MgB2 superconductor

Addition of nonsuperconducting phases, such as carbon nanotubes, can modify the superconducting properties of MgB 2 samples, improving the critical current density and upper critical ﬁeld. A full understanding of the ﬂux creep mechanism involved is crucial to the development of superconducting magnets in persistent mode, one of the main thrusts for the development of MgB 2 wires. In this paper we present a creep study in bulk MgB 2 samples, pure and with different amounts of carbon nanotubes additions. We conclude that the most consistent picture at low temperatures is a single vortex pinning regime, where the correlation length is limited by the grain size. We introduce a novel analysis that allows us to identify the region where the Anderson-Kim model is valid. V C 2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4813132]


I. INTRODUCTION
Since the discovery of the superconductor MgB 2 , 1 a lot of work was devoted to understand and to improve its pinning properties. The great potential for technological applications of this material is due to its high critical temperature T c ($39 K), low cost of raw materials, 2 chemical simplicity, and absence of weak-link limitations to the critical current density J c . 3 In the last years, several groups have achieved a good improvement of the transport properties. In bulk polycrystalline samples, good results for increasing J c have been achieved by chemical doping or by the inclusion of nonsuperconducting phases, in particular with carbon nanotubes (CNT) inclusions, [4][5][6][7][8][9] with diameters close to the MgB 2 coherence length n. On the other hand, theoretical results predict and experimental studies confirm that the presence of two superconducting gaps produces significant H c2 enhancement in the dirty limit. 10 In a previous publication we have reported results in samples with double wall CNTs additions showing both effects: an important increase in the critical current density and the upper critical field H c2 . 7,8 However, it is not clear if this two effects are connected. Recent works have studied the activation energy in dirty MgB 2 samples, but the results are not conclusive. 11 In fact, the vortex physics underlying this performance improvement has not been so far understood.
In this paper we perform a deeper analysis of the role of naturally formed and artificially introduced pinning centers by studying the magnetic relaxation in bulk MgB 2 samples. The time decay rate of the supercurrents (flux creep rate), beyond its basic interest, is an important parameter for applications, particularly those involving the use of superconducting magnets in persistent mode, such as MRI, which is currently one of the main thrusts for the development of MgB 2 wires. 12 As it is well known, if a superconducting sample in the critical state is fully penetrated by the magnetic field B, the magnetization M is proportional to the current density J flowing in the sample. The "measurable" critical current density J c is lower than the "true" J c0 due to the thermal activated motion. In a magnetic relaxation experiment the measured J c decays in time as À @J c @t ¼ C j e ÀUðJc;T;BÞ kT ; (1) where C j $ J c =s 0 can be approximated as a constant and UðJ c ; T; BÞ is the activation barrier.
In oxide High Temperature Superconductors (HTS), the creep drastically drops the measurable density current, so J c ( J c0 , an effect highly detrimental for applications. A large number of glassy creep regimes in this limit have been proposed theoretically, 13,14 and many of them observed experimentally. 15 In those cases the temperature dependence of the measurable critical current J c is dominated by creep while the intrinsic J c0 ðTÞ dependence is much weaker. In recent years it has also became clear that iron-arsenide superconductors suffer from creep rates approximately as high as oxide HTS. 16 On the other hand, in traditional type II low temperature superconductors (LTS) superconductors the current decay is very small, so J c0 À J c ( J c0 ; therefore, the activation barrier takes the traditional Anderson perspective, this puts MgB 2 in the rather unique situation of being the only known superconductor with high enough T c to be easily operated in liquid-He free systems and at the same time sharing some important "LTS-like" advantages such as low creep rates, low anisotropy, and the absence of weaklinks problems. The influence of thermal fluctuations in the vortex physics is measured by the Ginzburg number where H c is the thermodynamic critical field and e < 1 is the effective mass anisotropy. For YBa 2 Cu 3 O 7Àd this is as large as Gi $ 10 À2 and even larger for the more anisotropic Bi-based compounds, while for NbTi, the paradigmatic strong-pinning conventional superconductor, Gi $ 10 À8 . For MgB 2 , depending on the doping level we have Gi $ 10 À4 À 10 À5 . This is just in the middle between the extreme cases. The first consequence is that J c is expected to be in the intermediate range J c0 À J c Շ J c0 and the A-K limit would be only valid at low temperature.
