# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.1 Without Illumination
\[u(\textbf{r},t)=\Psi(\textbf{r}).e^{-2\pi iEt/h},\] (5)
3. The simplified wave function, which is now only space-dependent, is further divided into two components, according to [PERSON]'s formulas. As shown in (6) for the \(j\)-th cell. \(A_{j}\) and \(B_{j}\) can be determined through the boundary conditions of the wave function, and \(p_{j}(x)\) is a complex function whose value can be determined from ([PERSON], 1968) (6). \[\Psi_{j}(x)=A_{j}.e^{p_{j}(x)}+B_{j}.e^{-p_{j}(x)},\] (6)  
Yet further, the \(A_{j}\) and \(B_{j}\) values of adjacent cells can be related using the matrix relation depicted in (7). Applying this framework for all successive cells will lead to the solution of the considered physical problem. Thus, the tunneling likelihood for an electron \(D(E_{x})\) is found, then the process will be similar to that adopted by [PERSON] (1992). \[ \begin{bmatrix}A_{j+1}\\ B_{j+1}\end{bmatrix} =M_{j}. \begin{bmatrix}A_{j}\\ B_{j}\end{bmatrix}  \tag{7}\]  
After solving this set of equations, the transmission probability of an electron \(T(E_{x})\) can be found from (8), where \(k_{0}=\sqrt{2m_{e}m_{L}eE_{x}}/\hbar\) and \(k_{N+1}=\sqrt{2m_{e}m_{R}eE_{x}}/\hbar\) such that \(m_{e}\), \(m_{L}\) and \(m_{R}\) are the electron effective mass in the insulator, left electrode metal and right electrode metal, respectively. \[T(E_{x})=\frac{k_{N+1}}{k_{0}}\cdot\frac{|A_{N+1}^{+}|^{2}}{|A_{0}^{+}|^{2}}, \tag{8}\]  
The transfer matrix method has gained increasing interest over the last two decades starting from its introduction by [PERSON] and [PERSON] (1992). The algorithm is inherently simple as it uses the widespread TMM concept and applies it to a simplified version of the Schrodinger equation. Even so, its performance is superior to that of WKB in particular for low barrier MIM diodes ([PERSON], 1992). Another added value for this method is its adaptability to simulate multi-insulator diodes as shown by [PERSON] et al. [PERSON] _et al._ (2007). Moreover, the algorithm simulation output is in close agreement with the more computer-intensive algorithms such as the quantum transmitting boundary method (QTBM) ([PERSON], 1992) and the non-equilibrium Green function method (NEGF) ([PERSON] _et al._, 2007), which will be encountered later on. On the other hand, the method has some drawbacks that have been referred to much earlier. [PERSON] and [PERSON] and [PERSON] (1992) state that the accuracy of this method is three to four computer digits using the computer technology of the time. Trials have been performed to improve its accuracy by introducing a linear potential barrier increase inside the cell instead of a constant value [PERSON] _et al._ (2007), as shown in Figure 7b. This can also be referred to as the Airy function transfer matrix method (AF-TMM) [PERSON] _et al._ (2007), which gives identical results to that obtained by the NEGF algorithm as stated by [PERSON] et al. [PERSON] _et al._ (2007) for their model. Another criticism of this method is the dependence on [PERSON]'s approximation such that the wave function envelope approach has been postulated rather than getting derived from the microscopic Schrodinger equation ([PERSON], 1992). Quantum tunneling boundary method:  
Also referred to as QTBM, it was presented by [PERSON] _et al._ (2007). It also relies on a time-independent Schrodinger form. Another simplification is made by assuming that the voltage value does not change along the leads that connect the device to the outside world. There are other assumptions that are imposed to simplify the Schrodinger solution. However, it solves for the equation in a two-dimensional approach, whereas the device is divided into smaller mesh cells, and the wave equation value inside the device is the main target to solve for in the algorithm as shown in (9).