拓扑节线与节面金属的研究进展

王珊珊; 吴维康; 杨声远

doi:10.7498/aps.68.20191538

摘要

拓扑节线与节面金属指的是在费米能附近存在能带交叉, 并且这些交叉点在动量空间分别形成一维曲线和二维曲面的金属材料. 这种特殊的能带结构可以带来很多奇异的物理性质, 使得这两类体系在近几年得到了广泛关注. 本文着重讨论了节线与节面金属相关概念的发展, 回顾了有关的研究工作, 包括节线与节面的特征与分类以及相应材料的预测.

关键词:

拓扑材料 /
节线 /
节面 /
能带结构

Electronic band crossing can not only form zero-dimensional nodal points, but also one dimensional nodal lines and two dimensional nodal surfaces. These topological band features have been attracting significant research interest, as they may lead to many special physical properties. In this article, we review the progress in this field, including the conceptual development, the character and classification of these nodal structures, and the material realization.

Keywords:

作者及机构信息

1.
东南大学物理学院, 南京　211189

2.
新加坡科技与设计大学, 量子材料理论研究室, 新加坡　487372

通信作者: 王珊珊, 101012564@seu.edu.cn

基金项目: 东南大学至善青年学者计划A类和Singapore Ministry of Education Academic Research Fund Tier 2 (Grant No. MOE-2017-T2-2-108)资助的课题

Authors and contacts

1.
School of Physics, Southeast University, Nanjing 211189, China

2.
Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore

Corresponding author: Wang Shan-Shan, 101012564@seu.edu.cn

Funds: Project supported by the Excellent Scholar Project of Southeast University, Class A, China and the Singapore Ministry of Education Academic Research Fund Tier 2 (Grant No. MOE-2017-T2-2-108)

