I. INTRODUCTION In recent years, the study of several quantum mechanical systems within the framework of an effective position-dependent mass (PDM) has received increasing attention in the literature. Posi- tion dependent effective masses enter, for example, in the dynamics of electrons in semiconductor hetero-structures [1], and when describing the properties of hetero-junctions and quantum dots [2]. In non-relativistic quantum mechanics when the mass becomes dependent on the position coordi- nate the mass and momentum operators no longer commute, thereby making the generalization of the non-relativistic Hamiltonian (kinetic energy operator) to the PDM case highly non trivial [3, 4], because of the ambiguities in the choice of a correct ordering of mass and momentum operator [5]. Another important issue is that of Galilean invariance [6]. The investigation of relativistic effects is of course important in those systems containing heavy atoms or heavy ion doping [7]. Therefore for these type of materials the investigations of the properties of the Dirac equation in circumstances where the mass becomes a function of the position is certainly of great interest. In addition the problems posed by the ambiguities of the mass and momentum operator ordering are absent in the Dirac equation. An effort in this direction has been reported in some recent literature [7–15]. For example the authors of ref. [7] have reported an interesting numerical investigation of the scattering problem for the three-dimensional Dirac hamiltonian within a position dependent mass with a costant asymptotic limit, studying the energy resonance structure. In Ref. [16] the scattering problem is solved for a smooth potential and a mass step but in the non-relativistic regime. The authors of ref. [17] reported an approximated solution of the one-dimensional Dirac equation with a position dependent mass for the generalized Hulth´ en potential. To the best of our knowledge few attempts have been reported that study the Dirac equation in an external potential with position dependent effective masses. In ref. [18] the author studies Dirac equation in 3+1 dimension in the Coulomb field and with a spherically symmetric singular mass distribution. In Ref [10] the author reports an exact solution for the Dirac equation with central potential and mass distribution both inversely proportional to the distance from the center. It is worth pointing out that graphene (single atomic layer of graphite), a recently discovered material [19, 20] which is receiving a lot of attention, exhibits several properties whose explanation involve the Dirac equation for massless fermions. For a comprehensive review see for example [21] Recent reports studying these effects [22, 23] attest the use of the Dirac equation in explaining the properties of single-layer graphene . 2