Analytical Prediction Formula of Random Variation in High Frequency Performance of Weak Inversion Scaled MOSFET

Transcription

1 Analytical Prediction ormula of Random Variation in Hih reuency Performance of Weak Inversion Scaled MOSET Rawid Banchuin * and Rounsan Chaisricharoen * Department of Computer Enineerin, am University, Bankok, Thailand. rawid.ban@siam.edu Tel: School of Information Technoloy, Mae ah Luan University, Chianrai, Thailand. rounsan.cha@mfu.ac.th Tel: Abstract This paper proposes an analytical prediction formula of probability distribution of random variation in hih freuency performance of weak inversion reion operated scaled MOSET where the manufacturin process induced physical defects of MOSET have been taken into account. urthermore, the correlation between process parameter included random variables which contribute such variation, and the effects of physical differences between N-type and P-type MOSET have also been considered. As the scaled MOSET is of interested, the up to dated formula of physical defects induced random variation in parameter of such scaled device has been used as the basis. The proposed formula can accurately predict the probability distribution of random variation in hih freuency performance of weak inversion scaled MOSET with a confidence level of 99%. So, it has been found to be an efficient alternative approach for the variability aware desin of various weak inversion scaled MOSET based sinal processin circuits and systems. I. INTRODUCTION Weak inversion reion operated scaled MOSET has been adopted in many low voltae/low power hih freuency circuits such as low power LNA [1], low power receiver [] and active inductor for low power oscillator [3] which are often included in sinal processin circuits and systems, because its cutoff freuency is extremely hih []. Of course, the performances of these circuits are stronly affected by hih freuency performance of the intrinsic MOSETs. Such performance can be determined by the total capacitance seen at the ate terminal of MOSET, C. Practically, the physical defects of MOSET induced by its manufacturin process e.. random dopant fluctuation, line ede rouhness and ate lenth random fluctuation etc., cause random variations in its electrical parameters [4] such as drain current, I d etc. This motivates many previous derivations of formulas for predictions of such variations e.. [5-10] which weak inversion MOSET has also been considered. Unfortunately, only I d and other DC parameters such as transconductance etc., have been concerned without any mentionin on random variation in C, ΔC which also exists and causes the undesired variations in performances of the MOSET based hih freuency circuits. By this motivation, a formula for the prediction of ΔC of weak inversion reion operated MOSET has been proposed in [11]. However, this previous work has nelected the effects of physical differences between N-type and P-type MOSET caused by their difference structures which can be icted as follows [1] i. 1 The physical structure of MOSET: N-type (left) and P-type (riht) [1] Obviously, N-type MOSET has p-doped silicon bulk with n-doped drain and source and vice versa for P-type device [1] which yields their physical differences e.. difference in types of chared carriers etc., which in turn have been inored by [11]. Moreover, [11] also nelected the correlation between the process parameter [4] involved random variables which contribute ΔC even thouh it has stron influence. inally, the scaled MOSET which is often cited nowadays has not been focused in [11]. Hence, the analytical prediction formula of probability distribution of ΔC of weak inversion reion operated scaled MOSET has been proposed in this paper by takin the physical defects of MOSET into account where the formerly nelected physical differences and correlation have also been considered. nce scaled MOSET has been now focused, the up to dated formula of physical defects induced variation in parameter of scaled device [13], has been used as the basis. Accordin to the verification by usin the 65nm level BSIM4 based benchmarks obtained from Monte-Carlo simulations and the oodness of fit test, it has been found that the proposed formula can accurately predict the probability distribution of ΔC of weak inversion scaled MOSET of both N-type and P-type with 99% confidence and sinificantly hiher accuracy than that of [11]. By usin this formula, various statistical parameters of ΔC of such device can be analytically predetermined without any necessity of APSIPA APSIPA 014

