Research Article | Open Access
Hugues Murray, Patrick Martin, Serge Bardy, "Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral", Active and Passive Electronic Components, vol. 2010, Article ID 268431, 11 pages, 2010. https://doi.org/10.1155/2010/268431
Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral
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.
The Metal Oxide Semiconductor Field Effect Transistor (MOSFET), first proposed in 1926 by Lilienfeld , 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 is above a threshold voltage , 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 , 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 . In 1995, Persi and Gildenblat proposed a computational calculation of the double integral by numerical treatment of carrier concentration and surface potential . Recently, a complete history of MOSFET modeling  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  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  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 [9–11], 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 .
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 in the form of the well known Poisson-Boltzmann equation. Under the Gradual Channel Approximation (GCA) , this equation is solved analytically and gives the exact 1-D electric field . The gate voltage versus surface potential 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 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 . In this method, the electric field magnitude in the direction parallel to the conducting channel is assumed to be much smaller than in the vertical direction toward silicon bulk allowing the formulation and . The Poisson-Boltzmann equation can then be solved analytically according to the 1-D model of Nicollian and Brews  from the complete charge density: .
In the following, electrons distribution in the channel will be considered out of thermal equilibrium with the introduction of the quasi-Fermi energy . 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 is defined relative to flat band.
The presentation relates to a -type semiconductor (-MOSFET), but this is not restrictive and could easily be extended to -type semiconductor. Upon usual notation conventions (see Nomenclature section), the bulk Fermi level is . is the potential in every points of the semiconductor material and is the channel voltage which defines the quasi-Fermi level offset from equilibrium for electrons. Upon application of a drain bias , ranges from at the source side to at the drain side is the thermal voltage.
In the following, we use the reduced potentials and instead of and respectively. are shorthand notations for surface quantities.
In the first MOSFET modeling, charge densities have been written by introducing the quasi-Fermi level only in with the expressions:
After solving the 1-D Poisson equation , these expressions give the 1-D electric field by and its derivative
As in the channel region, this expression is equivalent to
This later expression gives a simple path to the Pao-Sah integral .
With this exact formulation of the donor density, the electric field gradient from the 1-D Poisson equation (5) now reads which gives the electric field where we have defined
In this form, the derivative is no longer with a numerator proportional to 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 is given by substituting and by in (10)-(11).
3. The Surface Potential Dependence to Gate Bias
From electrostatic considerations , the gate voltage is expressed by
The exponential expression in (13) allows to separate and with the derivative in which we introduce the dimensionless quantity
Several approximations to (13) have been reported. For instance, the equation is often used in practice in a large variety of surface potential-based models with the band-bending: . It provides a sufficiently accurate solution for the surface potential in inversion mode but generally lacks accuracy near the flat-band conditions: .
4. The Surface Potential Dependence to Drain Bias
The explicit relation 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 and to the inverse function 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 compatible with the denominator
4.1. Boundary Limits of Surface Potential at Constant Gate Voltage
Equation (14) has only physical meaning when . At a constant value, the reduced surface potential ranges between (lower value), solution of (13) with and between (upper value), solution of (13) with leading to and are defined by implicit relations to the gate voltage . Following a methodology previously described in , and can easily be obtained iteratively from first order Taylor expansion by setting , in which is the sample step size and an integer. For a given step size and for varying from to , and at iteration are calculated values at previous iteration according to
In (20), reaches the threshod voltage when: with the function int = (number down to the nearest integer). As a result, if and nm and . when the step is and the corresponding band bending is .
As this has been previously shown , the error induced by first order Taylor expansion depends on the step size . Equations (20) and (22) show that the relative error is in the same order of magnitude as the step . For instance, a step gives a relative error less than .
Figure 2 shows and plots versus . 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 and are used as integration limits in Pao-Sah integral (see Section 7).
4.2. The Inversion of
Equation (14) is an explicit relationship between and . As this has been done in previous section with the calculation of the lower and upper limits of the surface potential, the inversion of (14) can be obtained by a first order Taylor expansion of the inverse function by setting in which is the sample step size and an integer:
Note that this representation is versus and not which would need the knowledge of the variation of versus . 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 versus is not needed for generating the main device current voltage and other related characteristics.
Figure 4 shows a complete set of curves versus from up to V. is calculated from (24) and from in (14). This figure illustrates the pinch-off voltage in strong inversion which appears when becomes to decrease.
In terms of surface electron density (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 gradient.
In strong inversion, the transition between and the regimes happens for a voltage . The inset plots in Figure 5 shows that when is equal to the threshold voltage V defined in the charge sheet model .
