diff --git a/src/ch/misc.tex b/src/ch/misc.tex index cd9a1f5..f8d83ec 100644 --- a/src/ch/misc.tex +++ b/src/ch/misc.tex @@ -71,15 +71,6 @@ } \end{formula} - \begin{formula}{covalent_bond} - \desc{Covalent bond}{}{} - \desc[german]{Kolvalente Bindung}{}{} - \ttxt{ - \eng{Bonds that involve sharing of electrons to form electron pairs between atoms.} - \ger{Bindungen zwischen Atomen die durch geteilte Elektronen, welche Elektronenpaare bilden, gebildet werden.} - } - \end{formula} - \begin{formula}{grotthuss} \desc{Grotthuß-mechanism}{}{} \desc[german]{Grotthuß-Mechanismus}{}{} diff --git a/src/cm/charge_transport.tex b/src/cm/charge_transport.tex index c91ba34..1fed869 100644 --- a/src/cm/charge_transport.tex +++ b/src/cm/charge_transport.tex @@ -71,21 +71,40 @@ \ger{Boltzmann-Transport} ]{boltzmann} \begin{ttext} - \eng{Semiclassical description using a probability distribution (\fRef{stat:todo:fermi_dirac}) to describe the particles.} - \ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{stat:todo:fermi_dirac}).} + \eng{Semiclassical description using a probability distribution (\fRef{cm:sc:fermi_dirac}) to describe the particles.} + \ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{cm:sc:fermi_dirac}).} \end{ttext} \begin{formula}{boltzmann_transport} - \desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{stat:todo:fermi-dirac}} + \desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{cm:sc:fermi_dirac}} \desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{} \eq{ \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}}} } \end{formula} +\Subsection[ + \eng{Magneto-transport} + \ger{Magnetotransport} +]{mag} + \begin{formula}{cyclotron_frequency} + \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}} + \desc[german]{Zyklotronfrequenz}{Ladungstraäger bewegen sich in einem Magnetfeld auf einer Kreisbahn}{} + \eq{w_\txc = \frac{qB}{m}} + \end{formula} + + \TODO{TODO} + % \begin{formula}{cyclotron_resonance} + % \desc{}{}{} + % \desc[german]{}{}{} + % \eq{} + % \end{formula} + \TODO{move hall here} + + \Subsection[ \eng{misc} \ger{misc} - ]{misc} +]{misc} \begin{formula}{tsu_esaki} \desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_potential} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$} \desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_potential} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$} @@ -94,6 +113,12 @@ } \end{formula} + \begin{formula}{diffusion} + \desc{Diffusion current}{Equilibration of concentration gradients}{\QtyRef{diffusion_coefficient}, \ConstRef{charge}, $n,p$ \qtyRef{charge_carrier_density}} + \desc[german]{Diffunsstrom}{Ausgleich von Konzentrationsgradienten}{} + \eq{\vec{j}_\text{diff} = -\abs{e} D_n \left(-\Grad n\right) + \abs{e} D_p \left(-\Grad p\right)} + \end{formula} + \begin{formula}{continuity} \desc{Charge continuity equation}{Electric charge can only change by the amount of electric current}{\QtyRef{charge_density}, \QtyRef{current_density}} \desc[german]{Kontinuitätsgleichung der Ladung}{Elektrische Ladung kann sich nur durch die Stärke des Stromes ändern}{} diff --git a/src/cm/cm.tex b/src/cm/cm.tex index 83a2315..05c8aef 100644 --- a/src/cm/cm.tex +++ b/src/cm/cm.tex @@ -10,13 +10,84 @@ \quantity{D}{\per\m^3}{s} \eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))} \end{formula} - \begin{formula}{dos_parabolic} - \desc{Density of states for parabolic dispersion}{Applies to \fRef{cm:egas}}{} - \desc[german]{Zustandsdichte für parabolische Dispersion}{Bei \fRef{cm:egas}}{} - \eq{ - D_1(E) &= \frac{1}{2\sqrt{c_k(E-E_0)}} && (\text{1D}) \\ - D_2(E) &= \frac{\pi}{2c_k} && (\text{2D}) \\ - D_3(E) &= \pi \sqrt{\frac{E-E_0}{c_k^3}}&& (\text{3D}) - } + + + +\Section[ + \eng{Bonds} + \ger{Bindungen} +]{bond} + \begin{formula}{metallic} + \desc{Metallic bond}{}{} + \desc[german]{Metallbindung}{}{} + \ttxt{\eng{ + \begin{itemize} + \item Delocalized electrons form a cloud + \item High \qtyRef[electrical]{conductivity} and \qtyRef[thermal]{thermal_conductivity} conductivity + \item No internal electric field + \end{itemize} + }\ger{ + \begin{itemize} + \item Elektronen delokalisiert und bilden Wolke + \item Hohe \qtyRef[elektrische]{conductivity} und \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit + \item Kein internes elektrisches Feld + \end{itemize} + }} + \end{formula} + + \begin{formula}{covalent} + \desc{Covalent bond}{}{} + \desc[german]{Kolvalente Bindung}{}{} + \ttxt{\eng{ + \begin{itemize} + \item \fRef{cm:band:hybrid_orbitals} of shared electrons + \item Highly directional + \item Varying \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity + \end{itemize} + }\ger{ + \begin{itemize} + \item \fRef{cm:band:hybrid_orbitals} geteilter Elektronen + \item Richtungsabhängige Bindung + \item Verschiedene \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeiten + \end{itemize} + }} + \end{formula} + + \begin{formula}{ionic} + \desc{Ionic bond}{}{} + \desc[german]{Ionenbindung}{}{} + \ttxt{\eng{ + \begin{itemize} + \item Charge transfer from anion to cation + \item Non.directional bonding + \item Strong bond + \item Low \qtyRef[electrical]{conductivity} and high \qtyRef[thermal]{thermal_conductivity} conductivity + \item Always in combination with a \fRef{:::covalent} + \end{itemize} + }\ger{ + \begin{itemize} + \item Ladungstransfer von Anion zu Kation + \item Richtungsunabängig + \item Starke Bindung + \item Geringe \qtyRef[elektrische]{conductivity} und hohe \qtyRef[thermische]{thermal_conductivity} Leitfähigkeit + \item Immer in Kombination mit einer \fRef[kovalenten Bindung]{:::covalent} + \end{itemize} + }} + \end{formula} + + \begin{formula}{van-der-waals} + \desc{Van der Waals bond}{}{} + \desc[german]{Van-der-Waals Bindung}{}{} + \ttxt{\eng{ + \begin{itemize} + \item Dipole-dipole interaction from local charge fluctuations + \item Weak bond + \end{itemize} + }\ger{ + \begin{itemize} + \item Dipol-Dipol Wechselwirkung durch lokale Ladungsfluktuationen + \item Schwache Bindung + \end{itemize} + }} \end{formula} diff --git a/src/cm/crystal.tex b/src/cm/crystal.tex index 60cf5a0..64d10d3 100644 --- a/src/cm/crystal.tex +++ b/src/cm/crystal.tex @@ -1,203 +1,311 @@ \Section[ - \eng{Crystals} - \ger{Kristalle} + \eng{Crystals} + \ger{Kristalle} ]{crystal} \Subsection[ - \eng{Bravais lattice} - \ger{Bravais-Gitter} + \eng{Bravais lattice} + \ger{Bravais-Gitter} ]{bravais} + \Eng[lattice_system]{Lattice system} + \Ger[lattice_system]{Gittersystem} + \Eng[crystal_family]{Crystal system} + \Ger[crystal_family]{Kristall-system} + \Eng[point_group]{Point group} + \Ger[point_group]{Punktgruppe} + \eng[bravais_lattices]{Bravais lattices} + \ger[bravais_lattices]{Bravais Gitter} + \newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}} + \renewcommand\tabularxcolumn[1]{m{#1}} + \newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X} - \Eng[lattice_system]{Lattice system} - \Ger[lattice_system]{Gittersystem} - \Eng[crystal_family]{Crystal system} - \Ger[crystal_family]{Kristall-system} - \Eng[point_group]{Point group} - \Ger[point_group]{Punktgruppe} - \eng[bravais_lattices]{Bravais lattices} - \ger[bravais_lattices]{Bravais Gitter} + \begin{bigformula}{2d} + \desc{2D}{In 2D, there are 5 different Bravais lattices}{} + \desc[german]{2D}{In 2D gibt es 5 verschiedene Bravais-Gitter}{} + \begin{adjustbox}{width=\textwidth} + \begin{tabularx}{\textwidth}{||Z|c|Z|Z||} + \hline + \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4} + & & \GT{primitive} (p) & \GT{centered} (c) \\ \hline + \GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline + \GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline + \GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline + \GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline + \end{tabularx} + \end{adjustbox} + \end{bigformula} + - \newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}} - \renewcommand\tabularxcolumn[1]{m{#1}} - \newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X} + \begin{bigformula}{3d} + \desc{3D}{In 3D, there are 14 different Bravais lattices}{} + \desc[german]{3D}{In 3D gibt es 14 verschiedene Bravais-Gitter}{} + % \newcolumntype{g}{>{\columncolor[]{0.8}}} + \begin{adjustbox}{width=\textwidth} + \begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||} + \hline + \multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7} + & & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline + \multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline + \multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline + \multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline + \multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline + \multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7} + & \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline + \multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline + \end{tabularx} + \end{adjustbox} + \end{bigformula} - \begin{bigformula}{2d} - \desc{2D}{In 2D, there are 5 different Bravais lattices}{} - \desc[german]{2D}{In 2D gibt es 5 verschiedene Bravais-Gitter}{} - \begin{adjustbox}{width=\textwidth} - \begin{tabularx}{\textwidth}{||Z|c|Z|Z||} - \hline - \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{2}{c||}{5 \gt{bravais_lattices}} \\ \cline{3-4} - & & \GT{primitive} (p) & \GT{centered} (c) \\ \hline - \GT{monoclinic} (m) & $\text{C}_\text{2}$ & \bvimg{mp} & \\ \hline - \GT{orthorhombic} (o) & $\text{D}_\text{2}$ & \bvimg{op} & \bvimg{oc} \\ \hline - \GT{tetragonal} (t) & $\text{D}_\text{4}$ & \bvimg{tp} & \\ \hline - \GT{hexagonal} (h) & $\text{D}_\text{6}$ & \bvimg{hp} & \\ \hline - \end{tabularx} - \end{adjustbox} - \end{bigformula} - + \begin{formula}{lattice_constant} + \desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{} + \desc[german]{Gitterkonstante}{Parameter (Länge oder Winkel) der die Einheitszelle beschreibt}{} + \quantity{a}{}{s} + \end{formula} - \begin{bigformula}{3d} - \desc{3D}{In 3D, there are 14 different Bravais lattices}{} - \desc[german]{3D}{In 3D gibt es 14 verschiedene Bravais-Gitter}{} - % \newcolumntype{g}{>{\columncolor[]{0.8}}} - \begin{adjustbox}{width=\textwidth} - \begin{tabularx}{\textwidth}{||Z|Z|c|Z|Z|Z|Z||} - \hline - \multirow{2}{*}{\GT{crystal_family}} & \multirow{2}{*}{\GT{lattice_system}} & \multirow{2}{*}{\GT{point_group}} & \multicolumn{4}{c||}{14 \gt{bravais_lattices}} \\ \cline{4-7} - & & & \GT{primitive} (P) & \GT{base_centered} (S) & \GT{body_centered} (I) & \GT{face_centered} (F) \\ \hline - \multicolumn{2}{||c|}{\GT{triclinic} (a)} & $\text{C}_\text{i}$ & \bvimg{tP} & & & \\ \hline - \multicolumn{2}{||c|}{\GT{monoclinic} (m)} & $\text{C}_\text{2h}$ & \bvimg{mP} & \bvimg{mS} & & \\ \hline - \multicolumn{2}{||c|}{\GT{orthorhombic} (o)} & $\text{D}_\text{2h}$ & \bvimg{oP} & \bvimg{oS} & \bvimg{oI} & \bvimg{oF} \\ \hline - \multicolumn{2}{||c|}{\GT{tetragonal} (t)} & $\text{D}_\text{4h}$ & \bvimg{tP} & & \bvimg{tI} & \\ \hline - \multirow{2}{*}{\GT{hexagonal} (h)} & \GT{rhombohedral} & $\text{D}_\text{3d}$ & \bvimg{hR} & & & \\ \cline{2-7} - & \GT{hexagonal} & $\text{D}_\text{6h}$ & \bvimg{hP} & & & \\ \hline - \multicolumn{2}{||c|}{\GT{cubic} (c)} & $\text{O}_\text{h}$ & \bvimg{cP} & & \bvimg{cI} & \bvimg{cF} \\ \hline - \end{tabularx} - \end{adjustbox} - \end{bigformula} + \begin{formula}{lattice_vector} + \desc{Lattice vector}{}{$n_i \in \Z$} + \desc[german]{Gittervektor}{}{} + \quantity{\vec{R}}{}{\angstrom} + \eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}} + \end{formula} - \begin{formula}{lattice_constant} - \desc{Lattice constant}{Parameter (length or angle) describing the smallest unit cell}{} - \desc[german]{Gitterkonstante}{Parameter (Länge oder Winkel) der die Einheitszelle beschreibt}{} - \quantity{a}{}{s} - \end{formula} + \begin{formula}{primitive_unit_cell} + \desc{Primitve unit cell}{}{} + \desc[german]{Primitive Einheitszelle}{}{} + \ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}} + \end{formula} - \begin{formula}{lattice_vector} - \desc{Lattice vector}{}{$n_i \in \Z$} - \desc[german]{Gittervektor}{}{} - \quantity{\vec{R}}{}{\angstrom} - \eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}} - \end{formula} + \Eng[miller-point]{Point} + \Ger[miller-point]{Punkt} + \Eng[miller-direction]{Direction} + \Ger[miller-direction]{Richtung} + \Eng[miller-direction-family]{Family of directions} + \Ger[miller-direction-family]{Familie von Richtungen} + \Eng[miller-plane]{Plane} + \Ger[miller-plane]{Ebene} + \Eng[miller-plane-family]{Family of planes} + \Ger[miller-plane-family]{Familie von Ebenen} + \begin{formula}{miller} + \desc{Miller indices}{}{ + Miller planes: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ give intersection with $x$/$y$/$z$ axes\\ + Miller family: planes that are equivalent due to crystal symmetry + } + \desc[german]{Millersche Indizes}{}{ + Miller-Ebenen: $(hkl)$, $\frac{1}{h}$/$\frac{1}{k}$/$\frac{1}{l}$ geben die Schnittpunkte mit den $x$/$y$/$z$-Achsen\\ + Miller-Familien: Ebenen, die durch Kristallsymmetrie äquivalent sind + } + \centering + \newFormulaEntry + \begin{tabularx}{\textwidth}{clcl} + $(h,k,l)$ & \GT{miller-point} & & \\ + $hkl$ & \GT{miller-direction} & $\langle hkl \rangle$ & \GT{miller-direction-family} \\ + $(hkl)$ & \GT{miller-plane} & $\{hkl\}$ & \GT{miller-plane-family} + \end{tabularx} + \pgfmathsetmacro{\rectX}{2} + \pgfmathsetmacro{\rectZ}{2} + \newFormulaEntry + \begin{tikzpicture}[3d view={100}{20},perspective={p={(-55,0,0)},q={(0,25,0)},r={(0,0,-30)}}] + % <100> direction family + \begin{scope} + \drawRectCS{1.4*\rectX}{1.4*\rectZ} + \setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX} + \setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX} + \drawRectBack{R1} + \drawRectConnectionsBack{R1}{R2} + \draw[miller dir] (0,0,0) -- ++( \rectX,0,0) node[anchor=east] {$[100]$}; + \draw[miller