PH 671 - Spring 2016, Due 5pm on Friday, Week 2
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:
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):
Electrons in a solid have a dispersion relation between the energy and wave vector of $E\left( {\vec k} \right)$. Begin with semiclassical equations $\vec p = \hbar \vec k$ and $\vec F = \hbar {{d\vec k} \over {dt}}$. Show that the group 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 ${m^*} = {{{\hbar ^2}} \over {\nabla _k^2E}}$. Verify that this gives the expected results for the free electron dispersion relation and also apply it the tight binding dispersion relation for a 1-d chain of s-orbitals that you learned in PH575.
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
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: