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近年来, 基于拓扑绝缘体对声波操控已成为声学领域的研究热点, 在声通讯、声信息处理及集成声学器件等方面具有重要的应用前景. 引入环形旋转气流类比电子自旋, 模拟声学等效磁场可以有效打破时间反演对称性, 进一步理论实现了声拓扑绝缘体[1-4], 并实验验证拓扑绝缘体的边缘模式[5], 然而, 精确控制单元结构中的流体速度仍然存在较大的困难. 利用声学三聚体中空腔的声容对时间进行调制[6], 或引入支持顺/逆时针传输声波模式的环形谐振腔波导[7-9]能够设计实现声Floquet拓扑绝缘体. 将电子体系中的能谷态概念引入到二维声子晶体体系中[10-16], 利用不同谷霍尔相的声子晶体设计实现了支持谷态输运的拓扑波导结构. 引入C6晶体对称性与时间反演算符构成赝时间反演算符, 可以模拟声量子自旋霍尔效应. 基于该物理机理, 利用三角[17,18]/蜂窝[19-23]晶格声子晶体的四重偶然简并狄拉克点及声拓扑相变, 设计实现了赝自旋相关的声拓扑绝缘体. 除此之外, 三维声学系统中的Weyl点拓扑态[24-27]与高阶声学拓扑绝缘体[28-30]也成为声学领域的前沿热点. 从上述研究工作可以看出, 声拓扑绝缘体的物理机理与设计方法已取得较大的进展, 然而, 现有的声拓扑绝缘体均局限于单个工作频带, 在设计多频带声学器件[31-34]方面仍然存在着很大的挑战.
本文提出一种基于蜂窝晶格声子晶体赝自旋相关的双频带声拓扑绝缘体. 选取不同的散射体结构参数, 可以获得蜂窝晶格声子晶体的四重偶然简并狄拉克点及声拓扑相变. 在此基础上, 利用两种不同拓扑相的蜂窝晶格声子晶体设计实现声拓扑波导结构, 并实验验证了赝自旋相关的边缘模式鲁棒性. 此外, 保持蜂窝晶格声子晶体的结构不变, 同时可以在高频区域获得四重偶然简并的狄拉克点与声拓扑相变.
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如图1(a), 蜂窝晶格声子晶体由齿轮形散射体组成(散射体间距b = 20 mm, 晶格常数
$a = \sqrt 3 b$ ), 齿轮形散射体由光敏树脂材料3D打印制备, 其内外半径、齿角及旋转角分别为r, R, θ及α. 本文采用有限元软件COMSOL Multiphysics模拟声拓扑绝缘体特性, 数值模拟的材料参数为: 光敏树脂密度ρ = 1050 kg/m3, 杨氏模量E = 5.08×109 Pa, 泊松比μ = 0.35; 空气密度ρ0 = 1.21 kg/m3, 声速c0 = 343 m/s.
图 1 (a)基于齿轮形散射体组成的蜂窝晶格声子晶体原胞; (b)声子晶体I(r = 0.3671b, R = 0.455b, θ = 20°和α = 15.67°)的色散关系
Figure 1. (a) Schematic of an airborne honeycomb-lattice sonic crystal (SC) composed of gear-like rods; (b) dispersion relation of SC-I.
调整散射体结构参数r, R, θ及α, 可以设计实现具有四重偶然简并双狄拉克锥的声子晶体. 图1(b)为蜂窝晶格声子晶体I的色散关系, 其中齿轮形散射体的结构参数为r = 0.3671b, R = 0.455b, θ = 20° 和α = 15.67°. 可以看出, 声子晶体I的带隙在9.11 kHz处闭合, 形成四重偶然简并的狄拉克点(蓝点), 此时, 在布里渊区中心表现出双狄拉克锥特征. 进一步改变参数r, R, θ及α, 则会破坏声子晶体I的四重偶然简并特性, 在布里渊区中心得到两个双重简并态. 如图2(a), 声子晶体II (r = 0.375b, R = 0.477b, θ = 26°和α = 22°)对应的带隙打开, 在布里渊区中心出现两种类似于电子p和d轨道的双重简并态, 其中p态位于d态下方, 表现为拓扑平庸相. 图2(b)为声子晶体III (r = 0.35b, R = 0.44b, θ = 20°和α = 0°)对应的色散关系, 与声子晶体II的特征相反, 此时 p态位于d态上方, 即声子晶体能带出现反转, 表现为拓扑非平庸相.
图 2 声子晶体(a) II (r = 0.375b, R = 0.477b, θ = 26°和α = 22°)与(b) III (r = 0.35b, R = 0.44b, θ = 20°和α = 0°)对应的色散关系; (c)声子晶体II与III在布里渊区中心的本征模式声压场分布
Figure 2. Dispersion relations of (a) SC-II and (b) SC-III; (c) distributions of pressure field for eigenmodes at Brillouin zone center of SC-II and SC-III.
