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DiffusionDiffusion is a process of mass transport that involves
the movement of
one atomic species into another. It occurs by random atomic
jumps
from one position to another and takes place in the gaseous,
liquid,
and solid state for all classes of materials.
partial mixing homogenization
time
water
adding dye
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What is Diffusion? Diffusion is material transport by atomic
motion. Inhomogeneous materials can become homogeneous by
diffusion. For an active diffusion to occur, the temperature
should
be high enough to overcome energy barriers to atomic motion.
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Diffusion Mechanisms. There are two main mechanisms of
diffusion
of atoms in a crystalline lattice:
•the vacancy or substitutional mechanism
•the interstitial mechanismAtoms move from
concentrated regions to
less concentrated regions.
Vacancy diffusion.
To jump from lattice site to
lattice site, atoms need
energy to break bonds with
neighbors, and to cause the
necessary lattice distortions
during jump. This energy
comes from the thermal
energy of atomic vibrations
(Eav ~ kT).
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Interstitial diffusion:
Interstitial diffusion is
generally faster than
vacancy diffusion because
bonding of interstitials to
the surrounding atoms is
normally weaker and there
are many more interstitial
sites than vacancy sites to
jump to.
Requires small impurity
atoms (e.g. C, H, O) to fit
into interstices in host.
Materials flow (the atom) is opposite the vacancy flow
direction.
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Generation of Point Defects
Point defects are caused by:
1. Thermal energy
)](exp[kT
EC
n
nX defect
site
defect
defect −==
kTECX defectdefect /ln]ln[ −=
Ln[X]
1/T
Edefect/k
*
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Example
If, at 400oC, the concentration of vacancies in aluminum is 2.3
x 10-5,
what is the excess concentration of vacancies if the aluminum
is
quenched from 600oC to room temperature? What is the number
of
vacancies in one cubic µm of quenched aluminum?
Given, Es = 0.62 eV
k = 86.2 x 10-6 eV/K,
rAl = 0.143 nm
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Diffusion Flux
The flux of diffusing atoms, J, is used to quantify how fast
diffusion
occurs. The flux is defined as either in number of atoms
diffusing
through unit area and per unit time (e.g., atoms/m2-second) or
in terms
of the mass flux - mass of atoms diffusing through unit area per
unit
time, (e.g., kg/m2-second).
At
MJ =
t
M
AJ
δ
δ1=
(Kg m-2 s-1); where M is the mass of
atoms diffusing through the area A
during time t.
Area Ain out
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Steady-State Diffusion
Flux is proportional to the concentration gradient and the
diffusion coefficient, D (m2/s), by Fick’s first law:
x
C
δ
δ
•Negative sign indicates direction of
gradient
•It is the “driving force”
•[m2/s (kg/m3)/m] = kg/(m2 As)
x
CDJ
δ
δ−=
x
C
δ
δ
Flux does not change with time
Concentration profile – concentration gradient is
maintained constant.
Concentration is expressed in terms of mass of diffusing
species per unit volume of solid (kg/m3)
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AB
AB
xx
cc
x
c
−
−=
δ
δ
Fick’s First Law of Diffusionx
CCDJor
dx
dCDJ
∆
−=−= 21
Where
J: the number of atom diffusing down the concentration gradient
per second per unit area, unit: atoms/cm2⋅s
C: the concentration of molecules (or the number of diffused
molecules per unit volume), unit: atoms/cm3
x: atomic jump distance
D: diffusion coefficient, unit: cm2/s
Ji units[ ] =g
s ⋅ cm2
J i units?[ ] = Dcm2
s
⋅
∂C
∂x
g
cm4
, i = x, y, z( )
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Example: (Fick’s 1st Law) : A thin plate of BCC Fe, T=1000K
carbon concentration:
C1=0.2wt%; C2=0%
CO/CO2
Oxidizing
atmosphereFe
t=0.1cm
Density of Fe: ρ = 7.9g/cm3
D = 8.9×10-7 cm2/s at 1000K
Calculate: the number of carbon atoms transport to
back surface per second through an area of 1cm2
Solution:
The concentration of carbon (atoms/cm3): AC
Fe NA
wtC ⋅
⋅=
ρρρρ%
scmatoms
cm
cmatomsscm
t
CCD
dx
dCDJ
C
cmatoms
molatomsmolg
cmgC
⋅×=
××=
−=−=
=
×=
×⋅×
=
−
215
3202721
2
320
233
1
/109.6
1.0
/1092.7/107.8
0
/1092.7
/10023.6/01.12
/9.7%2.0
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Diffusivity -- the proportionality constant between flux and
concentration gradient depends on:
� Type of bonding
� Diffusion mechanism. Substitutional vs interstitial.
� Temperature.
� Type of crystal structure of the host lattice. Interstitial
diffusion easier in BCC
than in FCC.
