Wave Functions in the Neighborhood of a Toroidal Surface; Hard vs. Soft Constraint

Abstract

The curvature potential arising from confining a particle initially in three-dimensional space onto a curved surface is normally derived in the hard constraint q → 0 limit, with q the degree of freedom normal to the surface. In this work the hard constraint is relaxed, and eigenvalues and wave functions are numerically determined for a particle confined to a thin layer in the neighborhood of a toroidal surface. The hard constraint and finite layer (or soft constraint) quantities are comparable, but both differ markedly from those of the corresponding two dimensional system, indicating that the curvature potential continues to influence the dynamics when the particle is confined to a finite layer. This effect is potentially of consequence to the modelling of curved nanostructures.

Thin layer quantization in higher dimensions and codimensions

“…Recent progress in nanotechnology caused a great activity in the field. Free particle energy spectrum was investigated for thin layers around cylinders [9], tori [10] and arbitrary surfaces of revolution [11]. In this article we clarify the general properties of the thin layer method and establish its equivalence with Prokhorov quantization procedure [12,13] for codimension 1 surfaces.…”

Wave functions for a toroidal quantum dot in the presence of an axially symmetric magnetic field: transition from ring to bulk states as a function of aspect ratio

“…with Y n a Bessel function of the second kind. Previous work regarding solutions on a toroidal surface indicate the angular dependence of those eigenfunctions will be more complicated [15,16].…”

Dipole and solenoidal magnetic moments of electronic surface currents on toroidal nanostructures

“…1 further by demanding conservation of the norm in the correct limit q → 0 [19][20][21][22][23][24][25][26]. There is a second geometric potential that arises from the coupling of the SCD to the normal part A N of the vector potential in this limit [7,13,27,28]. Equation 1 may be solved by routine methods.…”

“…Toroidal structures allow for electron motion around the major radius R (azimuthal or dipole mode) and the minor radius a (solenoidal modes) of a circularly symmetric torus, which is subject to specific boundary conditions which are different from those for flat two-dimensional rings. Generally one finds that for a hollow torus electrons are thought to be localized near the surface of the object, which has interesting theoretical consequences: a surface-dependent geometric potential V c which acts as an effective potential for the electron motion has to be included [12,13]. In this theoretical study we want to focus on electronic currents on the surface of a nanometer-scale torus T 2 .…”