The development of scanning probe microscopy has allowed the emergence of a powerful tool for material property characterization at the micro- and nanoscale. Atomic force microscopy (AFM) is now widely used to image the material surface as well as to study their physical properties. The originality of AFM comes from its ability to measure the physical properties of nanometer-sized objects

Figure 1: Topographic image of a nanowire crossing a pore.

 

 

 

The resonance spectrum of the cantilever in contact with the suspended nanowire is recorded (figure 2). Three peaks are observed: two of them correspond to flexural cantilever vibrations (F1, F2) and the third one is due to torsional vibrations (T1). Analysis of these resonance frequencies clearly indicates that the nanowires behave as springs. Thus, the suspended nanowires can be modelled as simple springs in these experiments.

 

 

The stiffness of the probed nanowires can then be obtained from the first flexural resonance frequency. In order to deduce the elastic modulus from the measured stiffness, it is however necessary to describe the boundary conditions of the suspended nanowires. As the nanowire adhesion on the membrane was sufficiently high, the use of the clamped-beam model was assumed to calculate the elastic modulus.

The results show that the measured elastic modulus increases strongly when the nanowire diameter decreases (figure 3). Moreover, the obtained values of the elastic modulus measured for nanowires having their diameter higher than 80nm are comparable to the elastic modulus measured on silver films. At the opposite, for the silver nanowires with the smaller diameters (i.e below 80nm), the elastic modulus increases continuously when the diameter decreases. 

Figure 3: Variation of the elastic modulus of silver nanowires as a function of their diameter.

 

This increase of the elastic modulus may be explained by taking into account surface tension effects. Indeed, at nanometer length scales, due to the increasing surface-to-volume ratio, surface effects may become predominant. Therefore, a calculation of the stiffness of the suspended nanowires due to the elastic modulus and surface tension is developed.

Assuming a force F applied at the nanowire midpoint and inducing a deflection δ, an expression for the total energy U of the bent nanowire is :

 

where δ is the nanowire elastic stiffness, ΔS is the surface increase and γ is the surface tension of the material.

In this equation, the first term is the work of the applied force, the second one represents the elastic deformation energy and the third one corresponds to the deformation energy of the surface resulting from nanowire extension.

The surface deformation energy term has a quadratic dependence on the central deflection, giving rise to an additive surface contribution to the nanowire stiffness. Therefore, the total energy of the bent nanowire can be rewritten:

 

Avec:

La raideur apparente k_app d

The apparent stiffness of the nanowire, k_app, is the sum of the elastic stiffness (kt) and of the additive surface stiffness (ksurf).

Finally, we obtain for the nanowires the following expression :

By applying this model to the experimental data, a linear regression allows the determination of the intrinsic elastic modulus, E_r, and the surface tension of the probed material. . 

 

 

In figure 4, a nonzero intercept with the ordinate axis is expected to be due to the surface tension contribution whereas the slope of the linear regression is related to the elastic modulus. The obtained elastic modulus is equal to 67.5 ± 2.1 GPa comparable to the value of the modulus of silver (76 GPa). The value of the surface tension for the Ag nanowires is of 3.09 ± 0.33 J.m^‑2, which is comparable to the value already published.

This good agreement shows that the increase of the apparent elastic modulus for the smaller diameters is attributed to surface tension effects.

 

 

 

Figure 4: Product of the apparent stiffness and L/D as a function of D³/L².