Theory of biexcitons in mixed and segregated stack charge-transfer solids, molecular crystals and π-conjugated polymers

ELSEVIER

Synthetic Metals 78 (1996) 187-193

Theory of biexcitons in mixed and segregated stack charge-transfer solids, molecular crystals and 7r-conjugated polymers

s. Mazumdar, F. Guo 1 Department of Physics and Optical Sciences Center. University ofArizona. Tucson. AZ 85721. USA

Received 3 July 1995; accepted 5 September 1995

Abstract

We discuss biexcitons within three different theoretical models in one dimension with excitonic interactions. We develop a general physical picture for optical absorption from the one-photon allowed exciton to the biexciton and to the two-exciton continuum in one dimension. Based on this picture, we propose a criterion ofbiexciton stability. The validity of our criterion of biexciton stability is proved in the strong-coupling limits of models describing mixed and segregated stack charge-transfer solids, while studies by other investigators have shown that the criterion also remains valid within the Frenkel exciton model with moderate exciton-exciton binding. Based on the same criterion of biexciton stability, we present theoretical evidence for stable biexcitons within the Coulomb-correlated model for 1T-conjugated polymers. High-energy picosecond photoinduced absorption in 1T-conjugated polymers is explained as a transition from the optical exciton to the biexciton. Keywords: Theoretical study; Biexcitons; Charge-transfer solids; Molecular crystals; Conjugated polymers

1. Introduction Beginning with molecular crystals [ 1], the exciton concept has been applied to a wide variety of organic solid-state sys• tems. The concept is especially important in one dimension, where confinement greatly enhances exciton binding. Optical absorption is described within the exciton model in both half• filled bands segregated stack charge-transfer (CT) solids [2] , and in mixed stack CT solids [3] . The CT exciton model has been applied to other related quasi-one-dimensional sol• ids (for example, the metal-halide MX chains, the one-band model [4] which is similar to the models of mixed stack CT solids [5]). Many recent theoretical [6] and experimental [7] investigations indicate the applicability of the exciton model to 1T-conjugated polymers. The theoretical discussions in many cases can be considered as continuations of those found in the early exciton literature of polyenes [8]. The discussion of excitons in the polysilanes [9] have paralleled those in the 1T-conjugated polymers. Given the above, it is surprising there has been relatively little discussion of the biexciton, or the excitonic molecule, in quasi-one-dimensional organic systems. The biexciton concept has existed for a long time in conventional semicon• ductors [10], and biexcitons in these systems have been I Present address: Department of Physics. National Chung Hua University of Education. 1 Chin Te Road, Chunghua City 500, Taiwan, ROC.

0379-6779/96/$15.00 «:> 1996 Elsevier Science S.A. All rights reserved SSDI0379-6779(95)03609-1

studied extensively [11]. In the quasi-one-dimensional organics though, the authors are aware of only a few isolated applications of the biexciton concept: in the areas of optical nonlinearity in the J- and H-aggregates [12], photoinduced absorption (PA) and two-photon absorption (TPA) in the polysilanes [9], PA in the mixed stack CT solids [13], and, very recently, PA and TPA in 1T-conjugated polymers [14,15]. Theoretical work in these areas has largely focused on explaining specific experiments in individual material sys• tems. It is the purpose of the present paper to review the various theoretical models simultaneously, and to present a somewhat unified picture of the biexciton in one dimension. It is also our goal to establish that biexcitons exist in 1T• conjugated polymers and play an important role in the pho• tophysics of these systems. The specific feature of the biexciton we will focus on is the optical absorption from the optical exciton, which in its tum is generated by ground-state absorption. Ground-state absorption within one-dimensional electronic Hamiltonians is well understood. In the absence of exciton binding, ground• state absorption reflects the square-root singularity at the band edge. As the exciton binding increases, the oscillator strength of the transition from the ground state progressively concen• trates in the lowest exciton, at the expense of transitions to the continuum band with free electrons and holes. In the present work, we show that there exists a similar universality

188

S. Mazumdar. F. Guo/Synthetic Metals 78 (1996) 187-193

in the absorption from the exciton to the two-exciton states. Within the theoretical model appropriate for 'iT-conjugated polymers, we then demonstrate stable biexcitons. Brief pres• entation of this work has been made elsewhere [14].

