#### Abstract

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.

#### 1. Introduction

Although MOSFET modeling is now well covered and addressed in BSIM, EKV, and PSP compact models [1], it is always interesting to present a semianalytic resolution of 1D Poissons equation which can be implemented in popular computers with usual software giving most physical results (potential and charges distribution) instantaneously. New approaches of MOSFET surface potential modeling were performed from analytic treatment and have brought a renewal in analytic resolution of surface potential [2–5]. We previously used a similar method in the analytic description of surface potential by Taylor expansion [6].

Oguey and Cserveny [7] proposed as early as 1982 a complete analytic model based on the gate and drain source voltages. An important step was reached in modeling by Enz et al. in 1995 [8], Iniguez et al. [9] in 1996, and Cheng [10] in 1998 who gave analytic expression of the inversion charge. We certainly do not pretend to provide an alternative method to the compact models implemented on the simulators for CAD, but are simply trying to provide analytical support to the understanding of strategic components of microelectronics.

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 (,) and the transconductance . The originality is based on a model in which the threshold voltage does not appears explicitly, but is replaced in the analytical expression by a parameter dependent on the surface potential at zero drain bias.

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) [11] which assumes that the electric field in the direction perpendicular to the channel is much greater than in the direction parallel to the channel and allows a 1D model of Poisson-Boltzmann equation. The different explicit equations (gate voltage and channel potential versus surface potential) are inverted using Taylor expansion, and we solve all equations until the point analytic calculations can be done then calculate the single integrals by Simpson algorithm encoded in simple C programs.

#### 2. Basic Assumptions in MOSFET

##### 2.1. The Surface Potential Equation

Under the gradual channel approximation [11], with the introduction of the reduced channel voltage as quasi-Fermi potential [12] and the correct charges densities are [4]

The Poisson-Boltzmann equation can be analytically solved using the 1Dmodel of Nicollian and Brews [13] from the charge density:

The surface electric field, , along (Figure 1) solution of (2) in , is by setting:

The gate voltage relative to flat band is

##### 2.2. The Surface Potential Dependence to Gate and Drain Bias

The gate voltage is an explicit function of and . Several solutions of were reported to express the band bending as an analytic function of the gate and channel voltages [3, 14]. Gildenblat et al. have given in [5, 14] a noniterative expression of the surface potential which serves as a reference for surface potential-based models. In the following, we generated by first-order Taylor expansion as previously done in [6].

versus at a constant drain bias is generated by , ( sample step and integer).

And at a constant is generated by:

( sample step and integer) is expressed as a function of at a constant by an analytic model previously developed by Baccarani et al. [15]: with the introduction of the dimensionless quantity:

#### 3. Analytic Model of Inversion Charges

##### 3.1. The Inversion Charges Dependence to Gate and Drain Bias

In an *-*MOSFET, the inversion charges are defined by the integral of electrons density over the “physical” thickness :

The “physical” inversion starts at the silicon surface with the surface potential = and ends at the abscissa corresponding to and = . The inversion charge dependence with channel potential at a constant noted can be written in terms of potential as follows:

From (8), becomes a single integral of with the limits only dependent of . Figure 2 shows versus with as a parameter in linear (strong inversion) and log scale (weak inversion).

A threshold voltage of inversion charges can be defined by the interpolation of the linear part of with the axis. = plots (Figure 3) give at a threshold voltage which differs from by a factor . is used in (12) instead of .

##### 3.2. Analytic Expression of the Inversion Charges Dependence to Gate and Drain Bias

The simplest analytic approximate expression of in the whole range of gate and drain bias is well represented by

*≈*1.2 is the slope factor defined by the exponential law if .

However, this formulation should contain adjustment coefficients to reduce the error between (11) and (12). This was done in 1996 by Iniguez et al. [9] with the introduction of adjustment coefficients based on the threshold voltage in an expression of inversion charges similar with (12).

We propose an alternative method by introducing a “charge linearization factor,” = which fits the slope in strong inversion, and a preexponential parameter which fits (12) with (11) in :

Another definition of the “charge linearization factor” was introduced by Sallese et al. [16] in strong inversion and gives results different from as shown in Figure 4. >1 increases when decreases. The difference between and results from which can be calculated from (5). and are interdependent and will be estimated in order to minimize the error between and .

Figure 4 shows plots calculated from and are well represented by a smoothing function as follows:

is the asymptotic value of at high gate voltages. is a slope factor which minimizes the error between and in a large range . Under this condition becomes with and is written as follows: with

The coefficient is dependent on the gate voltage by solution of (5) in = . Equation (17) gives, respectively, in strong and weak inversion the simplified expressions:

is a monotonic function in all inversion modes (Figure 5). The originality of the correction by is to give an expression of the inversion charges in which the threshold voltage is not explicit but included in and appears in plots.

