Emergence of intrinsically isolated flat bands and their topology in fully relaxed twisted multilayer graphene

We study the electronic structure and band topology of fully relaxed twisted multilayer graphene (TMLG). Isolated flat bands emerge in TMLG with the number of layers $[M+N$ with $M$ the layer number of the bottom few-layer graphene (FLG)] up to 10 and with various stacking orders, and most of them are on the hole side. The touched bands of FLGs around the Fermi level are split by the moiré coupling through the electron-hole asymmetry in low-energy bands of FLGs and by the vertical hopping between next-nearest layers. The full structural relaxation leads to global gaps that completely isolate a flat band. For TMLG with given $M$ and $N$, the highest magnitude of Chern numbers ($\left|C\right|$) of the separable flat bands reaches $M+N-1$ and can be hosted by certain isolated bands. The $\left|C\right|=9$ occurs in the isolated flat valence band of several configurations with 10 layers. Such high $\left|C\right|$ originates from the lifting of the band-state degeneracy in the weak regime of moiré coupling or from the topological phase transitions induced by the strong moiré coupling. Moreover, large orbital magnetic moments arise in isolated flat bands with high $\left|C\right|$ and depend on the structural configurations of TMLG.

Figure 1

The geometry and electronic band structures of TMLG. (a) The schematic view of the TMLG with the top FLG (tFLG) rotated by $\theta$ counterclockwise with respect to the bottom FLG (bFLG). The side view shown is around the AA stacking between layers 1 and $\tilde{1}$. The layer numbers of bFLG and tFLG are denoted by $M$ and $N$, respectively. The sublattices in each layer are labeled by $n\alpha$ with $n$ the layer index and $\alpha =\mathrm{A}$, B. [(b)–(e)] The band structures with isolated flat valence bands (red lines) for typical configurations of the four stacking types of TMLG. As bands in the two valleys are related by the time reversal symmetry, only bands in the $\xi =+$ valley are displayed. The isolated valence band is separated from other bands by global gaps ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_h$ labeled in (b). The Chern numbers of the isolated bands in the $\xi =+$ valley are labeled. The Fermi levels are shown by dashed lines.

Figure 2

The properties of isolated flat valence bands in configurations with different stacking types. (a) ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_h$ for all configurations with isolated flat valence bands. Each configuration of TMLG is characterized by the stacking and $\theta$. (b) Width ($W_v$) of the isolated valence band versus $\theta$ of these configurations.

Figure 3

The band structures and state composition for TMLG with the (BA)-(ABCA) stacking [(a)–(e)] and the (CA)-(AB)(AB) stacking [(f)–(j)] at $\theta =1.{35}^{\circ }$. Starting from the rigid superlattice without moiré coupling between the FLGs and without the ${\gamma }_2$ hopping in the FLGs (a, e), the moiré coupling [(b), (g)], the ${\gamma }_2$ hopping terms [(c), (h)], the corrugation effect [(d), (i)], and the in-plane relaxation effect [(e), (j)] are included in the Hamiltonian successively. The band energies are directly calculated by diagonalizing the Hamiltonian without energy shifting, so the Fermi levels do not necessarily lie at the zero energy here. The energy (${\varepsilon }_{1B}$) of $|1B,K_{+}\rangle$ is indicated in (b) and (g).

Figure 4

Variations of $W_v$ (a), ${\mathrm{\Delta }}_h$ (b), ${\mathrm{\Delta }}_0$ (c), and the Chern number ($C_v$) of the highest valence band (d) with $\theta$ for the (BA)-(ABCA) and (CA)-(AB)(AB) stackings considering full relaxation (solid symbols) or only corrugation (open symbols).

Figure 5

(a) The electronic and topological properties of TMLG with $M+N\le 10$. The red (blue) squares represent the existence of configurations with isolated (separable but not isolated) flat bands hosting $\left|C\right|=N+M-1$ for given $M$ and $N$, and the empty squares denote the absence of such flat bands. (b) The variations of global gap ${\mathrm{\Delta }}_0$ at $E_F$ and the largest local gap between the highest valence band and the lower bands (local ${\mathrm{\Delta }}_h$) of configurations with $\left|C\right|=N+M-1$ and the largest local ${\mathrm{\Delta }}_h$ with $M$ at $N=2$ and $\theta =1.{41}^{\circ }$. The stackings at $M=8$ and $M=13$ are labeled.

Figure 6

The $|C_v|, {\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_h$ of configurations with separable flat valence bands at given $M$ and $N$. $M+N=6$ in [(a)–(b)] and $M+N=10$ in [(c)–(d)].

