Comparative Analysis of the Irradiation with Medium Fluences of High-Energy Electrons and Pr Doping on the Fluctuation Conductivity of YBa₂Cu₃O_7–δ Single Crystals

Abstract

Medium-fluence fast electron irradiation (10¹⁹ e/cm² to 10²⁰ e/cm²) or the changes in the praseodymium concentration in the range of 0.0 ≤ z ≤ 0.5 on the excess conductivity of YBa₂Cu₃O_7–δ single crystals ware investigated. These can lead to a wider range of the temperature interval of excess conductivity which narrows the interval of linearity in the ab plane. At fluences 0 ≤ Φ ≤ 6.5 × 10¹⁹ e/cm², there was a threefold increase in the transverse coherence length ξ_c(0) with an increase in Φ of more than four times as the praseodymium concentration increased to z ≈ 0.42. The two-dimensional–three-dimensional (2D–3D) crossover point shifted upward in temperature. Conversely, to irradiation with low fluences (Φ ≤ 10¹⁹ e/cm²) or low praseodymium doping (z ≤ 0.39), irradiation with medium fluences or high praseodymium doping led to a non-monotonic dependence of ξ_c(0) on the irradiation fluence, with characteristic maxima at Φ~(7–8) × 10¹⁹ e/cm² and z ≈ 0.42, likely due to the suppression of the superconducting characteristics.

1. Introduction

The data obtained during the investigation of the impact of high-energy electron irradiation or praseodymium doping on the magnetoresistive characteristics of high-temperature superconducting cuprates (HTSCs) are important experimental material, the processing of which makes it possible to achieve success in determining the microscopic mechanism of superconductivity and improving the functional characteristics of existing HTSC compounds [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Superconducting, high-temperature, single crystals are the technological basis of fundamental research aimed at establishing the mechanisms regarding the interaction of layered structures with radiation [16,17,20].

Irradiation with high-energy electrons makes it possible to create radiation defects while at the same time retaining the composition of the material. These defects change the electrical resistance in both normal and superconducting states [16,17,18,19,20]. Doping with praseodymium also has a similar effect on the resistance of YBa₂Cu₃O_7–δ, which creates impurity defects [6,10,18,19,21,22,23] without changing the content of labile oxygen [24,25]. Comparing the resistance of electron-irradiated YBa₂Cu₃O_7–δ with the resistance of Y_1–zPr_zBa₂Cu₃O_7–δ will allow us to better understand the difference in the ensemble of defects in both cases, especially in the region of the superconducting transition.

Considering the prospect of using high-temperature superconductors as supersensitive sensors and electric current transmission lines with low energy losses operating in the region of liquid nitrogen’s boiling temperature, the creation of a so-called “controlled” defect structure [9,10,15] in a superconductor has important fundamental and practical significance.

Compounds of the YBa₂Cu₃O_7–δ system are among the most promising HTSC materials because of the following reasons: (a) these compounds have a fairly high superconducting transition temperature, Tc ≈ 90 K (higher than the boiling point of liquid nitrogen); (b) single-crystalline samples of these compounds are relatively easy to grow, and their physical properties can be varied quite easily by modifying the oxygen concentration; and (c) they hold technological importance so they need to be stable under extreme external influences.

The partial replacement of yttrium with praseodymium allows, unlike rare-earth elements, for a change in the Tc from the maximum value to 0 (antiferromagnetic insulator) while maintaining the degree of oxygen content at the optimal level (praseodymium anomaly). Thus, the application of electron irradiation or praseodymium doping makes it possible to tune the critical and electrical transport properties of Y_1–zPr_zBa₂Cu₃O_7–δ single crystals in a wide range.

Note that, despite intensive experimental studies of the influence of various kinds of external influences on electric transport in the YBa₂Cu₃O_7–δ system [1,2,3,4,5,6,7,8,9,10,15,16,17,19,20,21,22,23,24], there is only a small amount of scientific work devoted to comparative studies of the influence of electron irradiation and Pr doping on the pseudogap and fluctuation conductivity anomalies. At the same time, according to contemporary ideas, it is these “unusual” physical phenomena observed in HTSC compounds in the non-superconducting state that are critical to understanding the microscopic nature of HTSC, which is presently not fully explained, in spite of more than three decades of systematical experimental work.

