MOSFET transconductance parameter

The constant that scales how much drain current a MOSFET produces for a given drive. Two flavours: a process value fixed by the fabrication technology, and a device value that also folds in the transistor’s geometry.

Process transconductance parameter:

$k_n^{\prime }={\mu }_n{C}_{ox}$

where ${\mu }_n$ is the electron mobility in the channel (how easily electrons move under a field) and $C_{ox}$ is the Gate oxide capacitance per unit area. Both are set by the fabrication process (the chemistry of the silicon, the thickness of the oxide), so $k_n^{\prime }$ is the same for every transistor made on a given process. Units are A/V², since it multiplies a squared voltage in the drain-current law.

Device transconductance parameter:

$k_n=k_n^{\prime }\frac{W}{L}={\mu }_n{C}_{ox}\frac{W}{L}$

where $W$ is the channel width and $L$ the channel length. This folds in the one thing the circuit designer controls: the aspect ratio $W/L$. A wider or shorter device (larger $W/L$) carries more current at the same overdrive. Still A/V².

Where it appears

Both flavours sit at the front of every drain-current equation:

• Triode: $i_D=k_n[V_{OV}{V}_{DS}-V_{DS}^2/2]$.
• Saturation (square-law): $i_D=\frac{1}{2}{k}_n{V}_{OV}^2$, with $V_{OV}=V_{GS}-V_t$ the Overdrive voltage.
• [[MOSFET transconductance|Small-signal $g_m$]]: $g_m=k_n{V}_{OV}=\sqrt{2k_n{I}_D}$.

$k_n$ ties the geometry and physics of the device to its current strength. It’s the MOSFET counterpart of the BJT’s saturation current $I_S$ in setting the absolute current scale.

Why it matters for design

$k_n$ spreads between nominally identical transistors and with temperature, just like the Threshold voltage. A given $V_{GS}$ can therefore produce wildly different $I_D$ in two “identical” parts. That spread is why you must not set the operating point by fixing $V_{GS}$ (see MOSFET biasing), and why good bias circuits use Negative feedback so that $I_D$ depends on stable external resistors rather than on $k_n$.