# Imaging the Permittivity of Thin Film Materials by Using Scanning Capacitance Microscopy
## 1 Introduction  
Since the invention of the scanning tunneling microscope (STM) by [PERSON] et al. in August 1981, [1] it has been widely used in the research of physics, chemistry, biology, material science and other fields. Additionally, a large family, called scanning probe microscopy (SPM), has been derived. Atomic Force Microscope (AFM) was a kind of SPM. AFM could be used to explore the nanomechanical properties of soot particle layers [2]. A developed ultrasonic AFM was used to measure the sub-surface microstructures [3]. Scanning capacitance microscope (SCM) was an important member of the SPM family [4]. In September 1981, [PERSON] filed a patent application related to SCM [5]. In 1998, [PERSON] et al. used tapping AFM to realize intermittent contact mode SCM. It could image the metal lines buried under a 1-um thick dielectric film [6]. Great advances have been made by using SPM to quantify the relative permittivity \(\varepsilon_{r}\) of thin film materials on the nanometer scale [7]. In 2015, [PERSON] et al. used SCM to determine the subsurface of electronic devices in silicon layer. The electronic device was covered with 10-100-nm thick silicon [8]. In 2019, [PERSON] et al. measured the capacitance gradient between the tip and the sample through electrostatic force [9]. It was an instrument that could not only image the surface, but also image the internal inhomogeneity of semiconductors or dielectric materials. It was considered to be a promising tool for high-resolution and high-precision analysis of doping concentration and dielectric properties in semiconductor material. In recent ten years, people had made great efforts to achieve this goal. Then, by using image method, they established the theoretical mode, and deduced the permittivity of the sample [10]. There were previous applications of measuring the dielectric properties of films by means of SCM or electrostatic force combined with a theoretical model. However, there was no direct method to measure the relative permittivity \(\varepsilon_{r}\) of dielectric films by SCM. Furthermore, a variety of studies have been conducted, but the imaging techniques of \(\varepsilon_{r}\) for thin film materials with SPM, especially for photoelectric materials, had not been fully researched until now. In this paper, the experiment and numerical fittings were used to directly measure the \(\varepsilon_{r}\) distribution of the film by using SCM at room temperature under an atmospheric condition. This method combined the quantitative measurement by using SCM with the capacitance gradient-distance fitting curve to obtain the two-dimensional (2D) image of \(\varepsilon_{r}\). There were four steps in the method as following:  
Step 1. A capacitance gradient \(\frac{\text{d}C^{total}_{\text{total}}}{\text{d}z}\)-distance \(z\) curve was acquired on a metal ground. The capacitance gradient was the gradient of the capacitance between the tip and the metal ground. The distance was the tip-metal ground distance. Then, the effective tip parameters (such as tip radius \(R=R_{0}\), one half angle \(\theta=\theta_{0}\), and stray capacitance constant \(k^{metal}_{stray}\)) could be determined by means of fitting [11; 12]. Step 2. A capacitance gradient \(\frac{\text{d}C^{total}_{\text{total}}}{\text{d}z}\)-distance \(z\) curve was acquired on a monocrystalline silicon (Si) substrate ground. The capacitance gradient was the gradient of the capacitance between the tip and the Si substrate ground. The distance was the tip-Si substrate ground distance. The curve was fitted with \(R=R_{0}\) and \(\theta=\theta_{0}\), the stray capacitance constant \(k^{ground}_{stray}\) could be determined. Usually, there were \(k^{ground}_{stray}\
eq k^{metal}_{stray}\), and \(k^{ground}_{stray}\approx k^{dielectric}_{stray}\), where \(k^{dielectric}_{stray}\) was the stray capacitance constant when an insulating film was coated on the Si substrate ground. Step 3.