An Acoustic Platform for Twistronics

An acoustic platform for twistronics

S. Minhal Gardezi,1 Harris Pirie,2 Stephen Carr,3 William Dorrell,2 and Jennifer E. Hoffman1, 2, ∗ 1School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA 2Department of Physics, Harvard University, Cambridge, MA, 02138, USA 3Brown Theoretical Physics Center and Department of Physics, Brown University, Providence, RI, 02912-1843, USA (Dated: December 3, 2020) Twisted van der Waals (vdW) heterostructures have recently emerged as a tunable platform for studying correlated electrons. However, these materials require laborious and expensive effort for both theoretical and experimental exploration. Here we present a simple platform that reproduces twistronic behavior using acoustic metamaterials composed of interconnected air cavities in two stacked steel plates. Our classical analog of twisted bilayer graphene perfectly replicates the band structures of its quantum counterpart, including mode localization at a magic angle of 1.12◦. By tuning the thickness of the interlayer membrane, we reach a regime of strong interlayer tunneling where the acoustic magic angle appears as high as 6.01◦, equivalent to applying 130 GPa to twisted bilayer graphene. In this regime, the localized modes are over five times closer together than at 1.12◦, increasing the strength of any emergent non-linear acoustic couplings. Our results enable greater inter-connectivity between quantum materials and acoustic research: twisted vdW metamaterials provide simplified models for exploring twisted electronic systems and may allow the enhancement of non-linear effects to be translated into acoustics.

1 Van der Waals (vdW) heterostructures host a di- and fabricate, acoustic metamaterials have straightfor- verse set of useful emergent properties that can be cus- ward governing equations, continuously tunable proper- tomized by varying the stacking configuration of sheets ties, and fast build times, making them attractive models of two-dimensional (2D) materials, such as graphene, to rapidly explore their quantum counterparts. Cutting- other xenes, or transition-metal dichalcogenides [1–4]. edge ideas in several areas of theoretical condensed mat- Recently, the possibility of including a small twist angle ter are demonstrated using phononic analogs, such as chi- between adjacent layers in a vdW heterostructure has ral Landau levels [15], higher-order topology [16, 17], and led to the growing field of twistronics [5]. The twist an- fragile topology [18]. The utility of phononic metamate- gle induces a moirépattern that acts as a tunable po- rials also spans a large range of vdW heterostructures: tential for electrons moving within the layers, promoting the Dirac-like electronic bands in graphene have been enhanced electron correlations when their kinetic energy mimicked using longitudinal acoustics [19–21], surface is reduced below their Coulomb interaction. Even tradi- acoustics waves [22], and mechanics [23, 24]. Further, tional non-interacting materials can reach this regime, as it was recently discovered that placing a thin membrane exemplified by the correlated insulating state in twisted between metamaterial layers can reproduce the coupling bilayer graphene [6]. Already, vdW heterostructures effects of vdW forces, yielding acoustic analogs of bilayer with moirésuperlattices have led to new platforms for and trilayer graphene [25, 26]. The inclusion of a twist Wigner crystals [7], interlayer excitons [8–10], and uncon- angle between metamaterial layers has the potential to ventional superconductivity [11–13]. But the search for further expand their utility. In addition to electronic sys- novel twistronic phases is still in its infancy and there are tems, moiréengineering has recently been demonstrated countless vdW stacking and twisting arrangements that in systems containing vibrating plates [27], spoof surface- remain unexplored. Theoretical investigations of these acoustic waves [28], and optical lattices [29]. However, new arrangements is limited by the large and complex without simultaneous control of in-plane hopping, inter- moirépatterns created by multiple small twist angles. layer coupling, and twist angle, rapidly prototyping next- Meanwhile, fabrication of vdW heterostructures is re- generation twistronic devices using acoustic metamateri- stricted to the symmetries and properties of the few free- als remains an elusive goal. standing monolayers available today. It remains pressing to develop a more accessible platform to rapidly proto- 3 Here we introduce a simple acoustic metamaterial type and explore new twistronic materials to accelerate that precisely recreates the band structure of twisted bi- their technological advancement. layer graphene, including mode localization at a magic ◦ 2 The development of acoustic metamaterials over angle of 1.12 . We start with a monolayer acoustic the last few years has unlocked a compelling platform metamaterial that implements a tight-binding model de- to guide the design of new quantum materials [14]. scribing the low-energy band structure of graphene. By Whereas quantum materials can be difficult to predict combining two of these monolayers with an intermedi- ate polyethylene membrane, we recreate both stacking configurations of bilayer graphene in our finite-element simulations using comsol multiphysics. Our acoustic ∗ jhoff[email protected] analog of bilayer graphene hosts flat bands at the same 2

