Chap3 - $ ECE432 Junctions Chapter 3& Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 0 The Ohio State University $ Lecture 7& Patrick Roblin

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Unformatted text preview: ' $ ECE432 Junctions Chapter 3 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 0 The Ohio State University % ' $ Lecture # 7 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 1 The Ohio State University % ' $ Introduction I P N V Applications: & recti cation, ampli cation, switching, regulation, LED, lasers Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 2 The Ohio State University % ' $ Topic Covered Fabrication of pn junctions pn junction theory (equilibrium, steady-state, transiant) Metal-semiconductor diodes Heterojunctions Light emetting diodes & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 3 The Ohio State University % ' $ 5.1 Fabrication of p-n Junctions Processes involved in fabrication: Thermal oxidation Thermal di usion Rapid thermal processing Ion implantation Photolithography Chemical Vapor Deposition Etching Metallization & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 4 The Ohio State University % ' $ Thermal oxidation heating coils Silica tube O 2 or H2O SiO2 Si wafers Wafers are placed in a silica tube heated to 800-1000o C dry O2 or H2 O is owed in the silica tube Chemical reactions involved: Si + O2 ! SiO2 (dry oxidation) Si + 2 H2 O ! SiO2 + 2 H2 (wet oxidation) For every micron of oxide, 0.44 m of Silicon is consumed. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 5 The Ohio State University % ' $ Di usion (Older Process) heating coils Impureties Silica tube Si −n Si −p SiO2 wafers Doping Na Nd Wafers are placed in a silica tube Dt Position heated to 800-1100o C (minutes to hours) to creates vacancies Dopants: B, P or As are owed in the silica tube using a dry or vapor source and di use in the silicon wafers & p The di usion length Dt is di cult to control as it is strongly temperature OHIO S ATE T sensitive: D = D0 exp ; EA =(kT )] ! supplanted by ion implantation Patrick Roblin T.H.E UNIVERSITY 6 The Ohio State University % ' $ Rapid Thermal Processing gold plated reflectors Tungsten−halogen infrared lamps Temperature Process 1000 oC Temp ramp ~100o C/s Quartz window Cool−down Gas flow Gas in Wafer 27 o C Gas out Gas flow stabilization Time (s) Low thermal mass provides fast heating and cooling (seconds) by IR radiation Used with oxidation, annealing after ion implantation, chemical vapor deposition & RTP reduces exposure time to high temperature for improved control of dopant OHIO di usion Patrick Roblin T.H.E S ATE T UNIVERSITY 7 The Ohio State University % ' $ Ion Implentation Doping Iionized Accelerated Mass separator Rp ΔRp High voltage source Magnet Wafer Gas source (Impureties) Distance from surface Impurities are ionized and accelerated to kinetic energies of several keV to MeV and directed to the semiconductor surface Concentration pro le: & N (x) = p Patrick Roblin 2 Rp 2 !23 exp 4; 1 x ; Rp 5 2 Rp T.H.E OHIO S ATE T UNIVERSITY 8 The Ohio State University % ' $ Advantages Advantages Low temperature process (avoid di usion of previously implanted dopants) Precise control of doping pro le High uniformity (beam scanning) across wafer Doping & Patrick Roblin Sum of gaussians T.H.E Distance from surface OHIO S ATE T UNIVERSITY 9 The Ohio State University % ' $ Other Issues with Impact Ionization Problem: damages are created in the semiconductor crystal due to the collision. Solution: Annealing which is the self-repair of crystal at 1000o C Consequence: unintended di usion of the implanted dopants N (x) = p q 2 2 Rp + 2Dt & Patrick Roblin 3 20 2 12 1 (x ; R ) 5 4 exp 6; @ 2 p A 7 2 Rp + 2Dt T.H.E OHIO S ATE T UNIVERSITY 10 The Ohio State University % ' $ Chemical Vapor Deposition (CVD) heating coils Silica tube Gas flow SiO2 Si wafers Whereas oxidation consumes Si from the substrate and requires high temperatures, CVD uses low pressure and low temperature and does not consume Si from the substrate. In CVD a chemical reaction leads to the deposition of SiO2 on Si. Can be used to grow SiO2 , silicon nitride oxide (Si3 N4 ), polycrystaline and amorphous Si CVD can be plasma enhanced (PECVD) (plasma to be discussed later on) Di erent from VPE (MOCVD) which grows epitaxial crystal layer and isOHIO more challenging S ATE T & Patrick Roblin T.H.E UNIVERSITY 11 The Ohio State University % ' $ Photolithography Patterns used for devices and circuits are formed using photolithography. This involves the following key components: A reticle (mask) which is a transparent silica plate containing the pattern for a single chip (die). Opaque regions are realized using iron oxide A photosensistive (or electron beam sensitive) resist which is a material