2. Theoretical models We will consider three different one-dimensional theoret• ical models. The first is the standard Peierls-extended Hub• bard model, J1?'sow' written as

J1?'SDW= VEn/.tn/,J. j

+ ~1:
-t}2[ 1- (-1)/5] [ctuC/+l,O'+ci~I,uC/,O']

(1)

1.0'

Here ctO' creates an electron of spin u on site 1, t is the one• electron hopping matrix element, U and V are on-site and nearest-neighbor Coulomb interactions, and /) is a bond-alternation parameter. We are interested only in the half-filled band, in the parameter region where the optical gap is dominated by U and V. Furthermore, we will be inter• ested only in the parameter space U> 2V, for which the half• filled band is a superposition of long-range bond order wave (BOW) and short-range spin density wave (SDW) [16]. J1?'SDW has been applied to both half-filled band segregated stack CT solids [2] and to 'iT-conjugated polymers [16]. It is believed that Ulltl =8-10, Vlltl =2~ in segregated stack CT solids [2], while the estimates for these parameters in the 'IT-conjugated polymers are Ulltl =2.5-3.5, VI It I = 1-1.5. Although many of the polymers are not linear chains and contain aromatic rings, we will assume that the excitons (and consequently the biexcitons) in these systems can also be described within the linear chain model, albeit with an effecti ve /) that is larger than that in the polyacetylenes [17] . Excitons and biexcitons within Eq, (1) can be most easily visualized in the strong coupling limit, U» V» t. In this limit, the ground state of the half-filled band has all sites singly occupied. The corresponding configuration can be written as .... 1111 ... , where the numbers denote site occupan• cies. Optical absorption within Eq. (I) is a nearest-neighbor CT process, so that the lowest exciton states are odd-parity and even-parity linear combinations of configurations .... 11120111 ... , that occur at energy U - V. The doubly occu• pied site and the empty site correspond to the particle and the hole, which are bound by V. The one electron-one hole ( Ie• Ih) continuum then consists of all states in which the double occupancy and the empty site are separated by more than one site (for example, ... 11211...1011. .. ). The continuum is cen• tered at energy U. The biexciton states consist of configura• tions ... 111202011 L~, which occur at energy 2U - 3V, while the lowest two-exciton continuum states have occupancies ... 120 11.. .120 11..., and occur at energy 2 U - 2 V. Thus, in the strong coupling limit ofEq. (1), the biexciton binding energy

and the exciton binding energy are the same, namely, V. In addition to the above, there exist free 2e-2h states ... 11211101111211110... , that occur above the two-eXCito~ states. These have vanishing dipole coupling with the oPtical exciton and are not of interest. The second model is applicable to the half· filled hand charge-density wave systems, J1?'cow, written as

J1?'cow = }2 ( - 1) lecTuCIO' 1,0'

1

-t[/cluCl+I.O'+cl+l.uCIO') +2'}2V/mPIPm 1.0' I,m

(2)

Eq. (2) has been applied to mixed stack CT solids [3,13] with the convention that the donor (D) sites occupy odd site~ with site energy - e, and the acceptor (A) sites occupy even sites with site energy + e. The quantity

P,=Z- EcTuCIO' 0'

where z = 2 for a D site and z = 0 for an A site. Vim = V"' with n = 1/- m I, is the intersite Coulomb interaction between sin•

gly charged sites. Doubly charged D + + and A - - are excluded from our description, because of the very high ener_ gies necessary to create these species. This is equivalent to a Hubbard U -+ 00 approximation, where the on-site repulsion is between like charges on a D or A site. The strong coupling limit of Eq. (2) is once again Useful for a visual picture. The ground state of the strongly coupled J1?'cow, 1el » I tl, has the site occupancies ... 202020.... Now nearest-neighbor CT leads to the exciton ... 2020112020.:. at energy 2e - VI' The particle and the hole both correspond to site occupancies of 1 (they are distinguished by whether they occupy even or odd sites). The le-lh continuum, the biexciton and the two-exciton continuum have occupancies ... 201020210... (separated single occupancies), ... 202011112020... (a pair of nearest-neighbor excitons) and ...201120... 20112020... (separated excitons), respectively. Finally, we are interested in the spin Hamiltonian, .:e' • wntten as

•

J1?'s= Wo}2S}-UzE(Sj+ 1I2)(Sj+1 + 1/2) j

-J[)st Sj+ I +Sj- S/~ I)

