1. Introduction. Rawid Banchuin

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1 011 Inernaional Conerence on Inormaion and Elecronics Engineering IPCSIT vol.6 (011 (011 IACSIT Press, Singapore Process Induced Random Variaion Models o Nanoscale MOS Perormance: Eicien ool or he nanoscale regime analog/mixed signal CMOS saisical/variabiliy aware design Rawid Banchuin Deparmen o Compuer Engineering, Siam Universiy Absrac. In his research, he novel models o random variaion in I which is a key parameer o any MOS ransisor, have been proposed in his research as he probabiliy densiy uncions. Boh riode and uraion regions have been explored. Unlike he previous researches, his research has been perormed based upon he up o daed nanoscale regime MOS equaions. The proposed models or boh regions have been veriied a 65 nm echnology by using he Mone Carlo simulaion and he Kolmogorov-Smirno goodness o i es (KS-es. These models are very accurae since hey can i he Mone Carlo based disribuion wih 99% conidence. Hence, he proposed models have been ound o be eicien or he saisical/variabiliy aware design o various CMOS analog/mixed signal circuis and sysems in he nanoscale regime. Keywor: Nanoscale, CMOS, analog, mixed signal, saisical design, variabiliy aware design 1. Inroducion Process induced random variaions in MOS ransisor level parameers play a very imporan role in he saisical/variabiliy aware design o CMOS analog/mixed signal circuis and sysems. These variaions produce he random mismaches in MOS perormances which are releced as random variaions in circui level parameers or example, I, gm, gmb and r ec. Obviously, I has been ound o be he key parameer since i is he basis or he derivaions o he ohers. For he MOS ransisor level parameers, one o he major variaions is ha in he hreshold volage which is radiionally modeled as a normally disribued random variable wih zero mean where as many models have been proposed or he variance or example [1], [], [3] and [4] ec. For he analysis/design simpliciy, random variaions in MOS circui level parameers have been explored and modeled as proposed in many previous researches or example [5] and [6] ec. In [6], he percenage o variaions in I and relaed circui level parameers have been modeled as he uncion o he percenage o variaion in hreshold volage. However, he model derivaion basis adoped in [6] rely on he simple sensiiviy analysis which only he sandard deviaions o he hreshold volage and he modeled parameers have been menioned, where as he corresponding acual disribuion uncions have no been discussed. Furhermore, hese previous researches have been perormed based upon he convenional MOS equaions which are invalid in he nanoscale regime. Hence, he novel models o random variaion in I which is a key parameer as menioned above have been proposed in his research as he complee probabiliy densiy uncions no he sandard deviaions. Obviously, means, variance, momens and oher saisical parameers can be deermined by using he proposed model. Boh riode and uraion regions have been explored. Unlike he previous researches, his research has been perormed based upon he up o daed nanoscale regime MOS equaions. The proposed models or boh regions have been veriied a 65 nm echnology by using he Mone Carlo simulaion and he Kolmogorov-Smirno goodness o i es (KS-es. These models are very accurae since hey can i he Mone Carlo based disribuion wih 99% conidence according o he KS-es resuls. Hence, he proposed 6

2 models have been ound o be eicience or he saisical/variabiliy aware design o various CMOS analog/mixed signal circuis and sysems in he nanoscale regime.. The Proposed Models In his secion, he proposed models or boh riode and uraion regions will be discussed respecively. Beore proceed urher, i should be menioned here ha he Pelgrom s model or he variaion in hreshold volage which has been proposed in [1] is adoped or his research since i is he mos oen cied. A he nanoscale regime, he devices are closely spaced. Hence, he probabiliy densiy uncion o he random hreshold volage variaion ( V can be given by ( δv 1 WL WLδV exp π A A V V ΔV where A V, δv, W and L denoe he proporional consan o random hreshold volage variaion, any sampled value o V, channel widh and channel lengh respecively. In he upcoming subsecion, he proposed model or he riode region will be discussed..1. Triode region model In he ideal siuaion which process induced random variaion can be negleced, I o he nanoscale ransisor operaed in he riode region o operaion can be given by (1 ( V V ( ideal WC V + μ Lv I v ( where C, µ and v denoe gae ide capaciance, device mobiliy and uraion velociy respecively. O course, he above I (ideal is a deerminisic variable. Including he eec o V which is a random variable as saed earlier, I becomes a random variable which can be given as ollows I [ V ( V + ΔV ] V WC V + μ Lv v (3 So, he resuling random variaion in I which denoed by can be simply deermined rom ( and (3 as a simple linear uncion o V as ollows WCVv V + μ Lv ΔV (4 Obviously, is a random variable and is probabiliy densiy uncions which is our proposed model or he riode region o operaion can be simply given by ( V + μ Lv WL 4 π A WC V v V ( V + μ Lv WLδI exp 16AVWCVv (5 A his poin, mean o DS can be evaluaed as ollows δi dδi 0 (6 7

