# A Unified Channel Charges Expression for Analytic MOSFET Modeling

Authors: Hugues Murray; Patrick Martin
Journal: Active and Passive Electronic Components
 (2012)

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## 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.


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## Body

## 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 currentID (Vg,VD) and the transconductance  g(Vg,VD). 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  b(Vg) 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  ξ(y)=V(y)/UT as quasi-Fermi potential [12] and the correct charges densities are [4]
(1)n(x,y)=nieu(x,y)-ξ(y),p(x,y)=pie-u(x,y),NA=pie-ub,ND=nieub-ξ(y).The Poisson-Boltzmann equation can be analytically solved using the 1Dmodel of Nicollian and Brews [13] from the charge density:
(2)dF(x)dx=qεS[p(x,y)-n(x,y)+ND-NA].The surface electric field,FS(y)=Fx(0,y), along  y (Figure 1) solution of (2) in x=0, is
(3)FS(y)=2KTniεSe-ξ(y)H[uS(y)]+G[uS(y)],
by setting:
(4)H(u)=eu+eub[ub-u-1],G(u)=e-u-e-ub[ub-u+1].Figure 1
Then-MOSFET.The gate voltageVg relative to flat band is
(5)Vg-Vfb=-QS(y)C0+ΨS(y)=γ0e-ξ(y)H[uS(y)]+G[uS(y)]+UT[uS(y)-u(b)].


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

The gate voltage is an explicit function ofuS(y) and ξ(y). Several solutions of Vg=f[uS(y),ξ(y)] were reported to express the band bending  ΨS=UT[uS(y)-ub] 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  uS(y) by first-order Taylor expansion as previously done in [6].u
S
(
y
)   versus  Vg at a constant drain bias  V(y) is generated by
(6)uS(l+1)=uS(l)+δ(dusdVg)us[l],Vg(l)=l·δ,  (δ sample step and  l integer).AnduS(y)=uS,mversusV(y) at a constant  Vg is generated by:
(7)uS,m+1=uS,m+h[duSdξ]uS=uS,m.ξ
(
y
)
=
m
h (h sample step and m integer) is expressed as a function of  uS(y) at a constant Vg by an analytic model previously developed by  Baccarani et al.  [15]:
(8)ξ(y)=lnH[uS(y)]E[uS(y)]-G[uS(y)]
with the introduction of the dimensionless quantity:
(9)E(u)=[Vg-UT(u-ub)γ0]2.


## 2.1. The Surface Potential Equation

Under the gradual channel approximation [11], with the introduction of the reduced channel voltage  ξ(y)=V(y)/UT as quasi-Fermi potential [12] and the correct charges densities are [4]
(1)n(x,y)=nieu(x,y)-ξ(y),p(x,y)=pie-u(x,y),NA=pie-ub,ND=nieub-ξ(y).The Poisson-Boltzmann equation can be analytically solved using the 1Dmodel of Nicollian and Brews [13] from the charge density:
(2)dF(x)dx=qεS[p(x,y)-n(x,y)+ND-NA].The surface electric field,FS(y)=Fx(0,y), along  y (Figure 1) solution of (2) in x=0, is
(3)FS(y)=2KTniεSe-ξ(y)H[uS(y)]+G[uS(y)],
by setting:
(4)H(u)=eu+eub[ub-u-1],G(u)=e-u-e-ub[ub-u+1].Figure 1
Then-MOSFET.The gate voltageVg relative to flat band is
(5)Vg-Vfb=-QS(y)C0+ΨS(y)=γ0e-ξ(y)H[uS(y)]+G[uS(y)]+UT[uS(y)-u(b)].


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

The gate voltage is an explicit function ofuS(y) and ξ(y). Several solutions of Vg=f[uS(y),ξ(y)] were reported to express the band bending  ΨS=UT[uS(y)-ub] 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  uS(y) by first-order Taylor expansion as previously done in [6].u
S
(
y
)   versus  Vg at a constant drain bias  V(y) is generated by
(6)uS(l+1)=uS(l)+δ(dusdVg)us[l],Vg(l)=l·δ,  (δ sample step and  l integer).AnduS(y)=uS,mversusV(y) at a constant  Vg is generated by:
(7)uS,m+1=uS,m+h[duSdξ]uS=uS,m.ξ
(
y
)
=
m
h (h sample step and m integer) is expressed as a function of  uS(y) at a constant Vg by an analytic model previously developed by  Baccarani et al.  [15]:
(8)ξ(y)=lnH[uS(y)]E[uS(y)]-G[uS(y)]
with the introduction of the dimensionless quantity:
(9)E(u)=[Vg-UT(u-ub)γ0]2.


