# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling  
The tunneling category represents the quantum mechanism that describes the transition of charge carriers based on the Schodinger wave equation, which can be written in its generic form as follows:  
\[i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2} \Psi}{\partial x^{2}}+V\Psi, \tag{2}\]  
\[\hat{H}=\hat{T}+\hat{V} \tag{3}\]  
Figure 5: Categorization of the theory of operation.  
In (2), \(\hbar\) stands for [PERSON]'s constant divided by \(2\pi\), \(m\) is the particle effective mass, \(i\) is the imaginary number, \(V\) is the potential energy the particle has and \(\Psi\) is the wave function. The \(\hat{H}\) shown in (3) represents the Hamiltonian operator, which consists of both the static energy \(V(\mathbf{r},t)\) and the kinetic energy operator represented by \(\hat{T}\). Although we will not attempt any theoretical work regarding these equations in this paper, it is important to understand some of their implications because they impact the adopted models for simulation as will become clear later. The simulation approaches may be divided, based on (2), into simulations for a time-independent wave function and a time-dependent wave function; which correspond to the no illumination and the under illumination cases shown in Figure 5, respectively.  
For this category and regardless of the tunneling theory, the procedure is the same such that the Schrodinger equation is used to calculate the transmission probability of the electron, and then, the current density can be found from multiplying this probability by the number of available electrons. This theoretical framework is well explained in [44; 45]. The reader, while reading other literature, might encounter tunneling types such as Fowler-Nordheim tunneling (FT), resonant tunneling (RT), step tunneling (ST), direct tunneling (DT), etc. These are _types_ that obey the same tunneling theoretical framework and rely on insulators thicknesses, biasing voltage and potential barrier height. So, we prefer not to explain them discretely as they can happen in any tunneling diode based on the aforementioned parameters. The current is then simply the current density multiplied by the contact surface area. Although many models adopt this approach, [PERSON] et al. [46] argue that the tunneling effect happens at the surface edges rather than the center part of the junction. The experiment result of [PERSON] et al. [28] as shown in (Figure 29b) of their work supports this assumption where the polarization-dependent response component \(V_{p}\) seems to level out with increasing the contact area in an asymptotic approach.  
Following Figure 6, the diagram shows the energy band of one type of tunneling diode, the metal-insulator-metal (MIM) diode; when the diode is forward biased, the electron will tunnel from the higher barrier region identified as \(\Phi_{L}\), to the lower barrier region identified as \(\Phi_{R}\). This is an asymmetric diode where the barrier heights are different at both ends, and this happens when two metal types are used. Conversely, a symmetric diode utilizes electrodes of the same metal at both ends. It is important to notice that the tunneling will occur anyway, with or without an illuminated junction, as long as there is enough forward bias represented as displacement between Fermi levels of the left and right regions by \(eV_{D}\) and in the same direction shown in the figure. Illumination of the junction would affect the tunneling phenomenon differently as experimentally represented [40].  
Figure 6: Energy band diagram of a metal-insulator-metal (MIM) tunneling diode (Adapted with permission from [47], [PERSON], 2012).