# Application of Areal Topography Parameters in Surface Characterization
## 3 Assessment of the Quality of Measurement Results Obtained via an Atomic Force Microscope
### Influence of Surface Profile Geometry on the Parameters in AFM Measurements: A Monte Carlo Approach
Although both \(Sdq\) (root mean square gradient) and \(Sdr\) (developed interfacial area ratio) are classified as hybrid parameters combining amplitude and spatial characteristics of surface topography, the results obtained in this simulation do not confirm the expected trends. The literature consistently reports that increases in slope, surface rotation, or reductions in step pitch lead to higher values of both parameters, reflecting greater surface complexity and microstructural articulation. However, in this study, the small-scale variations applied to the simulated rectangular profiles did not produce any clear or systematic increase in either \(Sdq\) or \(Sdr\). The observed fluctuations appear to be random rather than systematic. For more conclusive insights, it would be necessary to analyze more pronounced geometric or instrumental influences. In the simulations, an idealized rectangular profile was used without surface imperfections or noise, resulting in the \(Sz\) value matching the nominal step height (97.6 nm). However, in the case of the real reference standard, microstructural variations, local protru \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value/Range** \\ \hline Step pitch & (2.98–3.12) \(\upmu\)m \\ \hline Step width & (0.25–0.35) \(\upmu\)m \\ \hline Surface rotation & up to 10\({}^{\circ}\) \\ \hline Surface slope & linear increase in the y-direction \\ \hline Lateral noise & \(\pm\)1 to \(\pm\)100 nm \\ \hline Step height & fixed at 97.6 nm \\ \hline Sampling area & random sampling over a 20 \(\upmu\)m \(\times\) 20 \(\upmu\)m area \\ \hline Vertical noise (z-axis) & \(\pm\)3 nm (to mimic measurement uncertainty) \\ \hline Base profile & rectangular, 3.0 \(\upmu\)m pitch, 0.3 \(\upmu\)m step width \\ \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters. \begin{table}
\begin{tabular}{l l l l l l l} \hline
**Step Pitch (\(\upmu\)m)** & **Step Width (\(\upmu\)m)** & **Simulated \(Sq\) (\(\upmu\)m)** & **Simulated \(Sz\) (\(\upmu\)m)** & **Simulated \(Sdq\) (\(\upmu\)m)** & **Simulated \(Sdr\) (\(\upmu\)m)** \\ \hline
2.98 & 0.25 & 15.65 & 22.67 & 100.44 & 0.22 & 2.12 \\ \hline
3.02 & 0.28 & 17.18 & 28.96 & 100.39 & 0.21 & 2.11 \\ \hline
3.08 & 0.30 & 18.41 & 29.98 & 100.40 & 0.20 & 1.82 \\ \hline
3.12 & 0.35 & 21.92 & 32.71 & 100.49 & 0.20 & 1.82 \\ \hline
2.98 & 0.35 & 21.06 & 32.09 & 100.99 & 0.24 & 2.09 \\ \hline
3.12 & 0.25 & 15.68 & 27.67 & 101.35 & 0.22 & 2.14 \\ \hline \end{tabular}
\end{table} 
Table 5: Simulated surface parameter values for varying step geometry. sions, indentations, or measurement noise are always present to some extent. These factors can cause the measured \(Sz\) value to exceed the nominal step height, even when the step height itself is accurately measured. To validate the simulation model, the simulated parameter values were compared with those obtained from actual measurements performed on the standard. This comparison was made possible by simulating a reference standard with a rectangular profile using the certified values of step height and step width. The results are presented in Table 6.  
The simulated and experimental results show good agreement for all the parameters, confirming the validity of the model. The deviation of 7 nm between the simulated and measured values of the \(Sz\) parameter is attributed to surface noise and microstructural irregularities, which are inevitably present in real physical standards but are absent from the idealized simulation model. These findings provide a good basis for interpreting the behavior of individual parameters and evaluating their robustness in both simulated and real measurement conditions. To support more detailed empirical validation, future research should aim to develop a standardized intercomparison protocol.