# Novel Complete Probabilistic Models of Random Variation in High Frequency Performance of Nanoscale MOSFET

Authors: Rawid Banchuin
Journal: Journal of Electrical and Computer Engineering
 (2013)

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## Abstract

The novel probabilistic models of the random variations in nanoscale MOSFET's high frequency performance defined in terms of gate capacitance and transition frequency have been proposed. As the transition frequency variation has also been considered, the proposed models are considered as complete unlike the previous one which take only the gate capacitance variation into account. The proposed models have been found to be both analytic and physical level oriented as they are the precise mathematical expressions in terms of physical parameters. Since the up-to-date model of variation in MOSFET's characteristic induced by physical level fluctuation has been used, part of the proposed models for gate capacitance is more accurate and physical level oriented than its predecessor. The proposed models have been verified based on the 65 nm CMOS technology by using the Monte-Carlo SPICE simulations of benchmark circuits and Kolmogorov-Smirnov tests as highly accurate since they fit the Monte-Carlo-based analysis results with 99% confidence. Hence, these novel models have been found to be versatile for the statistical/variability aware analysis/design of nanoscale MOSFET-based analog/mixed signal circuits and systems.


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## Body

## 1. Introduction

Nanoscale MOSFET has been adopted in many recently proposed analog/mixed signal circuits and systems such as high speed amplifier [1–3], millimeter wave components [4–6], and analog-to-digital converter [7–9]. Of course, the high frequency performances of these circuits and systems can be determined by those of their intrinsic MOSFET which can be defined by two major MOSFET’s parameters, entitled gate capacitance, Cg, and transition frequency, fT.Obviously, imperfection in MOSFET’s properties at the physical level, for example, random dopant fluctuation, line edge roughness, and so forth, causes the fluctuations in many of its characteristics such as threshold voltage,Vt, channel width, W, and channel length, L, which in turn yields the random variations in its circuit level parameters, for example, drain current, Id, transconductance, gm, and so forth. These variations are crucial in the statistical/variability aware design of MOSFET-based analog/mixed signal circuits and systems particularly at the nanometer regimes since their magnitudes become relatively large percentages. Hence, there are many previous researches on analytical modeling of such variations, for example, [10–13] and so forth, on which the nanoscale CMOS technology has been focused. However, they did not mention anything about variations in Cg and fT even though they also exist and greatly affect the high frequency performances of the MOSFET-based circuits and systems. For these variations, results of studies as graphical plots and numerical measurements have been reported in previous articles, for example, [14–16] and so forth. In [17], the relationship between the variance of Cg and that of fT has been derived. Unfortunately, formulation of each of these variations has not been mentioned. Later, the physical level oriented analytical model of random variation in Cg has been originally proposed in [18]. However, this model has been derived based on the classical model of physical level fluctuation-induced MOSFET’s characteristic variation [19] which is being replaced by its more up-to-date successors such as that in [20]. Furthermore, the model in [18] is incomplete since the modeling on fT which is as important as Cg has not been performed. To the knowledge of the author, such modeling on fT has never been performed.Hence, the novel probabilistic models of the random variations in nanoscale MOSFET’s high frequency performances have been proposed in this paper. Beside the variation inCg, that in fT has also been focused on. So, these models are complete. Both random dopant fluctuation and process variation effects such as line edge roughness, which are the major causes of the random variations in the MOSFET’s high frequency characteristics [16], have been taken into account. The nanoscale MOSFET equation [21] has been used as the mathematical basis similarly to [11, 12, 18]. As the precise mathematical expressions in terms of physical parameters, these models have been found to be both analytic and physical level oriented. Unlike [18], the state of the art physical level fluctuation-induced characteristic variation model of MOSFET [20] has been adopted instead of the classical one [19]. As a result, part of the models for variation in Cg is more accurate and physical level oriented than its predecessor [18]. These novel models have been verified based on the 65 nm CMOS process technology by using the Monte-Carlo SPICE simulations of the benchmark circuits and the Kolmogorov-Smirnov goodness of fit tests as very accurate since they fit the results obtained from Monte-Carlo SPICE simulations with 99% confidence. So, they have been found to be the versatile for the statistical/variability aware analysis/design of nanoscale MOSFET-based analog/mixed signal circuits and systems.


