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# SOLID EE 203

UCR

GPA 3.92

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This 51 page Class Notes was uploaded by Ana Hilpert on Thursday October 29, 2015. The Class Notes belongs to EE 203 at University of California Riverside taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/231767/ee-203-university-of-california-riverside in Electrical Engineering at University of California Riverside.

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Date Created: 10/29/15

nMOSFET Schematic Drain Contact Source contact Field oxide FOX ptype silicon substrate 2 Vbv El Four structural masks Field Gate Contact Metal El Reverse doping polarities for pMOSFET in Nwell i I nMOSFET Schematic polysilicon gate oxide gate Vg Vds nJr source depletion region inversion channel ptype substrate l39Vias Source terminal Ground potential Gate voltage Vg Drain voltage VdS Substrate bias voltage Vbs gt lX y Band bending at any point xy gt Vy QuasiFermi potential along the channel gtVy0 0 VyL Vds Drain Current Model 2 i V kT eqw N a 1 2 Electron concentration nxy Electric field 2 dry 2 ZkTN WW qw nz W W W 2 1 4 1 W 1 5 x y dx 8st e kT Naz e e kT Condition for surface inversion w0y Vy21B Maximum depletion layer width at inversion 28siVy 2913 Gradual Channel Approximation Assumes that vertical field is stronger than lateral field in the channel region thus 2D Poisson s eq can be solved in terms of 1D vertical slices Current density eq both drift and diffusion Jnxy qlunnx Integrate in X and zdirections dV dV 1 W W V dsy e e Q1 where Qiy qjoxinxydx is the inversion chargearea Current continuity requires Ids independent of y integration with respect to yfrom O to L yields Ids rte7 gJOVdX Q1VdV PaoSah s Double Integral Change variable from Xy to WV quot392 qw VkT 71XaynlVN e v13 dx 1 n2 N qu VW V V d a 039 Q gt qle no gt01 w ILB aw v Substituting into the current expression W m 11 n2 N eWVW I d d d mg7 LJ0 5WV W V where WSW is solved by the gate voltage eq for a vertical slice of the MOSFET 12 VzgsikTNa rqlG quot1392 eqwSJykT l C t Q3 V V V g fb V s Cm fb V s kT Nf l 0X Charge Sheet Approximation Assumes that all the inversion charges are located at the silicon surface like a sheet of charge and that there is no potential drop across the inversion layer After the onset of inversion the surface potential is pinned at 9 213 VY Depletion charge Qd qNade 2 1283ina 2913 V Total charge Q CaCVg Vfb 14 CoxVg Vfb 2vB V lnv charge Q1 2 Q5 Qd C0xVg V 21B V128sina21IB V W Vdr Substituting In Ids rte7f QlVdV and integrate 0 W V 2 28sina 1d e Cax fVg Vfb 2 ijdg 3C2vB Vds ZrB W1 Linear Region IV Characteristics For VdS ltlt Vg W V48sinaW W Ids lue rcox Vfb 2W3 1 3de 9sz Il Vds 0x 1i4 N where V V 2VIB is the MOSFET threshold voltage 0x 1E0 08 I I 132 A a E 06 E 1E4i E a E g g 04 5 A 1E 6E s 3 02 E 1E 8 39 q 1E10 g I 39 I 0 0 05 1 15 2 25 3 Gate Voltage VgV Von Vt Saturation Region IV Characteristics Keeping the 2nd order terms in Vds Ids ye CoxIg VtVdS Vd52 J N 4 where mlw1ffdm1Ii IS the bodyeffect coeffICIent ax 0x dm Vdmz Idml l 1 I Vg4 W Vg V2 f2m Ids dsat e cox when V Vdsat Vg Vtm Dram Current ds I I I I I I x I Vgl X Drain Voltage Typically m z 12 Pinchoff Condition From inversion charge density point of view QKV 40ng Vt WV while Ids ye ngdJ QiVdV 0 am At VdS Vdsat Vg Vtm Q O and Ids max CoxUgVt Source Drain 0 Vi VM V Vg Pinchoff from Potential Point of View V V V V 2 V V Vyg t g t 21 g t VdslVdsz m m L m L my 39 tthe pinchoff pomt VgVt E L gt Gradual channel E Vx m IQImc of 1 N apprOXImatlon breaks 1 39 down I I Current IS Injected Into the bulk depletion Vo V region at 0 0 L yr Source Drain Beyond Pinchoff vxgtv v3 small Dcplclion mgmn m I Subthreshold Region V 7 V 7 E t V015 e I m I I I I I I I I I I I I Saturatron I I reg10n Sub I I threshold I I I re ion I I g I I I I I I I I L1near I I I regron I I I II V Vt V Drain Current arbitrary units 1E0 lEl 1E2 1E3 1E4 1E5 1E6 1E7 1E8 0 Component Diffusion Component 1 Gate Voltage V 15 Subthreshold Currents 2 12 QS 255155 ngikTNa MJrLzequxrmkr kT Na Power series expansion 1st term Qd 2nd term Q 2 Q kT eqw ykr I 2 q Na 2 2 W gsina I ll d 3 It 2 r wt a 2 W kT m or ds ue C m eqVg Vt kT1equJkT q oxL Inverse subthreshold slope 71 S d1 g1ds 23 39 kT 23k T 1 dVg q q C ox Body Effect Dependence of Threshold Voltage on Substrate Bias If vbs 0 W V 228qua Ids Iue COx Vg Vfb 2W3 A Vds ms Vds32 Ibs32 L 2 3C U 18 A 16 E Vzgsina2lBVbs S KZVfb2VlBC 0x 33147 E U g 12 I Na3gtlt1015 cm 3 1 d I gsina 1711 5 7 dVbS Cox 08 7 06 Substrate Bias Voltage Vbs V Dependence of Threshold Voltage on Temperature For n poly gated nMOSFET Vfb Eg2q JB E 1 48N gt IIg B szq allB 261W C ox WI 1 ZEng VasinaV BW B 1 dEglt2 1de m dT 2g dT L Cox J dT 2g dT dT gt d 2m 1 3 m 1 CHE g d Na 2 dT From Table 21 dEgdT 27gtlt10 4 eVK and NCNV 2 z 24gtlt1019 cm3 For Na 1016 cm 3 and m 11 dVIIdT is typically 1 mVK MOSFET Channel Mobility l0 ynnltxgtdx My 2 xi jo nxdx It was empirically found that when ue is plotted against an effective normal field ge there exists a universal relationship independent of the substrate bias doping concentration and gate oxide thickness Sabnis and Clemens 1979 1 1 Here ge QdEQi SinceQd 48sCINaVB C0xVt Vb 2VBand lQil N C0XVg Vt szr gt geff 3er 6t w 3t 6t 0x 0x For n poly gated nMOSFET 36f M17 cmZVs 103 Nchannel MOSFET Mobility I I l I Nbcrn 3 o 39x1015 0 20x1016 El 72gtlt10m I 30x10397 A 77x10 A 24x1018 U l a El If 5233 0600 390 U I 0Q g Z bi bi l W 300 K DD mac 13 Effective eld 917 MVcm Low field region low electron density Limited by impurity or Coulomb scattering screened at high electron densities Intermediate field region Limited by phonon sca enng ue z 32500gtlt513 High field region gt 1 MVcm Limited by surface roughness scattering less temp dependence Drain Current Arpm Temperature Dependence of MOSFET Current 5E4 4E4 2E 4 OEO 4quot u I O 05 I I I 1 1 Gate Voltage V awv5 Pchannel MOSFET Mobility Nb cnf 1 1 5WZEiQdi lQi i 01 rmch eld ad MVcm In general pMOSFET mobility does not exhibit as universal behavior as nMOSFET Electron and Hole Mobilities vs Field 1000 03 A tOX35 tOX7O A 500 39 Electron quotquotquotquotquot 7 300 2 E 200 3 is 100 39 0 1 50 I l 1 MIT CMOS I01 meCMOS I 30 39 39 39 01 02 03 05 1 2 3 Eeff MVcm