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

Abstract

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Hugues Murray 1 and Patrick Martin 1 and Serge Bardy 2

Recommended by Ashok Goel

1, LaMIPS, Laboratoire commun NXP-CRISMAT, UMR 6508 CNRS ENSICAEN, UCBN, 2, rue de la Girafe BP 5120, F-14079 Caen Cedex 5, France
2, NXP 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

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 [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 [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 [straight phi](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 _{[straight phi]S} 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 div F=dF/dx and F[x,y]=-d[[straight phi](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)+_ND -_NA ] .

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.

[figure omitted; refer to PDF]

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 _{[straight phi]b} . [straight phi](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 =_{[straight phi]b} /_UT ,u(x,y)=[straight phi](x,y)/_UT and ξ(y)=V(y)/_UT instead of _{[straight phi]b} ,[straight phi](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: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

After solving the 1-D Poisson equation dF(x,y)/dx=ρ(x,y) , these expressions give the 1-D electric field by [figure omitted; refer to PDF] and its derivative [figure omitted; refer to PDF]

As ^e_ub <<^eu(x,y) in the channel region, this expression is equivalent to [figure omitted; refer to PDF]

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[arrow right]∞)=_ND , (4) must be corrected by [figure omitted; refer to PDF]

With this exact formulation of the donor density, the electric field gradient from the 1-D Poisson equation (5) now reads [figure omitted; refer to PDF] which gives the electric field [figure omitted; refer to PDF] where we have defined [figure omitted; refer to PDF]

In this form, the derivative [figure omitted; refer to PDF] 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 by [figure omitted; refer to PDF]

The exponential expression in (13) allows to separate u(s) and ξ(y) [figure omitted; refer to PDF] with the derivative [figure omitted; refer to PDF] in which we introduce the dimensionless quantity [figure omitted; refer to PDF]

Several approximations to (13) have been reported. For instance, the equation [figure omitted; refer to PDF] is often used in practice in a large variety of surface potential-based models with the band-bending: Ψ(s)=[straight phi](s)-_{[straight phi]b} . 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)=^f-1 [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)=0 [figure omitted; refer to PDF] and between _uUp (upper value), solution of (13) with ξ(y)[arrow right]∞ leading to ^e-ξ(y) [arrow right]0 [figure omitted; refer to PDF] _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=k·δ , 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 [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

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 =1⁰¹⁷ c^m-3 and _tox =10 nm and _kT =12819 . _uLow,kT =15.712=-_ub when the step is δ=1^0-4 and the corresponding band bending is _ΨS =_UT (_uLow,kT -_ub )= -2_{[straight phi]b} .

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=1^0-10 gives a relative error less than 1^0-10 .

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 =1⁰¹⁷ c^m-3 and _tox =10 nm. Vg=_VT when (a) crosses the line _uLow =-_ub .

[figure omitted; refer to PDF]

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)=^f-1 [ξ(y)] by setting ξ(y,m)=m·h in which h is the sample step size and m an integer: [figure omitted; refer to PDF] [figure omitted; refer to PDF]

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)/h_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 h=0.01 ). Curves (a) and (b) merge with an error less than 1^0-4 .

Figure 3: [_uS (y),V(y)] plots along the channel. (a) equation (14) and (b) equation (24) (step h=0.01 ). _NA =1⁰¹⁷ c^m-3 ,_tox =10 nm and Vg=5 V.

[figure omitted; refer to PDF]

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 =1⁰¹⁷ c^m-3 and _tox =10 nm.

[figure omitted; refer to PDF]

In terms of surface electron density _nS (y)=_ni^{e_uS (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 d_nS /dy gradient.

Figure 5: The electron density [_nS (y),V(y)] plots along the channel from (1) and (14). _NA =1⁰¹⁷ c^m-3 and _tox =10 nm.

