Porous-B18: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states

doi:10.1088/1674-1056/ae41a2

中国物理B ›› 2026, Vol. 35 ›› Issue (5): 57110-057110.doi: 10.1088/1674-1056/ae41a2

Porous-B₁₈: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states

Xiao-Jing Gao(高晓晶), Yanfeng Ge(盖彦峰), and Yan Gao(高炎)†

State Key Laboratory of Metastable Materials Science and Technology & Hebei Key Laboratory of Microstructural Material Physics, School of Science, Yanshan University, Qinhuangdao 066004, China

收稿日期:2025-12-13 修回日期:2026-01-01 接受日期:2026-02-04 发布日期:2026-05-15
通讯作者: Yan Gao E-mail:yangao9419@ysu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 12304202), the Natural Science Foundation of Hebei Province, China (Grant No. A2023203007), the Science Research Project of Hebei Education Department (Grant No. BJK2024085), and the Cultivation Project for Basic Research and Innovation of Yanshan University (No. 2022LGZD001).

Porous-B₁₈: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states

Xiao-Jing Gao(高晓晶), Yanfeng Ge(盖彦峰), and Yan Gao(高炎)†

State Key Laboratory of Metastable Materials Science and Technology & Hebei Key Laboratory of Microstructural Material Physics, School of Science, Yanshan University, Qinhuangdao 066004, China

Received:2025-12-13 Revised:2026-01-01 Accepted:2026-02-04 Published:2026-05-15
Contact: Yan Gao E-mail:yangao9419@ysu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 12304202), the Natural Science Foundation of Hebei Province, China (Grant No. A2023203007), the Science Research Project of Hebei Education Department (Grant No. BJK2024085), and the Cultivation Project for Basic Research and Innovation of Yanshan University (No. 2022LGZD001).

1. 2026-057110-SI.pdf(1402KB)

摘要/Abstract

摘要： Topological semimetals (TSMs) featuring symmetry-protected band degeneracies have attracted considerable attention due to their exotic quantum properties and potential applications. While nodal line (NL) and nodal surface (NS) semimetals have been extensively studied, the realization of a material where both NL and NS coexist and are intertwined, particularly with an ideal electronic band structure, remains a significant challenge. Here, we predict via first-principles calculations and symmetry analysis a metastable boron allotrope, porous-B$_{18}$ (space group $P6_3/m$, No. 176), as a pristine TSM hosting a NS and two straight NLs near the Fermi level. The structure, a honeycomb-like porous 3D framework, exhibits excellent dynamical, thermal (stable up to 1000 K), and mechanical stability. Its electronic band structure is remarkably clean: only the highest valence band (HVB) and the lowest conduction band (LCB) cross linearly within a large energy window of 1.84 eV, free from trivial-band interference. The nodal surface lies on the $k_z = \pm \pi$ planes, protected by combined time-reversal symmetry ($T$) and twofold screw-rotational symmetry ($S_{2z}$), yielding a full-plane Kramers-like degeneracy. The two nodal lines along $K$-$H$ and $K'$-$H'$ are protected by inversion and time-reversal symmetries, carry a quantized Berry phase of $\pm \pi$, and connect orthogonally to the nodal surface, forming an intertwined nodal network. Drumhead surface states on the $(1\bar{1}0)$ surface further confirm the nontrivial topology. Porous-B$_{18}$ thus provides an ideal platform for investigating the interplay between nodal-line and nodal-surface fermions and exploring novel quantum transport phenomena.

