磁性拓扑材料中贝利曲率驱动的非常规电输运行为

杨金颖; 王彬彬; 刘恩克

doi:10.7498/aps.72.20230995

摘要

近年来, 磁性拓扑材料特别是磁性Weyl半金属越来越多地被发现, 为研究拓扑输运行为提供了重要载体. 磁性拓扑半金属材料具有动量空间的强贝利曲率, 显著增强了电子的常规横向输运行为, 也使得曾经被忽略或无法观测的输运效应逐渐浮现出来, 导致当前广泛采用的经典输运方程不能准确地描述磁性拓扑电子的输运行为. 本文从半经典输运方程出发, 介绍磁性拓扑材料中新近出现的非常规电输运行为, 内容涉及化学掺杂、磁场调制拓扑电子态、贝利曲率相关的线性正磁电阻及磁场线性依赖的输运行为. 这些行为为磁性与拓扑相互作用下的电输运行为提供新的理解和思考. 最后, 对非常规电输运的发展进行总结和展望.

关键词:

Abstract

In recent years, more and more magnetic topological materials, especially magnetic Weyl semimetals, have been discovered, providing a platform for studying the electronic transport behavior. The strong Berry curvature of magnetic topological materials can significantly enhance the conventional transverse transport behaviors, and can also make the transport phenomena that have been overlooked or unobserved appear gradually. In this review, the semi-classical equation is used to understand the anomalous transport behaviors in magnetic topological materials. The intrinsic anomalous Hall conductivity is obtained by integrating the Berry curvature of the occupied states, which is determined by the electronic band structure. The topological electronic state can be modulated by magnetic field and doping, and the anomalous Hall conductivity was changed with the evolution of the Berry curvature. A linear positive magnetoresistance behavior associated with the Berry curvature and magnetic field is introduced, which establishes the relation between the Berry curvature and the longitudinal transport. Due to the presence of tilted Weyl cone, the conductivity terms related to the first power of magnetic field are observed in magnetic Weyl systems. These behaviors under the interaction of topology and magnetic provide a new understanding and insight for the electric transport behaviors. At last, this review also provides a viewpoint on the field of magnetic topological physics.

Keywords:

作者及机构信息

1.
中国科学院物理研究所, 磁学国家重点实验室, 北京　100190

2.
中国科学院大学, 北京　100049

通信作者: 刘恩克, ekliu@iphy.ac.cn

基金项目: 国家重点基础研究发展计划(批准号: 2022YFA1403800, 2022YFA1403400)、国家自然科学基金(批准号: 11974394)、中国科学院战略性先导科技专项B类(批准号: XDB33000000)和综合极端装置(SECUF)中国科学院建制化科研项目资助的课题.

Authors and contacts

1.
State Key Laboratory for Magnetism, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

2.
University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Liu En-Ke, ekliu@iphy.ac.cn

Funds: Project supported by the State Key Development Program for Basic Research of China (Grant Nos. 2022YFA1403800, 2022YFA1403400), the National Natural Science Foundation of China (Grant No. 11974394), the Strategic Priority Research Program (B) of Chinese Academy of Sciences (Grant No. XDB33000000), and the Synergetic Extreme Condition User Facility (SECUF), Chinese Academy of Sciences.

