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@ -71,15 +71,6 @@
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}
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\end{formula}
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\begin{formula}{covalent_bond}
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\desc{Covalent bond}{}{}
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\desc[german]{Kolvalente Bindung}{}{}
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\ttxt{
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\eng{Bonds that involve sharing of electrons to form electron pairs between atoms.}
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\ger{Bindungen zwischen Atomen die durch geteilte Elektronen, welche Elektronenpaare bilden, gebildet werden.}
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}
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\end{formula}
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\begin{formula}{grotthuss}
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\desc{Grotthuß-mechanism}{}{}
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\desc[german]{Grotthuß-Mechanismus}{}{}
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@ -71,17 +71,36 @@
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\ger{Boltzmann-Transport}
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]{boltzmann}
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\begin{ttext}
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\eng{Semiclassical description using a probability distribution (\fRef{stat:todo:fermi_dirac}) to describe the particles.}
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\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{stat:todo:fermi_dirac}).}
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\eng{Semiclassical description using a probability distribution (\fRef{cm:sc:fermi_dirac}) to describe the particles.}
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\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{cm:sc:fermi_dirac}).}
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\end{ttext}
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\begin{formula}{boltzmann_transport}
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\desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{stat:todo:fermi-dirac}}
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\desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{cm:sc:fermi_dirac}}
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\desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{}
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\eq{
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\pdv{f(\vec{r},\vec{k},t)}{t} = -\vec{v} \cdot \Grad_{\vec{r}} f - \frac{e}{\hbar}(\vec{\mathcal{E}} + \vec{v} \times \vec{B}) \cdot \Grad_{\vec{k}} f + \left(\pdv{f(\vec{r},\vec{k},t)}{t}\right)_{\text{\GT{scatter}}}
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}
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\end{formula}
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\Subsection[
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\eng{Magneto-transport}
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\ger{Magnetotransport}
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]{mag}
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\begin{formula}{cyclotron_frequency}
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\desc{Cyclotron frequency}{Moving charge carriers move in cyclic orbits under applied magnetic field}{$q$ \qtyRef{charge}, \QtyRef{magnetic_flux_density}, m \qtyRef[effective]{mass}}
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\desc[german]{Zyklotronfrequenz}{Ladungstraäger bewegen sich in einem Magnetfeld auf einer Kreisbahn}{}
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\eq{w_\txc = \frac{qB}{m}}
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\end{formula}
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\TODO{TODO}
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% \begin{formula}{cyclotron_resonance}
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% \desc{}{}{}
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% \desc[german]{}{}{}
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% \eq{}
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% \end{formula}
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\TODO{move hall here}
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\Subsection[
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\eng{misc}
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\ger{misc}
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@ -94,6 +113,12 @@
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}
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\end{formula}
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\begin{formula}{diffusion}
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\desc{Diffusion current}{Equilibration of concentration gradients}{\QtyRef{diffusion_coefficient}, \ConstRef{charge}, $n,p$ \qtyRef{charge_carrier_density}}
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\desc[german]{Diffunsstrom}{Ausgleich von Konzentrationsgradienten}{}
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\eq{\vec{j}_\text{diff} = -\abs{e} D_n \left(-\Grad n\right) + \abs{e} D_p \left(-\Grad p\right)}
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\end{formula}
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\begin{formula}{continuity}
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\desc{Charge continuity equation}{Electric charge can only change by the amount of electric current}{\QtyRef{charge_density}, \QtyRef{current_density}}
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\desc[german]{Kontinuitätsgleichung der Ladung}{Elektrische Ladung kann sich nur durch die Stärke des Stromes ändern}{}
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@ -10,13 +10,84 @@
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\quantity{D}{\per\m^3}{s}
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\eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))}
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\end{formula}
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\begin{formula}{dos_parabolic}
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\desc{Density of states for parabolic dispersion}{Applies to \fRef{cm:egas}}{}
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\desc[german]{Zustandsdichte für parabolische Dispersion}{Bei \fRef{cm:egas}}{}
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\eq{
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D_1(E) &= \frac{1}{2\sqrt{c_k(E-E_0)}} && (\text{1D}) \\
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D_2(E) &= \frac{\pi}{2c_k} && (\text{2D}) \\
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D_3(E) &= \pi \sqrt{\frac{E-E_0}{c_k^3}}&& (\text{3D})
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}
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\Section[
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\eng{Bonds}
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\ger{Bindungen}
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]{bond}
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\begin{formula}{metallic}
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\desc{Metallic bond}{}{}
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\desc[german]{Metallbindung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Delocalized electrons form a cloud
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\item High \qtyRef[electrical]{conductivity} and \qtyRef[thermal]{thermal_conductivity} conductivity
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\item No internal electric field
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item Elektronen delokalisiert und bilden Wolke
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\item Hohe \qtyRef[elektrische]{conductivity} und \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit
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\item Kein internes elektrisches Feld
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\end{itemize}
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}}
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\end{formula}
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\begin{formula}{covalent}
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\desc{Covalent bond}{}{}
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\desc[german]{Kolvalente Bindung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item \fRef{cm:band:hybrid_orbitals} of shared electrons
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\item Highly directional
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\item Varying \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item \fRef{cm:band:hybrid_orbitals} geteilter Elektronen
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\item Richtungsabhängige Bindung
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\item Verschiedene \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeiten
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\end{itemize}
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}}
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\end{formula}
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\begin{formula}{ionic}
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\desc{Ionic bond}{}{}
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\desc[german]{Ionenbindung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Charge transfer from anion to cation
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\item Non.directional bonding
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\item Strong bond
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\item Low \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity
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\item Always in combination with a \fRef{:::covalent}
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item Ladungstransfer von Anion zu Kation
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\item Richtungsunabängig
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\item Starke Bindung
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\item Geringe \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit
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\item Immer in Kombination mit einer \fRef[kovalenten Bindung]{:::covalent}
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\end{itemize}
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}}
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\end{formula}
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\begin{formula}{van-der-waals}
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\desc{Van der Waals bond}{}{}
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\desc[german]{Van-der-Waals Bindung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Dipole-dipole interaction from local charge fluctuations
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\item Weak bond
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item Dipol-Dipol Wechselwirkung durch lokale Ladungsfluktuationen
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\item Schwache Bindung
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\end{itemize}
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}}
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\end{formula}
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@ -6,8 +6,6 @@
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\eng{Bravais lattice}
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\ger{Bravais-Gitter}
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]{bravais}
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\Eng[lattice_system]{Lattice system}
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\Ger[lattice_system]{Gittersystem}
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\Eng[crystal_family]{Crystal system}
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\ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}}
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\end{formula}
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\Eng[miller-point]{Point}
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\Ger[miller-point]{Punkt}
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\Eng[miller-direction]{Direction}
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\Ger[miller-direction]{Richtung}
