Determining the mechanical properties of AlN films using micromechanical membranes

Micromechanical membranes are an important platform for a wide range of electro- and optomechanical experiments. This goes all the way from applications like ultra-sensitive scanning force microscopy,¹ radio frequency signal amplification,² and microwave to optics conversion,³ to fundamental studies of cavity optomechanical backaction,⁴ exceptional points,⁵ and radiative heat transfer mediated via Casimir fluctuations.⁶ Their size in the 100-micrometer range allows the study of, e.g., mode-hybridization⁷ and the spatial structure of nonlinear dynamics.^8,9 Another important aspect of these very thin membranes is that they allow to study the properties of materials placed on top.¹⁰ This provides an interesting route to measuring the mechanical properties of a variety of materials that can be deposited or grown on top of membranes.

Here, we employ high-stress silicon nitride (SiN) membranes to study the properties of aluminum nitride (AlN) grown on top of it. AlN is not only an important piezo-electric material for classical devices, but it is an emerging material for quantum technology¹¹ with applications ranging from mechanical resonator–qubit coupling,¹² microwave to optics conversion,¹³ on-chip isolation,¹⁴ photon-pair generation using spontaneous parametric downconversion,¹⁵ frequency shifting of single photons,¹⁶ to hybrid photonics.¹⁷ For many of the aforementioned applications, knowing the mechanical properties of the AlN is crucial.^18,19 Furthermore, the mechanical and optical properties are sensitive to variations in the deposition conditions^20,21 and are often correlated. The growth may also vary spatially²² and thus make it desirable to locally determine the mechanical properties in situ on the same chip without negatively impacting the intended application. We show that this is possible with bilayer membranes, which are readily fabricated, robust, and easy to measure. The stress in the AlN top layer can be determined as a function of the film thickness from the membrane's mechanical resonance frequency shift, in combination with mechanical band structure calculations.

Figure 1(a) shows optical micrographs of resulting membranes. The different colors result from the different AlN thicknesses, $hAlN$⁠, which are determined using optical reflectometry.¹⁷ Each chip is glued onto a piezo-electric actuator and mounted in a vacuum chamber. The driven motion of its 120 membranes is detected optically in a scanning interferometric setup with a probe laser (wavelength 633 nm) as detailed in Refs. 7 and 25. A brief setup description can also be found in the supplementary material. Figure 1(b) shows the responses of a typical bilayer membrane and a SiN-only membrane. There, the fundamental (1,1) modes appear at different frequencies, showing the influence of AlN. Higher modes are also visible. Membranes dominated by bending rigidity will show different fundamental-to-higher-mode frequency ratios than tension-dominated membranes. In our case, the higher modes match with the $f(m,n)$ calculated with Eq. (1) and the extracted $f(1,1)$⁠, thus confirming that these membranes are indeed in the tension-dominated regime. Still, to avoid that bending rigidity (which becomes more important for smaller membranes and/or higher modes¹⁰) starts to influence the results,²⁵ we limit the analysis to membranes with size $≥100 μm$⁠, of which there are 53 on each chip. For all membranes, the exact resonance frequency and quality factor are obtained semi-automatically from detailed driven response measurements in the linear or weakly nonlinear regime.²⁵ This is repeated for all chips with different AlN thicknesses. We observe that the more AlN is deposited, the lower the resonance frequency is. Figure 1(b) already showed that $f(1,1)=1.36 MHz$ for the SiN-only membrane doped to $f(1,1)=1.00 MHz$ when depositing 77 nm AlN. Such a comparison can be done for all 120 membranes on each chip. These differ in the numbers of holes $Nx,y$ as well as in lattice constant A and thus have different dimensions. Hence, $f(1,1)$ varies strongly,²⁵ making it difficult to see subtle trends when comparing frequencies directly. Equation (1) shows that by multiplying $f(1,1)$ with twice the effective membrane size $Leff≡(1/Lx2+1/Ly2)−1/2$⁠, the speed of sound c is obtained. Its value depends on the bilayer properties as outlined above, but—importantly—it should be size-independent. Given the close relation between the quantity $2Lefff(1,1)$ and c, we will also call the former the speed of sound.²⁵  Figure 1(c) shows that, although the frequencies from which c was extracted varied by more than a factor of 3, c is indeed independent of $Leff$⁠. Moreover, the decrease in c with increasing AlN thickness is now much more visible than when plotting $f(1,1)$ directly.²⁵  Table I shows that the change in c is not simply proportional to the AlN thickness; there is, e.g., a large decrease in c for the 45 nm layer, whereas the difference between the 77 and 140 nm films is quite small. The in-plane tensile stress of the AlN films clearly affects the tension of the membrane.

