Homework #4

Due 5pm on Friday of Week 4

Journal reading (5 pts)

Write a one paragraph summary (3 or 4 sentences) about an experimental or theoretical solid state physics paper from 2010 or 2011 that contains one or more of the following:

  • Measurement of current (or resistance) in a magnetic field
  • Measurements of current (or resistance) in a nanoscale system
  • An STM spectroscopy measurement
  • Measurement of electronic properties near a phase transition (for example, a metal-insulator transition).

Include bibliographic information (journal name, volume number, page number) for the paper you choose. Please limit yourself to the following journals:

  • Science
  • Nature
  • Proceedings of the National Academy of Sciences (PNAS)
  • Nature Physics (you will need to request an interlibrary loan to access Nature Physics)
  • Physical Review Letters
  • Nano Letters

Anderson Localization (5 pts)

Figure 1. Experimental dependence of the low voltage resistance vs length for CVD-grown SWNTs (triangles and stars) and HipCo SWNTs (squares). The experimental set up is illustrated in the central inset of the figure. A metallic AFM tip is moved along the nanotube probing its electrical conductance as a function of the tip-electrode distance. The upper inset is an AFM image of a long nanotube partially covered with gold. The nanotubes directly grown on surface present low disorder and hence the conductance is quasiballistic; the low increment of the resistance with length is a consequence of the interaction with acoustic phonons and disorder. The HipCo SWNT presents a high density of defects resulting in an electronic transport regime governed by the Anderson localization phenomenon. - Sundqvist et al. Nano Letters (2007)

Use the graph above to find the elastic scatting length in HipCo single-walled nanotubes (SWNTs) and the inelastic scattering length for CVD-grown SWNTs. Show your working.

Note for next year: “The L = 0 data points do not agree with the theoretical model for a CNT channel, i.e. R(L=0) is larger than 6.5 kOhm. The extra resistance is called R_contact. The contact resistance behaves like a resistor in series with the CNT channel.”

Your answer will have some uncertainty because you are “eye-balling” the graph rather than working with real data. However, eye-balling a graph is an important skill. When you give your answers, please estimate the uncertainty in your answer.

Potential well has a bound state? (5 pts)

Note for next year: Part a should say, “Using the position uncertainty of the wavefunction, write an approximate expression for kinetic energy…”

The Mott transition can be examined in the framework of Thomas-Fermi screening. In this framework, the metal-to-insulator transition occurs when the attractive potentials around ionic cores are so well screened bound wavestates no longer exist.

Consider an electron that may (or may not) be bound to a weak potential minimum, as shown above.

a) From the curvature of the wavefunction, write an approximate expression for kinetic energy (the order of magnitude should be correct, and it should scale with W in the correct way).

b) Argue that the expectation value for potential energy is: PE ~ -V_0(a/W)^d where d is the dimensionality (assuming W » a)

c) Using these results show that, in 1D, there is always a bound state, while in 3d there is not (assuming W » a)

Metal-Insulator Transition in doped silicon (5pts)

Note for next year: Change the rule of thumb to 4a_Bohr (instead of 5)

Using the rule of thumb,

where n is the critical density of free carriers, estimate the doping density that makes silicon degenerately doped. “Degenerately doped” means that the sample will remain conducting even at zero temperature (when you might expect the free electrons to become bound to the ionized dopant atoms).

  • Note: a_Bohr for electrons in Si is not equal to a_Bohr for electrons in vacuum.

Sheet density changes with magnetic field (10 pts)

Note added on Tuesday night: To simplify the problem, assume there is no Zeeman splitting: i.e. the spin up and spin down electrons have degenerate energy in a magnetic field.

Consider a 2DEG that is connected to an electron reservoir. The chemical potential inside the 2DEG stays constant, even when a B-field is applied to the system.

A magnetic field is applied perpendicular to the 2DEG and the number of electrons in the 2DEG fluctuate due to Landau levels crossing the chemical potential.

Make a plot of electron sheet density as a function of B, similar to the plot in the class notes. Find expressions for the B fields where sudden switching events occur. Also find expressions for key values of the electron density.