1. Lets see.. Do we know these?
• Activation Energy
• Concentration gradient
• Diffusion VS Solid soln.
• Diffusion Coefficient
• Diffusion Flux
• Self Diffusion
• Interdiffusion (impurity
diffusion)
• Steady state Diffusion
• Non-steady state diffusion
• Fick’s 1st and 2nd laws
• Vacancy diffusion
• Interstitial Diffusion
• Error Function
• Carburizing, Nitriding
CHAPTER 5:
DIFFUSION IN SOLIDS
Dr. Mir M. Atiqullah
(Includes materials from Callister and other sources)
What is diffusion?
Diffusion of:
Solids: solid solution, needs high temp.
Liquids: mixing of two liquids.. Solution..
Gases: gas diffusing into solid (carburizing), liquid,
another gas, smoke diffusing into atmosphere,
pollutant into air, …
Diffusion- is dispersion, spreading,….phenomenon of
material transport by atomic motion.
During diffusion, there is commotion among the atoms to
make their way to lower density area.. Ready to take any
twisted, back alley, to keep moving….kind of confusion..
2
• Glass tube filled with water.
• At time t = 0, add some drops of ink to one end
of the tube.
• Measure the diffusion distance, x, over some time.
• Compare the results with theory.
to
t1
t2
t3
xo x1 x2 x3
time (s)
x (mm)
DIFFUSION needs time
100%
Concentration Profiles
0
Cu Ni
3
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
Needs favorable conditions- vacancies, temperature..
Initially After some time
100%
Concentration Profiles
0
Adapted from Figs. 5.1 and 5.2, Callister 6e.
DIFFUSION: THE PHENOMENA (1)
2. 4
• Self-diffusion: In an elemental solid, atoms
also migrate.
Conditions:
Temperature has to be ‘sufficiently’ high.
Some vacancy must exist (high temp helps)
Label some atoms
After
some
time
A
B
C
D A
B
C
D
DIFFUSION: THE PHENOMENA (2)
Existence of Defects – any effect?
The atoms have moved..
5
Substitutional Diffusion:
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
1. number of vacancies
2. activation energy to exchange.
increasing elapsed time
DIFFUSION MECHANISMS
6
• Simulation of
interdiffusion?
across an interface:
• Rate of substitutional
diffusion depends on:
--vacancy concentration
(how much space available?)
--frequency of jumping
(how fast moving?
Is it Temperature ?).
(Courtesy P.M. Anderson)
DIFFUSION (subst.) SIMULATION
7(Courtesy P.M. Anderson)
• Applies to interstitial impurities.
• More rapid than vacancy diffusion.
• Simulation:
--shows the jumping of a smaller atom
(gray) from one interstitial site to
another in a BCC structure. The
interstitial sites considered here are
at midpoints along the unit cell edges.
INTERSTITIAL SIMULATION
Larger amount of interstitial diffusion is
probable because of abundance of interstitial sites
3. • Case Hardening (carburizing):
--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
8
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
Midland-
Ross.)
PROCESSING USING DIFFUSION (1)
• Doping Silicon for n-type semiconductors:
• Process:
9
1. Deposit P rich
layers on surface.
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
silicon
magnified image of a computer chip
0.5mm
light regions: Si atoms
light regions: Al atoms
Fig. 18.0,
Callister 6e.
PROCESSING USING DIFFUSION (2)
• Flux:dispersion rate of atoms : thru unit area per unit time.
10
J =
1
A
dM
dt
⇒
kg
m2
s
or
atoms
m2
s
• Directional Quantity
• Flux can be measured for:
--vacancies
--host (A) atoms
--impurity (B) atoms
--anything else ?
Jx
Jy
Jz x
y
z
x-direction
Unit area A
through
which
atoms
move.
MODELING DIFFUSION: FLUX
• Concentration Profile, C(x): [kg/m3]
11
• Fick's First Law:
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Position, x
Cu flux Ni flux
• The steeper the concentration profile,
the greater the flux!
..applicable at which location ?
Adapted
from Fig.
5.2(c),
Callister 6e.