(3)

j

Jordan-Wigner transformation of )E's leads to a description of Frenkel excitons [12,18] with molecular excitation energy Wo, nearest-neighbor exciton-exciton interaction U z , and nearest-neighbor exciton hopping with matrix element J. In contrast to Eqs. (1) and (2), exciton-exciton binding is already built into the description here. J1?'s is relevant for the description of J- and H-aggregates [12], and has also been applied to the mixed stack CT solids [18], where a neigh_ boring (DA) unit was assumed to behave like a Frenkel exciton. Multiexcitons in this model have been discussed both for attractive and repulsive excitons [18], but we are inter_

S. Mazumdar. F. Guo/Synthetic Metals 78(1996) 187-193

ested in the attractive case only (Jz > 0). Note that there is no equivalent of the le-lh continuum within Eq. (3). There is a single dispersive exciton state which corresponds to the one-magnon state. Two-exciton states are bound and free two-magnon states [19]. 3. Excited state absorption, a physical picture

189

First exciton

(a)

(b)

a(E)

\

/

Second exciton

Energy

Optical excitations from the exciton can be broadly clas• sified into two types: (i) those leading to a physical separation of the electron and the hole, generating either a higher exciton or a state in the le-lhcontinuum [20], and (ii) those leading to two excitons [14,15,18]. We are only interested in process (ii) here, which is possible within all three models. Before we discuss absorption from the exciton within the interacting models, we remind the reader of the interband absorption expected from the lowest optical state in the rigid• band one-dimensional semiconductor in the absence of inter• actions that favor exciton and biexciton binding. This would be the case within Eq. (1) for U = V = 0, but 1):1= O. In this case, optical transitions from the ground state are simple le• Ih transitions, from the valence band (v .b.) to the conduction band (c.b.). Interband transitions from the lowest le-lh state are to 2e-2h states, and involve the same v.b. and c.b. levels that are involved in ground-state absorption. The ground-state and the excited-state absorption spectra are then identical and absorption from the lowest optical state is strongest to the lowest 2e-2h state. The same description is valid for the noninteracting limit ofEq. (2), namely, Vlm=O. A slightly modified description is required for Eq. (3) in the limitJz = 0, when the solutions are n-exciton bands. For J>O ground• state absorption is strongest to the lowest one-exciton state, while absorption from this state is again strongest to the lowest state in the two-exciton band. We now argue that, once binding occurs, transition from the exciton to the biexciton (the lowest two-exciton state) gradually becomes weaker than that to the threshold of the two-exciton continuum. The physical reasoning we present here is independent of the nature of the interaction that binds the excitons and is therefore equally valid for all three theo• retical models of interest. Consider a long chain with the first exciton already formed. For stable biexcitons to exist, the second 'exciton can occur in only one of two 'spots', to the immediate left or to the right of the first exciton, at a distance such that binding can occur (see Fig. 1(a». In contrast, the second exciton in a two-exciton continuum can appear at arbitrary relative distance from the first exciton. Thus, in a long chain, the two-exciton continuum states are superposi• tions of a very large number of basis functions, while the biexciton is constructed out of a few. The density of states is consequently much larger at the continuum edge, and we expect that the strength of the absorption from the exciton to the two-exciton continuum edge is larger than the strength of the absorption to the biexciton. A schematic of the transitions from the exciton, based on this simple principle, is shown in

Fig. I. (a). Location of the second exciton (one of two dashed ellipses). relative to the first (solid ellipse) in a biexciton. In a two-exciton continuum state the second exciton occurs at arbitrary distance from the first. (b) Schematic of the predicted absorption from the exciton to the two-exciton states in the case of biexciton formation. Transition to the edge of the two• exciton continuum is stronger than that to the biexciton.

Fig. l(b). We expect therefore a gradual, but qualitative change of the absorption from the lowest optically excited state as exciton binding increases. Furthermore, this change is opposite to that in the case of ground-state absorption, where the exciton acquires strength at the expense of the continuum. The difference arises from the difference in the natures of the le-lh continuum and the two-exciton contin• uum (see discussions of their wavefunctions in the strong coupling limit). Conversely, exciton absorption (in the proper energy range) resembling Fig. 1(b) is a definite sig• nature of bound biexcitons. In the next section we show numerically the validity of this intuitive argument. Since our numerical work is for short chains, it is also useful to understand how the absorption of Fig. l(b) is expected to vary with chain length within our physical picture. The density of states at the two-exciton continuum threshold increases rapidly with chain length, while the density of states at the biexciton level quickly sat• urates. Therefore, we would predict that the relative absorp• tion strength to the biexciton, compared to that to the two-exciton continuum, should decrease with chain length for fixed binding. 4. Validity of the physical picture, numerical demonstrations