## 2. The Proposed Models

In this section, the proposed models will be presented. Before proceeding further, it is worthy to give some foundation on the nanoscale MOSFET’s equation. It can be seen from [21] that Id of the nanoscale MOSFET can be given with the gate oxide capacitance expressed in term of gate oxide permittivity εox and thickness tox as
(1)Id=Wεoxtox(Vgs-Vt)vsat,
where vsat and Vgs denote the saturation velocity [20] and gate to source voltage. So, gm of the nanoscale MOSFET is given by using (1) as follows:
(2)gm=Wεoxtoxvsat.At this point, the derivation of the proposed models will be given by starting from finding the analytical expression ofCg and fT. Similar to [18], Cg can be defined as [22]
(3)Cg≜dQgdVgs,whereQg denotes the gate charge [22] which can be determined as a function of Id by using the approach adopted from [23] as
(4)Qg=-W2LεoxαdIdVttox∫0Vgs-Vt(Vgs-Vc-Vt)2dVc-QB,max,
where αd and QB,max denote the coulomb scattering coefficient and the maximum bulk charge, respectively.So,Qg of the nanoscale MOSFET can be found by using (1) and (4) as
(5)Qg=-WL(Vgs-Vt)23αdvsatVt-QB,max.As a result,Cg can be given by using (3) and (5) as follows:
(6)Cg=23WLαdvsatVt-VgsVt.At this point, the analytical expression offT can be immediately given by using (2) and (6) as follows:
(7)fT=34αdεoxvsat2toxLVtVt-Vgs.By taking random dopant fluctuation and process variation effects into account, fluctuations inVt, W, L, and so forth, denoted by ΔVt, ΔW, ΔL, and so on, existed which yield random variations in Cg and fT denoted by ΔCg and ΔfT, respectively. This can be stated alternatively that ΔCg and ΔfT are functions of ΔVt, ΔW, ΔL, and so forth. With this in mind, ΔCg and ΔfT can be mathematically defined as
(8)ΔCg≜Cg(ΔVt,ΔW,ΔL,…)-Cg,ΔfT≜fT(ΔVt,ΔW,ΔL,…)-fT,whereCg(ΔVt,ΔW,ΔL,…) and fT(ΔVt,ΔW,ΔL,…) denoted the fluctuated Cg and fT as functions of ΔVt, ΔW, ΔL, and so on which are distributed in the Gaussian fashion. By using (6), (7), and (8), ΔCg and ΔfT can be given in terms of physical level variables as follows:
(9)ΔCg=23(W+ΔW)(L+ΔL)αdvsat(Vt+ΔVt)-VgsVt+ΔVt-23WLαdvsatVt-VgsVt≈2WLVgs(VFB+2φB-Vtf)3αdvsat(VFB+2φB)2,ΔfT=34αdεoxvsat2(tox+Δtox)(L+ΔL)Vt+ΔVt(Vt+ΔVt)-Vgs-34αdεoxvsat2toxLVtVt-Vgs≈3αdεoxvsat2Vgs(VFB+2φB-Vtf)4πtoxL(Vgs-VFB-2φB)2,whereVFB, Vtf, and φB denote flat band voltage, fluctuated threshold voltage, and half of surface band bending at inversion, respectively. Like other variations, Δtox denotes random variation in tox. The summation of VFB and φB approximates the threshold voltage which its fluctuation is the dominant one. It can be seen that the deterministic behaviors of ΔCg and ΔfT can be analytically modeled by using (9).For the probabilistic modeling of our interested the probability density functions ofΔCg and ΔfT, denoted by pdfΔCg(δCg) and pdfΔfT(δfT), where δCg and δfT stand for any sampled value of ΔCg and that of ΔfT, respectively, must be derived. In order to do so, the up-to-date model of physical level fluctuations-induced variation in MOSFET’s characteristic [20] has been adopted. At this point, pdfΔCg(δCg) and pdfΔfT(δfT) can be, respectively, derived by using (9) and the basis model as in (10) and (11), where Neff and Wd denote effective doping concentration and depletion region width, respectively [20].Consider,(10)pdfΔCg(δCg)=4.338παdεoxvsat(VFB+2φB)2qtoxNeffWdWLVgs×exp[-2.344αd2εox2vsat2(VFB+2φB)4δCg2qtoxNeffWdWLVgs2],(11)pdfΔfT(δfT)=π(Vgs-VFB-2φB)2qαdvsat2VgstoxWL30.54NeffWd×exp[-1.852π2(Vgs-VFB-2φB)4toxWL3δfT2q2αd2vsat4Vgs2NeffWd].Finally, the novel complete probabilistic models of the random variations in nanoscale MOSFET’s high frequency performances have been now derived. By using these models and conventional statistical mathematics, the probabilistic/statistical behaviors ofΔCg, ΔfT and even variations in other nanoscale MOSFET’s high frequency parameters can be analytically explained in terms of related physical level variables. These models can be the mathematical tool for the statistical/variability aware optimization of MOSFET’s high frequency performance and computer aid sensitivity analysis-based variability aware simulations of any circuit/system’s high frequency parameters. The required computational effort has been found to be lower than that of the traditional Monte-Carlo analysis based on variation of MOSFET’s characteristic which is time expensive [24]. Even though these models are oriented to variations in a single device, they can also be the modeling basis of mismatches in high frequency performances between transistors. These points will be discussed later.Compared with [17] which proposed the relationship between the variance of Cg and that of fT as [17]
(12)(fT,mean)-2σfT2=(gm,mean)-2σgm2+(Cg,mean)-2σCg2.The models proposed in this paper, that is,pdfΔCg(δCg) and pdfΔfT(δfT) shown in (10) and (11), respectively, are more precise as they are the probability density functions in terms of physical level parameters such as W, L, and VFB.Furthermore, compared with the previous model in [18] composed of only one expression which models ΔCg as [18]
(13)pdfΔCg(δCg)=3αdvsatVt22πAVTWLexp[-9αd2vsat2Vt2δCg28AVT2WL].The models proposed in this research are more up-to-date and complete because they are derived based on newer basis model [20], and the previously overlooked ΔfT has now been modeled as pdfΔfT(δfT) in (11). Moreover, pdfΔCg(δCg) in (10) is more physical level oriented than its predecessor [18] as shown above since it contains additional physical level variables such as Neff, Wd, and φB. The proposed pdfΔCg(δCg) is more accurate than its predecessor as can be seen from its better goodness of fit test results shown in the upcoming section in which verification of both pdfΔCg(δCg) and pdfΔfT(δfT) will be given. Hence, these novel models have been found to be the potential mathematical tool for the variability aware analysis/design of various MOSFET-based analog/mixed signal circuits and systems at the nanometer regime.


## 3. The Verification

Verification of the proposed models has been performed in both qualitative and quantitative aspects. In the qualitative sense,pdfΔCg(δCg) and pdfΔfT(δfT) have been graphically compared with probability distributions of ΔCg and ΔfT obtained from the Monte-Carlo SPICE simulations of the benchmark circuits affected by normally distributed fluctuations in characteristics of MOSFET such as Vt, W, and L. For verification in the quantitative manner, goodness of fit tests have been performed to pdfΔCg(δCg) and pdfΔfT(δfT). Since it has been suspected that ΔCg and ΔfT are normally distributed via observation of the proposed models, the appropriate goodness of fit test has been found to KS test Here, overview of the KS test and its application to this research will be given. Strategy of the KS test is to performed the comparison of the K-S test statistic (KS) and the critical value (c) where it can be stated that any model fits its target data set if and only if its KS does not exceed its c [25, 26]. For this research, KS can be defined as
(14)KS=maxδx{|CDFΔx′(δx)|-|CDFΔx(δx)|},whereΔx can be either ΔCg or ΔfT so δx can be either δCg or δfT. As a result, CDFΔx(δx) which represents the cumulative distribution function of any Δx can be either CDFΔCg(δCg) for Δx=ΔCg and δx=δCg or CDFΔfT(δfT) for Δx=ΔfT and δx=δfT. Obviously, CDFΔCg(δCg) can be obtained by using pdfΔCg(δCg) as shown in (15) whereas CDFΔfT(δfT) can be derived by using pdfΔfT(δfT) as in (16). Note that erf()denotes the error function which can be mathematically defined for arbitrary variable y as erf(y)=(2/π)∫0yexp(-u2)du.On the other hand,CDFΔx′(δx) denotes the benchmark cumulative distribution function of any Δx which can be either ΔCg or ΔfT, calculated by integration of its corresponding probability distribution obtained from simulations of benchmark circuits. Since the confidence level of the test is 99%, c can be given by (17), where n denotes the number of runs of Monte-Carlo SPICE analysis, which is chosen to be 3000 for this research. As a result, c=0.0297596.Consider,(15)CDFΔCg(δCg)=∫-∞δCgpdfΔCg(u)du=12{1+erf[2.344αdεoxvsat(VFB+2φB)2δCgqtoxNeffWdWLVgs]},(16)CDFΔfT(δfT)=∫-∞δCgpdfΔfT(v)dv=12{1+erf[1.852toxWL3NeffWdπ(Vgs-VFB-2φB)2δfTqαdVgsvsat2]},(17)c=1.63n.At this point, other issues such as verification basis, benchmark circuits, and simulation methodologies will be mentioned. As the nanoscale MOSFET is focused, the 65 nm CMOS process technology has been adopted as the verification basis. Similar to [18], parameterization of the proposed models has been performed by using the 65 nm CMOS process parameters, and the BSIM4-based benchmark circuits at 65 nm level with all necessary parameters extracted by Predictive Technology Model (PTM) [27] have been used. Of course, both NMOS and PMOS technologies have been considered.In order to verify the accuracy ofpdfΔCg(δCg), the benchmark circuit and methodology used in [18] can be adopted. The core of this circuit for NMOS technological basis can be depicted in Figure 1, and Cg can be found as [18]
(18)Cg=Yinjω=1jωIinVin,Figure 1
NMOS-based circuit for verifyingpdfΔCg(δCg) [18].whereYin, Iin, and Vin denote the input admittance, input current, and input voltage of the benchmark circuit, respectively. Similar to [18], the Monte-Carlo SPICE simulation of this circuit gives the benchmark probability distribution of ΔCg for verification of pdfΔCg(δCg) based on NMOS technology. For PMOS technology, the similar circuit with the NMOS device replaced with PMOS one and the similar methodology can be used as done in [18].For verifyingpdfΔfT(δfT), core of the benchmark circuit can be depicted in Figure 2 for NMOS-based verification. From this circuit, fT can be defined as the frequency in which the current gain is unity; that is, the magnitude of the small signal input current, |iin| is equal to that of the output one, |iout|. Of course, the Monte-Carlo SPICE simulation of this circuit gives the benchmark probability distribution of ΔfT for verification of pdfΔfT(δfT) based on NMOS technology. Of course, the similar circuit with the NMOS device replaced with PMOS one and the similar methodology can be used for PMOS technology-based verification. It should be mentioned here that all circuits operate under 1 V. supply voltage with L=60 nm which is closed to the minimum allowable one, and W=1.2 μm. In the upcoming subsections, NMOS technology-based verification and PMOS-based one will be, respectively given; then it will be shown that pdfΔCg(δCg) is more accurate than its predecessor.Figure 2
NMOS-based circuit for verifyingpdfΔfT(δfT).