whose material properties changes when exposed to light. The following steps are used: The resist is spin-coated (3000 rpm) on the wafer to form a coat of (0.5 m) The resist is exposed using an optical stepper The exposed resist is etched away using NaOH (Sodium hydroxide). This transfers the patterns to the semiconductor. The resist is baked (125o C) to harden it and prepare it for subsequent processing (metal deposition, plasma etching, ion implantation). OHIO S ATE T Note: The reticle/mask is created itself using a resist. & Patrick Roblin T.H.E UNIVERSITY 12 The Ohio State University % ' & Patrick Roblin $ Optical Stepper T.H.E OHIO S ATE T UNIVERSITY 13 The Ohio State University % ' $ Limitation of Lithography Resolution is limited by di raction to the wavelength of the light used for the resist exposure This has pushed for the use of shorter wavelength from UV mercury lamp (0.365 m), to argon uoride (ArF) excimer lasers (0.193 m) or extreme UV (EUV) (0.154 m). Xray (0.01-1 nm) is been studied as an option. Improved resolution is obtained using Fourier optics (e.g., fuzzy edge) Depth of focus (DOF) decreases with decreasing wavelength and it is necessary to planarize circuits by chemical and mechanical polishing. Direct e-beam exposure of resists requires no mask and achieves down to 0.01 m=10 nm resolution which is ideal for nanotechnology research but is for IC production. too slow & OSU has recently acquired such an e-beam lithographic system: http://www.ece.osu.edu/%7Eberger/ebl.html/ Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 14 The Ohio State University % ' $ Etching In early days of Si technology selective etching of SiO2 was done with HF (hydro uoric) acid Wet etched is isotropic and is used now mostly for wafer cleaning Plasma etching is now prevalent as it is anisotropic (important for small geometry devices) The most popular plasma etching is reactive ion etching (RIE) as it is both anisotropic and selective (chemical etch by highly reactive radical) & What is a plasma? Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 15 The Ohio State University % ' $ Plasma State of Matter Plasma is a state of matter featuring ionized particles with high kinetic energy. It is often referred as the 4th state of matter but was really the rst state formed after the Big Bang (universe creation theory). Plasmas are conductive and screen radiation (skin depth). & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 16 The Ohio State University % ' & Patrick Roblin $ Ubiquitous Plasma T.H.E OHIO S ATE T UNIVERSITY 17 The Ohio State University % ' $ Plasma Etching Wafers Etch gas in Plasma Radicals & Ions out Anode (+) Cathode (−) RF gen Results from plasma physics: A high DC voltage (100-1000 V) develops near the cathode This builtin voltage accelerates positive ions toward the OHIO S T wafers and induces an etch normal to the surface (anisotropic)ATE & Patrick Roblin T.H.E UNIVERSITY 18 The Ohio State University % ' $ Metallization Al RF generator Al Al Cathode (−) Ar gas in Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al Al target Ar Plasma out Wafers 1111 0000 1111 0000 1111 0000 Al 1111 0000 1111 0000 11111 00000 11111 00000 11111 00000 Ar 11111 00000 11111 00000 Ar+ Anode (+) Devices needs to be connected together and with the IC package using metallization. Metal deposition techniques used include: Thermal or E-beam evaporation in vacuum (Au on GaAs) or sputtering with OHIO low-pressure Argon (Ar) plasma (Al and Cu on Si) S ATE T & Patrick Roblin T.H.E UNIVERSITY 19 The Ohio State University % ' $ PN Junction Fabrication http://jas.eng.buffalo.edu/education/fab/pn/diodeframe.html & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 20 The Ohio State University % ' $ Lecture # 8 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 21 The Ohio State University % ' $ 5.2 Equilibrium Conditions P N P ε Concentration pp= Na hole diffusion φp nn= Nd Log scale electron diffusion np= & N Patrick Roblin n 2/ i pp pn= ni2/ n n φn Vn x Vp T.H.E OHIO S ATE T UNIVERSITY 22 The Ohio State University % ' $ Equilibrium Diode Band Diagram P N P N Energy Ec Ec Ecp EF Ecn EF Ev & Ev EFn Evn Fermi level is constant in equilibrium Patrick Roblin EFp Evp T.H.E OHIO S ATE T UNIVERSITY 23 The Ohio State University % ' $ Sailing, Biking (or Skying) on the Band Diagram Diffusion E cp * E cn Drift E Fn E Fp E vp Drift * & Patrick Roblin E vn Diffusion T.H.E OHIO S ATE T UNIVERSITY 24 The Ohio State University % ' & Patrick Roblin $ Sailing with a Laser T.H.E OHIO S ATE T UNIVERSITY 25 The Ohio State University % ' $ Built-in Contact Potential P N P N ε (electric field) Concentration pp= Na hole diffusion φp nn= Nd Log scale electron diffusion np= n 2/ p p i pn= ni2/ nn φn Vn V0 x Vp Energy Ec Ec Ecp EF Ecn & Patrick Roblin