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Polarons and their stability in poly(phenylenevinylene)
SYIfilTI"I TIIC ELSEVIER
Synthetic Metals 68 (1995) 145-151
Polarons and their stability in poly(phenylenevinylene) H.A. Mizes, E.M. Conwell Xerox Corporation, 800 Phillips Road, Building 114-2319, Webster, NY 14580, USA Center for Photoinduced Charge Transfer, University of Rochester, Rochester, NY 14624, USA Received 27 May 1994; accepted 10 August 1994
Abstract To study properties of polarons in poly(phenylenevinylene) (PPV) we set up a tight-binding Hamiltonian along the lines of the SSH H a m i l t o n i a n for polyacetylene. The parameters were determined by fitting the bandgap, valence bandwidth and vinylene geometry to better than 1%. We used the resulting Hamiltonian to calculate the geometry of the polaron and the distance of the polaron level from the closest band edge as a function of chain length. To describe interchain coupling we took into account the actual separation of each atom f i o m the closest atom on the neighboring chain which, unlike polyacetylene, is different for each atom in the P P V monomer. It was found that interchain coupling does not destabilize the polaron in PPV.
Keywords: Polarons; Stability; Poly(phenylenevinylene)
1. I n t r o d u c t i o n
Light-emitting diodes, which appear to be a promising application for poly(phenylenevinylene) (PPV) and some other conducting polymers, rely on the recombination of electrons and holes injected at opposite contacts. It has been expected that an injected electron or hole would cause chain relaxation or deformation, forming a polaron [1]. Recently, however, it has been pointed out that, with sufficient interchain coupling, the polaron could be unstable [2]. Calculations for polyacetylene showed that polarons are unstable in the P21/a structure [3] for long chains [4]. However, it has been shown that the presence of defects tends to stabilize the polaron [4]. Such extrinsic stabilization is expected to be even more effective in PPV because of poor interchain registry. In any case it is important to have a good description of a single-chain polaron in PPV and to determine whether it is intrinsically stable. As a preliminary to studying the polaron we determined the ground-state geometry for a 4-mer of PPV using modified neglect of differential overlap M N D O , known to be accurate for this purpose. We found that the bond lengths agree with earlier quantum-chemical calculations [5,6]. Because M N D O is known to give poor results for open-shell systems, we set up a Hamiltonian similar in spirit to the SSH Hamiltonian to study the polaron [7]. The SSH Hamiltonian has been
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widely used for studying polaron properties in polyacetylene, as well as other properties where elect r o n - p h o n o n coupling is more important than Coulomb effects. We set up the Hamiltonian in Section 2 and discuss our choice of the parameters. SSH-type Hamiltonians for PPV were also set up by Choi and Rice [8] and Shuai et al. [9]. We compare their choices of parameters, and some of the consequences, with ours. In Section 3 we describe the properties of the PPV polaron, specifically geometry and energy levels, obtained with our Hamiltonian. Finally, in Section 4 we discuss the interchain coupling and show that the polaron in PPV is stable.
2. The H a m i l t o n i a n
For the calculations the backbone was assumed to be planar, which should be essentially true in the solid. The Hamiltonian was taken in the form: M
Z = ~
~[-(to-Oat-C)2+H.C.]
rn = 1 (i i)
(1) Here M is the number of monomers in the chain and (ij) indicates one of the pairs of nearest-neighbor carbon atoms in the monomer. The sum is taken over all nine
146
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
pairs, to is the electronic coupling, i.e., the transfer integral, between adjacent p orbitals and u is the change in length of the (ij) bond, referred to an imagined initial state with all bonds equal in length, a is the ratio between the electronic coupling change and bond length change, and K is the effective spring constant due to the cr bonds. C is a stiffness constant adjusted to give the correct chain length in a self-consistent calculation [10]. In principle a and K should be different for different bonds. With complete band-structure information from the local density functional (LDF) calculations [11] and the ground-state geometry determined by MNDO, we have in principle enough information to solve for a and K values for every bond. However, with the form (1) for the Hamiltonian we were not successful in obtaining a set of a and K values that led to physically reasonable results. As indicated earlier, we determined the parameters to, a and K by fitting three properties of PPV. One of these properties was the valence bandwidth (VBW), 5.47 eV, obtained in the L D F calculations [11]. A second was the energy gap, taken as the sum of the observed optical absorption edge, 2.4 eV, and the calculated exciton binding energy, 0.4 eV [12]. The third was the dimerization, or the difference between single and double bond lengths of the vinyl group obtained from M N D O calculations. These calculations gave the double bond as 1.355 A, and the single bond as 1.474 ,~. The difference of 0.119 ~ is almost 40% larger than the 0.08 .~ difference found in polyacetylene, but less than the difference between double and single bonds in nonconjugated polymers and other organic compounds. For each possible set of to, a and K, a self-consistent calculation on a long chain was performed in order to determine the values of the three physical properties chosen. The parameters were adjusted until the VBW, energy gap and dimerization were matched to better than 1%. We found that choosing a = 10.29 eV/A, K = 99.0 e g / ~ 2 and to = 2.66 eV gave a dimerization of 0.118 ~ , an energy gap of 2.77 eV and a valence bandwidth of 5.49 eV. With these parameters in the Hamiltonian (1), features of the L D F electronic structure and M N D O geometric structure that were not specifically fitted are matched fairly well. PPV has eight 7r-bands corresponding to the eight carbon atoms in the monomer, four comprising the valence band and four the conduction band. The midpoints of the L D F bands, relative to the almost flat band on the rings, are 0.84, 0.00, - 1 . 5 0 and - 3 . 2 7 eV. The Hamiltonian (1) calculated band midpoints are 0.70, 0.00, - 1 . 2 4 and - 3 . 3 3 eV, respectively. The L D F calculated bandwidths are 2.39, 0.01, 1.49 and 0.32 eV. The Hamiltonian (1) calculated bandwidths are 2.05, 0.00, 1.81 and 0.46 eV, respectively. The M N D O ground-state structure gives a difference of 0.015 ~ between the upper and side bonds in the
