Twistronics and moiré excitonic physics in van der Waals heterostructures

Abstract

Heterostructures composed of two-dimensional van der Waals (vdW) materials allow highly controllable stacking, where interlayer twist angles introduce a continuous degree of freedom to alter the electronic band structures and excitonic physics. Motivated by the discovery of Mott insulating states and superconductivity in magic-angle bilayer graphene, the emerging research fields of “twistronics” and moiré physics have aroused great academic interests in the engineering of optoelectronic properties and the exploration of new quantum phenomena, in which moiré superlattice provides a pathway for the realization of artificial excitonic crystals. Here we systematically summarize the current achievements in twistronics and moiré excitonic physics, with emphasis on the roles of lattice rotational mismatches and atomic registries. Firstly, we review the effects of the interlayer twist on electronic and photonic physics, particularly on exciton properties such as dipole moment and spin-valley polarization, through interlayer interactions and electronic band structures. We also discuss the exciton dynamics in vdW heterostructures with different twist angles, like formation, transport and relaxation processes, whose mechanisms are complicated and still need further investigations. Subsequently, we review the theoretical analysis and experimental observations of moiré superlattice and moiré modulated excitons. Various exotic moiré effects are also shown, including periodic potential, moiré miniband, and varying wave function symmetry, which result in exciton localization, emergent exciton peaks and spatially alternating optical selection rule. We further introduce the expanded properties of moiré systems with external modulation factors such as electric field, doping and strain, showing that moiré lattice is a promising platform with high tunability for optoelectronic applications and in-depth study on frontier physics. Lastly, we focus on the rapidly developing field of correlated electron physics based on the moiré system, which is potentially related to the emerging quantum phenomena.

Graphical abstract

Keywords

moiré superlattice / twistronics / van der Waals heterostructure / moiré exciton / correlated electronic state

References

[1] M. Osada , T. Sasaki . Two‐dimensional dielectric nanosheets: Novel nanoelectronics from nanocrystal building blocks. Adv. Mater., 2012, 24(2): 210

[2] M. Xu , T. Liang , M. Shi , H. Chen . Graphene-like two-dimensional materials. Chem. Rev., 2013, 113(5): 3766

[3] K. S. Novoselov , D. Jiang , F. Schedin , T. J. Booth , V. V. Khotkevich , S. V. Morozov , A. K. Geim . Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. USA, 2005, 102(30): 10451

[4] A. K. Geim , K. S. Novoselov . The rise of graphene. Nat. Mater., 2007, 6(3): 183

[5] P. Vogt , P. De Padova , C. Quaresima , J. Avila , E. Frantzeskakis , M. C. Asensio , A. Resta , B. Ealet , G. Le Lay . Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon. Phys. Rev. Lett., 2012, 108(15): 155501

[6] K. Watanabe , T. Taniguchi , H. Kanda . Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal. Nat. Mater., 2004, 3(6): 404

[7] L. Song , L. Ci , H. Lu , P. B. Sorokin , C. Jin , J. Ni , A. G. Kvashnin , D. G. Kvashnin , J. Lou , B. I. Yakobson , P. M. Ajayan . Large scale growth and characterization of atomic hexagonal boron nitride layers. Nano Lett., 2010, 10(8): 3209

[8] H. Liu , A. T. Neal , Z. Zhu , Z. Luo , X. Xu , D. Tománek , P. D. Ye . Phosphorene: An unexplored 2D semiconductor with a high hole mobility. ACS Nano, 2014, 8(4): 4033

[9] L. Li , Y. Yu , G. J. Ye , Q. Ge , X. Ou , H. Wu , D. Feng , X. H. Chen , Y. Zhang . Black phosphorus field-effect transistors. Nat. Nanotechnol., 2014, 9(5): 372

[10] C.AtacaM.TopsakalE.AktürkS.Ciraci, A comparative study of lattice dynamics of three- and two-dimensional MoS₂, J. Phys. Chem. C 115(33), 16354 (2011)

[11] R. Ganatra , Q. Zhang . Few-layer MoS₂: A promising layered semiconductor. ACS Nano, 2014, 8(5): 4074

[12] Q. H. Wang , K. Kalantar-zadeh , A. Kis , J. N. Coleman , M. S. Strano . Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol., 2012, 7(11): 699

[13] C. Gong , H. Zhang , W. Wang , L. Colombo , R. M. Wallace , K. Cho . Band alignment of two-dimensional transition metal dichalcogenides: Application in tunnel field effect transistors. Appl. Phys. Lett., 2013, 103(5): 053513

[14] H. J. Chuang , X. Tan , N. J. Ghimire , M. M. Perera , B. Chamlagain , M. M. C. Cheng , J. Yan , D. Mandrus , D. Tománek , Z. Zhou . High mobility WSe₂ p- and n-type field-effect transistors contacted by highly doped graphene for low-resistance contacts. Nano Lett., 2014, 14(6): 3594

[15] Y. Ye , Z. J. Wong , X. Lu , X. Ni , H. Zhu , X. Chen , Y. Wang , X. Zhang . Monolayer excitonic laser. Nat. Photonics, 2015, 9(11): 733

[16] Y. K. Luo , J. Xu , T. Zhu , G. Wu , E. J. McCormick , W. Zhan , M. R. Neupane , R. K. Kawakami . Opto-valleytronic spin injection in monolayer MoS₂/few-layer graphene hybrid spin valves. Nano Lett., 2017, 17(6): 3877

[17] B. Radisavljevic , M. B. Whitwick , A. Kis . Integrated circuits and logic operations based on single-layer MoS₂. ACS Nano, 2011, 5(12): 9934

[18] J. S. Ross , P. Rivera , J. Schaibley , E. Lee-Wong , H. Yu , T. Taniguchi , K. Watanabe , J. Yan , D. Mandrus , D. Cobden , W. Yao , X. Xu . Interlayer exciton optoelectronics in a 2D heterostructure p–n junction. Nano Lett., 2017, 17(2): 638

[19] C. Gao , Q. Nie , C. Y. Lin , F. Huang , L. Wang , W. Xia , X. Wang , Z. Hu , M. Li , H. W. Lu , Y. C. Lai , Y. F. Lin , J. Chu , W. Li . Touch-modulated van der Waals heterostructure with self-writing power switch for synaptic simulation. Nano Energy, 2022, 91: 106659

[20] G. D. Wenbiao Niu , Z. Jia , X. Q. Ma , J. Y. Zhao , K. Zhou , S. T. Han , C. C. Kuo , Y. Zhou . Recent advances in memristors based on two-dimensional ferroelectric materials. Front. Phys., 2024, 19(1): 13402

[21] A. Chaves , J. G. Azadani , H. Alsalman , D. R. da Costa , R. Frisenda , A. J. Chaves , S. H. Song , Y. D. Kim , D. He , J. Zhou , A. Castellanos-Gomez , F. M. Peeters , Z. Liu , C. L. Hinkle , S.-H. Oh , P. D. Ye , S. J. Koester , Y. H. Lee , P. Avouris , X. Wang , T. Low . Bandgap engineering of two-dimensional semiconductor materials. npj 2D Mater. Appl., 2020, 4: 29

[22] Y. Wang , Z. Nie , F. Wang . Modulation of photocarrier relaxation dynamics in two-dimensional semiconductors. Light Sci. Appl., 2020, 9(1): 192

[23] S. Wu , J. S. Ross , G. B. Liu , G. Aivazian , A. Jones , Z. Fei , W. Zhu , D. Xiao , W. Yao , D. Cobden , X. Xu . Electrical tuning of valley magnetic moment through symmetry control in bilayer MoS₂. Nat. Phys., 2013, 9(3): 149

[24] I. Lee , S. Rathi , D. Lim , L. Li , J. Park , Y. Lee , K. S. Yi , K. P. Dhakal , J. Kim , C. Lee , G. H. Lee , Y. D. Kim , J. Hone , S. J. Yun , D. H. Youn , G. H. Kim . Gate-tunable hole and electron carrier transport in atomically thin dual-channel WSe₂/MoS₂ heterostructure for ambipolar field-effect transistors. Adv. Mater., 2016, 28(43): 9519

