Bound states in waveguides and bent quantum wires. I. Applications to waveguide systems

Abstract

It has been shown that in quantum wires which contain bends there will be one or more bound states for electrons placed in such systems. Bound states have been observed in quantum wires, but detailed mapping of such states is difficult. However, there is a one-to-one correspondence between wave functions of free electrons in two-dimensional ͑2D͒ systems, and electric fields of TE modes in rectangular waveguides with the same cross section as the 2D system. We therefore construct bent waveguides, find the frequencies at which confined EM fields occur, and map out the electromagnetic energy density there. We compare the experimental results with theoretical predictions of bound state energies and eigenfunctions. The geometry has been chosen to correspond to two-dimensional systems for which quantum wire experiments have been carried out. In such systems, we can predict the number and location of the bound states in the system; in addition, we can predict the electric and magnetic fields for the confined TE modes in this system. We show very good agreement between our predictions and experiment for bent waveguides in this geometry.

Weakly bound states in heterogeneous waveguides

“…In particular, Exner and Seba [5] have proved that the existence of a bound state for an electron confined to a planar waveguide, with curvature decaying at infinity and obeying Dirichlet boundary conditions at the border; Goldstone and Jaffe [6] have proved that an electron confined to an infinite tube of constant cross section, in two or more dimensions, has always a bound state, when the tube is not perfectly straight. The effect of bound states in infinite non-straight waveguides has been studied in [7,8,9,10,11,12]. It is worth mentioning a recent pedagogical article by Londergan and Murdock [13], that illustrates different numerical methods for the solutions of confined systems, in particular two-dimensional waveguides.…”

Resonance scattering in quantum wave guides

“…If the function w in (29) satisfies the conditions (9), then we multiply both parts of (29) by (G 0 (t) − exp(−t(λ + iε))) −1 and pass to the limit as ε → 0. The radiation conditions and Lemma 2 allow us to compute this limit and obtain (30). If equation (30) holds, then the validity of equation (29) follows from Lemma 5, while the validity of the radiation conditions follows from Lemma 2.…”

Confined modes in two-dimensional tubes

“…The advantage of using the particular solutions method here is that it can be applied to a wire of arbitrary shape and it is not necessary to approximate the sections as having finite length. This method was extended by Tuegel 35 to cover the case of a bent wire (i.e., a wire with an arbitrary value for the interior angle), which was initially calculated by Carini et al 16,17,31 using series expansion and relaxation methods.…”

“…The relaxation method was applied to the crossed wires geometry in Schult et al 15 and to quantum wires similar to the L geometry but with bend angles other than 90 in Carini et al 16,17,31 The problems with stability for the simple relaxation method given above can be avoided with the CrankNicholson relaxation method (described in Section 17.2 of Numerical Recipes 30 ). This method combines forward and backward "time" propagation to give better behavior for convergence, at the expense of a more complicated operator that produces Wðr; s þ DsÞ from Wðr; sÞ.…”