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.