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Physics and Design of Nonclassical Nanoscale CMOS Devices with Ultra-Thin Bodies

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Physics and Design of Nonclassical Nanoscale CMOS Devices with Ultra-Thin Bodies
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TRIVEDI, VISHAL P.
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2008

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Ballistics ( jstor )
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Design engineering ( jstor )
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Electrons ( jstor )
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University of Florida
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Copyright Vishal P. Trivedi. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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PHYSICS AND DESIGN OF NONCLASSICAL NANOSCALE CMOS DEVICES WITH ULTRA-THIN BODIES By VISHAL P. TRIVEDI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Vishal P. Trivedi

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-Tomy parents

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iv ACKNOWLEDGMENTS Iwishtoexpressmyheart-feltgratitudetoProfessorJerryG.Fossum,my advisor,forhisinvaluableguidance,constantencouragement,andgeneroussupport, combinedwithcountlesslymanyinsightfuldiscussions,throughoutthecourseofthis work.Ialsoappreciatehiscriticalreadingofthisdissertation.Itrulyadmirehis patienceandhiscommitmenttohisworkand,especially,tohisstudents.Itwasa greathonor,andapleasure,tohaveworkedwithhim.Throughoutmylife,Iwill benefit from this exceptional experience, and the knowledge he has given me. IwouldalsoliketothankProfessorsGijsBosman,ScottThompson,and KevinIngersentforparticipatinginmysupervisorycommitteeandfortheirinterest inthiswork.IwouldespeciallyliketothankProfessorIngersentforthesystematic wayhehasintroducedtomeQuantumMechanics,anessentialingredientof nanoelectronics,anditsapplications/formulation(intermsoftheGreen’sFunctions) in the field of modern condensed matter physics. IalsoappreciateProfessorF.GmizofUniversityofGranada(in Granada,Spain)forperformingMonteCarlosimulationsneededforpartofthis dissertation.IamgratefultoFreescaleSemiconductor,SamsungElectronics,theU. S.NationalScienceFoundation,theSemiconductorResearchCorporation,andthe UniversityofFloridafortheirfinancialsupport.IthankErlindaLanefortakingcare of all of my paperwork associated with trips to numerous conferences.

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v Ihavebeenfortunatetohaveinteractedwithmyfellowgraduatestudents, Mr.MurshedM.Chowdhury,Mr.Seung-HwanKim,andMr.WeiminZhang,aswell asformerstudentsDr.LixinGe,Mr.Guan-ShyanLin,andDr.Ji-WoonYang.Ihave learnedmuchfrommanystimulatingdiscussionswiththem,especiallyMurshed,and I greatly appreciate their friendship over the past few years. Iamgratefultomyimmediatefamily,includingmygrandparents,uncles, aunts,andcousins,fortheirsupport.Iwouldespeciallyliketoexpressmygenuine appreciationtomyuncle,BipinAdhyaru,andmyaunt,IlaAdhyaru,fortheircare throughoutmycollegeyears.Ialsoextendspecialthankstomycousins,Bhavinand Manan,whowerealsomyroommatesforcoupleofyears,fortheirwarmcompany, and for putting up with my good, bad, and the ugly. Lastly,butmostimportantly,Iwouldliketoacknowledgemyparents,my brother,andmysister-in-law.Ibelievethatconversationswithmybrother,Vaibhav, onsolid-stateelectronicsinitiatedmyinterestinthefield.Hehasbeenasourceof inspirationforme,andIamgladtohavehimbymyside.Iamdeeplythankfultomy sister-in-law,Krushangi,forherconstantlove,care,andsupport.Finally,nowords candescribemyfeelingsforeverythingthatmyfather,PareexitTrivedi,andmy mother,HansaTrivedi,haveprovided,andtheendlesssacrificestheyhavemadefor my education and well-being. This work is dedicated to them.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi KEY TO ABBREVIATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi i CHAPTER 1INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1Classical CMOS Devices and Their Scaling Limit. . . . . . . . . . . . . . . . . . . . . .1 1.2Nonclassical CMOS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 1.3Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 2SCALING FULLY DEPLETED SOI CMOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 2.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 2.22-D Effect in the BOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 2.32-D Effects in the SOI Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 2.3.1Current Flow in Nonclassical MOSFETs . . . . . . . . . . . . . . . . . . . .23 2.3.2Subthreshold Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 2.3.3Drain-Induced Barrier Lowering. . . . . . . . . . . . . . . . . . . . . . . . . . .28 2.4Insights from the 2-D Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 2.5Threshold Control via Channel Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 2.6Threshold Control via Gate Work Function(s). . . . . . . . . . . . . . . . . . . . . . . .37 2.6.1Ultra-Thin Bodies for SCE Control. . . . . . . . . . . . . . . . . . . . . . . . .38 2.6.2Quantum-Mechanical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 2.6.3HP and LP CMOS Designs with a Midgap Gate. . . . . . . . . . . . . . .41 2.6.4UFDG/Spice3-Based Performance Projections. . . . . . . . . . . . . . . .48 2.6.5HP and LP CMOS Designs with Dual Metal Gates . . . . . . . . . . . .50 2.7Scalability of FD/SOI CMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

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vii 2.8Scalability of Other DG CMOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 2.8.1Symmetrical DG MOSFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 2.8.2Asymmetrical DG MOSFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 2.8.3DG MOSFETs with Independently Biased Gates. . . . . . . . . . . . . .56 2.9Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 3ON THE "NONCLASSICAL" PHYSICS OF UNDOPED DG MOSFETS . . . . . . .61 3.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.2Subthreshold Analysis: Generalized Charge Coupling and Threshold Voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 3.2.1The Depletion Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 3.2.2Classical Theory of Charge Coupling and Threshold Voltage . . . .65 3.2.3Generalized Charge Coupling and Threshold Voltage . . . . . . . . . .66 3.3Random Dopant Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 3.4Subthreshold Quantum-Mechanical Effects. . . . . . . . . . . . . . . . . . . . . . . . . .76 3.4.1Physical Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77 3.4.2Model Verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 3.4.3Model Applications and Physical Insights. . . . . . . . . . . . . . . . . . . .83 3.5Short-Channel Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 3.6Bulk-Inversion Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 3.6.1Inversion-Layer Capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 3.6.2Effective Carrier Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 3.6.3Velocity Saturation and Overshoot . . . . . . . . . . . . . . . . . . . . . . . .101 3.7Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 4EFFECTIVE CHANNEL LENGTH AND SOURCE/DRAIN-EXTENSION ENGINEERING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 4.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 4.2Effective Channel Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 4.2.1Bias Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 4.2.2Source/Drain-Doping Profile Dependence . . . . . . . . . . . . . . . . . .111 4.3Source/Drain Series Resistance and Optimal Design Insights. . . . . . . . . . .120 4.4Source/Drain-Extension Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 4.5Model for Relating SDE design and Effective Channel Length. . . . . . . . . .126 4.5.1Simplified Source/Drain-Doping Profile. . . . . . . . . . . . . . . . . . . .126 4.5.2What is Leff(weak)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 4.5.3Model Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 4.5.4Model Verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 4.6Source/Drain-Doping Profile Requirements. . . . . . . . . . . . . . . . . . . . . . . . .136 5QM-BASED MODELING OF EFFECTIVE CARRIER MOBILITY . . . . . . . . . .140 5.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 5.2Coulomb-Limited Mobility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141

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viii 5.3Phonon-Limited Mobility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 5.3.1Scattering Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 5.3.2Momentum Relaxation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145 5.3.3Monte Carlo Simulations of SDG and ADG nMOSFETs. . . . . . .147 5.3.4Physical Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 5.3.5Physics-Based Compact Model. . . . . . . . . . . . . . . . . . . . . . . . . . .160 5.4Surface-Roughness-Limited Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168 5.4.1Scattering Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168 5.4.2Monte Carlo Simulations and Physical Insights . . . . . . . . . . . . . .169 5.4.3Physics-Based Compact Model. . . . . . . . . . . . . . . . . . . . . . . . . . .173 5.5Compact Model for Effective Carrier Mobility . . . . . . . . . . . . . . . . . . . . . .181 5.6Temperature Dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182 5.7Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184 5.8Mobility in {110}-Si Surface MOSFETs. . . . . . . . . . . . . . . . . . . . . . . . . . .188 5.9UFDG Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 5.10Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 5.11Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196 6THERMAL INJECTION-, OR BALLISTIC-LIMIT CURRENT. . . . . . . . . . . . . .198 6.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198 6.2Ballistic-Limit Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199 6.2.1Current and Inversion-Charge Density . . . . . . . . . . . . . . . . . . . . .199 6.2.2Drain-Bias Dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 6.2.3Simple Alternative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 6.2.4Conductivity Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 6.2.5Fermi-Dirac Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211 6.3UFDG-Based Analysis and Physical Insights. . . . . . . . . . . . . . . . . . . . . . . .211 6.3.1Effective Channel Length and UTB Thickness Dependences. . . .211 6.3.2Ballistic-Limit Current in Nonclassical Devices. . . . . . . . . . . . . .212 6.3.3Effect of Source/Drain Series Resistance . . . . . . . . . . . . . . . . . . .219 6.3.4Comparison to ITRS Projections. . . . . . . . . . . . . . . . . . . . . . . . . .219 6.3.5Simplified Implementation for UFDG. . . . . . . . . . . . . . . . . . . . . .224 6.4Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226 7SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . . . . . . . . . . . . .227 7.1Summary and Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227 7.2Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .234 APPENDIX AUPGRADES/REFINEMENTS TO THE WEAK-INVERSION CURRENT FORMALISM IN UFDG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237 A.1Issues with the Existing Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237 A.2Overview of the Iwk Formalism in UFDG2.3. . . . . . . . . . . . . . . . . . . . . . . .240

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ix A.3Causes of Errors in the Existing Formalism. . . . . . . . . . . . . . . . . . . . . . . . .246 A.4New Iwk Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246 A.5UFDG2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 BA NOVEL 2-D SPLINE FOR UFDG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257 B.1Moderate-Inversion Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257 B.2Existing 1-D Spline in UFDG and Its Deficiencies . . . . . . . . . . . . . . . . . . .260 B.3Concept of 2-D Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261 B.4VTW and VTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263 B.4.1Inversion-Carrier Density at VTW and VTS. . . . . . . . . . . . . . . . .263 B.4.2VTW and VTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265 B.4.3Extension to Heavily-Doped Bodies . . . . . . . . . . . . . . . . . . . . . . .267 B.52-D Cubic Spline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .268 CCONVERGENCE OF THE 2-D QM SOLUTION TO THE 3-D CONTINUUM CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .286

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x LIST OF TABLES Table page 1.1Scaling issues and their solutions currently being considered. . . . . . . . . . . . . . .6 3.1Valleys, valley degeneracy, and confinement (mx) and DOS (md) effective masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 4.1MEDICI-predicted characteristics for the nFinFET designs . . . . . . . . . . . . . .115 4.2Comparison of the model-predicted sL to the actual sL . . . . . . . . . . . . . . . . .137 5.1Valleys, valley degeneracy, and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179 5.2Values of b for surface-roughness-limited mobility . . . . . . . . . . . . . . . . . . . .180 5.3Phonon-limited mobility model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . .183 6.1Conductivity effective masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .210 6.2Comparison of ITRS-projected Ion versus UFDG-predicted Ion(lim) . . . . . . . .222 7.1Comparison of how some major scaling issues are, or can be, resolved . . . . .233 A.1Predominant computation in Iwk formalism. . . . . . . . . . . . . . . . . . . . . . . . . . .254

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xi LIST OF FIGURES Figure page 1.1Cross-sectional view of classical MOSFET structures . . . . . . . . . . . . . . . . . . . .2 1.2Model-predicted Wdm requirements, and the corresponding gate swing. . . . . . .4 1.3Nonclassical MOSFET structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.4Nonclassical MOSFET structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 1.5Generic (five-terminal) double-gate structure . . . . . . . . . . . . . . . . . . . . . . . . . .11 2.1Conventional, single-gate fully depleted SOI MOSFET structure. . . . . . . . . . .18 2.2Numerically siimulated equipotential contours and electric field vectors. . . . .20 2.3Potential along the channel at some x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 2.4Qualitative difference between the realistic dependence. . . . . . . . . . . . . . . . . .29 2.5Comparison of SCEs predicted by our analytic models versus MEDICI . . . . .34 2.6MEDICI-predicted tSi and NB requirements . . . . . . . . . . . . . . . . . . . . . . . . . . .36 2.7Quasi-2-D estimates of S and DIBL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.8Quasi-2-D estimates of Si film-thickness requirements. . . . . . . . . . . . . . . . . . .42 2.9UFDG-predicted mobility dependence on tSi and effective transverse field. . .44 2.10UFDG-predicted tSi dependence of the off-state and on-state currents. . . . . . .46 2.11UFDG-predicted Ioff(tSi) and tolerable variation in tSi. . . . . . . . . . . . . . . . . . . .47 2.12UFDG/Spice3-predicted delay/stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 2.13Dual-metal-gate FD/SOI CMOS design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 2.14UFDG/Spice3-predicted delay/stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

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xii 2.15SCHRED-predicted threshold voltage and quantiztion-governed Vt-shift . . . .57 3.1Qualitative depiction of VGbS-dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 3.2MEDICI-predicted IDS-VGfS for variaous VGbS . . . . . . . . . . . . . . . . . . . . . . . .68 3.3MEDICIand model-predicted threshold voltage and reff. . . . . . . . . . . .73 3.4MEDICI-predicted IDS-VGS andVt(NB). . . . . . . . . . . . . . . . . . . . . . . . .75 3.5SCHRED-(symbol)andmodel-(line)predictedEj(tSi)(open/dashed)andEj’(tSi) (lled/solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81 3.6SCHREDand model-predicted D Vt QM versus tSi. . . . . . . . . . . . . . . . . . . . . . .84 3.7Model-predicted D VtQM(tSi) in SDG and ADG nMOSFETs. . . . . . . . . . . . . . .85 3.8Energy diagram for weakly inverted SDG and ADG nMOSFETs . . . . . . . . . .86 3.9Model-predicted D Vt QM versus tSi in SDG and ADG nMOSFETs. . . . . . . . . .88 3.10SCHREDand model-predicted Ninv = |Qi|/q . . . . . . . . . . . . . . . . . . . . . . . . . .89 3.11Model-predicted variation in D Vt QM per variation in tSi versus tSi . . . . . . . . . .91 3.12 MEDICI-predicted electron concentration along the body-direction. . . .94 3.13MEDICI-predicted IDS-VGS in log (left) and linear (right) scale . . . . . .96 3.14SCHRED-predicted inversion-carrier density versus gate bias . . . . . . . . . . . . .97 3.15 SCHRED-predicted relative increase in Ninv versus tox . . . . . . . . . . . . .99 3.16 MEDICI-predicted IDS-VGS in log (left) and linear (right) scale. . . . . .100 3.17 MEDICI-predicted IDS-VDS and IDS-VGS . . . . . . . . . . . . . . . . . . . . . . .102 4.1Top cross-sectional view of a FinFET structure. . . . . . . . . . . . . . . . . . . . . . . .106 4.2MEDICI-predicted inverse weak-inversion channel current . . . . . . . . . . . . . .110 4.3MEDICI-predicted average electron velocity. . . . . . . . . . . . . . . . . . . . . . . . . .112 4.4Gaussian source/drain doping proles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

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xiii 4.5MEDICI-predicted electron density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 4.6UFDG-evaluated Leff(weak) versus straggle for the nFinFET designs . . . . . . .119 4.7MEDICI-predicted channel current (per unit hSi)-versus-gate voltage . . . . . .123 4.8MEDICI-predicted low-frequency, low-VDS gate capacitance . . . . . . . . . . . .125 4.9A depiction of the actual S/D lateral doping profile. . . . . . . . . . . . . . . . . . . . .127 4.10MEDICI-predicted magnitude of the lateral field along the channel. . . . . . . .128 4.11A qualitative representation of Leff(weak) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130 5.1Measured room-temperature mco(Ninv) in FD/SOI nMOSFETs . . . . . . . . . . .143 5.2Monte Carlo-predicted room-temperature mph(Ninv, tSi). . . . . . . . . . . . .148 5.3Monte Carlo predicted room-temperature mph(tSi) . . . . . . . . . . . . . . . . .150 5.4Monte Carlo-predicted ground-state form factor versus tSi. . . . . . . . . . . .152 5.5SCHRED-predicted (fractional) carrier population . . . . . . . . . . . . . . . .154 5.6Monte Carlo-predicted mph in thick-tSi SDG nMOSFETs. . . . . . . . . . . .155 5.7Weak-inversion transverse electric-eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 5.8SCHRED-predicted electron distribution along the UTB. . . . . . . . . . . .158 5.9Model-predicted ground-state form factors versus tSi . . . . . . . . . . . . . . . .163 5.10 Ratio of carrier population at the middle of the UTB and at x = xav(ec) .165 5.11 Model(Eq. (5.21)) and Monte Carlo-predicted mph. . . . . . . . . . . . . . . .167 5.12 Monte Carlo-predicted surface-roughness-limited mobility . . . . . . . . . .170 5.13Monte Carlo-predicted msr at low Ninv in ADG nMOSFETs. . . . . . . . . . . . . .171 5.14 MEDICI-predicted electric potential distribution along the UTB. . . . . .172 5.15 Monte Carlo predicted surface-roughness-limited mobility . . . . . . . . . .174 5.16 Calibrated electron-mobility model predictions compared with measured meff(Ninv) in SDG and FD/SOI nMOSFETs . . . . . . . . . . . . . . . . . . . . . .186

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xiv 5.17 Calibrated hole-mobility model predictions compared with measured meff(Ninv) in FD/SOI pMOSFETs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187 5.18 Calibratedelectron-mobilitymodelpredictionscomparedwithMonteCarlopredicted meff(Ninv) in ADG nMOSFETs. . . . . . . . . . . . . . . . . . . . . . . .189 5.19 Model-predicted ratio of electron mobility in {110}and {100} Si . . . .191 5.20 UFDG-predicted IDS-VDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194 6.1IDS(lim)(VDS) predicted by Eq. (6.19). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207 6.2UFDG-predicted tSi dependence of IDS(lim)-VDS. . . . . . . . . . . . . . . . . . . . . . .213 6.3UFDG-predicted IDS(lim)-VDS and IDS(sct)-VDS for SDG nMOSFETs . . . . . .214 6.4UFDG-predicted usat(eff) and D L versus VDS. . . . . . . . . . . . . . . . . . . . . . . . . .216 6.5UFDG-predicted IDS(lim)-VDS and IDS(sct)-VDS for ADG nMOSFETs. . . . . .217 6.6UFDG-predicted IDS(lim)-VDS and IDS(sct)-VDS for SDG pMOSFETs . . . . . .218 6.7UFDG-predicted IDS(lim)-VDS and IDS(sct)-VDS in SDG nMOSFETs. . . . . . .220 6.8UFDG-predicted IDS-VDS with IDS(sct) subjected to . . . . . . . . . . . . . . . . . . . .225 A.1UFDG2.3-predicted IDS-VGS for an ADG nMOSFET . . . . . . . . . . . . . .238 A.2Qualitative picture of the exact potential distribution. . . . . . . . . . . . . . .239 A.3Integration paths in x-y and potential profiles along x. . . . . . . . . . . . . .243 A.4Integration paths in x-y and potential profiles along x. . . . . . . . . . . . . .245 A.5UFDG-predicted IDS-VGS for ADG nMOSFET . . . . . . . . . . . . . . . . . . .247 A.6Le at the front and back surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248 A.7SDG MOSFET with four UTB-partitions. . . . . . . . . . . . . . . . . . . . . . . .250 A.8UFDG2.4-predicted Iwk for Leff = 32 nm FD/SOI nMOSFET. . . . . . . . .253 A.9UFDG2.4and UFDG2.3-predicted IDS-VGS. . . . . . . . . . . . . . . . . . . . .256 B.1Partitioning of the MOSFET IDS-VGS characteristics. . . . . . . . . . . . . . .258

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xv B.2Mechanics of our novel 2-D spline . . . . . . . . . . . . . . . . . . . . . . . . . . . .262 B.3The existing 1-D spline as a subset of our new 2-D spline. . . . . . . . . . .264

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xvi KEY TO ABBREVIATIONS MOSFETmetal-oxide-semiconductor field-effect transistor CMOScomplementary MOS SOIsilicon-on-insulator FDfully depleted UTBultra-thin body SGsingle gate SDGsymmetrical double gate ADGasymmetrical double gate IGindependent gate UFDGUniversity of Florida double gate HPhigh performance LPlow power SCEshort-channel effect DIBLdrain-induced barrier lowering QMquantum mechanical SCstructural confinement ECelectrical confinement DOSdensity of states SDEsource/drain extension

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xvii AbstractofDissertationPresentedtotheGraduateSchoolofthe UniversityofFloridainPartialFulfillmentoftheRequirements for the Degree of Doctor of Philosophy PHYSICS AND DESIGN OF NONCLASSICAL NANOSCALE CMOS DEVICES WITH ULTRA-THIN BODIES By Vishal P. Trivedi May 2005 Chairman: Jerry G. Fossum Major Department: Electrical and Computer Engineering Thisdissertationpresentsthephysics,anditsapplicationtodesign,of nonclassicalnanoscalecomplementarymetal-oxide-semiconductorfield-effect transistors(CMOSFETs)havingultra-thinbodies(UTBs).Improvementstothe University of Florida Double-Gate (UFDG) MOSFET model are also developed. Physicalinsightsonthedependenceofshort-channeleffectsaregained usingquasi-2-Danalysesandnumericaldevicesimulations.Nanoscalefully depleted(FD)silicon-on-insulator(SOI)CMOSdevicesaredesignedusingUFDG, andtheirperformancepotentialisprojected.Theresultsshowtheneedforundoped UTBsandatleastamidgapgatematerial,anddefinescalability,intermsofan effectiveelectricalchannellength(Leff),ofFD/SOIandothernonclassicalCMOSas predicatedbyapragmaticlowerlimitontheUTBthickness(tSi).Withundoped UTBs,manyfundamentalassumptionsofMOSFETanalysesarefoundtobe

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xviii physicallyinvalid.Thus,theimpactofundopedbodyontheelectrostaticsofthegenericDGMOSFETisexamined.Carrierdistributioninthebodyandinthe quantizedenergystatesisfoundtohaveprofoundeffectsinbothweakandstrong inversion.Quantum-mechanical(QM)effects,dependentontSi,transverseelectric field (ex), and crystal orientation, are also physically modeled. WithanundopedUTB,theneedforgate-source/drain(G-S/D)underlapis emphasizedasthegatelength(Lgate)approaches7nm.LeffisrelatedtoLgatefor designswithG-S/Dunderlap.Usingnumericaldevicesimulations,physicalinsights onthebiasdependenceandtheS/DlateraldopingprofiledependenceofLeffare gained,relatingthenotedscalabilityintermsofLefftoLgate.TheextrinsicS/Dseries resistance(RS/D)andtheparasiticG-S/Dcapacitance(CGS/D)arealsoexamined. TradeoffbetweenLeffandCGS/DversusRS/Disfound,andcanbeoptimizedviaS/D extensionengineering.AphysicalmodelforLeffisdevelopedtoaidthisengineering. ForUTBs,MonteCarlosimulationsandexperimentalobservationshave shownsubstantivedependenceoftheeffectivecarriermobility( meff)ontSi.Thus, physicalinsightsgainedviaMonteCarlosimulationsareappliedtodevelopaQMbasedcompactmodelfor meff,physicallyaccountingforphononandsurfaceroughnessscattering,bothwithdependencesontSi,ex,andcrystalorientation.The modelisverifiedusinglargesetsofexperimentaldata.Inundopednonclassical MOSFETs, meffisfoundtobeupto4xhigherthaninclassicalMOSFETs.Finally, withour meffmodel,UFDGyieldschannelcurrenthigherthantheballistic-limit value.Thus,amodelforthelimitcurrentisdoneandimplementedinUFDG,and usedforpreliminarystudyoftheassociatedperformancelimitationofDGMOSFETs.

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1 CHAPTER 1 INTRODUCTION Afterthefabricationofthefirstmetal-oxide-semiconductorfield-effect transistor(MOSFET)onsiliconsubstratewithSiO2gateoxidebyKhangandAtalla in1960[Kha60],followedbytheinventionofCMOS(complementaryMOS)by WanlassandSahin1963[Wan63],integrated-circuits(ICs)havegonefromhaving fewtransistorstohundredsofmillionsoftransistors[Sti03].Thisrapidgrowthinto thevery-large-scale-integration(VLSI)erahasbeen,andcontinuestobe,drivenby reducedcost-per-function,andisenabledbycontinuedscalingoftheMOSFETgate length(Lgate)[Tau98].Overthepast3-4decades,Lgatehasbeenscaledfrom~10 m m in the early 1970s into the nanoscale regime, i.e., <100nm, for today’s technology. 1.1 Classical CMOS Devices and Their Scaling Limit Theclassoftransistorsthatpredominantlyenabledthistransitionintothe VLSIeraarewhatwecharacterizehereasclassicalCMOSdevices.Theyincludethe bulk-SiMOSFETs( Fig.1.1 (a))andthepartiallydepletedsilicon-on-insulator(PD/ SOI)MOSFETs( Fig.1.1 (b)).Whilethesetwotechnologieshavetheirdifferences, duelargelytothefloatingbodyofthePD/SOIMOSFETs[Kri98],bothtechnologies requireengineeringthechannel/bodydopingdensityandgradientstosimultaneously controlthresholdvoltage(Vt)andshort-channeleffects(SCEs)[Tau98].Thelatter isachievedviathinningthedepletionwidth(Wdm)definedbythedopinggradient,

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2 Figure 1.1 Cross-sectionalviewofclassicalMOSFETstructures:(a)bulk-Si MOSFETand(b)partially-depletedsilicon-on-insulatorMOSFETwith thick buried oxide (BOX). Si Substrate (Body) Gate Source Drain Si Substrate Gate Source Drain Body BOX(a) (b)

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3 andtheformerviaadjustingthetransverseelectric-fieldattheSiO2/Siinterface(es) definedbythedopingdensity.FromTaurandNing[Tau98],reasonableSCEcontrol resultswhentheratioofaneffectiveelectricalchannellength(Leff)andWdmis roughlytwicethebodyfactor(m),wherem @ (1+ esitox/ eoxWdm)forgateoxidewith electricalpermittivity eoxandthicknesstox,andSichannelwithelectrical permittivity eSi. For SiO2 gate oxide, we then need .(1.1) Also,fromthesolutionofthe2-DPoisson’sequationintheLeffxWdmrectangular depletion region [Tau98], the subthreshold gate swing (S) can be derived as .(1.2) NotethatalthoughLgateisthefeatureofinteresttechnologically,MOSFETelectrical characteristicsareingeneraldefinedbyLeff,whichwhilescalingwithLgate, typically differs from Lgate [Tau98]. UsingEq.(1.1)andEq.(1.2),weplotin Fig.1.2the neededWdmscaling inthenanoscaleregimewithtoxassumed(i)fromtheInternationalTechnology RoadmapforSemiconductors(ITRS)[ITR01]and(ii)tobelimitedat1nm.Also shownin Fig.1.2isthe correspondingSforeachdesignpoint.Notethatwithtypical gate-source/drain(G-S/D)overlap[Tho98],Leff
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4 F igure 1.2Model-predictedWdmrequirements,andthecorrespondinggateswing, versusLeff= LgatefortoxassumedfromITRS(solid)andlimitedto1.0nm (dashed). 5.010.015.020.025.030.035.040.045.050.0Leff [nm] 1.0 6.0 11.0 16.0 21.0Wdm [nm] tox as indicated tox = 1.0nm 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0S [mV] tox=1.5nm 1.1nm 1.1nm 0.5nm 0.9nm

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5 requirements,andthelargeS,implybothtechnologicalandelectrical/physical barriers to the (conventional) scaling of classical CMOS. Fromtechnologicalperspective,thestatistical/randomnatureofdopant placementisthefundamentalbarriertoachievingsuchultra-thinWdm.Electrically, thethin-Wdm-governedlargeSinFig.1.2meansasmallratiooftheon-statecurrent (Ion)totheoff-statecurrent(Ioff),especiallyforthelowsupplyvoltages(VDD) requiredfornanoscaleCMOS.Infact,forclassicaldevices,Stendstobetraded-off forbetterVt(Leff)andDIBLcontrol,assuggestedbyEq.(1.1)andEq.(1.2).In addition,tomaintainreasonableVtfordesignswithITRS-recommendedtox,higheres(orchanneldoping)isneeded,whichsubstantively( es 2)degradescarrier mobility,andhenceIon.Thehighesandtheultra-thintoxenhancethepoly-depletion effect(inthegate),andhence,furtherundermineIon.Also,sinceitreducesthe effectiveoxidebarrier,thehigheresexacerbatesanincreasinglysevereproblemof gatetunnelingcurrent(Ig)throughtheultra-thingateoxide,increasingIoff[Fra01]. IoffisfurtherincreasedasthecombinationofabruptS/Djunctionsandincreasingly higherchanneldoping(e.g.,viahaloimplant)substantivelyincreasetheS/D-body junction-tunnelingcurrent[Fra01],includinggate-induceddrainleakage(GIDL) current.Last,butnottheleast,therequiredultra-shallowS/Dextension,reflectedby WdminFig.1.2,unacceptablyincreasestheexternalS/Dseriesresistance(RS/D) even when S/D doping is equal to that limited by solid solubility [Plu01]. Table1.1liststhenotedscalingissues,andthesolutionscurrentlybeing considered[45nm04].Inadditiontohavingnosolutionfortherandomdopant effects,junction-tunnelingcurrents,andlargeS,solutionsforotherscalingissues

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6 Scaling IssuesPossible Solution(s) Mobility degradationUni-, or bi-axially strained-Si channel (via SiGe) Poly-depletion effectMetal gates (e.g., Ru, Ta) Gate tunneling currentHighk gate dielectric (e.g., HfO2) Source/drain series resistanceSiGe, or Schottky, source/drain, ? Junction tunneling current? Large (> 100mV) gate swing? Random dopant effect? Table 1.1 Scalingissuesandtheirsolutionscurrentlybeingconsidered[45nm04]toextend scaling of classical CMOS to Lgate< 45nm.

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7 requireintroducingmanynewmaterialsintotheprocess,whichisaformidabletask. Hence,wesurmise,fromFig.1.2and[Fra01],thescalinglimitofhigh-performance (HP)classicalCMOSwithreasonableIofftobeLeff~30nm,orLgate~45nmfor typicaltotalG-S/Doverlapof @ 0.3Lgate[Tho98].Forlow-power(LP)CMOS,the limit is much longer at Leff>@ 50nm. 1.2 Nonclassical CMOS Devices WiththenotedscalinglimitofviableclassicalCMOS,newdevice architecturesarebeingearnestlyconsideredtoextendCMOSscalingtoLgate~7nm. (BeyondLgate~7nm,source-draindirecttunnelingyieldsunacceptableIon/Ioff[Wan02].)WecharacterizethesenovelCMOSarchitectureshereasnonclassical CMOS.Theyinclude,asillustratedin Fig.1.3,the single-gate(SG)FD/SOI MOSFETs[Dor02],(planar)double-gate(DG)MOSFETs[Kim01b],triple-gate (TG)MOSFETs[Doy03],andgate-all-around(GAA)MOSFETs[Col90].Also, anothernonclassicalstructurethatseemstobetechnologicallymostpromisingisthe FinFET[His91],[His00],[Ked01],shownin Fig.1.4. Withathicktop-gateoxide,the FinFETcanbeessentiallytreatedasa(quasi-planar)DGMOSFET.Further,when thetwogatesofaFinFETarephysicallyisolatedandoperatedindependently,the resultingdeviceiscalledanindependent-gate(IG)FinFET[Fri04],ormultiple independent-gateFET(MIGFET)[Mat04].Althoughthenamesforthese nonclassicalstructuresvary,mostofthemwereconceptuallyproposedmorethanone or two decades ago [Col04].

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8 F igure 1.3 NonclassicalMOSFETstructures:cross-sectionalviewof(a)single-gate FD/SOIMOSFETand(b)double-gateMOSFETwithraisedsource/drain andthinsource/drain-extensions,and(c)triple-gateMOSFETand(d)gateall-around MOSFET, latter two from point of view of source/drain.(a) (b) Gate Gate Drain Source BOX Substrate (Gate) Gate BOX Substrate (Gate)(c) (d)Gate Body Drain Source Body Body Gate BOX Substrate (Gate) Body

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9 Figure 1.4 NonclassicalMOSFETstructure:(a)nFinFETwithraisedandwidesource/ draincontactregionandthin-nsource/drain-extension.(b)A2-DcrosssectionalviewoftheFinFETstructurefromthepointofviewofthesource/ drain.Thetopcross-sectionalviewoftheFinFETisthecross-sectional view of the planar DG MOSFET as shown in Fig. 1.3(b).(b) n+ Source n+ Drain Gate eD LeS Lg h tSi x y n+ Source n+ Drain Gate eD LeS Lg h tSi x y Gate BOXSubstrate (Gate) Body (a)

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10 1.3 Dissertation Outline Manytechnologicaldifferencesexistamongthenotednonclassical MOSFETs.Anobviousdifferenceisthenumberofactivegates.However,theyall haveacommonfeatureofafullydepleted(FD)ultra-thinbody(UTB),andasa result,physicsoftheiroperationisverysimilar.Forexample,becauseoftheFD UTB,allnonclassicaldevicesshowninFigs.1.3-1.4tendtohaveanear-idealS, yieldingveryhighIon/Ioff.Further,becauseoftheexcellentdevice-substrate isolation,thesenonclassicaldevicesareimmunetolatch-up,havereducedsoft-error rate, and have low parasitic junction capacitance. Unfortunately,althoughtheTGandGAAdeviceshavethebestSCE control,theysufferfromparasiticcornereffects[Fos03b],whichuponelimination resultsinlessflexibilityinchoosingthebodydimensionsthanforSGandDG devices[Fos04b],[Yan05].Then,becauseoftheirsmallbodyvolume,theyoccupy alargerlayoutarea,relativetoSGorDGMOSFETs,forgiven(total)Ion,andhence do not seem viable for VLSI applications [Fos04b], [Yan05]. Thus,ourmainfocusherewillbeonnonclassicalSGandDGMOSFETs shownin Fig.1.3 (a), Fig.1.3 (b),and Fig.1.4.Thesedevicestructurescanbecategorized undera genericDGstructure illustratedin Fig.1.5 withfiveterminals(source(S), drain(D),body(B),frontgate(Gf),andbackgate(Gb)),afront-(back-)gateoxide ofthicknesstoxf(toxb),andaFDSi-bodyofthicknesstSi.Whenbothofthegatesare tiedtogetherandarethesamematerial(Gb=Gf)andtoxb=toxf,theresultingdevice isknownasasymmetricalDG(SDG)MOSFET[Kim01b].Ifthetwogatesaretied togetherandeithertoxb toxfand/orGb Gf,thenthedeviceiscalledan

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11 Figure 1.5 Generic (ve-terminal) double-gate structure. D S Gf Gb toxftoxb x tSi y LgateB

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12 asymmetricalDG(ADG)MOSFET[Kim01b].Hence,unlessspecified,DG MOSFETthroughoutthisdissertationreferstothegenericstructureinFig.1.5with the indicated coordinate system. ThefundamentalsofoperationoftheDGMOSFET(withlongLgate)were studiedandmodeledbyLimandFossum[Lim83]abouttwodecadesago.Thismodel waslaterimprovedtorigorouslyaccountforSCEsinweakinversion[Yeh95],and carrierdistribution[Chi01a]andquantum-mechanical(QM)effects[Ge02a]in stronginversion.(Thelargetemporalgapbetweentheoriginalmodelandthe subsequentimprovementsisduetotheprevalentinterestinclassicalCMOSas definedbytheirrelativelyeasytechnology.)However,becauseoftheir comprehensivenature,thelatterworksfocusedonmodeldevelopment,andtheystill needtobeapplied,inconjunctionwithexperimentalstudies,foroptimaland/or pragmaticdesignofnanoscaleDGCMOS,aswellastogainphysicalinsightson noveleffectsinnonclassicaldevices.Further,becausetheclosedformofthese modelsisbasedonassumptionsconsideringCMOSarchitectureanddesign approachesofthetimeoftheirdevelopment,theymayneedtoberefinedasawider rangeofdevicearchitecturesandnoveldesignideasareexploredinthenanoscale regime.Forexample,theclassiccharge-couplinganalysisin[Lim83],whichisstill usedincompactmodelstodefinetheboundariesofweak-andstrong-inversion conditions(andhence,theuseofthementionedimprovedmodels),assumesa predominantfrontchannel,whichisinvalidfordevicessuchasIGFinFETs.Also, theQMmodelingin[Ge02a]focusespredominantlyonthestrong-inversion condition.

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13 Further,withrespecttoCMOSdesign,althoughmanynumericalsimulation-basedstudieshavebeendonetoexamineSCEsinvariousnonclassical devices,e.g.,[Won98],thesestudieseitherlackadequatephysicalinsights,ordonot accountforothercrucialeffectssuchasQMeffects,forviablenanoscaleCMOS design.Also,whileMonteCarlosimulationsofcarriertransportinnonclassical MOSFETshavebeendone[Gm99b],[Gm01b],theyfocusonlyonFD/SOIand SDGnMOSFETs.Inaddition,nophysicalcompactmodelaccountsforthenovel dependenceoftheeffectivecarriermobility( meff)ontSi<20nmpredictedbythe noted Monte Carlo simulations, and observed experimentally [Ess00], [Ess01a]. Thus,thisdissertationfocusesonthephysicsanddesign,andmodeling, of nanoscale DGMOSFETs.(ManyofthephysicalinsightsarevalidforTGand GAAMOSFETsaswell.)Thephysicalinsightsandmodelsareappliedtotheongoingdevelopmentofaprocess/physics-basedMOSFETmodel(UFDG)for nonclassicaldeviceandcircuitdesigns[Fos04a],whichislargelybasedonthenoted models in [Lim83], [Yeh95], [Chi01a], and [Ge02a]. WebegininChapter2withgainingphysicalinsightsonSCEs,andtheir dependenceontSiand(uniform)bodydoping(NB),inFD/SOIMOSFETsviaquasi2-Danalyses.TheresultsshowtheneedforundopedUTBs(i.e.,tSi<20nm).Next, weuseUFDGtooptimallydesignnanoscaleFD/SOICMOS(withregardsto,for example,Vtcontrol)andtoprojectperformanceofHPandLPFD/SOICMOSwith Leff=35nm.Basedonourquasi-2-Danalysesandinsightsfromthenoted performanceprojections,wedefinethescalabilityofFD/SOItechnologyintermsof Leff.WethenextendouranalysestoothernonclassicalCMOStechnologies,and

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14 examinetheirscalability.Basedoninsightsfromourquasi-2-Danalyses,wealso upgrade UFDG, as described in Appendix A, to improve its accounting for SCEs. OneofourfindingsinChapter2isthatnonclassicalnanoscaledevices mustuseundopedUTBs.WithundopedUTBs,however,wefindthattheunderlying classicalassumptions,i.e.,charge-sheetanddepletionapproximation,ofthe fundamentalanalysisin[Lim83]and[Yeh95]becomeinvalid,andthemodel applicationislimited.Hence,inChapter3,westudyandmodel"nonclassical" physicsofundopedDGMOSFETs.Bynonclassicalphysics,wemeantheeffectsof inversionchargedistributioninthebodyandinthequantizedcarrier-energystates ontheweak-andstrong-inversioncharacteristics.Fortheweak-inversioncondition, welargelyfocusongeneralizingthecharge-couplinganalysisof[Lim83]and modelingeffectsofcarrier-energyquantization.Forthestrong-inversioncondition, wemainlyexaminehowcarrierdistributioninthebodyaffectsIoninSDG MOSFETs.Further,becauseUFDGusesthemodelin[Lim83]todefinethe boundariesofweak-andstrong-inversionconditions,andpolynomialsplinesfor modelingmoderate-inversionconditions[Fos04a],itsapplicationislimitedto deviceswithapredominantfrontchannel.Thus,torenderUFDGtrulygenericfor DGMOSFETs,wedevelop,basedonournotedgeneralization,anovel2-D polynomialsplineformodelingmoderate-inversionconditions.Thisnovel2-D spline is described in Appendix B. Next,withundopedUTBs,wefindthattheconventionaldesignapproach withG-S/DoverlapisnolongeraviableoptionasLgate 7nm,andthatG-S/D underlapwillbeneeded.WhilethephysicsofLeffdefinedbyG-S/Doverlapiswell

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15 understood[Tau98],thatofG-S/Dunderlapisnot.Thus,inChapter4wepresentthe physicsofLeffasdefinedbyG-S/Dunderlap,andinparticular,theLeffdependence onthegatebiasandthelateraldopingprofileintheS/Dextension(SDE).Basedon ourphysicalinsights,weproposeasystematicapproachtooptimal,doablenanoscale CMOSdesignviaengineeringoftheSDEs,includingtheSDElength,lateraldoping profileintheSDE,andtheUTBthickness.ToaidthisSDEengineering,wedevelop amodelforLeff.WithLeff,whichdefinestheMOSFETelectricalcharacteristics, nowphysicallycorrelatedtoLgate,whichisthequantityofinteresttechnologically, ouranalysesofnonclassicalCMOSinChapter2isnowlinkedtoLgate,andthus,to real technology. Inthelastpartofthisdissertation,wefocusoncarriertransportinUTBs, whichdiffersfromthatinclassicaldevices[Ge02a].WebegininChapter5with QM-basedcompactmodelingof meffforUFDG,usingphysicalinsightsfromMonte Carlosimulations.WeconsiderCoulomb,phonon,andsurface-roughnessscattering. Themodelaccountsforthenoveldependenceof meffontheUTBthickness,the transverseelectric-field,andtheSi-surfaceorientation.Itisverifiedusinglargesets ofexperimentaldata.Also,afterimplementingthemodelinUFDG,weapplyitto gain insights on CMOS design. Furtherapplyingour meffmodelusingUFDG,wefind,forsome nonclassicalnMOSFETs,thatUFDGpredictschannelcurrenthigherthanthat limitedbythethermalinjectionvelocityatthesource,orballistictransport.Thus, wedescribeinChapter6a1-Dmodelfortheballistic-limitcurrentasdevelopedby Natori[Nat94],butwithadditionalanalysisofissues,suchastheconductivity

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16 effectivemass,notadequatelyaddressedinliterature.Wethenimplementthismodel inUFDGandexaminehowclosetotheballisticlimitdosomenonclassicalCMOS devicesoperate.Wealsodopreliminaryassessmentoftheassociatedperformance limitations of SDG MOSFETs. Finally,wesummarizethisworkinChapter7.Suggestionsforfuture research are also discussed.

