Emergence of intrinsically isolated flat bands and their topology in fully relaxed twisted multi-layer graphene

The emergence of low-energy flat bands Bistritzer and MacDonald (2011); Lopes dos Santos et al. (2012); Fang and Kaxiras (2016); Tarnopolsky et al. (2019) and the observation of associated superconductivity and correlated-insulator phasesCao et al. (2018a, b); Lu et al. (2019); Xie et al. (2019); Kerelsky et al. (2019); Jiang et al. (2019); Uri et al. (2020); Nuckolls et al. (2020) in magic-angle twisted bilayer graphene (TBG) have inspired great interest in exploring the peculiar electronic structure of graphene moiré systemsAndrei and MacDonald (2020); Andrei et al. (2021). Finite Berry phases around the Dirac cones of monolayer graphene are maintained in TBGSong et al. (2019); Po et al. (2019); Ahn et al. (2019); Liu et al. (2019a), while the breaking of $C_{2}T$ symmetry is required to obtain separable flat bands with nontrivial Chern numbers ($C$)Bultinck et al. (2020); Zhang et al. (2019a); Liu and Dai (2020), which can be achieved by carefully aligning TBG with hexagonal boron nitride to commensurate twist angles between themLin and Ni (2020); Cea et al. (2020); Lin et al. (2021); Shi et al. (2021); Mao and Senthil (2021). In such heterostructures, quantized anomalous Hall conductivity (QAHC) $\sigma_{xy}=Ce^{2}/h$ is observed at an odd filling of a flat band together with spontaneous orbital ferromagnetismSerlin et al. (2020); Sharpe et al. (2019). QAHC with $|C|=2$ has been realized in chirally stacked trilayer graphene aligned with BNChen et al. (2020). In contrast to TBG, intrinsic flat Chern bands in a valley arise in BA-AB stacked twisted double bilayer graphene (TDBG) and their $|C|$ can reach 3Lee et al. (2019); Zhang et al. (2019b); Liu et al. (2019b); Koshino (2019); Chebrolu et al. (2019); Lin et al. (2020). This suggests that higher-order flat Chern bands may occur in thicker twisted multi-layer graphene (TMLG) composed of two few-layer graphene (FLGs) with relative rotation. Moreover, complete isolation of a flat Chern band by global gaps from other bands is also essential to obtain QAHC that is contributed purely by edge statesShi et al. (2021). Therefore, it is important to explore TMLG systematically to identify isolated flat bands with high-order topology in graphene moiré systems.

The electronic structure of TMLG depends on the stacking orders of the FLGs and the twist angle ($\theta$) Vela et al. (2018); Liu et al. (2019b); Cea et al. (2019); Tritsaris et al. (2020); Zhang et al. ; Cao et al. ; Ma et al. . The flat bands around the Fermi level ($E_{F}$) of TMLG were found to be entangled with each other or with other dispersive bands by band crossingsLiu et al. (2019b); Tritsaris et al. (2020); Zhang et al. ; Cao et al. ; Ma et al. . Most studies introduced external electric field to separate the flat bands so that their band topology becomes well defined, while the produced $C$ is limited to small valuesZhang et al. ; Ma et al. . For TMLG composed of chirally stacked FLGs, the flat valence and conduction bands in a valley can be separated from other bands and their total $C$ were demonstrated to increase with the layer numbers of TMLGLiu et al. (2019b). This suggests that high $C$ may occur in one flat band if it can become isolated. We note that only rigid moiré superlattices of TMLG were considered in these previous studies Liu et al. (2019b); Tritsaris et al. (2020); Zhang et al. ; Cao et al. ; Ma et al. , while full relaxation has been shown to be able to enhance the band separation in TBG and TDBG Nam and Koshino (2017); Yoo et al. (2019); Lucignano et al. (2019); Guinea and Walet (2019); Choi and Choi (2019); Lin et al. (2020). In addition, the interlayer coupling was limited to that between adjacent layers Liu et al. (2019b); Tritsaris et al. (2020); Zhang et al. ; Cao et al. ; Ma et al. . The coupling between next-nearest layers may also play an important role to isolate the flat bands.

Here, we have identified various stacking orders of fully relaxed TMLG with isolated flat bands. Certain isolated flat valence bands in configurations with $M+N$ layers can host $|C|$ as large as $M+N-1$ and also large orbital magnetic moments for $M+N$ up to 10. The mechanism behind the emergence of isolated flat bands and their significant band topology have been revealed.