� Type of crystal imperfections.
(a) Diffusion takes place faster along grain boundaries than
elsewhere in
a crystal.
(b) Diffusion is faster along dislocation lines than through
bulk crystal.
(c) Excess vacancies will enhance diffusion.
� Concentration of diffusing species.
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Diffusion coefficient D
depends on the temperature
RT
Q
o
d
eDD−
=
T R
Q - D = D
dolnln
D is the Diffusivity or Diffusion Coefficient
(m2 / sec )
Dois the prexponential factor or Diffusion
constant (m2 / sec )
Qdis the activation energy for diffusion (joules /
mole )
R is the gas constant ( joules / (mole deg) )
T is the absolute temperature ( K in Kelvin )
– Q/R
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Non Steady State Diffusion
Diffusion flux and the concentration gradient at some particular
point in a solid vary
with time, with a net accumulation of depletion of the diffusing
species resulting
•Fick’s second law apples (when D is independent of
composition)
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Fick’s 2nd Law
Chigh Clow
dxdA
Jin Jout
dV=dA⋅dx
dAJJdVt
Coutin )( −=
∂
∂
Fick’s 2nd Law:
The rate of change of the number of
atoms in the slice dVThe rate that atoms entering the slice –the
rate that atoms leaving the slice
=
2
2
)(
x
CD
x
CD
x
x
J
dV
dAJJ
t
Coutin
∂
∂=
∂
∂−
∂
∂−=
∂
∂−=−=
∂
∂
⇒2
2
x
CD
t
C
∂
∂=
∂
∂
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In words: The rate of change of composition at
position x with time, t, is equal to the rate of
change of the product of the diffusivity, D, times
the rate of change of the concentration gradient,
dCx/dx, with respect to distance, x.
2
2
x
CD
t
C
∂∂∂∂
∂∂∂∂====
∂∂∂∂
∂∂∂∂
�Solutions to the DE are possible when physically meaningful
boundary conditions are specified
�Particularly important solution – semi-infinite solid in which
surface concentrations are constant, diffusing species is usually a
gas, and the partial pressure is maintained at a constant value
Second order differential equations are nontrivial and difficult
to solve.
Consider diffusion in from a surface where the concentration of
diffusing species
at the surface is always constant. This solution applies to gas
diffusion into a solid
as in carburization of steels or doping of semiconductors.
Boundary Conditions
• For t = 0, C = Co
at 0 < x
• For t > 0 C = CS
at x = 0
and C = Co
at x =∞∞∞∞
-
Dt2
x erf - 1 =
C - C
C - C
os
ox
where
CS = surface concentration
Co = initial uniform bulk concentration
Cx = concentration of element at distance x from surface at time
t
x = distance from surface
D = diffusivity of diffusing species in host lattice
t = time
erf = error function = erf (x/2ooooDt) is the Gaussian error
function – this is like a
continuous probability density function from 0 to x/2ooooDt
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�The equation below demonstrates the relationship between
concentration, position, and time
� Cx being a function of the dimensionless parameter x/2ooooDt
may be determined at any time and position if the parametes Co, Cx,
and D are known
Dt2
x erf- 1 =
C- C
C- C
os
ox
Special Case
Desired to achieve some specific
concentration of solute, C1in an alloy,
then
constant= C- C
C- C
os
ox
constant=Dt2
x
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Example
The carburization of a steel gear at a temperature of 1000oC in
gaseous CO/CO2mixture, took 10hours. How long will take to
carburize the steel gear to attain
similar concentration conditions at 1200oC?
For C in γγγγ – iron D = 0.2 exp{ - 34000 / 2T} cm2/s
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Example: (Fick’s 2nd Law)
Determine the time it takes to obtain a carbon concentration of
0.24% at depth 0.01cm beneath the surface of an iron bar at 1000oC.
The initial concentration of carbon in the iron bar is 0.20% and
the surface concentration is maintained at 0.40%.
The Fe has FCC structure and the diffusion coefficient is
D = 2××××10-5 m2/s ⋅⋅⋅⋅exp( ).
Known: T=1000oC, depth x = 0.01cm, CX = 0.24%
CO = 0.2%, CS = 0.4%
D=2××××10-5 m2/s ⋅⋅⋅⋅exp( )
R = 8.314 J/K
Find: time t = ?
RT
molJ /000,142−
RT
molJ /000,142−
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Solution:
D1273K = 2×10-5 m2/s ⋅exp
D1273K = 2.98 ×10-11 m2/s= 2.98 ×10-7 cm2/s
)1273314.8
000,142(
×−
−===
−
−=
−
−
Dt
xerf
CC
CC
OS
OX
212.0
2.0
04.0
2.04.0
2.024.0
⇒ erf(z) = 0.8, where z = Dt
x
2erf(z) = 0.8
12
1
12
1
)()(
)()(
zz
zz
zerfzerf
zerfzerf
−
−=
−
−
90.095.0
90.0
7970.08209.0
797.08.0
−
−=
−
− z
-
⇒ z = 0.906 ⇒ = 0.906Dt
x
2
⇒ t = [x / (2 × 0.906)]2/D = .min73.11041098.2
)812.1/01.0(7
2
==× −
s
t = 1.73min.
