Fig.5.10 The variation of the drain saturation current as a function of the gate voltage for three different values of the series source resistance

Fig.5.11 The drain current drain-to-source voltage characteristics for different values of

The series source resistance reduces the drain current, and the series drain resistance increases the drain-to-source saturation voltage.
Both series source resistance and series drain resistance reduce the drain conductance at low drain-to-source voltages.

Velocity Saturation Effects in MOSFETs

In modern day MOSFETs, the channel length is very small, the electric field in the channel is very high, and the velocity saturation effects are very important.
The measured electron and hole mobilities in the inversion layer may be quite different than those measured in the bulk.
Note: the channel, in reality, is under a two-dimensional electric field, one directed longitudinally from the gate to the substrate, and the other directed laterally along the length of the channel.
The effective inversion layer thickness is approximately given by thus, a large vertical field creates a narrow inversion layer, and vice versa.

Fig.5.12 The random path of electrons in the channel, undergoing surface scattering, which is more intense in narrow channels.
Electrons in the channel move in random directions, undergoing surface scattering, which increases for narrow channels thus their mobility drops.

Fig.5.13 The variation of the electron and hole mobilities in the channel as a function of the gate electric field.
The dependence of the electron and hole mobilities on the gate field can be crudely approximated by

where n0 and p0 are the electron and hole mobilities for
It is very interesting to note that in highly constricted channels or at low temperatures, the carrier mobility is seen to get enhanced.
This is because for these cases, the electron motion in the direction perpendicular to the interface gets quantized, and the channel electrons behave like a two-dimensional electron gas (2DEG).
Thus, the surface scattering is not that important, and the impurity scattering is screened by a high density of electrons in the channel.
Such enhancement of electron mobility was observed in GaAs, and is exploited in high electron mobility transistors (HEMTs) or modulation-doped field effect transistors (MODFETs).

Effects of Velocity Saturation on the I-V Characteristic

For this derivation, a simple two-piece linear approximation for the electron velocity is used:
where is the electric field required for velocity saturation, and is the saturation limited thermal velocity.
Recall: in the linear region, the I-V characteristic can be given by:

where
The saturation current can now be found by assuming that the current saturation occurs when the electric field at the drain side of the channel exceeds the critical field required for velocity saturation.
This is a much more realistic assumption than the Shockley model, which assumes saturation occurs when
The constant mobility model is still used for drain voltages below the saturation voltage.
The absolute value of the electric field in the channel at drain voltages below the saturation voltage can be obtained from Eq.(5.21):
Integrating Eq.(5.35) from 0 to x, the following equation for the channel potential is obtained for drain voltages below the saturation voltage:
The solution of this equation is given by
Substituting Eq.(5.37) into Eq.(5.35), the following expression for the electric field as a function of distance is obtained:

and the electric field F(L) at the drain side of the channel (where it is the largest),
From the condition the drain saturation current can now be found as
At very large values of the term in the brackets in Eq.(5.40) may be expanded into Taylor series, which gives the following expression for the saturation drain current for long channel devices: , which does not take into account the velocity saturation effects.
For long channel devices, as predicted by the constant mobility model, hence, the velocity saturation effects are not too important for long channel devices.
Example: assume then for channel length velocity saturation effects on the drain saturation current may be neglected.
However, for modern day MOSFETs, the typical gate length is much smaller than (recently, Intel has introduced processors using technology), where the velocity saturation effects are extremely important.
In the limiting case for short channel devices, when from Eqs.(5.40) and (5.41), it is seen that
Note: for short channel device, the drain saturation current is times smaller than the value predicted by the constant mobility model; and it becomes linearly dependent on instead of the familiar square law relation.
while plotted as a function of for a long channel device, shows a linear behavior; however, for short channel devices, it shows a significant departure from linearity a measure of whether the device is a short-channel or a long-channel device.
The drain saturation voltage is also much smaller than that predicted by the constant mobility model.

Fig.5.14 The variation of the drain saturation current as a function of the gate length for three different values of the gate voltage (3 V, 5 V, and 7 V). The drain saturation current predicted by the constant mobility model (shown by the dashed line) is also shown for comparison.
The effects of source/drain series resistance, for these cases, can be accounted for (as done earlier for long channel devices), and the following expressions for the drain saturation current and the drain saturation voltage are obtained:

Interpolated Relation

The following interpolation formula for the MOSFET I-V characteristic has been proposed by Shur, which describes both limiting cases correctly:
This was one of the earlier formulas, and a huge amount of work has been done in this area for the last ten years or so, in order to further refine the description of the behavior of short-channel MOSFETs.
In practical devices, the I-V characteristics do not completely saturate at large drain-to-source voltages, and this is related to the short channel and other nonideal effects in MOSFETs.
In order to account for the finite slope of the output characteristics in saturation, the following modification to the drain current expression has been proposed:

where is referred to as the channel-length modulation parameter (an extremely important parameter for short channel device a measure of the nonidealities present in the device)