We study TMLG with the top FLG (tFLG) rotated by $\theta$ counterclockwise and the bottom FLG (bFLG) fixed, as seen in Fig. 1(a). The layer numbers of bFLG and tFLG are denoted by $M$ and $N$, respectively. We consider the strictly periodic moiré superlattices of TMLG with $M+N$ up to 10 and $\theta$ from 1.890^∘ to 1.018^∘. Starting from the twist interface, the layers in bFLG and tFLG are indexed by $n=1\sim M$ and $\tilde{n}=\tilde{1}\sim\tilde{N}$, respectively. Besides the tunable $\theta$, there exist $2^{M+N-3}$ inequivalent stacking orders for TMLG with $M+N$ layers, which are represented by the stackings of tFLG and bFLG. For example, a configuration composed of tFLG with the BA stacking and bFLG with the ABCA stacking is denoted by BA-ABCA. The geometry of moiré superlattices in TMLG is detailed in the Supplemental Material (SM).

Within a FLG, a pair of sites from two adjacent layers with one site directly above the other are referred to as dimer sites, and there is a relatively strong interlayer hopping ($\gamma_{1}$) between them. In a chirally stacked FLG, the sites in inner layers are all dimer sites, and the non-dimer sites which constitute the low-energy band states are on the surface layersKoshino and McCann (2009); Zhang et al. (2010). For a general stacking of FLG, the layers can be decomposed into a sequence of chiral subsets Min and MacDonald (2008a, b); Zhang et al. ; Cao et al. , which are represented as chiral stacking orders in parentheses, such as (AB)(ABC). Such chiral decomposition of tFLG and bFLG in TMLG determines some characters of the electronic structure.

In TMLG, the local stacking between layer $\tilde{1}$ of tFLG and layer $1$ of bFLG varies continuously across the moiré superlattice. Then the optimal local spacing between these layers determined by the local stacking also varies with the in-plane position in the superlattice, leading to the corrugation of the layers. More importantly, the spatially varying potential at the twist interface can drive the in-plane structural relaxation. Full relaxation has been performed for each configuration of TMLG employing the continuum elastic theory, as detailed in the SM.

For each relaxed configuration of TMLG, we have built a tight-binding Hamiltonian ($\hat{H}$) taking into account the effects of out-of-plane corrugation and in-plane structural deformation. The Hamiltonian parameters and computational approaches of the electronic structure are given in the SM. It is noted that the vertical next-nearest-layer hopping ($\gamma_{2}$) within the top and bottom FLGs is included in $\hat{H}$. To diagonalize $\hat{H}$, the plane-wave-like basis functions are adopted and denoted by $|n\alpha,\mathbf{k}\rangle$, where $n$ is the layer index, $\alpha=A,B$, and $\mathbf{k}$ represents the momentum of the state in the reciprocal space of the pristine FLGs.

By calculating the two-dimensional ($2D$) energy bands in the entire Brillouin zone (BZ) for relaxed TMLG with different $\theta$ and stackings, we find that completely isolated flat bands emerge around $E_{F}$ for various configurations of TMLG, even for those with 10 layers. We consider flat bands with widths smaller than 10 meV. For an isolated band, global gaps are opened above and below this band. The bands around $E_{F}$ show rather strong electron-hole asymmetry and most isolated flat bands are on the hole side. The stacking orders that host isolated flat valence bands in a relatively large range of $\theta$ are listed in Table I and those with flat valence bands that are only isolated at some $\theta$ can be seen in Table SI of the SM. Figure 1(b-e) show the band structures of four typical configurations with different types of stacking orders and their DOS can be seen in Fig. S3. Since only very few configurations have isolated flat conduction bands (see Table SII), we focus on isolated flat valence bands in the following.

