**4. Theoretical investigations: Impact of surface effects and native oxide layers**

**Figure 10.** Measurement results of pull-in voltage versus the length for an array of cantilevers with different thickness‐

Above 150 nm the effective modulus is constant and converges to the bulk value of [110] silicon elastic modulus. Somewhere near 150 nm it starts to decrease monotonically. It can be seen that at 40 nm, the *Ê* is about half of the bulk value. So the question to be answered is: what

es and lengths.

causes this scale-dependent behavior?

168 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

The influence of surface effects at scales higher than micrometers on the overall elastic behavior is negligible and the elastic behavior can be explained by the bulk properties. However, in nano systems, due to their small size, the surface-to-volume ratio is quite large and therefore, the surface effects cannot be neglected. This has motivated major research to try to explain the scale-dependence by the surface effects [41, 47, 20, 17]. The surface effects can be categorized into two types; the extrinsic effects such as surface contamination, native oxide layers and environmental adsorptions, and the intrinsic effects which originate from the difference in the atomic configurations near or at the surface and in the core of the bulk. The latter are known as surface stress and surface elasticity effects. The energy associated with the atoms near the surface is different from the energy of atoms in the core. The surface atoms have lower coordination numbers and electron densities and therefore tend to adopt equilibrium lattice spacing differently from the bulk ones [48, 23]. On the other hand, the epitaxial relationship from bulk to surface has to be maintained, therefore, bulk atoms strain the atoms near the surface and create the so-called surface stress [13, 23, 36, 49]. Consequently, the surface atoms like to reconstruct and can deviate easily from their original ideal situations, and therefore, have different elasticity compared to the bulk, known as surface elasticity [41, 47, 20, 23]. In order to predict the surface effects on *Ê*, we have developed a semi-continuum, two-dimen‐ sional plane-stress framework [23], which takes into account the influence of surface elasticity on the elastic behavior of nanocantilevers. One of the main modifications to the previous models [47, 20] is the inclusion of coupling stiffness that arises from a surface stress and surface elasticity imbalance caused by either surface reconstruction [50] or molecular adsorption [51] at the top and bottom surfaces [22]. According to this framework, the bending mode of *Ê* can be calculated as

$$
\hat{E} = E\_b \left( \mathbf{1} + \frac{\mathbf{3} \Sigma S}{E\_b t} \right) \tag{3}
$$

where *Eb* is the bulk value of Young's modulus, known to be 169 GPa for [110] mono-crystalline silicon, *∑S* is the sum of the surface elasticity at the top and the bottom surfaces and *t* is the thickness of the cantilever. In order to calculate *Ê* the contribution of the surface elasticity (*∑S)* is required. Because the surface region is only a few atomic layers thick, atomistic calculations are necessarily involved in order to calculate the contribution of the surface elasticity, and in general to include the surface effects in the modeling of nanosystems. We carried out molecular dynamics calculations (MDC) using a recently developed modified embedded atom method (MEAM) potential [52]. Silicon nanobeams and plates were created with initial atomic positions corresponding to the bulk diamond-cubic crystal. The simulation cells were then fully relaxed to a minimum energy state at room temperature and zero pressure. After relaxation, surfaces, such as the one shown in Fig. 12, display a (2×1)–type reconstruction. The details of MDC can be found in [49]. After reaching the equilibrium configurations, the nanoplates were subjected to a quasi-static loading and were again fully relaxed after each strain increment. *Ê* in extensional mode was defined as

$$
\hat{E} = \left(\bigvee\_{V\_0} \bigvee\_{d\mathcal{E}}^2 \bigvee\_{\mathcal{E}}^2 \right) \tag{4}
$$

where *V0* is the volume of the nanoplate in fully relaxed zero-strain configuration (undeformed configuration), *U* is the total energy of the system and ε is the applied strain.

the surface effects cannot be neglected. This has motivated major research to try to explain the scale-dependence by the surface effects [41, 47, 20, 17]. The surface effects can be categorized into two types; the extrinsic effects such as surface contamination, native oxide layers and environmental adsorptions, and the intrinsic effects which originate from the difference in the atomic configurations near or at the surface and in the core of the bulk. The latter are known as surface stress and surface elasticity effects. The energy associated with the atoms near the surface is different from the energy of atoms in the core. The surface atoms have lower coordination numbers and electron densities and therefore tend to adopt equilibrium lattice spacing differently from the bulk ones [48, 23]. On the other hand, the epitaxial relationship from bulk to surface has to be maintained, therefore, bulk atoms strain the atoms near the surface and create the so-called surface stress [13, 23, 36, 49]. Consequently, the surface atoms like to reconstruct and can deviate easily from their original ideal situations, and therefore, have different elasticity compared to the bulk, known as surface elasticity [41, 47, 20, 23]. In order to predict the surface effects on *Ê*, we have developed a semi-continuum, two-dimen‐ sional plane-stress framework [23], which takes into account the influence of surface elasticity on the elastic behavior of nanocantilevers. One of the main modifications to the previous models [47, 20] is the inclusion of coupling stiffness that arises from a surface stress and surface elasticity imbalance caused by either surface reconstruction [50] or molecular adsorption [51] at the top and bottom surfaces [22]. According to this framework, the bending mode of *Ê* can

