the difference between K coefficient and gm in the mosfet
The letter \$K\$ is typically used to isolate the process-dependent parameters from the design parameters. It is not the same as the transconductance gain \$g_m\$.

For example, the most simple model equations of a mosfet can be stated in the triode and saturation region as

$$\begin{align}
i_{DS} &amp;= K\cdot\frac{W}{L}\left((v_{GS}-v_{TH})v_{DS} - \frac{v_{DS}^2}{2}\right) &amp;\text{triode}\\
i_{DS} &amp;= K\cdot \frac{W}{2\cdot L}(v_{GS}-v_{TH})^2 &amp; \text{saturation}
\end{align}$$

In these equations, the letter \$K\$ means the following

$$K = \mu\cdot C_{OX}' = \mu\cdot \frac{\epsilon_{ox}}{t_{ox}}$$

where \$\mu\$ is the electron or hole mobility, \$\epsilon_{ox}\$ is the permittivity of the gate material (usually Silicon-dioxide) and \$t_{ox}\$ is the oxide thickness. None of these parameters can be changed by the IC designer.

The transconductance gain \$g_m\$ is defined as the derivative of the current to the gate-source voltage, or

$$g_m = \frac{\partial i_{DS}}{\partial v_{GS}}$$

If you used the model equations from before, you can calculate them in the linear and saturation region

$$\begin{align}
g_m &amp;= K\cdot \frac{W}{L}\cdot v_{DS} &amp; \text{triode}\\
g_m &amp;= K\cdot \frac{W}{L}\cdot (v_{GS}-v_{TH}) &amp; \text{saturation}
\end{align}$$

The transconductance gain \$g_m\$ will depend on its biasing and size and is similar to a BJT's \$\beta\$.