Section 7 Diffusion Jaeger Chapter 4 EE143 Ali Javey Surface Diffusion Dopant Sources a Gas Source AsH3 PH3 B2H6 BN Si BN Si b Solid Source c Spin on glass SiO2 dopant oxide d Liquid Source EE143 Ali Javey Fick s First Law of Diffusion N J D x D diffusion coefficient EE143 Ali Javey Fick s Second Law of Diffusion Continuity Equation for Particle Flux Rate of increase of concentration is equal to the negative of the divergence of the particle flux N J t x in one dimension Fick s Second Law of Diffusion Combine First Law with Continuity Eqn N 2N D 2 t x Assumes D is concentration independent which isn t true in many situations in modern devices EE143 Vivek Subramanian Slide 4 4 Diffusion Coefficients of Impurities in Si Substitutional Diffusers D DO e EA Interstitial Diffusers 10 6 Cu B P As EE143 Ali Javey kT Au Diffusion Coefficients E D DO exp A Arrhenius Relationship kT E A activation energy k Boltzmann s constant 1 38 x10 23 J K T absolute temperature EE143 Ali Javey Diffusion Mechanisms in Si a Interstitial Diffusion Example Cu Fe Li H Fast Diffusion Cu 10 6 cm2 sec EE143 Ali Javey Au Diffusion Mechanisms in Si b Substitutional Diffusion c Interstitialcy Diffusion Example Dopants in Si e g B P As Sb EE143 Ali Javey Constant Source Diffusion Complementary Error Function Profiles Concentration x N x t N 0 erfc 2 Dt Total Dose Q N x t dt 2 N 0 Dt 0 N 0 Surface Concentration D Diffusion Coefficient erfc Complementary Error Function EE143 Vivek Subramanian Slide 4 9 Limited Source Diffusion Gaussian Profiles Concentration x 2 x 2 Q exp N x t N 0 exp Dt 2 Dt 2 Dt Q N 0 Surface Concentration N 0 Dt D Diffusion Coefficient Gaussian Profile EE143 Ali Javey Two Step Dopant Diffusion dopant gas 1 Predeposition dose control SiO2 SiO2 Si 2 Drive in profile control junction depth concentration Doped Si region Turn off dopant gas or seal surface with oxide SiO2 SiO2 SiO2 Si Note Note Predeposition Predepositionby bydiffusion diffusion can canalso alsobe be replaced replacedby byaashallow shallowimplantation implantationstep step EE143 Ali Javey Normalized Concentration versus depth Drive in Drive in Predeposition Predeposition EE143 Ali Javey Diffusion of Gaussian Implantation Profile EE143 Ali Javey Successive Diffusions Thermal Budget Thermal Budget Temp t Dt effective Dt Example i step i Dttotal of Well drive in well drive in step S D Anneal step and time S D annealing For Foraacomplete completeprocess processflow flow only onlythose thosesteps stepswith withhigh highDt Dt values valuesare areimportant important EE143 Ali Javey Solid Solubility Limits There is a limit to the amount of a given impurity that can be dissolved in silicon the Solid Solubility Limit At high concentrations all of the impurities introduced into silicon will not be electrically active EE143 Ali Javey High Concentration Diffusion Effects 1 E Field Enhanced Diffusion 2 Charged point defects enhanced diffusion Log C x High conc profile D gets larger when C x is large Log C x J large Low conc profile Erfc or gaussian J small x x C x looks flatter at high conc regions EE143 Ali Javey Electric field Enhancement Example Acceptor Diffusion Na x p x Na x Na x hole gradient Hole diffusion tendency E build in x Complete Completeacceptor acceptor ionization ionizationat atdiffusion diffusiontemperature temperature At Atthermal thermalequilibrium equilibrium hole holecurrent current 0 0 Hole Holegradient gradientcreates createsbuild in build in electric electricfield fieldto tocounteract counteractthe thehole hole diffusion diffusiontendency tendency EE143 Ali Javey Electric Field Enhancement p holes tend to move away due to hole concentration gradient Ebuild in B B acceptors experience an additional drift force Enhanced Diffusion for B acceptor atoms EE143 Ali Javey Electric Field Enhancement Substrate Perturbation As diffusion caused by As conc gradient Uniform B conc in substrate B EE143 Ali Javey
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