Eur. Phys. J. B (2012) 85: 176
https://doi.org/10.1140/epjb/e2012-21038-y

Regular Article

Trapped modes in finite quantum waveguides

¹ Mathematical Department of the Faculty of Physics, Moscow State University, 119991 Moscow, Russia
² Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS – Ecole Polytechnique, 91128 Palaiseau, France
³ Laboratoire Poncelet (UMI 2615), CNRS – Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
⁴ Chebyshev Laboratory, Saint Petersburg State University, 14th line of Vasil’evskiy Ostrov 29, Saint Petersburg, Russia

^a e-mail: denis.grebenkov@polytechnique.edu

Received: 16 December 2011
Received in final form: 29 February 2012
Published online: 30 May 2012

Abstract

The eigenstates of an electron in an infinite quantum waveguide (e.g., a bent strip or a twisted tube) are often trapped or localized in a bounded region that prohibits the electron transmission through the waveguide at the corresponding energies. We revisit this statement for resonators with long but finite branches that we call “finite waveguides”. Although the Laplace operator in bounded domains has no continuous spectrum and all eigenfunctions have finite L₂ norm, the trapping of an eigenfunction can be understood as its exponential decay inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped or, equivalently, the waveguide state from conducting to insulating. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two strips, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for applications to finite-size microscopic quantum devices.