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弹性拓扑材料研究进展

陈毅 张泉 张亚飞 夏百战 刘晓宁 周萧明 陈常青 胡更开

陈毅, 张泉, 张亚飞, 夏百战, 刘晓宁, 周萧明, 陈常青, 胡更开. 弹性拓扑材料研究进展. 力学进展, 2021, 51(2): 189-256 doi: 10.6052/1000-0992-21-015
引用本文: 陈毅, 张泉, 张亚飞, 夏百战, 刘晓宁, 周萧明, 陈常青, 胡更开. 弹性拓扑材料研究进展. 力学进展, 2021, 51(2): 189-256 doi: 10.6052/1000-0992-21-015
Chen Y, Zhang Q, Zhang Y F, Xia B Z, Liu X N, Zhou X M, Chen C Q, Hu G K. Research progress of elastic topological materials. Advances in Mechanics, 2021, 51(2): 189-256 doi: 10.6052/1000-0992-21-015
Citation: Chen Y, Zhang Q, Zhang Y F, Xia B Z, Liu X N, Zhou X M, Chen C Q, Hu G K. Research progress of elastic topological materials. Advances in Mechanics, 2021, 51(2): 189-256 doi: 10.6052/1000-0992-21-015

弹性拓扑材料研究进展——

doi: 10.6052/1000-0992-21-015
基金项目: 感谢赵玉臣提供连续介质边界态相关素材. 国家自然科学基金(11632003, 1173207, 11872111, 11972080, 11972083, 11991030, 12002030, 12072108)资助项目.
详细信息
    作者简介:

    陈毅, 主要开展声/弹性波超材料/拓扑材料研究. 2012年、2018年在北京理工大学分别获学士、博士学位, 博士导师胡更开教授. 2019年至今获德国洪堡基金会资助, 于德国卡尔斯鲁厄理工学院开展博士后研究, 合作导师德国科学院院士Martin Wegener教授. 获博士后创新人才支持计划、国家自然科学基金青年基金项目支持. 以第一作者或通讯作者在《Nature Communications》、《Physical Review Letters》、《Journal of the Mechanics and Physics of Solids》等期刊发表SCI论文近20篇

    通讯作者:

    hugeng@bit.edu.cn

  • 中图分类号: O343, O424

Research progress of elastic topological materials

More Information
  • 摘要: 拓扑绝缘体起源于量子波动系统, 因其单向传输、能量无耗散等新奇物理性质, 近年逐渐被拓展到电磁波、声波、弹性波等经典波动领域, 为经典波的调控提供了新思路. 本文将系统介绍拓扑绝缘体理论及其在弹性波领域的相关研究进展. 首先以一维、二维离散点阵系统为例, 阐释拓扑物理研究中的基本数学、物理概念, 如狄拉克锥、能带翻转、贝里曲率、拓扑数等. 随后, 依次讨论弹性系统谷霍尔绝缘体、陈绝缘体、自旋霍尔绝缘体的设计思想及目前研究进展, 并讨论了近年来逐渐受关注的高阶拓扑现象. 最后, 讨论了静力学中拓扑孤立子、拓扑零能模式现象.

     

  • 图  1  拓扑基本概念与拓扑绝缘体性质. (a) 球形橡皮泥可光滑连续变形至圆饼, 表明球和圆饼在几何上拓扑等价. 球形橡皮泥被撕裂产生1个孔洞, 则拓扑发生变化, 此时其与甜甜圈在几何上拓扑等价. 孔洞数可以看作分类几何的拓扑数; (b) 拓扑绝缘体内部不导电(传波), 但边界导电(传波), 其边界导电性是一种拓扑性质, 不易受到杂质、缺陷等影响

    图  2  (a) 示例二维周期系统; (b) 由于周期系统在实空间的周期性, 菱形布里渊区四条边两两等价, 将对应边界拼接起来后, 布里渊区在几何上与圆环表面拓扑等价. 圆环表面每一处有对应的波矢及波函数

