3. LATTICE VIBRATIONS. 3.1 Sound Waves

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1 3. LATTIC VIBRATIONS Atoms in lattice are not stationary even at T 0K. They vibrate about particular equilibrium positions at T 0K ( zero-point energy). For T > 0K, vibration amplitude increases as atoms gain thermal energy. 3. Sound Waves Sound waves travel through solids with typical speeds ~ 5(km/s). At low frequencies (f < THz), λ~50a o, one can treat solid as continuum, ie. ignore discrete nature. At any frequency & in a given direction in crystal, 3 sound waves possible: One longitudinal and two transverse. 3. Continuum Model Propagation of an elastic wave in a sample of long bar follows the dispersion relation : ω v s q Where v s is the wave velocity (sound velocity) which equal in this case to v s Y / ρ Y & ρ are Young s modulus and density of the sample.

2 3.3 Discrete Lattice As the wavelength decreases and q increases, the atoms begin to scatter the wave, and hence to decrease its velocity. Thus, one expects the dispersion relation to be as shown ω Continuum Discrete 0 A- Monatomic Chain Consider D chain of identical atoms each of mass M, where atoms joined to nearest neighbors by 'spring' of force constant α, and spacing a apart. q Look for wavelike solution: u n Ae i( qx ω t) Where X n na is the equilibrium position. Substitute into eq. of motion get: n 4α ω sin M qa

3 Hence, the Dispersion relation, which Determines how wave vector q is related to ω,in D is : ω ω m sin qa Where ω m (4α/M) /. Note: ω is periodic in q & all values of ω contained in range: -π/a < q < π/a. * Range of q related to st Brillouin zone. Values of q outside range have no physical meaning. * N does not appear in dispersion relation, ie. equations of motion of all atoms lead to the same algebraic relation between ω and q.. Long wavelength limit: As q 0, sinθ θ, and then ω a m ω q

4 Or ω v ma s Hence, α Ya. short wavelength limit: As q increases, the dispersion relation begins to deviate from the straight line. It will saturate at qπ/a with maximum frequency ω m 4α M 3. Phase & Group velosity : v p ω q defines the velocity of a pure wave with frequency ω and wave vector q. v g ω q defines the velocity of a wave pulse with average frequency ω and average wave vector q. Thus, at q 0, phase velocity group velocity. defines velocity of sound, which can be identified with the appropriate Young s modulus. And, at qπ/a, group velocity is zero, ie. standing waves.

5 B- Diatomic Chain Consider D chain of two different atoms of masses M and M, where the distance between two neighbouring atoms is a. Dispersion relation,that determines how wave vector q is related to ω, is ω α( + ) ± α M M ( M + M ) 4sin ( qa) M M

6 3.5. Lattice Vibrations in 3D If we xtend properties of D chain to 3D crystals: - For crystal lattice with unit cell containing only one atom, * 3 dispersion relations, one in each direction of vibration. * All three branches are acoustic. - Crystals with unit cell containing different atoms, * 3 acoustic branches & 3 optical branches. 3- Generalize to crystals with s different atoms in unit cell. * 3 acoustic branches & 3(s - ) optical branches. 3.6 Phonons Define: Phonons: quantum of vibrational (elastic) energy. photons: quantum of electromagnetic energy. * The energy of phonon is: With a momentum equal to: ε hω p hq

7 * The number of phonons is given by Bose-instein distribution: n This number depends on temperature, at T0 n0. At very high temperature, n KT h ω Fact: The number of phonons in a system is not conserved. Interaction of phonons with other excitation (or with themselves) governed by conservation laws. hω / KT e

8 3.6. Density of States In calculating physical properties of solids, the no. of modes in a given frequency (or energy, or q- space) range is required. D density of States A 'better' way of obtaining this is to apply the periodic boundary condition: u(xl) u(x0) e iql ql nπ, ie. q nπ/l Thus, 'length' occupied by one mode in q-space is π/l. Hence the No. of modes in the range dq (i.e. the density of states in q-space) is: L/π dq To obtain density of states in frequency space, need dispersion relation. Since there is the same No. of states in g(q)dq & g(ω)dω, ie. or g( ω) d( ω) g( q) d( q) g( ω) L dq π dω L dq π Since the modes lying in( ve) q-region must be included as well, then:

