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Chapter 2

Crystal imperfections

Introduction

You have been introduced to the lattice structures and types of bonding in crystalline

solids in earlier lectures. You know that the behavior of electrons determine the way the

atoms interact - the type of bonding (metallic, ionic, covalent, and van der Waals) that holds

atoms in a solid together. But is the knowledge of bonding and crystal structure sufficient to

predict the macroscopic properties of materials? In this lecture we will discuss different types

of imperfections or defects in the ideal arrangement of atoms in a crystal. We will see that the

presence of a relatively small number of defects have a profound impact on the macroscopic

properties of materials, and the control (and intentional introduction) of defects is important

in many kinds of material processing.

Examples of the relevance of defects in crystals for life in general or materials science

in particular: When you buy a diamond ring, it is mostly the number and type of defects in

the diamond crystal that define the amount of money you pay for a given crystal size.

Production of advanced semiconductor devices requires not only a rather perfect Si crystal as

starting material, but also involve introduction of specific defects in small areas of the

sample. Forging a metal tool introduces defects … and increases strength and elasticity of the

tool. Note, that in this case the required properties are achieved without changes in

composition of the material, but just by manipulating the crystal defects.

Session 7: Introduction; Classification of defects, point

defects & line-defects

30 Minutes

Classification of defects & point defects (Board Teaching and PPT)

Perfect crystals do not exist. Defects exist in crystals which result in departure from

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periodicity. Some defects are the result of crystal growth and handling (dislocations, grain

boundaries), others are intrinsic properties of the crystalline state at non-zero (Kelvin)

temperature (vacancies or intersticials). Various defects affect various type of physical or

chemical properties of crystalline solids. It is thus extremely important to understand the

origin and nature of defects in crystals. In this chapter we will discuss the three basic classes

of defects in crystals.

i. Point defects – 0 dimensional defect (atoms missing or in irregular places in the lattice

(lattice vacancies, substitutional and interstitial impurities, self-interstitials)

ii. Line defects – 1 dimensional defect (groups of atoms in irregular positions (e.g. screw

and edge dislocations)

iii. Planar defects – 2 dimensional defect (the interfaces between homogeneous regions of

the material (grain boundaries, stacking faults, external surfaces).

iv. Volume defects – 3 dimensional defect

Point Defects:

– Vacancies: Diffusion, Color Center

– Interstitials: Mechanical Properties, Diffusion

– Impurity Atoms: Electrical Properties

Line Defects:

– Dislocations: Mechanical Properties

Planar Defects:

– Grain Boundaries: Fabrication, Corrosion

– Stacking Faults: Mechanical Properties

Volume Defects:

– Voids: Porosity, Precipitation

– Second Phase: Mechanical and Magnetic Properties

Point Defects:

A perfect crystal with regular arrangement of atoms cannot exist. There are always defects,

and the most common defects are point defects. This is especially true at high temperatures

when atoms frequently and randomly change their positions leaving behind empty lattice

sites, called vacancies. Point defects include (a) vacancies (b) self-interstitial atoms (c)

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interstitial impurity atoms (d) substitutional atoms (e) Frenkel defect (f) Schottky defect.

Intrinsic point defects: Interstitials – atoms that are squeezed in between regular lattice sites.

If the interstitial atom is of the same species as the lattice atoms, it is called self-interstitial.

Creation of a self-interstitial causes a substantial distortion in the surrounding lattice and

costs more energy as compared to the energy for creation of a vacancy (Ei > EV), under

equilibrium conditions, self-interstitials are present in lower concentrations than vacancies.

Foreign, usually smaller atoms (carbon, nitrogen, hydrogen, oxygen) are called interstitial

impurities. They introduce less distortion to the lattice and are more common in real materials

and more mobile. As shown in the schematic drawing, all point defects introduce local

distortions to the lattice, and due to these distortions they can feel each other (interact) and

feel external stresses. The external stresses or stresses from a larger defects can give a

directionality to an otherwise random jumps of atoms.

In ionic crystals (e.g. table salt – Na+Cl-) the bonding is provided by coulombic forces

between positively and negatively charged ions. Point defects in ionic crystals are charged as

well. The Columbic forces are very large and any charge imbalance has a very strong

tendency to balance itself. To maintain charge neutrality several point defects can be created.