5. The Pao-Sah Double Integral
Under the gradual channel approximation  the drift drain-source current density varies from bulk silicon toward gate oxide-silicon interface and along the channel:
Or, in terms of potential
With the introduction of the inversion charge density where we recall and the inversion length in direction; the expression for the drain-source drift current follows in which a total channel width is assumed. The Pao-Sah double integral then reads
Since can be expressed in terms of or alternatively defined as function of , 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
The Pao-Sah double integral can be expressed in terms of two single iterated integrals , and the drift current is given by
The integral in braces is integrated first and gives a function of and the final integral is integrated with respect to from to
In (32):(i) is the surface potential along the channel.(ii) is a function of from (14).(iii) is given by (15).(iv) is the surface potential in . At a constant bias, it corresponds to solution of (20) with (v) is the surface potential in . At a constant , it corresponds to solution of (24) with (vi) is the effective mobility which has extensively been studied previously [20, 21]. The correction over a constant mobility used in this paper, principally depends on and is easy to implement in the integral.
5.2. Solution from Channel Potential
In the drift current (30), it is possible to define the surface potential versus from (24). In this case, the channel potential is sampled according to Taylor expansion, and the integral on must be replaced by a discrete summation
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 and 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%.
Figures 7(a) and 7(b) show the simulation results in linear scale. Curve (a) is compared to data reported in  and curve (b) with more recent data reported in  . All these calculations have been performed using a constant mobility: .
The transconductance of the drift current is defined from the derivative 
The mathematical definition of the integral operator  gives the condition and, consequently
The transconductance is which from (31) gives a single integral of the surface potential with
Figure 8(a) shows plots in log scale when the source is grounded 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 is a linear function versus .
In the E.K.V. model , the authors introduce the normalized transconductance: with ; is the slope factor and Figure 8(c) shows plots using (39) and (32) . The function the asymtotes and 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 and given by iterative equations (20) and (24). Figure 9 shows the variations of these integration limits versus respectively, in strong (a) and weak inversion (b). The difference between and 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.
The accuracy of Pao-Sah equation can be increased if the step in (20) is decreased. As an example, in weak inversion, gives an accurate value of which leads to equal to with an absolute error less than
8. The Diffusion Current
8.1. Integral of Current Diffusion
The diffusion current density of an n-MOSFET is given by
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 density
Equation (45) is equivalent to the well known expression
Figure 10 shows plots according to (13) and (28) for different bias. corresponds to and corresponds to the drain bias . is rapidly negligible in (45). Figure 11 shows plots in weak inversion calculated from (45) and compared with (46) from in (28).
In weak inversion, the diffusion current is well captured by with which is in good agreement with the slope of plots (Figure 10).
8.2. Diffusion Plots
A complete graph can be plotted by sampling from (24). Figure 12 shows the contribution of the diffusion current compared with the drift current in strong and weak inversion and . The diffusion current is negligible in strong inversion and dominant in weak inversion.
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 versus bulk silicon, the reduced drain-source potential along the channel varies from to . Then, the surface potential starts at given from (24) with and stops at .
Equation (32) can be rewritten with index in
10. Transfer Characteristics
As a comparison, Figure 14 shows the curves with the same parameters reported in . The conditions are flatband for and for The oxide thickness is nm and the drain bias is volt. Our simulations are in good agreement with these results.
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 , 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 , oxide thickness , and gate and drain voltages and .
12. Annex: Taylor Polynomials of Inverse Functions
If a function has continuous derivatives up to order , then this function can be expanded in the following Taylor polynomials  where is called the remainder after terms.
This expansion converges over a certain range of , if lim, and the expansion is called the Taylor Series of expanded about .
In inverse function, if is a one-to-one function, then is continuous and, if has continuous derivatives up to th order, can be expanded in the Taylor polynomials. These conditions have been previously verified in the inversion of surface potential , with an accuracy compatible with the integral simulations.
The limits of Paoh-Sah integral are based on the inverse function , The first derivative variations given by (25) are shown in Figure 16 for different bias. The first derivative is define in all the region and varies from 1 to ≈1 without discontinuity. The strong decrease observed for a value depending on is due to the asymptotic value in the surface potential when reaches solution of
Nomenclature: Voltage gate with bulk silicon grounded and : Silicon and silicon oxide permittivity: Oxide thickness: Channel length: Channel width: Intrinsic carrier concentration in and : Dopant concentrations in : Thermal voltage: Bulk potential of -doped silicon: Reduced potential: Threshold voltage in strong inversion: Intrinsic body factor with : -type semiconductor body factor.
Numerical applications use SI units, excepted dopant concentration in , and in farads..
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