Through examination of variations of the electronic properties with different configurations of TMLG, we category the stacking orders into four types, i.e. cases with chirally stacked FLGs (type I), those with Bernal-stacked FLGs that have even numbers of layers (type II), other cases without a single layer in the stacking decomposition of FLGs (type III) and with a single layer in decompositions (type IV), as listed in Tables I and SI. The isolated flat valence band is separated from other bands by global gaps at the charge neutrality point ($\Delta_{0}$) and just below it ($\Delta_{h}$). $\Delta_{0}$ is smaller than 4 meV for all cases, while $\Delta_{h}$ can have a rather large value, especially for type-I cases, as shown in Fig. 2(a). Most type-IV configurations have extremely narrow $\Delta_{h}$ as some characters of the low-energy linear dispersions contributed by the decomposed single layer in the pristine FLGs are maintained. The $\Delta_{h}$ of most type-II cases are quite large, while those of many type-III cases are rather small. The widths ($W_{v}$) of isolated valence bands begin to become smaller than 10 meV at $\theta\approx 1.5^{\circ}$, as shown in Fig. 2(b). For type-I cases, $W_{v}$ can be narrower than 5 meV only when $\theta$ is around or below $1.1^{\circ}$, while the $W_{v}$ of type-III and type-IV cases can reach such narrow $W_{v}$ at about $1.3^{\circ}$. The electronic behavior of different stacking types also depends on the thickness of TMLG. Only when $N+M\leq 7$, isolated flat bands can emerge for type-I stacking, as seen in Table II. For $N+M\geq 9$, only type-III and type-IV cases can have isolated flat bands. No isolated flat bands exist for systems with $N=1$. In addition, similar to pristine FLGsKoshino and McCann (2009); Zhang et al. (2010), the 2D energy dispersions of the isolated flat bands exhibit trigonal warping, as seen in Fig. S2.

To reveal the mechanism behind the emergence of isolated flat bands in TMLG, we have compared the band structures and state composition of some configurations with different parts of the Hamiltonian, as shown in Fig. 3 for the type-I (BA)-(ABCA) stacking and the type-II (CA)-(AB)(AB) stacking at $\theta=1.35^{\circ}$. Starting from the rigid superlattice without moiré coupling between FLGs and without the $\gamma_{2}$ hopping within FLGs, the moiré coupling, the $\gamma_{2}$ hopping, the corrugation effect, and the in-plane relaxation effect are included in the Hamiltonian successively.

Without moiré coupling, the low-energy bands of bFLG (tFLG) touch at the corner $\bar{K}$ ($\bar{K}^{\prime}$) [see Fig. S1(b)] of the supercell BZ, and the bands of bFLG cross with those of tFLG, as seen in Figs. 4(a), 4(f) and S4. The band crossings can be eliminated by the moiré hopping between layers 1 and $\tilde{1}$ in a way similar to TBG, as these degenerate states at crossings have contributions from these interface layers. In contrast to TBG always with touched valence and conduction bands at the BZ corners, state splitting occurs around $E_{F}$ upon inclusion of the moiré coupling for TMLG. In the pristine bFLG, the states composed of non-dimer sites have zero energy at $\bar{K}$. These states are just the non-dimer basis functions with the momentum of $K_{+}$, which is at the corner of the bFLG BZ [see Fig. S1(a)]. When bFLG is coupled to tFLG, the energy of the non-dimer state ($|1B,K_{+}\rangle$) in the first layer of bFLG rises above zero and the other non-dimer states still have zero energy. We will show that the positive energy of $|1B,K_{+}\rangle$ is due to the electron-hole asymmetry in low-energy bands of tFLG.

For the chiral tFLG whose zero-energy states are at $\bar{K}^{\prime}$, three degenerate conduction states ($|\psi_{c,n}\rangle$ with $n=1-3$) and three degenerate valence states ($|\psi_{v,n}\rangle$) have low energies ($\varepsilon_{c,n}$ and $\varepsilon_{v,n}$) at $\bar{K}$, as seen in Fig. S4. By second-order perturbation approximation, the energy ($\varepsilon_{1B}$) of $|1B,K_{+}\rangle$ can be expressed as

Considering only the interlayer hopping $\gamma_{1}$ between dimer sites, the bands of tFLG are electron-hole symmetric. Then all $|\langle 1B,K_{+}|\hat{H}|\psi_{c,n}\rangle|$ and $|\langle 1B,K_{+}|\hat{H}|\psi_{v,n}\rangle|$ have the same value. One of $|\psi_{c,n}\rangle$ is expanded in the basis functions with momentum at $K_{+}$ including $|\tilde{1}\alpha,K_{+}\rangle$ ($\alpha=A,B$) in layer $\tilde{1}$ of tFLG. For the rigid superlattice, the moiré coupling between layers $1$ and $\tilde{1}$ produces the same value ($u$) of the Hamiltonian elements between $|1B,K_{+}\rangle$ and $|\tilde{1}\alpha,K_{+}\rangle$. For the BA stacked tFLG, $|\langle 1B,K_{+}|\hat{H}|\psi_{c,n}\rangle|$ becomes $u/\sqrt{2}$ as layer $\tilde{1}$ contributes half of the squared norm of $|\psi_{c,n}\rangle$. For other tFLG, $|\langle 1B,K_{+}|\hat{H}|\psi_{c,n}\rangle|$ decreases slowly with its number of layers. With the electron-hole symmetry, $\varepsilon_{v,n}=-\varepsilon_{c,n}$, then $\varepsilon_{1B}$ is still zero.