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17 CHAPTER 2 SCALING FULLY DEPLETED SOI CMOS 2.1 Introduction Conventionalfullydepleted(FD)SOICMOS,thedevicestructureof whichisillustratedinFig.2.1,wasofmuchinterestadecadeagobecauseofits projectedsuperiorityoverthepartiallydepleted(PD)andthebulk-Sicounterparts [Col04].Itsadvantages,duemainlytothe(front)gate-substratechargecoupling enabledbythethinFDSi-filmbodyonathickburiedoxide(BOX)[Lim83],[Fos92], includedhigherdrivecurrent/transconductance,near-idealsubthresholdslope,and suppressionofthefloating-bodyeffects.However,inthedeep-submicronregime, these advantages diminished [Fos92], and the interest in FD/SOI CMOS subsided. Now,asclassicalCMOStechnologyisbeingrapidlyscaled[ITR01],and approachingitstechnologicallimits,theconventionalFD/SOICMOShasonceagain becomeacenterofattention[Suz00],[Cho00],[Dor02],[Cha01].Thisinterestis augmentedbyitsrelativelysimplerandmatureprocessingtechnologycomparedto, forexample,SDGCMOS.Whiletheseriesresistanceinthethin-Sifilmofextremely scaledFD/SOIdevicesisaseriousissue,raisingtheS/Dregionshasbeenshownto alleviatetheproblem[Cha01].Further,theexperimentaldevicestudiesin[Suz00], [Cho00],and[Dor02]showthattheSCEscanbewellcontrolledinFD/SOI nMOSFETswithLgate<50nmbythinningtheSi-filmbody/channeltotSi<10nm, i.e.,byusingUTBs.Whilethedevicesin[Suz00],[Cho00],[Dor02]and[Cha01]had

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18 Figure 2.1Conv entional,single-gatefullydepletedSOIMOSFETstructure.For nanoscaledevices,thesource/drainregionswouldberaised(notshown). Thex-ycoordinatesystemusedinthequasi-2-DanalysesinSec.2.3is superimposed.The y=0originindicatedcorrespondstothelocationofthe effective source contact region as dened in Sec. 2.3.1. Gf Substrate (Gb)DBSBOX tSi tBOX toxb Lgate toxf x y

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19 varyingstructuresforVtcontrol,e.g.,lowandhighbody/channeldoping(NB)with metalandpolysilicongates,publishedphysicalandanalyticinsightsonthebehavior ofsuchnanoscaledevicesareinadequatetoascertainifthedevicesareoptimally designed.PreviousworkshavefocusedmainlyonSCEs,withoutanyattentionpaid tocarrier-energyquantization[Ge02b]andcarriermobility[Gm99b],[Ess01b]in UTBs.Further,VtcontrolforbothCMOSdevices,governedbythequantizationand theSCEs,aswellasthegateworkfunction,NB,andtSi[Lim83],hasnotbeen addressed comprehensively. Thus,inthischapter,wefirstpresentquasi-2-Danalysessupportedby 2-DnumericaldevicesimulationstogaincrucialinsightsonSCEsforoptimal nanoscaleFD/SOICMOSdesignandscaling;theanalysescanapplytoDG MOSFETsingeneral.Then,usingUFDG-baseddevicesimulationstorigorously accountfor,inadditiontotheSCEs,thequantizationeffects,wepresentstructural requirementsforviableHPandLPFD/SOICMOSdesign.Thisisfollowedby UFDG-basedcircuitsimulationstoprojectperformancesofLeff=35nmFD/SOI CMOSandtoexaminethetechnology’sscalinglimit,whichweshowispredicated bypragmaticultra-thintSiandtheSCEsandquantizationeffectstherebyimplied. Finally, we extend our analyses to project the scaling limit of other DG CMOS. 2.2 2-D Effect in the BOX Oneofthemajorissuesthatcontributedtothedeclinedinterestinscaled FD/SOICMOSwastheelectric-fieldfringing,a2-Deffect,inthethickBOX [Yeh95].AsdepictedinFig.2.2,duetothelargedistancebetweenthesubstrateand

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20 Figure 2.2Numericallysimulatedequipotentialcontoursandelectricfield vectors(arrows)inLmet=0.2 m mFD/SOInMOSFET[Yeh95]; tSi=100nm,tBOX=350nm,VDS=50mVemphasizingthenatureof electric-field fringing in the BOX. S D BOX Distance [ m m] Distance [ m m]

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21 theS/Djunctions,theelectricfieldintheBOX,emanatingfromtheS/Ddepletion charge,tendstoterminateintheSi-filmchannelratherthaninthesubstrate,thereby possiblyinducingsubstantiveinversionchargeatthebacksurfaceanddegradingthe MOSFETsubthresholdcharacteristics[Yeh95].Atrivialsolutionforreducingsuch degradationistomerelydecreasetheBOXthickness.However,theBOXmustbe scaledsubstantively(tBOX<50nm)togetanynoticeablesuppressionofthis2-D effect[Num02],andtBOX( toxb)mustbeontheorderofthefrontoxidethickness (toxf) to completely eliminate it. WebelievethatthinningtheBOXisnotaviableoptionbecause(i)the effectivebodycapacitance(CB(eff))isincreased,negatingtheuniquebenefitsofthe intrinsicFD/SOIMOSFET[Lim83],[Fos92],[Vee88],e.g.,thenear-ideal subthresholdslope,(ii)substratedepletioneffects,e.g.,morefield-fringinginthe BOXandincreaseintheS/Djunctioncapacitance,areintensified[Yeh95],and,as intimatedabove,(iii)thedevicebecomesunconventionalanddifficultto manufacture.Furthermore,forpMOSFETs,thegroundedsubstrateusuallygives substrate-to-sourcebiasof-VDD,whichtendstofurtherincreasetheback-surface conductionastBOXisthinned.Also,thedeviceasymmetryresultingfromthevery thinBOXdefinesahightransversefieldthat,whilesuppressingthefieldfringingin theBOXandthe2-Deffectsinthebody,degradesthecarriermobility,andhenceIon[Tri05].ThecombinationofthelowerIonandthehigherCB(eff)andparasitic capacitance in turn yield slower CMOS [Tri05]. Hence,theeffectsofBOXfieldfringingmustbecontrolledbyother means.Experimentalresultsin[Dor02]and[Bur97]showthatusingahaloimplant

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22 (orincreasingNB)couldbeapartialsolution,subjectofcoursetotheSi-filmbody becomingpartiallydepleted.Theseresults,alongwith2-Dnumericaldevice simulationsin[Van02],alsoshowthatthinningtSisuppressestheBOXfield-fringing effects.IncreasingNBresultsinmoreeffectivescreeningofthefringingfieldsinthe BOXaswellasthelateralfieldsintheSOIfilmassociatedwiththeS/Djunctions, anddecreasingtSitendstoenablebettergatecontroloverthebacksurfacesincethe 1-Dchargecoupling[Lim83],whichindeedunderliestheuniqueadvantagesof FD/SOICMOS[Lim83],[Fos92],isenhanced.Thesimulationresultsin[Van02] furthershowthatwhentSiisultra-thin,thenotedbenefitofthinningtBOXislessened. Hence,forUTBs,whichareneededforscaledFD/SOIMOSFETsasweshowherein, the2-DeffectsthatgovernthemajorityoftheSCEsarethoseintheSOIfilm/body, even for thick BOX. 2.3 2-D Effects in the SOI Film Togainphysicalinsightsonthesubthresholdgateswing(S),andthe drain-inducedbarrierlowering(DIBL)ofnanoscaleFD/SOIMOSFETs,weadopta DICE-likeapproach[Vee88]forquasi-2-Danalysesoftheelectricpotentialinthe body/channel.SimilaranalysesforDGMOSFETshavebeendonein[Kim01a]; however,theyassumedchargesheetsandnegligiblechangeinthe(integrated) inversionchargeduetothe2-Deffects.Incontrast,thebasicSCEanalysispresented hereisgenerallyapplicabletoanythinSi-filmMOSFETs,withcarrierdistribution throughout the body.

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23 2.3.1 Current Flow in Nonclassical MOSFETs InnonclassicaldeviceswithUTBs,thecurrentcanflowthroughoutthe entirebody,i.e.,body=channel,andinsomecasesthedistributioncanbeuniform. Also,aswewillshownext,S/Dencroachmentvariesalongthebody(oralongxin Fig.2.1).Hence,SCEsevaluatedfromexperimentalorsimulatedIDS-VGScharacteristicsaredefinedbythe2-Deffectsintheentirebody,weightedbythelocal carrierconcentration.Thus,toaccuratelypredictSCEs,wewouldneedtoanalyze the2-DeffectsintheentireSi-film(alongx).WhilethisisessentialforaphysicsbasedMOSFETmodelsuchas[Yeh95],itisnotpragmaticforourpurposeshere, whichistogainphysicalinsightsandadequatelyquantifythem(viaquasi-2-D analysis)soastoprovideviabledevicedesigns.So,wefocusonthe2-Deffectsata "surface"(atx)correspondingtothemaximumleakage-currentflow,orthe maximumcarrierconcentration.Forexample,aswewillshowlater,inashortchannelundopedSDGMOSFET,theleakiestpathislocatedatx=tSi/2,andfor FD/SOIMOSFETs,itistypicallyeitherthefront(x=0)ortheback(x=tSi)surface, depending on NB. 2.3.2 Subthreshold Slope Tounderstandhowthepotential( f )inthebody/channelrespondstothe appliedgatebias,wewrite f0( x,y) =f1( x )+Df1( x,y ), where f1( x ) isthesolutionof the1-DPoisson’sequation(neglectingfreecarriers)andtheVDS-dependent Df1( x,y ) istheincreaseinthepotentialdueto2-Deffects,satisfying,forweak inversion,

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24 .(2.1) Notethattheneglectoffreecarriersinweakinversionlinearizesthe2-DPoisson’s equation,andthus,Eq.(2.1)isexact.Wethengetaclosed-formsolutionforEq.(2.1) by approximating it as (2.2) where h1isaspatialconstant.IntegratingEq.(2.2)oncealongthechannelfromthe (effective)sourcecontactregion(y=0+)satisfying Df1( y=0+) = fbi,thesource-tobodybuilt-inpotential,toy=yssuchthat Dey1(y=ys)<< Dey1(y=0+)(Fig.2.3), we get .(2.3) Now,integratingEq.(2.2)onceacrossthefilm(inxaty=ys@ ymin)yieldsarelation betweentheperturbedfront(sf)-andback(sb)-surfacetransverseelectricfields,i.e., ,(2.4) whileintegratingtwicecouplestheperturbationsintheminimum(iny)front-and back-surface potentials [Vee88]: .(2.5) x2 2 f1D y2 2 f10 = D + x2 2 f1D y2 2 f1h1– = D – @ h1eDy10+()eDy1ys() – ys--------------------------------------------------Dey10+() ys----------------------@ = De1sb ()De1sf ()h1tSi+ = Df1sb ()Df1sf ()– De1sf ()tSi– h1tSi 22 ------------ – =

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25 Figure 2.3Potentialalongthechannelatsomexforalong-channelanda short-channelMOSFETwith f0(x,y)= f1(x)+ Df1(x,y)andVDS@ 0V. AsVDSincreases,theadditionalperturbationduetoitwillchange the2-Dpotentialprofileleadingtofurtherincreasein Df1(ys=ymin), which then defines DIBL . Source Drain y f0 f1y = 0+Df1(y = 0+) = fbi ys = yminLeff Df1(ys)Short-Channel Device Long-Channel Device

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26 Finally,usingtheboundaryconditionsateachinterfacetorelatethecorresponding perturbedfieldandperturbedminimum(iny)potential,Eq.(2.4)andEq.(2.5)give (2.6) and (2.7) whereCSi(= eSi/tSi),Coxf(= eox/toxf),andCoxb(= eox/toxb)arethebody,front-oxide, andback/buried-oxidecapacitances,respectively.(Wenotethatwhereas Df1(sb)and Df1(sf)in[Vee88]and[Kim01a]werederivedfromEq.(2.4)andEq.(2.5)usingthe charge-sheetanalysisin[Lim83],wehaveallowedforcarrierstobeanywhereinthe bodysoastobeconsistentwithourinsightsinSec.2.3.1.)Hence,forCoxf>Coxb, Eq.(2.6)andEq.(2.7)showthat Df1(sb)> Df1(sf).However,thesignificanceofeither perturbation depends on the total potentials at the respective surfaces. From the mathematical definition of S [Tau98], we write it as (2.8) wherekBistheBoltzmannconstant,Tisthetemperature,qistheelectroncharge, and f0(max)representsthe“surface”potentialoftheleakiestsource-to-drainpath.In the approximate expression in Eq. (2.8), f1sf ()D 2CSiCoxb+ CSiCoxfCoxb+ () CoxbCoxf+ ----------------------------------------------------------------------eSitSih12 ------------------= f1sb ()D 2CSiCoxf+ CSiCoxfCoxb+ () CoxbCoxf+ ----------------------------------------------------------------------eSitSih12 ------------------= S 10 () lnkBTq d f0max ()dVGS---------------------------------------------------10 () lnkBTq d dVGS------------f1max ()Df1max ()+ () ---------------------------------------------------------------== m10 () lnkBTq 1m Df1max ()() d VGSd ----------------------------+ ------------------------------------------@

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27 .(2.9) ForFD/SOIMOSFETswiththickBOX,Coxb< f0( sf ), thenSwillbedeterminedby Df1(sb),andsimilarlyfor f0( sf )> f0( sb ). Later we show that Q(Df0(sb) Df0(sf)) is governed by NB. To complete the estimation for S in Eq. (2.8), we approximate ,(2.11) whereKrepresentsthetermsotherthan h1inEq.(2.10);inEq.(2.11)wehave assumed(i)ys@ Leff/2,whereLeffwillbeclearlydefinedinthenextsub-section(and furtheraddressedinChapter4),(ii) Dey1(0+) @Dj /ys,with Dj beingthedifference betweentheperturbedpotentialatthesourceandatys,and(iii),which weinferredfrom2-DnumericalsimulationsdonewithMEDICI[MED04].The negativesigninEq.(2.11)isconsistentwithourphysicalintuitionthatthe2-D effectsarediminishedasVGSisincreased.Finally,substitutingEq.(2.11)inEq. (2.8), with eSi/ eox@ 3, we get m f1max ()d d VGS1 CSiCoxbCoxfCoxbCSi+ () ----------------------------------------+1 CBeff ()Coxf---------------+ = @ = f1max ()D eSitSih1Coxf------------------1 Qf0sb ()f0sf ()– () Coxf2CSi----------+ @ Df1max ()() d VGSd -----------------------------K h1() d VGSd ------------K Leff2 ()2----------------------Dj () d VGSd -------------K Leff2 ()2----------------------1.4 – () @ = Dj () d VGSd -------------1.4 –

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28 (2.12) A qualitative description of our model for S (and for DIBL) is shown in Fig. 2.4. 2.3.3 Drain-Induced Barrier Lowering TomodelDIBL,wewritethepotentialforhighVDSas f( x,y) =f0( x,y )+ Df0( x,y ) ,where f0( x,y ) isthe2-DsolutionforVDS@ 0Vdefinedinthepreceding section,and Df0( x,y ), independentofVGS,istheincreaseinthepotentialdueto drain bias, which, for weak inversion, satisfies .(2.13) AnalogoustoEq.(2.2),weseparatethetwopartialderivativesinEq.(2.13), assuming ,(2.14) where h0isanotherspatialconstant.Then,integratingtwicealongthechannel,with the boundary conditions Df0(y = 0+) = 0 and Df0(y = Leff) = VDS, yields ,(2.15) wherefortheapproximationweassumenegligiblechangein Dey0(0+)withVDS,i.e., Dey0(0+)variesmuchslowerwithVDSthantheaveragelateralfieldVDS/LeffS 10 () lnkBTq 1 17tSitoxfLeff 2-------------------1 Qf0sb ()f0sf ()– () tSi6toxf----------+ – ---------------------------------------------------------------------------------------------------------@ x2 2 f0D y2 2 f00 = D + x2 2 f0D y2 2 f0h0– = D – @ h02 Leff 2--------VDSDey00+() Leff+ () 2 Leff 2--------VDS@ =

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29 Figure 2.4 QualitativedifferencebetweentherealisticdependenceofSorDIBLon NBandtheapproximatedone(Eq.(2.12)orEq.(2.19))usingthe Heavsidefunctioninthequasi-2-Danalysis.Whentheoff-statecurrentis notconnedtoeitherthefrontorthebacksurface,2-Deffectsare characterizedbythetransitionregionshownfortherealisticcase.This occurswhentheperturbationsoftheback-surfacepotentialare comparable to ( f1(sf) f1(sb)) in Eq. (2.21). NB [cm-3]S or DIBL Realistic

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30 [Vee88].WeemphasizethatLeff,definedhereandinEq.(2.11),isaneffective electricalchannellengththatgovernsthe2-DeffectsintheSi-filmchannel,andthat itisdefinedbywheretheS/Ddopingdensitiesaresufficientlyhightovalidatethe notedboundaryconditions(seeforexampleFig.2.3).Hence,Leffisdependenton theS/Ddopingprofile.Inactualdevices,Leff,whichcanbenearlyequaltothat definedin[Tau04]forchannelcurrent,willclearlyexceedthemetallurgicalchannel length(Lmet)becauseoffiniteS/Ddoping-densitygradient,and,infact,will typicallyexceedLgatewhenpunch-throughissafelyavoided(perhapsvialong spacersorS/Dextensions,whichcouldresultinfiniteG-S/Dunderlap).Also,we notethatEq.(2.14)isnotvalidwhenthereissignificantcouplingofthetwopartial derivatives. With h0inEq.(2.15),wederive,analogoustoEq.(2.6)andEq.(2.7), from Eq. (2.14) ,(2.16) and (2.17) where Df0(sf)and Df0(sb)aretheVDS-inducedperturbationsoftheminimumsurface potentials(iny).NotethatEq.(2.16)andEq.(2.17)areapplicabletoSDGandADG MOSFETsaswellasFD/SOIMOSFETs.Fordeviceswithtoxb=toxf=tox, Df0(sb)= Df0(sf)@ tSi 2(CSi/2Cox) h0inEq.(2.16)andEq.(2.17),whichissameasthatin f0sf ()D 2CSiCoxb+ CSiCoxfCoxb+ () CoxbCoxf+ ----------------------------------------------------------------------eSitSih02 ------------------= f0sb ()D 2CSiCoxf+ CSiCoxfCoxb+ () CoxbCoxf+ ----------------------------------------------------------------------eSitSih02 ------------------=

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31 [Kim01a]forSDGandADGMOSFETs.However,forFD/SOIdevices,Eq.(2.16) andEq.(2.17),likeEq.(2.6)andEq.(2.7),implythat Df0(sb)Df0(sf)always, whichsimplyreflectsthefactthatthebacksurfaceisfartherfromthegateandhence lesscontrolledbyit.So,thebacksurfacewillcontroltheDIBLwhen fsb>fsf. In anycase,forFD/SOIMOSFETswiththickBOXand eSi/ eox@ 3,Eq.(2.16)andEq. (2.17) yield .(2.18) WithEq.(2.18)andthemodelforSinEq.(2.12),wecanexpress,ingeneral,the reduction of Vt due to the VDS-induced increase in potential, or DIBL: ,(2.19) where Q( r ) istheHeavsidefunctionusedinEq.(2.10),whichaccountsforthe dependenceofDIBL( D Vt/VDS)on Df0(sf)or Df0(sb)asillustratedinFig.2.4.In Eq.(2.19),weuse f0ratherthan f becauseinmostcases,thesignofthedifferences ( f0(sb) f0(sf)) and ( fsb fsf) is the same. 2.4 Insights from the 2-D Analyses Equations(2.12)and(2.19)approximatethesubthresholdslopeand DIBL,respectively,fornanoscaleFD/SOICMOS.Thesemodelsquantifythe behavioroftheexperimentaldevicesin[Suz00],[Cho00],and[Dor02],whichshow smallerSandDIBL,resultinginlowerIoff,astSiisreduced.However,before f0sb ()D tSitSi6toxf+ () Leff 2---------------------------------VDSf0sf ()1 tSi6toxf----------+ D = @ D VtS 10 () lnkBTq --------------------------------6tSitoxfLeff 2----------------1 Qf0sb ()f0sf ()– () tSi6toxf----------+VDS@

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32 applyingEq.(2.12)andEq.(2.19)toanyparticulardevice,weneedtoknowthe relationbetweenthetotalfrontandbacksurfacepotentials, fsfand fsb,whichas notedpreviouslycanbereflectedviathatbetween f0(sf)and f0(sb).Thisrelationcan bederivedbyfirstintegratingtwicethe1-DPoissonequationsatisfiedby f1,i.e.,for the nMOSFET, .(2.20) AssumingnegligibletransversefieldatthebacksurfaceduetothethickBOX,we get from Eq. (2.20) ,(2.21) whereesfisthetransversefieldatthefrontsurface.Now,superimposingthe correspondingperturbedpotentials, Df1and Df0,wecanunderstandtherelation betweenthetotalfrontandbacksurfacepotentials.ForhighNB(>~1018cm-3fortSi<10nm),hightransversefieldinEq.(2.21)definesalargeseparationbetweenthe two1-Dpotentialssuchthatevenaftersuperimposingtheperturbedpotentials,the front-surfacepotentialremainshigherthanthatattheback,i.e.,carriersare predominantlylocatedatthefrontsurface,and Q(f0(sb)f0(sf))=0 inEq.(2.12)and Eq.(2.19).However,forlowNB(i.e.,negligibledepletioncharge),Eq.(2.21)results inanegligibledifferencebetweenthetwo1-Dpotentials,andtheperturbed potentialsbecomethedominantterms.(Notethatthezerofieldforthiscaseimplies thatthecarriersinlong-channelundopedFD/SOICMOSareuniformlydistributed x2 2d d f1x2dx t Si0 t SiqNBe Si -----------x2dx t Si0 t Si= f1sf ()f1sb ()–esftSi2 ------------qNBtSi 22 eSi---------------==

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33 inthefilm,orthatthereissignificantinversionchargeinthebulkoftheSi-film.This isknownasvolume,orbulkinversion,asopposedtosurfaceinversion.)Then,with f0( sb )higherthan f0( sf),asdefinedbyEq.(2.7)andEq.(2.17),weconcludethat larger2-Deffectsatthebacksurfacedefinethedevicesubthresholdcharacteristics, i.e., Q(f0(sb)f0(sf))=1 inEq.(2.12)andEq.(2.19).Thisinsightisreflectedby Fig.2.5,whichcomparesthepredictionsofourmodelswithcorrespondingonesof MEDICIforFD/SOInMOSFETswithlowNB.Notethatthemodelerrorsaresmall (<5%).Qualitatively,thelargerSCEsinundopedMOSFETscanalsobethoughtof asresultingfromaneffectivelythickertoxfdefinedbytheinversion-chargecentroid locatedatthebacksurface;thethickertoxf,however,isnotsimplydefinedbythe ratio of eSi and eox due to 2-D electrostatics. Theseinsightfulresultsexplaintheneedforultra-thintSi,orUTBs,for scaledFD/SOICMOS,whichhasbeenpreviouslyinferredfromexperimental [Suz00],[Cho00],and[Dor02]aswellasnumericaldevicestudiesin[Van02].For lowNB,thescalingismoredemandingbecausetheSCE-controlling Dfsbisthe dominatingperturbation.Contrarily,forhighNB, f0( sf )> f0( sb),andSandDIBLare determinedbythecorrespondingperturbationsatthefrontsurface,whichinfactcan besignificantlysmallerthanthoseattheback.Ourinsightsaresupportedbythe experimentalresultsof[Dor02],whichshowsignificantlylessSCEsinFD/SOI MOSFETswithstronghalo-implantcomparedtothoseindeviceswithmildhaloimplant,bothwithoutanyattentionpaidtomaintainthesameVtinlong-channel devices.ThisisanalogoustoSCEsinsurface-channelversusburied-channelbulk-Si MOSFETs,wherethesurface-channeldevicehasbetterimmunitytoSCEsthanthe buried-channel counterpart [Tau98].

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34 Figure 2.5ComparisonofSCEspredictedbyouranalyticmodelsversus MEDICI-predictedS,DIBL,and Df0(sb)(dividedbyVDS=1.0V)for low-NBFD/SOInMOSFETs.ForeachLeff,(tSi,toxf)wereassumedas indicated.Abruptsource/drainjunctionswereassumedinMEDICI,so Leff@ Lgate=Lmet.Thesmallmodelerrors(<5%)validateourinsights and analysis for nanoscale FD/SOI CMOS. 25.030.035.040.045.050.0 55.0Leff [nm] 70.0 90.0 110.0 130.0 150.0 170.0 190.0Df0(sb)/VDS & DIBL [mV/V] MEDICI Models 60.0 65.0 70.0 75.0 80.0 85.0 90.0 95.0 100.0S [mV] Df0(sb)/VDSDIBL(7nm, 1.1nm) (9nm, 1.1nm) (10nm, 1.5nm)

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35 NotealsothatthemodelsinEq.(2.12)andEq.(2.19)canbeused,in conjunctionwithmeasurements,todeterminetheeffectivechannellengthofactual deviceswithnon-abruptsource/draindopingprofiles.Forexample,forFD/SOI CMOS with thick BOX and low NB, Eq. (2.12) gives (2.22) whichapproaches for D Vt 0 , asinlong-channeldevices.Asnotedpreviously, Leff>Lmet,and,forscaleddevices,itistypicallylongerthanLgatedependingonthe S/Ddopingprofile.WestressLeffherebecauseitistheeffectivechannellengththat governstheSCEsinscaledMOSFETs.Thus,oursubsequentanalysesarebasedon Leff;itsconnectiontoLgateandtheITRSnodes[ITR01]canbeinferredfromthe S/D doping profile (and will be further discussed in Chapter 4). 2.5 Threshold Control via Channel Doping PhysicalinsightsgainedfromouranalysesofSandDIBLimplythathigh Si-filmchannel/bodydopingdensityisadvantageousforscaledFD/SOICMOS. HighNBimpliesthintSitoensurefulldepletion,which,asnotedintheprevious section,isalsoneededforSCEcontrol.TochecktheneededNBandtSivaluesfor acceptableSCEsindeviceswithscaledLeff,weuseMEDICIandUFDG[Fos02a], [Fos04a](bothwithoutaccountingforjunctiontunnelinghere).Usingheavilydoped (n+andp+)polysilicongatesandspecifyingVt@ 0.25VandDIBL @ 100mV/Vyield theNBandtSivs.LeffdesigncurvesshowninFig.2.6,.ForLeff=28nm,extremely highNB~1019cm-3withtSi<6nmisneeded.ApplyingUFDG,whichisconsistent Leff6tSitof1 tSi6tof--------+ 12.8 D VtVDS () + D VtVDS ---------------------------------------------@

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36 Figure 2.6MEDICI-predictedtSiandNBrequirementsforFD/SOICMOSwith abruptsource/draindopingprole(Leff@ Lgate=Lmet),Vt=0.25V (controlledviaNB),andDIBL @ 100mV/V.UFDG-basedanalysesimply similarrequirements,whichbecomemoresevereforshorterLeff.The noted values of toxf track the SIA ITRS [ITR01]. 5.010.015.020.025.030.035.040.045.050.0Leff [nm] 1.0 6.0 11.0 16.0tSi[nm] UFDG MEDICI 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0NB (x1018) [cm-3] toxf = 1.5nm 1.1nm 1.1nm 0.9nm 0.5nm

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37 withMEDICIresultsforLeff=28-50nmasevidentinFig.2.6todeviceswith Leff<28nmandscaledtoendoftheITRS[ITR01]revealsadramaticincreaseinthe requiredNBforLeff<20nm.ForLeff=9nm,NB=6.5x1019cm-3withtSi=3nm wouldberequired.Inaddition,sinceVtisdirectlyrelatedtobothNBandtSiviathe depletioncharge,bothstructuralparametersmustbewell-controlled,whichisa challengingtechnologicaltask[Dor02],[All02].Interestingly,theserequirements regardingNBandtSiareanalogoustothoseofchanneldopingdensityandgradients in classical MOSFETs, which in fact are limiting the scaling of these devices. InconjunctionwithNBcontrol,theeffectsofspatialrandomnessoffinite numbersofdopantatoms[Fra99]canseverelylimitscalingaswell,asintimatedin [Dor02].Also,highNBseverelydegradesthecarriermobilityandvelocity[Ge01], resultinginlowercurrentandlongerpropagationdelays.Finally,althoughneglected here,band-to-bandtunnelingduetoextremelyhighNBwillpreventsuchFD/SOI CMOSscalingforlow-powerapplications.Hence,despitethebettercontrolofSCEs afforded,webelievethattheconventionalapproachtoVtcontrolviahighchannel doping and polysilicon gates is no longer a viable option. 2.6 Threshold Control via Gate Work Function(s) Alternatively,employingmetalgateswithtunedworkfunctions( FG) [Par01],ortuning FGoffully-silicidedgatesbydopingthepre-silicidedpolysilicon gate[Ked02],forVtcontrolallowstheSOIchannelstobeleftundoped(i.e.,with inherent low NB ~ 1015cm-3), eliminating the noted drawbacks of high NB.

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38 2.6.1 Ultra-Thin Bodies for SCE Control BasedonourinsightsinSec.2.4,Eq.(2.12)andEq.(2.19)forSand DIBL, respectively, in undoped FD/SOI CMOS become (2.23) ,(2.24) wherewehaveassumedm @ 1and eSi/ eox@ 3.UsingEq.(2.23)andEq.(2.24)with toxf=1nm,weplotinFig.2.7,theestimatedSandDIBLasfunctionsofLefffor severalvaluesoftSi.Theseplotsshow,asaresultofenhancedgatecontroloverthe backsurface,dramaticsuppressionofthe2-DeffectswhentSiisscaledtoultra-thin values.Moreimportantly,itisclearlyevidentinFig.2.7thatUTBs(andultra-thin toxf)willbeneededfornanoscaleundopedFD/SOICMOS;therequirementis 15-20%thinnerthanthatforthehigh-NBdesign(with10%thickertoxf)discussedin Sec. 2.5, as can be inferred from Fig. 2.6 and Fig. 2.7. 2.6.2 Quantum-Mechanical Effects ForUTBs,quantum-mechanical(QM)carrierconfinementduetothe devicestructureaswellas(transverse)electricfieldleadstocarrier-energy quantization,resultinginincreasedVtanddegradedgatecapacitance(CG)[Ge02b]. WecharacterizetheseeffectsasQMeffects.DuetothelowNBinourFD/SOIdevice design,thestructuralconfinementpredominatesinweak/moderateinversion,andthe S 10 () lnkBTq 1 17tSitoxfLeff 2-------------------1 tSi6toxf----------+ – -------------------------------------------------------------@ DIBL S 10 () lnkBTq --------------------------------6tSitoxfLeff 2----------------1 tSi6toxf----------+ @

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39 Figure 2.7 Quasi-2-DestimatesofSandDIBLinlow-NBFD/SOIMOSFETsas functionsofLefffortSi=15nm,10nm,7nm,and5nm,usingEq.(2.23) andEq.(2.24)withtoxf=1nm.Notethedramaticreductionin2-Deffects astSiisreduced;e.g.,DIBLisreducedfrom140mV/Vto40mV/VandS isloweredfrom83mVto66mVwhentSiisscaledfrom10nmto5nmfor Leff = 40nm. 20.0 40.060.080.0100.0120.0140.0Leff [nm] -450.0 -400.0 -350.0 -300.0 -250.0 -200.0 -150.0 -100.0 -50.0 0.0-DIBL [mV/V] tSi=10nm tSi=7nm tSi=5nm tSi=15nm 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0S [mV]

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40 Vtincreasecanberoughlyapproximatedusingtheinfinite(square)potential-well expression for the lowest-energy subband, i.e., (2.25) whereSisthegateswinggivenbyEq.(2.23),mx/m0istheratioofthecarrier effectivemassinthedirectionofconfinementtothefreeelectronmass(e.g.,0.92for electronsand0.29forholes),andtSiisgiveninnm.( D Vt QMwillbetreated rigorouslyinChapter3.)WiththeincreaseinVtgivenbyEq.(2.25),Ioffnow dependson,inadditiontotheSCEs,theQMeffects.Infact,sincethelong-channel Vt[Lim83](andhenceIoff)forlowNBisonlyweaklydependentontSi(andtoxf)in FD/SOIdevices(aswillbediscussedinSec.3.2.3),Ioffinnanoscaledeviceswith undopedUTBsischaracterizedpredominantlybySCEsandQMeffects,bothof whichdependontSi.ThedependenceofIononSCEsisrelativelyweak,butitis affectedsignificantlybyquantizationthroughtheCGdegradation,evenif D Vt QMis negligible. InadditiontotheirnoteddependencesontheQMeffects,IoffandIonalso dependonthecarriermobility,which,forUTBs,hasbeenfoundtodecreasewith decreasingtSi[Gm99b],[Ess01b].Thismobility( meff)degradationinUTBsis explainedviahigherphononscatteringratesfor(spatially)confinedcarriers,aswell asmoresurface-roughnessscattering[Gm99b],anditcanunderminethe advantagesofFD/SOICMOSwithlowNB.(CarriermobilityinUTBswillbe discussedrigorouslyinChapter5.)Wecategorizethis meffdegradationasan D Vt QMS kBTq () 10 () ln -------------------------------------0.3763 mxm0 () tSi 2---------------------------@

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41 additionalQMeffectsinceitresultsfromthecarrierconfinementandstrongly dependsonQMbehaviorofelectrons,orholes.Wenote,however,thatdespitethis degradation,carriermobilityinundopedUTBMOSFETstendstobemuchhigher than those in bulk-Si counterparts (as we will show in Chapter 5). 2.6.3 HP and LP CMOS Designs with a Midgap Gate AlthoughundopedSOIchannelseliminatetheproblemsregardinghigh NB,viabilityofthisapproachisstronglysubjectedtotheavailabilityofneededgate materialsandtheeasewithwhichtheycanbeintegratedintotheprocessflow.Of course,onenewgatewouldbeeasiertointegratethantwo.Thus,wefirstconsidera midgapgatematerial( FG=4.6eV)forwhichVtinbothCMOSdeviceswouldbe comparable.TooptimizetSi,wefirstnotethatalthoughtheQMeffectsandtheSCEs shiftVtinoppositedirections,theQMeffectsdonotaffectSandDIBL.So, nanoscaledevicedesignsbasedonSandDIBL,ratherthanVt,isolatethe2-Deffects onVt.Hence,todefinetheoptimaltSiforaspecifiedLeff,weuseEq.(2.23)andEq. (2.24),whichhavebeenverifiedbyMEDICIasexemplifiedinFig.2.5withtoxfinferredfromtheITRS.Figure2.8showstheestimated,oroptimal,tSivs.LeffneededforDIBLofabout100mV/VinnanoscaleFD/SOIdevices.Includedinthe figureistheestimatedS,whichremainslessthan80mVandapproximatelythesame foralldesigns.Interestingly,unlikeinclassicalMOSFETs,wheresuppressingDIBL viathinnerdepletionwidthyieldsdegradationinS,suppressingDIBLviathinnertSiinnonclassicalMOSFETsalsoreducesS,definingarelativelylargerdesignspace fornonclassicalCMOS.ThisDIBL-Sbehaviorisfurtherverifiedby3-Dnumerical

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42 Figure 2.8Quasi-2-DestimatesofSilm-thicknessrequirementsforDIBL @ 100 mV/V,innanoscaleFD/SOIMOSFETswithlowNB.Thenoted(ITRS) valuesoftoxfatcorrespondingLeffareusedinEq.(2.24)todeterminethe needed values of tSi. 60.0 65.0 70.0 75.0 80.0 85.0 90.0S [mV] 10.0 20.030.040.050.0 60.0Leff [nm] 1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0tSi [nm] 0.5nm 0.9nm 1.1nm 1.1nm toxf = 1.5nm

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43 devicesimulationsofnanoscalesingle-andmulti-gateMOSFETs[Foss04b], [Yan05]. Then assuming 6toxf ~ tSi, we find, from Eq. (2.22), that (2.26) isneededforviableFD/SOICMOSdevices,consistentwithFig.2.8.Also,notethis ratio will increase (decrease) with increase (decrease) in toxf. With FG=4.6eV,DIBL @ 100mV/V,andconstantSofabout78mV,we roughlypredict,foralldesignsofFig.2.8,Ioff@ 0.3nA/ m musingtheconstant-current (IDS=(Weff/Leff)100nA/ m m)Vt,approximated,withoutquantization,bythe1-D expressionin[Lim83](plustwothermalvoltages);IoffreductionduetoQMeffects canbedeterminedforaspecificdesignusingEq.(2.25).Inanycase,thisoff-state currentistoohighforLPapplications,andsinceIonisunknown,itreflectsadevice thatisquestionableforHPapplications.Thus,tofurtherinvestigateHPandLP FD/SOICMOSdesigns,weuseUFDG[Fos02a],whichinessenceisacompact Poisson-SchrdingersolverforDGMOSFETs.Inadditiontoitsrigorous accountingsofSCEsintheentirefilm[Yeh95]andQMeffects[Ge02b],themobility dependenceontSi[Gm99b],[Ess01b]ismodeledinUFDGbyextendingthe commonlyusedempiricalmodelfor meff[Chi01a],[Chi01b].(Aphysicalmodelfor meffinUTBswillbedevelopedinChapter5.)TheUFDG-predicted meff(tSi,eeff)in Fig. 2.9 exemplifies the noted mobility modeling as used herein. UsingUFDG,wenowfocusonLeff=35nmFD/SOICMOShavinga singlemidgapgate.Asnotedpreviously,LgatecorrespondingtoLeffdependsonthe S/Ddopingprofile.FromFig.2.8,our35nmFD/SOICMOSdesignwiththickBOX LefftSi--------5.0 ~

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44 Figure 2.9UFDG-predictedmobilitydependenceontSiandeffectivetransverseeld forFD/SOInMOSFETswiththickBOX.Regionswherethemobilityis denedpredominantlybyphononscatteringandsurface-roughness scattering are indicated. 104105106eeff [V/cm] 100.0 200.0 300.0 400.0 500.0 600.0 700.0meff [cm2/V-s] tSi = 50nm tSi = 25nm tSi = 15nm tSi = 10nm tSi = 7nm tSi = 5nm Phonon scattering-limited Surface-roughness scattering-limited

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45 (200nm)comprisestoxf=1.1nmandtSi=7nm,which,with D Vt QM@ 10mV,is estimatedtogiveIoff@ 0.2nA/ m m.BecauseofdifferencesinSCEmodelinginUFDG (whichaccountsfor2-DeffectsintheentireUTB)andinEq.(2.12)andEq.(2.19) (whichaccountsfor2-Deffectsonlyatthebacksurface),wefind,bycalibrating UFDGtothenotedDIBLandS,thatLeffshorter(32nm)butcomparabletothe35nm defined by Eq. (2.22) is needed. UFDG-predictedIoff(tSi)andIon(tSi),withandwithoutQMeffects,forthe noted35nmnMOSFETareplottedinFig.2.10.Withoutquantization,Ioff(tSi)is definedpredominantlybythe2-Deffects,whiletheconstantIon(tSi)reflectsthe1-D Vt,whichisvirtuallyindependentoftSi.ThesignificanceofQMeffectsonIonand Ioff,whicharehigherinthepMOSFETsduetosmallermx/m0forholes,isnow apparentinFig.2.10.ThelargeIon/Ioffratio(~106)isadirectresultofthelowS inherent in FD/SOI devices and velocity overshoot [Ge01]. TheresultsinFig.2.10furthersuggestthatour35nmFD/SOICMOSwith tSiscaleddowntoabout5nm(notethattSiwillbeamultipleoftheatomiclayer, 0.28nm)couldbeusefulforLPapplications.Incontrast,therelativelysmallincrease inIonfortSi>7nminFig.2.10meansthatbetterHPCMOScannotbeachievedby increasingtSi.Further,asexemplifiedinFig.2.11,theconstantslope( @ 0.83decade/ nm)ofIoff(tSi)inFig.2.10meansthatanacceptablevariationintSi( D tSi/tSi),defined bymaximum-allowedvariationinIoff,isinverselyrelatedtothenominalvalueoftSiused,andthat,becauseoftheinsensitivityofthe1-DVttovariationsinnegligibly smalldepletioncharge,itcanbesubstantiallylargerthangenerallypresumed [All02].Forexample,ournominalHPnMOSFETwithtSi=7nmcantolerateabout

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46 Figure 2.10UFDG-predictedtSidependenceoftheoff-stateandtheon-statecurrents atVDS=1.0VinourLeff=35nmFD/SOInMOSFETwith(solid)and without (dashed) QM effects. 0.5 0.6 0.7 0.8 0.9 1.0 1.1Ion [mA/ m m] 4.05.06.07.08.09.010.0tSi [nm] 10-1210-1110-1010-910-810-710-6Ioff [A/ m m] @ 0.83 decade/nm @ 0.04 (mA/ m m)/nm

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47 Figure 2.11UFDG-predictedIoff(tSi)andtolerablevariationintSi( D tSi/tSi)foran allowedorder-of-magnitudeincreaseinIofffortheLeff=35nmFD/SOI nMOSFETatVDS=1.0V.AlsoshownisthetSirequirementforLP CMOSifthesamegatematerialistobeemployedforbothHPandLP technologies. 10.0 15.0 20.0 25.0 30.0D tSi/tSi [%] 4.05.06.07.08.09.0 10.0tSi [nm] 10-1310-1210-1110-1010-910-810-710-6Ioff [A/ m m] LP

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48 15%variationintSiforanallowedincreaseinIoffofanorderofmagnitude,and relativevariationintSiasexcessiveas25%canbeaffordedforourLPnMOSFET withtSi=5nm.Also,notethattheabsolutevariationintSifor10xincreaseinIoffremainsconstantat D tSi@ 1.5nm.Finally,fortheHPnMOSFETwithtSi=7nmin Fig.2.10,thevariationinIonwithtSiissmall @ 0.04(mA/ m m)/nm,definedlargelyby the tSi dependence of the QM effects. 2.6.4 UFDG/Spice3-Based Performance Projections Toexaminetheviabilityofour35nmFD/SOICMOSwithtSi=7nmand midgapgateforHPapplications,aswellastodefinethespeed-powertradeofffor theLPdesignwithtSi=5nm,speedevaluationsvia9-stageunloadedCMOS-inverter ring-oscillator(RO)simulationsusingUFDG/Spice3aredone.WeassumeS/D parasiticcapacitancesdefinedbytoxfandrepresentativeofG-S.Doverlapsof20% ofLeff,andS/Dresistancecomparabletobulk-CMOSvalues(200 W-m mfor nMOSFETsand350 W-m mforpMOSFETs).TheUFDG/Spice3-predicted propagationdelay( td)forseveralsupplyvoltages(VDD)andthetwonotedSi-film thicknessesareshowninFig.2.12.AtVDD=1.0V, td=4.3psreflectsgoodspeed performanceforourHPCMOSdeviceswithsingle,midgap-gatedesign,andthe predictedspeedisgoodforlowerVDDaswell.Figure2.12alsoreflectsgood speed-powertradeofffortheLPdesign,withthepredicted tdbeingnotmuchlonger thantheHPdelayforVDD= 1.0V.Thepredicteddelaysareroughlyconsistentwith the(longer)measuredonesreportedin[Dor02]wherethe(equivalent)gateoxideis twiceasthickaswhatweassumed.Thelonger tdfortSi=5nmisexpectedfromthe lowerIon(Fig.2.10)duetohigherQMeffectswhichbecomemorepronouncedfor

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49 Figure 2.12 UFDG/Spice3-predicteddelay/stageof9-stageunloadedCMOS-inverter ringoscillatorsusingourLeff=35nmFD/SOICMOSdevicedesignsfor HP and LP applications.VDD [V]td [ps/stage] 0.650.700.750.800.850.900.951.001.05 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 tSi = 5nm tSi = 7nm Low-Power High-Performance

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50 lowerVDD.Therefore,tSinotonlygovernsSCEs(andhenceIoff)innanoscale FD/SOI CMOS but also controls, via the quantization, Ion and speed performance. 2.6.5 HP and LP CMOS Designs with Dual Metal Gates Althoughourmidgap-gatedesignyieldsacceptableHPaswellasLPFD/SOI CMOSwithultra-thintSi,wenotethatbothtechnologiescouldbefurtheroptimized,i.e., higherIonforHPandthickertSiforLPCMOS,byusingdualmetalgateswithtunedwork functions.Infact,sinceIoffinourDIBL-basedmidgap-gatedesignis ~ 10-10A/ m m (Fig.2.10),twonewmetalgates,symmetricaboutthemidgapvalue,selectedforHP applicationscanalsobeappliedtorealizeLPdevices,orviceversa,simplybyswitching thegatesonthetwoCMOSdevices,which,ifneeded,canbefollowedbyscalingoftSito achievethedesiredIoff.ThisisillustratedinFig.2.13forour35nmFD/SOIdesign.Such designexibilityimpliesviabilityofscaledFD/SOICMOSforallapplications(including system-on-chip) while minimizing the number of new gate materials needed. UsingUFDGandrequiringIoff@ 100nA/ m mforHPnMOSFETs,wefind that( FGn, FGp)=(4.42eV,4.88eV)isneeded,anditrendersIon@ 1.0mA/ m mat VDS=1.0VfornMOSFETs;IDSinpMOSFETsisabouthalfthatofnMOSFETs. Then,ifweexchangethetwogatematerialsonthetwoCMOSdevices,UFDGpredictsIoff@ 2.2pA/ m mandIon@ 0.42mA/ m matVDS=1.0VfornMOSFETs,which isacceptableforLPtechnologies.Hence,dualmetalgateswith FG=4.6eV0.18eV have increased Ion for HP CMOS and allowed tSi = 7nm for LP CMOS. Tocompareperformanceofthedual-metal-gatedesignswiththatof midgapgate,wesimulate9-stageunloadedCMOS-inverterROusingUFDG/Spice3. Figure2.14showsthepredicteddelayversusVDDsuperimposedonthoseofmidgap-

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51 Figure 2.13Dual-metal-gateFD/SOICMOSdesignsbasedonUFDG-predicted Ioff(tSi).For( FGn, FGp)neededforHPnMOSandpMOSdevices, respectively,LPnMOSandpMOSdevicescanberealizedbyswitching thegatesonthetwoCMOSdevices,i.e,using( FGp, FGn),followedby little tSi-scaling to tune to desired Ioff. ( FGn, FGp) ( FGp, FGn) 4.05.06.07.08.09.0 10.0tSi [nm] 10-1310-1210-1110-1010-910-810-710-6Ioff [A/ m m]

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52 Figure 2.14UFDG/Spice3-predicteddelay/stageof9-stageunloadedCMOS-inverter ringoscillatorsusingourLeff=35nmHPandLPFD/SOICMOSdevice designshavingdual-metalgates(solid)andmidgapgates(dashed).All designs except that of LP CMOS with midgap gate have tSi = 7nm. 0.650.700.750.800.850.900.951.001.05VDD [V] 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0td [ps/stage] LP HP Dual metal gates Midgap gate tSi = 5nm

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53 gatedesigns.WhiletheHPtechnologywithdualmetalgatesshowsmuchlower delaysthatarealsolessdependentonVDD,power-speedtradeoffforLPtechnology issignificantlydegraded.ThisdegradationisaresultofachievingIoffinthedualmetal-gatedesignviarigidlyshiftingtheIDS-VGScharacteristics,i.e.,increasingthe 1-DVt,ratherthansuppressingSCEsandexploitingQMeffectsasinthe midgap-gatedesign.Wealsonotethatthisdegradationisonlyrelative,becausethe power-speedcharacteristicsforthedual-metal-gatedesignarecomparabletothatof LPbulk-SiandPD/SOICMOS.Thus,forLPtechnology,thereexistsatradeoff betweencontrollingVt,andhenceIoff,viatSi,whichresultsinbetterperformance butscalinglimitatlongerLeff(aswewillshownext),versusVtcontrolbydualmetal gates,whichresultsinlowerperformancebutextendedscalinglimit.ForHPdevices, however,usingdual-metalgatesistheoptimalchoicewithrespecttoboththescaling limit and CMOS performance. 2.7 Scalability of FD/SOI CMOS Figures2.7and2.8clearlyshowtheneedforUTBsinnanoscaleFD/SOI CMOS,and,foroursinglemidgap-gatedesign,Figs.2.9-2.12reflectthesensitivity ofperformancetovariationsintSi.Indeed,itisthefilmthicknessthatcontrolsand definestheviabilityofFD/SOICMOS.ForourLeff=35nmandnominaltSi=7nm CMOSdesign,Fig.2.10showsthata15%increaseintSiwouldbeacceptablefor anallowedorder-of-magnitudeincreaseinIoff,andFig.2.12showsthata15% decreaseintSiwouldincreasetheRO tdbylessthan20%.Hence,atthisnode, substantive D tSi/tSi=15%wouldbetolerable.FornominaltSi=5nm,whereasthe