In this paper we present an experimental study in bulk MgB 2 samples, pure and synthesized with different amount of carbon nanotubes, under the A-K approximation. The absence of week links and the relatively low anisotropy allow us to consider the system under study as a bulk 3D superconductor with orientation-averaged J c and U c . A creep analysis was performed considering these "mean" isotropic variables. In Sec. II experimental details are described. In Sec. III the results are presented and discussed in Sec. IV. In Sec. V we present an analysis that allows us to identify the region where the A-K model is valid. Conclusions are summarized in Sec. VI.

II. EXPERIMENTAL
MgB 2 samples used were prepared by solid-state reaction with magnesium (À325 mesh, 99%), amorphous boron (99%), and double-walled carbon nanotubes (DWCNT; diameter 1.3-5 nm, length 50 lm, 90%, Aldrich) as starting materials. 8 The powders were ground inside a glove box and pressed under $500 MPa into small pellets with dimensions of 6 mm in diameter and $4 mm in thickness, wrapped together with extra 20 at % Mg turnings (99.98% Puratronic) in Ta foil and then placed in an alumina crucible inside a tube furnace in flowing Ar=H 2 at 900 C for 30 min.
Samples with 0%, 1%, 2.5%, 5%, 7.5%, and 10% of CNT addition have been characterized. AC susceptibility sample characterization was carried out in a 9 T Quantum Design Physical Properties Measurement System (PPMS). The first harmonics of the AC susceptibility were normalized in such a way that the in-phase components v 0 vary between -1 (ac field fully screened) and 0 (ac full penetration).
The magnetization measurements were carried out in a Quantum Design Magnetic Properties Measurement System (MPMS) model MPMS XL 7T SQUID based magnetometer. A scan length of 3 cm was used in order to minimize the effects due to the nonuniformity of the applied magnetic field (H), which was applied parallel to the longest axis of the sample. Isothermal magnetization loops M(H) were measured to calculate J c using the critical state model. Timedependent data were taken with a protocol similar to that described in Ref. 18. For each relaxation measurement, the samples were first cooled and stabilized at the measurement temperature. Then the field was first raised up to a high field (5-6 T) and then lowered to the measuring field to assure that the sample was fully penetrated. Additional measurements were performed in the upper and lower magnetization branches to subtract the reversible magnetization and any background signal. We checked for and ruled out any effects due to the superconducting magnet relaxations (H variations during the measurement time) that could lead to spurious changes in the magnetization of the samples.

III. RESULTS
Samples have been characterized by AC susceptibility and magnetization measurements. In Figure 1, as an example, the first harmonic of the ac susceptibility is plotted as a function of temperature for a pure sample and for a sample with 10% addition of CNTs. These measurements were performed with a fixed AC amplitude h a at various DC fields H ( Fig. 1(a)) and at different AC fields ( Fig. 1(b)). Even if there is a clear decrease in T c with doping, the transitions at H ¼ 0 are sharp in all the samples, indicating homogeneous superconducting properties in the macroscopic scale. No structure in the transition is observed when increasing both the AC and the DC field, indicating that the grain boundaries do not significantly destroy the superconducting properties in these ceramic samples. This fact allows us, in principle, to apply the theory, and data analysis developed to study vortex dynamics in homogeneous superconductors such as single crystals having in mind that, in the present case, grain with different orientations will surely have different properties and that our results are valid for mean properties in the macroscopic scale.
