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One- and two-dimensional bilayer systems are examples of ultra-tunablequantum materials that are considered as the basis for the new generation ofelectronic and photonic devices. Here we develop a general theory of theelectron band structure for such commensurate and incommensurate bilayer carbonstructures within the tight binding approximation. To model the band structureof commensurate twisted bilayer graphene (TBLG), we apply the classic zonefolding theory. The latter leads us to the construction of TBLG Hamiltonians inthe basis of shifted Bloch wave functions (SBWF), which, in contrast to theusual 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 Tis a number of vertices Q_i of the folded reciprocal space falling into theoriginal first Brillouin zone of any of the layers. We propose and discuss amethod for choosing a reduced set of SBWFs to construct effective Hamiltoniansthat correctly describe the low-energy spectrum of commensurate TBLG. Theflattening of low-energy bands with a decrease in twist angle is discussed. Aswe show, this spectrum results from interactions between the lowest-energymodes of the folded dispersion curves. The effective Hamiltonians forcalculating the low-energy band structure of incommensurate TBLG anddouble-walled carbon nanotubes (DWCNTs) are constructed in a similar way. Totest the developed theory, we calculate the energies of 105 intra-tube opticaltransitions in 29 DWCNTs and compare them with previously published and novelexperimental data. We also apply the theory to calculate the energies ofrecently discovered inter-tube transitions. Geometrical conditions allowingthis type of transitions are discussed. We show that these transitions occur inDWCNTs which layers have close chiral angles and the same handedness.

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