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

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

$E_{\pm }(k)={\frac {1}{2}}(\Delta ^{2}\pm (\Delta ^{2}+8t^{2}cos^{2}(2ka))^{1/2}).$

c) Use periodic boundary conditions to show that for a finite system of length L, the allowed values of the wavevector are $k=2\pi n/L,$, with Failed to parse (syntax error): {\displaystyle n=0, \pm1, \pm 2,…,\pm N} , i.e. lie in the first Brilloun zone. Explain why the wavevector cannot assume larger values.

d) Sketch the band structure. Determine the band gap separating the two bands. Show that for $\Delta \gg t$ the band gap $E_{gap}\approx \Delta$, while for $\Delta \rightarrow 0$ the gap also vanishes.

e) Assume that you have one electron per atom and $\Delta >0$. Explain why this is an insulator? What happens if we have only one electron per unit cell? Explain why this is a metal. What happens for $\Delta =>0$? Is this a metal or an insulator?

f) Imagine that we start with a band insualtor with two electrons per unit cell. The bottom (so called valence) band s then fully occupied, while the top (so-called conduction) band is empty. now imagine that we chemically dope the system and add 5% more carriers, which start to populate the bottom of the conduction band. Sketch the band occupation.

Problem 3

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.