The order parameter may be considered as an average of the localized quantity over a few lattice spacings, or a microscopic length scale a. Only for distance much larger then a can a field-theoretic description of the system in terms of order field be meaningful.
Below the microscopic length a microscopic properties of the material begin to become relevant. Such that one can classified the thermal fluctuation to the microscopic thermal fluctuations and two kinds of mesoscopic thermal fluctuations.
On the microscopic level, temperature modifies properties of the electron gas and the pairing interaction responsible for the creation of Cooper pairs. When “integrating out” the microscopic (electronic) degrees of freedom, one obtains an effective mesoscopic GL free energy with temperature dependent parameters, m∗, α and β, in terms of the distributions of the order parameter Ψ(x) with a lower limit on the length scale, a, over which the order parameter field can vary, or an upper limit to the momentum , Λ = 2π/a, of the Fourier components of the field. In practice, if one decompose Ψ (x) to a bases, for example Fourier components, with amplitude ϕk, the measure, DΨ,of a path integral in partition function of the system can be written as
DΨ = Y
k>0;|k|<Λ
d2ϕk (3.70)
The remaining part of the statistical sum can be view as the mesoscopic fluctuations.
Therefore, unlike the theory proposed in ref.[80], for measurable quantity such as mag-netization, the U V cut-off dependence should cancel out with the by renormalization of the bare mass, i.e. the mean field transition temperature.
The mean field lowest Landau level theory of thermal fluctuation is able to describe magnetization curves near Tc, including the intersection point, in both the quasi 2D superconductors like BSCCO and the 3D superconductors like Y BCO. It is therefore expected that the natural generalization of the model to include the coupling between layers, the Lawrence - Doniach model, should describe sufficiently well the thermal fluc-tuations in a wide range of layered materials, which exhibit neither the 2D nor the 3D behavior. A characteristic general feature is that the magnetization curves intersect always below Tc. While we show that the theory is consistent with the recent very de-tailed studies on HgBCCO and earlier studies on LaSCO, the results on the strongly underdoped LaSCO, which show the intersection point above Tc, are incompatible with the theory. Despite the fact that the theory has a number of assumptions like effects of disorder and contributions of higher Landau levels, the discrepancy is real. Since the de-scription of the layered superconductors by the Lawrence - Doniach is a very important part of the physics of the high Tc superconductors this question should be addressed experimentally.
Chapter 4
Structural Phase transition in Fourfold Symmetric
Superconductors
4.1 Introduction
The symmetry of the order parameter in superconductor is strongly related to the crys-tallographic symmetry group of the material, the structure of the Fermi surface and the nature of the pairing mechanism. In Y BCO, the in-plane O(2) is breaking due to the d-wave character of pairing. It has both a square and a rhombic phases [18]. In overdoped LaSCO, at low temperatures, the square and rhombic lattices were observed using SANS by Gilardi et al [16]. Asymmetry is not always related to the non s-wave nature of pairing. In borocarbides (RE : Y, Lu, Er) N i2B2C [14], N b [13], and V3(Si) [15], square vortex lattice is observed using techniques such as decoration, STM,SANS or µSR etc.
The precise location in the T −H plane of the square-rhombohedral SPT in the vortex crystal is still a matter of discussion. Earlier experiments on LuN i2B2C indicate a very small positive slope of the transition line in the T − H plane H2(T ) till it reaches the Hc2(T ) region. According to some experiments, it abruptly turns up and even acquires a
negative slope at high fields, while in other experiments experiments in a closely related material Y N i2B2C,it continues the gradual increase even near H2(T ). The results at low temperatures was first explained in the framework of the nonlocal London N LL theory proposed by Kogan et al [20].The N LL theory includes four derivative terms which bring in the anisotropy effects essential to trigger the SPT between the vortex lattice phases. The more symmetric square vortex crystal, stable at a stronger magnetic field higher density of vortices , transforms into a less symmetric rhombic vortex crystal as the magnetic field decreases. Thus the square-to-rhombus SPT is associated with a spontaneous breaking of the fourfold symmetry of the system. The transition has also been understood theoretically on the basis of the Ginzburg-Landau functional which had to be extended in a similar fashion, by including an asymmetric four derivative term. Such higher derivatives Ginzburg- Landau HDGL theories are applicable, strictly speaking, not far from Tc, but they generally work well in a much larger part of the T − H plane, including fields and temperatures well below the Hc2(T ) line. Although temperature might be introduced into these phenomenological models via a temperature dependence of their coefficients, the resulting slope of the transition line is typically very small and, more importantly, is of higher order in the relevant expansion parameter and therefore cannot be predicted.
It should be emphasized at this point that both N LL and HDGL were solved at the mean-field level only in the papers mentioned above and only recently in Refs. [20]and [24] , respectively, were attempts made to take into account thermal fluctuations on the
“mesoscopic” scale. Although thermal fluctuations are dominant in high-Tc supercon-ductors, leading, they are negligible in low-Tc materials for which the Ginzburg number, characterizing the strength of the thermal fluctuations, is several orders of magnitude lower. In high-Tc superconductors the square-to-rhombus transition was observed di-rectly via neutron scattering in Y Ba2Cu3O7+δ ,and La2Sr1−xCuxO4,and indirectly via the peak effect in LaSCO.
In this study, we will consider both thermal fluctuations and disorder in both per-turbation and nonperper-turbation method. We suggest that for low-Tc material such as
borocarbides, the disorder influence is pronounced and result in the reentrance of rhom-bic latter near the Hc2.