The interlayer hopping $\gamma_{4}$ between a non-dimer site and a dimer site introduces electron-hole asymmetry in the bands of FLGs. The supercell momentums of the basis functions expanding $|\psi_{v,n}\rangle$ and $|\psi_{c,n}\rangle$ all have the same length $k_{\theta}\approx 4\pi\theta/3a$. With only $\gamma_{1}$ and $\gamma_{4}$ as well as the intralayer nearest-neighbor hopping (-$t_{0}$) for chiral tFLG with $N$ layers, $\varepsilon_{v,n}$ and $\varepsilon_{c,n}$ can be expressed analytically asKoshino and McCann (2009); Zhang et al. (2010)

which lead to $-\varepsilon_{v,n}<\varepsilon_{c,n}$. Numerical calculations show that $\varepsilon_{v,n}$ is also closer to zero than $\varepsilon_{c,n}$ for non-chiral FLGs. In addition, the norms of all the Hamiltonian elements in Eq. (1) are still approximately equal when including $\gamma_{4}$. Therefore, $\varepsilon_{1B}$ has a positive value. For the BA stacked tFLG, $\varepsilon_{1B}$ is given analytically by

As $\gamma_{4}$ is much smaller than $t_{0}$, we have $\varepsilon_{1B}\approx 9\gamma_{1}\gamma_{4}u^{2}/(2\pi^{2}t_{0}^{3}\theta^{2})$.

At $\bar{K}^{\prime}$, similar analysis shows that the positive energy of the non-dimer state $|\tilde{1}B,K_{+}^{\prime}\rangle$ in layer $\tilde{1}$ of tFLG can also be attributed to the electron-hole asymmetry in the bands of bFLG.

When only moiré coupling is included in the Hamiltonian of the (CA)-(AB)(AB) configuration with non-chiral bFLG, three states contributed by the non-dimer sites in the bottom three layers of bFLG still have zero energy at $\bar{K}^{\prime}$, as seen in Fig. 3(g). The turning on of the $\gamma_{2}$ hopping between the 2A and 4A sites can split these degenerate states, as shown in Fig. 3(h). Then only the $|3B,K_{+}^{\prime}\rangle$ state is located at almost zero energy.

For the rigid superlattice with all interlayer hopping, the flat bands around $E_{F}$ remain overlapped with other dispersive bands. The corrugation effect reduces such band overlapping by making the flat valence band narrower, as shown in Figs. 3(d) and 3(i). The in-plane relaxation has a more significant impact on the band dispersions. The flat valence band can become completely gapped from other bands with full relaxation, while $\Delta_{0}$ may be decreased by the in-plane relaxation, especially for configurations with a non-chiral FLG, as shown in Figs. 3(e) and 3(j).

To further demonstrate the importance of full relaxation for the emergence of isolated flat bands, the variations of $W_{v}$, $\Delta_{h}$ and $\Delta_{0}$ with $\theta$ considering both out-of-plane and in-plane relaxation are compared with those considering only corrugation in Fig. 4. For (BA)-(ABCA) with only corrugation effect, $W_{v}$ is overestimated at $\theta$ around 1.1^∘, $\Delta_{h}$ is underestimated at all $\theta$, and $\Delta_{0}$ is also underestimated at small $\theta$. For (CA)-(AB)(AB), only with full relaxation can $\Delta_{h}$ and $\Delta_{0}$ become positive at small $\theta$. The trends of these electronic properties with $\theta$ are also rather different for the two stackings. In particular, the maximum value of $\Delta_{h}$ is located at about 1.5^∘ for (BA)-(ABCA), while it is at about 1.2 ^∘ for (CA)-(AB)(AB). In addition, the first local minimum of $W_{v}$ is also at a larger $\theta$ for (BA)-(ABCA).

