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

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 cir- cuits/systems. As a result, the statistical/variability aware analysis and designing strate-gies 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. 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 f max has also been proposed. This revised model has been found to be more accurate and detailed than the previous one.


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 highfrequency characteristics of intrinsic subthreshold MOSFET, i.e., gate capacitance, C g , transition frequency, f T , and maximum frequency of oscillation, f max . 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, I D and transconductance, g m , 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, C g , f T , and f max 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 f T derived as a function of the variation in C g 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 C g and f T , 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 C g , f T , and f max 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 C g in Section 3. The models of f T and f max will, respectively, be reviewed in Sections 4 and 5 where an improved model of variation in f max 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.

Foundations on subthreshold MOSFET
Unlike the strong inversion MOSFET in which I d is a polynomial function of the gate to source voltage, V gs , I d of the subthreshold MOSFET is an exponential function of V gs and can be given as follows: where C dep 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 g m ¼ dI d =dV gs , g m of subthreshold MOSFET can be given by

Variation in gate capacitance (C g )
Before reviewing the models of variation in C g of subthreshold MOSFET, it is worthy to introduce the mathematical expression of C g as it is the mathematical basis of such models.
Here, C g 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, Q g as [15] where It is noted that Q B,max stands for the maximum bulk charge [15]. By using Eq. (1), Q g of the subthreshold MOSFET can be found as As a result, the expression of C g 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 V t , W, L, etc., denoted by ΔV t , ΔW, ΔL, and so on existed. These variations yield the randomly varied C g i.e. C g (ΔV t , ΔW, ΔL,…) [15]. Thus, the variations in C g , ΔC g can be mathematically defined as [15] ΔC g ¼ Δ C g ðΔV t , ΔW, ΔL, …Þ À C g where C g stands for the nominal gate capacitance in this context.
With this mathematical definition and the fact that ΔV t is the most influential in subthreshold MOSFET [18], the following comprehensive analytical expression of ΔC g has been proposed in [15] ΔC g ¼ 2 where N eff , V FB , W dep , and ϕ s denote the effective values of the substrate doping concentration N sub (x), the flat band voltage, depletion width, and surface potential, respectively. Moreover, N eff can be obtained by weight averaging of N sub (x) as [15] As ΔC g 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 ΔC g , Var[ΔC g ] can be analytically obtained as follows [15] Var where ε 0 stands for the permittivity of free space. At this point, it can be seen that the comprehensive analytical model of ΔC g proposed in [15] is composed of Eqs. (8) and (10) where the latter has been derived based on the former. In [15], (Var[ΔC g ]) 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[ΔC g ]) 0.5 and the benchmark have been found. The average deviation from the benchmark obtained from the entire range of V gs used for simulation given by 0-100 mV has been found to be 9.42565 and 8.91039% for N-type and P-type MOSFETbased comparisons, respectively [15].
Later, an improved model of ΔC g 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 ΔC gN and ΔC gP are ΔC g of N-type and P-type MOSFETs, respectively. Moreover, N a , N d , V sb , and ϕ F denote 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] [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 Ptype MOSFETs have also been taken into account.

Variation in transition frequency (f T )
Apart from that of ΔC g , the comprehensive analytical model of variation in f T of subthreshold MOSFET, Δf T has also been proposed in [16]. Before reviewing such model, it is worthy to show the definition of f T and its comprehensive analytical expression derived in [16]. According to [21], f T 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 C g by the following equation [13] By using Eqs. (2) and (6), the following comprehensive analytical expression of f T can be obtained [16] Similar to ΔC g , Δf T can be mathematically defined as [16] Δf where f T stands for the nominal transition frequency in this context.
By also keeping in mind that ΔV t is the most influential, the following comprehensive analytical expression of Δf T 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 Δf TN and Δf TP are Δf T 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] Var Δf TN Â Ã ¼ At this point, it can be stated that the comprehensive analytical model of Δf T proposed in [16] is composed of Eqs. (18), (19), (20), and (21) [16]. Moreover, it has been proposed in [16] that there exists a very strong statistical relationship between ΔC g and Δf T of any certain subthreshold MOSFET as it has been found by using the proposed model that the magnitude of the statistical correlation coefficient of ΔC g and Δf T is unity for both N-type and P-type devices.

