# Tunneling Current Variations in Small-Sized Devices Based on a Compact Threshold Voltage Model
## 3 Tunneling Current Model  
When MOSFET devices are scaled down proportionally, tunneling current significantly deteriorates the devices' performances. To illustrate the impact of tunneling current, it is further explained through Figures 4-6. Figure 4 depicts the device's structural model,  
Figure 3: Cross-sectional diagram of a narrow-channel device in the width direction. while Figures 5 and 6 illustrate the principles of tunneling current and the composition of tunneling current, respectively. There are three direct tunneling mechanisms in strained silicon MOSFET devices: (1) electron tunneling from the strained silicon conduction band to the gate conduction band (CBET), (2) hole tunneling from the strained silicon valence band to the gate valence band (VBET), and (3) hole tunneling from the strained silicon valence band to the gate conduction band (VBHT). However, in a strained silicon device with a Poly-Si/SiO\({}_{2}\)/SSI system, electrons cannot tunnel from the valence band to the conduction band, and holes also cannot tunnel. Therefore, the VBET and VBHT current components are negligible and can be ignored. The  
Figure 4: Schematic diagram of a strained Si/SiGe MOSFET’s structure. Figure 5: Subthreshold characteristics of tunneling current. Figure 6: Composition of tunneling current. gate direct tunneling current in a strained silicon MOSFET device is mainly due to CBET and can be modeled as such [15]:  
\[J_{DT}=J_{G}\approx J_{CBET}=A\left(\frac{V_{ox}}{T_{ox}}\right)^{2}\cdot\exp \left[-B\left(\frac{V_{ox}}{T_{ox}}\right)^{-1}\times\left(1-\left(1-\frac{V_{ox }}{T_{ox}}\right)^{\frac{3}{2}}\right)\right], \tag{34}\]  
where \(J_{DT}\) represents the direct tunneling current density, \(\phi_{ox}\) represents the barrier height, \(T_{ox}\) represents the oxide layer thickness, and A and B are physical parameters with values of \(A=q^{3}/16\pi\hbar\phi_{ox}\) and \(B=\frac{4\sqrt{2m^{*}}\phi_{ox}^{3/2}}{3\hbar q}\), respectively. Here, \(\hbar\) represents the Planck constant, and \(m^{*}\) represents the effective mass of electrons in the strained silicon conduction band. Therefore, when assuming \(1-\left(1-\frac{V_{ox}}{\phi_{xx}}\right)^{\frac{3}{2}}\approx 1\), the gate tunneling current at the point can be approximated as:  
\[J_{G}(x)\approx AE_{ox}^{2}e^{-B/E_{ox}}\approx AE_{oxx}^{2}e^{-BT_{ox}/(V_{ xx}-V(x))}\equiv J_{G0}e^{-B^{*}V(x)}, \tag{35}\]  
In the equation, \(J_{G0}\) represents the gate tunneling current density at \(V_{DS}=0\); \(B^{*}=\frac{pBT_{ox}}{V_{ds}^{2}}\), where \(p\) is an adjustment parameter with a default value of 1; \(V_{ox}=V_{GS}\) represents the gate-to-source voltage at \(x=0\); \(E_{ox}=V_{ox}/T_{ox}\). If the gate leakage current is much smaller than the drain current, then:  
\[ \begin{split} V(x)=V_{GS}-V_{TH}-\sqrt{\left(V_{GS}-V_{TH} \right)^{2}-2\left(V_{GS}-V_{TH}-\frac{V_{DS}}{2}\right)V_{DS}\cdot\frac{x}{L} }\\ \approx(V_{GS}-V_{TH}-V_{DS}/2)\cdot V_{DS}/(V_{GS}-V_{TH})L \cdot x,\end{split}  \tag{36}\]  
In Equation (36), an approximate form is used to describe the characteristics of small-sized devices. This approximation is correct and reasonable when the value is very small or huge. To more accurately represent tunneling current, the total gate tunneling current can be expressed in integral form as:  
\[I_{G}=W\int_{0}^{L}J_{G}(x)dx=(WL)\cdot\frac{\int_{0}^{L}J_{G}(x)dx}{\int_{0} ^{L}dx}=(WL)\cdot\langle J_{G}(x)\rangle, \tag{37}\]  
According to the mathematical definition of the average value, \(\langle x\rangle=\int_{0}^{y}xdx/\int_{0}^{y}dx\), the current intensity is equal to the product of current density and area. From Equation (37), we can obtain \(I_{G}=\left\langle W\int_{0}^{L}J_{G}(x)dx\right\rangle=\langle I_{G}(x)\rangle\). For the method of integrating to solve tunneling current, its physical interpretation is that the total current intensity is the product of the gate oxide interface area \((W\cdot L)\) and the average current density \(\langle I_{G}(x)\rangle\) passing through the interface.