Open access peer-reviewed chapter

# Comprehensive Analytical Models of Random Variations in Subthreshold MOSFET’s High-Frequency Performances

By Rawid Banchuin

Submitted: October 6th 2017Reviewed: November 24th 2017Published: December 25th 2017

DOI: 10.5772/intechopen.72710

## Abstract

Subthreshold MOSFET has been adopted in many low power VHF circuits/systems in which their performances are mainly determined by three major high-frequency characteristics of intrinsic subthreshold MOSFET, i.e., gate capacitance, transition frequency, and maximum frequency of oscillation. Unfortunately, the physical level imperfections and variations in manufacturing process of MOSFET cause random variations in MOSFET’s electrical characteristics including the aforesaid high-frequency ones which in turn cause the undesired variations in those subthreshold MOSFET-based VHF circuits/systems. As a result, the statistical/variability aware analysis and designing strategies must be adopted for handling these variations where the comprehensive analytical models of variations in those major high-frequency characteristics of subthreshold MOSFET have been found to be beneficial. Therefore, these comprehensive analytical models have been reviewed in this chapter where interesting related issues have also been discussed. Moreover, an improved model of variation in maximum frequency of oscillation has also been proposed.

### Keywords

• gate capacitance
• maximum frequency of oscillation
• subthreshold MOSFET
• transition frequency
• VHF circuits/systems

## 1. Introduction

Subthreshold MOSFET has been extensively used in many VHF circuits/systems, e.g., wireless microsystems [1], low power receiver [2], low power LNA [3, 4] and RF front-end [5], where performances of these VHF circuits/systems are mainly determined by three major high-frequency characteristics of intrinsic subthreshold MOSFET, i.e., gate capacitance, Cg, transition frequency, fT, and maximum frequency of oscillation, fmax. Clearly, the physical level imperfections and manufacturing process variations of MOSFET, e.g., gate length random fluctuation, line edge roughness, random dopant fluctuation, etc., cause the variations in MOSFET’s electrical characteristics, e.g., drain current, ID and transconductance, gm, etc. These variations are crucial in the statistical/variability aware analysis and design of MOSFET-based circuits/systems. So, there exist many previous studies on such variations which some of them have also focused on the subthreshold MOSFET [1, 6, 7, 8, 9, 10, 11, 12]. Unfortunately, Cg, fT, and fmax have not been considered even though they also exist and greatly affect the high-frequency performances of such MOSFET-based circuits/systems. Therefore, analytical models of variations in those major high-frequency characteristics have been performed [13, 14, 15, 16, 17]. In [13], an analytical model of variation in fT derived as a function of the variation in Cg has been proposed where only strong inversion MOSFET has been focused. However, this model is not comprehensive, as none of any related physical levels variable of the MOSFET has been involved. In [14], the models of variations in Cg and fT, which are comprehensive as they are in terms of the related MOSFET’s physical level variables, have been proposed. Again, only the strong inversion MOSFET has been considered in [14].

According to the aforementioned importance and usage of subthreshold MOSFET in the MOSFET-based VHF circuits/systems, the comprehensive analytical models of variations in Cg, fT, and fmax of subthreshold MOSFET have been proposed [15, 16, 17]. Such models have been found to be very accurate as they yield smaller than 10% the average percentages of errors. In this chapter, the revision of these models will be made where some foundations on the subthreshold MOSFET will be briefly given in the subsequent section followed by the revision on models of Cg in Section 3. The models of fT and fmax will, respectively, be reviewed in Sections 4 and 5 where an improved model of variation in fmax will also be introduced. Some interesting issues related to these models will be mentioned in Section 6 and the conclusion will be finally drawn in Section 7.

## 2. Foundations on subthreshold MOSFET

Unlike the strong inversion MOSFET in which Id is a polynomial function of the gate to source voltage, Vgs, Id of the subthreshold MOSFET is an exponential function of Vgs and can be given as follows:

Id=μCdepWLkTq2expVgsVtnkT/q1expVdskT/qE1

where Cdep and n denote the capacitance of the depletion region under the gate area and the subthreshold parameter, respectively.

