1

Defects in Solids

Defects in Solids

• A defect is a break in the pattern

• Defects make things work and make them beautiful

– Pentium chips

– Rubies

Parameters Affecting the Conductivity of Metals

•Electrons are scattered by the thermal vibration of ionic cores, impurities and defects. This scattering reduces the conductivity (enhances resistivity).

In a very perfect (no defects) and pure (no impurities) copper crystal conductivity is 105 times larger at T=4K than at room temperature.

铜 中掺杂对电导率的影响

Defects have a profound impact on the macroscopic properties of materials

Bonding

+

Structure

+

Defects

Properties

“Crystals are like people, it is the defects in them which tend to make them interesting!”

Colin Humphreys

Many properties of a material are related to the defects

• Electronic conductivity, particularly semiconductors

• Colors and luminescence

• Mechanical strength

• Ionic conductivity, usually in ionic crystals

• Chemical reactivity

Importance of Defects

2

the Processing Determines the Defects

Composition

Bonding Crystal Structure

Thermomechanical Processing

Microstructure

defects introduction and manipulation

0D, Point defects vacancies interstitials impurities

1D, Dislocations edge screw

2D, Stacking Faults and Grain Boundaries

mosaic structure high angle grain boundary tilt grain boundary twist grain boundary

3D, Bulk or Volume defects precipitates second phase particles voids

Defects Types

Real crystals are never perfect, there are always defects

Defects in Solids

Relative Size Ranges of Defects

10-6 10-8 102 101 100 10-2 10-4 10-12

Atomic point defect

Line defect Bulk defect

Interfacial defect

cm

Methods of Introducing Point Defects

Nonintentional during growth (host lattice defects, impurities coming from contamination) during processing (ion implantation) as a result of radiation damage Intentional by changing crystal growth parameters by annealing by irradiation by implantation by diffusion

Point Defects

Intrinsic defects: interstitials and vacancies Interstitials: Self-interstitial (host atom in interstitial

position) complexes of interstitial: di-interstitial and tri-interstitial

Vacancies: lack of an atom complexes of vacancies: di-vacancy, tri-vacancy etc

Extrinsic defects: chemical impurities

Substitutional and interstitial

Point Defects

vacancy: the site of the missing atom

Substitutional atom interstitial atom

self-interstitial atom

disturbances in a crystal ~ a few interatomic distances

Vacancy a lattice position that is vacant because the atom is missing.

Interstitial an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity interstitial atom.

3

• Vacancies: vacant atomic sites in a structure.

Vacancydistortion

of planes

• Self-Interstitials: "extra" atoms in between atomic sites.

self-

interstitialdistortion

of planes

Point Defects

Point defects in ionic crystals are charged. Coulombic forces are large and any charge imbalance has wants to be balanced. Charge neutrality several point defects created:

Frenkel defect: pair of cation vacancy and a cation interstitial or an anion vacancy and anion interstitial. (Anions are larger so it is not easy for an anion interstitial to form).

Schottky defect: pair of anion and cation vacancies

Frenkel and Schottky Defects

Frenkel defect Schottky defect

Frenkel Defects

Frenkel defect: pair of cation vacancy and a cation interstitial

Schottky Defects

Schottky defect: pair of anion and cation vacancies

Defects in Ionic Crystals

Kröger-Vink Notation: a standard notation used to describe the point defects

Main body M : V Vacancy; M central ion Superscript x : the effective charge or the relative charge of the defect with respect to the original species . positive effective charge , negative effective charge x neutrality, zero charge

x

yM Charge

Position site Defect type Position site i: interstitial

x

Defect Notations

Vacancies: the effective charge of a vacancy is the opposite sign of a missing ion charge

NaCl: MgO:

Interstitials: the effective charge of an

interstitial ion is the same sign of the ion charge

NaCl: MgO:

Cl

'

NaVV

O

"

Mg VV

'iiClNa ''

ii OMg

4

Kröger-vink Notation for MX Crystals

Frenkel defect: O VM + Mi

Schottky defect: O VM + Vx

Where VM: void at the site of M Mi: interstitial atom, M Vx: void at the site of X

Xi: interstitial atom, X Point defects

thermodynamic equilibrium (concentration) (temperature) line defects interface defects

Non-equilibrium

Boltzmann's constant

(1.38 x 10 -23 J/atom K)

(8.62 x 10 -5 eV/at om K)

ND

N exp

QD

kT

No. of defects

No. of potential

defect sites.

Activation energy

Temperature

Each lattice site

is a potential

vacancy site

• Equilibrium concentration varies with temperature!

