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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 a chain of alternating atoms A and B. The site energies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_A = 0\; } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_B = \Delta\; } (with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta > 0;} ) alternate along the chain, and the hopping elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( -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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=2Na} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 wavefunction of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi> = \sum_{i=1}^N (C_i^A|i, A> + C_i^B |i, B>)} ,

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i^{\alpha} = C^{\alpha} e^{ikR_i}} .

Solve the corresponding eigenvalue problem and show that we find two bands of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\pm} (k) = \frac{1}{2} (\Delta^2 \pm (\Delta^2 + 16t^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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=2\pi n/L,} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \gg t} the band gap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{gap} \approx \Delta} , while for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \rightarrow 0} the gap also vanishes.

e) Assume that you have one electron per atom and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta=> 0} ? Is this a metal or an insulator?

f) Consider the electronic states near the bottom of the lowest band (corresponding to k very small). Show that here the dispersion assumes the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(k) \approx \frac{\hbar^2k^2}{2m^*}} , i.e. is quadratic in k.

Determine the effective band mass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^*} , and show that for very narrow bands (very small hopping t) one finds very heavy band masses. The electrons then move as much heavier particles. This situation is found in some compounds containing rare-earth elements such as Cerium or Uranium, often called heavy fermion compounds.

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.