The appearance of a fluctuation tail above the critical temperature in the Nernst signal was observed in strongly Type-II superconductors, both low-Tc like NbSe2 and NbSi films [24] and several different high-temperature materials [17–20, 23]. The related Etting-shausen effect was detected as well [16]. In particular, the Nernst effect was observed well above Tc2(H) and even above Tc in Bi2Sr2CaCu2O8+δ (Bi2212) [23] , strongly un-derdoped YBCO [20, 21, 23] and LaSCO [19, 20, 22, 23]. With the overdoped regime (LaSCO with x = 0.20 and Tc=28 K) in Fig. 2.3, the signal rises steeply at each temper-ature T , attaining a prominent maximum before decreasing. The total data set defines experimentally the region in H and T where vorticity is strongly present. At high fields, all the curves below 14 K are observed to follow a common curve towards zero (dashed line). Hence all the low-T curves vanish at the intercept of the common curve with the field axis (45-50 T), which corresponds to Hc2(0). Going to higher T , we immediately encounter an anomaly we immediately encounter an anomaly. Conventionally, the Hc2
line goes linearly to zero at Tc. Hence, ey ought to be finite in a field interval that → 0 as T → Tc. In sharp contrast, we find that, close to Tc, but the magnitude of ey remains
Figure 2.3: The field dependence of ey at indicated T in samples LaSCO.
large and nearly unchanged up to intense fields for close to Tc. The anomalous features of the Nernst signal become more pronounced when we go to the underdoped regime.
The results in underdoped YBCO (with y = 6.50 and Tc=50 K) are showed in Fig. 2.4.
As H increases above melting line, ey rises rapidly, but attains a very broad maximum that extends undiminished to 30 T. These layered materials are highly anisotropic and
Figure 2.4: The field dependence of ey at indicated T in samples YBCO.
effect of thermal fluctuations is enhanced. However in less anisotropic materials like the hole-doped cuprate Nd2−xCexCuO4 (NCCO) [23] and weakly anisotropic and overdoped or fully oxidized YBCO6.99[23] the effect persists. Fluctuations in these materials cannot be described by a 2D model and generalization to anisotropic 3D model is required. The quasiparticle contribution to the Nernst signal attains a magnitude comparable to the vortex signal in the superconducting state. More recently, in experiment on amorphous thin films of the conventional low temperature superconductor Nb0.15Si0.85[24], a Nernst signal generated by short-lived Cooper pairs in the normal state in Fig. 2.5. In these
Figure 2.5: (Color) Nernst signal (N ) as a function of magnetic field for temperatures ranging from 0.180 K to 0.360 K (upper left panel) and from 0.56 to 4.3 K (upper right panel) measured on thin films of Nb0.15Si0.85 (with Tc=380 mK and thicknesses 35 nm).
amorphous films, the contribution of free electrons to the Nernst signal is negligible. In-deed, the Nernst coefficient of a metal scales with electron mobility. The extremely short mean free path of electrons in amorphous Nb0.15Si0.85 damps the normal-state Nernst ef-fect and allows a direct comparison of the data with theory. In the zero-field limit and close to Tc, the magnitude of the Nernst coefficient was found to be in quantitative agree-ment with a theoretical prediction [12] by Ussishkin et al, invoking the superconducting correlation length as its single parameter. At high temperature and finite magnetic field,
the data were found to deviate from the theoretical expression. In electron-doped cuprate NCCO the quasiparticle contribution to the Nernst signal is large [23]. The quasiparticle contribution actually dominates the Nernst signal far below Tc. Nevertheless, the vortex signal retains its characteristic tilted-hill profile which is easily distinguished from the monotonic quasiparticle contribution.
The observation of the Nernst effect above Tc along with other strong fluctuation effects was interpreted as a support for the preformed pairs scenario for the mechanism of the transition to the superconducting state. At the same time thermal fluctuations in high-Tc materials lead to many other remarkable phenomena, most notably vortex lattice melting and thermal depinning well studied both experimentally and theoretically over the last two decades, so that the theory of the Nernst effect should be consistent with the theory of these phenomena. Most importantly, the material parameters determining the fluctuation strengths can be determined from these better studied effects since in many recent experiments at least the melting line was measured on the same samples.
