Models and Simulations

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Models and Simulations
Numerical Solutions of the Diffusion Equations.
No predetermined boundary conditions - no
analytical solutions
Planar density of atoms
Hopping to a vacancy or exchange places with
frequency vb
vd - Debye frequency 1013sec-1 in Si
Atoms jump between planes (left-right
probabilityright-left) They have to surmount
the energy barrier Eb
For stable numerical solution
Max. numerical interval means that we cannot have
more than available number of atoms to jump in ?t
Average volume concentration CiNi?xcm-3
Result
Adjacent concentrations calculated ? SUPREM etc.
solve for any dopant profile
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Modifications Of Fick's Laws
A. Electric Field Effects
When the doping is higher than ni, E field
effects become important. E field induced by
higher mobility of electrons and holes
compared with dopant ions. E field enhances the
diffusion of dopants causing the field (see
derivation in text).
SUPREM simulation at 1000C. Note the boron
profile (h 2 for the As but E field effects
dominate the B diffusion - B loss into S/D
n regions). Field effects can dominate the
doping distribution near the source/drain of
a MOS device.
B
Uniform B
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Modification to Ficks Loss to Account for
Electric Field Effects.
Built-in E-field enhances diffusion
Very strong effect of ? at low concentrations
Set up by the dopants _at_ increased concentrations
As
The potential
Donors will drift
B
Flux due to E-field
F.-II Law
The enhancement of diffusion by the electric
field is by a factor of two _at_ high concentrations.
This equation is solved in simulators such as
SUPREM under equilibrium conditions Useful when E
is due to multiple dopants
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B. Concentration Dependent Diffusivity
BORON
At high doping concentrations, the
diffusivity appears to increase. Fick's
equation must then be solved numerically
since D ? constant.
Experiment.
erfc
Boxlike
5x1020cm-3
1018cm-3
erfc
Isoconcentration experiments indicate the
dependence of D on concentration e.g. B10 in
a B11 background.
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Modifications to Ficks Laws to Account for
Concentration Dependent Diffusion
ISO Concentration experiments B11 (substrate)
B10 diffusant give DAeff
Original Ficks Law depended on gradient not on
C, now Df(C) ?Deff(C)
Often, D is well described by
The n and n2 (p and p2 for P type dopants)
terms are due to charged defect diffusion
mechanisms.
Neutral and charged point defects
For intrinsic conditions
Different Activation Energies
Ex As at 1000C For 11018cm-3?DAs 1.4310-15
cm2 sec-1 For 11020cm-3?DAs
1.6510-14 cm2 sec-1
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SUPREM Simulation
1000C/30min
CED Boxlike
BJT
As
Emitter
Base
Due to E-field
Boron
Collector
Phosphorus
SUPREM simulation including E field and
concentration dependent D effects. Note
E field effects around junctions (As determines
the field). Steep As profile -
concentration dependent effects. Boron in
the base region does not show concentration
dependent effects. Boron inside As is
slowed down by concentration dependent effects.
Phosphorus diffusing up inside As profile is
affected by concentration effects.
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Segregation of Dopants at Interfaces
Dopants segregate at interfaces This gives an
interface flux of point defects
N-type dopants tend to pile-up while boron
depletes (SUPREM simulation).
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Segregation of Dopants at Interfaces
Oxidation of a uniformly doped boron substrate
depletes the boron into the growing SiO2 (SUPREM
simulation).
A
B
Ex Oxide Silicon
_at_ the interface
A ?B
Flux
B segregates into the oxide P piles up at the Si
Surface ? Snowplow during oxidation.
Segregation h is the interface
transfer coefficient. h and ko are exponentially
activated
This gives an interface flux of
SIMS - for thick oxides can give ko (difficult
measurements). Matrix effect, intermixing, yield.
Effect of dopant diffusion in the cap, ambient
role on diffusion and segregation. - for thin
oxides SIMS is inadequate (no steady state
reached). So use VT or C-V areas.
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Segregation Interfacial Dopant Pileup.
DAS ltlt Dp ? steeper profile close to the
interface.
Dopants may also segregate to an interface layer,
perhaps only a monolayer thick. Interfacial
dopant dose loss or pile-up may consume up to 50
of the dose in a shallow layer
In the experiment (right) 40 of the dose was
lost in a 30 sec anneal.
Kasnavi et. al.
Dopant Loss at the surface - SIMS does not detect
that (see the interface).