    图  3  (a) 一维双原子链模型; (b) 三种典型情况的频散曲线, 插图表示相应点处的模态相位. 箭头表示质点的运动方向. 波数q = π/a处, γ = 0.2的声学模态与γ = −0.2时光学模态一样, 且 γ = 0.2的光学模态与γ = −0.2时声学模态一样. 即参数γ由正变负时, 出现了模态顺序交换, 称之为能带翻转

    图  4  参数γ由正变负时, 复平面上z(q)对应的曲线η跨过坐标轴Re(z) = 0, 系统缠绕数υ由0变为1, 产生拓扑变化

    图  5  (a)含400个单胞(γ < 0, 即υ = 1)的点阵结构, 其固定边界上支持1支边界态模式, 如图(b)所示; (c) 含400个单胞(γ > 0, 即υ = 0)的点阵结构, 其固定边界上不支持边界态模式; (d) 图(b)所示边界态对应位移分布; (e) 右端固定的含400个单胞的点阵结构在不同刚度参数下的特征频谱

    图  6  (a) 含界面的点阵结构, 其两侧各有200个单胞; (b) γ = −0.5时的界面模态; (c) γ = 0.5时的界面模态; (d) 两侧各有200个单胞的含界面点阵结构在不同刚度参数下的特征频谱

    图  7  (a) 二维蜂窝质量弹簧系统示例, 每一个单胞包含两个质点, 最近邻质点之间由弹簧连接; (b) 第一布里渊区及角点K′和K. 图中三个K点相互之间通过倒格矢量b1b2平移得到, 等价于同一个K点, 同理K′也类似. 来自文献(Chen et al. 2019)

    图  8  二维蜂窝质量弹簧系统第2、第3支色散曲面. (a) 参数mp = mq = 1.0, t = 1.0, 第2、第3支色散曲面在K′和K处呈线性简并特征, 局部色散曲面在K′和K处呈现双锥, 称为狄拉克锥(Dirac cone); (b) 参数mp = 0.9, mq = 1.1, t = 1.0, 色散曲面在布里渊区角点K′和K处取不同极值, 系统存在完全禁带

    图  9  常见二维拓扑绝缘体色散曲线及波传播规律. (a) 霍尔绝缘体色散曲线, 灰色区域对应禁带, 斜线为界面态色散曲线, 该界面态只能单向传播, 如图(d); (b) 自旋霍尔绝缘体带结构曲线, 禁带内存在斜率相反的两条色散曲线, 对应于沿界面相反方向传播的上、下自旋界面态, 如图(e); (c) 正负谷霍尔绝缘体组成的界面传播色散曲线, 禁带区域有斜率相反的两条色散曲线, 表明该界面支持双向传播的界面态

    图  10  (a) 带结构曲线, 其中离散点表示精确解, 谷点KK′附近的连续曲线为微扰模型得出的解析解; (b)(c) 胞元中两质量不等时(mp = 0.9, mq = 1.1, t = 1)的精确贝里曲率及由微扰模型得出的解析贝里曲率. 来自文献(Chen et al. 2019)

    图  11  (a) 正负谷霍尔相组成的界面示例, 界面具体几何形式与方位角度θ有关; (b)(c) 锯齿形/扶手形界面对应于方位角θ = 0o/30o. 来自文献(Chen et al. 2019)

    图  12  (a) 正负谷霍尔相组成的条状几何结构; (b) 条状几何结构沿y方向的波传播带结构曲线, 计算中沿y方向施加Bloch波连续条件; (c)(e) 图(b)中谷点K′上下两个体态对应的质点位移场, 颜色表示水平方向位移幅值, 红色代表幅值更大; (d) 图(b)中界面态对应的质点位移场. 来自文献(Chen et al. 2019)

    图  13  (a)谷点K′/K界面态对应的质点位移场; (b)(c) 界面附近质点的运动轨迹, 黑色圆点代表质点的位置, 红色/蓝色代表绕平衡位置顺时针/逆时针旋转; (d)(e) q/p质点水平位移幅值沿着x方向的分布. 来自文献(Chen et al. 2019)