In this section we present numerical results that confirm the validity of the physical criterion developed in Section 3. In the case of Eq. (1), exact numerical results are limited to rather short chains, which do not possess the two-exciton continuum states necessary for the verification of our conjec• ture for moderate Coulomb interactions. We therefore work in the strong coupling limit first, where all exciton and con• tinuum states have converged already at the chain lengths studied. In Fig. 2(a) we show the exact energy states for a periodic ring of N= 10 sites for Ulltl =50, Vlltl = 15, and I) =0.1. Finite size effects are minimal for these very large Coulomb parameters, and exciton states at U- V, the le-lh continuum centered at U, the biexciton at 2U - 3V and the two-exciton continuum at 2U - 2V are all distinct. Only one

190

S. Mazumdar, F. Guo/Synthetic Metals 78 (1996) 187-193 (a) 2U-IV

2U-SV

~

r.:I

Z

u-v

2 ....lOIIbond(4.54.V)

' - BieJcitona (4.40 eV)

BIEXClTOHS

_

U

p:;

(a)!!!!!!!!!~J -

2-EXClroH CONTIHUUII

IIHH BAND

===~J-IO - Ib bond (2.62.V) ' - Excitona (2.27 oV)

EXClroNS

_ _ _ Oroaad Stole (0.0 eV)

r.:I

9' 1.0 (b) ................ ·......-

~Q

GROUND STATE + SPIN-WAVE

0

~

c

B

0.5

j

~~L ................................

00

I

1

1.85

!,../!.

I

2U - 3V

2U - 2V

Energy

Fig. 2. (a) Total energies of the N-tO periodic ring, within Eq. (1), for VIlli =50, VIlli-IS and c5=0.1. (b) Normalized dipole couplings between the optical lB. exciton and the two-exciton A, states of (a). The normalization is with respect to the dipole coupling between the ground state (IA,) and lB. exciton. Dipole coupling to the continuum state C is weakly larger than that to the biexciton B. The oscillator strength of the transition to C is much larger.

of the exciton states at U - V has large dipole coupling with the ground state. This is the opticallBu exciton. In Fig. 2(b) we show the normalized dipole couplings between the optical exciton and all two-exciton states. Note that the dipole cou• pling with the biexciton B at 2U - 3V is smaller than the dipole coupling with the state C at the edge of the two-exciton continuum, in agreement with our prediction. We expect the relative dipole moment of C to become even larger for larger system sizes (see below). Note that the oscillator strength of the transition to C (as opposed to the dipole moment) is already much larger than that of the transition to B. Calculations for Eq. (2) were performed for the periodic ring of 14 atoms. The particular parameter values that we have chosen, It I =0.1 eV, VI =0.5 eV, V,,=VI/n, e= 1.38 eV, are those that were used to fit the absorption spectrum of a strongly neutral mixed-stack cr solid, anthracene-pyro• mellitic acid anhydride [ 13,21] . The large e/ It I corresponds to the strong coupling limit. The energy spectrum for these parameters within Eq. (2) is shown in Fig. 3(a). Note that once again the exciton states, the le-lh continuum, the biex• citon and the two-exciton continuum are clearly visible because of strong coupling. The very narrow widths of the le-lh continuum and the two-exciton continuum are due to the U = 'XI approximation and the inequivalence of the sites. In Fig. 3(b) we show the normalized dipole couplings from the optical exciton to the two-exciton states. The results are identical to those shown in Fig. 2(b). Similar results are obtained [21] with It I =0.15 eV and 0.2 eV, and several

2.00

Fig. 3. (a) Total energies of the N-14 periodic ring. within Eq. (2). for It I =0.1 eV. ~-1.3S eV, VI-O.S eV. V.. - VI/n. The COW energy states are similar to the SOW states of Fig. 2(a), with the only difference that th spin wave states are missing here. The relatively narrow widths of the 1:• Ih band and the two-exciton band are due to the U-oo restriction and the inequivalence of consecutive sites. (b) Normalized dipole COUPling between the optical exciton (ex) and the two-photon (2ph) twO-exCito S states of (a). Note that the dipole moment is largest to the edge state of thn two-exciton continuum at a normalized energy of 2.0. e

10

~l ....

i

(0)

• •

1

~

0.1

1

• • • • •••• 10

41

10r---~~~~··~r-1--~~~~

(b)

~l ....

