Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration

TL;DR

This paper analyzes how complex Robin boundary conditions and magnetic fields affect wave propagation and bound states in guiding structures, revealing resonance behaviors and potential experimental observations in superconductors.

Contribution

It introduces a detailed analysis of complex Robin boundary conditions in magnetic fields, showing their impact on eigenvalues, quasibound states, and transport properties in guiding structures.

Findings

Eigenvalues become complex with nonzero imaginary parts due to complex boundary conditions.

Magnetic field tends to restore real eigenvalues approaching Landau levels.

Resonance effects are pronounced at zero magnetic field and are suppressed at high magnetic fields.

Abstract

Solutions of the scalar Helmholtz wave equation are derived for the analysis of the transport and thermodynamic properties of the two-dimensional disk and three-dimensional infinitely long straight wire in the external uniform longitudinal magnetic field $\bf B$ under the assumption that the Robin boundary condition contains extrapolation length $\Lambda$ with nonzero imaginary part $\Lambda_i$. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues $E$ become complex too and the discrete bound states of the disk characteristic for the real $\Lambda$ turn into the corresponding quasibound states with their lifetime defined by the eigenenergies imaginary parts $E_i$. Accordingly, the longitudinal flux undergoes an alteration as it flows along the wire with its attenuation/amplification being $E_i$-dependent too. It is shown that, for zero magnetic field, the component $E_i$ as a function of the Robin imaginary part exhibits a pronounced sharp extremum with its magnitude being the largest for the zero real part $\Lambda_r$ of the extrapolation length. Increasing magnitude of $\Lambda_r$ quenches the $E_i-\Lambda_i$ resonance and at very large $\Lambda_r$ the eigenenergies $E$ approach the asymptotic real values independent of $\Lambda_i$. The extremum is also wiped out by the magnetic field when, for the large $B$, the energies tend to the Landau levels. Mathematical and physical interpretations of the obtained results are provided; in particular, it is shown that the finite lifetime of the disk quasibound states stems from the $\Lambda_i$-induced currents flowing through the sample boundary. Possible experimental tests of the calculated effect are discussed; namely, it is argued that it can be observed in superconductors by applying to them the external electric field $\cal E$ normal to the surface.