    图  14  (a) 局域共振型谷霍尔绝缘体离散模型; (b) 对应的第一布里渊区以及两个布里渊区角点K′和K. 来自文献(Zhang et al. 2020)

    图  15  (a) 单胞中两个局域振子完全相同时的频散曲线(灰色点线)以及不含局域振子的蜂窝点阵的频散曲线(红色实线); (b) 单胞中两个局域振子的质量存在小幅差异时的频散曲线; (c) 图(b)中前两条频散分支对应的贝里曲率. 来自文献(Zhang et al. 2020)

    图  16  (a) 设计的局域共振型谷霍尔绝缘体微结构; (b) 单胞内两个局域振子完全相同时的频散曲线, 颜色表征极化模式, 值为1对应出平面极化振动; (c) 单胞内两个局域振子的质量存在小幅差异时的频散曲线; (d) 单胞内两个局域振子的质量存在小幅差异时的出平面等效密度. 来自文献(Zhang et al. 2020)

    图  17  (a) 包含界面的条带状超胞; (b) 条带状超胞的频散曲线, 绿色标注的为界面态分支; (c) f = 2500 Hz时的特征模态; (d) (e) (f) f = 2500 Hz时三种有限尺寸结构(不包含界面, “直线”形界面路径, “Z”形界面路径)的稳态位移场; (g) 图(e)中标注的蓝色虚线上的振幅分布; (h) 三种有限尺寸结构(不包含界面的体态, “直线”形界面路径, “Z”形界面路径)的透射率曲线. 来自文献(Zhang et al. 2020)

    图  18  制备的局域共振型谷霍尔绝缘体以及实验测试系统. 来自文献(Zhang et al. 2020)

    图  19  (a) 图18中A点和B点的实验测得频响曲线; (b) (c) (d) 频率1500 Hz, 2045 Hz和2500 Hz时的均方根速度场. 来自文献(Zhang et al. 2020)

    图  20  左侧: 图19(a)中的频响曲线; 右侧: 从实验带隙位置反演得到微结构梁的真实刚度后, 重新计算的图17(a)所示条带状超胞的带结构. 来自文献(Zhang et al. 2020)

    图  21  (a) 单胞不含磁流体时, 布里渊区角点K处发生狄拉克简并; (b) 单胞其中一个空腔充满磁流体时, 空间反演对称性破坏, 使得狄拉克简并退化并形成带隙(蓝色频散分支); (c) 在图(b)中蓝色频散分支带隙范围内, 存在拓扑界面态传播模式; (d) 利用设计的可编程磁场可以控制每个单胞中磁流体的分布; (e) 因此, 通过改变单胞中磁流体的分布状态, 可以调整用于传播拓扑界面态的界面路径的形状; (f) 设计的单胞的几何参数. 来自文献(Zhang et al. 2019a)

    图  22  (a) 条带状超胞; (b) 出平面极化振动模式频散曲线; (c) f = 1234 Hz下界面态频散分支上的特征模态; (d) (e) (f) f = 1234 Hz下三种界面路径的稳态位移场, 分别对应直线形界面路径、“L”形界面路径、“Z”形界面路径. 来自文献(Zhang et al. 2019a)

    图  23  (a) 制备的16 × 16测试样件; (b)(c)(d) 实验测试的f = 1450 Hz下三种界面路径的稳态位移场, 分别对应直线形界面路径、“L”形界面路径、“Z”形界面路径. 来自文献(Zhang et al. 2019a)

    图  24  设计的可编程磁铁升降阵列系统. (a) 可编程控制软件; (b) 16通道继电器开关; (c) 磁铁升降阵列, 绿色圆圈标注的磁铁已由软件控制升起; (d) 磁铁升降阵列的侧视图. 来自文献(Zhang et al. 2019a)

    图  25  将离散蜂窝系统放置于旋转基座上可以打破时间反演对称, 实现霍尔绝缘体; 在与旋转平台固定的坐标系上, 质点受到科里奥利力及离心力, 科氏力与旋转基座角速度大小|Ω|线性相关, 离心力与角速度平方|Ω|2线性相关