Effective penetration distance: xeff
(for 50% of concentration)
5.02/)(2
,2
),(
0
0
0
00
0
0
0
=−
−=
−
−+
=−
−
+=
CC
CC
CC
CCC
CC
CC
CCtxC
s
s
s
s
s
seff
Fick’s 2nd Law: )2(15.0
0
0
Dt
xerf
CC
CC eff
s
−==−
−
erf (0.5) ≈ 0.5 ⇒ xeff ≈≈≈≈ Dt
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Effective penetration distance
In general, for most diffusion problems
xeff =
where γγγγ: a geometry-dependent parameter
γγγγ = 1 for a flat plate
γγγγ = 2 for cylinders
Dtγγγγ
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Thermal Diffusion of Impurities into Silicon
The ability to modify the properties of a semiconductor through
the
addition of controlled amounts of impurity atoms is an important
aspect of
silicon device and IC manufacture.
There are two principal methods which are used to introduce
impurities
into silicon, thermal diffusion and ion implantation.
We will discuss the basic equations describing the impurity
profiles below
the surface of the wafer using the thermal diffusion method.
Thermal diffusion is a high temperature process where the dopant
atoms
are deposited on to or near the surface of the wafer from the
gas phase.
Wafers can be batch-processed in furnaces. The impurity profile
or
distribution is determined mainly by the diffusion temperature
and time,
and decreases monotonically from the surface. The maximum
concentration of a particular diffusing impurity is always found
at the
surface.
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The impurity concentration C(x,t) as a function of depth below
the wafer surface, x,
and diffusion time, t is determined from Fick's diffusion
law;
D is the diffusion coefficient and varies markedly from one
impurity to another;
some impurities diffuse quickly through silicon (fast
diffusants), while others move
more slowly (slow diffusants). of impurities in silicon.
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D depends on the temperature of diffusion and can be expressed
in the generalized
form as D(T) = Do exp(-EA / kBT) where Do is the diffusion
coefficient extrapolated
to infinite temperature and EA is an activation energy (usually
quoted in eV).
Thus, a plot of log D(T) (µm2 / hr) vs 1/T (K-1) will give a
straight line with slope
EA.
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Diffusion:
Smaller atoms diffuse more readily than big ones, and diffusion
is
faster in open lattices or in open directions
Self-diffusion coefficients
for Ag depend on the
diffusion path. In general
the diffusivity if greater
through less restrictive
structural regions – grain
boundaries, dislocation
cores, external surfaces.
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Example
(A)For an ASTM grain size of 6, approximately how many grains
would there
be per square inch at a magnification of 100?
(B)The diffusion coefficients for copper in aluminum at 500 and
600oC are
4.8x10-14 and 5.3x10-13 m2s-1, respectively. Determine the
approximate
time at 500oC that will produce the same diffusion results (in
terms of
concentration of Cu at some specific point in Al) as a 10 hour
heat
treatment at 600oC.
(C) For the problem (B) compute the activation energy for the
diffusion of Cu
in Al.
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(A) This problem asks that we compute the number of grains per
square inch for
an ASTM grain size of 6 at a magnification of 100x. All we need
do is solve for
the parameter N in the equation below, inasmuch as n = 6.
Thus
N = 2n−1
= 26 −1
= 32 grains/in2
(B) Fick’s second law, as it is desired to achieve some specific
concentration
conditions.
( )( )( )
hourssmx
hourssmx
D
tDt
tDtD
tconsDt
4.110108.4
10.103.5
tan
1214
1213
500
600600
500
600600500500
===
=
=
−−
−−
-
(C) Using the equation
−−
−
−
−
−
−−
===
==
500600
500
600
500
600
500600
500
600
500600 and
RT
Q
RT
Q
RT
Q
RT
Q
RT
Q
o
RT
Q
o
RT
Q
o
RT
Q
o
dd
d
d
d
d
dd
e
e
e
eD
eD
D
D
eDDeDD
( ) ( )
( ) ( )( )
( ) ( ) ( )[ ]
1
1213121411
500600
600500
500600500600
500600
500
600
.7.134
773
1
873
1
.103.5ln.108.4ln..31.8
11
lnln
11lnlnln
−
−−−−−−
=
−
−=
−
−=
−−=
−−=−=
molkJQ
KK
smxsmxKmolJQ
TT
DDRQ
TTR
Q
RT
Q
RT
QDD
D
D
d
d
d
ddd