research article analytical model of random variation in...

Research ArticleAnalytical Model of Random Variation in DrainCurrent of FGMOSFET

Rawid Banchuin

Department of Computer Engineering, Siam University, 235 Petchakasem Road, Bangkok 10163, Thailand

Correspondence should be addressed to Rawid Banchuin; rawid [email protected]

Received 15 March 2015; Revised 4 June 2015; Accepted 11 June 2015

Academic Editor: Jiun-Wei Horng

Copyright © 2015 Rawid Banchuin. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The analytical model of random variation in drain current of the Floating Gate MOSFET (FGMOSFET) has been proposed inthis research. The model is composed of two parts for triode and saturation region of operation where the process induced devicelevel random variations of each region and their statistical correlations have been taken into account. The nonlinearity of floatinggate voltage and dependency on drain voltage of the coupling factors of FGMOSFET have also been considered. The model hasbeen found to be very accurate since it can accurately fit the SPICE BSIM3v3 based reference obtained by using Monte-CarloSPICE simulation and FGMOSFET simulation technique with SPICE. It can fit the BSIM4 based reference if desired by using theoptimally extracted parameters. By using the proposed model, the variability analysis of FGMOSFET and the analytical modelingof the variation in the circuit level parameter of any FGMOSFET based circuit can be performed. So, this model has been found tobe an efficient tool for the variability aware analysis and design of FGMOSFET based circuit.

1. Introduction

FGMOSFETs have been extensively utilized in various ana-log/digital circuits such as [1–9]. Similarly to the MOSFETbased circuits, the performances of the FGMOSFET basedcircuits have been deteriorated by process induced devicelevel random variations [10–12]. This is because these devicelevel variations yield random variations in circuit levelparameters, for example, drain current (𝐼

𝐷) and transcon-

ductance. These variations yield variations in parametersof FGMOSFET based circuit such as transconductance ofthe FGMOSFET based voltage to current converter [1] andequivalent resistance of the FGMOSFET based voltage con-trolled resistor [2, 3]. For handling this issue, variability awareanalysis/design concept has been applied in the designing ofmany FGMOSFET based circuits, for example, [4–6].

Similarly to MOSFET, 𝐼𝐷

has been found to be thekey circuit level parameter of FGMOSFET as it is directlymeasurable and can be the basis for determining the others.According to its importance, the analytical models of processinduced random variation and mismatch in 𝐼

𝐷of MOSFET

have been proposed without regard to certain circuit inmany previous researches, for example, [13–15]. So, their

results are applicable to any MOSFET based circuit. For theFGMOSFET, most of the previous work has been orientedto certain FGMOSFET circuits, for example, [16–19]. Someprevious studies have not been devoted to any circuit, forexample, [20, 21]. However, the results of these works areincomplete since the statistical correlations between devicelevel random variations have been neglected. Moreover,nonlinearity of floating gate voltage and dependency on drainvoltage of the coupling factors, which are the importantfeatures of FGMOSFET [22], have not been considered.

So, the analytical model of process induced randomvariation in 𝐼

𝐷(Δ𝐼𝐷) of the FGMOSFET has been proposed

in this research. The proposed model is composed of twoparts for triode and saturation regions of operation where theprocess induced device level randomvariations of each regionand their statistical correlations have been taken into account.The nonlinearity of floating gate voltage and dependency ondrain voltage of the coupling factors of FGMOSFET have alsobeen considered. This model has been formulated withoutregard to any circuit. So, it is applicable to all FGMOSFETbased circuits. It has been found to be very accurate sinceit can fit the SPICE BSIM3v3 based reference obtained byusing FGMOSFET simulation techniquewith SPICE [23] and

Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2015, Article ID 315105, 12 pageshttp://dx.doi.org/10.1155/2015/315105

2 Active and Passive Electronic Components

p-substrate

Source Drain

Floatinggate

Inputgates

N-input voltagesV1 V2 V3 VN

C1 C2 C3 CN

· · ·

N+ N+

Figure 1: A cross-sectional view of N-type 𝑁 inputs FGMOSFET[2].

Monte-Carlo SPICE simulation with very high accuracy. Ifdesired, it can fit the BSIM4 based reference by applyingthe optimum parameters extracted by using the optimizationalgorithm [24]. The variability analysis of FGMOSFET andthe analytical modeling of process induced random variationin circuit level parameter of any FGMOSFET based circuitcan be performed by using the proposed model as the basis.Hence, this model has been found to be an efficient tool forthe variability aware analysis/design of any circuit involvingFGMOSFET.

2. The Overview of FGMOSFET

FGMOSFET is a special type of MOSFET with an additionalgate isolated within the oxide, namely, the floating gate [2]. Across-sectional view of an N-type FGMOSFET with 𝑁 inputsimplemented as 𝑁 discrete input gates where 𝑁 > 1 can beshown in Figure 1.

It should be mentioned here that the dimension (widthand length) of any input gate determines the magnitude of itscorresponding input capacitance.The symbol and equivalentcircuit of 𝑁 inputs FGMOSFET are shown in Figure 2. Suchequivalent circuit is composed of a MOSFET, 𝑁 input capac-itances (𝐶

1, 𝐶2, 𝐶3, . . . , 𝐶

𝑁), overlap capacitance between

floating gate and drain (𝐶fd), overlap capacitance betweenfloating gate and source (𝐶fs), and parasitic capacitancebetween floating gate and substrate (𝐶fb) [2].

Let {𝑖} = {1, 2, 3, . . . , 𝑁} and let any 𝑖th input capacitancebe denoted by 𝐶

𝑖; the floating gate voltage, 𝑉FG, can be given

by [2]

𝑉FG =∑𝑁

𝑖=1 [𝐶𝑖𝑉𝑖] + 𝐶fd𝑉𝐷 + 𝐶fs𝑉𝑆 + 𝐶fb𝑉𝐵𝐶𝑇

, (1)

where 𝑉𝑖is the input voltage at any 𝑖th input gate, 𝑉

𝐷is the

drain voltage, 𝑉𝑆is the source voltage, and 𝑉

𝐵is the bulk

voltage. Moreover, 𝐶𝑇denotes the total capacitance of the

floating gate which can be defined as [2]

𝐶𝑇

=

𝑁

∑

𝑖=1[𝐶𝑖] + 𝐶fd + 𝐶fs + 𝐶fb. (2)

Let 𝑘𝑖, 𝑘fd, 𝑘fs, and 𝑘fb denote the coupling factor of any

𝑖th input gate, drain, source, and bulk and let them be definedas 𝑘𝑖

= 𝐶𝑖/𝐶𝑇, 𝑘fd = 𝐶fd/𝐶𝑇, 𝑘fs = 𝐶fs/𝐶𝑇, and 𝑘fb = 𝐶fb/𝐶𝑇,

respectively; 𝑉FG can be alternatively given as follows:

𝑉FG =𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fs𝑉𝑆 + 𝑘fb𝑉𝐵. (3)

From either (1) or (3), it can be seen that 𝑉FG dependson 𝑉𝑖, 𝑉𝐷, 𝑉𝑆, and 𝑉

𝐵. According to [22], 𝑉FG of FGMOSFET

in triode is a nonlinear function of 𝑉𝐷and 𝐶

𝑇depends on

𝑉𝐷as 𝐶fb does. So, 𝑘𝑖, 𝑘fd, 𝑘fs, and 𝑘fb are dependent on 𝑉𝐷

since they are functions of 𝐶𝑇. For FGMOSFET in saturation

region, 𝐶fb is constant and so are 𝐶𝑇, 𝑘𝑖, 𝑘fd, 𝑘fs, and 𝑘fb. Asa result, 𝑉FG become a linear function of 𝑉𝐷.

