# Passive Intermodulation at Contacts of Rough Conductors
## 2 Electrical Nonlinearities of Conductor Joints  
PIM at contacts of conductors with rough surfaces is a nonlinear multiphysics process with several distinct time scales, as illustrated in Figure 1. The fastest nonlinearities in contacts of good conductors are associated with the _electrical effects_ of charge tunnelling and current constriction at the MIM junctions [17]. The charges are funnelled through the contact spots of rough surfaces as shown in Figure 2a. The current magnitude depends on the size of a contact spot and thickness of an insulating layer, which are determined by the applied pressure, local temperature and deformations of the surface asperities as illustrated by the links in Figure 1. The high-power carriers are mixed and modulated at the contact nonlinearities due to variations of the contact size, temperature and resistance [27, 38, 39, 40]. The thermal effects and mechanical deformations develop much slower than the oscillations of the RF carriers and the tunnelling current. However, both fast and slow nonlinearities remain intrinsically coupled to each other despite their different time scales.  
Figure 1: Main types of contact nonlinearities of rough conductor surfaces. Links between different mechanisms of nonlinearities are colour coded in relation to the primary source.  
Figure 2: **(a)** Sketch of contacts between conductors with rough surfaces. Tips of the collided asperities are deformed and the constriction current is funnelled through the contact areas. Only the displacement current (broken lines) is shown at the non-contact asperities. **(b)** Equivalent circuit of an asperity pair: \(R_{v1}\) and \(R_{v2}\) are the resistances of bulk conductors outside asperities; \(R_{cm}\) and \(C_{cm}\) are the resistance and capacitance of either the collided asperities or a pair of isolated asperities.  
Current flow through the contact asperities of rough surfaces can be described by the equivalent circuit shown in Figure 2b, where \(R_{v1}\) and \(R_{v2}\) are the resistances of solid conductors outside of the asperities. The resistance and capacitance of a pair of compressed or non-touching asperities are represented by \(R_{cm}\) and \(C_{cm}\). The linear resistance \(R_{cm}\) of the thin oxide film is very high even in contacts of good conductors. For example, Al\({}_{2}\)O\({}_{3}\) film of thickness \(\delta s\) = 1 nm, area 10 \(\upmu\)m\({}^{2}\) and the macroscopic resistivity \(\rho_{\text{Al2O3}}=1\times 10^{14}\)\(\Omega\cdot\)cm has its Maxwell resistance at \(R_{cm}=10^{14}\)\(\Omega\). Such a high value of \(R_{cm}\) is typical for good insulators and suggests that the charge tunneling must be responsible for the resistance of conductor junctions with thin insulating films. It is also necessary to note that the capacitive reactance \(1/j\omega C_{cm}\) of the conductor joint can noticeably influence the contact impedance. Indeed, the reactance of this junction is \(1/j\omega C_{cm}=-\beta 89.9\)\(\Omega\) that affects the impedance of the contact joint. Therefore, the resistance of the electrical contacts of rough conductors and their basic properties have to be examined in more detail.