NSE201Lec05

Restricted
Dimensional
Assemblies
Solid
Solutions
Solid Solutions
• Many pure elements dissolve large quantities of other
elements to form “solid solutions” or “alloys” (Latin
alligare, “bind”)
• Substitutional (Cu-Ni, Au-Ag, Ti-Zr, K-Rb, SiGe, Se-Te, NaCl-KCl)
• Interstitial (H, C, B, O, N)
• Solid solutions may be disordered or ordered
Systematics
• William Hume-Rothery
(1899 —1968)
• Oxford University, first chair
of Metallurgy, 1958
• the “Hume-Rothery Rules”
Hume-Rothery Rule 1
• SIZE EFFECT
• If the difference between atomic radii of the
elements forming an alloy exceeds 15%, solid
solubility is restricted (a “negative” rule)
• Hume-Rothery used the closest distance of approach
of atoms in the structure of each pure element as
measure of atomic size
Complete Solubility
• Cu + Ni
• rCu = 0.128 nm, rNi = 0.125nm
• Δr/r = 2.3% (<15%), H-R #1 applies
• Cu and Ni comprise a “binary isomorphous” system;
complete miscibility for any composition
°C
1800
10
20
30
40
50
1700
1600
at. % Ni
60
70
80
90
L
1500
1455°
1400
1300
1200
1100
1084.87°
1000
L+!
!
900
800
700
Cu
10
20
30
40
50
60
wt. % Ni
70
80
90
Ni
Restricted Solubility
• Cu + Sn
• rCu = 0.128 nm, rNi = 0.158nm
• Δr/r ≈ 23% (>15%), H-R #1 does NOT applies
• Cu and Sn exhibit limited solubility
1100
at % Sn
10 20 30 40 50 60 70 80 90
1085°
1000
900
800
T (°C)
700
600 "
500
400
L
799°
!
586°
520°
756°
#
'
640°
&
$
415°
350°
300
200
100
%
189°
232°
186°
%’
!(Sn
0
Cu 10 20 30 40 50 60 70 80 90 Sn
wt % Sn
Hume-Rothery Rule 2
• ELECTRONEGATIVE VALENCE EFFECT
• The likelihood of formation of a stable phase is
increased as one of the elements becomes more
electronegative and the other more electropositive
Galvanic Series
Noble
(cathodic)
Active
(anodic)
Platinum
Gold
Graphite
Titanium
Stainless Steel
Copper
Brass
Nickel
Tin
Steel
Aluminum
Zinc
Magnesium
Electronegative
(gains electrons)
Electropositive
(loses electrons)
1100
1064.43°
1000
900
L
800
rAu = 0.144 nm
700
600
rSn = 0.158 nm
500
Δr/r = 9.7%
(Au)
86.2
418°
490°
95.8
400
°C 300
309°
231.9681°
200
0.3 10
100
! Sn
0
Sn 10
252°
280°
80
217°
"
#
$
20 30 40 50 60 70
wt % Au
%
80 90 Au
Hume-Rothery Rule 3
• RELATIVE VALENCE EFFECT
• Elements with lower valence dissolve more readily in
elements of higher valence
• Observation: monovalent Cu, Ag and Au dissolve in
B-subgroup elements having valences >1
• For polyvalent elements, relative valence rule is less
general
Extended Packing
• When size difference exceeds
15% (notably 1.225:1), Laves
phases, composition AB2, may
be favored
• Known examples ≈ 360
• Basic layer of smaller atoms
stacked in either cubic or
hexagonal arrangement
• Larger atoms have CN = 16
C14
Laves Phases
• Three basic structures
• C15 (cubic, MgCu2)
• C14 (hexagonal, MgZn2)
• C36 ( hexagonal MgNi2)
• Smaller atoms sometimes in
icosahedral coordination
with larger atoms
C15
Strukturbericht
• A Elements (uniary systems)
• B 1:1 stoichiometry
• E Perovskites
• C 1:2 stoichiometry
• H Spinels
• D 1:3 stoichiometry
• L long-period ordering
Ref: Strukturbericht, Akademische Verlagsgesellschaft M.B.H.,
Leipsig, Germany (1913 - 1939); continued as Structure Reports,
International Union of Crystallography, (1940 - present)
DO3 Structure
Ni at !
Ni at 0,1
Ni at !
Sn at 0,1
Sn at !
Sn at 0,1
Ni at ",#
Ni at ",#
Ni at 0,1
Ni at !
Ni at 0,1
Sn at !
Sn at 0,1
Sn at !
Ni at ",#
Ni3Sn
Ni at ",#
Ni at !
Ni at 0,1
Ni at !
Sn at 0,1
Sn at !
Sn at 0,1
rNi = 0.125 nm
rSn = 0.158 nm
1500
at % Sn
20 30 40 50 60 70 80 90
10
1455°
1400
L
1300
1266°
1200
19.0
1100
T
(°C)1000
900
1132°
1176°
32.5
"
!
cubic DO3
978°
922°
38.0
851°
43.1
800
796°
700
600
Ni3Sn
!’
