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Crystal Vibration Equilibrium Hot Cold Energetic atomic vibrations Conduction of heat in insulators involves the generation and propogation of atomic vibrations through the bonds that couple the atoms. (An intuitive figure.) Crystal Vibration Interatomic Bonding Spring constant (C) Energy Parabolic Potential of Harmonic Oscillator ro Eb Distance s-1 s s+1 x Mass (M) Transverse wave: 3 ur ur1 ur+1 x a Atoms executing longitudinal vibrations parallel to x. Crystal Vibration of a Monoatomic Linear Chain Longitudinal wave of a 1-D Array of Spring Mass System Spring constant, g M Mass, m Equilibrium Position a Deformed Position xun-1 s-1 usx n x n+1 us+1 us: displacement of the sth atom from its equilibrium position 5 6 Solution of Lattice Dynamics s-1 s s+1 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: 7 w-K Relation: Dispersion Relation K = 2/l lmin = 2a Kmax = /a -/a<K< /a l: wavelength 2a 8 Polarization and Velocity w 2 M = C2 exp iKa exp iKa = 2C 1 cos Ka 1 2C 1 cos Ka 2 w= M Group Velocity: Frequency, w dw vg = dK Speed of Sound: dw vs = lim K 0 dK 0 Wave vector, K /a 9 Two Atoms Per Unit Cell M2 Lattice Constant, a M1 yn-1 xn 2 M1 M2 d xn dt 2 d 2 yn dt 2 yn = f yn yn1 2 xn xn+1 f: spring constant = f xn1 xn 2 yn Solution: Ka 10 Acoustic and Optical Branches Ka 1/µ = 1/M1 + 1/M2 What is the group velocity of the optical branch? What if M1 = M2 ? K 11 Polarization TA & TO Lattice Constant, a LA & LO yn-1 xn xn+1 yn Total 6 polarizations Optical Vibrational Modes Frequency, w LO TO LA 0 TA Wave vector, K /a 12 Dispersion in Si 13 Dispersion in GaAs (3D) LO Frequency (1012 Hz) 8 LO TO TO 6 LA LA 4 2 0 L TA 0.4 0.2 0 (111) Direction TA 0.2 Ka/ 0.4 0.6 (100) Direction 0.8 1.0 X 14 Allowed Wavevectors (K) A linear chain of N=10 atoms with two ends jointed a Solution: us ~uK(0)exp(-iwt)sin(Kx), x =sa B.C.: us=0 = us=N=10 x K=2n/(Na), n = 1, 2, …,N Na = L 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 15 Allowed Wave Vectors in 3D 2 4 N K x , K y , K z = 0; ; ;...; L L L N3: # of atoms Kz Ky Kx 2/L 16 Phonon Energy •The linear atom chain can only have N discrete K w is also discrete Distance • The energy of a lattice vibration mode at frequency w was found to be 1 u = n w 2 hw • where ħw 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 w, and follows the Bose-Einstein statistics: 1 n = w 1 exp k BT Equilibrium distribution 17 Total Energy of Lattice Vibration El = p 1 n w K , p 2 w K , p K p: polarization(LA,TA, LO, TO) K: wave vector 18 Reflected waves Backward wave -k k B' A' C' Forward wave x A B C a An electron wave propagation through a linear lattice. For certain k values the reflected waves at successive atomic planes reinforce each other to give rise to a reflected wave travelling in the backward direction. The electron then cannot propagate through the crystal. c 2 k=±/a Energy = Ec s2 Energy = Es Forward and backward waves in the crystal with k = ± /a give rise to two possible standing waves, c and s. Their probability density distributions, c2 and s2 , have maxima either at the ions or between the ions.