Reaching silicon-based NEMS performances with 3D printed nanomechanical resonators

The extreme miniaturization in NEMS resonators offers the possibility to reach an unprecedented resolution in high-performance mass sensing. These very low limits of detection are related to the combination of two factors: a small resonator mass and a high quality factor. The main drawback of NEMS is represented by the highly complex, multi-steps, and expensive fabrication processes. Several alternatives fabrication processes have been exploited, but they are still limited to MEMS range and very low-quality factor. Here we report the fabrication of rigid NEMS resonators with high-quality factors by a 3D printing approach. After a thermal step, we reach complex geometry printed devices composed of ceramic structures with high Young’s modulus and low damping showing performances in line with silicon-based NEMS resonators ones. We demonstrate the possibility of rapid fabrication of NEMS devices that present an effective alternative to semiconducting resonators as highly sensitive mass and force sensors.


Thermal noise
An additional evaluation of material Young's modulus is performed by extrapolating the effective stiffness (or spring constant) of the devices from the measurement of thermal noise spectra. From the effective stiffness is then possible to obtain the material Young's modulus.
The effective stiffness of the device can be extrapolated using the equipartition theorem from the mean squared displacement of the resonator's thermal motion 〈 ℎ 2 〉 as: where kB is the Boltzmann constant and T the absolute temperature. 〈 ℎ 2 〉 is experimentally measured integrating the area under the resonance peak of the thermal noise spectrum 1 .
Then by comparing the theoretical resonance frequency f0 of the resonator from the Eulero-Bernoulli beam theory (Equation 1 of the main article): with the theoretical resonance frequency from a lumped-element model resonator 2 : where meff is the effective resonator mass (for a cantilever at fundamental resonance mode is 0.25LWTρ), the dependence of the Young's modulus from the effective stiffness results as: The evaluation of Young's modulus from thermal noise spectra has been conducted on less devices than the driven analysis, because the measurement of resonance peak from thermal noise is experimentally complex on small devices with resonance frequencies in the MHz regime. Very small amplitude vibration and elevate contribution of instrumental noise reduce the detection of thermal noise peak and introduce high uncertainty in the Lorentzian peak fitting. We obtained a Young's modulus from thermal noise spectra evaluation of 268 ± 59 GPa (see Supplementary Figure S7) which is in line with the Nd:YAG tabulated material value (290 GPa) and the results of drive measurement analysis (fit in Figure 3b).

Supplementary Figure S7
Young's modulus for different devices evaluated by thermal noise method.

Nanoindentation
The Young's modulus was also determined by carrying out a static mechanical analysis by means of The deflection is measured in Volts and converted into force as = ⋅ ⋅ while is obtained as = − ( ⋅ ). Given that the measured specimens are stiff and there is negligible adhesion between the tip and the sample, the Hertz model 3,4 was adopted for fitting the retract part of the curves and determining the Young's modulus. The Hertz formula that describes the force as a function of the tip penetration into the sample is: where is the tip radius and is the reduced Young's modulus, given by: where is the Young's modulus, is the Poisson ratio and the subscripts and refer to the tip and sample, respectively. For the tip, we used = 1140 GPa and = 0.2. For the analysis, we used the linearized model and fitted the curves as so that the fitting function is a line, and the reduced modulus is obtained from the slope, avoiding the need to precisely determining the contact point. We always fitted the retract curves in the 25%-90% range to take as much as possible into account only the elastic deformations of the sample and to avoid possible mechanical adjustments between the tip and the sample during the approach or during the final part of the retract curve.
Supplementary Figure S8a reports a topography map obtained on a membrane. Since the nanoindenting tip is not meant for imaging, the topography contains artefacts due to the convolution between the sample and the tip shape. Indeed, the height of the tip is 50 μm, the cantilever is slightly inclined with respect to the surface of the sample and the scan was performed by acquiring 256 horizontal lines starting from the bottom. Therefore, it can be noticed that on the sides but especially on the top edge of the membrane the apparent slowly decreasing height of the sample is due to the fact that the lateral face of the pyramidal diamond tip is touching the edge of the sample. The map was merely used for helping to select the points on which to acquire the force curves, which were taken close to the edges of the sample. Supplementary Figure S8b reports the Young's modulus value obtained by averaging over 30 different values obtained from as many force curves on the YAG membrane. The result was = 272 ± 89 GPa, by assuming a Poisson ratio equal to = 0.275. is therefore in good agreement with that reported in literature for Nd:YAG 5,6 . The rather large uncertainty, calculated as the standard deviation, is due to the fact that the measurement is local and can be particularly sensitive to the presence of crystalline grains and grain boundaries (see Figure S6). Some force curves were also acquired on the sapphire substrate close to the membrane. Also in this case the obtained value, = 408 ± 64 GPa (with = 0.27), is in the correct range for this material 7 , while the significant uncertainty is mainly given to the fact that, the higher the Young's modulus, the more sensitive is

Supplementary note 3 -Quality factor contribution
For the evaluation of the theoretical quality factor of our resonators we take in account all the possible dissipation mechanism which are involved in the damping of the mechanical vibration of our structures. The total contribution will be extrapolated by the sum of all the damping mechanism, by considering their contribution as the inverse of Q. The dissipation mechanism in our devices are gas damping (Qgas), clamping loss (Qclamp), internal friction (Qfrict), surface loss (Qsurf) and thermoelastic damping (QTED) 2 .
Gas damping (Qgas) in high vacuum regime is the dissipation mechanism due by the interaction of air particles with the resonator device. Two separate contribution are present, drag-force damping where ρ, ω, L are the resonator density, eigenfrequency and length, respectively, p the air pressure, T the temperature, d0 the gap between surface and the device, M the gas molar mass and R the universal gas constant. Both damping mechanisms give a negligible Q contribution since their values are always above 10 10 for our resonators 9 .
Clamping loss (Qclamp) are related to the energy loss at the clamping of the resonating device with the bulk. Since we have a sufficiently thick supporting structures, the damping contribution can be evaluated as: where w is the device width. Even if the clamping loss follow a t -4 dependence, the contribution in our device is always far below the other damping mechanism (Qclamp>10 5 ) 10 .
Internal friction (Qfrict) are due to the atoms motions during the device vibrations and thus to material viscoelasticity 11 . Friction losses are computed directly by the ratio between the real (E) and imaginary (E') part of Young's modulus which is the definition of the inverse of the loss tangent (tan δ): Polymeric materials have loss tangent in the range 10 -1 -10 -2 and then friction loss results as the main dissipation mechanism in standard 3D printed device and in our NEMS device before thermal curing.
Ceramic materials like aluminum oxides and thus garnet (YAG) exhibits loss factor of around 10 -5 which give a Q contribution of 10 5 . Therefore, after thermal curing, the contribute to overall damping given by internal friction in our resonators becomes very small.
Surface loss (Qsurf) and thermoelastic damping (QTED) mechanisms are described in detail in the manuscript.
All the dissipation mechanism contributes in our devices are reported in Supplementary Figure S9 evidencing as Qsurf and QTED dominates over the other loss factors.   Figure S10 and S11 (some works presented more than one type of resonator).