aromatic ring. With the Hamiltonian (1) the difference is also 0.015 A. PPV was also described with a SSH Hamiltonian by Choi and Rice (CR) [8] and by Shuai et al. (SBB) [9]. A comparison of their parameters with those we are using, and of the electronic structure predicted by these parameters, is summarized in Table 1. CR were interested in the energetics of the excited states in PPV. They fixed to=2.5 eV and K = 3 5 . 0 e V / ~ to fit the experimental gap between the two bipolaron levels as measured by Bradley et al. [13]. However, these parameters give a bandgap 0.3 eV too large and overestimate the dimerization of the vinyl group by a factor of two. The large dimerization occurs because of the large electron-phonon coupling (proportional to the ratio a/K), which also gives deep polaron and bipolaron levels, as shown in Table 1. SBB chose the SSH-type parameters to reproduce the geometric and 7r-band electronic structure of the PPV chain obtained from V E H (valence effective Hamiltonian) calculations. T o do this they used a bond o r d e r - b o n d type relationship of the Coulson type, which depends exponentially on distance. However, for the displacements that occur, linearization of the bond order, as done in the SSH approximation, results in a deviation from the exponentials of less than 1%. SBB also generalize the problem so that the bond order-bond type relationships for the aromatic rings are different from those for the vinyl groups. Their expression for the couplings can be put in terms of equivalent SSH a and K, which are given in Table 1. The SBB parameters give properties closer to our parameters. However, they obtain a larger VBW due to the larger to. This makes the electron-phonon coupling less effective and they obtain energy levels for the polaron and bipolaron that are closer to the band edge than ours, as seen in Table 1. The distance of the top bipolaron level from the conduction band edge has been found by Colaneri et al. [14] using photoinduced absorption to be 0.6 eV. Although our parameters have been fitted to groundstate properties, they predict the top bipolaron level to be 0.54 eV below the conduction band edge, in much better agreement with experiment than the CR or SBB results. However, our calculation predicts the distance from the lower bipolaron to the conduction band edge to be 2.23 eV, in poor agreement with the experimental value of 1.40 eV [14]. The disagreement is not unexpected. It is due to the neglect of the repulsion between the two electrons, or holes, in the bipolaron and to the symmetry of the SSH-type Hamiltonian [15]. 3. Calculated properties of the PPV polaron Electron microscope measurements have shown that PPV contains crystalline regions of about 50 or 100
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
147
Table 1 Parameters for three SSH-like theories and their predictions for PPV Mizes-Conwell
Choi-Rice"
Shuai-Beljonne-Br6das b
to (eV)
2.66
2.72
3.47 (ring) 3.24 (vinyl)
a (eV/~)
10.29
6.53
5.72 (ring) 5.40 (vinyl)
K ( e V / ~ 2)
99.0
35.0
50.8 (ring) 31.4 (vinyl)
Gap (eV) VBW (eV) Dimerization (/~) c Polaron E (eV) a
2.79 5.46 0.118 0.189
3.12 4.64 0.239 0.322
2.65 6.40 0.120 0.090
Ref. 8. b Ref. 9. c Difference between single and double bond lengths of the vinyl group. a Distance of polaron energy level from the band edge.
in diameter, or about 8 to 16 monomers, separated by disordered regions [16]. X-ray measurements give coherence lengths of 50 [17] to 90 ~ [18]. A number of other measures of conjugation length suggests about seven monomers as an average. It has been pointed out by several researchers that interchain coherence of PPV is poor [16,18]. In Fig. 1 we plot the calculated energy levels for anionic oligomers of different lengths. For the longest chains the polaron levels are seen as a pair of discrete levels detached from a continuum. For these chains the distance of the polaron levels from the closest band edge is 0.19 eV. As the chains grow shorter the location of the polaron levels does not change much, but the distance between the polaron levels and the adjacent 'band' levels increases quite considerably, reflecting the strong increase of the energy gap with decreasing chain length. The increase of polaron level to band edge spacing is shown on a larger scale in Fig. 2. The effect of the polaron on a chain may be visualized through plots of the chain distortion, the total charge 10
5
:111 -I!i
v
aJ E L,J
-5
-10
- -
i
m
m
m
J
i
i
i
2
3
5
10
Number
of
i
20
mm
i
30
m
I
60
monomers
Fig. 1. "rr-Band energy levels calculated using the Hamiltonian (1) for various chain lengths. The chains have a single negative charge.
2.0
& .c_ ! . 5 u 0 (:L (/I
0 1.0
E O
I 1-
8 0.5
O
O
o.. 0.0
0
0
0
i
i
I
I
I
i
I
2
3
5
10
20
30
60
Number
0
of m o n o m e r s
Fig. 2. Distance from polaron level to the band edge as a function of chain length.
per atom or the potaron wave function. These three quantities are plotted versus position for a ten-monomer long chain in Fig. 3. To identify the carbon atoms in the monomer we use the numbering scheme shown in Fig. 4. In Fig. 3(a) where bond lengths are plotted versus chain position, the two single bonds per monomer show up as the pair of repeating peaks, while the double bond per monomer corresponds to the local minimum. One sees that near the center of the chain where the charge is localized the single bonds become shorter and the double bonds become longer. It is also seen that the top and bottom bonds on the ring decrease in length at the center of the polaron, i.e., the ring becomes slightly quinoidal. Fig. 3(b) shows that most of the charge resides on the vinyl group. The amplitude of the distortion falls to 10% at a distance 4.2 monomers from the center of the polaron, while 90% of the charge is contained within 5.8 monomers. From comparison of Fig. 3(b) and (c) it is seen that the extent of the wave function of the polaron is somewhat larger than that of the charge distribution. Comparing the extent
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
148
(a) 1.50
.
.
.
.