EF Ev Ev EFp Evp EFn T.H.E Evn OHIO S ATE T UNIVERSITY 26 The Ohio State University % ' $ Built-in Contact Potential In equilibrium there is no electron and hole currents: Jp = Jp(drift) + Jp(diffusion) = 0 Jn = Jn(drift) + Jn(diffusion) = 0 Using the holes's current we have dp J0 = 0 = q ppE ; qDp dx 0 = ; pp dV ; Dp dp dx dx 1 dp p dV ); Dp dx = p dx q dV = 1 dp ); kT p & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 27 The Ohio State University % ' $ Built-in Contact Potential q dV = 1 dp ; kT p q Z Vn dV = Z pn 1 dp ); pp p kT Vp q ) ; (Vn ; Vp ) = Ln(pn) ; Ln(pp) kT q V = Ln pp ! ) kT 0 pn The built-in contact voltage is: ! kT 0 n 1 kT 0 N N 1 p a V0 = kT Ln pp = q Ln @ nn A = q Ln @ n2 d A q n p i using nppp = nnpn = n2. i & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 28 The Ohio State University % ' $ Built-in Contact Voltage 0 1 kT Ln @ NaNd A V0 = 2 q cannot be measured. Why? ni Diffusion E cp * E cn Drift E Fn E Fp E vp & Patrick Roblin Drift * E vn Diffusion T.H.E OHIO S ATE T UNIVERSITY 29 The Ohio State University % ' $ Non-Measurable Contact Potential The voltage VD across the diode in equilibrium is 0. Only di erence in Fermi levels can be measured as voltage ;qVD . 0 1 EF = Ec(0) ; qV (x) + kT Ln @ n(x) A | {z } {z Nc } | electrostatic potential energy = n (x) ; qV (x) = constant chemical potential The variation in electrostatic potential is compensated by variation in chemical potential: & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 30 The Ohio State University % ' $ Non-Measurable Contact Potential Diffusion E cp * E cn Drift E Fn E Fp E vp Drift * E vn Diffusion The variations in electrostatic potential energy is compensated by the variations in chemical potential energy: & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 31 The Ohio State University % ' $ Proof P V0 = ) eqV0=(kT ) = = & = = )1 = Patrick Roblin ) kT Ln pp ! q pn Ecp pp pn EFp Nv e(Evp;EF p)=(kT ) Evp (Evn ;EF n )=(kT ) Nv e e(Evp;Evn)=(kT ) e(EF n;EF p)=(kT ) eqV =(kT )e(EF n;EF p)=(kT ) e(EF n;EF p)=(kT ) EFn = EFp N qV0 Ecn E Fn Evn 0 T.H.E OHIO S ATE T UNIVERSITY 32 The Ohio State University % ' $ Space Charge at a Junction Equations to solve: dE = (x) with dx (x) = q p(x) + Nd(x) ; n(x) ; Na(x)] P N Ecp qV0 Ecn & Patrick Roblin EFp Evp E Fn T.H.E Evn 33 OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Depletion Approximation P ρ=0 E cp N A ρ= ρ= ρ=0 −qN a qNd qV0 E cn EFp E vp E Fn E vn & −x p0 0 Macroscopic space charge neutrality: qAxp0Na = qAxn0Nd Patrick Roblin 34 xn0 x T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Depletion width W P ρ=0 N A ρ= ρ=0 ρ= −qN a qNd −x p0 0 xn0 x W & xp0 ! = x 1 + Nd ! W = xn0 + xp0 = xn0 1 + x n0 Na n0 W = Na W ) xn0 = Nd Na + Nd 1 + Na Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 35 The Ohio State University % ' $ Charge, Electric Field, Potential P N A ρ qNd −x p0 xn0 0 −qN a dE = qNd dx Z0 qNd Z xn dx dE = E 0 qNd x ;E = −x p0 ε xn0 0 V 0 & Patrick Roblin x ε0 0 0 x n0 qV0 T.H.E −x p0 0 xn0 x OHIO S ATE T UNIVERSITY 36 The Ohio State University % ' $ V0 = & ; Z xn0 ;xp0 Contact Potential and Depletion Width E (x)dx ! 1 1 = ; xp0E0 + xn0E0 2 2 1E W =; 0 2 = 1 qNd xn0W 2 1 q NaNd W 2 = 2 Na + Nd v u 2 (N + N ) u a W = t qN N d V0 ad Patrick Roblin P N A ρ qNd −x p0 xn0 0 x −qN a −x p0 ε xn0 0 x ε0 V T.H.E qV0 −x p0 37 0 xn0 OHIO S ATE Tx UNIVERSITY The Ohio State University % ' $ PN Junction Formation http://jas.eng.buffalo.edu/ & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 38 The Ohio State University % ' $ Lecture # 9 Forward & Reverse Biased Junctions & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 39 The Ohio State University % ' Charge and Band Diagram when a Voltage is Applied V +− P −+ V −+ N P −+ −+ P N −+ Ec Ec −qV E F Ev & $ Forward Biased Patrick Roblin E F Ev N Ec E F Ev − − ++ −− ++ −− ++ E F Equilibrium −qV E F T.H.E Reverse Biased OHIO S ATE T UNIVERSITY 40 The Ohio State University % ' $ Band Diagram in Equilibrium Diffusion E cp * E cn Drift E Fn E Fp E vp Drift * & E vn Diffusion Drift and di usion currents are cancelling each other. Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 41 The Ohio State University % ' $ Band Diagram in Forward Bias Diffusion E cp Drift E Fp E vp E cn E Fn −qV Drift E vn & Diffusion Di usion currents dominate over drift currents. Patrick Roblin 42 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Band Diagram in Reverse Bias E cp Diffusion Drift E Fp E vp −qV E cn E Fn Drift Diffusion & E vn Drift currents dominate over di usion currents but both are neglegible. Only the rare Olympic electrons doing the high jump can make it over the increased OHIO S ATE T potential barrier. Patrick Roblin T.H.E UNIVERSITY 43 The Ohio State University % ' $ All Biases Diffusion Drift E Fp E vp Diffusion E cp E cp E cn E Fn −qV E vn Diffusion Diffusion * Drift E cn Drift E Fp E vp Drift E cp Drift E Fn E Fp E vp −qV E cn E Fn * E vn Diffusion Drift Diffusion Forward Bias & Patrick Roblin Equilibrium E vn Reverse Bias T.H.E OHIO S ATE T UNIVERSITY 44 The Ohio State University % ' $ Carrier Injection In equilibrium there is no electron and hole currents: Jp = Jp(drift) + Jp(diffusion) = 0 Jn = Jn(drift) + Jn(diffusion) = 0 Using the holes's current we have dp J0 = 0 = q ppE ; qDp dx Let us evaluate the magnitude of the di usion/drift currents dp Jp(diffusion) = qDp dx 19 5 ;3 ;19 (C) 10(cm2 /s) 10 ; 10 (cm ) = 1:6 10 4 10;{zcm | } 1m OHIO = 1:6 105A/cm2 Very Large! S ATE T & Patrick Roblin T.H.E UNIVERSITY 45 The Ohio State University % ' $ Assumption Since Jp(diffusion) = 1:6 105A/cm2 is very large we can safely assume that when a bias is applied Jp = Jp(drift) + Jp(diffusion) << Jp(diffusion) or equivalently Jp = Jp(drift) + Jp(diffusion) ' 0 like in equilibrium. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 46 The Ohio State University % ' $ Carrier Injection So like in equilibrium we start from: Jp = Jp(drift) + Jp(diffusion) = 0 Using the holes's current we have dp Jp = 0 = q ppE ; qDp dx 0 = ; pp dV ; Dp dp dx dx 1 dp p dV ); Dp dx = p dx q dV = 1 dp ); kT p & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 47 The Ohio State University % ' $ Carrier Injection q dV = 1 dp ; kT p q Z Vn dV = Z p(xn ) 1 dp ); p(;xp ) p kT Vp q (V ; V ) = Ln p(;x )] ; Ln p(x )] ) p0 n0 kT n p 2 3 q (V ; V ) = Ln 4 p(;xp0) 5 ) kT 0 p(xn0) Resulting in: 2 3 p(;xp0) = exp 4 q(V0 ; V ) 5 p(xn0) kT 0 0 & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 48 The Ohio State University % ' $ Range of Integration P ρ=0 E cp N A ρ= ρ= ρ=0 −qN a qNd qV0 E cn EFp E vp E Fn E vn & Patrick Roblin −x p0 0 xn0 x T.H.E OHIO S ATE T UNIVERSITY 49 The Ohio State University % ' $ Electrostatic Potential when Diode is Biased V>0 +− P V<0 +− N −+ P −+ −+ −+ N P − − ++ −− ++ −− ++ N Vn Vn Electrostatic Potential Vn V0 −V Vp with V>0 Vp Vp −q(V 0−V) with V0 −V V0 −qV 0 −q(V 0−V) V>0 with −qV V<0 −qV & Patrick Roblin T.H.E Forward Bias Equilibrium 50 Reverse Bias OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Carrier Injection We have found that when the 3 diode is under a bias we have: 2 p(;xp0) = exp 4 q(V0 ; V ) 5 p(xn0) kT Using the equilibrium relation we have: pp = exp " qV0 # ) p(;xp0) = pp exp ;qV ! pn kT p(xn0) pn kT Low injection approximation: p(xn0) = pn + pn minority carrier p(;xp0) = pp + pp ' pp majority carrier It results that: qV ! ) p = p "exp qV ! ; 1# p(xn0) = pn exp kT n n kT & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 51 The Ohio State University % ' $ Minority Carrier We derived that the excess minority carrier is: " q V ! ; 1# pn = pn exp kT Variation of the excess minority carriers: 8 >V =0 > pn = 0 > < for > V >> 0 pn ' pn exp qV kT > > : V << 0 pn = ;pn & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 52 The Ohio State University % ' $ Lecture # 10 Carrier Injection and Current & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 53 The Ohio State University % ' $ Band Diagram and Excess Carriers 1.5 1 Energy (eV) 0.5 0 −0.5 V=0.5 V V=0 V −1 E v E V=−0.5 V c & −1.5 −1.5 −1 −0.5 0 position qV ! ; 1# and pn = pn exp kT Patrick Roblin " 54 0.5 1 " 1.5 qV ! ; 1#OHIO np = np exp kT S ATE T −7 x 10 T.H.E UNIVERSITY The Ohio State University % ' $ Doping Concentration 25 10 20 10 hole concentration p Concentration (cm−3) electron concentration n 15 V=0.5 V V=0.5 V V=0 V V=0 V V=−0.5 V 10 V=−0.5 V 10 10 5 10 & Patrick Roblin T.H.E 0 10 −1 −0.5 0 position 0.5 1 −7 x 10 OHIO S ATE T UNIVERSITY 55 The Ohio State University % ' Excess Carriers Concentration Δ np Δ pn diffusion & recombination Ln P diffusion & recombination −x p0 − x n0 0 − + neutral − + depletion +− np(x) = npe(x+xp )=Ln and 0 & qV ! ; 1# and np = np exp kT Patrick Roblin Lp " 56 Position + Qn Qd Qd Qp − with $ N + neutral pn(x) = pne;(x;xn )=Lp 0 " qV ! ; 1#OHIO pn = pn exp kT S ATE T T.H.E UNIVERSITY The Ohio State University % ' $ Current Distribution Current total current I Ip diffusion & recombination diffusion & recombination In Ln −x p0 0 x n0 − + Qn − + Qp − + Position + − P − Lp + N +− & We shall rst assume that there is neglegible recombination in the depletion region. Patrick Roblin 57 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' Diode Forward Biased Ln P Neutral W Space Charge 1 0 Excess electrons 1 0 Diffusion 1 0 1 0 1 0 In 1 0 1 0 1 0 1 0 1 0 1 0 Recombination 1 0 1 0 1 0−qv Di 1 0 1 0 1 0 1 0 E cp −qvD= EFp E vp & Patrick Roblin $ vD Ohmic loss Drift In +Ip Diffusion Ip + R 58 E Lp Neutral 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 N 1 0 1 0 1 0 I +I 1 0 np 1 0 Drift 1 0 1 0 1 0 1 0 1 0 E cn 