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\title{Quantum Transport in Nanoporous Graphene}
\subtitle{Bachelor defense}
\author{\scshape\centering Christoffer Vendelbo Sørensen (s163965) \vskip\vspace{7.5mm} \includegraphics[width=1cm]{Figures/DTU3CMYK.eps}}
\date{\scshape June 25, 2019}

%\AtBeginSection[]
%{
%  \begin{frame}<beamer>
%    \frametitle{Outline for section \thesection}
%    \tableofcontents[currentsection]
%  \end{frame}
%}

\begin{document}

\begin{frame}[plain]
	\titlepage
\end{frame}

\begin{frame}
	\frametitle{Outline}
	\tableofcontents
\end{frame}

\section{Introduction}
\subsection{Project aim and nanoporous graphene}
\begin{frame}{Project aim}
	\centering
	\begin{beamercolorbox}[sep=1em,wd=15cm]{redbox}
		``Development of tight-binding routines in Python in order to understand electron transport in novel nanoporous graphene devices (NPGs)''
	\end{beamercolorbox}
\end{frame}

\begin{frame}{Nanoporous graphene}
	\centering
	\begin{columns}[c]
		\column{.7\textwidth}
		\begin{itemize}
			\item Planar graphene sheets
			\item Periodically removed atoms
			\item Ribbons and bridges
			      %\item Bridge chemistry \(\rightarrow\) current control
			      \note{How do electrons move in these materials?}
			\item Ballistic electron movement
			\item Potential for controlling currents on nanoscale
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[width=.6\textwidth]{Figures/NPGintroGraphic.eps}
	\end{columns}
\end{frame}

\section{Tight Binding}
\subsection{\mathinhead{\pi}{\pi}-orbitals, \mathinhead{\pi}{\pi}-electrons and the TB approximation}
\begin{frame}{\mathinhead{\pi}{\pi}-orbitals and \mathinhead{\pi}{\pi}-electrons}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.7\textwidth}
		\begin{itemize}
			\item How are the electric orbitals structured?
			\item \(\sigma\)- and \(\pi\)-systems
			\item 1 \(p_z\)-electron per site
			\item ``Tightly bound'' hops between sites
		\end{itemize}
		\column{.7\textwidth}
		\resizebox{.4\textwidth}{!}{
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\end{frame}

\begin{frame}{TB approximation}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.5\textwidth}
		\begin{itemize}
			\item What is Tight Binding?
			\item Electrons tightly bound to sites
			\item Hops with potential
			\item Average electron energy on site
			\item The Hamiltonian is a hop matrix
		\end{itemize}
		\column{.7\textwidth}
		\begin{align}
			V_{pp\pi}  & = \bra{\phi_{\pi}(m)}\vu{H}\ket{\phi_{\pi}(n)}\nonumber \\
			\epsilon_0 & = \bra{\phi_{\pi}(i)}\vu{H}\ket{\phi_{\pi}(i)}\nonumber
		\end{align}
	\end{columns}
	\note{\(\epsilon_0 = 0\)}
	\note{\(\Psi_{\mathrm{MO}} &= \sum_{\alpha,R}c_{\alpha,R}\phi_{\alpha}(R)\)}
\end{frame}

\begin{frame}{Hamiltonian for benzene}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.7\textwidth}
		\begin{align}
			\mqty{                            \\ \\ \\ \vb{H} = V_{pp\pi}\\ \\ \\} \ \mqty{						&  \mqty{1 & 2 & 3 & 4 & 5 & 6} \\
				\mqty{1                           \\ 2 \\ 3 \\ 4 \\ 5 \\ 6} &	\mqty*(0 & 1 & 0 & 0 & 0 & 1 \\
			1 & 0 & 1 & 0 & 0 & 0             \\
			0 & 1 & 0 & 1 & 0 & 0             \\
			0 & 0 & 1 & 0 & 1 & 0             \\
			0 & 0 & 0 & 1 & 0 & 1             \\
			1 & 0 & 0 & 0 & 1 & 0)} \nonumber
		\end{align}
		\column{.5\textwidth}
		\begin{tikzpicture}
			\chemfig{1*6(-2-3-4-5-6-)}
		\end{tikzpicture}
	\end{columns}
\end{frame}