Intrinsic MOSFET Capacitance 1 El Subthreshold region Cg WLLCLCLJ z WLCd 0x 1 El Linear region Cg WLCox El Saturation region y Vyii QiyC0xVgVt 1 L Vgth L m ii39s39 39v 2 iQimCa Ei I gt C WLC 4 3 I Vltygt 0 Vi gt 0 L y Drain Source Inversion Layer Capacitance Gate 0 Vg QS 0x inversion d l i Qd Qz39 low freq I T V Ptype n substrate channel In the chargesheet model C 00 and Q COXVg Vt In reality inversion layer has a finite thickness and finite capacitance CoxCi N C 1 dV Cox C Cd 0x 1C Cox 3 Inversion Charge Density Qt pCcmz 08 06 04 02 Inversion Layer Capacitance 1st order approximation C N Q2kTq and Qi z COXVg Vt therefore CCOX Vg Vt2kTq r Qz CoxlVg 4071 2kT 1 Na5x1016 cm 3 105100 A Vb0 l 05 1 15 2 25 3 Gate Voltage Vg V W Vgti 2kT l Note Linearly extrapolated threshold voltage is typically 24kTq higher than the threshold voltage Vt at lSan 21B Short Channel Effect WV K2 IfL 1 1t C g T ds dsa Iueff oxL 2m And CgWLC0x But 1E gatecontrolled barrier 29k drain I w FE L marmme V mamm wrug th r Pm mmcg Ur rmg m f lm SourceDrain Current Alum 02 04 06 08 Gate Voltage V 39 39 0 12 P 05 00 SourceDrain Current mApm N ShortChannel Vt Rolloff W I I I 9 nMOSFET 4 08 6 no A g I DA g 06 o A E 0 4 A o Linear threshold IgsOJV 1 quotquot S A A Saturation threshold 13 3V E VI0V 4 bJ l 0 0 1 2 3 4 Leff Hm a 10 I a pMOSFET i 08 0 00 53 Q g 706 ooZg I U 6 0 4 A 0 Linear threshold Igs01V n E A A Saturation threshold Zis3V 1 02 KUZOV 0 I I I 2D Potential Contours Same gate voltage Gate L W0 05 pm Lona channel Short channel Surface potential DrainInduced Barrier Lowering 1E1 1E0 Curve A E L6m5 pun E 1132 vir 05 v A 1133 7 Cum C L 1125 11m r 1 1E5 1E6 1E7 yL 1E8quotquotquot 19 05 0 Drain current A cm 1 5 kTq l l I I I I I I I n 02 03 05 1 Gate voltage V lsvl Vcm ital rvm Lateral Field Penetration L2un mun 2D Poisson s Eq agx gy qzva 5x y 8 8 850 EX0 X gate controlled depletion charge 85550 V SD controlled depletion charge Note that the characteristic length of exponential decay is independent of channel length 2D Analysis in a Simplified MOSF ET Geometry i N V Substrate A 2D boundaryvalue problem with Poisson s equation Gate Source 40x np01y Drain T L K T ix Na In AFGH 6 01 0 xwfdgv B C D R X oxide 5x2 yz V In ABEF 0 52w qNa Substrate X silicon 5x2 ayz gm 11gt Tj r11 Boundary conditions To eliminate the boundary condition at the Sioxide interface the oxide 39r 3 ox7y Z Vg Vfb along GH region is replaced by an equivalent V x0 2 Wm along AB Si region gsgun z 3tOX thick W W Vds along EF llWdiy 0 along CD General approach to a 2D boundary value problem Gate Sorce Kim 77 p 1y A Drin G y F I e Let Way may umcy uROcy uBOcy vXy is a solution to the inhomogeneous equation and satisfies the top boundary condition uL uR uB are solutions to the homogeneous equation such that 1Xy satisfies the other BC s Gate np01y Drain Substrate Source 39tox For satisfying the Boundary conditions A Smh m Wd 3tox mh n7239x 3t mzL W 310x n71 uLxy 2b 7 sinh sinh My Ha zici Wd3tox Sin n7rx3tox quot 811 mzL Wd3rox Note that for usm kx Wd3 ox d2udx2k2u Smhmrx3tm And that for usi nhky1 L d2udy2k2u uBxy 2d Sinhn7rL3t0x L Shortchannel threshold rolloff