## 3. Analytic Model of Inversion Charges

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

In ann-MOSFET, the inversion charges are defined by the integral of electrons density over the “physical” thickness  dinv:
(10)Qinv=q∫0di‍n(x,y)dx.The “physical” inversion starts at the silicon surface with the surface potentialu(0,y) = uS(y) and ends at the abscissa  x=dinv corresponding to  n(x,y)=p(x,y) and  u(dinv,y) = ξ(y)/2. The inversion charge dependence with channel potential ξ(y) at a constant  Vg noted Qinv(ξ(y))|V(g)  can be written in terms of potential as follows:
(11)Qinv(ξ(y))|V(g)=λqni∫ξ(y)/2u(s)‍eu-ξ(y)due-ξ(y)H(u)+G(u).From (8), Qinv(ξ(y))|V(g) becomes a single integral of u with the limits only dependent of m. Figure 2 shows Qinv(V(y))|V(g)  versus V(y)=ξ(y)·UT  with Vg as a parameter in linear (strong inversion) and log scale (weak inversion).Figure 2
Normalized [Qinv(V(y))|Vg,V(y)] plots in logarithm and linear scales.  (−)  (11), (+) (17).A threshold voltage of inversion chargesVDT|V(g) can be defined by the interpolation of the linear part of  Qinv(VD)|V(g) with the VD axis.  VDT|V(g) = f(Vg) plots (Figure 3) give at  VD=0 a threshold voltage VT which differs from  VT0 by a factor  ≈1.1.  VT is used in (12) instead of VT0.Figure 3
The threshold voltagesVDT|V(g)=f(Vg).


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

The simplest analytic approximate expression ofQinv(Vg,VD) in the whole range of gate and drain bias is well represented by
(12)Qinv(Vg,VD)=ηC0UTln[1+exp(Vg-VTη0UT-VDUT)].η
0
(
≈1.2) is the slope factor defined by the exponential law Qinv(Vg,0) if Vg<VT0.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,”η  =  η(Vg) which fits the slope dQinv(VD)|V(g)/dV  in strong inversion, and a preexponential parameter  a(Vg) which fits (12) with (11) in  y=0:
(13)Q~inv0(Vg,VD)=η(Vg)C0UTln[1+a(Vg)exp(Vg-VTη0UT-VDUT)].Another definition of the “charge linearization factor”η(ΨS) was introduced by Sallese et al. [16] in strong inversion and gives results different from η(Vg) as shown in Figure 4. η(ΨS)(>1) increases when Vg decreases. The difference between η(ΨS) and η(Vg) results from dΨS/dVD=duS/dξ which can be calculated from (5). η(Vg) and a(Vg) are interdependent and will be estimated in order to minimize the error between  Qinv(VD)|V(g)  and  Q~inv0(Vg,VD).Figure 4
η
(
V
g
) modeling by exponential functions.  (η2f = 1.08;   N = 5.5.)Figure4 shows [η(Vg),Vg] plots calculated from Qinv(v(y))|V(g)  and  η(Vg) are well represented by a smoothing function η21as follows:
(14)η(Vg)≈η21=η2f21+exp((Vg-VT0)/NUT)1+0.5exp((Vg-VT0)/NUT).η
2
f is the asymptotic value of η(Vg) at high gate voltages. N is a slope factor which minimizes the error between η(Vg) and η21 in a large range [NA,tox]. Under this condition a(Vg) becomes
(15)a(Vg)|VD=0=exp(Qinv(0)|Vg/η(Vg)C0UT)-1exp((Vg-VT)/η0UT)
with
(16)Qinv(0)|Vg=Qinv(VD=0)|Vg=λqni∫0uS(0)euH(u)+G(u)du,
and  Q~inv(Vg,VD) is written as follows:
(17)Q~inv0(Vg,VD)=η(Vg)C0UTln[1+b(Vg)exp(-VDUT)]
with
(18)b(Vg)=a(Vg)|VD=0exp(Vg-VTη0UT)=exp(Qinv(0)|Vgη(Vg)C0UT)-1.The coefficientb(Vg) is dependent on the gate voltage Vg by uS(0) solution of (5) in y = 0. Equation (17) gives, respectively, in strong and weak inversion the simplified expressions:
(19)Q~inv0(Vg,V(y))=Qinv(0)|Vg-η(Vg)C0V(y),Q~inv0(Vg,V(y))=Qinv(0)|Vgexp-V(y)UT.b
(
V
g
) is a monotonic function in all inversion modes (Figure 5). The originality of the correction by  b(Vg)  is to give an expression of the inversion charges  Q~inv0(Vg,VD) in which the threshold voltage is not explicit but included in uS(0) and appears in [ln(1+b(Vg)),f(Vg)] plots.Figure 5
b
(
V
g
) from (18) (○○) and fT(Vg)=exp(Vg-VT/η0UT)(××). (A): [ln(1+b(Vg)),Vg] plots.The parameterb(Vg) varies from  1025 to 10-5 and as shown in Figure 5 is different from fT(Vg)=exp(Vg-VT/η0UT). Nevertheless, in usual applications (Section 7), the derivative db(Vg)/dVg can be approximated by
(20)db(Vg)dVg=b(Vg)η0UT.


