《Microelectronics Process》Some useful equations for Quiz

3155J/6.152J, Fall tern,2003 Some useful equations for Quiz# 2 Constants:kB=1.38×1023J/K=862×105eV/K,e=16×101Coul I atm=760 Torr=10 Pa Ideal gas: Pv=NkBT N/=n=C(concentration), J Gas kinetics: v= Mean free path: n d= molecular diameter ndp v2mk, Im m=species mass Energy transfer in collision of Mi and M, AE=E 4M, M2-c0520 (M+M2) Reliability: mean time to failure: MTTF =t f(ndt or acJ-mne* Electromigration J=cv,=c DF DqP (number/m* x sec) Ja is flux of species A, having a concentration of CA, where flux is due to current J in a medium of electrical resistivity is p. at temperature, T, by grain-boundary diffusion D=Dexp//(kgT/ Change in concentration due to electromigration and temperature gradient dt o eT Electrical: Resistivity, R-PIA(Ohm), mobility, Capacitance, C= KEA/d(Farad), &=8.85 x 10(F/m) Crystal growth: vacancy concentration: n,c =no expF-Eg /kg TEg=2.6ev no=5x- Si/cm Oxygen concentration 2×102exp-1.03e/k2T Maximum crystal growth velocity: v k. aTI solid

3.155J/6.152J, Fall Term, 2003 Some useful equations for Quiz # 2 Constants: kB = 1.38 × 10-23 J/K = 8.62 × 10-5 eV/K, e = 1.6 × 10-19 Coul. 1 atm = 760 Torr = 105 Pa x Ideal gas: PV = NkBT, N/V = n = C (concentration), J = nv . x 2 kBT Gas kinetics: v = 2kBT , Mean free path: λ = 2πd2 P , d = molecular diameter. x πm P J = 2πkBTm , m = species mass. ∆E = E1 4 M1M2 ) 2 cos Energy transfer in collision of M1 and M2: (M1 + M2 2 θ ∞ ∝ J − n e + E a / kB T Reliability: mean time to failure: MTTF = ∫ t ⋅ f ( t ) dt or 0 Electromigration: J DAqZA Jρ A = cAvA = cA DAF = cA * (number/m2 x sec) RT RT JA is flux of species A, having a concentration of cA, where flux is due to current J in a medium of electrical resistivity is ρ.. at temperature, T, by grain-boundary diffusion obeying: DA = DA o exp[− Ea /(kBT)] Change in concentration due to electromigration and temperature gradients: ∂JA dcA dT = − ∂JA − dt ∂x ∂T dx v σ eτ = = Electrical: Resistivity, R =ρ l/A (Ohm), mobility, µ = E ne m * Capacitance, C = κε0A/d (Farad), ε0 = 8.85 x 10-12 (F/m) Crystal growth: vacancy concentration: n = n0 exp[− E /k vac g BT ] Eg = 2.6 eV n0 = 5 x 1022 Si/cm3 . Oxygen concentration: Coxy = 2 × 1022 exp[−1.03eV /kBT ] Maximum crystal growth velocity: vmax = ks ∂T  Lρ z m ∂  solid 1