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, we can neglect n in (N-n); so that, n=N exp(-Ev /kB. T) If Ev =1 eV and T= 1000K; then, 푛 -11.6 = e 푁 푛 -6 -5 = 9.1x10 =10 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 -40.4 =e 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.