# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.1 Without Illumination  
When a charge carrier electron is expected to traverse through a tunnel, the likelihood of this occurrence is a function of the wave equation denoted in (2). However, the oscillating potential energy component of the electron imposed by the outside voltage will be independent of time if no outside photon (illumination, in other words) is encountered across the barrier. This is usually referred to as no illumination or the dark current situation. It would also yield a simplified solution to (2) since \(V(\mathbf{r},t)\) becomes \(V(\mathbf{r})\). Having said that, finding an accurate solution to the Schrodinger equation has always been a challenge, and there are many proposals with different features that have been tried so far to approach an accurate solution to it. Following is a list of the commonly-used approaches in the literature:  
Simmons model:  
This model has been proposed by [PERSON] [44] to find the \(I(V)\) curve of a diode consisting of a thin insulating film between two metal electrodes. In his paper, [PERSON] assumes a potential barrier that takes into account the image forces [48] and formulates its approximation using a continuous function. The resulting current of the applied voltage is calculated by finding the probability of an electron passing through the tunnel, using the Wentzel, Kramers and Brillouin (WKB) approach [49], and multiplying it by the number of the available free electrons. The likelihood function is developed form the Schrodinger equation and is shown in (4) assuming that the electron transfer happens only in the \(x\) direction; where \(h\) is [PERSON]'s constant, \(m\) is the electron's effective mass, \(V(x)\) is the potential difference between the two electrodes and \(E_{x}\) is the energy component of the incident electron in the \(x\) direction. The presence of the image forces lowers the potential barrier as shown by the dashed curve in Figure 6. The intersection between this curve and the higher Fermi level of the cathode identifies the potential tunnel width \(\Delta S=S_{2}-S_{1}\) which is smaller than the _physical_ tunnel width of the trapezoid--\(L\) in Figure 6--which is the insulator thickness; thus facilitating the tunneling of electrons. The presence of the image forces is largely affected by the dielectric value of the insulating material such that lower values lead to lower barrier heights. Thus, when a higher response is needed, it is favorable to use insulators with a low dielectric value. \[D(E_{x})=exp\left\{-\frac{4\pi}{\hbar}\int_{s_{1}}^{s_{2}}\sqrt{2m(V(x)-E_{x} )}\,dx\right\}, \tag{4}\]  
The major drawback of this model is that it leads to inaccurate current values many orders of magnitude lower than the experimental results. This has been noted by [PERSON] et al. [50] where the metal work function \(\Phi_{M}\) values used in the model to give comparable results to measurements are lower than those found by direct physical measurements. However, curve fitting may by used with the model equation and leads to good agreement with measurements. Moreover, the value difference between the fitted work functions is equal to that of measured values. The reader may consult [50] (Table I-II) for further information. There have been many trials to improve the performance of the [PERSON] model. In his Master's thesis, [PERSON] [51] tries a rigorous mathematical framework to improve the performance of this model, but the resulting model also needs curve fitting to experimental results; in which case, there is little point to optimizing this model. The reason for this behavior, as suggested by [PERSON], is the unphysical divergence of the barrier at the metal surface. Albeit that the model he presents overcomes the discontinuity problem of the potential barrier curve between the metal-insulator interfaces, it still needs curve fitting. [PERSON] and [PERSON] [47] state that the WKB approach used by the [PERSON] model gives different results with respect to other approaches at lower barrier heights.