1 Introduction

It is known that electrical transport characteristics determine the important structural and operational properties of high-temperature superconductors (HTSCs) used as sensors or electric power transmission lines [1]. Electron irradiation is a very effective tool for creating a significant number of defects without changing the composition of the irradiated object [2, 3]. This makes it possible to separate the effects that are directly related to defects that may occur both during controlled exposure [4, 5] and during the operation of systems based on HTSC compounds [6]. In particular, the determined influence of electron irradiation on the processes of scattering of charge carriers on defects and phonons, on fluctuating conductivity, the phase separation and the stability of characteristics make it possible to predict the behavior of the specified materials in extreme conditions and create samples with the desired functional characteristics [2, 3, 7].

Despite a fairly large number of scientific works devoted to the study of the influence of various factors on electrical transport in the YBa2Cu3O7−δ system [1,2,3,4,5,6,7,8,9,10], there is very limited scientific work regarding the study of the influence of irradiation on the anisotropy of charge carrier scattering processes both in the normal state and near the superconducting transition, the pseudogap and fluctuation anomalies, as well as regarding the incoherent electrical transport. According to modern ideas [8], it is precisely these unusual physical phenomena observed in HTSC compounds in the normal (non-superconducting) state that are important for understanding the physical essence of the microscopic nature of HTSC, which still remains unclear, despite more than thirty-seven years history of intensive theoretical and experimental research conducted in this field of solid-state physics.

Notably, until now the contradictions of even such classic experiments as measuring the temperature dependence of the Hall effect and electrical resistance in the normal state and near the transition to the superconducting state have not been completely resolved. There is the presence of a noticeable maximum when approaching the SC transition [9], while the electrical resistivity in a fairly wide range is a linear function of temperature [10]. Experiment indicates the presence of at least two different scattering mechanisms in the system, or a possible significant anisotropy of scattering of current carriers.

Therefore, the study of the effect of electron irradiation on the electrical resistance of the Y–Ba–Cu–O system in a wide range of temperatures, from the superconducting transition to room temperatures, can provide important information about the interaction of charge carriers with phonon and defect subsystems. In particular, adequate allocation of fluctuating conductivity is possible only in the case when the conductivity of the normal state up to the superconducting transition is reliably known. Previous studies [11,12,13,14,15,16] reported the influence of small fluences of irradiation by fast electrons on longitudinal and transverse electrical transport in YBa2Cu3O7−δ single crystals with optimal oxygen content and a high critical temperature, Tc ≈ 90 K.

This work presents the results of the study of the influence of rather high fluences of irradiation by fast electrons on the conductivity of the normal state in the basal plane of HTSC YBa2Cu3O7−δ single crystals, which lead to a significant degradation of the SC characteristics.

2 Experimental techniques

The growth and the thermal treatment of the studied in this work YBa2Cu3O7−δ single crystals were performed according to the method described in [12, 13]. Irradiation was carried out with electrons with energies of 0.5–2.5 MeV at temperatures, T ≤ 10 K at various fluences [12], the largest of which was 230 × 1018 cm–2. The fluence of irradiation Φ = 1018 cm–2 by electrons with an energy of 2.5 MeV corresponds to the concentration of defects averaged over all sublattices of 10–4 displacement/atom (dpa) [4]. A specially designed helium cryostat [12] made it possible to measure the resistivity after irradiation in the temperature range 10 < T < 300 K. The resistivity was measured by the four-contact method [15, 16]. The characteristic sizes of the crystals for the measurements were 1.4 × 0.5 × 0.04 mm3, where the smallest size corresponded to the c-axis. All resistivity measurements were performed at a fixed temperature. The temperature was measured with a platinum resistance thermometer, and the temperature stability was about 5 mK.

3 Results and discussion

Figure 1 shows the temperature dependence of the electrical resistivity of the same sample, measured in the ab-plane, ρab(T), before (curve 1) and after (curves 2–6) irradiation. The inset shows the corresponding derivatives, dρab/dT, in the region of the superconducting transition.

