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Moores Law and material processing Defects in crystals Eutectic phase diagram Solid solubility Homogeneous nucleation
Heterogeneous Nucleation Growth processes
INTRODUCTION
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High
Low
FailureRate
Co
st
1 mm
1-5 nm High
Low
Minim
um
FeatureSize
Comp
lexity
1930-
1950s
1960s
1970-90
VACUUM TUBES
SEMI CONDUCTOR BASEDTRANSI STORS
PLANAR TRANSI STORS
I Cs LSI , VLSI
SMART STRUCTURES1995-
NANO STRUCTURES
2010-
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Intel Roadmap for VLSI/ULSI
300300300200/
300
200200Wafer (mm)
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Moores LawProcessor power will keep doubling every two years.
Aft er I ntel CorpAft er I ntel Corp
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Aft er I ntel CorpAft er I nt el Corp
Moores Law
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Aft er I ntel CorpAft er I nt el Corp
Moores Law
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Aft er I ntel CorpAft er I ntel Corp
Moores Law
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Processed Silicon Wafer in 2000
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EUV Lithography
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Silicon Crystal
Crystal Structure :
Diamond cubicTwo interpenetrating FCC latticesa = 0.543 nmd = distance between atoms
= 3a/4Atomic density = 8 / a3
Closed packed plane (111)
http://jas.eng.buffalo.edu/education/solid/unitCell/home.html
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Silicon Crystal
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Silicon Crystal
100 = 6.8 x 1018 at/m2
111 = 9.6 x 1018 at/m2
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Silicon Crystal
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Silicon Crystal
K300atSiform/10n
eVT10x6.321.1)T(Ewhere
m/)Tk2
)T(Eexp(T10x9.3n
316i
4g
3
B
g2/322i
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Silicon CrystalCharge Carriers in Doped Si
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Silicon Crystal
SinforEvvelocityDrift
pqnq
nd
pn
Conductivity and Mobility :
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
1E+19 1E+21 1E+23 1E+25 1E+27at/m3
m2/V-sec
n
p1E+19
1E+21
1E+23
1E+25
1E-05 1E-03 1E-01 1E+01
Resistivity, ohm-m
at/m3 p-type
n-type
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Mobility in Silicon
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
1.E+19 1.E+21 1.E+23 1.E+25 1.E+27
at/m3
m2/V-sec
n
p
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1E+19
1E+21
1E+23
1E+25
1E-05 1E-03 1E-01 1E+01
Resistivity, ohm-m
at/m
3
p-type
n-type
Resistivity in Silicon
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Defects in CrystalsTypes of Defects:
Point Defects vacancy, interstitial, V-I pairs, clusters Line Defects Dislocations edge & screw Planar Defects stacking fault, twin, grain boundary
Point Defects:
Interstitial
Impurity Vacancy
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Defects in CrystalsDislocations:
Extra planeof atoms
Edge
dislocation Screwdislocation
Screw Dislocation:b ||le to l
Edge Dislocation:Extra plane of atom is
present.b r to l
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Defects in Crystals
Screw Dislocation
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Defects in CrystalsPlanar Defects: Stacking Faults Twins
Grain Boundaries
Layer ordering in silicon:A B C A B C along (111) plane.
Stacking fault:A B C B C A B C A B C
Twin :A B C A B A C B A
Grain Boundary:
Twin plane
C
A
B
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Phase ChangeLiquid state : High entropy and High enthalpySolid state : Low entropy and low enthalpy
A phase change, say liquid to solid, occurs, if
F = H - T S is negative
where F = free energy change for solidificationH = enthalpy change for solidification = latent heat of fusionS = entropy change during solidification
Above melting point (Tm), F > 0At melting point, F = 0 gives H = Tm SBelow melting point, F = S(Tm-T) < 0
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Phase DiagramComplete Solid Solution:Liquid of composition x onsolidifying at temp T gives solid of
composition xs and liquid ofcomposition xl.
s
l
xx
xx
os
ol
liquidofVol
solidofVol
Eutectic :At TeutecticA + B = L
or + = L
Temp
compA B
L
L+S
Sxs xl
s o l
x
T
TmA
TmB
Temp
compA B
L
A+L
A+B
Teutectic
B+L
TmA TmB
Eutectic with nosolid solubility
Temp
compA B
L
+L
+
Teutectic
+L
TmBTm
A
Eutectic withlimited solid
solubility
Gibbs Phase Rule :F + P = C + 2
P no. of phasesC no. of componentsF degree of freedom
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SegregationWhen a liquid having composition xlfreezes, the solid that first freezes has
composition xs.
This gives a purification onsolidification since xs < xl.
Segregation coefficient = ko
0citliminc
ck l
l
so
= ratio of slopes of solidus and
liquidus at origin
ko < 1 purificationko > 1 no purification
Temp
Comp(%B)
L
L+S
S
clcs
liquidus
solidus
ks1
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Solid SolubilitySolubility of B in A depend on the bond energies of A-A, B-B and A-B bondsand the entropy of the mixture of A and B atoms.
