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Active and Passive Electronic Components
Volume 2010, Article ID 268431, 11 pages
Research Article

Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral

1LaMIPS, Laboratoire commun NXP-CRISMAT, UMR 6508 CNRS ENSICAEN, UCBN, 2, rue de la Girafe BP 5120, F-14079 Caen Cedex 5, France
2NXP Semiconductors 2, Esplanade Anton Philips Campus Effiscience, Colombelles, BP 20 000 14906 Caen Cedex 9, France

Received 16 October 2009; Revised 24 March 2010; Accepted 21 April 2010

Academic Editor: Ashok Goel

Copyright © 2010 Hugues Murray et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We propose a simple model, derived from Pao-Sah theory, valid in all modes from weak to strong inversion, to calculate the drain current in Metal Oxide Semiconductor Field Effect Transistor (MOSFET). The Pao-Sah double integral is decomposed into single integrals with limits of integration calculated from Taylor polynomials of inverse functions. The solution is presented analytically wherever possible, and the integration is made from simple numerical methods (Simpson, Romberg) or adaptative algorithms and can be implemented in simple C-program or in usual mathematical software. The transconductance and the diffusion current are also calculated with the same model.

1. Introduction

The Metal Oxide Semiconductor Field Effect Transistor (MOSFET), first proposed in 1926 by Lilienfeld [1], is considered one of the most widely used electronic devices, particularly in digital integrated circuits due to the attractivity of Complementary Metal Oxide Semiconductor (CMOS) logic, a device using energy only during transition states. Since the early days and its simplified model of a conductive channel when the gate voltage 𝑉𝑔 is above a threshold voltage 𝑉𝑇 [2], the MOSFET has certainly been one of the electronic devices which received the most extensive attention from the microelectronic design community. The requirement in the knowledge of the MOSFET working with a gate voltage just below threshold, when the device is still conducting, has induced a complete modeling of drain current in all conductive modes from strong to weak inversion. The first attempt was in 1966, from Pao and Sah [3], who gave an expression of the drain current in a double integration format derived from the equation of the inversion charges  versus surface potential. Pierret and Shields gave in 1983 a single integral expression based on the derivative of the electric field [4]. In 1995, Persi and Gildenblat proposed a computational calculation of the double integral by numerical treatment of carrier concentration and surface potential [5]. Recently, a complete history of MOSFET modeling [6] has given a reference for users of MOS transistors.

The MOSFET modeling is now so much well covered and addressed in BSIM, EKV, and PSP compact models [8] that yet-another-paper on the topic of MOSFET models among the thousand of previous works could appear unnecessary. But, if the state-of-the-art MOSFET models [8] are always the reference in Computer Aided Design (CAD), it could also be interesting to present a semianalytic resolution of Pao-Sah integral which can be implemented in usual commercial software, giving most results quite instantaneously.

Although most of the resolutions of Poisson-Boltzmann equation use finite element software [911], we prefer to focus our attention on the physics of the surface potential in order to estimate the effect of gate/drain bias. In the same time, an analytic resolution has the advantage to highlight the influence of the different physical parameters on the derived characteristics. We previously used a similar method in the analytic description of scanning capacitance microscopy based on silicon surface potential [12].

Under thermal equilibrium conditions, mobile charge densities are exponential functions of the potential distribution and this leads to a nonlinear differential equation for the potential 𝜑(𝑥) in the form of the well known Poisson-Boltzmann equation. Under the Gradual Channel Approximation (GCA) [13], this equation is solved analytically and gives the exact 1-D electric field 𝐹(𝑥). The gate voltage 𝑉𝑔  versus surface potential 𝜑𝑆 explicit equation is inversed using Taylor expansion with an iterative step compatible with the necessary accuracy of integral simulation. The same algorithm is used to estimate the surface potential  versus the channel potential 𝑉(𝑦) induced by drain bias.

We qualify our approach being “semianalytic” in the sense that we solve all equations analytically up to the point where analytic calculations can be done, then decompose the Pao Sah double integral into single integrals and finally we converge to an approximated solution by using simple iterative algorithms which can easily be implemented in usual commercial mathematical worksheet tools or even encoded in simple C programs. This approach is equally well suited for calculation (however not restricted to) of the drift current, the transconductance and the diffusion current, as is further discussed in next sections, since those quantities share the same type of integration methodologies.

2. Surface Potential in MOSFET

In this paper, like in most other papers dealing with analytic description of surface potential, we base our derivations on the gradual channel approximation [13]. In this method, the electric field magnitude in the 𝑦 direction parallel to the conducting channel is assumed to be much smaller than in the vertical 𝑥 direction toward silicon bulk allowing the formulation div𝐹=𝑑𝐹/𝑑𝑥 and 𝐹[𝑥,𝑦]=𝑑[𝜑(𝑥)]/𝑑𝑥. The Poisson-Boltzmann equation can then be solved analytically according to the 1-D model of Nicollian and Brews [14] from the complete charge density: 𝜌(𝑥,𝑦)=𝑞[𝑝(𝑥,𝑦)𝑛(𝑥,𝑦)+𝑁𝐷𝑁𝐴].

In the following, electrons distribution in the channel will be considered out of thermal equilibrium with the introduction of the quasi-Fermi energy [15]. The drift and diffusion electron currents are shown in Figure 1. Source and bulk will always be considered to be grounded (except in Section 9) and gate voltage 𝑉𝑔 is defined relative to flat band.

Figure 1: The channel scheme in the Gradual Channel Approximation.

The presentation relates to a 𝑝-type semiconductor (𝑛-MOSFET), but this is not restrictive and could easily be extended to 𝑛-type semiconductor. Upon usual notation conventions (see Nomenclature section), the bulk Fermi level is 𝜑𝑏. 𝜑(𝑥,𝑦) is the potential in every points of the semiconductor material and 𝑉(𝑦) is the channel voltage which defines the quasi-Fermi level offset from equilibrium for electrons. Upon application of a drain bias 𝑉𝐷, 𝑉(𝑦) ranges from 0 at the source side (𝑦=0) to 𝑉𝐷 at the drain side (𝑦=𝐿).𝑈𝑇=𝐾𝑇/𝑞 is the thermal voltage.

