Carbon Nanomaterials and Technology EE 6909 Dr Md

Carbon Nanomaterials and Technology EE 6909 Dr. Md. Sherajul Islam Assistant Professor Department of Electrical and Electronics Engineering Khulna University of Engineering & Technology Khulna, Bangladesh

LECTURE - 4 Band Structure: Lattice Vibrations

Lattice dynamics above T=0 • Crystal lattices at zero temperature posses long range order – translational symmetry (e. g. , generates sharp diffraction pattern, Bloch states, …). • At T>0 ions vibrate with an amplitude that depends on temperature – because of lattice symmetries, thermal vibrations can be analyzed in terms of collective motion of ions which can be populated and excited just like electrons – unlike electrons, phonons are bosons. Thermal lattice vibrations are responsible for: → Thermal conductivity of insulators is due to dispersive lattice vibrations (e. g. , thermal conductivity of diamond is 6 times larger than that of metallic copper). → They reduce intensities of diffraction spots and allow for inellastic scattering where the energy of the scatter (e. g. , neutron) changes due to absorption or creation of a phonon in the target. → Electron-phonon interactions renormalize the properties of electrons (electrons become heavier). → Superconductivity (conventional BCS) arises from multiple electron-phonon scattering between time-reversed electrons. 3

Crystal Dynamics Atomic motions are governed by the forces exerted on atoms when they are displaced from their equilibrium positions. To calculate the forces it is necessary to determine the wavefunctions and energies of the electrons within the crystal. Fortunately many important properties of the atomic motions can be deduced without doing these calculations.

Hooke's Law • One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law. It takes twice as much force to stretch a spring twice as far. Spring constant k

SOUND WAVES waves which propagate a • Mechanical It corresponds to theare atomic vibrations with a longthrough λ. material medium (solid, liquid, or gas) at a wave speed which depends on the elastic and inertial properties of that • Presence of atoms significance this wavelength medium. There arehas twonobasic types ofinwave motion for limit, since λ>>a, there will nowaves scattering due to the mechanical waves: solongitudinal and transverse presence waves. of atoms. Longitudinal Waves Transverse Waves

SOUND WAVES • Sound waves propagate through solids. This tells us that wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The detailed atomic structure is unimportant for these waves and their propagation is governed by the macroscopic elastic properties of the crystal. • We discuss sound waves since they must correspond to the low frequency, long wavelength limit of the more general lattice vibrations considered later in this chapter. • At a given frequency and in a given direction in a crystal it is possible to transmit three sound waves, differing in their direction of polarization and in general also in their velocity.

This equation is called the secular equation, whose eigenvalues Ei(k) give the energy band structure

Tight Binding for Graphene

Calculation of π Bands Using TB We start with transfer matrix

The calculations of the tight binding method can be summarized in the following steps