SOLID STATE PHYSICS AND MATERIAL SCIENCE

Transcription

1 M.Sc. (FINAL) PAPER II BLOCK I SOLID STATE PHYSICS AND MATERIAL SCIENCE Writer: Editor: Dr. Meetu Singh Dr. Purnima Swarup Khare

2 SOLID STATE PHYSICS AND MATERIAL SCIENCE Unit -1 Unit- Unit-3 Lattice dynamics and polarization Band theory of solid Magnetism

3 BLOCK -I PAPER II SOLID STATE PHYSICS AND MATERIAL SCIENCE

4 CONTENTS UNIT-1 Lattice Dynamics and Polarization Page No 1. INTRODUCTION OBJECTIVE 3 1. POLARIZATION LORENTZ RIELD IONIC POLARIZABILITY ORIENTATION POLARIZABILITY DEBYE EQUATION FOR GASES THE COMPLEX DIELECTRIC CONSTANT DIELECTRIC LOSSES DIELECTRIC RELAXATION TIME SUMMARY CHECK YOUR PROGRESS 19 Unit- Band Theory of Solid.1 INTRODUCTION. OBJECTIVE.3 KRONIG-PENNEY MODEL.4 EFFECTIVE MASS OF AN ELECTRON.5 QUANTUM FREE ELECTRON THEORY 5.6 FERMI DIRAC STATISTICS, FERMI FACTOR AND FERMI ENERGY 5.7 CLASSIFICATION OF SOLIDS ON THE BASIS OF BAND THEORY 6.8 HALL EFFECT 8.9 BLOCH THEOREM 31.1 SUMMARY 33.11CHECK YOUR PROGRESS 33

5 Unit-3 Magnetism 3.1 INTRODUCTION OBJECTIVE MAGNETIC FIELD AND ITS STRENGTH: MAGNETIC DIPOLE MOMENT: ELEMENTARY IDEAS OF CLASSIFICATION: QUANTUM THEORY OF PARA MAGNETISM: THEORY OF FERROMAGNETISM: QUANTUM THEORY OF FERROMAGNETISM: DOMAIN THEORY OF FERROMAGNETISM: MAGNETIC RESONANCE SUMMARY CHECK YOUR PROGRESS 6

6 1. INTRODUCTION UNIT-1 Lattice Dynamics and Polarization A crystal lattice is consisting of a special long range order. This yield a sharp direction patterns in 3-d. lattice vibrations are important. They contribute in many things like, the thermal conductivity of insulators is due to dispersive lattice vibrations, and it can be quite large (in fact, diamond has a thermal conductivity which is about 6 times that of metallic copper). In scattering they reduce of the intensities, and also allow for inelastic scattering where the energy of the scattered (i.e. a neutron) changes due to the absorption of a phonon in the target. Electron-phonon interactions renormalize the properties of electrons. 1.1 OBJECTIVE Lattice deformation can be studied in detail if one has the knowledge of the dielectric constant. For that, the basic starting point is the Maxwell equations. 1. POLARIZATION According to the dielectric properties, we deal more often with dipoles instead of isolated charges. In electrostatics, as we know that a dipole with charges +e and e displaced by distance d has the dipole moment as p (3.1) ed and the electric field due to this dipole at a point r is 3( p. r) r r ( r) 5 4 r (3.) In case of insulators, under the influence of an electric field the forces acting upon the charges bring about a small displacement of the electrons relative to the nuclei, as the field tends to shift the positive and the negative charges in opposite directions. This is the state of electric polarization, in which a certain amount of charge is transported through every plane element in the dielectric. This transport is called the displacement current. After reaching the state of equilibrium in an applied field, every volume element of the dielectric has acquired an induced dipole moment. The induced dipole moment in a volume element V will be given by p PV (3.3) e d i N V i i Where polarization as N i V is the average number of charges ei with displacement d i. This gives the electric

7 (3.4) P N e d i i i i Alternatively, one can calculate the charge density induced at the ends of the dielectric specimen by the displacement d i. This is simply the amount of charge per unit area which is separated by the displacement from charge of the opposite sign or P (3.5) P i N e d i i i Comparison of these two equations gives (3.6) P l l P The sign of the polarization surface charge is positive, where P is directed out of the body and negative where it is directed in ward. In fact, (3.7) P n P Where n is the unit normal to the surface, drawn outward from the dielectric into the vacuum. The electric field P r ) produced by the polarization is equal to the field produced by the fictitious ( charge density on the surface of the specimen as shown in Fig. 1.1 (a). The total macroscopic field inside the specimen is then (3.8) P Where is the applied electric field.

8 1.3 LORENTZ FIELD The field due to the polarization-induced surface charges on the surface of the fictitious s spherical cavity around the point A (where the field s is to be. Calculated) was investigated by Lorentz. The apparent surface charge density on the part of the spherical surface between and d is (4.1) Pcos Choose the x-axis in the direction of the electric field. The contribution of all the surface charges between and d to the y-and z-contribution of local field cancel each other for reasons of symmetry. The contribution of a surface charge to the x-component of local field is cos and the total surface charge between and d is a sin d. Combining these a results, the electric field at the centre of the spherical cavity of radius a is 1 s 4 loc Pcos.a sind a 3 P (4.) Fig.1.1. Geometry for the determination of Lorentz fields. The Lorentz field of dipoles inside the spherical cavity depends on the crystal structure. In order to evaluate this contribution, we consider a cubic lattice structure and divide the sphere into a very

9 large number of small volume elements of equal size. If all dipoles are parallel to the x-axis and have dipole moment p, then the Lorentz field of dipoles inside the spherical cavity d is given by i i i i i z i i i i i i i x d r z x P r y x Py r r x P (4.3) Lorentz showed that by summing over all points of a sphere that are distributed symmetrically over a sphere, ; 3 4 ; 3 4 ; i i i i i z i i i i i i i r z x P r y x Py r r x (4.4) This gives d Thus, in a cubic lattice the local field according to the Lorentz method of evaluation is 3 3 P P P loc (4.5) For non-cubic lattices, the procedure for evaluation of the local field is not that straight forward. Mueller has worked out d tetragonal and hexagonal lattices. 1.4 IONIC POLARIZABILITY Ionic Polarizability i a is due to the displacement of adjacent ions of opposite sign and is only found in ionic substances. In an electric field the resultant torque lines up the dipole parallel to the field at the absolute zero of temperature. The field produces forces on the charges of opposite sign so that the distance between them is changed by some amount as shown in Fig. 1.. The balance between the electrostatic force and the inter atomic force due to stretching or compressing gives the value of the change in the distance between two ions of Opposite charge.

10 Fig. 1. Ionic Polarization, the field distorts the lattice. From this change in the distance, the ionic-dipole moment and hence the ionic Polarization i can be determined. Like electronic Polarization, ionic Polarization is also independent of temperature at moderate temperatures. 1.5 ORIENTATION POLARIZABILIY Orientation Polarization i occurs in liquids and solids which have asymmetric molecules whose permanent dipole moments can be aligned by the electric field, molecules whose permanent dipole moments can be aligned by the electric field, as shown in Fig.1.3. Let us consider a system of N permanent dipoles, with dipole moment P P, subjected to an external field Parallel to the x-axis. The work required to bring one of the dipole molecules into a position where with, as shown in Fig. 1.4, is given by P makes an angle P (5.1) P P P P cos According to Boltzmann s energy distribution law, the various positions of the dipoles are not equally probable when the uniform field is applied. Without this field the number of dipoles, inclined to x- axis between and d is equal to Fig.1.3 Orientation polarization, the field orients the orients the permanent dipoles.

11 (5.) a sin ad N( ) d A A sind a Where the constant A is determine from the total number of dipoles. If an external uniform electric field is applied, then Boltzmann s law introduces a factor of changing (5.3)into T k B e /, Fig. 1.4 The couple produced on the dipole due to applied field. (5.4) N( ) d A sinde P cos / k T P B cos The x component of each dipole, making an angle with the x-axis, will be P and therefore the x component of all the dipoles within the range and d will be P cosn ( ) d. The net x component P due to all N dipoles will be the sum of equation (5.5) over all angles. Fig.1.5. To calculate number of dipoles in range d as a function of.

12 (5.6) PP B P A sin P cose d coslk T The total number of dipoles N is N (5.7) N( ) d Which provides the value of the constant A. Substituting this un equation (5.6) we get P N Pp sin cosepp coslk T sindepp (5.8) coslk T B d B Use the abbreviations P cos y k T B and P x P k T B P NPp x x x x ye e dy y x y dy NPp x y x ye y x e x x e NPp e x x e e x x 1 1 NPp coth x x x NPpL(x) (5.9) L (x) is called the Langevin function, since this formula was first derived by Langevin in 195. A plot of L (x) is shown in Fig It is obvious that L (x) has a limit unity. If x is very small, the value of L (x) is nearly equal to 3 x. In fact, the expansion is given by

13 . Fig.1.6 Langevin function L( x 1 1 x 3 45 (5.1) x x x 7...) x Therefore, using the approximation L( x), the orientation polarization is 3 P NPp 3k T B (5.11) This gives the orientation l polarization per molecule as Pp 3k T B (5.1) At room temperature, the orientation polarization is of the same order as the electronic polarization 1.6 DEBYE EQUATION FOR GASES The total polarization for dilute gas can now be written as the sum of the above discussed three components as P p e i 3kBT (6.1)

14 Fig.1.7. Curve between ( 1) and (1/T) for a polar gas. If this equation (1.4) is substituted in the Clausius -Mossotti relation (1.6) one obtains r (6.) r r N j e i j 3 P p kbt j This is the Debye equation for the determination of dipole moments and polarization from measurements on gases. There is a very slight difference between r and 3 and therefore, a plot between ( r 1) and (1/T) for a gas will be straight line as shown in Fig The intercept of the 1 line for T provides the value of ) and the slope often line yields Pp and hence. this ( e i unit usually used for the dipole moment is Debye coulomb. 1.7 THE COMPLEX DIELEX DIELECTRIC CONSTANT Till now, we were concerned with the dielectric constant when a dielectric was subjected to a static electric field. Let us now consider the dielectric under an alternating electric field. Two situation exist: (I) when there is no measurable phase difference between the alternating electric field and D and then polarization is in phase with D is a valid relation and (II) when there is a phase difference between D and, then polarization P is not in phase with the alternating field and D is not a valid relation. The basic difference between these two situations is that in the first possibility no energy is absorbed by the dielectric from the electric field, whereas in the second possibility energy is

15 absorbed by the dielectric, which is known as dielectric loss. In order to see this, let us apply an alternating voltage to dielectric between the plates f plane capacitor (7.1) cost The true surface charge density on the capacitor plates, which is equal to D, gives the current density as (7.) D J t In the first case, when the electric displacement D is in phase with, then giving (7.3) (7.4) D D cost J D sin t Thus, the electric current density is out of phase by volume per second of the dielectric is from. The dissipated energy per unit W / J Substituting j and from (1.45) and (1.4) respectively, we get (7.5) W / ( D sint)( cost) dt Thus there is no dissipation of energy when, the electric displacement will then be D and are phase. When they are out of phase by D D cos( t ) (7.6) D cos cost D sin sint

16 Substituting in (1.43) gives current density J D t cos D cos t sin (7.7) This will give the energy dissipated per unit volume per second as / W cost[ D cost sin D sint cos ] dt Dsin (7.8) This is the dielectric loss and therefore the term sin is called the loss factor (or power factor) and the loss angle (or phase angle). In case of phase difference between D and to use complex notation where D is the real part of phase pg, then the ratio between the complex quantities dielectric constant as If the real and imaginary parts of are and on comparison of real and imaginary parts The equation (7.1) will give D i( t ) D e and, D e * is the real part of, it is useful to useful i t e. When i( t ) i t D e and e will give a complex i (7.9) respectively, then equation (1.51) will give, D cos and sin (7.1) tan and (7.11) D Substituting equation (7.1) in equation (7.8) will give dissipated energy per unit volume per second as W (7.1)

17 1.8 DIELECTRIC LOSSES In this section we show that the energy absorbed per second per unit volume (or the energy loss) in a dielectric medium is proportional to the imaginary part " of the dielectric constant. The relationship among vectors E, P and D clearly indicates that on the application of an alternating electric field in a dielectric, relative to that of E. Defining vectors magnitudes as E D t E exp it t where is the phase angle, giving the measure of phase lag. (8.1) D exp i t (8.) In view of (8.1) and (8.) we express the dielectric function in the following form, being aware that it is a complex quantity in the present situation: ' i" t D t (8.3) E On substituting E and D form (8.1) and (8.), respectively, in (8.3) and then rationalizing the obtained relation for, we get D ' cos (8.4) E D sin " (8.5) E Relation (8.6) establishes the frequency dependence of the phase angle. " tan (8.6) ' Let us now take the example of a parallel plate capacitor filled with a dielectric material and bearing a surface charge density t on its plates at any time t. Then the current density in the capacitor at that moment of time is t d dd t jt dt dt D cos sint D sin cost Since j(t) is real physical quantity, only the real part of D(t) is considered in (8.7) The energy dissipated per unit time in one cubic meter of the dielectric is equal to / t Et (8.7) W j dt (8.8)

18 Using (8.6) and (8.1) (taking the real part as W is real), we obtain 1 W " E (8.9) Showing thereby that the energy losses in the dielectric are proportional to Relation (8.9) can also be put in the form ". W 1 sin E D (8.1) The tan, given by (8.7), is often referred to as the loss factor, But this terminology is relevant only when is small, so that tan sin, and the usage may thus be held justified. In the interpretation of optical phenomena it is a practice to use the complex index of refraction n instead of. Therefore, a brief discussion in this regard is very much in order. The development of Maxwell s equation for the electromagnetic field shows that the velocity of electromagnetic waves in a medium is given by 1/ In free space, and are both equal to unity and, therefore, If medium is non-magnetic, I and then. v (8.11) 1 c (8.1) c 1/ v (8.13) Since the ratio c/v by definition equals the index of refraction n. n (8.14) Further the electromagnetic waves in a dielectric medium are described by an electric field. E E exp i t nx/ c (8.15) Where the index of refraction n is complex function Hence, we get n n ik (From 1.8) (8.16) n k ' (8.17) " nk (8.18)

19 The signs of the exponent in (8.18) and in the decomposition of and n are chosen such that " and k (the extinction coefficient have positive signs, i.e. that wave amplitude decreases in the +xdirection. Had we taken a positive sign in the exponent, we would have been required to write ' andn n ikwithk There is nothing sacred about the choice of sign in question as both representations are in common use DIELECTRIC RELAXATION TIME It has already been that the total polarization P in a static field comes from electronic, ionic (together called atomic) and orientation polarization. When a dielectric is subjected to an external static field, a certain time is required for polarization to reach its final value. It is observed that the electronic and the ionic polarization is attained instantaneously, if we consider high frequencies ( sec1) and not the optical frequencies. At these frequencies, the dielectric loss is mainly due to the relaxation effect of the permanent dipoles. Therefore, first we will consider the transient effects, in which this relaxation effect of the permanent dipoles is characterized by a relaxation time and then we go on further to discuss the situation of applying an alternating field. Let suffix s denote the static electric field case, so that equation (8.) D s rs, Ps Ds ( rs 1) (9.1) The total polarization can be written as the sum of only two terms, one in which the polarization is attained instantaneously, denoted by P and the other in which relaxation effects are important, denoted by P os. P P s P os (9.) P ( 1) (9.3) Instantaneously on the application of static field, the polarizations is denoted by P and then let us consider that in time t, Ps part out P os is build up. So that at certain time t, we have Pt P P s (9.4) In general, in relaxation processes, one assumes that the increase d P ) /d t is proportional to the difference between the final value P os and the actual value P s i.e. dps 1 ( Pos Ps ) dt (9.5) Where is a constant, known as relaxation time, a measure of the time log. Integration of (1.59), using the initial boundary condition that at t, P, we obtain s ( s

20 t / Ps Pos(1 e ) (9.6) In case of an alternating field also, it is assumed that equation (1.59) is valid. Denoting for a complex quantity, the super script *, we have 1 * * os Ps * dps [ P ] (9.7) With the help of equations (9.5), (9.6) and (9.7), * * it Pos Ps * P ( rs ) e (9.8) Substituting (9.3) in (9.4) and integrating, we obtain * t / ( rs ) i t Ps Ce e 1 i (9.9) After some time, the first term on the right hand side is small that it can be neglected. Now, the total polarization in an alternating field is (9.1) This will give the displacement as ( ) 1 i * rs i t E P ( 1) e ( ) 1 i (9.11) thus, the complex dielectric constant in case of alternating field, is ( rs ) * 1 i (9.1) Separation into real and imaginary parts provide rs i D* * P* e ( rs ) ( ) (1 ) (9.13) ( rs ) ( ) (1 ) (9.14) These equations are often referred to as Debye equations. These can be rewritten as ( ( ) ) 1 ( rs ) (1 ) (9.15) ( ) and ( ) (1 ) rs (9.16) The right hand side of these equations is plotted as a function of in Fig It should be mentioned that these relations are in satisfactory agreement with the experimental observations.

21 Fig.1.8. Frequency dependence of the real and the imaginary part of the dielectric constant. 1.1 SUMMARY:- This chapter explain the concept of Polarization, which is a property of certain types of waves that describes the orientation of their oscillations and show that Ionic polarization occurs when an electric field is applied to an ionic material then cations and anions get displaced in opposite directions giving rise to a net dipole moment. The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. This chapter describe the Debye equation for the determination of dipole moments and polarization and derive the formula for dissipated energy per unit volume per second. Next part shows that the energy loss in a dielectric medium is proportional to the imaginary part " of the dielectric constant. When a dielectric is subjected to an external static field, a certain time is required for polarization to reach its final value. At high frequencies, the dielectric loss is mainly due to the relaxation effect of the permanent dipoles.

22 1.11 CHECK YOUR PROGRESS Q. 1. Define Polarization and Lorentz field Hint: - Refer to topic no. 1. & 1.3 Q.. Explain Orientation Polarizability Hint: - Refer to topic no. 1.5 Q. 3. Explain Debye Equation for Gases Hint: - Refer to topic no. 1.6 Q. 4. Drive the Complex Dielectric Constant Hint: - Refer to topic no. 1.7

23 Unit- Band theory of solid.1 Introduction During the discussion of the free electro theory of metals, the conduction electrons behave like a classical free particle of a gas obeying Fermi-Dirac statistics. But this could not be made clear that why in metals the electrical conductivity is quite low. Band theory describes the behavior of electrons in solids, by postulating the existence of energy bands. It uses a material's band structure to explain many physical properties of solids, such as electrical resistivity and optical absorption. A solid creates a large number of closely spaced molecular orbital, which appear as an energy band.. Objective It is the concept of electronic energy bands which provides the basis for the classification of solids as good conductors, semiconductors and poor conductors of electricity..3 Kronig-Penny Model The free electron model of solids considered the electrons to be free inside the solid. It was able to explain the electrical and thermal conductivity of metals but could not explain the same for semiconductors and insulators. Kronig and Penney considered the electrons to be moving in a variable potential region in the crystal instead of being free. The potential was approximated by a square well periodic potentials as shown in Fig..1. Fig..1 The time independent Schrodinger wave equation in one dimension is, d m E V dx Or d E V m dx In region < x < a, where v =, the general solution for wave equation is, ikx ikx Ae Be (3.1) 1 and the energy is, K E.(3.) m

24 in the region b < x <, V = V. The solution of wave equation is, Qx Qx Ce De..(3.3) The continuity of wave function at x = requires that the values of and be equal at this at this point...(3.4) d is also continuous at x = dx 1 x x A B C D d 1 dx x d dx x ik A B QC D...(3.5) By Bloch theorem, ikab a x a b b x e Where k is the wave vector. The condition for continuity of at x = a with Bloch theorem is ikab e 1 x xb ika ika Qb Qb ikab Ae Be Ce De e d For continuity of at x = a, dx (3.6) d 1 d. e dx dx x ika ika Qb Qb ikab ik Ae Be QCe De e xb ik ab..(3.7) Equation (3.4), (3.5), (3.6) and (3.7) have solutions only if the determinant of coefficients of A, B, C and d vanishes. This leads to the condition. P sin ka cos Ka cos ka.. Ka (3.8) Q ba Where P The R.H.S of equation (3.8) is cos ka which lies between 1 and + 1. Hence solutions are obtained P only when the L.H.S lies between 1 and +1. The graph of sin ka cos Ka cos ka plotted for Ka different values of Ka is shown in Fig... In regions of solution for do not exist. The corresponding energies which are forbidden can be obtained using equation (3.). Thus, there are bands of allowed energies, which are separated by energies which are not allowed and hence known as forbidden bands.

