# moritz/spinoptics

mostly(?) finish the MPI slides

 @@ -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}