Ginzburg-Landau Theory

In 1950, Ginzburg and Landau introduced the so-called superconducting order parameter $\Psi (\vec{r})$ to describe the nonlocality of the superconducting properties. $\Psi (\vec{r})$ can be thought of as a measure of the order in the superconducting state at position $\vec{r}$below T_c. It has the properties that $\Psi (\vec{r}) \rightarrow 0$ as $T \rightarrow T_{c}$, and $\vert \Psi (\vec{r}) \vert^{2} = n_{s}(\vec{r})$(i.e. the local density of superconducting electrons). At the time Ginzburg-Landau (GL) theory was developed, the nature of the superconducting carriers was yet to be determined. Interpreting m^* as the effective mass and q as the charge of the fundamental superconducting particles (whatever they may be), they derived the penetration depth as:

$$\lambda (T) = \sqrt{\frac{m^{*} c^{2}}{4\pi q^{2} \vert \Psi_{\circ} \vert^{2}}}$$ (6)

where $\vert \Psi_{\circ} \vert^{2}$ is the value of $\vert \Psi \vert^{2}$deep inside the superconductor (i.e. its equilibrium value). The coherence length defined in the GL-formalism is:

$$\xi (T) = \sqrt{\frac{\hbar^{2}}{2m^{*} \vert \alpha (T) \vert}}$$ (7)

where $\alpha(T)$ is a temperature-dependent coefficient in the series expansion of the free energy (see [19]). As written, (2.6) and (2.7) suggest that the penetration depth $\lambda (T)$ and the coherence length $\xi (T)$ are temperature dependent quantities. Eq. (2.7) is closely related to the Pippard coherence length $\xi_{\circ}$ defined in Eq. (2.4). In GL-theory, the coherence length $\xi (T)$is the characteristic length for variations in $\vert \Psi (\vec{r}) \vert^{2} = n_{s}(\vec{r})$. One noteable difference is that Pippard's coherence length $\xi_{\circ}$ is independent of temperature. In fact in the dirty limit $\xi (T=0)$ reduces to Eq. (2.4). Near the transition temperature T_c both $\lambda (T)$ and $\xi (T)$ vary as (1-T/T_c)^-1/2, prompting the introduction of the Ginzburg-Landau parameter $\kappa$, where:

$$\kappa = \frac{\lambda (T)}{\xi (T)}$$ (8)

The variation of $\lambda (T)$ and $\xi (T)$ with temperture near T_ccomes from an expansion of the free energy density to first-order in the field $\vec{H}$. It follows that $\kappa$ in (2.8) is temperature independent to this order. An exact calculation from microscopic theory gives a weak temperature dependence for $\kappa$, with $\kappa$increasing for decreasing temperature T. Through energy considerations, Ginzburg and Landau characterized a type-I superconductor as one in which $\kappa < 1/\sqrt{2}$. For $\kappa \gg 1$, GL theory reduces to the London model.