532 lines
30 KiB
TeX
532 lines
30 KiB
TeX
\def\txL{\text{L}}
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\def\gl{\text{GL}}
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\def\GL{Ginzburg-Landau }
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\def\Tcrit{T_\text{c}}
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\def\Bcth{B_\text{c,th}}
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\Section{super}
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\desc{Superconductivity}{
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Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
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Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
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}{}
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\desc[german]{Supraleitung}{
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Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
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Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.
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}{}
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\begin{formula}{type1}
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\desc{Type-I superconductor}{}{}
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\desc[german]{Typ-I Supraleiter}{}{}
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\ttxt{\eng{
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Has a single critical magnetic field, $\Bcth$.
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\\$B < \Bcth$: \fRef{:::meissner_effect}
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\\$B > \Bcth$: Normal conductor
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\\ Very small usable current density because current only flows within the \fRef{cm:super:london:penetration_depth} of the surface.
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}}
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\end{formula}
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\begin{formula}{type2}
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\desc{Type-II superconductor}{}{}
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\desc[german]{Typ-II Supraleiter}{}{}
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\ttxt{\eng{
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Has a two critical magnetic fields.
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\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
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\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
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\\$B > B_\text{c2}$: Normal conductor
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\\ In \fRef{:::shubnikov_phase} larger usable current density because current flows within the \fRef{cm:super:london:penetration_depth} of the surface and the penetrating flux lines.
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}}
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\end{formula}
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\begin{formula}{perfect_conductor}
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\desc{Perfect conductor}{}{}
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\desc[german]{Ideale Leiter}{}{}
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\ttxt{
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\eng{
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In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
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(\fRef{ed:em:induction:lenz})
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}
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\ger{
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Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
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(\fRef{ed:em:induction:lenz})
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}
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}
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\end{formula}
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\begin{formula}{meissner_effect}
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\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
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\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
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\ttxt{
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\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.}
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\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.}
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}
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\end{formula}
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\begin{formula}{bcth}
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\desc{Thermodynamic cricitial field}{for \fRef[type I]{::type1} and \fRef[type II]{::type2}}{}
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\desc[german]{Thermodynamisches kritische Feldstärke}{für \fRef[type I]{::type1} und \Ref[type II]{::type2}}{}
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\eq{g_\txs - g_\txn = - \frac{\Bcth^2(T)}{2\mu_0}}
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\end{formula}
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\begin{formula}{shubnikov_phase}
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\desc{Shubnikov phase}{in \fRef{::type2}}{}
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\desc[german]{Shubnikov-Phase}{}{}
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\ttxt{\eng{
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Mixed phase in which some magnetic flux penetrates the superconductor.
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}\ger{
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Gemischte Phase in der der Supraleiter teilweise von magnetischem Fluss durchdrungen werden kann.
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}}
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\end{formula}
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\begin{formula}{condensation_energy}
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\desc{Condensation energy}{}{\QtyRef{free_enthalpy}, \ConstRef{magnetic_vacuum_permeability}}
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\desc[german]{Kondensationsenergie}{}{}
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\eq{
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\d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
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G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
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}
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\end{formula}
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\Subsection{london}
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\desc{London Theory}{}{}
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\desc[german]{London-Theorie}{}{}
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\begin{formula}{description}
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\desc{Description}{}{}
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\desc[german]{Beschreibung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Phenomenological theory
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\item Quantitative description of the \fRef{cm:super:meissner_effect}.
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\item Assumies uniform charge density $n(\vecr,t) = n(t)$ (London-approximation).
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\item Does not work near $T_\txc$
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item Phänomenologische Theorie
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\item Quantitative Beschreibung des \fRef{cm:super:meissner_effect}s.
