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# Interconnect Simulation

1. The classical long-channel Pao and Sah model 2. The charge-sheet based models 3. Bulk Charge Model 4. Square law model 5. Modified charge sheet model

INTERCONNECT SIMULATION

1. The classical long-channel Pao and Sah model

The Pao-Sah model, published in 1966, was the first advanced long channel MOSFET model to be developed. While it retained the GCA, it didn’t invoke the depletion approximation and permitted carrier transport in the channel by both drift and diffusion current.

The formulation of the drain current equation is therfore general, but as a result requires numerical integration in two dimensions, which limits its application in CAD tools.

Approximations:

i . Gradual Channel Approximation is used. ii . Constant mobility is assumed.

iii . Uniform substrate doping is considered.

i . It is physically based.

ii . It gives a continuous representation of the device characteristics from weak to strong inversion even to the saturation mode of operation.

i. It requires excessive computational requriments since it requires numerical integration in two dimension, rendering it unsuitable to be used for circuit CAD.

2. The charge-sheet based models

The limited practical utility of the Pao-Sah model motivated a search for an approximate advanced analytical model, that is still accurate over a wide range of operating conditions.

The charge sheet model, introduced separately by Bacarani and Brews in 1978, has become the most widely adopted long channel MOSFET model that is accurate over the entire range of inversion.

In this model the inversion layer is supposed to be a charge sheet of infinitesimal thickness (charge sheet approximation).

The inversion charge density Qi can then be calculated in terms of the surface potential ψs. The drain current is then expressed in terms of the surface potential at the source and drain boundaries of the channel.

Approximations:

i . Gradual Channel Approximation is used.

ii          . The mobility is assumed to be proportional to the electric field and is constant with position along the channel.

iii        . Uniform substrate doping is considered.

i . It is physically based.

ii . It gives a continuous representation of the device characteristics from weak to strong inversion even to the saturation mode of operation.

ii . The charge sheet approximation introduces negligibly small error, and it is more computationally efficient than the classical model.

i . The boundary surface potentials cannot be expressed explicitly in terms of the bias voltages applied to the device, but must be found by a numerical process.

ii . The model is not valid in depletion or accumulation.

Different approaches have been introduced to circumvent this disadvantage. In it is shown that accurate numerical solutions for these surface potentials can be obtained with negligible computation time penalty.

In the surface potentials are computed using cubic splines functions. In the implicit equation including the surface potential is replaced by an approximate function.

Although all of these approaches have given good results, they have neglected the effect of the interface trap charge which is important in determining the subthreshold characteristics of the device, namely the subthreshold swing (the gate voltage swing needed to reduce the current by one decade).

3. Bulk Charge Model

The Bulk Charge model also known as variable depletion charge model, was developed in 1964, describes the MOSFET drain current only in strong inversion but of course has less computational requirements.

Approximations :

i . Drift current component only is considered

ii       . Constant surface potential is assumed

iii        . Id considered zero below threshold

i . Less computational time than the charge sheet model

i . The subthreshold region not defined

4. Square law model

This model has great popularity, when a first estimate to device operation, or simulating a circuit with a large number of devices is required. This model is obtained from the bulk charge model, on the assumption that VDS << 2φf+VBS .

i . Drift current component only is considered ii . Constant surface potential is assumed

iii        . Id considered zero below threshold

iv       . VDS << 2φf+VBS

i . Very small computational time than any other model ii . Suitable for hand calculations

i . The subthreshold region is not defined

ii . Overestimates the drain current in saturation region

Approximate models

There exists a large number of introduced approximate models. All of these models originate from Brews' charge sheet model, where approximations to the surface potentials in various operating regions of the device have been used.

This leads to different current equations each valid only in a specific region. The resulting equations are then empirically joined using different mathematical conditions of continuity.

i . They have good accuracy in the desired region of operation.

ii . They are very efficient from the point of view of computational time.

i . The error increases in the transition regions between different modes of operations. ii . They include many non-physical fitting parameters.

5. Modified charge sheet model

The last discussed MOSFET models, have a common illness, no interface charges are included which play a great role in subthreshold region. So a modified model to the charge sheet model, which include the effect of interface charges is carried out in ICL, and will be presented now. The derivation begins by rewriting equation (2.3.1) in the following form :

I D = I D1+ I D2 (2.3.6.1)

where ID1 is due to the presence of drift: after mathematical manipulation and following the same approximations as charge sheet model we reach the following drain current equations: where ψs0 is the surface potential at the source end of the channel, ψsL is the surface potential at the drain end of the channel, both are referred to the bulk. And their values are computed from the following two implicit equations.  Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
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