Flux Dynamics and Time Dependent Effects in Superconducting MgB₂

Abstract

We have investigated the time dependent flux motion in a polycrystalline superconducting MgB₂ sample by means of current-voltage (I–V curves) and transport relaxation (V–t curves) measurements. The transport measurements were carried out as functions of temperature (T), transport current (I), and applied magnetic field (H). The time effects in the I–V curves were investigated with the help of the sweep rate of transport current (dI/dt). The I–V curves exhibit nearly reversible behavior for both slow and high current sweep rates (dI/dt) upon cycling the transport current. It was observed that the evolution of the I–V curves is nearly independent of dI/dt. The small instabilities, voltage jump, and drops appearing at low dissipation levels were interpreted as a kind of plastic flow of the vortices evolving in the form of stripe or cluster in different sizes and local fluctuations in the superconducting order parameter. It was shown that the experimental I–V curves are in good agreement with a power law behavior, V(I)∼I ⁿ. The pinning potential U ₀ extracted from the fitting procedure is found to be approximately independent of dI/dt. In addition to the standard procedure, a reverse procedure was employed to study how the transport current (or associated vortices) penetrates from the surface into the sample. It was found that there is no marked difference between the I–V curves obtained in the standard and reverse procedures. These results suggest that, in polycrystalline MgB₂, the weak links between grains (standard procedure) and surface weak links (reverse procedure) do not have a considerable effect on the evolution of the I–V curves and also on the other transport measurements. In order to understand better the flux dynamics, the experimental results for polycrystalline MgB₂ sample were compared to previous similar studies on superconducting Y₁Ba₂Cu₃O_7−x (YBCO) and Bi_1.7Pb_0.3Sr₂Ca₂Cu₃O _x (BSCCO) polycrystalline samples.

1 Introduction

Detailed investigations on type-II superconductors reveal that the vortex dynamics are governed by two distinct mechanisms. One of them is the repulsive interaction between vortices and the other is the interplay between vortices and pinning centers [1–4]. It is well known that the type and distribution of the pinning centers inside the superconducting sample influence strongly the motion of vortices and the measured dissipation induced by them. Therefore, the pinning and depinning of vortices are considered as the main mechanisms of the dissipation, provided that the thermal fluctuations are negligible compared to pinning [3–11]. Pinning which arises from defects and inhomogeneities inside the sample promotes disorder by decreasing the effective interaction between vortices; however, depinning favors spatially the ordering of vortices [2, 12, 13]. The effects of pinning and depinning can be observed in details by magnetic and transport measurements via the response of the moving entity to the applied current and magnetic field [2, 3, 14–17]. Several unusual and interesting phenomena, such as low frequency noise [18, 19], slow voltage oscillations [20], history dependent dynamic response [1], and memory effect [2, 21] were observed in type-II superconductors.

Many experimental [22–27] and numerical [28–30] studies show that the sweep rate of the applied current (dI/dt) is a very important parameter in the I–V measurements of type-II superconductors. Ding et al. [24] found that the I–V curves were affected apparently by dI/dt, i.e. by decreasing the critical current with increasing dI/dt. Zhang et al. [28] reported that the numerical solution of the nonlinear diffusion equation in one dimension gives current sweep rate dependence of electric field (E) associated with transport current density J. Numerical results confirmed that the voltage dissipation appears at smaller current values with increasing dI/dt in the I–V curves. Xiao et al. [22] found that the depinning process governs the formation of contiguous channels of mobile vortices connecting the sample edges. They suggested that the formation time of such channels gives rise to delays observed in the voltage response to the variation of dI/dt [22]. In the light of these studies, it can be suggested that the evolution of the I–V curves depends on the sweeping rate of transport current.

Hysteresis effects in the I–V curves of type-II superconductors have been reported in many studies [1, 15, 31–42] in which the driving current was swept up and down. Two kinds of hysteresis loops in I–V curves have been observed, i.e., clockwise (cw) [34, 37, 38] and counter-clockwise (ccw) [1, 15, 31–42]. Different interpretations have been made to explain these behaviors. For instance, Samoilov et al. [37] observed that ccw hysteresis effects develop in the I–V curves of superconducting amorphous Mo₃Si film at high magnetic fields, whereas cw hysteresis effects appear at low magnetic fields. They explained the ccw hysteresis loops in the I–V curves on the basis of the Larkin–Ovchinnikov (LO) theory [43], while the cw hysteresis loops were attributed to enhancement in the inelastic electron scattering rate as the superconductivity is suppressed by the presence of large electric field.

Fendrich et al. [33] reported sudden jumps and time dependent ccw hysteresis loops in the I–V curves of untwined single-crystal sample of Y₁Ba₂Cu₃O_7−x , which were interpreted as a relaxation-pinning effect. Danckwerts et al. [31] observed large ccw hysteresis effects in the I–V curves of low-T _c superconducting Pb/In film, which diminish with increasing magnetic field. These authors [31] interpreted the hysteresis effects in the I–V curves with a model corresponding to the vortex flow in the form of filaments and channels. History-dependent irreversibility on the I–V curves of a single crystal sample of Bi₂Sr₂CaCu₂O₈ (Bi-2212 phase) were reported by Sas et al. [40]. Similar irreversibility effects were observed in superconducting MgB₂ single crystal by Lee et al. [42] and Welp et al. [44]. Numerical studies [38, 45] confirm the presence of hysteresis effects in the I–V curves in type-II superconductors under suitable conditions.

The time effects on the evolution of vortex dynamics is another interesting subject in type-II superconductors. The magnetization measurements (M–H) in type-II superconductors exhibit many interesting and fascinating features. One of them is the observation of the fast relaxation of magnetization (M) with time (t) in high temperature superconductors (HTSCs) [46–49]. There are several detailed studies on M–t measurements in type-II superconductors [46, 47, 50–53]. In recent years, several studies on transport relaxation measurements (V–t curves) in HTSCs were reported [23, 29, 54–63]. The transport measurements show that the dynamic effects in polycrystalline superconducting bulk Y₁Ba₂Cu₃Cu_7−x (YBCO) samples can be monitored experimentally by the time evolution of the sample voltage [4, 23, 32, 64]. Unusual time effects and relaxation of the sample voltage observed in transport measurements were discussed in terms of the superconducting glass model [59], modified flux flow, and resistive weak link models [61], and two-kind of flux creep models [54, 55].

The time effects in transport measurements were observed experimentally in current-voltage measurements (I–V curves) of single-crystalline [18, 33] and polycrystalline HTSCs samples [34, 65, 66] and in low temperature superconductors (LTSCs) [15, 22]. For instance, it was found that there is a significant current sweep-rate (dI/dt) dependence in the I–V curves of single-crystalline 2H-NbSe₂ sample [15, 22]. In recent studies, significant hysteretic and time effects were observed in the I–V curves of polycrystalline bulk HTSCs as a function of dI/dt for both low and moderate dissipation levels [35, 36, 59]. It was found that the cycling of the transport current up and down leads to the formation of hysteresis loops in the I–V curves. Thus, increasing/decreasing of the width of hysteresis loops and also the magnitude of measured dissipation in the I–V curves with dI/dt could be considered as evidence of the time effects. The numerical studies support that the current sweep rate can control the degree of order of the moving vortices and cause significant time effects in transport measurements [29, 57, 67, 68].

It has been shown both experimentally and theoretically that the flux dynamics in pure single-crystal sample of MgB₂ is different from that in type-I and type-II superconductors [69, 70]. Babaev and Speight [71] predicted theoretically a semi-Meissner state in two band systems that produce superconductivity. The two bands compete with each other, and one of them becomes responsible for the superconductivity at high temperatures [72, 73]. Now, it is well established that MgB₂ has a two-band structure composed by π and σ bands, which are responsible for the superconductivity. Moshcalkov et al. [69] showed experimentally that the vortex structure in pure single-crystal sample of MgB₂ evolves in the form of highly inhomogeneous vortex clusters. The size of clusters increases with increasing the applied magnetic field so that they merge with other small clusters by forming larger ones. In such a vortex response which is now called as type-1.5 superconductivity, the effective interaction between the vortices is a combination of short-range repulsion and long-range attraction [69]. In a type-I superconductor (π-bands of MgB₂), neighboring vortices attract each other and tend to merge by causing nonsuperconducting regions, while, in a type-II superconductor (σ-bands of MgB₂), the vortices repel each other and distribute themselves by trying to form a regular lattice structure inside the sample [69].

