Synthetic Metals. 17(1987) 115 121
THEORETICAL DESIGN OF POLYMERIC CONDUCTORS
J. L. BREDAS* Laboratoire de Chimie Throriquc Appliqure, Centre de Recherches sur les Matrriaux Avancrs, Facultfs Universitaires Notre-Dame de la Paix, B-5000 Namur (Belgium)
ABSTRACT We focus on organic polymers that could possess very small bandgaps and therefore display high conductivities intrinsically (i.e., without the need of doping). We concentrate on aromatic polymers such as polythiophene and its derivatives. We establish for those compounds the complete relationship between bandgap and bond-length alternation along the polymer backbone. We discuss certain types of substitution which can provoke an effective quinoid-like contribution to the electronic structure. We describe in particular the electronic properties of polyisothianaphthene as well as polythieno[3,4-c]thiophene and polyisonaphtothiophene. We present the conditions under which the latter two compounds might present a vanishingly small bandgap.
INTRODUCTION Major efforts are currently devoted to the search for organic polymers that would have very small or even vanishing bandgaps, i.e. that would be intrinsically good electrical conductors without the need of doping. One way to accomplish this goal is to consider some of the polymers that are already well known in the field of synthetic metals and to try and modify their electronic properties by the action of substituents. In that framework, a reasonable approach is to consider as parent polymers those which already have a rather small bandgap and whose chemical nature leads to substitution reactions. Polythiophene, which has an intrinsic bandgap of about 2 eV, appears to be an almost ideal target because of the ease of substitution on the carbons in ~ positions without necessarily resulting in causing strong steric interactions. These characteristics have led Wudl et al. [ 1, 2] to prepare polyisothianaphthene, a polymer of a "nonclassical" thiophene, see Fig. 1. Wudl a al. have found that polyisothianaphthene has an energy gap which is one full eV lower than that of
polythiophene itself .
* Chercheur Qualifi6 of the Belgian National Fund for Scientific Research (FNRS). 0379-6779/87/$3.50
© Elsevier Sequoia/Printed in The Netherlands
Fig. 1. Sketch of the geometric structures of: (a) polyisothianaphthene, (b) polythieno[3,4c]thiophene, and (c) polyisonaphtothiophene.
In this contribution, we first set the theoretical framework in which it is possible to rationalize the bandgap lowering when going from polythiophene to polyisothianaphthene. We establish the relationship that exists in aromatic polymers such as polythiophene or its derivatives, between the bandgap value and the degree of bond length alternation along the chain. We then discuss the electronic structure of polyisothianaphthene and two other polythiophene derivatives, polythieno[3,4-c]thiophene and polyisonaphtothiophene (Fig. 1).
RELATIONSHIP BETWEEN BANDGAP AND BOND LENGTH ALTERNATION IN AROMATIC POLYMERS We have performed VEH calculations on polythiophene, first considering the neutral polymer and then varying the geometry from aromatic (which corresponds to the ground-state optimal structure) to quinoid-like [3, 4]. The geometries between which variations are made are taken from ab initio Hartree-Fock optimizations on undoped (aromatic) and 50 % n-doped (quinoid) structures for quaterthiophene . For each geometry, the degree of bond length alternation, Ar, is calculated as the largest difference between the carbon-carbon bond lengths. By definition, we set the Ar values to be negative (positive) for a strongly aromatic (quinoid) structures. The evolution of the bandgap as a function of increasing quinoid contributions to the geometry is depicted in Fig. 2. The bandgap is a direct bandgap at the Brillouin zone center (k=0). A most important result is that the bandgap evolves linearly as a function ofzlr (in agreement with the solidstate physics concepts developed by Brazovskii and Kirova ). The evolution is such that, at first, the bandgap decreases linearly when Ar increases. As Ar equal to +0.06 ,~ is approached, the bandgap is calculated to become very small. A careful analysis of the symmetries of the HOMO and LUMO bands indicates that they belong to the same irreducible representation. Therefore, the bandgap is not calculated at the VEH level to completely vanish. However, an accidental degeneracy
I ~ 1
" / " .....
