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ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology Lecture 4: Crystal Vibration and Phonon

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. Outline.

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ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology Lecture 4: Crystal Vibration and Phonon

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

  3. K’: wavevector of refracted X ray Reciprocal Lattice K: wavevector of Incident X ray Real 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 Construction refraction occurs only when DKK’-K=ng1+mg2 Diffraction pattern or reciprocal lattice

  4. Reciprocal Lattice Points

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

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

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

  8. FCC in Real Space • Kittel, P. 13 • Angle between a1, a2, a3: 60o

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

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

  11. Primitive Translation Vectors: • Kittel, p. 13 BCC in Real Space • Rhombohedron primitive cell 0.53a 109o28’

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

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

  14. Crystal Vibration of a Monoatomic Linear Chain Longitudinal wave of a 1-D Array of Spring Mass System M us-1 us us+1 us: displacement of the sth atom from its equilibrium position

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

  16. w-K Relation: Dispersion Relation l: wavelength K = 2/l lmin = 2a Kmax = /a -/a<K< /a 2a

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

  18. Two Atoms Per Unit Cell Lattice Constant, a xn+1 yn-1 xn yn M2 M1 f: spring constant Solution: Ka

  19. Acoustic and Optical Branches Ka optical branch 1/µ = 1/M1 + 1/M2 acoustic branch What is the group velocity of the optical branch? What if M1= M2? K

  20. Polarization Lattice Constant, a xn+1 yn-1 xn yn TA & TO LA & LO Optical Vibrational Modes Total 6 polarizations LO TO Frequency, w TA LA p/a 0 Wave vector, K

  21. Dispersion in Si

  22. Dispersion in GaAs (3D)

  23. Allowed Wavevectors (K) A linear chain of N=10 atoms with two ends jointed x a Solution: us ~uK(0)exp(-iwt)sin(Kx), x =sa B.C.: us=0 = us=N=10 K=2np/(Na), n = 1, 2, …,N Na = L Only N wavevectors (K) are allowed (one per mobile atom): K=-8p/L -6p/L -4p/L -2p/L 0 2p/L 4p/L 6p/L 8p/Lp/a=Np/L

  24. Allowed Wave Vectors in 3D N3: # of atoms Kz Ky Kx 2p/L

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

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

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