Time dependent Ginzburg-Landau equations with thermal noise describing strong thermal fluctuations on the mesoscopic scale is used to describe strongly Type-II superconductor in the vortex-liquid regime both in 2D (describing strongly layered high-Tc supercon-ductors) and 3D (less layered superconductors like optimally doped YBCO). Using GL theory developed earlier we estimated the region in the parameter space in which, on one hand vortex crystal is effectively destroyed by thermal fluctuations and, on the other hand disorder (significantly “weakened” by thermal fluctuations) is not strong enough to significantly affect the transport. Under these conditions we obtained explicit expressions for the transverse thermoelectric conductivity αxy and the Nernst signal eN including all Landau levels were obtained using a gaussian approximation. It is very similar to the Hartree-Fock approximation utilized in Ref. [11], but has a virtue of being a variational principle.
The results are presented using both the strength of the thermal fluctuation η and more often used Ginzburg number Gi in the 2D and 3D. The applicability region con-sidered coincides with domain on the phase diagram in which the signal is large. We compared the results to the available 2D numerical simulations of the same model and the experiments on high-Tc materials. Our results in 2D are significantly lower than the available numerical simulation in Ref. [14] below the irreversibility line at which theory should be modified by including both pinning and crystalline correlation effects. How-ever within the applicability region theory is in good qualitative and even quantitative agreement with experimental data on both overdoped La1.8Sr0.2CuO4 and underdoped YBa2Cu3O6.5 Ref. [21] for temperatures close to Tc.
We also compared the values of eN calculated in the three dimensions with experiment data for temperature close to Tc on overdoped YBa2Cu3O6.99, and this comparison is also in good quantitative agreement. The Ginzburg numbers Gi were taken out from the fitting of melting lines of La1.8Sr0.2CuO4, YBa2Cu3O6.5 and YBa2Cu3O6.99 on the same
samples. The Ginzburg numbers Gi are consistent with the adjusted values of η when we fit the transverse thermoelectric conductivity and the Nernst signal. The irreversibility line of La1.8Sr0.2CuO4 was fitted as well with the same set of parameters.
Electrical conductivity beyond a linear response in layered
superconductors under a magnetic field
3.1 Introduction
Electric response of a HTSC under magnetic field has been a subject of extensive exper-imental and theoretical investigation for years. Magnetic field in these layered strongly Type-II superconductors create magnetic vortices, which, if not pinned by inhomogeneities, move and let the electric field to penetrate the mixed state. The dynamic properties of fluxons appearing in the bulk of a sample are strongly affected by the combined effect of thermal fluctuations, anisotropy (dimensionality) and the flux pinning [29, 43]. Thermal fluctuations in these materials are far from negligible and in particular are responsible for existence of the first-order vortex lattice melting transition separating two thermo-dynamically distinct phases, the vortex solid and the vortex liquid. Magnetic field and reduced dimensionality due to pronounced layered structure (especially in materials like
IN LAYERED SUPERCONDUCTORS UNDER A MAGNETIC FIELD
Bi2Sr2CaCuO8+δ) further enhance the effect of thermal fluctuations on the mesoscopic scale. On the other hand the role of pinning in high-Tc materials is reduced significantly compared to the low temperature one, leading to smaller critical currents. At elevated temperatures the thermal depinning [43] further diminishes effects of disorder.
Linear response to electric field in the mixed state of these superconductors has been thoroughly explored experimentally and theoretically over the last three decades. These experiments were performed at very small voltages in order to avoid effects of nonlinearity.
Deviation from linearity however are interesting in their own right. These effects have also been studied in low-Tc superconductors experimentally[44, 45] and theoretically [46, 47]
and recently experiments were extended to HTSC compounds [48, 49].
Since thermal fluctuations in the low-Tcmaterials are negligible compared to the inter-vortex interactions, the moving inter-vortex matter is expected to preserve a regular lattice structure (for weak enough disorder). On the other hand, as mentioned above, the vortex lattice melts in HTSC over large portions of their phase diagram, so the moving vortex matter in the region of vortex liquid can be better described as an irregular flowing vortex liquid. In particular the nonlinear effects will also be strongly influenced by the thermal fluctuations.