文章全文 : translate this paragraph

参考文献

[1]	Chang C Z, Zhang J S, Feng X, Shen J, Zhang Z C, Guo M H, Li K, Ou Y B, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S H, Chen X, Jia J F, Dai X, Fang Z, Zhang S C, He K, Wang Y Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167 Google Scholar
[2]	Deng Y J, Yu Y J, Meng Z S, Guo Z X, Xu Z H, Wang J, Chen X H, Zhang Y B 2020 Science 367 895 Google Scholar
[3]	Liu E K, Sun Y, Kumar N, Muechler L, Sun A L, Jiao L, Yang S Y, Liu D F, Liang A J, Xu Q N, Kroder J, Suss V, Borrmann H, Shekhar C, Wang Z S, Xi C Y, Wang W H, Schnelle W, Wirth S, Chen Y L, Goennenwein S T B, Felser C 2018 Nat Phys 14 1125 Google Scholar
[4]	Ilya Belopolski K M, Sanchez D S, Chang G Q, Ernst B, Yin J X, Zhang S T, Cochran T, Shumiya N, Zheng H, Singh B, Bian G, Multer D, Litskevich M, Zhou X T, Huang S M, Wang B K, Chang T R, Xu S Y, Bansil A, Felser C, Lin H, Hasan M Z 2019 Science 365 1278 Google Scholar
[5]	Li P, Koo J, Ning W, Li J, Miao L, Min L, Zhu Y, Wang Y, Alem N, Liu C X, Mao Z, Yan B 2020 Nat. Commun. 11 3476 Google Scholar
[6]	Guin S N, Vir P, Zhang Y, Kumar N, Watzman S J, Fu C, Liu E, Manna K, Schnelle W, Gooth J, Shekhar C, Sun Y, Felser C 2019 Adv. Mater. 31 e1806622 Google Scholar
[7]	Guin S N, Manna K, Noky J, Watzman S J, Fu C, Kumar N, Schnelle W, Shekhar C, Sun Y, Gooth J, Felser C 2019 NPG Asia Mater. 11 16 Google Scholar
[8]	Xing Y Q, Shen J L, Chen H, Huang L, Gao Y X, Zheng Q, Zhang Y Y, Li G, Hu B, Qian G J, Cao L, Zhang X L, Fan P, Ma R S, Wang Q, Yin Q W, Lei H C, Ji W, Du S X, Yang H T, Wang W H, Shen C M, Lin X, Liu E K, Shen B G, Wang Z Q, Gao H J 2020 Nat. Commun. 11 5613 Google Scholar
[9]	Liu D F, Liang A J, Liu E K, Xu Q N, Li Y W, Chen C, Pei D, Shi W J, Mo S K, Dudin P, Kim T, Cacho C, Li G, Sun Y, Yang L X, Liu Z K, Parkin S S P, Felser C, Chen Y L 2019 Science 365 1282 Google Scholar
[10]	Okamura Y, Minami S, Kato Y, Fujishiro Y, Kaneko Y, Ikeda J, Muramoto J, Kaneko R, Ueda K, Kocsis V, Kanazawa N, Taguchi Y, Koretsune T, Fujiwara K, Tsukazaki A, Arita R, Tokura Y, Takahashi Y 2020 Nat. Commun. 11 4619 Google Scholar
[11]	Wang Q Y, Zeng Y, Yuan K, Zeng Q Q, Gu P F, Xu X L, Wang H R, Han Z, Nomura K, Wang W H, Liu E K, Hou Y L, Ye Y 2022 Nat. Electron. 6 119 Google Scholar
[12]	Yang S Y, Noky J, Gayles J, Dejene F K, Sun Y, Dorr M, Skourski Y, Felser C, Ali M N, Liu E K, Parkin S S P 2020 Nano Lett. 20 7860 Google Scholar
[13]	Howard S, Jiao L, Wang Z, Morali N, Batabyal R, Kumar-Nag P, Avraham N, Beidenkopf H, Vir P, Liu E K, Shekhar C, Felser C, Hughes T, Madhavan V 2021 Nat. Commun. 12 4269 Google Scholar
[14]	Tanaka M, Fujishiro Y, Mogi M, Kaneko Y, Yokosawa T, Kanazawa N, Minami S, Koretsune T, Arita R, Tarucha S, Yamamoto M, Tokura Y 2020 Nano Lett. 20 7476 Google Scholar
[15]	Zeng Q Q, Gu G X, Shi G, Shen J L, Ding B, Zhang S, Xi X K, Felser C, Li Y Q, Liu E K 2021 Sci. China Phys. Mech. Astron. 64 287512 Google Scholar
[16]	Sanchez D S, Chang G, Belopolski I, Lu H, Yin J X, Alidoust N, Xu X, Cochran T A, Zhang X, Bian Y, Zhang S S, Liu Y Y, Ma J, Bian G, Lin H, Xu S Y, Jia S, Hasan M Z 2020 Nat. Commun. 11 3356 Google Scholar
[17]	Shen J L, Gao J C, Yi C J, Li M, Zhang S, Yang J Y, Wang B B, Zhou M, Huang R J, Wei H X, Yang H T, Shi Y G, Xu X H, Gao H J, Shen B G, Li G, Wang Z J, Liu E K 2023 The Innovation 4 100399 Google Scholar
[18]	Yao Y G, Kleinman L, MacDonald A H, Sinova J, Jungwirth T, Wang D S, Wang E G, Niu Q 2004 Phys. Rev. Lett. 92 037204 Google Scholar
[19]	Shen J L, Yao Q S, Zeng Q Q, Sun H Y, Xi X K, Wu G H, Wang W H, Shen B G, Liu Q H, Liu E K 2020 Phys. Rev. Lett. 125 086602 Google Scholar
[20]	Zhang S, Wang Y, Zeng Q Q, Shen J L, Zheng X, Yang J, Wang Z, Xi C, Wang B, Zhou M, Huang R, Wei H, Yao Y, Wang S, Parkin S S P, Felser C, Liu E K, Shen B 2022 Proc. Natl. Acad. Sci. USA 119 e2208505119 Google Scholar
[21]	Xiao D, Shi J R, Niu Q 2005 Phys. Rev. Lett. 95 137204 Google Scholar
[22]	Ma D, Jiang H, Liu H, Xie X C 2019 Phys. Rev. B 99 115121 Google Scholar
[23]	Das K, Agarwal A 2019 Phys. Rev. B 99 085405 Google Scholar
[24]	Jiang B Y, Wang L J Y, Bi R, Fan J W, Zhao J J, Yu D P, Li Z L, Wu X S 2021 Phys. Rev. Lett. 126 236601 Google Scholar
[25]	Zeng Q Q, Yi C, Shen J L, Wang B B, Wei H, Shi Y G, Liu E K 2022 Appl. Phys. Lett. 121 162405 Google Scholar
[26]	Berry M V 1997 Proc. Math. Phys. Eng. Sci. 392 45 Google Scholar
[27]	Chang M C, Niu Q 1995 Phys. Rev. Lett. 75 1348 Google Scholar
[28]	Chang M C, Niu Q 1996 Phys. Rev. B 53 7010 Google Scholar
[29]	Sundaram G, Niu Q 1999 Phys. Rev. B 59 14915 Google Scholar
[30]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[31]	Shen J L, Zeng Q Q, Zhang S, Sun H Y, Yao Q S, Xi X K, Wang W H, Wu G H, Shen B G, Liu Q H, Liu E K 2020 Adv. Funct. Mater. 30 2000830 Google Scholar

