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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

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

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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

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4 Reciprocal Lattice Points

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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, ….

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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

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

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

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9 Kittel pg. 43 Reciprocal Lattice of the FCC Lattice

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

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

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

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

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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

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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

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16 -K Relation: Dispersion Relation K = 2 / min a K max = /a - /a

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

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

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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

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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

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21 Dispersion in Si

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22 Dispersion in GaAs (3D)

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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

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

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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

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

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