3. Formulation of the Proposed Model

3.1. Triode Region Part. Firstly, 𝐼𝐷of FGMOSFET in triode

region will be formulated by letting the gate to source voltage(𝑉GS) in the analytical model of 𝐼𝐷 of MOSFET in suchregion be replaced by 𝑉FG − 𝑉𝑆. The second order effectssuch as mobility degradation and short channel effect havebeen taken into account for making the model applicable todeeply scaled technology. In order to do so, the linear modelof mobility degradation [15] has been adopted. Hence, 𝐼

𝐷of

the MOSFET in triode can be given by

𝐼𝐷

= 𝛽 [1− 𝜃 (𝑉GS − 𝑉TH)] [(𝑉GS − 𝑉TH) 𝑉DS −12

𝑉2DS] ,

(4)

where 𝑉DS, 𝑉TH, 𝛽, and 𝜃 denote drain to source voltage,threshold voltage, current factor, and mobility degradationcoefficient, respectively.

Since 𝑉FG of FGMOSFET in this region is a nonlinearfunction of 𝑉

𝐷, 𝑉FG can be given by

𝑉FG =𝑁

∑

𝑖=1[𝑘𝑖(𝑉𝐷

) 𝑉𝑖] + 𝑘fd (𝑉𝐷) 𝑉𝐷 + 𝑘fs (𝑉𝐷) 𝑉𝑆

+ 𝑘fb (𝑉𝐷) 𝑉𝐵,

(5)

Active and Passive Electronic Components 3

B

S

DFloating gate

Input gates

N-inputvoltages

V1

V2V3

VN

C1C2

C3

CN

···

(a)

B

S

D

V1

V2

V3

VN

C1

C2

C3

CN

···

VFG

Cfd

Cfs

Cfb

(b)

Figure 2: The symbol (a) and equivalent circuit model (b) of N-type 𝑁 inputs FGMOSFET [2].

where 𝑘𝑖(𝑉𝐷

), 𝑘fd(𝑉𝐷), 𝑘fs(𝑉𝐷), and 𝑘fb(𝑉𝐷), which are,respectively, 𝑉

𝐷dependent 𝑘

𝑖, 𝑘fd, 𝑘fs, and 𝑘fb, can be

modeled as power series of 𝑉𝐷as follows:

𝑘𝑖(𝑉𝐷

) =

∞

∑

𝑚=0[𝛼𝑖𝑚

𝑉𝑚

𝐷] , (6)

𝑘fd (𝑉𝐷) =∞

∑

𝑚=0[𝛼𝑑𝑚

𝑉𝑚

𝐷] , (7)

𝑘fs (𝑉𝐷) =∞

∑

𝑚=0[𝛼𝑠𝑚

𝑉𝑚

𝐷] , (8)

𝑘fb (𝑉𝐷) =∞

∑

𝑚=0[𝛼𝑏𝑚

𝑉𝑚

𝐷] , (9)

where 𝛼𝑖𝑚, 𝛼𝑑𝑚

, 𝛼𝑠𝑚, and 𝛼

𝑏𝑚are coefficients of the power

series representation of 𝑘𝑖(𝑉𝐷

), 𝑘fd(𝑉𝐷), 𝑘fs(𝑉𝐷), and 𝑘fb(𝑉𝐷),respectively.

As 𝑉𝐷can be very low in recent CMOS technology, high

order terms of 𝑘𝑖(𝑉𝐷

), 𝑘fd(𝑉𝐷), 𝑘fs(𝑉𝐷), and 𝑘fb(𝑉𝐷) can beneglected. So, (6)–(9) become (11)–(13) where 𝛼

𝑖0, 𝛼𝑑0, 𝛼𝑠0,and 𝛼

𝑏0 and 𝛼𝑖1, 𝛼𝑑1, 𝛼𝑠1, and 𝛼𝑏1 are 𝛼𝑖𝑚, 𝛼𝑑𝑚, 𝛼𝑠𝑚, and 𝛼𝑏𝑚with 𝑚 = 0 and 𝑚 = 1:

𝑘𝑖(𝑉𝐷

) = 𝛼𝑖0 + 𝛼𝑖1𝑉𝐷, (10)

𝑘fd (𝑉𝐷) = 𝛼𝑑0 + 𝛼𝑑1𝑉𝐷, (11)

𝑘fs (𝑉𝐷) = 𝛼𝑠0 + 𝛼𝑠1𝑉𝐷, (12)

𝑘fb (𝑉𝐷) = 𝛼𝑏0 + 𝛼𝑏1𝑉𝐷. (13)

By using (4), (10)–(13), and themethodology stated above,𝐼𝐷of the triode region operated FGMOSFET is given by (14)

as 𝛼𝑠0 + 𝛼𝑠1𝑉𝐷 ≪ 1:

𝐼𝐷

= 𝛽 [1− 𝜃 (𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖]

+ (𝛼𝑑0 + 𝛼𝑑1𝑉𝐷) 𝑉𝐷 + (𝛼𝑏0 + 𝛼𝑏1𝑉𝐷) 𝑉𝐷 − 𝑉𝑆

− 𝑉TH)] [(𝑁

∑



− 𝑉TH) 𝑉DS −12

𝑉2DS] .

(14)

It can be seen from (14) that the process induced devicelevel random variations of FGMOSFET in triode are randomvariations in 𝑉TH, 𝛼𝑖0, 𝛼𝑖1, 𝛼𝑑0, 𝛼𝑑1, 𝛼𝑏0, 𝛼𝑏1, 𝛽, and 𝜃 denotedby Δ𝑉TH, Δ𝛼𝑖0, Δ𝛼𝑖1, Δ𝛼𝑑0, Δ𝛼𝑑1, Δ𝛼𝑏0, Δ𝛼𝑏1, Δ𝛽, and Δ𝜃,respectively. So, Δ𝐼

𝐷of FGMOSET in triode region can be

given by (15) where all derivatives given by (16)–(24) can befound by using (14):

Δ𝐼𝐷

= (𝜕𝐼𝐷

𝜕𝑉TH) Δ𝑉TH + (

𝜕𝐼𝐷

𝜕𝛽) Δ𝛽 + (

𝜕𝐼𝐷

𝜕𝜃) Δ𝜃

+ (𝜕𝐼𝐷

𝜕𝛼𝑑0

) Δ𝛼𝑑0 + (

𝜕𝐼𝐷

𝜕𝛼𝑑1

) Δ𝛼𝑑1 + (

𝜕𝐼𝐷

𝜕𝛼𝑏0

) Δ𝛼𝑏0

+ (𝜕𝐼𝐷

𝜕𝛼𝑏1

) Δ𝛼𝑏1 +𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖0

) Δ𝛼𝑖0]

+

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖1

) Δ𝛼𝑖1] ,

(15)


𝜕𝐼𝐷

𝜕𝑉TH=

𝛽

2𝑉DS [4𝜃 (𝛼𝑏1𝑉𝐵𝑉𝐷 + 𝛼𝑏0𝑉𝐵

+ (𝛼𝑑0 + 𝛼𝑑1𝑉𝐷) 𝑉𝐷 +

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])

− 𝜃 (4 (𝑉𝑆

+ 𝑉TH) + 𝑉DS) − 2] ,

(16)

𝜕𝐼𝐷

𝜕𝛽= [1− 𝜃 (

𝑁

∑



− 𝑉TH)] [(𝑁

∑



− 𝑉TH) 𝑉DS −12

𝑉2DS] ,

(17)

𝜕𝐼𝐷

𝜕𝜃= 𝛽 [𝑉

𝑆+ 𝑉TH − 𝑉𝐵 (𝛼𝑏0 + 𝛼𝑏1𝑉𝐷) − 𝑉𝐷 (𝛼𝑑0

+ 𝛼𝑑1𝑉𝐷) −

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖]] −

12

𝑉DS (𝑉DS

+ 2 (𝑉𝑆

+ 𝑉TH)) + 𝑉DS (𝛼𝑏1𝑉𝐵𝑉𝐷 + 𝛼𝑏0𝑉𝐵 + (𝛼𝑑0

+ 𝛼𝑑1𝑉𝐷) 𝑉𝐷 +

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖]) ,

(18)

