C.
Polop
*^{a},
E.
Vasco
^{b},
A. P.
Perrino
^{b} and
R.
Garcia
^{b}
^{a}Departamento de Física de la Materia Condensada and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: celia.polop@uam.es
^{b}Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain

Received
3rd February 2017
, Accepted 9th June 2017

First published on 12th June 2017

From aircraft to electronic devices, and even in Formula One cars, stress is the main cause of degraded material performance and mechanical failure in applications incorporating thin films and coatings. Over the last two decades, the scientific community has searched for the mechanisms responsible for stress generation in films, with no consensus in sight. The main difficulty is that most current models of stress generation, while atomistic in nature, are based on macroscopic measurements. Here, we demonstrate a novel method for mapping the stress at the surface of polycrystals with sub-10 nm spatial resolution. This method consists of transforming elastic modulus maps measured by atomic force microscopy techniques into stress maps via the local stress-stiffening effect. The validity of this approach is supported by finite element modeling simulations. Our study reveals a strongly heterogeneous distribution of intrinsic stress in polycrystalline Au films, with gradients as high as 100 MPa nm^{−1} near the grain boundaries. Consequently, our study discloses the limited capacity of macroscopic stress assessments and standard tests to discriminate among models, and the great potential of nanometer-scale stress mapping.

Three features make intrinsic stress particularly harmful for these systems. (i) It is unavoidable: even a single-crystal solid has non-zero defect density in its lattice due to entropy under conditions of thermodynamic equilibrium. The defect density is higher in films due to kinetic limitations caused by the deposition conditions and substrate constraints. (ii) It is reversible and cumulative: intrinsic stress can be regenerated by residual stress in the system (the unreleased fraction under normal temperature and pressure – NTP), and can accumulate under conditions of heating, overpressure, and periodic loads. (iii) It exhibits a non-uniform spatial distribution at the scale of the solid lattice defects. In polycrystals, this scale (∼10–100 nm) is far below the resolution of standard stress tests. Macroscopic assessments are ineffective at detecting these steep stress gradients, which can be higher than the mechanical strengths required for commercial use (including safety margins). Consequently, stress mapping at the nanoscale is an irreplaceable tool for the study of material resistance and nanomechanics.

The failure of current technology to investigate stress at the inherent spatial scales of polycrystals is mirrored in the academic world, where the mechanisms responsible for stress generation during film deposition and processing have generated intense conjecture and scientific activity.^{5–12} However, no consensus has been reached so far. The main difficulty is that current models of stress generation, most of which are atomistic in nature, are only supported by data with, at best, sub-micron resolutions. For example, techniques such as curvature-based measurements, Raman spectroscopy, and X-ray diffraction cannot reveal the stress distribution in films on nanometer scales.

Atomic force microscopy (AFM) is a suitable tool for determining the mechanical properties of solids at the nanoscale.^{13–15} In this work, we develop a method to map the stress on the surface of polycrystals with sub-10 nm spatial resolution. Our method maps the elastic modulus of the surface by two AFM techniques (Force Modulation Microscopy and bimodal AFM), then transforms these data into a stress map via the local stress-stiffening effect. The validity of our method is supported by Finite Element Modeling (FEM) simulations. Applying this method to Au films reveals a highly heterogeneous distribution of intrinsic stress along grain diameters, with stress gradients as high as 100 MPa nm^{−1} near the grain boundaries (GBs). Consequently, our results call into question the validity of stress assessments based on standard tests and micrometer-scale characterization techniques.

Fig. 2 shows the results of the FMM experiments measured at dissimilar modulation voltage ΔV on Au films with two different thicknesses: 600 nm in (a, c, e) and 1200 nm in (b, d, f, g, h). For each experiment, we try three different modulation voltages ΔV driving the probe vibration. The Methods section describes how we estimate the corresponding force modulation ΔF. Comparing the topography of both films (Fig. 2a and b), we see that grain size increases with film thickness as expected (note the different length scales). The grains with flat tops are surrounded by deeper regions where GBs intercept the surface.^{16} The overlapping height distributions for different values of ΔV (the histograms of Fig. 2a and b) demonstrate that the topography measurements are independent of ΔF, and hence that the average tip–sample contact geometry is preserved.