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\Eng[miller-direction-family]{Family of directions}
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\Ger[miller-direction-family]{Familie von Richtungen}
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\Eng[miller-plane]{Plane}
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\Ger[miller-plane]{Ebene}
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\Eng[miller-plane-family]{Family of planes}
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\Ger[miller-plane-family]{Familie von Ebenen}
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\begin{formula}{miller}
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\desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry}
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\desc[german]{Millersche Indizes}{}{}
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\eq{
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(hkl) & \text{\GT{plane}}\\
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[hkl] & \text{\GT{direction}}\\
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\{hkl\} & \text{\GT{millerFamily}}
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\desc{Miller indices}{}{
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Miller planes: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ give intersection with $x$/$y$/$z$ axes\\
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Miller family: planes that are equivalent due to crystal symmetry
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}
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\desc[german]{Millersche Indizes}{}{
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Miller-Ebenen: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ geben die Schnittpunkte mit den $x$/$y$/$z$-Achsen\\
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Miller-Familien: Ebenen, die durch Kristallsymmetrie äquivalent sind
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}
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\centering
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\newFormulaEntry
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\begin{tabularx}{\textwidth}{clcl}
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$(h,k,l)$ & \GT{miller-point} & & \\
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$hkl$ & \GT{miller-direction} & $\langle hkl \rangle$ & \GT{miller-direction-family} \\
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$(hkl)$ & \GT{miller-plane} & $\{hkl\}$ & \GT{miller-plane-family}
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\end{tabularx}
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\pgfmathsetmacro{\rectX}{2}
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\pgfmathsetmacro{\rectZ}{2}
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\newFormulaEntry
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\begin{tikzpicture}[3d view={100}{20},perspective={p={(-55,0,0)},q={(0,25,0)},r={(0,0,-30)}}]
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% <100> direction family
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\begin{scope}
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\drawRectCS{1.4*\rectX}{1.4*\rectZ}
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\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
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\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
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\drawRectBack{R1}
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\drawRectConnectionsBack{R1}{R2}
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\draw[miller dir] (0,0,0) -- ++( \rectX,0,0) node[anchor=east] {$[100]$};
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\draw[miller dir] (0,0,0) -- ++(-\rectX,0,0) node[anchor=west] {$[\bar{1}00]$};
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\draw[miller dir] (0,0,0) -- ++(0, \rectX,0) node[anchor=south] {$[010]$};
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\draw[miller dir] (0,0,0) -- ++(0,-\rectX,0) node[anchor=south] {$[0\bar{1}0]$};
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\draw[miller dir] (0,0,0) -- ++(0,0, \rectX) node[anchor=east] {$[001]$};
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\draw[miller dir] (0,0,0) -- ++(0,0,-\rectX) node[anchor=west] {$[00\bar{1}]$};
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\drawRectFront{R1}
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\drawRectBack{R2}
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\drawRectConnectionsFront{R1}{R2}
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\drawRectFront{R2}
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\node at (1.5*\rectX,1.5*\rectX, 0) {$\langle100\rangle$};
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\end{scope}
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\pgfmathsetmacro{\rectDistance}{4.5}
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% {100} plane family
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\begin{scope}[shift={(0,\rectDistance,0)}]
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\drawRectCS{1.4*\rectX}{1.4*\rectZ}
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\setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX}
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\setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX}
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\drawRectBack{R1}
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\drawRectConnectionsBack{R1}{R2}
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\drawRectFront{R1}
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\drawRectBack{R2}
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\drawRectConnectionsFront{R1}{R2}
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\drawRectFront{R2}
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\fill[miller plane] (R1-C) -- (R1-D) node[anchor=north,midway] {$(100)$} -- (R2-D) -- (R2-C) -- cycle;
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\fill[miller plane] (R1-A) -- (R1-D) node[anchor=west,midway] {$(010)$} -- (R2-D) -- (R2-A) -- cycle node[anchor=north east] {$(010)$};
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\fill[miller plane] (R2-A) -- (R2-B) node[midway,anchor=south] {$(001)$} -- (R2-C) -- (R2-D) -- cycle;
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\node at (1.5*\rectX,1.5*\rectX, 0) {$\{100\}$};
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\end{scope}
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\end{tikzpicture}
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% describe how to construct miller planes
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\end{formula}
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\begin{formula}{miller-hexagon}
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\desc{Hexagonal miller indices}{}{}
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\desc[german]{Hexagonale Millersche Indizes}{}{}
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\eq{ (hkil) && \tGT{with}\quad i = h + k }
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\centering
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\newFormulaEntry
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\begin{tikzpicture}[3d view={0}{20}]
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\pgfmathsetmacro{\hexxY}{1.5}
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\begin{scope}
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\drawHexagonCS{1}{\hexxY}
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\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
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\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
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\drawHexagonBack{H1}
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\drawHexagonConnectionsBack{H1}{H2}
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\drawHexagonFront{H1}
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\drawHexagonBack{H2}
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\drawHexagonConnectionsFront{H1}{H2}
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\drawHexagonFront{H2}
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\end{scope}
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\pgfmathsetmacro{\hexDistance}{3.5}
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% 1121
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\begin{scope}[shift={(\hexDistance,0,0)}]
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\drawHexagonCS{1}{\hexxY}
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\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
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\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
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\drawHexagonBack{H1}
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\drawHexagonConnectionsBack{H1}{H2}
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\fill[miller plane] (H1-A) -- (H2-M) -- (H1-E) -- cycle;
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\drawHexagonFront{H1}
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\drawHexagonBack{H2}
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\drawHexagonConnectionsFront{H1}{H2}
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\drawHexagonFront{H2}
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\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1211)$};
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\end{scope}
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% 1010
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\begin{scope}[shift={(2*\hexDistance,0,0)}]
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\drawHexagonCS{1}{\hexxY}
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\setHexagonPoints{H1}{(0,0,0)}{1}{1}{1}
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\setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1}
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\drawHexagonBack{H1}
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\drawHexagonConnectionsBack{H1}{H2}
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\drawHexagonFront{H1}
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\drawHexagonBack{H2}
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\drawHexagonConnectionsFront{H1}{H2}
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\drawHexagonFront{H2}
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\fill[miller plane] (H1-F) -- (H2-F) -- (H2-E) -- (H1-E) -- cycle;
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\node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1010)$};
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\end{scope}
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\end{tikzpicture}
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\end{formula}
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@ -200,4 +309,3 @@
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}
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\end{formula}
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@ -2,10 +2,14 @@
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\eng{Free electron gas}
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\ger{Freies Elektronengase}
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]{egas}
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\begin{ttext}
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\begin{formula}{desc}
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\desc{Description}{\GT{see_also}: \fRef{td:id_qgas}}{}
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\desc[german]{Beschreibung}{}{}
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\ttxt{
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\eng{Assumptions: electrons can move freely and independent of each other.}
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\ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.}
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\end{ttext}
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}
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\end{formula}
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\begin{formula}{drift_velocity}
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\desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity}
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@ -14,6 +18,7 @@
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\end{formula}