𝜕𝐼𝐷

𝜕𝛼𝑑0

=𝛽

2𝑉𝐷

𝑉DS [2+ 𝜃𝑉DS + 4𝜃 (𝑉𝑆 + 𝑉TH)

− 4𝜃 (𝛼𝑏1𝑉𝐵𝑉𝐷 + 𝛼𝑏0𝑉𝐵 + (𝛼𝑑0 + 𝛼𝑑1𝑉𝐷) 𝑉𝐷

+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] ,

(19)

𝜕𝐼𝐷

𝜕𝛼𝑏0

=𝛽

2𝑉𝐵



+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] ,

(20)

𝜕𝐼𝐷

𝜕𝛼𝑑1

=𝛽

2𝑉DS𝑉

2𝐷

[2+ 𝜃𝑉DS + 4𝜃 (𝑉𝑆 + 𝑉TH)


+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] ,

(21)

𝜕𝐼𝐷

𝜕𝛼𝑏1

=𝛽

2𝑉𝐵

𝑉𝐷



+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] ,

(22)

𝜕𝐼𝐷

𝜕𝛼𝑖0

=𝛽

2𝑉DS

𝑁

∑

𝑖=1[𝑉𝑖] [2+ 𝜃𝑉DS + 4𝜃 (𝑉𝑆 + 𝑉TH)


+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] ,

(23)

𝜕𝐼𝐷

𝜕𝛼𝑖

=𝛽

2𝑉𝐷

𝑉DS

𝑁

∑

𝑖=1[𝑉𝑖] [2+ 𝜃𝑉DS + 4𝜃 (𝑉𝑆 + 𝑉TH)


+

𝑁

∑

𝑖=1[(𝛼𝑖0 + 𝛼𝑖1𝑉𝐷) 𝑉𝑖])] .

(24)

Since Δ𝑉TH, Δ𝛼𝑖0, Δ𝛼𝑖1, Δ𝛼𝑑0, Δ𝛼𝑑1, Δ𝛼𝑏0, Δ𝛼𝑏1, Δ𝛽,and Δ𝜃 are random variables, so are Δ𝐼

𝐷and the statistical

behavior ofΔ𝐼𝐷must be analyticallymodeled for its complete

modeling. In order to do so, its average (Δ𝐼𝐷,avr) and variance

(𝜎2Δ𝐼𝐷

) must be formulated. As a result, Δ𝐼𝐷,avr = 0 similar to

those ofΔ𝑉TH,Δ𝛼𝑖0,Δ𝛼𝑖1,Δ𝛼𝑑0,Δ𝛼𝑑1,Δ𝛼𝑏0,Δ𝛼𝑏1,Δ𝛽, andΔ𝜃and 𝜎2

Δ𝐼𝐷of FGMOSFET in triode can be given by taking the

statistical correlations of Δ𝑉TH, Δ𝛼𝑖0, Δ𝛼𝑖1, Δ𝛼𝑑0, Δ𝛼𝑑1, Δ𝛼𝑏0,Δ𝛼𝑏1, Δ𝛽, and Δ𝜃 into account as follows:

𝜎2Δ𝐼𝐷

= (𝜕𝐼𝐷

𝜕𝑉TH)

2𝜎2Δ𝑉TH

+ (𝜕𝐼𝐷

𝜕𝛽)

2𝜎2Δ𝛽

+ (𝜕𝐼𝐷

𝜕𝜃)

2𝜎2Δ𝜃

+ (𝜕𝐼𝐷

𝜕𝛼𝑑0

)

2𝜎2Δ𝛼𝑑0

+ (𝜕𝐼𝐷

𝜕𝛼𝑑1

)

2𝜎2Δ𝛼𝑑1

+ (𝜕𝐼𝐷

𝜕𝛼𝑏0

)

2

⋅ 𝜎2Δ𝛼𝑏0

+ (𝜕𝐼𝐷

𝜕𝛼𝑏1

)

2𝜎2Δ𝛼𝑏1

+

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖0

)

2𝜎2Δ𝛼𝑖0

]


+

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖1

)

2𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝑉TH

) (𝜕𝐼𝐷

𝜕𝛽)

⋅ 𝜌Δ𝑉TH ,Δ𝛽

√𝜎2Δ𝑉TH

√𝜎2Δ𝛽

+ 2( 𝜕𝐼𝐷𝜕𝑉TH

) (𝜕𝐼𝐷

𝜕𝜃)

⋅ 𝜌Δ𝑉TH ,Δ𝜃

√𝜎2Δ𝑉TH

√𝜎2Δ𝜃


) (𝜕𝐼𝐷

𝜕𝛼𝑑0

)

⋅ 𝜌Δ𝑉TH ,Δ𝛼𝑑0

√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑑0


) (𝜕𝐼𝐷

𝜕𝛼𝑑1

)


√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑑1


) (𝜕𝐼𝐷

𝜕𝛼𝑏0

)

⋅ 𝜌Δ𝑉TH ,Δ𝛼𝑑0

√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑏0


) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)


√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑉TH) (

𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ 𝜌Δ𝑉TH ,Δ𝛼𝑖0

√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑉TH)

⋅ (𝜕𝐼𝐷

𝜕𝛼𝑖1

) 𝜌Δ𝑉TH,Δ𝛼𝑖1

√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝛽

)

⋅ (𝜕𝐼𝐷

𝜕𝜃) 𝜌Δ𝛽,Δ𝜃√𝜎

2Δ𝛽

√𝜎2Δ𝜃

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝛼𝑑0

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑑0

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑑0

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝛼𝑑1

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑑1

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑑1

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝛼𝑏0

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑏0

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑏0

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑏1

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛽) (

𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑖0

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛽) (

𝜕𝐼𝐷

𝜕𝛼𝑖1

)

⋅ 𝜌Δ𝛽,Δ𝛼𝑖1

√𝜎2Δ𝛽

√𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝛼𝑑0

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑑0

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑑0

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝛼𝑑1

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑑1

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑑1

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝛼𝑏0

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑏0

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑏0

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑏1

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝜃) (

𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑖0

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝜃) (

𝜕𝐼𝐷

𝜕𝛼𝑖1

)

⋅ 𝜌Δ𝜃,Δ𝛼𝑖1

√𝜎2Δ𝜃

√𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝛼𝑑0

) (𝜕𝐼𝐷

𝜕𝛼𝑑1

)

⋅ 𝜌Δ𝛼𝑑0,Δ𝛼𝑑1

√𝜎2Δ𝛼𝑑0

√𝜎2Δ𝛼𝑑1

+ 2( 𝜕𝐼𝐷𝜕𝛼𝑑0

) (𝜕𝐼𝐷

𝜕𝛼𝑏0

)

⋅ 𝜌Δ𝛼𝑑0,Δ𝛼𝑏0

√𝜎2Δ𝛼𝑑0

√𝜎2Δ𝛼𝑏0

+ 2( 𝜕𝐼𝐷𝜕𝛼𝑑0

) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)


√𝜎2Δ𝛼𝑑0

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑑0

) (𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ 𝜌Δ𝛼𝑑0,Δ𝛼𝑖0

√𝜎2Δ𝛼𝑑0

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑑0

) (𝜕𝐼𝐷

𝜕𝛼𝑖1

)


√𝜎2Δ𝛼𝑑0

√𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝛼𝑑1

) (𝜕𝐼𝐷

𝜕𝛼𝑏0

)


√𝜎2Δ𝛼𝑑1

√𝜎2Δ𝛼𝑏0

+ 2( 𝜕𝐼𝐷𝜕𝛼𝑑1

) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)


√𝜎2Δ𝛼𝑑1

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑑1

) (𝜕𝐼𝐷

𝜕𝛼𝑖0

)


√𝜎2Δ𝛼𝑑1

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑑1

) (𝜕𝐼𝐷

𝜕𝛼𝑖1

)


√𝜎2Δ𝛼𝑑1

√𝜎2Δ𝛼𝑖1

] + 2( 𝜕𝐼𝐷𝜕𝛼𝑏0

) (𝜕𝐼𝐷

𝜕𝛼𝑏1

)

