**5. Impact of fabrication-induced defects**

( ( ) ( ) )

(5)

( )

2

*t t*

+

*s Ox Ox Ox Ox*

*<sup>S</sup> Et E t tt tt*

++ ++ ç ÷

*b*

*E t*

æ ö S

è ø <sup>=</sup>

172 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

is still a considerable gap between experiments and theory.

varies between 2 to 5 nm and the thickness of the silicon cantilever.

ˆ

*E*

3 2 3 2

<sup>3</sup> 1 8 6 12

*Ox*

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

**Figure 13.** The effective Young's modulus of silicon cantilever as a function of native oxide layers thickness, which

3

There are still some remaining issues that require resolution in order to close the gap between experiments and theory [25]. Among them, accounting for the fact that experimentally fabricated nanosystems are not defect-free, has been recognized as the most important effect, which has not yet been investigated and quantified [25].

As described in Section *2*, reducing the cantilever thickness was done by alternating thermal oxidation and etching in HF solution. It was observed that the density of fabrication-induced defects increased when reducing the thickness of the silicon device layer of SOI wafers. During the thermal oxidation process, the lattices of silicon and silicon oxide mismatch, thus intro‐ ducing stress in both layers [53]. The precipitated oxygen and local strains in Si-SiO2 interface are likely sources of defects [29]. Further reduction of the thickness of the silicon layer would result in the initiation of defects and cracks, which can extend through the whole layer thickness similar to a pinhole, reaching the underlying BOX layer [53, 29]. These are called HF defects, since the existence of these defects is tested and observed by using the HF etching [38]. The HF solution that reaches the underlying BOX layer etches isotropically the SiO2 under‐ neath, resulting in a suspended membrane-like layer with a tiny defect in the middle. An optical microscope image of a HF defect at a 14 nm thick silicon device layer is shown in Fig. 15.a. The figure clearly shows a buckled membrane with a HF defect in the middle. The buckling is due to the residual stress between silicon and silicon oxide.

**Figure 15.** (a) Optical microscope image of a HF defect. (b) SEM of a HF defects. The BOX layer below the HF defects is etched by the HF solution, resulting in the formation of a membrane with a defect in the middle.

The existence of HF defects was also confirmed by SEM, which is observed as a circle (mem‐ brane) with the defect at the center (Fig. 15.b). Actually, the HF modifies the defect size and shape, and makes it visible enough to locate it. Using AFM the defect can be observed without HF modifications. The AFM image is shown in Fig. 16.a.

**Figure 16.** (a) AFM image of a defect without HF modification (shape is caused by a AFM probe). (b) Extending the HF etching of the defects for a longer time causes suspension of relatively large membranes, which finally collapse to the layer below and create a Swiss cheese-like structures.

Continuing the HF etching of the defects results in formation of membranes with a much larger diameter, which are no longer stable and collapse to the layer underneath. This has been shown in Fig. 16.b. What is ultimately remaining, is a structure with big holes, similar to a "Swiss cheese" shape.

thickness similar to a pinhole, reaching the underlying BOX layer [53, 29]. These are called HF defects, since the existence of these defects is tested and observed by using the HF etching [38]. The HF solution that reaches the underlying BOX layer etches isotropically the SiO2 under‐ neath, resulting in a suspended membrane-like layer with a tiny defect in the middle. An optical microscope image of a HF defect at a 14 nm thick silicon device layer is shown in Fig. 15.a. The figure clearly shows a buckled membrane with a HF defect in the middle. The

**Figure 15.** (a) Optical microscope image of a HF defect. (b) SEM of a HF defects. The BOX layer below the HF defects is

The existence of HF defects was also confirmed by SEM, which is observed as a circle (mem‐ brane) with the defect at the center (Fig. 15.b). Actually, the HF modifies the defect size and shape, and makes it visible enough to locate it. Using AFM the defect can be observed without

**Figure 16.** (a) AFM image of a defect without HF modification (shape is caused by a AFM probe). (b) Extending the HF etching of the defects for a longer time causes suspension of relatively large membranes, which finally collapse to the

etched by the HF solution, resulting in the formation of a membrane with a defect in the middle.

HF modifications. The AFM image is shown in Fig. 16.a.

layer below and create a Swiss cheese-like structures.

buckling is due to the residual stress between silicon and silicon oxide.

174 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

The appearance of defects on the surface of the cantilevers can be observed with white light interferometry. As an example, Fig. 17.a shows a white light interferometry image of 40 nm thick silicon cantilevers with defects. The depth profile of one of the defects on the cantilever is shown in Fig. 17.b.

**Figure 17.** (a) Reconstructed 3D measurements of 40 nm thick silicon cantilevers with defects, using a white light in‐ terferometer. (b) The profile measurement of a defect taken along the line 1 in (a), showing the depth and geometry of the defect.

