Planar Dirac Fermions in External Electromagnetic Fields

We study the electron propagator in two spatial dimensions in the presence of external electromagnetic fields, this is, we focus in (2+1)-dimensional quantum electrodynamics (QED), where a third spatial dimension is suppressed. This is not a mere theoretical simplification, and we explain ourselves: back in time, some twenty years ago, it was shown that the low-energy effective theory of graphene in a tight-binding approach is the theory of two species of massless Dirac electrons in a (2+1)-dimensional Minkowski spacetime, each on a different irreducible representation of the Clifford algebra. The isolation of graphene samples in 2004 and 2005, has given rise to the new paradigm of relativistic condensed matter, bringing a new boost, both theoretical and experimental, to the matching of common interests of the condensedmatter and high energy physics communities. Thus, themassless limit of our findings is of direct relevance in this subject. We assume the electrons moving in a magnetic field alone pointing perpendicularly to their plane of motion. We first develop the general case and then, we present a couple of examples: themotion of electrons in a uniformmagnetic field, which is a canonical example to present the Ritusmethod and the case of a static magnetic field which decays exponentially along the x-axis (Murguia et al, 2010; Raya & Reyes, 2010). There are many problems relating electrons in non-uniform magnetic fields of relevance in graphene. In particular, it has been established the possibility to confine quasiparticles in magnetic barriers (DeMartino et al, 2007; Ramezani et al, 2009). This could be feasible creating spatially inhomogeneous, but constant in time, magnetic fields depositing ferromagnetic layers over the substrate of a graphene sample layer (Reijniers et al, 2001). The physical properties of graphene make it a promising novel material to control the transport properties in nanodevices. It has been considered to be used in electronics and spintronics applications, like in single-electron transistors (Ponomarenko et al, 2008; Wu et al, 2008), in the so called magnetic edge states (Park & Sim, 2008), which may play an important role in the spin-polarized currents along magnetic domains, and in quantum dots and antidots magnetically confined. Moreover, the quantum Hall effect in graphene has been observed at room temperature (Novoselov et al, 2007), evidence which confirms the great potential of graphene as the material to be used in carbon-based electronic devices. The effects of the exponentially decaying magnetic field can hardly be considered with other approaches, 13