3 And he corresponding variance can be given by σ 4AVWCV ( V + μ Lv μ v WL dδi (7 In he nex subsecion, he proposed model or he uraion region o operaion will be inroduced... Sauraion region model In he ideal siuaion, I o he nanoscale ransisor operaes in he uraion region is a deerminisic variable which can be given by I V v (8 ( ideal WC ( V Including he eec o V, I becomes a random variable and can be given as ollows I WC [ V ( V + ΔV ] v (9 So, he resuling Δ I can be simply deermined rom (8 and (9 as ollows Δ I WC v ΔV (10 As a uncion o V, is a random variable and is probabiliy densiy uncions which is our proposed model or he uraion region o operaion can be simply given by WL π ( WC v A V WLδI exp ( WCv AV (11 So, mean o can be evaluaed as ollows δi dδi 0 (1 And he corresponding variance can be given by σ ( I δ μ ( WCv AV WL dδi (13 Beore leaving his secion, since he KS-es which relies on he cumulaive disribuion uncion has been adoped in his research, i is worhy o evaluae he proposed models in heir cumulaive disribuion uncion orms. For he riode region, he proposed model in is cumulaive disribuion uncion orm ( F δi can be ound by using (5 as ollows Δ I ( 8

4 F δi ( u du 1 ( V + μ Lv WLδI 1 + er A V 8WCVv (14 For he uraion region, F δi can be obained by using (11 as ollows ( F δi ( u du er WLδI A WC V DS v (15 where er ( x denoes he error uncion o any arbirary variable, x which can be mahemaically deined x as er ( x exp( u du. π 0 3. The Veriicaion The veriicaions o he proposed models have been perormed in boh qualiaive and quaniaive aspecs. In he qualiaive aspecs, he esimaed probabiliy densiy uncions o obained rom boh models have been graphically compared o heir acual counerpars obained rom he Mone Carlo simulaion o es circuis. On he oher hand or he quaniaive aspecs, he KS-ess have been perormed by using he similar daa o ha adoped in he qualiaive veriicaion. According o [7], KS-es can be perormed by he comparison o he K-S es saisic (KS and he criical value (c. Any model can i is arge daa se i and only i is KS is no larger han is c [7]. According o [7], KS can be deined or his research as KS max{ F F } (16 δi acual mod el where FΔ I (δi and F (δi denoe he cumulaive disribuion uncion orms o he esimaed mod el acual probabiliy densiy uncions obained rom he proposed models and he acual cumulaive disribuion uncions obained rom he es circuis respecively. On he oher hand, as he conidence level o he es is 99% or α 0.01 in he oher wor, c can be given by [8] 1.63 c n (17 where n denoes he number o samples. Beore proceed o he veriicaions, i should be menioned here ha he veriicaions o he proposed models or boh regions have been perormed based upon he CMOS echnology a 65 nm level by using n 150 which yiel c In he upcoming subsecion, he veriicaion o he model or he riode region will be discussed Triode region model veriicaion For he veriicaion o he riode region model, a single MOS ransisor acive resisor a 65 nm echnology has been adoped as he es circui. This circui is depiced in Fig.1 where R(V denoes is resuling resisance which can be elcronically conrolled via V. O course, R(V can be simply given by 9