[25] J. Lee , K. F. Mak , J. Shan . Electrical control of the valley Hall effect in bilayer MoS₂ transistors. Nat. Nanotechnol., 2016, 11: 421

[26] L. A. Jauregui , A. Y. Joe , K. Pistunova , D. S. Wild , A. A. High , Y. Zhou , G. Scuri , K. De Greve , A. Sushko , C. H. Yu , T. Taniguchi , K. Watanabe , D. J. Needleman , M. D. Lukin , H. Park , P. Kim . Electrical control of interlayer exciton dynamics in atomically thin heterostructures. Science, 2019, 366(6467): 870

[27] A. Chernikov , A. M. van der Zande , H. M. Hill , A. F. Rigosi , A. Velauthapillai , J. Hone , T. F. Heinz . Electrical tuning of exciton binding energies in monolayer WS₂. Phys. Rev. Lett., 2015, 115(12): 126802

[28] P. V. Nguyen , N. C. Teutsch , N. P. Wilson , J. Kahn , X. Xia , A. J. Graham , V. Kandyba , A. Giampietri , A. Barinov , G. C. Constantinescu , N. Yeung , N. Hine , X. Xu , D. H. Cobden , N. R. Wilson . Visualizing electrostatic gating effects in two-dimensional heterostructures. Nature, 2019, 572(7768): 220

[29] Z. Peng , X. Chen , Y. Fan , D. J. Srolovitz , D. Lei . Strain engineering of 2D semiconductors and graphene: From strain fields to band-structure tuning and photonic applications. Light Sci. Appl., 2020, 9(1): 190

[30] H. J. Conley , B. Wang , J. I. Ziegler , R. F. Jr Haglund , S. T. Pantelides , K. I. Bolotin . Bandgap engineering of strained monolayer and bilayer MoS₂. Nano Lett., 2013, 13(8): 3626

[31] M. G. Harats , J. N. Kirchhof , M. Qiao , K. Greben , K. I. Bolotin . Dynamics and efficient conversion of excitons to trions in non-uniformly strained monolayer WS₂. Nat. Photonics, 2020, 14(5): 324

[32] Z. Li , Y. Lv , L. Ren , J. Li , L. Kong , Y. Zeng , Q. Tao , R. Wu , H. Ma , B. Zhao , D. Wang , W. Dang , K. Chen , L. Liao , X. Duan , X. Duan , Y. Liu . Efficient strain modulation of 2D materials via polymer encapsulation. Nat. Commun., 2020, 11(1): 1151

[33] G. Luo , X. Lv , L. Wen , Z. Li , Z. Dai . Strain induced topological transitions in twisted double bilayer graphene. Front. Phys., 2022, 17(2): 23502

[34] J. M. Fitzgerald , J. J. P. Thompson , E. Malic . Twist angle tuning of moiré exciton polaritons in van der Waals heterostructures. Nano Lett., 2022, 22(11): 4468

[35] Y. Hoshi , T. Kuroda , M. Okada , R. Moriya , S. Masubuchi , K. Watanabe , T. Taniguchi , R. Kitaura , T. Machida . Suppression of exciton−exciton annihilation in tungsten disulfide monolayers encapsulated by hexagonal boron nitrides. Phys. Rev. B, 2017, 95(24): 241403

[36] A. K. M. Newaz , Y. S. Puzyrev , B. Wang , S. T. Pantelides , K. I. Bolotin . Probing charge scattering mechanisms in suspended graphene by varying its dielectric environment. Nat. Commun., 2012, 3(1): 734

[37] A. Raja , A. Chaves , J. Yu , G. Arefe , H. M. Hill , A. F. Rigosi , T. C. Berkelbach , P. Nagler , C. Schüller , T. Korn , C. Nuckolls , J. Hone , L. E. Brus , T. F. Heinz , D. R. Reichman , A. Chernikov . Coulomb engineering of the bandgap and excitons in two-dimensional materials. Nat. Commun., 2017, 8(1): 15251

[38] N. P. Wilson , W. Yao , J. Shan , X. Xu . Excitons and emergent quantum phenomena in stacked 2D semiconductors. Nature, 2021, 599(7885): 383

[39] K. F. Mak , C. Lee , J. Hone , J. Shan , T. F. Heinz . Atomically thin MoS₂: A new direct-gap semiconductor. Phys. Rev. Lett., 2010, 105: 136805

[40] Y. Li , A. Chernikov , X. Zhang , A. Rigosi , H. M. Hill , A. M. van der Zande , D. A. Chenet , E. M. Shih , J. Hone , T. F. Heinz . Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS₂, MoSe₂, WS₂, and WSe₂. Phys. Rev. B, 2014, 90(20): 205422

[41] T. C. Berkelbach , M. S. Hybertsen , D. R. Reichman . Theory of neutral and charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B, 2013, 88(4): 045318

[42] D. Xiao , W. Yao , Q. Niu . Valley-contrasting physics in graphene: Magnetic moment and topological transport. Phys. Rev. Lett., 2007, 99(23): 236809

[43] W. Yao , D. Xiao , Q. Niu . Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B, 2008, 77(23): 235406

[44] T. Cao , G. Wang , W. Han , H. Ye , C. Zhu , J. Shi , Q. Niu , P. Tan , E. Wang , B. Liu , J. Feng . Valley- selective circular dichroism of monolayer molybdenum disulphide. Nat. Commun., 2012, 3(1): 887

[45] W. Jin , P. C. Yeh , N. Zaki , D. Chenet , G. Arefe , Y. Hao , A. Sala , T. O. Mentes , J. I. Dadap , A. Locatelli , J. Hone , R. M. Osgood . Tuning the electronic structure of monolayer graphene/MoS₂ van der Waals heterostructures via interlayer twist. Phys. Rev. B, 2015, 92(20): 201409

[46] H. Yu , Y. Wang , Q. Tong , X. Xu , W. Yao . Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers. Phys. Rev. Lett., 2015, 115(18): 187002

[47] J. P. Eisenstein , A. H. MacDonald . Bose–Einstein condensation of excitons in bilayer electron systems. Nature, 2004, 432(7018): 691

[48] C. R. Dean , L. Wang , P. Maher , C. Forsythe , F. Ghahari , Y. Gao , J. Katoch , M. Ishigami , P. Moon , M. Koshino , T. Taniguchi , K. Watanabe , K. L. Shepard , J. Hone , P. Kim . Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature, 2013, 497(7451): 598

[49] H. Yu , G. B. Liu , J. Tang , X. Xu , W. Yao . Moiré excitons: From programmable quantum emitter arrays to spin‒orbit coupled artificial lattices. Sci. Adv., 2017, 3(11): e1701696

[50] H. Guo , X. Zhang , G. Lu . Shedding light on moiré excitons: A first-principles perspective. Sci. Adv., 2020, 6(42): eabc5638

[51] P. Rivera , K. L. Seyler , H. Yu , J. R. Schaibley , J. Yan , D. G. Mandrus , W. Yao , X. Xu . Valley- polarized exciton dynamics in a 2D semiconductor heterostructure. Science, 2016, 351(6274): 688

[52] Y. Wang , Z. Wang , W. Yao , G. B. Liu , H. Yu . Interlayer coupling in commensurate and incommensurate bilayer structures of transition-metal dichalcogenides. Phys. Rev. B, 2017, 95(11): 115429

[53] W. Ren , S. Lu , C. Yu , J. He , Z. Zhang , J. Chen , G. Zhang . Impact of moiré superlattice on atomic stress and thermal transport in van der Waals heterostructures. Appl. Phys. Rev., 2023, 10(4): 041404

[54] Y. Jiang , S. Chen , W. Zheng , B. Zheng , A. Pan . Interlayer exciton formation, relaxation, and transport in TMD van der Waals heterostructures. Light Sci. Appl., 2021, 10(1): 72

