GLEquations

Derivation of Ginzburg Landau equations

This section may be skipped, at least at first reading. Following the general GL theory for inhomogeneous orderer parameters {$\psi$} (complex) in the presence of magnetic field, {$\mathbf B = \boldsymbol \nabla\times\mathbf A=\mu_0(\mathbf H + \mathbf M)$}, written in terms of the vector potential {$\mathbf A$}, the Gibbs free energy is {$$ G_s(T,H) = G_n(T,H) +\int d\mathbf r g_s(T;\mathbf r) $$} The Gibbs free energy density {$g_s(T;\mathbf r)$}, contains two modifications with respect to the {$H=0$} case. We must apply the minimal substitution, {$-i\hbar\boldsymbol\nabla\rightarrow-i\hbar\boldsymbol\nabla-q\mathbf A$} to the gradient term (and {$q$} turns out to be equal to {$-2e$} for the Cooper pair). We must also add the density of work done by the field, the opposite of the corresponding energy density {$-B^2/2\mu_0$}. For ease of notation we rename {$a$} the whole quadratic coefficient, normally {$a(T_c-T)$}. We obtain

{$$ g_s(T;\mathbf r) = a|\psi|^2+\frac b 2 |\psi|^4 + \frac 1{2m^*}|(-i\hbar\boldsymbol\nabla-q\mathbf A)\psi|^2 + \frac 1 {2\mu_0}|\boldsymbol\nabla \times A|^2 $$}

The variational principle must be applied for three independent parameters:

• {$\psi^*\rightarrow\psi^*+\delta\psi^*$}, that yields the Ginzburg-Landau equation for the complex order parameter {$\psi$},
• {$\psi\rightarrow\psi+\delta\psi$}, that yields the conjugate equation (thus we may skip it)
• {$\mathbf A \rightarrow \mathbf A + \delta \mathbf A$}, that yields the equation for the source of the magnetization, the current density {$J_{ext}+J_s$}; with boundary conditions on an infinite superconducting cylinder {$J_s$}, since {$J_{ext}=0$} in the volume.

We require that {$\delta\int d\mathbf r g_s(T;\mathbf r)=0$} for each independent variation. As usual the trick is to write the integral kernel as a coefficient times the selected variation. Then the integral will always vanish, for any variation, if and only if the coefficient vanishes, which yields the GL equation.

For the order parameter, the variation of the density is {$\delta g_s = g_s(\psi^*+\delta\psi^*)-g_s(\psi^*)$}, i.e.

{$$ \delta g_s = \left[a\psi + b |\psi|^2\psi + \frac 1 {2m^*}(iq\hbar\boldsymbol\nabla \psi\cdot \mathbf A+ q^2 A^2 \psi)\right]\delta \psi^* + \frac 1 {2m^*} (\hbar^2\boldsymbol\nabla \psi-iq\hbar\mathbf A \psi)\cdot\boldsymbol\nabla \delta\psi^* $$}.

The second part of the RHS must be integrated by parts, with {$\int d\mathbf r\, \mathbf g \cdot\boldsymbol\nabla f=\int_\Sigma \,f\mathbf g \cdot d\mathbf \sigma - \int d\mathbf r f \boldsymbol\nabla\cdot\mathbf g$}. The surface integral on the boundary of the superconductor vanishes since the order parameter there, hence we can rewrite

{$$ \delta g_s = \left(a\psi + b |\psi|^2\psi + \frac 1 {2m^*}\left[-\hbar^2\nabla^2 \psi +iq\hbar(\boldsymbol\nabla \psi\cdot \mathbf A+ \boldsymbol\nabla \cdot\mathbf A\psi) +q^2 A^2 \psi\right]\right)\delta \psi^* $$}.

First GL equation

Hence the first GL equation is {$$ \begin{equation} \left[\frac 1 {2m^*}\left(-i\hbar\boldsymbol \nabla-q\mathbf A\right)^2 + a + b |\psi|^2 \right]\psi=0 \end{equation} $$}

where {$(-i\hbar\boldsymbol\nabla-q\mathbf A)^2=-\hbar^2\nabla^2+iq\hbar\boldsymbol\nabla\cdot\mathbf A+iq\hbar\mathbf A \cdot\boldsymbol\nabla +q^2A^2$}.