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54 acceptabletSivariationforIoffcontrol(Fig.2.10)isactuallylarger,thatfor acceptablespeedperformance(Fig.2.12)andIon(Fig.2.10)issmaller(butdoable [All02]).However,fortSi<5nm,thequantizationduetostructuralconfinement [Ge02b]issevere,and,withVDD<1V,itsdegradingeffectonspeedandits sensitivityto D tSiwouldquicklybecomeprohibitiveasimpliedbyFig.2.12.Hence, basedontheseinsights,andontheultra-thintSimanufacturingburdenimplied,we inferapragmaticlowerlimitontSiofabout5nm.Then,forHPapplications, Figs.2.7and2.8showthescalinglimitofconventional,single-gateFD/SOICMOS withmidgapgatetobeLeff@ 25nm.ForLPapplications,thelimitisabitlongerat Leff@ 35nmforwhichIoffcontrolviatSiimpliesneedfortSi=5nmasnoted previously.Theuseoftwodifferentgatematerialswithworkfunctions4.6eV 0.18eVtobettercontrolVt(andIoff)significantlyenhancestheHPCMOS,andit alsoextendsthescalinglimitoftheLPCMOStoLeff@ 25nm,butwithadegraded (relative to midgap-gate design) power-speed tradeoff. 2.8 Scalability of Other DG CMOS WiththescalinglimitofFD/SOICMOSnearLeff@ 25nm,wenowextend ouranalysisforFD/SOICMOStodefinethescalabilityofSDGandADGCMOS, includingFinFETs.WealsodefinethescalabilityofIGFinFETs,orMIGFETs,or GP CMOS. 2.8.1 Symmetrical DG MOSFETs Inasense,SDGMOSFETsaresimilartotheconventionalFD/SOI devices.Forexample,esb@ 0forFD/SOICMOS,andforSDGdevices,thesymmetry

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55 defineszerotransversefieldatthecenterofthebody,i.e.,ec=0.Notethatwhilethe symmetryrendersec=0,ingeneral,itdoesnotmeanthatthepotentialismaximum there.Infact,forheavilydopedSDGdevices,highesfyieldstwopredominant surfacechannelswithecstillzero.However,whenthebodyisleftundoped,thelongchannelpotential( f1(x))alongthebodyisuniform(i.e.,bulkinversion)analogous totheFD/SOIMOSFETs,andthe2-Dperturbationsdefinethepotentialdistribution inshortchanneldevices.Sinceec=0always,SCEstendtobehighestatthecenter ofthefilm,andhence,theleakiestpathoccursatx=tSi/2,definingaburiedchannel. Using Dec=0duetothedevicesymmetry,wefindthatSandDIBLinEq. (2.23)andEq.(2.24),respectively,withtSi tSi/2characterizeSCEsinSDG CMOS.Then,assuming6tox@ tSi/2,wefindthatforDIBL @ 100mV/V(S @ 80mV), (2.27) isneeded.NotethatbecauseoftheincreasedimmunitytoSCEs,SDGMOSFETscan alsoemploythickertox.Finally,withapragmaticlowerlimitontSiof5nm,defined by QM effects, Eq. (2.27) defines the scaling limit of SDG CMOS at Leff ~ 13nm. 2.8.2 Asymmetrical DG MOSFETs Dependingonthedeviceasymmetry,the1-DPoissonequationrenders hightransversefieldinADGdevices.Forexample,inundopedADGMOSFETwith n+-andp+-polysilicongates,toxf=toxb=tox=1nm,andtSi=20nm,wegetesf=esb@ 5x105V/cm.Suchhigh"built-in"fielddefines,analogoustohighNBinFD/SOIand SDGdevices,apredominantsurfacechannel.ThismeansthattheSCE-controlling perturbation is that at the front surface, which for toxf = toxb = tox is LefftSi--------2.5 ~

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56 (2.28) from Eq. (2.16). Also, we find that S is given by .(2.29) MultiplyingEq.(2.28)withEq.(2.29),anddividingbyln(10)kBT/qgivestheDIBL in ADG MOSFETs. Then, assuming 6tox@ tSi, we find that (2.30) is needed for DIBL @ 100mV/V and S @ 80mV. WhilecomparisonofEq.(2.30)toEq.(2.27)impliesthatADGCMOScan bescaledfartherthanSDGCMOS,the1-DVtoflong-channelADGCMOShas muchstrongertSidependencethroughthedeviceasymmetry.Thisdependenceis augmentedbylargerQMeffectsasdefinedbythehightransversefield.Thisis exemplifiedbySCHRED-[Vas00]predictedVtand D VtQMversustSiinFig.2.15(a) andFig.2.15 (b),respectively,forSDGandADG nMOSFETswith{100}-Sisurface. However,assumingthatthestructuralvariationscanbecontrolledappropriately, Eq. (2.30) with tSi = 5nm defines scaling limit of ADG CMOS at Leff ~ 11nm. 2.8.3 DG MOSFETs with Independently Biased Gates InSec.2.2,wearguedthatscalingtheBOXisnotaviableoption.While thisisthecasewhenconsideringperformanceandmanufacturabilityrelativeto conventionalFD/SOICMOS,recentlytherehasbeenanewinterestinDGFinFET f0sf ()D eSitSih02Cox------------------3tSitoxLeff 2--------------VDS@ = S 10 () lnkBTq 18.4tSitoxLeff 2 () – ----------------------------------------------@ LefftSi--------2.2 ~

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57 Figure 2.15SCHRED-predicted(a)thresholdvoltageand(b)quantization-governed Vt-shiftversusthebodythicknessinSDGandADGnMOSFETswith {100}-Sisurfaceandtoxf=toxb=1nmforbothstructures;midgapgate (n+and p+-polysilicon gates) is (are) specied for SDG (ADG) devices. (a) (b) 0.04.08.012.016.020.0tSi [nm] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Vt [V] SDG ADG 0.04.08.012.016.020.0tSi [nm] 0.0 0.1 0.2 0.3 0.4D Vt QM SDG ADG w/o QM w/ QM

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58 structureswithbothgatesbiasedindependentlyfornovel(analog)circuitdesignsas wellastoreducelayoutareawhileachievingperformancecomparabletoclassical CMOS.ThesenewstructuresarereferredtoasIGFinFETs[Fri04],orMIGFETs [Mat04].Inthissection,wedefinethescalabilityofsuchDGstructuresby considering a SDG MOSFET with a grounded back gate. Withundopedbodies,wefind,fromthe1-DPoisson’sequation,esfVGfS.Then,forrealisticVtwithacceptableIoff,wefindthatthetransverse fieldatVtishighenoughthatevenaftersuperimposing Df1and Df0,thefrontsurfacepotentialremainshighest,i.e.,wegetapredominantsurfacechannel.(We notethatwithesfVGfS,thelocationoftheleakiestpath,orthechargecentroid, caningeneralvarysignificantlyinsubthreshold.Thus,ouranalysishereonly reflectsatrend.)Hence,SCEsinIGFinFETswillbesmallerthanthatinSGFD/SOI MOSFET with thick BOX. From Eq. (2.16), we get, for toxb = toxf, ,(2.31) whichissameasthatforADGdevicesinEq.(2.28).However,becausethetwogates areoperatedindependently,thelargeCB(eff)inEq.(2.9)substantivelydegradesSof long-channel IG FinFETs, and so Eq. (2.8) and Eq. (2.11) give .(2.32) TheDIBLcannowbecomputedbymultiplyingEq.(2.31)andEq.(2.32)normalized byln(10)kBT/q,andduetothelarge1-DS(orm>1),itishigherthanthatinthe f0sf ()D eSitSih02Cox------------------3tSitoxLeff 2--------------VDS@ = S m10 () lnkBTq 1m8.4tSitoxLeff 2 () – ---------------------------------------------------@

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59 SDGFinFETcounterpart.Assuming6tox@ tSi,whichgive sm=1.33,likeinclassical MOSFETs, we find that (2.33) isneededforDIBL @ 100mV/V,orS @ 105mV,whichisworsethanthatforSDG and ADG MOSFETs due to the large CB(eff). WhilethehighertransversefieldinIGFinFETscandefinehigher immunitytoSCEsrelativetothatthaninSGFD/SOIMOSFETs,it,likeinADG MOSFETs,alsoincreasesQMeffectsanddevicesensitivitytovariationintSi. However,stillassumingalowerlimitontSi=5nm,Eq.(2.33)definesthescalability of independently-biased DG MOSFETs to Leff ~ 15nm. 2.9 Summary Quasi-2-Ddeviceanalyses,supportedby2-Dnumericaldevice simulations,andprocess/physics-basedcircuitsimulationsofconventionalFD/SOI CMOSweredonetoinvestigatescalabilityandperformanceofthetechnology.The quasi-2-Danalyses,whichareapplicabletoDGCMOSaswell,yieldedanalytic expressionsforDIBL,S,andtheeffectivechannellength(Leff)thatgovernsthe 2-Deffects(SCEs).Further,basedpredominantlyonthesensitivityofthedevice characteristicstothevariationsinNBandtSi,wearguedthatVtcontrolviahigh body/channeldopingandpolysilicongatesisnolongeraviableapproach,andhence thatlowNB(undopedSOI)withVtcontrolviagateworkfunctionsisessentialfor nanoscaleFD/SOICMOS.Then,ouranalyticmodelsrevealedandexplainedthe LefftSi--------3.0 ~

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60 needfortSi<10nminFD/SOICMOSwithLeff<50nm,whichpreviouslyhadonly beeninferredfromexperimentaland2-Dnumericaldevice-simulationresults. Further,thesignificanceofQMeffects,duetocarrierconfinementintheUTBs,and its importance in design were noted and stressed. UnloadedCMOS-inverterring-oscillatorsimulations,donewithUFDG, whichproperlyaccountsfortheSCEs,theQMeffects,andthemobilitydegradation inultra-thinSi-filmchannels,predictedveryshortdelaysforasinglemidgap-gate designofLeff=35nmFD/SOICMOSwithtSi=7nmaimedatHPapplications,while thesamedeviceswithathinnerSOIfilm(tSi=5nm)wereshowntobeusefulforLP applications.ThesecircuitsimulationsalsoshowedthatwhilereducingtSisuppressesSCEs,yieldingIoffcontrol,italsodegradestheswitchingspeedasaresult oftheQMeffects.Thequantizationeffect,anditssensitivitytovariationsinfilm thickness,werefoundtodefineapragmaticlowerlimitofabout5nmfortSi, implyingscalinglimitsforFD/SOICMOS,withasinglemidgapgate,ofLeff@ 25nm forHPapplications,andLeff@ 35nmforLPapplications.Wefurthernotedthatif dualmetalgatesareusedtosatisfyIoffrequirementsofLPapplications,thenthey canextendthescalinglimitofLPtechnologytoLeff@ 25nm,andcanalsobeused to realize HP devices by switching the gate materials on the two CMOS devices. Finally,weextendedourinsightsandanalysestoexaminethescalability ofotherDGCMOS.WefoundthatADGMOSFETshavethehighestimmunityto SCEs,followedbySDGMOSFETsandDGMOSFETswithindependentlybiased gates,allwithhigherimmunitytoSCEsthaninSGFD/SOIMOSFETs.Although neitherSDGnorADGCMOSwerefoundtobescalabletoLeff@ 7nmwithtSi 5nm, they can be scaled to Lgate@ 7nm if G-S/D underlap is employed.

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61 CHAPTER 3 ON THE "NONCLASSICAL" PHYSICS OF UNDOPED DG MOSFETs 3.1 Introduction In undoped UTBs,neededforviablenonclassicalnanoscaledevices,the naturalp-typeimpuritiesinundopedSidefinep-typeNB~1015cm-3.Interestingly, withsuchlowNB,theextremelysmallchannelvolumeofananoscaledevicewith UTBphysicallyforcesthenumberofimpuritydopantspresentinthebodytobe absolutelyzero(NB=0),yieldingan intrinsic body.Asasresult,the"classical" depletionapproximation[Sze81],[Lim83],[Tau98],whichhasbeenusedto characterizeMOSFETcharacteristicsinthepastbecomesinvalid,asdoesthe conventionaluseof2 fB(twicetheFermipotential)forVtcharacterization[Sze81], [Lim83],[Tau98].Further,asintimatedinChapter2,asubstantialportionofthe (integrated)inversionchargedensityislocatedinthebulkofthebodyforlongchannelundopedSDGandthick-BOXFD/SOIMOSFETs.This“nonclassical” condition,knownasvolume,or bulkinversion (asopposedtosurfaceinversion) [Bal87],[Kim04],invalidatestheclassicalcharge-sheetapproximation.Inaddition, becausethetransverseelectricfield(ex)inanundopedUTBismuchlowerthanina heavilydopedUTB,theclassicalassumptionofpredominantcarrieroccupationof the ground-state subband energy also becomes invalid, as we will show here. ThefundamentalphysicsofoperationofthegenericDGMOSFET (Fig.1.5),includingthegate-gatechargecouplingeffectsinanFDbody,werefirst

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62 modeledbyLimandFossum[Lim83]morethantwodecadesago.However,because itisbasedonthenotedclassicalassumptions,andneglectstheQMeffects,itsuse, andthatofitsderivatives,forundoped/intrinsic-UTBMOSFETshasphysical limitations. Thus,inthischapter,weanalyzeanddiscussthe“nonclassical”physicsof genericDGMOSFETswith undoped / intrinsic UTBs.Effectsofcarrierdistribution intheUTBandinthecarrier-energysubbandsonMOSFETcharacteristics (includingSCEs)constitutethenonclassicalphysicsdiscussedhere.Wefirst examinethenonclassicalsubthresholdcharacteristicsby(i)generalizingthe classicalchargecouplinganalysis[Lim83]forgeneric,intrinsic-bodyDG MOSFETs,includingindependent-gate(IG)FinFETs[Fri04],[Mat04],(ii) presentingphysicalinsightsonhowcarrierdistributionintheUTBaffectsthe couplingandtheSCEs,and(iii)consideringQMeffectsresultingfromtSi-induced structuralconfinement(SC)[Ge02b]andex-inducedelectricalconfinement(EC). Furthermore,immunitytorandomdopanteffectsisbrieflyexaminedintermsof variationintheaveragebodydopingdensity,orNB.Instronginversion,wemainly investigatetheeffectsofbulkinversion,augmentedbyQMeffects,ontheintegrated inversionchargedensity(Ninv)andIonofSDGnMOSFETsasdefinedbythe inversion-layercapacitance(Ci),effectivecarriermobility( meff),andvelocity saturationandovershoot.Finally,theeffectivedevicewidth(Weff)innonplanar multi-gateMOSFETs,suchasDGandTGFinFETs[Ked01],[Doy03],isaddressed in consideration of bulk inversion.

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63 3.2 Subthreshold Analysis: Generalized Char ge Coupling and Threshold V oltage Toexaminethenonclassicaleffectsinthesubthresholdregion,wefocus onthegate-gatechargecouplingandthethresholdvoltage,Vt.Therearetypically twodefinitionsofVt:thelinearlyextrapolatedVtandtheconstant-currentVt[Tau98].Thelatterismostoftenemployeddueitssimplicity.And,ouranalysisin thissection,basedonsubthresholdanalysisfacilitatedbythedepletion approximation,isconsistentwithit.However,asnotedearlier,becausethedepletion approximationbecomesinvalidforintrinsic-bodydevices,webeginwitha discussionontheviabilityofdepletionapproximation-basedmodelssuchasin [Lim83] for intrinsic-body devices. 3.2.1 The Depletion Approximation Forthesakeofclarity,weonlyconsiderSDGnMOSFETshere,andnote thattheresultsaregenerallyapplicable.Then,beginningwiththeclassicalcaseof nonzeroNB,applicationofthe1-DGauss’slawinthebodyandoneofthetwo surface-boundary conditions give ,(3.1) whereVGSisthegatebias, FMSisthegate-bodyworkfunctiondifference, fsfisthe front-surfacepotential,Cox= eox/tox,QBisthedepletionchargedensity,Qiisthe inversion-chargedensityatthevirtualsource,andthefactorof2inthelasttermis duetotheSDGdevicesymmetry;weareneglectingtheoxideandinterfacialcharges. (Wenotethatalthoughtheterms“inversion”and“accumulation”losetheirphysical VGSFMS– fsfQBQi+ 2Cox------------------- – =

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64 meaningforintrinsic-bodyMOSFETs,weusethemhereasamatterofconvention/ simplicity.)Thedepletionapproximationassumes|QB|>>|Qi|,whichisobviously invalidwhenthebodyisintrinsic.However,consideringthattypical|Qi|>tSi.Thissimplymeansthatfreecarrierscannotsufficientlyscreentheapplied potentialwithintSi,andhencethepotentialdistributionintheentirebodywillbe independentofQi,asalsoassumedinthedepletionapproximation.Finally,wenote thatsinceQianditseffectonthepotentialdistributionarenegligible,thechargesheet approximation is superfluous for subthreshold/weak-inversion analysis. QiCox-------fsf

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65 3.2.2 Classical Theory of Charge Coupling and Threshold Voltage FromourinsightsinSec.3.2.1,gate-gatechargecouplinginweakly invertedintrinsic-bodyDGMOSFETsiscorrectlycharacterizedbythetwowellknownequationsin[Lim83],derivedfromsolvingthe(1-D)Poissonequationwith the depletion approximation: (3.3) ,(3.4) whereVGfS(VGbS),VFBf(VFBb),andCoxf(Coxb)isthefront-(back-)gatebias, flatbandvoltage,andoxidethickness,respectively, fsf( fsb)isthefront-(back-) surfacepotential(referencetoahypotheticalneutralbody),andCb= eSi/tSi. Equations(3.3)and(3.4)clearlyshowthattheabilityofVGfStomodulate fsf(or fsb),andhenceQi,dependsonVGbS,andviceversa,whichistheessenceofgategatechargecouplinginFDDGMOSFETs.(Althoughthecharge-sheet approximationisassertedin[Lim83]forstrong-inversionanalyses,itdoesnotaffect the subthreshold/weak-inversion modeling, as argued in Sec. 3.2.1.) Previously,Eq.(3.3)with fsbfromEq.(3.4)and fsf=2 fBcharacterized the threshold voltage at the front surface (Vtf) [Lim83]: ,(3.5) where VGfSVFBf CbCoxf---------+ fsfCbCoxf---------fsb– = VGbSVFBb– CbCoxb----------fsf CbCoxb----------+ fsb+ = VGfSVtfVFBfrVFBb1r + () 2 fB++ Qb2 -----1 Coxf---------r Coxb----------+ rVGbS– – ==

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66 (3.6) andCB(eff)istheeffectivebodycapacitanceanalogoustothedepletioncapacitance (Cd)inbulk-SiandPD/SOIMOSFETs.Foraccumulatedorinvertedbacksurface, freecarrierseffectivelyscreenVGbS,andVtfbecomesindependentofVGbS[Lim83]. ThisVtf(VGbS)dependenceisshowninFig.3.1,whereanddefinethe back-surfaceaccumulationandinversiononsetconditions,respectively,fromEq. (3.4) with ( fsf = 2 fB and) fsb = 0 and fsb = 2 fB, respectively [Lim83]. However,whenthebodyisintrinsic,2 fB=0,andVtfinEq.(3.5)becomes nondefinitive.Also,duetothelackofanyimpurities,theassumptionof fsb=0for back-surfaceaccumulationbecomesnonphysical.Furthermore,whenthetwogates areoperatedindependently(asinIGFinFETs)andVGbS>inFig.3.1,Vtis definedbyapredominantbackchannel,andVt Vtf.Thus,tocharacterizeVtfor suchconfigurations,weneedtoaccountforcarrierdistributionthroughoutthebody, anddoingsowillalsoincorporatebulkinversion.Finally,wenotethatsinceVt VtfforVGbS>,thechargecouplingfromEq.(3.5)isalsoinapplicable,asis apparentfromMEDICI-predictedIDS-VGfSforanIGFinFETwithvariousVGbSin Fig.3.2.Hence,wenowgeneralizethenotedclassicalanalysisbyincorporatingthe nonclassical effects of carrier distribution in the SOI film. 3.2.3 Generalized Charge Coupling and Threshold Voltage Conventionally,Vtischaracterizedbybandbendingof2 fB.However, physically,Vtreflectsatransitionfromweakinversion,whereQiisexponentially dependentonthegatebias,tostronginversion,whereQiislinearlydependentonthe r CoxbCbCoxfCoxbCb+ () ---------------------------------------CBeff ()Coxf---------------= VGb AVGb IVGb IVGb I

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67 Figure 3.1QualitativedepictionoftheVGbSdependenceofthefront-surface thresholdvoltage,Vtf[Lim83].Fordepletedbacksurface, characterizedby
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68 Figure 3.2MEDICI-predictedIDS-VGfSforvariousVGbSforlong-channelIG FinFETwithtoxf=toxb=2nm,tSi=25nm,undopedbody,and n+-polysilicon gates; constant mobility is specified. -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5VGfS [V] 10-1210-1110-1010-910-810-710-610-5IDS [A/ m m] VGbS= -1.0V to 0.0V

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69 gatebias.ThistransitionregionofmoderateinversionwasfirstanalyzedbyTsividis [Tsi82]forbulk-SiMOSFETsintermsofCi,Cd,andCox.ForFDUTBMOSFETs, Cd CB(eff),andthemoderate-inversionregioncanbecharacterizedby0.1(Cox+ CB(eff))>Ci>10(Cox+CB(eff))[Tsi82].Thus,forourgenericmodel,wedefineVtsuch that ,(3.7) whichcorrespondstoaconstantQi,reasonablyquantifiedviasubthresholdanalysis, and is thus in line with the constant-current Vt. 3.2.3.1 Inversion-Charge Density at Threshold SinceQidoesnotaffectthepotentialdistributioninthesubthreshold condition,Poisson’sequationintheintrinsicbodyimpliesthatthetransversefieldin theundopedSi-filmbodyisapproximatelyconstant.Thus,thepotentialdistribution is given by ,(3.8) whereExCisthenotedconstanttransversefield.SubtractingEq.(3.4)fromEq.(3.3), and using Eq. (3.8), we find .(3.9) Using Eq. (3.8), we get, for nMOSFETs (and similarly for pMOSFETs), ,(3.10) CiCoxCBeff ()+ = f x ()fsfExCx – = ExCVGfSVGbS– () VFBfVFBb– () – eSieox () toxftoxb+ () tSi+ ----------------------------------------------------------------------------= Qiq –nx () x d0 tSinikBT ExC-------------- – q fsfkBT --------1 qExCtSikBT ------------------ – exp – exp ==

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70 where,classically,n( f ) @ niexp(q f /kBT)andniistheintrinsiccarrierconcentration. Interestingly,notethatwhenExCislarge,Eq.(3.10)becomesanalogoustothatfor classicaldevices[Tau98],andforExC 0,asintheSDGMOSFET,first-order expansion of the last term in Eq. (3.10) yields ,(3.11) i.e.,bulkinversion.(Also,sincetherearenoimpuritiesinthebody,thereexistsa weak-accumulationregion,liketheweak-inversionregion,thatisdescribedbyEq. (3.10)with fsf fsf,and,ofcourse,Qi Qa,theaccumulation-chargedensity, with a change in the sign.) TodetermineQiatVt,i.e.,atCi=(Cox+CB(eff)),wefirstderive,fromEq. (3.10), the inversion-layer capacitance as .(3.12) FromEq.(3.8)andEq.(3.10),wenotethat f inEq.(3.12)isarbitrary.Then,for (SiO2)gateoxidesthinnerthan3nm,Ci@ (Cox+CB(eff))givesQi(Vt)~-q1011cm-2. 3.2.3.2 Generalized Threshold Voltage Becauseofthepresenceofthesecondgate,thereare,ingeneral,countably infinitepairsof(VGfS,VGbS)thatsatisfythenotedchargecondition.FromEq.(3.3) and Eq. (3.4), these pairs satisfy ,(3.13) which is same as Eq. (3.5) with 2fBfc, where Qiqni– q f sfkBT --------tSiexpqntSi– == CidQid f -------- – dQid fsf--------- – q kBT --------Qi== VGfSVt= () rVGbS+1r + ()fcVFBfrVFBb++ =

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71 (3.14) isevaluatedfromEq.(3.10).SinceourmodelforQiisgeneric,theVtmodelinEq. (3.13) is applicable even for VGbS > in Fig. 3.1; we will illustrate this later. ForSDGandthick-BOXFD/SOIMOSFETs,ExC 0,andwefind,from Eq. (3.11) and Eq. (3.14), (3.15) and, from Eq. (3.13), .(3.16) Although fcdependsontSiduetobulkinversion,wefindthatforUTBs, fc 0.41V givesareasonableestimateforVt.Incontrast,forADGdeviceswithVFBfVFBb,e.g.,p+polysiliconforthefrontgateandn+polysiliconforthe backgate,thenExC~-105V/cm,and fc@ (0.5V+ExCtSi),whichreflectsthefactthat wehaveapredominantbackgateand fsb@ 0.5V.NotethatwhileVtinEq.(3.17) remainsthesameinbothconfigurationswhentoxb=toxf,theclassicalmodelin [Lim83]canonlypredictthatforVFBf
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72 thickBOX,CB(eff) 0,andthus,Eq.(3.13)impliesnear-idealS @ ln(10)kBT/q;for SDGandADGMOSFETs,VGbS=VGfSinEq.(3.13)impliesidea lS=ln(10)kBT/q. 3.2.3.3 Generalized Charge Coupling InadditiontoVt,Equation(3.13)alsodescribesthegeneralizedcharge couplinginDGMOSFETs.ItsgenericnaturebecomesapparentforanSOIstructure withthintoxbandVGbSoperatedindependentofVGfS,asforIGFinFETs.However, forsuchconfiguration,ExCinEq.(3.9)dependsonVGfS=Vt,whichinturndepends onExCthrough fc;i.e.,Vtmustbesolvediteratively/numerically.Thentoexemplify thetrulygenericnatureofourVtmodel,weconsidertheIGnFinFETofFig.3.2,i.e., toxb=toxf=2.0nm,tSi=25nm,andn+-polysilicongates.Themodel-predicted constant-chargeVt(VGbS)inFig.3.3(a)isingoodagreementwithMEDICIpredictedconstantcurrentVt(VGbS);constantmobilityisspecifiedinMEDICIfor allbiasestoeliminateanyeffect(s)ofmobilityonconstant-currentVt.Notethat theseresultsincludethepredominantback-channelcondition,unlikethoseinFig.3.1. WeshowinFig.3.3(b)aneffectiver(reff),definedconsistentlywithrin Eq.(3.13)asthevariationinVtwithVGbS(orVFBb),versusVGbS.Wealsoshowthe model-predicted fc(VGbS).Notethatreff=rand fc@ 0.46VindependentofVGbS< @ -0.2V,butitincreasessignificantlyasVGbSbeginstoinvertthechannel,i.e., forVGbS>.Thelatterisreflectedbythereductionin fc,or fsf,tomaintainthe givenQiastheback-channel-defining fsbincreases.ForincreasinglylargerVGbS, theinversioncarriersarepredominantlyconfinedatthebacksurfaceinachargesheetlikemanner,andreffsaturatesasshowninFig.3.3(b).Interestingly,wefind thatthesaturationvalueofreffis1/r’,wherer’=r(toxf toxb,toxb toxf)in VGb IVGb I

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73 Figure 3.3MEDICI-andmodel-predicted(a)thresholdvoltageand(b)reff=dVt/ dVGbSversusVGbSforIGFinFETofFig.3.2.Alsoshownin(b)isthe model-predicted fc. -1.0-0.8-0.6-0.4-0.20.00.20.4VGbS [V] -2.0 -1.5 -1.0 -0.5 0.0Vt [V] MEDICI Model -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5fc [V] -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0VGbS [V] 0.0 1.0 2.0 3.0 4.0 5.0 6.0reff MEDICI Model (a) (b) classical model classical model

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74 Eq.(3.6),anditcanbederivedbysubstitutinginEq.(3.3) fsf,ratherthan fsb,from Eq.(3.4).Forthecasehere,toxb=toxf,andthesaturationvalueisreff=1/r.Further, wefindthattheincreaseinthesubthresholdswing,S=(1+r)(kBT/q)ln(10),with VGbSinFig.3.2isinaccordwithr reff.Thus,thevariationinrefffromris physicallyexplainedbythefrontgatehavingtocontrolconductioninachannelaway fromthefrontsurface,i.e.,reff(VGbS)dependsonthelocationoftheinversionchargecentroid[Zha05],whichisphysicallyincorporatedinourVtmodelvia fc(VGbS). Effects of reff(VGbS) on IG FinFET design are discussed in [Zha05]. 3.3 Random Dopant Effects WhiletheUTBsinnanoscaledevicesaremostlikelytobeintrinsic,itis possibleforadopantatomortwotoberandomlyplacedinthebody.Insuchacase, NB,the average bodydoping,isdefinedbythebodyvolume,andcanbeashighas ~1017cm-3.(Forourbriefanalysishere,wedonotconsidereffectsofrandom dopant-placement[Bro02].)Sincethenominaldevicehasanintrinsicbody,this substantivechangeisthetotalfluctuationinNB.However,wefindthataslongas QB/Coxremainsnegligiblysmall(e.g.,<< fsfasinEq.(3.2)),theelectrostaticsare independentofNBandthedopanttype.Toexplainthisquantitatively,wenotethat while fsf(Qi@ -q1011cm-2)inEq.(3.13) increasesby fBforlow-dopedbodies, wherelowisquantifiedbytheimportanceofQB/Cox,theflatbandvoltagesdecrease by fB,therebyrenderingVtinEq.(3.13)independentoflown-andp-typeNB. Indeed,MEDICIpredictionsinFig.3.4(a)forlong-channelADGnMOSFETsand inFig.3.4(b)forshort-channelSDGnMOSFETssupportourinsights.Further,Vt

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75 Figure 3.4(a)MEDICI-predictedIDS-VGSforlong-channelADGnMOSFETs withtoxf=toxb=1nm,tSi=9nm,n+-andp+-polysilicongates,and variousNB.TheinsetshowsMEDICI-predicted f (x)forvariousVGSfor theNB=0case.(b)MEDICI-predictedVt(NB)for28nmlongSDG nMOSFETs with toxf = toxb = 1.5nm, tSi = 14nm, and midgap gate. 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-910-810-710-610-510-410-310-2IDS [A/ m m] NB = 0 NB = 1015cm-3(p-type) NB = 1015cm-3(n-type) 0.01.02.03.04.05.06.07.08.0x [nm] -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Potential [V] 9.0 0.8 VGS = 0.0VVGS = 1.0VNB = 0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Vt[V] n-type p-type 10151016101710181019NB [cm-3](a) (b)

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76 fromEq.(3.17)fortheADGstructureisingoodaccordwiththeMEDICI predictionin Fig.3.4(a),andsoare fcatVtfromEq.(3.10)andExCfromEq.(3.9) withtheinsetofFig.3.4(a).BecauseofSCEsintheshort-channelSDGdevices,the VtpredictionfromEq.(3.16)ishigher,buttheMEDICIpredictionsshowthatSCEs arealsoindependentoflowNB.Theseresultsfurtherindicatethatthenaturalp-type impuritydopantsinundopedpMOSFETsdonotyieldpunch-through.Finally,note thatdependingontheUTBandgate-oxidethicknesses,fluctuationsinNB~10171018cm-3can be tolerated in undoped UTB MOSFETs. 3.4 Subthreshold Quantum-Mechanical Effects ThemodelinginSec.3.2mustnowbeaugmentedbymodelingtheQM effects,whichcansignificantlyaffecttheMOSFETsubthresholdcharacteristics, e.g.,viaanincreaseinVt( D Vt QM).InderivingEq.(3.10)fortheintegratedinversion chargedensity,whichwenowrelabelasclassicalQi(Qi CL),weimplicitlyassumed thatthecarriersare3-D.However,theUTBneededforSCEcontrolstructurally confinesthecarriers,asdoesthehightransversefieldinADGMOSFETs(andIG FinFETs).Thisspatialconfinementresultsinsubstantivecarrier-energy quantization,andthus,forgivengatebias,theQMinversionchargedensity(Qi QM) islowerthanQi CL.Then,self-consistency,viaGauss’slaw,showsaQMchannel potential( fs QM)lowerthantheclassicalone( fs CL,i.e.,thatinQi CLinEq.(3.10)). Thedifferenceinthetwopotentials( Dfs QM)definestheQMeffectsinsubthreshold operation,e.g., D Vt QM,andcanbeinterpretedasareductionin fs CLneededtolower Qi CL to be the same as Qi QM.

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77 3.4.1 Physical Modeling Tocharacterize Dfs QM,weneedtosolvethePoissonandtheeffectivemassSchrdingerequationsself-consistently.WhileQi CLisgivenbyEq.(3.10)with fsfrelabeledas,forQi QM,wefirstsolvetheSchrdingerequationwiththelinear potentialdistributioninEq.(3.8)viathevariationalapproach[Sha94]togetthe quantized2-D-carriersubbandenergies(eigenvalues),whichwillbegivenlater. AlthoughSCEstendtodistortthelinearpotential,weassumethattheireffecton quantizationisnegligibleforviabledesignswithgoodSCEcontrol.Then,assuming parabolicbandstomodelthe2-Ddensityofstates(DOS),andusingMaxwellBoltzmann statistics which is valid for weak inversion, we find (for Si nMOSFETs) ,(3.18) whereisthereducedPlanck’sconstant,Ncisthe3-DeffectiveDOSforthe conductionband,Ej(Ej’)istheseparationbetweenthejthsubbandintheunprimed (primed)valleyandthebottomoftheconductionbandatthefrontsurface,andg(g’) andmd(md’)arethedegeneracyandDOSeffectivemassfortheunprimed(primed) valley,respectively.(Forholes,Nc Nv,the3-DeffectiveDOSforthevalence band,andunprimed(primed)valley heavy-hole(light-hole)band;thespin-orbit band is neglected.) EquatingQi CLtoQi QMnowyieldsananalyticmodelfor Dfs QMwithout any fitting parameters, thereby characterizing the QM effects: fsf C L Qi QMqnikBT Nc----------------- – q fsf QMkBT ------------exp gmdp h 2---------Ej– kBT --------expj0 =g md p h 2------------Ej – kBT ---------expj0 =+ = h

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78 ,(3.19) where E0 is the ground-state subband energy in the unprimed valley and .(3.20) SinceweassumethatSCEsdonotaffectQMeffectsmuch,bothcanbeaccounted for via fCL(x) ( fCL(x) Dfs QM) in a 2-D model for Qi CL. Then, .(3.21) Interestingly,notethat Dfs QMinEq.(3.19)isanalogoustothatforclassicaldevices forhighExCandpredominantoccupationofE0(i.e., g =1inEq.(3.20))[Tau98],and thatitdependsonlyontSiwhenExC 0,asinSDGandFD/SOIMOSFETs.Inthe caseExC@ 0,wemathematicallyproveinAppendixCthat Dfs QM 0astSi in Eq. (3.19), consistent with our physical intuition. Finally,wemodelEjandEj’inEq.(3.18)-Eq.(3.20),usingthevariational approach with a trial eigenfunction of the form ,(3.22) whereajisthenormalizationconstantandbjisthevariationalparameter.Then,for the 1-D effective-mass Schrdinger equation ,(3.23) Dfs QME0q ----kBT q --------gmdp h 2Nc---------------qExC1qExCtSi–kBT () exp – ----------------------------------------------------------g ln – = g 1 E0Ej– kBT ----------------expj1 =g md gmd------------E0Ej – kBT -----------------expj0 =++ = D Vt QMS kBTq () 10 ln --------------------------------Dfs QM= yjajj1 + ()p x tSi----ebjx2 –sin = h 22mx---------- – x2 2d d yjqExCx ()yj+Ejyj=

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79 where mx is the effective mass in the confinement direction, we find, for ExC 0, .(3.24) (Similaranalysisisdoneforeigenfunctionsandenergyeigenvaluesintheprimed valley.)Interestingly,thefirstterminEq.(3.24)correspondstoE0forpredominant SCandthesecondterm(withtwoinnerterms)forpredominantEC[Ste72].Thelast term(withtwoinnerterms)canthenbetreatedasacorrectionwhenneitherprevails. FortSi>3-4nm,wefindthatthesecondinnerterminthiscorrectioncanbe neglected,andthatthefirstinnertermofthiscorrectionhasrelativelyweak dependenceonb0.Hence,withdE0/db0definedpredominantlybythesecond,orthe EC term in Eq. (3.24), we approximate b0 as .(3.25) Substituting Eq. (3.25) in Eq. (3.24), we get ,(3.26) wherewehaveneglectedthesecondinnertermofthecorrectionterminEq.(3.24) asexplainedearlier.Notethatwiththisneglect,E0inEq.(3.26)isindependentof thesignofExC.Finally,wefindapproximatethehighersubbandenergiesviaEq. (3.26) with p (j+1) p and b0 b0(4j/3+1)1/3. E0h 22mx---------p tSi----2h 22mx---------b0 23 2 -qExCb0-----------+ qExC1 b0---1 b0tSip ()21 + [] ---------------------------------------tSie2b0tSi–1 – -----------------------+ – + = b03 4 -2mxqExCh 2----------------------13 @ E0h 22mx---------p tSi----2b0 23 4 3 -1 b0p tSi ()21 + [] ---------------------------------------- – + @

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80 3.4.2 Model Verification ToverifyourmodelforEj,Ej’,and Dfs QM,weconsider(long-Leff)SDG andADGnMOSFETswith{100}-Sisurfaces,anduseSCHRED[Vas00],a numerical,self-consistent1-DPoisson-Schrdingersolver.(Wenotethatalthough notseparatelyaddressedhere,subthresholdQMeffectsinFD/SOIMOSFETswith thickBOX(toxb~200nm)arevirtuallythesameasthoseinSDGdevices.)Weuse thenotation{hkl}tospecifysurfaceorientationalonganyoneofthefamilyofplanes equivalenttotheplaneorientatedalong(hkl).Then,Fig.3.5(a)andFig.3.5(b)show model-andSCHRED-predictedEj(tSi)andEj’(tSi)forSDGandADGnMOSFETs, respectively,with{100}-Sisurfacesandtoxf=toxb=1nm;theeffectivemassesand valleydegeneraciesassumedforSiarelistedinTable3.1.Also,amidgapgate ( FG=4.6eV)isspecifiedfortheSDGdevices,andn+-polysilicon( FG=4.05eV)and p+-polysilicon( FG=5.17eV)gatesarespecifiedfortheADGdevices.Themodel predictionsforSDGMOSFETsareinexcellentagreementwithSCHRED,andarein accordwithpredominantSC,i.e.,Ej [(j+1) p /tSi]2.ForADGMOSFETswith tSi>3nm,ECduetohighExCinEq.(3.9)predominates,andthemodelpredictions forE0andE0’areingoodagreementwithSCHRED.Althoughhighersubband energiesareoverestimatedinFig.3.5(b),theerrorisinconsequentialheresinceonly E0andE0’arepredominantlyoccupiedintheADGdevices,asweshowlater.Also, theerrorinE0(E0’)fortSi 3nmisduetooursimpleapproximationforb0inEq. (3.25);however,sincetSi 3nmisnotpragmatic,asweshowlater,weneednot refine our model.

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81 Figure 3.5 SCHRED-(symbol)andmodel-(line)predictedEj(tSi)(open/dashed)and Ej’(tSi)(lled/solid)in(a)SDGand(b)ADGnMOSFETswith{100}-Si surfaceandtoxf=toxb=1nm.AMidgapgate(n+-andp+-polysilicongates) is(are)speciedforSDG(ADG)nMOSFETs.SCHREDassumes mx=0.98m0fortheunprimedvalley,andthus,themodelalsousesthismx.(a) (b) 0.05.010.015.020.0tSi [nm] 0 100 200 300 400Ej, Ej’ [meV] E0-E3; E0’-E3’ 0.05.010.015.020.0tSi [nm] 90 190 290 390 490 590 690 790Ej, Ej’ [meV] E0-E2; E0’-E2’

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82 ElectronsHoles Surface {100}{110}{111}All Valley/BandUPPUPPALLHHLH Degeneracy 2442611 mx [m0]0.9160.1900.3150.1900.2580.2900.200 md[m0]0.1900.4170.3240.4170.3580.4330.169 Table 3.1 Valleys,valleydegeneracy,andconfinement(mx)andDOS(md)effectivemasses pervalleyforelectronsandholesinSiwithvarioussurfaceorientations[Ste72], [Mog86].Theeffectivemassesaregiveninunitsofthefreeelectronmass(m0= 9.11x10-31kg);UP=unprimedvalley,P=primedvalley,HH=heavy-holeband, andLH=light-holeband;spin-orbitbandisneglected.Anisotropyofholes[Fis03] is also neglected.

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83 Forlong-LgateSDGandADGdevices, D Vt QM= Dfs QMinEq.(3.21). Thus,weshowin Fig.3.6the model-andSCHRED-predicted D Vt QMversustSiinthe SDGandADGnMOSFETsofFig.3.5;alsoshownisSCHRED-predictedE0/q.Our modelisinagreementwithSCHREDforSDGdeviceshavinganytSiandforADG devices having tSi > 3nm, where this range of tSi for the latter is pragmatic. 3.4.3 Model Applications and Physical Insights Inadditiontoverifyingourmodel,Fig.3.6clearlyshowsthat D Vt QMin ADGnMOSFETsismuchlargerthanthatinSDGnMOSFETs.Moreimportantly, D Vt QMinSDGnMOSFETsisrelativelysmall(<50mV)fortSi>4nm.Whilewefind similardifferencesbetween D Vt QMofADGandSDGpMOSFETs, D Vt QMforboth devicestructuresissignificantlylarger(~1.5-3xdependingontSi)thanforthe respective nMOSFETs; i.e., QM effects are more pronounced in pMOSFETs. Further,weshowin Fig.3.7the model-predicted D Vt QM(tSi)with accountingforcarrierpopulationinvariousnumbersofsubbandsintheSDGand ADGnMOSFETsofFig.3.6.ThesepredictionsrevealthatonlyE0is(E0andE0’ are)sufficientforADGMOSFETswithtSi<15nm(tSi>15nm),supportingthe unimportanceofthemodelerrornotedinFig.3.5(b).ForSDGMOSFETs(or,when ExCissmall),atleastthefirsttwosubbandsintheunprimedandprimedvalleysare importantwhentSi>5nm,whichisalsoreflectedby D Vt QM>E0/qinFig.3.6.In contrast,becauseofhighExC,andfiniteinversion-layerthickness,inADG MOSFETs,theshiftinthecarrierenergyfromthebottomoftheconductionband alongxtoE0duetoquantizationiseffectivelysmallerthanE0,asillustratedin Fig.3.8,and thus D Vt QM
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84 Figure 3.6 SCHRED-andmodel-predicted D Vt QMversustSiinSDGandADG nMOSFETsofFig.3.5;carrierpopulationintherst10subbands(various subbands)ofbothprimedandunprimedvalleysisconsiderforthe SCHRED(model)predictions.AlsoshownistheSCHRED-predicted ground-state energy, E0/q (dashed line). 0.04.08.012.016.020.0tSi [nm] 0 100 200 300 400D Vt QM[mV] Symbol: SCHRED Solid Line: Model SDG ADG E0/q E0/q

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85 Figure 3.7Model-predicted D Vt QM(tSi)inSDGandADGnMOSFETsofFig.3.6with carrierpopulationaccountedforin(i)E0only,(ii)E0andE0’,(iii)E0-E1and E0’-E1’, and (iv) E0-E3 and E0’-E3’. 0.04.08.012.016.020.0tSi [nm] 0 100 200 300 400D Vt QM[mV] SDG ADG (i)-(iv) (i)-(iii)

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86 Figure 3.8EnergydiagramforweaklyinvertedSDGandADGnMOSFETs. Dfs QMisdefinedbyanaverage(alongx)increaseinthecarrier energy,shownbythearrows,fromthecontinuumstates(Ec)tothe quantized states (Ej and Ej’). E0E1E3E0’ E1’ Ec tSitSiE0E1E0’ Ec SDG ADG E2

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87 both D Vt QM>E0/qinSDGdevicesand D Vt QM/mx<110>)1/3@ 1.43fortSi>3-4nm.ForSDGnMOSFETs, D Vt QMincreasesbymx<100>/mx<110>@ 3fortSi<3nm;however,fortSi>5nm, D Vt QMin {110}-and{100}-Sisurfacedevicesiscomparable.Thisrathersurprisingresultfor tSi>5nmisexplainedbythecounteractingeffectsoflowermxversushighervalley degeneracyandheaviermd(i.e.,higherDOS)intheunprimedvalleyof{110}Si. Whiletheformerdefineshighersubband-energylevelsrelativetothosein{100}Si, thelatterforcescarrierstooccupylowersubband-energylevelsrelativetothosein {100}Si,andhencethecombinedeffectisrelativelynochangein D Vt QM.Thisonce againemphasizestheneedtoaccountforcarrierpopulationinsubbandsaboveE0in SDG MOSFETs. Next,weexemplifythegenericnatureofourphysicalmodelbyapplying ittoDGMOSFETswithindependentlybiasedgates,e.g.,IGFinFETs.Figure3.10 showsthemodel-andSCHRED-predictedNinv=|Qi|/qversusVGfSinanIG nFinFETwith{100}-Sisurfaces,toxf=toxb=1nm,tSi=10nm,midgapgates,and

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88 Figure 3.9Model-predicted D Vt QMversustSiinSDGandADGnMOSFETsofFig. 3.6butwith{110}-Sisurface;E0-E1andE0’-E1’(E0-E3andE0’-E3’)are consideredforADG(SDG)nMOSFETs.Forcomparison,alsoshownis D Vt QM for {100}-Si surface counterparts from Fig. 3.6. 0.0 4.08.012.016.020.0tSi [nm] 0 100 200 300 400D Vt QM[mV] SDG ADG solid: {110}-Si Surface dashed: {100}-Si Surface

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89 Figure 3.10 SCHRED-andmodel-predictedNinv=|Qi|/qwithandwithoutQMeffects inanIGnFinFETwith{100}-Sisurface,toxb=toxf=1.0nm,tSi=10nm, midgapgates,andVGbS=0.0V;alsoshownismodel-predictedNinvwith QMeffectsinthesameIGnFinFETbutwith{110}-Sisurface(dashed line).Carrieroccupationinthersttwosubbandsofunprimedandprimed valleys is considered. 0.00.10.20.30.40.50.60.70.80.91.0VGfS [V] 1031041051061071081091010101110121013Ninv [cm-2] Open Symbol: SCHRED w/o QM Lines: Model Filled Symbol: SCHRED w/QM

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90 VGbS=0.0V;alsoshownaremodelpredictionsfor{110}-Sisurfacecounterpart. ThemodelpredictionsarecalculatedusingEq.(3.10)with fs CL(VGfS)from[Lim83] whereQMeffectsareneglected,andusingEq.(3.10)with fs CL(VGfS) [ fs CL(VGfS)Dfs QM(VGfS)]toincludetheQMeffects.Interestingly,notethat carrier-energyquantizationincreasesVtand degradesSoftheIGFinFET.Thelatter isduetosignificantvariation(increase)inExCinEq.(3.9),andhencein Dfs QMin Eq.(3.19),withthegatebias,anditcanbeexacerbatedbysignificantreductionin carriermobilitywiththeincreasingExC.And,asmaybeexpected,themodelpredictedQMeffectsinFig.3.10fortheIGFinFETwith{110}-Sisurfacesare comparable to that with {100}-Si surfaces. Finally,weexaminehowcarrier-energyquantizationaffectsDGCMOS designandscalability.WhileasymmetricaldevicesarebetterforcontrollingSCEs viatheirhightransversefield,Fig.3.6andFig.3.10showthattheysufferfrom severeQMeffects,andhenceSDG(andFD/SOI)CMOSaremoreattractiveinthis regard.Further,fortSi<4nm, D Vt QMinSDGnMOSFETsalsobecomeslarge(Fig. 3.6),and D Vt QMfortherespectivepMOSFETsisupto3xlarger.Then,becauseVtisdefinedbythegateworkfunction(s),avarietyofgatematerialswillberequired forcomparableVtinbothCMOSdevices,which,however,isnotfeasible.Hence,tSi> @4 nmisneeded.ThislowerlimitontSi,basedon D Vt QM,isfurthersolidifiedby model-predictedvariationin D Vt QMpervariationintSiasshowninFig.3.11. Becauseofthe @1/ 3-timeslightermx,variationin D Vt QMpervariationintSiinSDG pMOSFETs(andinSDGnMOSFETswith{110}-Sisurfaces)ismuchworsethanin SDGnMOSFETswith{100}-Sisurfaces.TheseresultssetalowerlimitonviabletSi

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91 Figure 3.11 Model-predictedvariationin D Vt QMpervariationintSiversustSiinlongchannelSDGnMOSFETs,with{100}-and{110}-Sisurface,andlongchannel SDG pMOSFETs. 1.02.03.04.05.0tSi [nm] 0 10 20 30 40 50 60 70 80 90 100d( D Vt QM)/dtSi [mV/nm] {100} Si {110} Si pMOS nMOS

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92 ~ 4-5nmtoavoidlarge D Vt QMandexcessivevariationin D Vt QMwith D tSi.Then,as inChapter2and[Yan05],foracceptableSCEcontrol,scalingofSDG(andFD/SOI) CMOSislimitedtoaneffectivechannellengthof~8-10nm(~20-25nm).Indeed, theseresultsshowthatviabilityandscalabilityofanynewdevicearchitectureand/ orchannelmaterialinthenanoscaleregimemustbepredicatednotonlyonimmunity toSCEs,butalsooncontrolofQMeffectsandtheirvariationwiththedevice geometry. 3.5 Short-Channel Effects Astheeffectivechannellength(Leff)isscaled,controllingSCEs,or2-D effects,duetoS/Dencroachmentintotheweaklyinvertedchannellargelydefines nanoscaleCMOSdesign[Tau98].Classically,these2-Deffectsareconsideredby augmentingthelong-channelsurfacepotentialbyashiftinthepotential,whichis modeledeitherviasmallereffectiveNBbasedonthecharge-sharingconcept [Tau98],[Vee88],orbythesolutionofthe2-DPoissonequationatthesurface [Tau98],[Yan92].However,becausecarriersinUTBdevicesaredistributed throughoutthebody,suchcharge-sheettreatmentofSCEsisingeneralinsufficient. Infact,asweshowedinChapter2,carrierdistributionintheUTBhasasignificant impactonSCEs.WhiletheseeffectsaredetailedinChapter2,thegeneralconclusion isthatSCEsincreasewithincreasing(electrical)distanceoftheinversion-charge centroidfromtheinterface(s),whichcanbeconceptuallyinterpretedasbeingdueto aneffectivelythickergateoxide(s)renderedbythedeeperinversion-chargecentroid.

