Diffusion
Some introductory comments...
• Diffusion the redistribution of material in response to a
concentration gradient (more accurately: a chemical
potential gradient)
• The effect of diffusion is usually (but not always!) to
reduce the magnitude of a concentration gradient
• We will view diffusion from two different standpoints:
1. a unit manufacturing process for changing chemical
composition with depth important in modern
semiconductor manufacturing (but somewhat difficult
to achieve submicron geometries)
2. a parasitic effect that accompanies any thermal
process very important in semiconductor
manufacturing (alters desired concentration profiles)
Spin-on doped glass
Solid
Liquid (bubbler)
Gaseous
Ion Implant
Planar Sources
solid
silica tube
Carrier
gasSource
boatwafers
gas
dopant
Heated
bath
liquid
Planar sources
Fick’s First Law
where D is the diffusion coefficient and J is the net flux (cm-2
sec-1) of the diffusing species (note: negative sign shows that a positive flux will decrease the concentration profile)
J DC z t
z
( , ) xy
z
Problem: How do we observe or measure the flux
within a solid material?
Answer: We usually can’t.
Solution: Use a different equation!
Fick’s Second Lawx
y
z
dz
J1
J2
In the volume with length dz,
J
z
J J
z
2 1
J2 = flux leaving
J1 = flux entering
The concentration in the volume element
will be changing with time (J2 J1), and this
can be expressed as:
A J J AdzJ
zAdz
C
t2 1
or
C z t
t
J
z zD
C
z
,
C z t
tD
C z t
z
, ,
2
2
2nd order differential equation needs 2 independent boundary conditions
Analytical solutions of Fick’s Law (2)Case 2: “drive-in diffusion” an initial amount QT is deposited onto
the surface and allowed to diffuse into the bulk
C z C t
d C t
d zC z t QT
, ,
,,
0 0 0
00
0
Boundary conditions:
constant
Solution: C z tQ
Dte tT z Dt, ,
2 4 0
C tQ
Dt
T0,
The surface concentration decreases with time:
Analytical solutions of Fick’s Law (1)
Case 1: constant surface concentration (“pre-deposition diffusion”):
C z
C t C
C t
s
,
,
,
0 0
0
0
Boundary conditions:
in words: the surface concentration
is always Cs; the concentration at
t=0 everywhere or z= anytime = 0
Solution: C z t Cz
DttS, ,
erfc
20
where “efrc u” is the complimentary error function (i.e. 1-erf u) or
erfc u e dxx
u
2 2
“characteristic
diffusion length”Dt
Sidebar -- Useful expressions to the Error Function
erfc u e dxx
u
2 2
ux dxeu
0
22 erf
erfc u = 1 - erf u erf (0) = 0 erf () = 1
12
erf uuu
11
erfc
2
uu
eu
u
Analytical solutions of Fick’s Law (3)In both cases (pre-deposition and drive-in diffusion), the dose is the
total amount of materials diffused into the substrate
Q t C z t dz C t DtT ( ) , ,
02
0
Pre-deposition diffusion:
Drive-in diffusion:
C z t dz QT,0
dose increases with time
dose = constant
Diffusion doping for junction formation
• The junction depth xj is the depth where the concentration of a diffused dopant (n or p) equals the bulk concentration CB (p or n)
• CB will be either the intrinsic carrier concentration ni or the bulk doped concentration, which ever is larger, where
for Si (pre-factor of 4.21014 for GaAs)
1
10-1
10-2
-310
-410
CS
CB
0 xj
CS
C
distance
junction
depth
kTE
igeTn
22315)103.7(
The diffusion coefficient D
• The diffusion coefficient is a strong function of temperature:
where Eio is the activation energy (usually between 1 to 5 eV)
and Dio is a pre-exponential factor ( temperature independent)
that typically ranges from 10-2 to 10+2
• If diffusion occurs by vacancies, then the diffusion coefficient is
the sum of all possible diffusion coefficients, weighted by their
probability of existence:
kTE
ioiioeDD
3
3
2
2
3
3
2
2
Dn
pD
n
pD
n
p
Dn
nD
n
nD
n
nDD
iii
iii
i
Typical diffusion calculations (1)
The problem:
A p+-n junction is made by diffusing boron into silicon with a background n-type concentration of 1016 cm-3. A constant boron surface concentration Cs = 1020 cm-3 is maintained. For boron in silicon Do= 0.76 cm2/s and Ea = 3.46 eV. Calculate the time required to form the junction at a depth of 1.0 mm if the diffusion temperature is 1050°C.
