Comprehensive topography characterization of polycrystalline diamond coatings

Representative images of all three techniques are shown for the four materials in figure 2. The stylus profilometry data are shown with decreasing scan size in figure 2(a). It is clear that the roughness on the MCD has the largest amplitude (RMS height) while the pUNCD surface shows the smallest. Going from larger scans (figure 2(a₁)) to smaller scans (figure 2(a₃)), the amplitude of measured topography decreases for all four diamond species. For example, while a 5-mm scan of MCD spans a vertical range of 646 nm, a 0.5-mm scan of the same surface spans just 391 nm. In order to interpret the stylus data correctly, an estimate of the tip radius is needed because the tip introduces artifacts at and below this size scale. Figure 2(a₄) shows a scanning electron microscopy (SEM) image of the stylus tip. Fitting a circle to the tip yields a radius of R = 5.1 μm. The exact point where tip artifacts become dominant [37] will be estimated using the PSD in the following section.

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The AFM measurements are shown as topography maps of increasing resolution in figures 2(b)–(e). The NCD and MCD clearly show grain structure and faceting at scan sizes of 5 μm and 1 μm (figure 2(d_1-2) and (e_1-2)). The UNCD sample (figure 2(c_1-2)) also shows strong texture, but the size indicates that these are multi-grain clusters. The pUNCD shows topographic features, but no grain structure. At the highest resolution (figures 2(b₃)–(e₃)), all features look relatively smooth. The smoothness of the features is likely related to tip artifacts (caused by convolution between tip curvature and topography). Figures 2(b₄)–(e₄) show TEM images of the AFM tips after they were used to image the surfaces. The tip radii were measured as R = 17 nm for pUNCD, R = 36, 47 and 31 nm for UNCD, NCD and MCD respectively using the same procedure used for the stylus tip. As with the stylus tip, a circle was fitted to the tip profile for the extraction of the radius (figures 2(b₄), (c₄), (d₄), and (e₄)). In cases where the tip apex did not appear perfectly circular, a best-fit circle was fitted to the region of the tip that makes contact with the substrate.

Finally, the surfaces were analyzed using side-view TEM (figures 2(f)–(i)). Once again, the pUNCD surfaces had the lowest amplitude of topography and the MCD had the highest. The NCD and MCD materials showed clear faceting from the individual crystallites. At the scales accessible by the TEM, the UNCD surface also shows faceting (see figures 2(g₂), and (g₃)). However, the smooth pUNCD surface shows no indication of faceting but rather a smoothly varying surface topography. Surprisingly, despite significant differences in topography at larger scales, the smallest-scale topography is nearly identical between the microcrystalline diamond, the nanocrystalline diamond, and the ultrananocrystalline diamond. This finding is further discussed in the next paragraph.

To further investigate the similarities in roughness at the smallest scales, representative images of the three unpolished surfaces are shown in greater detail in figure 3. In all cases, significant roughness is visible on the scale of Angstroms to nanometers. This small-scale topography is visible even on a single facet of a single grain of the NCD and MCD materials. The atomic lattice of the diamond is clearly visible in the TEM whenever a grain is aligned with a zone axis lying near to the imaging axis. In these cases, the lattice is observed in many areas to extend to within 1 nm of the surface. Therefore, it is not simply a rough and potentially amorphous surface layer that is sitting on a flat diamond facet, rather the diamond crystal itself exhibits significant roughness at the small scale.