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93 3.6 Bulk-Inversion Effects Wenowfocusononeofthemostrecognizednonclassicaleffects(which wemodeledearlier)inundopedUTBMOSFETs:volume,orbulkinversion[Bal87], [Kim04].Figure3.12showsMEDICI-predictedelectronconcentration(n(x))across thebodyofalongchannelFD/SOIandSDGnMOSFETswithundopedbody,tSi= 20nm,amidgapgate,andlowVDS,clearlyexemplifyingbulkinversionwhich occursforallbiasconditions.AlsosuperimposedisMEDICI-predictedn(x)inlongchannelundoped-bodyTGnMOSFETwithamidgapgate[Fos03b].(Becausen(x)is uniforminallthreedevicesandtSi=30nmfortheTGnMOSFET[Fos03b],wehave normalizedthedistancealongxbytSi.)Interestingly,duetobulkinversion,allthree deviceshavevirtuallythesameQiinsubthreshold,independentofthenumberof activegates!Thisimpliesthattheeffectivedevicewidth(Weff)ofundopedSDGand FD/SOIMOSFETsshouldbethesame,andhence,Weff=hSiforFinFETs,whichis inagreementwiththatbasedonstrong-inversionanalysis[Kim04];WeffforTG devices cannot be physically defined [Kim04]. AsVGSisincreased,wenotefromFig.3.12thatQiintheSDGdevice (Qi(SDG))approachestwicethatintheFD/SOIdevice(Qi)!Thistransitionfrom Qi(SDG)=Qi(FD)inweakinversiontoQi(SDG)=2Qi(FD)instronginversionisdueto thescreeningofthefront-gateinfluenceontheinversionconditionat/neartheback surfacebytheincreasingconcentrationofinversioncarriers,whichintheDG devicesiscompensatedbythesecondgate.Further,noteinFig.3.12thatwhilethere isrelativelymoresurfaceinversionathighVGS,westillhavepredominantofthe inversionchargeinthebulk.Hence,wenowexaminebulk-inversioneffectsonNinv

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94 Figure 3.12MEDICI-predictedelectronconcentrationalongthebody-directionin FD/SOIandSDGnMOSFETswith1nmgateoxide,20nmthick undopedbody,midgapgate,andvariousgatebiases;thedistance alongthebodyisnormalizedbythebodythickness.The(electrical) thicknessoftheBOXofFD/SOIMOSFETis678cm;forBOX thicknessof200nm,n(x)maydifferdependingonthesubstratework functionviathetransversefieldatthebacksurface.Also superimposedaren(x)inTGnMOSFETfrom[Fos03b]with1.1nm gateoxide,30nmx30nmundopedbody,200nmthickBOX,anda midgap gate. 0.00.10.20.30.40.50.60.70.80.91.0Normalized distance along the body 10101011101210131014101510161017101810191020Electron concentration (cm-3) TG FD/SOI SDG VGS = 0.0V VGS = 1.0V

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95 andIonthroughtheinversion-layercapacitance,theeffectivecarriermobility,and velocitysaturationandovershoot.WeconsiderSDGnMOSFETswith(hypothetical) surfaceinversionversus(real)bulkinversion.Toinducepredominantsurface inversion,weusen+-polysilicongateandhighNBtosetVttothatofthebulkinversion device having a midgap gate and an undoped body. 3.6.1 Inversion-Layer Capacitance Bulkinversionisdefinedbypresenceoflargefractionofcarrier populationawayfromthesurface(s).Thismeansthattheaverageinversionlayeris thickerthanthatforsurfaceinversion,andhence,thesmallerinversion-layer capacitanceforbulkinversionunderminesthegatecapacitance(CG),whichthen yieldslowerNinvforagivengatedrive.ThisisreflectedbyMEDICI-predicted current-voltage(IDS-VGS)characteristicsin Fig.3.13forthe1 m mSDGnMOSFETs withbulkandsurfaceinversion;constant meffisspecifiedtodrawmeaningful conclusions.Becauseconstant meffisspecified,IonherereflectsNinv,anditis @ 7% lower in the bulk-inversion design due to the noted CG degradation. WhenQMeffectsareconsidered,Ciforbulkinversionisfurtherreduced. Althoughthestrongerconfinementinthesurface-inversiondesignyieldslarger D Vt QM,whenweadjustitsNBforsameIoff,orVt,asthebulk-inversiondesign,the noteddifferenceinNinvincreases.AsshowninFig.3.14,SCHREDpredicts,fora givenoff-stateNinv(orIoff), @ 15%lowerNinvinthebulk-inversiondesignattheon state.ThissuggeststhatwhiletheQMeffectonVtpredominatesthatonCifor classicalsurface-inversiondesigns,theQMeffectonCipredominatesthatonVtfor nonclassical bulk-inversion designs.

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96 Figure 3.13MEDICI-predictedIDS-VGSinlog(left)andlinear(right)scalefor 1 m mlongSDGnMOSFETswithtoxf=toxb=1nm,andtSi=10nm. Thebulk-inversiondesignusesmidgapgateandundopedbody,and surface-inversiondesignusesn+-polysilicongatewithNBadjusted tosetVtsameastheformerdesign.Toisolatetheeffectofgate capacitanceonthecurrent,constantmobilityof50cm2/V-sis specifiedforbothdesigns.Poly-depletioneffectsarenotconsidered. 0e+0 1e-5 2e-5 3e-5 4e-5 5e-5IDS [A/ m m] 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-1510-1410-1310-1210-1110-1010-910-810-710-610-510-4IDS [A/ m m] NB = 2.05x1019cm-3, n+ polysilicon gates Undoped body, midgap gate VDS=1.0V VDS=50mV

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97 Figure 3.14 SCHRED-predictedinversion-carrierdensityversusgatebiasinlog(left) andlinear(right)scalefor1 m mlongSDGnMOSFETsofFig.3.13.The bodydopingforsurface-inversiondesignisadjusted,afterQMeffectsare included, for the same Ioff as in the bulk-inversion design. 0.0 2.0 4.0 6.0 8.0 1.0 1.2 1.4Ninv [x 1013 cm-2] 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10410510610710810910101011101210131014Ninv [cm-2] NB = 1.6x1019cm-3, n+ polysilicon gates Undoped body, midgap gate

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98 Further,wenotethatthesmallerCiforbulkinversiondegradesIonenhancementwithgate-oxidescalingasreflectedbySCHRED-predictedrelative increaseinNinvwithdecreasingtoxinFig.3.15.Likethepoly-depletioneffecton NinvthroughCG[Tau98],thebulk-inversioneffectbecomesmorepronouncedfor thinner tox. 3.6.2 Effective Carrier Mobility Presenceoflargeconcentrationofcarriersawayfromtheinterfacesdueto thelowtransversefieldforbulkinversionmeanssignificantlyreducedsurfaceroughnessscattering,whichisthepredominantscatteringmechanismathighNinv. Hence, meffismuchlargerinthebulk-inversion(ortheundoped-body)design,and canpossiblyyieldhigherIondespitethelowerNinv.Toassessthispossibility,weuse MEDICI,nowwithfield-dependentmobility.Fromexperimentaldata[Ess00], [Ess03a],wenotethatinstronginversion, meffisvirtuallyindependentoftSi>45nm,andconsistentwiththeuniversalbehavior[Tak88]athighNinv.So,MEDICI withtheHPMOBmodel[MED04]isreasonableforourpurposehere.WithNBadjustedinthesurface-inversiondevicetomatchIoffofthebulk-inversiondevice, theMEDICI-predictedIDS-VGSforthe1 m mSDGnMOSFETin Fig.3.16 shows @ 70%higherIoninthebulk-inversiondevice.DuetoquestionableQMmodelingin MEDICI,thesepredictionsdonotincludetheQMeffectsonQi,however,basedon Fig.3.14andFig.3.15,weexpectthenoted meff-governedenhancementinIonto remain very high.

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99 Figure 3.15SCHRED-predictedrelativeincreaseinNinvversustoxfor1 m mlong SDGnMOSFETsofFig.3.13.Thesimulationsaredonefortoxf=toxb=1nm,1.5nm,and2nm,andtheidealincreaseiscalculatedfromthe ratioofthenominaltox(=2nm)tothenotedtox.Thebodydopingfor thesurface-inversiondesignisvariedforthe sameIoffasinthebulkinversion design. 1.01.21.41.61.82.0tox [nm] 1.0 1.2 1.4 1.6 1.8 2.0Relative increase in Ninv Ideal Surface Inversion Bulk Inversion

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100 Figure 3.16MEDICI-predictedIDS-VGSinlog(left)andlinear(right)scalefor 1 m mlongSDGnMOSFETsofFig.3.13.Mobilitydependenceonthe electricfieldsisnowconsideredusingtheHPMOBmodelwithlowfieldmobilityof600cm2/V-sandcritical(transverse)fieldat106V/ cm.Thebodydopingfor surface-inversiondesignisadjustedforthe same Ioffas in the bulk-inversion design. 0e+0 5.0e-5 1.0e-4 1.5e-4 2.0e-4 2.5e-4 3.0e-4 3.5e-4IDS [A/ m m] 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-1410-1310-1210-1110-1010-910-810-710-610-510-410-3IDS [A/ m m] NB = 1.955x1019cm-3, n+ polysilicon gates Undoped body, midgap gate VDS=1.0V VDS=50mV

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101 3.6.3 Velocity Saturation and Overshoot The meff-governedIonenhancementinthelong-channelbulk-inversion devicescanbeunderminedinshort-channeldevicesinwhichvelocitysaturation throughoutthemajorityofthechannel(andovershootnearthedrain[Ge01])govern theon-statecharacteristics.Tocheckthis,we,onceagain,useMEDICI.Indeed, MEDICI-predictedIDS-VDSandIDS-VGSinFig.3.17 supportourinsightandshow thattheIonenhancementof70%inFig.3.16isseverelyreducedto8%for50nmSDG nMOSFETs.However,wenotethatIonenhancementby8%isstillconsidered substantive.WealsonotethatalthoughQMeffects,reflectedbyFig.3.14andFig. 3.15,andnonzeroS/Dseriesresistancewillreducethenoted8%Ionenhancement, morevelocityovershoot,definedbyhigher meff,inbulk-inversiondevicesaugments theIonenhancement,andthus,maintainingthehigherIon.Inanycase,theIonenhancement is much lower than that implied by the higher meff. Finally,thecombinationof meff-governedenhancementinIon(Fig.3.17) andCi-governedlowergatecapacitance(impliedbyFig.3.15)canyieldfaster CMOS,anditcanfacilitatepragmaticDGCMOSdesignswiththickgateoxides, gate-source underlap, and midgap gates [Fos04]. 3.7 Summary Wehaveshownthatthedepletionapproximation-facilitatedanalysesare applicabletothemostlikelyintrinsic-UTBDGMOSFETs.Wegeneralizedthe classicalgate-gatecharge-couplinganalysisofLimandFossum[Lim83]by physicallyaccountingforcarrierdistributionandcarrier-energyquantizationinthe UTB.ThegenericnatureofthenewmodelwasexemplifiedviaapplicationstoIG

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102 Figure 3.17MEDICI-predicted(a)IDS-VDSand(b)IDS-VGSinlog(left)andlinear (right)scalefor50nmlongSDGnMOSFETswithtoxf=toxb=1nm, tSi=10nm,andvariousNBandgatematerials.MEDICImodelssame as that for Fig. 3.16 are used. 0.0e+0 5.0e-4 1.0e-3 1.5e-3 2.0e-3IDS [A/ m m] 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-1210-1110-1010-910-810-710-610-510-410-3IDS [A/ m m] NB = 2.0x1019cm-3, n+ polysilicon gates Undoped body, midgap gate VDS=1.0V VDS=50mV 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 0.5 1.0 1.5 2.0IDS [mA/ m m] NB = 2.0x1019cm-3, n+ polysilicon gates Undoped body, midgap gate VGS=1.0V (a) (b)

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103 FinFETs,showingenhanced(butnotnecessarilybeneficial)VGbS-dependenceofVt. Aphysicalmodelforquantizationeffectsinthesubthresholdconditionwasalso developed,incorporatingdependencesontSi,transversefield,andsurfaceorientation withoutanyfittingparameter.ItwasverifiedusingSCHRED.Theseverifications andfurtherapplicationsofthemodelrevealedthat(i)thecarrierpopulationin higher-energysubbandsinDGdevicesisimportant,(ii)QMeffectsinDGdevices with{110}-Sisurfacescanbecomparabletothosewith{100}-Sisurfaces,(iii)QM effectscandegradethegateswingofDGdevices.Also,thelowerlimitontSi,and thusscalabilityofDGCMOS,issetbyQMeffectsanditsvariationwithtSiin pMOSFETs(andnMOSFETswith{110}-Sisurface).Inaddition,wealsofoundthat undopedDGMOSFETscantoleratelargefluctuations(~1017cm-3)in|NB|,the average body doping. Further,examinationofvolume,orbulkinversioninSDGandFD/SOI MOSFETsshowedthatWeffforbothdevicesisthesame,andthatWeff=hSifor FinFETs;WeffforTGMOSFETscannotbephysicallydefined.Forstrong-inversion conditions,wefoundthatbulkinversionunderminesthegatecapacitanceduetothe thickinversionlayer,orsmallCi.However,highermobilityofless-confinedcarriers inbulk-inversiondesignmorethancompensatesforthisdegradationandyields higherIon,butseverelylimitedbyvelocitysaturation.Interestingly,thecombination oftheCi-governedsmallergatecapacitanceandthe meff-governedhigherIon,both definedbybulkinversion,implieshigherCMOSperformanceforundoped(i.e., bulk-inversion)CMOSoverdoped(i.e.,surface-inversion)CMOS,andthus,itcan facilitate pragmatic approach to DG CMOS design.

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104 CHAPTER 4 EFFECTIVE CHANNEL LENGTH AND SOURCE/DRAIN-EXTENSION ENGINEERING 4.1 Introduction InChapters2and3,westudiedDGMOSFETsintermsoftheeffective electricalchannellengthLeff,whichcandifferfromthephysicalgatelengthLgate, theprimarytechnology-parameterofinterest.Ingeneral,LeffandLgatearerelated viatheS/Dlateraldopingprofile[Tau98].IntypicalscaledclassicalMOSFETs, G-S/Doverlap(perside)of @ 20%ofLgate[Tho98],[Gha00]defineLeff<0.7Lgate. However,asLgate 7nm,implementingsuchoverlapinanintrinsic-body nonclassical(orheavilydopedclassical)MOSFETwithanonabruptS/Dlateral dopingprofileintheS/Dextension(SDE)canresultinsubstantialS/D-dopant diffusionintothebody,andhenceinS-Dpunch-through.Moreimportantly,the neededtSi 3.5nmforLgate@ 7nmisnotdoabletechnologically,andbasedonour insights in Sec. 3.4, not viable. Hence,asLgate 7nm,G-S/D underlap isneeded.Theconceptof“nonoverlapped”G-S/Dwithlow-dopedchannelwasrecentlysuggested[Boe01]to facilitatethescalingofbulk-SiMOSFETstoLgate<20nm.However,sinceSCEsin conventionallyscaledbulk-SiMOSFETsarecontrolledpredominantlyviahigh channeldoping[Tau98],largeunderlaps(>10nm)areneededforacceptableSCEcontrol[Boe01],[Gus03],andtheyresultinsuboptimalspeedperformance[Gus03]. Hence,theconceptdoesnotseempromisingforclassicalCMOS.Incontrast,arecent

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105 studybyFossumetal.[Fos03c]ofSDGFinFETs,inwhich(undoped)UTBscontrol theSCEs,intimatedperformancebenefitsofG-S/Dunderlap,definedbyabiasdependentLeff,andsuggesteditcouldunderlieoptimalnanoscaledesign.Although previoussimulation-basedstudies[Bal03],[She03],[Sch04]havealsoshownthatGS/DunderlapisoptimalwithregardstoDGCMOSspeedandRS/D,nophysical analysis of the underlying device physics was done. ExtendingtheanalysisofFossumetal.[F os03c]in thischapter,we thoroughlyexplorenanoscaleDGFinFETdesignexploitingtheG-S/Dunderlapand associatedbias-dependentLeff;theresultsarereadilyapplicabletoother nonclassicalundopeddevicesaswell,andpossiblytoheavilydopedbulk-Si MOSFETs.WestudythedependenceofLeffonthegatebiasand,moreimportantly, ontheS/Dlateraldopingprofile(NSD(y))withnonzerolateralstraggle( sL)inthe SDE.Incontrastto[Bal03],[She03],and[Sch04],wefocushereonLeff,ratherthan theunderlapthatdefinesit,sinceLeffisdirectlyrelatedtothedeviceperformance. ThephysicalinsightsonLeff(NSD(y))arethencombinedwiththedependenceofRS/DonNSD(y)todefineasystematicapproachtodesignoptimizationofnanoscaleUTB MOSFETsviaSDEengineering.Weexemplifysuchoptimization/engineeringviaa well-tempereddesignofLgate=18nmSDGFinFETs.Finally,toaidthisSDE engineering, we develop a (relatively simple) physics-based model for Leff. 4.2 Effective Channel Length Atopcross-sectionalviewoftheFinFETstructure(Fig.1.4)isshownin Fig.4.1.Thegate“wraps”overthethinSifin,orUTB,yieldingaquasi-planarSDG MOSFETwithtwochannelsthatarebeneficiallycharge-coupled[Lim83].Thegate

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106 Figure 4.1Topcross-sectionalviewofaFinFETstructure(Fig.1.4)withthinsource/ drainn-extensionsandwidesource/draincontactregions.Thenwidth is wSi, as indicated, and its height is hSi. D S G G tox Lgate toxwSi x y LextLext

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107 isnecessarilyseparatedfromtheflaredout(andraised)S/Dcontactregionsbythe SDEs(oflengthLext),whicharegenerallycoveredwithoxidespacers.Notevident inthefigureistheheightofthefin(hSi),whichisthewidthoftheDGdevicebut whichistechnologicallylimitedtobeontheorderofthefinthickness(wSi); typically, hSi/wSi ~ 4-5. 4.2.1 Bias Dependence Whiletheintrinsicbody/channelavoidsrandomdopanteffectsand provideshighermobility,itportends,asmentionedearlier,thepossibilityofS-D punch-throughwhentheSDEsareheavilydopedtominimizeRS/D.Hence,theSDE lateraldopingprofilewillhavetobewell-controlled.Fortheextremecaseinwhich portions(oflengthsLeSandLeD)oftheSDEsareleftundoped,Fossumetal. [Fos03c]previouslyshowedthatVGSmodulatesthesurfacepotential( fs(y))and(for thenFinFET)electrondensity(n(y))intheseundopedregions.Thismodulationisa resultofredistributionofgate-inducedelectronsinsupportofdrift-diffusionbalance inthechannel-length-(ory-)direction(Fig.4.1).Insightonthisredistributionis gained by approximating Poisson’s equation in the undoped extension region as ,(4.1) wheren( fs)=niexp(q fs/kBT)fromMaxwell-Boltzmannstatitstics,whichshowsthat n(y)isgovernedbythen-definedDebyelength(LD 1/).NeartheS/Dcontact regionswherenishighandLDisshort,theelectronstendtobeconfined.However, nearthegatenisloweranddependsdirectlyonVGS,andthusgate-controlled y2 2d d fsqn fs() eSi---------------@ n

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108 electronsextendintotheSDEs,i.e.,thegateencroachesintotheSDEs,and Leff>Lgate.IncontrasttotheLeff Lgate(orLmet)behaviorinaG-S/D-overlap device, this Leff Lgate behavior is physically different. Forconventionalbulk-SiMOSFETswithG-S/Doverlap,Tauretal. [Tau95]definedLeffasaneffectivelengthoverwhichthegatemodulatestheS-D channelcharge,orconductivity,anddescribedhowgate-inducedcarrier accumulationintheSDEswithnonzero sLresultsinthestrong-inversionLeff(Leff(strong)whichdefineschannelcurrentinthelinearregion)beinglongerthanthe metallurgicalchannellength(Lmet),wherethelatterwasassumedtobetheweakinversionLeff(Leff(weak)whichdefinestheSCEs).(NotethatthisincreaseinLeffwas thebasisfortheconclusionin[Tau95]thatabruptSDEdopingprofilesareoptimal forscaledMOSFETs).SincetheaccumulationintheSDEsinstronginversionis directlydefinedbyaparasiticMOSFETformedbythegate/oxide/SDEstructure,the elongationofLeff(strong)issimplythesumofthelengthsoftheaccumulationlayers inthetwoSDEs[Tau98].Hence,theactualdevicewithnonzero sLcanbe effectivelymodeledinstronginversionbythenotedLeff(strong)>Lmetwithassumed abruptS/Djunctions[Tau95].Similarly,ourinsightsfromEq.(4.1)fornonclassical deviceswithG-S/DunderlapsuggestthatLeff>Lgate,and,ingeneral,its characterizationwouldrequiresolvingEq.(4.1)intheSDEsaswellasunderthe gate.However,becausetheelongationofLefffromLgateisdefinedindirectlyvia gate-inducedperturbationofdrift-diffusionbalanceintheSDEs,theunderlying physics is different from that in the G-S/D-overlap device.

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109 AlthoughrigorousquantitativetreatmentofLeffisoutofthescopeofour studyhere,wecanstillgainsubstantiveinsightsregardingLeffbasedonthenoted dependenceonLD(n)ofthegateencroachmentintheSDEs.Forweakinversion,with n low and hence LD long, we can argue that .(4.2) Indeed,theplotsinFig.4.2ofinversechannelcurrents( Ich)forrelativelylong-LgateFinFETs(withthinwSi)inweakinversionversusLgate,predictedbyMEDICI[12] fordifferentLeS=LeD,supportEq.(4.2)becauseIch 1/Leff.(Wenotethat,for shortLgatewithnoticeableSCEs,Leff(weak)andwSidefineadepleted-bodyrectangle inwhichthe2-DelectrostaticscharacterizesSCEs[Tau98].Thismeansthatthe weak-inversioncurrent,definedaspredominantlydiffusion,ischaracterizedbyan effectivelengththatisshorterthanLeff(weak)duetoS/D-field(Ey)encroachments [Yeh95].However,intheabsenceofSCEsasinFig.4.2,Leff(weak)definesIch.)In contrast,forstronginversion,highnandshortLDlimittheextentofthegatecontrol to .(4.3) NotefromEq.(4.2)andEq.(4.3)thatLeffhasstrongbiasdependence,andit decreases with increasing VGS. ThisVGS-dependentbehaviorofLeff,unlikethatofLeff(VGS)intheG-S/ D-overlapdevice,appearstobeideal;thelongLeff(weak)tendstosuppressSCEsand limitIoff,asshownin[Fos03c],andtheshortLeff(strong)canyieldhighIon. Leffweak ()LgateLeSLeD++ @ Leffstrong ()Lgate@

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110 Figure 4.2MEDICI-predictedinverseweak-inversionchannelcurrentversusgate lengthofundopedDGnFinFETswithdifferentLeS=LeD;wSi=10nm. TheLgateinterceptsofthelinearextrapolationsindicateLeff(weak)=0and conrm Eq. (4.2). -300-200-100 0 100200300Lgate [nm] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 LeS = LeD = 0nm LeS = LeD = 50nm LeS = LeD = 100nm1/Ich (normalized)

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111 Unfortunately,asinthebulk-Sicase[Boe01],largenon-ohmicvoltagedrops,or RS/D,acrossLeSandLeDlimitIon[Fos03c].Thenon-ohmicnatureofRS/Disclearly reflectedinFig.4.3byMEDICI-predictedvelocitysaturationintheundoped portionsoftheSDEs,whichthusalsodefinesthesaturationcurrent.Hence,foran optimaldesign,wewouldliketomaintainthenovel,bias-dependentLefffeature, while reducing RS/D. 4.2.2 Source/Drain-Doping Profile Dependence FornonzeroandpositiveLeSandLeD,nowdefinedgenerallybyLeff(weak)inEq.(4.2)forSCEcontrol,RS/DcanbereducedbydopingtheSDEs,subjectof coursetothenotedpossibilityofS-Dpunch-through(andtechnologicalissues regardingthin-findoping).However,suchdopingcanunderminetheG-S/Dunderlap andthegatecontrolofthechargeintheLeSandLeDregions.Hence,weexamine,by 2-Dnumericalsimulations,thedependenceofLeffonNSD(y),andoptimization thereof.WeuseMEDICItosimulatetheDG(n)FinFET2-Dcross-sectioninFig.4.1 (withouttheflaredS/Dcontactregions),assumingLgate=18nm,Lext=30nm,tox= 1nm(butignoringgatetunneling),wSi=12nm,midgapgate,andthefourdifferent fin-SDEgaussiandopingprofilesshowninFig.4.4.The sL=1nmprofile(with differentoriginnearthegateedge)representsnearlyabruptS/Djunctionswith negligibleunderlap/overlap.Suchaprofileisnotpossibletechnologically,butis consideredhereforcomparison.Theotherthreeprofilesrepresentlateraldiffusion ofdopantsfromtheS/Dcontactregion,withvarying sL.Thisseemstobethemost viableapproachtodopingtheSDEsasdirectionimplantationintothethinfins wouldrequirehightiltangleandlowenergy[Ked03],followedbyrelativelyhigh-

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112 Figure 4.3MEDICI-predictedaverageelectronvelocity(a tx=0,wSi)variation betweenthesourceanddraincontactregionsofanLgate=17.5nmDG nFinFETforincreasingVDS[Fos03c];wSi=17nm,tox=1.7nm.Theentire S/Dn-extensionregions(LeS=LeD=27nm)wereleftundoped,aswas thebody/channel.TheS/Ddopingprolewasassumedtobeabrupt.With theenergy-balancetransportoptionon,MEDICIpredictsvelocity saturation, with some overshoot, in the undoped portions of the SDEs. 60708090100110120130140150160y [nm] 0.0 0.5 1.0 1.5Electron Velocity (107 cm/s) 1.2V VDS=0.4V LeSLeDLgate SD

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113 Figure 4.4Gaussiansource/draindopingproles(NSD exp(-y2/ sL 2))denedwith alateralstraggleasindicated;Lgate=18nm,Lext=30nm.The sL=1nm prole(withadifferentoriginnearthegate)yieldsnearlyabruptS/D junctions.AlsoindicatedarethegateandSDElengths.Notethatthe straggleisnotthelateralabruptnesstypicallyspeciedinnm/decade; however, the latter is generally not much smaller than the former. NSD [cm-3] Lgate LextLext 101010111012101310141015101610171018101910201021 2030405060708090100 sL=5nm 10nm 15nm 1nmy [nm]

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114 temperatureandfast,rapid-thermalannealtopreventS-Dpunch-through.The sL=5nmdopingprofilethatleavespartoftheSDEsvirtuallyundopedislikethat usedin[Fos03c],whiletheothertwoaremoreviable.Wenotethattheassumed NSD(y)andLextareusedheretogainphysicalinsights,andarenotnecessarily optimal for the assumed Lgate and wSi. AlthoughtheMEDICIaccountingsforwSi-dependentquantization [Ge02b]andcarrierscattering[Gm01b]arequestionable,thepredictedIon( sL), includinghydrodynamics(i.e.,velocityovershoot),andIoff( sL),bothgiven(perunit hSi)inTable4.1alongwiththepredictedSCEs(DIBLandS),stillprovide meaningfultrends.Interestingly,notethattheSCEsandIoffforthe sL=15nm NSD(y)aresignificantlysmallerthanthosefortheabruptNSD(y),implyingthat despitethesubstantialS/D-dopantdiffusionintothebody(yieldingn-typeNB~ 1017cm-3)apparentinFig.4.4,thereisnoS-Dpunch-through.Toexplainthis,we considerthevariationinthetransverseelectricfield(ex)causedbythechangeinthe depletion charge, QB = qNBwSi with NB = ND + NA -: .(4.4) Sinceex=0atthecenterofthefin(x=wSi/2),Eq.(4.4)impliesthatchangeinthe electrostaticsduetosomeeffectiven-typeNB,definedbyNSD(y),isnegligibleso long as ;(4.5) DexQB2 eSi--------QB2 eSi---------kBTq wSi2 ---------------<

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115 sL [nm]Ion[mA/ m m]Ioff[A/ m m]DIBL [mV/V]S [mV] 1 (abrupt)1.7 2.9x10-718094 50.20 1.3x10-113562 100.43 4.9x10-114666 151.0 1.6x10-97971 Table 4.1 MEDICI-predictedcharacteristicsforthenFinFETdesignsinFig.4.4.Band-to-band tunnelingandimpactionizationarenotaccountedfor,buttheenergy-balancetransport optionisturnedon.TheabsolutevaluesofIon(perunithSi)areequivocaldueto questionable physical modeling in MEDICI, but the relative values are meaningful.

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116 thatis,theUTBremainseffectivelyundopedwheneverEq.(4.5)issatisfied,asitis foralldopingprofilesinFig.4.4fortheassumeddevicestructure.Thismeansthat nanoscaleFinFETswithundoped(orunintentionallydoped)ultra-thinbodiesand gateoxidescantoleratelarge sLforLeff-baseddesignoptimization.(Notethatsuch devicesare,strictlyspeaking,accumulation-mode,andsuchdevicestypicallyshow poorSCEs.However,theSCEsarenotunderminedherebecauseofthenonabruptS/ Djunctions,aswellasthethinwSiandtox.Also,forVGS>Vt,thestrongaccumulationelectrostaticsremainsthesameasthatforstronginversion,andthus we do not treat these two conditions separately.) Further,becauseweknowLeff(weak)@ LgatefortheabruptNSD(y)(i.e., highlydopedSDE)[Tau98],andLeff(weak)>Lgateforthe sL=5nmNSD(y)(i.e., undopedSDE)[Fos03c],weinferthatLeff(weak)( sL)>Lgateunderliesthegood controlofSCEsseeninTable4.1fortheotherprofiles,implying effective gate-S/D underlap,i.e.,LeS/D>0,eventhoughtheSDEsaredoped.Toverifyourinsights,we show,inFig.4.5,MEDICI-predictedvariationinn(x,y)withVGSforthe sL=15nm design,atVDS=50mV.Thesepredictionsclearlyindicatethat,inthesubthreshold region,gatecontroloftheS-DchannelchargeextendsintothedopedSDEs,i.e., Leff(weak)>LgateasinEq.(4.2),butitislimitedbyshortLD(n)toaboutLgateinthe on-state[Fos03c]asapproximatedinEq.(4.3).(Theresultsalsoverifythatthereis no significant S-D punch-through.) ThecomparisonofNSD(y)andn(y)inFig.4.5furtherimpliesthatthe gate-inducedmodificationofdrift-diffusionbalanceinthey-direction,which governsLeff(weak)>Lgate,canbecharacterizedasgate-inducedcarrierdepletionin

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117 Figure 4.5MEDICI-predictedelectrondensity(atx=0an dx=wSi/2)variationwith VGS,atVDS=50mV,forthenFinFETdesignwith sL=15nm;notethat thethresholdvoltageis~0.4V.(Thetwodifferentnotedvaluesofxreect thelocationofmaximumelectronconcentrationintheUTB.)TheS/D dopingproleissuperimposedonthen(y)plots.(Theabnormallyhighn instronginversionisbelievedtobecausedbytheuseof3-Ddensityof states(DOS)insteadof2-DDOSinMEDICI,aswellasneglectofcarrierenergy quantization.) 2030405060708090 100y [nm] 1012101310141015101610171018101910201021n, NSD [cm-3] NSD(y) n(x=wSi/2, y) n(x=0, y) VGS=1.0V 0V Lgate

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118 theSDEs.(Similarly,notethatLeff(strong)canbesomewhatlongerthanLgatedueto gate-inducedcarrieraccumulationintheSDEs,theunderlyingphysicsofwhich,as explainedearlier,isdifferentfromthatintheconventionalMOSFETwithG-S/D overlap.)Interestingly,suchdepletionoftheSDEsinweakinversioncanalsooccur intheconventionalbulk-SiMOSFETwherethechannelisheavilydoped,andthe source/drain-bodyjunctionisthus“two-sided.”Thatis,thechanneldopingcan suppressSCEsvialengtheningLeff(weak)aswellaslimitingEyinthechannel [Tau98].Thisinsightexplainstheresultsofthesimulation-basedstudyin[Kwo02] that show SCE dependence on sL analogous to that in Table 4.1. ToquantifyLeff(weak),orLeSandLeDinEq.(4.2)fordopedSDEs,we calibratedUFDG[Fos03a]toMEDICI-predictedsubthresholdcurrent-voltage characteristicsoftheFinFETs.UFDGassumesabruptS/Ddopingprofiles,but rigorouslyaccountsfortheLeff-governedSCEs.TheUFDG-evaluatedLeff(weak)(withLeS=LeDinEq.(4.2))plottedversus sLinFig.4.6isingoodaccordwithwhat canbeinferredaboutLefffromFig.4.5for sL=15nm.AlsoshowninFig.4.6is Leff(weak)fortheabruptNSD(y),which,asexpected,isapproximatelyLgate.In addition,withreferencetotheUFDG-predictedLeff(weak)for sL=10nm,weused MEDICItosimulateannFinFEThavingLgate=Leff(weak)andabruptS/Djunctions, andwefoundthatthepredictedsubthresholdcharacteristicsarevirtuallythesameas thosereflectedinTable4.1forthe sL=10nmdesign.Thisfindingfurtherverifies Eq.(4.2)forshort-channeldevices.ThedecreaseinLeff(weak)withincreasing sLin Fig.4.6canbeexplainedbyconsideringLD(n),which,asnotedearlier,governsthe extentofthegatecontrolintheSDEs.As sLisincreased,NSD(y)andEynearthe

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119 Figure 4.6UFDG-evaluatedLeff(weak)versusstraggleforthenFinFETdesignsin Fig. 4.4,as indicatedintheinset.TheabruptNSD(y)(with sL=1nminFig. 4.1) yieldsLeff(weak)@ Lgate=18nmasindicated.(Duetothenon-physical electronmobilitydependenceonEyintheSDEsassumedinMEDICI, thereissomeuncertaintyinthevaluesofLeff(weak)shown,butonlyonthe order of 1nm.) 0.02.04.06.08.010.012.014.016.0sL [nm] 15.0 20.0 25.0 30.0 35.0 40.0Leff(weak) [nm] [abrupt NSD(y)] NSD [cm-3] Lgate LextLext 101010121014101610181020 20 30 40 50 60 70 80 90 100 y [nm] Leff(weak)

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120 gateincrease(seeFig.4.4),drawingmoreelectronsthatcaneffectivelyscreenthe gateencroachmentintotheSDEs,andthusshortenLDandLeff(weak).Finally,as shownintheinsetofFig.4.6,westressthatLeff(weak),andtheimpliedunderlap,are not defined by a specific value of NSD. 4.3 Source/Drain Series Resistance and Optimal Design Insights WhilehigherNSDnearthegateedgereducesLeff(weak),italsodrawsmore electronsintotheSDEinstronginversion,andtherebylowersRS/D.Thereisthen clearlyatradeoffbetweenSCEcontrol(orlongLeff(weak))andRS/Dlimitingin nanoscaleFinFETswithG-S/Dunderlap,asreflectedinTable4.1,andasshown empiricallyin[She03]forDGMOSFETsandin[Kwo02]forbulk-SiMOSFETs.The lowerRS/DunderliestheincreaseinIonwith sL,whiletheSCEsandIoffincreasetoo. Notethattheseresultsexplaintheexperimentaldata,e.g.,in[Hua99]and[Yu02], showingwell-controlledSCEs,withhighRS/D,inDGFinFETshavingwSi/Lgate 1.0.Moreimportantly,notethatsinceLeff(weak),governedbyNSD(y),definesthe SCEs,wSi/Lgateisnotanappropriatedesigncriterion.TheSCE-RS/Dtradeoff,and theimportanceofusingLeff(weak)ratherthanLgate,becomemoreapparentfromthe significantlylargerSCEsintheabrupt-NSD(y)design(Table4.1),which,inaddition tolowerRS/D,contributetothehigherIon.TosuppresstheSCEsinthisdesigntobe comparabletothewell-controlledonesofthe sL=15nmdesign,wefind,using MEDICI,thatthinnerwSi=8nm,i.e.,wSi/Leff(weak)@ 1/2.4,isneeded,whichalso causesIontobereducedbyabout20%(andmoreiftheenhancedwSi-induced quantization[Ge02b]isconsidered).AlthoughtheresultingIon(perunithSi)isstill

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121 higherthanthatfortheunderlapdesigns,thecurrentperpitchcouldbesignificantly lowerduetothesmallerhSiimpliedbythetechnology,andwebelievethatbothwSi=8nm,withgoodthicknesscontrol,and sL=1nm,withintrinsicUTBs,aredubious technologicalpossibilities.Thus,nanoscaleFinFETswillneedgate-S/Dunderlap that can be optimized via the Leff-based SCE-RS/D tradeoff. 4.4 Source/Drain-Extension Engineering Wenowexemplifysuchoptimizationviaawell-tempered,highperformanceLgate=18nmFinFETdesignbasedonourinsightsregarding Leff(weak)(NSD),RS/D(NSD),andtechnologicallimitationsonwSiand sL,i.e.,by engineeringtheSDEs.Wenotethatwhilesimilaroptimizationwasattempted empiricallyin[She03]forDGdevices,ournewphysicalinsightsregarding Leff(NSD)andRS/D(NSD)implyasystematicapproachtoSDEengineeringfor absolutedesignoptimizationofgenericnanoscaleUTBCMOS.Also,notethatsuch optimizationdependsontheCMOSapplication.Forourhigh-performancenFinFET design,wespecifyIonnear1.2mA/ m m(approximatelythatinITRS[ITRS03]scaled fortox=1.0nm)andIoffabout10nA/ m m.Tothisaim,wefirstinferbycomparingIonvs. sLinTable4.1andLeff(weak)vs. sLinFig.4.6thatRS/Dtendstobecomeseverely largeforLeS=LeDlongerthan~5nm,alimitthatwehavefoundtobeindependent ofnanoscalegatelengths.Whiletheabrupt-NSD(y)designwithLeS=LeD@ 0isideal whenonlyIon,orRS/D,isconsidered,itrequireswSi<10nm(andwSi<5nmforLgate@ 10nm)foracceptableSCEs,andhenceIoff,control.However,aswenotedearlier, suchthinfinsandabruptjunctionsaretechnologicallynotpossible.Then,tokeep wSiasthickaspossiblefortechnologicaleasewhilemaintaininglowRS/D,weneed

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122 LeS=LeD=4nm,orLeff(weak)@ 26nm,whichsignificantly(>40%)relaxesthewSirequirement to @ 14nm for acceptable SCE control. WithwSi=14nm,wenowtailorNSD(y)toachievethedesiredLeSand LeD.Notethatingeneral,engineeringofNSD(y)forneededLeSandLeDdependson technologicallyachievablewSi,sincewSialsogovernsSCEsandRS/D.Althoughthe sL=15nm(withLext=30nm)designinFig.4.4withwSi=14nmissufficientfor SCEcontrol,wefindthatitsIonisrelativelylow.ToincreaseIonusingthesame midgapgate,weneedtoreduceRS/D.BasedonourinsightsandTable4.1,increasing sLisameanstoachievethis,however,itislikelytoresultinS-Dpunch-throughas impliedbyEq.(4.5).Alternatively,scalingLextcanalsoreduceRS/D,butwith sLredesignedforthedesiredLeff(weak),aswellastoavoidS-Dpunch-throughby satisfyingEq.(4.5).NotingthelowerlimitofLexttobeaboutLeS=LeD=4nm (whichwouldrequireaprohibitive sL@ 1nmforLeff(weak)@ 26nm)asimpliedbyour physicalinsights,weletLext=20nm @ Lgate.Then,usingMEDICI,wefindthat doable sL=9.5nm(abruptness @ 7nm/decade)isneeded,andyields(viaUFDG) Leff(weak)=27nm.WenotethatforLextmuchshorterthan20nm, sLneededforLeS=LeD=4nmbecomesaquestionabletechnologicalachievement,whileforLextmuch longerthan20nm,lowerIonresults(asinthecaseofLext=30nmwith sL=15nm); i.e.,anabsoluteoptimal,doableSDEdesignliesinthevicinityof(Lext, sL)=(20nm, 9.5nm).MEDICI-predictedIDS-VGScharacteristicsofourwell-tempered18nm nFinFETwith(Lext, sL)=(20nm,9.5nm)areshowninFig.4.7,contrastedwiththe characteristicsforthecounterpartdesignwiththeabruptNSD(y).Thelatter,aswe haveexplained,reflectsevereSCEsandhighIoff.Tuningthegateworkfunctionof

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123 Figure 4.7MEDICI-predictedchannelcurrent(perunithSi)-versus-gatevoltage characteristicsforawell-temperedLgate=18nmnFinFETdesignwith effectiveG-S/Dunderlap.Alsoshownarethecharacteristicsforthe counterpartdevicewithabruptNSD(y)havingthesamewSiasthewelltempereddesign.ThehigherIoninthelatterdeviceisduetoSCEsaswell aslowerRS/D.ForequalIoff,thisdeviceyieldsabout10%-lowerIonrelative to that of the underlap design. 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-1010-910-810-710-610-510-410-310-2IDS [A/ m m] well-tempered NSD(y) abrupt NSD(y) VDS=50mV VDS=1.0V DIBL=110mV/V S=78mV DIBL=270mV/V S=102mV

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124 thelattertomatchIoff=14.1nA/ m moftheformer,wegetanon-statecurrentthatis about10%lowerthantheIon=1.16mA/ m mofthewell-tempereddesignwithG-S/D underlap.Wereiteratethattheinferior,abrupt-NSD(y)deviceisnotmanufacturable. Increasing sLformanufacturability,whileavoidingunderlapinaconventional overlap-designapproach,wouldresultinpunch-throughand/orsevereSCEssince LeffwouldbesubstantivelylessthanLgate.Indeed,theeffective-underlapFinFET design,withSDEengineering(i.e.,Lext,NSD( sL),andwSiengineering)forLeff(weak)control,isoptimalfornanoscaleCMOS.AlsonotethatsinceIonforlow-power applicationsismuchlowerthanthatinourdesign,low-powerFinFETscouldbe achieved by increasing LeS and LeD rather than changing the gate materials. Further,becauseoftherelativelylowNSD(~1018cm-3)nearthegateofthe G-S/D-underlapFinFET,gate-induceddrainleakage(GIDL)current(whichwasnot accountedforinourMEDICIsimulationshere)willbesignificantlyreduced,ifnot eliminated,aswasshownin[Shu03]forUTBFD/SOICMOS.Theoptimalnatureof theG-S/D-underlapFinFETdesign,asopposedtotheabrupt-NSD(y)device,isalso reflectedbytheMEDICI-predictedgate-capacitancecharacteristics(CGGvs.VGS) showninFig.4.8.Notethemuchhighercapacitanceoftheabrupt-NSD(y)deviceat lowVGS,whichisduetogatefieldfringing;thisimplieshigherGIDLcurrent relativetothatoftheunderlapdesign,aswellasreducedCMOSspeedasnotedin [Bal03].Also,noteinFig.4.8thatwhenthethresholdvoltageoftheabrupt-NSD(y) deviceisincreasedtocontrolIoff,thehigh-VGSCGG,andhenceIon,areclearly undermined,implyingthattheG-S/D-underlapdesignisoptimalwithrespectto CMOS speed as well.

