007 007 Explc Shor Channel Compac Model of Independen Double Gae MOSFET M. Reyboz, O. Rozeau, T. Poroux, P. Marn, M. Caveler and J. Jomaah
OUTLINE I Inroducon II Physcal & Mahemacal Dffcules of Compac Modelng of IDG MOSFET III Threshold olage-based Compac Model 007 I Inroducon n erlog-a and Resuls Concluson & Prospecs
INTRODUCTION General Conex NEW DEICES classcal CMOS echnologes + forecas of he ITRS Roadmap =? New devces appear: GAA, FnFET, SON & planar DG MOSFET Bulk CMOS FD SOI CMOS New devces 90 nm 65 nm 45 nm nm 004 007 00 06 (producon) 007 Excellen elecrosac conrol hanks o a beer couplng beween gae & channel (manly for SG MOS) beer I on /I off rao Almos wce more I on curren han a classcal MOSFET wh one gae More flexbly hanks he second gae whch can be ndependenly drven for an IDG MOSFET Undoped flm no more dopng flucuaon n he channel MODEL To ake advanage of hs new devce, desgners need a model, parcularly a compac model o desgn new crcus 3
INTRODUCTION Dfferen Knds of DG MOSFET T s : 0nm T ox & T ox :.nm L: 0nm o µm 007 Source Back gae Fron gae Dran IDG MOSFET M.ne e al, SSDM 004 Channel ADG MOSFET: dfferen gae oxde hcknesses and/or dfferen gae funcons -T ox 0 Source T s T s +T ox X g Fron Gae Back Gae IDG MOSFET: gaes are ndependenly drven L g Dran Y 4
OUTLINE I Inroducon II Physcal & Mahemacal Dffcules of Compac Modelng of IDG MOSFET 007 III Threshold olage-based Compac Model I Inroducon n erlog-a and Resuls Concluson & Prospecs 5
PHYSICAL & MATHEMATICAL DIFFICULTIES Basc Equaons of an IDG MOSFET I ds = Gauss heorem Q nv = ε ( E E ) s s s Dran curren W µ L s d Q nv dφ mref E Boundary condons c ox E s = ( g ψ s ) ε s c ox s = ( g ψ s ε s ) 007 E s Posson equaon & s frs negraon d ψ q n =. dx ε. q. u. n ψ s φ mref E = s exp exp ε s u s ψ s u φ mref To physcally derve I ds, surface poenals ψ s &ψ s should be known 6
PHYSICAL & MATHEMATICAL DIFFICULTIES Mahemacal Dffcules ASYMETRICAL or INDEPENDENT GATE DEICES no always a mnmum of poenal n he slcon flm: CASES should be dsngushed Ψ s Ψ s Ψ s Ψ s Ψs Ψ Ψ s s E ferm Ψ s Band Dagrams 007 ψ mn ψ mn Frs dffculy Two cases should be defned Second dffculy There s no exac soluon of ψ s & ψ s Numercal resoluon n he model code 4 unknown parameers: ψ ssource, ψ sdran & ψ ssource, ψ sdran To make physcal assumpons Smplfcaons Posson equaon 7
OUTLINE I Inroducon II Physcal & Mahemacal Dffcules of Compac Modelng of IDG MOSFET 007 III Threshold olage-based Compac Model I Inroducon n erlog-a and Resuls Concluson & Prospecs 8
THRESHOLD OLTAGE BASED COMPACT MODEL Long Channel Model PRINCIPLE Q nv = Q nv + Q nv Q WI nv Q ( ) g h mref x = q. N. u.exp n. u SI ( x) = C. n. φ ( x) nv ox n. φ ( ) g h mref q s he elecronc charge, N nrnsc concenraon of carrers, u s he hermal volage Φ mref he quas level of Ferm of elecrons n he channel, g are he fron and he back volages and C ox he fron and he back gae oxdes. = or 007 We know how o lnk he followng equaons Now, n (couplng facor) and h (hreshold volage) should be expressed analycally and explcly 9
THRESHOLD OLTAGE BASED COMPACT MODEL Long Channel Model / n expressons, he couplng facors g Physcal assumpon n weak nverson a boh nerfaces: Transverse elecrcal feld s unform because he channel s undoped ψ s C ox C s ψ = C C s ox s ox s g g CsCox + CoxCox + CsC ox CsCox + CoxCox + CsCox + C C 007 ψ s C ox g ψ s = n / h expressons, he hreshold volages g + n n g Boh nerfaces are n weak nverson, nverson charges Q nv & Q nv are analycally expressed as: example These expressons are denfed wh: Q Q S S mref S nv = q. N u exp u C eq g WI nv T ψ φ C anh C C n. φ ( ) g h mref x = q. N. u.exp n. u S eq g. u. u g g 0
