Figure 2: Sample fabrication.

(a) Optical image of the membrane microchip, which is mounted, bonded and clamped to a low loss printed circuit board (scale bar, 1 mm). The microchip contains four sets of four membranes. In this image the Si₃N₄ membranes of thickness 300 nm are semi-transparent purple, the aluminum coated regions are grey and the uncoated silicon substrate is green. The two bright regions in the middle of each membrane correspond to the two coil resonators coupled to each nanobeam resonator. (b) False colour scanning electron microscope (SEM) image of the centre part of the membrane depicting two aluminum planar coils (white) coupled to two sides of a single patterned phononic crystal nanobeam with stress pull-in cuts (black). Scale bar, 100 μm. (c) SEM image zoom-in of the spiral inductor (p=1 μm, n=42), showing the cross-overs needed to connect the inductor coil to the vacuum gap capacitor across the nanobeam resonator. Scale bar, 50 μm. (d) SEM image zoom-in of the released centre region of the nanobeam mechanical resonator and vacuum gap capacitors with gap size of s≈80 nm. Scale bar, 5 μm. (e) Tilted SEM image of the capacitor gap showing the etch profile of the nanobeam and the aluminum electrode thickness (≈65 nm). Scale bar, 200 nm. (f) Schematic of the main circuit fabrication steps: (i) LPCVD of stoichiometric Si₃N₄ on both sides of a 200 μm-thick silicon substrate, (ii) C₄F₈:SF₆ plasma etch through the nitride membrane defining the mechanical beam resonator and pull-in cuts on the top side and membrane windows on the bottom side, (iii) electron beam lithography, aluminum deposition and lift-off steps to pattern the microwave circuit and (iv) final release of the nitride membrane using a silicon-enriched tetramethylammonium hydroxide (TMAH) solution. (g) Simulation of the membrane relaxation during release. The image shows the regions of positive (red) and negative (blue) displacement, δy, of the membrane. The stress release cuts (white) are shaped at an angle α to controllably narrow the capacitor gaps s during release. The rounded shape of the pull-in cut end section has been optimized to minimize the maximal stress points to avoid membrane fracturing. (h) Plot of the simulated (solid red curve) and SEM-measured (blue solid circles) change in the slot gap (δy) versus slot-cut angle α. Error bars indicate the single s.d. uncertainty in SEM measurements of the gap size.