Asymptotic and numerical study of resonant tunneling in two-dimensional quantum waveguides of variable cross section

A waveguide is considered that coincides with a strip having two narrows of width $\varepsilon$. The electron wave function satisfies the Helmholtz equation with Dirichlet boundary conditions. The part of the waveguide between the narrows plays the role of a resonator, and there arise conditions for electron resonant tunneling. This phenomenon means that, for an electron of energy $E$, the probability $T(E)$ of passing from one part of the waveguide to the other through the resonator has a sharp peak at $E=E_{\mathrm{res}}$, where $E_{\mathrm{res}}$ is a “resonant” energy. To analyze the operation of electronic devices based on resonant tunneling, it is important to know $E_{\mathrm{res}}$ and the behavior of $T(E)$ for $E$ close to $E_{\mathrm{res}}$. Asymptotic formulas for the resonance energy and the transition and reflection coefficients as $\varepsilon\to0$ are derived. These formulas depend on the limit shape of the narrows. The limit waveguide near each narrow is assumed to coincide with a pair of vertical angles. The asymptotic results are compared with numerical ones obtained by approximately computing the waveguide scattering matrix. Based on this comparison, the range of $\varepsilon$ is found in which the asymptotic approach agrees with the numerical results. The methods proposed are applicable to much more complicated models than that under consideration. Specifically, the same approach is suitable for an asymptotic and numerical analysis of tunneling in three-dimensional quantum waveguides of variable cross section.