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44494-1.
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without permission in writing from the publisher.
Introduction toMicroelectronic Fabrication
by
Richard C. JaegerDistinguished University Professor
ECE Department
Auburn University
Chapter 4Diffusion
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Copyright Notice
• © 2002 Pearson Education, Inc., Upper Saddle River, NJ. Allrights reserved. This material is protected under all copyrightlaws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission inwriting from the publisher.
• For the exclusive use of adopters of the book Introduction to
Microelectronic Fabrication, Second Edition by Richard C.Jaeger. ISBN0-201-44494-1.
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Impurity Diffusion
•Diffusion Mechanisms
• Substitutional
• Interstitial
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DiffusionFick’s First Law
Particle flux J is proportional to the negativ
of the gradient of the particle concentratio
J = − D∂ N
∂ x
D = diffusion coefficient
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DiffusionFick’s Second Law
Continuity Equation for Particle Flux :
Rate of increase of concentration is equal to the
negative of the divergence of the particle flux
∂ N
∂ t = −
∂ J
∂ x
(in one dimension)
Fick's Second Law of Diffusion :
Combine First Law with Continuity Eqn.
∂ N
∂ t = D
∂ 2 N
∂ x 2
D assumed to be independent of concentration!
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Constant Source DiffusionComplementary Error Function Profiles
Concentration : N x, t ( )= N 0erfc x2 Dt
Total Dose : Q = N x, t ( )dt = 2 N 00
∞
∫ Dt
π
N 0 =Surface Concentration
D =Diffusion Coefficient
erfc = Complementary Error Function
erfc z( )=1− erf z( )
erf z( )=2
π exp − x
2[ ] dx0
z
∫
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Limited Source DiffusionGaussian Profiles
Concentration:
N x,t ( )= N 0 exp − x
2 Dt
2
=
Q
π Dt exp −
x
2 Dt
2
N 0 =Surface Concentration N 0 = Q
π Dt
D = Diffusion Coefficient
Gaussian Profile
Initial Impulse with Dose Q
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DiffusionProfile Comparison
Complementary Error Function and
Gaussian Profiles are Similar in Shape
erfc z( )=1− erf z( )
erf z( )=2
π exp − x 2[ ] dx
0
z
∫
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Diffusion Coefficients
Substitutional
Diffusers Interstitial
Diffusers
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Diffusion Coefficients
D = DO
exp − E A
kT
Arrhenius Relationship
EA = activation energy
k = Boltzmann's constant =1.38 x10-23 J/K
T = absolute temperature
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Successive Diffusions
• Successive Diffusions Using Different Times and
Temperatures
• Final Result Depends Upon the Total Dt Product
Dt ( )tot = Di
i
∑ t i
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DiffusionSolid Solubility Limits
• There is a limit to the amount of agiven impurity that can be “dissolved”
in silicon (the Solid Solubility Limit)
• At high concentrations, all of the
impurities introduced into silicon willnot be electrically active
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Diffusionp-n Junction Formation
x j =Metallurgical Junction Depth
P - n junction occurs at the point x j where the net impurity
concentration is zero
(i. e. p - type doping cancels out n - type doping)
Gaussian Profile : x j = 2 Dt ln N0
NB
Error Function profile : x j = 2 Dt erfc -1 N0
NB
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DiffusionResistivity vs. Doping
ρ =σ −1 = q µ n n + µ p p( )[ ]−1
n − type: ρ ≅ q µ n N D − N A( )[ ]−1
p − type: ρ ≅ q µ p N A − N D( )[ ]−1
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DiffusionTwo Step Diffusion
• Short constant source diffusion used toestablish dose Q (“Predep” step)
• Longer limited source diffusion drives
profile in to desired depth (“drive in”step)
• Final profile is Gaussian
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Diffusion CalculationExample 4.3 - Boron Diffusion
• A boron diffusion is used to form the base region of an npn
transistor in a 0.18 Ω-cm n-type silicon wafer. A solid-
solubility-limited boron predeposition is performed at 900o
C for 15 min followed by a 5-hr drive-in at 1100o
C. Findthe surface concentration and junction depth (a) after the
predep step and (b) after the drive-in step.
