Abstract
A simple one-dimensional differential equation in the centerline coordinate of an arbitrarily curved quantum waveguide with a varying cross section is derived using a combination of differential geometry and perturbation theory. The model can tackle curved quantum waveguides with a cross-sectional shape and dimensions that vary along the axis. The present analysis generalizes previous models that are restricted to either straight waveguides with a varying cross-section or curved waveguides, where the shape and dimensions of the cross section are fixed. We carry out full 2D wave simulations on a number of complex waveguide geometries and demonstrate excellent agreement with the eigenstates and energies obtained using our present 1D model. It is shown that the computational benefit in using the present 1D model to calculate both 2D and 3D wave solutions is significant and allows for the fast optimization of complex quantum waveguide design. The derived 1D model renders direct access as to how quantum waveguide eigenstates depend on varying cross-sectional dimensions, the waveguide curvature, and rotation of the cross-sectional frame. In particular, a gauge transformation reveals that the individual effects of curvature, thickness variation, and frame rotation correspond to separate terms in a geometric potential only. Generalization of the present formalism to electromagnetics and acoustics, accounting appropriately for the relevant boundary conditions, is anticipated.
Highlights
IntroductionEven a small number of coupled equations was shown to provide an accurate description of the groundstate, the first excited states and their frequencies
With the vast possibilities to manufacture complex topological structures in quantum technology, it becomes increasingly important to address how physical properties change due to shape, size etc
Nanowire technology provides a rich platform to discover novel physical properties related to curvature effects such as flexible electronics [1], battery [2] and nanoelectromechanical sensors and generators [3,4,5,6,7]
Summary
Even a small number of coupled equations was shown to provide an accurate description of the groundstate, the first excited states and their frequencies Both methods [25,27,28], assumed a straight centerline (cylinder structures) and cannot be applied to a more general class of nanowire geometries characterized by, simultaneously, a curved axis and a varying cross section. We present a new method using differential geometry and perturbation theory to determine eigenstates to the Schrödinger equation of a quantum-mechanical particle confined to a curved, varying-thickness waveguide with small cross-sectional dimensions relative to the waveguide length. Besides the insight that can be gained from the 1D equations, there is significant computational advantages, as discussed in Appendix A
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
