Novel Comprehensive Analytical Probabilistic Models of The Random Variations in MOSFET s High Frequency Performances

Transcription

1 WSEAS RANSAIONS on IRUIS SYSEMS Novel omprehensive Analytical Probabilistic Models of he Rom Variations in MOSFE s Hih Frequency Performances RAWID BANHUIN Department of omputer Enineerin Siam University 35 Petchakasem Rd., Phasicharoen, Bankok HAILAND rawid_b@yahoo.com Abstract: In this research, the probabilistic models of the rom variations in MOSFE s hih frequency performance defined in terms of variations in ate capacitance transition frequency, have been proposed. Both rom dopant fluctuation process variation effects which are the major causes of the MOSFE s hih frequency characteristic variations have been taken into account. he short channel MOSFE has been focused. he proposed models which take the form of the comprehensive analytical expressions have been derived by usin the alphapower law which is more comprehensive than the conventional square law. he up to dated physical level fluctuation model has been adopted as the basis instead of the classical one. So, the model of ate capacitance variation has been refined. hese probabilistic models have been verified based on the IBM 90nm RF MOS technoloy by usin the Montearlo SPIE simulations the Kolmoorov Smirnof oodness of fit tests. hey are very accurate since they can fit the Montearlo SPIE based data distributions with 99% confidence. Hence, the proposed models have been found to be the potential mathematical tool for the statistical/variability aware analysis /desin of various MOSFE based hih frequency applications. KeyWor: Gate capacitance, Hih frequency, MOSFE, Short channel, Statistical desin, ransition frequency, Variability aware desin. 1 Introduction Recently, the subthreshold (weak inversion reion MOSFE has been adopted in many hih frequency circuits, system applications such as active inductor, active transformer, hih frequency filter, optical front en, clock/data recovery circuits Bluetooth devices. Of course, the performances of these hih frequency equipments are mainly determined by the hih frequency performances of their intrinsic MOSFE which can be determined by two major MOSFE s parameter entitled ate capacitance, transition frequency f. Obviously, imperfection in MOSFE s properties for example rom dopant fluctuation, line ede rouhness ate lenth rom fluctuation cause the rom variations in MOSFE s physical parameters such as threshold voltae which are crucial in the statistical/variability aware desin of MOSFE based applications. his is because the MOSFE s circuit level parameters such as drain current, I d transconductance, m are romly fluctuated due to the cited rom variations at the physical level. By this motivation, there are many previous researches on the modelin of such variations in the circuit level parameters for example, [1. he resultin models take the form of the comprehensive analytical expressions which have been found to be the convenient mathematical tools for the statistical/variability aware analysis/desin of various MOSFE based applications. However, these researches did not mention anythin about the variations in f even thouh they also exist. Furthermore, such f reatly determine the hih frequency performances of the MOSFE as mentined above. So, it can be seen that these previous models lack of some important information for hih frequency applications. Fortunately, there are previous studies in the characteristics of rom variations in both f for example [57 etc. he results of these previous studies are the raphical plots numerical measurement of such variations. Unfortunately, such results are not eneric since they are obtained from a certain technoloical basis. Furthermore, none of any related comprehensive analytical expressions that display the variation characteristics of both f has been derived. However, it has been pointed out in [7 that rom dopant fluctuation process variation effects such as line ede rouhness ate lenth rom fluctuation are the major causes of rom EISSN: 66X 69 Issue 3, Volume 1, March 013