⋅ 𝜌Δ𝛼𝑏0,Δ𝛼𝑏1

√𝜎2Δ𝛼𝑏0

√𝜎2Δ𝛼𝑏1

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑏0

) (𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ 𝜌Δ𝛼𝑏0,Δ𝛼𝑖0

√𝜎2Δ𝛼𝑏0

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑏0

) (𝜕𝐼𝐷

𝜕𝛼𝑖1

)


√𝜎2Δ𝛼𝑏0

√𝜎2Δ𝛼𝑖1

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑏1

) (𝜕𝐼𝐷

𝜕𝛼𝑖0

)


√𝜎2Δ𝛼𝑏1

√𝜎2Δ𝛼𝑖0

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑏1

) (𝜕𝐼𝐷

𝜕𝛼𝑖1

)


√𝜎2Δ𝛼𝑏1

√𝜎2Δ𝛼𝑖1

] + 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖0

) (𝜕𝐼𝐷

𝜕𝛼𝑖1

)

⋅ 𝜌Δ𝛼𝑖0,Δ𝛼𝑖1

√𝜎2Δ𝛼𝑖0

√𝜎2Δ𝛼𝑖1

] +

𝑁

∑

𝑗=1𝑗 ̸=𝑖

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖0

)

⋅ (𝜕𝐼𝐷

𝜕𝛼𝑗0

) 𝜌Δ𝛼𝑖0,Δ𝛼𝑗0

√𝜎2Δ𝛼𝑖0

√𝜎2Δ𝛼𝑗0

]


+

𝑁

∑

𝑗=1

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖0

) (𝜕𝐼𝐷

𝜕𝛼𝑗1


√𝜎2Δ𝛼𝑖0

√𝜎2Δ𝛼𝑗1

]

+

𝑁

∑

𝑗=1

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖1

) (𝜕𝐼𝐷

𝜕𝛼𝑗0


√𝜎2Δ𝛼𝑖1

√𝜎2Δ𝛼𝑗0

]

+

𝑁

∑

𝑗=1𝑗 ̸=𝑖

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛼𝑖1

) (𝜕𝐼𝐷

𝜕𝛼𝑗1

)

⋅ 𝜌Δ𝛼𝑖1,Δ𝛼𝑗1

√𝜎2Δ𝛼𝑖1

√𝜎2Δ𝛼𝑗1

] ,

(25)

where 𝜌Δ𝑥,Δ𝑦

, for example, 𝜌Δ𝑉TH ,Δ𝛽

, 𝜌Δ𝛽,Δ𝜃

, and 𝜌Δ𝑉TH ,Δ𝜃

,denotes the correlation coefficient of Δ𝑥 and Δ𝑦 and displaystheir degree of statistical correlation. Moreover, 𝜎2

Δ𝑉TH, 𝜎2Δ𝛽,

𝜎2Δ𝛼𝑖0

, 𝜎2Δ𝛼𝑖1

, 𝜎2Δ𝛼𝑑0

, 𝜎2Δ𝛼𝑑1

, 𝜎2Δ𝛼𝑏0

, 𝜎2Δ𝛼𝑏1

, and 𝜎2Δ𝜃

are nonzero and,respectively, denote the variances of Δ𝑉TH, Δ𝛼𝑖0, Δ𝛼𝑖1, Δ𝛼𝑑0,Δ𝛼𝑑1, Δ𝛼𝑏0, Δ𝛼𝑏1, Δ𝛽, and Δ𝜃. Finally, it can be seen that the

1st up to 9th terms of (25) have been contributed by 𝜎2Δ𝑉TH

,𝜎2Δ𝛽, 𝜎2Δ𝛼𝑖0

, 𝜎2Δ𝛼𝑖1

, 𝜎2Δ𝛼𝑑0

, 𝜎2Δ𝛼𝑑1

, 𝜎2Δ𝛼𝑏0

, 𝜎2Δ𝛼𝑏1

, and 𝜎2Δ𝜃

where theothers have been caused by the statistical correlations.

3.2. Saturation Region Part. Firstly, 𝐼𝐷of FGMOSFET in this

region will be formulated in a similar manner to that ofFGMOSFET in triode where the second order effects havebeen taken into account as well. With the linear model ofmobility degradation, 𝐼

𝐷of MOSFET in saturation can be

given by

𝐼𝐷

=𝛽

2[1− 𝜃 (𝑉GS − 𝑉TH)] (1+ 𝜆𝑉DS) (𝑉GS − 𝑉TH)

2,

(26)

where 𝜆 stands for the channel length modulation coefficientwhich must be taken into account for MOSFET in saturationif the short channel effect has been considered.

For FGMOSFET in saturation region, the model formu-lation is simpler than that of triode FGMOSFET since 𝑘

𝑖, 𝑘fd,

𝑘fs, and 𝑘fb are constant. So, 𝑉FG can be given by (3) and 𝐼𝐷can be determined in a similarmanner to that of FGMOSFETin triode by using (3) and (26) as given in (27) since 𝑘fs ≪ 1:

𝐼𝐷

=𝛽

2(1+ 𝜆𝑉DS) [1

− 𝜃 (

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH)]

⋅ (

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖]

+ 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH)

2

.

(27)

As the process induced random variations in 𝜆(Δ𝜆) mustalso be considered for FGMOSET in saturation region, Δ𝐼

𝐷

is given by (28). By using (27), all derivatives can be found as(29)–(35):

Δ𝐼𝐷

= (𝜕𝐼𝐷

𝜕𝑉TH) Δ𝑉TH + (

𝜕𝐼𝐷

𝜕𝛽) Δ𝛽 + (

𝜕𝐼𝐷

𝜕𝜃) Δ𝜃

+ (𝜕𝐼𝐷

𝜕𝜆) Δ𝜆 + (

𝜕𝐼𝐷

𝜕𝑘fd) Δ𝑘fd + (

𝜕𝐼𝐷

𝜕𝑘fb) Δ𝑘fb

+

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑘𝑖

) Δ𝑘𝑖] ,

(28)

𝜕𝐼𝐷

𝜕𝛽=1 + 𝜆𝑉DS

2[1

− 𝜃 (

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH)]

⋅ (

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆

− 𝑉TH)

2

,

(29)

𝜕𝐼𝐷

𝜕𝜃=

𝛽

2(1+ 𝜆𝑉DS) (

𝑁

∑


− 𝑉TH)

3

,

(30)

𝜕𝐼𝐷

𝜕𝑉TH=

𝛽

2(1+ 𝜆𝑉DS)

⋅ [3𝜃 (𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH)

− 2] (𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH) ,

(31)

𝜕𝐼𝐷

𝜕𝜆=

𝛽

2𝑉DS [1

− 𝜃 (

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 −VTH)]

⋅ (

𝑁

∑


− 𝑉TH)

2

,

(32)


𝜕𝐼𝐷

𝜕𝑘fd= −

𝛽

2𝑉𝐷

(1+ 𝜆𝑉DS)

⋅ [3𝜃 (𝑁

∑


− 2] (𝑁

∑


(33)

𝜕𝐼𝐷

𝜕𝑘fb= −

𝛽

2𝑉𝐵

(1+ 𝜆𝑉DS)

⋅ [3𝜃 (𝑁

∑


− 2] (𝑁

∑


(34)

𝜕𝐼𝐷

𝜕𝑘𝑖

= −𝛽

2(

𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖]) (1+ 𝜆𝑉DS)

⋅ [3𝜃 (𝑁

∑


− 2] (𝑁

∑

𝑖=1[𝑘𝑖𝑉𝑖] + 𝑘fd𝑉𝐷 + 𝑘fb𝑉𝐵 − 𝑉𝑆 − 𝑉TH) .