J= 1.45
~:~ 1.40
1.35 1.30
i-
,
2
o
-
,
-
,
,
4 6 Chain position (monomers)
i
8
10
8
10
8
10
(b)
0.08 [C3 vinyl carbons (I&8) ~" ~-~,ring end carbons (2&7) o 0.06 I-o left side carbons (3 &4) "~ fo right sid. . . . b°ns (5&6) [ ~
0
2
[~ i~,
~
4
6
Choln position (monomers)
(c) 0.30 o. E
.co
0.20 0.10 0.00 -O.lO
-o.2o o
-0.50 2
4 6 Chain position (monomers)
Fig. 3. (a) Bond lengths of each of the bonds on a fully relaxed ten-monomer chain with one extra electron, calculated using the Hamiltonian (1). The bonds are identified by the symbols in the plot. The lines indicate which bonds adjoin a given bond. (b) Total charge on each atom for a negatively charged PPV ten-monomer chain. Note that the lines do not join nearest-neighbor atoms. The atoms are identified by the numbers in Fig. 4. (c) Amplitude of the H O M O , or polaron wave function, for the same chain, with symbols identifying the atoms given in (b) and Fig. 4.
/
l 1- - 2
\
3~5
\
/ 4
1 7 - - 8
6
Fig. 4. Number scheme used to identify the carbon atoms in the PPV m o n o m e r in the text and Fig. 3.
of the bond distortion obtained with the Hamiltonian (1) with that obtained by MNDO, we find the latter to be about half as large. As noted earlier, the disagreement is not surprising because M N D O is known to have poor accuracy for open-shell calculations. Significantly, a calculation of the polaron geometry using the unrestricted Hartree-Fock PM3 method gave results in good agreement with those from Eq. (1) for the four central monomers of the chain. This agreement is additional evidence for the validity of our Hamiltonian.
Interchain coupling effects on polaron stability in polyacetylene have been explored by a number of groups [19]. In crystalline form, the polyacetylene chains line up in a herringbone structure with a two-chain unit cell. Calculations that have tested the effects of interchain coupling have only considered the two neighboring nonequivalent chains in the unit cell. Each atom on one chain lies opposite its counterpart on the other chain. The interchain coupling has been incorporated into the SSH Hamiltonian by adding a term which couples the two carbon atoms opposite each other with a coupling that may be different for the two pairs of atoms in the unit cell. It should be noted, however, that calculations for polyacetylene have shown that the interchain coupling is not directly between two carbon 7r orbitals, but is instead mediated by the p level on the hydrogen bonded to one of these carbon atoms [3]. The direct coupling between the carbon ~- orbitals in the SSH calculation is thus an approximation to the true coupling. Earlier we calculated the stability of a polaron in polyacetylene in the P2~/a structure for a pair of chains and for a cluster of 19 chains [4]. To do this we added to the SSH Hamiltonian [7] the term: ~ e d_ = E (t 2_)i,j,k(Ci+,j Ci,k "~ Ci+,kCi,j) i,j,k
(2)
where (t±)ij.k represents the coupling between the ith carbon on chain j and that on chain k. This coupling was chosen to fit the splitting obtained in the LDF calculation [3]. We found, in agreement with Ref. [3], that for long chains the polaron was unstable at this interchain coupling. Conjugation breaks were found, however, to stabilize the polaron [4]. To obtain the maximum effect of interchain coupling on the polaron in PPV we assumed an ordered structure. The PPV crystalline solid also forms a herringbone structure with two nonequivalent chains in the unit cell [20]. Each of the 16 carbon atoms in the two-chain unit cell (eight from each chain) has a different spacing to the nearest hydrogen atom. The coupling should be exponentially dependent on the distance between the carbon on one chain and the hydrogen on the other chain, and should therefore be dominated by those pairs that are most closely spaced. From the crystal structure one can determine these spacings. For the P2fla structure the three most closely spaced pairs are shown in Fig. 5. The interchain coupling for PPV can be quantified according to the principles just stated by using the results of the LDF calculations. Because each chain has four inequivalent chains as neighbors, we take the calculated splitting at the band edge to be 4(t±), (t±) being defined as the average interchain transfer integral for a pair of atoms. The total coupling for a pair of monomers, to be denoted t . . . . is then 8(t±). To dis-
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
Fig. 5. Three strongest interchain couplings in PPV. The chain has been distorted for clarity. Note that the chains are not in the same plane and are in fact rotated in opposite directions from the plane. t~o, = 0.64 eV. (a) 1.50 v
1.45
~
1.40
,c
.
.
.
.
o 1.35[ 1.30
1
r
,
0
2
"
,
-
,
,
4 6 Choln I position (monomers)
8
]
10
(b) 1,50
.
.
. 2
.
.
.
J: 1.45 ~m co 1.40
o
1.35
F
1.30 t 0
. 4
.
1
. 6
8
1 10
Chain 2 position (monomers)
Fig. 6. B o n d lengths calculated w i t h the H a m i l t o n i a n ( I ) plus (2) for two coupled PPV chains with one extra electron. 90% of the charge sits on the top chain.
