The Dirac paradox in 1+1 dimensions and its realization with spin-orbit coupled nanowires

TL;DR

This paper investigates the Dirac paradox in one-dimensional systems, demonstrating that the paradox can be resolved by including massive Dirac modes, which enable tunable transmission in spin-orbit coupled nanowires under external fields.

Contribution

It introduces a model combining massless and massive Dirac modes to resolve the Dirac paradox in 1D and shows how this can be realized in nanowires with electrical control.

Findings

Massless Dirac modes alone lead to trivial solutions to the paradox.

Including massive Dirac modes allows for non-trivial scattering solutions.

Transmission in nanowires can be electrically tuned via interface coupling.

Abstract

At the interface between two massless Dirac models with opposite helicity a paradoxical situation arises: A transversally impinging electron can seemingly neither be transmitted nor reflected, due to the locking between spin and momentum. Here we investigate this paradox in one spatial dimension where, differently from higher dimensional realizations, electrons cannot leak along the interface. We show that models involving only massless Dirac modes lead to either no solutions or to trivial solutions to the paradox, depending on how the helicity change across the interface is modeled. However, non trivial scattering solutions to the paradox are shown to exist when additional massive Dirac modes are taken into account. Although these modes carry no current for energies within their gap, their interface coupling with the massless modes can induce a finite and tunable transmission. Finally, we show that such massless+massive Dirac model can be realized in suitably gated spin-orbit coupled nanowires exposed to an external Zeeman field, where the transmission coefficient can be controlled electrically.