(35)

Similarly to Δ𝐼𝐷of the FGMOSFET in triode, Δ𝐼

𝐷of

the FGMOSFET in saturation is also a random variable. So,its statistical behavior must be analytically modeled as well.SinceΔ𝑉TH,Δ𝛽,Δ𝑘𝑖,Δ𝑘fd,Δ𝑘fb,Δ𝜃, andΔ𝜆 have zeromeans,it has also been found that Δ𝐼

𝐷,avr = 0. Moreover, 𝜎2Δ𝐼𝐷

can begiven by taking the statistical correlations of Δ𝑉TH, Δ𝛽, Δ𝑘𝑖,Δ𝑘fd, Δ𝑘fb, Δ𝜃, and Δ𝜆 into account as given in (36) wherethe first seven terms have been, respectively, contributed by𝜎2Δ𝑉TH

, 𝜎2Δ𝛽, 𝜎2Δ𝑘𝑖

, 𝜎2Δ𝑘fd

, 𝜎2Δ𝑘fb

, 𝜎2Δ𝜃, and 𝜎2

Δ𝜆. On the other

hand, the other terms have been contributed by the statisticalcorrelations.

Before proceeding to the model verification, it should bementioned here that 𝜌

Δ𝑥,Δ𝑦can be obtained either by using

the scatter plot [25, 26] or by calculation as given in (37)which can be obtained based on the prior knowledge that theaverages ofΔ𝑥 andΔ𝑦 are 0 as they are process induceddevicelevel random variations, that is, Δ𝑉TH, Δ𝛽, and Δ𝜃. Moreover,𝐸[ ] stands for the expectation operator:

𝜎2Δ𝐼𝐷

= (𝜕𝐼𝐷

𝜕𝑉TH)

2𝜎2Δ𝑉TH

+ (𝜕𝐼𝐷

𝜕𝛽)

2𝜎2Δ𝛽

+ (𝜕𝐼𝐷

𝜕𝜃)

2𝜎2Δ𝜃

+ (𝜕𝐼𝐷

𝜕𝜆)

2𝜎2Δ𝜆

+ (𝜕𝐼𝐷

𝜕𝑘fb)

2𝜎2Δ𝑘fb

+ (𝜕𝐼𝐷

𝜕𝑘fd)

2𝜎2Δ𝑘fd

+

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑘𝑖

)

2𝜎2Δ𝑘𝑖

]

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝑉TH) 𝜌Δ𝛽,Δ𝑉TH

√𝜎2Δ𝛽

√𝜎2Δ𝑉TH

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝜃) 𝜌Δ𝛽,Δ𝜃√𝜎

2Δ𝛽

√𝜎2Δ𝜃

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝜆) 𝜌Δ𝛽,Δ𝜆√𝜎

2Δ𝛽

√𝜎2Δ𝜆

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝑘fd) 𝜌Δ𝛽,Δ𝑘fd

√𝜎2Δ𝛽

√𝜎2Δ𝑘fd

+ 2( 𝜕𝐼𝐷𝜕𝛽

) (𝜕𝐼𝐷

𝜕𝑘fb) 𝜌Δ𝛽,Δ𝑘fb

√𝜎2Δ𝛽

√𝜎2Δ𝑘fb

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝛽) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝛽,Δ𝑘𝑖

√𝜎2Δ𝛽

√𝜎2Δ𝑘𝑖

]

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝑉TH) 𝜌Δ𝜃,Δ𝑉TH

√𝜎2Δ𝜃

√𝜎2Δ𝑉TH

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝜆) 𝜌Δ𝜃,Δ𝜆

√𝜎2Δ𝜃

√𝜎2Δ𝜆

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝑘fd) 𝜌Δ𝜃,Δ𝑘fd

√𝜎2Δ𝜃

√𝜎2Δ𝑘fd

+ 2( 𝜕𝐼𝐷𝜕𝜃

) (𝜕𝐼𝐷

𝜕𝑘fb) 𝜌Δ𝜃,Δ𝑘fb

√𝜎2Δ𝜃

√𝜎2Δ𝑘fb

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝜃) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝜃,Δ𝑘𝑖

√𝜎2Δ𝜃

√𝜎2Δ𝑘𝑖

]


) (𝜕𝐼𝐷

𝜕𝜆) 𝜌Δ𝑉TH,Δ𝛼𝜆

√𝜎2Δ𝑉TH

√𝜎2Δ𝜆


) (𝜕𝐼𝐷

𝜕𝑘fd) 𝜌Δ𝑉TH,Δ𝑘fd

√𝜎2Δ𝑉TH

√𝜎2Δ𝑘fd


) (𝜕𝐼𝐷

𝜕𝑘fb) 𝜌Δ𝑉TH,Δ𝑘fb

√𝜎2Δ𝑉TH

√𝜎2Δ𝛼𝑘fb

+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑉TH) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝑉TH ,Δ𝑘𝑖

√𝜎2Δ𝑉TH

√𝜎2Δ𝑘𝑖

]

+ 2( 𝜕𝐼𝐷𝜕𝜆

) (𝜕𝐼𝐷

𝜕𝑘fd) 𝜌Δ𝜆,Δ𝑘fd

√𝜎2Δ𝜆

√𝜎2Δ𝛼𝑘fd

+ 2( 𝜕𝐼𝐷𝜕𝜆

) (𝜕𝐼𝐷

𝜕𝑘fb) 𝜌Δ𝜆,Δ𝑘fb

√𝜎2Δ𝜆


+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝜆) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝜆,Δ𝑘𝑖

√𝜎2Δ𝜆

√𝜎2Δ𝑘𝑖

]

+ 2( 𝜕𝐼𝐷𝜕𝑘fd

) (𝜕𝐼𝐷

𝜕𝑘fb) 𝜌Δ𝑘fd ,Δ𝑘fb

√𝜎2Δ𝑘fd


+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑘fd) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝑘fd ,Δ𝑘𝑖

√𝜎2Δ𝑘fd

√𝜎2Δ𝑘𝑖

]


+ 2𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑘fb) (

𝜕𝐼𝐷

𝜕𝑘𝑖

) 𝜌Δ𝑘fb,Δ𝑘𝑖

√𝜎2Δ𝑘fb

√𝜎2Δ𝑘𝑖

]

+

𝑁

∑

𝑗=1𝑗 ̸=𝑖

𝑁

∑

𝑖=1[(

𝜕𝐼𝐷

𝜕𝑘𝑖

) (𝜕𝐼𝐷

𝜕𝑘𝑗

) 𝜌Δ𝑘𝑖 ,Δ𝑘𝑗

√𝜎2Δ𝑘𝑖

√𝜎2Δ𝑘𝑗

] ,

(36)

𝜌Δ𝑥,Δ𝑦

=𝐸 [Δ𝑥Δ𝑦]

𝜎Δ𝑥

𝜎Δ𝑦

. (37)

4. Model Verification

Before proceeding further, it should be mentioned here thatthe model verification has been performed by assuming that𝐶𝑖

≫ 𝐶fd, 𝐶fs, 𝐶fb as has been assumed in many previousstudies on FGMOSFET, for example, [2, 7–9, 20]. As a result,𝑘fd, 𝑘fs, and 𝑘fb can be neglected as 𝑘𝑖 ≫ 𝑘fd, 𝑘fs, 𝑘fb and𝐶𝑇is now independent of 𝑉

𝐷because 𝐶fb which causes the

dependency of 𝐶𝑇on 𝑉𝐷

can be neglected. So, 𝑘𝑖, which

depends on𝐶𝑇, is now constant for both triode and saturation

regions as also assumed in [22].Moreover, the model verification has been performed

based on FGMOSFET of both N-type and P-type with 𝑁 = 2and 𝑘1

= 𝑘2

= 0.5 for both triode and saturation regions,W/L= 20/0.25, SPICE BSIM3v3, and 0.25 𝜇m level CMOS processtechnology of TSMC where all necessary parameters havebeen provided by MOSIS. For performing the verification,the root mean square (rms.) value of Δ𝐼

𝐷calculated by using

the model (Δ𝐼𝐷,rms,𝑀) has been graphically compared to its

SPICE BSIM3v3 based reference (Δ𝐼𝐷,rms,SPICE) obtained by

using theMonte-Carlo SPICE simulation with 1000 runs. Forconvenience, Δ𝑉TH, Δ𝛽, Δ𝑘1, Δ𝑘2, Δ𝜃, and Δ𝜆 have beenassumed to be normally distributed and all 𝜌

Δ𝑥,Δ𝑦’s have been

assumed to be 0.5 which is a reasonable estimation [26]. Sincecorrelations must be taken into account in the simulations,each of Δ𝑉TH, Δ𝛽, Δ𝑘1, Δ𝑘2, Δ𝜃, and Δ𝜆 has been expressedas a weighted sum of its correlated and uncorrelated com-ponents which are both normally distributed [26] and whichhave equal weights given by√0.5 due to the assumed 𝜌

Δ𝑥,Δ𝑦’s.

