EE143 F2010
Lecture 11
Electric-field Enhancement
Example 1 : Diffusing specie generates build-in E-field
Na(x)
Acceptor Boron
Na(x) = Na-(x)
hole gradient
x
Complete acceptor ionization at diffusion
temperature. Acceptor atoms are
negatively charged. Mobile + holes are
near vincinity of the acceptor atoms.
Professor N Cheung, U.C. Berkeley
p(x) =Na-(x)
E build-in
Hole diffusion
tendency
At thermal equilibrium, hole current =0
Hole gradient creates build-in
electric field to counteract the hole
diffusion tendency
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EE143 F2010
Lecture 11
p+
B-
Mobile holes tends to move
away due to hole
concentration gradient
Ebuild-in due to the hole gradient
B- acceptors
experience
an additional
drift force
Enhanced Diffusion for B- acceptor atoms
due to the Boron profile graduent
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Calculation of build-in Electric Field E
dp
J p  q   p  p  E  qDp 
0
dx
at thermal equilibrium.
kT 1 dp
E
 
q p dx
kT d ln p


q
dx
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Calculation of total acceptor flux F
Na-(x)
E
x
Total acceptor atom flux = (diffusion + drift) components
F = - D A dNA/dx
Professor N Cheung, U.C. Berkeley
-
A
 E  NA
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EE143 F2010
Since
Lecture 11
A
= DA/[kT/q] from Einstein’s Relationship
 dN A d ln p 

F   DA  

 NA
dx
 dx


From (1) p  n  N A charge Neutrality
(2) pn  ni
2
Law of Mass Action
NA 
2
2
N A  4ni
(1) & (2)  p 
2
d ln p  dN A
1



2
2
dx
dx
N A  4ni
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11

F  D
A
dN

dx
A




1


 1 
2 
4ni 

1
2


N
A
     

h  factor
If n i  N
n i  N
A
A
h  1
low dopant conc.
h  2
high dopant conc.
Value of h depends on concentration NA
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Intrinsic Carrier Concentration ni
If n or p >
ni(Tdiffusion), we will
expect to observe high
concentration diffusion
effects.
Professor N Cheung, U.C. Berkeley
Temp range for
dopant diffusion
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EE143 F2010
Lecture 11
Example 2: Diffusion enhanced by build-in E-Field of
other diffusion species
Illustration: Uniform substrate concentration profile is
perturbed by E-field of another diffusion specie
As
diffusion
caused by As
conc gradient
Uniform B
conc in
substrate
B-
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Dopant Diffusion Revisited
•Diffusion Mechanisms
•Dopant interactions with Point Defects of Si substrate
- Dopant Diffusivity affected by Dopant Concentration
- Dopant Diffusivity affected by Point Defect Concentration
- Point Defect Concentration affected by Dopant Concentration
- Point Defect Concentration affected by Processing Steps
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
For reference only
Si Native
Point
Defects
Si vacancy
1) Thermal-equilibrium values of Si
neutral interstitials and vacancies at
diffusion temperatures
<< doping concentration of interest
(1015 –1020 /cm3)
Si interstitial
At 1000oC, CIo* ~ 1012 /cm3
CVo* ~ 1013 /cm3
2) Diffusivity of Si interstitials and
Si vacancies >>
diffusivity of dopants
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Diffusion Mechanisms in Si
(A) No Si Native Point Defect Required
Example: Cu, Fe, Li, H
(a) Interstitial Diffusion
Fast Diffusion
Cu
10-6 cm2/sec
Professor N Cheung, U.C. Berkeley
Au
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EE143 F2010
Diffusion Mechanisms in Si
Lecture 11
(B) Si Native Point Defects Required (Si vacancy and Si interstitials)
Example: Dopants in Si ( e.g. B, P,As,Sb)
(a) Substitutional Diffusion
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(b) Interstitialcy Diffusion
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EE143 F2010
Lecture 11
(B) Si Native Point Defects Required (Si vacancy and Si interstitials)
continued
(c) Kick-Out Diffusion
(d) Frank Turnbull Diffusion
Slow Diffusion
B,P
10-12 cm2/sec
As
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Diffusivity Comparison:
Dopants, Si interstitial, and interstitial diffusers
Lecture 11
For reference only
108 times higher
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Diffusion Enhanced by Charged Point Defects
(Vacancies and Interstitials )
Si vacancy
Vo = neutral
V+ = + vacancy
V- = - vacancy
V= = - - vacancy
…..
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EE143 F2010
Lecture 11
Possible Point Defect Configurations
For reference only
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EE143 F2010
Lecture 11
How processing steps affect point defect concentrations
• Neutral interstitial and vacancy point defects present at
thermal equilibrium
At 1000oC, CIo* ~ 1012 /cm3
CVo* ~ 1013 /cm3
• Charged Point Defects enhanced by heavy doping; total point
defect concentrations enhanced by ~10x
I +, I o , I V+, Vo, V-, V=
•Point defects Injected by interfaces during oxidation
(total point concentrations enhanced by ~10x)
• Implantation collisions (total point defect concentration
enhanced by ~ 1000X)
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Derivation of charged vacancy concentration
Law of Mass Action
For the reaction :
aA + b  B  c  C
1) Negatively charged vacancies
Vo + r • electrons  V -r (r = 1,2,3…)
[Vo] n i r
[Vo] n r
Kc =
=
[V -r ]
[V i -r ]
[V -r ]
n
=[ ]
ni
[V i -r ]
r
same Kc for undoped Si
where n = electron concentration ,
[ ] denotes the concentrations ,
subscript i denotes condition for intrinsic Si
[A] a [B] b
= Kc
[C]c
[ ] denotes
concentration
Kc = equilibrium
Constant at Temp
T
2) Positively charged vacancies
Vo + r • holes  V +r (r =1,2,3…)
ni
[V +r ]
p
r
=[ ] =[ ] r
+r
ni
n
[V i ]
p =hole concentration
See Diffusion Handout for Law of Mass Action Discussion
Professor N Cheung, U.C. Berkeley
18
EE143 F2010
Lecture 11
Law of Mass Action Examples
1) Electron Hole recombination
e+h0
[e] [h] = K(T) or np =ni2
2) Generation of singly charged negative vacancies
e+VoV[e] [V o] / [V-] = K1(T) or [V-]  n
3) Generation of doubly charged negative vacancies
2e + V o  V =
[e]2  [V o] / [V=] = K2(T) or [V=]  n2
4) Generation of singly charged positive vacancies
h+VoV+
[h] [V o] / [V+] = K3(T) or [V+]  p  1/n
Professor N Cheung, U.C. Berkeley
19
EE143 F2010
Lecture 11
Charged Vacancy Enhanced Diffusion
2
Dtotal
 p   n    n 
 D  D    Di    Di   ...
 ni 
 ni 
 ni 
o

