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mostly(?) finish the MPI slides

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commit 0e51310d6110bce7b7cdac88364169220154096c 1 parent 4a92a25
Moritz Lenz authored

Showing 2 changed files with 80 additions and 28 deletions. Show diff stats Hide diff stats

  1. +80 28 slides/mpi.tex
  2. BIN  slides/setup-simple.pdf
108 slides/mpi.tex
@@ -17,6 +17,7 @@
17 17 \usepackage{array}
18 18
19 19 \usefonttheme{professionalfonts}
  20 +\setbeamertemplate{footline}[page number]
20 21 \setlength{\extrarowheight}{2mm}
21 22
22 23 \newcommand{\vect}[2]{\ensuremath{\inp{\hspace{-.8ex}\begin{array}{c}#1\\#2\end{array}\hspace{-.4ex}}}}
@@ -52,7 +53,7 @@
52 53 \author{Moritz Lenz}
53 54 \institute{Institut für Theoretische Physik und Astrophysik, Universität
54 55 Würzburg}
55   -\date{Max Planck Institut, 2010-02-24}
  56 +\date{Max Planck Institut, 2010-02-22}
56 57
57 58 \subject{Physics}
58 59
@@ -89,14 +90,6 @@
89 90 \begin{frame}
90 91 \titlepage
91 92
92   - \begin{center}
93   -{
94   - Diploma Thesis\\[1.5em]
95   -}
96   - Supervisor: Prof. Ewelina Hankiewicz
97   -
98   -\end{center}
99   -
100 93 % \begin{multicols}{2}
101 94 % \includegraphics[width=0.4\textwidth]{setup-79_reduced.jpg}
102 95 %
@@ -107,6 +100,10 @@
107 100
108 101 \end{frame}
109 102
  103 +\begin{frame}{Outline}
  104 + \tableofcontents
  105 +\end{frame}
  106 +
110 107 \section{Motivation}
111 108
112 109 \begin{frame}
@@ -127,7 +124,7 @@ \section{Motivation}
127 124 \subsection{Ferromagentic materials}
128 125
129 126 \begin{frame}{Motivation - Achievements of Spintronics}
130   - \textbf{Giant Magnetoresistance}
  127 + \textbf{Giant Magnetoresistance} used in reading heads of hard discs
131 128
132 129 \includegraphics[width=75mm]{storage-density.png}
133 130
@@ -263,9 +260,12 @@ \subsection{Landauer Formula}
263 260 \end{align*}
264 261 \pause
265 262 \begin{align*}
266   - G^R = ((E + i \eta) -H)^{-1}\quad & \textnormal{wave moving away from exitation}\\
267   - G^A = ((E - i \eta) -H)^{-1}\quad & \textnormal{wave moving
268   - towards exitation}
  263 + G^R = ((E + i \eta) -H)^{-1}\quad & \textnormal{Retarded Green's
  264 + function}\\
  265 + \qquad & \textnormal{wave moving away from exitation}\\
  266 + G^A = ((E - i \eta) -H)^{-1}\quad & \textnormal{Adveanced Green's
  267 + function}\\
  268 + \qquad & \textnormal{wave moving towards exitation}
269 269 \end{align*}
270 270
271 271 \begin{align*}
@@ -287,22 +287,29 @@ \subsection{Landauer Formula}
287 287 \end{frame}
288 288
289 289 \begin{frame}{Theory: Fisher-Lee Relation}
290   - \huge
  290 + {
  291 + \huge
  292 + \begin{align*}
  293 + T_{pq} = \textnormal{Trace}( \Sigma_p G^R \Sigma_q G^A )
  294 + \end{align*}
  295 + }
291 296 \begin{align*}
292   - T_{pq} = \textnormal{Trace}( \Sigma_p G^R \Sigma_q G^A )
  297 + \Sigma_p\qquad &\textnormal{Self-Energy matrix for lead $p$}\\
  298 + G^R \qquad &\textnormal{Retarded Green's function}\\
  299 + G^A \qquad &\textnormal{Advanced Green's function}\\
293 300 \end{align*}
294 301 \end{frame}
295 302
296 303 \section{Work done}
297   -\begin{frame}{Setup}
  304 +\begin{frame}{Model}
298 305 \begin{itemize}
299 306 \item 2D electron gas in quantum well
300 307 \item 2 bands considered
301   - \item $T = 0$
  308 + \item $T = 0K$
302 309 \item size: about 200nm
303 310 \item ballistic transport
304 311 \item coherent transport
305   - \item Interface between SO and normal regimes
  312 + \item Interface between "normal" (N) and Spin-orbit coupling (SO) regimes
306 313 \end{itemize}
307 314 \end{frame}
308 315
@@ -312,6 +319,11 @@ \subsection{Analytical calculations}
312 319 \begin{multicols}{2}
313 320 \includegraphics[width=55mm]{setup-simple}
314 321
  322 + \begin{minipage}{0.5\textwidth}
  323 + \textbf{N}: Normal regime, $\alpha = 0$\\
  324 + \textbf{SO}: Spin-orbit coupling regime, $\alpha \not= 0$
  325 + \end{minipage}
  326 +
315 327 \begin{align*}