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Now, a more quantitative analysis of the topography data is presented, and scalar roughness parameters are computed for the various surfaces. First, the root-mean-square (RMS) height ${h}_{{\rm{rms}}},$ RMS slope ${h}_{{\rm{rms}}}^{{\prime} },$ and RMS curvature ${h}_{{\rm{rms}}}^{{\prime\prime} }$ are computed in real-space from each line scan by numerically integrating the squared height data (or its derivatives) over the scan length $L$ [32]:

$$\begin{eqnarray}&&\begin{array}{l}{h}_{{\rm{rms}}}^{2}=\displaystyle \frac{1}{L}\displaystyle {\int }_{0}^{L}{h}^{2}\left(x\right)dx,{h}_{{\rm{rms}}}^{{\prime} 2}=\displaystyle \frac{1}{L}\displaystyle {\int }_{0}^{L}{\left(\displaystyle \frac{dh}{dx}\right)}^{2}dx,\\ {h}_{{\rm{rms}}}^{{\prime\prime} 2}=\displaystyle \frac{1}{L}\displaystyle {\int }_{0}^{L}{\left(\displaystyle \frac{{d}^{2}h}{d{x}^{2}}\right)}^{2}dx\end{array}\end{eqnarray} \tag{ 1 }$$

using the trapezoidal rule (equation (4) of [32]). Figure 4 shows the computed roughness parameters as a function of scan size $L$ (for RMS height, which depends on the larger-scale features) and measurement resolution $l$ (for RMS slope and curvature, which depend on smaller-scale features). Note that for TEM data, the 'measurement resolution' is different from the size of a pixel in the camera ('pixel size') (which can be sub-atomic at the highest magnifications). The measurement resolution $l$ is determined from the point spacing of the extracted profiles, as shown in figure 2. Computing the RMS height as a function of size is equivalent to an analysis of the surfaces' self-affine properties using a variable bandwidth method [47].

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Figure 4(a) shows that ${h}_{{\rm{rms}}}$ increases with $L$ for all the surfaces studied here at small $L,$ i.e. L less than 1 μm. There is a crossover to constant (independent of $L$) ${h}_{{\rm{rms}}}$ at a scan size of 1–10 μm. All of the surfaces studied here show this transition, which corresponds to the thickness of these coatings (2 μm). The amplitude of the pUNCD surface is much smaller than the other three surfaces. NCD and MCD roll off to the same constant value. The values in figure 4(a) at large $L$ correspond to the observation of the amplitude in stylus profilometry shown in figure 2, with pUNCD being the 'smoothest' and MCD being the 'roughest' surface at large scales.

From the analysis of ideal self-affine surfaces [22, 30, 32], it can be shown that ${h}_{{\rm{rms}}}\propto {L}^{H}$ while ${h}_{{\rm{rms}}}^{{\prime} }\propto {l}^{H-1}$ and ${h}_{{\rm{rms}}}^{{\prime\prime} }\propto {l}^{H-2}.$ Therefore, in figure 4(a), the small-scale data has been fit with a power-law function (solid lines), and the extracted values are used to determine the expected trends in figures 4(b), (c) (dashed lines). As argued above and first described in [36, 48], it is apparent from figure 4 that no single value of RMS height, RMS slope, and RMS curvature can be defined, because these parameters depend on $L$ or $l.$ This demonstrates a key difficulty that impedes efforts to link surface function to a single scalar roughness parameter; these common roughness parameters (including RMS roughness) are scanning-length dependent and do not describe an intrinsic property of the material.

As an alternative to analyzing the real-space measurements individually (as done in figure 4), the measurements can all be combined in frequency space to create a single PSD, denoted $C(q),$ that yields a complete statistical description of topography for each surface, as shown in figure 5. This analysis was carried as follows: first, the PSD of each measurement was computed, following the procedures laid out in [30, 32]. Second, a reliability cutoff [30, 32] due to tip artifacts was calculated for each PSD based on the measured tip radius (figure 2), and all data below this size scale was deemed unreliable and removed [32]. Third, all of the reliable portions of the many individual PSDs were combined by computing the arithmetic average of all measurements in logarithmically-spaced bins. The result is a single whole-surface PSD that describes the material across all length scales. There are no fitting parameters in this analysis; rather the PSD serves to separate the different size scales of topography, and the various techniques agree within experimental uncertainty. The only exception is for pUNCD, where the small-size stylus data lies below the AFM and TEM data, causing a dip around q = 10⁷ m⁻¹. This is believed to be an instrumental artifact, rather than resulting from the real topography. Overall, the value of these comprehensive PSDs is that they can be used in analytical and numerical models (such as [22, 25–27, 49–52]) to understand and predict surface properties (e.g. [17]).