Important problem in small devices (dominant).
Entrapped dopants can diffuse later. SUPREM
includes a trapping flux.
Very thin ? monolayer acts a sink for dopants
(inactive there) Stripping of the oxide removes
dopants.
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The Physical Basis for Atomic Scale Diffusion
Fick's first law macroscopically describes
dopant diffusion at low concentrations.
"Fixes" to this law to account for experimental
observations (concentration dependent
diffusion and ? field effects), are useful, but
at this point the complexity of the "fixes"
begins to outweigh their usefulness. Many
effects (OED, TED etc) that are very important
experimentally, cannot be explained by the
macroscopic models discussed so far. ? We turn to
an atomistic view of diffusion for a deeper
understanding.
Kick - Out
The role of interstitials-kick-out
Vacancy Assisted or Exchange Mechanism
Kick-out and Interstitial(cy) Assisted
Mechanisms (Identical from a mathematical
viewpoint.)
Typical diffusion in metals ? use XRD to measure
changes in lattice constant with T ? extract
vacancy concentration. For Si it is below the
detection limit of XRD. V models were used
earlier for diffusion in Si especially CED Nv
f(EF)
Dopant and Si-I diffuse as bound pairs ? split
Dopant stays in the substitusional position, I
is released.
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A. Modeling I And V Components Of Diffusion
Oxidation provides an I injection source.
Nitridation provides a V injection source.
Stacking faults serve as "detectors" as do
dopant which diffuse. Experiments like these
have "proven" that both point defects are
important in silicon. Therefore,
Thus dopant diffusion can be enhanced or
retarded by changes in the point defect
concentrations.

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OED and ORD
Measurements of enhanced or retarded diffusion
under oxidizing or nitriding conditions allow
an estimate of the I or V component of diffusion
to be made. Oxidation injects interstitials,
raises and reduces
through I-V recombination in the bulk
silicon. Nitridation does exactly the opposite.
TSUPREM IV simulations of oxidation
enhanced diffusion of boron (OED) and
oxidation retarded diffusion of antimony
(ORD) during the growth of a thermal oxide on
the surface of silicon. The two shallow
profiles are antimony, the two deeper
profiles are boron. Note that the
and profiles are relatively
flat indicating the stiff source
characteristic of the oxidation process.
Sb
B
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also OISF ? OISF ?.
Oxidation Enhanced or Retarded Diffusion
B, P enhanced by oxidation Sb retarded by
oxidation.
Two different mechanisms of diffusion by
interstitials and by vacancies.

• Oxidation injects I (relieve of larger volume)
• OE DB,P ( I ?)
• ORDSb (V ?)

B, P
Larger atoms by vacancy mechanism
OISF
Concentration of interstitials increases with
oxidation rate.
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Dopant Diffusion Occurs by Both I and V
Recent consensus confirmed by computer simulations
fI a fraction via interstitial type
mechanism. fV 1 fI a fraction via vacancy
type mechanism CI, CV ? CI, CV
Means equilibrium
Affected by point defects
Inert conditions
Surface oxidized
fIfv1
Sb retarded even deep in the Si substrate
Oxidation Retardation condition point defects
equilibrium
Oxidation As enhanced Sb
retarded
Because point defects (I in oxidation) affect D
constants
CI/CI1 For excess of I Cv/Cv0
Inert Ambient
Dopants, whose diffusivity decreases in oxidation
to 1/10 must diffuse at least 9/10 by a vacancy
mechanism
Nitridation B, P, As retarded diffusion
Sb enhanced diffusion
Injection of vacancies
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For Dopants which Show Oxidation Enhanced
Diffusion
_at_ low concentrations
Interstitials and vacancies assisted diffusion
have different. D0 and EA in respective diffusion
constants
Conservative Assumptions
Enhancement wont exceed I super saturation. CI
vs. CI
Diffusivities of mobile species
In steady state IV recombine
Estimations of CI and CV from OISF growth or
annihilation less accurate than from diffusion of
dopants.
computed
I I Dual V
Activation Energy for Self-Diffusion (Si) and
Dopant Diffusion. EAdopant (3-4 eV) lt EASi (4-5
eV)
Lowering of the activation barrier Coulombic
interaction, Dopant interaction, ex. relaxation
of Si lattice (atom size), electronic interaction
(charge transfer)
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B. Modeling Atomic Scale Reactions
Consider the simple chemical reaction This
contains a surprising amount of physics. For
example OED is explained because oxidation
injects I driving the equation to the right,
creating more AI pairs and enhancing the
dopant D.
