1. Crystal Vibration

3. 3 s-1ss+1 Mass (M) Spring constant (C) x Transverse wave: Interatomic Bonding

5. 5 Crystal Vibration of a Monoatomic Linear Chain Longitudinal wave of a 1-D Array of Spring Mass System u s : displacement of the s th atom from its equilibrium position u s-1 usus u s+1 M

6. 6

7. 7 Solution of Lattice Dynamics Identity: Time dep.: cancel Trig: s-1ss+1 Same M Wave solution: u(x,t) ~ uexp(-i  t+iKx) = uexp(-i  t)exp(isKa)exp(  iKa)  frequency K: wavelength

8. 8  -K Relation: Dispersion Relation K = 2  / min  a K max =  /a -  /a<K<  /a 2a : wavelength

9. 9 Polarization and Velocity Frequency,  Wave vector, K 0  /a Longitudinal Acoustic (LA) Mode Transverse Acoustic (TA) Mode Group Velocity: Speed of Sound:

10. 10 Lattice Constant, a xnxn ynyn y n-1 x n+1 Two Atoms Per Unit Cell Solution: Ka M2M2 M1M1 f: spring constant

11. 11 1/µ = 1/M 1 + 1/M 2 What is the group velocity of the optical branch? What if M 1 = M 2 ? Acoustic and Optical Branches K Ka

12. 12 Lattice Constant, a xnxn ynyn y n-1 x n+1Polarization Frequency,  Wave vector, K 0  /a LA TA LO TO Optical Vibrational Modes LA & LO TA & TO Total 6 polarizations

13. 13 Dispersion in Si

14. 14 Dispersion in GaAs (3D)

15. 15 Allowed Wavevectors (K) Solution: u s ~u K (0)exp(-i  t)sin(Kx), x =sa B.C.: u s=0 = u s=N=10 K=  2n  /(Na), n = 1, 2, …,N Na = L a A linear chain of N=10 atoms with two ends jointed x Only N wavevectors (K) are allowed (one per mobile atom): K= -8  /L -6  /L -4  /L -2  /L 0 2  /L 4  /L 6  /L 8  /L  /a=N  /L

16. 16 KxKx KyKy KzKz 2  /L Allowed Wave Vectors in 3D N 3 : # of atoms

17. 17Phonon Equilibrium distribution where ħ  can be thought as the energy of a particle called phonon, as an analogue to photon n can be thought as the total number of phonons with a frequency  and follows the Bose-Einstein statistics: The linear atom chain can only have N discrete K   is also discrete The energy of a lattice vibration mode at frequency  was found to be

18. 18 Total Energy of Lattice Vibration p: polarization(LA,TA, LO, TO) K: wave vector