dir] (0,0,0) -- ++(-\rectX,0,0) node[anchor=west] {$[\bar{1}00]$}; + \draw[miller dir] (0,0,0) -- ++(0, \rectX,0) node[anchor=south] {$[010]$}; + \draw[miller dir] (0,0,0) -- ++(0,-\rectX,0) node[anchor=south] {$[0\bar{1}0]$}; + \draw[miller dir] (0,0,0) -- ++(0,0, \rectX) node[anchor=east] {$[001]$}; + \draw[miller dir] (0,0,0) -- ++(0,0,-\rectX) node[anchor=west] {$[00\bar{1}]$}; + \drawRectFront{R1} + \drawRectBack{R2} + \drawRectConnectionsFront{R1}{R2} + \drawRectFront{R2} + \node at (1.5*\rectX,1.5*\rectX, 0) {$\langle100\rangle$}; + \end{scope} + \pgfmathsetmacro{\rectDistance}{4.5} + % {100} plane family + \begin{scope}[shift={(0,\rectDistance,0)}] + \drawRectCS{1.4*\rectX}{1.4*\rectZ} + \setRectPoints{R1}{(0.5*\rectX,0.5*\rectX,0)}{\rectX}{\rectX} + \setRectPoints{R2}{(0.5*\rectX,0.5*\rectX,\rectZ)}{\rectX}{\rectX} + \drawRectBack{R1} + \drawRectConnectionsBack{R1}{R2} + \drawRectFront{R1} + \drawRectBack{R2} + \drawRectConnectionsFront{R1}{R2} + \drawRectFront{R2} + \fill[miller plane] (R1-C) -- (R1-D) node[anchor=north,midway] {$(100)$} -- (R2-D) -- (R2-C) -- cycle; + \fill[miller plane] (R1-A) -- (R1-D) node[anchor=west,midway] {$(010)$} -- (R2-D) -- (R2-A) -- cycle node[anchor=north east] {$(010)$}; + \fill[miller plane] (R2-A) -- (R2-B) node[midway,anchor=south] {$(001)$} -- (R2-C) -- (R2-D) -- cycle; + \node at (1.5*\rectX,1.5*\rectX, 0) {$\{100\}$}; + \end{scope} + \end{tikzpicture} + % describe how to construct miller planes + \end{formula} - \begin{formula}{primitive_unit_cell} - \desc{Primitve unit cell}{}{} - \desc[german]{Primitive Einheitszelle}{}{} - \ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}} - \end{formula} - - \begin{formula}{miller} - \desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry} - \desc[german]{Millersche Indizes}{}{} - \eq{ - (hkl) & \text{\GT{plane}}\\ - [hkl] & \text{\GT{direction}}\\ - \{hkl\} & \text{\GT{millerFamily}} - } - \end{formula} - + \begin{formula}{miller-hexagon} + \desc{Hexagonal miller indices}{}{} + \desc[german]{Hexagonale Millersche Indizes}{}{} + \eq{ (hkil) && \tGT{with}\quad i = h + k } + \centering + \newFormulaEntry + \begin{tikzpicture}[3d view={0}{20}] + \pgfmathsetmacro{\hexxY}{1.5} + \begin{scope} + \drawHexagonCS{1}{\hexxY} + \setHexagonPoints{H1}{(0,0,0)}{1}{1}{1} + \setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1} + \drawHexagonBack{H1} + \drawHexagonConnectionsBack{H1}{H2} + \drawHexagonFront{H1} + \drawHexagonBack{H2} + \drawHexagonConnectionsFront{H1}{H2} + \drawHexagonFront{H2} + \end{scope} + \pgfmathsetmacro{\hexDistance}{3.5} + % 1121 + \begin{scope}[shift={(\hexDistance,0,0)}] + \drawHexagonCS{1}{\hexxY} + \setHexagonPoints{H1}{(0,0,0)}{1}{1}{1} + \setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1} + \drawHexagonBack{H1} + \drawHexagonConnectionsBack{H1}{H2} + \fill[miller plane] (H1-A) -- (H2-M) -- (H1-E) -- cycle; + \drawHexagonFront{H1} + \drawHexagonBack{H2} + \drawHexagonConnectionsFront{H1}{H2} + \drawHexagonFront{H2} + \node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1211)$}; + \end{scope} + % 1010 + \begin{scope}[shift={(2*\hexDistance,0,0)}] + \drawHexagonCS{1}{\hexxY} + \setHexagonPoints{H1}{(0,0,0)}{1}{1}{1} + \setHexagonPoints{H2}{(0,0,\hexxY)}{1}{1}{1} + \drawHexagonBack{H1} + \drawHexagonConnectionsBack{H1}{H2} + \drawHexagonFront{H1} + \drawHexagonBack{H2} + \drawHexagonConnectionsFront{H1}{H2} + \drawHexagonFront{H2} + \fill[miller plane] (H1-F) -- (H2-F) -- (H2-E) -- (H1-E) -- cycle; + \node[anchor=north] at (xyz cylindrical cs:radius=1.5,angle=270) {$(1010)$}; + \end{scope} + \end{tikzpicture} + \end{formula} + \Subsection[ - \eng{Reciprocal lattice} - \ger{Reziprokes Gitter} + \eng{Reciprocal lattice} + \ger{Reziprokes Gitter} ]{reci} - \begin{ttext} - \eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.} - \ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.} - \end{ttext} + \begin{ttext} + \eng{The reciprokal lattice is made up of all the wave vectors $\vec{k}$ that ressemble standing waves with the periodicity of the Bravais lattice.} + \ger{Das rezioproke Gitter besteht aus dem dem Satz aller Wellenvektoren $\vec{k}$, die ebene Wellen mit der Periodizität des Bravais-Gitters ergeben.