Curvature-induced bound states in quantum wires

Abstract

A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.

Figures (5)

• Figure 1: Four representatives of the curve family $\{\mathcal{X}_{\mathcal{M}_{\varepsilon}}\}_{\varepsilon \in \mathbb{R}^{+}}$ and the limiting degenerate curve $\mathcal{X}_{\mathcal{M}}$. $\theta$ is the opening angle of the latter. The subsets of Euclidean space converge in the pointwise sense. Parameters: $J = (-5, 5)\,\mathrm{nm}$, $\alpha = \frac{1}{2}$, $\theta = \frac{\pi}{8}$, $m = m_{e}$
• Figure 2: Curvature functions $\kappa_{\varepsilon}$ for four different representatives of the regularized curve family, and their divergent limit $\kappa$. The parameters are the same as in fig. \ref{['fig:ExampleCurves']}.
• Figure 3: The four lowest energy eigenvalues of the regularized Hamilton operators in dependence of the de-singularization parameter $\varepsilon$. In the limit, the ground state is negative, and resembles a bound state when $J = \mathbb{R}$. The parameters are the same as in fig. \ref{['fig:ExampleCurves']}.
• Figure 4: The probability density of the ground state for four regularized Hamilton operators, and their non-differentiable limit. The parameters are the same as in fig. \ref{['fig:ExampleCurves']}.
• Figure 5: The ground state energy of the limiting degenerate case in dependence of the turn of the curve. Other parameters are the same as in fig. \ref{['fig:ExampleCurves']}.

Theorems & Definitions (10)

• Definition 2.1: Parametrized differentiable (smooth) submanifold of Euclidean Space
• Remark
• Definition 2.2: Degenerate submanifold of Euclidean space (CPA)
• Theorem A.1
• proof
• Theorem A.2
• proof
• Definition A.3: Admissible Regularization
• Theorem A.4
• proof