By using Eq. (1) and keeping in mind that gm=dId/dVgs, gm of subthreshold MOSFET can be given by

gm=μnCdepWLkTq2expVgsVtnkT/q1expVdskT/qE2

## 3. Variation in gate capacitance (Cg)

Before reviewing the models of variation in Cg of subthreshold MOSFET, it is worthy to introduce the mathematical expression of Cg as it is the mathematical basis of such models. Here, Cg which can be defined as the total capacitance seen by looking in to the gate terminal of the MOSFET as shown in Figure 1, can be given in terms of the gate charge, Qg as [15]

Cg=dQgdVgsE3

where

Qg=μW2LCox2Id0VgsVtVgsVcVt2dVcQB,maxE4

It is noted that QB,max stands for the maximum bulk charge [15]. By using Eq. (1), Qg of the subthreshold MOSFET can be found as

Qg=WL2Cox2CdepkT/q2VgsVt331expVdskT/qexpqnkTVgsVtQB,maxE5

As a result, the expression of Cg can be obtained by using Eqs. (1) and (5) as follows

Cg=13WL2Cox2CdepkT/q23VgsVt2qnkTVgsVt3expqnkTVgsVtE6

By taking the physical level imperfections and manufacturing process variations of MOSFET into account, random variations in MOSFET’s parameters such as Vt, W, L, etc., denoted by ΔVt, ΔW, ΔL, and so on existed. These variations yield the randomly varied Cg i.e. CgVt, ΔW, ΔL,…) [15]. Thus, the variations in Cg, ∆Cg can be mathematically defined as [15]

ΔCg=ΔCgΔVt,ΔW,ΔL,CgE7

where Cg stands for the nominal gate capacitance in this context.

With this mathematical definition and the fact that ΔVt is the most influential in subthreshold MOSFET [18], the following comprehensive analytical expression of ∆Cg has been proposed in [15]

ΔCg=2WCdepLCoxkT/q2expVdskT/q11VgsVFBϕsNeffWdepVtVFBϕsNeffWdepE8

where Neff, VFB, Wdep, and ϕs denote the effective values of the substrate doping concentration Nsub(x), the flat band voltage, depletion width, and surface potential, respectively. Moreover, Neff can be obtained by weight averaging of Nsub(x) as [15]

Neff=30WdepNsubx1xWdep2dxWdepE9

As ∆Cg is a random variable, it is necessary to derive its statistical parameters for completing the comprehensive analytical modeling. Among various statistical parameters, the variance has been chosen as it determines the spread of the variation in a convenient manner. Based on the traditional analytical model of statistical variation in MOSFET’s parameter [19], the variances of ∆Cg, Var[∆Cg] can be analytically obtained as follows [15]

VarΔCg=8q4NeffWdepWLε02k2T2Cdep2expVdskT/q12VgsVFBϕsNeffWdep2E10

where ε0stands for the permittivity of free space. At this point, it can be seen that the comprehensive analytical model of ∆Cg proposed in [15] is composed of Eqs. (8) and (10) where the latter has been derived based on the former. In [15], (Var[∆Cg])0.5 calculated by using the proposed model has been compared to its 65 nm CMOS technology-based benchmarks obtained by using the Monte Carlo simulation for verification where strong agreements between the model-based (Var[∆Cg])0.5 and the benchmark have been found. The average deviation from the benchmark obtained from the entire range of Vgs used for simulation given by 0–100 mV has been found to be 9.42565 and 8.91039% for N-type and P-type MOSFET-based comparisons, respectively [15].