(a) (c) (e) 10 AB 2Δ θ > θmagic θ ≈ θmagic

R D Frequency (kHz) a r w T 2 Г K M Г (b) (d) (f) 10 10 AA AB

AA 6 6 M Frequency (kHz) Frequency (kHz) Г K

2 2 Г K M Г Г K M Г

FIG. 1. Acoustic vdW metamaterials. (a) In this acoustic metamaterial, sound waves propagate through connected air cavities in solid steel to mimic the way that electrons hop between carbon atoms in bilayer graphene. Our metamaterial graphene has cavity spacing a = 10 mm, cavity radius R = 3.5 mm, channel width w = 0.875 mm, and steel thickness D = 1 mm. The presence of smaller, unconnected cavities in the center of each unit cell improves the interlayer coupling of our bilayer metamaterial, but they do not act as additional lattice sites. (b) The C6 symmetry of the lattice protects a Dirac-like crossing at the K point of the monolayer acoustic band structure. (c-d) Two sheets of acoustic graphene coupled by an interlayer polyethylene membrane of thickness T = 2.35 mm accurately recreates the AB and AA configurations of bilayer graphene. (e) As the two sheets are twisted relative to one another, the two Dirac cones contributed by each layer hybridize, ultimately producing a flat band at small twist angles. (e) The twisted bilayer heterostructure has its own macroscopic periodicity with distinct AB and AA stacking regions. magic angle of ∼ 1.1◦ as its quantum counterpart [6, 30]. through our metamaterial. They allow nearest- and next- The advantage of our metamaterial is the ease with which nearest-neighbor coupling to be controlled independently it reaches coupling conditions beyond those feasible in by varying the width or length of separate air channels, atomic bilayer graphene. By tuning the thickness of the providing a platform to implement a broad class of tight- interlayer membrane, we design new metamaterials that binding models. host flat bands at several magic angles between 1.12◦ and 6.01◦. Our results demonstrate the potential for acoustic 5 To recreate bilayer graphene in an acoustic meta- vdW metamaterials to precisely simulate and explore the material, we started from a honeycomb lattice of air ever-growing number of quantum twistronic materials. cavities, with radius of 3.5 mm and a separation of 10 mm, in a 1-mm-thick steel sheet, encapsulated by 1-mm- 4 We begin by introducing a general framework to thick polyethylene boundaries. Each cavity is coupled to acoustically mimic electronic materials that are well de- its three nearest neighbors using 0.875-mm-wide chan- scribed by a tight-binding model, in which the electron nels, giving an s-mode bandwidth of 2t = 7.8 kHz (see wavefunctions are fairly localized to the atomic sites. Fig. 1(b)). This s manifold is well isolated from other In our metamaterial, each atomic site is represented by higher-order modes in the lattice, which appear above a cylindrical air cavity in a steel plate (see Fig. 1(a)). 25 kHz. The C6 symmetry of our metamaterial ensures a The radius of the air cavity determines the frequencies linear crossing at the K point, similar to the Dirac cone in of its ladder of acoustic standing modes. In a lattice graphene [32]. The frequency of this Dirac-like crossing of these cavities, the degenerate standing modes form and other key aspects of the band structure are controlled narrow bands, separated from each other by a large fre- by the dimensions of the cavities and channels. Build- quency gap. We focus primarily on the lowest, singly ing on previous work, we coupled two layers of acous- degenerate s band. Just as electrons hop from atom to tic graphene together using a thin interlayer membrane atom in an electronic tight-binding model, sound waves [26]. By adjusting only the stacking configuration, the propagate from cavity to cavity in our acoustic meta- same acoustic metamaterial mimics both the parabolic material through a network of tunable thin air chan- touching around the K point seen in AB-stacked bilayer nels. This coupling is always positive for s cavity modes, graphene, and the offset Dirac bands seen in AA-stacked but either sign can be realized by starting with higher- bilayer graphene [33, 34], as shown in Fig. 1(c-d). The order cavity modes [31]. Because sound travels much vertical span between the K-point eigenmodes is twice more easily through air than through steel, these chan- the interlayer coupling strength ∆, which is set by the nels are the dominant means of acoustic transmission membrane’s thickness [26]. We found that a 2.35-mm- 3