In the following, we use the reduced potentials 𝑢𝑏=𝜑𝑏/𝑈𝑇,𝑢(𝑥,𝑦)=𝜑(𝑥,𝑦)/𝑈𝑇 and 𝜉(𝑦)=𝑉(𝑦)/𝑈𝑇 instead of 𝜑𝑏,𝜑(𝑥,𝑦) and 𝑉(𝑦), respectively. 𝑢𝑆(𝑦)=𝑢(𝑥,𝑦)|𝑥=0,𝐹𝑆(𝑦)=𝐹(𝑥,𝑦)|𝑥=0,𝑛𝑆(𝑦)=𝑛(𝑥,𝑦)|𝑥=0 are shorthand notations for surface quantities.

In the first MOSFET modeling, charge densities have been written by introducing the quasi-Fermi level only in 𝑛(𝑥,𝑦) with the expressions:𝑛(𝑥,𝑦)=𝑛𝑖𝑒𝑢(𝑥,𝑦)𝜉(𝑦),(1)𝑝(𝑥)=𝑛𝑖𝑒𝑢(𝑥,𝑦)𝑁,(2)𝐴=𝑛𝑖𝑒𝑢𝑏𝑁,(3)𝐷=𝑛𝑖𝑒𝑢𝑏.(4)

After solving the 1-D Poisson equation 𝑑𝐹(𝑥,𝑦)/𝑑𝑥=𝜌(𝑥,𝑦), these expressions give the 1-D electric field by=𝐹(𝑥,𝑦)2𝐾𝑇𝑛𝑖𝜀𝑆×𝑒𝜉(𝑦)𝑒𝑢(𝑥,𝑦)𝑒𝑢𝑏+𝑒𝑢(𝑥,𝑦)[𝑒𝑢𝑏𝑒𝑢𝑏]𝑢𝑏𝑢(𝑥,𝑦)𝑒𝑢𝑏,(5) and its derivative𝑑𝐹(𝑥,𝑦)𝑑𝜉(𝑦)=2𝐾𝑇𝑛𝑖𝜀𝑆𝑒𝜉(𝑦)𝑒𝑢(𝑥,𝑦)𝑒𝑢𝑏2𝐹(𝑥,𝑦).(6)

As 𝑒𝑢𝑏𝑒𝑢(𝑥,𝑦) in the channel region, this expression is equivalent to𝑑𝐹(𝑥,𝑦)𝑑𝜉(𝑦)=2𝐾𝑇𝜀𝑆𝑛(𝑥,𝑦)2𝐹(𝑥,𝑦),(7)

This later expression gives a simple path to the Pao-Sah integral [4].

But, as this has recently been emphasized and extensively discussed in [16], since quasi-neutrality imposes that 𝑛(𝑥)=𝑁𝐷, (4) must be corrected by𝑁𝐷=𝑛𝑖𝑒𝑢𝑏𝜉(𝑦).(8)

With this exact formulation of the donor density, the electric field gradient from the 1-D Poisson equation (5) now reads𝑑𝐹(𝑥)=𝑑𝑥𝑞𝑛𝑖𝜀𝑆𝑒𝑢(𝑥,𝑦)𝑒𝑢(𝑥,𝑦)𝜉(𝑦)+𝑒𝑢𝑏𝜉(𝑦)𝑒𝑢𝑏,(9) which gives the electric field𝐹(𝑥,𝑦)=2𝐾𝑇𝑛𝑖𝜀𝑆𝑒𝜉(𝑦)𝐻[𝑢][𝑢](𝑥,𝑦)+𝐺(𝑥,𝑦),(10) where we have defined𝐻(𝑢)=𝑒𝑢+𝑒𝑢𝑏𝑢𝑏,𝑢1𝐺(𝑢)=𝑒𝑢𝑒𝑢𝑏𝑢𝑏.𝑢+1(11)

In this form, the derivative𝑑𝐹(𝑥,𝑦)𝑑𝜉(𝑦)=2𝐾𝑇𝜀𝑆𝑒𝜉(𝑦)𝐻[]𝑢(𝑥,𝑦)2𝐹(𝑥,𝑦),(12) is no longer with a numerator proportional to 𝑛(𝑥,𝑦) as in (7) and the Pao-Sah double integral must now be studied by other methods as outlined in Section 5. In the following, the surface electric field 𝐹𝑆(𝑦)=𝐹(𝑥,𝑦)|𝑥=0 is given by substituting 𝑢 and 𝑢(𝑥,𝑦) by 𝑢𝑆(𝑦) in (10)-(11).

3. The Surface Potential Dependence to Gate Bias

From electrostatic considerations [14], the gate voltage is expressed by𝑉𝑔=𝛾0𝑒𝜉(𝑦)𝐻𝑢𝑆𝑢(𝑦)+𝐺𝑆(𝑦)+𝑈𝑇𝑢𝑆(𝑦)𝑢(𝑏).(13)

The exponential expression in (13) allows to separate 𝑢(𝑠) and 𝜉(𝑦)𝐻𝑢𝜉(𝑦)=ln𝑆(𝑦)𝐸𝑢𝑆(𝑢𝑦)𝐺𝑆(𝑦),(14) with the derivative𝑑𝜉(𝑦)=𝑢𝑑𝑢(𝑠)𝑑𝐻𝑆(𝑦)/𝑑𝑢𝑆(𝑦)𝐻𝑢𝑆(𝑢𝑦)𝑑𝐸𝑆(𝑦)/𝑑𝑢𝑆𝑢(𝑦)𝑑𝐺𝑆(𝑦)/𝑑𝑢𝑆(𝑦)𝐸𝑢𝑆(𝑢𝑦)𝐺𝑆(,𝑦)(15) in which we introduce the dimensionless quantity𝐸(𝑢)=𝑉𝑔𝑈𝑇𝑢𝑢𝑏𝛾02.(16)

Several approximations to (13) have been reported. For instance, the equation𝑉𝑔2=Ψ(𝑠)+𝛾𝑛𝑖𝑁𝐴2expΨ(𝑠)𝑉(𝑦)𝑈𝑇+Ψ(𝑠)𝑈𝑇,(17) is often used in practice in a large variety of surface potential-based models with the band-bending: Ψ(𝑠)=𝜑(𝑠)𝜑𝑏. It provides a sufficiently accurate solution for the surface potential in inversion mode but generally lacks accuracy near the flat-band conditions: 𝑢(𝑠)=𝑢𝑏.