[55] Q. Tong , H. Yu , Q. Zhu , Y. Wang , X. Xu , W. Yao . Topological mosaics in moiré superlattices of van der Waals heterobilayers. Nat. Phys., 2017, 13(4): 356

[56] C. Zhang , C. P. Chuu , X. Ren , M. Y. Li , L. J. Li , C. Jin , M. Y. Chou , C. K. Shih , Interlayer couplings . Moiré patterns, and 2D electronic superlattices in MoS₂/WSe₂ hetero-bilayers. Sci. Adv., 2017, 3(1): e1601459

[57] Y. Pan , S. Fölsch , Y. Nie , D. Waters , Y. C. Lin , B. Jariwala , K. Zhang , K. Cho , J. A. Robinson , R. M. Feenstra . Quantum-confined electronic states arising from the moiré pattern of MoS₂–WSe₂ heterobilayers. Nano Lett., 2018, 18(3): 1849

[58] L. J. McGilly , A. Kerelsky , N. R. Finney , K. Shapovalov , E. M. Shih , A. Ghiotto , Y. Zeng , S. L. Moore , W. Wu , Y. Bai , K. Watanabe , T. Taniguchi , M. Stengel , L. Zhou , J. Hone , X. Zhu , D. N. Basov , C. Dean , C. E. Dreyer , A. N. Pasupathy . Visualization of moiré superlattices. Nat. Nanotechnol., 2020, 15(7): 580

[59] Y. Cao , V. Fatemi , A. Demir , S. Fang , S. L. Tomarken , J. Y. Luo , J. D. Sanchez-Yamagishi , K. Watanabe , T. Taniguchi , E. Kaxiras , R. C. Ashoori , P. Jarillo-Herrero . Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature, 2018, 556(7699): 80

[60] Y. Cao , V. Fatemi , S. Fang , K. Watanabe , T. Taniguchi , E. Kaxiras , P. Jarillo-Herrero . Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556(7699): 43

[61] L. Li , M. Wu , X. Lu . Correlation, superconductivity and topology in graphene moiré superlattice. Front. Phys., 2023, 18(4): 43401

[62] K. L. Seyler , P. Rivera , H. Yu , N. P. Wilson , E. L. Ray , D. G. Mandrus , J. Yan , W. Yao , X. Xu . Signatures of moiré-trapped valley excitons in MoSe₂/WSe₂ heterobilayers. Nature, 2019, 567(7746): 66

[63] B. Wu , H. Zheng , S. Li , J. Ding , J. He , Y. Zeng , K. Chen , Z. Liu , S. Chen , A. Pan , Y. Liu . Evidence for moiré intralayer excitons in twisted WSe₂/WSe₂ homobilayer superlattices. Light Sci. Appl., 2022, 11(1): 166

[64] F. Wu , T. Lovorn , A. H. MacDonald . Topological exciton bands in moiré heterojunctions. Phys. Rev. Lett., 2017, 118(14): 147401

[65] F. Wu , T. Lovorn , A. H. MacDonald . Theory of optical absorption by interlayer excitons in transition metal dichalcogenide heterobilayers. Phys. Rev. B, 2018, 97(3): 035306

[66] H. Yu , G. B. Liu , W. Yao . Brightened spin-triplet interlayer excitons and optical selection rules in van der Waals heterobilayers. 2D Mater., 2018, 5: 035021

[67] C. Jin , E. C. Regan , D. Wang , M. Iqbal Bakti Utama , C. S. Yang , J. Cain , Y. Qin , Y. Shen , Z. Zheng , K. Watanabe , T. Taniguchi , S. Tongay , A. Zettl , F. Wang . Identification of spin, valley and moiré quasi-angular momentum of interlayer excitons. Nat. Phys., 2019, 15(11): 1140

[68] J. He , K. Hummer , C. Franchini . Stacking effects on the electronic and optical properties of bilayer transition metal dichalcogenides MoS₂, MoSe₂, WS₂, and WSe₂. Phys. Rev. B, 2014, 89(7): 075409

[69] Q. Liu , L. Li , Y. Li , Z. Gao , Z. Chen , J. Lu . Tuning electronic structure of bilayer MoS₂ by vertical electric field: A first-principles investigation. J. Phys. Chem. C, 2012, 116(40): 21556

[70] P. K. Nayak , Y. Horbatenko , S. Ahn , G. Kim , J. U. Lee , K. Y. Ma , A. R. Jang , H. Lim , D. Kim , S. Ryu , H. Cheong , N. Park , H. S. Shin . Probing evolution of twist-angle-dependent interlayer excitons in MoSe₂/WSe₂ van der Waals heterostructures. ACS Nano, 2017, 11(4): 4041

[71] W. Choi , I. Akhtar , M. A. Rehman , M. Kim , D. Kang , J. Jung , Y. Myung , J. Kim , H. Cheong , Y. Seo . Twist-angle-dependent optoelectronics in a few-layer transition-metal dichalcogenide heterostructure. ACS Appl. Mater. Interfaces, 2019, 11(2): 2470

[72] J. Kunstmann , F. Mooshammer , P. Nagler , A. Chaves , F. Stein , N. Paradiso , G. Plechinger , C. Strunk , C. Schüller , G. Seifert , D. R. Reichman , T. Korn . Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures. Nat. Phys., 2018, 14(8): 801

[73] G. Scuri , T. I. Andersen , Y. Zhou , D. S. Wild , J. Sung , R. J. Gelly , D. Bérubé , H. Heo , L. Shao , A. Y. Joe , A. M. Mier Valdivia , T. Taniguchi , K. Watanabe , M. Lončar , P. Kim , M. D. Lukin , H. Park . Electrically tunable valley dynamics in twisted WSe₂/WSe₂ bilayers. Phys. Rev. Lett., 2020, 124(21): 217403

[74] S. Zheng , L. Sun , X. Zhou , F. Liu , Z. Liu , Z. Shen , H. J. Fan . Coupling and interlayer exciton in twist-stacked WS₂ bilayers. Adv. Opt. Mater., 2015, 3(11): 1600

[75] A. A. Puretzky , L. Liang , X. Li , K. Xiao , B. G. Sumpter , V. Meunier , D. B. Geohegan . Twisted MoSe₂ bilayers with variable local stacking and interlayer coupling revealed by low-frequency Raman spectroscopy. ACS Nano, 2016, 10(2): 2736

[76] J. Zhang , J. Wang , P. Chen , Y. Sun , S. Wu , Z. Jia , X. Lu , H. Yu , W. Chen , J. Zhu , G. Xie , R. Yang , D. Shi , X. Xu , J. Xiang , K. Liu , G. Zhang . Observation of strong interlayer coupling in MoS₂/WS₂ heterostructures. Adv. Mater., 2016, 28(10): 1950

[77] T. Jiang , H. Liu , D. Huang , S. Zhang , Y. Li , X. Gong , Y. R. Shen , W. T. Liu , S. Wu . Valley and band structure engineering of folded MoS₂ bilayers. Nat. Nanotechnol., 2014, 9(10): 825

[78] K. Wang , B. Huang , M. Tian , F. Ceballos , M. W. Lin , M. Mahjouri-Samani , A. Boulesbaa , A. A. Puretzky , C. M. Rouleau , M. Yoon , H. Zhao , K. Xiao , G. Duscher , D. B. Geohegan . Interlayer coupling in twisted WSe₂/WS₂ bilayer heterostructures revealed by optical spectroscopy. ACS Nano, 2016, 10(7): 6612

[79] Z. Ji , H. Hong , J. Zhang , Q. Zhang , W. Huang , T. Cao , R. Qiao , C. Liu , J. Liang , C. Jin , L. Jiao , K. Shi , S. Meng , K. Liu . Robust stacking-independent ultrafast charge transfer in MoS₂/WS₂ bilayers. ACS Nano, 2017, 11(12): 12020

