15 Crystal Defects

Prof. P. N. Kotru

epgp books

 

15.1    Introduction

 

In an ideal crystal the atoms composing it are supposed to be regularly arranged. However, in real crystals one finds some regions where the regular periodic arrangement of atoms breaks down. These regions are called defects, flaws or imperfections. The defects in crystals affect their properties. It is primarily because the crystals exhibit most of the properties which are structure sensitive. The presence of defects in them influence the structure sensitive properties, be it mechanical, optical, electronic or any other.

 

Defect in a solid is a deviation from periodicity. The primary defects in crystals include phonons , electrons and holes , excitons , vacant lattice sites and interstitial atoms ,foreign atoms in either interstitial or Substitutional positions and dislocations .These different types of defects can all interact with surface. However, they are usually found in the volume of the crystal and thereby contribute to the bulk properties of the solid.

 

Defects in solids have been extensively studied which makes it necessary to broaden their list by adding the following:

 

(a)    Quasi-particles such as the interacting electrons near the Fermi surface in a normal metal.

 

(b)   Quasi-particles viz., Cooper pairs in a superconductor where an attractive electron-electron interaction

is induced by phonon.

 

(c)    Quasi-particles such as polarons which consist of a conduction electron or a hole in an insulating ionic

crystal combined with a cloud of phonons.

 

(d)   Polaritons which are mixed states of photons and phonons or of photons and excitons in insulating

crystals.

 

(e)    Plasmons which are collective oscillations of interacting electrons in a solid.

 

(f)    Magnons or quantized spin waves in a ferromagnetic substance. (g) Fluxons or quantized flux loops in

superconductors.

 

 

15.2    Classification of Defects.

 

Depending on the type of defects they are classified as point defects, line defects, plane defects, three-dimensional defects, electronic imperfections. Let us describe these defects in a slightly more detail.

 

 

15.2.1 Point Defects.

 

If the defects are confined to a very small region of not more than a few lattice constants, they are called as point defects. Point defect is almost a Zero-dimensional defect. The type of defects that fall under this category of defects include the following:

 

i.         Simple vacancy or Schottky defect

 

ii.       Frenkel defect

 

iii.     Anti-Schottky defect

 

iv.      Substitutional impurity atom

 

A Schottky defect is just a plain simple vacancy. There are no corresponding interstitial atoms or ions. It is a defect in which an atom or ion is found to be missing from its normal position in the lattice as shown in figure 15.1 .Such defects occur in almost all crystals to some extent.

 

 

If an atom or ion is not at its normal site but is found to occupy a position somewhere in the interstice, it is called a Frenkel defect as shown in the figure 15.2.It, therefore, involves creation of two imperfections – an interstitial atoms or ion and a vacancy. The third kind of defect, the anti-Schottky defect is just a single interstitial atom or ion, without there being any vacancy. When an atom is wedged into a hole, which is generally too small for it, between the atoms which occupy their normal lattice positions creates a defect known as “interstitials” .However if an atom of the crystal is replaced by some foreign atom, the defect thus created is called “impurity”. If a foreign (impurity) atom is present in the lattice either at any interstitial position or at any substitutional position i.e. in place of any regular lattice site, the defect in the crystal falls under the category of “Substitutional impurity atom” as shown in figures 15.3 and 15.4

 

Figure 15.1 : A vacancy in ionic crystal

Figure 15.2: Frenkel defect formed by taking cation away from its lattice site and adjusting it into one of the nearby interstitial sites

Figure 15.3: An interstitial atom or ion

 

Figure 15.4: An impurity atom or ion

 

 

It may be noted that the presence of Schottky defects in a crystal reduces its density while the presence of Frenkel defects does not affect the density of crystals.The lattice will distort to some extent around the interstitial atom and the amount of distortion will depend mainly upon the repulsive energy terms and the space available in the interstices of the crystal structure. In solids, such as closed packed metals, interstitials atoms will have energy of formation. Energy to form Frenkel defect is the sum of the energy to form the interstitial and the energy to form the vacancy. Such defects could be form of the interior of crystals due to thermal vibrations.

 

It should be expected that some, though may be a small number, of the above said point defects to exist in thermal equilibrium and the number should increase as the temperature rises. However, which of these defects predominate will depend upon the relative energies of formation. A substantial number of defects are formed in any one of several ways:

 

1.      By quenching from high temperature

 

2.      By radiation damage, for example, by irradiating with high speed protons or neutrons.

 

3.      By deformation or severe cold working

 

4.      Interstitial foreign atoms are sometimes introduced by diffusion from the surface

 

Vacancies and interstitials have an important role in compound materials particularly ionic crystals. In ionic crystals of a binary compound like NaCl, the various defects have a tendency to be created in pairs so as to preserve the stoichiometry (by way of maintenance of equal proportion of sodium and chlorine). Secondly, the defects of one kind, as for example, creation of only vacancy of one type of ion, viz., Na+ ion or Cl– ion in NaCl would leave the crystal charged or would produce high electric fields between the interior and the surface of the crystal. So, in order to have charge neutrality, positive and negative ion vacancies must occur in pairs. Such an associated defect is neutral and so does not contribute to the electrical conductivity of the crystal. Type of defect involving pair of opposite charges is shown in figure 15.5

Figure 15.5: Schottky defect. Vacancy created by a pair of ions of opposite charge