4. The Surface Potential Dependence to Drain Bias

The explicit relation 𝜉(𝑦)=𝑓[𝑢𝑆(𝑦)] is given above from (14). According to the importance of this expression in the Pao-Sah double integral, we must pay a particular attention to 𝜉(𝑦)=𝑓[𝑢𝑆(𝑦)] and to the inverse function 𝑢𝑆(𝑦)=𝑓1[𝜉(𝑦)] which will be used in the second integration of the Pao-Sah double integral.

Unfortunately, (14) cannot be inversed by mathematical function and needs a special treatment which is presented next, after having fixed the limits of 𝑢𝑆(𝑦) compatible with the denominator 𝐸[𝑢𝑆(𝑦)]𝐺[𝑢𝑆(𝑦)].

4.1. Boundary Limits of Surface Potential at Constant Gate Voltage

Equation (14) has only physical meaning when 𝐸[𝑢𝑆(𝑦)]𝐺[𝑢𝑆(𝑦)]>0. At a constant 𝑉𝑔 value, the reduced surface potential 𝑢𝑆(𝑦) ranges between 𝑢Low (lower value), solution of (13) with 𝜉(𝑦)=0𝑉𝑔=𝛾0𝐻𝑢Low𝑢+𝐺Low+𝑈𝑇𝑢Low𝑢𝑏,(18) and between 𝑢Up (upper value), solution of (13) with 𝜉(𝑦) leading to 𝑒𝜉(𝑦)0𝑉𝑔=𝛾0𝐺𝑢Up+𝑈𝑇𝑢Up𝑢𝑏,(19)𝑢Low and 𝑢𝑈 are defined by implicit relations to the gate voltage 𝑉𝑔. Following a methodology previously described in [17], 𝑢Low and 𝑢Up can easily be obtained iteratively from first order Taylor expansion by setting 𝑉𝑔=𝑘𝛿, in which 𝛿 is the sample step size and 𝑘 an integer. For a given step size and for 𝑘 varying from 0 to 𝑉𝑔/𝛿, 𝑢Low and 𝑢Up at iteration 𝑘 are calculated values at previous iteration according to 𝑢Low,𝑘+1=𝑢Low,𝑘+𝛿𝑑𝑢||||𝑑𝑉𝑔𝑢=𝑢Low,𝑘,(20)𝑑𝑢||||𝑑𝑉𝑔𝑢=𝑢Low,𝑘=1𝑈𝑇𝛾1+0𝑈𝑇𝑒𝑢𝑏𝑒𝑢𝑏𝑒𝑢Low,𝑘+𝑒𝑢Low,𝑘2𝐻(𝑢Low,𝑘)+𝐺(𝑢Low,𝑘)1,(21)𝑢Up,𝑘+1=𝑢Up,𝑘+𝛿𝑑𝑢||||𝑑𝑉𝑔𝑢=𝑢Up,𝑘,(22)𝑑𝑢||||𝑑𝑉𝑔𝑢=𝑢Up,𝑘=1𝑈𝑇𝛾1+0𝑈𝑇𝑒𝑢𝑏𝑒𝑢Up,𝑘2𝐺(𝑢Up,𝑘)1,(23)

In equations (20) and (22), the initial value 𝑘=0 corresponds to flat band (𝑉𝑔=0), and the starting conditions are 𝑢Low,0=𝑢Up,0=𝑢𝑏.

In (20), 𝑢Low,𝑘 reaches the threshod voltage 𝑉𝑇 when: 𝑘=𝑘𝑇=int(𝑉𝑇/𝛿); with the function int = (number down to the nearest integer). As a result, 𝑉𝑇=1.2819 if 𝑁𝐴=1017cm3 and 𝑡ox=10 nm and 𝑘𝑇=12819.  𝑢Low,𝑘𝑇=15.712=𝑢𝑏 when the step is 𝛿=104 and the corresponding band bending is Ψ𝑆=𝑈𝑇(𝑢Low,𝑘𝑇𝑢𝑏)=2𝜑𝑏.

As this has been previously shown [12], the error induced by first order Taylor expansion depends on the step size 𝛿. Equations (20) and (22) show that the relative error is in the same order of magnitude as the step 𝛿. For instance, a step 𝛿=1010 gives a relative error less than 1010.

Figure 2 shows 𝑢Low and 𝑢Up plots  versus 𝑉𝑔. The surface potential has well defined upper and lower limits in strong inversion, however, those limits are almost confounded in weak inversion (small channel voltage drop), which might give issues when 𝑢Low and 𝑢Up are used as integration limits in Pao-Sah integral (see Section 7).

