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Interchain Pair Hopping of Solitons and Polarons via Dopants in Polyacetylene

Interchain hopping of solitons and polarons in polyacetylene is studied by numerical simulation of their motion under an electric field. Use is made of the Su-Schrieffer-Heeger model supplemented with intrachain electron-electron interactions and dopant potentials. We find that charged solitons can hop to the opposite chain by forming bound pairs (bipolarons). For the case of polarons also, hopping in a pair is more favorable than single polaron hopping. Interchain hopping of a polaron pair is more efficient than that of a soliton pair.


I. INTRODUCTION
The mechanism of charge hopping among chains is one of the intricate unsettled questions related to understanding of the electronic properties of conducting polymers. [1] It is expected that charged solitons or polarons play a major role in conduction processes in polyacetylene. Motion of such nonlinear excitations can contribute directly to the conduction. [2,3] Polyacetylene, however, has a complex structure composed of polymer chains with finite lengths and dopant ions (Fig. 1). A difficulty in the charged soliton model is that a single soliton cannot hop to neighboring chains, since there exists a large activation barrier for inversion of the phase of bond alternation over a large region. The difficulty can be removed by consideration of pairs of solitons, i.e. bipolarons. Interchain hopping of polarons, which is another conduction mechanism, is considered to be much easier. It is crucial to consider the dynamics of the nonlinear excitations by taking into account the three-dimensional effects, such as dopant potentials and interchain interactions.
Silva and Terai [4] investigated the effect of the interactions between two chains of polyacetylene on the dynamics of solitons, and found that the motion of a soliton is suppressed by a confinement effect and that the amount of the charge transfer between the two chains via charged and neutral solitons is quite small. The effect of a dopant has been usually incorporated as the Coulomb potential for electrons. Yamashiro and coworkers pointed out that interchain transfers are enhanced by the presence of the valence orbitals of the dopant ions. [5] We study here interchain hopping of solitons and polarons in polyacetylene by numerical simulation of their motion under an electric field. We consider two neighboring chains that are coupled with each other through dopant ions as shown in Fig. 1(b). One of these chains contains a pair of solitons or polarons in the initial state. The effects of a dopant are twofold: the Coulomb potential and enhancement of interchain transfer around the dopant. In our model, the interchain transfer is assumed to occur in a small region around the dopant, and is neglected in the other regions, for simplicity.
The effect of chain ends on the motion of the nonlinear excitations is also important.
For discussion of this effect the dopant is located near the end of one of the chains, chain a in Fig. 1(b). The other chain (chain b) is arranged so that the dopant is located near its midpoint. Thus we choose the structure shown in Fig. 1(b), and it seems reasonable to regard this structure as a typical example of part of the three-dimensional complex structure shown in Fig. 1(a).

II. MODEL AND NUMERICAL METHOD
The model Hamiltonian is given by (2.1) The first two terms, describing the two chains a and b, are the Su-Schrieffer-Heeger model [6] modified to include the electric field.
where λ = a, b and y λ,i = u λ,i+1 − u λ,i . Here, t 0 is the hopping integral between the nearest neighbor sites, α the electron phonon coupling constant, u λ,i the displacement of the ith with n λ,iσ = c † λ,iσ c λ,iσ and n λ,i = σ n λ,iσ . The parameters U and V are the on-site and nearest neighbor Coulomb repulsion, respectively. The two chains are of the same size. Chain b is shifted by i 0 sites with respect to chain a. The interaction between two chains is represented by where p and q denote the first and last site in the interaction region around the dopant. The last term of eq.(2.1) represents the dopant potential, which is approximated by a square-well potential, We use the following values of parameters: [6,8] [7,9] In solving the time dependent HF equation we use the method of fractal decomposition for exponential operators. [10,11] For the equation of motion the time differential equation is integrated with discretization of the time with an interval ∆t. The interval ∆t is chosen to be ∆t = 0.0025ω −1 Q throughout this work. Here in our simulations the unit of time is the inverse of the bare optical phonon frequency (ω Q = 4K/M ≈ 2.5 × 10 −14 s −1 ) and that of the electric field is E 0 =hω/ea (≈ 1.3 × 10 7 V/cm).

A. Case of a pair of solitons
In the following we show the results in terms of the smoothened quantities of the bond variable and the excess charge distribution, defined as ỹ λ,i = (−1) i (y λ,i−1 − 2y λ,i + y λ,i+1 )/4, where the excess charge density is calculated from the time-dependent wavefunctions The prime attached to the summation symbol denotes a sum over occupied states. We have performed numerical simulations for various strengths of the field for changing of the incident velocity of the free soliton at the collision with the pinned soliton. In Fig. 3 we display the time dependence of the excess charge density on chain a, which is defined by When the two polarons collide, the hopping occurs almost instantaneously. Therefore, the time of the hop varies monotonically with the field strength, not in a chaotic manner.
It seems that the back-and-forth motion of the pinned polaron before the collision is induced mainly by the external field. For the parameter set used in our simulations, the energy difference between the state of the polaron being almost localized on one chain and the state of delocalization over the two chains is quite small. Therefore the oscillation can be induced by weak disturbance. This is considered to be one of the reasons for the small potential barrier for the polaron pair hopping.
It is worth noting that the final states result in formation of a pair of solitons or a pair of polarons depending on whether the initial state is a pair of solitons or pair of polarons.
For a single chain with no electron-electron interaction, it has been shown by use of the TLM model [12] that a two-polaron state is unstable against a two soliton-state. [13] This is supported by the results of a numerical study on the dynamics of a polaron pair: it was observed that two polarons dissociate into a pair of solitons after their collision. [14] In our two-chain system, a two-polaron state, not a two-soliton state, appears after the collision (hopping) of a pair of polarons. One of the reasons for this may be that the delocalization of polarons over the two chains stabilizes the polaron pair state.
In summary, we have numerically investigated interchain hopping of a pair of solitons and a pair of polarons. We found that charged solitons can hop to the opposite chain by forming a composite (bipolaron) in the electric field parallel to the chains. The existence of both the dopant and the chain end assists the hopping. The bipolaronic lattice distortion on the opposite chain is necessary for the hop to occur. There exists a finite potential barrier for the hopping. For the case of polarons also, hopping in a pair is more favorable than single polaron hopping. Interchain hopping of a polaron pair is more efficient than that of a soliton pair.