C.1 IV Characteristics

The drain voltage along the channel is shown in the right side of Fig. C.1. The voltage drop across an elemental section $\mathrm{d}x$ of the channel can be obtained from (C.1), given by

$$\mathrm{d}V=I_\mathrm{D}\cdot\mathrm{d}R=\frac{I_\mathrm{D}\cdot\... ...rm{q}}\cdot\mu_n\cdot N_\mathrm{D}\cdot W_\mathrm{g}\cdot \left[a-W(x)\right]},$$ (C.4)

and the depletion-layer width at a distance $y$ from the source is given by

$$W(x)=\sqrt{\frac{2\varepsilon_s\cdot\left[V(x)+V_\mathrm{G}+V_\mathrm{bi}\right]}{{\mathrm{q}}\cdot N_\mathrm{D}}}.$$ (C.5)

The drain current $I_\mathrm{D}$ is constant, independent of $x$. One can rewrite (C.4) as

$$I_\mathrm{D}\cdot\mathrm{d}x= {\mathrm{q}}\cdot\mu_n\cdot N_\mathrm{D}\cdot W_\mathrm{g}\cdot \left[a-W(x)\right]\cdot\mathrm{d}V.$$ (C.6)

The differentiation of the drain voltage $\mathrm{d}V$ is obtained from (C.5)

$$\mathrm{d}V=\frac{{\mathrm{q}}\cdot N_\mathrm{D}}{\varepsilon_s}\cdot W\cdot\mathrm{d}W.$$ (C.7)

Substituting (C.7) into (C.6) and integrating from $x=0$ to $x=L_\mathrm{g}$ yields

$$I_\mathrm{D} = \frac{1}{L_\mathrm{g}}\int_{W_1}^{W_2}{\mathrm{q}}... ...cdot\left[a\left(W^2_2-W^2_1\right)-\frac{2}{3}\left(W^3_2-W^3_1\right)\right].$$ (C.8)

Replacing the depletion-layer width $W$ in (C.8) in terms of voltage from (C.5) gives

$$I=I_\mathrm{p}\left[\frac{V_\mathrm{D}}{V_\mathrm{p}}-\frac{2}{3}... ...2}{3}\left(\frac{V_\mathrm{G}+V_\mathrm{bi}}{V_\mathrm{p}}\right)^{3/2}\right],$$ (C.9)

where

$$I_\mathrm{p}=\frac{W_\mathrm{g}\cdot\mu_n\cdot{\mathrm{q}}^2\cdot N^2_\mathrm{D}\cdot a^3}{2\varepsilon_s\cdot L_\mathrm{g}},$$ (C.10)

and

$$V_\mathrm{p}=\frac{{\mathrm{q}}\cdot N_\mathrm{D}\cdot a^2}{2\varepsilon_s}.$$ (C.11)

The voltage $V_\mathrm{p}$ is called the pinch-off voltage, that is the total voltage ( $V_\mathrm{D}+V_\mathrm{G}+V_\mathrm{bi}$) at which $W_2=a$. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation