Time-dependent Quantum Waveguide Theory: A Study of Nano Ring Structures

Abstract

As electronic circuits get progressingly smaller to the nanometre scale, the quantum wave nature of the electrons starts to play a dominant role. It is thus possible for the devices to operate by controlling the phase of the quantum electron waves rather than the electron density as in present-day devices. This paper presents a highly accurate numerical method to treat quantum waveguides with arbitrarily complex geometry. Based on this model, a variety of quantum effects can be studied and quantified.

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

“…In a previous work [6] the Schrodinger equation for an electron on a toroidal surface T 2 in an arbitrary static magnetic field was developed and numerical results obtained. There however, the electron was restricted ab-initio to motion on T 2 , which precluded the appearance of the well known geometric potential V C ∝ (h 2 − k), [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] with h, k the mean and Gaussian curvatures, that arises via thin-layer quantization [43]. Furthermore, if the degree of freedom normal to Σ (labelled q in everything to follow) is included, then the component of the vector potential A N normal to Σ couples to the normal part of the gradient.…”

Elliptical tori in a constant magnetic field

“…(4) Defining γ = 0.263B 0 , with B 0 in tesla, which is the conversion factor for an R = 500 Å torus. (5) Performing the well-known procedure for obtaining V C (which will appear below as the scaled dimensionless function U C ) for which the reader is directed to [14][15][16][17][18][19][23][24][25][26][27][28][29][30]. (6) Noting that the azimuthal symmetry of the problem allows for the eigenfunction on E T 2 to be taken as…”

“…b. setting α = a/R, β = b/R, ε = 2Ea 2 , c. letting D(θ) = P (θ)/R, p(θ) = P (θ)/a and F (θ) = 1 + α cosθ, d. defining γ = .263B 0 , with B 0 in Teslas, which is the conversion factor for an R = 500 Å torus, e. performing the well-known procedure for obtaining V C (which will appear below as the scaled dimensionless function U C ) for which the reader is directed to the relevant references[14,15,16,17,18,19,23,24,25,26,27,28,29,30],…”

Electron wave functions on in a static magnetic field of arbitrary direction

“…There is a wealth of literature concerning curvature effects when a particle is constrained to a two-dimensional surface in three-space [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], including some dealing with the torus specifically [39], but the scope of this work will remain restricted to study of the Hamiltonian given by Eq. ( 9).…”