AC magnetic response of superconducting YBa₂Cu₃O₇/PrBa₂Cu₃O₇ superlattices

Vortex pinning potential of superconducting materials is widely investigated by measuring the DC magnetization relaxation over a relatively large time interval¹ (of the order of 10³ s), as well as by analyzing the AC magnetic response.² The DC relaxation results are well described by a vortex diffusion process, where, at a constant DC magnetic field H, the induced current density J and temperature T dependence of the vortex-creep activation energy U is given by the general vortex-creep relation,³ U(J,  T) = Tln(t/t₀), for the relaxation time t larger than the time scale for creep t₀. The latter is a complex, macroscopic quantity, depending of the sample size, as well.⁴ 

Alternatively, the AC magnetic response² offers a simple method for the extraction of the activation energy U_AC at short time scales based on the Arrhenius relation. AC magnetic measurements performed at usual frequencies f and amplitudes h_AC revealed a logarithmic U_AC(J) dependence and unexpectedly high U_AC values around the DC irreversibility line (IL),^5–12 where the critical-state related AC response appears. While it is often assumed that the flux creep process at long t extends at short time scales, the continuity of DC and AC magnetic data achieved just below IL points towards a change of magnetic relaxation.¹² 

In this work, the nature of the large U_AC values is addresses by investigating the AC magnetic response of [Y Ba₂Cu₃O₇]_n/[PrBa₂Cu₃O₇]₄ superlattices with n = 11 or 4 unit cells (Y11Pr4 and Y4Pr4, respectively) in perpendicular external magnetic fields. The presence of four-unit-cell thick nonsuperconducting layers leads to almost decoupled superconducting blocks.¹³ Such samples offer a precise measure of the length of the hopping vortex segment, since the vortex correlation along the c axis is limited by the small superconducting layer thickness s.¹⁴ Moreover, at least at low H, vortices in the superconducting layers are disentangled, whereas the “superentanglement” at the nonsuperconsucting layer level¹⁵ (if any) would bring in this case an insignificant contribution to the activation energy. It is argued that the repeatedly observed linear Arrhenius plots^5–12 reflect a thermally activated, non-diffusive vortex motion process at short t = 1/f scales, where the thermal smearing of the pinning potential below IL (or the smoothening of the vortex structure above IL)⁴ is not completed. In these conditions, the high U_AC values result from the large contribution of the pinning enhanced viscous drag and the dynamic critical current density as the relevant (relaxation free) critical current density for the AC magnetic response.

The superlattices were obtained by high pressure sequential sputter deposition on (100) oriented SrTiO₃, as described in Ref. 14. The overall thickness d_o = 200 nm, and the measured specimens were disk-shaped, with the radius R = 2 mm. The AC magnetic signal was registered with a Magnetic Property Measurement System (MPMS), in increasing temperature, after the sample was cooled down from the normal state to T = 70 K in H = 5 kOe or 500 Oe oriented perpendicular to the film surface, as usually does. The AC field was of the same orientation, with h_AC = 3 Oe (constant in this work) and f ranging in the interval 1 Hz – 500 Hz. For a comparative analysis, the DC magnetic moment m with the perpendicular H applied in zero-field cooling conditions has been measured with the same MPMS. Well below IL, the irreversible (relaxing) DC magnetic moment can be identified with m.

The critical temperature T_c (taken at the onset of the diamagnetic signal in H = 10 Oe) is 87 K for Y11Pr4 and 78.1 K in the case of Y4Pr4. The demagnetization factor D was accurately determined from the initial slope of the DC m(H) curves obtained by increasing H with a small step. In the case of Y11Pr4, for example, 1 – D = 1.02 × 10⁻⁴.

As known, the in-phase component m′ of the AC magnetic moment is proportional to the screening current density J, whereas the out-of-phase component m″ is a measure of dissipation.¹⁶ The main panel of Fig. 1 illustrates the temperature variation of m′ and m″ at different f for Y11Pr4 in H = 5 kOe. The peak in m″(T) at the peak temperature T_p is interpreted in terms of the critical state model, where the maximum dissipation corresponds to the first full penetration of the critical state, and by the influence of relaxation. At a given h_AC, T_p decreases by lowering f, where the time 1/f is longer. An important aspect is that at T_p(f) the screening current density J_p (affected by relaxation) is constant, as shown by the horizontal line segment from the main panel of Fig. 1. In the case of specimens with D close to unity, J_p depends on h_AC and the sample thickness only, and is precisely given by¹⁷ 

where here d = d_on/(n + 4) is the effective superlattice thickness. In the case of Y11Pr4, J_p = 1.68 × 10⁵ A/cm² (h_AC = 3 Oe), and this value will be used below for the rapid determination of J from |m′|, since in the conditions of full penetration the relation J ∝ |m′|^2/3 holds.¹⁷ The extraction of J with Eq. (32) from Ref. 17 gives similar results.