2 WSEAS RANSAIONS on IRUIS SYSEMS variations in hih ftrequency characteristics of MOSFE includin f. In [8, simple formulas on f the relationship between the variance of that of f have been iven. Unfortunately, these formulas are incomprehensive since they do not display the physical level variation causes their relationships with the resultin variations in f. Accordin to the importance of both f alon with their possibility to be romly fluctuated as mentioned above, the comprehensive analytical modelin of their variations similarly to the modelin of the other parameters proposed in the previous researches such as [1, must be urently performed. Fortunately, the cited comprehensive analytical model of rom variation in has been proposed in [9. However, this model has been derived based on the classical model of variation [10 which is bein replaced by the state of the art ones such as that proposed in [11, so, a refinement is necessary. Furthermore, the similar modelin on f which is as important as that of has not been performed yet. Hence, due to the above motivation, the probabilistic models of rom variations in MOSFE hih frequency performance defined in terms of variations in f have been proposed in this research. Both rom dopant fluctuation process variation effects which are the major causes of the rom viriations in the MOSFE s hih frequency characteristics as stated in [7, have been taken into account. he short channel MOSFE has been focused since it reins the universe of the MOS technoloy nowadays. he proposed models which take the form of the comprehensive analytical expressions have been derived by usin the alphapower law [1 which is more comprehensive than the conventional square law. his alphapower law has also been previously adopted in [1, [ [9. he up to dated physical level fluctuation model proposed in [11 has been adopted as the basis instead of the classical one proposed in [10 which has been adopted in [9 as aforementioned. So, the model of variation in has been refined. he proposed models have been verified by usin the Montearlo SPIE simulations based on the IBM 90nm RF MOS process technoloy the KolmoorovSmirnof oodness of fit tests. he verifications have been performed based on both NMOS PMOS technoloies. hese models are very accurate since they can fit the Montearlo SPIE based data distributions with 99% confidence. Hence, the proposed models have been found to be the potential mathematical tool for the statistical/variability aware analysis/desin of various MOSFE based hih frequency applications. he Proposed Model In this section, the proposed models will be discussed. Before procee further, it is worthy to ive some foundation on the alphapower law of MOSFE s characteristic. Accordin to [1, [, [9 [1, this law which is more comprehensive than the classical square law stated that I d can be iven by I µ W L α d (1 λv ( Vs Vt (1 where µ,, V t, α λ denote the carrier mobility, the ate ide capacitance of the transistor, practical thresold voltae which is affected by rom dopant fluctuation process variation, velocity saturation index [1 channel lenth modulation parameter respectively. With such I d, the alphapower law based m is µ α 1 m α (1 λv ( Vs Vt ( W L At this point, the derivation of the proposed model can be started. Firstly, the analytical expression of has been derived. Similarly to [9 accordin to [13, can be iven by dq (3 dvs where Q denotes the ate chare [13. As in [9, accordin to [1, Q can be iven by Q d Vst ( Vs c t dvc QB,max 0 µ W L ( I where Q B,max denotes the maximum bulk chare [1. By usin (1 (, alphapower law based Q is [9 Q WL ( Vs Vt 3 1 λv 3 α Q B,max (5 So, can be iven by usin (3 (5 as follows [9 WL (3 α ( Vs 3 1 λv t α (6 EISSN: 66X 70 Issue 3, Volume 1, March 013

3 WSEAS RANSAIONS on IRUIS SYSEMS Hence, the alphapower law based f can be iven by usin ( (6 as f 3 αµ 8π (1 λv ( V (3 α L s 3 α 3 t (7 By takin both rom dopant fluctuation which is assumed that it ives the number of dopant atoms that obeys the Poisson distribution [15, process variation effects into account, the variations in MOSFE s properties existed. hese fluctuations include the rom variations in both f [7. Such variations are denoted by f respectively. Of course, both f become romly varied due to these variations which can be defined as follows (8, Nom (9 f f f, Nom So, it can be seen that these variations can be defined in terms of the different between their actual romly varied values their correspondin nominal counterparts denoted by,nom f,nom respectively. hese nominal values which can be ideally obtained by nelectin all imperfections of the MOSFE are obviously deterministic. With (6 (9, Δf can be iven as follows α α WL [( Vs Vt ( Vs VH ( (1 λv (3 α f 3 αµ 8π [( V s (1 λv α 3 t ( V s (3 α L H 3 α 3 (11 where V H denotes the nominal threshold voltae. By usin the state of the art model of physical level variation proposed in [11 the principle of rom variable transformation with (10 (11 as mappin functions, the probability density functions