In previous studies [26], we studied the impact of relatively low irradiation fluences from 1.4 to 8.8 × 10¹⁸ e/cm², or of weak doping with praseodymium on the fluctuation conductivity (FC) and paraconductivity in YBa₂Cu₃O_7–δ single crystals with a stoichiometric composition. Here, we consider the effect of medium fluences (up to 100 × 10¹⁸ e/cm²) of irradiation with high-energy electrons or the effect of praseodymium impurities in a significant concentration range (0.0 ≤ z ≤ 0.5) on the fluctuation conductivity in Y_1–zPr_zBa₂Cu₃O_7–δ single crystals during the flow of a transport current in basic ab-plane.

2. Experimental Methods

The YBa₂Cu₃O_7−δ single crystals investigated in this study were grown by the melt solution method [1,3,5]. The crystals were subjected to an oxygen atmosphere at 430 °C for four days and this led to their saturation. The crystals contained twins, whose planes of which had a block structure. Resistivity measurements were taken with the standard 4-pin method. The measured crystal sizes were (1.5…2) × (0.2…0.3) × (0.01…0.02) mm³, where the third dimension was the c axis. To prepare crystals with partial replacement of Y by Pr, Y_1–zPr_zBa₂Cu₃O_7–δ, Pr₅O₁₁ was introduced to the initial mixture in the appropriate percentage.

The modes of growth and oxygen saturation of Y_1–zPr_zBa₂Cu₃O_7–δ crystals were the same as for undoped single crystals [5,20]. Y₂O₃, BaCO₃, CuO and Pr₅O₁₁ compounds were used in the crystal growth. A transport current of up to 10 mA passed through the largest sample size, where the typical distance of the potential contacts is 1 mm.

The grown single crystals exhibited characteristics of normal and superconducting states corresponding to a given oxygen and/or praseodymium content. The technological and experimental details for the synthesis of experimental samples, resistivity measurements and analysis of the transport properties were previously described [1,3,5,19,20].

Electron irradiation with energies in the range of 0.5 to 2.5 MeV was employed at temperatures below 10 K. The irradiation fluence Φ = 10¹⁸ e/cm² by electrons with an energy of 2.5 MeV corresponded to a defect concentration averaged over all sublattices of 10⁻⁴ displacement per atom [26]. The measurement technique was the following: first, the temperature dependences of the resistivity of the sample were measured. The sample was then cooled to 5 K and irradiated with electrons. The beam intensity was chosen such that the temperature of the sample during irradiation did not exceed 10 K. After irradiation, the sample was heated to a temperature of 300 K and, gradually lowering the temperature, the resistivity of the sample was measured. The same sample was successively irradiated at different fluences. To measure electrical resistance after irradiation in the temperature range of 4.2 < T < 300 K, a specially designed helium cryostat was used. All electrical resistance measurements were carried out at a fixed temperature; temperature stability was about 5 mK. Temperature was measured with a platinum resistance thermometer.

3. Results and Discussion

In Figure 1a, the dependences r_ab(T) obtained before and after irradiation with fast electrons at fluences from 0 (curve 1) to 86.3 × 10¹⁸ e/cm² (curve 8) are shown. These dependencies were analyzed in previous work [27]. The electrical resistivity of Y_1–zPr_zBa₂Cu₃O_7–δ single crystals when the praseodymium content, z, changed from zero (curve 1) to z = 0.5 (curve 8) is shown in Figure 1b. As can be seen from Figure 1, in both cases, the curves are characterized by quasi-metallic behavior of electrical resistivity with a characteristic linear area of the dependence ρ(T) at high temperatures. Both at maximum irradiation fluences and at maximum Pr content, the ρ(T) curves acquired a characteristic S-shape, which indicates the appearance of a thermally activated region on the ρ(T) dependences, which will be discussed in more detail below.

From Figure 1, it can be deduced that when the T dropped below T*, in the basal plane, on the ρ_ab(T) dependencies in the region of relatively high temperatures, an extended linear section was preserved even at for high irradiation fluences. Numerous theoretical models have been introduced to explain these dependences including the resonating valence band theory (RVB theory) [28] and the nearly antiferromagnetic Fermi liquid (NAFL) model [29]. In the RVB theory, scattering in HTSC compounds occurs through the interaction of carriers (spinons and holons) [28]. Herewith, the T dependence of the electrical resistivity additionally to the linear T term introduces an additional term, proportional to 1/T [30], in both cases, as in longitudinal and as in transverse electrical resistivity:

$$\rho \left(\mathrm{T}\right)=\mathrm{A}{\mathrm{T}}^{-1}+\mathrm{B}\mathrm{T}$$

(1)

From Figure 2 (note: irradiation fluences up to ≤ 70 × 10¹⁸ cm⁻² or a low level of Pr doping, z < 0.25), it was observed that the dependences of the products ρ_ab(T)⋅T on T² become close to linear.

