This dissertation demonstrates a new structure for topologically shielded phonon metamaterial wave propagation in a continuous, elastic solid using an analogue of the quantum spin Hall effect and the quantum valley quantum Hall effect based on finite element simulations. This enables the waveguide to support tunable topologically shielded mechanical waves that can be tuned for required frequencies ranging from kHz to over GHz either by selectively biasing the overlay membrane or by increasing the lattice parameter of the underlying substrate. In addition, the adaptive waveguide is based on topological phase transitions with respect to manipulating the substrate structures.
Second part, the structure for pseudospin dependent topologically protected waveguide based on translational symmetry with analogy to quantum spin Hall effect is proposed. Last, third part, the structure for pseudospin-dependent topologically protected waveguide based on space inversion symmetry with analogy with quantum valley Hall effect proposed. Each part of Hall effect analogy starts from a method for constructing pseudospin/valley-dependent topology phase transition using phononic metamaterials with mimicry symmetry that mimics lattice structure of graphene.
Introduction Introduction
Quantum Hall Family
- Quantum Hall effect
- Quantum spin Hall effect
- Quantum valley Hall effect
In conventional condensed matter physics, so-called solid state physics, phases of matter such as crystals and magnets are categorized and the phase transition is classified by the spontaneous breaking of symmetries2, 3. Since 1980, this traditional classification methodology for phases of matter has taken an innovative turn with epic discovery of the quantum Hall effect (QHE)4. The TKKN theory links the Chern number to the quantized Hall conductance, so that the distinguishing phases of matter can be defined in terms of topological state.
In terms of the energy band structure, SOC makes the spin-band lower in the conduction band and the spin-band higher in the valence band so that the band inversion that indicates the topological phase transition occurs . That is, since these spin-dependent chiral edge states are defined in terms of topological features, perturbations such as backscattering from impurities cannot affect these states, so the notion of topologically protected chiral edge states is allowed for QSHE. In terms of the band structure, the overlapping and interesting spin-dependent edge states in QSHE unfolded in the K and K' valleys, respectively, so the topological phase transition occurs by breaking the inversion symmetry, and each valley has topological order of opposite.
Topological Phase in Phononic Systems
- Phononic crystal
- QHE analogy in phononic system
- QSHE/QVHE analogy in phononic system
Elastic waves or mechanical waves that describe an oscillation in solid matter transmit its energy in terms of disorder state in a medium and have potential for wide engineering applications such as energy harvesting26, control of acoustics27, seismic waves28 even in data transmission29. Therefore, band structure can be established in momentum space and the control of band gap frequency is available as at highly symmetry point in Brillouin zone by modulating geometrical parameters such as lattice size, periodic characteristic and material parameters of elastic structures or environment33-35. The feasibility of modulation of phonic systems contributes to the development of future designs based on nanotechnology with fundamental mechanisms in acoustic wave such as negative refraction in a sonic crystal36-38, double negative material with negative modulus and negative mass density39-42 in terms of metamaterials .
In addition, these advantages are extended to an effective way to control mechanical wave propagation on elastic devices, such as energy trapping43, vibration isolation44 and elastic wave matching technique45, 46. At the same time, a similar approach using rotational sources of nonreciprocity in mechanical wave propagation such as rotating gyroscopic metamaterials on a honeycomb lattice, are proposed60 and experimentally realized with a DC motor61. Although differences from electrons, such as the absence of spin-orbit interactions, pose challenges in achieving topological order in phononic systems, recent work has successfully discovered analogues of the quantum Hall family, such as QSHE, for manipulating unit cells to create a double Dirac-cone dispersion using zone folding techniques and degenerate pseudospin intact TRS77, 78.
Research Objective
Background and Simulation Methods Background and Simulation Methods
Basic Notions in Band Theory
- Energy band theory
- Band structure in reciprocal space
- Elastic wave propagation in membrane
With the periodic characteristic of wave properties, the Fourier transform of periodicity in real space to a corresponding space called a reciprocal space (k-space). Therefore, the extended space with unit lattice vector 𝒃 is called reciprocal k-space or momentum space in crystal lattice. Since all vectors in real space can be mapped into k-space, so-called Brillouin zone, the solution of Eq. 2.1) fully demonstrate energy bands in momentum space and defined as band structure of crystal.
Based on the Bloch-Floquet theorem87, the transverse displacement in the periodic structure can be written as. Finally, the governing equation Eq. 2.2) derives from the conventional eigenvalue problem which can be solved using the framework of the finite element method by substituting the transverse displacement into Eq. Therefore, the analysis of the frequency dispersion in the membrane-based phononic crystal can be defined in terms of the reciprocal space.
Topological phase in phononic system
- QSHE analogy in phononic system
- QVHE analogy in phononic system
However, for mechanical wave propagation in phononic crystal, the existence of an additional polarization degree of freedom in continuous solid makes the analogy system need complexity. Valley-based information conveyed in phononic systems starts with sonic crystals for acoustic waves. That is, the dual Dirac pairs protected by C3v symmetry degenerate into two different extrema at the K and K points in the Brillouin zone, which refers to the concept of valley.
Furthermore, the acoustic energy flow at the K and K' valleys shows acoustic vortices with opposite topological charges. Compared to the above electronic system, a high accessibility of inversion-symmetry breaking is guaranteed in phonon system with additional degrees of freedom. As detailed above, in addition to the pseudo-spin, acoustic waves are capable of simulating a valley degree of freedom.
