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[tex] explicit form of \chi_N^\pm; comment on normalization

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1 parent 410d491 commit d388f4a1d4f50016bf1a36b53b4fd16530c171a4 @moritz committed Dec 16, 2009
Showing with 17 additions and 5 deletions.
  1. +13 −1 tex/khodas.tex
  2. +1 −2 tex/summary.tex
  3. +3 −2 tex/thesis.tex
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@@ -76,19 +76,31 @@ \section{Interface Between Normal and Spin-Orbit Coupling Regions}
\begin{align}
\chi_{SO}^{\pm} &= \frac{1}{n_{SO}^{\pm}}
\vect{-p_{x,SO}^{\pm} \pm p_{SO}^\pm}{p_z} \\
- (n_{SO}^{\pm})^2 &= |-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 + p_z^2
+ n_{SO}^{\pm} &= \sqrt{|-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 +
+ p_z^2}
\label{eq:chi-so-pm}
\end{align}
where the lower index $x$ means that the value is projected onto the
$x$ axis. The angle between the $x$ axis and the momentum of the
incident wave is called $\phi$, so that $p_x = p \cos \phi$.
+If one wants to expand $\chi_{SO}^\pm$ in powers of $\ta$, it is
+important to ensure that the normalization condition
+$\chi_{SO}^{\pm\dagger} \cdot \chi_{SO}^\pm$ still holds after the
+expansion. However, the following results have been derived for the
+full (and not expanded) form of $\chi_{SO}^\pm$.
+
Note that, in the N regime, $H$ is a diagonal matrix, and the direction
of the eigenvectors can be chosen with some freedom. We pick
$\chi_N^{\pm} = \lim_{\ta \mapsto 0} \chi_{SO}^{\pm}$ to ensure that
$<\chi_N^+|\chi_{SO}^+> = 1$ holds true at a vanishing interface.
+\begin{align}
+ \chi_N^{\pm} &= \frac{1}{n^{\pm}}
+ \vect{-p_x \pm p}{p_z} \\
+ n^{\pm} &= \sqrt{(-p_x \pm p)^2 + p_z^2}
+\end{align}
The overall wave function consists of an incident wave,
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@@ -19,14 +19,13 @@ \chapter{Summary and Outlook}
in $z$-direction.
We also discussed the experimental more accessible setup of two regions with
-different, non-zero strength of spin-orbit interaction, and found that such an
+different, non-zero strengths of spin-orbit interaction, and found that such an
interface can also be used to achieve some spin polarization, albeit
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 obtaining a decent spin polarization.
-
Future work in this area could involve a four-band model which includes both
the conductance and valance band for each spin direction, would
allow more precises modeling of a particular semiconductor, and thus be of
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@@ -183,11 +183,12 @@ \section{Transmission and Green's Functions}
\end{align}
where $H$ is the Hamiltonian operator. In a one-dimensional wire oriented
-along the $x$ axis, we expect an excitation of the form
+along the $x$ axis with Hamiltonian $H =
+-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$, we expect an excitation of the form
$\delta = \delta(x - x_0)$ to result in two waves propagating away from $x_0$,
so our ansatz is
-TODO: explicit form of $H$
+% TODO: explicit form of $H$
\begin{align}
G(x, x_0) = \begin{cases}

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