Confinement of Dirac electrons in graphene magnetic quantum dots

Abstract

We characterize the confinement of massless Dirac electrons under axially symmetric magnetic fields in graphene, including zero energy modes and higher energy levels. In particular, we analyze in detail the Aharonov-Casher theorem, on the existence of zero modes produced by magnetic fields with finite flux in two dimensions. We apply techniques of supersymmetric quantum mechanics to determine the confined states by means of the quantum number j associated to isospin and angular momentum. We focus on magnetic fields, regular at the origin, whose asymptotic behaviour is B(r) ∼ rα, with α a real number. A confinement of infinite zero-energy modes and excited states is possible as long as α ≥-1. When -2 ≤ α < -1 the quantum dot is able to trap an infinite number of zero modes but no excited states, while for α < -2 only a finite number of zero modes are confined.