of Δf can be derived with the emphasis on the short channel MOSFE. hese probability density functions are α 1 H 3 3ε (1 λv( Vs ( δ π qw L N W ( α(3 α INV sub α 7ε (1 λv ( Vs H δ exp 3 8qINVNsubW WL ( α (3 α (1 f 8 3πε W L (3 α ( δf [3 µ q (1 λv ( V N sub W exp{ [9µ INV (1 λv α 3 α 7 q INVNsubW 96π ε WL (3 α δf ( V s s ( α H ( α H } α 3 α (13 where δ, δf, q, ε, N sub, INV, W, W L denote any sampled value of, that of f, electron chare, ate ide permittivity, channel impurity concentration, electrical ate dielectric thickness, channel width, letion width channel lenth respectively [11. Hence, it can be stated that the proposed model which probabilistically describes the rom dopant fluctuation process variation effects induced rom variation in the similar variation in f can be iven by (1 (13 which many physical parameters have been incorporated. At this point, the probabilistic models of rom variations in MOSFE s hih frequency performance which can be defined in terms of variations in f have been derived in the form of the probability density functions which are the comprehensive analytical expressions includin many physical level parameters. ompared to the previous studies such as those in [57 which presents the study results of variation in in terms of raphical plots numerical measurements obtained from a certain technoloical basis, the proposed model is a more analytic, comprehensive eneric result as it is an analytical expression in term of many physical variables without any reference to a certain technoloy node but to the eneral MOS technoloy. hese models are more comprehensive than that proposed in [8 as they contain several physical level parameters so, the physical causes of such hih frequency performance variation have been revealed. his clarifies the exploration of the cited hih frequency performance variation. hese models are also as comprehensive as those in [ [3 but they emphasize on the hih frequency operation. Finally, the proposed models are more complete up to dated than that proposed in [9 which mention nothin on the variation in f, also relies on the outdated physical level fluctuation model [10. For fittin of the variation in, part of the proposed models for this variation is more accurate than that mentioned in [9 due to the better KolmoorovSmirnof test statistic obtained from the EISSN: 66X 71 Issue 3, Volume 1, March 013

4 WSEAS RANSAIONS on IRUIS SYSEMS data fittin usin a part of the proposed ones as will be seen later. As this model takes the form of the probability density function, much meaninful statistical information of the variations in f can be obtained. For examples, it can be seen that both variations in f have the followin means variances σ f σ where δ ( δ dδ 0 (1 ( δ 3 qinvnsubw WL ( α (3 α α 7ε (1 λv ( V f ( δf f 9µ q 19π, INV ε s ( δ dδ H (15 δ f ( δf dδf 0 (16 N f sub 7 W f ( δf dδf α 3 α WL (3 α ( V σ, f s, V (1 λv ( α H (17 σ f denote mean variance of alon with those of f respectively. Furthermore, the analytical expressions of many other meaninful statistical parameters such as moments of various orders, skewness kurtosis can also be obtained by simply apply the conventional mathematical statistic to the proposed model with the required mathematics can be performed even by h calculation. hese are the major benefits of this model. For the desinin with electronic AD tools, this model can serves as the mathematical basis for the desired simulations. he required computational effort is potentially smaller than that of the brute force Montearlo analysis based on the rom variations of the parameters at the physical level which is time expensive as mentioned in [16. Hence, the proposed models have been found to be the potential mathematical tool for the statistical/variability aware analysis/desin of various MOSFE based hih frequency applications. In the subsequent section, the verification of the proposed model will be discussed. 