Fig. 1
figure 1

Temperature dependencies of the electrical resistivity, ρab(T), in the ab-plane of the YBa2Cu3O7–δ single crystal, measured before and after irradiation with high-energy electrons, curves 1–6, fluences: (0, 13.1, 24.7, 38.9, 55.7, 68.1)·1018 cm–2, respectively. Points—experiment, solid lines—approximations according to (3)-(5), dashed lines—extrapolations of linear dependencies. T*1,6—low-temperature limits of linear dependences for curves 1 and 6, respectively. Inset: The corresponding derivatives, dρab/dT, in the region of the superconducting transition, numbering corresponds to the main panel

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All ρab(T) curves are characterized by the presence in the normal state of a section with “metallic” behavior, where the dependence of ρab(T) is linear, ρab(T) ≈ a + bT (dashed lines). The dependences of the parameters a, b, and the low-temperature limit of linear dependence, T*, on the electron fluence are shown in Fig. 2 and corresponding inserts. It can be seen that the free term of the linear temperature dependence of the resistivity, a, increases in proportion to the electron fluence (Fig. 2, main panel). Parameters b and T* depend on the fluence in a complicated way, demonstrated a tendency to increase by increasing the fluence (Fig. 2, insets). All this suggests that changes in the linear temperature dependence of the resistivity due to irradiation are caused by defects, including the non-irradiated sample, where oxygen deficiency, δ, can also be considered as a defect.

Fig. 2
figure 2

Dependencies of the parameters of the linear temperature dependence of the resistivity, ρab(T) ≈ a + bT and its low-temperature limit, T*, on the electron fluence, Φ

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At temperatures below T*, ρab(T) deviates from linearity. If at T < T* ρab(T) decreases (compared to the linear dependence), then it is often believed that the system realizes a pseudogap (PG) regime [17], which is characterized by the presence of a strong temperature variation of the band structure of quasiparticles, which should lead to the evolution of the Fermi surface and, as a result, to a decrease in scattering intensity.

Note that the fairly long linear section of ρab(T) is preserved even at significant irradiation fluences, shifting, however, toward higher temperatures. A number of different theoretical models have been proposed to explain the nature of the linear resistivity dependence, the most well-known of which are the so-called RVB theory [18] and the NAFL model [19]. According to the first of them, scattering in HTSC compounds is carried out by the interaction of charge carriers with two types of quasi-particle excitations—spinons and holons [18]. At the same time, the temperature dependence of electrical resistivity implies, in addition to the linear in temperature term, the presence of an additional term proportional to 1/T [18], in both, for longitudinal and transverse electrical resistivity cases:

$$\rho (T)=A{T}^{-1}+BT$$
(1)

According to the NAFL model [19], the scattering of carriers in HTSC systems is determined by the antiferromagnetic interaction Veff, which makes it possible to explain both, the anomalous temperature dependence of the Hall coefficient RH ∝ 1/T and the presence of a linear section on the dependences ρab(T). According to the NAFL theory, the linear dependence of ρab(T) at high temperatures can be considered as a reliable sign of the normal state of the system, which is characterized by the stability of the Fermi surface, and therefore by the stability of the scattering intensity of normal carriers. At temperatures below a certain T*, the magnetic behavior of the system realizes a pseudogap (PG) regime, characterized by a change in the spectrum of magnetic fluctuations, as well as the presence of a strong temperature variation of the band structure of quasiparticles, which should lead to the evolution of the Fermi surface [10] and, as a result, to a decrease in scattering intensity. As predicted in the NAFL model, the resistivity of cuprate superconductors in the normal state is due to the magnetic interaction between planar quasiparticles:

$$\rho_{\text{NAFL}} = \frac{{A_{1} T^{2} }}{{B_{1} + T}}$$
(2)

Notably, at high temperatures, both theories give a linear dependence of ρab(T) (\(T\gg \sqrt{\frac{A}{B}}\) for the RVB theory [18] and \(T\gg {B}_{1}\) for the NAFL model [19]). But none of these theoretical models, which explain this behavior of the ρ(T) curves in the region of high temperatures, could satisfactorily describe the downward deviation of electrical resistance from a linear dependence at temperatures below a certain characteristic value T*, which corresponds to the temperature of the opening of the pseudogap [20].

