Coupling curvature to a uniform magnetic field: An analytic and numerical study

Abstract

The Schrodinger equation for an electron near an azimuthally symmetric curved surface Σ in the presence of an arbitrary uniform magnetic field B is developed. A thin layer quantization procedure is implemented to bring the electron onto Σ, leading to the well known geometric potential V C ∝ h 2 − k and a second potential that couples A N , the component of A normal to Σ to mean surface curvature, as well as a term dependent on the normal derivative of A N evaluated on Σ. Numerical results in the form of ground state energies as a function of the applied field in several orientations are presented for a toroidal model.

Schrödinger Equation for a Particle on a Curved Surface in an Electric and Magnetic Field

“…For example, studying cylindrical geometries, the magnetic field parallel to the axis was introduced through the boundary conditions of the wave-function [14], on the other hand only the component of the field perpendicular to the surface has been considered in the Schrödinger equation [4,9,14]; also for toroidal surfaces the same approach has generally been adopted [15,16], while the simple geometry of the sphere does not allow to distinguish between the different approaches [2,3,17]. Nevertheless, the da Costa method is recognized as the one to be employed [18], but an analytical expression for the Schrödinger equation including the magnetic field has not been derived yet.In this Letter, we follow the procedure of da Costa including the effect of the magnetic field via the vector potential A and the electric field via the scalar potential V. We shall derive analytically a Schrödinger equation valid for any 2D geometry, that describes in the most appropriate way real curved nanostructures with an electric and magnetic field applied, given the above considerations. We shall show that there is no coupling between the field and the surface curvature and that the dynamics on the surface is decoupled from the transverse one with a proper choice of the gauge, without approximations.…”

Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders

“…From the theoretical perspective, many intriguing phenomena pertinent to electronic states [15,16,17,18,19,20,21,22], electron diffusion [23], and electron transport [24,25,26,27] have been suggested. In particular, the correlation between surface curvature and spin-orbit interaction [28,29] as well as with the external magnetic field [30,31,32] has been recently considered as a fascinating subject.Most of the previous works focused on noninteracting electron systems, though few have focused on interacting electrons [33] and their collective excitations. However, in a low-dimensional system, Coulombic interactions may drastically change the quantum nature of the system.…”

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.…”