Tight-binding approximation for bilayer graphene and nanotube structures: From commensurability to incommensurability between the layers

One- and two-dimensional bilayer systems are examples of ultra-tunable quantum materials that are considered as the basis for the new generation of electronic and photonic devices. Here we develop a general theory of the electron band structure for such commensurate and incommensurate bilayer carbon structures within the tight binding approximation. To model the band structure of commensurate twisted bilayer graphene (TBLG), we apply the classic zone folding theory. The latter leads us to the construction of TBLG Hamiltonians in the basis of shifted Bloch wave functions (SBWF), which, in contrast to the usual Bloch functions, have the wave vector q shifted by a set of vectors Q_i. The dimension of the considered Hamiltonians is equal to 4T, where the factor T is a number of vertices Q_i of the folded reciprocal space falling into the original first Brillouin zone of any of the layers. We propose and discuss a method for choosing a reduced set of SBWFs to construct effective Hamiltonians that correctly describe the low-energy spectrum of commensurate TBLG. The flattening of low-energy bands with a decrease in twist angle is discussed. As we show, this spectrum results from interactions between the lowest-energy modes of the folded dispersion curves. The effective Hamiltonians for calculating the low-energy band structure of incommensurate TBLG and double-walled carbon nanotubes (DWCNTs) are constructed in a similar way. To test the developed theory, we calculate the energies of 105 intra-tube optical transitions in 29 DWCNTs and compare them with previously published and novel experimental data. We also apply the theory to calculate the energies of recently discovered inter-tube transitions. Geometrical conditions allowing this type of transitions are discussed. We show that these transitions occur in DWCNTs which layers have close chiral angles and the same handedness.