PHY592 - Chapter 3 Lattice Vibration and Phonons PDF

Chapter 3 Phy592 Lattice Vibration and Phonons 3.1 elastic waves and dispersion 3.2 density of states of a lattice 3.3 specific heat model of einstein & debye 3.4 lattice waves...

Chapter 3 Phy592 Lattice Vibration and Phonons 3.1 elastic waves and dispersion 3.2 density of states of a lattice 3.3 specific heat model of einstein & debye 3.4 lattice waves 1 What is lattice vibrations? 1.Lattice vibrations can explain sound velocity, thermal properties, elastic properties and optical properties of materials. 2.Lattice Vibration is the oscillations of atoms in a solid about the equilibrium position. For a crystal, the equilibrium positions form a regular lattice, due to the fact that the atoms are bound to neighboring atoms. 3.The vibration of these neighboring atoms is not independent of each other. A regular lattice with harmonic forces between atoms and normal modes of vibrations are called lattice waves. suhaida dila 2020 2 Vibration of Monoatomic Lattices (1D) 1. Monoatomic lattice is the one with same type of atoms. The objectives in this section is to understand a detailed model of vibration in solid. 2. Let us consider a chain of identical atoms of mass m where the equilibrium spacing between atoms is ‘a’. suhaida dila 2020 3 Assume periodic condition un = u N + n so that all atoms are in identical environment a a a a a a un-2 un-1 un un+1 un+2 suhaida dila 2020 4 Lattice vibrations of one dimensional solids Now, consider only nearest neighbor interaction And find equation of motion for nth atom a a Spring 1 Spring 2 un-1 un un+1 +u Force exerted by spring 1: F1 = - K (un - un -1 ) Force exerted by spring 2: F2 = - K (un - un +1 ) Therefore, equation of motion for nth atom Mu!! = K (un +1 - 2un + un -1 ) 5 1. Total potential energy of the chain 2. The force on the nth mass on the chain is 3. Newton’s equation 4. Normal mode is defined to be a collective oscillation where all particles move at the same frequency. A solution to Newton’s equation that describes the normal modes as waves Where A is an amplitude of oscillation suhaida dila 2016 6 Plugging In general, a relationship between a frequency or energy and wavevector (or momentum) is known as dispersion relation 7 5. The dispersion relation for vibrations of the One Dimensional Monoatomic Harmonic Chain. The dispersion is periodic in 6. A system which is periodic in real space with a periodicity ‘a’ will be periodic in reciprocal space (k-space) with periodicity suhaida dila 2016 8 7. The periodic unit (the “unit cell”) in k-space is conventionally known as the Brillouin Zone. The “First Brillouin Zone” is a unit cell in k-space centered around the point k = 0. Vibration mode : 8. At shorter wavelength, there are 2 different velocities : group velocity ( the speed at which a wavepacket moves) 9. The phase velocity (the speed at which the individual maxima and minima move) suhaida dila 2016 9 Dispersion relation in the 1st BZ for a monoatomic lattice The group velocity Vg – the slope of 𝛚 versus k : vg = d 𝛚 /dk suhaida dila 2016 10 11 Vibrations of a One-Dimensional Diatomic Chain In previous sub chapter, we studied a 1-D model of a solid where every atom is identical to every other atom. In real materials, not every atom is the same. Sodium chloride NaCl has two types of atoms. Model system : 2 different types of atoms with two masses m1 and m2 which alternate along the one-dimensional chain. The spring connecting the atoms have spring constant k1 and k2 and also alternate. suhaida dila 2016 12 Vibrations of a One Dimensional Diatomic Chain let us focus on the case where all of the masses along our chain are the same m = 1 m = m but the two spring constants κ and κ are different 2 1 2 suhaida dila 2016 13 Now, consider only nearest neighbor interaction And find equation of motion, but now we have two different types of atoms a two equations of motion must be written; aOne for mass M(nth atom), and One for mass m((n-1)th atom) a a/2 M m M m un-2 un-1 un un+1 +u suhaida dila 2016 14 Chains of two types of atoms a a/2 M m M m un-2 un-1 un un+1 +u For mass M: Mu!!n = - K (un - un +1 ) - K (un - un -1 ) = K (un +1 - 2un + un -1 ) !!n -1 For mass m:mu = - K (un -1 - un ) - K (un -1 - un - 2 ) = K (un - 2un -1 + un - 2 ) suhaida dila 2016 15 The 1D diatomic lattice 1.The diatomic lattice in which the unit cell is composed of two atoms of masses M1 and M2 and the distance between two neighboring atoms is ‘a’. The unit cell has a length of 2a. 1.There are 2 different types of atoms, the equation of motion : suhaida dila 2016 16 Simplification: suhaida dila 2016 17 acoustic mode is any mode that has linear dispersion as k → 0. 18 suhaida dila 2016 19  Acoustic branch: long wavelength limit k g 0, sounds wave Transverse acoustic mode for diatomic chain Optical branch: a higher energy vibration need a certain amount of energy to excite this mode (oscillating charged particle creates electromagnetic Transverse optical mode for diatomic wave) chain 20 suhaida dila 2016 21 1. The atoms were introduced as the basic units placed on the lattice points to form the crystal structure. They were assumed to be stationary while studying the crystal planes and directions. However, in reality, the atoms are not at rest but vibrate about their mean position. 