Later, an improved model of ∆Cg has been proposed in [16] where the physical level differences between N-type and P-type MOSFETs, e.g., carrier type, etc., has also been taken into account. Such model is composed of the following equations

ΔCgN=2WCdepLCoxkT/q2expVdskT/q11VgsVFB2ϕFCox12SiNa2ϕF+Vsb×VtVFB2ϕFCox12SiNa2ϕF+VsbE11
ΔCgP=2[WCdepLCoxkT/q]2[exp[VdskT/q]1]1[VgsVFB+|2ϕF|+Cox12qεSiNd(|2ϕF|Vsb)]×[VtVFB+|2ϕF|+Cox12qεSiNd(|2ϕF|Vsb))]E12
VarΔCgN=12q6NeffWdepWL3Cdep2CoxkT41expVdskT/q2VgsVFB2ϕFCox12SiNa2ϕF+Vsb2Vt1VFB+2ϕF+Cox12SiNa2ϕF+VsbE13
VarΔCgP=12q6NeffWdepWL3Cdep2CoxkT41expVdskT/q2VgsVFB+2ϕF+Cox12SiNd2ϕFVsb2Vt1VFB2ϕFCox12SiNd2ϕFVsbE14

where ∆CgN and ∆CgP are ∆Cg of N-type and P-type MOSFETs, respectively. Moreover, Na, Nd, Vsb, and ϕFdenote acceptor doping density, donor doping density, source to body voltage, and Fermi potential, respectively [16]. Also, it is noted that Eqs. (13) and (14) have been, respectively, derived by using Eqs. (11) and (12) based on the up-to-date analytical model of statistical variation in MOSFET’s parameter [20] instead of the traditional one.

In [16], a verification similar to that of [15] has been made, i.e., (Var[∆CgN])0.5 and (Var[∆CgP])0.5 have been, respectively, compared with their 65 nm CMOS technology-based benchmarks. Both (Var[∆CgN])0.5 and (Var[∆CgP])0.5 have been calculated by using the proposed model, and the benchmarks have been obtained from the Monte Carlo simulation. The comparison results have been redrawn here in Figures 2 and 3 where strong agreements with their benchmarks of the model-based (Var[∆CgN])0.5 and (Var[∆CgP])0.5 can be seen for the whole range of Vgs. The average deviations determined from such range have been found to be 8.45033 and 6.53211%, respectively [16], which are lower than those of the previous model proposed in [15]. Therefore, the model proposed in [16] has also been found to be more accurate than its predecessor [15] apart from being more detailed as the physical level differences between N-type and P-type MOSFETs have also been taken into account.

## 4. Variation in transition frequency (fT)

Apart from that of ∆Cg, the comprehensive analytical model of variation in fT of subthreshold MOSFET, ∆fT has also been proposed in [16]. Before reviewing such model, it is worthy to show the definition of fT and its comprehensive analytical expression derived in [16]. According to [21], fT can be defined as the frequency at which the small-signal current gain of the device drops to unity, while the source and drain terminals are held at ground and can be related to Cg by the following equation [13]

fT=gm2πCgE15

By using Eqs. (2) and (6), the following comprehensive analytical expression of fT can be obtained [16]

fT=32μCdep2kT/q32nπL3Cox21expVdskT/q2exp2qnkTVgsVt3VgsVt2qnkTVgsVt3E16

Similar to ∆Cg, ∆fT can be mathematically defined as [16]

ΔfT=ΔfTΔVtΔWΔLfTE17

where fT stands for the nominal transition frequency in this context.

By also keeping in mind that ΔVt is the most influential, the following comprehensive analytical expression of ∆fT has been proposed in [16] where the aforesaid physical level differences between N-type and P-type MOSFETs have also been taken into account.

ΔfTN=μCdep2kT/q31expVdskT/q2VFB+2ϕF+Cox12SiNa2ϕF+VsbVtπnL3Cox2VgsVFB2ϕFCox12SiNa2ϕF+Vsb3E18
ΔfTP=μCdep2kT/q31expVdskT/q2VFB2ϕFCox12SiNd2ϕFVsbVt1πnL3Cox2VgsVFB+2ϕF+Cox12SiNd2ϕFVsb3E19

It is noted that ∆fTN and ∆fTP are ∆fT of N-type and P-type MOSFETs, respectively. By also using the up-to-date analytical model of statistical variation in MOSFET’s parameter, we have [16]

VarΔfTN=μ2Cdep4kT6q4NeffWdep1expVdskT/q4Vt1VFB+2ϕF+Cox12SiNa2ϕF+Vsb3π2n2WL7Cox6VgsVFB2ϕFCox12SiNa2ϕF+Vsb6E20
VarΔfTP=μ2Cdep4kT6q4NeffWdep1expVdskT/q4Vt1VFB2ϕFCox12SiNd2ϕFVsb3π2n2WL7Cox6VgsVFB+2ϕF+Cox12SiNd2ϕFVsb6E21