.

·S

1

~.-----.~.-----­ •••

~.

10

••

• 100

N Fig. 4. (a) Dependence of the ratio Wout.. /W_ on .1. where .1-J.1J in Eq. (3), for a periodic ring of 20 atoms. Here Wild.. is the oscillator strength of the transition from the exciton to the biexciton. and W_ is the oscillator strength of the transition from the exciton to the two-exciton continUum. (b) System size dependence ofW.lrin.!W_ for .1-2.0 (from [IS]).

different V". Note that (a) the dipole moment to the two• exciton continuum edge at the normalized energy of 2.0 is now much larger than that to the biexciton at lower energy. and (b) the normalized dipole moment to the continuum edge is now 1.0. Both results are consequences of the larger system size. In the case ofEq. (3), the existence of the biexciton (bound two-magnon state) is known [19]. The absorption strengths we are interested in have recently been calculated numerically for finite periodic rings by Ezaki et a1. [18]. We have repro_

S. Mazumdar. F. Guo! Synthetic Metals 78 (1996) 187-193

13B u (5.87) 6A. 2A. (4.17 (3.34)

1

IB" 3.00)

5B 2Bu (4.21 13.63)

n

14A. (5.52)

+2.85(0.20) +2.45(0.10) +2.25(0.10) + 1.90(0.10) +1.40(0.15) -1.40(0.15) -1.90(0.10) -2.25(0.10) -2.45(0.10) -2.85(0.20)

11 AI (4.63) Fig. 5. Decomposition of the QCI N= 10 total energies (in units of It I ) into approximate MOenergies forUI It I =3. Vlltl-l and c'l-0.5. The numbers against the MOs are their approximate single-particle energies, with the error bars included in parentheses. The vertical arrows indicate the excitations within the MO scheme. The numbers against these are the total excitation energies obtained by QCI. The 14A. is the lowest 2e-2h excitation for large c'l.

duced their results for the case of attractive exciton-exciton interaction in Fig. 4(a). In Fig. 4(a) Wstting is the oscillator strength of the transition from the exciton to the biexciton ('string exciton' in [18] ), and Wfree is the oscillator strength of the transition to the two-exciton continuum ('free exci• tons' in [18]). Notice that Wstting!Wfree becomes less than unity even for moderate binding of excitons. The results of Fig. 4(a) should be compared with those of Fig. 4(b), where the size dependence of Wsttingl Wfree is plotted, for a fixed binding. Note that the decrease of this ratio is in agreement with that anticipated from our physical picture. 5. '1T.Conjugated polymers

The consistency of the results among diverse theoretical models indicates the universal applicability of our criterion. The Coulomb parameters for the segregated stack CT solids are large enough that we can anticipate the results of Fig. 2 to persist. We therefore do not present these results here. Instead, we now investigate Eq. (1) for even smaller Cou• lomb interactions, UII tl = 3, VI It I = I, the parameter regime thought to be appropriate for the '1T-conjugated polymers [ 16]. The bond alternation parameter 8 is taken to be 0.10.3, to simulate both linear chain and aromatic polymers [ 17]. Dipole moment calculations within the periodic bound• ary conditions (see Fig. 2) are done using the velocity oper• ator, the application of which becomes complicated when the initial and final states are very close in energy [2], as is the case for pairs of excited states in finite periodic rings with small Coulomb interaction parameters. We have therefore done these calculations for open chains. Note th,at this has an added advantage. The exciton width for moderate Coulomb interactions is larger than the strong-coupling exciton of Sec• tion 4, and the open chain simply has more physical space necessary for the two-exciton continuum states. Our calcu-

