1. A closer look at Diffusion: Part II March 2001 D.G. Ast

2. Derivation of Diffusion Equation

3. A. Derivation of Diffusion Equation Gradient in Chemical Potential Activation Barrier Jump Distance

4. Forward jumps: f = o exp((S-A)/kT) Backward jumps: b = o exp((S-B)/kT) Net jumps: = f - b  E = A -B =  (net) = o exp(-E a /kT) 2 sinh(  /2kT))  G = 

5. 2 sinh (  /kT) =~ (  )/kT How good is this assumption ? Take: kT = 0.05 eV (about 600C) = 1E-8 cm = 1 Angstrom  = 1 eV/ 20Angstrom  G = 0.05 eV 2sinh (0.05/0.05) = 1.17 (  )/kT = 1  = 1 eV/ 2Angstrom  G = 0.5 eV (Delta Doping) 2sinh(10) = 11000 (  )/kT = 10 Will break down for delta doping

6. Using this expansion, we get = 0 (  )/(kT) For an ideal solution (no interactions) G =  * = kT * ln (C 2 /C 1 ) Velocity of jumping atom v = * = o 2  /(kT) The flux is F = v* C (where C = concentration) ;  = dG/dx F = V * C = (v o 2 C/kT)  (dG/dx)

7. dG/dx = (dG/dc)/(dc/dx) dG/dx = kT (dlnC/dC) = kT (1/C) F = - o exp(-E a /kT) 2 (dc/dx) F = - D (dc/dc) D = o 2 exp(-E a /kT) We now can calculate what the diffusion coefficient should be !

8. D = o 2 exp(-E a /kT) o = 1E14 (sec -1 ) = 3E-8 cm (three Angstroms) E a = 1.5 eV kT (1200 C) = 0.1 eV D = 0.09 exp (-1.5 eV/kT) For 1200 C: = 2.75E-8 cm 2 /sec Copper in Silicon: D = 0.04 exp(-1 eV/kT) Sodium in Silicon D= 0.0016exp(-0.72/kT) Summary Ficks First derivation requires  G/a 0 < kT Ideal solution behavior

9. II. Random walk model 1 D model Lattice site i Jumps to lattice site i + 1 Jumps to lattice site i -1 (dN i /dt) = - *N i + (1/2) N i+1 + (1/2) N i-1 Change in concentration is loss (proportional to numbers in cell) plus gain from neighbors ).

10. N i + (1/ )(dN i /dt) = + (1/2) N i+1 + (1/2) N i-1 New value Average over “old neighbors This is the origin of the relaxation approach to solve the diffusion equation. If we make the N a continuos function of position, than N i+1 = N i + (dN i /dx) |  x| N i-1 = N i - (dN i /dx) |  x| we see that the first terms in the Taylor expansion cancel. This leaves us with the second derivative dN/dt = D (d 2 N/dx 2 )

11. Yet an other approach is to look at the probability for an atom to jump out of a cell in time t (usually t = 1 sec). If this probability is called p, the # of atoms jumping out of the central cell, i, is equal to pN(i), and the number of atoms staying in the cell N(i) is equal to (1-p)N(i). As before the central cell will gain from atoms jumping in from left and right. The # of these atoms is (1/2) p N(i-1) + (1/2)p (i + 1) The new value of the cell is losses plus gain or N(i,  t) = (1/2) p N(i-1) + (1/2)p (i + 1) + (1-p)N(i). If you set p=1 you get the result of the previous slide. If you set p =0.5 you get a smoother, more stable numerical solution, at the price of doing twice as many steps.

12. Well, its hard to see but take my word, it is smoother ! ERFC

13. It is much clearer in the case of the Gaussian

14. Decay of an arbitrary diffusion profile N(x,t) = 0.25(N(x-1) +N(x+1) + 0.5(N(x)) (p= 0.5)

15. Diffusion Mechanisms

16. Vacancy mechanism: Exchange with vacancy Vacancy has higher D value than dopant atom When a vacancy becomes a neighbor dopant atom can exchange place with vacancy. Complications: Pair Formation (PV) “Back jumps” Dominant for Sb Important for As

17. Interstitial Cu, Fe, Au, H, …. Moves through “empty space” in silicon lattice Fast diffusers, as no vacancy or interstitial is involved. Some fast diffusers, notably Au have an equilibrium with Au sitting on lattice sites (substitutional Au)

18. Interstitialcy Interstitial diffuser kicks out Si lattice atom. If dwell time on Si site is short, difficult to separate experimentally from interstitial

19. Gradient dependent D = f(dc/dx) I/S diffusers (e.g. Au)

20. Interstitial(cy) / Substitutional Atom is ‘dissolved’ up to ‘solubility limit’ on lattice sites C s is high, D s is low A small fraction is dissolved as interstitial, C i is low, D i is large High diffusion gradient : Interstitial’s move away, V+ I = S mechanism delivers more I’s Low diffusion gradient: No conversion D becomes concentration gradientdependent

21. Fermi level effects “Extrinsic” Diffusion

22. ------------------ Ev -- (about 0.5 eV) + (about 0.35 eV) - (about 0.05 ev) -----------------------Ec [V - ] = [V t ] / (1 + exp((E - - E f )/kT) [V t ] = [Vo] + [V_] + [V__] + [V+] [V_ ] = [V o ] * exp ((E f - E_ )/kT) Fermi level effects

24. D = D o + D - (n/n i ) + D - - (n/n i ) 2 + D + (p/n i ) The D’s are function of the charge state, as are the E a since the bonding of the charged vacancy is different from that of the neutral vacancy. D as = 0.066exp(-3.44/kT) + 12 (n/n i ) exp(- 4.05 eV)

25. Extrinsic As diffusion

27. Electric Field effect Upper limit

29. J = -D(dc/dx) -  *c*E E = (1/e) (d/dx) (E f - E i ) E f - E i = kT ln (c/n i ) (assumes # charge = # dopants !!) E = (kT/e) * (d/dx) ln(c/n i ) = (kT/e) * (1/c)dc/dx  = (e/kT) D J = - D*(dc/dx) - D(e/kT) c*E = - D*(dc/dx) - D(e/kT) c (kT/e) * (1/c)dc/dx J = - 2D *(dc/dx) Upper limit of electric field effect on diffusion

30. Cluster Formation

31. Cluster Formation at high doping concentrations.

32. Dopant Segregation to Grain Boundaries

34. Electrical Activity of Dopants in Poly

35. At N<1E17 the dopant atoms do not provide mobile carriers.

36. The resistivity decreases with increasing film thickness. Control is epi - silicon

37. Dopant diffusivity in poly-Si

39. Experimental studies of point defects via diffusion experiments

40. Distance traveled by point defects

41. Range of interstitials is about 300 microns. Backside of wafer can influence front size of wafer

42. Stanford’s test structure

43. As diffusion

44. P diffusion

45. B diffusion

46. The Phosphorous Diffusion Profile

47. P profile is due to generation of interstititials by P. Interstitials outrun P, and kick up the tail profile. (Discarded model: P-V pairs)

48. Summary Point defects control diffusion of dopant atoms. Concentration of point defects is variable and influenced by long range effects (even events on the back side of the wafer) Point defects as well as dopants can be charged and interact with electric fields. In poly-silicon, both dopants and point defects interact with grain boundaries.