Beyond Thermal Budget: Simple Kinetic Optimization in RTP

1. Beyond Thermal Budget: Simple Kinetic Optimization in RTP Lecture 13a Text Overview

2. Two Approaches to Kinetic Modeling • “Sophisticated” • Detailed kinetics • Treatment integrated with uniformity, strain… • Computationally intensive • “Not-so-sophisticated” • Simplified kinetics • Heuristic integration of process constraints • Computationally simple Here we employ second approach

3. Thermal Budget: Basic Idea • Definition: varies in common usage • Product of T and t • Area under a T-t curve • Area under a D-t curve • Principle: minimum budget optimizes against unwanted rate processes • Diffusion • Interface degradation • Many others…

4. Merits of the Concept • Positives • Appealing metaphor: like fiscal budget • Successfully predicts kinetic advantages of RTP over conventional furnaces • Negatives • Definition varies, ambiguous treatment • Focuses excessively on initial, final states • Tends to ignore transformations during ramp • Ignores rate selectivity

5. Controlled Test of the Concept • Simultaneous Si CVD (desired) and dopant diffusion (undesired) • Undoped Si on B-doped Si • Undoped Si on Cu-doped Si • Fix CVD thickness, measure dopant profile by SIMS • See whether budget minimization works for • Eundes > Edes • Eundes < Edes See R. Ditchfield and E. G. Seebauer, JES40 (1997) 1842

6. T-t Curves: B on Si • Undoped Si grown epitaxially on B-doped Si(100) • T>730°C, 57.5 nm • Hi T  lowest budget • Ediff = 3.6 eV Edep = 0.52 eV

7. Boron Profiles • Lowest budget (Hi T) gives worst spreading • Thermal budget prediction fails

8. T-t Curves: Cu in Si • Undoped Si grown epitaxially on Cu-doped Si(100) • T<700°C, 57.5 nm • Hi T  lowest budget • Ediff = 1.0 eV Edep = 1.5 eV

9. Copper Profiles • Lowest budget (Hi T) gives best spreading • Thermal budget prediction OK

10. Summary of Experiments • T-t budget minimization fails completely when Eundes > Edes • D-t minimization works • x2 = 6Dt is always smallest for minimum budget but • Ignores rate selectivity: doesn’t give unique prediction for T-t profile

11. Ambiguity of Budget Minimization Minimize t Minimize T • These scenarios give different kinetic results • Need consideration of rate selectivity

12. Setting the Stage: Desiderata • Typical “desired” rate processes • CVD • Silicidation • Oxidation/nitridation • Post-implant annealing/activation • Typical “undesired” rate processes • TED • Interface degradation • Silicide agglomeration

13. Problem Formulation • Restrict attention appropriately • Ignore strain, uniformity, control… • Focus on one desired, one undesired rate • Assume suitable rate data exist • Focus on integrated (not instantaneous) rates  r(t,T) dt vs. r(t,T)

14. Phenomenological View of Activation Energy • Ignore notions of “activation barrier” • Applies to single, elementary steps only • Qualitative view • Describes strength of T dependence • Higher Ea stronger T variation of rate • Quantitative description Ea = – d(lnK)/d(1/kBT)

15. Effects of Activation Energy • Eact higher • t varies strongly with T • High slope • Eact lower • t varies weakly with T • Low slope

16. A Subtle Distinction • We use t-T curves to represent actual time and temperature • We use T-t curves to represent total process times for design purposes

17. Rate Selectivity • Principle: • Processing Rules: • If Eundes > Edes favor low T • If Eundes < Edes favor high T • Corollaries: • Get to and from soak T (or max T in spike) as fast as possible • No kinetic advantage to mixture of ramp and soak At high T, rate with stronger T dependence wins

18. Eundes < Edes • Use fast ramp and cool • High T best, limited only by process constraints on Tmax

19. Eundes > Edes • Use fast ramp and cool • Low T best, limited only by process constraints on tmax

20. Accounting for Constraints: Examples • T constraints • Tmax wafer damage, differential thermal exp. • Tmin thermodynamics (dopant activation) • t constraints • tmax throughput • tmin equipment limitations (maximum heating rate)

