## n-channel MOSFET

The expression for the drain current of a n-channel MOSFET in the linear regime $(V_{DS} < V_{GS} - V_T)$ is,

$\begin{equation} I_{D}=\frac{\mu WC}{L}\left( V_{GS}-V_T-\frac{V_{DS}}{2}\right)V_{DS}. \end{equation}$

Here $I_D$ is the drain current, $V_{DS}$ is the drain-source voltage, $V_{GS}$ is the gate-source voltage, $V_T$ is the threshold voltage, $L$ is the length of the transistor (in the direction that the current flows), $W$ is the width of the transistor, $C$ is the specific capacitance of the gate in F/m², and $\mu$ is the mobility.

For $(V_{DS} > V_{GS} - V_T)$, the channel is pinched off and the transistor is in the saturation regime. The expression for the drain current in the saturation regime is,

$\begin{equation} I_{D}=\frac{\mu WC}{2L}\left( V_{GS}-V_T\right)^2. \end{equation}$

MOSFET charateristics are typically drawn for several gate voltages. In the figure below, the threshold voltage is 1 V and the gate voltages are 2, 3, 4, 5 and 6 V.

 $I_{DS}$ [mA] $V_{DS}$ [V]
 W = m L = m C = F/m² μ = cm²/Vs VDS,max = V Vg = V Vg = V Vg = V Vg = V Vg = V Vg = V Vg = V Vg = V Vg = V Vg = V
 φm = eV χs = eV tox = nm εox = εsemi = Nc = 1/cm³ Nv = 1/cm³ NA = 1/cm³ Eg(T) = eV T = K φs = 2 eV Eg = 2 eV ni = 2 V Vfb = 2 V VT = 2 V

The quantities in the right column are used to calculate the threshold voltage $V_T$,

$$V_T = \frac{2t_{ox}}{\epsilon_{ox}}\sqrt{\epsilon_{\text{semi}}N_Ak_BT \ln \left (\frac{N_A}{n_i} \right )} +\frac{2k_BT}{e} \ln \left (\frac{N_A}{n_i} \right ) +V_{fb}.$$

Here φm is the work function of the metal, χs is the electron affinity of the semiconductor, φs is the work function of the semiconductor, $V_{fb}=$ φm - φs is the flat band voltage, $E_g$ is the band gap (It is possible to include the temperature dependence of the band gap.), $t_{\text{ox}}$ is the thickness of the oxide, $\epsilon_{\text{ox}}$ is the relative dielectric constant of the oxide, $\epsilon_{\text{semi}}$ is the relative dielectric constant of the semiconductor, and $n_i$ is the intrinsic carrier concentration. The temperature dependence of the device enters primarily through the temperature dependence of the threshold voltage.

$V_G$ [V] $V_D$ [V] $I_D$ [mA]