# A Review of Sharp-Switching Band-Modulation Devices
## Operating Principles and Models
### Switching Voltages  
Figure 10 shows the large variation of the electrostatic potential from depletion to inversion under the gate. The potential in the gated region \(\mathrm{\Psi_{G}}\) and in the ungated body \(\mathrm{\Psi_{UC}}\) can be formulated via an MOS-centric description. In depletion mode, the equivalent capacitor dividers in Figure 11a yield the corresponding potentials \(\mathrm{\Psi_{Gdep}}\) and \(\mathrm{\Psi_{UGdep}}\):  
\[\mathrm{\Psi_{G_{dep}}}=\frac{\mathrm{C_{BOX}}\cdot\mathrm{C_{Si}}\cdot( \mathrm{V_{BG}}-\mathrm{V_{FG}})}{\mathrm{C_{OX}}\cdot\mathrm{C_{Si}}+\mathrm{ C_{BOX}}\cdot\mathrm{C_{Si}}+\mathrm{C_{OX}}\cdot\mathrm{C_{BOX}}}+\mathrm{V_{FG}} \tag{5}\]  
\[\mathrm{\Psi_{UG_{dep}}}=\bigg{(}\frac{\mathrm{C_{BOX}}}{\mathrm{C_{BOX}}+ \mathrm{C_{Si_{UC}}}}\bigg{)}\mathrm{V_{BG}} \tag{6}\]  
The inversion mode is mathematically more complicated. The gated region is a classical FD-SOI MOS structure, where the explicit solution from the Leti-UTSOI model applies [31]. The ungated region behaves like a bulk device, with limited substrate depth and thick gate oxide (the BOX), mainly driven by the back-gate bias. The Poisson integration is simplified by considering only the back channel in the ungated region. The explicit solution proposed in [29] reads:  
\[\mathrm{\Psi_{UG}}=\mathrm{\Psi_{UG_{dep}}}-2\beta\cdot\mathrm{LW}\bigg{(} \frac{\sqrt{\mathrm{q}\cdot\epsilon_{Si}\cdot\mathrm{ni}}}{\sqrt{2\beta} \cdot\mathrm{C_{BOX}}}\cdot\mathrm{e}^{\frac{\mathrm{\Psi_{UG_{dep}}-\mathrm{V_ {C}}}}{2\beta}}\bigg{)} \tag{7}\]where the analytical formulation of the Lambert function is  
\[\text{LW(x)}=\ln(1+\text{x})\bigg{[}1-\frac{\ln(1+\ln(1+\text{x}))}{2+\ln(1+\text{ x})}\bigg{]}\]  
The quasi-Fermi level potential \(\text{V}_{\text{C}}\) is deduced from TCAD simulations. Figure 11b illustrates the complex variation of the electrostatic potentials \(\Psi_{\text{G}}\) and \(\Psi_{\text{UG}}\) with anode voltage. The validity of the proposed model is confirmed by a comparison with the TCAD results. The turn-on voltage \(\text{V}_{\text{ON}}\) corresponds to the sharp switching point in the increasing \(\text{V}_{\text{A}}\) sweep, that is the boundary voltage between the barrier collapse and negative differential resistance regimes (point B in Figure 9). The potential variation \(\Psi_{\text{UG}}-(\Psi_{\text{G}}-\text{V}_{\text{A}})\) monitors the difference between the pseudo build-in voltages in the junctions formed by the cathode and the ungated body (\(\Psi_{\text{UG}}\)) and by the anode and the gated region (\(\Psi_{\text{G}}-\text{V}_{\text{A}}\)). Figure 12a shows that the switch point occurs when this differences is close to 0, wherever this happens within the silicon film. An iterative calculation yields a first order approximation of \(\text{V}_{\text{ON}}\):  
\[\text{V}_{\text{ON}}=\psi_{\text{G}_{\text{dep}}}-0.5\text{b}_{\text{UG}} \Big{(}\psi_{\text{G}_{\text{dep}}}\Big{)}-0.5\text{b}_{\text{UG}}\Big{(} \psi_{\text{G}_{\text{dep}}}-0.5\text{b}_{\text{UG}}\Big{(}\psi_{\text{G}_{ \text{dep}}}\Big{)}\Big{)} \tag{8}\]  
where \(\Psi_{\text{G}\text{dep}}\) is the potential in the depleted gated region (Equation (5)) and \(\Psi_{\text{UG}}(\text{V}_{\text{A}})\) is the potential in the ungated region as a function of anode voltage; an analytical expression of \(\Psi_{\text{UG}}(\text{V}_{\text{A}})\) is derived in [29]. Conversely, the turn-off voltage \(\text{V}_{\text{OFF}}\) is the boundary voltage between the negative differential resistance and PIN regimes (point C in Figure 9). The device turns off when \(\text{V}_{\text{A}}\) is sufficiently low to allow the gate to regain control of the gated region, as shown in Figure 12b. The lower limit of the PIN regime occurs when the potential difference \(\Psi_{\text{G}}-\Psi_{\text{G}\text{inv}}\) reaches zero.