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Band Theory of Solids • This theory has been developed in three main stages. • 1. The Classical free electron theory: Drude and Lorentz developed this theory in the year 1900.According to this theory, the metals contains free electrons and obey the laws of classical mechanics. • 2.The Quantum free electron theory: Sommerfeld developed this theory during 1928.According to this theory, free electrons obey quantum laws. • 3.The Zone theory: Bloch stated this theory in 1928. According to this theory, the free electrons move in a periodic field provided by the lattice. This theory is also called Band theory of solids. • The energy band theory of solids is the basic principle of semiconductor physics and it is used to explain the differences in electrical properties between metals, insulators and semiconductors. • The concept of discrete allowed electron energies that occur in a single atom and large number of allowed energies of electrons are explained by band theory of solids Electron in a periodic potential – Bloch theorem: • According to Bloch a one dimensional lattice consists of large number of positive ion cores at regular intervals and the conduction electrons moves freely throughout the lattice. • The variation of potential inside the metallic crystal with the periodicity of the lattice is explained by Bloch theorem. It experiences zero Potential at centre of the ion core and maximum potential between the two ion cores. The motion of the electron through periodic lattice can be explained by Schrödinger's time independent equation. • The periodic potential V (x) may be defined by means of the lattice constant ‘a’ as V (x) = V ( x + a ) • From Schr odinger wav e equation d 2 8 2 m [ E V ] 0 2 2 dx h d 2 8 2 m [ E V ( x a )] 0 2 2 dx h • Bloch has shown that the one dimensiona l solution of the Schr odinger equation ( x) U k ( x) exp( ikx) Where U k (x) is a periodicit y of crystal lattice and can be written as U k (x) U k (x a) where k represents the state of motion of the electron Kroning – penney model • According to Kroning - Penney , the model consists of a series of potential wells and potential barriers alternately as follows. Potential barriers Square wells V0 + + -b + 0 + a + • The electrons move in a periodic potential field provided by the lattice between V = 0 and Vmax=V0. • Note: The potential of the solid varies periodically with the periodicity of space lattice.i.e. V(x) = V(x+a)……=V(x+Na). • For one dimensional periodic potential field , to describe the motion of electron, we have …. d 2 8 2 m 2 [ E V ( x)] 0 2 dx h We know that the potential experienced by the electron is Zero In square wells and becomes maximum in potential barriers. i.e, V ( x) o.... for,.0 x a and V ( x) V0 .. for,. b x 0 For Square wells the schrodinge r equation is reduced to d 2 8 2 m [ E ] 0 2 2 dx h where as for potential barriers, the equation t hat can be applied is d 2 8 2 m 2 [ E V0 ] 0 2 dx h d 2 2 0.....(1) 2 dx d 2 2 0.....(2) 2 dx where 8 2 mE h2 2 8 2 m [V0 E ] 2 h 2 The solutions of wave functions Ψ for equations (1) and (2) are of the form , according to Bloch ( x) U k ( x)eikx ....(3) Where Uk(x) is the periodicity of the lattice and can be written as U k ( x) U k ( x a) ..........U k ( x Na) Where N is an integer and K is wave propagation constant. 2 p k Differentiating equation (3) and substituting in equation (1) & (2) and And further solving we get 2 2 sinh b sin a cosh b cos a cos k (a b)...(4) 2 Kroning penney suggested that when V0 tends to infinity, b approaches in such a way that the product V0b remains finite and it is called barrier strength. Under such situations sinhβb βb & cosh βb =1 Equation (4) can be written as 2 2 b sin a cos a cos ka...(5) 2 Substituat ing for 8 2 m(V0 E ) 8 2 mE 2 and 2 2 In equation (5) 2 8 2 m(V0 E ) 8 2 mE ( )( ) 2 2 b sin a cos a cos ka 2 4 2 mV0 ab sin a ( ) cos a cos ka 2 a sin a P cos a cos ka......(6) a 4 2 ma where..P V0b 2 h p is called scattering power of the potential barrier wh ich is the measure of strength w ith which electrons are attracted by the positive ions. P sin a cos a a +1 +1 a -1 -1 Un allowed bands Allowed bands • Conclusions : 1. The motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions. 2. As the value of άa increases, the width of allowed energy bands also increases and the width of the forbidden bands decreases. 