↓†1gk↓†b2k↑†!u0&, and uk

&

D[(1/

A

2)

3(ak↑ † _d 2k↓ † _1a k↓ † _d 2k↑

† _)u0

_&

_{. The ground stateu0}

_&

_{is the state}

with no electrons and holes. The electron and hole are in the

FIG. 1.~a! The energies of the low-lying excited states «u1,2and

«g1,2~solid lines! are shown as functions of vertical interchain

dis-tance d. The horizontal shift Dh for CN-PPV is used. Exciton

en-ergies «EX6 ~lower two circle lines! and CT energies «CT ~upper

circle line! before configuration interaction are also shown. In

prac-tice, A512.8 eV,m51.18 Å are used for t_li. The result is almost

independent on m, the maximal number of interchain hoppings

from each site, as long as m.3. ~b! The CT fractions u^CT

2ug1&u2 _{and the off-diagonal matrix element}

^CT2uH˜iuEX2&~real! of ug1,2& for CN-PPV~solid! and

same chain for uk

&

A and uk

&

B, while they are in different chains foruk

&

Canduk

&

D. All the basis states have zero total momentum. They span four subspaces, which are decoupled from one another without H˜i. The matrix elements of H0

within each subspace are

A

^

k

8

uH0uk

&

A5B

^

k

8

uH0uk

&

B

5@2Ec~k!1DEg#dk8k2V~k

8

2k!, C

^

k

8

uH0uk

&

C5D

^

k

8

uH0uk

&

D

5@2Ec~k!1DEg#dk8k2Vab~k

8

2k!, where V~k

8

2k!5V1~k

8

2k!1V2~0! 51 N

F

n

(

Þ0 e2iR~k82k! V R~n!1U

G

, V_ab~k

8

2k!51 N

F

(

n e2iR~k82k! V R_'~n!

G

. ~8!

We have used the relation E_v(k)52Ec(k) for PPV band structure. R(n)@R_'(n)# is the distance between two unit cells in the same~opposite! chain~s! differing in indices by n. Anisotropy of the dielectric constant is also included.12The bandgap renormalization DEg5DEc1DEv, together with the on-site and off-site Coulomb energies U and V are treated as adjustable parameters. The lowest states, denoted byuA

&

,uB

&

uC

&

uD

&

in these subspaces can be expanded as

uA,B

&

5(kFEX(k)uk

&

A,B, uC,D

&

5(kFCT(k)uk

&

C,D. The energies are «EX for uA

&

,uB

&

and «CT for uC

&

,uD

&

. Parity

eigenstates can be constructed out of them: uEX6

&

[

A

1/2(uA

&

6uB

&

),uCT6

&

[

A

1/2(uC

&

6uD

&

). As mentioned before, there is no matrix element between 1 and 2 states. We can, therefore, consider them separately. For the diago-nal matrix elements, we have H0uEX6

&

5«EX6uE6

&

,

H0uCT6

&

5«CT6uE6

&

. Using Eq.~4!, the off-diagonal matrix element becomes

^

CT6uH˜iuEX6

&

5 1

2

(

k FCT

* _{~k!FEX~k!@2Tcc}*_~k!6Tvv*_~k!

6Tvv~k!2Tcc~k!#. ~9!

It turns out that

^

CT1uH˜iuEX1

&

50 because the transition amplitudes T_aa is real and the same for a5c,v. Conse-quently, there is no configuration interaction between EX and CT states for the1 subspace, and we need to diagonize the 232 matrix for the 2 subspace only:

S

«EX2

^

CT2uH˜iuEX2

&

^

CT2uH˜iuEX2

&

* «CT2

D

. ~10! Note that«_EX65«EXand«CT65«CT so far. Davydov splitting between «EX6 13 is phenomenologically included by the re-placement«_EX6→«EX6m2/d3, where m is the transition di-pole and d is the interchain distance.

Davydov splitting is a purely many-body Coulomb effect, which couples the exciton states uA

&

and uB

&

. In order to

coupleuA

&

anduB

&

, a term proportional tog†d†ab ~simul-taneous interchain transfer of an electron-hole pair due to Coulomb interaction! is required in H. The coefficient of such a term is hard to know without the detailed knowledge of some integrals involving the the atomic wave functions. Since our treatment of the Coulomb interaction is not very microscopic, with parameters U and V so chosen such that the intrachain exciton energy«EXis equal to the

experimen-tal value, the transition moment m for the «EX6 splitting can only be introduced as a fitting parameter. On the other hand, the interchain hopping Hamiltonian H_i is a purely one-body effect. The configuration interaction due to Hi are then treated much more microscopically. Note that, unlike H_i, Davydov splitting does not mixuEX

&

withuCT

&

. Therefore, it does not affect the CT fraction as much as Hi does.