Jx
= −D
dC
dx
Diffusion coefficient [m2/s]
concentration
gradient [kg/m4]
flux in x-dir.
[kg/m2-s]
CONCENTRATION PROFILES & FLUX
Why
–ve?
4. • Steady State: the concentration profile doesn't
change with time.
12
• Apply Fick's First Law:
• Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary
with position)! Essentially a st. line.
Jx(left) = Jx(right)
Steady State:
Concentration, C, in the box doesn’t change w/time.
Jx(right)Jx(left)
x
Jx = −D
dC
dx
dC
dx
left
=
dC
dx
right
• If Jx)left = Jx)right , then
STEADY STATE DIFFUSION
• Steel plate at
700C with
geometry
shown:
13
• Q: How much
carbon transfers
from the rich to
the deficient side?
J = −D
C2 − C1
x2 − x1
= 2.4 × 10−9 kg
m2
s
Adapted
from Fig.
5.4,
Callister 6e.
C1
= 1.2kg/m
3
C2
= 0.8kg/m
3
Carbon
rich
gas
10m
m
Carbon
deficient
gas
x1 x20
5m
m
D=3x10-11m2/s
Steady State =
straight line!
EXAMPLE: STEADY STATE DIFFUSION
Class Practice: Steady State Diffusion(6th ed.)
Problem 5.6 Palladium sheet for purifying hydrogen gas.
Palladium sheet 5 mm thick, area=0.2 m2, Temp 500°C, D=1.0e-8 m2/s
Concentration (high end) = 2.4 kg/m3 of palladium and (low end) = 0.6 kg/m3.
Assume steady state situation. Calculate H2 flux kg/hour.
( ) ..
005.
6.04.2
)3600(10*0.1 8
=
−
=
∆
∆
−= −
x
C
Dj 2.6x10-3 kg/hr.
• Concentration profile,
C(x), changes with time.
14
• To conserve matter:
• Fick's First Law:
• Fick’s 2nd Law of Diffusion:
Concentration,
C, in the box
J(right)J(left)
dx
dC
dt
= D
d2C
dx2
−
dx
= −
dC
dt
J = −D
dC
dx
or
J(left)J(right)
dJ
dx
= −
dC
dt
dJ
dx
= −D
d2C
dx2
(if D does
not vary
with x)
equate
NON STEADY STATE DIFFUSION
5. • Copper diffuses into a (semi-infinite) bar of aluminum.
15
• General solution for gas
atoms diffusing into solid:
"error function“Callister 7e .
Eqtn. at foot note p-116.
Also Table 5.1, Page 116
C(x,t) − Co
Cs − Co
= 1− erf
x
2 Dt
pre-existing conc., Co of copper atoms
Surface conc.,
Cs of Cu atoms bar
Co
Cs
position, x
C(x,t)
to
t1
t2
t3
EX: NON STEADY STATE DIFFUSION
3 assumptions
p-129.
Diffusion rate slows
down after some
time…
OK but Why?
Class Practice – Non steady state diffusion
Problem # 5.15 Page 163.
Write down all data given
Write down the formula to be used
Any data needed from table/figure ?
..
Must write units, if the quantity/number has one.
• Copper diffuses into a bar of aluminum.
• 10 hours at 600°C gives desired C(x).
• How many hours would it take to get the same C(x)
if we processed at 500C?
16
(Dt)500ºC =(Dt)600ºC
s
C(x,t)−Co
C − C
o
= 1− erf
x
2Dt
• Result: D1t1 = D2t2.
• So calc.
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
t500
=
(Dt)600
D500
= 110hr
4.8x10-14m2/s
5.3x10-13m2/s 10hrs
Example 5.3 Page 148
See also p-116-7 Eq. 5.6b
17
• The experiment: we recorded combinations of
t and x that kept C constant.
to
t1
t2
t3
x o x 1 x 2 x3
• Diffusion depth given by: xi ∝ Dti
C(xi,ti) − Co
Cs − Co
= 1− erf
xi
2 Dti
= (constant here)
DIFFUSION DEMO: ANALYSIS
Means: Depth of diffusion achieved is proportional to sq. root
of length of diffusing time.