### 3.1. NMOS-Based Verification

The graphical comparisons of the probability distributions are depicted in Figures3 and 4 for verification in the qualitative sense of pdfΔCg(δCg) and that of pdfΔfT(δfT), respectively. The horizontal axes are labeled by ΔCg/Cg,nom and ΔfT/fT,nom which display the amounts of ΔCg and ΔfT as percentages of nominal Cg,Cg,nom and nominal fT,fT,nom. A strong agreement between pdfΔCg(δCg) and its benchmark obtained from the Monte-Carlo simulation along with that between pdfΔfT(δfT) and its benchmark has been found. Furthermore, normal distributions of variations high frequency performances as predicted by the proposed models have been observed.Figure 3
NMOS-based graphical comparison for the probability distribution ofΔCg: pdfΔCg(δCg) (line) versus benchmark distribution from the circuit (histogram).Figure 4
NMOS-based graphical comparison for the probability distribution ofΔfT: pdfΔfT(δfT) (line) versus benchmark distribution from the circuit (histogram).In the quantitative point of view, it can be seen that the resulting KS from the test ofpdfΔCg(δCg) can be found as KS=0.021463, and that from the test of pdfΔfT(δfT) is KS=0.029223. These statistics do not exceed c=0.0297596. This means that the proposed models can fit ΔCg and ΔfT obtained from the Monte-Carlo SPICE simulation of NMOS-based circuits with 99% confidence. At this point, both pdfΔCg(δCg) and pdfΔfT(δfT) have been verified based on NMOS technology as highly accurate.


### 3.2. PMOS-Based Verification

The graphical comparisons of the probability distributions are depicted in Figures5 and 6 for verification in the qualitative sense of pdfΔCg(δCg) and that of pdfΔfT(δfT). Strong agreements along with normal distributions of variations high frequency performances similar to those of NMOS-based verification have been found.Figure 5
PMOS-based graphical comparison for the probability distribution ofΔCg: pdfΔCg(δCg) (line) versus benchmark distribution from the circuit (histogram).Figure 6
PMOS-based graphical comparison for the probability distribution ofΔfT: pdfΔfT(δfT) (line) versus benchmark distribution from the circuit (histogram).For verification in quantitative manner, on the other hand, KS from the test ofpdfΔCg(δCg) can be found as KS=0.016144 and that from the test of pdfΔfT(δfT) is KS=0.029192. Similar to the previous NMOS-based case, these KS values do not exceed c=0.0297596. This means that the proposed model can fit ΔCg and ΔfT obtained from the simulation of PMOS-based circuits with 99% confidence. At this point, pdfΔCg(δCg) and pdfΔfT(δfT) are verified based on PMOS technology as highly accurate.


### 3.3. Higher Accuracy ofpdfΔCg(δCg)

In order to show thatpdfΔCg(δCg) proposed in this research is more accurate than its predecessor, its KS statistics have been compared with those of such predecessor as given in [18]. Since the verification of the proposed pdfΔCg(δCg) and that of its predecessor are identical, the comparison of the obtained statistics is said to be unbiased.Obviously, the KS values obtained from verifying the proposedpdfΔCg(δCg) which are 0.025466 for NMOS-based verification and 0.016144 for PMOS based are one, respectively-lower than those of the previous one in [18] given by 0.028462 for NMOS-based verification and 0.018036 for PMOS-based one [18]. According to [25, 26], this means pdfΔCg(δCg) in this research is better fit to the simulated data than its predecessor [18] according to its lower KS statistics. This can be alternatively stated that the proposed pdfΔCg(δCg) is more accurate.


## 3.1. NMOS-Based Verification

The graphical comparisons of the probability distributions are depicted in Figures3 and 4 for verification in the qualitative sense of pdfΔCg(δCg) and that of pdfΔfT(δfT), respectively. The horizontal axes are labeled by ΔCg/Cg,nom and ΔfT/fT,nom which display the amounts of ΔCg and ΔfT as percentages of nominal Cg,Cg,nom and nominal fT,fT,nom. A strong agreement between pdfΔCg(δCg) and its benchmark obtained from the Monte-Carlo simulation along with that between pdfΔfT(δfT) and its benchmark has been found. Furthermore, normal distributions of variations high frequency performances as predicted by the proposed models have been observed.Figure 3
NMOS-based graphical comparison for the probability distribution ofΔCg: pdfΔCg(δCg) (line) versus benchmark distribution from the circuit (histogram).Figure 4
NMOS-based graphical comparison for the probability distribution ofΔfT: pdfΔfT(δfT) (line) versus benchmark distribution from the circuit (histogram).In the quantitative point of view, it can be seen that the resulting KS from the test ofpdfΔCg(δCg) can be found as KS=0.021463, and that from the test of pdfΔfT(δfT) is KS=0.029223. These statistics do not exceed c=0.0297596. This means that the proposed models can fit ΔCg and ΔfT obtained from the Monte-Carlo SPICE simulation of NMOS-based circuits with 99% confidence. At this point, both pdfΔCg(δCg) and pdfΔfT(δfT) have been verified based on NMOS technology as highly accurate.


## 3.2. PMOS-Based Verification

The graphical comparisons of the probability distributions are depicted in Figures5 and 6 for verification in the qualitative sense of pdfΔCg(δCg) and that of pdfΔfT(δfT). Strong agreements along with normal distributions of variations high frequency performances similar to those of NMOS-based verification have been found.Figure 5
PMOS-based graphical comparison for the probability distribution ofΔCg: pdfΔCg(δCg) (line) versus benchmark distribution from the circuit (histogram).Figure 6
PMOS-based graphical comparison for the probability distribution ofΔfT: pdfΔfT(δfT) (line) versus benchmark distribution from the circuit (histogram).For verification in quantitative manner, on the other hand, KS from the test ofpdfΔCg(δCg) can be found as KS=0.016144 and that from the test of pdfΔfT(δfT) is KS=0.029192. Similar to the previous NMOS-based case, these KS values do not exceed c=0.0297596. This means that the proposed model can fit ΔCg and ΔfT obtained from the simulation of PMOS-based circuits with 99% confidence. At this point, pdfΔCg(δCg) and pdfΔfT(δfT) are verified based on PMOS technology as highly accurate.