关键词: topological semimetal, nodal line, nodal surface, boron allotrope, first-principles calculations

Abstract: Topological semimetals (TSMs) featuring symmetry-protected band degeneracies have attracted considerable attention due to their exotic quantum properties and potential applications. While nodal line (NL) and nodal surface (NS) semimetals have been extensively studied, the realization of a material where both NL and NS coexist and are intertwined, particularly with an ideal electronic band structure, remains a significant challenge. Here, we predict via first-principles calculations and symmetry analysis a metastable boron allotrope, porous-B$_{18}$ (space group $P6_3/m$, No. 176), as a pristine TSM hosting a NS and two straight NLs near the Fermi level. The structure, a honeycomb-like porous 3D framework, exhibits excellent dynamical, thermal (stable up to 1000 K), and mechanical stability. Its electronic band structure is remarkably clean: only the highest valence band (HVB) and the lowest conduction band (LCB) cross linearly within a large energy window of 1.84 eV, free from trivial-band interference. The nodal surface lies on the $k_z = \pm \pi$ planes, protected by combined time-reversal symmetry ($T$) and twofold screw-rotational symmetry ($S_{2z}$), yielding a full-plane Kramers-like degeneracy. The two nodal lines along $K$-$H$ and $K'$-$H'$ are protected by inversion and time-reversal symmetries, carry a quantized Berry phase of $\pm \pi$, and connect orthogonally to the nodal surface, forming an intertwined nodal network. Drumhead surface states on the $(1\bar{1}0)$ surface further confirm the nontrivial topology. Porous-B$_{18}$ thus provides an ideal platform for investigating the interplay between nodal-line and nodal-surface fermions and exploring novel quantum transport phenomena.

Key words: topological semimetal, nodal line, nodal surface, boron allotrope, first-principles calculations

中图分类号: (Electron density of states and band structure of crystalline solids)

71.20.-b

73.20.At (Surface states, band structure, electron density of states) 71.90.+q (Other topics in electronic structure)

Xiao-Jing Gao(高晓晶), Yanfeng Ge(盖彦峰), and Yan Gao(高炎). Porous-B₁₈: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states[J]. 中国物理B, 2026, 35(5): 57110-057110.