Figure 2: Upper 𝑢Up and lower 𝑢Low limits of the reduced surface potential  versus gate voltage 𝑉𝑔. (a) equation (20) and (b) equation (22). 𝑁𝐴=1017cm3 and 𝑡ox=10 nm. 𝑉𝑔=𝑉𝑇 when (a) crosses the line 𝑢Low=𝑢𝑏.
4.2. The Inversion of 𝜉(𝑦)=𝑓[𝑢𝑆(𝑦)]

Equation (14) is an explicit relationship between 𝑢𝑆(𝑦) and 𝜉(𝑦). As this has been done in previous section with the calculation of the lower (𝑢Low) and upper (𝑢Up) limits of the surface potential, the inversion of (14) can be obtained by a first order Taylor expansion of the inverse function 𝑢𝑆(𝑦)=𝑓1[𝜉(𝑦)] by setting 𝜉(𝑦,𝑚)=𝑚 in which is the sample step size and 𝑚 an integer:𝑢𝑆,𝑚+1=𝑢𝑆,𝑚+𝑑𝑢𝑆||||𝑑𝜉𝑢𝑆=𝑢𝑆,𝑚,(24)𝑑𝑢𝑆||||𝑑𝜉𝑢𝑆=𝑢𝑆,𝑚=𝑒𝑢𝑆,𝑚𝑒𝑢𝑏𝐻𝑢𝑆,𝑚+2𝑈𝑇𝑉𝑔𝑈𝑇𝑢𝑆,𝑚𝑢𝑏/𝛾20+𝑒𝑢𝑏𝑒𝑢𝑆,𝑚𝐸𝑢𝑆,𝑚𝑢𝐺𝑆,𝑚1.(25)

In (24), initial value for 𝑢𝑆 is 𝑢𝑆,0=𝑢Low,𝑘; 𝑢Low,𝑘 from (20) with 𝑘=𝑉𝑔/𝑈𝑇, and iteration stops at 𝑚=𝑀=𝑉(𝑦)/𝑈𝑇.

Figure 3 shows [𝑢𝑆(𝑦),𝑉(𝑦)] plots along the channel in strong inversion (𝑉𝑔=5 V). (a) is derived from (14) by sampling 𝑢𝑆(𝑦) and (b) is derived from iterative (24) (step =0.01). Curves (a) and (b) merge with an error less than 104.

Figure 3: [𝑢𝑆(𝑦),𝑉(𝑦)] plots along the channel. (a) equation (14) and (b) equation (24) (step =0.01). 𝑁𝐴=1017cm3,𝑡ox=10 nm and 𝑉𝑔=5 V.

Note that this representation is 𝑢𝑆(𝑦)  versus 𝑉(𝑦) and not 𝑢𝑆(𝑦)=𝑔(𝑦) which would need the knowledge of the variation of 𝑉(𝑦)  versus 𝑦. Such variations suppose a complete 2-D resolution of the Poisson equation, but as is discussed later in Section 4, under the gradual channel approximation, a precise knowledge of 𝑉(𝑦)  versus 𝑦 is not needed for generating the main device current voltage and other related characteristics.

Figure 4 shows a complete set of curves 𝑢𝑆(𝑦)  versus 𝑉(𝑦) from 𝑉𝑔=1.2 up to 5 V. 𝑢𝑆(𝑦)=𝑢𝑆,𝑚 is calculated from (24) and 𝜉(𝑦) from 𝑢𝑆,𝑚 in (14). This figure illustrates the pinch-off voltage in strong inversion which appears when 𝑢𝑆(𝑦)𝜉(𝑦) becomes to decrease.

Figure 4: [𝑢𝑆(𝑦),𝑉(𝑦)] and [𝑢𝑆(𝑦)𝜉(𝑦),𝑉(𝑦)] plots for different 𝑉𝑔 bias. 𝑁𝐴=1017cm3 and 𝑡ox=10 nm.

In terms of surface electron density 𝑛𝑆(𝑦)=𝑛𝑖𝑒𝑢𝑆(𝑦)𝜉(𝑦) (Figure 5), the difference between strong and weak inversion is evident. The drift current dominates when surface electron density is almost constant and diffusion takes place when there is a 𝑑𝑛𝑆/𝑑𝑦 gradient.

Figure 5: The electron density [𝑛𝑆(𝑦),𝑉(𝑦)] plots along the channel from (1) and (14). 𝑁𝐴=1017cm3 and 𝑡ox=10 nm.

In strong inversion, the transition between 𝑛𝑆(𝑦)=𝑐𝑡𝑒 and the 𝑛(𝑦)𝛼𝑒𝜉(𝑦) regimes happens for a voltage 𝑉𝑡(𝑉𝑔). The inset plots in Figure 5 shows that 𝑉𝑡(𝑉𝑔)=0 when 𝑉𝑔 is equal to the threshold voltage 𝑉𝑇=2𝜑𝑏+𝛾|2𝜑𝑏|/𝑈𝑇1.28 V defined in the charge sheet model [18].

5. The Pao-Sah Double Integral

Under the gradual channel approximation [13] the drift drain-source current density varies from bulk silicon toward gate oxide-silicon interface and along the channel:𝐽(𝑥,𝑦)=𝑞𝜇𝑛𝑛(𝑥,𝑦)𝐹(𝑥,𝑦).(26)

Or, in terms of potential𝐽[]𝑢(𝑥,𝑦),𝜉(𝑦)=𝑞𝜇𝑛𝑛𝑖𝑒[𝑢(𝑥,𝑦)𝜉(𝑦)]𝐹[]𝑢(𝑥,𝑦),𝜉(𝑦).(27)

With the introduction of the inversion charge density𝑄inv=𝑞𝑥𝑑0𝑛(𝑥,𝑦)𝑑𝑥=𝑞𝑛𝑖𝑈𝑇𝑢𝑆0(𝑦)𝑒𝑢𝜉(𝑦)𝐹(𝑢,𝜉(𝑦))𝑑𝑢,(28) where we recall 𝐹=𝑈𝑇(𝑑𝑢/𝑑𝑥) and 𝑥𝑑 the inversion length in 𝑥 direction; the expression for the drain-source drift current follows𝐼drift=𝜇𝑛𝑊𝐿𝑉𝐷0𝑄inv𝑑𝑉(𝑦)=𝜇𝑛𝑊𝐿𝑈𝑇0𝜉(𝐿)𝑄inv𝑑𝜉(𝑦),(29) in which a total channel width 𝑊 is assumed. The Pao-Sah double integral then reads𝐼drift=𝑞𝑛𝑖𝜇𝑛𝑊𝐿𝑈2𝑇0𝜉(𝐿)𝑢𝑆0𝑒𝑢𝜉(𝑦)𝐹(𝑢,𝜉(𝑦))𝑑𝑢𝑑𝜉(𝑦).(30)

Since 𝜉(𝑦) can be expressed in terms of 𝑢𝑆(𝑦) or alternatively 𝑢𝑆(𝑦) defined as function of 𝜉(𝑦), the Pao-Sah double integral can be reduced into separate single integrals depending on the choice we make for the integration variable.

