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mostly(?) finish the MPI slides

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1 parent 4a92a25 commit 0e51310d6110bce7b7cdac88364169220154096c @moritz committed Feb 21, 2010
Showing with 80 additions and 28 deletions.
  1. +80 −28 slides/mpi.tex
  2. BIN slides/setup-simple.pdf
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108 slides/mpi.tex
@@ -17,6 +17,7 @@
\usepackage{array}
\usefonttheme{professionalfonts}
+\setbeamertemplate{footline}[page number]
\setlength{\extrarowheight}{2mm}
\newcommand{\vect}[2]{\ensuremath{\inp{\hspace{-.8ex}\begin{array}{c}#1\\#2\end{array}\hspace{-.4ex}}}}
@@ -52,7 +53,7 @@
\author{Moritz Lenz}
\institute{Institut für Theoretische Physik und Astrophysik, Universität
Würzburg}
-\date{Max Planck Institut, 2010-02-24}
+\date{Max Planck Institut, 2010-02-22}
\subject{Physics}
@@ -89,14 +90,6 @@
\begin{frame}
\titlepage
- \begin{center}
-{
- Diploma Thesis\\[1.5em]
-}
- Supervisor: Prof. Ewelina Hankiewicz
-
-\end{center}
-
% \begin{multicols}{2}
% \includegraphics[width=0.4\textwidth]{setup-79_reduced.jpg}
%
@@ -107,6 +100,10 @@
\end{frame}
+\begin{frame}{Outline}
+ \tableofcontents
+\end{frame}
+
\section{Motivation}
\begin{frame}
@@ -127,7 +124,7 @@ \section{Motivation}
\subsection{Ferromagentic materials}
\begin{frame}{Motivation - Achievements of Spintronics}
- \textbf{Giant Magnetoresistance}
+ \textbf{Giant Magnetoresistance} used in reading heads of hard discs
\includegraphics[width=75mm]{storage-density.png}
@@ -263,9 +260,12 @@ \subsection{Landauer Formula}
\end{align*}
\pause
\begin{align*}
- G^R = ((E + i \eta) -H)^{-1}\quad & \textnormal{wave moving away from exitation}\\
- G^A = ((E - i \eta) -H)^{-1}\quad & \textnormal{wave moving
- towards exitation}
+ G^R = ((E + i \eta) -H)^{-1}\quad & \textnormal{Retarded Green's
+ function}\\
+ \qquad & \textnormal{wave moving away from exitation}\\
+ G^A = ((E - i \eta) -H)^{-1}\quad & \textnormal{Adveanced Green's
+ function}\\
+ \qquad & \textnormal{wave moving towards exitation}
\end{align*}
\begin{align*}
@@ -287,22 +287,29 @@ \subsection{Landauer Formula}
\end{frame}
\begin{frame}{Theory: Fisher-Lee Relation}
- \huge
+ {
+ \huge
+ \begin{align*}
+ T_{pq} = \textnormal{Trace}( \Sigma_p G^R \Sigma_q G^A )
+ \end{align*}
+ }
\begin{align*}
- T_{pq} = \textnormal{Trace}( \Sigma_p G^R \Sigma_q G^A )
+ \Sigma_p\qquad &\textnormal{Self-Energy matrix for lead $p$}\\
+ G^R \qquad &\textnormal{Retarded Green's function}\\
+ G^A \qquad &\textnormal{Advanced Green's function}\\
\end{align*}
\end{frame}
\section{Work done}
-\begin{frame}{Setup}
+\begin{frame}{Model}
\begin{itemize}
\item 2D electron gas in quantum well
\item 2 bands considered
- \item $T = 0$
+ \item $T = 0K$
\item size: about 200nm
\item ballistic transport
\item coherent transport
- \item Interface between SO and normal regimes
+ \item Interface between "normal" (N) and Spin-orbit coupling (SO) regimes
\end{itemize}
\end{frame}
@@ -312,25 +319,32 @@ \subsection{Analytical calculations}
\begin{multicols}{2}
\includegraphics[width=55mm]{setup-simple}
+ \begin{minipage}{0.5\textwidth}
+ \textbf{N}: Normal regime, $\alpha = 0$\\
+ \textbf{SO}: Spin-orbit coupling regime, $\alpha \not= 0$
+ \end{minipage}
+