## 3.3. Higher Accuracy ofpdfΔCg(δCg)

In order to show thatpdfΔCg(δCg) proposed in this research is more accurate than its predecessor, its KS statistics have been compared with those of such predecessor as given in [18]. Since the verification of the proposed pdfΔCg(δCg) and that of its predecessor are identical, the comparison of the obtained statistics is said to be unbiased.Obviously, the KS values obtained from verifying the proposedpdfΔCg(δCg) which are 0.025466 for NMOS-based verification and 0.016144 for PMOS based are one, respectively-lower than those of the previous one in [18] given by 0.028462 for NMOS-based verification and 0.018036 for PMOS-based one [18]. According to [25, 26], this means pdfΔCg(δCg) in this research is better fit to the simulated data than its predecessor [18] according to its lower KS statistics. This can be alternatively stated that the proposed pdfΔCg(δCg) is more accurate.


## 4. Discussion

In this section, some discussions regarding to the applications of the proposed models introduced in the previous section and some verification results will be given.


### 4.1.  Analytical Explanation of the Probabilistic/Statistical Behaviors ofΔCg and ΔfT and Other Variations

The probabilistic/statistical behaviors ofΔCg, ΔfT and variations in other high frequency parameters can be analytically explained by using their probabilistic/statistical parameters which can be derived by using the proposed pdfΔCg(δCg) and pdfΔfT(δfT) along with traditional statistical mathematic as analytical expressions involving physical level parameters. Simple examples of these parameters are means and variances of ΔCg and ΔfT. They can be derived by applying statistical mathematic to pdfΔCg(δCg) and pdfΔfT(δfT) as follows:(19a)ΔCg¯=∫-∞∞δCgpdfΔCg(δCg)dδCg=0,(19b)σΔCg2=∫-∞∞(δCg-ΔCg¯)2pdfΔCg(δCg)dδCg=qtoxNeffWdWLVgs24.688αd2εox2vsat2(VFB+2φB)4,(20a)
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,whereΔCg¯, σΔCg2, ΔfT¯, and σΔfT2 denote mean and variance of ΔCg along with those of ΔfT, respectively. Analytically obtaining these parameters yields various benefits to the statistical/variability aware analysis/design of nanoscale MOSFET-based analog/mixed signal circuits and systems, for example, obtaining the design tradeoffs which are important issues and so forth. A simple example of such design tradeoffs is shrinking the transistor dimension reduces ΔCg with increasing of ΔfT as a penalty. This design tradeoff can be obtained from observations of (19b) and (20b) that σΔCg2∝WL and σΔfT2∝1/WL3, respectively. It should be mentioned here that more sophisticated parameters such as moments of various orders, skewness, and kurtosis can also be derived by applying the statistical mathematic to pdfΔCg(δCg) and pdfΔfT(δfT).An interesting joint parameter ofΔCg and ΔfT is their correlation coefficient denoted by ρΔCg,ΔfT, which defines their statistical relationship. ρΔCg,ΔfT can be given in terms of ΔCg¯, σΔCg2, ΔfT¯, and σΔfT2 as follows:
(21)ρΔCg,ΔfT=E[(ΔCg-ΔCg¯)(ΔfT-ΔfT¯)]σΔCg2σΔfT2,whereE[] denotes the expectation operator. By using (9) and (19a)–(20b) which can be obtained from pdfΔCg(δCg) and pdfΔfT(δfT), along with (21), magnitude of ρΔCg,ΔfT has been found as unity. This means that there exists a very strong statistical relationship between ΔCg and ΔfT.As mentioned above, the analytical explanation of the probabilistic/statistical behaviors of variations in other high frequency parameters can be performed by using thepdfΔCg(δCg) and pdfΔfT(δfT) as these high frequency parameters are functions of Cg and/or fT. Let any of such high frequency parameter and its variation be denoted by X and ΔX, respectively: ΔX can be given by
(22)ΔX=(∂X∂Cg)ΔCg+(∂X∂fT)ΔfT.Obviously,ΔX is a random variable where its probabilistic/statistical behavior can be analytically explained by using its probabilistic/statistical functions/parameters which can be derived by using either the proposed pdfΔCg(δCg) and pdfΔfT(δfT) or related parameters which are their functions such as ΔCg¯, σΔCg2, ΔfT¯, σΔfT2, and so forth. Simple examples of such functions/parameters of ΔX are its mean ΔX¯ and variance σΔX2. They can be formulated in terms of  ΔCg¯, σΔCg2, ΔfT¯, and  σΔfT2 as follows:
(23)ΔX¯=(∂X∂Cg)ΔCg¯+(∂X∂fT)ΔfT¯=0,(24)σΔX2=(∂X∂Cg)2σΔCg2+(∂X∂fT)2σΔfT2+2(∂X∂Cg)(∂X∂fT)σΔCg2σΔfT2.Obviously, the fact that magnitude ofρΔCg,ΔfT is unity obtained by using pdfΔCg(δCg) and pdfΔfT(δfT) as stated above has been used in the formulation of (24). By using (19a)–(20b) which can be obtained from the pdfΔCg(δCg) and pdfΔfT(δfT), (23) and (24), ΔX¯ and σΔX2 can be derived as analytical expressions in terms of physical level parameters.More sophisticatedly, the probability density function ofΔX denoted by pdfΔX(δX), where δX stands for the sample variable of ΔX, can be derived by using pdfΔCg(δCg) and pdfΔfT(δfT) along with a well-known statistical mathematic methodology entitled the transformation of random variables which can be found in the usual texts such as [28, 29], as ΔX is a function of ΔCg and/or ΔfT, since X is a function of Cg and/or fT as aforementioned. According to [28, 29], if ΔX is function of Δu which can be either ΔCg or ΔfT as X is function of u which can be either Cg or fT, pdfΔX(δX) can be given in a straightforward manner as
(25)pdfΔX(δX)=pdfΔu(δu(δX))[[|∂δu∂δX|]|δu→δu(δX)]-1,wherepdfΔu(δu(δX)) stands for the probability density function of Δu denoted by pdfΔu(δu) with δu expressed in terms of δX. Since Δu can be either ΔCg or ΔfT, pdfΔu(δu) can be either pdfΔCg(δCg) or pdfΔfT(δfT).According to [29], if ΔX depends on both ΔCg and ΔfT as X is a function of both Cg and fT, pdfΔX(δX) can be given by(26a)pdfΔX(δX)=∫-∞∞l(δX,δY)dδY,where(26b)l(δX,δY)=pdfΔCg,ΔfT(δCg(δX,δY),δfT(δX,δY))×J(δX,δY),(26c)J(δX,δY)=|∂[δCg(δX,δY)]∂δX∂[δCg(δX,δY)]∂δY∂[δfT(δX,δY)]∂δX∂[δfT(δX,δY)]∂δY|,(26d)δCg(δX,δY)=|δX∂X∂fTδY∂Y∂fT|[|∂X∂Cg∂X∂fT∂Y∂Cg∂Y∂fT|]-1,(26e)δfT(δX,δY)=|∂X∂CgδX∂Y∂CgδY|[|∂X∂Cg∂X∂fT∂Y∂Cg∂Y∂fT|]-1,whereδY denotes the sample variable of ΔY which stands for the variation in a conveniently chosen auxiliary function denoted by Y. Similar to X, Y is also a function of both Cg and fT. It should be mentioned here that pdfΔCg,ΔfT(δCg(δX,δY),δfT(δX,δY)) can be found by applying (26d) and (26e) to pdfΔCg,ΔfT(δCg,δfT) which stands for the joint probability density function of ΔCg and ΔfT. This function can be found from the pdfΔCg(δCg) and pdfΔfT(δfT) by simultaneously solving the following equations:(27a)pdfΔCg(δCg)=∫-∞∞pdfΔCg,ΔfT(δCg,δfT)dδfT,(27b)pdfΔfT(δfT)=∫-∞∞pdfΔCg,ΔfT(δCg,δfT)dδCg.The next subsection will discuss the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical tool of the statistical/variability aware optimization scheme for nanoscale MOSFET’s high frequency performance.


