\section*{Problems}
\label{sec10.8}

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

\item Consider two $\pi$ orbitals on two carbon atoms. When we assume
that the nearest neighbor tight binding parameters for the Hamiltonian and
overlap matrices are $\gamma_0$ and $s_0$, obtain the eigen-energies for
the bonding and anti-bonding orbitals.

\item Obtain a tight binding Hamiltonian matrix for single layer graphene.
Consider two cases: (1) consider only the nearest-neighbor interaction
and (2) consider up to the third nearest-neighbor interaction.

\item In the previous problem, also consider the overlap matrix elements.

\item Obtain a tight binding Hamiltonian matrix for bilayer graphene.
Consider the two cases: (1) consider only the $\gamma_1$ band parameter
and (2) consider the  $\gamma_1$, $\gamma_3$ and $\gamma_4$ band parameters.
In the case of bilayer graphene, we also need to consider the energy $E_0^e$.

\item Obtain a tight binding Hamiltonian matrix for trilayer
graphene in which we consider band parameters
from $\gamma_0$ to $\gamma_5$.  In the
case of trilayer graphene, we need to consider the energy $\Delta$
which differentiates A and B carbon sites from one another.

\item For the case of bilayer graphene, obtain the electronic energy
dispersion numerically. What is the role of $\gamma_3$ and $\gamma_4$
in the energy dispersion near the $K$ point?

\item In the case of single-layer graphene, show explicitly the
relationship of the force constant sum rule for $\phi_{ti}$ and
$\phi_{to}$.

\item Show that the in-plane and out-of-plane phonon modes are decoupled in the
dynamical matrix when we use the force constant models.

\item In the case of phonons around the K point, we can consider zone
folding in which the K point phonon modes are folded into the $\Gamma$
point if we consider a super-cell structure. What kind of super-cell
structure is needed for obtaining the zone folding for the K point
phonon mode?

\item Discuss the selection rule between
the atomic orbitals which are specified by
$\ell$, $m$, $n$ atomic quantum numbers.
Show the optically-allowed states for 2p orbitals as an initial state.

\item Show that the relationship between the electric field  ${\bf E}$
for light in vacuum and the vector potential ${\bf A}$ is given by
${\bf E}= i\omega{\bf A}$.

\item Using the Poynting vector, $I =E B/\mu_0=E^2/(\mu_0 c)$, obtain
Eq.\,(\ref{equ:A}).

\item Using the simplest tight binding Hamiltonian, obtain the
wavefunction coefficients in Eq.\,(\ref{eq:cf1-2}) around ${\bf
K}=(0,-4\pi/(3a))$.

\item Obtain the wavefunction coefficient around ${\bf
K'}=(0,4\pi/(3a))$, and the dipole vector $ {\bf D}^{\rm cv}({\bf K}'+{\bf
k})=[{3m_{\rm opt}}/{2k}](-k_y,k_x,0)$. Explain that the rotational
directions of ${\bf D}^{\rm cv}({\bf k})$ around the K and K$'$ points are
opposite to each other.

\item When the crystal potential (Eq.\,(\ref{eq:VofR})) is periodic,
show that the matrix elements of Eq.\,(\ref{potential1}) have non-zero
values only for ${\bf k}={\bf k}'$.

\item In Fig.\,\ref{matrix2}, another set of
 off-site deformation potential vectors
$\vec{\beta}_{p}$ are shown. By comparing
$\vec{\beta}_{p}$ with
$\vec{\alpha}_{p}$ in Fig.\,\ref{alphalambda}, show the
following relationships between $\vec{\beta}_{p}$ and $\vec{\alpha}_{p}$.
\begin{figure}
\includegraphics[width=6.5cm]{./eps/beta.eps}
\caption[]{The nine non-zero off-site deformation potential vectors.
$\vec{\beta}_{p}$. The dashed curves represent the atomic potential.}
\label{matrix2}
\end{figure}
\begin{equation}
\begin{array}{lllllll}
\vec{\beta}_{ss}&=&-\vec{\alpha}_{ss},& \vec{\beta}_{s\sigma}&=&\vec{\alpha}_{\sigma s},\\
\vec{\beta}_{\sigma s}&=&\vec{\alpha}_{s\sigma},& \vec{\beta}_{\sigma\sigma}
&=&-\vec{\alpha}_{\sigma\sigma},\\
\vec{\beta}_{\pi\pi}&=&-\vec{\alpha}_{\pi\pi},&
\vec{\beta}_{s\pi}&=& \vec{\alpha}_{\pi s},\\
\vec{\beta}_{\pi s}&=&\vec{\alpha}_{s\pi},&
\vec{\beta}_{\sigma\pi}&=&-\vec{\alpha}_{\pi\sigma},\\
\vec{\beta}_{\pi\sigma}&=&-\vec{\alpha}_{\sigma\pi}.
\end{array} \end{equation}

\item Consider the $E$ exciton wavefunction and show that $E$ exciton
states do not contribute to an optical absorption (or emission).

\item When we consider the phonon emission from a photo-excited
carrier by the electron-phonon interaction, show by illustrating in a
figure that there are 24 possible final states in graphene which
satisfy energy-momentum conservation.

\item In Eq.\,(\ref{hel-ph}), we consider the scattering of a
hole in the second term. Explain why the minus
sign in $-M^{\nu}_{{\bf k},{\bf k}+{\bf q}}(v)$
appears.


\item When the excitation energy is not
resonant with the energy separation between an
occupied and an unoccupied state, time-dependent
perturbation theory tells us that the optical
transition probabilities to possible real states are
non-zero. Then we can consider a virtual state
whose energy is determined by the energy of the light.
Then show that the virtual state can be expressed
by a linear combination of the real
states. Especially when the virtual state position
is close to a real state, the coefficient of the
real state is then close to unity and thus we can
approximate the virtual state by the real state.

\end{enumerate}
