Based on a 1D Poissons equation resolution, we present an analytic model of inversion charges allowing calculation of the drain current and transconductance in the Metal Oxide Semiconductor Field Effect Transistor. The drain current and transconductance are described by analytical functions including mobility corrections and short channel effects (CLM, DIBL). The comparison with the Pao-Sah integral shows excellent accuracy of the model in all inversion modes from strong to weak inversion in submicronics MOSFET. All calculations are encoded with a simple C program and give instantaneous results that provide an efficient tool for microelectronics users.
Although MOSFET modeling is now well covered and addressed in BSIM, EKV, and PSP compact models [
Oguey and Cserveny [
From the analytical expression of inversion charge as a function of gate and drain bias, we attempted to provide a single analytical expression that achieves explicit functions of the drain current
It became obvious to us that the influence of other parameters could be included in these equations by more complex developments based on quasi two-dimensional analysis that exceeded this paper. Thus, we have not considered the specific effects: ballistic transport, tunneling through the oxide gate, which alone account for modeling of complex developments and led to numerical 2D treatments.
The presentation is made under the Gradual Channel Approximation (GCA) [
Under the gradual channel approximation [
The Poisson-Boltzmann equation can be analytically solved using the 1Dmodel of Nicollian and Brews [
The surface electric field,
The
The gate voltage
The gate voltage is an explicit function of
And
In an
The “physical” inversion starts at the silicon surface with the surface potential
From (
Normalized [
A threshold voltage of inversion charges
The threshold voltages
The simplest analytic approximate expression of
However, this formulation should contain adjustment coefficients to reduce the error between (
We propose an alternative method by introducing a “charge linearization factor,”
Another definition of the “charge linearization factor”
Figure
The coefficient
The parameter
By using the mathematical properties of the function:
These parameters are available in a large range of
Normalized inversion charges
The term in braces in (
The general expression for the drain current
Using the inversion charges dependence to drain bias
By substituting
Equation (
Equation (
This expression describes the current-voltage characteristics in all inversion modes, insuring a continuous transition between weak and strong inversion. Unfortunately, there is no primitive function for the one defined by (
Following previous results we propose an analytic expression of the drain current in a square-logarithmic function of
is a dimensional factor.
The drain current, represented by a square-logarithmic function of gate and drain voltage, was proposed as early as 1982 by Oguey and Cserveny [
The inversion charge of this model is given by
Thereafter, the Oguey and Cserveny model has been simplified by Enz et al. [
Equations (
Analytic models.
Drain current | Inversion charges | ||
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Model 1 | Pao-Sah integral | ( |
( |
Model 2 | Single integral | ( |
( |
Model 3 | [ |
( |
( |
Model 4 | Analytic model | ( |
( |
The different drain currents
In order to insure carrier drift velocity to be less than the saturation velocity
Several expressions are introduced to evaluate the mean electric field
The expression of the electric field calculated in (
Figure
The correction factor
According to
The correction over constant mobility is introduced in the general expression of drain current by substituting
With mobility correction, the models of drain current
In this paper,
The saturation voltage
The drain current formulation with mobility
Equation (
The coefficient
Figures
Transfer characteristics
(
The transfer characteristics
The channel length modulation (CLM) is a shortening of the length of the inverted channel region
Figure
This approximation is analogous to the early voltage and has the advantage to be described by the single analytic function
Figure ( (∘) are ( The full line shows the single analytic function
The different (
The drain-induced barrier lowering (DIBL) was described as soon as 1979 by Troutman et al. [
In this paper, following the model of DIBL in [
In the present paper with
In [
The total current
Due to the simplifying assumptions in the derivative
In the case of
An example of SCIBE (
The Pao-Sah double integral gives an expression of the transconductance from the derivative,
A simplified expression of the transconductance can be obtained from (
In this case, the integral in
These expressions correspond to a long channel MOSFET with a constant mobility. The mobility model given by (
Each terms of this equation are calculated from (
A simple numerical calculation of the complete transconductance including VFMR, CLM, and DIBL is obtained from (
Figure
Transconductance versus
Normalized transconductance [
In this paper, we propose a solution of the Poisson-Boltzmann equation which describes the physical parameters of the MOSFET under gate and drain bias. The Taylor expansion of inverse functions is well suited in the case of implicit functions and gives an accurate solution of the channel potential
The simplified analytic drain current models.
Physical constants |
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Process parameters |
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Device voltages |
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Surface potential in |
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Surface potential |
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Mobility |
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Drain current from Pao-sah |
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Gate voltage factors |
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Analytic expression of drain current |
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Saturation voltage |
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Drain current with saturation |
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Channel length modulation |
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Drain-induced barrier lowering |
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The graphic user interface.