## 2 Modelling
### The Electromechanical Coupling Model of the Akiyama Probe
#### 2.2.2 The Verification of the Model  
For proving the validity of this electromechanical coupling model of an Akiyama probe, finite element analysis (FEA) software (ANSYS, 14.0, ANSYS Inc., Canonsburg, PA, USA, 2011), is used to simulate the contact between the tip and the sample [13]. The parameters of the Akiyama probe used to build the model are shown in Table 2.  
The contact between the tip and the sample is simulated by a spring as shown in Figure 12 and the spring constant represents the effective force gradient between the tip and the sample (\(k_{\text{eff}}\)). Where \(k_{\text{eff}}\) is  
\[k_{\text{eff}}=\langle\
abla F_{\text{bs}}\rangle \tag{16}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Tuning Fork** & **Cantilever** \\ \hline Length/\(\upmu\)m & 2400 & 175 \\ Thickness/\(\upmu\)m & 220 & 31.9 \\ Width/\(\upmu\)m & 103.7 & 3.7 \\ Density/kg:m\({}^{-3}\) & 2290 & 2300 \\ Young modulus/GPa & 78.7 & 180 \\ \hline \end{tabular}
\end{table} 
Table 2: The parameters of ANSYS simulation. Figure 12: The ANSYS model of an Akiyama probe. The \(F_{\text{ts}}\) is the force between the tip and sample, it can be described as follows in DMT model:  
\[F_{\text{ts}}=\left\{ \begin{array}{l}\frac{F_{0}}{(1-\frac{a_{0}-z}{F_{0}})^{2 }},\ z\geq a_{0}\\ -\frac{4}{3}E^{*}\sqrt{R}(a_{0}-z)^{\frac{3}{2}}+F_{0},\ z<a_{0}\end{array} \right. \tag{17}\]  
where \(F_{0}\) is the adhesive force, \(z\) is the distance between the tip and sample, \(a_{0}\) is the radius of the atom, R is the radius of the tip, and \(E^{*}\) is  
\[\frac{1}{E^{*}}=\frac{1-v_{\text{s}}^{2}}{E_{\text{s}}}+\frac{1-v_{\text{t}}^ {2}}{E_{\text{t}}} \tag{18}\]  
where \(E_{\text{s}}\), \(E_{\text{t}}\) are the Young's modulus of the sample and the tip, \(v_{\text{s}}\), \(v_{\text{t}}\) are the poisson ratio of the sample and the tip. In the simulation, a constant \(k_{\text{eff}}\) is used, which means the maximal \(F_{\text{ts}}\) is  
\[F_{\text{max}}=k_{\text{eff}}A \tag{19}\]  
where \(A\) is the amplitude of the cantilever. The results of the non-contact simulation are shown in Table 3.  
Coupling factors, \(p\) and \(n\), can therefore be calculated as:  
\[ \begin{array}{l}p=\frac{2v_{\text{eff}}}{1-\frac{2v_{\text{eff}}}{v_{\text{ eff}}}}=\frac{2v_{\text{eff}}}{v_{\text{eff}}}=6.76\times 10^{5}\ \text{m}/\text{C}\\ n=\frac{v_{\text{tip}}}{v_{\text{eff}}}=\frac{\omega v_{\text{tip}}}{\omega v_ {\text{eff}}}=\frac{D_{\text{tip}}}{D_{\text{tf}}}=26.67\end{array}  \tag{20}\]  
The tuning fork prong is always preserved by rectangular cantilever, the spring constant, and effective mass can be calculated as:  
\[ \begin{array}{l}k=\frac{1}{4}Ew\left(\text{$\int$}\right)^{3}=1571.55\ \text{N}/\text{m}\\ m=0.24 qvtl=3.01\times 10^{-8}\ \text{kg}\end{array}  \tag{21}\]  
The resonant frequency with contact can be expressed as:  
\[f^{\prime}=\frac{1}{2\pi}\sqrt{f_{0}^{2}+\frac{n^{2}k_{\text{eff}}}{2m_{ \text{probe}}}} \tag{22}\]  
The effective mass of the U-shaped cantilever is difficult to calculate. If this mass is \(0.001m_{\text{probe}}\) for example, the error in the resonant frequency after ignoring this mass will be:  
\[f_{1}^{2}-f_{2}^{2}=\frac{1}{4\pi^{2}}(\frac{n^{2}k_{\text{eff}}}{m_{\text{ probe}}}-\frac{n^{2}k_{\text{eff}}}{1.001m_{\text{probe}}})<0.001\times\frac{1}{4\pi^{ 2}}\frac{n^{2}k_{\text{eff}}}{m_{\text{probe}}} \tag{23}\]  
\[f_{1}-f_{2}=\frac{f_{1}^{2}-f_{2}^{2}}{f_{1}+f_{2}}<0.001\times\frac{1}{4\pi^{ 2}}\frac{n^{2}k_{\text{eff}}}{m_{\text{probe}}}\frac{1}{2f_{0}} \tag{24}\]  
When the value of \(k_{\text{eff}}\) is small enough, \(0.01\ \text{N}/\text{m}\) for example, the error is less than \(1\ \text{Hz}\), so the effective mass of the U-shaped cantilever can be ignored during the calculation of the resonant frequency and the spring constant of the probe is given by: