# A Review of Sharp-Switching Band-Modulation Devices
## Operating Principles and Models
### Regional Model  
With the \(Z^{2}\)-FET being a partially gated PIN diode, its I\({}_{\text{A}}\)-V\({}_{\text{A}}\) characteristic can be formulated using the standard PIN diode current model, given by [30]:  
\[\text{I}_{\text{A}}=\text{I}_{\text{Diff}}\bigg{(}\text{e}^{\frac{\text{V}_{ \text{A}}-\text{R}_{\text{PIN}}\text{I}_{\text{A}}}{\text{NP}}}-1\bigg{)}+ \text{I}_{\text{Rec}}\bigg{(}\text{e}^{\frac{\text{V}_{\text{A}}-\text{R}_{ \text{PIN}}\text{I}_{\text{A}}}{\text{NP}}}-1\bigg{)}^{2} \tag{1}\]  
Figure 11: (**a**) Definition of the \(Z^{2}\)–FET electrical parameters and equivalent capacitor circuit. (**b**) Electrostatic potential in gated and ungated regions vs. anode voltage in the OFF and barrier-collapsing (BC) regimes; the results derived from the model and TCAD are in good agreement (adapted from [29]).  
where \(\mathrm{I_{Diff}}\) and \(\mathrm{I_{Rec}}\) respectively stand for diffusion and recombination currents, \(\mathrm{R_{PIN}}\) represents the intrinsic body resistance, \(\beta=kT/q\), and \(\mathrm{N}\) is the diode ideality factor dependent on the operation regime (see [29] for details).  
In the \(\mathrm{I_{OFF}}\) regime, the applied voltage \(\mathrm{V_{A}}\) is small, so the square term in Equation (1) can be neglected. As the drain current in this regime is insignificant, the term \(\mathrm{R_{PIN}}\mathrm{I_{D}}\) can also be ignored. The anode current in the \(\mathrm{I_{OFF}}\) region reduces to:  
\[\mathrm{I_{A_{OFF}}}=\mathrm{I_{Diff_{OFF}}}\bigg{(}\mathrm{e}^{\frac{\mathrm{ V_{A}}}{\mathrm{N\beta}}}-1\bigg{)} \tag{2}\]  
where \(\mathrm{I_{Diff_{OFF}}}\) is a fitting parameter related to the diffusion current.  
In the 'barrier collapse' (BC) regime, \(\mathrm{V_{A}}\) cannot be neglected, but the drain current is low enough and the term \(\mathrm{R_{PIN}}\mathrm{I_{D}}\) is still negligible. The first term of Equation (1) can be dropped, leading to a well-defined expression for the current \(\mathrm{I_{A\text{-}BC}}\):  
\[\mathrm{I_{A_{BC}}}=\mathrm{I_{Rec_{BC}}}\bigg{(}\mathrm{e}^{\frac{\mathrm{V_{ A}}}{\mathrm{N\beta}}}-1\bigg{)}2 \tag{3}\]  
where \(\mathrm{I_{Rec_{BC}}}\) is a fitting parameter connected to the recombination current and \(\mathrm{N}\approx 2\).  
The 'negative differential resistance' regime is too complex for a physics-based model. However, noting the linear variation of the anode current logarithm in Figure 9, we can construct an empirical equation for \(\mathrm{I_{A\text{-}NEG}}\):  
\[\mathrm{I_{A_{NEG}}}=\frac{\mathrm{I_{ON}}-\mathrm{I_{OFF}}}{\mathrm{V_{ON}}- \mathrm{V_{OFF}}}\mathrm{V_{A}}+\frac{\mathrm{V_{ON}}\mathrm{I_{OFF}}- \mathrm{V_{OFF}}\mathrm{I_{ON}}}{\mathrm{V_{ON}}-\mathrm{V_{OFF}}} \tag{4}\]  
where \(\mathrm{V_{ON}}\) and \(\mathrm{V_{OFF}}\) are the switching voltages under forward and reverse bias sweeps, while \(\mathrm{I_{ON}}\) and \(\mathrm{I_{OFF}}\) are the corresponding drain currents, just before the switch.  
The keys aspect of the regional model is the calculation of the boundary conditions between each regime, namely the switching voltages \(\mathrm{V_{ON}}\) and \(\mathrm{V_{OFF}}\)[29].