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[tex] rewrite second half of the introduction

Also include a plot of the parameters that influence T_S
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1 parent 2d9d937 commit 27b102ff4a92ea76abec46afdfa4b3071ae37aa7 @moritz committed Dec 16, 2009
Showing with 55 additions and 21 deletions.
  1. +1 −1
  2. +28 −14 tex/intro.tex
  3. +24 −6 tex/numerics.tex
  4. BIN tex/parameters-abcd.pdf
  5. BIN tex/parameters-chi.pdf
  6. +1 −0 tex/summary.tex
  7. +1 −0 tex/thesis.tex
@@ -28,7 +28,7 @@
my %defaults = (
-b => 0,
-e => 2.0,
- -r => 0.1,
+ -r => 0.2,
-p => 10,
-n => 21,
-s => 0.5,
@@ -64,20 +64,34 @@ \chapter{Introduction}
experimentally accessible in quantum wells at heterojunctions in GaAs,
HgTe and other semiconductors.
-We present both analytical calculations and numerical simulations based on
-tight-binding approximation. The analytical calculations show an analogy to
-light optics: an interface between two media with different optical densities
-can result in polarized light, and polarization-dependent reflection. In
-analogy, an interface between two regions of different spin-orbit coupling
-strength splits up an electron beam into two beams of defined chirality.
-The fact that the beams are of different chirality -- as opposed to
-linear polarization, as in the case of light optics -- makes it much hard to
-build an efficient spin polarizer.
-The numerical results show that a certain degree of spin separation can be
-achieved for appropriate interface angles and spin-orbit strengths, even in
-the new case where there is non-zero spin-orbit interaction on both sides of
+In chapter \ref{sec:theory} we present the basic theoretical underpinning for
+the calculations to come: the Landauer Formula which relates conductance to
+the transmission matrix $T$, the Fisher-Lee relation which allows calculation
+of $T$ based on the Green's function in sample and lead, the Rashba-Bychkov
+spin-orbit coupling which causes all the interesting effects discussed in this
+thesis, and finally we present a tight binding model which allows numerical
+calculation of the Green's functions and thus $T$.
+In chapter \ref{sec:analytical} we present an analytical model of an electron
+wave traveling from a normal region to a region with spin-orbit coupling. The
+wave is decomposed into two parts of opposite chirality, and we
+analyze the transmission and reflection coefficients resolved by chirality.
+We expand this model to a system where both sides of the interface have
+non-zero (but different) spin-orbit interaction.
+Chapter \ref{sec:numerics} explains the numerical calculations in depth. We
+present the used algorithm and possible alternatives, considerations regarding
+the run time performance and numerical errors, and of course results from
+these calculations. We find that decent spin polarizations can be achieved
+for appropriate interface angles and spin-orbit strengths, even in the new
+case where there is non-zero spin-orbit interaction on both sides of
the interface.
+We also explain how the results of the analytical calculations can be compared
+to the numerical results, and how projection from the chiral bases to the
+spin-up/spin-down bases diminishes some of effects of the interfaces.
+Chapter \ref{sec:summary} finally summarizes our achievements, and shows up
+possible directions in which our models could be expanded.
% vim: spell
@@ -1,4 +1,5 @@
\chapter{Numerical Calculations and Results}
Numerical transport calculations have a huge advantage over the analytical
calculation: once a formalism is implemented, it works for arbitrary
@@ -420,6 +421,18 @@ \subsection{Comparison to Analytical Results}
cosine form that the tight binding model implies. This explains why there is a
shift between the two curves in figure \ref{fig:a-n-matching-phi}.
+ \begin{center}
+ \includegraphics[width=0.45\textwidth]{parameters-abcd.pdf}%
+ \hspace{0.05\textwidth}%
+ \includegraphics[width=0.45\textwidth]{parameters-chi.pdf}
+ \end{center}
+ \caption{The parameters appearing in eq. \ref{eq:simplfied-t} as function
+ of $\phi$. \textbf{Left: } $a, b, c, d$ \textbf{Right: } components of
+ $\chi_{SO}^+$ and $\chi_{SO}^-$ for $\ta = 0.1$.}
+ \label{fig:simplfieid-parmas}
For a rather small spin-orbit coupling strength of $\frac{\tso}{2 t a} = 0.02$
we already get a quite respectable relative spin-polarization of $20\%$.
@@ -428,6 +441,7 @@ \subsection{Comparison to Analytical Results}
ignore the global phases that the $\exp$ functions provide, and obtain a
simplified expression for $a$ and $b$.
a &= \frac{1-\sin \phi}{1 + \sin\phi} \nonumber\\
b &= \sqrt{1-a^2}
@@ -440,16 +454,18 @@ \subsection{Comparison to Analytical Results}
T_{2\uparrow,1\uparrow} &\approx \left|a \chi_{SO}^{+U} t_{++}
- + b \chi_{SO}^{-U} t_{--} \right|^2\\
+ + b \chi_{SO}^{-U} t_{--} \right|^2\nonumber\\
T_{2\downarrow,1\downarrow} &\approx \left|c \chi_{SO}^{-D} t_{--}
+ d \chi_{SO}^{+D} t_{--} \right|^2
+ \label{eq:simplfied-t}
where the superscript index $U$ means \emph{upper component of}, and $D$ means
\emph{lower component of}.
When $\phi$ is varied in the range of 0 to $\pi/2$, the coefficients $a, b, c,
-d$ and the absolute values of the spinor components cover the range of 0 to 1.
+d$ and the absolute values of the spinor components cover the range of 0 to 1
+(compare figure \ref{fig:simplfieid-parmas}).
The critical phenomena and the variation of $t_{++}$ and $t_{--}$ drown in
eight parameters which oscillate roughly with the same frequency and
magnitude, washing out a clear signature from the chiral waves.
@@ -477,7 +493,7 @@ \section{Interface Between Two Spin-Orbit Coupling Regions}
-When a sample contains a 2-dimensional electron gas with Rashba spin-orbit
+When a sample contains a two-dimensional electron gas with Rashba spin-orbit
coupling, it is very hard to create a region without any spin-orbit coupling.
While it is hard to switch it off entirely, it is quite possible to tune the
strength of an individual region by using a gate electrode on top of the
@@ -493,9 +509,11 @@ \section{Interface Between Two Spin-Orbit Coupling Regions}
- \caption{Relative spin polarization $T_s^{rel}$ as function of $q
-=\frac{\ta_A}{\ta_B}$ (ratio of spin-orbit coupling strength) for a fixed
- $\phi=80^\circ$ and $\tso{}_B = 0.1$.}
+ \caption{Relative spin polarization $T_s^{rel}$ as function of
+ $ \frac{\tso{}_A}{\tso{}_B} =\frac{\ta_A}{\ta_B}$ (ratio of spin-orbit
+ coupling strengths) for a fixed
+ $\phi=80^\circ$ and $\tso{}_B = 0.1$.
+ }
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@@ -1,4 +1,5 @@
\chapter{Summary and Outlook}
We analyzed the possibilities of achieving spin polarization in a
non-magnetic, microscopic semiconductor. We found that an interface between
@@ -98,6 +98,7 @@
Consider a two-dimensional conductor of width $W$ and length $L$. If these two
dimensions are large enough, the conductance is

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