## 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:

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* of subthreshold MOSFET can be given by

## 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]

where

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

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

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 Δ*Vt*, Δ*W*, Δ*L*, and so on existed. These variations yield the randomly varied *Cg* i.e. *Cg*(Δ*Vt*, Δ*W*, Δ*L*,…) [15]. Thus, the variations in *Cg*, ∆*Cg* can be mathematically defined as [15]

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

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

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]

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]

where *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

where ∆*CgN* and ∆*CgP* are ∆*Cg* of N-type and P-type MOSFETs, respectively. Moreover, *Na*, *Nd*, *Vsb*, and

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]

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

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

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

By also keeping in mind that Δ*Vt* 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.

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]

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 Δ*Cg* and Δ*fT* of any certain subthreshold MOSFET as it has been found by using the proposed model that the magnitude of the statistical correlation coefficient of Δ*Cg* and Δ*fT* 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

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

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

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.

It is noted that Eq. (25) has been derived by also keeping in mind that Δ*Vt* 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

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

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 *C*_{g} 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 2–7. Moreover, it can be implied that there exists a strong correlation between Δ*fmax* and Δ*fT* as *fmax* is related to *fT* by Eq. (31). An implication of strong correlation between Δ*fmax* and Δ*Cg* can be similarly obtained by observing Eq. (22) that is given as

## 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 *L* can reduce Δ*Cg* of the subthreshold MOSFET of any type with the increasing Δ*fT* and Δ*fmax* as penalties. Moreover, we have also found that *fT* and Δ*fmax* by lowering *T* with higher Δ*Cg* 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 Δ*fT* has been shown in [16]. In general, let any high-frequency parameter of the subthreshold MOSFET be *P*, the amount of its variation, Δ*P*, can be determined given the amounts of Δ*Cg*, Δ*fT*, and Δ*fmax* if *P* depends on *Cg, fT*, and *fmax*. It is noted that the amounts of Δ*Cg, ΔfT*, and Δ*fmax* can be predetermined by using the reviewed comprehensive analytical models. Mathematically, Δ*P* can be expressed in terms of Δ*Cg, ΔfT*, and Δ*fmax* as follows

Therefore, the variance of Δ*P*, *Var*[Δ*P*] can be given by keeping the aforementioned strong statistical relationships among Δ*Cg, ΔfT*, and Δ*fmax* in mind as follows

Noted also that the *Var*[Δ*Cg*], *Var*[Δ*fT*], and *Var*[Δ*fmax*] 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 Δ*Cg*, Δ*fT*, and Δ*fmax* 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 Δ*C*_{g12}, Δ*f*_{T12}, and Δ*f*_{max12}, respectively, their variances, i.e., *Var*[Δ*C*_{g12}], *Var*[Δ*f*_{T12}], and *Var*[Δ*f*_{max12}], can be respectively related to *Var*[Δ*Cg*], *Var*[Δ*fT*], and *Var*[Δ*fmax*] of M1 and M2, which can be determined by using those reviewed models, via the following equations

It is noted that Δ*Cgi*, Δ*fTi*, Δ*fmaxi*, *Var*[Δ*Cgi*], *Var*[Δ*fTi*], and *Var*[Δ*fmaxi*], respectively, denote Δ*Cg*, Δ*fT*, Δ*fmax*, *Var*[Δ*Cg*], *Var*[Δ*fT*], and *Var*[Δ*fmax*] of M*i* where {*i*} = {1, 2}. Moreover, *X* and *Y* where {*X*} = {Δ*C*_{g1}, Δ*f*_{T1}, Δ*f*_{max1}} and {*Y*} = {Δ*C*_{g2}, Δ*f*_{T2}, Δ*f*_{max2}}. For closely spaced MOSFETs with positive correlation,

If we assume that both M1 and M2 are statistically identical, we have *Var*[Δ*C*_{g1}] = *Var*[Δ*C*_{g2}] = *Var*[Δ*Cg*], *Var*[Δ*f*_{T1}] = *Var*[Δ*f*_{T2}] = *Var*[Δ*fT*], and *Var*[Δ*f*_{max1}] = *Var*[Δ*f*_{max2}] = *Var*[Δ*fmax*]. Thus, Eqs. (34), (35), and (36) become

From these equations, it can be seen that *Var*[Δ*C*_{g12}], *Var*[Δ*f*_{T12}], and *Var*[Δ*f*_{max12}] can all be approximately given by 0 if those statistically identical devices are closely spaced and positively correlated as all

### 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

where *C*_{g1}, *g*_{m1}, and *g*_{m2} are gate capacitance of M1, transconductance of M1, and transconductance of M2, respectively.

By using Eq. (40), the variation in *l*, Δ*l* due to the variation in *C*_{g1}, Δ*C*_{g1} can be immediately given by [16]

Therefore, we have the following relationship between the variances of Δ*l* and Δ*C*_{g1}

It is noted that *Var*[Δ*C*_{g1}] can be determined by using those reviewed models. It can also be seen that *l* by reducing *g*_{m1} and *g*_{m2} as they are electronically controllable unlike Δ*C*_{g1}, 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.

It is noted that the magnitude of *X*} = {Δ*Cgi*, Δ*fTi*, Δ*fmaxi*}, {*Y*} = {Δ*Cgj*, Δ*fTj*, Δ*fmaxj*}, and the subscripts *i* and *j* refers to the arbitrary i^{th} and j^{th} MOSFET, respectively, in this scenario, approaches 1 when *i* = *j* as it determines the correlation of the same device. Moreover, *Z* to *Cg*, *fT*, and *fmax* of i^{th} (j^{th}) 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 Δ*Cg*, Δ*fT*, and Δ*fmax* 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 Δ*fmax* has also been proposed. This revised model has been found to be more accurate and detailed than the previous one.