施引文献

图 1 贝利曲率相关的非常规电输运行为

Fig. 1. The unconventional electric transport behaviors related to the Berry curvature.

下载: 全尺寸图片幻灯片

图 2 Co_3–xNi_xSn₂S₂的能带结构与内禀反常霍尔电导^[19]

Fig. 2. The band structure and intrinsic anomalous Hall conductivity in Co_3–xNi_xSn₂S₂^[19].

下载: 全尺寸图片幻灯片

图 3 Co₂MnAl中随着外磁场转动贝利曲率分布的演化^[5]

Fig. 3. The evolution of Berry curvature distribution in Co₂MnAl with the rotation of magnetic field^[5].

下载: 全尺寸图片幻灯片

图 4 EuB₆实空间磁矩方向、k空间能带结构及输运行为演化示意图^[17]

Fig. 4. Schematic diagram of the evolution of the real space magnetic moment direction, the k-space band structure and the transport behavior in EuB₆^[17].

下载: 全尺寸图片幻灯片

图 5 CoS₂线性正磁电阻行为及含温度纵向横向输运实验数据拟合结果^[20]

Fig. 5. The linear positive magnetoresistance behavior in CoS₂ and experimental data fitting results of longitudinal and transverse with temperature^[20].

下载: 全尺寸图片幻灯片

图 6 EuB₆中PHE和AMR中关于磁场奇对称输运行为^[25]

Fig. 6. The antisymmetric transport behavior of PHE and AMR in EuB₆^[25].