5.1. Solution from Surface Potential 𝑢(𝑠)

𝜉(𝑦) is a function of 𝑢𝑆(𝑦) from (14). In (30), 𝑑𝜉(𝑦) is replaced by𝑑𝜉(𝑦)=𝑑𝜉(𝑦)𝑑𝑢𝑆(𝑦)𝑑𝑢𝑆(𝑦),(31) in which 𝑑𝜉(𝑦)/𝑑𝑢𝑆(𝑦) from (15), is function of 𝑢𝑆(𝑦).

The Pao-Sah double integral can be expressed in terms of two single iterated integrals [19], and the drift current is given by𝐼drift=𝑞𝑛𝑖𝑊𝐿𝑈2𝑇×𝑢𝑆𝑢(𝐿)𝑆(0)𝑢𝑆0(𝑦)𝜇ne𝑒𝑢𝜉(𝑦)𝐹(𝑢,𝜉(𝑦))𝑑𝑢𝑑𝜉(𝑦)𝑑𝑢𝑆(𝑦)𝑑𝑢𝑆(𝑦).(32)

The integral in braces is integrated first and gives a function of 𝑢𝑆(𝑦) and the final integral is integrated with respect to 𝑢𝑆(𝑦) from 𝑢𝑆(0) to 𝑢𝑆(𝐿).

In (32):(i)𝑢𝑆(𝑦) is the surface potential along the channel.(ii)𝜉(𝑦) is a function of 𝑢𝑆(𝑦) from  (14).(iii)𝑑𝜉(𝑦)/𝑑𝑢𝑆(𝑦) is given by  (15).(iv)𝑢𝑆(0) is the surface potential in 𝑦=0. At a constant 𝑉𝑔 bias, it corresponds to 𝑢Low,𝑘 solution of (20) with 𝑘=𝑉𝑔/𝛿.(v)𝑢𝑆(𝐿) is the surface potential in 𝑦=𝐿. At a constant 𝑉𝑔, it corresponds to 𝑢𝑆,𝑚 solution of (24) with 𝑚=𝑉𝐷/𝑈𝑇.(vi)𝜇ne is the effective mobility which has extensively been studied previously [20, 21]. The correction over a constant mobility (𝜇𝑛=550cm2V1s1) used in this paper, principally depends on 𝐹[𝑢,𝜉(𝑦)] and is easy to implement in the integral.

5.2. Solution from Channel Potential 𝜉(𝑦)=𝑉(𝑦)/𝑈𝑇

In the drift current (30), it is possible to define the surface potential 𝑢𝑆  versus 𝜉 from (24). In this case, the channel potential is sampled according to Taylor expansion, and the integral on 𝜉=𝑚 must be replaced by a discrete summation𝐼drift=𝑞𝑛𝑖𝑈2𝑇𝑊𝐿𝑉𝐷/𝑈𝑇𝑚=0𝑢𝑆,𝑚0𝜇ne𝑒𝑢𝑚𝐹(𝑢,𝜉(𝑦))𝑑𝑢.(33)

This expression contains only one integral and is more suitable for numerical treatment with a simple worksheet; but this leads to a longer calculation time due to multiple loops in the summation.

5.3. Drift Current-Voltage Characteristics

Equation (32) has been calculated with a C-program in a large range of drain and gate voltages. The integration is made from Simpson algorithm.

Figure 6 shows the simulation results in log scale with 𝑁𝐴=1017cm3 and 𝑡ox=10 nm. The solid lines correspond to (32) and markers (+) to (33). The comparison between (32) and (33), in weak and strong inversion gives numerical values with accuracy better than 1%.

Figure 6: 𝐼drift  versus drain voltage 𝑉𝐷 from (32) in log scale. 𝑁𝐴=1017cm3,𝑡ox=10 nm and 𝑊/𝐿=10. Markers (+) corresponds to (33).

Figures 7(a) and 7(b) show the simulation results in linear scale. Curve (a) is compared to data reported in [4] (𝑁𝐴=1015cm3,𝑡ox=50nm) and curve (b) with more recent data reported in [7] (𝑁𝐴=5.1017cm3,𝑡ox=5nm). All these calculations have been performed using a constant mobility: 𝜇𝑛=550cm2V1s1.

Figure 7: 𝐼drift  versus drain voltage 𝑉𝐷 from (32) in linear scale compared with [4] and [7]. (a) 𝑁𝐴=1015cm3,𝑡ox=50 nm and 𝑊/𝐿=10, () corresponds to [4]. (b) 𝑁𝐴=5.1017cm3 and 𝑡ox=5 nm, () corresponds to [7].