\section{The Hamiltonian}
\subsection{Onsites, hops and the full TB Hamiltonian}
\begin{frame}{Creating the first Hamiltonian}
	\begin{center}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\begin{itemize}
				\item How do obtain the Hamiltonian
				\item Atom coordinates
				\item Interatomic distances
				\item Find nearest neighbours
				\item Fill out Hamiltonian
				\item Subtract diagonal
			\end{itemize}
			\note{Outer product: Nested lööps}
			\column{.9\textwidth}
			\includegraphics[width=.6\textwidth]{Figures/atomreffig.eps}
		\end{columns}
	\end{center}
	\begin{columns}[t]\column{.05\textwidth}\column{.9\textwidth}\im{Listings/Functions.py}{33}{38}\end{columns}
\end{frame}

\begin{frame}{Hopping matrices}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.4\textwidth}
		\begin{itemize}
			\item What is hopping matrices and how do we get them?
			\item Shift by lattice vector
			\item Resulting matrices: \(\vb{h_0}, \vb{V}, \vb{V}^{\dagger}\)
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[width=.8\textwidth]{Figures/stitch.eps}
	\end{columns}
\end{frame}

\begin{frame}{Full Hamiltonian and first band plots}
	\begin{overprint}
		\onslide<1>
		\begin{align}
			\note{We want to solve the S.E. eigen vector/value problem to get the eigen energies for band plots.                                               \\ How do solve it? }
			\vb{H}(k_x,k_y) \vb*{\phi}_k & = \vb*{\epsilon}_n\pqty{k_x,k_y}\vb*{\phi}_k\nonumber                                                               \\
			\vb{H}(k_x,k_y) = \vb{h}_0   & + (\vb{V}_{x}e^{-ik_x} + \vb{V}_{x}^{\dagger}e^{ik_x} + \vb{V}_{y}e^{-ik_y} + \vb{V}_{y}^{\dagger}e^{ik_y}\nonumber \\
			                             & + \vb{V}_{xy}e^{-ik_x}e^{-ik_y} + \vb{V}_{xy}^{\dagger}e^{ik_x}e^{ik_y})\nonumber
		\end{align}
		\im{Listings/Functions.py}{73}{80}
		\onslide<2>
		\vspace{-1cm}
		\begin{columns}[c]
			\column{.1\textwidth}
			\column{.6\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/BetaStruct.eps}
			\column{.25\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/Beta1.eps}
		\end{columns}
		\vspace{-.5cm}
		\begin{columns}[c]
			\column{.7\textwidth}
			\begin{align}
				\qq{X:} \vb{H}_{X} = \vb{h}_0 + ( & \vb{V}_{x}e^{ik_x} + \vb{V}_{x}^{\dagger}e^{-ik_x} + \vb{V}_{y} + \vb{V}_{y}^{\dagger}\nonumber \\
				+\                                & \vb{V}_{xy}e^{ik_x} + \vb{V}_{xy}^{\dagger}e^{-ik_x}) \nonumber                                 \\
				\qq{Y:} \vb{H}_{Y} = \vb{h}_0 + ( & \vb{V}_{x} + \vb{V}_{x}^{\dagger} + \vb{V}_{y}e^{-ik_y} + \vb{V}_{y}^{\dagger}e^{ik_y}\nonumber \\
				+\                                & \vb{V}_{xy}e^{-ik_y} + \vb{V}_{xy}^{\dagger}e^{ik_y}) \nonumber
			\end{align}
			\column{.001\textwidth}
			\column{.25\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/Beta2.eps}
		\end{columns}
	\end{overprint}
\end{frame}