Fig. 2c and d show the corresponding FMM amplitude maps, together with their A_{stiff}-normalized histograms. The meaning of A_{stiff} is discussed below. In general, we observed only slight variations in A_{FMM} within grains (light areas). However, the amplitude A_{FMM} decreases dramatically near the GBs (dark areas) and peaks inside the GBs (white areas). The FMM amplitude images also reveal some small-scale morphological features such as vicinal surfaces (arrows in Fig. 2c and d) with better resolution than the topography images. A possible explanation for this enhanced FMM resolution is provided in the Methods section.

Once the FMM amplitude was measured, we mapped the effective elastic modulus E_{eff} using the following novel procedure. For an FMM probe excited by a piezoelectric actuator coupled to the cantilever base, the A_{FMM} response is described by the equation:^{17–21}

(1) |

k_{eff} = [6F_{L}E_{eff}^{2}R_{eff}]^{1/3} | (2) |

At this point, we need to remove the topography contribution from the FMM amplitude maps, because the rough surface of the samples has an impact on the FMM amplitude images (by changing the contact area of the probe). If we assume that the intrinsic mechanical properties of the sample are independent of its morphology (i.e., we reject potential finite-size effects), the spatial dependence of E_{eff} can be described as a perturbation around its bulk value E^{bulk}_{eff}, thus . By substituting this form of into eqn (2), we can express k_{eff} in terms of separable functions: . Here, describes exclusively the morphology dependence of k_{eff} due to the local curvature of the surface. Thus, the effective elastic modulus at each point on the image is calculated as follows:

(3) |

Fig. 2e and f show the E_{eff} maps for the Au samples together with their histograms. Fig. 2g plots typical profiles across a GB for the three mapped magnitudes (topography, A_{FMM} and E_{eff}). The profile paths are the solid straight lines overlaid on the maps. The E_{eff} maps reveal that the inner grain regions do have homogeneous mechanical properties, with E_{eff} variations being no more than 10% (region Z_{1} in Fig. 2g). The regions near a GB are softer than the grain interior, with E_{eff} decreasing to 48% of E^{bulk}_{eff} (region Z_{2}). The values of E_{eff} inside the GBs (where A_{FMM} peaks) are influenced by the difficulty of accessing these narrow gaps with the probe tip and we do not analyze them. The decrease in E_{eff} near the GBs is significant and reproducible for different R_{tip} (we used both standard and ultrasharp tips, with nominal R_{tip} = 10 and 2 nm) and different scanning angles.

Examining Fig. 2g, we see that the A_{FMM} and E_{eff} profiles (middle and bottom panels) do not follow the shape of the height profile (top) around the GB. In particular, the positions where the topography slope changes (marked by red dashed lines) do not coincide with major variations in A_{FMM}. The fact that both profiles are scanned simultaneously rules out the possibility that this shift is caused by potential artifacts of measurement or topography. Additionally, we calculated the normalized topography contribution to the FMM amplitude, A^{topo}_{FMM} = k^{topo}_{eff}/(k_{1} + k^{topo}_{eff}), and compared it with the experimental A_{FMM} maps. The corresponding histograms are shown in Fig. 2h. The degree of overlap between the histograms indicates the amount of contrast in the A_{FMM} maps that originates from topography effects. If the majority of the A_{FMM} contrast is due to topography, we would also obtain flat E_{eff} maps (fixed to the value E^{bulk}_{eff}) after removing the topography contribution. Fig. 2h reveals that the hypothesis fails: the A_{FMM} histogram is significantly wider and preferentially spread to lower (i.e., softer) values. The following conclusions can be drawn: (i) the topography contribution predicts a more homogeneous mechanical response than what we actually measured; and (ii) the surface measured by FMM is more compliant than expected from the topography contribution alone. Consequently, we can conclude that topography plays a minor role in the contrast of the A_{FMM} maps, which is removed in the E_{eff} maps.

