\begin{equation} I_D(\mathrm{sub}) \propto\left[\exp \left(\frac{e V_{G S}}{k T}\right)\right] \cdot\left[1-\exp \left(\frac{-e V_{D S}}{k T}\right)\right] \qquad (11.1) \end{equation} \begin{equation} x_p=\sqrt{\frac{2 \epsilon_s \phi_{f p}}{e N_a}} \qquad (11.2) \end{equation} $$ x_p=\sqrt{\frac{2 \epsilon_s}{e N_a}\left(\phi_{f p}+V_{D S}\right)} \qquad (11.3) $$ $$ \begin{gathered} \Delta L=\sqrt{\frac{2 \epsilon_s}{e N_a}}\left[\sqrt{\phi_{f p}+V_{D S}(\mathrm{sat})+\Delta V_{D S}}-\sqrt{\phi_{f p}+V_{D S}(\mathrm{sat})}\right] \qquad (11.4)\\ \Delta V_{D S}=V_{D S}-V_{D S}(\mathrm{sat}) \qquad (11.5) \end{gathered} $$ $$ \frac{d \mathrm{E}}{d x}=\frac{\rho(x)}{\epsilon_s} \qquad (11.6) $$ $$ \mathrm{E}=-\frac{e N_a x}{\epsilon_s}-\mathrm{E}_{\mathrm{sat}} \qquad (11.7) $$ $$ \phi(x)=-\int \mathrm{E} d x=\frac{e N_a x^2}{2 \epsilon_s}+\mathrm{E}_{\mathrm{sat}} x+C_1 \qquad (11.8) $$ $$ V_{D S}=\frac{e N_a(\Delta L)^2}{2 \epsilon_s}+\mathrm{E}_{\mathrm{sat}}(\Delta L)+V_{D S}(\mathrm{sat}) \qquad (11.9) $$ $$ \begin{gathered} \Delta L=\sqrt{\frac{2 \epsilon_s}{e N_a}}\left[\sqrt{\phi_{\text {sat }}+\left[V_{D S}-V_{D S}(\mathrm{sat})\right]}-\sqrt{\phi_{\text {sat }}}\right] \qquad (11.10)\\ \phi_{\text {sat }}=\frac{2 \epsilon_s}{e N_a} \cdot\left(\frac{\mathrm{E}_{\mathrm{sat}}}{2}\right)^2 \end{gathered} $$ $$ I_D^{\prime}=\left(\frac{L}{L-\Delta L}\right) I_D \qquad (11.11) $$ $$ I_D^{\prime}=\frac{k_n^{\prime}}{2} \cdot \frac{W}{L} \cdot\left[\left(V_{G S}-V_T\right)^2\left(1+\lambda V_{D S}\right)\right] \qquad (11.12) $$ $$ r_o=\left(\frac{\partial I_D^{\prime}}{\partial V_{D S}}\right)^{-1}=\left\{\frac{k_n^{\prime}}{2} \cdot \frac{W}{L} \cdot\left(V_{G S}-V_T\right)^2 \cdot \lambda\right\}^{-1} \qquad (11.13a) $$ $$ r_o \cong \frac{1}{\lambda I_D} \qquad (11.13b) $$ $$ \mathrm{E}_{\mathrm{eff}}=\frac{1}{\boldsymbol{\epsilon}_s}\left(\left|Q_{S D}^{\prime}(\max )\right|+\frac{1}{2} Q_n^{\prime}\right) \qquad (11.14) $$ $$ \mu_{\text {eff }}=\mu_0\left(\frac{E_{\text {eff }}}{E_0}\right)^{-1 / 3} \qquad (11.15) $$ $$ V_{D S}=V_{D S}(\mathrm{sat})=V_{G S}-V_T \qquad (11.16) $$ $$ I_D(\mathrm{sat})=W C_{\mathrm{ox}}\left(V_{G S}-V_T\right) v_{\mathrm{sat}} \qquad (11.17) $$ $$ \mu=\frac{\mu_{\mathrm{eff}}}{\left[1+\left(\frac{\mu_{\mathrm{eff}} \mathrm{E}}{v_{\mathrm{sat}}}\right)^2\right]^{1 / 2}} \qquad (11.18) $$ $$ g_{m s}=\frac{\partial I_D(\mathrm{sat})}{\partial V_{G S}}=W C_{\mathrm{ox}} v_{\text {sat }} \qquad (11.19) $$ $$ f_T=\frac{g_m}{2 \pi C_G}=\frac{W C_{\mathrm{ox}} v_{\mathrm{sat}}}{2 \pi\left(C_{\mathrm{ox}} W L\right)}=\frac{v_{\mathrm{sat}}}{2 \pi