It is easier at first to neglect the second order coefficient {$b$}. In this case the GL equation is identical to a Schroedinger equation for a boson wave function, in the presence of a magnetic field (as granted by the minimal substitution). The square modulus of the boson ground state wave function (not normalized to 1, since more bosons may occupy the same ground state) represents the density of superconducting carriers. The inclusion of the term in {$b$} makes this a non-linear Schroedinger equation, complicating greatly the solution, but preserving the meaning of {$|\Psi|^2=n_s$}.

For the vector potential the same procedure, {$\delta g_s = g_s(\mathbf A+\delta\mathbf A)-g_s(\mathbf A)$}, yields

{$$ \delta g_s = \left[\frac {iq\hbar}{2m^*}(\psi^*\boldsymbol\nabla\psi - \psi\boldsymbol\nabla\psi^* ) + \frac {q^2}{m^*}|\psi|^2 \mathbf A\right]\cdot\delta\mathbf A + \frac {q^2}{2m^*}|\psi|^2 \delta A^2 + \frac 1 {\mu_0}\boldsymbol\nabla \times\mathbf A\cdot\boldsymbol\nabla \times\delta\mathbf A $$}

The term quadratic in the variation vanishes, and the last term must be again manipulated to isolate the variation, using the identity {$\boldsymbol\nabla\cdot(\mathbf f \times \mathbf g)=\mathbf g\cdot\boldsymbol\nabla\times\mathbf f-\mathbf f\cdot\boldsymbol\nabla\times\mathbf g$}, with {$\mathbf f=\boldsymbol\nabla\times\mathbf A $}, to yield

{$$ \boldsymbol\nabla\cdot (\boldsymbol\nabla\times\mathbf A \times \delta\mathbf A) = \delta\mathbf A\cdot\boldsymbol\nabla\times\boldsymbol\nabla\times \mathbf A - \boldsymbol\nabla\times\mathbf A\cdot\boldsymbol\nabla\times\delta\mathbf A $$}

Another known identity transforms the parenthesis on the LHS into {$\mathbf A\boldsymbol\nabla\cdot\delta\mathbf A - \delta\mathbf A\boldsymbol\nabla\cdot\mathbf A$}, hence the LHS vanishes identically. Furthermore {$\boldsymbol\nabla\times\boldsymbol\nabla\times \mathbf A = \boldsymbol\nabla\times\mathbf B = \mu_0\mathbf J$}, by Ampére equation. As we saw it reduces to {$\mu_0 J_s$} in the infinite cylindrical superconductor case. Hence the variation of the free Gibbs energy density is

{$$ \delta g_s = \left[\frac {iq\hbar}{2m^*}(\psi^*\boldsymbol\nabla\psi - \psi\boldsymbol\nabla\psi^* ) + \frac {q^2}{m^*}|\psi|^2 \mathbf A + \mathbf J_s\right]\cdot\delta\mathbf A $$}

Requiring that the coefficient of {$\delta \mathbf A$} is zero, we get the second GL equation, that defines the supercurrents

{$$ \begin{equation} \mathbf J_s = -\frac {iq\hbar}{m^*}(\psi^*\boldsymbol\nabla\psi - \psi\boldsymbol\nabla\psi^* ) - \frac {q^2}{m^*}|\psi|^2 \mathbf A \end{equation} $$}

This defines the relation between the supercurrents {$J_s$}, the order parameter {$\Psi|^2$} and the vector potential {$\mathbf A$}. The two GL equations must be solved simultaneously. (Tinkham, p.117)

Ginzburg-Landau phase stiffness is defined by Doniach Phys Rev B 41 6668 for high Tc cuprates as {$C$} in {$J=Cd|\psi|^2$}, where {$d$} is a lengthscale (see also Emery Nature 374 434 1995) to replace the gradient of eq. 2 by a finite difference {$\nabla_{ij}/b$} where {$b^2d$} is the unit cell volume (i.e. {$d$} is the c lattice spacing?).

Perhaps introduced (without naming it) in Deutscher Phys. Rev. B 10 4598: it is effectively the GL free energy coefficient {$c$} in front of the gradient term, multiplied by the lengthscale {$d$} over which phase varies (the grain size in this case), to yield {$C=cd$}. One has {$\xi(T)^2 = C/2da(T_c-T)$} below {$T_c$}, with {$a(T-T_c)$} the coefficient of the leading term in the GL free energy. In this way

{$$ C = \frac {\hbar^2 d}{2m^*}$$}

as long as the phase is varying over {$d\ll \xi$} (i.e. we consider {$|\psi|$} constant).