Theory of the electronic and the heat transport (including the Nernst effect) based on the phenomenological TDGL equations with thermal noise describing strongly fluctuating superconductors was developed long time ago [8, 11]. More recently within the same framework I. Ussishkin et al. [12] calculated perturbatively the low-field Nernst effect for T > Tc due to contribution of Gaussian fluctuations and obtained results in agreement with a microscopic Aslamazov-Larkin [8] calculation. They obtained the result for αSCxy , which diverges as the conductivity, and in reasonable agreement with experimental data on LaSCO in Fig. 2.6.
αSCxy ∝ σSCxy ∝ 1
(T − Tc)(d−4)/2. (2.4)
If only Gaussian fluctuations are considered then, αxy, diverges at the mean-field transi-tion, in conflict with the experimental results. One of important conclusions that inter-actions between the fluctuations must be considered in order to obtain even qualitative
Figure 2.6: Points are σxx(ν − νn) for different samples of LaSCO, with x = 0.12 (underdoped, Tc=29 K), x = 0.17 (near optimal doping, Tc=36 K), and x = 0.2 (overdoped, Tc=27 K). The solid line is the theoretical value of αxy/B, using ξ=30 ˚A and an anisotropy of γ = 20. The dashed line is obtained using a Hartree approximation.
agreement with the experimental results. S. Ullah and A. T. Dorsey [11] applied the Hartree approximation to treat the quartic term in the GL Hamiltonian within LLL.
In the limit of high magnetic fields, they found a smooth crosser from a regime dom-inated by two-dimensional Gaussian fluctuation for T > Tc2(H), to mean-field results for T < Tc2(H), with no intervening divergence, in agreement with the experimental re-sults. The absence of such a divergence is due to the one-dimensional character of the fluctuations-fluctuations transverse to the applied magnetic fields.
S. Mukerjee et al. [14] numerically simulated the two dimensional TDGL equation with Langevin thermal noise for T < Tcand obtained results in reasonable agreement with experimental data on LaSCO [21] at lower temperature, but the transverse thermoelectric conductivity became independent of magnetic field at higher temperatures in contrast to experiment. The simulation of this system, even in 2D, is difficult and it was one of our goals to supplement it with a reliable analytical expression in the region of the vortex liquid, namely in the region above the melting line (see Fig. 2.7) at which the vortex matter becomes homogeneous on a scale of several lattice spacings and the crystalline symmetry is lost. In this phase the pinning is ineffective and, unlike in the vortex glass
phase, vortices actively promote the Nernst effect. Recent understanding of the vortex matter phase diagram is summarized in Fig. 2.7. There are four phases separated by two transition lines [33]: the first order melting line (sometimes called the order-disorder line at lower temperatures, Hm(T ) line in Fig. 2.7) and the irreversibility (or glass) continuous transition. The melting line separates crystalline phases from a homogeneous phases,
Figure 2.7: The thermodynamic phase diagram of BSCCO accommodates four distinct phases, separated by a first order melting line Hm(T ) (open circles), which is intersected by the second-order glass line Hg(T ) (solid dots). The inset plots an equivalent phase diagram, calculated based on Ref. [29], consisting of a second-order replica symmetry breaking lines Hg(T ) both above (dotted line) and below (dashed line) the first-order transition Hm(T ) (solid line).
while the glass line (Hg(T ) line in Fig. 2.7) separates pinned phases from the unpinned ones. The mean field Hc2(T ) line in strongly fluctuating superconductors becomes a crossover. Both pinning and crystalline order lead to a strong reduction of the Nernst signal and therefore these phases will not be considered here. We concentrate on the vortex liquid phase (see Fig. 2.7) and discuss the melting line and disorder only as limits of applicability of the theory and for determining the material parameters. The quantitative GL theory of the vortex liquid have been developed recently and it was established that the Hartree-Fock approach for the thermodynamic is close to the convergent Borel-Pade one in the wide region of the vortex liquid phase [29].