Akira Terai

Motion of Charged Soliton in Polyacetylene Due to Electric Field

Real time dynamics of a moving charged soliton in polyacetylene is studied numerically by using Su-Schrieffer-Heegers model. An electric field is introduced to the system through a time dependent vector potential which can be included into the Hamiltonian through the Peierls substitution of the phase factor to the transfer integral. Several interesting properties of the moving soliton are obtained, e.g. , the Brownian-like motion of the soliton due to the soliton-phonon interaction, the saturation of the soliton velocity, the saturation velocity being independent of the applied electric field strength, while the time needed to attain the saturation velocity is, roughly speaking, linearly dependent on the logarithm of the applied field.

Phonons around a soliton and a polaron in Su-Schrieffer-Heeger's model of trans-(CH)x

Linear modes around a soliton and a polaron are calculated after static self-consistent solutions for a soliton and a polaron are numerically determined in Su-Schrieffer-Heegers (SSH) model of trans -polyacetylene [ t -(CH) x ] with finite size. The linear chain is assumed to satisfy the periodic boundary condition. In the case of a soliton, four localized modes are found, three of which correspond to the localized optical modes obtained in the continuum version of the SSH model and the fourth appears because of the discreteness. In the case of a polaron, the number of the localized modes is seven; five have been already obtained in the continuum model and the remaining two are generated due to the discreteness.

Linear Mode Analysis around a Soliton in Trans-Polyacetylene (CH)x

Linear modes around the polaron solution are analyzed numerically in the model of Takayama, Lin-Liu, and Maki, which is a continuum model of trans -polyacetylene (CH) x . The polaron generates five localized phonon modes, one of which with \(\omega{\cong}0\) corresponds to the Goldstone mode. The phase shifts of extended phonon modes are obtained, which show that effective potential for them is reflectionless. The creation energy of a single polaron is found to be \(\sqrt{2}\varDelta_{0}/\pi-0.80\omega_{0}\), the second term being the quantum correction and Δ 0 and ω 0 denoting the half of the gap in the electronic energy spectrum and the frequency of the optical phonon with vanishing wave number, respectively, in the uniformly dimerized state.

MOTION OF CHARGED SOLITON IN POLYACETYLENE DUE TO ELECTRIC FIELD. II, BEHAVIOR OF WIDTH

By numerical simulations treating the motion of a charged soliton in polyacetylene, we analyze the width of a moving soliton. The motion of the soliton is induced by applying an electric field as in the previous work. The soliton width is estimated by two methods; one is from the excess charge density profile and the other from the distortion in the lattice displacement. Both results indicate that the width decreases as the soliton velocity increases. Quantitatively, however, the amount of decrease is different in the two cases; the maximum change of the width is about 10% in the former case whereas it is 20% in the latter. Furthermore, the width obtained from the lattice distortion is found to oscillate as a function of time, consistently with the excitation of the amplitude mode around a soliton.

Reexamination of Finite-Lattice Extrapolation of Haldane Gaps

We propose two methods of estimating a systematic error in extrapolation to the infinite-size limit in the study of measuring the Haldane gaps of the one-dimensional Heisenberg antiferromagnet with the integer spin up to S =5. The finite-size gaps obtained by numerical diagonalizations based on Lanczos algorithm are presented for sizes that have not previously been reported. The changes of boundary conditions are also examined. We successfully demonstrate that our methods of extrapolation work well. The Haldane gap for S =1 is estimated to be 0.4104789 ±0.0000013. We successfully obtain the gaps up to S =5, which make us confirm the asymptotic formula of the Haldane gap in S →∞.

Infrared Activity of Phonons around a Soliton in trans-(CH)x and (CD)x–The Origin of the New IR Active Modes–

The dynamical conductivity due to phonons around a charged soliton is calculated, using Horovitzs three phonon band model for trans-polyacetylene. We find three weak i.r. modes in addition to the three intense ones which were known. The weak modes have their origins in the third localized mode which was found in the Takayama-Lin-Liu-Maki model. The vibration frequencies and relative optical intensities are calculated to give a good agreement with a recent photo-induced absorption experiment. This strongly indicates that there actually is the third localized mode.

Fractionally Charged States in One-Dimensional Electron-Phonon Systems with Commensurability 3

Numerical analyses of the various states with nonlinear excitations in one-dimensional nearly one-third filled electron-phonon systems are performed by using Su-Schrieffer-Heegers model originally proposed for polyacetylene. The states are determined by solving numerically the self-consistent equations satisfied by the lattice displacement and the electronic wave functions. Different solutions are obtained for different values of the excess electron number δ n = N e -2 N /3 with N the total number of the lattice sites and N e the total electron number; the cases with δ n =±1/3, ±2/3, ±1, ±4/3, ±5/3 and ±2 are considered. Through comparison of the energies for δ n =0 and δ n ≠0, the creation energies of various nonlinear excitations are calculated and shown to depend not only on the absolute value but also on the sign of the charge of the excitations.

Electric Field Induced Depinning of a Charged Soliton from an Impurity Center in Polyacetylene

The dynamics of a charged soliton pinned by an impurity in the presence of an electric field is numerically studied by using Su-Schrieffer-Heegers model for trans-polyacetylene. The field is introduced in terms of a time-dependent vector potential which is included in the Hamiltonian in the form of a phase of the transfer integral. The time-dependent Schrodinger equation for the electronic wave functions and the equation of motion for the lattice displacements are numerically integrated in a self-consistent way. When the field is lower than a certain threshold value, the soliton remains trapped and oscillates periodically around the impurity position. As the field is increased, the time period of the oscillation becomes longer. When the field exceeds the threshold value, the soliton is depinned and begins to move in one direction. The threshold field is estimated to be 10 5 -10 6 V/cm, which is in rough agreement with the value obtained in experiments on polyacetylene lightly doped with AsF 5 .

Solitons and Their Dynamics in One-Dimensional SDW Systems

The dynamics of solitons in spin-density-wave systems is studied numerically using the one-dimensional Hubbard model. The motion of solitons is induced by an electric field, which is introduced into the model in the form of a time-dependent vector potential. The unrestricted Hartree-Fock approximation is used. An improved method of the decomposition of exponential operators is used to solve the time-dependent Hartree-Fock equation. First, the motion of a single soliton is studied, with emphasis on discrete lattice effects. It is found that the discreteness effects reduce the soliton saturation velocity which was found in a previous paper. Second, processes of collisions between a soliton and an antisoliton are studied

Motion of Charged Soliton in Polyacetylene Due to Electric Field.III. : Energetics

In numerical simulations on a moving charged soliton in polyacetylene, the behavior of the total, electric, lattice potential and lattice Kinetic energies are analyzed. In order to induce physically natural motion, the soliton is accelerated by a uniform electronic field as in the previous papers of this series. During the acceleration the electronic energy shows an almost monotone decrease while the lattice potential energy increases in an almost monotone way. These two energies have also an oscillating component which is related to the previously reported oscillation of the soliton width. On the contrary the total and the lattice kinetic energies show no oscillation. The relation between the total energy e tot and the soliton velocity υ is well fitted to a functional form e tot (υ)=e tot (0)-( M s υ m 2 /2) ln (1-υ 2 /υ m 2 ) where υ m is the saturation velocity of about three times the sound velocity and M s the soliton effective mass of four to five times the bare electron mass.