} + \end{ttext} - \begin{formula}{vectors} - \desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell} - \desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle} - \eq{ - \vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\ - \vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\ - \vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2} - } - \end{formula} - \begin{formula}{reciprocal_lattice_vector} - \desc{Reciprokal attice vector}{}{$n_i \in \Z$} - \desc[german]{Reziproker Gittervektor}{}{} - \quantity{\vec{G}}{}{\angstrom} - \eq{\vec{G}_{{hkl}} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}} - \end{formula} + \begin{formula}{vectors} + \desc{Reciprocal lattice vectors}{}{$a_i$ real-space lattice vectors, $V_c$ volume of the primitive lattice cell} + \desc[german]{Reziproke Gittervektoren}{}{$a_i$ Bravais-Gitter Vektoren, $V_c$ Volumen der primitiven Gitterzelle} + \eq{ + \vec{b_1} &= \frac{2\pi}{V_c} \vec{a_2} \times \vec{a_3} \\ + \vec{b_2} &= \frac{2\pi}{V_c} \vec{a_3} \times \vec{a_1} \\ + \vec{b_3} &= \frac{2\pi}{V_c} \vec{a_1} \times \vec{a_2} + } + \end{formula} + \begin{formula}{reciprocal_lattice_vector} + \desc{Reciprokal attice vector}{}{$n_i \in \Z$} + \desc[german]{Reziproker Gittervektor}{}{} + \quantity{\vec{G}}{}{\angstrom} + \eq{\vec{G}_{{hkl}} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}} + \end{formula} - \Subsection[ - \eng{Scattering processes} - \ger{Streuprozesse} - ]{scatter} - \begin{formula}{matthiessen} - \desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}} - \desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{} - \eq{ - \frac{1}{\mu} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\mu_i} \\ - \frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\tau_i} - } - \end{formula} + \Subsection[ + \eng{Scattering processes} + \ger{Streuprozesse} + ]{scatter} + \begin{formula}{matthiessen} + \desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}} + \desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{} + \eq{ + \frac{1}{\mu} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\mu_i} \\ + \frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\tau_i} + } + \end{formula} \Subsection[ - \eng{Lattices} - \ger{Gitter} + \eng{Lattices} + \ger{Gitter} ]{lat} - \begin{formula}{sc} - \desc{Simple cubic (SC)}{Reciprocal: Simple cubic}{\QtyRef{lattice_constant}} - \desc[german]{Einfach kubisch (SC)}{Reziprok: Einfach kubisch}{} - \eq{ - \vec{a}_{1}=a \begin{pmatrix} 1\\0\\0 \end{pmatrix},\, - \vec{a}_{2}=a \begin{pmatrix} 0\\1\\0 \end{pmatrix},\, - \vec{a}_{3}=a \begin{pmatrix} 0\\0\\1 \end{pmatrix} - } - \end{formula} - \begin{formula}{bcc} - \desc{Body centered cubic (BCC)}{Reciprocal: \fRef{::fcc}}{\QtyRef{lattice_constant}} - \desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fRef{::fcc}}{} - \eq{ - \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\, - \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\, - \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\-1 \end{pmatrix} - } - \end{formula} + \begin{formula}{sc} + \desc{Simple cubic (SC)}{Reciprocal: Simple cubic}{\QtyRef{lattice_constant}} + \desc[german]{Einfach kubisch (SC)}{Reziprok: Einfach kubisch}{} + \eq{ + \vec{a}_{1}=a \begin{pmatrix} 1\\0\\0 \end{pmatrix},\, + \vec{a}_{2}=a \begin{pmatrix} 0\\1\\0 \end{pmatrix},\, + \vec{a}_{3}=a \begin{pmatrix} 0\\0\\1 \end{pmatrix} + } + \end{formula} + \begin{formula}{bcc} + \desc{Body centered cubic (BCC)}{Reciprocal: \fRef{::fcc}}{\QtyRef{lattice_constant}} + \desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fRef{::fcc}}{} + \eq{ + \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\, + \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\, + \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\-1 \end{pmatrix} + } + \end{formula} - \begin{formula}{fcc} - \desc{Face centered cubic (FCC)}{Reciprocal: \fRef{::bcc}}{\QtyRef{lattice_constant}} - \desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fRef{::bcc}}{} - \eq{ - \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\, - \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\, - \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\0 \end{pmatrix} - } - \end{formula} + \begin{formula}{fcc} + \desc{Face centered cubic (FCC)}{Reciprocal: \fRef{::bcc}}{\QtyRef{lattice_constant}} + \desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fRef{::bcc}}{} + \eq{ + \vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\, + \vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\, + \vec{a}_{3}=\frac{a}{2} \begin{pmatrix} 1\\1\\0 \end{pmatrix} + } + \end{formula} - \begin{formula}{diamond} - \desc{Diamond lattice}{}{} - \desc[german]{Diamantstruktur}{}{} - \ttxt{ - \eng{\fRef{:::fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$} - \ger{\fRef{:::fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$} - } - \end{formula} - \begin{formula}{zincblende} - \desc{Zincblende lattice}{}{} - \desc[german]{Zinkblende-Struktur}{}{} - \fsplit{ - \centering - \includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png} - }{ - \ttxt{ - \eng{Like \fRef{:::diamond} but with different species on each basis} - \ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen} - } - } - \end{formula} - \begin{formula}{rocksalt} - \desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{} - \desc[german]{Kochsalz-Struktur}{}{} - \fsplit{ - \centering - \includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png} - }{ - - } - \end{formula} - \begin{formula}{wurtzite} - \desc{Wurtzite structure}{hP4}{} - \desc[german]{Wurtzite-Struktur}{hP4}{} - \fsplit{ - \centering - \includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png} - }{ - - } - \end{formula} + \begin{formula}{diamond} + \desc{Diamond lattice}{}{} + \desc[german]{Diamantstruktur}{}{} + \ttxt{ + \eng{\fRef{:::fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$} + \ger{\fRef{:::fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$} + } + \end{formula} + \begin{formula}{zincblende} + \desc{Zincblende lattice}{}{} + \desc[german]{Zinkblende-Struktur}{}{} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png} + }{ + \ttxt{ + \eng{Like \fRef{:::diamond} but with different species on each basis} + \ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen} + } + } + \end{formula} + \begin{formula}{rocksalt} + \desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{} + \desc[german]{Kochsalz-Struktur}{}{} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png} + }{ + + } + \end{formula} + \begin{formula}{wurtzite} + \desc{Wurtzite structure}{hP4}{} + \desc[german]{Wurtzite-Struktur}{hP4}{} + \fsplit{ + \centering + \includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png} + }{ + } + \end{formula} diff --git a/src/cm/egas.tex b/src/cm/egas.tex index 893c0eb..a8096af 100644 --- a/src/cm/egas.tex +++ b/src/cm/egas.tex @@ -2,10 +2,14 @@ \eng{Free electron gas} \ger{Freies Elektronengase} ]{egas} -\begin{ttext} - \eng{Assumptions: electrons can move freely and independent of each other.} - \ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.} -\end{ttext} +\begin{formula}{desc} + \desc{Description}{\GT{see_also}: \fRef{td:id_qgas}}{} + \desc[german]{Beschreibung}{}{} + \ttxt{ + \eng{Assumptions: electrons can move freely and independent of each other.} + \ger{Annahmen: Elektronen bewegen sich frei und unabhänig voneinander.} + } +\end{formula} \begin{formula}{drift_velocity} \desc{Drift velocity}{Velocity component induced by an external force (eg. electric field)}{$v_\text{th}$ thermal velocity} @@ -14,6 +18,7 @@ \end{formula} \begin{formula}{mean_free_path} + \abbrLabel{mfp} \desc{Mean free path}{}{} \desc[german]{Mittlere freie Weglänge}{}{} \eq{\ell = \braket{v} \tau} @@ -26,11 +31,20 @@ \eq{\mu = \frac{q \tau}{m}} \end{formula} +\Subsection[ + \eng{3D electron gas} + \ger{3D Elektronengas} +]{3deg} + \begin{formula}{dos} + \desc{Density of states}{}{} + \desc[german]{Zustandsdichte}{}{} + \eq{D_\text{3D}(E) = \frac{1}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}} + \end{formula} + \Subsection[ \eng{2D electron gas} \ger{2D Elektronengas} ]{2deg} - \begin{ttext} \eng{Lower dimension gases can be obtained by restricting a 3D gas with infinetly high potential walls on a narrow area with the width $L$.} \ger{ @@ -51,6 +65,12 @@ \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{$z$}} \end{formula} + \begin{formula}{dos} + \desc{Density of states}{}{} + \desc[german]{Zustandsdichte}{}{} + \eq{D_\text{2D}(E) = \frac{m}{\pi\hbar^2}} + \end{formula} + \Subsection[ \eng{1D electron gas / quantum wire} \ger{1D Eleltronengas / Quantendraht} @@ -61,12 +81,25 @@ \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} + diff --git a/src/cm/misc.tex b/src/cm/misc.tex index 0dfbea6..2bdde1d 100644 --- a/src/cm/misc.tex +++ b/src/cm/misc.tex @@ -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} diff --git a/src/cm/semiconductors.tex b/src/cm/semiconductors.tex index 643c132..6a754ac 100644 --- a/src/cm/semiconductors.tex +++ b/src/cm/semiconductors.tex @@ -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}{}{} - \desc[german]{Intrinsische Ladungsträgerdichte}{}{} + + +\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_\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_\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_\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} diff --git a/src/cm/superconductivity.tex b/src/cm/superconductivity.tex index 4f13dbd..2f629a8 100644 --- a/src/cm/superconductivity.tex +++ b/src/cm/superconductivity.tex @@ -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\\ diff --git a/src/cm/techniques.tex b/src/cm/techniques.tex index 801705a..9255374 