Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters

We analyze a description of twisted graphene bilayers that incorporates the deformation of the layers using state-of-the-art interlayer atomic potentials and a modification of the hopping parameters between layers in the light of the classic Slonczewski-Weiss-McClure parametrization. We obtain narrow bands in all cases, but their nature can be rather different. We will show how to describe the results by equivalent continuum models. Even though such models can be constructed, their complexity can vary, requiring many coupling parameters to be included, and the full in-layer dispersion must be taken into account. The combination of all these effects will have a large impact on the wave functions of the flat bands, and modifications in details of the underlying models can lead to significant changes. A robust conclusion is that the natural strength of the interlayer couplings is higher than usually assumed, leading to shifts in the definition of the magic angles. The structure at the edges of the narrow bands, at the $\mathrm{\Gamma }$ point of the Brillouin zone is also strongly dependent on parametrization. As a result, the existence, and size, of band gaps between the flat bands and the neighboring ones are changed. Hence, the definition of Wannier functions, and descriptions based on local interactions are strongly dependent on the description of the model at the atomic scale.

Figure 1

Examples of a graphene bilayer in (approximate) $AA$ and $AB$ alignment.

Figure 2

The definition of the hopping parameters ${\gamma }_i$ as used in the SWM model for $B_1{A}_2$ aligned layers. We denote ${\gamma }_0$ by a black line, ${\gamma }_1$ by a red line, ${\gamma }_3$ in green and ${\gamma }_4$ in purple.

Figure 3

Graphical representation of the terms used in Eqs. (4) and (5). The first diagram (a) is for $AA$ alignment, the last two (b, c) define two situations in $AB$ alignment. The blue dotted circles are the positions of the blue upper layer carbon atoms inverted relative to the central one.

Figure 4

Lattice coordination for a lattice with sides $32{\mathit{a}}_1+31{\mathit{a}}_2$ lattice: (a) no deformation, (b) Nam and Koshino parameters, (c) LCBOP+KC, (d) AIREBO-M+ILP. In each case the color map ranges from dark green for $AA$ alignment to purple for the $AB$ case. White indicates equal $AA$ and $AB$ alignment, and the scale is the same in all figures. Each of these figures shows four unit cells, with AA registration at the corners and midpoints.

Figure 5

Alignment for a $32{\mathit{a}}_1+31{\mathit{a}}_{\mathit{2}}$ (a), $50{\mathit{a}}_1+49{\mathit{a}}_{\mathit{2}}$ (b), and $100{\mathit{a}}_1+99{\mathit{a}}_{\mathit{2}}$ (c) bilayer graphene lattice described by the AIREBO-M+ILP potential.

Figure 6

Results for the interface soliton for a bilayer graphene lattice with periodicity $50{\mathit{a}}_1+49{\mathit{a}}_{\mathit{2}}$ (green), $100{\mathit{a}}_1+99{\mathit{a}}_{\mathit{2}}$ (blue), and $150{\mathit{a}}_1+149{\mathit{a}}_{\mathit{2}}$ (red) described by the AIREBO-M+ILP potential. In each case the strain soliton has a full width at half maximum of $2.3\mathrm{nm}$.

Figure 7

Alignment for a $32{\mathit{a}}_1+31{\mathit{a}}_{\mathit{2}}$ (a), $50{\mathit{a}}_1+49{\mathit{a}}_{\mathit{2}}$ (b), and $100{\mathit{a}}_1+99{\mathit{a}}_{\mathit{2}}$ (c) bilayer graphene lattice described by the LCBOP+KC potential.

Figure 8

Results for the interface soliton for a $50{\mathit{a}}_1+49{\mathit{a}}_{\mathit{2}}$ (green), $100{\mathit{a}}_1+99{\mathit{a}}_{\mathit{2}}$ (blue), graphene lattice described by the LCBOP+KC potential. In the largest case the soliton has a full width at half maximum of $3.1\mathrm{nm}$.

