\section*{Problem}

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

\item In Eq.\,(\ref{eq:scatter-r}), if $V$ is periodic in the crystal,
show that the matrix elements vanish except for
$k^{\prime}=k$ in the first Brillouin zone. Here
use the fact that $\langle \phi
(R_{s^{\prime}})|V|\phi (R_s)\rangle$ depends only
on $R_{s^{\prime}}-R_s$. Show that the matrix
elements have a value for $k^{\prime}=k+G$,
where $G$ is a reciprocal lattice vector.

\item Obtain the next order correction to the wavefunction
of~Eq.\,(\ref{eq:correction-WF}) by substituting $\Phi(k')$
iteratively for $\Psi ^{c}(k')$ in the last term
of~Eq.\,(\ref{eq:correction-WF}).

\item Using the previous results, obtain Eq.\,(\ref{eq:Mkk}).

\item Higher order correction terms can be obtained similarly by
substituting  $\Phi(k')$ iteratively for $\Psi ^{c}(k')$ in the
last term of Eq.\,(\ref{eq:correction-WF}). Using this approach,
get a general expressions for $\Phi(k')$ and $M_{k^{\prime} k}$.

\item After consulting some textbook on scattering theory, explain
 that the T-matrix is expressed by $T = V + VG_0T$ or $T = V + VGV$,
 where $V\equiv H - H_0$ is an impurity potential operator and where
 $H$ and $H_0$, respectively, denote the perturbed and non-perturbed
 Hamiltonians, while $G = (E - H)^{-1}$ and $G_0 = (E - H_0)^{-1}$
 denote perturbed and non-perturbed Green's function. Using proper
 basis functions, we can make a matrix for each operator and get the
 T-matrix.

\item In scattering theory, the S-matrix is also frequently used.
Explain the difference between the T-matrix and the S-matrix. What are
the merits and demerits for using the T and S matrices?

\item Considering the electron and phonon dispersions near the K
point in graphene, derive a quantitative description which
explains why $\omega_{G^{\prime}} \neq 2\omega_{D}$ for  one- vs.
two-phonon processes in the double resonance of the iTO phonon
$\omega(q)$ branch near the K point, where $q$ denotes the phonon
wavevector.

\item Draw all the possible defect-induced double-resonance
one-phonon processes in monolayer graphene for Stokes and
anti-Stokes, intra-valley and inter-valley processes.

\item Consider the electronic dispersion of graphene in the linear
approximation, i.e., $E^\pm(k)=\pm \hbar v_F |k|$, where $v_F$ is
the Fermi velocity of the electrons given by $v_F =\sqrt{3}
(\gamma_0 a / 2 \hbar)$, and $a = \sqrt{3} a_{\rm C-C}$ is the lattice
constant of graphene and $a_{\rm C-C}=1.42$\AA\, is the
nearest-neighbor carbon-carbon distance. Calculate the pair of
$(q,k)$ wavevectors which would fulfill the one-phonon
defect-induced double-resonance conditions for $E_{\rm laser} =
1.98$\,eV and 2.41\,eV. Considering the phonon dispersion in
Fig.\,\ref{f-DR-phonondispersion}(b), give the approximate
frequency for all the defect-induced Raman peaks that should be
observed for these two laser lines. Check your results against the
values in Fig.\,\ref{f-DR-phonondispersion}(a).

\item Calculate the number of Ar$^+$ ion-induced defects per C atom
expected for a $10^{11}$\,Ar$^+$/cm$^2$ and a $10^{15}$\,Ar$^+$/cm$^2$
ion bombardment dose on a monolayer of graphene.

\item In our phenomenological model for the D-band intensity
dependence on Ar$^+$ ion impacts, consider $S_S$, $S_A$ and $S_T$ as the
structurally damaged, activated and total areas of the graphene sheet,
respectively, and $N$ as the number of Ar$^+$ ion collisions. Derive
Eq.\,(\ref{IDIGcomplex}) using rate equations (for $dS_S/dN$ and $dS_A/dN$)
and considering the initial conditions $f_S = 0$ and $f_A = 0$ for
$\sigma = 0$.

\item Derive Eq.\,(\ref{eq-IDIGoneD}). Suppose
that the width $\delta$  of the boundary is responsible for the
$E_{\rm laser}^4$ dependence observed experimentally for $I_D/I_G$.
How should $\delta$ vary with $E_{\rm laser}$ to obtain this relation?
In this case, what would be the expected $E_{\rm laser}$ dependence
of $I_D/I_G$ for point defects inducing the D band?

\item Show that $(560/E_{\rm laser}^4) = (2.4 \times 10^{-10}\lambda^4)$,
as shown in Eq.\,(\ref{eq-IDIGlambda}).

\item Explain why Eq.\,(\ref{eqe.7}) is responsible for the polarization
dependence observed for the D band in armchair edges, and show that a
$I_D \propto \cos{\theta}^4$ dependence is expected for the D-band
intensity when both the incident and scattered light are analyzed
along an angle $\theta$ with the edge direction.

\item Calculate the inter-valley matrix element $M_{k^{\prime}k}$ for
an armchair edged nanoribbon of width $L_a$. Here
we assume that an electron is reflected at
the armchair edge. Show that this assumption
predicts a width dependence for the D/G intensity
ratio given by $I_D/I_G \propto L_a^{-2}$.  Such a
relation has not been observed experimentally,
showing that more work (both experimental and
theoretical) is still needed.

\item The $G^{\prime}$ band was observed to shift to higher
frequencies under acceptor ($p$-type) doping, and to lower frequencies
under donor ($n$-type) doping.  Explain qualitatively which doping-induced
changes should happen to
the electronic and/or phonon dispersion relations to generate such a
shift.

\end{enumerate}