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To compute scale-invariant scalar roughness parameters, the full stitched-together PSD was used to compute RMS height, RMS slope, and RMS curvature as:

$$\begin{eqnarray}&&\begin{array}{l}{\left({h}_{{\rm{rms}}}\right)}^{2}=\displaystyle \frac{1}{\pi }\displaystyle {\int }_{0}^{\infty }C\left(q\right)dq,{\left({h}_{{\rm{rms}}}^{{\prime} }\right)}^{2}=\displaystyle \frac{1}{\pi }\displaystyle {\int }_{0}^{\infty }{q}^{2}C\left(q\right)dq,\\ {\left({h}_{{\rm{rms}}}^{{\prime\prime} }\right)}^{2}=\displaystyle \frac{1}{\pi }\displaystyle {\int }_{0}^{\infty }{q}^{4}C\left(q\right)dq\end{array}\end{eqnarray} \tag{ 2 }$$

Table 2 shows the computed RMS parameters. Also computed from the whole-surface PSD is the area ratio, which is the increase in full surface area, per unit apparent area, due to the roughness. The area ratio is a critical parameter in investigations of soft-material adhesion, including [17], and it is calculated as described in [17]. While MCD is the roughest in terms of RMS height, the unpolished UNCD is the steepest in terms of RMS slope. However, while table 2 shows the mathematically correct or 'true' values of RMS parameters for a surface, any individual application or surface property may depend only on a certain range of length-scales. In that case, scale-dependent parameters would be recomputed by integrating only across the relevant size scales.

The Hurst exponent (which can be related to the fractal dimension, as described in [53]), can be calculated from the PSD, which is commonly separated (somewhat arbitrarily) into the 'self-affine' region, where the topography appears to be described by a power law relationship of $C\propto {q}^{\beta }$ where $\beta$ is the power-law exponent, and the 'roll-off' region, where the PSD appears to be flatter. The Hurst exponent $H$ is typically extracted from the self-affine region as $H=\left(\beta -1\right)/2$ [53, 54]. Using this procedure, and the region of the curves between $q=6.3\times {10}^{6}$ m⁻¹ and $1.8\times {10}^{10}$ m⁻¹, the resulting value was $H=0.62\pm 0.09$ and $0.77\pm 0.06$ for pUNCD and UNCD respectively; and $H=0.89\pm 0.04$ and $0.87\pm 0.03$ for NCD and MCD respectively.

There are two alternative methods of extracting the Hurst exponent: from the real-space data using the variable bandwidth method (VBM) [55]; and from the roll-off region by assuming that the full PSD is described by Fractional Gaussian Noise (FGN) [53], a hypothesis put forth by some of us in [32]. The VBM is nothing more than an analysis of the functional dependence of RMS height ${h}_{{\rm{rms}}}$ as a function of scan size $L$ (that scales as ${h}_{{\rm{rms}}}\left(L\right)\propto {L}^{H}$ for self-affine surfaces), as shown in figure 4(a). Note that even for an individual scan, the RMS height could be computed over a subsection of that scan (yielding an estimate of the Hurst exponent for a single realization of the topography [33, 47, 55]) but here full-size scans were used for calculation. As shown in figure 4(a), RMS height ${h}_{{\rm{rms}}}\left(L\right)$ can be accurately fit with a power-law form over the range from the smallest size ($L$ ∼ 10 nm ) up to approximately $L$ = 1 − 10 μm. Over this region, the Hurst exponents were ${H}_{{\rm{VBM}}}$ = 0.73 ± 0.18 for pUNCD; ${H}_{{\rm{VBM}}}$ = 0.74 ± 0.05 for UNCD; ${H}_{{\rm{VBM}}}$ = 0.93 ± 0.09 for NCD and ${H}_{{\rm{VBM}}}$ = 1.02 ± 0.10 for MCD. The Hurst exponents from this method (${H}_{{\rm{VBM}}}$) yield similar results to the above Hurst exponents ($H$) that are extracted from the more common method of fitting the 'self-affine' region of the PSD. The VBM is a useful technique because it is simpler to perform—requiring only a straightforward calculation of the root-mean-square height, calculated in real-space, from a series of multi-resolution measurements. Additionally, it can be readily performed on reentrant surfaces, while the calculation of the PSD requires surfaces to be functions (one height value for each lateral position).

