Atomistic picture of diffusion See web site for movies. http://www.tf.uni-kiel.de/matwis/amat/def_en/index.htm En 4 most important is vacancy diffusion. ○○○○○○ ●○00○ Initial and final states have same energy Also possible is direct exchange (x =broken bond) Higher energy barrier or break more bonds = lower value of d= d ex T Atoms that bond with si are substitutional impurities p, b, As, Al, Ga, Sb, Ge 6.155/3.155J9/29/
6.155/ 3.155J 9/29/03 7 Atomistic picture of diffusion See web site for movies: http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html Most important is vacancy diffusion. Ea i f En Initial and final states have same energy Also possible is direct exchange (¥ = broken bond) Higher energy barrier or break more bonds => lower value of D = D0 exp -E kT ( ) Atoms that bond with Si are substitutional impurities P, B, As, Al, Ga, Sb, Ge ¥ ¥ ¥ ¥
Atomistic picture of diffusion eps for diffusion: 1) create vacancy 2)achieve energy ea 2.6 ex Vo ex KT AG E VF一 S EVE eM→>e kT = ee k o kT e +e daxv=a'voexpl kT Vacancy diffusion D=d ex kT Contains vo= Debye frequency 9×10 10 6.155/3.155J9/29/
6.155/ 3.155J 9/29/03 8 2 steps for diffusion: 1) create vacancy 2) achieve energy Ea nv = Nv N0 = exp - 2.6 kT È ÎÍ ˘˚˙ nv = n0 exp - Ea kT ÊËÁ ˆ¯˜ D = D0 exp - EVD kT È Î ˘ ˚ D ~ a x v = a2n 0 exp - Ev + Ea kT ÈÎÍ ˘˚˙ cm2 s Ê ËÁ ˆ¯˜ Contains n0 § Debye frequency ª 3 2 kBT h = 9 ¥1012 s-1 @1013 s-1 e - DG kT Æ e - EVF -TS kT = e S k e - EVF kT Ê ËÁ ˆ¯˜ Atomistic picture of diffusion EVD a n0 Vacancy diffusion
Analytic Solution to Diffusion Equations, Fick ll: ac dt 0 Steady state dC/at=0 Conversely, C(z C(z=a+b if C(z)is curved dc/dt≠0 We assumed this to be the case in oxidation 02 diffusion through SiO2, where flux J C -Dd-=-Db, is same everywhere 6.155/3.155J9/29/
6.155/ 3.155J 9/29/03 9 Analytic Solution to Diffusion Equations, Fick II: ,C ,t = D ,2C ,z 2 Steady state, dC/dt = 0 C(z) = a + bz 0 z C(z) We assumed this to be the case in oxidation: O2 diffusion through SiO2 , where flux ,J = -D is same everywhere ,C,z = -Db, Conversely, if C(z) is curved, dC/dt 0
For other solutions, consider classical experiment Diffusion couple: thin dopant layer on rod face press 2 identical pieces together, heat Then study diffusion profile in sections symmetry dC(o. =0 C(x0)=0(x≠ 6.155/3.155J9/29/
6.155/ 3.155J 9/29/03 10 Diffusion couple: thin dopant layer on rod face, press 2 identical pieces together, heat. Then study diffusion profile in sections. For other solutions, consider classical experiment: symmetry 0 z t = 0 dC( ) 0,t dz = 0 C( ) •,t = 0 C z( ) ,0 = 0 ( ) z 0 C z( ) ,t -• • Ú dz = Q = const. (# /area)
Analytic Solution to Diffusion Equations J=-Dco d-c ot dz ."Drive in"of small, fixed amount of dopant, solution is Gaussian dC(o. _o d Predeposition is C(∞,)=0 delta function, 8(2). Xt=0 c(i a)=exp Units lTDt 4DE Width of Gaussian= a=2Dt=diffusion length a (a is large relative to width of predeposition) Dose, Q, amount of dopant in sample, is constant. 6.155/3.155J9/29/
6.155/ 3.155J 9/29/03 11 Analytic Solution to Diffusion Equations J = -D,C,z , ,C,t = D,2C,z 2 t = 0 dC( ) 0,t dz = 0 C( ) •,t = 0 C z( ) ,0 = 0 ( ) z 0 0 z I. “ Drive in” of small, fixed amount of dopant, solution is Gaussian Predeposition is delta function, d(z). C z( ) ,t = Q pDt exp - z 2 4Dt È ÎÍ ˘ ˚˙ t > 0 Units Width of Gaussian = = diffusion length a (a is large relative to width of predeposition) 1 2 a = 2 Dt Dose, Q, amount of dopant in sample, is constant