# Passive Intermodulation at Contacts of Rough Conductors
## 2 Electrical Nonlinearities of Conductor Joints
### Contact and Constriction Resistances  
Contacts of conductors with rough surfaces contain the MIM and metal-to-metal junctions. Their resistance depends on the number of touching asperities, size of each contact spot and thickness of an insulating layer in MIM junctions [60]. The mechanical and thermal deformations of individual asperities determine an overall size of the contact area and its contact resistance. The resistance \(R_{c}\) of a single contact spot with an equivalent radius \(a\) is usually approximated as [61]  
\[R_{c}=\frac{\rho_{c}}{2a}\left[f\left(\frac{\lambda}{a}\right)+\frac{8\lambda }{3\pi a}\right] \tag{1}\]  
where \(\rho_{c}\) is an average electrical resistivity of a pair of contact asperities, \(\lambda\) is an electron-free path and \(f(\lambda/a)\) is an interpolation function describing a contribution of Maxwell resistance \(R_{M}=\rho_{c}/2a\). The approximation of \(f(\lambda/a)\) was proposed in [61] and has a maximum error less than 1% at any \(\lambda/a\)  
\[f(\lambda/a)=\frac{1+0.83\cdot\lambda/a}{1+1.33\cdot\lambda/a} \tag{2}\]  
The values of \(f(\lambda/a)\) vary in a relatively narrow range between \(f(\lambda/a)\approx 0.624\) at \(\lambda>>a\) and \(f(\lambda/a)\approx 1\) at \(\lambda<<a\). The second term in (1) represents Sharvin resistance [62]: \(R_{S}=\frac{4\rho_{c}\lambda}{3\pi a^{2}}\) that is associated with the collision-free motion of charges. It plays an important role when the size \(a\) of a contact spot is smaller than \(\lambda\). Then Sharvin resistance \(R_{S}\) can exceed Maxwell resistance \(R_{M}\) and its contribution to the constriction current becomes significant.  
Current constriction by asperities is an inherent feature of the contacts of conductors with rough surfaces [50, 54, 63]. The constriction resistance is determined by the number of the compressed asperities, sizes of their contact spots and thicknesses of the insulating layer. The constriction current decreases when the size of the contact spot is larger than the mean free path of electrons, \(a>\lambda\), as evident from (2). Then the charge transport is predominantly diffusive and is determined by Maxwell resistance \(R_{M}\), defined in (1). In high-quality conductor contacts, e.g., in MEMS, the constriction current is much smaller than the conduction current and is usually combined with the conduction or tunnelling currents [64].  
The effective resistance \(R_{u}\) of the whole junction surface containing \(M\) asperities can be averaged and approximated as suggested in [50]  
\[R_{u}=\left(\sum_{m=1}^{M}R_{cm}^{-1}\right)^{-1}\approx\frac{\rho_{av}}{2a_{ eff}}\left[f\left(\frac{\lambda}{a_{eff}}\right)+\frac{8\lambda}{3\pi a_{eff}}\right] \tag{3}\]  
where \(M\) is the number of compressed asperity pairs, and \(R_{cm}\) is defined by (1) and describes the contact resistance of \(m\)th pair of asperities. It was suggested in [50] that \(R_{u}\) could be evaluated with (1) where \(a\) and \(\rho\) are replaced by an effective radius \(a_{\mathit{eff}}\) and average resistivity \(\rho_{av}\). Such an approximation given in (3) proved to be fairly accurate when the contact area is large at \(a_{\mathit{eff}}\)\(>\)\(\lambda\). Then the contribution of Sharvin resistance is small and deviations of resistivities \(\rho_{cm}\) from their average value \(\rho_{av}\) remain small.  
The contact resistances \(R_{cm}\) of individual contact pairs and the whole ensemble, \(R_{u}\), are the main electrical parameters used for characterising the performance of MEMS switches. They include the effects of both the electrical properties of the contact surfaces and the mechanical deformations. The electro-mechanical models of the contacts in the MEMS switches have been discussed in detail in [50, 51, 52, 64].