# Effect of non-ideal clamping shape on the resonance frequencies of silicon nanocantilevers

Samuel Guillon, Daisuke Saya, Laurent Mazenq, Sorin Perisanu, Pascal

Vincent, Arnaud Lazarus, Olivier Thomas, Liviu Nicu

► To cite this version:

Samuel Guillon, Daisuke Saya, Laurent Mazenq, Sorin Perisanu, Pascal Vincent, et al., Effect of non-ideal clamping shape on the resonance frequencies of silicon nanocantilevers. Nanotechnology, 2011, 22 (24), pp.id 245501. 10.1088/0957-4484/22/24/245501. hal-01343625

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# Effect of non-ideal clamping shape on the resonance frequencies of silicon nanocantilevers

Samuel Guillon<fnref n="1" />,<fnref n="2" />, Daisuke Saya<fnref n="1" />,<fnref n="2" />, Laurent Mazenq<fnref n="1" />,<fnref n="2" />, Sorin Perisanu<fnref n="3" />, Pascal Vincent<fnref n="3" />, Arnaud Lazarus<fnref n="4" />, Olivier Thomas<fnref n="4" /> and Liviu Nicu<fnref n="1" />,<fnref n="2" />

E-mail: sguillon@laas.fr

## Abstract

In this paper, we investigate the effects of non-ideal clamping shapes on the dynamic behavior of silicon nanocantilevers. We fabricated silicon nanocantilevers using silicon on insulator (SOI) wafers by employing stepper ultraviolet (UV) lithography, which permits a resolution of under 100 nm. The nanocantilevers were driven by electrostatic force inside a scanning electron microscope (SEM). Both lateral and out-of-plane resonance frequencies were visually detected with the SEM. Next, we discuss overhanging of the cantilever support and curvature at the clamping point in the silicon nanocantilevers, which generally arises in the fabrication process. We found that the fundamental out-of-plane frequency of a realistically clamped cantilever is always lower than that for a perfectly clamped cantilever, and depends on the cantilever width and the geometry of the clamping point structure. Using simulation with the finite-elements method, we demonstrate that this discrepancy is attributed to the particular geometry of the clamping point (non-zero joining curvatures and a flexible overhanging) that is obtained in the fabrication process. The influence of the material orthotropy is also investigated and is shown to be negligible.

## 1. Introduction

The transition from the micro- to the nanoscale, or particularly from microelectromechanical systems (MEMS) to nanoelectromechanical systems (NEMS) arouses great interest in both science and technology [1]. The downscaling progress allows many benefits such as reduced size devices for large scale integration, higher performances, as in the case of mass detection, and decreased energy consumption.

Two main methods exist for the manufacture of nanosystems: the bottom-up approach, involving self-assembly building of nanosystems, and the top-down method,

that uses micro- and nanofabrication processes to shape in the bulk material [1–11, 16–20]. The bottom-up method provides a relatively easy implementation and the ability to manufacture a large number of systems simultaneously; however, it is difficult and more or less random to manipulate the manufactured devices. With the top-down approach, systems are directly realized at the desired location.

Usually, the top-down approach uses optical lithography or electron beam lithography for structuring patterns on a substrate [1–11]. Conventional optical lithography, generally limited to micrometer resolution, does not allow high resolution for NEMS. In this context, we developed an optical

<footnote n="1">¹ CNRS, LAAS, 7 Avenue du Colonel Roche, F-31077 Toulouse Cedex 4, France</footnote>

<footnote n="2">² Université de Toulouse, UPS, INSA, INP, ISAE, UT1, UTM, LAAS, F-31077 Toulouse Cedex 4, France</footnote>

<footnote n="3">³ LPMCN, Université Claude Bernard Lyon 1 et CNRS, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France</footnote>

<footnote n="4">⁴ Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers, 2 rue Conté, 75003 Paris, France</footnote>
method for manufacturing NEMS, achieving resolutions below
those of conventional lithography and even laser lithography.
We used a UV stepper photo repeater (I Line CANON FPA
3000 i4/i5 NA 0.63) equipped with a strongly converging lens
(i5) which drastically reduces the size of the insolated patterns.
This method provides submicron size patterns while using
photoresists which are less sensitive to the electron emissions
frequently occurring in NEMS manufacturing processes.

