**1. Introduction**

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154 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

Nowadays, mechanical designers and engineers of elastic structures at ultra-small scales face an interesting challenge; traditional flexures with several components, mechanical joints/ welds and linkages are almost impossible to manufacture with existing microsystem fabrica‐ tion technologies. Therefore, the majority of mechanical components, elements and building blocks are based on micro/nano machined elastic flexures [1]. Despite the complexity of modeling and analyzing these systems, their design uses the averaged relation between stress and strain, which requires a relatively accurate knowledge of their effective elastic properties, specifically the effective elastic modulus. Experimental results show that the elastic properties are constant at length scales of meters down to micrometers [2]. However, in order to increase the performance, i.e. sensitivity and dynamic range, the dimensions of mechanical devices have been scaled down towards a few nanometers. Consequently, high performances such as single-electron tunneling [3], sub-attonewton force sensing [4], and sub-femtometer displace‐ ment sensing [5] have been successfully achieved. Unlike at micron and higher length scales, at sub-micron and nanometer length scales the effective elastic behavior shows strong scaledependent behavior, meaning that the elastic properties are no longer constant, but a function of length scale. For all the application examples above, the high performance was achieved due to the mechanical response of a nanosystem, which strongly depends on the effective elastic properties. Therefore, a clear understanding of the scale-dependent behavior is important for the design and performance of nanosystems.

### **1.1. Scale-dependence**

The existence of scale-dependent behavior has been confirmed by experimental measure‐ ments, including resonance frequency tests [6], tensile testing in scanning electron microscope

© 2013 Sadeghian et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sadeghian et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(SEM) [7], transmission electron microscope (TEM) [8, 9], atomic force microscope (AFM) [10, 11] and nanoindenter [12] and also theoretical investigations, including *ab initio* and density functional theory (DFT) [13, 14, 15], molecular dynamics (MD) [16, 17, 18, 19] and modifica‐ tions tocontinuumtheory[20,21,22,23].Althoughscale-dependencehasbeenobservedbyboth theory and experiment, a considerable discrepancy still remains between the experiments and models. As an example, in mono-crystalline [110] silicon nanowires and nanocantilevers the experimental scale-dependence has been observed at about 150 nm [24, 11, 8, 6, 25], whereas theoreticalstudiesdonotagreewithanyscale-dependencelargerthan10nm.Inordertohighlight the inescapablediscrepancybetweenexperiment andtheory,the resultsofrecent experimental‐ lymeasuredeffectiveYoung'smodulus*Ê*andthose extractedfromtheories for siliconnanocan‐ tilevers and nanowires were collected from relevant literature and are illustrated in Fig. 1. The figure demonstrates that a large gap exists between the theoretical predictions and experimen‐ tal observations. The smallest silicon nanowire with a diameter of about 12 nm that has been experimentally tested by Han *et al.* [26] shows an effective elastic modulus which is about one third of the bulk value, while the largest theoretically modeled (mainly atomistic simulations) silicon nanowire (diameter ~ 10 nm) shows an effective elastic modulus that is closer to the bulk value [26]. It can also be observed from Fig. 1, that there is lack of data, both in experiment and theory; 1) there is no computational data for structures in the range from 10 to 150 nm, so no direct comparison between experiments and theory at about 100 nm can be performed. The reason is that it is computationally extremely expensive and impossible to model the atomistic systemsas largeas theones experimentallytested.Althoughquasi-continuumapproacheshave been developed as computationally efficient methods [27, 28], they are lacking important phenomena, such as surface reconstruction, defects emitted from surfaces and surface stress induced phase transformations. Thus, one of the biggest challenges in the field of nanomechan‐ ics is the development of a multi-scale modeling framework, capable of performing simula‐ tionsatvariouslengthscales.2)Thereisnoexperimentallymeasureddatabelow12nm,therefore for structures smaller than 10 nm a direct comparison between theory and experiment cannot be performed either. The main reason is that it is extremely challenging and difficult to per‐ form nanoscale experiments with sufficient accuracy and resolution, while introducing minimum disturbances to the measured nanostructure [24]. Therefore, experimental capabili‐ ties need to be improved, enabling measurements of structures with a size of less than 10 nm.