The superconducting magnesium diboride (MgB₂) has some advantages when compared to HTSCs due to its structural and superconducting properties, although its critical temperature is quite low. For instance, unlike HTSCs, MgB₂ has a lower anisotropy and transparency of the grain boundaries to current flow. Such features of this superconducting material make it attractive to study. The aim of this study is to investigate the time dependent flux dynamics in a polycrystalline superconducting MgB₂ sample by means of current-voltage (I–V curves) and transport relaxation (V–t curves) measurements. The transport and transport relaxation measurements were carried out as functions of temperature (T), transport current (I), and applied magnetic field (H). In addition to the V–t curves, the time effects in the I–V curves were investigated with the help of the current sweep rate of transport current (dI/dt). The experimental results for polycrystalline MgB₂ sample were compared to similar studies on superconducting Y₁Ba₂Cu₃O_7−x (YBCO) and Bi_1.7Pb_0.3Sr₂Ca₂Cu₃O _x (BSCCO) polycrystalline samples.

2 Experimental

Polycrystalline Mg/MgB₂ bulk sample was prepared by the conventional solid state reaction method using high purity powders of MgB₂ (99.9%) and Mg (99.99%). All powders which had 325-mesh particle size were purchased from Alfa–Aesar. In order to minimize the oxidation, Mg was added to the superconducting matrix. The weight ratio of Mg to MgB₂ was taken as 20 wt% of MgB₂. The powders of constituents were weighted in appropriate ratio by using an electronic balance (Sartorius CP2245), and they were mixed and grinded in an agate mortar for 2 hours under argon (Ar) atmosphere. The Mg/MgB₂ powder was pressed into pellet of 15.6 mm diameter and 0.9 mm thickness with a pressure of 5 tons/cm² (Lightpath Optical (UK) LTD) in a stainless steel die. The pellet within the metallic die was sealed and introduced into a furnace (Nabertherm L5/S27) and then heat treatment was performed at 500 °C. The pellets was kept at 500 °C for 2 hours in air, then the furnace was switched off and allowed to cool down to the room temperature.

X-Ray Diffraction (XRD) measurements of bulk MgB₂ sample was carried out to identify the phases forming inside the material and to analyze the microstructure. An X-ray diffractometer (Rigaku D/Max-IIIC) was used to determine the phase type, phase composition and the lattice parameters a and c of the MgB₂ sample used in this study. The target material and filter were selected to be CuK _α (λ=1.5418 Å) during the measurements. The measurements were taken at room temperature using 35 kV tube voltage and 0.1 mA tube current in the range of 2θ=20°–80° with a scan speed of 3°/min.

The XRD measurements of the commercial MgB₂ and Mg powders, which were used in preparing the MgB₂ sample, were also made to compare the prepared bulk MgB₂ sample. The corresponding XRD patterns are given in Figs. 1(a)–1(c). In Fig. 1(a), the main phase marked by the symbol * is MgB₂. Some Mg peaks marked by # also appear in the XRD pattern of bulk MgB₂ sample. The presence of Mg peaks in XRD pattern of bulk MgB₂ sample can be attributed to low annealing temperature (i.e., 500 °C). It is well known that Mg begins to melt at 650 °C. This implies that there is no major interaction between MgB₂ and Mg during the annealing process at 500 °C. On the other hand, in Fig. 1(a), we also observe a small amount of MgO impurity phase marked on XRD pattern by “o.” The presence of MgO can be expected since the annealing was carried out in air. The recent studies show that oxidation of MgB₂ begins at about 400 °C and becomes much stronger as temperatures higher than 700 °C. We note that the MgO impurity phase is also seen in the XRD pattern of commercial MgB₂ powder (Fig. 1(b)). Therefore, some part of the MgO phase in our MgB₂ sample may be originated from starting MgB₂ powder. In the XRD patterns of the MgB₂ sample, no secondary phase formations such as MgB₄ were found.

Fig. 1

XRD patterns of (a) the MgB₂ sample, (b) commercial MgB₂ powder, and (c) Mg powder

The crystal symmetry of the MgB₂ sample was found to be hexagonal and the unit cell parameters of the sample were determined as a=3.0895 Å and c=3.5307 Å from the analysis of XRD patterns of sample. The results obtained for the unit cell parameters of our sample are close to the typical values of MgB₂: a=3.0834 Å and c=3.5213 Å, (Joint Committee on Powder Diffraction Standards (JCPDS) file, card number 74-0982).

The average grain size of the MgB₂ sample was also estimated from XRD pattern using the Debye–Scherrer formula $L_{hkl} =\tfrac{0.9\lambda}{B_{\mathit{hkl}}\cos \theta _{\mathit{hkl}}}$, where L _hkl is the grain size for (hkl) plane, λ is the X-ray wavelength of radiation for CuKα, B _hkl is the full-width at half maximum (FWHM) at (hkl) peak in radian, and θ _hkl is the diffraction angle for (hkl) plane. By using the Debye-Scherrer formula, the calculated average grain size L ₍₀₀₂₎ is found as ∼9.2 nm for the MgB₂ sample.

The surface morphology, grain size estimation of the MgB₂ sample was investigated by the JEOL 6390-LV scanning electron microscope (SEM). The instrument is also equipped with Energy Dispersive Spectroscopy (EDS). The EDS provides qualitative and semiqualitative identifications of the chemical elements existing in the sample. SEM and EDS measurements were carried out at room temperature under an accelerating voltage of 20 kV for ×6000 and ×10,000 magnifications. Figure 2 shows SEM images of the MgB₂ sample. The pores within the sample are observed partially in both SEM images taken in 6000 and 10,000 magnification of the bulk MgB₂ sample. The EDS analysis was also carried out to investigate the stoichiometry and the chemical contents of our MgB₂ sample. It was found that the amounts of oxidation and unreacted Mg in our sample are 8.852 and 9.306 wt%, respectively. These EDS results are compatible with XRD patterns in which Mg and MgO peaks were observed.

Fig. 2

The SEM images of polycrystalline superconducting MgB₂ sample taken for (a) 6000 and (b) 10,000 magnification

The polycrystalline superconducting MgB₂ pellet was cut carefully in the shape of a rectangular prism by using a diamond saw. We note that the MgB₂ samples were extracted from the middle region of the pellets. In order to eliminate possible Joule heating effects during the transport measurements, small sizes for the sample were preferred. The sample has the dimensions of length l∼5.3 mm, width w∼1 mm, and thickness t∼0.2 mm. Transport measurements were carried out using a four point probe technique. Fine copper wires of diameter 0.5 mm and good quality silver-paint were used in all contacts. The contact resistance measured by using a three point method was in the order of 0.01 Ω below T _c and 0.1 Ω at room temperature.

After making the necessary contacts, the MgB₂ sample was placed into a closed cycle He-refrigerator (Oxford Instruments (OI) CCC1104). All measurements were carried out under PC control via IEEE-488 interface. Temperature stability better than 0.01 K was maintained during the measurements (OI, ITC-503 temperature controller). The temperature was measured by using a calibrated 27 Ohm–rhodium–iron thermocouple (OI, Calibration number 31202). Keithley-182 (nano-voltmeter) with resolution 1 nV and Keithley-220 (current source) were used to measure the sample voltage and supplying the current, respectively. The nano-voltmeter was triggered for a maximum resolution of 61/2 digits, and its buffer was read directly within the integration time of 100 ms in order not to cause any artificial effects. The measured sample voltage is the average value of 5 readings for each data point. In all measurements under external magnetic field, the magnetic field was generated by an electromagnet (Oxford Instruments, N100).

The resistivity of the MgB₂ sample at room temperature is ρ (292 K)=25.78 μΩ-cm, which is reasonable when compared to the values (10–100 μΩ-cm) given in literature for very pure polycrystalline MgB₂ samples. The critical temperature T _c of the polycrystalline superconducting MgB₂ sample used in this work was determined as 38.8 K from the resistivity versus temperature measurements. In order to determine the current carrying capacity of the MgB₂ sample, the I–V characteristics were measured at temperatures near the critical temperature of the sample (T _c =38.8 K). The critical current density of MgB₂ sample was 12.33 A/cm² at 30 K for a current sweep rate dI/dt=1.25 mA/s. In determining the critical current (I _c ), the standard value of 1 μV/cm criterion was used.