6. Transconductance

The transconductance of the drift current is defined from the derivative [3]𝑔𝑚=𝑑𝐼drift||||𝑑𝑉𝑔𝑉𝐷=𝑑𝐼drift𝑑𝑢𝑆||||𝑉𝐷𝑑𝑢𝑆||||𝑑𝑉𝑔𝑉𝐷.(34)

The mathematical definition of the integral operator [22] gives the conditionIf𝑔(𝑥)=𝑥0𝑓(𝑢)𝑑𝑢,then𝑑𝑔(𝑥)𝑑𝑥=𝑓(𝑥),(35) and, consequently𝑑𝑑𝑢𝑆𝑢𝑆0𝑒𝑢𝜉=𝑒𝐹(𝑢,𝜉)𝑑𝑢𝑢𝑆𝜉𝐹𝑢𝑆,𝜉.(36)

The transconductance is calculated at a constant 𝑉𝐷 bias. The derivative of the drain current is only on 𝑢𝑆 and (36) in (32) results in𝑑𝐼drift𝑑𝑢𝑆=𝑞𝑛𝑖𝑈2𝑇𝑊𝐿𝑉𝐷/𝑈𝑇0𝜇ne𝑒𝑢𝑆𝜉𝐹𝑢𝑆,𝜉𝑑𝜉.(37)

The transconductance is𝑔𝑚=𝑞𝑛𝑖𝑈2𝑇𝑊𝐿𝑉𝐷/𝑈𝑇0𝜇ne𝑒𝑢𝑆𝜉𝐹𝑢𝑆,𝜉𝑑𝑢𝑆𝑑𝑉𝑔𝑑𝜉,(38) which from (31) gives a single integral of the surface potential 𝑢𝑆(𝑦)𝑔𝑚=𝑞𝑛𝑖𝑈2𝑇𝑊𝐿×𝑢𝑆𝑢(𝐿)𝑆(0)𝜇ne𝑒𝑢𝑆𝜉𝐹𝑢𝑆,𝜉𝑑𝑢𝑆(𝑦)𝑑𝑉𝑔𝑑𝜉(𝑦)𝑑𝑢𝑆(𝑦)𝑑𝑢𝑆(𝑦),(39) with𝑑𝑢𝑆(𝑦)=1𝑑𝑉𝑔𝑈𝑇𝛾1+0𝑈𝑇𝑒𝜉𝑑𝐻/𝑑𝑢+𝑑𝐺/𝑑𝑢2𝑒𝜉|||||𝐻+𝐺𝑢=𝑢𝑆(𝑦)1.(40)

Figure 8(a) shows [𝑔𝑚,𝑉𝑔] plots in log scale when the source is grounded (𝑉𝑆=0,𝑉𝐷=5 V) and Figure 8(b) shows the simulation results in linear scale using (39) compared to the charge sheet model in the quadratic region when 𝑔𝑚=𝜇𝑛𝐶0𝑉𝐷(𝑊/𝐿) is a linear function  versus 𝑉𝐷.

Figure 8: The transconductance from (39). (a) 𝑔𝑚=𝑓(𝑉𝑔)(𝑉𝑆=0 and 𝑉𝐷=5 V), (b) 𝑔𝑚=𝑓(𝑉𝐷)(𝑉𝑆=0 and 𝑉𝑔=5 V) (o) linear charge sheet model,. (c) normalized transconductance [𝐺(𝑖𝐹),𝑖𝐹] plots, parameters are 𝑁𝐴=1017cm3,𝑡ox=10 nm and 𝑊/𝐿=10.

In the E.K.V. model [23], the authors introduce the normalized transconductance: 𝐺(𝑖𝐹)=𝑔𝑚𝑛𝑈𝑇/𝑖𝐹 with 𝑖𝐹=𝐼𝐹/2𝑛𝛽𝑈2𝑇; 𝑛 is the slope factor and 𝛽=𝜇𝑛𝐶ox𝑊/𝐿. Figure 8(c) shows [𝐺(𝑖𝐹),𝑖𝐹] plots using (39) and (32) (𝐼𝐹=𝐼drift). The function 𝐺(𝑖𝐹), the asymtotes 𝐺(𝑖𝐹)=1 and 𝐺(𝑖𝐹)=(𝑖𝐹)1/2, respectively, in weak and strong inversion are in good agreement with this model.

7. The Accuracy of Pao-Sah Integral

We have integrated the Pao-Sah equation between the limits 𝑢𝑆(0) and 𝑢𝑆(𝐿) given by iterative equations (20) and (24). Figure 9 shows the variations of these integration limits  versus 𝑉𝐷, respectively, in strong (a) and weak inversion (b). The difference between 𝑢𝑆(0) and 𝑢𝑆(𝐿) is sufficient large in strong inversion. However, the situation is not the same in weak inversion. In this regime, the Pao-Sah integral will be integrated on a very small range and the accuracy of the simulation heavily depends on the number of digits used in the floating point arithmetic.

Figure 9: The limits of surface potentials 𝑢𝑆(0) and 𝑢𝑆(𝐿)  versus 𝑉𝐷. (a) strong inversion, (b) weak inversion. 𝑁𝐴=1017cm3,𝑡ox=10 nm,𝑊/𝐿=10.

The accuracy of Pao-Sah equation can be increased if the step 𝛿 in (20) is decreased. As an example, in weak inversion, 𝛿=1010 gives an accurate value of 𝑢Low,𝑘 which leads to 𝑉𝑔[𝑢Low,𝑘] equal to 𝑉𝑔 with an absolute error less than 1010.

8. The Diffusion Current

8.1. Integral of Current Diffusion

The diffusion current density of an n-MOSFET is given by𝐽di(𝑥,𝑦)=𝜇𝑛𝐾𝑇𝑑𝑛(𝑥,𝑦)𝑑𝑦.(41)

As shown in Figure 1, the gradient of concentration gives a diffusion of electrons moving from source to drain. The diffusion current is in the same direction as the drift current. For the same reason, it is evident that the diffusion current will have a relative contribution higher in weak inversion than in strong inversion. For instance, in strong inversion, a large region of the channel has a constant electrons concentration which does not contribute to the diffusion current.

