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外尔半金属是指三维能带结构具有手性拓扑点简并特征的无能隙固体材料, 并且简并点附近的色散关系遵从外尔方程的描述. 它具有很多独特的电子输运性质, 比如: 费米弧表面态、负磁阻效应、手性朗道能级等. 类比电子系统的外尔半金属材料, 人们设计出理想外尔超构材料, 在电磁波体系里实现了频率一致的外尔点简并. 本文打破这种超构材料的镜面对称性, 通过数值计算发现了原本频率一致的外尔点出现了依赖手性的频移, 频移的正负由外尔点的手性决定, 因此实现了手性不同的外尔点在频率上的分离, 同时也检验了
$\left\langle {001} \right\rangle $ 晶面上外尔点之间的费米弧表面态.Weyl semimetal has the massless and chiral low-energy electronic excitation charateristic, and its quasi-particle behavior can be described by Weyl equation, and may lead to appealing transport properties, such as Fermi arc surface state, negative magnetic resistance, chiral Landau level, etc. By analogous with Weyl semimetal, one has realized Weyl point degeneracy of electromagnetic wave in an ideal Weyl metamaterial. In this article, by breaking the mirror symmetry of the saddle-shaped meta-atom structure, we theoretically investigate chirality-dependent split and shift effect of Weyl point frequencies which would otherwise be identical. The frequency shift can be tuned by the symmetry-broken intensity. Finally, we study the Fermi arc surface state connecting two Weyl points on$\left\langle {001} \right\rangle $ crystal surface.-
Keywords:
- Weyl point /
- Chiral /
- frequency shift /
- Fermi arc
[1] Ashcroft N W, Mermin N D 1976 Solid State Physics (New York: Holt, Rinehart and Winston) pp284–311
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图 1 (a)马鞍形超构原子, 灰色阴影为z方向的截面; (b)−(d)圆柱在截面上的正方形分布到菱形分布的变化过程
Fig. 1. (a) Unit cell where the gray surface is a cut plane along z direction; (b)−(d) the process of changing the positions of four metallic rods from square to rhombus structure.
图 2 (a)
$ \theta $ = 77.31°的晶胞结构; (b)第一布里渊区示意图, 蓝色点和红色点代表不同手性的外尔点, 分布在$ {k}_{z} $ = 0的切面; (c)数值计算的能带结构, 蓝色阴影代表只有外尔点简并能带出现的频率区间Fig. 2. (a) Unit cell where
$ \theta $ = 77.31°; (b) the first Brillouin zone where blue dots and red dots represent, respectively, the Weyl points of different chirality; (c) the numerically calculated band structure, where shaded region denotes the frequency range with only Weyl degenerate bands appears.
图 3 (a)−(d) 不同
$ \theta $ 角度下的能带结构, 分别对应$ \theta $ = 90°, 81.18°, 77.31°, 60°, 其中示意插图是超构原子的顶视图, 蓝色阴影代表只有外尔点简并能带出现的频率区间Fig. 3. (a)−(d) Band structure with
$ \theta $ = 90°, 81.18°, 77.31° and 60°, respectively. The inset is the top view of meta-atom, and the shaded region denotes the frequency range with only Weyl degenerate bands appears.
图 4 (a) 第一布里渊区内两个切面示意图, 黄色平面代表
$ {k}_{x} $ = 0.432的切面, 虚线代表两个切面的交线; (b)$ {k}_{x} $ = 0.432时的体投影能带(灰色曲线)和$\left\langle {001} \right\rangle $ 晶面的表面态(蓝色和红色曲线); (c) 相应的电场强度剖面图, 标记对应于(b)里的位置, 水平方向为z方向; (d) 第一布里渊区内两个切面示意图, 黄色平面代表$ {k}_{x}={k}_{y} $ 的切面, 虚线代表两个切面的交线; (e) Γ →M方向的体投影能带(灰色曲线)和$\left\langle {001} \right\rangle $ 晶面的表面态(桔色曲线); (f)相应的电场强度剖面图, 标记对应于(e)里的位置, 水平方向为z方向Fig. 4. (a) Two cut planes in the first Brillouin zone where the yellow plane represents the plane of
$ {k}_{x} $ = 0.432 and the dashed line denotes the intersecting line of two planes; (b) the projected bulk bands (grey curves) and the associated$\left\langle {001} \right\rangle $ surface state (blue and red curves) at$ {k}_{x} $ = 0.432; (c) the magnitude of E -fields of the eigenmodes for different marks in (b), where z axis is horizontal; (d) two cut planes in the first Brillouin zone where the yellow plane represents the plane of$ {k}_{x}={k}_{y} $ and the dashed line denotes the intersecting line of two planes; (e) the projected bulk bands (grey curves) and the associated$\left\langle {001} \right\rangle $ surface state (orange curves) along Γ → M; (f) the magnitude of E -fields of the eigenmodes for different marks in (e), where z axis is horizontal.表 1 外尔点的频移效应
Table 1. Frequency shift of Weyl points.
