We report on the analyses of fluctuation induced excess conductivity in the ρ-T behavior in the in situ prepared MgB_{2} tapes. The scaling functions for critical fluctuations are employed to investigate the excess conductivity of these tapes around transition. Two scaling models for excess conductivity in the absence of magnetic field, namely, first, Aslamazov and Larkin model, second, Lawrence and Doniach model, have been employed for the study. Fitting the experimental ρ-T data with these models indicates the three-dimensional nature of conduction of the carriers as opposed to the 2D character exhibited by the HTSCs. The estimated amplitude of coherence length from the fitted model is ~21 Å.
1. Introduction
Since the discovery of superconductivity in MgB_{2} there has been enormous research owing to its speculated potential applications. Though there has been extensive research on both scientific and technical aspects, but yet little attention was paid to the fluctuation induced [1–5] enhanced conductivity in MgB_{2}. Superconductor transition broadening, induced by these fluctuations in the superconducting order parameter, is observed in various kinds of superconductors [6, 7]. Such broadening is normally due to superconducting fluctuation derived from a low dimensionality, short coherence length, and high TC. High TC, short coherence length, and low carrier densities impart an excess conductivity due to thermal fluctuation in the superconducting order parameter. There have been numerous studies on excess conductivity and order parameter fluctuations in YBa2Cu3O7-y (Y-123) [8–17], Bi_{2}Sr_{2}CaCu_{2}O8+y (Bi-2122), Bi_{2}Sr_{2}Ca_{2}Cu_{2}Cu_{3}O10+y (Bi-2223) [18–22], Tl_{2}Ba_{2}CaCu_{2}O8+y (Tl-2212) [23, 24], HgBa_{2}Ca_{2}Cu_{3}O8+y (Hg-1223), and HgBa_{2}CaCu_{2}O6+y (Hg-1212) [25] systems, but only few reports exist on the fluctuation studies [26–28] in MgB_{2}, and those are performed in the presence of magnetic field.
The superconducting transition in the absence of fluctuations is characterized by a mean non zero value of the order parameter below the transition temperature and zero above the transition temperature. There are fluctuations in the order parameter both above and below the transition temperature, that is, there is a probability of Cooper pair formation even above the transition temperature. These fluctuations contribute to the various physical properties like electrical conductivity, diamagnetism and specific heat both above and below TC [29]. The effect of thermodynamic fluctuations on most of these properties is in the small range around TC and the study of fluctuation conductivity may also provide information regarding the critical region close to TC. We now first provide a brief description of the model describing the fluctuation effects in a superconductor based on its resistivity behaviour. We shall than use this framework/model to understand the transport behaviour observed for the MgB_{2} tape samples near the critical region around their TC. We would present the results of the fluctuation induced conductivity in MgB_{2} tape via resistivity versus temperature measurement. The analyses of dimensionality of the conduction of the carriers and the estimation of the coherence length amplitude from fluctuation studies are also presented.
2. Experimental Details
For studying the fluctuation induced excess conductivity the in situ stainless steel sheathed tape sample synthesized at 800°C, for 1 hr, described elsewhere [30], with the stoichiometric (Mg : B = 1 : 2) starting composition was prepared [30]. Silver pads for making four contacts on the sample were made via dc-sputtering method. After that the current and voltage contacts were made using copper wire and silver paste. The resistivity versus temperature data were recorded via the four-probe method while warming the sample with an accuracy of ±1 mK and at a rate of 0.1 K/min around the transition, at the rate of 0.25 K/min in the 50–100 K range, and at 0.5 K/min in 100–300 K range. The temperature was recorded using a programmable temperature controller (Model-34 from Cryo-Con). A programmable current source (Model-224 from Keithley Inc.) was used to provide a constant current through the sample. Digital nanovoltmeter (Model-181 from Keithley Inc.) having a high input impedance was employed to measure the potential difference developed when sample was slowly warmed/cooled between 15 K and room temperature.
3. Result and Discussion3.1. Fluctuation Induced Excess Conductivity in the Absence of Magnetic Field
The initial studies on the fluctuation effects were performed by carrying out the measurement of excess conductivity above TC, commonly known as the paraconductivity. Using the microscopic approach in the mean field region (MFR), Aslamazov and Larkin [31, 32], considering acceleration of the short lived superconducting charge carrier pairs which form in thermal equilibrium above TC in an electrical field, provided the expressions for 3- and 2-dimensional excess conductivity as follows:
(1)Δσ3D=(e232ℏξ(0))×1ε1/2,(2)Δσ2D=(e216ℏd)×1ε,
where, ξ(0) is amplitude of isotropic coherence length, “d” is the two-dimensional characteristic length of the sample, such that if d≪ξε1/2, then 2-dimentional excess conductivity dominates in the superconductor, and ε is the reduced temperature defined as
(3)ε=T-TCmfTCmf.