Finally, the SPICE BSIM3v3 basedmodelling of FGMOS-FET with 𝑁 = 2 can be performed by using the two inputs’version of the equivalent circuit of FGMOSFET in Figure 2where the core MOSFET has been modelled by using theSPICE BSIM3v3 and the simulation methodology proposedin [23] has been adopted for solving the convergence problemof the simulator. Both Δ𝐼

𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE have beenexpressed as percentage of deterministic 𝐼

𝐷and compara-

tively plotted against the magnitude of the voltage of the1st and 2nd inputs of FGMOSFET which are ranged from 3to 8V and denoted by |𝑉1| and |𝑉2|, respectively. It shouldbe mentioned here that |𝑉

2| = 0V in the comparative plots

against |𝑉1| and vice versa. At this point, verification of theproposed model will be presented.

4.1. Verification of Triode Region Part. Since 𝜆 and of courseΔ𝜆 have not been considered in triode region, only Δ𝑉TH,

30

25

20

15

10

5

03 4 5 6 7 8

ΔI D

,rms

(%)

|V1| (V)

Figure 3: Triode and saturation regions N-type FGMOSFET basedcomparative plots of Δ𝐼

𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE against |𝑉1| where|𝑉2| = 0.

30

25

20

15

10

5

0

ΔI D

,rms

(%)

3 4 5 6 7 8

|V2| (V)

Figure 4: Triode and saturation regions N-type FGMOSFET basedcomparative plots of Δ𝐼


Δ𝛽, Δ𝜃, Δ𝑘1, and Δ𝑘

2have been taken into account. We let

𝜎Δ𝑉TH

/𝑉TH = 𝜎Δ𝛽/𝛽 = 𝜎Δ𝜃/𝜃 = 𝜎Δ𝑘1/𝑘1 = 𝜎Δ𝑘2/𝑘2 for makingthe device level variations be bias-free and let these quantitiesbe 0.01 as it has been assumed that 𝑉TH, 𝛽, 𝜃, 𝑘1, and 𝑘2have 1% variation. For N-type FGMOSFET, the comparativeplots of Δ𝐼

𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE against |𝑉1| where |𝑉2| =0 and vice versa can be, respectively, shown in Figures 3 and4 whereas those of P-type FGMOSFET are shown in Figures5 and 6. In these figures, highly strong agreements betweenΔ𝐼𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE which are, respectively, drawn as

blue normal curves and red dotted curves can be observedand the average deviations of Δ𝐼

𝐷,rms,𝑀 from Δ𝐼𝐷,rms,SPICEhave been found as 3.74725% and 3.70765% for N-typeand P-type FGMOSFET, respectively.These deviations whichcan be determined from triode region N-type and P-typeFGMOSFET based comparative plots shown in Figures 3 and4 and in Figures 5 and 6, respectively, are notably very small.So, it can be seen that triode region part of the proposedmodel has been found to be very accurate.

4.2. Verification of Saturation Region Part. In saturationregion, 𝜆 and Δ𝜆 must be considered. For making the device


30

25

20

15

10

5

0

ΔI D

,rms

(%)

3 4 5 6 7 8

|V1| (V)

Figure 5: Triode and saturation regions P-type FGMOSFET basedcomparative plots of Δ𝐼


30

25

20

15

10

5

0

ΔI D

,rms

(%)

3 4 5 6 7 8

|V2| (V)

Figure 6: Triode and saturation regions P-type FGMOSFET basedcomparative plots of Δ𝐼


level variations be bias-free and following the assumption that𝑉TH, 𝛽, 𝜃, 𝑘1, 𝑘2, and 𝜆 have 1% variation, we let 𝜎Δ𝑉TH/𝑉TH =𝜎Δ𝛽

/𝛽 = 𝜎Δ𝜆

/𝜆 = 𝜎Δ𝜃

/𝜃 = 𝜎Δ𝑘1

/𝑘1

= 𝜎Δ𝑘2

/𝑘2

= 0.01.TheN-type FGMOSFETbased comparative plots ofΔ𝐼

𝐷,rms,𝑀and Δ𝐼

𝐷,rms,SPICE against |𝑉1| where |𝑉2| = 0 and vice versacan also be, respectively, depicted in Figures 3 and 4 wherethose based on P-type device are shown in Figures 5 and6 as well. However, Δ𝐼

𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE have been,respectively, drawn as green normal curves and black dottedcurves instead in this case. From Figures 3–6, highly strongagreements between Δ𝐼

𝐷,rms,𝑀 and Δ𝐼𝐷,rms,SPICE can also beseen in this case. The average deviations between Δ𝐼

𝐷,rms,𝑀and Δ𝐼

𝐷,rms,SPICE have been, respectively, found as 2.6901%and 3.6681% for N-type and P-type FGMOSFET.These devi-ations which can be determined from the saturation regionFGMOSFET based comparative plots are also considerablyvery small. As a result, the accuracy of saturation region partof the model has been verified.

From Figures 3–6, it has been found that both Δ𝐼𝐷,rms,𝑀

and Δ𝐼𝐷,rms,SPICE are increased with decreasing |𝑉1| and |𝑉2|

because they have been expressed as percentage of their

corresponding deterministic 𝐼𝐷’s which are increased with

increasing |𝑉1| and |𝑉2| but with larger degree than the grossvalues of both rms. values do. Such increasing of Δ𝐼

𝐷,rms,𝑀andΔ𝐼

𝐷,rms,SPICE with decreasing |𝑉1| and |𝑉2|means thatΔ𝐼𝐷become more critical in low voltage/low power condition.

By considering the magnitude of both Δ𝐼𝐷,rms,𝑀 and

Δ𝐼𝐷,rms,SPICE, it has also been found that the FGMOSFET in

triode is more robust than that in saturation region when |𝑉1|and |𝑉

2| become sufficiently large.This can be obviously seen

in P-type device. On the other hand, the device in saturationis more robust when |𝑉1| and |𝑉2| become adequately low.This is more obvious in the N-type FGMOSFET. If desired,Δ𝐼𝐷,rms,𝑀 can fit Δ𝐼𝐷,rms,SPICE simulated by using BSIM4. In

order to do so, the optimum parameters of MOSFET’s 𝐼𝐷

equations, that is, (6) and (14), extracted from BSIM4’s 𝐼𝐷

model by using the optimization algorithm [24]must be used.

5. Discussions

Firstly, insight into device level variability of FGMOSFETwill be discussed. By using the proposed model, the rate ofchange of per unit Δ𝐼

𝐷(Δ𝐼𝐷

/𝐼𝐷) of FGMOSFET in triode

region with respect to per unit Δ𝑉TH, Δ𝛼𝑖0, Δ𝛼𝑖1, Δ𝛼𝑑0, Δ𝛼𝑑1,Δ𝛼𝑏0, Δ𝛼𝑏1, Δ𝛽, and Δ𝜃 denoted by Δ𝑉TH/𝑉TH, Δ𝛼𝑖0/𝛼𝑖0,

Δ𝛼𝑖1

/𝛼𝑖1, Δ𝛼𝑑0

/𝛼𝑑0, Δ𝛼𝑑1

/𝛼𝑑1, Δ𝛼𝑏0

/𝛼𝑏0, Δ𝛼𝑏1

/𝛼𝑏1, Δ𝛽/𝛽, and

Δ𝜃/𝜃, respectively, can be approximately given under lowvoltage/low power condition as follows:

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝑉TH/𝑉TH=1 + 2𝜃𝑉TH1 + 𝜃𝑉TH

, (38)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛽/𝛽= 1, (39)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝜃/𝜃=

