# AFM Characterization of Track-Etched Membranes:
## 3 Instrumental Response Function of the AFM  
One of the key parameters characterizing the experimental setup of the AFM is the instrumental function of the probe. It directly affects the quality of the obtained topography map of the sample, for example, for a single pore of a TM (Figure 1a).  
To determine the instrumental response function, the calibration silicon periodic structure TGZ was measured, which is a grating with a period of 3 \(\upmu\)m and a step height of 110 nm (see Figure 1b).  
Figure 1: (**a**) AFM image (topography map) of a single TM pore. (**b**) AFM image of the calibration periodic structure TGZ2. (**c**) Measured profile of a single step item in the TGZ2 structure (red dashed line) and the result of modeling this profile using the specified instrumental function (see Equation (1)) of the AFM (black solid line). (**d**) Single pore profile measured in PETP TM (black solid line) and the result of its approximation by the Gaussian function (green dash-dotted line).  
The measured profile of a single step item in the TGZ2 structure (Figure 1c, black solid line) was fitted with the convolution of Heaviside \(H(x)\) and Gaussian functions \(G(x)\):  
\[F(x)=C\cdot\int_{-\infty}^{\infty}H\big{(}x^{\prime}\big{)}\cdot G\big{(}x-x^{ \prime}\big{)}dx^{\prime} \tag{1}\]  
Here, \(x\) is the coordinate along the cut of the sample surface. The Heaviside function, which describes the profile of the real single step item in the TGZ2 structure, was used as a model. The Gaussian function was selected as an approximation of the instrumental response function of the AFM probe:  
\[H(x)=\left|\begin{array}{cc}1&if&x\geq 0\\ 0&if&x<0\end{array}\right.\text{ and }G(x)=\frac{1}{\gamma\sqrt{2\pi}}e^{ \frac{-(x-m)^{2}}{2\gamma^{2}}} \tag{2}\]  
The fitting result using expression (1) of the experimentally measured TGZ2 profile structure (red dashed line in Figure 1c) is shown in Figure 1c by the black solidline. The value of the width parameter of the Gauss function \(\gamma\) was found to be equal to 40 nm. Thus, the tilt of the measured curve can be characterized by the value of full width at half maximum of the fitting curve, which was equal to 40 nm (Figure 1c) for the TGZ2 profile step with the depth of 100 nm. This value represents the effective width of the AFM tip at some height; the deeper the hole we characterize, the larger the effective width is. Thus, if a narrow hole is profiled, a smoothed peak shape will be found in the experiment (Figure 1a). At the same time, theposition of the pore edge (point E on Figure 1c) can be found with extremely high precision (higher than 5 nm).  
To simplify and automate the procedure of pore recognition in AFM images and pore characterization, a two-dimensional Gaussian function (2D-Gauss) can be used to describe the profile of a pore, so the AFM profile of each pore is fitted by 2D-Gauss:  
\[f(x,y)=f_{0}+\frac{A}{\Gamma}\;\exp\left(-\frac{\left(x-X_{c}\right)^{2}}{ \Gamma^{2}}\right)\;\exp\left(-\frac{\left(y-Y_{c}\right)^{2}}{\Gamma^{2}} \right), \tag{3}\]  
Here, \(f_{0}\) is the base, \(A/\Gamma\) is the depth of the measured pore profile, \(2\Gamma\) is the width of the measured pore profile at 1/e height, \(X_{c}\) and \(Y_{c}\) are the lateral coordinates of the geometrical center of the pore. This original software has been developed and used for the analysis of fluorescence images of single molecules [25]. An example of the approximation of a pore profile measured in PETP TM by the one-dimensional Gaussian function is shown in Figure 1d.  
Further, it was necessary to find the coefficient for recalculating the width of the measured pore profile \(\Gamma\) with the true value of pore diameter \(D\) (see Figure 1d). After the analysis of the data obtained for pores with true diameters in the range of 183 to 1430 nm, we found this coefficient as high as 4.2 (\(\pm\)5%); thus, for each pore, we determined the diameter as \(D=4.2\Gamma\). The parameters \(X_{c}\), \(Y_{c}\), and \(D\) were further analyzed.