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Asymptotic theory of resonant tunneling in 3D quantum waveguides of variable cross-section. (English) Zbl 1202.35061

Summary: Resonant tunneling can occur as an electron propagates in a quantum waveguide of variable cross-section. The waveguide narrows play the role of effective potential barriers for the longitudinal electron motion. The part of the waveguide between two narrows becomes a “resonator,” and there can arise conditions for electron resonant tunneling. This phenomenon consists of the fact that, for an electron with energy \(E\), the probability \(T(E)\) to pass from one part of the waveguide to the other through the resonator has a sharp peak at \(E=E_{\text{res}}\), where \(E_{\text{res}}\) denotes a “resonant” energy. Such reentrant quantum resonators can find applications as elements of nanoelectronics devices and provide some advantages in regard to operation properties and production technology. To analyze their operation, it is important to know \(E_{\text{res}}\), the behavior of \(T(E)\) for \(E\) close to \(E_{\text{res}}\), the height of the resonant peak, and its width at the half-height (the so-called resonant quality factor). We consider electron propagation in an infinite waveguide with two cylindric outlets to infinity and two narrows of small diameters \(\varepsilon_1\) and \(\varepsilon_2\). The electron motion is described by the Helmholtz equation. We derive asymptotic formulas (and estimate the remainders) for the resonant energy, the shape of the resonant peak, and the transition and reflection coefficients as \(\varepsilon_1\) and \(\varepsilon_2\) tend to zero. Such formulas depend on the limiting shape of the narrows; we assume that the limiting waveguide in a neighborhood of each narrow coincides with two cones intersecting only at their common vertex. Our approach (based on the compound asymptotics method) can be employed for sound resonators and super-high-frequency resonators.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs
35C20 Asymptotic expansions of solutions to PDEs
81U20 \(S\)-matrix theory, etc. in quantum theory
81V10 Electromagnetic interaction; quantum electrodynamics
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