3. If V0b is large ,i.e. if p is large ,the function described by the left hand side of the equation crosses +1 and -1 region as shown in figure.. As P infinite, the allowed energy bands are compressed into energy levels and a line spectrum is resulted. p sin a 0 p 0 a n a n a 8 2 mE As we know that 2 n 2 8 2 mE ( ) a 2 n2h2 E 8ma 2 2 It clearly explains that the particle (electron) is confined to One dimensional infinitely deep square well potential box , Where we can find desecrate energies ( En = n2E1). cos a cos ka 4. If P tends to zero n No energy levels exist: k all energies are allowed for the electrons. a 2 k2 p0 2mE k 2 2 2 E ( )k 2m h 2 2 2 E ( 2 )( ) 8 m h2 1 E( ) 2 2m h2 p2 p 2 1 2 E ( ) 2 mv 2m h 2m 2 2 a All energies are possible for electron. The energy spectrum is continuous 2 Brillouin zones: • The Brilouin zone is a representation of permissive values of k of the electrons. 2k 2 p k & E 2m p 2 n k a (or ) n 2a which reflects Braggs condition for normal incidence Therefore the electron suffers Bragg s reflection s at the n values k which results in discontinu ities in the E - K curve. a E-k diagram : E For free electron Energy gap Allowed bands Energy gap 3 a 2 a a a 2 a 3 a k n note : k is the boundary condition for a electron motion and at these values , the energy is discontinu ous The region between - a and a is called first Brillouin zone 2 2 the region between - , and , is called second a a a a Brillouin zone. llly the other Brillouin zones are explained for electron motion. • Origin of energy band formation in solids: • When we consider isolated atom, the electrons are tightly bound and have discrete, sharp energy levels. • When two identical atoms are brought closer the outer most orbits of these atoms overlap and interact. • If more atoms are brought together more levels are formed and for a solid of N atoms , each of the energy levels of an atom splits into N levels of energy. • The levels are so close together that they form an almost continuous band. • The width of this band depends on the degree of overlap of electrons of adjacent atoms and is largest for outer most atomic electrons. E1 E1 E2 E1 E2 E3 N atoms ΔE N energy levels • The energy bands in solids are important in determining many of physical properties of solids. The allowed energy bands are (1) Valance band (2) Conduction band • The band corresponding to the outer most orbit is called conduction band and the inner band is known as valence band. The gap between these two allowed bands is called forbidden energy gap. • • • • Classifications of materials into Conductors, Semiconductors & Insulators:On the basis of values of forbidden ( band ) gap, the solids are classified into insulators, semiconductors and conductors. Insulators:In case of insulators, the forbidden energy band is very wide as shown in figure below. Due to this fact the electrons cannot move from valance band to conduction band even at high values of energies. In insulators at 00k , the energy gap is of the order of 5eV to 10eV. Conduction band Forbidden gap INSULATORS Valance band Conduction band SEMI CONDUCTORS Forbidden gap Conduction band Valance band Valance band CONDUCTORS SEMI CONDUCTORS: • In semi conductors, the forbidden band is very small as shown in figure above. • Ge and Si are the best examples of semiconductors. • Forbidden ( band ) gap is of the order of 0.7ev &1.1ev. CONDUCTOS: • In conductors there is no forbidden gap and both valence and conduction bands overlaps each other as shown in figure above. • The electrons moves freely from valance band to conduction band and vice versa. • The value of energy gap is zero. • Effective mass of an electron: • The effective mass of an electron arises due to periodic potential provided by the lattice. • When an electron in a periodic potential of lattice is accelerated by the electric field, the mass of the electron varies and this mass is called effective mass of the electron ( m* ). • Consider an electron of charge q and mass m acted upon by electric field. f qE • The acceleration ma qE qE a m • Acceleration ( a ) is not a constant in the periodic lattice of the crystal so mass of the electron is replaced by its effective mass m* when it is moving in a periodic potential or crystal lattice. • Now we can find a relation for m* in terms of ‘e’ and wave vector “k”. d vg dk where • Consider the free electron as a wave packet moving with a velocity Vg, called group velocity and is defined as change in angular velocity with respect to wave propagation 2 angular . frequency T k wave.vector d vg dk d v g 2 dk where.. 