As shown in the upper inset of Fig. 1, ln(f) vs. 1/T_p is accurately linear, supporting a thermally activated vortex hopping process. This has led to the determination of the effective vortex activation energy U_AC at short time scales with the Arrhenius law, which at constant H and h_AC is

where f₀ is the characteristic attempt frequency (∼10⁹–10¹² Hz),⁴ and J_p is given by Eq. (1). It was reported by many authors^5–12 that the fit with Eq. (2) (see the inset of Fig. 1) is in agreement with f₀ in the above range if a linear decrease of U_AC with increasing T is considered, U_AC(J, T) = U₀(J)(1 − T/T_c), where U₀ is the apparent U_AC at T = 0 and J = J_p. The fit in the Arrhenius plot leads for H = 5 kOe to U₀ ∼ 10⁴ K (Y11Pr4, with f₀ ∼ 1 GHz), and to U₀ ∼ 2.4 × 10³ K in the case of Y4Pr4 (where f₀ slightly decreases), which are similar to those extracted from DC transport measurements.^13,U_AC(T) is obtained from U₀ and the linear U_AC(T) dependence. It results U_AC(T = 73.5 K) = 1550 K (Y11Pr4, H = 5 kOe), and U_AC(T = 55.5 K) = 706 K (Y4Pr4, H = 5 kOe). These values are surprisingly high, since Y11Pr4 is just below IL, whereas Y4Pr4 is above IL (with the vortex system in the liquid state), as revealed by the DC magnetic hysteresis curves from the lower inset of Fig. 1. Moreover, determined with the Bean model¹⁸ from the irreversible moment for Y11Pr4 at H = 5 kOe, J (DC) is roughly one order of magnitude lower that J_p, and U is a non-diverging function at low J.¹⁹ 

Thus, a necessary condition for linear Arrhenius plots appears to be a constant J, and one can take other horizontal lines [m′(f) = constant] in the full penetration range of Y11Pr4 in H = 5 kOe to construct Arrhenius plots at J < J_p with the correspondent T and f values. As shown in the main panel of Fig. 2, the plots are linear, while f₀ remains very close to 1 GHz, whereas the U₀(J) dependence is close to logarithmic²⁰ (see the inset of Fig. 2), as obtained using AC measurements at various h_AC.¹¹ In this context, one can recall the DC relaxation, because the general vortex-creep relation allows, in principle, to obtain Arrhenius plots ln(1/t) vs. 1/T at long t, with the same condition (constant J). We measured the DC m(t) curves at long t for Y4Pr4 in H = 5 kOe (with enhanced two-dimensional vortex fluctuations) for a relatively narrow temperature interval (similar to the T_p variation for the investigated f range), where t₀ ∼ constant. The DC relaxation curves |m| vs. t in the log-log plot (Fig. 3, main panel) exhibit a strong downward curvature, as a signature of plastic (dislocated mediated) flux creep.¹⁹ More importantly, the Arrhenius plot at long t constructed with the (T, t) pairs for |m| = 2 × 10⁻³ emu, for example, is nonlinear, as shown in the inset of Fig. 3. If a linear fit is still performed, the obtained U₀ is much lower than that extracted with the Arrhenius plot from the upper inset of Fig. 1, while t₀ is in the range of a macroscopic time scale for creep (orders of magnitude higher than the microscopic attempt time 1/f₀). The downward curvature in the Arrhenius plot at long t reflects the smearing of the pinning potential by thermally induced vortex fluctuations,⁴ which is more effective at high temperatures, reducing drastically the critical current density. The conclusion here is that the linear Arrhenius plots at short t = 1/f scales indicate a negligible effect of vortex fluctuations on pinning, which will be discussed later.

To eliminate the influence of vortex entanglement as a possible source for large plastic pinning barriers,²¹ H was lowered to 500 Oe (Fig. 4), where T_p(f = 500 Hz) = 83.3 K and decreases at f = 11 Hz by ∼1.1 K only, meaning a larger U_AC. The plots from the inset give for Y11Pr4 U₀(J_p) ∼ 2.35 × 10⁴ K, and a lower value (∼9.1 × 10³ K) for Y4Pr4. For quantitative estimations, we measured m′(f) at T = 83.3 K for Y11Pr4 in H = 500 Oe. As can be seen in Fig. 5, ln(J) vs. ln(t = 1/f) is linear, in agreement with a logarithmic U_AC(J), which can be written as U_AC(J) = U_cln(J_c0/J), where J_c0 is the true (relaxation free) critical current density at short t = 1/f scales, and U_c is a characteristic pinning energy scale. In terms of t = 1/f, J(t) ∝ t^−0.125, whereas −TΔln(t)/Δln(J) = U_c = 666 K. Having J_p, precisely given by Eq. (1), U_AC(J_p,  T = 83.3 K) ∼ 10³ K from U₀(J_p), and U_c, the above U_AC(J) relation leads to J_c0 ∼ 7.5 × 10⁵ A/cm². Since the linear ln(J) vs. ln(t) plots have been observed over a wide f range,⁶ the extrapolation in the main panel of Fig. 5 in the high f (short t) region indicates that the induced J would attain J_c0 at f of the order of 10⁹ Hz ∼ f₀. Indeed, according to Eq. (2), U_AC should vanish at f = f₀. On the other hand, it is well known that this is the frequency domain where the dynamic critical current density J_d is reached in Y Ba₂Cu₃O₇ films,²² and above J_d the flux motion is non-thermally activated. Thus, U_AC(J) = U_cln(J_d/J).