The total diffusion current is the integral of the diffusion current density𝐼di=𝑊0𝑥𝑑𝐽di(𝑥,𝑦)𝑑𝑥=𝜇𝑛𝑑𝐾𝑇𝑊𝑑𝑦0𝑥𝑑𝑛(𝑥,𝑦)𝑑𝑥.(42)

Equation (28) gives the equivalence to potentials0𝑥𝑑𝑛(𝑥,𝑦)𝑑𝑥=𝑛𝑖𝑈𝑇0𝑢(𝑠)𝑒𝑢𝜉𝐹(𝑢,𝜉)𝑑𝑢,(43) and after integration on 𝑦, (42) becomes𝐼di=𝑞𝜇𝑛𝑊𝐿𝑛𝑖𝑈2𝑇𝐿0𝑑𝑢𝑆0(𝑦)𝑒𝑢𝜉𝐹(𝑢,𝜉)𝑑𝑢,(44) and reduces to single integrals𝐼di=𝑞𝜇𝑛𝑊𝐿𝑛𝑖𝑈2𝑇×𝑢𝑆0(0)𝑒𝑢𝜉(0)𝐹[]𝑢,𝜉(0)𝑑𝑢𝑢𝑆0(𝐿)𝑒𝑢𝜉(𝐿)𝐹[].𝑢,𝜉(𝐿)𝑑𝑢(45)

Equation (45) is equivalent to the well known expression𝐼di=𝜇𝑛𝑊𝐿𝑈𝑇𝑄inv(0)𝑄inv(𝐿).(46)

This expression is easily calculated from the exponential variations of 𝑄inv  versus 𝑉𝑔 in weak inversion. Equation (13) gives the exact relation 𝑉𝑔=𝑓[𝑢𝑆(𝑦),𝜉(𝑦)], and the inversion charge 𝑄inv  versus 𝑢𝑆(𝑦) is given from (28).

Figure 10 shows [𝑄inv,𝑉𝑔] plots according to (13) and (28) for different 𝑉𝐷 bias. 𝑄inv(0) corresponds to 𝜉(𝑦)=0 and 𝑄inv(𝐿) corresponds to the drain bias 𝜉(𝐿)=𝑉𝐷/𝑈𝑇. 𝑄inv(𝐿) is rapidly negligible in (45). Figure 11 shows 𝐼di(𝑉𝑔) plots in weak inversion calculated from (45) and compared with (46) from 𝑄inv in (28).

Figure 10: [𝑄inv,𝑉𝑔] plots in log scale for different channel voltages. 𝑁𝐴=1017cm3 and 𝑡ox=10 nm.
Figure 11: [𝐼di,𝑉𝑔] plots in weak inversion (𝑉𝐷=0.3 V) (–-) from (46) and (■) from (45). 𝑁𝐴=1017cm3 and 𝑡ox=10 nm.

In weak inversion, the diffusion current is well captured by𝐽di=𝐽0𝑒𝑉𝑔/𝑁𝐾𝑇,(47) with 𝑁=1.38, which is in good agreement with the slope of [𝑄inv,𝑉𝑔] plots (Figure 10).

8.2. Diffusion Plots 𝐼𝑑𝑖𝑓𝑓(𝑉𝐷)

A complete graph 𝐼di(𝑉𝐷) can be plotted by sampling 𝑢𝑆(𝐿) from (24). Figure 12 shows the contribution of the diffusion current (𝐼di) compared with the drift current (𝐼𝑑) in strong and weak inversion (𝐼𝑑1>𝐼di1 and 𝐼𝑑2<𝐼di2). The diffusion current is negligible in strong inversion and dominant in weak inversion.

Figure 12: The drift 𝐼𝑑 (–-) and diffusion 𝐼di (....) currents  versus 𝑉𝐷 in log scale. 𝐼𝑑1 and 𝐼di1 strong inversion: 𝐼𝑑2 and 𝐼di2 weak inversion. 𝑁𝐴=1017cm3,𝑡ox=10 nm, and 𝑊/𝐿=10.

9. Effects of Source Bias

In Section 5, the Pao-Sah integral is calculated with a zero bias source. If the source is biased with a voltage 𝑉𝑆𝐵  versus bulk silicon, the reduced drain-source potential 𝜉(𝑦) along the channel varies from 𝜉(0)=𝑉𝑆𝐵/𝑈𝑇 to 𝜉(𝐿)=𝑉𝐷/𝑈𝑇. Then, the surface potential 𝑢𝑆,𝑚 starts at 𝑢𝑆,𝑚𝑆 given from (24) with 𝑚𝑆=𝑉𝑆𝐵/𝑈𝑇 and stops at 𝑚=𝑀=𝑉𝐷𝐵/𝑈𝑇.

The drift and the diffusion currents are always given by (32) and (45) by substituting 𝑢𝑆,0 by 𝑢𝑆,𝑚𝑆 solution of (24) with 𝑚𝑆=𝑉𝑆𝐵/𝑈𝑇.

By setting𝐺𝑢𝑆=𝑞𝑛𝑖𝑊𝐿𝑈2𝑇𝑢𝑆0𝜇ne𝑒𝑢𝜉𝐹(𝑢,𝜉)𝑑𝑢.(48)

Equation (32) can be rewritten with 𝑚 index in 𝑢𝑆𝐼𝑑=𝑢𝑆,𝑀𝑢𝑆,0𝐺[]𝑢(𝑠)𝑑𝜉𝑑𝑢𝑆𝑑𝑢𝑆𝑢𝑆,𝑚𝑠𝑢𝑆,0𝐺[]𝑢(𝑠)𝑑𝜉𝑑𝑢𝑆𝑑𝑢𝑆,𝐼(49)𝑑=𝐼𝐹+𝐼𝑅.(50)

The integrals in (49) correspond, respectively, to the forward 𝐼𝐹 and reverse 𝐼𝑅 currents introduced in the E.K.V model [23]. The same expressions can also be defined in the diffusion current in (45).

10. Transfer Characteristics

The transfer characteristics represented in Figure 13 shows 𝐼=𝐼drift+𝐼di=𝑓(𝑉𝑔) calculated from (32) and (45) for constant 𝑉𝐷𝑆 and different 𝑉𝑆𝐵. The parameters of 𝐼=𝑓(𝑉𝑔) are 𝑉FB(flatband)=0, 𝑁𝐴=1017cm3,𝑡ox=10 nm, 𝑉𝐷𝑆=5 volt and 𝑉𝑆𝐵 from 0 to 1 volt.