文章全文 : translate this paragraph

参考文献

[1]	von Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[2]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[3]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015 Google Scholar
[4]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Google Scholar
[5]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[6]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183 Google Scholar
[7]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[8]	Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310 Google Scholar
[9]	Yao Y, Ye F, Qi X L, Zhang S C, Fang Z 2007 Phys. Rev. B 75 041401
[10]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[11]	König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766 Google Scholar
[12]	Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803 Google Scholar
[13]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[14]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[15]	Essin A M, Moore J E, Vanderbilt D 2009 Phys. Rev. Lett. 102 146805 Google Scholar
[16]	Schnyder A P, Ryu S, Furusaki A, Ludwig A W 2008 Phys. Rev. B 78 195125 Google Scholar
[17]	Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2006 Rev. Mod. Phys. 88 035005
[18]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[19]	Niu Q, Chang M C, Wu B, Xiao D, Cheng R 2017 Physical Effects of Geometric Phases (Singapore: World Scientific)
[20]	Hořava P 2005 Phys. Rev. Lett. 95 016405 Google Scholar
[21]	Zhao Y X, Wang Z D 2013 Phys. Rev. Lett. 110 240404 Google Scholar
[22]	Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101 Google Scholar
[23]	Nielsen H B, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[24]	Volovik G E 2003 The Universe in A Helium Droplet (Vol. 117) (Oxford: Oxford University Press on Demand)
[25]	Guan S, Yu Z M, Liu Y, Liu G B, Dong L, Lu Y, Yao Y, Yang S A 2017 npj Quant. Mater. 2 23 Google Scholar
[26]	Yang S A 2016 SPIN 6 1640003 Google Scholar
[27]	Young S M, Zaheer S, Teo J C, Kane C L, Mele E J, Rappe A M 2012 Phys. Rev. Lett. 108 140405 Google Scholar
[28]	Xu G, Weng H, Wang Z, Dai X, Fang Z 2011 Phys. Rev. Lett. 107 186806 Google Scholar
[29]	Weng H, Fang C, Fang Z, Dai X 2016 Phys. Rev. B 93 241202 Google Scholar
[30]	Zhu Z, Winkler G W, Wu Q, Li J, Soluyanov A A 2016 Phys. Rev. X 6 031003
[31]	Chang G, Xu S Y, Huang S M, Sanchez D S, Hsu C H, Bian G, Yu Z M, Belopolski L, Alidoust N, Zheng H, Chang T R, Jeng H T, Yang S A, Neupert T, Lin H, Hasan M Z 2017 Sci. Rep. 7 1688 Google Scholar
[32]	Bradlyn B, Cano J, Wang Z, Vergniory M G, Felser C, Cava R J, Bernevig B A 2016 Science 353 aaf5037 Google Scholar
[33]	Wieder B J, Kim Y, Rappe A M, Kane C L 2016 Phys. Rev. Lett. 116 186402 Google Scholar
[34]	Kennett M P, Komeilizadeh N, Kaveh K, Smith P M 2011 Phys. Rev. A 83 053636 Google Scholar
[35]	Roy B, Smith P M, Kennett M P 2012 Phys. Rev. B 85 235119 Google Scholar
[36]	Chen C, Wang S S, Liu L, Yu Z M, Sheng X L, Chen Z, Yang S A 2017 Phys. Rev. Mater. 1 044201 Google Scholar
[37]	Liu Q, Zunger A 2017 Phys. Rev. X 7 021019
[38]	Yu W C, Zhou X, Chuang F C, Yang S A, Lin H, Bansil A 2018 Phys. Rev. Mater. 2 051201 Google Scholar
[39]	Zhu Z, Liu Y, Yu Z M, Wang S S, Zhao Y X, Feng Y, Sheng X L, Yang S A 2018 Phys. Rev. B 98 125104 Google Scholar
[40]	Son D T, Spivak B Z 2013 Phys. Rev. B 88 104412 Google Scholar
[41]	Gao Y, Yang S A, Niu Q 2017 Phys. Rev. B 95 165135 Google Scholar
[42]	Luo Y, McDonald R D, Rosa P F S, Scott B, Wakeham N, Ghimire N J, Bauer E D, Thompson J D, Ronning F 2016 Sci. Rep. 6 27294 Google Scholar
[43]	Li Y, Li L, Wang J, Wang T, Xu X, Xi C, Cao C, Dai J 2016 Phys. Rev. B 94 121115 Google Scholar
[44]	Wang Z, Sun Y, Chen X Q, Franchini C, Xu G, Weng H, Dai X, Fang Z 2012 Phys. Rev. B 85 195320 Google Scholar
[45]	Wang Z, Weng H, Wu Q, Dai X, Fang Z 2013 Phys. Rev. B 88 125427 Google Scholar
[46]	Weng H, Fang C, Fang Z, Bernevig B A, Dai X 2015 Phys. Rev. X 5 011029
[47]	Huang S M, Xu S Y, Belopolski I, Lee C C, Chang G, Wang B K, Alidoust N, Bian G, Neupane M, Zhang C, Jia S, Bansil A, Lin H, Hasan M Z 2015 Nat. Commun. 6 7373 Google Scholar
[48]	Tang F, Po H C, Vishwanath A, Wan X 2019 Nature 566 486 Google Scholar
[49]	Zhang T, Jiang Y, Song Z, Huang H, He Y, Fang Z, Weng H, Fang C 2019 Nature 566 475 Google Scholar
[50]	Vergniory M G, Elcoro L, Felser C, Regnault N, Bernevig B A, Wang Z 2019 Nature 566 480 Google Scholar
[51]	Yang S A, Pan H, Zhang F 2014 Phys. Rev. Lett 113 046401 Google Scholar
[52]	Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X, Kawazoe Y 2015 Phys. Rev. B 92 045108 Google Scholar
[53]	Mullen K, Uchoa B, Glatzhofer D T 2015 Phys. Rev. Lett. 115 026403 Google Scholar
[54]	Yu R, Weng H, Fang Z, Dai X, Hu X 2015 Phys. Rev. Lett. 115 036807 Google Scholar
[55]	Kim Y, Wieder B J, Kane C L, Rappe A M 2015 Phys. Rev. Lett. 115 036806 Google Scholar
[56]	Chen Y, Xie Y, Yang S A, Pan H, Zhang F, Cohen M, Zhang S 2015 Nano Lett. 15 6974 Google Scholar
[57]	Bian G, Chang T R, Sankar R, Xu S Y, Zheng H, Neupert T, Chiu C K, Huang S M, Chang G, Belopolski I, Sanchez D S, Neupane M, Alidoust N, Wang B, Lee C C, Jeng H T, Bandil A, Chou F, Lin H, Hasan M Z 2015 arXiv: 1505.03069
[58]	Fang C, Chen Y, Kee H Y, Fu L 2015 Phys. Rev. B 92 081201 Google Scholar
[59]	Zhong C, Chen Y, Xie Y, Yang S A, Cohen M L, Zhang S B 2016 Nanoscale 8 7232 Google Scholar
[60]	Liang Q F, Zhou J, Yu R, Wang Z, Weng H 2016 Phys. Rev. B 93 085427 Google Scholar
[61]	Li S, Yu Z M, Liu Y, Guan S, Wang S S, Zhang X, Yao Y, Yang S A 2017 Phys. Rev. B: Rapid Comm. 96 081106 Google Scholar
[62]	Lu Y, Zhou D, Chang G, Guan S, Chen W, Jiang Y, Jiang J, Wang X S, Yang S A, Feng Y P, Kawazoe Y, Lin H 2016 npj Comput. Mater. 2 16011 Google Scholar
[63]	Wu W, Liu Y, Li S, Zhong C, Yu Z M, Sheng X L, Zhao Y X, Yang S A 2018 Phys. Rev. B 97 115125 Google Scholar
[64]	Zhang X, Yu Z M, Lu Y, Sheng X L, Yang H Y, Yang S A 2018 Phys. Rev. B 97 125143 Google Scholar
[65]	Xu Q, Yu R, Fang Z, Dai X, Weng H 2017 Phys. Rev. B 95 045136 Google Scholar