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\item Annahme: uniforme Ladungsdichte $n(\vecr,t) = n(t)$ (London-Näherung)
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\item Funktioniert nicht nahe $T_\txc$
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\end{itemize}
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}}
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\end{formula}
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% \begin{formula}{coefficient}
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% \desc{London-coefficient}{}{}
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% \desc[german]{London-Koeffizient}{}{}
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% \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
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% \end{formula}
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\Eng[of_sc_particle]{of the superconducting particle}
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\Ger[of_sc_particle]{der Supraleitenden Teilchen}
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\begin{formula}{first}
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% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
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\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}}
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\desc[german]{Erste London-Gleichun-}{}{}
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\eq{
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\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
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}
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\end{formula}
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\begin{formula}{second}
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\desc{Second London Equation}{Describes the \fRef{cm:super: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_flux_density}}
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\desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:super:meissner_effect}}{}
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\eq{
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\Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
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}
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\end{formula}
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\begin{formula}{penetration_depth}
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\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}}
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\desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
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\eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
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\end{formula}
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\begin{formula}{penetration_depth_temp}
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\desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
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\desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
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\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
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\end{formula}
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\Subsubsection{macro}
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\desc{Macroscopic wavefunction}{}{}
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\desc[german]{Makroskopische Wellenfunktion}{}{}
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\begin{formula}{ansatz}
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\desc{Ansatz}{}{}
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\desc[german]{Ansatz}{}{}
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\ttxt{\eng{Alternative derivation of London equations by assuming a macroscopic wavefunction which is uniform in space}\ger{Alternative Herleitung der London-Gleichungen durch Annahme einer makroskopischen Wellenfunktion, welche nicht Ortsabhängig ist}}
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\eq{\Psi(\vecr,t) = \Psi_0(\vecr,t) \e^{\theta(\vecr,t)}}
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\end{formula}
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\begin{formula}{energy-phase_relation}
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\desc{Energy-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, $\phi_\text{el}$ \qtyRef{electric_scalar_potential}, \QtyRef{chemical_potential}}
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\desc[german]{Energie-Phase Beziehung}{}{}
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\eq{\hbar \pdv{\theta(\vecr,t)}{t} = - \left(\frac{m_\txs}{n_\txs^2 q_\txs^2} \vecj_\txs^2(\vecr,t) + q_\txs\phi_\text{el}(\vecr,t) + \mu(\vecr,t)\right)}
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\end{formula}
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\begin{formula}{current-phase_relation}
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\desc{Current-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, \QtyRef{magnetic_vector_potential}}
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\desc[german]{Strom-Phase Beziehung}{}{}
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\eq{\vecj_\txs(\vecr,t) = \frac{q_\txs^2 n_\txs(\vecr,t)}{m_\txs} \left(\frac{\hbar}{q_\txs} \Grad\theta(\vecr,t) - \vecA(\vecr,t)\right) }
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\end{formula}
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\Subsubsection{josephson}
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\desc{Josephson Effect}{}{}
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\desc[german]{Josephson Effekt}{}{}
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\begin{formula}{1st_relation}
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\desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\phi$ phase difference accross junction}
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\desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\phi$ Phasendifferenz zwischen den Supraleitern}
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\eq{\vecj_\txs(\vecr,t) = \vecj_\text{C}(\vecr,t) \sin\phi(\vecr,t)}
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\end{formula}
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\begin{formula}{2nd_relation}
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\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\phi$ phase differnce accross junction, \ConstRef{flux_quantum}, \QtyRef{voltage}}
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\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\phi$ Phasendifferenz, \ConstRef{flux_quantum}, \QtyRef{voltage}}
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\eq{\odv{\phi(t)}{t} = \frac{2\pi}{\Phi_0} U(t)}
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\end{formula}
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\begin{formula}{coupling_energy}
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\desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction}
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\desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
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\eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)}
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\end{formula}
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\Subsection{gl}
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\desc{\GL Theory (GLAG)}{}{}
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\desc[german]{\GL Theorie (GLAG)}{}{}
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\begin{formula}{description}
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\desc{Description}{}{}
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\desc[german]{Beschreibung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Phenomenological theory
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\item Improvement on the Landau-Theory of 2nd order phase transitions
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% which introduces an order parameter that is $0$ in the normal state and rises to saturation in the superconducting state.
