# Design and Analysis of Heavily Doped n+
## 3 Investigation Results and Discussion  
Figure 5 shows the variation of gate voltage concerning drain current for different dielectric constants with changes in the length of the cavity region. The lengths of the cavity region considered were 3 nm, 5 nm, 7 nm. It was observed that the threshold voltage changed for different dielectric materials. Dielectric materials taken into consideration were air (k = 1), SiO\({}_{2}\) (k = 3.9), S\({}_{3}\)N\({}_{4}\). (k = 7.5) and HfO\({}_{2}\)(k = 25). The threshold voltage for AJ DG MOSFET is shown in Equation (1) [21]. Figure 4: (a) n\({}^{+}\) pocket AJ DG MOSFET with a cavity region; (b) n\({}^{+}\) pocket AJ DG MOSFET with a cavity region over a _SiO\({}_{2}\)_ layer. \[V_{th}=\frac{E_{g}}{2Q}+\frac{\frac{\varepsilon_{Si}}{\varepsilon_{ox}}t_{ox}}{t_{ si}+\frac{\varepsilon_{Si}}{\varepsilon_{ox}}2t_{ox}}\Delta\Phi_{2}+\frac{t_{si}+ \frac{\varepsilon_{Si}}{\varepsilon_{ox}}t_{ox}}{t_{si}+\frac{\varepsilon_{Si}} {\varepsilon_{ox}}2t_{ox}}\Delta\Phi_{1} \tag{1}\]  
Therefore,  
\[C_{ox}=\frac{\varepsilon_{ox}}{t_{ox}}=\frac{k.\varepsilon_{o}}{t_{ox}} \tag{2}\]  
and  
\[\Delta\Phi_{2}=\Delta\Phi_{1} \tag{3}\]  
Evaluating Equations (1) and (2):  
\[V_{th}=\frac{E_{g}}{2Q}+\frac{\varepsilon_{Si}}{t_{si}C_{ox}+2\ \varepsilon_{Si}} \Delta\Phi_{2}+\frac{t_{si}C_{ox}+\varepsilon_{Si}}{t_{si}C_{ox}+2\varepsilon _{Si}}\Delta\Phi_{1} \tag{4}\]  
where \(V_{th}\) is the threshold voltage of AJ DG MOSFET, \(\varepsilon_{Si}\) is the permittivity of the silicon wafer, \(\varepsilon_{ox}\) is the permittivity of the gate oxide, \(\Delta\Phi\) is the work function of the gate contact, \(t_{si}\) is the thickness of silicon wafer, \(t_{ox}\) is the thickness of gate oxide, \(\varepsilon_{o}\) is the permittivity of vacuum, and k is the dielectric constant of the material when the cavity region is not introduced in it. When the cavity region in the gate oxide is introduced, the capacitance of the cavity region is also considered. Changing the dielectric material in the cavity region of the MOSFET changes the total capacitance of the gate oxide and the cavity region and hence can be written as  
\[\frac{1}{C_{AJ\ DG\ MOSFET}}=\frac{1}{C_{oxide}}+\frac{1}{C_{dielectric}} \tag{5}\]  
\[\frac{1}{C_{AJ\ DG\ MOSFET}}=\frac{t_{oxide}}{k_{oxide}\.\ \varepsilon_{0}}+ \frac{t_{dielectric}}{k_{dielectric}\.\ \varepsilon_{0}} \tag{6}\]  
where \(C_{AJ\ DG\ MOSFET}\) is the total capacitance of the gate oxide along with the cavity region having dielectric material, \(C_{dielectric}\) is the capacitance due to dielectric material in the cavity region, \(t_{dielectric}\) is the thickness of the dielectric medium, \(k_{dielectric}\) is the dielectric constant in the cavity region, \(t_{oxide}\) is the thickness of the oxide and \(k_{oxide}\) is the dielectric constant of the gate oxide. From Equations (4)-(6) it can be observed that:  
\[V_{th}\propto\ C_{dielectric},\ \text{therefore,}\ V_{th}\propto k_{dielectric} \tag{7}\]  
From Equation (7), it can be seen that the threshold voltage is directly proportional to the constant dielectric present in the cavity region. In Figure 5a-c, the threshold voltage increases with increases in the dielectric constant of the material introduced in the cavity region. Increasing the length of the cavity also shows the variation in threshold voltage, which results in the shifting of channel inversion at a higher threshold voltage. Figure 5a, with the length of the cavity region as 3 nm, shows less variation in the threshold voltage than Figure 5b,c. Therefore, with increases in the dielectric constant, the threshold voltage significantly reduces. Hence, this device is highly sensitive to change in dielectric constant and helps detect biomolecules. Figure 6a-c shows the graph of gate voltage concerning drain current keeping the dielectric material constant and varying the oxide thickness. The oxide thicknesses considered are 1 nm, 3 nm and 5 nm. The length of the cavity is kept constant at 5 nm.