[80] L. Wu , C. Cong , J. Shang , W. Yang , Y. Chen , J. Zhou , W. Ai , Y. Wang , S. Feng , H. Zhang , Z. Liu , T. Yu . Raman scattering investigation of twisted WS₂/MoS₂ heterostructures: Interlayer mechanical coupling versus charge transfer. Nano Res., 2021, 14(7): 2215

[81] X. Wang , K. Yasuda , Y. Zhang , S. Liu , K. Watanabe , T. Taniguchi , J. Hone , L. Fu , P. Jarillo-Herrero . Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides. Nat. Nanotechnol., 2022, 17: 367

[82] J. Sung , Y. Zhou , G. Scuri , V. Zólyomi , T. I. Andersen , H. Yoo , D. S. Wild , A. Y. Joe , R. J. Gelly , H. Heo , S. J. Magorrian , D. Bérubé , A. M. M. Valdivia , T. Taniguchi , K. Watanabe , M. D. Lukin , P. Kim , V. I. Fal’ko , H. Park . Broken mirror symmetry in excitonic response of reconstructed domains in twisted MoSe₂/MoSe₂ bilayers. Nat. Nanotechnol., 2020, 15(9): 750

[83] J. Michl , C. C. Palekar , S. A. Tarasenko , F. Lohof , C. Gies , M. von Helversen , R. Sailus , S. Tongay , T. Taniguchi , K. Watanabe , T. Heindel , B. Rosa , M. Rödel , T. Shubina , S. Höfling , S. Reitzenstein , C. Anton-Solanas , C. Schneider . Intrinsic circularly polarized exciton emission in a twisted van der Waals heterostructure. Phys. Rev. B, 2022, 105(24): L241406

[84] J. Shi , Y. Li , Z. Zhang , W. Feng , Q. Wang , S. Ren , J. Zhang , W. Du , X. Wu , X. Sui , Y. Mi , R. Wang , Y. Sun , L. Zhang , X. Qiu , J. Lu , C. Shen , Y. Zhang , Q. Zhang , X. Liu . Twisted-angle-dependent optical behaviors of intralayer excitons and trions in WS₂/WSe₂ heterostructure. ACS Photonics, 2019, 6(12): 3082

[85] J. E. Zimmermann , M. Axt , F. Mooshammer , P. Nagler , C. Schüller , T. Korn , U. Höfer , G. Mette . Ultrafast charge-transfer dynamics in twisted MoS₂/WSe₂ heterostructures. ACS Nano, 2021, 15(9): 14725

[86] D. Luo , J. Tang , X. Shen , F. Ji , J. Yang , S. Weathersby , M. E. Kozina , Z. Chen , J. Xiao , Y. Ye , T. Cao , G. Zhang , X. Wang , A. M. Lindenberg . Twist-angle-dependent ultrafast charge transfer in MoS₂-graphene van der Waals heterostructures. Nano Lett., 2021, 21(19): 8051

[87] A. M. van der Zande , J. Kunstmann , A. Chernikov , D. A. Chenet , Y. You , X. Zhang , P. Y. Huang , T. C. Berkelbach , L. Wang , F. Zhang , M. S. Hybertsen , D. A. Muller , D. R. Reichman , T. F. Heinz , J. C. Hone . Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist. Nano Lett., 2014, 14(7): 3869

[88] S. Huang , X. Ling , L. Liang , J. Kong , H. Terrones , V. Meunier , M. S. Dresselhaus . Probing the interlayer coupling of twisted bilayer MoS₂ using photoluminescence spectroscopy. Nano Lett., 2014, 14(10): 5500

[89] Y. Wang , Z. Su , W. Wu , S. Nie , N. Xie , H. Gong , Y. Guo , J. Hwan Lee , S. Xing , X. Lu , H. Wang , X. Lu , K. McCarty , S. Pei , F. Robles-Hernandez , V. G. Hadjiev , J. Bao . Resonance Raman spectroscopy of G-line and folded phonons in twisted bilayer graphene with large rotation angles. Appl. Phys. Lett., 2013, 103(12): 123101

[90] C. C. Lu , Y. C. Lin , Z. Liu , C. H. Yeh , K. Suenaga , P. W. Chiu . Twisting bilayer graphene superlattices. ACS Nano, 2013, 7(3): 2587

[91] W. Ren , J. Chen , G. Zhang . Phonon physics in twisted two-dimensional materials. Appl. Phys. Lett., 2022, 121: 140501

[92] Y. Dai , P. Qi , G. Tao , G. Yao , B. Shi , Z. Liu , Z. Liu , X. He , P. Peng , Z. Dang , L. Zheng , T. Zhang , Y. Gong , Y. Guan , K. Liu , Z. Fang . Phonon-assisted upconversion in twisted two-dimensional semiconductors. Light Sci. Appl., 2023, 12(1): 6

[93] D. Huang , J. Choi , C. K. Shih , X. Li . Excitons in semiconductor moiré superlattices. Nat. Nanotechnol., 2022, 17(3): 227

[94] D. Xiao , G. B. Liu , W. Feng , X. Xu , W. Yao . Coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides. Phys. Rev. Lett., 2012, 108(19): 196802

[95] K. F. Mak , K. He , J. Shan , T. F. Heinz . Control of valley polarization in monolayer MoS₂ by optical helicity. Nat. Nanotechnol., 2012, 7: 494

[96] G. Sallen , L. Bouet , X. Marie , G. Wang , C. R. Zhu , W. P. Han , Y. Lu , P. H. Tan , T. Amand , B. L. Liu , B. Urbaszek . Robust optical emission polarization in MoS₂ monolayers through selective valley excitation. Phys. Rev. B., 2012, 86(8): 081301

[97] K. F. Mak , K. L. McGill , J. Park , P. L. McEuen . The valley Hall effect in MoS₂ transistors. Science, 2014, 344: 1489

[98] J. Lee , Z. Wang , H. Xie , K. F. Mak , J. Shan . Valley magnetoelectricity in single-layer MoS₂. Nat. Mater., 2017, 16: 887

[99] A. Srivastava , M. Sidler , A. V. Allain , D. S. Lembke , A. Kis , A. Imamoğlu . Valley Zeeman effect in elementary optical excitations of monolayer WSe₂. Nat. Phys., 2015, 11(2): 141

[100] Z. Gong , G. B. Liu , H. Yu , D. Xiao , X. Cui , X. Xu , W. Yao . Magnetoelectric effects and valley- controlled spin quantum gates in transition metal dichalcogenide bilayers. Nat. Commun., 2013, 4(1): 2053

[101] A. Ebnonnasir , B. Narayanan , S. Kodambaka , C. V. Ciobanu . Tunable MoS₂ bandgap in MoS₂-graphene heterostructures. Appl. Phys. Lett., 2014, 105: 031603

[102] P. Rivera , J. R. Schaibley , A. M. Jones , J. S. Ross , S. Wu , G. Aivazian , P. Klement , K. Seyler , G. Clark , N. J. Ghimire , J. Yan , D. G. Mandrus , W. Yao , X. Xu . Observation of long-lived interlayer excitons in monolayer MoSe₂–WSe₂ heterostructures. Nat. Commun., 2015, 6(1): 6242

[103] P. Rivera , H. Yu , K. L. Seyler , N. P. Wilson , W. Yao , X. Xu . Interlayer valley excitons in heterobilayers of transition metal dichalcogenides. Nat. Nanotechnol., 2018, 13(11): 1004

[104] J. R. Schaibley , H. Yu , G. Clark , P. Rivera , J. S. Ross , K. L. Seyler , W. Yao , X. Xu . Valleytronics in 2D materials. Nat. Rev. Mater., 2016, 1(11): 16055

[105] K. F. Mak , D. Xiao , J. Shan . Light–valley interactions in 2D semiconductors. Nat. Photonics, 2018, 12(8): 451