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\item Additional complex, position-dependent order parameter is introduced $\Psi(\vecr)$
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\item Only valid close to $T_\txc$.
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\item Does not have time dependancy
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\end{itemize}
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}\ger{
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\begin{itemize}
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\item Phänomenologische Theorie
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\item Weiterentwicklung der Landau-Theorie für Phasenübergänge zweiter Ordnung,
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% in der ein Ordnungsparameter in the normalen Phase 0 ist und ein der supraleitenden Phase bis zur Sättigung ansteigt.
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\item Zusätzlicher, komplexer, ortsabhängiger Ordnungsparameter $\Psi(\vecr)$
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\item Nur nahe $T_\txc$ gültig.
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\item Beschreibt keine Zeitabhängigkeit
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\end{itemize}
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}}
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\end{formula}
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\begin{formula}{expansion}
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\desc{Expansion}{Expansion of free enthalpy of superconducting state}{
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$g_{\txs/\txn}$ specific \qtyRef{free_enthalpy} of superconducting/normal state,
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$\Psi(\vecr) = \abs{\Psi_0(\vecr)} \e^{\I\theta(\vecr)}$ order parameter,
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$n(\vecr) = \abs{\Psi}^2$ Cooper-Pair density,
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\QtyRef{magnetic_flux_density},
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\QtyRef{magnetic_vector_potential},
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$\alpha(T) = -\bar{\alpha} \left(1-\frac{T}{T_\txc}\right)^2$,
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% $\alpha > 0$ for $T > T_\txc$ and $\alpha < 0$ for $T< T_\txc$,
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$\beta = \const > 0$
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}
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% \desc[german]{}{}{}
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\begin{multline}
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g_\txs = g_\txn + \alpha \abs{\Psi}^2 + \frac{1}{2}\beta \abs{\Psi}^4 +
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\\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(-\I\hbar\Grad - q_\txs \vecA\right)\Psi}^2 + \dots
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\end{multline}
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\end{formula}
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\begin{formula}{first}
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\desc{First Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\Psi$ in \fRef{::expansion}}{
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$\xi_\gl$ \fRef{cm:super:gl:coherence_length},
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$\lambda_\gl$ \fRef{cm:super:gl:penetration_depth}
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}
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\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
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\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
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\end{formula}
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\begin{formula}{second}
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\desc{Second Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\vec{A}$ in \fRef{::expansion}}{}
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\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
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\eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
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\end{formula}
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\begin{formula}{coherence_length}
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\desc{\GL Coherence Length}{Depth in the superconductor where $\abs{\Psi}$ goes from 0 to 1}{}
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\desc[german]{\GL Kohärenzlänge}{Tiefe im Supraleiter, bei der $\abs{\Psi}$ von 0 auf 1 steigt}{}
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\eq{
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\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
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\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
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}
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\end{formula}
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\begin{formula}{penetration_depth}
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\desc{\GL Penetration Depth}{Field screening length\\Depth in the supercondcutor where $B_\text{ext}$ decays}{}
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\desc[german]{\GL Eindringtiefe}{Tiefe im Supraleiter, bei der $B_\text{ext}$ abfällt}{}
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\eq{
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\lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
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\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
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}
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\end{formula}
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\begin{formula}{boundary_energy}
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\desc{Boundary energy}{Negative for \fRef{:::type2}, positive for \fRef{:::type1}}{$\Delta E_\text{B}$ energy gained by expelling the external magnetic field, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}}
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\desc[german]{Grenzflächenenergie}{Negativ für \fRef{:::type2}, positiv für \fRef{:::type1}}{}
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\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda_\gl) \frac{B_\text{c,th}^2}{2\mu_0}}
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\end{formula}
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\begin{formula}{parameter}
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\desc{Ginzburg-Landau parameter}{}{}
