**3.3 Shallow trap-controlled carrier mobility and conductivity**

The surface potential experimental results of samples charged by negative corona discharging and positive corona discharging as a function of time are shown in **Figure 5** [29]. Negative and positive charges are deposited on the surface of polyimide, and electric field is established inside the polyimide during the charging process. After charging, surface charges are injected into polyimide, and the migration of charges toward the grounded electrode in the bulk leads to the decay of surface potential. The decay rate of surface potential varies before and after the injected charge carriers flow out of the dielectric material, as shown in **Figure 5** [29]. The time when the front charge carriers arrive at the grounded electrode is defined as transit time *t*T. The transit point existing at the beginning of the potential decay curve can represent the mobility of carriers controlled by shallow traps [29]. In order to obtain the transit time *t*T, the potential decay results were fitted by an exponential function, and then we obtained the relation between *t*d*ϕ*s/dt and *t*. The time corresponding to the peaks can be regarded as transit time *t*T, and it is used to calculate carrier mobility controlled by shallow traps according to the following Eq. (1) [29, 32]:

$$
\mu\_{\rm 0(e,h)} = d^2 / \phi\_{\rm s0} t\_{\rm T} \tag{1}
$$

Here, *μ*0 is the carrier mobility controlled by shallow traps in m<sup>2</sup> V<sup>−</sup><sup>1</sup> s −1 , *d* is the thickness of sample in m, and *ϕ*s0 represents the initial surface potential in V. The subscripts have the following meaning: (*e*) for electrons and (*h*) for holes. By calculation, the hole and electron mobilities controlled by shallow traps are 1.80 × 10<sup>−</sup><sup>14</sup> and 3.67 × 10<sup>−</sup>14 m2 V<sup>−</sup><sup>1</sup> s −1 , respectively.

For studying the surface and volume charge transportation properties, the isothermal surface potential decay (ISPD) experiment on space-grade polyimide was carried out at various temperatures from 298 to 338 K. In high vacuum, the charge was accumulated on the surface of polyimide under low-energy electron irradiation. After irradiation, the charge was transferred to the grounding electrode on the surface through the volume. Through the three parameters of surface resistivity, volume ohmic resistivity and charge carrier mobility, the leakage rate of electron was determined. And the three parameters were revealed by a two-dimensional ISPD model established by using genetic algorithm (GA), as shown in **Figure 6** [33].

As shown in **Figure 6**, the carrier mobility increased with temperature, while the surface resistivity and volume ohmic resistivity of polyimide decreased with temperature. The surface resistivity, volume ohmic resistivity and carrier mobility were obtained. For example, at 298 K, they were 1.02 × 1019 Ω, 2.87 × 1017 Ω m and 1.49 × 10<sup>−</sup>19 m2 /V s, respectively. The calculated errors were all not more than 0.9%, which showed that there was a good consistency between the experimental and simulated 2D ISPD results [33].

### **3.4 Electrical breakdown properties**

The influencing mechanism of sample thickness on electrical breakdown of polyimide is not very clear until now. The dc electrical breakdown fields of polyimide films with different thicknesses from 25 to 250 μm were measured using a computer-controlled voltage breakdown test device. The dc electrical breakdown experiments were carried out under 30°C using spherical copper electrodes with a diameter of 25 mm in transformer oil. The rate of dc voltage increase is 1 kVs<sup>−</sup><sup>1</sup> . For

### **Figure 5.**

*Surface potentials of polyimide charged by negative corona discharging and positive corona discharging as a function of time at room temperature [29].*

### *Charging and Discharging Mechanism of Polyimide under Electron Irradiation and High Voltage DOI: http://dx.doi.org/10.5772/intechopen.92251*

each thickness of the sample, at least 15 times breakdown tests are performed. The average value of all data is taken as the breakdown electric field of the sample.

**Figure 7** shows the experimental results of the dc electrical breakdown field of the polyimide film, *Fb*, as a function of thickness, *d*, at room temperature [29]. As shown in **Figure 7(a)**, the dc electrical breakdown field of polyimide films decreases with an increase in sample thickness. In addition, the derivative of dc electrical breakdown field with respect to sample thickness d*Fb*/d*d* decreases with the increase in sample thickness. The relation between the dc electrical breakdown field and sample thickness looks like an inverse power function. Accordingly, we change the linear coordinates in **Figure 7(a)** to double logarithmic coordinates in **Figure 7(b)**. It can be seen from **Figure 7(b)** that the dc electrical breakdown field of polyimide is linear with sample thickness under double logarithmic coordinates [29].

The influence of sample thickness on polymer breakdown can be explained by electron avalanche breakdown, electromechanical breakdown, free volume breakdown and space charge modulated electrical breakdown [29, 34]. In electron avalanche breakdown, the energy gain of electron can be obtained by free electron movement in the conduction band of dielectric material under the action of electric field [34]. When the energy exceeds the band gap energy, the electrons in the valence band may be excited to the conduction band, resulting in the chemical bond breaking. The avalanche effect is caused by further collision and ionization of the released electrons with other matrix atoms, which results in the doubling of local current and finally triggers the breakdown. The electric breakdown field decreases

### **Figure 6.**

*Arrhenius plot of surface resistivity (a), volume ohmic resistivity (b), and charge carrier mobility (c) of polyimide. The linear fitting errors, R2 , were respectively 0.959, 0.986 and 0.991 from (a) to (c) [33].*

### **Figure 7.**

*Experimental results of dc electrical breakdown field of polyimide at various thicknesses in linear coordinates (a) and in double logarithmic coordinates (b) [29].*

### *Polyimide for Electronic and Electrical Engineering Applications*

with the increase of sample thickness, which is due to the critical number of electrons produced in the whole sample thickness by collision ionization [29, 34]. The Stark-Garton model of mechanical and electrical breakdown has been widely used to predict the breakdown strength of thermoplastics, while the Young's modulus and dielectric constant of temperature-sensitive polymers determine the mechanical and electrical breakdown strength [34]. Because the thickness of the sample determines the electrostatic compressive stress and the opposite elastic stress produced by the electrostatic attraction of the two electrodes, the electric breakdown field is a decreasing function of the sample thickness. In the theory of free volume breakdown, it is assumed that the electric breakdown field of polymer depends on the longest mean free path of electron. Electrons are accelerated in the free volume, and their average free path depends on the maximum length of the free volume. When enough energy is obtained by electrons in the free volume to overcome the potential barrier, the local current will be multiplied, so that the material is heated to a very high temperature and finally causes the phenomenon of electrical breakdown. From a statistical point of view, the longest free path is a function of sample size, so the electric breakdown strength is related to sample thickness.
