固体物理Chapter 4Phonons I. crystal vibration

1、固体物理固体物理Chapter 4Phonons I. Chapter 4Phonons I. crystal vibrationcrystal vibrationPhonon is one kind of elementary excitations in solidsNameFieldElectronPhotonElectromagnetic wavePhononElastic wavePlasmonCollective electron waveMagnonMagnetization (spin) wavePolaronElectron + elastic deformationExci

2、tonPolarization waveImportant elementary excitations in solidsVibrations of crystals with monoatomic basisConsider the elastic vibrations of a crystal with one atom in the primitive cell.Describe with a single coordinate us the displacement of the plane s from its equilibrium position.Three mode for

3、 each wavevector, one of longitudinal polarization and two of transverse polarization.longitudinal modetransverse modeNote: for N atoms there are 3N mode, N L-modes and 2N T-modes.Assume that the elastic response of the crystal in linear function of the force. (Hookes law)i.e. the elastic energy is

4、a quadratic function of the relative displacement of any two points in the crystal.Neglect cubic and higher order terms.From Hookes law, the force on the s plane caused by the displacement of the plane s+p is proportional to the difference of their displacement,i.e. Fsp = Csp (us+p us)Consider only

5、the nearest neighbor interactions. The total force on s comes from the plane s1Fs = C (us+1 us) + C (us-1 us) = C (us+1 + us-1 2us)The equation of the motion is:)2(1122ssssuuuCdtudMa traveling wave solution:us = u expi(sKa t),where a is the spacing between planes and K is the wavevector.Then we have

6、:)cos1 (22)exp()exp(2KaCiKaiKaCMthe dispersion relation (色散关系):2sin4)cos1)(/2(22KaMCKaMCor2sin42/1KaMCFor Ka), KAt the boundary of first Brillouin zone (K = /a),d/dK = 0First Brillouin zoneOnly K in the first Brillouin zone is physically significant for the elastic waves.The range of independent val

7、ues of K is specified by:aKaKaor ,We may treat a value of K outside the 1st Brillouin zone by subtracting a appropriate reciprocal lattice vector in order to obtain an equivalent wavevector in the 1st Brillouin zone.zone.Brilouin 1st where)/2 dimension, one(in KanKKGKKAt the Brillouin zone boundarie

8、s, K = /a,us = u expi( s t) = (1)s u exp(it) (standing wave)The critical value, K = /a, satisfies the Bragg condition.Note: for x-ray, it is possible to have other n which cause the wavevector K outside 1st Brillouin zone.Group velocityThe group velocity is the transmission velocity of a wave packet

9、, which is also the velocity of energy propagation in the medium.dimension onein /dimension in three )(dKdvKvgKgWith the particular dispersion relation2sin42/1KaMC2cos2/12KaMCavgLong wavelength limitWhen a i.e. Ka 122222sin4KMCaKaMCKaMC2/1 i.e.aMCdKdvg2/1/Derivation of force constants from experimen

10、tConsidering p nearest planes, the force on s plane)(201spjjsjuuCFThe equation of motion is022)(2pjsjsjsuuCdtudMThe dispersion relation012)cos1 (2pjjjKaCMaaKppKadKMaC/2cos2Two atoms per primitive basisThe dispersion relation shows new features in crystals with two or more atoms per primitive basis.a

11、coustical and optical branches (声学支和光学支)With p atoms in the primitive cell and N primitive cells, there are pN atoms, 3pN degrees of freedom, N LA branches, 2N TA branches, (p1)N LO branches and (2p2)N LO branches.If there are p atoms in the primitive cell, there are 3p branches to the dispersion re

12、lation: 3 acoustical branches (1 LA and 2 TA) and 3p3 optical branches (p1 LO and 2p2 LO ).Assume that each plane interacts only with the nearest-neighbor planes. The equations of motion is)2()2(12221221ssssssssvuuCdtvdMuvvCdtudMthe solution in a form of a traveling wave:)(exp)(exptsKaivvtsKaiuusssu

13、bstitute the solution in the equationsCviKaCuvMCuiKaCvuM2)exp(1 2)exp(1 2212The homogenous linear equations have a solution only if the determinant of the coefficients of the unknown u, v vanishes.02 )exp(1 )exp(1 22221MCiKaCiKaCMCor0)cos1 (2)(22221421KaCMMCMMthe dispersion relationKaMMMMMMMMCMMKaMM

14、CMMCMMCcos2)( 2)cos1 (8)(2)(2212221212121212221212at long wavelength limit (Ka 1)branch) l(acoustica )(2branch) (optical 11222122212KMMCaMMC12MMvuFor the optical branchThe atoms vibrate against each other, but their center of mass is fixed.vu For the acoustical branchThe atoms and their center of ma

15、ss move together.at the 1st Brillouin zone boundary (Ka = )2212/2 ;/2MCMCThere is a frequency gap at boundary of the 1st Brillouin zone.Quantization of the elastic wavesThe energy of a lattice vibration is quantized. The quantum of the energy is called phonon in analogy with the photon of the electr

16、omagnetic wave.Thermal vibrations in crystals are thermally excited phonons.The energy of the elastic mode)21( nthis mode occupied by n phonons with a frequency the zero point energy21The amplitude of the elastic vibration)/()21(420VnuConsider a standing wave modetKxuucoscos0The time average kinetic

17、 energy is)21(2181202nuVEtKxuMtuM222022sincos2121The kinetic energy of an atom isWhat is the sign of ?The energy of a phonon must be positive.It is conventional and suitable to view as positive.A mode with imaginary (negative 2) will be unstable.Phonon momentumA phonon of wavevector K will interact

18、with particles such as photons, neutrons, and electrons as if it has a momentum .KHowever a phonon does not carry physical momentum.For most practical purpose a phonon acts as if its momentum were , which is called the crystal momentum.KIn crystal there exists wavevector selection rules for allowed

19、transition between quantum states.The true momentum of the whole system always is rigorously conserved.For the elastic scattering of a photon by a crystal, the wavevector selection rule isGkkGkKk If the scattering of the photon is inelastic, with the creation of a phonon , the wavevector selection r

20、ule becomesKIf a phonon is absorbed in the inelastic scattering, the wavevector selection rule becomesKGKkkInelastic scattering by phononsPhonon dispersion relations (K) are most often determined by inelastic scattering of neutrons with the emission or absorption of a phonon.A neutron sees the crystal lattice chiefly by interact