The evolution of the sample voltage in time (i.e., V–t curves) is one of the best ways to monitor preciously the dynamic effects, which are sensitive to the external parameters such as temperature and applied magnetic field [1–4, 19, 20, 48]. Both slow and fast transport relaxation measurements of the ${V \mbox{--} t}$ curves provide an accurate method to investigate the details of the time effects and to determine the metastable states, motional reorganization, dynamic transitions, and dynamics of driven vortices etc., in superconducting samples at temperatures just below T _c [1–4, 19, 20, 64].

In this study, the time evolution of the sample voltage (${V \mbox{--} t}$ curves) of the MgB₂ sample corresponds to the slow transport relaxation regime. In the ${V \mbox{--} t}$ measurements, the voltage developing in the direction of the transport current in the superconducting sample was measured as a function of time after a dc initial driving current I ₁ is applied. The initial current I ₁ was maintained for a period of 60 s to achieve a steady state. To create a quenched state, the initial dc current I ₁ was reduced to a lower finite value I ₂ (or interrupted completely) during the time evolution of the sample voltage, and I ₂ was kept at the same value in the rest of the relaxation process, i.e., up to 120 s. In the mean time, the sample voltage was recorded continuously. Thus, it becomes possible to examine the time evolution of the quenched state and the details of current dependent reorganization of flux lines [74].

3 Experimental Results

3.1 Current–Voltage (I–V curves) Characteristics

Figure 3 presents a set of I–V curves for the MgB₂ sample measured at T=34 K for a current sweep rate dI/dt=1.25 mA/s, and under different external magnetic fields (H=0, 10, 15, 20, 25, and 30 mT). The magnetic field was oriented perpendicular to the transport current ($\vec{H} \bot\vec{I}$). Each of the I–V curves consists of two branches: The current-increase (CI) branch on which the current is increased from zero to a maximum current value I _max (=40 mA) and the current-decrease (CD) branch on which the current is decreased from I _max to zero. All the I–V curves in Fig. 3 are nearly reversible and exhibit a nonlinear behavior as the transport current is cycled up and down. A very small hysteresis effect begins to appear on the I–V curve at H=0 for I>20 mA. It should be noted that the I–V curve at H=0 exhibits a stepwise structure for I<10 mA. As H is increased from 0 to 30 mT, the measured sample voltage shifts to high values and the nonlinearity of the I–V curves is still observed even at H=30 mT. The ${I \mbox{--} V}$ curves measured at magnetic fields applied higher than 15 mT are practically independent of H. This finding implies that the measured voltage dissipation begins to enter the flux flow region.

Fig. 3

A set of I–V curves for the MgB₂ sample measured at T=34 K for dI/dt=1.25 mA/s and at different external magnetic field values of H=0, 10, 15, 20, 25, and 30 mT. CI and CD with the arrows indicate the current-increase and current-decrease branches of the I–V curves upon cycling the transport current, respectively. The inset shows the V–H curves of the MgB₂ sample at different transport currents (dI/dt=1.25 mA/s, T=34 K). The V–H curves were extracted from the CI branch of the I–V curves given in main panel

In order to assess the variation of the measured voltage dissipation with the applied magnetic field, the voltage values measured at some fixed values of transport current (I=10, 20, 30, and 40 mA) were extracted from the CI branches of I–V curves in Fig. 1 and plotted as a function of H as shown in the inset of Fig. 3. It is seen that the resulting V–H curves tend to saturate for the magnetic field values of H>15 mT.

Figures 4(a)–4(b) show the results of the transport measurements made at T=32.5 K and zero magnetic field (H=0), for the two extreme values of the current sweep rate dI/dt=5.00 and 0.625 mA/s up to I _max=60 mA. We observe that the line shape of the I–V curves are not influenced much by the variation of dI/dt, and keep its reversible character upon cycling the transport current. We also note that the maximum sample voltage measured at I _max=60 mA is approximately the same for both sweep rates. The stepwise behavior was observed at the currents below 30 mA for dI/dt=0.625 mA/s. Figure 4(c) shows some part of the data in Fig. 4(b) on an expanded scale between 0–30 mA.

Fig. 4

The I–V curves of the MgB₂ sample measured at T=32.5 K and zero field (H=0) for (a) rapid (dI/dt=5 mA/s), and (b) slow (dI/dt=0.625 mA/s) current sweep rate values. (c) A low current zoom of the data given in (b)

In order to study the effect of external magnetic field on the I–V curves, the measurements were made at T=32.5 K and H=10 mT, by applying the magnetic field perpendicular to the transport current ($\vec{H} \bot\vec{I}$). Figure 5 presents the I–V curves measured for the selected current sweep rates dI/dt=0.312, 0625, and 1.25 mA/s. Except the increase in the measured dissipation, the line shape of the I–V curves does not change due to the presence of external magnetic field as seen in Figs. 5(a)–5(c): The reversible behavior is preserved, as in case of the zero field I–V curves (Fig. 4), and the stepwise behavior on the CD and CI branches of the I–V curve (Fig. 5(b) and Fig. 5(c)) is still observed at the currents below 20 mA. The I–V curves do not exhibit any instability under the experimental conditions studied.

Fig. 5

The I–V curves of the MgB₂ sample measured at T=32.5 K and H=10 mT for different current sweep rates: (a) dI/dt=1.25 mA/s, (b) dI/dt=0.625 mA/s, and (c) dI/dt=0.312 mA/s

Figure 6 shows a set of I–V curves of the MgB₂ sample measured at a lower temperature T=30 K and zero field (H=0) for different values of dI/dt ranging from 5.00 to 0.625 mA/s, by increasing I _max to 80 mA. Except the expected decrease in the measured dissipation, the line shape of the I–V curves is almost the same as in the previous I–V curves (Figs. 4 and 5) and there is no marked difference between the voltage dissipations measured for the CI and CD branches. The small voltage difference (about ∼1–2 μV) between the CI and CD branches for all I–V curves in Fig. 6 is negligible. Thus, the reversible behavior in the I–V curves still continues for both slow and fast current sweep rates. It is also seen from Figs. 6(a)–6(d) that the voltage dissipation is essentially independent of the value of dI/dt. Figure 7 shows the variation of the maximum sample voltage V _max with dI/dt obtained at selected transport currents in the range 48–80 mA, as extracted from the data in Fig. 6. For instance, for I _max=80 mA, the difference between the maximum voltage values measured for dI/dt=0.625 and 5.00 mA/s is about 1.8 μV. It appears that, under the experimental conditions considered, the I–V curves measured at T=30 K, and 32.5 K are nearly independent of dI/dt.

Fig. 6

A set of I–V curves of the MgB₂ sample measured at T=30 K and H=0 for different current sweep rate values of (a) dI/dt=5.00, (b) dI/dt=2.50, (c) dI/dt=1.25, and (d) dI/dt=0.625 mA/s, by taking I _max=80 mA. The arrows show the direction of sweeping of the applied current

Fig. 7

The variation of the sample voltage obtained at I=48, 56, 64, 72, and 80 mA with dI/dt. The data are extracted from the I–V curves in Fig. 6

To further investigate the effect of the external magnetic field H, I–V measurements were carried out at selected values of H and lower temperature value T=28 K, by taking I _max=70 mA. Figure 8 shows the I–V curves measured at H=20, 40, and 60 mT, for dI/dt=5.00 mA/s. As expected, the increasing magnetic field results in an increase in the measured dissipation. The I–V curves exhibit nearly reversible behavior. We note that the voltage difference between the CI and CD branches of I–V curves is smaller than 2 μV even at high currents.

Fig. 8

A set of I–V curves of the MgB₂ sample measured at T=28 K and applied magnetic field values of H=20, 40, and 60 mT. The current sweep rate dI/dt=5.00 mA/s and the maximum current I _max=70 mA. The arrows show the direction of sweeping of the applied current

However, Fig. 9 clearly demonstrates the hysteresis effects in the I–V curves as the current is cycled up and down at T=28 K and H=0, for dI/dt=0.625 and 0.312 mA/s, in contrast to the I–V curves presented above. In the I–V curves presented in Figs. 3, 5, 6, 8, the voltage dissipation level was much greater than the voltage difference measured between the CI and CD branches of I–V curve at a fixed current, therefore, we could easily say that the I–V curves exhibit a reversible behavior as the current is cycled up and down. However, note that the measured maximum voltage dissipation in Fig. 9 is below 1 μV, and, since the height of the hysteresis is comparable to the voltage dissipation itself, the observed hysteresis effect in the I–V curves should be considered to be a consequence of the experimental conditions studied. As dI/dt is decreased from 0.625 to 0.312 mA/s, the height of the hysteresis loops decreases, and I–V curves become more reversible. In addition, the measured dissipation increases slightly with decreasing dI/dt.

