# Measurement of Sub-Surface Microstructures Based on a Developed Ultrasonic Atomic Force Microscopy
## 3 Results and Discussion  
### Finite Element Simulation  
Subsurface imaging technology can be realized by detecting various aspects of interaction between the probe tip and the sample, but it is necessary to analyze multi-influence coupling parameters, it will be too complicated to calculate if the modulus is calculated by analytical formula. In this study, the physical model based on the contact between the probe tip and the sample are used to simulate the contact resonance frequency of the vibrating probe scanned over the sample. The physical model and the involved calculation formula have been presented in Section 2.2. In order to compare the simulation results with the specific experimental results, a schematic diagram of the simulation model in Figure 4 is established.  
As shown in Figure 4a, the probe is scanned with a distance of 8 \(\upmu\)m over the surface of the sample with subsurface microstructures formed by a photolithographic grating with a HOPG film. The width and spacing of the lithographic grooves are 2 \(\upmu\)m, and the depth of the grooves is 500 nm. Through the finite element simulation, the contact resonance frequency of the probe can be calculated when the probe scans over each point on the scanning path. The calculated values can then be numerically plotted, which can be used for estimation of the detected sample subsurface topography. The geometric modeling and the meshing of this FEM is shown in Figure 4b. The experimental parameters used in the simulation have also been set as the same as the actual experiment, and shown in Table 2[17].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Tip Radius & 10 nm \\ Tip Height & 5 \(\upmu\)m \\ Tip Spring Constant & 300 N/m \\ Tip Length & 110 \(\upmu\)m \\ Tip Width & 50 \(\upmu\)m \\ Contact Force & 10 nm \\ Si Density & 2330 kg/m\({}^{3}\) \\ HOPG Density & 2270 kg/m\({}^{3}\) \\ Elastic Modulus of Si & 190 Gpa \\ Elastic Modulus of HOPG & 18 Gpa \\ Poisson’s Ratio of Si & 0.2782 \\ Poisson’s Ratio of HOPG & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental parameters used in the simulation.  
Figure 4: (**a**) Schematic of the simulation model. (**b**) shows the geometry model of finite element simulation and (**c**) shows the meshing for finite element simulation.  
The finite element simulation results are revealed in Figure 5. When scanning along the sample surface with different HOPG film thicknesses, the resonance frequencies of the probe-sample contact at each point on the \(X\)-axis are shown in Figure 5a. It can be seen clearly that with the increase of the buried depth of the subsurface microstructure, the resonant frequency gradually decreased. The relationship between the buried depth and the maximum frequency difference is shown in Figure 5b. It is easy to find that when the tip is directly above the geometric center of the defect, the resonant frequency of the probe-sample contact is the smallest. Conversely, the probe-sample contact resonance frequency is maximum when the tip is in the absence of defects. The difference between these two frequencies reflects the detection capability of the CR-AFM for defects. When this difference tends to zero, it indicates that the probe is difficult to detect the existence of defects. From Figure 5b, it can be seen that as the buried depth increases, the maximum frequency difference become smaller, showing an exponential downward trend. This phenomenon will be further discussed in conjunction with the experimental results later.