Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac lines

We review the energy spectrum and transport properties of several types of one-dimensional superlattices (SLs) on single-layer and bilayer graphene. In single-layer graphene, for certain SL parameters an electron beam incident on a SL is highly collimated. On the other hand there are extra Dirac points generated for other SL parameters. Using rectangular barriers allows us to find analytic expressions for the location of new Dirac points in the spectrum and for the renormalization of the electron velocities. The influence of these extra Dirac points on the conductivity is investigated. In the limit of δ-function barriers, the transmission T through, conductance G of a finite number of barriers as well as the energy spectra of SLs are periodic functions of the dimensionless strength P of the barriers, Pδ(x) = V (x)/~vF, with vF the Fermi velocity. For a Kronig-Penney SL with alternating sign of the height of the barriers the Dirac point becomes a Dirac line for P = pi/2 +npi with n an integer. In bilayer graphene, with an appropriate bias applied to the barriers and wells, we show that several new types of SLs are produced and two of them are similar to type I and type II semiconductor SLs. Similar as in single-layer graphene extra “Dirac ” points are found. Non-ballistic transport is also considered