[106] A. Singh , K. Tran , M. Kolarczik , J. Seifert , Y. Wang , K. Hao , D. Pleskot , N. M. Gabor , S. Helmrich , N. Owschimikow , U. Woggon , X. Li . Long-lived valley polarization of intravalley trions in monolayer WSe₂. Phys. Rev. Lett., 2016, 117(25): 257402

[107] M. Ge , H. Wang , J. Wu , C. Si , J. Zhang , S. Zhang . Enhanced valley splitting of WSe₂ in twisted van der Waals WSe₂/CrI₃ heterostructures. npj Comput. Mater., 2022, 8: 32

[108] W. Hu , J. Yang . Two-dimensional van der Waals heterojunctions for functional materials and devices. J. Mater. Chem. C, 2017, 5(47): 12289

[109] V. O. Özçelik , J. G. Azadani , C. Yang , S. J. Koester , T. Low . Band alignment of two-dimensional semiconductors for designing heterostructures with momentum space matching. Phys. Rev. B, 2016, 94(3): 035125

[110] X. Hong , J. Kim , S. F. Shi , Y. Zhang , C. Jin , Y. Sun , S. Tongay , J. Wu , Y. Zhang , F. Wang . Ultrafast charge transfer in atomically thin MoS₂/WS₂ heterostructures. Nat. Nanotechnol., 2014, 9(9): 682

[111] H. Zhu , J. Wang , Z. Gong , Y. D. Kim , J. Hone , X. Y. Zhu . Interfacial charge transfer circumventing momentum mismatch at two-dimensional van der Waals heterojunctions. Nano Lett., 2017, 17(6): 3591

[112] A. F. Rigosi , H. M. Hill , Y. Li , A. Chernikov , T. F. Heinz . Probing interlayer interactions in transition metal dichalcogenide heterostructures by optical spectroscopy: MoS₂/WS₂ and MoSe₂/WSe₂. Nano Lett., 2015, 15(8): 5033

[113] Y. Yu , S. Hu , L. Su , L. Huang , Y. Liu , Z. Jin , A. A. Purezky , D. B. Geohegan , K. W. Kim , Y. Zhang , L. Cao . Equally efficient interlayer exciton relaxation and improved absorption in epitaxial and nonepitaxial MoS₂/WS₂ heterostructures. Nano Lett., 2015, 15(1): 486

[114] F. Liu , Q. Li , X. Y. Zhu . Direct determination of momentum-resolved electron transfer in the photoexcited van der Waals heterobilayer WS₂/MoS₂. Phys. Rev. B, 2020, 101(20): 201405

[115] P. Merkl , F. Mooshammer , P. Steinleitner , A. Girnghuber , K. Q. Lin , P. Nagler , J. Holler , C. Schüller , J. M. Lupton , T. Korn , S. Ovesen , S. Brem , E. Malic , R. Huber . Ultrafast transition between exciton phases in van der Waals heterostructures. Nat. Mater., 2019, 18(7): 691

[116] H. Wang , J. Bang , Y. Sun , L. Liang , D. West , V. Meunier , S. Zhang . The role of collective motion in the ultrafast charge transfer in van der Waals heterostructures. Nat. Commun., 2016, 7(1): 11504

[117] S. K. Behura , A. Miranda , S. Nayak , K. Johnson , P. Das , N. R. Pradhan . Moiré physics in twisted van der Waals heterostructures of 2D materials. Emergent Mater., 2021, 4(4): 813

[118] F. Wu , T. Lovorn , E. Tutuc , A. H. MacDonald . Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett., 2018, 121(2): 026402

[119] H. Wang , S. Ma , S. Zhang , D. Lei . Intrinsic superflat bands in general twisted bilayer systems. Light Sci. Appl., 2022, 11(1): 159

[120] Z. Ma , S. Li , M. M. Xiao , Y. W. Zheng , M. Lu , H. Liu , J. H. Gao , X. C. Xie . Moiré flat bands of twisted few-layer graphite. Front. Phys., 2023, 18(1): 13307

[121] N. Zhang , A. Surrente , M. Baranowski , D. K. Maude , P. Gant , A. Castellanos-Gomez , P. Plochocka . Moiré intralayer excitons in a MoSe₂/MoS₂ heterostructure. Nano Lett., 2018, 18(12): 7651

[122] B. Remez , N. R. Cooper . Leaky exciton condensates in transition metal dichalcogenide moiré bilayers. Phys. Rev. Mater., 2022, 4: L022042

[123] R. Bistritzer , A. H. MacDonald . Moiré bands in twisted double-layer graphene. Proc. Natl. Acad. Sci. USA, 2011, 108(30): 12233

[124] G. Chen , L. Jiang , S. Wu , B. Lyu , H. Li , B. L. Chittari , K. Watanabe , T. Taniguchi , Z. Shi , J. Jung , Y. Zhang , F. Wang . Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys., 2019, 15(3): 237

[125] G. Chen , A. L. Sharpe , P. Gallagher , I. T. Rosen , E. J. Fox , L. Jiang , B. Lyu , H. Li , K. Watanabe , T. Taniguchi , J. Jung , Z. Shi , D. Goldhaber-Gordon , Y. Zhang , F. Wang . Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature, 2019, 572(7768): 215

[126] G. Chen , A. L. Sharpe , E. J. Fox , Y. H. Zhang , S. Wang , L. Jiang , B. Lyu , H. Li , K. Watanabe , T. Taniguchi , Z. Shi , T. Senthil , D. Goldhaber-Gordon , Y. Zhang , F. Wang . Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature, 2020, 579(7797): 56

[127] R. Su , M. Kuiri , K. Watanabe , T. Taniguchi , J. Folk . Superconductivity in twisted double bilayer graphene stabilized by WSe₂. Nat. Mater., 2023, 22(11): 1332

[128] L. Wang , E. M. Shih , A. Ghiotto , L. Xian , D. A. Rhodes , C. Tan , M. Claassen , D. M. Kennes , Y. Bai , B. Kim , K. Watanabe , T. Taniguchi , X. Zhu , J. Hone , A. Rubio , A. N. Pasupathy , C. R. Dean . Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater., 2020, 19(8): 861

[129] Y. Tang , L. Li , T. Li , Y. Xu , S. Liu , K. Barmak , K. Watanabe , T. Taniguchi , A. H. MacDonald , J. Shan , K. F. Mak . Simulation of Hubbard model physics in WSe₂/WS₂ moiré superlattices. Nature, 2020, 579(7799): 353

[130] Y. Xu , S. Liu , D. A. Rhodes , K. Watanabe , T. Taniguchi , J. Hone , V. Elser , K. F. Mak , J. Shan . Correlated insulating states at fractional fillings of moiré superlattices. Nature, 2020, 587(7833): 214

[131] Y. Shimazaki , I. Schwartz , K. Watanabe , T. Taniguchi , M. Kroner , A. Imamoğlu . Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature, 2020, 580(7804): 472

[132] Y. Liu , C. Zeng , J. Yu , J. Zhong , B. Li , Z. Zhang , Z. Liu , Z. M. Wang , A. Pan , X. Duan . Moiré superlattices and related moiré excitons in twisted van der Waals heterostructures. Chem. Soc. Rev., 2021, 50(11): 6401

[133] M. H. Naik , S. Kundu , I. Maity , M. Jain . Origin and evolution of ultraflat bands in twisted bilayer transition metal dichalcogenides: Realization of triangular quantum dots. Phys. Rev. B, 2020, 102(7): 075413

[134] L. Yuan , B. Zheng , J. Kunstmann , T. Brumme , A. B. Kuc , C. Ma , S. Deng , D. Blach , A. Pan , L. Huang . Twist-angle-dependent interlayer exciton diffusion in WS₂–WSe₂ heterobilayers. Nat. Mater., 2020, 19(6): 617

[135] L. Zhang , Z. Zhang , F. Wu , D. Wang , R. Gogna , S. Hou , K. Watanabe , T. Taniguchi , K. Kulkarni , T. Kuo , S. R. Forrest , H. Deng . Twist-angle dependence of moiré excitons in WS₂/MoSe₂ heterobilayers. Nat. Commun., 2020, 11(1): 5888

