Ginzburg-Landau theory of superconductivity

Vitaly Lazarevich Ginzburg and Lev Landau applied an order parameter description to superconductivity in 1950. A spatially varying complex amplitude extends the fixed-amplitude description of the London equations. Their initial charge assumption preceded the later identification of the universal pair charge.^[ref]

Consider the following isotropic, local SI Helmholtz free energy model near a continuous superconducting transition temperature:Here $\beta >0$, $m_{\ast }>0$ is a phenomenological mass parameter, and $q=-2e$ fixes the electron-pair convention. The uniform superconducting minimum for $\alpha <0$ has $|{\psi }_0{|}^2=-\alpha /\beta$. The gradient term penalizes changes of the amplitude and the gauge-invariant quantum phase; the magnetic term penalizes magnetic field penetration. The gauge-coupling sign in Ginzburg-Landau superconductivity keeps the electric current and field equation consistent. This model omits material anisotropy, microscopic nonlocality and critical fluctuations.

The gradient and quadratic terms of the stated model balance at the amplitude-variation scale $\xi$. It is a superconducting length scale, distinct from optical coherence length. The weak-field London penetration depth in the same normalization obeysA core suppresses the pairing amplitude over a scale set by $\xi$; screening electric currents and the magnetic field involve $\lambda$. Neither scale is a universal particle size.

In the isotropic single-component model, $\kappa <1/\sqrt{2}$ gives the type-I regime and $\kappa >1/\sqrt{2}$ the type-II regime; the interface-energy distinction belongs to the model, rather than a purely geometric test of core overlap.^[ref]

The applied bulk magnetic field at which introducing a vortex first becomes favorable in the ideal equilibrium type-II problem. Surface barriers, sample shape and flux pinning can change the apparent entry field.

Linearize the stated model about $\psi =0$ in a uniform magnetic field. The kinetic operator has the Landau level ground energy $\hslash |q|B/(2m_{\ast })$. Its cancellation with $\alpha <0$ givesThis is the bulk orbital field scale of this model; it excludes spin-polarization limits and uses the normal-state relation $B_{c2}={\mu }_0{H}_{c2}$. For the chosen $\xi =10\mathrm{nm}$, it is about $3.29\mathrm{T}$. With $\lambda =100\mathrm{nm}$, $\kappa =10$. Counting triangular cells at $B_{c2}$ would give $a/\xi =\sqrt{4\pi /\sqrt{3}}\simeq 2.69$, but it is the linearized field equation, rather than disks of an arbitrarily chosen core radius touching, that supplied the transition criterion.