At the same time, the linear dependence of ρab(T) is characteristic of the elastic scattering of electrons on phonons at T >  > θ (θ—the Debye temperature). Then, the downward deviation of the electrical resistivity from the linear dependence is associated with the transition to inelastic scattering of electrons on phonons when the temperature decreases to T ≤ θ. This transition may coincide with the region of the pseudogap opening.

For the scattering of electrons on phonons, the dependence ρab(T) has the form:

$$\rho \left(T\right)= \frac{1}{\frac{1}{{\rho }_{0}+{\rho }_{ph}}+\Delta {\sigma }_{sc}}$$
(3)

Here, ρ0 is the residual resistivity caused by the scattering of charge carriers on defects and ρph is the contribution of phonons to the scattering of charge carriers:

$$\uprho_{ph}= {A}_{r}{\left(\frac{T}{\theta }\right)}^{r}{\int }_{0}^{\theta /T}\frac{{e}^{x}{x}^{r}dx}{{\left({e}^{x}-1\right)}^{2}}$$
(4)

Here, Ar is a parameter; r = 3 and/or 5 (the Bloch- Grüneisen ratio [21]), or r = 2 and/or 4 (spectrum of the phonon density of states for superconducting compounds β-W type, obtained in superconducting tunneling experiments [22, 23]). Δσcs is the excess conductivity due to the approach of the superconducting transition [24, 25]:

$$\Delta \sigma_{sc} = \left[ {\frac{{e^{2} }}{16\hbar d}} \right]\frac{1}{{\sqrt {\varepsilon_{0} \cdot \sinh \left( {\frac{2\varepsilon }{{\varepsilon_{0} }}} \right)\left[ {J + \varepsilon_{0} \cdot \sinh \left( {\frac{2\varepsilon }{{\varepsilon_{0} }}} \right)} \right]} }}$$
(5)

Here, e is the electron charge; ħ is Planck’s constant; d is the distance between the conductive layers; \(\varepsilon = \text{ln}\frac{T}{{T}_{c}}\); \({\varepsilon }_{0}= \text{ln}\frac{{T}_{0}}{{T}_{c}}\); T0 is the characteristic temperature that determines, together with the coherence length along the c-axis, ξc(0), the collapse of superconducting fluctuations [25]; and \(J=4\frac{{\upxi }_{c}^{2}\left(0\right)}{{d}^{2}}\) is the anisotropy parameter [24].

We approximated the experimental data by relations (3)–(5) with r = 2 [22, 23] in the entire investigated temperature interval Tc − 300 K. The corresponding curves are shown by solid lines 1–6 in Fig. 1. The parameters that varied were ρ0, θ, A2, d, J, and ε0. The values of these parameters, providing the adjusted coefficient of determination (Adj. R2) at the level of 0.997 and an average error of no worse than 0.3%, are shown in the Table 1.

Table 1 Approximation parameters ρab(T) of the YBa2Cu3O7–δ single crystal by relations (3)–(5) with r = 2 in the normal state and in the region of the superconducting transition at different electron fluences Φ
Full size table

Thus, the experimental curves ρab(T) can be approximated by relations (3)–(5) in the entire studied temperature range Tc–300 K within the framework of the model [21,22,23], which assumes a crossover from elastic to inelastic scattering (with decreasing temperature) charge carriers on phonons. Note that with decreasing temperature ρph(T) decreases monotonically, ρ0 = const., and Δσsc increases rapidly near Tc. Therefore, experimental curves ρab(T) can be approximated by relations (3)–(4), if only with decreasing temperature ρab(T) decreases or remains approximately constant (before Tc). This occurs at fluences Φ < 70⋅1018 cm–2 (curves 1–6, Fig. 1). For higher fluences, ρab(T) shows a minimum that cannot be described by relations (3)–(4). In this case, for an adequate approximation of ρab(T), it is necessary to take into account additional mechanisms of charge transfer or scattering.

At sufficiently high temperatures, θ/T <  < 1:

$${\rho }_{0}+{\rho }_{ph}={A}_{2}{\left(\frac{T}{\theta }\right)}^{2}{\int }_{0}^{\theta /T}\frac{{e}^{x}{x}^{2}dx}{{\left({e}^{x}-1\right)}^{2}})\approx {\rho }_{0}+{A}_{2}\frac{T}{\theta }$$
(6)

That is, among the parameters of the linear temperature dependence of the resistivity (see Fig. 2), the free term a corresponds to the residual resistance ρ0, and the slope b corresponds to the high-temperature derivative of the phonon resistivity, A2/θ.

The dependencies of charge carrier scattering parameters, ρ0, θ, and A2 from electron fluence are shown in Fig. 3.

Fig. 3
figure 3

Dependence of the approximation parameters ρab(T) by relations (3)–(4) on the electron fluence. 1—ρ0; 2—A2; 3—θ. Inset: connection between A2 and θ (see (6))

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The change of ρ0 depending on fluence is close to linear growth, which indicates the accumulation of defects due to irradiation. Certain violations of this regularity are associated, apparently, with the annealing of defects during measurements up to temperatures of ~ 300 K. The value of the “residual resistivity ratio,” rrr = (ρph + ρ0)/ρ0 ≈ ρ(300 K)/ρ0, which is a criterion for metal defects, decreases in our case from rrr ~ 30 (ρph >  > ρ0) to rrr ~ 1 (ρph <  < ρ0).

The values of the Debye temperature, θ, are quite low, which may be related to the significant anisotropy of YBa2Cu3O7−δ—the interaction between the layers is much less than the interaction within the layers, so θ1 is associated with transverse oscillations propagating along the c-axis, much smaller than θ2, which is associated with transverse oscillations propagating in the layers [26]. Longitudinal oscillations correspond to the Debye temperature θ3 ~ θ2 >  > θ1, because longitudinal oscillations are always associated with deformations in all three directions. Fitting the heat capacity of YBa2Cu3O7–δ at low temperatures gives values θ1 = 90 K, θ2 = 850 K, and θ3 = 295 K [26]. Thus, during irradiation, we observe a transition from scattering of charge carriers on oscillations characterized by the Debye temperature θ1, to scattering on oscillations characterized by the Debye temperature θ2.

Since Δθ/θ ≈ – αΔV/V + βΔf/f (ΔV is the change in the volume, Δf is the change in the force constants), then the non-monotonic behavior Δθ/θ = F(Φ) is due to the competition of both terms, but ultimately, the change in Debye temperature due to irradiation is caused by the isotropization of the phonon spectrum due to the increase in the defectivity of the sample.

Parameter A2, which characterizes the intensity of charge carriers’ scattering on phonons, also changes with increased defectiveness of the sample, which is consistent with the data [27] for transition metals and is probably associated with the deformation of the phonon spectrum of the sample (see, e.g., [28]). The inset to Fig. 3 shows the relationship between A2 and θ: A2 ≈ bθ in the considered interval of fluences. Since A2 and θ varied independently, the specified relationship testifies to the adequacy of the relations (3)–(4) regarding the experimental values of ρab(T).

The temperatures of the superconducting transitions, Tc, were determined by the positions of the low-temperature maxima of the derivatives, dρab/dT (see the inset to Fig. 1), that is, by the positions of the low-temperature inflections of the ρab(T) curves. These points are the only experimental points at which it is possible to determine the temperature of the superconducting transition, especially in conditions of its significant broadening.

The transition broadening means that there are areas in the sample with different Tc, the values of which lie in the interval Tc − ΔTc ≤ Tc ≤ Tc + ΔTc. In HTSC, the existence of these areas is associated with variations in the concentration of defects. Similar regions in ferroelectrics have size from 10–5 to 10–6 cm [29, 30]. At the same time, turning the resistivity to zero can be associated with the formation of a superconducting cluster, which extends over the entire sample, shunting both areas, with smaller Tc and the normal ones.

In Fig. 4 (curve 1) and from the Table 1, it is clear that the temperature dependence of the superconducting transition Tc, as a function of the electron fluence, Φ, is almost linear, which is consistent with [31]. This means that the defects responsible in our case for the reduction of Tc do not anneal at T ≤ 300 K. Such defects are, first of all, nonmagnetic interstitial atoms [31,32,33], displaced from their regular positions by the impacting electrons. The widths of dρab/dT maxima increase due to irradiation (curve 4), which indicates broadening of the superconducting transition due to an increase in the number of defects.