\section{Green's functions and recursion}
\subsection{Green's matrix, recursion routine and LDOS}

\begin{frame}{Green's matrix}
	\begin{center}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\begin{itemize}
				\item What is it?
				\item Solution to the Schr\"{o}dinger Equation
				\item Propagator
				\item Why do we need it?
				\item How do we obtain it?
			\end{itemize}
			\column{.7\textwidth}
			\begin{align*}
				 & [(E+i\eta)\vb{1}-\vb{H}]\vb{G}(E)=\vb{1}        \\
				 & \downarrow                                      \\
				 & \vb{G}(E)=\vb{1}([(E+i\eta)\vb{1}-\vb{H}])^{-1}
			\end{align*}
		\end{columns}
	\end{center}
\end{frame}

\begin{frame}{Recursion routine}
	\begin{overprint}
		\onslide<1>
		\begin{center}
			\begin{columns}[c]
				\column{.5\textwidth}
				\begin{itemize}
					\item Solution to the system requires recursion
					\item Semi-infinite chain
				\end{itemize}
				\begin{align*}
					 & \begin{pmatrix}
						z\mathbf{1}-\mathbf{H}_c & -\mathbf{V}^{\dagger} \\ -\mathbf{V} & (z-\varepsilon')\mathbf{1}
					\end{pmatrix}
					\begin{pmatrix}
						\mathbf{X}      & \mathbf{G}_{0c} \\
						\mathbf{G}_{c0} & \mathbf{G}_{00}
					\end{pmatrix}
					=
					\begin{pmatrix}
						\mathbf{1} & \mathbf{0} \\
						\mathbf{0} & \mathbf{1}
					\end{pmatrix}
				\end{align*}
				\begin{align*}
					\mathbf{G}_{00}(z) & = \left[(z-\varepsilon')-\vb{V}(z\vb{1}-\vb{H}_c)\vb{V}^{\dagger}\right]^{-1} \\
					                   & = (z-\varepsilon'-\Sigma(z))^{-1}                                             \\
					                   & \qq{where}                                                                    \\
					z                  & = E+i\eta                                                                     \\
				\end{align*}
				\column{.4\textwidth}
				\begin{align*}
					a_0           & = \vb{V}^{\dagger}, \quad b_0 = \vb{V}                        \\
					e_{s0}        & = \vb{h}_s, \quad e_0\ = \vb{h}                               \\
					              & \qq{in loop:}                                                 \\
					a_1           & = a_0 \times g_0 \times a_0                                   \\
					b_1           & = b_0 \times g_0 \times b_0                                   \\
					e_1           & = e_0 + a_0 \times g_0 \times b_0 + b_0 \times g_0 \times a_0 \\
					e_{1s}        & = e_{0s} + a_0 \times g_0 \times b_0                          \\
					g_1           & = \pqty{z-e_1}^{-1}                                           \\
					\vb{\Sigma}_R & = e_s - h                                                     \\
					\vb{\Sigma}_L & = e - h - \vb{\Sigma}_R                                       \\
					\vb{G00}      & = \pqty{z-e_s}^{-1}
				\end{align*}
			\end{columns}
		\end{center}
		\onslide<2>
		\begin{columns}[c]
			\column{.2\textwidth}
			\column{\textwidth}
			\im{Listings/Functions.py}{92}{109}
		\end{columns}
	\end{overprint}
\end{frame}

\begin{frame}{LDOS}
	\centering
	\includegraphics[width=.55\textwidth]{Figures/BetaimrealTE.eps}
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{\textwidth}
		\im{Listings/SelfEnergyByRecursion.py}{64}{68}
	\end{columns}
\end{frame}

\section{Transmission}
\subsection{Device Green's functions, left/right geometry and rate matrices.}

\begin{frame}{Transmission}
	\centering
	\vspace{.05\textwidth}
	\begin{beamercolorbox}[sep=1em,wd=15cm]{redbox}
		Transmission is the probability of an electron being transported through a specific region for a specific range of energies.
	\end{beamercolorbox}
\end{frame}

\begin{frame}{Translating from system to matrices}
	\begin{columns}[c]
		\column{.3\textwidth}
		\column{.9\textwidth}
		\includegraphics[width=.64\textwidth]{Figures/illu.eps}
	\end{columns}
\end{frame}

\begin{frame}{The left and right self-energy}
	\begin{center}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\begin{itemize}
				\item Device Hamiltonian
				\item Left \textit{and} right self-energy
				\item Device Green's matrix
			\end{itemize}
			\column{.9\textwidth}
			\includegraphics[width=.6\textwidth]{Figures/2DHam.eps}
		\end{columns}
	\end{center}
	\begin{columns}[t]\column{.05\textwidth}\column{.9\textwidth}\im{Listings/Functions.py}{210}{212}\end{columns}
\end{frame}

