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

doi:10.1088/1674-1056/ae23ac

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).

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

1. .pdf(1239KB)

[1]	Electronic structure study of the intermetallic compound GdRhIn₅ Yulu He(贺宇璐), Bo Wang(王博), Xiangfei Yang(杨向飞), Dengpeng Yuan(袁登鹏), Wei Feng(冯卫), Yaobo Huang(黄耀波), and Qiuyun Chen(陈秋云). Chin. Phys. B, 2026, 35(3): 037101.
[2]	First-principles insights into NaMgPO₃S oxysulfide solid electrolyte Jian Sun(孙健), Shaohui Ding(丁少辉), Daquan Yang(杨大全), Kan Zhang(张侃), and Huican Mao(毛慧灿). Chin. Phys. B, 2026, 35(2): 027105.
[3]	Two-dimensional kagome semiconductor Sc₆S₅X₆ (X = Cl, Br, I) with trilayer kagome lattice Jin-Ling Yan(闫金铃), Xing-Yu Wang(王星雨), Gen-Ping Wu(吴根平), Hao Wang(王浩), Ya-Jiao Ke(柯亚娇), Jiafu Wang(王嘉赋), Zhi-Hong Liu(刘志宏), and Jun-Hui Yuan(袁俊辉). Chin. Phys. B, 2026, 35(2): 027102.
[4]	Machine learning-assisted optimization of MTO basis sets Zhiqiang Li(李志强) and Lei Wang(王蕾). Chin. Phys. B, 2026, 35(1): 016301.
[5]	First-principles insights into strain-mediated He migration and irradiation resistance in W-Ta-Cr-V complex alloys Mengdie Wang(王梦蝶), Chao Zhang(张超), Xinyue Lan(兰新月), Biao Hu(胡标), and Xuebang Wu(吴学邦). Chin. Phys. B, 2026, 35(1): 016102.
[6]	Doping-induced magnetic and topological transitions in Mn₂X₂Te₅ (X = Bi, Sb) bilayers Wei Chen(陈威), Chuhan Tang(唐楚涵), Chao-Fei Liu(刘超飞), and Mingxing Chen(陈明星). Chin. Phys. B, 2025, 34(9): 097304.
[7]	Site occupation of Al doping in Lu₂SiO₅: The role of ionic radius versus chemical valence Xuejiao Sun(孙雪娇), Yu Cui(崔宇), Feng Gao(高峰), Zhongjun Xue(薛中军), Shuwen Zhao(赵书文), Dongzhou Ding(丁栋舟), Fan Yang(杨帆), and Yi-Yang Sun(孙宜阳). Chin. Phys. B, 2025, 34(9): 096101.
[8]	Superconductivity and band topology of double-layer honeycomb structure M₂N₂ (M = Nb, Ta) Jin-Han Tan(谭锦函), Na Jiao(焦娜), Meng-Meng Zheng(郑萌萌), Ping Zhang(张平), and Hong-Yan Lu(路洪艳). Chin. Phys. B, 2025, 34(9): 097402.
[9]	First-principles calculations on strain tunable hyperfine Stark shift of shallow donors in Si Zi-Kai Zhou(周子凯) and Jun Kang(康俊). Chin. Phys. B, 2025, 34(8): 087102.
[10]	Theoretical investigation of potential superconductivity in Sr-doped La₃Ni₂O₇ at ambient pressure Lei Shi(石磊), Ying Luo(罗颖), Wei Wu(吴为), and Yunwei Zhang(张云蔚). Chin. Phys. B, 2025, 34(7): 077403.
[11]	Layer-dependent structural stability and electronic properties of CrPS₄ under high pressure Jian Zhu(朱健), Dengman Feng(冯登满), Liangyu Wang(王亮予), Liang Li(李亮), Fangfei Li(李芳菲), Qiang Zhou(周强), and Yalan Yan(闫雅兰). Chin. Phys. B, 2025, 34(6): 066102.
[12]	Unveiling the role of high-order anharmonicity in thermal expansion: A first-principles perspective Tianxu Zhang(张天旭), Kun Zhou(周琨), Yingjian Li(李英健), Chenhao Yi(易晨浩), Muhammad Faizan, Yuhao Fu(付钰豪), Xinjiang Wang(王新江), and Lijun Zhang(张立军). Chin. Phys. B, 2025, 34(4): 046301.
[13]	Quantum anomalous Hall effect in twisted bilayer graphene Wen-Xiao Wang(王文晓), Yi-Wen Liu(刘亦文), and Lin He(何林). Chin. Phys. B, 2025, 34(4): 047301.
[14]	Fabrication of two-dimensional van der Waals moiré superlattices Zihao Wan(万子豪), Chao Wang(王超), Hang Zheng(郑航), Wenna Tang(唐文娜), Zihao Fu(付梓豪), Weilin Liu(刘伟林), Zhenjia Zhou(周振佳), Jun Li(李骏), Guowen Yuan(袁国文), and Libo Gao(高力波). Chin. Phys. B, 2025, 34(4): 047302.
[15]	Electron momentum spectroscopy study on trifluorobromomethane: Electronic structure and electron correlation Guangqing Chen(陈广庆), Shanshan Niu(牛珊珊), Yaguo Tang(唐亚国), Yuting Zhang(张雨亭), Zhaohui Liu(刘朝辉), Chunkai Xu(徐春凯), Enliang Wang(王恩亮), Xu Shan(单旭), and Xiangjun Chen(陈向军). Chin. Phys. B, 2025, 34(4): 043401.

