Hall conductivity

The influence of superconducting fluctuations on off-diagonal components of the magne-toconductivity tensor (usually denoted as the excess Hall effect) in HTSC has received considerable experimental and theoretical attention over the past few years [8, 63–78] . Though a general consensus seems to be achieved now regarding the existence and the temperature dependence of the excess Hall effect, theoretical predictions of its sign are still controversial. Experimentally, the Hall resistivity shows a peculiar temperature de-pendence. Specifically, as the temperature is decreased through the fluctuation region, the Hall resistivity decreases and changes its sign relatively to the normal-state one, ex-hibits a negative minimum, and eventually reaches zero at low temperatures. This simple sign change was detected in many different HTSC in Refs. [70, 73, 79, 80] and even in

RESPONSE IN LAYERED SUPERCONDUCTOR UNDER A MAGNETIC FIELD

conventional superconductors Ref. [71, 72]. Furthermore, a double-sign reversal, which is a subsequent return of the Hall resistivity to the positive value before vanishing, has been observed in highly anisotropic HTSC, such as Bi2212 crystals [81] and films [82], Tl2Ba2CaCu2Ox films [83], or HgBa2CaCu2O6 films [84]. Recently, the existence of the second sign change was also reported in YBCO films, either at high current densities [85]

or in the strong pinning limit at low magnetic fields [86]. Finally, even a triple-sign rever-sal was reported in HgBa₂CaCu₂O₆ films with columnar defects induced by high-density ion irradiation [87].

Several theoretical approaches have attempted to explain the complex features of the Hall resistivity temperature dependence, but no consensus has been achieved. The Hall anomaly might originate from the pinning force [66], nonuniform carrier density in the vortex core [88, 89], or can be calculated in the TDGL model [90, 91]. Most recent theories claim to predict the double or triple-sign reversal, based either on entirely intrinsic mechanism of vortex motion and electronic spectrum [92], or on hydrodynamic interaction between vortices and the superconducting and normal-state fluids [93]. Some theories invoke superconducting fluctuations alone to account for the Hall effect sign reversal [11, 94], while others present a more extended picture based on the same foundations of TDGL using both the hydrodynamics and the vortex charging effect, arising from the difference in electron density between the core and the far outside region of the vortex [88, 89, 95].

Thus, the Hall effect in the mixed state of HTSC reflects a complex interplay between electronic properties of quasiparticles, thermodynamic fluctuations, hydrodynamic effects of vortices, and pinning.

From a considerable part of the published theoretical work, it appears that at least the first sign reversal, which occurs near the critical region, where vortex pinning is negligible and the superconducting order-parameter fluctuations play an important role, should be ascribed to a microscopic origin of superconductivity [68, 96, 97]. From the viewpoint of the TDGL formalism [11, 91, 94], to which any theory of vortex dynamics must reduce near the critical temperature T_c [92, 98], the Hall anomaly is a consequence

of the difference in sign between the normal (quasiparticle) part and the superconducting fluctuation (or vortex flow) part of the total Hall conductivity. These two components have opposite signs, if the energy derivative of the density of states averaged over the Fermi surface is positive when the carriers are holes in the normal state [99]. Thus, the sign reversal can be intrinsic and depends on the details of the structure of the normal-state electronic spectrum. Such notion is further supported by the fact that in several HTSC, the sign reversal disappears when the material is strongly overdoped and the band structure approaches that of a conventional metal [100].

The possibility of the Hall angle sign change in the critical region was first discussed by Fukuyama, Ebisawa, and Tsuzuki (FET) [101], who pointed out that the origin of a nonvanishing Hall current due to fluctuating Cooper pairs could come from a hole-particle asymmetry, which reveals a complex relaxation time in the TDGL theory. In this early work, it was implicitly assumed that the fluctuations did not interact; that is, only Gaus-sian fluctuations were considered. Accordingly, the fluctuation parts of the conductivity tensor elements were predicted to diverge at T_cin the presence of magnetic field. However, this predicted divergence has not been observed. A great improvement was obtained when the interaction between fluctuations was taken into account by incorporating the quartic term |Ψ|⁴ from the GL expression of the free energy. Such a treatment was performed by Ullah and Dorsey [11] (UD) in the frame of a simple Hartree approach of the TDGL theory. More recently, Nishio and Ebisawa [94] (NE) extended the FET calculations of the weak (Gaussian) fluctuation contribution of the Hall conductivity to the strong (non-Gaussian) fluctuation regime, based on more sophisticated renormalization theory by Ikeda, Ohmi, and Tsuneto [102] (IOT). The renormalized, non- Gaussian fluctuation regime connects therefore the weak (Gaussian) fluctuation regime in the paraconducting region above T_c2(H) to the vortex liquid (flux-flow) regime below the mean-field transi-tion, interpolating smoothly without the Tc divergence predicted by the Gaussian theory.

The comparison between experimentally observed Hall anomaly in HTSC and the full quantitative application of the TDGL theory was done well by Puica et. al. [65].

RESPONSE IN LAYERED SUPERCONDUCTOR UNDER A MAGNETIC FIELD

However, all theories mentioned above is linear response. Recently, non-Ohmic fluc-tuation Hall conductivity was treated theoretically [58] and experimentally [49]. The non-Ohmic fluctuation Hall conductivity calculated in [58] was in good qualitative cor-respondence with the theoretical one [49], but the magnitude of the effect is somewhat smaller. The authors in [58] believe that the two quantities, layer distance and thick-ness in the Lawrence-Doniach for HTSC, are equal. They also made assumption that the imaginary part of the relaxation time γ₁ in the TDGL equation is small in comparison with the real one γ as calculating the fluctuation Hall conductivity.

In this study the non-Ohmic fluctuation Hall conductivity of the moving vortex liquid in a layered superconductor under magnetic field perpendicular to the layers is studied using the TDGL approach. The layered structure is modeled via the Lawrence-Doniach discretization in the magnetic field direction. We consider layer distance and thickness in the Lawrence-Doniach as two independent parameters. The self-consistent Gaussian approximation mentioned in section 3.3.1 is used to treat the model while Ref. [58] used the Hartree approximation. A main contribution of this study is an explicit form of the Green’s function incorporating all Landau levels. This allows to obtain explicit formulas without need to cutoff higher Landau levels and any assumption about the imaginary part of the relaxation time γ₁ in the TDGL equation. In Ref. [58], a nontrivial matrix inversion (of infinite matrices) or cutting off the number of Landau levels is required. The renormalization of the models is also different from Ref. [58]. One of the main result of our work is that the Hall conductivity formula is independent of UV cutoff (unlike in Ref.

[58]).

4.1.3 Dissipative dynamics of vortices and electric fields in the