MOIRÉ FLAT BANDS BEYOND MATBG

The emergence of correlated states in MATBG has spurred extensive exploration of flat bands graphene-based twisted systems. In particular, the twisted multilayer graphene (TMG) superlattices, formed by integrating additional twisted layer, or altering the “parent” Dirac cones, can serve as a new playground of electronic correlation, superconductivity and band topology. A fundamental distinction from MATBG is that TMG hosts more tunable and markedly different moiré flat bands, arising from the modified symmetry constraints and more complex interlayer couplings associated with additional layers. Here, we summarize several widely studied TMGs systems, briefly outlining their low-energy band structures and highlighting key features of their electronic correlation, superconductivity, and topology.

Twisted M + N multilayer graphene

Inherited from the electrically tunable band structure of the parent crystalline graphene, twisted double layer system of crystalline M-layer and N-layer graphene can exhibit flat bands with a tuning knob of displacement field D, as induced by the broken inversion symmetry [101,102]. The widely investigated twisted M + N graphene includes twisted double bilayer graphene (TDBG, 2 + 2), twisted monolayer-bilayer graphene (TMBG, 1 + 2), etc. [103–106]. The displacement field can effectively change the bandwidth and adjust the energy of van Hove singularity where the DOS is divergent, producing a rich phase diagram in n-D space (here n is the carrier density). In a specific range of displacement field D, the low-energy conduction band is flat and well isolated from other bands by gap openings at charge neutrality and full fillings (Fig. 6a). Electronic correlated insulators appear consequently (Fig. 6b). The half-filling correlated insulator in TDBG or TMBG can be different from MATBG, showing a spin-polarized behavior under in-plane magnetic field [103–105]. With an optimal doping away from the half filling, the resistance of TDBG exhibits an abrupt drop as the temperature is lowered, which is believed to be associated with spontaneous symmetry breaking rather than superconductivity [107]. TDBG in proximity to a sheet of tungsten diselenide (WSe2) exhibits superconductivity near the van Hove singularities of conduction and valence bands (Fig. 6b) [108].

Figure 6.: Band structure and electronic properties of twisted multilayer graphene: (a) Schematic of TDBG and calculated band structure at an optimal displacement field and twist angle 1.33° [103]. The C1, C2 (V1, V2) denote the first and second conduction (valence) band, respectively. The first conduction band C1 is flat and isolated from other bands. (b) Resistivity of 1.37° TDBG proximitized to WSe2 as a function of top and bottom gates [108]. The half filling (ν = 2) of C1 band shows resistive state at optimal displacement field where both charge neutrality point (CNP, ν = 0) and full filling of C1 band (ν = 4) are gapped. Superconductivity appears in small pockets of the phase diagram near the van Hove singularities. (c) Schematic phase diagram of topological Wigner crystal in twisted bilayer-trilayer graphene [75]. The vertical axis denotes perpendicular magnetic field. The Chern numbers at fractional filling factors ν = 1/4 and ν = 1/3 are C = ±1 and tuned by perpendicular magnetic field. (d) Schematic of ATTG and calculated band structure at zero displacement field [111]. (e) Schematic phase diagram of ATTG [110]. The dark blue shadings denote superconducting phase which is bounded by the van Hove singularity denoted by the blue lines.

A large family of twisted M + N multilayer graphene superlattices has been found to exhibit exotic anomalous Hall (AH) or QAH effects that are markedly different from MATBG [102]. In twisted bilayer-trilayer graphene superlattice, TEC states at zero magnetic field are demonstrated in a general formation where discrete rather than continuous translation symmetry is broken (Fig. 6c) [75]. In addition, since twisted M + N multilayer graphene has the intrinsic C2z sublattice symmetry breaking, the AH or QAH effect appears without additional alignment of graphene layers to the hBN substrate [60,73,106]. Another completely different feature from MATBG is the higher Chern numbers in twisted M + N graphene, which is of high importance to establish FQAH states that do not resemble the FQH state—a typical counterpart of FQAH in high magnetic field. TMBG exhibited Chern number C = ±2 for spin and valley projected flat bands. At moiré fillings of ν = 1 and ν = 3, QAHE with Hall resistance approximately equal to h/2_e_2 were observed. Fractionalizing the spontaneous valley-polarized flat bands at fractional fillings of ν = 3/2 results in the topological charge density wave state with Chern number C = 1 in TMBG—an electronic state with both the chiral edge modes and unit cell doubling [73].

