Quantitative Hardness Measurement by Instrumented AFM-indentation

In this work, the bending stiffness of the cantilever k was calculated according to the geometrical beam theory¹⁹. For the particular diamond-coated cantilever used in this work, we found k = 55.69 N/m. Note that we neglected the diamond coating; the thickness of the diamond coating is one to two orders of magnitude smaller than the cantilever thickness and thus does not significantly increase its bending stiffness (although its Young's modulus is significantly larger than that of silicon).

In order to avoid sample deformation effects, the sensitivity of the photodiode was determined by recording the force-distance curve with the previously-calibrated force sensor on a smooth nano-crystalline diamond surface with a Young's modulus E = 759 GPa²². The force signal was recorded in volt units (the unit of the photodiode signal) and over a small range of repulsive forces in order to avoid tip deformation and damage. The repulsive part of the force-distance curve was then fitted with a linear function, the inverse slope of which corresponds to the photodiode sensitivity S. In this particular experiment, the sensitivity of the photodiode was determined to be S = 23.903 nm/V. The assumption of a linear response of the photodiode is limited to when the base displacement of the cantilever is smaller than 500 nm. For larger displacements Z, the non-linearity of the photo-sensitive detector needs to be considered, in which case the Z-V_PD response is a third-order polynomial¹². For the calibration, the base displacement was set to 50 nm, while in our experiments, the base displacement was 150 nm. In these cases, we considered the response of the photodiode to be linear.

Figure 1: Surface topography of a gold thin-film surface. (Left) Non-contact AFM topography image of a 5 x 5 µm² and (right) of a 1.25 x 1.25 µm² Au thin-film surface area displaying micrometer-sized grains, each of which exhibits an atomically flat Au(111) surface consisting of large terraces and monoatomic steps. Please click here to view a larger version of this figure.

Figure 1 shows non-contact AFM topography images of a gold thin-film surface. The thin-film surface is found to consist of grains in the micrometer range. Each grain exhibits an atomically flat Au(111) surface consisting of large terraces and monoatomic steps. Figure 2 shows the indents caused during indentation measurements by an AFM tip with a maximal vertical force of 7.2 µN applied on the same Au(111) thin-film surface as in Figure 1. Also, the topography difference between the imaged area before and after a series of four indentations at distinct locations is displayed in Figure 2(c). It is worth noting how similar all remaining indents look. This similarity attests to the stability of the tip and the reproducibility of the measurements.

Figure 2: AFM indents on an atomically smooth gold thin-film surface. (a) Non-contact AFM topography image of a 1 x 1 µm² Au thin-film surface area selected for AFM indentation measurements. (b) Non-contact AFM topography image of the same surface area in (a) after four consecutive AFM indentation measurements up to a vertical force F_n = 7.2 μN. (c) Topography difference between images in (a) and (b). (d – f) Non-contact AFM topography images of three individual AFM indents shown in (b). Please click here to view a larger version of this figure.

Figure 3: Calculation of the projected area of an AFM indent on an atomically smooth gold thin-film surface. (Left) Non-contact AFM topography images of individual AFM indents shown in Figure 2. (Right) Same topography image as in the left panel after cropping and with an overlaid mask used to calculate the projected area using the free SPM data analysis software Gwyddion. The projected area is found to be A_p = 4703.52 nm²; this yields a hardness value H_AFM = 1.53 GPa. Please click here to view a larger version of this figure.

Figure 3 demonstrates the procedure to determine the projected area of an indent by masking the area with negative topography values relative to the unmarred surface. From this measurement, the projected area of the indent is found to be A_p = 4703.52 nm². The indentation was performed with a maximal load F_n,max = 7.2 µN (see Figure 4). Accordingly, the hardness can be calculated as . The measured A_p-value is likely to be underestimated by tip convolution effects during imaging, on the one hand, and by elastic recovery effects upon unloading²³, on the other.

Figure 4: Indentation curves from force-distance curves measured by AFM. (a) Typical loading part of a force-distance curve measured by AFM on an atomically smooth Au thin-film surface. (b) Force-displacement curve calculated according to Equation (3) (blue line) and the Hertzian fit (red line) of the elastic part up to the first observable plasticity event (pop-in) at F_n = 0.908 µN, with tip displacement δ_el = 3.786 nm (the length of the first pop-in is measured to be λ_pop-in = 543 pm) according to Equation (4). The corresponding fit parameter is determined to be , where R is the indenter radius and E^* is the reduced modulus of elasticity. Note that the Hertzian fit curve is extended beyond the elastic regime so as to calculate the work of plasticity W_plasticity from the integrated difference between the Hertzian fitting curve and the experimental result; W_plasticity = 11.44 x 10^-15 J. (c) Series of four consecutive force-penetration curves. (d) Magnified view of the force-penetration curve shown in (b) showing pop-ins with lengths in the range of several 100 pm (indicated by arrows). Please click here to view a larger version of this figure.