𝜃𝑉TH1 + 𝜃𝑉TH

, (40)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑑0/𝛼𝑑0

= 0, (41)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑑1/𝛼𝑑1

= 0, (42)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑏0/𝛼𝑏0

= 0, (43)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑏1/𝛼𝑏1

= 0, (44)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑖0/𝛼𝑖0

= 0, (45)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛼𝑖1/𝛼𝑖1

= 0. (46)

For the saturation region, a similar rate of changes can beapproximately found under a similar condition as

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝛽/𝛽= 1, (47)


𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝜃/𝜃=

𝜃𝑉TH1 + 𝜃𝑉TH

, (48)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝑉TH/𝑉TH=2 + 3𝜃𝑉TH1 + 𝜃𝑉TH

, (49)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝜆/𝜆= 0, (50)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝑘fd/𝑘fd= 0, (51)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝑘fb/𝑘fb= 0, (52)

𝜕Δ𝐼𝐷

/𝐼𝐷

𝜕Δ𝑘𝑖/𝑘𝑖

= 0. (53)

From these equations, it can be seen that all processinduced device level randomvariations are relatively insignif-icant compared to Δ𝛽, Δ𝜃, and Δ𝑉TH under low voltage/lowpower condition. So, Δ𝛽, Δ𝜃, and Δ𝑉TH must be mostlyconsidered in the variability aware design of low voltage/lowpower FGMOSFET based circuit. Moreover, the effects of Δ𝜃and Δ𝑉TH can be reduced by lowering either 𝜃 or 𝑉TH or bothas can be seen from (38), (40), (48), and (49).

Secondly, application of the proposed model will bediscussed. This model can be used as the basis for theanalytical modeling of process induced random variation inthe circuit level parameter of any FGMOSFET based circuit.Let the circuit level parameter of interest of any FGMOSFETbased circuit which is composed of𝑀FGMOSFETs be𝑋.Theprocess induced random variation in 𝑋 (Δ𝑋) caused by thecombined effect of Δ𝐼

𝐷of each FGMOSFET can be given by

Δ𝑋 =

𝑀

∑

𝑚=1[

𝜕𝑋

𝜕𝐼𝐷𝑚

Δ𝐼𝐷𝑚

] , (54)

where 𝐼𝐷𝑚

and Δ𝐼𝐷𝑚

denote the 𝐼𝐷

and Δ𝐼𝐷

of any 𝑚thFGMOSFET. Since Δ𝐼

𝐷𝑚can be analytically determined by

using the proposed model due to its definition, so does Δ𝑋via (54).

Since the average of Δ𝑋 is zero as those of Δ𝐼𝐷𝑚

’s are, itis convenient to analytically model the statistical behavior ofΔ𝑋 by using its variance (𝜎2

Δ𝑋) as those ofΔ𝐼

𝐷𝑚’s are nonzero.

According to the above definition of Δ𝐼𝐷𝑚

, the varianceof Δ𝐼𝐷𝑚

(𝜎2Δ𝐼𝐷𝑚

) can be determined by using the proposedmodel. So, 𝜎2

Δ𝑋can be determined by using the proposed

model as well as with the aid of the following equation:

𝜎2Δ𝑋

=

𝑀

∑

𝑚=1[(

𝜕𝑋

𝜕𝐼𝐷𝑚

)

2𝜎2Δ𝐼𝐷𝑚

] +

𝑀

∑

𝑚=1𝑚 ̸=𝑘

𝑀

∑

𝑘=1[(

𝜕𝑋

𝜕𝐼𝐷𝑚

)

⋅ (𝜕𝑋

𝜕𝐼𝐷𝑘

) 𝜌𝐼𝐷𝑚 ,𝐼𝐷𝑘

√𝜎2Δ𝐼𝐷𝑚

𝜎2Δ𝐼𝐷𝑘

] .

(55)

As a practical illustrative example, let 𝑋 be the transcon-ductance (𝐺

𝑚) of the FGMOSFET based voltage to current

converter (VIC) [1]. The equivalent circuit of its core circuit

VinCin

M1

CF

VF

Cfd1

Cfb1

Cfs1

Figure 7: Equivalent circuit of FGMOSFET based transconductorwhich is the core circuit of FGMOSFET based VIC [1].

which is the FGMOSFET based transconductor can bedepicted as in Figure 7 where𝐶

𝐹and𝐶in are the feedback and

input capacitance of the transconductor [1]. Moreover, 𝐶fd1,𝐶fs1, and 𝐶fb1 are, respectively, 𝐶fd, 𝐶fs, and 𝐶fb of 𝑀1.

Obviously, 𝐺𝑚can be given by

𝐺𝑚

=√2𝛽1𝐶in

𝐶in + 𝐶𝐹 + 𝐶fd1 + 𝐶fs1 + 𝐶fb1√𝐼𝐷1, (56)

where 𝐼𝐷1

and 𝛽1denote 𝐼

𝐷and 𝛽 of 𝑀

1.

Since 𝑀1is in saturation region and has 𝑁 = 2, 𝐼

𝐷1can

be given as follows:

𝐼𝐷1 =

𝛽12

(1+ 𝜆1𝑉𝑓)

⋅ [1− 𝜃1 (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 −VTH1)]

⋅ (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 −VTH1)2

,

(57)

where 𝑉TH1, 𝜃1, and 𝜆1 denote 𝑉TH, 𝜃, and 𝜆 of 𝑀1.Moreover, 𝑘in, 𝑘𝐹, and 𝑘fd1 can be defined as

𝑘in =𝐶in

𝐶in + 𝐶𝐹 + 𝐶fd1 + 𝐶fs1 + 𝐶fb1,

𝑘𝐹

=𝐶𝐹

𝐶in + 𝐶𝐹 + 𝐶fd1 + 𝐶fs1 + 𝐶fb1,

𝑘fd1 =𝐶fd1

𝐶in + 𝐶𝐹 + 𝐶fd1 + 𝐶fs1 + 𝐶fb1.

(58)

At this point, the analytical model of process inducedrandom variation in 𝐺

𝑚(Δ𝐺𝑚) can be given by using (54)

and (56) as follows:

Δ𝐺𝑚

=𝑘in (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 − VTH1)

−1

√(1 + 𝜆1𝑉𝐹) [1 − 𝜃1 (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 − VTH1)]

⋅ Δ𝐼𝐷1,

(59)


10

8

6

4

2

00 1 2 3 4 5

√𝜎2 Δ

Gm

(%)

Vin (V)

Figure 8: √𝜎2Δ𝐺𝑚

(%) of the FGMOSFET based transconductoragainst |𝑉in|.

where Δ𝐼𝐷1

denotes the process induced random variation in𝐼𝐷1

and can be determined by using (28)–(35) which belongto the saturation region part of the proposedmodel, with𝑁 =2.

Moreover, it is necessary to model the statistical behaviorof Δ𝐺

𝑚by using its variance (𝜎2

Δ𝐺𝑚) because its mean is equal

to zero similarly to that of Δ𝐼𝐷1. By using (55) and (56), 𝜎2

Δ𝐺𝑚

can be formulated as

𝜎2Δ𝐺𝑚

= [𝑘2in (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 − VTH1)

−2

(1 + 𝜆1𝑉𝐹) [1 − 𝜃1 (𝑘in𝑉in + (𝑘𝐹 + 𝑘fd1) 𝑉𝐹 − VTH1)]]

2

⋅ 𝜎2Δ𝐼𝐷1

,

(60)

where 𝜎2Δ𝐼𝐷1

denotes the variance of Δ𝐼𝐷1

and can be ana-lytically determined by using (29)–(36) which belong to thesaturation region part proposedmodel, with 𝑁 = 2. It shouldbe mentioned here that there is no correlation related termin (60) since 𝐺

𝑚is solely contributed by 𝐼

𝐷1. Finally, we can

plot √𝜎2Δ𝐺𝑚

against |𝑉in| as shown in Figure 8 where √𝜎2Δ𝐺𝑚has been expressed as percentage of the nominal 𝐺

𝑚.

