Magnetic field barriers in graphene: an analytically solvable model

We study the dynamics of carriers in graphene subjected to an inhomogeneous magnetic field. For a magnetic field with a hyperbolic profile the corresponding Dirac equation can be analyzed within the formalism of supersymmetric quantum mechanics, and leads to an exactly solvable model. We study in detail the bound-state spectrum. For a narrow barrier the spectrum is characterized by a few bands, except for the zero energy level that remains degenerated. As the width of the barrier increases we can track the band's evolution into the degenerated Landau levels. In the scattering regime a simple analytical formula is obtained for the transmission coefficient, this result allows us to identify the resonant conditions at which the barrier becomes transparent.