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Crystal Defects
The crystal lattices = an idealized, simplified system
used to understand important principles governing the behavior of solids
Real crystals - contain large numbers of defects, e.g., variable amounts of
impurities, missing or misplaced atoms or ions.
These defects occur for three main reasons:
1) It is impossible to obtain any substance in 100% pure form. Some impurities are
always present.
2) Even if a substance were 100% pure, forming a perfect crystal would require
cooling the liquid phase infinitely slowly to allow all atoms, ions, or molecules to
find their proper positions. Cooling at more realistic rates usually results in one or
more components being trapped in the “wrong” place in a lattice or in areas where
two lattices that grew separately intersect.
3) Applying an external stress to a crystal, such as a hammer blow, can cause
microscopic regions of the lattice to move with respect to the rest, thus resulting
in imperfect alignment.
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Crystal Defects
Perfect crystals - every atom of the same type in the correct equilibrium
position (does not exist at T > 0 K)
Real crystals - all crystals have some imperfections - defects
most atoms are in ideal locations, a small number are out of place
• Intrinsic defects – present for thermodynamic reasons
• Extrinsic defects – not required by thermodynamics, can be controlled
by purification or synthetic conditions
• Chemical defects (foreign atom, mixed crystals, nonstoichiometry)
• Geometrical defects (vacancy, interstitials, dislocations, boundaries,
surface)
Defects dominate the material properties:
Mechanical, Chemical, Electrical, Diffusion
Defects can be added intentionally
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Crystal Defects
Perfect crystal Real crystal
does not exist at T > 0 K
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Classes of Crystal Defects
Point defects (0D) places where an atom is missing or
irregularly placed in the lattice structure – lattice vacancies, selfinterstitial
atoms, substitution impurity atoms, interstitial impurity
atoms
Linear defects (1D) groups of atoms in irregular positions –
dislocations
Planar defects (2D) interfaces between homogeneous regions
of the material - grain boundaries, stacking faults, external
surfaces
Volume defects (3D) spaces of foreign matter – pores,
inclusions
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Classes of Crystal Defects
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Point Defects
Point defects - an atom is
missing or is in an irregular
position in the lattice
• self interstitial atoms
• interstitial impurity atoms
• substitutional impurity atoms
• vacancies
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Point Defects – Ionic Compounds
perfect crystal lattice AB
interstitial imputity
cation vacancy anion vacancy
substitution of a cation substitution of an anion
BA antisite defect
AB antisite defect
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Point Defects – Ionic Compounds
• Vacancy
• Interstitial
• Substitutional
• Frenkel
• Schottky
Schottky: a pair of vacancies, missing cation/anion moved to the
surface, equal numbers of vacancies at both A and B sites preserving
charge balance, found in compounds where metal ions are able to
assume multiple oxidation states
Frenkel: ions moved to interstitial positions, vacancies, found in open
structures (wurtzite, sphalerite, etc) with low coordination numbers,
open structure provides room for interstital sites to be occupied
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Vacancies
There are naturally occurring vacancies in all crystals
Equilibrium defects – thermal oscillations of atoms at T > 0 K
The number of vacancies grows as the temperature increases
The number of vacancies:
• N = the total number of sites in a crystal
• Nv = the number of vacancies
• Ha = the activation energy for the formation of a vacancy
• R = the gas constant
Nv goes up exponentially with temperature T
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Crystal Energies
Point defects = equilibrium concentration
Enthalpy H is positive but configurational
entropy S is positive – defects = disorder
Minimum on free energy G – equilibrium conc.