[66]	Chan Y H, Chiu C K, Chou M Y, Schnyder A P 2016 Phys. Rev. B 93 205132 Google Scholar
[67]	Du Y, Tang F, Wang D, Sheng L, Kan E J, Duan C G, Savrasov S Y, Wan X 2017 npj Quantum Mater. 2 3 Google Scholar
[68]	Huang H, Liu J, Vanderbilt D, Duan W 2016 Phys. Rev. B 93 201114 Google Scholar
[69]	Li S, Liu Y, Fu B, Yu Z M, Yang S A, Yao Y 2018 Phys. Rev. B 97 245148 Google Scholar
[70]	Li R, Ma H, Cheng X, Wang S, Li D, Zhang Z, Li Y, Chen X Q 2016 Phys. Rev. Lett. 117 096401 Google Scholar
[71]	Bian G, Chang T R, Sankar R, Xu S Y, Zheng H, Neupert T, Chiu C K, Huang S M, Chang G, Belopolski I, Sanchez D S, Neupane M, Alidoust N, Liu C, Wang B, Lee C C, Jeng H T, Zhang C, Yuan Z, Jia S, Bansil A, Chou F, Lin H, Hasan M Z 2016 Nat. Commun. 7 10556 Google Scholar
[72]	Young S M, Kane C L 2015 Phys. Rev. Lett. 115 126803 Google Scholar
[73]	Wang S S, Liu Y, Yu Z M, Sheng X L, Yang S A 2017 Nat. Commun. 8 1844 Google Scholar
[74]	Li S, Liu Y, Wang S S, Yu Z M, Guan S, Sheng X L, Yao Y, Yang S A 2018 Phys. Rev. B 97 045131 Google Scholar
[75]	Bzdušek T, Wu Q, Rüegg A, Sigrist M, Soluyanov A A 2016 Nature 538 7623
[76]	Sato T, Wang Z, Nakayama K, Souma S, Takane D, Nakata Y, Iwasawa H, Cacho C, Timur K, Takahashi T, Ando Y 2018 Phys. Rev. B 98 121111 Google Scholar
[77]	An L, Zhang H, Hu J, Zhu X, Gao W, Zhang J, Xi C, Ning W, Mao Z, Tan M 2018 Phys. Rev. B 97 235133 Google Scholar
[78]	Soluyanov A A, Gresch D, Wang Z, Wu Q, Troyer M, Dai X, Bernevig B A 2015 Nature 527 495 Google Scholar
[79]	Xu Y, Zhang F, Zhang C 2015 Phys. Rev. Lett. 115 265304 Google Scholar
[80]	Zhang X, Jin L, Dai X, Liu G 2017 J. Phys. Chem. Lett. 8 4814 Google Scholar
[81]	Chang T R, Pletikosic I, Kong T, Bian G, Huang A, Denlinger J, Kushwaha S K, Sinkovic B, Jeng H T, Valla T, Xie W, Cava R J 2018 Adv. Sci. 6 1800897
[82]	Li Z H, Wang W, Zhou P, Ma Z S, Sun L Z 2019 New J. Phys. 21 033018 Google Scholar
[83]	Yu Z M, Wu W, Sheng X L, Zhao Y X, Yang S A 2019 Phys. Rev. B. Rapid Comm. 99 121106 Google Scholar
[84]	Zhong C, Wu W, He J, Ding G, Liu Y, Li D, Yang S A, Zhang G 2019 Nanoscale 11 2468 Google Scholar
[85]	Wang S S, Yu Z M, Liu Y, Jiao Y, Guan S, Sheng X L, Yang S A 2019 Phys. Rev. Mater. 3 084201 Google Scholar
[86]	Zeng M, Fang C, Chang G, Chen Y A, Hsieh T, Bansil A, Lin H, Fu L 2015 arXiv: 1504.03492
[87]	Kobayashi S, Yamakawa Y, Yamakage A, Inohara T, Okamoto Y, Tanaka Y 2017 Phys. Rev. B 95 245208 Google Scholar
[88]	Gong C, Xie Y, Chen Y, Kim H S, Vanderbilt D 2018 Phys. Rev. Lett. 120 106403 Google Scholar
[89]	Sheng X L, Yu Z M, Yu R, Weng H, Yang S A 2017 J. Phys. Chem. Lett. 8 3506 Google Scholar
[90]	Zhong C, Chen Y, Yu Z -M, Xie Y, Wang H, Yang S A Zhang S 2017 Nat. Commun. 8 15641 Google Scholar
[91]	Chang G, Xu S Y, Zhou X, Huang S M, Singh B, Wang B, Belopolski I, Yin J, Zhang S, Bansil A, Lin H, Hasan M Z 2017 Phys. Rev. Lett. 119 156401 Google Scholar
[92]	Kim K S, Zhao Y, Jang H, Lee S Y, Kim J M, Kim K S, Ahn J H, Kim P, Choi J Y, Hong B H 2009 Nature 457 706 Google Scholar
[93]	Lee C, Wei X, Kysar J W, Hone J 2008 Science 321 385 Google Scholar
[94]	Wu W, Jiao Y, Li S, Sheng X L, Yu Z -M, Yang S A 2019 Phys. Rev. Materials 3 054203 Google Scholar
[95]	You J Y, Chen C, Zhang Z, Sheng X L, Yang S A, Su G 2019 Phys. Rev. B 100 064408 Google Scholar
[96]	Guan S, Liu Y, Yu Z M, Wang S S, Yao Y, Yang S A 2017 Phys. Rev. Materials 1 054003 Google Scholar
[97]	Jin Y J, Wang R, Zhao J Z, Du Y P, Zheng C D, Gan L Y, Liu J F, Xu H, Tong S Y 2017 Nanoscale 9 13112 Google Scholar
[98]	Zhou P, Ma Z S, Sun L Z 2018 Journal of Materials Chemistry C 6 1206 Google Scholar
[99]	Feng B, Fu B, Kasamatsu S, Ito S, Cheng P, Liu C C, Feng Y, Wu S, Mahatha S K, Sheverdyaeva P, Moras O, Arita M, Sugino O, Chiang T C, Shimada K, Miyamoto K, Okuda T, Wu K, Chen L, Yao Y 2017 Nat. Commun. 8 1007 Google Scholar
[100]	Gao L, Sun J T, Lu J C, Li H, Qian K, Zhang S, Zhang Y Y, Qian T, Ding H, Lin X, Du S, Gao H J 2018 Adv. Mater. 30 1707055 Google Scholar
[101]	Li R, Xu Y, He J, Ullah S, Li J, Liu J M, Li D, Franchini C, Weng H, Chen X Q 2016 arXiv: 1610.07142
[102]	Shi Y, Guo Y, Wang X, Princep A J, Khalyavin D, Manuel P, Michiue Y, Sato A, Tsuda K, Arai M 2013 Nature Mater. 12 1024 Google Scholar
[103]	Chang G, Xu S Y, Zheng H, Singh B, Hsu C H, Bian G, Alidoust N, Belopolski I, Sanchez D S, Zhang S, Lin H, Hasan M Z 2016 Sci. Rep. 6 38839 Google Scholar
[104]	Zhang R W, Zhang Z, Liu C C, Yao Y 2019 arXiv: 1907.01130
[105]	Wang R, Zhao J Z, Jin Y J, Du Y P, Zhao Y X, Xu H, Tong S Y 2018 Phys. Rev. B 97 241111 Google Scholar
[106]	Chen C, Yu Z M, Li S, Chen Z, Sheng X L, Yang S A 2019 Phys. Rev. B 99 075131 Google Scholar
[107]	Wang J 2017 Phys. Rev. B 96 081107 Google Scholar
[108]	Hosen M M, Dhakal G, Dimitri K, Maldonado P, Aperis A, Firoza K, Christopher S, Riseborough P, Oppeneer P M, Kaczorowski D, Durakiewicz T, Neupane M 2018 Sci. Rep. 8 13283 Google Scholar
[109]	Zhou X, Zhang R W, Zhang Z, Ma D S, Feng W, Mokrousov Y, Yao Y 2019 J. Phys. Chem. Lett. 10 3101 Google Scholar
[110]	Feng B, Zhang R W, Feng Y, Fu B, Wu S, Miyamoto K, He S, Chen L, Wu K, Shimada K, Okuda T, Yao Y 2019 Phy. Rev. Lett. 123 116401 Google Scholar
[111]	Heikkilä T T, Kopnin N B, Volovik G E 2011 JETP Lett. 94 233 Google Scholar
[112]	Liu J, Balents L 2017 Phys. Rev. B 95 075426 Google Scholar
[113]	Bzdušek T, Sigrist M 2017 Phys. Rev. B 96 155105 Google Scholar
[114]	Xiao M, Sun X Q, Fan S 2019 Phys. Rev. B 99 094206 Google Scholar
[115]	Zhang X, Yu Z M, Zhu Z, Wu W, Wang S S, Sheng X L, Yang S A 2018 Phys. Rev. B 97 235150 Google Scholar
[116]	Zhao Y X, Schnyder A P 2016 Phys. Rev. B 94 195109 Google Scholar
[117]	Yu Z M, Wu W, Zhao Y X, Yang S A 2019 Phys. Rev. B 100 041118 Google Scholar
[118]	Nielsen H B, Ninomiya M 1981 Nucl. Phys. B 185 20 Google Scholar
[119]	Nielsen H B, Ninomiya M 1981 Nucl. Phys. B 193 173 Google Scholar
[120]	Rhim J W, Kim Y B 2015 Phys. Rev. B 92 045126 Google Scholar
[121]	Liu Y, Yang S A, Zhang F 2018 Phys. Rev. B 97 035153 Google Scholar