Using the same m′(t = 1/f) data at T = 83.3 K, one can also extract the electric field at the sample edge, E ∼ (R/2)(1 − D) dM′/dt), where M′ = m′/V is the volume magnetization, although this offers only a limited view of a complex picture.²³ The determined E (Y11Pr4, H = 500 Oe, T = 83.3 K) is plotted vs. t = 1/f in Fig. 5 (double logarithmic scales). A linear fit in the representation from Fig. 5 supplies E(t = 1/f) ∝ t^−1.13, J(t = 1/f) ∝ t^−0.125, and the linear resistivity ρ(t) = E/J ∝ 1/t. This behavior was confirmed by increasing temperature and has been observed for Y4Pr4, as well, meaning that the critical state is self-organized.^4,24 Since for an inductive process one has, roughly, dJ/dtρJ ∝ J/t,²⁵ the linear ln(J) vs. ln(t) plot [leading to a logarithmic U_AC(J)] becomes obvious, and the analysis is self-consistent.

An important point is that with the vortex velocity v = E/μ₀H at the sample edge and the vortex hopping time t_h ∼ 1/(2f) (vortex moving in and out during an AC cycle), in the investigated f range the hopping length l_h is of the order of 10² nm. This is expected to be larger than the mean distance between the pinning centers in our superlattices, with a high density of point-like quenched disorder. By considering a non-diffusive vortex hopping at short t = 1/f scales, as predicted in Ref. 23, one can understand the occurrence of high U_AC values in the vicinity of IL through a relatively large l_h and a short pinning time, by difference with the flux creep process at long t (the DC relaxation). It is known that the non-thermally activated flow of pinning free vortices (with the Bardeen-Stephen result for the viscous drag coefficient due to the dissipation in the vortex core²⁶ is present only for J ≥ J_d, where any influence of pinning becomes negligible. In the usually performed AC measurements J < J_d, and a pinning enhanced viscous drag coefficient²⁷ has to be taken into account. The pinning enhanced viscosity results from rapid, successive pinning and depinning events²⁸ during the main vortex hoping over l_h, and, owing to the short time involved in such an event, the thermal smearing of the pinning potential or smoothening of the vortex structure is weak. Thus, the activation energy U_AC should balance the pinning enhanced viscous drag w over the hopping distance l_h. At low H (nearly isolated vortices), with a linear U_AC(J) approximation for a limited f range, this is w ∼ Φ₀l_hs(J_d − J).¹² With J_d ∼ 7.5 × 10⁵ A/cm², s ∼ 12 nm, an averaged J from Fig. 5, and the above l_h value, one obtains w ∼ 1.4 × 10⁻²⁰ J, corresponding to an activation energy ∼1.1 × 10³ K, of the same order of magnitude with U_AC(J_p) for Y11Pr4 (H = 500 Oe, T = 83.3 K). In the case of Y4Pr4 in H = 500 Oe, U_AC and l_h determined at T = 69.3 K = T_p(f = 500 Hz) are close to those for Y11Pr4, the increase of J_d in Y4Pr4 being compensated by the s decrease.

In summary, by difference with the DC relaxation at long t scales, where the influence of thermally induced vortex fluctuations on pinning is strong, the common linear Arrhenius plots at short t = 1/f scales in AC magnetic measurements reflect a thermally activated, non-diffusive vortex motion process, where the thermal smearing of the pinning potential or the smoothening of the vortex structure is weak.

In these conditions, the high U_AC values in the AC magnetic response result from the large contribution of the pinning enhanced viscous drag over the vortex hopping length larger that the mean spacing between the pinning centers and the dynamic critical current density as the relevant (relaxation free) critical current density. The logarithmic variation of U_AC with the screening current density, reported for various specimens, appears to be generated by a self-organized critical state.

Work supported by the Romanian Ministry of Education, Executive Unit for Funding High Education, Research, Development and Innovation, under the Core Programs PN16-480102 and 4N/2016, Grants PNII PCCA No. 138/2012, and Grant PNII PCCA No. 7/2012. The authors thank G. Jakob and A. El Tahan for sample preparation. The kind assistance of the Alexander von Humboldt Foundation is gratefully acknowledged.