(a) 9.43˚ 3.89˚ 1.12˚ 1.12˚ 1.47˚ 2.00˚

0.4 2.35 mm 1.0 0.05 7.15 7.15 7.00 0.5 0.2

0 0 0 7.10 7.10 9.95 Frequency (kHz) Graphene

Energy (eV) -0.5 -0.2 K Г M K S S S S -1.0 -0.05 -0.4 1.67 mm K Г M K K Г M K K Г M K 7.25 7.10 S S S S S S S S S S S S (b) 7.20 7.05 7.4 7.2 K Г M K 7.2 7.1 7.15 S S S S

7.0 1.21 mm 7.0 6.8 7.20

Metamaterial 6.6 6.9 7.10 Frequency (kHz) 7.15 6.4 6.8 2.35 mm 2.35 mm 2.35 mm K Г M K K Г M K K Г M K K Г M K S S S S S S S S S S S S S S S S (c) AB 3 FIG. 3. Interlayer coupling tunes the magic angle. Reduc- AB AB ing the thickness of the polyethylene membrane enhances the coupling between layers of our acoustic metamaterial. Con- AA sequently, an acoustic flat band (orange) can been formed either by decreasing the twist angle with a fixed membrane AB AB thickness (along rows), or by decreasing the thickness at fixed Acoustic Pressure (Pa) -3 ◦ AB angle (columns). For example, to realize flat bands at 2 , we 10.5a 25.5a 88.5a need to reduce the membrane thickness to 1.21 mm.

