SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

8.3. CURRENT-VOLTAGE CHARACTERISTICS 367

and the saturated channel current becomes, from equation 8.3.7 (the drain current does not
change in our simple model onceVDS≥VDS(sat)),

ID(sat)=go

[

Vp

3

−Vbi+VGS+

2(Vbi−VGS)^3 /^2

3 Vp^1 /^2

]

(8.3.10)

This expression will be reexamined with a better approximation later.
An important parameter of the device is the transconductancegm, which defines the control
of the gate on the drain current. From equation 8.3.7, the transconductance becomes

gm=

dID

dVGS

∣∣

∣

∣

VDS=constant

=go

(VDS+Vbi−VGS)^1 /^2 −(Vbi−VGS)^1 /^2

Vp^1 /^2

(8.3.11)

From equation 8.3.5 and equation 8.3.11 we can see that the transconductance is improved
by using a higher-mobility material as well as a shorter channel length. An improved transcon-
ductance means the gate has a greater control over the channel. This results in higher gain and
high-frequency capabilities, as will be discussed later.
When the source-drain voltage is small, the expression for the current can be simplified by
using
VDSVbi−VGS (8.3.12)

Using the Taylor series, we then get from equation 8.3.7,

ID=go

[

1 −

(

Vbi−VGS

Vp

) 1 / 2 ]

VDS (8.3.13)

The device is ohmic in this regime, as shown in figure 8.9, with a transconductance

gm=

goVDS

2 Vp^1 /^2 (Vbi−VGS)^1 /^2

(8.3.14)

In the saturation regime the transconductance is, from equation 8.3.10,

gm(sat)=go

[

1 −

(

Vbi−VGS

Vp

) 1 / 2 ]

(8.3.15)

In the model discussed here (known as the Shockley model) the current cannot be calculated
beyond pinch-off. At pinch-off, the channel width becomes zero so that the electron velocity
must, in principle, go to infinity to maintain constant current. This, of course, does not happen.
We will now discuss, using physical arguments, what happens in the saturation region.