\begin{frame}{Getting Transmission}
	\centering
	\begin{overprint}
		\onslide<1>
		\begin{itemize}
			\item Considering corrections from both contact regions
		\end{itemize}
		\begin{align*}
			\vb{G}_D  = \bqty{\vb{1}\pqty{E+i\eta}-\vb{H}_D - \vb{\Sigma}_L(E)-\vb{\Sigma}_R(E)}^{-1}
		\end{align*}
		\begin{itemize}
			\item How to account for states going in/out?
		\end{itemize}
		\begin{align*}
			\vb{\Gamma}_{L,R} & = i\pqty{\vb{\Sigma}_{L,R} - \vb{\Sigma}_{L,R}^{\dagger}}
		\end{align*}
		\im{Listings/Functions.py}{225}{228}
		\onslide<2>
		\begin{itemize}
			\item States propagating through device
			\item Entering/Exiting by rate \(\Gamma\)
		\end{itemize}
		\begin{align*}
			T(E) = \mathrm{Tr}\bqty{\vb{\Gamma}_R\vb{G}_D\vb{\Gamma}_L\vb{G}_D^{\dagger}}(E)
		\end{align*}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.9\textwidth}
			\im{Listings/Functions.py}{240}{243}
		\end{columns}
		\onslide<3>
		\vspace{-.5cm}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.25\textwidth}
			\includegraphics[width=.75\textwidth]{Figures/Beta1.eps}
			\column{.5\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/BetaStruct.eps}
		\end{columns}
		\vspace{-1cm}
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.25\textwidth}
			\includegraphics[width=.75\textwidth]{Figures/Beta2.eps}
			\column{.5\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/BetaTE.eps}
		\end{columns}
	\end{overprint}
\end{frame}

\subsection{Transmission in 2D}

\begin{frame}{Transmission in 2D}
	\centering
	\begin{columns}[c]
		\column{.6\textwidth}
		\begin{itemize}
			\item Periodic boundary conditions
			\item Shift of cells in transverse direction
			\item Bloch phase added
		\end{itemize}
		\begin{align*}
			\vb{H} = \vb{h} + \vb{V}\me^{ik_{\perp}} + \vb{V}^{\dagger} \me^{i\pqty{-k_{\perp}}}
		\end{align*}
		\column{.6\textwidth}
		\includegraphics[width=.7\textwidth]{Figures/2DTrans.eps}
	\end{columns}
	\begin{columns}[c]
		\column{.9\textwidth}
		\im{Listings/Functions.py}{250}{253}
	\end{columns}

\end{frame}

\begin{frame}{Code validity}
	\centering
	\begin{overprint}
		\onslide<1>
		\centering
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/NPG_synthesized.png}
			\column{.5\textwidth}
			\begin{align*}
				k = \frac{\pi}{2}
			\end{align*}
		\end{columns}
		\begin{columns}[t]
			\column{.6\textwidth}
			\includegraphics[width=.9\textwidth]{Figures/NPGNormal_pi-half.eps}
			\column{.6\textwidth}
			\includegraphics[width=.87\textwidth]{Figures/txy_pi-half.eps}
		\end{columns}
		\onslide<2>
		\centering
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/NPG_synthesized.png}
			\column{.5\textwidth}
			\begin{align*}
				k = \pi
			\end{align*}
		\end{columns}
		\begin{columns}[t]
			\column{.6\textwidth}
			\includegraphics[width=.9\textwidth]{Figures/NPGNormal_pi.eps}
			\column{.6\textwidth}
			\includegraphics[width=.87\textwidth]{Figures/txy_pi.eps}
		\end{columns}
		\onslide<3>
		\centering
		\begin{columns}[c]
			\column{.05\textwidth}
			\column{.5\textwidth}
			\includegraphics[width=.8\textwidth]{Figures/NPG_synthesized.png}
			\column{.5\textwidth}
			\begin{align*}
				k = \mathrm{AVG}\pqty{0,\frac{\pi}{2},\pi}
			\end{align*}
		\end{columns}
		\begin{columns}[t]
			\column{.6\textwidth}
			\includegraphics[width=.9\textwidth]{Figures/NPGNormal_AVER.eps}
			\column{.6\textwidth}
			\includegraphics[width=.87\textwidth]{Figures/txy_AVER.eps}
		\end{columns}
	\end{overprint}
\end{frame}

