【作者】 王健；

【导师】 梅军；

【作者基本信息】 华南理工大学 ， 凝聚态物理， 2018， 硕士

【摘要】 最近几年,受到电子系统中量子自旋霍尔效应和量子谷霍尔效应的启发,经典波的拓扑性质研究成为了一个新的研究热门。量子系统中受拓扑保护的波的传输这一特性很快被推广到其它的经典物理系统。利用这些系统的拓扑性质,开发出的材料具有一些独特的功能,比如单向传输、背向散射抑制和缺陷免疫等等。本文研究了声学系统中的量子自旋霍尔效应和兰姆波中的量子谷霍尔效应。具体研究内容如下:1.在第二章中,我们构建了一种简单的二维声子晶体:由两个横截面为三角形的钢柱所组成的复式元胞按三角点阵的形式排列在空气中,等效地形成了一个蜂巢点阵结构。当三角形钢柱的取向与三角点阵的高对称方向一致时,整个体系具有₆对称性。研究发现:在保持钢柱填充率不变的条件下,只需要将所有三角柱绕着自己的中心旋转180^o,就可实现二重简并的p态和d态在布里渊区中心Γ点处的频率反转,且该能带反转过程实质上是一个拓扑相变过程。通过利用Γ点的p态和d态的空间旋转对称性,构造了一个赝时反演对称性,并在声学系统中实现了类似于电子系统中量子自旋霍尔效应的赝自旋态。随后通过?微扰法导出了Γ点附近的有效哈密顿量,并分别计算了拓扑平庸和非平庸系统的自旋陈数,揭示了能带反转和拓扑相变的内在联系。最后通过数值模拟演示了声波边界态的单向传输,以及边界态对一般缺陷比如弯曲、空腔和失序的鲁棒性。我们的设计提供了一种新的方式去实现声波的自旋霍尔效应,在实际应用上具有较大的优势和潜力,使人们可以以一种更有效的方式去操纵声波。2.在第三章中,我们构建了一种声子晶体薄板,它由不规则的六边形硅柱子按三角晶格的形式嵌入在硅橡胶薄板中形成,硅柱子可以绕自己的中心旋转,当旋转角度为0时,体系结构布里渊区中的K点具有C_3v对称性,计算能带发现布里渊区上的K点会出现一个兰姆波A₀模式的狄拉克锥,而且在锥的附近还有第三条带。然后旋转介质柱打破ΓK方向的镜面对称性,当旋转角度为π/6时,K点的对称性降为C₃,这时K点上的狄拉克锥会打开一个带隙,虽然这个带隙不是完全带隙,但与带隙重合的部分是弹性波的S₀和SH₀模式。但在低频范围下,A₀模式与S₀、SH₀模式是解耦和,所以这个带隙可以看成是A₀模式的完全带隙。然后我们给出了体系的有效哈密顿量,计算了K点附近A₀模式这三条带的陈数,发现上下两条带的陈数为±₂¹,而中间那条带的陈数为0。然后我们得到了受谷保护的拓扑边界态,证明了边界态的手性。当谷间散射较弱时,边界态是单向传播的并且不受缺陷的影响。因此我们在兰姆波中以一种简单的方式实现了量子谷霍尔效应。更多还原

【Abstract】 The study of the topological properties of classical waves in materials and metamaterials is an active area of research,inspired by quantum spin Hall effect and quantum valley Hall effect in electronic system.Recent years,topologically protected wave propagation in quantum systems has quickly extended to other classical systems of physics.The topological properties of these systems can be exploited to achieve unique and exciting functionalities,such as unidirectional transmission,backscattering suppression and defects of immune.We study the quantum spin Hall effect in sound waves and the quantum valley Hall effect in Lamb waves.The specific research is as follows:1.In chapter 2,we design a two-dimensional acoustic crystal（AC）to obtain topologically protected edge states for sound waves.The AC is composed of a triangular array of a complex unit cell consisting of two identical triangle-shaped steel rods arranged in air.The steel rods are placed on the vertices of the hexagonal unit cell so that the whole lattice possesses the₆symmetry.We show that by simply rotating all triangular rods around their respective centers by 180 degree,a topological phase transition can be achieved,and more importantly,such a transition is accomplished with no need of changing the fill ratios or changing the positions of the rods.By utilizing the spatial symmetry of the p and d states in the AC,we can construct the pseudo-time reversal symmetry and find pseudospin states in the interface between topologically trivial and nontrivial ACs,which are very similar to the real spin states of quantum spin Hall Effect in electronic systems.We also develop an effective Hamiltonian for the associated bands to characterize the topological properties of the AC around the Brillouin zone center with the help of the k?p perturbation method.We calculate the spin Chern numbers of the ACs,and reveal the inherent link between the band inversion and the topological phase transition.With full-wave simulations,we demonstrate the one-way propagation of sound waves along the interface between topologically distinct ACs,and demonstrate the robustness of the edge states against different types of defects including bends,cavity and disorder.Our design provides a new way to realize acoustic spin Hall Effect.Potential applications and acoustic devices based on our design are expected,where people can manipulate and transport sound waves in a more efficient way.2.In chapter 3,we design a phononic crystal thin plate（PCP）,the PCP is composed of a triangular array of hexagonal silicon pillars embedded in a rubber plate.The silicon pillars can rotate around its center,when the rotation angle is zero,the point group of point K in the first Brillouin zone is C_3v,and the energy band has a Dirac-cone of A₀ mode at point K and also has third band in the vicinity of this Dirac-cone.When the rotation angle isπ/6,inversion symmetry is broken and the symmetry of point K is reduced to C₃,and the Dirac-cone will be opened to form a band gap,valley pseudospin degree of freedom is formed around K and K’valleys for the A₀ Lamb mode,which is decoupled from the S₀ and SH₀ modes in the low frequency regime.Then we develop an effective Hamiltonian to describe the dispersion relation of A₀ mode and calculate spin Chern number,the Chern number of upper and lower band is±1/2,the Chern number of middle band is 0.Then we can obtain topologically valley-protected edge states.Chiral edge states are explicitly demonstrated,which are immune to defects and exhibit unidirectional transport behaviors when inter-valley scattering is weak.Quantum valley Hall effect is thus simulated in a simple way in the context of Lamb waves.更多还原