Solid State Electronic Devices, 7th Edition

MOS-n Channel

$${\text{Oxide: C}}_i=\frac{{ϵ}_i}{d}{\text{ Depletion: C}}_d=\frac{{ϵ}_s}{W}\text{ MOS: C}=\frac{{\text{C}}_i{\text{C}}_d}{{\text{C}}_i+{\text{C}}_d}$$(6–36)

Threshold: $$V_T=\underset{\text{Flat band}}{\underbrace{{\Phi }_{ms}-\frac{{\text{Q}}_i}{{\text{C}}_i}}}-\frac{{\text{Q}}_d}{{\text{C}}_i}+2{\text{ϕ}}_F$$ (6–38)

Inversion: $${\varphi }_s\left(\text{inv}.\right)=2{\varphi }_F=2\frac{kT}{q}\text{ln}\frac{N_a}{n_i}$$ (6–15)

$$W={\left[\frac{2{ϵ}_s{\varphi }_s}{q{N}_a}\right]}^{1/2}$$(6–30)

$${\text{Q}}_d=-q{N}_a{W}_m=-2{({ϵ}_{s}q{N}_a{\varphi }_F)}^{1\text{/}\text{2}}$$(6–32)

$$\text{At }{V}_{FB}{\text{: C}}_{FB}=\frac{{\text{C}}_i{\text{C}}_{\text{debye}}}{{\text{C}}_i+{\text{C}}_{\text{debye}}}$$

Debye screening length: $$L_D=\sqrt{\frac{{ϵ}_{s}kT}{q^2{p}_0}}$$ (6–25)

$${\text{C}}_{\text{debye}}=\frac{{ϵ}_s}{L_D}$$(6–40)

Substrate bias: $$\Delta V_T\simeq \frac{\sqrt{2{ϵ}_{s}q{N}_a}}{{\text{C}}_i}{\left(-V_B\right)}^{1/2}\left(\text{n channel}\right)$$ (6–63b)

$$I_D=\frac{\overline{{\text{μ}}_n}Z{\text{C}}_i}{L}\left\{\left(V_G-V_{FB}-2{\text{Φ}}_F-\frac{1}{2}{V}_D\right)V_D-\frac{2}{3}\frac{\sqrt{2{ϵ}_{s}q{N}_a}}{{\text{C}}_i}[{\left(V_D+2{\text{ϕ}}_F\right)}^{3/2}-{\left(2{\text{ϕ}}_F\right)}^{3/2}\right\}$$(6–50)

$$I_D\simeq \frac{{\overline{\mu }}_{n}Z{\text{C}}_i}{L}[\left(V_G-V_T\right)V_D-\frac{1}{2}{V}_D^2]$$(6–49)

Saturation: $$I_D\left(\text{sat}.\right)\simeq \frac{1}{2}{\overline{\mu }}_n{\text{C}}_i\frac{Z}{L}{\left(V_G-V_T\right)}^2=\frac{Z}{2L}{\overline{\mu }}_n{\text{C}}_i{V}_D^2\left(\text{sat}\text{.}\right)$$ (6–53)

$$g_m=\frac{\partial I_D}{\partial V_G};g_m\left(\text{sat}\text{.}\right)=\frac{\partial I_D\left(\text{sat}\text{.}\right)}{\partial V_G}\simeq \frac{\text{Z}}{\text{L}}{\overline{\text{μ}}}_{\text{n}}{\text{C}}_i\left(V_G-V_T\right)$$(6–54)

For short L: $$I_D\left(\text{sat}.\right)\simeq {\text{ZC}}_i\left(V_G–V_T\right){\text{ v}}_s$$ (6–60)

Subthreshold slope: $$S=\frac{d{V}_G}{d\left(\text{log }{I}_D\right)}=\frac{kT}{q}\text{ In 10 }[1+\frac{{\text{C}}_d+{\text{C}}_i}{}$$