图2(c)显示声子晶体II与III在布里渊区中心的双重简并点(图2(a)和图2(b)中的红点和黑点)对应的本征模式声压场分布. 可以看出, 两种声子晶体均存在一对偶极与四极模式, 且能带发生反转. 此时, 由于声子晶体II和III仅具有C6晶体对称性, 则对应的偶极(px, py)和四极(dxy,
${d_{{x^2} - {y^2}}}$ )模式不是关于x与y轴完美对称. 然而, 声子晶体的C6对称性仍然满足赝时间反演算符对称性, 从而可以实现声学量子自旋霍尔效应[35]. 声子晶体体模式的赝自旋–1/2态可以通过杂化声压场${p_ \pm } = \left( {{p_x} \pm {\rm{i}}{p_y}} \right)$ 与${d_ \pm } = \left( {{d_{xy}} \pm {\rm{i}}{d_{{x^2}}}_{ - {y^2}}} \right)$ 得到[19]. -
图3(a)为声子晶体II与III组成的超原胞结构, 数值模拟超原胞的色散关系如图3(b)所示. 可以看出, 声子晶体II与III重叠的体带隙中(阴影区域, 8.74 —9.26 kHz)存在一对边缘模式(红线与蓝线), 其对应的色散曲线斜率为赝自旋相关的边缘模式群速度. 图3(c)为k|| = 0.05 (图3(b)中N点)对应的超原胞对称模S与反对称模A的场分布, 将S模与A模分别杂化为S-iA和S+iA, 可以得到M点与N点对应的赝自旋相关的边缘模式. 图3(d)显示k|| = ± 0.05 (图3(b)中M点与N点)对应的超原胞的声能流场分布(蓝箭头). 可以看出, 在超原胞的分界面处, 存在着两个方向相反的声涡旋, 其中M点与N点的声涡旋手性为逆时针与顺时针, 分别对应赝自旋+(蓝圆形箭头)和赝自旋-(红圆形箭头). 从图中可以看出, 两种赝自旋相关的边缘模式速度大小相同, 方向相反, 从而实现赝自旋相关的声单向传输.
图 3 (a)声子晶体II与III组成的超原胞结构, 红虚线表示两种声子晶体之间的界面; (b)超原胞对应的色散关系; (c)超原胞的对称模S与反对称模A声压本征场分布(图(b)中N点, k|| = 0.05); (d)超原胞的边缘模式声能流场分布(图(b)中M点与N点, k|| = ± 0.05)
Figure 3. (a) Schematic of a supercell composed of SC-II and SC-III. Red dashed line represents the domain wall between two SCs; (b) dispersion relation of the supercell; (c) simulated pressure eigenfunctions (S and A modes) of the supercell at point N (k|| = 0.05) in Fig. (b); (d) simulated acoustic energy flow for pseudospin-dependent edge modes at points M and N (k|| = ± 0.05) in Fig. (b).
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图4(a)为3 D打印的基于声子晶体II与III的声拓扑波导样品. 为了验证边缘模式的鲁棒性, 在所设计的声拓扑波导中引入两种不同类型的缺陷: 无序与弯曲. 数值模拟频率为9.1 kHz的声波分别通过无缺陷、包含无序与弯曲缺陷的三种类型声拓扑波导产生的声能量密度场分布, 分别如图4(b)—图4(d). 可以看出, 赝自旋相关的边缘模式能够绕过拓扑波导中的无序与弯曲缺陷(图4(c)与图4(d)), 高效通过声拓扑波导, 与无缺陷声拓扑波导对应的声传播特性基本相同.
图 4 (a)基于声子晶体II与III的声拓扑波导样品, 红虚线表示两种声子晶体的界面. 数值模拟频率为9.1 kHz (体带隙内)的声波通过三种类型的拓扑波导产生的声能量密度场分布, 三种拓扑波导分别对应(b)无缺陷, 含内置(c)无序与(d)弯曲缺陷. 绿色星与箭头分别表示声源位置和声传播方向
Figure 4. (a) Photograph of a topological waveguide composed of SC-II and SC-III. Red dashed line represents the domain wall of two SCs. Simulated distributions of acoustic intensity field at 9.1 kHz (within bulk band gap) through topological waveguides with three different configurations, corresponding to (b) without defects, and with (c) a disorder and (d) a bend. Green star and arrows represent source positions and propagation directions.