Here, TCmf is mean field temperature. In general, HTSCs are the highly anisotropic materials, and for such systems, Lawrence and Doniach [33] suggested a model with layered conduction in which conduction occurs mainly in two-dimensional planes, and these planes are coupled through the well-known Josephson coupling. Within each layer, the superconductivity can be very well described by the GL theory. Lawrence and Doniach provided an expression for excess conductivity as
(4)ΔσLD=(e216ℏs)×ε-1/2(ε+4L)-1/2.
Here L=[ξC(0)/s]2 with ξC(0) as the coherence length amplitude perpendicular to the planes, and s is the distance between the planes. For ε≫4L, that is, when T≫TCmf, this expression varies as ε-1, thus showing the 2D behavior predicted by Aslamazov and Larkin (2). On the other hand, closer to TCmf, that is, when ε≪4L, this expression varies as ε-1/2, thus presenting the 3D behavior (1). In other words, this formula interpolate between 2D formulation for ξ(ε)≪s (the interaction between layers is weak) and 3D (1) results for ξ(ε)≫s, and a crossover 2D/3D is predicted at ξ(ε)~s/2.
3.2. Resistivity-Temperature Behavior
Figure 1(a) presents the resistivity versus temperature ρ(T,0) curve for the in situ prepared stainless steel sheathed MgB_{2}-tape sample S800. Figure 1(b) shows the enlarged view of the resistivity curve around the transition temperature and its derivative with the temperature. The single peak in the dρ/dT versus T curve indicates that the sample is single phasic. The various parameters obtained from the curve in Figures 1(a) and 1(b) are the ρ(300 K) = 7.92 × 10^{−6} Ωm, ρ(42 K) = 2.68 × 10^{−6} Ωm, RRR = 2.95, ΔT (90%-10% criteria) = 0.50 K, TC (90% drop criterion) = 40.663 K, TC (10% drop criterion) = 41.189 K, and TC (from peak in dρ/dT) = 40.901 K. Here, RRR means residual resistivity ratio, namely, ρ(300)/ρ(42) and TC (90% drop criterion), and TC (10% drop criterion) corresponds to the temperature at which ρ(42 K) becomes 90% and 10%, respectively.
(a) Resistivity versus temperature curve for the in situ prepared stainless steel sheathed MgB_{2} tape sample. In the inset, experimentally observed data for resistivity with temperature has been shown by the dotted (…) curve, and the fitted data (ρn(T)=a+bTc) in the normal state has been shown by the solid (—) curve. (b) Variation of resistivity (diamond symbol with continuous line) and the differential of the resistivity dρ/dT (triangle) with the temperature for the in situ prepared stainless steel sheathed MgB_{2} tape sample. (c) Δσ versus T-1/2 plot for determining TCmf, the linear extrapolated line hits the temperature where Δσ starts tending to zero.
The normal state behavior of the resistivity versus temperature curve of MgB_{2} tape S800 was fitted to ρn(T)=a+bTc above 1.5TC to 260 K (inset of Figure 1(a)) so as to obtain the Δσfl (i.e., fluctuation induced enhanced conductivity), which was found from the experimentally observed resistivity and the ρn as
(5)Δσfl(T)=1ρ(T)-1ρn(T).
From the fitting of the observed normal state ρn(T) data, the constants a, b, and c are found to be 2.6680 × 10^{−6}, 2.9982 × 10^{−12}, and 2.5373 with msd (mean square deviation) between the experimental and the fitted data of 4.9327 × 10^{−15} (Ωm)^{2} (see inset of Figure 1(a)). The msd for any variable f (varying as a function of x) is obtained by the following formula:
(6)msd=∑n(fexperimental(x)-ffitting)2n.
The fitting of the observed excess conductivity Δσ(T), obtained from the ρ-T data with different models described previously, requires the estimation of the mean field temperature, TCmf.
3.2.1. Mean Field Temperature
The choice of the mean field temperature is a well-known problem in the analysis of the fluctuation study. Various criteria for determining the TCmf have been considered in the literature [9, 34–36] as (i) midpoint of the resistive transition (ii) from the 90% drop in normal state resistivity near TC and (iii) the temperature at which dρ/dT versus T curve exhibits a peak to use TCmf as a fitting parameter in minimizing the least square deviation between the observed and the predicted model. Notably, sometimes, TCmf is estimated from the linear extrapolation of (Δσ)-2=0 in the (Δσ)-2 versus T plot (or Δσ versus T-1/2 plot). In the present work on MgB_{2}, we have determined the TCmf from the dρ/dT versus T curve and also from the Δσ(T) data as suggested by Oh et al. [9], that is, from the Δσ versus T-1/2 plot (see Figure 1(c)). The linear extrapolation of the plot to Δσ=0 has been considered to infer the corresponding temperature TCmf.