Fig. 9

The I–V curves of the MgB₂ sample measured at T=28 K and H=0, by taking I _max=40 mA. (a) dI/dt=0.625 mA/s, and (b) dI/dt=0.312 mA/s. The arrows show the direction of sweeping of the applied current

Figures 10(a)–10(c) represent the I–V curves measured at T=28 K and H=18 mT for dI/dt=5.00, 0.625, and 0.312 mA/s. It is seen that the voltage dissipation increases to about 9 μV when the magnetic field of 18 mT is applied. The I–V curves measured for dI/dt=0.625 and 0.312 mA/s exhibit hysteresis effect together with small voltage jumps and drops. However, the I–V curve for dI/dt=5.00 mA/s shows a negligible hysteresis effect and does not show any voltage jump or drop. It seems that the current sweep rate and the external magnetic field play an important role in the evolution of I–V curves at 28 K. On the other hand, negative differential resistance develops in some regions in the CD branch of I–V curves measured for dI/dt=0.625 and 0.312 mA/s.

Fig. 10

The I–V curves of the MgB₂ sample measured at T=28 K and H=18 mT, by taking I _max=40 mA. (a) dI/dt=5.00 mA/s, (b) dI/dt=0.625 mA/s, and (c) dI/dt=0.312 mA/s. The arrows show the direction of sweeping of the applied current

The I–V measurements in the reverse procedure provide a useful tool to study the surface effects, the penetration of flux lines and also the penetration of the transport current (or the associated vortices) from the outer surface of the MgB₂ sample. In the reverse procedure, contrary to the standard route, the applied current is first decreased from I _max to 0 (the reverse current decrease (RCD) branch), and, then, increased from 0 to I _max (the reverse current increase (RCI) branch). Figures 11(a)–11(d) show the results of such an experiment performed on the MgB₂ sample at T=30 K and H=0 for different current sweep rates varied from 5.00 to 0.625 mA/s. As in the case of standard procedure, the RCD and RCI branches fall approximately on the same curve, and thus nearly reversible I–V curves are obtained upon cycling transport current in the reverse procedure (see Fig. 6 for a comparison). At first sight, one can easily say that there is not any remarkable change in the shape of the I–V curves, and, moreover, it is very difficult to distinguish the I–V curves obtained in the reverse and the standard procedures. Further, it is seen from Fig. 11 that when I _max=80 mA is applied to the sample, the sample voltage achieves a value of ∼45 μV for all dI/dt values. This behavior will be important when the I–V curves of MgB₂ sample are compared to the ones of other polycrystalline HTSC samples.

Fig. 11

(a)–(d) The I–V curves of the MgB₂ sample measured at T=30 K and H=0 by the reverse procedure for various current sweep rates (dI/dt) from 5.00 to 0.625 mA/s. The arrows show the direction of sweeping of the applied current

Figure 12 shows the I–V curves measured at T=30 K and H=0, for different current sweep rates varied from 1.25 to 0.312 mA/s by using the reverse procedure. The maximum current value I _max was reduced from 80 mA to 30 mA. It is seen from the Figs. 12(a)–12(c) that the hysteresis loops become narrow with decreasing dI/dt. When I _max=30 mA is applied to the sample, a voltage of about 0.05 μV develops for dI/dt=1.25 mA/s (see Fig. 12(a)), the sample voltage does not start from zero. For the other current sweep rates dI/dt=0.625 and 0.312 mA/s (see Figs. 12(b) and 12(c)), the voltage value measured starts from ∼0.1 μV. The measured dissipation corresponding to the RCD branch increases gradually, and passes through a maximum voltage value which shifts to higher currents with decreasing dI/dt. Finally, the sample voltage decreases with decreasing transport current and recovers the zero resistance state within our experimental accuracy. We note that the zero resistance state develops at higher currents as the current sweep rate is decreased. On the other hand, the measured dissipation for the RCI branches of I–V curves increases somewhat with decreasing dI/dt, in particular, at high currents. If I _max=30 mA, the voltage dissipation becomes 0.1 μV for dI/dt=1.25 mA/s and 0.14 μV and 0.13 μV for dI/dt=0.625 mA/s and 0.312 mA/s, respectively.

Fig. 12

(a)–(c) The I–V curves of the MgB₂ sample measured at T=30 K and H=0 by the reverse procedure for various current sweep rates (dI/dt) from 1.25 to 0.312 mA/s. I _max was reduced from 80 mA to 30 mA The arrows show the direction of sweeping of the applied current

3.2 Time Evolution of the Sample Voltage (V–t Curves)

In order to study the time evolution of the sample voltage of the MgB₂ sample, V–t curves are obtained by applying a dc driving current with two steps. The initial current I ₁ was applied for a first period of 60 s to achieve a steady state and then the initial current I ₁ was reduced to a lower finite I ₂ value, and then for the remaining time of the relaxation process, I ₂ was kept constant up to 120 s.

Figure 13 shows the V–t curves of the MgB₂ sample measured at T=34 K and H=0 for I ₁=50 mA and I ₂=20, 30, and 45 mA. The sample voltage evolves in two stages before the driving current is reduced from I ₁ to I ₂. There is a very sharp rise in sample voltage within a time less than about 1 s at the beginning of the relaxation process. This implies that the transport current organizes itself quickly along the sample and causes the sample voltage to level off at around 32 μV. Such a fast organization of the transport current can be expected because the temperature at which the measurements were performed is close to the critical temperature T _c =38.8 K of the MgB₂ sample and also the initial current I ₁=50 mA is relatively high. As I ₁ is reduced to I ₂, the sample voltage drops abruptly to a finite value depending on the value of I ₂ and levels off within very short time. The variation of the sample voltage with time is nearly constant for all current values considered, except the time intervals just at which the driving current I ₁ is first applied and reduced to I ₂.

Fig. 13

The V–t curves of the MgB₂ sample measured at T=34 K and H=0 for I ₁=50 mA and I ₂=20, 30, and 45 mA. The bold solid curves are calculated from the relation of V(t)∼exp(−t/t ₀)^α. The inset in main panel shows the schematic representation of operating current. The solid lines are a guide for the eyes.

Figure 14 shows the evolution of the voltage response of the sample measured at T=32 K and H=22 mT, for I ₁=10 mA and I ₂=7 and 5 mA. In this measurement, the magnetic field was selected high enough to observe the dissipation, due to low values of both I ₁ and I ₂. As is seen from Fig. 14, under these experimental conditions, a pronounced change in the line shape of the V–t curve is observed. The sample voltage, following a sharp rise, increases gradually with time and levels off at about 2.2 μV within about 10 s. The results suggest that the flux lines along the sample organize themselves with the help of the current within ∼10 s. When the initial current I ₁=10 mA is reduced to I ₂=7 mA (or 5 mA), following a sharp drop to a low dissipation level, the sample voltage decreases smoothly as the time evolves and finally levels off at ∼1 μV (or ∼0.6 μV) at t∼80 s.

Fig. 14

The V–t curves of the MgB₂ sample measured at T=32 K and H=22 mT for I ₁=10 mA and I ₂=5 and 7 mA. The bold solid curves are calculated from the relation of V(t)∼exp(−t/t ₀)^α

Figure 15 shows the V–t curves taken for I ₁=50 mA and I ₂=30, 20, and 10 mA at T=28 K and H=16 mT. At the beginning of the relaxation process, it is seen that there is a sharp rise in sample voltage with time. The sample voltage levels off at ∼11 μV within a very short time as in Fig. 12. As the current is reduced from I ₁=50 mA to I ₂=30 and 20 mA at t=60 s, the sample voltage drops abruptly to different dissipation levels for both I ₂ values. It should be noted that the zero resistance state evolves abruptly when the current is reduced from 50 mA to 10 mA. Further, as the driving current is interrupted (that is, I ₂=0), a similar zero resistance state is observed as for I ₂=10 mA (see the inset in Fig. 15). This finding implies that no residual voltage is left on the sample to be relaxed within the time response of the experimental set-up. This observation also rules out the thermal relaxation due to possible local heating effects.