FIG. 2. Trapping sound by twisting. (a) As the twist angle between two layers of graphene decreases, the Dirac bands Our acoustic flat bands correspond to real-space pres- are compressed around the Fermi level, eventually becoming sure modes that are primarily located on the AA region, completely flat at a magic angle of 1.12◦, as shown in these electronic tight-binding calculations of unrelaxed twisted bi- see Fig. 2(b). This AA localization agrees with calcula- layer graphene. (b) The same trend occurs in our bilayer tions of the local density of states in magic-angle twisted acoustic metamaterial, spawning acoustic flat bands at 1.12◦. bilayer graphene [6, 38]. In our acoustic system, it rep- (c) For large angles, the acoustic Dirac-like modes at the Ks resents sound waves that propagate with a low group ve- point are itinerant and persist across the entire supercell. But locity of 0.05 m/s, compared to 30 m/s in the untwisted at the magic angle, they become localized on the AA-stacked bilayer. region in the center of the supercell. 7 The magic angle of twisted bilayer graphene can be tuned by applying vertical pressure to push the graphene layers closer together and increase the interlayer coupling thick polyethylene membrane (density 950 kg/m3 and [39]. Consequently, the Dirac bands begin to flatten at speed of sound 2460 m/s) accurately matched the dimen- higher angles under pressure than at ambient conditions. sionless coupling ratio ∆/t ≈ 5% in bilayer graphene. However, a substantial vertical pressure of a few GPa ◦ ◦ 6 Introducing a twist angle between two graphene lay- is required to move the magic angle from 1.1 to 1.27 , ers creates a moirépattern that grows in size as the angle corresponding to only a 20% increase in the interlayer decreases. At small angles, the Dirac cones from each coupling strength [12]. In our metamaterial, no such layer are pushed together and hybridize due to the inter- physical restrictions apply: the interlayer coupling can layer coupling [35, 36], as shown in Fig. 1(e). Eventually, be tuned continuously over two orders of magnitude sim- they form a flat band with a vanishing Fermi velocity ply by changing the thickness of the coupling membrane (vF ) at a so-called magic angle [6, 30, 37], see Fig. 2(a). [26]. To demonstrate this flexibility, we reproduced the ◦ We searched for the same band-flattening mechanism by band flattening at a fixed angle of 2 simply by incre- introducing a commensurate-angle twist to our acoustic mentally reducing the thickness of the interlayer mem- bilayer graphene metamaterial. Strikingly, our metama- brane to 1.21 mm (Fig.3). In other words, the flat band terial mimics its quantum counterpart even down to the condition can be approached from two directions: either magic angle, producing acoustic flat bands at 1.12◦, as reduce the twist angle at fixed interlayer thickness, or re- shown in Fig. 2(a-b). Importantly, there are no other duce the thickness at fixed angle. For comparison, it is acoustic bands near the Dirac point that can fold and in- expected to require about 9 GPa of vertical pressure for ◦ terfere with these flat bands (see Fig. 1(b)). The flat twisted bilayer graphene to reach flat bands at 2 [39]. bands appear upside down in our acoustic model be- 8 Our acoustic metamaterial provides a simple plat- cause its interlayer coupling (between s cavity modes) has form to explore twistronics in extreme coupling regimes, the opposite sign from graphene’s (between pz orbitals). well beyond the experimental capability of its electronic 4

(a) 3.89˚ 5.09˚ 6.01˚ ceptible to non-linear eﬀects if tuned correctly, akin to a 8.0 8.2 phonon-phonon interaction. In principle, any twist angle 7.7 can become a magic angle that hosts a dense array of 7.9 8.1 such localized modes, by choosing the correct interlayer 7.6 7.8 8.0 thickness (Fig. 4(c)). 7.7 7.9 7.5 Frequency (kHz) 9 Although we focused on recreating twisted bi- 7.8 0.58 mm 7.6 0.43 mm 0.35 mm layer graphene, our metamaterial can be easily ex- K Г M K K Г M K K Г M K (b) S S S S S S S S S S S S tended to capture other vdW systems. For example, by breaking the sublattice symmetry in our unit cell, 3 one can explore the localized modes shaped from twisted quadratic bands, imitating semiconductors like hexagonal boron nitride [40] or transition-metal dichalgogenides [41]. Our metamaterial platform could even mimic twistronic Hamiltonians that cannot yet be realized in Acoustic Pressure (Pa) -3 a a a condensed matter experiments, which are restricted to 25.5 19.5 16.5 (c) the primarily-hexagonal set of 2D materials available to- 12 2.21 mm day. In principle, a twistronic heterostructure can be 10 1.67 mm constructed from any lattice symmetry to spawn diverse 1.21 mm ﬂat bands with distinct topologies [42]. Further, by tex- 8 0.58 mm 0.51 mm turing the interlayer membrane, our metamaterial also 6 0.43 mm allows independent control of the AA and AB interlayer (m/s) F