2., 2 dE vg h dk 1 dE vg dk E h a dv g dt 1 d 2E a dkdt 1 d 2 E dk a dk 2 dt sin ce., k p and .. dp F dt p ) 1 d E ) a ( dk 2 dt 1 d 2 E dp a 2 ( ) 2 dk dt 1 d 2E a 2 F 2 dk 2 d( F a 2 d 2E 2 dk F a * m 2 where., m* 2 d E 2 dk is called effective mass of an electron in periodic lattice E a. Variation of E with K b. Variation of v with K c. Variation of m* with K (a ) (b ) Point of inflection 0 V 0 m d. Variation of fk with K (c ) The degree of freedom of an electron is generally defined by a factor. fk 2 m m d E fk 2 { 2 } m dk (d ) a 0 k k0 a BONDING IN SOLIDS Introduction: •A solid is composed of billions of atoms packed closely together and solids have usually strong elastic structures. •According to strength and directionality the inter atomic forces or bonds in solids are grouped into primary bonds and secondary bonds. CLASSIFICATION OF BONDS IN SOLIDS BONDS Bond energy Bond range energy rang 0.1-10ev 0.01-0.5ev PRIMARY SECONDARY nter atomic Inter molec VANDER-W HYDROGEN IONIC COVALENT METALLIC H o 2 Neon Nacl H2,clNa,Al 2 in the form of ice • Primary bonds are the strongest bonds which hold atoms together. The three types of primary bonds are 1.Ionic 2.Covalent 3.Metallic • Secondary bonds are much weaker than primary bonds and they are 1.Hydrogen bond 2.Vander waal’s bond Ionic bond: • An ionic bond is an attractive force existing between a positive ion and a negative ion when they are brought into close proximity. Example:1 Na Na+ + eCl + eclNa+ + clNacl Example:2 Mg Mg2+ + 2e2cl + 2e2clMg2+ + 2clMgcl2 Mgcl2, KOH and Al2o3 Formation of ionic bond in Nacl Na Cl Na+ and Cl- ions formed by ionic bonding mechanism Na+ Cl- Characteristics of ionic bond: • It is a very strong bond and ionic crystals are rigid. • These solids having high melting and boiling point values. • Conductivity is very less and these solids are soluble in water and liquid ammonia. • Transparent to visible light. • Non-directional because the charge distribution is spherical in nature. • Ionic crystals have close packed structure. • Melting points of ionic solids: 1. NaF 2. Nacl 3. NaBr 4. NaI 1270k 1073k 1023k 924k Covalent bond: • In this type of bonding the valence electron are not transferred from one atom to the other atom but their neighboring atoms share their valence electrons under the formation of a covalent bond. + H + H + + H H • Characteristics of covalent bond: • These crystals are very hard and brittle. • Bonding energy is very high, so MP & BP values are high. • These crystals are soluble in non-polar solvents like benzene. • Conductivity increasing with increasing temperature. • Covalent bonds are highly directional in character. • These bonds have saturation property. • These crystals are transparent to longer wavelengths but opaque to shorter wavelengths. • Melting points of covalent solids: • Diamond 3280k Comparison between ionic and covalent solids: Ionic solids: Covalent solids: • The bonds are non directional. • Bonds are relatively stronger. • Soluble in polar solvents. • Not very hard. • Possess high melting and boiling points. • The bonds are directional • The bonds are relatively weaker. • Soluble in non polar solvents. • Very hard. • Comparatively lower MP and BP values. Metallic bonding: Positive ion co + + + + + + + + The valence electrons from all the metallic atoms belonging to the crystal are free to move through out the crystal. The crystal may be considered as an array of positive metal ions embedded in a “cloud” or “sea” of free electrons as shown in figure. The bond formed between these positively charged metal ions to the negatively charged electron cloud is called metallic