Fig. 1. (Color online) (a) Moiré pattern obtained by the overlapping of two similar fringes with different displacement and twist angles^[27]. (b) Crystallographic superlattice and moiré superlattice in (5,7)-tBLM withθ = 10.99°. The green solid parallelogram represents the crystallographic superlattice unit cell and the red dashed parallelogram represents the moiré superlattice unit cell. (c) The reciprocal lattice of (5,7)-tBLM. The green and red regular hexagons correspond to the Wigner-Seitz primitive cells of the crystallographic superlattice and moiré superlattice, and the orange and blue hexagons represent the first Brillouin zone of the bottom and top MoS₂, respectively. Reproduced with permission from Ref. [19]. Copyright 2018, ACS Publications.

Fig. 2. (Color online) Raman spectra of tBLMs and monolayer MoS₂. The region of (a) is 50−365 cm⁻¹ and (b) is 370–425 cm⁻¹. Raman modes in various phonon branches are labeled by different colors and symbols. (c, d) The experimental and calculated frequencies of moiré phonons vary with |g| and θ. The solid lines and the scatter symbols are theoretical and experimental results, respectively. (e, f) The peak position of FA'1 mode vary with |g| and θ. The peak position of A'1-related branch of the monolayer MoS₂ (pink square) and various stacked BLMs (crossed circles) are also shown in (e). In (f), the theoretical phonon dispersion of A'1- related branch along the Γ–M and Γ–K directions in the monolayer MoS₂ are shown as gray lines and the phonon dispersion of A'1- related branch of the monolayer MoS₂ along g is shown as a dashed line. The stars are the experimental peak position of FA'1 in tBLMs. Reproduced with permission from Ref. [19]. Copyright 2018, ACS Publications.

Fig. 3. (Color online) (a, b) Calculated the low-energy evolution of phonon modes developed twist angle θ at Γ. Color bars based on optical activity project the phonon eigenmodes onto the central Γ point. Optically inactive modes are shown by grey lines in (a) entirely originating at neighboring Γ points. Reproduced with permission from Ref. [21]. Copyright 2021, Springer Nature.

Fig. 4. (Color online) (a) The moiré pattern as seen in the tBLG. The first Brillouin zones of layers 1 and 2 are shown as two large hexagons, and small hexagons are moiré Brillouin zones of the tBLG. (b) Schematic illustrating the coupling of interlayer from the initial k in layer 1 (given by as K+(1)) to the three k' points in layer 2 in an undistorted tBLG, where k is a two-dimensional Bloch wave vector. Reproduced with permission from Ref. [30]. Copyright 2020, American Physical Society. (c) Phonon dispersion in the inner asymmetric mode ( u−) under different twist angles θ. (d) The gap width between the third and second branches of the phonon of u− mode as a function of θ. The blue dashed line indicates the linear dependence on θ. (e) Group velocities of the second and first u−phonon modes dependent on θ. The velocities of transverse ( vt) and longitudinal ( vl) phonons in monolayer graphene are represented by horizontal dashed lines. Reproduced with permission from Ref. [20]. Copyright 2018, American Physical Society.

Fig. 5. (Color online) Illustration of the electron–phonon interaction in moiré superlattices. The springs representing phonons and a periodic electronic potential is on the top of the picture.

Fig. 6. (Color online) EPC strength g¯n as a function of the phonon band indexn in various twist angles. Reproduced with permission from Ref. [30]. Copyright 2020, American Physical Society.

Fig. 7. (Color online) (a) Electron–phonon scattering in the extended Brillouin zone. The first Brillouin zone is shown as dashed hexagon marks; the blue circles are the Fermi surface. Yellow and orange arrows represent the umklapp processes and the purple arrow represents the normal processes. Both of them contribute to scattering. (b) Temperature dependence of the power of electron-lattice cooling for two different Wannier orbitals radii. (c) System resistivity and the resistivity from the sum of contributions from different phonon branches varying with the temperature of different electron–phonon processes. (d) At twist angles θ = 1.20° and 1.05°, shape factors are shown for diamond and disk respectively, calculated of the continuum model at K points from the Wannier function. With different Wannier function ξ, the grey lines are shape factors gq proportional to Gaussian functions exp(−ξ2q2/4). (e) Shape factors at θ = 1.05° for different electron wave numbers k. Reproduced with permission from Ref. [32]. Copyright 2021, ACS Publication.

Fig. 8. (Color online) (a) PL spectra obtained measured from 1.356 to 1.377 eV in case of near-resonant excitation. Lorentzian functions filled are shown as black lines. (b) 2D PLE intensity map. The excess energy at 24 and 48 meV is shown with sloping black dashed lines. (c) Lorentzian fitted PLE spectra as a function of the excess energy of the PL spectra. The gray areas mean excess energy at 48 and 24 meV. Reproduced with permission from Ref. [35]. Copyright 2021, ACS Publication.

Fig. 9. (Color online) (a) Cross-plane thermoelectricity with different twist angles. The dotted lines show the linear T dependence of thermoelectric power. The solid lines show the fit of the thermoelectric power driven by phonons. Reproduced with permission from Ref. [26]. Copyright 2020, American Physical Society. (b) The thermal conductivity depending on twist angles of different temperatures. Reproduced with permission from Ref. [25]. Copyright 2021, AIP Publishing. (c) Calculated lattice thermal conductivities of tbBPs at 300 K with various twist angles at 300 K. Reproduced with permission from Ref. [37]. Copyright 2022, Wiley Publishing.

Fig. 10. (Color online) (a) Schematic of the tBLG devices fabricated on SiO₂/Si substrates. (b) Current–voltage curves of two devices M1 and M2 measured in graphene superlattices and at different temperatures. ResistanceR_xx was measured in two devices, with θ = 1.05° and 1.16°, respectively. Reproduced with permission from Refs. [7,40]. Copyright 2018, Nature. (c) Real-space map of pair amplitudes Δ(r) that satisfy s-wave linearized gap equations. (d) Real-space map of pair amplitudes Δ(r) that satisfy d-wave linearized gap equations. In (c) and (d), chemical potential μ= −0.3 meV and twist angle θ= 1.05°. Reproduced with permission from Ref. [22]. Copyright 2018, American Physical Society.

Fig. 11. (Color online) (a) Moiré surface state DOS with sharp higher-order Van hove singularities-like peaks atC₆ potential with various potential U^[23]. (b) Transition temperature Tc at vF/L = 435 K and εD = 80 K for different potential U. The blue axis on the right and red axis on the left correspond to the Tc, which shows broad peaks corresponding to higher-order Van Hove singularities around potentials. Reproduced with permission from Ref. [23]. Copyright 2021, American Physical Society.

Table 0. T_c andD calculated for tBLG, tTLG, tDBLG, and tMBLG. Reproduced with permission from Ref. [24]. Copyright 2021, American Physical Society.

Table 0. Electron–phonon coupling strengthλ_i, mode-resolved. Reproduced with permission from Ref. [24]. Copyright 2021, American Physical Society.