Lecture 10 & 11: Lattice Waves in 2D and 3D

6.730 Physics for Solid State Applications

Outline

•Algebra for Bond Stretching in 3-D • Example: 2-D Lattice Waves with Bond Stretching • Algebra for Bond Bending in 3-D • Example: 1-D and 2-D Lattice Waves with Bond Stretching and Bending •3-D Examples

February 25 and 27, 2004

Lattice Waves in 3-D Crystals

Displacements of basis atoms along three directions…

1 Lattice Waves in 3-D Crystals

Second order Taylor series expansion for total potential energy:

Harmonic Matrix:

Equation of motion for lattice atoms assuming ‘plane wave’ solutions:

Dynamical Matrix:

Lattice Waves in 3-D Crystals

2 L Bond Stretching

L + ∆L

uj bi,j bi,j + uj - ui

Example: 2-D Lattice with Bond Stretching

α1 7 3 5 2 0 1 α 648 2

3 Example: 2-D Lattice with Bond Stretching Elements of the Dynamical Matrix

α1 7 3 5 2 0 1 α 648 2

Example: 2-D Lattice with Bond Stretching Dynamical Matrix

4 2d Dispersion Relation

Dispersion Relation on special directions

0 L L 0 ω/ω

T ω/ω T

ΓΧΓ W L

0 W T ky ω/ω Γ Χ

Γ Χ

5 Example: 2-D Lattice with Bond Stretching Dispersion Relation

2D Modes

6 Example: 2-D Lattice with Bond Stretching Relating Microscopic and Macroscopic Models

Bond Bending L φ

uj y bi,j φ x ui

7 Bond Stretching and Bending

Example: 1-D Diatomic Lattice with Bond Stretching and Bending Potential Energy

M u [Rp,t] y 1 M2 1 x u2[Rp,t] A B

8 Example: ‘1-D’ Diatomic Lattice with Bond Stretching and Bending Potential Energy

M1 M u1[Rp,t] y 2 x u2[Rp,t] A B

1-D DiatomicLattice with Bond Stretching and Bending Dispersion Relation

M M u1[Rp,t] y 1 2 x u2[Rp,t] A B

Optical modes

Acoustic modes

k -π/a π/a

9 2d rectangular monatomic crystal

L M ky Γ Χ kx X Γ L ML

Phonon Dispersion in 3D Cubic Crystal

aluminum

http://debian.mps.krakow.pl/phonon/Public/phrefer.html

10 Phonon Dispersion in FCC with 2 Atom Basis

http://debian.mps.krakow.pl/phonon/Public/phrefer.html

Phonon Dispersion in FCC with 2 Atom Basis

http://debian.mps.krakow.pl/phonon/Public/phrefer.html