Class-3_111601652.pdfVIP

Tight Binding model ε N fold Bands in levels of free solids N fold electrons N fold a • Atomic levels spread into bands as the atoms come together • What is the band width? • Small a, big overlap, wide band; large a, small overlap, narrow band • Why the band width is smaller for lower energy bands? Summary of band theory • Electrons in periodic lattice: Bloch theorem and formation of band • Band gap: Bragg diffraction for k at the Brillouin zone boundary • Band structure explanation of metals vs. insulators • Nearly free electron model vs. tight binding model • Band theory is a major triumph of solid state physics, but can it explain everything regarding the electronic properties of solid? • Under what circumstances will it fail? Lattice vibrations and Phonons • So far we treat lattice as purely static, why is it ok? • In fact the lattice dynamics is very interesting and important, but in general it is very difficult to describe lattice vibrations in solids Lattice vibrations and Normal modes The problem can be significantly simplified by two approximations: • we only consider nearest neighbor interaction and ignore long range interaction • we treat the interaction as harmonic potential: V(a+δx) = V + ½ β(δx)2 0 • The resulting lattice can be visualized as a system of balls connected by springs Lattice vibrations and Normal modes • In classical wave mechanics, with Harmonic approximation, the lattice vibrations can be represented by independent normal modes, in which all the lattice ions osci