Homework #2

Due 5pm Friday, end of Week 2

Write a one paragraph summary (3 or 4 sentences) about an experimental or theoretical solid state physics paper from 2012 or 2013 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 current (or resistance) near a phase transition (for example, normal metal to superconductor).

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

Effective mass (5 pts)

When describing electron transport is often useful to relate dv/dt to force, F, in terms of an effective mass, m_eff

.

Use the semiclassical dynamics relations (discussed in class) to show that, for a 1d material,

Note for next year: Do not assume free electron dispersion relations, this relationship is more general

Boltzman transport equation (15 pts)

based on text by Ibach and Luth

The linearized Boltzmann Equation is used to derive the steady-state distribution function for conduction electrons in a 3d material under the influence of a small electric field:

For “free electrons” (quadratic dispersion) and τ independent of k, show that the linearized Boltzmann result is equivalent to the Drude formula:

Hints

• Use the fact that eτE/ħ is much less than k_F (typical values of τ range from 10^-13 s at low temperature to 10^-15 s at room temperature)
• To calculate current at a certain electric field, and therefore conductivity, you will encounter a tricky integral. Simplify this integral by setting T = 0 so the Fermi-Dirac function is a step function.

Phonons in graphene (5 pts)

It takes surprisingly high force to stretch a sheet of graphene, a single sheet of atoms. If the sheet was 1 cm x 1 cm, and you attached a pair of rods to opposite edges of the sheet, the stretching force would be (3 Newtons)*strain, where strain is the ratio (change in length)/(original length).

Estimate the energy of the highest frequency phonon in graphene (this phonon mode will show up in many types of experiments, including Raman spectroscopy of graphene). I'm looking for an answer in units of electron volts. Use the following simplifying assumptions:

• Treat the graphene lattice as square grid (simplifies the math).
• Choose an interatomic spacing that gives the correct area per unit mass for graphene, i.e. 1500 m^2/gram.
• You can check your answer by reading this Phys. Rev. Lett. paper, where the authors observe that a phonon with energy 160 meV is responsible for current saturatation in CNTs.