### 3.3. Equivalent Expression of Inversion Charge

By using the mathematical properties of the function:(21)f2(x)=2ln[1+exp(x/2)]1+2exp(-x/2)=2exp(x/2)2+exp(x/2)ln[1+exp(x2)]
which has some similarities with f1(x)=ln[1+exp(x)] in the range ]-∞,+∞[ then (17) can be rewritten by setting
(22)X(Vg,VD)=b(Vg)exp(-VD2UT),Γ(Vg)=1+b(Vg)1+0.5b(Vg),S(Vg)=13exp(Vg-VTσ)2,(23)Q~inv1(Vg,VD)=η(Vg)C0UTΓ(Vg)1-S(Vg)×{X(Vg,VD)1+X(Vg,VD)ln[1+X(Vg,VD)]}.(i)
Γ(Vg) is an adaptive factor which varies between 0.5 (b(Vg)≫1) and 1 (b(Vg)≪1);(ii)
1-S(Vg)  and  σ2=145  are fitting factors which minimize the error between Q~inv1(Vg,VD) and Q~inv0(Vg,VD) at b(Vg)=1.These parameters are available in a large range of[NA,tox]. Equation (23) gives an expression similar to the Unified MOSFET Channel Charge Model given by (7a) and (7b) 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 Q~inv(0,1)(Vg,VD)/η0C0UT at V(y)=0 as a function of the gate voltage.Figure 6
Normalized inversion chargesQinv(Vg,VD = 0)∕η0C0UT versus Vg in logarithm and linear scales.  (−)  (11), (■) (17), (•) (23), and (▲) (7a) and (7b) in [10].The term in braces in (23) can be integrated versus VD and gives an analytic expression of the drain current similar to Oguey and Cserveny model [7].


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

In ann-MOSFET, the inversion charges are defined by the integral of electrons density over the “physical” thickness  dinv:
(10)Qinv=q∫0di‍n(x,y)dx.The “physical” inversion starts at the silicon surface with the surface potentialu(0,y) = uS(y) and ends at the abscissa  x=dinv corresponding to  n(x,y)=p(x,y) and  u(dinv,y) = ξ(y)/2. The inversion charge dependence with channel potential ξ(y) at a constant  Vg noted Qinv(ξ(y))|V(g)  can be written in terms of potential as follows:
(11)Qinv(ξ(y))|V(g)=λqni∫ξ(y)/2u(s)‍eu-ξ(y)due-ξ(y)H(u)+G(u).From (8), Qinv(ξ(y))|V(g) becomes a single integral of u with the limits only dependent of m. Figure 2 shows Qinv(V(y))|V(g)  versus V(y)=ξ(y)·UT  with Vg as a parameter in linear (strong inversion) and log scale (weak inversion).Figure 2
Normalized [Qinv(V(y))|Vg,V(y)] plots in logarithm and linear scales.  (−)  (11), (+) (17).A threshold voltage of inversion chargesVDT|V(g) can be defined by the interpolation of the linear part of  Qinv(VD)|V(g) with the VD axis.  VDT|V(g) = f(Vg) plots (Figure 3) give at  VD=0 a threshold voltage VT which differs from  VT0 by a factor  ≈1.1.  VT is used in (12) instead of VT0.Figure 3
The threshold voltagesVDT|V(g)=f(Vg).


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

The simplest analytic approximate expression ofQinv(Vg,VD) in the whole range of gate and drain bias is well represented by
(12)Qinv(Vg,VD)=ηC0UTln[1+exp(Vg-VTη0UT-VDUT)].η
0
(
≈1.2) is the slope factor defined by the exponential law Qinv(Vg,0) if Vg<VT0.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,”η  =  η(Vg) which fits the slope dQinv(VD)|V(g)/dV  in strong inversion, and a preexponential parameter  a(Vg) which fits (12) with (11) in  y=0:
(13)Q~inv0(Vg,VD)=η(Vg)C0UTln[1+a(Vg)exp(Vg-VTη0UT-VDUT)].Another definition of the “charge linearization factor”η(ΨS) was introduced by Sallese et al. [16] in strong inversion and gives results different from η(Vg) as shown in Figure 4. η(ΨS)(>1) increases when Vg decreases. The difference between η(ΨS) and η(Vg) results from dΨS/dVD=duS/dξ which can be calculated from (5). η(Vg) and a(Vg) are interdependent and will be estimated in order to minimize the error between  Qinv(VD)|V(g)  and  Q~inv0(Vg,VD).Figure 4
η
(
V
g
) modeling by exponential functions.  (η2f = 1.08;   N = 5.5.)Figure4 shows [η(Vg),Vg] plots calculated from Qinv(v(y))|V(g)  and  η(Vg) are well represented by a smoothing function η21as follows:
(14)η(Vg)≈η21=η2f21+exp((Vg-VT0)/NUT)1+0.5exp((Vg-VT0)/NUT).η
2
f is the asymptotic value of η(Vg) at high gate voltages. N is a slope factor which minimizes the error between η(Vg) and η21 in a large range [NA,tox]. Under this condition a(Vg) becomes
(15)a(Vg)|VD=0=exp(Qinv(0)|Vg/η(Vg)C0UT)-1exp((Vg-VT)/η0UT)
with
(16)Qinv(0)|Vg=Qinv(VD=0)|Vg=λqni∫0uS(0)euH(u)+G(u)du,
and  Q~inv(Vg,VD) is written as follows:
(17)Q~inv0(Vg,VD)=η(Vg)C0UTln[1+b(Vg)exp(-VDUT)]
with
(18)b(Vg)=a(Vg)|VD=0exp(Vg-VTη0UT)=exp(Qinv(0)|Vgη(Vg)C0UT)-1.The coefficientb(Vg) is dependent on the gate voltage Vg by uS(0) solution of (5) in y = 0. Equation (17) gives, respectively, in strong and weak inversion the simplified expressions:
(19)Q~inv0(Vg,V(y))=Qinv(0)|Vg-η(Vg)C0V(y),Q~inv0(Vg,V(y))=Qinv(0)|Vgexp-V(y)UT.b
(
V
g
) is a monotonic function in all inversion modes (Figure 5). The originality of the correction by  b(Vg)  is to give an expression of the inversion charges  Q~inv0(Vg,VD) in which the threshold voltage is not explicit but included in uS(0) and appears in [ln(1+b(Vg)),f(Vg)] plots.Figure 5
b
(
V
g
) from (18) (○○) and fT(Vg)=exp(Vg-VT/η0UT)(××). (A): [ln(1+b(Vg)),Vg] plots.The parameterb(Vg) varies from  1025 to 10-5 and as shown in Figure 5 is different from fT(Vg)=exp(Vg-VT/η0UT). Nevertheless, in usual applications (Section 7), the derivative db(Vg)/dVg can be approximated by
(20)db(Vg)dVg=b(Vg)η0UT.


## 3.3. Equivalent Expression of Inversion Charge

By using the mathematical properties of the function:(21)f2(x)=2ln[1+exp(x/2)]1+2exp(-x/2)=2exp(x/2)2+exp(x/2)ln[1+exp(x2)]
which has some similarities with f1(x)=ln[1+exp(x)] in the range ]-∞,+∞[ then (17) can be rewritten by setting
(22)X(Vg,VD)=b(Vg)exp(-VD2UT),Γ(Vg)=1+b(Vg)1+0.5b(Vg),S(Vg)=13exp(Vg-VTσ)2,(23)Q~inv1(Vg,VD)=η(Vg)C0UTΓ(Vg)1-S(Vg)×{X(Vg,VD)1+X(Vg,VD)ln[1+X(Vg,VD)]}.(i)
Γ(Vg) is an adaptive factor which varies between 0.5 (b(Vg)≫1) and 1 (b(Vg)≪1);(ii)
1-S(Vg)  and  σ2=145  are fitting factors which minimize the error between Q~inv1(Vg,VD) and Q~inv0(Vg,VD) at b(Vg)=1.These parameters are available in a large range of[NA,tox]. Equation (23) gives an expression similar to the Unified MOSFET Channel Charge Model given by (7a) and (7b) 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 Q~inv(0,1)(Vg,VD)/η0C0UT at V(y)=0 as a function of the gate voltage.Figure 6
Normalized inversion chargesQinv(Vg,VD = 0)∕η0C0UT versus Vg in logarithm and linear scales.  (−)  (11), (■) (17), (•) (23), and (▲) (7a) and (7b) in [10].The term in braces in (23) can be integrated versus VD 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 currentID(VD) (including drift and diffusion) with a constant mobility  μn follows:
(24)ID(VD)=μnWLUT∫0ξ(L)Qinv(V(y))|Vgdξ.


### 4.1. The Pao-Sah Double Integral

Using the inversion charges dependence to drain biasQinv(V(y))|Vg (developed in Section 3.1), the Pao-Sah double integral then reads
(25)IDPS(VD)=qniμnWLUT2∫0ξ(L)‍{∫ξ(y)/2uS(y)‍eu-ξ(y)F(u,ξ(y))du}dξ(y).By substitutingdξ(y) by dξ(y) = [dξ(y)/duS]duS, and grouping  e-ξ(y)  with dξ(y)/duS, the Pao-Sah double integral has no singular point and, (25) can be solved into iterated integrals from surface potential uS(y):
(26)IDPS(VD)=qniμnWLUT2∫uS(0)uS(L){∫ξ(y)/2uSeuF(u,ξ(y))du}×[dduSG(uS)-E(uS)H(uS)]duS.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:
(27)IQ~inv0(VD)=μnη(Vg)C0UTWL×∫0VDln[1+b(Vg)exp(-V(y)UT)]dV(y).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 ofVD based on the adaptive coefficient  b(Vg) by integration of (23). 								(28)Iap(Vg,VDSVS)=IDAΓ(Vg)1-S(Vg)×{ln2[1+X(Vg,VS)]-ln2[1+X(Vg,VD)]},(29)X(Vg,VDVS)=b(Vg)exp(-VD/VS2UT),IDA=μnη(Vg)C0UT2WLis 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 VC derived from the gate voltage  Vg and from drain source functions  fw(VD/VS),  fh(VD/VS):
(30)IOC(Vg,VDS)=μnC0UT2WL[y(VC,VS)-y(VC,VD)],y(VC,VDVS)=ln2[1+expfw(VDVS)+expfh(VDVS)].The inversion charge of this model is given by(31)Qinv|OcVg=1μn(W/L)dIOC(Vg,VDS)dVD.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 b(Vg) instead of fT(Vg) = exp(Vg-VT/η0UT).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.Table 1
Analytic models.
Drain current
Inversion charges
Model 1
Pao-Sah integral
(25)
(11)
Model 2
Single integral
(27)
(13)
Model 3
[7]
(30)
(31)
Model 4
Analytic model
(28)
(23)Figure 7
The different drain currentsID  versus  Vg from models (1–4). (A): in the vicinity of the threshold voltage  VT0.


