Contact scaling is a major challenge in nano complementary metal–oxide–semiconductor (CMOS) technology, as the surface roughness, contact size, film thicknesses, and undoped substrate become more problematic as the technology shrinks to the nanometer range. These factors increase the contact resistance and the nonlinearity of the current–voltage characteristics, which could limit the benefits of the further downsizing of CMOS devices.

where W is the workfunction of the metal, kT is the thermal voltage, and A* is the effective Richardson constant. Schottky later modified the expression to include the effect of an electric field, and its final form is now usually expressed in terms of an applied voltage, V, as follows [32,33][16][17]:
where q is the electron charge, and ΦSB is the Schottky barrier.
In the original proposal by Schottky and Mott [32[16][17],33], the Schottky barrier is as follows”



where is the χ mean barrier height value between the insulator and the semiconductor and δ is the tunneling oxide thickness. It can define a new effective Richardson constant as follows:
Note that the current conduction is, in fact, dominated by the direct tunneling of the carriers over the thin insulating layer. The approximation of the Schottky equation involves modifying the Richardson coefficient with a transmission coefficient, which depends on the barrier height between the silicon and the insulator and the insulator thickness. These two factors are the key parameters that govern the conduction current. In addition, some effects, such as the minority carrier injection, image force, and quantum mechanism reflection, are also neglected. This approximation is only valid for a forward current with V > 3kT/q, because the metal/insulator barrier differs from the silicon/insulator barrier. Some authors have ignored these assumptions and treated the MIS diode as a regular Schottky diode.
Instead of treating the n factor as an “ideality factor” in (8), it was proposed that the nonunity n factor could be calculated from the surface potential of the barrier [42][26]. Defining n as a change in the surface potential, ΔVs, with respect to the applied voltage (i.e., 𝑛=−𝑉/Δ𝑉𝑠), and further taking the interface states into consideration, the electrostatic calculation leads to the following:
where εi and εs are the permittivities of the insulator and semiconductor, respectively; W is the depletion layer in the semiconductor surface; and Ds and Dm are the surface state densities in the equilibrium with the semiconductor and metal, respectively. If the surface state is negligible, (10) reduces to the following:
Thus, large n values, in some cases, are not mysterious; it is only because the electrostatic thickness of the insulating layer, 𝛿/𝜀𝑖, is much larger than the electrostatic thickness of the depletion layer width, 𝑊/𝜀𝑠. If neglecting the surface states on the metal side, which is the case for most MIS diodes, Equation (10) reduces to the following:
In this expression, n is a parameter that describes how effectively the barrier surface potential varies with the applied voltage. It has nothing to do with the generation and recombination of carriers or the validity of the Schottky current model. Therefore, it is not an indicator of the ideality of the junction.
In fact, the validity of (8) for the MIS diode is limited. The conduction current should be more precisely modeled by the direct tunneling current (see Equation (13)) if the insulator layer and barrier height are in the direct tunneling range [43,44][27][28].
where mi is the electron mass in the oxide; ℏ is the reduced Planck’s constant; and ΦB is the barrier height (in electron volts) between the emitting electrode and the oxide.
For the case of thicker insulators or with a smaller voltage applied, tunneling over the triangular edge (see Figure 32c) is also possible. In this situation, the current conduction is better described with the Fowler–Nordheim (FN) equation, as shown below [45][29]:
where me is the electron mass in the free space.
The FN formula has been widely used to explain the conduction behavior of various thin dielectric films [45][29]. However, in contact studies including the so-called MIS Schottky diode, the current–voltage characteristics were often fitted with the modified Schottky Equation (8) or even the simple Schottky equation in (7) instead of the FN relationship. It has to be pointed out that the barrier height extracted from these fitting should be inaccurate, and the ideality factor does not carry any technical implications. The effective Richardson coefficient is also inappropriate in these cases. The current–voltage characteristics are better described with the FN equation.



The unavailability of a heavily doped contact results in a wider barrier. As shown in Figure 75a, this reduces the tunneling efficiency. Following the method developed by Yu [47][31], it can calculate the contact resistance depending on the dopant concentration. Figure 86 shows the contact resistance as a function of the dopant concentration. For high dopant concentrations (>2.5 × 1019 cm−3), the contact resistance decreases to the range of 10−5 to 10−6 Ω-cm2. The current conduction is mainly due to field emission.



