handout19 - Handout 19 Lattice Waves (Phonons) in 3D...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 19 Lattice Waves (Phonons) in 3D Crystals Group IV and Group III-V Semiconductors and Macroscopic Models of Acoustic Phonons in Solids In this lecture you will learn: • Lattice waves (phonons) in 3D crystals • Phonon bands in group IV and group III-V Semiconductors • Macroscopic description of acoustic phonons from elasticity theory • Stress, strain, and Hooke’s law ECE 407 – Spring 2009 – Farhan Rana – Cornell University Counting the Number of Phonon bands in 3D Crystals 3 3 2 2 1 1 b b b q r r r r α + + = { { { 2 2 where 2 2 where 2 2 where 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 N m N - N m N m N - N m N m N - N m < = < = < = There are N 1 N 2 N 3 allowed wavevectors in the FBZ There are N 1 N 2 N 3 phonon modes per phonon band Counting degrees of freedom and the number of phonon bands: Monoatomic Basis • There are 3 N 1 N 2 N 3 degrees of freedom corresponding to the motion in 3D of N 1 N 2 N 3 atoms The number of phonon bands must be 3 (two TA bands and one LA band) Counting degrees of freedom and the number of phonon bands: Diatomic Basis • There are 6 N 1 N 2 N 3 degrees of freedom corresponding to the motion in 3D of 2 N 1 N 2 N 3 atoms The number of phonon bands must be 6 (two TA bands and one LA band for acoustic phonons and two TO bands and one LO band for optical phonons) Periodic boundary conditions for a lattice of N 1 x N 2 x N 3 primitive cells imply:
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Phonon Bands in Silicon z q y q q x Silicon has a FCC lattice with two basis atoms in one primitive cell FBZ of Silicon The number of phonon bands must be 6; two TA bands and one LA band for acoustic phonons and two TO bands and one LO band for optical phonons ECE 407 – Spring 2009 – Farhan Rana – Cornell University Phonon Bands in Silicon z q y q q x Calculations Data (Neutron scattering) ( ) ( ) meV 64 0 0 = = x TO x LO q q ω h h X Frequency (THz)
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Phonon Bands in GaAs z q y q q x GaAs has a FCC lattice with two basis atoms in one primitive cell FBZ of GaAs The number of phonon bands must be 6; two TA bands and one LA band for acoustic phonons and two TO bands and one LO band for optical phonons ECE 407 – Spring 2009 – Farhan Rana – Cornell University Calculations Data (Neutron scattering) z q y q q x ( ) () meV 34 0 meV 36 0 = = x TO x LO q q ω h h Frequency (THz) LO TO LA TA TO TO LO LA TA TA TA LA TO Γ∆ X K ΣΓ Λ L Phonon Bands in GaAs
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4 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Phonon Bands in Diamond Frequency (cm -1 ) Γ∆ X ΓΛ L LO TO/TO LA TA/TA LO TO/TO LA TA/TA Calculations Data (Neutron scattering) z q y q q x ( ) ( ) meV 165 0 0 = = x TO x LO q q ω h h () ( ) 1 - cm 1330 2 0 2 0 = = c q c q x TO x LO π
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handout19 - Handout 19 Lattice Waves (Phonons) in 3D...

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