The second alternative method for computing the Hurst exponent uses the 'roll-off' region and the fractional Gaussian noise (FGN) approach. Here, the Hurst exponent is given by ${H}_{{\rm{FGN}}}=\left(\alpha +1\right)/2$ [32, 53], where $\alpha$ is the scaling exponent $C\left(q\right)\propto {q}^{\alpha }$ at low $q.$ This results in ${H}_{{\rm{FGN}}}=1.10\pm 0.04$ for pUNCD; ${H}_{{\rm{FGN}}}=0.82\pm 0.04$ for UNCD; and ${H}_{{\rm{FGN}}}=0.62\pm 0.04$ and 0.70 ± 0.05 for NCD and MCD, respectively. A prior paper by the present authors ([32]) speculated that there may be a connection between ${H}_{{\rm{FGN}}}$ and $H$ (from the self-affine region). This observation would be extremely useful as it suggests that the small-scale behavior could be predicted from large-scale measurements. Unfortunately, when these four different surfaces are compared, there is no clear relationship that emerges.

Even in the traditional 'self-affine' portion of the PSD, a range of values can be extracted for Hurst exponent for a single surface depending on the window used for fitting $H.$ This is particularly true for MCD and NCD where Hurst exponents can be calculated by dividing the 'self-affine' region into two parts. For $q$ in the range of $2.8\times {10}^{7}$ to $1.7\times {10}^{8}$ m⁻¹, the extracted values are ${H}_{larger\lambda }$ = 1.27 ± 0.27 and 1.32 ± 0.09 for NCD and MCD, respectively; for q in the range of $1.7\times {10}^{8}$ to 1.8 $\times {10}^{10}$ m⁻¹, the extracted Hurst exponents are ${H}_{smaller\lambda }$ = 0.75 ± 0.04 and 0.78 ± 0.05 for NCD and MCD, respectively. Indeed, the roughness in these two portions of the curve seems to be qualitatively different, with the upper portion having a scaling behavior near $C\propto {q}^{-4},$ corresponding to $H$ = 1.5. The origin of these differences in scaling behavior between different length scales is discussed in detail in the next section.

Because there can be so much variability in the measurement of a single surface, the whole practice of assuming self-affinity and assigning a single Hurst exponent to describe a surface must be done with caution. It is mathematically convenient to assume self-affinity as this simplifies numerical and analytical models, and it is common practice in experiments to use assumptions of self-affinity to extrapolate to small scales where the topography is not easily measured. However, at least for the diamond materials investigated here, the best-fit value for $H$ depends on the region over which it is measured, and it is strongly influenced by other factors such as grain size (see next section). Instead, where possible, it is preferable to measure surface topography across all size scales and to use the whole-surface PSD as the primary descriptor for the surface, rather than any scalar parameter.

As discussed in the previous section, the larger-grain-size materials (NCD and MCD) demonstrate a region where the PSD scaling is similar to ${q}^{-4}$ in the larger-wavelength portion of the 'self-affine' region (see solid black lines in figures 5(d) and (e)). This scaling is characteristic of 'kinks' in the real-space line scan, such as sharp peaks or valleys. Note that such kinks can arise in topography as an artifact of the nonvanishing tip radius [30, 37], so one must first rule out this as the cause. However, in the present work, this ${q}^{-4}$ scaling is clearly observed both in the reliable portion of the AFM measurement and in the TEM measurement, both of which are free from tip-based artifacts. Therefore, this ${q}^{-4}$ scaling in the PSDs of the MCD and NCD is a feature of the measured topography, rather than emerging from an artifact. This behavior of the PSD corresponds to kinks in the surface topography that are directly observable in the TEM imaging, as shown in figures 2 and 3.