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Diffusion CalculationExample 4.3 - Boron Diffusion
Predeposition step is solid- solubility limited.
T1 = 900
o
C =1173K →NO =1.1x10
20
/cm
3
D1 =10.5exp −
3.69eV
8.614 x10−5eV / K ( )1173K
=1.45 x10
−15cm2 /sec
t 1 =15 min = 900 sec D1t 1 =1.31 x10−12 cm2
N x( )=1.1 x1020
erfc x
2.28 x10
−6
cm
/ cm
3
Dose: Q = 2 N O D1t 1
π =1.42 x10
14 / cm
2
T2 =1100oC =1373K
D2 =10.5exp −3.69eV
8.614 x10−5 eV / K ( )1373K
= 2.96 x10
−13cm2 /sec
t 2 = 5 hr =18000 sec D2
t 2 = 5.33 x10
−9cm2
N 2 x( )=1.42 x10
14 / cm2
π 5.33 x10−9cm2( )
exp − x
2 D2t 2
2
=1.1 x1018 exp − x
1.46 x10−4
2
/ cm3
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Diffusion CalculationExample 4.3 (cont.)
N 1 x( )=1.1 x1020 erfc x2.28 x10−6 cm
/ cm3
x j1 = 2 D
1t 1erfc−1 N o
N B
= 2.28 x10
−6cm( )erfc−1 3 x1016
1.1 x1020
= 2.28 x10
−6cm( )erfc−12.73 x10
−4( )
x j1 = 2.28 x10−6cm( )2.57( )= 5.86 x10
−6cm = 0.058
N 2 x( )=1.1 x1018 exp − x
1.46 x10−4
2
/ cm3
x j 2 =1.46 x10
−4 cm ln1.1 x1018
3 x1016
= 2.77 x10
−4 cm = 2.77 µ m
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Diffusion CalculationExample 4.3 (cont.)
Starting Wafer : n - type 0.18Ω− cm
n - type 0.18 Ω− cm →ND = 3 x 1016 /cm3
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Two Step Diffusion
• Short constant source diffusion used toestablish dose Q (“Predep” step)
• Longer limited source diffusion drives
profile in to desired depth (“drive in”
step)
• Final profile is Gaussian
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Lateral DiffusionUnder Mask Edge
• Diffusion is really a 3-D process.
As impurities diffuse vertically,
they also diffuse horizontally in
both directions.
• Diffusion proceeds laterally under
the edge of the mask opening
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Lateral DiffusionUnder Mask Edge
Original Mask
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Concentration DependentDiffusion
Second Law of Diffusion
∂ N
∂ t = ∂
∂ x D x( )
∂ N
∂ x
Profiles More Abrupt at High Concentrations
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Concentration DependentDiffusion
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ResistorsSheet Resistance
A =W • t
R = ρ
t
L
W
= RS
L
W
RS = ρ
t = Sheet Resistance [Ohms per Square]
L
W
=Number of Squares of Material
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ResistorsCounting Squares
Figure 4.14
• Top and Side Views ofTwo Resistors of Different
Size
• Resistors Have Same
Value of Resistance
• Each Resistor is 7 in
Length
• Each End ContributesApproximately 0.65
• Total for Each is 8.3
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ResistorsContact and Corner Contributions
• Effective SquareContributions of Various
Resistor End and Corner
Configurations
Figure 4.15
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Sheet ResistanceIrvin’s Curves
• Irvin Evaluated this Integral andPublished a Set of NormalizedCurves Plot SurfaceConcentration Versus AverageResistivity
• Four Sets of Curves
– n-type and p-type– Gaussian and erfc
ρ =1
σ =1
1
x j
σ x( )dx0
x j
∫
RS = ρ x j
= 1
σ x( )dx0
x j
∫
RS ≅ q µ N x( )dx0
x j
∫
−1
ρ = RS x j
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Sheet ResistanceIrvin’s Curves
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Sheet ResistanceIrvin’s Curves (cont.)