[136] Z. Li , X. Lu , D. F. Cordovilla Leon , Z. Lyu , H. Xie , J. Hou , Y. Lu , X. Guo , A. Kaczmarek , T. Taniguchi , K. Watanabe , L. Zhao , L. Yang , P. B. Deotare . Interlayer exciton transport in MoSe₂/WSe₂ heterostructures. ACS Nano, 2021, 15(1): 1539

[137] X. Zhao , J. Qiao , X. Zhou , H. Chen , J. Y. Tan , H. Yu , S. M. Chan , J. Li , H. Zhang , J. Zhou , J. Dan , Z. Liu , W. Zhou , Z. Liu , B. Peng , L. Deng , S. J. Pennycook , S. Y. Quek , K. P. Loh . Strong moiré excitons in high-angle twisted transition metal dichalcogenide homobilayers with robust commensuration. Nano Lett., 2022, 22(1): 203

[138] R. Ribeiro-Palau , C. Zhang , K. Watanabe , T. Taniguchi , J. Hone , C. R. Dean . Twistable electronics with dynamically rotatable heterostructures. Science, 2018, 361(6403): 690

[139] A. Weston , Y. Zou , V. Enaldiev , A. Summerfield , N. Clark , V. Zólyomi , A. Graham , C. Yelgel , S. Magorrian , M. Zhou , J. Zultak , D. Hopkinson , A. Barinov , T. H. Bointon , A. Kretinin , N. R. Wilson , P. H. Beton , V. I. Fal’ko , S. J. Haigh , R. Gorbachev . Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol., 2020, 15(7): 592

[140] M. R. Rosenberger , H. J. Chuang , M. Phillips , V. P. Oleshko , K. M. McCreary , S. V. Sivaram , C. S. Hellberg , B. T. Jonker . Twist angle-dependent atomic reconstruction and moiré patterns in transition metal dichalcogenide heterostructures. ACS Nano, 2020, 14(4): 4550

[141] T. I. Andersen , G. Scuri , A. Sushko , K. De Greve , J. Sung , Y. Zhou , D. S. Wild , R. J. Gelly , H. Heo , D. Bérubé , A. Y. Joe , L. A. Jauregui , K. Watanabe , T. Taniguchi , P. Kim , H. Park , M. D. Lukin . Excitons in a reconstructed moiré potential in twisted WSe₂/WSe₂ homobilayers. Nat. Mater., 2021, 20(4): 480

[142] J. Quan , L. Linhart , M. L. Lin , D. Lee , J. Zhu , C. Y. Wang , W. T. Hsu , J. Choi , J. Embley , C. Young , T. Taniguchi , K. Watanabe , C. K. Shih , K. Lai , A. H. MacDonald , P. H. Tan , F. Libisch , X. Li . Phonon renormalization in reconstructed MoS₂ moiré superlattices. Nat. Mater., 2021, 20(8): 1100

[143] B. H. Lin , Y. C. Chao , I. T. Hsieh , C. P. Chuu , C. J. Lee , F. H. Chu , L. S. Lu , W. T. Hsu , C. W. Pao , C. K. Shih , J. J. Su , W. H. Chang . Remarkably deep moiré potential for intralayer excitons in MoSe₂/MoS₂ twisted heterobilayers. Nano Lett., 2023, 23(4): 1306

[144] S. Li , H. Zheng , J. Ding , B. Wu , J. He , Z. Liu , Y. Liu . Dynamic control of moiré potential in twisted WS₂‒WSe₂ heterostructures. Nano Res., 2022, 15(8): 7688

[145] B. Wu , H. Zheng , S. Li , J. Ding , Y. Zeng , Z. Liu , Y. Liu . Observation of moiré excitons in the twisted WS₂/WS₂ homostructure. Nanoscale, 2022, 14(34): 12447

[146] E. M. Alexeev , D. A. Ruiz-Tijerina , M. Danovich , M. J. Hamer , D. J. Terry , P. K. Nayak , S. Ahn , S. Pak , J. Lee , J. I. Sohn , M. R. Molas , M. Koperski , K. Watanabe , T. Taniguchi , K. S. Novoselov , R. V. Gorbachev , H. S. Shin , V. I. Fal’ko , A. I. Tartakovskii . Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature, 2019, 567(7746): 81

[147] K. Tran , G. Moody , F. Wu , X. Lu , J. Choi , K. Kim , A. Rai , D. A. Sanchez , J. Quan , A. Singh , J. Embley , A. Zepeda , M. Campbell , T. Autry , T. Taniguchi , K. Watanabe , N. Lu , S. K. Banerjee , K. L. Silverman , S. Kim , E. Tutuc , L. Yang , A. H. MacDonald , X. Li . Evidence for moiré excitons in van der Waals heterostructures. Nature, 2019, 567(7746): 71

[148] Y. Tang , J. Gu , S. Liu , K. Watanabe , T. Taniguchi , J. Hone , K. F. Mak , J. Shan . Tuning layer- hybridized moiré excitons by the quantum-confined Stark effect. Nat. Nanotechnol., 2021, 16(1): 52

[149] S. Brem , K. Q. Lin , R. Gillen , J. M. Bauer , J. Maultzsch , J. M. Lupton , E. Malic . Hybridized intervalley moiré excitons and flat bands in twisted WSe₂ bilayers. Nanoscale, 2020, 12: 11088

[150] D. A. Ruiz-Tijerina , V. I. Fal’ko . Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides. Phys. Rev. B, 2019, 99(12): 125424

[151] C. Jin , E. C. Regan , A. Yan , M. Iqbal Bakti Utama , D. Wang , S. Zhao , Y. Qin , S. Yang , Z. Zheng , S. Shi , K. Watanabe , T. Taniguchi , S. Tongay , A. Zettl , F. Wang . Observation of moiré excitons in WSe₂/WS₂ heterostructure superlattices. Nature, 2019, 567(7746): 76

[152] H.ZhengB.WuS.LiJ.HeK.ChenZ.LiuY.Liu. Evidence for interlayer coupling and moiré excitons in twisted WS₂/WS₂ homostructure superlattices. Nano Res. 16(2), 3429 (2023)

[153] O. Karni , E. Barré , V. Pareek , J. D. Georgaras , M. K. L. Man , C. Sahoo , D. R. Bacon , X. Zhu , H. B. Ribeiro , A. L. O’Beirne , J. Hu , A. Al-Mahboob , M. M. M. Abdelrasoul , N. S. Chan , A. Karmakar , A. J. Winchester , B. Kim , K. Watanabe , T. Taniguchi , K. Barmak , J. Madéo , F. H. da Jornada , T. F. Heinz , K. M. Dani . Structure of the moiré exciton captured by imaging its electron and hole. Nature, 2022, 603(7900): 247

[154] M. Dandu , G. Gupta , P. Dasika , K. Watanabe , T. Taniguchi , K. Majumdar . Electrically tunable localized versus delocalized intralayer moiré excitons and trions in a twisted MoS₂ bilayer. ACS Nano, 2022, 16(6): 8983

[155] S. Brem , E. Malic . Bosonic delocalization of dipolar moiré excitons. Nano Lett., 2023, 23(10): 4627

[156] C. Lagoin , F. Dubin . Key role of the moiré potential for the quasicondensation of interlayer excitons in van der Waals heterostructures. Phys. Rev. B, 2021, 103(4): L041406

[157] N. Götting , F. Lohof , C. Gies . Moiré-Bose-Hubbard model for interlayer excitons in twisted transition metal dichalcogenide heterostructures. Phys. Rev. B, 2022, 105(16): 165419

[158] J. Choi , W. T. Hsu , L. S. Lu , L. Sun , H. Y. Cheng , M. H. Lee , J. Quan , K. Tran , C. Y. Wang , M. Staab , K. Jones , T. Taniguchi , K. Watanabe , M. W. Chu , S. Gwo , S. Kim , C. K. Shih , X. Li , W. H. Chang . Moiré potential impedes interlayer exciton diffusion in van der Waals heterostructures. Sci. Adv., 2020, 6(39): eaba8866