Fig. 2. Evolution of the bandgap value, Eg (in eV), as a function of increasing quinoid contributions to the geometry of a polythiophene chain. Ar (in/~) is the degree of bond length alternation along the carbon backbone; it corresponds to the largest difference between the length of a carbon-carbon bond parallel to the chain axis and that of a bond inclined with respect to that axis.
might occur at a point of high symmetry in the Brillouin zone, such as the zone center (k=0) or the zone edge (k=n/a). For Ar values larger than +0.06/~, the bandgap then increases linearly with Ar. An interesting aspect is that, with respect to the situation prevailing for Ar lower than +0.06 ,~, the HOMO and LUMO characteristics have exchanged for Ar larger than +0.06/~. Thus, the HOMO [LUMO] electronic pattern for what we can define as an aromatic-type geometry (Ar < +0.06/~) is similar to the LUMO [HOMO] electronic pattern for a strongly quinoid-type geometry (Ar > +0.06/~). It becomes then easy to understand the evolution of the bandgap when starting from an aromatic geometry. Indeed, as the quinoid contributions are increased, the HOMO becomes destabilized since it is equivalent to the LUMO of the quinoid structure whereas the LUMO becomes stabilized. This evolution results in a net decrease of the bandgap . On Figure 2, we can divide the V-shaped curve into two parts: (i) the left part (for Ar < +0.06 ,~) which corresponds to an aromatic-type electronic structure (in this case the HOMO has no contributions coming from the sulfur n atomic orbital) and (ii) the right part (for Ar > +0.06/~) which corresponds to a quinoidtype electronic structure (the HOMO has contributions from the sulfur n atomic orbital).
The bandgap---bond length alternation relationship that we have derived can be used as a nualitative ~uideline to design new organic polymers with a ~mall intrinsic bandgap and is very important because it provides a framework within which one can work. What we need is a compound in which some quinoid contributions are stabilized in the ground state. Note, however, that this relationship predicts that the corresponding polymeric chains will generally not have a metallic behavior (except as a result of accidental degeneracy) but would constitute semiconductors with possibly (very) small bandgaps. Furthermore, it must be borne in mind that, for systems lying near the bottom of the V-shaped curve of Figure 2, correlation and nuclear relaxation phenomena  may become important and lead to a larger bandgap than that predicted at the VEH level. (We will come back to these points later in the discussion.) Polysulfur nitride, however, demonstrates the possibility that a conjugated polymer be intrinsicallyhighly conducting .
ELECTRONIC STRUCTURE OF POLYISOTHIANAPHTHENE, POLYTHIENO[3,4-c]THIOPHENE, AND POLYISONAPHTOTHIOPHENE For the VEH polymer calculations to be presented here, we have considered the geometries of the polymer unit cells to be equal to the MNDO-optimized geometries for the corresponding monomers or dimers [9, 10]. The polymer chains are assumed to be coplanar since no steric interactions are present between adjacent monomer units. The VEH-calculated electronic properties are collected in Table 1. We first discuss polyisothianaphthene. The bandgap of polyisothianaphthene is calculated at the VEH level to be 0.54 eV. This is 1.17 eV lower than the VEH bandgap calculated for polythiophene (1.71 eV) in the same framework, i.e. using an MNDO-optimized geometry for the thiophene monomer unit . This 1.17 eV lowering of the bandgap when going from polythiophene to polyisothianaphthene is in excellent agreement with experimental data since optical absorption measurements lead to a bandgap around 1 eV in polyisothianaphthene, about 1 eV lower than the 2.0-2.2 eV bandgap observed in polythiophene. We have compared the evolution of the top of the HOMO (ionization potential) and the bottom of the LUMO (electron affinity) when going from thiophene to isothianaphthene and from polythiophene to polyisothianaphthene . The fusion of a benzene ring to form the isothianaphthene structure leads to an upward shift of the HOMO by 0.77 eV in the polymeric chain and by 1.29 eV in the molecule. The LUMO's undergo a downward shift, although by a smaller extent (0.38 eV in the polymer, 0.81 eV in the molecule). The decrease in bandgap when going from polythiophene to polyisothianaphthene can therefore be rationalized by realizing that the fusion of a benzene ring onto the thiophene ring effectively increases the quinoid contributions to the electronic structure. As described in the previous paragraph, this destabilizes the HOMO and stabilizes the LUMO. In polyisothianaphthene, the increased quinoid contributions to the electronic structure cut the bandgap in half with respect to that of the parent polymer, polythiophene. It is also interesting to note that the width of the highest occupied band in polyisothianaphthene is large, 2.53 eV. This value is even 0.17 eV bigger than the width calculated for the polythiophene HOMO on the basis of the MNDO-optimized geometry for the thiophene monomer unit . We
can thus expect maximum conductivities in doped polyisothianaphthene of the same order as those in doped polythiophene, which is what is experimentally observed . We have also investigated the electronic structure of three simply