2 numerical analysis and the reduced computational effort variability aware analysis of the weak inversion scaled MOSET based low voltae/low power hih freuency circuits can be conveniently performed. So, it can be seen that the proposed formula has been found to be an efficient alternative approach for the variability aware desin of various weak inversion reion operated scaled MOSET based sinal processin circuits and systems. II. THE ORMULATION irstly, C of the weak inversion reion operated MOSET can be iven as follows [11] WL C ) exp s Vt ktt / 3 s Vt ) s Vt ) By the influences of manufacturin process induced physical defects of MOSET, random fluctuations in its parameters such as V t, W and L etc., exist where that in V t which is normally distributed has been found to be dominant for MOSET operated in weak inversion reion [14], and mainly causes C. By takin the physical differences between N-type and P-type MOSET into account, C can be iven under low voltae/low power condition which is typical for weak inversion scaled MOSET as follows C 3 WL kt T / V B V B ) ) exp V B ) where V stands for the fluctuated threshold voltae. urthermore, υ can be iven by C C 1 1 N ( d a N ( V V );for N type MOSET );for P type MOSET Note that N a, N d, V B, V and ø stand for acceptor dopin density, donor dopin density, flat band voltae, source to body voltae and ermi potential respectively. or the weak inversion scaled MOSET of our interest, its related voltaes such as V, V B and V etc., are extremely small so that V (1) () (3) VB (4) even at the ambient temperature which is a most typical operatin temperature of any circuit. As a result, C can be apprimately iven by where u v (5) C WL u VB ) (6) t C kt L v VB ) (7) t kt Obviously, u and v are random variables with process parameters for example, t and V B [4]. Their probability density functions can be determined based on the up to dated prediction formula of physical defect induced variation in V t of scaled MOSET [13] by usin (6) and (7) as follows 1.5kTC 1.5( ktcu) f ( u) exp (8) WL WL 1.5WkT 1.5( ktw v) ( v) exp (9) L L where δu and δv denote any sampled value of u and v respectively. Moreover, ξ can be iven by N ( a d N ( V V );for N type MOSET );for P type MOSET (10) or explorin the correlation between u and v, their correlation coefficient, ρ defined in (11) where E[ ] denotes the expectation operator, must be determined. E[( u E[ u])( v E[ v])] (11) 0.5 [(E[ u ] (E[ u]) )(E[ v ] (E[ v]) )] As a result, ρ can be found as WL 4 t t 9W L (1) nce ρ 0, u and v are correlated. At this point, the probability density function of C will be formulated. Accordin to (5), the Rohati s theorem [15] has been found to be applicable for solvin this problem. This theorem states that if Z = X Y where X and Y are random variables with k(x) and l(y) as their probability density functions, Z is also a random variable with its probability density function, p(z) iven by

3 p 1 ( z) h( x, y / x)( x ) dx (13) where h(x, y/x) denotes the joint probability density function of X and Y with y substituted by y/x which can be found by usin k(x) and l(y). By usin (8) and (9), the probability density function of C which is the proposed prediction formula of the distribution of C can be analytically formulated with the aid of [16] and keepin in mind that ρ 0 as follows ( kt) ( kt) C ( kt) C p( C ) exp K (14) 0 3. L (1 ) L (1 ) L 5 where δc and K 0 [ ] denote any sampled value of C and the 0 th order modified Bessel function of the second kind which can be iven in term of any variable, r by cos( rt) K0[ r] dt (15) 1/ ( t 1) It can be seen that the proposed formula which predicts that C employs a renowned normal product distribution [17], contains ξ which is differently iven for each type of MOSET, and ρ that determines the correlation between those process parameter based random variables which contribute C. This is not the case in the previous attempt [11] which inores such type different and correlation. So, the proposed formula is more complete and also more accurate as can be seen in the verification to be presented in the subseuent section. A very straihorward application of this formula is the analytical predetermination of statistical parameters of C by usin the conventional mathematical techniues without any numerical analysis. As simple examples, mean and variance of C can be predetermined as in (16) and (17). A more sophisticated parameter such as the characteristic function can be predetermined as in (18). C ( C ) C p( C ) dc C p( C ) dc C p( C ) dc L ( kt) (1 ) L 4 9C ( kt) 7 (16) (17) C ( ) p( C ) exp[ jc ] dc (18) L L 1 j 4 9C ( kt ) ( kt ) An interestin application of the proposed formula is bein the basis of the reduced computational effort variability aware analysis of any weak inversion scaled MOSET based low voltae/low power hih freuency circuit. or any circuit, the desired outcome of the variability aware analysis is simply the variance of ΔP, P where ΔP stands for variation in the performance metric of interested, P. Conventionally, P can be determined by usin the Monte-Carlo simulation. Unfortunately, this conventional methodoloy reuires very lare amount of computational effort [6] as it typically needs at least hundreds up to thousands runs for any circuit. So, a faster non-monte-carlo sensitivity based analysis has been considered for increasin the computational efficient. In summarize, the principle of such analysis is that if ΔP is influenced by the variation in parameter Q of certain kind of device, ΔQ and there are totally M devices of this kind within the circuit of interested, P can be numerically computed as M Q, i (19) P S i i1 where S i denotes the sensitivity of ΔP to ΔQ of any i th device of such kind and can be determined in many different computationally efficient ways such as heuristic approach and sensitivity analysis [6] which reuire much fewer number of runs than the conventional Monte-Carlo simulation. After obtainin the necessary sensitivities, can be immediately computed by usin such sensitivities and Q,i which is the variance of ΔQ of the i th device. However, a drawback of this methodoloy is that Q,i must be predetermined. or the weak inversion scaled MOSET based low voltae/low power hih freuency circuit, the predetermination of Q,i become convenient by applyin the proposed formula because ΔQ of such circuit is ΔC. So, Q,i = C,i which is C of any i th weak inversion scaled MOSET in such circuit, which in turn can be conveniently predetermined by usin the proposed formula without any numerical analysis as shown in (17). The non-monte-carlo sensitivity based variability aware analysis of weak inversion scaled MOSET based low voltae/low power hih freuency circuit with this formula will be practically demonstrated in the subseuent section. P III. THE VERIICATION As the scaled MOSET is focused, the proposed formula has been verified at 65nm level. or each type of MOSET, the probability distribution of ΔC evaluated by usin the formula has been comparatively plotted with respected to ΔC expressed in percentae of C aainst its benchmark which is the probability distribution of C obtained by usin the BSIM4 (SPICE LEVEL 54) based Monte-Carlo simulation. In order to demonstrate the accuracy improvement of the proposed formula over the previous one [11], the probability distribution of C evaluated by usin such previous formula,