191

lations are for the open chain of N = 10 atoms within the quadruples configuration interaction (QCI) approximation. This is the largest system whose entire energy spectrum (as opposed to the lowest states) can be currently investigated without loss of accuracy. We have verified that exact N = 8 results are almost indistinguishable from the QCI N = 10 results. For Ull tl and VI I tl as small as 3 and 1, not only are finite chain energy spectra discrete, but also there are considerable overlaps between the energies of the exciton states, the Ie• Ih continuum and the two-exciton states, which are so clearly distinguishable from energies alone in the strong coupling limit (Fig. 2(a». The very identification of the lowest two• exciton state is difficult now. In order to overcome this prob• lem, we start our calculations with very large 8 ( 8 = O.S). For the relatively small U and V and such large 8, all excitations can be described within an effective single-particle picture, and the lowest 2e-2h state can be easily identified. We prove this by determining approximate molecular orbital (MO) energies that can be used to reproduce all total excitation energies, already known from the QCI calculation. In Fig. 5 we have shown N = 10 total excitation energies against sev• eral representative odd-parity (Bu) and even-parity (Ag) excitations. The approximate MO energies, with error bars in parentheses, are shown against the individual v.b. and c.b. levels. From Fig. 5, the 14A, state is the lowest 2e-2h state in N= 10 with Ulltl =3, Vlltl = 1,8=0.5. In Fig. 6(a) we have plotted the normalized dipole cou• plings between the 1Bu and all excited AI states against their normalized energies, for the same parameters. The level labeled 2 is the 2A , the level m is the mAs state discussed previously in the c~ntext of third-order optical nonlinearity

. . . . . . .l'i . . . . . . . . . . (a)

2

(b)

;2

6=0.5

JL .............................. . 6=0.4

..............\.......rn............... B' ............................ . 6=0.3

3.0 EkA/E 1B•

Fig. 6. Dipole couplings between the lB. and all excited A, states, normal• ized by the dipole coupling between the ground state and !he lB., plotted against the normalized energies of the A. states, for different c'l and UI Itl .. 3, VII tl "" 1. The emergence of a level C above B for 6 S 0.3, that is more strongly dipole coupled to the 1B. than B. indicates that B is a biexciton (see text).

192

S. Mazumdar, F. Guo/Synthetic Metals 78 (1996) 187-193

[20], while the level B is the 14Ag state of Fig. 5, the lowest 2e-2h excitation. The state B and states above it to which the dipole moments from the lBu are relatively strong (see Fig. 6(a» constitute a nearly independent electron-like band, as evidenced by the magnitudes of the dipole moments in this region. The occurrence of B below 2XE(1B u ), where E( lBu) is the energy of the 1Bu state, is therefore significant. Having identified the lowest 2e-2h state for the weakly correlated case, we now increase the correlation contribution to the optical gap by gradually reducing 8. For smaller 8, the wavefunctions are highly correlated and simple descriptions of the kind in Fig. 5 are not at all possible. For each 8, we again plot the normalized dipole couplings of all excited Ag states with the lBu against their normalized energies. These results are shown in Figs. 6(b)-(d). The gradual reduction in 8, along with the result of Fig. 6(a), allows us to continue to demarcate between 1e-l hand 2e-2h excitations. It is clear from Fig. 6 that the major effect of reducing 8 (increasing correlations) is a redistribution of dipole moments of Ag states with the lBu: the relative shifts of energy are small within this range of parameters. There should be no doubt whatsoever that in all cases the level labeled B is the lowest two-exciton state. In long chains, as the effective correlations increase, we expect the level B to be lowered in energy, in contrast to what is observed in Fig. 6. This increase in energy of all two• exciton states is a finite size effect that is understandable. For 8=0.5, the excitations are band-like. As 8 decreases, the ground state begins to have strong covalent contributions, and thus the exciton begins to resemble the single ion pair discussed for the strong-coupling limitofEq. (1). Unlike the strong coupling limit though, the particle-hole pair is not localized to nearest neighbors but is spread out over several sites. Since the particle and the hole also tend to avoid the ends of an open chain, the two-exciton states in weakly cor• related short chains are effectively 'squeezed', which raises their energies relative to the lBu with a single ion pair (in other words, finite size effects associated with various eigen states increase with their energies [20]). What is more sig• nificant in the present context, however, is the clear. emer• gence of a level C above B, for 8:s 0.3, which is more strongly dipole-coupled to the 1Bu than B. As discussed above, this is a clear signature of stable biexcitons in the long chain'limit.

