Thermal Fluctuation Induced Excess Conductivity Analysis of Au Nanoparticles and CuTl-1223 Superconducting Phase Composites

M Mumtaz?? ¹ , Arshid Khan ¹ , M Rahim ¹ , Mubasher ²

1 Department of Physics, Faculty of Sciences (FS), International Islamic University (IIU), 44000, Pakistan

2 Department of Physics, Faculty of Engineering and Sciences (FEAS), Riphah International University, Islamabad, I­14, 44000, Pakistan

Abstract

Fluctuation induced conductivity (FIC) analysis of (Au)_y/CuTl-1223 (y = 0, 0.5, 1.0 and 1.5 wt. %) nanoparticles-superconductor composites was carried out via Aslamazov-Larkin (AL) and Lawrence-Doniach (LD) models in different fluctuation regions near and above the critical temperature T_c in the transition region. Different superconductivity parameters, like coherence length along c-axis (ξ_c), inter-layer coupling constant (J), phase relaxation time (τ_φ), Fermi velocity (V_F), Fermi energy (E_F), penetration depth (????_p.d), critical magnetic field (B_c), lower critical magnetic field (B_c1), upper critical field B_c2 and critical current density (J_c) were calculated with the help of FIC analysis. Almost all these superconducting parameters were increased with the increasing concentration of Au nanoparticles in these composites. The enhancement in these parameters can be attributed to the conducting nature of Au nanoparticles (NPs) residing on the grain boundaries of the host CuTl-1223 superconducting matrix. The suppression of some critical parameters like B_c, B_c1 and J_c with the addition of Au NPs reflected the paramagnetic nature of Au NPs.

Keywords

Gold Nanoparticles, CuTl­1223 superconducting phase, Composites, Fluctuation induced conductivity (FIC) analysis

INTRODUCTION

Thermal fluctuations arise in the oxide high temperature superconductors (HTSCs) due to their anisotropic behavior and short coherence length along c-axis (ξ_c) 1, 2, 3, 4, 5. A lot of experimental as well as theoretical work has been reported on critical effects of thermal fluctuations e.g. the fluctuation induced excess conductivity and magnetic conductivity. Thermal fluctuations are of immense interest in studying the phase transition from normal state to superconducting state at the onset temperature T_c ^onset (K) of the superconductivity. These fluctuations result in the formation of Cooper pairs in the transition region from T_c ^onset (K) to T_c (0), which becomes the source of excess conductivity in the transition region of HTSCs 6, 7, 8, 9, 10. The FIC analysis of the bulk as well as the thin films samples of different HTSCs have already been reported by many groups 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. The FIC analysis helps to estimate the different superconducting microscopic critical parameters of superconducting materials that are very diffi?cult to measure directly by the experiments 21. This analysis provides information about the fluctuation in pairing and scattering mechanism near T_c ^onset (K) 22.

To investigate the effects of these fluctuations on excess conductivity, Aslamazov-Larkin (AL) model, Lawrence-Doniach (LD) model, Maki–Thompson (MT) model and Hikami–Larkin (HL) model. The FIC (Δσ) were used 23, 24, 25, 26, 27.

1 ???=1?(?)-1??(?)

Where ρ(T) is the actual resistivity, ρ_n(T) is the extrapolated normal state resistivity to 0 K following the straight-line equation α is the intercept and β is the slope that is obtained from fitting of the resistivity versus temperature plot.

AL model is used to explain superconducting fluctuations in intra-grain and inter-grain regions of layered superconductors 16. The analysis of AL model is generally valid for the single crystalline samples but it can be applied to polycrystalline materials with certain limitations 9. The excess conductivity according to AL model can be written as;

2 ???=????

Where A, ε and λ_D are fluctuation amplitude, reduced temperature and critical exponent respectively. The mathematical expressions for A and ε are given by 16;

3 ?=?-????????

4 ?=?2/(32??? (0))  ??? 3? ?2/16??  ??? 2? 

