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Center for Nonlinear Studies |
When do electrons on nanowires trap themselves—interact with the wire in such a way that they become localized? Electrons that are not self-trapped have wave functions substantially larger than the size of the cells on a nanotube (e.g. carbon), so in this case the discrete structure of the wire—that it is made up of cells – may be ignored. But when an electron becomes self-trapped, its wave function has width on the order of the width of these cells, so a model of the wire reflective of its discrete structure must be used. In this talk, we will discuss our work to determine when such a model is required. The electron’s wave function is governed by an effective non-linear Schrödinger equation. Since this equation demonstrates self-trapping in 2-d but not in 1-d, we think for a quasi-1-d wire (much thinner than it is long), we will not see self-trapping, but as we gradually scale up the width a critical value will be obtained at which trapping begins to occur. In addition to developing the analogy between our system and the non-linear Schrödinger equation, in this talk we will go into some detail on the numerical method we use to solve both the non-linear Schrödinger equation and the more general coupled system of pde’s governing the electron-nanowire system. Specifically, we will provide a brief derivation of and discussion of the symmetrized split-step Fourier method. We will conclude by presenting the results we have obtained so far, one of which in particular is in interesting contrast to analytical predictions for the non-linear Schrödinger equation.
Host: Cristiano Nisoli