\section{Problems}

\begin{enumerate}
\renewcommand{\labelenumi}{ [\arabic{chapter}-\arabic{enumi}] }

\item Calculate the period of the
oscillation for the vibration of 1580\,cm$^{-1}$. Also give the frequency in THz.
(Use the fact that 1eV = 8650\,cm$^{-1}$).

\item Estimate the Raman spectral width in cm$^{-1}$
by using the uncertainty relation when the lifetime of the
photo-excited carrier is 500fs. Repeats the calculation for 50\,fs.

\item A typical length of a carbon nanotube is 1$\mu$m. How long does it take for the light
to go 1$\mu$m. How many times do carbon atoms oscillate at the G band frequency during
this time?

\item What is the maximum velocity or acceleration
 of the atomic vibration for a phonon of 1580\,cm$^{-1}$?
In this calculation, we should consider the number of phonons to be $n$.

\item When we use the maximum acceleration in the previous problem,
evaluate the force that a $\pi$ electron feels in this acceleration.
Compare this force with the Coulomb force in a carbon atom.
You can use $Z=4$ for the screened ion core and $r=0.5$\AA\  for the radius.
Check if the Coulomb potential is sufficiently strong to keep the $\pi$
electron bound to the carbon atom.

\item (Peierls instability) Consider a linear carbon chain in which
the nearest neighbor transfer parameters have alternating values:
$t_1, t_2, t_1, t_2, \ldots$.  Show that in this case, the energy gap
is opened at the zone boundary and the value of the energy gap is
proportional to $t_1 - t_2$.

\item In the previous problem, let us consider the electron-phonon
parameter $\alpha$ such that $t_i = t_0 - \alpha(x_{i+1} - x_{i})$,
where $x_i$ denotes the lattice distortion.  By an alternative lattice
distortion, $x_i = x_0 (-1)^i$, the total energy of the electron
decreases because of opening an energy gap at the Fermi energy. On
the other hand, because of the lattice distortion, the system loses
lattice energy which is proportional to $Kx_0^2/2$ per bond ($K$ is
the spring constant).  By minimizing the total energy, obtain the
optimized $x$ and the energy gap.

\item When the temperature of the nanotubes is either high or very
low, how does the phonon softening change as a function of the Fermi
energy? Using the Fermi distribution function, and explain your
result qualitatively.

\item The G-band phonon becomes soft when the temperature becomes high. Explain the mechanism of phonon softening for high temperature.


\item When the Fermi energy changes, how is the electron-phonon
interaction suppressed? Explain and plot qualitatively the phonon
frequency as a function of the Fermi energy.

\item Show that, for metallic SWNTs, only the phonon causes an electronic
gap opening, independent on tube chiral angle (this was shown in
Fig.\,\ref{f-GPeierls} for an armchair SWNT).  Here you should
show the above result for a zigzag and a chiral SWNT.

\item The RBM (radial breathing mode) frequency can be expected to
produce phonon softening by changing the Fermi energy. However, the
shift of the RBM frequency is known not to be large ($\sim$
1cm$^{-1}$). Consider why the phonon softening of the RBM is
small. How about the phonon softening for the  D or $G'$ bands?

\end{enumerate}
