PH 671 - Spring 2016, Due 5pm on Friday, Week 2

Under construction - will remove when ready

Write a one-paragraph summary (4 or 5 sentences) about an experimental or theoretical solid state physics paper from 2010 or later that contains one or more of the following:

• Measurement of current (or resistance) in a nanoscale system
• Measurements of current (or resistance) in a magnetic field
• Measurement of current (or resistance) near a phase transition (for example, normal metal to superconductor).
• An STM spectroscopy measurement

Include full bibliographic information (journal name, volume number, page number, article title). Please limit yourself to the following journals (this list can be augmented with class consensus; and you may need to request an interlibrary loan to access some):

• Science
• Nature
• Proceedings of the National Academy of Sciences (PNAS)
• Nature Physics
• Physical Review Letters
• Nano Letters
• Applied Physics Letters
• Physical Review X

Electrons in a solid have a dispersion relation between the energy and wave vector of $E\left( {\vec k} \right)$. Begin with semiclassical equations $\vec F = \hbar {{d\vec k} \over {dt}}$. Show that the velocity $\vec v\left( {\vec k} \right)$ and effective mass ${m^*}$ of an electron packet with wave vector k are, respectively, $\vec v\left( {\vec k} \right) = {1 \over \hbar }{\nabla _k}E$ and ${m^*} = {{{\hbar ^2}} \over {\nabla _k^2E}}$.

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 ${\vec \varepsilon }$:

$$f\left( {\vec k} \right) = {f_0}\left( {\vec k} \right) + {\tau _k}{{e\vec \varepsilon } \over \hbar } \cdot {\nabla _k}{f_0}$$

For “free electrons” (quadratic dispersion) and τ independent of k, show that the linearized Boltzmann result is equivalent to the Drude formula with the scattering time evaluated at the Fermi wavevector:

$$\sigma = {{n{e^2}\tau \left( {{k_F}} \right)} \over {{m^*}}}$$

Hints

• Use the fact that ${{{\tau _k}e\vec \varepsilon } \over \hbar } < < {k_F}$ (typical values of τ range from 10^-13 s at low temperature to 10^-15 s at room temperature)
• Make the simplification that the electric field in in the x-direction and that the system is isotropic (i.e. a field applied in one direction induces a current in that direction only).
• 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.

It takes surprisingly large 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 in eV of the highest frequency phonon in graphene (this phonon mode will show up in many types of experiments, including Raman spectroscopy of graphene). Use the following simplifying assumptions:

• Treat the graphene lattice as square grid.
• Choose an interatomic spacing that gives the correct area per unit mass for graphene, i.e. 1500 m³/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.