3 he Verification Similarly to [, 3, 9, the verification of the proposed model has been performed in both qualitative quantitative aspects. In the qualitative sense, the estimated distributions of rom variations in f obtained from the models have been raphically compared to their counterparts obtained from the Montearlo SPIE simulations of the benchmark circuits. On the other h for the quantitative point of view, numbers of KolmoorovSmirnof oodness of fit test (KStest have been performed by usin the cumulative distributions of such rom variations obtained from the same set of Montearlo SPIE simulations for the qualitative verifications. It should be mentioned here that both NMOS PMOS technoloies have been considered. he verification has been performed based on the up to dated IBM 90nm RF MOS technoloy by the model parameterization with the IBM 90nm RF MOS process parameters the usae of the BSIM.3 based benchmark circuits parameterized by the SPIE parameters extracted from such IBM 90nm RF MOS process. hese necessary parameters are provided by the MOSIS. Since the KStest relies on the cumulative distributions, it is worthy to derive the cumulative distribution function forms of the proposed models at this point. With the conventional mathematical statistic, they can be iven by DF ( δ δ ( u du 3 1 8qINV N subwwl ( α 1 erf 7ε (1 λv ( Vs DF ( δf δf f f ( u du (3 α α H 1 9µ q INVNsubW α 3 α (1 λv 1 erf 7 ( α 96π ε WL (3 α ( Vs VH 1 δ 1 δf (18 (19 Note that erf ( x denotes the error function of any arbitrary variable, x which can be mathematically defined as x erf ( x exp( u du. (0 π 0 EISSN: 66X 7 Issue 3, Volume 1, March 013

5 WSEAS RANSAIONS on IRUIS SYSEMS Accordin to [17, 18, the stratey of the KStest is to performed the comparison of the KS test statistic (KS the critical value (c where it can be stated that any model fits its taret data set if only if its KS is not exceed its c [17, 18. For this research, KS can be defined as KS max{ DF ( δx DF ( δx } (1 δx x circuit x model where x can be either or f while DF x (δx DF model x(δx represent the circuit cumulative distribution of any x obtained from the proposed models the counterpart obtained from the simulation of the benchmark circuit respectively. Furthermore, as the confidence level of the test is 99%, c can be iven by [ c ( n where n denotes the number of runs for the Montearlo SPIE analysis. Similarly to [9, it should be mentioned here that the model verification has been performed by usin n 3000 which yiel c In the upcomin subsections, the verification of the model based on NMOS technoloy the PMOS counterpart will be respectively discussed. 3.1 NMOS based model verification As in [9, for the verification the model s accuracy in the prediction of rom variation in, the chosen benchmark circuit is a sinle MOSFE with both drain source have been ac rounded where as the ate terminal has been fed by the ac input voltae. he input admittance which takes into account can be seen by lookin into the input terminal of this benchmark circuit. his methodoloy of findin the MOSFE s input admittance by usin SPIE analysis has also been used in [19. For NMOS technoloical basis, the core of this circuit can be icted in Fi.1. As the input conductance is extremely small for conventional MOSFEs, can be apprimately iven as follows [9 Y 1 I in in (3 jω jω Vin where Y in, I in V in denote the input admittance, input current input voltae respectively. Similarly to [9, the variation in the formerly defined above obtained from the Montearlo SPIE simulation of this circuit at 1 GHz has been adopted as the circuit based Δ for NMOS technoloy. On the other h, the chosen benchmark circuit for verifyin the model s accuracy in the prediction of rom variation in f is also a sinle MOSFE but both drain ate have been fed with their own ac voltaes where as the source terminal has been ac rounded. hese ac voltaes applied at the ate drain are V s V respectively. For NMOS technoloical basis, the core of this circuit can be icted in Fi.. From this circuit, f can be defined as the frequency which the current ain is unity. Alternatively, it is the frequency which the manitude of the small sinal input current is equal to that of the output one. his methodoloy has also been adopted in findin f of EP s enhancement mode GaN power transistor with SPIE simulation [0. Of course, the correspondin variation in f obtained from the Montearlo SPIE simulation of this circuit has been adopted as the circuit based Δf for NMOS technoloy. It should be mentioned here that these benchmark circuit uses the nominal channel lenth (L of 100nm which is closed to the minimum allowable value alon with the nominal aspect ratio (W/L of 50. For both circuits, the dc supplies of ±0.5 v have been used. Finally, these simulation strateies, parameters settin biasin condition have been used for the PMOS based model verification. By proceed in the similar manner to the model verifications in [, 3, 9, the raphical comparisons for the distributions Δ Δf with respected to their nominal values, denoted by /,Nom f /f,nom respectively are icted in Fi. 3 Fi. for the qualitative NMOS based verification. For this NMOS case, it can be seen