# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Geometric Devices  
While most of the THz-rectifying diodes' behavior can be explained with quantum physics alone, geometric diodes seem to have comparably accurate simulation results, using the classical Drude model, to those obtained through quantum simulations without their complexity [72]. Geometric diodes are structurally different from other types and have some beneficial modeling features as will be explained. These devices depend on physical asymmetry to simplify the movement of charge carriers into one direction more than the other. They are primarily ballistic devices that use long mean-free path length (MFPL) material to achieve this goal [64]. The recent development in the graphene industry and its properties has fostered its use for this goal such that a tapered graphene structure, as that shown in Figure 8, would facilitate the movement of electrons from the wider side to the narrower side. Technically, graphene devices have a very special feature compared to metals and semiconductors due to the linear energy dispersion relation near the Dirac point [73]. Modeling a device with this special energy and carrier transport features has also many approaches. However, there seems fewer theoretical frameworks and little agreement between experiment results and theoretical models' output. Therefore, it is worth reviewing some model types that witnessed many trials, whereas their outputs have been benchmarked against practical results. Dirac equation:  
This equation was developed by [PERSON] [75] based on the known wave equation to account for the relativistic particles. [PERSON] and [PERSON] [74] develop a model for the \"2 DEG (2 DEG stands for two-dimensional electron gas) nanodevice with a linear tapered geometry working in the ballistic regime, where the transport of charge carriers is described by either the Schrodinger or the Dirac equation.\" Their model is based on Figure 8 where the device is sliced up into N cells such that the Dirac equation for each cell is given by (18). \[ \begin{split}\Psi_{j}(x,y)&=\sum\limits_{n=1}^{N} \left[A_{n,j}exp\left(ik_{n,j}x\right)\right.\\ &+\left.B_{n,j}exp\left(-ik_{n,j}x\right)\right]sin\left(2n\pi \gamma/d_{j}\right),\end{split}  \tag{18}\]  
In a similar style to other models, \(A_{n,j}\) and \(B_{n,j}\) are the incoming and outgoing wave components in slice \(j\); \(k_{n,j}\) is the wave vector component and can be found by the knowledge of the potential \(v\) and particle energy \(E\). Working out some math and replacing the \(j\) with the incident and outgoing indices in the following equation, the transmission probability \(T\) for slice \(j\) can be obtained from (19). Based on the knowledge of transmission probability, we can find the expected current value. \[T=\frac{\sum\limits_{n=1}^{N_{out}}k_{n,out}\left|A_{n,out}\right|^{2}}{\sum \limits_{n=1}^{N_{in}}k_{n,in}\left|A_{n,in}\right|^{2}}, \tag{19}\]  
Although this method accounts for relativistic occurrences, the output results do not agree with measurements. This has been reported by [PERSON] et al. [76] where they reported that the  
Figure 8: Schematic representation of the ballistic rectifier (Reproduced with permission from [74], IOP Publishing, 2013). rectification region is less compared to that indicated by the simulation, and the measured current is a bit higher. Monte Carlo simulation:  
This simulation assumes electron movement and collision in geometric diode according to Drude model. In his work, [PERSON] (2017) has developed a model where the electron collides with the structure boundaries, phonons, other electrons, etc., such that the electron moves within the device with velocity \(v_{tot}\), which is a composite of Fermi and current drift velocities. [PERSON] (1966) has also tried the Monte Carlo method for the simulation of his model. Although there have been some important results from the work of [PERSON] (2017), no agreement has been obtained between the modeled and measured current values.