We have focused on TMLG without potential differences between layers. It is noted that equal potential differences ($\Delta$) between adjacent layers produced by applied out-of-plane electric field may separate crossed bands locally for some configurations, while only a rather small $\Delta$ can enhance the gaps around the isolated flat bands slightly and a larger $\Delta$ tends to close these global gaps, as shown for example in Fig. S5 for the (BA)-(ABCA) and (BA)-(ABCA) configurations.

At an odd filling of an isolated flat band, spontaneous valley polarization may occur due to the electron-electron interactionZhang et al. (2019b). If such a valley polarized band hosts a non-zero Chern number, the TMLG can support QAHC. The Chern numbers $C$ of flat bands in the $\xi=+$ valley have been obtained explicitly by integral of the Berry curvature ($\Omega_{z}$) in the supercell BZ as detailed in the SM, and the $C$ of a band in the $\xi=-$ valley is just the opposite of that for $\xi=+$. Chern numbers are also calculated for separable flat bands which are separated from nearby bands by local gaps larger than 0.5 meV to characterize the dependence of $C$ on stacking orders of TMLG.

We also focus on the Chern numbers ($C_{v}$) of flat valence bands in the $\xi=+$ valley. Among all configurations with $M+N$ layers, the largest $|C_{v}|$ is $M+N-1$. The largest $|C_{v}|$ can occur in certain configurations with isolated flat bands, such as those shown in Figs. 1(b) and 1(d) and listed in Table SI. We note that the isolated flat valence band in a configuration with 10 layers [see Fig. 1(d)] has $C_{v}=-9$, which is the largest magnitude of $C_{v}$ for all considered TMLG with $N+M\leq 10$.

Systematic calculations of all configurations with different stackings and $\theta$ show that most flat valence bands with high $C_{v}$ slightly overlap with other bands and such band overlapping is related to the layers numbers ($M$ and $N$) of FLGs in TMLG, as illustrated in Fig. 5. The appearance of highest $C_{v}$ generally becomes less likely with increasing N+M. For $N+M=6$, more cases with $|C_{v}|=5$ appear for $M=4$ than for $M=3$ as type-II stacking is only possible for $M=4$. In contrast, there are much fewer cases with the largest $|C_{v}|$ for $M=6$ than for $M=5$ when $M+N=10$, which is related to the occurrence of type-III configurations only for $M=5$. We notice that it is mainly the overlapping of the valence and conduction bands around EF, i.e. negative $\Delta_{0}$ that keeps most flat valence bands from being isolated for large $N+M$.

By examining the evolution of $\Omega_{z}$ maps for the middle bands with the strength of the moiré coupling, we find that the large $|C_{v}|$ originates from the splitting of degenerate states by the weak moiré coupling or from the topological transitions induced by the strong coupling. With the decreasing $\theta$ the coupling between FLGs can be tuned from the weak to the strong regime. At a fixed $\theta$, the coupling strength can be further reduced by rigidly increasing the average spacing ($h_{0}$) between FLGs.

Figure 6 displays the $\Omega_{z}$ maps of the valence and conduction bands around $E_{F}$ for the (BA)-(ABCA) and (BA)-(ABCA) stacked configurations with different $\theta$ and $h_{0}$. For (BA)-(ABCA) with $M=4$ and $N=2$, $\Omega_{z}$ peaks appear around $\bar{K}$ and $\bar{K}^{\prime}$ in the weak coupling regime. The highest valence band states around $\bar{K}$ ($\bar{K}^{\prime}$) are mainly contributed by the chiral bFLG (tFLG), whose $\Omega_{z}$ is integrated to be about $-M\pi$ ($-N\pi$)Koshino and McCann (2009); Zhang et al. (2010). Such integral values of $\Omega_{z}$ correspond to the Berry phases of the band states in a circle around a corner of the hexagonal BZ for the pristine FLGs, and the sign of the Berry phases is fixed by the fact that the valence state at $\bar{K}$ ($\bar{K}^{\prime}$) is composed of non-dimer states in the bottom layer of bFLG (the top layer of tFLG) as discussed above. These Berry phases around $\bar{K}$ and $\bar{K}^{\prime}$ contribute $-(M+N)/2=-3$ to the total $C_{v}$ for (BA)-(ABCA), and the Chern number ($C_{c}$) of the lowest conduction band gains a contribution of $3$. Moreover, the degeneracy lifting at the band crossings between the middle bands and the other bands gives rise to negative $\Omega_{z}$ peaks along three curves beginning from the $\bar{\Gamma}$ point for both the valence and conduction bands, and both $C_{v}$ and $C_{c}$ gain a contribution of -2 from these $\Omega_{z}$ peaks. Then $C_{v}$ and $C_{c}$ amount to -5 and 1, respectively. Similar to the 2D band dispersions, the distributions of $\Omega_{z}$ around $\bar{K}$ and $\bar{K}^{\prime}$ exhibit trigonal warping inherited from the pristine chiral FLGs. With the decreasing $\theta$ or $h_{0}$, the feathers of the $\Omega_{z}$ maps undergo great changes with the peaks merging, while the local gaps between the middle bands remain opened and the Chern numbers are maintained in a large range of $\theta$, as seen in Fig. 4(d). Such evolution of $\Omega_{z}$ maps suggests that exploration of the electronic structure of TMLG in the weak regime of the moire coupling is essential to interpret the origin of the Chern numbers at small $\theta$. For some configurations with non-chiral FLGs, the largest $|C_{v}|$ can also be already present at large $\theta$, such as the (BA)-(ABCA)(CB) stacking with $C_{v}=-7$ [see Fig. S6(a)] for $\theta$ from 1.89^∘ to 1.47^∘ at which the flat valence band becomes isolated.

