Theory of surface-plasmon excitation in metal-insulator-metal tunnel junctions

A new calculation of surface-plasmon excitation in tunnel junctions is described. The tunnel junction is divided into three regions of complex dielectric function ${\epsilon }_L\left(\omega \right)$, ${\epsilon }_0\left(\omega \right)$, and ${\epsilon }_R\left(\omega \right)$ 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, $\overrightarrow{\mathrm{J}}=\left(\frac{\mathrm{ie}\hslash }{2m}\right)(\psi _{}^{*}{}_R{}^{}\nabla {\psi }_L-{\psi }_L\nabla \psi _{}^{*}{}_R{}^{})$ and $\rho =-e\psi _{}^{*}{}_R{}^{}{\psi }_L$, for an electronic transition from a state ${\psi }_L$ in the left electrode to a state ${\psi }_R$ in the right. The transition rate is given by $(-\frac{2}{\hslash \omega })\mathrm{Re}\int {\overrightarrow{\mathrm{E}}}^{*}·\overrightarrow{\mathrm{J}}{d}^{3}r$ where $\hslash \omega =E_L-E_R$. This new formulation avoids the need to quantize the electromagnetic fields and allows the use of complex dielectric functions. Current-carrying orthogonal eigenfunctions are used for ${\psi }_L$ and ${\psi }_R$ in place of the nonorthogonal basis set associated with the transfer-Hamiltonian theory of inelastic tunneling. The transition rate calculated from the latter theory differs from the present calculation by a factor as large as 10 to 100 in some instances. Numerical estimates of the rate of surface-plasmon excitation in Al-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$-Ag junctions are given. The inelastic tunneling rate is found to be ∼0.1 of the elastic rate (for electrons tunneling into Ag). Excitation of electromagnetic modes which can be made radiative by roughening the electrodes is discussed.