Fig. 15

The V–t curves of the MgB₂ sample measured at T=28 K and H=16 mT, for I ₁=50 mA and selected values of I ₂ from 10 to 50 mA. The inset shows the V–t curve measured at T=28 K and H=16 mT, for I ₁=50 mA and I ₂=0

4 Discussion

Now, we first discuss the heating effects which can be an important problem causing hysteresis effects in transport measurements of superconducting materials [75]. In order to give a measure of heating effects, it is necessary to calculate the power dissipated at the current leads. By taking the contact resistance (∼10⁻²Ω) measured below the transition temperature T _c , and I _max=60 mA (see Fig. 4) we find the power dissipation at contacts as about 3.6×10⁻⁵ Watt. We consider that this power loss would cause no effects on the shape of the I–V curves and should not be a source of instability seen rarely on I–V curves of our MgB₂ sample.

4.1 Evolution of I–V Curves of MgB₂ Depending on the Current Sweep Rate

The irreversible behavior of the I–V curves of superconductors is an unwanted case from the point of their practical use. The superconducting sample should have only one J _c value and the I–V curves should be independent of dI/dt within a certain range. In another words, the time effects during the measurements, which are related to how fast the data are collected for a specific experiment, must be negligible. The time effects can appear in transport measurements in which the current is cycled, by causing significant irreversibilities.

We now focus on the I–V curves (see Figs. 3–6) obtained at different experimental conditions as the transport current is cycled up and down. One of the important observations in these I–V curves is the nearly absence of hysteresis effects, in spite of using slow and fast sweep rates of the transport current during the experiments. Nearly absence of hysteresis effects in the I–V curves of the MgB₂ sample implies that the effective pinning seen by the flux lines is nearly the same along the sample and the mobile vortices move in the same flow channels. That is, different flow patterns or different dynamic vortex phases for the CI and CD branches of the I–V curves do not evolve upon cycling of driving current. Further, nearly reversible behavior in the I–V curves can be taken as an indication of the fact that there are no closing or opening new flow channels for CI and CD branches.

The physical origin of the nonlinear behavior of I–V curves for MgB₂ sample can be examined by a simple power-law relation V(I)∼I ⁿ [68]. Here, the exponent n=U ₀/k _B T depends on the temperature T, magnetic field H and flux pinning strength, and varies from 1 to infinity (∞). When the exponent n is equal to 1, the I–V relation exhibits an ohmic behavior. If n>1, the I–V curve describes nonlinear flux creep. As the exponent n goes to infinity, the I–V curve represents the Bean critical state. A typical example for the curve fitting procedure which considers only the CI branch of the I–V curves in Fig. 6 is shown in Figure 16(a). The solid line calculated by using V(I)∼I ⁿ shows a reasonable agreement with the data corresponding to the experimental parameters dI/dt=1.25 mA/s, T=30 K, and H=0. We note that reasonable agreements were also obtained for the data obtained using other sweep rates.

Fig. 16

(a) A comparison of the experimental I–V curve (open circles) of the MgB₂ sample measured at T=30 K and H=0 for dI/dt=1.25 mA/s (see Fig. 6) with that calculated using the power-law relation V(I)∼I ⁿ. (b) The variation of the critical current I _c and the exponent n with the current sweep rate dI/dt. All the I _c and n data were extracted by fitting V(I)∼I ⁿ to the I–V curves given in Fig. 6. (c) Variation of the pinning potential U ₀ with current sweep rate dI/dt. The data were extracted by fitting V(I)∼I ⁿ to the I–V curves given in Fig. 6

The values of the critical current I _c and the exponent n corresponding to each dI/dt were extracted by fitting the experimental I–V curves to V(I)∼I ⁿ. The results thus obtained for I _c and n are plotted in Fig. 16(b) as a function of dI/dt. It is evident that the variation of both I _c and n with dI/dt is almost negligible. The exponent n takes a value about 3.27 for all current sweep rates. I _c values extracted from the power law relation is found to be nearly equal to of real macroscopic critical current value of 26.33 mA at 30 K for a current sweep rate dI/dt=1.25 mA/s. We note that the sample inhomogeneities have drastic effects on the critical current value, whereas the power law relation is an general expression in characterizing the superconductors and considers the ideal cases.

Figure 16(c) shows the variation of the pinning energy U ₀ with the current sweep rate dI/dt. The data were extracted from the I−V curves given in Fig. 6 by fitting to V(I)∼I ⁿ. As is seen from Fig. 16(c), although the pinning energy tends to decrease with increasing dI/dt, the variation between the U ₀ values determined for the current sweep rates 5.00 mA/s and 1.25 mA/s is about 0.1%. The results indicate that U ₀ is essentially independent of dI/dt.

In fact, the pinning potential can be correlated to the sweep rate of transport current via the relation [76] dU(J)/dt=(dJ/dt)(dU/dJ). And further, for a slab geometry, by solving the diffusion equation in one dimension, it can be easily shown that dU(J)/dt is written as dU(J)/dt=A(dU(J)/dJ)exp[−U(J)/k _B T]. Here, A is a constant and k _B is the Boltzmann constant. Geshkenbein and Larkin [77] solved this equation analytically and showed that dU(J)/dt=k _B Tln(t/t ₀), where t ₀ is the characteristic time. It is seen that the pinning potential has a time dependence in the simplest case and there is a direct relation between dJ/dt and dU(J)/dt. Although the variation of U ₀ with dI/dt presented in Fig. 16(c) is very small, the decrease in U ₀ with decreasing dI/dt (or dJ/dt) can explained as follows. dJ/dt is proportional to the rate of the generation of the self-field and the rate of magnitude of Lorentz force acting on flux lines. The fast sweep rates permit to apply higher currents within a very short time interval. This leads to the fact that the work done by Lorentz force at high values of dI/dt can lower the effective height of pinning barrier much more as compared to that of the slow sweep rates. Anyway, as is seen from Fig. 16(c), the variation of U ₀ with dI/dt is negligible and consequently, U ₀ can be taken to be independent of dI/dt. Finally, it can be suggested that the experimental observations establish the absence of relaxation effects in the I–V curves of the MgB₂ sample.

The CI and CD branches of the I–V curves in Figs. 3–6 follow almost the same line shape without exhibiting any remarkable voltage instability. However, if it is focused on the low-current region of the I–V curves in Fig. 4(b), it will be seen that there are small steps as the current is cycled up and down. Similar instabilities are observed in the I–V curves of Figs. 5(c), 5(d), 6(d), 10(b), and 10(c), especially at the slow rates of 0.625 and 0.312 mA/s. As shown experimentally by Moshcalkov et al. [69], the vortices in pure single-crystal sample of MgB₂ evolve in the form of clusters and stripes at very low magnetic field values. Olson et al. [78] showed numerically by solving the overdamped Langevin equation of motion in two-dimension that the stripe phase formation originates from the dynamic competition of between long-range repulsion and short-range attraction. Of course, the physical case associated with flux distribution and its dynamic will be different in a polycrystalline sample having grains of different size and different orientations. Here, the grain boundaries can be assumed as the edges of a single-crystalline sample. At very low magnetic fields, it is expected that the flux dynamic inside an independent grain should mimic nearly the observations of Moshcalkov et al. [69]. Thus, the vortex clusters and stripe phase can be expected to form inside the grains at low and moderate magnetic fields. Such vortex patterns in motion should include the easy motion channels around the strong pinning centers in a complicated way. Therefore, we interpret these steps in the sample voltage (see for instance in Fig. 4(b)) as a kind of plastic flow of the vortices evolving in the form of stripe or cluster in different sizes. In such a plastic motion, it can be suggested that the local order parameter decreases or increases suddenly, depending on the topological distribution of the pinning centers.

Interestingly, it is seen from Figs. 4(a), 4(b), 5(a), 6(a)–6(c), and 10(a) that there are no significant jumps or drops evolving in the form of steps in the sample voltages measured for dI/dt≥1.25 mA/s. It seems that the increasing rate of the current is not faster than the time needed for the vortices to move. Note that the measured dissipation is nearly the same for all values of dI/dt when the current reaches I _max. Similarly, we observe negligible hysteresis effects and instabilities on the I–V curves in Figs. 10(b) and 10(c). At this magnetic field and temperature regions, such instabilities (jumps and drops) develop at low dissipation levels and also for the low values of current sweep rates of about 0.625 mA/s.