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\desc[german]{Ginzburg-Landau Parameter}{}{}
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\eq{\kappa \equiv \frac{\lambda_\gl}{\xi_\gl}}
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\eq{
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\kappa \le \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type1}} \\
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\kappa \ge \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type2}}
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}
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\end{formula}
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\begin{formula}{ns_boundary}
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\desc{Normal-superconductor boundary}{}{}
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\desc[german]{Normal-Supraleiter Grenzfläche}{}{}
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\eq{
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\abs{\Psi(x)}^2 &= \frac{n_\txs(x)}{n_\txs(\infty)} = \tanh^2 \left(\frac{x}{\sqrt{2}\xi_\gl}\right) \\
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B_z(x) &= B_z(0) \Exp{-\frac{x}{\lambda_\gl}}
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}
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\fig{img/cm_super_n_s_boundary.pdf}
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% \TODO{plot, slide 106}
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\end{formula}
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\begin{formula}{bcth}
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\desc{Thermodynamic critical field}{}{}
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\desc[german]{Thermodynamisches kritisches Feld}{}{}
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\eq{\Bcth = \frac{\Phi_0}{2\pi \sqrt{2} \xi_\gl \lambda_\gl}}
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\end{formula}
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\begin{formula}{bc1}
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\desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda_\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}}
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\desc[german]{Unteres kritisches Magnetfeld}{Über $B_\text{c1}$ dringt erstmals Fluss in die supraleitende Phase ein}{}
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\eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda_\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth}
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\end{formula}
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\begin{formula}{bc2}
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\desc{Upper critical magnetic field}{Above $B_\text{c2}$, superconducting phase is is destroyed}{\ConstRef{flux_quantum}, $\xi_\gl$ \fRef{::coherence_length}}
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\desc[german]{Oberes kritisches Magnetfeld}{Über $B_\text{c2}$ ist die supraleitende Phase zerstört}{}
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\eq{B_\text{c2} = \frac{\Phi_0}{2\pi\xi_\gl^2}}
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\end{formula}
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\begin{formula}{proximity_effect}
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\desc{Proximity-Effect}{}{}
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% \desc[german]{}{}{}
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\ttxt{\eng{
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Superconductor wavefunction extends into the normal conductor or isolator
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}}
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\end{formula}
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\Subsection{micro}
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\desc{Microscopic theory}{}{}
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\desc[german]{Mikroskopische Theorie}{}{}
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\begin{formula}{isotop_effect}
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\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby on the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
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\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
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\eq{
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\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{bcs}
|
|
\desc{BCS-Theory}{}{}
|
|
\desc[german]{BCS-Theorie}{}{}
|
|
\begin{formula}{description}
|
|
\desc{Description}{}{}
|
|
\desc[german]{Beschreibung}{}{}
|
|
\ttxt{\eng{
|
|
\begin{itemize}
|
|
\item Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state
|
|
\item The wave function spans the whole material, which makes it conduct without resistance
|
|
\item The exchange bosons between the electrons are phonons
|
|
\end{itemize}
|
|
}\ger{
|
|
\begin{itemize}
|
|
\item Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
|
|
\item Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert
|
|
\item Die Austauschbosononen zwischen den Elektronen sind Bosonen
|
|
\end{itemize}
|
|
}}
|
|
\end{formula}
|
|
|
|
\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}{ansatz}
|
|
\desc{BCS ground state wave function Ansatz}{\fRef{comp:est:mean_field} approach\\Coherent fermionic state}{}
|
|
\desc[german]{BCS Grundzustandswellenfunktion-Ansatz}{\fRef{comp:est:mean_field} Ansatz\\Kohärenter, fermionischer Zustand}{}
|
|
\eq{\Ket{\Psi_\BCS} = \prod_{\veck=\veck_1,\dots,\veck_M} \left(u_\veck + v_\veck \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger\right) \ket{0} }
|
|
\end{formula}
|
|
|
|
\begin{formula}{coherence_factors}
|
|
\desc{BCS coherence factors}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ probability that pair state is $(\veck\uparrow,\,-\veck\downarrow)$ is empty/occupied, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
|
\desc[german]{BCS Kohärenzfaktoren}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ Wahrscheinlichkeit, dass Paarzustand $(\veck\uparrow,\,-\veck\downarrow)$ leer/besetzt ist, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
|
\eq{
|
|
u_\veck &= \frac{1}{\sqrt{1+\abs{\alpha_\veck}^2}} \\
|
|
v_\veck &= \frac{\alpha_\veck}{\sqrt{1+\abs{\alpha_\veck}^2}}
|
|
}
|
|
\end{formula}
|
|
|
|
\begin{formula}{potential}
|
|
\desc{BCS potential approximation}{}{}
|
|
\desc[german]{BCS Potentialnäherung}{}{}
|
|
\eq{
|
|
V_{\veck,\veck^\prime} =
|
|
\left\{ \begin{array}{rc}
|
|
-V_0 & k^\prime > k_\txF,\, k<k_\txF + \Delta k\\
|
|
0 & \tGT{else}
|
|
\end{array}\right.