4 has also been included in the comparative plot. It should be mentioned here that all necessary parameters have been provided by Predictive Technoloy Model (PTM) [18] where W/L of 50/3 has been chosen for MOSET of both N-type and P-type. milarly to [11], the number of for each Monte-Carlo simulation runs (n r ) has been chosen to be Moreover, for each type of MOSET, the oodness of fit test of the proposed formula has also been performed. The Kolmoorov-Smirnov test (KS test) has been chosen since it is a powerful test with simple statistic. The stratey of the KS test is to compare the KS test statistic, KS to the critical value, c where it can be stated that the formula under is accurate at certain confidence level if KS c [19, 0].nce such confidence level has been chosen to be 99% and n r = 1000, c has been found to be [0]. In order to show the accuracy improvement of the proposed formula over the previous one, its KS must be compared to that of the previous formula, KS p. or N-type MOSET, such comparative plot can be icted in i. where a very stron areement between the formula based distribution and its benchmark can be observed. Obviously, such very stron areement is stroner than that between the previous formula based distribution and the benchmark. rom the KS test, it has been found that KS = < c = This means that the proposed formula can accurately predict the probability distribution of ΔC of N-type MOSET with 99% confidence. It has also been found that KS = < KS p = which means that the proposed formula can predict the distribution of ΔC of N-type MOSET with hiher accuracy than that of the previous one. or P-type MOSET, the similar comparative plot can be icted in i. 3 where a very stron areement between the formula based distribution and its benchmark can also be observed. Obviously, such very stron areement is stroner than that between the previous formula based distribution and the benchmark as well. rom the KS test, it has been found that KS = < c = This means that the proposed formula can accurately predict the probability distribution of ΔC P-type MOSET with 99% confidence. It has also been found that KS = < KS p = This means that the proposed formula can predict the distribution of ΔC of P-type MOSET with hiher accuracy than the previous one does. By careful consideration of i. and i. 3, it can be seen that ΔC of the weak inversion scaled MOSET employs normal product distribution as predicted by the formula with nonzero mean as predetermined by (0). Such nonzero mean can be observed from the asymmetries about the vertical axes of benchmark distributions which can also be seen in the formula based ones. On the other hand, the previous formula based distributions are symmetrical about the vertical axes. This means that such previous formula predicts a zero mean for ΔC which is wron as it contravenes the result observed from the benchmarks. The reason of such failure is the aforementioned inorance of correlation. Moreover, ΔC of P- type MOSET is sinificantly lower than that of N-type MOSET. So, it can be stated that C of P-type MOSET is more robust to the physical defects than that of N-type device. Probability distribution ΔC (%) i. Comparative plot of the proposed formula based probability distribution of ΔC of weak inversion scaled N-type MOSET (line), the benchmark (dot) and the previous formula [11] based distribution (dashed) Probability distribution ΔC (%) i. 3 Comparative plot of the proposed formula based probability distribution of ΔC of weak inversion scaled P-type MOSET (line), the benchmark (dot) and the previous formula [11] based distribution (dashed) or the practical demonstration of the aforementioned non- Monte-Carlo sensitivity based variability aware analysis of the weak inversion scaled MOSET based low voltae/low power hih freuency circuit with the usae of the proposed formula, a weak inversion MOSET based Wu current-reuse active inductor proposed in [3] which can be icted in i.4, will be considered. i. 4 Weak inversion MOSET based active inductor [3]