Figure 13: The transfer characteristics 𝐼𝐷=𝑓(𝑉𝑔) for constant 𝑉𝐷𝐵=5 volt. 𝑁𝐴=1017cm3,𝑡ox=10 nm and 𝑉𝑆𝐵=[01] volt.

As a comparison, Figure 14 shows the curves 𝐼=𝑓(𝑉𝑔) with the same parameters reported in [4]. The conditions are 𝑉FB(flatband)=0.92 for 𝑁𝐴=1014cm3and 𝑉FB=0.86 for 𝑁𝐴=1015cm3. The oxide thickness is 𝑡ox=13 nm and the drain bias is 𝑉𝐷=1 volt. Our simulations are in good agreement with these results.

Figure 14: Transfer characteristics 𝐼𝐷=𝑓(𝑉𝑔). (a) 𝑁𝐴=1014cm3,𝑉FB(flat-band)=0.92 V. (b) 𝑁𝐴=1015cm3,𝑉FB=0.86 V. Markers (■) and () are from [4].

11. Conclusion

In previous papers, we presented an analytic resolution of Poisson-Boltzmann equation applicable in semiconductor junctions [12, 24]. We showed that the analytic method, which can be lead as far as possible without approximation (except the hypothesis of Channel Gradual Approximation), is able to illustrate the influence of physical parameters in surface potential and carriers density. By following the same methodology, the drift and diffusion current and the transconductance in MOSFET are given by iterated integrals easily solved quite instantaneously. The iterative treatment by Taylor expansions leads to a reasonable computation efficiency and simulation speed. All simulations using the flowchart as described in Section 5.1 are almost instantaneous with a simple C-program.

The excellent agreement of our results with the standard models [8], can be considered as an accurate tools for users in the complete knowledge of the MOSFET without access to specific CAD software. Moreover, we are well aware that the present work can not be a complete model of the MOSFET in terms of equivalent circuit with resistances and capacitances. But, by a simple calculation, available to a large community, it could give an overview of the complete MOSFET in all inversion modes with a single integral formulation.

Figure 15 shows the user interface for current calculation. The parameters are doping 𝑁𝐴, oxide thickness 𝑡ox, and gate and drain voltages 𝑉𝑔 and 𝑉𝐷.

Figure 15: The user interface of current and transconductance calculations. 𝑉𝑔=𝑉𝑔(𝑢Low,𝑘) calculated from 𝑢Low,𝑘 in (20).

12. Annex: Taylor Polynomials of Inverse Functions

If a function 𝑦=𝑓(𝑥) has continuous derivatives up to (𝑛)th order 𝑓(𝑛)(𝑥), then this function can be expanded in the following Taylor polynomials [25]𝑥𝑓(𝑥)=𝑓0+𝑛1𝑥𝑥0𝑛𝑓𝑛!(𝑛)(𝑥)+𝑅(𝑛),(51) where 𝑅(𝑛) is called the remainder after 𝑛+1 terms.

This expansion converges over a certain range of 𝑥, if lim𝑛𝑅(𝑛)=0, and the expansion is called the Taylor Series of 𝑓(𝑥) expanded about 𝑥0.

In inverse function, if 𝑦=𝑓(𝑥) is a one-to-one function, then 𝑓1 is continuous and, if 𝑓1 has continuous derivatives up to (𝑛)th order, 𝑓1 can be expanded in the Taylor polynomials. These conditions have been previously verified in the inversion of surface potential [12], with an accuracy compatible with the integral simulations.

The limits of Paoh-Sah integral are based on the inverse function 𝑢𝑆(𝑦)=𝑓1[𝜉(𝑦)], The first derivative variations given by (25) are shown in Figure 16 for different 𝑉𝑔 bias. The first derivative is define in all the [0,𝑉𝐷] region and varies from 1 to 1013 without discontinuity. The strong decrease observed for a 𝑉𝐷 value depending on 𝑉𝑔 is due to the asymptotic value in the surface potential 𝑢𝑆(𝑦) when 𝑢𝑆(𝑦) reaches 𝑢Up, solution of 𝐸(𝑢)𝐺(𝑢)=0.

Figure 16: The first derivative 𝑑𝑢𝑆(𝑦)/𝑑𝜉(𝑦)  versus drain voltage 𝑉𝐷 for different 𝑉𝑔 bias. 𝑁𝐴=1017cm3 and 𝑡ox=10 nm.


𝑉𝑔: Voltage gate with bulk silicon grounded𝜀𝑆 and 𝜀ox: Silicon and silicon oxide permittivity𝑡ox: Oxide thickness𝐿: Channel length𝑊: Channel width𝑛𝑖: Intrinsic carrier concentration in cm3𝑁𝐴 and 𝑁𝐷: Dopant concentrations in cm3𝑈𝑇=𝐾𝑇/𝑞: Thermal voltage𝜑(𝑏)=𝜑𝑏=𝑈𝑇ln(𝑁𝐴/𝑛𝑖): Bulk potential of 𝑝-doped silicon𝑢(𝑥)=𝜑(𝑥)/𝑈𝑇: Reduced potential𝑉𝑇=2𝜑𝑏+𝛾|2𝜑𝑏|/𝑈𝑇: Threshold voltage in strong inversion𝛾0=(1/𝐶ox)2𝐾𝑇𝜀𝑆𝑛𝑖: Intrinsic body factor with 𝐶ox=𝑡ox/𝜀ox𝛾=(1/𝐶ox)2𝐾𝑇𝜀𝑆𝑁𝐴: 𝑝-type semiconductor body factor.

Numerical applications use SI units, excepted dopant concentration in cm3, 𝜀𝑆 and 𝜀ox in farads.cm1.


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