25 Fig.. The graph of energy E for different wave vectors k is shown in Fig..3. There are discontinuities at P k,,... which correspond to the condition that sin Ka cos Ka 1. a a Ka Thus according to Kronig-Penney model, the motion of electrons in a periodic potential in crystals gives rise to certain allowed energy bands separated by forbidden energy bands..4 Effective Mass of an Electron Fig..3 An electron in a crystal is not free. When an external field is applied, the electron in a crystal behaves as if it had a mass different from its actual mass. Consider an electron described as a wave packet having wave function in the region of wave vector k. Let the electron be in a crystal to which an electric field is applied. The group velocity of the wave packet will be d vg (4.1) dk If E is the energy of the wave packet,

26 E hv h.v E E v g d dk v g 1 d E dk dv dt dv dt g g 1 d E dk dt 1 d E dk dk dt (4.) The work done de by the electric field E in a time interval dt is de = - e E v g dt (The force on the electron is e E and the displacement is v g dt) As de = (4.3) and From equation (4.1) (4.4) From equations (4.3) and (4.4), E vg de v g dk dk edt de dkdk dk e dt (4.5) The above equation describes the force e E due to an external field E on an electron in terms of the rate of change of wave vector k. Hence we can write dk F dt dk dt F Substituting in equation (4.) dv g 1 d E F. dt dk dv F d E dt dk g (4.3)

27 dv g As has dimensions of acceleration, the quantity is defined as the effective mass (m * ) of dt d E dk an electron, m* d E dk (4.7) d E Thus the curvature of the energy band decides the effective mass of an electron in a crystal. dk d E The curvature is negative for an electron at the top of the valence band and positive at the dk bottom of conduction band as shown in Fig..5. Hence the effective mass of an electron is negative near the top of valence band and positive near the bottom of conduction band. The motion of valence band electrons with negative charge and negative mass can be equivalently described by motion of holes having positive charge and positive effective mass in same direction. For a free electron, P k k E m d E dk m E m Fig..5 P h h P From equation (4.7) m* = m i.e., the effective mass is same as its mass for a free electron..5 Quantum free Electron Theory

28 The quantum free electron theory developed by Summerfield retained some of the assumptions of the classical free electron theory and introduced a few new assumptions. The quantum free electron theory was successful in eliminating certain drawback of the classical theory. The assumptions in quantum free electron theory are listed below. Assumptions 1) The valence electrons are free to move inside the metal. ) The electrons are confined to the metal by potential barrier at the boundaries. The potential is constant inside the metal. 3) The electrostatic forces of attraction between the free electrons and the ion cores are negligible. 4) The electrostatic forces of repulsion amongst the free electrons are negligible. 5) The energies of electrons are quantized and the distribution of electrons in the allowed discrete energy levels is according to Pauli s exclusion principle which prohibits more than one electron in single quantum state..6 Fermi Dirac Statistics, Fermi Factor and Fermi Energy Different types of particles have different probabilities of occupying the available energy states. Statistically, there are three different types of particles: i) Identical particles which are so far apart that they can be distinguished and their wave function do not overlap. The Maxwell Boltzmann distribution function is applicable to such particles. For example, molecules of a gas. ii) Identical particles with or integer spins with overlapping wave functions which cannot be distinguished. Such particles are called bosons and obey the Bose Einstein probability distribution for energy. For example, photons. iii) Identical particles for which the spin is an odd integer multiple of half 1 3 5,,... which cannot be distinguished form one another. These particles are called fermions and obey the Fermi - Dirac probability distribution function. Electrons are example of this type. The Fermi Dirac probability distribution function, also known as Fermi function, is 1 f ( E) ( EE 1 e F ) / kt (6.1) Where f (E) Probability of an electron occupying the energy state E E F k = Fermi energy = Boltzmann constant and T = Absolute temperature For T = K, if E > E F 1 f ( E) 1 e 1 1 f ( E) i.e. no electron can have energy greater than the Fermi energy at K, It means that all energy states above the Fermi energy are empty an K.

29 For T = K, if E < E F, 1 f ( E) 1 e 1 1 f ( E) 1 i.e. all electrons occupy energy states below the Fermi energy at K. Thus, all energy states below Fermi energy are filled and energy states above Fermi energy are empty at K. Hence Fermi energy as the highest occupied energy state at K. For T > K, if E = E F, 1 1 f (E F ) = 1 e 11 1 f (E F ) = i.e. the Fermi energy level represents the energy state with a 5 % probability of being filled if forbidden gap does not exist as in the case of good conductors. The Fermi functions described by equation (6.1) is shown for different temperatures in Fig..6 Fig..6 Valence band: it is an energy band which contains the outermost valence electrons. Conduction band: it is an allowed energy band next to the valence band which contains free electrons that take part in conduction. Forbidden band: it is an energy band between the valence and conduction band. The energies in this band are forbidden, i.e. not allowed, for the electron. To raise the electrons from valence band to conduction band, energy equivalent to the forbidden energy gap has to be supplied to the electrons..7 Classification of Solids on the Basis of Band Theory Solids can be classified into conductors, insulators and semiconductors based on their energy band structure 1) Conductors: In conductors, the valence band and the conduction band overlap. There is no forbidden band. The electrons can be made to move and constitute a current by applying a small potential different. The resistivity of conductors is very low and increases with temperature. Hence the conductors are said to have a positive temperature coefficient of resistance. The energy band structure is shown in Fig..7 Metals like copper, silver, gold, aluminum etc. are good conductors of electricity.

30 Fig..7 ) Insulators: insulators have a completely filled valence band and an empty conductions band which are separated by a large forbidden band. The band gap energy is large (of the order of 5 ev). Hence large amount of energy is required to transfer electrons from valence band to conduction band. The insulators have very low conductivity and high resistivity. Diamond, wood glass etc. are insulators. 3) Semiconductors: In semiconductors, the valence band is completely filled and the conduction band is empty at absolute zero temperature. The valence band and conductions band are separated by a small forbidden band of the order of 1 ev. Hence, compared to insulators, smaller energy is required to transfer the electrons from valence band to conduction band. Hence the conductivity is better than insulators but not as good as the conductors. Silicon and germanium are semiconductors having band gap energies of 1.1 ev and.7 ev respectively. Some compounds formed between group III and group V elements like gallium arsenide (GaAs) are also semiconductors. As temperature is increased, elements from valence band jump to conduction band leaving a vacancy in valence band which is known as hole. The free elements in conduction band and the holes in valence band take part in conduction. Hence conductivity increase and resistivity decreases with increase in temperature. The semiconductors are said to have a negative temperature coefficient. Conductivity in a Semiconductor In a semiconductor, the current is due to free electrons as well as holes. The current due to electrons can be written as, I e = n e a v e where v e = Drift velocity of holes n = number density of holes Similar the current due to holes is I pea Where = Drift velocity of holes and p = number density of holes The total current is I I I The current density Also, e I ea n p J 1 a J ene p J E e en p E E e

31 e p, the mobility of electrons E and p, the mobility of holes. E e n p e For intrinsic semiconductors, n = p = n i is called the density of intrinsic charges carries. n For n-type semiconductors, n >> n p ne e As each donor atom contributes one free electron, n is also the density of donor impurity atoms. For p-type semiconductors, p >> n pe p Where nh is the density of holes which is same as the density of acceptor impurity atoms. p i e e p.8 Hall Effect When magnetic field is applied perpendicular to direction of current in a conductor, a potential different develops along as axis perpendicular to both current and magnetic field. This effect is known as hall effect and the potential difference developed is known as Hall voltage. Force on a charge q moving with velocity due to a magnetic field B is given by, F q B (8.1) for an electron, q = - e F e B (8.) F eb The forces on positive and negative charge carriers and the corresponding Hall voltages developed are shown in Fig (a) and (b) respectively. The magnetic field is directed into the plane of the paper and the current is flowing upwards.

32 From Fig.6.14 (a) and (b) it is clear that opposite polarity of hall voltage will be developed for the types of charge carries for the same direction s of current and magnetic field. Therefore this effect can be used to find the polarity of charge carriers and hence to find whether a given semiconductor is p- type or n-type. Hall voltage and Hall coefficient Consider a conductor of rectangular cross section of dimensions w d in which current I flows along x-axis, magnetic field is applied along z-axis and Hall voltage develops along y-axis which is measured across terminals 1 and as shown in Fig..8. Fig..8 The dimension w is parallel to the direction of magnetic field and d is parallel to the axis along which hall voltage develops. Let V H = Hall voltage and E H, the corresponding electric fields and = Drift velocity of charges Under equilibrium conditions, force on charge carriers due to magnetic field will be balanced by the force on them due to E H. qe qb From equation (8.1) E E H H V d V H H B V d H B Bd (8.3)

33 Substituting is equation (8.3), I nqa V V H I nqa IBd nqa a d H IB nqw 1 The quantity is the reciprocal of charge density and is defined as the Hall coefficient RH, nq 1 R H (8.6) nq From equation (8.6) IBd V H RH a (8.7) As V H, B, d and a are measurable quantities, RH and hence charge density nq can be determined using equations (8.7) Once charge density is known, we can determine mobility of charge carriers using nq Conductivity can be determined using 1 l Ra Thus the Hall Effect can be used to determine i) Whether charge carriers are positive or negative which in turn determines whether semiconductor is n-type or p-type. ii) Density of charge carriers iii) Mobility of charge carriers..9 BLOCH THEOREM In the quantum mechanical description of an electron in a crystal, a realistic view is of a single electron in a perfectly periodic potential which has the periodicity of the crystal. The Bloch theorem defines the form of the one electron wave functions for this perfectly periodic potential. For simplicity, we consider one dimensional crystal of lattice parameter a, shown in fig.9, with the potential energy of the electron (x) being periodic with period a i.e. ( x) ( x a) (9.1) The Schrodinger equation of an electron moving in one dimensional electrostatic potential field with potential energy (x) is d m [ E (x)] (9.) dx Since (x) is periodic, the solution of equation (9.) can be easily written if we solve a general differential equation d f ( x) ( x) (9.3) dx Where f (x) has a period a i.e.

34 f ( x) f ( x a) (9.4) Since equation (9.3) represents a second order differential equation, it will have the general solution as Fig..9 Potential in a perfectly periodic crystal Surface potential barrier is shown at the ends. ( x) Cg ( x) Dh( x) ( 9.5) Where g (x) and h (x) are solution of equation (9.3), Also g (x + a) and h (x + a) will be solutions of equation (8.3) because f (x )= f (x + a). These solutions g(x + a) and h (x + a) also can be expressed as a linear combination of g (x) and h (x) equation (9.5), as g( x a) A1 g( x) B1 h( x) h( x a) A g( x) Bh( x) (9.6) Substitution in equation (9.5) will give ( x a) ( CA1 DA ) g( x) ( CB1 BD ) h( x) (9.7) Since ( x a) can always be expressed in form ( x a) ( x) (9.8) Where is a constant, Comparing (9.7) and (9.8), we get C( A1 ) DA CB 1 D( B ) (9.9) Solution of equation (9.9) is the solution of the determinant E or ( A1 B ) ( A1 B A B1 ) (9.1) This quadratic equation (9.1) gives two values of as 1and Now if these constants 1 and are taken as a e ik 1 1 and e ik a (9.11) and let us define u 1 (x) and u (x) as k 1x u ( x) e ( ) 1 x k x u( x) e ( x) (9.1) then, use of equation (9.11) and (9.8) yields k ) 1 ( xa) k1 ( xa) u x a e ( x a) e ( x ) 1( 1

35 k 1 ( xa) ik1a k1x e e ( x) e ( x) u1( x ) (9.13) Similarly, u ( ) will be periodic with period a. equation (9.1) can be rewritten in the form x ikx k ( x ) uk ( x) e (9.14) Where u k (x) has the same periodicity as the (x). This is Bloch s function, which on extension to three-dimensional case is i k r k ( r ) uk ( r) e (9.15) and the Bloch theorem can be stated that has the same form as a plane wave of vector k modulated by a function u ( k r ) that depends on k and has the periodicity of crystal potential. Let us now try to find the probability density * using the Bloch function given by equation th (9.14). In the N unit cell, ikx( Na) k ( x Na) e uk ( x Na) (9.16) ikna ikx e e u (x) [From equation (9.13)] k ikna e k (x) [From equation (9.16)] Similarly * ikna * k ( x Na) e k ( x) (9.17) * * This gives k ( x Na) k ( x Na) k ( x) k( x) (9.18) So we obtain the same probability density in each unit cell of the crystal. The same is true for a three dimensional wave function. If the crystal is finite, as the practical case is, then suitable boundary conditions must be satisfied at the surfaces. For example, in a crystal of N atoms, if the wave function has to be single valued, then we must have from equation (9.16) ikna or ( x Na) e ( x) ( x) k ikna e 1 n or k n, n, 1,, N Na (9.19) So the solutions, which satisfy the Schrodinger equation, are found only for certain discrete energy Eigen values corresponding to values of k n given by equation (9.19). Since N is large, there will be many allowed values of kn and they may be thought of forming a quasi-continuous range, hence the notion of bands of energy Eigen values in solids..1 SUMMARY:- This chapter describes the behavior of electrons in solids. Kronnig penny model describe that the motion of electrons in a periodic potential in crystals gives rise to certain allowed energy bands separated by forbidden energy bands. Also show the concept of effective mass according to which mass of the electrons changes inside the solids due to interaction of electron with atoms. This chapter k k

36 includes the classification of solids on the basis of band theory, which explains how some solids are insulator some are semiconductor and other are metals. Hall Effect can be used to find the polarity of charge carriers and also helped on finding which type of semiconductor we have used. Bloch theorem defines the form of one electron wave function for perfectly periodic potential..11 CHECK YOUR PROGRESS Q. 1. Explain Kronig-Penney Model Hint: - Refer to topic no..3 Q.. Explain the concept of Effective Mass of an Electron Hint: - Refer to topic no..4 Q. 3. Explain Quantum Free Electron Theory Hint: - Refer to topic no..6 Q. 4. Explain what do you understand by Hall Effect in detail and what is Hall Coefficient. Hint: - Refer to topic no..8 Q. 5. Explain Bloch Theorem Hint: - Refer to topic no..9

37 UNIT 3 MAGNETISM 3.1Introduction The phenomenon of magnetism attracts everybody. The following aspects of magnetism are generally familiar to you- A compass needle always points north, an observation reportedly made around 5 BC by the Chinese. The stickers or alphabets with magnet sticks on the iron fridge or cupboard but falls down from the aluminum window frames or copper, stainless steel objects. Magnets have south and north poles. The like poles repel and unlike poles attract. The magnetism is produced by the electrical current in a solenoid or by an electronic revolution in a permanent magnet i.e. always due to charge in motion. The magnets have wide range of applications starting from a minute magnetism generated by our brain, heart waves to huge magnets used in dock yards or particle accelerators. In our day-to-day life we encounter with audio-video tapes, computer disks, motors, generators etc. 3. Objective Define or explain the magnetism, you will find it difficult to put in proper words. St the post-graduate level, let us just review some of the basic concepts learned by you during the college courses. 3.3 Magnetic field and its strength: One of the most fundamental ideas in magnetism is the concept of magnetic field. A field is generated whenever there is a change in the energy within a volume of space. In a most familiar way the presence of the field is sensed by the forces (attractive or repulsive) or by the torque. Thus, the attractive force on magnetic stickers and the torque on compass needle are the manifestation of the magnetic field. The region of space where the force or torque is experienced is known as magnetic field. A magnetic field is produced whenever there is electrical charge in motion. It was first observed by H. C. Oersted in the year 1819 that the electric current flowing in a conductor produces certain force. In case of permanent magnets, there is no conventional current. However, the electrons orbiting around nucleus and spinning around them create so called Ampere currents. These currents are responsible for the magnetism therein. Although the electrons are mandatory constituents of all materials, the magnetism is not exhibited by all of them. Few materials have adhoc magnetism very few have permanent magnetism. The reasons for this variation will be clear to you as we proceed through this course. The magnetic force is expressed in terms of the magnetic field strength (H). Its magnitude, obviously, depend on the current, length of current carrying conductor and the distance at which it is measured. Thus, for elemental conductor, the magnetic field strength is given by 1 H. il u (3.1) 4r

38 Where i is the current in ampere flowing through an elemental length 1of a conductor, r is the radial distance and u is the unit vector. There for the field strength is A/m. Magnetic Flux ( ) : In a conventional way, the presence of magnetic field is indicated by the magnetic flux lines, as shown in Fig Fig. 3.1 The flux lines are closed loops i.e. there is no source or sinks of magnetic flux. The magnetic flux is measured in terms of Weber. The way magnetic field is created by the current, the changing magnetic flux can generate e.m.f. Thus the Weber is defined as the amounts of magnetic flux which, when reduced to zero in one second produces an e.m.f. of 1 volt in a turn of coli. Magnetic Induction (B) Whenever magnetic field is generated in a medium, it responds in a certain way. As a result some induction is shown by the medium. The magnetic induction can be defined in terms of flux density. According, the flux density of one Weber per Square meter is equivalent to the magnetic induction of one Tesla. Alternatively, the magnetic induction is said to be one Tesla, when a force of one Newton per meter is generated by one ampere current in a perpendicular direction. Generally, for a nonmagnetic media the induction is proportional to the applied field strength. i.e. B= H (3.) Where, is known as the permeability. The permeability of a free space ( ) is a universal constant 7 having value 4 1 Henry/m. For the magnetic media, the equation (3.) is not valid as the response of the material is modified through a quantity called Magnetization. 3.4 Magnetic Dipole Moment:

39 The electric charge is the fundamental unit of electricity. We conveniently indicate the flow of charge through a completed circuit, where we assume a source and link of charge. In case of magnetism, we adopt a pole view. Note that the pole is a fictitious just conceived for the simplicity. Any smallest magnet has a south and a north poles. Thus we cannot have a monopole like the charge. Instead, the dipole is the fundamental unit of magnetism. A closed current loop having area a and current I, generates magnetic dipole moment given by- m =i.a (4.1) The dipole moment is always directed perpendicular to the plane of loop as shown in Fig.3.. The unit of magnetic dipole moment is A.m. In individual atoms, the magnetic dipole moments are due to angular, spin motions of electron as well as spin motion of nucleus. Unless these moments cancel each other, each atom will behave as a magnetic dipole. Magnetization (M): In general, the magnetic dipoles inside a material are oriented randomly and there is no (or very less) net magnetic moment. When external magnetic field is applied, these dipoles respond by aligning themselves along the field direction. Then there can be bet magnetic dipole moment. The number of such magnetic moments per unit volume is termed as magnetization. Thus, M=N m /V (4.) From equation (4.) the unit of magnetization is A/m. Now, the total number of magnetic flux lines will have two contributions: one from applied field (H) and second from magnetization (M). The magnetic induction in a free space as per equation 4. is. Similarly the induction due to the magnetization will be B (4.3) M ( H M H M The quantity M H. Therefore: the net magnetic induction is- ) = 1 is often termed as magnetic polarization or intensity of magnetization. It is noteworthy that the units H and M are the effect of magnetic field on magnetizations whereas B is more convenient for the effect on currents. The distinction between B and H is really important hen magnetic materials are present. Magnetic Susceptibility: In the presence of the magnetic field, different materials respond differently. It is mostly depends on the presence and alignment of the magnetic dipole moments within. As we increase the strength of applied magnetic field, more dipoles will be aligned or even some more will be created. It means. M H M H i.e. M / H.