Figure 4 shows indentation curves calculated from force-distance curves measured by AFM. It is important to note how the curves overlap, which further attests to the reproducibility of the measurements. In Figure 4(b), a force-displacement curve calculated from a force-distance curve (Figure 4(a)) according to Equation (3) is fitted with the Hertzian fit (Equation (4)) over its elastic part. The elastic limit was determined from the first observable plasticity event (pop-in) at F_n,el = 0.908 µN and at the tip displacement δ_el = 3.786 nm (the length of the first pop-in is measured to be λ_pop-in = 543 pm). The corresponding fit parameter is determined to be , where R is the indenter radius and E^* is the reduced modulus of elasticity. While it may be tempting to extract the indentation modulus of gold M_Au(111) from the fitting parameter, the radius at the tip apex and the indentation modulus of the diamond-coated tip remain uncertain. In principle, the indenter properties can be calibrated by indenting on a calibration sample. The determination of the indentation modulus of Au(111) is beyond the scope of this work. Assuming an elasticity modulus for Au(111) E_Au = 80 GPa, a Poisson's ratio ν_Au = 0.45, and a tip E_nc-diamond = 759 GPa and ν_nc-diamond = 0.003, we calculate from a tip radius R ≈ 1 nm. The derived value is far too low to be credible, as already pointed out in Ref. 8. It has been suggested that the elasticity modulus of metals decreases in near the surface region¹⁰. Using the value suggested in Ref. 8 (E = 30 GPa), we obtain R = 5.5 nm. Also, the Hertzian fit function used in Figure 4(b) assumes a spherical tip geometry. However, this assumption only applies to the very apex of the indenting tip, i.e., for tip displacement within the elastic deformation regime. As seen below, for larger displacements, the tip can no longer be considered as spherical but rather resembles a Berkovich tip. Note further that the Hertzian fit curve is extended beyond the elastic regime so as to calculate the work of plasticity W_plasticity from the integrated difference between the Hertzian fitting curve and the experimental result²¹; W_plasticity = 11.44 x 10^-15 J. A magnified view of the force-penetration curve shown in Figure 4(b) further demonstrates the outstanding resolution of the method to detect single atomistic plasticity events with pop-in lengths of the same order of magnitude as the Burger's vector of gold.

Moreover, the shape of the AFM tip was estimated from the non-contact AFM images, shown in Figure 2(d – f), using the free SPM data analysis software Gwyddion (see Figure 5(a – c)). Subsequently, an averaged tip shape was calculated, from which the half-opening angle of the indenter was determined to be α = 67.21° (see Figure 5(d)). Together with the tip displacement values shown in Figure 4, a hardness value was determined, where δ_max = 18 nm is the maximal tip displacement. Both hardness calculations deliver virtually the same value: H_Au(111) = 1.5 GPa. This result is in good agreement with reported values for gold thin-films as measured by nanoindentation, H_NI/Au = 1 – 2.5 GPa^{24, 25}. The mean pressure at the first plasticity event during AFM indentation on Au(111) has been found to be GPa⁴. From our experimental values and together with the estimated tip radius, we find 13.7 GPa. This value corresponds to a critical shear stress ²¹. From our data, we find that τ = 6.3 GPa, which is in the range of values found by Asenjo et al.⁸ but is much larger than those found in previous studies, where τ = 1.7 – 3.4 GPa^{4, 26, 27}. However, this value is overestimated by the lower value of the assumed tip radius, and it is reasonable to assume that the critical shear stress at the first plasticity event is bounded by the theoretical strength τ_theo,Au = 4.3 GPa.     

Figure 5: Tip reconstruction from non-contact AFM topography images of AFM indents on an atomically smooth gold thin-film surface. (a – c) Reconstructed tip shapes calculated from the non-contact AFM images shown in Figure 2(d – f) using the free SPM analysis software Gwyddion. (d) Averaged tip shape from the images shown in (a – c). From (d), the half-opening angle of the indenter is determined to be α = 67.21°; together with the tip displacement values shown in Figure 4, a hardness value was determined, where F_n,max = 7.2 µN is the maximal vertical force and δ_max = 18 nm is the maximal tip displacement. Please click here to view a larger version of this figure.