M.Sc. Physics

Study Material for Lattice Defects Subject: Condensed Matter-Physics Semester: III Paper: 1 Unit: IV By: Dr. (Mrs.) K.L.Pandey

Lattice Defects In a perfect crystal lattice the atoms are supposed to be in a periodic arrangement and none of the atom need to be out of place at any moment. But practically it is impossible to get this perfection in a crystal. Germanium is a common impurity in silicon. It prefers the same tetrahedral bonding as silicon and readily substitutes for silicon atoms. Similarly, silicon is a common impurity in germanium. No large crystal can be made without impurities; the purest large crystal ever grown was made of germanium. It had about 1010 impurities in each cubic centimetre of material, which is less than one impurity for each trillion atoms.

Impurities often make crystals more useful. In the absence of impurities, α- alumina is colourless. Iron and titanium impurities impart to it a blue colour, and the resulting gem-quality mineral is known as sapphire. Chromium impurities are responsible for the red colour characteristic of rubies, the other gem of α- alumina. Pure semiconductors rarely conduct electricity well at room temperatures. Their ability to conduct electricity is caused by impurities. Such impurities are deliberately added to silicon in the manufacture of integrated circuits. In fluorescent lamps the visible light is emitted by impurities in the phosphors (luminescent materials).

Crystal defects, imperfections in the regular geometrical arrangement of the atoms in a crystalline solid may occur due to deformation of the solid, rapid cooling from high temperature, or high-energy radiation (X-rays or neutrons) striking the solid. The properties of crystal or material may be divided in two categories with respect to these Crystal defects. 1. Crystal Structure Insensitive Properties: Density, Specific Heat, Stiffness. 2. Crystal Structure Sensitive Property: Mechanical Strength, Ductility, Crystal growth, Magnetic Hysteresis, Dielectric Strength, Conduction in Semi-Conductors. Classification of Lattice Defects -Structural Imperfections in crystals has been studied on the basis of geometry and shown by chart. Also they can be classified as the periodic regularity is interrupted in three (point), two (line) and one (surface/plane) dimensions.

Point Defects- A lattice defect which spreads out in all the three dimensions is called point defect. These have following sub-classifications shown below:

Line Defects/Dislocation- When a lattice defect is confined to a small region in two dimensions, it is called line defect. In this type of defect, part of lattice undergoes a shearing strain equal to one lattice vector (called a Burgers Vector) and have been classified in two categories.

Surface/Plane Defects: When a lattice defect is confined to only in one dimension, it is called a plane defect. When the defects cluster in a plane, they can form defects which can be classified in following four types:

Point Defects: Vacancies A crystal is never perfect; a variety of imperfections can damage the ordering. The simplest type of defect is a missing atom and is called a vacancy.

When an atom is missing from its lattice site in a crystal structure of a metal, it is called a vacancy (or vacant lattice site) as illustrated above. The atoms surrounding a vacancy experience a slight displacement into the empty lattice site. Such defects may arise either from imperfect packing during original crystallisation or from thermal vibration of the atoms at higher temperatures. In case of thermal vibrations internal energy increases which increases the possibility of jumping of atom from position of lowest energy. For most of the crystal the thermal energy is of the value of 1 eV per vacancy. Vacancies may be single or two (di-vacancy) or tri-vacancy or more. Interstitialcies This is an extra atom inserted into void between the regularly occupied sites thus such an atom does not occupy regular lattice sites. This extra atom may be an impurity atom or an atom of the same type as on the regular lattice sites. It is reverse to vacancy phenomenon. This is known as interstitials.

An atom can enter the interstitial void or space between the regularly positioned atoms only when it is substantially smaller than the parent atoms, otherwise it will produce atomic distortion. Interstitials may be mono-interstitial, di- interstitial or tri-interstitial.

Schottky Defect and Frenkel Defects

When the atoms of perfect lattice escape away (after absorption of energy) from lattice site to surface of crystal lattice then such type of vacancy is called as Schottky defect.

On the other hand if charge neutrality is maintained by having a positive ion in an interstitial position, it is Frenkel defect.