In Table 1, we presents the various parameters obtained by fitting the measured resistivity data to different AL and LD models using (1)–(6). Figures 2(a) and 2(b) show the fitting of the experimental obtained resistivity versus temperature data with various models outlined previously using two values of mean field temperature TCmf, respectively.
Summary of the results obtained from fitting the resistivity data with AL and LD models with the normal state was fitted to ρn(T)=a+bTc.
Criterion for TCmf
TCmf
Fitting model
d or s(Å)
ξ(0) (Å)
msd (σS) (Ωm)^{2}
From linear extrapolation of Δσ=0
40.889 K
AL-2D
29.70
—
2.1×10-12
AL-3D
—
23.40
8.2×10-14
LD-Layered
40.87
35.3
1.3×10-13
Peak in dρ/dT versus T curve
40.901 K
AL-2D
30.76
—
2.1×10-12
AL-3D
—
21.05
6.1×10-14
LD-Layered
35.20
31.99
8.5×10-14
(a) Variation of resistivity versus temperature for MgB_{2} tape sample, experimental curve, and the fit to AL-2D, AL-3D, and LD-layered expressions with TCmf=40.889 K are shown near the superconducting transition. The inset shows the same curves over the wide range of temperature—(15–250 K). (b) Variation of resistivity versus temperature for MgB_{2} tape sample, experimental curve, and the fit to AL-2D, AL-3D, and LD-layered expressions with TCmf=40.901 K are shown near the superconducting transition. The inset shows the same curves over the wide range of temperature—(15–250 K).
It is evident from Table 1 that the resistivity fit using AL model yields higher value of msd (namely, msd = 2.1 × 10^{−12} (Ωm)^{2} using TCmf = 40.889 and 40.901 K) for 2D formulation of excess conductivity compared to the AL-3D model (namely, msd = 8.2 × 10^{−14} (Ωm)^{2} and 6.1 × 10^{−14} (Ωm)^{2} using TCmf = 40.880 and 40.901 K, resp.) for both the values of TCmf employed for fitting purpose. On the other hand, fit to LD-layered model yielded the msd value (namely, msd = 1.3 × 10^{−13} (Ωm)^{2} and 8.5 × 10^{−14} (Ωm)^{2} using TCmf = 40.880 and 40.901 K, resp.) higher than that achieved in the case of AL-3D fit but lower than that for AL-2D fit. ε values [37] corresponding to both the TCmf values have been found using specific heat jump = 64 mJ/(mol K) [38] and ξC(0) (from Table 1) and Coherence length anisotropy, γξ~6 [39]. It is found that ε = 1.94 × 10^{−2} and ε = 2.6 × 10^{−2} corresponding to TCmf = 40.889 K and 40.901 K, respectively. Further, the corresponding ξ(ε) comes out to be ξ(ε)≫s (namely, for TCmf = 40.889 K, ξ(ε)~253.9Å), which clearly indicates the 3D fluctuation in the MgB_{2} tape sample. All these results show that the different observables around the MgB_{2} superconducting transition may be explained, in most of the experimentally accessible range, by considering Gaussian fluctuation in the mean-field Ginzburg-Landau like approaches. Thus, in the absence of magnetic field, the full critical region must then be expected only for temperatures close to mean field temperature (for |T-TCmf|≤1 K). Here, it is worth mentioning that the msd values presented in Table 1 for various fitting are evaluated for the data in the previous range.
Thus, it may be concluded that fluctuation induced enhanced conductivity in MgB_{2} tapes fit to the 3D relation for enhancement of the conductivity near the transition regime. The fluctuation study in the zero field yields the amplitude of coherence length ξ(0)~21Å which agrees very closely to the value (~26 Å) reported by Kim et al. [34]. These authors carried out the fluctuation study in the specific heat behavior in the bulk Mg^{11}B_{2} sample in absence of field, neglecting the inhomogeneity of the sample.
4. Conclusions
We have experimentally observed the fluctuation induced enhancement of conductivity in MgB_{2} in the absence of magnetic field. Study of fluctuation induced excess conductivity in the MgB_{2} sample via fitting of resistivity versus temperature data with Aslamazov and Larkin model and Lawrence and Doniach model indicates the three-dimensional nature of conduction of the carriers as opposed to the 2D character exhibited by the HTSCs. The estimated amplitude of coherence length from the fitted model is ~21 Å.
Conflict of Interests
It should be noted that the authors (S. Rajput and S. Chaudhary) do not have any conflict of interests regarding the content of the paper.
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