9 g( ω ) L π dω / dq () Note the relation between the density of states and the group velocity. * At long wavelength limit: and hence, ω v s q, ie. dω v s dq g( ω ) L π v s () * At Dispersive region: ω ω m sin qa or q a sin ω ( ) ω m Differentiating: dq dω a ( ω m ω ) -/ Hence density of states in ω-space is: L g( ω ) ( ) aπ ω ω m -/ (3)

10 3D density of States Volume occupied by one mode in q-space is: (π/l) 3 (8π 3 )/V Thus, No. of modes contained in a small spherical shell in q-space is: 4πq dq / 8π 3 /V Hence, density of states in ω-space is g ( ω) V 3 8π 4πq dq dω V dq or g( ω) q (4) d This is the density of state if we assume that each value of q associated with one single mode. But, in fact for each q there are 3 modes ( longitudinal & transverse ), then π ω g ( ω) For non-dispersive limit; g( ω ) 3V π 3V π q dq dω ω ( ) v s v s (5) or, 3V ω g ω ) π ( v 3 s

11 3.7. Specific Heat of Lattice General definition of specific heat is; ΔQ C Δ T Where ΔQ is the heat required to increase the temperature of one mole by ΔT. At constant volume all internal energy converts to heat, i.e. ΔQ Δ, then one can define the specific heat at constant volume as; C v T v () xperimentally, it was found that there is a relation between C v & T as is shown in the figure. Theoretically, three different approaches were used to determine C v : A- Classical Theory: Vibration of lattice may be treated as that of a harmonic oscillator. Thus, the average energy for D oscillator is ε kt Therefore, the total thermal energy per mole in 3D is C v T

12 3 N AkT 3RT where N A & R are Avogadro s no. and the universal gas constant respectively. Substituting into (), we find C v 3R () This relation predicts a constant value to C v whatever the temperature is. Although it is in agreement with experiment at high T, it fails totally at low T. B- instein model: * Considering the atoms as independent oscillators, vibrating with a common frequency. * Treating the oscillator quantum mechanically, where the average energy for D oscillator is ε h kt e ω hω / At high T, ε kt (and the classic value is reached) At low T,total energy is; 3 N hω / kt e hω Where the common frequency of the oscillators, ω, is known as instein frequency. Substituting into (), we find

13 ( ) / / 3 kt kt v e e kt R C ω ω ω h h h Introducing instein temperature, k / θ hω, the above expression can be reduced to: ( ) / / 3 T T v e e T R C θ θ θ (3) This formula is in a good agreement with the experimental results as it shown in the following figure: At very low temperature, T<< θ, and hence, T v e T R C / 3 θ θ Which indicates that C v approaches zero exponentially. But careful measurements show that it does that slower.

14 Although the fact that (C v 0 as T 0 ) agrees with experiments, the manner in which this is approached is not. C- Debye model: * Treating the atoms not as independent oscillators, but as coupled oscillators vibrating collectively as sound wave follows the dispersion relation: ω v s q Total energy in such system is, ε ( ω) g( ω) dω where g(ω) is the density of state, and hence the integration is taken over all allowed frequencies. Or; 3V hω ω dω 3 ω / kt π v h e s Debye stated the integral limit : (from 0 to ω D ), where ω D is Debye frequency wich given by ω D v s ( 6π n) / 3 Specific heat C v then can be found to be,

15 ( ) dx e e x T R C T x x D v D / θ θ With kt x / hω and k D D / θ hω Here θ D is Debye temperature. At very low temperature, T<< θ D, and hence D v T R C θ π

16 4. FR LCTRON THORY OF MTALS Classical theory : Drude & Lorentz (based on freeelectron model) gave explanations for conductivity and optical properties in metal. Quantum theory: Summerfield included Pauli's exclusion principle into classical theory. 4. Conduction electrons A valence electron in a free atom, becomes a conduction electron in a solid. The core electrons do not contribute anything to the electric current. Hence, No. of conduction electrons No. of atoms x atomic valence N Z v ρ m N M A Where M`is the molar mass of the element (gm/mole). 4. The Free-lectron Gas Assumptions: a. lectrons free to move in a background of uniform (+ve) charge provided by ions (jellium model), ie. electrons behave like a gas. b. No interactions between electrons, because of: - Pauli exclusion.

