TWIST AS A BAND-ENGINEERING KNOB

Twisting two periodic lattices against each other at a small angle creates large-scale periodic interference-like patterns, the so-called moiré pattern, shown in Fig. 2a. The moiré structures exhibit a wavelength λ that is inversely proportional to the twist angle θ, typically given by

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} \lambda \approx \frac{a}{{2\sin \left( {\frac{\theta }{2}} \right)}}, \end{eqnarray*}\end{document}

where a ≈ 0.246 nm is the lattice constant of graphene.

Figure 2.: Emergence of flat band in MATBG [5]. (a) Moiré pattern formed in MATBG, where θ is the twist angle between two graphene layers. (b) Mini-Brillouin zone arising from the relative rotation of the two Dirac cones, defined by the mismatch between the K points of the top and bottom layers. (c) Illustration of the effect of interlayer hybridization. When the hybridization energy 2w is smaller than the kinetic energy scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbar {{v}0}{{k}\theta }$\end{document}, the two layers remain effectively decoupled. As 2w approaches \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbar {{v}0}{{k}\theta }$\end{document}, significant interlayer hybridization occurs, leading to band reconstruction and the emergence of the flat band. (d) Local density of states (LDOS) under magic angle condition. Electron density is strongly localized in the AA stacking region, while significantly suppressed in the AB and BA stacking areas. (e) Nano-ARPES measured energy band structure of MATBG, in which the flat band and multiple hybridization gaps are marked by red and black arrows respectively [17]. (f) Atomic topography of MATBG measured at ν = 2 in MATBG by STM [25]. The dashed circle surrounds the AA region, and the radial dotted lines indicate the bridge regions that separate AB and BA regions.

To the zeroth order, the low-energy band structure of twisted bilayer graphene can be considered as two sets of monolayer-graphene Dirac cones rotated about the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{\Gamma }}$\end{document} point in the Brillouin zone by the twist angle θ. The difference between the two K (or K′) wavevectors gives rise to the mini-Brillouin zone, as shown in Fig. 2b [13]. The resulting band structure behavior of twisted bilayer graphene is governed by the interplay between two energy scales: the strength of interlayer coupling w and the band separation energy. One can introduce a dimensionless parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha = \frac{w}{{\hbar {{v}0}| {{{k}\theta }} |}}$\end{document} to characterize the degree of band reconstruction, where w is the interlayer potential, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{v}0}$\end{document} is the Fermi velocity of graphene and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{k}\theta }$\end{document} is the wavevector of the moiré pattern. Two distinct regimes of α are associated with different behaviors. At large twist angle (or small α), the layers remain weakly coupled and preserve linear dispersion of each individual Dirac cone (Fig. 2c left). As the twist angle decreases, the Dirac cones near either the K or K’ valley mix through interlayer hybridization, whereas interactions between distant Dirac cones are suppressed exponentially (Fig. 2c right). As a result, Fermi velocity gets renormalized. The renormalized Fermi velocity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{v}_f}( \alpha )$\end{document} can be expressed as:

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} {{v}_f} = \left| {\frac{{\partial E( k)}}{{\partial k}}} \right| = v_f^0\frac{{1 - 3{{\alpha }^2}}}{{1 + 6{{\alpha }^2}}}, \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_f^0$\end{document} is the monolayer graphene Fermi velocity. As can be seen, for small values of α, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{v}_f}$\end{document} decreases slightly, indicating a minor modification to the linearly dispersive bands. However, as α keeps increasing, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{v}_f}$\end{document} decreases rapidly until it reaches zero at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha = \frac{1}{{\sqrt 3 }}$\end{document}. The special angle corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{v}_f} = 0$\end{document} is the so-called magic angle where flat band physics emerge. At the first magic angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\theta }_1} = \frac{{\sqrt 3 w}}{{\hbar {{v}_F}{{G}_k}}} \approx 1.1$\end{document}°, the Fermi velocity at the mini-Brillouin zone corners drops to zero, resulting in electronic states becoming highly localized in momentum space [13,14]. These electronic states, associated with the flat bands, also exhibit real-space localization at AA stacking regions within the moiré superlattice [13,15,16], as illustrated in Fig. 2d.

Direct experimental evidence for these flat bands has been provided by nano angle-resolved photoemission spectroscopy (ARPES) measurements and scanning tunneling microscopy (STM). Nano ARPES experiments reveal a strong concentration of spectral weight near the moiré Brillouin zone corners, as well as multiple hybridization gaps that signal the formation of moiré minibands [17]. The bandwidth of the flat bands is found to be on the order of tens of millielectronvolts, in good agreement with theoretical expectations (indicated by the red arrows in Fig. 2e) [17–19]. The STM topographies of MATBG, shown in Fig. 2f, reveal a moiré superlattice in which the bright (dark) regions correspond to the AA (AB/BA) stacking regions that are associated with high (low) local density of states (LDOS). By tuning the carrier concentration via the gate modulation, STM enables observation of the dynamic evolution of moiré Bloch bands with respect to charge filling.

At charge filling where the flat bands are fully occupied or empty, STM resolves the van Hove singularities associated with nearly flat conduction and valence bands as two sharp peaks whose energy separation is consistent with a noninteracting model [20–24]. Once the chemical potential is tuned into the flat bands, STM spectra exhibit pronounced energy broadening of the flat band features, reflecting the strong Coulomb interaction (Fig. 3a) [21–24].

_Figure 3.: Characterization of electronic correlation in the flat band of MATBG: (a) STM measured differential conductance dI/dV of the cascade of transitions in MATBG [26]. Reorganization of the low-energy excitations of MATBG happens near each integer fillings of the moiré flat bands. (b) Inverse compressibility of the cascade of transition in MATBG, measured using scanning single-electron transistor [27]. The characteristic sawtooth signal reflects chemical potential resets near integer fillings. (c) STM topographic image acquired at filling factor ν = −2 in MATBG reveals atomic-scale signatures of the IVC ground state [25]. A high-resolution region of the image is analyzed to extract the local FFT amplitude and phase, which are then decomposed into three IVC order parameters. (d) Momentum resolved spectroscopy of MATBG measured by QTM. Key momenta are marked on the top axis [45]. At \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{K}T}$\end{document}, the measurement discloses two extremely flat bands, separated by a large energy gap. The gap appears at almost all momentum except near the Γ point, where the bands are gapless. (e) Temperature-dependent thermoelectric response for a MATBG device at 10 K (top), 15 K (middle) and 20 K (bottom) [46]. As lattice temperature decreases, the response evolves from conventional sign-preserving towards sign-preserving thermoelectricity at integer fillings, indicating electron-hole asymmetry in electron-doped correlated states.