On the other hand, both experimental observations and numerical calculations reveal that the stripe vortex phase or vortex clusters can change their character with increasing magnetic field. Large voids between stripes of vortices decrease and bigger vortex clusters can form inside the sample. At high dissipation levels and in the presence of pinning, we suggest that the magnitude of Lorentz force and polycrystalline structure of superconducting material can destroy the vortex clusters or stripe phase and cause to formation of a usual flux distribution along the sample.

It can also be suggested that the stepwise structure in I–V curves (see for instance Figs. 4(c), 5(b), 5(c), 10(b), and 10(c)) originates from the phase slip centers due to the presence of metallic Mg, or other impurities along the intergranular region. The magnitude of the order parameter at these regions can be weakened locally by imperfections and, even, can go to zero. The phase difference of order parameter can vary at these regions from one place to another and can manifest itself as a stepwise structure in I–V curves. We note that such a case appears, in particular, at low values of dI/dt and also at low dissipation levels. At high current sweep rates, the Lorentz force acting on vortices becomes greater than the one at low values of dI/dt, and, the distance traveled by the vortices will be relatively greater than the correlation length which is generally comparable to the size of the phase slip centers. Therefore, the vortices moving at high current sweep rates will pass rapidly those centers without causing any stepwise structure in I–V curves. Elmuradov et al. [79] observed similar stepwise structure and ccw hysteresis effects in I–V curves of NbN wire with multicontact configurations. The stepwise structure in those I–V curves was attributed to the phase slip lines and was confirmed by numerical simulations using time-dependent Ginzburg–Landau theory.

4.2 A Comparison of the I–V Curves of the MgB₂ Sample to Those of YBCO and BSCCO

A comparison of the I–V curves of the MgB₂ sample to those of Y₁Ba₂Cu₃O_7−x (YBCO) and Bi_1.7Pb_0.3Sr₂Ca₂Cu₃O _x (BSCCO) will help to understand better and also to distinguish some important properties of superconducting MgB₂. Figure 17 shows typical I–V curves for YBCO and BSCCO samples measured with dI/dt=1.25 mA/s at zero magnetic field. We see that pronounced hysteresis effects develop in the I–V curves upon cycling the transport current, as compared to that of observed in polycrystalline sample of MgB₂ (see, for instance, Fig. 6(c)). In addition, the evolution of the I–V curves of both YBCO and BSCCO samples strongly depends on the current sweep rate (dI/dt). The irreversibility appeared in the I–V curves in Fig. 17 could be due to the different degree of the inhomogeneous flux motion between CI and CD branches [34, 36, 39, 41, 42, 44–46, 69, 80–88].

Fig. 17

(a) Hysteresis effect observed in I–V curve of polycrystalline sample of YBCO measured at T=88 K and zero magnetic field for dI/dt=1.25 mA/s. (b) Hysteresis effect observed in I–V curve of polycrystalline sample of BSCCO measured at T=102 K and zero magnetic field for dI/dt=1.25 mA/s

The above comparison between the I–V data of the YBCO and BSCCO samples and the MgB₂ sample, and, also the analysis of the I–V data of the MgB₂ sample presented above suggest that the strong hysteresis effects observed in YBCO and BSCCO can be related to the presence of a weak-link structure in HTSCs. At this point, we need further support to this suggestion, and the first question to be asked is whether a great weak-link problem exists in polycrystalline MgB₂ sample or not.

It is well known that the presence of weak-links is a great problem in HTSC cuprates and can cause a significant decrease in the current carrying capacity of the superconducting material. Therefore, many attempts have been made to improve the weak-links connecting the grains and also to produce weak-link free superconducting materials. In addition, it is widely assumed that the weak-link structure in HTSCs cuprates behaves like a type-II superconductor having poor superconducting properties with its own penetration depth, first $H_{c1}^{W}$ and second ($H_{c2}^{W}$) critical magnetic fields [23, 32, 64, 89, 90].

In a granular structure including weak-links as HTSC cuprates, when the transport current (or external magnetic field) is applied to the sample, the flux lines induced by the current (or by the external magnetic field) must first overcome the surface barriers which can be considered as surface weak-links, and also the de-magnetizing field of the corresponding material. The surface weak-link resists against the injection of the flux lines to the interior of the superconducting material in order to keep the sample in the Meissner state. In fact, it is well known that a surface potential barrier in Type-II superconductors called as Bean–Livingstone barriers appears for the flux lines which enter or leave the sample [91–93]. The surface barriers together with bulk pinning play an important role in the irreversible properties of cuprates. Surface pinning can prevent the flux penetration, and, thus, can retard the flux entry.

Owing to the magnitude of transport current or external magnetic field, the motion of flux lines starts from the weakest points of the material until they meet the weak-link structure between the grains and, finally, the flux lines must also overcome the latter weak-link structure to end in highly superconducting grains composing the material. Therefore, it is possible that the evolution of pronounced hysteresis effects observed in the I–V curves of YBCO and BSCCO originates mainly from the complex weak-link structure and, therefore, inhomogeneous motion of vortices.

However, it can be suggested that the weak-links between the grains and the surface weak-links do not play a remarkable role in the evolution of the I–V curves of our MgB₂ sample, and hence nearly absence of hysteresis effects in the I–V curves can be attributed to the lack of strong weak links in MgB₂. The transport current (or the associated vortices) or external magnetic field easily penetrates into the MgB₂ sample from its surface and redistributes itself very rapidly without encountering any obstruction, so that the weak links between the grains do not affect the motion of flux lines which follows the same easy flow channels. We note that Larbalestier et al. [94] first suggested the absence of weak link problem in MgB₂ sample by considering the similarity between the pinning curves of MgB₂ and Nb₃Sn.

To further evaluate the I–V curves of the MgB₂, it is useful to compare the I–V curves of MgB₂ sample given in Fig. 11 with those of YBCO and BSCCO samples obtained by the reverse procedure. Figures 18(a) and 18(b) present the I–V curves obtained at T=85 K and T=102 K and zero magnetic field with dI/dt=1.25 mA/s for the polycrystalline superconducting YBCO and BSCCO samples, respectively.

Fig. 18

(a) The I–V curve of polycrystalline superconducting YBCO sample measured at T=85 K and H=0 by the reverse procedure for dI/dt=1.25 mA/s. (b) The I–V curve of polycrystalline superconducting BSCCO sample measured at T=102 K and H=0 by the reverse procedure for dI/dt=1.25 mA/s. The arrows show the direction of sweeping of the applied current

In the I–V curve of the YBCO sample (Fig. 18(a)), contrary to the one observed in the I–V curve of the MgB₂ sample, the voltage starts from zero and increases nonlinearly with decreasing the applied current. It passes through a peak at about 30 μV, and then decreases with further reduction of current. Upon cycling the transport current in the reverse procedure, there is a large hysteresis effect in I–V curve of YBCO. The I–V curve of the BSCCO sample (Fig. 18(b)) nearly mimics the behavior of YBCO, but there are some differences. It is seen from Fig. 18(b) that the voltage attains a value of ∼10 μV as I _max=40 mA is applied. Marked (but not much as in the YBCO sample) hysteresis effect in the I–V curve of the BSCCO sample evolves upon cycling the transport current. We note that such I–V curves obtained for YBCO and BSCCO samples are similar to the I–V curves observed at low dissipation levels for MgB₂ sample (see Fig. 12). It seems that the surface pinning centers work actively at low dissipation levels and at high current sweep rates by preventing the penetration of current induced flux lines. The nonzero voltage values observed in I–V curves of Fig. 12 can be evaluated as flux motion from outer surface of the sample to the interior and a partial accumulation of the flux lines near the sample surface.

When I _max is applied to the YBCO sample, the zero voltage observed on the I–V curve of YBCO implies that the flux lines accumulate on the periphery of the sample, and are prevented completely to penetrate into the sample by the surface pinning centers and also by the weak-link structure of the material. A similar case is observed in the I–V curve of BSCCO sample (Fig. 18(b)), although a sample voltage (∼10 μV) develops as soon as I _max is applied to the BSCCO sample. However, it seems that the roles of the surface weak links and the weak links between grains in our MgB₂ sample are not so deterministic as much as in the YBCO and BSCCO samples. Thus, the dramatic difference between the line shape of RCD and RCI branches observed in reverse procedure of I–V curves of HTSC cuprates and MgB₂ can be mainly attributed to the strength of surface pinning seen by the flux lines. Finally, based on this discussion and also the comparison of I–V curves of MgB₂ to those of cuprates, we note that the I–V curves obtained by the reverse procedure enable us to study how the applied current (or the associated vortices) penetrates from the sample surface to its interior.