Simulation Methods
- Structural design
- Phononic crystal dispersion
- Topological invariants
- Full-field simulation
To achieve frequency spreading in terms of mutual k-space, the natural frequency analysis is defined with the Floquet boundary condition on the unit cell boundaries described as. where 𝒂 is the lattice parameter of the primitive unit cell and k is the lattice vector in the reciprocal k-space. Therefore, under the discretization in the mutual k-space, the Berry curvature, Eq. where 𝑛 represents yet another Bloch band. In frequency domain response analysis using a commercial software package, COMSOL Multiphysics, the governing equation of the frequency domain perturbation without dumping term is described as where M and K represent the mass and stiffness matrix, respectively.
In general, the equation of motion applies to the propagation of waves without the effect of damping. where U is the displacement field, M, K and F are lumped mass matrices corresponding to the mass density, the stiffness matrix corresponds to the elastic properties of the structure or the external force. Then, the wave equation of motion can be calculated numerically under an explicit central difference formulation defined as . where 𝑈¦ is the displacement of the Nth element, i represents the number of increments in the explicit scheme. 2.21), the time increment is a key parameter for obtaining an exact solution of the equation, i.e. Therefore, the following three conditions are met to ensure the stability of explicit dynamic analysis, which are described as
Pseudospin-dependent phononic topological waveguide
Pseudospin-dependent phononic topological waveguide
- Introduction
- Zone folding
- Phononic dispersion analysis
- Topological analysis
- Topological invariants
- Topological protected edge states
- Full-field simulation
- Adaptable structure design
- Summary
- Valley-dependent phononic topological waveguide
In Figure 3.1a, an inner hexagonal unit cell (yellow) has a lattice constant a0 and the enlarged unit cell (red) has a lattice constant a, where a = a0 3. In Figure 3.3, we plot the output of constant of the band in relation to in the change of the value of gd. First, the frequency response in the straight line path constructed by the zigzag interface edge is discussed with 15x10 unit cells as shown in Figure 3.12a.
The wave propagation generated from below at a Dirac frequency of 87.3 MHz propagates intact to the upper unstructured membrane in the case of our topological waveguide, as shown in Figure 3.15b. Nevertheless, the intact wave propagation to the upper unstructured membrane is still not available at the Dirac frequency (Figure 3.15d). The excitation at Dirac frequency 87.3 MHz successfully propagates to the upper unstructured membrane through our topological edge interface, as shown in Figure 3.16b.
Valley-dependent phononic topological waveguide
- Introduction
- Phononic dispersion analysis
- Frequency dispersion curve
- Valley polarization
- Topological analysis
- Topological invariants
- Topologically protected edge states
- Full-field simulations
- Frequency domain analysis
- Real-time wave propagation
- Advanced extension
- Tunability
- Adaptable structure design
- Summary
- Conclusion Conclusion
Basically, our veinless phononic crystal structure has a constant lattice parameter, a = 4.5 µm and symmetric radius, R0 = 3 µm as shown in Figure 4.1a. R1 and R2 are changed in a complementary manner so that R1 increases as R2 decreases to maintain a constant lattice parameter as shown in Figure 4.1b. In the case of veined phononic crystal structure, it has a constant lattice parameter, a = 4.5 µm, constant vein width, L=1.5 µm, and symmetrical radius, R0 = 1.5 µm as shown in Figure 4.2a.
In the same way with coreless structure, bandgap emerges through SIS breaking caused by changing radius as shown in Figure 4.2b. Consequently, the topological phase transition occurs at point G and wave propagation has opposite value within positive G and negative G as shown in Figure 4.10b. It has gapless phase transition in high frequency band gap region and shows localization as shown in Figure 4.10c, but there exists a relatively large band gap in low frequency region.
Although the intrinsic ARM edge states in the low-bandgap region still support a topological phase transition, we add lattice components with radius R0 = 3 µm to improve symmetry and enforce a gapless phase transition, as shown in Figure 4.11a. In addition, the edge states of the vascular structure in the same methods are discussed and analyzed how the relatively large breaking order of the SIS affects the topological phase transition, as shown in Figure 4.12. The edge of the ARM interface consists of 16x10 unit cells as shown in Figure 4.16a and shows the same propagation results (Figure 4.16 b-g).
If we excite ZIG-L (ZIG-S) with symmetric (anti-symmetric) force, the wave does not propagate even in frequency in the bandgap region, as shown in Figure 4.17a and Figure 4.17b. For all edge cases, trough-dependent wave propagation corresponds to phased excitation. Unidirectional wave propagation refers to any trough condition in both frequency band gaps, as shown in Figure 4.18 and Figure 4.19. Waves generated from the red (blue) point propagate in a topologically protected manner, first along the ZIG-L (ZIG-S) edge, and then along the ARM edge (Figure 4.20b).
25b-c and Figure 4.26b-c show the real-time snapshot of the wave propagation displacement field with the same time period.
Qiao, Z., et al., Two-dimensional topological insulator state and topological phase transition in bilayer graphene. Feng, L., et al., Acoustic feedback negative refraction in the second band of an audio crystal. Wu, X., et al., Direct observation of valley-polarized topological edge states in designer surface plasmon crystals.
Miniaci, M., et al., Experimental observation of topologically protected helical edge modes in patterned elastic plates.
Acknowledgement