下载: 全尺寸图片幻灯片

图 7 Co₃Sn₂S₂中关于磁场奇对称的纵向和横向电阻^[24]

Fig. 7. The antisymmetric longitudinal and transverse electric resistivity in Co₃Sn₂S₂^[24].

下载: 全尺寸图片幻灯片

表 1 各类输运效应与材料体系对照表

Table 1. Comparison of various transport effects and material systems.

输运效应	物理机制	材料体系
反常霍尔效应	掺杂引起局域无序导致拓扑能带被调制	Ni-doped Co₃Sn₂S₂
反常霍尔效应	外磁场调制外尔点的产生和湮灭	Co₂MnAl
反常霍尔效应	外磁场诱发非线性磁结构调制拓扑能带	EuB₆
纵向磁电阻	贝利曲率和外磁场共同作用	CoS₂
平面霍尔效应和各向异性磁电阻	倾斜外尔锥导致的关于磁场奇对称行为	Co₃Sn₂S₂, EuB₆

下载: 导出CSV

[1]	Chang C Z, Zhang J S, Feng X, Shen J, Zhang Z C, Guo M H, Li K, Ou Y B, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S H, Chen X, Jia J F, Dai X, Fang Z, Zhang S C, He K, Wang Y Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167 Google Scholar
[2]	Deng Y J, Yu Y J, Meng Z S, Guo Z X, Xu Z H, Wang J, Chen X H, Zhang Y B 2020 Science 367 895 Google Scholar
[3]	Liu E K, Sun Y, Kumar N, Muechler L, Sun A L, Jiao L, Yang S Y, Liu D F, Liang A J, Xu Q N, Kroder J, Suss V, Borrmann H, Shekhar C, Wang Z S, Xi C Y, Wang W H, Schnelle W, Wirth S, Chen Y L, Goennenwein S T B, Felser C 2018 Nat Phys 14 1125 Google Scholar
[4]	Ilya Belopolski K M, Sanchez D S, Chang G Q, Ernst B, Yin J X, Zhang S T, Cochran T, Shumiya N, Zheng H, Singh B, Bian G, Multer D, Litskevich M, Zhou X T, Huang S M, Wang B K, Chang T R, Xu S Y, Bansil A, Felser C, Lin H, Hasan M Z 2019 Science 365 1278 Google Scholar
[5]	Li P, Koo J, Ning W, Li J, Miao L, Min L, Zhu Y, Wang Y, Alem N, Liu C X, Mao Z, Yan B 2020 Nat. Commun. 11 3476 Google Scholar
[6]	Guin S N, Vir P, Zhang Y, Kumar N, Watzman S J, Fu C, Liu E, Manna K, Schnelle W, Gooth J, Shekhar C, Sun Y, Felser C 2019 Adv. Mater. 31 e1806622 Google Scholar
[7]	Guin S N, Manna K, Noky J, Watzman S J, Fu C, Kumar N, Schnelle W, Shekhar C, Sun Y, Gooth J, Felser C 2019 NPG Asia Mater. 11 16 Google Scholar
[8]	Xing Y Q, Shen J L, Chen H, Huang L, Gao Y X, Zheng Q, Zhang Y Y, Li G, Hu B, Qian G J, Cao L, Zhang X L, Fan P, Ma R S, Wang Q, Yin Q W, Lei H C, Ji W, Du S X, Yang H T, Wang W H, Shen C M, Lin X, Liu E K, Shen B G, Wang Z Q, Gao H J 2020 Nat. Commun. 11 5613 Google Scholar
[9]	Liu D F, Liang A J, Liu E K, Xu Q N, Li Y W, Chen C, Pei D, Shi W J, Mo S K, Dudin P, Kim T, Cacho C, Li G, Sun Y, Yang L X, Liu Z K, Parkin S S P, Felser C, Chen Y L 2019 Science 365 1282 Google Scholar
[10]	Okamura Y, Minami S, Kato Y, Fujishiro Y, Kaneko Y, Ikeda J, Muramoto J, Kaneko R, Ueda K, Kocsis V, Kanazawa N, Taguchi Y, Koretsune T, Fujiwara K, Tsukazaki A, Arita R, Tokura Y, Takahashi Y 2020 Nat. Commun. 11 4619 Google Scholar