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125 Figure 4.8MEDICI-predictedlow-frequency,low-VDSgatecapacitanceversusgate voltageforthewell-temperednFinFETdesignwithG-S/Dunderlap,and thoseforthecounterpartdevicewithabruptNSD(y),withandwithouta tuned gate work function to match the Ioff of the underlap design. 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 0.2 0.4 0.6 0.8 1.0 1.2CGG [fF/ m m] well-tempered NSD(y) abrupt NSD(y) abrupt NSD(y) w / same Ioff

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126 4.5 Model for Relating SDE design and Effective Channel Length SDEengineeringforournotedexampleinSec.4.4isdonevianumerical devicesimulations,withfewiterations.Tobetteraidthisengineering,wedevelop hereamodelrelatingtheSDEdesign(i.e.,Lextand sL)andLeff(weak),withLeS= LeD,foragivenchannel/bodydesign(i.e.,Lgate,tox,andwSi).Themodelis applicabletoundopedDGMOSFETs,andweverifyitforSDGdevicesviaour earlier simulation results as well as new ones. 4.5.1 Simplified Source/Drain-Doping Profile BasedonourphysicalinsightsfromFig.4.5,Leff(weak)>LgateduetogateinduceddepletionintheSDEs.Further,Fig.4.5indicatesthatthedepletionwidthin theSDEs(wdS/D)isontheorderofLDdefinedbytheS/Ddopingnearthegateedge. Henceforourmodel,wesimplifythegenericNSD(y)toaneffective,uniformS/D doping(NSD(eff)),asillustratedinFig.4.9,definedbyanaverageNSD(y)within @ 2LDfromthegateedge.NoteinFig.4.9thatabruptS/Djunctionatthegateedge isassumedforthesimplifiedS/Ddopingprofile.Thisassumptionisreasonablesince theS/D junction occursatthegateedgeinbothcases.Whilethisisobviousforthe simplifiedprofile,toillustrateitfortheNSD(y)case,wefirstnotethata junction physicallycorrespondstoapointcorrespondingtozerochargedensity,withcharge densityononesidehavingpolarityoppositethatontheotherside(asinap-n junction),andthattheelectricfieldismaximumatthejunction[Sze81].Then,we showinFig.4.10MEDICI-predictedmagnitudeofthelateralelectric-field(ey)for variousVGSinourearlierFinFETdesignwith sL=15nm.Notethetwomaximasat the two gate edges, defining the source-body and drain-body junctions.

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127 Figure 4.9AdepictionoftheactualS/Dlateraldopingprole,withpeakdensity NSD(peak),andasimpliedabruptprolewithuniformS/Ddopingof NSD(eff)denedasanaverageoftheactualprolewithincoupleofDebye lengths from the gate edge. Lgate LextLext NSD(eff)NSD(peak)y NSDActual Simplified

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128 Figure 4.10MEDICI-predictedmagnitudeofthelateralelectric-eldalongthe channel(atx=wSi/2)forvariousVGSinFinFETdesigninFig.4.4with sL=15nm.Thegateplacementisalsoindicated.Thetwopeaksineyreectthesource-anddrain-bodyjunctions,eybetween20-40nm(80100nm)reectthebuilt-ineldduetoNSD(y),andeybetween40-50nm (70-80nm) reect source (drain) depletion. 102030405060708090100110 Distance along the channel [nm] 0.0 1.0 2.0 3.0 4.0| ey| x 105 [V/cm] VGS=50mV, 0.1V, 0.3V, 0.4V Lgate

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129 4.5.2 What is L eff(weak) ? WhendefiningLeff(weak)(Leff(strong))viaUFDG,wearemappingthe originaldevicestructureinFig.4.9ontoonehaving(a)Lgate=Leff(weak)(Leff(strong)) inweak(strong)inversion,and(b)heavily(~1020cm-3)anduniformlydopedSDEs withabruptjunctionsattheLgate=Leff(weak)(Leff(strong))edges.Thiseffective devicestructureisillustratedinFig.4.11(a)forweakinversion;theactual(and simplified)devicestructureisshowninFig.4.11(b)forcomparison.Physically,this mappingshouldensurethattheweak-inversionchannelcurrentintheeffective device is same as that in the actual device. Then since (4.6) intheabsenceofSCEs,Leff(weak)ensuresthisequivalence,anditcanbeevaluated from the x-intercept of 1/Ich versus Lgate plot for given VGS, as in Fig. 4.2. However,whenSCEsarepresent,wecannotensurethenotedIchequivalencesimplyviaLeff(weak).Toexplainthis,weaugment,inrathersimple terms, Eq. (4.6) to account for SCEs: ,(4.7) where D Vt SCEistheVt-shiftduetoSCEs,dependentonLeff(weak)andS/Ddoping, and Le is the length over which predominant carrier diffusion occurs [Yeh95], i.e., (4.8) 1 Ich-----Leffweak () LgateLeSLeD++ = 1 Ich-----LeD Vt SCEkBTq ---------------- – exp LeLgateLeSLeD++ () LSLD– – =

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130 Figure 4.11(a)AqualitativerepresentationofLeff(weak)andtheresultingeffective devicewithheavilyanduniformlydopedabruptS/D.(b)Theactualdevice withspatiallyvaryingS/Ddopingprole;alsoshownisthesimplied prole from Fig. 4.9. Lgate LextLext NSD(eff)NSD(peak)yNSDActual Simplified Lgate = Leff(weak)LeS ~1020 cm -3yNSD LeD(a) (b)

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131 withLSandLDbeingthesourceanddrainencroachments,respectively,asdefined inAppendixA,andtheydependonLeff(weak)andS/Ddoping[Yeh95].Now,because LS,LD,and D Vt SCEdependontheS/Ddoping(viathebuilt-involtage,Vbi[Yeh95]), whichvariesspatiallyintherealdevice,Leff(weak)alonecannotensureIchinthe effectivedeviceofUFDGtobethesameasthatintherealdevice.However,from the2-Danalysisin[Yeh95],wefindthatSandDIBLpredominantlydependon Leff(weak),whiletheS/Ddopingonlyaffectstheabsolute D Vt SCE,i.e.,onlyrigidly shiftsIch.And,sincenanoscaleMOSFETswillhaveSCEs,e.g.,S @ 80mV,wemodel Leff(weak)inshort-channeldevicesastoensureSandDIBLintheeffectivedeviceto bethesameasthatintherealdevice,whichinconjunctionwithanappropriate change in Vbi [Cho04] ensures equal Ich in the two devices. 4.5.3 Model Development IncreaseinSwithdecreasingLeff(weak)reflectslossofgatecontrolover thechannelcharge,orpotential.Locally,thislostgatecontrolisreflectedbya reductionind fmin(x)/dVGScausedbyey(x),where fmin(x)istheminimum(iny) potentialatsomexinthebody.Further,because fminisindependentofeyinalongchanneldevice,therelativechangeinthegatecontrolofashort-channeldeviceis also reflected by dey(x)/dVGS. Thus,toensureequalSandDIBL,dey(x)/dVGSbetweeny=LeSandy= (Leff(weak)-LeD)inFig.4.11(a)shouldbecomparabletothatbetweeny=0andy= LgateinFig.4.11(b);inbothcases,thex-yoriginisdefinedatthegateedgeonthe sourceside.(Notethat f (x,y)intheeffectivedeviceofFig.4.11(a)isnonphysical fory=0(Leffweak-LeD)toy=LeS(Leff(weak))becauseofthepresenceofthegate

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132 overtheseregionsunlikethatintheactualdeviceinFig.4.11(b).)Forourmodel,we only require .(4.9) Then, from 2-D analysis in [Yeh95], we find, for low VDS, ,(4.10) whereLgateisthegatelengthoftheactualdeviceand l [Yeh95]reflectstheextent ofS/Dencroachmentintothechannel.Forthederivativeontheleft-handsideofEq. (4.9), we use the chain rule to write .(4.11) SinceweassumeuniformS/DdopingofNSD(eff),thefirstterminEq.(4.11)issimply .(4.12) The last term is ,(4.13) whereSisthegateswing.Forthemiddleterm,weusethebasicp-njunctionanalysis VGSd dey y0 =in Fig. 4.11(b)VGSd dey yLeS=in Fig. 4.11(a)= VGSd dey yLeS=in Fig. 4.11(a)1 l -- – Lgate2 l ----------sinh h Leffweak ()2 l ----------------------cos ----------------------------------------= VGSd dey y0 =in Fig. 4.11(b)wdSD d dey y0 =in Fig. 4.11(b)fmind d wdSD VGSd d fmin = wdSD d dey y0 =in Fig. 4.9(b)qNSDeff ()eSi----------------------= VGSd d fminkBTq () 10 () ln S -------------------------------------=

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133 [Sze81] to approximateey at the source-body junction as ,(4.14) where f (y=0)isthepotentialatthejunction.Evaluating f (y=0)via1-Danalysisinthe S/D depletion region, we get a quadratic equation which can be solved analytically to get ,(4.15) with the extrinsic Debye length .(4.16) Then, with Eq. (4.10)-Eq. (4.13) and Eq. (4.15), Eq. (4.9) becomes .(4.17) WenotethatalthoughwdS/DvarieswithVGSthrough fmininEq.(4.15),becauseof therelativelystrongercosh-dependenceforLeff(weak)ontheright-handsideofEq. (4.17),Leff(weak)isvirtuallyindependentofVGSinweakinversion,consistentwith our earlier insights. BecauseLeff(weak)isvirtuallyinsensitivetovariationinwdS/D,weusein Eq.(4.17)wdS/DfromEq.(4.15)with fmincorrespondingtoVGS=Vt/2,i.e.,an average wdS/D. From [Yeh95], we find qNSDeff ()wdSD eSi--------------------------------------2 f y0 = ()fmin– () Lgate2 ---------------------------------------------@ wdSD Lgate4 ----------- – Lgate4 ----------22 Vbifmin– kBTq -----------------------LD()2+ + = LDeSikBT q2NSDeff ()------------------------= kBTq () 10 () ln S -------------------------------------1 wdSD Lgate4 ----------+ ---------------------------------1 l -Lgate2 l ----------sinh h Leffweak ()2 l ----------------------cos ----------------------------------------=

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134 ,(4.18) wherethelong-channel,or1-D fmin()atVGS=Vt/2canbeeasilydetermined fromthe1-DsolutionofthePoisson’sequation(orfromcombinationofEq.(3.8), Eq.(3.9)andEq.(3.13)).Further,wenotethatEq.(4.17)andEq.(4.18)dependon xthrough l and,implyingthatLeff(weak)dependsonx.Thisisphysical,because gatecontroloverthechannelcharge,andhencethedegreetowhichitperturbslateral drift-diffusionbalance,varieswithx.However,forourmodel,weonlyconsider,as inChapter2,xcorrespondingtotheleakiestS-Dconductionpath,e.g.,x=tSi/2for undoped SDG MOSFETs. Finally,evaluatingEq.(4.17)withwdS/DfromEq.(4.15)andEq.(4.18) attheleakiestS-DpathquantitativelyrelatesLeff(weak)andNSD(eff),andviceversa, inundopednonclassicalnanoscaleMOSFETs.TorelateittotheactualSDEdesign (i.e., Lext and sL) with a Gaussian doping profile, we define, as noted earlier, ,(4.19) where l =2LDandNSD(peak)isthepeakS/Ddopingdensity.WecanevaluateEq. (4.19) using the error function (erf) as .(4.20) Thus,oneneedstosolveEq.(4.17)andEq.(4.20)self-consistentlytorelate Leff(weak) to SDE design. Vbifmin–Vbifmin 1D– () 12 Leffweak ()2 l ----------------------- – exp – = fmin 1Dfmin 1DNSDeff ()1 l -NSDpeak ()Lexty – sL-----------------2– exp0 l NSDeff ()NSDpeak ()sLl -----------------------------p 2 -----erf LextsL--------erf Lextl – sL----------------– =

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135 4.5.4 Model Verification Whileourmodelisgeneric,itispredominantlyaimedatestimating sL(or Lext)foraUFDG-evaluated/desiredLeff(weak)foradevicewithgivenLgate,wSi,tox, andLext(or sL).Thus,usingEq.(4.15)andEq.(4.18),werearrangeEq.(4.17)toget .(4.21) ForaknownLeff(weak),withgivenLgate,wSi,andtox,equatingEq.(4.21)andEq. (4.20)definestheneeded sL(Lext)forgivenLext( sL);wefindthatassumingVbi@ 0.5V,independentofNSD(eff)<1020cm-3,isreasonable.(WenotethatforLeff(weak)RS/D optimization, Lext ( sL) would be determined based on acceptable RS/D.) WithEq.(4.21)andEq.(4.20),wenowverifyourmodelforintrinsicbodySDGMOSFETsusingourearliernumericaldevicesimulationsofFinFETs,and fewnewones.Inparticular,wecomparethemodel-predicted sLtothatactually employedinMEDICI.Forintrinsic-bodySDGnMOSFETswithmidgapgates,we first note that (4.22) in Eq. (4.21), and .(4.23) NSDeff ()2 eSiVbifmin 1D– () 12 Leffweak ()2 l ----------------------- – exp – q l kBTq () 10 () ln S -------------------------------------h Leffweak ()2 l ----------------------cos Lgate2 l ----------sinh -----------------------------------------2Lgate4 ----------2– -------------------------------------------------------------------------------------------------------------------------------= fmin 1DVGSVFB–Vt2 == l x tSi2 ----= tSi2 ----1 2 -14 eSieox------toxtSi-----+ =

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136 Next,wesolveforNSD(eff)usingEq.(4.21)withUFDG-predictedLeff(weak),andthe assumedLgate,wSi,andtox.WethenuseasimpleNewton-Raphsonmethodtosolve for sLfromEq.(4.20),withinitialguessimpliedbylettingNSD(eff)betheS/D doping at the gate edge; typically, 2-3 iterations are needed for accuracy of 0.1nm. WecompareinTable4.2(a)themodel-predictedandactual sLforFinFET designsinTable4.1.Ourmodelisingoodaccordwiththeactual sLemployedin MEDICIsimulations,exceptforthecaseof sL=5nm.Themodelerrorfor sL=5nm isexpectedsinceitisdoesnothavemuchSCEs,andmoreimportantly,itisthe designwhereportionoftheSDEisleftundoped(seeFig.4.4).Thelatterimpliesthat ourmodelisonlyvalidwhenactualS/Ddopingnearthegateedgeissufficientlyhigh (>~1015cm-3).Hence,inpractice,themodel-predictedNSD(eff),and sL,shouldbe cross-checked with UFDG-evaluated RS/D in strong inversion. Tofurtherverifyourmodel,weuseMEDICItosimulatenanoscale FinFETswithLgate=18nm,tox=1nm,wSi=14nm,midgapgate,andvarious combinationofLextand sL.BothLextand sLarevariedtomaintainS/Ddoping ~1018cm-3atthegateedge.Asbefore,weevaluateLeff(weak)viaUFDG,whichis thenusedforourmodeltopredict sLforgivenLext.Themodel-predictedandthe actual sLarelistedinTable4.2(b).Notethattheseconddesignisouroptimal FinFETfromSec.4.4.Moreimportantly,wenotethatourmodelisinexcellent agreement with the actual sL. 4.6 Source/Drain-Doping Profile Requirements Finally,weuseourmodeltoprojecttheneeded sLforHP22(Lgate=9nm) andLSTP22(Lgate=13nm)technologynodeinITRS2003,assumingLext=Lgate.As

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137 sL [nm]Lext[nm]S [mV]Leff(weak)[nm] sL(model) [nm] 12.525792413.1 9.52078279.2 7.51583238.0 2.5592.4212.6 Table 4.2(b) Comparisonofthemodel-predicted sL(right-mostcolumn)totheactual sLemployed inMEDICI (left-mostcolumn)forLgate=18nmSDGFinFETswithtox=1.0nm, midgapgate,andthegivenLext,MEDICI-predictedS,andcorrespondingUFDGpredicted Leff(weak). sL [nm] sL(model) [nm] 159.1 1011.0 1514.1 Table 4.2(a) Comparisonofthemodel-predicted sLtotheactual sLemployedinMEDICIforour earlierLgate=18nmSDGFinFETdesignsofTable4.1withcorresponding Leff(weak) from Fig. 4.6.

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138 reportedin[Fos04b],forHP22nodewithtox=1.2nm(forgateleakagecontrol)and wSi=6nm(duetotechnologicallimitation),LeS=LeD=1.0nm(orLeff(weak)=11nm) isneededtosatisfytheITRSIoffrequirements;S=102mV.Similarly,forLSTP22 nodewithtox=1.5nmandwSi=6nm,LeS=LeD=4.2nm(orLeff(weak)=21.4nm)is needed ;S=66mV.ThenassumingLext=LgateandGaussianNSD(y)withNSD(peak)=1020cm-3,wefind,viaourmodel,that sL=7.3nm(withNSD(eff)@ 3x1019cm-3) and sL=5.4nm(withNSD(eff)@ 3x1018cm-3)isneededforHP22andLSTP22nodes, respectively.Unfortunately,while sL=7.3nmmaybedoableforHP22node,it implies,inconjunctionwithNSD(eff)@ 3x1019cm-3,S-Dpunch-through.Toavoid punch-through,wefindthatthinnerLext@ 5-6nmandsteeper sL@ 4nmareneeded. Incontrast,forLSTP22node,longerLextcanbeemployedtorelaxthe sL=5.4nm requirement, e.g., with Lext = 17nm, sL = 7.6nm would be needed. 4.7 Summary Wehaveshown,via2-Dnumericaldevicesimulations,thateffectiveG-S/ DunderlapwillbeneededtoyieldoptimallydesignednanoscaleFinFETs.Wefound thattheunderlap,whichisnotdefinedbyaspecificSDEdopingdensity,resultsin abias-dependentLeffthatislonginthesubthresholdregion,significantlyrelaxing thefin-thicknessrequirementforSCE-control,andshortintheon-state. DependencesofLeffandRS/Donthelateraldopingprofileinthefin-SDEsclearly revealedadesigntradeoffregardingSCEsandRS/D,whichstronglydependson technologicallyachievablewSi.Thistradeoffwasexemplifiedbyawell-tempered, high-performanceLgate=18nmnFinFETdesignviaproperengineeringofSDEs

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139 usingalarge,doablestraggle(lateralabruptness)of9.5nm( @ 7nm/decade)without causingS-Dpunch-through.Theeffectiveunderlapalsoimpliessuppression,ifnot elimination,ofleakagecomponentssuchasGIDLcurrent.Finally,weshowedthat designswiththeeffectiveunderlaphaveminimalCGS/Dinweakinversionandhigh CGG in the on-state that will further optimize CMOS speed and drivability. ToaidtheSDEengineering,wedevelopedamodelforLeff(weak), dependingontheSDEdesign(i.e.,Lextand sL)andthechannel/bodydesign(i.e., Lgate,tox,andtSi,orwSi).Themodelwasverifiedvianumericaldevicesimulations. Wealsoappliedourmodeltoprojecttheneeded sLrequirementsforHPandLSTP CMOS near the end of ITRS, and found that sL@ 5-7nm will be needed. Finally,thenotedSCE-RS/Dtradeoffcanbefurtheroptimizedbyflaring out,i.e.,widening,theSDEs.Whileawell-temperedNSD(y)definesLeff(weak),and henceSCEs,alargerareaoftheSDEswillyieldlowerRS/D.Hence,theLeff-based underlapdesignoptimizationisapplicableevenwhenthefinextensionsareflared outtoreduceRS/D.Ingeneral,thenotedLeff-basedoptimizationremainsapplicable foranyotherplausiblesolutionstolowerRS/DprovidedthattheneededNSD(y)for the desired Leff(weak) can be achieved.

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140 CHAPTER 5 QM-BASED MODELING OF EFFECTIVE CARRIER MOBILITY 5.1 Introduction MOSFETcharacteristicsarelargelydefinedbycombinationof electrostaticsandcarriertransport.Thusfar,wehavepredominantlydiscussed electrostatics,andoptimaldesignsimplied.However,liketheelectrostatics,carrier transportinnonclassicalMOSFETsalsodifferssignificantlyfromthatinclassical MOSFETs[Ge02a].Forscattering-dominatedtransport,carriermobilityisthemost importanttransportvariable.FornonclassicaldeviceswithUTBs,MonteCarlo simulations[Gm98],[Sho99],[Gm01b],[Fis03],[Gm03]andexperimental observations[Ess00],[Ess01a],[Ren02],[Uch03]showthatinadditiontothetypical effective(transverse)field(eeff)dependence[Sab79],[Tak88],theeffective,or average,carriermobility( meff)alsodependenceontSi.Thisadditionaldependence mustbeincorporatedintocompactmodelsformeaningfulCMOSdevice/circuit designsandperformancepredictions.Previously,Ge[Ge02a]developeda preliminaryphysicalmodelfor meff(tSi,Ninv)inDGdevices,withfocusmainlyon theeffectsofvolume/bulkinversion(VI)on meffinSDGMOSFETs.However,we findthatthismodelisinsufficientforgenericUTBCMOSdevices,anditbecomes impractical for compact models upon generalization to such an end. Thus,inthischapter,wefirstgainphysicalinsightsonthetSi-dependence of meffusingMonteCarlosimulationsandSCHRED.(WeacknowledgeProfessor

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141 F.GmizofUniversityofGranadainGranada,Spainforperformingtheneeded MonteCarlosimulations.)Basedontheseinsights,weuseQM-basedanalysesto developaphysicalcompactmodelfor meffinviable(undopedwithtSi> @ 4-5nm) genericUTBsiliconCMOSdevices.Themodel,implementedinUFDG,focuses mainlyonCoulomb-,phonon-,andsurface-roughness-limitedmobilityinUTB devices,anditisbasedontheBoltzmanntransportequationwiththerelaxation-time approximation[Lun00].Themodelaccountsforelectronandhole meffdependence ontheUTBthickness,effectivetransverseelectric-field(includingNinv),andthe crystalorientation.Weverifythemodelusinglargesetsofexperimentaldataand MonteCarlopredictions.WethenapplyittocompareelectronmobilityinUTB deviceswith{100}-and{110}-Sisurfaceorientations.Weconcludethischapter withdiscussionofsomeoftheinterestingimplicationsof meffpredictedbyourmodel andmeasuredviaexperiments,includingcomparisonof meffinSDG,FD/SOI,and bulk-Si MOSFETs. 5.2 Coulomb-Limited Mobility Coulombscattering(CS)isoneofthemostimportantscattering mechanismatlowinversioncarrierdensity[Tak88].Whileintrinsic-UTBMOSFETs avoidCSduetodopantimpurities,CSduetointerfacetraps(orsurfacestates)is unavoidable,andhasbeenexperimentallyfoundtohavenoticeabletSidependence [Uch03].Thescatteringpotential(Vs(x,y,z))duetoaCoulombchargecenterwas modeled by Stern and Howard [Ste67], and it satisfies the 1-D Poisson’s equation ,(5.1) e x () Vsxyz ,, () 2rextrsc+ () – =

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142 where e (x)isthespace-dependentelectricpermittivity, rextistheexternalcharge densityresponsibleforCS,and rscisthescreeningchargedensityinducedby Vs(x,y,z).Theinter-dependenceoftheinducedchargedensityandthescattering potentialinEq.(5.1)[Ste67]typicallyrequiresnumericalevaluationofthescattering potential, unless a delta-function electron wavefunction/eigenfunction is assumed. ThesolutionofEq.(5.1)iscomprehensivelydetailedin[Gm94]for classicaldevicesandin[Gm02]fornonclassicaldevices.However,atlow inversion-carrierdensity(Ninv< @ 5x1011cm-2), rsc,orcarrierscreening,is negligible.Then,Vs(x,y,z)isvirtuallythebareCoulombpotential,andhence,itstSidependenceisdefinedbythedistanceoftheinversion-chargecentroidfromthetwo interfaces,i.e.,Vs(x,y,z)increaseswithtSi,whichinturndegradesthecarrier mobility.ForNinv> @ 1012cm-2,Vs(x,y,z)isscreenedbyfreecarriers,andthenoted tSi-dependenceissignificantlyreduced.Ourinsightsaresupportedby experimentallyextractedroom-temperatureCoulomb-limitedmobility( mco)[Uch03] shownin Fig.5.1for FD/SOInMOSFETs.Althoughthemeasureddatashows noticeabletSi-dependencefor mco,todeterminetheinuenceofthistSi-dependenceon meff,weconsideraworst-casescenariobyadding(viaMathiessen’srule)themeasured mco(tSi)forvariousNinvtoMonteCarlo-predictedlow-eeffphonon-limitedmobilityin UTBs.AsalsoshowninFig.5.1,wendthat,forrelativelycleaninterfaces(Nit<1011cm-2),theeffectof mco(tSi)on meffisnegligiblysmall(<10%). Hence,wedonotexplicitly model mco here.

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143 Figure 5.1 Measuredroom-temperature mco(Ninv)inFD/SOInMOSFETswithtSi= 60nm(solid)andtSi=5.9nm(dash-dot) [Uch03];the dottedlinesforNinv<2x1011cm-2projectNinv-independent mcoduetonegligiblescreening.For aworst-casescenario,thesmalleffectofthis mco(tSi)dependenceon meffis exempliedbyadding,viaMathiessen’srule,thetwo mco(Ninv) characteristicstoMonteCarlo-predictedlow-eeff,thin-tSiphonon-limited mobility( mph0@ 580cm2/V-sfortSi=5.9nm),gettingthelowertwo, comparable curves. 450 500 550 600 650 700( mco -1 + mph0 -1)-1 [cm2/V-s] 101110121013Ninv [cm-2] 103104mco [cm2/V-s] Nit(front interface) @ 1x1011cm-2Nit(back interface) @ 3x1010cm-2

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144 5.3 Phonon-Limited Mobility Carrierscatteringduetophonons,orquantizedthermalvibrationofa crystal,isaninherentscatteringprocess.Atlowinversioncarrierdensity(Ninv)and inabsenceofstrongcoulombscattering,carriermobilityispredominantlydefined byphononscattering[Tak88].Whilephonon-limitedmobility( mph)inconventional MOSFETshasbeenextensivelystudied[Tak88],[Fis93],showing @ Ninv -1/3dependence,recentMonteCarlosimulations[Gm98],[Sho99],[Gm01b],[Fis03], [Gm03]shownoveldependenceof mphontSi.Thus,inthissection,wefirstgain physicalinsightson(room-temperature) mph(tSi)ingenericUTBdevicesviaMonte CarlosimulationsofSDGandADGnMOSFETs.Wethendevelop,usingQM-based analysis,aunified,compactmodelfor mphthatisapplicabletogenericDGdevices, leading to a compact model for meff in Sec. 5.5. 5.3.1 Scattering Hamiltonian Foracousticphonons,thetheoryofdeformationpotential,proposedby BardeenandShockley[Bar50],andlaterextendedbyHerringandVogt[Her56]for anisotropicandmulti-valleyscattering,ismostcommonlyused[Fis93].Thetheory assumesthatalocaldeformationofthelatticeduetothermalvibrationbehaves similartoauniformlydeformedlattice,andthus,intermsofthelatticedilation ( D ( r )),thescatteringHamiltonian,tofirstorder,forisotropicscatteringbecomes [Bar50] .(5.2) InEq.(5.2),Eacisthedeformationenergy/potentialandu( r ,t)isthedisplacement Hacr t , () EacD r () Eacu r t , () ==

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145 vector describing the harmonic oscillations of an atom about its origin, i.e., ,(5.3) where b isthephononmomentumvectorandAbisthemagnitudeofthedisplacement inthepolarizationdirection eb.Typically,withitsmomentumclosetothecenterof theBrillouinzone[Lun00],energyofacousticphonons( vsb withsoundvelocity vs) is small such that it predominantly assists intravalley, intrasubband scattering. Similarly,thescatteringHamiltonianforoptical,orintervalley,phonons is [Lun00] ,(5.4) whereDivisadeformationfield,orelectron-phononcouplingfactor.Generally, therearetwotypesofintervalleyphononmodes:theg-typemodesassistscattering toavalleydirectlyacrosstheoriginalvalleyintheBrillouinzone,andthef-type modesscattercarrierstotheremainingvalleys[Lun00].Finally,sincemaximum energyofthesef-andg-typephonons( wiv,aconstantfrequency)is @ 60meV [Fis93]andoptical/intervalleyphononscatteringisinelastic,energyconservation requiresseparationbetweentheinitialandthefinalstatetobe 60meVfor intravalley and intervalley, intersubband scattering to occur. 5.3.2 Momentum Relaxation Rate Acousticphononscatteringratefor2-Dcarriersoccupyingonlyasingle bandwasdeterminedbyPrice[Pri81],andlaterformulti-subbandoccupationand intersubbandscatteringbyRidley[Rid82].Fromtheirwork,theintravalley, u r t , () Abei b r wbt – ()ei – b r wbt – ()+ () eb= Hivr t , () Divu r t , () =

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146 intrasubband(i.e.,acoustic-phonon)scatteringrate( tii -1)andtheintravalleyand intervalley,intersubband(i.e.,intervalley-phonon)scatteringrate( tfi -1,wherevalley indexesareneglectedforsimplicityandfdenotesthefinalsubband)forcarriersin the ith subband can be written as (5.5) and ,(5.6) where nfisthevalleydegeneracyofthefinalstate, rm=2329kg/m3isthemass densityofsilicon,vs=9037m/sisthelongitudinalsoundvelocity,g2Df(=mdf/ p2) isthe2-Ddensity-of-states(DOS)inthefinalstate,Nw ivisthephononoccupation numbergivenbyBose-Einsteinstatistics,and(Wfi)-1istheformfactorwithWfidescribinganeffectivephonon-carrierinteractionlength[Lun00].Ingeneral,the formfactorreflectstheoverlapbetweentheinitialandthefinalstate,andisgivenby ,(5.7) where yfand yiaretheelectronwavefunctionsinthefinalandtheinitialstates, respectively.Also,the inEq.(5.6)representsphononabsorptionandemission, respectively,andtheHeavsidefunction( Q (Ei-Efwiv))assertsenergy conservation.WefurthernotethattypicalvaluesofEacvaryfrom9-14eV[Jac83], 1 tii----niEach ------2kBT2 rmvs 22 ------------------h p Wii-------g2Di2 ---------= 1 tfi----nfDiv 2h wivrm-----------------Nwiv1 2 -1 2 -+ h p Wfi-------g2Df2 ---------Q EfEih wiv+ - – () = h 1 Wfi-------yfx ()2yix ()2x d0 tSi= h

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147 [Jun93],[Tak96],[Ess03b],whichsubstantivelyvaries tii Eac -2,andthevalueof Divdependsonthephononmode,i.e.,3g-typeand3f-typeforelectronsinSi [Lun00], [Fis93]. Finally,thetotalphonon-limitedmobilityisdeterminedbythesumof mobilityineachsubbandofeachvalley,givenbytotal(momentum-averaged [Lun00])scatteringrateforthatsubbandandconductivityeffectivemassforthat ( nth)valley(mC n),weightedbythefractionofcarrierpopulationinthatsubband,i.e., .(5.8) NotethatsincethescatteringratesinEq.(5.5)andEq.(5.6)areindependentof carrierenergy,andmomentum,theyarethemomentum-averaged[Lun00]scattering rates. Further, note that mph Eac 2, Div 2, and Wfi(/ form factor). 5.3.3 Monte Carlo Simulations of SDG and ADG nMOSFETs Thephysicalmodelfor mphinEq.(5.8)isasimplifiedversionofthat employedinMonteCarlosimulations[Fis93],butitisstillnotpracticalforcompact models.However,todevelopaphysics-basedcompactmodelfor mphinUTBCMOS devices,wefirstgainphysicalinsightsbasedonitsevaluationviaMonteCarlo simulations.Wefocuson mphinundopedSDGandADGnMOSFETswithundoped {100}-SisurfaceandtSivaryingfrom50nmto1.5nm.Themodelfor mphemployed for these simulations is analogous to that in [Fis93] with Eac and Div from [Can75]. MonteCarlo-predicted mph(Ninv)inundopedSDGandADGnMOSFETs withvarioustSiareshownin Fig.5.2;theADG structureconsistsofn+-andp+mphNiNinv---------mi inNiNinv---------q mC n---------1 tii----1 tfi----+ 1 – in==

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148 Figure 5.2MonteCarlo-predictedroom-temperature mph(Ninv,tSi)in(a)SDG [Gm01]and(b)ADGnMOSFETswith{100}-Sisurfacesandtoxf= toxb=1nm;forADGdevice,n+-andp+-polysilicongateswithNpoly= 1020cm-2 are specified. (a) (b) 10121013Ninv [cm-2] 500 700 900 1100 1300mph [cm2/V-s] tSi = 30nm tSi= 20nm tSi= 15nm tSi = 10nm tSi = 5nm tSi= 4nm tSi=3nm tSi = 1.5nm 10121013Ninv [cm-2] 450 500 550 600 650 700 750mph [cm2/V-s] tSi = 50nm tSi = 35nm tSi = 25nm tSi = 20nm tSi = 17nm tSi = 14nm tSi = 10nm tSi = 8nm tSi = 5nm tSi= 3nm

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149 polysilicongatesand1nmthickgateoxides.While mph(Ninv)inADG,andtSi 10nm SDG,devicesisvirtuallyindependentofNinv,notethatinthick-tSiSDGdevicesit exhibitsthetypicalNinv -1/3dependence.Toexamine mph(tSi),weplotin Fig.5.3 mph(tSi)from Fig.5.2atNinv@ 1011cm-2and1013cm-2.Clearly, mphinSDG MOSFETsexhibitsstrongtSi-dependence,especiallyatlowNinv,whilethatinADG MOSFETsisvirtuallyindependentoftSi>4nm.Further,inSDGdeviceswithlow Ninv, mphbeginstosaturatefortSi>20nmandtSi<10nm,anditisapproximately proportionaltotSiintheintermediaterange.Suchsaturationcanalsobeobservedfor mph(Ninv)inthick-tSi(e.g.,tSi=30nm)SDGdevicesinFig.5.2,andthesaturation valuesarecomparable.FromFig.5.3,wealsonotethat mphinstrongly-invertedSDG MOSFETsincreasesfortSi<15nm,reachinga(local)maximaneartSi@ 8nm. Further,fortSi@ 3nm, mphinbothSDGandADGMOSFETs,atbothlowandhigh Ninv,attains(anotherlocal)maximum,followedbyasharpreductionwithtSi. Finally,althoughnotshownhere,wenotethatMonteCarlo-predicted mph(Ninv,tSi) inSGFD/SOIMOSFETs[Gm98]isanalogoustothatinSDGMOSFETsexceptfor the enhancement around tSi@ 8nm at high Ninv. 5.3.4 Physical Insights Tounderstandthephysicsunderlying mph(Ninv,tSi)inFig.5.2andFig.5.3, wefirststress,fromEq.(5.5)-Eq.(5.8),thatthedevice-structuredependenceof mphisdefinedbytheformfactor(reflectingcarrierdistributioninrealspace)andcarrier populationineachsubband(reflectingcarrierdistributioninenergyspace),bothof whicharegovernedbyQMcarrierconfinement.Thecarrierconfinementeffecton thelatteristoincreasethesubbandseparation(reflectedin,forexample,Eq.(3.26)),

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150 Figure 5.3MonteCarlopredictedroom-temperature mph(tSi)inSDGandADG nMOSFETsofFig.5.2(a)andFig.5.2(b),respectively,withlowNinv@ 1011cm-2 (dashed) and high Ninv = 1013cm-2 (solid). 0.010.020.030.0tSi [nm] 400 600 800 1000 1200 1400mph [cm2/V-s] SDG ADG tSi 5.0 15.0 25.0

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151 reducingpopulationinthehigher-energysubbands,andthus,reducingscatteringin andtothesehigher-energysubbands.Thisischaracterizedhereasthesubbandmodulationeffect.Thereducedscatteringthenyieldshighermobility.The confinementeffectontheformercanbeunderstoodbythefactthatwithincreasing spatialconfinement,theHeisenberguncertaintyprinciple[Sha94]requireslarger spreadinthecarriermomentum,increasingthenumberofstatestoandfromwhich acarriercanscatter,andthus,reducingmobility.FromEq.(5.5)andEq.(5.6),this spread,or"fuzzyness"inthecarriermomentumisreflectedbytheterminvolvingthe form factor. Basedonthisdependenceoncarrierconfinement,thetendencyforthe low-NinvmphofSDGMOSFETinFig.5.3tosaturatefortSi>20nmcanbephysically explainedbythefactthatwithExC@ 0,carriersbecomefree,or3-D.Thisinsightis supportedbytheMonteCarlo-predictedsaturationvalueof mph@ 1350cm2/V-s, which is the bulk phonon mobility ( mph(bulk)) for electrons in Si [Tau98]. AstSiisthinnedbelow20nm,carriersinweakly-invertedSDGMOSFET begintobespatiallyconfinedviastructuralconfinement(SC),asnotedinChapter 3.Hence,theSC-governedincreaseintheformfactorwiththinnertSi,showninFig. 5.4forthegroundstateaspredictedbyMoteCarlosimulations,underliesthelinear dependenceoflow-Ninvmphon10nm
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152 Figure 5.4MonteCarlo-predicted[Gm03]ground-stateformfactorversustSiforlowNinv=1011cm-2(dashed)andhighNinv=1013cm-2(solid)in SDG nMOSFETs with {100}-Si surface. 0.0 10.020.030.0tSi [nm] 0.0 2.0 4.0 6.0 8.0 10.0Form Factor [x 106 cm-1] Ninv tSi

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153 Figure5.5showsSCHRED-predicted(fractional)carrierpopulationinall theunprimedandprimedsubbandsof{100}-SisurfaceSDGnMOSFETforvarious NinvandtSi.ConsistentwithourresultsinChap.3,atlowNinvandfor10nm
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154 Figure 5.5SCHRED-predicted(fractional)carrierpopulationinallunprimed (solid/filledsymbols)andprimed(dashed/emptysymbols)subbands of{100}-SisurfaceSDGnMOSFETsatroomtemperatureandwith various UTB thicknesses. 101110121013Ninv [cm-2] 0.0 20.0 40.0 60.0 80.0 100.0Subband Population [%] tSi = 3nm tSi = 5nm tSi = 10nm, 15nm, 20nm

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155 Figure 5.6MonteCarlo-predicted[Gm01] mphinthick-tSiSDGnMOSFETsof Fig.5.2(a)versusfractionofxavthatisequivalenttotSiinits contributiontodefiningtheground-stateformfactor.Alsoshownis the corresponding Ninv. 0.05.010.015.020.0(8/3)xav [nm] 400 600 800 1000 1200 1400mph [cm2/V-s] 101110121013Ninv [cm-2] xav

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156 confinedviaEC,whichlikeSC,increasestheformfactor,anddegrades mph.AsNinvincreasesfurther,subbandmodulationoccurs,and mphlevelsoff.Thus,fromthe similarityof mph(xav)inFig.5.6and mph(tSi)inFig.5.3,weconcludethatwhile mphisgovernedbycarrierconfinement,itsbehaviorisvirtuallyindependentofthe confinement mechanism. Basedonouraboveconclusion,the(virtually)tSi-independent mphof ADGMOSFETswithtSi 4nminFig.5.2(b)andFig.5.3canbeexplainedbythe verythinxavdefinedbyhighExC(tSi)(Eq.(3.9)),showninFig.5.7,andNinv.(We notethatalthoughtheExC-dependencetendstodecreaseathighNinv,ECisunaltered due to the high Ninv.) Wenowdiscussthepeaksin mph(tSi)inFig.5.3.Basedonourphysical insightsonthe mph-saturationneartSi=10nm,theincreaseintheformfactorinFig. 5.4,andthesubbandmodulationinFig.5.5,wefindthatthepeakneartSi=3nmin bothSDGandADGnMOSFETsatlowandhighNinvisduetotheprevailing subband-modulationeffects.Onceallthecarriersareinthelowestsubband, mphis predominantlydefinedbytheformfactor,which,asinFig.5.4,sharplyincreasesfor tSi<3nm,definingthesevere mph-degradationinFig.5.3.NotethatthisstrongtSidependencefortSi<4nmyieldssuchtSiimpractical,asweconcludedinChapter2 basedontSi-dependenceofFD/SOICMOSelectrostaticsanddelay,andinChapter 3 based on QM effect on Vt. The mphenhancementattSi@ 8nminstrongly-invertedSDGnMOSFETin Fig.5.3isduetovolume,orbulkinversion[Gm01].Asexemplifiedin Fig.5.8(a) bySCHRED-predictedelectrondistributionalongthebodyofSDGnMOSFETswith

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157 Figure 5.7Weak-inversiontransverseelectric-eldpredictedbyEq. (3.9)forv arious tSiinintrinsic-bodyADGMOSFETswithn+-andp+-polysilicongatesand toxf=toxb=1nm.Fermilevelinthepolysiliconisassumedtobeatthe conduction, or valence, band edge. 0.010.020.030.040.050.0tSi [nm] 105106ExC [V/cm]

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158 Figure 5.8SCHRED-predictedelectrondistributionalongtheUTBin{100}-Si surface(a)SDGand(b)FD/SOInMOSFETwithNinv=1013cm-3and variousUTBthicknesses.Thedistancealongthebodyisnormalized by the UTB thickness. (a) (b) 0.00.10.20.30.40.50.60.70.80.91.0Normalized distance along the body 0.0 1.0 2.0 3.0 4.0 5.0 6.0Electron Concentration [x1019 cm-3] tSi = 10nm, 8nm, 5nm Ninv = 1013cm-2 0.00.10.20.30.40.50.60.70.80.91.0Normalized distance along the body 0.0 1.0 2.0 3.0Electron Concentration [x1019 cm-3] tSi = 10nm, 8nm, 5nm Ninv = 1013cm-2

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159 varioustSiandNinv=1013cm-3,volumeinversionisphysicallycharacterizedbythe increaseintheelectronconcentration(orholeconcentrationforpMOSFETs)atthe centerofthebody.And,sincethetransversefieldatthecenteriszero,volume inversionreducesEC[Ge02a],whichincreases mph,asinFig.5.3.Thisisconsistent withthereductioninthehigh-NinvformfactoraftertSi<12nminFig.5.4.Wenote thatduetothedevicesymmetry,theelectronwavefunctionwillhaveevenorodd parity,andvolumeinversioncanonlyoccurinthosewithevenparity[Sho99],i.e., volume-inversioneffectsdependonsubbandmodulation.Inanycase,astSiis thinnedbelow8nm,thenotedreductioninECisnegatedbyincreaseinSC,as reflectedbytheoverlapofhigh-Ninvformfactorandlow-NinvformfactorsfortSi< 8nm,wherethelatterisdefinedonlybySC.Thus, mphisreducedinFig.5.3despite thehighertendencyofvolumeinversioninthintSiobviousin Fig.5.8. Hence,while carrierdistributioninthe(subband)energyspacedefinesthepeakattSi=3nm, carrierdistributionintherealspace,reflectedbytheformfactor,underliesthepeak at tSi@ 8nm. Further,independentofvolumeinversion,weakercarrierconfinementin SDGMOSFETrelativetothatinFD/SOIMOSFETforagivenNinvisapparentby comparingSCHRED-predictedelectrondistributioninFD/SOInMOSFETshownin Fig.5.8(b)tothatofSDGMOSFETinFig.5.8(a).Thisweakerconnementmeansthat mph, and meff, in SDG MOSFETs will always be higher than that in FD/SOI MOSFETs. Finally,forholes,theheavierconductivityeffectivemassintheground staterelativelyweakensthesubband-modulationeffect,andaugmentsthe mphreductionduetolargerformfactor.However,fortSi 7nm(<10nmforelectrons),

