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 $\mathbf{q}$ shifted by a set of vectors ${\mathbf{Q}}_i$. The dimension of the considered Hamiltonians is equal to $4T$, where the factor $T$ is a number of vertices ${\mathbf{Q}}_i$ 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 moiré pattern.

Figure 1

The first two commensurate bilayer structures (2,1) and (3,1) in the (a) and (b) direct and (c) and (d) reciprocal spaces. (a) and (b) Large hexagons and arrows show the unit cell and the basis translations of superlattices, respectively. Dots represent the nodes of the primitive hexagonal lattice; the positions of carbon atoms in both the top and bottom layers coincide with some of the nodes. The existence of this parent lattice simplifies the calculation of the matrix elements of the bilayer Hamiltonian in Eq. (5). The translation length of the parent lattice is $\sqrt{T}$ times shorter than the radius of the carbon hexagon. (c) and (d) Superposition of the top and bottom layer first Brillouin zones on a honeycomb hexagonal lattice reflecting the translational symmetry of the folded reciprocal space.

Figure 2

Band structures calculated using the complete and effective Hamiltonians. The electron spectra are calculated within a reduced first Brillouin zone. The point $\overline{\mathrm{\Gamma }}$ is at its center, the point $\overline{\mathbf{K}}$ is one of the vertices and coincides with the point $\mathbf{K}$, the point $\overline{\mathbf{M}}$ lies in the middle of the hexagon edge. The dispersion curves obtained within the Hamiltonian in Eq. (5) are shown in black, and the ones calculated within the effective Hamiltonian in Eq. (6) are shown in red. The light blue dashed line shows the spectrum of the effective Hamiltonian ${\mathbf{H}}_{24}$. (a) and (b) The band structures for twisted bilayer graphene (TBLG) (2,1) and (3,1) are shown in the ranges $\pm 6$ and $\pm 4$ eV, respectively. On the scale of (b), there is no visible difference in the range from −3 to 2.5 eV between the spectra calculated within the complete Hamiltonian and the effective one ${\mathbf{H}}_{24}$. (c) and (d) The band structure calculated near the $\mathbf{K}$ point for the same superstructures. А gap between the bands typical of commensurate TBLG appears [15, 32]. On the scale of (c) and (d), the spectrum of ${\mathbf{H}}_{24}$ coincides with the one of the complete Hamiltonian. The origin in (c) and (d) is chosen in the $\overline{\mathbf{K}}$ point. To the left from the origin, the curves in the direction $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathbf{K}}$ are shown, while to the right from the origin, the calculations in the direction $\overline{\mathbf{K}}\text{−}\overline{\mathbf{M}}$ are presented.

Figure 3

Flattening of the lowest-energy bands in twisted bilayer graphene with a small twist angle within the framework of zone folding theory. (a) Schematic band structure produced by the coupling between two pairs of Dirac cones. The cone location points ${\overline{\mathbf{K}}}_1$ and ${\overline{\mathbf{K}}}_2$ are separated by a minimum translation with length $G$. Purple dotted lines show the band structure when the coupling is switched off. The cones of the top and bottom layers are rotated by an angle $\theta$ and, therefore, coincide only in the vicinity of the points ${\overline{\mathbf{K}}}_1$ and ${\overline{\mathbf{K}}}_2$. The band structure (calculated with the account of van der Waals interaction between the layers) is shown by black solid lines. This interaction leads to the merging and repulsion of previously intersecting dispersion curves. (b) and (c) Spectra of the effective Hamiltonian ${\mathbf{H}}_{24}$ calculated for commensurate structures (15,1) and (27,1). The corresponding twist angles θ are 6.40 ° and 3.61 °. The maximum distance between the lowest energy bands (which is near the point $\overline{\mathrm{\Gamma }}$) decreases from 1 to 0.4 eV.

Figure 4

The direct and reciprocal spaces of the incommensurate twisted bilayer graphene (TBLG). (a) Two superimposed hexagonal lattices with a twist angle $\theta =3.5{}^{\circ }$. The arrows show the basis translations of the moiré lattice. Near the nodes, the TBLG structure is practically the same; however, the translations of the moiré lattice connect structurally nonequivalent points of TBLG. (b) Superimposed reciprocal lattices of the top and bottom monolayers; $\theta =3.5{}^{\circ }$. (c) and (d) Selected regions of the panel (b) at a larger scale. Far from the origin, points of $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ types can be arbitrarily close; an example is shown in panel (c). (d) Around the point Γ, there are three pairs of $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points.

Figure 5

Origin of intertube optical transitions in double-walled carbon nanotubes (DWCNTs) with the specific geometry. (a) Typical reciprocal space of a DWCNT, where intertube optical transitions are possible. Superimposed extended Brillouin zones of the inner and outer tubes are shown in red and black, respectively. Parallel gray lines represent the common cutting lines of the DWCNT. The projections of the points $K_1^{{}^{\prime }}$ and $K_1$ on these lines have very close $k$ coordinates, which correspond to the proximity of the band extrema originating from different pristine SWCNTs. As an example, DWCNT (10,6)@(14,13) is chosen. (b) Schematic rearrangement in the DWCNT band structure caused by inter-layer coupling. Black and red solid curves correspond, respectively, to the bands of the inner and outer tubes; $\mathrm{\Delta }k\ll K_1$, where $K_1$ is the $k$-coordinate of the point ${\mathit{K}}_1$; $E_{\mathrm{in}\left(\mathrm{out}\right)}^{+}$ and $E_{\mathrm{in}\left(\mathrm{out}\right)}^{-}$ are the energies of the conduction and valence bands in non-interacting tubes; $E_1^{\mathrm{DW}}, E_2^{\mathrm{DW}}, E_3^{\mathrm{DW}}, E_4^{\mathrm{DW}}$ are the corresponding energies of DWCNT bands. Double arrows demonstrate the emerging intertube transitions.

Figure 6

Rayleigh scattering spectra for double-walled carbon nanotubes (10,6)@(14,13) and (14,8)@(19,14). The dashed vertical lines show the energies of optical transitions predicted by our theory. The labels of the transitions originating from the inner and outer single-walled carbon nanotubes are shown at the top and bottom parts of the panels, respectively. Red dotted lines correspond to the intertube transitions.

Figure 7

Moiré patterns (MPs) in double-walled carbon nanotubes (DWCNTs) (14,2)@(15,13), (10,6)@(14,13), and (12,11)@(17,16). (a) The unrolled direct space of DWCNT (14,2)@(15,13). In this nanotube, $\mathrm{\Delta }\phi =21.052{}^{\circ }$, and intertube optical transitions are forbidden according to our theory. One can see an aperiodic motif in the pattern, and the regions of local similarity are very small. (b) The unrolled direct space of DWCNT (10,6)@(14,13). In this tube, intertube transitions are possible, and the points $K_1$ and $K_1^{{}^{\prime }}$ are quite close. The regions of local similarity become elongated in comparison with those in the panel (a). (c) The unrolled direct space of DWCNT ((17),16)@(12,11). The chirality vectors of the outer and inner tubes are almost collinear, which corresponds to the extremely elongated MP. Moreover, the points $K_1$ and $K_1^{{}^{\prime }}$ almost coincide in the scheme of superimposed zones.