为了进一步验证声拓扑波导的鲁棒性, 实验测量三种不同类型声拓扑波导的声透射谱. 实验装置如图5(a)所示, 高度为8 mm的实验样品放置在两个平行玻璃板(尺寸2 m × 2 m)之间, 在平行玻璃板的四周放置楔形吸音海绵实现准消声波导环境. 扬声器(Knowles, 7.2 × 9.5 × 4.1 mm3)由功率放大器驱动, 放置在样品左侧, 麦克风(B&K-4961型)放置在样品右侧5 mm处扫描测量声信号, 利用PULSE Labshop软件记录红虚线上不同位置的声信号幅值谱, 对同一频率下不同位置声信号幅值的平方进行积分求和, 并利用声源不同频率的幅值平方进行归一化, 即可得到实验测量的声透射谱.
图 5 (a)实验装置示意图; (b)实验测量无缺陷与内置无序与弯曲缺陷的声拓扑波导对应的声透射谱; (c)超原胞在室温为303 K对应的色散关系
Figure 5. (a) Experimental setup; (b) measured transmission spectra for topological waveguides without and with two types of defects; (c) dispersion relation of the supercell at 303 K.
图5(b)为实验测量三种类型声拓扑波导对应的声透射谱. 可以看出, 与无缺陷声拓扑波导(红实线)对应的测量结果相比, 内置无序与弯曲缺陷的声拓扑波导的声透射谱几乎不变, 从而说明边缘模式几乎不受缺陷诱导的背向散射影响, 实验验证了赝自旋相关的边缘模式鲁棒性. 此外, 与图3(b)中的工作频带(阴影区域)相比, 图5(b)中的工作频带(阴影区域, 8.89—9.41 kHz)向高频区域偏移150 Hz左右, 这主要与实验测量的室温(303 K)相关. 为了验证该因素, 数值模拟室温为303 K对应的超原胞色散关系(空气密度ρ0 = 1.165 kg/m3, 声速c0 = 348.9 m/s), 结果如图5(c)所示. 可以看出, 此时工作频带同样向高频区域偏移150 Hz左右, 与图5(b)中对应的实验结果符合很好.
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如图6(a), 声子晶体I在频率24.44 kHz处, 同时存在另一个四重偶然简并的狄拉克点, 与图1(b)结果类似, 声子晶体在布里渊区中心表现出双狄拉克锥特征. 当改变散射体结构参数r, R, θ及α时, 声子晶体II和III在高频区间对应的带隙同样打开, 并出现能带反转. 如图6(b), 在布里渊区中心处, 声子晶体II对应的f态位于g态下方; 而声子晶体III的f态则位于g态上方(图6(c)). 图6(d)显示高频区间对应的声子晶体II和III在布里渊区中心双重简并点(图6(b)和6(c)中红点与黑点)的本征声压场分布. 可以看出, 两种声子晶体均存在一对八极与十极模式, 能带同样发生反转. 然而, 与低频区间对应的场分布相比, 声子晶体II和III对应的本征声压场分布的差异明显增大.
图 6 声子晶体(a) I, (b) II和(c) III在高频区间(24.44 kHz左右)的色散关系; (d)高频区间对应的声子晶体II与III在布里渊区中心的本征声压场分布
Figure 6. Dispersion relations of (a) SC-I, (b) SC-II and (c) SC-III in high-frequency region (around 24.44 kHz); (d) distributions of pressure field for eigenmodes at Brillouin zone center of SC-II and SC-III in high-frequency region.
图7为图3(a)的超原胞在高频区间的色散关系. 与图3(b)类似, 在高频区间, 两种类型声子晶体重叠的体带隙中同样存在一对边缘模式. 然而, 与图3(b)的结果相比, 图7中边缘模式的带隙明显增大, 这主要由声子晶体II与III在高频区间对应的本征场分布具有更大的差异引起, 如图6(d)所示. 因此, 所提出的基于齿轮形散射体的声拓扑绝缘体可以同时在两个工作频带中同时实现赝自旋相关的边缘模式.
图 7 超原胞在高频区间(24.44 kHz左右)的色散关系
Figure 7. Dispersion relations of a supercell in high-frequency region (around 24.44 kHz).
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本文基于两种不同拓扑相的蜂窝晶格声子晶体, 设计实现了赝自旋相关的双频带声拓扑绝缘体. 结果表明: 调节齿轮形散射体的结构参数, 可以获得蜂窝晶格声子晶体的四重偶然简并狄拉克点与声拓扑相变. 在此基础上, 利用两种不同拓扑相的蜂窝晶格声子晶体设计实现声拓扑波导结构, 并实验验证了赝自旋相关的边缘模式对无序和弯曲两种类型缺陷的鲁棒性. 此外, 保持蜂窝晶格声子晶体的结构不变, 数值模拟得到高频区域的四重偶然简并狄拉克点与声拓扑相变. 所设计的声拓扑绝缘体可以在两个不同频带中同时实现赝自旋相关的边缘模式, 在多频带声通讯与声信息处理方面具有重要的应用前景.