tribute this total coupling among the eight atom pairs, we took the effective coupling between the ith carbon on the jth chain and that on the neighboring chain, the kth, to be
149
reside almost entirely on one chain. The charge on the chain with the distortion is 0.897 electrons, with 0.103 electrons on the other chain. The result is in contrast to that for polyacetylene in the e21/a structure, where the instability is manifest in the charge being distributed equally on the two chains. The electronic structure for two 20-monomer coupled PPV chains with one extra electron is shown in Fig. 7. The coupled chain calculation clearly shows a level localized in the gap. As a result of the coupling, however, the spacing between the polaron level and the conduction band minimum is only 0.13 eV instead of the 0.19 eV found for the single chain. Our conclusion about the intrinsic stability of the polaron depends, of course, on the parameters of our calculation. It is possible, for example, to question whether the correct value for the gap is 2.8 eV. Experimental evidence to back this value was obtained by Marks et al. [21] from a study of the photovoltaic response in PPV thin-film devices. Their measurements led them to an exciton binding energy of about 0.4 eV [21], thus a gap of 2.8 eV. However, Lee et al. [22] suggest that the gap is 2.4 eV, corresponding to the absorption edge. Leng et al. [23], on the other hand, claim that the exciton binding energy is 1.1 eV, which would make the gap 3.5 eV. We believe, however, that such a large binding energy is consistent only with a highly confined exciton (Frenkel exciton) rather than one that is known to be spread over at least six or seven monomers [24,25]. The conclusion of Lee et al. [22] is based on the fact that photoconductivity of PPV is observed at the absorption edge, their argument being that if excitons were formed rather than free electrons and holes, photoconductivity would require photons with energy greater than that of the absorption edge
(t ±)id.k = Z exp( ~lij.ik) (3) where du.ik is the distance between the ith carbon on -
j and the hydrogen coupled to the ith carbon on k, is the inverse decay length for the change in the effective coupling with do, ~ and A is an adjustable parameter chosen to give the correct effective coupling. We choose /x = 1.18 ~ - a, which is the rate of decay of an electron in a carbon p orbital. We assume the hydrogen p level decays much less rapidly. With this value of ~ and the crystallographically measured interatom distances [20], the relative values of the coupling for the three closest pairs are as shown in Fig. 5. To complete the determination of t± we use tmon=0.64 eV [11]. The chain distortion of two coupled ten-monomer PPV chains with one extra electron, calculated selfconsistently from Eqs. (1) and (2) with the t± values of Fig. 5, is shown in Fig. 6. The polaron is seen to
Ld
-1
-2 coupled chains
uncoupled chains
Fig. 7. Energy levels near the gap for two 20-monomer PPV chains. For the left set, the chains were coupled as described with tmo, = 0.64 eV, while the right set of levels is for uncoupled chains.
150
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
by the binding energy of the exciton. It must be pointed out, however, that the ps photoconductivity, large as it is, requires that only about 1% of the photons give rise to carriers in the case of polyacetylene [26]. This is also expected to be true for PPV [27]. The observed photoconductivity could be accounted for by dissociation of a small number of excitons, which could occur at the surface or at defects or impurities, or a small number of directly photogenerated polarons [28]. Nevertheless, we have carried out some calculations for Eg = 2.4 eV, arbitrarily keeping the same values of the other two properties we used to determine to, a and K. The resulting a and K values were considerably smaller, and the polaron was found to be unstable for this set of parameters. However, as discussed above, we do not believe Eg = 2.4 eV is correct. Photogeneration experiments on PPV do not give direct information about creation of polarons, apparently because a large number of these polarons, perhaps almost all, are bound in polaron pairs on adjacent chains [28,29]. The large majority of these recombine geminately [28]. The fact that no bipolarons are seen in more ordered samples [17] could be due to there being more polaron pairs formed in such samples, leaving fewer free polarons to ultimately find each other and combine into bipolarons. The one experiment that has given convincing evidence for the existence of polarons was that of Ziemelis et al. [30], who observed photoinduced absorption due to carriers injected at the contacts. The observation is possible because polaron pairs could not be formed except in the small region where the positive polarons injected at one contact meet the negative polarons injected at the other. This experiment gave the distance of the polaron level from the valence band as 0.4 eV [30]. The experiment was done on poly(3-hexylthiophene) (P3HT), but there is every reason to believe a similar result would be obtained for PPV. If it had been PPV, the average chain length, according to Fig. 2, would have been five or six monomers, or about 40 .A, which appears reasonable "for P3HT.
4. Conclusions
The parameters for PPV obtained by fitting our SSHtype Hamiltonian to the bandgap, valence bandwidth and vinylene geometry are: to = 2.66 eV, a = 10.29 eV/ /~ and K = 99.0 eV//k 2. Larger values of these parameters than those for polyacetylene are expected because the bandgap is twice as large, the vinyl dimerization 40% larger and the valence bandwidth 10% larger. With these values of the parameters the features of the LDFcalculated 7r bands and the M N D O calculated groundstate geometry are all matched well or fairly well. Further evidence for the validity of our Hamiltonian
is the fact that the PPV polaron geometry calculated using the unrestricted H a r t r e e - F o c k PM3 method agrees well with that calculated using our Hamiltonian. For the long-chain limit the energy difference found between the polaron level and the nearest band edge is 0.19 eV. The energy difference is much greater for short chains because the bandgap increases rapidly as the chains grow shorter. The distortion due to the polaron is spread over about eight monomers, its amplitude at a distance of four monomers from the center having decreased to about 10% of its value at the center. The interchain coupling due to the eight pairs of carbon atoms in a pair of monomers is determined by relating it to the splitting due to interchain interactions found in the L D F calculations [11]. This coupling is apportioned among the eight atom pairs by taking into account its exponential dependence on the distance between the atoms of a pair, the latter known from crystallographic studies. Only three pairs out of the eight have sufficient interchain coupling to affect any calculated results, t . for these three ranging from 0.33 to 0.12 eV. With the interchain coupling so determined, and the other parameters we believe to be correct for PPV, the polaron is found to be intrinsically stable.
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Synthetic Metals 68 (1995) 145-151
Polarons and their stability in poly(phenylenevinylene) H.A. Mizes, E.M. Conwell Xerox Corporation, 800 Phillips Road, Building 114-2319, Webster, NY 14580, USA Center for Photoinduced Charge Transfer, University of Rochester, Rochester, NY 14624, USA Received 27 May 1994; accepted 10 August 1994
Abstract To study properties of polarons in poly(phenylenevinylene) (PPV) we set up a tight-binding Hamiltonian along the lines of the SSH H a m i l t o n i a n for polyacetylene. The parameters were determined by fitting the bandgap, valence bandwidth and vinylene geometry to better than 1%. We used the resulting Hamiltonian to calculate the geometry of the polaron and the distance of the polaron level from the closest band edge as a function of chain length. To describe interchain coupling we took into account the actual separation of each atom f i o m the closest atom on the neighboring chain which, unlike polyacetylene, is different for each atom in the P P V monomer. It was found that interchain coupling does not destabilize the polaron in PPV.
