6.730 Physics for Solid State Applications

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Colin Lester
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1 6.730 Physics for Solid State Applications Lecture 8: Lattice Waves in 1D Monatomic Crystals Outline Overview of Lattice Vibrations so far Models for Vibrations in Discrete 1-D Lattice Example of Nearest Neighbor Coupling Only Relating Microscopic and Macroscopic Quantities February 20, 2004 Continuum Models 1-D D Wave Equation ω κ Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia Periodic Boundary Conditions: Traveling Waves 1

2 Continuum Models T 3 Specific Heat (hyperphysics.phy-astr.gsu.edu) The Atomistic Perspective Arrangement of Atoms and Bond Orientations 2

3 Diamond Crystal Structure: Silicon The Atomistic Perspective Arrangement of Atoms and Bond Orientations Bond angle = 109.5º Add 4 atoms to a FCC Tetrahedral bond arrangement Each atom has 4 nearest neighbors and 12 next nearest neighbors The Atomistic Perspective Vibrational Motion of Nuclei (Energy - I H ) / I H spring constant 3

4 equilibrium n-3 n-2 n-1 n n+1 n+2 n+3 a u n-3 u n-2 u n-1 u n u n+1 u n+2 u n+3 strained is the discrete displacement of an atom from its equilibrium position General Expansion The potential energy associated with the strain is a complex function of the displacements. A Taylor series expansion in the displacements gives where and the force on each lattice atom 4

5 Harmonic Matrix Spring Constants Between Lattice Atoms Harmonic Matrix: Dynamics of Lattice Atoms Force on the j th atom (away from equilibrium) 5

6 Solutions of Equations of Motion Convert to Difference Equation Time harmonic solutions Plugging in, converts equation of motion into coupled difference equations: Solutions of Equations of Motion We can guess solution of the form: This is equivalent to taking the z-transform 6

7 Solutions of Equations of Motion Consider Undamped Lattice Vibrations We are going to consider the undamped vibrations of the lattice: Solutions of Equations of Motion Dynamical Matrix 7

8 Solutions of Equations of Motion Dynamical Matrix equilibrium n-3 n-2 n-1 n n+1 n+2 n+3 a u n-3 u n-2 u n-1 u n u n+1 u n+2 u n+3 strained 8

9 Harmonic matrix: Dynamical matrix: Dispersion Relation 9

10 2nd Brillouin zone 1st Brillouin zone ω 2nd Brillouin zone k=-2π/a k=-π/a A k=π/a B k=2π/a k From what we know about Brillouin zones the points A and B (related by a reciprocal lattice vector) must be identical This implies that the wave form of the vibrating atoms must also be identical. ω A: k=-0.7π/a κ=-2π/a κ=-π/a A κ=π/a B κ=2π/a κ B: k=1.3π/a n-5 n-4 n-3 n-2 n-1 n n+1 n+2 n+3 n+4 n+5 But: note that point B represents a wave travelling right, and point A one travelling left 10

11 ω c Consider point C at the zone boundary κ=-2π/a κ=-π/a κ=π/a κ=2π/a κ When k=π/a, λ=2a, and motion becomes that of a standing wave (the atoms are bouncing backward and forward against each other λ=2a n-5 n-4 n-3 n-2 n-1 n n+1 n+2 n+3 n+4 n+5 In the limit of long-wavelength, we should recover the continuum model Linear dispersion, just like the sound waves for the continuum solid Connects the microscopic with the macroscopic 11

Solid State Physics. Lecturer: Dr. Lafy Faraj

Solid State Physics. Lecturer: Dr. Lafy Faraj Solid State Physics Lecturer: Dr. Lafy Faraj CHAPTER 1 Phonons and Lattice vibration Crystal Dynamics Atoms vibrate about their equilibrium position at absolute zero. The amplitude of the motion increases