The MgB₂ material has many advantages from the view point of practical applications as compared to low temperature superconductors (LTSCs) and also HTSC cuprates. For instance, the grain boundaries in MgB₂ act as effective pinning centers without weak-link problem, and, further, MgB₂ does not exhibit fast flux creep which limit the performances of some LTSCs such as Nb₃Sn and HTSC cuprates. In addition, this makes MgB₂ very competitive to Nb₃Sn in practical applications. The experimental studies reveal that the critical current density of MgB₂ is comparable to that of Nb₃Sn at 4.2 K, i.e., J _c ∼10⁶ A/cm² at its highest field, but its T _c is much higher [95]. Therefore, it is more relevant to the magnet applications as compared to both HTSC cuprates and conventional superconductors. Indeed, the experimental studies on MgB₂ are very promising and reveal that its optimized critical parameters can surpass the many capabilities of LTSCs [95].

4.3 Time Evolution of Sample Voltage (V–t Curves)

It is seen from the V–t curves of the MgB₂ sample in Figs. 13–15 that the increase in dissipation is quite sharp for both H=0 and H≠0 at the initial stage of the relaxation process. This supports the suggestion that the transport current (or the associated flux lines) and the flux lines in the presence of magnetic field distribute themselves very rapidly along sample. Such an organization of the flux lines involves complex resistive flow patterns in the bulk of the sample where the effective superconducting order parameter is suppressed as the time progresses [10]. The flux lines (or current induced flux lines) will move from the sample edges towards the center of sample by following the easy motion channels (i.e., weak pinning centers). Here, a fast increase in the measured dissipation is an indication of the fact that the de-pinning is more effective. After the completion of organization of flux lines inside the sample, a steady state is reached where nearly constant flux motion develops across the sample. In this case, the measured dissipation becomes essentially constant with time. Lee et al. [42] observed that a two-level telegraph noise appears in the V–t curves of single crystalline sample of MgB₂ at zero field cooled (ZFC) state. They showed that the evolution of the V–t curves depends strongly on the magnitude of the transport current and, at the early stage of relaxation process, the sample voltage increases abruptly at ZFC state. In field cooled (FC) state, depending on the magnitudes of transport current and magnetic field, the authors [42] demonstrated that the sample voltage increases very rapidly and saturates as in the ZFC state as the time progresses. However, at magnetic fields below a certain value where the peak effect [12, 24] appears, the voltage increases gradually and non-linearly as a function of time until a peculiar abrupt jump. We note that the relaxation effects observed in single crystalline sample of MgB₂ [42] and the time effects seen in the V–t curves for polycrystalline MgB₂ sample (Figs. 13–15) are quite similar with those of obtained under the magnetic field for relatively lower current values but different from those observed in the YBCO [10] and BSCCO samples [96]. In the presence of external magnetic field, the V–t curves given in Fig. 15 show a similar behavior as compared to the zero field V–t curves (see Fig. 13). It is also seen that the dissipation rate at the beginning of relaxation process is quite high. This suggests that some of the flux lines inside the sample are subjected to weak pinning and some of them are partly free, and thus are ready to move along the easy motion channels.

However, the V–t curves in Fig. 14 exhibit a nonlinear voltage increase following the abrupt increase at the beginning of the relaxation process. We attribute the nonlinearity observed in V–t curves to the complex interplay between pinning and depinning. In this case, the flux lines (or current-induced flux lines) organize themselves gradually with the help of the transport current by suppressing the order parameter. The enhancement in the measured dissipation indicates that the depinning wins the competition.

Fast transport relaxation measurements in an undoped single crystal of 2H-NbSe₂ showed that the time evolution of voltage response in high velocity vortex regime could be characterized by a stretched exponential relation given as

$$ V(t) = V_{0} \bigl[1- \exp(-t/t_{0})^{\alpha} \bigr],$$

(1)

where V ₀ is the saturation voltage, t ₀ is the characteristic time, and the exponent α is a constant [1, 97–99]. In those experiments on single crystal of 2H-NbSe₂, the rise in sample voltage is recorded continuously as a function of time [99]. Li et al. [97, 98] and Du et al. [99] found that (1) describes well the nonlinear time evolution of voltage response. We note that there is a similarity between the nonlinear V–t curves given in Figs. 13–15 and the ones observed in the studies of Li et al. [97, 98] and Du et al. [99]. The bold solid lines in Figs. 13 and 15 are the calculated curves by using (1) which shows a reasonable agreement with the experimental data. Best fits are obtained between (1) and all V–t data given in Figs. 13 and 15 when the exponent α is ∼1. We note that the fitting is related to the time evolution of sample voltage before the quenched state is created. The characteristic time value t ₀ related to the experimental data given in Fig. 13 is found as 0.38 s for I ₁=50 mA. In Fig. 15, the agreement between the experimental data and (1) is satisfied for t ₀=0.26 s. Low values of t ₀ found from fitting procedure is a measure of the rapid reorganization of flux lines [or transport current (or associated vortices)] along the sample depending on the magnitude of the transport current.

4.4 Time Evolution of the Quenched State for I ₁>I ₂ at Different Magnetic Fields and Temperatures

We now discuss the quenched state produced by switching the applied current from I ₁ to I ₂ (see Figs. 13–15). At a given magnetic field and temperature, as the initial current I ₁ is reduced to I ₂, a quenched state develops inside the sample depending on the magnitude of I ₂. The zero magnetic field V–t measurements in Fig. 13 reveal that the voltage drops evolve very abrupt to low dissipation values. Such a time evolution of quenched state is also observed in the V–t curves presented in Fig. 15 for H≠0. The results suggest that, as I ₁ is reduced to I ₂, the flux lines find accessible states very rapidly and easily inside sample, and do not need to explore any new ones. Furthermore, it can be suggested that the effective size of dissipative non-superconducting flow channels decreases in the same rate as compared to the switching rate of the applied current from I ₁ to I ₂.

However, the physical case observed in the V–t curves in Fig. 14 is somewhat different for relatively low current values. As the initial current is reduced from 10 mA to 7 mA (or 5 mA), the quenched state relaxes smoothly towards low dissipation values before leveling off as the time progresses. We note that the maximum voltage values measured in V–t curves in Fig. 14 is about 2.2 μV. It seems that despite the sudden reduction in the Lorentz force, the average velocity of the flux lines does not decrease in the same rate as compared to reducing rate of the Lorentz force, and the motion of flux lines is gradually prevented by the pinning centers. However, similar abrupt drops without any smooth decrease are observed in the sample voltage at higher dissipation levels as in the V–t curves Figs. 13 and 15), so that the quenched state enters directly into the voltage-leveling state.

At this point, we may assume that the driving current serves as an effective temperature as in the usual sense in statistical mechanics, i.e., the dynamic annealing effect [11, 23]. When the transport current is applied to the sample, it anneals dynamically the corresponding states and may lead to a raise in the internal energy of the sample, and hence the measured dissipation. Therefore, as the driving current is suddenly reduced from I ₁ to I ₂, the physical case associated with the corresponding sharp voltage drop in V–t curves can be correlated to supercooling. On the other hand, as the driving current is interrupted completely, i.e., I ₂=0 (see the inset in Fig. 15), the sample voltage becomes immediately zero indicating that there is no residual voltage to be relaxed on the sample. This observation also rules out the thermal relaxation due to possible local heating effects [32, 75, 100–102]. A similar case is also observed in the V–t curve obtained for I ₂=10 mA (see the V–t curve in Fig. 15), where the measured voltage becomes zero without showing any relaxation in time. This observation implies that the sample enters into the superconducting state by excluding very rapidly the flux lines at the same rate as compared to the switching rate of the applied current from I ₁=50 mA to I ₂=10 mA. During the relaxation process in the quenched state (60<t<120 s), the dynamic annealing effect of the driving current does not cause a measurable voltage dissipation and does not destroy the formed coherent state between the grains, which is relatively strong.

