# Imaging the Permittivity of Thin Film Materials by Using Scanning Capacitance Microscopy
## 2 Materials and Methods
### Principles
From Equation (10), the capacitance gradient \(\partial C/\partial z\) was represented with the vibration amplitude. In the region where the effect of the apex capacitance was negligible, the measured capacitance gradient \(\frac{\mathrm{d}C_{meas}}{\mathrm{d}z}\) should be  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]3  
With tip radius \(R\) and cone half angle \(\theta\), [PERSON] et al. studied the capacitance between the tip and a metallic surface. The apex capacitance was given by [18]  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]4  
Its capacitance gradient was  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]5  
Then, the \(\frac{\mathrm{d}C_{meas}^{metal}}{\mathrm{d}z}\) should be fitted in the apex capacitance region by  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]6  
Detecting the capacitance gradient \(\frac{\mathrm{d}C_{meas}^{metal}}{\mathrm{d}z}\) vs. setting height \(z\) between the tip and a metal ground, the tip radius \(R=R_{0}\), cone half angle \(\theta=\theta_{0}\), and stray capacitance constant \(k_{stray}^{metal}\) could be obtained by fitting \(\frac{\mathrm{d}C_{meas}^{metal}}{\mathrm{d}z}\)\(\sim\)\(z\) curve with Equation (17). When an insulating film with thickness \(h\) and relative permittivity \(\epsilon_{r}\) was coated on the metal ground, [PERSON] et al. proposed to use \(z+h/\epsilon_{r}\) equivalent instead of \(z\) in Equation (15) to calculate the capacitance. The apex capacitance was proposed to be [7]  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]7  
Its apex capacitance gradient was  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]8  
Then, the \(\frac{\mathrm{d}C_{meas}^{dielectric}}{\mathrm{d}z}\) could be fitted in the apex capacitance region by  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]9When an insulating film was coated on the metal (or monocrystalline silicon substrate) ground, the value of \(k_{stray}\) for ground (marked with \(k_{stray}^{ground}\)) could be obtained by fitting \(\frac{\mathrm{d}\mathrm{C}_{\mathrm{bias}}^{ground}}{\mathrm{d}z}\)\(\sim\)\(z\) curve with tip radius \(R_{0}\), cone half angle \(\theta_{0}\), and Equation (17) above. The value of \(k_{stray}^{idictoric}\) could be approximately replaced by that of \(k_{stray}^{ground}\). The value of \(k_{stray}^{ground}\) was usually different from that of \(k_{stray}^{metal}\). Double scanning technology was used to acquire the topography image and SCM image (with the tip lifted to a proper height \(z^{\prime}\)). A procedure of 1 Order (line-by-line subtraction of the first order approximating curve) was selected, the detection signal was corrected for the tilt contribution. The value of \(h/\varepsilon_{r}\) of every point could be obtained from the SCM image by using \(R_{0}\), \(\theta_{0}\), \(k_{stlight}^{idictoric}\), and Equation (20) above for data reconstruction. The value of \(h\) of the insulating film at every point could be obtained from the topography image. Then, the value of relative permittivity \(\varepsilon_{r}\) of every point could be calculated, and the 2D distribution of \(\varepsilon_{r}\) on the nanometer scale could be acquired.