Ids Vs Vds Characteristics for NMOS

Consider an NMOS device. When V_gst the transistor is off, whatever the drain voltage i.e. I_ds = 0. When V_gs>V_t, the device conducts.

If V_gs is constant and V_ds is variable then the resulting I_ds Vs V_ds curve have two region-

a)     Resistive Region:

When V_dsgs-V_t, V_gs>V_t,

Voltage across SiO₂ at source, V_gs = V_g-V_s

Voltage across SiO₂ at drain, V_gd = V_g-V_d

But, voltage across SiO₂ is not constant. Let’s assumed that channel is linear so voltage along the channel increases from the source to the drain. Since, the voltage across SiO₂ increases from drain to source. So there are infinite capacitances.

Consider one capacitance of length dx situated at a distance x meters from the drain, as shown in figure below-

The capacitance, C = Wεdx/D

The voltage v in excess of V_t across the capacitor is,

v = V_gd + x(V_ds - V_t)/L = V_gs – V_ds + xV_ds/L - V_t

The charge, q, induced on this capacitor is,

q =Cv = Wdx (V_gs – V_ds + xV_ds/L - V_t)/D

The total charge, Q, induced in the channel is-

Now, Q = tI_ds => I_ds = Q/t

Where, t is time to move across the channel.

And,

t = channel length, L/electron velocity

Again,

μ_n = electron velocity/electric field = electron velocity/(V_ds/L)

so,

Since, V_ds < V_gs - V_t   

so,

A linear relationship between I_ds and V_ds for a constant V_gs.

b)     Saturation Region:

When V_ds ≥ V_gs –V_t. As the drain voltage rises, the voltage across SiO₂ at the drain decreases.
At V_ds = V_gs - V_t

V_gd = V_gs – V_ds = V_gs – V_gs + V_t

       =>V_gd = V_t

This is the voltage necessary to just support inversion, this point on I_ds Vs V_ds curve is called “Pinch-off”.

At V_ds>V_gs – V_t, the point at which inversion ceases moves away from the drain.

The voltage difference along the inversion channel from the source to where it ceases is V_gs – V_t and excess potential V_ds – V_gs + V_t is dropped between the end of the inversion channel and the drain. It’s creates a high electric field across this very short distance and electrons are quickly swept across this area.

At and above pinch-off, the voltage between a and b is constant at V_gs – V_t. The channel length also be considered to be constant.

Thus current flow is constant and the device is said to be saturated.

So, V_ds = V_gs - V_t

But, in practical situation, L decreases when V_ds increases so I_ds slightly increases.