WCM-MSM007 Workshop on Compact Modeling 10th International Conference on Modeling and Simulation of Microsystems Santa Clara, California, USA Unified Compact Model for Generic Double-Gate MOSFETs Xing Zhou 1, Guan Huei See 1, Guojun Zhu 1, Karthik Chandrasekaran, Zhaomin Zhu 1,3, Subhash Rustagi,3, Shihuan Lin 1, Chengqing Wei 1, and Guan Hui Lim 1 1 Nanyang Technological University, Singapore Advanced Micro Devices, USA 3 Institute of Microelectronics, Singapore (exzhou@ntu.edu.sg May, 007 X. ZHOU 1 007
Outline Generic Poisson solution and voltage equation for DG Motivation: Unified Regional Modeling (URM Unified s solutions in s-dg scalable with N A and T Si Correct voltage-equation s solution in undoped s-dg Summary and conclusions X. ZHOU 007
The Generic Double-Gate Structure K ox1 X j V s (X o K ox T Si V g1 0 L V c V r V c s1 x V g o NA ND s T ox1 T ox y V d Zero-field potential: o [ o' (X o = 0] Imref split: V cr = Fn Fp V cr = V cb (bulk; V cr = V cs (DG Symmetric/common-DG V g1 = V g =V g : two gates with one bias All 1 = : s-dg (X o = T Si / Full-depletion: V FD = V g (X d =T Si / Some 1 = : ca-dg (X o < T Si Asymmetric/independent-DG V g1 V g : ia-dg, biased independently Zero-field location may be outside body Consider two independent gates; linked through full-depletion condition: X d1 + X d = T Si Unification of MOS ia-dg ca-dg s-dg bulk X. ZHOU 3 007
Poisson Equation and First Integral Poisson equation (First integral d dex de = = E x dx d d Field equation Es Eo qn A = F s ε Si x d dx ρ ( + ( n p N N D A = = = + A εsi εsi εsi qn A ( Fp Vcr vth v ( th Fp + Vcr vth qn A = e e + 1 e G, V ε ε Si ( vth Fn n = ne i ( Fp p = ne i v th q p n N N N D = n = ne A i i N = p = ne ' ( 0, y s( y, Ex( 0, y Es( y = s( 0, y ' ( x, y ( y, E ( x, y E ( y = ( x, y o o x o o o o Fn Fp vth v th q cr Fn Si V = Fp F Fp D ( cr ( Imref split Non-equilibrium ' Eo =, 0 o xo y x o : Zero-field location 1 s Fs( s, o, Vcr = sgn ( s G(, Vcr d sgn ( s f ( s, o, Vcr = sgn ( ϕs vth fϕ ( ϕs, ϕo, v cr o ( s vth o v th Fp + Vcr vth s vth o vth = sgn ( s { vth( e e + ( s o + e vth( e e ( s o } ϕo { ( } 1 ϕs ϕ o ϕ ( s ϕ ϕfp + v o cr ϕs = sgn ( ϕs v th e e + ( ϕs ϕo + vthe e e ( p ( N ( n A X. ZHOU 4 007 ( N D 1 ( ϕ v th
Gauss Law and Voltage Equation Gauss law + Field equation ε V E = =+, V V V ( ' ox1 gf 1 s1 s1 s1 gf1 gs1 FB1 ε Si tox 1 ε V E = =, V V V ( ' ox gf s s s gf gs FB ε Si tox Two boundary conditions = Two voltage equations (second integral not possible for doped body (Two unknowns with one equation unless assume one relation for s and o Undoped (pure body (most of the current literature follows this solution d = q ( n p q ( n = q ne q i dx εsi εsi εsi εsi p 0 Most ignored one carrier ( v ( V ne Fn th cr th Fp = 0 F i v All assumed majority carrier at equilibrium θ { } ( x = c+ bln cos θ( x, a ( x, a = arccos( α d( x a ' ( a, ( a B.C. s: Second integral of Poisson possible leading to exact implicit s solution. However, it cannot be extended to DG with unintentional doping. X. ZHOU 5 007