Variation in maximum frequency of oscillation (f max )
Before reviewing the model of variation in f max of subthreshold MOSFET, it is worthy to introduce its definition and mathematical expression. The f max , 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], f max can be given under an assumption that C g is equally divided between drain and source by where R g stands for the resistance of gate metallization [17].
By substituting g m and C g as respectively given by Eqs. (2) and (6) into Eq. (22), we have Similar to the other variations, Δf max can be mathematically defined as [17] [16].
where f max stands for the nominal maximum frequency of oscillation in this context.
In [17], the comprehensive analytical model of Δf max 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 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[Δf max ]) 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[Δf max ]) 0.5 and the benchmark can be observed from the whole simulated range of V gs given by 0-100 mV. The average deviation has been found to be 9.17682 and 8.51743% for N-type and Ptype 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 Δf maxN and Δf maxP are Δf max 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   [17]. Therefore, our improved model Δf max 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 Δf max and Δf T as f max is related to f T by Eq. (31). An implication of strong correlation between Δf max and ΔC g 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, ΔC g , Δf T , and Δf max must be minimized. It has been found from Eqs. (13), (14), (20), (21), (29), and (30) that Var ΔC g Â Ã ∝ L 3 , Var Δf T Â Ã ∝ L À7 and Var Δf max Â Ã ∝ L À1 for both types of MOSFET. Therefore, it can be seen that shrinking L can reduce ΔC g of the subthreshold MOSFET of any type with the increasing Δf T and Δf max as penalties. Moreover, we have also found that This means that we can reduce Δf T and Δf max by lowering T with higher ΔC g 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.

Variation in any high-frequency parameter
Occasionally, determining the variation in other high-frequency parameters apart from C g , f T , and f max e.g., bandwidth, f BW , etc., has been found to be necessary. The determination of variation in f BW as a function of Δf T 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 ΔC g , Δf T , and Δf max if P depends on C g , f T , and f max . It is noted that the amounts of ΔC g , Δf T , and Δf max can be predetermined by using the reviewed comprehensive analytical models. Mathematically, ΔP can be expressed in terms of ΔC g , Δf T , and Δf max as follows Therefore, the variance of ΔP, Var[ΔP] can be given by keeping the aforementioned strong statistical relationships among ΔC g , Δf T , and Δf max in mind as follows Noted also that the Var[ΔC g ], Var[Δf T ], and Var[Δf max ] can be known by applying those reviewed models.

High-frequency parameter mismatches
The amount of mismatches in C g , f T , and f max of multiple subthreshold MOSFETs can be determined by applying those reviewed comprehensive analytical models of ΔC g , Δf T , and Δf max even though they are dedicated to a single device. As an illustration, the mismatches in C g , f T , and f max 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 C g , f T , and f max of M1 and M2 be denoted by ΔC g12 , Δf T12 , and Δf max12 , respectively, their variances, i.e.,  For closely spaced MOSFETs with positive correlation, r XY can 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, r XY become À1 and the mismatches are minimized [16]. For distanced devices, we have, r XY ¼ 0 as the correlation is very weak and can be neglected. ] can all be approximately given by 0 if those statistically identical devices are closely spaced and positively correlated as all r XY '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.

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, Δl due to the variation in C g1 , ΔC g1 can be immediately given by [16] Δl ¼ ΔC g1 g m1 g m2 (41) Therefore, we have the following relationship between the variances of Δl and ΔC g1 Var It is noted that Var[ΔC g1 ] can be determined by using those reviewed models. It can also be seen that Var Δl ½ ∝ Var ΔC g1 Â Ã and Var Δl ½ ∝ 1=g m1 g m2 [16]. Therefore, it is far more convenient to minimize Δl by reducing g m1 and g m2 as they are electronically controllable unlike ΔC g1 , which must be minimized at the physical level by lowering L as stated above.

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.