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Chin. Phys. B, 2026, Vol. 35(2): 026801    DOI: 10.1088/1674-1056/ae23ac
SPECIAL TOPIC — Moiré physics in two-dimensional materials Prev   Next  

Three-dimensional flat bands and possible interlayer triplet pairing superconductivity in the alternating twisted NbSe2 moiré bulk

Shuang Liu(刘爽)1,2, Peng Chen(陈鹏)2,3,4, and Shihao Zhang(张世豪)1,†
1 School of Physics and Electronics, Hunan University, Changsha 410082, China;
2 Department of Physics, Guangdong Technion-Israel Institute of Technology, Shantou 515063, China;
3 Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel;
4 Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology, Shantou 515063, China
Abstract  Moiré superlattices hosting flat bands and correlated states have become a central focus in condensed matter research. Using first-principles calculations, we investigate three-dimensional flat bands in alternating twisted NbSe$_2$ moiré bulk structures, which exhibit stronger interlayer interactions than twisted bilayer configurations. Our results show that moiré bulks undergo spontaneous large-scale structural relaxation, leading to the formation of remarkably flat energy bands at twist angles $\leq 7.31^\circ$. The $k_z$-dependent dispersion of these flat bands across different moiré bulks highlights their intrinsic three-dimensional character. Moreover, the presence of out-of-plane mirror symmetry in these moiré bulk structures indicates potential interlayer triplet superconducting pairing mechanisms, distinct from those in twisted bilayer systems. This work opens new avenues for exploring three-dimensional flat bands in other moiré bulk systems.
Keywords:  moiré superlattices      first-principles calculations      electronic structure  
Received:  20 September 2025      Revised:  13 November 2025      Accepted manuscript online:  25 November 2025
PACS:  68.65.-k (Low-dimensional, mesoscopic, nanoscale and other related systems: structure and nonelectronic properties)  
  71.15.Mb (Density functional theory, local density approximation, gradient and other corrections)  
  73.21.Cd (Superlattices)  
  74.20.-z (Theories and models of superconducting state)  
Fund: This work was supported by the National Key Research and Development Program of China (Grant No. 2024YFA1410300), the National Natural Science Foundation of China (Grant No. 12304217), the Natural Science Foundation of Hunan Province (Grant No. 2025JJ60002), and the Fundamental Research Funds for the Central Universities of China (Grant No. 531119200247). We gratefully acknowledge HZWTECH for providing computational facilities.
Corresponding Authors:  Shihao Zhang     E-mail:  zhangshh@hnu.edu.cn

Cite this article: 

Shuang Liu(刘爽), Peng Chen(陈鹏), and Shihao Zhang(张世豪) Three-dimensional flat bands and possible interlayer triplet pairing superconductivity in the alternating twisted NbSe2 moiré bulk 2026 Chin. Phys. B 35 026801