Close packed structures have fewer interstitialcies or Frenkel defects than Schottky defects, as additional energy is required to force the atom in their new position on surface in case of Schottky defect. From the study of Ionic conductivity and the density measurements it is concluded that in pure alkali halides, Schottky vacancies are more common, where as in pure silver halides, Frenkel vacancies are more common.

Due to Schottky defects volume of the crystals decrease without any change in the mass and consequently, production of this defect lowers the density of the crystal. On the other hand the production of Frenkel defects does not change the volume of the crystal so that the density of the crystal remains constant.

Fig: Schottky & Frenkel Defect

Expression for Equilibrium Concentration of the Schottky Defects

Although care is taken in the preparation of the crystals, vacancies are always present in all crystals. In fact, as a result of thermal fluctuations, vacancies are produced and destroyed constantly in the crystal. Initially such a defect arises after plucking an interior atom out of its regular lattice position and moving it on the surface of crystal. This act requires energy. Moreover, the disorder increases resulting in an increase in the entropy.

Let us assume N be the total number of atoms present in crystal lattice and n be the number of Schottky defects created at temperature T. If Ev is the energy required to take an atom from a lattice site inside the crystal to a lattice site on the surface. The total amount of energy required to create n vacancy = nEv.

The total number of ways in which we can pick up n-atoms from the crystal consisting of N atoms is given by: 푁! P = (N − n)! n! Since disorder increases due to creation of n vacancies, the corresponding increase in entropy is given by: S= kB log P 푁! S= kB log (N−n)!n! This in turn produces a change in free energy F, F = U-TS 푁! F=nEv – kB T log (N−n)!n!

The second term on the right and side can be simplified by using Sterling approximation: log x! =x log x-x Consequently, F= nEv – kB T [N log N- (N-n) log (N-n) -n log n]

Free energy in thermal equilibrium at constant volume must be minimum with respect to changes in n; 훿퐹 푁−푛 i.e., [ ] =0 = Ev – kB T log [ ] 훿푛 푇 n

퐸 푁−푛 Thus 푣 = log 푘퐵.푇 푛

n= (N- n) exp(-Ev /kB T)

If n<

n=N exp(-Ev /kB. T)

If Ev =1 eV and T= 1000K; then, 푛 = e-11.6 푁

푛 = 9.1x10-6 =10-5 푁 The equilibrium concentration of vacancies decreases as the temperature decreases. In ionic crystals, the formation of paired vacancies is most favoured i.e., an equal number of positive and negative ion vacancies are produced. The formation of pairs make it possible to keep the surface of the crystal electrostatically neutral. The number of pairs can be related to total number of atoms present in the crystal on following the same procedure as adopted in pure atomic crystals.

The different ways in which n separated pairs can be formed are: 푁! P = [ ]2 (N−n)!n!

Increase in entropy is given by: S=kB log P

푁! 2 = kB log [ ] (N−n)!n!

With corresponding change in free energy: F=U-TS 푁! 2 = nEP –kB T log [ ] (N−n)!n!

Where EP is the energy of formation of pair.

As before we now apply Stirling’s approximation to simplify the factorial term.

푁! i.e. log[ ]2 =2 [log N!-log (N- n)! – log n! ] (N−n)!n!

= 2[N log N- N- (N- n) log (N- n) + (N- n) - n log n + n]

= 2[N log N – (N- n) log (N - n) – n log n]

Thus free energy, F= n EP – 2kB T [N log N – (N-n) log (N-n) – n log n]

Differentiating the above equation with respect to n, we get: 훿퐹 [ ] = EP – 2 kB T [0 + log (N- n) + 1 – log n – 1] 훿푛 푇 푁−n = EP – 2kB T log 푛 At equilibrium, the free energy is constant, so that: 푁−n EP -2 kB T log [ ] = 0 푛 푁−n EP /2 kB T = log [ ] 푛 푁−n = exp( EP /2 kB T) 푛

n = N exp(-EP /2 kB T)

Provided n<< N. In NaCl crystal, EP =2.02 eV and at random temperature:

n = exp(-EP /2 kB T) 푁

n = exp(-2.02 /2 x 0.025) 푁

n =e-40.4 푁 n i.e. = 2.8 푋 10 -18 푁

Expression for Equilibrium Concentration of the Frankel Defects

Another vacancy defect is the Frenkel defect in which an atom is transferred from a lattice site to interstitial position, a position not normally occupied by an atom. The calculation of the equilibrium number of Frenkel defects proceeds along the lines followed in Schottky defect case.

