Poisson’s ratio of individual metal nanowires

The measurement of Poisson’s ratio of nanomaterials is extremely challenging. Here we report a lateral atomic force microscope experimental method to electromechanically measure the Poisson’s ratio and gauge factor of individual nanowires. Under elastic loading conditions we monitor the four-point resistance of individual metallic nanowires as a function of strain and different levels of electrical stress. We determine the gauge factor of individual wires and directly measure the Poisson’s ratio using a model that is independently validated for macroscopic wires. For macroscopic wires and nickel nanowires we find Poisson’s ratios that closely correspond to bulk values, whereas for silver nanowires significant deviations from the bulk silver value are observed. Moreover, repeated measurements on individual silver nanowires at different levels of mechanical and electrical stress yield a small spread in Poisson ratio, with a range of mean values for different wires, all of which are distinct from the bulk value.


Supplementary note 1: AFM length measurement accuracy
The spanning length of the NW between the trench edges i.e. the clamped length, is measured by AFM before and after manipulation. The accuracy of the AFM tip in measuring the NW span length is critical in understanding the error in the measured values of Poisson's ratio.

Supplementary Discussion
Relationship between resistance and resistivity changes in a nanowire under normal loading.

Bending
We first consider the contribution due to bending, according to classical Euler-Bernoulli beam theory -this applies to high aspect ratio NW (length/width). All strain components vary linearly with z (coordinate normal to the applied load direction). These do not contribute to the relationship between the measured net change in resistance and resistivity because they balance symmetrically across the thickness -their average (total) effect over the beam volume is zero, due to their linear dependence on z. In reality, shear deformation will be present for a beam of finite thickness-to-length ratio, and is expected to yield a nonzero contribution. This will induce a net strain Dz is the beam displacement, h its thickness and L its length, which for a long and narrow beam will (normally) be smaller than any contribution due to stretching, as we discuss below.

Stretching
As the beam deflects normal to its long axis, strain develops along the beam due to stretching: where w is the deflection normal to the beam axis. This induces strains in two orthogonal directions: (1) (2) where ν is Poisson's ratio.
It is shown by Ngo et al 1 that the relationship between the induced uniaxial tensile force, T, due to stretching and the centre beam deflection, Δz, is well approximated by: where and I is the areal moment of inertia of the cross section, and A is its area. This formula is accurate to ~2%, in comparison to the exact solution.
From Hooke's law, it follows that the strain along the beam axis is given by: Equations (2) and (5) completely describe the strain tensor in the beam as all other strain components are zero.
The resistance, R, is related to the resistivity, ρ, by It directly follows that the relationship between the change in resistance, ΔR, and the change is resistivity, Δρ, is This can be written in terms of the strain tensor components: Substituting equations (2) and (5) into equation (8) results in: Equation (9) allows the relative change in resistivity, Δρ/ρ, to be determined from the measured change in resistance, ΔR/R, the centre displacement and the beam dimensions.
If we consider the case where Δz is much smaller than the beam radius, then equation (9) simplifies: Thus, the resistance and resistivity are related by the square of the beam displacement. If Δz is greater than the beam radius, i.e. in the case of large beam deflection, we again obtain a quadratic dependence with strain, albeit with a slightly different coefficient: Shear deformation induces a correction of higher order in the thickness-to-length ratio correction, however, this should only be observable at very small deflections, i.e.
Measurements are performed well away from this limit, so shear deformation effects does not affect the theory.

Young's Modulus Measurement
Supplementary figure 1 shows a typical force-displacement curve for an AgNW. The elastic modulus is found to be 86 ± 10 GPa for this 90 nm diameter nanowire, calculated form the well-known generalised model 2 ; where F is the applied force, E is Young's modulus, I is the areal moment of inertia, L is the length, Δz center is the lateral displacement and the f(α) term accounts for tensile deformation along the nanowire. A full description of the mechanical model is described elsewhere 2 .
Generally, we found E to be between 40 -160 GPa for individual pentagonal AgNWs agreeing with previously reported values [3][4][5]  The NW is ohmic with a resistance, R = 15.5 Ω. In general, R ~15 -70 Ω, depending on the diameter and length of the NW, with ρ ~ 17 -35 nΩ m. The NW resistivity is found to increase with decreasing radius, consistent with the general trend in the literature 6 . In addition, the resistance and resistivity values agree with previously reported measurements on similar nanowires 7 .

NiNW Electrical Preparation: a Resistive Switching (RS) Process
NiNWs display a physical phenomenon known as resistive switching (RS), so that a number of electrical forming steps must be performed before any 4-point measurements are possible.
The forming steps produce a conductive filament (CF), through the NiO layer that coats the wire, resulting in an ohmic connection between the Ni core and the contacting electrode. The RS process has been described extensively elsewhere [8][9][10] .
Supplementary figure 7 outlines the CF forming process. Supplementary figure 7a shows a schematic of the NW contacted with four blue electrodes labelled 1-4. Firstly, the area designated as "FORM 1" is made conductively active across electrodes 1 and 2 by the forming process, given by the IV curve in supplementary figure 7b. Between 0 -2.8 V the NW is in the high-resistance state as there are no CFs at the NW/contact-1 or NW/contact-2 junctions. At a threshold voltage of 2.8 V the resistance has an abrupt decrease to the low resistance state. Here, the current is limited by a set compliance current of 500 nA. The NW remained in the low resistance state, signifying that a non-volatile CF had formed. The same procedure is applied to the "FORM 2" section between electrodes 3 and 4, given by the IV curve in supplementary figure 7c. We see the same characteristics as supplementary figure 7b but with a higher compliance current of 10 µA. The higher compliance current is necessary for the NW to remain in the high resistance state at 0 V, more than likely due to a slightly thicker oxide at one, or both, electrodes 3 and 4. After the forming process standard 4-point electrical characterisation could be performed, given by the 2-and 4-point IV curve shown in supplementary figure 7d. The voltage is sourced and the current is measured across the outer electrodes 1 and 4, the voltage is measured across the inner electrodes 2 and 3. The IV curve displays typical linear behaviour observed in metals, showing that the forming process produces secure metallic filaments between the Ag electrode and NiNW core.
Typically the 4-point resistance is found to be between 60 -200 Ω, depending on the dimensions of the NW, with resistivities between 70 -190 nΩ m comparable with the bulk value (69 nΩ m).