Similar to the short channel device, the threshold voltage of a narrow channel (along the width) device increases with a reduction in the effective device width Weff due to the fringing fields outside the gate region, and the change in the threshold voltage as a function of Weff can be given by

where is a constant.

Fig.5.19 Variation of the threshold voltage with the channel width.
Another non-ideal effect that may be especially important for short-channel devices is the injection of electrons from the channel directly to the gate dielectric, where these electrons get trapped => hot electron effect.
This phenomenon takes place because the carriers gain sufficient energy while traversing the drain depletion region, which contains a high electric field, and has been used to advantage in the FAMOS (Floating gate avalanche MOS) structures used in memories.
Avalanche breakdown of the drain-substrate junction can cause a sharp increase in the drain current, and can damage the device unless it is controlled by some external means.
Typically, avalanche breakdown for a heavily doped drain-moderately doped substrate junction takes place at approximately 8 to 10 V.
Another very important nonideal and potentially hazardous situation may arise due to punchthrough, where the drain and source depletion regions touch each other and cause abnormally large current to flow through the device: this effect is particularly severe for short channel devices.
Punchthrough effect creates a superlinear increase in the drain current with the drain voltage, even at gate voltages below the threshold voltage.

Subthreshold Conduction

So far, we have considered current flow in a MOSFET only when the gate voltage exceeds the threshold voltage.
However, in reality, a finite (nonzero) current does flow in a MOSFET even for gate voltages below the threshold voltage, and this effect is more marked for short channel length devices than their long channel counterparts.
This current is referred to as the subthreshold current, and it flows for when the surface potential lies between the ranges of the onset of weak inversion and the onset of strong inversion.
The mechanism responsible for subthreshold current is quite different for long-channel and short-channel devices. 5.6.1 Subthreshold Current in a Long Channel Device
In a long channel device, the situation is similar to a BJT, where the source plays the role of the emitter, the drain is equivalent to the collector, and the substrate is the base.
The drain voltage drops almost entirely across the drain-substrate depletion region.
Thus, the component of the electric field parallel to the interface is small, and the subthreshold current is contributed primarily by diffusion, just as the case for BJTs.

Fig.5.20 The depletion regions associated with a (a) long channel and (b) short channel device.
Thus, the subthreshold current can be evaluated as

where is the region where most electrons are located) is the effective cross-sectional area.
The electron density n at the surface is proportional to , and it decreases with y (perpendicular to the interface) proportionally to
where is the vertical electric field, given by
Thus, the effective depth where most of the electrons are concentrated, can be estimated as where y = 0 corresponds to the interface.
If the diffusion length of electrons in the substrate is much greater than the channel length L, then the electron density n should be a linear function of x, decreasing from the source towards the drain (just like the linear distribution of minority carriers in the base of a BJT):

where the volume concentrations for electrons at the source and the drain sides of the channel are given by

where V(y) is the potential given by is the length of the undepleted portion of the channel.
For long channel devices, it is assumed that the depletion widths at the source and the drain sides of the channel are small compared to the channel length L, and
Also, note that since
Using all the relations given above, the subthreshold current for a long channel MOSFET can be given by
The surface potential at the source can be expressed as a function of the gate voltage by noting that thus,

where
Note: For the subthreshold current becomes independent of the drain voltage.
This is expected since in a long channel device, most of the applied drain voltage drops at the drain-substrate depletion region, and since the current is diffusive in nature, there is no change in the current with the drain voltage.
Also, for large since the gradient of n is not affected by the drain voltage: a situation similar to BJTs, where the collector current in the forward active mode is independent of the collector-to-emitter voltage.
Note: the subthreshold current is almost independent of the drain voltage
The substrate bias shifts the threshold voltage to a more positive value, affects the surface potential, and thus the subthreshold current changes.

Fig.5.21 The subthreshold characteristics for a long channel device as a function of the gate voltage for different values of drain and substrate voltages.

Subthreshold Current in a Short Channel Device

In a short channel device, the source and drain depletion widths may be a significant portion of the channel length L, and, hence, can not be neglected.
To account for this effect, the term L in Eq.(5.67) is replaced by another term Leff, where where

where is the built-in voltage of the source/drain-substrate junction, and the surface potential is now found from the solution of the following equation:

where
The curves clearly show shifts in the subthreshold current for different values of drain voltages, a characteristic typical of short channel devices.
The subthreshold current is a strong function of temperature as well

Fig.5.22 The subthreshold characteristics for a short channel device as a function of gate voltage for different values of drain and substrate voltages.

Fig.5.23 The subthreshold characteristics as a function of gate voltage for two different temperatures (77 K and 300 K).

MOSFET Capacitances and Equivalent Circuit

Note: in a MOSFET, the charges in the depletion region and the inversion layer depend on the gate, source, drain, and substrate potentials; and the derivatives of these charges with respect to the terminal voltages give rise to MOSFET capacitances.
The small signal equivalent circuit shown in Fig.5.24 is the one used by the popular circuit simulation package called SPICE, and it contains:
the drain-to-source current source IDS,
two resistances (due to the quasi-neutral region resistances of the source and drain respectively)