that stron areements between the predicted distributions obtained from the proposed model the counterparts obtained from the Montearlo SPIE analysis of the benchmark circuits can be observed. Hence, proposed model has been qualitatively verified as hihly accurate for NMOS technoloy. For the quantitative verification, it can be seen by usin (19 that the resultin KS for x can be found as KS that for x f is KS which are both smaller than c his means that the proposed model can fit the variations in f obtained from the NMOS based benchmark circuits with 99% confidence. Furthermore, a ood areement can also be seen from the comparative plots for the cumulative distributions of /,Nom f /f,nom icted in Fi.5 Fi.6 respectively. At this EISSN: 66X 73 Issue 3, Volume 1, March 013

6 WSEAS RANSAIONS on IRUIS SYSEMS point, the proposed model verified as hihly accurate for NMOS technoloical basis. 3. PMOS based model verification For the model verification based on PMOS technoloy, the benchmark circuits similarly to those icted in Fi.1 Fi. can be used. hese circuits can be icted in Fi.7 Fi.8 which are adopted for verifyin the model s accuracy to predict the variation the accuracy for the prediction of such variation in f respectively. It should be mentioned here that, f their variations in this case can be determined from the benchmark circuits in the similar manner to those of the NMOS based verification. Furthermore, the similar verification methodoloy, parameters settin biasin condition applied to the previous NMOS based scenario can also be used for this PMOS based case. Also similarly to the NMOS based case, the raphical comparisons for the distributions of /,Nom f /f,nom respectively are icted in Fi.9 Fi.10 for the qualitative PMOS based verification which stron areements between those obtained from the proposed model the counterparts obtained from the Montearlo SPIE analysis of the benchmark circuits has also been observed. Hence, proposed model has been qualitatively verified as hihly accurate for PMOS technoloy. For the quantitative verification, it can be seen that KS for x can be found as KS that for x f is KS which are also smaller than c his means that the proposed model can fit the variations in f obtained from the PMOS based benchmark circuits with 99% confidence. Similarly to the NMOS based case, a ood areement can also be seen from the comparative plots for the cumulative distributions of /,Nom f /f,nom icted in Fi.11 Fi.1 respectively. At this point, the proposed model is verified as also hihly accurate for PMOS technoloy. Before leavin this section, it should be mentioned here that the resultin KS statistics for the model of variation in proposed in this research iven by KS for the NMOS based verification KS for the PMOS based one as mentioned above are lower than their correspondin counterparts formerly proposed in [9 iven by KS KS for NMOS based data fittin PMOS based one respectively. Accordin to [17, 18, this means that the refined model for the variation of proposed in this research is better fit to both NMOS based data PMOS based one due to its lower statistics so, it is more accurate than the previous model for the variation in proposed in [9. his is a result of the refinement in the model for variation. Fi.1. NMOS benchmark circuit for verifyin the model of I out V s Fi.. NMOS benchmark circuit for verifyin the model of f Probability I in V /,Nom(% Fi.3. NMOS based comparative distribution plots for /,Nom : he model based (line v.s. he benchmark circuit based (historam EISSN: 66X 7 Issue 3, Volume 1, March 013

7 WSEAS RANSAIONS on IRUIS SYSEMS Probability I in Y in f /f,nom(% Fi.. NMOS based comparative distribution plots for f /f,nom : he model based (line v.s. he benchmark circuit based (historam Fi.7. PMOS benchmark circuit for verifyin the model of I out umulative probability I in V s /,Nom(% Fi.5. NMOS based comparative cumulative distribution plots for /,Nom : he model based (line v.s.he benchmark circuit based (dots Fi.8. PMOS benchmark circuit for verifyin the model of f Probability umulative probability /,Nom(% f /f,nom(% Fi.6. NMOS based comparative cumulative distribution plots for f /f,nom : he model based (line v.s. he benchmark circuit based (dots Fi.9. PMOS based comparative distribution plots for /,Nom : he model based (line v.s. he benchmark circuit based (historam EISSN: 66X 75 Issue 3, Volume 1, March 013

