# moritz/spinoptics

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[tex] more fixes by Signe++

 @@ -106,22 +106,22 @@ \chapter{Theory} G = \sigma \frac{W}{L} \end{align} -where $\sigma$ is a material specific parameter, and independent of the +where $\sigma$ is a material specific parameter and independent of the geometry of the conductor. \emph{Large enough} means in this context specifically that both lengths are large compared to all of three characteristic lengths: the Fermi wavelength, the mean free path and the phase-relaxation length. The mean free path is the average distance which a charge carrier can travel -before it is scattered (by an impurity, electrons or phonons), and thus loses +before it is scattered (by an impurity, electrons or phonons) and thus loses momentum. If the length of the conductor is smaller than the mean free path, most electrons travel through it without scattering, and one could naively assume that this means the resistance is zero. -Still experiments show that a finite resistance can be observed. That's -because the sample can't be measured in isolation; it is attached to the +Still experiments show that a finite resistance can be observed. That is +because the sample cannot be measured in isolation; it is attached to the macroscopic measuring system through \emph{leads}. Even if the leads are very good conductors themselves, a contact resistance arises. So the theory has to take into account both the sample and the leads. @@ -144,8 +144,8 @@ \subsection*{Landauer Formula} without reflection. This means that the $k_x$ states in the left lead are occupied by electrons -coming from contact 1, and thus have the same electrochemical potential as -the contact, $\mu_1$. Likewise the $-k_x$ states in the right lead have the +coming from contact 1 and thus have the same electrochemical potential as +the contact $\mu_1$. Likewise the $-k_x$ states in the right lead have the potential $\mu_2$. At zero temperature, only electrons with energies between $\mu_1$ and $\mu_2$ @@ -164,14 +164,14 @@ \subsection*{Landauer Formula} G = \frac{I |e|}{\mu_1 - \mu_2} = \frac{2 e^2}{h} MT \end{align} -\subsection*{Transmission and Green's functions} +\subsection*{Transmission and Green's Functions} To calculate the matrix $T$, one can make use of the so-called \emph{Green's functions}. A Green's function $G$ is, loosely speaking, an inverse of a -differential operator $D$. More precisely if the relation between an +differential operator $D$. More precisely, if the relation between an excitation $\delta$ and a response $R$ is $D R = \delta$, then every -operator $G$ for -which the equation $R = G \delta$ holds is called a Green's function. +operator $G$, for +which the equation $R = G \delta$ holds, is called a Green's function. To calculate the wave function $\psi$ in response to an excitation $\delta$ at a given energy $E$ we can use the inhomogeneous Schrödinger Equation