6. Conclusion

In this research, the analytical model of Δ𝐼𝐷of FGMOSFET

has been proposed. The model is composed of two partsfor triode and saturation regions of operation. The processinduced device level random variations of each region, theirstatistical correlations, nonlinearity of 𝑉FG, and dependencyon 𝑉𝐷of the coupling factors have been taken into account.

The proposedmodel has been found to be very accurate sinceit can fit the SPICE BSIM3v3 based reference obtained byusing Monte-Carlo simulation and FGMOSFET simulationtechnique with SPICE with very high accuracy. If desired,it can also fit the BSIM4 based reference if the optimallyextracted parameters obtained by using optimization algo-rithm have been applied.

By using the proposed model, the insight into the devicelevel variability of FGMOSFET can be provided and theanalytical model of process induced random variation incircuit level parameter of any FGMOSFET based circuitcan be obtained. It has been shown that Δ𝐼

𝐷become more

critical in low voltage/low power operating condition wherethe influences of Δ𝛽, Δ𝜃, and Δ𝑉TH become relatively largecompared to those of other variations. Furthermore, theeffects of Δ𝜃 and Δ𝑉TH can be reduced by lowering either𝜃 or 𝑉TH or both. So, this model has been found to be anefficient tool for the variability aware analysis/designing ofany FGMOSFET based circuit.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The author would like to acknowledge Mahidol University,Thailand, for the online database service.

References

[1] A. S.Medina-Vazquez, J. de la Cruz-Alejo, F. Gomez-Castañeda,and J. A. Moreno-Cadenas, “Low-voltage linear transconductorand a memory current using the MIFGMOS Transistor,” Inter-national Journal of Electronics, vol. 96, no. 9, pp. 895–914, 2009.

[2] R. Pandey and M. Gupta, “FGMOS based voltage-controlledgrounded resistor,” Radioengineering, vol. 19, no. 3, pp. 455–459,2010.

[3] M. Gupta and R. Pandey, “Low-voltage FGMOS based analogbuilding blocks,” Microelectronics Journal, vol. 42, no. 6, pp.903–912, 2011.

[4] J. M. Miguel, A. J. Lopez-Martin, L. Acosta, J. Ramı́rez-Angulo,and R. G. Carvajal, “Using floating gate and quasi-floatinggate techniques for rail-to-rail tunable CMOS transconductordesign,” IEEE Transactions on Circuits and Systems. I. RegularPapers, vol. 58, no. 7, pp. 1604–1614, 2011.

[5] A. J. Lopez-Martin, J. M. Algueta, C. Garcia-Alberdi, L. Acosta,R. G. Carvajal, and J. Ramirez-Angulo, “Design of micropowerclass AB transconductors: a systematic approach,” Microelec-tronics Journal, vol. 44, no. 10, pp. 920–929, 2013.

[6] G.-C. Lizeth, T. Asai, and M. Motomura, “Application ofnonlinear systems for designing low-power logic gates basedon stochastic resonance,”NonlinearTheory and Its Applications,IEICE, vol. 5, no. 4, pp. 445–455, 2014.

[7] E. Rodriguez-Villegas, Low Power and Low Voltage CircuitDesign with the FGMOS Transistor, vol. 20 of IET Circuits,Devices & Systems Series, The Institution of Engineering andTechnology, London, UK, 2006.

[8] V. S. Babu, R. S. Devi, A. Sekhar, and M. R. Baiju, “FGMOSFETcircuit for neuron activation function and its derivative,” inProceedings of the 4th IEEE Conference on Industrial Electronicsand Applications (ICIEA ’09), pp. 739–744, IEEE, Xi’an, China,May 2009.

[9] M. Gupta, R. Srivastava, and U. Singh, “Low voltage floatinggate MOS transistor based differential voltage squarer,” ISRNElectronics, vol. 2014, Article ID 357184, 6 pages, 2014.


[10] M. J. M. Pelgrom, A. C. J. Duinmaijer, and A. P. G. Welbers,“Matching properties ofMOS transistors,” IEEE Journal of Solid-State Circuits, vol. 24, no. 5, pp. 1433–1440, 1989.

[11] S. K. Saha, “Modeling process variability in scaled CMOStechnology,” IEEE Design & Test of Computers, vol. 27, no. 2, pp.8–16, 2010.

[12] K. Kuhn, “Variability in nanoscale CMOS technology,” ScienceChina Information Sciences, vol. 54, no. 5, pp. 936–945, 2011.

[13] P. R. Kinget, “Device mismatch and tradeoffs in the design ofanalog circuits,” IEEE Journal of Solid-State Circuits, vol. 40, no.6, pp. 1212–1224, 2005.

[14] H. Masuda, T. Kida, and S.-I. Ohkawa, “Comprehensive match-ing characterization of analog CMOS circuits,” IEICE Trans-actions on Fundamentals of Electronics, Communications andComputer Sciences, vol. 92, no. 4, pp. 966–975, 2009.

[15] F. Hong, B. Cheng, S. Roy, and D. Cumming, “An analyticalmismatchmodel of nano-CMOS device under impact of intrin-sic device variability,” in Proceedings of the IEEE InternationalSymposium of Circuits and Systems (ISCAS ’11), pp. 2257–2260,May 2011.

[16] S. Vlassis and S. Siskos, “Design of voltage-mode and current-mode computational circuits using floating-gate MOS tran-sistors,” IEEE Transactions on Circuits and Systems I: RegularPapers, vol. 51, no. 2, pp. 329–341, 2004.

[17] A. E. Mourabit, P. Pittet, and G. N. Lu, “A low voltage, highlylinear CMOS OTA,” in Proceedings of the 16th InternationalConference on Microelectronics (ICM ’04), pp. 700–703, tun,December 2004.

[18] Y. Zhai andP.A.Abshire, “Adaptive log domain filters for systemidentification using floating gate transistors,” Analog IntegratedCircuits and Signal Processing, vol. 56, no. 1-2, pp. 23–36, 2008.

[19] V. S. Babu, A. Sekhar, R. S. Devi, andM. R. Baiju, “Floating gateMOSFET based operational transconductance amplifier andstudy of mismatch,” in Proceedings of the 4th IEEE Conferenceon Industrial Electronics and Applications (ICIEA ’09), pp. 127–132, IEEE, Xi’an, China, May 2009.

[20] A. E. Mourabit, G.-N. Lu, and P. Pittet, “Wide-linear-range sub-threshold OTA for low-power, low-voltage, and low-frequencyapplications,” IEEE Transactions on Circuits and Systems I:Regular Papers, vol. 52, no. 8, pp. 1481–1488, 2005.

[21] L.-L. Xiao, K. Liu, and D.-P. Han, “CMOS low data rateimaging method based on compressed sensing,” Optics andLaser Technology, vol. 44, no. 5, pp. 1338–1345, 2012.

[22] S. K. Saha, “Non-linear coupling voltage of split-gate flashmemory cells with additional top coupling gate,” IET Circuits,Devices & Systems, vol. 6, no. 3, pp. 204–210, 2012.

[23] J. Ramirez-Angulo, G. Gonzlrlez-Altamirano, and S. C. Choi,“Modeling multiple-input floating-gate transistors for analogsignal processing,” in Proceedings of the IEEE InternationalSymposiumonCircuits and Systems (ISCAS ’97), vol. 3, pp. 2020–2023, IEEE, Hong Kong, June 1997.

[24] P. E. Allen and D. R. Holdberg, CMOS Analog Circuit Design,The Oxford Series in Electrical and Computer Engineering,Oxford University Press, Oxford, UK, 2011.

[25] W. Lü, G. Wang, and L. Sun, “Analytical modeling of cutoff fre-quency variability reserving correlations due to random dopantfluctuation in nanometer MOSFETs,” Solid-State Electronics,vol. 105, pp. 63–69, 2015.

[26] K. Khu, “Statistical modeling for Monte Carlo simulation usingHspice,” in Proceedings of the Synopsys Users Group Conference,pp. 1–10, 2006.

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