)]x1ln()x1(xlnx[Tk)x1(zxNN
ABBBBBBB
BA
where
NA no. of A atoms & NB no. of B atomsxB = NB/ (NA + NB)9uhg
= AB (AA + BB)/2AA, BB, AB are energies of A-A, B-B and A-B bonds respectivelyz = no. of nearest neighbours of A
A = Helmholtz Free energy of forming solid solution of B in A = E - TS
Slope ( A/ xB) near xB=0 is - . It is impossible to purify silicon to100%.
We have to spend more and more
energy to purify as material gets purer.Solubility limit
xBA >0
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TkEexpxx,1xIf
]Tk
)x21(zexp[
x1
x
0x1
x
lnTk)x21(z
.itlimilitylubsogives0x
])NN/[A(
B
BoBB
B
B
B
B
B
B
bB
B
BASolid Solubility
This equation applies to solutes, vacancies, interstitials and allother thermal defects present in solids.
exampleforerstitialsintforTk
Eexpxx
exampleforvacanciesforTk
Eexpxx
B
IIoI
B
VVoV
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nucliicriticalinmoleculesof.no)gg(3
n2n
nucliiofradiuscritical)gg(
v2r
0r8v
)gg(r4gives0
r
G
area/energysurfacewhere,n)v(36)gg(n
)areasurface(G)volchemcial(Gr4
v
gr
3
4G
,growingis'r'radiusofnucleisphericalaSince
)p
pln(TkT)ss()TT)(ss()ss(Thhggg
ioncondensat0G,TTAt
ioncondensatno0G,TTAt
3
vlc
vl
l
c
cl
vl2c
r
32
3 2lvl
sv2
l
lv3
iBs
vlv
vlvlvlvllv
v
v
c
Homogeneous NucleationCondensation of Liquid from Vapour Phase:r
rc r
G
Gc
Gv ~r3
Gs ~r2
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2s
3
2c3
2
vl
lc
T
K
energysurfaceofrd3/13
r4
gg
v
3
16G
Homogeneous NucleationCondensation of Liquid from Vapour Phase:For r < rc, the nuclei will reduce in size.For r > rc, the nuclei will grow.
An energy barrier of Gc has to be overcome for nucleation toproceed.
rc
where, Ts is the supersaturation, i.e., (Tv -T)
2sB
2
o
B
co
)T(TkKexpI
TkGexpIInucleationofRate
Rate of nucleation increases with supersaturation.
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Homogeneous NucleationNucleation of Solid from Vapour : = f(orientation of the crystal plane) = anisotropic
Singular surface, is minimum
Non-singular surface consists
of steps of singular surface
2
3s
c
s
s
32
3 2svs
ib
vsc
)T(
)(KG
area/energysurfaceaveragewheren)v(36)gg(nG
,growingismolecules'n'containingofnucleiaSince
)p
pln(TkggG
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Homogeneous NucleationNucleation of Solid from Liquid :
Tk
g
)T(T
KKlnIln
Tk
)gG(expIIrateNucleation
)T(
1
)gg(
1G
)ss(T)ss)(TT(
)ss(T)ss(T)ss(Thhgg
B
#
2
21
B
#c
o
22lsc
lslsm
lslsm
lslsls
where g# = energy required to break bonds in liquid
Below a temperature, nucleation rate reduces so that glassformation takes place at that point.
Tm
I
glassformation
T
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Heterogeneous Nucleation
4
CosCos32
gg
v
3
16
)gg(
v)(27
4
G
)(rv
ggrG
anglecontactwhere,Cos
32
vl
l3lv
2vll
2l
3
lssvlslvlv
c
lssvlslvlv2
l
vl
3l
lvslsv
Gc
=0
Gc decreases with smaller contact angle.
Hence nucleation rate increases if there isheterogeneous nucleation.
Nucleation is still easier inside a cavity.
lv
sv
sl surface
Condensation of Liquid on a Surface :
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Nucleation will proceed by arrival ofmolecule on the surface and then itsmovement to the nucleating site bysurface diffusion.If surface diffusion is slow,nucleation rate, I, will become
dependent on surface diffusion.
Heterogeneous Nucleation
Tk
gGexp
Tk3
GANI
B
sdc
B
3c
cs
surface
surfacediffusion
I = Nucleation rate for heterogeneous nucleationgsd = activation energy for surface diffusion.
Condensation of Liquid on a Surface :
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Heterogeneous NucleationNucleation of a Solid inside Another Solid : Phase is growing inside phase.
2
3c
gg
v4G
Cos2
Cos
Gc
GcH
0
1 boundary
edge
cornerStrain :
gs = strain energyIf g
s
is small, surface energydominates.
If gs is large, nucleation takesplace to minimize the strain energy.
bsn A)ggg(nG
1
y/RE(y/R)
sphere
needle
platelet
0
R
y
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