Figure 7

The band structures and ${\mathrm{\Omega }}_z$ maps of the middle valence and conduction bands around $E_F$ for the (BA)-(ABCA) [(a)–(b)] and the (BA)-(ABCA) [(c)–(d)] stackings with different $\theta$ and $h_0$. In [(a), (c)], the middle bands are displayed in red and their Chern numbers are labeled, and the Fermi levels are shown by dashed lines. In [(b), (d)], the ${\mathrm{\Omega }}_z$ maps are illustrated in the supercell BZ and the zero level of ${\mathrm{\Omega }}_z$ is represented by gray contour lines.

Figure 8

(a) The orbital magnetic moments ($m$) contributed by the isolated flat valence band for configurations with different layer numbers ($M+N$). (b) $m$ versus $C_v$ for all configurations with isolated flat valence bands.

Figure 9

(a) The schematic reciprocal lattice of the moiré superlattice in TMLG. Small hexagons are periodic BZs of the superlattice, spanned by ${\mathbf{b}}_{\mathbf{1}}^{\left(\mathbf{s}\right)}$ and ${\mathbf{b}}_{\mathbf{2}}^{\left(\mathbf{s}\right)}$. Large hexagons are BZ of the fixed bFLG (solid), spanned by ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, and BZ of the twisted tFLG (dashed). The band-touching points at the zero energy of the fixed ($K_{\xi }$) and twisted ($K_{\xi }^{\prime }$) FLGs are located at corners of the supercell BZs, with $\xi =\pm 1$ the valley index. ${\mathbf{k}}_{\xi }$ is the center of one of the supercell BZs containing the BN corners of the fixed and twisted FLGs at their corners. (b) The hexagonal BZ of the superlattice. The k-point lines for plotting band structures are displayed in red. In the $\xi =+$ valley, the $K_{+}$ and $K_{+}^{\prime }$ points in the large BZs of FLGs are folded to $\overline{K}$ and ${\overline{K}}^{\prime }$, respectively.

Figure 10

(a) The top view of a carbon hexagon in graphene. A and B denote the two sublattices in graphene and C indicates the hollow site. [(b), (c)] The schematic side views of the infinite sequence of graphene layers with the chiral stacking ($\cdots \mathrm{ABCABC}\cdots$) (b) or ($\cdots \mathrm{CBACBA}\cdots$) (c).

Figure 11

[(a), (b)] The $|\langle 1A,K_{+}+{\mathbf{k}}^{\prime }|\widehat{H}|\tilde{1}A,K_{+}+{\mathbf{k}}^{\prime }\rangle |$ (a) and $|\langle 1A,K_{+}+{\mathbf{k}}^{\prime }|\widehat{H}|1B,K_{+}+{\mathbf{k}}^{\prime }+{\mathbf{b}}_{\mathbf{1}}^{\left(\mathbf{s}\right)}\rangle |$ (b) as a function of $k_x^{\prime }$ and $k_y^{\prime }$ for the (CA)-(AB)(AB) configuration at $\theta =1.{35}^{\circ }$. [(c), (d)] The effect of in-plane relaxation on the the band structure of TMLG with the (BA)-(ABCA) stacking (c) or the (CA)-(AB)(AB) stacking (d). The solid lines show bands with both the intralayer and interlayer hopping terms modified by the in-plane relaxation, while the dashed lines represent bands with only the interlayer hopping modified by the in-plane relaxation and thus the fixed intralayer nearest-neighbor hopping.

Figure 12

The 2D energy dispersion of the isolated flat valence band for two configurations corresponding to the band structures shown in Figs. 1 and 1.

Figure 13

The density of states (DOS) corresponding to the band structures shown in Figs. 1–1.

Figure 14

The schematic side view and the band structures of the rigid superlattice without the moiré coupling between the FLGs and without the ${\gamma }_2$ hopping within the FLGs in a larger energy range than those shown in Figs. 3 and 3 for the (BA)-(ABCA) (a) and (CA)-(AB)(AB) (b) stackings at $\theta =1.{35}^{\circ }$. The low-energy three degenerate conduction states ($|{\psi }_{c,n}\rangle$) and three degenerate valence states ($|{\psi }_{v,n}\rangle$) of tFLG at ${\overline{K}}^{\prime }$ are labeled by red circles.

Figure 15

The band structures for the (BA)-(ABCA) (a) and (CA)-(AB)(AB) (b) stackings at $\theta =1.{35}^{\circ }$ under varying potential differences ($\mathrm{\Delta }$) between adjacent layers. The middle valence and conduction bands are represented by red lines.

Figure 16

The ${\mathrm{\Omega }}_z$ maps of the middle valence and conduction bands for the (BA)-(ABCA)(CB) (a) and (CBA)(BA)-(AB)(ABC) (b) stackings at varying $\theta$.