The critical current density J c ðtÞ has been calculated from the measured magnetic moment using the relationship for a sample with rectangular cross section in the critical state following the Bean model. 19  and the strong dependence of J c ðBÞ produce an asymmetric M(H) loop with higher magnetization values in the lower branch. As has been suggested in the earlier work of Kim, 20 the average magnetic field in the upper and lower branches can be estimated by hBi ' l 0 H 6 MðHÞ=2 and a quasisymmetric loop MðhBiÞ can be obtained. For fields l 0 H $ 1T and higher, where the creep studies are performed, the difference between hBi and l 0 H in all the samples becomes negligible; therefore, the relaxation study and sample comparison can be done at fixed H.
As discussed in previous works, [7][8][9]24 CNT additions are capable of H c2 and J c simultaneous improvement. On one hand, they partially dilute into the MgB 2 matrix, acting as a source of C that increases H c2 . On the other hand, the remaining CNT's increase the amount of pinning centers, being the optimum CNT amount temperature and field dependent. In the range of CNT additions between 1% and 10%, the "measured" critical current density at temperatures far below T c is highest and quite similar for different amounts of CNTs. For x > 0:10%, J c decreases due to both a larger decrease in T c and a probable deterioration of interconnectivity between grains denoted by higher resistivity values. Analogous results had been reported by Dou et al. 21 This fact is illustrated in Figure 2(a), where the experimental J c obtained in samples with different densities of CNT, is plotted as a function of l 0 H at T ¼ 5 K and 20 K. It can be observed that the response for samples with 1% (light gray triangles) and 10% (gray squares) addition of CNT is very similar and robust. On the other hand, pure MgB 2 samples are not so robust, and after large periods of time (several month or years) the critical current density can slightly decrease. At low temperature (5 K), CNT addition enhances J c for all the measured range of H while at higher temperature (20 K) the response of pure and CNT samples is quite similar at low H (up to l 0 H ¼ 2 T).
The magnetic relaxation has been measured in pure MgB 2 and samples with CNT contents between 1% and 10% in the range between 5 K and 25 K at l 0 H ¼ 1 T. The corresponding relaxation rate S has been obtained as the slope of lnJ vs lnt. An example of the fitting procedure is shown in the inset of Figure 3(a). Results are shown in the main panel of Figure 3(a) (full symbols). Samples with 2.5% and 5% of carbon nanotubes additions have also been measured and display similar results, not shown in the figure for clarity.
The first, expected result is that, as was described in the introduction, the S values are intermediate between those measured in low and high T c materials. Regarding the effect of CNT addition, the S(T) curves display approximately the same behavior for all the samples with CNTs. However, as an unexpected result, it can be observed that at l 0 H ¼ 1T pure samples (circles and triangles in Figure 3(a)) that have a lower J c (associated with a lower pinning) have also a lower relaxation (associated with a higher pinning energy U c ).
The behavior of the relaxation S as a function of the applied magnetic field at fixed temperature T ¼ 10 K is shown in Figure 2(b). The relaxation increases with CNTs addition at low fields but decreases notably at the higher   Figure 3(a). In Figure 3(b) J c for pure and 10% CNT sample is also plotted as a function of temperature at l 0 H ¼ 1 T and 3 T. CNT addition enhances J c at both fields bellow T ¼ 15 K.

IV. DISCUSSION
The increase of S upon introduction of additional defects to increase pinning is not typically observed in other superconductors. For instance, irradiation of iron-based superconductors such as BaðFeCoÞ 2 As 2 produces increases in J c but the relaxation rates either decrease 25 or remain constant. 26 On the other hand, YBa 2 Cu 2 O 7 (YBCO) samples with vastly different J c values show very similar relaxation S(T). One exception is the case of YBCO crystals with columnar defects where, together with the increase in J c , a "peak" in S(T) occurs. 15,18 However, that increase is observed over limited T and H ranges. The increase in S(T,H) that we observe in our MgB 2 samples is a more general effect, taking place over a much broader T-H range. To get an understanding of the causes of the puzzle presented in the above section, the relaxation data in a pure and a 10% CNT samples were analyzed and compared.