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Two Step DiffusionSheet Resistance - Predep Step
Initial Profile
N o =1.1 x1020 / cm3
N B = 3 x1016 / cm3
x j = 0.0587 µ m
p− type erfc profile
RS x j = 50 Ω - µ m
RS =32 Ω - µ m
0.0587 µ m= 850 Ω /Square
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Two Step DiffusionSheet Resistance - Drive-in Step
Final Profile
N o =1.1 x1018 / cm3
N B = 3 x1016 / cm3
x j = 2.73 µ m
p− type Gaussian profile
RS x j = 700 Ω - µ m
RS =700 Ω - µ m
2.73 µ m= 260 Ω /Square
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Resistivity MeasurementFour-Point Probe
ρ = 2π sV
I [Ω - m] for t >> s
ρ = π t
ln2
V
I
[Ω - m] for s >> t
RS = ρ
t = π
ln2
V
I ≅ 4.53
V
I [Ω /square]
Four Terminal Resistance Measurement
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Four-Point ProbeCorrection Factors
Correction Factors
(a) Wafers Thick Relative to the Probe Spacing
(b) Wafers of Finite Diameter
ρ = F ρ measured
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Sheet Resistancevan der Pauw’s Method Van der Pauw's Theory
Any Four - Terminal Region without Holes
exp -π t R AB ,CD
ρ
+ exp -π t
R BC , DA
ρ
=1
R AB,CD =
V CD
I AB and R BC , DA =
V DA
I BC
For symmetrical structure R AB ,CD = R AB ,CD
RS = ρ
t
= π
ln2
V CD
I AB
Four Terminal Resistance Measurement
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Junction Depth Measurement
• Groove and StainMethod
x j
= a+ b( ) a− b( )
2 R
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Junction Depth Measurement
• Angle Lap Technique
x j = d tanθ = N λ 2
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Impurity ProfilingSpreading Resistance
• Region of Interest is Angle-
Lapped
• Two-Point Probe Resistance
Measurements vs. Depth• Profile Extracted
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Impurity ProfilingSecondary Ion Mass Spectroscopy (SIMS)
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Diffusion Systems
Open Furnace Tube Systems(a) Solid source in platinum
source boat
(b) Liquid Source - carrier gaspassing through bubbler
(c) Gaseous impurity source
Wafers in Quartz Boat
Scrubber at Output
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Diffusion SystemsPhosphorus Diffusion
Gaseous Source : Phosphine PH3 (Extremely Toxic)
2P H 3 + 4O2 → P2O5 + 3 H 2O
All systems need careful scrubbing!
P2O5 POCl3 Cl2
Surface Reaction:
2P2O5 + 5Si↔ 4P + 5SiO2
Solid Sources:
Phosphorus Pentoxide
Ammonium monophosphate NH4H2PO4
Ammonium diphosphate NH4( )2H2PO4
Liquid Source : Phosphorus Oxychloride POCl3
4POCl3 + 3O2 → 3P2O5 + 6Cl2
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Diffusion SystemsArsenic & Antimony Diffusion
Arsenic Surface Reaction
2 As2O3 + 3Si↔ 3SiO2 + 4 As
Solid Sources : Possible - Low Surface Concentations
Gaseous Source: Arsine AsH3 (Extremely Toxic)
Ion− Implantation Is Normally Used for Deposition
Antimony Surface Reaction
2Sb2O3 + 3Si↔ 3SiO2 + 4Sb
Liquid Source : Antimony Pentachloride Sb3Cl5
Ion− Implantation Is Normally Used for Deposition
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DiffusionToxicity of Gaseous Sources
Silane and Dichlorosilane Used for
Polysilicon Deposition
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DiffusionGettering
• Improves Quality of Wafers
– Removes Metallic Impurities: Cu, Au, Fe, Ni (Rapid Diffusers)
– Removes Crystal Defects: Dislocations
• Backside Treatment
– Surface Damage e. g. Sandblasting
– Phosphorus Diffusion
• Argon Implantation
• Internal Stress
• Crystal Defects
• Oxygen Incorporation
– During Growth
– Implantation
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DiffusionReferences
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End of Chapter 4
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Introduction to
Microelectronic Fabrication
by
Richard C. JaegerDistinguished University Professor
ECE Department
Auburn University
Chapter 5
Ion Implantation
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Copyright Notice
• © 2002 Pearson Education, Inc., Upper Saddle River, NJ. Allrights reserved. This material is protected under all copyrightlaws as they currently exist. No portion of this material may bereproduced, in any form or by any means, without permission in
writing from the publisher.