CV=

Equilibrium Concentration of Point Defects

• We can get Q from an experiment.

• Measure this... • Replot it...

1/T

N

NDln

1

-QD/k

slope

Measuring Activation Energy Estimating Vacancy Concentration

Find the Equilibrium number of vacancies in 1m3 of Cu at 10000C.

• Given:

• Solve:

106 cm3 = 1 m3

8.62 x 10 -5 eV/atom-K

0.9eV/atom

1273K

ND

N exp

QD

kT

For 1m 3, N =NA

ACu x x 1m 3 = 8.0 x 10 28 sites

= 2.7 ?10 -4CV= X 410x2.7

vacancies10x2.2ssite10x0.8x10x7.2N 25284D

Defect Equilibrium Concentration (Intrinsic Defects)

The change in free energy H associated with the introduction of n vacancies or interstitial G= nEf TS Ef: the formation energy of one defect S: the change in entropy n: the number of defects

Equilibrium condition G/n=0 Defect concentration:

N: the total number of atoms

fE-

kTn=Ne

Defect Equilibrium Concentration

Interstitial concentration:

Vacancy concentration:

Vacancies in ionic crystals

Schottky defects:

E+: formation energy of a cation vacancy

E: formation energy of an anion vacancy

Frenkel defects:

N: number of lattice sites; Ni: number of interstitial site

kT

E

i

i

Nen

kT

E

V

V

Nen

kT2

EE

V Nen

kT

E

iv

i

NeNn

5

Point Defects in Metals

Intrinsic defects Vacancies are predominant

Ev ~ 1 eV for Cu, Ag, Au

Concentration at T=1000C n/N~10-5

Ei ~ 3 eV at T=1000C n/N~10-16

Extrinsic defects Small atomic radius form the interstitial solid solutions with metals, such as H, B, C, N, O

Frenkel or Schottky Defects: no change in cation to anion ratio compound is stoichiometric

Non-stoichiometry (composition deviates from the one predicted by chemical formula) may occur when one ion type can exist in two valence states, (e.g. Fe2+, Fe3+). In FeO, usual Fe valence state is 2+. If two Fe ions are in 3+ state, then a Fe vacancy is required to maintain charge neutrality fewer Fe ions non-stoichiometry

Imperfections induce Non-stoichiometry

FeO

• Replacement of Na+ by a Ca2+ introduces 1 excess positive (+1).

Q=0 only if • a single positive charge is eliminated. (Make a Na+ vacancy. –1) • •a single negative charge is added. (Make a Cl- interstitial, –1)

Consider NaCl. (use charge neutrality) • What point defects are possible when a Ca2+ substitutes for Na+? • How many of these defects exits for every Ca2+ ion?

Na

Cl

Ca2+ VNa

Cl

Extrinsic Defects in Ionic Crystals： Interstitial, anion-substitutional,

cation-substitutional

• With multiple valances, it is possible not to have correct cation/anion ratio. • e.g., NaCl or FeO are 1-to-1 ratio. Consider Fe2+ O2- and add Al3+ (via Al2O3) to replace Fe2+ .

Al3+

Fe2+

O2–

Add Al3+ in place Fe2+ (not neutral), so excess charge of +1 must be offset. What happens? With two Al3+ defects, need one Fe2+ vacancy for neutrality, i.e., 2(+1) - (+2) = 0.

Extrinsic Defects in Ionic Crystals： Interstitial, anion-substitutional,

cation-substitutional

Defect Structure of Ceria

Defects in ceria – intrinsic or extrinsic (1)Intrinsic defects – due to thermal disorder or by the redox process (2)Extrinsic defects – by impurities or by the introduction of aliovalent dopents. Three possible thermally generated intrinsic disorder in ceria

,,,, ..

Ce O Ce O 2

.... ,,,,

Ce i Ce

,, ..

O i O

Ce + 2 O V +2V +CeO

Ce Ce +V

O O +V

E = 3.53 eV Schottky

E = 3.53 eV Frenkel

E = 3.20 eV Frenkel

• From variation in E, it is evident that the predominant defect category is the anion Frenkel-type.

• Results obtained from X-ray, neutron diffraction and combined dilatometric and X-ray lattice parameter measurements proved that the predominant defects in ceria are anion vacancies.

,,,, ..

Ce O Ce O 2

.... ,,,,

Ce i Ce

,, ..