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\begin{formula}{mean_free_path}
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\abbrLabel{mfp}
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\desc{Mean free path}{}{}
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\desc[german]{Mittlere freie Weglänge}{}{}
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\eq{\ell = \braket{v} \tau}
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@ -26,11 +31,20 @@
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\eq{\mu = \frac{q \tau}{m}}
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\end{formula}
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\Subsection[
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\eng{3D electron gas}
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\ger{3D Elektronengas}
|
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]{3deg}
|
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\begin{formula}{dos}
|
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\desc{Density of states}{}{}
|
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\desc[german]{Zustandsdichte}{}{}
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\eq{D_\text{3D}(E) = \frac{1}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}}
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\end{formula}
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\Subsection[
|
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\eng{2D electron gas}
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\ger{2D Elektronengas}
|
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]{2deg}
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\begin{ttext}
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\eng{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.}
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\ger{
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@ -51,6 +65,12 @@
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\eq{E_n = \underbrace{\frac{\hbar^2 k_\parallel^2}{2\masse}}_\text{$x$-$y$: \GT{plain_wave}} + \underbrace{\frac{\hbar^2 \pi^2}{2\masse L^2} n^2}_\text{$z$}}
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\end{formula}
|
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\begin{formula}{dos}
|
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\desc{Density of states}{}{}
|
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\desc[german]{Zustandsdichte}{}{}
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\eq{D_\text{2D}(E) = \frac{m}{\pi\hbar^2}}
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\end{formula}
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\Subsection[
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\eng{1D electron gas / quantum wire}
|
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\ger{1D Eleltronengas / Quantendraht}
|
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@ -61,12 +81,25 @@
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\desc[german]{Energie}{}{}
|
||||
\eq{E_n = \frac{\hbar^2 k_x^2}{2\masse} + \frac{\hbar^2 \pi^2}{2\masse L_z^2} n_1^2 + \frac{\hbar^2 \pi^2}{2\masse L_y^2} n_2^2}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{1D}(E) = \frac{1}{\pi\hbar} \sqrt{\frac{m}{2}} \frac{1}{\sqrt{E}}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{condunctance}
|
||||
|
||||
\Subsection[
|
||||
\eng{0D electron gas / quantum dot}
|
||||
\ger{0D Elektronengase / Quantenpunkt}
|
||||
]{0deg}
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{}
|
||||
\desc[german]{Zustandsdichte}{}{}
|
||||
\eq{D_\text{0D}(E) = 2\delta(E-E_C)}
|
||||
\end{formula}
|
||||
|
||||
\TODO{TODO}
|
||||
|
||||
|
||||
|
||||
@ -12,29 +12,41 @@
|
||||
\end{ttext}
|
||||
|
||||
% chemmacros package
|
||||
\begin{formula}{sp}
|
||||
\desc{sp Orbital}{\GT{eg} \ce{C2H2}}{}
|
||||
\desc[german]{sp Orbital}{}{}
|
||||
\ttxt{\eng{Linear with bond angle \SI{180}{\degree}}\ger{Linear mit Bindungswinkel \SI{180}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 1\text{p} = \text{sp}
|
||||
\orbital{sp}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sp2}
|
||||
\desc{sp2 Orbital}{\GT{eg} \ce{C2H4}}{}
|
||||
\desc[german]{sp2 Orbital}{}{}
|
||||
\ttxt{\eng{Trigonal planar with bond angle \SI{120}{\degree}}\ger{Trigonal planar mit Bindungswinkel \SI{120}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 2\text{p} = \text{sp2}
|
||||
\orbital{sp2}
|
||||
% \\ \ket{p} = \cos\theta \ket{p_x} + \sin\theta \ket{p_y}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{sp3}
|
||||
\desc{sp3 Orbital}{\GT{eg} \ce{CH4}}{}
|
||||
\desc[german]{sp3 Orbital}{}{}
|
||||
\ttxt{\eng{Tetrahedral with bond angle \SI{109.5}{\degree}}\ger{Tetraedisch mit Bindungswinkel \SI{109.5}{\degree}}}
|
||||
\eq{
|
||||
1\text{s} + 3\text{p} = \text{sp3}
|
||||
\orbital{sp3}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{sp2}
|
||||
\desc{sp2 Orbital}{}{}
|
||||
\desc[german]{sp2 Orbital}{}{}
|
||||
\eq{
|
||||
1\text{s} + 2\text{p} = \text{sp2}
|
||||
\orbital{sp2}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{sp}
|
||||
\desc{sp Orbital}{}{}
|
||||
\desc[german]{sp Orbital}{}{}
|
||||
\eq{
|
||||
1\text{s} + 1\text{p} = \text{sp}
|
||||
\orbital{sp}
|
||||
}
|
||||
|
||||
\begin{formula}{wave_function}
|
||||
\desc{Wave function}{of a hybrid orbital}{$N$ number of involved $p$ orbitals}
|
||||
\desc[german]{Wellenfunktion}{eines Hybridorbitals}{$N$ Anzahl der beteiligten $p$ Orbitale}
|
||||
\eq{\ket{h_{1\dots N+1}} = \frac{1}{\sqrt{N+1}} \left(\ket{s} + \sqrt{N} \ket{p}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
@ -117,3 +129,18 @@
|
||||
\eng{\textit{Gain medium} is energized \textit{pumping energy} (electric current or light), light of certain wavelength is amplified in the gain medium}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum ranges}{}{}
|
||||
\desc[german]{Vakuumklassen}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{Rough vacuum}: \SI{1}{\atm} - \SI{10e-2}{\milli\bar} \\ viscous flow
|
||||
\item \textbf{Process vacuum}: \SI{10e-2}{\milli\bar} - \SI{10e-4}{\milli\bar} \\ \abbrRef{mfp} $\le$ chamber size
|
||||
\item \textbf{High vacuum}: \SI{10e-5}{\milli\bar} - \SI{10e-9}{\milli\bar} \\ \abbrRef{mfp} $>$ chamber size, mostly residual \ce{H20} vapor
|
||||
\item \textbf{Ultra-high vacuum}: $<$ \SI{10e-9}{\milli\bar} \\ \abbrRef{mfp} $\gg$ chamber size, mostly residual \ce{H2}
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\TODO{translate}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -1,44 +1,97 @@
|
||||
\def\meff{m^{*}}
|
||||
\Section[
|
||||
\eng{Semiconductors}
|
||||
\ger{Halbleiter}
|
||||
]{sc}
|
||||
\begin{formula}{types}
|
||||
\desc{Intrinsic/extrinsic}{}{$n,p$ \fRef{cm:sc:charge_density_eq}}
|
||||
\desc[german]{Intrinsisch/Extrinsisch}{}{}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{$n,p$ \fRef{cm:sc:charge_carrier_density:equilibrium}}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
Intrinsic: pure, electron density determiend only by thermal excitation and $n_i^2 = n_0 p_0$\\
|
||||
Extrinsic: doped
|
||||
Materials with an electrical conductivity that can be modified through \fRef[doping]{::doping}.\\
|
||||
\textbf{Intrinsic}: pure, electron density determined only by thermal excitation and $n_i^2 = n_0 p_0$\\
|
||||
\textbf{Extrinsic}: doped
|
||||
}
|
||||
\ger{
|
||||
Intrirnsisch: Pur, Elektronendichte gegeben durch thermische Anregung und $n_i^2 = n_0 p_0$ \\
|
||||
Extrinsisch: gedoped
|
||||
Materialien, bei denen die elektrische Leitfähigkeit durch \fRef[Dotierung]{::doping} verändert werden kann.\\
|
||||
\textbf{Intrinsisch}: Pur, Elektronendichte gegeben durch thermische Anregung und $n_i^2 = n_0 p_0$ \\
|
||||
\textbf{Extrinsisch}: dotiert
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge_density_eq}
|
||||
\desc{Equilibrium charge densitites}{Holds when $\frac{\Econd-\EFermi}{\kB T}>3.6$ and $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\desc[german]{Ladungsträgerdichte im Equilibrium}{Gilt wenn $\frac{\Econd-\EFermi}{\kB T}>3.6$ und $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\begin{formula}{fermi_dirac}
|
||||
\desc{Fermi-Dirac distribution}{For electrons and holes}{}
|
||||
\desc[german]{Fermi-Dirac Verteilung}{Für Elektronen und Löcher}{}
|
||||
\eq{
|
||||
n_0 &\approx N_\text{c}(T) \Exp{-\frac{E_\text{c} - \EFermi}{\kB T}} \\
|
||||
p_0 &\approx N_\text{v}(T) \Exp{-\frac{\EFermi - E_\text{v}}{\kB T}}
|
||||
f_\txe(E) &= \frac{1}{\Exp{\frac{E-\EFermi}{\kB T}+1}}\\
|
||||
f_\txh(E) &= 1-f_\txe(E)
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{charge_density_intrinsic}
|
||||
\desc{Intrinsic charge density}{}{}
|
||||
|
||||
|
||||
\begin{formulagroup}{charge_carrier_density}
|
||||
\desc{Charge carrier density}{}{}
|
||||
\desc[german]{Ladungsträgerichte}{}{}
|
||||
\begin{formula}{general}
|
||||
\desc{Charge carrier density}{General form}{$D$ \qtyRef{dos}, $f$ \fRef{:::fermi_dirac}}
|
||||
\desc[german]{Ladungsträgerdichte}{Allgemeine Form}{}
|
||||
\eq{
|
||||
n &= \int_{\Econd}^\infty D_\txe f_\txe(E)\d E\\
|
||||
p &= \int_{-\infty}^{\Evalence} D_\txh f_\txh(E)\d E
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{equilibrium}
|
||||
\desc{Equilibrium charge carrier densities}{\fRef{math:cal:integral:list:boltzmann_approximation}, holds when $\frac{\Econd-\EFermi}{\kB T}>3.6$ and $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\desc[german]{Ladungsträgerdichte im Equilibrium}{\fRef{math:cal:integral:list:boltzmann_approximation}, gilt wenn $\frac{\Econd-\EFermi}{\kB T}>3.6$ und $\frac{\EFermi-\Evalence}{\kB T} > 3.6$}{}
|
||||
\eq{
|
||||
n_0 &\approx N_\txC(T) \Exp{-\frac{\Econd - \EFermi}{\kB T}} \\
|
||||
p_0 &\approx N_\txV(T) \Exp{-\frac{\EFermi - \Evalence}{\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{intrinsic}
|
||||
\desc{Intrinsic charge carrier density}{}{$N$ \fRef{:::band_edge_dos}}
|
||||
\desc[german]{Intrinsische Ladungsträgerdichte}{}{}
|
||||
\eq{
|
||||
n_\text{i} \approx \sqrt{n_0 p_0} = \sqrt{N_\text{c}(T) N_\text{v}(T)} \Exp{-\frac{E_\text{gap}}{2\kB T}}
|
||||
n_\txi \approx \sqrt{n_0 p_0} = \sqrt{N_\txC(T) N_\txV(T)} \Exp{-\frac{E_\text{gap}}{2\kB T}}
|
||||
}
|
||||
\end{formula}
|
||||
\end{formulagroup}
|
||||
|
||||
\begin{formula}{band_edge_dos}
|
||||
\desc{Band edge density of states}{}{$\meff$ \qtyRef{effective_mass}, \ConstRef{boltzmann}, \QtyRef{temperature}, \ConstRef{planck2pi}}
|
||||
\desc[german]{Bandkanten-Zustandsdichte}{}{}
|
||||
\eq{
|
||||
N_\txC &= 2\left(\frac{\meff_\txe\kB T}{2\pi\hbar^2}\right)^{3/2} \\
|
||||
N_\txV &= 2\left(\frac{\meff_\txh\kB T}{2\pi\hbar^2}\right)^{3/2}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{mass_action}
|
||||
\desc{Mass action law}{Charge densities at thermal equilibrium, independent of doping}{}
|
||||
\desc{Mass action law}{Charge densities at thermal equilibrium, independent of doping}{$n_0/p_0$ \fRef{::charge_carrier_density:equilibrium}, $n_i/p_i$ \fRef{::charge_carrier_density:intrinsic}}
|
||||
\desc[german]{Massenwirkungsgesetz}{Ladungsträgerdichten im Equilibrium, unabhängig der Dotierung }{}
|
||||
\eq{np = n_i^2}
|
||||
\eq{n_0p_0 = n_i^2 = p_i^2 \text{\TODO{check if ni=pi}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bandgap}
|
||||
\desc{Bandgap}{}{}
|
||||
\desc[german]{Bandlücke}{}{}
|
||||
\ttxt{\eng{
|
||||
Energy gap between highest occupied (HO) and lowest unoccupied (LU) band/orbital\\
|
||||
\begin{itemize}
|
||||
\item \textbf{direct}: HO and LU at same $\veck$
|
||||
\item \textbf{indirect} HO and LU at different $\veck$
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Energielücke zwischen höchstem besetztem (HO) und niedrigsten unbesetzten (LU) Band/Orbital
|
||||
\begin{itemize}
|
||||
\item \textbf{direkt}: HO und LU bei gleichem $\veck$
|
||||
\item \textbf{indirekt}: HO und LU bei unterschiedlichem $\veck$
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bandgaps}
|
||||
\desc{Bandgaps of common semiconductors}{}{}
|
||||
@ -71,8 +124,184 @@
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{effective_mass}
|
||||
\desc{Effective mass}{}{}
|
||||
\desc[german]{Effektive Masse}{}{}
|
||||
\quantity{\ten{\meff}}{\kg}{t}
|
||||
\eq{\left(\frac{1}{\meff}\right)_{ij} = \frac{1}{\hbar^2} \pdv{E}{k_i,k_j}}
|
||||
\ttxt{\eng{
|
||||
Approximate effects using a effective mass. \TODO{more detail}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
\TODO{effective mass approx}
|
||||
|
||||
\Subsection[
|
||||
\eng{Doping}
|
||||
\ger{Dotierung}
|
||||
]{dope}
|
||||
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
Modification of charger carrier densities through defects.
