Graphene-based superlattice (SL) formed by a periodic gap modulation is studied theoretically using a Dirac-type Hamiltonian. Analyzing the dispersion relation we have found that new Dirac points arise in the electronic spectrum under certain conditions. As a result, the gap between conduction and valence minibands disappears. The expressions for the positions of these Dirac points in k-space and threshold value of the potential for their emergence were obtained. Also, the dispersion law and renormalized group velocities around the new Dirac points were calculated. At some parameters of the system, we have revealed interface states which form the top of the valence miniband.

We study transport properties of graphene nanostructures consisted of alternating slabs of gapless (∆ = 0) and gapped (∆ = 0) graphene in the presence of piecewise constant external potential equal to zero in the gapless regions. The transmission through single-, double-barrier structures and superlattices has been studied. It was revealed that any n-barrier structure is perfectly transparent at certain conditions defining the positions of new Dirac points created in the superlattice. The conductance and the shot noise were as well computed and investigated for the considered graphene systems. In a general case, existence of gapped graphene fraction leads to decrease of the conductance and increase of the Fano factor. For two barriers formed by gapped graphene and separated by a long and highly doped region the Fano factor rises up to 0.5 in contrast to the similar gapless structure where the Fano factor is close to 0.25. Similarly to a gapless graphene superlattice, creation of each new Dirac point manifests itself as a conductivity resonance and a narrow dip in the Fano factor. However, gapped graphene inclusion into the potential-barrier regions in the superlattice leads to more complicated dependence of the Fano factor on the potential height compared to pseudo-diffusive behaviour (with F = 1/3) typical for a gapless superlattice.

We investigate the Goos-Hänchen shift for ballistic electrons (i) reflected from a step-like inhomogeneity of the potential energy and (or) effective mass, and (ii) transmitted through a ferromagnetic barrier region in monolayer silicene or gapped graphene. For the electrons reflected from a single interface we found that the Goos-Hänchen shift is valley-polarized for gapped graphene structure, and valley-and spin-polarized for silicene due to the spin-valley coupling. Incontrast, for example, to gapless graphene the lateral beam shift in gapped structures occurs not only in the case of total, but also of partial, reflection, i.e. at the angles smaller than the critical angle of total reflection. We have also demonstrated that the valley-and spin-polarized displacement of the electron beam, transmitted through a ferromagnetic silicene barrier, resonantly depends on the barrier width. The resonant values of the displacement can be controlled by adjusting the electric potential, the external perpendicular electric field, and the exchange field induced by an insulating ferromagnetic substrate.

We study the effect of weak disorder on the delocalization properties of gapped graphene superlattice (SL) formed by periodically located rectangular potential barriers. We consider two types of the SLs: the SLs with uniform gap and SLs consisting of alternating layers of gapped and gapless graphene regions. Using the perturbative approach we obtain an analytical expression for the inverse localization length (ILL) derived for the case of randomly fluctuating geometric and energetic parameters. In the first case, when the barrier (well) width fluctuates around its mean value, the corresponding equation for the ILL reveals the presence of the Fabry-Perot resonances, at which the localization length diverges. These resonances are exact, i.e., are stored in any degree of disorder. It has been found that the localization properties manifest stronger for the particles with energies lying in the non-resonant bands where our approach is extremely sensitive to the degree of disorder. For the case of weakly fluctuating both barrier and well widths we analytically obtain ILL taking correlations into account. The main effect of the correlations, which lead to an increase (or decrease) in the localization length, was revealed near the double resonance arising at coincidence of two Fabry-Perot resonances associated with barrier and well widths. The random fluctuations of the potential strength also lead to the delocalization resonances. However, they exist only in a weak-disorder approximation. We found that, for an array composed of alternating strips of gapless and gapped graphene modifications these resonances can appear only for normally incident particles in contrast to the SL with a uniform gap. For such particles, the delocalization resonances occur also in the purely random potential. This means, in particular, that in the one-dimensional case, not all the states of the massive Dirac particles are localized in the presence of weak disorder.

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