Increasing ΔV beyond a certain threshold, which is defined by the condition ΔF > F_{L}/2, causes an abnormal broadening of the E_{eff} histograms (see arrows in Fig. 2e and f). We know that this broadening is not related to changes in the average contact geometry, such as tip deformation or plastic regime, because those would also be visible in the topographic images (note that the height histograms still overlap). Two possible explanations for the histogram broadening are anharmonic distortions in the mechanical response of the sample^{17} (probably responsible for the high-E_{eff} tail in the histogram of Fig. 2e) and/or small slips of the tip on steep regions^{18} (the low-E_{eff} tail in the histogram of Fig. 2f).

To validate our method, Fig. 3 compares the inferred E_{eff} results obtained by FMM (b) with E_{eff} maps obtained independently by bimodal AFM (a), for different regions of the 1200 nm thick Au film. The latter method determines the effective elastic modulus without any prior assumptions.^{23–26} The general features of the E_{eff} maps obtained by both techniques agree with regard to: (i) the low dispersion of mechanical properties within the grain interiors (E_{eff} ≈ E^{bulk}_{eff} ± 6% for bimodal AFM), (ii) the fact that regions close to a GB are softer (E_{eff} decreases to 42% of E^{bulk}_{eff} in the bimodal AFM map) and (iii) the observed peak in E_{eff} at the GBs. This qualitative agreement between the three types of regions and their behaviors is supported by the two GB-crossing profiles depicted in Fig. 3c. Fig. 3d compares the corresponding E_{eff} histograms. The difference in the modes, which are 60 GPa and 63 GPa for bimodal AFM and FMM, respectively, is within the error generated during calibration of the cantilever force constants. The good agreement between the results obtained by the two different AFM techniques with dissimilar tip–sample interactions (while the FMM requires a strong continuous contact, the bimodal AFM is based on weaker intermittent contact), using two different setups (see the Methods section) supports our proposed procedure for E_{eff} mapping by FMM experiments on stiff polycrystalline films.

The high variability of E_{eff} observed in our study suggests that the elastic moduli measured by FMM and bimodal AFM are affected by the tip–sample contact (note that this does not happen with optical techniques). In this case, the spatial variation in E_{sample} is mostly due to a stress-stiffening effect (also known as geometric nonlinearity). This process is typical of stressed membranes when the deformation produced by a normal load generates out-of-plane contributions of the stress force that counteract such a load. In our case, the membrane corresponds to the outermost sublayer of the film, the normal load is F_{L}, the out-of-plane deformation is the indentation depth d (as defined by the Hertz model), and the stress in the membrane corresponds to the biaxial intrinsic stress in the film σ. Thus, as sketched in Fig. 4a and b, the indentation of a region under compression (with σ < 0, dashed red arrows) creates a stress force F_{σ} that strengthens F_{L}. On the other hand, the indentation of a region under traction (σ > 0) creates a stress force F_{σ} that counters F_{L}. The fact that we calculate E_{eff} from F_{L} instead of F_{L} ± F_{σ} implies an underestimation (overestimation) of the applied load, and means that the regions under compression (traction) are displayed as softer (stiffer): i.e., E_{eff} < E^{bulk}_{eff} (E_{eff} > E^{bulk}_{eff}).

Fig. 4 Stress-stiffening effect. (a, b) Diagrams of the tip–film contact geometry in regions under stress of compression and traction respectively, showing the parameters of the stress-stiffening model. (c) Load-indentation curves computed by FEM (symbols) and by the stress-stiffening model (continuous lines) for σ-stressed isotropic Au films under a load exerted by a Si tip.^{29} The inset shows the σ-dependencies of E_{eff} computed from the fit of the FEM results to the updated Hertz model (symbols), the stress-stiffening model (solid line), and the lattice anharmonicity effect estimated for Au <111> films (dashed curve).^{30} |

