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Silicon Carbide

Silicon Carbide. Phonons. Ruth D’Oleo February 28, 2002. Lattice. Zincblende structure (two fcc lattices superposed). Displacements from equilibrium. u i [ R n ]. i runs over degrees of freedom and type of atom R n is vector to nth cell. Potential. Expand potential in Taylor series.

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Silicon Carbide

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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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