The first consideration is that with the moderate relaxation rates measured, the strong temperature dependence J c ðTÞ observed in Figure 3(b) cannot be explained assuming a T independent J c0 at low temperature. Thus, an intrinsic J c0 ðTÞ dependence must be taken into account, and the measured J c would be similar to the true critical current density J c0 ðTÞ: Therefore, experimental data were analyzed at each temperature assuming the validity of the A-K approximation J c $ J c0 . A novel procedure that is presented in Sec. V has been developed to check the consistence of this assumption.
In the general case, 13 the relaxation rate S ¼ ÀdðlnðJÞÞ=dðlnðtÞÞ is related to the pinning energy by where t ) t 0 and ln t 0 $ lnðs 0 kT=J c0 j@J c =@tjÞ is approximated as a constant. This expression should be valid in all the current density range, including the A-K regime. 17 At low temperature U c =kT ) 1 and, as a good approximation, In Figure 4 the resulting pinning energies obtained from Eq. (4) are plotted as a function of temperature for pure and 10% CNT samples, at l 0 H ¼ 1 T and 3 T. In the dashed regions marked in Figures 3(b) and 4, the results are not consistent with the A-K approximation. In favor of a continuity in the argument, let for the time being accept this claim as a fact; it will be justified in Sec. V. Figure 4 shows that, in the regions where the A-K approximation is valid, pinning energies remain nearly constant. Consistent with the relaxation rates, at l 0 H ¼ 1 T, there is a clear decrease of U c after CNT addition, whereas at l 0 H ¼ 3 T the pinning energies match up. The field dependence of the pinning energies is displayed in the inset, showing that the difference between 0% and 10% CNT samples decreases with magnetic field up to B $ 3 T.
In the following we present a discussion and a possible picture to explain our experimental results. The parameter U c is associated with the pinning energy of a pinning volume that is determined by the competition between pinning and elastic energies. In a single vortex pinning regime the pinning length L c is defined by the balance between the pinning and the line elastic energy of a vortex segment. 13 The Lorentz force over a vortex segment of length L c is F L $ ð1=cÞ/ 0 J L c . On the other hand, in a pinning regime of vortex bundles, the pinning volume V c $ L c R 2 c involves the competition between pinning forces and lattice elastic moduli; in that case R c is the distance where the autocorrelation function of the lattice displacement exceeds the range of the pinning potential r p (Ref. 22) and the Lorentz force over a volume V c is F L $ ð1=cÞBJ V c . The B independence of pinning energies and critical currents has been often considered as the fingerprint of single vortex pinning. Obviously if pinning energies and critical currents were field independent we would be in the presence of a single vortex pinning regime. However, the inverse is not necessarily true: if pinning is strong enough the correlation volume could be small enough to involve only one vortex, but if the vortex distance is smaller than the penetration depth the effective pinning force acting over a vortex segment could be modified by the neighboring vortices. Therefore, a possible way to investigate this issue is by estimating the correlation volumes and lengths involved.
In a rough estimate, the pinning force is F p $ U c =r p . When J reaches the critical current density J c0 , the pinning where r p is generally a fraction of the lattice constant (r p $ a 0 =4) and in the limit of a dilute vortex lattice reaches the coherence length nðTÞ.
In previous work, 7 we have presented H c2 measurements in MgB 2 with different amounts of CNTs, and we have found that H c2 ðTÞ fits very well the function proposed in Ref. 23. We have now estimated nðTÞ from H c2 ðTÞ for the as grown samples and for samples with 10%CNT. Again, both H c2 and n are related with mean values, averaged over the various grain orientations. Using this nðTÞ dependence (inset in Figure 5), and considering J c0 ¼ J c we have estimated L c ðT; BÞ from our data using Eq. (5). Results are shown in the main panel of Figure 5, where L c is plotted as a function of temperature at l 0 H ¼ 1 T and 3 T for both samples. Error bars take into account the possible difference between J c and J c0 (we will admit a 10% difference to validate the A-K regime). Horizontal dashed lines indicate the spread of the reported grain size L g . (150 nm < L g < 800 nm in virgin samples and 100 nm < L g < 500 nm in samples with CNTs). 24 It can be seen that if a single vortex pinning regime were present, the correlation length would be similar to grain size, slightly smaller in the samples with CNTs. L c increases with temperature and magnetic field, until it exceeds the grain size, situation that striking matches with the limit of the validity of the A-K regime. Note that L c is estimated solely from measurements of critical fields and vortex pinning and, in principle, could take any value between n to 1 (in practice, the sample size). The coincidence of L c with the grain sizes is remarkable, strongly suggesting that the pinning domains are constrained to single grains.
