Fabry–Perot interferometric calibration of van der Waals material-based nanomechanical resonators

One of the challenges in integrating nanomechanical resonators made from van der Waals materials in optoelectromechanical technologies is characterizing their dynamic properties from vibrational displacement. Multiple calibration schemes using optical interferometry have tackled this challenge. However, these techniques are limited only to optically thin resonators with an optimal vacuum gap height and substrate for interferometric detection. Here, we address this limitation by implementing a modeling-based approach via multilayer thin-film interference for in situ, non-invasive determination of the resonator thickness, gap height, and motional amplitude. This method is demonstrated on niobium diselenide drumheads that are electromotively driven in their linear regime of motion. The laser scanning confocal configuration enables a resolution of hundreds of picometers in motional amplitude for circular and elliptical devices. The measured thickness and spacer height, determined to be in the order of tens and hundreds of nanometers, respectively, are in excellent agreement with profilometric measurements. Moreover, the transduction factor estimated from our method agrees with the result of other studies that resolved Brownian motion. This characterization method, which applies to both flexural and acoustic wave nanomechanical resonators, is robust because of its scalability to thickness and gap height, and any form of reflecting substrate.

Nanomechanical resonators (NMRs) made from van der Waals materials like graphene 1-3 and transition metal dichalcogenides 4-7 offer plausible platforms for measuring minuscule forces 3,8 and ultralight masses 9 with unprecedented sensitivity and motion responsivity 5,7,10 to external stimuli. Typical NMRbased sensing platforms involve monitoring changes in features of the driven frequency response, primarily the shis of the mechanical frequency and variations of the linewidth, in the presence of an actuating force. 11 The calibrated vibrational amplitude at resonance, together with the resonator frequency, provides direct quantication of the Young's elastic modulus, effective mass, and driving force felt by the NMR from a single response spectrum. 12 From a practical standpoint, monitoring the vibrational amplitude is essential for avoiding dynamic pullin instability 13 when driving the NMR at high amplitude, which limits the actuation ranges for electromotively driven NMRs. Furthermore, the vibrational resonant amplitude enables understanding of the nonlinear regime of motion 14,15 and optomechanical effects on driven mechanics, 8,16,17 both of which are accessible for mechanical resonators with nanoscale dimensions, and embedded in motion-sensitive interferometric platforms.
Calibration of the motional amplitude of NMRs using Fabry-Perot (FP) interferometry has been challenging because it requires pre-determination of device parameters such as thickness and spacer gap, which are difficult to ascertain and vary from device to device. Previous attempts of this calibration were shown through the resolving of the Brownian motion, 7,18,19 analyzing the high-amplitude Duffing response, 3 measuring the driven motion at higher harmonics, 20 and monitoring photodetector responses. 12 The calibration methods used in these previous studies, however, rely on specic device conditions such as optically transparent cross-sections, 2,7 access to high amplitude nonlinear motion, and high optical-to-displacement responsivities 10 from an optimal vacuum gap. These characteristics are specic to monolayer, bilayer, trilayer, and few-layer van der Waals materials, whose unique optical characteristics, ultralight mass, and mechanics more closely resemble neartransparent, prestressed membranes. The methods would not be accurate if one or several of these criteria are not met.
Also, there are applications where a smaller vacuum gap between the NMR and the substrate has advantages over the optimal gap for interferometric detection. These include large frequency tunability in electromotively driven NMRs, 9,21 stronger electromechanical coupling between NMR and microwave cavities, 8,16 and introducing optomechanics in an electromechanical system. [22][23][24] For these applications, van der Waals materials of intermediate thickness (between 10 L and 100 L) offer comparable (if not better) optical reectance-todisplacement responsivity 10 and electromechanical coupling, 8,14,16,25 and near strain-free mechanical frequencies 26 as compared to their few-layer counterparts. Furthermore, they are easier to prepare via micromechanical exfoliation, and polymeric contaminants have a negligible contribution to their masses.
