## 3 Results
### Correction Factor for Euler Beam Equation for Plan-View Trapezoidal Cantilevers
A comparison between our data in Figure 2 and the data plotted in [PERSON] [57] graph shows a maximum difference of approximately 1.6%. However, it is also important to note that for the calibration curve displayed in the [PERSON] [57] work, there are no details given on how their plot was generated. Given that the analysis performed in this work is traceable to established analytical calculations, the authors recommend using Equation (6), or Equation (7) combined with Figure 2, to determine the spring constant of trapezoidal cantilevers. If the spring constant of the cantilever is determined using another experimental technique (e.g., refs [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]) and the dimensions are determined using SEM, for example, then Equation (7) can also be  
Figure 3: Plan-view SEM image of trapezoidal-shaped Bruker FastScan C cantilever. Figure 2: A calibration curve for trapezoidal plan-view cantilevers. The correction factor, K, can be determined from the _w/w\({}_{0}\)_, and then substituted into Equation (7) to determine the cantilever spring constant. used to determine [PERSON]'s modulus of the AFM cantilever or indeed any cantilevered structure with a trapezoidal plan view, provided \(w\) is less than \(w_{0}\). This technique has been used to determine [PERSON]'s modulus of wool fibres and polymer actuators [58; 59]. Deriving Solutions to the Torque Correction Factor for Plan-View Triangular and Trapezoidal Cantilevers  
[PERSON] et al. [55] defines the general expression for determining the torque, \(T_{i}\), for a cantilever of arbitrary shape with a sharp tip, and we use the same notation that [PERSON] et al. [55] use in their work. It is given as:  
\[T_{i}=1-\rho_{i}\;(tana)D \tag{8}\]  
where \(i\) refers to \(z\) or \(\theta\), which are the normal and longitudinal directions of the cantilever, respectively. The torque we are interested in is the normal direction therefore \(T_{z}\) is the correction factor we focus on for the various plan-view geometries. The term \(\rho_{i}\) (Equation (17) from [PERSON] et al. [55]) can be expressed as:  
\[\rho_{z}=k_{z}/k_{z\theta} \tag{9}\]  
where \(k_{z}\) is the normal spring constant for the cantilever, and \(k_{z\theta}\) is the longitudinal spring constant of the cantilever. For a beam-shaped cantilever, we get \(k_{z}\) from Equation (1) and \(k_{z\theta}\) ([PERSON] et al. [55]) is defined as:  
\[K=(E_{c}t^{3}w_{0}/(4L^{3}))\cdot\{(w/w_{0}-1)^{3}/(3w/w_{0})\;((1-w/w_{0})-(1- w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})\;ln(w/w_{0}))^{-1}\} \tag{6}\]0  
So if we divide \(k_{z}\) by \(k_{z\theta}\) for a beam-shaped cantilever we can obtain \(\rho_{z}\). If we then substitute the result of \(k_{z}/k_{z\theta}\) back into Equation (8), we get the familiar term for the torque for a beam-shaped cantilever, \(T_{beam}\), which is given in Equation (3). For a trapezoidal cantilever, the situation is more complex since while we have an expression for \(k_{z}\) for a trapezoidal cantilever (refer to Equations (6) and (7)), we do not have one for \(k_{z\theta}\). However, we can assume that the torque, \(T_{trap,}\), for a trapezoidal cantilever will be in between that for a beam-shaped cantilever, \(T_{beam}\), and a triangular-shaped cantilever, \(T_{tri,}\), with the same base width, \(w_{0}\), and length, \(L\). This is represented in \[K=(E_{c}t^{3}w_{0}/(4L^{3}))\cdot\{(w/w_{0}-1)^{3}/(3w/w_{0})\;((1-w/w_{0})-(1- w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})\;ln(w/w_{0}))^{-1}\} \tag{6}\]1  
\[K=(E_{c}t^{3}w_{0}/(4L^{3}))\cdot\{(w/w_{0}-1)^{3}/(3w/w_{0})\;((1-w/w_{0})-(1- w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})\;ln(w/w_{0}))^{-1}\} \tag{6}\]2 Shows that the trapezoidal cantilever shape, given in Figure 4b, is a trapezoidal shaped cantilever with base width \(w_{o}\) and top width \(w\). Its geometry is in between that for a beam-shaped cantilever (**a**) and a triangular cantilever (**c**) with the same base width \(w_{0}\). Therefore, if we determine \(T_{bum}\) and \(T_{tri. }\), we can determine an upper and lower limit for \(T_{trap}\).