8 WSEAS RANSAIONS on IRUIS SYSEMS Probability f /f,nom(% Fi.10. PMOS based comparative distribution plots for f /f,nom : he model based (line v.s. he benchmark circuit based (historam umulative probability Discussion In this section, some discussion reardin to the proposed model, some interestin applications of this model the results obtained from the model verification will be iven. Firstly, it can be obviously seen from the proposed models that many physical level parameters have been incorporated. Since these models describe the behavior of hih frequency performance variations these physical level parameters are ended on the fabrication technoloy of the MOSFE, it can be seen that such technoloy sinificantly affects the hih frequency performance variation. As mentioned above, f become romly varied due to f. Mathematically, they become rom variables. By usin (8, (9 the proposed models, their probability density functions can be simply derived as follows 3 3ε (1 λv ( V 7ε (1 λv ( V exp 8qINVNsubW α 1 s H ( c π INVqW L NsubW ( α(3 α ( s 3 ( c α H WL ( α (3 α, Nom /,Nom(% Fi.11. PMOS based comparative cumulative distribution plots for /,Nom : he model based (line v.s. he benchmark circuit based (dots umulative probability 8 3πε W L (3 α f( φ [3 µ q (1 λv ( V N sub W INV 96π εwl (3 α ( φ f exp{ [9µ q N W (1 λv α 3 α INV 0.5 ( V 7 s 3.5 sub s ( α H ( α H, Nom } α 3 α (5 where c denotes any sample values of f respectively. Furthermore, it can be observed from (15 (17 which have been derived from the proposed model that 3 σ (6 WL 1 σ (7 (1 λv 1 f 7 σ (8 WL f /f,nom(% Fi.1. PMOS based comparative cumulative distribution plots for f /f,nom : he model based (line v.s. he benchmark circuit based (dots σ f ( 1 λv (9 So, it can be seen that shrikin the transistor dimension can reduce the variation in with EISSN: 66X 76 Issue 3, Volume 1, March 013

9 WSEAS RANSAIONS on IRUIS SYSEMS increasin of variation in f as a penalty. Furthermore, application of the trendy low voltae desinin methodoloy which uses low supply voltae yiel the reduction in f variation with increasin in variation as a price to pay. So, these tradeoff issues must be taken into account in the desinin of MOSFE based hih frequency applications. For the in th exploration of the statistical relationship between the variation in the one in f, their correlation coefficient denoted by ρ must be investiated. Such, f ρ, f can be defined as E[( ( f f ρ, f (30 σ σ f where E[( ( f f is the covariance of both variations denoted by ov(δ, Δf which in turn can be defined as maximum oscillation frequency which is also an important hih frequency parameter can be precisely understood by usin the proposed model since such frequency is a function of f. his point can be demonstrated as follows, the maximum oscillation frequency denoted by f max is a critical hih frequency parameter when the ate parasitic resistance has been taken into account. Let such ate parasitic resistance be denoted by R, f max can be iven as a function of f as follows f f max (3 8π d R where d denotes the capacitance from drain to ate which is totally differrent from that from ate to drain denoted by d [1. he reason for this is that d d can be iven accordin to [1 by Q d d (35 V ov(, f E[( ( f ( δ ( δf f ( δ, δf dδ dδf, f f (31 d Q (36 V d ( δ, δf which denotes the joint, f probability density function of both variations can be found from the proposed model by solvin the followin equations δ f ( δ, δf dδf (, (3 f δ f f ( δ, δf dδ (, (33 By usin the proposed model alon with (1(17 which are obtained from this model for, σ, f, σ respectively, the manitude of such correlation coefficient has been found to be unity. his means that there exists a very stron statistical relationship between both variations. Secondly, apart from obtainin meaninful statistical parameters of f as formerly mentioned, the proposed model can serve as a basis for understin the behavior of the similar variations in other hih frequency parameters. For example, the behavior for such variation in the f Note that Q d denotes the drain chare where as V V d denote the ate drain voltaes respectively [1. It can be obviously seen from (35 (36 that d d. Since f is undeniably arised due to both rom dopant fluctuation process variation effects, the maximum oscillation frequency can be practically iven by f f f, Nom max ( f (37 8πd R where, fmax ( f denotes such practical maximum oscillation frequency which is undeniably fluctuated accordin to f induced by rom dopant fluctuation process variation effect while f,nom denotes the nominal value of f. So, the rom variation in the square of such f max denoted by can be simply found as f max f f max (38 8π d R By usin (17 which obtained from the proposed model, the variance of can be simply iven by f max EISSN: 66X 77 Issue 3, Volume 1, March 013