O i O

Ce + 2 O V +2V +CeO

Ce Ce +V

O O +V

E = 3.53 eV Schottky

E = 3.53 eV Frenkel

E = 3.20 eV Frenkel

6

Examples of Crystals with Defects

Crystal Crystal

Structure

Predominant

Intrinsic Defect

Alkali halides (not Cs) Rock salt, NaCl Schottky

Alkaline earth oxides Rock salt Schottky

AgCl, AgBr Rock salt Cation Frenkel

Cs halides, TlCl CsCl Schottky

BeO Wurtzite, ZnS Schottky

Alkaline earth fluorides,

CeO2, ThO2

Fluorite, CaF2 Anion Frenkel

Example:

TiO2 lose part of oxygen in reduced atmosphere

and produce TiO2-x

or

, ..

2 Ti O O 2

..

Ti O Ti O O 2

12TiO 2Ti +V +3O + O

2

12Ti +4O 2Ti +V +3O + O

2

，

Faber et al. examined the electron density

distribution using XRD and concluded that the

amount of interstitial Ce is less than 0.1% of the

total defect concentration in CeO1.91.

.. ,

O Ce O Ce 2

1O +2Ce V +2Ce + O (gas)

2

In the case of H2 reduction:

The process of ceria reduction may be written as:

.. ,

O Ce 2 O Ce 2O +2Ce +H (gas) V +2Ce +H O(gas)

Already existing oxide vacancies may be removed

by doping with oxides of higher valency than 4

2

2

CeO ,, ..

Ce O O

CeO , ..

2 3 Ce O O

CaO Ca +V +O

Gd O 2Gd +V +3O

Oxide vacancies may also be introduced by doping

with oxides of metals with lower valences, e.g.

dissolution of CaO and Gd2O3

2CeO.. .

2 5 O Ce ONb O +V 2Nb +O

Extrinsic defects:

chemical impurities Substitutional and interstitial

Two Possibilities for Solid Solutions: B atoms in A atoms Substitutional Interstitials ‘new element replaces host atoms’ ‘new element goes in holes’

Can we roughly estimate what atoms will form solid solutions?

e.g. semiconductor devices: doped-Si, C in Fe

e.g. Ni in Cu, steels

7

Solid Solutions

Solid solutions are made of a host (the solvent or matrix) which dissolves the minor component (solute). The ability to dissolve is called solubility.

Solvent: the element or compound present in greater amount

Solute: the element or compound present in lesser amount

Solid Solution: homogeneous maintain crystal structure Contain randomly dispersed impurities (substitutional or

interstitial) Second Phase: as solute atoms are added, new

compounds/ structures are formed, or solute forms local precipitates

Whether the addition of impurities results in formation of solid solution or second phase depends the nature of the impurities, their concentration and temperature, pressure…

A solid solution is a crystalline phase that has a variable compositions.

By forming solid solutions, certain properties of materials such as conductivity, ferromagnetism can be systematically modified.

Examples Ferromagnetism can be tuned in ferrites (MFe2O4 )

by using different combinations of divalent transition metal ions Co, Fe, & Ni

Bi2Te3 can be doped with Sb or Se to improve its thermoelectric properties

Solid Solutions

When atom sizes differ greatly, substitution of the smaller atom on a crystal structure site may be energetically unstable. ex) carbon in Fe

Interstitial atoms : C, N, H, B

Interstitial site in BCC: 1/2, 0, 1/2 (octahedral) 1/4, 1/2, 0 (tetrahedral)

-Fe (BCC) -Fe (FCC)

Interstitial site in FCC: 1/2, 1/2, 1/2 (octahedral) 1/4, 1/4, 1/4 (tetrahedral)

Interstitial solid solution Substitutional Solid Solutions

Factors for high solubility: Atomic size factor: atoms need to “fit” solute and

solvent atomic radii should be within ~ 15% Crystal structures of solute and solvent should be the

same Electronegativities of solute and solvent should be

comparable (otherwise new inter-metallic phases are encouraged)

Generally more solute goes into solution when it has higher valence than solvent

Ni

Cu

Substitutional Solid Solutions

A1.0Z0.0 A0.8Z0.2 A0.6Z0.4 A0.4Z0.6 A0.2Z0.8

A0.0Z1.0

Solid Solutions with Limited Compositional Range

In most cases, only limited ranges of solid solutions can be formed. In such cases, it is not necessary for the two end members to be isostructural.

For example, Zn2SiO4 (Zn tetrahedral) and Mg2SiO4 (Mg octahedral) have very different structures.

Two different solid solutions are possible.

Mg2SiO4 doped with Zn to give Mg2-xZnxSiO4 solid solution and the structure is the same as Mg2SiO4.

Zn2SiO4 doped with Mg to give Zn2-xMgxSiO4 solid solution and the structure is the same as Zn2SiO4.