• For the exclusive use of adopters of the book Introduction toMicroelectronic Fabrication, Second Edition by Richard C.
Jaeger. ISBN0-201-44494-1.
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Ion ImplantationHigh Energy Accelerator
1. Ion Source
2. Mass Spectrometer
3. High-Voltage Accelerator (Up to 5 MeV)
4. Scanning System
5. Target Chamber
Force on charged particle F = q v x B( )
Magnetic Field B =2mV
qr 2
Implanted Dose Q =1
mqA I t ( )
0
T
∫ dt
m =mass
v = velocity
V = acceleration potential
A = wafer area
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Ion ImplantationOverview
• Wafer is Target in High Energy Accelerator
• Impurities “Shot” into Wafer
• Preferred Method of Adding Impurities to Wafers
– Wide Range of Impurity Species (Almost Anything)
– Tight Dose Control (A few % vs. 20-30% for high temperaturepre-deposition processes)
– Low Temperature Process
• Expensive Systems
• Vacuum System
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Ion ImplantationMathematical Model
Gaussian Profile
N x( )= N p exp − x − R p( )
2
2∆ R p
2
R p = Projected Range
∆Rp =Straggle
Dose Q = N x( )dx = 2π 0
∞
∫ N p∆ R p
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Ion ImplantationProjected Range
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Ion ImplantationStraggle
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Ion ImplantationSelective Implantation
N x, y( )= N x( )F y( )
F y( )=1
2erfc
y − a
2∆ R⊥
− erfc
y + a
2∆ R⊥
∆ R⊥ = transverse straggle
N x( ) is one - dimensional solution
Figure 5.4
Contours of equal ion concentration for an implantationinto silicon through a 1-µm window. The profiles are
symmetrical about the x-axis and were calculated using
the equation above taken from Ref. [3].
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Ion ImplantationSelective Implantation
• Desire Implanted Impurity
Level to be Much Less
Than Wafer Doping
N(X0) << NB
or
N(X0) < NB /10
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Ion ImplantationSelective Implantation
X 0 ≥ R p + ∆ R p 2ln 10 N p
N B
= R p + m∆ R p
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Ion ImplantationJunction Depth
N x j( )= N B
N p exp − x j − R p( )
2
2∆ R p
2
= N B
x j = R p ± ∆ R p 2ln N p
N B
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Ion ImplantationLattice Damage and Annealing
• Implantation Causes
Damage to Surface
• Typically Removed by
Annealing Cycle 800-
1000
o
C for 30 min.• Rapid Thermal Annealing
(RTA) Now Used for
Lower Dt Product
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Ion ImplantationDeviation from Gaussian Theory
• Curves Deviate from
Gaussian for Deeper
Implantations (> 200 keV
• Curves Fit Four-Moment(Pearson Type-IV)
Distribution Functions
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Ion ImplantationShallow Implantation
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Ion Implantation
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Ion ImplantationReferences
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End of Chapter 5
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