4.5 Exponential Decay of the V–t Curves and the Quenched State for I ₁>I ₂

Xiao et al. [3] showed that the quenched state in fast transport relaxation measurements of single crystalline sample of 2H-NbSe₂ is closely related to the glassy state relaxation and fits a stretched exponential time dependence in the form V(t)∼exp(−t/t ₀)^α, where t ₀ is a characteristic time and the exponent α is a temperature dependent constant [10]. This expression describes well the time decay in the remnant magnetic moment in magnetic materials [59]. The time evolution and degree of ordering of individual magnetic moments can be determined via the exponential term. In the glassy state, it is considered that in an energy landscape, there exists a broad distribution of both long-lived and short-lived metastable states separated by several energy barriers for the flux lines [59]. The energy barriers between the neighboring barriers are small while the energy difference between the more distant barriers is higher. We suggest that the voltage decays at low dissipation levels observed in the V–t curves of the MgB₂ sample (see Fig. 14) can be described by an empirical relation for a glassy state, i.e., V(t)∼exp(−t/t ₀)^α [3]. The bold dashed lines superimposed on the V–t curves of Fig. 14 in the quenched state are calculated by using V(t)∼exp(−t/t ₀)^α by taking t ₀ as fitting parameter. Good agreement between the experimental data and the calculated curve is obtained for α∼1 and t ₀=5.1 s and 4.4 s for I ₂=7 and 5 mA, respectively.

Marchevsky et al. [103] proposed a model so-called as vortex capacitor model which explains the dynamic effects related to the contamination and annealing of vortices by driving current. It was shown that releasing and accumulation of energy by driving current resemble the charging/discharging process of an electrical capacitor with a resistive load. The model gives an exponential relation of ∼exp(−t/t ₀) for the time decay of sample voltage when the applied current is smaller than the critical current value I _H for high pinning regime, but larger than the critical current value I _L for low pinning regime. The edges of a single crystalline sample can be taken as grain boundaries in a polycrystalline sample. In our case, the grains can correspond to high pinning regime, and intergranular region and grain boundaries can be taken low pinning regions. We suggest that vortex capacitor model can be applied to explain the exponential decays in sample voltage, provided that the applied current is switched to a value less than the critical current of grains (or intergranular region).

4.6 V–t Curves Calculated by Stretched Exponential

Figure 19(a) shows the V–t curves calculated by using (1) to represent the time evaluation of voltage rise for different values of t ₀=0.5, 2.5, 6.0, 10, 15, and 20 s. In the calculations, α is taken as 1 and the parameter A is chosen arbitrarily. The line shape of the V–t curves depends strongly on the magnitude of t ₀. Note that the sharp increase in voltage developing at low values of t ₀ mimics directly the V−t curves of Figs. 13 and 15 taken for 0<t<60 s. Figure 19(b) represents the V–t curves calculated by using (1) for the selected values of α=1, 3, 5, 7, 9, and 11 and t ₀=15 s. It is seen that the line shape of calculated V–t curves changes significantly as α increases. For the values of α>1, the calculated voltage does not immediately grow up with time but begins to increase smoothly from zero to a any value after a waiting time t _w which increases with increasing α. The inset in Fig. 19(b) reveals clearly the influence of t ₀ on the calculated V–t curves. The parameter t ₀ is taken as 1 s and α is chosen as 1 and 3. We obtain two important results for these parameters considered: first, the waiting time t _w decreases with decreasing t ₀ from 15 s to 1 s, and, second, the rise in voltage with time becomes extremely sharp as compared to that of calculated for t ₀=15 s. As is discussed above, the low values of t ₀ corresponds to high applied currents or high magnetic fields in the experimental V–t measurements and the rapid reorganization of flux lines with the help of the current along the sample.

Fig. 19

(a) The calculated V–t curves by using (1) for the time evolution of voltage rise. The characteristic time values, t ₀, are taken as 0.5, 2.5, 6.0, 10, 15, and 20 s. In the calculations, α is taken as 1 and the parameter A is chosen arbitrarily. (b) The calculated V–t curves by using (1) for the time evolution of voltage rise. In the calculations, t ₀ and α are taken as 15 s, and 1, 2, 3, 5, 7, 9, 11, respectively. The inset shows the calculated V–t curves for t ₀=1 s, and α=1 and 3. In all calculations, the parameter A is chosen arbitrarily

4.7 A Comparison of the V–t Curves of the MgB₂ Sample with Those of YBCO and BSCCO Samples

Earlier V–t measurements carried out on superconducting polycrystalline samples of YBCO [10] and BSCCO [96] demonstrate that the increase in the measured voltage is always highly nonlinear, and their V–t curves exhibit a different behavior as compared to the ones observed for the MgB₂ sample. Typical examples for the V–t curves of the YBCO and BSCCO samples are represented in Figs. 20(a) and 20(b), respectively.

Fig. 20

(a) The V–t curve for the polycrystalline YBCO sample measured at T=87 K and H=0 for I ₁=20 mA with I ₂=10 mA. (b) The V–t curve for the polycrystalline BSCCO sample measured at T=102 K and H=0 for I ₁=40 mA with I ₂=30 mA

It is observed from the V–t curves of the YBCO and BSCCO samples that the variation of the sample voltage with time is quite smooth just after I ₁ is switched to I ₂. We suggest that the different behavior in the V–t curves of YBCO, BSCCO, and MgB₂ originates from their crystalline structure, structural composition, chemical, and anisotropic states. The smooth transitions in V–t curves of YBCO and BSCCO samples as the applied current is switched from I ₁ to I ₂ can be related to the presence of weak-link structure in these polycrystalline. We correlate the abrupt rises and drops in the sample voltage of the MgB₂ (see Figs. 13–15) to nearly absence of weak link structure of MgB₂ sample as is discussed in the I–V curves obtained for different sweep rates.

5 Conclusion

In this work, the time dependent flux dynamics in a polycrystalline superconducting sample of MgB₂, below the critical temperature T _c , was investigated in details via measurements of the current – voltage (I–V curves) with different current sweep rates (dI/dt) and transport relaxation (V–t curves). The transport measurements were carried out as functions of temperature (T), transport current (I), external magnetic field (H) and orientation of H with respect to I.

One of the important observations in the I–V curves of the MgB₂ sample was nearly absence of hysteresis effects when the transport current was cycled up. These measurements reveal that the I–V curves are practically independent of dI/dt. It was shown that the experimental I–V curves at high dissipation levels are in good agreement with a power law behavior, V(I)∼I ⁿ. The pinning potential U ₀ extracted by the fitting the experimental data to the power law behavior is approximately independent of dI/dt. This means that the effective pinning seen by the vortices is nearly the same along the MgB₂ sample within the measurement accuracy. The small instabilities, voltage jump, and drops appearing in I–V curves were interpreted in terms of plastic flow of the vortices evolving in the form of stripe or cluster in different sizes and local fluctuations in the superconducting order parameter and also presence of possible phase slip centers. In addition to the standard procedure, a reverse procedure was employed to study how the transport current (or associated vortices) penetrates from the surface into the sample. It was found that there is no marked difference between the I–V curves obtained in the standard and reverse procedures. A comparison of the I–V curves of the MgB₂ sample with those of the YBCO and BSCCO samples reveals that the role of the weak-links between the grains (standard procedure) and surface weak-links (reverse procedure) in our MgB₂ sample is not so deterministic in contrast to the YBCO and BSCCO samples. Thus, the MgB₂ sample can be taken as a weak link free material which is desired for practical applications.

The sharp and nonlinear increases in sample voltage of V–t curves were interpreted in terms of re-organization of flux lines and suppression of the effective superconducting order parameter as the time progresses. The time variation of sample voltage associated with sharp and non-linear increase gives good agreements with a relationship of V(t)∼[1−exp(−t/t ₀)^α]. At relatively high dissipation levels, the measured dissipation increases very rapidly just after the transport current is applied. This finding implies that some of the flux lines are ready to move along easy motion channels in the sample.

At high dissipation levels, a sudden decrease of the transport current from an initial value I ₁ to a lower value I ₂ causes a quenched state which reflects itself as a sharp voltage drop in V–t curves. It can be suggested that the sample enters into the superconducting state by excluding very rapidly the flux lines (or the current-induced flux lines). This quenched state observed in V–t curves behavior was correlated to the supercooling effect caused by the dynamic annealing effect of the transport current. At relatively low dissipation levels, the voltage drop in the quenched state shows a decay which is consistent with an exponential relationship, ∼exp(−t/t ₀), as in the case of a glassy state. As the transport current was interrupted (i.e., I ₂=0), the observed zero voltage rules out any thermal relaxation due to possible local heating effects.