4.3 Fluctuation Hall conductivity
4.3.2 Comparison with experiment and discussion
Hall effect measurements on an optimally doped YBCO films of thickness 50 nm and Tc = 86.8 K was done in Ref. [49] in which the resistivity of the same sample was fitted in section 3.4.2. The parameters, namely, ξ, κ, s0, k, γ, σn, remained the same, as used in the fits shown in Fig. 3.1 The comparison is presented in Fig. 4.2. The Hall conductivity curves were fitted to Eq. (4.26) with the normal-state conductivity measured in Ref. [49]
to be σxyn = 42 (Ωcm)−1. We found that the best fits were obtained with ϑ = −0.0017, and inferred empirically. The absolute value of ϑ obtained from our fitting is consistent with its value [74]. The negative value of the hole-particle asymmetry parameter ϑ (this means
RESPONSE IN LAYERED SUPERCONDUCTOR UNDER A MAGNETIC FIELD
75 80 85 90 95 100 105
-500 -400 -300 -200 -100 0 100
E=50 V/cm
E=100 V/cm
E=150 V/cm
xy
[cm)
-1 ]
T[K]
B=0.8 T
Figure 4.2: Points are Hall conductivity for different electric fields of an optimally doped YBCO in Ref. [49]. The solid line is the theoretical value of resistivity calculated from Eq. (4.26) with fitting parameters (see text).
a negative σsxy) implies a positive energy-derivative of the density of states at εF when the carriers are holes in the normal state. As suggested by Kopnin and Vinokur [105],one possibility to explain this behavior is that the Fermi surface of a metal in the normal state has both hole-like and electronic pockets. The Hall anomaly may thus depend on the doping level, as it was reported by Nagaoka et al. [100]. Very recently, Angilella et al. [106] have found that, close to an electronic topological transition of the Fermi surface, in the hole-like doping range, the fluctuation Hall conductivity has indeed an opposite sign with respect to the normal-state one, giving additional strong support that the Hall resistivity sign reversal is intrinsic and depends on the details of the structure of the electronic spectrum.
4.4 Summary
We have calculated the fluctuation Hall conductivity for a layered superconductor in an arbitrary in-plane electric field and perpendicular magnetic field in the frame of the TDGL
theory with thermal noise describing the thermal fluctuations using the self-consistent Gaussian approximation. We have obtained explicit formulas including all Landau levels without any assumption about the imaginary part of the relaxation time γ00 in the TDGL equation. It is then easy to get the expression for the fluctuation Hall conductivity under assumption that the imaginary part of the relaxation time is very small.
The renormalization of the critical temperature is calculated and is strong in layered high-Tc materials. The results were compared to the experimental data on HTSC. Our the fluctuation Hall conductivity results are in good qualitative and even quantitative agreement with experimental data on YBCO in strong electric fields.
Chapter 5
Fluctuation ac conductivity in linear response
5.1 Introduction
The analysis of fluctuations conductivity has stimulated in the past years a considerable amount of work. Theoretical investigations of the dc as well as the finite-frequency con-ductivity have been a subject for years and development of this topic proceeded until the discovery of HTSC. Experimental investigations have been reported showing clear signs of fluctuations in both the real and imaginary parts of the ac conductivity in zero mag-netic field [107–110]. The real part σ1 of the complex conductivity (σ = σ1+ iσ2) has a sharp peak at Tc , which is not observed in, e.g., Nb as representative of low temperature classical superconductors [111]. The salient feature of the ac case is that the fluctuation conductivity does not diverge at Tc because a finite frequency provides a limit to the ob-servation of the critical slowing down near Tc. The determination of Tc from the peak in σ1 is a reasonable choice [107]. It is also important to note that σ1 and σ2 have individually different temperature and frequency dependence, even though they result from the same underlying physics. Testing a given theoretical model becomes more stringent when two curves have to be fitted with the same set of parameters. Recently, the high-frequency
electromagnetic response of vortices has been investigated [112–115].
