Theory of Surface-Plasmon Excitation by Electron Tunneling

Abstract

A new calculation of surface-plasmon excitation in tunnel junctions is described. The tunnel junction is divided into three regions of complex dielectric function ε_L(ω), ε_O(ω), and ε_R(ω) which correspond to the left electrode, the barrier, and the right electrode respectively. Maxwell’s equations are solved for the classical electromagnetic fields. The source terms are given by the quantum-mechanical transition current and charge, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} = \frac{{ie\hbar }}{{2m}}\left( {\psi _R^*\nabla {\psi _L} - {\psi _L}\nabla \psi _R^*} \right){\mkern 1mu} {\mkern 1mu} and{\mkern 1mu} \rho = - e\psi _R^*{\psi _L}$, for an electron transition from a state ψ_L in the left electrode to a state ψ_R in the right. The transition rate is given by $\frac{{ - 2}}{{\hbar w}}{\mkern 1mu} {\mkern 1mu} \operatorname{Re} \smallint E*{\mkern 1mu} {\mkern 1mu} \cdot{\mkern 1mu} {\mkern 1mu} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} \;{d^3}r\;{\mkern 1mu} {\mkern 1mu} where\;{\mkern 1mu} {\mkern 1mu} \hbar w = {E_L} - {E_R}$. This new formulation avoids the need to quantize the electromagnetic fields and allows the use of complex dielectric functions. Numerical estimates of the rate of surface plasmon excitation in Aℓ-Aℓ₂O₃-Ag junctions are given.

Surface plasmons are a popular subject now. They show up in a number of experiments. Examples include optical experiments, photoemission, and electron energy loss experiments. Almost any time an electron crosses a boundary between a metal and an insulator it has a good chance of interacting with surface plasmons. My interest in them came about because their importance in tunneling, in particular because of the experiments by JOHN LAMBE and SHAUN McCARTHY at Ford [1]. In these experiments surface plasmons are excited in metal-insulator-metal junctions. The surface plasmons radiatively decay and light from the tunnel junctions is detected. In place of exciting a molecular vibration, the tunneling electrons excite surface plasmons. Of course, there are some tricks to convert them into photons. I am interested in trying to estimate the probability of surface-plasmon emission, particularly which surface plasmons will be emitted because we know that certain ones of them have a good chance to radiate and some of them (that have too high a momenta) cannot radiate. In addition, there are several questions relating to tunneling theory which I want to ask. These relate to what are the electron wave functions, and what is the region of interaction. This goes back to the old question about the transfer Hamiltonian and exact wave functions that Dr. KIRTLEY has already discussed. There are some very simple physical ideas which answer these questions.