Figure 15.6: Divalent impurity− vacancy complex in an ionic crystal

 

Another type of defect occurs in ionic crystals when a multivalent impurity ion such as Cd ++ is substituted for a lattice ion. Divalent cations such as Sr++ , Ca++ , and Cd++ etc. are reported to enter the lattice together with positive ion vacancies, thereby neutralizing the extra charge of the impurity. The vacancy may be either free in the crystal or associated with the cation to form a complex as shown in figure 15.6.Some of each type of these defects will generally be present in a real crystal. At any temperature above absolute zero there will be some equilibrium number of these defects. At any given temperature there will be a possibility of an equilibrium condition set up within a solid when the number of defects of any particular type which are created per second is the same as the number of defects of that type which disappear each second. The shape of the curve showing the number of defects present at different temperatures can be shown to be proportional to exp(− E/kT) ,where E is the energy required to create a defect of that type and k is Boltzman’s constant. The dependence of defects on temperature follows a curve of the shape shown in figure 15.7

Figure 15.7: Dependence of number of defects on temperature

 

15.3. Equilibrium number of Schottky & Frenkel Defects

 

Let Ev represent energy required to remove an atom from a lattice site to the surface. Then n Ev is the increase in internal energy associated with the creation of n isolated vacancies. Application of statistical mechanics gives the total number of ways in which we can create these defects:

 

 

Here N stands for total number of atoms in the crystal.

 

From thermodynamics we know that the Helmholtz free energy F is given by:

 

 

S stands for the entropy and is connected to probability and hence, W through an equation:

 

S = k log W

 

Substituting for S in equation 15.2 and using equation 15.1 we have:

 

F= n Ev – kT log.       N!    . …………………..15.3

( N ─ n ) ! n!

Using Stirling ‘s approximation log N! ≈ Nlog N ─ N for very large N and making substitution in equation no.15.3

 

 

F = n Ev ─ kT [ N log N ─ ( N ─ n ) log ( N ─ n ) ─ n log n ]

 

 

(∂F/∂n)T  = Ev  ─ kT[ log ( N ─ n ) ─ log n ]

 

In equilibrium at a given temperature T, the free energy is constant and so,

This equation gives the approximate number of Schottky defects present at any temperature T. One can similarly work out equilibrium number of Frenkel defects in crystals.

 

15.3.1 Equilibrium number of Frenkel Defects

 

Let Ef represent energy required to remove an atom from a lattice site to an interstitial position. n Ef is the increase in internal energy associated with the production of n isolated Frenkel defects. Application of Statistical mechanics provides the total number of ways in which one can create n Frenkel defects:

 

Using the relation between entropy S and the probability i.e., S = klog W

 

 

As already explained that in order to have charge neutrality in ionic crystals, positive and negative ion vacancies must occur in pairs. In the case of Schottky defect occurring in pairs, the expression at no. 15.4 for equilibrium number of defects should get modified to:

 

np≈ N exp ( ─ Ep/2kT ) ……………………..15.9

 

where np stands for the number of vacancy pairs and Ep is the energy of a vacancy pair which is the energy required to remove a molecule ( anion and cation ) from within the crystal and carry it to its surface.

 

For a metal Ep ≈ 1 eV and at T = 1000⁰K ,

np≈ N e─12

or,  np/N ≈ e─12  = 10─5.

 

you can view video on Crystal Defects

References.

  1. Cracknell, A.P. “ Crystals and Their Structures”, Pergamon Press, Oxford(1969).
  2. Brown, F.C. “ The Physics Of Solids”, W.A.Benjamin,Inc.,N.Y.(1967).
  3. Kittel, C. : “ Introduction to Solid State Physics”, Wiley,N.Y.,1971.
  4. McKelvey, J.P. “ Solid State & Semiconductor Physics”, Harper & Row, N.Y.,1966.
  5. Dekker,A.J. “ Solid State Physics”, Macmillan, London ,1958.

 

Interesting Reading.

  1. Ramasamy, R. “ Elementary Solid State Physics”, Laxmi Publishers, Madurai,1971.
  2. Epifanov,G.I.” Solid State Physics”,Mir Publishers, Moscow, 1979.
  3. N.Kato: “ Crystal Imperfection & X-ray Diffraction” in “ Crystal Growth & Characterization”(Eds.Ueda,R& Mullin,J.B.,Proc.ISSCG2 Spring School, Japan,),North Holland/Oxfor Am. Elsevier,N.Y.,1974.
  4. Laudise,R.A. “ Crystal Characterization”( rest of the particulars same as in 3)
  5. Kelly,A., Grovers, G.W.: “ Crystallography & Crystal Defects”,1970.

 

For Additional Information On Defects in Solids

  1. Van Bueren,H.G. “ Imperfections In Crystals”, North Holland, Amsterdam,1960
  2. Proceedings of the conference on Crystal Lattice Defects,Tokyo& Kyoto,March1963,Vol.1,2,3 Published by The Physical Society Of Japan.
  3. Proceedings international School of Physics “Enrico Fermi’vol.XVIII “ Radiation Damage in Solids”,Corso conference1962, Academic Press, N.Y. 1963.
  4. Shockley,W.,Hollomon,S.H.,Maurer,R andSeitz,F.” Imperfections In Nearly Perfect Crystals”, John Wiley,N.Y.1952.