In this work, we show that the motion of an NMR can be calibrated by considering multilayer wave interference occurring on FP structures. To demonstrate the robustness of the technique, a thick van der Waals material, 2H-NbSe 2 , is used as the drumhead. 2H-NbSe 2 , at room temperature, is a semimetallic material, whose number of layers, when increased, is difficult to differentiate with Raman spectroscopy signatures 27 from the bulk, unlike MoS 2 . 28 This material makes it a unique model for assessing the number of layers based on the layerdependent refractive index. Our approach, implemented with laser scanning confocal microscopy, allows robust, non-contact and in situ determination of the layer thickness, spacer height, and device responsivity of each translucent exible mirror. Our calibration scheme, using a 532 nm laser wavelength, reveals a subnanometer displacement response of NMRs with thickness exceeding 50 nm. Furthermore, calibrated spatial imaging of the driven fundamental mode of circular and elliptical NbSe 2 drums enables direct investigation of the modal properties (i.e. effective mass and Young's elastic modulus) of the drumheads.

Experimental
Device fabrication P-type doped Si chips (7 mm Â 7 mm Â 0.65 mm), with a thermally grown 543 nm thick SiO 2 layer, were lithographically patterned with 20 nm Cr and 40 nm Au electrodes. The chips are cleaned, and then spin-coated with a CSAR-62 electron-beam resist to create the spacer. The resist is exposed in an electron-beam writer with drum holes and contact window patterns. Aer the development, the resist is baked at 180 C for 9 minutes to harden the spacer. The spacer thickness, as measured with a commercial stylus prolometer, is 295 AE 10 nm. Bulk NbSe 2 akes purchased from HQ Graphene are exfoliated and transferred onto the patterned drum and contact windows using a dry polydimethylsiloxane stamp transfer process. Additional details on the ake transfer process are narrated in other studies. 29,30 Electromotive actuation and interferometric detection The sample is placed in a vacuum box at ambient temperature (20-22 C), with optical and electrical access, which is evacuated to a base pressure of 6 Â 10 À7 mbar to minimize viscous losses. The NbSe 2 ake is galvanically connected to the output of the function generator, which creates DC and AC voltages between the ake and the grounded Au/Cr electrode forming a parallelplate capacitor. This voltage produces an attractive force between the capacitor plates, which actuates the drums when the driving frequency matches the resonance frequency. The drums experience an electromotive force arising from the DC and AC voltage inputs of the function generator. A custom-built confocal microscope, with a continuous wave 532 nm laser (Lighthouse Photonics Sprout-G) and a spot size of roughly 1.9 mm, probes the out-of-plane position and motional amplitude of the drums interferometrically. A fast avalanche photodetector (Thorlabs APD130A/M) measures both the average and modulated intensities of light reected from the drums, and converts them to the output voltages V andṼ . The data acquisition (DAQ) system records V at coordinates of X and Y. A highfrequency lock-in amplier (Stanford Research SRS-844) processesṼ with the signal frequency referenced from the function generator. See Fig. S1 † for a detailed schematic diagram of the interferometer setup.

Results and discussion
Determination of the thickness and vacuum gap via optical contrast Fig. 1(a) shows device A, a circular drum with a hole diameter of 7 mm, and device B, an elliptical drum with hole diameters of 8 mm (X, major axis) and 7 mm (Y, minor axis). The devices share the same ake, ground electrode, and driving voltages. The NbSe 2 ake and ground electrodes are separated by an insulating layer (electron-beam resist CSAR-62) and vacuum spacers, and hence form Fabry-Perot cavities for detection, and capacitors for actuation. A large rectangular opening, located tens of microns below the drum centers, allows the ake to connect to the voltage-controlled Au/Cr electrodes. The motion of the electromotively driven drumheads is detected interferometrically in a high vacuum environment.