100644 --- a/src/cm/techniques.tex +++ b/src/cm/techniques.tex @@ -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} diff --git a/src/cm/vib.tex b/src/cm/vib.tex index 5d0a0b7..db96e39 100644 --- a/src/cm/vib.tex +++ b/src/cm/vib.tex @@ -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} diff --git a/src/comp/est.tex b/src/comp/est.tex index 2b72d7d..b91aef9 100644 --- a/src/comp/est.tex +++ b/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} diff --git a/src/img/cm/sc_junction_metal_n_sc.tex b/src/img_static/cm/sc_junction_metal_n_sc.tex similarity index 100% rename from src/img/cm/sc_junction_metal_n_sc.tex rename to src/img_static/cm/sc_junction_metal_n_sc.tex diff --git a/src/img/cm/sc_junction_metal_n_sc_separate.tex b/src/img_static/cm/sc_junction_metal_n_sc_separate.tex similarity index 100% rename from src/img/cm/sc_junction_metal_n_sc_separate.tex rename to src/img_static/cm/sc_junction_metal_n_sc_separate.tex diff --git a/src/img/cm/sc_junction_ohmic.tex b/src/img_static/cm/sc_junction_ohmic.tex similarity index 100% rename from src/img/cm/sc_junction_ohmic.tex rename to src/img_static/cm/sc_junction_ohmic.tex diff --git a/src/img/cm/sc_junction_ohmic_separate.tex b/src/img_static/cm/sc_junction_ohmic_separate.tex similarity index 100% rename from src/img/cm/sc_junction_ohmic_separate.tex rename to src/img_static/cm/sc_junction_ohmic_separate.tex diff --git a/src/img/cm/sc_junction_pn.tex b/src/img_static/cm/sc_junction_pn.tex similarity index 100% rename from src/img/cm/sc_junction_pn.tex rename to src/img_static/cm/sc_junction_pn.tex diff --git a/src/main.tex b/src/main.tex index 06fb97d..ac9ade8 100644 --- a/src/main.tex +++ b/src/main.tex @@ -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} diff --git a/src/math/calculus.tex b/src/math/calculus.tex index 7584ad7..ff36b94 100644 --- a/src/math/calculus.tex +++ b/src/math/calculus.tex @@ -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} diff --git a/src/mechanics.tex b/src/mechanics.tex index 4527215..5176aa5 100644 --- a/src/mechanics.tex +++ b/src/mechanics.tex @@ -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} diff --git a/src/pkg/mqformula.sty b/src/pkg/mqformula.sty index 229b30f..a9fe99c 100644 --- a/src/pkg/mqformula.sty +++ b/src/pkg/mqformula.sty @@ -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 diff --git a/src/pkg/mqfqname.sty b/src/pkg/mqfqname.sty index e863991..fe182ce 100644 --- a/src/pkg/mqfqname.sty +++ b/src/pkg/mqfqname.sty @@ -3,7 +3,6 @@ \RequirePackage{mqlua} \RequirePackage{etoolbox} - \begin{luacode} sections = sections or {} diff --git a/src/pkg/mqtranslation.sty b/src/pkg/mqtranslation.sty index 3e0eaaa..d3ff67e 100644 --- a/src/pkg/mqtranslation.sty +++ b/src/pkg/mqtranslation.sty @@ -171,7 +171,7 @@ % (temporarily change fqname to the \fqname: 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}% diff --git a/src/qm/misc.tex b/src/qm/misc.tex new file mode 100644 index 0000000..179235c --- /dev/null +++ b/src/qm/misc.tex @@ -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} + diff --git a/src/qm/qm.tex b/src/qm/qm.tex index d792a9f..3c62cd2 100644 --- a/src/qm/qm.tex +++ b/src/qm/qm.tex @@ -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} - - diff --git a/src/quantum_computing.tex b/src/quantum_computing.tex index 0f9d50c..e1a06a2 100644 --- a/src/quantum_computing.tex +++ b/src/quantum_computing.tex @@ -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{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$} diff --git a/src/statistical_mechanics.tex b/src/statistical_mechanics.tex index 2ee2faf..4351b15 100644 --- a/src/statistical_mechanics.tex +++ b/src/statistical_mechanics.tex @@ -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} @@ -393,25 +393,25 @@ \desc{Isobaric-isothermal}{Gibbs ensemble}{} % \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{const_variables} {$N$, $p$, $T$} + \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) \\ - &= \left(\frac{V}{N}\right)^N \left(\frac{4\pi m E}{3 h^2 N}\right)^{\frac{3N}{2}} \e^\frac{5N}{2} + \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} - diff --git a/src/util/environments.tex b/src/util/environments.tex index baf7ae6..a612ed7 100644 --- a/src/util/environments.tex +++ b/src/util/environments.tex @@ -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}) } } }{ @@ -115,7 +123,11 @@ \hline \directlua{ for _, kv in ipairs(entries) do - tex.print("\\GT{" .. kv.key .. "} & " .. kv.value .. "\\\\") + if kv.raw == true then + tex.print(kv.key .. " & " .. kv.value .. "\\\\") + else + tex.print("\\GT{" .. kv.key .. "} & " .. kv.value .. "\\\\") + end end } \hline diff --git a/src/util/math-macros.tex b/src/util/math-macros.tex index a00bf57..c8706ca 100644 --- a/src/util/math-macros.tex +++ b/src/util/math-macros.tex @@ -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 diff --git a/src/util/tikz_macros.tex b/src/util/tikz_macros.tex index 9ad5717..14e0c7b 100644 --- a/src/util/tikz_macros.tex +++ b/src/util/tikz_macros.tex @@ -1,4 +1,3 @@ - \tikzset{ % bands sc band con/.style={ draw=fg0, thick}, @@ -8,9 +7,9 @@ sc fermi level/.style={draw=fg-aqua,dashed,thick}, % electron filled sc occupied/.style={ - pattern=north east lines, - pattern color=fg-aqua, - draw=none + pattern=north east lines, + pattern color=fg-aqua, + draw=none }, % materials sc p type/.style={ draw=none,fill=bg-yellow!20}, @@ -18,25 +17,28 @@ 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]{ - \draw[<->] (#1,#2) -- (#1,#3) node[midway,right] () {#4}; + \draw[<->] (#1,#2) -- (#1,#3) node[midway,right] () {#4}; } % Band bending down at L-R interface: BendH must be negative % need two functions for different out= angles, or use if else on the sign of BendH \newcommand\leftBandAuto[2]{ \directlua{ if \tkLBendH == 0 then - tex.print([[(\tkLx,#2) \ifblank{#1}{}{node[anchor=east] \detokenize{{#1}}} -- (\tkLW,#2) ]]) + tex.print([[(\tkLx,#2) \ifblank{#1}{}{node[anchor=east] \detokenize{{#1}}} -- (\tkLW,#2) ]]) else - if \tkLBendH > 0 then - angle = 180+45 - else - angle = 180-45 - end - tex.sprint([[(\tkLx,#2) \ifblank{#1}{}{node[anchor=east] \detokenize{{#1}}} - -- (\tkLW-\tkLBendW,#2) to[out=0,in=]], angle, [[](\tkLW,#2+\tkLBendH)]]) + if \tkLBendH > 0 then + angle = 180+45 + else + angle = 180-45 + end + tex.sprint([[(\tkLx,#2) \ifblank{#1}{}{node[anchor=east] \detokenize{{#1}}} + -- (\tkLW-\tkLBendW,#2) to[out=0,in=]], angle, [[](\tkLW,#2+\tkLBendH)]]) end } % % \ifthenelse{\equal{\tkLBendH}{0}}% @@ -48,21 +50,21 @@ } \newcommand\rightBandAuto[2]{ \directlua{ - if \tkRBendH == 0 then - %-- tex.print([[\rightBand{#1}{#2}]]) - tex.print([[(\tkRx,#2) -- (\tkW,#2)]]) %-- \ifblank{#1}{}{node[anchor=west] \{#1\}}]]) - else - if \tkRBendH > 0 then - angle = -45 - else - angle = 45 - end - tex.sprint([[(\tkRx,#2+\tkRBendH) to[out=]], angle, [[,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) ]]) - %-- \ifblank{#1}{}{node[anchor=west] \{#1\}} ]]) - end - if "\luaescapestring{#1}" \string~= "" then - tex.print([[node[anchor=west] \detokenize{{#1}} ]]) - end + if \tkRBendH == 0 then + %-- tex.print([[\rightBand{#1}{#2}]]) + tex.print([[(\tkRx,#2) -- (\tkW,#2)]]) %-- \ifblank{#1}{}{node[anchor=west] \{#1\}}]]) + else + if \tkRBendH > 0 then + angle = -45 + else + angle = 45 + end + tex.sprint([[(\tkRx,#2+\tkRBendH) to[out=]], angle, [[,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) ]]) + %-- \ifblank{#1}{}{node[anchor=west] \{#1\}} ]]) + end + if "\luaescapestring{#1}" \string~= "" then + tex.print([[node[anchor=west] \detokenize{{#1}} ]]) + end } % \ifthenelse{\equal{\tkRBendH}{0}}% % {\rightBand{#1}{#2}} @@ -75,28 +77,28 @@ % } } \newcommand\leftBandDown[2]{ - (\tkRx,#2+\tkRBendH) to[out=45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) - \ifblank{#1}{}{node[anchor=west]{#1}} + (\tkRx,#2+\tkRBendH) to[out=45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) + \ifblank{#1}{}{node[anchor=west]{#1}} } \newcommand\rightBandDown[2]{ - (\tkRx,#2+\tkRBendH) to[out=45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) - \ifblank{#1}{}{node[anchor=west]{#1}} + (\tkRx,#2+\tkRBendH) to[out=45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) + \ifblank{#1}{}{node[anchor=west]{#1}} } % Band bending down at L-R interface: BendH must be positive \newcommand\leftBandUp[2]{ - (\tkLx,#2) \ifblank{#1}{}{node[anchor=east]{#1}} - -- (\tkLW-\tkLBendW,#2) to[out=0,in=180+45] (\tkLW,#2+\tkLBendH) + (\tkLx,#2) \ifblank{#1}{}{node[anchor=east]{#1}} + -- (\tkLW-\tkLBendW,#2) to[out=0,in=180+45] (\tkLW,#2+\tkLBendH) } \newcommand\rightBandUp[2]{ - (\tkRx,#2+\tkRBendH) to[out=-45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) - \ifblank{#1}{}{node[anchor=west]{#1}} + (\tkRx,#2+\tkRBendH) to[out=-45,in=180] (\tkRx+\tkRBendW,#2) -- (\tkW,#2) + \ifblank{#1}{}{node[anchor=west]{#1}} } % Straight band \newcommand\leftBand[2]{ - (\tkLx,#2) \ifblank{#1}{}{node[anchor=east]{#1}} -- (\tkLW,#2) + (\tkLx,#2) \ifblank{#1}{}{node[anchor=east]{#1}} -- (\tkLW,#2) } \newcommand\rightBand[2]{ - (\tkRx,#2) -- (\tkW,#2) \ifblank{#1}{}{node[anchor=west]{#1}} + (\tkRx,#2) -- (\tkW,#2) \ifblank{#1}{}{node[anchor=west]{#1}} } \newcommand\drawAxes{ @@ -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); +}