|
|
}
|
|
\end{formula}
|
|
|
|
\begin{formula}{gap_at_t0}
|
|
\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\\
|
|
\frac{\Delta(T=0)}{\kB T_\txc} &= \frac{\pi}{\e^\gamma} = 1.764
|
|
}
|
|
\end{formula}
|
|
|
|
|
|
\begin{formula}{cooper_pair_binding_energy}
|
|
\desc{Binding energy of Cooper pairs}{}{$E_\txF$ \absRef{fermi_energy}, \QtyRef{debye_frequency}, $V_0$ retarded potential, $D$ \qtyRef{dos}}
|
|
\desc[german]{Bindungsenergie von Cooper-Paaren}{}{}
|
|
\eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0 D(E_\txF)}}}
|
|
\end{formula}
|
|
|
|
\Subsubsection{excite}
|
|
\desc{Excitations and finite temperatures}{}{}
|
|
\desc[german]{Anregungen und endliche Temperatur}{}{}
|
|
\begin{formula}{description}
|
|
\desc{Description}{}{}
|
|
\desc[german]{Beschreibung}{}{}
|
|
\ttxt{\eng{
|
|
The ground state consists of \fRef{cm:super:micro:cooper_pairs} and the excited state of Bogoliubov quasi-particles (electron-hole pairs).
|
|
The states are separated by an energy gap $\Delta$.
|
|
}\ger{
|
|
Den Grundzustand bilden \fRef{cm:super:micro:cooper_pairs} und den angeregten Zustands Bogoloiubons (Elektron-Loch Quasipartikel).
|
|
Die Zustände sind durch eine Energielücke $\Delta$ getrennt.