A Frenkel defect is a pair of cation (positive ion) vacancy and a cation interstitial. Or it may

also be an anion (negative ion) vacancy and anion interstitial. However anions are much

larger than cations and it is not easy for an anion interstitial to form. A Schottky defect is a

pair of anion and cation vacancies. In both Frenkel and Schottky defects, the pair of point

defects stay near each other because of strong columbic attraction of their opposite charges.

Extrinsic point defects: If the foreign atom replaces or substitutes for a matrix atom, it is

called a substitutional impurity. The extrinsic point defects are foreign atoms, which are

called solutes if they are intentionally added to the material and are called impurities. The

foreign atom may occupy a lattice site, in which case it is called a substitutional solute (or

impurity) or it may fill an interstitial site, in which case it is called an interstitial solute.

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Fig 1: Point defect and their types.

Fig 2: Frenkel and Schottky defect

Effect of Temperature:

The higher is the temperature, more often atoms jump from one equilibrium position to

another and larger number of vacancies can be found in a crystal. Actually, the number of

vacancies, Nv, increases exponentially with the absolute temperature, T, and can be estimated

using the equation (Boltzmann Distribution):

where, Ns is the number of regular lattice sites, k is the Boltzmann constant, and Ev is the

energy needed to form a vacant lattice site in a perfect crystal. Using this simple equation we

can estimate that at room temperature in copper there is one vacancy per 1015

lattice atoms.

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10 Minutes:

Verify

1. Define a defect in Crystal?

2. State the possible reasons for formation of defects in Crystals?

3. Explain classification of defects in Crystals.

30 Minutes

Line defects: Dislocations, Edge and Screw Dislocations (Board Teaching

and PPT) Dislocations are linear defects; they are lines through the crystal along which

crystallographic registry is lost. Their principle role in the microstructure is to control the

yield strength and subsequent plastic deformation of crystalline solids at ordinary

temperatures. Dislocations also participate in the growth of crystals and in the structures of

interfaces between crystals. They act as electrical defects in optical materials and

semiconductors, in which they are almost always undesirable.

The concept of a dislocation in a solid was introduced by Volterra in the nineteenth

century. Since the 1950's it has been possible to observe and study dislocations directly using

such techniques as transmission electron microscopy and x-ray topography. While

dislocations influence many aspects of physical behavior, they are studied almost exclusively

in Materials Science.

Fig3: Edge dislocation: the force to break

all the bonds is much higher than the force

needed to cause deformation in a plane.

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Fig 4: This picture shows the inserted half

plane dislocation going through the center

Edge dislocation:

The edge defect can be easily visualized as an extra half-plane of atoms in a lattice.

The dislocation is called a line defect because the locus of defective points produced in the

lattice by the dislocation lie along a line. This line runs along the top of the extra half-plane.

The inter-atomic bonds are significantly distorted only in the immediate vicinity of the

dislocation line.

Fig 5. Edge dislocations

Understanding the movement of a dislocation is key to understanding why

dislocations allow deformation to occur at much lower stress than in a perfect crystal.

Dislocation motion is analogous to movement of a caterpillar. The caterpillar would have to

exert a large force to move its entire body at once. Instead it moves the rear portion of its

body forward a small amount and creates a hump. The hump then moves forward and

eventual moves all of the body forward by a small amount.

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Fig 6. Burger vectors – Edge dislocations

As shown in the set of images above, the dislocation moves similarly moves a small amount

at a time. The dislocation in the top half of the crystal is slipping one plane at a time as it

moves to the right from its position in image (a) to its position in image (b) and finally image

(c). In the process of slipping one plane at a time the dislocation propagates across the crystal.

The movement of the dislocation across the plane eventually causes the top half of the crystal

to move with respect to the bottom half. However, only a small fraction of the bonds are

broken at any given time. Movement in this manner requires a much smaller force than

breaking all the bonds across the middle plane simultaneously.

Screw dislocation:

There is a second basic type of dislocation, called screw dislocation. The screw

dislocation is slightly more difficult to visualize. The motion of a screw dislocation is also a

result of shear stress, but the defect line movement is perpendicular to direction of the stress

and the atom displacement, rather than parallel. To visualize a screw dislocation, imagine a

block of metal with a shear stress applied across one end so that the metal begins to rip. This

is shown in the upper right image. The lower right image shows the plane of atoms just above

the rip. The atoms represented by the blue circles have not yet moved from their original

position. The atoms represented by the red circles have moved to their new position in the

lattice and have re-established metallic bonds. The atoms represented by the green circles are

in the process of moving. It can be seen that only a portion of the bonds are broke at any

given time. As was the case with the edge dislocation, movement in this manner requires a

much smaller force than breaking all the bonds across the middle plane simultaneously.

