A matrix method for treating the coupling between an electron and a surface plasmon: a dynamical image potential in model tunnelling junctions

In this paper, we present a method for treating in a general manner the coupling between a single electron and boson fields represented by harmonic oscillators. The (general) solution of the corresponding many-body Schrodinger equation in real space is obtained by means of a propagation matrix method without using any ansatz for the many-body wave functions. We study the particular case of coupling between electrons and surface plasmons in tunnelling junctions. The electron is coupled inside the tunnelling barrier to a few surface plasmon modes. We present results for the dynamical effective potential felt by an electron tunnelling in model one-dimensional tunnelling junctions as well as for the various traversal times. As expected, significant differences from the corresponding static image potential are obtained when the tunnelling times tau are shorter than the characteristic response time of the surface charge (i.e. the inverse of the surface plasmon frequency omega). Examples of such dynamical effective potentials are given for various typical tunnelling conditions in the presence and absence of an applied bias voltage. The behaviour of the potential is studied versus the tunnelling times tau. It is also shown that the apparent barrier height that can be deduced from experiments does not always contain useful information about the dynamics of the coupled electron-plasmon system. On the other hand, the absolute values of the current in the tunnelling junctions are strongly dependent on the characteristic parameters of this dynamics.