1、Crystal Lattice Vibrations: PhononsIntroduction to Solid State Physics 1Lattice dynamics above T=0 Crystal lattices at zero temperature posses long range order translational symmetry (e.g., generates sharp diffraction pattern, Bloch states, ).At T0 ions vibrate with an amplitude that depends on temp

2、erature because of lattice symmetries, thermal vibrations can be analyzed in terms of collective motion of ions which can be populated and excited just like electrons unlike electrons, phonons are bosons (no Pauli principle, phonon number is not conserved). Thermal lattice vibrations are responsible

3、 for: Thermal conductivity of insulators is due to dispersive lattice vibrations (e.g., thermal conductivity of diamond is 6 times larger than that of metallic copper). They reduce intensities of diffraction spots and allow for inellastic scattering where the energy of the scatter (e.g., neutron) ch

4、anges due to absorption or creation of a phonon in the target. Electron-phonon interactions renormalize the properties of electrons (electrons become heavier). Superconductivity (conventional BCS) arises from multiple electron-phonon scattering between time-reversed electrons.PHYS 624: Crystal Latti

5、ce Vibrations: Phonons2Vibrations of small amplitude: 1D chain Classical Theory: Normal ModesQuantum Theory: Linear Harmonic Oscillator for each Normal Mode3421PHYS 624: Crystal Lattice Vibrations: Phonons3Normal modes of 4-atom chain in picturesPHYS 624: Crystal Lattice Vibrations: Phonons4Adiabati

6、c theory of thermal lattice vibrations Born-Oppenheimer adiabatic approximation: Electrons react instantaneously to slow motion of lattice, while remaining in essentially electronic ground state small electron-phonon interaction can be treated as a perturbation with small parameter: PHYS 624: Crysta

7、l Lattice Vibrations: Phonons5Adiabatic formalism: Two Schrdinger equations (for electrons and ions)The non-adiabatic term can be neglected at T100K!PHYS 624: Crystal Lattice Vibrations: Phonons6Newton (classical) equations of motion Lattice vibrations involve small displacement from the equilibrium

8、 ion position: and smaller harmonic (linear) approximationN unit cells, each with r atoms 3Nr Newtons equations of motionPHYS 624: Crystal Lattice Vibrations: Phonons7Properties of quasielastic force coefficientsPHYS 624: Crystal Lattice Vibrations: Phonons8Solving equations of motion: Fourier Serie

9、sPHYS 624: Crystal Lattice Vibrations: Phonons9Example: 1D chain with 2 atoms per unit cellPHYS 624: Crystal Lattice Vibrations: Phonons101D Example: Eigenfrequencies of chainPHYS 624: Crystal Lattice Vibrations: Phonons111D Example: Eigenmodes of chain at q=0Optical Mode: These atoms, if oppositely

10、 charged, would form an oscillating dipole which would couple to optical fields with Center of the unit cell is not moving!PHYS 624: Crystal Lattice Vibrations: Phonons122D Example: Normal modes of chain in 2D spaceConstant force model (analog of TBH) : bond stretching and bond bendingPHYS 624: Crys

11、tal Lattice Vibrations: Phonons133D Example: Normal modes of SiliconL longitudinalT transverseO optical A acousticPHYS 624: Crystal Lattice Vibrations: Phonons14Symmetry constraintsRelevant symmetries: Translational invariance of the lattice and its reciprocal lattice, Point group symmetry of the la

12、ttice and its reciprocal lattice, Time-reversal invariance. PHYS 624: Crystal Lattice Vibrations: Phonons15Acoustic vs. Optical crystal lattice normal modesAll harmonic lattices, in which the energy is invariant under a rigid translation of the entire lattice, must have at least one acoustic mode (s

13、ound waves)3 acoustic modes (in 3D crystal)PHYS 624: Crystal Lattice Vibrations: Phonons16Normal coordinatesThe most general solution for displacement is a sum over the eigenvectors of the dynamical matrix:In normal coordinates Newton equations describe dynamics of 3rN independent harmonic oscillato

14、rs! PHYS 624: Crystal Lattice Vibrations: Phonons17Quantum theory of small amplitude lattice vibrations: First quantization of LHOFirst Quantization:PHYS 624: Crystal Lattice Vibrations: Phonons18Second quantization representation: Fock-Dirac formalismPHYS 624: Crystal Lattice Vibrations: Phonons19Q

15、uantum theory of small amplitude lattice vibrations: Second quantization of LHOSecond Quantization applied to system of Linear Harmonic Oscillators:Hamiltonian is a sum of 3rN independent LHO each of which is a refered to as a phonon mode! The number of phonons in state is described by an operator:P

16、HYS 624: Crystal Lattice Vibrations: Phonons20Phonons: Example of quantized collective excitationsCreating and destroying phonons:Lattice displacement expressed via phonon excitations zero point motion!Arbitrary number of phonons can be excited in each mode phonons are bosons:PHYS 624: Crystal Latti

17、ce Vibrations: Phonons21Quasiparticles in solids Electron: Quasiparticle consisting of a real electron and the exchange-correlation hole (a cloud of effective charge of opposite sign due to exchange and correlation effects arising from interaction with all other electrons).Hole: Quasiparticle like e

18、lectron, but of opposite charge; it corresponds to the absence of an electron from a single-particle state which lies just below the Fermi level. The notion of a hole is particularly convenient when the reference state consists of quasiparticle states that are fully occupied and are separated by an

19、energy gap from the unoccupied states. Perturbations with respective to this reference state, such as missing electrons, are conveniently discussed in terms of holes (e.g., p-doped semiconductor crystals).Polaron: In polar crystals motion of negatively charged electron distorts the lattice of positi

20、ve and negative ions around it. Electron + Polarization cloud (electron excites longitudinal EM modes, while pushing the charges out of its way) = Polaron (has different mass than electron).PHYS 624: Crystal Lattice Vibrations: Phonons22Collective excitation in solids In contrast to quasiparticles,

21、collective excitations are bosons, and they bear no resemblance to constituent particles of real system. They involve collective (i.e., coherent) motion of many physical particles.Phonon: Corresponds to coherent motion of all the atoms in a solid quantized lattice vibrations with typical energy scale of Exciton: Bound state of an electron and a hole with binding energy Plasmon: Collective excitation of an entire electron gas relative to the lattice of ions; its existence is a manifestation of the long-range nature of the Cou