Normal modes are eigenfunctions of T

Transcription

1 Quasiparticles

2 Phonons N atom atoms in crystal 3N atom normal modes p atoms in the basis N atom /p unit cells N atom /p translational symmetries N atom /p k-vectors 3p modes for every k vector 3 acoustic branches and 3p-3 optical branches

3

4 Normal modes are eigenfunctions of T x x u u exp i lka mka nka t lmn These are eigenfunctions of T. k y y u u exp i lka mka nka t lmn k z z u u exp i lka mka nka t lmn k T u u i lk a pa mk a qa nk a ra t x exp x i lpk a1 qmk a2 rnk a3 u exp i lk a k 1 mk a2 nk a3 t x pqr lmn k exp x exp ilpka qmka rnka u lmn

5 fcc 2 x dulmn 2 C m u u u u u u u u dt 2 x x x x x x x x l1mn lmn l1mn lmn lm1n lmn lm1n lmn x x x x x x x x ul 1mn 1ulmn ul 1mn1ulmn ulm1n 1ulmn ulm 1n 1ulmn y y y y y y y y ul 1mn ulmn ul 1mn ulmn ulm1n 1ulmn ulm 1n 1ulmn u z z lm z z 1 1 z z 1 1 z z n ulmn ulm n ulmn ul mn ulmn ul 1mn 1ulmn and similar expressions for the y and z motion

6 fcc Substitute the eigenfunctions of T into Newton's laws. l nkya x x x l m kxa m n kza ulmn uexp i lka1mka2nka3 uexp i. k k

7 3N degrees of freedom fcc phonons

8 Phonon dispersion Au From Springer Materials: Landholt Boernstein Database

9 Materials with the same crystal structure will have similar phonon dispersion relations Cu Au

10 fcc phonons energy spectral density internal energy density specific heat

11 NaCl 2 atoms/unit cell 6 equations 3 acoustic and 3 optical branches u x u x x x exp ika ka ka t v v exp ika ka ka t nml k nml k 1 2 3

12 Two atoms per primitive unit cell NaCl Si

13 GaAs Hannes Brandner

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15 Phonon quasiparticle lifetime Phonons are the eigenstates of the linearized equations, not the full equations. Phonons have a finite lifetime that can be calculated by Fermi's golden rule. 2 2 f H i E E i f ph ph f i Occupation is determined by a master equation (not the Bose-Einstein function). dp 0 0 i 1 0 N 0 dt i0 P0 dp i N1 P 1 i1 dt P N dp N 0N 1N Ni i N dt

16 Acoustic attenuation The amplitude of a monocromatic sound wave decreases as the wave propagates through the crystal as the phonon quasiparticles decay into phonons with other frequencies and directions.

17 Raman Spectroscopy Inelastic light scattering k k K G C. V. Raman Phonons, magnons, plasmons, polaritons, excitons K 0

18 Raman Spectroscopy 0 X cos( t) X P 0Ecos( t) 0 X cos( t) Ecos( t) X There are components of the polarization that oscillate at.

19 Raman Spectroscopy Stokes: I( ) n k 1 anti-stokes: I ( ) nk

20 Raman spectroscopy

21 Magnons Magnons are excitations of the ordered ferromagnetic state

22 Longitudinal plasma waves nm 2 d y dt 2 nee E ney 0 d y nm dt 2 n e y d y p y dt 0 Plasma frequency p 2 ne m There is no magnetic component of the wave. 0 Kittel Plasma waves can be quantized like any other wave

23 Electron energy loss spectroscopy E n p EELS is often used to measure phonons

24 Electron energy loss spectroscopy Aluminum Plasmons 15.3 ev Surface plasmons 10.3 ev Magnesium Plasmons 10.6 ev Surface plasmons 7.1 ev

25 Transverse optical plasma waves The dispersion relation for light For a free electron gas 1 2 p p 2 c k 2 ck Plasmons Kittel p ck

26 Surface Plasmons Waves in the electron density at the boundary of two materials. Surface plasmons have a lower frequency that bulk plasmons. This confines them to the interface.

27 Surface Plasmons High-resolution surface plasmon imaging of gold nanoparticles by energy-filtered transmission electron microscopy PHYSICAL REVIEW B 79, R 2009 Surface plasmons on nanoparticles are efficient at scattering light. Green and blue require different sized particles.

28 Nature Photonics Surface plasmons are used for biosensors.

29 Plasmon filter Plasmon modes on the other side of the metal films are excited. SURFACE%20PLASMON%20POLARITONS%20AT%20%20METALINSULATOR%20INTERFACES/Lecture_on_the_Web_SURFAC E-PLASMONS-POLARITONS.pdf

30 Polaritons Transverse optical phonons will couple to photons with the same and k. photon dispersion avoided crossing Light Bragg reflects off the sound wave; sound Bragg reflects off the light wave.

31 Polaritons The dispersion relation for light c k 2 For an insulator The description of polaritons is already built into the dielectric function.

32 Polaritons Ignore the loss term i Use a common denominator Define L L 2 2 L 2 2 0

33 Polaritons L c k There are two solutions for every k, one for the upper branch and one for the lower branch. Polaritons are the normal modes near the avoided crossing. A gap exists in frequency.

34 Polaritons allow us to study the properties of phonons using optical measurements = 4.7E13 = 3.1E13 Kittel By looking at the reflectance in different crystal directions, you can determine the frequencies of the transverse optical phonons.

35 Polaritons and optical properties

36 Excitons Bound state of an electron and a hole in a semiconductor or insulator Mott Wannier excitons (like positronium)

37 Mott-Wannier Excitons Bound state of an electron and a hole in a semiconductor or insulator (like positronium) E Hydrogenic model e E * nk, g * * 32 0 n 2 mh me K Kittel

38 Excitons Gross & Marx

39 Excitons Biexcitons H 2? Kittel Metallic plasma droplets Observe with an infrared camera See: C. D. Jeffries, Electron-Hole Condensation in Semiconductors, Science 189 p. 955 (1975).