The parameter varies from to and as shown in Figure 5 is different from . Nevertheless, in usual applications (Section 7), the derivative can be approximated by

##### 3.3. Equivalent Expression of Inversion Charge

By using the mathematical properties of the function: which has some similarities with in the range then (17) can be rewritten by setting (i) is an adaptive factor which varies between 0.5 and 1 ;(ii) and are fitting factors which minimize the error between and at .

These parameters are available in a large range of . Equation (23) gives an expression similar to the Unified MOSFET Channel Charge Model given by () and () in [10] and used in BSIM model [17]. Moreover, (17) and (23) are the synthesis between the expression of inversion charges given in [9, 10] in agreement with the theoretical model (11). Figure 6 shows the normalized expressions of the inversion charges at as a function of the gate voltage.

The term in braces in (23) can be integrated versus and gives an analytic expression of the drain current similar to Oguey and Cserveny model [7].

#### 4. Analytic Model of the Drain Current

The general expression for the drain current (including drift and diffusion) with a constant mobility follows:

##### 4.1. The Pao-Sah Double Integral

Using the inversion charges dependence to drain bias (developed in Section 3.1), the Pao-Sah double integral then reads

By substituting by = , and grouping with , the Pao-Sah double integral has no singular point and, (25) can be solved into iterated integrals from surface potential :

Equation (26) was previously calculated in a large range of drain and gate voltages and presented in [6].

##### 4.2. Simplified Expression of the Drain Current

Equation (17) gives a simplified drain current expression in a single integral:

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 (27) which must be numerically calculated by classical integration methods.

##### 4.3. Explicit Equation of the Drain Current

Following previous results we propose an analytic expression of the drain current in a square-logarithmic function of based on the adaptive coefficient by integration of (23).

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 [7] in an analytic model based on a control voltage derived from the gate voltage and from drain source functions , :

The inversion charge of this model is given by

Thereafter, the Oguey and Cserveny model has been simplified by Enz et al. [8]. The main difference in this paper is the use of the coefficient instead of = .

Equations (27) and (28) (models 2 and 4) coincide with the double integral of Pao-Sah (model 1). The analytic models (Figure 7) are summarized in Table 1.

#### 5. Mobility Model

In order to insure carrier drift velocity to be less than the saturation velocity at high electric field, a correction over constant mobility can be implemented in the drain current [18]. In the following, we use the mobility model developed by Roldan et al. [19]:

##### 5.1. Correction by the Transverse Electric Field

Several expressions are introduced to evaluate the mean electric field in relation with the channel inversion charges. BSIM models introduce the voltage defined by () and () in [10]. An excellent approximation of can be obtained from the equivalent gate voltage defined from (17) as follows:

The expression of the electric field calculated in (3) allows calculating as the mean electric field in the inversion region with a dimensionless adaptive coefficient .

Figure 8 shows the correction factors , compared with simplified BSIM 4.6.4 [17].

##### 5.2. Correction by the Lateral Electric Field

According to -MOSFET models in [20, 21], we use : with and the average of the lateral electric field:

The correction over constant mobility is introduced in the general expression of drain current by substituting by in (24) as follows:

##### 5.3. The Saturation Voltage

With mobility correction, the models of drain current present a maximum (Figure 9) at a saturation voltage defined, according to the mobility model by [17], [20], or [8]. curves are presented with the same model of correction by transverse electric field. The adaptive parameter in must then be applied to give the same current and to minimize the error between measured and calculated data.

In this paper, represented on Figure 10 is calculated from the iterative definition of drain current (Section 4.1) with substituted by . The saturation voltage is a linear function of in strong inversion and becomes constant in weak inversion.

#### 6. Short Channel Drain Current

##### 6.1. Correction of Saturation Voltage

The drain current formulation with mobility given from (39) is now written as follows:

Equation (40) leads to an unphysical , which must be clamped at . Gildenblat et al. [20] proposed to replace by a smoothing function with a parameter . From the analytical and explicit drain current expressions and , we can define a new function which includes the effect of velocity saturation by introducing the saturation voltage in (27):

The coefficient fits with ): and (28) becomes:

##### 6.2. Current-Voltage Characteristics

Figures 11 and 12 show the simulation results in strong and weak inversion with a mobility model deduced from (39).

The transfer characteristics (Figure 11) show linear variations in strong inversion and exponential variations in weak inversion. Figure 12 shows that the smoothing functions (41) and (43) give a unified formulation in the complete range of drain voltage.