10 WSEAS RANSAIONS on IRUIS SYSEMS 9µ q INVNsubW α 3 α (1 λv [ f max 7 ( α 188π drε WL (3 α ( Vs H Var (39 Hence, it can be seen at this point that the behavior of rom variation in f max is now well understood via the above information on the variation of f max which has been obtained by usin the proposed model. For obtainin an in th insiht, the probability density function of fmax ( f denoted by ( φ max which completely describes the fluctuation in the maximum oscillation frequency analytically, can also be obtained by usin the proposed model. Let ø max denotes any sampled value of fmax ( f, by usin the principle of rom variable transformation with (37 as a mappin function, φ can be iven by ( φ max ( max π drφmax ( φmax f, Nom/8π dr 16 exp[ (0 σ ( σ /8π R f f where σ f can be found by usin (17 which can be obtained from the proposed model. At this point, it can be obviously seen that ( φ max which describes the fluctuation in the maximum oscillation frequency, can be simply obtained by usin the proposed model. Furthermore, the proposed model is applicable as the mathematical foundation for the modelin of the hih frequency performance mismatch of the lobally spaced MOSFEs for example, the mismatch in f of such spaced devices. Let these mismatches be denoted by D Df respectively. Since these transistors have similar local variations low correlation as they are lobally spaced, the distributions of D Df can be simply iven by usin the proposed model as follows d α 1 H 3 3ε (1 λv( Vs D( d πinvqw L NsubW ( α(3 α (1 α 7ε (1 λ ( V Vs VH d exp 3 16qINVNsubW WL ( α (3 α Df 3πε W L (3 α ( df [3µ q (1 λv ( V N sub W INV (1 λv α 3 α 7 INV NsubW 0.5 ( V s 3.5 s ( α H ( α H 8π ε WL (3 α df exp{ [9µ q } α 3 α ( where d df denote any sampled values of D Df respectively. Now, the discussion reardin to the results obtained from the model verification will be iven. It can be seen from the formerly shown Monte arlo SPIE analysis results that the variations in both ate capacitance transition frequency follows the Gaussian distribution as icted in Fi.3, Fi., Fi.9 Fi.10. Similarly to the results proposed in [9, the maximum percentae of variation in denoted by Max[ / obtained from the NMOS based benchmark circuit for verifyin the model of icted in Fi.1 is hiher than its counterparts obtained from the PMOS based one shown in Fi.7. hese maximum percentaes are similar to those proposed in [9 since they have been enerated from the ceteris paribus scenario. Furthermore, the similar relationship can be seen amon the maximum percentae of variation f denoted by Max[ f /f obtained from the NMOS based benchmark circuit for verifyin the model of f shown in Fi. its counterpart obtained from the PMOS based circuit icted in Fi.8. All of these maximum percentaes are concluded in able I. It can be observed from these percentaes that PMOS technoloy is more robust to the rom dopant fluctuation process variation effects than its NMOS counterpart due to its lower variations. At this point, the validity of the similar conclusion proposed in [9 has been emphasized. Furthermore, it has been observed that the KS test statistics for the PMOS based verification are smaller than their NMOS based counterparts as can be seen in able II. his means that the proposed model is better fit to the PMOS based data than the NMOS based one due to the smaller statisitics. So, it can be seen that this model can predict the variations for PMOS transistor with hiher accuracy than the similar prediction for the NMOS one. Finally, the one of the tentative further researches is to perform the similar modelin with specific orientation to the nanometer level MOS technoloy as the modelin for other parameters such as I d m of the nanoscale MOSFE which have been proposed in [, 3. EISSN: 66X 78 Issue 3, Volume 1, March 013

