# Passive Intermodulation at Contacts of Rough Conductors
## 2 Electrical Nonlinearities of Conductor Joints
### Nonlinearity of MIM Contacts  
The tunnelling current in MIM contacts subjected to RF power is the main source of their fast nonlinearity [22, 23, 24]. When a thin insulating film of a few nanometre thick separates asperities of rough conductors, the tunnelling current flows through. The timescale of the tunnelling effects in the conductor junctions allows an efficient interaction of the high-power RF carriers with free charges. However, when the thickness of the insulating layer exceeds the free path of electrons, charges cannot penetrate the potential barrier and the tunnelling current rapidly decays. The tunnelling current becomes practically negligible at the insulator between asperities thicker than 10 nm. The charge tunnelling between two _smooth_ conductor surfaces was studied first by [PERSON] and [PERSON] in the cases of the electric bias being either very weak or very strong. [PERSON] has extended their model to the range of intermediate voltages [22] to characterise the tunnelling effect at a moderate electric field applied to MIM junction. For this purpose, [PERSON] introduced an additional corrective term into the transition function, which determined the likelihood that electrons can pass through a thin insulator layer. However, [PERSON] showed that [PERSON]'s approximation was valid for the symmetric structure only and failed in asymmetric cases. [PERSON] analysed a general case of a potential barrier, assuming smooth variations of the barrier height and fairly small changes of its average value [23]. In the widely accepted [PERSON] model, the current density \(J(V_{g})\) in MIM junction is represented as  
\[J\big{(}V_{g}\big{)}=J_{0}\Big{[}\overline{\varphi}e^{-A\sqrt{\overline{ \varphi}}}-\big{(}\overline{\varphi}+eV_{g}\big{)}e^{-A\sqrt{\overline{\varphi }+eV_{g}}}\Big{]} \tag{4}\]  
where \(J_{0}=\frac{e}{2\pi\hbar(\beta\Delta s)^{2}}\) and \(A=\frac{4\pi\beta\Delta s}{\hbar}\sqrt{2n}\); \(\overline{\varphi}\) is an average height of potential barrier inside an insulator layer, \(V_{g}\) is the voltage between the contact conductors, \(e\) is the electron charge, \(h\) is the Planck constant, \(\beta\) is the correction factor used in the approximation, \(m\) is the electron mass, and \(\Delta s\) is an effective thickness of insulator film which is usually smaller than the actual thickness. The value of the parameter \(\beta\) is close to 1, and at insulator thicknesses \(\Delta s\)-4-5 nm, the error of the approximation \(\beta\) = 1 is less than a few percent [23]. The accuracy of the Simmons model (4) of MIM junctions was recently examined for thin insulator films in [26, 27, 28, 29]. The simulated characteristics of \(J(V_{g})\) in the MIM junction are shown in Figure 3 for an insulating film with a thickness of 1 nm. Comparison of the Simmons model [23] with the self-consistent model (SCM) [26] demonstrates their good correlation. The Simmons model also proved to be fairly accurate when the insulator thicknesses varied between 1 nm and 10 nm. It was also found that the equivalent resistivity \(V_{g}/J\) inferred from (4) is slightly overestimated at sub-nanoscale \(\Delta s\) and underestimated at \(\Delta s\) > 5-7 nm. The uncertainty of the Simmons model in these cases increases to a few percent. The effect of current crowding at the conductor edges and the nonuniform current distribution at the metal plates of partially overlapped parallel MIM contacts were modelled in [27, 28, 29]. The nonlinear tunnelling resistivity of MIM junction, defined as the ratio of voltage to current density, was evaluated in dependence of several parameters such as contact length, permittivity and thickness of an insulating layer and the sheet resistance. It was observed that at the sub-nanometre insulator thicknesses 0.5 nm and 0.65 nm, the contact resistivity reduced and remained nearly linear at low applied voltages (\(v<0.3\) V).