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.

A phenomenological near-transition relaxation equation for a charged complex order parameter, with a positive real relaxation coefficient in its simplest dissipative form. Under $\Psi \mapsto e^{iq\chi /\hslash }\Psi$, $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi -{\partial }_t\chi$, the displayed time derivative transforms covariantly, as does the spatial derivative in the Ginzburg-Landau theory of superconductivity. Microscopic applicability depends on regime and scattering processes; this equation is distinct from a universal conservative condensate wave equation.

Expanding the BCS theory near the transition in small pair amplitude and slow spatial gradients produces a local complex order-parameter free energy. The microscopic theory fixes its normalization and coefficients, including the pair's twice-electron charge. This establishes a controlled near-transition relation rather than an exact all-temperature replacement for BCS theory.^[ref]

A pairing-channel Hubbard–Stratonovich transformation replaces an attractive fermion interaction by a complex Cooper pair field coupled to fermion pairs. A Grassmann Gaussian integral then gives a trace-log effective action, with its trace convention fixed to avoid doubling independent spin-pair blocks. The uniform stationary equation is the BCS gap equation. A small-field, slow-gradient expansion near the transition yields Ginzburg-Landau theory of superconductivity; the normalization of the phenomenological field need not equal that of the microscopic gap.

For a clean isotropic three-dimensional weak-coupling s-wave model, use the energy-valued gap $\Delta$, a density of states $N(0)$ per spin, and free-energy density ${\alpha }_{\Delta }|\Delta {|}^2+{\beta }_{\Delta }|\Delta {|}^4/2+K_{\Delta }|D\Delta {|}^2$. Near $T_c$, ${\alpha }_{\Delta }\simeq N(0)(T-T_c)/T_c$. The quartic coefficient follows fromThe integral is $\pi /(2|{\omega }_n{|}^3)$ and the positive odd-integer sum is $7\zeta (3)/8$, yielding the displayed coefficient. The isotropic Fermi-velocity average gives $K_{\Delta }={\beta }_{\Delta }{\hslash }^2{v}_F^2/6$. The pair charge is $q=-2e$ in $D=\nabla -iq\mathbf{A}/\hslash$. These values change under a rescaling of $\Delta$, different density-of-states convention or different material model; the local expansion is a near-transition approximation.^[ref]

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]

Combining $B_{c1}\simeq {\Phi }_0\ln \kappa /(4\pi {\lambda }^2)$ and $B_{c2}={\Phi }_0/(2\pi {\xi }^2)$ gives $r=B_{c1}/B_{c2}=\ln \kappa /(2{\kappa }^2)$. Its derivative is $(1-2\ln \kappa )/(2{\kappa }^3)$: the ratio peaks at $\kappa =\sqrt{e}$, with maximum $1/(4e)$. Thus each $0<r<1/(4e)$ has two solutions with $\kappa >1$. The large-$\kappa$ branch is the one consistent with using a leading-log approximation; the small branch cannot be selected or rejected by the inequality $\kappa >1$ alone. Omitted core-energy terms limit numerical accuracy.

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.