This preview shows pages 1–11. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4. Phonons Lecture 6 1/20/11 Last Lecture: Chapter 4 Phonons Hookes Law force, displacement Newtons Law force, mass, acceleration Oscillation modes longitudinal, transverse Dispersion relation Group velocity Brillouin zone boundaries and limit Diatomic chains This Lecture Chapter 4 continued Reminder: Homework #3 due Feb. 2 Tuesday 4.1: For a monatomic chain, detemine total energy in an elastic wave. 4.3 For a diatomic basis, calculate wave amplitude ratio , u/v at K=/a . Extra what is the dispersion relation for a diatomic basis if m1=m2? Sketch. 4.5 For diatomic basis, find dispersion relationship at K=0 and K=/a. Phonon What are they? Propagation lattice vibrations that carry energy through a crystal Crystal can be considered an array of mass centers connected by springs Harmonic Oscillator Hookes law M oscillates with kinetic and potential energy. n=0 n=1 n=2 w k E M E n E n = ( + 1 2 29 mode number n = 2 = 2 , , = / , k = 2 We use Newtons law to derive (29 Energy in a Crystal Definition: Kinetic energy is the sum of the individual kinetic energies each of the form E 1 2 2 What is u? How do we relate F to E? How do we relate u to (amplitude to frequency)? Force between atoms s and s+1 is C(u s u s +1) Potential energy associated with the stretching of this bond is C(u s u s+1 ) 2 Newtons Law F Ma = 2 2 ( ) d r t M dt = v ( , ), r t = v X=0 X r x = v ( , ) ( , ) d F r t r t dx = v v ( ) is position of atom r t v In one dimension: potential energy ( 0) x = = ( ) x x k x = = arises from interaction between atom and rest of crystal. displacement 2 3 ( ) 1 ... r r r r = + + + + Liner nearest neighbor interaction Monatomic chain  Phonons Derive equation of motion S S+1 S1 S+2 u S u S+1 u S1 u Su S 1 u S u S+1 2 1 1 2 ( 2 ) S S S S d u C u u u M dt + + = 2.) Equation of motion Hooke meets Newton 1 ( ) S S S S nn F C u u = = , + 1 (  + 1 29 + , 1 (  1 29 spring constant C S,S+1 =C S,S1 =C 1.) Force similar to Hookes law We assumed Nearest neighbors Uniform force constant u S ( t ) ( 29 d 2 u S dt 2 =  2 3.) Write wave solution Find Dispersion Relation monatomic chain. 2 1 1 2 ( 2 ) S S S S d u C u u u M dt + + = d 2 u S dt 2 =  2 C ( u S + 1 + 1 2 29 =  2 1 exp( )exp( ) S u u isKa iKa = 5.) Use wave solution: a: spacing between atomic planes K : the wavevector 4.) Combine equation of motion and wave solution: Acoustic, Optical Phonons  contd....
View Full
Document
 Winter '11
 DianaHuffaker

Click to edit the document details