40 The proportionality constant () or the ration of magnetization to the magnetic field strength is knows as magnetic susceptibility. Since, M and H have the same unit is a unit-less quantity. It is the basic parameter on the basis of which the materials are classified. 3.5 Elementary ideas of classification: According to the classification of magnetic materials diamagnetic, Paramagnetic and ferromagnetic is based on how the material reacts to a magnetic moment induced in them that opposes the direction of the magnetic field. This property is now understood to be a result of electric currents that are induced in individual atoms and molecules. These currents produce magnetic moments in opposition to the applied field. Many materials are diamagnetic: the strongest ones are metallic Bismuth and organic molecules, such as benzene, that have a cyclic structure, enabling the easy establishment of electric currents. Paramagnetic behavior results when the applied magnetic field lines up all the existing magnetic moments of the individual atoms or molecules that makes up the material. This results in an overall materials moment that adds to the magnetic field. Paramagnetic materials usually contain transition metals or rare earth elements that possess unpaired electrons. Para magnetism in non-metallic substances is usually characterized by temperature dependence; that is, the size of an induced magnetic moment varies inversely with the temperature. This is a result of the increasing difficulty of ordering the magnetic moments of the individual atoms along the direction of the magnetic field as the temperature is raised. A ferromagnetic substance is one that, like iron, a magnetic moment even when the external magnetic field is reduced to zero. This effect is a result of a strong interaction between the magnetic moments of the individual atoms or electrons in the magnetic substance that causes them to line up parallel to one another. In ordinary circumstances, ferromagnetic materials are divided into regions called domains; in each domain, the atomic moments are aligned parallel to one another. Separate domains have total moments that do not necessarily point in the same direction. Thus, although an ordinary piece of iron might not have an overall magnetic field. Therefore aligned the moments of all the individual domain. The energy expended in reorienting the domains from the magnetized back to the demagnetized state manifests itself in a lag in response, known as hysteresis. Ferromagnetic materials, when heated, eventually lose their magnetic properties. This loss becomes complete above the Curie temperature, named after the French physicist Pierre Curie, who discovered it in (The Curie temperature of metallic iron is about 77 o C/1418 o F.) In recent years, a greater understanding of the atomic origins of magnetic properties has resulted in the discovery of types of magnetic ordering. Substances are known in which the magnetic moments interact in such a way that it is energetically favorable for them to line up anti-parallel; such materials are called anti ferromagnetism. There is a temperature analogous to the Curie temperature called the Neel temperature, above which anti ferromagnetic order disappears. Other, more complex atomic arrangements of magnetic moments have also been found. Ferromagnetic substances have at least two different kinds of atomic magnetic moment, which are oriented anti-parallel to one another. Because the moments are of different size, a net magnetic moment remains, unlike the situation in an anti ferromagnetic, where all the magnetic moments cancel out. Interestingly, lodestone is a ferromagnetic rather than a ferromagnetic; two types of iron ion, with different magnetic moments, occur in the material. Even more complex arrangements have been found in which the magnetic moments are arranged in spirals. Studies of these arrangements have provided much information on the interactions between magnetic moments in solids. A representative list of various types of magnetic materials is given in Table 3.1

41 Table 3.1 Types of magnetic materials

42 Theory of Paramagnetism: Atoms and ions with unfilled shells have non-zero magnetic moments, which, may be aligned by a magnetic field. This alignment is off-set by the randomizing action of thermal agitation and the analysis of these competing processes leads to an expression for magnetic susceptibility as a function of temperature. Before the advent of the quantum theory Langevin analyzed this problem classically, this entails considering that all orientations are possible in an applied field. This Langevin analysis is applicable to the description of the magnetic behavior of systems containing units, which large values of magnetic moment. In fact, there are number of possible explanations for the paramagnetic behavior. These are mainly, 1. Langewin s theory of non-interacting magnetic moments.. Van-vlack model of Localized moment. 3. Weiss theory of molecular field. 4. Pauli s model of paramagnetic. 5. Quantum theory of paramagnetic. Besides, there are some laws based on the experimental observations like Curie law and Curie-Weiss law, which indicate the temperature dependence of the susceptibility. Here, we will discuss only the quantum theory of paramagnetic. 3.6 Quantum theory of Para magnetism: Unlike the classical theories, the quantum theory of par magnetism is based on the assumption that the permanent magnetic dipole moments are not free rotating but are restricted to a finite set of orientations relative to the applied field. Let N be the number of atoms per unit volume and J be the total angular momentum quantum number such that J = L + S with L, S as orbital and spin quantum numbers respectively. The magnetic moment of an atom is proportional to the total angular momentum J i.e. (6.1) J J Where, is called gyro magnetic ratio and is given by- (6.) (6.3) g B J( J 1) S( S 1) L( L 1) g 1 J( J 1) Where, g is Lande s g factor or spectroscopic splitting factor given by, and Magnetron B =eh/m is the Bohr (6.4) Thus, J g B J

43 In the presence of magnetic field H, the magnetic moment J will presses about the field direction such that the resolved component of the magnetic moment in field direction is M j g B where M J is magnetic quantum number having values M J =-J,-J-1,-(J-),,1,, J-,J-1,J. The potential energy will be- H g M E B J (6.5) The average value of the magnetic moment in the field direction is given by- kt E kt E j j j j j ava / exp / exp (6.6) J j B j B j J j B J kt H g M kt H g M g M ) / exp( ) / exp( (6.7) At normal temperature, T K H g M B B J i.e. 1 / T K H g M B B J Or, ) / exp( T K H g M B B J =1+ T K H g M B B J / Therefore, J J T k g M J J T k g M B J ava B B J B B J g M ) (1 ) (1 J J J J J B B J J J J J B B J B M T k H g M T k H g M g 1 But, J J J J J J J J J J J M M J 3 1) 1)( ( ; 1; 1 T k J H J g B B ava 3 1) (. (6.8)

44 Therefore, total magnetization due to N number of atoms is M=N ava (6.9) M N g H. J( J 1) B 3k B T The paramagnetic susceptibility will be M H (6.1) N g J( J 1) B. 3k B T or (6.11) NB. 3k T B C T where g. J( J 1) is known as Effective Bohr Magnetron number Thus, the susceptibility has form C/T and C= NB / 3KB is known as the Curie constant. The equation (3.17) is found true in the cases of monatomic gases. However, distinct discrepancies arise for the transition group elements. According to van Vleck, it may be due to the fact that all atoms may not have the same values of L, S and J. At low temperature or strong fields, the situation will be rather different. In this case the magnetization will be given by- N M (6.1) j j M j j J g exp exp B M J M Now let gbh / KBT, Then B J g H / kt B g H / kt M Ng j B j J J M J exp( M exp( M J x) J x) J d ( Ng B) log e exp( M J x) dx J Using the values of M J =J,J-1,J-,.,-(J-1),-J d dx J x ( J 1) x Jx M ( Ng B) log e( e e... e

45 )... (1 log ) ( Jx x Jx e B e e e dx d ng x x J Jx e B e e e dx d Ng 1 1 log ) ( 1) ( x x Jx Jx e B e e e e dx d Ng 1. log ) ( ) / sinh( 1 sinh log ) ( x x J dx d Ng e B ) / sinh( log 1 sinh log ) ( x x J dx d Ng e e B ) / coth( 1 1 coth 1 ) ( x x J J Ng B ) / coth( 1 1 coth 1 ) ( J y J y J J J J JNg B Jx y where... (y) B NgJ j B Where ) / coth( 1 1 coth 1 ) ( J y J y J J J J y B j is called Brillouin function and y=jx=jg B H/K B T. This is a general equation for the par magnetism and the equation (6.1) is a special case for low field and normal temperature. The Brillouin function varies from zero when the applied field is zero to unity when the field is infinite. The saturation value of the magnetization is M s =NgJ B 3.7Theory of Ferromagnetism: In diamagnetic materials, the magnetic moments are induced by the application of external field whereas in paramagnetic materials, already exist ion dipoles are aligned in the field direction. In ferromagnetic materials, the dipoles exist and oriented even in the absence of external field. The spontaneous existence of magnetic dipoles can be attributed to the uncompensated electron spins. For example, Fe with atomic number 6 has electronic configuration- 1s s p 6 3s 3p 6 3d 6 4s. These electrons are arranged in various orbitals in accordance with the Hand s rule as follows-

46 Note that in 3d orbital 6 electrons are arranged in such a way that two electrons are paired with spin up and down while the other four electrons are in spin up configurations. The paired electrons cancel magnetic moments of each other. However, net spin magnetic moments of 4 Bohr magnetrons is always present due to the 4 unpaired electrons. In the bound states of atoms, the net spin magnetic moments are affected due to the proximity of other atoms. As a result, the average spin moment is reduced to. Bohr magnetrons. This magnitude of the magnetic moment is of the same order of the paramagnetic materials. It means that the large magnetization of ferromagnetic substance is not only due to the moments of individual atoms. There are various theories of ferromagnetism based on two mutually exclusive approaches- 1. Localized moment model. Itinerant electron model. The localized moment model assumes that the magnetic moments of atoms are due to electrons localized to that particular atom and the magnetic properties of the solids are merely the perturbation of the magnetic properties of the individual atoms. The theories based on this approach include Weiss Mean Field theory, Weiss Domain theory, Heisenberg s model of Exchange interaction and Quantum theory of Ferromagnetism. The approach works well for the rare earth metals. However, for the elements of 3d series eg. Fe, CO,Ni) the outer electrons are relatively free to move through the solid. In such cases, the itinerant electron model is more realistic. The Pauli s free electron theory and Slater s Band theory are examples of this second approach. The fundamental calculations are extremely difficult with the itinerant electron theories. therefore, in spite of its realistic nature, they are less preferred and the interpretations of magnetic properties are more conveniently made on the basis of localized moment models. 3.8 Quantum theory of Ferromagnetism: A paramagnetic material can behave as a ferromagnetic, if there is some internal interaction to alight the magnetic moment. Weiss proposed such internal field that couples the magnetic moment of adjacent atoms. Such interaction is called the exchange or Molecular or Weiss Field (B E ). The orientation effect of this field is opposed by the thermal agitation. At elevated temperature the alignment is destroyed completely and the material becomes paramagnetic. According to Weiss Mean Field approximation, exchange field is proportional to the magnetization. B E M or M (8.1) B E Where is known as Weiss constant, which determines the strength of interaction between magnetic dipoles and it is temperature independent. Thus, each magnetic moment experiences a field due to magnetization (alignment) of all other magnetic moments. Therefore if B is the applied magnetic field, then the total field will be B T = B + B E or H T = (H+ M) (8.) Now, the quantum theory of ferromagnetism can be derived from the quantum theory of par magnetism. A perturbation in the form of exchange field M has to be introduced in this case. According to the quantum theory of paramagnetic, the energy of electron in the magnetic field B T will be E = -M j g B T. Thus, with the perturbation term of the exchange field the energy is, B

47 E = -M j g B (H+ M) (8.3) Moreover, the magnetization at normal temperature i.e. in the limit E<< K B T will be M Ng B. J( J 1).( H M ) (8.4) 3K T B Ng 1 B. J( J 1) M 3KBT Ng B. J( J 1). H 3K T B Therefore the ferromagnetic susceptibility, M Ng BJ.( J 1) C (8.5) H K T Ng J.( J 1) T T B This equation is similar to the Curie Weiss Law with B C Ng J.( J 1)/ 3 B K B e and T Ng J.( J 1)/ 3K B B Thus, the quantum theory also leads qualitatively the similar results to the classical theory. 3.9 Domain Theory of Ferromagnetism: One of the most celebrated theories of ferromagnetism is the domain theory. It was originally proposed by Weiss in the year and was based on the ideas of Ampere, Weber and Ewing about magnetism. It can be understood through the concept of domains the origin of domains. The concept of Domains: According to Weiss proposal, the ferromagnetic solids are divided into a large number of small regions termed as Domains. The dimensions of these domains can be from few microns to the size of the crystal and typically it consists of 1 1 to1 15 magnetic dipole moments aligned in a single direction. It means, different domains have different directions of magnetization so that net magnetic moment is zero. Thus, the immediate consequences of the domain theory are: 1. The magnetic dipole moments exist permanently.. There is alignment of these moments (ordered state) even in the demagnetized state. 3. The demagnetized state is characterized by the random alignments of the domains only. 4. The process of magnetization consists of reorientation of the domains. In weak applied field, the volume of domain having magnetization in the field direction increases whereas in strong applied field the magnetization of the domain is rotated in the field direction. Origin of Domains: We know that the Ferro magnets do not get magnetized spontaneously. Instead, the magnetization has to be done by the application of external magnetic field. The empirical explanation for this fact was given by Weiss through the postulation of the domains. The existence of the domains was further

48 confirmed several experiments like Barkhausen effect, Bitter patterns, faraday Effect, Kerr effect and also through Magneto-optic and transmission electron microscopy (TEM) techniques. One such typical domain pattern observed through Kerr Effect is shown in Fig.3.3 The first explanation for the origin of domains was given by Landau and Lifschitz in 193. They showed that the existence of the domains is the consequence of the energy minimization. There are mainly three contributions to the potential energy viz- 1. Magneto static or exchange energy. Anisotropy energy 3. Magneto striation energy The magneto static energy ideas to the interaction of the magnetic dipole moments, which keep them, aligned. The anisotropy energy is the natural consequence of the preferred directions of magnetization. It is found that the ferromagnetic crystals have easy and hard directions of magnetization i.e. higher fields are required to magnetize the crystal in a particular direction. E.g. for iron crystal (1) is easy and (111) is hard direction whereas for Nickel (111) is easy and (1) is hard direction. The excess of energy required for the magnetization along hard direction is called the anisotropy energy. The process of magnetization can induce a slight change in the dimensions of the samples. This change is obtained by the work done against the elastic restoring forces. The associated energy is known as magnetostrictive energy. The origin of domains can be clearly understood by considering the domain structures of a single crystal as shown in Fig. 3.4 In Fig 3.4a), the entire specimen has a single magnetic domain with the magnetic poles (S, N) formed on the surfaces of the crystal. The magneto static energy of such configuration is ( 1/8 ) B dv. Its value is quite high of the order of 1 6 erg/cm 3. This much energy is required to assemble the atomic magnets into single domain. This energy is reduced by approximately one half, if the crystal is divided into two domains as shown in Fig 3.4b). In this case the two domains are magnetized in opposite directions and the flux lines are completed on the same surfaces. The subdivision of domains, then the magneto static energy will be reduced approximately by the factor 1/N. Further, there is another possible configuration as shown in Fig 3.4d). In this case, there are triangular domains near the end faces of crystals. The magnetizations in the vertical and the triangular domains are at an angle of 9 o and the boundaries of the domains bisect this angle by making equal angles of 45 o with both directions of magnetization. The surface domains complete the flux circuit and therefore are referred as domains of closure. In such configuration, there are no free poles and the magneto static energy is zero. The domains of closure are nucleated at the boundary of the specimen or at certain

49 defects inside. During magnetization processes, those domains are swept out certain defects inside. During magnetization processes, those domains are swept out at higher fields only. Thus, the origin of the domain structure is attributed to the possibility of lowering the energy of the system by going from a saturated configuration of high energy (Fig 3.4a) to a domain configuration of the lowest energy (Fig. 3.4d). The introduction of a domain raises the overall energy of the system, therefore the division into domains only continues while the reduction in magneto static energy is greater than the energy required to form the domain wall. The energy associated a domain wall is proportional to its area. The schematic representation of the domain wall, is shown in Fig 3.5. It illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction of magnetization and hence are in a higher energy state. In addition, the atomic dipoles within the wall are not at 18 o to each other and so the exchange energy is also raised within the wall. Therefore, the domain wall energy is an intrinsic property of a material depending on the degree of magnetocrystalline anisotropy and the strength of the exchange interaction between neighboring atoms. The thickness of the wall will also vary in relation to these parameters as a strong magneto-crystalline Anisotropy will favor a narrow wall, whereas strong exchange interaction will favor a wider wall. A minimum energy can therefore be achieved with a specific number of domains within a specimen. This number of domains will depend on the size and shape of the sample (which will affect the magneto static energy) and the intrinsic magnetic properties of the material (which will affect the magneto static energy and the domain wall energy). 3.1 Magnetic Resonance The course material, so far, is related to the response of materials to the static magnetic field. However, there are many dynamical magnetic effects, which as frequency dependent. These effects are particularly associated with the spin angular momentum of the electrons and the nuclei. The wide known such phenomena can be identified as follows- Nuclear Magnetic Resonance (NMR) Electron Paramagnetic (ESR) Nuclear Quadric pole Resonance (NQR) Ferromagnetic Resonance (FMR) Spin Wave Resonance (SWR) Anti ferromagnetic Resonance (AFMR) The first observation of the magnetic resonance was made by E. Zaviosky kin 1945 through electron spin resonance absorption in the paramagnetic salt MnSo 4 using.75 Ghz field. The magnetic resonance can provide significant information about the samples. It can be categorized as follows.