    图  26  (a) 拓扑相分布图, 横坐标与基座旋转强弱相关, 纵坐标与胞元内两质点质量差相关; (b) 当参数处于A和C相的公共边界时, 如图中黄色圆点所示, 系统对应的带结构曲线. 来自文献(Chen et al. 2019)

    图  27  6种拓扑相组成的锯齿界面波传播色散曲线; 棕色代表体波色散曲线, 蓝色代表界面态色散曲线. 来自文献(Chen et al. 2019)

    图  28  弹性波沿6种拓扑界面传播时瞬态结果; 颜色代表位移幅值大小, 红色表示位移更大; 数值模拟区域大致包含78 × 90个单胞. 来自文献(Chen et al. 2019)

    图  29  (a) 夹杂六角排布的二维弹性陀螺复合结构示意图; (b) 单胞剖面图, 上下为滑移边界, 基体为弹性体, 夹杂为刚体, 其内部耦合一个陀螺转子; (c) 运动状态下的夹杂侧视图和顶视图. 来自文献(Zhao et al. 2020)

    图  30  (a) 非互易瑞利波传播仿真结果, 激励形式为上下振动点激励; (b) 归一化波速及表面附近质点位移场, 当α = 3.0时, 仅支持左行波; (c) 表面质点极化轨迹曲线. 来自文献(Zhao et al. 2020)

    图  31  能带折叠产生双重狄拉克锥. (a) 蜂窝排布质量弹簧点阵系统, 其中六边形超胞包含6个质点, 连接胞元内质点的弹簧的刚度为ti, 连接胞元间质点的弹簧的刚度为to; (b) 较小的六边形为图(a)中六边形超胞对应的第一布里渊区; 较大的六边形区域为刚度均匀分布时, 即ti = to对应第一布里渊区, 最简单胞为图(a)中菱形区域; (c) Γ点对应dp特征模态的位移分布示意图, 蓝色圆圈表示质点振动最大位置, 空心圆为质点平衡时位置. 来自文献(Chen et al. 2019)

    图  32  自旋霍尔相变. (a) ti > to时蜂窝质量弹簧系统的带结构曲线, 系统为平凡绝缘体; (b) ti = to对应的带结构曲线, Γ点带隙完全闭合, 系统处于拓扑相变临界状态; (c) ti < to对应的带结构曲线, 系统为自旋霍尔绝缘体; 颜色表示特征模态包含的pd特征模态成分多少. 来自文献(Chen et al. 2019)

    图  33  (a) 左侧自旋霍尔绝缘体与右侧平凡绝缘体组成的条状超胞, 界面沿y方向; (b) 超胞计算得到的带结构曲线, 红色/蓝色表示上/下自旋界面态, 灰色表示体态; (c) 上/下自旋界面态对应的质点位移幅值; (d)(e) 上/下自旋界面态中界面附近质点振动轨迹示意图, 红色/蓝色表示质点振动为顺/逆时针方向. 来自文献(Chen et al. 2019)

    图  34  自旋霍尔绝缘体单向传播模拟结果. (a)(c) 施加上/下自旋激励时, 胞元内各质点振动轨迹示意图; (b)(d) 上/下自旋激励时弹性波传播瞬态模拟结果, 红色代表质点位移幅值较大, 模拟区域共包含78 × 90个胞元. 来自文献(Chen et al. 2019)

    图  35  自旋霍尔绝缘体不同边界选取及其能带结构. (a) (b) 边界为完整胞元的条状超胞及其能带结构, 红/蓝色曲线对应自旋边界态; (c) (d) 边界为不完整胞元的条状超胞及其能带结构, 带结构中不包含边界态. 来自文献(Chen et al. 2019)

    图  36  弹性波中的类自旋自由度. (a) 弹性体表面瑞利波的色散曲线, 红/蓝表示向右/左传播色散曲线, 箭头表示质点旋转方向; (b) 边界施加逆/顺时针圆极化激励激发向右/左传播的瑞利波. 来自文献(Long et al. 2018)