For medium and high fluences of Φ > 100 × 10¹⁸ cm^–2 or in the case of medium and heavily Pr-doped samples z ≥ 0.25, the experimental curves ρ_c(T) could not sufficiently be described by the dependence from Equation (1).

Figure 3 shows the dependencies of the linearity parameters A and B in Equation (1) from fluence (Φ) or praseodymium concentration (z).

As inferred from the figure, the linearity of the ρ(T)·T–T² dependence was directly related to the defectiveness of the structure of the experimental samples, since both the (linear) parameters increased rapidly by increasing Φ or z, i.e., by increasing the concentration of irradiation and impurity defects.

From Figure 4, it could be concluded that increasing Φ led to a rapid decrease in the interval of linearity or shift toward high temperatures (measurements were carried out up to 300 K); regarding the increase in z, the situation was even more diverse, and namely, for concentration number 8 (of the experimental measurements) there was no linearity (up to 300 K), and the derivative decreased by increasing the temperature. That is, there was a contradiction such that by (with) increasing defectiveness, the parameters of the linear dependence ρ(T)·T–T² increased, but the linear dependence itself showed a tendency to disappear due to the narrowing of the interval of its existence.

As noted above, there were noticeable differences in the evolution of resistivity curves under irradiation or doping with praseodymium. Figure 1 reveals that irradiation resulted in an anomalously strengthened (in comparison to the change in composition [1,16,22]) decrease in T_c in YBa₂Cu₃O_7–δ. Nevertheless, the nature of the change in the electrical and superconducting properties of HTSC with a change in composition [1,22] and under the action of irradiation was somewhat different. The primary difference was for Y_1–zPr_zBa₂Cu₃O_7–δ single crystals with T_c < 85 K, where as a rule, a change in the shape of the ρ(T) curves was observed from metallic to the so-called “S-shaped” with a characteristic thermally activated deflection [1,17,18] (z > 0.2, Figure 1b); during irradiation, a similar “S-shaped” shape of the ρ(T) curves was observed for T_c < 35 K (Φ > 60 e/cm², Figure 1a).

A key reason for the strong decrease in T_c in irradiated samples was the appearance of dielectric inclusions under the influence of irradiation because of the redistribution of oxygen between the O(4) and O(5) positions and the formation of local regions with a tetragonal structure.

With increased fluence, the T_c decreased from~92 to~24 K, and ρ_ab(300 K) increased from ρ~158 to 384 μOhm·cm, respectively, which is consistent with previous studies [26,30]. Clearly similar effects were avoided when supplemented with praseodymium [1,18]. The dependencies of T_c and ρ(300 K) on Φ and z are shown in the insets in Figure 1a,b.

As shown from Figure 1, below, for the characteristic temperature T*, the dependencies for all ρ_ab(T) curves were “rounded off”, which may have been caused by the emergence of excess conductivity Δσ_ab. Lawrence and Doniach [13] used the following relation for the fluctuation conductivity in the plane of layers in the immediate vicinity of the SC transition:

$$\Delta {\sigma }_{ab}={\displaystyle \frac{e^2}{16\mathrm{ħ}d}}{\displaystyle \frac{1}{\sqrt{\left[\epsilon \left(\epsilon +r\right)\right]}}},$$

(2)

where d = 11.7 Å and is the interlayer distance [31].

$$\epsilon ={\displaystyle \frac{T-Tc}{Tc}}\ll 1; r={\displaystyle \frac{4{\mathrm{\xi }}_{\mathrm{c}}^2\left(0\right)}{d^2}}.$$

(3)

Equation (1) describes a 2D–3D crossover that occurs in a certain temperature interval: when ε << r σ_ab^AL∝(ε r)^−1/2 (3D mode), but at ε >> r σ_ab^AL∝ε⁻¹ (2D mode). According to the theory [12], the 2D–3D crossover should happen when

T₀ = T_c{1 + 2[ξ_c(0)/d]²},

(4)

where it is assumed that α = 1/2, i.e., the following:

ξ_c(0) = (d/2)ε₀^1/2.