The solution:
Calculate the diffusion coefficient at 1050°C and solve for C(z,t) under pre-deposition conditions
Solution to (1)
1. Calculate D:
o 5
14 2 1
3.46(0.76)exp
8.617 10 1050 273
5 10 cm s
aE
kTD D e
2. Determine depth where n = p:
416 20
14
4
10( , ) erfc 10 10 erfc
2 2 (5 10 )
223.610 erfc 6612
s
zC z t C
Dt t
tt
seconds (~1 hr 50 min)
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201-44494-1.
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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
DiffusionProfile Comparison
Complementary Error Function and
Gaussian Profiles are Similar in Shape
erfc z 1 erf z
erf z 2
exp x 2 dx
0
z
Typical diffusion calculations (2)The problem:A two-step boron diffusion is performed into n-type silicon with an impurity concentration of 1016 cm-3. The pre-deposition diffusion is conducted at 950°C for 15 minutes with a surface concentration of Cs = 1021 cm-3. The drive-in temperature is 1100°C. 1. How long should the drive-in last to form a junction 2.0 mm deep?2. What is the impurity concentration at the surface after the drive-in diffusion?The solution:• Calculate the diffusion coefficients at 950°C and 1100°C • Determine the pre-deposition dose QT(t)• Use this dose as input to the drive-in diffusion time calculation
Solution to (2)1. Calculate D’s:
o 5
15 2 1 o 13 2 1 o
3.46(0.76)exp
8.617 10 950 273
4 10 cm s (950 C); 1.5 10 cm s (1100 C)
aE
kTD D e
2. Determine the pre-deposition dose:
21 15
15 2
2 2pre-dep. 0, 10 (4 10 )(15 60)
2.14 10 cm
TQ C t Dt
3. Calculate time for the drive-in diffusion profile to reach the background:
22 2
4
2
15 2 4 219
16 2 13 13
2( , ) ln ln
4( , )
(2.14 10 ) 2(2 10 ) 1ln ln 8200s; 3.44 10
(10 ) (1.5 10 ) 4(1.5 10 )
z DtT T
S
Q Q zC z t e t
DtDt C z t D
t t Ct
Typical diffusion calculations (3)
The problem:
At high dopant concentrations the diffusion process may be altered by the effect of doping on the Fermi level position. What is the diffusion coefficient of arsenic in silicon doped with 11020 As/cm3 at 1400K?
The solution:
Calculate the diffusion coefficient at 1050°C and solve for C(z,t) under pre-deposition conditions
Solution to (3)1. Determine mechanism: according to Table 3.2, As diffuses according to both
uncharged (Di) and –1 (D- )charged vacancies. Using the data in the Table,
the diffusion coefficient is written as:
3.44 4.05(0.066) (12.0)kT kT
As i
i i
n nD D D e e
n n
2. Determine the effect of doping on the bandgap and hence ni:
5
2 4 2
0
23 2 15 3 2 0.715 2(8.617 10 )(1400) 19
0
2014 14
19
14 2 1
(4.73 10 )(1400)1.17 0.715 eV
636 1400
(7.3 10 )(1400) 2 10
1 105 (2.7 10 ) (5 3.2 10 )
2 10
1.9 10 cm s
g
g g
E kT
i i
As
i
TE E
T
n n T e e
nD
n
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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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
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Diffusion CalculationExample - Boron Diffusion
• A boron diffusion is used to form the base region of an npn
transistor in a 0.18 W-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 1100oC. Find
the surface concentration and junction depth (a) after the
predep step and (b) after the drive-in step.
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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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Diffusion CalculationExample - Boron Diffusion
Predeposition step is solid - solubility limited.
