1. ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology Lecture 4: Crystal Vibration and Phonon Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu

2. 2 Outline  Reciprocal Lattice Crystal Vibration Phonon Reading: 1.3 in Tien et al References: Ch3, Ch4 in Kittel

3. 3 Reciprocal Lattice The X-ray diffraction pattern of a crystal is a map of the reciprocal lattice. It is a Fourier transform of the lattice in real space It is a representation of the lattice in the K space K: wavevector of Incident X ray Real lattice Diffraction pattern or reciprocal lattice K’: wavevector of refracted X ray Construction refraction occurs only when  K  K’-K=ng 1 +mg 2

4. 4 Reciprocal Lattice Points

5. 5 Reciprocal lattice & K-Space a 1-D lattice K-space or reciprocal lattice: Lattice constant Periodic potential wave function: Wave vector or reciprocal lattice vector: G or g = 2n  /a, n = 0, 1, 2, ….

6. 6 Reciprocal Lattice in 1D a The 1 st Brillouin zone: Weigner-Seitz primitive cell in the reciprocal lattice Real lattice Reciprocal lattice k 0 2  /a 4  /a -2  /a-4  /a-6  /a x -  /a  /a

7. 7 Kittel pg. 38 Reciprocal Lattice of a 2D Lattice

8. 8 FCC in Real Space Angle between a 1, a 2, a 3 : 60 o Kittel, P. 13

9. 9 Kittel pg. 43 Reciprocal Lattice of the FCC Lattice

10. 10 Special Points in the K-Space for the FCC 1 st Brillouin Zone

11. 11 BCC in Real Space Primitive Translation Vectors: Rhombohedron primitive cell 0.5  3a 109 o 28 ’ Kittel, p. 13

12. 12 Real: FCC Reciprocal: BCC 1 st Brillouin Zones of FCC, BCC, HCP Real: HCP Real: FCC Reciprocal: BCC

13. 13 Crystal Vibration s-1ss+1 Mass (M) Spring constant (C) x Transverse wave: Interatomic Bonding

14. 14 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

15. 15 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

16. 16  -K Relation: Dispersion Relation K = 2  / min  a K max =  /a -  /a<K<  /a 2a : wavelength

17. 17 Polarization and Velocity Frequency,  Wave vector, K 0  /a Longitudinal Acoustic (LA) Mode Transverse Acoustic (TA) Mode Group Velocity: Speed of Sound:

18. 18 Lattice Constant, a xnxn ynyn y n-1 x n+1 Two Atoms Per Unit Cell Solution: Ka M2M2 M1M1 f: spring constant

19. 19 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

20. 20 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

21. 21 Dispersion in Si

22. 22 Dispersion in GaAs (3D)

23. 23 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

24. 24 KxKx KyKy KzKz 2  /L Allowed Wave Vectors in 3D N 3 : # of atoms

25. 25Phonon 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

26. 26 Total Energy of Lattice Vibration p: polarization(LA,TA, LO, TO) K: wave vector