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GinzburgLandau

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Two different approaches exist to justify superconductivity: microscopic models, starting from the electron Hamiltonian, and phenomenological mean field theories. The BCS theory is an example of the first kind, whereas the second type of approach is given by the Ginzburg-Landau treatment. They are both beyond the scope of this brief introduction, but we shall simply mention the framework of the second, which is the usual starting point for a description of any superconductor in a magnetic field, as is the case in a µSR experiment.

Free energy: Landau described phase transitions identifying an order parameter (e.g. the magnetization of a ferromagnet) and writing the free energy of the system as a power series expansion in this parameter. In order to apply the same scheme, the superconducting order parameter is identified in the density of superconducting electrons, represented as the square modulus of a wave function, {$|\psi|^2$}. The difference of the superconducting free energy density to that of the normal metal at {$B=0$} is written as the the sum of a linear, a quadratic and a kinetic energy term for an order parameter {$|\psi|^2$}

{$(1a)\qquad\qquad \Delta f= \alpha |\psi|^2 + \frac \beta 2 |\psi|^4 + \frac 1 {2m} \left|-i\hbar \nabla\psi\right|^2 $}

and in a finite magnetic field {$\mathbf{B}=\nabla\times\mathbf{A}$} as

{$(1b)\qquad\qquad \Delta f=\frac {B^2}{2\mu_0} + \alpha |\psi|^2 + \frac \beta 2 |\psi|^4 + \frac 1 {2m} \left|\left(-i\hbar \nabla - e\mathbf{ A}\right)\psi\right|^2 $}

All Ginzburg-Landau (G-L) results are accurate only close to {$T_c$}, where the order parameter is small, otherwise more terms in the expansion may be required.

Let's consider {$B=0$}, Eq. 1a. If {$\alpha>0$} the free energy is minimum for {$|\psi|^2=0$}, i.e. the order parameter vanishes. If instead {$\alpha<0$}, deep inside the superconductor, where {$\psi_\infty$} does not vary spatially and the last term in Eq. 1a is zero, the free energy is minimum for {$|\psi_\infty|^2=-\frac \alpha \beta$} .

This is the usual Landau treatment of the phase transition, which takes place when the first order coefficient {$\alpha(T)$} changes sign. For {$\alpha<0$}, i.e. below {$T_c$}, we can substitute the density {$|\psi_\infty|^2$} back into Eq. 1b. and note that there is a critical field, {$B_c^2=\mu_0 \frac {\alpha^2} {\beta}$} for which {$\Delta f$} vanishes. This means that above {$B_c$} the normal metal free energy is lower than the superconductor's and the system goes back to the normal state.

Ginzburg-Landau equations: The importance of the G-L theory derives from a variational treatment of Eq. 1, which leads to a non linear (due to the term in {$\beta$}) differential equation

{$ (2)\qquad \qquad\alpha \psi + \beta |\psi|^2 \psi + \frac 1 {2m} \left(\frac\hbar i \nabla - e \mathbf A \right)^2 \psi=0$}

The magnetic field in Eq. 2 is linked self-consistently the current density, whose quantum mechanica delfinition is {$\mathbf{J}_s=e\psi\nabla\psi$}. Writing {$\psi=|\psi| e^{i\phi(r)}$} one obtains a readily recognizable expression {$\mathbf{J}_s=e |\psi|^2 \mathbf{v}_s$}, where {$m\mathbf{v}_s=-i\hbar\nabla\phi-e\mathbf{ A}$}. This yields a second coupled differential equation. A general analytic solution is not known and one has to resort to approximations, either analytical or numerical.

Linearized case and G-L coherence length: A natural length scale emerges considering the special case when Eq. (2) can be linearized, i.e. dropping the {$\beta|\psi|^2$} non linear term. This can be done very close to T_c, where the wave function is much smaller than the stable solution {$\psi \ll -\alpha/\beta$}, so that the nonlinear term is much smaller than the linear one. Then, considering for simplicity {$\mathbf{B}=0$}, hence {$\mathbf{A}=0$}

{$ (2a)\qquad \qquad - \frac{\hbar^2} {2m} \nabla^2 \psi=-|\alpha| \psi$}

that reads like the Schroedinger equation. Let us consider the radial part: solutions decay with a constant {$1/\xi=\sqrt{2m\alpha}/\hbar$}, i.e. over a new lengthscale

{$ (3) \qquad \qquad \xi(T)= \frac {\hbar} {\sqrt{2m|\alpha(T)|}}, $}

which is called the G-L coherence length, an entirely new quantity that can be shown to coincide with {$\xi_0$} only for {$T=0$}. However the G-L coherence length clearly diverges at {$T=T_c$}, where {$\alpha$} vanishes, whereas the electrodynamical {$\xi$} does not. The second important length scale is the penetration depth, which can be rewritten {$\lambda^2=\frac {m} {\mu_0|\psi|^2 e^2}$}, having identified {$n$} with {$|\psi|^2$}.

The linearized G-L equation in a field reads

{$ (2b) \qquad\qquad\frac {\hbar^2} {2m} \left(-i\nabla - \frac{2 e\mathbf{ A}} \hbar\right)^2\psi=-\,\alpha \psi $}

and introducing the quantum of magnetic flux {$\qquad\qquad\Phi_0=\frac h {2e},$}

{$(2c) \qquad\qquad\frac {\hbar^2} {2m}\left(\frac\nabla i - \frac{2\pi} {\phi_0} \mathbf{ A} \right)^2\psi=-\frac {\hbar^2}{2m\xi(T)^2} \psi $}

that is like the Schrödinger equation for a particle of mass {$m$}, charge {$2e$}, in a potential {$2\pi\mathbf{ A(r)} /{\phi_0}$}, with eigenvalue {$-\alpha=-\hbar^2/2m\xi^2$}.

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