[tex] more polishing

tex/bib.bib

@@ -164,6 +164,22 @@ @article{datta-das
 doi = {10.1063/1.102730}
 }
 
+@Article{ishe-ew,
+  title = {Charge Hall effect driven by spin-dependent chemical potential gradients and Onsager relations in mesoscopic systems},
+  author = {Hankiewicz, E. M. and Li, Jian  and Jungwirth, Tomas  and Niu, Qian  and Shen, Shun-Qing  and Sinova, Jairo },
+  journal = {Phys. Rev. B},
+  volume = {72},
+  number = {15},
+  pages = {155305},
+  numpages = {5},
+  year = {2005},
+  month = {Oct},
+  doi = {10.1103/PhysRevB.72.155305},
+  publisher = {American Physical Society}
+}
+
+
+
 @article{ISHE,
 author = {E. Saitoh and M. Ueda and H. Miyajima and G. Tatara},
 collaboration = {},

tex/intro.tex

@@ -42,7 +42,7 @@ \chapter{Introduction}
 
 However, building ferromagnetic contacts or devices on
 the nano scale is a serious technological challenge, and combining millions of
-ferromagnetic structures on a single seems hardly possible.
+ferromagnetic structures on a single device seems hardly possible.
 
 In this Diploma Thesis, we therefore investigate how a spin polarized electron
 beam can be achieved by using only non-magnetic materials. The Rashba 
@@ -64,15 +64,15 @@ \chapter{Introduction}
 experimentally accessible in quantum wells at heterojunctions in GaAs,
 HgTe and other semiconductors.
 
-In chapter \ref{sec:theory} we present the basic theoretical underpinning for
+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
+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.

tex/khodas.tex

@@ -222,7 +222,7 @@ \section{Interface Between Normal and Spin-Orbit Coupling Regions}
 it follows that
 
 \begin{align}
-    \phi_c          &= -\sin ^{-1}\left(a-\sqrt{a^2+1}\right)
+    \phi_c          &= -\sin ^{-1}\left(\ta-\sqrt{\ta^2+1}\right)
 \end{align}
 
 Figure \ref{fig:critical-angle} shows the critical angle as a function
@@ -307,14 +307,22 @@ \section{Generalization to two Spin-Orbit Regions}
 \ref{fig:plots-nonzero}, and in the region $\phi < \phi+_{B,c}$ the various transmission and reflection
 coefficients look very similar to the case with $\ta_A = 0$.
 
-The second gray line is critical angle $\phi^+_{c2}$ associated with
+The second gray line is the critical angle $\phi^+_{c2}$ associated with
 $\ta_A$. For
 $\phi > \phi^+_{A,c}$ the momentum in $x$ direction $p_{x,A}$ is again
 imaginary, just like if we had another interface to a normal region
 left of the $A$ region. 
 
-It follows the same $\ta$ dependency, namely $\phi^+_{c2} =
--\sin^{-1}(\ta_A - \sqrt{1+\ta_A^2})$.
+$p_{x,A}$ has the same structural dependence on $\ta_A$ as $p_{x,B}$
+has on $\ta_B$:
+
+\begin{align}
+    p_{x,A}^+ = \sqrt{ p_A^2 - p_z^2} = p \sqrt{(\sqrt{1+\ta_A^2} -
+            \ta_A)^2 - \sin^2 \phi^+}
+\end{align}
+
+It follows that $\phi^+_{c2}$ has a similar form as $\phi^+_{c1}$,
+namely $\phi^+_{c2} = -\sin^{-1}(\ta_A - \sqrt{1+\ta_A^2})$.
 
 %
 %\begin{figure}

tex/numerics.tex

@@ -287,7 +287,14 @@ \subsection{Comparison to Analytical Results}
     \begin{center}
     \includegraphics[width=0.8\textwidth]{adapting-pic.pdf}
     \end{center}
-    \caption{Injecting one spin-up charge carrier into the system}
+    \caption{Injecting one spin-up charge carrier into the system notionally
+        splits up the wave into two wave functions with $+$ and $-$ chirality.
+        At the interface a part of the wave is scattered, and
+        $\mathbf{\psi^\pm}$
+        describe the injected and reflected wave functions left of the
+        interface. The transmitted fraction of the wave function is then
+        projected onto the $\uparrow,\downarrow$ bases when it reaches the right
+        lead.}
 \end{figure}
 
 % Since there is no dissipation in our model, and we assume that the mere
@@ -438,7 +445,7 @@ \subsection{Comparison to Analytical Results}
 (figure \ref{fig:n-so-rel}).
 
 To understand better why we don't get a very clear picture of the critical
-angle the numerical results, we look at \ref{eq:a-n-left} and for a moment
+angle, we look at \ref{eq:a-n-left} and for a moment
 ignore the global phases that the $\exp$ functions provide, and obtain a
 simplified expression for $a$ and $b$.
 
@@ -457,7 +464,7 @@ \subsection{Comparison to Analytical Results}
     T_{2\uparrow,1\uparrow}     &\approx \left|a \chi_{SO}^{+U} t_{++}
             + 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
+            + d \chi_{SO}^{+D} t_{++} \right|^2
     \label{eq:simplfied-t}
 \end{align}
 
@@ -533,19 +540,21 @@ \section{Relation to experiments}
     \begin{center}
         \includegraphics[width=0.7\textwidth]{beamsplitter2.jpg}
     \end{center}
-    \caption{Experimental realization of a beam splitter, with two
+    \caption{Experimental realization of a beam splitter in a HgTe/CdTe quantum
+        well, with two
         collimating quantum point contact on top and bottom. In the middle
         there is a strip at angle $\phi = 45^\circ$ where an electric field
         can be applied by a gate. Image courtesy of M. Mühlbauer,
-        Physikalisches Institut, Universität Würzburg}
+        Physikalisches Institut, Universität Würzburg\cite{mathias}}
     \label{fig:experiment}
 \end{figure}
 
 It is possible to realize interfaces between two spin-orbit coupling regimes
 in experiments. Figure \ref{fig:experiment} shows such a sample as produced
 by the group of H. Buhmann in our department \cite{mathias}.
 
-It has two wide contacts on the left and right side, and two collimating point
+It consists of a two-dimensional electron gas in a HgTe/CdTe quantum well. It has
+two wide leads on the left and right side, and two collimating point quantum
 contacts on the top and bottom. At an angle of $45^{\circ}$ there is a stripe
 across the sample. Inside the strip the electric field, and thus the strength
 of the spin-orbit coupling strength can be tuned by a gate electrode that is
@@ -558,7 +567,7 @@ \section{Relation to experiments}
 scatters electrons too.
 
 The spin polarization can be measured via the Inverse Spin-Hall Effect
-\cite{ISHE} as an electrical current.
+\cite{ishe-ew,ISHE} as an electrical current.
 
 %For $\phi > \phi_c$, the wave $\exp{i p_x^+ x}\exp{i p_z z} t_{++}\chi_{SO}^+$
 %does not propagate, because $p_x^+$ is imaginary. That means that the relative