We can calculate the number of Frenkel defects in equilibrium at a temperature T. Let in a perfect crystal,

Ei = Energy required to displace an atom from a regular lattice site to an interstitial position.

Ni = Number of interstitial atoms.

N = Number of atoms.

Now the total number of ways in which n Frenkel defects can be formed will be given by: 푁! 푁 ! P = [ ] x [ 푖 ] (N−n)!n! (Ni−n)!n!

The corresponding increase in entropy due to the creation of Frenkel defect is given by:

푁! 푁푖! S= kB T log[ . ] (N−n)!n! (Ni−n)!n!

Which in turn produces a change in free energy:

F= U – TS

푁! 푁푖! F = n Ei - kB T log[ . ] (N−n)!n! (Ni−n)!n! Using Stirling’s approximation for logarithmic term, we get:

푁! 푁! 푁! 푁 ! log[ . ] = log + log 푖 (N−n)!n! (Ni−n)!n! (N−n) !n! (Ni−n)!n!

= N log N + Ni log Ni – (N-n) log (N-n) – (Ni –n) log (Ni –n)-2n log n

Substituting this value of a logarithmic term in the expression for free energy and then differentiating with respect to n, we get: 훿퐹 2 [ ] = Ei – kB T log [(N- n) (Ni –n)/ n ] 훿푛 푇 At equilibrium, the free energy is constant, so that:

훿퐹 [ ] = 0 훿푛 푇

(푁−푛)(푁푖−푛) or Ei = kB T log n2

NNi Ei = kB T log (Taking N >> n and Ni >>n) 푛2

Thus, Ei = kB T log [(NNi) –2 log n]

1 Ei or log n = log (NNi ) - 2 2 kB 푇

1/2 (- Ei or n = (NNi ) exp ) 2 kB 푇

1/2 Showing that n should be proportional to (NNi )

Compositional Defects (Substitutional and Interstitial Impurity) Compositional defects arises from impurity atoms during original crystallisation. Impurity atoms considered as defects in a perfect lattice are responsible for the functioning of most semiconductor devices. They occur on a lattice point as a substitutional impurity or as an interstitial impurity.

A substitutional impurity is created when a foreign atom substitutes for or replaces a parent atom in the lattice. Substitutional impurity atoms are usually close in size (within approximately 15%) to the bulk atom. An example of substitutional impurity atoms is the zinc atoms in brass. In brass (an alloy of copper and Zinc) zinc atoms with a radius of 0.133 nm replaces some of the copper atoms, which have a radius of 0.128 nm. Brass has 70% copper and 30% Zinc. By this replacement of the copper atom Brass, an alloy is created, whose properties are different than Copper. Hence Zn is substitutional atom in copper lattice.

An interstitial impurity is a small sized foreign atom occupying an interstitial space, i.e. space between the regularly positioned atoms and that is why it is called interstitial impurity. Substitutional atom size is smaller than size of original atoms of lattice. An example of interstitial impurity atoms is the Carbon atoms that are added to Iron to make Steel. Carbon atoms, with a radius of 0.071 nm, fit nicely in the open spaces between the larger (0.124 nm) iron atoms. A self- interstitial atom is an extra atom that has crowded its way into an interstitial void in the crystal structure. Self- interstitial atoms occur only in low concentrations in metals because they distort and highly stress the tightly packed lattice structure.

Electronic Defect These are defects in electronic structure e.g. 1. Conduction Electron 2. Hole Which are excited thermally from filled bands or impurity levels. These defects are responsible for important electrical and magnetic properties. Colour Centre Ionic crystals such as NaCl, KCl etc. in the pure state without imperfections, are transparent throughout the visible region of spectrum. However, the impure salts are often coloured because energy is absorbed in certain regions (called absorption bands) of the visible spectrum owing to the presence of various impurities and point defects. As mentioned above, presence of vacancies in the crystal causes the absorption band to appear in the range of visible spectrum. When the vacancies capture electrons or holes, they are capable of absorbing light and since the crystal appears coloured because of absorption of light, these are called coloured centres. A colour centre is therefore a lattice defect, which absorbs visible light.