|
|
}}
|
|
\end{formula}
|
|
\begin{formula}{bogoliubov-valatin}
|
|
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{
|
|
$\xi_\veck = \epsilon_\veck-\mu$ Energy relative to the \qtyRef{chemical_potential},
|
|
\\ $E_\veck$ \fRef{::excitation_energy},
|
|
\\ $\Delta$ Gap
|
|
\\ $g_\veck$ \fRef{::pairing_amplitude},
|
|
\\ $\alpha / \beta$ create and destroy symmetric/antisymmetric Bogoliubov quasiparticles
|
|
}
|
|
\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 \alpha_\veck^\dagger \alpha_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
|
|
}
|
|
\end{formula}
|
|
|
|
\begin{formula}{pairing_amplitude}
|
|
\desc{Pairing amplitude}{}{}
|
|
\desc[german]{Paarungsamplitude}{}{}
|
|
\eq{g_\veck \equiv \Braket{\hat{c}_{-\veck\downarrow} \hat{c}_{\veck\uparrow}}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{excitation_energy}
|
|
\desc{Excitation energy}{}{}
|
|
\desc[german]{Anregungsenergie}{}{}
|
|
\eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{coherence_factors_energy}
|
|
\desc{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}}
|
|
\desc[german]{Energieabhängigkeit der \fRef{:::bcs:coherence_factors}}{}{}
|
|
\eq{
|
|
\abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\
|
|
\abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\
|
|
u_\veck^* v_\veck &= \frac{\Delta_\veck}{2E_\veck}
|
|
}
|
|
\end{formula}
|
|
|
|
|
|
\begin{formula}{gap_equation}
|
|
\desc{Self-consistend gap equation}{}{}
|
|
\desc[german]{Selbstkonsitente Energielückengleichung}{}{}
|
|
\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}{gap_t}
|
|
\desc{Temperature dependence of the BCS gap}{}{}
|
|
\desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{}
|
|
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \sqrt{1-\frac{T}{T_\txC}}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{dos}
|
|
\desc{Quasiparticle density of states}{}{}
|
|
\desc[german]{Quasiteilchen Zustandsdichte}{}{}
|
|
\eq{D_\txs(E_\veck) = D_\txn(\xi_\veck) \pdv{\xi_\veck}{E_\veck} = \left\{
|
|
\begin{array}{ll}
|
|
D_\txn(E_\txF) \frac{E_\veck}{\sqrt{E^2_\veck -\Delta^2}} & E_\veck > \Delta \\
|
|
& E_\veck < \Delta
|
|
\end{array}
|
|
\right.}
|
|
\end{formula}
|
|
|
|
\begin{formula}{Bcth_temp}
|
|
\desc{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{}
|
|
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{Sprung bei $T_\txc$, denn exponentieller Abfall}{}
|
|
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] }
|
|
% \TODO{empirical relation, relate to BCS}
|
|
\end{formula}
|
|
|
|
\begin{formula}{heat_capacity}
|
|
\desc{Heat capacity in superconductors}{}{}
|
|
\desc[german]{Wärmekapazität in Supraleitern}{}{}
|
|
\fsplit{
|
|
\fig{img/cm_super_heat_capacity.pdf}
|
|
}{
|
|
\eq{c_\txs \propto T^{-\frac{3}{2}} \e^{\frac{\Delta(0)}{\kB T}}}
|
|
}
|
|
\end{formula}
|
|
|
|
\Subsubsection{pinning}
|
|
\desc{Flux pinning}{}{}
|
|
\desc[german]{Haftung von Flusslinien}{}{}
|
|
\begin{formula}{description}
|
|
\desc{Description}{}{}
|
|
\desc[german]{Beschreibung}{}{}
|
|
\ttxt{\eng{
|
|
If a current flows in a \fRef{cm:super:type2}s in the \fRef{cm:super:shubnikov_phase} perpendicular to the penetrating flux lines,
|
|
the lines experience a Lorentz force. This leads to ohmic behaviour of the superconductor.
|
|
The flux lines can be pinned to defects, in which the superconducting order parameter is reduced.
|
|
To move the flux line out of the defect, work would have to be spent overcoming the \fRef{cm:super:micro:pinning:potential}.
|
|
This restores the superconductivity.
|
|
}\ger{
|
|
Wenn ein Strom in einem \fRef{cm:super:type2}s in der \fRef{cm:super:shubnikov_phase} senkrecht zu den eindringenden Flusslinien fließt, erfahren die Linien eine Lorentzkraft.
|
|
Dies führt zu einem ohmschen Verhalten des Supraleiters.
|
|
Die Flusslinien können an Defekten festgehalten werden, in denen der supraleitende Ordnungsparameter reduziert ist.
|
|
Um die Flusslinie aus dem Defekt zu bewegen, müsste Arbeit aufgewendet werden, um das \fRef{cm:super:micro:pinning:potential} zu überwinden.
|
|
Dies stellt die Supraleitfähigkeit wieder her.
|
|
}}
|
|
\end{formula}
|