# Compact Model for L-Shaped Tunnel Field-Effect Transistor Including the 2D Region
## 2 Model Development
### 2D Model: \(C_{\text{nonoverlap}}\) Model  
The 2D model is based on the work in [3]. The following assumptions are used in this model: (1) Electric fields are assumed to be completely circular and terminate from gate to source. This helps in obtaining a convenient expression for the tunneling length (\(W_{\text{t}}\)); (2) Gate dielectric is treated as the same material as the channel with an equivalent dielectric thickness \(t^{\prime}_{\text{ox}}\) given by \(t^{\prime}_{\text{ox}}=t_{\text{ox}}\varepsilon_{\text{si}}/\varepsilon_{ \text{ox}}\), where \(t_{\text{ox}}\), \(\varepsilon_{\text{si}}\), and \(\varepsilon_{\text{ox}}\) are the physical dielectric thickness, silicon dielectric permittivity, and oxide dielectric permittivity, respectively. This is necessary to ensure a continuous and perfectly circular electric field from the gate-to-channel/gate dielectric interface, and finally terminate at the source. Without this assumption, the electric field will be discontinuous, that is, not perfectly circular in its path from gate to source. In this work, \(W_{\text{t}}\) is conveniently calculated as the length along the perfectly circular electric field arc, from gate to source, as will be shown below. If a discontinuity arises in the circularity of the electric field arc, such convenient calculation of \(W_{\text{t}}\) will not be possible; (3) The source is assumed to be completely depleted, and depletion length is ignored. The source is heavily doped. Depletion length is inversely proportional to doping concentration [3]. This makes the source depletion length negligible. Including the source depletion length would add complexity to the model without significantly increasing the accuracy of the model; (4) The source is assumed to be touching the gate. Table 1 mentions most of the symbols used in the equations below. With these assumptions, the boundary conditions at source and gate can be given by  
\[\varphi(x,0)=\varphi_{\text{source}}\ \ \ \text{and}\ \ \ \varphi(0,y)=\varphi_{\text{g}}. \tag{1}\]  
Figure 2: (a) \(G_{\text{tun}}\) along the surface of \(C_{\text{overlap}}\) and \(C_{\text{nonoverlap}}\) (left axis), and \(\varphi_{\text{s}}\) along the channel (right axis) at \(V_{\text{gs}}\) = 0.15 V and \(V_{\text{ds}}\) = 0.5 V. (b) Integrated band-to-band tunneling (BTBT) tunneling rates in \(C_{\text{overlap}}\) (black triangles) and \(C_{\text{nonoverlap}}\) (red circles) as a function of \(V_{\text{gs}}\) (left axis), and \(I_{\text{ds}}\)–\(V_{\text{gs}}\) characteristics of LTFET (right-axis). BTBT threshold voltages for \(C_{\text{overlap}}\) and \(C_{\text{nonoverlap}}\) are indicated by arrows. where \(\varphi_{\rm source}\) is assumed to be 0 V and \(\varphi_{\rm g}\) is the gate-source bias minus the flat band voltage (\(V_{\rm fb}\)), that is, \(\varphi_{\rm g}=V_{\rm gs}-V_{\rm fb}\). The solution of potential in polar co-ordinates is given by  
\[\varphi(x,y)=\frac{2\theta\varphi_{\rm g}}{\pi},\qquad 0\leq\theta\leq\frac{\pi}{2}. \tag{2}\]  
The above boundary condition along with \(\theta_{0}\) is shown in Figure 3a. Thanks to assumptions 1 and 2, \(W_{\rm t}\) is calculated as the length along a perfectly circular electric field line as follows:  
\[W_{\rm t}=r_{0}\theta_{0} \tag{3}\]  
where \(\theta_{0}\) is given by \(\theta_{0}=\pi E_{\rm g}/(2q\varphi_{\rm g})\), (where \(E_{\rm g}\) is the bandgap, and \(q\) is the charge on an electron) and is the angle when the potential difference between the source edge and some point along the electric field line becomes equal to \(E_{\rm g}/q\). \(\theta_{0}\) is obtained by substituting the BTBT condition, that is, \(\varphi=E_{\rm g}/q\) in (2), and inverting it. Since \(\theta_{0}\) is bias-dependent, \(\theta_{0}\) decreases, and \(W_{\rm t}\) decreases as \(V_{\rm gs}\) bias increases. This is illustrated in Figure 3b. \(r_{0}\) is the radius of the electric field arc and is given by \(r_{0}=t^{\prime}_{\rm ox}/cos\left(\theta_{0}\right)\).