Moreover, the observed discrepancies are not only between the simulations and experiments but also between various experimental measurements. The observed discrepancies [25, 29, 24] could be explained by: some exterior factors such as ill-defined boundary conditions and uncertainties in experiments [30], loading modes (extensional vs. bending) [31, 23], calibration and required input accuracy (i.e. variations in geometry) limitations [11] as well as intrinsic nature of inaccuracy in the experimental methods (i.e. mass effects in resonance based tests) [32, 33, 34] and the environmental effects such as surface contaminations, measurement induced-errors and native oxide layers could all be influencing the measurement at sub-micron scales, inducing a large disagreement between measurement data. On the other hand, there are intrinsic effects which become significant at sub-micron scales such as surface stress, surface elasticity, nonlinear bulk effects, surface reconstructions, crystal defects and fabrication induced imperfections that make the scale-dependent phenomena more complicated.

Due to their small sizes, nanosystems predominantly have high surface-to-volume-ratio, thus, unlike micron size systems, the influence of surface stress cannot be neglected. Therefore, the scale-dependence of elastic behavior is generally attributed to surface effects, including surface elasticity [35, 20, 23] and surface stress [36, 37], surface oxidation [31, 29, 38] and surface contamination [39]. Besides the intrinsic surface effects, it was speculated that surface geom‐ etry and cross section are important factors. Most simulations have been performed on nanowires with rectangular cross section, which is significantly simplified compared to actual cross sections that are closer to rhombic or pentagonal configurations [40]. In order to examine to what extent these effects are significant, McDowell *et al.* [31] using atomistic simulations, studied the scale-dependent elastic behavior of nanowires with various cross sections and with various surface steps. The conclusion was that, although these factors influence *Ê* to some extent, they alone are insufficient to explain the experimentally observed trends.

In addition to surface geometry, the loading method can also influence *Ê*. The extracted *Ê* in the extensional mode is different from the one extracted in the bending mode [16]. Conse‐ quently, there are two types of experimental characterizations: those measuring *Ê* in the extensional mode, such as uniaxial tensile loading tests [41, 42, 9], and those in the bending mode such as resonance frequency based [6] tests or bending tests inside an atomic force microscopes [10, 11]. Most of the research today focuses on the bending mode of *Ê* because it is more sensitive to surface stress and surface elasticity effects and because of its importance in sensing and actuating applications. Again the question arises: at which scale does the difference between bending and extensional modes become significant? To answer this question, McDowell *et al.* [16] investigated the effect theoretically and found out that it is influential only for dimensions below approximately 8 nm. This indicates that other factors beyond the above effects play a role in experimentally observed nanosystem scale-dependent elastic behavior. Therefore, the question remains: why does *Ê* start to decrease already at sizes between 10 nm and 150 nm?

A couple of issues have been raised by researchers in order to reduce the gap between the experiments and the theory [24, 11, 39, 16, 25].

A recently recognized substantial effect, which has not yet been considered, is accounting for the fact that experimentally measured nanosystems are not perfect (such as existence of surface defects, native oxide layers, contaminations etc.) and not defect free (such as single crystal defects, fabrication induced imperfections etc.) in contrast to the perfect mono-crystalline structures studied using atomistic simulations [19, 17, 14, 13]. The goal of this chapter is to try to determine to what extent these effects influence the effective elastic modulus of nanosys‐ tems, to try to reduce the existing gap between the experiments and theory of scale-depend‐ ence, and to determine the origin of the scale-dependence in silicon nanosystems.