of defects
The concentration of vacancies grows as the
temperature increases
Extended defects = no equilibrium concentration
Enthalpy is HIGHLY positive, configurational
entropy cannot outweight
No minimum on free energy
Metastable defects – dislocations, grain
boundaries, surface
Heating = minimize free energy:
polycrystalline  single crystal grain growth
Grains with high dislocation density consumed
Atoms move across grain boundary
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Typical Point Defects in Crystals
Alkali halides Schottky (cations and anions)
Alkaline earth oxides Schottky (cations and anions)
Silver halides Frenkel (cations)
Alkaline earth fluorides Frenkel (anions)
Typical activation energies for ion diffusion
Na+ in NaCl  0,7 eV
Cl- in NaCl  1 eV
Schottky pair  2,3 eV
(1 eV/molecule = 96.49 kJ/mol)
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The addition of the dopant (an impurity) into a perfect crystal =
point defects in the crystal
NaCl heated in Na vapors
Na is taken into the crystal and changes its compostion
NaCl  Na1+ x Cl
Na atoms occupy cation sites
an equivalent number of unoccupied anion sites
Na atoms ionize, Na+ ions occupy the cation sites,
the electrons occupy the anion vacancies – F centers – color
This solid is now a non-stoichiometric compound as the ratio of
atoms is no longer the simple integer
Violet color of Fluorite (CaF2) = missing F anions replaced by e
Extrinsic Defects
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Non-Stoichiometric Compounds
Non-stoichiometry can be caused by
• introducing an impurity (doping)
• the ability of an element to show multiple valencies
Vanadium oxide
varies from VO0.79 to VO1.29
other examples: TiOx, NixO, UOx and LixWO3
Covalent compounds - held to together by very strong covalent
bonds which are difficult to break, do not show a wide range of
composition
Ionic compounds - do not show a wide range because a large
amount of energy is required to remove / add ions
What oxidation states?
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Non-Stoichiometric Compounds
Non-stoichiometric ionic crystals
a multi-valent element - changes in the number of ions can be
compensated for by changes in the charge on the ions, therefore
maintaining charge balance but changing the stoichiometry
non-stoichiometric compounds have formulae with non-integer ratios
and can exhibit a range of compositions.
The electronic, optical, magnetic and mechanical properties of nonstoichiometric
compounds can be controlled by varying their
composition.
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Non-Stoichiometric Compounds
Non-stoichiometric superconductor YBCO
YBa2Cu3O6.5
a multi-valent element = Cu
YBa2Cu3O6.8−7.0 90 K superconductor
YBa2Cu3O6.45−6.7 60 K superconductor
YBa2Cu3O6.0−6.45 antiferromagnetic semiconductor
Oxygen content
Tcritical
Kelvin
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Dislocations
Line imperfections in a 3D lattice
• Edge
• Screw
• Mixed
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Edge Dislocation
• Extra plane of atoms
• Burgers vector
– Deformation direction
– For edge dislocations it is perpendicular to the dislocation line
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Edge Dislocation
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Screw Dislocation
• A ramped step
• Burgers vector
– Direction of the displacement of the atoms
– For a screw dislocation it is parallel to the line of the dislocation
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Deformation
When a shear force is applied to a material, the dislocations move
Plastic deformation = the movement of dislocations (linear defects)
The strength of the material depends on the force required to make the
dislocation move, not the bonding energy
Millions of dislocations in a material - result of plastic forming operations
(rolling, extruding,…)
Any defect in the regular lattice structure (point, planar defects, other
dislocations) disrupts the motion of dislocation - makes slip or plastic
deformation more difficult
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Deformation
Dislocation movement produces additional dislocations
Dislocations collide – entangle – impede movement of other dislocations the
force needed to move the dislocation increases - the material is
strengthened
Applying a force to the material increases the number of dislocations
Called “strain hardening” or “cold work”
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Slip
• When dislocations move slip occurs
– Direction of movement – same as the Burgers vector
• Slip is easiest on close packed planes
• Slip is easiest in the close packed direction
• Affects
– Ductility
– Material Strength
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Schmidt’s Law
• In order for a dislocation to move in its slip system, a
shear force acting in the slip direction must be produced
by the applied force.
