Lecture 5 Andrei Sirenko, NJIT 1Phys 446: Solid State Physics / Optical PropertiesLattice vibrations: Thermal, acoustic, and optical propertiesFall 20072Solid State Physics Lecture 5 (Ch. 3)Last week: • Phonons• Today:Einstein and Debye models for thermal capacity Thermal conductivityHW2 discussion3Material to be included in the testOct. 12th2007• Crystalline structures. 7 crystal systems and 14 Bravais lattices• Crystallographic directions and Miller indices • Definition of reciprocal lattice vectors: • What is Brillouin zone• Bragg formula: 2d·sinθ = mλ ; ∆k = G21222222⎟⎟⎠⎞⎜⎜⎝⎛++=clbkahndhkl4•Factors affecting the diffraction amplitude:Atomic scattering factor (form factor): reflects distribution of electronic cloud.In case of spherical distribution •Structure factor•Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law):•Elastic wave equation: rdenflia3)(rkr⋅∆∫=()∫⋅⋅=002∆∆sin)(4radrrkrkrnrfπ∑++=jlwkvhuiajjjjefF)(2πklklijklijCεσ∑=klklijklijSσε∑=2222xuCtuxeff∂∂=∂∂ρsound velocityρeffCv =5• Lattice vibrations: acoustic and optical branches In three-dimensional lattice with s atoms per unit cell there are 3s phonon branches: 3 acoustic, 3s - 3 optical• Phonon - the quantum of lattice vibration. Energy ħω; momentum ħq• Concept of the phonon density of states • Einstein and Debye models for lattice heat capacity.Debye temperatureLow and high temperatures limits of Debye and Einstein models • Formula for thermal conductivity • Be able to obtain scattering wave vector or frequency from geometry and data for incident beam (x-rays, neutrons or light)CvlK31=6Summary of the Last Lecture  Elastic properties – crystal is considered as continuous anisotropic medium Elastic stiffness and compliance tensors relate the strain and the stress in a linear region (small displacements, harmonic potential)Hooke's law: Elastic waves sound velocity Model of one-dimensional lattice: linear chain of atoms More than one atom in a unit cell – acoustic and optical branches All crystal vibrational waves can be described by wave vectors within the first Brillouin zone in reciprocal spaceWhat do we need? 3D case considerationPhonons. Density of statesklklijklijCεσ∑=klklijklijSσε∑=2222xuCtuxeff∂∂=∂∂ρρeffCv =7Vibrations in three-dimensional lattice. PhononsPhonon Density of statesSpecific heat(Ch. 3.3-3.9)8Three-dimensional latticeIn general 3D case the equations of motion are: )()(1122nnnnnuuCuuCFtuM −+−==∂∂−+In simplest 1D case with only nearest-neighbor interactions we hadequation of motion solution)(),(tqxinAetxuω−=∑=∂∂ββααα,22mmnnuFtMN unit cells, s atoms in each →3N’s equations)()(1),(tiiinneuMtxuωααα−⋅=rqqFortunately, have 3D periodicity ⇒Forces depend only on differencem-nWrite displacements as90)(1)(,matrix dynamical - )()(2=−−∑∑−⋅qqqrrqmnmnmjjDijiiueFMMujiβββαβααβαωsubstitute into equation of motion, get 0)()()(,2=+−∑qqqjjjiiuDuβββααω⇒{}0)(2=− 1qDetωβαjiD- dispersion relation3s solutions – dispersion branches3 acoustic, 3s - 3 opticaldirection of u determines polarization(longitudinal, transverse or mixed)Can be degenerate because of symmetryphonon dispersion curves in Ge10Phonons• Quantum mechanics: energy levels of the harmonic oscillator are quantized• Similarly the energy levels of lattice vibrations are quantized.• The quantum of vibration is called a phonon(in analogy with the photon - the quantum of the electromagnetic wave)Allowed energy levels of the harmonic oscillator:where n is the quantum numberA normal vibration mode of frequency ω is given bymode is occupied byn phonons of energy ħω; momentum p = ħqNumber of phonons is given by :(T – temperature)The total vibrational energy of the crystal is the sum of the energies of the individual phonons:(p denotes particular phonon branch))( tieω−⋅=rqAu1(,)1kTnTeωω=−=11 1213 14Density of statesConsider 1D longitudinal waves. Atomic displacements are given by: iqxAeu =Boundary conditions: external constraints applied to the endsPeriodic boundary condition: 1=iqLeThen⇒ condition on the admissible values of q:..210 where2, ., , n nLq ±±==πregularly spaced points, spacing 2π/LdqdωωqNumber of modes in the interval dq in q-space :dqLπ2Number of modes in the frequency range (ω, ω + dω): D(ω) - density of statesdetermined by dispersion ω = ω(q)dqLdDπωω2)( =15Density of states in 3D caseNow havePeriodic boundary condition: 1===LiqLiqLiqzyxeee⇒ l, m, n - integersPlot these values in a q-space, obtain a 3D cubic meshnumber of modes in the spherical shell between the radii q and q + dq: V = L3– volume of the sample⇒ Density of states16Few notes:•Equation we obtained is valid only for an isotropic solid, (vibrational frequency does not depend on the direction of q) •We have associated a single mode with each value of q.This is not quite true for the 3D case: for each q there are 3 different modes, one longitudinal and two transverse. •In the case of lattice with basis the number of modes is 3s, where s is the number of non-equivalent atoms. They have different dispersion relations. This should be taken into account by index p =1…3s in the density of states.17Lattice specific heat (heat capacity)dTdQC =Defined as (per mole) If constant volume V0)(,==∑qqq pppnEω=The total energy of the phonons at temperature T in a crystal: (the zero-point energy is chosen as the origin of the energy). 11−=kTenω=- Planck distributionThen replace the summation over q by an integral over frequency:Then the lattice heat capacity is:Central problem is to find the density of states18Debye model•assumes that the acoustic modes give the dominant contribution to the heat capacity •Within the Debye approximation the velocity of sound is taken a constant independent of polarization (as in a classical elastic continuum) The dispersion relation: ω = vq, v is the velocity of sound. In this approximation the density of states is given by: 322222221212)(vVvVqdqdVqDπωπωπω===Need to know the limits of integration over ω. The lower limit is 0.How about the upper limit ? Assume N unit cells is the crystal, only one atom in per cell ⇒ the total number of phonon modes is 3N