\def\txL{\text{L}}
\def\gl{\text{GL}}
\def\GL{Ginzburg-Landau }
\def\Tcrit{T_\text{c}}
\def\Bcth{B_\text{c}}

\Section[
    \eng{Superconductivity}
    \ger{Supraleitung}
]{sc}
    \begin{ttext}
        \eng{
            Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
            Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
            \\\textbf{Type I}: 
            \\\textbf{Type II}: Has two critical 
        }
        \ger{
            Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
            Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.

        }
    \end{ttext}

    \begin{formula}{type1}
        \desc{Type-I superconductor}{}{}
        \desc[german]{Typ-I Supraleiter}{}{}
        \ttxt{\eng{
            Has a single critical magnetic field, $\Bcth$.
            \\$B < \Bcth$: \fRef{:::meissner_effect}
            \\$B > \Bcth$: Normal conductor
        }}
    \end{formula}

    \begin{formula}{type2}
        \desc{Type-II superconductor}{}{}
        \desc[german]{Typ-II Supraleiter}{}{}
        \ttxt{\eng{
            Has a two critical magnetic fields.
            \\$B < B_\text{c1}$: \fRef{:::meissner_effect}
            \\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
            \\$B > B_\text{c2}$: Normal conductor
        }}
    \end{formula}

    \begin{formula}{perfect_conductor}
        \desc{Perfect conductor}{}{}
        \desc[german]{Ideale Leiter}{}{}
        \ttxt{
            \eng{
                In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
                (\fRef{ed:fields:mag:induction:lenz})
            }
            \ger{
                Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
                (\fRef{ed:fields:mag:induction:lenz})
            }
        }
    \end{formula}

    \begin{formula}{meissner_effect}
        \desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
        \desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
        \ttxt{
            \eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.} 
            \ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.} 
        }
    \end{formula}

    \begin{formula}{condensation_energy}
        \desc{Condensation energy}{}{\QtyRef{gibbs_energy}}
        \desc[german]{Kondensationsenergie}{}{}
        \eq{
            \d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
            G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
        }
    \end{formula}


    \Subsection[
        \eng{London equations}
        \ger{London-Gleichungen}
    ]{london}
        \begin{ttext}
            \eng{
                Quantitative description of the \fRef{cm:sc:meissner_effect}.
            }
            \ger{
                Quantitative Beschreibung des \fRef{cm:sc:meissner_effect}s.
            }
        \end{ttext}
        % \begin{formula}{coefficient}
        %     \desc{London-coefficient}{}{}
        %     \desc[german]{London-Koeffizient}{}{}
        %     \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
        % \end{formula}
        \Eng[of_sc_particle]{of the superconducting particle}
        \Ger[of_sc_particle]{der Supraleitenden Teilchen}
        \begin{formula}{first}
            % \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
            \desc{First London Equation}{}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{electric_field}}
            \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}
            \desc{Second London Equation}{Describes the \fRef{cm:sc:meissner_effect}}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{magnetic_field}}
            \desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:sc:meissner_effect}}{}
            \eq{
                \Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
            }
        \end{formula}
        \begin{formula}{penetration_depth}
            \desc{London penetration depth}{Depth at which $B$ is $1/\e$ times the value of $B_\text{ext}$}{$m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}}
            \desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
            \eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
        \end{formula}
        \begin{formula}{penetration_depth_temp}
            \desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
            \desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
            \eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
        \end{formula}



    \Subsection[
        \eng{\GL Theory (GLAG)}
        \ger{\GL Theorie (GLAG)}
    ]{gl}
        \begin{ttext}
            \eng{
                \TODO{TODO}
            }
        \end{ttext}

        \begin{formula}{boundary_energy}
            \desc{Boundary energy}{}{$\Delta E_\text{boundary}$ \TODO{TODO}}
            \desc[german]{Grenzflächenenergie}{}{}
            \eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda) \frac{B_\text{c,th}^2}{2\mu_0}}
        \end{formula}

        \begin{formula}{coherence_length}
            \desc{\GL Coherence Length}{}{}
            \desc[german]{\GL Kohärenzlänge}{}{}
            \eq{
                \xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
                \xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
            }
        \end{formula}
        \begin{formula}{penetration_depth}
            \desc{\GL Penetration Depth / Field screening length}{}{}
            \desc[german]{\GL Eindringtiefe}{}{}
            \eq{
                \lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
                \lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
            }
        \end{formula}
        \begin{formula}{first}
            \desc{First Ginzburg-Landau Equation}{}{$\xi_\gl$ \fRef{cm:sc:gl:coherence_length}, $\lambda_\gl$ \fRef{cm:sc:gl:penetration_depth}}
            \desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
            \eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
        \end{formula}
        \begin{formula}{second}
            \desc{Second Ginzburg-Landau Equation}{}{}
            \desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
            \eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
        \end{formula}
        

        \TODO{proximity effect}

    \Subsection[
        \eng{Microscopic theory}
        \ger{Mikroskopische Theorie}
    ]{micro}

        \begin{formula}{isotop_effect}
            \desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby of the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
            \desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
            \eq{
                \Tcrit \propto \frac{1}{\sqrt{M}} \\
                \omega_\text{ph} \propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
            }
        \end{formula}

        \begin{formula}{cooper_pairs}
            \desc{Cooper pairs}{}{}
            \desc[german]{Cooper-Paars}{}{}
            \ttxt{
                \eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
            }
        \end{formula}
    
        \Subsubsection[
            \eng{BCS-Theory}
            \ger{BCS-Theorie}
        ]{bcs}
            \begin{ttext}
                \eng{
                    Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state. 
                    The wave function spans the whole material, which makes it conduct without resistance.
                    The exchange bosons between the electrons are phonons.
                }
                \ger{
                    Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können. 
                    Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert.
                    Die Austauschbosononen zwischen den Elektronen sind Bosonen.
                }
            \end{ttext}
            \def\BCS{{\text{BCS}}}
            \begin{formula}{hamiltonian}
                \desc{BCS Hamiltonian}{for $N$ interacting electrons}{
                    $c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
                    First term: non-interacting free electron gas\\
                    Second term: interaction energy
                }
                \desc[german]{BCS Hamiltonian}{}{}
                \eq{
                    \hat{H}_\BCS = 
                    \sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
                    + \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime} 
                        \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
                        \hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
                }
            \end{formula}

            \begin{formula}{bogoliubov-valatin}
                \desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:sc:micro:bcs:hamiltonian} to derive excitation energies}{}
                \desc[german]{Bogoliubov-Valatin transformation}{}{}
                \eq{
                    \hat{H}_\BCS  - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
                }
            \end{formula}

            \begin{formula}{gap_equation}
                \desc{BCS-gap equation}{}{}
                \desc[german]{}{}{}
                \eq{\Delta_\veck^* = -\sum_\veck^+\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
            \end{formula}

            \begin{formula}{tcrit_temp}
                \desc{Temperatur dependance of the crictial temperature}{}{}
                \desc[german]{Temperaturabhängigkeit der kritischen Temperatur}{}{}
                \eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{t}{T_\txc}\right) \right] }
                \TODO{empirical relation, relate to BCS}
            \end{formula}
