Crystal Vibration
3 s-1ss+1 Mass (M) Spring constant (C) x Transverse wave: Interatomic Bonding
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
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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 -K Relation: Dispersion Relation K = 2 / min a K max = /a - /a<K< /a 2a : wavelength
9 Polarization and Velocity Frequency, Wave vector, K 0 /a Longitudinal Acoustic (LA) Mode Transverse Acoustic (TA) Mode Group Velocity: Speed of Sound:
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 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 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 Dispersion in Si
14 Dispersion in GaAs (3D)
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 KxKx KyKy KzKz 2 /L Allowed Wave Vectors in 3D N 3 : # of atoms
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 Total Energy of Lattice Vibration p: polarization(LA,TA, LO, TO) K: wave vector