
\section*{Problems}

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

\item Obtain Eq.\,(\ref{eq:C-M}) from Eq.\,(\ref{eq:defchi}).

\item Consider the equation for a damped harmonic oscillator driven
by a force $F_0$

\begin{equation}
m\frac{\partial^2 x}{\partial t^2} + \gamma\frac{\partial x}{\partial t} + Kx
= F_0\exp{i(\omega t + \phi)}.
\end{equation}

(a) Obtain and plot the oscillator response $x$ as a function of the
driving frequency~$\omega$.

(b) Show that the power response ($|x|^2$) exhibits
a Lorentzian lineshape when the natural
oscillator frequency is much larger than the peak full width at half
maximum intensity. Obtain an expression for  the
time dependence of the full width at half
maximum intensity.

\item Show that 1\,eV corresponds to 8065\,cm$^{-1}$. What is the
energy in eV for the G-band Raman spectrum of graphite at
1580\,cm$^{-1}$?

\item Obtain the energy in eV of a laser with a wavelength of 633\,nm.
What is the conversion formula from nm to eV, and from eV to nm?

\item What is the wavelength of the Stokes and anti-Stokes
scattered light for the graphene (1590 cm$^{-1}$)
 G-band for 514\,nm laser light?

\item When we consider the room temperature Raman spectrum
for a carbon nanotube at 300K,
what is the intensity ratio of the anti-Stokes and Stokes intensities
$I_{AS}/I_S$ for the G-band (1590cm$^{-1}$) and for the radial breathing
mode for a 1nm diameter SWNT, with
$\omega_{RBM} = 250$\,cm$^{-1}$? When the temperature is
changed to 4K and to 2000K, what are the values of $I_{AS}/I_S$ for
these cases?

\item Calculate the wave vectors for 488nm, 514nm, 633nm lasers and
compare them with the wavevector of the $K$ and $M$ points of graphene.

\item Consider the linear dispersion of energy for graphene,
$$
E(k) = \pm \frac{\sqrt{3}\gamma_0}{2}\left(\sqrt{k_x^2 + k_y^2}\right)a,
$$ where $\pm$ denote the valence and conduction bands, respectively,
and $\gamma_0 = 2.9$ eV (nearest neighbor transfer energy for an
optical transition) and $a = 0.246$nm. Show that this energy
dispersion has the shape of two cone structures with the apexes of
their Dirac cones that touch each other at the Dirac point.  Calculate
the $|\vec{k}|$ value (distance from the $\Gamma$ point in $k$ space)
for an optical transition observed with a 514\,nm laser.

\item In problem [4-7], when an optical phonon (LO branch) with energy
1580cm$^{-1}$ is emitted, show the energy-momentum
conserved final states in the energy dispersion
and show the possible $q$ vectors in a
figure. (Hint: the final states should be on an
equi-energy contour of $E_{\rm laser}-E_{q}).$

\item In problems [4-7] and [4-8],
show $\vec{q}$ for the back-scattering geometry for
the detection of the Raman scattering signal?
How about for forward scattering?

\item In the case of graphene, the Dirac cone shape appears not only
at the K point but also at the $K'$ point. Explain in what ways $K$
and $K'$ are not equivalent to each other. If we consider the
inelastic scattering from $\vec{k}$ in the K Dirac cone to $\vec{k}-\vec{q}$
in the $K'$ Dirac cone, plot the possible $\vec{q}$ vectors measured
from the $\Gamma$ point in the two dimensional Brillouin zone.

\item In the previous problem, for a given laser excitation, the
$\vec{k}$ wave vector for photo-excited electrons exists on an equi-energy
contour. Plot the possible $\vec{q}$ vectors in the two dimensional
Brillouin zone.

\item The condition for the resonance Raman effect is given by $E_{\rm
laser} = E_{\rm gap}$, where $E_{\rm gap}$ denotes the energy separation
between the conduction and valence energy bands. If the resonance
condition is applied for the scattered resonance for phonon energy
$E_q$, show the resonance condition for the scattered light, which we
call the scattered light resonance condition. Explain why the scattered
light resonance condition depends on the phonon energy while the
incident light resonance condition does not.

\item When we plot the Raman intensity as a function of $E_{\rm
laser}$ (Raman excitation profile), we expect two peaks, one for the
incident resonance and one for the scattered resonance conditions. When we have
two different phonons with different phonon energies, illustrate the
expected Raman excitation profiles.

\item Explain the resonance conditions for anti-Stokes shifts.
Illustrate two Raman excitation profiles for the Stokes and anti-Stokes
Raman signals. Explain why the two Raman excitation profiles do not
appear at the same excitation energy values.

\item If we consider second-order Raman scattering for a  $q\ne 0$ phonon for
graphene, we have two distinct $q$ vectors for a $q\ne 0$ scattering
event. Explain this situation by using the two Dirac cone model. We denote
these two scattering events by intra-valley scattering and
inter-valley scattering.

\item Explain the results in problem [4-15] with an illustration of
inter-valley scattering for the forward ($q$) and backward ($-q$) scattering
geometries.

\item Explain in the phonon dispersion of graphene
(Fig.\,\ref{f-graphene-phonon}),
which phonons are selected for intra-valley and inter-valley $q\ne 0$
scattering. Explain what happens
for the case of forward and backward scattering.

\item When we consider two laser energies $E_1 < E_2$, show that the
$q$ vectors are larger for $E_{\rm laser} = E_2$. When the laser
energy increases, how is the scattered $q$ vector selected in the
phonon energy dispersion?

\item If $E_{\rm laser}$ is selected in the Raman measurement of
single wall carbon nanotubes (SWNTs), we can get a Raman signal only
from SWNTs which satisfy the resonance condition $E_{\rm laser} =
E_{ii}$. When we observe the radial breathing mode (RBM) whose frequency
is inversely proportional to the diameter, how can we determine
$(n,m)$ values from the known calculated values of $E_{ii}$ and
$\omega_{\rm RBM}$? Explain a procedure
that could be used to get $(n,m)$ values.

\item In order to obtain all the $(n,m)$ values for SWNTs in a given
sample, we need an almost continuously tunable energy output from
$E_{\rm laser}$. Suppose that you have a laser system with continuously
tunable energy, how
should you arrange the experimental schedule to obtain all these
$(n,m)$ values?  Explain your purpose and the method you will use.

\item Suppose that we have only one SWNT on a Si substrate. The position
of the SWNT can not be seen by optical microscopy. Explain why
obtaining $(n,m)$ for the specified SWNT by Raman spectroscopy is
difficult.  In order to overcome this difficulty, how should we prepare our
samples or arrange our experimental set up?

\item Try to correlate the modifications in the $G^{\prime}$ feature,
as shown in Fig.\,\ref{f-manylayers} with the changes in the
electronic structure, as discussed in \S\ref{s2.15}.

\item How should the $I_D/I_G$ intensity ratio depend on the
defect density  in a graphene layer? Discuss a physical picture for some 
defect density condition with use of some typical lengths of the system.


\end{enumerate}
