PHZ3400-09 Problem Set 7

Problem 1

Describe the classifications of solids according to the band theory. Pay particular attention to materials on the borderlines of different classifications. Specifically, discuss the electrical conduction at 0K and finite temperatures, and the type of charge carriers involved.

Problem 2

Consider a one-dimensional tight-binding model describing the dimerized state of polyacetylene, where the p-electron orbitals hopping elements $t_{1}\;$ and $t_{2}\;$ (with $t_{1}>t_{2}\;$) alternate along the chain. Derive the band structure for this model, and discuss what happens in the limit when $t_{1}\rightarrow t_{2}\;$. Compare this result with that obtained for a simple chain (discussed in class) where $t_{1}=t_{2}=t\;$. Explain the phenomenon of band-folding. Sketch the density of states for this model.

Problem 3

In the Feynman model of energy bands, the energy dispersion of electrons in a cubic lattice is given by

$E(k)=E_{o}-2Acos(|\mathbf {k} |a)$,

where $a\;$ is the lattice constant.

a) Derive an expression for the effective mass for this band.

b) Prove that the effective mass at the bottom of the band is inversely proportional to the band width.

c) Derive an expression for the electronic DOS in this band.

d) Prove that at the bottom of the band the DOS has the same energy dependence as that for free electrons.

Problem 4

Our understanding of the electrical conductivity of metals evolved from the classical free electron gas to quantum free electron gas to the band theory. Write a short essay describing this evolution. Specify the new physics introduced in each model.