Electronic Devices and Circuits Engineering Sciences 154

 
See The MOS Capacitor: The Basic Ingredient of MOSFETs

The "bottom line" of our consideration of the MOS capacitor is the relationship:

(source) (local copy)

 
From this relationship, lets now "derive" the all important characteristic relating drain current and drain-source voltage.  Consider the following configuration:

At some position x along the source-drain channel:

But since I_D is constant throughout conducting channel:

At the peak  so that stored charge at the drain vanishes and the equation is no longer valid.  At this point the drain current has reached its "saturation" value as indicated in the figure above.  The drain current at maximum follows the parabolic equation

.

Beyond the peak the current is injected into the insulating region and travels ballistically!

Freeze Frames from the SUNY-Buffalo applet MOSFET Operation - II
(click on images to enlarge)

This sequence illustrates how the channel distribution of n-type conductivity looks
at a series on points on the  I_d  vs. V_ds output characteristic.

Therefore in the "linear" regime:

(source) (local copy)

and in the "saturation" regime:

Important bottom line: To the extent that channel reduction can be ignored (output resistance infinite), in saturation the drain current of a MOSFET depends only on the gate-source voltage!

(source)(local copy)

For small signals in the "saturation" regime:

If we include the channel reduction effect (see figures above), we can write

which modifies the saturation of drain-Source characteristic as shown in the following (exaggerated) figure:

An additional compication:

Depletion Mode vs. Enhancement Mode MOSFETs

The threshold voltage depends on many factors and may be either positive or negative
(see the composite figure our earlier discussion)

Click on images to enlarge

For more see the following Commentary on the SUNY-Buffalo MOSFET Applets

pi-equivalent circuit: t-equivalent circuit:

The "Body Effect"
Substrate Voltage not equal to Source Voltage

Click on image to enlarge

When the body (substrate) - inversion channel junction is reverse biased the body voltage has no effect on the operation of the device!  However, when it is forward biased it tends to increase the threshold voltage (usually a "bad thing") according to the relationship:

where g  is the process-dependent "body-effect coefficient."

Click on image to enlarge (source)

As might be expected, the forward bias effectively acts to increase the body doping!  The following figures illustrate the effect:

Click on images to enlarge   Variation in the threshold voltage with varying substrate doping levels when the body-source voltage is zero.

Variation in the threshold voltage with varying body-source voltage at fixed substrate doping levels. (source)

Therefore, for a MOSFET to operate properly the body must be biased to the largest back bias in the circuit - viz., the most negative power suppy rail for NMOST and the most positive power supply rail for PMOS.  From a small signal point of view, the body should be at signal ground and, thus, a small signal variation in the gate voltage produces a variation in the drain current.

However, from above we see that

Small signal equivalent pi-circuit modified to included the effect of body-source signal voltages