[1] Aoki H 2020 J. Supercond. Nov. Magn. 33 2341
[2] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E and Jarillo-Herrero P 2018 Nature 556 43
[3] Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez- Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C and Jarillo-Herrero P 2018 Nature 556 80
[4] Tang Y, Li L, Li T, Xu Y, Liu S, Barmak K, Watanabe K, Taniguchi T, MacDonald A H, Shan J and Mak K F 2020 Nature 579 353
[5] Regan E C, Wang D, Jin C, Bakti Utama M I, Gao B, Wei X, Zhao S, Zhao W, Zhang Z, Yumigeta K, Blei M, Carlström J D, Watanabe K, Taniguchi T, Tongay S, Crommie M, Zettl A and Wang F 2020 Nature 579 359
[6] Wang L, Shih E M, Ghiotto A, Xian L, Rhodes D A, Tan C, Claassen M, Kennes D M, Bai Y, Kim B,Watanabe K, Taniguchi T, Zhu X, Hone J, Rubio A, Pasupathy A N and Dean C R 2020 Nat. Mater. 19 861
[7] Pasquale G, Sun Z, Č er?evicš K, Perea-Causin R, Tagarelli F, Watanabe K, Taniguchi T, Malic E, Yazyev O V and Kis A 2022 Nano Lett. 22 8883
[8] Wilhelm P, Lang T C and Läuchli A M 2021 Phys. Rev. B 103 205146
[9] Zhang Y, Tan Y W, Stormer H L and Kim P 2005 Nature 438 201
[10] Törmä P, Peotta S and Bernevig B A 2022 Nat. Rev. Phys. 4 528
[11] Lee J Y, Khalaf E, Liu S, Liu X, Hao Z, Kim P and Vishwanath A 2019 Nat. Commun. 10 5333
[12] Wu F, MacDonald A H and Martin I 2018 Phys. Rev. Lett. 121 257001
[13] Codecido E, Wang Q, Koester R, Che S, Tian H, Lv R, Tran S, Watanabe K, Taniguchi T, Zhang F, Bockrath M and Lau C N 2019 Sci. Adv. 5 eaaw9770
[14] Lu X, Stepanov P, YangW, Xie M, AamirMA, Das I, Urgell C,Watanabe K, Taniguchi T, Zhang G, Bachtold A, MacDonald A H and Efetov D K 2019 Nature 574 653
[15] Stepanov P, Das I, Lu X, Fahimniya A,Watanabe K, Taniguchi T, Koppens F H L, Lischner J, Levitov L and Efetov D K 2020 Nature 583 375
[16] Saito Y, Ge J,Watanabe K, Taniguchi T and Young A F 2020 Nat. Phys. 16 926
[17] Liu X, Wang Z, Watanabe K, Taniguchi T, Vafek O and Li J 2021 Science 371 1261
[18] Cao Y, Rodan-Legrain D, Park J M, Yuan N F, Watanabe K, Taniguchi T, Fernandes R M, Fu L and Jarillo-Herrero P 2021 Science 372 264
[19] Kang J and Vafek O 2019 Phys. Rev. Lett. 122 246401
[20] Seo K, Kotov V N and Uchoa B 2019 Phys. Rev. Lett. 122 246402
[21] Xie M and MacDonald A H 2020 Phys. Rev. Lett. 124 097601
[22] Bultinck N, Chatterjee S and Zaletel M P 2020 Phys. Rev. Lett. 124 166601
[23] Zhang S, Dai X and Liu J 2022 Phys. Rev. Lett. 128 026403
[24] Zhang S, Lu X and Liu J 2022 Phys. Rev. Lett. 128 247402
[25] Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A and Zaletel M P 2020 Phys. Rev. X 10 031034
[26] Liu J and Dai X 2021 Phys. Rev. B 103 035427
[27] Zhang Y, Jiang K, Wang Z and Zhang F 2020 Phys. Rev. B 102 035136
[28] Hejazi K, Chen X and Balents L 2021 Phys. Rev. Res. 3 013242
[29] Kang J and Vafek O 2020 Phys. Rev. B 102 035161
[30] Chen B B, Liao Y D, Chen Z, Vafek O, Kang J, Li W and Meng Z Y 2021 Nat. Commun. 12 5480
[31] Da Liao Y, Kang J, Breiø C N, Xu X Y, Wu H Q, Andersen B M, Fernandes R M and Meng Z Y 2021 Phys. Rev. X 11 011014
[32] Bernevig B A, Song Z D, Regnault N and Lian B 2021 Phys. Rev. B 103 205413
[33] Lian B, Song Z D, Regnault N, Efetov D K, Yazdani A and Bernevig B A 2021 Phys. Rev. B 103 205414
[34] Xie F, Cowsik A, Song Z D, Lian B, Bernevig B A and Regnault N 2021 Phys. Rev. B 103 205416
[35] Soejima T, Parker D E, Bultinck N, Hauschild J and Zaletel M P 2020 Phys. Rev. B 102 205111
[36] Zhang X, Pan G, Zhang Y, Kang J and Meng Z Y 2021 Chin. Phys. Lett. 38 077305
[37] Parker D E, Soejima T, Hauschild J, Zaletel M P and Bultinck N 2021 Phys. Rev. Lett. 127 027601
[38] Xu C, Li J, Xu Y, Bi Z and Zhang Y 2024 Proc. Natl. Acad. Sci. USA 121 e2316749121
[39] Mao N, Xu C, Li J, Bao T, Liu P, Xu Y, Felser C, Fu L and Zhang Y 2024 Commun. Phys. 7 262
[40] Zhang Y, Liu T and Fu L 2021 Phys. Rev. B 103 155142