|
||||
\begin{itemize}
|
||||
\item $N_\txA \gg N_\txD$ \Rightarrow p-type semiconductor
|
||||
\item $N_\txA \ll N_\txD$ \Rightarrow n-type semiconductor
|
||||
\item Else: compensated semiconductor, acceptors filled by electrons from donors:
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Modifizierung der Ladungsträgerichten durch Einbringung von Fremdatomen.
|
||||
\begin{itemize}
|
||||
\item $N_\txA \gg N_\txD$ \Rightarrow p-Typ Halbleiter
|
||||
\item $N_\txA \ll N_\txD$ \Rightarrow n-Typ Halbleiter
|
||||
\item Sonst: Kompensierter Halbleiter, Akzeptoren nehmen Elektronen der Donatoren auf
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{charge_neutrality}
|
||||
\desc{Charge neutrality}{Fermi level must adjust so that charge neutrality is preserved}{$N_{\txd/\txa}^{+/-}$ ionized donor/acceptor density, $n,p$ \fRef{cm:sc:charge_carrier_density}}
|
||||
\desc[german]{Ladungsneutralität}{Fermi-Level muss sich so anpassen, dass Ladungsneutralität erhalten ist}{$N_{\txd/\txa}^{+/-}$ Dichte der ionisierten Donatoren/Akzeptoren , $n,p$ \fRef{cm:sc:charge_carrier_density}}
|
||||
\eq{0 = N_\txd^+ + p - N_\txa^- -n}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ionization_ratio}
|
||||
\desc{Fraction ionized donors/acceptors}{At thermal equilibrium}{$N_{\txd/\txa}^{+/-}$ ionized donor/acceptor density, $N_{\txd/\txa}$ donor/acceptor density, $E_{\txd/\txa}$ donor/acceptor energy level, $g$ spin degeneracy}
|
||||
\desc[german]{Anteil ionisiserter Akzeptoren/Donatoren}{Im thermischen Equilibrium}{$N_{\txd/\txa}^{+/-}$ ionisierte Donor/Akzeptordichte, $N_{\txd/\txa}$ Donor/Akzeptordichte, $E_{\txd/\txa}$ Energie der Donatoren/Akzeptoren, $g$ Spindegenierung}
|
||||
\eq{
|
||||
\frac{N_\txd^+}{N_\txd} &= 1- \frac{1}{1+\frac{1}{g}\Exp{\frac{E_\txD-\Efermi}{\kB T}}} \\
|
||||
\frac{N_\txa^-}{N_\txa} &= \frac{1}{1+g\Exp{\frac{E_\txA-\Efermi}{\kB T}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electron_density}
|
||||
\desc{Charge carrier density}{In a doped semiconductor}{}
|
||||
\desc[german]{Ladungsträgeridchte}{In einem dotierten Halbleiter}{}
|
||||
\fig[width=0.5\textwidth]{img_static/cm_sc_doped_TODO.png}
|
||||
\TODO{plot}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Defects}
|
||||
\ger{Defekte}
|
||||
]{defect}
|
||||
\Subsubsection[
|
||||
\eng{Point defects}
|
||||
\ger{Punktdefekte}
|
||||
]{point}
|
||||
\begin{formula}{vacancy}
|
||||
\desc{Vacancy}{}{}
|
||||
\desc[german]{Fehlstelle}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Lattice site missing an atom
|
||||
\item Low formation energy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Unbesetzter Gitterpunkt
|
||||
\item Geringe Formationsenergie
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{interstitial}
|
||||
\desc{Interstitial}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Extranous atom between lattice atoms
|
||||
\item High formation energy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Zusätzliches Atom zwischen Gitteratomen
|
||||
\item Hohe Formationsenergy
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{schottky}
|
||||
\desc{Schottky defect}{}{}
|
||||
\desc[german]{Schottky-Defekt}{}{}
|
||||
\ttxt{\eng{
|
||||
Atom type A \fRef{:::vacancy} + atom type B \fRef{:::vacancy}.
|
||||
Only in (partially) ionic materials.
|
||||
}\ger{
|
||||
\fRef{:::vacancy} von Atomsorte A und \fRef{:::vacancy} von Atomsorte B.
|
||||
Tritt nur in ionischen Materialiern auf.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{frenkel}
|
||||
\desc{Frenkel defect}{}{}
|
||||
\desc[german]{Frenkel Defekt}{}{}
|
||||
\ttxt{\eng{
|
||||
\fRef{:::vacancy} + \fRef{:::interstitial}
|
||||
}\ger{
|
||||
\fRef{:::vacancy} + \fRef{:::interstitial}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Line defects}
|
||||
\ger{Liniendefekte}
|
||||
]{line}
|
||||
\begin{formula}{edge}
|
||||
\desc{Edge distortion}{}{}
|
||||
\desc[german]{Stufenversetzung}{}{}
|
||||
\ttxt{\eng{
|
||||
Insertion of an extra plane of atoms
|
||||
}\ger{
|
||||
Einschiebung einer zusätzliche Atomebene
|
||||
}}
|
||||
\TODO{images}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{screw}
|
||||
\desc{Screw distortion}{}{}
|
||||
\desc[german]{Schraubenversetzung}{}{}
|
||||
\ttxt{\eng{
|
||||
\TODO{TODO}
|
||||
}\ger{
|
||||
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{burgers_vector}
|
||||
\desc{Burgers vector}{Magnitude and direction of dislocation}{}
|
||||
\desc[german]{Burgers-Vektor}{Größe und Richtung einer Versetzung}{}
|
||||
\quantity{\vecb}{units}{ievs}
|
||||
\eq{
|
||||
\TODO{TODO}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Area defects}
|
||||
\ger{Flächendefekte}
|
||||
]{area}
|
||||
\begin{formula}{grain_boundary}
|
||||
\desc{Grain boundary}{}{}
|
||||
\desc[german]{Korngrenze}{}{}
|
||||
\ttxt{\eng{
|
||||
Lead to
|
||||
\begin{itemize}
|
||||
\item Secondary phases
|
||||
\item Charge carrier trapping, recombination
|
||||
\item High mass diffusion constants
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
Führen zu
|
||||
\begin{itemize}
|
||||
\item Sekundärphasen
|
||||
\item Separierung, Trapping und Streuung von Ladunsträgern
|
||||
\item Hohe Massendiffusionskonstante
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Devices and junctions}
|
||||
@ -92,8 +321,8 @@
|
||||
\desc{Schottky barrier}{Rectifying \fRef{cm:sc:junctions:metal-sc}}{}
|
||||
% \desc[german]{}{}{}
|
||||
\centering
|
||||
\resizebox{0.49\textwidth}{!}{\input{img/cm/sc_junction_metal_n_sc_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img/cm/sc_junction_metal_n_sc.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_metal_n_sc_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_metal_n_sc.tex}}
|
||||
\TODO{Work function electron affinity sind doch Energien und keine Potentiale, warum wird also immer $q$ davor geschrieben?}
|
||||
\end{bigformula}
|
||||
\begin{formula}{schottky-mott_rule}
|
||||
@ -106,15 +335,15 @@
|
||||
\desc{Ohmic contact}{}{}
|
||||
\desc[german]{Ohmscher Kontakt}{}{}
|
||||
\centering
|
||||
\resizebox{0.49\textwidth}{!}{\input{img/cm/sc_junction_ohmic_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img/cm/sc_junction_ohmic.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_ohmic_separate.tex}}
|
||||
\resizebox{0.49\textwidth}{!}{\input{img_static/cm/sc_junction_ohmic.tex}}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{pn}
|
||||
\desc{p-n junction}{}{}
|
||||
\desc[german]{p-n Übergang}{}{}
|
||||
\centering
|
||||
\input{img/cm/sc_junction_pn.tex}
|
||||
\input{img_static/cm/sc_junction_pn.tex}
|
||||
\resizebox{0.49\textwidth}{!}{\tikzPnJunction{1/3}{0}{0}{1/3}{0}{0}{}}
|
||||
\resizebox{0.49\textwidth}{!}{\tikzPnJunction{1/2}{0.4}{-0.4}{1/2}{-0.4}{0.4}{}}
|
||||
\end{bigformula}
|
||||
|
||||
@ -128,7 +128,6 @@
|
||||
\desc[german]{Erste London-Gleichun-}{}{}
|
||||
\eq{
|
||||
\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
|
||||
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
@ -415,7 +414,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gap_at_t0}
|
||||
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, \TODO{gamma}}
|
||||
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, $\gamma$ Sommerfeld constant}
|
||||
\desc[german]{BCS Lücke bei $T=0$}{}{}
|
||||
\eq{
|
||||
\Delta(T=0) &= \frac{\hbar\omega_\txD}{\Sinh{\frac{2}{V_0\.D(E_\txF)}}} \approx 2\hbar \omega_\txD\\
|
||||
|
||||
@ -1,15 +1,19 @@
|
||||
\Section[
|
||||
\eng{Techniques}
|
||||
\ger{Techniken}
|
||||
]{tech}
|
||||
|
||||
\Subsection[
|
||||
\eng{Measurement techniques}
|
||||
\ger{Messtechniken}
|
||||
]{meas}
|
||||
|
||||
\newcommand\newTechnique{\hline}
|
||||
\Eng[name]{Name}
|
||||
\Ger[name]{Name}
|
||||
\Eng[application]{Application}
|
||||
\Ger[application]{Anwendung}
|
||||
|
||||
\Subsection[
|
||||
\Subsubsection[
|
||||
\eng{Raman spectroscopy}
|
||||
\ger{Raman Spektroskopie}
|
||||
]{raman}
|
||||
@ -62,7 +66,7 @@
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\Subsubsection[
|
||||
\eng{ARPES}
|
||||
\ger{ARPES}
|
||||
]{arpes}
|
||||
@ -71,7 +75,7 @@
|
||||
how?