Keywords: Polarons; Stability; Poly(phenylenevinylene)
1. I n t r o d u c t i o n
Light-emitting diodes, which appear to be a promising application for poly(phenylenevinylene) (PPV) and some other conducting polymers, rely on the recombination of electrons and holes injected at opposite contacts. It has been expected that an injected electron or hole would cause chain relaxation or deformation, forming a polaron [1]. Recently, however, it has been pointed out that, with sufficient interchain coupling, the polaron could be unstable [2]. Calculations for polyacetylene showed that polarons are unstable in the P21/a structure [3] for long chains [4]. However, it has been shown that the presence of defects tends to stabilize the polaron [4]. Such extrinsic stabilization is expected to be even more effective in PPV because of poor interchain registry. In any case it is important to have a good description of a single-chain polaron in PPV and to determine whether it is intrinsically stable. As a preliminary to studying the polaron we determined the ground-state geometry for a 4-mer of PPV using modified neglect of differential overlap M N D O , known to be accurate for this purpose. We found that the bond lengths agree with earlier quantum-chemical calculations [5,6]. Because M N D O is known to give poor results for open-shell systems, we set up a Hamiltonian similar in spirit to the SSH Hamiltonian to study the polaron [7]. The SSH Hamiltonian has been
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widely used for studying polaron properties in polyacetylene, as well as other properties where elect r o n - p h o n o n coupling is more important than Coulomb effects. We set up the Hamiltonian in Section 2 and discuss our choice of the parameters. SSH-type Hamiltonians for PPV were also set up by Choi and Rice [8] and Shuai et al. [9]. We compare their choices of parameters, and some of the consequences, with ours. In Section 3 we describe the properties of the PPV polaron, specifically geometry and energy levels, obtained with our Hamiltonian. Finally, in Section 4 we discuss the interchain coupling and show that the polaron in PPV is stable.
2. The H a m i l t o n i a n
For the calculations the backbone was assumed to be planar, which should be essentially true in the solid. The Hamiltonian was taken in the form: M
Z = ~
~[-(to-Oat
rn = 1 (i i)
(1) Here M is the number of monomers in the chain and (ij) indicates one of the pairs of nearest-neighbor carbon atoms in the monomer. The sum is taken over all nine
146
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
pairs, to is the electronic coupling, i.e., the transfer integral, between adjacent p orbitals and u is the change in length of the (ij) bond, referred to an imagined initial state with all bonds equal in length, a is the ratio between the electronic coupling change and bond length change, and K is the effective spring constant due to the cr bonds. C is a stiffness constant adjusted to give the correct chain length in a self-consistent calculation [10]. In principle a and K should be different for different bonds. With complete band-structure information from the local density functional (LDF) calculations [11] and the ground-state geometry determined by MNDO, we have in principle enough information to solve for a and K values for every bond. However, with the form (1) for the Hamiltonian we were not successful in obtaining a set of a and K values that led to physically reasonable results. As indicated earlier, we determined the parameters to, a and K by fitting three properties of PPV. One of these properties was the valence bandwidth (VBW), 5.47 eV, obtained in the L D F calculations [11]. A second was the energy gap, taken as the sum of the observed optical absorption edge, 2.4 eV, and the calculated exciton binding energy, 0.4 eV [12]. The third was the dimerization, or the difference between single and double bond lengths of the vinyl group obtained from M N D O calculations. These calculations gave the double bond as 1.355 A, and the single bond as 1.474 ,~. The difference of 0.119 ~ is almost 40% larger than the 0.08 .~ difference found in polyacetylene, but less than the difference between double and single bonds in nonconjugated polymers and other organic compounds. For each possible set of to, a and K, a self-consistent calculation on a long chain was performed in order to determine the values of the three physical properties chosen. The parameters were adjusted until the VBW, energy gap and dimerization were matched to better than 1%. We found that choosing a = 10.29 eV/A, K = 99.0 e g / ~ 2 and to = 2.66 eV gave a dimerization of 0.118 ~ , an energy gap of 2.77 eV and a valence bandwidth of 5.49 eV. With these parameters in the Hamiltonian (1), features of the L D F electronic structure and M N D O geometric structure that were not specifically fitted are matched fairly well. PPV has eight 7r-bands corresponding to the eight carbon atoms in the monomer, four comprising the valence band and four the conduction band. The midpoints of the L D F bands, relative to the almost flat band on the rings, are 0.84, 0.00, - 1 . 5 0 and - 3 . 2 7 eV. The Hamiltonian (1) calculated band midpoints are 0.70, 0.00, - 1 . 2 4 and - 3 . 3 3 eV, respectively. The L D F calculated bandwidths are 2.39, 0.01, 1.49 and 0.32 eV. The Hamiltonian (1) calculated bandwidths are 2.05, 0.00, 1.81 and 0.46 eV, respectively. The M N D O ground-state structure gives a difference of 0.015 ~ between the upper and side bonds in the