The methods of characterization of NEMS are numerous,
depending on the manufacturing process. Excitation and
detection may be thermoelastic, piezoresistive [11] or
optical [2-5]. Excitation by field effect emission inside a
scanning electron microscope (SEM) and visual detection
of resonance frequencies by the SEM is often used as a
characterization tool for bottom-up devices [12-15]. Davis
and coworkers have already developed the excitation of top-
down poly Si nanocantilevers by DC and AC electrostatic force
and visual detection by optical microscope and SEM [16-18].
Here, the nanocantilever is driven by the electrostatic force
approaching a tungsten nanotip by an XYZ positioner inside
the SEM. The resonance frequencies are visually detected by
the SEM.

In this paper, we describe the fabrication process of
silicon nanocantilevers and their mechanical characterization
by electrostatic force inside an SEM. The fundamental out-
of-plane and in-plane resonance frequencies are measured
and compared to the analytical theory of a perfectly clamped
cantilever beam, showing discrepancies. Then, some finite-
element numerical simulations are proposed to take into
account the particular clamping point geometry and the
material orthotropy.

## 2. Fabrication of the silicon nanocantilevers

Silicon on insulator (SOI) wafers were used as the starting
material. The thickness of the top silicon layer was 340 nm
with a 1 µm-thick buried SiO₂ layer on a 525 µm-thick silicon
substrate. The buried SiO₂ layer (which is etched for releasing
nanostructures) served as sacrificial layer.

As the lithography tool, a stepper photo repeater was
employed to define the nanocantilevers' shapes. By regulating
the photo repeater exposure power, a minimum width of
100 nm was achieved. As with any optical lithography
equipment, the photo repeater allows the use of various types
of photoresist for an adequate etching process. Moreover,
the essential of the photo repeater is the ability to obtain
precise alignment (better than 50 nm). After the photoresist
development, the top silicon layer was vertically etched by
reactive ion etching (RIE) until an intermediate SiO₂ layer
appeared. Then the intermediate SiO₂ was etched by dipping
the entire wafer into a buffered hydrofluoric acid (BHF)
solution for liberation of the structures. In order to avoid
irreversible collapse and subsequent substrate sticking of
the nanostructures, the water used to rinse the wafers was
progressively changed to ethanol which was evaporated by
heating. The fabricated cantilevers with lengths varying from 1
to 14 µm, widths varying from 100 nm to 1 µm and thickness
of 340 nm are shown in figure 1.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** The silicon nanocantilevers of various sizes after liberation. The structures were defined by a stepper photo repeater (I Line CANON FPA 3000 i4/i5 NA 0.63).</figcaption></figure>

## 3. Measurement of resonance frequency of the Si nanocantilevers inside an SEM

Excitation of cantilevers was performed individually via
coupling between an AC voltage of a few volts brought near
the apex by an etched tungsten tip and an induced electric
charge on the cantilever. We induced a DC charge on the
cantilever by adding a DC voltage of a few tens of volts to
the excitation. The electric excitation force had thus a strong
component at the excitation frequency. Each nanocantilever
started vibrating when the frequency of the AC voltage
matched its eigenfrequency. The resonances were detected
by visual inspection on the SEM, by selecting the excitation
frequency so that the motion amplitude was maximum. An
XYZ nanromanipulator was used to bring the excitation tungsten
tip to few tens of nanometers from the sample [12, 13]
(figure 2). By this method of excitation and detection, both
out-of-plane and in-plane resonances could be detected.

The main experimental challenge was to deal with the
poor quality SEM image due to the applied AC voltage
(see figure 3). Significant DC voltages had to be used in
order to enhance the vibration amplitude for detection of the
resonance frequencies. We obtained vibration amplitudes of
the cantilever's end in the micron range, as shown in the out-
of-plane and in-plane polarizations of the resonance in figure 3.