## 4.1. The Pao-Sah Double Integral

Using the inversion charges dependence to drain biasQinv(V(y))|Vg (developed in Section 3.1), the Pao-Sah double integral then reads
(25)IDPS(VD)=qniμnWLUT2∫0ξ(L)‍{∫ξ(y)/2uS(y)‍eu-ξ(y)F(u,ξ(y))du}dξ(y).By substitutingdξ(y) by dξ(y) = [dξ(y)/duS]duS, and grouping  e-ξ(y)  with dξ(y)/duS, the Pao-Sah double integral has no singular point and, (25) can be solved into iterated integrals from surface potential uS(y):
(26)IDPS(VD)=qniμnWLUT2∫uS(0)uS(L){∫ξ(y)/2uSeuF(u,ξ(y))du}×[dduSG(uS)-E(uS)H(uS)]duS.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:
(27)IQ~inv0(VD)=μnη(Vg)C0UTWL×∫0VDln[1+b(Vg)exp(-V(y)UT)]dV(y).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 ofVD based on the adaptive coefficient  b(Vg) by integration of (23). 								(28)Iap(Vg,VDSVS)=IDAΓ(Vg)1-S(Vg)×{ln2[1+X(Vg,VS)]-ln2[1+X(Vg,VD)]},(29)X(Vg,VDVS)=b(Vg)exp(-VD/VS2UT),IDA=μnη(Vg)C0UT2WLis 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 VC derived from the gate voltage  Vg and from drain source functions  fw(VD/VS),  fh(VD/VS):
(30)IOC(Vg,VDS)=μnC0UT2WL[y(VC,VS)-y(VC,VD)],y(VC,VDVS)=ln2[1+expfw(VDVS)+expfh(VDVS)].The inversion charge of this model is given by(31)Qinv|OcVg=1μn(W/L)dIOC(Vg,VDS)dVD.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 b(Vg) instead of fT(Vg) = exp(Vg-VT/η0UT).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.Table 1
Analytic models.
Drain current
Inversion charges
Model 1
Pao-Sah integral
(25)
(11)
Model 2
Single integral
(27)
(13)
Model 3
[7]
(30)
(31)
Model 4
Analytic model
(28)
(23)Figure 7
The different drain currentsID  versus  Vg from models (1–4). (A): in the vicinity of the threshold voltage  VT0.


## 5. Mobility Model

In order to insure carrier drift velocity to be less than the saturation velocityvsat 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]:
(32)μneff=μ1(F~x)[1+(F~y/Fsat)β]1/β.


### 5.1. Correction by the Transverse Electric FieldFx

(33)
μ
1
(
F
~
x
)
=
μ
0
[
1
+
(
F
~
x
/
F
0
)
]
.Several expressions are introduced to evaluate the mean electric fieldF~x in relation with the channel inversion charges. BSIM models introduce the voltage Vgsteff  defined by (7a) and (7b) in [10]. An excellent approximation of Vgsteff can be obtained from the equivalent gate voltage Vgeff defined from (17) as follows:
(34)Vgeff=Q~inv(Vg,0)C0=η(Vg)UTln[1+b(Vg)].The expression of the electric field calculated in (3) allows calculating F~x as the mean electric field in the inversion region with a dimensionless adaptive coefficient ua.(35)F~x=ua22KTniεS(H[uS(y=0)]+G[uS(y=0)]+H(0)+G(0)H[uS(y=0)]+G[uS(y=0)]).Figure8 shows the correction factors  F~x, compared with simplified BSIM 4.6.4 [17].Figure 8
The correction factorFx/F0. (A) (FS(uS(y))+Fx(0))/2F0, (B)  Ua(Vgsteff+2Vth)/tox, and (C) Ua(Vgeff+2Vth)/tox. (F0=0.67·106V·cm-1.)


### 5.2. Correction by the Lateral Electric FieldFy

According ton-MOSFET models in [20, 21], we use β=2:
(36)μneff=μ1(F~x)1+(F~y/Fsat)2
with
(37)Fsat=μ1(F~x)vsat,
and F~y the average of the lateral electric field:
(38)F~y=VDL.The correction over constant mobility is introduced in the general expression of drain current by substitutingμn by μneff  in (24) as follows:
(39)μneff=μ0[1+(F~x/F0)]1+{F~yvsat[1+(F~x/F0)]/μ0}2.


### 5.3. The Saturation VoltageVDsat

With mobility correction, the models of drain current[ID,VD]  present a maximum (Figure 9) at a saturation voltage VDsat defined, according to the mobility model by ID,=IDsat=WQinvvsat [17], dID/dΨS [20], or dID/dVD [8]. [ID,VD] curves are presented with the same model of correction by transverse electric field. The adaptive parameter in F~x must then be applied to give the same current and to minimize the error between measured and calculated data.Figure 9
(
I
D
,
V
D) plots with velocity saturation. (A) [17] (β=1), (B) mobility correction with β=2, (C) [20], and (D) saturation current  IDsat=WQinvvsat.In this paper,VDsat represented on Figure 10 is calculated from the iterative definition of drain current (Section 4.1) with dID/dVD substituted by ID,m-ID,m-1=0. The saturation voltage is a linear function of Vg in strong inversion and becomes constant in weak inversion.Figure 10
The saturation voltageVDsat versus Vg. (L=90nm;  W/L=10.)