施引文献

图 1 关于拓扑绝缘体中拓扑边缘态的简单图像

Fig. 1. A schematic figure for the topological boundary state in a topological insulator.

下载: 全尺寸图片幻灯片

图 2 节线的示意图　(a)由两条能带交叉所形成的节线; (b)绿色的环代表节线, $\ell $是环绕节线一个路径, 沿着$\ell $走一圈的贝利相为π^[61]

Fig. 2. Schematic figure of a nodal loop: (a) Nodal loop formed by two crossing bands; (b) the Berry phase of a closed path $\ell $ circling the nodal loop (green circle) is π^[61].

下载: 全尺寸图片幻灯片

图 3 在狄拉克超导体中出现的由手征对称性保护的节线　(a)当空间反演或者时间反演破坏时, 一个狄拉克点会变为一个节环或两个外尔点; (b)−(d)刻画了(a)中几种简并点的拓扑保护机制, 其中节环是由拓扑绕数所保护^[51]

Fig. 3. Chiral symmetry protected nodal line in a Dirac superconductor: (a) A Dirac node can evolve into a nodal ring or two Weyl nodes under different symmetry breaking; (b)−(d) illustrate the different topological protection for the degeneracies in (a). Here, the nodal ring is protected by the winding number^[51].

下载: 全尺寸图片幻灯片

图 4 在三种碳材料中发现的节线　(a) Mackay-Terrones结构的三维碳和节线在动量空间的表示^[52]; (b) hyperhoneycomb结构的三维碳和节线在动量空间的表示^[53]; (c)三维的石墨烯网络结构和节线在动量空间的表示^[56]

Fig. 4. Nodal lines found in three carbon allotropes: (a) 3D carbon with Mackay-Terrones crystal structur^[52]; (b) 3D hyperhoneycomb carbon^[53]; (c) 3D graphene network structure^[56].

下载: 全尺寸图片幻灯片

图 5 滑移镜面所保护的节线　(a) O和X是滑移镜面上对应两个不同配对类型的TRIM点; (b)展示了沿着连接O和X的一条路径L上的能带特征, 这里每四条能带都会形成一种沙漏形的结构; 沙漏脖子处的交叉点P在滑移镜面上会形成一条节线

Fig. 5. Nodal line protected by the glide mirror symmetry: (a) Shows the glide-mirror-invariant plane in Brillouin zone, O and X are two TRIM points with different glide mirror eigenvalues; (b) shows the band structure along a path L connecting O and X (as in (a)); it displays an hourglass shaped spectrum. The degeneracy point P in the hourglass traces out a nodal loop in the glide mirror plane.

下载: 全尺寸图片幻灯片

图 6 具有滑移镜面所保护的节线的例子　(a) ReO₂的晶体结构和能带结构, 可以看到高对称线上的沙漏型色散^[73]; (b) X₃SiTe₆(X = Ta, Nb)的晶体结构和能带结构, 以及在高对称线上的沙漏型能量色散^[74]

Fig. 6. Material examples with glide-mirror-protected nodal rings: (a) ReO₂^[73]; (b) X₃SiTe₆(X = Ta, Nb)^[74], the hourglass dispersions can be observed in their band structures.

下载: 全尺寸图片幻灯片

图 7 三种不同色散类型的节线　(a) type-I节线; (b) type-II节线; (c) hybird节线; (d)−(f)三种节线的等能面^[64]

Fig. 7. Three types of nodal lines classified by the energy dispersion: (a) Type-I nodal line; (b) type-II nodal lines; (c) hybrid nodal lines; (d)−(f) show the typical shapes of the constant energy surface for the three types^[64].

下载: 全尺寸图片幻灯片

图 8 Type-II节线和hybrid节线的特殊物理性质　(a) Type-II节线和type-I节线的光学性质的比较^[61]; (b) hybrid节线导致的磁坍塌效应和磁振荡中的各向异性^[64]

Fig. 8. Unique properties of type-II and hybrid nodal lines: (a) Comparison between type-I and type-II nodal lines in terms of JDOS and optical absorption rate^[61]; (b) the magnetic breakdown and its feature in anisotropic magnetic oscillation for a hybrid nodal loop^[64].

下载: 全尺寸图片幻灯片

图 9 (a)按照节线的色散次数进行分类的示意图; (b)−(d)展示了一个具有二次节线的材料ZrPtGa, (c)是ZrPtGa的能带结构, 蓝色实线标记了二次节线, (d)是这个节线在垂直于Γ-A的平面上的色散, 可以清楚地看到是二次色散^[83]

Fig. 9. (a) Schematic figure for the higher order nodal lines; (b)−(d) show the quadratic nodal line in ZrPtGa: (c) the band structure of ZrPtGa, the blue solid curve indicates the quadratic nodal line; (d) shows the band dispersion in the plane perpendicular to Γ-A, which clearly demonstrates a quadratic dispersion^[83].

下载: 全尺寸图片幻灯片

图 10 具有不同形态的节线　(a)穿越布里渊区的一对节线^[56]; (b)局域在布里渊区某个点周围的节线^[69]

Fig. 10. Nodal lines with different kinds of distribution in Brillouin zone: (a) Nodal lines in a carbon allotrope, which traverse the Brillouin zone^[56]; (b) nodal line in CuTeO₃, which is located around a point in Brillouin zone^[69].

下载: 全尺寸图片幻灯片

图 11 节环可以形成的一些复杂结构　(a)笼子状的结构^[38]; (b)骨架状的结构^[89];(c)三能带形成的结状节线^[90]; (d) Hopf链环^[91]; (e)外尔链; (f)狄拉克链^[73]

Fig. 11. Different structures formed by nodal lines: (a) Crossed nodal rings^[38]; (b) nodal box^[89]; (c) inter-connected nodal loops^[90]; (d) nodal Hopf link^[91]; (e) weyl chain; (f) dirac chain^[73].

下载: 全尺寸图片幻灯片

图 12 二维材料中在SOC作用下仍然稳定的节线　(a)−(c)二维GaTeI中的节线^[94]; (d)−(f)单层MnN中的节线, 单层MnN是一个铁磁材料, 在费米面处只存在一个自旋通道, 因此这里的节线是完全自旋极化的^[85]

Fig. 12. Stable nodal lines under SOC in 2D: (a)−(c) GaTeI family materials^[94]; (d)−(f) MnN monolayer, here MnN is a half metal, so the nodal loops are fully spin^[85].