|
||||
plot
|
||||
|
||||
\Subsection[
|
||||
\Subsubsection[
|
||||
\eng{Scanning probe microscopy SPM}
|
||||
\ger{Rastersondenmikroskopie (SPM)}
|
||||
]{spm}
|
||||
@ -128,7 +132,7 @@
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
\Section[
|
||||
\Subsection[
|
||||
\eng{Fabrication techniques}
|
||||
\ger{Herstellungsmethoden}
|
||||
]{fab}
|
||||
@ -173,7 +177,7 @@
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\Subsubsection[
|
||||
\eng{Epitaxy}
|
||||
\ger{Epitaxie}
|
||||
]{epitaxy}
|
||||
@ -206,7 +210,7 @@
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/cm_mbe_english.pdf}
|
||||
\includegraphics[width=\textwidth]{img_static/cm_mbe_english.pdf}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
@ -39,6 +39,13 @@
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{petit-dulong}
|
||||
\absLabel
|
||||
\desc{Petit-Dulong law}{Empirical heat capacity at high temperatures}{$C_\txm$ molar \qtyRef{heat_capacity}, \ConstRef{avogadro}, \ConstRef{boltzmann}, \ConstRef{gas}}
|
||||
\desc[german]{Petit-Dulong Gesetz}{Empirische Wärmekapazität bei hohen Temperaturen}{}
|
||||
\eq{C_\txm = 3\NA \kB = 3R \approx \SI{25}{\joule\per\mol\kelvin}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Einstein model}
|
||||
\ger{Einstein-Modell}
|
||||
|
||||
117
src/comp/est.tex
117
src/comp/est.tex
@ -249,123 +249,6 @@
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{bigformula}{comparison}
|
||||
\desc{Comparison of DFT functionals}{}{}
|
||||
\desc[german]{Vergleich von DFT Funktionalen}{}{}
|
||||
% \begin{tabular}{l|c}
|
||||
% \fRef[Hartree-Fock]{comp:est:dft:hf:potential} & only exchange, no correlation \Rightarrow upper bound of GS energy \\
|
||||
% \abbrRef{lda} & understimates e repulsion \Rightarrow Overbinding \\
|
||||
% \abbrRef{gga} & underestimate band gap \\
|
||||
% hybrid & underestimate band gap
|
||||
% \end{tabular}
|
||||
|
||||
\TODO{HFtotal energy: upper boundary for GS density $n$}
|
||||
|
||||
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
|
||||
% \begin{tabular}{|P{0.15\textwidth}|P{0.2\textwidth}|P{0.1\textwidth}|P{0.2\textwidth}|P{0.1\textwidth}|P{0.1\textwidth}|P{0.15\textwidth}|}
|
||||
% \hline
|
||||
% \textbf{Method} & \textbf{Description} & \textbf{Mean Absolute Error (eV)} & \textbf{Band Gap Accuracy} & \textbf{Computational Cost} & \textbf{Usage} & \textbf{Other Notes} \\
|
||||
% \hline
|
||||
% Hartree-Fock (HF) &
|
||||
% $E_C \sim E_C^{HF\text{theory}}$
|
||||
% $E_X \sim E_X^{FOCK}$
|
||||
% & 3.1 (Underbinding) & \tabitem no SIE \tabitem correct long-range behaviour \tabitem nonlinear chemical potential (missing DD) \tabitem positive correlation effects & High & Reference for exact exchange, useful for small molecules. & Self-interaction free, but lacks correlation. \\
|
||||
% \hline
|
||||
% Local Density Approximation (LDA) &
|
||||
% $E_x \sim n(r)$
|
||||
% $E_c \sim n(r)$
|
||||
% & 1.3 (Overbinding) & \tabitem SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Low & Basic solids and metallic systems, where accuracy is not critical. & Simple and computationally cheap. \\
|
||||
% \hline
|
||||
% Generalised Gradient Approximation (GGA) &
|
||||
% $E_x \sim n(r), \nabla n(r)$
|
||||
% $E_c \sim n(r), \nabla n(r)$
|
||||
% & 0.3 (Mostly overbinding) & \tabitem SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Moderate & More accurate for molecules and chemical bonding studies. & Better than LDA for chemical bonding. \\
|
||||
% \hline
|
||||
% Hybrid Functionals &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_x = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
||||
% \tabitem Add expensive non-local Fock term to reduce self-interaction
|
||||
% & Lower than GGA (Improved balance) & \tabitem reduced SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Higher & Molecular chemistry, solid-state physics requiring better accuracy. & Balances accuracy and cost. \\
|
||||
% \hline
|
||||
% Range-Separated Hybrid (RSH) &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_{X,SR} = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
||||
% $E_{X,LR} = E_x^{GGA}$
|
||||
% \tabitem Mix-in expensive Fock term only for short-range interactions $\rightarrow$ since for LR the Coulomb interaction gets screening in dielectric substances ($\epsilon > 1$), such as crystalline materials.
|
||||
% & Lower than Hybrid (Even better balance) & \tabitem reduced SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Very High & Semiconductors, materials with screened Coulomb interactions. & Used for dielectric materials. \\
|
||||
% \hline
|
||||
% Optimally Tuned RSH (OT-RSH) &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_{X,SR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
||||
% $E_{X,LR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
||||
% \tabitem More advanced tuning between Fock and GGA. So that set also have the correct asymptotic behaviour of $1/r$ (Coulomb e.g. Fock) instead of $e^{-r}$ from GGA
|
||||
% & Lowest & \tabitem reduced SIE \tabitem better long-range behaviour \tabitem /+ better chemical potential - they include non-multiplicative, orbital dependent terms. Hence, in principle they allow for including a DD. & Extremely High & Precise calculations for band gap predictions and electronic properties. & Most flexible but computationally expensive. \\
|
||||
% \hline
|
||||
% \end{tabular}
|
||||
|
||||
% \begin{tabularx}{\textwidth}{lXlllll}
|
||||
% \toprule
|
||||
% \textbf{Method} & \textbf{Description} & \textbf{Mean Absolute Error (eV)} & \textbf{Band Gap Accuracy} & \textbf{Computational Cost} & \textbf{Usage} & \textbf{Other Notes} \\
|
||||
% \midrule
|
||||
% Hartree-Fock (HF) & $E_C \sim E_C^{HF\text{theory}}$ $E_X \sim E_X^{FOCK}$ & 3.1 (Underbinding) & Overestimates
|
||||
% \tabitem no SIE
|
||||
% \tabitem correct long-range behaviour
|
||||
% \tabitem nonlinear chemical potential (missing DD)
|
||||
% \tabitem positive correlation effects
|
||||
% & High & Reference for exact exchange, useful for small molecules. & Self-interaction free, but lacks correlation. \\
|
||||
% \midrule
|
||||
% Local Density Approximation (LDA) &
|
||||
% $E_x \sim n(r)$
|
||||
% $E_c \sim n(r)$
|
||||
% & 1.3 (Overbinding) & Underestimates
|
||||
% \tabitem SIE
|
||||
% \tabitem wrong long-range behaviour
|
||||
% \tabitem nonlinear chemical potential (missing DD)
|
||||
% & Low & Basic solids and metallic systems, where accuracy is not critical. & Simple and computationally cheap. \\
|
||||
% \midrule
|
||||
% Generalised Gradient Approximation (GGA) &
|
||||
% $E_x \sim n(r), \nabla n(r)$
|
||||
% $E_c \sim n(r), \nabla n(r)$
|
||||
% & 0.3 (Mostly overbinding) & Improved over LDA
|
||||
% \tabitem SIE
|
||||
% \tabitem wrong long-range behaviour
|
||||
% \tabitem nonlinear chemical potential (missing DD)
|
||||
% & Moderate & More accurate for molecules and chemical bonding studies. & Better than LDA for chemical bonding. \\
|
||||
% \midrule
|
||||
% Hybrid Functionals &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_x = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
||||
% \tabitem Add expensive non-local Fock term to reduce self-interaction
|
||||
% & Lower than GGA (Improved balance) & Better than GGA
|
||||
% \tabitem reduced SIE
|
||||
% \tabitem wrong long-range behaviour
|
||||
% \tabitem nonlinear chemical potential (missing DD)
|
||||
% & Higher & Molecular chemistry, solid-state physics requiring better accuracy. & Balances accuracy and cost. \\
|
||||
% \midrule
|
||||
% Range-Separated Hybrid (RSH) &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_{X,SR} = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
||||
% $E_{X,LR} = E_x^{GGA}$
|
||||
% \tabitem Mix-in expensive Fock term only for short-range interactions $\rightarrow$ since for LR the Coulomb interaction gets screening in dielectric substances ($\epsilon > 1$), such as crystalline materials.