Solid Solutions with Limited Compositional Range

8

Random and Ordered Solid Solutions

a: random solid solution

b and c: partly ordered solid solution

d:ordered solid solution

Substitutional solid solution of metals

• Random solid solution

• Ordered solid solution : Solute atoms preferentially occupy particular sites in unit cell below the ordering temperature. ex) AuCu3.

T > 390C disordered

T < 390C ordered

Interstitial an element goes into holes in an orderly arrangement

e.g., Ni3Al (hi-T yield strength), Al3(Li,Zr) (strengthening)

e.g., small impurities, clays ionic crystals, ceramics.

Solid Solutions and Ordered Compounds

Ordered Substitutional and Interstititials Compounds

Substitutional an element replaces host atoms in an orderly arrangement

8

• Solid solution of B in A plus particles of a new phase (usually for a larger amount of B)

Second phase particle --different composition --often different structure.

Solid Solution phase B atoms in A

Particles of New Phase in Solid-Solution Alloys

The Copper-Gold system Random mixture

Single phases Mixed phases

(fcc) (fcc)

实际情况都是比较复杂的~

1. Temperature

Cation disorder in a solid solution increases the configurational entropy: solid solution is stabilized at high temperature

Cation-size mismatch increases the enthalpy (structure must strain to accommodate cations of different size): solid solution is destabilized at low temperatures

Extent of solid solution tolerated is greater at higher temperatures

Factors Controlling the Extent of Solid Solution

G H T S

9

2. Structural flexibility

Cation size alone is not enough to determine the extent of solid solution, it also depends on the ability of the structural framework to flex and accommodate differently-sized cations e.g. there is extensive solid solution between MgCO3 and CaCO3 at high temperature.

3. Cation charge

Complete solid solution is usually only possible if the substituting cations differ by a maximum of ± 1. Heterovalent substitutions often lead to complex behaviour at low temperatures due to the need to maintain local charge balance.

Factors Controlling the Extent of Solid Solution

2. Crystal Structure Like elemental crystal structures are better For appreciable solubility, the crystal structure for metals must be the same.

%15%100xr

rrR

solvent

solventsolute

Hume-Rothery Rules for Mixing Metals

Empirical rules for substitutional solid-solution formation were identified from experiment that are not exact, but give an expectation of formation.

1. Atomic Size Factor The 15% Rule If "size difference" of elements are greater than ±15%, the lattice distortions (i.e. local lattice strain) are too big and solid-solution will not be favored.

will not disallow formation.

3) Electronegativity E ~ 0 favors solid-solution.

The more electropositive one element and the more electronegative the other, then "intermetallic compounds" (order alloys) are more likely.

4) Valences Higher in lower alright. Lower in higher, it’s a fight.

A metal will dissolve another metal of higher valence more than one of lower valence.

4%%100xr

rrR

solvent

solventsolute

Hume-Rothery Empirical Rules In Action

Example Applications:Si-Ge semiconductor, Cu-Ni and Cu-Ag metal alloys. Is solid-solution favorable, or not?

Rule 1: rSi = 0.117 nm and rGe= 0.122 nm.

Rule 2: Si and Ge have the diamond crystal structure.

Rule 3: ESi = 1.90 and EGe= 2.01. Thus, E%= 5.8%

Rule 4: Valence of Si and Ge are both 4

Expect Si and Ge to form Solid Solution over wide composition range. In fact, Solid Solution forms over entire composition at high temperature.

Hume-Rothery Empirical Rules In Action

Is solid-solution favorable, or not?

Cu-Ag Alloys

Rule 1: rCu = 0.128 nm and rAg= 0.144 nm.

%4.9%100xr

rrR

solvent

solventsolute

Rule 2: Ag and Cu have the FCC crystal structure

Rule 3: ECu = 1.90 and EAg= 1.80. Thus, E%= -5.2%

Rule 4: Valence of Cu is +2 and Ag is +1. NOT favorable

Expect Ag and Cu have limited solubility

In fact, the Cu-Ag phase diagram (T vs. c) shows that a solubility of only 18% Ag can be achieved at high T in the Cu-rich alloys.

Vegard’s Law

x 1-xP[(A B O)] P[x(AO)] P[(1-x)(BO)]

Any property (P) of a solid-solution member is the atom fraction weighted average of the end-members.

P: lattice parameter, band gap….

Example: In metallurgy, Vegard's law is an approximate empirical rule which holds that a linear relation exists, at constant temperature, between the crystal lattice parameter of an alloy and the concentrations of the constituent elements.

The basic assumptions of Vegard’s law are: (1) the solid solution is formed by simple substitution. (2) unit cell sizes are governed by relative sizes of atoms or ions.