Our hypothesis to describe this local ${q}^{-4}$ scaling is that it is characteristic of topography at scales approximately equal to the grain size of the material. At these scales, features are dominated by the crystal facets and kinks between them: adjacent grains with different orientations will create concave kinks where the grains meet, while edges between crystal facets will create convex kinks. It is therefore assumed that typical topography line scans will pass through approximately two kinks (one concave and one convex) per grain. At sizes much below the grain size, the topography of MCD and NCD reverts to random, self-affine behavior, showing roughness very similar to UNCD (see figure 3), where the PSD scales as ${q}^{-1-2H}$ (see blue dashed lines in figures 5(b)–(e)). This finding agrees with prior work [56] on fracture surfaces in sandstone. In that work, stylus profilometry measurements showed a transition around the grain size; however sub-grain features were mostly inaccessible there due to the aforementioned tip artifacts. Computer-generated profiles were used to verify that facets can cause ${q}^{-4}$-scaling at a wavevector related to the average kink spacing, designated ${l}_{k}$. The mathematical basis for this hypothesis linking kink spacing to ${q}^{-4}$ scaling in the PSD is given in the appendix.

In order to demonstrate how the PSD is affected by the superposition of facets from a characteristic grain size and random roughness below that size, artificial one-dimensional surfaces were created that were composed of: a superposition of triangular peaks (figure 6(a₁)); self-affine random roughness (figure 6(a₂)); and the summation of those two into a single surface (figure 6(a₃)). The piecewise linear surface (figure 6(a₁)) has kinks with uncorrelated heights drawn from a Gaussian distribution and lateral distances between kinks drawn from a Rayleigh distribution. The surface is scaled in order to have an RMS slope of 1 and an average kink spacing ${l}_{k}.$ The small-scale self-affine random roughness (figure 6(a₂)) is generated using a Fourier-filtering algorithm [30, 57] with a Hurst exponent $H=0.8$ and RMS-slope of 1.2.

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When the PSDs (figure 6(b)) are computed from these surfaces, there is a transition from a flat PSD at the largest sizes to scaling as $C\propto {q}^{-4}$, and the transition point occurs at a wavelength of ${\lambda }_{k}=4{l}_{k}$, which corresponds to ${q}_{k}=\frac{2{\rm{\pi }}}{{\lambda }_{k}}=\frac{{\rm{\pi }}}{2{l}_{k}}$. (The transition is somewhat gradual; the choice of this particular value for the transition point is motivated from the mathematical consideration in the appendix). Importantly, the summed-surface PSD follows the kinked-surface PSD and displays ${q}^{-4}$ scaling at larger scales, and then transitions to self-affine scaling at smaller scales, in this case corresponding to a Hurst exponent $H$ of 0.8. To ensure that these results were not unique to the particular way that the kinked surface was generated, this analysis was repeated using kinked surfaces with uniform and exponential distributions of kink spacings, as well as a surface with slopes alternating between $-1$ and 1. These analyses are shown in the supplemental section 1 (available online at stacks.iop.org/STMP/9/014003/mmedia), but the results are similar, differing only in the sharpness of transitions. The key finding from this analysis is that the kink spacing, which corresponds approximately to the grain size, introduces a signature in the scaling of the topography that causes deviations from the commonly assumed fractal-like self-affine scaling.