At this point, it can be stated that the comprehensive analytical model of ∆fT proposed in [16] is composed of Eqs. (18), (19), (20), and (21). For verification, (Var[∆fTN])0.5 and (Var[∆fTP])0.5 calculated by using the proposed model have also been compared with their corresponding 65 nm CMOS technology-based benchmarks obtained from the Monte Carlo simulation. The results have been redrawn here in Figures 4 and 5 where strong agreements to the benchmarks of the model-based (Var[∆fTN])0.5 and (Var[∆fTP])0.5 can be observed. The average deviations have been found to be 8.22947 and 6.25104%, respectively [16]. Moreover, it has been proposed in [16] that there exists a very strong statistical relationship between ΔCg and ΔfT of any certain subthreshold MOSFET as it has been found by using the proposed model that the magnitude of the statistical correlation coefficient of ΔCg and ΔfT is unity for both N-type and P-type devices.

## 5. Variation in maximum frequency of oscillation (fmax)

Before reviewing the model of variation in fmax of subthreshold MOSFET, it is worthy to introduce its definition and mathematical expression. The fmax, which takes the effect of the resistance of gate metallization into account, can be defined as the frequency at which the power gain of MOSFET becomes unity. Such gate metallization belonged to the extrinsic part of MOSFET. According to [17], fmax can be given under an assumption that Cg is equally divided between drain and source by

fmax=14πCg2gmRgE22

where Rg stands for the resistance of gate metallization [17].

By substituting gm and Cg as respectively given by Eqs. (2) and (6) into Eq. (22), we have

fmax=2μnexpVgsVnkT/q11expVdskT/q4π3WCdepRgL2.5Cox2kT/q3VgsVt2VgsVt3nkT/qE23

Similar to the other variations, ∆fmax can be mathematically defined as [17]

Δfmax=ΔfmaxΔVtΔWΔLfmaxE24

where fmax stands for the nominal maximum frequency of oscillation in this context.

In [17], the comprehensive analytical model of ∆fmax have been proposed. Such model is composed of the following equations.

Δfmax=12πμnRg121expVdskT/q12kTqexpVgsVt2nkT/q[CdepWL12+1expVdskT/q1WLCdep32CoxkT/q2×VgsVFBϕsNeffWdep×VtVFBϕsNeffWdep]E25
VarΔfmax=μq4NeffWdepW2π2nCdepRgε02k2T2expVdskT/q11expVgsVtnkT/qkTq2VgsVFBϕsNeffWdep2E26

It is noted that Eq. (25) has been derived by also keeping in mind that ΔVt is the most dominant. Moreover, Eq. (26) has been formulated based on Eq. (25) and the traditional model of statistical variation in MOSFET’s parameter. The model-based (Var[∆fmax])0.5 has been compared with its 65 nm CMOS technology-based benchmarks obtained by the Monte Carlo simulation for verification. The strong agreements between the model-based (Var[∆fmax])0.5 and the benchmark can be observed from the whole simulated range of Vgs given by 0–100 mV. The average deviation has been found to be 9.17682 and 8.51743% for N-type and P-type subthreshold MOSFETs, respectively, [17].

Unfortunately, the model proposed in [17] did not take the physical level differences between N-type and P-type MOSFETs into account. By taking such physical level differences into consideration, we have

ΔfmaxN=12π(μnRg)12[1exp[VdskT/q]]12(kTq)exp[VgsVt2nkT/q][(CdepWL)12+[1exp[VdskT/q]]1(WLCdep)32(CoxkT/q)2×[VgsVFB2ϕFCox12qεSiNa(2ϕF+Vsb)]×[VtVFB2ϕFCox12qεSiNa(2ϕF+Vsb)]]E27
ΔfmaxP=12π(μnRg)12[1exp[VdskT/q]]12(kTq)exp[VgsVt2nkT/q][(CdepWL)12+[1exp[VdskT/q]]1(WLCdep)32(CoxkT/q)2×[VgsVFB+|2ϕF|+Cox12qεSiNd(|2ϕF|Vsb)]×[VtVFB+|2ϕF|+Cox12qεSiNd(|2ϕF|Vsb)]]E28

where ∆fmaxN and ∆fmaxP are ∆fmax of N-type and P-type MOSFETs, respectively. By using the up-to-date analytical model of statistical variation in MOSFET’s parameter, we have