[1]	YE HONG-JUAN (叶红娟), HUANG YE-XIAO (黄叶肖), FU ROU-LI (傅柔励), LU XIAO-FENG (陆晓峰), SHI GUO-LIANG (史国良), CAI PEI-XIN (蔡培新), GU WEI-FANG (顾为芳), RAN ZHONG-YUAN (冉中原), LIU JING-QING (刘竞青), ZHAO ZHONG-XIAN (赵忠贤), YANG BING-CHUAN (杨秉川), YAN SHAO-LIN (阎少林). THE SCREENING EFFECT OF PHONON MODES IN HIGH-T_c OXIDE SUPERCONDUCTOR[J]. Acta Physica Sinica (Overseas Edition), 1993, 2(10): 754 -763 .
[2]	Chen Xiao-Fan (陈小凡), Yang Xue-Dong (杨学栋), Han Ling (韩玲). Two-pion interferometry at small relative momentum for pion sources with transverse and longitudinal expansions in relativistic heavy ion collisions[J]. Chinese Physics, 2004, 13(3): 341 -343 .
[3]	Xu Wen-Cheng (徐文成), Gao Jie-Li (高洁丽), Liang Zhan-Qiang (梁湛强), Chen Qiao-Hong (陈巧红), Liu Song-Hao (刘颂豪). Supercontinuum spectra generation in the single-mode optical fibre with concave dispersion profile[J]. Chinese Physics, 2006, 15(4): 715 -720 .
[4]	Xue Zheng-Yuan (薛正远), Yi You-Min (易佑民), Cao Zhuo-Liang (曹卓良). Scheme for sharing classical information via tripartite entangled states[J]. Chinese Physics, 2006, 15(7): 1421 -1424 .
[5]	Chen Jia(陈佳), Li Sheng(李晟), and Ma Hong-Ru(马红孺). Quasispecies distribution of Eigen model[J]. Chinese Physics, 2007, 16(9): 2600 -2607 .
[6]	Ren Gang(任刚), Zheng Wan-Hua(郑婉华), Wang Ke(王科), Du Xiao-Yu(杜晓宇), Xing Ming-Xin(邢名欣), and Chen Liang-Hui(陈良惠) . Design of a compact polarization beam splitter based on a deformed photonic crystal directional coupler[J]. Chin. Phys. B, 2008, 17(7): 2553 -2556 .
[7]	Meng Qing-Miao(孟庆苗), Wang Shuai(王帅), Jiang Ji-Jian(蒋继建), and Deng De-Li(邓德力). Thermal radiation and nonthermal radiation of the slowly changing dynamic Kerr--Newman black hole[J]. Chin. Phys. B, 2008, 17(8): 2811 -2816 .
[8]	Zhang Yun(张云), Zuo Yu-Hua(左玉华), Guo Jian-Chuan(郭剑川), Ding Wu-Chang(丁武昌), Cheng Bu-Wen(成步文), Yu Jin-Zhong(余金中), and Wang Qi-Ming(王启明). Optical matching layer structures in evanescent coupling photodiodes at a wavelength of 1.55μm: physics, design and simulation[J]. Chin. Phys. B, 2009, 18(1): 225 -230 .
[9]	Li Zheng-Yan(李政言),Xie Zheng-Wei(谢正伟), Chen Tong(陈同), and Ouyang Qi(欧阳颀). Noise effect in metabolic networks[J]. Chin. Phys. B, 2009, 18(12): 5544 -5551 .
[10]	Long Xue(龙雪), Li Xaing(李祥), Lin Peng-Ting(蔺彭婷), Cheng Xing-Wang(程兴旺), Liu Ying(刘颖), and Cao Chuan-Bao(曹传宝). Mg doping reduced full width at half maximum of the near-band-edge emission in Mg doped ZnO films[J]. Chin. Phys. B, 2010, 19(2): 27202 -027202 .

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

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

Cite this article:

Online attention

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