### 4.2.  Mathematical Tool of Statistical/Variability Aware Optimization Scheme

Obviously, target of the optimization has been found to be minimizingΔCg and ΔfT. Simple objective functions can be obtained by using σΔCg2 and σΔfT2 which can be derived by using pdfΔCg(δCg) and pdfΔfT(δfT) as follows:
(28)minimize[σΔCg2],minimize[σΔfT2].More sophisticatedly, the objective functions can be given in terms ofPr{ΔCg=0} and Pr{ΔfT=0} which stand for the probability of obtaining ΔCg=0 and that of obtaining ΔfT=0, respectively. These probabilities can be determined from pdfΔCg(δCg) and pdfΔfT(δfT) by using the mathematical foundation in [28] as follows:
(29)Pr{ΔCg=0}=limχ→0[∫-χ0pdfΔCg(δCg)dδCg],Pr{ΔfT=0}=limφ→0[∫-φ0pdfΔfT(δfT)dδfT],whereχ≠0 and φ≠0; moreover, they may or may not be equal to each other. In this case, the optimal MOSFET’s parameters can be found by searching for those which maximize both Pr{ΔCg=0} and Pr{ΔfT=0}. As a result, the objective functions can be now given by
(30)maximize[Pr{ΔCg=0}],maximize[Pr{ΔfT=0}].If a single objective function optimization scheme is required,Pr{ΔCg=0,ΔfT=0} which is the probability to obtain ΔCg=0 and ΔfT=0 simultaneously, can be used. This probability can be defined as in (31), where pdfΔCg,ΔfT(δCg,δfT) can be determined from pdfΔCg(δCg) and pdfΔfT(δfT) as shown in (27a) and (27b). In this case, the optimal MOSFET’s parameters can be found by searching for those which maximize Pr{ΔCg=0,ΔfT=0}. As a result, the objective function can be obtained as in (32).Consider,(31)Pr{ΔCg=0,ΔfT=0}=limφ→0[limχ→0[∫-φ0∫-χ0pdfΔCg,ΔfT(δCg,δfT)dδCgdδfT]],(32)
maximize
[
Pr
{
Δ
C
g
=
0
,
Δ
f
T
=
0
}
]
.The next subsection will give a discussion on the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical tool of the variability aware simulation of nanoscale MOSFET’s high frequency performance with reduced computational effort.


### 4.3.   Mathematical Tool for Reduced Computational Effort Variability Aware Simulation of High Frequency Parameters

Obviously, outcome of any variability aware simulation is often expressed as the standard deviation of the simulated parameter. Let any interested high frequency parameter of variability aware simulation beZ, the desired outcome in terms of σZ can be conventionally determined by using the Monte-Carlo simulation which requires numerous runs [24]. So, many of the computational efforts are consumed.On the other hand,σZ can be computed in a more computationally efficient manner by using the sensitivity of Z with respect to Cg and that with respect to fT. These sensitivities can be determined by using the sensitivity analysis for which the targeted circuit/system is needed to be solved only once [24, 30]; then σZ can be simply computed by using σΔCg2 and σΔfT2 of MOSFETs within the targeted circuit/system. It can be seen that much of the computational effort for determining σZ is significantly reduced by using such sensitivity analysis-based method.At this point, the sensitivity analysis-based computation ofσZ will be explained. Let any circuits/systems compose M MOSFETs; σZ can be given in terms of σΔCg2 of any ith MOSFET,σΔCg,i2, that of any jth MOSFET,σΔCg,j2,σΔfT2 of any ith MOSFET σΔfT,i2, and that of any jth MOSFET, σΔfT,j2, as
(33)σZ=[∑i=1M[(SCgZ|i)2σΔCg,i2+(SfTZ|i)2σΔfT,i2]k+∑i=1i≠jM∑j=1M[(SCgZ|i)(SCgZ|j)ρΔCg,i,ΔCg,jσΔCg,i2σΔCg,j2mmmmma+(SfTZ|i)(SfTZ|j)ρΔfT,i,ΔfT,jσΔfT,i2σΔfT,j2]k+2∑i=1M∑j=1M[σΔCg,i2σΔfT,j2(SCgZ|i)(SfTZ|j)mmmmmma×ρΔCg,i,ΔfT,jσΔCg,i2σΔfT,j2(SCgZ|i)(SfTZ|j)]∑i-1M]1/2,whereρΔCg,i;ΔCg,j, ρΔfT,i;ΔfT,j and ρΔCg,i;ΔfT,j denote correlation coefficient between ΔCg of ith and jth MOSFET, that between ΔfT of ith and jth MOSFET, and that between ΔCg of ith MOSFET and ΔfT of jth MOSFET, respectively. Their magnitudes can be approximated by unity when i=j. Furthermore, SCgZ|i,  SCgZ|j,  SfTZ|i, and SfTZ|j denote sensitivity of Z to Cg of ith MOSFET, that to Cg of jth MOSFET, that to fT of ith MOSFET, and that to fT of jth MOSFET, respectively. These sensitivities can be determined by using the aforementioned sensitivity analysis. Finally, according to their definitions, σΔCg,i2 (σΔCg,j2) and σΔfT,i2 (σΔfT,j2) can be simply determined by substitution of the ith (jth) MOSFET’s parameters into (19b) and (20b) which have been derived by using the proposed pdfΔCg(δCg) and pdfΔfT(δfT).In the next subsection, the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical basis for analytical modeling of the high frequency performances mismatches between theoretically identical nanoscale MOSFETs will be discussed.


### 4.4.  Basis for Analytical Modeling of High Frequency Performances Mismatches

For illustration, the mismatch inCg and that in fT will be considered by letting such mismatches between theoretically identical devices, denoted by δCgij=Cgi-Cgj and δfTij=fTi-fTj for mismatch in Cg and that in fT, respectively. It should be mentioned here that Cgi, Cgj, fTi, and fTj denote Cg of ith MOSFET, Cg of jth MOSFET, fT of ith MOSFET, and fT of jth MOSFET.Since the behaviors of mismatches can be conveniently modeled by their variances [31, 32], the modeling of δCgij and δfTij is to derive their variances, that is, σδCgij2 and σδfTij2. Obviously the variances of Cgi, Cgj, fTi, and fTj which are fluctuated by the effects of random dopant fluctuation and process variation can be given by σΔCg,i2, σΔCg,j2, σΔfT,i2, and σΔfT,j2 due to (8). Of course, σΔCg,i2,σΔCg,j2, σΔfT,i2 and σΔfT,j2 can be determined by using pdfΔCg(δCg) and pdfΔfT(δfT) which are single devices oriented. In terms of these variances, σδCgij2 and σδfTij2 can be given as follows:
(34)σδCgij2=σΔCg,i2+σΔCg,j2-2ρΔCgi,ΔCgjσΔCg,iσΔCg,j,σδfTij2=σΔfT,i2+σΔfT,j2-2ρΔfTi,ΔfTjσΔfT,iσΔfT,j.Obviously, transistors are highly correlated if they are closely spaced [31, 32]. So, σδCgij2 and σδfTij2 are minimized if the correlations are in the positive manner and maximized if devices are negatively correlated. On the other hand, correlation terms can be neglected from (34) if devices are distanced [31, 32]. In the final subsection, discussion regarding observations from the verification results will be given.


### 4.5.  Observations from Verification Results

It can be observed from the formerly shown Monte-Carlo SPICE simulation results that the maximum percentages ofΔCg and ΔfT denoted by Max[ΔCg/Cg,nom] and Max[ΔfT/fT,nom], respectively, obtained from the NMOS-based circuits are higher than their counterparts from the PMOS ones. These maximum percentages are listed in Table 1. This means that high frequency performances of the nanoscale PMOS transistor is more robust to the random dopant fluctuation and process variation effects than that of NMOS one due to its lower Max[ΔCg/Cg,nom] and Max[ΔfT/fT,nom].Table 1
Max
(
Δ
C
g
/
C
g
,
nom
) and Max(ΔfT/fT,nom).
Max
(
Δ
C
g
/
C
g
,
nom
) (%)
Max
(
Δ
f
T
/
f
T
,
nom
) (%)
NMOS
PMOS
NMOS
PMOS
14.037
10.6
7.676
6.023Furthermore, it has been observed that the KS values obtained from the PMOS-based verification are smaller than those obtained from the NMOS-based one. These statistics are repeated in Table2 for convenience. This means that the proposed pdfΔCg(δCg) and pdfΔfT(δfT) model ΔCg and ΔfT of nanoscale PMOS transistor more accurately than they do to ΔCg and ΔfT of NMOS device according to such smaller statistics.Table 2
K-S test statistics.
From verification of
FromΔfT verification of
pd
f
Δ
C
g
(
δ
C
g)
pd
f
Δ
f
T
(
δ
f
T)
NMOS
PMOS
NMOS
PMOS
0.021463
0.016144
0.029223
0.029192