21. Formulation of Constraints • Half-window: upper or lower limit • Most undesired phenomena: upper limit • Degree of interface degradation • Extent of TED • Some desired phenomena: lower limit • Defect annealing • Silicidation • Full window: upper and lower limit • Some desired phenomena • Film deposition • Oxidation

22. Window Collapse • Hopefully shaded area is nondegenerate! min max

23. Mapping of Constraints: Half Window • Eundes > Edes • Eundes < Edes

24. Mapping of Constraints: Full Window • Eundes > Edes • Eundes < Edes

25. Superposing other Process Constraints • Eundes > Edes

26. Final Optimization • From a kinetic perspective, it’s usually best to operate… • Better: along an edge of allowed window • Best: at a corner • Example: Eundes > Edes • Lowest T gives best selectivity • Lowest t gives best throughput • Alternatives: • Highest T gives best rate at cost of selectivity

27. Spike Anneals • Characteristics • No “soak” period • Sometimes very fast ramp (> 400°/C) • Motivation • Takes selectivity rules for high T to their logical extreme • Improved kinetic behavior, esp. in post-implant annealing

28. An Idealized Spike • Assume: • Linear ramp up at rate  (C/s) • Cooling by radiation only • Constant emissivity • Surroundings at negligible temperature • Assumptions satisfactory only for semiquantitative results

29. Mathematical Analysis • Simplified kinetic expressions • Desired differential: r = A exp(–Ed /kT) integral: R=  r dt • Undesired integral: x 2= 6Dt = 6Do exp(–Eu /kT)t

30. Integrated Rates during Ramp Up • Ramp up: • Desired • Undesired • These integrals need an approximation to evaluate analytically

31. Laplace Asymptotic Evaluation • Integrals like have the form: for y >> 1 • Let so that • Thus and so

32. Behavior of Laplace Approximation • With E/kTM  30, approximation is good to ~7% • More accurate (1%) approximation comes from the Incomplete Gamma Function • See E. G. Seebauer, Surface Science, 316 (1994) 391-405.

33. Laplace Approximations to Rates during Ramp Up • Ramp up • Desired • Undesired • Note: 1/ trades off with E the way t does in non-ramp expressions

34. Effects of Activation Energy • Eact higher • t varies strongly with T • High slope • Eact lower • t varies weakly with T • Low slope

35. Laplace Approximations to Rates during Cool-Down • Cool down: • Desired • Undesired

36. Total Integrated Rates • Desired • Undesired • Control variables:  and TM only • If  >> CTM4, increasing  brings little extra return

37. Mapping of Constraints: Half Window • Eundes > Edes • Eundes < Edes

38. Mapping of Constraints: Full Window • Eundes > Edes • Eundes < Edes

39. Mapping of Constraints: Full Window • Optimal point shown to give most throughput

40. Summary • Concept of “thermal budget” problematic • Rate selectivity more reliable • Simple graphical procedure helps conceptualize 2-rate problems, including constraints • Framework can be generalized to 3 or more rates • Laplace approximation useful for variable-T applications

41. For Further Reference • R. Ditchfield and E. G. Seebauer, “General Kinetic Rules for Rapid Thermal Processing,” Rapid Thermal and Integrated Processing V (MRS Vol. 429, 1996), 133-138. (General rules, child metaphor) • E. G. Seebauer and R. Ditchfield, “Fixing Hidden Problems with Thermal Budget,” Solid State Technol.40 (1997) 111-120. (Review, expt’l data, mapping concepts) • R. Ditchfield and E. G. Seebauer, “Rapid Thermal Processing: Fixing Problems with the Concept of Thermal Budget,” J. Electrochem. Soc., 144 (1997) 1842-1849. (Detailed expt’l data) • R. Ditchfield and E. G. Seebauer, “Beyond Thermal Budget: Using Dt in Kinetic Optimization of RTP,” Rapid Thermal and Integrated Processing VII (MRS Vol. 525, 1998), 57-62. (More mapping concepts) • E. G. Seebauer, “Spike Anneals in RTP: Kinetic Analysis,” Advances in Rapid Thermal Processing (ECS Vol. 99-10, 1999) 67-71. (Extension of concepts to spikes)