i
Why each diffusivity written with ni as normalization factor ?
Consider the +charge vacancy first , D+  [V+].
From Law of Mass Action: [V+]  p
Therefore , D+  p and can be written as D+ = K p
where K is a proportionality constant
Let us multiply K by ni and call it Di+, then D+ = Di+ (p / ni )
The physical meaning of Di+ is diffusivity D+ for undoped Si
because p =n= ni for undoped silicon
Similarly, Di- and Di= represent diffusivity values for undoped Si
due to –charge and = charge vacancies
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Diffusion Profile with enhanced D at high concentration
Log C(x)
High conc. profile:
D gets larger
when C(x)
is large
Log C(x)
J large
Low conc profile:
Erfc or gaussian
J small
x
x
* C(x) goes deeper but the shape looks “flatter”at high conc. regions
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Example : High Concentration Arsenic
diffusion profile becomes “box-like”
Professor N Cheung, U.C. Berkeley
Lecture 11
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EE143 F2010
Lecture 11
Summary of High-Concentration Diffusion
First, Check if doping concentration is > n i(T) or < ni(T)
If doping conc < ni:
D= Do
Use constant diffusivity solutions
(diffusion profile is erfc or half-gaussian)
If doping conc > ni:
Use D = Dv = h [ Do + D+ + D- + D= + ...]
(Solution requires numerical techniques)
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Oxidation Enhanced Diffusion (OED)
Diffusivity is enhanced for B and P underneath a growing oxide
Example:
Channel-stop implant diffuses faster during field oxidation
Field Oxide
SiO2
Si[I]
Si
Injected Si[I] interstitials
enhance dopant diffusion
DOED = D + D
D  [dXox / dt]n where 0.4 < n < 0.6
Note: The dopant Sb shows retarded diffusivity during oxidation
=> it diffuses primarily via the vacancy mechanism
Professor N Cheung, U.C. Berkeley
24
EE143 F2010
Lecture 11
Transient Enhanced Diffusion (TED)
Dopant Implantation
C(x)
Si
substrate
Implantation
induced excess
point defects
Implantation creates
large number of excess Si
interstitials and vacancies.
(1000X than thermal process).
After seconds of annealing,
the excess point defects will
be reduced substantially by
recombination .
Professor N Cheung, U.C. Berkeley
no annealing
900oC, several minutes
(After excess point defects
recombine. normal diffusion)
x
900oC, several sec (TED)
Extremely rapid diffusion due
to excess point defects
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EE143 F2010
Lecture 11
Junction Depth versus post-implantation annealing time
Junction
Depth xj
Normal diffusion
(with high concentration effects)
TED
seconds
Annealing
time
•Difficult to make ultra-shallow (< 100nm) junctions
with implantation and annealing
Professor N Cheung, U.C. Berkeley
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EE143 F2010
Lecture 11
Summary of Diffusion Module
•Diffusion Sources (gas, solid, liquid)
•Diffusion Constant D
•Concentration Profiles with D independent of Concentration
•The Thermal Budget
•Junction Depth and Sheet Resistance
•Irvin’s Curves
•Two-Dimensional diffusion profiles – qualitative
•Enhanced Diffusion due to electric field
•Point Defects in Si
•Diffusion due to point defects :
-Charged point defects,
-Oxidation induced point defects ( OED/ORD),
-Implantation induced point defects (TED)
Professor N Cheung, U.C. Berkeley
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