In the weak coupling regime for (CA)-(AB)(AB), the $\Omega_{z}$ peaks around $\bar{K}^{\prime}$ originating from tFLG contribute +1 to $C_{v}$, while there are both positive and negative peaks around $\bar{K}$ as bFLG is non-chiral, then $C_{v}$ has a relatively small value of 2. With decreasing of $\theta$, dipole-like pairs of positive and negative peaks form and band inversion between the middle bands occur at about 1.30^∘, leading to the largest $C_{v}$ of 5. The next topological phase transition occurs at about 1.16^∘. We note that full relaxation is required to drive the topological phase transition with $C_{v}$ rising to 5 in the strong coupling regime, as shown in Fig. 4(d). The $C_{v}=-9$ for the (CBA)(BA)-(AB)(ABC) stacking shown in Fig. 1(e) also occurs at small $\theta$, and the variation of $\Omega_{z}$ maps with $\theta$ can be seen in Fig. S6(b).

The non-trivial topology in isolated flat bands suggests that spontaneous orbital magnetic moments ($m$) may be observed at odd fillings of the flat bandsZhu et al. (2020); Repellin et al. (2020). We evaluate the magnitude of $m$ contributed by the isolated flat valence band by setting the chemical potential at the middle of the $\Delta_{0}$ gap. The magnitude of $m$ can reach 10$\mu_{B}$ per supercell for $\theta$ around 1.1^∘, as shown in Fig. 7(a). For configurations with positive $m$, its value is related to the layer numbers of TMLG with the $N+M=8$ cases generally having the largest $m$ at a $\theta$. $m$ is roughly proportional to $C_{v}$, while the spanning of its value is still rather large for a given $C_{v}$, especially for small $|C_{v}|$. We note that for certain cases with a small $|m|$ but a large $|C_{v}|$, the sign of the total orbital magnetic moment may be inverted by tuning the chemical potential across the gapZhu et al. (2020).

For TMLG with $M+N$ layers, full relaxation has been performed for the $2^{M+N-3}$ inequivalent stacking orders at varying $\theta$. Isolated flat bands emerge in relaxed TMLG with $M+N$ up to 10 and with various stackings, and most of them are on the hole side. The stacking orders that host isolated flat bands can be categorized into four types based on the electronic behavior of flat bands and the stacking decomposition. For type-I configurations with both chiral FLGs, the touched bands of FLGs around $E_{F}$ are split by the moiré coupling through the electron-hole asymmetry in low-energy bands of FLGs while the hopping $\gamma_{2}$ between the next-nearest layers is further required for the state splitting in other types with non-chiral FLGs. The corrugation effect reduces the band overlapping around $E_{F}$ and the full structural relaxation leads to global gaps that completely isolate a flat band. For TMLG with given $M$ and $N$, the highest Chern number $|C|$ of the separable flat bands reaches $M+N-1$ and can be hosted by certain isolated bands. The $|C|=9$ occurs in the isolated flat valence band of several configurations with 10 layers. Such high $|C|$ 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 $m$ arise in isolated flat bands with high $|C|$ and depend on the structural configurations of TMLG.