## 3 Results  
### Correction Factor for Euler Beam Equation for Plan-View Trapezoidal Cantilevers  
Figure 1 defines the plan-view parameters for a trapezoidal cantilever, and we assume the cantilever has a uniform thickness, \(t\). The base width of the cantilever is \(w_{0}\) and the top width is \(w\). We shall use the same nomenclature as used by [PERSON] and [PERSON] [1]. Therefore, Equation A3a in [PERSON] and [PERSON] [1] describes the equation for the angled side of the trapezoidal cantilever and is given by:  
\[y=f(x)=a(1-mx)=a-amx \tag{4}\]  
where [PERSON] and [PERSON] [1] define \(a\) and \(m\) as adjustable gradient parameters. Therefore, the y-intercept for this function is the variable \(a\), and the absolute value of the slope of the function is \(am\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**AFM Probe** & \begin{tabular}{c} **Spring Constant,** \\ \(\mathbf{k_{tip}}\) (N/m)** \\ \end{tabular}  & \begin{tabular}{c} \(w_{0}\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(w\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(L_{tip}\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(L\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(t\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(D\) \\ \(\upmu\)m \\ \end{tabular}  & \begin{tabular}{c} \(T_{beam}\) \\ \end{tabular}  & \begin{tabular}{c} \(T_{tri}\) \\ \end{tabular}  & \ \begin{tabular}{c} \(T_{trap.}\) \\ \end{tabular}  & \begin{tabular}{c} \(K\) \\ \end{tabular}  \\ \hline PSA1 & 15.81 \({}^{*}\) & 30.53 & 10.7 \({}^{*}\) & 0.35 & 24.46 & 26.61 & 0.57 & 4.55 & 0.19 & 0.94 & 0.92 & 0.93 & 1.23 \\ \hline FSA2 & 14.24 & 31.86 & 10.7 & 0.34 &  
We can see from Figure 1 that half the base width of the cantilever equals to \(a\) (i.e., \(a=w_{0}\)/2) and the slope of the angled side equals to \((w_{0}-w)\)/(2L). Therefore, \(am=(w_{0}-w)\)/(2L), and given \(a=w_{0}\)/2, then \(m=(w_{0}-w)\)/(\(Lw_{0}\)). [PERSON] and [PERSON] [1] derived the following formula (which is Equation A3b in their article) for a trapezoidal cantilever:  
\[k=2(D^{\prime}am^{3})/(mL-1)[mL-m^{2}L^{2}/(2(1-mL))+(1-mL)ln(1+mL)]^{-1} \tag{5}\]  
where \(D^{\prime}=E_{c}t^{3}/12\)  
Therefore, if we substitute \(a=w_{0}\)/2, \(m=(w_{0}-w)\)/(\(Lw_{0}\)), and \(D^{\prime}=E_{c}t^{3}\)/12 into Equation (5), it can be rewritten as:  
\[K=(E_{c}t^{3}w_{0}/(4L^{3}))\cdot\{(w/w_{0}-1)^{3}/(3w/w_{0})\;((1-w/w_{0})-(1- w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})\;ln(w/w_{0}))^{-1}\} \tag{6}\]  
If we let \(K=3(w/w_{0})/(w/w_{0}-1)^{3}((1-w/w_{0})-(1-w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})ln (w/w_{0}))\), then we can rewrite Equation (6) as  
\[k=E_{c}t^{3}w_{0}/(4 KL^{3}) \tag{7}\]  
For AFM cantilevers, where the tip is offset from the very end of the cantilever \(L=L_{tip}\). Therefore, the \(K\) term can also be considered a correction factor to the Euler beam equation (given in Equation (1)), which is dependent on the ratio of the top width to the base width, i.e., \(w/w_{0}\) for trapezoidal cantilevers. We can, therefore, plot \(K\) versus \(w/w_{0}\), and this is displayed in Figure 2.  
An advantage of the calibration curve in Figure 2 is that it relies on the ratio of \(w\)/\(w_{0}\) which means that any error in the calibration of the instrument measuring \(w\) or \(w_{0}\) will effectively cancel. For practical cantilevers with a trapezoidal plan-view, the width, \(w\), should be measured through the tip apex position since this is where the force is applied to the cantilever when in operation. Figure 3 shows a plan-view SEM image of a trapezoidal-shaped cantilever (Bruker FastScan C) with the dimensions \(w_{0}\), \(w\), \(L\) and \(L_{tip}\) defined. Figure 1: Schematic of the top-down view of a trapezoidal cantilever using the same notation as [PERSON] and [PERSON] [1]. It should be noted that an engineering text authored by [PERSON] [57] also has a correction factor versus _w/w\({}_{0}\)_ graph for plan-view trapezoidal cantilevers displayed in Figure 6.43 of the [PERSON] text [57].