Fig. 4
figure 4

Dependence on electron fluence: 1—superconducting transition temperature, Tc; 2 –temperature of superconducting fluctuations collapse, T0; 3—crossover temperature, Tcross; 4—width of superconducting transitions, ΔTc

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The temperature of superconducting fluctuations collapse [25] is determined by the parameter ε0: T0 = Tc⋅exp(ε0). The temperature of the 2D-3D crossover, Tcross, is taken to be the temperature at which the transverse coherence length ξc(T) = ξc(0)⋅ε–1/2 becomes equal to the interplanar distance d, i.e., d = ξc(0) /\(\sqrt{\text{ln}({T}_{cross}/{T}_{c})}\). Then, ln(Tcross) = ln(Tc) + [ξc(0)/d]2, i.e., Tcross is slightly larger than Tc—we have it by about 6 K at all the investigated fluences.

Formula (5) describes the 2D mode when ε0sinh(2ε/ε0) >  > J, then Δσsc ∝ [ε0sinh(2ε/ε0)]−1; at ε0sinh(2ε/ε0) <  < J, then Δσs ∝ [Jε0sinh(2ε/ε0)]–1/2 − 3D-mode (\(J=4\frac{{\upxi }_{c}^{2}\left(0\right)}{{d}^{2}}\)). Notably, the errors in determining the temperatures T0 and Tcross ~ 2ΔTc /Tc, that is, the temperatures T0 and Tcross practically coincide (Fig. 4, curves 2 and 3). This is due, both, to the fact that Eq. (5) is empirical and that the linear dependence ρ(T) is an asymptote, i.e., there is no sharp end of this linear dependence and no beginning of deviation from it—at least, at high temperatures, the temperature derivatives of resistivity change smoothly, without showing any jumps. Thus, the fit is not very sensitive to small changes in T* [25] and/or T0.

From the Table 1, it can be seen that with increasing the fluence, the values of the J parameter demonstrate a tendency to increase. The distance between the conducting layers, d, is in the range of 3–5 Å. For an unirradiated sample, this is approximately a quarter of the size of the unit cell along the c-axis and coincides with the data of [34]. The authors [29, 34, 35] draw attention to the fact that in YBa2Cu3O7, there are two CuO2 layers which are responsible for the superconductivity and the distances between these layers are different: d1 = 8.4 Å and d2 = 3.3 Å (d1 + d2 = 11.7 Å), but the relation 2ξc(0)/d = J1/2 does not depend on the choice of d. These relations are given in the last row of the Table 1. They are close to the data [34]. A particular role in this can be played by structural and kinematic anisotropy in the system [36,37,38,39].

For the values of d given in the Table 1, ξc(0) varies from 0.94 Å for the unirradiated sample to 2.2 Å for Φ = 68.1 × 1018 cm–2; for d = 11.7 Å (unit cell size along the c-axis), ξc(0) changes from 3.6 to 8.8 Å under the same conditions.

4 Conclusions

Summarizing the obtained results, we can conclude that electron irradiation leads to the appearance of a significant number of defects, which cause a significant decrease in anisotropy, a significant increase in scattering by phonons; reduction of Tc and broadening of the superconducting transition.

In the temperature interval Tc − 300 K and electron fluences Φ ≤ 70⋅1018 cm–2, the temperature dependence of the electrical resistivity in the base plane can be approximated with high accuracy, taking into account the scattering of charge carriers on defects and phonons and the fluctuating conductivity in the Lawrence-Doniach (LD) model.

The approximation parameters that provide the minimum error indicate the following.

In the normal state, with an increase in the irradiation fluence, an increase in the Debye temperature is observed due to the isotropization of the phonon spectrum caused by the accumulation of defects; the residual resistance increases as the fluence increases.

When approaching the superconducting transition, the temperature of the superconducting transition decreases with increasing fluence almost linearly; the width of the transition demonstrates (shows) a tendency to increase; the characteristic temperatures of the fluctuating conductivity model—the temperature of superconducting fluctuations collapse and the 2D-3D crossover temperature—are very close to each other; the distance between the conductive layers lies in the interval 3–5 Å; the corresponding transverse coherence length changes from 0.94 Å to 2.2 Å when the fluence is increasing.