Modifying the bonding character of coupled states of thin-plate resonators via prestress modulation
In diatomic molecules, the anti-bonding eigenstate is the higher energy state as the chemical bond is weakened between the two atoms due to the destructive interference of electronic wavefunctions. It is counterintuitive for the anti-bonding state to have lower energy given the bonding state has stronger chemical bond and is more stable. However, in this study we demonstrate that the analog of the "anti-bonding" eigenstate can be lower than the "bonding" eigenstate in coupled thin-plate resonators. In fact, the reversal of such two states has been demonstrated in photonics [1], quantum dots [1], and superconductors [3], achieved by tuning the coupling or the tunneling parameter. One distinct advantage in the case of thin-plate resonator lattices is that the reversal can be achieved dynamically by modulating the pre-stress to the bending stiffness ratio, allowing intriguing phenomena like stopping and reversing a sound pulse [4]. A moving sound pulse can be arrested by dynamic dispersion tuning towards the degeneracy between the two states occurring at a flat dispersion band and reversed by further tuning the band structure from acoustic-like to optical-like or vice-versa [4]. Lattice symmetry of the 2D thin-plate resonator Kagome lattice creates flat bands of a singular kind that hosts robust boundary modes protected by the real-space topology of the underlying lattice [5].
We used the general form PDE module in COMSOL MULTIPHYSICS to define an eigenvalue problem for a fourth-order partial differential equation describing thin-plate elasticity. Dirichlet and Neumann boundary conditions (clamped boundary condition) were used for edges that were not coupled. Simulation results were compared against the analytical solutions for the square Laplacian plate (setting the bending stiffness equal to zero) and circular clamped biharmonic plate (setting the pre-stress equal to zero). For the plate geometry, triangular mesh was used with following parameters: maximum element size = 0.08, minimum element size = 1.5E-04, maximum element growth rate = 1.2, curvature factor = 0.25, and resolution of narrow regions = 1. Band structures were calculated by setting Bloch periodicity at appropriate edges of the unit cell. The continuum thin-plate resonator model has infinitely many bands, of which a subset was obtained numerically. The full-wave dynamical simulations were performed by adding the second-order time derivative term to the partial differential equation using time-dependent physics in the general form PDE. A time step of 0.01 was used in the time-dependent solver called the generalized alpha.
In this study, we used COMSOL MULTIPHYSICS to create a custom model describing the fourth order thin-plate elastic resonator system. Using the model, we demonstrated that the “bonding” and “anti-bonding” eigenstates become degenerate at a pre-stress value that is dependent on the coupling strength. The ground state can be switched from the bonding to the anti-bonding eigenstate and vice-versa by modulating the pre-stress around the degeneracy. We calculated band structures for the 1D and 2D continuum lattices and presented dispersion bands associated with first eight eigenstates. Finally, we used dynamical simulations to show that higher energy eigenstates evolve towards the ground state.