##### 6.3. Channel Length Modulation

The channel length modulation (CLM) is a shortening of the length of the inverted channel region due to inversion layer in the drain junction. An accurate calculation of requires solving the 2D Poisson equation near the drain. An 1D approach may be used for standard expression of the depletion layer in the abrupt junction approximation [22]

Figure 13 shows an illustration of CLM with from (41) modified by (46). The drain current formulation is

This approximation is analogous to the early voltage and has the advantage to be described by the single analytic function .

Figure 14 gives a complete summary of the different as follows: (*⋄*) are data from Pao-Sah double integral from (40) with correction mobility in the range ; (∘) are data from the saturation current corrected by the channel length modulation (46) ; () are data from (41); The full line shows the single analytic function from (47).

##### 6.4. Drain-Induced Barrier Lowering

The drain-induced barrier lowering (DIBL) was described as soon as 1979 by Troutman et al. [23]. The MOSFET is a three-terminal device in which source-channel drain is a −− (or −−) double junction. We described in a previous paper the complete potential distribution in double junction from a 1D resolution of Poissons equation [24]. If this analytic description gives an accurate description of the potential in an unbiased double junction, the 1D resolution cannot be extrapolated with drain biased, which supposes a 2D device simulation. Most models describe the DIBL by a linear lowering of threshold voltage [21] = with the DIBL parameter .

In this paper, following the model of DIBL in [25], we propose to insert the increase of inversion charge by a quasi 1D calculation. With the same method, Cheng and Hu [26] calculated the threshold shift when

In the present paper with , , is relatively a small correction in .

In [25], the authors propose, as shown on Figure 15, to add in the square logarithm with the new expressions:

Due to the simplifying assumptions in the derivative , such a model gives a phenomenological description of DIBL, but must include fitting parameter to agree with experimental data. A new study is in progress in order to obtain a more accurate expression of and apply this model to inversion charges in (41) and (43) taking into account the lateral field to provide a complete expression of DIBL.

In the case of -MOSFETs, we have to add the Substrate Current-Induced Body Effect (SCIBE) which is the result of impact ionization by hot electrons coming from the source [23]. The expression of SCIBE is given by

and are adaptive parameters resulting from measurements. In the present work, this effect must be added to (51) from and and gives the total current. Figure 16 shows an example of SCIBE with .

#### 7. Analytic Model of TransConductance

The Pao-Sah double integral gives an expression of the transconductance from the derivative, in (26) [6]:

A simplified expression of the transconductance can be obtained from (27) by a derivative under the integral using from (20) as follows:

In this case, the integral in appears as , and is given by an analytical expression versus and :

These expressions correspond to a long channel MOSFET with a constant mobility. The mobility model given by (39) introduces a second term in the transconductance due to Vertical Field Mobility Reduction (VFMR):

is less than and appears as a corrective term in the transconductance. This contribution, negligible in long channel MOSFET, must be introduced as a corrective factor in the transconductance from

Each terms of this equation are calculated from (32), (33), (35), and (5). The contribution of CLM and DIBL in transconductance can be, respectively, deduced from (46), (47), (49), and (50).

A simple numerical calculation of the complete transconductance including VFMR, CLM, and DIBL is obtained from (51) by

Figure 17 shows transconductance plots, and Figure 18 shows the “normalized” ratio , versus drain current.

#### 8. Conclusion

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 . We introduce an analytic function of the inversion charge giving an expression of the drain current insuring a continuous transition between weak and strong inversion associated with a simple expression of the transconductance. Furthermore, the method gives a good approach of drain current with the velocity saturation. All the equations have been solved with a simple C-encoding program available on all personal computers. This program, associated with a graphic user interface (Figure 19), generates a graph (Figure 20) with different bias. The excellent agreement of the results obtained by an analytic continuous function of the inversion charge compared with those of standard models [1] can be considered as an accurate tool for microelectronics without access to specific CAD software and can provide a comprehensive overview of the complete MOSFET available in all inversion modes (Table 2).

#### Nomenclature

and : Boltzmann constant and temperature (Kelvin) | |

and : silicon and silicon oxide permittivity | |

: oxide thickness | |

: normalized oxide capacitance | |

: intrinsic carrier concentration in | |

and : dopant concentrations in | |

: thermal voltage | |

: bulk potential of p-doped silicon | |

: reduced potential | |

: band bending | |

: charge sheet threshold voltage | |

: intrinsic body factor | |

: Debye length (cm) | |

channel width and channel length. | |

*Numerical applications use SI units, except for the following:*

, , , and , in . | |

inversion charges , in C· | |

and , in farads·. |