VarΔfmaxN=3q2NeffWdepW3L1μ/nRgkT/q22π2Vt1VFB+2ϕF+Cox12SiNa2ϕF+Vsb1expVdskT/qexpVgsVt2nkT/q2×CdepWL12+1expVdskT/q1WLCdep32CoxkT/q2×VgsVFB2ϕFCox12SiNa2ϕF+VsbE29
VarΔfmaxP=3q2NeffWdepW3L1μ/nRgkT/q22π2Vt1VFB+2ϕF+Cox12SiNa2ϕF+Vsb1expVdskT/qexpVgsVt2nkT/q2×CdepWL12+1expVdskT/q1WLCdep32CoxkT/q2×VgsVFB+2ϕF+Cox12SiNd2ϕFVsbE30

At this point, it can be seen that the improved model of ∆fmax is composed of Eqs. (27), (28), (29), and (30). For verification, the model-based (Var[∆fmaxN])0.5 and (Var[∆fmaxP])0.5 have been compared with their corresponding 65 nm CMOS technology-based benchmarks obtained by using the Monte Carlo simulation. The results are as shown in Figures 6 and 7 where strong agreements to the benchmarks of the model-based (Var[∆fmaxN])0.5 and (Var[∆fmaxP])0.5 can be observed. The average deviations from the benchmarks have been found to be 6.11788 and 5.85574% for (Var[∆fmaxN])0.5 and (Var[∆fmaxP])0.5, respectively, which are lower than those of the model proposed in [17]. Therefore, our improved model ∆fmax is also more accurate than the previous one apart from being more detailed as the physical level differences between N-type and P-type MOSFETs have also been taken into account.

Before proceeding further, it should be mentioned here that Cg has more severe variations compared to the other high-frequency characteristics and the P-type subthreshold MOSFET is more robust than the N-type as can be seen from Figures 27. Moreover, it can be implied that there exists a strong correlation between Δfmax and ΔfT as fmax is related to fT by Eq. (31). An implication of strong correlation between Δfmax and ΔCg can be similarly obtained by observing Eq. (22) that is given as

fmax=fT2gmRgE31

## 6. Some interesting issues

### 6.1. Statistical/variability aware design trade-offs

For the optimum statistical/variability aware design of any MOSFET-based VHF circuit, ∆Cg, ∆fT, and ∆fmax must be minimized. It has been found from Eqs. (13), (14), (20), (21), (29), and (30) that VarΔCgL3, VarΔfTL7and VarΔfmaxL1for both types of MOSFET. Therefore, it can be seen that shrinking L can reduce ΔCg of the subthreshold MOSFET of any type with the increasing ΔfT and Δfmax as penalties. Moreover, we have also found that VarΔCgT2, VarΔfTT6, and VarΔfmaxT2. This means that we can reduce ΔfT and Δfmax by lowering T with higher ΔCg as a cost. These design trade-offs must be taken into account in the statistical/variability aware design of any subthreshold MOSFET-based VHF circuits/systems.

### 6.2. Variation in any high-frequency parameter

Occasionally, determining the variation in other high-frequency parameters apart from Cg, fT, and fmax e.g., bandwidth, fBW, etc., has been found to be necessary. The determination of variation in fBW as a function of ΔfT has been shown in [16]. In general, let any high-frequency parameter of the subthreshold MOSFET be P, the amount of its variation, ΔP, can be determined given the amounts of ΔCg, ΔfT, and Δfmax if P depends on Cg, fT, and fmax. It is noted that the amounts of ΔCg, ΔfT, and Δfmax can be predetermined by using the reviewed comprehensive analytical models. Mathematically, ΔP can be expressed in terms of ΔCg, ΔfT, and Δfmax as follows