aromatic ring. With the Hamiltonian (1) the difference is also 0.015 A. PPV was also described with a SSH Hamiltonian by Choi and Rice (CR) [8] and by Shuai et al. (SBB) [9]. A comparison of their parameters with those we are using, and of the electronic structure predicted by these parameters, is summarized in Table 1. CR were interested in the energetics of the excited states in PPV. They fixed to=2.5 eV and K = 3 5 . 0 e V / ~ to fit the experimental gap between the two bipolaron levels as measured by Bradley et al. [13]. However, these parameters give a bandgap 0.3 eV too large and overestimate the dimerization of the vinyl group by a factor of two. The large dimerization occurs because of the large electron-phonon coupling (proportional to the ratio a/K), which also gives deep polaron and bipolaron levels, as shown in Table 1. SBB chose the SSH-type parameters to reproduce the geometric and 7r-band electronic structure of the PPV chain obtained from V E H (valence effective Hamiltonian) calculations. T o do this they used a bond o r d e r - b o n d type relationship of the Coulson type, which depends exponentially on distance. However, for the displacements that occur, linearization of the bond order, as done in the SSH approximation, results in a deviation from the exponentials of less than 1%. SBB also generalize the problem so that the bond order-bond type relationships for the aromatic rings are different from those for the vinyl groups. Their expression for the couplings can be put in terms of equivalent SSH a and K, which are given in Table 1. The SBB parameters give properties closer to our parameters. However, they obtain a larger VBW due to the larger to. This makes the electron-phonon coupling less effective and they obtain energy levels for the polaron and bipolaron that are closer to the band edge than ours, as seen in Table 1. The distance of the top bipolaron level from the conduction band edge has been found by Colaneri et al. [14] using photoinduced absorption to be 0.6 eV. Although our parameters have been fitted to groundstate properties, they predict the top bipolaron level to be 0.54 eV below the conduction band edge, in much better agreement with experiment than the CR or SBB results. However, our calculation predicts the distance from the lower bipolaron to the conduction band edge to be 2.23 eV, in poor agreement with the experimental value of 1.40 eV [14]. The disagreement is not unexpected. It is due to the neglect of the repulsion between the two electrons, or holes, in the bipolaron and to the symmetry of the SSH-type Hamiltonian [15]. 3. Calculated properties of the PPV polaron Electron microscope measurements have shown that PPV contains crystalline regions of about 50 or 100
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
147
Table 1 Parameters for three SSH-like theories and their predictions for PPV Mizes-Conwell
Choi-Rice"
Shuai-Beljonne-Br6das b
to (eV)
2.66
2.72
3.47 (ring) 3.24 (vinyl)
a (eV/~)
10.29
6.53
5.72 (ring) 5.40 (vinyl)
K ( e V / ~ 2)
99.0
35.0
50.8 (ring) 31.4 (vinyl)
Gap (eV) VBW (eV) Dimerization (/~) c Polaron E (eV) a
2.79 5.46 0.118 0.189
3.12 4.64 0.239 0.322
2.65 6.40 0.120 0.090
Ref. 8. b Ref. 9. c Difference between single and double bond lengths of the vinyl group. a Distance of polaron energy level from the band edge.
in diameter, or about 8 to 16 monomers, separated by disordered regions [16]. X-ray measurements give coherence lengths of 50 [17] to 90 ~ [18]. A number of other measures of conjugation length suggests about seven monomers as an average. It has been pointed out by several researchers that interchain coherence of PPV is poor [16,18]. In Fig. 1 we plot the calculated energy levels for anionic oligomers of different lengths. For the longest chains the polaron levels are seen as a pair of discrete levels detached from a continuum. For these chains the distance of the polaron levels from the closest band edge is 0.19 eV. As the chains grow shorter the location of the polaron levels does not change much, but the distance between the polaron levels and the adjacent 'band' levels increases quite considerably, reflecting the strong increase of the energy gap with decreasing chain length. The increase of polaron level to band edge spacing is shown on a larger scale in Fig. 2. The effect of the polaron on a chain may be visualized through plots of the chain distortion, the total charge 10
5
:111 -I!i
v
aJ E L,J
-5
-10
- -
i
m
m
m
J
i
i
i
2
3
5
10
Number
of
i
20
mm
i
30
m
I
60
monomers
Fig. 1. "rr-Band energy levels calculated using the Hamiltonian (1) for various chain lengths. The chains have a single negative charge.
2.0
& .c_ ! . 5 u 0 (:L (/I
0 1.0
E O
I 1-
8 0.5
O
O
o.. 0.0
0
0
0
i
i
I
I
I
i
I
2
3
5
10
20
30
60
Number
0
of m o n o m e r s
Fig. 2. Distance from polaron level to the band edge as a function of chain length.
per atom or the potaron wave function. These three quantities are plotted versus position for a ten-monomer long chain in Fig. 3. To identify the carbon atoms in the monomer we use the numbering scheme shown in Fig. 4. In Fig. 3(a) where bond lengths are plotted versus chain position, the two single bonds per monomer show up as the pair of repeating peaks, while the double bond per monomer corresponds to the local minimum. One sees that near the center of the chain where the charge is localized the single bonds become shorter and the double bonds become longer. It is also seen that the top and bottom bonds on the ring decrease in length at the center of the polaron, i.e., the ring becomes slightly quinoidal. Fig. 3(b) shows that most of the charge resides on the vinyl group. The amplitude of the distortion falls to 10% at a distance 4.2 monomers from the center of the polaron, while 90% of the charge is contained within 5.8 monomers. From comparison of Fig. 3(b) and (c) it is seen that the extent of the wave function of the polaron is somewhat larger than that of the charge distribution. Comparing the extent
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
148
(a) 1.50
.
.
.
.
J= 1.45
~:~ 1.40
1.35 1.30
i-
,
2
o
-
,
-
,
,
4 6 Chain position (monomers)
i
8
10
8
10
8
10
(b)
0.08 [C3 vinyl carbons (I&8) ~" ~-~,ring end carbons (2&7) o 0.06 I-o left side carbons (3 &4) "~ fo right sid. . . . b°ns (5&6) [ ~
0
2
[~ i~,
~
4
6
Choln position (monomers)
(c) 0.30 o. E
.co
0.20 0.10 0.00 -O.lO
-o.2o o
-0.50 2
4 6 Chain position (monomers)
Fig. 3. (a) Bond lengths of each of the bonds on a fully relaxed ten-monomer chain with one extra electron, calculated using the Hamiltonian (1). The bonds are identified by the symbols in the plot. The lines indicate which bonds adjoin a given bond. (b) Total charge on each atom for a negatively charged PPV ten-monomer chain. Note that the lines do not join nearest-neighbor atoms. The atoms are identified by the numbers in Fig. 4. (c) Amplitude of the H O M O , or polaron wave function, for the same chain, with symbols identifying the atoms given in (b) and Fig. 4.