The accuracy of the method also allows for the detection of small trends. Interestingly, Table I shows that c depends on the lattice constant A. This is not expected from the “ideal”-membrane model presented above. Looking back at the micrographs of Fig. 1(a) shows that the actual geometry is not a perfect rectangular membrane. First of all, the membranes are not continuous but have periodically arranged holes through which the buffered hydrofluoric acid (BHF) etches the oxide, forming cylindrical “drums” of suspended SiN. After 130 min, the undercut of $10 μm$ is so large that fully suspended membranes are formed but with slightly “wavy” edges.²⁹ The SiN is also partially etched. On the top, it thins uniformly, but at the bottom side, longer BHF exposure closer to the holes results in conical profiles.^25,30 All these effects can be accounted for with computationally expensive three-dimensional finite element (FEM) simulations.²⁵ However, since the entire structure is periodic, mechanical band structure simulations provide a good alternative as shown there. Figure 2(b) shows the calculated band structure $f(k)$ of the unit cell shown in Fig. 2(a); the color indicates the polarization of the modes. In addition to the flexural modes, the longitudinal and transverse modes were also observed. Further details about the simulations can be found in the supplementary material. In the following, we will only consider the experimentally observable flexural modes.⁷ For small k, their frequency depends linearly on the wavenumber. For increasing k, a slightly higher slope (i.e., an increased group velocity) is observable; this is due to the bending stiffness. Furthermore, near the edges of the Brillouin zone gaps open, as visible in the zoom of the area near the X point. These effects become important for large $k∼π/A$⁠, whereas we are interested in the opposite limit. Note that the wavevector $k$ is fixed by the membrane size and mode: $kx=mπ/Lx=mπ/(NxA)$ and likewise for y. Since only the fundamental mode $(m,n)=(1,1)$ of large membranes $Lx, Ly≫A$ is studied here, the band structure near the $Γ$-point is the most relevant. There, $f(k)$ is linear and the group velocity $2π∂f/∂k$ matches the speed of sound c. Modes propagating in different directions may have different speeds of sound,²⁵ but for the lowest mode, c is independent of the direction of $k$ [cf. circular shape of Fig. 2(c)]. Therefore, the membrane can be treated as isotropic and just a single $k$ value needs to be simulated to obtain c for all combinations of $Lx$ and $Ly$⁠. With this simplification, one could think that the patterning has little impact, but this is not true, as the experimental values in Table I already showed. When a unit cell of an idealized membrane, i.e., one without release holes and with a uniform thickness $h SiN$⁠, is simulated [Fig. 2(d)], the results match with Eq. (4). However, for the more realistic geometry with holes and partly etched SiN, simulations must be used to connect the speed of sound to the bilayer properties.²⁵ We note that using a geometry close to the “ideal” membrane is also possible, e.g., by using commercial membranes.³¹ It is the convenient combination of our in situ measurement, front-side-only processing, and easy interferometric detection that results in this geometry, which requires this modest computational step. This approach is illustrated in Fig. 2(e), where we plot the dependence of the speed of sound on the stress in the AlN film for the four different thicknesses studied in the experiment. For zero stress in the AlN layer, c decreases with increasing $hAlN$⁠, due to the increased mass $μ$⁠. For $σAlN≈σ SiN =1105.1MPa$ (right side), the speed of sound converges to a single value as also predicted by the simple model of Eqs. (2)–(4). Note that in the stress-dominated regime, bending rigidity is unimportant, but this may no longer be the case for very thick membranes. This will then be signaled by a nonlinear $c(k)$ dependence in the simulations or $c(Leff)$ dependence in the experiments. Returning to the experimental data, the sample without AlN is considered first. Here, the simulations are fitted to the experimental values for c by varying the SiN film stress.^32, Table I shows the $σ SiN$ value that is obtained using this method. In contrast to the simple model of Eq. (4), this now yields a consistent $σ SiN$ for all lattice constants, despite the observed variation of c with A. This shows that the mechanical band structure calculations are essential for quantitatively understanding the membranes. Combining all SiN-only data gives a weighted mean value $σ SiN =1101.5MPa$ for the tensile in-plane stress, which is used for the subsequent simulations of bilayer membranes. By adjusting the stress in the AlN film, these simulations are fitted to the experimental c. The resulting $σAlN$ for every set of A and $hAlN$ is listed in Table I, whereas the combined (i.e., with all A for a given $hAlN$⁠) fits are shown in Fig. 3(a). The FEM model again reproduces the upward trends with increasing lattice constant, although experimentally, the dependence $c(A)$ is a bit flatter than in the simulations. The stress values obtained by fitting the data are shown in Fig. 3(b), with negative stress values for the 45 nm and the 77 nm thick film. This indicates a compressive strain in the AlN, translating into a reduced in-plane stress component of the bilayer membrane. For the thickest AlN film, the strong reduction in c with increasing thickness has halted and the 140 nm thick AlN film has a tensile stress. Such a transition has also been observed in other AlN thin films.^27,33,34 The increased tension from the strained AlN film outweighs the additional increase in mass. These measurements thus show the feasibility of using micromechanical membranes to determine the stress (and with the elastic constants also the strain) in the top layer. The method is nondestructive, easy to use, and unlike wafer-bending^{22,27,33–35} and bulge-test^22,36 techniques, it is fully compatible with pre-patterned samples like our hybrid AlN-on-SiN photonic integrated circuits¹⁷ and will also work with samples where the second film is not continuous.³⁷ Furthermore, every membrane probes the local material properties and the uniformity can be studied over a chip or wafer. To demonstrate this, the spatial dependence is studied in Fig. 4. Panel (a) shows the speed of sound. The colorscale shows that c decreases with the vertical position on the chip. This is, however, mainly due to variation of A in the chip design [cf. Fig. 3(a)]. Still, devices on the right-hand side of the chip appear to have slightly higher c than the left side. To make this more apparent, $σAlN$ is determined for every membrane individually and plotted in Fig. 4(b). To avoid any residual systematic trends with the lattice constant, the average value over all membranes with the same A is subtracted. This map clearly shows the gradient of the stress over the chip. The variations are only about a few percent of the stress in the SiN membrane, highlighting the sensitivity. Further improvements will focus on reducing the variations and fit uncertainty and also on extracting the elastic moduli, e.g., by probing different modes and/or smaller membranes, and the dissipation.¹⁰ 

In summary, we demonstrated the use of micromechanical membranes to probe the mechanical properties of thin films, specifically the stress in AlN on SiN. Additionally, we showed that the speed of sound can vary depending on the geometry of the release holes. Finally, we presented the use of these membranes as local probes to examine the mechanical properties across a centimeter-large sample.