V MOSFET Scale Length 24t IzLZ AV 2 Wd l WinW131 Vdse WWW 08 06 04 02 O O l i 2 Lmin 3on Define scale length 25 tax To keep shortchannel effect under control me should be kept larger than about 21 m MamS 1 3tOXde Maximum Depletion Width pm Depletion Width Scaling W 0 4gsikT1nNa nl dm 2 V q Na l 0 001 39 10E14 10E15 10E16 10E17 10E18 10E19 Substrate Doping Concentration cm 3 Generalized Scale Length In the oneregion model the eigenvalues are k L a a quot Wd 31 Source 5 v For two regions assume eigenvalues k l n in L D Frank etal EDL 1098 L 1 u11xybm Slnh7r 1 Vsm n 80 at X 0 uL1 uL2 duL1dy duL2dy w W umxybnz sinhsm x d and glduHdx 58duL2dx E Itwanvf39tIJXn gtan7r Wdiln 0 h S S Normalized Gate Insulator Thicknes 1 0 oo 0 on 04 02 Generalized Scale Length Lowest eigenvalue asjltan7ztiit1 qtanm VIM11 0 SiSSF 13 SiOZ 3 10 02 04 06 08 Normalized Si Depletion Depth Wdxl1 A jSCEoceXp 7zL2t1 Lmin 211 Note that 21gtWd and 21gtt 12Wd 2t is always a point of symmetry regardless of 5 58 I 8igsh 31Wd ti MOSFET Body Effect Depletion boundary Gate 85 IEOX 3 m AVgAys 1 3tOXde Gate Oxide Thickness nm MOSFET Design Space with SiO2 A o 00 Si02 Wd 1 Otozg 3 5 1o 15 20 Depletion Depth in Si nm In the intercept region 2L W 3tOX is a good approximation To obtain a good subthreshold slope the bodyeffect coefficient m AvgAw 1 3tOXde should be kept close to unity I Lmin 15 20tOX L In samp 0 0 0 0 y N h 05 CD 0 Normalized Gate Insulator Thicknes Highk Gate Insulator 88 Highk gate insulator I SI is an active area of 7 1 13 83902 Si research because 3 it may replace SiO2 10 thereby circumventing the a tunneling problem 7 But 3 Wd gsQM is valid only when tltlt A i i i i i i i i 02 04 06 08 1 Normalized Si Depletion Depth Wdk O In general requires lt regardless of Ids A 0016 0012 0008 0004 0000 0 Velocity Saturation Long Channel Behavior 025 pm nMOSFET V 25V 20V Vds Because of velocity saturation the saturation of drain current in a short channel device occurs at a much lower voltage than Vdsat Vg Vtm for long channel devices This causes the saturation current Idsat to deviate from the oc Vg Vt2 behavior and from the 1L dependence VelocityField Relationship Il leffg At low fields v ue g Ohm s law v I AS 5 00 V Vsat Iue gc39 Critical Field 5 e It is commonly believed that n 2 for electrons n 1 for holes vsat is independent of ue vertical field but so depends on 617 Only the n 1 case can be solved analytically Analytical Solution for n1 MAW01y 1 W VmdV dy Current continuity requires that Ids be a constant independent of y I 01V 1118 S gt ZZ wMZV jijzgj sat ds WQV WQV Multiplying dy on both sides and integrating from y O to L and from V O to Vds one solves for Ids ue W of QiltVdV L ds 1y Vgv sat Chargesheet model QiV C0xVg K mV WCMW LgtltVg Vans m 2W 1ye VdSv L Therefore I d5 sat Saturation Drain Voltage and Current The saturation voltage Vdsat can 2V Vm be found by solvmg dIdSdVds 0 g V 5 1 J1 2Vg Vmv satL V V sze wg KgtltmvmL 1 g t 39 39 I C Wv And the saturation current IS m ox J12WVgKWWL1 06 v t0x100A 0 U1 I 0 J I Dashed longch model solid velocity sat model Drain saturation current mAum Gate Overdrive Vg V V