bond. Electron clouds Characteristics of metallic bond: • Metallic solids are malleable and ductile. • They have high electrical and thermal conductivities. • Metallic solids are not soluble in polar and non polar solvents. • These metals have high optical reflection and absorption coefficients. • Due to the symmetrical arrangement of the positive ions in space lattice, metals are crystalline. • The metals are opaque to all electromagnetic radiation from very low frequency to the middle ultraviolet, where they become transparent. • Examples: Na, Aluminum. Hydrogen bonds: • The electrostatic force of attraction between the hydrogen atom of one molecule and more electronegative atom of the same or another molecule is called hydrogen bond. Example: H2o molecule in the form of ice. Hydrogen bonds in H2O molecule H O H H Hydrogen bonds O H H O H Characteristics of hydrogen bond: 1. Weak binding 2. Low electrical and thermal conductivities. 3. Hydrogen bonded crystals are transparent. 4. These crystal have peculiar directional properties. 5. Lose structure. Vander - waal’s bond: • This type of bonding arises due to mutual polarization of atoms due to each other or fluctuating dipole bonds between hydrogen atoms are known as vander – waal’s bonding and they are non directional. • These bonds especially takes place in noble gases which are cooled to very low temperature. Example: solid Ne & Ar When an electron cloud density occurs at one side of an atom or molecule during the electron flight about the nucleus, vander - waal’s forces are generated. Atom-A + Atom-B _ + _ High density clouds High density clouds Attraction Characteristics; 1. Weak bonding and soft crystals. 2. Low melting and boiling points. 3. Poor electrical conductors. 4. Closed packed structure usually transparent to electromagnetic radiation. 5. Non directional. Forces between atoms: • Electrical forces are responsible in binding the atoms giving different solid structures. • The forces between atoms can be of two types 1. Attractive forces 2. Repulsive forces • Attractive force which keeps the atoms together. And repulsive force which comes into existence when the distance between atoms is small. • The resultant force is equal to sum of the both factors. Variation F (r)of F (r) Attractive for inter Resultant force atomic r F max ospacing (r r0 B r ) Repulsive fo • If two atoms are separated by the distance r, the net force between the atoms is given as A f (r ) M r N Where A,B,M and N are constants • The first term in the above equation represents attractive force which according to inverse square law. • The second term represents the repulsive force which is very strong at small distances and the value of N≈9. • When the separation between atoms is equal to zero. the separation r0 is called equilibrium distance. f (r ) A ro M B rNo equilibrium position f (r ) 0 A ro M B rNo ro N B r0 M A B r0 A B N 1M r0 [ ] A N M • Cohesive energy: • It is defined as the minimum amount of energy is require to form bond in between atoms or ions. • It is also called as bond energy or bond dissociation energy. • Calculation of cohesive energy : • To calculate the cohesive energy let us consider the general situation of two identical atoms. • The potential energy = decrease in potential energy due to attraction + increase in potential energy due to repulsion. • Work done on the system is stored as potential energy and work done moving through small distance dr is given by U (r ) du ( r ) f ( r ) dr A B [ M N ]dr r r [ Ar Ar M M M 1 Br N ]dr dr Br N dr N 1 Ar Br c M 1 N 1 1 M 1 N Ar Br c 1 M 1 N A 1 B 1 [ M 1 ] [ N 1 ] c M 1 r N 1 r a b M 1 N 1 c r r a b U (r ) m n c r r Where a and b are new constants related to A and B as a a = A/M-1 b = B/N-1 n = N-1 a b U (r ) m n and m = M-1 When r =∞,U=0 hence c=0 r r t a equilibrium position r=r0 the potential energy is minimum URepulsive energy U (r ) o a b n m r r U (r ) b rn r0 r Attractive energy U (r ) a rm du [ ]r r0 0 dr For d a b minimu { [ ]} dr r r m am bn 0 potentia r r am bn l energy m n m 1 0 r0 r0 n 1 0 m 1 nm r0 bn am n 1 r r0 0 bn r0 [ ] am 1 nm a b U ( r0 ) m n ro r0 This energy corresponding to the equilibrium position (r=r0) is called the bonding energy or the energy of cohesion of the molecule. • This is also called the energy of dissociation this can be calculated as follows. bn r0 [ ] am 1 nm 1 1 am m[ ] n r0 r0 b n a b U ( r0 ) m n ro r0 a 1 am m b{ m [ ]} ro r0 b n a a m m { m } r0 r0 n a m U (r0 ) m [1 ] r0 n Thus the minimum value of energy U (ro) is negative. The positive quantity U (ro) is the dissociation energy the molecule since m≠n the attractive and repulsive energies are not equal though the attractive and repulsive force’s are equal in equilibrium. The total binding is essentially determined by the energy of attraction. 