Our Approach: Unified Regional Model (URM Start with s-dg with doped body (two unknowns, s and o ' ε V ox gf s Es = s =+ s1 = s = s εsi t = sgn ( ϒ, ϒ= ε, = ε ox ϕ = sgn ϕ γ, γ ϒ, ϕ, E o = 0 ( Vgf Vgr VFB, Vr = Vs V f q N C C t gf s s Si A ox ox ox ox ( v f v v v V v gf s s ϕ th th th ϕ ( s ϕ ϕfp + v o s o (,, ( c r ϕ ϕ fϕ ϕs ϕo vcr = e e + ϕs ϕo + e e e ϕs ϕo ( bulk ϕ o = 0 ( Fp vcr s ( e ϕs s e ϕ ϕ ϕ 1+ ϕ + e 1 s + Extend URM approach to s solution finding the full-depletion voltage V FD Scale with N ch, T Si, T ox, K ox in all regions s readily applicable in bulk I ds ( s Bulk model for short-channel/high-order effects can be easily extended Extending to a-dg two independent s-dg coupled by one V FD (V g1,v g Without solving coupled a-dg solutions, which is impossible for doped body Recoverable to s-dg, FD/PD-SOI, undoped, bulk unification of MOS models X. ZHOU 6 007
Regional s Solutions and Full-Depletion Voltage Surface Potential, s (V seff FD condition: 1. 1.0 0.8 0.6 0.4 0. Symbols: Medici Lines: Model (Xsim s seff fd dv str acc ds ( + ( = dfd( gfd, X V X V T d1, FD g1, FD d, FD g, FD Si V FB V FD V t -0. - -1 0 1 = + ac c ds Common Gate Voltage, V g (V fd qn X X V T, = Si A o s o = o = Si ε Si ( X T qn X A d, FD ϒ ϒ = = + + V ε Si 4, ; dep = ϑeff sub fd δ d = V + ϒ dv gf { } = V + v L W acc gbr s-dg: ( ( Fp + Vcr vth s v th o vth Vgf s = ϒ vt he e e + dp e ds th eff { } dep { d v, s tr; } = ϑ δ ( ϒ ϒ = + + 4 gf, FD sub V gbf X. ZHOU 7 007
Surface-Potential Derivatives and Regional Components d s /dv g (V/V 1.4 1. 1.0 0.8 0.6 0.4 0. -0. Symbols: Medici Lines: Model (Xsim s seff fddv acc str ds - -1 0 1 Common Gate Voltage, V g (V X. ZHOU 8 007
Surface-Potential Scaling Over Body Doping Surface Potential, s (V 1.0 0.5-0.5-1.0 Symbols: Medici Lines: Model (Xsim s-dg (V g1 = V g = V g T ox1 = T ox = 3 nm T Si = 50 nm - -1 0 1 Common Gate Voltage,V g (V N A (cm 3 n i 10 14 10 16 10 18 X. ZHOU 9 007
Surface-Potential Derivatives Scaling Over Body Doping 1. 1.0 Symbols: Medici Lines: Model (Xsim s-dg (V g1 = V g = V g T ox1 = T ox = 3 nm T Si = 50 nm d s /dv g (V/V 0.8 0.6 0.4 0. N A (cm 3 n i 10 14 10 16 10 18 - -1 0 1 Common Gate Voltage, V g (V X. ZHOU 10 007
Evaluation in Surface-Potential Potential-Based I ds ( s Model Drain Current, I ds (ma/μm 1.0 0.8 0.6 0.4 0. Symbols: Medici Lines: Model (Xsim N A = 10 18 cm 3 T Si = 50 nm T ox = 3 nm L = 1 μm Vds = 5 V Vds = 1. V log(i ds -0.5 0.5 1.0 1.5 Common Gate Voltage, V g (V I ds - -3-4 -5-6 -7-8 -9-10 -11-1 -13-14 -15-16 -17-18 Drain current, log(i ds (A/μm X. ZHOU 11 007