    图  37  压电主动调节实现双重狄拉克锥. (a) 包含压电片的单胞; (b) 压电片可接入(c)负电容电路. (d) 电路为开路时带结构; (e) 接入负电容电路后的能带结构, 四条能带在Γ处简并; (f) 四个简并模态. 来自文献(Li et al. 2020)

    图  38  弹性波赝自旋态的构建. (a) 实验样品由两块弹性波绝缘体拼接构成, 左侧和右侧绝缘体的带隙范围一致, 但顶带和底带对应的特征模态互为反转(两侧绝缘体的能带具有不同的拓扑不变量); (b) 在两侧绝缘体的体带隙范围内存在两条界面态模式; (c) 由四种简并态构建的赝自旋基矢(S与A); (d) 实验探测到的由下向上传输的界面态, 出面位移场呈现出“+S → +A → −S ···”的时域特征, 对应于赝自旋态S + iA; (e) 实验探测到的由上向下传输的界面态, 出面位移场呈现出“+S → − A → − S ···”的时域特征, 对应于赝自旋态S − iA. 来自文献(Yu et al. 2018)

    图  39  对缺陷和拐角免疫的界面态传输功能. (a) 界面路径不含任何缺陷及拐角; (b) 界面路径上含有一个由孔洞缺失构成的“空位”缺陷; (c) 界面路径上含有一个由孔洞错位构成的“位错”缺陷; (d) 包含两个120°拐角的“Z”形界面路径; (e) 上述四种界面路径的传输率实验测试结果. 来自文献(Yu et al. 2018)

    图  40  弹性拓扑环形谐振器. (a) 实验样品包含一条平直、一条闭合环形界面; (b) 在左侧平直界面下端激发的pseudospin+赝自旋态向上传输; (c) 环形谐振腔中的能量谱实验结果, 可观测到两个关于狄拉克频率对称分布的共振峰分布; (d)(e) 图(c)中两个谐振频率下的出面位移场及能流分布. 来自文献(Yu et al. 2018)

    图  41  (a) 蜂窝点阵的复合元胞; (b) 蜂窝弹性声子晶体板的局部截图, 菱形框标记的单胞为原始单胞, 正六边形标记的单胞为复合元胞. 蓝色横梁表示胞间耦合梁, 定义为linter. 红色横梁表示胞内耦合梁, 定义为lintra; (c) linter = lintra时, 复合元胞的能带结构, 在1517 Hz处具有双狄拉克点; (d) lintra < linter时, 复合元胞的能带结构. 来自文献(Fan et al. 2019)

    图  42  (a) 第一不可约布里渊区Γ点本征模态频率关于lintra/L的函数; (b) lintra/L = 0.836和lintra/L = 1.2时, 第一不可约布里渊区Γ点本征模态的位移场图. 来自文献(Fan et al. 2019)

    图  43  (a) 膨胀复合元胞组成的正六边形样件; (b) 收缩复合元胞组成的正六边形样件; (c) 带有缺陷的收缩复合元胞组成的样件, 红色虚线框标记为缺陷; (d) 膨胀复合元胞组成的正六边形样件的本征频率; (e) 收缩复合元胞组成的正六边形样件的本征频率, 绿色、红色、蓝色和黑色圆点分别表示带隙边缘模态、拓扑角模态、平庸角模态和体模态; (f) 带有缺陷的收缩复合元胞组成的样件的本征频率; (g) ~ (j) 体模态(1750.2 Hz)、拓扑角模态(1555.8 Hz)、平庸角模态(1529.1 Hz)和带隙边缘模态(1600.1 Hz)的位移场图. 来自文献(Fan et al. 2019)

    图  44  (a) 平庸正六边形样件的体(黑色)、边缘(绿色)和角(红色)传输谱; (b) 拓扑正六边形样件的体(黑色)、边缘(绿色)和角(红色)传输谱. 来自文献(Fan et al. 2019)