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160 significantreductionofscatteringinandtothehighersubbandscounteractsthis increased mph-degradation,andeventuallyrenders mphenhancementneartSi=2nm [Fis03](analogoustothatneartSi=3nmforelectrons).Hence, mph(tSi)forholesis analogoustothatforelectronswithsmallermagnitudeandashifttowardsthinnertSi. 5.3.5 Physics-Based Compact Model 5.3.5.1 Form Factor Tomodel mph,wefocusontheformfactordefinedinEq.(5.7).Wefirst considerthetwoextremecasesofpredominantECandSCseparately.ForEC,we usethetrialeigenfunctionofFangandHoward[Fan66],i.e., yj x,andfind ,(5.9) wherewehaveusedb0from[Ge02b]andExCisasinEq.(3.9).Similarly,using yj sin((j+1) p x/tSi) for SC, we get .(5.10) Further,wenotethatxavineachcaseisproportionaltothedistanceofthepeakin | y0|2(atx=xpeak)fromthesurface(a tx=0),i.e.,xav(ec)=3xpeak(ec)/2andxav(sc)= xpeak(sc). Then, Eq. (5.9) and Eq. (5.10) can be written in terms of xpeak as (5.11) and ebjx2 –W00ec ()16 3b0-------2 3 -8 3 -xavec () 2ExC5 6 -qNinveSi------------+ 13 / – == W00sc ()2 3 -tSi4 3 -xavsc ()== W00ec ()4 3 -2xpeakec ()() =

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161 ,(5.12) respectively.TheextrafactoroftwoinEq.(5.11),relativetoEq.(5.12),isbelieved to be due to the asymmetry of charge distribution for EC. Now,forthegeneralcase,wherebothECandSCcanoccur,weconsider the composite trial eigenfunction [Ge02b] ,(5.13) where h describesthesymmetryoftheinversionchargedistributionabou tx=tSi/2, e.g., h =0forFD/SOIMOSFETsand h =1forSDGMOSFETs,andbjisa variational parameter given by [Ge02b] (5.14) withtheeffectivemassintheconfinementdirectionofmx;ananalyticexpressionfor h willbegivenlater.RatherthancalculatingtheformfactordirectlyfromEq.(5.13), however,foracompactmodel,weexploitthenotedW00(xpeak)inEq.(5.11)andEq. (5.12).FortheeigenfunctioninEq.(5.13),wefirstfind,neglectingthesecondterm, .(5.15) (Ourneglectofthesecondtermistantamounttoneglectingvolumeinversion,which wewillincorporatelater.)NotethatwhenSCpredominates,i.e.,tSi<<(1/b0),Eq. W00sc ()4 3 -xpeaksc ()= yjx () j1 + ()p tSi------------------x sinebj2 ---x –h ebj2 ---tSix – () –+ bj2mxj1 + () h 2---------------------2ExC5 6 -qNinveSi---------------+ 13 /= xpeaktSip ----2 p tSib0----------atan =

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162 (5.15)givesxpeak=xpeak(sc),andwhenECpredominates,i.e.,tSi>>(1/b0),firstorderexpansionoftheinversetangentinEq.(5.15)yieldsxpeak=xpeak(ec).Then conformingtotheW00-asymptotesforpredominantECandSCinEq.(5.11)andEq. (5.12), respectively, we approximate the generic W00 as ,(5.16) wheretheargumentoftheinversetangentinEq.(5.15)isscaledbyfactoroftwoto accountforthefactorof2inEq.(5.11)duetotheasymmetryintheinversioncharge distribution.Further,because mphisvirtuallyindependentoftheconfinement mechanism,wealsodefineinEq.(5.16)aneffectivetSi(tSi(eff))thatgivesW00asif SCwerepredominant.Thisfacilitatesphysicalmodelingof mph(tSi,ExC,Ninv)in terms of a single variable. UsingEq.(5.9),Eq.(5.10),andEq.(5.16),weplot,inFig.5.9,themodel predictedformfactorsversustSiforvariousb0.ComparingtotheMonteCarlo predictionsforSDGnMOSFETsinFig.5.4,wenotethatthemodelpredictionsat low-NinvareinexcellentagreementwithMonteCarlopredictions,andthoseathigh Ninvareunderestimatedbyabout30%.However,ingeneral,forsuchhighNinv,or strongEC,thecarriermobilityispredominantlylimitedbysurface-roughness scattering,andhence,theeffectofthisrelativelylargedifferenceinW00on meffwill berathersmall.Finally,wenotethattheMonteCarlopredictionsforhighandlow NinvcoincidefortSi<8nm,whereasinourmodelpredictionsinFig.5.9,thisdoes notoccuruntiltSi@ 1.0nm.Thisisduetoourneglectofvolumeinversion,whichwe now address. W004 3 -tSip ----4 p tSib0----------atan 2 3 -tSieff () @

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163 Figure 5.9Model-predictedground-stateformfactorsversustSicorrespondingto b0=3.15x108cm-1(square)andb0=1.46x109cm-1(diamond),orNinv=1011cm-2andNinv=1013cm-2,respectively,inSDGandFD/SOI nMOSFETs.AlsoshownareformfactorsforpredominantSC (dashed) and EC (solid). 0.010.020.030.040.0 50.0tSi [nm] 0.0 2.0 4.0 6.0 8.0 10.0Form Factor [x 106 cm-1] b0 = 3.15 x 108cm-1b0 = 1.46 x 109cm-1

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164 5.3.5.2 Volume/Bulk Inversion Toincorporatevolume/bulkinversion(VI)inourmodel,weuseEq. (5.13),andourinsightsfromFig.5.8(a)todefinethedegreeofVIastheratioof carrier concentration at x= tSi/2 and at x = xav(ec) = 3/b0: .(5.17) We plot in Fig. 5.10 RVI versus tSi/2xav(ec). Note that VI becomes significant for .(5.18) Further, comparing Fig. 5.10 and Fig. 5.3, we find that the peak in mph occurs at (5.19) withRVI@ 0.4,whichisconsistentwithSCHREDpredictionsinFig.5.8(a). Interestingly,thetSirequirementforVI-governedmobilityenhancementinEq. (5.19)iscomparabletothatderivedin[Ge02b]viarigorousmathematicalanalysis. Thus,basedonFig.5.10andourphysicalinsightsonVIasreducingEC,we incorporateitseffecton mphinourmodelbysmoothingb0 0suchthattheECdefinedformfactorapproachestheSC-definedformfactorasinFig.5.4. Specifically, we redefine b0 for SDG MOSFETs as RVIy0tSi2 ()2y03b0 ()2----------------------------1 p 2 -2 tSi----3 b0---sin2------------------------------4e3 tSi2 ----b03 ---- –e3 –2e3 tSi2 ----b03 ---- –e6 tSi2 ----b03 ---+++ ---------------------------------------------------------------= tSi8xavec ()< 24 5 3 -mxh 2-----q2eSi-----Ninv 13 /= tSi4xavec ()@ 12 5 3 -mxh 2-----q2eSi-----Ninv 13 /=

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165 Figure 5.10RatioofcarrierpopulationatthemiddleoftheUTBandatx=xav(ec)versustSi/2xav(ec).ThepointAcorrespondstotheminimuminthe formfactorinFig.5.4forNinv=1013cm-2,andthepeakinhigh-Ninvmph near tSi = 8nm in Fig. 5.3 for SDG nMOSFET. 1.02.03.04.0tSi/2xav(ec) 0.0 0.5 1.0RVI A

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166 ,(5.20) where A = 1.0 and K = 4, from Eq. (5.16), for now. 5.3.5.3 A Compact Model For2-Dcarriersinasinglesubband, mph tSi(eff)withtSi(eff) -3 ExC,NinvforpredominantEC,andtSi(eff)=tSiforpredominantSC. However,toaccountforthe saturationin mph(tSi),and mph(xav),atthicktSi(xav)duetocarriersbecomingfreeand atthintSi(xav)duetosubbandmodulation,wedefineacompactmodelfor mphin terms of tSi(eff) as ,(5.21) where mmin@ 550cm2/V-sforelectronsin{100}-SisurfaceUTBfromFig.5.2and Fig.5.3,and mmax= mph(bulk).CalibratingthiscompactmodeltoMonteCarlopredicted mph(tSi)inSDGnMOSFETsatlowNinv(Fig.5.3),wefindtSi(ref)=16nm and a =4.TosupportouruseoftSi(eff)tomodelECandSC,andoursimplemodel forvolumeinversion,weshow,inFig.5.11,model-andMonteCarlo-predicted mph(Ninv)forvarioustSi.Thereasonableagreementbetweenthetwo,exceptathigh NinvintSi=10nmand5nm,whichisduetoourneglectofVI-dependenceonsubband modulation supports our model and its predictive nature. b0b0A tSi2 K3b0 () – 3b0 ---------------------------------------exp 1A tSi2 K3b0 () – 3b0 ---------------------------------------exp + --------------------------------------------------------------------= mphtSieff ()()mminmmaxmmin– 1 tSiref ()tSieff ()--------------a+ --------------------------------+ =

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167 Figure 5.11Model-(Eq.(5.21))andMonteCarlo-predicted mphversusNinvfor SDG nMOSFETs of Fig. 5.2 with various UTB thicknesses. 101110121013Ninv [cm-2] 400 600 800 1000 1200 1400mph [cm2/V-s] tSi = 30nm tSi = 20nm tSi = 15nm tSi = 10nm tSi = 5nm Symbols: Monte Carlo Line: Model

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168 5.4 Surface-Roughness-Limited Mobility Whencarriersarestronglyconfinednearthesurface,carrierscatteringdue tothesurfaceroughness(SR),oroxide-semiconductorinterfaceroughness, dominatesthescatteringprocess.InconventionalMOSFETs,EC-governedSR scatteringisimportantforeffectivetransversefield(eeff)higherthan5x105V/cm [Tak88],[Tau98].ForUTBMOSFETs,however,SCcanleadtostrongtSidependence[Ess04],[Uch02],[Sak87].Inthissection,wegainphysicalinsights fromMonteCarlosimulationstodevelopananalyticmodelforSR-limitedmobility ( msr) in generic UTB MOSFETs. 5.4.1 Scattering Hamiltonian AlthoughthetheoryofSRscattering,likethatforphononscattering, remainspremature[Fis93],awidelyusedscatteringHamiltonianduetointerface roughnessintheUTB(orx-)andthechannel-length-(ory-)directions,describedby D (x,y), is [Gm99] ,(5.22) where f (x)istheunperturbedelectricpotentialinthebody.ThisHamiltonianis questionableasitimplicitlyassumesthatthereisnosignificantvariationinthe electroneigenfunctions(andenergyeigenvalues),whereas,intuitively,changeinthe structuralconfinementduetoSRcanyieldsubstantivechangeintheeigenfunction andenergyeigenvalues.Indeed,experimentalresults[Uch02]showseveretSidependence( tSi 6)fortSi<4-5nmthataregovernedbyvariationinthequantization duetoSR-inducedvariationintSi[Ren02],[Uch02],[Sak87].Thisisyetanother Hsrxy , () q f x D xy , () + []f x () – {} – =

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169 reasonforapragmaticlowerlimitof~4-5nmontSi.However,sinceweareinterested in msr for pragmatic tSi > @ 4nm, using the Hamiltonian in Eq. (5.22) is reasonable. 5.4.2 Monte Carlo Simulations and Physical Insights Togainphysicalinsights,wefirstevaluate msrviaMonteCarlo simulations. The assumed scattering Hamiltonian is ,(5.23) where DmisthermsvalueoftheSRinthex-direction.Thecompletemodelfor msrbasedonEq.(5.23)isdetailedin[Gm99],[Gm01].(Thesimulationsinclude phononandSRscattering,and msrisevaluatedfromthepredicted meffusingthe Mathiessen’s rule.) MonteCarlo-predicted msr(Ninv)forSDGandADGnMOSFETsofFig. 5.2areshownin Fig.5.12; msrisextractedfrom meff,includingSRandphononscattering,viaMatthiesen’srule.Notethat msrinADGdevicesissignificantlylower thanthatinSDGdevicesforallNinv,withthedifferencebeing~102atlowNinvand ~3-5athighNinv.Thelarge(~102)differenceatlowNinvcanbeexplainedbythe highExC(tSi)forADGdevicesasexemplifiedinFig.5.7.Indeed,theplotof msrversusExC(tSi)inFig.5.13forNinv@ 1010cm-2showsthat msrhasstrongdependence onExC.Interestinglythough,thisdependencedeviatessignificantlyfromtheExC -2dependenceexpectedfromthescatteringHamiltonianinEq.(5.23)andtheMEDICIpredicted f (x)inFig.5.14forADGMOSFETs.This"discrepancy"canbeattributedto signicantvariationinthe effectiveconductioneffectivemasswithtSiinADG Hsrxy , () q f x Dm+ ()f x () – Dm----------------------------------------D xy , () – =

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170 Figure 5.12MonteCarlo-predictedsurface-roughness-limitedmobilityversus NinvforvarioustSiin(a)SDGand(b)ADGnMOSFETswith{100}Sisurfaceandtoxf=toxb=1nm.ForADGdevices,n+-andp+polysilicon gates with Npoly = 1020cm-2 are specified. (a) (b) 10121013Ninv [cm-2] 102103104105msr [cm2/V-s] tSi = 35nm tSi = 25nm tSi = 20nm tSi = 14nm tSi = 10nm tSi = 8nm tSi = 5nm tSi = 3nm Dm = 0.5nm, Lm = 1.5nm 10121013Ninv [cm-2] 102103msr [cm2/V-s] Dm = 0.5nm, Lm = 1.5nm tSi = 50nm tSi = 35nm tSi = 25nm tSi = 20nm tSi = 17nm tSi = 14nm tSi = 10nm tSi = 8nm tSi = 5nm tSi= 3nm Ninv -2

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171 Figure 5.13 MonteCarlo-predicted msratlowNinvinADGnMOSFETsofFig.5.12 versusExC(tSi)fromEq. (3.9). Alsoshownisalinecorrespondingto ExC -2 dependence. [ ExC(tSi)]-2 105106ExC [V/cm] 102103104msr [cm2/V-s] Dm = 0.2nmDm = 0.5nm

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172 Figure 5.14MEDICI-predictedelectricpotentialdistributionalongtheUTBfor variousgatebiasesinADGnMOSFETwithn+-andp+-polysilicon gates,toxf=toxb=1nm,andtSi=9nm.Theslopeofthisdistributionin weak inversion, e.g., VGS < 0.35V here, denes ExC. 0.01.02.03.04.05.06.07.08.0x [nm] -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Potential [V] 9.0 0.8 VGS = 0.0VVGS = 1.0VNB = 0

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173 MOSFETs[Gm03],aswellastheuseoftheMatthiesen’sruleforextracting msr. Further,asreflectedin Fig.5.14,thetransverseelectric-eldatthebacksurface(esb) decreaseswithincreasingNinv,reducingtheExC-dependenceof msr.Thus,thedifference between msrinSDGandADGdevicesreducesto~3-5athighNinv.And,for sufficientlyhighgatebias,orNinv,theADGdevicewilleventuallyresembleanSDG device with msr Ninv -2 and independent of ExC. FromFig.5.12,weshowin Fig.5.15thetSi-dependenceof msrinSDGand ADGMOSFETswithlowandhighNinv. msr(tSi)inADGdevicesisgovernedby ExC(tSi)andNinvasnotedearlier.ForweaklyinvertedSDGnMOSFETs, msr(tSi)is explainedbyverysmallvariationsinthepotentialdefinedbytheinversioncharge density.And,thelargerdistanceofthechargecentroidduetovolumeinversionin stronginversion(seeFig.5.8)underliesthehigher msrneartSi@ 8nmathighNinv. Aswewillshowlater,however,this msrenhancementduetovolumeinversionisnot observedexperimentally,andhence,forouranalyticmodelingof msrinthenext section, we neglect volume inversion. 5.4.3 Physics-Based Compact Model Toanalyticallymodel msr(tSi,ExC,Ninv)forgenericUTBdevices,wefirst approximate the scattering Hamiltonian in Eq. (5.22) as ,(5.24) where Hsrxy , () q x d d f D xy , () – @ qeeffD xy , () DsrD xy , () ==

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174 Figure 5.15MonteCarlopredictedsurface-roughness-limitedmobilityversustSiinSDGnMOSFETswith(relatively)lowNinv@ 1012cm-2(dashed) andhighNinv=1013cm-2(solid),andinADGnMOSFETswithlow Ninv@ 1010cm-2 (dashed) and high Ninv = 1013cm-2 (solid). 0.010.020.030.040.0tSi [nm] 102103104105msr [cm2/V-s] SDG ADG

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175 (5.25) for SDG nMOSFETs, and (5.26) forADGandSGdeviceswithback-surfacetransversefieldesb 0.Foraunified model,however,weuse0 h 1inthetrialeigenfunctioninEq.(5.13),which reflects the symmetry in Ninv about x = tSi/2, to define ,(5.27) where (5.28) and, we approximate h as .(5.29) ForUFDG,"smooth"definitionsofesb +and h areimplementedtoavoid discontinuities. Physically,whenesb>0,thereisrelativelynoinversionchargenearthe backsurface.Infact,thepositiveesbforcesthecarrierstobenearthefrontsurface,eeffx d d f nx d0 tSi2 qNinv---------------------------- – qNinv4 eSi------------= eeffx d d f nx d0 tSiqNinv----------------------- – qNinv2 eSi-------------esb+ = eeffqNinv21 h + ()eSi---------------------------esb ++ esb +esbifesb0 0ifesb0 < = h 0ifesb0 2 – eSiesbqNinv-------------ifesb0 < @

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176 contributingtoeeff,andhence,esb +=esb.Also,asthepositiveesb"pushes"carriers towardthefrontsurface,itcreateslargeasymmetryinthechargedistribution,and hence, h@ 0.Incontrast,whenesb<0,carriersare"pulled"awayfromthefront surfaceandtowardthebacksurface,increasingthechargedistributionsymmetry(or h >0)andretardingeeff.However,inthiscase,wecannotsimplysubtractesbfromeeff,asitwouldresultineeff=0forSDGdevices,contradictingEq.(5.25).Instead, theroleofesb<0mustbetreatedasreducingtheamountofinversioncarriers,which indirectlyreduceeeff,thatscatterofftheroughnessatthefrontinterface.Thisis accomplishedvia h >0inEq.(5.27),with h physicallyapproximatedinEq.(5.29) astheratiooftheinversionchargeassociatedwithesbtothetotalinversioncharge inthebody.ThefactoroftwoinEq.(5.29)ensures h =1.0forperfectsymmetry about x = tSi/2, as in SDG MOSFETs. ForweaklyinvertedADGMOSFETs,wenote,from Fig.5.14, thateeff ExC,andwithesb 0withVGS,thisdependencediminishes,consistentwithour physicalinsights.Notethatthisresultsin msr ExC -2inweakinversion,incontrast toMonteCarlopredictionsinFig.5.13.However,asnotedearlier,webelievethis discrepancyisequivocal,andexperimentaldataareneededforassurance.Also,we haveneglectedanyVI-inducedmobilityenhancementinSDGdevicesbased predominantlyonitsabsenceinexperimentallyextractedmobility[Ess01a], [Uch03], which will be presented later. Now,usingtheFermiGoldenRule[Lun00],[Sha94]andneglectingthe SR-induced intersubband scattering, the scattering rate due to SR is [Gm99]

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177 ,(5.30) where D (Q)isthe2-Dfouriertransformof D (x,y),andQ=2| k |sin( q /2)withtheangle q betweentheincidentandthescatteredwaveofamplitude| k |.Experimentalwork ofGoodnicketal.[Goo85]hasshownthattheSRcanbebestdescribedbyan exponential model, which gives [Goo85] ,(5.31) where Dmisthermsvalueof D (x,y),asnotedearlier,andLmistheauto-covariance, orthecharacteristicdecaylengthdefiningtheextentoftheSRinthey-direction. Also,inEq.(5.30), e (Q)accountsforscreening[Gm99].Defining f = q /2and simplifying e (Q)byassumingeigenfunctionoftheform d (x-xav),Eq.(5.30)becomes ,(5.32) where s = q2g2D/4 eSi is the screening factor [Gm99]. WefindthatingeneraltheintegralinEq.(5.32)isnotdoableanalytically. However,assumingtypicalLm=1.5nm[Goo85]andapproximating|k|asinHess [Hes79], i.e., ,(5.33) thevalueofthisintegralcanbedeterminednumerically.Interestingly,withEq. 1 tsr----2 p h -----1 2 p -----Dsr 2D Q ()2q d e Q () ---------------------------------0 2 pg2D2 -------= D Q ()2pDm 2Lm 21Q2Lm 22 + ()32 ------------------------------------------= 1 tsr----2 p h -----4k f sin 2k f sins + () 12Lm 2k2f sin22 + ()32 -----------------------------------------------------------------------------------------------f d0 pDmLmDsr2 ------2g2D= k 2mDkBT h 2--------------------@ 2 p g2DkBT =

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178 (5.33)basedonassumingaveragekineticenergyofkBTinthey-zplane,wefindthat anelectroninthegroundstateof{100}-Sisurfacedevicehas|k| @ 3.6x108m-1,oran electronwavelengthofabout17nm,whichiscomparabletothechannellengthinthe nanoscale regime! Finally,lettingAfbethenumericallyevaluatedvalueoftheintegralinEq. (5.32), and using Dsr from Eq. (5.24), the scattering rate due to SR is given by (5.34) and the SR-limited mobility by (5.35) with mC that in the ground state. BasedontheeffectivemassesgiveninTable5.1forelectronsandholes inSiwith{100}-,{110}-,and{111}-Sisurface,welist b inTable5.2fortypical Dm=0.5nmandLm=1.5nm;Af=2.7554,2.1477,and1.8181forelectronsin{100}Sisurfacebody,electronsin{110}-Sisurfacebody,andholes,respectively,atroom temperature.Interestingly,notethat b ,andhence msr,forelectronsin{100}-Si surfacebodyishighest,withthatin{110}-Sibody @ 2xlowerandthatofelectrons in{111}-Sisurfacebodyanholes~3-3.5xlower.Thisimpliesrelativevaluesof meffforelectronsandholes,andelectronsinvariousSi-surfaceorientedbodies,at sufficientlyhighNinvforwhich msristhedominantterm.(Finally,wenotethatifone treatsSRasperturbingtSi(eff),thenthevariationintheSC-definedground-state 1 tsr----2 p h -----AfDmLmDsr2 ------2g2Dq2pDm 2Lm 2g2D2h ---------------------------------Af Eeff 2== msrq tsrmC--------b Eeff 2--------==

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179 Electronsa,bHoles Surface {100}{110}{111}All Valley/BandUPPUPPALLHHLH Degeneracy 2442611 mx [m0]0.9160.1900.3150.1900.2580.2900.200 my [m0]0.1900.9160.5530.9160.674-mz [m0]0.1900.1900.1900.1900.190-md[m0]0.1900.4170.3240.4170.3580.4330.169 mC[m0]0.1900.3150.2830.3150.2960.4330.169 Table 5.1 Valleys,valleydegeneracy,andconfinement(mx),DOS(md),andconductivity (mC)effectivemassespervalleyforelectronsandholesinSiwithvarioussurface orientations[Ste72],[Mog86].Theeffectivemassesareintheunitsoffreeelectron mass(m0=9.11x10-31kg),andUP=unprimedvalley,P=primedvalley,HH= heavy-holeband,andLH=light-holeband.Thespin-orbitbandforholesis neglected.aFor 3D electrons: md = (mxmymz)1/3 bFor 2D electrons: md = (mymz)1/2 and mC = 2mymz/(my+mz).

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180 carrier: surface orientation b [V/s] Electron: {100} Si 3.152x1014Electron: {110} Si 1.592x1014Electron: {111} Si 1.074x1014Hole 9.196x1013 Table 5.2 Valuesof b forsurface-roughness-limitedmobilityinEq.(5.35)forelectronsand holesinSiwithvarioussurfaceorientations;effectivemassesinUPfromTable 5.1andtypicalSRparametervaluesof Dm=0.5nmandLm=1.5nmareassumed.

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181 energywithtSi=tSi(eff)givesDsr tSi(eff) 3,or eeffforECand tSi 3forSC;the formerisconsistentwithourresultshere,andthelatterwithexperimentalresultsin [Ren02], [Uch02], [Sak87] for tSi < 4nm.) 5.5 Compact Model for Effective Carrier Mobility Wenowuseourmodelsfor mphand msrinEq.(5.21)andEq.(5.35), respectively,todefinethecompositeeffectivemobilityingenericUTBCMOS devices.Physically, meffisdefinedsimilartoEq.(5.8)withthetotalscatteringrate includingthoseofallpossiblescatteringmechanisms.Whenallthescatteringrates havethesameenergydependence,asinourcompactmodels, meffcanbecalculated using the Mathiessen’s rule [Lun00], i.e., ,(5.36) where the sum is over all scattering mechanisms. For a compact model, we thus get ,(5.37) where mothersisthecarriermobilitylimitedbyallscatteringmechanismsotherthan phonon,surface-roughness,andCoulombscattering,andisassumedtoberelatively independentoftSi(eff).However,since mcoand mothersaretypicallyunknown,weadd and subtract, via Mathiessen’s rule, mph(bulk) to express meff as ,(5.38) 1 meff-------1 ml----l= 1 meff-------1 mph------1 msr-----1 mco------1 mothers--------------+ ++ = meffUO 1 UO mphbulk ()-------------------mphbulk ()mphtSieff ()() ----------------------------1 – q UO msr-------++ --------------------------------------------------------------------------------------------=

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182 where (5.39) isthelow-fieldmobilityinthick-tSidevices,andatuningparameterinourmodel. Asnotedearlier,thetemperaturedependenceofUO[Wor99]isassumedtobethe sameasthatofcarrierbulkmobility[Aro82],andthatofuph(bulk)anduph(tSi(eff))as discussedinSec.5.3.5.3; msr(T)isnotmodeled(Sec.5.4.3).InEq.(5.38),wehave alsointroducedanothertuningparameter q toaccountforuncertaintyinour msrmodel, e.g., the SR parameters. Finally,wefinetune mmin,tSi(ref),and a inour mph(tSi(eff))model(Eq. (5.21))abouttheirMonte-Carlovaluesusingexperimentallyobserved meff(tSi)in FD/SOICMOSatlowNinv[Ess00].Theresultingparametervalues,withthose estimatedforelectronsin{110}SiandholesviadifferenceintheirmCfromthatof electronsin{100}Si,arelistedinTable5.3alongwith b for msrfromTable5.2. Thesevaluesnowfullycharacterize mphinEq.(5.21)and msrinEq.(5.35).Hence, ourphysics-based meffmodelhaspredominantlytwotuningparameters,UOand q ; fornow,wealsoletKinEq.(5.20),whichaccountsforVIinSDGMOSFETs,bea tuning parameter, and later eliminate it based on measured data. 5.6 Temperature Dependence Tocompletethemodelformalism,weincorporatethetemperature(T) dependenceof meffasfollows.Forphononscatteringboththeacousticandoptical/ intervalleyphononscatteringisdirectlyrelatedtothetemperaturethroughtheBoseUO 1 mco------1 mothers--------------1 mphbulk ()-------------------++ 1 –=

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183 ElectronsHoles surface {100}{110} mmax[cm2/Vs] 13501350475 mmin [cm2/Vs] 550392235 tSi(ref) [cm2/Vs] 141411 a 444 b 3.152x10141.592x10149.196x1013 Table 5.3 Phonon-limitedmobilitymodelparameters,with mmax= mph(bulk),evaluatedbased onMonteCarlosimulations,andfine-tunedusingmeasuredlow-Ninvmeff(tSi)in FD/SOICMOS[Ess00].NotethatthethinnertSi(ref)forholesisconsistentwith strongerconfinementneededforsignificanthole-subbandmodulation[Fis03],and that mminfor{110}Siisestimatedfromthatof{100}SiviadifferencesinmC. Thesevaluesnowcharacterize mphinUTBCMOSdeviceswith{100}-and{110}Si surface orientation. Also listed is msr parameter b from Table 5.2.

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184 Einsteinstatistics,i.e.,throughthenumberofphonons.Also,MonteCarlo-based studyof mph(T)inUTBs[Gm04]showsadependencesimilartothatof mph(bulk)(T). Forsurface-roughnessscattering(Eq.(5.32)),thetemperaturedependenceis governedbytheelectronwavevector|k|giveninEq.(5.33).Thewavevectordefines theelectronwavelengthinthey-zdirection,andhencetheeffectiveproximityofthe carrierstotheinterfaces.Then,with|k|decreasingwithTfromEq.(5.33), tsr( msr) generallydecreases(increases)withtemperature.This msr(T)alsodependsonLm,as impliedbyEq.(5.32);however,fortypicalLm=1.5nmandTvaryingfrom77K400K, we find, via Eq. (5.32), that msr(T) is relatively weak. Althoughthetemperaturedependenceof meff(T)aftersummingthoseof mphand msrisnotevident,experimentaldata[Ess00]showselectron meff T-1.4(for Ninv=2x1012cm-3)inUTBnMOSFETs,whichissimilarto meff T-1.7forclassical, orthick-tSi,devicedata[Tak88].Hence,weaccountforthetemperaturedependence inourmodelbyassuming meff(T)inUTBstobethesameasthatinthick-tSidevices [Wor99].Thispreliminary,andpessimistic,modelfortemperaturedependence involvesassuming mmax(T), mmin(T),andUO(T)sameas mph(bulk)(T)[Aro82], [Wor99],andneglectingthetemperaturedependenceof msr.Wenote,however,that ifneeded,themodelcanberefinedwhenstudyinginmoredetailthetemperature dependence of UTB MOSFET characteristics. 5.7 Model Verification Wenowverifyourcompactmodelfor meffinEq.(5.38)using experimentallyevaluated meffinFD/SOIandSDGCMOSdevices[Ess00],[Ess01a],

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185 [Ren02],[Uch03].DuetothelackofmeasureddataforADGMOSFETs,weverify themodelforitusingMonteCarlopredictions.WefirsttuneUOand q basedon measured meffinathick-tSidevice.Then,asshownin Fig.5.16andFig.5.17 withUO and q ,andKforSDGdevices,asindicated,wefindthatthemodelpredictionsarein goodagreementwiththemeasured meff(Ninv)forallultra-thintSi;thediscrepancyin Fig.5.17(b)is believedtobeduetotheanisotropyofholes[Fis03],neglectedinour model,andthering-devicestructureusedin[Ren02].Wenotethatotherthanthe ring-devicestructurein[Ren02],allothermeasurementsaremadeindeviceswith {100} surface orientation and current flow along a <100> direction. Our mph(tSi(eff))modelpredictswellthetSi-andNinv-dependenceof electronmobilityinFD/SOIandSDGnMOSFETs[Ess01a],[Uch03]( Fig.5.16),as wellasofholemobilityinFD/SOIpMOSFETs[Ess00],[Ren02]( Fig.5.17).Asfor our msrmodel,wefindthatwriting q UO/ msrinEq.(5.38)as( q ’Eeff)2gives,fromFig. 5.16,Fig.5.17,andTable5.3(for b ), q ’~1.6x10-6cm/V,or1/ q ’~6x105V/cm, whichisconsistentwiththe"critical"field[Tak88],[Tau98]beyondwhichSR scatteringissignificant.WealsostressthatthesameUOand q foreachdevice technologyreflecttheintegrityofour meffmodel,aswellasitsphysical/predictive nature. Inaddition,wenotethatthemeasureddataforSDGnMOSFETsin [Ess01a]and[Uch03]showsvirtuallynoVI-governed meff-enhancement,whichis alsoreflectedbysmallvalueofK=1.5,andjustifiesourneglectofVIin msr.From Eq.(5.20),thesmallvalueofK=1.5canonlyfacilitateVIatrelativelylowNinv,as canbeseeninFig.5.16andFig.5.17bycomparinglow-NinvmeffinFD/SOIand

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186 Figure 5.16Calibratedelectron-mobilitymodelpredictions(curves)compared withmeasured(symbols) meff(Ninv)in{100}-SisurfaceSGFD/SOI andSDGnMOSFETswithvarioustSi:(a)withexperimentaldatafrom [Ess01a], and (b) with experimental data from [Uch03]. (a) (b) 10121013Ninv [cm-2] 200 300 400 500 600 700me ff [cm2/V-s] tSi = 21nm tSi = 9.4nm tSi = 5.2nm UO = 1100cm2/Vs q = 0.833 FD/SOI SDG 10121013Ninv [cm-2] 200 300 400 500 600 700meff [cm2/V-s] FD/SOI SDG tSi = 22nm tSi =14.9nm tSi = 7.4nm UO = 1200cm2/Vs q = 0.714 K = 1.5K = 1.5

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187 Figure 5.17Calibratedhole-mobilitymodelpredictions(curves)comparedwith measured(symbols) meff(Ninv)inSGFD/SOIpMOSFETswithvarious tSi:(a)withexperimentaldatafrom[Ess00]for{100}-Sisurface,and (b) with experimental data from [Ren02] for ring-device structure. (a) (b) 10121013Ninv [cm-2] 50 70 90 110 130 150 170meff [cm2/V-s] 10121013Ninv [cm-2] 50 70 90 110 130 150meff [cm2/V-s] tSi =6.8nm tSi = 18nm tSi = 10nm tSi =6nm UO = 190cm2/Vs q = 1.13 UO = 160cm2/Vs q = 1.0 tSi = 22.5nm tSi =17.5nm tSi = 11.1nm

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188 SDGdeviceswithtSi<20nm.Further,sincethevalueofKforallSDGdevicesin [Ess01a]and[Uch03]isthesame,Kdoesnotneedtobeconsideredasatuning parameter.Infact,neglectofVIaltogetherdoesnotchangethehigh-Ninvmeffmuch, andaffectsthelow-Ninvforthick-tSicasesonly.Hence,forimplementationin UFDG, we do not include VI, it but can be incorporated via Eq. (5.20) if needed. ForADGMOSFETs,wefirstuseUFDGtoevaluateesb(Ninv)foreeffin Eq.(5.27).Then,themodel-andMonteCarlo-predictedlow-Ninvandhigh-NinvmeffcharacteristicsinADGnMOSFETsareshownin Fig.5.18(a)andFig.5.18(b), respectively. Sinceonly mphand msrareincludedintheMonteCarlosimulations,we assumeUO=1350cm2/V-s.Forhigh-Ninvmeff,wefind q =2.3,however,forlowNinv,wefindthat q needstobevariedfrom1-3fortSivaryingfrom5nmto50nm. Thisvariationin q withtSiatlow-NinvisduetothenoteddeviationoftheMonte Carlo predictions from ExC -2 dependence. 5.8 Mobility in {110}-Si Surface MOSFETs Beforediscussingsomeoftheimplicationsof meffinFig.5.16-Fig.5.18, weuseourmodelinEq.(5.38)tobrieflyexamine meffinSDGFinFETswith{110} surfaceorientation.Althoughholesingeneralareanisotropicandshowasignificant surface-orientationeffecton meff[Fis03],duetothelackofknowledgeofthe effectiveholemassesforvariouscrystalorientationwepredominantlyfocuson electronmobilityhere.Wenotehoweverthatexperimentaldatashows @ 2xhigher holemobilityalong<110>channeldirectionin{110}-SisurfacepMOSFETsrelative tothatin{100}-Sisurfacecounterpart[Yan03];ourmodelmaybecalibratedtoitby tuning UO and q .

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189 Figure 5.18Calibratedelectron-mobilitymodelpredictions(solidlines) comparedwithMonteCarlo-predicted(dashedlines)(a) meff(tSi)at lowNinv@ 1010cm-2and(b) meff(Ninv,tSi)in{100}-SisurfaceADG nMOSFETs with n+and p+-polysilicon gates and toxf = toxb = 1nm. 1013Ninv [cm-2] 50 150 250meff [cm2/V-s] tSi = 50nm, 25nm, 14nm, 8nm, 5nm 5.010.015.020.025.0tSi [nm] 150 200 250 300 350 400 450meff [cm2/V-s] (a) (b) Ninv@ 1010cm-2

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190 Weshowin Fig.5.19the model-predictedratiooftheeffectiveelectron mobilityinSDGnMOSFETswith{110}-Siand{100}-SisurfacesversusNinv.The modelpredictionsfromFig.5.16areassumedfor{100}Si,and mefffor{110}Siis predictedbyusingtheappropriatevaluesfromTable5.3forparametersinour mphand msrmodels.Further,UO< mph(bulk)inthe{100}-Sidevices(Fig.5.16)implies some mcoand/or mother,whichwethusscaleaccordingtothedifferenceinmCfor {110}and{100}SiintheunprimedvalleyasgiveninTable5.1.Thisyieldsthe lowerUOgivenin Fig.5.19for{110}Si.(We notethatwhiletheassumedvaluesin Table5.3for{110}Sidevicesmayneedtoberefinedbasedonreliablemeasured data,thepredictedtrendsaremeaningfulhere.)Then, Fig.5.19showsthat meffin stronglyinverted{110}-SisurfaceSDGnMOSFETsis~0.66-0.75xthatin{100}-Si surfacecounterpart,anditapproaches @ 0.5xasdefinedbythecorrespondingdifference in b (for msr)inTable5.3.Thisasymptoteat @ 0.5canbealternativelyreachedatlowerNinvif relatively more roughness occurs for {110}-Si surfaces. Physically,thelower meffin{110}-SisurfaceMOSFETscanbeexplained bytheheavierDOSeffectivemass,highervalleydegeneracy,andheavier conductivityeffectivemassinunprimedvalleysof{110}Si(Table5.1).The1.7x heaviermdmeanshigherDOS,whichinconjunctionwith2xhighervalley degeneracy,increasestheavailabilityofinitialandfinalscatteringstatesinthe unprimedvalleys(whichincludethegroundstate),andtherebyreducesmobility. Notethatalthoughthevalleydegeneracyofprimedvalleysisreduced,changesinthe propertiesoftheunprimedvalleydominatebecauseamajorityofcarriersoccupies subbandsintheunprimedvalleyathighNinv(e.g.,seeFig.5.5),especiallyfor

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191 Figure 5.19Model-predictedratioofelectronmobilityin{110}-and{100}-Si surface SDG nMOSFETs. 1013Ninv [cm-2] 0.60 0.70 0.80 0.90 1.00meff(110)/ meff(100) tSi = 21nm tSi = 9.4nm tSi = 5.2nm UO = 1000cm2/Vs q = 0.833 K = 1.5

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192 {110}-Sisurfacedevices.Finally,this meff-degradationduetoincreasedscattering rate is further exacerbated by the heavier mC in the unprimed valley (Table 5.1). 5.9 UFDG Implementation ForUFDG,weimplementthe meffmodelinEq.(5.38)withuser-defined parameters UO and q ( THETA ).BasedonthemodelcalibrationforADGMOSFETs intheprevioussection,wescaletheuser-specified THETA internallyinUFDGby 2.3instronginversionandbyatSi-dependentfunctionf(tSi)inweakinversionsuch thatthetypicalvalueof THETA =1.0forgenericDGMOSFETs.Thisspecial treatmentof THETA isdoneonlywhen|VFBf-VFBb|>0.2,i.e.,onlyforADG devices. The typical value of UO = 1100 (190) cm2/V-s for electrons (holes). Thephonon-limitedmobility mph(tSieff)in meffinEq.(5.38)isasgivenin Eq.(5.21)with mmax, mmin,tSi(ref),and a aslistedinTable5.3,andtSi(eff)asdefined inEq.(5.16).Basedonthemeasured meffinFig.5.16,wedonotincorporatethe volumeinversioneffectforSDGMOSFETsinUFDG,andhence,b0inthedefinition oftSi(eff)inEq.(5.16)isthatgivenbyEq.(5.14)withj=0.Thebulk-phonon mobility mph(bulk)inEq.(5.38)islistedinTable5.3as mmax= mph(bulk).Thesurfaceroughness-limitedmobility msrin meffinEq.(5.38)isgivenbyEq.(5.35)with b as listedinTable5.3andeeffasdefinedbyEqs.(5.27)-(5.29).Wenote,however,thatesb +inEq.(5.28)and h inEq.(5.29)areimplementedviasmoothingfunctionsto yieldcontinuouseeffinEq.(5.27).Finally,thetemperaturedependenceof meffis implementedasdescribedinSec.5.6,i.e.,with mmax(T), mmin(T),and UO (T)same as mph(bulk)(T) given in [Wor99], and neglecting msr(T).

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193 5.10 Discussion ToconcludeouranalysisofcarriermobilityingenericUTBMOSFETs, weaddress,inthissection,someramificationsoftheresultsinFig.5.16-Fig.5.19. Wefirstnotethatwithnovolume-inversioneffecton(measured) meffinSDG MOSFETs, meffinSDG,andFD/SOIMOSFETsisvirtuallyindependentoftSiathigh Ninv.Then,the meffdegradationwithdecreasingtSioccurringonlyatlowNinvisin factdesirableforhighIon/Ioff,as meffinsubthresholdisloweredwithtSi.Next,ofthe threedevicestructuresconsideredhere,SDGMOSFEThasthehighest meff,followed byFD/SOIandADGMOSFETswith meff@ 0.67xand~0.25xthatintheSDG counterpart,respectively,atNinv=1013cm-2.Thisdifferenceisduetotherelatively smalleeff,orcarrierconfinement,inSDGMOSFETs(e.g.,seeFig.5.8).However, because meffinSDGMOSFETswith{110}-Sisurfaceis~0.5-0.7xthatin{100}-Si surfacecounterpart,SDGnFinFETshaving{110}-Sisurfacewillhave meffcomparableto,orlowerthan,thatinplanarFD/SOInMOSFETswith{100}-Si surface. Further,UOforholesinFD/SOIpMOSFETs(Fig.5.17(a))is considerably(~5-6x)lowerthanthatforelectrons(Fig.5.16),and meffis @ 3.75x loweratNinv=1013cm-2.ThelargedifferenceathighNinvislargelydefinedby @ 3.5xlower msrforholes,asreflectedby b inTable5.3,duetotherelativelyheavier holeeffectivemasses.Despitesuchlargedifferencein meff,UFDGpredictsIon(n)/ Ion(p)=1.9duetovelocitysaturation,asshownin Fig.5.20(a)with usat=7x106cm/s and 6 x106cm/sforelectronsandholes,respectively[Tau98]. Althoughvelocity

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194 Figure 5.20UFDG-predictedIDS-VDSinLeff=28nmFD/SOIn-andp-MOSFETs withtSi=5nm,toxf=1nm,toxb=200nm,midgapgate,and(a)without velocity overshoot, and (b) with velocity overshoot. 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 0.2 0.4 0.6 0.8IDS [mA/ u m] |VGS| = 1.0V nMOS pMOS 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0IDS [mA/ m m] |VGS| = 1.0V nMOS pMOS(a) (b)RS/D = 0.0 W m m RS/D = 0.0 W m m RS/D = 0.0 W m m RS/D = 160.0 W m m RS/D = 300.0 W m m

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195 overshootinnMOSFETtendstoincreasethisratioto3.94,asshownin Fig.5.20 (b), whentypicalRS/D=160 W m mand300 W m mfornMOSFETandpMOSFET, respectively,areconsidered,Ion(n)/Ion(p)@ 3.0;forhigherRS/D,theratiofallsbelow 3.0.Further,sinceIoninFig.5.20(b)hasnot beensubjectedtothethermal-injection/ballistic-limitcurrent[Ass00],whichmaybesurpassedinnMOSFETsduetheir high meff,thisratiocanbeevenlower(andmayapproach2.0);wewilladdressthe limit current in the next chapter. Interestingly,wenotefromFig.5.20(a)thatvelocitysaturationoccursatrather lowVDS~0.1V.Infact,with meff@ 280cm2/V-s(420cm2/V-s)atNinv=1013cm-3in FD/SOI(SDG)nMOSFETs,velocitysaturationoccursaroundVDS=140mV(93mV)for Leff=28nmandRS/D=0.0 W m m,asimpliedbyecritical=2 usat/ meff=VDS/Leffwith usat=7x106cm/sforelectrons[Tau98].And,forSDGnMOSFETswithLeff~10nmand RS/D = 0.0 W m m, velocity saturation would occur at even lower VDS = 33mV! Finally,wenotethatalthoughthe meff(eeff)behaviorismostwidely consideredwhenexaminingmobility,itis meff(Neff)thatultimatelymatters,because IonisdefinedbygivenNinvandnoteeff.Thedifferencebetweenusing meff(eeff)and meff(Neff)isespeciallyimportantwhencomparing meffindifferentdevice architectures,wheredependenceofeeffonNinvcanvarysignificantly.Thus, comparing meffatNinv@ 1013cm-2inFD/SOIandSDGnMOSFETsfromFig.5.16to bulk-SinMOSFETwithdoping @ 1018cm-3forviableVt[Tau98],wefindthat meffin FD/SOInMOSFETisabout2.8xhigherthanthatinbulk-Sicounterpart,and meffin SDGnMOSFETisabout4.2xhigherthanthatinbulk-Sicounterpart!(Forsameeeff, meffinallthreedevicesiscomparable.)AlthoughholemobilityinFD/SOI

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196 pMOSFETsis @ (1/3.75)xelectronmobilityinFD/SOInMOSFETs,wefindthatitis stillsignificantly( @ 1.5x)higherthanholemobilityinbulk-SipMOSFETwith doping @ 1018cm-3forviableVt.Also,basedlargelyonnegligibledepletioncharge and/orrelativelyweakerNinv-dependenceofeeff, meffinundopedFD/SOIandSDG MOSFETsisexpectedtoremainhigherthanindopedbulk-SiMOSFETswith strained-Sichannels.Thisimpliesthatstrainengineeringmaynotbeessentialfor nonclassical nanoscale MOSFETs. 5.11 Summary Physicalinsightsoneffectivecarriermobilityinnonclassicaldevices werepresentedbasedonMonteCarlosimulationsofphonon-andsurface-roughnesslimitedcarriermobilityinSDGandADGMOSFETs,andmeasuredCoulomblimitedmobilityinFD/SOIMOSFETs.ThetSi-dependenceof meffwasfoundtobe definedpredominantlybyphononscatteringsolongastheinterfacesarerelatively clean(Nit<1011cm-2).Further,wefoundthatwhilephonon-limitedmobilityis definedbythecarrierconfinement,itisvirtuallyindependentoftheconfinement mechanism. Thephysicalinsightswerethenusedtodevelopaphysics-basedcompact modelfor meffingenericUTBCMOSdeviceswithpragmatictSi> @ 4nm.Themodel, implementedinUFDG,accountsforthedependencesontSi,eeff(includingNinv), andtheSi-surfaceorientation,anditinvolvesonlytwotuningparametersthatcanbe calibratedviameasurementsfromathick-tSidevice.Themodelwasverifiedusing

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197 largesetsofexperimentaldatafromFD/SOICMOSFETsandSDGnMOSFETs,and Monte Carlo predictions for ADG nMOSFETs. Themeasureddatashowedvirtuallyno meff-enhancementduetovolume/ bulkinversioninSDGdevices,andnotSi-dependenceathighNinv.Incontrast,the datashowedreductioninlow-NinvmeffwithtSi,asgovernedbyenhancedphonon scattering.Wealsoappliedourmodeltopredict meffinDGFinFETswith{110}-Si surface,andfoundthatelectronmobilityinitathighNinvis~0.5-0.7xthatin{100}Sisurfacecounterparts.Finally,wefoundthatforgivenNinv,andhenceon-state condition,electron(hole)mobilityinnonclassicaldevicescanbeupto4.2x(2.3x) higherthatinclassicaldevices,andisexpectedtoremainhigherthanclassical devices with strained-Si channels.

