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PHYSICAL SCIENCES / PHYSICS
Multiple self-localized electronic states in trans-polyacetylene
*Departments of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139; and Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210
Edited by Esther M. Conwell, University of Rochester, Rochester, NY, and approved April 26, 2006 (received for review February 15, 2006)
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conducting polymer | self-localization | soliton
bands, except for the study by Strafstrom and Chao (22) showing that the local DOS of the SSH soliton depletes the local DOS of bands at the same sites. In this work, we show that electron correlations lead to additional, previously undescribed localization effects, regardless of the absence or presence of counterions, in a study of the t-PA system by using first-principles Hartree–Fock (HF) and density functional theory (DFT) methods. Fig. 2 shows the DOS for the 
and electrons obtained by HF/3-21G and DFT hybrid BHandHLYP/3-21G (23), for the positive-charge soliton S+. In contrast to the single gap state in Fig. 1, several states appear in Fig. 2. To our knowledge, a demonstration of multiple self-localized states in conjugated polymers has not been described previously. To investigate the nature of these additional states in the gap, we analyze an extended Hubbard model as extension of the SSH Hamiltonian to include electron–electron interactions. The additional localized states are found to arise from local shifts in the valence and conduction bands induced by the presence of charged solitons, similar to the band-bending mechanism well known from semiconductor heterojunctions. Such localizations persist in the presence of counterions.
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In Eqs. 1 and 2, U and V are the on- and off-site Coulomb repulsion strengths, respectively; p is the atomic momentum; µ is the mass of a carbon–hydrogen (CH) unit; K is the spring constant representing the bonding between adjacent CH units; u is the atomic displacement with respect to the undimerized chain; t0 is the hopping integral of the undimerized chain; 
is the linear electron–phonon coupling constant; and cn,
and cn, are creation and annihilation operators for -electron of spin at site n, respectively. In view of the importance of Coulomb charge interactions as explained in detail below, we choose to solve the extended Hubbard model (1) under the unrestricted HF approximation (12).
By switching on U and V interactions, we find that both the valence and conduction bands respond strongly to the localized charge distributions in solitons with positive and negative charges, S+ and S–. Although this process leads to eigenstate hybridization in the Hilbert space of electrons, little hybridization occurs between the valence and conduction subspaces across the 
1.5-eV band gap (1 eV = 1.602 x 10–19 J). In the case of S+ where the localized net charge distribution is positive, the Hamiltonian (1) creates a locally attractive potential acting on the electrons of the neighboring CH sites. This change causes the energies of the bottom valence and conduction band states to move into the forbidden regions, thereby forming new interfacial misfit states, whose wavefunctions tunnel into the +1 and –1 phases with characteristic exponentially decaying tails as described by the complex band structure theory (24, 25). The effect is similar to the band-bending phenomena known to occur at semiconductor heterojunctions and surfaces (26), except that here the much shorter localization length scale is caused by the primary self-trapping soliton state carrying a single elementary charge. The overall band bending computed by using standard tight-binding parameters from the literature (6, 12, 27) is shown in an energy-site plot, Fig. 3, that combines energy information with spatial distribution of the wavefunction (juxtaposed local DOS plots). Five localized gap states are seen to stand out (labeled a–e and further displayed in Fig. 4). Their wavefunctions are in sharp contrast to the reconstructed delocalized valence and conduction band states, which also are shown in Fig. 3.
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Through parametric studies we find the behavior just described to persist over a wide range of U and V values. For a reasonable set of parameters, U = 4.0 eV (12), V01 = (U/2)·(7.55/11.0) (27), and V02 = (U/2)·(5.2/11.0) (27); the primary gap state (Fig. 3b) appears at 0.23 eV below the gap center, which would correspond to the 0.5-eV peak [the so-called low-energy (LE) band that was 0.25 eV below the gap center] observed in photo-induced absorption (PA) experiments; it was assigned to charged solitons S± (28). Another excitation would appear at 1.30 eV (Fig. 3a) from the valence band edge, overlapping in energy with two distinct experimental PA features. One feature is the so-called high-energy (HE) band, which has been assigned to neutral solitons S0 because of different responses to magnetic excitation (29, 30), temperature (9), optical polarization (31), and disorder of the sample (31) from the LE band. Another feature is the near-edge oscillatory structure observed in the PA spectra (8), believed to be due to strong electric field polarization of the surrounding medium caused by charged solitons (electroabsorption effect at the microscopic length scale). This explanation has the same physical origin as our secondary localized states.