[11]	Wang Q Y, Zeng Y, Yuan K, Zeng Q Q, Gu P F, Xu X L, Wang H R, Han Z, Nomura K, Wang W H, Liu E K, Hou Y L, Ye Y 2022 Nat. Electron. 6 119 Google Scholar
[12]	Yang S Y, Noky J, Gayles J, Dejene F K, Sun Y, Dorr M, Skourski Y, Felser C, Ali M N, Liu E K, Parkin S S P 2020 Nano Lett. 20 7860 Google Scholar
[13]	Howard S, Jiao L, Wang Z, Morali N, Batabyal R, Kumar-Nag P, Avraham N, Beidenkopf H, Vir P, Liu E K, Shekhar C, Felser C, Hughes T, Madhavan V 2021 Nat. Commun. 12 4269 Google Scholar
[14]	Tanaka M, Fujishiro Y, Mogi M, Kaneko Y, Yokosawa T, Kanazawa N, Minami S, Koretsune T, Arita R, Tarucha S, Yamamoto M, Tokura Y 2020 Nano Lett. 20 7476 Google Scholar
[15]	Zeng Q Q, Gu G X, Shi G, Shen J L, Ding B, Zhang S, Xi X K, Felser C, Li Y Q, Liu E K 2021 Sci. China Phys. Mech. Astron. 64 287512 Google Scholar
[16]	Sanchez D S, Chang G, Belopolski I, Lu H, Yin J X, Alidoust N, Xu X, Cochran T A, Zhang X, Bian Y, Zhang S S, Liu Y Y, Ma J, Bian G, Lin H, Xu S Y, Jia S, Hasan M Z 2020 Nat. Commun. 11 3356 Google Scholar
[17]	Shen J L, Gao J C, Yi C J, Li M, Zhang S, Yang J Y, Wang B B, Zhou M, Huang R J, Wei H X, Yang H T, Shi Y G, Xu X H, Gao H J, Shen B G, Li G, Wang Z J, Liu E K 2023 The Innovation 4 100399 Google Scholar
[18]	Yao Y G, Kleinman L, MacDonald A H, Sinova J, Jungwirth T, Wang D S, Wang E G, Niu Q 2004 Phys. Rev. Lett. 92 037204 Google Scholar
[19]	Shen J L, Yao Q S, Zeng Q Q, Sun H Y, Xi X K, Wu G H, Wang W H, Shen B G, Liu Q H, Liu E K 2020 Phys. Rev. Lett. 125 086602 Google Scholar
[20]	Zhang S, Wang Y, Zeng Q Q, Shen J L, Zheng X, Yang J, Wang Z, Xi C, Wang B, Zhou M, Huang R, Wei H, Yao Y, Wang S, Parkin S S P, Felser C, Liu E K, Shen B 2022 Proc. Natl. Acad. Sci. USA 119 e2208505119 Google Scholar
[21]	Xiao D, Shi J R, Niu Q 2005 Phys. Rev. Lett. 95 137204 Google Scholar
[22]	Ma D, Jiang H, Liu H, Xie X C 2019 Phys. Rev. B 99 115121 Google Scholar
[23]	Das K, Agarwal A 2019 Phys. Rev. B 99 085405 Google Scholar
[24]	Jiang B Y, Wang L J Y, Bi R, Fan J W, Zhao J J, Yu D P, Li Z L, Wu X S 2021 Phys. Rev. Lett. 126 236601 Google Scholar
[25]	Zeng Q Q, Yi C, Shen J L, Wang B B, Wei H, Shi Y G, Liu E K 2022 Appl. Phys. Lett. 121 162405 Google Scholar
[26]	Berry M V 1997 Proc. Math. Phys. Eng. Sci. 392 45 Google Scholar
[27]	Chang M C, Niu Q 1995 Phys. Rev. Lett. 75 1348 Google Scholar
[28]	Chang M C, Niu Q 1996 Phys. Rev. B 53 7010 Google Scholar
[29]	Sundaram G, Niu Q 1999 Phys. Rev. B 59 14915 Google Scholar
[30]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[31]	Shen J L, Zeng Q Q, Zhang S, Sun H Y, Yao Q S, Xi X K, Wang W H, Wu G H, Shen B G, Liu Q H, Liu E K 2020 Adv. Funct. Mater. 30 2000830 Google Scholar

