The problem of fluctuation smearing of the superconducting transition was not even considered during the first half of the century after the discovery of superconductivity. In bulk samples of traditional superconductors the critical temperature Tc sharply divides the superconducting and the normal phases. It is worth mentioning that such behavior of the physical characteristics of superconductors is in perfect agreement with both the GL phenomenological theory (1950) [2] and the BCS microscopic theory of superconductivity (1957) [4].
The characteristics of high temperature and organic superconductors, low-dimensional and amorphous superconducting systems studied today strongly differ from those of the traditional superconductors discussed in textbooks. The transitions turn out to be much more smeared out. The appearance of superconducting fluctuations above the critical temperature leads to precursor effects of the superconducting phase occurring while the system is still in the normal phase, sometimes far from Tc. The conductivity, the heat capacity, the diamagnetic susceptibility, the sound attenuation, etc. may increase consid-erably in the vicinity of the transition temperature.
The first numerical estimation of the fluctuation contribution to the heat capacity of a superconductor in the vicinity of Tc was done by Ginzburg in 1960 [5]. In that paper he showed that superconducting fluctuations increase the heat capacity even above Tc. In this way fluctuations change the temperature dependence of the specific heat in the vicinity of the critical temperature where, according to the phenomenological Landau
atures where the fluctuation correction to the heat capacity of a bulk, clean, conventional superconductor is relevant was estimated by Ginzburg to be
Gi ∼ 10−12÷ 10−14, (1.1)
The correction occurs in a temperature range δT many orders of magnitude smaller than that accessible in real experiments.
In the 1950s and 1960s the formulation of the microscopic theory of superconductivity, the theory of Type-II superconductors, and the search for HTSC attracted the attention of researchers to dirty systems, superconducting films and filaments. In 1968, in papers by Aslamazov and Larkin [6], and Maki [9], and a little later in a paper by Thompson [10], the fundament of the microscopic theory of fluctuations in the normal phase of a super-conductor in the vicinity of the critical temperature were formulated. This microscopic approach confirmed Ginzburg’s evaluation [5] for the width of the fluctuation region in a bulk clean superconductor. Moreover, it was found that the fluctuation effects increase drastically in thin dirty superconducting films and whiskers. In the cited papers it was demonstrated that fluctuations affect not only the thermodynamical properties of a super-conductor but its dynamics too. Simultaneously the fluctuation smearing of the resistive transition in bismuth amorphous films was found experimentally by Glover [7], and it was perfectly fitted by the microscopic theory.
In the BCS theory [4] only the Cooper pairs forming a Bose-condensate are considered.
Fluctuation theory deals with the Cooper pairs out of the condensate. In some phenomena these fluctuation Cooper pairs behave similarly to quasiparticles but with one important difference. While for the well defined quasiparticle the energy has to be much larger than its inverse life time, for the fluctuation Cooper pairs the “binding energy” E0 turns out to be of the same order. The Cooper pair life time τGL is determined by its decay into two free electrons. Evidently, at the transition temperature the Cooper pairs start to condense and τGL= ∞. Therefore it is natural to suppose from dimensional analysis that
τGL∼ ~/kB(T − Tc). The microscopic theory confirms this hypothesis and gives the exact coefficient:
τGL= π~
8kB(T − Tc). (1.2)
Another important difference of the fluctuation Cooper pairs from quasiparticles lies in their large size ξ(T ). This size is determined by the distance by which the electrons forming the fluctuation Cooper pair move apart during the pair lifetime τGL. In the case of an impure superconductor the electron motion is diffusive with the diffusion coefficient Ddif f ∼ vF2τscatt (τscatt is the electron scattering time), and ξdir(T ) = p
Ddif fτGL ∼ vF√
τscattτGL. In the case of a clean superconductor, where kBT τscatt À ~ , impurity scattering no longer affects the electron correlations. In this case the time of electron ballistic motion turns out to be less than the electron-impurity scattering time τscatt and is determined by the uncertainty principle: τbal ∼ ~/kBT . Then this time has to be used in this case for the determination of the effective size instead of τscatt: ξcl(T ) ∼ vFp
~τGL/kBT . In both cases the coherence length grows with the approach to the critical temperature as ²−1/2, where
² = ln T Tc
≈ T − Tc Tc
, (1.3)
is the reduced temperature. The coherence length can be written in the unique way (ξ = ξcl,dir):
ξ(T ) = ξ
√². (1.4)
Finally it is necessary to recognize that fluctuation Cooper pairs can really be treated as classical objects, but that these objects instead of Boltzmann particles appear as clas-sical fields in the sense of Rayleigh–Jeans. That is why the more appropriate tool to study fluctuation phenomena is not the Boltzmann transport equation but the GL equation for
classical fields. Nevertheless, at the qualitative level the treatment of fluctuation Cooper pairs as particles. it was demonstrated [8], in the framework of both the phenomenological GL theory and the microscopic BCS theory, that in the vicinity of the transition.
The complete description of its thermodynamic properties can be done through the calculation of the partition function [8]:
Z = tr (
exp Ã
−Hb T
!)
. (1.5)
In the vicinity of the superconducting transition, side by side with the fermionic electron excitations, fluctuation Cooper pairs of a bosonic nature appear in the system. As already mentioned, they can be described by means of classical bosonic complex fields Ψ(r) which can be treated as “Cooper pair wave functions”. Therefore the calculation of the trace in (1.5) can be separated into a summation over the “fast” electron degrees of freedom and a further functional integration carried out over all possible configurations of the “steady flow” Cooper pairs wave functions:
Z = Z
DΨDΨ∗(r) exp µ
−F [Ψ(r)]
T
¶
, (1.6)
where F [Ψ(r)] is GL functional.