The expressions for the ac fluctuation conductivity in zero magnetic field in the Gaus-sian regime have been deduced within the TDGL theory of Schmidt [116]. Using general physical arguments, Fisher, Fisher, and Huse [117] provided a formulation for the scaling of the complex ac conductivity as
σ(ω) ∝ ξz+2−dS±(ωξz), (5.1)
where ξ is the correlation length, z is the dynamical critical exponent, d is the dimen-sionality of the system, and S±(ωξz) are some complex scaling functions above and below Tc, with the correlation length diverging at T = Tc as ξ ∼ ²−ν , where sufficiently close to Tc, ² = (T /Tc− 1). This form of fluctuation conductivity was claimed to hold in both the Gaussian and critical regimes. Dorsey [118] has deduced the scaling functions in the Gaussian regime above Tc and verified the previous results of Schmidt [116]. More re-cently, Wickham and Dorsey [119] have shown that even in the critical regime, where the quartic term in the GL free energy plays a role, the scaling functions preserve the same form as in the Gaussian regime.
In high-Tc cuprates at relatively high temperatures, vortices move and vibrate due to thermal fluctuations to the extent that the lattice can melt becoming a “vortex liquid”
which was introduced in 2.5.1. Recently, Lin and Lipavsky [120] used the TDGL equation to calculate the far infrared conductivity in the Abrikosov vortex lattice state of a Type-II superconductor, and the results were in good agreement with experiment data. Thermal fluctuations neglected. This work is complementary.
Measurements of the complex conductivity as a function of frequency [108] analyzed in terms of the above-mentioned theory have revealed a somehow puzzling behavior: in fact, the complex conductivity σ(ω) does exhibit a scaling behavior close to the expected one, but the so-obtained critical exponents, ν ' 1.2 and z = 2.6, are quite different with respect to the Gaussian values, ν = 0.5 and z = 2; the critical exponent ν is
also in conflict with the prediction for the 3DXY uncharged fluid, ν = 2/3 [121]. The determination of the critical exponents close to Tc is uncertain: in fact, a different scaling analysis of measurements of the frequency-dependent conductivity up to 2 GHz in zero magnetic field gave large exponents, ν ' 1.7 and z = 5.6 [109], in contrast with those previously obtained. The authors [110] extended the renormalized-fluctuations theory in zero magnetic field developed by Dorsey [118] by introducing an anisotropic mass tensor, and they compared the results to resistive transitions in zero field obtained in dc and at high frequency, above the critical temperature. The temperature dependence of the resistivity in zero magnetic field at all the frequencies investigated could be well described by the theory [110] from slightly above Tc, with values of the parameters in good agreement with common values. Far from above Tc the theory did not apply, and more extensions are needed.
The electromagnetic response to microwaves in the mixed state of YBCO was mea-sured [114] in order to investigate the electronic state inside and outside the vortex core.
The magnetic-field dependence of the complex surface impedance at low temperatures was in good agreement with a general vortex dynamics description assuming that the field-independent viscous damping force and the linear restoring force were acting on the vortices. In other words, both real and imaginary parts of the complex resistivity, δρ1, and δρ2, were linear in B. However, at higher temperatures, there is a clear deviation from the B-linear behavior. This deviation became more prominent with increasing temperature.
In the high temperature region, thermal effects on the vortices cannot be neglected, and the low temperature approximation will be no longer accurate. The microscopic calcula-tion of the Larkin-Varlamov included lowest Landau level and high Landau levels (HLL) [8], but the result of HLL is cumbersome and cutoff dependence.
In this study we will calculate the complex conductivity and resistivity in linear re-sponse in a layered superconductor under magnetic field by using TDGL with thermal fluctuations conveniently modeled by the Langevin white noise. The interaction term in dynamics is treated in self-consistent Gaussian approximation. The results is compared
with experimental data [114] at high temperature where thermal effects on the vortices should be included.