Ginzburg-Landau Theory
We have seen in Sect.2 that a superconductor often resides in a state of high spatial inhomogeneity. Here the intermediate state and the situation near a domain wall are just an example showing spatial variations of the order parameter. Spatial inhomogene
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We have seen in Sect.2 that a superconductor often resides in a state of high spatial inhomogeneity. Here the intermediate state and the situation near a domain wall are just an example showing spatial variations of the order parameter. Spatial inhomogeneity is displayed perhaps even more importantly in the case of the vortex state in type-II superconductors. A phenomenological theory particularly suited for dealing with such inhomogeneous situations has been developed by GINZBURG and LANDAU [1.4J. The Ginzburg-Landau (GL) theory is based on LANDAU's [1.2J theory of secondorder phase transitions, in which LANDAU introduced the important concept of the order parameter. This concept, originally developed for treating structural phase transitions, has since proved extremely useful in many systems where phase transitions take place. Depending on the physical system, the order parameter can have different dimensions. In order-disorder phase transitions the order parameter is a scalar. In a ferromagnetic and ferroelectric phase transition the order parameter is a vector, namely the magnetization and the polarization, respectively. In the superconducting phase transition the order parameter is the density of the superconducting electrons (pair wave function) or the energy-gap parameter. More recently, the concept of the order parameter has been extended to nonequilibrium phase transitions [3.1J. Here an example is the laser near threshold pumping power which can be treated using the light field or the photon number as the order parameter. In the original GL theory the order parameter is a complex quantity, namely a pseudo wave function ~(r). The absolute value 1~(r)1 is connected with the local density of superconducting electrons, 1~(r)12 = ns(r). The phase of the order parameter is needed for describing supercurrents. The free-energy density is then expanded in powers of 1~12 and Iv~12, assuming that ~ and V~ are small. The minimum energy is found from a variational method leading to a pair of coupled differential equations for ~(r) and the vector potential ~(r). It is important to note that the GL theory is based on purely phenomenological concepts. However, GOR'KOV [1.5,6J has shown that, for both very pure and very impure superconductors, the GL theory follows
R. P. Huebener, Magnetic Flux Structures in Superconductors © Springer-Verlag Berlin Heidelberg 2001
34 rigorously from the BCS theory for the temperature region near Tc ' provided that spatial variations of the order parameter and of the magnetic field are slow. In GOR'KOV's reformulation of the GL theory the quantity ~(r) is denoted by ~(r), which often has been called the gap parameter. However, in general there is no simple relationship between this function ~(r) and a gap in the excitation spectrum.
3.1 Free Energy and the Ginzburg-Landau Equations Starting with the simplest case, we assume the order parameter ~(r) to be constant and the local magnetic flux density ~ to be zero throughout the superconductor. For small values of ~(r), i.e., for T ~ Tc '
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