The gate-to-body capacitance

The source-to-substrate capacitance

The drain-to-substrate capacitance

Note: in the presence of series source/drain resistances the intrinsic (internal to the device) conductance and transconductances are related to the extrinsic (measured) transconductances and conductance by the following equation:

EXAMPLE 5.3: An n-channel MOSFET has Determine

SOLUTION: The intrinsic body transconductance

The coefficient

Therefore, and respectively. Thus, significant degradation in the transconductances and drain conductance may take place for large values of source/drain series resistances.

The two conductance terms appearing in the equivalent circuit shown in Fig.5.26(a) are the reverse-bias conductances of the source-substrate and drain-substrate diodes, and their values are very small (tending to zero).

Fig.5.26(b) The simplified equivalent circuit of a MOSFET.
A simplified equivalent circuit is shown in Fig.5.26(b).
For the circuit shown in Fig.5.26(b), the small signal voltage gain expression can be given by:
Note: at low frequencies, when the effects of the capacitances can be neglected, the voltage gain can be given by as expected.
Another simplified equivalent circuit, suitable for the calculation of the current gain, is shown in Fig.5.26(c).

Fig.5.26(c) The alternate simplified equivalent circuit for a MOSFET suitable for the calculation of the short circuit current gain.
From Fig.5.26(c), the short circuit current gain can be easily found to be:
Thus, the unity gain cutoff frequency (i.e., the frequency at which the absolute value of the short circuit current gain is equal to unity) can be given by

where
Now, note that in the strong inversion region.
Also, the drain current
Thus,
Hence,

where is the transit time of electrons in the channel.
This equation gives the theoretical maximum value for
Assuming the characteristic switching time for a MOSFET is obtained as
In reality, the measured switching times for MOSFETs are at least several times larger than that predicted above due to the parasitic and fringing capacitances that has to be added to the gate capacitance leading to the following modified expression for :

EXAMPLE 5.4: Calculate the unity-gain cutoff frequency for the MOSFET considered in Example 5.2. Compare this value with theoretical maximum value for , assuming

SOLUTION: The unity-gain cutoff frequency

The theoretical maximum value for = = 7.96 GHz.

An actual device would show a cutoff frequency, which is smaller of the two, thus, the actual unity-gain cutoff for the device considered in Example 5.2 would be 2.82 GHz.

Types of MOSFETs

Broadly, MOSFETs can be categorized into two types: enhancement and depletion.
Enhancement type devices are normally off, i.e., channel does not exist for and the applied must be greater than for the device to turn on.
On the other hand, depletion type devices are normally on, i.e., channel does exist even for and the applied must be reduced below for the device to turn off.
To put it simply, an n-channel enhancement type device has a positive , whereas an n-channel depletion type device has a negative .
Similarly, a p-channel enhancement type device has a negative , whereas a p-channel depletion type device has a positive .
The threshold voltage can be changed either by doping or by ion implantation, where high energy ions are made to bombard the surface and get embedded into it: since these are charged, they can change the charge state of the surface, and, hence, the threshold voltage.
The shift in the threshold voltage is related to the ion density by the relation: eg., negative ions (like Boron) implanted in a p-channel (n-substrate) device will compensate some of the positive depletion charges and make the threshold voltage less negative, however, note the same ions would shift the threshold voltage to more positive for n-channel (p-substrate) device.

EXAMPLE 5.5: An n-channel MOSFET with has a threshold voltage Determine the type and dose of ion implantation required to make it a depletion mode device with

SOLUTION: The oxide capacitance per unit area

The dose of ion implantation required

Since the threshold voltage is shifting towards negative value, hence, obviously, the type of implant required is positive ions (e.g., P, As, Sb, etc.), which would compensate the negative depletion charge of the substrate and push the threshold voltage towards negative direction.

Some Advanced Models

Unified Charge Control Model for MOSFETs

For MOSFETs, the UCCM equation for MIS capacitors [Eq.(4.24)] has to be modified to account for the channel potential, thus, the inversion charge is related to the gate-source and channel potential as follows:

where is the quasi-Fermi (electrochemical) potential measured relative to the Fermi potential at the source side of the channel, and the parameter accounts for the dependence of the threshold voltage on the channel potential in strong inversion, and, hence, on the position along the channel.
In order to get a better understanding of the term first consider the simplified version of the charge control model, given by
Now, in reality, the threshold voltage depends on the depletion charge.
Taking into account the dependence of this charge on the channel potential, one can write the corresponding position dependent threshold voltage as
This makes the charge control equation nonlinear and difficult to use in device modeling.
However, if Eq.(5.90) is linearized with respect to V, one can write where now is the value of the threshold voltage at the source side of the channel.
Thus, one obtains
A generalized solution for ns is used in UCCM, given by
This equation allows the direct determination of the carrier distribution along the channel as a function of

Saturation Region: The Region of the Channel with Velocity Saturation

Of late, area of considerable interest, since an accurate modeling of the pinch-off region is essential in order to obtain an exact drain current model in saturation.
Important to find a solution for the longitudinal field in the channel.
The model relies on the fundamental assumption that the carrier velocity in the saturated part of the channel is constant and equal to the saturation velocity, which implies that the carrier sheet density in the saturated part of the channel is also constant.
Another assumption made is that the substrate is lowly doped: this assumption oversimplifies the true physics of the saturation region, however, it also leads to a manageable theory with qualitatively correct features, which gives a fairly good fit to experimental data with a judicious choice of parameters such as the saturation velocity and the effective channel thickness.
The intrinsic saturation voltage can be defined as the intrinsic drain-source voltage for which the longitudinal electric field at the drain end of the channel just becomes equal to the saturation field
For the location in the channel where marks the boundary between the saturated and the non-saturated regions.
The boundary point moves towards the source with increasing drain-source voltage: this effect is called the channel length modulation.
Another important parameter is the channel potential at the boundary point
The two parameters and on the intrinsic gate-source voltage and have to be determined self-consistently using the models for the two regions with the requirement that the potential, the electric field, and the velocity be continuous at
For a description of the saturated region, it is necessary to consider a two-dimensional Poisson's equation of the form