Useful equations kg=1.381023J/K=86210°eVK Gas kinetics: Pre Cave=(8kB T/ms I atm=760 torr: 10 Pa P=nbT an free path Arrival rate at a surface J=P/(2kBT m)0.=3.5 102P(torr)(MT(g/mol K). s m= mass of a molecule. M= molar mass Oxidation: thickness xo after time t 2+Axo=B(t+t) )+A(xo-Xi= Bt B/A=cok/N, B= 2DoxCo/N Co= conc. of oxidant at surface, ks= rate constant for oxidation, Dox= diffusivity of oxidant in oxide n= number of oxidant molecules needed to make unit volume of oxide. t= time offset X: initial oxide thickness Chemical vapor deposition v=(cg/NChg k/(hg+ k)) Reaction flux Transport flux Fd=hg(cg-Cs) Gas phase transport coefficient h2=3/2(D2/L)(puL/n)03=D26e Boundary layer thickness δ=(xn/pu Reaction coefficient k=kexp(-△G/kgT cg is concentration of reactant species in bulk of gas; cs is concentration on the surface, N is density of film(atoms/cm ), Dg is the diffusivity in the gas, L is a characteristic length, x is distance along wafer, p is gas density, n is viscosity, u is gas velocity. kB=1.38 10-J/K 86210eV/K Semiconductor conductivity G=( Hn=electron mobility, Hh=hole mobility, n=number of electrons, p=number of holes,e Diffusion Source with constant surface concentration: C(z, t)=Csurf erfc(-z/(2v(D) This introduces dose Q Q=2CsurfV(Dt)/ Fixed amount of dopant diffusing into wafer: C(z, t)=Q/(rDt)exp(-27/4Dt) z=distance, t=time, C= concentration(atoms/cm), Csurf=conc. at surface, Q=dose (atoms/cm) Effective Intrinsic Diffusivity Deff=D+D+D+D+ where D=D°exp(-E/kn,D=D。exp(-E/kT,etc Extrinsic Diffusivity DefT=D+(n/ni)D+(n/ni-D +(p/ni)DUseful equations kB = 1.38 10-23 J/K = 8.62 10-5 eV/K Gas kinetics: Pressure P = nkBT 1 atm = 760 torr ≈ 105 Pa Average speed cave = (8kBT/πm) 0.5 Mean free path λ = 1/(πd2 n√2) Arrival rate at a surface J = P/(2πkBT m) 0.5 = 3.5 1022 P(torr)/(MT (g/mol.K))0.5 m = mass of a molecule, M = molar mass. Oxidation: thickness xo after time t xo 2 + A xo = B (t + τ) or (xo 2 – xi 2 ) + A(xo – xi) = Bt B/A = coks/N, B = 2Doxco/N co = conc. of oxidant at surface, ks = rate constant for oxidation, Dox = diffusivity of oxidant in oxide, N = number of oxidant molecules needed to make unit volume of oxide, τ = time offset, xi = initial oxide thickness. Chemical vapor deposition film growth rate v = (cg / N)(hg k/(hg + k)) Reaction flux Fr = k cs Transport flux Fd = hg (cg -cs) Gas phase transport coefficient hg = 3/2 (Dg /L) (ρ u L/η)0.5 = Dg /δave Boundary layer thickness δ = (x η/ρ u ) 0.5 Reaction coefficient k = ko exp (-∆G/kBT) cg is concentration of reactant species in bulk of gas; cs is concentration on the surface, N is density of film (atoms/cm3 ), Dg is the diffusivity in the gas, L is a characteristic length, x is distance along wafer, ρ is gas density, η is viscosity, u is gas velocity. kB = 1.38 10-23 J/K, = 8.62 10-5 eV/K Semiconductor conductivity σ = (µnn + µhp)e µn = electron mobility, µh = hole mobility, n = number of electrons, p = number of holes, e = electronic charge 1.6 10-19 C. Difffusion: Source with constant surface concentration: C(z,t) = Csurf erfc (-z/(2√(Dt))) This introduces dose Q Q = 2Csurf√(Dt)/√π Fixed amount of dopant diffusing into wafer: C(z,t) = Q/√(πDt) exp (-z2 /4Dt) z = distance, t = time, C = concentration(atoms/cm3 ), Csurf = conc. at surface, Q = dose (atoms/cm2 ) Effective Intrinsic Diffusivity Deff = Do + D- + D= + D+ … where Do = Do o exp (-Eo /kT), D+ = D+ o exp (-E+ /kT), etc 2 D= Extrinsic Diffusivity Deff = Do + (n/ni)D- + (n/ni) + (p/ni)D+ …