T1 = 900oC 1173KNO 1.1x1020/cm3
D1 10.5exp 3.69eV
8.614x105eV /K 1173K
1.45x1015cm2 /sec
t1 15 min 900 sec D1t1 1.31x1012cm2
N x 1.1x1020erfcx
2.28x106cm
/cm
3
Dose : Q 2NOD1t1
1.42x1014 /cm2
T2 =1100oC 1373K
D2 10.5exp 3.69eV
8.614x105eV /K 1373K
2.96x1013cm2 /sec
t2 5 hr 18000 sec D2t2 5.33x109cm2
N2 x 1.42x1014 /cm2
5.33x109cm2 exp
x
2 D2t2
2
1.1x1018 exp x
1.46x104
2
/cm3
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201-44494-1.
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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Diffusion CalculationExample (cont.)
N1 x 1.1x1020erfcx
2.28x106cm
/cm
3
x j1 2 D1t1erfc1 No
NB
2.28x106cm erfc1 3x1016
1.1x1020
2.28x106cm erfc1 2.73x104
x j1 2.28x106cm 2.57 5.86x106cm 0.058
N2 x 1.1x1018 exp x
1.46x104
2
/cm3
x j2 1.46x104cm ln1.1x1018
3x1016
2.77x104cm 2.77mm
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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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Diffusion CalculationExample (cont.)
Starting Wafer : n - type 0.18 W cm
n - type 0.18 W cm ND 3 x 1016/cm3
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201-44494-1.
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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Two Step Diffusion
• Short constant source diffusion used to
establish dose Q (“Predep” step)
• Longer limited source diffusion drives
profile in to desired depth (“drive in”
step)
• Final profile is Gaussian
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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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Sheet Resistance
Irvin’s Curves
• Irvin Evaluated this Integral and Published a Set of Normalized Curves Plot Surface Concentration Versus Average Resistivity
• Four Sets of Curves
– n-type and p-type
– Gaussian and erfc
1
1
1
x j x dx
0
x j
RS
x j
1
x dx0
x j
RS qmN x dx0
x j
1
RSx j
For the exclusive use of adopters of the book Introduction to
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201-44494-1.
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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Sheet Resistance
Irvin’s Curves (cont.)
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201-44494-1.
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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Two Step DiffusionSheet Resistance - Predep Step
Initial Profile
No 1.1x1020 /cm3
NB 3x1016 /cm3
x j 0.0587 mm
p type erfc profile
Square/ 850 0587.0
- 32
- 32
WW
W
m
mR
mxR
S
jS
m
m
m
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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copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,
without permission in writing from the publisher.
Two Step DiffusionSheet Resistance - Drive-in Step
Final Profile
No 1.1x1018 /cm3
NB 3x1016 /cm3
x j 2.73 mm
p type Gaussian profile
RSx j 700 W -mm
RS 700 W -mm
2.73 mm 260 W /Square
Some basic concepts in diffusion
mechanisms
• Diffusion is a thermally activated process -- energy is required to
displace an atom from a given site in a crystal
• Impurities can occupy either interstitial or substitutional sites
• Interstitial impurities
– atoms that don’t bond readily with host atoms
– can diffuse rapidly via interstitial sites
– do not contribute to doping (electrically inactive)
• Substitutional impurities
– bond with the host atoms
– slower diffusion
– contribute to doping (electrically active)
Atomistic models of diffusion (1)
Direct exchange -- requires multiple bonds to be broken
for the host and impurity atoms to exchange
Atomistic models of diffusion (2)
Vacancy exchange -- requires fewer bonds to be broken
and is a dominant mechanism for substitutional impurities
but
the mechanism requires the presence of vacancies
Atomistic models of diffusion (3)
Interstitialcy mechanism -- a host-atom interstitial
displaces the impurity, which can then diffuse as an
interstitial to another lattice site; occurs simultaneously
with vacancy diffusion
Atomistic models of diffusion (4)
Mechanisms for fast-diffusing impurities to reincorporate
into the lattice:
Kick-out mechanism -- direct replacement of a host atom
Frank-Turnbull mechanism -- impurity combines with a
host atom vacancy