1 0 1 0 1 0 1 0Ohmic loss E Fn 1 0 1 0 1 0 Recombination 1 0 1 0 1 0 1 0 1 0 E vn 1 0 Excess holes − A T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Current Calculation Using pn(x) = pne;(x;xp0)=Lp and np(x) = npe(x+xn0 )=Ln with " qV ! ; 1# and n = n "exp qV ! ; 1# pn = pn exp kT p p kT the total diode current is evaluated from: Ip = AJp(xn0) = ;AqDp dp(x) = AqDp Lpn dx xn0 p np dn(x) = AqDn In = AJn(;xp0) = AqDn dx Ln ;xp0 0 1" Dn n + Dp p A exp qV ! ; 1# I = In + Ip = Aq @ L p L n kT n {z p} | & Patrick Roblin I0 T.H.E OHIO S ATE T UNIVERSITY 59 The Ohio State University % ' $ Diode I-V Characteristic " q V ! ; 1# I = I0 exp kT I0 is the leakage current (very small): 1 0 I0 = Aq @ Dn np + Dp pnA Ln Lp For 8 > I0 exp qV > for V >> 0 kT > < I=>0 for V = 0 > > : ;I0 for V << 0 & Patrick Roblin I − I0 V<< V0 V0 V T.H.E OHIO S ATE T UNIVERSITY 60 The Ohio State University % ' $ p+n Diode Dp p "exp qV ! ; 1# = Qp(V ) I = Aq L n kT p p N P ρ=0 ρ=−qN a ρ=qNd −x p0 0 & A ρ=0 x xn0 W The hole minority carrier in the n side dominates Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 61 The Ohio State University % ' $ High Injection Limit Solving the system p(;xp0) = pp + pp = pp exp ;qV ! p(xn0) pn + pn pn kT n(xn0) = nn + nn = nn exp ;qV ! n(;xp0) np + np np kT using pn = hnn we obtainithe current exp qV ; 1 kT I = qA 1 ; exp 2q(V ;V0) kT 2 0 !1 D 0 n2 !13 2 4 Dn np @1 + ni exp qV A + p pn @1 + i exp qV A5 Ln n2 kT Lp p2 kT n p The current is proportional to exp 2 qV at moderate injection kT The current is in nite when V = V0 (contact potential) OHIO V0 is approximately the bandgap for strong doping S ATE T & Patrick Roblin T.H.E UNIVERSITY 62 The Ohio State University % ' $ Quasi Fermi Levels for Forward Bias 1.2 1 0.8 Energy (eV) 0.6 E v 0.4 Ec F n 0.2 F p 0 −0.2 −0.4 −0.6 −1.5 −1 −0.5 0 position 0.5 1 1.5 −7 x 10 In which regions are the carriers in thermal equilibrium with themselves? OHIO 2 eqV= n(x)p(x) = nie(Fn;Ei)=(kT ) nie(Ei;Fp)=(kT ) = n2e(Fn;Fp)=(kT ) = nTATE (kT ) i Si & Patrick Roblin T.H.E UNIVERSITY 63 The Ohio State University % ' $ Recombination-Generation (RG) Current Using n(x)p(x) = n2eqV=(kT ) we can evaluate the recombination i current: Z xn JRG = q ;xp0 r (n(x)p(x) ; n2]dx i =q Z xn 2 qV=(kT ) ; 1]dx r ni e ;xp0 2 qV=(kT ) ; 1] W (V ) r ni e =q where we assumed that r is the same on the N and P sides. JRG contribution is more important for narrow bandgap semiconductors (Si, 1.11 eV and GaAs, 1.43 eV), high temperature (ni(T )) and low voltages and negative p The depletion width dependence introduces a V ; V0 dependence which dominates in reverse bias. & Patrick Roblin 64 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Non Ideal E ects in Diode IV Characteristics To t the IV of real diodes we need to introduce an ideality factor n which is typically between 1 and 2 log |I| 20 13 Ohmic effects q ( V ; R I ) A ; 15 I = I0 4exp @ nkT n=2 High injection Recombination in the depletion region (n = 2) n=1 Ohmic loss V 0 = V ; R I Diffusion current High injection (n = 2) Recombination current n=2 Recombination current & Patrick Roblin I0 V T.H.E OHIO S ATE T UNIVERSITY 65 The Ohio State University % ' $ JRG, High-Injection and Ohmic Loss −2 10 10 10 Johmic with Rd=0.001 Ω 8 10 −3 10 J J 6 10 ohmic R =1 Ω d high−injection Jideal −4 10 4 10 2 J (A/m ) 2 J (A/m ) 2 10 −5 10 −6 10 0 10 −2 10 −4 −7 10 10 −6 10 −8 10 Jtotal J −8 10 ideal JRG −9 10 −8 −10 −6 −4 −2 0 2 4 6 8 V/(kT/q) & −30 −20 −10 0 10 20 30 V/(kT/q) Contribution of JRG Patrick Roblin 10 High-Injection and Ohmic Loss T.H.E OHIO S ATE T UNIVERSITY 66 The Ohio State University % ' $ Quasi Fermi Levels for Reverse Bias 1.5 1 E v E c Fn 0.5 Energy (eV) Fp 0 −0.5 −1 & −1.5 −2 −1.5 −1 −0.5 0 position 0.5 1 1.5 −7 x 10 Note: Fp in the conduction band is of no consequence. Why? Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 67 The Ohio State University % ' $ Breakdown by Zener E ect Ec I Ev E F E F − + −Vbr V −Vbr Requires strong doping (degenerate) and an abrupt junction to increase the electric eld and enhance interband tunneling. OHIO & Patrick Roblin T.H.E S ATE T UNIVERSITY 68 The Ohio State University % ' $ Breakdown By Avalanche E ect Ec I K=3 Eg /2 E F Ev K=3 Eg /2 −Vbr V E F −Vbr − + Carrier muliplication factor and empirical formula: nout = 1 + P + P 2 + P 3 + = 1 ' 1 Mn = n 1 ; P |1 ; jV=Vbr jn} in {z n=3-6 & Patrick Roblin 69 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Breakdown Voltage 3 10 Ge Si GaAs 2 Vbr (V) 10 1 10 Tunneling Starts 0 10 14 10 15 10 16 10 17 −3 Nd (cm ) 10 18 10 For a Si p+n diode the breakdown voltage is related to the maximum electric eld Em using 2 1 E W ' Em with E = 4 105 Vbr ' 2 m m 2qN m 1 log (N =1016 ) (V=cm) 1 ; 3 10 d d ;3 and Wm the depletion width at breakdown. OHIO S ATE T with Nd in