Our method relies on different contrasts of light elastically reected from each zone, as shown in Fig. 1(b). The ake (pink bar) acts as a translucent movable mirror with thickness h, which is separated from the ground electrode by a spacer of height s. For convenience, the reected intensity is expressed in terms of reectance R, which is the ratio of the total reected light intensity to the incident intensity. Stationary mirrors have only DC component R ¼ R, while movable mirrors have both R and AC componentR. Zones 1 and 2 represent two stationary mirrors: stacks of gold, orange, green and blue bars having reectance R 1 ¼ R 1 and a mirror covered with a spacer (light gray) having reectance R 2 ¼ R 2 , respectively. Zone 3 represents two stationary mirrors separated by a dielectric gap (clamp) with reectance R 3 ¼ R 3 . Finally, zone 4 is the main FP cavity composed of one stationary and one movable mirror, which are separated by a vacuum gap with reectance R 4 . Here, zones 1 and 2 are references for zones 4 and 3, respectively. Scanning mirrors in the measurement setup move the laser spot in each zone a distance X and Y away from the center of the drums.
Application of DC and AC voltages to the ake exerts an attractive force; the NMR responds with an out-of-plane motional amplitude z at a driving frequency f d . Due to the position and motion of the movable mirror in zone 4, the main FP cavity has reectance Fig. 1(d) shows the photodetector output signal V acquired from R 4 . Both the DC component V and the AC componentṼ of the output signal are proportional to R 4 andR 4 , respectively. Amplitude z is determined aer obtaining h and s.
Though we calculate R 1-4 using the multilayer interference approach 31-33 (MIA), the reectance of FP cavities with four interfaces 34 R 3,4 captures the stationary reections occurring for each drum. Here, we assume that the coherent probe light, having wavelength l, originates from a point source and propagates from a semi-innite vacuum layer. The drum and the bottom mirror have complex refractive indicesn h (ref. 27) and n m , respectively, while the spacers have real refractive indexn s (n s,drum for the vacuum spacer andn s,clamp for the CSAR-62 spacer). In this geometry, the vacuum-NMR, NMR-spacer, and spacer-mirror interfaces contribute signicantly to the cavity's overall reectance. The total reectance is dened as where d h ¼ 2pn h h/l is the optical phase thickness of the NMR, d s ¼ 2pn s s/l is the optical phase thickness of the spacer, r h ¼ (1 Àn h )/(1 +n h ) is the Fresnel coefficient of the vacuum-NMR interface, r s ¼ (n h Àn s )/(n h +n s ) is the Fresnel coefficient of the NMR-spacer interface, and G m ¼ (n s Àn m )/(n s +n m ) is the equivalent Fresnel coefficient of the spacer-mirror interface. For convenience, G m is computed using MIA. † Fig. 1(c) shows the confocal image constructed from the DC voltage V of the photodetector. The image reveals the topographical features of the drum and its surroundings. The measured voltages that correspond to zones 1 and 2, mentioned in Fig. 1(b), are taken from the areas shown with the arrows in Fig. 1(c) and they are represented by V 1 and V 2 , respectively. Following the white dashed line in Fig. 1(c), the measured voltage outside the boundary is dened by V 3 , while the measured voltage inside is represented by V 4 . It is important to note that the mismatch between the drum boundaries in Fig. 1(a) and (c) is caused by the deformation of the edge of the drum holes during the elastomeric stamp step of the ake transfer.
Since R is susceptible to scattering losses, 35 we circumvent this issue by normalizing the Michelson contrast 34,36 of each FP cavity to its reference. Having dened the experimental and calculated reectance, the cavity's optical contrast, C, is quan- is the stationary reectance of the FP cavity, and R 2,1 is the stationary reectance of the cavity's reference. Apparently, C ranges between À1 and 1, with zero denoting no difference from the reference. If C is positive, then the cavity is brighter than the reference. Otherwise, the cavity is darker than its reference.