The appearance of fluctuating Cooper pairs above Tc leads to the opening of a “new channel” for charge transfer. In the Introduction the fluctuation Cooper pairs were treated as carriers with charge 2e while their lifetime τGL was chosen to play the role of the scattering time in the Drude formula. The generalization of the phenomenological GL functional approach to transport phenomena was presented in [8]. Dealing with the fluctuation order parameter, it is possible to describe correctly the paraconductivity type fluctuation contributions to the normal resistance and magnetoconductivity, Nernst effect, Hall effect, thermoelectric power and thermal conductivity at the edge of the transition [8].
Theory of Nernst effect in high-T c superconductor
2.1 Introduction
The electric field is induced in a metal under magnetic field by the temperature gradient
∇T perpendicular to the magnetic field H, phenomenon known as Nernst effect [8] (di-rection of the electric field being perpendicular to both ∇T and H). Recently the Nernst effect in high-Tc superconductors and low-temperature superconductor attracted atten-tion both theoretically [8, 11–15] and experimentally [16–24]. In these materials effect of thermal fluctuations is very strong leading to depinning of Abrikosov vortices created by the magnetic field in Type-II superconductor below second critical field Hc2(T ). In the mixed state the Nernst effect is large due to vortex motion, while in the normal state and in the vortex lattice or glass states it is typically smaller. The Nernst effect therefore is a probe of thermal fluctuations phenomena in the vortex matter, but in principle could shed some light on the underlying microscopic mechanism of superconductivity in cuprates. In the vortex-liquid state, a gradient −∇T drives the vortices to the cooler end of the sample because a normal vortex core has a finite amount of entropy relative to the zero-entropy condensate Fig. 2.1 [25, 26]. Because of the 2π phase singularity at each vortex core,
Figure 2.1: The vortex-Nernst effect in a Type-II superconductor. Concentric circles represent vortices.
vortex motion induces phase slippage [27]. By the Josephson equation 2eVJ = ~ ˙θ, the time derivative of the phase ˙θ produces an electrochemical potential difference VJ. We have ˙θ = 2π ˙Nυ, where ˙Nυ is the number of vortices crossing a line kˆy. per second. The Josephson voltage VJ may be expressed as a transverse electric field E = B × v which is detected as the Nernst signal.
Figure 2.2 shows the setup in the Nernst experiment [23]. One end of the crystal is glued with silver epoxy onto a sapphire substrate, which is heat-sunk to a copper cold finger. A thin-film heater, silver-epoxied to the top edge of the crystal, generates the heat current flowing in the ab plane of the crystal. The temperature difference ∆T is measured by a pair of fine-gauge Chromel-Alumel thermocouples. A pair of Ohmic contacts are prepared on the edge of the sample by annealing different kinds of conductive materials.
After the bath temperature is stabilized, the gradient is turned on. The Nernst voltage is preamplified and measured by a nanovoltmeter as the magnetic field is slowly ramped up.
To remove stray longitudinal signals due to misalignment of the contacts, the magnetic field is swept in both directions. Only the field-asymmetric part of the raw data is taken as the Nernst signal.
Figure 2.2: Crystal mounting geometry in the Nernst experiment.
Measurements of eN in fields H up to 45 T [23] reveal that the vortex Nernst sig-nal eN has a characteristic “tilted-hill” profile, which is qualitatively distinct from that of quasiparticles. The hill profile, which is observed above and below Tc, underscores the continuity between the vortex-liquid state below Tc and the Nernst region above Tc. Recently, the study of the Nernst effect in NbSe2 reveals a large quasiparticle contribu-tion with a magnitude comparable and a sign opposite to the vortex signal [24]. A large negative Nernst coefficient, persisting at temperatures well above Tc=7.2 K, was found in this metal. However, we will concentrate on the vortex-Nernst effect in Type-II su-perconductor of the overdoped La2−xSrxCuO4 (LaSCO) [21], underdoped and overdoped YBa2Cu3Oy (YBCO) [21], where eN is intrinsically strongly nonlinear in H and generally much larger than in nonmagnetic normal metal.
In this study we revisit the calculation in TDGL originally performed in Ref. [11]
to obtain explicit expressions for the transverse thermoelectric conductivity αxy and the Nernst signal eN in both a two dimensional 2D) and a three dimensional (3D) model.
Typically only the lowest Landau level (LLL) contribution was investigated [28]. We extend it to higher Landau levels necessary for exploring the experimentally accessible
made, disorder and crystallization. In this theory the strength of the thermal fluctuations is described by just one dimensionless adjustable parameter η (closely related to the Ginzburg number Gi). This parameter determines simultaneously the location of the melting line measured on the same samples in recent experiments on Nernst effect. The expression of Ref. [29] for the melting line is in good agreement with many experiments in very wide range of materials (as was established recently in [30]) and MC simulation. Then fitting of the transverse thermoelectric conductivity and related quantities practically has no free parameters (of course there is a certain freedom in determining mean field parameters like Hc2 and Tc, but the range is limited by experimental values). The value fitted from the Nernst effect turns out to be consistent with that derived from the melting line calculated in [29]. We will present the fitting of the melting line for the overdoped LaSCO [21], underdoped and overdoped YBCO [21].