Let us turn to the inversion symmetry~parity! of the prob-lem now. Consider the symmetry of a single chain, say, a, first. The lattice structure is invariant under inversion with respect to the point P_a, the center of benzene @see inset of Fig. 1~b!#. For Bloch states, we have uk,a

&

↔hu2k,a

&

, h

561. ↔ means inversion with respect to Pa. In terms of the Wannier functions, the symmetry manifests itself as

z_l¯a(k)5hz_la(2k)5hz_la*(k), with l↔l¯ under inversion. More explicitly, (ll¯)5(1,6),(2,5),(3,4),(7,0), and (8,21).

l less than 1 means carbon site in the previous unit cell.

Explicit calculation ofz_la(k) shows thath51 fora5v, and

h521 fora5c, which implies uk

&

A↔2u2k

&

A. Combined with the fact that the envelope functionsFEX,CT(k) are both even functions of k, we haveuA

&

↔2uA

&

, i.e., the exciton is of odd parity~dipole allowed!. In the case of two chains, the point of inversion symmetry becomes the middle point P of the symmetry points Pa of chain a and Pb of chain b. The parity of the states can be easily seen to be uA

&

⇔

2uB

&

, uC

&

⇔2uD

&

. ⇔ means inversion with respect to P. Finally, we have uEX,CT1

&

⇔2uEX,CT1

&

and

uEX,CT2

&

⇔uEX,CT2

&

. The interesting point here is that the 2 subspace, which contains the lowest state, is of even parity and dipole forbidden to decay radiatively to the ground state. Quantum lattice fluctuations must be consid-ered to yield a finite radiative lifetime. In fact, similar situa-tion has been reported for a pair of thiophene oligomers.14

In the following we present our results. First of all, since the energies of intrachain excitations have been studied intensively9 and is not our main concern here, we simply choose V(5U) such that the exciton creation energy «EXis 2.43 eV. The renormalized bandgap E_e(0)1E_h(0) is chosen to be 3.4 eV. We consider two PPV derivatives CN-PPV and MEH-PPV, with packing geometries determined from com-puter simulations.8 The horizontal shift Dh is (2,

A

3)b for CN-PPV and (9/4,2

A

3/4)b for MEH-PPV.8 b51.4 Å is

the carbon bond length. Consider CN-PPV first. In Fig. 1~a!, we plot the energies«u1,2and«g1,2of the lowest four excited states uu1,2

&

and ug1,2

&

for a range of vertical interchain distance d with fixed Dh. u,g means ungerude ~odd! and gerude ~even!. At large d, we have uu1,2

&

.uEX1

&

,uCT1

&

and ug1,2

&

.uEX2

&

,uCT2

&

.ug1

&

is always the lowest ex-cited state. Even though it is dipole forbidden, it must be responsible for the PL photon emission. By definition, an excimer is formed if such a state has a significant CT component.6,7The CT fraction ofug1

&

is shown in Fig. 1~b!.

Indeed, the excimer character is clear when d<3 Å, which is not far from the the ground state equilibrium interchain distance d053.3 Å.8Since«g1(d) decreases rapidly with d, the two chains must come closer to each other due to the attractive force when excited. The new equilibrium distance

d* depends on both the exciton density and the molecular force field between these two chains. Two characteristic en-ergy differences are of particular interest here: D«1

[«g1(`)2«g1(d*) and D«2[«g1(d0)2«g1(d*). Experi-mentally,D«1 is the difference in solution and film PL pho-ton energies, while D«2 is the net Stokes shift due to inter-chain lattice relaxation, i.e., the apparent Stokes shift minus the optical phonon energy. For CN-PPV,D«1. 0.35 eV and

D«2. 0.25 eV.2,3_{Since the transition dipole m responsible}

for Davydov splitting is unknown in our model, we adjust it such that «g1(`)2«g1(d0)5(0.3520.25) eV. The

corre-sponding transition diople length is 1.6 Å. The actual final lattice configuration in films is, however, very difficult to predict due to two reasons.~a! Polymers are heavy molecules with large atomic weight. It is not likely that a significant shrinkage of interchain distance can be caused by the exci-tation of a single electron-hole pair. A finite exciton densi-ty~exciton number per monomer! is required. d*is therefore determined by the exciton density, instead of a single exci-ton. Because the magnitude of exciton density is presently unknown, a reliable prediction seems unlikely. ~b! In the solution we need only to consider a pair of polymers in order to determine d*. In films, however, the force experienced by a particular chain is affected by many neighboring chains and is much more complicated. Strictly speaking, our two-chain approach is accurate in predicting only the initial ten-dency toward excimer formation, but not the final configura-tion, which may contain distortions involving more than two chains, or relaxations other than vertical distance shrinkage, e.g., rotation and horizontal displacement. We can, however, give a simple estimate of d* based on the Leanard-Jones form of the molecular force field U(d)5C@(1/2)(d₀/d)12 2(d0/d)6#. d*is the minimum point of«g1(d)1U(d). C is so chosen such that D«250.25 eV, from which we predict

d*.2.35 Å, where the CT fraction is about 15 %. The

ac-tual equilibrium configuration may be even lower in energy and have an even larger CT component.