The Colour centres can be produced in a number of ways:  By introduction of chemical impurities;  By introduction of an excess of the metal ion(we may heat the crystal in the vapour of the alkali metal and then cool it quickly for example; an NaCl crystal heated in the presence of sodium vapour becomes yellow; a KCl crystal heated in potassium vapour becomes magenta);  By x-ray, γ-rays, neutron and electron bombardment; and  By electrolysis  By any above methods energy is given (UV or X-rays or gamma rays, Vapour heating) are used to promote an electron from the valence band to the trap. Example of Self Darkening Sun Glasses Colour change can occur merely while the glasses are illuminated, leading to optical bleaching. Glasses may fade in the dark at room temperature when the ultraviolet energy present in sunlight produces darkening, and room temperature leads to fading as soon as ultraviolet light is no longer present.

F Centres The F centre comes from German word Farbe which means colour. We usually produce F centres by heating the crystal in excess alkali vapour or by x- irradiation. When crystals of NaCl are heated in an atmosphere of sodium vapour, the sodium atoms are deposited on the surface of the crystal. The Cl- ions diffuse from the crystal to its surface and combine with Na atoms, forming NaCl. During this process, the Na atom on the surface of the crystal lose electrons. These released electrons diffuse into the crystal and occupy the vacant anionic site and creates F –Centre.

Figure: F Centre

The absorption band associated with F-centres in several alkali halides have been given below.

Figure: The F band for several Alkali Halides: Optical absorption Vs, wavelength for crystals that contain F centres.

Pohl investigated the following experimental properties of F centre: 1. It is noted that band in KCl or NaCl is exactly same whether the excess metal is added by heating the crystal in presence of Na or K vapour. This implies F-absorption band is independent of added metal and it is the characteristic of the crystal. Also the same band is formed when the stoichiometric crystals are irradiated with UV, x-ray or other type of radiation which produces free electron. Such free electrons are ultimately Trapped at negative ion vacancies forming F-centres. 2. It was noted that the density of the crystal decreases when excess metal is introduced or in other words coloured crystals are less dense than uncoloured crystals 3. Width of F- band increases with temperature, the peak shifting at same time to lower energies. Coagulation of F Centres: If crystals containing F centres are irradiated with light in F band, there is a suitable temperature range(for NaCl and KCl room tempr) in which number of bands appear on the long wave length side of F band. This is because of (-) ion vacancy introduced during creation of F centre, combine with existing (+) ion vacancies forming pairs and therefore combination of F centres occurs. R1 Centre: F centre combined with a vacant anion site is called R1 Centre. R2 Centre: When two F centres combine it is R2 centre.

M Centre: Two Adjacent F centres (with pair of vacancies of opposite spin) form M centre.

Fig: M Centre

R Centre: Three Adjacent F centres form R centres (a group of three negative Ion vacancies in a plane).

Fig: R Centre

FA Centre: In the FA Centre one of the six nearest neighbours of an F Centre has been replaced by different alkali ion, shown below.

Fig:FA Centre

The electron trapped can absorb only certain colours of light. F-centres in sodium chloride absorb only blue light, giving the solid a yellow-orange colour. Sodium chloride is usually colourless, however, because its electrons are not free to move to vacancies formed by removal of negative chloride ions from the solid, X-rays striking is needed.

V Centres: This type of colour centres result from an excess of halogen in alkali halides. Mollwo observed new absorption band for KBr when heated in Br2 vapour. The excess bromine is incarporated in the lattice in the form of negative ions, occupying normal lattice sites. As result of the introduction of extra bromine atoms, positive holes are formed. These holes are most likely to be situated near a poitive ion vacancy where they can be trapped. A hole trapped at a positive ion vacancy is called V centre.

Line Defects /Dislocations A dislocation occurs when the periodicity of atomic lattice array is interrupted along certain direction in the crystal. The dislocation is deformation and it is related to shear and stress applied on it. Take a look at this graph. The elastic deformation is shown by red while plastic deformation is shown by blue.