ΔP=PCgΔCg+PfTΔfT+PfmaxΔfmaxE32

Therefore, the variance of ΔP, VarP] can be given by keeping the aforementioned strong statistical relationships among ΔCg, ΔfT, and Δfmax in mind as follows

VarΔP=PCg2VarΔCg+PfT2VarΔfT+Pfmax2VarΔfmax+2PCgPfTVarΔCgVarΔfT+2PCgPfmaxVarΔCgVarΔfmax+2PfT×PfmaxVarΔfTVarΔfmaxE33

Noted also that the VarCg], VarfT], and Varfmax] can be known by applying those reviewed models.

### 6.3. High-frequency parameter mismatches

The amount of mismatches in Cg, fT, and fmax of multiple subthreshold MOSFETs can be determined by applying those reviewed comprehensive analytical models of ΔCg, ΔfT, and Δfmax even though they are dedicated to a single device. As an illustration, the mismatches in Cg, fT, and fmax of two deterministically identical subthreshold MOSFETs, i.e., M1 and M2, will be determined. Traditionally, the magnitude of mismatch can be measured by using its variance [22]. Let the mismatches in Cg, fT, and fmax of M1 and M2 be denoted by ΔCg12, ΔfT12, and Δfmax12, respectively, their variances, i.e., VarCg12], VarfT12], and Varfmax12], can be respectively related to VarCg], VarfT], and Varfmax] of M1 and M2, which can be determined by using those reviewed models, via the following equations

VarΔCg12=VarΔCg1+VarΔCg22ρΔCg1ΔCg2VarΔCg1VarΔCg2E34
VarΔfT12=VarΔfT1+VarΔfT22ρΔCg1ΔCg2VarΔfT1VarΔfT2E35
VarΔfmax12=VarΔfmax1+VarΔfmax22ρΔCg1ΔCg2VarΔfmax1VarΔfmax2E36

It is noted that ΔCgi, ΔfTi, Δfmaxi, VarCgi], VarfTi], and Varfmaxi], respectively, denote ΔCg, ΔfT, Δfmax, VarCg], VarfT], and Varfmax] of Mi where {i} = {1, 2}. Moreover, ρXYstands for the correlation coefficient of X and Y where {X} = {ΔCg1, ΔfT1, Δfmax1} and {Y} = {ΔCg2, ΔfT2, Δfmax2}. For closely spaced MOSFETs with positive correlation, ρXYcan be given by 1 as the statistical correlation between closely spaced devices is very strong [22]. As a result, the mismatches are maximized. If the negative correlation is assumed on the other hand, ρXYbecome −1 and the mismatches are minimized [16]. For distanced devices, we have, ρXY=0as the correlation is very weak and can be neglected.

If we assume that both M1 and M2 are statistically identical, we have VarCg1] = VarCg2] = VarCg], VarfT1] = VarfT2] = VarfT], and Varfmax1] = Varfmax2] = Varfmax]. Thus, Eqs. (34), (35), and (36) become

VarΔCg12=2VarΔCg1ρΔCg1ΔCg2E37
VarΔfT12=2VarΔfT11ρΔfT1ΔfT2E38
VarΔfmax12=2VarΔfmax11ρΔfmax1Δfmax2E39

From these equations, it can be seen that VarCg12], VarfT12], and Varfmax12] can all be approximately given by 0 if those statistically identical devices are closely spaced and positively correlated as all ρXY’s are given by 1. This implies that the high-frequency parameter mismatches of statistically identical, closely spaced, and positively correlated subthreshold MOSFETs can be neglected.

### 6.4. Variation in any VHF circuit/system

By using the reviewed models, the variation in the crucial parameter of any subthreshold MOSFET-based VHF circuit/system can be analytically formulated. As a case study, the subthreshold MOSFET-based Wu current-reuse active inductor proposed in [1] will be considered. This active inductor can be depicted as shown in Figure 8. According to [1], the inductance, l, of this active inductor can be given by

l=Cg1gm1gm2E40

where Cg1, gm1, and gm2 are gate capacitance of M1, transconductance of M1, and transconductance of M2, respectively.