cm & Patrick Roblin T.H.E UNIVERSITY 70 The Ohio State University % ' $ Punch-Through P A N ρ=0 ρ= qNd ε −x p0 0 L xn0 x VPT εm Wm Punch-Through occurs when the depletion reaches the diode edge in a lightly doped diode (e.g., p+n diode). Breakdown occurs then for the same electric eld Em but for lower voltage: VPT = L 2 ; L ! where W is the depletion width m Vbr Wm Wm OHIO at which breakdown would occur if the diode was longer. S ATE T & Patrick Roblin T.H.E UNIVERSITY 71 The Ohio State University % ' $ Zener Diode I I − V + Vbr V Application: & voltage regulation limiters Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 72 The Ohio State University % ' $ Lecture # 11 Transient and AC Conditions & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 73 The Ohio State University % ' $ Variation of the Stored Charged Continuity Equations for holes: @ p = ; p ; 1 @Jp(x) ) ; @Jp(x) = q p + @ p @t @x @t p q @x p Integration for 0 to 1 Z1@ p Z1 p A Jp(0) ; Jp({z )] = qA 0 dx + A 0 @t dx | 1} p !0 i(t) = Qp(t) + dQp(t) dt p For example if the current is removed abrupty the charge decreases gradually: Qp(t) = I pe;t= p A p+n diode with short (L < Lp) or Au doped n-region will OHIO S ATE T have a reduced storage e ect. & Patrick Roblin T.H.E UNIVERSITY 74 The Ohio State University % ' $ Reverse Recovery Transient V(t) I E I If vD Time If =E/R − I0 vD 0 −Ir −E Time 0 −Ir =−E/R t sd + vD − Δ pn N P & Patrick Roblin V(t) + − current x n0 x n0 I δ pn (x,t) x R − pn T.H.E OHIO S ATE T UNIVERSITY 75 The Ohio State University % ' $ Analysis of Step Recovery Transient Solving for the time evolution of Qp(t) using i(t) = Qp(t) + dQp(t) dt p for a current switching between E=R and ;E=R i(t) = E 1 ; u(t)] ; E u(t) with u(t) the step function R R we obtain: h ;t= p i u(t) Qp(t) = pIf 1 ; u(t)] + p ;Ir + (If + Ir )e The time tsd where the diode still conduct is reversed bias is then obtained from: If ! Qp(tds) = 0 ) tds = p Ln 1 + I r Application: Comb generator up to 50 GHz with 10 MHz tone OHIO S ATE T spacing. & Patrick Roblin T.H.E UNIVERSITY 76 The Ohio State University % ' Diode Forward Biased Ln P Neutral W Space Charge 1 0 Excess electrons 1 0 Diffusion 1 0 1 0 1 0 In 1 0 1 0 1 0 1 0 1 0 1 0 Recombination 1 0 1 0 1 0−qv Di 1 0 1 0 1 0 1 0 E cp −qvD= EFp E vp & Patrick Roblin $ vD Ohmic loss Drift In +Ip Diffusion Ip + R 77 E Lp Neutral 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 N 1 0 1 0 1 0 I +I 1 0 np 1 0 Drift 1 0 1 0 1 0 1 0 1 0 E cn 1 0 1 0 1 0 1 0Ohmic loss E Fn 1 0 1 0 1 0 Recombination 1 0 1 0 1 0 1 0 1 0 E vn 1 0 Excess holes − A T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' Transient Recovery at t = 0+ Ln P Neutral W Space Charge 1 0 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 1 0 11 00 1 0 11 00 Recombination 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 11 00 1 0 1 0 E cp −qvD= EFp E vp Patrick Roblin R E − 78 Neutral Lp N 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Recombination 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Excess electrons 1 0 1 0 & $ Excess holes + A E cn E Fn E vn T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Capacitance of p-n Junctions Concentration Δ np Δ pn diffusion & recombination Ln −x p0 − P diffusion & recombination − x n0 0 + Lp + Qn Qd Qd Qp − neutral − + depletion +− Position N + neutral The capacitance for a diode can be calculated from: C = dQ dV where Q is the total diode charge. There are 2 contributions to Q: Qd depletion charge in the space charge region Qn and Qp charges from excess carriers in the neutral region OHIO S ATE T & Patrick Roblin T.H.E UNIVERSITY 79 The Ohio State University % ' $ Depletion Charge P N A ρ= ρ=0 ρ= −qN a qNd ρ=0 −x p0 0 x xn0 W The depletion charge is: = qAxn0Nd = qA Na Nd W (V ) Na + Nd Na W using xn0 = Na + Nd & jQdj = qxp0 Na Patrick Roblin 80 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Depletion Charge P N A In equilibrium the depletion width is: ρ= ρ=0 ρ=0 ρ= v −qN qN u 2 (N + N ) u a −x 0 x W (0) = t qN N d V0 ad W When a voltage V is applied the builtin potential varies and W is: v u 2 (N + N ) u a W (V ) = t qN N d (V0 ; V ) ad The depletion charge is then: v u NaNd W (V ) = Au2q (V ; V ) NaNd t jQdj = qA 0 Na + Nd Na + Nd a p0 & Patrick Roblin d T.H.E OHIO S ATE T UNIVERSITY 81 x n0 The Ohio State University % ' $ Depletion Capacitance Using the depletion charge v u u NaNd jQdj = At2q (V0 ; V ) Na + Nd the depletion capacitance is evaluated to be: dQd Cd = dV v u A u 2q Na Nd t = 2 (V0 ; V ) Na + Nd =A W (V ) This is the expression for a parallel plate capacitor where the plate spacing W is voltage dependent. OHIO This capacitance dominates for reversed bias. S ATE T & Patrick Roblin T.H.E UNIVERSITY 82 The Ohio State University % ' Storage (di usion) Capacitance Concentration Δ np Δ pn diffusion & recombination Ln P diffusion & recombination −x p0 − − x n0 0 + Lp neutral − + depletion +− Position + Qn Qd Qd Qp − N + neutral The storage charge Qp is given by: Qp = pI ' pI0eqV=kT for V > 26 mV The storage capacitance is then: q Cs = j dQp j = p kT I dV & Patrick Roblin $ 83 T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Storage (di usion) Capacitance Qn P Neutral Qp Space Charge Neutral N A Cd Cs Gs Excess electrons V + Excess holes − We can rewrite the storage capacitance as: q Cs = p kT I = Gs p where Gs is the diode conductance: dI q Gs = dV = kT I This capacitance dominates for forward bias. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 84 The Ohio State University % ' $ Simulation, Measurement and Modeling of the CV Characteristics (A) (B) (C) 0.0897 V 1..234 mA XY output X Rs In Drive Y L P1 Probe Anode P2 Gate P NP N Thyristor Rd Cathod C Diode (A) Experimental setup equivalent to the PISCES C-V simulations (B) The microwave measurement of the C-V of a diode (C) The equivalent circuit of a general diode. & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 85 The Ohio State University % ' $ PISCES simulated C-V characteristics associated The C-V characteristics associated with the forward biased (A) P1/N1, (B) N1/P2 and (C) P2/N2 diodes −12 2 x 10 (A) −12 x 10 (B) −10 0.5 0 x 10 (C) 0 0 −2 −0.5 −1 −6 −8 C3 in (F/um) −1 C2 in (F/um) C1 in (F/um) −4 −2 −1.5 −2 −3 −10 −2.5 −12 −4 −3 −14 0 & 0.5 1 V1 in (V) −5 0 0.5 1 (−1) x V2 in (V) 0 0.5 1 V3 in (V) The capacitances become negative when the DC biased voltage is OHIO greater than the correspondent built-in voltage! Is this physical?TATE S Patrick Roblin T.H.E UNIVERSITY 86 The Ohio State University % ' $ Extracted Data Table 1: Rs , Rd , C , and L chosen for reproducing the measured S11 curves & Patrick Roblin DC biased voltage (V ) Rs ( ) Rd ( ) L ( H ) C (pF ) 0.2 23.90 948.53 4.5 2.3 0.3 23.90 294.21 4.5 4.0 0.4 19.23 128.33 4.5 9.0 0.5 15.85 51.54 4.5 50.0 0.6 8.69 19.19 8.5 100.0 0.7 2.94 4.92 9.1 800.0 0.8 1.92 1.37 9.4 600.0 0.9 1.42 0.53 9.4 200.0 1.0 1.27 0.33 9.4 1.0 T.H.E OHIO S ATE T UNIVERSITY 87 The Ohio State University % ' $ Fit of Microwave Measurement The C−V extracted from the measured S11 800 700 Capacitance in (pF) 600 500 400 300 200 100 & 0 0.2 0.4 0.5 0.6 0.7 DC biased voltage in (V) 0.8 0.9 1 T.H.E Fitted S11 parameter Patrick Roblin 0.3 OHIO Extracted C-V characteristics S ATE T UNIVERSITY 88 The Ohio State University % ' $ Conclusion The measured capacitance of the diodes increases when the DC forward biased voltage increases toward the built-in voltage (0.7 V) decreases to zero when the DC forward biased voltage increases and is greater than the built-in voltage (0.7 V) eventually decays to zero when the DC forward biased voltage is much larger than the built-in voltage (0.7 V) & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 89 The Ohio State University % ' $ Lecture # 12 Schottky Diodes & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 90 The Ohio State University % ' $ Formation of Schottky Diode Vacuum Level qφ s qφ m EFm & 11111111 00000000 11111111 00000000 11111111 00000000 111 000 1 0 111 000 1 0 11 Metal 00 000 111 How do we form the band diagram? Patrick Roblin qχ Ec EFs Ei 1 0 10 01 11111111 00000000Ev 11111111 00000000 11 00 1 0 111111111 000000000 11 00 1 0 11 00 1 0 Semiconductor 11 00 1 0 T.H.E OHIO S ATE T UNIVERSITY 91 The Ohio State University % ' $ Band Diagram for n-type Schottky Diode Depletion Vacuum Level Metal qφ s qχ qφ m EFm 00000000 11111111 11 00 00000000 11111111 11 00 11 00 11 Metal 00 11 00 & Patrick Roblin Ec EFs Ei qφ B = qφ m− qχ EFm N qV0 = qφ m− qφ s Ec EFs 111111111 000000000 11 00 111111111 000000000 11 00 11 00 11 Metal 00 00 11 E 11111111 00000000 v 11111111 00000000 Semiconductor 11111111 00000000Ev 11 00 11111111 00000000 1 1111111 0 0000000 11 00 0 1 1 0 11 00 11111111 00000000 1 0 11 00 1 0 Semiconductor 11 00 1 0 N−type T.H.E OHIO S ATE T UNIVERSITY 92 The Ohio State University % ' $ Band Diagram for p-type Schottky Diode Depletion Vacuum Level P Metal qφ m EFm 11111111 00000000 11111111 00000000 11 00 11111111 00000000 11 00 11 00 11 Metal 00 11 00 11 00 11 00 11 00 11 00 00 11 & Patrick Roblin qφ s qχ Ec Ec 111111111 000000000 EFm 00000000 011111111 1 111 000 000000000 111111111 111 000 111 000 111 Metal 000 111 000 111 000 111 000 111 000 111 000 111 000 Ei 1 000000000EFs 111111111Ev 111 000 111111111 000000000 111 000 111111111 000000000 111 000 111 000Semiconductor 111111111 000000000 1 0 P−type 111 000 1 0 EFs 1111 0000Ev 1111 0000 1111 0000 = qφ − qφ qV0 s m T.H.E OHIO S ATE T UNIVERSITY 93 The Ohio State University % ' $ Forward and Reverse Biased Schottky Diode +− V>0 N Metal −qV N Metal Ec EFs EFm N Metal Depletion Depletion