Vortex activation energy *U*_{AC} in the critical-state related AC magnetic response of superconductors (appearing in the vicinity of the DC irreversibility line) takes large values, as often reported, which is not yet understood. This behavior is essentially different from that of the vortex-creep activation energy at long relaxation time scales, and may become important for AC applications of superconductors. To elucidate this aspect, we investigated the AC signal of almost decoupled [Y Ba_{2}Cu_{3}O_{7}]_{n}/[PrBa_{2}Cu_{3}O_{7}]_{4} superlattices (with n = 11 or 4 units cells) in perpendicular DC and AC magnetic fields. In these model samples, the length of the hopping vortex segment is fixed by the thickness of superconducting layers and vortices are disentangled, at least at low DC fields. It is shown that the high *U*_{AC} values result from the large contribution of the pinning enhanced viscous drag in the conditions of thermally activated, non-diffusive vortex motion at short time scales, where the influence of thermally induced vortex fluctuations on pinning is weak.

## I. INTRODUCTION

Vortex pinning potential of superconducting materials is widely investigated by measuring the DC magnetization relaxation over a relatively large time interval^{1} (of the order of 10^{3} 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*) = *T*ln(*t*/*t*_{0}), for the relaxation time *t* larger than the time scale for creep *t*_{0}. The latter is a complex, macroscopic quantity, depending of the sample size, as well.^{4}

Alternatively, the AC magnetic response^{2} 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 Ba_{2}Cu_{3}O_{7}]_{n}/[PrBa_{2}Cu_{3}O_{7}]_{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 level^{15} (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.

## II. EXPERIMENTAL

The superlattices were obtained by high pressure sequential sputter deposition on (100) oriented SrTiO_{3}, 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}.

## III. RESULTS AND DISCUSSION

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*decreases by lowering

_{p}*f*, where the time 1/

*f*is longer. An important aspect is that at

*T*(

_{p}*f*) the screening current density

*J*(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

_{p}*D*close to unity,

*J*depends on

_{p}*h*

_{AC}and the sample thickness only, and is precisely given by

^{17}

where here *d* = *d*_{o}n/(n + 4) is the effective superlattice thickness. In the case of Y11Pr4, *J _{p}* = 1.68 × 10

^{5}A/cm

^{2}(

*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 *f*_{0} is the characteristic attempt frequency (∼10^{9}–10^{12} 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

*f*

_{0}in the above range if a linear decrease of

*U*

_{AC}with increasing

*T*is considered,

*U*

_{AC}(

*J*,

*T*) =

*U*

_{0}(

*J*)(1 −

*T*/

*T*), where

_{c}*U*

_{0}is the apparent

*U*

_{AC}at

*T*= 0 and

*J*=

*J*. The fit in the Arrhenius plot leads for

_{p}*H*= 5 kOe to

*U*

_{0}∼ 10

^{4}K (Y11Pr4, with

*f*

_{0}∼ 1 GHz), and to

*U*

_{0}∼ 2.4 × 10

^{3}K in the case of Y4Pr4 (where

*f*

_{0}slightly decreases), which are similar to those extracted from DC transport measurements.

^{13}

^{,}

*U*

_{AC}(

*T*) is obtained from

*U*

_{0}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 model

^{18}from the irreversible moment for Y11Pr4 at

*H*= 5 kOe,

*J*(DC) is roughly one order of magnitude lower that

*J*, and

_{p}*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

*f*

_{0}remains very close to 1 GHz, whereas the

*U*

_{0}(

*J*) dependence is close to logarithmic

^{20}(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*variation for the investigated

_{p}*f*range), where

*t*

_{0}∼ 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

*U*

_{0}is much lower than that extracted with the Arrhenius plot from the upper inset of Fig. 1, while

*t*

_{0}is in the range of a macroscopic time scale for creep (orders of magnitude higher than the microscopic attempt time 1/

*f*

_{0}). 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

*U*

_{0}(

*J*) ∼ 2.35 × 10

_{p}^{4}K, and a lower value (∼9.1 × 10

^{3}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*ln(

_{c}*J*

_{c0}/

*J*), where

*J*

_{c0}is the true (relaxation free) critical current density at short

*t*= 1/

*f*scales, and

*U*is a characteristic pinning energy scale. In terms of

_{c}*t*= 1/

*f*,

*J*(

*t*) ∝

*t*

^{−0.125}, whereas −

*T*Δln(

*t*)/Δln(

*J*) =

*U*= 666 K. Having

_{c}*J*, precisely given by Eq. (1),

_{p}*U*

_{AC}(

*J*,

_{p}*T*= 83.3 K) ∼ 10

^{3}K from

*U*

_{0}(

*J*), and

_{p}*U*, the above

_{c}*U*

_{AC}(

*J*) relation leads to

*J*

_{c0}∼ 7.5 × 10

^{5}A/cm

^{2}. 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 10

^{9}Hz ∼

*f*

_{0}. Indeed, according to Eq. (2),

*U*

_{AC}should vanish at

*f*=

*f*

_{0}. On the other hand, it is well known that this is the frequency domain where the dynamic critical current density

*J*is reached in Y Ba

_{d}_{2}Cu

_{3}O

_{7}films,

^{22}and above

*J*the flux motion is non-thermally activated. Thus,

_{d}*U*

_{AC}(

*J*) =

*U*ln(

_{c}*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*/*μ*_{0}*H* at the sample edge and the vortex hopping time *t _{h}* ∼ 1/(2

*f*) (vortex moving in and out during an AC cycle), in the investigated

*f*range the hopping length

*l*is of the order of 10

_{h}^{2}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*and a short pinning time, by difference with the flux creep process at long

_{h}*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 core

^{26}is present only for

*J*≥

*J*, where any influence of pinning becomes negligible. In the usually performed AC measurements

_{d}*J*<

*J*, and a pinning enhanced viscous drag coefficient

_{d}^{27}has to be taken into account. The pinning enhanced viscosity results from rapid, successive pinning and depinning events

^{28}during the main vortex hoping over

*l*, 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

_{h}*U*

_{AC}should balance the pinning enhanced viscous drag

*w*over the hopping distance

*l*. At low

_{h}*H*(nearly isolated vortices), with a linear

*U*

_{AC}(

*J*) approximation for a limited

*f*range, this is

*w*∼ Φ

_{0}

*l*(

_{h}s*J*−

_{d}*J*).

^{12}With

*J*∼ 7.5 × 10

_{d}^{5}A/cm

^{2},

*s*∼ 12 nm, an averaged

*J*from Fig. 5, and the above

*l*value, one obtains

_{h}*w*∼ 1.4 × 10

^{−20}J, corresponding to an activation energy ∼1.1 × 10

^{3}K, of the same order of magnitude with

*U*

_{AC}(

*J*) for Y11Pr4 (

_{p}*H*= 500 Oe,

*T*= 83.3 K). In the case of Y4Pr4 in

*H*= 500 Oe,

*U*

_{AC}and

*l*determined at

_{h}*T*= 69.3 K =

*T*

_{p}(

*f*= 500 Hz) are close to those for Y11Pr4, the increase of

*J*in Y4Pr4 being compensated by the

_{d}*s*decrease.

## IV. CONCLUSIONS

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.

## ACKNOWLEDGMENTS

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.