1 diffusion part 2 chapter 7 dr. wanda wosik ece 6466, f2012

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DIFFUSIONPart 2

Chapter 7

Dr. Wanda WosikECE 6466, F2012

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Models and SimulationsNumerical Solutions of the Diffusion Equations. No predetermined boundary conditions - no analytical solutions

Planar density of atoms

Atoms jump between planes

They have to surmount the energy barrier Eb

Hopping to a vacancy or exchange places with frequency vb

For stable numerical solution:

Max. numerical interval (Imax) means that we cannot have more than available number of atoms to jump in t

Adjacent concentrations calculated SUPREM etc. solve for any dopant profile

Average volume concentration Ci=Nix[cm-3]

Result:

vd - Debye frequency ~1013sec-1 in Si

F=-vb/2(N2-N1)=-DC/x

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_3/backbone/r3_2_1.html



© 2000 by Prentice HallUpper Saddle River NJ

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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).


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Modification to Fick’s Loss to Account for Electric Field Effects.

Set up by the dopants @ increased concentrations

Very strong effect of at low concentrations

The enhancement of diffusion by the electric field is by a factor of two @ high concentrations.

F.-II Law

Built-in E-field enhances diffusion

This equation is solved in simulators such as SUPREM under equilibrium conditionsUseful when E is due to multiple dopants

Flux due to E-field

The potential

As

BDonors will drift


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• SUPREM simulation at 1000˚C. Note the boron profile (h ≤ 2 for the As but field effects dominate the B diffusion).• Field effects can dominate the doping distribution near the source/drain of a MOS device.

E-field Effect - SLVACO simulation

NMOS transistor with B implanted channel region

Diffusion of As doped N+ S/D affect B diffusion in the channel

Simulations: with w/o field induced enhancement


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B. Concentration Dependent Diffusivity

• At high doping concentrations, the diffusivity appears to increase. Fick's equation must then be solved numerically since D ≠ constant.

• Isoconcentration experiments indicate the dependence of D on concentration; e.g. B10 in a B11 background.

• The n and n2 (p and p2 for P type dopants) terms are due to charged defect diffusion mechanisms.

erfc

erfc

Boxlike5x1020cm-3

1018cm-3

Boron diffusion ideal theoretical and with diffusion enhancement (match experiments).


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Modifications to Fick’s Laws to Account for Concentration Dependent Diffusion

ISO Concentration experiments: B11 (substrate) B10 diffusant give DA

eff

Neutral and charged point defects

Different Activation Energies

Ex: As at 1000°C For 1*1018cm-3DAs = 1.43*10-15 cm2 sec-1

For 1*1020cm-3DAs = 1.65*10-14 cm2 sec-1

Often, D is well described by

For intrinsic conditions

Original Fick’s Law depended on gradient not on C, now D=f(C) Deff(C)

• The n and n2 (p and p2 for P type dopants) terms are due to charged defect diffusion mechanisms.


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• 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.

Emitter Base

Collector

CED Boxlike

As

Due to E-field

Boron

Phosphorus

1000°C/30min

SUPREM Simulation

BJT


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• Dopants segregate at interfaces.

N-type dopants tend to pile-up while boron depletes (SUPREM simulation).

Segregation of Dopants at Interfaces

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Segregation; Interfacial Dopant Pileup.

DAS << Dp steeper profile close to the interface.

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.

• In the experiment (right) ~40% of the dose was lost in a 30 sec anneal

Kasnavi et. al.

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

Q=integrated profile

As

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Ex : Oxide Silicon

Segregation

h is the interface transfercoefficient.

SIMS

- for thick oxides can give ko (difficult measurements).

- for thin oxides SIMS is inadequate (no steady state reached). So use VT or C-V areas.

B segregates into the oxide

P piles up at the Si Surface

Snowplow during oxidation.

Flux:

Segregation of Dopants at Interfaces

• This gives an interface flux of interstitials

Oxidation of a uniformly doped boron substrate depletes the boron into the growing SiO2 (SUPREM simulation).

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• 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.

The Physical Basis for Atomic Scale Diffusion

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.

Both are “Interstitial –” assisted diffusion




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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,

(22)

• Thus dopant diffusion can be enhanced or retarded by changes in the point defect concentrations.

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• 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 Sb, the two deeper profiles are B.

• Note that the and profiles are relatively flat indicating the “stiff source” characteristic of the oxidation process.

B

Sb

OED and ORD

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Oxidation – Enhanced or Retarded Diffusion

B, P – enhanced by oxidation

Sb – retarded by oxidation.

