Pseudospin Splitting of the Energy Spectrum of Planar Polytype Graphene-Based Superlattices
The energy spectrum of planar polytype graphene-based superlattices has been investigated. It is shown that their energy spectrum undergoes pseudospin splitting due to the asymmetry of quantum wells forming the superlattice potential profile.
Currently, researchers pay much attention to graphene-based superlattices. In particular, methods of molecular dynamics were used to calculate graphene-based superlattices with periodically located rows of vacancies (Chernozatonskii 2006). Then superlattices of single-atom thickness, formed by lines of pairs of hydrogen atoms adsorbed on graphene, were calculated within the density functional theory (Chernozatonskii 2007). Rippled graphene, which can be considered as a superlattice with a one-dimensional periodic potential of ripples, was investigated in (Isacsson 2008, Guinea 2008). Analytical studies were performed on superlattices obtained by applying a periodic electrostatic potential (Bai 2007, Barbier 2008, Park 2008, Park 2008) or periodically arranged magnetic barriers (Masir 2008, Masir 2009, Ghosh 2009) to graphene. A detailed review of graphene-based semiconductor heterostructures can be found in (Sorokin 2013).
We investigate planar superlattices based on gapless graphene and its gapped modifications. The main concepts of these superlattices were formulated in (Ratnikov 2009), where a dispersion relation for charge carriers in these structures was derived. Then a very simple example of such superlattices was considered: two-type superlattices composed of alternating strips of gapless graphene and its band gap modification.
In (Ratnikov 2012) we studied graphene-based quantum wells (QWs): planar heterostructures based on graphene, where gapped graphene modifications play a role of potential barriers. In particular, it was shown that the energy spectrum of asymmetric QWs (containing different gapped graphene modifications) is split with respect to pseudospin: the dispersion curves in different valleys do not coincide. This result suggests that pseudospin splitting should also occur in polytype graphene-based superlattices. This splitting is similar in many respects to the spin splitting of the energy spectrum in narrow-gap heterostructures (Kolesnikov 1997).
Note also that recently we proposed and investigated another version of planar superlattices, which is peculiar for graphene (Ratnikov 2014). Using the Fermi velocity engineering in gapless graphene, one can fabricate structures with periodically modulated Fermi velocity.
In this paper, we discuss the pseudospin splitting of the energy spectrum of a polytype superlattice. As an example of polytype graphene-based superlattices, we consider a three-type superlattice in the form \(A\)—\(B\)—\(C\), where \(A\) and \(C\) are gapped graphene modifications with different bandgap widths, and \(B\) is gapless graphene. This is the simplest example of a polytype superlattice which retains fundamental features of polytype superlattices.
To implement a three-type graphene-based superlattice, we purpose a version involving gapless graphene and its gapped modifications obtained as a result of deposition of a gapless graphene sheet on a particular substrate (e.g., hexagonal boron nitride h-BN) or due to deposition of particular atoms or molecules (for example, CrO\(_3\)) on the surface of gapless graphene. This version is presented pictorially in Fig. 1.
Let region I be a layer of gapped graphene modification with a bandgap half-width \(\Delta_I\) and work function \(V_I\); the thickness of this region is \(d_I\). Region II is a layer of ga