A simpler case of a zero or very small magnetic field in the case of strong thermal fluctuations was in fact comprehensively studied theoretically [8] albeit in linear response only. In any superconductor there exists a critical region around the critical temperature
|T − Tc| ¿ Gi · Tc, in which the fluctuations are strong (the Ginzburg number charac-terizing the strength of thermal fluctuations is just Gi ∼ 10−10− 10−7 for low Tc, while Gi ∼ 10−5 − 10−1 for HTSC materials). Outside the critical region and for small elec-tric fields, the fluctuation conductivity was calculated by Aslamazov and Larkin [50] by considering (noninteracting) Gaussian fluctuations within BCS and within a more phe-nomenological GL approach. In the framework of the GL approach (restricted to the lowest Landau level approximation), Ullah and Dorsey [11] computed the Ettingshausen coefficient by using the Hartree approximation. This approach was extended to other
transport phenomena like the Hall conductivity [11] and the Nernst effect [51].
The fluctuation conductivity within linear response can be applied to describe suffi-ciently weak electric fields, which do not perturb the fluctuations’ spectrum [52]. Physi-cally at electric field, which is able to accelerate the paired electrons on a distance of the order of the coherence length ξ so that they change their energy by a value corresponding to the Cooper pair binding energy, the linear response is already inapplicable [8]. The resulting additional field dependent depairing leads to deviation of the current-voltage characteristics from the Ohm’s law. The non-Ohmic fluctuation conductivity was calcu-lated for a layered superconductor in an arbitrary electric field considering the fluctuations as noninteracting Gaussian ones [53, 54]. The fluctuations’ suppression effect of high elec-tric fields in HTSC was investigated experimentally for the in-plane paraconductivity in zero magnetic field [55–57], and a good agreement with the theoretical models [53, 54] was found. Theoretically the nonlinear fluctuation conductivity in HTSC has been treated by Puica and Lang [58]. Below we compare their approach and results to ours.
In this study the nonlinear electric response of the moving vortex liquid in a layered superconductor under magnetic field perpendicular to the layers is investegated using the TDGL approach. The layered structure is modeled via the Lawrence-Doniach discretiza-tion in the magnetic field direcdiscretiza-tion. In the moving vortex liquid the long range crystalline order is lost due to thermal fluctuations and the vortex matter becomes homogeneous on a scale above the average inter-vortex distances. Although sometimes motion tends to suppress the fluctuations, they are still a dominant factor in flux dynamics. The TDGL approach is an ideal tool to study a combined effect of the dissipative (overdamped) flux motion and thermal fluctuations conveniently modeled by the Langevin white noise. The interaction term in dynamics is treated in self-consistent Gaussian approximation which is similar in structure to the Hartree approximation [8, 11, 58, 59].
Firstly the model of Ref. [58], is physically different from ours. The authors in Ref.
[58] believe that the two quantities, layer distance and thickness in the Lawrence-Doniach for HTSC are equal (apparently not the case in HTSC), while we consider them as two
IN LAYERED SUPERCONDUCTORS UNDER A MAGNETIC FIELD
independent parameters. Another difference is we use so called self-consistent Gaussian approximation to treat the model while Ref. [58] used the Hartree approximation.
A main contribution of our paper is an explicit form of the Green’s function incorporat-ing all Landau levels. This allows to obtain explicit formulas without need to cutoff higher Landau levels. In Ref. [58], a nontrivial matrix inversion (of infinite matrices) or cutting off the number of Landau levels is required. Note that the exact analytical expression of Green function of the linearized TDGL equation in DC field can be even generalized also to AC field. The method is very general, and it allow us to study transport phenomena beyond linear response of Type-II superconductor like the Nernst effect, Hall effect. The renormalization of the models is also different from Ref. [58]. One of the main result of our work is that the conductivity formula is independent of UV cutoff (unlike in Ref. [58]) as it should be as the standard |Ψ|4 theory is renormalizable. Furthermore self-consistent Gaussian approximation used in this paper is consistent to leading order with perturba-tion theory, see Ref. [40] in which it is shown that this procedure preserved a correct the ultraviolet (UV) renormalization (is RG invariant). Without electric field the issue was comprehensively discussed in a textbook Kleinert [40]. One can use Hartree procedure only when UV issues are unimportant. We can also show, if there is no electric field, the result obtained using TDGL model and self-consistent Gaussian approximation will lead the same thermodynamic equation using self-consistent Gaussian approximation.