L} \qquad (11.20) $$ \begin{equation} x_D=\sqrt{\frac{2 \epsilon\left(V_{b i}+V_D\right)}{e N_a}} \qquad (11.21) \end{equation} $$ \frac{I_D}{W}=\frac{\mu_n \epsilon_{\mathrm{ox}}}{2 t_{\mathrm{ox}} L}\left(V_G-V_T\right)^2 \rightarrow \frac{\mu_n \epsilon_{\mathrm{ox}}}{2\left(k t_{\mathrm{ox}}\right)(k L)}\left(k V_G-V_T\right)^2 \approx \mathrm{constant} \qquad (11.22) $$ $$ V_T=V_{F B}+2 \phi_{f p}+\frac{\sqrt{2 \epsilon e N_a\left(2 \phi_{f p}\right)}}{C_{\mathrm{ox}}} \qquad (11.23) $$ $$ V_{T N}=\left(\left|Q_{S D}^{\prime}(\max )\right|-Q_{s s}^{\prime}\right)\left(\frac{t_{\mathrm{ox}}}{\epsilon_{\mathrm{ox}}}\right)+\phi_{m s}+2 \phi_{f p} \qquad (11.24) $$ $$ x_s \approx x_d \approx x_{d T} \equiv x_{d T} \qquad (11.25) $$ $$ \left|Q_B^{\prime}\right| \cdot L=e N_a x_{d T}\left(\frac{L+L^{\prime}}{2}\right) \qquad (11.26) $$ $$ \begin{gathered} \frac{L+L^{\prime}}{2 L}=\left[1-\frac{r_j}{L}\left(\sqrt{1+\frac{2 x_{d T}}{r_j}}-1\right)\right] \qquad (11.27)\\ \left|Q_B^{\prime}\right|=e N_a x_{d T}\left[1-\frac{r_j}{L}\left(\sqrt{1+\frac{2 x_{d T}}{r_j}}-1\right)\right] \qquad (11.28) \end{gathered} $$ $$ \Delta V_T=-\frac{e N_a x_{d T}}{C_{\text {ox }}}\left[\frac{r_j}{L}\left(\sqrt{1+\frac{2 x_{d T}}{r_j}}-1\right)\right] \qquad (11.29) $$ $$ \Delta V_T=V_{T(\text { short channel })}-V_{T(\text { long channel })} \qquad (11.30) $$ $$ Q_B=Q_{B 0}+\Delta Q_B \qquad (11.31) $$ $$ \begin{gathered} \left|Q_{B 0}\right|=e N_a W L x_{d T} \qquad (11.32)\\ \Delta Q_B=e N_a L x_{d T}\left(\xi x_{d T}\right) \qquad (11.33) \end{gathered} $$ $$ \begin{aligned} \left|Q_B\right| & =\left|Q_{B 0}\right|+\left|\Delta Q_B\right|=e N_a W L x_{d T}+e N_a L x_{d T}\left(\xi x_{d T}\right) \\ & =e N_a W L x_{d T}\left(1+\frac{\xi x_{d T}}{W}\right) \qquad (11.34) \end{aligned} $$ $$ \Delta V_T=\frac{e N_a x_{d T}}{C_{\mathrm{ox}}}\left(\frac{\xi x_{d T}}{W}\right) \qquad (11.35) $$ $$ I_C=\alpha I_E+I_{C B 0} \qquad (11.36) $$ $$ I_C=\alpha I_C+I_{C B 0} \qquad (11.37) $$ $$ I_C=M\left(\alpha I_C+I_{C B 0}\right) \qquad (11.38) $$ $$ I_C=\frac{M I_{C B 0}}{1-\alpha M} \qquad (11.39) $$ $$ M=\frac{1}{1-\left(V_{C E} / V_{B D}\right)^m} \qquad (11.40) $$ \begin{equation} \Delta V_T=+\frac{e D_I}{C_{\mathrm{ox}}} \qquad (11.41) \end{equation} $$ x_{d T}=\sqrt{\frac{2 \epsilon_s}{e N_a}}\left[2 \phi_{f p}-\frac{e x_I^2}{2 \epsilon_s}\left(N_s-N_a\right)\right]^{1 / 2} \qquad (11.42) $$ $$ V_T=V_{T 0}+\frac{e D_I}{C_{\mathrm{ox}}} \qquad (11.43) $$ $$ D_I=\left(N_s-N_a\right) x_I \qquad (11.44) $$ $$ V_{T 0}=V_{F B 0}+2 \phi_{f p 0}+\frac{e N_a x_{d T 0}}{C_{\mathrm{ox}}} \qquad (11.45) $$