Alternating twisted multilayer graphene

One family of graphene moiré system that mostly resembles MATBG is the alternating twisted multilayer graphene (ATMG), where monolayer graphene layers (layer number N > 3) are successively stacked with alternating twist angle θ and - θ. As the even or odd-numbered layers are strictly aligned, ATMG hosts C2z symmetry. Unlike most of other moiré systems, ATMG has [N/2] bands at the lowest energy which can be flattened at the corresponding magic angle

(9) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \theta _{{\mathrm{magic}}}^{{\mathrm{ATMG}}} = 2\theta _{{\mathrm{magic}}}^{{\mathrm{TBG}}}\cos \left( {\frac{{\pi k}}{{N + 1}}} \right), \end{eqnarray*}\end{document}

where k = 1, 2, 3…[N/2], [N/2] is the largest integer less than N/2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta _{{\mathrm{magic}}}^{{\mathrm{TBG}}} \approx 1.1$\end{document}° is the magic angle of TBG [109]. When N is odd, there is a dispersive Dirac cone concomitant with the flat bands. For instance, in alternating twisted trilayer graphene, the low-energy band structure is decomposed into a MATBG-like nondispersive band with the scaled interlayer coupling by a factor of \sqrt 2, and dispersive Dirac cones with Dirac point located at K points in the moiré Brillouin zone [110–112] as shown in Fig. 6d. The TBG-like band is extremely flat at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta \approx 1.5$\end{document}° that can host pronounced electronic correlation effects like in MATBG [112,113]. ATTG has a mirror symmetry when the displacement field D is zero. With an applied external displacement field to break the mirror symmetry, the electronic potential difference shifts Dirac cones up in energy and hybridizes them into the flat band sector [112].

Of the critical importance, ATMG can have larger magic twist angles which can mitigate one of the most severe experimental challenges—twist angle control. It has been revealed to be more structurally stable than MATBG, enabling more robust superconductivity and higher _T_C [110,111,114,115]. Similar to MATBG, alternating twisted trilayer graphene also exhibits multiple features of unconventional superconductivity, as discussed in Section 5. In its n-D phase diagram, the superconductivity is connected to the symmetry-broken phase and bounded by the van Hove singularity, which cannot reconcile with the weak-coupling BCS theory (Fig. 6e) [110].

Moiré quasicrystal and supermoiré superlattice

Beyond the single periodical moiré pattern, the moiré design now has expanded to the multi-moiré patterns by constructing multiple twist angles that play with the stacking chirality and angular difference. The representative case is twisted trilayer graphene with two twist angles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TM}}}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{MB}}}}$\end{document} (here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TM}}}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{MB}}}}$\end{document} are twist angle between the top and middle layers, middle and bottom layers, respectively). When the ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TM}}}}/{{\theta }{{\mathrm{MB}}}}$\end{document} is away from plus or minus one, two mutually incommensurate moiré patterns will lead to a quasiperiodic structure, named moiré quasicrystal. Moiré quasiperiodicity is defined on the moiré length and doesn’t exhibit rotation symmetries like in the usual quasicrystal. Recent experiments and theoretical predictions point to the presence of flat bands that can induce electronic correlation and superconductivity in twisted trilayer graphene moiré quasicrystals [116]. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TM}}}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{MB}}}}$\end{document} have the same sign, namely graphene is helically stacked, or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TM}}}} \approx - {{\theta }{{\mathrm{MB}}}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }{{\mathrm{TB}}}} \ll - {{\theta }{{\mathrm{TM}}}}$\end{document}, numerical calculations and experiments indicate the presence of long-wavelength supermoiré structure arising from the interference of the two moiré patterns. The supermoiré effect can impact strongly on the electronic correlation and band topology, leading to symmetry-broken ground states and anomalous Hall effect [117,118].