The output voltages measured for each pixel along the dashed lines in Fig. 1(c) are converted into contrast values for devices A and B, as depicted in Fig. 2(a). The experimental contrast C exp represents the ratio of voltages acquired from different zones in the confocal image of each device while the modelled contrast C mod is derived using MIA. † See Fig. S5 † for Fig. 2 (a) Diagram for determining h and s for the clamp and drum zones. Minimization of the difference between the experimental contrast (C exp ) and the modelled contrast (C mod ) results in h and s profiles for device A (b) and device B (c). Colored dashed lines refer to the hole radius set in Fig. 1(a and c), separating the drum (white fills) and clamp (gray fills) zones.
the correspondence between the modelled contrast and its h and s pairs for devices A and B. Fig. 2(b and c) show the resulting h and s cross-sectional proles acquired from minimizing the difference between the experimental contrast values and the contrasts generated by MIA. The mean resonator thicknesses and spacer heights are in excellent agreement with the mean values listed in Table 1. The thickness of the drums measured by our method agrees well with atomic force microscopy measurements for different areas of the ake, as shown in Fig. S2. † The spacer height for both drums and clamps agrees well with the stylus prolometer measurements. From the ake thickness of about 55 nm, we deduce 92 layers of NbSe 2 sheets assuming a single layer thickness of 0.6 nm. 37 The h proles in Fig. 2(b and c) show a hundred picometer variation between the drum and clamp zones. Meanwhile, buckling is observed in the s proles in Fig. 2(b and c) as s drum for both devices is greater than s clamp by 1.2-1.4 nm. We see the drumheads bulge [38][39][40] presumably due to the pressure difference between the trapped air in the drum hole and the vacuum environment. The surface roughness of the movable mirror likely originates from the thermally grown oxide 41 on the surface of the stationary mirror.

Conversion of optical signals to motional amplitudes
Having determined the mean h drum and s drum , we evaluate the optical reectance-to-displacement responsivity jd R 4 /dsj of each drum. This quantity is obtained fromR 4 The AC component reected from zone 4 and characterized byR 4 , being purely due to mechanical motion, is insensitive to any scattering losses as this wave is amplitude-modulated. Eqn (1) is then corrected by a prefactor of 0.28 that accounts for the 1.9 mm spot diameter of the probe Gaussian beam. See Fig. S3(a) † for more details on the scaling prefactor.
Estimation of jd R 4 /dsj requires the calculation of the gradient of the corrected R 4 with respect to s. Fig. 3(a) shows R 4 and its gradient as functions of s. As our NbSe 2 plate is considered bulk, 27 the R 4 versus s dependence shows a periodic yet non-sinusoidal behaviour. Yet, this dependence exhibits l/2 periodicity, though the peak-to-dip and dip-to-peak spacings are asymmetric. The minima and maxima in the jd R 4 /dsj versus s response are shied by about AEl/12 with respect to the dip in R 4 versus s, deviating from the periodic l/4 spacing expected for FP cavities with a partially transparent moving mirror. We also nd that the R 4 versus s dependence for a monolayer NbSe 2 ake follows a distorted, sinusoidal behaviour as shown in Fig.-S4(a). † This is unlike the regular sinusoid shape for those mechanical resonators of optically thin cross-sections. 2,7 Evaluating jd R 4 /dsj at s ¼ s drum (black dotted line) yields a device responsivity of 0.40 Â 10 À3 nm À1 .