On the other hand, under the supposition of a pinning regime of vortex bundles and assuming r p $ a 0 =4, V c can be estimated from Eq. (6). Results are shown in Figure 6(a) and compared with the grain size and the volume of a cube of side a 0 . In this picture V c should be smaller in samples with CNTs and decreases at higher fields, contrary to the expectation in the theory of vortex bundles in the presence of random pinning centers. 13 That is not surprising because in our samples strong pinning centers are present, and grain boundaries probably influences the correlation volume. Grains are however larger than the estimated correlation volumes. Therefore, let us assume that the grain size limits only the longitudinal correlation length L c . The correlation volume can be expressed in term of the transverse and longitudinal correlation lengths as V c $ L c R 2 c . The area of a the correlation volume R 2 c can be estimated by considering L c $ L g . The estimated correlation areas are shown in Figure 6(b) for both samples as a function of temperature and compared with the area occupied by a single vortex at l 0 H ¼ 1 T and 3 T. Error bars take into account the B distribution inside the sample (DB $ pM) and the spread in L g . The resulting areas are in all the cases smaller than the region occupied by one vortex and, therefore, inconsistent with a vortex bundle pinning regime. Again, once the estimated area becomes larger than a 2 0 results are inconsistent with the A-K approximation (dashed region; see Sec. V).  From the above results, a pinning regime of small isotropic bundles containing a few vortices, in which the correlation volume anomalously decreases with B (Figure 6(a)), cannot be discarded. However, the most consistent picture indicates that in the region of small creep, where the A-K approximation is valid, a single vortex pinning regime with correlation length limited by the grain size ( Figure 5) holds. As vortices are close enough, vortex-vortex interactions are not negligible, and the effective pinning energy U c diminishes with B. This effect is stronger in pure samples. Because CNT addition reduces the mean grain size, the pinning correlation length is reduced in these samples. However, CNT addition not only decreases the grain size and correlation length but also reduces the coherence length n. This fact is probably the main responsible for the increase in the critical current density.

V. A-K MODEL CONSISTENCE
In this section we present a method to check the consistence of the A-K approximation. The procedure is applied to our results and allows us to discard, within an experimental criterion, data where the A-K model is not consistent (dashed regions in Figures 3, 4, and 6).
In the A-K condition J c0 À J c ( J c0 , the function U(J) can be linearized to the first order in ðJ c0 À J c Þ to give the expression (2). The integration of Eq. (1) using Eq. (2) gives the well known logarithmic decay of the experimental critical current density with time. A mayor difficulty to directly decide the validity of the A-K approximation from the relaxation data at a single temperature is the extremely large time needed to reliably determine this logarithmic decay.
An alternative method, similar to that developed by Maley 27 for the case of giant creep, consists of taking the logarithmic expression of Eq. (1) together with the A-K expression for the activation barrier (Eq. (2)), to obtain a linear dependence between J c and @J c =@t that results in Due to the numerical differentiation, this method has a greater error in the calculation of the pinning energy but allows estimating J c0 ðT; BÞ. The resulting fitting parameters must be consistent with the A-K conditions and with those obtained by using Eq. (4). If these two conditions are not fulfilled some of the assumptions will be wrong, and the A-K analysis should not be valid. At each temperature J c ðtÞ has been plotted as a function lnðÀ@J=@tÞ (an example is shown the inset of Figure 7). By estimating ln C j , the true J c0 ðTÞ can be extracted from the intercept using Eq. (8). The easier assumption, using the fact that ln C j ( kT=U c , is to neglect ln C j . This procedure allows a reasonable estimation of J c0 ðTÞ but will underestimate the small difference ðJ c0 À J c ÞðTÞ. It is possible to roughly estimate C j from the expression in Ref. 13 (p. 1350), C j $ J c =s 0 with s 0 $ pd 2 =2c 2 H 2 q FF , being d the characteristic sample dimension, c the speed of light, q FF the flux flow resistivity, and H expressed in Oe. This gives s 0 $ 10 À4 s for our experimental situation, although this value could drop to 10 À6 s, depending on the reported resistivity.