Fig. 4c (symbols) shows the load-indentation curves computed by FEM for σ-stressed isotropic Au films under a normal load F_{L} exerted by a Si tip.^{29} The curves for different values of σ follow a power law of the form F_{L} ∝ d^{1.5±0.1}, in agreement with the Hertz model. As previously discussed, the indentation depth for a given F_{L} increases as σ decreases (considering the sign). The σ-dependence of E_{eff} is calculated by fitting each curve to the Hertz model. This dependence is plotted in the inset of Fig. 4c (symbols), together with the E_{eff} variations due to changing σ expected from the lattice anharmonicity effect in Au(111) films^{30} (dashed curve). The FEM results show a stronger σ-dependence (e.g., E_{eff} changes up to 20% for σ ≤ ±1 GPa) than that estimated for the lattice anharmonicity effect, which predicts changes in E_{eff} only up to 4%. Consequently, the E_{eff} variation with σ found by FEM appears to be consistent with the spatial dispersions in the E_{eff} maps (Fig. 2 and 3), since local residual stress σ of at most a few GPa is expected. It should be remembered that the Au films are macroscopically relaxed (as measured by MOSS, see the ESI†). Therefore, the stresses causing the E_{eff} dispersion correspond to local residual fractions of the growth intrinsic stress that survives at NTP.

By taking into account the σ-dependence of E_{eff} found by FEM, we can now transform the E_{eff} maps into σ maps using the following analytical model of stress stiffening. The film indentation caused by the tip pressure breaks the in-plane film symmetry. The biaxial intrinsic stress σ in the film therefore contributes an amount P_{σ} = β(σ·ẑ) to the normal pressure, where β is a factor describing the stress field geometry^{31} and ẑ is the unit vector normal to the film plane. This contribution can be estimated as , where is the radius of the contact surface A (shaped like a spherical cap). P_{σ} generates a stress force on the tip:

(4) |

Hence, the resulting force becomes F_{T} = F_{L} − F_{σ}. Updating the Hertz model to take stress stiffening into consideration, we obtain . Now, if we interpret the spatial F_{T} variation due to the stress field in terms of the E_{eff} variation, we get:

(5) |

Eqn (5) allows us to transform the E_{eff} maps into . Then, by substituting F_{σ} into the updated Hertz model, we can compute the σ maps shown in Fig. 5a and b for the 600 nm and 1200 nm thick Au films, respectively. Since the σ contribution to F_{σ} (eqn (4)) depends on the indentation depth, deeper indentations induced by higher force modulations ΔF of the tip load are required to sense lower stresses. However, increasing ΔF beyond a certain threshold also produces abnormal broadening of the E_{eff} histograms, as discussed above. Thus, the practical ΔF maximum determines the amount of uncertainty in the stress resolution δσ. This uncertainty is defined as the minimum σ variation required to produce a change in d greater than the experimental error in determining the indentation depth δd (see the Methods section).

Because the intrinsic stress modifies the indentation depth nonlinearly, the analytical model determines a series of uncertainties δσ_{i} for different measurement ranges. The uncertainty varies with both the sign and the magnitude of the intrinsic stress. These uncertainties are used to determine the statistical properties of the σ maps, and correspond to the bin widths of the histograms in Fig. 5c and d. As specified in the histograms, the intrinsic stress measurements are grouped into five levels (colors): 0-green, A-dark red, A′-light red, B-dark blue and B′-light blue. The green bins represent the relaxed area, while the red (blue) bins collect measurements from areas under compression (traction). Dark (light) colors correspond to the areas under low (high) stress. Fig. 5e and f redraw the σ maps using this discrete color scale, to improve the contrast between regions with different stress levels. In addition, these figures illustrate the GB mesh with white lines, calculated by applying a tessellation filter to the topography images.^{32} The line thickness in the figure is similar to the diameter of the tip–sample contact area (≈5.5–5.7 nm), providing a visual estimation of the areas inaccessible to the AFM tip. We call this simplified representation a “compression–traction map” hereafter.