\begin{frame}{Summary of code structure}
	\centering
	\includegraphics[width=1\textwidth]{Figures/Flowchart.eps}
\end{frame}

\section{Exploring GNR bridges}
\subsection{Para-O\mathinhead{_4}{_4}-NPG, Para-(OH)\mathinhead{_4}{_4}-NPG, Meta-O\mathinhead{_2}{_2}-NPG, Meta-(OH)\mathinhead{_2}{_2}-NPG}

\begin{frame}{Para and meta bridges}
	\centering
	\begin{overprint}
		\onslide<1>
		\centering
		\begin{columns}[t]
			\column{.55\textwidth}
			\includegraphics[height=.7\textwidth]{Figures/Parametagraphic.eps}
			\column{.55\textwidth}
			\includegraphics[height=.7\textwidth]{Figures/Metaparagraphic.eps}
		\end{columns}
		\onslide<2>
		\begin{columns}[c]
			\column{.4\textwidth}
			\begin{itemize}
				\item Path length difference
				\item Quantum interference
				\item Transmission
				\item coupling/decoupling of GNR
			\end{itemize}
			\column{.7\textwidth}
			\includegraphics[height=.55\textwidth]{Figures/metapararesultdraft.eps}
		\end{columns}
	\end{overprint}
\end{frame}

\begin{frame}{Para-O\mathinhead{_4}{_4}-NPG}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.5\textwidth}
		\begin{itemize}
			\item Functionalisation with oxygen
			\item Quantum inteference at valence band
			\item Reproducing DFT result
			\item Decoupling of GNR
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[height=.8\textwidth]{Figures/fig19.eps}
	\end{columns}
\end{frame}

\begin{frame}{Para-(OH)\mathinhead{_4}{_4}-NPG}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.4\textwidth}
		\begin{itemize}
			\item Hydrogenation of oxygen
			\item Band splitting of DFT
			\item GNR coupling and resemblance with Para-NPG
			\item Reproducing DFT results
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[height=.8\textwidth]{Figures/fig20.eps}
	\end{columns}
\end{frame}

\begin{frame}{Meta-O\mathinhead{_2}{_2}-NPG}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.5\textwidth}
		\begin{itemize}
			\item Split in the valence for DFT
			\item Similarity in valence/conduction difference for meta/para
			\item Trying different potentials
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[height=.8\textwidth]{Figures/fig21.eps}
	\end{columns}
\end{frame}

\begin{frame}{Meta-(OH)\mathinhead{_2}{_2}-NPG}
	\centering
	\begin{columns}[c]
		\column{.05\textwidth}
		\column{.5\textwidth}
		\begin{itemize}
			\item Hydrogenation lowering potential of oxygen
			\item Decoulping of oxygen = decouplig of GNR
			\item Resemblance with pristine Meta-NPG
			\item In agreement with DFT
		\end{itemize}
		\column{.7\textwidth}
		\includegraphics[height=.8\textwidth]{Figures/fig22.eps}
	\end{columns}
\end{frame}

\begin{frame}{Conclusion}
	\centering
	\begin{beamercolorbox}[sep=1em,wd=15cm]{redbox}
		``A tight-binding routine has successfully been developed in Python. This makes it easier to, qualitatively, understand electron transport in nanoporous graphene devices (NPG)''\huge\checkmark
	\end{beamercolorbox}
\end{frame}

\section*{Questions}
\title{Questions}
\subtitle{}
\begin{frame}
	\titlepage
\end{frame}
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% \appendix
% \section{}
% \begin{frame}{}
% 	\begin{itemize}
% 		\item Appendix
% 	\end{itemize}
% \end{frame}

\end{document}