\begin{align*}
H_r &= \frac{p^2}{2m} + (-\vec y \times \vec \sigma) \cdot
\alpha(x) \vec p\\
E_{\pm} &= \frac{p^2}{2m} \pm \alpha \\
v_{\pm} &= \frac{\partial E_{\pm}}{\partial p} = \frac{p}{m} \pm \alpha
\end{align*}
+ $\vec \sigma$ is the vector of Pauli matrices and describes the Spin
+
\end{multicols}
\end{frame}
-\begin{frame}{Analytical calculations - Wave functions}
+\begin{frame}{Analytical calculations - Eigenstates}
\begin{align*}
- \chi_{SO}^{\pm} &= \frac{1}{n_{SO}^{\pm}}
- \vect{-p_{x,SO}^{\pm} \pm p_{SO}^\pm}{p_z} \\
- n_{SO}^{\pm} &= \sqrt{|-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 +
- p_z^2}
+ \chi_{SO}^{\pm} &= \frac{1}{n^{\pm}}
+ \vect{-p_{x}^{\pm} \pm p^\pm}{p_z} \\
+ n_{SO}^{\pm} &= \sqrt{|-p_{x}^{\pm} \pm p^\pm|^2 + p_z^2}
\end{align*}
- \pause
+\end{frame}
+\begin{frame}{Analytical calculations - Wave Function}
\begin{align*}
\Psi^+ = e^{i p_z z} * \left\{
\begin{array}{ll}
@@ -341,12 +355,19 @@ \subsection{Analytical calculations}
\end{array} \right.
\end{align*}
+ \begin{align*}
+ r_{\pm+} \qquad & \textnormal{Reflection coefficients}\\
+ t_{\pm+} \qquad & \textnormal{Transmission coefficients}
+ \end{align*}
+
\end{frame}
\begin{frame}{Transmission coefficients}
- \includegraphics[width=7.0cm]{zero-plus.pdf}
+ \begin{center}
+ \includegraphics[width=10.0cm]{zero-plus.pdf}
- \includegraphics[width=7.0cm]{zero-minus.pdf}
+ For $\phi > \phi_C$ the $e^{i p_{x,SO}^+ x }$ part vanishes
+ \end{center}
\end{frame}
\begin{frame}{Critical angle for $+$ wave}
@@ -359,6 +380,23 @@ \subsection{Analytical calculations}
\end{center}
\end{frame}
+\begin{frame}{Figure of merit: Spin polarization}
+ Each lead is assumed to consist of a spin-up ($\uparrow$) and a
+ spin-down ($\downarrow$) sub-lead\\[2em]
+
+ {
+ \huge
+ \begin{align*}
+ T_S = T_{2\uparrow, 1\uparrow} + T_{2\uparrow, 1\downarrow}
+ - T_{2\downarrow, 1\uparrow} - T_{2\downarrow, 1\downarrow}\\
+ \end{align*}
+ }
+
+ $T_S$: Spin polarization perpendicular to the plane of 2-dimensional
+ electron gas
+
+\end{frame}
+
\begin{frame}{Adapting to $\uparrow, \downarrow$ bases}
\begin{center}
\includegraphics[width=\textwidth]{adapting-pic.pdf}
@@ -378,6 +416,19 @@ \subsection{Analytical calculations}
\includegraphics[width=\textwidth]{comparison-over-phi.pdf}
\end{frame}
+\begin{frame}{Limits of analytical calculations}
+
+ \begin{center}
+ \begin{itemize}
+ \item Limited to a single mode (typically 8 to 12 in experiment)
+ \item Limited to simple geometry
+ \item Hard to incorporate scattering centers, boundary conditions,
+ finite size effects
+ \end{itemize}
+ \end{center}
+
+\end{frame}
+
\subsection{Numerical calculations}
\begin{frame}{Numerical Calculations - The Plan}
\begin{center}
@@ -521,9 +572,10 @@ \section{Summary}
\begin{itemize}
\item Spintronics is a successful and interesting field (GMR,
Datta-Das transistor)
- \item Non-magnetic materials necessary for scaling
- \item Rashba SO-Coupling: filtering with critical phenomena
+ \item Rashba SO-Coupling: spin filtering with critical phenomena
\item Up to $20\%$ spin separation
+ \item rough agreement between analytical and numeric calculations
+ \item Generalization to two different SO regions
\end{itemize}
\end{frame}
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