Typical values of δσ_{i} obtained under our experimental conditions are a few hundred MPa. For example, the average uncertainties in the σ maps around the relaxed state (level 0) in Fig. 5a and b are δσ_{0} ≈ 170 and 158 MPa, respectively. This stress uncertainty also reduces the spatial (lateral) resolution of the σ maps. The spatial resolution is theoretically limited by the diameter of the contact area ( for both maps), but worsens to δσ_{i}/∇σ where ∇σ is the magnitude of the stress gradient to be resolved. For example, given the statistics of the σ maps in Fig. 5a and b, we can estimate their average stress gradients as 〈∇σ〉 = 26.6 and 25.8 MPa nm^{−1}, respectively. This implies that the maps have average spatial resolutions of δσ_{0}/〈∇σ〉 ≈ 6.4 and 6.1 nm, respectively. Since these resolutions (below 10 nm) are smaller than the inherent length scales of the lattice imperfections in polycrystals (∼10–100 nm), the method of stress mapping proposed here is good enough to image residual stress gradients in polycrystalline films. In particular, the method is suitable to sense the stress within the outermost sublayer with thickness in the order of the indentation depth d (∼1 nm for F_{L} of a few hundred nN). Note that this sublayer plays a key role in the mechanical properties of systems with a high surface-to-volume ratio.

The compression–traction maps reveal that the stress distribution in polycrystals is highly heterogeneous: relaxed areas alternate with regions under compression and traction. Some regions exemplifying the different stress regimes are highlighted in Fig. 5e and f. While the inner vicinal surfaces of the grains are mostly relaxed (0-green areas), most of the regions near the grain boundaries are under compression (A and A′-red areas). Annular areas with traction stress (B-blue areas) appear frequently in between the previous two regions. Fig. 5g shows a typical stress profile measured across a GB, performed along the black line in Fig. 5a. The corresponding 1D gradient (right axis) demonstrates that residual stress gradients as high as 100 MPa nm^{−1} persist along the grain diameters in macroscopically relaxed films.

These results are directly connected to the generation of compressive stress at the post-coalescence stage (i.e., once the GBs are formed) during the deposition of polycrystalline films, which is extensively discussed in the literature.^{5–12} Although we are mapping residual stresses rather than in situ growth stresses, it is reasonable to assume that the two quantities are related in a straightforward way by stress-relaxation thermodynamics. Thus, after deposition stops, the accumulated growth stress relaxes progressively until it reaches a steady state (residual stress), wherein the strain energy generated by the residual stress is lower than the activation energy of the relaxation mechanism. Subsequently, two preliminary conclusions can be drawn from our results. (i) Neither force–dipole interactions between morphology features^{5} nor adatom insertion between ledges^{6} at vicinal surfaces are responsible for post-coalescence compression, since those areas are mostly relaxed. (ii) The fact that compression regions mostly decorate GB edges indicates that GBs are involved in the generation of the post-coalescence compression, as proposed in ref. 7, 10 and 12. We will address the physical origin of the stress gradients along the grain diameter in a forthcoming work.

Finally, note that the heterogeneous distribution of stress over the surface of polycrystalline films, as resolved here in the sub-10 nm stress maps, is undetectable by the standard techniques and tests used for stress analysis. Such techniques, which have sub-micron spatial resolutions at best, are only sensitive to the average stress over the entire displayed areas in Fig. 5a and b: 〈σ〉 ≈ −33 MPa and −14 MPa, respectively. Besides, these values correspond to the macroscopically relaxed samples, as discussed above. Furthermore, average stresses hide the existence of steep gradients (as high as 100 MPa nm^{−1} in Au) which can be greater than the mechanical strengths required by many applications.^{33} Consequently, we hope that stress mapping at the nanoscale becomes an irreplaceable tool for the study of material resistance. More generally, this and other nanomechanical technologies will change our perception about the atomistic nature of stress in crystalline solids.