The eigenfrequencies of the nanocantilever series shown
in figure 2 were measured and the results are given in table 1.
The effective spring constant for the in-plane polarization
increases with the width of the nanocantilever and thus makes
visual detection in the SEM more difficult because of the
lower amplitudes. That is the reason why values of in-plane
frequencies with larger width are not included in table 1.
<table><caption>**Table 1.** Calculation and measurement results of the eigenfrequencies of the nanocantilevers with variation of width. Dimensions were measured by SEM observation.</caption><thead><tr><th colspan="2">Designed dimensions</th><th colspan="3">Measured dimensions</th><th colspan="2">In-plane eigenfrequency</th><th colspan="2">Out-of-plane eigenfrequency</th></tr><tr><th>Length (μm)</th><th>Width (nm)</th><th>Length (μm)</th><th>Width (nm)</th><th>Thickness (nm)</th><th>Calculated (MHz)</th><th>Measured (MHz)</th><th>Calculated (MHz)</th><th>Measured (MHz)</th></tr></thead><tbody><tr><td>15</td><td>60</td><td>14.5</td><td>129</td><td>340</td><td>0.857</td><td>0.876</td><td>2.21</td><td>2.30</td></tr><tr><td>15</td><td>80</td><td>14.5</td><td>183</td><td>340</td><td>1.22</td><td>1.12</td><td>2.21</td><td>2.18</td></tr><tr><td>15</td><td>100</td><td>14.5</td><td>233</td><td>340</td><td>1.55</td><td>1.43</td><td>2.21</td><td>2.18</td></tr><tr><td>15</td><td>120</td><td>14.5</td><td>272</td><td>340</td><td>1.77</td><td>1.74</td><td>2.21</td><td>2.15</td></tr><tr><td>15</td><td>140</td><td>14.6</td><td>288</td><td>340</td><td>1.85</td><td>Not detected</td><td>2.18</td><td>2.13</td></tr><tr><td>15</td><td>160</td><td>14.6</td><td>325</td><td>340</td><td>2.08</td><td>Not detected</td><td>2.18</td><td>2.11</td></tr><tr><td>15</td><td>180</td><td>14.6</td><td>335</td><td>340</td><td>2.15</td><td>Not detected</td><td>2.18</td><td>2.10</td></tr><tr><td>15</td><td>200</td><td>14.6</td><td>421</td><td>340</td><td>2.70</td><td>Not detected</td><td>2.18</td><td>2.06</td></tr><tr><td>15</td><td>300</td><td>14.7</td><td>598</td><td>340</td><td>3.78</td><td>Not detected</td><td>2.15</td><td>2.04</td></tr><tr><td>15</td><td>400</td><td>14.7</td><td>743</td><td>340</td><td>4.70</td><td>Not detected</td><td>2.15</td><td>2.01</td></tr><tr><td>10</td><td>80</td><td>9.8</td><td>196</td><td>340</td><td>2.79</td><td>Not detected</td><td>4.84</td><td>4.91</td></tr><tr><td>10</td><td>100</td><td>9.8</td><td>235</td><td>340</td><td>3.34</td><td>3.24</td><td>4.84</td><td>4.89</td></tr><tr><td>10</td><td>120</td><td>9.8</td><td>279</td><td>340</td><td>3.97</td><td>3.86</td><td>4.84</td><td>4.79</td></tr><tr><td>10</td><td>140</td><td>9.8</td><td>316</td><td>340</td><td>4.50</td><td>4.31</td><td>4.84</td><td>4.78</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** (a) Relative positioning of the tip and the sample, (b) excitation circuit.</figcaption></figure>

Higher order modes were also difficult to observe. The charged apex of the cantilever moves in a varying electric potential which can affect the spring constant, the eigenfrequency and even induce nonlinear effects, as for an atomic force microscopy tip. We varied the distance between the excitation tip and the cantilever by more than a factor of two and we saw no measurable change in the mechanical eigenfrequencies. We concluded that the change was smaller than the natural Lorentzian width of the resonance, which determines the accuracy of our measurements.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** (a) In-plane and (b) out-of-plane resonance of a nanocantilever.</figcaption></figure>

We saw nonlinear effects characterized by jumps and hysteresis for high excitation amplitudes. When we lowered the excitation amplitude, the jumps and hysteresis were no
<figure><img src="image_5.png" /><figcaption>Figure 4. The ratio of in-plane frequency to out-of-plane frequency, $f_h/f_v$, and the ratio of width to thickness, $b/e$, as a function of the width $b$.</figcaption></figure>

longer observable. This means that the nonlinearities change the eigenfrequency by less than the Lorentzian width given by the Q-factor ($\Delta f/f = 1/Q \approx 0.05\%$). Unfortunately, the quality of our SEM images was strongly affected by the electric potentials we used for the measurements and thus our curve of the response amplitude versus excitation frequency is not very accurate. For a cantilever beam, made of homogeneous material of Young's modulus $E$ and density $\rho$ and with a rectangular cross-section, the out-of-plane $f_v$ (i.e. perpendicular to the wafer plane) and in-plane $f_h$ (i.e. parallel to the wafer plane) eigenfrequencies are given by

$$f_v = \frac{\beta^2 e}{4\pi L^2} \sqrt{\frac{E}{3\rho}}, \quad f_h = \frac{\beta^2 b}{4\pi L^2} \sqrt{\frac{E}{3\rho}} \quad (1)$$

where $L$, $e$ and $b$ denote respectively the length, the thickness and the width of the beam and $\beta = 1.875$ for the fundamental mode.