下载: 全尺寸图片幻灯片

图 13 节线对应的拓扑表面态　(a)狄拉克超导体中节线导致的鼓膜态^[51]; (b)碳的同素异形体中的鼓膜态^[52]; (c), (d) ReO₂^[73]和Ta₃SiTe₆^[74]中的双鼓膜态; (e)对应着三次节线的遍布布里渊区的环面表面态^[83]

Fig. 13. Surface states of nodal line metals: (a) Drumhead surface states for nodal rings in superconductors^[51]; (b) drumhead surface states in a 3D carbon allotrope^[52]; (c), (d) show the double drumhead surface states in ReO₂^[73]and Ta₃SiTe₆^[74]; (e) surface states of cubic nodal line, which spreads over the whole BZ^[83].

下载: 全尺寸图片幻灯片

图 14 两种不同的节面　(a)三维碳材料中的节面^[63]; (b) BaMX₃中的节面^[60]

Fig. 14. Two kinds of nodal surfaces: (a) Nodal surfaces in a 3D carbon allotrope^[63]; (b) nodal surface in BaMX₃^[60].

下载: 全尺寸图片幻灯片

图 15 具有第二类节面的材料　(a) K₆YO₄; (b) TlMo₃Te₃^[63]

Fig. 15. Materials with Class-II nodal surfaces: (a) K₆YO₄; (b) TlMo₃Te₃^[63].

下载: 全尺寸图片幻灯片

图 16 SOC作用下稳定的节面　(a)展示了Ta₃TeI₇晶体结构; (b)是Ta₃TeI₇在考虑SOC时的能带结构; 能带在考虑SOC时没有打开能隙^[63]

Fig. 16. Nodal surface robust against SOC: (a) Crystal structure of Ta₃TeI₇; (b) is the band structure of Ta₃TeI₇ in the presence of SOC with no gap opening^[63].

下载: 全尺寸图片幻灯片

图 17 磁性材料中的节面　(a) CsCrI₃晶体结构; (b)不考虑SOC时的能带结构; (c), (d)考虑SOC时, 磁矩分别沿面内和面外时的能带结构^[63]

Fig. 17. Nodal surface in magnetic materials: (a) The crystal structure of CsCrI₃; (b) the band structure of CsCrI₃ without SOC; (c) and (d) band structures with magnetic moment along x and z directions respectively^[63].

下载: 全尺寸图片幻灯片

图 18 存在多个节面的材料　(a) Cu₃Se₂; (b) Rb₂Se₅; 布里渊区中的节面分布用橙色标记^[63]

Fig. 18. Materials with multiple nodal surfaces: (a) Cu₃Se₂; (b) Rb₂Se₅. The location of the nodal surfaces is indicated by the orange color^[63].

下载: 全尺寸图片幻灯片

图 19 绕过Nielson-Ninomiya不可行定理的方法　(a)一个单独外尔点的示意图; (b)贝利曲率分布; (c), (d)显示了在表面上不存在连接单外尔点的费米弧表面, 白色点标记了体内外尔点在表面的投影^[117]

Fig. 19. A method to circumvent the Nielson-Ninomiya no-go theorem: (a) Schematic figure showing the single Weyl point; (b) Berry curvature distribution; (c), (d) show that there is no surface Fermi arc emitted from the Weyl point, the white dot labels the surface projection of the Weyl point^[117].