## 4.1.  Analytical Explanation of the Probabilistic/Statistical Behaviors ofΔCg and ΔfT and Other Variations

The probabilistic/statistical behaviors ofΔCg, ΔfT and variations in other high frequency parameters can be analytically explained by using their probabilistic/statistical parameters which can be derived by using the proposed pdfΔCg(δCg) and pdfΔfT(δfT) along with traditional statistical mathematic as analytical expressions involving physical level parameters. Simple examples of these parameters are means and variances of ΔCg and ΔfT. They can be derived by applying statistical mathematic to pdfΔCg(δCg) and pdfΔfT(δfT) as follows:(19a)ΔCg¯=∫-∞∞δCgpdfΔCg(δCg)dδCg=0,(19b)σΔCg2=∫-∞∞(δCg-ΔCg¯)2pdfΔCg(δCg)dδCg=qtoxNeffWdWLVgs24.688αd2εox2vsat2(VFB+2φB)4,(20a)
Δ
f
T
¯
=
∫
-
∞
∞
δ
f
T
pd
f
Δ
f
T
(
δ
f
T
)
d
δ
f
T
=
0
,
(20b)
σ
Δ
f
T
2
=
∫
-
∞
∞
(
δ
f
T
-
Δ
f
T
¯
)
2
pd
f
Δ
f
T
(
δ
f
T
)
d
δ
f
T
=
q
2
α
d
2
v
sat
4
V
gs
2
N
eff
W
d
3.704
π
2
(
V
gs
-
V
FB
-
2
φ
B
)
4
t
ox
W
L
3
,whereΔCg¯, σΔCg2, ΔfT¯, and σΔfT2 denote mean and variance of ΔCg along with those of ΔfT, respectively. Analytically obtaining these parameters yields various benefits to the statistical/variability aware analysis/design of nanoscale MOSFET-based analog/mixed signal circuits and systems, for example, obtaining the design tradeoffs which are important issues and so forth. A simple example of such design tradeoffs is shrinking the transistor dimension reduces ΔCg with increasing of ΔfT as a penalty. This design tradeoff can be obtained from observations of (19b) and (20b) that σΔCg2∝WL and σΔfT2∝1/WL3, respectively. It should be mentioned here that more sophisticated parameters such as moments of various orders, skewness, and kurtosis can also be derived by applying the statistical mathematic to pdfΔCg(δCg) and pdfΔfT(δfT).An interesting joint parameter ofΔCg and ΔfT is their correlation coefficient denoted by ρΔCg,ΔfT, which defines their statistical relationship. ρΔCg,ΔfT can be given in terms of ΔCg¯, σΔCg2, ΔfT¯, and σΔfT2 as follows:
(21)ρΔCg,ΔfT=E[(ΔCg-ΔCg¯)(ΔfT-ΔfT¯)]σΔCg2σΔfT2,whereE[] denotes the expectation operator. By using (9) and (19a)–(20b) which can be obtained from pdfΔCg(δCg) and pdfΔfT(δfT), along with (21), magnitude of ρΔCg,ΔfT has been found as unity. This means that there exists a very strong statistical relationship between ΔCg and ΔfT.As mentioned above, the analytical explanation of the probabilistic/statistical behaviors of variations in other high frequency parameters can be performed by using thepdfΔCg(δCg) and pdfΔfT(δfT) as these high frequency parameters are functions of Cg and/or fT. Let any of such high frequency parameter and its variation be denoted by X and ΔX, respectively: ΔX can be given by
(22)ΔX=(∂X∂Cg)ΔCg+(∂X∂fT)ΔfT.Obviously,ΔX is a random variable where its probabilistic/statistical behavior can be analytically explained by using its probabilistic/statistical functions/parameters which can be derived by using either the proposed pdfΔCg(δCg) and pdfΔfT(δfT) or related parameters which are their functions such as ΔCg¯, σΔCg2, ΔfT¯, σΔfT2, and so forth. Simple examples of such functions/parameters of ΔX are its mean ΔX¯ and variance σΔX2. They can be formulated in terms of  ΔCg¯, σΔCg2, ΔfT¯, and  σΔfT2 as follows:
(23)ΔX¯=(∂X∂Cg)ΔCg¯+(∂X∂fT)ΔfT¯=0,(24)σΔX2=(∂X∂Cg)2σΔCg2+(∂X∂fT)2σΔfT2+2(∂X∂Cg)(∂X∂fT)σΔCg2σΔfT2.Obviously, the fact that magnitude ofρΔCg,ΔfT is unity obtained by using pdfΔCg(δCg) and pdfΔfT(δfT) as stated above has been used in the formulation of (24). By using (19a)–(20b) which can be obtained from the pdfΔCg(δCg) and pdfΔfT(δfT), (23) and (24), ΔX¯ and σΔX2 can be derived as analytical expressions in terms of physical level parameters.More sophisticatedly, the probability density function ofΔX denoted by pdfΔX(δX), where δX stands for the sample variable of ΔX, can be derived by using pdfΔCg(δCg) and pdfΔfT(δfT) along with a well-known statistical mathematic methodology entitled the transformation of random variables which can be found in the usual texts such as [28, 29], as ΔX is a function of ΔCg and/or ΔfT, since X is a function of Cg and/or fT as aforementioned. According to [28, 29], if ΔX is function of Δu which can be either ΔCg or ΔfT as X is function of u which can be either Cg or fT, pdfΔX(δX) can be given in a straightforward manner as
(25)pdfΔX(δX)=pdfΔu(δu(δX))[[|∂δu∂δX|]|δu→δu(δX)]-1,wherepdfΔu(δu(δX)) stands for the probability density function of Δu denoted by pdfΔu(δu) with δu expressed in terms of δX. Since Δu can be either ΔCg or ΔfT, pdfΔu(δu) can be either pdfΔCg(δCg) or pdfΔfT(δfT).According to [29], if ΔX depends on both ΔCg and ΔfT as X is a function of both Cg and fT, pdfΔX(δX) can be given by(26a)pdfΔX(δX)=∫-∞∞l(δX,δY)dδY,where(26b)l(δX,δY)=pdfΔCg,ΔfT(δCg(δX,δY),δfT(δX,δY))×J(δX,δY),(26c)J(δX,δY)=|∂[δCg(δX,δY)]∂δX∂[δCg(δX,δY)]∂δY∂[δfT(δX,δY)]∂δX∂[δfT(δX,δY)]∂δY|,(26d)δCg(δX,δY)=|δX∂X∂fTδY∂Y∂fT|[|∂X∂Cg∂X∂fT∂Y∂Cg∂Y∂fT|]-1,(26e)δfT(δX,δY)=|∂X∂CgδX∂Y∂CgδY|[|∂X∂Cg∂X∂fT∂Y∂Cg∂Y∂fT|]-1,whereδY denotes the sample variable of ΔY which stands for the variation in a conveniently chosen auxiliary function denoted by Y. Similar to X, Y is also a function of both Cg and fT. It should be mentioned here that pdfΔCg,ΔfT(δCg(δX,δY),δfT(δX,δY)) can be found by applying (26d) and (26e) to pdfΔCg,ΔfT(δCg,δfT) which stands for the joint probability density function of ΔCg and ΔfT. This function can be found from the pdfΔCg(δCg) and pdfΔfT(δfT) by simultaneously solving the following equations:(27a)pdfΔCg(δCg)=∫-∞∞pdfΔCg,ΔfT(δCg,δfT)dδfT,(27b)pdfΔfT(δfT)=∫-∞∞pdfΔCg,ΔfT(δCg,δfT)dδCg.The next subsection will discuss the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical tool of the statistical/variability aware optimization scheme for nanoscale MOSFET’s high frequency performance.


## 4.2.  Mathematical Tool of Statistical/Variability Aware Optimization Scheme

Obviously, target of the optimization has been found to be minimizingΔCg and ΔfT. Simple objective functions can be obtained by using σΔCg2 and σΔfT2 which can be derived by using pdfΔCg(δCg) and pdfΔfT(δfT) as follows:
(28)minimize[σΔCg2],minimize[σΔfT2].More sophisticatedly, the objective functions can be given in terms ofPr{ΔCg=0} and Pr{ΔfT=0} which stand for the probability of obtaining ΔCg=0 and that of obtaining ΔfT=0, respectively. These probabilities can be determined from pdfΔCg(δCg) and pdfΔfT(δfT) by using the mathematical foundation in [28] as follows:
(29)Pr{ΔCg=0}=limχ→0[∫-χ0pdfΔCg(δCg)dδCg],Pr{ΔfT=0}=limφ→0[∫-φ0pdfΔfT(δfT)dδfT],whereχ≠0 and φ≠0; moreover, they may or may not be equal to each other. In this case, the optimal MOSFET’s parameters can be found by searching for those which maximize both Pr{ΔCg=0} and Pr{ΔfT=0}. As a result, the objective functions can be now given by
(30)maximize[Pr{ΔCg=0}],maximize[Pr{ΔfT=0}].If a single objective function optimization scheme is required,Pr{ΔCg=0,ΔfT=0} which is the probability to obtain ΔCg=0 and ΔfT=0 simultaneously, can be used. This probability can be defined as in (31), where pdfΔCg,ΔfT(δCg,δfT) can be determined from pdfΔCg(δCg) and pdfΔfT(δfT) as shown in (27a) and (27b). In this case, the optimal MOSFET’s parameters can be found by searching for those which maximize Pr{ΔCg=0,ΔfT=0}. As a result, the objective function can be obtained as in (32).Consider,(31)Pr{ΔCg=0,ΔfT=0}=limφ→0[limχ→0[∫-φ0∫-χ0pdfΔCg,ΔfT(δCg,δfT)dδCgdδfT]],(32)
maximize
[
Pr
{
Δ
C
g
=
0
,
Δ
f
T
=
0
}
]
.The next subsection will give a discussion on the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical tool of the variability aware simulation of nanoscale MOSFET’s high frequency performance with reduced computational effort.