316 328 H_r &= \frac{p^2}{2m} + (-\vec y \times \vec \sigma) \cdot
317 329 \alpha(x) \vec p\\
@@ -319,18 +331,20 @@ \subsection{Analytical calculations}
319 331 v_{\pm} &= \frac{\partial E_{\pm}}{\partial p} = \frac{p}{m} \pm \alpha
320 332 \end{align*}
321 333
  334 + $\vec \sigma$ is the vector of Pauli matrices and describes the Spin
  335 +
322 336 \end{multicols}
323 337 \end{frame}
324 338
325 339
326   -\begin{frame}{Analytical calculations - Wave functions}
  340 +\begin{frame}{Analytical calculations - Eigenstates}
327 341 \begin{align*}
328   - \chi_{SO}^{\pm} &= \frac{1}{n_{SO}^{\pm}}
329   - \vect{-p_{x,SO}^{\pm} \pm p_{SO}^\pm}{p_z} \\
330   - n_{SO}^{\pm} &= \sqrt{|-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 +
331   - p_z^2}
  342 + \chi_{SO}^{\pm} &= \frac{1}{n^{\pm}}
  343 + \vect{-p_{x}^{\pm} \pm p^\pm}{p_z} \\
  344 + n_{SO}^{\pm} &= \sqrt{|-p_{x}^{\pm} \pm p^\pm|^2 + p_z^2}
332 345 \end{align*}
333   - \pause
  346 +\end{frame}
  347 +\begin{frame}{Analytical calculations - Wave Function}
334 348 \begin{align*}
335 349 \Psi^+ = e^{i p_z z} * \left\{
336 350 \begin{array}{ll}
@@ -341,12 +355,19 @@ \subsection{Analytical calculations}
341 355 \end{array} \right.
342 356 \end{align*}
343 357
  358 + \begin{align*}
  359 + r_{\pm+} \qquad & \textnormal{Reflection coefficients}\\
  360 + t_{\pm+} \qquad & \textnormal{Transmission coefficients}
  361 + \end{align*}
  362 +
344 363 \end{frame}
345 364
346 365 \begin{frame}{Transmission coefficients}
347   - \includegraphics[width=7.0cm]{zero-plus.pdf}
  366 + \begin{center}
  367 + \includegraphics[width=10.0cm]{zero-plus.pdf}
348 368
349   - \includegraphics[width=7.0cm]{zero-minus.pdf}
  369 + For $\phi > \phi_C$ the $e^{i p_{x,SO}^+ x }$ part vanishes
  370 + \end{center}
350 371 \end{frame}
351 372
352 373 \begin{frame}{Critical angle for $+$ wave}
@@ -359,6 +380,23 @@ \subsection{Analytical calculations}
359 380 \end{center}
360 381 \end{frame}
361 382
  383 +\begin{frame}{Figure of merit: Spin polarization}
  384 + Each lead is assumed to consist of a spin-up ($\uparrow$) and a
  385 + spin-down ($\downarrow$) sub-lead\\[2em]
  386 +
  387 + {
  388 + \huge
  389 + \begin{align*}
  390 + T_S = T_{2\uparrow, 1\uparrow} + T_{2\uparrow, 1\downarrow}
  391 + - T_{2\downarrow, 1\uparrow} - T_{2\downarrow, 1\downarrow}\\
  392 + \end{align*}
  393 + }
  394 +
  395 + $T_S$: Spin polarization perpendicular to the plane of 2-dimensional
  396 + electron gas
  397 +
  398 +\end{frame}
  399 +
362 400 \begin{frame}{Adapting to $\uparrow, \downarrow$ bases}
363 401 \begin{center}
364 402 \includegraphics[width=\textwidth]{adapting-pic.pdf}
@@ -378,6 +416,19 @@ \subsection{Analytical calculations}
378 416 \includegraphics[width=\textwidth]{comparison-over-phi.pdf}
379 417 \end{frame}
380 418
  419 +\begin{frame}{Limits of analytical calculations}
  420 +
  421 + \begin{center}
  422 + \begin{itemize}
  423 + \item Limited to a single mode (typically 8 to 12 in experiment)
  424 + \item Limited to simple geometry
  425 + \item Hard to incorporate scattering centers, boundary conditions,
  426 + finite size effects
  427 + \end{itemize}
  428 + \end{center}
  429 +
  430 +\end{frame}
  431 +
381 432 \subsection{Numerical calculations}
382 433 \begin{frame}{Numerical Calculations - The Plan}
383 434 \begin{center}
@@ -521,9 +572,10 @@ \section{Summary}
521 572 \begin{itemize}
522 573 \item Spintronics is a successful and interesting field (GMR,
523 574 Datta-Das transistor)
524   - \item Non-magnetic materials necessary for scaling
525   - \item Rashba SO-Coupling: filtering with critical phenomena
  575 + \item Rashba SO-Coupling: spin filtering with critical phenomena
526 576 \item Up to $20\%$ spin separation
  577 + \item rough agreement between analytical and numeric calculations
  578 + \item Generalization to two different SO regions
527 579 \end{itemize}
528 580
529 581 \end{frame}
BIN  slides/setup-simple.pdf
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