We dene the average jd R 4 /dsj to account for spatial variations in s drum across the plate due to the pressure difference and DC voltage. Note that each complex-valued refractive index is dependent on the probing wavelength; this translates to the wavelength-dependent r h , r s and G m . We modeled jd R 4 /sj avg. by the chain rule jD R 4 (l)/Dl FP kDl FP /Dsj s ¼ s drum , where D R 4 /Dl FP is the change of R 4 with regards to the wavelength shi in the FP cavity, and Dl FP /Ds is the wavelength shi of the FP cavity caused by the change of the spacer gap. Here, we dene l FP as the wavelength at which R 4 (l) is maximum in the range of l ¼ 500-900 nm. The resulting dependences are shown as a waterfall plot in Fig. 3(b) with a gap range exceeding the uncertainty of our stylus prolometer. Fig. 3(b) demonstrates larger R 4 at near-infrared wavelengths. Fig. 3(c) shows the peak wavelength of the cavity, falling in the near-infrared range, shiing linearly with a slope of 1.543 nm/nm as s drum increases from 285 nm to 305 nm. Fig. 3(d) shows how the shi consequently increases R 4 (l) linearly, with a slope of 0.28 Â 10 À6 nm À1 . The product of these two slopes, jd R 4 /dsj avg ¼ 0.43 Â 10 À3 nm À1 , agrees with the evaluation in Fig. 3(a). The continuous behaviour seen in Fig. 3(b and c) is different from the discontinuous dependence observed for thin membranes in the same ranges of s, as shown in Fig. S4(b-d). † Their difference is mainly due to sharper features of the device responsivity of thin membranes as compared to thicker plates.  We use the average responsivity together with the interferometer system gain S(l) (V/W) (see the ESI † for more detailed calculations) and the laser probe power P in to dene the displacement amplitude z as whereṼ pk is the frequency and position-dependent peak voltage response of the NMR. The denominator in eqn (2), when squared, represents the transduction factor a (in V 2 m À2 ) that one can deduce from the measured Brownian motion of a mechanical resonator probed by an interferometric system. 18,19 This quantity accounts for the device responsivity and the detection parameters in the interferometric setup. We deduce ffiffiffi a p factors of 0.20 mV pm À1 for device A and 0.22 mV pm À1 for device B for probe powers listed in Fig. 4(a and b).
Calibration of motional amplitude in the frequency and spatial domains Fig. 4(a and b) show the measured voltage response and its corresponding z for devices A and B. The measured z response prole agrees well with a driven mechanical resonator model in the linear regime: 42 where f m is the fundamental mode frequency, Q m is the mode quality factor, and A eff is the amplitude expressed as effective acceleration. F(X, Y) is the mode shape of the device described as where J 0 and I 0 are the zeroth Bessel and modied Bessel functions of the rst kind, respectively, b ¼ 3.1961 is the fundamental root of the frequency equation for a clamped circular plate, and K 0 ¼ 0.947 is a normalization constant. Here, represents the normalized coordinates away from the maximum of z, where a and b represent the NMR major (X axis) and minor (Y axis) modal radii, respectively. By setting F(X, Y) ¼ 1, we measure z A ¼ 158 AE 2 pm for device A and z B ¼ 259 AE 3 pm for device B. Their magnitudes are three orders of magnitude smaller than h drum and s drum . By driving the plates at f m , and probing their spatial mode shape with scanning mirrors, we observe surface plots of z for devices A and B as shown in Fig. 4(c and d). Fig. 4(e and f) show X and Y axes cuts, with both axes intersecting at z max of Fig. 4(c  and d). They reveal z proles that agree with eqn (4), with a and b acting as free parameters. z max , a, and b of the two plates are listed in Table 2. The discrepancy in the values of z B and z max of device B is due to the location of the laser spot that probed Fig. 4(b). While z A lies at X ¼ Y z 0, z B lies at X, Y z 1 mm from the spatial peak. Both a and b for devices A and B are smaller than the hole radii (set as cyan and yellow dashed lines in Fig. 4(e and f)), making f m for both devices higher than the designed values. A plausible explanation for this is that an initial slack is introduced in the system during the fabrication process. 43,44 First, the ake is anchored to the spacer at the hole edge by van der Waals forces. Second, at room temperature, the annealed spacer, CSAR-62, which acts as a suspension support to the drumhead, is soer than typical NMR supports like SiO 2 1,4,5 and Au. 1,2,9 These factors, which add up to clamping losses, contribute to the low Q m measured for devices A and B. Furthermore, both devices suffer from imperfect, non-uniform clamping boundaries. 