# Passive Intermodulation at Contacts of Rough Conductors
## 4 Mechanical Nonlinearities at Conductor Contacts
### Contact Area Expansion  
Pressure, applied to the contacts of conductors with rough surfaces, causes asperity deformations. The compressed asperities expand laterally, and their contact areas increase with the applied pressure and mechanical stress, depending on the material stiffness and surface coatings. When exposed to the RF power, the contact spots are heated due to conductor losses and the compressed asperities spread gradually to reduce their strain. The thermal expansion and softening of the contact spots also cause plastic deformations and creep of the colliding asperities. These effects were examined in [71] for RF MEMS with frustoconical contact asperities. At a small radius \(r_{1}\) of the undeformed frustum tip, contact radius \(r_{c}(t)\) of the compressed asperity varies with time \(t\) and is approximated as [71]  
\[r_{c}(t)=\left[r_{1}^{1/\alpha}+t\frac{LC}{\alpha\tan\beta}\left(\frac{F}{ \pi}\right)^{(1/\alpha-1)/2}\exp\left(-\frac{Q_{c}}{kT}\right)\right]^{\alpha} \tag{14}\]  
where \(L\) is asperity initial height, \(C\) is a constant dependent on the contact material and creep mechanism, \(\beta\) is a slant angle of the asperity frustum, \(F\) is contact load, \(Q_{c}\) is the activation energy for creep, \(\alpha=1/(1+2p)\), and \(p\) is the stress exponent defined in (12). The value of \(\alpha\) is specifically related to the creep coefficient, and it is fitted to the experimental data. Values of \(\alpha\) are usually small, and \(\alpha<0.1\) in the example presented in [71].  
Maxwell contact resistance \(R_{M}\) of conical frustums with the time-dependent radius \(r_{c}(t)\) and resistivity \(\rho_{c}\) of the contact spot can be calculated with (1). The simulated and measured time-dependent resistances \(R_{M}(t)\) are shown in Figure 8. Approximated by (1) with constant \(\rho_{c}\) and the time variable contact size \(r_{c}(t)\) defined in (14), \(R_{M}(t)\) fits the measurement results very well when the switch is closed the first time. When the contacts are closed for the second and third times (middle and top plots), the measured resistance \(R_{M}(t)\) is flatter than the calculated ones at the first \(\delta t=4\) s and \(\delta t=9\) s, respectively. But \(R_{M}(t)\) is well correlated with the simulations later on. It is also necessary to note that Sharvin resistance showed no visible effect on the contact resistance and its time variations here. This is a manifestation of the incommensurate time scales of the charge tunnelling effects and the mechanical deformation in the contacts of conductors with rough surfaces.  
Figure 8: Symbols are the experimental data; solid lines are the fit of equation \(\rho=At^{-a}+B\) to each entire data set used in [71]. Reproduced from [71] with the permission of AIP Publishing.  
The dynamics of MEMS contacts was alternatively examined in [51; 52] where the contact area was evaluated in the frequency domain. This approach required less computational resources but its accuracy was lower, and it did not allow the analysis of the temporal variations of the contact area and its resistance. These works were applicable to the very slow contact deformations that allowed calculations of the size of the expanded contact area, which determined the contact resistance only.