Dirac and Klein-Gordon particles in one-dimensional periodic potentials

Abstract

We evaluate the dispersion relation for massless fermions, described by the Dirac equation, and for zero-spin bosons, described by the Klein-Gordon equation, moving in two dimensions and in the presence of a one-dimensional periodic potential. For massless fermions the dispersion relation shows a zero gap for carriers with zero momentum in the direction parallel to the barriers in agreement with the well-known "Klein paradox". Numerical results for the energy spectrum and the density of states are presented. Those for fermions are appropriate to graphene in which carriers behave relativistically with the "light speed" replaced by the Fermi velocity. In addition, we evaluate the transmission through a finite number of barriers for fermions and zero-spin bosons and relate it with that through a superlattice.

One-dimensional Coulomb problem in Dirac materials

“…Indeed, in the case of 2D Dirac-Weyl systems like graphene, screening does not alter the long-range functional dependence of the Coulomb interaction [22], while screening is further reduced in carbon nanotubes [26]. Additionally, in a similar framework to this work, transmission problems through periodic potentials [38] as well as linear [39] and smooth step potentials [40] have been treated.…”

Transport and localization in periodic and disordered graphene superlattices

“…3a, which consists of a discrete set of incidence angles, is the result of the degeneracy caused by the high symmetry of the structure (u 1 = −u 2 , ε = 0, and δ 1 = δ 2 ). Any symmetrybreaking splits the degeneracy and the spectrum takes the form usual for ordinary periodic structures: a set of conducting zones of non-zero width separated by band gaps 31 . In Fig.…”

“…Among the vast amount of publications on graphene, a significant and ever increasing part belongs to papers devoted to the charge transport in graphene superlattices formed by a periodic external potential (see, e.g., 3,31,32,33,34 ). This is not only due to its theoretical interest but also because of the possibility of experimental realization and potential applications 33 .…”

Spin-resolved scattering through spin-orbit nanostructures in graphene

“…We now briefly summarize the transfer matrix approach employed in this paper to solve the DW scattering problem. [35][36][37][38] We assume translational invariance in the y-direction, thus the scattering problem for the Hamiltonian (1) reduces to an effectively one-dimensional (1D) one. The wave function factorizes as Ψ(x, y) = e iky y χ(x), where k y is the conserved y-component of the momentum, which parameterizes the eigenfunctions of the Hamiltonian of given energy E.…”