By using Eq. (40), the variation in l, Δl due to the variation in Cg1, ΔCg1 can be immediately given by [16]

Δl=ΔCg1gm1gm2E41

Therefore, we have the following relationship between the variances of Δl and ΔCg1

VarΔl=VarΔCg1gm1gm2E42

It is noted that VarCg1] can be determined by using those reviewed models. It can also be seen that VarΔlVarΔCg1and VarΔl1/gm1gm2[16]. Therefore, it is far more convenient to minimize Δl by reducing gm1 and gm2 as they are electronically controllable unlike ΔCg1, which must be minimized at the physical level by lowering L as stated above.

### 6.5. Reduced computational effort simulation

If we let the key parameter of any subthreshold MOSFET-based VHF circuit/system with M MOSFETs under consideration be Z, its variance, Var[Z], which is the desired statistical/variability aware simulation result, can be given by.

Var[Z]=i=1M[(SCgiZ)2σΔCgi2+(SfTiZ)2σΔfTi2+(SfmaxiZ)2σΔfmaxi2]+ijMj=1M[(SCgiZ)(SCgjZ)ρΔCgiΔCgjσΔCgi2σΔCgj2+(SfTiZ)(SfTjZ)ρΔfTiΔfTjσΔfTi2σΔfTj2+(SfmaxiZ)(SfmaxjZ)ρΔfmaxiΔfmaxjσΔfmaxi2σΔfmaxj2]+2i=1Mj=1M[(SCgiZ)(SfTiZ)ρΔCgiΔfTjσΔCgi2σΔfTj2+(SCgiZ)(SfmaxjZ)ρΔCgiΔfmaxjσΔCgi2σΔfmaxj2+(SfTiZ)(SfmaxjZ)ρΔfTiΔfmaxjσΔfTi2σΔfmaxj2]E43

It is noted that the magnitude of ρXY, where {X} = {ΔCgi, ΔfTi, Δfmaxi}, {Y} = {ΔCgj, ΔfTj, Δfmaxj}, and the subscripts i and j refers to the arbitrary ith and jth MOSFET, respectively, in this scenario, approaches 1 when i = j as it determines the correlation of the same device. Moreover, SCgiZSCgjZ, SfTiZSfTjZ, and SfmaxiZSfmaxjZdenote the sensitivity of Z to Cg, fT, and fmax of ith (jth) MOSFET, respectively. By using Eq. (43) and the reviewed comprehensive analytical models for predetermining all Var[X]‘s and Var[Y]‘s, Var[Z] can be numerically determined in a reduced computational effort manner as those sensitivities can be obtained by using the sensitivity analysis [23], which required much less computational effort compared to the conventional Monte Carlo simulation. This is because the circuit/system of interest is needed to be solved only once for obtaining the sensitivities and then Var[Z] can be immediately determined unlike the Monte Carlo simulation that requires numerous runs in order to reach the similar outcome [16]. Therefore, much of the computational effort can be significantly reduced.

## 7. Conclusion

In this chapter, the comprehensive analytical models of ΔCg, ΔfT, and Δfmax of subthreshold MOSFET, which serves as the basis of many VHF circuits/systems, have been reviewed. Interesting issues related to these models i.e., statistical/variability aware design trade-offs of subthreshold MOSFET-based VHF circuit/system; determination of variation in any high-frequency parameter and mismatch in Cg, fT, and fmax; determination of variation in any subthreshold MOSFET-based VHF circuit/system; and the computationally efficient statistical/variability aware simulation with sensitivity analysis have been discussed. Moreover, a modified version of the comprehensive analytical model of Δfmax has also been proposed. This revised model has been found to be more accurate and detailed than the previous one.

## How to cite and reference

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Rawid Banchuin (December 25th 2017). Comprehensive Analytical Models of Random Variations in Subthreshold MOSFET’s High-Frequency Performances, Complementary Metal Oxide Semiconductor, Kim Ho Yeap and Humaira Nisar, IntechOpen, DOI: 10.5772/intechopen.72710. Available from:

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We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

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