# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.1 Without Illumination
\[\Psi_{0}(\mathbf{r})=\sum_{i}\Psi_{i}(\mathbf{r})\phi_{i}(\mathbf{r}), \tag{9}\]  
where \(\Psi_{0}(\mathbf{r})\) is the wave equation value inside the device, the diode insulator in our case, \(\Psi_{i}(\mathbf{r})\) is the wave function value inside cell iand \(\phi_{i}(\mathbf{r})\) is the \(\mathbf{r}\)-dependent global shape function that takes the value of one inside the cell \(i\) and zero in all other cells to suppress the values of \(\Psi_{i}(\mathbf{r})\) outside their assigned cells. The problem is defined as solving for the vector of unknowns \(\Psi_{i}(\mathbf{r})\), which is done through a system of linear equations as shown in [58] (54). The algorithm has many interesting features. Based on the concept it was built on, it is a multiport simulation algorithm, and so, it is ready for more complicated device shapes with more than one port input. The simulation output can provide more location-based information because the wave function is described by two dimensions. In the simulation performed in [58] (Figure 4a), the figure shows the wave function value surface plot of a rectangular cavity along a quantum waveguide. The authors in [59] provide another development to the algorithm to account for three-dimensional modeling for band-to-band tunneling in a nanowire. Three-dimensional QTBM simulation, also referred to as 3 DQTBM, opens up better flexibility and accuracy options for the method. In spite of the benefits of this algorithm, it is more computer intensive than the predecessor and still time independent as opposed to NEGF as will be shown below. Non-equilibrium Green function:  
This is frequently quoted in the literature as NEGF. The algorithm attempts to solve the wave function, without involving directly the Schrodinger equation, using the following definition for the Green function as shown in (10) taken from [60]. \[(E-\hat{H})G(E)=\mathbb{I}, \tag{10}\]  
Assuming a wave function, the time independent Schrodinger equation can be expressed as:  
\[\hat{H}\Psi(\mathbf{r})=E\Psi(\mathbf{r})+V, \tag{11}\]  
where \(V\) is the small perturbation introduced to help find a solution to this equation according to perturbation theory [61]. Now, the wave function can be found directly from (12):  
\[\Psi(\mathbf{r})=-G(E)V, \tag{12}\]  
The theoretical framework is rigorous and will not be attempted here. The algorithm is frequently quoted for its accuracy [56; 62]. An outstanding feature of this modeling scheme is that it can be applicable to \"uniform illumination\" as stated by [PERSON] and [PERSON] [63] where the oscillating field is expressed as \(V(z,t)=V(t)=V_{0}cos\left(\omega t\right)\) or if it takes a phase-dependent form, such that \(V(z,t)\sim V_{0}\cos\left(\omega t-qz\right)\). More importantly, it is applicable to other devices than tunneling ones such as the geometric diode simulated by [PERSON] et al. [64]. Yet further, the algorithm can account for time-modulated potentials, and so, it becomes extendible for time-dependent equations [63] as shown in Figure 5.  
On the contrary, its complicated math makes it computer intensive since it can be used to describe tunneling, non-tunneling and time-dependent voltage scenarios.