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Fig 7. Screw dislocation

If the shear force is increased, the atoms will continue to slip to the right. A row of the

green atoms will find there way back into a proper spot in the lattice (and become red) and a

row of the blue atoms will slip out of position (and become green). In this way, the screw

dislocation will move upward in the image, which is perpendicular to direction of the stress.

Recall that the edge dislocation moves parallel to the direction of stress. As shown in the

image below, the net plastic deformation of both edge and screw dislocations is the same,

however.

Fig 8: to show edge and screw dislocations

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The dislocations move along the densest planes of atoms in a material, because the

stress needed to move the dislocation increases with the spacing between the planes. The

screw dislocation is parallel to the direction in which the crystal is being displaced (Burgers

vector is parallel to the dislocation line).

Burgers Vector: To describe the size and the direction of the main lattice distortion caused

by a dislocation we should introduce so called Burgers vector, b.

Fig 9. Burger Vector: A Burgers circuit closes in a {100} plane of a cubic crystal, but fails to

close by the Burgers vector, b, when the same circuit encloses an edge dislocation.

To find the Burgers vector, we should make a circuit from atom to atom counting the same

number of atomic distances in all directions. If the circuit encloses a dislocation it will not

close. The vector that closes the loop is the Burgers vector b. Dislocations that have been

considered until now have Burgers vector directed perpendicular to the dislocation line.

These dislocations are called edge dislocations. There is a second basic type of dislocation,

called screw dislocation. The screw dislocation is parallel to the direction in which the crystal

is being displaced (Burgers vector is parallel to the dislocation line). The exact structure of

dislocations in real crystals is usually more complicated than the ones shown in this pages.

Edge and screw dislocations are just extreme forms of the possible dislocation structures and

even they usually would be split in "partial" dislocations. Partial dislocations have their cores

spread out over a larger area and look much more complicated.

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10 Minutes

Verify

1) Write types of line defects.

2) State the direction of Burger vector in Edge and Screw defects.

3) Define dislocation line.

Session 8: Surface, volume defects and effect of

imperfections on properties of crystalline solids.

30 Minutes:

Surface defects (Board Teaching and PPT)

These are two dimensional imperfections that lies in the metal with polycrystalline structures

or they are also called as interfacial defects can be defined as boundaries that have two

dimensional imperfections in crystalline solids, and have different crystal structures and/or

crystallographic orientations on either side of them. They refer to the regions of distortions

that lie about a surface having thickness of a few atomic diameters. For example: external

surfaces, grain boundaries, twin boundaries, stacking faults, and phase boundaries. These

imperfections are not thermodynamically stable, rather they are meta-stable imperfections.

They arise from the clustering of line defects into a plane.

External surface: The environment of an atom at a surface differs from that of an atom in the

bulk; especially the number of neighbors (coordination) at surface is less. Thus the

unsaturated bonds of surface atoms give rise to a surface energy. This result in relaxation (the

lattice spacing is decreased) or reconstruction (the crystal structure changes). To reduce the

energy, materials tend to minimize, if possible, the total surface area.

Grain boundaries: Crystalline solids are, usually, made of number of grains separated by

grain boundaries. Grain boundaries are several atoms distances wide, and there is mismatch

of orientation of grains on either side of the boundary as shown in figure-3.6. When this

misalignment is slight, on the order of few degrees (< 10°), it is called low angle grain

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boundary. These boundaries can be described in terms of aligned dislocation arrays. If the

low grain boundary is formed by edge dislocations, it is called tilt boundary, and twist

boundary if formed of screw dislocations. Both tilt and twist boundaries are planar surface

imperfections in contrast to high angle grain boundaries. For high angle grain boundaries,

degree of disorientation is of large range (> 15°). Grain boundaries are chemically more

reactive because of grain boundary energy. In spite of disordered orientation of atoms at grain

boundaries, polycrystalline solids are still very strong as cohesive forces present within and

across the boundary.