50 1. The fine structure of absorption can reveal the electronic structure of defects.. The changes in line width of absorption pattern indicate the spin motion. 3. The position of resonance line reveals the internal magnetic field. 4. It can elaborate the collective spin resonance. i. Nuclear Magnetic Resonance: Theory: The atomic nuclei have an angular momentum due to the4 nuclear spin in the case of electrons, the total angular momentum is the result of spin and orbital quantum number (I) its total angular momentum is Ih. The spinning nuclei will give rise to nuclear magnetic moment. hl (1.1) Where is called the gyro magnetic ratio. In the presence of applied magnetic field (B a ) along direction the magnetic moment will process about the field direction with resolved component z hml (1.) Where the allowed values of m l are I, I-1, I-,.-I The potential energy of this interaction will be give by U= z. Ba hmi Ba (1.3) 1 rhba The nucleus with m I = will have two energy level viz., u I = and u rba. The splitting of energy levels of nucleus is shown in Fig 3.6. Fig.3.6 Nuclear energy levels The energy levels of these two levels can be denoted in terms of frequency such tatt, U U 1

51 e.i. B B a (1.4) The equation (1.5) is the fundamental condition for magnetic resonance absorption. a It means that the resonance can be observed only if an alternating magnetic field of frequency is applied. For Proton, 8.675x1 (s -1 tesla -1 ) =.675 x 1 8. B (s -1 ) Orv =w/ =4.58 x 1 6 B (s -1 ) Thus, the frequency (v) is of the order of few MH Z, which is in radio frequency range. Experimental: When a sample of magnetically active nuclei is placed into an external magnetic field, the magnetic fields of these nuclei align themselves with the external field into various orientations. Each of these spin-states will be nearly populated with a slight excess in lower energy levels. During the experiment, electromagnetic radiation is applied to the sample with energy exactly equivalent to the energy separation id two adjacent spin states. Some of the energy is absorbed and the alignment of one nucleus magnetic field reorients from a lower energy to a higher energy alignment (spin transition). By sweeping the frequency, and hence the energy, of the applied electromagnetic radiation, a plot of frequency versus energy absorption can be generated. This plot is the NMR spectrum as shown in Fig 3.7. Fig.3.7 In a homogeneous system with only one kind of nucleus, the NMR spectrum will show only a single peak at a characteristic frequency. In real samples the nucleus is influenced by its environment. Some environments will increase the energy separation of the spin-states giving a spin transition at a higher frequency. Others will lower the separation consequent lowering the frequency at which the spin transition occurs. These changes in frequency are called the chemical shift of the nucleus and can be examined in more detail. By examining the exact frequencies (chemical shift) at which the spin transitions occur conclusions about the nature of the various environment can be made. This simply type of experiment, where the frequency is swept across a range, is know as a continuous wave (CW) experiment. One simple variation on this experiment is to hold the frequency

52 of the electromagnetic radiation constant and to sweep the strength of the applied magnetic field instead. The energy separation of the spin states will increase as the external field becomes stronger. At some point, this energy separation matches the energy of the electromagnetic radiation and absorption occurs. Plotting energy absorption versus external magnetic field strength produces the identical NMR spectrum as shown in Fig Fig. 3.8 In fact, the MNR spectrum obtained by plotting magnetic field increasing to the right will be a mirror image of the spectrum where frequency is plotted increasing to the right. Low energy transitions (to the left) in a frequency swept experiment will not occur until very high magnetic fields (to the right) in a magnetic field swept experiment. Early NMR spectrometers swept the magnetic field since it was too difficult to build the very stable swept RF sources that NMR required. Even today where this is no longer required, NMR spectra are still plotted with magnetic field increasing to the right. Technological advances have made the CW experiment obsolete and today virtually all NMR experiments are conducted using pulse methods. These methods are inherently much more sensitive and this explains part of their popularity. A simplified block diagram of the NMR apparatus is shown in Fig The diagram does not show all the functions of each module, but it does represent the most important functions of each modular component of the spectrometer. The Pulse Programmer creates the pulse stream that gates the synthesized oscillator into radio frequency pulse bursts, as well triggering the oscilloscope on the appropriate pulse. The pulse bursts are amplified and sent to the transmitter coils in the sample probe. The current bursts in these coil produce a homogeneous 1 Gauss rotating magnetic field at the sample. These are the time dependent B, fields that produce the precession of the magnetization, referred to as the 9 o or 18 o pulses. The transmitter coils are wound in a Helmholtz configuration to optimize rf magnetic field homogeneity.

53 Nuclear magnetization processing in the direction transverse to the applied constant magnetic field (the so called x-y plane) induces an EMF in the receiver coil, which is then amplified by the receiver circuitry. This amplified radio frequency (15 MH Z ) signal can be detected (demodulated) by two separate and different detectors. The Amplitude Detector rectified the signal and has an output proportioned to the peak amplitude of the processionals signal. This detector is used to record both the free induction decays and the spin echoes signals. The second detector is a Mixer, which effectively multiplies the precession signal from the sample magnetization with the master oscillator. Its output frequency is proportional to the difference between the two frequencies. This Mixer is essential for determining the proper frequency of the oscillator. The magnet and the nuclear Magnetic moment of the protons uniquely determined the processionals frequency of the nuclear magnetization. The oscillator is tuned to this precession frequency when a aero-beat output signal of the mixers obtained. A dual channel scope allows simultaneous observations of the signals from both detectors. The field of the permanent magnet is temperature dependent so periodic adjustments in the frequency are necessary to keep the spectrometer on resonance. There are both an analog and a digital (sampling) oscilloscope for the measurements. Both have their strengths and weaknesses. The digital oscilloscope samples its input signals at a fixed high frequency, which, if the signal you are measuring has similar frequency components, can lead to spurious displays. The analog oscilloscope samples continuously, but it is slightly less convenient for making numerical measurements of pulse heights or widths. The NMR apparatus are widely used in the hospitals with a common name as Magnetic Resonance Imaging (MRI). It can detect the minute magnetic signals generated by organs. By using sophisticated instrumentation and image processing software, it can produce a three dimensional color. A schematic view of a typical MRI scanner and a brain scan is shown in Fig.3.1. ii Electron Paramagnetic Resonance:

54 Electron paramagnetic resonance (EPR) and/or electron spin resonance (ESR) is defined as the form of spectroscopy concerned with microwave-induced transitions between magnetic energy levels of electrons having a net spin and orbital angular momentum. The term electron paramagnetic resonance and the symbol EPR are Preferred and should be used for primary indexing. The correspondence between NMR and ESR is very close, of ESR it is necessary to have an unpaired electron instead of an unpaired nuclear spin (as in NMR). Further, it is also necessary to provide an external static magnetic field to generate the ground and excited state energy levels. The major difference is that ESR spectroscopy has a higher absorption frequency than NMR spectroscopy. Consequently, the sensitivity of EPR is considerably higher. However, the absorptions lines are also significantly broader. The electron spin resonance spectrum of a free radical or coordination complex with one unpaired electron is the simplest of all forms of spectroscopy. The degeneracy of the electron spin states characterized by the quantum number, ms 1/, is lifted by the application of a magnetic field and transitions between the spin levels are induced by radiation of the appropriate frequency, as shown in Fig If unpaired electron in radicals were indistinguishable from free electrons, the only information content of an ESR spectrum would be the integrated intensity, proportional to the radical concentration. Fig.3.11 An unpaired electron interacts with its environment, and the details of ESR spectra depend on the nature of those interactions. There are two kinds of environmental interactions which are commonly important in the ESR spectrum of a free radical: (i) To the extent that the unpaired electron has unquenched orbital angular momentum, the total magnetic moment is different from the spin-only moment (either larger or smaller, depending on how the angular momentum vectors couple). It is customary to ump the orbital and spin angular moment together in an effective spin and to treat the effect as a shift in the energy of the spin transition. (ii) The electron spin energy levels are split by interaction with nuclear magnetic moments the nuclear hyperfine interaction. Each nucleus of spin I splits the electron spin levels into ( I 1) sublevels. Since transitions are observed between sublevels with the same values of mi, nuclear spin splitting of energy levels is mirrored by splitting of the resonance line. When an electron is placed magnetic field, the degeneracy of the electron spin energy levels is lifted as shown in Figure 3.11 and as described by the spin

55 Hamiltonian: H g B. S (1.5) s B z In equation (1.5), g is called the g-va;ie(g=.3 fro a free electron), B is the Bohr magnetron (9.74x1-8 JG -1 ), B is the magnetic field strength in Gauss, and Sz is the z-component of the spin angular momentum operator (the field defines the z direction). Energy level splitting in a magnetic field is called the Zeeman effect, and the Hamiltonian of equation (1.5) is sometimes referred to as the electron Zeeman Hamiltonian. The electron spin energy levels are easily found by application of H S to the electron spin Eigen functions corresponding to m 1/ : E 1/ ( gbb) (1.6) The difference in energy between the two levels is. s E E E g B B (1.7) It corresponds to the energy of a photon required to cause a transition, i.e. hy g B (1.8) B or v gb B/ h (1.9) where g B / h =.9348 x1-4 cm -1 G -1. Magnetic fields of up to 15 KG are easily obtained with an iron-core electromagnet; thus we could use radiation with up to 1.4 cm -1 (y < 4 GH or >.71cm). Radiation with this kind of wavelength is in the microwave region. Microwaves are normally handled using wave guides designed to transmit over a relatively narrow frequency range. Wave guides look like rectangular cross-section pipes with dimensions on the order of the wavelength to be transmitted. The ESR sensitivity (net absorption) increases with decreasing temperature and with increasing magnetic field strength. Since field is proportional to microwave frequency, in principle sensitivity should be greater for K-band or Q-band spectrometers than for X-band. However, since the K- or Q- band waveguides are Smaller, samples are also necessarily smaller, usually more than canceling the advantage of a more favorable Boltzmann factor. Under ideal conditions, a commercial X-band spectrometer can detect the order f 1 1 spins (1-1 moles) at room temperature. The ESR is a remarkably sensitive technique, especially compared with NMR, Because the spin levels are so nearly equally populated, magnetic resonance suffers from a problem not encountered in higher energy forms of spectroscopy: An intense radiation field will tend to equalize the populations, leading to a decrease in net absorption; this effect is called saturation. A spin system returns to thermal equilibrium via energy transfer to the surroundings, a rate process

56 called spin-lattice relaxation, with a characteristic time, T1, the spin-lattice relaxation time (rate constant = 1/T1). Systems with a long T1(i.e., spin systems weakly coupled to the surroundings) will be easily saturated; those with shorter T1 will be more difficult to saturate. Experimental Although many spectrometer designs have been produced over the years, the vast majority of laboratory instruments are based on the simplified block diagram shown in figure 3.1. Microwaves are generated by the Klystron tube and the power level is adjusted with the Attenuator. The Circulator behaves like a traffic circle: microwaves entering from the Klystron are routed toward the Cavity where the sample is mounted. Microwaves reflected back from the cavity (less when power is being absorbed) are routed to the diode detector, and any power reflected from the diode is absorbed completely by the Load. The diode is mounted along the E-vector of the plane-polarized microwaves and thus produces a current proportional to the microwave power reflected from the cavity. Thus in principle, the absorption of microwaves by the sample could be detected by noting a decrease in current in the micro ammeter. Fig.3.1 In practice, of course, such a d.c. measurement would be far too noisy to be useful. The solution to the signal-to-noise ratio problem is to introduce small amplitude field modulation. An oscillating magnetic field is superimposed on the d.c. field by means of small coils, usually built into the cavity walls. When the field is in the vicinity of a resonance line, it is swept back and forth through part of the line, leading to an a.c. component in the diode current. This a.c. component is amplified using a

57 frequency selective amplifier, thus eliminating a great deal of noise. The modulation amplitude is normally less than the line width. Thus the detected a.c. signal is proportional to the change in sample absorption. As shown in Figure 3.13, this amounts to detection of the first derivative of the absorption curve. It takes a little practice to get used to looking at first-derivative spectra, but there is a distinct advantage; first derivative spectra have much better apparent resolution than do absorption spectra. Indeed, second-derivative spectra are even better resolved (though the signal-to-noise ratio decreases on further differentiation). Fig b. Mossbauer spectroscopy: The Mossbauer spectroscopy is a versatile technique that can be used to provide information in many areas of science such as Physics, Chemistry, Biology and Metallurgy. It can give very precise information about the chemical, structural, magnetic and time-dependent properties of a material. The discovery of recoilless gamma ray emission and absorption, is referred as the Mossbauer Effect, after its discoverer Rudolph Mossbauer, who first observed the effect in 1957 and received the Nobel Prize in Physics in 1961 for his work. ii. The Mossbauer Effect: In a free nucleus during emission or absorption of a gamma ray it recoils due to conservation of momentum, just like a gun recoils when firing a bullet, with a recoil energy E R greater than the transition energy due to the recoil of the absorbing nucleus. To achieve resonance the loss of the recoil energy must be overcome in some way.

58 As the atoms will be moving due to random thermal motion the gamma-ray energy has a spread of values E D caused by the Doppler effect. This produces a gammaray energy profile as shown in Fig To produce a resonant signal the two energies need to overlap and this is shown in the shaded area. This area is shown exaggerated as in reality it is extremely small, a millionth or less of the gamma-rays are in this region, and impractical as a technique. What Mossbauer discovered is that when the atoms are within a solid matrix the effective mass of the nucleus is very much greater. The recoiling mass is now effective the mass of the whole system. Making E R and E D very small. If the gamma-ray energy is small enough the recoil of the nucleus is too low to be transmitted as a phonon (vibration in the crystal lattice) and so the whole system recoils, making the recoil energy practically zero: a recoil-free event. In this situation, as shown in Fig 3.16, the emitted and absorbed gamma-ray have the same energy: resonance! If emitting and absorbing nuclei are in identical, cubic environments then the transition energies are identical and this produces a spectrum as shown in Fig a single absorption line.

59 Now for achieving resonant emission and absorption can we use it to probe the tiny hyperfine interaction between an atom s nucleus and its environment? The natural line width of the excited nuclear state is related to the average lifetime of the excited state before it decays by emitting the gamma-ray. For the most common Mossbauer isotope, 57 Fe, this line width is 5x1-9 ev. Compared to the Mossbauer gamma-ray energy is 14.4 KeV this gives a resolution of 1 in 1 1 or the equivalent of one sheet of paper in the distance between the Sun and the Earth. As resonance only occurs when the transition energy of the emitting and absorbing nucleus match exactly this effect is isotope specific. The relative number of recoil-free events (and hence the strength of the signal) is strongly dependent upon the gamma-ray energy and so the Mossbauer effect is only detected in isotopes with very low lying excited states. Similarly the resolution is dependent upon the lifetime of the excited state. These two factors limit the number of isotopes that can be used successfully for Mossbauer spectroscopy e.g. Fe, Ru, Sn, Sb, Te,I, W, Au, Eu, Gd. Dy, etc. The most used is 57 Fe, which has both a very low energy gamma-ray and long-lived excited state. iii. Fundamentals of Mossbauer Spectroscopy The energy changes caused by the hyperfine interactions are very small, of the order of billionths of an ev. Such miniscule variations of the gamma-ray are quite easy to achieve by the use of the Doppler effect i.e. by moving the gamma-ray source towards and away from the absorber. This is most often achieved by oscillating a radioactive source with a velocity of a few mm/s and recording the spectrum in discrete velocity steps. Fractions of mm/s compared to the speed of light (3x1 11 mm/s) give the minute energy shifts necessary to observe the hyperfine interactions. For convenience the energy scale of a Mossbauer spectrum is quoted in terms of the source velocity, as

60 With an oscillating source we can now modulate the energy of the gamma-ray in very small increments. With the modulated gamma-ray energy matches precisely the energy of a nuclear transition in the absorber the gamma-rays are resonantly absorbed and we see a peak. As we re seeing this in the transmitted gamma-rays the sample must be sufficiently thin to allow the gamma-rays to pass through, the relatively low energy gamma-rays are easily attenuated. In Fig 3.18 the absorption peak occurs at mm/s, where source and absorber are identical. The energy levels in the absorbing nuclei can be modified by their environment in three main ways: by the Isomer Shift, Quadruple Splitting and Magnetic Splitting. The isomer shift arises due to the non-zero volume of the nucleus and the electron charge density due to s-electrons within it. This leads to a monopole (Coulomb) interaction, altering the nuclear energy levels. Any difference in the s-electron environment between the source and absorber thus produces a shift in the resonance energy of the transition. This shifts the whole spectrum positively or negatively depending upon the s-electron density, and sets the spectrum. The isomer shift is useful for determining valence states, bonding states, electron shielding and the electron-drawing power of electronegative groups. The nuclei in states with an angular momentum quantum number I>1/ have a non-spherical charge distribution. This produces a nuclear quadruple moment. In the presence of an asymmetrical electric field (produced by an asymmetric electronic charge distribution or ligand arrangement) this splits the nuclear energy levels. The magnitude of splitting is related to the nuclear quadruple moment and electronic charge distribution. Thus, additional energy levels are available for the absorption spectra. In the presence of a magnetic field the nuclear spin moment exp experiences a dipolar interaction with the magnetic field ie Zeeman splitting. The magnetic field splits nuclear levels with a spin of I into (I+1) sub-states. The transitions between the excited state and ground state can only occur wherem 1 changes by or1. The line positions are related to the splitting of the energy levels, but the line intensities are related to the angle between the Mossbauer gamma-ray and the intensities can give information about moment

61 orientation and magnetic ordering. These interactions, Isomer Shift, Quadruple Splitting and Magnetic Splitting, alone or in combination are the primary characteristics of many Mossbauer spectra. iv. Application to impure crystal The impurities, which can be displaced from their regular positions, in a crystal lattice are termed as off-center impurities. They can be considered as existing in an asymmetric double potential well. Such atoms can change their position as the temperature changes. Unfortunately there are often many other phenomena in such systems that can mask the off-centering effect. The Mossbauer spectroscopy provides a good tool for observing this effect. Firstly the movement of the off-center atom within the lattice will change the symmetry of the electric field. Mossbauer spectroscopy is also isotope and site specific, meaning we can observe the off-center single component without any masking from other elements or effects. A compound which was thought to exhibit off-centering is Pb.8 Sh., with tin as an off-center atom. The typical spectra of this sample at K and K are shown in Fig There are two components: one from an off-center site and one from a normal single-potential site. It can be seen in the highlighted region that the small component develops from a single line to a (broad) doublet. The quadrupole splitting is increasing, indicating the electric field environment around these particular atoms has become more asymmetrical. This is consistent with atom moving within an asymmetric potential well. The other component shows no variation in quadruple splitting. A series of spectra were taken in a temperature cycle and a hysteresis was observed in the values of quadruple splitting. These results show that tin is an off-center atom in this compound and that there are two tin sites within it: one normal and one off-center. 3.11SUMMARY:- This chapter describes the concept of magnetism, according to which magnetic field is produced due to motion of electic charge and define the various type of magnetic materials like diamagnetic, Paramagnetic and ferromagnetic, which are based on how the material reacts to a magnetic moment induced in them that opposes the direction of the magnetic field. This chapter also explains the theory of Para magnetism, in which we have derived the formula for magnetization. It also cover the quantum theory of ferromagnetism Which explain the origin of domain, which are formed when ferromagnetic materials are divided in to small region Magnetic resonance is related to the response of materials to the static magnetic field. It elaborates the fundamental condition for magnetic resonance absorption and experimental verification and their application. 3.1CHECK YOUR PROGRESS

62 Q.1. Explain Magnetic Field And Its Strength. Hint: - Refer to topic no. 3.3 Q.. Explain Magnetic Dipole Moment. Hint: - Refer to topic no. 3.4 Q.3. Explain the Theory of Ferromagnetism. Hint: - Refer to topic no. 3.7 Q.4. Explain Quantum Theory of Ferromagnetism. Hint:- Refer to topic no. 3.8 Q.5. Define Magnetic Resonance Hint: - Refer to topic no. 3.1 Books Recommended 1. Solid State Physics : A.J. Dekker. Introduction to Solide State Physics : C. Kittel 3. Solid State Physics : Azaroff. 4. Thin Film Technology : K.L. Chopra