6. Discussion and conclusions In the first part of this paper we presented the physical arguments that lead to a general conjecture for the relative strengths of the absorption from the optical exciton to the biexciton and to the two-exciton continuum states. The valid• ity of the conjecture was proved numerically for strong cou• pling limits of Hamiltonians that describe CT excitons, while the results obtained by other investigators [18] for a model describing Frenkel excitons agree with the conjecture. Using this criterion we are able to prove the occurrence of stable

biexcitons in 'iT-conjugated polymers within Eq. (1) for real_ istic parameters. The occurrence of stable biexcitons below the two-exciton continuum can explain a long-standing mys_ tery in the photophysics of conjugated polymers. In many conjugated polymers a high energy (1.4-1.8 eV) ps PA that is distinct from PA due to bipolarons or triplet excitons has been observed [22]. No such PA is expected within single_ particle electron-phonon coupled models. Within the Cou• lomb correlated model, a natural explanation of the p A. emerges as the optical transition from the exciton to the biex_ citon [14]. This assignment has recently been challenged [23], based on the lack of correlation between PA and ph 0toluminescence (PL) in poly(phenylene vinylene) (PPV) at long times. An alternate explanation to ps PA in PPV has been suggested using a two-polaron mechanism [23], within which dissociation of the exciton occurs immediately after its creation due to interchain migration of the electron or the hole, creating two polarons in short polymer segments. P A. within this picture corresponds to polaron absorptions of short chains. The observation of very similar ps PA in the polydi_ acetylenes with long chains and large interchain separations would seem to support the biexciton model. Recent experi_ ments with poly (phenyleneacetylene) [24] find that PA and PL in this system remain correlated up to very long times also supporting our assignment of ps PA. The biexciton' unlike the two-polaron state is an even-parity state that ca~ be observed in direct two-photon absorption (TPA) [11,13],. We propose high-energy TPA measurements in PPV to dis• tinguish between the proposed mechanisms for PA.

Acknowledgements We are grateful to A.J. Epstein and L.J. Rothberg for their kind invitation to contribute to the Proceedings of the Con_ ference on Charge Transport in Electronic Polymers, held in honor of Esther Conwell. Conwell's work in the areas of CT solids as well as 'iT-conjugated polymers has continued to be a source of inspiration for our work for a number of years now. We acknowledge many useful discussions with Profes• sors E. Hanamura, M. Kuwata-Gonokarni, N. Peyghambarian and Z.V. Vardeny, and are particularly grateful to Professor Hanamura for his kind permission to reproduce Fig. 4 from Ref. [18]. This work was supported by the NSF (Grant No: ECS-9 408810), AFOSR (Grant No. F496 209 310 199) and the ONR (Grant No. NOOO 149410322).

References [1] A.S. Davydov. Theory 01 Molecular Excitons, Plenum, New York, 1971. . [2] See, for instance: S. MazumdarandZ.G. Soos,Phys. Rev. B, 23 (1981) 2810. [3] Z.G. S005. S. KuwajimaandR.H. Harding,J. Chem. Phys., 85 (1986) 601 and Refs. therein; N. NagaosaandJ. Takimoto,J. Phys. Soc. Jpn., 55 ( 1986) 2754; A. Painelli and A. Girlando. Phys. Rev. B, 37 ( 1988)

S. Mazumdar, F. Guo / Synthetic Metals 78 (1996) 187-193

[4] [5] [6]

[7]

[8] [9]

[IO] [ 1I]

5748 and Refs. therein; D. Haarer and M.R. Philpott, in V.M. Agranovich and R.M. Hochstrasser (eds.), Spectroscopy and Excitation Dynamics 0/ Condensed Molecular Systems, North• Holland, Amsterdam, 1983, p. 27 and Refs. therein. D. Baeriswyl and A.R. Bishop, Phys. Scr., 19 (1987) 239. A. Painelli and A. Girlando, Synth. Met., 29 (1989) F181. See, for instance: S. Abe, in T. Kobayashi (ed.), Relaxation in Polymers, World Scientific, Singapore, 1994, p. 215; D. Varon and R. Silbey, Phys. Rev. B, 45 (1992) 11 655; D. Guo, S. Mazumdar, S.N. Dixit, F. Kajzar, F. larka, Y. Kawabe and N. Peyghambarian, Phys. Rev. B, 48 (1993) 1433 and Refs. therein; M.I. Rice and Yu.N. Gartstein, Phys. Rev. Lett.. 73 (1994) 2504-2507; D. Mukhopadhyay, G.W. Hayden and Z.G. Soos, Phys. Rev. B,51 (1995) 9476 and Refs. therein. See, for instance: G. Weiser,Phys. Rev. B, 45 (1992) 14067 and Refs. therein; H. Bassler, in D. Bloor and R.B. Chance (eds.), Polydiacetylenes, NATO ASI Series E, No. 102, Martinus Nijhoff, Dordrecht, 1985, pp. 135-153; U. Rauscher, H. Bassler, D.D.C. Bradley and M. Hennecke,Phys. Rev. B, 42 (1990) 9830; R. Kersting, U. Lemmer, M. Deussen, H.I. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler and E.O. Gobel, Phys. Rev. Lett., 73 (1994) 1440. W.T. Simpson, J. Am. Chem. Soc., 77 (1955) 6164; H.C. Longuet• Higgins and I.N. Murrell, Proc. Phys. Soc. London, Ser. A, 68 (1955) 601. I.R.G. Thome, S.T. Repinec, S.A. Abrash, I.M. Zeigler and R.M. Hochstrasser, Chem. Phys., 146 (1990) 315; Z.G. Soos and R.G. Kepler, Phys. Rev. B, 43 (1991) 11 908; T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto and S. Abe, Phys. Rev. Lett.. 668 (1992) 668. O. Akimoto and E. Hanamura, J. Phys. Soc. Jpn., 33 (1972) 1537; E. Hanamura, Solid State Commun., 12 (1973) 951; W.F. Brinkman, T.M. Rice and B. Bell, Phys. Rev. B, 8 (1973) 1570. See, for instance: M. Veta, H. Kanzaki, H. Kobayashi, Y. Toyozawa and E. Hanamura, Excitonic Processes in Solids, Springer, Berlin, 1986; N. Peyghambarian, S.W. Koch and A. Mysyrowicz.lntroduction to Semiconductor Optics, Prentice-Hall, Englewood Cliffs, NJ, 1993.

