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Understanding Silicon Carbide Phonons with Ruth D’Oleo on February 28, 2002

Explore the lattice structure of silicon carbide, analyzing phonons, displacements, potential energy, and harmonic responses. Dive into dynamic matrix equations, dispersion relations, and bond bending in the context of the tetrahedral nature of silicon carbide.

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Understanding Silicon Carbide Phonons with Ruth D’Oleo on February 28, 2002

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  1. Silicon Carbide Phonons Ruth D’Oleo February 28, 2002

  2. Lattice • Zincblende structure (two fcc lattices superposed) • Displacements from equilibrium ui[Rn] • i runs over degrees of freedom and type of atom • Rn is vector to nth cell

  3. Potential • Expand potential in Taylor series • Force at equilibrium is zero • Define Harmonic Matrix:

  4. Dynamical Equations • Potential becomes • Force on lth atom in nth cell: • Using we get • Assuming a harmonic response

  5. Matrix Equation • We get • Put equation in matrix form: Displacement vector Mass matrix

  6. Dynamical Matrix • Matrix equation: • Use Bloch’s Theorem: • Can write • Dynamical matrix

  7. Summary of Main Result • Dynamical matrix • Eigenvalue equation • Obtain dispersion relation (depends on direction of k in BZ)

  8. Bond Bending and Stretching The Tetrahedral Nature of Silicon Carbide. The Lennard-Jones Potential • Minimum number of constants for Born model = 2 • Can be infinite number of constants if higher order terms are considered • (further away from minimum of Lennard-Jones potential) • Use to get the potential

  9. Bending and Stretching • Stretching energy: • Bending energy: • Total potential:

  10. Obtain Dynamical Matrix from Potential • Use potential to find dynamical matrix

  11. Dynamical Matrix • Dynamical matrix becomes • Recall eigenvalue equation: • Use MATLAB to solve • along some symmetry • directions of k in Brillouin • Zone

  12. Phonon Dispersion Curves Brillouin Zone:

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