2 d U [ 2 ]r r0 0 dr Since U has a minimum at r=r0 dU am(m 1) bn(n 1) [ 2 ]r r0 0 m 2 n2 dr r0 r0 2 am(m 1)r0 bn(n 1)r0 n2 m 2 bn(n 1)r0 m2 0 am(m 1)r0 n2 Substituting for ro b n bn(n 1) am(m 1)( )( ) a m (n 1) (m 1) nm Hence we understand that the forces acting between the atoms are mostly electro static in Calculation of Cohesive energy of Ionic solids: • The bond energy of diatomic ionic molecules different from the cohesive energy of an ionic crystal. • The ionic molecules being stable because of presenting the following balancing forces. 1. Electro static coulomb attractive forces 2. Negligible vander - waal’s forces of attraction. 3. Inter atomic repulsive forces • The resultant force of attractive and repulsive forces brings the system to an equilibrium state where the minimum potential energy is present. • Let us consider ions of charges Z1e and Z2e separated by a distance ‘r’ 2 z1 z2e Attractiveforce 2 4 0 r For complete crystal coulomb potential energy Where A is Mode lung's constant Az1 z2e 4 0 r Potential energy due to repulsive force b n r The potential energy of a crystal can be expressed as 2 Az1 z2 e b U (r ) n 4 0 r r 2 For univalent alkali halides Z1=Z2=Z3…there fore Ae 2 b U (r ) n 4 0 r r The total energy of on kile mole could be written as b Ae2 U (r ) N A [ n ] (1) r 4 0 r r r0 andU (r ) U min dU [ ]r r0 0 dr Ae 2 bn 2 n 1 4 0 r0 r0 2 n 1 0 Ae r b (2) 4 0 n Substituting equation 2 in 1 [U (r )]r r0 U min 2 n 1 0 n 0 0 2 Ae r Ae N A{ } 4 nr 4 0 r0 2 U min Ae N A 1 [ 1] 4 0 r0 n Where Umin represents cohesive energy of an ionic compound. • Made lung constant (A): • The Made lung constant A is function of crystal structure. • It can be calculated from the geometrical arrangement of ions in the crystal. • Let us consider the equilibrium positions of ions in Nacl structure as shown in a figure. • Let us choose the central Na+ ion as the reference ion having a single positive charge on it. • Six cl- ions are surrounding this Na+ ion first nearest neighbors. • Let us consider them at unit distance.12 Na+ ions are the second nearest neighbors at a distance √2. + Na Cl 1 unit + Na 1 unit √2 units Cl + Na 1 unit √3 units √2 units Cl • Eight cl- ions are the third nearest neighbors at a distance √3 and so on. • The Made lung constant for the Nacl structure can be written as a summation series. 6 12 8 6 24 24 A ............. 1 2 3 4 5 6 This converges to a value 1.74756 Made lung constants for some typical Ionic crystals: • • • • Nacl 1.74756 Cscl 1.76267 Fluorite 2.51939 Zinc 1.638 Bond energy: Bond energy is defined as the energy of the formation of one kilo mole of a substance from its atoms or ions. Bond energy of Nacl molecule: Consider Na and Cl atoms are infinitely separated. • The ionization energy of a Na atom is 5.1ev. Na + 5.1ev = Na+ + e-------------1 • The electron affinity of a chlorine is 3.6ev cl + e- = cl- + 3.6ev--------2 • Thus the net energy required to form one sodium ion and one cl ion is equal to 5.1ev-3.6=1.5ev • Electrostatic attraction between Na+ ion and cl- ion brings them together to the equilibrium spacing r0 = 0.24nm and forms an ionic bond resulting Nacl molecule. • Potential energy of Na+ and cl- ions at equilibrium position e 2 4 0 r0 (1.602 1019 ) 2 4 8.85 1012 0.24 1091.602 1019 6ev • The negative sign indicates that there is electrostatic attraction between the ions. • The potential energy of resultant Nacl molecule => 1.5 - 6 = - 4.5ev • Bond energy of Nacl molecule = -potential energy of Nacl molecule = 4.5ev • Thus the formation of sodium chloride molecule from their ions can be written as Na+ + cl- Na+ + cl- + 4.5ev Nacl