角度/(°) 外尔点频率/GHz 频差/GHz 带宽/GHz $ {f}_{1} $ 中心频率 $ {f}_{2} $ 90 15.70 15.70 15.70 0 3.07 81.18 15.90 15.69 15.48 0.43 2.29 77.31 16.07 15.70 15.33 0.74 1.93 60 16.40 15.585 14.77 1.63 0 
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[1] Ashcroft N W, Mermin N D 1976 Solid State Physics (New York: Holt, Rinehart and Winston) pp284–311
[2] 张智强, 蒋庆东, 陈垂针, 江华 2018 物理学进展 38 101
Google Scholar
Zhang Z Q, Jiang Q D, Chen C Z, Jiang H 2018 Progr. Phys. 38 101
Google Scholar
[3] Wan X G, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101
Google Scholar
[4] Burkov A A, Balents L 2011 Phys. Rev. Lett. 107 127205
Google Scholar
[5] Xu G, Weng H M, Wang Z J, Dai X, Fang Z 2011 Phys. Rev. Lett. 107 186806
Google Scholar
[6] Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001
Google Scholar
[7] Xu S Y, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C L, Sankar R, Chang G Q, Yuan Z J, Lee C C, Huang S M, Zheng H, Ma J, Sanchez D S, Wang B K, Bansil A, Chou F C, Shibayev P P, Lin H, Jia S, Hasan M, Zahid 2015 Science 349 613
Google Scholar
[8] Weng H M, Fang C, Fang Z, Bernevig B A, Dai X 2015 Phys. Rev. X 5 0110291
[9] Soluyanov A A, Gresch D, Wang Z J, Wu Q S, Troyer M, Dai X, Bernevig B A 2015 Nature 527 495
Google Scholar
[10] Huang L N, Mccormick T M, Ochi M, Zhao Z Y, Suzuki M T, Arita R, Wu Y, Mou D X, Cao H B, Yan J Q, Trivedi N, Kaminski A 2016 Nat. Mater. 15 1155
Google Scholar
[11] Lu L, Joannopoulos J D, Soljacic M 2014 Nat. Photonics 8 821
Google Scholar
[12] Lu L, Wang Z Y, Ye D X, Ran L X, Fu L, Joannopoulos J D, Soljacic M 2015 Science 349 622
Google Scholar
[13] Chen W J, Xiao M, Chan C T 2016 Nat. Commun. 7 13038
Google Scholar
[14] Noh J, Huang S, Leykam D, Chong Y D, Chen K P, Rechtsman M C 2017 Nat. Phys. 13 611
Google Scholar
[15] Yang B, Guo Q H, Tremain B, Barr L E, Gao W L, Liu H C, Beri B, Xiang Y J, Fan D Y, Hibbins A P, Zhang S 2017 Nat. Commun. 8 7
Google Scholar
[16] Yang Y H, Gao Z, Xue H R, Zhang L, He M J, Yang Z J, Singh R J, Chong Y D, Zhang B L, Chen H S 2019 Nature 565 622
Google Scholar
[17] Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903
Google Scholar
[18] Zhang X J, Wang H X, Lin Z K, Tian Y, Xie B Y, Lu M H, Chen Y F, Jiang J H 2019 Nat. Phys. 15 582
Google Scholar
[19] Yang B, Guo Q, Tremain B, Liu R, Barr L E, Yan Q, Gao W, Liu H, Xiang Y, Chen J, Fang C, Hibbins A, Lu L, Zhang S 2018 Science 359 1013
Google Scholar
[20] Nguyen V H, Charlier J C 2018 Phys. Rev. B 97 235113
Google Scholar
[21] Guan S, Yu Z M, Liu Y, Liu G B, Dong L, Lu Y, Yao Y, Yang S A 2017 NPJ Quantum Mater. 2 23
Google Scholar
[22] Westström A, Ojanen T 2017 Phys. Rev. X 7 041026
[23] Pendry J B, Schurig D, Smith D R 2006 Science 312 1780
Google Scholar
[24] Urzhumov Y A, Smith D R 2010 Phys. Rev. Lett. 105 163901
Google Scholar
[25] Lu H Z 2019 Natl. Sci. Rev. 6 208
Google Scholar
[26] Zhang C, Zhang Y, Yuan X, Lu S H, Zhang J L, Narayan A, Liu Y W, Zhang H Q, Ni Z L, Liu R, Choi E S, Suslov A, Sanvito S, Pi L, Lu H Z, Potter A C, Xiu F X 2019 Nature 565 331
Google Scholar
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