Background.
FMM (also called ‘modulated nanoindentation’^{34}) is an AFM technique used to study materials with intermediate stiffness (1 GPa < E_{sample} < hundreds of GPa). Using a standard AFM set-up, FMM measures the amplitude of vibration in a cantilever whose tip is held in continuous contact with the surface of the sample (see Fig. 1d). As it moves across the surface, the cantilever vibrates at a frequency f lower than its first resonance f_{01}.^{17–21,34} This vibration is induced by applying a modulation voltage ΔV to a piezoelectric element at the base of the cantilever or the sample (acoustic excitation). This process results in a load modulation ΔF around the static load F_{L} used to hold the continuous contact. The amplitude of the cantilever vibration A_{FMM} is measured by a four-sector photodiode operating in a frequency-locked loop (FLL). Relatively high static load values (F_{L} ∼ several hundred nN) are required to ensure that adhesion forces can be ruled out. The Hertz model predicts for a spherical tip. From a Taylor expansion of the Hertz model for small force modulations, the force balance corresponding to harmonic FMM quasi-static vibration can be estimated as k_{1}A_{FMM} ≈ k_{eff}Δd for an indentation amplitude Δd = A_{stiff} − A_{FMM}. Thus, a decrease (increase) in the measured value of A_{FMM} is expected on more compliant (stiffer) areas (see Fig. 1d).

The resulting FMM equation k_{eff}/k_{1} = 1/(A_{stiff}/A_{FMM} − 1), as clarified in eqn (1), relates the contrast in the vibration amplitude map to the ratio between the force constants of the cantilever and the tip–sample contact. This ratio depends on the mechanical properties of the sample and tip. This equation offers a straightforward interpretation of the mechanical properties of soft materials with flat surfaces. In systems with stiffness much lower than that of the tip and negligible roughness, k_{eff} ≈ {6F_{L}[E_{sample}/(1 − ν_{sample}^{2})]^{2}R_{tip}}^{1/3} is independent of the sample topography, and any deformation of the tip can be neglected. Consequently, contrast in the FMM images can be attributed almost exclusively to gradients in the mechanical properties of the sample, in particular, to variations of E_{sample} since the spatial dependence of ν_{sample} is smoother.

However, when the target surface is a metal or ceramic polycrystalline film grown by the Volmer–Weber (V–W) mechanism, interpreting the FMM images is a more complex task. These films are composed of grains with non-negligible roughness, which implies high curvature gradients. Ceramic and metal grains may also have stiffnesses comparable to that of the tip. Therefore, the sample topography and tip deformation become non-negligible factors, as we describe in the main text. Our procedure to remove the topography contribution from the FMM amplitude maps and inferred material properties of the surface is related to other research studies^{35} with similar aims, in particular with respect to features with sizes in the order of the tip radius.

Experimental.
The FMM experiments were performed with a commercial AFM (Nanotec Electronica S.L.) in a dry N_{2(g)} atmosphere.^{36} The environmental humidity was held below 10% in order to avoid capillary forces. The modulation voltage ΔV was applied to a piezoelectric element at the base of the cantilever. Si cantilevers (PPP-NCHR Nanosensors) with k_{1} = 40 N m^{−1}, R_{tip} = 10 nm and f_{01} = 300 kHz were used. Topographic images and FMM amplitude images were acquired simultaneously. In order to remain within the linear elastic regime of the material, the static load F_{L} and modulation voltage ΔV were chosen to produce indentations of only a few Å. The relationship between ΔV and ΔF for each experiment was estimated from the static and dynamic calibrations of the photodiode (namely, photodiode response in nm V^{−1} and signal gain at the modulation frequencies). Specifically, for the 600 nm and 1200 nm thick Au films, we obtain F_{L} ± ΔF ≈ 180[nN] ± 131[nN V^{−1}] × ΔV[V] and F_{L} ± ΔF ≈ 200[nN] ± 839[nN V^{−1}] × ΔV[V] respectively. The frequency-locked loop (FLL) fixed to the modulation frequencies (f = 47 kHz for 600 nm and 80.6 kHz for 1200 nm) allowed us to determine the harmonic indentation amplitude Δd with an experimental error of δd = 0.2 Å, as shown in Fig. 1 for systems with low stiffness contrast. The FLL mode improves the FMM resolution by attenuating inelastic responses to the tip–sample interactions. Also, in FLL mode, FMM exhibits enhanced resolution on abrupt hollow features^{18} (e.g., GBs and steps at vicinal surfaces) where the effective radius R_{eff} of contact diverges for . The data were processed assuming the following mechanical properties: E_{Au} = 78 GPa and ν_{Au} = 0.44 for the sample, and E_{Si} = 170 GPa and ν_{Si} = 0.28 for the tip material.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7nr00800g |

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