VelocitySaturationLimited Current At the drain end of the channel when VdS Vdsat Qiy C0xVg It desat and Idsat statQy L ie carriers move at the saturation velocity This implies that dVdy gt 00 at the drain Therefore the gradual channel approximation breaks down and the carriers are no longer confined to the surface channel J1 2wng VM mvmzL 1 I C W V V dsal 0x vsal g 11 Zlue Vg mvaL 1 When Vg Vt ltlt mvsatLZye In the limit of L gt 0 W V V2 Idmt Ue CongZ nlt dsat CovasatVg Long channel limit Velocity saturation limited current Velocity Overshoot Monte Carlo Simulation Velocity saturation is derived from the drift and diffusion model which assumes that carriers are always in thermal equilibrium with the silicon lattice But if the MOSFET is only a few mean free path 10 nm long carriers do not travel enough distance to establish equilibrium gt velocity overshoot ie carrier velocity at the high field region near the drain can exceed the saturation velocity Velocity Overshoot A 25x1o7 gt nFEl39 pFE39I39 001 I 002 003 004 005 Position Along Channel pm MonteCarlo simulation Even in a 30 nm device nFETpFET velocity and therefore current ratio is still z 2 because of the difference in effective masses gt The velocity at the source does not greatly exceed 107 cms therefore current does not greatly exceed Idsa 2 Covang K Distribution Function FermiDirac distribution under equilibrium fE 2 1 eE EfkT The standard semiclassical transport theory is based on the Boltzmann transport equation BTE v W f w ZSltk39kfrk39t1 frkt Sltkk39frkt1 frk39tl where r is the position k is the momentum frkt is the distribution function v is the group velocity 5 is the electric field Skk is the transition probability between the momentum states k and k The summation on the right hand side is the collison term which accounts for all the scattering events The terms on the left hand side indicate respectively the dependence of the distribution function on time space explicitly related to velocity and momentum explicitly related to electric field Velocities at the Source and at the Drain Ids WQiV Ids WCOXVg 39 VtVs Source Ids WCOXVg 39 Vt 39 VdsatVd Drain r El Inversion charge density at the source is given by CoxVgVt El Inversion charge density at the drain is much lower because of the drain bias El Current continuity is maintained consistent with band bending Scattering theory Source At high drain bias T O Ids W Let rns39ns the backscattering coefficient Then l r Ids W C0xVg IIVTEJ Ref Lundstrom EDL p361 1997 time that 39i tte39pemts cm the 39raootir rdl ddlit y ween thescans In the ballistic limit no collisions in the channel ie r 0 and Ids C0xVg VtvT Carrier Thermal Injection Velocity For 2D nondegenerate carriers f ENEfEdE 7 gt J f NE f EdE 39 7 rms m For unidirectional injection fvx eXp mvx2 2kTdvx 2kT VT 2 2 2 f eXp mvx 2kTdvx 72m Injection Velocity in the Degenerate Case At 0 K all states below the Fermi energy are filled In 2D define a Fermi Circle with velocity VF Vdexdvy j v gt0 4 ltvxgt x VF Hdvxdvy 372 r vxgt0 1 m 1 2 C0xV Vt Since 2 F W 2 F S q 4h 2CoxVgK vT 3m q7z

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