### **1.2. Outline of the chapter**

(SEM) [7], transmission electron microscope (TEM) [8, 9], atomic force microscope (AFM) [10, 11] and nanoindenter [12] and also theoretical investigations, including *ab initio* and density functional theory (DFT) [13, 14, 15], molecular dynamics (MD) [16, 17, 18, 19] and modifica‐ tions tocontinuumtheory[20,21,22,23].Althoughscale-dependencehasbeenobservedbyboth theory and experiment, a considerable discrepancy still remains between the experiments and models. As an example, in mono-crystalline [110] silicon nanowires and nanocantilevers the experimental scale-dependence has been observed at about 150 nm [24, 11, 8, 6, 25], whereas theoreticalstudiesdonotagreewithanyscale-dependencelargerthan10nm.Inordertohighlight the inescapablediscrepancybetweenexperiment andtheory,the resultsofrecent experimental‐ lymeasuredeffectiveYoung'smodulus*Ê*andthose extractedfromtheories for siliconnanocan‐ tilevers and nanowires were collected from relevant literature and are illustrated in Fig. 1. The figure demonstrates that a large gap exists between the theoretical predictions and experimen‐ tal observations. The smallest silicon nanowire with a diameter of about 12 nm that has been experimentally tested by Han *et al.* [26] shows an effective elastic modulus which is about one third of the bulk value, while the largest theoretically modeled (mainly atomistic simulations) silicon nanowire (diameter ~ 10 nm) shows an effective elastic modulus that is closer to the bulk value [26]. It can also be observed from Fig. 1, that there is lack of data, both in experiment and theory; 1) there is no computational data for structures in the range from 10 to 150 nm, so no direct comparison between experiments and theory at about 100 nm can be performed. The reason is that it is computationally extremely expensive and impossible to model the atomistic systemsas largeas theones experimentallytested.Althoughquasi-continuumapproacheshave been developed as computationally efficient methods [27, 28], they are lacking important phenomena, such as surface reconstruction, defects emitted from surfaces and surface stress induced phase transformations. Thus, one of the biggest challenges in the field of nanomechan‐ ics is the development of a multi-scale modeling framework, capable of performing simula‐ tionsatvariouslengthscales.2)Thereisnoexperimentallymeasureddatabelow12nm,therefore for structures smaller than 10 nm a direct comparison between theory and experiment cannot be performed either. The main reason is that it is extremely challenging and difficult to per‐ form nanoscale experiments with sufficient accuracy and resolution, while introducing minimum disturbances to the measured nanostructure [24]. Therefore, experimental capabili‐ ties need to be improved, enabling measurements of structures with a size of less than 10 nm.

156 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

Moreover, the observed discrepancies are not only between the simulations and experiments but also between various experimental measurements. The observed discrepancies [25, 29, 24] could be explained by: some exterior factors such as ill-defined boundary conditions and uncertainties in experiments [30], loading modes (extensional vs. bending) [31, 23], calibration and required input accuracy (i.e. variations in geometry) limitations [11] as well as intrinsic nature of inaccuracy in the experimental methods (i.e. mass effects in resonance based tests) [32, 33, 34] and the environmental effects such as surface contaminations, measurement induced-errors and native oxide layers could all be influencing the measurement at sub-micron scales, inducing a large disagreement between measurement data. On the other hand, there are intrinsic effects which become significant at sub-micron scales such as surface stress, surface elasticity, nonlinear bulk effects, surface reconstructions, crystal defects and fabrication

induced imperfections that make the scale-dependent phenomena more complicated.

This chapter is organized as follows: Section 2 describes the fabrication process for micro/nano cantilevers, used for experimental investigations of scale-dependent *Ê*, including the inspec‐ tion of the fabricated structures. In Section 3 the experimental determination of *Ê* is presented and discussed. Section 4 is focused on the theoretical investigations of surface effects and native

**Figure 1.** (a) Illustration of the scale-dependence of mono-crystalline silicon effective elastic modulus *Ê* obtained via computations and experimental observations [29]. The inset shows results for the scale less than 10 nm, indicating that all simulations are limited to scales of less than 10 nm due to complexity and expense of computations. (b) A logarithmic plot of (a) to further illustrate the differences between the simulation and the experimental results. Both show sharp drops near the down-scaling side; the question is why discrepancies exist? (Reprinted with permission from [29]. Copyright 2010 IOP Publishing Ltd.)

oxide layers on the scale-dependent *Ê*, including molecular dynamics calculations as well as semi-continuum approaches, to show to what extent these effects are responsible for the observed scale-dependence. Section 5 discusses the possible effects of defects that may contribute to the reduction of *Ê* at larger scales. In order to show the extent that defects play a role in scale dependence, molecular dynamics is utilized to quantify the effects of defects on *Ê*. Finally in Section 6, a summary, conclusions and recommendations for future work on scaledependence of *Ê* are presented.