10

In1-xGaxN bandgap (room temp)

Define Emission Color via Band Engineering

InGaN growth 780C 760C 720C 690C 630C

Indium (%) ~5% ~10% ~20% ~30% ~35%

ΔΕ(eV) 3.18 2.95 2.64 2.38 2.14

Emission (nm) 390 420 470 520 580

FWHM (nm) 7 27 30 48 61

)x1(x43.1)x1(77.0x42.3)x(EG

Wu, et al, Superlattices Microstruct., 2003,34, 63

For Ceramics Solid Solutions

If forming Substitutional Solid Solution: x-2x-1x OZrCa

Adding CaO in ZrO2:

If forming Interstitials Solid Solution: 2y-12y OZrCa

Oo''Zr

ZrO VOCaCaO 2

x

x x

''ZrOi

ZrO CaO2CaCaO2 2

2y y y

. . .

.

. ( )

Ca Zr O

i

M M MWW

x g

2 4 2

23

23

0 15 0 85 1 854 4 8

1 1 2

6 022 10

75 18 10

Unit cell volume V=a3=135.1×10-24cm3

3

24

23

s cm/g565.5101.135

1018.75

V

W

1. Substitutional Solid Solution CaxZr1-xO2-x

ZrO2: Fluorite structure: Z=4，

XRD give a=5.131Å

Unit cell weight

density

15% CaO-doped ZrO2

Ca0.15Zr0.85O1.85 2+ 4+ 2-

3

-23=81.25x10

. / . . / . /

.

( )

Ca Zr O

i

M M xMW W

g

2

0 3 1 85 1 7 1 85 2 24 4 8

1 1 16 022 10

3

24

23

i cm/g014.6101.135

1025.81

V

W

2. Interstitials Solid Solution:

21.7/1.850.3/1.85 OZrCa

2y-12y OZrCa

If the measured density is 3

m 5.477g/cm

So, it is a Substitutional Solid Solution

Theoretical Density of the CaO-doped ZrO2

vs the amount of CaO doping

CaO%

Substitutional

De

nsit

y(

)

Interstitials

Oo''Zr

ZrO VOCaCaO 2

''ZrOi

ZrO CaO2CaCaO2 2

不等价置换型固溶体组分缺陷

High valence

substitute low

valence

(a) cation vacancy Substitutional

Solid solution

(b) anion interstitial Interstitial

Solid solution

Low valence

substitute high

valence

(a) anion vacancy-type Substitutional

Solid solution

(b) Cation interstitial Interstitial

Solid solution

O,,

Mg.Mg

MgO32 3OV2AlOAl

O,i

.Mg

MgO32 3OO2AlOAl

O..O

,,Zr

ZrO OVCaCaO 2

O..i

,,Zr

ZrO 2OCaCa2CaO 2

11

1、anion vacancy-type

61

2

2

2

-

O..o

o

21

o2..

o..O

..o

,

o

21

o2,..

O

221..

OO

O221..

OTiOTi

O221..

O,TiOTi

P][V

][O

P])](2[V[VK

]2[V][e

][O

P]][e[VK

OV2eO

3OOV2e2Ti4O2Ti

3OOV2Ti4O2Ti

）（

，

，

如 TiO2-x、ZrO2-x

important is pressureO,V,PWith 2..OO2

2. Cation interstitial -type

2

221,.

i

221,..

i

OeZnZnO

1O2eZnZnO

Experiments confirmed that (2) is feasible, then

41

2

21

2

)(P][Zn

][e][Zn

)](P][e[ZnK

O.i

,.i

O,.

i

2. Cation interstitial -type 如Zn1+xO、Cd1+xO

torsemiconductypenZ,PWith .iO2

,n

3. anion interstitial -type

Example，UO2+x

61

2

21

2

)(P][O

]2[O][h

)/(P]][h[OK

2hOO

O,,i

,,i

.

O2.,,

i

.,,i22

1

torsemiconductypep,O,PWith ,,iO2

cation vacancy

4. cation vacancy-type

61

2

21

2

)(P][h

]2[V][h

)(P]][h][V[OK

VO2h(g)O

3OV2Fe(g)O2O2Fe

O.

,,Fe

.

O2.,,

Feo

,,FeO

.22

1

O,,

Fe.Fe22

1OFe

Examples: Cu2-xO，Fe1-xO

tyconductivi,h,PWith .O2

Summary: Solid Solutions

Solid Solution

Interstitial Substitutional

Random Ordered

Color Centers

Electrons trapped in vacant sites give rise to colored materials

color centers color arises due to transitions between

electron in a box levels

Trapped electrons can be produced by irradiation of the sample treatment with an electron donor like

sodium or potassium vapor

12

Color Centers

Exposure to radiation can induce defects When crystals of alkali halides were exposed to X-rays (or

other high energy radiation such as UV), they became brightly colored.

