## 2 Model Presentation  
### Surface Potential Model  
This section proposes an accurate analytical approximation of surface potential for FDSOI MOSFETs to achieve fast convergence. This reduces the number of iterations or even eliminates the need for iterative estimation. By defining the coordinates as shown in Figure 1 and using the gradual channel approximation (GCA), a one-dimensional Poisson's equation with only the inversion charge term for the silicon region can be written as [13]  
\[\frac{\partial^{2}\psi}{\partial x^{2}}=\frac{qN_{c}}{\varepsilon_{Si}}exp \bigg{[}\frac{\psi-V_{ch}(y)}{V_{tm}}\bigg{]} \tag{1}\]where \(\psi\) is defined in [13] as the potential in the silicon body and \(V_{ch}(y)\) is the channel voltage, \(\varepsilon_{Si}\) is the permittivity constant of silicon, \(N_{c}\) is the conduction band density of states, \(V_{tm}\) is the thermal voltage. The boundary conditions result from the continuity of the electric displacement at the front and the back interfaces as  
\[\varepsilon_{Si}E_{S1,Si}=\varepsilon_{OX}E_{S1,OX1}=\varepsilon_{OX}\frac{ \left(V_{fg}^{*}-\psi_{S1}\right)}{T_{OX1}} \tag{2}\]  
\[\varepsilon_{Si}E_{S2,Si}=\varepsilon_{OX}E_{S1,OX2}=-\varepsilon_{OX}\frac{ \left(V_{bg}^{*}-\psi_{S2}\right)}{T_{OX2}} \tag{3}\]  
where \(\varepsilon_{OX}\) is the dielectric permittivity of the front and back gate, \(V_{fg}^{*}=V_{fg}-\Delta\phi_{MS1}\)\(V_{bg}^{*}=V_{bg}-\Delta\phi_{MS2}\), other terms are defined in Table 1.  
After the integration of Equation (1) in the silicon body [14], we obtain  
\[E_{S1}{}^{2}-E_{S2}{}^{2}=K\bigg{\{}exp\bigg{[}\frac{\psi_{S1}-V_{ch}}{V_{tm}} \bigg{]}-exp\bigg{[}\frac{\psi_{S2}-V_{ch}}{V_{tm}}\bigg{]}\bigg{\}} \tag{4}\]  
where \(K=2 qN_{c}V_{tm}/\varepsilon_{Si}\). Assume the back gate to be in weak inversion and approximate the back surface potential \(\psi_{S2}\) as a linear expression of \(\psi_{S1}\):  
\[\psi_{S2}=k1\cdot\psi_{S1}+k2 \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Description** \\ \hline \(\psi_{S1}(\psi_{S2})\) & Front (Back) surface potential \\ \(E_{S1}(E_{S2})\) & Front (Back) surface electric field \\ \(V_{fg}\Big{(}V_{bg})\) & Front (Back) gate voltage \\ \(T_{OX1}(T_{OX2})\) & Front (Back) gate oxide thickness \\ \(\Delta\phi_{MS1}(\Delta\phi_{MS2})\) & Work function of front (back) gate \\ \(T_{Si}\) & Body thickness \\ \hline \end{tabular}
\end{table} 
Table 1: Description of symbols. Figure 1: Schematic diagram of independent double gate FDSOI MOSFET. where the expressions of \(k1\) and \(k2\) are given in Table 2. The approximate expression of \(\psi_{S2}\) is substituted into the implicit Equation (4) and the equation is normalized by \(V_{tm}\) as  
\[\left(x_{eff}-x\right)^{2}-spD=spC\cdot exp(x-x_{Vch}) \tag{6}\]  
where \(x=\psi_{S1}/V_{tm}\), \(x_{Vch}=V_{ch}/V_{tm}\) and the expressions of other symbols are given in Table 2. Reference [13] uses a simplified approximation for the back gate electric field and the initial solution needs to be refined using a perturbation technique. Reference [15] provides the initial solution assuming a parabolic potential profile in the body. The form of Equation (6) is similar to that in the literature [16], and a non-iterative evaluation of surface potential in [16] is adopted and modified for the case of Equation (6). A relatively small refinement \(u\) is donated by \(x-\eta\), where \(\eta\) is a rough estimated value of the solution of (6) The value of \(\eta\) is given in weak and strong inversion region, respectively. In the weak inversion region, the exponential term on right hand side of Equation (6) is negligibly small and \(\eta_{winv}\) is solved using Equation (6) as follows:  
\[\eta_{winv}=x_{eff}-\sqrt{spD} \tag{7}\]  
In the strong inversion region, \(\eta\) cannot closely track \(x_{eff}\) because the exponential term dominates.