[1]	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
[2]	祝鑫强, 王剑, 朱璨, 罗丰, 陈树权, 徐佳辉, 徐峰, 王嘉赋, 张艳, 孙志刚. Co₃Sn₂S₂单晶的磁性和电-热输运性能. 物理学报, 2023, 72(17): 177102. doi: 10.7498/aps.72.20230621
[3]	刘畅, 王亚愚. 磁性拓扑绝缘体中的量子输运现象. 物理学报, 2023, 72(17): 177301. doi: 10.7498/aps.72.20230690
[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]	孙慧敏, 何庆林. 层状磁性拓扑材料中的物理问题与实验进展. 物理学报, 2021, 70(12): 127302. doi: 10.7498/aps.70.20210133
[6]	王朝, 张铭, 张持, 王如志, 严辉. n = 2 Ruddlesden-Popper Sr₃B₂Se₇ (B = Zr, Hf) 非常规铁电性的第一性原理研究. 物理学报, 2021, 70(11): 116302. doi: 10.7498/aps.70.20202142
[7]	刘雨亭, 贺文宇, 刘军伟, 邵启明. 二维材料中贝里曲率诱导的磁性响应. 物理学报, 2021, 70(12): 127303. doi: 10.7498/aps.70.20202132
[8]	李建新. 自旋涨落与非常规超导配对. 物理学报, 2021, 70(1): 017408. doi: 10.7498/aps.70.20202180
[9]	胡江平. 探索非常规高温超导体. 物理学报, 2021, 70(1): 017101. doi: 10.7498/aps.70.20202122
[10]	刘祥, 米文博. Verwey相变处Fe₃O₄的结构、磁性和电输运特性. 物理学报, 2020, 69(4): 040505. doi: 10.7498/aps.69.20191763
[11]	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
[12]	谭丛兵, 钟向丽, 王金斌. 铁电材料中的极性拓扑结构. 物理学报, 2020, 69(12): 127702. doi: 10.7498/aps.69.20200311
[13]	李宏, 张斯淇, 郭明, 李美萱, 宋立军. Fabry-Perot腔与光学参量放大复合系统中实现可调谐的非常规光子阻塞. 物理学报, 2019, 68(12): 124203. doi: 10.7498/aps.68.20190154
[14]	赵国栋, 杨亚利, 任伟. 钙钛矿型氧化物非常规铁电研究进展. 物理学报, 2018, 67(15): 157504. doi: 10.7498/aps.67.20180936
[15]	孙晓晨, 何程, 卢明辉, 陈延峰. 人工带隙材料的拓扑性质. 物理学报, 2017, 66(22): 224203. doi: 10.7498/aps.66.224203
[16]	齐伟华, 马丽, 李壮志, 唐贵德, 吴光恒. 金属价电子结构对磁性和电输运性质的影响. 物理学报, 2017, 66(2): 027101. doi: 10.7498/aps.66.027101
[17]	程金光. 高压调控的磁性量子临界点和非常规超导电性. 物理学报, 2017, 66(3): 037401. doi: 10.7498/aps.66.037401
[18]	张志东. 磁性材料的磁结构、磁畴结构和拓扑磁结构. 物理学报, 2015, 64(6): 067503. doi: 10.7498/aps.64.067503
[19]	李诚迪, 赵敬龙, 仲崇贵, 董正超, 方靖淮. 量子顺电EuTiO3材料基态磁性的第一性原理研究. 物理学报, 2014, 63(8): 087502. doi: 10.7498/aps.63.087502
[20]	张永祥, 孔贵芹, 俞建宁. 振动筛系统的两类余维三分岔与非常规混沌演化. 物理学报, 2008, 57(10): 6182-6187. doi: 10.7498/aps.57.6182

计量

文章访问数: 3950
PDF下载量: 336
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