v 0.35 mm coupling, which may unlock exactly flat bands at the 4 magic angle [43]. Beyond two-layer systems, our meta- 2 material provides an experimental platform to explore the intricate moirépatterns created by multiple arbitrary 0 twist angles. Such multilayer systems may quickly sur- ˚ pass theoretical electronic calculations, which are made 1.12˚ 1.47˚ 2.00˚ 3.89˚ 4.41˚ 5.09 6.01˚ nearly impossible by the highly incommensurate geome- FIG. 4. Reproducing flat bands at high magic angles. (a) By try, even in the three-layer case [44]. reducing the thickness of the interlayer membrane to 0.58 mm, 10 Our twisted bilayer metamaterial translates the 0.43 mm, and 0.35 mm, we discover flat bands at 3.89◦, 5.09◦, field twistronics to acoustics. By introducing a twist an- ◦ and 6.01 twist angles. (b) In each case, the Ks-point eigen- gle to vdW metamaterials, we discovered flat bands that modes appear predominantly around the AA-stacked region precisely mimic the behavior of twisted bilayer graphene in the center of the supercell. As the supercell shrinks, these at 1.1◦ and that slow transmitted sound by a factor of localized modes form a dense array, increasing the possibility 600 (20 times slower than a leisurely walk). The precision of interactions with similar localized modes in neighboring su- percells. (c) By correctly choosing the interlayer membrane’s of our results gives confidence that our twisted acoustic thickness, any angle can become a magic angle with a vanish- metamaterials can be an accurate platform for more gen- ing Ks-point velocity. eral quantum material design. The dense lattice of localized modes we found at higher magic angles may lead to new paths for acoustic emitters and ultrasound imagers. counterpart. By further reducing the interlayer mem- Finally, our design of magic-angle twisted vdW metama- brane thickness, we searched for flat bands at high com- terials directly translates to photonic systems under a mensurate angles of 3.89◦, 5.09◦, and 6.01◦, equivalent to simple mapping of variables [19]. pressures of 45 GPa, 85 GPa, and 129 GPa that would need to be applied to the graphene system [39]. In each case, we discovered flat bands similar to those in twisted ACKNOWLEDGEMENTS bilayer graphene (Fig. 4(a)). These flat bands all correspond to collective pressure modes that are localized We thank Alex Kruchkov, Haoning Tang, Clayton De- on the AA-stacked central regions (Fig. 4(b)). But these Vault, Daniel Larson, Nathan Drucker, Ben November, localized modes are over five times closer to each other Walker Gillett, and Fan Du for insightful discussions. at 6.01◦ than they are at 1.12◦. Generally speaking, the This work was supported by the National Science Foun- modes interact more strongly as they become closer to- dation Grant No. OIA-1921199 and the Science Tech- gether, potentially allowing interactions to dominate over nology Center for Integrated Quantum Materials Grant the reduced kinetic energy at a high magic angle. Conse- No. DMR-1231319. W.D. acknowledges support from a quently, a high-magic-angle metamaterial could be sus- Herchel Smith Scholarship. 5

