# Imaging the Permittivity of Thin Film Materials by Using Scanning Capacitance Microscopy
## 2 Materials and Methods  
### Principles  
By considering the tip as a point-mass spring, the equation of motion for the tip could be represented as [14]:  
\[m\frac{d^{2}\xi}{dt^{2}}+\frac{m\omega_{0}}{Q}\frac{d\xi}{dt}+K\xi=F_{ts}+F_{0 }\cos(\omega_{d}t), \tag{1}\]  
where \(m\) denoted the effective mass, \(K\) the spring constant, \(\xi\) the displacement of the tip from the equilibrium position, \(Q\) the quality factor, \(\omega_{0}\) the free resonance angular frequency, \(F_{ts}\) the tip-surface interaction, \(F_{0}\) and \(\omega_{d}\) the amplitude and angular frequency of the driving force, respectively. The tip-surface interaction \(F_{ts}\) may consist of various contributions, such as short range repulsive and chemical binding forces, van der Waals force \(F_{vdW}\), and long-range electrostatic and magnetic forces, \(F_{d}\) and \(F_{mug}\), respectively. When the tip was lifted to a proper height \(z\), the influence of van der Waals force on the tip could be negligible, [15] and the tip vibrated completely under the long-range electrostatic and magnetic forces, \(F_{el}\) and \(F_{mug}\). When both the tip and the sample were conductive materials, the tip and the sample formed a capacitance \(C\). If a DC voltage \(U_{0}\) and an AC voltage \(U_{1}\sin(\omega t)\) were applied between the tip and the sample, the voltage between the tip and the sample was  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]  
where \(\varphi(x,y)\) was the surface potential of the sample. The energy stored in this capacitor was  
\[E=\frac{C{U}^{2}}{2}. \tag{3}\]  
The force on the tip was  
\[F=-\frac{\partial E}{\partial z}=-\frac{1}{2}U^{2}\frac{\partial C}{\partial z}. \tag{4}\]  
Then the force between the tip and the sample was [15]  
\[F_{el}=-\frac{1}{2}\frac{\partial C}{\partial z}[(U_{0}-\varphi(x,y))+U_{1} \sin(\omega t)]^{2}=F_{\rm dc}\ +\ F_{\omega}\ +\ F_{2\omega}, \tag{5}\]  
where the spectral components were:  
\[F_{dc}=-\frac{1}{2}\frac{\partial C}{\partial z}\left[(U_{0}-\varphi(x,y))^{ 2}+\frac{1}{2}U_{1}^{2}\right], \tag{6}\]  
\[F_{\omega}=-\frac{\partial C}{\partial z}(U_{0}-\varphi(x,y))U_{1}\sin(\omega t), \tag{7}\]  
\[F_{2\omega}=\frac{1}{4}\frac{\partial C}{\partial z}U_{1}^{2}\cos(2\omega t). \tag{8}\]  
When the tip was lifted to a proper height \(z\), the tip vibrated completely under the electric field force. If the frequency of the AC voltage applied to the tip was half of the resonance frequency of the cantilever, the tip was under the drive of \(F_{2\omega}\). The vibration equation was  
\[m\frac{d^{2}\xi}{dt^{2}}+\frac{m\omega_{0}}{Q}\frac{d\xi}{dt}+K\xi=\frac{1}{4 }\frac{\partial C}{\partial Z}U_{1}^{2}\cos\left[2\left(\frac{1}{2}\omega_{0} \right)t\right], \tag{9}\]  
When the amplitude of the tip was very small, the influence of vibration on \(\partial C/\partial z\) could be ignored, where \(C\) was the capacitance between the tip and the sample. The solution of Equation (9) was  
\[\xi=-\frac{QU_{1}^{2}}{4K}\frac{\partial C}{\partial Z}\cos\left(\omega_{0}t+ \frac{\pi}{2}\right), \tag{10}\]  
It could be seen from Equation (10) that the amplitude of the cantilever was proportional to the capacitance gradient \(\partial C/\partial z\) when the amplitude was very small. The measured capacitance variation with respect to reference distance \(z_{0}\) was [7]  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]0  
and  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]1where \(\Delta C_{apex}(z)\) was the apex capacitance contribution including both the dielectrics and the tip shape effect, and \(\Delta C_{stray}(z)\) was the stray capacitance between the cantilever and the ground. The stray contribution could be precisely quantified as [16; 17]  
\[U=(U_{0}-\varphi(x,y))+U_{1}\sin(\omega\ t), \tag{2}\]2  
by linearly fitting the region where the effect of the apex capacitance was negligible. In Equation (13), stray capacitance constant \(k_{stray}\) was determined by the structure between the cantilever and the ground.