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[tex] another citation; fix some labels and references

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1 parent b39a2a4 commit 410d49157ab1f6dcd714e88f107ddcc6ec69c8ee @moritz committed Dec 16, 2009
Showing with 67 additions and 10 deletions.
  1. +21 −0 tex/bib.bib
  2. +1 −1 tex/khodas.tex
  3. +37 −3 tex/numerics.tex
  4. +8 −6 tex/summary.tex
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@@ -163,3 +163,24 @@ @article{datta-das
url = {http://link.aip.org/link/?APL/56/665/1},
doi = {10.1063/1.102730}
}
+
+@article{ISHE,
+author = {E. Saitoh and M. Ueda and H. Miyajima and G. Tatara},
+collaboration = {},
+title = {Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect},
+publisher = {AIP},
+year = {2006},
+journal = {Applied Physics Letters},
+volume = {88},
+number = {18},
+eid = {182509},
+numpages = {3},
+pages = {182509},
+keywords = {spin Hall effect; platinum; metallic thin films; electric potential; electron spin polarisation},
+url = {http://link.aip.org/link/?APL/88/182509/1},
+doi = {10.1063/1.2199473}
+}
+
+
+
+
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@@ -187,7 +187,7 @@ \section{Interface Between Normal and Spin-Orbit Coupling Regions}
\frac{\pi}{2}$, the momentum $p_{x,SO}^+$ is imaginary and no current flows
anymore with $+$ chirality.
-\begin{figure}
+\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.7\textwidth]{critical-angle.pdf}
\end{center}
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@@ -434,7 +434,8 @@ \subsection{Comparison to Analytical Results}
\end{figure}
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 $20\%$.
+we already get a quite respectable relative spin-polarization of $20\%$
+(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
@@ -490,7 +491,7 @@ \section{Interface Between Two Spin-Orbit Coupling Regions}
with additional peaks going up to about $40\%$.
($150 \times 150$ sample of $100~nm \times 100~nm$ at $E_F = 2 t$)
}
- \label{fig:n-so-rel}
+ \label{fig:so-so-rel}
\end{figure}
When a sample contains a two-dimensional electron gas with Rashba spin-orbit
@@ -499,7 +500,7 @@ \section{Interface Between Two Spin-Orbit Coupling Regions}
strength of an individual region by using a gate electrode on top of the
sample to apply an electric field.
-Figure \ref{fig:n-so-rel} shows the relative spin polarization as a function
+Figure \ref{fig:so-so-rel} shows the relative spin polarization as a function
of the interface angle $\phi$, for $\ta_B = 2 \ta_A$. Again a spin
polarization of $20\%$ can be observed, with some few spikes going up as high
as $40\%$ (but with stronger SO interaction on the right-hand side than in the
@@ -526,6 +527,39 @@ \section{Interface Between Two Spin-Orbit Coupling Regions}
the envelope function decreases for $q \mapsto 1$, because the regions become
similar and thus the effect of the interface less pronounced.
+\section{Relation to experiments}
+
+\begin{figure}[htb]
+ \begin{center}
+ \includegraphics[width=0.7\textwidth]{beamsplitter2.jpg}
+ \end{center}
+ \caption{Experimental realization of a beam splitter, 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}
+ \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.
+
+It has two wide contacts on the left and right side, and two collimating point
+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
+located above the sample.
+
+In this experimental setup two things disagree with our model, which
+means we can't compare experimental results directly with ours: firstly the
+electron sees two interfaces (one on entering the stripe, one on leaving) and
+secondly the electric field also introduces a potential barrier, which
+scatters electrons too.
+
+The spin polarization can be measured via the Inverse Spin-Hall Effect
+\cite{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
%spin polarization caused by the $\mathbf{\Psi^+}$ wave
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@@ -6,13 +6,15 @@ \chapter{Summary and Outlook}
normal and Rashba spin-orbit coupling areas splits an electron beam into
components with different chirality, and if the angle between the interface
and the incident beam exceeds a critical angle, the beam with $+$ chirality
-ceases to propagate. This effect can be used to achieve a spin polarization.
+ceases to propagate. This effect can be used to achieve spin polarization.
-We modeled such a system both analytically and numerically, in order to
-achieve both a good understanding, and having maximal flexibility with the
-choice of parameters and system details.
+We modeled such a system analytically in order to obtain a good understanding
+for the physics, and in an extensible numerical simulation in order to have
+maximal flexibility with the choice of parameters and system details.
-We found that the measurable spin polarization is diminished by fact that the
+We developed a method to compare the analytical and numerical results, and in
+that course we found that the measurable spin polarization is partially
+absorbed by fact that the
interface separates electron waves by chirality, not by spin component
in $z$-direction.
@@ -22,7 +24,7 @@ \chapter{Summary and Outlook}
of decreasing magnitude when the spin-orbit coupling strengths become similar.
In both cases a large angle between the incident beam the interface is
-essential for a decent spin polarization.
+essential for obtaining a decent spin polarization.
Future work in this area could involve a four-band model which includes both

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