Fig 10: Schematic presentation of grain boundaries.

Tilt boundaries: If the low grain boundary is formed by edge dislocations, it is called tilt

boundary. A tilt boundary, between two slightly misaligned grains as an array of edge

dislocations. Rotation axis is parallel to the boundary plane.

Fig 11: Tilt boundaries

Twist boundaries: If the low grain boundary is formed by screw dislocations, it is called

twist boundary. Rotation axis is perpendicular to the boundary plane

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Twin boundaries: It is a special type of grain boundary across which there is specific mirror

lattice symmetry. Twin boundaries occur in pairs such that the orientation change introduced

by one boundary is restored by the other (figure-). The region between the pair of boundaries

is called the twinned region. Twins which forms during the process of recrystallization are

called annealing twins, whereas deformation twins form during plastic deformation.

Twinning occurs on a definite crystallographic plane and in a specific direction, both of

which depend on the crystal structure. Annealing twins are typically found in metals that have

FCC crystal structure (and low stacking fault energy), while mechanical/deformation twins

are observed in BCC and HCP metals. Annealing twins are usually broader and with

straighter sides than mechanical twins. Twins do not extend beyond a grain boundary.

Fig 12. A pair of twin boundaries

Stacking faults: They are faults in stacking sequence of atom planes. Stacking sequence in

an FCC crystal is ABC ABC ABC …, and the sequence for HCP crystals is AB AB AB….

When there is disturbance in the stacking sequence, formation of stacking faults takes place.

Two kinds of stacking faults in FCC crystals are: (a) ABC AC ABC…where CA CA represent

thin HCP region which is nothing but stacking fault in FCC, (b) ABC ACB CABC is called

extrinsic or twin stacking fault. Three layers ACB constitute the twin. Thus stacking faults in

FCC crystal can also be considered as submicroscopic twins. This is why no microscopic

twins appear in FCC crystals as formation of stacking faults is energetically favorable.

Stacking fault energy varies in range 0.01-0.1 J/m2. Lower the stacking fault energy, wider

the stacking fault, metal strain hardens rapidly and twin easily. Otherwise, metals of high

stacking fault energy i.e. narrower stacking faults show a deformation structure of banded,

linear arrays of dislocations.

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Phase boundaries exist in multiphase materials across which there is sudden change in

physical/chemical characteristics.

The top and bottom faces of the unit cell consist of six atoms that form regular

hexagons and surround a single atom in the center. Another plane that provides three

additional atoms to the unit cell is situated between the top and bottom planes. The atoms in

this mid-plane have as nearest neighbors atoms in both of the adjacent two planes. The

equivalent of six atoms is contained in each unit cell; one-sixth of each of the 12 top and

bottom face corner atoms, one-half of each of the 2 center face atoms, and all 3 mid-plane

interior atoms. If a and c represent, respectively, the short( basal) and long 9 height)

parameters of unit cell dimensions of Fig a, the c/a ratio should be 1.633 . The coordination

number and the atomic packing factor for the HCP crystal structure are the same as for FCC:

12 and 0.74, respectively

Figure 13. HCP structure.

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Figure 14: Staking Sequence.

Figure 15: Representation of atomic planes.

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Figure 16: Atomic planes

Volume defects

As name suggests are defects in 3-dimensions. These include pores, cracks, foreign

inclusions and other phases. These defects are normally introduced during processing and

fabrication steps. All these defects are capable of acting as stress raisers, and thus deleterious

to parent metal’s mechanical behavior. However, in some cases foreign particles are added

purposefully to strengthen the parent material. The procedure is called dispersion hardening

where foreign particles act as obstacles to movement of dislocations, which facilitates plastic

deformation. The second-phase particles act in two distinct ways – particles are either may be

cut by the dislocations or the particles resist cutting and dislocations are forced to bypass

them. Strengthening due to ordered particles is responsible for the good high-temperature

strength on many super-alloys. However, pores are detrimental because they reduce effective

load bearing area and act as stress concentration sites.

10 Minutes

Verify

1. Describe surface defect.

2. Describe the types of surface defects

3.Describe tilt, twist and twin boundaries.

4. Explain stacking defect.

5.Discuss stacking fault.

30 Minutes

Effects of imperfections on crystal properties (Board Teaching & PPT)

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The crystal imperfections can lead to variation of mechanical, electrical, magnetic and

optical properties in a material. The different types of imperfections have their significant

impact on the base property of the material. This variation of the properties is utilized to the

suitable application in the industry.

