PHZ3400-11 Problem Set 5

Problem 1.

a) Derive the expressions for the Fermi energy, Fermi velocity, and electronic density of states for a two-dimensional free electron gas.

b) A 2D electron gas formed in a GaAs/AlGaAs quantum well has a density of $2{\rm {x}}10^{11}cm^{-2}$. Assuming that the electrons there have the free electron mass, calculate the Fermi energy and Fermi velocity.

Problem 2

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 zero temperature and finite temperatures, and the type of charge carriers involved.

Problem 3

Consider a one-dimensional tight-binding model describing the a chain of alternating atoms A and B. The site energies $\epsilon _{A}=0\;$ and $\epsilon _{B}=\Delta \;$ (with $\Delta >0;$) alternate along the chain, and the hopping elements $(-t)$ are the same between all nearest neighbors (same model as discussed in class).

a) Divide the chain in unti cells consisting of two atoms (A and B) in each cell. Assume that there are exactly N such unit cells, so that the chain length is $L=2Na$, where $2a$ is the unit cell length (inter-atomic spacing is a).

b) Calculate the band structure of such a chain with one orbital per atom. Within the tight-binding approximation, look for a the wavefunction of the form

$|\psi >=\sum _{i=1}^{N}(C_{i}^{A}|i,A>+C_{i}^{B}|i,B>)$,

with

$C_{i}^{\alpha }=C^{\alpha }e^{ikR_{i}}$.

Solve the corresponding eigenvalue problem and show that the band dispersion takes the form of two band with

Failed to parse (syntax error): {\displaystyle E_(\pm} (k) = \frac{1}{2} (\Delta^2 \pm (\delta^2 + 8t^2 cos^2(2ka))^{1/2}).}

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.