5 As this active inductor is stand alone without any connection to other circuits in this scenario, the extra current source, J does not exist. Moreover, there are totally 3 MOSETs in this active inductor as the biasin and power supplyin are performed by the voltae sources. nce P of this circuit is the resultin inductance, L, ΔP is the variation in L L, ΔL and P is the variance of ΔL, which can be determined via the non-monte-carlo sensitivity based analysis by usin C of M1, M and M3. These variances can be predetermined without any numerical analysis by usin (1) which can be derived by applyin the proposed formula. Moreover, all sensitivities can be heuristically calculated by usin the finite different approach which uses n+1 runs for any n transistors circuit [6]. nce there are totally 3 MOSETs in this circuit, only 4 runs are sufficient for numerically calculatin the sensitivities. As a result, much faster computational speed than that of the Monte-Carlo simulation under the similar environment can be expected. In order to verify this proposed formula aided non-monte- Carlo analysis, ( L,nMC L obtained by usin this methodoloy ) and that from the conventional Monte-Carlo simulation with 1000 runs ( L,MC ) have been comparatively plotted aainst I d of M1, I d1 which is eual to those of M and M3 since J does not exist, as shown below. (%) L 8 6 determinin L,MC. IV. CONCLUSIONS The analytical prediction formula of probability distribution of ΔC of weak inversion reion operated scaled MOSET has been proposed in this paper by takin the physical defects of MOSET induced by the manufacturin process, type induced physical differences and correlation between process parameter based random variables which contributes such ΔC into account. The up to dated formula of physical defects induced variation in the parameter of scaled MOSET [13] has been adopted as the formulation basis as the scaled device has been focused. It has been found that the proposed formula can accurately predict the probability distribution of ΔC of weak inversion scaled MOSET of both types which exhibits nonzero mean normal product behavior, with 99% confidence and sinificantly hiher accuracy than that of [11]. Moreover, analytical predetermination of various statistical parameters of ΔC of such weak inversion scaled device without any numerical analysis and reduced computational effort variability aware analysis of weak inversion scaled MOSET based low voltae/low power hih freuency circuits can be conveniently performed by applyin this formula. Hence, the proposed formula has been found to be an efficient alternative approach for the variability aware desin of various weak inversion reion operated scaled MOSET based sinal processin circuits and systems. ACKNOWLEDGMENT The first author would like to acknowlede Mahidol University, Thailand for online database service μ μ μ μ μ 10-6 i. 5 Comparative plots of (line) and L,MC (dot) v.s. I d1,nmc L L,nMC L,MC I d1 (A) Noted that the unit of both and are percentae of L. It can be seen that a stron areement between and can be observed which L,nMC L,MC verifies the accuracy of this non-monte-carlo analysis. This stron areement also verifies the proposed formula in the practical application point of view. Moreover, based on the similar computational environment, much faster speed can be achieved by usin the non-monte-carlo sensitivity based analysis as expected since the speed of determinin L,nMC has been found to be times faster than that of REERENCES [1] B. G. Perumana, S. Chakraborty, C.-H. Lee, and J. Laskar, A fully monolithic 60-μW, 1-GHz subthreshold low noise amplifier, IEEE Microw. Wireless Compon. Lett., vol. 15, pp , June 005. [] B. G. Perumana, R. Mukhopadhyay, S. Chakraborty, C-H Lee and J. Laskar, A low power fully monolithic subthreshold CMOS receiver with interated LO eneration for.4 GHz wireless PAN application, IEEE J. Solid-State Circ., vol. 43, pp. 9-38, October 008 [3] Y. Zhou and. Yuan, Subthreshold CMOS active inductor with applications to low-power injection-locked oscillators for passive wireless Microsystems, Proc. 53 rd IEEE MWSCAS, pp , 010 [4] P. G. Drennan and C.C. McAndrew, Understandin MOSET mismatch for analo desin, IEEE J. Solid-State Circ., vol. 38, pp , March 003. [5] K. Papatanasiou, A desiner s approach to device mismatch: Theory, modelin, simulation techniues, scriptin, applications and examples, Analo Inter. Circ.. Process., vol. 48, pp , 006. [6] G. Cijan, T. Tuma and A. Burmen, Modelin and simulation of MOS transistor mismatch, Proc. 6 th EUROSIM, pp. 1-8, 007.

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