|
||||
% & Lower than Hybrid (Even better balance) & Strongly underestimates
|
||||
% \tabitem reduced SIE
|
||||
% \tabitem wrong long-range behaviour
|
||||
% \tabitem nonlinear chemical potential (missing DD)
|
||||
% & Very High & Semiconductors, materials with screened Coulomb interactions. & Used for dielectric materials. \\
|
||||
% \midrule
|
||||
% Optimally Tuned RSH (OT-RSH) &
|
||||
% $E_x = E_x^{GGA}$
|
||||
% $E_{X,SR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
||||
% $E_{X,LR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
||||
% \tabitem More advanced tuning between Fock and GGA. So that set also have the correct asymptotic behaviour of $1/r$ (Coulomb e.g. Fock) instead of $e^{-r}$ from GGA
|
||||
% & Lowest & Most accurate
|
||||
% \tabitem reduced SIE
|
||||
% \tabitem better long-range behaviour
|
||||
% \tabitem /+ better chemical potential - they include non-multiplicative, orbital dependent terms. Hence, in principle they allow for including a DD.
|
||||
% & Extremely High & Precise calculations for band gap predictions and electronic properties. & Most flexible but computationally expensive. \\
|
||||
% \bottomrule
|
||||
% \end{tabularx}
|
||||
\end{bigformula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Basis sets}
|
||||
|
||||
@ -8,6 +8,8 @@
|
||||
\usepackage{adjustbox}
|
||||
\usepackage{colortbl} % color table
|
||||
\usepackage{tabularx} % bravais table
|
||||
\usepackage{array} % more array options
|
||||
\newcolumntype{C}{>{$}c<{$}} % math-mode version of "c" column type
|
||||
\usepackage{multirow} % for superconducting qubit table
|
||||
\usepackage{hhline} % for superconducting qubit table
|
||||
% TOOLING
|
||||
@ -28,8 +30,6 @@
|
||||
% \setlist[enumerate]{labelsep=*, leftmargin=1.5pc} % horizontal indent of items
|
||||
|
||||
\usepackage{titlesec} % colored titles
|
||||
\usepackage{array} % more array options
|
||||
\newcolumntype{C}{>{$}c<{$}} % math-mode version of "c" column type
|
||||
% \usepackage{sectsty}
|
||||
% GRAPHICS
|
||||
\usepackage{pgfplots}
|
||||
@ -38,6 +38,8 @@
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing} % braces
|
||||
\usetikzlibrary{calc}
|
||||
\usetikzlibrary{3d}
|
||||
\usetikzlibrary{perspective} % 3d view
|
||||
\usetikzlibrary{patterns}
|
||||
\usetikzlibrary{patterns}
|
||||
\input{util/tikz_macros}
|
||||
@ -146,11 +148,11 @@
|
||||
\Input{cm/egas}
|
||||
\Input{cm/charge_transport}
|
||||
\Input{cm/vib}
|
||||
\Input{cm/superconductivity}
|
||||
\Input{cm/semiconductors}
|
||||
\Input{cm/misc}
|
||||
\Input{cm/techniques}
|
||||
\Input{cm/topo}
|
||||
\Input{cm/superconductivity}
|
||||
\Input{cm/mat}
|
||||
|
||||
\Input{particle}
|
||||
|
||||
@ -320,5 +320,16 @@
|
||||
\eq{\txB(x; z_1,z_2) = \int_0^x t^{z_1-1} (1-t)^{z_2-1} \d t}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fermi_dirac}
|
||||
\desc{Fermi-Dirac integral}{}{$\Gamma$ \fRef{::gamma_function}}
|
||||
\desc[german]{Fermi-Dirac-Integral}{}{}
|
||||
\eq{F_j(x)= \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{\Exp{t-x}+1}\d t}
|
||||
\end{formula}
|
||||
\begin{formula}{boltzmann_approximation}
|
||||
\desc{Boltzmann approximation}{$-x\gg1$}{$F$ \fRef{::fermi_dirac_integral}}
|
||||
\desc[german]{Boltzmann-Näherung}{}{}
|
||||
\eq{F_{1/2}(x) \approx \Exp{x}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{differential equation solutions}
|
||||
|
||||
@ -43,6 +43,12 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{centripetal_force}
|
||||
\desc{Centripetal force}{Force that must act to keep a mass on an arc trajectory}{}
|
||||
\desc[german]{Zentripetalkraft}{Kraft die auf einen Körper wirken muss, damit er sich auf einer gegrümmten Bahn bewegt}{}
|
||||
\eq{\vecF_\txc = m v^2 (-\vece_r) = m \vec{\omega}\times\vecv = -m\omega^2\vecr}
|
||||
\end{formula}
|
||||
|
||||
\def\lagrange{\mathcal{L}}
|
||||
\Section[
|
||||
\eng{Lagrange formalism}
|
||||
|
||||
@ -106,10 +106,10 @@
|
||||
##2%
|
||||
\end{alignat}
|
||||
}
|
||||
\newcommand{\fig}[1]{
|
||||
\newcommand{\fig}[2][]{
|
||||
\newFormulaEntry
|
||||
\centering
|
||||
\includegraphics{##1}
|
||||
\includegraphics[##1]{##2}
|
||||
}
|
||||
% 1: content for the ttext environment
|
||||
\newcommand{\ttxt}[2][text]{
|
||||
@ -224,11 +224,56 @@
|
||||
\end{formulainternal}
|
||||
}
|
||||
|
||||
% GROUP
|
||||
\newenvironment{formulagroup}[1]{
|
||||
\mqfqname@enter{#1}
|
||||
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\ifblank{##2}{}{\dt{##1}{##2}}
|
||||
\ifblank{##3}{}{\dt[desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\dt[defs]{##1}{##4}}
|
||||
}
|
||||
|
||||
\par\noindent
|
||||
\begin{minipage}{\textwidth} % using a minipage to now allow line breaks within the bigformula
|
||||
\mqfqname@label
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
% \vspace{0.5\baselineskip}
|
||||
\textbf{
|
||||
\raggedright
|
||||
\GT{\fqname}
|
||||
}
|
||||
\IfTranslationExists{\fqname:desc}{
|
||||
: {\color{fg1} \GT{\fqname:desc}}
|
||||
}{}
|
||||
\hfill
|
||||
\par
|
||||
}{
|
||||
\IfTranslationExists{\fqname:defs}{%
|
||||
\smartnewline
|
||||
\noindent
|
||||
\begingroup
|
||||
\color{fg1}
|
||||
\GT{\fqname:defs}
|
||||
% \edef\temp{\GT{#1_defs}}
|
||||
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
|
||||
\endgroup
|
||||
}{}
|
||||
\end{minipage}
|
||||
\separateEntries
|
||||
% \textcolor{fg3}{\hrule}
|
||||
% \vspace{0.5\baselineskip}
|
||||
\ignorespacesafterend
|
||||
\mqfqname@leave
|
||||
}
|
||||
|
||||
\newenvironment{hiddenformula}[1]{
|
||||
\begin{formulainternal}{#1}
|
||||
\renewcommand{\eq}[1]{}
|
||||
\renewcommand{\eqFLAlign}[2]{}
|
||||
\renewcommand{\fig}[2][1.0]{}
|
||||
\renewcommand{\fig}[2][]{}
|
||||
\renewcommand{\ttxt}[2][#1:desc]{}
|
||||
% 1: symbol
|
||||
% 2: units
|
||||
|
||||
@ -3,7 +3,6 @@
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
|
||||
\begin{luacode}
|
||||
sections = sections or {}
|
||||
|
||||
|
||||
@ -171,7 +171,7 @@
|
||||
% (temporarily change fqname to the \fqname:<env arg> to allow
|
||||
% the use of \eng and \ger without the key parameter)
|
||||
% [1]: key
|
||||
\newenvironment{ttext}[1][desc]{%
|
||||
\newenvironment{ttext}[1][ttext]{%
|
||||
\mqfqname@enter{#1}%
|
||||
}{%
|
||||
\GT{\fqname}%
|
||||
|
||||
41
src/qm/misc.tex
Normal file
41
src/qm/misc.tex
Normal file
@ -0,0 +1,41 @@
|
||||
\Section[
|
||||
\eng{Other}
|
||||
\ger{Sonstiges}
|
||||
]{misc}
|
||||
\begin{formula}{RWA}
|
||||
\desc{Rotating Wave Approximation (RWS)}{Rapidly oscilating terms are neglected}{$\omega_\text{L}$ light frequency, $\omega_0$ transition frequency}
|
||||
\desc[german]{Rotating Wave Approximation / Drehwellennäherung (RWS)}{Schnell oscillierende Terme werden vernachlässigt}{$\omega_\text{L}$ Frequenz des Lichtes, $\omega_0$ Übergangsfrequenz}
|
||||
\eq{\Delta\omega \coloneq \abs{\omega_0 - \omega_\text{L}} \ll \abs{\omega_0 + \omega_\text{L}} \approx 2\omega_0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{adiabatic_theorem} \absLabel