Depletion EFm + V>0 − Ec EFm EFs 00000000 11111111 00000000 11111111 11 00 111111111 000000000 111111111 100000000 111111111 000000000 000000000 011111111 11 00 111 000 1 0 111111111 111111111 11111111 000000000 000000000 00000000 11 00 1 0 11 00 11 1 00 0 111 000 11 00 11 00 111 11 00 00 0 11 00 11111111 00000000 v Metal 00 Metal 000 Metal 11 E 11 111 00 11111111 00000000 111111111 000000000 v E 11111111 00000000 100000000 011111111 000000000 111111111 & Forward Bias Equilibrium 94 Ec EFs E 11111111 00000000 v 11111111 00000000 Reverse Bias The Schottky diode is a majority carrier device ) fast! Patrick Roblin −qV T.H.E OHIO S ATE T UNIVERSITY The Ohio State University % ' $ Derivation of Schottky Diode Current The diode current density is calculated by summing the current contribution of all the electron clearing the top of the conduction band edge. Using Boltzman approximation this gives: DkB T q Z 1 exp " EFm ; qVD ; E # ; exp " EFm ; E #! dE JD = 2 h S kB T! ! kB T "q # 2 exp ; B exp q VD ; 1 = AT (1) kB T kT where m D = h2 is the 2DEG density of state q B = S ; EFm is the barrier height A is the so-called Richardson constant 2 2 DkB q = 4 m qkB OHIO A=2 h h3 S ATE T & Patrick Roblin T.H.E UNIVERSITY 95 The Ohio State University % ' $ Non-Ideal E ects To account for tunneling and image force (barrier lowering due to the repulsive Coulombic potential of the electron) and ohmic loss the non-ideal current voltage characteristic of a Schottky diode of area SG can be modeled using the modi ed expression ! 0 2 q(V ; RI ) 3 1 IG = SGA T 2 exp ; q B0 @exp 4 G k T G 5 ; 1A (2) kB T nB where B0 is the zero voltage barrier height, n is the ideality factor and R the series resistance. OHIO & Patrick Roblin T.H.E S ATE T UNIVERSITY 96 The Ohio State University % ' $ Ohmic Contact for N-Type Semiconductors Accumulation Vacuum Level N Metal qφ s qχ qφ m 111111111 000000000 EFm 00000000 1 011111111 & 111 000 111111111 000000000 111 000 111 000 111 Metal 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 100 011 111 000 EFm Ec EFs Ei 1 0 111 000 E 111 000 10000000 v 01111111 111 000Semiconductor 1 00000000 0 11111111 111 000 N−type 1 0 11111111 00000000 10000000 01111111 11 00 11111111 00000000 11 00 11 00 11 Metal 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 Ohmic contact is obtained for metal/n-type junction if Patrick Roblin Ec EFs 1111 0000 Ev 1111 0000 qφ s − qφ m < s.OHIO S ATE T T.H.E m UNIVERSITY 97 The Ohio State University % ' $ Ohmic Contact for P-Type Semiconductors Accumulation Vacuum Level P Metal qφ s qχ qφ m Ec Ei EFs 11111111 00000000Ev 11 00 11111111 00000000 EFm 00000000 111111111 0 & 111111111 000000000 11 00 1 0 11 00 11 Metal 00 11 00 11 00 11111111 00000000 11 00 Semiconductor 11 00 P−type 11 00 11 00 11 00 11 00 qφ m− qφ s Ec EFm 111111111 000000000 EFs 11111111111111111 00000000000000000 v E 11 00 1 0 Metal 11 00 11111111 00000000 11 00 Semiconductor 00 11 Ohmic contact is obtained for metal/p-type junction if Patrick Roblin m > s. T.H.E OHIO S ATE T UNIVERSITY 98 The Ohio State University % ' $ Ohmic Contact by Tunneling in n+ -metal Junction Depletion Vacuum Level N Metal qφ s qχ qφ m Ec EFs Ei 11111111 EFm 00000000 11111111 00000000 11111111 00000000 11 00 11 00 11 Metal 00 11 00 111111111 000000000Ev 11 00 11111111 00000000 Tunneling qφ m− qφ s EFm 111111111 000000000 Ec EFs 000000000 111111111 000000000 111111111 111 000 11 1 00 0 111 000 1 0 Metal 000 111 E 11111111 00000000 v 11111111 00000000 Semiconductor 111111111 000000000 11 00 11 00 Semiconductor 11 00 11 00 N−type 11 00 11 00 11 00 A Schottky diode realized with a n+ or p+ semiconductor becomes & an ohmic contact when the depletion is small enough to allow for OHIO S ATE T tunneling through the barrier. Patrick Roblin T.H.E UNIVERSITY 99 The Ohio State University % ' $ Heterojunctions Energy Ec Ec Ec Ec EF Ev Ev + n AlGaAs & Patrick Roblin iGaAs EF Ev Position Ev a) b) T.H.E OHIO S ATE T UNIVERSITY 100 The Ohio State University % ' $ Two Dimensional Electron Gas Energy Energy 22 22 Ep + h k Ep + h k Sub-band 3 EF Sub-band 2 Sub-band 1 kz kz ky ky a) b) Large concentration of carrier No collision with donors ) high mobility (HEMT) & Patrick Roblin T.H.E OHIO S ATE T UNIVERSITY 101 The Ohio State University % ' $ The Ferrari of Transistors are realized with Heterojunctions Heterojunctions are used to built fast devices (gain up to 500 GHz): HEMTs (High Electron Mobility Transistor) OHIO HBT (Heterojunction Bipolar transistor) S ATE T & Patrick Roblin T.H.E UNIVERSITY 102 The Ohio State University % ...
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This note was uploaded on 01/11/2012 for the course ECE 432 taught by Professor Lu during the Fall '08 term at Ohio State.

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