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Beyond Thermal Budget: Simple Kinetic Optimization in RTP. Lecture 13a Text Overview. Two Approaches to Kinetic Modeling. “Sophisticated” Detailed kinetics Treatment integrated with uniformity, strain… Computationally intensive “Not-so-sophisticated” Simplified kinetics

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two approaches to kinetic modeling
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

thermal budget basic idea
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…
merits of the concept
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
controlled test of the concept
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

t t curves b on si
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
boron profiles
Boron Profiles
  • Lowest budget (Hi T) gives worst spreading
  • Thermal budget prediction fails
t t curves cu in si
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
copper profiles
Copper Profiles
  • Lowest budget (Hi T) gives best spreading
  • Thermal budget prediction OK
summary of experiments
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
ambiguity of budget minimization
Ambiguity of Budget Minimization

Minimize t

Minimize T

  • These scenarios give different kinetic results
  • Need consideration of rate selectivity
setting the stage desiderata
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
problem formulation
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)

phenomenological view of activation energy
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)

effects of activation energy
Effects of Activation Energy
  • Eact higher
  • t varies strongly with T
  • High slope
  • Eact lower
  • t varies weakly with T
  • Low slope
a subtle distinction
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
rate selectivity
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

e undes e des
Eundes < Edes
  • Use fast ramp and cool
  • High T best, limited only by process constraints on Tmax
e undes e des1
Eundes > Edes
  • Use fast ramp and cool
  • Low T best, limited only by process constraints on tmax
accounting for constraints examples
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)
formulation of constraints
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
window collapse
Window Collapse
  • Hopefully shaded area is nondegenerate!

min

max

mapping of constraints half window
Mapping of Constraints: Half Window
  • Eundes > Edes
  • Eundes < Edes
mapping of constraints full window
Mapping of Constraints: Full Window
  • Eundes > Edes
  • Eundes < Edes
final optimization
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
spike anneals
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
an idealized spike
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
mathematical analysis
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

integrated rates during ramp up
Integrated Rates during Ramp Up
  • Ramp up:
    • Desired
    • Undesired
  • These integrals need an approximation to evaluate analytically
laplace asymptotic evaluation
Laplace Asymptotic Evaluation
  • Integrals like have the form:

for y >> 1

  • Let

so that

  • Thus

and

so

behavior of laplace approximation
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.
laplace approximations to rates during ramp up
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
effects of activation energy1
Effects of Activation Energy
  • Eact higher
  • t varies strongly with T
  • High slope
  • Eact lower
  • t varies weakly with T
  • Low slope
total integrated rates
Total Integrated Rates
    • Desired
    • Undesired
  • Control variables:  and TM only
  • If  >> CTM4, increasing  brings little extra return
mapping of constraints half window1
Mapping of Constraints: Half Window
  • Eundes > Edes
  • Eundes < Edes
mapping of constraints full window1
Mapping of Constraints: Full Window
  • Eundes > Edes
  • Eundes < Edes
mapping of constraints full window2
Mapping of Constraints: Full Window
  • Optimal point shown to give most throughput
summary
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
for further reference
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)