EE143 F2010 Lecture 10 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 SiO 2 SiO2 SiO2 Si Note Predeposition by diffusion can also be replaced by a shallow implantation step Professor N Cheung U C Berkeley 1 EE143 F2010 Lecture 10 Dopant Diffusion Sources a Gas Source AsH3 PH3 B2H6 b Solid Source BN Si BN Si B oxide SiO2 c Spin on glass Professor N Cheung U C Berkeley SiO2 dopant oxide 2 EE143 F2010 Lecture 10 d Liquid Source Professor N Cheung U C Berkeley 3 EE143 F2010 Lecture 10 Solid Solubility of Common Impurities in Si oC C0 cm 3 Professor N Cheung U C Berkeley 4 EE143 F2010 Lecture 10 Diffusion Coefficients of Impurities in Si D DO e EA kT 10 6 Cu 10 12 10 13 10 14 Professor N Cheung U C Berkeley B P Au As 5 EE143 F2010 Lecture 10 Temperature Dependence of D D D 0e E A E A activation energy k Boltzman 8 6 10 D 0 Arrhenius Relationship kT E Professor N Cheung U C Berkeley A in eV constant 5 eV are tabulated kelvin 6 EE143 F2010 Lecture 10 Mathematics of Diffusion C x J x Fick s First Law C x t J x t D x D diffusion D cm Professor N Cheung U C Berkeley constant 2 sec 7 EE143 F2010 Lecture 10 From the Continuity Equation C x t J x t 0 t C x t J x t C x t D t x x x C x t C x t D t x x Diffusion Equation Professor N Cheung U C Berkeley 8 EE143 F2010 Lecture 10 Concentration independence of D If D is independent of C i e D is independent of x C x t C x t D 2 t x 2 Concentration Independent Diffusion Equation State of the art devices use fairly high concentrations causing variable diffusivity and other significant sideeffects transient enhanced diffusion for example Professor N Cheung U C Berkeley 9 EE143 F2010 Lecture 10 A Predeposition Diffusion Profile Boundary Conditions C x 0 t C0 solid solubility of the dopant C x t 0 Justification Si wafers are 500um thick doping depths of interest are typically several um Initial Condition C x t 0 0 Professor N Cheung U C Berkeley At time 0 there is no diffused dopant in substrate 10 EE143 F2010 Lecture 10 Diffusion under constant surface concentration 2 C x t C 0 1 x 2 Dt 0 e y2 dy x C 0 erfc 2 Dt 2 Dt Characteristic distance for diffusion C 0 Surface Concentration solid solubility limit C0 t3 t2 t1 x 0 Professor N Cheung U C Berkeley t2 t1 x 11 EE143 F2010 Lecture 10 Properties of Error Function erf z and Complementary Error Function erfc z z 2 y2 e dy erf z 0 erf 0 0 2 erf z z for z 1 d erf z d erfc z 2 z 2 e dz dz d2 erf z 4 z 2 ze dz2 z erfc z 1 erf z erf 1 2 1 e z erfc z for z 1 z 1 2 z erfc y dy z erfc z 1 e 0 Professor N Cheung U C Berkeley erf 1 1 erfc z dz 0 12 EE143 F2010 Lecture 10 Practical Approximations of erf and erfc The value of erf z can be found in mathematical tables as build in functions in calculators and spread sheets If you have a programmable calculator this approximation is accurate to 1 part in 2 107 erf z 1 a1T a2T2 a3T 3 a 4T4 a5T 5 e z 1 where T 1 P z and P 0 3275911 a1 0 254829592 a2 0 284496736 a3 1 421413741 a4 1 453152027 a5 1 061405429 1 10 1 exp z 2 10 2 erfc z 10 3 10 4 10 5 10 6 10 7 0 0 2 0 4 0 6 0 8 1 Professor N Cheung U C Berkeley 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 3 3 2 3 4 3 6 13 EE143 F2010 Lecture 10 1 Predeposition dose Q t 0 C x t dx C 0 2 Dt t 2 Concentration gradient C Co e x Dt Professor N Cheung U C Berkeley x2 4 Dt 14 EE143 F2010 Lecture 10 B Drive in Profile Boundary Conditions C x t 0 C x 0 x 0 Physical meaning of C x 0 No diffusion flux in out of the Si surface Therefore dopant dose is conserved Initial Conditions C x t 0 Co erfc 2 x Dt C x x 0 x Predep s Dt Professor N Cheung U C Berkeley 15 EE143 F2010 Lecture 10 C x t 0 Q x Solution of Drive in Profile with Shallow Predeposition Approximation Q C0 2 Dt predep C x t 0 Approximate predep profile as a delta function at x 0 x C x t t1 t2 Q C x t e Dt drive in x 2 4 Dt drive in x Professor N Cheung U C Berkeley 16 EE143 F2010 Lecture 10 How good is the x approximation Let C x C0 R Dt Dt Professor N Cheung U C Berkeley predep drive in Approximation over estimates conc here R 1 Good agreement For reference only R 0 25 Exact solution Delta function Approximation Approximation under estimates concentration here x 17 EE143 F2010 Lecture 10 Summary of Predeposition Drive in D1 Diffusivity at Predeposition temperature t1 Predeposition time D2 Diffusivity at Drive in temperature t2 Drive in time 2C 0 C x D 1t1 D 2t2 1 2 x2 e 4 D 2t2 This will be the overall diffusion profile after a shallow predeposition diffusion step followed by a drive in diffusion step Professor N Cheung U C Berkeley 18 EE143 F2010 Lecture 10 Semilog Plots of normalized Concentration versus depth Predeposition Professor N Cheung U C Berkeley Drive in 19 EE143 F2010 Lecture 10 Diffusion of Gaussian Implantation Profile Note is the implantation dose Professor N Cheung U C Berkeley 20 EE143 F2010 Lecture 10 Diffusion of Gaussian Implantation Profile arbitrary Rp C 0 a t x 0 i e n o d o p a n t lo s s th r o u g h s u r fa c e x c a n b e c o n s tr u c te d b y a d d in g a n o th e r fu ll g a u s s ia n p la c e d a t R p M e th o d o f Im a g e s T h e e x a c t s o lu tio n s w ith C x t 2 2 R p 2 D t 1 2 x R p 2 2 e 2 R p 2 D t x R p …
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