Relations (1) indicate that the in-plane resonance frequencies linearly increase with the width while not depending on the thickness; meanwhile, the reverse behavior is noted for the out-of-plane resonance frequencies.

The ratio of in-plane frequency to out-of-plane frequency, $f_h/f_v$, should always be equal to the width to thickness ratio, $b/e$, independently of any homogeneous variation of the material parameters. It is, for example, independent of the Young's modulus, and thus of the Si crystalline direction and also independent of the density. We can see that experimental data are in excellent agreement with this prediction in figure 4.

## 4. Discussion of the experimental data

The measured resonance frequencies are generally well consistent with calculation within a 10% error. Imperfection in measuring the dimensions in the SEM can explain the slight difference from the calculation.

It should be noted that, in the out-of-plane mode, the measured resonance frequencies decrease with the cantilever

<figure><img src="image_6.png" /><figcaption>Figure 5. Top view SEM image of the nanocantilever. The SiO$_2$ layer is seen under the top Si layer due to the penetration of the electron beam. Curvature at the clamping point and overhanging at the cantilever clamping are also seen.</figcaption></figure>

width value at supposedly constant length, contrary to the theoretical prediction (equation (1)). For instance, the out-of-plane resonant frequency of the 129 nm wide cantilever is 2.30 MHz while in the case of the 743 nm wide cantilever the out-of-plane resonant frequency value is 2.01 MHz. Due to the limitations in the precision of the photolithography technique, the measured length from one cantilever to another varies slightly. However this difference is limited to a few hundreds of nanometers which, according to the out-of-plane resonant frequency calculation using equation (1), cannot explain the decrease of the measured resonance frequency as can be seen in table 1. Hence, this must certainly be explained by non-ideal shapes of the fabricated nanocantilever structures.

One of these imperfections is the overhanging of the cantilever clamping point (see figure 5). The clamping zones of the cantilevers are subjected to undercut in releasing the cantilevers' structure by BHF etching of the SiO$_2$ sacrificial layer. As a whole, in top-down fabrication of cantilevers and bridges with nanometer scales by surface micromachining, release of the structures is performed by the sacrificial layer etching underneath with generation of the overhanging of the cantilever clamping [1–11, 16–19]. This overhanging can affect the mechanical characteristics [19]. The out-of-plane resonance frequency may be shifted towards lower values because of the increase in the effective length of the cantilever, which decreases the spring constant. Meanwhile, it is probable that the overhanging lowers the quality factor value due to support loss through the flexible fixation of the cantilevers [20].

Another parameter to consider is the curved shape at the cantilever clamping point (see figure 5). Due to the limitation in precision of the lithography technique, the clamping point of the cantilevers has a curved shape with a radius of about 0.5 µm. Usually, in the case of such nanostructures defined by optical lithography, curvature on a corner of a structure is difficult to avoid. As the structures are miniaturized to nanometer scale, the clamping point curvatures become more dominant.
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Geometry of the nanocantilever used in the finite-element simulations.</figcaption></figure>

Recently, Ilic and coworkers mentioned that due to overhanging of the cantilever support, the resonance frequencies are lower than predicted by ideal beam theory [2, 3]. They found that the in-plane vibration is less sensitive to the overhanging. The same authors also demonstrated by experiment and simulation that, with the overhanging of the cantilever support, both the out-of-plane and in-plane resonance frequencies decrease as a function of the width [3].