A V ? AV AI ?AI
mobile bound pair or a substitutional Dopant
kicked out by I
Not mobile in a substitutional position
I? Increased diffusion of a Dopant A (P, B) I?
decreased diffusion of Dopant A P, B)
increased diffusion of Sb I V ?SiS
Oxidation Nitridation
Pumping of point defects
AI AI
I supersaturation CED
P concentration high
B CED due to I
Phosphorus diffuses with I, and releases them in
the bulk. This enhances the tail region D.
"Emitter push" is also explained by this
mechanism.
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Dopant-Defect Interaction
Chemical Equilibrium Formulation for
Dopant-Defect Interaction
In chemical equilibrium mobile species form
quickly
AI AI
measured
Intrinsic equilibrium value (measured)
Applying Ficks law to the mobile species
Applying the chain rule from calculus, Thus,
gradients in defects as well as gradients in
dopant concentrations can drive diffusion
fluxes. The overall flux equation solved by
simulators like SUPREM (see text) is
(written for boron diffusing with neutral and
positive interstitials as an example).
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Thus there are several distinct effects that
drive the dopant diffusion inert, low
concentration diffusion, driven by the dopant
gradient the interstitial
supersaturation high concentration
effects on the dopant diffusivity the
electric field effect Compare the above
expression with Ficks Law, which is where we
started.
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Coupling Between Dopant and Defects
Continuity equation
Interstitials flux
1100C
experimental
800C
EASi 4.8 eV, EAdopant3-4 eV
The effect of I supersaturation is larger at T ?
and increases for dopants of large DA( ex. P
compared to As)
Interstitial component of the self diffusion flux
TSUPREM IV simulations of the interstitial
supersaturations generated by a phosphorus
versus an arsenic diffusion to the same
depth. The fast diffusing phosphorus profile
has a larger effect on than the
slow diffusing arsenic profile. DpgtDAs
P tail
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Scaled Down Devices - Role of Defects on Doping
2D SUPREM simulation of small MOS
transistor. Ion implantation in the S/D
regions generates excess I. These diffuse
into the channel region pushing boron
(channel dopant) up towards the surface.
Effect is more pronounced in smaller devices.
Result is that VTH depends on channel
length (the "reverse short channel effect"
only recently understood).
(See text for more details on these examples.)
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Charge State Effects
Simplified Expressions for Modeling.
Continuity Equation
Solved in simulators
Potential included
Fully Kinetic Description
Effective diffusivity
Chemical equilibrium not reached in Ion
Implantation RTP
Effects of Electric Field
Concentrations of VI depend on EF
Total flux of mobile dopant
Simulations will include more physico-chemical
parameters to accurately model dopant diffusions.
Enhancements due to Fermi level
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Future Projections - Shallow Junctions
Assuming ND or NA 2 x 1020 cm-3 and µ 52
cm2volt-1 sec-1 an ideal box-shaped profile
limits the xJrS product to values outside the red
area. The ITRS goals for S/D extension regions
are not physically achievable towards the end
of the roadmap without metastable doping
concentrations gt 2 x 1020 cm-3.
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Future Projections - Shallow Junctions
Flash annealing - ramp to intermediate T (
800 C) then msec flash to high T ( 1300
C). Recent flash annealing results with
boron are much better than RTA.
flash data points Well see why this works in
the next set of notes on ion implantation and
damage annealing.
msec Flash
Overall, the shallow junction problem seems
manageable through process innovation.
Spike Anneal
RTA
Furnace
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Summary of Key Ideas
Selective doping is a key process in
fabricating semiconductor devices. Doping atoms
generally must sit on substitutional sites to be
electrically active. Both doping concentration
and profile shape are critical in device
electrical characteristics. Ion
implantation is the dominant process used to
introduce dopant atoms. This creates damage
and thermal annealing is required to repair this
damage. During this anneal dopants diffuse much
faster than normal (Chapter 8 - TED).
Atomistic diffusion processes occur by pairing
between dopant atoms and point defects. In
general diffusivities are proportional to the
local point defect concentration. Point defect
concentrations depend exponentially on
temperature, and on Fermi level, ion implant
damage, and surface processes like oxidation.
As a result dopant diffusivities depend on time
and spatial position during a high
temperature step. Powerful simulation tools
exist today which model these processes and which
? can predict complex doping profiles.