[159] F. Mahdikhanysarvejahany , D. N. Shanks , C. Muccianti , B. H. Badada , I. Idi , A. Alfrey , S. Raglow , M. R. Koehler , D. G. Mandrus , T. Taniguchi , K. Watanabe , O. L. A. Monti , H. Yu , B. J. LeRoy , J. R. Schaibley . Temperature dependent moiré trapping of interlayer excitons in MoSe₂‒WSe₂ heterostructures. npj 2D Mater. Appl., 2021, 5: 67

[160] J. Choi , M. Florian , A. Steinhoff , D. Erben , K. Tran , D. S. Kim , L. Sun , J. Quan , R. Claassen , S. Majumder , J. A. Hollingsworth , T. Taniguchi , K. Watanabe , K. Ueno , A. Singh , G. Moody , F. Jahnke , X. Li . Twist angle-dependent interlayer exciton lifetimes in van der Waals heterostructures. Phys. Rev. Lett., 2021, 126(4): 047401

[161] H. Cai , A. Rasmita , Q. Tan , J. M. Lai , R. He , X. Cai , Y. Zhao , D. Chen , N. Wang , Z. Mu , Z. Huang , Z. Zhang , J. J. H. Eng , Y. Liu , Y. She , N. Pan , Y. Miao , X. Wang , X. Liu , J. Zhang , W. Gao . Interlayer donor-acceptor pair excitons in MoSe₂/WSe₂ moiré heterobilayer. Nat. Commun., 2023, 14(1): 5766

[162] H. Zheng , B. Wu , C. T. Wang , S. Li , J. He , Z. Liu , J. T. Wang , J. Duan , Y. Liu . Exploring the regulatory effect of stacked layers on moiré excitons in twisted WSe₂/WSe₂/WSe₂ homotrilayer. Nano Res., 2023, 16(7): 10573

[163] H. Zheng , B. Wu , S. Li , J. Ding , J. He , Z. Liu , C. T. Wang , J. T. Wang , A. Pan , Y. Liu . Localization- enhanced moiré exciton in twisted transition metal dichalcogenide heterotrilayer superlattices. Light Sci. Appl., 2023, 12(1): 117

[164] L. Zhang , R. Gogna , G. W. Burg , J. Horng , E. Paik , Y. H. Chou , K. Kim , E. Tutuc , H. Deng . Highly valley-polarized singlet and triplet interlayer excitons in van der Waals heterostructure. Phys. Rev. B, 2019, 100(4): 041402

[165] M. Brotons-Gisbert , H. Baek , A. Molina-Sánchez , A. Campbell , E. Scerri , D. White , K. Watanabe , T. Taniguchi , C. Bonato , B. D. Gerardot . Spin-layer locking of interlayer excitons trapped in moiré potentials. Nat. Mater., 2020, 19(6): 630

[166] X. Wang , J. Zhu , K. L. Seyler , P. Rivera , H. Zheng , Y. Wang , M. He , T. Taniguchi , K. Watanabe , J. Yan , D. G. Mandrus , D. R. Gamelin , W. Yao , X. Xu . Moiré trions in MoSe₂/WSe₂ heterobilayers. Nat. Nanotechnol., 2021, 16(11): 1208

[167] Y. Bai , L. Zhou , J. Wang , W. Wu , L. J. McGilly , D. Halbertal , C. F. B. Lo , F. Liu , J. Ardelean , P. Rivera , N. R. Finney , X. C. Yang , D. N. Basov , W. Yao , X. Xu , J. Hone , A. N. Pasupathy , X. Y. Zhu . Excitons in strain-induced one-dimensional moiré potentials at transition metal dichalcogenide heterojunctions. Nat. Mater., 2020, 19(10): 1068

[168] W. Zhao , E. C. Regan , D. Wang , C. Jin , S. Hsieh , Z. Wang , J. Wang , Z. Wang , K. Yumigeta , M. Blei , K. Watanabe , T. Taniguchi , S. Tongay , N. Y. Yao , F. Wang . Dynamic tuning of moiré excitons in a WSe₂/WS₂ heterostructure via mechanical deformation. Nano Lett., 2021, 21(20): 8910

[169] M. Anđelković , S. P. Milovanović , L. Covaci , F. M. Peeters . Double moiré with a twist: Supermoiré in encapsulated graphene. Nano Lett., 2020, 20(2): 979

[170] F. Tagarelli , E. Lopriore , D. Erkensten , R. Perea-Causín , S. Brem , J. Hagel , Z. Sun , G. Pasquale , K. Watanabe , T. Taniguchi , E. Malic , A. Kis . Electrical control of hybrid exciton transport in a van der Waals heterostructure. Nat. Photonics, 2023, 17(7): 615

[171] Z. Lian , D. Chen , Y. Meng , X. Chen , Y. Su , R. Banerjee , T. Taniguchi , K. Watanabe , S. Tongay , C. Zhang , Y. T. Cui , S. F. Shi . Exciton superposition across moiré states in a semiconducting moiré superlattice. Nat. Commun., 2023, 14(1): 5042

[172] L. Li , M. Wu . Binary compound bilayer and multilayer with vertical polarizations: Two-dimensional ferroelectrics, multiferroics, and nanogenerators. ACS Nano, 2017, 11(6): 6382

[173] F. Li , J. Fu , M. Xue , Y. Li , H. Zeng , E. Kan , T. Hu , Y. Wan . Room-temperature vertical ferroelectricity in rhenium diselenide induced by interlayer sliding. Front. Phys., 2023, 18(5): 53305

[174] Y. Wang , C. Cong , W. Yang , J. Shang , N. Peimyoo , Y. Chen , J. Kang , J. Wang , W. Huang , T. Yu . Strain-induced direct–indirect bandgap transition and phonon modulation in monolayer WS₂. Nano Res., 2015, 8(8): 2562

[175] X. He , H. Li , Z. Zhu , Z. Dai , Y. Yang , P. Yang , Q. Zhang , P. Li , U. Schwingenschlogl , X. Zhang . Strain engineering in monolayer WS₂, MoS₂, and the WS₂/MoS₂ heterostructure. Appl. Phys. Lett., 2016, 109(17): 173105

[176] J. Feng , X. Qian , C. W. Huang , J. Li . Strain-engineered artificial atom as a broad-spectrum solar energy funnel. Nat. Photonics, 2012, 6(12): 866

[177] A. Castellanos-Gomez , R. Roldán , E. Cappelluti , M. Buscema , F. Guinea , H. S. J. van der Zant , G. A. Steele . Local strain engineering in atomically thin MoS₂. Nano Lett., 2013, 13(11): 5361

[178] W. Wang , X. Ma . Strain-induced trapping of indirect excitons in MoSe₂/WSe₂ heterostructures. ACS Photon., 2020, 7: 2460

[179] X. Liu , J. Zeng . Gap solitons in parity–time symmetric moiré optical lattices. Photon. Res., 2023, 11(2): 196

[180] Y. Liu , C. Ouyang , Q. Xu , X. Su , Q. Yang , J. Ma , Y. Li , Z. Tian , J. Gu , L. Liu , J. Han , Y. Shi , W. Zhang . Moiré-driven electromagnetic responses and magic angles in a sandwiched hyperbolic metasurface. Photon. Res., 2022, 10(9): 2056

[181] E. C. Regan , D. Wang , C. Jin , M. I. Bakti Utama , B. Gao , X. Wei , S. Zhao , W. Zhao , Z. Zhang , K. Yumigeta , M. Blei , J. D. Carlström , K. Watanabe , T. Taniguchi , S. Tongay , M. Crommie , A. Zettl , F. Wang . Mott and generalized Wigner crystal states in WSe₂/WS₂ moiré superlattices. Nature, 2020, 579(7799): 359