## 5.1. Correction by the Transverse Electric FieldFx

(33)
μ
1
(
F
~
x
)
=
μ
0
[
1
+
(
F
~
x
/
F
0
)
]
.Several expressions are introduced to evaluate the mean electric fieldF~x in relation with the channel inversion charges. BSIM models introduce the voltage Vgsteff  defined by (7a) and (7b) in [10]. An excellent approximation of Vgsteff can be obtained from the equivalent gate voltage Vgeff defined from (17) as follows:
(34)Vgeff=Q~inv(Vg,0)C0=η(Vg)UTln[1+b(Vg)].The expression of the electric field calculated in (3) allows calculating F~x as the mean electric field in the inversion region with a dimensionless adaptive coefficient ua.(35)F~x=ua22KTniεS(H[uS(y=0)]+G[uS(y=0)]+H(0)+G(0)H[uS(y=0)]+G[uS(y=0)]).Figure8 shows the correction factors  F~x, compared with simplified BSIM 4.6.4 [17].Figure 8
The correction factorFx/F0. (A) (FS(uS(y))+Fx(0))/2F0, (B)  Ua(Vgsteff+2Vth)/tox, and (C) Ua(Vgeff+2Vth)/tox. (F0=0.67·106V·cm-1.)


## 5.2. Correction by the Lateral Electric FieldFy

According ton-MOSFET models in [20, 21], we use β=2:
(36)μneff=μ1(F~x)1+(F~y/Fsat)2
with
(37)Fsat=μ1(F~x)vsat,
and F~y the average of the lateral electric field:
(38)F~y=VDL.The correction over constant mobility is introduced in the general expression of drain current by substitutingμn by μneff  in (24) as follows:
(39)μneff=μ0[1+(F~x/F0)]1+{F~yvsat[1+(F~x/F0)]/μ0}2.


## 5.3. The Saturation VoltageVDsat

With mobility correction, the models of drain current[ID,VD]  present a maximum (Figure 9) at a saturation voltage VDsat defined, according to the mobility model by ID,=IDsat=WQinvvsat [17], dID/dΨS [20], or dID/dVD [8]. [ID,VD] curves are presented with the same model of correction by transverse electric field. The adaptive parameter in F~x must then be applied to give the same current and to minimize the error between measured and calculated data.Figure 9
(
I
D
,
V
D) plots with velocity saturation. (A) [17] (β=1), (B) mobility correction with β=2, (C) [20], and (D) saturation current  IDsat=WQinvvsat.In this paper,VDsat represented on Figure 10 is calculated from the iterative definition of drain current (Section 4.1) with dID/dVD substituted by ID,m-ID,m-1=0. The saturation voltage is a linear function of Vg in strong inversion and becomes constant in weak inversion.Figure 10
The saturation voltageVDsat versus Vg. (L=90nm;  W/L=10.)


## 6. Short Channel Drain Current

### 6.1. Correction of Saturation Voltage

The drain current formulation with mobilityμneff given from (39) is now written as follows:
(40)ID(VD)=μneffWLUT∫0ξ(L)Qinv(V(y))|Vgdξ(y).Equation (40) leads to an unphysical  ID,VD, which must be clamped at  VDsat. Gildenblat et al. [20] proposed to replace VD by a smoothing function with a parameter ax:Vde=VD[1+(VD/VDsat)ax]-1/ax. From the analytical and explicit drain current expressions IQ~inv0(VD)  and  Iap(Vg,VD), we can define a new function which includes the effect of velocity saturation by introducing the saturation voltage VDsat in (27):
(41)ID11(VD)=WLΓ1η21C0UT×∫0VDln[1+exp(VDsat-V(y)UT)]dV(y).The coefficientΓ1 fits  ID11(VD) with  IQinv(VD):
(42)Γ1=μneff(VDsat)×∫0VDsatln[1+b(Vg)exp(-V(y)/UT)]dV(y)∫0VDsatln[1+exp((VDsat-V(y))/UT)]dV(y)
and (28) becomes:
(43)ID22(VDsat,VD)=WLΓ1η21C0UT2Γ(Vg)1-S(Vg)×{Y022(VDsat)-YD22(VDsat,VD)},(44)Y02(VDsat)=ln[1+exp(VDsat2UT)],(45)YD2(VDsat,VD)=ln[1+exp(VDsat-VD2UT)].


