\section*{Problem}

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

\item Polarization {\bf P} is expressed by {\bf P}=$\alpha${\bf E},
where is $\alpha$ is polarization tensor which is given by
Eq.\,(\ref{eq-polariznlinear}).  When {\bf E}={\bf E}$_0$$\exp (i\omega
t)$, $Q_i=Q_i^0\exp (i\omega_i t)$, ($i=1,2$), show that the last term
of Eq.\,(\ref{eq-polariznlinear}) gives frequency terms of $\omega \pm
\omega_1 \pm \omega_2$.

\item In an optical process such as photo-absorption or
phonon-emission, the probability for the occurrence of the optical
process is proportional to the square of the matrix elements of the
electromagnetic perturbation in time-dependent perturbation
theory. Explain how we get energy conservation and momentum
conservation from perturbation theory. Especially show that these
conservation rules are is not always necessary because of the
uncertainty principle which becomes important for short times or short
lengths.

\item The $\pm$ symbols in Eq.\,(\ref{eq-En2phon}) correspond to
phonon absorption and emission with the wavevectors $q_1$ and
$q_2$, respectively. Then explain, in the corresponding momentum
conservation rule of Eq.\,(\ref{eq-k2phon}), why we should use not
$\pm$ but rather we should use $\mp$ by illustrating the
scattering process in the two-dimensional Brillouin zone.

\item Using Eqs.\,\ref{eq02a-again} and \ref{eq02-again}, analyze how
the intensity of a second-order Raman process
involving two phonons with $q$ and $-q$
wavevectors changes when one, two or three
denominators are minimized (select a constant
value for $\Gamma$ and draw the resonance
window). Make a comparative analysis of the
intensities for the six processes depicted in
Fig.\,\ref{f-DRpossibilities}.

\item Discuss how the triple-resonance process in
Figs.\,\ref{f-DRpossibilities}(c) and (f) depends on the symmetry
between the valence and conduction band. For this discussion, show how the total
intensity would change when the slopes for the electron and hole
dispersions are different.  Analyze the results as a function of
$\Gamma$.

\item Since the electronic dispersion in graphene near the K
point is characterized by a Fermi velocity of $v_{\rm F}=1\times
10^6$m/s and the $G^{\prime}$-band dispersion is given by $(\partial
\omega_{G^{\prime}} /\partial E_{\rm laser} ) \simeq
88$\,cm$^{-1}$/eV, find the phonon dispersion for the iTO phonon
branch in graphene. Compare the result you find with the result
obtained by the force constant model described in \S\ref{s7.1g}.  You
will see why the addition of electron-phonon coupling, not accounted
for in the force constant model, is very important for describing the
phonon dispersion near the K point and, consequently,
for explaining the dispersive behavior of the $G'$ band.

\item Explain qualitatively how to obtain the circles around the K and
K$'$ points in Fig.\,\ref{f-boia} by using the electron and phonon
energy dispersion. Estimate the diameters of these circles for $E_{\rm
laser}=2.41$eV, and a phonon frequency of 1350cm$^{-1}$ and compare these
lengths with the $\Gamma K$ distance.

\item Plot the phonon density of states as a function of energy for
possible $q$ vectors, as shown in Fig.\,\ref{f-boia}, by assuming that
all circles are regular (not deformed) circles. Here assume that the
phonon dispersion is proportional to the distance of $q$ from the K
point in the two dimensional Brillouin zone.

\item Considering the electron and phonon dispersions near the K point, calculate the difference in frequency
between the Stokes and anti-Stokes processes for
the $G^{\prime}$-band (intervalley process), equivalent
to the Stokes and anti-Stokes processes depicted
in Fig.\,\ref{f-DRpossibilities}(a) and (d) for
the intravalley scattering process.

\item Considering the double resonance condition $q \approx 0$, do you
expect a dispersive ($E_{\rm laser}$ dependent) behavior and why?

\item Draw a picture similar to Fig.\,\ref{f-boia}, but considering
the trigonal warping effect.  What differences do we expect to find in
the double resonance features when the trigonal warping effect is
included?

\item Derive Eq.\,(\ref{Gprime-freq}).

\item Consider: (a) the average $G^{\rm \prime}$ dispersion $(\partial
\omega_{G^{\prime}} /\partial E_{\rm laser} ) = 106$\,cm$^{-1}$/eV;
(b) the $G^{\rm \prime}$ diameter dependence given by
Eq.\,(\ref{eq-Gprimedt}); (c) the Kataura plot. Describe quantitatively
the oscillatory behavior expected for the $G^{\rm \prime}$ band in
SWNT bundles, considering a diameter distribution $1.2 \leq d_t \leq
1.6$\,nm for the above three cases. For which diameter distribution
would the oscillatory behavior become an averaged linear dispersion?
Analyze your result as a function of the $G^{\rm \prime}$-band Raman peak
width for one given SWNT.

\item Explain how the inclusion of
trigonal warping in the phonon dispersion relation
would generate a chiral angle ($\theta$) dependence for the $G^{\rm
\prime}$ band of a hypothetical SWNT with the same $d_t$ and as $\theta$
is changed from 0 to 30$^{\circ}$.

\item There is a mirror symmetry for $n$ layer graphene ($n$ is odd number)
in which a mirror is parallel to the graphene
layer. The $\pi$ energy band and phonon modes are
either symmetric (S) or anti-symmetric (AS) with regard to
the mirror operation.  Discuss the selection rules
for optical transitions and for the electron-phonon
interaction for four possible combination of S and AS energy bands.
How about the case that $n$ is even number?

\item When you look at Figs.\,\ref{f-DRpossibilities}(a) and (b), it
is easy to see that the incident and scattered double resonance processes
in graphene occur with the same $(q,-q)$ wavevector pair. Explain why
two peaks are observed in the $G^{\rm \prime}$ band of SWNTs when both the
incident and scattered light are in resonance with two different
optical transition energies.

\item Explain why there is no possibility of a second-order Raman
process with the combination modes of two phonons, one causing an
intra-valley and one causing an inter-valley scattering event.

\item Explain that the density of states (DOS) of $E_{ii}^M$ is split
 into two peaks from the two cutting lines near the K point except for
 armchair nanotubes. Please specify the cutting line in
the two dimensional Brillouin zone which
corresponds to the DOS peak at the higher
energy for each $i$ of $E_{ii}^M$.

\end{enumerate}