63 UNIT IV DEFECTS IN CRYSTALS Structure I. 4.Introduction 4.1 Objectives 4. Point Defect in ionic crystals and metals 4.3 Diffusion in solids Type of Diffusion 1.3. Diffusion Mechanisms Diffusion Coefficient Applications 4.4 Ionic Conductivity 4.5 Colour Centres F- Centres 1.5. V-Centres 4.6 Excitions 4.7 General Idea of Luminescence 4.8 Dislocations & Mechanical Strength of Crystals 4.9 Plastic Bahaviour 4.1 Type of Dislocations 4.11 Stress field of Dislocations 4.1 Grain Boundaries 4.13 Etching- Types of Etching 4.14 Let Us Sum Up 4.15 Check Your Progress: The Key

64 II. 4. INTRODUCTION Up to now, we have described perfectly regular crystal structures, called ideal crystals and obtained by combining a basis with an infinite space lattice. In ideal crystals atoms were arranged in' a regular way. However, the structure of real crystals differs from that of ideal ones. Real crystals always have certain defects or imperfections, and therefore, the arrangement of atoms in the volume of a crystal is far from being perfectly regular. Natural crystals always contain defects, often in abundance, due to the uncontrolled conditions under which they were formed. The presence of defects which affect the colour can make these crystals valuable as gems, as in ruby (chromium replacing a Defects in crystals and Elements of Thin Films small fraction of the aluminium in aluminium oxide: Al 3 ). Crystal prepared in laboratory will also always contain defects, although considerable control may be exercised over their type, concentration, and distribution. The importance of defects depends upon the material, type of defect, and properties, which are being considered. Some properties, such as density and elastic constants, are proportional to the concentration of defects, and so a small defect concentration will have a very small effect on these. Other properties, e.g. the colour of an insulating crystal or the conductivity of a semiconductor crystal, may be much more sensitive to the presence of small number of defects. Indeed, while the term defect carries with it the connotation of undesirable qualities, defects are responsible for many of the important properties of materials and much of material science involves the study and engineering of defects so that solids will have desired properties. A defect free, i.e. ideal silicon crystal would be of little use in modern electronics; the use of silicon in electronic devices is dependent upon small concentrations of chemical impurities such as phosphorus and arsenic which give it desired properties. Some simple defects in a lattice are shown in Fig. 1. There are some properties of materials such as stiffness, density and electrical conductivity which are termed structure-insensitive, are not affected by the presence of defects in crystals while there are many properties of greatest technical importance such as mechanical strength, ductility, crystal growth, magnetic

65 Key a = vacancy (Schottky defect) b = interstitial c = vacancy interstitial pair (Frenkel defect) d = divacancy e = split interstitial = vacant site Fig. 1 Some Simple defects in a lattice Defects in Crystal Hysteresis, dielectric strength, condition in semiconductors, which are termed structure sensitive are greatly affected by the-relatively minor changes in crystal structure caused by defects or imperfections. Crystalline defects can be classified on the basis of their geometry as follows: (i) Point imperfections (ii) Line imperfections (iii) Surface and grain boundary imperfections (iv) Volume imperfections The dimensions of a point defect are close to those of an interatomic space. With linear defects, their length is several orders of magnitude greater than the width. Surface defects have a small depth, while their width and length may be several orders larger. Volume

66 defects (pores and cracks) may have substantial dimensions in all measurements, i.e. at least a few tens of A. We will discuss only the first three crystalline imperfections. 4.1 OBJECTIVES III. The Main aim of this unit is to study defect in crystals after going through the unit you should be able to Describe the type of defects Explain the diffusion in crystal Explain the color center and excitations Explain the type of dislocation IV. 4. POINT DEFECT IN IONIC CRYSTALS AND METALS The point imperfections, which are lattice errors at isolated lattice points, take place due to imperfect packing of atoms during crystallisation. The point imperfections also take place due to vibrations of atoms at high temperatures. Point imperfections are completely local in effect, e.g. a vacant lattice site. Point defects are always present in crystals and their present results in a decrease in the free energy. One can compute the number of defects at equilibrium concentration at a certain temperature as, n = N exp [-E d / kt] (1) Where n - number of imperfections, N - number of atomic sites per mole, k - Boltzmann constant, E d - free energy required to form the defect and T - absolute temperature. E is Defects in crystals and Elements of Thin Films typically of order l ev since k = 8.6 X 1-5 ev /K, at T = 1 K, n/n = exp[-1/(8.6 x 1-5 x 1)] 1-5, or 1 parts per million. For many purposes, this fraction would be intolerably large, although this number may be reduced by slowly cooling the sample. (i) Vacancies: The simplest point defect is a vacancy. This refers to an empty (unoccupied) site of a crystal lattice, i.e. a missing atom or vacant atomic site [Fig. (a)] such defects may arise either from imperfect packing during original crystallisation or from thermal vibrations of the atoms at higher temperatures. In the latter case, when the thermal energy due to vibration is increased, there is always an

67 increased probability that individual atoms will jump out of their positions of lowest energy. Each temperature has a Fig. Point defects in a crystal lattice corresponding equilibrium concentration of vacancies and interstitial atoms (an interstitial atom is an atom transferred from a site into an interstitial position). For instance, copper can contain 1-13 atomic percentage of vacancies at a temperature of -5 C and as many as.1 % at near the melting point (one vacancy per 1 4 atoms). For most crystals the-said thermal energy is of the order of I ev per vacancy. The thermal vibrations of atoms increases with the rise in temperature. The vacancies may be single or two or more of them may condense into a di-vacancy or trivacancy. We must note that the atoms surrounding a vacancy tend to be closer together, thereby distorting the lattice planes. At thermal Defects in Crystal equilibrium, vacancies exist in a certain proportion in a crystal and thereby leading to an increase in randomness of the structure. At higher temperatures, vacancies have a higher concentration and can move from one site to another more frequently. Vacancies are the most important kind of point defects; they accelerate all processes associated with displacements of atoms: diffusion, powder sintering, etc. (ii) Interstitial Imperfections: In a closed packed structure of atoms in a crystal if the atomic packing factor is low, an extra atom may be lodged within the crystal structure. This is known as interstitial position, i.e. voids. An extra atom can enter the interstitial space or void between the regularly positioned atoms only when it is substantially smaller than the parent

68 atoms [Fig. (b)], otherwise it will produce atomic distortion. The defect caused is known as interstitial defect. In close packed structures, e.g. FCC and HCP, the largest size of an atom that can fit in the interstitial void or space have a radius about.5% of the radii of parent atoms. Interstitialcies may also be single interstitial, di-interstitials, and tri-interstitials. We must note that vacancy and interstitialcy are inverse phenomena. (iii) Frenkel Defect: Whenever a missing atom, which is responsible for vacancy occupies an interstitial site (responsible for interstitial defect) as shown in Fig. (c), the defect caused is known as Frenkel defect. Obviously, Frenkel defect is a combination of vacancy and interstitial defects. These defects are less in number because energy is required to force an ion into new position. This type of imperfection is more common in ionic crystals, because the positive ions, being smaller in size, get lodged easily in the interstitial positions. (iv) Schottky Defect: These imperfections are similar to vacancies. This defect is caused, whenever a pair of positive and negative ions is missing from a crystal [Fig. (e)]. This type of imperfection maintains charge neutrality. Closed-packed structures have fewer interstitialcies and Frenkel defects than vacancies and Schottky defects, as additional energy is required to force the atoms in their new positions.

69 Defects in crystals and Elements of Thin Films Check Your Progress 1 Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit Explain Frenkel and Schottky defects?.... (v) Substitutional Defect: Whenever a foreign atom replaces the parent atom of the lattice and thus occupies the position of parent atom (Fig. (d)], the defect caused is called substitutional defect. In this type of defect, the atom which replaces the parent atom may be of same size or slightly smaller or greater than that of parent atom. (vi) Phonon: When the temperature is raised, thermal vibrations takes place. This results in the defect of a symmetry and deviation in shape of atoms. This defect has much effect on the magnetic and. electric properties. All kinds of point defects distort the crystal lattice and have a certain influence on the physical properties. In commercially pure metals, point defects increase the electric resistance and have almost no effect on the mechanical properties. Only at high concentrations of defects in irradiated metals, the ductility and other properties are reduced noticeably. In addition to point defects created by thermal fluctuations, point defects may also be created by other means. One method of producing an excess number of point defects at a given temperature is by quenching (quick cooling) from a higher temperature. Another method of creating excess defects is by severe deformation of the crystal lattice, e.g., by hammering or rolling. We must note that the lattice still retains its general crystalline nature, numerous defects are introduced. There is also a method of creating excess point defects is by external bombardment by atoms or high-energy particles, e.g. from the beam of the cyclotron or the neutrons in a nuclear reactor. The first particle collides with the lattice atoms and displaces them, thereby causing a

70 Defects in Crystal point defect. The. number of point defects created in this manner depends only upon the nature of the crystal and on the bombarding particles and not on the temperature. Check Your Progress Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit What are crystal defects and how are they classified? 4.3 DIFFUSION Diffusion refers to the transport of atoms through a crystalline or glassy solid. Many processes occurring in metals and alloys, especially at elevated temperatures, are associated with self-diffusion or diffusion. Diffusion processes play a crucial 'role in many solid-state phenomena and in the kinetics of micro structural changes during metallurgical processing and applications; typical examples include phase transformations, nucleation, recrystallization, oxidation, creep, sintering, ionic conductivity, and intermixing in thin film devices. Direct technological uses of diffusion include solid electrolytes for advanced battery and fuel cell applications, semiconductor chip and microcircuit fabrication and surface hardening of steels through carburization. The knowledge of diffusion phenomenon is essential for the introduction of a very small concentration of an impurity in a solid state device: V Types of Diffusion (i) Self Diffusion: It is the transition of a thermally excited atom from a site of crystal lattice to an adjacent site or interstice. (ii) Inter Diffusion: This is observed in binary metal alloys such as the Cu-Ni system. iii) Volume Diffusion: This type of diffusion is caused due to atomic movement

71 in bulk in materials. (iv) Grain Boundary Diffusion: This type of diffusion Defects in crystals and Elements of Thin Films is caused due to atomic movement along the grain boundaries alone. (v) Surface Diffusion: This type of diffusion is caused due to atomic movement along the surface of a phase. VI Diffusion Mechanisms Diffusion is the transfer of unlike atoms which is accompanied with a change of concentration of the components in certain zones of an alloy. Various mechanisms have been proposed to explain the processes of diffusion. Almost all of these mechanisms are based on the vibrational energy of atoms in a solid. Direct-interchange, cyclic, interstitial, vacancy etc. are the common diffusion mechanisms. Actually, however, the most probable mechanism of diffusion is that in which the magnitude of energy barrier (activation energy) to be overcome by moving atoms is the lowest. Activation energy depends on the forces of interatomic bonds and crystal lattice defects, which facilitate diffusion transfer (the activation energy at grain boundaries is only one half of that in the bulk of a grain). For metal atoms, the vacancy mechanism of diffusion is the most probable and for elements with a small atomic radius (H, N and C), the interstitial mechanism. Now, we will study these mechanisms. (i) Vacancy Mechanism: This mechanism is a very dominant process for diffusion in FCC, BCC and HCP metals and solid solution alloy. The activation energy for this process comprises the energy required to create a vacancy and that required to move it. In a pure solid, the diffusion by this mechanism is shown in Fig. 3(a). Diffusion by the vacancy mechanism can occur by atoms moving into adjacent sites that are vacant. In a pure solid, during diffusion by this mechanism, the atoms surrounding the vacant site shift their equilibrium positions to adjust for the change in binding that accompanies the removal of a metal ion and its valency electron. We can assume that the vacancies move through the lattice and produce random shifts of atoms from one lattice position to another as a result of atom jumping. Concentration changes takes place due to diffusion over a period of time. We must note that vacancies are continually being created and destroyed at the surface, grain boundaries and suitable interior positions, e.g. dislocations. Obviously, the rate of diffusion increases rapidly with increasing temperature.

72 Defects in Crystal Fig. 3. Various Diffusion mechanism (a) Vacancy mechanisms (b) Interstitial mechanisms (c) Two atoms interchange mechanisms

73 If a solid is composed of a single element, i.e. pure metal, the movement of thermally excited atom from a site of the crystal lattice to an adjacent site or interstice is called self diffusion because the moving atom and the solid are the same chemical-element. The selfdiffusion in metals in which atoms of the metal itself migrate in a random fashion throughout the lattice occurs mainly through this mechanism. We know that copper and nickel are mutually soluble in all proportions' in solid state and form substitutional solid solutions, e.g., plating of nickel on copper. For atomic diffusion, the vacancy mechanism is shown in Fig. 4. Fig. 4. Vacancy mechanism for atomic diffusion (a) Pure solid solution, and

74 Defects in crystals and Elements of Thin Films (ii) The Interstitial Mechanism: The interstitial mechanism where an atom changes positions using an interstitial site does not usually occur in metals for elfdiffusion but is favored when interstitial impurities are present because of the low activation energy. When a solid is composed of two or more elements whose atomic radii differ significantly, interstitial solutions may occur. The large size atoms occupy lattice sites where as the smaller size atoms fit into the voids (called as interstices) created by the large atoms. We can see that the diffusion mechanism in this case is similar to vacancy diffusion except that the interstitial atoms stay on interstitial sites (Fig. 3(b)). We must note that activation energy is associated with interstitial diffusion because, to arrive at the vacant site, it must squeeze past neighbouring atoms with energy supplied by the vibrational energy of the moving atoms. Obviously, interstitial diffusion is a thermally activated process. The interstitial mechanism process is simpler since the presence of vacancies is not required for the solute atom to move. This mechanism is vital for the following cases: (a) The presence of very small atoms in the interstices of the lattice affect to a great extent the mechanical properties of metals. (b) At low temperatures, oxygen, hydrogen and nitrogen can be diffused in metals easily. (iii) Interchange Mechanism: In this type of mechanism, the atoms exchange places through rotation about a mid point. The activation energy for the process is very high and hence this mechanism is highly unlikely in most systems. Two or more adjacent atoms jump past each other and exchange positions, but the number of sites remains constant (Fig. 3 (c) and (d)). This interchange may be two-atom or four-atom (Zenner ring) for BCC. Due to the displacement of atoms surrounding the jumping pairs, interchange mechanism results in severe local distortion. For jumping of atoms in this case, much more energy is required. In this mechanism, a number of diffusion couples of different compositions' are Defects in Crystal produced, which are objectionable. This is also termed as Kirkendall's effect. Kirkendall was the first person to show the inequality of diffusion. By using an ά brass/copper couple, Kirkendall showed that Zn atoms diffused out of brass into

75 Cu more rapidly than Cu atoms diffused into brass. Due to a net loss of Zn atoms, voids can be observed in brass. From theoretical point of view, Kirkendall's effect is very important in diffusion. We may note that the practical importance of this effect is in metal cladding, sintering and deformation of metals (creep) Diffusion Coefficient: Fick s Laws of Diffusion Diffusion can be treated as the mass flow process by which atoms (or molecules) change their positions relative to their neighbours in a given phase under the influence of thermal energy and a gradient. The gradient can be a concentration gradient; an electric or magnetic field gradient or a stress gradient. We shall consider mass flow under concentration gradients only. We know that thermal energy is necessary for mass flow, as the atoms have to jump from site to site during diffusion. The thermal energy is in the form of the vibrations of atoms about their mean positions in the solid. The classical laws of diffusion are Fick's laws which hold true for weak solutions and systems with a low concentration gradient of the diffusing substance, dc/dx (= C C 1 /X X 1 ), slope of concentration gradient. (i) Fick's First Law: This law describes the rate at which diffusion occurs. This law states that dc dn D a dt () dx i.e. the quantity dn of a substance diffusing at constant temperature per unit time t through unit surface area a is proportional to the concentration gradient dc/dx and the coefficient of diffusion (or diffusivity) D (m /s). The 'minus' sign implies that diffusion occurs in the reverse direction to concentration gradient vector, i.e. from the zone with a higher concentration to that with a lower concentration of the diffusing element. The equation () becomes: dn D dt Defects in crystals and Elements of Thin Films dc dx dn dc J 1 D (3) a dt dx where J is the flux or the number of atoms moving from unit area of one plane to unit area of another per unit time, i.e. flux J is flow per unit cross sectional a

76 area per unit time. Obviously, J is proportional to the concentration gradient. The negative sign implies that flow occurs down the concentration gradient. Variation of concentration with x is shown in Fig. 5. We can see that a large negative slope corresponds to a high diffusion rate. In accordance with Fick's law (first), the B atoms will diffuse from the left side. We further note that the net migration of B atoms to the right side means that the concentration will decrease on the left side of the solid and increase on the right as diffusion progress. Fig.5 Model for illustration of diffusion : Fick s first law. We note that the concentration of B atoms in the direction indicates the concentration profile This law can be used to describe flow under steady state conditions. We find that it is identical in form to Fourier's law for heat flow under a constant temperature gradient and Ohm's law for current flow under a constant electric field gradient. We may see that under steady state flow, the flux is independent of time and remains the same at Defects in Crystal any cross-sectional plane along the diffusion direction. Diffusion coefficient (diffusivity) for a few selected solute solvent systems is given in Table.1.