### 6.2. Current-Voltage Characteristics

Figures11 and 12 show the simulation results [ID,VD] in strong and weak inversion with a mobility model deduced from (39).Figure 11
Transfer characteristicsID11(Vg) in logarithm and linear scales. VS=0;VD=2V. (L=90nm;  W/L=10.)Figure 12
(ID,VD):  (−)  (41), (•) (43), (⋯) (40). (L=90nm;  W/L=10.)The transfer characteristicsID11(Vg) (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 regionLeff=L-ΔL due to inversion layer in the drain junction. An accurate calculation of ΔL 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]
(46)ΔL=2εSqNA[εSqNA(VDL)2+VD-εSqNA(VDL)2].Figure13 shows an illustration of CLM with ID(VD) from (41) modified by (46). The drain current formulation is
(47)ICLM(VD)=LL-ΔLID22(VD).Figure 13
I
D(VD):  (−)   (43), (- -) (47). (L=90nm;  W/L=10.)This approximation is analogous to the early voltage and has the advantage to be described by the single analytic functionICLM(VD).Figure14 gives a complete summary of the different ID(VD) as follows:(⋄) are ID(VD) data from Pao-Sah double integral from (40) with correction mobility in the range  0<VD<VDsat;(∘) areID(VD) data from the saturation current corrected by the channel length modulation (46)  (VDsat<VD);(⋯) are  ID11(VD) data from (41);The full line shows the single analytic functionICLM(VD) from (47).Figure 14
The different (ID,VD) plots with CLM. (L=90nm,  W/L=10.)


### 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 n−p−n (or p−n−p) 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 φ(x) 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] VT = VT0-σVD with the DIBL parameter σ.In this paper, following the model of DIBL in [25], we propose to insert the increase of inversion charge εSdFy(x,y)/dy  by a quasi 1D calculation. With the same method,  Cheng and Hu  [26] calculated the threshold shift when  L≫l=εStoxXdep/εoxκ(48)ΔVth=[2(Vbi-2φb)+VD][exp(-L2l)+2exp(-Ll)].In the present paper withL=90nm,  L/l≈10, ΔVth is relatively a small correction in ID(VD).In [25], the authors propose, as shown on Figure 15, to add ΔVth in the square logarithm with the new expressions:
(49)Y02DB(VDsat)=ln[1+exp(VDsat+ΔVth(0)2UT)](50)YD2DB(VDsat,VD)=ln[1+exp(VDsat-VD+ΔVth(L)2UT)],(51)ICLM+DIBL(VD)=Γ(Vg)1-S(Vg)WL-ΔLΓ1η21C0UT2×{Y02DB2(VDsat)-YD2DB2(VDsat,VD)}.Figure 15
The total currentICLM+DIBL(VD)  (−)  from (51). (L=90nm,L/l=9.5,andW/L=10).Due to the simplifying assumptions in the derivativedFy(x,y)/dy, 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 dFy(x,y)/dy 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 ofn-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
(52)Isub=ABID(VD-VDsat)exp(-BlVD-VDsat).A and B are adaptive parameters resulting from (ID,VD) measurements. In the present work, this effect must be added to (51) from A and B and gives the total current. Figure 16 shows an example of SCIBE with L=90nm.Figure 16
An example of SCIBE (L=90nm;W/L=10).


## 6.1. Correction of Saturation Voltage

The drain current formulation with mobilityμneff given from (39) is now written as follows:
(40)ID(VD)=μneffWLUT∫0ξ(L)Qinv(V(y))|Vgdξ(y).Equation (40) leads to an unphysical  ID,VD, which must be clamped at  VDsat. Gildenblat et al. [20] proposed to replace VD by a smoothing function with a parameter ax:Vde=VD[1+(VD/VDsat)ax]-1/ax. From the analytical and explicit drain current expressions IQ~inv0(VD)  and  Iap(Vg,VD), we can define a new function which includes the effect of velocity saturation by introducing the saturation voltage VDsat in (27):
(41)ID11(VD)=WLΓ1η21C0UT×∫0VDln[1+exp(VDsat-V(y)UT)]dV(y).The coefficientΓ1 fits  ID11(VD) with  IQinv(VD):
(42)Γ1=μneff(VDsat)×∫0VDsatln[1+b(Vg)exp(-V(y)/UT)]dV(y)∫0VDsatln[1+exp((VDsat-V(y))/UT)]dV(y)
and (28) becomes:
(43)ID22(VDsat,VD)=WLΓ1η21C0UT2Γ(Vg)1-S(Vg)×{Y022(VDsat)-YD22(VDsat,VD)},(44)Y02(VDsat)=ln[1+exp(VDsat2UT)],(45)YD2(VDsat,VD)=ln[1+exp(VDsat-VD2UT)].


## 6.2. Current-Voltage Characteristics

Figures11 and 12 show the simulation results [ID,VD] in strong and weak inversion with a mobility model deduced from (39).Figure 11
Transfer characteristicsID11(Vg) in logarithm and linear scales. VS=0;VD=2V. (L=90nm;  W/L=10.)Figure 12
(ID,VD):  (−)  (41), (•) (43), (⋯) (40). (L=90nm;  W/L=10.)The transfer characteristicsID11(Vg) (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 regionLeff=L-ΔL due to inversion layer in the drain junction. An accurate calculation of ΔL 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]
(46)ΔL=2εSqNA[εSqNA(VDL)2+VD-εSqNA(VDL)2].Figure13 shows an illustration of CLM with ID(VD) from (41) modified by (46). The drain current formulation is
(47)ICLM(VD)=LL-ΔLID22(VD).Figure 13
I
D(VD):  (−)   (43), (- -) (47). (L=90nm;  W/L=10.)This approximation is analogous to the early voltage and has the advantage to be described by the single analytic functionICLM(VD).Figure14 gives a complete summary of the different ID(VD) as follows:(⋄) are ID(VD) data from Pao-Sah double integral from (40) with correction mobility in the range  0<VD<VDsat;(∘) areID(VD) data from the saturation current corrected by the channel length modulation (46)  (VDsat<VD);(⋯) are  ID11(VD) data from (41);The full line shows the single analytic functionICLM(VD) from (47).Figure 14
The different (ID,VD) plots with CLM. (L=90nm,  W/L=10.)