THRESHOLD OLTAGE BASED COMPACT MODEL Long Channel Model 007 / h expressons, he hreshold volages h h = n u = n nc oxu ln q. N TS ncoxu u ln q. NTS ( n ) 3/ Srong nverson descrpon ( n ) g g n u n 0.65 0.60 Ceq g anh CS u ln Ceq g g CS u g Ceq g anh CS u u ln Ceq g g CS u g T ox = T ox =.nm, T s =5nm We enlarge when boh nerfaces are a hreshold h,lm s defned h () 0.55 0.50 0.45 g =. hlm : back nerface s srongly nvered 0.40 0.0 0. 0.4 0.6 0.8.0. g ()
THRESHOLD OLTAGE BASED COMPACT MODEL Long Channel Model 3/ Srong nverson descrpon We add a correcon facor o model he non oal screenng of he channel by he nverson charge Q SI nv ( x) = C ( n. φ ( x) )( ε ( x) ) ox g h mref ε represens he dependence of a srong nvered nerface versus s gae volage Q nv expressons, boundary condons & he fac ha he weak nverson charge can be negleced, we ge an explc ε 007 W 4/ Dran curren expressons Ids = µ Qnv ( φmref ) L ds 0. dφ mref I ds = I ds + I ds We have a unfed expresson of he dran curren
007 A THRESHOLD OLTAGE BASED COMPACT MODEL Shor Channel Model evanescen analyss mode s used o ge expressons of he elecrcal poenals n he slcon flm ψ ( x, y) = ψ ( x) + ψ (x, D Correcon facor: ald n weak nverson! ψ (x, y) = y) π b sh λ ( L y) sh + c π L λ To explcly calculae he dran curren, we assume ha for he correcon facor, he curren s domnan n x max & y mn * sh Explc expressons π λ y cos π λ * x * I ds W = µ q n L X. Lang and Y. Taur, A -D analycal soluon for SCEs n DG MOSFETs, IEEE TED, vol.5, n 8, 004. T s u ψ s ψ s exp exp u u ( ψ ψ ) s s ψ exp ( x, y ) max u mn exp u ds 3
THRESHOLD OLTAGE BASED COMPACT MODEL Shor Channel Model 007 I ds W = µ L Compac Modelng The dran curren s wren as he sum of a fron & a back dran curren We wan o wre as: By denfcaon, we ge explc expressons of h,sce and of n,sce n c ox Fron hreshold volage () I u ds 0.7 0.68 0.66 0.64 0.6 0.6 0.58 0.56 0.54 0.5 ( e ds u W = µ L ) e n c T ox = T ox = nm T s = 0nm ds = 5m aldy lm of he model g n u ox u h ( e e ψ ( x u u ds max, g = 0 y mn ) e ) g h, n, sce u sce IDG Model Alas smulaon 0.5 0 30 50 70 90 0 30 L (nm) 4
OUTLINE I Inroducon II Physcal & Mahemacal Dffcules of Compac Modelng of IDG MOSFET 007 III Threshold olage-based Compac Model I Inroducon n erlog-a and Resuls Concluson & Prospecs 5
INTRODUCTION IN ERILOG-A Unfcaon of he dfferen operang modes on he dran curren off =ε ( g - h ) 007 Unfcaon n g Weak Inverson g h = g un exp un off g = u n g h exp un ln + g + exp un Srong Inverson g = g h off off h Unfcaon n ds ( ) = + + dseff, dsa, dsa, ds δ dsa, ds δ 4δdsa, 6
007 I ds (A) 0-04 0-06 0-08 0-0 0-0 -4 RESULTS Shor Channel Model ds = 5 m 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.. g = 0 g from 0.0 o. by sep of 0. g () Alas smulaon IDG Model.5x0-4.0x0-4.5x0-4.0x0-4 5.0x0-5 I ds (A) I ds (A) 5x0-3 4x0-3 3x0-3 x0-3 x0-3 T ox =T ox =.nm Ts=0nm L=30nm W=µm µ=consan Alas smulaon IDG Model g =. g from 0.0 o. by sep of 0. I ds (A) 5x0-3 4x0-3 3x0-3 Alas smulaon IDG Model g from 0.0 o. by sep of 0. 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.. ds () x0-3 x0-3 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.. ds () 7
OUTLINE I Inroducon II Physcal & Mahemacal Dffcules of Compac Modelng of IDG MOSFET 007 III Threshold olage-based Compac Model I Inroducon n erlog-a and Resuls Concluson & Prospecs 8
CONCLUSION & PROSPECTS Our compac model akes no accoun ndependen gaes Threshold volage based compac model Explc shor channel effecs Included effecs: R seres, GIDL, gae leakage, mobly degradaon Resuls correspond very closely o numercal smulaons 007 Effecs o add Quanum Effecs Ballsc Transpor 9
007 0