77 Parentheses indicate that the phase is metastable (ii) Fick s second Law: This is an extension of Fick s first law to non steady flow. Frick s first law allows the calculation of the instaneous mass flow rate (Flux) past any plane in a solid but provides no information about the time dependence of the concentration. However, commonly available situations with engineering materials are non-steady. The concentration of solute atom changes at any point with respect to time in non-steady diffusion. If the concentration gradient various in time and the diffusion coefficient is taken to be independent of concentration. The diffusion process is described by Frick s second law which can be derived from the first law: dc dt dc dc D dx dx (4) Equation 4 Fick s second law for unidirectional flow under non steady conditions. A solution of Eq. (4)given by A c( x, t) exp ( x / 4Dt) (4a) Dt Where A is constant Let us consider the example or self diffusion or radioactive nickel atoms in a non-radioactive nickel specimen. Equation (4a) indicates that the concentration at x = falls with time as r -1 and as time increases the radioactive penetrate deeper in the metal block [Fig.6 ] At time t 1 the concentration of radioactive atoms at x = is c 1 = A/(Dt 1 ) 1/. At a distance x 1 = (Dt 1 ) 1/ the concentration falls to 1/e of c 1. At time t. the concentration at x =

78 Defects in crystals and Elements of Thin Films is c = A/(Dt ) 1/ and this falls to 1/e and x = (Dt ) 1/. These results are in agreement with experiments. Fig.6. The radioactive sheet of Nickel (shown by shaded section) is kept in contact with a block of nonradioactive nickel. Radioactive atoms diffuse from the sheet to the bulk metal and can be detected as a function of time. In figure, the diffusion of atoms is shown (i) for t= (ii) for t 1, and (iii) t with t > t 1 If D is independent of concentration, Eq. (4) simplifies to dc d c D (5) dt dx Even though D may vary with concentration, solutions to the differential Eq. 5 are quite commonly used for practical problems, because of their relative simplicity. The solution to Eq.5 for unidirectional diffusion from one medium to another a cross a common interface is of the general form. c( x, t) A Berf ( x/ Dt (5a) Where A and B are constant to be determined from the initial and boundary conditions of a particular problem. The two media are taken to be semi-infinite i.e. only one end of each of them, which the interface is defined. The other two ends are at an infinite distance The initial uniform concentrations of the diffusing species in the two media are different, with an

79 abrupt change in concentrations at the interface erf in eqn.5 (a) stands for error function, which is erf x Dt x / Dt exp( ) d (5a) is an integration variable, that gets deleted as the limits of the integral are substituted. The lower limits of the integral is always zero, while the upper limit of the integral is the quantity, whose function is to be determined is a normalization factor. The diffusion coefficient D (m /s) determines the rate of diffusion at a concentration gradient equal to unity. It depends on the composition of alloy, size of grains, and temperature. Solutions to Fick s equations exist for a wide variety of boundary conditions, thus permitting an evaluation of D from c as a function of x and t. A schematic illustration of time dependence of diffusion is shown in fig7. The curve corresponding to the concentration profile at a given instant of time t 1 is marked by t 1. We can see from fig.7 at a later time t, the concentration profile has changed. We can easily see that this changed in concentration profile is due to the diffusion of B atoms that has occurred in the time interval t -t 1 The concentration profile at a still later time t 3 is marked by t 3. Due to diffusion, B atoms are trying to get distributed uniformaly throughout the solid salutation. From Fig. 7 Its is evident that the concentration gradient becoming less negative as time increases. Obviously, the diffusion rate becomes slower as the diffusion process progress. Defects in Crystal

80 Fig. 7. Time dependence of diffusion (Fick s second law) Dependence of Diffusion Coefficient on Temperature The diffusion coefficient D (m /s) determines the rate of diffusion at a oncentration gradient equal unity. It depends on the composition of alloy, size of grains, and temperature. The dependence of diffusion coefficient on temperature in a certain temperature range is described by Arrhenius exponential relationship Defects in crystals and Elements of Thin Films D = D exp (-Q/RT) (6) Where D is a preexponential (frequency) factor depending on bond force between atoms of crystal lattice Q is the activation energy of diffusion: where Q = Q v +Q m, Q v and Q m are the activation energies for the formation and motion of vacancies respectively, the experimental value of Q for the diffusion of carbon in -Fe is.1 k cal/mole and that of D is 1-6 m /s and R is the gas constant. Factors Affecting Diffusion Coefficient (D) We have mentioned that diffusion co-efficient is affected by concentration. However, this effect is small compared to the effect of temperature. While discussion diffusion mechanism, we have assumed that atom jumped from one lattice position to another. The rate at which atoms jumped mainly depends on their vibrational frequency, the crystal structure. Activation energy and temperature we may note that at the position. To overcome this energy barrier, The energy required by the atom is called the activation of diffusion (Fig. 8)

81 Fig. 8. Activation energy for diffusion (a) vacancy mechanism (b) interstitial mechanism The energy is required to pull the atom away from its nearest atoms in the vacancy mechanism energy is also required top force the atom into closer contact with neighbouring atoms as it moves along them in interstitial diffusion. If the normal inter- atomic distance is either increases or decrease, addition energy is required. We may note that the activation energy depends on the size of the atom. i.e. it varies with the size of the atom, strength of bond and the type of the diffusion mechanism. It is reported that the activation energy required is high for large- sized atoms, strongly bonded material, e.g. corundum and

82 Defects in Crystal tungsten carbide (since interstitial diffusion requires more energy than the vacancy mechanism.) Applications of Diffusion Diffusion processes are the basis of crystallization recrystallization, phase transformation and saturation of the surface of alloys by other elements, Few important applications of diffusion are : (i) (ii) (iii) (iv) (v) Oxidation of metals Doping of semiconductors. Joining of materials by diffusion bonding, e.g. welding, soldering, galvanizing, brazing and metal cladding Production of strong bodies by sintering i.e. powder metallurgy. Surface treatment, e.g. homogenizing treatment of castings, recovery, recrystallization and precipitation of phases. (vi) Diffusion is fundamental to phase changed e.g. y to -iron. Now, we may discuss few applications in some detail. A common example of solid state diffusion is surface hardening of steel, commonly used for gears and shafts. Steel parts made in low carbon steel are brought in contact with hydrocarbon gas like methane (CH 4 ) in a furnace atmosphere at about 97 C temperature. The carbon from CH 4 diffuses into surface of steel part and theory carbon concentration increases on the surface. Due to this, the hardness of the surface increase. We may note that percentage of carbon diffuses in the surface increases with the exposure time. The concentration of carbon is higher near the surface and reduces with increasing depth Fig. (9)

83 Defects in crystals and Elements of Thin Films Fig. 9. C gradient in 1 steel carburized in 1.6% CH 4, % CO and 4%H. Check Your Progress 3 Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit What is diffusion and on what variable it depends? IONIC CONDUCTIVITY It is known that the dominant lattice defect responsible for the ionic conductivity in pure and doped lead chloride is the anion vacancy (Jost 195). The activation energy for migration of the anion vacancy has been measured by Simkovich (1963), Seith (De Vries 1965) and Gylai (De Vries 1965) in powder samples and is found to range from -48 ev to - 4 ev. The measurements on single crystals of pure and doped lead chloride, however, show that the energy of formation of vacancies is 1-66 ev and that for migration of the anion vacancies is -35 ev (De Vries and Van Santen 1963; De Vries 1965). Theroles of various point defects in this material are not yet clearly understood. Simkovich, fox example, concluded that in the extrinsic region half of the anion vacancies are associated with cation vacancies to form charged pairs. Barsis and Taylor (1966), on the other hand, proposed that appreciable number of inteistitials, i.e., unassociated Frenkel defects, are present in the extrinsic region as seen from the analysis of isotherms obtained by them from the data of De Vries and Van Santen. The recent experiments by Van den Brom etal (197)

84 on the dielectric relaxation in pure lead chloride suggest that in this region dipole species such as anion vacancy-impurity associates are piesent. In this paper, we shall present the results of self-diffusion and ionic conductivity measurements made on pure crystals of lead chloride, and show that ir this material Defect in Crystal Schottky defects are mainly responsible for the observed ionic transport and that the impurity anion vacancy associates, particularly the oxygen ions, influence it markedly in the extrinsic region. 4.5 COLOUR CENTRES Colour centres: Becquerel discovered that a transparent NaCl crystal was coloured yellowish when it was placed near a discharge tube. The colouration of the NaCl and other crystals was responsible for the study of colour centres. Actually, rocksalt should have an infrared absorption due to vibrations of its ions and an ultraviolet absorption due to the excitation of the electrons. A perfect NaCl crystal should not absorb visible light and so it should be perfectly transparent. This leads us to the conclusion that the colouration of crystals is due to defects in the crystals. It is also found that exposure of a coloured crystal to white light can result in bleaching of the colour. This gives further clues to the nature of absorption by crystals. Experiments show that during the bleaching of the crystal the crystal becomes photoconductive. i.e., electrons are excited to the conduction band. Photoconductivity tells us about the quantum efficiency (number of free electrons produced per incident photon) of the colour centres. It is known that insulators have large energy gaps and that they are transparent to visible light. Ionic crystals have the forbidden energy gap of about 6eV which corresponds to a wavelength of about A in the ultraviolet region. From dielectric properties we know that the ionic polarizability resonates at a wavelength of 6 microns in the far infrared region. It is why these crystals are expected to be transparent over a wide range of spectrum including the visible region. Due to such a good transparency, the crystals of KCl, NaCl, LiF and other alkali halides are used for making prisms, lenses and optical windows in optical and infrared spectrometers. However, due to different reasons, absorption bands may occur in the visible, near ultraviolet and near infrared regions in these crystals. If the absorption band is in the visible region and the band is quite narrow, it gives a characteristic colour to

85 the crystal. When the crystal gets coloured, it is said to have colour centres. Thus a colour Defects in crystals and Elements of Thin Films centre is a lattice defect, which absorbs light. It is possible to colour the crystals in a number of different ways as described below: (i) Crystals can be coloured by the addition of suitable chemical impurities like transition element ions with excited energy levels. Hence alkali halide crystals can be coloured by ions whose salts are normally coloured. (ii) The crystals can be coloured by introducing stoichiometric excess of the cation by heating the crystal in the alkali metal vapour and then cooling it quickly. The colours produced depend upon the nature of the crystals e.g., LiF heated in Li vapour colours it pink, excess of K in KCl colours it blue and an excess of Na in NaCl makes the crystal yellow. Crystals coloured by this method on chemical analysis show an excess of alkali metal atoms, typically 1 16 to 1 19 per unit volume. (iii) Crystals can also be coloured or made darker by exposing them to high energy radiations like X-rays or ϒ-rays or by bombarding them with energetic electrons or neutrons F Centres: The simplest and the most studied type of colour centre is an F centre. It is called an F centre because its name comes from the German word Farbe which means colour. F centres are generally produced by heating a crystal in an excess of an alkali vapour or by irradiating the crystal by X rays, NaCl is a very good example having F centres. The main absorption band in NaCl occurs at about

86 465A and it is called the F band. This absorption in the blue region is said to be Defect in Crystal responsible for the yellow colour produced in the crystal. The F band is characteristic of the crystal and not of the alkali metal used in the vapour i.e., the F band in KCI or NaCl will be the same whether the crystal is heated in a vapour of sodium or of potassium. The F bands associated with the F centres of some alkali halide crystals are shown in fig. 1, in which the optical absorption has been plotted against wavelength or energy in ev Formation of F-Centres: Colour centres in crystals can be fanned by their non-stoichiometric properties i.e., when crystals have an excess of one of its constituents. NaCl crystal can therefore be coloured by heating it in an atmosphere of sodium vapour and then cooling it quickly. The excess sodium atoms absorbed from the vapour Split up into electrons and positive ions in the crystal (fig. 11). The crystal becomes slightly non-stoichiometric, with more sodium ions than chlorine ions. This results in effect in CI - vacancies. The valence electron of the alkali atom is not bound to the atom, it diffuses into the crystal and becomes bound to a vacant negative ion site at F because a negative ion vacancy in a perfect periodic lattice has the effect of an isolated positive charge. It just traps an electron in order to maintain local charge neutrality. The excess electron captured in this way at a negative ion vacancy in an alkali halide crystal is called an F centre. This electron is shared largely by the six positive metal ions adjacent to the vacant negative lattice site as shown in -dimensions by the dotted circle in fig. 11. The figure shows an anion vacancy and an anion vacancy with an associated electron, i.e., the centre. This model was first suggested by De-Boer and was further developed by Mott and Gurney. Change of Density: Since some Cl - vacancies are always present in a NaCl crystal in thermodynamic equilibrium, any sort of radiation which will cause electrons to be knocked

87 Defects in crystals and Elements of Thin Films into the Cl - vacancies will cause the formation of F centres. This explains Becquerel's early results also. With that the generation of vacancies by the introduction of excess metal can be experimentally demonstrated by noting a decrease in the density of the crystal. The change of density is determined by X-ray diffraction measurements. Energy Levels of F -centres: Colour centres are formed when point defects in a crystal trap electrons with the resultant electronic energy levels spaced at optical frequencies. The trapped electron has a ground; state energy determined by the surroundings of the vacancy. These energy levels lie in the forbidden energy gap and progress from relatively widely spaced levels to an almost continuous set of levels just below the bottom of the conduction band. When the crystal is exposed to white light, a proper component of energy excites the trapped electron to a higher energy level, it is absorbed in the process and a characteristic absorption peak near the visible region appears in the absorption spectrum of the crystal having F-centres. The peak does not change when an excess of another metal is introduced in the crystal if the foreign atoms get substituted for the metal atoms of the host crystal. This justifies the assumption that the absorption peak is due to transitions to excited states close to the conduction band-determined by the trapped electron. Fig. 1 shows the energy level diagram for an F centre. It also shows that the F absorption band is produced due to a transition from the ground state to the first excited state below the conduction band. Effect of temperature on F-band: We have seen above that the energy levels of an F- centre depend upon the atomic surroundings of vacancy. This means that the absorption peak should shift to shorter wavelengths i.e., higher energies when the interatomic distances in the crystal are decreased. This shift is actually observed on varying the temperature of the crystal. The absorption maximum has a finite breadth even at very low temperature, which increases on increasing the temperature of the crystal. It can be explained by studying the dependence of the energy of-a colour

88 Defect in Crystal centre on temperature. Fig. 13. Shows a graph plotted between the changes in energy of an electron in F centre and the coordination of a vacancy i.e., the distance from centre of vacancy to nearest ions surrounding it E denotes the excited state of the electron bound to a CI - vacancy and G is for the ground state of that electron. At any finite temperature the ground state is not at, the minimum of curve G but lies above it by about ke because the coordinating ions vibrate between A and B due to thermal energy. Hence the energy of the absorbed radiation can range between that of transition A A` or B B`. The difference between energies, corresponding to A` and B` gives the width of the absorption peak. AB represents the amplitude of vibration of ions at a lower temperature but as the temperature rises it moves to a higher energy position so that CD represents the amplitude of vibration at the higher temperature and thus the width of the absorption peak- the F band increases. Klcinschord observed that the F band instead of being exactly like a bell, ossesses a shoulder and a tail on the short wavelength side. Seitz called the shoulder as a K-band and it may be considered to be due to transitions of the electron to excited states, which lie between the first excited state and the conduction band. The tail may be supposed to be due to the transition from the ground state of F-centre to the conduction band. Magnetic Properties of F-Centres: In fig. 13, the upper curve E is determined by the change in the surroundings of a vacancy when the trapped electrons is in the excited state. This is usually expressed by a change of the effective dielectric constant in the neighbourhood of such a vacancy. An alkali halide crystal is normally diamagnetic because the ions have closed outer shells. Since an F-centre contains an unpaired trapped electron, crystals additively coloured with a metal have some paramagnetic

89 behavior. Thus the structure of F-centres can be studied by electron paramagnetic resonance experiments which tell us about the wave-functions of the trapped electron V Centres :.Till now we had been considering the electronic properties associated with an excess of alkali metal. It is, however, quite natural to think what will happen if we have an excess of halogen in alkali halides. Thus if an alkali halide crystal is heated in a halogen vapour, a stoichiometric excess of halogen ions is introduced in it, the accompanying cation vacancies trap holes just as the anion vacancies trap electrons in F centres. Thus we should expect a whole new series of colour centres, which are produced by excess alkali metal atoms. The new centres have holes in place of electrons. The colour centres produced in this way are called V centres and the crystals having these centres show several absorption maxima which are called as V 1, V bands and so on. Mollwo was able to introduce access halogen into KBr and KI and found that it is was not possible in case of KCl. He shows that by heating KI in iodine vapour,new absorption bands are obtain in the ultraviolet.the bands obtain by Mollwo for KBr when heated in Br vapour are shown in fig. 14, having V 1, V and V 3 bands. The formation of V centres can be explained on the same lines as for F centres. The excess bromine enters the normal lattice positions as negative ions. Positive holes are thus formed which are situated near a positive ion vacancy where they can be trapped. Fig.15 Proposed models of V centres after Seitz Nagamiya A hole trapped at a positive ion vacancy forms a V centre as shown in fig. 15. The optical absorption associated with a trapped hole may be due to the transition of an electron from the filled band into the hole.

90 It can be understood that the strong peak observed by Mollwo in KBr as shown in fig. 14 is however, not of the above type. Mollwo's experiment proves that the saturation density of colour centres is proportional to the number of bromine molecules at a particular temperature. By the law of mass action, we know that one colour centre should be produced by each molecule absorbed from the vapour. Hence it was proposed by F. Seitz that the centres associated with the strong peak are of molecular nature, i.e., two holes are trapped by two positive ion vacancies. Such a centre is called a V centre and is shown in fig. 15. As is evident from the figures 13 and 15, the V 1 centre is the counterpart of the F-centre, V and V 3 are those of the R centres and V 4 is the counterpart of the M centre. However, the identification of the V 1 centre with the V 1 band is uncertain because the spin resonance results of Kaenzig suggest that a centre having the symmetry of the V 3 centre produces the V 1 band. The detailed properties of V centres have not yet been properly understood. Production or Colour Centres by X-rays or Particle Irradiation: The colour centres can also be produced in crystals by irradiating them with very high energy radiation like X -rays or ϒ rays. An X-ray quantum when passes through an ionic crystal produces fast photo electrons having the energy nearly equal to that of the incident quantum. These high energy electrons interact with the valence electrons in the crystal and lose their energy by producing free electrons and holes, excitons (electron hole pairs) and phonons. These free electrons and holes diffuse into the crystal and come across vacancies present in the crystal where they may be caught producing trapped electrons and holes. In this way both F and V types of colour centres are produced in crystals irradiated with high energy radiations. However, these are not permanent like those produced in non stoichiometric crystals in which there is an internally produced excess of electrons and holes. Their colours cannot be removed permanently without changing them chemically. The colour centres produced by X-ray radiation are easily bleached by visible light or by heating because the excited electrons and holes ultimately recombine with each

91 Defects in crystals and Elements of Thin Films other. The F and V centres produced by irradiation with 3 kev X -rays at room temperature ( C) have been shown in fig. 16 in the absorption spectrum of KCl taken by Dorendorf and Pick. Check Your Progress 4 Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit What are color centers and how do they affect electric conductivity of solids? EXCITIONS

92 The most obvious point defects consist of missing ions (vacancies), excess ions (interstitials), or the wrong kind of ions (substitution impurities). A more subtle possibilitials is the case of an ion in a perfect crystal, that differs from its colleagues only by being in an excited electronic state. Such a defect is called a Frenkel exciton. Since any ion is capable of being so excited, and since the coupling between Defects the in ions Crystal outer electronic shells is strong, the excitation energy can actually be transferred from ion to ion. Thus the Frenkel exciton can move through the crystal wit\hout the ions themselves having to change places, as a result of which it is (like the polaron) for more mobile than vacancies, interstitials, or substitutional impurities. Indeed, for more accurate to describe the electronics structure of a crystal containing an exciton, as a quantum mechanical superposition of states, in which it is equally probable that the excitation is associated with any ion in the crystal. This latter view bears the same relation to specific excited ions, as the Bloch tight binding levels (Chapter 1) bear to the individual atomic levels, in the theory of band structures. Thus the exciton isprobably better regarded as one of the more complex manifestions of electronic band structure that as a crystal defect. Indeed, once one recognizes that the proper description of an exciton is really a problem in electronic band structure, one can adopt a very different view of the same phenomenon: Suppose we have calculated the electronic ground state of an insulator in the independent electron approximation. The lowest excited state of the insulator willevidently be given by removing one electron from the highest level in the highest occupied band 9the valence band) and placing it into the lowest lying level of the lowest unoccupied band (conduction band). Such a rearrangement of the distribution of electrons does not alter the self- consistent periodic potential in which they move. This is because the Bloch electron are not localized (since nk (r) is periodic), and therefore the change in local charge density produced by changing the level of a single electron will be of order 1/N (since only an Nth of the electron's charge will be in any given cell) i.e. negligibly small. Thus the electronic energy levels do not have to be recomputed for the excited configuration and the first excited state will lie an energy c - v above the energy of the ground state, where c is the conduction band minimum and v the valence band maximum. However, there is another way to make an excited state. Suppose we form a one-electron level by superposing enough level near the conduction band minimum to form a welllocalized wave packet. Because we need levels in the neighborhood of the minimum to

93 Defects in crystals and Elements of Thin Films produce the wave packet, the energy c of the wave packet will be somewhat grater than c. Suppose in addition that the valence band level we depopulate is also wave packet., formed of levels in the neighborhood of the valence band maximum (so that its energy v is somewhat less than v ) and chosen so that the center of the wave packet is spatially very near the center of the conduction band wave packet. If we ignored electron electron interactions, the energy required to move an electron from valence to conduction band wave packet. If we ignored electron- electron interactions, the energy required to move an electron from valence to conduction band wave packets would be c - v > c - v, but because the levels are localized, there will, in addition, be a non negligible amount of negative Coulomb energy due to the electrostatic attraction of the (localized) conduction band electron and (localized) valence band hole. This additional negative electrostatic energy can reduce the total excitation energy to an amount that is less than c - v, so the more complicated type of excited state, in which the conduction band electron is spatially correlated with the valence band hole it left behind, is the true lowest excited state of the crystal. Evidence for this is the onset of optical absorption at energies below the inter band continuum threshold the following elementary theoretical argument, indicating that one always does better by exploiting the electron hole attraction: Let us consider the case in which the localized electron and hole levels extend over many lattice constants. We may then make the same type of semi classical argument that we used to deduce the form of the impurity levels in semiconductors. We regard the electron and hole as particles of mass m c and m v (the conduction and valence band effective masses, which we take, for simplicity, to be isotropic). They interact through an attractive Coulomb interaction screened by the dielectric constant of the crystal. Evidently this is just the hydrogen atom problem, with the hydrogen atom reduced mass (1/ = 1/M proton + 1/m electron 1/m electron ) replaced by the reduced effective mass m* (1/m* = 1/m c + 1/m v ), and the electronic charge replaced by e /. Thus there will be bound states, the lowest of which extends over a Bohr radius given by:. Defects in Crystal a ex m *( e / ) m a m *

94 the energy of the bound state will be lower than the energy ( c - v ) of the non-interacting electron and hole by E ex ( e / ) m * 1 * a m m * 1 m (13.6) ev e a The validity of this model requires that a ex be large on the scale of the lattic (i.e., a ex >>a ), but since insulators with small energy gaps tend to have small effective masses and large dielectric constants, that is no difficult to achieve, particularly in semiconductors. such hydrogenic spectra have in fact been observed in the optical absorption that occurs below the inter and threshold. The exciton described by this model is known as the Mott- Wannier exciton Evidently as the atomic levels out of which the band levels are formed become more tightly bound will decrease m* will increases, a * will decrease, the exciton will become more localized, and the Mott- Wannier picture will eventually break down. The Mott- Wannier exciton and the Frenkel exciton are opposite extremes of the same phenomenon. In the Frenkel case, based as it is on a single excited ionic level, the elelctron and hole are sharply localized on the atomic scale. The exciton spectra of the solid range gases fall in this class. 4.7 GENERAL IDEA OF LUMINESCENCE When a substance absorbs energy in some form or other, a fraction of the absorbed energy may be re-emitted in the form of electromagnetic radiation in the visible or near-visible region of the spectrum. This phenomenon is called luminescence, with the understanding that this term does not include the emission of blackbody radiation, which obeys the laws of Kirchhoff and Wien. Luminescent solids are usually referred to as phosphors. Luminescence is a process, which involves at least two steps: the excitation of the electronic system of the solid and the subsequent emission of photons. These steps may or may not be separated by intermediate processes. Excitation may be achieved by bombardment with photons (photoluminescence: with

95 Defects in crystals and Elements of Thin Films electrons (cathodo luminescence), or with other particles. Luminescence can also be induced as the result of a chemical reaction (chemi luminescence) or by the application of an electric field (electro luminescence) When one speaks of fluorescence, one usually has in mind the emission of light during excitation; the emission of light after the excitation ha ceased is then referred to as phosphorescence or afterglow. These definitions are not very exact since strictly speaking there is always a time la between a particular excitation and the corresponding emission of photon, even in a free atom. In fact, the lifetime of an atom in an excite state for which the return to the ground state is accompanied by dipole radiation is 1-8 second. For forbidden transitions, involving quadrupole or higher-order radiation, the lifetimes may be 1-4 second or longer. One frequently takes the decay time of ~1-8 second as the demarcation line between fluorescence and phosphorescence. Some authors define fluorescence as the emission of light for which the decay time is temperature independent, and phosphorescence as the temperature-dependent part.in many cases the latter definition is equivalent to the former, but these are exceptions. One of the most important conclusions reached already in the early studies of luminescence, is that frequently the ability of a material to exhibit luminescence is associated with the presence of activators. These activators may be impurity atoms occurring in relatively small concentrations in the host material, or a small stoichiometric excess of one of the constituents of the material. In the latter case one speaks of self-activation. The presence of a certain type of impurity may also inhibit the luminescence of other centers, in which case the former are referred to as "killers." Since small amounts of impurities may play such an important role in determining the luminescent properties of solids, studies aimed at a better understanding of the mechanism of luminescence must be carried out with materials prepared under carefully controlled conditions. A great deal of progress has been made in this respect during the last two decades. Defects in Crystal A number of important groups of luminescent crystalline solids may be mentioned here. (i) Compounds which luminesce in the "pure" state. According to Randall, such compounds should contain one ion or ion group Per unit cell with an incompletely filled shell of electrons which is well screened from its surroundings. Examples are probably the manganous halides, samarium and gadolinium sulfate, molybdates, and platinocyanides.

96 (ii) The alkali halides activated with thallium or other heavy metals. (iii) ZnS and CdS activated with Cu, Ag, Au, Mn, or with an excess of one of their constituents (self-activation). (iv) The silicate phosphors, such as zinc orthosilicate (willernite, ZnSi4) activated with divalent maganese, which is used as oscilloscope screens. (v) Oxide phosphors, such as self-activated ZnO and Al 3 activated with transition metals. (vi) Organic crystals, such as anthracene activated with naphtacene these materials are often used as scintillation counters. A. 4.8 DISLOCATIONS & MECHANICAL STRENGTH OF CRYSTALS The first idea of dislocations arose in the nineteenth century by observations that the plastic deformation of metals was caused by the formation of slip bands in which one portion of the material sheared with respect to the other. Later with the discovery that metals were crystalline it became more evident that such slip must represent the shearing of one portion of a crystal with respect to the other upon a rational crystal plane. Volterra and Love while studying the elastic behaviour of homogene-ous isotropic media considered the elastic properties of a cylinder cut in the forms shown in Figs. 17 (a) to (d), some of the deformation operations correspond to slip while some of the resulting configurations correspond to dislocation. The work on crystalline slip was then left out till dislocations were postulated as crystalline defects in the late 193's. The configuration (a) shows the cylinder as originally cut (b) and (c) correspond to edge dislocations while (d) corresponds to screw dislocation. Defects in crystals and Elements of Thin Films

97 After the discovery of X-rays, Darwin and Ewald found that the intensity of X-ray beams reflected from actual crystals was about times greater than that expected from a perfect crystal. In a perfect crystal, the intensity is low due to long absorption path given by multiple internal reflections. Also, the width of the reflected beam from an actual crystal is about 1 to 3 minutes of an are as compared with that expected for a perfect crystal which is only about a few seconds. This discrepancy was explained by saying that the actual crystal consisted of small, roughly equiaxed crystallites, 1-4 to 1-5 cm. In diameter, slightly misoriented with respect to one another, with the boundaries between them consisting of amorphous material. This is the "mosaic block" theory in which the size of the crystallites limits the absorption path and increases the intensity. The misorientation explains the width of the beam. It was however found recently that the boundaries of the crystallites are actually arrays of dislocation lines. The presence of dislocation lines is also proved by the study of crystal growth. Volmer's and Gibbe's theoretical study on nucleation of new layers showed that the layer growth of perfect crystals is not appreciable until supersaturation of about 1.5 were attained. However, experimental work of Volmer and Schultze on iodine showed that crystals grew under nearly equilibrium conditions. Frank removed this

98 Defects in Crystal discrepancy by saying that the growth of crystals could take place at low supersaturations by the propagation of shelves associated with the production of a dislocation at the surface. The development of the theory of dislocations was given a great impetus by the consideration of the strength of a perfect crystal. A crystal can be deformed elastically by applying stresses on it but it can regain its original condition when the stresses are removed. If the stresses applied be very large, of the order of about dynes per cm then a small amount of deformation will be left on removing these stresses and the crystal is said to suffer a plastic deformation. It will be seen that the atomic interpretation of the plastic flow of crystals requires the postulation of a new type of defect called dislocations. Mechanical Strength of Crystal : The weakness of good crystals was a mystery for many years, in part, no doubt, because the observed data easily led one to the wrong conclusion. Relatively poorly prepared crystal were found to have yield strengths close to the high value we first estimate for the perfect crystal. However, as the crystals were improved (for example, by annealing) the yield strengths were found to drop drastically, falling by several orders of magnitude in very well prepared crystals. It was natural to assume that the yield strength was approaching that of a perfect crystal as specimens were improved, but, in fact, quite the opposite was happening. Three people independently came up with the explanation in 1943, inventing the dislocation to account for the data. They suggested that almost all real crystals contain dislocations, and that plastic slip occurs through their motion as described above. There are then two ways of making a strong crystal. one is to make an essentially perfect crystal, free of all dislocation. This is extremely difficult to achieve. Another way is to arrange to impede the flow of dislocations, for although dislocations move with relative ease in a perfect crystal, if they work required to move them can increase considerably. Thus the poorly prepared crystal is hard because it is infested with dislocations and defects, and these interfere so seriously with each other's motion that slip can occur only by the more drastic means described earlier. However, as the crystal is purified and improved, dislocation largely move out of the crystal, vacancies and interstitials are reduced to their (low) thermal equilibrium concentrations, and the unimpeded motion of those dislocations that remain makes it possible for the crystal to deform with c\ease. At this point the crystal is very soft. If one could continue the process of refinement to the point where all dislocations were removed, the crystal would again become hard.

99 Defects in crystals and Elements of Thin Films 4.9 PLASTIC BEHAVIOUR Plastic deformation takes place in a crystal due to the sliding of one part of a crystal with respect to the other. This results in slight increase in the length of the crystal ABCD under the effect of a tension FF applied to it as shown in fig. 18. The 'process of sliding is called slip. The direction and place in which the sliding takes place are called respectively the slip direction as shown by the arrow P and slip plane. The outer surface of the single crystal is deformed and a slip band is formed, as is seen in the figure, which may be several thousand Angstroms wide. This can be observed by means of an optical microscope, but when observed by an electron microscope a slip band is found to consist of several slip lamellae. The examination of slips by an electron microscope reveals that these extend over several tens of lattice constants. The slip lines do not run throughout the crystal but end inside it, showing that slips do not take place simultaneously over the whole Slip planes but occur only locally. The study of slips in detail tells us that plastic deformation is inhomogeneous i.e., only a small number of those atoms take part in the slip which form layers on either side of a slip plane. In the case of elastic deformation all atoms in the crystal are affected and its properties can be understood in terms of interatomic forces acting in a perfect lattice. On the other hand, plastic deformation cannot be studied by simply extending elasticity to large stresses and strains or on the basis of a perfect lattice. We will now prove below that for plastic flow in a perfectly periodic lattice, we have to apply very much larger stresses (~ 1 1 dynes per cm ) than those required for the normal plastic flow observed in actual crystals (~1 6 dynes per cm ). Shear Strength Crystals: J. Frenkel in calculating the theoretical shear strength of a perfect crystal. The model proposed by him is given in fig. 19, showing a cross-section through two

100 adjacent atomic planes separated by a distance d. The full line circles indicate the equilibrium positions of the atoms without any external force. Defects in Crystal Let us now apply a shear stress Ʈ in the direction shown in fig.19 (a). All the atoms in the upper plane are thus displaced by an amount x from the original positions as shown by the dotted circles. In fig. (b), the Fig. 19 shear stress has been plotted as a function of the relative displacement of the planes from their equilibrium positions and this gives the periodic behavior of as supposed by Frenkel. ' is found to become zero for x =, a/, a etc., where a is the distance between the atoms in the direction of the shear. Frenkel assumed that this periodic function is given by

101 x c sin (7) a where the amplitude c denotes the critical shear strees which we have to calculate For x << a, we have as usual, x c (8) a In order to calculate the force required to shear the two planes of atoms, we from the definition of shear modulus G Stress Strain x c y x / d a where G is the shear module and y x d is the elastic strain G Defects in crystals and Elements of Thin Films x G (9) d Comparing it with equation (1), we have x x G a c G or c. a d x b or G G c, if a d x 6 (1) This gives the maximum critical stress above which the crystal becomes unstable. It is about one sixth of the shear modulus. In a cubic crystal, G c 44 = 1 11 dynes per cm. for a shear in the <1> direction. Hence the theoretical value of the critical shear stress on Frenkel's model is c = 1 1 dynes per sq..cm. which is much larger than the observed values for pure crystals. However., the experimental values for the maximum resolved shear stress required to start the plastic flow in metals were of the order of 1-3 to 1-4 G at that time and it was not a agreement with the results of eqn.. (1) Later it was considered that eqn (1) gave a higher value as the different semi-inter atomic force of Fig.19 (b) as taken by Frenkel. The above disagreement may also be due to other special configuration of mechanical stability which the lattice may develop when it is sheared. Mackenzie in 1949, using central forces in the case of close packed lattices found that c could be reduced to a value G/3, corresponding to a critical shear strain of about

102 This value, however, is supposed to be an underestimate due to the neglect of the small directional force which are also present in such lattices. The contributions of thermal stresses also reduce c below G/3 only near the melting point. Thus at room temperature we should have G/5 > c > G/3 i.e. = G/15. In the case of whiskers only,. the experimental value of c for various metals has been found to be of this order which is in excellent agreement with the theoretical result. Recent experimental work on bulk copper and zinc has shown that plastic deformation being at stresses of the order of 1-9 G. Hence, except for whiskers the disagreement is even larger than before.. It is therefore clear that agreement between theory and experiment be obtained on the basis of Frenkel's model where atomic plant glide past each other assuming fig. 19 (a) that the atoms of the upper atomic plane move simulantaneously relative to the lower plane. Defects in Crystal This assumption is based on the supposition of a perfect lattices, and that is the main cause of difficulty. We have, therefore, to consider the presence of imperfections which act as sources of mechanical weakness in actual crystals and which may proud a slip by the consecutive motion of the atoms but not by simultaneous motion of the atoms of one plane relative to another. After Frenkel theory Masing and Polanyi, Pradtl and Dehlinger proposed different defects but in 1934, Orowan, Polanyi, and Taylor proposed edge dislocation, while in 1939 Burgers gave the description of screw dislocation to explain the discrepancy between c theoretical and experimental. It has now been established that the new type of defect called dislocation exists in almost all crystals and it is responsible for producing slip by the application of small stresses only 4.1 TYPE OF DISLOCATION The Edge Dislocation: Dislocation is a more complicated defect than any of the point defects. A dislocation is a region of a crystal in which the atoms are not arranged in the perfect crystal lattice. There are two extreme types of dislocations viz., the edge type and the screw type. Any particular dislocation is usually a mixture of these two types. An edge dislocation is the simplest one and a cross-sectional view of the atomic arrangement of atoms in it and the distortion of the crystal structure is shown in fig.. The part of the crystal above the slip plane at ABC has one more plane of atoms DB than the part below it. The line normal to the paper at B is called the dislocation

103 line and the symbol at B is used to indicate the dislocation. The distortion is mostly present about the lower edge of the half plane of extra atoms and so the dislocation is that line of distortion which is near the end of the half plane. Hence a dislocation is a line imperfection as compared to the point imperfections considered before. In all the dislocations, the distortion is very intense near the dislocation line where the atoms do not have the correct number of neighbours. This region is called the core of dislocation. A few atom distances away from the centre, the distortion is very small and the crystal is almost perfect locally. At the core, the local strain is very high whereas it is so small at distances away form the core that the elasticity theory can be applied and it is called the Defects in crystals and Elements of Thin Films elastic region. Another characteristic of the distortion of atomic arrangement in an edge dislocation is that the atoms just above the end of the extra half plane are in compression but just below the half plane the two rows of atoms to the right and left of the extra plane BD are farther apart from each other and the structure is expanded. This local expansion round an edge dislocation is called a dislocation. Besides the expansion and contraction near the dislocation, the structure is sheared also and this shear distortion is quite complicated. Dislocations are produced when the crystal solidifies from the melt. Plastic deformation of cold crystals also produces dislocations. 'Dislocations are of importance in determining the strength of ductile metals. These dislocations can be experimentally observed by many techniques. Electron microscopes can be used to study dislocations in their specimens of the order of a few angstroms which may transmit 1 kv electrons. It can be studied by the precipitation of impurities because the region of dilatation along a dislocation line is very suitable for their precipitation. Optical microscopes can be used to study them if the dislocations are first decorated by precipitating metallic impurities along the dislocation lines e.g., silver decoration of alkali halides. The intersection of dislocation lines with the surface of a crystal can be revealed by the etch pitch technique.

104 Fig. Fig. 1 Defects in Crystal The Screw Dislocation: The screw dislocation was introduced by Burger in It is also called Burger's dislocation. To understand this, let a sharp cut be made part way through a perfect crystal and let the crystal on one side of the cut be moved down by one atomic spacing relative to the other so that the rows of atoms are placed back into contact as shown in fig. 1. A line BD of distortion exists along the edge of the cut, which is called the screw dislocation. In this case complete planes of atoms normal to the dislocation do not exist any longer but all atoms lie on a single surface which spirals from one end of the crystal to the other and so it is called screw dislocation. The pitch of the screw may be left-handed or right-handed and one or more atom distances per rotation. The distortion is very little in regions away from the screw dislocation of while atoms near the centre are in regions of high distortion so much so that the local symmetry in the crystal is completely destroyed. In

105 this case, the atoms near the centre of the screw dislocation are not in a dilatation as in edge dislocation but are on a twisted or sheared lattice. Motion of a Dislocation: Dislocations can move just like the point defects move in the lattice but these are more constrained in motion because a dislocation must always be a continuous line. Motion of a dislocation is possible either by a climb or by a slip or by a glide. The motion of dislocation can give rise to a slip by a mechanism shown in fig.. When the upper half is pushed sideways by an amount b, then under the shear the motion of a dislocation tends to move the upper surface of the specimen to the right. Edge dislocations for which the extra half plane DB lies above the slip plane are called positive. If it is below the slip plane it is called negative edge dislocation. When an edge dislocation moves from one lattice site to another on the Fig. slip plane, the atoms in the core move slowly so that the extra half plane at one lattice position becomes connected to a plane of atoms below the slip plane and the nearby plane of atoms becomes the new extra half plane. When finally the extra half plane BD reaches the right hand side of the block, the upper half of the block has completed the slip or glide by an amount b. Defects in crystals and Elements of Thin Films Climb of a dislocation corresponds to its motion up or down from the slip plane. If the dislocation absorbs additional atoms from the crystal, it moves downward by substituting these atoms below B in the lattice. If the dislocation absorbs vacancies it moves up as the atoms are removed one by one from above B from the lattice sites. Fig.