At yield stress point deformation converts from elastic to plastic. Elastic deformation is one in which, material regains its shape perfectly, after removing stress. But after reaching to yield stress as per stress strain diagram above, after removal of applied stress, deformation becomes plastic deformation and shape does not comes back to normal position. This point corresponding to yield stress on the graph is called yield point, where irreversible Plastic deformation starts. This plastic deformation occurs due to “slip” of one part of crystal on other part due to stress applied. The direction in which slip occurs is called Slip Direction and the plane in which slip occurs is called Slip Plane.

Edge Dislocation

Defects in the crystal structure are present in almost all metals and these defects are responsible for most deformations. One of the most common crystal structure line defects is known as an edge dislocation. This occurs when there are extra atoms inserted into a plane in the crystal lattice (shown below).

When there are lot of edge dislocations in a metal, it will deform much more easily than a similar metal containing fewer edge dislocations. It is because they stretch out bonds between adjacent atoms and make it easier for the atoms to slide along one another. Movements of dislocations give rise to their plastic behaviour. When a metal bar is cold-worked by rolling or hammering, dislocations and grain boundaries are introduced; this causes the hardening. Edge dislocation is responsible for the ductility and malleability.

As shown in the set of images a, b, and c below, after application of shear force from top and bottom shown by arrow, the dislocation moves in three stages shown in fig. a, b, and c. 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. Unit step of slip shown in figure (c) above, is Berger Vector, which is perpendicular to dislocation line.

Burgers Vector: In material science, named after Dutch physicist Jan Burgers, is a vector, often denoted as b, that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice.

The Burgers vector associated with a dislocation is a measure of the lattice distortion caused by the presence of the line defect. The diagram shows the convention for measuring the Burgers vector. A circuit is made around a dislocation line in a clockwise direction (Fig. 1) with each step of the circuit connecting lattice sites that are fully coordinated. This circuit is then transferred to a perfect lattice of the same type, where there is no dislocation. This circuit fails to close on itself on perfect lattice of same type, finally the vector linking the end of the circuit to the starting point is the Burgers vector, b = QM.

The Burgers vector defined in this way is a unit vector of the lattice if the dislocation is a unit dislocation, and a shorter stable translation vector of the lattice if the dislocation is a partial dislocation.

Burger Vector in Edge Dislocation

Fig: 1 Fig: 2

Burger Vector in Screw Dislocation

Fig: 3 Fig: 4 In edge dislocations, the Burgers vector and dislocation line are perpendicular to each other. In screw dislocations, Burger vector and dislocation line are parallel. Screw Dislocation There is a second type of dislocation, called screw dislocation. A screw dislocation is a topological (property of a crystal structure such as stretching, twisting, bending, but not tearing) defect of a crystal lattice. If one moves around the dislocation, the lattice plane shifts by one layer (or more layers), like a spiral staircase. The Burgers vector of a screw dislocation is parallel to the dislocation line.

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 (upper image).

The lower 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.

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

Another Example Showing Screw Dislocation.

Surface Defects A surface Defect is a discontinuity of the perfect crystal structure across a plane.

Grain Boundaries A Grain Boundary is a general planar defect that separates regions of different crystalline orientation (i.e. grains) within a polycrystalline solid. The atoms in the grain boundary will not be in perfect crystalline arrangement. Grain boundaries are usually the result of uneven growth when the solid is crystallising. Grain sizes vary from 1 µm to 1 mm.

Tilt /Lineage Boundaries It is boundary between two adjacent perfect regions in the same crystals that are slightly tilted with respect to each other.

Twin Boundaries

A Twin Boundary happens when the crystals on either side of a plane are mirror images of each other. The boundary between the twinned crystals will be a single plane of atoms. There is no region of disorder and the boundary atoms can be viewed as belonging to the crystal structures of both twins. Twins are either grown-in during crystallisation, or the result of mechanical or thermal work.

Stacking Faults A stacking fault is a surface imperfection that arise from the stacking of one atomic plane out of sequence on another, while the lattice on either side of fault is perfect.

Volume Defects Volume defects such as cracks may arise when there is only small electrostatic dis-similarity between the stacking sequences of close packed planes in metals. Further when clusters of atoms are missing, a large vacancy is seen, known as volume imperfection. Foreign particle inclusions, large voids or non- crystalline regions which have dimensions of the order of 0.20 nm are also called volume imperfections.