下载: 全尺寸图片幻灯片

[1]	von Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[2]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[3]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015 Google Scholar
[4]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Google Scholar
[5]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[6]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183 Google Scholar
[7]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[8]	Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310 Google Scholar
[9]	Yao Y, Ye F, Qi X L, Zhang S C, Fang Z 2007 Phys. Rev. B 75 041401
[10]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[11]	König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766 Google Scholar
[12]	Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803 Google Scholar
[13]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[14]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[15]	Essin A M, Moore J E, Vanderbilt D 2009 Phys. Rev. Lett. 102 146805 Google Scholar
[16]	Schnyder A P, Ryu S, Furusaki A, Ludwig A W 2008 Phys. Rev. B 78 195125 Google Scholar
[17]	Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2006 Rev. Mod. Phys. 88 035005
[18]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[19]	Niu Q, Chang M C, Wu B, Xiao D, Cheng R 2017 Physical Effects of Geometric Phases (Singapore: World Scientific)
[20]	Hořava P 2005 Phys. Rev. Lett. 95 016405 Google Scholar
[21]	Zhao Y X, Wang Z D 2013 Phys. Rev. Lett. 110 240404 Google Scholar
[22]	Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101 Google Scholar
[23]	Nielsen H B, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[24]	Volovik G E 2003 The Universe in A Helium Droplet (Vol. 117) (Oxford: Oxford University Press on Demand)
[25]	Guan S, Yu Z M, Liu Y, Liu G B, Dong L, Lu Y, Yao Y, Yang S A 2017 npj Quant. Mater. 2 23 Google Scholar
[26]	Yang S A 2016 SPIN 6 1640003 Google Scholar
[27]	Young S M, Zaheer S, Teo J C, Kane C L, Mele E J, Rappe A M 2012 Phys. Rev. Lett. 108 140405 Google Scholar
[28]	Xu G, Weng H, Wang Z, Dai X, Fang Z 2011 Phys. Rev. Lett. 107 186806 Google Scholar
[29]	Weng H, Fang C, Fang Z, Dai X 2016 Phys. Rev. B 93 241202 Google Scholar
[30]	Zhu Z, Winkler G W, Wu Q, Li J, Soluyanov A A 2016 Phys. Rev. X 6 031003
[31]	Chang G, Xu S Y, Huang S M, Sanchez D S, Hsu C H, Bian G, Yu Z M, Belopolski L, Alidoust N, Zheng H, Chang T R, Jeng H T, Yang S A, Neupert T, Lin H, Hasan M Z 2017 Sci. Rep. 7 1688 Google Scholar
[32]	Bradlyn B, Cano J, Wang Z, Vergniory M G, Felser C, Cava R J, Bernevig B A 2016 Science 353 aaf5037 Google Scholar
[33]	Wieder B J, Kim Y, Rappe A M, Kane C L 2016 Phys. Rev. Lett. 116 186402 Google Scholar
[34]	Kennett M P, Komeilizadeh N, Kaveh K, Smith P M 2011 Phys. Rev. A 83 053636 Google Scholar
[35]	Roy B, Smith P M, Kennett M P 2012 Phys. Rev. B 85 235119 Google Scholar
[36]	Chen C, Wang S S, Liu L, Yu Z M, Sheng X L, Chen Z, Yang S A 2017 Phys. Rev. Mater. 1 044201 Google Scholar
[37]	Liu Q, Zunger A 2017 Phys. Rev. X 7 021019
[38]	Yu W C, Zhou X, Chuang F C, Yang S A, Lin H, Bansil A 2018 Phys. Rev. Mater. 2 051201 Google Scholar
[39]	Zhu Z, Liu Y, Yu Z M, Wang S S, Zhao Y X, Feng Y, Sheng X L, Yang S A 2018 Phys. Rev. B 98 125104 Google Scholar
[40]	Son D T, Spivak B Z 2013 Phys. Rev. B 88 104412 Google Scholar
[41]	Gao Y, Yang S A, Niu Q 2017 Phys. Rev. B 95 165135 Google Scholar
[42]	Luo Y, McDonald R D, Rosa P F S, Scott B, Wakeham N, Ghimire N J, Bauer E D, Thompson J D, Ronning F 2016 Sci. Rep. 6 27294 Google Scholar
[43]	Li Y, Li L, Wang J, Wang T, Xu X, Xi C, Cao C, Dai J 2016 Phys. Rev. B 94 121115 Google Scholar
[44]	Wang Z, Sun Y, Chen X Q, Franchini C, Xu G, Weng H, Dai X, Fang Z 2012 Phys. Rev. B 85 195320 Google Scholar
[45]	Wang Z, Weng H, Wu Q, Dai X, Fang Z 2013 Phys. Rev. B 88 125427 Google Scholar
[46]	Weng H, Fang C, Fang Z, Bernevig B A, Dai X 2015 Phys. Rev. X 5 011029
[47]	Huang S M, Xu S Y, Belopolski I, Lee C C, Chang G, Wang B K, Alidoust N, Bian G, Neupane M, Zhang C, Jia S, Bansil A, Lin H, Hasan M Z 2015 Nat. Commun. 6 7373 Google Scholar
[48]	Tang F, Po H C, Vishwanath A, Wan X 2019 Nature 566 486 Google Scholar
[49]	Zhang T, Jiang Y, Song Z, Huang H, He Y, Fang Z, Weng H, Fang C 2019 Nature 566 475 Google Scholar
[50]	Vergniory M G, Elcoro L, Felser C, Regnault N, Bernevig B A, Wang Z 2019 Nature 566 480 Google Scholar
[51]	Yang S A, Pan H, Zhang F 2014 Phys. Rev. Lett 113 046401 Google Scholar
[52]	Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X, Kawazoe Y 2015 Phys. Rev. B 92 045108 Google Scholar
[53]	Mullen K, Uchoa B, Glatzhofer D T 2015 Phys. Rev. Lett. 115 026403 Google Scholar
[54]	Yu R, Weng H, Fang Z, Dai X, Hu X 2015 Phys. Rev. Lett. 115 036807 Google Scholar
[55]	Kim Y, Wieder B J, Kane C L, Rappe A M 2015 Phys. Rev. Lett. 115 036806 Google Scholar
[56]	Chen Y, Xie Y, Yang S A, Pan H, Zhang F, Cohen M, Zhang S 2015 Nano Lett. 15 6974 Google Scholar
[57]	Bian G, Chang T R, Sankar R, Xu S Y, Zheng H, Neupert T, Chiu C K, Huang S M, Chang G, Belopolski I, Sanchez D S, Neupane M, Alidoust N, Wang B, Lee C C, Jeng H T, Bandil A, Chou F, Lin H, Hasan M Z 2015 arXiv: 1505.03069
[58]	Fang C, Chen Y, Kee H Y, Fu L 2015 Phys. Rev. B 92 081201 Google Scholar
[59]	Zhong C, Chen Y, Xie Y, Yang S A, Cohen M L, Zhang S B 2016 Nanoscale 8 7232 Google Scholar
[60]	Liang Q F, Zhou J, Yu R, Wang Z, Weng H 2016 Phys. Rev. B 93 085427 Google Scholar
[61]	Li S, Yu Z M, Liu Y, Guan S, Wang S S, Zhang X, Yao Y, Yang S A 2017 Phys. Rev. B: Rapid Comm. 96 081106 Google Scholar
[62]	Lu Y, Zhou D, Chang G, Guan S, Chen W, Jiang Y, Jiang J, Wang X S, Yang S A, Feng Y P, Kawazoe Y, Lin H 2016 npj Comput. Mater. 2 16011 Google Scholar
[63]	Wu W, Liu Y, Li S, Zhong C, Yu Z M, Sheng X L, Zhao Y X, Yang S A 2018 Phys. Rev. B 97 115125 Google Scholar
[64]	Zhang X, Yu Z M, Lu Y, Sheng X L, Yang H Y, Yang S A 2018 Phys. Rev. B 97 125143 Google Scholar
[65]	Xu Q, Yu R, Fang Z, Dai X, Weng H 2017 Phys. Rev. B 95 045136 Google Scholar