**5. Numerical simulations**

In order to investigate how these non-ideal shapes mentioned above affect the resonance frequencies, simulations with the finite-element method were conducted for nanocantilevers with curvature at the clamping point and overhanging of the cantilever support. The finite-element code Cast3M, developed at CEA Saclay (France) [21], was used. The model geometry, shown in figure 6, is meshed with 21 651 ten-node tetrahedral elements. The mesh can be seen in figures 8 and 11. The numerical values of the geometrical parameters used in the simulations were L = 14.3 µm and e = 340 nm for the beam length and thickness, and an overhanging width B = 42 µm, large enough so that it had no noticeable influence on our numerical results. In addition to the non-ideal clamping shapes, it has been noticed that the cantilevers present a slight variation of the width between the apex and the clamping point: taking a closer look at figure 5, it can be seen that the width at the apex is slightly larger than at the end of the curved shape. To take this into account in the simulations, the beam's width was chosen as an affine function of the beam's axial coordinate in a region of length L/2 from the end of the curved shape (b' denotes the width at this point) and then constant and equal to b for the remaining part of the beam. In the simulations, the beam's widths b and b', the curvature radius at clamping point r and the overhanging depth H were varied.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Simulation results of natural frequency evolution as a function of width, for various curvature radii at the clamping point (r = 0, 0.5, 1 µm) and a zero overhanging depth (H = 0). A uniform width is considered (b = b'). (a) In-plane mode, (b) out-of-plane mode.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Mode shapes in the case of no overhanging depth (H = 0). The finite-element mesh is also shown. (a) In-plane mode shape, (b) out-of-plane mode shape.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Simulation results of natural frequency evolution as a function of width, for various overhanging depths ($H = 0$ and $1.3~\mu\text{m}$) and no curvature radius at the clamping point ($r = 0$). A uniform width is considered ($b = b'$). (a) In-plane mode, (b) out-of-plane mode.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Simulation results of natural frequency evolution as a function of width, for various curvature radii at the clamping point ($r = 0, 0.5, 1~\mu\text{m}$) and an overhanging depth $H = 1.3~\mu\text{m}$. A uniform width is considered ($b = b'$). (a) In-plane mode, (b) out-of-plane mode.</figcaption></figure>

The silicon used here for manufacturing the nanocantilevers was an anisotropic crystalline material [22]. As a consequence, one can expect that the mechanical properties of the nanocantilevers depend on the relative orientation of their neutral axis with respect to the crystal lattice. To evaluate the influence of the material as well as to identify the right choice for the material constants and model, we performed several simulations. The cantilever's neutral axis was oriented in the [110] direction of the silicon crystal lattice. In the first simulation, an orthotropic material was used. The material constants were those of a standard (100) silicon wafer with frame of reference [110], $\bar{[110]}$, [001], obtained in [22]: $E_x = E_y = 169$ GPa, $E_z = 130$ GPa, $\nu_{yz} = 0.36$, $\nu_{zx} = 0.28$, $\nu_{xy} = 0.064$, $G_{yz} = G_{zx} = 79.6$ GPa, $G_{xy} = 50.9$ GPa. In

a second simulation, an isotropic material of Young's modulus $E = 169$ GPa (the one in the [110] direction), Poisson's ratio $\nu = 0.3$ and density $\rho = 2330$ kg m$^{-1}$ was used. This was motivated by the fact that because the cantilevers are slender, the main deformation mechanism that governs their transverse oscillations is the compression/extension of the fibers in the [110] neutral axis direction, so that only the Young's modulus in this direction has an effect. Both simulations give the same results for the in-plane and out-of-plane natural frequencies with a deviation of less than 1.3%.

This result shows that an isotropic model with the right value of the Young's modulus is sufficient for simulations of transverse vibrations of silicon beam-like structures. However,
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Mode shapes in the case of a non-zero overhanging depth. The finite-element mesh is also shown. (a) In-plane mode shape, (b) out-of-plane mode shape.</figcaption></figure>

the following numerical results have been obtained with the orthotropic finite-element model, extremely easy to implement with the Cast3M software.

### 5.1. Effect of curvature at the clamping point of the cantilevers

Figure 7 shows the simulation results for the natural frequency evolution with respect to width and various values of curvature radius at the clamping point ($r = 0, 0.5, 1$ µm), with no overhanging depth ($H = 0$) and no width variation ($b' = b$). Figure 8 shows the corresponding mode shapes as well as the mesh of the finite-element model. Clamping surface 1 of figure 6 was then used as a boundary condition in the finite-element model. As the curvature radius at the clamping point becomes larger, the natural frequencies are increased both for the in-plane and out-of-plane motion. The main reason is the effective length, which decreases because of the curvature. In the out-of-plane mode, without curvature at the clamping point, the natural frequency is independent of the width as predicted by equation (1). Figure 7(b) shows that, in the case where the clamping point has curvature, the natural frequency of the out-of-plane mode can decrease with increasing width. Nevertheless, the widest cantilevers are less sensitive to the curvature at the clamping point in the sense that the out-of-plane natural frequency remains almost unchanged.

