Analysis of Membrane Phononic Crystals with Wide Band Gaps and Low-Mass Defects

We present techniques to model and design membrane phononic crystals with low-mass defects, optimized for force sensing. Further, we identify the importance of the phononic crystal mass contrast as it pertains to the size of acoustic band gaps and to the dissipation properties of defect modes. In particular, we quantify the tradeoff between high-mass-contrast phononic crystals, with their associated robust acoustic isolation, and a reduction of soft clamping of the defect mode. We fabricate a set of phononic crystals with a variety of defect geometries out of high-stress stoichimetric silicon-nitride membranes and measure at both room temperature and 4 K in order to characterize the dissipative pathways across a variety of geometries. Analysis of these devices highlights a number of design principles integral to the implementation of low-mass low-dissipation mechanical modes into optomechanical systems.

Figure 1

(a),(b) An optical-microscope image of high- and low-mass-contrast PnC devices containing a defect respectively. The displacement of the devices measured both in the crystal bulk (blue) and at the defect (orange). (c),(e) Thermal noise spectra of the high-contrast device on a logarithmic scale. The defect mode shown in (e) is the fundamental symmetric mode. (d),(f) Mechanical spectra for the low-contrast device on a logarithmic scale. The device is driven with sufficient white noise in order to observe all expected defect modes. The spectrum shown in (f) contains five separate defect modes within the band gap, including the second symmetric mode examined in this study. The defect-mode frequencies in both (e),(f) agree with predictions from 2D FEM simulations. Mechanical modes that appear inside the band gap in (d) have quality factors less than 100, and therefore are most likely hybridized modes between the membrane and the silicon chip or mounting assembly.

Figure 2

The critical dimensions for the unit cells of all fabricated devices. The dimensions in the table are depicted above. $n_x$ and $n_y$ are the unit cell numbers of the PnC along the $X$ and $Y$ axes. The table entries are omitted where they do not apply. The fabricated devices for A and C include fillets of radii $2.5\mu \mathrm{m}$ at the sharp corners of the pad.

Figure 3

The band-gap widths as a function of the mass contrast as determined from the 1D model: a comparison between different models of the band-gap to mid-gap ratio ($\mathrm{\Delta }$) versus the PnC contrast ($V$). The analytic prediction of the 1D model [Eq. (8)] (solid blue line), the FEM simulations of equivalent 2D structures (blue circle markers), and the experimental results of device A (red star marker) and device B (yellow square).

Figure 4

The effects of contrast for 1D PnCs with a constant defect geometry. (a) The orange points are the fundamental defect-mode frequencies across all simulated structures. The shaded region denotes the frequency range of the simulated band gaps. The insets in (a) are schematics of the simulated 1D unit cell at the lower extreme (left) and upper extreme (right) of the band-gap width. (b) The normalized-mode amplitude-decay length (blue) and quality factors (gray) for 1D PnC strings with a constant defect. We define $n_0=l_0/a$, where $l_0$ is the exponential decay length. At large contrasts, $n_0$ approaches $1$ because the defect mode does not begin to decay until the onset of the PnC.

Figure 5

The kinetic energy in the PnC frame, normalized to the total energy, in both high-contrast (red points, $V=2.3$) and low-contrast (yellow points, $V=1.3$) PnCs as calculated from FEM simulations. The insets are the unit cells used in the simulation. The dashed lines are fits assuming exponential energy decay for the defect mode. The decay length for the high-contrast unit cell (red fit) is 0.54 unit cells. The decay length from the low-contrast unit cell (yellow fit) is 1.24 unit cells.

Figure 6

A schematic of the experimental apparatus used for measuring both the mechanical spectra and the ringdowns. The mechanics chip (green) is affixed atop a stack consisting of a piezoelectric transducer (“piezo,” dark blue) and a highly reflective mirror (light blue). This stack is mounted onto a stage linked to the sample stage (cold stage) for room-temperature (4-K) measurements. The solid box represents the vacuum shroud for both the room-temperature and the 4-K apparatus, while the dashed box represents the radiation shield present for 4-K measurements. The light reflected off the stack is sent to a photodiode via a polarizing beam splitter.

Figure 7

A compilation of all the devices presented in this work. In the first row are optical-microscope images of suspended phononic structures. Above are device labels that correspond to device properties plotted in Fig. 8. The second row presents FEM simulations of symmetric defect-mode shapes. The third row shows FEM simulations of the static stress distribution normalized to the film stress. The fourth row displays the normalized bending loss density.

Figure 8

(a) The dependence of the quality factor on the loss factor $L_s$ for all fabricated devices, measured at room temperature (orange) and at 4 K (blue). The symbols correspond to the key presented in Fig. 7. The dashed lines represent calculated $Q$ values from Eq. (9), with $E_2=80$ MPa for the room-temperature data and $E_2=17$ MPa for 4 K. (b) The calculated force sensitivity at room temperature (orange) and at cryogenic conditions (blue). The box is to emphasize the similarity in defect design and performance between devices A and C.

Figure 9

The geometry of the 2D-to-1D conversion for a pad-tether unit cell. The blue box (orange box) defines the pad region (tether region) of the PnC unit cell.

Figure 10

A schematic of 1D collapse for a low-contrast unit cell. (a) The definition of the pad and tether regions inside the low-contrast unit cell. (b) The 1D geometry that results from performing the collapse: the gray regions indicate the high-velocity tether regions used to define $V$ for this unit cell.

Figure 11

The distribution of contrast values derived from different partitioning of the 1D geometry.