Lattice Dynamics

1. Physical properties of solids Lattice Dynamics related to movement of atoms about their equilibrium positions determined by electronic structure • Sound velocity • Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors) • Hardness of perfect single crystals (without defects)

2. Uniform Solid Material There is energy associated with the vibrations of atoms. They are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. (X-1) (X) (X+1)

3. Wave-Particle Duality They don’t like to be seen together  Just as light is a wave motion that is considered as composed of particles called photons, we can think of the normal modes of vibration in a solid as being particle-like. Quantum of lattice vibration is called the phonon.

4. Phonon: A Lump of Vibrational Energy Propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material. Roughly how big is ? Phonon: Sound Wavepackets The different possible oscillations are limited by the size of the system. Crystals are big! Result?

5. Cx X=A cos ωt X Reminder to the physics of oscillations and waves (a few slides) Harmonic oscillator in classical mechanics Equation of motion: C Example: vertical springs Cx or Hooke’s law where Solution with where Epot = ½ Cx2 x

6. Displacement as a function of time and k Traveling plane waves: Y or X (Phonon wave vector also often given as q or K instead of k) Useful to mention where approach not necessary Consider a particular state of oscillation (long) Yconstant solves wave equation

7. 10-10m good for Like diffraction, simple for large wavelengths λ>10-8m Large wavelength λ Crystal can be viewed as a continuous medium: Speed of longitudinal wave: where Bs:bulk modulus (ignoring anisotropy of the crystal) Bs determines elastic deformation energy density (click for details in thermodynamic context) dilation E.g.: Steel Bs=160 109N/m2 ρ=7860kg/m3

8. > interatomic spacing continuum approach fails phonons vibrational modes quantized Phonons are responsible for the majority of the thermal capacity of solids (a later lecture). Also important for electrical conductivity

9. Vibrational Modes of a Monatomic Lattice Linear chain: Remember: two coupled harmonic oscillators Anti-symmetric mode Symmetric mode Superposition of normal modes Today 1D, Next time: 3D

10. a un+1 un+1 un-2 un-2 un-1 un-1 un+2 un+2 un generalization Infinite linear chain ? How to derive the equation of motion in the harmonic approximation n n-2 n-1 n+1 n+2 C un fixed

11. n n Total force driving atom n back to equilibrium equation of motion Old solution for continuous wave equation was . Use similar? ? Let us try! approach for linear chain , ,

12. Note: here pictures of transversal waves although calculation for the longitudinal case k Continuum limit of acoustic waves:

13. Technically, only have longitudinal modes in 1D (but transverse easier to see what’s happening) x = na un un (a) Chain of atoms in the absence of vibrations. (b) Coupled atomic vibrations generate a traveling longitudinal (L) wave along x. Atomic displacements (un) are parallel to x. (c) A transverse (T) wave traveling along x. Atomic displacements (un) are perpendicular to the x axis. (b) and (c) are snapshots at one instant.

14. k Region is called first Brillouin zone Slope at boundary? Results in same motion of the atoms Arbitrary origin What was the slope equal to for low k? , here h=1 1-dim. reciprocal lattice vector Same as before! Means we only have to consider a small range of k values rather than all possible wavelengths!

15. What is going on at the edge of the Brillouin zone? Equivalent to a Bragg reflection of x-rays. When the Bragg condition is satisfied, the wave does not travel but reflects back and forth forming a standing wave.

16. a un+1 un+1 un-2 un-2 un-1 un-1 un+2 un+2 un What is we had more than nearest neighbor interactions? ? How to derive the equation of motion in the harmonic approximation n n-2 n-1 n+1 n+2 C un fixed

17. a C u2n u2n+1 u2n-2 u2n-1 u2n+2 Vibrational Spectrum for structures with 2 or more atoms/primitive basis Linear diatomic chain: 2n Note different convention for a than Kittel. I have a good reason! 2n-2 2n-1 2n+1 2n+2 2a Equation of motion for atoms on even positions: Equation of motion for atoms on odd positions: Solution with: and

18. , What happens? • Click on the picture to start the animation M->m note wrong axis in the movie 2 2

19. Transverse optical mode for diatomic chain Amplitudes of different atoms A/B=-m/M Transverse acoustic mode for diatomic chain A/B=1

20. Longitudinal Eigenmodes in 1D What if the atoms were opposite charged? Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with λ~a

21. Relation to stress/strain • Whether phonons or stress/strain is taught first in a solid state text varies, but they are normally taught in close proximity. • What do you think is the relation?

22. Phonon Dispersion in 3D 1D model • The 1D model can be extended to 3D if the variables u refer not to displacements of atoms but planes of atoms. • Need to include motions that are perpendicular to the wave vector. • These are called transverse acoustic modes (TA), as opposed to longitudinal acoustic modes (LA). 3D model

23. 3D Dispersion curves “Primary wave” = faster wave LA Every 3D crystal has 3 acoustic branches, 1 longitudinal and 2 transverse Are the branches degenerate? “Secondary wave” = slower wave TA No, the perpendicular displacements will have different force (“spring”) constants from the longitudinal force constants. Example: Earthquake waves

24. Summary: What is a phonon? • Consider the regular lattice of atoms in a uniform solid material. • There should be energy associated with the vibrations of these atoms. • But they are tied together with bonds, so they can't vibrate independently. • The vibrations take the form of collective modes which propagate through the material. • Such propagating lattice vibrations can be considered to be sound waves. • And their propagation speed is the speed of sound in the material.

25. z y # atoms in primitive basis # of primitive unit cells no translations, no rotations Intuitive picture: 1atom 3 translational degrees of freedom x 3+3=6 degrees of freedom=3 translations+2rotations +1vibraton 3p N vibrations 3D Solid: p N atoms

26. Analogy with classical mechanical pendulums attached by spring Amplitude of vibration is strongly exaggerated!