[66]	Chan Y H, Chiu C K, Chou M Y, Schnyder A P 2016 Phys. Rev. B 93 205132 Google Scholar
[67]	Du Y, Tang F, Wang D, Sheng L, Kan E J, Duan C G, Savrasov S Y, Wan X 2017 npj Quantum Mater. 2 3 Google Scholar
[68]	Huang H, Liu J, Vanderbilt D, Duan W 2016 Phys. Rev. B 93 201114 Google Scholar
[69]	Li S, Liu Y, Fu B, Yu Z M, Yang S A, Yao Y 2018 Phys. Rev. B 97 245148 Google Scholar
[70]	Li R, Ma H, Cheng X, Wang S, Li D, Zhang Z, Li Y, Chen X Q 2016 Phys. Rev. Lett. 117 096401 Google Scholar
[71]	Bian G, Chang T R, Sankar R, Xu S Y, Zheng H, Neupert T, Chiu C K, Huang S M, Chang G, Belopolski I, Sanchez D S, Neupane M, Alidoust N, Liu C, Wang B, Lee C C, Jeng H T, Zhang C, Yuan Z, Jia S, Bansil A, Chou F, Lin H, Hasan M Z 2016 Nat. Commun. 7 10556 Google Scholar
[72]	Young S M, Kane C L 2015 Phys. Rev. Lett. 115 126803 Google Scholar
[73]	Wang S S, Liu Y, Yu Z M, Sheng X L, Yang S A 2017 Nat. Commun. 8 1844 Google Scholar
[74]	Li S, Liu Y, Wang S S, Yu Z M, Guan S, Sheng X L, Yao Y, Yang S A 2018 Phys. Rev. B 97 045131 Google Scholar
[75]	Bzdušek T, Wu Q, Rüegg A, Sigrist M, Soluyanov A A 2016 Nature 538 7623
[76]	Sato T, Wang Z, Nakayama K, Souma S, Takane D, Nakata Y, Iwasawa H, Cacho C, Timur K, Takahashi T, Ando Y 2018 Phys. Rev. B 98 121111 Google Scholar
[77]	An L, Zhang H, Hu J, Zhu X, Gao W, Zhang J, Xi C, Ning W, Mao Z, Tan M 2018 Phys. Rev. B 97 235133 Google Scholar
[78]	Soluyanov A A, Gresch D, Wang Z, Wu Q, Troyer M, Dai X, Bernevig B A 2015 Nature 527 495 Google Scholar
[79]	Xu Y, Zhang F, Zhang C 2015 Phys. Rev. Lett. 115 265304 Google Scholar
[80]	Zhang X, Jin L, Dai X, Liu G 2017 J. Phys. Chem. Lett. 8 4814 Google Scholar
[81]	Chang T R, Pletikosic I, Kong T, Bian G, Huang A, Denlinger J, Kushwaha S K, Sinkovic B, Jeng H T, Valla T, Xie W, Cava R J 2018 Adv. Sci. 6 1800897
[82]	Li Z H, Wang W, Zhou P, Ma Z S, Sun L Z 2019 New J. Phys. 21 033018 Google Scholar
[83]	Yu Z M, Wu W, Sheng X L, Zhao Y X, Yang S A 2019 Phys. Rev. B. Rapid Comm. 99 121106 Google Scholar
[84]	Zhong C, Wu W, He J, Ding G, Liu Y, Li D, Yang S A, Zhang G 2019 Nanoscale 11 2468 Google Scholar
[85]	Wang S S, Yu Z M, Liu Y, Jiao Y, Guan S, Sheng X L, Yang S A 2019 Phys. Rev. Mater. 3 084201 Google Scholar
[86]	Zeng M, Fang C, Chang G, Chen Y A, Hsieh T, Bansil A, Lin H, Fu L 2015 arXiv: 1504.03492
[87]	Kobayashi S, Yamakawa Y, Yamakage A, Inohara T, Okamoto Y, Tanaka Y 2017 Phys. Rev. B 95 245208 Google Scholar
[88]	Gong C, Xie Y, Chen Y, Kim H S, Vanderbilt D 2018 Phys. Rev. Lett. 120 106403 Google Scholar
[89]	Sheng X L, Yu Z M, Yu R, Weng H, Yang S A 2017 J. Phys. Chem. Lett. 8 3506 Google Scholar
[90]	Zhong C, Chen Y, Yu Z -M, Xie Y, Wang H, Yang S A Zhang S 2017 Nat. Commun. 8 15641 Google Scholar
[91]	Chang G, Xu S Y, Zhou X, Huang S M, Singh B, Wang B, Belopolski I, Yin J, Zhang S, Bansil A, Lin H, Hasan M Z 2017 Phys. Rev. Lett. 119 156401 Google Scholar
[92]	Kim K S, Zhao Y, Jang H, Lee S Y, Kim J M, Kim K S, Ahn J H, Kim P, Choi J Y, Hong B H 2009 Nature 457 706 Google Scholar
[93]	Lee C, Wei X, Kysar J W, Hone J 2008 Science 321 385 Google Scholar
[94]	Wu W, Jiao Y, Li S, Sheng X L, Yu Z -M, Yang S A 2019 Phys. Rev. Materials 3 054203 Google Scholar
[95]	You J Y, Chen C, Zhang Z, Sheng X L, Yang S A, Su G 2019 Phys. Rev. B 100 064408 Google Scholar
[96]	Guan S, Liu Y, Yu Z M, Wang S S, Yao Y, Yang S A 2017 Phys. Rev. Materials 1 054003 Google Scholar
[97]	Jin Y J, Wang R, Zhao J Z, Du Y P, Zheng C D, Gan L Y, Liu J F, Xu H, Tong S Y 2017 Nanoscale 9 13112 Google Scholar
[98]	Zhou P, Ma Z S, Sun L Z 2018 Journal of Materials Chemistry C 6 1206 Google Scholar
[99]	Feng B, Fu B, Kasamatsu S, Ito S, Cheng P, Liu C C, Feng Y, Wu S, Mahatha S K, Sheverdyaeva P, Moras O, Arita M, Sugino O, Chiang T C, Shimada K, Miyamoto K, Okuda T, Wu K, Chen L, Yao Y 2017 Nat. Commun. 8 1007 Google Scholar
[100]	Gao L, Sun J T, Lu J C, Li H, Qian K, Zhang S, Zhang Y Y, Qian T, Ding H, Lin X, Du S, Gao H J 2018 Adv. Mater. 30 1707055 Google Scholar
[101]	Li R, Xu Y, He J, Ullah S, Li J, Liu J M, Li D, Franchini C, Weng H, Chen X Q 2016 arXiv: 1610.07142
[102]	Shi Y, Guo Y, Wang X, Princep A J, Khalyavin D, Manuel P, Michiue Y, Sato A, Tsuda K, Arai M 2013 Nature Mater. 12 1024 Google Scholar
[103]	Chang G, Xu S Y, Zheng H, Singh B, Hsu C H, Bian G, Alidoust N, Belopolski I, Sanchez D S, Zhang S, Lin H, Hasan M Z 2016 Sci. Rep. 6 38839 Google Scholar
[104]	Zhang R W, Zhang Z, Liu C C, Yao Y 2019 arXiv: 1907.01130
[105]	Wang R, Zhao J Z, Jin Y J, Du Y P, Zhao Y X, Xu H, Tong S Y 2018 Phys. Rev. B 97 241111 Google Scholar
[106]	Chen C, Yu Z M, Li S, Chen Z, Sheng X L, Yang S A 2019 Phys. Rev. B 99 075131 Google Scholar
[107]	Wang J 2017 Phys. Rev. B 96 081107 Google Scholar
[108]	Hosen M M, Dhakal G, Dimitri K, Maldonado P, Aperis A, Firoza K, Christopher S, Riseborough P, Oppeneer P M, Kaczorowski D, Durakiewicz T, Neupane M 2018 Sci. Rep. 8 13283 Google Scholar
[109]	Zhou X, Zhang R W, Zhang Z, Ma D S, Feng W, Mokrousov Y, Yao Y 2019 J. Phys. Chem. Lett. 10 3101 Google Scholar
[110]	Feng B, Zhang R W, Feng Y, Fu B, Wu S, Miyamoto K, He S, Chen L, Wu K, Shimada K, Okuda T, Yao Y 2019 Phy. Rev. Lett. 123 116401 Google Scholar
[111]	Heikkilä T T, Kopnin N B, Volovik G E 2011 JETP Lett. 94 233 Google Scholar
[112]	Liu J, Balents L 2017 Phys. Rev. B 95 075426 Google Scholar
[113]	Bzdušek T, Sigrist M 2017 Phys. Rev. B 96 155105 Google Scholar
[114]	Xiao M, Sun X Q, Fan S 2019 Phys. Rev. B 99 094206 Google Scholar
[115]	Zhang X, Yu Z M, Zhu Z, Wu W, Wang S S, Sheng X L, Yang S A 2018 Phys. Rev. B 97 235150 Google Scholar
[116]	Zhao Y X, Schnyder A P 2016 Phys. Rev. B 94 195109 Google Scholar
[117]	Yu Z M, Wu W, Zhao Y X, Yang S A 2019 Phys. Rev. B 100 041118 Google Scholar
[118]	Nielsen H B, Ninomiya M 1981 Nucl. Phys. B 185 20 Google Scholar
[119]	Nielsen H B, Ninomiya M 1981 Nucl. Phys. B 193 173 Google Scholar
[120]	Rhim J W, Kim Y B 2015 Phys. Rev. B 92 045126 Google Scholar
[121]	Liu Y, Yang S A, Zhang F 2018 Phys. Rev. B 97 035153 Google Scholar

[1]	江龙兴, 李庆超, 张旭, 李京峰, 张静, 陈祖信, 曾敏, 吴昊. 基于拓扑/二维量子材料的自旋电子器件. 物理学报, 2024, 73(1): 017505. doi: 10.7498/aps.73.20231166
[2]	鲍昌华, 范本澍, 汤沛哲, 段文晖, 周树云. 量子材料的弗洛凯调控. 物理学报, 2023, 72(23): 234202. doi: 10.7498/aps.72.20231423
[3]	巴佳燕, 陈复洋, 段后建, 邓明勋, 王瑞强. 拓扑材料中的平面霍尔效应. 物理学报, 2023, 72(20): 207201. doi: 10.7498/aps.72.20230905
[4]	王欢, 何春娟, 徐升, 王义炎, 曾祥雨, 林浚发, 王小艳, 巩静, 马小平, 韩坤, 王乙婷, 夏天龙. 拓扑半金属及磁性拓扑材料的单晶生长. 物理学报, 2023, 72(3): 038103. doi: 10.7498/aps.72.20221574
[5]	邱子阳, 陈岩, 邱祥冈. 拓扑材料BaMnSb₂的红外光谱学研究. 物理学报, 2022, 71(10): 107201. doi: 10.7498/aps.71.20220011
[6]	许佳玲, 贾利云, 刘超, 吴佺, 赵领军, 马丽, 侯登录. Li(Na)AuS体系拓扑绝缘体材料的能带结构. 物理学报, 2021, 70(2): 027101. doi: 10.7498/aps.70.20200885
[7]	姜天舒, 肖孟, 张昭庆, 陈子亭. 周期与非周期传输线网络的物理与拓扑性质. 物理学报, 2020, 69(15): 150301. doi: 10.7498/aps.69.20200258
[8]	顾开元, 罗天创, 葛军, 王健. 拓扑材料中的超导. 物理学报, 2020, 69(2): 020301. doi: 10.7498/aps.69.20191627
[9]	杨雯, 宋建军, 任远, 张鹤鸣. 光器件应用改性Ge的能带结构模型. 物理学报, 2018, 67(19): 198502. doi: 10.7498/aps.67.20181155
[10]	逯瑶, 王培吉, 张昌文, 冯现徉, 蒋雷, 张国莲. Fe, S共掺杂SnO2材料第一性原理分析. 物理学报, 2012, 61(2): 023101. doi: 10.7498/aps.61.023101
[11]	宫丽, 冯现徉, 逯瑶, 张昌文, 王培吉. Ta掺杂对ZnO光电材料性能影响的研究. 物理学报, 2012, 61(9): 097101. doi: 10.7498/aps.61.097101
[12]	陈懂, 肖河阳, 加伟, 陈虹, 周和根, 李奕, 丁开宁, 章永凡. 半导体材料AAl2C4(A=Zn, Cd, Hg; C=S, Se)的电子结构和光学性质. 物理学报, 2012, 61(12): 127103. doi: 10.7498/aps.61.127103
[13]	逯瑶, 王培吉, 张昌文, 蒋雷, 张国莲, 宋朋. 第一性原理研究In,N共掺杂SnO2材料的光电性质. 物理学报, 2011, 60(6): 063103. doi: 10.7498/aps.60.063103
[14]	逯瑶, 王培吉, 张昌文, 冯现徉, 蒋雷, 张国莲. 第一性原理研究Fe掺杂SnO2材料的光电性质. 物理学报, 2011, 60(11): 113101. doi: 10.7498/aps.60.113101
[15]	马小凤, 王懿喆, 周呈悦. a-Si ∶H/SiO2多量子阱材料制备及其光学性能和微结构研究. 物理学报, 2011, 60(6): 068102. doi: 10.7498/aps.60.068102
[16]	于峰, 王培吉, 张昌文. Al掺杂SnO2 材料电子结构和光学性质. 物理学报, 2011, 60(2): 023101. doi: 10.7498/aps.60.023101
[17]	马建立, 张鹤鸣, 宋建军, 王冠宇, 王晓艳. (001)面任意方向单轴应变硅材料能带结构. 物理学报, 2011, 60(2): 027101. doi: 10.7498/aps.60.027101
[18]	邬云文, 海文华, 蔡丽华. Paul阱中一维两离子系统的能带结构. 物理学报, 2006, 55(2): 583-589. doi: 10.7498/aps.55.583
[19]	陈德艳, 吕铁羽, 黄美纯. BaSe的准粒子能带结构. 物理学报, 2006, 55(7): 3597-3600. doi: 10.7498/aps.55.3597
[20]	郭宝增. 用全带Monte Carlo方法模拟纤锌矿相GaN和ZnO材料的电子输运特性. 物理学报, 2002, 51(10): 2344-2348. doi: 10.7498/aps.51.2344

计量

文章访问数: 25470
PDF下载量: 1381
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板

拓扑节线与节面金属的研究进展