170 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

<sup>3</sup> <sup>ˆ</sup> <sup>1</sup> *<sup>b</sup>*

strain increment. *Ê* in extensional mode was defined as

*<sup>S</sup> E E*

*b*

(3)

*E t* æ ö S = + ç ÷ è ø

where *Eb* is the bulk value of Young's modulus, known to be 169 GPa for [110] mono-crystalline silicon, *∑S* is the sum of the surface elasticity at the top and the bottom surfaces and *t* is the thickness of the cantilever. In order to calculate *Ê* the contribution of the surface elasticity (*∑S)* is required. Because the surface region is only a few atomic layers thick, atomistic calculations are necessarily involved in order to calculate the contribution of the surface elasticity, and in general to include the surface effects in the modeling of nanosystems. We carried out molecular dynamics calculations (MDC) using a recently developed modified embedded atom method (MEAM) potential [52]. Silicon nanobeams and plates were created with initial atomic positions corresponding to the bulk diamond-cubic crystal. The simulation cells were then fully relaxed to a minimum energy state at room temperature and zero pressure. After relaxation, surfaces, such as the one shown in Fig. 12, display a (2×1)–type reconstruction. The details of MDC can be found in [49]. After reaching the equilibrium configurations, the nanoplates were subjected to a quasi-static loading and were again fully relaxed after each

> 2 <sup>2</sup> <sup>0</sup> <sup>ˆ</sup> <sup>1</sup> , *d U <sup>E</sup> <sup>V</sup> <sup>d</sup>*

æ ö <sup>=</sup> ç ÷

e

è ø (4)

be calculated as

**Figure 12.** Snapshot of a fully relaxed nanoplate at room temperature. Coloring is according to energy.

The value of *Ê* was calculated from total energy for simulations with various thicknesses. The results are shown in Fig. 14 (triangles). In order to determine the *∑S* using MD, similar to the calculation of *Ê* described above, the nanoplate is subject to a quasi-static loading and is fully relaxed after each strain increment. The *∑S* is extracted in extensional mode as the *2nd*-order derivative of surface energy with respect to the strain, and is about -1 Nm-1 [29, 49]. This value was used as an input for Equation 3. Here, it is assumed that the surface elasticity in bending and extensional modes are very close [16].Considering only this surface elasticity effect, a scaledependence can be calculated and the results are plotted in Fig. 14 as a solid line. The thickness is shown on a logarithmic scale in order to visualize clearly the difference between the experimental results and theoretical investigations. The results of the semi-continuum approach indicate that the surface effects are significant and influence the scale-dependence below 15 nm. It is therefore clear that surface effects can only partially explain the observed scale-dependence.

The native oxide, as an extrinsic surface effect, influences *Ê* via both its own elastic response and the complex interactions between the oxide and the silicon at the interface [29]. However, it has been shown that the influence of substantial elastic behavior differences between silicon and native oxide is much more significant than the interface elasticity of Si-SiO2. Moreover, during the oxidation process, for every thickness unit of silicon oxide 0.44 units of the silicon surface is consumed, resulting in reducing the thickness of silicon and, consequently, decreas‐ ing *Ê*. By taking into account the surface elasticity of the original silicon and the elastic modulus of native oxide layers *EOx*, *Ê* can be estimated as [29]

$$\hat{E} = \frac{E\_s t^3 + E\_{\rm Ox} \left( 1 + \frac{\text{ZES}}{E\_b t} \right) \left( 8 \left( t\_{\rm Ox} \right)^3 + 6t^2 t\_{\rm Ox} + 12t \left( t\_{\rm Ox} \right)^2 \right)}{\left( t + 2t\_{\rm Ox} \right)^3} \tag{5}$$

where *EOx* and *tOx* are the native oxide Young's modulus and layers' thickness at the top and bottom of the cantilever. The thickness of the native oxide layers have been reported to vary between 2 to 5 nm [11] and its Young's modulus reported between 50 to 75 GPa [11, 29]. These variations, especially the thickness (since it has a higher influence on the effective elasticity) have been used to calculate the *Ê* of silicon cantilevers with different thicknesses. The result is shown in Fig. 13. The dotted and dashed lines in Fig. 14 show a comparison between the resultant *Ê* as a function of cantilever thickness considering the various native oxide scenarios' influence (ranges of thicknesses and Young's modulus reported in the literature) and the measured values. It is clear from the figure that taking the native oxide layer scenarios into account reduces the difference and partially explains the distinctive reduction in *Ê*, yet there is still a considerable gap between experiments and theory.

**Figure 13.** The effective Young's modulus of silicon cantilever as a function of native oxide layers thickness, which varies between 2 to 5 nm and the thickness of the silicon cantilever.

**Figure 14.** Comparison between experiments and models including surface elasticity and native oxide layers. Hexa‐ gons are the measured data via EPI for different thicknesses. The triangles are the results of direct MD calculations in extensional mode. The dashed line illustrates the prediction of scale-dependence when considering surface elasticity via the semi-continuum framework. The dotted-dashed and dotted lines show the prediction when different native oxide scenarios are considered.

One of the most important issues, which has not previously been taken into account, is considering the fact that experimentally tested nanosystems are not defect-free in contrast to the perfect single crystal structures studied using atomistic simulations. In the next section, we discuss the existence of defects and their influence on the scale-dependence.