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Based on honeycomb-lattice sonic crystals with gear-like scatterers, we study and design a pseudospin-dependent dual-band acoustic topological insulator. Compared with cylindrical scatterers with only a single tunable structure parameter (radius), there exist four tunable parameters for the gear scatterer, which enables the sonic crystal to realize four-fold accidental degeneracy at two different frequencies simultaneously. By changing structure parameters of the gear-like scatterers, we can obtain topological phase transitions between two sonic crystals. Based on this, we design acoustic topological waveguides based on two honeycomb-lattice sonic crystals with different topological phases, and introduce two kinds of defects (a lattice disorder and a bend) into the topological waveguide near the domain wall. Numerical simulations show that pseudospin edge states almost immune to two types of defects and can pass through the topological waveguides with negligible backscatterings. Compared with the results for the topological waveguide without defects, the measured transmission spectra are almost unchanged with the two types of defects, which further experimentally verify the robustness of pseudospin-dependent edge states. Additionally, by keeping the structure of the sonic crystals unchanged, we can also obtain another four-fold accidental degenerate Dirac point and the corresponding topological sound phase transitions in the high-frequency region. The simulations show that there also exists a pair of edge states in the overlapped bulk bandgap of the two sonic crystals in the high-frequency region. It is worth noting that the tiny gap between two edge states is larger than that in the low-frequency region, which may arise from the greater difference between the distributions of pressure eigenfunction of two sonic crystals. The proposed dual-band acoustic topology insulator has potential applications in multi-band sound communication and sound information processing.
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Keywords:
- sonic crystal /
- Dirac point /
- topological insulator /
- dual-band
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图 1 (a)基于齿轮形散射体组成的蜂窝晶格声子晶体原胞; (b)声子晶体I(r = 0.3671b, R = 0.455b, θ = 20°和α = 15.67°)的色散关系
Fig. 1. (a) Schematic of an airborne honeycomb-lattice sonic crystal (SC) composed of gear-like rods; (b) dispersion relation of SC-I.
图 2 声子晶体(a) II (r = 0.375b, R = 0.477b, θ = 26°和α = 22°)与(b) III (r = 0.35b, R = 0.44b, θ = 20°和α = 0°)对应的色散关系; (c)声子晶体II与III在布里渊区中心的本征模式声压场分布
Fig. 2. Dispersion relations of (a) SC-II and (b) SC-III; (c) distributions of pressure field for eigenmodes at Brillouin zone center of SC-II and SC-III.
图 3 (a)声子晶体II与III组成的超原胞结构, 红虚线表示两种声子晶体之间的界面; (b)超原胞对应的色散关系; (c)超原胞的对称模S与反对称模A声压本征场分布(图(b)中N点, k|| = 0.05); (d)超原胞的边缘模式声能流场分布(图(b)中M点与N点, k|| = ± 0.05)
Fig. 3. (a) Schematic of a supercell composed of SC-II and SC-III. Red dashed line represents the domain wall between two SCs; (b) dispersion relation of the supercell; (c) simulated pressure eigenfunctions (S and A modes) of the supercell at point N (k|| = 0.05) in Fig. (b); (d) simulated acoustic energy flow for pseudospin-dependent edge modes at points M and N (k|| = ± 0.05) in Fig. (b).
图 4 (a)基于声子晶体II与III的声拓扑波导样品, 红虚线表示两种声子晶体的界面. 数值模拟频率为9.1 kHz (体带隙内)的声波通过三种类型的拓扑波导产生的声能量密度场分布, 三种拓扑波导分别对应(b)无缺陷, 含内置(c)无序与(d)弯曲缺陷. 绿色星与箭头分别表示声源位置和声传播方向
Fig. 4. (a) Photograph of a topological waveguide composed of SC-II and SC-III. Red dashed line represents the domain wall of two SCs. Simulated distributions of acoustic intensity field at 9.1 kHz (within bulk band gap) through topological waveguides with three different configurations, corresponding to (b) without defects, and with (c) a disorder and (d) a bend. Green star and arrows represent source positions and propagation directions.
图 5 (a)实验装置示意图; (b)实验测量无缺陷与内置无序与弯曲缺陷的声拓扑波导对应的声透射谱; (c)超原胞在室温为303 K对应的色散关系
Fig. 5. (a) Experimental setup; (b) measured transmission spectra for topological waveguides without and with two types of defects; (c) dispersion relation of the supercell at 303 K.
图 6 声子晶体(a) I, (b) II和(c) III在高频区间(24.44 kHz左右)的色散关系; (d)高频区间对应的声子晶体II与III在布里渊区中心的本征声压场分布
Fig. 6. Dispersion relations of (a) SC-I, (b) SC-II and (c) SC-III in high-frequency region (around 24.44 kHz); (d) distributions of pressure field for eigenmodes at Brillouin zone center of SC-II and SC-III in high-frequency region.
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