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198 CHAPTER 6 THERMAL INJECTION-, OR BALLISTIC-LIMIT CURRENT 6.1 Introduction InChapter5,wefoundthatvelocitysaturationinnanoscaleSDG nMOSFETscanoccuratverylowVDS~kBT/q.Inaccordwiththisfinding,the channelresistanceinthelinearregionofoperation(Rch,givenin W m m)approaches zeroasLeff/ meff 0.Forexample,foranSDGnMOSFETwithNinv@ 1013cm-2, whichcorrespondsto meff@ 420cm2/V-sfromFig.5.16,Rch@ 15 W m mwhen Leff=10nm!However,channelcurrentinsuchlow-resistanceregimemustbe subjectedtotheaveragethermalinjectionvelocity( uinj)atwhichcarriersfromthe source,andthedrain,diffuse(orareemitted)intothechannel.Similarly,evenwith highRch,Ion,orIDSforVDS>VDS(sat),canbelimitedby uinjduetovelocity overshoot,dependingon meff[Ge01].Ineithercase,whenthechannelcurrentis limitedby uinj,itiscalledthethermalinjection-limitcurrent(IDS(lim)).Althoughthe carriertransportleadingtothethermalinjectionlimitviavelocityovershootcannot bereadilyequatedtoballistictransport(i.e.,noscattering)[J.G.Fossum,private communication],thethermalinjection-limitcurrentistheballistic-limitcurrent,and is the limit current independent of carrier transport in the channel. Basedonouraboveinsights,theballisticlimitmaybereachedinboth linearandsaturationregionsofoperationviarelativelyhigh meffandnanoscaleLeff. Hence,nanoscaleFD/SOIandSDGnMOSFETs,whichhavehigh meff(Fig.5.16),

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199 canpossiblyoperatenear,orat,theballisticlimit.Thus,inthischapter,wefirst describeamodelforIDS(lim)asderivedbyNatori[Nat94],withadditionalanalyses ofissuesofthismodelthathavenotbeenadequatelyaddressedinliterature.More interestingly,wewillimplementthismodelinUFDGandapplyittoexaminehow closetotheballisticlimitdosomeofthenonclassicalCMOSdevicesoperate. Finally,wewillconcludethisstudybycomparingtheon-stateIDS(lim)(Ion(lim))to the current projected in the ITRS [ITR03]. 6.2 Ballistic-Limit Current Inthissection,wedescribetheballistic-limitcurrentmodelofNatori [Nat94]inthe1-Delectrostaticsregime.By1-D,wemeanthattheinversioncharge satisfiesthe1-DGauss’slaw,orPoissonequation,inthebody.Althoughthismodel iswidelyknown,wealsoaddressheresomesubtleissueswithitthathavenot receivedadequateattentioninliterature,andwhichareessentialforproper implementationinaphysics-basedMOSFETmodel,andformeaningfulpredictions. 6.2.1 Current and Inversion-Charge Density Thetreatmentoflimitcurrentandinversion-chargedensityforballistic transportisthesameasthatforthermionicemission[Sze81],asbotharephysically equivalent.Sincecarriersareballistic,thelimitcurrentissimplygivenbythesum of carrier flux from the source and the drain [Nat94], i.e., ,(6.1) IDSlim ()q uyDE () fFDEFE , () fFDEFqVDS–E , () – [] Eydnzjvalleys=

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200 wherethesecondsummationisoverallthesubbandsinagivenvalley,thethird summationisoverallthequantizedstatesduetopossibleconfinementinthe channel-width(orz)direction,fFDistheFermi-Diracdistributionfunction, uyisthe carrier group velocity in the channel-length (or y) direction given by (6.2) undertheparabolic-bandapproximationwithmyandEybeingthey-directional effectivemass(Table5.1)andthekineticenergy,respectively,inthegivenvalley, D(E) is the 1-D DOS, and (6.3) isthetotalcarrierenergy,whereEjistheseparationofjthsubbandfromthebottom oftheconductionband(Ec)atthefrontsurface,mzistheeffectivemassinthe channel-widthdirection,Wisthechannelwidth,andnz=1,2,..., .Forsufficiently widechannels,thesummationovernzinEq.(6.1)canbeconvertedtoanintegral, and the resulting double integral in Eq. (6.1) can be evaluated to get [Nat94] ,(6.4) whereg2D=md/ p2isthe(constant)2-DDOSwithDOSeffectivemassmD= (mymz)1/2, h = (EF Ec), uy2Eymy--------= EEcq fsf–Ej+ () h 22mz--------nzp W -------2Ey++ = IDSlim ()Wq g2DkBT 2 ------------------uT12 h q fsfEj– + kBT ------------------------------j valleys= 12 h q fsfEj–qVDS– + kBT --------------------------------------------------– h

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201 (6.5) is the average unidirectional thermal velocity for nondegenerate carriers, and (6.6) istheFermi-Diracintegraloforder1/2.NotethatEq.(6.4)isvalidforanywide MOSFET structure in all regions of operation. Similarly,theinversioncarrierdensityatthevirtualsourceisgivenbythe sum of that injected from the source (NS) and that from the drain (ND) [Nat94]: (6.7) . NotetheVDSdependenceandthefactorof1/2thatareabsentinthetypicalmodel for Ninv, e.g., Eq. (3.18) or [Tau98], [Ge02a]. 6.2.2 Drain-Bias Dependence FromEq.(6.4),wenotethatIDS(lim)=0.0AforVDS=0.0V,andinthe absenceofany2-Delectrostaticeffects,IDS(lim)=Ion(lim)>0,independentofVDS, forVDS>3-4kBT/q.Tophysicallyexplainthisdrain-biasdependence,weuseEq. uT2kBT p my------------= 12 u () 2 p ------z 1zu – () exp + ----------------------------------z d0 = Ninvg2DkBT 2 ------------------1 h q fsfEj– + kBT ------------------------------exp + ln jvalleys= 1 h q fsfEj–qVDS– + kBT --------------------------------------------------exp + ln + Nj SNj D+ ()jvalleysNj jvalleys= =

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202 (6.7) to express IDS(lim) in Eq. (6.4) as (6.8) (6.9) wherethe(degenerate)carriersfromthesource()andthedrain()areinjected into the channel at the thermal injection velocity (6.10) ,(6.11) respectively.ThenegativesigninEq.(6.8)representsthedirectionofcarrier injectionfromthedrain.Then,atlowVDS(<
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203 Eventually,forVDS>3-4kBT/q, 0,and.(Interestingly,the resultingIDS-VDSofaballisticMOSFETislikethatdefinedbyvelocitysaturation in scattering-dominated MOSFET.) Further,becausethecarrierdensityinjectedfromthedrainreduceswith VDS,andbecausethetotalinversioncarrierdensityatthevirtualsourceisthesumofthat injectedfromthesourceandthatfromthedrain,NinvinEq.(6.7)decreaseswithVDS. However,thisdecreaseinNinvincreases fsfinordertosatisfythe1-DGauss’slaw inthebodyforagivengatebias,whichinturnincreasesNinv.Thisselfconsistency ultimatelyyieldsanincreasein fsfby Dfsf(whichisoffsetbynegligiblysmall (relativetoNinv)decreaseinNinvtosatisfy1-DGauss’slaw)andNinvremains virtuallyindependentofVDS[Ass00].Physically,theincreasein fsfwithVDS,which eventuallysaturatesforVDS>3-4kBT/q,meansthatthedensityofcarriersinjected fromthesourceincreaseswithVDS,compensatingthereductionofcarrierinjection fromthedrain,andfordegeneratecarriers,italsoincreases.Thus,inthe remainingofthissection,wedescribeamodelfor Dfsf(VDS),whichisphysically different from DIBL in subthreshold. Tobetterexplaintheaboveargument,andtogainphysicalinsighton Dfsf(VDS), we consider SDG nMOSFETs, for which 1-D Gauss’s law requires .(6.12) SubstitutingNinvfromEq.(6.7)intoEq.(6.12),anddifferentiatingwithrespectto VDS, we get Nj Duavgj ()uinjj () S= uinjj () SVGSVFB– fsfqNinv2Cox------------+ =

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204 ,(6.13) whereandaretheFermi-Diracdistributionfunctionsevaluatedatthesource andthedrainedge,respectively.Equation(6.13)showsthat fsfincreaseswithVDS, whichcountersthedecreaseinNinvwithVDSinEq.(6.7)byallowingmorecarriers fromthesourcetobeinjectedintothechannel,andthussatisfyingEq.(6.12).AsVDSbecomeshigherthan3-4kBT/q, 0,andthisincreasein fsf(andthedecreasein Ninv)withVDStendstosaturateinaccordwithEq.(6.13)(Eq.(6.7)).Further,we findthatfor(SiO2)tox>~0.5nm,thefirstterminthedenominatorofEq.(6.13)is negligible.Interestinglythen,theresultingapproximationford fsf/VDSneglecting thistermisthesameasthatrequiringdNinv/dVDS=0fromEq.(6.7);i.e.,for sufficiently thick tox, Ninv is virtually independent of VDS. Unfortunately,evenwiththisapproximation,theresultingdifferential equationfor fsf(VDS)cannotbesolvedanalyticallyfordegeneratecarriers;for nondegenerate carriers, we get .(6.14) Inordertoreasonablymodel Dfsf(VDS)fordegeneratecarriersin any MOSFET structure,wefirstnotethatsince(a) fsfbecomesindependentofVDSforhighVDS, and(b)NinvisvirtuallyindependentofVDS, DfsfathighVDSwillbesuchthatthe valueofthefirstterminEq.(6.7),i.e.,thedensityofcarriersinjectedfromthe VDSd d fsfg2DkBT 2 ------------------fFD D jvalleys2Coxq ----------g2DkBT 2 ------------------fFD SfFD D+jvalleys+ -----------------------------------------------------------------------------------= fF D SfF D DfF D DDfsfVDS() kBT q --------2 1qVDSkBT – () exp + -----------------------------------------------------ln =

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205 source,athighVDSistwiceitsVDS=0.0Vvalue.Next,inordertoarriveatan analyticmodel,weeliminatethesecondsuminEq.(6.7)bydefininganaverage energy: .(6.15) Finally,equatingtheresultingEq.(6.7)withVDS=0.0Vand fsffsf0= fsf(VDS= 0.0V) to that with VDS >> 3-4kBT/q and fsffsf0 + Dfsf, where we get Dfsf as .(6.16) WecannowgeneralizeEq.(6.16)foranyVDSbynotingthatthetermraisedto2in Eq.(6.16)willberaisedto a, where a variesbetween1to2dependingonVDS,and isthefractionalincreaserequiredinthefirsttermofEq.(6.7)tomaintainNinvindependentofVDS.Interestingly,theargumentofthelnterminEq.(6.14)isthis VDS-dependentfractionalincreasefornondegeneratecarriers,andhence,we approximate ,(6.17) and generalize Dfsf(VDS) as .(6.18) EavgEjNjNinv----------jEj Nj Ninv----------j + = h q fsf0Eavg–q Dfsf++ kBT -----------------------------------------------------------1 h q fsf0Eavg– + kBT --------------------------------------exp +21 – ln @ a 2 1qVDSkBT – () exp + -----------------------------------------------------@ DfsfVDS() kBT q --------1 h q fsf0Eavg– + kBT --------------------------------------exp +a1 – ln h q fsf0Eavg– + () – @

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206 6.2.3 Simple Alternative An alternative to modeling Dfsf(VDS) is to use IDS(lim) in Eq. (6.9), i.e., (6.19) with Nj from Eq. (6.7) with VDS = 0.0V, i.e., ,(6.20) and (6.21) with fsfindependentofVDS[Ass00].WhileneglectingtheVDSdependenceofNjis reasonableasnotedearlier,neglectofVDSdependenceof fsfin uavg(j)impliesthat theresultingIDS(lim)willbelowerthanthatifthisdependencewasincorporated. Nevertheless,asexemplifiedin Fig.6.1, thisdifferenceistypicallysmall(andmay be neglected for devices with realistic RS/D). 6.2.4 Conductivity Effective Mass Formultipleequivalentvalleys,thefirstsummationinEq.(6.4)andEq. (6.7)isingeneralreducedtoasumoverallnonequivalentvalleyswitheachnew termofthesummationmultipliedbythecorrespondingvalleydegeneracy.For example, for Si, Eq. (6.4) reduces to IDSlim ()WqNjuavgj () jvalleys= Njg2DkBT1 h q fsfEj– + kBT ------------------------------exp + lnjvalleys= uavgj ()uT12 h q fsfEj– + kBT ------------------------------12 h q fsfEj–VDS– + kBT -----------------------------------------------– 1 h q fsfEj– + kBT ------------------------------exp + ln1 h q fsfEj–VDS– + kBT -----------------------------------------------exp + ln + -----------------------------------------------------------------------------------------------------------------------------------------------------------=

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207 Figure 6.1IDS(lim)(VDS)predictedbyEq.(6.19)withandwithoutaccountingforVDSdependenceofNjand uavg(j)forLeff=18nmSDGnMOSFETwith tSi=7nm,tox=1.0nm,midgapgate,VGS=1.0V,andvariousRS/D; fsfand Ej are evaluated using UFDG. 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 0.5 1.0 1.5 2.0 2.5 3.0IDS(lim) [mA/ m m] w/ VDS dependence w/o VDS dependence RS/D = 100 W m m RS/D = 0 W m m VGS = 1.0V

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208 ,(6.22) where d ( d ’)istheunprimed(primed)valleydegeneracy,andg2D(g2D’), uT( uT’), andF(F’)arethe2-DDOS,averageunidirectionalthermalvelocityfor nondegeneratecarriers,andtheremainingsummationtermsinEq.(6.4)for all d ( d ’) unprimed(primed)valleys,respectively.However,whilevalleyequivalence(for2Dconfinedsystem)requiresthattheconfinementmassmx,whichdefinesthe subbandenergies,ofall d equivalentvalleysbethesame,itdoesnotassertany constraintonthe uT-definingmy.Hence, uT(and uT’)inEq.(6.22)isingeneralnot that given by Eq. (6.5). Instead, ,(6.23) where mC is the conductivity effective mass for all d equivalent valleys. InaccordwiththereductionofEq.(6.4)toEq.(6.22)forIDS(lim),mCin Eq. (6.23) is defined via Eq. (6.5) as ,(6.24) wheremyiismyinithequivalentvalley.Interestingly,wenotethatmCdefinedinEq. (6.24)isnotthesameasthatusedforscattering-dominatedtransport.Inthelatter case, uy is the drift velocity defined as (6.25) IDSlim ()W d qg2DkBT 2 ---------------------uTF d qg2D kBT 2 -----------------------uT F + = uT2kBT p mC------------= d 1 mC----------1 myi------------i1 = d= uyq tmmy---------ey=

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209 withanaveragemomentum-relaxationtime tmandlateralelectricfieldey,andthus, .(6.26) ComparingEq.(6.24)andEq.(6.26),itisobviousthatmCforballistictransportis notthesameasthatforscattering-dominatedtransport,unlessalltheequivalent valleys have the same my. Toexemplifythenoteddifference,weconsiderelectrontransportinSi alonga<100>direction.WhentheSi-surfaceorientationis{100},therearetwo equivalentunprimedvalleys,bothwithmy=mt(thetransversemass),andfour equivalentprimedvalleys,twowithmy=mtandtwowithmy=ml(thelongitudinal mass)[Ste72].Sincebothunprimedvalleyshavethesamemy,mCforballisticand scattering-dominatedtransportisequivalent.However,inthecaseofthefourprimed valleys, we get, from Eq. (6.24), (6.27) for ballistic transport, which compared to (6.28) forscattering-dominatedtransport,fromEq.(6.26),isabout1.14xheavier.Table6.1 listselectronandholemCforballisticandscattering-dominatedtransportinSiwith various surface orientations and a <100> transport direction; also given is uT. d 1 mC------1 myi--------i1 = d= mC4mtmlmtml+ ()2---------------------------------= mC2mtmlmtml+ -----------------=

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210 carrier: surf. orientationvalleymC(ballsitic)uTmC(scattering)electrons: {100} SiUP0.19 1.24 x 1070.19 P0.359 0.9 x 1070.315 electrons: {110} SiUP0.302 0.98 x 1070.282 P0.359 0.9 x 1070.315 HolesHH0.433 0.82 x 1070.433 LH0.169 1.31 x 1070.169 Table 6.1 Conductivityeffectivemasses(intheunitsofm0=9.11x10-31kg)forballisticand scattering-dominatedelectronandholetransportinSiwithvarioussurface orientationandtransportalonga<100>direction.Alsolistedis uT(incm/s),using mC for ballistic transport.

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211 6.2.5 Fermi-Dirac Integral Tocompleteour1-DanalyticmodelforIDS(lim)inEq.(6.4),weneedto approximatetheFermi-Diracintegraloforder1/2inEq.(6.6).Therearemany differentapproximationstothisintegral,andagoodcomparisonofsomeofthemis presentedin[Won94].Sincetheargumentoftheintegralcanvaryoverwiderange astSiisvariedfrom1nmto20-30nm,weuseanapproximationproposedby BednarczykandBednarczyk[Bed78],whichhasgeneralrangefromto+ with maximum relative error of only 0.4%. This approximation is of the form (6.29) with, from [Bed78], .(6.30) 6.3 UFDG-Based Analysis and Physical Insights ToexaminehowclosetotheballisticlimitdononclassicalMOSFETs operate,andtogainphysicalinsights,weimplementtheIDS(lim)(VDS)modelinEq. (6.22)inUFDG,with uTandmCinEq.(6.23)andEq.(6.24),respectively, Dfsf(VDS) inEq.(6.18),andtheFermi-DiracintegralinEq.(6.29).Boththesubbandenergies and zero-VDSfsf are evaluated from the QM model developed in [Ge02a]. 6.3.1 Effective Channel Length and UTB Thickness Dependences Sincewedonotaccountfor2-DelectrostaticsinourmodelforIDS(lim)in Eq.(6.22),thelimitcurrenthasnodirectdependenceonLeff.However,sincetSiis 12 u () 1 eu –x u () + -----------------------= x u () 3 4 -p 50u433.6u10.68e0.17 –u1 + ()2– () ++ []3 =

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212 thinnedtomaintainacceptableSCEcontrolwithscalingofLeff,IDS(lim)indirectly dependsonLeffthroughtSi,becausetSiaffectscarrier-energyquantization. Figure6.2 showsUFDG-predictedIDS(lim)(VDS)inSDGnMOSFETswith{100}-Sisurface,tox=1nm,midgapgate,VGS=1.0V,andtSirangingfrom20nmto3nm.Asimpliedby Fig.5.5andhigher uTintheunprimedvalleyinTable6.1,theon-stateIDS(lim)(Ion(lim))significantlyincreaseswithtSi-governedincreaseincarrierpopulationof theunprimedsubband,especiallyfortSi<5nm.Forexample,Ion(lim)fortSi=3nmis @ 1.74xthatfortSi=20nm.Alternatively,significantcarrierpopulationintheprimed subband(Fig.5.5)underminesIon(lim)inSDG(andFD/SOI)MOSFETswithviable tSi>4nm.DuetosignificantincreaseinVtduetoquantizationfortSi<3nm(Fig. 3.9),wefindthatIon(lim)decreaseswithtSi<3nm(notshown).Finally,similar IDS(tSi)arepredictedforSDGnMOSFETswith{110}-Sisurface,SDGpMOSFETs, and FD/SOI CMOS devices. 6.3.2 Ballistic-Limit Current in Nonclassical Devices Inthissection,wecompareIDS(lim)tothatpredictedassumingscatteringdominatedtransport(IDS(sct)).Thelatterisdefinedviaour meffmodeldescribedin Sec.5.9,inconjunctionwithaphysics-basedvelocityovershootmodel[Ge01].For SDGnMOSFETswith meffasinFig.5.16(a), Fig.6.3 comparesUFDG-predicted IDS(lim)andIDS(sct)indeviceswithLeff(strong)=45nm,18nm,and7nm,allwithtox=1nm,tSi=7nm,midgapgate,VGS=1.0V,and,ideally,RS/D=0.0 W m m. Interestingly,IDS(sct)>IDS(lim)forLeff(strong)<45nm.And,withRch 0.0 W m m, IDS(sct)forthe7nmSDGnMOSFETishigherthanIDS(lim)eveninthelinearregion ofoperation.Wealsofind,usingUFDG,thatIDS(sct)>IDS(lim)forLeff(strong)<45nm

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213 Figure 6.2UFDG-predictedtSidependenceofIDS(lim)-VDSofSDGnMOSFETswith {100}-Sisurface,tox=1nm,midgapgate,VGS=1.0V,andRS/D=0.0 W m m. 0.00.10.20.30.40.50.60.70.80.91.0 VDS [V] 0.0 1.0 2.0 3.0 4.0IDS(lim) [mA/ m m] tSi = 20nm, 15nm, 10nm, 8nm, 5nm, 4nm, and 3nm

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214 Figure 6.3UFDG-predictedIDS(lim)-VDSandIDS(sct)-VDSforSDGnMOSFETswith Leff=45nm,18nm,and7nm,tSi=7nm,tox=1nm,midgapgate, VGS = 1.0V,RS/D = 0.0 W m m, and{100}-Si surface. 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDS [mA/ m m] IDS(lim)Leff(strong) = 18nm Leff(strong) = 7nm Leff(strong) = 45nm

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215 SDGnMOSFETswith{110}-Sisurface.However,UFDGpredicts @ 1.32xhigher Ion(lim)inSDGnMOSFETswith{100}-Sisurfacesthanthatwith{110}-Sisurfaces duetothehigher uTintheunprimedvalleyoftheformer.Further,forFD/SOI nMOSFETswith meffinaccordwithFig.5.16,UFDGpredicts,analogoustoFig.6.3, IDS(sct)higherthanIDS(lim)forLeff<45nm.However,asreflectedin Fig.6.4forLeff=18nmand7nmSDGnMOSFETsofFig.6.3,UFDG-predictedeffectivesaturation velocity usat(eff)> usat,definedbyvelocityovershoot[Ge01],isrelativelyhigh,and thelengthofthehighfieldregion( D L),definedbychannellengthmodulation [Chi01a],iscomparabletoLeff,especiallyfortheLeff=7nmdevice.( usat(eff)issame forbothdevicesbecauseNinv,andhence meff,isthesameforboth,anditisexpected tobethesamefortheLeff=45nmSDGnMOSFETofFig.6.3forthesamereason.) Nevertheless,theimportanceofsubjectingIDStothethermalinjection/ballisticlimit is clear. ForADGnMOSFET,UFDGpredictionsexemplifiedin Fig.6.5 showthat IDS(sct)iswellbelowtheballisticlimit,asmaybeexpectedfromtheverylow electron meffinFig.5.18(b).Similarly,becauseoflowholemobility(seeFig. 5.17(a)),wefindthatIDS(sct)IDS(lim)forSDG pMOSFETsasLeff(strong) 7nm.NotethatifSDGandFD/SOInMOSFETswere doped,thentheywouldnotbeat,orapproach,theballisticlimitasthehigh-NBdefinedtransversefielddegradescarriermobility,analogoustohowthedevice asymmetry-defined transverse field degrades carrier mobility in ADG nMOSFETs.

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216 Figure 6.4UFDG-predicted usat(eff)and D LversusVDS,withtheformerdenedby velocityovershootandthelatterbychannellengthmodulation,in(a)Leff= 18nm and (b) Leff = 7nm SDG nMOSFET of Fig. 6.3.(a) (b) 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 3.60 3.65 3.70 3.75 3.80 3.85usat(eff) [x107 cm/s] usat(eff) 0.0 2.0 4.0 6.0 8.0 10.0 12.0D L [nm] D L VGS = 1.0V Ninv = 1.22x1013cm-2meff = 390 cm2/V-s 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0D L [nm] D L 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 3.60 3.65 3.70 3.75 3.80 3.85usat(eff) [x107 cm/s] usat(eff) VGS = 1.0V Ninv = 1.22x1013cm-2meff = 390 cm2/V-s

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217 Figure 6.5UFDG-predictedIDS(lim)-VDSandIDS(sct)-VDSforADGnMOSFETs withLeff=18nm,tSi=7nm,tox=1nm,n+-andp+-polysilicongates, VGS = 1.0V,RS/D = 0.0 W m m, and{100}-Si surface. 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 1.0 2.0 3.0 4.0IDS [mA/ m m] IDS(lim)Leff(strong) = 18nm

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218 Figure 6.6UFDG-predictedIDS(lim)-VDSandIDS(sct)-VDSforSDGpMOSFETswith Leff=18nmand7nm,tSi=7nm,tox=1nm,midgapgate,VGS=1.0V, RS/D = 0.0 W m m, and{100}-Si surface 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 1.0 2.0 3.0IDS [mA/ m m] IDS(lim)Leff(strong = 18nm Leff(strong) = 7nm

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219 6.3.3 Effect of Source/Drain Series Resistance Intheprevioussection,weexaminedthechannelcurrentinintrinsic,i.e., RS/D=0.0 W m m,nonclassicalCMOSdevices.However,weneedtoconsiderRS/Ddueto,forexample,theneededunderlapinrealnanoscaleUTBdevices.Toexamine howmuchdoesRS/DaffectsIDS(lim)andIDS(sct),weshowinFig.6.7(a)andFig. 6.7(b)UFDG-predictedIDS(lim)andIDS(sct)inSDGnMOSFETsofFig.6.3butwith RS/D=100 W m mandRS/D=200 W m m,respectively.AlthoughRS/Dtendstoreduce thedifferencebetweenthepredictedIDS(sct)andIDS(lim),thedifferenceisstill noticeableaton-state.Moreimportantly,RS/DhassignificantlyloweredIon(lim).For example,Ion(lim)forRS/D=200 W m mis @ 0.44xthatforRS/D=0.0 W m m,andeven Ion(lim)forRS/D=100 W m misbelowtheIon=1.9mA/ m mITRSprojection[ITRS03] atLgate=Leff(strong)=18nmnode.Theseresultsclearlyshowthattheon-state current,andhencetheCMOSspeed,ofnanoscaleSDG(andthick-BOXFD/SOI) nMOSFETs is limited by RS/D. 6.3.4 Comparison to ITRS Projections Inthissection,wecompareUFDG-predictednanoscalemidgap-gateSDG CMOScharacteristicstotheprojectionsintheITRSforHPCMOS.WeassumeLeff=Lgate,tox,VDD,andRS/D(=RSD/2)asrecommendedbyITRS,andweassumetSi=Leff/2.WenotethatalthoughtherequiredtSiforLeff< @ 10nmisnotbedoable technologically,neitheristheITRSrecommendedRS/D,especiallywhenG-S/D underlapisemployed.Further,theIonprojectionsinITRSshouldbetreatedmerely astrends,sincetheyarebasedonneededimprovementintheCV/Imetricthatcanbe

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220 Figure 6.7UFDG-predictedIDS(lim)-VDSandIDS(sct)-VDSinSDGnMOSFETsof Fig.6.3butwith(a)RS/D=RSD/2=100.0 W m mand(b)RS/D= 200.0 W m m; Leff(strong) = 45nm device is not considered. 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 1.0 2.0 3.0 4.0IDS [mA/ m m] IDS(lim)Leff(strong) = 18nm Leff(strong) = 7nm RS/D = 100 W m m 0.00.10.20.30.40.50.60.70.80.91.0VDS [V] 0.0 1.0 2.0IDS [mA/ m m] IDS(lim)Leff(strong) = 18nm Leff(strong) = 7nm RS/D = 200 W m m(a) (b)

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221 misleading,especiallyfornonclassicalCMOS[Fos02b],[Fos04b].Hence,the comparisonsinthissectionsimplyreflectperformancelimitationofmidgap-gate SDG CMOS. Table6.2liststheITRSrecommendedtox,VDD,RS/D,andIon,andthe correspondingUFDG-predictedIon(lim)inSDGnMOSFETswith{100}-and{110}Sisurfaces;whenconsideringFinFETs,theUFDG-predictedIon(lim)isthatperhSi. Further,incaseswhereIon(lim)doesnotmeettheITRSrequirement,wegiveinTable 6.2theUFDG-predictedrequiredincreaseinthegatedrive(i.e.,(VGS-Vt)), reflectingeithertheneededincreaseinVDDordecreasein FG,tomeettheITRS projections;VDDwasvariedinUFDGtofindthisadditionalgatedrive.Wenotethat inballisticMOSFETs[Nat94],[Ass00],unlikeinlong-channelandvelocitysaturated MOSFETs, .(6.31) Theadditional~(VGS-Vt)1/2factorresultsfromtheNinv-dependenceof uinjatthe source for degenerate carriers (see Eq. (6.10)). Interestingly,theITRSprojectionscannotbemetbymidgap-gateSDG nMOSFETswith{100}-SisurfacesforLgate<18nm,andwith{110}-Sisurfacesfor Lgate<25nm.FurtherinvestigationshowedthatthisisduemainlytoRS/D(and somewhattothesignificantcarrierpopulationintheprimedsubbands),solidifying ourearlierconclusion(Sec.6.3.3)thatRS/Dcanseverelylimittheperformanceof nanoscaleSDG(andFD/SOI)MOSFETs.FortheLgate=7nmnode,verylowVDD(andbulk-inversion-inducedCGdegradation)prohibitshighIon(lim)evenwith Ionlim ()~VGSVt– ()32

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222 Lgate [nm]452518137 tox[nm]1.30.90.7 0.6 0.5 VDD[V]1.21.11.0 0.9 0.7 RS/D[ W m m]908167.553.530 Ion [mA/ m m]0.981.511.92.052.19 Ion(lim) [mA/ m m] ({100}/{110}) 1.97/1.692.06/1.701.96/1.521.68/1.270.84/0.56 D( VGSVt) [V] ({100}/{110}) ---/0.050.07/0.200.19/0.30 Table 6.2 ComparisonofITRS-projectedIonversusUFDG-predictedIon(lim)formidgapgateSDGnMOSFETswith{100}-and{110}-Sisurfaces;fornFinFETs,the UFDG-predictedIon(lim)isthatperhSi.AlsogivenistheUFDG-predicted additional gate drive needed to meet the ITRS projections.

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223 RS/D=0.0 W m m.ForSDGdeviceswith{100}-Sisurface,dualmetalgates,e.g., FG(n)@ 4.42V(4.6V-0.18V)fromChapter2,wouldbeneededtomeettheITRS projections.For{110}-Sisurface,thenoteddualmetalgatesmaysufficeonlyifVDDforLgate 13nmisappropriatelyincreased.However,SDGnMOSFETswith{100}Sisurfaceisanoptimalchoice.(AlthoughwehavenotgivenIoffhere,sincewekeep tSi=Lgate/2andusemidgapgate,ITRSIoffrequirementsareeasilymet.InfactIoffisanorder,ortwo,magnitudeslowerthantheITRSprojections.)Interestingly,note thatsincetheUFDG-predictedIon(lim)inTable6.2isthatperhSiforFinFETs,Ion(lim)per2hSi,whichiscommonly(butnonphysically)doneinliterature,cannotmeetthe ITRS projections for Lgate 45nm! WhiletheCV/ImetricimpliesunacceptableCMOSdelayforourmidgapgateSDGCMOSwithIon(lim)lessthanIoninITRS,especiallywhenIon(lim)is normalizedby2hSiforFinFETs,UFDGpredictsCMOSNAND-gatedelay( tNAND) of10.4ps(11.2ps)for{100}-Si({110}-Si)surfaceLgate=18nmtechnology,andof 7.5ps(8.4ps)for{100}-Si({110}-SisurfaceLgate=13nmtechnology.Comparedto theITRS-projected tNANDof9.9psand6.5psfor18nmand13nmtechnology, respectively,theseUFDG-predicteddelaysareinareasonablerangeofITRS projectionsdespitethesubstantiallylowerIon(lim)and2xhigherparasitic capacitance.(WenotethattheUFDG-predicted tNANDisdeterminedusingthesame methodasinITRS[ITR03],i.e., tNAND=3 1.3 2 tinv,where tinvistheunloaded CMOS-inverterdelay.)Thisinteresting(andrathersurprising)result,whichreflect theinadequacyoftheCV/Imetric,isduetothenegligiblelow-VGSintrinsic-CGin DG MOSFETs [Fos02b], [Fos04b].

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224 6.3.5 Simplified Implementation for UFDG Figure6.3impliesthatweneedtosubjectIDSinnonclassicalMOSFETs totheballisticlimitforallVDS>0.0V.However,whenweconsiderrealdeviceswith RS/D>0.0 W m m,Fig.6.7showsthatIDS(sct)@ IDS(lim)inthelinearregion,i.e.,RS/Ddominatesthelinearregioncharacteristicsindependentofcarriertransportinthe nanoscalechannel.Further,asnotedinSec.6.2.3,wecanexpressIDS(lim)suchthat theVDSdependenceofNinvand fsfcanbeneglected,whileincorporatingthe essentialphysics.Hence,toreducethecomputationalburdenforcompactMOSFET model such as UFDG, we only subject IDS(sct) to Ion(lim), i.e., for Si, .(6.32) WithNj(Nj’)inEq.(6.20)and()inEq.(6.10),Eq.(6.32)simplifiesto (6.33) with fsf independent of VDS. Finally,whenIDS(sct)exceedsIon(lim),itislimitedtoIon(lim)viaa smoothing function ,(6.34) wher eA=5isa constantthatcontrolsthesmoothnessofthetransition.Thisis exemplified in Fig. 6.8 for the SDG nMOSFETs of Fig. 6.7(b). Ionlim ()WqNjuinjj () S jqNjuinjj () Sj+ = uinjj () Suinjj () S Ionlim ()WqkBTg2DuT12 h q fsfEj– + kBT ------------------------------jg2DuT12 h q fsfEj – + kBT --------------------------------j+ = IDSVDS() IDSsct ()VDS() 1AIDSsct ()VDS() Ionlim ()– () [] exp + {} ln A ---------------------------------------------------------------------------------------------------------- – =

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225 Figure 6.8UFDG-predictedIDS-VDSofSDGnMOSFETsofFig.6.7(b)withIDS(sct)subjectedtothehigh-VDSballistic-limitcurrentIon(lim)inEq.(6.33)using the smoothing function in Eq. (6.34). 0.00.10.20.30.40.50.60.70.80.91.0 VDS [V] 0.0 0.5 1.0 1.5IDS [mA/ m m] Leff(strong) = 18nm Leff(strong) = 7nm Ion(lim)

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226 6.4 Summary A1-Dmodelforthethermal-injection-/ballistic-limitcurrentfrom [Nat94]wasoverviewed,andanalysisofadditionalissuespertainingtoits implementationinaphysics-basedmodel,i.e.,UFDG,weredone.Comparisonof UFDG-predictedIDS-VDSwithscattering-dominatedtransporttothatwithballistic transportshowedthatnanoscaleSDGandFD/SOInMOSFETswithLeff<45nm operateattheballisticlimitduetotheirsignificantlyhigh meff.Incontrast,because ofmuchlowerholemobility,theballisticlimitforSDGandFD/SOIpMOSFETsis reachedatmuchshorterLeff~7nm.Similarly,IDSinADGnMOSFETsremainswell belowtheballisticlimitforallLeffconsideredhereduetotheirsubstantivelylow meff.Finally,UFDG-predictionsshowedthattheperformanceofSDG(andFD/SOI) CMOSislimitedbyRS/D,andthatIon(lim)inmidgap-gateSDGMOSFETscannot meettheITRS-projectedIonwhenITRS-recommendedtox,VDD,andRS/Dare considered;higherVDDand/ordualmetalgatesmaybeneeded.However,despitethe lowerIon(lim),UFDG-predictedSDGspeedwaswithinareasonablerangeofITRS projectionduetothenegligibleintrinsicgatecapacitanceforDGMOSFETsinweak inversion, reflecting the inadequacy of the CV/I metric assumed for the ITRS.

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227 CHAPTER 7 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 7.1 Summary and Conclusions Wehavepresentedphysicalinsightsontheoperationofnonclassical nanoscaleMOSFETswithundopedUTBs,andhaveappliedtheseinsightsfor optimaldesignsandphysicalmodeling,emphasizingtheeffectsofundopedand ultra-thinbodiesonMOSFETelectrostaticsandcarriertransport,andofsource/drain lateraldopingprofileonLeff,RS/D,andCGS/D.Thephysicalinsightsandmodels werealsoappliedtoimproveUFDG,aprocess/physics-basedcompactmodelfor nonclassical CMOS device and circuit design. InChapter2,wedidquasi-2-DanalysestogaininsightsonSCEs,andits dependenceonNB,tSi,andthedeviceasymmetry.WefoundthatUTBsareneeded tocontrolSCEsindeviceswithLeff<50nm.WealsofoundthatalthoughhighNBeffectsbetterSCEcontrol,unacceptablesensitivityofthedevicecharacteristicsto variationsinNBandtSiyieldVtcontrolviahighchanneldopingandpolysilicon gatesimpractical,andhencethatundopedUTBswithVtcontrolviagatework functionsisnecessary.ThedependenceofSCEsonNBandthedeviceasymmetry wasrelatedtothelocationofthepredominantcurrentflow,ortheinversion-charge centroidinthebody/channel,showingthatcarrierdistributionisimportantevenin weakinversion.Performanceprojections,usingUFDG,fornanoscaleFD/SOI CMOSshowedthatmidgapgateisacceptableforhigh-performanceapplications,and

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228 thatusingtSiforIoffcontrol,viasuppressingSCEsandinducingQMeffects,allows forlow-powerCMOSusingthesamemidgapgate.UFDG-basedstudiesalsoshowed thatmoderate(~15%-25%)variationin4nm
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229 CMOSwith{110}-Sisurfaceorientation.Withregardstocarrierdistributioninthe body,wefoundthatinweaklyinvertedlong-channelFD/SOIandSDGMOSFETs (andTGMOSFETs),theinversioncarriersaredistributeduniformlythroughoutthe body,showingthatsubthresholdcharacteristicsoftheselong-channeldevicesare independentofthenumberofactivegates.ThisdefinedWeff=hSiforSDGFinFETs (andfurtherargumentsimpliedthatnoWeffcanbephysicallydefinedforTG MOSFETs).Althoughnotuniformlydistributedinstronginversion,carrierdensity inthebulkofthebodywasfoundtobesubstantive,whichunderminedCG,andthus, Ninv.However,thisdegradationwasfoundtobemorethancompensatedby substantial increase in meff due to bulk inversion, but limited by velocity saturation. InChapter4,wefound,via2-Dnumericaldevicesimulations,that effectiveG-S/DunderlapisneededtoyieldoptimalnanoscaleMOSFETs.We explainedthattheunderlap,whichisnotdefinedbyaspecificSDEdopingdensity, resultsinabias-dependentLeff.Specifically,LeffcanbelongerthanLgateinweak inversion,relaxingtheUTB-thicknessrequirementforSCEcontrol,andthatLeff Lgateintheon-state.However,Ionwasfoundtobelimitedby(possibly)non-ohmic RS/Ddefinedbytheunderlap.DependencesofLeffandRS/Donthelateraldoping profileintheSDEsclearlyrevealedadesigntradeoffregardingSCEsandRS/D, whichstronglydependsontechnologicallyachievablewSi,andwhichcanbe optimizedviaSDEengineering.Weexemplifiedsuchoptimization/engineeringby designingawell-tempered,high-performanceLgate=18nmnFinFET.ThiswelltempereddesignconsistedofproperengineeringofSDEsusingalarge,doable straggle(lateralabruptness)of9.5nm( @ 7nm/decade)withoutcausingS-Dpunchthrough.WefoundthattheeffectiveunderlapalsoyieldsminimalCGS/Dinweak

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230 inversionandhighCGGintheon-statethatwillfurtheroptimizeCMOSspeedand drivability.Inaddition,theeffectiveunderlapisexpectedtosuppress,ifnot eliminate,leakagecomponentssuchasGIDLcurrent.Finally,toaidthenotedSDE engineeringforoptimaldesign,wedevelopedamodelforrelatingLefftotheSDE design(i.e.,Lextand sL)forgivenchannel/bodydesign(i.e.,Lgate,tox,andtSi,or wSi). InChapter5,wepresentedphysicalinsightsonthetSi-dependenceof meffinUTBDGMOSFETs,usingMonteCarlosimulationsandmeasureddata.Wefound thatsolongastheinterfacesarerelativelyclean(i.e.,Nit<1011cm-2),phonon scattering,governedbycarrierconfinement,predominantlydefinesthetSidependenceof meffinUTBswithpragmatictSi> @ 4-5nm.Wefoundthatwhile phonon-limitedmobilityisdefinedbythecarrierconfinement,itisvirtually independentoftheconfinementmechanism.Basedonthisinsight,wedevelopeda compactmodelforphonon-limitedmobility,andcombineditwithaphysics-based analyticmodelforsurface-roughness-limitedmobilitytodefineacompactmodelfor meffinUTBs.Themodel,intendedforUFDG,accountsfordependencesontSi,eeff(includingNinv),andthecrystalorientation,andhasonlytwotuningparametersthat canbecalibratedviameasurementsfromathick-tSidevice.Weverifiedourmodel usinglargesetsofexperimentaldata,aswellasMonteCarlosimulations.The measureddatashowedstrongtSi-dependenceforlow-Ninvmeff,however,itshowed virtuallynotSi-dependenceforhigh-Ninvmeff,implyingthathigherIoff/Ioncanbe achievedviathinnertSi.Wealsoappliedourmodeltopredict meffinDGFinFETs with{110}-Sisurface,andfoundthatelectronmobilityinitathighNinvis~0.5-0.7x

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231 thatin{100}-Sisurfacecounterparts.Inaddition,wefoundthatforgivenNinv,and henceon-statecondition,electron(hole)mobilityinnonclassicaldevicescanbeup to4.2x(2.3x)higherthanthatinclassicaldevices,andmaybeexpectedtoremain higherthan,orcomparable,tothatinclassicaldeviceswithstrained-Sichannels. Finally,because meffathighNinvremainsvirtuallyindependentoftSi,suchhigh meffinSDGandFD/SOIMOSFETswillremainthesameforallfuturetechnologynodes, unlike meff in classical MOSFETs. Withsignificantlyhigher meff,nonclassicalMOSFETsarelikelyto operateclosertotheballistic,orthethermal-injectionlimit.Hence,inChapter6,we examinedtheballistic-limitcurrentinDGMOSFETs,usingthe1-DmodelofNatori [Nat94].Afteroverviewingthenotedmodel,weanalyzedtheconductivityeffective massandthedrain-biasdependenceofballistictransport.Weimplementedthemodel inUFDGandfoundthattheUFDG-predictedchannelcurrentwithscatteringdominatedtransportcanexceedthethermalinjection,orballistic-limitcurrentin SDGandFD/SOInMOSFETswithLeff<45nm,implyingballistictransportinthese devices.Incontrast,wefoundthattheballisticlimitisnotreachedinSDGandFD/ SOIpMOSFETsuntilLeff~7nmduetothelowerholemobility.Similarly,IDSin ADGMOSFETsremainswellbelowtheballisticlimitduetotheirsubstantivelylow meff.Further,UFDGpredictionsshowedthatnanoscaleSDGandFD/SOICMOS performanceislimitedbyRS/D.Finally,weappliedUFDGtoexamineIonlimitation ofSDGnMOSFETswithmidgapgatebycomparingthepredictedballistic-limit currenttoITRSprojectionsatLgate=45nm-7nmtechnologynodes;devicestructures wereassumedtobethatrecommendedinITRS.Interestingly,wefoundthathigher

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232 VDDand/ordualmetalgatesareneededtomeettheITRSprojectionfor Lgate < 13nm. Basedonourwork,welistinTable7.1howthescalingissuesofclassical CMOSinTable1.1areaddressedwithnonclassicalCMOS.Interestingly,theuseof undopedbodyalleviates,ifnoteliminates,manyofthescalingissues.Since meffin stronginversionisofthemaininteresthereandsince meffathighNinvisvirtually independentoftSi, meffinnonclassicalMOSFETs,whichisalreadymuchhigherthan thatinclassicaldevices,doesnotdegradewithLgate(andtSi)scaling.Asforpolydepletioneffects,theyareinherentlyeliminatedasmetalgatesareneededforVtcontrolinundoped-UTBMOSFETs;themetalgatesmaybeachievedviaeither sillicidation[Ked02]orusingTaNSiorTiN[Par01].Further,withundopedbody,the transversefieldismuchlowerinnonclassicaldevices(whichunderliesthehigh meff), definingrelativelyhighereffectiveoxide-barrier,andthus,relativelylower tunnelingcurrent[Cha02].Also,wecanemployG-S/Dunderlaptorelaxthetoxrequirement[Fos03c].Thismayenableusingoxy-nitridedowntoLgate~7nm. Similartogatetunnelingcurrent,usingundopedbodiesandeffectiveunderlapcan alsosubstantivelyreducejunctiontunnelingcurrent.AswefoundinChapter2,since SandDIBLinFDMOSFETsscaleinsimilarfashionwithtSi,theissueoflargeS resultingfrombetterDIBLcontrolisinherentlyeliminatedinnonclassicaldevices withFDbodies.Also,thesevereproblemofrandomdopanteffectinclassical MOSFETs is resolved in nonclassical MOSFETs via employing undoped bodies. TheissueofhighRS/Dstillremainseveninnonclassicaldevices;however, moresystematicexperimentalandtheoreticalinvestigationsareneededtoassessthis

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233 Scaling IssuesClassical CMOSNonclassical CMOS Mobility degradationUni-, or bi-axially strained-Si channelalready very high undoped body Poly-depletion effectMetal gates (e.g., Ru, Ta)Metal gates for Vt control Gate tunneling currentHighk gate dielectric (e.g., HfO2) relatively lower lowerex undoped body; avoid thin tox via effective underlaps S/D series resistanceSiGe S/D, Schottky S/D, ?thicker wSi effective underlap, SiGe S/D, ? Junction tunneling current?undoped body, effective underlap Large gate swing?fully depleted body Random dopant effect?undoped body wSi < Lgate (for FinFETs)N/Anew patterning techniques Table 7.1 Comparison of how some major scaling issues are, or can be, resolved in classical and nonclassical CMOS.