17 - To minimize the system energy, electrons tend to stay away of each other. The two above reasons lead to the hole concept: ach electron is surrounded by a spherical region which is lacking of other electrons Known as a hole. 4.3 lectrical Conductivity Apply external electric field (ε) on a wire has a length L and cross section A, we find: J I V ε and R A L Lρ A Where J is the current density, and ρ is the electrical resistivity /σ (σ is the electrical conductivity). According to Ohm s Law: J σε () When external electric field is applied two main things happen:. Coulomb force causes electrons to accelerate, ie., where v d is the drift velocity.

18 . Scattering (with other electrons, ions, impurities etc) produces a decay of v d back to zero. This retardation is represented by: -v d /τ, where τ is the relaxation time. Thus, adding up total force on electron get: m dv dt d eε m In steady state, dv/dt0, and hence vd τ v d eτ ε m But since J represents the amount of charge crossing a unit area per unit time, then Ne τ J ( Ne) v d ε m () Comparing () with () leads to; Ne τ σ m Since τ is the time between two following collisions, it can be expressed as;

19 τ l v r ( ~0-4 s in metals ) Where l is distance between two following collisions (electron s mean free path) and v r is the random velocity. Using these terms (σ) can be rewritten as σ Ne l m v r In metals : N ~0 9 m -3 & v r ~0 6 m.s - σ ~ 0 7 Ω -.m - In semiconductors: N ~0 0 m -3 & v r ~0 4 m.s - σ ~ Ω -.m - The mean free path, l, for electrons in metals may be estimated via: v r τ l. This give l ~ 0-8 m ~ 0 A o, which is 0 times larger than the interatomic spacing. i.e., electrons can move freely in a perfect periodic structure. scattering occurs when periodicity is interrupted by phonons or impurities.

20 4.4 Temperature dependence of lectrical Resistivity In metals, the temperature dependence of ρ is determined by τ, according to the relation: ρ m Ne τ But total scattering rate τ given by addition of two individual scattering rates: + τ τ ph τ i Which known as Matthiesens' Rule. Therefore; ρ ρ + ρ i ph m Ne τ i + m Ne τ ph Impurity Scattering ρ i, the residual resistivity, which is independent of T. It is determined by distance between impurities. lectron-phonon scattering ρ ph, the ideal resistivity, which is dependent of T. * At very low temperature, scattering by phonons is negligible. So τ ph & ρ ph 0, and hence ρ ρ i * At high temperature, no. of phonons increases as T. Hence scattering increases as T. τ proportional to /T (true for T > ¼θ D ).

21 4.5. Heat Capacity of Conduction lectrons Based on Drude-Lorentz model, the average energy of electrons per mole is : 3 3 N kt RT A Hence, the electrons heat capacity is 3 C e R The total heat capacity in metals should then be; C C ph + C e This means that at high temperature C4.5 R ~ 9 cal/mole K But, experiments show that C~3R, and C e is smaller than the classical value by a factor of 0 -. Quantum explanation * Pauli's exclusion principle implies that only electrons of opposite spin orientation are allowed in each state. * At T 0K, electrons occupy all energy levels from the ground state upward. The energy of the highest filled state is known as the Fermi energy. * At T 0K, electrons distribution function is f ( { 0, > ) F, < F

22 Thermal Distribution * For T > 0, electrons with energy ~ F can gain thermal energy and occupy energy states > F. * In many cases, it is the electrons near F that dominate the properties of the metal. * For T > 0, electrons distribution function is f ( ) ( - F )/ e KT + i.e. Fermi-Dirac distribution. * Using this distribution function, one can find energy of electron gas, hence the heat capacity of the electrons. electron per mole as N ( kt / ) A F Which leads to an approximate thermal energy N A ( kt ) F Hence, the electrons heat capacity is, kt Ce R F

23 Or T Ce R T F With F kt F, where T F is Fermi temperture. * The exact value of electrons heat capacity is then, C e π R kt F

24 4.6. Fermi Surface Because of the random motion, the energy of an electron is mainly kinetic. Hence: m v r In the velocity space, all conduction electrons confined in a sphere of radius v F which is called Fermi velocity. nergy at the surface of this sphere is; Fermi Surface m v F v F 6 v F is very large ~ 0 m.s -. v v y F v x Fermi surface (FS) is temperature Independent, and so the Fermi velocity. The value of F is determined by the electron concentration N according to : F ( 3π N ) / 3 h m ~ 5 ev 4.7. lectrical Conductivity (with respect to FS): σ Ne l m v F F

