Professor N Cheung, U.C. BerkeleyLecture 10EE143 F20101Dopant Diffusion(1) Predepositiondopant gasSiO2SiO2Sidose control(2) Drive-inTurn off dopant gasor seal surface with oxideSiO2SiO2SiSiO2Doped Si regionprofile control(junction depth;concentration)Note: Predeposition by diffusion can also bereplaced by a shallow implantation step.Note: Predeposition by diffusion can also bereplaced by a shallow implantation step.Professor N Cheung, U.C. BerkeleyLecture 10EE143 F20102Dopant Diffusion Sources(a) Gas Source: AsH3, PH3, B2H6(b) Solid SourceBN Si BN SiB oxide +SiO2(c) Spin-on-glassSiO2+dopant oxideProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F20103(d) Liquid Source.Professor N Cheung, U.C. BerkeleyLecture 10EE143 F20104Solid Solubility of Common Impurities in SioCC0(cm-3)Professor N Cheung, U.C. BerkeleyLecture 10EE143 F20105Diffusion Coefficients of Impurities in Si10-1410-1310-12B,PAs10-6AuCukTEOAeDDProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F20106Temperature Dependence of D.,/106.8050tabulatedareEDkelvineVconstantBoltzmankeVinenergyactivationEeDDAAkTAEArrhenius RelationshipProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F20107Mathematics of DiffusionFick’s First Law: sec][:,,2cmDconstantdiffusionDxtxCDtxJJC(x)xProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F20108From the Continuity Equation x)t,x(CDxt)t,x(Cx)t,x(CDxx)t,x(Jt)t,x(C0t,xJtt,xC“Diffusion Equation”Professor N Cheung, U.C. BerkeleyLecture 10EE143 F20109If D is independent of C(i.e., D is independent of x).C x ttDC x tx, ,22Concentration Independent Diffusion EquationConcentration Independent Diffusion EquationConcentration independence of DState of the art devices use fairly high concentrations,causing variable diffusivity and other significant side-effects (transient-enhanced diffusion, for example.)State of the art devices use fairly high concentrations,causing variable diffusivity and other significant side-effects (transient-enhanced diffusion, for example.)Professor N Cheung, U.C. BerkeleyLecture 10EE143 F201010-- Boundary ConditionsInitial ConditionC x t C solid solubility of the dopantC x tC x t::,,,000 00A. Predeposition Diffusion ProfileJustification:Si wafers are ~500um thick, dopingdepths of interest are typically < several umAt time =0, there is no diffused dopantin substrateProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201011 002002221,2CDtDtxerfcCdyeCtxCDtxyC0t3>t2t2>t1x=0xt1Characteristic distance for diffusion.Surface Concentration (solid solubility limit)Diffusion under constant surface concentrationProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201012erf (z) =2 0 z e-y2 dy erfc (z) 1 - erf (z)erf (0) = 0 erf() = 1 erf(- ) = - 1erf (z)2 z for z <<1 erfc (z)1e-z2z for z >>1d erf(z)dz = -d erfc(z)dz =2 e2-zd2 erf(z)dz2 = -4 z e2-z0z erfc(y)dy = z erfc(z) +1 (1-e-z2 )0 erfc(z)dz =1Properties of Error Function erf(z)and Complementary Error Function erfc(z)Professor N Cheung, U.C. BerkeleyLecture 10EE143 F201013The value of erf(z) can be found in mathematical tables, as build-in functions in calculators andspread sheets. If you have a programmable calculator, this approximation is accurate to 1 part in107: erf(z) = 1 - (a1T + a2T2+a3T3+a4T4+a5T5) e-z2where T =11+P z and P = 0.3275911a1= 0.254829592 a2= -0.284496736 a3= 1.421413741 a4= -1.453152027 a5= 1.06140542910-710-610-510-410-310-210-110 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6erfc(z)exp(-z^2)Practical Approximations of erf and erfcProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201014 DtxeDtCoxCtDtCdxtxCtQ422,00[1] Predeposition dose[2] Concentration gradientProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201015B. Drive-in Profile --DtxerfcCotxCConditionsInitialxCtxCConditionsBoundaryx20,:00,:0xC(x)x=0Predep’s (Dt)Physical meaning of C/x =0:No diffusion flux in/out of the Sisurface. Therefore, dopant dose isconservedProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201016 xQtxC 0,C(x,t=0)xSolution of Drive-in Profilewith Shallow Predeposition Approximation:C x tQDtedrive inxDtdrive in, 24C(x,t)xt1t2QC Dtpredep02Approximate predep profileas a delta function at x=0Professor N Cheung, U.C. BerkeleyLecture 10EE143 F201017indrivepredepDtDtRLetxApproximation over-estimates conc. hereApproximationunder-estimatesconcentration here.GoodagreementC(x)/C0R=1R=0.25Exact solutionDelta functionApproximationHow good is the (x) approximation ?For reference onlyFor reference onlyProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201018Summary of Predeposition + Drive-in 2224212211022112tDxetDtDCxCtDtDDiffusivity at Predeposition temperaturePredeposition timeDiffusivity at Drive-in temperatureDrive-in time*This will be the overall diffusion profile after a “shallow” predepositiondiffusion step, followed by a drive-in diffusion step.Professor N Cheung, U.C. BerkeleyLecture 10EE143 F201019PredepositionPredepositionDrive-inDrive-inSemilog Plots of normalized Concentration versus depthProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201020Note: is the implantation doseNote: is the implantation doseDiffusion of Gaussian Implantation ProfileProfessor N Cheung, U.C. BerkeleyLecture 10EE143 F201021T h e e x a c t s o lu tio n s w ith C x= 0 a t x = 0 (.i.e . n o d o p a n t lo s s th ro u g h s u r fa c e ) c a n b e c o n s tru 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 -Rp[M e th o d o fIm a g e s ].C (x , t) =2 ( R2p + 2 D t)1 /2 [ e-(x - Rp)22 ( R2p + 2 D t) + e-(x + Rp)22 ( R2p + 2 D t) ]W e c a n s e e th a t in th e lim it (D t)1 /2 > > Rp a n d Rp ,C (x ,t) e- x2/4 D t(D t)1 /2(th e h a lf-g a u s s ia n
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