Surprisingly, when we apply the same procedure to MEH-PPV, with Dh5(9/4,2

A

3/4)b, the story becomes totally dif-ferent. For the same range of d as before, the off-diagonal element in Eq.~10! is nearly zero, and the ug1

&

state remains basically uEX2

&

@see Fig. 1~b!#. Consequently, the light-emitting state remains basically the exciton as in the solution (d5`). Since the slope of «g1(d) at d054.05Å is vanish-ingly small, d0 and d*almost coincide with each other. This

explains the very small solution-film PL difference and Stokes shift observed in MEH-PPV.15Similar results are ob-tained for PPV, with Dh5(21.79,3.17) Å and d053.26 Å,16 agreeing with the fact that excimers are also absent in the PL spectrum of PPV.2The difference between these three PPV derivatives becomes conspicuously transparent when we scan the off-diagonal term

^

CT2uH˜iuEX2

&

, which is real due to certain symmetry, and the CT fraction of ug1

&

, over a range of D_h for some fixed d @see Figs. 2~a!,2~b!#. Since

^

CT2uH˜iuEX2

&

is the superposition of hopping

terms, which involves the complex Wannier wave fuctions

zla(k), between many different pairs of carbon orbitals, large

variations are expected due to the quantum interferences

~cancelations!. The landscapes of all the figures changes little

as long as d is in the physical range of 2.524.5 Å. CN-PPV happens to be close to a local maximum, while MEH-PPV is close to the contour of

^

CT2uH˜iuEX2

&

50. PPV is already in the flat region away from all the peaks. D

^

CT2uH˜_iuEX

2

&

/Dd can be viewed as the effective attractive force due to one excitation, which is equal to20.094 eV/Å for CN-PPV and 20.0095 eV/Å for MEH-PPV, if Dd is chosen to be 0.5 Å. In addition to the spectra, our model also explains the lifetimes of PL and photoinduced absorption ~PA!,17 in par-ticular the large difference in PL lifetimes between MEH-PPV ~1.2 ns! ~Ref. 18! and CN-PPV ~17 ns!.3

The symmetry-forbidden transition rate rg from ug1

&

to

u0

&

, via vibrational coupling touu1

&

, is given by19

rg5ru

S

1 «u12«g1

D

2

(

i

^

q_i2

&

S

] ]qi

^

u1uDH~q!ug1

&

D

0 2 . ~11!

FIG. 2. ~a! The CT fraction u^CT2ug1&u2_{of the light-emitting}

stateug1&are shown for an area of Dh. d is fixed at 3.3 Å. The CT

fraction for CN-PPV is about 10 times larger than that of

MEH-PPV. Surface elements at Dh of CN-PPV and MEH-PPV are

re-moved for clarity. The central peak at~0,0! corresponds to the

ge-ometry that two chains are right on top of each other. ~b! The

off-diagonal matrix element ^CT2uH˜iuEX2& for the2 subspace

are plotted for the same area of Dh. The contour of zero is shown to

emphasize that MEH-PPV happens to be in a geometry of vanishing CT-EX mixing due to inteference.

q is the collective lattice coordinates. ru is the transition rate fromuu1

&

tou0

&

, which is actually equal to the rate fromuA

&

tou0

&

for a single chain.«u1 and«g1 are excited state ener-gies at the relaxed lattice configuration.

^

q_i2

&

is the mean square lattice quantum fluctuation. Consider, for example, the lattice displacements in chain a only. Since they effect the electronic states in chain a but not in chain b, the exact inversion symmetry of uu1

&

is broken, and the emission is slightly dipole allowed. ruis inversely proportional to the PL radiative lifetime t_PLf in films, while rg is inversely propor-tional to the PL radiative lifetimet_PLs in solutions. Using the relation 1/tPL

f _;(«

u12«g1)22, we can predict the relativetPL

f of the derivatives. For MEH-PPV, we simply take d*5d0

54.05 Å, at which «u12«g150.17 eV from our calculation. For CN-PPV, the lattice is relaxed substantially and the ac-tual value of d* is hard to find. However, we can approxi-mate «u12«g1 by 2D«150.7 eV, assuming that the even and odd states split evenly off the the unperturbed value

«g1(`). The ratio between their tPL

f

is predicted to be (0.7/0.17)2517, consistent with the experimental value 17 ns/1.2 ns514.2.

The author is grateful for the support of the National Science Council of Taiwan, R.O.C., under Contract No. NSC 86-2112-M-009-001. The hospitality and support of the Center for Theoretical Science in Taiwan is also appre-ciated.

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