Figure 1: Device design.

(a) Schematic of the membrane electromechanical circuit. (b) Unit cell of the phononic crystal nanobeam. (c) Acoustic band diagram of the phononic crystal nanobeam with a=W=2.23 μm, W_x=W_y=1.52 μm and W_Al=170 nm. The nitride membrane thickness and aluminum wire thickness are t_mem=300 nm and t_Al=65 nm, respectively. The acoustic bandgap is shaded in blue, with the localized breathing mode frequency indicated as a dashed line. (d) Plot of the finite-element method (FEM)-simulated breathing mode profile. Mechanical motion is indicated by an exaggerated displacement of the beam structure and by colour, with red (blue) colour indicating regions of large (small) amplitude of the motion. (e) Electrical circuit diagram, where I_c is the current through the reflective coupler, L is the coil inductance, C_l is the coil capacitance, C_s is additional stray capacitance and C_m is the motional capacitance. The simulated displacement of the in-plane fundamental flexural mode of the beam is shown. (f) Inductance (L) and capacitance (C_l) of a planar square coil inductor of constant area A_coil=87 μm × 87 μm and variable wire-to-wire pitch p. Wire width and thickness are 500 and 120 nm, respectively. Method of moments³⁰ numerically simulated values are shown as open circles (inductance) and open squares (capacitance). Calculations using an analytical model of the planar coil inductor³¹ are shown as a solid line. Vertical lines are shown for coils with a characteristic impedance of Z₀=1, 5 and 20 kΩ, with the coil self-resonance frequency indicated in brackets. (g) FEM simulations of the modulated capacitance C_m (blue symbols) and the electromechanical coupling g₀/2π (red symbols) of the in-plane fundamental flexural mode (circles) and the phononic crystal breathing mode (squares) as a function of the capacitor gap size s. Solid curves indicate a 1/s fit for C_m and 1/s^d with d≈1.4 for g₀.