/
l 1- - 2
\
3~5
\
/ 4
1 7 - - 8
6
Fig. 4. Number scheme used to identify the carbon atoms in the PPV m o n o m e r in the text and Fig. 3.
of the bond distortion obtained with the Hamiltonian (1) with that obtained by MNDO, we find the latter to be about half as large. As noted earlier, the disagreement is not surprising because M N D O is known to have poor accuracy for open-shell calculations. Significantly, a calculation of the polaron geometry using the unrestricted Hartree-Fock PM3 method gave results in good agreement with those from Eq. (1) for the four central monomers of the chain. This agreement is additional evidence for the validity of our Hamiltonian.
Interchain coupling effects on polaron stability in polyacetylene have been explored by a number of groups [19]. In crystalline form, the polyacetylene chains line up in a herringbone structure with a two-chain unit cell. Calculations that have tested the effects of interchain coupling have only considered the two neighboring nonequivalent chains in the unit cell. Each atom on one chain lies opposite its counterpart on the other chain. The interchain coupling has been incorporated into the SSH Hamiltonian by adding a term which couples the two carbon atoms opposite each other with a coupling that may be different for the two pairs of atoms in the unit cell. It should be noted, however, that calculations for polyacetylene have shown that the interchain coupling is not directly between two carbon 7r orbitals, but is instead mediated by the p level on the hydrogen bonded to one of these carbon atoms [3]. The direct coupling between the carbon ~- orbitals in the SSH calculation is thus an approximation to the true coupling. Earlier we calculated the stability of a polaron in polyacetylene in the P2~/a structure for a pair of chains and for a cluster of 19 chains [4]. To do this we added to the SSH Hamiltonian [7] the term: ~ e d_ = E (t 2_)i,j,k(Ci+,j Ci,k "~ Ci+,kCi,j) i,j,k
(2)
where (t±)ij.k represents the coupling between the ith carbon on chain j and that on chain k. This coupling was chosen to fit the splitting obtained in the LDF calculation [3]. We found, in agreement with Ref. [3], that for long chains the polaron was unstable at this interchain coupling. Conjugation breaks were found, however, to stabilize the polaron [4]. To obtain the maximum effect of interchain coupling on the polaron in PPV we assumed an ordered structure. The PPV crystalline solid also forms a herringbone structure with two nonequivalent chains in the unit cell [20]. Each of the 16 carbon atoms in the two-chain unit cell (eight from each chain) has a different spacing to the nearest hydrogen atom. The coupling should be exponentially dependent on the distance between the carbon on one chain and the hydrogen on the other chain, and should therefore be dominated by those pairs that are most closely spaced. From the crystal structure one can determine these spacings. For the P2fla structure the three most closely spaced pairs are shown in Fig. 5. The interchain coupling for PPV can be quantified according to the principles just stated by using the results of the LDF calculations. Because each chain has four inequivalent chains as neighbors, we take the calculated splitting at the band edge to be 4(t±), (t±) being defined as the average interchain transfer integral for a pair of atoms. The total coupling for a pair of monomers, to be denoted t . . . . is then 8(t±). To dis-
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
Fig. 5. Three strongest interchain couplings in PPV. The chain has been distorted for clarity. Note that the chains are not in the same plane and are in fact rotated in opposite directions from the plane. t~o, = 0.64 eV. (a) 1.50 v
1.45
~
1.40
,c
.
.
.
.
o 1.35[ 1.30
1
r
,
0
2
"
,
-
,
,
4 6 Choln I position (monomers)
8
]
10
(b) 1,50
.
.
. 2
.
.
.
J: 1.45 ~m co 1.40
o
1.35
F
1.30 t 0
. 4
.
1
. 6
8
1 10
Chain 2 position (monomers)
Fig. 6. B o n d lengths calculated w i t h the H a m i l t o n i a n ( I ) plus (2) for two coupled PPV chains with one extra electron. 90% of the charge sits on the top chain.
tribute this total coupling among the eight atom pairs, we took the effective coupling between the ith carbon on the jth chain and that on the neighboring chain, the kth, to be
149
reside almost entirely on one chain. The charge on the chain with the distortion is 0.897 electrons, with 0.103 electrons on the other chain. The result is in contrast to that for polyacetylene in the e21/a structure, where the instability is manifest in the charge being distributed equally on the two chains. The electronic structure for two 20-monomer coupled PPV chains with one extra electron is shown in Fig. 7. The coupled chain calculation clearly shows a level localized in the gap. As a result of the coupling, however, the spacing between the polaron level and the conduction band minimum is only 0.13 eV instead of the 0.19 eV found for the single chain. Our conclusion about the intrinsic stability of the polaron depends, of course, on the parameters of our calculation. It is possible, for example, to question whether the correct value for the gap is 2.8 eV. Experimental evidence to back this value was obtained by Marks et al. [21] from a study of the photovoltaic response in PPV thin-film devices. Their measurements led them to an exciton binding energy of about 0.4 eV [21], thus a gap of 2.8 eV. However, Lee et al. [22] suggest that the gap is 2.4 eV, corresponding to the absorption edge. Leng et al. [23], on the other hand, claim that the exciton binding energy is 1.1 eV, which would make the gap 3.5 eV. We believe, however, that such a large binding energy is consistent only with a highly confined exciton (Frenkel exciton) rather than one that is known to be spread over at least six or seven monomers [24,25]. The conclusion of Lee et al. [22] is based on the fact that photoconductivity of PPV is observed at the absorption edge, their argument being that if excitons were formed rather than free electrons and holes, photoconductivity would require photons with energy greater than that of the absorption edge
(t ±)id.k = Z exp( ~lij.ik) (3) where du.ik is the distance between the ith carbon on -
j and the hydrogen coupled to the ith carbon on k, is the inverse decay length for the change in the effective coupling with do, ~ and A is an adjustable parameter chosen to give the correct effective coupling. We choose /x = 1.18 ~ - a, which is the rate of decay of an electron in a carbon p orbital. We assume the hydrogen p level decays much less rapidly. With this value of ~ and the crystallographically measured interatom distances [20], the relative values of the coupling for the three closest pairs are as shown in Fig. 5. To complete the determination of t± we use tmon=0.64 eV [11]. The chain distortion of two coupled ten-monomer PPV chains with one extra electron, calculated selfconsistently from Eqs. (1) and (2) with the t± values of Fig. 5, is shown in Fig. 6. The polaron is seen to
Ld
-1
-2 coupled chains
uncoupled chains
Fig. 7. Energy levels near the gap for two 20-monomer PPV chains. For the left set, the chains were coupled as described with tmo, = 0.64 eV, while the right set of levels is for uncoupled chains.