The color is associated with a defect known as F-center.

KBr KCl NaCl

Useful for imaging Useful for dating

F, H and V Centers

F Center – electron trapped in anion vacancy

H Center –interstitial Cl atom

bonds to lattice Cl-

V Center – electron removed

from lattice anion site, resulting Cl atom pairs with neighboring Cl-

• Alkali-halides made from Groups I and VII

• F-center is an electron in place of a halogen – Long studied – Not completely

understood

The F-center in Alkali-Halides

e-

F-center Applications

Tunable solid-state lasers

optical performance

BaF-Br:Eu材料

色心可以被用来存储X射线的图像

Color characteristic of host crystal Color shifts to red as the anion size increases Color does not shift if Na or K is added to NaCl ESR (Electron Spin Resonance) indicates F-center is

an electron trapped in an anion vacancy: electron in an octahedral box problem

Number of F-centers 1 in 10,000 halide ions

Ways of Creating Color Centers:

Main Features of Color Centers:

Heating the crystal in the vapor of the metal Introduction of impurities (extrinsic defects) X-ray, -ray, neutron or electron beam irradiation Electrolysis

Optical Absorption of

F-centers

Electron in an octahedral box (halide ion vacancy) of size L

Difference between energy levels of the box proportional to

Absorption takes place between energy levels of the box,

corresponds to the energy of UV-Vis-NIR photons

The larger L is, the lower the absorption energy. i.e. Absorption red

shifts with the lattice parameter

Pressure causes box to shrink and absorption to blue shift

Color of crystals results from the light they reflect or transmit.

Reflection + Transmission + Absorption=1

Example: KBr absorbs red, it looks blue-green

2

22

mL8

hnE

2

1

L

13

Absorption spectra obtained from color centers in halide salts exhibit a clear trend in the variation of the wavelength with the size of the halide vacancy (as estimated by the lattice parameter, the length of an edge of the cubic unit cell).

2

22

mL8

hnE

Deformation of Solids

Motion of a dislocation (line of missing particles) in a crystalline solid results in a permanent change in the shape of the solid.

Dislocations—Linear Defects

Dislocations are linear defects: the interatomic bonds are significantly distorted only in the immediate vicinity of the dislocation line. This area is called the dislocation core. Dislocations also create small elastic

deformations of the lattice at large distances.

Dislocations are very important in mechanical properties of material. Introduction/discovery of dislocations in 1934 by Taylor, Orowan and Polyani marked the beginning of our understanding of mechanical properties of materials.

DISLOCATIONS

•Material permanently deforms as dislocation moves through the crystal. • Bonds break and reform, but only along the dislocation line at any point in time, not along the whole plane at once. • Dislocation line separates slipped and unslipped material.

Linear Defects Dislocation

Edge dislocation Screw dislocation

A dislocation which may be regarded as the result of inserting an extra plane of atoms, terminating along the line of the

dislocation.

Two kinds of dislocations:

A dislocation in which atomic planes form a spiral ramp winding around the line of the dislocation.

Screw dislocation

Scanning tunneling micrograph of a screw dislocation on a GaN crystal surface of wurtzite structure

14

Migration aids ductile deformation

Motion of many of these dislocations will result in plastic deformation

Edge dislocations move in response to shear stress applied perpendicular to the dislocation line.

Edge dislocation

GaN

Pd Existence of dislocation even without deformation 106/cm3 in usual

Edge dislocation

E

E’

Extra half plane

Edge Dislocation

Slip System

Preferred planes for dislocation movement (slip planes)

Preferred crystallographic directions (slip directions)

Slip planes + directions (slip systems) highest packing density.

Distance between atoms shorter than

average; distance perpendicular to plane longer than average. Far apart planes can slip more easily. BCC and FCC have more slip systems compared to HCP: more ways for dislocation to propagate FCC and BCC are more ductile than HCP.

11

• are line defects,

• cause slip between crystal plane when they move,

• produce permanent (plastic) deformation.

Dislocations:

Schematic of a Zinc Crystal (HCP):

• before deformation • after tensile elongation

slip steps

LINE DEFECTS

Slip in a Single Crystal

Each step (shear band) results from the generation

of a large number of dislocations and their

propagation in the slip system Zn

In Situ Observation of the Electrochemical Lithiation of a Single SnO2 Nanowire Electrode

Chong Min Wang*, Science, 2010, 330, 1515

Time-lapse structure evolution of a SnO2

nanowire anode during charging at –3.5 V against a LiCoO2 cathode. The single-crystal

nanowire was elongated 60% and the diameter increased 45% (resulting in a 240% volume expansion) after charging for 1860 s.