(5)

Having determined the value of ε₀ and applying the literature data on the dependence of T_c and interplanar spacing on δ [31], it was feasible to calculate the value ξ_c(0).

In Figure 5 are shown the Δσ_ab(T) dependencies in lnΔσ(lnε) coordinates for different r values, experimentally derived from the following relation:

Δσ_abexp(T) = 1/ρ_exp(T) − 1/ρ_nab(T),

(6)

before and after irradiation. With a temperature range between T_c and ≤ 1.1T_c (depending upon the oxygen concentration), these dependencies were adequately approximated by straight lines with a slope angle of α₁ ≈ −0.5, thus indicating a three-dimensional character of fluctuation superconductivity in this temperature regime. by increasing the T, the rate of decrease in Δσ increased significantly (α₂ ≈ −1). This is an indication of a change in the dimensionality of fluctuation conductivity (FC). This change corresponds to a 2D–3D crossover. Notably, in previous studies of the FC in YBa₂Cu₃O_7–δ HTSC systems of various compositions using the Kouvel–Fisher method [4,6,10], a whole chain of crossovers was repeatedly recorded, and this was interpreted as a sequence of transitions of 1D–2D, 2D–3D and 3D–critical fluctuations and intermediate types between them.

Taking into account these results, we considered the following picture of superconducting pairing in HTSC. Fluctuation pairs, nucleated inside the CuO₂ planes at T ≤ T*, led to an increase in nsc. Since at T >> T_c the values of nsc and especially the ξ_c(T) are very small, there is likely to be no interaction between the pairs. The corresponding electronic state of the fluctuation pairs can be considered as 0-D, which is not represented by the existing FC theories. At T ≤ T_2D, fluctuation pairs begin to overlap, but still only within the CuO₂ planes, forming a 2D electronic state, which was described by the Mackie–Thompson regime (MT) contribution of the Hikami–Larkin (HL) theory [14]. At T ≤ T_3D, the increasing ξ_c(T) becomes greater than d and connects the conducting planes by paired tunnel interactions of Josephson type. Now, the fluctuation pairs interact through the whole volume of the superconductor and form a 3D electronic state, which is well described by the 3D contribution of the Aslamazov–Larkin (AL) theory [12]. In fact, only now is the system entirely ready to complete the transition to the superconducting state.

Note that for the lnΔs(lne) dependences obtained at maximum irradiation fluences (F = 7.92 and 8.63 × 10¹⁹ e/cm²), a non-monotonic progression of the curves was observed, which may have indicated an additional crossover at temperatures e ≥ e₀ with a sharp decrease in slope, α. This behavior was already observed previously in [8] and may have indicated the possible presence of the so-called Mackie–Thompson (MT) regime of behavior of the temperature dependences of paraconductivity Δs in the system [14].

In the 2D region, two-particle tunneling between layers is excluded, resulting in superconducting and the normal carriers being located directly in the planes of the leading layers. The dominant contribution to the FC in this regime is made by an additional contribution, justified by Mackie–Thompson [14] and determined as a result of the interaction of the fluctuation pairs with the normal charge carriers. This contribution depends on the lifetime of fluctuation pairs and is determined by the processes of pairing for each specific sample. Importantly, the degree of heterogeneity of the sample structure has to be taken into account. According to [8], for samples of a perfect structure:

$${\sigma }_{MT}={\displaystyle \frac{e^2}{8\hslash d\left(1-\alpha /\delta \right)}}\ln \left\{\left({\displaystyle \frac{\delta }{\alpha }}\right){\displaystyle \frac{1+\alpha +{\left(1+2\alpha \right)}^{1/2}}{1+\delta +{\left(1+2\delta \right)}^{1/2}}}\right\}{\epsilon }^{-1}$$

(7)

Here,

$$\alpha =2{\left[{\xi }_c\left(0\right)/d\right]}^2{\epsilon }^{-1} \mathrm{and} \delta =1.203\left(l/{\xi }_{ab}\left(0\right)\right)\left(16/\pi \hslash \right){\left[{\xi }_c\left(0\right)/d\right]}^2{k}_{b}T{\tau }_{\varphi }$$

(8)

are the communication and depairing parameters, respectively. Here, l is the mean free path, ξ_ab is the coherence length in the ab-plane and τ_ϕ is the existence time of fluctuation pairs. In the presence of inhomogeneities in the structure, the Δσ(T) dependence is determined by the Lawrence–Doniach (LD) model [13].