#
89.3
hexagonal DO19
$
500
400
Ni 10 20 30 40 50 60 70 80 90 Sn
wt % Sn
Perovskite Unit Cell
z
• Lattice = simple cubic (lattice
points at corners of cube)
• Motif = 1 Ca2+ at 0,0,0; 1
Ti4+ at ½,½,½; and 3 O2- at
½,½,0, 0,½,½, and ½,0,½
y
x
Perovskites
NaNbO3
CaTiO3
CaSnO3
BaPrO3
YAlO3
KMgF3
KNbO3
SrTiO3
SrSnO3
SrHfO3
LaAlO3
PbMgF3
NaWO3
BaTiO3
BaSnO3
BaHfO3
LaCrO3
KNiF3
CdTiO3
CaCeO3
BaThO3
LaMnO3
KZnF3
PbTiO3
SrCeO3
CaZrO3
BaCeO3
SrZrO3
CdCeO3
BaZrO3
PbCeO3
PbZrO3
LaFeO3
Spinel
• Formula A2+B3+2O4
• Eight (8) formula units per cubic unit cell (eight
divalent cations, 16 trivalent cations, 32 oxygen
anions), with oxygen anions in close-packed
configuration
• “Normal” spinel: divalent cations on tetrahedral
sites; trivalent cations on octahedral sites
• “Inverse” spinel: trivalent cations on tetrahedral
sites and 1/2 of the octahedral sites; divalent cations
on remaining 1/2 of octahedral sites.
Top
View
tetrahedral sites
octahedral sites
Spinel
• Lattice = face-centered cubic
• Basis (motif) = 14 ions
• two divalent cations
• four trivalent cations
• eight oxygen anions
Spinels
Normal
SnMg2O4 MgAl2O4
FeV2O4 SrAl2O4
WNa2O4 CrAl2O4
MoAl2O4
FeAl2O4
CoAl2O4
NiAl2O4
CuAl2O4
ZnAl2O4
CoFe2O4 ZnK2(CN)4
NiFe2O4 CdK2(CN)4
AlFe2O4 HgK2(CN)4
PbFe2O4
MgCo2O4
TiCo2O4
CuCo2O4
CoCo2O4
Inverse
CuCr2S4
CoCo2S4
CuRh2S4
FeCr2S4
MnCr2S4
MgFe2O4
TiMg2O4
VMg2O4
SnZn2O4
MgGa2O4
MgIn2O4
FeIn2O4
CoIn2O4
NiIn2O4
Ordering
z
• CuAuI is tetragonal, c/a ≈ 0.93
• Orders below Tc = 385°C
• Ordering produces change in Bravais
lattice (cubic to tetragonal), P4/mmm
• Strukturbericht L10
y
x
L
α
Cu3Au CuAu
400
I
Cu
II
CuAu3
Au
Long Period Order
• CuAuII is a “long-period superlattice”
• Ordering occurs 410°C - 385°C
• Orthorhombic; b/a = 10.02; c/a ≈ 0.92
• Large unit cell comprised of 5 each CuAuI cells sideby-side, shifted by c/2
L
c
α
Cu3Au CuAu
400
b
a
I
Cu
II
CuAu3
Au
Clustering
L
ΔGm
α
A
β
%B
B
CLUSTERING
L
“spinodes”
∂ 2 (∆Gm )
= 0
2
∂x
ΔGm
SI
A
SII
%B
B
Spinodal
Decomposition
100 nm
Cu-Ni-Fe
100 nm
Phase Separation
• Decomposition of a solid solution can occur by
• nucleation & growth, either
• homogenous or
• heterogeneous, or by
• spinodal decomposition
• Control of thermodynamics and kinetics essential
Nano Assembly
• Formation of solid solutions governed by HumeRothery rules
• Solid solutions are susceptible to decomposition by
phase separation (clustering) or disordering
• Ordered solid solutions generally more resistant to
decomposition
Development of
Interfacial
Structure
Epitaxy
Greek: epi “upon” + taxis “arrangement”
Epitaxial Growth
• Occurs on a substrate (“substratum”) to induce a
specific crystalline arrangement within the growing
crystal
• Homoepitaxy (crystal A on substrate A)
• Heteroepitaxy (crystal B on substrate A)
• Deposition from vapor phase (VPE, MBE);
deposition from liquid phase (LPE); reaction in solid
phase (SPE)
Epitaxial Growth
• E. Bauer, Z. Kristallogr., 110, 423 (1958)
• W.A. Jesser and J.H. van der Merwe, in
Dislocations in Solids, F.R.N. Nabarro, Ed.,
Elsevier, Amsterdam (1989), p. 41
• G.H. Gilmer, in Handbook of Crystal
Growth, Vol. 1, D.T.J. Hurle, Ed., Elsevier,
Amsterdam (1993), p. 584
Frank-Van der Merwe
Film
Substrate
Stranski-Krastanov
Film
Substrate
Volmer-Weber
Film
Substrate
Columnar
Film
Substrate
Interfacial Structure
• Exerts a profound influence on the morphology of
thin films during epitaxial growth
• Control of original substrate/film interface
• Structural (“lattice matching”)
• Compositional (interphase formation)
End
Lecture 05

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