The PSD of the summed surface (figure 6(b)) reproduces the three regions that are visible in the PSDs of MCD and NCD: flat behavior at small $q$ (large sizes); scaling like ${q}^{-4}$ at intermediate values; and self-affine scaling ($H\sim 0.8$) at large $q\,$(small sizes). However, this does not yet explain the self-affine scaling behavior that is observed above the grain size in polished and unpolished UNCD. To account for this, another synthetic surface was created that is similar to the first, but this time with spatially correlated kink heights (figure 6(c)). The spatial correlation of the kink heights is enforced using the method of [56] as follows: a self-affine random surface was generated using the Fourier-filtering algorithm; then facets were computed by interpolating linearly between random points (the kinks); finally this faceted surface was summed with the same self-affine random roughness as before (figure 6(a₂)). While this does not substantially alter the behavior at small wavelengths, this adds in an additional self-affine region above the grain size (figure 6(d)).

To verify this proposed link between ${q}^{-4}$ scaling of the PSD and the kink spacing of the material, an analysis of kink spacing and grain size was performed on the real materials using the AFM and TEM measurements. For the MCD and NCD, concave kinks between grains were readily visible, and therefore were quantified using feature sampling, as is used in metallographic analysis [58] (supplemental section 2.1). The mean lineal intercept between concave kinks, which is of the same order as the grain diameter [58], was measured as 839 ± 68 nm for MCD and 647 ± 42 nm for NCD. Because the mean kink spacing used in the mathematical analysis involves both convex and concave kinks, it was computed as half of this value, i.e. 419 ± 34 nm for MCD and 323 ± 21 nm for NCD. These values can be converted to frequency-space using ${q}_{k}=\tfrac{\pi }{2{l}_{k}},$ as discussed above, and the values calculated are ${q}_{k}=3.7\times {10}^{6}$ m⁻¹ for MCD and ${q}_{k}=4.9\,\times {10}^{6}$ m⁻¹ for NCD. The same approach could not be applied to UNCD where the kink spacing was at or below the reliability cut-off due to tip artifacts. Therefore, the mean kink spacing for UNCD was estimated as half of the grain size, which was computed by averaging a sample of 20 different grains observed in the TEM (see supplemental section 2.2). The mean grain size of the UNCD was determined to be 14 ± 3 nm, which corresponds to a kink spacing for UNCD of 7 ± 2 nm and ${q}_{k}=\,2.2\times {10}^{8}$ m⁻¹. The mean kink spacings of these three materials are indicated as vertical bars in figures 5(c)–(e). Indeed, the MCD and NCD demonstrate that the scaling transitions to $C\sim {q}^{-4}$ right around the mean kink spacing. For UNCD, the mean kink spacing is too small to observe a clear ${q}^{-4}$ scaling regime; however, the self-affine scaling behavior is confirmed for sizes much larger than the characteristic grain size. The remainder of the paper will frame results in terms of grain size (again, approximately twice the value of kink spacing), since grain size is more widely measured and reported for polycrystalline materials.

In summary, based on the experimental measurements of these four polycrystalline diamond surfaces, and based on the computed PSDs of artificially generated surfaces, four characteristic regimes of topography scaling have been identified. (1) At the smallest size scales (if significantly smaller than the grain size) power-law scaling may be observed that is characteristic of self-affine random roughness, e.g., $C\sim {q}^{-1-2H},$ corresponding to H in the range of 0.6–0.9. (2) At size scales similar to and slightly smaller than the average grain size, the PSD displays characteristic scaling of $C\sim {q}^{-4}$ due to grain facets and kinks. (3) At sizes larger than the grain size, but smaller than the film thickness, there is another region of power-law scaling corresponding to random roughness ($C\sim {q}^{-1-2H}$). (4) Finally, at sizes larger than the film thickness, the PSD flattens out, with scaling in the range of ${q}^{0}$ to ${q}^{-1}.$ These four regions of topography scaling accurately describe the polycrystalline diamond surfaces and computer-generated surfaces investigated here. These four regions and their boundaries may be broadly generalizable to other materials, but further investigation is required. For instance, the boundary between regions (3) and (4) corresponded to the thickness of the diamond coatings in these measurements, but this thickness was not varied to explicitly investigate this connection. The future application of comprehensive topography characterization on other materials will elucidate the applicability of these four regions to other polycrystalline materials.