11 WSEAS RANSAIONS on IRUIS SYSEMS able 1. Max[ / Max[ f /f Max[ / (% Max[ f /f (% NMOS PMOS NMOS PMOS able. he comparison of KS test statistics From verification From f verification NMOS PMOS NMOS PMOS onclusion he probabilistic models of rom variations in MOSFE s hih frequency performance defined in terms of variations in f have been proposed. Both rom dopant fluctuation process variation effects which are the crucial causes of the MOSFE s hih frequency characteristic variations [7 have been taken into account. he state of the art short channel MOS technoloy has been focused. hese models have been derived by usin the alphapower law which has been proposed in [1 has also been previously adopted in [1, [ [9. he modern physical level variation model proposed in [11 has been adopted as the modelin basis instead of the classical one in [10. his ives the sinificant refinement in the model of variation in compared to the previous one proposed in [9. he proposed probabilistic models have been verified with both NMOS PMOS technoloies based on the IBM 90nm RF MOS process by usin the Montearlo SPIE simulations alon with the KS tests. hese models are very accurate since they can fit the Montearlo SPIE based data distributions with 99% confidence. Hence, the proposed models have been found to be the convenient mathematical tool for the statistical/variability aware analysis/desin of various MOSFE based hih frequency applications such as active inductor, active transformer, hih frequency filter, optical front en, clock/data recovery circuits many Bluetooth devices. Acknowledement he author would like to acknowlede Mahodol University, hail for online database service. References: [1 H. Masuda,. Kida S. Ohkawa, omprehensive matchin characterization of analo MOS circuits, IEIE rans. Fundamental, Vol. E9A, No., 009, pp [ R. Banchuin, Process induced rom variation models of nanoscale MOS performance: efficient tool for the nanoscale reime analo/mixed sinal MOS statistical/variability aware desin, Proceedins of the 011 International onference on Information Electronic Enineerin, 011, pp. 61 [3 R. Banchuin, omplete circuit level rom variation models of nanoscale MOS performance, International Journal on Information Electronic Enineerin, Vol. 1, No.1, 011, pp [ K. Haseawa, M. Aoki,. Yamawaki, S. anaka, Modelin transistor variation usin a power formula its application to sensitivity analysis on harmonic distortion in differential amplifier, Analo Interated ircuits Sinal Processin, 011. [5 A. R. Brown A. Asenov., apacitance fluctuations in bulk MOSFEs due to rom discrete dopants, Journal of omputer Electronic (008, Vol.7, 008, pp [6 Y. Li.H. Hwan, Hihfrequency characteristic fluctuations of nanomosfe circuit induced by rom dopants, IEEE ransaction on Microwave heory echniques, Vol. 56, No.1, 008, pp [7 M.H. Han, Y. Li.H. Hwan, he impact of hihfrequency characteristics induced by intrinsic parameter fluctuations in nano MOSFE device circuit, Microelectronics Reliability, Vol.50, 010, pp [8 H.S. Kim,. hun, J. Lim, K. Park, H. Oh, H.K. Kan, haracterization modelin of RFperformance (f fluctuation in MOSFEs, IEEE Electronic Devices Letters, Vol. 30, No.8, 009, pp [9 R. Banchuin, he novel analytical probabilistic model of rom variation in MOSFE s hih frequency performance, Proceedins of the IASED Asian onference on Modelin, Identification ontrol, 01, pp. to be published. [10 M. J. M. Pelrom, A.. J. Duinmaijer, A. P. G.Welbers, Matchin properties of MOS transistors, IEEE Journal of SolidState ircuits, Vol., No.5, 1989, pp EISSN: 66X 79 Issue 3, Volume 1, March 013

12 WSEAS RANSAIONS on IRUIS SYSEMS [11 K. akeuchi, A. Nishida. Hiramoto, Rom fluctuations in scaled MOS Devices, Proceedins of the 009 International onference on Simulation of Semiconductor Processes Devices, 009, pp [1. Sakurai A. R. Newton, Alphapower law MOSFE model its applications to MOS inverter delay other formulas, IEEE Journal on SolidState ircuits, Vol. 5, No., 1990, pp [13 H. Abebe, H. Morris, E. umberbatch V. yree, ompact ate capacitance model with polysilicon letion effect for MOS device, Journal of Semiconductor echnoloy Science, Vol 7, No.3, 007, pp [1 R.. Howe. G. Sodini, Microelectronics: An Interated Approach, Prentice Hall, [15 L. Weifen S. Linlin, Modelin of current mismatch induced by rom dopant fluctuation in nanomosfes, Journal of Semiconductor, Vol. 3, No.8, 011, pp [16 G. ijan,. uma A. Burmen, Modelin simulation of MOS transistor mismatch, Proceedins of the 6 th Eurosim onress on Modellin Simulation, 007, pp. 18 [17. Altiok B. Melamed, Simulation Modelin Analysis with ARENA, Academic Press, 007. [18 S.A. Kluman, H.H Panjer G.E. Willmot, Loss Models: From Data to Decisions, Wiley, 008 [19 H hoi, J.S. Goo,.Y. Oh, R.W. Dutton, A. Bayoumi, M. ao, P.V. Voorde, D. Vook.H. Diaz, MOS V characterization of ultrathin ate ide thickness ( nm, IEEE Electronic Devices Letters, Vol. 0, No.6, 1999, pp. 96. [0 R. Beach, A. Babakhani R. Strittmatter, ircuit simulation usin EP device models, Application notes: ircuit Simulation, Device Models, 011, pp. 11. [1 D.L. Pulfrey, Understin Modern ransistors Diodes, ambride University press, 010 EISSN: 66X 80 Issue 3, Volume 1, March 013