[1] S.-J. Liang, B. Cheng, X. Cui, and F. Miao, Van [15] V. Peri, M. Serra-Garcia, R. Ilan, and S. D. Huber, Axial- der Waals Heterostructures for High-Performance Device field-induced chiral channels in an acoustic Weyl system, Applications: Challenges and Opportunities, Advanced Nature Physics 15, 357 (2019). Materials 32, 1903800 (2020). [16] M. Serra-Garcia, V. Peri, R. Süsstrunk, O. R. Bilal, [2] A. K. Geim and I. V. Grigorieva, Van der Waals het- T. Larsen, L. G. Villanueva, and S. D. Huber, Obser- erostructures, Nature 499, 419 (2013). vation of a phononic quadrupole topological insulator, [3] P. Ajayan, P. Kim, and K. Banerjee, Two-dimensional Nature 555, 342 (2018). van der Waals materials, Physics Today 69, 38 (2016). [17] X. Ni, M. Weiner, A. Alù,and A. B. Khanikaev, Observa- [4] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. tion of higher-order topological acoustic states protected Castro Neto, 2D materials and van der Waals het- by generalized chiral symmetry, Nature Materials 18, 113 erostructures, Science 353, aac9439 (2016). (2019). [5] S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, [18] V. Peri, Z.-D. Song, M. Serra-Garcia, P. Engeler, and E. Kaxiras, Twistronics: Manipulating the electronic R. Queiroz, X. Huang, W. Deng, Z. Liu, B. A. Bernevig, properties of two-dimensional layered structures through and S. D. Huber, Experimental characterization of fragile their twist angle, Physical Review B 95, 075420 (2017). topology in an acoustic metamaterial, Science 367, 797 [6] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, (2020). J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, [19] J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, First- T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- principles study of Dirac and Dirac-like cones in phononic Herrero, Correlated insulator behaviour at half-filling and photonic crystals, Physical Review B 86, 035141 in magic-angle graphene superlattices, Nature 556, 80 (2012). (2018). [20] D. Torrent and J. Sánchez-Dehesa, Acoustic Analogue of [7] E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, Graphene: Observation of Dirac Cones in Acoustic Sur- B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu- face Waves, Physical Review Letters 108, 174301 (2012). migeta, M. Blei, J. D. Carlström, K. Watanabe, [21] J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, Dirac T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and cones in two-dimensional artificial crystals for classical F. Wang, Mott and generalized Wigner crystal states in waves, Physical Review B 89, 134302 (2014). WSe2/WS2 moirésuperlattices, Nature 579, 359 (2020). [22] S.-Y. Yu, X.-C. Sun, X. Ni, Q. Wang, X.-J. Yan, C. He, [8] C. Jin, E. C. Regan, A. Yan, M. Iqbal Bakti Utama, X.-P. Liu, L. Feng, M.-H. Lu, and Y.-F. Chen, Surface D. Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi, phononic graphene, Nature Materials 15, 1243 (2016). K. Watanabe, T. Taniguchi, S. Tongay, A. Zettl, and [23] D. Torrent, D. Mayou, and J. Sánchez-Dehesa, Elastic F. Wang, Observation of moiréexcitons in WSe2/WS2 analog of graphene: Dirac cones and edge states for flex- heterostructure superlattices, Nature 567, 76 (2019). ural waves in thin plates, Physical Review B 87, 115143 [9] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray, (2013). D. G. Mandrus, J. Yan, W. Yao, and X. Xu, Signatures [24] T. Kariyado and Y. Hatsugai, Manipulation of Dirac of moiré-trapped valley excitons in MoSe2/WSe2 hetero- Cones in Mechanical Graphene, Scientific Reports 5, bilayers, Nature 567, 66 (2019). 18107 (2016). [10] K. Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim, [25] J. Lu, C. Qiu, W. Deng, X. Huang, F. Li, F. Zhang, A. Rai, D. A. Sanchez, J. Quan, A. Singh, J. Emb- S. Chen, and Z. Liu, Valley topological phases in bi- ley, A. Zepeda, M. Campbell, T. Autry, T. Taniguchi, layer sonic crystals, Physical Review Letters 120, 116802 K. Watanabe, N. Lu, S. K. Banerjee, K. L. Silverman, (2018). S. Kim, E. Tutuc, L. Yang, A. H. MacDonald, and X. Li, [26] W. Dorrell, H. Pirie, S. M. Gardezi, N. C. Drucker, and Evidence for moiréexcitons in van der Waals heterostruc- J. E. Hoffman, van der Waals Metamaterials, Physical tures, Nature 567, 71 (2019). Review B 101, 121103(R) (2020). [11] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, [27] M. Rosendo, F. Peñaranda,J. Christensen, and P. San- E. Kaxiras, and P. Jarillo-Herrero, Unconventional super- Jose, Flat bands in magic-angle vibrating plates, conductivity in magic-angle graphene superlattices, Na- arXiv:2008.11706. ture 556, 43 (2018). [28] Y. Deng, M. Oudich, N. J. Gerard, J. Ji, M. Lu, [12] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- and Y. Jing, Magic-angle Bilayer Phononic Graphene, abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, arXiv:2010.05940. Tuning superconductivity in twisted bilayer graphene, [29] P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, Science 363, 1059 (2019). L. Torner, V. V. Konotop, and F. Ye, Localization and [13] G. Chen, A. L. Sharpe, P. Gallagher, I. T. Rosen, E. J. delocalization of light in photonic moirélattices, Nature Fox, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, 577, 42 (2020). J. Jung, Z. Shi, D. Goldhaber-Gordon, Y. Zhang, and [30] R. Bistritzer and A. H. MacDonald, Moiré bands in F. Wang, Signatures of tunable superconductivity in a twisted double-layer graphene, Proc. Natl. Acad. Sci. trilayer graphene moirésuperlattice, Nature 572, 215 USA 103, 12233 (2011). (2019). [31] K. H. Matlack, M. Serra-Garcia, A. Palermo, S. D. Hu- [14] H. Ge, M. Yang, C. Ma, M.-H. Lu, Y.-F. Chen, N. Fang, ber, and C. Daraio, Designing perturbative metamate- and P. Sheng, Breaking the barriers: Advances in acous- rials from discrete models, Nature Materials 17, 323 tic functional materials, National Science Review 5, 159 (2018). (2018). 6

