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

One- and two-dimensional twisted bilayer structures are examples of ultratunable quantum materials that are considered the basis for the next generation of electronic and photonic devices. Here, we develop a general theory of the electron band structure for such commensurate and incommensurate bilayer graphene structures within the framework of 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 (SBWFs) 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 intratube optical transitions in 29 DWCNTs and compare them with experimental data. We also apply the theory to calculate the energies of recently discovered intertube transitions. Geometrical conditions allowing this type of transition are discussed. We show that these transitions occur in DWCNTs whose layers have close chiral angles and the same handedness or, in the structural context, in DWCNTs with a large unit cell of the periodic moire pattern.