193

[12] F.C. Spano, V. Agranovich and S. Mukamel, J. Chem. Phys., 95 (1991) 1400. [13] M. Kuwata-Gonokami, N. Peyghambarian, K. Meissner, B. Fluegel, Y. Sato, K. Ema, R. Shimano, S. Mazumdar, F. Guo, T. Tokihiro, H. Ezaki and E. Hanamura, Nature, 367 (1994) 47. [14] F. Guo, M. Chandross and S. Mazumdar, Phys. Rev. Len., 74 (1995) 2086; F. Guo, M. Chandross and S. Mazumdar, Mol. Cryst. Liq. Cryst., 256 (1994) 53. ' [15] V.A. Shakin and S. Abe, Phys. Rev. B, 50 (1994) 4306; V.A. Shakin and S. Abe, Mol. Cryst. Liq. Cryst., 256 ( 1994) 643. [16] D. Baeriswyl, D.K. Campbell and S. Mazumdar, in H. Kiess (ed.), Conjugated Conducting Polymers, Springer, Berlin, 1992. [17] Z.G. Soos, S. Ramasesha, D.S. Galvao and S. Etemad, Phys. Rev. B, 47 (1993) 1742. [18] H. Ezaki, T. Tokihiro and E. Hanamura, Phys. Rev. B, 50 (1994) 10506. [19] D.C. Mattis, The Theory o/Magnetism, Vol.r, Springer Series in Solid State Sciences, Vol. 17, Springer, Berlin, 1988, p. 155. [20] D. Guo, S. Mazumdar and S.N. Dixit, Nonlinear Opt., 6 (1994) 337; D. Guo, S. Mazumdar, S.N. Dixit, F. Kajzar, F. larka, Y. Kawabe and N. Peyghambarian, Phys. Rev. B, 48 (1993) 1433 and Refs. therein. [21] S. Mazumdar, F. Guo, K. Meissner, B. Fluegel, N. Peyghambarian, M. Kuwata-Gonokami, Y. Sato, K. Ema, R. Shimano, T. Tokihiro, H. Ezaki and E. Hanamura, (1996) in press. [22] T. Kobayashi, M. Yoshizawa, U. Stamm, M. Taiji and M. Hasegawa, J. Opt. Soc. Am.. PartB, 7 (1990) 1558; M.B. Sinclair, D. McBranch, T.W. Hagler and A.I. Heeger, Synth. Met., 50 (1992) 593; I.M. Leng, S. leglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo and S. Mazumdar, Phys. Rev. Len., 72 (1994) 156. [23] I.W.P. Hsu, M. Yan, T.M. ledju, L.I. Rothberg and B.R. Hsieh, Phys. Rev. B, 49 (1994) 712; H.A. Mizes and E.M. Conwell, Phys. Rev. B. 50 (1994) 11 243. [24] I.M. Leng, X. Wei, Z.V. Vardeny, Y.W. Ding and T.1. Barton, Mol. Cryst. Liq. Cryst., 256 (1994) 697.