# Tunneling Current Variations in Small-Sized Devices Based on a Compact Threshold Voltage Model
## 2 Theoretical Model Derivation
### Long-Channel Threshold Voltage Model
#### 2.2.1 Establishment and Solution of the Quasi-Two-Dimensional Equations  
As shown in Figure 2, a Gaussian box is established within the depleted region of \(SiGe\), with a thickness of \(X_{dep}\). The influence of the free electron concentration is neglected (depletion approximation). By integrating the four surfaces within the plane, the following equations can be obtained:  
\[\varepsilon_{ox}\frac{V_{GS}-V_{FB}-V_{2}-V_{S}(y)}{T_{ox}}-\varepsilon_{SiGe} \frac{X_{dep}}{\eta}\frac{dE_{s}(y)}{dy}=qN_{SiGe}X_{dep}, \tag{23}\]  
The first term on the left side of the above equation represents the electric flux entering the Gaussian box from the upper surface, and the second term represents the total electric flux entering the Gaussian box along the \(y\)-direction. The whole edge electric flux entering the Gaussian box is equal to the total ionized charge within the Gaussian box, as shown on the right side of the equation. At the bottom of the depleted region, the potential and electric field are both zero due to the connection with the substrate.  
Here, \(V_{2}\) is given by \(V_{2}=\frac{qN_{Si}T_{SSi}^{2}}{2\varepsilon_{Si}}+\frac{qN_{Si}T_{SSi}T_{xx}}{ \varepsilon_{ox}}\), where \(\eta\) is a fitting parameter related to the process and can be obtained through parameter extraction. The expression for \(X_{dep}\) can be obtained by replacing \(\varphi_{s}\) with \(\phi_{s}-V_{bs}\) Equation (14) as follows:  
\[X_{dep}=\sqrt{\frac{2\varepsilon_{SiGe}}{qN_{SiGe}}(\phi_{s}-V_{bs})+\frac{ \varepsilon_{SiGe}}{\varepsilon_{SSi}}\bigg{(}\frac{\varepsilon_{SiGe}}{ \varepsilon_{SSi}}-\frac{N_{SSi}}{N_{SiGe}}\bigg{)}T_{SSi}^{2}}-\frac{ \varepsilon_{SiGe}}{\varepsilon_{SSi}}T_{SSi}, \tag{24}\]  
The boundary conditions for Equation (23) are \(V_{s}(0)=V_{bi}\) and \(V_{s}(L)=V_{bi}+V_{DS}\). The solution is given by:  
\[V_{s}(y)=V_{SL}+(V_{bi}+V_{DS}-V_{SL})\frac{\sinh(y/l)}{\sinh(L/l)}+(V_{bi}-V_ {SL})\frac{\sinh\left(\frac{L-y}{l}\right)}{\sinh\left(L/l\right)}, \tag{25}\]  
Figure 2: Schematic diagram of the strained \(SiGe\) layer depletion region in a Gaussian box.  
The equation \(V_{bi}=\frac{kT}{q}\log\left(\frac{N_{\text{SD}}N_{\text{SiGe}}}{nt_{SiGe}^{2}}\right)\) represents the built-in voltage, where \(N_{\text{SD}}\) is the doping concentration of the source/drain region and \(ni_{SiGe}\) is the doping concentration of the relaxed \(SiGe\) layer. In this equation, \(V_{SL}=V_{GS}-V_{FB}-V_{2}-\frac{qN_{\text{SiGe}}X_{dep}T_{ex}}{e_{ex}}\), \(k\) represents the Boltzmann constant, \(l=\sqrt{\frac{\varepsilon_{SiGe}T_{ex}X_{dep}}{\varepsilon_{ex}\eta}}\), \(T\) is the temperature, and \(q\) is the charge of an electron. In the strained Si NMOSFET with the strained Si layer, the surface potential is \(V_{s}(y)\) at the surface of the depletion layer in the relaxed \(SiGe\) layer. Assuming the strained Si layer is thin, we can approximately treat it as a one-dimensional structure:  
\[V_{S}^{\prime}(y)=V_{SL}^{\prime}+(V_{bi}+V_{DS}-V_{SL})\frac{\sinh(y/l)}{ \sinh(L/l)}+(V_{bi}-V_{SL})\frac{\sinh\left(\frac{L-y}{l}\right)}{\sinh\left(L /l\right)}, \tag{26}\]  
where \(V_{SL}^{\prime}=V_{GS}-V_{TH0}=\phi_{s}\) represents the surface potential of the long-channel strained Si NMOSFET.