25 Although the formula of σ is the same as the classical case, the actual picture is different. No. of electrons responsible about conductivity Velocity of those electrons Lorntz Model Summerfield model N N(v d /v F ) v d v F 4.8 Hall ffect in Metals Applying a magnetic field B normally on a wire carries a current of density J x will produce an electric field normal to both J x and B known as Hall field ε H. In steady state;

26 Or, Force of Hall field Lorentz force -e ε H -e v x B ε H v x B ε H Ne J x B Since J x N(-e)v x,where N is the electron density. The proportionality constant, i.e. ε H / J x B,is known as Hall constant R, therefore, H R H Ne 4.9 Failure of Free-lectron Model - The model suggests that electrical conductivity is proportional to electron concentration, but it is found that the divalent metals (Zn, Cd,..) and even trivalent metals (Al, In) are less conductive than the monovalent metals such as (Cu,Na,Ag,..). - Some metals show positive Hall constant (Zn, Cd,..). 3- Fermi surface was found to be often nonspherical in shape.

27 5.. nergy Bands in Solids * Solving Schrodiger equation for a free atom gives a series of discrete energy levels. * When two atoms come together to form a molecule, each of the atomic levels will split into two closely spaced levels. * The amount of splitting depends strongly on: - interatomic distance; the closer the atoms, the greater the splitting. - atomic orbital; the higher the energy, the greater the splitting.

28 This is an eigenvalue equation which leads to many solutions. Therefore, for each value of k there are a large number of solutions (i.e. a set of discrete energies,k,,k,.). * ach band covers a certain range of energy, extending from the lowest to the highest value it takes when plotted in k-space. * Within each band the electron states can be classified according to their momentum (pħk). 3-The crystal potential V(r) is the crystal potential seen by the electron. The most important property of this potential is its periodicity. This potential is composed of two parts; ionic & electronic: V ( r) V ( r) V ( r) i + e (4)

29 * When N atoms come together to form a solid, each of the atomic levels will split into N very closely spaced sublevels; i.e. energy band. * The energy spectrum in a solid is composed of a set energy bands. Regions separating these bands are energy gaps, i.e. regions of forbidden energy. The wave function describing the electronic states in a band, known as the crystal orbital, can extend throughout the solid, unlike the atomic orbital which is localized. 5.. The Bloch Theorem The wavelike properties of an electron in a crystalline solid can be described using the following form of Schrodinger equation: h + V ( r) ψ ( r) ψ ( r) m () Where V(r) is the crystal potential and ψ(r) & are the state function and energy of this electron.

30 -The Bloch Function The solution of () for a periodic potential is given by ψ ( r ) f ( r) u( r) () Where both the function u(r) and the probability density ψ(r) must have the same translational symmetry as the lattice, i.e, and u( r f ( r + R) u( r) + R) f ( r) The only function that satisfies this is e ikr, hence () can be written as ik. r ψ k ( r) e u ( r) (3) Which known as Bloch function. Note: *This state function has the form of a traveling plane wave so it propagates through the crystal like a free particle. *This function represents the crystal orbital of the solid. -The nergy Bands Substituting from (3) into () gives h ( + ik) m k + V ( r) uk ( r) k u k ( r)

The first term represents the interaction with the ion cores, and given by V ( r) v ( r R ) i j i j Since electron-electron interaction is very weak and has some difficulties to be calculated, the

31 The first term represents the interaction with the ion cores, and given by V ( r) v ( r R ) i j i j Since electron-electron interaction is very weak and has some difficulties to be calculated, the second term in (4) can be dropped out. The major effect of this interaction is to screen the ions from other electrons and hence to weaken the elecrton-ion interaction. Thus, V ( r) V ( r) v ( r R ) i Where v s (r-r j ) is the potential of the screened ion located at R j. j s j 5.3. Number of States in the Band Since each Bloch function ψ(r) n,k defines a specific electron state, let see how many states there are in each band. ik. x For D case: ψ k ( x) e u ( x) k

32 Applying the periodic boundary condition on this function shows that the allowed number of k is ; π k n L For the first zone, the number of states is π π L ( ) /( ) N a L a Where N is the number of unit cells. Hence, we end with these two important facts: - In each band the number of states (orbitals) inside the first zone is equal to the number of unit cells in the crystal N. - In each band the maximum number of electrons is equal to N.

Unit III Free Electron Theory Engineering Physics

Unit III Free Electron Theory Engineering Physics . Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals