# moritz/spinoptics

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

1 parent 410d491 commit d388f4a1d4f50016bf1a36b53b4fd16530c171a4 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
 @@ -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,
 @@ -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
 @@ -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}