150
H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151
by the binding energy of the exciton. It must be pointed out, however, that the ps photoconductivity, large as it is, requires that only about 1% of the photons give rise to carriers in the case of polyacetylene [26]. This is also expected to be true for PPV [27]. The observed photoconductivity could be accounted for by dissociation of a small number of excitons, which could occur at the surface or at defects or impurities, or a small number of directly photogenerated polarons [28]. Nevertheless, we have carried out some calculations for Eg = 2.4 eV, arbitrarily keeping the same values of the other two properties we used to determine to, a and K. The resulting a and K values were considerably smaller, and the polaron was found to be unstable for this set of parameters. However, as discussed above, we do not believe Eg = 2.4 eV is correct. Photogeneration experiments on PPV do not give direct information about creation of polarons, apparently because a large number of these polarons, perhaps almost all, are bound in polaron pairs on adjacent chains [28,29]. The large majority of these recombine geminately [28]. The fact that no bipolarons are seen in more ordered samples [17] could be due to there being more polaron pairs formed in such samples, leaving fewer free polarons to ultimately find each other and combine into bipolarons. The one experiment that has given convincing evidence for the existence of polarons was that of Ziemelis et al. [30], who observed photoinduced absorption due to carriers injected at the contacts. The observation is possible because polaron pairs could not be formed except in the small region where the positive polarons injected at one contact meet the negative polarons injected at the other. This experiment gave the distance of the polaron level from the valence band as 0.4 eV [30]. The experiment was done on poly(3-hexylthiophene) (P3HT), but there is every reason to believe a similar result would be obtained for PPV. If it had been PPV, the average chain length, according to Fig. 2, would have been five or six monomers, or about 40 .A, which appears reasonable "for P3HT.
4. Conclusions
The parameters for PPV obtained by fitting our SSHtype Hamiltonian to the bandgap, valence bandwidth and vinylene geometry are: to = 2.66 eV, a = 10.29 eV/ /~ and K = 99.0 eV//k 2. Larger values of these parameters than those for polyacetylene are expected because the bandgap is twice as large, the vinyl dimerization 40% larger and the valence bandwidth 10% larger. With these values of the parameters the features of the LDFcalculated 7r bands and the M N D O calculated groundstate geometry are all matched well or fairly well. Further evidence for the validity of our Hamiltonian
is the fact that the PPV polaron geometry calculated using the unrestricted H a r t r e e - F o c k PM3 method agrees well with that calculated using our Hamiltonian. For the long-chain limit the energy difference found between the polaron level and the nearest band edge is 0.19 eV. The energy difference is much greater for short chains because the bandgap increases rapidly as the chains grow shorter. The distortion due to the polaron is spread over about eight monomers, its amplitude at a distance of four monomers from the center having decreased to about 10% of its value at the center. The interchain coupling due to the eight pairs of carbon atoms in a pair of monomers is determined by relating it to the splitting due to interchain interactions found in the L D F calculations [11]. This coupling is apportioned among the eight atom pairs by taking into account its exponential dependence on the distance between the atoms of a pair, the latter known from crystallographic studies. Only three pairs out of the eight have sufficient interchain coupling to affect any calculated results, t . for these three ranging from 0.33 to 0.12 eV. With the interchain coupling so determined, and the other parameters we believe to be correct for PPV, the polaron is found to be intrinsically stable.
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H A . Mizes, E.M. Conwell / Synthetic Metals 68 (1995) 145-151 [18] D.A. Halliday, P.L. Burn, D.D.C. Bradley, R.H. Friend, O.M. Gelsen, A.B. Holmes, A. Kraft, H.F. Martens and K. Pichler, Adv. Mater., 5 (1993) 40. [19] Yu.N. Gartstein and A.A. Zakhidov, Synth. Met., 28 (1989) D501; D. Baeriswyl and K. Maki, Synth. Met., 28 (1989) D507; S. Stafstr6m, Phys. Rev. B, 32 (1985) 4060; P.L. Danielson, Synth. Met., 20 (1987) 125; K. Fesser, Phys. Rev. B, 40 (1989) 1962; J.A. Blackman and M.K. Sabra, Phys. Rev. 13, 47 (1993) 15 437. [20] D. Chen, M.J. Winokur, M.A. Masse and F.K. Karasz, Phys. Rev. B, 41 (1990) 6759. [21] R.N. Marks,J.M. Hails, D.D.C. Bradley, R.H. Friend and A.B. Holmes, J. Phys. Condens. Matter, 6 (1994) 1379. [22] C.H. Lee, D. Moses and A.J. Heeger, Phys. Rev. B, 49 (1994) 2396.
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[23] J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner and Z.V. Vardeny, Phys. Rev. Lett., 72 (1994) 156. [24] S.C. Graham, D.D.C. Bradley, R.H. Friend and C. Spangler, Synth. Met., 41-43 (1991) 1277. [25] We have other objections to the conclusions of Ref. [23] that have been submitted for publication. [26] M. Sinclair, D. Moses and A.J. Heeger, Solid State Commun, 59 (1986) 343. [27] D.D.C. Bradley, Y.Q. Shen, H. Bleier and S. Roth, J. Phys. C, 21 (1988) L515. [28] J.W.P. Hsu, M. Yan, T.M. Jedju, L.J. Rothberg and B.R. I-Isieh, Phys. Rev. t3, 49 (1994) 712. [29] E.M. Conwell and I-I.A. Mizes, Mol. Cryst. Liq. Cryst., in press. [30] K.E. Ziemelis, A.T. Hussain, D.D.C. Bradley, R.H. Friend, J. Riihe and G. Wegner, Phys. Rev. Lett., 66 (1991) 2231.