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Optical and Electromechanical Properties of Carbon Nanotubes via a Two-Field Elastic Description

Optical and Electromechanical Properties of Carbon Nanotubes via a Two-Field Elastic Description. Cristiano Nisoli Vincent H. Crespi Penn State University. Elasticity of Continua: Basics . elastic displacement field. strain tensor. Elastic free energy…. ...in harmonic limit,

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Optical and Electromechanical Properties of Carbon Nanotubes via a Two-Field Elastic Description

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  1. Optical and Electromechanical Properties of Carbon Nanotubes via a Two-Field Elastic Description Cristiano Nisoli Vincent H. Crespi Penn State University

  2. Elasticity of Continua: Basics elastic displacement field strain tensor Elastic free energy… ...in harmonic limit, lower derivatives… • Linear spectra • No Brillouin zone • No acoustic modes …for isotropic media. L. D. Landau, E. M. Lifshitz “Theory of Elasticity” Pergamon Press Oxford (1986)

  3. Two-Field Description: In plane Elastic free energy… …cross term. Direct term is isotropic. (accounts for NNNs) …direct terms …

  4. Coupling between sub-lattice strains Cross Term Cost for displacing sub lattices apart Internal displacement to strain coupling Similar formalism can be introduce for out-of-plane deformation fields.

  5. Uniform Deformations

  6. Phonons The equations of motion: • Admit analytical solution • Predict a Brillouin zone • Predict optical modes

  7. * ** *** Carbon Nanotubes. Raman Modes Eigenvectors can be obtained: Coupling between pure breathing and optical graphite-like mode** One-field result * in plane displacement, B- mode ***

  8. * Electro Mechanical Effects Hopping Integrals Band Gap Softening of the Longitudinal optical modes in metallic CNTs *

  9. Softening in Zig-Zag Softening of B modes in metallic tubes Phonon Softening Softening of speed of sound of Twist mode in metallic tubes

  10. * Self-Trapped Electrons. Correction of Yang formula * More general solutions, not angularly invariant can be found…

  11. 2-Field formalism explains many observed features • Analytical results for complete phonon spectra. • Coupling between Raman modes seen in DFT. • Eigenvectors previously observed in DFT. • Phonon softening in metallic CNTs, observed in DFT. • Framework for electromechanical effects. • Correction terms for accepted formulas.

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