polyisothianaphthenes: polydimethylisothianaphthene,polydimethoxyisothianaphthene, and polydicyanoisothianaphthene. Our aim is to find whether substitution with electron donating or electron accepting groups is sufficient to have a profound impact on the electronic structure and, in particular, on the size of the bandgap. The main result (see Table 1) is, however, that the effect of that kind of substitution on the major electronic properties is very small. The bandgap is modified by less than 0.05 eV with respect to polyisothianaphthene. It is therefore necessary to examine other types of substitutions if we wish a strong effect on the bandgap value to appear. This is the reason why we have investigated the electronic structure of two other polythiophene derivatives, polythieno[3,4-c]thiophene and polyisonaphtothiophene (Fig. 1). The VEH band structure of polythieno[3,4-c]thiophene indicates that the bandgap is 1.02 eV, a value 0.48 eV lareer than in polyisothianaphthene. An analysis of the HOMO and LUMO band characteristics shows that the electronic structure of polythieno[3,4-c]thiophene is actually of quinoid type. In other words, polythieno[3,4-c[thiophene effectively stands to the right of the minimum in the V-shaped curve of Figure 2. These results mean that polythieno[3,4-c]thiophene constitutes a compound in which the quinoid contributions to the electronic structure are too important: we have gone too far with that compound in the direction of quinoid stabilization. A way to decrease the quinoid contributions is to provoke the appearance of a torsion angle between adjacent rings, for instance by putting appropriate substituents. The presence of a torsion angle along the polymeric chains results in a decrease of the electronic interactions between consecutive tings and thus reduces the quinoid contributions. Considering a torsion angle of 30 ° along the chains of polythieno[3,4-c]thiophene, we calculate at the VEH level that the bandgap drops to 0.51 eV. The HOMO and LUMO characteristics indicate that we are still on the right side of the V-curve. By forcing a 45* torsion, the bandgap becomes very small, 0.09 eV, and we stand now on the left side of the V-curve. (These results are qualitatively consistent with those obtained on similar systems at the simple Hiickel level by Wennerstrrm .) It seems to us that these results are conceptually important because, in our opinion, they support and validate the relationship we have established between the bandgap value and the amount of quinoid contributions. Mastering the precise torsion angle value which is required to minimize the bandgap value is experimentally a difficult task, however. Therefore, we have attempted to conceive another polythiophene derivative which could present a very small gap in a coplanar conformation. We have chosen polyisonaphtothiophene whose geometry, as optimized by MNDO, is very similar to that of polyisothianaphthene, a localization of the single and double bonds occurring in almost the same way in both benzene tings . From the VEH band structure, we obtain a bandgap for polyisonaphtothiophene which is vanishingly small, 0.01 eV. PolvisonaDhtothioDhene could thus represent an example of an organic oolvmer with a vanishin~Iv small band~ar~. As we pointed out in the first part of this paper, electronic correlation and nuclear relaxation effects could play an important role in such a situation and cause a bandgap opening, which would
TABLE 1 VEH
polydimethylisothianaphthene, polydimethoxyisothianaphthene, and polydicyanoisothianaphtene. All values are in eV and refer to bandgap (Eg), HOMO bandwidth (BW) and ionization potential (IP) Eg
increase the bandgap value with respect to what is calculated at the VEH level. It is most difficult to estimate these effects precisely. We can, however, add the following aspects to the discussion. The high polarizability of molecules such as isonaphtothiophene will contribute to reduce the electronelectron interactions between two n electrons located on the same site. The rather large width of the HOMO band in polyisonaphtothiophene (1.73 eV) will tend to the same result. We have attempted to obtain a rough idea of the importance of the nuclear relaxation effects by performing bandstructure calculations on polyisonaphtothiophene chains where we vary the length of the inter-ring bond. On one hand, this bond is elongated from 1.46/~ to 1.50/~, so that the quinoid Contributions to the electronic structure are reduced. The result is that the bandgap goes merely up to 0.20 eV and the HOMO and LUMO characteristics indicate we are on the aromatic side of the V-curve. If, on the other hand, the inter-ring bond is shortened to 1.42 A, thereby favoring an increase in the quinoid contributions, the bandgap raises to a similar value, 0.21 eV, but we are now located on the quinoid side of the V-curve from Figure 6. Finally, it is important pointing out that strong Peierls-type relaxation effects are not expected since we are dealing with a low bandgap semiconductor and not with a real metal. In any event, it will he most informative to confront the theoretical predictions with experimental data when the synthetic efforts which are currently carried out will have (hopefully) succeeded.
ACKNOWLE-73GEMENTS It is a great pleasure to thank J.M. Andr6 and B. Thrmans in Namur, and A.J. Heeger and F. Wudl in Santa Barbara. This work has been supported by the Belgian FNRS. The collaborative work with UCSB has been supported by NATO through Research Grant No. 407/84.
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