It can be speculated that HF defects contribute to a decrease in *Ê* and thus, play a role in the gap between experiments and theory of scale-dependence. We utilized MD to examine the extent that the defects contribute to the scale-dependent behavior. Similarly to Section 4, silicon nanoplates were created along the [100] and [110] directions. Defects with cylindrical shapes of different sizes were created by removing the atoms inside the cylinder. Fig. 18 shows the relaxed [100] and [110] nanoplates with defects. After reaching equilibrium configurations, the nanoplates underwent a quasi-static tensile loading. The virial stresses [54] were calculated for each level of strain and *Ê* was calculated from the slope of the resulting stress-strain curve. Details of the calculations were published in [38]. The MD calculations were repeated for different nanoplates (small, medium and large size) with different dimensions of the defects. The ratio of effective elastic modulus with and without defects, *Ed/ E0*, versus the ratio of the defects' surface area and total surface area, *Ad/A0* are plotted in Fig. 19. The inset shows the results as a function of the ratio of the number of removed atoms and total number of atoms, *Nd/NT*. It is noteworthy that creating the defect did not change the average total energy per atom significantly, except for the atoms near the surface of the defects, which have higher excess energy. An example is shown in Fig. 18.e. The coloring is according to the energy. Apart from the two layers closest to the defect, the energy of the other atoms did not substantially change.

**Figure 18.** Snapshots of (a) a [100] silicon nanoplate with one defect at the centre, (b) and (c) patterned defects in a [100] silicon nanoplate and (d) [110] nanoplate with one defect at the centre. (e) a mid-plane of a [100] nanoplate with a defect in the centre, indicating the atoms with different energy due to the presence of a defect. Coloring is according to energy.

The reason that the first two layers have a different energy is because removing atoms creates free surfaces inside the defects, causing atoms inside the defect to reconstruct resulting in changes in energy. In order to determine the contribution of each atom to the reduction of *Ê*, the changes in the virial stress of each individual atom as a function of strain, which is the stiffness contribution of each atom, was calculated. The results are shown in Fig. 20, indicating substantial changes when the size of the defects is increased.

**Figure 19.** Reduction of the effective Young's modulus versus the size of a defect for different dimensions and direc‐ tions of nanoplates. The inset shows the re-plotting with respect to the number of removed atoms.

Besides MD, an analytical solution based on continuum theory was used to approximate the effects of defects on *Ê*. It is based on the approximation described in [55] and is as follows:

**Figure 18.** Snapshots of (a) a [100] silicon nanoplate with one defect at the centre, (b) and (c) patterned defects in a [100] silicon nanoplate and (d) [110] nanoplate with one defect at the centre. (e) a mid-plane of a [100] nanoplate with a defect in the centre, indicating the atoms with different energy due to the presence of a defect. Coloring is

The reason that the first two layers have a different energy is because removing atoms creates free surfaces inside the defects, causing atoms inside the defect to reconstruct resulting in changes in energy. In order to determine the contribution of each atom to the reduction of *Ê*, the changes in the virial stress of each individual atom as a function of strain, which is the stiffness contribution of each atom, was calculated. The results are shown in Fig. 20, indicating

substantial changes when the size of the defects is increased.

176 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

according to energy.

$$E\_d = \frac{E\_0 \left(1 - \frac{A\_d}{A\_0}\right)}{\left(1 + 2\frac{A\_d}{A\_0}\right)}\tag{6}$$

The result is plotted in Fig. 19 as a dashed line. A remarkably good agreement between MD and analytical solutions can be seen. The results show that *Ê* decreases monotonically with increasing defect density (defect volume fraction). The decrease is significant and therefore, it has to be considered as an additional factor contributing to the scale-dependence.

**Figure 20.** Snapshots of middle layers of silicon (a) [110] nanoplate without defect, (b) [110] nanoplate with a defect in the centre, (c) and (d) [100] nanoplates with patterned defects. Each individual atom is coloured by its stiffness con‐ tribution. Blue colours denote atoms of higher stiffness.

The experimental observations, which were described earlier in this section, indicate that the defect density (*Ad/A0*) is between 0.005% to 0.12%, depending on the thickness of the silicon device layer [38]. Taking into account the range of experimentally observed defect density and the results presented in Fig. 19, one can include the effect of defects in the scale-dependent *Ê*. The results including the effects of surface elasticity, native oxide layers and fabricationinduced defects were calculated for the thicknesses that were experimentally measured. The results are presented in Fig. 21. Due to variations in the defects' density, an upper and lower limit of the influence of defects is introduced in the figure. The shaded area in the figure demonstrates the extent that defects influence *Ê*. Further quantification of the defects' density versus the thickness of nanocantilevers has to be done. It can be seen, that taking the defects into account, could explain the observed gap between experimental measurements and theoretical calculations.

**Figure 21.** Scale-dependent *Ê* in bending mode, taking into account the surface elasticity, native oxide layers and the fabrication-induced defects. In order to take into account the variations in the defect density, an upper and a lower limit are introduced. Thus, the scale-dependence observed in silicon nanostructures can be explained by a contribu‐ tion of surface effects, native oxide layers and fabrication-induced defects.