## 4.3.   Mathematical Tool for Reduced Computational Effort Variability Aware Simulation of High Frequency Parameters

Obviously, outcome of any variability aware simulation is often expressed as the standard deviation of the simulated parameter. Let any interested high frequency parameter of variability aware simulation beZ, the desired outcome in terms of σZ can be conventionally determined by using the Monte-Carlo simulation which requires numerous runs [24]. So, many of the computational efforts are consumed.On the other hand,σZ can be computed in a more computationally efficient manner by using the sensitivity of Z with respect to Cg and that with respect to fT. These sensitivities can be determined by using the sensitivity analysis for which the targeted circuit/system is needed to be solved only once [24, 30]; then σZ can be simply computed by using σΔCg2 and σΔfT2 of MOSFETs within the targeted circuit/system. It can be seen that much of the computational effort for determining σZ is significantly reduced by using such sensitivity analysis-based method.At this point, the sensitivity analysis-based computation ofσZ will be explained. Let any circuits/systems compose M MOSFETs; σZ can be given in terms of σΔCg2 of any ith MOSFET,σΔCg,i2, that of any jth MOSFET,σΔCg,j2,σΔfT2 of any ith MOSFET σΔfT,i2, and that of any jth MOSFET, σΔfT,j2, as
(33)σZ=[∑i=1M[(SCgZ|i)2σΔCg,i2+(SfTZ|i)2σΔfT,i2]k+∑i=1i≠jM∑j=1M[(SCgZ|i)(SCgZ|j)ρΔCg,i,ΔCg,jσΔCg,i2σΔCg,j2mmmmma+(SfTZ|i)(SfTZ|j)ρΔfT,i,ΔfT,jσΔfT,i2σΔfT,j2]k+2∑i=1M∑j=1M[σΔCg,i2σΔfT,j2(SCgZ|i)(SfTZ|j)mmmmmma×ρΔCg,i,ΔfT,jσΔCg,i2σΔfT,j2(SCgZ|i)(SfTZ|j)]∑i-1M]1/2,whereρΔCg,i;ΔCg,j, ρΔfT,i;ΔfT,j and ρΔCg,i;ΔfT,j denote correlation coefficient between ΔCg of ith and jth MOSFET, that between ΔfT of ith and jth MOSFET, and that between ΔCg of ith MOSFET and ΔfT of jth MOSFET, respectively. Their magnitudes can be approximated by unity when i=j. Furthermore, SCgZ|i,  SCgZ|j,  SfTZ|i, and SfTZ|j denote sensitivity of Z to Cg of ith MOSFET, that to Cg of jth MOSFET, that to fT of ith MOSFET, and that to fT of jth MOSFET, respectively. These sensitivities can be determined by using the aforementioned sensitivity analysis. Finally, according to their definitions, σΔCg,i2 (σΔCg,j2) and σΔfT,i2 (σΔfT,j2) can be simply determined by substitution of the ith (jth) MOSFET’s parameters into (19b) and (20b) which have been derived by using the proposed pdfΔCg(δCg) and pdfΔfT(δfT).In the next subsection, the application ofpdfΔCg(δCg) and pdfΔfT(δfT) as the mathematical basis for analytical modeling of the high frequency performances mismatches between theoretically identical nanoscale MOSFETs will be discussed.


## 4.4.  Basis for Analytical Modeling of High Frequency Performances Mismatches

For illustration, the mismatch inCg and that in fT will be considered by letting such mismatches between theoretically identical devices, denoted by δCgij=Cgi-Cgj and δfTij=fTi-fTj for mismatch in Cg and that in fT, respectively. It should be mentioned here that Cgi, Cgj, fTi, and fTj denote Cg of ith MOSFET, Cg of jth MOSFET, fT of ith MOSFET, and fT of jth MOSFET.Since the behaviors of mismatches can be conveniently modeled by their variances [31, 32], the modeling of δCgij and δfTij is to derive their variances, that is, σδCgij2 and σδfTij2. Obviously the variances of Cgi, Cgj, fTi, and fTj which are fluctuated by the effects of random dopant fluctuation and process variation can be given by σΔCg,i2, σΔCg,j2, σΔfT,i2, and σΔfT,j2 due to (8). Of course, σΔCg,i2,σΔCg,j2, σΔfT,i2 and σΔfT,j2 can be determined by using pdfΔCg(δCg) and pdfΔfT(δfT) which are single devices oriented. In terms of these variances, σδCgij2 and σδfTij2 can be given as follows:
(34)σδCgij2=σΔCg,i2+σΔCg,j2-2ρΔCgi,ΔCgjσΔCg,iσΔCg,j,σδfTij2=σΔfT,i2+σΔfT,j2-2ρΔfTi,ΔfTjσΔfT,iσΔfT,j.Obviously, transistors are highly correlated if they are closely spaced [31, 32]. So, σδCgij2 and σδfTij2 are minimized if the correlations are in the positive manner and maximized if devices are negatively correlated. On the other hand, correlation terms can be neglected from (34) if devices are distanced [31, 32]. In the final subsection, discussion regarding observations from the verification results will be given.


## 4.5.  Observations from Verification Results

It can be observed from the formerly shown Monte-Carlo SPICE simulation results that the maximum percentages ofΔCg and ΔfT denoted by Max[ΔCg/Cg,nom] and Max[ΔfT/fT,nom], respectively, obtained from the NMOS-based circuits are higher than their counterparts from the PMOS ones. These maximum percentages are listed in Table 1. This means that high frequency performances of the nanoscale PMOS transistor is more robust to the random dopant fluctuation and process variation effects than that of NMOS one due to its lower Max[ΔCg/Cg,nom] and Max[ΔfT/fT,nom].Table 1
Max
(
Δ
C
g
/
C
g
,
nom
) and Max(ΔfT/fT,nom).
Max
(
Δ
C
g
/
C
g
,
nom
) (%)
Max
(
Δ
f
T
/
f
T
,
nom
) (%)
NMOS
PMOS
NMOS
PMOS
14.037
10.6
7.676
6.023Furthermore, it has been observed that the KS values obtained from the PMOS-based verification are smaller than those obtained from the NMOS-based one. These statistics are repeated in Table2 for convenience. This means that the proposed pdfΔCg(δCg) and pdfΔfT(δfT) model ΔCg and ΔfT of nanoscale PMOS transistor more accurately than they do to ΔCg and ΔfT of NMOS device according to such smaller statistics.Table 2
K-S test statistics.
From verification of
FromΔfT verification of
pd
f
Δ
C
g
(
δ
C
g)
pd
f
Δ
f
T
(
δ
f
T)
NMOS
PMOS
NMOS
PMOS
0.021463
0.016144
0.029223
0.029192


## 5. Conclusion

The novel complete probabilistic models of the random variations in nanoscale MOSFET’s high frequency performances have been proposed in this research aspdfΔCg(δCg) and pdfΔfT(δfT), shown in (10) and (11) as respectively. They have been found to be both analytic and physical level oriented as many physical level variables have been involved. Obviously, the formerly overlooked ΔfT has been now focused on. Both random dopant fluctuation and process variation effects have been taken into account. The nanoscale MOSFET equation [21] has been used as the mathematical basis, and the state of the art physical level fluctuation-induced characteristic variation model of MOSFET [20] has been adopted. So, pdfΔCg(δCg) is more accurate and physical level oriented than its predecessor [18]. Both pdfΔCg(δCg) and pdfΔfT(δfT) have been verified based on the 65 nm CMOS process technology by using the Monte-Carlo SPICE simulations of the benchmark circuits and the Kolmogorov-Smirnov goodness of fit tests as very accurate since they fit the results of Monte-Carlo SPICE simulations with 99% confidence. Furthermore, they have various utilities such as analytical explanation of probabilistic/statistical behaviors of such random variations and others, as mentioned above. So, they have been found to be versatile for the statistical/variability aware analysis/design of nanoscale MOSFET-based analog/mixed signal circuits and systems.



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*Source: 189436-2013-03-14.xml*