Double depth can be achieved if time allowed is __?_times.
6. 5.5 FACTORS THAT
INFLUENCE DIFFUSION
• Diffusion Species (combination)– nature of materials and
type of diffusion
– Substitutional -
– Interstitial – more vacancies
• Temperature – most profound effect
−=
RT
Q
DD d
exp0
D = Diffusion Coeeficient T = Abs Temp. (K)
D0 = pre-exponential (depends on T)
Qd = Activation Energy for diffusion
R = Gas const. 8.31 J/mol, 8.62E-5 eV/atom-K
Diffusivity increases with T.
And lower value of Qd
• Experimental Data:
1000K/T
D (m2/s) C in α-Fe
C
in
γ-Fe
Alin
Al
Cu
in
Cu
Zn
in
Cu
Feinα-Fe
Feinγ-Fe
0.5 1.0 1.5 2.0
10-20
10-14
10-8
T(C)
1500
1000
600
300
D has exp. dependence on T
Recall: Vacancy does also!
19
Dinterstitial >> Dsubstitutional
C in α-Fe
C in γ-Fe Al in Al
Cu in Cu
Zn in Cu
Fe in α-Fe
Fe in γ-Fe
Adaptedfrom Fig. 5.7, Callister6e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook
(Ed.) SmithellsMetals ReferenceBook, 7th ed., Butterworth-Heinemann,Oxford, 1992.)
DIFFUSION AND TEMPERATURE
Class Exercise #5.27
Problem: Given D at 2 Temp.,
Find: D at a third temp.
Strategy:
1. Set up 2 equations – 2 unknowns. Use
2. Calc. D0 and Qd
3. Use these to calc. D at 3rd temperature.
⋅
−= ⋅
TR
Q
DD d
exp0
In-class exercise
Diffusivity at two temperatures are known as follows. Find D0 and Qd. and
then D1100
Given: T1=1473° K D1=2.2E-15 m2/s
T2=1673° K D2=4.8E-14 m2/s
Find: First D0 and Qd ? Then D1300=?
⋅
−= ⋅
TR
Q
DD d
exp0
Hints:
(a) Eliminate D0 using 2 equations at 2 temperatures, and solve for Qd
Then calculate D0 using any one equation.
First use eq. in ex. prob 5.5
(b) Find D1300 using the above results. (T3=1573° F)
7. Fabrication of IC Chips: Diffusion
IC Chip- packaging removed
Chip # of transistors
8-core i7 (2012 ?) 2.6 Billion
22-core Xeon (2016) 7.2 Billion
SPARC M7 (2015) 10 Billion
Foreign/impurity material/atoms are diffused on
the Si wafer surface, by heating, to build the
circuits.
Two heat treatment/diffusion are used:
1. Predeposition – diffuse sufficient quantity of
impurity onto the surface, to some depth.
900-1000°C
2. Drive-in – ‘drive’ the impurity atoms deeper
without adding more. Higher temp. 1200°C,
longer time, oxidizing atmosphere = SiO2
Size of the ‘die’ at the
center is about 6x6 mm.
Semiconductor (contd.)
1. Predeposition stage:
Concentration Cx at depth x:
2. Drive-in stage:
Concentration Cx at depth x:
Note:Surface conc. Cs remains same.
Note: Surface conc. Cs depletes, but
depth xj increases over time, merging at
CB. Practically no loss of atoms from
surface, due to oxidation later.
CB
xj
C0
Semiconductor (contd.)
Stage 1:Where total amount Q0
deposited in time tp is given by:
Stage 2: The depth xj to which a
predetermined conc. CB achieved:
(just solving for x)
Expl. 5.6 Page 156 9th ed.
20
Diffusion FASTER for...
• open crystal structures
• lower melting T materials
• materials w/secondary
bonding
• smaller diffusing atoms
• cations /ˈkatīən/ +ve charged
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• higher melting T materials
• materials w/covalent
bonding
• larger diffusing atoms
• anions -ve charged
• higher density materials
Recap: DIFFUSION