Two different mechanisms of diffusion: by interstitials and by vacancies.

also

OISF

OISF .

Oxidation injects I (relieve of larger volume)

OEDB,P ( I )

ORDSb (V )

Larger atoms by vacancy mechanism

Concentration (supersaturation) of interstitials increases with oxidation rate.

B, P

OISF

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Dopant Diffusion Occurs by Both I and V

Recent consensus confirmed by computer simulations

Surface oxidized Sb retarded even

deep in the Si substrate

Oxidation: As enhanced

Sb retardedBecause point defects (I in oxidation) affect D constants

Inert Ambient:

Nitridation: B, P, As retarded diffusion

Sb enhanced diffusion

Injection of vacancies

fI a fraction via interstitial type mechanism.

fV =1 – fI a fraction via vacancy type mechanism

CI*, CV* CI, CV

Oxidation Retardation condition: point defects: equilibrium

* Means equilibrium

fI+fv=1Affected by point defects

Inert conditions

Dopants, whose diffusivity decreases in oxidation to 1/10 must diffuse at least 9/10 by a vacancy mechanism

CI/CI*=1

For excess of I:Cv/Cv

*=0

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For Dopants which Show Oxidation Enhanced Diffusion

Enhancement won’t exceed I super saturation. CI vs. CI

*

Conservative Assumptions:

Interstitials and vacancies assisted diffusion have different. D0 and EA in respective diffusion constants

Estimations of CI and CV from OISF growth or annihilation less accurate than from diffusion of dopants.

Activation Energy for Self-Diffusion (Si) and Dopant Diffusion.

EAdopant (3-4 eV) < EASi (4-5 eV)

Lowering of the activation barrier: Coulombic interaction, Dopant interaction, ex. relaxation of Si lattice (atom size), electronic interaction (charge transfer)

@ low concentrations

Diffusivities of mobile speciesIn steady state I&V recombine

computed

I

I

Dual

V

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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.

Phosphorus diffuses with I, and releases them in the bulk. This enhances the tail region D.

"Emitter push" is also explained by this mechanism.

AI A+I

I supersaturation CED

P concentration high

B CED due to I

A + V AV

A+I AI

mobile bound pair or a substitutional Dopant “kicked out” by I

Not mobile in a substitutional position

Oxidation:

Nitridation:

I: Increased diffusion of a Dopant A (P, B)

I: decreased diffusion of Dopant A P, B);

increased diffusion of Sb

I +V SiS

Pumping of point defects


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• If we assume “chemical equilibrium” between dopants and defects in Eqn. (23), then from the law of mass action,

(24)

• Applying Fick’s law to the mobile species (25)

•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

(26)

(27)

(written for boron diffusing with neutral and positive interstitials as an example).

(23)


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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 Fick’s Law, which is where we started.

(4)

(27)


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Coupling Between Dopant and Defects

Interstitials’ flux

EASi = 4.8 eV, EAdopant=3-4 eV

The effect of I supersaturation is larger at T and increases for dopants of large DA( ex. P compared to As)

• 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. Dp>DAs

Continuity equation

Interstitial component of the self diffusion flux

P tail

800°C

1100°C


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• TSUPREM IV simulations of boron diffused from a polysilicon source for 8 hours at 850˚C, with experimental data (diamonds) from [7.29]. The simulations use identical coefficients, but with and without full coupling between dopants and defects. The curve (right axis) is for the fully coupled case. Without full coupling, = 1. • No coupling produces a “boxier” profile because of concentration dependent diffusion.

experimental

Boron: Coupling Between Dopant and Defects


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• 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.)

As


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Charge State Effects

Simplified Expressions for Modeling.

Continuity Equation

Solved in simulators

Potential included

Fully Kinetic Description

Simulations will include more physico-chemical parameters to accurately model dopant diffusions.Enhancements

due to Fermi level

Effective diffusivity

Concentrations of V&I depend on EF

Total flux of mobile dopant

Chemical equilibrium not reached in Ion Implantation & RTP Effects of Electric Field


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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 > 2 x 1020 cm-3.


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• 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

• We’ll see why this works in the next set of notes on ion implantation and damage annealing.

Overall, the shallow junction problem seems manageable through process innovation.

Furnace

RTA

Spike Anneal

msec Flash

Future Projections - Shallow Junctions


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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.

1 diffusion part 2 chapter 7 dr. wanda wosik ece 6466, f2012

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