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234 issue.Also,fortheFinFET,whichisthebestoptionfornanoscaledevices,realizing wSi
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235 optimizingtheunderlap.Whilewehavequalitativelyexaminedtheirbehaviorwith NSD(y),quantitativeanalysisisessential.SimilartoourmodelforLeff(weak),these modelsshouldbeabletoprovidereasonableestimates.Wefurtherrecommend applicationofthesemodelsformoredefinitiveoptimizationusingUFDG-based performance projections. Third,werecommendimplementingourmodelforsubthresholdQM effectsfromSec.3.4inUFDG.Initsexistingweak-inversionformalism,UFDG assumespredominantoccupationoftheground-stateenergy,andmodels Dfs QM= E0/q[Ge02a],bothofwhichareinadequatebasedonouranalysesinSec.3.4.For example,fromFig.3.6, Dfs QMisabout20mVlargerthanE0/qforSDGMOSFETs with4nm 4nm. FourthandcontinuingwithQMeffects,werecommendstudyofcrystalorientationdependenceofpMOSFETs,andnMOSFETs.Inparticular,anisotropyof holesneedstobeincorporatedinUFDG.MonteCarlosimulations[Fis03]have shownthatcarrier-energyquantizationvarieswiththesurfaceorientation,andso doescarrierpopulationinheavy-hole,light-hole,andspin-orbitbands.Theformer impliessurface-orientationdependenceoftheconfinementeffectivemass,andthe latteroftheDOSeffectivemass.TheseMonteCarlosimulationresultscanbeused inconjunctionwithSCHRED,orforsimplicity,variationalapproach-basedmodels, toextractthenotedeffectivemasses.Further,thesesimulationsandexperimental observations[Yan03]alsoshowthattheholemobilityinpMOSFETswith{110}-Si surfaceorientationandcurrentflowalonga<110>directioncanbesignificantly(up

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236 to2-3x)higher,suggestingvariationintheconductivityeffectivemass,mC. However,unliketheconfinementandDOSeffectivemasses,extractionofmCis formidable,because meffstronglydependsonthescatteringmodelsappliedand becauseofthecomplexvalence-bandstructure.Hence,whenexaminingsuch pMOSFETs,werecommendcalibratingour meffmodeltotheexperimentaldatavia appropriatelytuningUOand q .However,forelectrons,detailedexaminationofthe current-flow direction on mC, and hence, on meff [Sat71] and Ion(lim), is suggested. Fifthandconcerningcarriertransport,werecommend,basedonFig.6.4, reevaluating,andpossiblyupgrading/refining,theexistingvelocityovershootand channel-lengthmodulationmodelinginUFDG.Wefurtherrecommendapplication oftheballistic-transportmodelinChapter6,inconjunctionwiththesuggestedRS/DandCGS/Dmodeling,toexaminetheperformancelimitationsofSDGMOSFETs, possibly with various surface orientations and current-flow directions. Sixthandfinally,werecommendassessmentof2-Deffectsinstrong inversion,i.e.,DICE[Vee88].SincethetransverseelectricfieldinundopedDG MOSFETsisverylow,underlyingitshigh meff,some2-Deffectsmaybepresent eveninstronginversion.Ifso,these2-Deffectsshouldbemodeled,withbulkinversiontakeninconsideration,andincorporatedinboththescattering-dominated and the ballistic transport models in UFDG.

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237 APPENDIX A UPGRADES/REFINEMENTS TO THE WEAK-INVERSION CURRENT FORMALISM IN UFDG Wedescribeheretheupgrades/refinementsmadetotheweak-inversion formalismofUFDG(Ver.2.3)formorephysicalpredictionsandtoavoid(i) discontinuityintheweak-inversionchannelcurrent(Iwk),(ii)negativevaluesforthe effectivechannellengthoverwhichthecarriersdiffuseinweakinversion(Le),and (iii) complex smoothing functions in weak inversion. A.1 Issues with the Existing Formalism InFig.A.1,we showUFDG2.3-predictedIDS-VGScharacteristicswithand withoutthequantummechanical(QM)optionturnedonforanADGnMOSFET havingtoxf=toxb=2nm,tSi=10nm,andn+-andp+-polysilicongates.WhentheQM optionison,adiscontinuityoccursinthehigh-VDSIDS-VGSpredictions,andakink appears in the low-VDS IDS-VGSpredictions. Inaddition,thepotentialdistributionin,forexample,nanoscaleundoped SDGnMOSFETs,depictedin Fig.A.2, definestheleakiestpathatthecenterofbody. However,aswewillexplainlater,theintegratedinversioncharge(Qn)iscurrently determinedbylinearlyextrapolatingthepotentialbetweenthetwosurfacesandthe centerofthebody,asexemplifiedbythestraightlinesin Fig.A.2. Clearly,thiswill

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238 Figure A.1UFDG2.3-predictedIDS-VGSforanADGnMOSFETwithVGfS= VGbS=VGS,toxf=toxb=2nm,tSi=10nm,andn+-andp+-polysilicon gates.NotethediscontinuityandthekinkwhentheQMoptionis turned on. -0.2-0.10.00.10.20.30.40.50.60.70.80.9 1.0VGS [V] 10-1210-1110-1010-910-810-710-610-510-410-3IDS [A/ m m] w/QM w/o QM VDS = 50mV VDS = 1.0V

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239 Figure A.2Qualitativepictureoftheexactpotentialdistribution(dashed)ina nanoscaleundopedSDGnMOSFET.Thepotentialdistributioninthe solidlinesisanalogoustothelinearextrapolationdoneinUFDG2.3 for computing the integrated inversion charge density. xtSi0f

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240 resultinUFDGunderestimatingIwk,ashortcomingthatisexacerbatedbyanother approximation for the length over which the carrier diffuse in weak inversion. Hence,wenowdescribethecauseof,andsolutionto,theseissuesinthe existingformalisminUFDGVer.2.3(UFDG2.3).However,foradequate presentation, we first briefly overview the Iwk-formalism in UFDG2.3. A.2 Overview of the I wk Formalism in UFDG2.3 AssumingthatIwkisdefinedpredominantlybycarrierdiffusionovera nonzero length Le, Iwk is given by ,(A.1) ,(A.2) whereDn=(kBT/q) meffistheelectrondiffusivity,Wisthechannelwidth,Le(avg)is anaverageLeinthebody,Qn(Ls)istheelectronchargedensityatthevirtualsource (y=Ls),andVDSisthedrain-to-sourcebias.InUFDG2.3,Iwkiscomputedbyfirst solvingthe2-DPoissonequation,withthedepletionapproximation,forthe2-D potentialdistributioninthebody[Yeh96],andthusn(Ls).From[Yeh96],the resulting 2-D potential distribution ( f (x,y)) has the form ,(A.3) where (K(x))isthelong-channel1-Dpotentialdistribution.SinceIwkisdefinedby IwkWq Dnx () nLsx () () Lex () -----------------------------------x d0 tSi 1qVDS–kBT () exp – () = WkBTq ()meffQnLs() Leavg ()------------------------------------------------------1VDS–VT () exp – () = f xy , ()c1x () y l x () ----------exp c2x () y l x () ----------- – expKx () – + =

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241 carrierdiffusion,thepotentialthatisneededforQninEq.(A.2)istheminimum potentialalongthey-direction.FromEq.(A.3),thisx-dependentminimumoccursat ,(A.4) and the corresponding potential is .(A.5) Further, Le, the length over which carrier diffuse, is defined as ,(A.6) whereLeff=L-DL(orL+LeS+LeDinthecaseofunderlap),andLS(x)andLD(x) arethex-dependentlengthsofsourceanddrainencroachmentsintothechannel, respectively.Asin[Yeh96],LSandLDaremodeledsimplyasthedepletionwidthin the lightly doped side of an one-sided p-n junction, i.e., ,(A.7) and ,(A.8) whereey(x)isthex-dependentlateralfieldderivedfromEq.(A.3).Also,Leis limitedtotheSilatticeconstant(5.23)if(LS+LD) Leff,i.e.,punch-through current is not modeled physically based on space-charge limited flow [Sze81]. yminx () 1 2 l x () -------------c2x () c1x () ------------ln = fminx () 2 c1x ()c2x () Kx () – = Lex () LeffLSx () –LDx () – = LSx () 2 fbifminx () – []eyx ()y0 =--------------------------------------= LDx () 2 fbifminx () –VDS+ []eyx ()yLeff=-------------------------------------------------------=

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242 With fmin(x)inEq.(A.5),Le(x)inEq.(A.6),andatypicalempirical modelfor meff,Yeh[Yeh96]computedIwkinEq.(A.1)bypartitioningitintofrontand back-channel components: (A.9) ,(A.10) wherexmincorrespondstoaminimuminpotentialalongx,andthesubscripts"f"and "b"correspondtothefront(x=0)andback(x=tSi)surface,respectively.InEq. (A.10),QnfandQnbareevaluatedassumingconstanttransversefieldfro mx=0tox = xmin and x = xmin to x = tSi, i.e., (A.11) and .(A.12) Assumingnon-degeneratecarriers,theintegrationinbothEq.(A.11)andEq.(A.12) istrivial.However,caremustbetakeninevaluatingthefieldtermoutsidethe integralinEq.(A.12),sincetheintegralisquasi-2-Dthroughymin(x).Thisismore apparentin Fig.A.3 (a),wherethedarkdashedlinerepresentstheexactpathof integrationandthedarksolidlinerepresentstheintegrationpathforEq.(A.11)and Eq.(A.12);xmininEq.(A.9)correspondstoeitherxmin(f)orxmin(b)inFig.A.3 IwkIwkf0xxmin () IwkbxminxtSi () + = W kBT q --------mefff ()QnfLsf() Lef-----------------------------------meffb ()QnbLsb() Leb-------------------------------------+ 1 qVDSkBT ------------- – exp – = Qnfq f d d x n f ()f dfsfmin ()f xminyminf (), ()= Qnbq f d d x n f ()f df xminyminf (), () f xminyminb (), ()=

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243 Figure A.3(a)Integrationpathsinx-yand(b)potentialprofilesalongxassumed toevaluateQninEq.(A.11)andEq.(A.12).Thedarksolidlines approximatetheactualpath(profile)shownbydarkdashedlines.For thisexample,xmin(b)=xminbecause f (xmin(b),ymin(b))< f (xmin(f), ymin(f)).Notetheneedtocomputesixpotentialstodeterminexmin, and two charge components.(a) (b) xtSi0ff (0,ymin(b)) f (tSi,ymin(f)) f (tSi,ymin(b)) f (0,ymin(f)) f (x,ymin(x)) Approx. f (x,ymin(x)) xmin(f)xmin(b) =xmin y xxmin(f) xmin(b)ymin(f)ymin(b) 0 LefftSi f (x,ymin(b)) f (x,ymin(f))f (x,ymin(x))Approx. f (x,ymin(x))

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244 dependingthecorrespondingpotential[Yeh96].Toaccountforthisquasi-2-D nature,Yeh[Yeh96]appliedthePythagoreantheoremtodeterminetheconstantfield term outside the integrals. AlthoughthisschemewassufficientforFD/SOIMOSFETs,Chong [Cho98]foundittocausediscontinuitiesinIwkofDGMOSFETs.The discontinuitiesin(DC)IDS-VGS,whichcanalsocausetransientconvergence problems,werecorrelatedtodiscontinuitiesinxminanditsderivative.Toresolve this,Chong[Cho98]evaluatedtheintegralsinEq.(A.11)andEq.(A.12)twice,once withymin=ymin(f)(andxmin(f))andsecondtimewithymin=ymin(b)(andxmin(b)),as depictedin Fig.A.4 (a)-(b).Thetwochargetermsforeachpartitionisthen (arithmetically)averagedtogivethefinalsolution,e.g.,Qnf=[Qnf(ymin(f))+ Qnf(ymin(b))]/2.Thisapproachavoidshavingtocomparethetwoxmin’sandgoing back-and-forthbetweenthetwovalues,whichin[Yeh96](or Fig.A.3( a)-(b))would haveoccurredabruptly,causingthenoteddiscontinuity.Also,itisclearin Fig.A.4(a) thattheintegralsalongthetwopathsarenolongerquasi-2-D,andhence,thereisno needtousethePythagoreantheorem.Finally,in[Cho98],smoothingfunctionswere alsoimplementedtoforcexmin(f)andxmin(b)tozero(tSi)whentheyarefoundtobe negative(>tSi);thiscanoccur,forexample,inADGMOSFETs. Theserenements byChong [Cho98],e xtensionof mefftoaccountfortSidependencebyChiang[Chi01a], andGe’s[Ge02a]additionoftheshift/decreaseintheminimumpotentialinEq.(A.5)due to carrier-energy quantization predominantly constitute the Iwk-formalism in UFDG2.3.

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245 Figure A.4(a)Integrationpathsinx-yand(b)potentialprofilesalongxusedto evaluateQninEq.(A.11)andEq.(A.12).Anarithmeticaverageof integrationalongthetwopaths(profiles)depictedbydarksolidlines approximatetheactualpath(profile)shownbydarkdashedlines. Notetheneedtocomputesixpotentialsinordertodeterminethefour charge components.(a) (b) xtSi0ff (0,ymin(b)) f (tSi,ymin(f)) f (tSi,ymin(b)) f (0,ymin(f)) f (x,ymin(x)) Approx. f (x,ymin(x))’s xmin(f)xmin(b) y xxmin(f) xmin(b)ymin(f)ymin(b) 0 LefftSi f (x,ymin(b)) f (x,ymin(f))f (x,ymin(x))Avg. = Approx. f (x,ymin(x))

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246 A.3 Causes of Errors in the Existing Formalism From Fig.A.4 (b),wenotethattheIwkcalculationinUFDGrequires computingsixdifferent fmin.Thus,toproperlyaccountforQMeffectsin subthreshold,theQM-shiftinthepotentialmustbeincorporatedinallsixofthese fminvalues.However,wefindthattheQM-shiftisonlyincorporatedinfiveofthese potentials,whichunderliesthediscontinuityinADGIDS-VGScharacteristicsin Fig. A.1. Unfortunately,asshownin Fig.A.5,the kinkinlow-VDSIDS-VGSstillremains. Asreflectedby Fig.A.6 (a),thiskinkisattributedtoLe 0withincreasingVGS,and eventuallybeinglimitedbythelatticeconstant.WefindthatUFDG2.3predictssuch nonphysicalbehaviorbecausethenotedQM-shiftisincorporatedbeforeLeis determined,whichis,however,supposetodefinetheregionforwhichtheQM-shift isapplicable.And,sincetheQM-shiftreduces fmin,thereversebiasinEq.(A.7)and Eq.(A.8)isincreased,elongatingLSandLD,andhenceshorteningLe.Notethat physically,2-Deffectsdecreasewiththegatebias,andhence,Leshouldincreaseand approachLeff.Thus,properlyimplementingtheQM-shiftaftercomputingLe,the kink is eliminated as can be seen in Fig. A.6 (b). A.4 New I wk Formalism WhileproperimplementationoftheQMmodelremovesthediscontinuity andthekinkinIDS-VGS,thecurrent,andcharge,partitioningschemesdescribedin Sec.A.2arestillinsufficienttoproperlyaccountfortheeffectsofcarrierdistribution inthebody.Thus,wenowdefineacompletelynewformalismforevaluatingIwkin

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247 Figure A.5UFDG-predictedIDS-VGSforADGnMOSFETof Fig.A.1 beforeand after appropriate implementation of the QM shift in all six fmin’s. -0.2-0.10.00.10.20.30.40.50.60.70.80.9VGS [V] 10-1210-1110-1010-910-810-710-610-510-410-3IDS [A/ m m] UFDG2.3: VDS = 50mV VDS = 1.0V VDS = 1.0V UFDG2.3 w/all QM shifts: VDS = 50mV 1.0

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248 Figure A.6(a)LeatthefrontandbacksurfaceforvariousVGSinADG nMOSFETof Fig.A.1.(b)IDS-VGSaftercorrectimplementationof the QM shift. -0.10-0.050.000.050.100.15VGS [V] 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Lef/b [nm] Lef, VDS = 50mV Leb Lef, VDS = 1.0V Leb 0.20 -0.2-0.10.00.10.20.30.40.50.60.70.80.9VGS [V] 10-1210-1110-1010-910-810-710-610-510-410-3IDS [A/ m m] UFDG2.3 w/Le fix: VDS = 50mV VDS = 1.0V VDS= 1.0V UFDG2.3 w/o Le fix: VDS = 50mV 1.0 (a) (b)

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249 UFDG.ThenewapproachcomputesIwkbyevaluatingQn"numerically",definingan averageLeovertheentirebody,andredefiningtheeffectivetransversefieldfor meff. FortheevaluationofQn,wepartitiontheUTBinto even numberof sections(m),whichrequiresoddnumber(m+1)ofnodes;usingevennumberof sectionsmaintainssymmetryalongthebody.Assumingthatthetransversefieldis constant in each individual section, Qn is given by ,(A.13) where Qni is (A.14) with (A.15) and .(A.16) Schematicrepresentationsofthismodelingapproachexemplifiedin Fig.A.7(a)and Fig.A.7(b) clearlyshowthekeydifferencesfromthatofYeh( Fig.A.3( a)-(b)) [Yeh96]andChong( Fig.A.4 (a)-(b))[Cho98].However,whereastheintegrationpath QnQni i1 = m= Qniq f d d x n f ()f df xiyminxi() , () f xi1 +yminxi1 +() , ()= x d d f f xi1 +yminxi1 +() , ()f xiyminxi() , () – tSim -------------------------------------------------------------------------------------------= n f () ni 2NB------q f kBT --------exp =

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250 Figure A.7SDGMOSFETwithfourUTB-partitions.(a)Integrationpathinx-y and(b)potentialprofilealongxusedtoevaluateQninEq.(A.13)-Eq. (A.16).Alsoshownin(b)istheapproximatedprofileforQninEq. (A.11)andEq.(A.12),anditisapparentthatitwillunderestimateQn. Thevolumeinversionin(b)isdueto2-Deffectsinthelow-doped UTB and not QM effects.(a) (b) y xx1x2ymin(f)ymin(b)xtSi0fApprox. f (x,ymin(x)) 0 LefftSi x3x4x5 # of node tSi/4 x1x2x3x4x5 Actual f (x,ymin(x)) Approx. f (x,ymin(x)) in UFDG2.3 # of node f (x,ymin(x)) Approx. f (x,ymin(x))

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251 inFig.A.7(a) reflectsaquasi-2-Dintegration,Eq.(A.15)isan1-Dintegration.The validityofEq.(A.15),despitethequasi-2-DnatureoftheintegralinEq.(A.14),can bearguedbasedontheassumptionthatthecarrierdiffusionoveralengthLedefines Iwk.Sincepredominantcarrierdiffusionmeansnegligiblechangeinthepotential(in they-direction),thepotentialovertheentirelengthofLeisapproximatelyconstant andequalsto f (x,ymin(x)).Hence,aslongasallymin(x)liewithinthe(average)Le, anypathcanbeusedfortheintegrationinEq.(A.14).Alongchanneldeviceisan exampleofthis.YetanotherargumentforEq.(A.16)isthatifxiandxi+1areclose enough,thenthevariationinthetwoyminwillbeverysmallsuchthatthequasi-2-D naturecanbeneglected.(WewilllatershowthatpartitioningtheUTBintofour sections suffices.) Next,comparingEq.(A.1)andEq.(A.2),andassumingaveragingof mnindependent of that for Le, we define ,(A.17) wheretheindexjdenotesthenumberofnode,andLe(x)isgivenbyEq.(A.6)-Eq. (A.8).NotethatunlikeLeforthecurrentpartitioningschemesin[Yeh96]and [Cho98],ournewschemecanbetteraccountfortheleakiestsource-to-drain conduction path being anywhere in the body. Finally,tocompleteournewIwk-formalism,weredefinetheeffective transversefield(eeff)governing meff.Although,thefunctionalformof meffisnot changed,eeff is now defined more physically as 1 Leavg ()---------------qnLsx () () Lex () -----------------------x d0 tSi QnLs() nLsxj() () Lexj() -----------------------j1 = m1 +Qn =

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252 ,(A.18) whereexi is the constant field in the ith section of the body. A.5 UFDG2.4 Theupgrades/refinementsdetailedinSec.A.4predominantlyconstitutea newversionofUFDG,UFDG2.4.Wenotethatbecauseoftherelativelysimple natureofthenewIwkformalism,UFDG2.4(internally)providesthefreedomto choosenumberofpartitionsoftheUTB(aswellasthecapabilitytoturnoffthe2D effects).TodeterminetheoptimumnumberofUTB-partitions,weplot,in Fig.A.8, UFDG2.4-predictedIwkforFD/SOInMOSFETwithnumberofUTB-partitions varyingfrom1to16;alsoshownaretheUFDG2.3predictions,whichare comparabletothatofUFDG2.4with1-2partitions.NotethatforUFDG2.4with morethan4partitions,thechangeinthepredictedIwkisnegligiblefromthatwith4 partitions.Hence,wesetthedefaultnumberofUTB-partitionsto4,whichtranslates into5nodes( Fig.A.7 (a)).WiththisdefaultvalueofUTB-partitions,wecompare,in TableA.1,theweak-inversioncomputationalburdeninUFDG2.3versusUFDG2.4. WhileUFDG2.4with5nodesrequirescomputingthreemoreLe’sandonemoreset ofstructuralparameters,itavoidscomputingonelesspotential,twoxmin’s,andtwo smoothingfunctionsforthetwoxmin’s.Hence,despitethenumericalevaluationof Qn, UFDG2.4 performs, with respect to run-time, just as efficiently as UFDG2.3.eeffexx () nLsx () () x d0 tSi QnLs() Qniexi i1 = mQn ==

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253 Figure A.8UFDG2.4-predictedIwkforLeff=32nmFD/SOInMOSFETwith UTB-partitionsvaryingfrom1to16,andtSi=7nm,toxf=1.1nm,toxb=200nm,gateworkfunctionof4.42V(4.6Vbeingmidgap),and substratedopingof1015cm-3.AlsoshownisIwkpredictedby UFDG2.3. 0.00.10.2VGfS [V] 10-810-710-610-510-4IDS [A/ m m] UFDG2.3 UFDG2.4 w/1 partition UFDG2.4 w/2 partitions UFDG2.4 w/4 partitions UFDG2.4 w/8 partitions UFDG2.4 w/16 partitions 0.3

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254 UFDG2.3UFDG2.4 Structure-dependent parameters45 Potentials65 Charges44 xmin20 Smoothing functions4 (2 at any given bias) 0 Le25 Table A.1 Predominant computations in Iwk formalism of UFDG2.3 and UFDG2.4.

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255 Finally,in Fig.A.9,we showUFDG2.4-andUFDG2.3-predictedIDS-VGSinSDGnMOSFET.WefindthatUFDG2.3noticeablyunderestimatesIwk,asclaimed earlierbasedon Fig.A.2. ThisisalsoverifiedbyreducingtheUTB-partitionstotwo in UFDG2.4, which is somewhat representative of the two partitions in UFDG2.3.

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256 Figure A.9UFDG2.4-andUFDG2.3-predictedIDS-VGSforLeff=28nmSDG nMOSFEThavingtSi=13.7nm,toxf=toxb=tox=1.1nmandmidgap gates. Also shown are UFDG2.4 predictions with 2 UTB-partitions. 0.00.10.20.30.40.50.60.70.80.91.0VGS [V] 10-1010-910-810-710-610-510-410-310-2IDS [A/ m m] UFDG2.4 UFDG2.3 UFDG2.4 w/2 partitions VDS = 1.0V VDS = 50mV

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257 APPENDIX B A NOVEL 2-D SPLINE FOR UFDG Anovelcubicsplineintermsofboththefront-andback-gatebiases,i.e., 2-D,for"linking"therigorousweak-andstrong-inversionformalismsinUFDGis described.Itdefinesthemoderate-inversionformalisminUFDG,replacingthe previous1-Dforth-ordersplineintermsofonlythefront-gatebias.Thisupgrade,in conjunctionwithanotheronebyZhang[W.Zhang,Ph.D.Dissertation,in preparation], renders UFDG a truly generic MOSFET model for DG MOSFETs. B.1 Moderate-Inversion Region Forphysics-basedcompactmodeling,theMOSFETIDS-VGScharacteristicsaregenerallypartitioned,asillustratedin Fig.B.1, intofourregions ofoperation:accumulation,weakinversion,moderateinversion,andstrong inversion.ThisdivisionoftheIDS-VGScharacteristicsfacilitatesdeveloping rigorousanalyticmodelsbasedonapproximationsvalidonlyinthegivenregion.For example,intheweak-inversionregion,i.e.,VGS VTS,allowsforsolvingthe1-DPoisson’sequationwiththefree-carrierterm.UFDG issucharegionalmodel,wheretheweak-andstrong-inversionconditionsare rigorouslymodeledasdescribedin[Yeh96],[Chi01a],and[Ge02a].Becauseboth

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258 Figure B.1PartitioningoftheMOSFETIDS-VGScharacteristicsintovarious regions of operation.VGSlog IDS VTW VTS Moderate Inversion Strong Inversion Weak Inversion Accumulation

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259 the2-Deffectsandtheinversioncarrierdensity(aswellasboththedriftand diffusioncurrentcomponents)areimportantinmoderateinversion,analytic modelingofthisregionisformidable.Hence,ingeneral,itismodeledviaeither smoothingfunction(s)orapolynomialsplinethatlinkstherigorousweak-and strong-inversionformalisms.(Weak-inversion-andaccumulation-regionformalisms aretypicallylinkedviasmoothingfunction(s).)Forapolynomialspline,oneneeds to know the boundaries VTW and VTS, and IDS and gm at those boundaries. ForclassicalCMOSdevices,Tsividisquantitativelycharacterizedthe notedboundariesofthemoderate-inversionregion[Tsi82]intermsoftheinversion capacitance(Ci),thedepletioncapacitance(Cd),andthegate-oxidecapacitance (Cox).Itisbasedonexponentialdependenceoftheinversionchargeonthegatebias fortheweak-inversionconditionandonthelineardependenceforthestronginversion condition. At VTW [Tsi82], ,(B.1) and at VTS, .(B.2) Thus, the moderate-inversion region is characterized by .(B.3) Thegatebiasatthecorrespondingboundary,i.e.,VTWandVTS,areimplicitinCi, andtypicallyrequiresiterative/numericalevaluation[Tsi82],[Yeh96].Wenotethat Ci0.1CoxCd+ () @ Ci10CoxCd+ () @ 0.1CoxCd+ () C
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260 forregionalmodelssuchasUFDG,VTWandVTSmustalwaysbeevaluatedinorder to determine which region of operation a given gate bias corresponds to. B.2 Existing 1-D Spline in UFDG and Its Deficiencies AlthoughtheanalysisbyTsividis[Tsi82]considersclassicalMOSFETs, itisalsoapplicabletononclassicalMOSFETswithCd CB(eff).Duetotherigorof thestrong-inversionformalisminUFDG,andtoavoidnumericalevaluation,VTW andVTSinUFDGaremodeled[Chi01a],[Ge02a]usingthecharge-coupling analysisofLimandFossum[Lim83]overviewedinChapter3.Anydiscrepancywith theinequalitiesinEq.(B.1)andEq.(B.2)isresolvedviatuningparametersWFACT andSFACTforVTWandVTS,respectively.Then,VGfS=VTWandVGfS=VTS, andIDSandgmatthesebiaspointsdefineaforth-orderpolynomialsplineinterms ofVGfS[Ge02a];VGbSiskeptattheuserspecifiedbias.Becausethesplinehasonly one variable VGfS, we characterize it as an 1-D spline. ThereasonwhythesplineinUFDGisdoneintermsofonlyVGfS,despite thepresenceofthebackgate,isthecharge-couplinganalysisin[Lim83]that assumesapredominantfrontchannel.This,however,limitsUFDGapplications, eventhoughtheweak-andstrong-inversionformalismsarevirtuallyindependentof thenotedassumption.Forexample,UFDGcannotbegenerallyappliedtoIG FinFETs,orMIGFETs,inwhichbothgatesarebiasindependently.Further,forsuch devices,UFDGassumesVTW(andVTS)proportionaltoVGbS,whichasweshowed inChapter3isnotvalidintheabsenceofapredominantfrontchannel.Hence,we applyourgeneralizedcharge-couplinganalysispresentedinChapter3togeneralize

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261 thenotedsplineviaincorporatingtheVGbS-dependence.Thisresultsina2-Dspline, i.e.,polynomialsplineintermsofbothVGfSandVGbS.Also,thetwoboundariesare now defined vectorially as VTW = (VTWf, VTWb) and VTS = (VTSf, VTSb). B.3 Concept of 2-D Spline SimilartoourgeneralizedVt-modelinChapter3,therearecountably infinitepairsofVGfSandVGbSthatsatisfyEq.(B.1),orEq.(B.2).And,fromEq. (3.12),theseinequalitiesintermsofCicanbetranslatedintoinversion-carrier density,i.e.,theweak-inversionconditionforNinvNinvS,andthemoderate-inversionconditionforNinvW
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262 Figure B.2Mechanicsofournovel2-Dspline.Foraspecificbiaspoint(Vf,Vb),the originisfirstshiftedto(Vf,Vb),and VTW and VTS aredeterminedasthe interceptsofaline45ofromthenewVGfS-axisandtheNinvWandNinvScontour, respectively. VGfSVGbS NinvWNinvS Strong Inversion Weak Inversion (Vf,Vb) VGbS’ VGfS’ (VTSf, VTSb) (VTWf, VTWb) 45o rtwrts

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263 VTW (rtw),and/orthedistancebetweentheneworiginand VTS (rts)canbe negative,which,however,simplycorrespondstoanactualangleof(45o+180o).For example, rtw inFig. B.2 is negative. Finally,ifweletVTWb=VTSb=Vb,i.e.,splinehorizontallywith45o 0o,wegettheexisting1-DsplineintermsofVGfSinUFDG,asindicatedin Fig.B.2. Note,however,thatbecauseofthenonphysicalVGbSdependenceofVTWandVTS intheexisting1-Dsplinemodel,VTWandVTSmaynotlieontheNinvWandNinvScontours, as exemplified in Fig. B.3. B.4 VTW and VTS Inthissection,wedescribeamodelforevaluating VTW and VTS in accordwithourschemedepictedin Fig.B.2.We firstdescribethemodelforintrinsicbodyDGMOSFETs,andthenextendittothedoped-bodycase.However,inorder tomodel VTW and VTS ,weneedtoknowNinvWandNinvS,respectively,andhence, we begin by characterizing these inversion carrier densities. B.4.1 Inversion-Carrier Density at VTW and VTS For NinvW, Eq (3.12) gives Ci as .(B.4) CombiningEq.(B.4)andEq.(B.1),latterwithCox CoxfandCd CB(eff),defines NinvW.However,sincetheproportionalityconstantinEq.(B.1)isanapproximation CidQid f -------- – dQid fsf--------- – q kBT --------Qi==

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264 VGfSVGbS NinvWNinvSVTWf VTSf Strong Inversion Weak Inversion (Vf,Vb) Figure B.3Theexisting1-Dsplineasasubsetofournew2-Dsplineillustrated inFig. B.2.

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265 andCB(eff)
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266 whereExC,giveninEq(3.8),isthespatiallyconstantfieldthatcharacterizesthe linearpotentialdistribution.Asnotedearlier,whileEq.(B.6)isaccuratefor VTW , it is a rough approximation for VTS . Now,tocalculate VTW ( VTS )forabiaspoint(Vf,Vb),wefirstdetermine fsfcorrespondingtoNinvW(NinvS)usingEq.(B.6),withExCgivenbyEq.(3.8)for (Vf,Vb).Wethensubstituteitand fsb= fsf-ExC tSiinEq.(B.5),withVGfS=VTWf (VTSf)andVGbS=VTWb(VTSb).(WenotethatalthoughExC,ingeneral,should bedeterminedusing(VTWf(VTSf),VTWb(VTSb)),requiringiterative/numerical evaluationfortheboundaries,wefindthatforourpurposehereExCfrom(Vf,Vb)is sufficient.) Next, in order to be consistent with our scheme in Fig. B.2, we define (B.7) ,(B.8) wherertw/sisasdefinedearlierandshowninFig.B.2.Finally,solvingforrtw/sin Eq. (B.5), we get, for a specific bias point (Vf, Vb), ,(B.9) whichviaEq.(B.7)andEq.(B.8)renders VTW and VTS consistentwithourscheme in Fig. B.2. ToaccountfortheQMeffectsandresolveanydiscrepancycausedviaour simplifying assumptions, we let (B.10) VTW/SfVfrtws p 4 () cos + VTW/SbVbrtws p 4 () sin + rtws CoxffsfVFBfVf– + () CoxbfsfExCtSi–VFBbVb– + () qNinvWS ++ Coxfp 4 () cosCoxbp 4 () sin + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------= fsffsf0Dfsf QMWFACTkBTq () – + =

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267 for VTW , and (B.11) for VTS ,inEq.(B.9),where fsf0isthatdeterminedfromEq.(B.6), Dfsf QMisthe potentialshiftduetoQMeffects,andWFACTandSFACTaretheuser-defined tuning parameters; SCEs are neglected. B.4.3 Extension to Heavily-Doped Bodies Whenthebodyisheavilydoped,ourmodelinEq.(B.7)-Eq.(B.11)is insufficient.Toextendittoaccountforthebodydoping,weneedtoaddthedepletion charge,QB=-qNBtSitotheleft-handsideofEq.(B.5),whichderivesfromthe Gauss’slawinthebody.Next,wenotethatwithsignificantNB,thepotential distributionisnolongerlinear,butparabolic,invalidatingEq.(B.6).However, becauseEq.(B.6)isbeingusedtoestimate fsfat VTW (or VTS ),westilluseitfor doped-body device but with ,(B.12) where fF is the Fermi potential, and (B.13) where, from the solution of the 1-D Poisson’s equation with high NB, fsffsf0Dfsf QMSFACTkBTq () ++ = fsffsffF– ExCExwk

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268 (B.14) such that .(B.15) Wenotethatanyerrorintheresulting fsfcanberesolvedviathetuningparameters WFACTandSFACTinEq.(B.10)andEq.(B.11),respectively.Then,Eq.(B.9) becomes ,(B.16) where the channel-charge density at VTW/S (NchW/S) is ,(B.17) and fsfisasinEq.(B.10),orEq.(B.11),with fsf0determinedfromEq.(B.6)with the noted changes. B.5 2-D Cubic Spline Withourmodelforrtw/sinEq.(B.9)definingthemoderate-inversion regionboundaries,wenowconcludeournovelformalismbydefininga2-Dthirdorder,orcubic,spline.Asnotedearlier,becausewefixtheangleof"spline"inFig. B.2 to 45o, rtw/s can be negative. Then, from Fig. B.2, we find that ExwkExCqNBtSi2 eSi---------------1 Coxf---------1 Coxb----------- – 1 Coxf---------1 Coxb----------1 CSi------++ ------------------------------------------- – = fsffsb–ExwktSi= rtws CoxffsfVFBfVf– + () CoxbfsfExwktSi–VFBbVb– + () qNchWS ++ Coxfp 4 () cosCoxbp 4 () sin + -----------------------------------------------------------------------------------------------------------------------------------------------------------------------= qNchWS qNinvWS qNBtSi+ =

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269 (B.18) characterizes the weak-inversion condition, for which the spline is not used, (B.19) characterizes the strong-inversion condition, for which the spline is not used, and (B.20) characterizes the moderate-inversion condition, for which UFDG used the spline. Tolinktherigorousweak-andstrong-inversionformalismsinmoderate inversion,wedefineacubicsplineintermsofbothVGfSandVGbS,i.e.,2-Dspline. For the drain-source current, our 2-D cubic spline is given by ,(B.21) whereristhedistancebetweentheoriginoftheVGfS’-VGbS’axesinFig.B.2and (Vf,Vb).Sincethenotedoriginisalwayschosentobeat(Vf,Vb),r=0alwaysin Eq.(B.21),exceptwhenevaluatingthecoefficients bi,i=0,1,2,3. These coefficients are determined via IDS and gm at VTW and VTS : ,(B.22) ,(B.23) rtw0 rts0 rtw0rts<< IDSVfVb, () ln b0b1rrtw– ()b2rrtw– ()2b3rrtw– ()3+++ = b0IDSrrtw= () lnIDSVGfSVTWf =VGbSVTWb = , () ln == b1r d d IDSlnrrtw=IDSrrtw= () lnIDSrrtwdr – = () ln – dr --------------------------------------------------------------------------------------==

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270 ,(B.24) ,(B.25) where ,(B.26) ,(B.27) ,(B.28) and ,(B.29) .(B.30) WenotethatinEq.(B.24)andEq.(B.25),(rts-rtw)>0always.Also,drinEq.(B.28) ischosentorendertheapproximationofthederivatives(andessentiallyofgm)inEq. (B.23) and Eq. (B.27) reasonable. Finally,wedefinesimilar2-Dsplineforallthe(terminal-)charge components in UFDG. b23rtsrtw– ()2A b0– () rtsrtw– ()3B2 b1+ () – rtsrtw– ()4---------------------------------------------------------------------------------------------------------= b32rtsrtw– () A b0– () rtsrtw– ()2B b1+ () – rtsrtw– ()4---------------------------------------------------------------------------------------------------- – = AIDSrrts= () ln = B r d d IDSlnrrts=IDSrrtsdr + = () lnIDSrrts= () ln – dr ------------------------------------------------------------------------------------== dr 0.1mV p 4 () cos ----------------------= IDSrrtwdr – = () lnIDSVTWfdr p 4 -cos –VTWbdr p 4 -sin – , ln = IDSrrtsdr + = () lnIDSVTSfdr p 4 -cos +VTWbdr p 4 -sin + , ln =

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271 APPENDIX C CONVERGENCEOFTHE2-DQMSOLUTIONTOTHE3-DCONTINUUMCASE We mathematically prove here that Qi QM in Eq. (3.19), i.e., ,(C.1) converges to the 3-D continuum case in Eq. (3.9), i.e., ,(C.2) atlowfieldsandthicktSi.SinceourmodelforthequantizedenergiesinEq.(3.27) isanalogoustothatforclassicaldeviceswhentSiisverythickandExCisvaried,this convergencewithExC 0canbeprovedasin[Tau98].Hence,weonlyshowhere the proof for ExC = 0, as in SDG MOSFETs, and tSi . Because we assume ExC = 0, Qi CL in Eq. (C.2) becomes ,(C.3) and Ej (and similarly Ej’) in Eq. (3.27) simplifies to .(C.4) Next, we note that Qi QMqnikBT Nc----------------- – q fsfkBT --------exp gmdp h 2---------Ej– kBT --------expj0 =g md p h 2------------Ej – kBT ---------expj0 =+ = Qi CLnikBT ExC-------------- – q fsfkBT --------1 qExCtSikBT ------------------ – exp – exp = Qi CLqni– q fsfkBT --------exptSi= Ejj1 + ()2h 2p22mxtSi 2---------------=

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272 (C.5) for De 0.Then,thefirstsuminEq.(C.1)(andsimilarlythesecondsum),withtSi in Eq. (C.4) and (C.6) in Eq. (C.5), converges to .(C.7) Finally, with [Tau98] ,(C.8) we get .(C.9) FromFig.3.6,wenotethattSi astSibecomesthickerthan20nm,andhence,we characterize UTB as a body with tSi < 20nm. en De ()2–Den0 = ee2–e d0 p 2 ------= De h p 2mxtSi-------------------1 kBT ------------= Ej– kBT --------expj0 =p 2 -----kBT 2mxh p -------------tSi Nc2gg + () mxmymzkBT 2 p h 2-----------32 /= Qi QMqni– q f sfkBT --------exptSiQi CL==

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273 REFERENCES [45nm04] 45nmCMOSTechnology ,Shortcoursepresentedat2004IEEEIEDM. [All02]L.P.Allen,S.Caliendo,N.Hofmeester,E.Harrington,M.Walsh,M. Tabat,T.G.Tetreault,E.Degenkolb,andC.Santeufemio,“SOI uniformityandsurfacesmoothnessprocessing,” Proc.IEEEInternat. SOI Conf. , pp. 192-193, Oct. 2002. [Aro82]N.D.Arora,J.R.Hauser,andD.J.Roulston,"Electronandhole mobilitiesinsiliconasafunctionofconcentrationandtemperature," IEEE Trans. Electron Devices , vol. ED-29, pp. 292-295, Feb. 1982. [Ass00]F.Assad,Z.Ren,D.Vasileska,S.Datta,M.S.Lundstrom,"Onthe performancelimitsforSiMOSFET’s:Atheoraticalstudy," IEEE Trans. Electron Devices , vol. 47, pp. 232-240, Jan. 2000. [Bal03]S.Balasubramanian,L.Chang,B.Nikolic,andT.-J.King,“Circuitperformanceimplicationsfordouble-gateMOSFETscalingbelow 25nm,” Proc.SiliconNanoelectronicsWorkshop ,June2003,pp.1617. [Bal87]F.Balestra,S.Cristoloveanu,M.Benachir,J.Brini,andT.Elewa, “Double-gatesilicon-on-insulatortransistorwithvolumeinversion:A newdevicewithgreatlyenhancedperformance,” IEEEElectron Device Lett, pp. 410-412, Sept. 1987. [Bar50]J.BardeenandW.Shockley,"Deformationpotentialsandmobilitiesin non-polar crystals," Phys. Rev. , vol. 80, pp. 72-80, Oct. 1950. [Bed78]D.BednarczykandJ.Bednarczyk,Phys.Lett.,vol.64A,p.409,1978. [Boe01]F.Boeuf,T.Skotnicki,S.Monfray,C.Julien,D.Dutartre,J.Martins, P.Mazoyer,R.Palla,B.Tavel,P.Ribot,E.Sondergard,andM. Sanquer,nmplanarNMOSFETmanufacturablewithinstate-ofthe-artCMOSprocessthankstospecificdesignandoptimisation,” IEEE IEDM Tech. Dig ., Dec. 2001, pp. 637-640.

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286 BIOGRAPHICAL SKETCH VishalP.TrivediwasbornonMarch12,1981,inAhemdabad,India.He cametoJacksonville,FL(USA)in1993,wherehereceivedhismiddle-andhighschooleducation.HereceivedtheB.S.andM.S.degreesinelectricalengineering fromtheUniversityofFlorida(UF;Gainesville,FL)inDecember2001andAugust 2002,respectively.HeiscurrentlypursuingaPh.D.degreeinelectricalengineering at UF. Inthesummerof2001,hewasaninternatHewlett-Packard’sSystems& VLSITechnologyOperationsResearchandDevelopmentsiteinFortCollins,CO. HeworkedonL3-cachecharacterizationfortheItanium2processor,alsoknowasthe McKinleyprocessor.Healsodidsomepreliminaryinvestigationofroutingschemes forafuture-generationItaniumprocessor.Intheyear2004-2005,Trivediwas awardedthePittmanFellowshipfromtheDepartmentofElectricalandComputer EngineeringatUF.Hehasalsoparticipatedasareviewerforseveralinternational journals in his field. Hisresearchinterestisinthefieldofnanoelectronics,inparticular, nanoelectronicdevicephysics,design,andmodeling.Currently,hisresearchfocuses onanalysis,design,andphysicalmodelingofnonclassicalnanoscaleCMOSdevices, suchasfullydepletedsingle-andmulti-gateMOSFETs.Upongraduation,hewill jointheCMOSDevelopmentGroupatFreescaleSemiconductor,Austin,TX,asa device/integration engineer working on 65nm SOI CMOS and beyond.