|
||||
\desc{Adiabatic theorem}{}{}
|
||||
\desc[german]{Adiabatentheorem}{}{}
|
||||
\ttxt{
|
||||
\eng{A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.}
|
||||
\ger{Ein quantenmechanisches System bleibt in im derzeitigen Eigenzustand falls eine Störung langsam genug wirkt und der Eigenwert durch eine Lücke vom Rest des Spektrums getrennt ist.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{slater_det}
|
||||
\desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
|
||||
\desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}
|
||||
\eq{
|
||||
\Psi(q_1, \dots, q_N) = \frac{1}{\sqrt{N!}}
|
||||
\begin{vmatrix}
|
||||
\phi_a(q_1) & \phi_a(q_2) & \cdots & \phi_a(q_N) \\
|
||||
\phi_b(q_1) & \phi_b(q_2) & \cdots & \phi_b(q_N) \\
|
||||
\vdots & \vdots & \ddots & \vdots \\
|
||||
\phi_z(q_1) & \phi_z(q_2) & \cdots & \phi_z(q_N)
|
||||
\end{vmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{de-borglie_wavelength}
|
||||
\desc{De-Broglie wavelength}{Matter wavelength}{\ConstRef{planck}, \QtyRef{momentum}}
|
||||
\desc[german]{De-Broglie Wellenlänge}{Materiewellenlänge}{}
|
||||
\eq{\lambda = \frac{h}{p}}
|
||||
\end{formula}
|
||||
|
||||
@ -178,6 +178,7 @@
|
||||
\eng{Schrödinger equation}
|
||||
\ger{Schrödingergleichung}
|
||||
]{se}
|
||||
\abbrLink{se}{SE}
|
||||
\begin{formula}{energy_operator}
|
||||
\desc{Energy operator}{}{}
|
||||
\desc[german]{Energieoperator}{}{}
|
||||
@ -565,39 +566,3 @@
|
||||
+ \frac{\hbar\Omega}{2} (\hat{a}\hat{\sigma^\dagger} + \hat{a}^\dagger \hat{\sigma})
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Other}
|
||||
\ger{Sonstiges}
|
||||
]{other}
|
||||
\begin{formula}{RWA}
|
||||
\desc{Rotating Wave Approximation (RWS)}{Rapidly oscilating terms are neglected}{$\omega_\text{L}$ light frequency, $\omega_0$ transition frequency}
|
||||
\desc[german]{Rotating Wave Approximation / Drehwellennäherung (RWS)}{Schnell oscillierende Terme werden vernachlässigt}{$\omega_\text{L}$ Frequenz des Lichtes, $\omega_0$ Übergangsfrequenz}
|
||||
\eq{\Delta\omega \coloneq \abs{\omega_0 - \omega_\text{L}} \ll \abs{\omega_0 + \omega_\text{L}} \approx 2\omega_0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{adiabatic_theorem} \absLabel
|
||||
\desc{Adiabatic theorem}{}{}
|
||||
\desc[german]{Adiabatentheorem}{}{}
|
||||
\ttxt{
|
||||
\eng{A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.}
|
||||
\ger{Ein quantenmechanisches System bleibt in im derzeitigen Eigenzustand falls eine Störung langsam genug wirkt und der Eigenwert durch eine Lücke vom Rest des Spektrums getrennt ist.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{slater_det}
|
||||
\desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
|
||||
\desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}
|
||||
\eq{
|
||||
\Psi(q_1, \dots, q_N) = \frac{1}{\sqrt{N!}}
|
||||
\begin{vmatrix}
|
||||
\phi_a(q_1) & \phi_a(q_2) & \cdots & \phi_a(q_N) \\
|
||||
\phi_b(q_1) & \phi_b(q_2) & \cdots & \phi_b(q_N) \\
|
||||
\vdots & \vdots & \ddots & \vdots \\
|
||||
\phi_z(q_1) & \phi_z(q_2) & \cdots & \phi_z(q_N)
|
||||
\end{vmatrix}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
@ -407,7 +407,6 @@
|
||||
\draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
|
||||
\node at (1,-0.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
\end{formula}
|
||||
|
||||
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
|
||||
|
||||
@ -358,8 +358,8 @@
|
||||
\entry{partition_sum} {$\Omega = \sum_n 1$ }
|
||||
\entry{probability} {$p_n = \frac{1}{\Omega}$}
|
||||
\entry{td_pot} {$S = \kB\ln\Omega$ }
|
||||
\entry{pressure} {$p = T \pdv{S}{V}_{E,N}$}
|
||||
\entry{entropy} {$S = \kB = \ln\Omega$ }
|
||||
\rentry{\qtyRef{pressure}} {$p = T \pdv{S}{V}_{E,N}$}
|
||||
\rentry{\qtyRef{entropy}} {$S = \kB = \ln\Omega$ }
|
||||
\end{minipagetable}
|
||||
\end{bigformula}
|
||||
|
||||
@ -371,8 +371,8 @@
|
||||
\entry{partition_sum} {$Z = \sum_n \e^{-\beta E_n}$ }
|
||||
\entry{probability} {$p_n = \frac{\e^{-\beta E_n}}{Z}$}
|
||||
\entry{td_pot} {$F = - \kB T \ln Z$ }
|
||||
\entry{pressure} {$p = -\pdv{F}{V}_{T,N}$ }
|
||||
\entry{entropy} {$S = -\pdv{F}{T}_{V,N}$ }
|
||||
\rentry{\qtyRef{pressure}} {$p = -\pdv{F}{V}_{T,N}$ }
|
||||
\rentry{\qtyRef{entropy}} {$S = -\pdv{F}{T}_{V,N}$ }
|
||||
\end{minipagetable}
|
||||
\end{bigformula}
|
||||
|
||||
@ -382,10 +382,10 @@
|
||||
\begin{minipagetable}{mvt}
|
||||
\entry{const_variables} {$T$, $V$, $\mu$ }
|
||||
\entry{partition_sum} {$Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ }
|
||||
\entry{probability} {$p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$}
|
||||
\entry{probability} {$p_n = \frac{\e^{-\beta (E_n - \mu N_n)}}{Z_\text{g}}$}
|
||||
\entry{td_pot} {$ \Phi = - \kB T \ln Z$ }
|
||||
\entry{pressure} {$p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ }
|
||||
\entry{entropy} {$S = -\pdv{\Phi}{T}_{V,\mu}$ }
|
||||
\rentry{\qtyRef{pressure}} {$p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ }
|
||||
\rentry{\qtyRef{entropy}} {$S = -\pdv{\Phi}{T}_{V,\mu}$ }
|
||||
\end{minipagetable}
|
||||
\end{bigformula}
|
||||
|
||||
@ -394,24 +394,24 @@
|
||||
% \desc[german]{Kanonisches Ensemble}{}{}
|
||||
\begin{minipagetable}{npt}
|
||||
\entry{const_variables} {$N$, $p$, $T$}
|
||||
\entry{partition_sum} {}
|
||||
\entry{probability} {$p_n ? \frac{\e^{-\beta(E_n + pV_n)}}{Z}$}
|
||||
\entry{td_pot} {}
|
||||
\entry{pressure} {}
|
||||
\entry{entropy} {}
|
||||
\entry{partition_sum} {$Z = \sum_{n}\e^{-\beta(E_n+pV)}$}
|
||||
\entry{probability} {$p_n = \frac{\e^{-\beta(E_n + pV_n-TS)}}{Z}$}
|
||||
\entry{td_pot} {$G = -\kB \ln Z$}
|
||||
\rentry{\qtyRef{volume}} {$V = \pdv{G}{V}_{T,N} $}
|
||||
\rentry{\qtyRef{entropy}} {$S = - \pdv{G}{T}_{p,N}$}
|
||||
\end{minipagetable}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{nph}
|
||||
\desc{Isonthalpic-isobaric ensemble}{Enthalpy ensemble}{}
|
||||
\desc{Isoenthalpic-isobaric ensemble}{Enthalpy ensemble}{}
|
||||
% \desc[german]{Kanonisches Ensemble}{}{}
|
||||
\begin{minipagetable}{nph}
|
||||
\entry{const_variables} {}
|
||||
\entry{partition_sum} {}
|
||||
\entry{probability} {}
|
||||
\entry{td_pot} {}
|
||||
\entry{pressure} {}
|
||||
\entry{entropy} {}
|
||||
\entry{const_variables} {$N$, $p$, $H$}
|
||||
% \entry{partition_sum} {$ $}
|
||||
% \entry{probability} {$ $}
|
||||
\entry{td_pot} {$H$}
|
||||
% \rentry{\qtyRef{pressure}} {$ $}
|
||||
% \rentry{\qtyRef{entropy}} {$ $}
|
||||
\end{minipagetable}
|
||||
\end{bigformula}
|
||||
|
||||
@ -443,7 +443,7 @@
|
||||
\end{formula}
|
||||
\begin{formula}{enthalpy}
|
||||
\desc{Enthalpy}{}{}
|
||||
\desc[german]{Enthalpie}{}{}
|
||||
\desc[german]{Enthalpie}{früher "Wärmeinhalt"}{}
|
||||
\eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
|
||||
\hiddenQuantity{H}{\joule}{s}
|
||||
\end{formula}
|
||||
@ -497,10 +497,16 @@