Where T_c ^mf in equation is the mean field critical temperature determined from the peak of dρ/dT plot, ? is the reduced Plank’s constant and d is the effective interlayer spacing. The critical exponents are determined by the following relation (5) 28, 29, 30, 31, 32;

5 ??=?2-2

Where D is the dimensionality of fluctuations i.e. D = 3 for 3 dimensional fluctuations, 2 for 2 dimensional fluctuations and 0 for zero dimensional fluctuations. Values of λ_D are -1/3, -1/2, -1 and -2 for critical, 3D, 2D and 0D fluctuations, respectively. The LD model is used to calculate ξ_c(0) for bulk superconductors at cross-over temperature from 3D to 2D regime (T_3D-2D), as follows 33.

6 ?3?-2?=??1+2??0?2

In this equation, 2??0?2=? , is the interplane coupling constant. The anisotropy parameter ‘γ’ is defined as, ?=???/?? , where ??? and ?? are the coherence lengths along the superconducting ab-planes and along c-axis, respectively 33. The values of ξ_ab for CuTl-based HTSCs vary between 10 to 20 Å 34.

The excess conductivity according to MT model is caused by the interaction of fluctuating Cooper pairs with normal electrons 23. Although the MT contribution is minimal but can’t be ignored. This model is dependent on the phase relaxation time and can be used to investigate the influence of the superconducting electrons on the normal electrons 24. The expression for excess conductivity according to MT model is given by 26;

7 ????=?28???(1-?/?)×????1+?+(1+2?)1/21+?+(1+2?)1/2     

Where δ = 4??0?2??????? represents pair breaking parameter, k_B is the Boltzmann constant and ?? is phase relaxation time. The MT contribution depends on ?? and in 2D fluctuations region, its importance becomes dominant due to moderate breaking of Cooper pairs 30. The expression for phase relaxation time of fluctuating Cooper pairs can be written as 29:

8 ??=??8?????        

Where ?? is the cross over temperature from 2D (LD) to 0D (MT) fluctuations regime at which δ ≈? 35. The 0D region exists at temperatures much higher than T_c, where the thermal and pairing energy of electrons compete, so the pairs do not have sufficient time to conduct but exist. The importance of this region is that the Cooper pair’s formation starts at temperatures much higher than T_c but due to much higher thermal energy available for breaking of the pairs, they can’t establish superconductivity 2.

The Fermi velocity (V_F), Fermi energy (E_F) and coupling constant (λ) of cooper pairs can be calculated from the following equations 9, 36:

9 ???=5???????(0)2??

Where K is co-efficient of proportionality and its value is K ≅ 0.12 18.

10   ?=???-12????  

11  ??=???1.6×10-19?? 

At the critical-3D crossover temperature (T_G), we can use Ginsburg–Landau (GL) theory to calculate some important superconductivity parameters 37. To find the thermodynamic critical magnetic field B_c(0), we use the Ginsburg number N_G, as follows 38, 39;

12 ??=??-????=12??????(0)2?2??(0)32  

13 ??=??22???.????(0)

14 ??1=?????2? 

15 ??2=2??? 

16 ??=4???133??.???? 

Where, ??=?2? and ?=??.?? are the flux quantum and GL parameter, respectively 40.

In this article, we have presented the results of the fluctuation induced conductivity analysis of (Au)_z/CuTl-1223 nanoparticles-superconductor composites. The objective behind the inclusion of Au NPs in the CuTl-1223 matrix was to fill the voids between the superconducting grains and to enhance the inter-grains connectivity with the conducting natured NPs to improve the superconducting transport properties. In order to further elaborate the experimental study, we have carried out the FIC analysis of the resistivity data. From this analysis it was observed that the Josephson’s coupling parameter J and ???? were increased with the addition of Au NPs. This shows that the superconductivity is enhanced due to the improved inter-grain coupling caused by the conducting Au NPs added across the grain-boundaries.