[182] J. G. Bednorz , K. A. Müller . Possible high T_c superconductivity in the Ba−La−Cu−O system. Zeitschrift für Physik B Condensed Matter, 1986, 64: 189

[183] T. Li , S. Jiang , L. Li , Y. Zhang , K. Kang , J. Zhu , K. Watanabe , T. Taniguchi , D. Chowdhury , L. Fu , J. Shan , K. F. Mak . Continuous Mott transition in semiconductor moiré superlattices. Nature, 2021, 597(7876): 350

[184] S. Miao , T. Wang , X. Huang , D. Chen , Z. Lian , C. Wang , M. Blei , T. Taniguchi , K. Watanabe , S. Tongay , Z. Wang , D. Xiao , Y. T. Cui , S. F. Shi . Strong interaction between interlayer excitons and correlated electrons in WSe₂/WS₂ moiré superlattice. Nat. Commun., 2021, 12(1): 3608

[185] D. Chen , Z. Lian , X. Huang , Y. Su , M. Rashetnia , L. Yan , M. Blei , T. Taniguchi , K. Watanabe , S. Tongay , Z. Wang , C. Zhang , Y. T. Cui , S. F. Shi . Tuning moiré excitons and correlated electronic states through layer degree of freedom. Nat. Commun., 2022, 13(1): 4810

[186] M. H. Naik , E. C. Regan , Z. Zhang , Y. H. Chan , Z. Li , D. Wang , Y. Yoon , C. S. Ong , W. Zhao , S. Zhao , M. I. B. Utama , B. Gao , X. Wei , M. Sayyad , K. Yumigeta , K. Watanabe , T. Taniguchi , S. Tongay , F. H. da Jornada , F. Wang , S. G. Louie . Intralayer charge-transfer moiré excitons in van der Waals superlattices. Nature, 2022, 609(7925): 52

[187] X. Wang , C. Xiao , H. Park , J. Zhu , C. Wang , T. Taniguchi , K. Watanabe , J. Yan , D. Xiao , D. R. Gamelin , W. Yao , X. Xu . Light-induced ferromagnetism in moiré superlattices. Nature, 2022, 604(7906): 468

[188] Z. Zhang , E. C. Regan , D. Wang , W. Zhao , S. Wang , M. Sayyad , K. Yumigeta , K. Watanabe , T. Taniguchi , S. Tongay , M. Crommie , A. Zettl , M. P. Zaletel , F. Wang . Correlated interlayer exciton insulator in heterostructures of monolayer WSe₂ and moiré WS₂/WSe₂. Nat. Phys., 2022, 18(10): 1214

[189] G. Rohringer , H. Hafermann , A. Toschi , A. A. Katanin , A. E. Antipov , M. I. Katsnelson , A. I. Lichtenstein , A. N. Rubtsov , K. Held . Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. Rev. Mod. Phys., 2018, 90(2): 025003

[190] J. Quintanilla , C. A. Hooley . The strong-correlations puzzle. Physics World, 2009, 22(06): 32

[191] A. Ghiotto , E. M. Shih , G. S. S. G. Pereira , D. A. Rhodes , B. Kim , J. Zang , A. J. Millis , K. Watanabe , T. Taniguchi , J. C. Hone , L. Wang , C. R. Dean , A. N. Pasupathy . Quantum criticality in twisted transition metal dichalcogenides. Nature, 2021, 597(7876): 345

[192] A. Szasz , J. Motruk , M. P. Zaletel , J. E. Moore . Chiral spin liquid phase of the triangular lattice Hubbard model: A density matrix renormalization group study. Phys. Rev. X, 2020, 10(2): 021042

[193] T. Senthil . Theory of a continuous Mott transition in two dimensions. Phys. Rev. B, 2008, 78(4): 045109

[194] M. Imada , A. Fujimori , Y. Tokura . Metal−insulator transitions. Rev. Mod. Phys., 1998, 70(4): 1039

[195] A. M. Jones , H. Yu , J. S. Ross , P. Klement , N. J. Ghimire , J. Yan , D. G. Mandrus , W. Yao , X. Xu . Spin–layer locking effects in optical orientation of exciton spin in bilayer WSe₂. Nat. Phys., 2014, 10(2): 130

[196] R. Xiong , J. Nie , S. Brantly , P. Hays , R. Sailus , K. Watanabe , T. Taniguchi , S. Tongay , C. Jin . Correlated insulator of excitons in WSe₂/WS₂ moiré superlattices. Science, 2023, 380(6647): 860

[197] S. Brem , C. Linderälv , P. Erhart , E. Malic . Tunable phases of moiré excitons in van der Waals heterostructures. Nano Lett., 2020, 20(12): 8534

[198] Z. Zhang , Y. Wang , K. Watanabe , T. Taniguchi , K. Ueno , E. Tutuc , B. J. LeRoy . Flat bands in twisted bilayer transition metal dichalcogenides. Nat. Phys., 2020, 16(11): 1093

[199] Q. Hu , Z. Zhan , H. Cui , Y. Zhang , F. Jin , X. Zhao , M. Zhang , Z. Wang , Q. Zhang , K. Watanabe , T. Taniguchi , X. Cao , W. M. Liu , F. Wu , S. Yuan , Y. Xu . Observation of Rydberg moiré excitons. Science, 2023, 380(6652): 1367

[200] Q. Tan , A. Rasmita , Z. Zhang , K. S. Novoselov , W. Gao . Signature of cascade transitions between interlayer excitons in a moiré superlattice. Phys. Rev. Lett., 2022, 129(24): 247401

[201] D. Halbertal , N. R. Finney , S. S. Sunku , A. Kerelsky , C. Rubio-Verdú , S. Shabani , L. Xian , S. Carr , S. Chen , C. Zhang , L. Wang , D. Gonzalez-Acevedo , A. S. McLeod , D. Rhodes , K. Watanabe , T. Taniguchi , E. Kaxiras , C. R. Dean , J. C. Hone , A. N. Pasupathy , D. M. Kennes , A. Rubio , D. N. Basov . Moiré metrology of energy landscapes in van der Waals heterostructures. Nat. Commun., 2021, 12(1): 242

[202] H. Li , S. Li , E. C. Regan , D. Wang , W. Zhao , S. Kahn , K. Yumigeta , M. Blei , T. Taniguchi , K. Watanabe , S. Tongay , A. Zettl , M. F. Crommie , F. Wang . Imaging two-dimensional generalized Wigner crystals. Nature, 2021, 597(7878): 650

[203] C. H. Stansbury , M. I. B. Utama , C. G. Fatuzzo , E. C. Regan , D. Wang , Z. Xiang , M. Ding , K. Watanabe , T. Taniguchi , M. Blei , Y. Shen , S. Lorcy , A. Bostwick , C. Jozwiak , R. Koch , S. Tongay , J. Avila , E. Rotenberg , F. Wang , A. Lanzara . Visualizing electron localization of WS₂/WSe₂ moiré superlattices in momentum space. Sci. Adv., 2021, 7(37): eabf4387

[204] Y. Z. Chen , S. G. Yu , T. Jiang , X. J. Liu , X. B. Cheng , D. Huang . Optical two-dimensional coherent spectroscopy of excitons in transition-metal dichalcogenides. Front. Phys., 2024, 19: 23301

[205] V. V. Enaldiev , F. Ferreira , S. J. Magorrian , V. I. Fal’ko . Piezoelectric networks and ferroelectric domains in twistronic superlattices in WS₂/MoS₂ and WSe₂/MoSe₂ bilayers. 2D Mater., 2021, 8: 025030

[206] F. Xiao , K. Chen , Q. Tong . Magnetization textures in twisted bilayer CrX₃ (X = Br, I). Phys. Rev. Research., 2021, 3: 013027

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

The authors are grateful for financial support from the National Natural Science Foundation of China (Nos. 62105364 and 62075240); the Science and Technology Innovation Program of Hunan Province (No. 2021RC2068); and the Scientific Researches Foundation of National University of Defense Technology (No. ZK22-16).

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