# Tunneling Current Variations in Small-Sized Devices Based on a Compact Threshold Voltage Model
## 2 Theoretical Model Derivation
### Long-Channel Threshold Voltage Model
#### 2.1.1 Zero Substrate Bias Threshold Voltage Model (\(V_{sb}=0\))  
The device structure shown in Figure 1 establishes and solves a one-dimensional Poisson's equation along the x-direction, perpendicular to the channel, when the substrate bias is zero, i.e., \(V_{sb}=0\). Therefore, the threshold voltage model for strained Si NMOSFETs can be derived with long channels under zero substrate bias. In the relaxed \(SiGe\) layer, the strain from the \(Si/SiGe\) quantum well causes it to relax. In the relaxed \(SiGe\) layer, the electron concentration can be disregarded, and the [PERSON]'s equation can be established as follows:  
\[\frac{d^{2}\phi}{dx^{2}}=\frac{qN_{SiGe}}{\varepsilon_{SiGe}},\ (T_{SSi}<x<T_{SSi}+W_{d}), \tag{1}\]  
Figure 1: Cross-sectional structure of a strained Si NMOSFET. Among them, \(\phi\) represents the potential with substrate B as the reference point and \(\varepsilon_{SiGe}\) represents the dielectric constant of the relaxed \(SiGe\) layer. The boundary conditions are as follows:  
\[E=-\left.\frac{d\phi}{dx}\right|_{x=T_{SSi}+W_{D}}=\left.\phi\right|_{x=T_{SSi}+ W_{D}}=0, \tag{2}\]  
where \(E\) is the electric field intensity. By integrating Equation (1), we can obtain the electric field and electric potential of the layer as follows:  
\[E(x)=-\left.\frac{d\phi}{dx}\right.=-\frac{qN_{SiGe}}{\varepsilon_{SiGe}}(x-T _{SSi}-W_{D}), \tag{3}\]  
\[\phi(x)=\left.\frac{qN_{SiGe}}{2\varepsilon_{SiGe}}(x-T_{SSi}-W_{D})^{2},\right. \tag{4}\]  
Therefore, when \(x=T_{SSi}\), the electric field and electric potential at the surface of the relaxed \(SiGe\) layer are, respectively:  
\[E(x)|_{x=T_{SSi}^{+}}=-\varepsilon_{SiGe}\frac{d\phi}{dx}\bigg{|}_{x=T_{SSi}^ {+}}=\frac{qN_{SiGe}W_{D}}{\varepsilon_{SiGe}}, \tag{5}\]  
\[\phi_{1}=\left.\phi\right|_{x=T_{SSi}}=\frac{qN_{SiGe}W_{D}^{2}}{\varepsilon_ {SiGe}}, \tag{6}\]  
For the strained \(Si\) layer, using the depletion approximation and ignoring the electron concentration, we can establish the one-dimensional [PERSON]'s equation as:  
\[\frac{d^{2}\phi}{dx^{2}}=\frac{qN_{SSi}}{\varepsilon_{SSi}},\ (0<x<T_{SSi}), \tag{7}\]  
where \(\varepsilon_{SSi}\) is the dielectric constant of the strained \(Si\) layer. The boundary conditions are as follows:  
\[\varepsilon_{SSi}\frac{d\phi}{dx}\bigg{|}_{x=T_{SSi}^{-}}=\varepsilon_{SiGe} \frac{d\phi}{dx}\bigg{|}_{x=T_{SSi}^{+}}=-qN_{SiGe}W_{D}, \tag{8}\]  
\[\phi_{1}=\left.\phi\right|_{x=T_{SSi}}=\frac{qN_{SiGe}W_{D}^{2}}{2\varepsilon_ {SiGe}}, \tag{9}\]  
By integrating Equation (7), we can obtain the electric field and electric potential of the strained \(Si\) layer as follows:  
Figure 1:0  
Figure 1:1  
The surface electric field and surface potential of the strained \(Si\) layer at the Si/SiO\({}_{2}\) interface are determined by the boundary conditions as:  
Figure 1:2  
Figure 1:3The depletion layer thickness in the relaxed \(SiGe\) layer can be obtained from Equation (13) as follows:  
Figure 1:4  
Along the x-direction perpendicular to the channel, the applied gate voltage can be decomposed into the voltage drop across the gate oxide due to the equivalent total depletion charge, surface potential \(\varphi_{s}\), and flat-band voltage \(V_{FB}\)[13]:  
Figure 1:5  
The total depletion charge \(Q_{B}\) is the sum of the depletion charges in the strained Si layer and the relaxed SiGe layer, i.e., \(Q_{B}=Q_{si}+Q_{SiGe}\):  
Figure 1:6  
By substituting Equations (14) and (16) into Equation (15), we can simplify and obtain:  
Figure 1:7  
where the values of \(A_{1}\), \(A_{2}\), and \(\gamma_{2}\) can be calculated using the following formulas:  
Figure 1:8  
When \(\varphi_{S}\) is equal to the threshold surface potential, \(V_{GB}=V_{TH0}\), where \(V_{TH0}\) is the threshold voltage at zero substrate bias. Its expression is as follows:  
Figure 1:9