[figure omitted; refer to PDF]

In strong inversion, the transition between _nS (y)=cte and the n(y)α^e-ξ(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 =-2_{[straight phi]b} +γ|2_{[straight phi]b} |/_UT [approximate]1.28 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: [figure omitted; refer to PDF]

Or, in terms of potential [figure omitted; refer to PDF]

With the introduction of the inversion charge density [figure omitted; refer to PDF] where we recall F=-_UT (du/dx) and _xd the inversion length in x direction; the expression for the drain-source drift current follows [figure omitted; refer to PDF] in which a total channel width W is assumed. The Pao-Sah double integral then reads [figure omitted; refer to PDF]

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 by [figure omitted; refer to PDF] in which dξ(y)/d_uS (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 by [figure omitted; refer to PDF]

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)/d_uS (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 /h_UT .

(vi) _μneff is the effective mobility which has extensively been studied previously [20, 21]. The correction over a constant mobility (_μn =550 c^m2 ·^V-1s-1 ) 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=mh must be replaced by a discrete summation [figure omitted; refer to PDF]

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 =1⁰¹⁷ c^m-3 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 =1⁰¹⁷ c^m-3 ,_tox =10 nm and W/L=10. Markers (+) corresponds to (33).

[figure omitted; refer to PDF]

Figures 7(a) and 7(b) show the simulation results in linear scale. Curve (a) is compared to data reported in [4] (_NA =1⁰¹⁵ c^m-3 ,_tox =50 nm) and curve (b) with more recent data reported in [7] (_NA =5.1⁰¹⁷ c^m-3 ,_tox =5 nm) . All these calculations have been performed using a constant mobility: _μn =550 c^{m2 V-1s-1} .

_Idrift versus drain voltage _VD from (32) in linear scale compared with [4] and [7]. (a) _NA =1⁰¹⁵ c^m-3 ,_tox =50 nm and W/L=10, ([composite function] ) corresponds to [4]. (b) _NA =5.1⁰¹⁷ c^m-3 and _tox =5 nm, ([composite function] ) corresponds to [7].

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

6. Transconductance

The transconductance of the drift current is defined from the derivative [3] [figure omitted; refer to PDF]

The mathematical definition of the integral operator [22] gives the condition [figure omitted; refer to PDF] and, consequently [figure omitted; refer to PDF]

The transconductance is calculated at a constant _VD bias. The derivative of the drain current is only on _uS and (36) in (32) results in [figure omitted; refer to PDF]

The transconductance is [figure omitted; refer to PDF] which from (31) gives a single integral of the surface potential _uS (y) [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

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 .

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 =1⁰¹⁷ c^m-3 ,_tox =10 nm and W/L=10.

[figure omitted; refer to PDF]

[figure omitted; refer to PDF]

In the E.K.V. model [23], the authors introduce the normalized transconductance: G(_iF )=_gm n_UT /_iF with _iF =_IF /2nβ^UT2 ; n is the slope factor and β=_μnCox W/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.

The limits of surface potentials _uS (0) and _uS (L) versus _VD . (a) strong inversion, (b) weak inversion. _NA =1⁰¹⁷ c^m-3 ,_tox =10 nm, W/L=10 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The accuracy of Pao-Sah equation can be increased if the step δ in (20) is decreased. As an example, in weak inversion, δ=1^0-10 gives an accurate value of _uLow,k which leads to Vg[_uLow,k ] equal to Vg with an absolute error less than 1^0-10 .

8. The Diffusion Current

8.1. Integral of Current Diffusion

The diffusion current density of an n- MOSFET is given by [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

Equation (28) gives the equivalence to potentials [figure omitted; refer to PDF] and after integration on y, (42) becomes [figure omitted; refer to PDF] and reduces to single integrals [figure omitted; refer to PDF]

Equation (45) is equivalent to the well known expression [figure omitted; refer to PDF]

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 _Idiff (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 =1⁰¹⁷ c^m-3 and _tox =10 nm.