In Figure 7 the estimated true critical current density J c0 is compared with the measured critical current density J c (the first point measured in the J c ðtÞ relaxation) by plotting ðJ c0 À J c Þ=J c0 as a function of temperature for a pure sample (sample A in Figure 3) and a 10% CNT sample, at l 0 H ¼ 1 T and 3 T. Error bars take into account the uncertainty in C j (from ln C j ¼ 0 to lnðC j $ J c ½A=cm 2 10 6 Þ discussed in the above paragraph. Taking a 15% criterion to determine the limit of the A-K condition ðJ c0 À J c Þ ( J c0 , the resulting J c0 ðtÞ is consistent below the temperature where each experimental curve crosses the doted horizontal line. From the slope A ¼ kTJ c0 =U c in Eq. (8) the pinning energy U c =k ¼ AJ c0 T can be extracted and compared with T/S for these samples. Results are shown with asterisk and stars in Figure 4. As can be observed (with exception of the lower temperatures data in the pure sample at l 0 H ¼ 1 T) pinning energies obtained from both methods are very similar, and this validates the fitting procedures, including the estimation of J c0 ðTÞ from Eq. (8).
The unexpected drop in U c at the lower temperatures data in the pure sample at l 0 H ¼ 1 T, together with the inconsistence of the analyzed data, could imply the presence of another relaxation mechanism, perhaps associated with macroscopic flux jumps.

VI. CONCLUSIONS
Magnetic relaxation in bulk MgB 2 samples with and without CNT additions has been measured at various temperatures and magnetic fields, and the corresponding mean critical current density has been calculated. The current decay in time is quite similar in all the samples with CNT amounts between 1% and 10%.
A detailed creep analysis has been carried out in a pure sample and in one with 10% of carbon nanotubes. The analysis has been performed under the Anderson-Kim approximation, valid in the limit where the measured critical current density J c is similar to the "true" one J c0 that would be present in the absence of creep phenomena. In these samples, even at the lower temperatures, it is necessary to include an explicit field and temperature dependence for true critical current densities J c0 ðT; BÞ.
Mean pinning energies have been calculated. By estimating approximately the volumes and/or correlated pinning lengths we conclude that the most consistent picture at low temperatures is a single vortex pinning regime, where the correlation length is limited by the grain size. Despite that, pinning energies decreases with magnetic field. This fact is not surprising because vortex densities are high enough to produce an effective pinning potential influenced by vortex interaction.
At low magnetic fields, CNT addition produces a decrease in the pinning energies (which increases the relaxation rates). However, the critical current density increases.
We conclude that the puzzle solution to explain the changes in the pinning properties with CNT addition is the combination of the reduction of the grain size (that decreases pinning energies) and coherence length (that increases the critical current density).
The region of validity of the A-K approximation has been tested by obtaining the pinning parameters with an alternative method. The consistence of the fitting parameters with the A-K limit has been required. The coincidence between the region where the A-K approximation is valid and that one where the correlation length is smaller than the grain size (see Figure 6) suggest that probably the grain boundaries modify the creep mechanism and pinning regime. A method to analyze the relaxation data in the high temperature region of compounds with intermediate creep, beyond the A-K approximation, as well as a model to describe pinning and relaxation in the presence of grain boundaries would be suitable to understand relaxation processes in the whole phase diagram of these materials.