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, Si-Ge, 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

wt. % Ni

at. % Ni

10Cu Ni20 4030 70 806050 90

10 20 4030 70 806050 90

°C

1600

1500

700

800

900

1000

1100

1200

1300

1400

1700

1800

1455°

1084.87°

!

L + !

L

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

10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 901100

1000

900

800

700

600

500

400

300

200

100

0

wt % Sn

at % Sn

SnCu

T (

°C)

L

!

"

#

$

%

%’

415°

189°

232°

186°

350°

520°586°

640°

756°

799°

1085°

&'

!(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

PlatinumGold

GraphiteTitanium

Stainless SteelCopperBrassNickel

TinSteel

AluminumZinc

Magnesium

Galvanic Series

Electronegative(gains electrons)

Electropositive(loses electrons)

Noble(cathodic)

Active(anodic)

Sn 10 20 30 40 50 60 70 80 90 Au

500

600

700

800

900

1000

1100

400

300

200

100

0

°C

wt % Au

! Sn " # $ %

L

(Au)

217°252°

309°

418°

231.9681°

280°

490°86.2 95.8

80

100.3

1064.43°

rAu = 0.144 nm

rSn = 0.158 nm

Δr/r = 9.7%

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

• C 1:2 stoichiometry

• D 1:3 stoichiometry

• E Perovskites

• H Spinels

• 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

rNi

= 0.125 nm rSn

= 0.158 nm

Sn at 0,1 Sn at !

Ni at !

Ni at !

Ni at !

Ni at !Ni at !

Sn at ! Sn at !

Sn at !

Ni at ",# Ni at ",#

Ni at ",# Ni at ",#

Sn at 0,1

Ni at 0,1

Ni at 0,1 Ni at 0,1

Ni at 0,1

Sn at 0,1

Sn at 0,1

Sn at 0,1

Ni3Sn

10Ni Sn20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 901500

1400

1300

1200

1100

1000

900

800

700

600

500

400

wt % Sn

at % Sn

T

(°C)

19.01132°

922°

851°

978°

1176°

796°

1266°

1455°

38.0

32.5

43.189.3

L

!’

"!

# $

Ni3Sn

cubic DO3

hexagonal DO19

Perovskite Unit Cell

x

z

y

• 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

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 LaFeO3

CaZrO3 BaCeO3

SrZrO3 CdCeO3

BaZrO3 PbCeO3

PbZrO3

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

octahedral sitestetrahedral sites

Spinel

• Lattice = face-centered cubic

• Basis (motif) = 14 ions

• two divalent cations

• four trivalent cations

• eight oxygen anions

SpinelsNormal Inverse

SnMg2O4 MgAl2O4 CoFe2O4 ZnK2(CN)4 CuCr2S4 MgFe2O4

FeV2O4 SrAl2O4 NiFe2O4 CdK2(CN)4 CoCo2S4 TiMg2O4

WNa2O4 CrAl2O4 AlFe2O4 HgK2(CN)4 CuRh2S4 VMg2O4

MoAl2O4 PbFe2O4 FeCr2S4 SnZn2O4

FeAl2O4 MgCo2O4 MnCr2S4 MgGa2O4

CoAl2O4 TiCo2O4 MgIn2O4

NiAl2O4 CuCo2O4 FeIn2O4

CuAl2O4 CoCo2O4 CoIn2O4

ZnAl2O4 NiIn2O4

Ordering

• 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

x

z

y

Cu3AuCuAu

I

II400

L

α

Cu Au

CuAu3

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 side-by-side, shifted by c/2

a

c

bCu3Au

CuAu

I

II400

L

α

Cu Au

CuAu3

Clustering

A B%B

ΔGm α β

L

CLUSTERING

A B%B

ΔGm

L

SISII

“spinodes”∂2(∆Gm)

∂x2= 0

100 nm 100 nm

Spinodal Decomposition

Cu-Ni-Fe

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 Hume-Rothery 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

Substrate

Film

Stranski-Krastanov

Substrate

Film

Volmer-Weber

Substrate

Film

Columnar

Substrate

Film

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)

EndLecture 05