45 Table 2 lists other NMR-related quantities that are derived from Fig. 4 such as the effective mass m eff , acceleration A eff , force F eff , and Young's elastic modulus E Y . These quantities are   46,47 for bulk NbSe 2 akes. These quantities are obtained without inducing damage on the ake, and are independent of the actuation scheme. Eqn (4) does not explain the asymmetric sinusoidal waves propagating beyond the drum edges seen in Fig. 4(e and f). These waves are spatial signatures of support losses due to imperfect ake clamping at the edges. 48 Discussing the waves' origin goes beyond the scope of this study, though resolving the waves' amplitude, which is 1/3 of z max , demonstrates the capability of our method to visualize acoustic waves in NMRs. 49,50 Merits of the method Our motional amplitude calibration works as an ab initio technique because our model-based approach presupposes nonsinusoidal yet periodic Fabry-Perot reectance versus spacer height which arises from the thickness-dependent optical properties of the suspended ake. With an extensive database that accounts for the effects of the number of layers on the complexvalued refractive indices of 2H-NbSe 2 , the proposed method can distinguish in great detail the thickness and the number of layers present in NbSe 2 NMRs. The resolution in differentiating 1L, 2L, 3L, and multilayer 2H-NbSe 2 is shown in Fig. S6(a). † Moreover, its corresponding s is traced as simulated in Fig. S6(b). † The high level of detail and contrast between the drum and clamps in the h and s proles, as shown in Fig. 2(b and c), makes the confocal microscope an indispensable tool for the optical contrast scheme. Such microscopy, as compared to Raman spectroscopy, has been demonstrated previously to be faster and covers a larger mapping area in distinguishing the number of graphene layers supported by SiO 2 (ref. 51) and SiC 52 substrates.
The calibration method also complements the photodetector-based approach 12 as the resonator does not need to be driven to a nonlinear regime of motion to trace its linear motion. Moreover, the values of ffiffiffi a p acquired from our approach agree with those extracted from the measurement of Brownian motion from other studies as shown in Table 3. Also, see Fig. S7(a and b) † for the estimated power spectral density and Brownian motion of devices A and B. The simulated results conrm the validity of our method for both Brownian and linearly driven motion, which is sufficient for preliminary testing of integrated NMR devices.
Applying MIA for simulating optical-to-motional responsive 2H-NbSe 2 resonators with the same substrate, probe laser wavelength, and vacuum gap, we interpret from Fig. S3(c and d) † that resonators with thicknesses below 30 nm have higher device responsivities than the experimental samples. For a vacuum gap of 85 nm, the smallest vacuum gap achieved for suspended van der Waals materials is conducive to microwave optomechanics 8,14 and NMR-mediated cavity-qubit systems; 53 these resonators, except for monolayer NbSe 2 , show higher device responsivity. 2H-NbSe 2 , apart from being a superconducting van der Waals material used for single-photon superconducting detectors, 54 has the potential of being integrated with optoelectromechanical platforms.

Conclusions
In summary, we have demonstrated an in situ, non-invasive method of calibrating motional displacement of NbSe 2 NMRs by exploiting wave interference phenomena in FP cavities. Using a probe laser beam, and applying MIA, we have rst determined the cross-sectional proles of the NMR thickness and spacer height through contrast minimization. Then the transduction factors of NbSe 2 plate resonators are extracted to convert the measured signal to the actual motional amplitudes, which turned out to be in the order of hundreds of picometers. This information provides access to the fundamental mode properties of the drumheads. These include the femtogram effective mass of the drumheads, the 135 AE 13 GPa Young's modulus of the drum material, and the hundreds of piconewton driving force felt by the drumheads.
We note that this method has been instrumental in investigating the superposition of vibrational modes in plate NMRs induced by off-resonant frequency driving. 30 We foresee that this method, not limited to van der Waals materials, can be extended to exural NMRs and acoustic wave resonators. Lastly, this work may help establish FP laser interferometry as a noninvasive tool for evaluating NMR specications that are integrated into other solid-state components like superconducting qubits, photonic cavities, and NMR arrays.

Conflicts of interest
There are no conicts to declare.