参考文献

[1] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[2] Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[3] Armitage N P, Mele E J and Vishwanath A 2018 Rev. Mod. Phys. 90 015001
[4] Burkov A A 2016 Nat. Mater. 15 1145
[5] Chiu C K, Teo J C Y, Schnyder A P and Ryu S 2016 Rev. Mod. Phys. 88 035005
[6] Gao H, Venderbos J W, Kim Y and Rappe A M 2019 Annu. Rev. Mater. Res. 49 153
[7] Wan X, Turner A M, Vishwanath A and Savrasov S Y 2011 Phys. Rev. B 83 205101
[8] Weng H, Fang C, Fang Z, Bernevig B A and Dai X 2015 Phys. Rev. X 5 011029
[9] Young S M, Zaheer S, Teo J C Y, Kane C L, Mele E J and Rappe A M 2012 Phys. Rev. Lett. 108 140405
[10] Wang Z, Sun Y, Chen X-Q, Franchini C, Xu G, Weng H, Dai X and Fang Z 2012 Phys. Rev. B 85 195320
[11] Zhu Z, Winkler G W, Wu Q, Li J and Soluyanov A A 2016 Phys. Rev. X 6 031003
[12] Bradlyn B, Cano J, Wang Z, Vergniory M G, Felser C, Cava R J and Bernevig B A 2016 Science 353 aaf5037
[13] Fang C, Chen Y, Kee H Y and Fu L 2015 Phys. Rev. B 92 081201
[14] Gao Y, Chen Y, Xie Y, Chang P Y, Cohen M L and Zhang S 2018 Phys. Rev. B 97 121108
[15] Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X and Kawazoe Y 2015 Phys. Rev. B 92 045108
[16] Wu W, Liu Y, Li S, Zhong C, Yu Z M, Sheng X L, Zhao Y X and Yang S A 2018 Phys. Rev. B 97 115125
[17] Liang Q F, Zhou J, Yu R, Wang Z and Weng H 2016 Phys. Rev. B 93 085427
[18] Zhong C, Chen Y, Xie Y, Yang S A, Cohen M L and Zhang S 2016 Nanoscale 8 7232
[19] Chen S Z, Li S, Chen Y and Duan W 2020 Nano Lett. 20 5400
[20] Yu R, Weng H, Fang Z, Dai X and Hu X 2015 Phys. Rev. Lett. 115 036807
[21] Bian G, Chang T R, Sankar R, Xu S Y, Zheng H, Neupert T, Chiu C K, Huang S M, Chang G, Belopolski I, et al. 2016 Nat. Commun. 7 10556
[22] Xie L S, Schoop L M, Seibel E M, Gibson Q D, Xie W and Cava R J 2015 APL Mater. 3 083602
[23] Chen Y, Xie Y, Yan X, Cohen M L and Zhang S 2020 Phys. Rep. 868 1
[24] Wang J T, Weng H and Chen C 2019 Adv. Phys. X 4 1625724
[25] Zhang X, Yu Z M, Zhu Z, Wu W, Wang S S, Sheng X L and Yang S A 2018 Phys. Rev. B 97 235150
[26] Song C, Jin L, Song P, Rong H, Zhu W, Liang B, Cui S, Sun Z, Zhao L, Shi Y, et al. 2022 Phys. Rev. B 105 L161104
[27] Yang T, Jin L, Liu Y, Zhang X and Wang X 2021 Phys. Rev. B 103 235140
[28] Li W L, Chen X, Jian T, Chen T T, Li J and Wang L S 2017 Nat. Rev. Chem. 1 0071
[29] Sun X, Liu X, Yin J, Yu J, Li Y, Hang Y, Zhou X, Yu M, Li J and Tai G 2017 Adv. Funct. Mater. 27 1603300
[30] Zhang Z, Penev E S and Yakobson B I 2017 Chem. Soc. Rev. 46 6746
[31] An Q, Reddy K M, Xie K Y, Hemker K J and Goddard W A 2016 Phys. Rev. Lett. 117 085501
[32] Cui L, Song T, Cai J, Cui X, Liu Z and Zhao J 2020 Phys. Rev. B 102 155133
[33] Gao Y, Xie Y, Chen Y, Gu J and Chen Z 2018 Phys. Chem. Chem. Phys. 20 23500
[34] Gao Y, Wu W, Guo P J, Zhong C, Yang S A, Liu K and Lu Z Y 2019 Phys. Rev. Mater. 3 044202
[35] He X L, Shao X, Chen T, Tai Y K, Weng X J, Chen Q, Dong X, Gao G, Sun J, Zhou X F, Tian Y and Wang H T 2019 Phys. Rev. B 99 184111
[36] Weng X J, Wu Q, Shao X, Yazyev O V, He X L, Dong X, Wang H T, Zhou X F and Tian Y 2022 Phys. Rev. B 105 235410
[37] Han S, Li C, Cui L, Cui X, Song T and Liu Z 2021 Phys. Rapid Res. Lett. 15 2100324
[38] Kresse G and Furthmuller J 1996 Comput. Mater. Sci. 6 15
[39] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[40] Blochl P E 1994 Phys. Rev. B 50 17953
[41] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188
[42] Togo A, Oba F and Tanaka I 2008 Phys. Rev. B 78 134106
[43] Car R and Parrinello M 1985 Phys. Rev. Lett. 55 2471
[44] Le Page Y and Saxe P 2002 Phys. Rev. B 65 104104
[45] Mostofi A A, Yates J R, Lee Y S, Souza I, Vanderbilt D and Marzari N 2008 Comput. Phys. Commun. 178 685
[46] Wu Q, Zhang S, Song H F, Troyer M and Soluyanov A A 2018 Comput. Phys. Commun. 224 405
[47] Mouhat F and Coudert F X 2014 Phys. Rev. B 90 224104
[48] Dong W H, Pan J, Sun J T and Du S 2023 Phys. Rev. B 108 115153

Porous-B₁₈: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states

Porous-B₁₈: An ideal topological semimetal with symmetry-enforced orthogonal nodal-line and nodal-surface states

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