Slip direction
Normal
to slip
plane

Ao

F
r Fr / A - Resolved Shear
Stress in the slip direction
 = F/Ao = unidirectional stress
applied to the cylinder
Fr = F cos()
A = A0/cos()
 =  cos() cos()
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Surface and Grain Boundaries
• The atoms at the boundary of a grain or on the surface
are not surrounded by other atoms – lower CN, weaker
bonding
• Grains line up imperfectly where the grain boundaries
meet
• Dislocations can not cross grain boundaries
• Tilt and Twist boundaries
• Low and High angle boundaries
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Grain Boundaries
High resolution STEM image from a grain boundary in gold at
the atomic level, imaged on an FEI Titan STEM 80-300.
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Low Angle Tilt Boundary
Low Angle Tilt Boundary =
Array of Edge dislocations
D = dislocation spacing
b = Burgers vector
 = misorientation angle
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Low Angle Twist Boundary
Low Angle Twist Boundary = a
Screw dislocation
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Stacking Faults
Low Angle Twist Boundary = a
Screw dislocation
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Effect of Grain Size on Strength
• Material with a small grain = a dislocation moves to the
boundary and stops – slip stops
• Material with a large grain = the dislocation can travel
farther
• Small grain size = more strength
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Hall-Petch Equation
y = 0 + K d –1/2
y = yield strength
(stress at which the material
permanently deforms)
d = average grain diameter
0 = yield stress for single
crystal
K = unpinning constant
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Control of the Slip Process
• Strain hardening
• Solid Solution strengthening
• Grain Size strengthening
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Amorphous Structures
• Cooling a material off too fast
- it does not have a chance to
crystallize
• Forms a glass
• Easy to make a ceramic
glass
• Hard to make a metallic glass
• There are no slip planes,
grain boundaries in a glass
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Interdiffusion: atoms migrate from regions of large to lower concentration
Initial state (diffusion couple) After elapsed time
Diffusion
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Diffusion Couple Experiments
La2O3 CoO
LaCoO3
Experimental conditions: T = 1370 – 1673 K
pO2 = 40 Pa – 50 kPa
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Diffusion
CaTiO3-NdAlO3 diffusion couple fired at 1350 °C/ 6 h
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Diffusion - Fick’s First Law
J = diffusion flux
[mol s1 m2]
D = diffusion coefficient
diffusivity
[m2 s1]
dc/dx = concentration
gradient [mol m3 m1]
A = area [m2]x
diffusion flux
Fick’s first law describes steady-state diffusion
Velocity of diffusion of particles (ions, atoms ...) in a solid
mass transport and concentration gradient for a given point in a solid
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Typical diffusion coefficients for ions (atoms) in a solid at
room temperature
10-13 cm2 s-1
In solid state ionic conductors (e.g. Ag-ions in -AgI) the
values are greater by orders of magnitude ( 10-6 cm2 s-1)
Diffusion - Fick’s First Law
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Conditions for diffusion:
• an adjacent empty site
• atom possesses sufficient energy to break bonds with its
neighbors and migrate to adjacent site (activation energy)
The higher the temperature, the higher is the probability that an
atom will have sufficient energy
Diffusion rates increase with temperature
Mechanisms of Diffusion
Diffusion = the mechanism by which matter is transported into or
through matter
Diffusion at the atomic level is a step-wise migration of atoms from
lattice site to lattice site
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Mechanisms of Diffusion
• Along Defects = Vacancy (or Substitutional) mechanism
– Point Defects
– Line Defects
• Through Interstitial Spaces = Interstitial mechanism
• Along Grain Boundaries
• On the Surface
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Vacancy Mechanisms of Diffusion
• Vacancies are holes in the matrix
• Vacancies are always moving
• An impurity can move into the vacancy
• Diffuse through the material
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Atoms can move from one site to another if there is sufficient
energy present for the atoms to overcome a local activation
energy barrier and if there are vacancies present for the atoms
to move into.
The activation energy for diffusion is the sum of the energy
required to form a vacancy and the energy to move the vacancy.