Figure 9

The atomic positions in the soliton region (range as in Fig. 8). This should be compared to Figs. 3 and 3 from Ref. [51].

Figure 10

The values of the SWM $\gamma$ parameters for bilayer graphene in eV as a function of interlayer distance for each of our hopping parameters. The blue line is the Koster-Slater parametrization, the yellow line is the screened-1 hopping [35], and the solid green line is our screened-2 modification; see Table 2. For the two-body Koster-Slater choice we always have ${\gamma }_3={\gamma }_4$.

Figure 11

Band structure (left) and density of states (right) of a Moiré commensurate superlattice of lattice parameter ${\mathit{b}}_1=32{\mathit{a}}_1+31{\mathit{a}}_2$. The twist angle is $\theta \approx 1.{05}^{\circ }$. All of these figures have a constant nearest-neighbor in-layer coupling. (a) Undeformed lattice with an exponential Koster-Slater interlayer coupling; (b) LKC deformed lattice with the Koster-Slater coupling; (c) AILP deformed lattice with the Koster-Slater coupling; (d) Undeformed lattice with our screened-1 inter-layer coupling; (e) LKC deformed lattice with the screened-1 coupling; (f) AILP deformed lattice with the screened-1 coupling; (g) LKC deformed lattice without vertical corrugation with the screened-1 coupling; (h) AILP deformed lattice without vertical corrugation with the screened-1 coupling; (i) Undeformed lattice with our screened-2 interlayer coupling; (j) LKC deformed lattice with the screened-2 coupling; (k) AILP deformed lattice with the screened-2 coupling; (l) LKC deformed lattice without vertical corrugation with the screened-2 coupling; (m) AILP deformed lattice without vertical corrugation with the screened-2 coupling.

Figure 12

Bands in an undeformed graphene bilayer for a Koster-Slater coupling. Panel (a) is for our choice of hopping parameters, panel (b) shows the effect of reducing the Fermi velocity by 10%, panel (c) shows the effect of reducing the interlayer hopping by 10%, and panel (d) shows the effect of both changes simultaneously.

Figure 13

The $K$ points of the two layers, with the expansion point in the middle. These fold onto the $\overline{K}$, ${\overline{K}}^{\prime }$, and $\overline{M}$ points.

Figure 14

The magnitude of the matrix elements Eq. (18) as a function of the momentum transfer $\mathit{k}$. Each hexagon—or rather its midpoint—denotes a single superlattice vector $\mathit{G}$, and the color shows the absolute value of the relevant matrix element. The green circles are the points ${\mathit{K}}^{\left(1\right)}$ and ${\mathit{K}}^{\left(2\right)}$. The plots correspond to the spectra shown in Fig. 11, and are labeled accordingly. In each case the entries on the left are $AA$ couplings, and the ones on the right the $AB$ ones. Note that the color-scale used is nonlinear to better show differences between small matrix elements.

Figure 15

Spectrum for various form of the continuum model for the case of the SWM model without lattice deformation. Panel (a) is the “standard” Bistritzer-MacDonald truncation, with only three interlayer matrix elements and Dirac in-layer dispersion; panel (b) is the same model now with the in-layer tight-binding dispersion, and the full $k$ dependence of the interlayer matrix elements. Panels (c) and (d) are similar figures, but now including the next group of intralayer matrix elements as well; panels (e) and (f) finally include all the matrix elements that give nonperturbative effects. The red and blue curves are the two valleys of the model; the gray lines are the exact diagonalization.

Figure 16

Spectrum for various form of the continuum model for the case of the second SWM model with LKC lattice deformation. See Fig. 15 for details of the results presented.

Figure 17

An example of a supercell in an $(m,n)=(5,4)$ grid. Red points show the + lattice and blue points the $-$ one. The circles are the average positions. Note the reflection symmetries in the two green lines, which are broken for the average positions.

Figure 18

points $q$ used in Table 4.