University of California Berkeley MSE 200A Fall 2017 Jumps of Interstitial

University of california berkeley mse 200a fall 2017

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University of California, Berkeley MSE 200A Fall, 2017 Jumps of Interstitial Atoms Atom motion Must overcome activation barrier G m to move from site to site (~ 1eV) Attempts barrier with vibrational frequency ν e ~ 10 14 /sec Number of jumps per unit time ( ω ) ω = (# attempts/time)(probability of jump/attempt) = ν e exp Δ G m kT = ν exp Q m kT (of the order of 10 5 /sec at room T) D G m G x
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Jumps of Substitutional Atoms For a substitutional atom to jump There must be a neighboring vacancy to permit the jump The atom must overcome the activation barrier to perform the jump ω = [P(vacant site)][P(jump given vacant site)] = [ c v ] ν exp Q m kT = exp Q v kT ν exp Q m kT = ν exp ( Q v + Q m ) kT
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Concentration-Driven Diffusion Diffusion by interchange of atoms on adjacent planes J = J 12 J 21 = nD dc dx J 12 = N 1 ω x = nc 1 a ω x N 1 = atoms of type 1 per unit area n = atom sites per unit volume a = volume per unit area of plane ω x = jumps in the x-direction/time J = J 12 J 21 = na ω x ( c 1 c 2 ) = 1 6 na ω ( c 1 c 2 ) = 1 6 na 2 ω dc dx J 21 c 1 c 2 J 12 a D = 1 6 a 2 ω
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Concentration-Driven Diffusion J 21 c 1 c 2 J 12 a Diffusion by interchange of atoms on adjacent planes Random-walk diffusion Atoms do not jump preferentially in either direction Net flux because there are more atoms on plane 1 than on 2 Diffusivity governs random-walk diffusion D = 1 6 a 2 ω = 1 6 a 2 ν exp Q D kT D = D 0 exp Q D kT Q D = Q m interstitial Q V + Q m substitutional J = J 12 J 21 = nD dc dx
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Random-Walk Diffusion Let an atom move by random steps of length a Its position at time t is X (t) X X ( t ) = 0 X 2 ( t ) = X X = na 2 = ω a 2 t x 2 = 1 3 X 2 = 1 3 ω a 2 t = 2 Dt x = x 2 1/ 2 = 2 Dt ( X and - X are equally likely)
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Application: Homogenization time Solidification usually results in chemical heterogeneities Represent heterogeneity with a sinusoid of wavelength, λ Composition should homogenize when The approximate homogenization time is: t > t h = 1 2 D λ 2 2 = λ 2 8 D t h = 1 8 D 0 λ 2 exp Q D kT Homogenization time - increases with λ 2 - decreases exponentially with T x > λ / 2
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J.W. Morris, Jr. University of California, Berkeley MSE 200A Fall, 2017 Microelectronic devices have built-in heterogeneities Can function only as long as these doped regions survive To estimate the limit on service life , t s Let doped island have dimension, λ Device is dead when ~ λ /2 , hence Application: Service Life of a Microelectronic Device t s << t h = 1 2 D λ 2 2 = λ 2 8 D t h = 1 8 D 0 λ 2 exp Q D kT Service life
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  • Fall '08
  • Staff
  • Thermodynamics, University of California, Phase transition, J.W. Morris

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