## 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 n−p−n (or p−n−p) 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 φ(x) 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] VT = VT0-σVD with the DIBL parameter σ.In this paper, following the model of DIBL in [25], we propose to insert the increase of inversion charge εSdFy(x,y)/dy  by a quasi 1D calculation. With the same method,  Cheng and Hu  [26] calculated the threshold shift when  L≫l=εStoxXdep/εoxκ(48)ΔVth=[2(Vbi-2φb)+VD][exp(-L2l)+2exp(-Ll)].In the present paper withL=90nm,  L/l≈10, ΔVth is relatively a small correction in ID(VD).In [25], the authors propose, as shown on Figure 15, to add ΔVth in the square logarithm with the new expressions:
(49)Y02DB(VDsat)=ln[1+exp(VDsat+ΔVth(0)2UT)](50)YD2DB(VDsat,VD)=ln[1+exp(VDsat-VD+ΔVth(L)2UT)],(51)ICLM+DIBL(VD)=Γ(Vg)1-S(Vg)WL-ΔLΓ1η21C0UT2×{Y02DB2(VDsat)-YD2DB2(VDsat,VD)}.Figure 15
The total currentICLM+DIBL(VD)  (−)  from (51). (L=90nm,L/l=9.5,andW/L=10).Due to the simplifying assumptions in the derivativedFy(x,y)/dy, 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 dFy(x,y)/dy 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 ofn-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
(52)Isub=ABID(VD-VDsat)exp(-BlVD-VDsat).A and B are adaptive parameters resulting from (ID,VD) measurements. In the present work, this effect must be added to (51) from A and B and gives the total current. Figure 16 shows an example of SCIBE with L=90nm.Figure 16
An example of SCIBE (L=90nm;W/L=10).


## 7. Analytic Model of TransConductance

The Pao-Sah double integral gives an expression of the transconductance from the derivative,g=dID/dVg in (26) [6]:
(53)gmPS=qniμneffUT2WL×∫uS(0)uS(L)euS-ξF(uS(y),ξ)(duS(y)dVg)(dξ(y)duS(y))duS(y).A simplified expression of the transconductance can be obtained from (27) by a derivative under the integral using  db(Vg)/dVg from (20) as follows:
(54)gma=dIDdVg|VD=μneffη21C0UTWL×∫VSVD(b(Vg)/η0UT)exp(-V(y)/UT)1+b(Vg)exp(-V(y)/UT)dV(y).In this case, the integral ingma appears as f′(ν)/f(ν), and gma is given by an analytical expression versus  VD and Vg:
(55)gma=μneffC0UTη21ηWLln{1+b(Vg)exp(-VS/UT)1+b(Vg)exp(-VD/UT)}.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):
(56)gm,μ=dμneffdVgIDμneff.g
m
,
μ is less than gma 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
(57)dμneffdVg=dμneffdμ1dμ1dF~xdF~xduS(0)duS(0)dVg.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
(58)g=ΔIDΔVg|VD.Figure17 shows transconductance [g,Vg] plots, and Figure 18 shows the “normalized” ratio [g/ID], versus drain current.Figure 17
Transconductance versusVg (L=90nm;W/L=4).Figure 18
Normalized transconductance [g/ID,ID]. VDS=1V; Vg-Vfb=1,1.1,…2V (L=90nm;  W/L=4).


## 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 potentialξ(y)=f(uS(y)). 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 Vg 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).Table 2
The simplified analytic drain current models.
Physical constants
ε
S,εox,KT/q
Process parameters
N
A
,
t
ox
,
μ
n
,
W
/
L
Device voltages
V
g
,
V
D
,
V
S
,
V
fb
Surface potential inx=0,   y=0
u
S
(
0
)
Surface potential
u
S
(
y
)
=
u
S
,
m
Mobility
μ
n
eff
(
V
g
,
V
D
)
Drain current from Pao-sah
I
D
PS
(
V
D
)
Gate voltage factors
η
(
V
g
),  b(Vg)
Analytic expression of drain current
I
ap
(
V
g
,
V
D
)
Saturation voltage
V
D
sat
Drain current with saturation
I
D
22
(
V
D
sat
,
V
D
)
Channel length modulation
I
CLM
(
V
D
)
Drain-induced barrier lowering
I
CLM
+
DIBL
(
V
D
)Figure 19
The graphic user interface.Figure 20
I
D(VD) plots from (51).  Vg-Vfb=1.5,1.6,…2V.



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*Source: 652478-2012-12-25.xml*