Control of Material Damping in High-$Q$ Membrane Microresonators

We study the mechanical quality factors of bilayer aluminum–silicon-nitride membranes. By coating ultrahigh-$Q$ ${\mathrm{Si}}_3{\mathrm{N}}_4$ membranes with a more lossy metal, we can precisely measure the effect of material loss on $Q$’s of tensioned resonator modes over a large range of frequencies. We develop a theoretical model that interprets our results and predicts the damping can be reduced significantly by patterning the metal film. Using such patterning, we fabricate Al-${\mathrm{Si}}_3{\mathrm{N}}_4$ membranes with ultrahigh $Q$ at room temperature. Our work elucidates the role of material loss in the $Q$ of membrane resonators and informs the design of hybrid mechanical oscillators for optical-electrical-mechanical quantum interfaces.

Figure 1

Geometry of membrane modes. (a) Image of a 1 mm membrane in its silicon frame. (b) Illustrations of the $(m,n)=(2,2)$ and $(15,15)$ modes. (c) Image of a patterned Al film on top of ${\mathrm{Si}}_3{\mathrm{N}}_4$. The central square is Al, and the rim is ${\mathrm{Si}}_3{\mathrm{N}}_4$ suspended on a Si frame. (d) Schematic diagram of the $\mathrm{Al}/{\mathrm{Si}}_3{\mathrm{N}}_4$ bilayer membrane (50 nm Al and 50 nm ${\mathrm{Si}}_3{\mathrm{N}}_4$).

Figure 2

Radiation loss for a square membrane. Measurements of quality factors for many different modes of an (a) $0.5\times 0.5$ and (b) $1\times 1\mathrm{mm}$ ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane. The symmetric modes generally have higher $Q$ than asymmetric modes, as predicted by a radiation-loss model [5].

Figure 3

Extracting the material-loss-limited $Q$. (a) Measured quality factors of a square ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane before (green circles) and after (blue squares) adding a 50 nm film of Al. The modes limited (not limited) by radiation loss are marked by open (closed) circles. The data are plotted as a function of frequency measured after adding the Al. (b) Mechanical linewidth of the modes limited by the material loss of Al for $0.5\times 0.5$ (red diamonds) and $1\times 1\mathrm{mm}$ (blue squares) membranes. To compare to theory, we calculate the damping rate ${\gamma }_{mn}/2\pi$ for each mode, and the points are connected with the displayed lines.

Figure 4

Ultrahigh-$Q$ metal-covered membranes. Measured quality factors of two ${\mathrm{Si}}_3{\mathrm{N}}_4$ membranes with Al everywhere but near the edge of the membrane (blue circles) [see Fig. 1c] and a full film of Al as a control experiment (red squares). Asymmetric modes (with one mode index less than or equal to 2) are marked by open circles or open squares. Calculated quality factors for each geometry are shown by the two lines; a single loss modulus is used for both. (Inset) The square of the curvature of a stressed mode as a function of distance along one coordinate of the membrane. This function decays exponentially near the membrane edge.