Linear/Saturation Transconductance Matching Transconductance, g ms (ms/μm 80 60 40 0 0 1.4 1. 1.0 0.8 0.6 0.4 0. g m0 (μs/μm Symbols: Medici Lines: Model (Xsim log(g m g m0 g ms -0. -0.5 0.5 1.0 1.5 Common Gate Voltage, V g (V N A = 10 18 cm 3 T Si = 50 nm T ox = 3 nm L = 1 μm Vds = 5 V Vds = 1. V - -3-4 -5-6 -7-8 -9-10 -11-1 -13-14 -15-16 -17-18 Transconductance, log(g m (S/μm X. ZHOU 1 007
Hole Imref at Non-equilibrium Imref split: V Fn Fp Fp NA N D NA N D = vth ln + + e ni ni V v th V = 0 (MOS C F 1 NA N D = vthsinh ni Quasi-Fermi Potential, Fp (V 0.5-0.5 n i -1 0 1 3 4 5 6 7 8 9 10111131415161718190 Channel Doping, log(n A N D (cm 3 F B V = 0 V = 0.3 V V = 0.6 V V = 0.9 V V = -0.3 V V = -0.6 V v NA N D N A ln v ln ni ni B th th (Highly doped Fp = V NA ND >> ni Undoped (pure body: N = N = 0 Fn A = + V This channel-voltage dependent imrefs for both carriers is missing in all existing undoped body DG! D X. ZHOU 13 007
Absolution Error in Missing One Carrier in Poisson Gate Capacitance, C gg (pf/cm 0.15 0.10 5 s (nv 1.5 Dotted Line: electron only ln{cos(arccos(α} solution -1.5-1 0 1 V gf (nv V = 0 T ox = nm electron + hole T Si = 0 nm electron only 0-50 0 50 V gf = V gs V FB (mv Solid Line: electron + hole Elliptic integral solution Both N R iterative solutions, equation residue < 10 15 V A constant error of 1.5 nv in s and a finite max. error in s derivative (C gg @ V ds = 0 A singularity in C gg at V gf = 0, which gives a glitch in C gg ' due to missing one carrier. X. ZHOU 14 007
Correct Voltage-Equation s Solution in Pure Silicon Surface Potential, s (V.0 1.5 1.0 0.5-0.5-1.0-1.5 V = 0 V V = 0.1 V V = 1 V V = 0.1 V V = 1 V V gf = V gs V FB (V T ox = nm, T Si = 0 nm Open Symbols: Hole imref = 0 Solid Symbols: Hole imref = V/ -1.5-1.0-0.5 0.5 1.0 1.5 Open Symbols: Fp = 0 No change in s for holes Holes assumed equilibrium Solid Symbols: Fp = V/ Reduced s for electrons Same s variation for holes Holes not at equilibrium The actual current transport (I ds formulation depends on type of S/D structure (n + /p + and contacts; and bipolar current may exist in belowintrinsically-doped DG MOS. X. ZHOU 15 007
Conclusions Approach to modeling generic DG Importance in modeling DG with doped body: tough but scalable to undoped URM: extension from bulk formulations physics embedded in the regional (but single-piece surface-potential solutions, including all SCEs/PDEs/QMEs s-dg: similar in formulation to bulk-mos; extendable to ca-dg, and ia-dg by two independent but coupled DG s Modeling undoped DG Starting equation simpler and integrable, but not really practical and nonextendable to doped body (i.e., need new models for doped device Missing one carrier and assuming majority carrier at equilibrium gave inaccurate (or even incorrect voltage-equation solutions in pure-body DG Unification of MOS models It is possible to arrive at one solution for generic doped a-dg that is extendable to all s-dg (FinFET, UTB-FD/PD-SOI, and bulk MOSFETs X. ZHOU 16 007