[figure omitted; refer to PDF]

Figure 11: [_Idiff ,Vg] plots in weak inversion (_VD =0.3 V) (--) from (46) and (...) from (45). _NA =1⁰¹⁷ c^m-3 and _tox =10 nm.

[figure omitted; refer to PDF]

In weak inversion, the diffusion current is well captured by [figure omitted; refer to PDF] 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 _Idiff (_VD ) can be plotted by sampling _uS (L) from (24). Figure 12 shows the contribution of the diffusion current (_Idiff ) compared with the drift current (_Id ) in strong and weak inversion (_Id1 >_Idiff1 and _Id2 <_Idiff2 ) . The diffusion current is negligible in strong inversion and dominant in weak inversion.

Figure 12: The drift _Id (--) and diffusion _Idiff (....) currents versus _VD in log scale. _Id1 and _Idiff1 strong inversion: _Id2 and _Idiff2 weak inversion. _NA =1⁰¹⁷ c^m-3 ,_tox =10 nm, and W/L=10.

[figure omitted; refer to PDF]

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 /hUT and stops at m=M=_VDB /hUT .

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 /h_UT .

By setting [figure omitted; refer to PDF]

Equation (32) can be rewritten with m index in _uS [figure omitted; refer to PDF] [figure omitted; refer to PDF]

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 +_Idiff =f(Vg) calculated from (32) and (45) for constant _VDS and different _VSB . The parameters of I=f(Vg) are _VFB ( flatband)=0 , _NA =1⁰¹⁷ c^m-3 ,_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 =1⁰¹⁷ c^m-3 ,_tox =10 nm and _VSB =[0-1] volt.

[figure omitted; refer to PDF]

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 =1⁰¹⁴ c^m-3 and _VFB =0.86 for _NA =1⁰¹⁵ c^m-3 . 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 =1⁰¹⁴ c^m-3 ,_VFB ( flat-band)=0.92 V. (b) _NA =1⁰¹⁵ c^m-3 ,_VFB =0.86 V. Markers (...) and ([open diamond] ) are from [4].

[figure omitted; refer to PDF]

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. V^g* =Vg(_uLow,k ) calculated from _uLow,k in (20).

[figure omitted; refer to PDF]

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] [figure omitted; refer to PDF] where R(n) is called the remainder after n+1 terms.

This expansion converges over a certain range of x , if lim_{n[arrow right]∞} R(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 ^f-1 is continuous and, if ^f-1 has continuous derivatives up to (n) th order, ^f-1 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)=^f-1 [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 [approximate] 1^0-13 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 d_uS (y)/dξ(y) versus drain voltage _VD for different Vg bias. _NA =1⁰¹⁷ c^m-3 and _tox =10 nm.

[figure omitted; refer to PDF]

Nomenclature

Vg : Voltage gate with bulk silicon grounded

_{[straight epsilon]S} and _{[straight epsilon]ox} : Silicon and silicon oxide permittivity

_tox : Oxide thickness

L : Channel length

W : Channel width

_ni : Intrinsic carrier concentration in c^m-3

_NA and _ND : Dopant concentrations in c^m-3

_UT =KT/q : Thermal voltage

[straight phi](b)=_{[straight phi]b} =-_UT ln (_NA /_ni ) : Bulk potential of p -doped silicon

u(x)=[straight phi](x)/_UT : Reduced potential

_VT =-2_{[straight phi]b} +γ|2_{[straight phi]b} |/_UT : Threshold voltage in strong inversion

_γ0 =(1/_Cox )2KT_{[straight epsilon]Sni} : Intrinsic body factor with _Cox =_tox /_{[straight epsilon]ox}

γ=(1/_Cox )2KT_{[straight epsilon]SNA} : p -type semiconductor body factor.

Numerical applications use SI units, excepted dopant concentration in c^m-3 , _{[straight epsilon]S} and _{[straight epsilon]ox} in farads .c^m-1 .

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