Electron Trapping by solitons: Classical versus Quantum Mechanical Approach

In two previous letters [Velarde et al., 2005, 2006] a new form of electric conduction mediated by solitons was shown to be possible in an anharmonic one-dimensional (1D) lattice. In the first, a strictly classical approach was adopted hence treating the lattice dynamics classically and the electron-(ion) lattice interaction with classical electrodynamics. In the second, the electron-lattice interaction was considered within the tight-binding approximation [Ashcroft & Mermin, 1976] while maintaining the classical approach to the lattice dynamics. The lattice interactions were of Toda or Morse type akin to the Lennard–Jones interaction [Choquard, 1967; Toda, 1989], hence allowing for phonon — and soliton — longitudinal vibrations with compressions governed by the repulsive part of the potential [Chetverikov et al., 2005, 2006a, 2006b]. These compressions were shown to be responsible for electron trapping by the lattice excitations thus leading to the formation of dynamic bound states (solectrons) of the electron with the soliton (the same phenomenon is valid also for the solitonic peaks of a cnoidal wave moving through the lattice). In the present letter, we proceed deeper in the analysis and further explore the analogies and differences between the classical electrostatic trapping and the quantum mechanical tight-binding approximation using the Morse interaction. Thus, we consider a 1D anharmonic lattice with dynamics dictated by the following Hamiltonian describing nearest-neighbor Morse interactions: