Fund Project:Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303702), the National Natural Science Foundation of China (Grant Nos. 119224071, 11834008, 11874215, 11674172, 11574148), and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20160018)
Received Date:09 September 2019
Accepted Date:04 November 2019
Available Online:19 November 2019
Published Online:20 November 2019
Abstract:The manipulation of surface acoustic wave (SAW) in phononic crystal plays an important role in the applications of SAW. The introduction of topological acoustic theory has opened a new field for SAW in phononic crystals. Here we construct pseudospin modes of SAW and topological phase transition along the surface of phononic crystal. The local SAW propagation is realized by air cylindrical holes in honeycomb lattice arranged on rigid substrate, and the Dirac cone is formed at the K point of the first Brillouin zone. Furthermore, using the band-folding theory, double Dirac cones can be formed at the center Гs point in the Brillouin zone of compound cell that contains six adjacent cylindrical air holes. The double Dirac cone can be broken to form two degenerated states and complete band gap by only shrinking or expanding the spacing of adjacent holes in the compound cell. It is found that the direction of energy is in a clockwise or counterclockwise direction, thus the pseudospin modes of SAW are constructed. The shrinkage-to-expansion of the compound cell leads to band inversion, and the system changes from trivial state to nontrivial state, accompanied by the phase transition. According to the bulk-boundary correspondence, the unidirectional acoustic edge states can be found at the interface between trivial system and nontrivial system. Then we can construct a topologically protected waveguide to realize the unidirectional transmission of surface waves without backscattering. This work provides a new possibility for manipulating the SAW propagating on the surface of phononic crystals and may be useful for making the acoustic functional devices based on SAW. Keywords:surface acoustic wave/ double Dirac cone/ acoustic pseudospin/ topological edge mode
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2.1.理论模型和狄拉克锥的形成
在钢板(密度ρ1 = 7800 kg/m3, 纵声速c1 = 6100 m/s)上排布圆柱状空气孔(密度ρ0 = 1.29 kg/m3, 纵声速c0 = 343 m/s), 如图1(a)中的二维平面图所示. 蓝色圆形代表空气孔, 将两个相邻圆形空气孔构成的最小元胞(如红色菱形区域所示, 由矢量a1和矢量a2定义)按照三角晶格点阵排列, 就形成了蜂巢晶格声子晶体. 图1(b)为最小元胞的三维图, 在本文的模拟计算中, 元胞的晶格常数为|a1| = |a2| = a = $ \sqrt {\rm{3}} $ cm, 其中相邻空气圆柱孔的中心间距为R = 1 cm, 圆柱孔的底面半径和高度分别为R0 = 0.35 cm, L = 0.5a. 在COMSOL multiphysics中计算结构能带图时, 由于钢板的声阻抗ρ1c1远大于空气声阻抗ρ0c0, 可将空气圆柱孔的侧面和底面设置为硬边界. 空气层上表面边界条件设置为平面波辐射条件, 因此该模型可等效为在无限延伸的空气层中计算能带结构. 如图1(c)中的能带图所示, 由于蜂窝晶格中的反转对称性保护, 在第一布里渊区的高对称角点K处形成了狄拉克锥(f = 7611.7 Hz). 在能带图中, 黑色实线代表空气中声色散线, 位于声色散线下方声波的模式均为表面模式. 由于狄拉克锥位于声线下方, 因此对应特征频率下的声波能量可以很好地局域在结构表面而不泄漏到空气中. 图 1 (a)晶格在xy平面截面图及最小元胞(红色箭头)和复合胞(黄色箭头)的矢量表示; (b) 最小元胞的三维图; (c) 最小元胞第一布里渊区的能带图, 插图为晶格的第一布里渊区 Figure1. (a) Cross section of the phononic crystal lattice in xy-plane and the vector representation of the minimum cell (red arrows) and compound cell (yellow arrows); (b) three-dimensional view of the minimum cell; (c) band structure of the first Brillouin zone of the minimum cell. The inset shows the first Brillouin zone.
基于能带折叠理论, 可以构造四重简并的狄拉克锥. 如图2(a)所示, 构建一个由6个相邻空气圆柱孔组成的复合胞, 复合胞在图1(a)中由黄色菱形(矢量b1和矢量b2定义)围成的区域表示. 此时, 复合胞的晶格常数|b1| = |b2| = b = $ \sqrt {\rm{3}}a$ = 3 cm. 图2(b)中红色六边形区域代表最小元胞的第一布里渊区(用Bz表示), 黄色六边形区域代表复合胞的第一布里渊区(用Bzs表示). 由折叠理论可知区域Bz可通过折叠得到区域Bzs. 由于布里渊区的对称性, 仅取区域的1/12进行观察. 由图2(b)可知三角形区域③可由区域①折叠两次得到. 区域①先沿KKs所在直线折叠得到区域②, 将区域②沿MsKs所在直线折叠即可得到区域③. 由于改变了元胞形状和布里渊区的大小, 原来处于布里渊区能带能谷位置处的狄拉克锥折叠到了新的布里渊区的中心. 因此由能带折叠理论可知, 两个布里渊区域的色散曲线满足: $ {M_{\rm s}}{\varGamma_{\rm s}}{|_{Bz_{\rm s}}} = MK{|_{Bz}}+$${M_{\rm s}}K{|_{Bz}}+{M_{\rm s}}\varGamma{|_{Bz}}, {\varGamma_{\rm s}}{K_{\rm s}}{|_{Bz_{\rm s}}}\!=\! K{K_{\rm s}}{|_{Bz}} + K{K_{\rm s}}{|_{Bz}}+ $$ \varGamma{K_{\rm s}}{|_{Bz}}$, $ {K_{\rm s}}{M_{\rm s}}{|_{Bz_{\rm s}}} $ = KsM|Bz + KsMs|Bz + KsMs|Bz. 计算得到复合胞的能带图如图2(c)所示. 图 2 (a) 复合胞的三维图; (b) 从Bz到Bzs的折叠机制, 红色和黄色六边形区域代表了最小元胞和复合胞的第一布里渊区, 分别用Bz和Bzs表示; (c) 复合胞第一布里渊区的能带图 Figure2. (a) Three-dimensional view of the compound cell; (b) the folding mechanism from Bz to Bzs. Red and yellow hexagon region represents the first Brillouin region of minimum cells and compound cells, respectively represented by Bz and Bzs; (c) band structure of the compound cell.
22.2.能带反转和赝自旋态 -->
2.2.能带反转和赝自旋态
从图2(c)中复合胞的色散曲线可发现在新的布里渊区中心Гs点处出现四重简并, 形成了双狄拉克锥. 此时整个晶格具备C6v对称性, 而C6v点群有两个二维不可约表象E1和E2. E1代表二重简并奇宇称的偶极子态, 以(x, y)为基函数表示; E2代表二重简并偶宇称的四极子态, 以(x2 – y2, xy)为基函数表示. 因此E1和E2分别对应于量子体系中电子的p态和d态. 根据E1和E2基函数的旋转对称性, 在此类声子晶体中可构造赝时间反演算符[25]T = UK = –iσyK, $ {{{T}}^{{2}}}{{{p}}_{{ \pm }}}{{ = T}}\left( { \mp {{\rm i}}{{{p}}_ \mp }} \right)= $$- {{{p}}_{{ \pm }}} $ 其中U为反幺正算符, K为复共轭算符且算符T满足T2 = –I. 因此, 构造的赝时间反演算符类似于电子系统中的时间反演对称, 由此, 我们可构造出类似于p态的赝自旋$ {p_{\pm}}=\left( {{p_x}{\pm {\rm{i}}}{p_y}} \right)/\sqrt {\rm{2}}$和类似于d态的赝自旋$ {{{d}}_{{ \pm }}}=\left( {{{{d}}_{{{{x}}^{{2}}} - {{{y}}^{{2}}}}}{{ \pm {\rm{i}}}}{{{d}}_{{{xy}}}}} \right)/\sqrt {{2}}$. 由于狄拉克点附近的色散关系近似满足线性关系[49], 在四重简并狄拉克锥附近很容易打破晶体的时间反演对称性, 将p态和d态分离, 形成非平庸带隙, 从而实现受拓扑保护的边缘传输. 本文通过改变复合胞内相邻单元间的耦合强度, 即改变图2(a)中R1的大小, 可以打开双狄拉克锥, 形成完全带隙并实现能带反转. 图3(a)和图3(c)分别表示R1 = 0.32b和R1 = 0.345b的能带图. 图 3 拓扑平庸 (a) R1 = 0.32b和非平庸(c) R1 = 0.345b复合胞的能带图; 插图给出了带隙频率下方能流顺时针流动时, 相应晶格在Гs点附近的声压场分布. 拓扑平庸(b)和非平庸(d) 复合胞能带图中Гs点两个双重简并态p态和d态的声压场分布图. 黑色箭头表示能流的运动方向. 能流顺时针转动, 对应向下赝自旋态, 用红色箭头表示; 能流逆时针转动, 对应向上赝自旋态, 用蓝色箭头表示 Figure3. Band structure of the compound cell for the case of (a) topologically trivial R1 = 0.32b and (c) topologically nontrivial R1 = 0.345b. The insets show the pressure field below the band gap around Гs in corresponding lattice when the energy flow rotating clockwise. The pressure filed of the double degenerated state at Гs point in the band structure of topologically (b) trivial and (d) nontrivial compound cell. The black arrows indicate the direction of energy flow. The energy flow rotating clockwise (anticlockwise) corresponds to the pseudospin-down state (pseudospin-up) represented by red (blue) arrow.