![• 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?](https://image.slidesharecdn.com/ch05-diffusion-updatedsept2016-sentfall2016-161102134156/85/Ch05-diffusion-updated-sept2016-sent-fall2016-3-320.jpg)
Ch05 diffusion-updated sept2016-sent fall2016
- 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