    图  45  (a) 无缺陷正三角形样件图; (b) 含缺陷的正三角形样件图; (c) 无缺陷正三角形样件的本征频率; (d) 含缺陷的正三角形样件的本征频率, 绿色、蓝色和黑色圆点为带隙边缘模态、角模态和体模态; (e) ~ (g) 带隙边缘(1610.4 Hz)、角(1498.9 Hz)和体(1719.1 Hz)模态的位移场图; (h) 无缺陷正三角形样件(红色)和含缺陷的正三角形样件(黑色)的角传输谱. 来自文献(Fan et al. 2019)

    图  46  (a) 正三边形结构π/3锐角的四个零模态; (b) 正六边形结构2π/3钝角的三个零模态; 绿色和紫色圆点表示的手征价(chiral charge)为+1和−1. 来自文献(Fan et al. 2019)

    图  47  (a) Scott摆链系统中拓扑孤子(kink/antikink)的激发构型; (b) 类摆链系统中拓扑孤子与孤子晶格的激发构型; (c) 在摆链系统多重简并基态的能谱结构中表征的拓扑与非拓扑孤子; (d) 类摆链系统拥有双重简并基态的能谱结构和拓扑孤子(孤子晶格), 其中antikink须紧随kink而激发. 来自文献(Zhang et al. 2019c)

    图  48  (a)(b) 软质力学超材料实验模型及元胞; (c) 准静态位移载荷下超材料的初始构型, 应变率约为έy = 3.1 × 10−5 s−1; (d)(e) 实验和有限元模拟中激发的静态孤子晶格, kink与antikink周期性地交替呈现于两种机械极化区域之间. 对应的宏观压缩应变约为εy = −0.11. 来自文献(Zhang et al. 2019c)

    图  49  (a) 代表性元胞几何及嵌刻其中的简化模型, 圆点表示两类颈弹簧; (b) 元胞简化模型(初始倾角θ0)及其变形构型(当前倾角θ), 转角α = θθ0; (c) 几何空间参数κ对原位势Ucell的基态和对称性的分类; (d) 准1D超材料结构(Nx × 1)及其简化模型(κ = 1时, θ0 = 0, α = θ); (e) 1D“球−链”机理模型, 弹簧链与演化的原位势场(单基态→双基态)相互作用, 以刻画超材料承受位移压缩过程; (f) 实验、模拟与φ4理论解相互论证了静态孤子晶格的激发. 来自文献(Zhang et al. 2019c)

    图  50  (a) 基于等效原位势特征的超材料分类相图; (b) 非平凡超材料等效原位势随应变的演化特征; (c) 超材料经历结构相变、对称性破缺至静态孤子激发的普适物理框架; (d) 基于普适性框架, 在“方块” 、 “杆系”与“圆−椭圆”多孔超材料中激发出静态拓扑孤子; (e) 由超材料序参量表征的拓扑孤子的实验、模拟与理论结果(εy = −0.01). 来自文献(Zhang et al. 2019c)

    图  51  杆数nb = 9, 节点数ns = 6, 满足Maxwell准则的有限桁架结构. (a)等静定桁架结构, N0 = 3, M = 0, NSS = 0; (b) 非等静定桁架结构, N0 = 4, M = 1, NSS = 1

    图  52  (a) 规则Kagome结构, 同时显示了周期边界条件下机构模式和自应力模式; (b) 规则Kagome结构第一支色散曲面的零频率等频线; (c) 扭曲Kagome结构; (d) 扭曲Kagome结构第一支色散曲面的零频率等频线; (e) 一般Kagome结构的参数化; (f) Kagome结构连续变形的拓扑相图

    图  53  (a) 具有不同拓扑相的Kagome桁架结构界面处的零能模式, 箭头表示机构模式的无限小位移, 红绿线条显示自应力边界态, 红、绿色分别代表拉、压内力; (b) 沿界面平行方向的色散关系. 来自文献(Kane et al. 2014)

    表  1  二维对称系统中狄拉克简并情况总结

    晶体对称性K点对称性K点位置简并类型
    C6v或C3vC3v布里渊区角点确定性
    C6C3布里渊区角点确定性
    C3v或C3C3布里渊区角点偶发性
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-03-29
  • 录用日期:  2021-05-25
  • 网络出版日期:  2021-06-07
  • 刊出日期:  2021-06-25

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