|
||||
\desc{Phase space volume}{$3N$ sphere}{$N$ \#particles, $h^{3N}$ volume of a microstate, $N!$ particles are undifferentiable}
|
||||
\desc[german]{}{$3N$ Kugel}{$N$ \#Teilchen, $h^{3N}$ Volumen eines Mikrozustandes, $N!$ Teilchen sind ununterscheidbar}
|
||||
\eq{
|
||||
\Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right) \\
|
||||
\begin{split}
|
||||
\Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \\
|
||||
&\qquad\qquad \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right)
|
||||
\end{split}\\
|
||||
\begin{split}
|
||||
&= \left(\frac{V}{N}\right)^N \left(\frac{4\pi m E}{3 h^2 N}\right)^{\frac{3N}{2}} \e^\frac{5N}{2}
|
||||
\end{split}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{entropy}
|
||||
\desc{Entropy}{}{}
|
||||
\desc[german]{Entropie}{}{}
|
||||
@ -833,13 +839,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity}{\gt{low_temps}}{differs from \fRef{td:TODO:petit_dulong}}
|
||||
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fRef{td:TODO:petit_dulong}}
|
||||
\desc{Heat capacity}{\gt{low_temps}}{differs from \absRef{petit-dulong}}
|
||||
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \absRef{petit-dulong}}
|
||||
\fig{img/td_fermi_heat_capacity.pdf}
|
||||
\eq{C_V = \pdv{E}{T}_V = N\kB \frac{\pi}{2} \left(\frac{T}{T_\text{F}}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{Entartung und Sommerfeld}
|
||||
\TODO{DULONG-PETIT Gesetz}
|
||||
|
||||
|
||||
@ -88,7 +88,7 @@
|
||||
% 2: entry text
|
||||
\newcommand{\entry}[2]{
|
||||
\directlua{
|
||||
table.insert(entries, {key = "\luaescapestring{##1}", value = [[\detokenize{##2}]]})
|
||||
table.insert(entries, {key = \luastring{##1}, value = \luastringN{##2}})
|
||||
}
|
||||
}
|
||||
% Translation entry
|
||||
@ -100,7 +100,15 @@
|
||||
##2
|
||||
\edef\fqname{\tmpFqname}
|
||||
\directlua{
|
||||
table.insert(entries, {key = "\luaescapestring{##1}", value = "\\gt{" .. table_name .. ":\luaescapestring{##1}}"})
|
||||
table.insert(entries, {key = \luastring{##1}, value = "\\gt{" .. table_name .. ":\luaescapestring{##1}}"})
|
||||
}
|
||||
}
|
||||
% Entry with raw field, for example a reference
|
||||
% 1: field text
|
||||
% 2: entry text
|
||||
\newcommand{\rentry}[2]{
|
||||
\directlua{
|
||||
table.insert(entries, {key = \luastring{##1}, value = \luastringN{##2}, raw = true})
|
||||
}
|
||||
}
|
||||
}{
|
||||
@ -115,8 +123,12 @@
|
||||
\hline
|
||||
\directlua{
|
||||
for _, kv in ipairs(entries) do
|
||||
if kv.raw == true then
|
||||
tex.print(kv.key .. " & " .. kv.value .. "\\\\")
|
||||
else
|
||||
tex.print("\\GT{" .. kv.key .. "} & " .. kv.value .. "\\\\")
|
||||
end
|
||||
end
|
||||
}
|
||||
\hline
|
||||
\end{tabularx}
|
||||
|
||||
@ -38,8 +38,8 @@
|
||||
\newcommand\NA{N_\text{A}} % avogadro
|
||||
\newcommand\EFermi{E_\text{F}} % fermi energy
|
||||
\newcommand\Efermi{E_\text{F}} % fermi energy
|
||||
\newcommand\Evalence{E_\text{v}} % val vand energy
|
||||
\newcommand\Econd{E_\text{c}} % cond. band nergy
|
||||
\newcommand\Evalence{E_\text{V}} % val vand energy
|
||||
\newcommand\Econd{E_\text{C}} % cond. band nergy
|
||||
\newcommand\Egap{E_\text{gap}} % band gap energy
|
||||
\newcommand\Evac{E_\text{vac}} % vacuum energy
|
||||
\newcommand\masse{m_\text{e}} % electron mass
|
||||
|
||||
@ -1,4 +1,3 @@
|
||||
|
||||
\tikzset{
|
||||
% bands
|
||||
sc band con/.style={ draw=fg0, thick},
|
||||
@ -18,6 +17,9 @@
|
||||
sc metal/.style={ draw=none,fill=bg-purple!20},
|
||||
sc oxide/.style={ draw=none,fill=bg-green!20},
|
||||
sc separate/.style={ draw=fg0,dotted},
|
||||
% crystal
|
||||
miller dir/.style={->,color=fg-purple,draw=fg-purple, thick},
|
||||
miller plane/.style={fill=bg-purple,fill opacity=0.6,draw=fg-purple,color=fg-purple},
|
||||
}
|
||||
|
||||
\newcommand\drawDArrow[4]{
|
||||
@ -112,3 +114,70 @@
|
||||
\pgfmathsetmacro{\tickwidth}{0.1}
|
||||
\draw (-\tickwidth/2, #1) -- (\tickwidth/2,#1) node[anchor=east] {#2};
|
||||
}
|
||||
|
||||
% 3D HEXAGON
|
||||
% 1: name
|
||||
% 2: center
|
||||
% 3-5: a1-3
|
||||
\newcommand{\setHexagonPoints}[5]{
|
||||
\coordinate (#1-M) at #2;
|
||||
\coordinate (#1-A) at ($#2+(xyz cylindrical cs:radius=#3,angle=0)$);
|
||||
\coordinate (#1-C) at ($#2+(xyz cylindrical cs:radius=#4,angle=120)$);
|
||||
\coordinate (#1-E) at ($#2+(xyz cylindrical cs:radius=#5,angle=240)$);
|
||||
\coordinate (#1-D) at ($#2+(xyz cylindrical cs:radius=#3,angle=180)$);
|
||||
\coordinate (#1-F) at ($#2+(xyz cylindrical cs:radius=#4,angle=300)$);
|
||||
\coordinate (#1-B) at ($#2+(xyz cylindrical cs:radius=#5,angle=60)$);
|
||||
}
|
||||
\newcommand\drawHexagon[1]{ \draw (#1-A) -- (#1-B) -- (#1-C) -- (#1-D) -- (#1-E) -- (#1-F) -- (#1-A); }
|
||||
\newcommand\drawHexagonBack[1]{ \draw (#1-A) -- (#1-B) -- (#1-C) -- (#1-D); }
|
||||
\newcommand\drawHexagonFront[1]{ \draw (#1-D)-- (#1-E) -- (#1-F) -- (#1-A); }
|
||||
% 1: r
|
||||
% 2: z
|
||||
\newcommand\drawHexagonCS[2]{
|
||||
\draw[->] (0,0,0) -- ++(xyz cylindrical cs:radius=1.5*#1,angle=000) node[anchor=west]{$a_1$};
|
||||
\draw[->] (0,0,0) -- ++(xyz cylindrical cs:radius=1.5*#1,angle=120) node[anchor=south east]{$a_2$};
|
||||
\draw[->] (0,0,0) -- ++(xyz cylindrical cs:radius=1.5*#1,angle=240) node[anchor=north east]{$a_3$};
|
||||
\draw[->] (0,0,0) -- ++(0,0,1.5*#2) node[anchor=south]{$a_4$};
|
||||
}
|
||||
% vertically connect two hexagons
|
||||
\newcommand\drawHexagonConnectionsBack[2]{
|
||||
\draw (#1-A) -- (#2-A);
|
||||
\draw (#1-B) -- (#2-B);
|
||||
\draw (#1-C) -- (#2-C);
|
||||
\draw (#1-D) -- (#2-D);
|
||||
}
|
||||
\newcommand\drawHexagonConnectionsFront[2]{
|
||||
\draw (#1-E) -- (#2-E);
|
||||
\draw (#1-F) -- (#2-F);
|
||||
}
|
||||
% 3D RECTANGLES
|
||||
% 1: name
|
||||
% 2: center
|
||||
% 3,4: w,l
|
||||
\newcommand{\setRectPoints}[4]{
|
||||
\coordinate (#1-M) at #2;
|
||||
\coordinate (#1-A) at ($#2+(-#3*0.5, #4*0.5,0)$);
|
||||
\coordinate (#1-B) at ($#2+(-#3*0.5,-#4*0.5,0)$);
|
||||
\coordinate (#1-C) at ($#2+( #3*0.5,-#4*0.5,0)$);
|
||||
\coordinate (#1-D) at ($#2+( #3*0.5, #4*0.5,0)$);
|
||||
}
|
||||
\newcommand\drawRect[1]{ \draw (#1-A) -- (#1-B) -- (#1-C) -- (#1-D) -- cycle; }
|
||||
\newcommand\drawRectBack[1]{ \draw (#1-D) -- (#1-A) -- (#1-B) -- (#1-C); }
|
||||
\newcommand\drawRectFront[1]{ \draw (#1-C)-- (#1-D); }
|
||||
% Coordinate System
|
||||
% 1: x,y
|
||||
% 2: z
|
||||
\newcommand\drawRectCS[2]{
|
||||
\draw[->] (0,0,0) -- ++(#1, 0, 0) node[anchor=west]{$x$};
|
||||
\draw[->] (0,0,0) -- ++( 0,#1, 0) node[anchor=south east]{$y$};
|
||||
\draw[->] (0,0,0) -- ++( 0, 0,#2) node[anchor=south]{$z$};
|
||||
}
|
||||
% vertically connect two rects
|
||||
\newcommand\drawRectConnectionsBack[2]{
|
||||
\draw (#1-A) -- (#2-A);
|
||||
\draw (#1-B) -- (#2-B);
|
||||
}
|
||||
\newcommand\drawRectConnectionsFront[2]{
|
||||
\draw (#1-C) -- (#2-C);
|
||||
\draw (#1-D) -- (#2-D);
|
||||
}
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user