[32] E. Kogan and U. V. Nazarov, Symmetry classification of [39] S. Carr, S. Fang, P. Jarillo-Herrero, and E. Kaxiras, Pres- energy bands in graphene, Physical Review B 85, 115418 sure dependence of the magic twist angle in graphene (2012). superlattices, Physical Review B 98, 085144 (2018). [33] P. L. de Andres, R. Ram´ırez,and J. A. Vergés,Strong [40] L. Xian, D. M. Kennes, N. Tancogne-Dejean, covalent bonding between two graphene layers, Physical M. Altarelli, and A. Rubio, Multiflat bands and strong Review B 77, 045403 (2008). correlations in twisted bilayer boron nitride: Doping- [34] E. McCann and M. Koshino, The electronic properties induced correlated insulator and superconductor, Nano of bilayer graphene, Reports on Progress in Physics 76, Letters 19, 4934 (2019). 056503 (2013). [41] M. H. Naik and M. Jain, Ultraflatbands and shear soli- [35] S. Shallcross, S. Sharma, E. Kandelaki, and O. A. tons in moirépatterns of twisted bilayer transition metal Pankratov, Electronic structure of turbostratic graphene, dichalcogenides, Phys. Rev. Lett. 121, 266401 (2018). Physical Review B 81, 165105 (2010). [42] T. Kariyado and A. Vishwanath, Flat band in twisted [36] P. Moon and M. Koshino, Optical Absorption in Twisted bilayer bravais lattices, Phys. Rev. Research 1, 033076 Bilayer Graphene, Physical Review B 87, 205404 (2013). (2019). [37] E. SuárezMorell, J. D. Correa, P. Vargas, M. Pacheco, [43] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Ori- and Z. Barticevic, Flat bands in slightly twisted bilayer gin of magic angles in twisted bilayer graphene, Phys. graphene: Tight-binding calculations, Physical Review B Rev. Lett. 122, 106405 (2019). 82, 121407(R) (2010). [44] Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxiras, [38] K. Kim, A. DaSilva, S. Huang, B. Fallahazad, S. Laren- Twisted trilayer graphene: A precisely tunable platform tis, T. Taniguchi, K. Watanabe, B. J. LeRoy, A. H. Mac- for correlated electrons, Phys. Rev. Lett. 125, 116404 Donald, and E. Tutuc, Tunable moirébands and strong (2020). correlations in small-twist-angle bilayer graphene, Proc. Natl. Acad. Sci. USA 114, 3364 (2017).