EXPERIMENTAL

Au NPs were extracted from gold colloidal solution, while CuTl-1223 superconducting matrix was synthesized via two-step solid-state reaction method. In first step, the calculated amounts of Ca(NO₃)₂. 4H₂O, Ba(NO₃)₂ and Cu₂(CN)₂. H₂O were mixed and ground for about two hours in an agate mortar and pestle. After grinding, the ground material was kept in quartz boat and then placed in a chamber furnace at 860 ^oC for 24 hours and then cooled down the material to room temperature. Further, the material was again ground for an hour and again heat-treated at 860 ^oC for 24 hours in pre-heated chamber furnace. In this way, the precursor Cu_0.5Ba₂Ca₂Cu₃O_10-material was obtained. In the second step, the calculated amount of Thallium Oxide (Tl₂O₃), Au NPs and precursor material were mixed and ground one hour. The ground material was pressed under a high pressure of 3.8 tons / cm² using a hydraulic press to obtain the samples in pellets form. These samples were enclosed in gold capsules for sintering at 860 ^oC for about 10 minutes in pre-heated chamber furnace followed by quenching to room temperature to get (Au)_z/CuTl-1223 composites.

The resistivity versus temperature measurements of the samples have been carried out via four probe technique by using Quantum Design Physical Properties Measurement System (PPMS). The FIC analysis of the samples was performed by AL and LD models to deduced different superconducting parameters.

RESULTS & DISCUSSION

The combined graph of resistivity versus temperature of (Au)_z/CuTl-1223 composites is shown in Figure 1. In the inset of Figure 1, the variation of T_c and ρ_300K versus (vs) Au nanoparticles (z) contents is also given. The values of T_c for (Au)_z/CuTl-1223 samples with z = 0, 0.5, 1.0 and 1.5 wt.% was found around 87.28 K, 97.32 K, 101.33 K and 95.31 K, respectively. It was observed that the value of T_c is increased with the addition of Au NPs up to certain optimum level (i.e. z = 1.0 wt.%) and then started to decrease beyond this optimum contents of Au NPs. The initial enhancement of T_c(0) is attributed to the improvement of inter-grains connectivity by healing up the voids and pores present in the host CuTl-1223 matrix with conducting Au NPs, while the decrease in T_c beyond the optimum level may be due to segregation and agglomeration of non-superconducting Au NPs at the grain boundaries of the host CuTl-1223 superconducting matrix. This segregation and agglomeration result into the suppression of superconducting volume fraction and density of charge carriers in the conducting CuO₂ planes of CuTl-1223 superconductor. The resistivity vs temperature measurements show that the normal state resistivity at 300 K (ρ_300K) is decreased up to optimum concentration level of Au NPs. The suppression of ρ_300K may be due to the suppression of weak-links among the superconducting grains, which may promote the charge carriers transport processes and decrease the energy losses at the grain boundaries. Beyond the optimum level, enhancement in the normal state resistivity may arise due to higher scattering and trapping of charge carriers with agglomerated Au NPs present at the grain boundaries 37, 38. The variation of T_c and ρ_300K is somehow random with Au NPs addition. This random behavior may be due to non-uniform distribution of these nanoparticles at the grain-boundaries of the host CuTl-1223 superconductor 41.

Figure 1: Resistivity versus temperature measurements of (Au)_z/CuTl-1223composites.

The FIC analysis of resistivity versus temperature data of (Au)_z/CuTl-1223 composites have been performed via various theoretical models such as AL and LD models along with GL theory in different transition regions. The log-log plots of excess conductivity (?σ) versus reduced temperature (ε) for (Au)_z/CuTl-1223 composites are shown in Fig. 2 (a-d). The combined plot of actual measured resistivity (ρ), extrapolated normal state resistivity (ρ_n) from room temperature to 0 K and dρ/dT for all samples are given in the insets of Figure 2 (a-d).

Figure 2: (a) log-log plot of ?σ vs ε for (Au)_z/CuTl-1223 composite with z = 0, (b) z = 0.5 wt.%, (c) z = 1.0 wt.%, (d) z = 1.5 wt.%.

The fitting of experimental curves shows the presence of different fluctuation regions such as critical, 3D, 2D and 0D regions in these samples. The numerical values of different superconducting parameters such as T_c(0), T_c ^mf, all cross over temperatures (T_G=T_CR-3D, T_LD=T_3D-2D, T_MT=T_2D-0D), T^*, α, ?T_c, ξ_c(0) and J extracted from FIC analysis of resistivity versus temperature data of (Au)_z/CuTl-1223 composites are given in Table 1.

