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

μ =

cm²/Vs 

VDS,max =

V

 

Vg[1] =

V

Vg[2] =

V

Vg[3] =

V

Vg[4] =

V

Vg[5] =

V

Vg[6] =

V

Vg[7] =

V

Vg[8] =

V

Vg[9] =

V

Vg[10] =

V

 

φm =

eV

χs =

eV

tox =

nm

εox =

εsemi =

Nc =

1/cm³

Nv =

1/cm³

NA =

1/cm³

Eg(T) =

eV

T =

K
 

C =

2 F/m²

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]