Vacancy Mechanisms of Diffusion
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Interstitial Mechanisms of Diffusion
• There are holes between the
atoms in the matrix
• If the atoms are small enough,
they can diffuse through the
interstitial holes
• Fast diffusion
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Interstitial Atoms
• An atom must be small to fit into the interstitial voids
• H and He can diffuse rapidly through metals by moving through
the interstitial voids
• Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc.
must squeeze through openings between interstitial sites to
diffuse around in a crystal
• The activation energy for diffusion is the energy required for
these atoms to squeeze through the small openings between
the host lattice atoms.
• Interstitial C is used to strengthen Fe = steel, it distorts the
matrix
• The ratio of r/R is 0.57 – needs an octahedral hole
• Octahedral and tetrahedral holes in both FCC and BCC –
however the holes in BCC are not regular polyhedra
• The solubility of C in FCC-Fe is much higher than in BCC-Fe
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Interstitial Atoms
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Interstitial Atoms
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Interstitial Atoms
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Activation Energy
• All the diffusion mechanisms require a certain minimum
energy to occur
• The activation energy
• The higher the activation energy, the harder it is for
diffusion to occur
• The highest energy is for volume diffusion
– Vacancy
– Interstitial
• Grain Boundary diffusion requires less energy
• Surface Diffusion requires the least
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Activation Energy
Energy barrier for diffusion
Initial state Final stateIntermediate state
E Activation energy
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Energy Barrier for Diffusion
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Diffusion in Perovskites ABX3
A cation diffusion B cation diffusion
B B
B B
BBO
OO
O O
O
O
O
AA
EA = 379
Activation energies (kJ mol-1)
The A cation diffusion is easier
EA = 1420 EA = 746
(100)Cubic plane
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Diffusion Rate
D = the diffusivity, which is proportional to the diffusion rate
D = D for T 
Q = the activation energy
R = the gas constant
T = the absolute temperature
D is a function of temperature
Thus the flux (J) is also a function of temperature
High activation energy corresponds to low diffusion rates
The logarithmic representation of D verus 1/T is linear, the slope
corresponds to the activation energy and the intercept to D
Diffusion coefficients show
an exponential temperature
dependence (Arrhenius type)
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surface
grain boundaries
volume
Ag in Ag
C in Fe
Diffusion Coefficients
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Velocity of diffusion of particles (ions, atoms ...) in a solid
- mass transport and concentration gradient for a given point in a
solid
Ji = -Di  ci/ x [ mol cm-2 s-1] (const. T)
Ji: flow of diffusion (mol s-1 cm-2); Di: diffusion coefficient (cm2 s-1)
ci/ x: concentration gradient (mol cm-3 cm-1) (i.e. change of
concentration along a line in the solid)
Diffusion
Knowledge of D allows an estimation of the average diffusion
length for the migrating particles:
<x2> = 2Dt (<x2>: average square of diffusion area; t: time)
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Diffusion
Diffusion FASTER for:
• open crystal structures
• lower melting T materials
• materials w/secondary
bonding
• smaller diffusing atoms
• lower density materials
Diffusion SLOWER for:
• close-packed structures
• higher melting T materials
• materials w/covalent
bonding
• larger diffusing atoms
• higher density materials
Non-Steady-State Diffusion
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Fick's Second Law of Diffusion
The rate of change of composition at position x with time, t, is
equal to the rate of change of the product of the diffusivity, D, times
the rate of change of the concentration gradient, dCx/dx, with
respect to distance, x.
Fick's Second Law of Diffusion
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Second order differential equations are nontrivial.
Consider diffusion in from a surface where the concentration of
diffusing species is always constant. This solution applies to gas
diffusion into a solid as in carburization of steels or doping of
semiconductors.
Boundary Conditions
For t = 0, C = Co at 0  x
For t > 0 C = Cs at x = 0
C = Co at x = 
Fick's Second Law of Diffusion
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Cs = surface concentration
Co = initial uniform bulk concentration
Cx = concentration of element at distance x from surface at time t
x = distance from surface
D = diffusivity of diffusing species in host lattice
t = time
erf = error function
The solution to Fick's second law is the relationship between
the concentration Cx at a distance x below the surface at time t
Fick's Second Law of Diffusion
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