[41] Xu C, Mao N, Zeng T and Zhang Y 2025 Phys. Rev. Lett. 134 066601
[42] Xu F, Sun Z, Jia T, Liu C, Xu C, Li C, Gu Y,Watanabe K, Taniguchi T, Tong B, Jia J, Shi Z, Jiang S, Zhang Y, Liu X and Li T 2023 Phys. Rev. X 13 031037
[43] Fan F R, Xiao C and Yao W 2024 Phys. Rev. B 109 L041403
[44] Tong Q, Yu H, Zhu Q, Wang Y, Xu X and Yao W 2017 Nat. Phys. 13 356
[45] Reddy A P, Devakul T and Fu L 2023 Phys. Rev. Lett. 131 246501
[46] Sheng D N, Reddy A P, Abouelkomsan A, Bergholtz E J and Fu L 2024 Phys. Rev. Lett. 133 066601
[47] Xu F, Chang X, Xiao J, Zhang Y, Liu F, Sun Z, Mao N, Peshcherenko N, Li J, Watanabe K, Taniguchi T, Yang H, Zhou B, Yang L, Zhang Y, Tong W Y and Liu X 2025 Nat. Phys. 21 542
[48] Zhang XW,Wang C, Liu X, Fan Y, Cao T and Xiao D 2024 Nat. Commun. 15 4223
[49] Reddy A P, Alsallom F, Zhang Y, Devakul T and Fu L 2023 Phys. Rev. B 108 085117
[50] Xie Y M, Zhang C P, Hu J X, Mak K F and Law K T 2022 Phys. Rev. Lett. 128 026402
[51] Pan H, Xie M,Wu F and Das Sarma S 2022 Phys. Rev. Lett. 129 056804
[52] Li B and Wu F 2025 Phys. Rev. B 111 125122
[53] Qiu W X, Li B, Luo X J and Wu F 2023 Phys. Rev. X 13 041026
[54] Wu F, Lovorn T, Tutuc E, Martin I and MacDonald A H 2019 Phys. Rev. Lett. 122 086402
[55] Li B, Qiu W X and Wu F 2024 Phys. Rev. B 109 L041106
[56] Cai J, Anderson E, Wang C, Zhang X, Liu X, Holtzmann W, Zhang Y, Fan F, Taniguchi T, Watanabe K, Yazdani A, Xu C, Zhang Y and Li T 2023 Nature 622 63
[57] Zhang L, Liu K, Wong A B, Kim J, Hong X, Liu C, Cao T, Louie S G, Wang F and Yang P 2014 Nano Lett. 14 6418
[58] Ci P, Zhao Y, Sun M, Rho Y, Chen Y, Grigoropoulos C P, Jin S, Li X and Wu J 2022 Nano Lett. 22 9027
[59] Fan X, Jiang Y, Zhuang X, Liu H, Xu T, Zheng W, Fan P, Li H, Wu X, Zhu X, Duan X and Hu W 2017 ACS Nano 11 4892
[60] Chen J, Bai Y, Qi M, Zhang W, Qin C, Fan X and Xiao L 2025 Adv. Mater. 37 2415214
[61] Lu X, Xie B, Yang Y, Zhang Y, Kong X, Li J, Ding F, Wang Z J and Liu J 2024 Phys. Rev. Lett. 132 056601
[62] Staley N E, Wu J, Eklund P, Liu Y, Li L and Xu Z 2009 Phys. Rev. B 80 184505
[63] Ozaki T 2003 Phys. Rev. B 67 155108
[64] Ozaki T and Kino H 2004 Phys. Rev. B 69 195113
[65] Grimme S, Antony J, Ehrlich S and Krieg H 2010 J. Chem. Phys. 132 154104
[66] Michaud-Rioux V, Zhang L and Guo H 2016 J. Comput. Phys. 307 593
[67] Kresse G and Hafner J 1993 Phys. Rev. B 47 558
[68] Kresse G and Hafner J 1994 Phys. Rev. B 49 14251
[69] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[70] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758
[71] Moon P and Koshino M 2012 Phys. Rev. B 85 195458
[72] Mele E J 2010 Phys. Rev. B 81 161405
[73] Cheung C T S, Goodwin Z A H, Han Y, Lu J, Mostofi A A and Lischner J 2024 Nano Lett. 24 12088
[74] Guinea F and Walet N R 2019 Phys. Rev. B 99 205134
[75] Miao W, Li C, Han X, Pan D and Dai X 2023 Phys. Rev. B 107 064501
[76] Uchida K, Furuya S, Iwata J I and Oshiyama A 2014 Phys. Rev. B 90 155451
[77] Xie B and Liu J 2023 Phys. Rev. B 108 094115
[78] Wickramaratne D, Khmelevskyi S, Agterberg D F and Mazin I I 2020 Phys. Rev. X 10 041003
[79] Enaldiev V, Zólyomi V, Yelgel C, Magorrian S and Fal’ko V I 2020 Phys. Rev. Lett. 124 206101
[80] Kim H, Choi Y, Lantagne-Hurtubise E, Lewandowski C, Thomson A, Kong L, Zhou H, Baum E, Zhang Y, Holleis L, Watanabe K, Taniguchi T, Young A F, Alicea J and Nadj-Perge S 2023 Nature 623 942
[81] Shen C, Ledwith P J, Watanabe K, Taniguchi T, Khalaf E, Vishwanath A and Efetov D K 2023 Nat. Mater. 22 316
[82] Christos M, Sachdev S and ScheurerM S 2022 Phys. Rev. X 12 021018
[83] Han Z, Herzog-Arbeitman J, Bernevig B A and Kivelson S A 2024 Phys. Rev. X 14 041004
[84] Gani Y S, Steinberg H and Rossi E 2019 Phys. Rev. B 99 235405
[85] Xie Y M and Law K T 2023 Phys. Rev. Lett. 131 016001
[86] Qin W and MacDonald A H 2021 Phys. Rev. Lett. 127 097001
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