106 4.11 STRESS FIELD OF DISLOCATION The Burger's Vector: The Burger's vector b denotes actually the dislocation-displacement vector. A dislocation can be very well described by a closed loop surrounding the dislocation line. This loop, called the Burger's circuit is formed by proceeding through the undisturbed region surrounding a dislocation in steps which are integral multiples of a lattice translation. The loop is completed by going an equal number of translation in a positive sense and negative sense in a plane normal to the dislocation line. Such a loop must close upon itself if it does not enclose a dislocation, or fail to do so by an amount called a Burger's vector s = n a a + n b b + n c c Where n a, n b, n c are equal to integers or zero and a, b, c are the three primitive lattice translations. Fig. 3 Defects in Crystal The Burger's circuit S134F is shown by dark line in fig. 3 for a screw dislocation. Starting at some lattice point S at the front of the Fig. 3 crystal, the loop fails to close on itself by one unit translation parallel to the dislocation line. This is the Burger's vector which always

107 points in a direction parallel to the screw dislocation. If the loop is continued, it will describe a spiral path around the Burger's dislocation just like the thread of a screw. In the figure, the height of the step on the top surface is one lattice spacing i.e., b, thus b is a vector giving both the magnitude and the vector of the dislocation. It must be some multiple of the lattice spacing so that an extra plane of atoms could be inserted to produce a dislocation. The dilatation at a point near an edge dislocation can now be described to be given by V b sin V r where b is the Burger's vector which measures the strength of the distortion caused by the dislocation, r is the radial distance from the point to the dislocation line and is the angle between the radius vector and the slip plane.as shown in fig.. Similarly, the atoms which are on a sheared lattice in a screw dislocation being on a spiral ramp, are displaced from their original positions in the perfect crystal according to the equation of a spiral ramp i.e. b u z where the z-axis lies along the dislocation and u, is the displacement in that direction. The angle is measured from one axis perpendicular to the dislocation. Thus when increases by the displacement increases by a quantity b, the Burger's vector, which measures the strength of the dislocation. The Burger s vector of a screw dislocation is parallel to the dislocation line while that of an edge dislocation, it is Perpendicular to the dislocation line and lies in the slip plane. In general cases, the Burger's vector may have other directions with respect to the dislocation and for these cases the dislocation is a mixture of both edge and screw types. Thus the mixed dislocation is defined in terms of the direction of the Defects in crystals and Elements of Thin Films Burger's vector. Stress Fields around Dislocations : Stress field of a screw dislocation: We know that the core of dislocations is a region within a few lattice constants of the centre of dislocation and that it is a "bad" region where the atomic arrangement of the crystal is severely changed from the regular state. The regions outside the core are "good" regions and the strains in these regions are elastic strains and so these can be treated by the theory of elasticity as an elastic continuum and the core region can be added later as a proper correction term. The

108 calculation of dislocation energy is simple for a straight screw dislocation but similar results are obtained for edge dislocation. Let us have a cylindrical shell of a material surrounding an axial screw dislocation. Let the radius of the shell be r and the thickness dr, The circumference of the shell is be sheared by an amount b, so that the shear strain r and let it Fig. 4 b e (11) r and the corresponding shear stress in the good region is, G. b G. e. (1) r where G is the shear modulus or modulus of rigidity of the material. A distribution of forces is exerted over the surface of the cut for producing a displacement b and the work done by the forces to do it gives the energy E s of the screw dislocation.

109 Hence, Defects in Crystal E S F. b. da (13) where F is the average force per unit area at a point on the surface during the displacement and the integral extends over the surface area of the cut. The average value is to be taken because the force at a point builds up linearly from zero to a maximum value as the displacement is produced. Thus the average force F (1/ ) is half the final value when the displacement is b i.e., G b F (14) 4r Putting it in (13), we get E S Gb da. 4 r But da = dz sr and so for a dislocation of length l, we have E S R Gb da r 4r Gb 1 R log 4 r (15) (16) Thus, total elastic energy per unit length of a screw dislocation is given by E S Gb R log (17) 4 r where R and r the proper upper and lower limits of r. The energy depends upon the values taken for R and r is suitable when it is equal to about the Burger's vector b or equal to one or two lattice constants and the value of R is not more than the size of the crystal. Actually, however in most cases K is very much smaller than the size of the crystal. The value of R/r is not important as it occurs in the logarithmic term. Stress field of an edge dislocation: The calculation of the stress field is done on the assumption that the medium is isotropic having a shear modulus G and Poisson's ratio ʋ Let us consider the cross-section of a cylindrical material of radius R whose axis is along the z-

110 axis and in which a cut has been in the plane y =, which becomes the slip plane. The portion above the cut is now slipped to the left by an amount b, the Burger's vector along Defects in crystals and Elements of Thin Films the x-axis so that the new position assumes the shape shown dotted in fig. 5. Thus, a positive edge dislocation has been produced along the z-axis. Let σ rr be the radial tensile stress, ie, compression or tension along the radius r and let σ θθ be the circumferential tensile stress i.e., compression or tension acting in a plane perpendicular to r. Let τ r θ denote the shear stress acting in a radial direction. As seen from fig., it is an odd function of x, considering the plane y = and is found to be proportional to (cos /r). In an isotropic elastic continuum σ rr and σ θθ compression or tension acting in a plane perpendicular to r. are found to be proportional to (sin /r) because we require a function which varies as 1/r and which changes sign when y changes sign. Also it can be shown dimensionally that the constants of proportionality in the stress vary as G and b. Without giving the details of calculations here, the stress field of the edge dislocation in terms of r and are given by the following :. rr Gb sin. (18) 4 (1 v) r and r Gb cos. (1 v) r..(19) where the positive values of σ are for tension and negative values for compression. Above the slip plane σ rr is negative giving a compression, below the slip plane, it corresponds to a tensile stress. It may be noted that for r =, the stresses become infinite and so a small cylindrical region of radius r o around the dislocation must be excluded. This is necessary because in the bad region, the theory of elasticity

111 Fig. 5 does not hold as the stresses near a dislocation are very large. To know the value of r o, let us put r o = b, the magnitude of the strain there is then of the order of 1/π(1-v) 1/4 which is too much large to be treated by Hooke's law. We shall now calculate the energy of Defects in Crystal formation of an edge dislocation of unit length. The final shear stress in the plane y = is given by (19) by putting θ =. For a cut along z-axis in a unit length, the strain energy for edge dislocation will be given by E e 1 1 R Gb Gb stress x strain dr r r(1 v) 4 r(1 log v) R r.() This shows that the energy of formation becomes infinite if R becomes infinite. But even in large crystals the stress field are actually displaced some distance by other dislocation so that R = 1-3 cm. Assuming r = cm. for a dislocation in copper E e = 31-4 erg/cm = ev/atom plane Since G = dynes/cm b =.51-8 cm. and v =.34 In the case of screw dislocations its value is about (/3) of this. The core energy of edge dislocation should be added to the elastic strain energy but it is of the order of 1eV per atom plane which is much less than the elastic strain energy and can be neglected. For a screw dislocation in the Z-direction in a cylindrical material, the stress field is given by a shear stress, according to (1). z Gb z.(1) r

112 Fig. 6 Low angle grain boundary (a) Two crystals Joined Together (b) Grain boundary formed with rows of dislocations. There is no tensile and compress ional stress in this expression and this is perhaps due to the fact that there is no extra half plane in a screw dislocation. Also in this case the stresses are independent of θ expecting thereby that the stress field is cylindrically Defects in crystals and Elements of Thin Films symmetric. We can also explain the free energy of a dislocation. The contribution to the free energy by entropy, in a dislocation, is very small as compared to the strain energy and so the free energy in crystals of ordinary size at room temperature can be assumed to be nearly equal to the strain energy. Since the strain energy is positive, the free energy increases by the formation of dislocation. Hence no dislocation can exist as a thermodynamically stable lattice defect 4.1 GRAIN BOUNDARIES Burger suggested that the boundaries of two crystallites or crystal grains at a low angle inclination with each other can be. Considered to be a regular array of dislocations. Two such crystallites placed close together at a small angle θ have been shown in fig. 6 (a). There are simple cubic crystals with

113 their axes perpendicular to the plane of the paper and parallel. The crystals have been rotated by θ / left and right of these axes. The results of joining the two crystals together is shown in fig. 6 (b). A grain boundary of the simple example of Burger's model is formed. The boundary plane contains a crystal axis common to the two crystals. Such a boundary is called a pure tilt boundary. Crystal orientations on both sides of the boundary plane are symmetric with each other such a boundary has a vertical arrangement of more than two edge dislocations of same sign. This arrangement is also stable as that for two dislocations. From the figure it is seen that the interval D between the dislocations so formed is given by tan b b or D D.() Where b is the Burger s vector of the dislocations and is small Burger s model of low angle grain boundary has been was confirmed experimentally by Vogel and co-workers for germanium single crystals. A germanium crystal was grown from a seeded melt along <1> direction. When the surface of this crystal was etched with a suitable chemical (acid), the terminus of a dislocation at the surface become a nucleus of the etching. action and a row of each pits was formed. It is shown diagrammatically in fig. 7. On Defects in Crystal examining these boundaries under very high optical magnification they were found to consist of regularly spaced conical pits. By counting the number of these etch pits, we are able to find out the number of dislocations in the crystal grain boundaries. The distance D between the pits is obtained by counting. The relative inclination angle was also measured by means of X ray diffraction experiments. From this value of and knowing the value of b = 4. A in germanium, the value of D was calculated theoretically. This was found to be in very good agreement with the experimental etch pit interval.

114 Fig. 7 Diagram of optical micrograph of a low angle grain boundary in Ge If is less than 5, the value of D is quite large as compared with the interatomic distance and so each dislocation can be considered as isolated. If is about 15 than D is only a few interatomic distance and we get a collection of irregular, diffused and deformed vacancies. At present the etch pit method is the most direct method of determining the dislocation density. The density of dislocations is the number of dislocation lines which intersect a unit area in the crystal. It ranges from in the best germanium and silicon crystals to dislocations/cm in heavily deformed metal crystals. The density of dislocations can be estimated in solids by the following methods: (i) By plastic deformation of crystals, just like the bending of a pack of playing cards. (ii) By X-ray transmission method. (iii) By X-ray reflection. Defects in crystals and Elements of Thin Films (iv) By electron microscopy (v) By measuring the increase in the electrical resistivity produced by the dislocations in heavily cold worked metals (vi) By measurement on magnetic saturation of cold- worked fermagnetic materials. (vii).by decoration methods. Decorated helical dislocations can be produced in calcium fluoride by decorating particles of CaO. (viii) By etch pit methods ETCHING- TYPES OF ETCHING In order to form a functional Micro-Electro-Mechanical Systems (MEMS) structure on a substrate, it is necessary to etch the thin films previously deposited and/or the substrate itself. In general, there are two classes of etching processes: 1. Wet etching where the material is dissolved when immersed in a chemical solution

115 . Dry etching where the material is sputtered or dissolved using reactive ions or a vapor phase etchant In the following, we will briefly discuss the most popular technologies for wet and dry etching. Wet etching: This is the simplest etching technology. All it requires is a container with a liquid solution that will dissolve the material in question. Unfortunately, there are complications since usually a mask is desired to selectively etch the material. One must find a mask that will not dissolve or at least etches much slower than the material to be patterned. Secondly, some single crystal materials, such as silicon, exhibit anisotropic etching in certain chemicals. Anisotropic etching in contrast to isotropic etching means different etch rates in different directions in the material. The classic example of this is the <111> crystal plane sidewalls that appear when etching a hole in a <1> silicon wafer in a chemical such as potassium hydroxide (KOH). The result is a pyramid shaped hole instead of a hole with rounded sidewalls with a isotropic etchant. The principle of anisotropic and isotropic wet etching is illustrated in the figure below. Defects in Crystal This is a simple technology, which will give good results if you can find the combination of etchant and mask material to suit your application. Wet etching works very well for etching thin films on substrates, and can also be used to etch the substrate itself. The problem with substrate etching is that isotropic processes will cause undercutting of the mask layer by the same distance as the etch depth. Anisotropic processes allow the etching to stop on certain crystal planes in the substrate, but still results in a loss of space, since these planes cannot be vertical to the surface when etching holes or cavities. If this is a limitation for you, you should consider dry etching of the substrate instead. However, keep in mind that the cost per wafer will be 1- orders of magnitude higher to perform the dry etching If you are making very small features in thin films (comparable to the film thickness), you may also encounter problems with isotropic wet etching, since the undercutting will be at least equal to the film thickness. With dry etching it is possible etch almost straight down without undercutting, which provides much higher resolution.

116 Figure 8: Difference between anisotropic and isotropic wet etching. Dry etching: The dry etching technology can split in three separate classes called reactive ion etching (RIE), sputter etching, and vapor phase etching. In RIE, the substrate is placed inside a reactor in which several gases are introduced. A plasma is struck in the gas mixture using an RF power source, breaking the gas molecules into ions. The ions are accelerated towards, and reacts at, the surface of the material being Defects in crystals and Elements of Thin Films etched, forming another gaseous material. This is known as the chemical part of reactive ion etching. There is also a physical part which is similar in nature to the sputtering deposition process. If the ions have high enough energy, they can knock atoms out of the material to be etched without a chemical reaction. It is a very complex task to develop dry etch processes that balance chemical and physical etching, since there are many parameters to adjust. By changing the balance it is possible to influence the anisotropy of the etching, since the chemical part is isotropic and the physical part highly anisotropic the combination can form sidewalls that have shapes from rounded to vertical. A schematic of a typical reactive ion etching system is shown in the figure below. A special subclass of RIE which continues to grow rapidly in popularity is deep RIE (DRIE). In this process, etch depths of hundreds of microns can be achieved with almost vertical sidewalls. The primary technology is based on the so-called "Bosch process", named after the German company Robert Bosch which filed the original patent, where two different gas compositions are alternated in the reactor. The first gas composition creates a polymer on the surface of the substrate, and the second gas composition etches the substrate. The polymer is immediately sputtered away by the physical part of the etching, but only on the horizontal surfaces and not the sidewalls. Since the polymer only dissolves very slowly in the chemical part of the etching, it builds up on the sidewalls and protects them from etching. As a result, etching aspect ratios of 5 to 1 can be achieved. The process can easily be used to etch completely through a silicon substrate, and etch rates are 3-4 times higher than wet etching.

117 Sputter etching is essentially RIE without reactive ions. The systems used are very similar in principle to sputtering deposition systems. The big difference is that substrate is now subjected to the ion bombardment instead of the material target used in sputter deposition. Vapor phase etching is another dry etching method, which can be done with simpler equipment than what RIE requires. In this process the wafer to be etched is placed inside a Defects in Crystal chamber, in which one or more gases are introduced. The material to be etched is dissolved at the surface in a chemical reaction with the gas molecules. The two most common vapor phase etching technologies are silicon dioxide etching using hydrogen fluoride (HF) and silicon etching using xenon diflouride (XeF), both of which are isotropic in nature. Usually, care must be taken in the design of a vapor phase process to not have bi-products form in the chemical reaction that condense on the surface and interfere with the etching process. The first thing you should note about this technology is that it is expensive to run compared to wet etching. If you are concerned with feature resolution in thin film structures or you need vertical sidewalls for deep etchings in the substrate, you have to consider dry etching. If you are concerned about the price of your process and device, you may want to minimize the use of dry etching. The IC industry has long since adopted dry etching to achieve small features, but in many cases feature size is not as critical in MEMS. Dry etching is an enabling technology, which comes at a sometimes high cost. Figure 9: Typical parallel-plate reactive ion etching system.

118 Defects in crystals and Elements of Thin Films 4.14 LET US SUM UP Like anything else in this world, crystals inherently possess imperfections, or what we often refer to as 'crystalline defects'. The presence of most of these crystalline defects is undesirable in silicon wafers, although certain types of 'defects' are essential in semiconductor manufacturing. Engineers in the semiconductor industry must be aware of, if not knowledgeable on, the various types of silicon crystal defects, since these defects can affect various aspects of semiconductor manufacturing - from production yields to product reliability. Crystalline defects may be classified into four categories according to their geometry. These categories are: 1) zero-dimensional or 'point' defects; ) one-dimensional or 'line' defects; 3) two-dimensional or 'area' defects; and 4) three-dimensional or 'volume' defects. Table presents the commonly-encountered defects under each of these categories. Table. Examples of Crystalline Defects Defect Type Examples Vacancy Defects Point or Zero-Dimensional Interstitial Defects Defects Frenkel Defects Extrinsic Defects Straight Dislocations (edge or Line or One-Dimensional screw) Defects Dislocation Loops Area or Two-Dimensional Twins Defects Stacking Faults Grain Boundaries Volume or Three-Dimensional Precipitates Defects

119 Voids There are many forms of crystal point defects. A defect wherein a silicon atom is missing from one of these sites is known as a 'vacancy' defect. If an atom is located in a non-lattice site within the crystal, then it is said to be an 'interstitial' defect. If the interstitial defect involves a silicon atom at an interstitial site within a silicon crystal, then it is referred to as a 'self-interstitial' defect. Vacancies and self-interstitial defects are classified as intrinsic point defects. Defectsin Crystal If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, then it becomes a 'Schottky' defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a 'Frenkel' defect. The formation of a Frenkel defect therefore produces two defects within the lattice - a vacancy and the interstitial defect, while the formation of a Schottky defect leaves only one defect within the lattice, i.e., a vacancy. Aside from the formation of Schottky and Frenkel defects, there's a third mechanism by which an intrinsic point defect may be formed, i.e., the movement of a surface atom into an interstitial site. Extrinsic point defects, which are point defects involving foreign atoms, are even more critical than intrinsic point defects. When a non-silicon atom moves into a lattice site normally occupied by a silicon atom, then it becomes a 'substitutional impurity.' If a nonsilicon atom occupies a non-lattice site, then it is referred to as an 'interstitial impurity.' Foreign atoms involved in the formation of extrinsic defects usually come from dopants, oxygen, carbon, and metals. The presence of point defects is important in the kinetics of diffusion and oxidation. The rate at which diffusion of dopants occurs is dependent on the concentration of vacancies. This is also true for oxidation of silicon. Crystal line defects are also known as 'dislocations', which can be classified as one of the following: 1) edge dislocation; ) screw dislocation; or 3) mixed dislocation, which contains both edge and screw dislocation components.

120 An edge dislocation may be described as an extra plane of atoms squeezed into a part of the crystal lattice, resulting in that part of the lattice containing extra atoms and the rest of the lattice containing the correct number of atoms. The part with extra atoms would therefore be under compressive stresses, while the part with the correct number of atoms would be under tensile stresses. The dislocation line of an edge dislocation is the line connecting all the atoms at the end of the extra plane. Defects in crystals and Elements of Thin Films Fig. 8. An edge dislocation; note the insertion of atoms in the upper part of the lattice If the dislocation is such that a step or ramp is formed by the displacement of atoms in a plane in the crystal, then it is referred to as a 'screw dislocation.' The screw basically forms the boundary between the slipped and unslipped atoms in the crystal. Thus, if one were to trace the periphery of a crystal with a screw dislocation, the end point would be displaced from the starting point by one lattice space. The dislocation line of a screw dislocation is the axis of the screw. Figure 9. A screw dislocation; note the screw-like 'slip' of atoms in the upper part of the lattice