To verify the predictions of the extended Hubbard model (Eq. 2), we return to the original HF and DFT results. We find that in the case of S+, additional localized states are deeply buried in the bands, below the valence band, which were not shown in the DOS plot (Fig. 2). These low-lying localized states are in agreement with the extended-Hubbard predictions (Fig. 4). Moreover, full geometry optimizations in both HF and DFT computations lead to a strong S± soliton-induced carbon backbone bending. It is important to note that this result is not a straightforward consequence of electrostatic repulsion among the charged CH sites (4).
We have thus far demonstrated that the secondary localized electronic states are intrinsically induced by photo-generated solitons (28, 32). Now we consider how counterions, carrying opposite charges to the SSH solitons, affect the localizations in chemically doped polymers. We have performed HF and DFT calculations explicitly treating several counterions species anions, F–, Cl–, ClO4–, PF6–, and N(SO2CF3)2–, and cations, Li+ and Na+. We find in all these cases at least one additional secondary localized state stands out from the valence and conduction bands (Fig. 5). Fig. 5 shows two characteristic properties of the secondary localized states in the case of Cl– counterion, their energies being located in the forbidden regions and their wavefunctions containing exponentially decaying wave tails. Detailed comparisons with the SSH soliton wavefunction indicate that the lowest additional localized state below the valence band Sv is as localized as the SSH soliton state, whereas the lowest additional localized state below the conduction band Sc is less localized. We also have investigated solvent screening effects by using the Polarizable Continuum Model (23), and no discernable effects on either the energy or wavefunction localization were found. It may be useful to emphasize that the secondary localized states discussed in the present work are an intrinsic manifestation of electron correlation effects on extended bands, distinct from extrinsic effects such as counterions and solvent molecules, which may affect many important physical properties of conjugated polymers (14, 15).
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-band gap, self-localized states also exist below the valence band (S+) and above the conduction band (S–). As shown in Fig. 2, discrete states below the valence band are buried in the continuum band and therefore cannot be seen in terms of excitation energy only. However, in view of the selection rules based on orbital symmetry for transitions among states, these intrinsic excitations should be detectable by photo-scattering measurements on highly stretch-aligned films (34) or layer-by-layer-assembled films (35). In this connection we estimate the excitation energy from the lowest localized state (Fig. 3e) to the conduction band edge to be 6.1 eV. Lastly, as can be seen in Figs. 2b and 5, the presence of counterions causes the energy of secondary localized states to shift toward the corresponding band edge, which demonstrates the sensitivity of these states to external fields. For example, our calculations find the width of the primary localized state does not change appreciably in the presence of F– or PF6– counterions, but the width of the secondary localized state of the former is 30% wider than the latter. This sensitivity to the presence of local external electric fields may provide opportunities for ion sensing.
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Abbreviations: CH, carbon–hydrogen; DFT, density functional theory; DOS, density of states; HF, Hartree–Fock; SSH, Su, Shrieffer, and Heeger; t-PA, trans-polyacetylene
To whom correspondence should be addressed. E-mail: syip{at}mit.edu
Author contributions: X.L., J.L., and S.Y. designed research; X.L. performed research; X.L. and J.L. contributed new reagents/analytic tools; X.L., J.L., and C.J.F. analyzed data; and X.L., J.L., C.J.F., and S.Y. wrote the paper.
Conflict of interest statement: No conflicts declared.
This paper was submitted directly (Track II) to the PNAS office.
© 2006 by The National Academy of Sciences of the USA
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n = (–1)nun showing the domain wall (a), DOS showing the single gap state (b), and eigenfunctions 
(n) showing the spatial self-localization of the soliton (c), where n labels the CH unit sites. Original parameter values are used as follows: K = 21 eV/Å2, t0 = 2.5 eV, and 
1/301. The blank regions between two adjacent dots reflect the corresponding nodal structure of an eigenfunction, where the lowest state (e) has no node, the next lowest state (d) has one node, c has two, b has 150, and a has 151 nodes (see 