See also movie S1. (A) Schematic of the experimental setup. The initially straight

nanowire (B and C) became significantly twisted and bent after charging (D to S). The chemical reaction front progressed along the

nanowire’s longitudinal direction, with the front clearly visible, as pointed out by

arrowheads in (E) to (S). The red line in (B) to (O) marks a reference point to track the change of the nanowire length. (P) to (S) are

sequential high-magnification images showing the progressive migration of the

reaction front, swelling, and the twisted morphology of the nanowire after the reaction front passed by. The big dark

particle in the middle of (O) is an island of gelled ILE. Because of the long cumulative

electron beam exposure time during the recording of TEM images, the ILE front became gelled (with high viscosity) at this

spot.

一套世界上最小的锂离子电池，其中负极材料仅是一根直径100纳米10微米长的单晶二氧化锡纳米线

15

(A) TEM micrograph of the nanowire containing

a reaction front (“dislocation cloud”) separating the reacted (“amorphous”) and nonreacted

(“single-crystal SnO2”) sections. (B to E) EDPs from the different sections of the nanowire. The

pristine nanowire was single crystalline and the corresponding EDP (B) can be indexed as the

zone axis of rutile SnO2. The EDP from the dislocation zone (C) shows a spot pattern superimposed on a diffuse scattering background.

The EDP from an area immediately after the reaction front (D) shows an amorphous halo. The

EDP from an area far away from the reaction front (E) shows diffraction rings superimposed

on a diffuse amorphous halo. The diffraction rings can be indexed as tetragonal Sn (black

indices) and a LixSn compound such as hexagonal Li13Sn5 (orange indices). (F) A

HRTEM image from a charged nanowire showing Sn nanoparticles dispersed in an amorphous

matrix. (G to H) Low-loss and core-loss EELS from a large area of the nanowire after reaction

(red line profile) and a pristine nanowire (blue line profile). The pristine SnO2 shows two characteristic core-loss peaks at 515 and 524 eV,

corresponding to the Sn-M4,5 edge riding on a delayed edge. The peaks at 532 and 538 eV arise

from the O-K edge. Note that Li is present in the charged nanowire (G). The plasmon loss peaks at

20 eV, 24 eV, and 14 eV are in excellent agreement with SnO2, Li2O, and pure Sn,

respectively.

Structural and phase

characterization of another SnO2 nanowire anode

during charging at –3.5 V against the LiCoO2 cathode.

反应前端产生高密度流动位错云

TEM images revealed a high density of dislocations emerging from the reaction front (marked by chevron-shaped dotted lines). As the dislocation front propagated, the crystalline contrast changed to gray amorphous contrast instantaneously, and the nanowire diameter increased immediately. See also movies S2 to S5. (A to F) and (G and H) Two sets of time-lapsed TEM images showing the high density of dislocations that appeared at the reaction front and the migration of the reaction front.

(A) Plot of the reaction front migration distance L versus the square root of time for 11 nanowires. (B) Representative Li+ migration energy barrier in crystalline and amorphous Li2O from DFT calculations. (C) Schematic drawing showing the high Li diffusion flux in Li2O.

Stacking Faults and Grain Boundaries

mosaic structure

high angle grain boundary

tilt grain boundary

twist grain boundary

2D Defects in Solids

Stacking Faults

Stacking faults occur in wide variety of materials not just simple metals.

Consider a structure to be built up from successive layers of atoms or other units, if the regular stacking of these units is interrupted, we have a stacking fault.

Close packed metals provide simple examples Perfect FCC has a ABCABCABCABCABC sequence The sequence ABCABCBCABCABC has a stacking

fault Perfect HCP is ABABABABABABAB ABABABCABABABABAB has a stacking fault Faults that put two of the same layers together AA

BB or CC are unlikely due to their very high energy

Undulating Slip in Laves Phase and Implications for Deformation in Brittle Materials

Structures of the C14 Laves phase and a stacking

fault of it. (a) Model of the C14 structure viewed along the [11-20] and [0001] directions,

respectively. The c-type triple layers are separated by single layers of kagome network. (b)

HRTEM of an undeformed C14 Laves phase viewed along the [11-20] direction, recorded with

the aberration-corrected high-resolution transmission electron microscope. (c) An HRTEM

image of a stacking fault in the deformed C14

Laves phase. (a) An aberration-corrected HRTEM

image of the partial dislocation. The closure failure of the Burgers circuit

implies that the dislocation has a Burgers vector, b1, of <1100>/3 and

moves from right to left. The simulated image of the stacking fault is shown as

an inset with a thickness of 2.4 nm and a focus value of 9 nm. (b) A magnified image of the dislocation core shown in

the red box in (a).

Zhang*， Phys. Rev. Lett., 2011, 106, 165505

利用球差校正电镜发现在Laves

相金属间化合物中，位错通过反复地在上下两个不同的滑移面间来回跳跃，从而以波浪形状的路径向前滑移。这种位错滑移机制的产生归结于Laves相中不同原子层之间结合力的不同。这种特殊的变形机制将有利于解释金属间化合物在高温变形时存在脆-

韧转变的特性。

16

Results of the strain field analysis. (a) " strain map obtained from an experimental HRTEM image of the dislocation by the LADIA method. The triangle denotes the center of the dislocation core and circles mark the location of measurements shown in Fig. 4(b) with distances of 1.5 nm (solid line) and 2.5 nm (dotted line), respectively, from the core. (b) Angular variation of strain fields measured experimentally and calculated with the Foreman model.

Zhang*， Phys. Rev. Lett., 2011, 106, 165505

Grain Boundaries

Mosaic structure

High angle grain boundary

Low angle grain boundary Left: tilt Right: twist

Grain boundary : the region of mismatch between two adjacent single crystals meeting at different orientation

Polycrystalline Materials

Grain Boundaries • regions between

crystals • transition from lattice

of one region to that of the other

• slightly disordered • low density in grain

boundaries – high mobility

– high diffusivity

– high chemical reactivity

Crystal Defects under HRTEM

Steel spheres:

a)Regular packed array with 3 point defects

b)Point and line defects

c)Mosaic (or domains) separated by defect boundaries

These are not twins!

Tilt Grain Boundaries

Low angle grain boundary is an array of aligned edge dislocations. This type of grain boundary is called tilt boundary (consider joint of two wedges)

Transmission electron microscope image of a small angle tilt boundary in Si. The red lines mark the edge dislocations, the blue lines indicate the tilt angle.

Tilt boundary

Low angle symmetrical tilt boundary

< 10~15

22sin

D

2b

Low Angle Symmetrical Tilt Boundary

The number of dislocations per unit length, 1/D

bD

1

Interfacial energy,

D

1

17

Twist Grain Boundaries

Twist boundary the boundary region consisting of arrays of screw dislocations (consider joint of two halves of a cube and twist an angle around the cross section normal)

Twin boundary

孪晶界也分为两类，共格孪晶界与非共格孪晶界，如图所示共格孪晶界就是孪生面，两侧晶体以此面为对称面，构成镜面对称关系。在孪晶面上的原子同时位于两个晶体点阵的结点上，为两晶体所共有，自然地完全匹配，使此孪晶面成为无畸变的完全共格界面。它的能量很低，很稳定。

Twin boundary 孪晶界

当孪生切变区与基体的界面不和孪生面重合时，这种界面称为非共格孪生面，它是孪生过程中的运动界面。随非共格孪生面的移动，孪晶长大。非共格孪晶界是一系列不全位错组成的位错壁，孪晶界移动就是不全位错的运动。

单晶突然折断时会产生孪晶

Adv. Funct. Mater., 2011, 20, 3982

HRTEM tensile test of a short

nanowire with twins: a) initial state with three twin boundaries;

b) surface non-uniformity near the bottom twin (marked by

white dotted line) under tension; c–d) further loading cause stress

concentration at the intersection between bottom twin boundary

and free surface; e) right before

fracture, a groove was formed near the marked twin boundary;

f) after fracture along the bottom twin boundary, the top two twin

boundaries remained inside the nanowire; g) a nanowire

fractured in brittle-like mode, corresponding FFT insert clearly

shows the remaining twin structures (all scale bars 5 nm).

金线拉伸得太细(<20nm)将变得非常易碎

Chemistry in Two Dimensions: Surfaces

Model of a heterogeneous solid surface, depicting different surface sites. These sites are distinguishable by their number of nearest neighbors

Surfaces & Grain Boundaries

External Surfaces Surface atoms have unsatisfied atomic bonds, and higher

energies than the bulk atoms Surface energy, (J/m2) Surface areas tend to minimize (e.g. liquid drop) Solid surfaces can “reconstruct” to satisfy atomic bonds at

surfaces.

Grain Boundaries Polycrystalline material comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

Surfaces and interfaces are reactive and impurities tend to segregate there. Since energy is associated with interfaces, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.