### 5.2. Effect of overhanging of the cantilever support

Simulation results for the natural frequency evolution as a function of the beam's width, for nanocantilevers without curvature at the clamping point ($r = 0$ µm) and no width variation ($b' = b$), are shown in figure 9 to evaluate the effect of a non-zero overhanging depth $H$ ($H = 0$ and $1.3$ µm are presented). Clamping surface 2 of figure 6 was then used as a boundary condition in the finite-element model. It shows that the natural frequencies decrease because of the overhanging both in the in-plane mode and in the out-of-plane mode. This trend is amplified by the increase of width [2, 3].

Figure 10 shows the simulation results combining both the effects of the overhanging depth ($H = 1.3$ µm) and various values of the curvature radius at the clamping point ($r = 0, 0.5, 1$ µm) on the evolution of the natural frequency as a function of the cantilever's width. The width is considered uniform ($b' = b$). It is found that, with increasing width, depending on the curvature radius, the natural frequencies are dramatically lower than those of figure 7, where the overhanging is not considered.

Finally, figure 12 shows simulation results for the natural frequency variation as a function of the overhanging depth for different cantilever widths, with a non-zero curvature radius at the clamping point (fixed at $r = 0.5$ µm) and a uniform width ($b = b'$). As the length of the overhanging increases, both the in-plane and the out-of-plane natural frequency decrease. The decrease in natural frequency with the overhanging length is more dominant in wider cantilevers.

### 5.3. Comparison with experiments

In figure 13, a comparison between simulation and experimental data for both the out-of-plane and in-plane modes' natural frequencies as a function of width is shown. The curvature radius at the clamping point was chosen as $r = 0.5$ µm and the overhanging depth of the cantilever's support was $H = 1.3$ µm; these values correspond to the reality of the fabricated nanostructures. To improve the match between theory and experiments, various slight variations of width between the clamping point and the apex were considered: $(b - b')/b = 0\%, 4\%$ and $8\%$. Those values roughly correspond to what can be estimated with the SEM images of figure 5. In the case of the in-plane mode (figure 13(a)), the resonance frequencies of the experimental results correspond rather well to the simulation results. In the case of the out-of-plane mode (figure 13(b)), the decrease rate in the natural frequencies as a function of width is higher in the measured results and the measured frequency values are smaller than the theoretical ones. Considering the imperfect clamping point (non-zero radius of curvature and overhanging depth) leads to a good trend of the natural frequencies that decrease as a function of width, but an error of about 10% on average remains. Then, adding to those imperfections a non-uniform width improves the simulations by reducing the error to about 5% on average.
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Simulation results of natural frequency evolution as a function of width, for various overhanging depths ($H = 0, 1, 2, 3, 4, 5$ µm) and a fixed curvature radius at the clamping point, $r = 0.5$ µm. A uniform width is considered ($b = b'$). (a) In-plane mode, (b) out-of-plane mode.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Comparison between experimental and simulation results for the natural frequency evolution as a function of width, with curvature radius at clamping point $r = 0.5$ µm and with overhanging depth of the cantilever support $H = 1.3$ µm. Various width variations along the beam's length have been considered ($\frac{(b-b')}{b} = 0\%, 4\%$ and $8\%$). (a) In-plane mode, (b) out-of-plane mode.</figcaption></figure>

**6. Conclusion**

We fabricated silicon nanocantilevers with a smallest width of 100 nm by stepper UV lithography. To measure their vibratory characteristics, the nanocantilevers were excited by electrostatic force by approaching a tungsten tip inside an SEM. Their fundamental resonance frequencies, in directions both parallel to the wafer plane (in-plane mode) and orthogonal to the wafer plane (out-of-plane mode) were detected visually with the SEM and compared to theoretical and numerical results. Slight decreases in resonance frequencies as a function of nanocantilever width, for the out-of-plane mode,

were observed. By considering various finite-element models, it is shown that this is mainly due to various geometrical imperfections of the cantilevers: a curved shape, an overhanging at the clamping point as well as non-uniform width along the axial coordinate.
## Acknowledgments

The French National Agency for Research (program ANR/PNANO 2008, project NEMSPIEZO 'ANR-08-NANO-015') is gratefully acknowledged for financial support.

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