5 R( V Lv μ WC ( V V (18 The obained graphical comparison o he probabiliy densiy uncions which is he qualiaive model veriicaion as saed above, is depiced in Fig.. I can be seen ha a srong agreemen beween he esimaed and acual probabiliy densiy uncions can be observed. Hence, proposed riode region model has been graphically veriied as highly accurae. Fig.1: Tes circui or riode region model veriicaion: a single MOS acive resisor PDF (δi DS /I DS 100% Fig.: Triode region PDF Comparison: esimaed PDF rom he model (line, acual PDF rom he es circui (. For he quaniaive veriicaion, i can be seen ha he resuling KS can be ound rom (16 as KS which is smaller han c This means ha he proposed riode region model can i he obained rom he es circui wih 99% conidence. A his poin, he riode region model is also quaniaively veriied as highly accurae. 3.. Sauraion region model veriicaion On he oher hand, or he case o he uraion region model, a diode conneced 65 nm MOS ransisor has been chosen as he es circui and can be depiced in Fig.3. The similar graphical comparison o he probabiliy densiy uncions is depiced in Fig. 4. In his case, a srong agreemen beween he esimaed and acual probabiliy densiy uncions can also be observed. Hence, proposed uraion region model has been qualiaively veriied as highly accurae. For he veriicaion in he quaniaive aspec, he resuling KS can be ound as KS which is smaller han c This means ha he proposed uraion region model can also i he obained rom he es circui wih 99% conidence. Hence, he uraion region model has also been quaniaively veriied as highly accurae. 10

6 Fig.3: Tes circui or uraion region model veriicaion: a diode conneced ransisor PDF (δi DS /I DS 100% Fig.4: Sauraion region PDF comparison: esimaed PDF rom he model (line, acual PDF rom he es circui ( 4. Discussion Beore proceed o he conclusion, some discussions are worhy o be menioned. Since he volage condiions or riode and uraion region are V < V V and V V V respecively, (5 and (11 can be combined ino a single probabiliy densiy uncion as ollows ( V + μ Lv WL 4 π AVWCVv ( V + μ Lv WLδI exp ; V V 16AVWCVv WL π ( WCv AV WLδI exp ; V V V ( WCv AV (19 Furhermore, a ew examples o urher sudies are also worhy o be discussed. These examples are rying o apply he oher models o variaion in ΔV such as hose proposed in [], [3] and [4] in order o obain he alernaive resuls, inclusion o he oher process induced random variaions apar rom ΔV such as he random variaion in µ (Δµ ec., in order o obain more precise models, perorming he similar sudy o he weak inversion operaed ransisor as his sudy has been perormed by assuming he srong inversion operaion and derivaion o he similar models or he random variaions in he oher relaed parameers or example, g m, g mb and r ec., similarly o [6]. 5. Conclusion 11

7 The probabilisic models o random variaion in I which is a key parameer o any MOS ransisor, or boh riode and uraion regions have been proposed in his research by using he up o daed nanoscale regime MOS equaions as bases. The proposed models have been veriied a 65 nm echnology by using he Mone Carlo simulaion and he KS-es. The chosen es circuis are a single ransisor acive resisor and a diode conneced ransisor or he riode and uraion region model respecively. These models are very accurae since hey can i he random variaion in I obained rom he es circuis wih 99% conidence. Hence, he proposed models are obviously eicien or he saisical/variabiliy aware design o various nanoscale CMOS analog/mixed signal circuis and sysems. 6. Acknowledgemen The auhor would like o acknowledge Mahodol Universiy, Thailand or online daabase service. 7. Reerrences [1] M. J. M. Pelgrom, A. C. J. Duinmaijer, A. P. G.Welbers. Maching properies o MOS ransisors. IEEE J. Solid- Sae Circuis. 1989, 4(5: [] K.Takeuchi e al. Channel Engineering or he Reducion o Random-Volage-Induced Threshold Volage Variaion. Proc. IEEE Inernaional Elecron Device Meeing. IEEE Press. 1997, pp [3] P.A. Solk, F.P. Widdershoven, D.B.M Klaassen. Modeling saisical dopan lucuaions in MOS ransisors. IEEE Trans. Elecron. Devices. 1998, 45(9: [4] S.K. Saha. Modeling process variabiliy in Scaled CMOS echnology. IEEE Design & Tes o Compuers. 010, 7(: [5] P.R. Kinge. Device mismach and radeos in he design o analog circuis. IEEE J. Solid-Sae Circuis. 005, 40(6: [6] H. Masuda, T. Kida, S. Ohkawa. Comprehensive maching characerizaion o analog CMOS circuis,. IEICE Trans. Fundamenal. 009, E9-A(4: [7] T. Aliok and B. Melamed. Simulaion Modeling and Analysis wih ARENA. Academic Press, 007. [8] S.A. Klugman, H.H Panjer, G.E. Willmo. Loss Models: From Daa o Decisions. John Wiley and Sons, 008 1