The dependence of the coherence length ξ_c(0) on the fluence of electron irradiation is shown in the inset (b) in Figure 5a.

On the main panels and inserts (a) in Figure 5a,b shows the dependencies of excess conductivity in the ab-plane in lnΔσ-lnε coordinates for YBa₂Cu₃O_7–δ (Figure 5a and insert (a)) and for Y_1–zPr_zBa₂Cu₃O_7–δ (Figure 5b and insert (a)). The dotted lines show the approximation of the experimental curves by straight lines with slopes α₁ ≈ −0.5 (3D regime) and α₂ ≈ −1.0 (2D regime). Arrows show 2D–3D crossover points.

Insets (b) show the values of ξ_c(0), calculated using Equation (5) for different Φ or z. It can be seen that ξ_c(0) passes through a maximum at Φ_m ≈ 80 × 10¹⁸ e/cm² (Figure 5a) or z_m ≈ 0.43 (Figure 5b).

Figure 6 shows the dependencies of ξ_c(0) on T_c for all investigated samples. Dark squares show the data obtained previously for YBa₂Cu₃O_7–δ film samples at different volumes of δ [8]. As is known from the general theory of superconductivity (Bogoliubov–de Gennes—BdG [32]), the relationship between ξ_c(0) and T_c in superconducting compounds obeys the following relation:

ξ₀ ~ ℏv_F/[πΔ(0)],

(9)

where Δ(0) is the order parameter at T = 0 K. Since for YBa₂Cu₃O_7–δ the value 2Δ(0)/k_BT_c ≈ 5, then, taking ξ₀ = ξ_c(0), this can be written as

ξ_c(0) = G/T_c

(10)

where G = 2K$\hslash$v_F/(5πk_B) and the proportionality coefficient is K ≈ 0.12.

The dependence ξ_c(0) as a function of T_c is shown in Figure 6 with the red solid line, which indicates that the pairing mechanisms in HTSC films, in this range of temperatures and praseodymium concentrations, obey to a large degree the general theory of superconductivity. Qualitatively similar behavior of analogous dependencies was observed for YBa₂Cu₃O_7–δ (circles) and Y_1–zPr_zBa₂Cu₃O_7–δ (triangles) single-crystal samples at T_c ≥ 65 K and 55 K, respectively. Note that for the lnΔσ(lnε) dependences obtained at maximum irradiation fluences (√ = 7.92 and 8.63 × 10¹⁹ e/cm²), a non-monotonic progression of the curves was observed, which may have indicated an additional crossover at temperatures ε ≥ ε₀ with a sharp decrease in the slope, α. This behavior was already observed previously in [8] and may indicate the possible presence of the so-called Mackie–Thompson (MT) regime of behavior of the temperature dependences of paraconductivity Δσ in a system [14]. Structural and kinematic anisotropy in the system can play a particular role [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59].

4. Conclusions

The presented data and their analysis allowed us to draw a general conclusion that for optimally oxygen-doped YBa₂Cu₃O_7–δ single crystals, irradiation with medium fluences of high-energy electrons or an increase in the degree of doping with praseodymium leads to similar changes in the temperature dependences of electrical resistivity, ρ(T), in the ab plane. In particular, there is a significant expansion of the temperature interval of excess conductivity Δσ(T) existence. In both cases, there is a multiple increase in the transverse coherence length ξ_c(0) and the 2D–3D crossover point significantly shifts by temperature. It was for the first time established that, in contrast to the case of irradiation with small fluences of high-energy electrons (Φ ≤ 10¹⁹ cm⁻²) or with praseodymium doping to concentrations z ≤ 0.39, irradiation with medium fluences or praseodymium doping at higher concentrations leads to a non-monotonic dependence of the transverse coherence length ξ_c(0) from the critical temperature T_c, with characteristic maxima at Φ ~ (7–8) × 10¹⁹ e/cm⁻² or z ≈ 0.42, which may be associated with a suppression of superconducting characteristics.