## Explain gradual channel approximation used to calculate the MOSFET current-voltage characteristics.

Ans. Gradual Channel Approximation – Gradual channel approximation (GCA) used to calculate the currenvoltage characteristics of MOSFET. It was proposed by William Shockley. GCA is based on the assumption that the electric charge density related to the variation of the electric field in the channel in the direction parallel to the semiconductor- insulator interface is much smaller than the electric charge density related to the variation of the electric field in the direction perpendicular to the semiconductor insulator interface. Therefore, it is assumed that the channel potential is a gradually changing function of position and is varying very little over the distance of the order of the insulator thickness (t_{ox}) as shown in fig. 2.6. The two-dimensional Poissons equation for the semiconductor is –

Where, F_{x} and F_{y} are the x and y components of the electric field in the channel, _{s}is the dielectric permittivity of the semiconductor and is the charge density in the channel. GCA is valid under the assumption

The cross sectional view of an n-channel MOSFET is shown in fig. 2.7. At the surface end x = 0 and at the drain end x = L. The potential at the source i.e., the channel potential is represented by V_{c}(x) and Q_{s} is the total surfacecharge density induced into the semiconductor layer considering a situation in which the device operates in the above threshold region and an inversion layer is present, the induced surface charge density Q_{s} is given by

Q_{s} = -C_{ox}[V_{G} – 2_{F} – V_{FB }– V_{C}(x)] ……..(iii)