Table 1: Superconducting parameters extracted from FIC analysis of (Au)_z/CuTl-1223 composites.

Table 2: Numerical values of dimensional exponents (λ_D)in different fluctuation regions for (Au)_z/CuTl-1223 composites.

Table 3: Temperature ranges of different fluctuationregions for (Au)_z/CuTl-1223 composites.

Table 4: Numerical values of V_F, ?? , λ and E_F of (Au)_z/CuTl-1223 composites.

Table 5: Numerical values of N_G, B_c, ??,? , κ, B_c1, B_c2 and J_c of (Au)_z/CuTl-1223 composites.

It has been observed that the values of T_c (0), T_c ^mf, T^*, ?T_c, ξ_c (0), J and all cross over temperatures were increased up to a certain optimum concentration level with the addition of Au NPs. The enhancement in the afore mentioned parameters may arise due to the strong inter-grains coupling caused by addition of conducting Au NPs in the host CuTl-1223 matrix. The value of residual resistivity was decreased with Au NPs addition, which may be due to enhancement of connectivity between the superconducting grains of the host superconducting matrix. The enhanced values of ξ_c(0) show that the samples become more isotropic in nature after addition of Au NPs 40. The dimensional exponent (λ_D) tells about the dimensionality of superconducting fluctuations. The numerical values of λ_D for different fluctuation regions such as critical, 3D, 2D and 0D regions are listed in Table 2, which have been calculated from the slopes of the lines in different fluctuation regions. The corresponding temperature ranges for λ_D in different fluctuation regions are given in Table 3. It was found that the values of temperature ranges are shifted towards higher values with Au NPs addition as compared to pure sample, which indicates an improvement of the non-superconducting grain boundaries in the host CuTl-1223 superconductor after inclusion of these conducting nanoparticles. The numerical values of V_F, ?? , λ and E_F for all samples are listed in Table 4. It has been observed that ?? is decreased, while values of V_F, λ and E_F are enhanced, which indicate that the addition of Au NPs promotes the charge carriers transport processes and enhances the coupling between the superconducting grains of host CuTl-1223 matrix. The numerical vales of N_G, B_c, ??.? , κ, ??1 , ??2 and J_c are displayed in Table 5. It can be seen that the values of B_c, B_c1 and J_c are decreased while the values of N_G, ??.? and κ are increased with the addition of Au NPs. The suppression of critical magnetic fields may be due to paramagnetic behavior of added Au NPs. The value of J_c is also suppressed, which may be due to the increase of the number of mobile carriers scattering sources with the addition of these non-superconducting metallic Au NPs nanoparticles 24.

It has been concluded that most of the superconducting parameters extracted from the FIC analysis have been increased with the addition of Au NPs up to certain optimum level. So it is proposed that the addition of metallic nanoparticles in different granular superconducting matrices is favorable for the improvement of most of the microscopic superconducting parameters.

CONCLUSIONS

Series of (Au)_z/CuTl-1223 nanoparticles-superconductor composites were successfully synthesized via solid-state reaction method. The resistivity versus temperature measurements of the samples was carried out via four probe technique and FIC analysis of resistivity vs temperature curves of all these samples was performed by using AL and LD models in different fluctuation regions above T_c. It was found that the values of T_c were increased while ρ(300K) was decreased with Au NPs addition. The enhancement of T_c may be due to improvement of inter-grains connectivity by filling the voids and pores present in the host CuTl-1223 superconductor after inclusion of Au NPs. The values of different superconducting microscopic parameters extracted from FIC analysis such as T_c ^mf, ξ_c(0), J, V_F, E_F, ????_p.d and all cross-over temperatures were found to increase with increasing concentration of Au NPs in these composites. The enhancement in these parameters is attributed to the conducting nature of Au NPs residing on the grain-boundaries of the host superconducting matrix. The enhanced values of ξ_c(0) show that the samples become more isotropic as a whole in nature after Au NPs addition. The magnetic properties such as B_c, B_c1 and J_c were decreased with the addition of Au NPs in CuTl-1223 matrix, which may be attributed to the paramagnetic nature of Au NPs. Most of the superconducting parameters, extracted from FIC analysis were increased with the addition of Au NPs up to certain optimum level.