A perfect crystal, with every atom of the same type in the correct position, does not

exist. All crystals have some defects. Defects contribute to the mechanical properties of

metals. In fact, using the term “defect” is sort of a misnomer since these features are

commonly intentionally used to manipulate the mechanical properties of a material. Adding

alloying elements to a metal is one way of introducing a crystal defect. Nevertheless, the term

“defect” will be used, just keep in mind that crystalline defects are not always bad.

Extrinsic point defects affect almost all engineering properties, but they are particularly

important in semiconducting crystals, where extrinsic defects are used to control electrical

properties, and in structural metals and alloys, where extrinsic defects are added to increase

mechanical strength.

Donors and acceptors in semiconductors: Point defects are intentionally added to

semiconductors to control the type and concentration of charge carriers. Consider, for

example, boron (valence 3) as a substitutional solute in elemental silicon. The saturated

covalent bonds in silicon are shown schematically in Fig 17a, and depend on the availability

of four valence electrons per silicon atom. Since the bonds are saturated, silicon has very low

conductivity in its pure state; pure silicon can only conduct electricity when electrons are

excited into high energy electron states. If boron is added, as in Fig 17b, a valence electron is

missing from the immediate environment of the boron atom, causing a hole in the bonding

pattern. Electrons can then move from bond to bond by exchanging with the hole. The

exchange requires some energy to separate the hole from the boron ion core, but this energy

is small compared to that required to excite an electron from a Si-Si bond into a high-energy

state. The room-temperature conductivity of Si increases significantly when a small amount

of B is added. Electron-deficient solutes like boron that cause holes in the configuration of

bonding electrons are called acceptors.

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Fig 17: (a) Tetrahedral bonding configuration in Si. (b) Bonding around a B solute, showing a

hole. (c) Bonding around a P solute, showing an electron (e) in a loose excited orbital.

The conductivity also rises when a solute with an excess of electrons is added to a

semiconductor with saturated bonds. For example, let phosphorous (valence 5) be added to

Si, as in Fig 17c. The 5 valence electrons of P are sufficient to fill the local covalent bonds

with one electron left over. This electron can only go into an excited state, and orbits about

the P ion core somewhat as shown in the figure. It requires a relatively small energy

increment to free this electron from the P core, in which case it can transport current by

moving through the lattice. The conductivity of Si rises dramatically if a small amount of P is

added. Electron-excess solutes such as P in Si are called donors. Semiconductors whose

electrical properties are controlled by electrically active solutes are called extrinsic

semiconductors. Almost all of the semiconductors that are used in engineering devices are

extrinsic.

Solution hardening in structural materials: The addition of solute atoms almost always

increases the mechanical strength of a solid. The phenomenon is called solution hardening. It

is due to the fact that the solute atom is always a bit too large or a bit too small to fit perfectly

into the crystal lattice site it is supposed to occupy, and distorts the crystal lattice in its

attempt to fit as well as possible. This distortion impedes the motion of the linear defects

(dislocations) that are responsible for plastic deformation and, consequently, hardens the

crystal. The distortion due to a substitutional solute is relatively small, though the associated

hardening may be large enough to be useful in the engineering sense. The distortions due to

interstitial atoms such as carbon and nitrogen are normally much greater because of the small

size of the interstitial void in which they must fit. The hardening effect of interstitial solutes is

large and technologically important; for example, high strength structural steels are alloys of

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Fe and C.

Plastic deformation: It is important to note at this point that plastic deformation in a material

occurs due to the movement of dislocations (linear defects). Millions of dislocations result for

plastic forming operations such as rolling and extruding. It is also important to note that any

defect in the regular lattice structure disrupts the motion of dislocation, which makes slip or

plastic deformation more difficult. These defects not only include the point and planer

defects. Dislocation movement produces additional dislocations, and when dislocations run

into each other it often impedes movement of the dislocations. This drives up the force

needed to move the dislocation or, in other words, strengthens the material.

10 Minutes

Verify

1. Recognize the structure property relations in crystals with some examples.

2. Identify structure sensitive and insensitive properties of crystals.

3. Explain the process of tuning electrical conductivity of semiconductor, mechanical strength

metals by introducing defects in them.