2. The vibration of all atoms constitute lattice vibrations. These vibrations are set by external forces such as sound waves. The vibrations set by heat conduction are significant. The energy associated is used in the calculation of specific heat of solid. 3. If an amount of heat (dQ) is given to the system, the internal energy of the system increases by dE. According to 1st law of thermodynamic : dQ = dE + PdV 4. Factors that contribute to specific heat of solids: a. atomic vibrations in the crystal. This require an input energy which is generally heat energy. Atoms vibrates more with an increase in temperature b. the electronic contributions. This is the case in semiconductor and metals. This is relatively smallsuhaida compared dila 2016 with that of lattice vibrations 22 As far as back 1819, it had been known that for many solids, the heat capacity is given by Which is known as Law of Dulong Petit. 1896- Boltzmann constructed a model that accounted for this law. In his model – each atom in the solid is bound to neighboring atoms. 1907- Einstein assumption was similar to that of Boltzmann. He assumed that every atom is in a harmonic well created by the interaction with its neighbours. He also assumed that every atom is in an identical harmonic well and has an oscillation frequency (ω). suhaida dila 2016 23 Debye’ s calculation 1912- Peter Debye discovered how to better treat the quantum mechanics of oscillations of atoms. Debye realized that oscillation of atoms is the same thing as sound, and sound is a wave, so it should be quantized the same way as Planck had quantized light waves in 1900 suhaida dila 2016 24 suhaida dila 2016 25 Specific heat : Model of Einstein and Debye 1. The atoms in the Einstein model were assumed to oscillates independently of each other. 2. Real : the motion of these in turn effects their neighbors and so forth 3. Need to consider the motion of the lattice as a whole, not a single independent atom – consider the collective lattice models. 4. Example : sound waves a) Sound waves propagates in a solid, the atoms do not oscillate independently. b) Debye assumed that the lattice modes have a similar characteristics with sound waves as they obey the same dispersion relation. 5. The Debye model : frequency of the lattice vibration covers a wide range of values. The lowest frequency value ω = 0. 6. Einstein : covers only a single frequency. suhaida dila 2016 26 Chains of two types of atoms a a/2 M m M m un-2 un-1 un un+1 +u For mass M: Mu!!n = - K (un - un +1 ) - K (un - un -1 ) = K (un +1 - 2un + un -1 ) !!n -1 For mass m:mu = - K (un -1 - un ) - K (un -1 - un - 2 ) = K (un - 2un -1 + un - 2 ) suhaida dila 2016 27 Lattice vibrations of one dimensional solids w C B A p p 2p - 0 k a a a ¶w At B, the group velocity > 0 a wave propagating to the right ¶k ¶w At A and C, the group velocity < 0 a wave propagating to the left ¶k At A and C, the atomic displacement, frequency, group velocity are all same. The k values of point A and C differ by 2p/a a adding any multiple of 2p/a to k does not alter physical properties of wave. p p The restricted range of - >a can be represented by elastic traveling waves with wavevector, q and frequency 𝛚. State the solution of the wave equation for a three- dimensional monoatomic cubic solid length : Lx, Ly and Lz and apply the periodic boundary condition to show that qx, qy and qz are discreet. b) State the dispersion equation 𝛚 versus q for a lattice wave with wavelength 𝛌➝2a travelling through a one-dimensional monoatomic chain. Plot the curve and identify the 1st BZ. If the chain has N = 21 atoms, find the two q values closest to the 1st BZ boundaries. suhaida dila 2016 43 DEC 2019 a) Lattice vibration in a one dimensional (1D) monoatomic solid can be described using group velocity Vg and phase velocity Vp. Draw 𝛚 versus q diagram and analyze both Vg and Vp for q ➝a and q ➝𝛑/a. b) The phonon dispersion curves for a diatomic one-dimensional lattice exhibits an optical branch and acoustic branch. State the expression for 𝜟𝛚 between the optical and acoustic branches at q = 0 in terms of masses M1 and M2 and the interatomic force constant 𝒂. Also, if the solid is ionic, state the basic difference between transverse vibrations in the optical and acoustic branches. Sketch a simple diagram to illustrate. 44 FEB 2020 In solids, due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions give rise to a set of vibration waves propagating through the lattice. The basic model used for the study of phonon involves a 1D monoatomic chain of length L consisting of N identical atoms. Draw the phonon dispersion curves 𝛚 vs q for the 1st and 2nd BZ for the solid. Using the dispersion equation, find the group velocity Vg at the boundary of the 1st BZ. 45 For a one-dimensional alternating chain of two types of atoms of masses M1 and M2 repeated periodically at a distance a, connected by a spring of spring constant 𝑎, two modes of vibrations exists, i.e the acoustic and optical modes. i) Sketch the phonon dispersion curves and analyze the frequency of vibration 𝛚 for the optical mode at q = 0. explain the vibration. ii) Based on the 𝛚 vs q plot, compute the maximum frequency gap 𝜟𝛚max between the optical branch and the acoustic branch in terms of masses M1 and M2. 46