Since the discovery of superconductivity at 39 K in MgB2 (Nagamatsu et al., 2001), considerable progress has been made in the understanding of the fundamental properties and the development of commercial applications of this material.
The strong potential for technological uses of MgB2 is due to a unique combination of characteristics, such as a high transition temperature Tc ~ 39K, chemical simplicity, lightweight and low cost of the raw materials (Buzea et al., 2001). In addition, the absence of weak-link behavior at grain boundaries in polycrystalline samples (Larbalestier et al., 2001) allows the use of simple Powder in Tube (PIT) methods to fabricate wires and tapes (Flükiger et al., 2003). One of the most important issues for MgB2 magnet applications is the simultaneous enhancement of its critical current density (Jc) and the upper critical field (Hc2).
Thus, on one hand, the pinning force may be improved by the incorporation of defects (nano particle doping, chemical substitutions, etc.). On the other hand, the doping level affects the intraband scattering coefficients and the diffusivity of the two bands of this peculiar superconductor, and these changes may cause a significant Hc2 variation. Carbon or C-compounds additions have been very successful to improve Jc and/or Hc2, and the effects of carbon doping on superconductivity in MgB2 has been extensively studied. The Jc-H performance can be greatly improved by adding different carbon sources, such as carbon doped MgB2 filaments (Wilke et al., 2004, 2005a), nanocarbon (Soltanian et al., 2003; Ma et al., 2006; Yeoh et al., 2006; Häβler et al., 2008), amorphous carbon (Senkowicz et al., 2005), diamond (Cheng et al., 2003), B4C (Wilke et al., 2005b; Ueda et al., 2005; Yamamoto et al., 2005a, 2005b), carbon nanohorn (Ban et al., 2005) and, particularly, carbon nanotubes (Dou et al., 2003; Yeoh et al., 2004, 2005, 2006; Serquis et al., 2007; Serrano et al., 2008, 2009); Shekhar, 2007; Vajpayee et al., 2010; etc).
In this chapter we present a review of recent developments in the study of the effect of carbon nanotubes (CNT) on the superconducting properties of MgB2 bulk and wire samples, based on the known literature data and our own results.
It is well known that pinning of vortex lines to defects in superconductors plays an extremely important role in determining their properties. CNT inclusions, with diameters close to the MgB2 coherence length (ab(0) 3.7-12nm; c(0) 1.6-3.6 nm, depending on the doping level, as discussed below in section 2.1.2) (Buzea et al., 2001) may be very good candidates for vortex pinning if they do not completely dissolve in the matrix but remain as tubes acting as columnar defects. Therefore, the presence of CNT may result in a critical current density improvement under applied fields that will depend on the synthesis parameters. Besides, theoretical models predict that the presence of two superconducting gaps could allow tuning of different upper critical fields by controlling diverse defect sublattices relative to orthogonal hybrid bands (Golubov et al., 2002; Gurevich, 2003). These models predict a significant Hc2 enhancement in the dirty limit and an anomalous Hc2(T) upward curvature. Several reports indicate that Hc2 can be significantly increased by introducing disorder through oxygen alloying, carbon doping or He-ion irradiation (Brinkmann et al., 2002; Gurevich et al., 2004; Putti et al., 2004; Braccini et al., 2005). Thus, it is important to analyze the effect of carbon-doping through CNT or other C-sources to the Hc2(T) in bulk MgB2 samples, where extrapolated values of Hc2(0) between 29 and 44 T have been reported (Wilke et al., 2004; Senkowicz et al., 2005; Serquis et al., 2007; Serrano et al., 2008).
In section 2 of this work, we describe the influence of C-incorporation into the MgB2 structure by using different CNT type over the microstructural and superconducting properties (critical current density, critical fields and critical temperature). In particular. we present the influence of synthesis parameters in the superconductivity of bulk samples prepared with single-walled (SW), double-walled (DW) and multi-walled (MW). In the first subsection, we compare the evolution of Tc, and the lattice parameters with the amount of CNT, analyzing if C is replacing B in the Mg(B1-xCx)2 structure or remains as CNT in each case. In the following subsections, we present a review of Jc determined by magnetization, and Hc2(T) by transport measurements, which may require measurements performed using high fields (i.e. 50 T) pulsed magnets. In the last subsection, we also present an analysis of the distinctive effect of C addition in samples with optimum contents of single-walled and double-walled carbon nanotubes SiC, that simultaneously increase Jc and Hc2.
In section 3, we offer a review of the use of CNT in the production of PIT MgB2 wires and tapes required for applications. The fabrication and processing conditions strongly affect microstructures and current carrying capability of PIT conductors, making the question of grain connectivity more relevant. The standard and low-cost fabrication PIT method involves filling a metallic tube with superconducting powder (ex-situ) or precursors (in-situ) and drawing it into a wire and/or rolling into a tape (Flükiger et al., 2003). We show the results obtained in MgB2 wires and tapes prepared by PIT using different kind of CNT and treated at different temperatures between 600 and 900 °C, using several sheath materials (e.g. iron, stainless steel, Fe/Nb, etc), to establish a relationship among annealing conditions, modifications in the microstructure, and changes in Jc.
In Section 4 we go deeper in the role of doping by studying the magnetic relaxation of MgB2 with and without DWCNT bulk samples. In the first subsection, we introduce the main equations contained in a very recent work (Pasquini et al., 2011) necessary to investigate the current decay, that relates parameters such as the magnetic field B, the magnetization m, current density J flowing in the sample, and the activation barrier U(J,T,B) that governs the creep of vortices. In the rest of this section we present the experimental relaxation rates (measured using a DC magnetization technique), and the general behaviour is described and analyzed under the Anderson-Kim frame model (Anderson, 1964). The pinning energies Uc, true critical current densities Jc0, and correlation volumes Vc are estimated and compared.
Finally, we summarize the main conclusions in section 6.
2. Effect of different kind of CNT addition to superconducting properties of MgB2 bulk samples
There are many reports in the literature that study the effect of CNT on the superconducting properties of MgB2. However, many of them mentioned the effect of CNT-doping without clarifying the meaning of this word. Doping is generally the practice of adding impurities to something, but it specifically means the replacement of one element within the same crystalline structure, creating a substitutional point defect. In the case of CNT additions to MgB2 it is possible that part of the carbon replaces boron giving place to the compound Mg(B1-xCx)2. As a consequence, in many works, the nominal amount of CNTs added during the synthesis is calculated as the stoichiometric amount according to this formula or, in other cases as some extra %at or %wt of carbon added to MgB2. In the later case, if part of carbon replaces boron, some extra magnesium should be added that may also evaporate or form MgO during the synthesis process. In this chapter, unless it is specified, we will use “x” as the nominal amount of carbon added according to Mg(B1-xCx)2, and the term “doping” when carbon is actually replacing boron.
2.1. Carbon doping and critical temperature (Tc)
Several systematic carbon doping studies of Mg(B1-xCx)2 have been performed in single crystals (Lee et al., 2003; Kazakov et al., 2005), polycrystalline wires fabricated by chemical vapor deposition (CVD) (Wilke et al., 2004, 2005a), and B4C-doped (Avdeev et al., 2003; Wilke et al., 2005b). All these studies indicate a monotonic decrease of Tc and the lattice parameter a for increasing x, while the lattice parameter c remains constant.
For single crystals, the solubility limit of C in MgB2 was estimated to be about 15 ± 1%, which is substantially larger than that reported for the polycrystalline samples. A dramatic decrease in Tc was also observed with C-substitution, followed by complete suppression of superconductivity for x>0.125. This solubility limit is usually not reached in bulk samples. A comparison of the critical temperature as a function of the nominal C-content for different carbon sources is displayed in Figure 1.
The reduction in Tc resulting from C addition using CNT sources is much lower than in the previous cases, suggesting that C substitution is lower than the concentration given by Mg(B1-xCx)2. The Tc decrease is even lower when using SWCNT as a source of carbon indicating a very low doping level.
As an example, it is possible to calculate the actual C content substituting B into the MgB2 structure, using the lattice parameters obtained from XRD: the shift in the a-axis lattice parameter can be used as a calibration for the actual amount of C (x) in the Mg(B1-xCx)2 structure in comparison with the fitting of neutron diffraction data (Avdeev et al., 2003) or single crystal data (Kazakov et al., 2005). The dependence of a-axis lattice parameter as a function of the nominal DWCNT %at content is plotted in Figure 2. The corrected x values obtained were used to plot the same data as open squares in Figure 1, indicating that not all C is incorporated into the MgB2 structure. A similar behavior is observed for all CNT samples, indicating that actual C-substitution tends to saturate for nominal CNT contents larger than 10%at. The actual C-doping varies according to the CNT type, synthesis temperature or other synthesis parameters (i.e. pressure or magnetic field). This C-substitution level determines not only the Tc but other superconducting properties such as Hc2, as will be described in section 2.2. Nevertheless, we will continue using the nominal content as the parameter to describe most effects to make it easier the comparison with other author’s results.
It is important to mention that not only the nominal carbon content x but also other synthesis parameters affect the superconducting properties like Tc due to the amount of C substituting B in the structure. All parameters studied up to now are listed at the beginning of next section.
2.2. Simultaneous enhancement of critical currents and fields
2.2.1. Critical currents (Jc)
Since the first work devoted to the study of the effect of the addition of CNT on the critical current density of MgB2 (Dou et al., 2003), several parameters have been studied:
the effect of sintering conditions such as temperature (Yeoh et al., 2004), pressure (Yuan et al., 2005), ultrasonication of precursors (Yeoh et al., 2006) or the application of magnetic field during sintering (Li et al., 2007)
Most researchers performed magnetization loops measurements to determine Jc(H) in bulk samples using the Bean critical state model (Bean, 1962) assuming full penetrated samples for a long parallelepiped:
where a and b are the lengths of the parallelepiped edges perpendicular to the magnetic field and ΔM is the width of the magnetization characteristic at the applied magnetic field H.
Figure 3 shows the dependence of Jc with the applied magnetic field at 5 and 20 K for samples prepared with different amount of single (SW) (Serrano et al., 2009), double (DW) (Serquis et al., 2007) and multi-walled (MW) (Shekhar et al., 2007) CNT (see inset for the later case). Some flux jumps were frequently reported at 5 K in regions up to 2 T, and these data were not included in the corresponding figures. It can be observed that in all cases the addition of CNT progressively improves the overall Jc(H) performance, reaching an optimum for the x=0.10 sample, where Jc is higher than for any other composition in all the reported temperatures and fields. When the amount of added CNT is over 10%at the performance deteriorates. However, many other authors also reported that the optimum addition is around 10%at, although in some cases the critical current densities are not higher than for other compositions in the whole range of temperatures or applied fields.
Figure 4 displays the dependence of Jc with different amount of MWCNT (Dou et al., 2003), DWCNT (Serquis et al., 2007) and SWCNT (Vajpayee et al., 2010) added to MgB2 samples under an applied field of 4 T and temperatures of 5 and 20 K. It is interesting to note that the critical current densities are very similar between x = 0,025 and x = 0,10 for each type of sample, but the Jc improvement tends to be more notorious for x = 0,10 at higher temperatures and fields. The best composition for a certain field and temperature may also change with the sintering temperature. Samples included in Figure 4 were prepared at 800, 900 and 850 °C for MW, DW and SW, respectively. A study about the dependence with sintering temperature (Ts) in MgB2 samples with MW CNT (Yeoh et al., 2004) indicated that when the Ts is 900 °C or higher, the nanotubes tend to dissolve and are incorporated into the MgB2 matrix, decreasing Tc but increasing the irreversibility field (Hirr). This effect will be discussed in the next subsection, since it is correlated with the Hc2 improvement. Therefore, the larger the amount of C-doping the smaller is the Jc(H) slope improving the performance at higher fields. On the contrary, the Jc decrease for x > 0,10 may be due to both an even larger decrease in Tc and a probable deterioration of interconnectivity between grains. This deterioration of grain connectivity was denoted by a large normal resistivity value (i.e. ρ(40K) ~ 200 μΩcm) for the DW sample with x = 0,125.
Figure 5 shows the dependence of Jc with the applied magnetic field of samples prepared with the same amount of single (SW), double (DW) and multi-walled (MW) CNT.
For comparing the absolute values it is important to take into account the pure MgB2 sample used as reference since this can be a good parameter to check the influence of variables other than composition, such as connectivity, that may result from the sintering process. Consequently, the best performances (i.e. Jc values and Jc(H) dependences) are obtained for MW (diameter 20-30 nm) (Yeoh et al., 2005) and DW (diameter 1.3–5 nm) (Serquis et al., 2007). These two samples (both heat treated in flowing high purity Ar at 900 for 30 minutes) presented the largest C-doping (smallest a lattice parameter) and Yeoh et al. suggested that it is not the diameter but the MWCNT length the responsible for allowing a more homogeneous C-incorporation by avoiding the nanotubes agglomeration. This explanation may not apply to DW CNTs, which have a length ≤ 50 μm. However, it is highly probable that synthesis parameters that promote a more homogeneous CNT distribution and improve connectivity (avoiding agglomerates at grain boundaries) are the key for Jc performance.
2.2.2. Upper critical fields (Hc2)
The Hc2(T) dependences are usually determined from four probe transport measurements. Since the involved fields are frequently very high for standard laboratory test equipments, several groups performed measurements at large High Magnetic Field facilities. As an example, measurements of DW and SW CNT samples were performed in the mid-pulse magnet of NHMFL-LANL, capable of generating an asymmetric field pulse up to 50 T. Figure 6 exhibits the temperature dependences of Hc2 and Hirr, defined as the onset (extrapolation of the maximum slope up to the normal state resistivity) and the beginning of the dissipation, respectively, of the R versus H data for samples with several DWCNT contents, at temperatures between 1.4 and 34 K. The criteria to define Hc2 and Hirr are exemplified in the inset of figure 6.
In all superconducting materials, an Hc2 enhancement is expected when disorder is increased (e.g., by doping). The reason is that the reduction in the electronic mean free path produces a decrease of the effective coherence length, driving the superconductor to the “dirty limit”. This effect is especially large in C-doped MgB2, and additionally an anomalous Hc2(T) upward curvature is also noted in the present data and reported many times. This positive curvature is related to the existence of two-gaps in MgB2, (π-band and σ-band) and predicted by some theoretical models.
These models for a two gap superconductor in the dirty limit (Golubov et al., 2002; Gurevich, 2003) consider that nonmagnetic impurities affect the intraband electron diffusivities D and D, and the interband scattering rates and . An implicit Hc2 dependence on temperature can be obtained from the Usadel equations, derived taking into account only the intraband effect.
where, is the reduced temperature -,
a0, a1, and a2 were determined in ref. (Gurevich, 2003).
Another equation was derived when taking also into account the interband effects:
In this case we optimized the diffusivity ratio =D / D and interband scattering parameter to fit the measurements using the equation (Eq. 3).
Figure 7 shows the dependence of Hc2 as a function of the reduced temperature t = T/Tc0, where Tc0 = 39 K, of some selected samples with and without CNT addition. Other C-doped
The upward curvature signaled as a characteristic of the presence of two gaps is apparent in these Hc2(T) data.
Figure 8 displays the Hc2(0) extrapolations as a function of x for DW (Serquis et al. 2007) and SW (Serrano et al. 2007) in comparison with data for other C-doped samples (Wilke et al., 2004). We observe that Hc2(0) increases with x and has a maximum for 10 at% for DW and 5 at% for SW. For a larger x values a decrease in Hc2 was found, probably due to a larger Tc decrease and an increase in the resistivity of the samples. Earlier MgB2 carbon doped data (Wilke et al. 2005) also indicates an initial rapid rise for lower C contents that then slows down, reaches a maximum at intermediate carbon concentrations, and decreases for larger C contents and the same behavior was reported for MgB2 single crystals.
The CNT additions produce a larger C incorporation than SiC, probably because of the higher synthesis temperature, resulting in samples with lower Tc. The DWCNT 10 at% sample has the highest C content into the lattice, indicating that using this kind of inclusions is an easier way to incorporate C. This allows to reach a record Hc2 value for this sample. Earlier MgB2 carbon doped data from Wilke et al (Wilke et al., 2005b) also indicates an initial rapid rise for lower C contents that then slows down, reaches a maximum at intermediate carbon concentrations, and decreases for larger C contents and the same behavior was reported for MgB2 single crystals. The Hc2(0) of CNTdw10 is close to the maximum Hc2 value as a function of x. A decrease in Hc2(0) was also observed for a larger x value (see Fig. 8)), in agreement with other reported data (Senkowicz et al., 2007).
3. Effect of different kind of CNT addition on the superconducting properties of MgB2 PIT wires and tapes
The standard and low-cost fabrication powder in-tube (PIT) method involves filling a metallic tube with superconducting powder (ex-situ) or precursors (in-situ) and drawing it into a wire and/or rolling into a tape (Flükiger et al., 2004).
MgB2 crystallizes in the hexagonal AlB2 type structure (space group P6 /mmm), and the anisotropic structure has given the motivation to investigate formation of texture by different deformation processes. A key issue on which there is no agreement in the literature is the optimization of the heat treatment parameters. Although some post-annealing appears to be necessary to achieve higher Jc, some authors reported detrimental effects of heat treatments in the performance of MgB2 wires or tapes (Serquis et al., 2003; Goldacker, 2003; Civale et al., 2003). However, most results obtained in MgB2 wires and tapes prepared by PIT using different kind of CNT and treated at different temperatures using several sheath materials (e.g. iron, stainless steel, Nb/Fe) indicate that the final sintering temperature is very important to improve the superconducting properties allowing carbon to be incorporated in MgB2.
Only one work (Xu et al., 2007) studied the influence of CNT amount, reporting the effect of the “doping level” (x) in the field dependence of critical current density for MWCNT Fe-sheathed MgB2 wires and tapes. Similarly to what was reported for bulk MgB2 samples, they found that there is an optimum composition for all fields and temperatures studied, but the best composition for the nominal Mg(B1-xCx)2 was x=0.05 and Jc decreased for x = 0.10 (see Figure 9). However, many other researchers focused in which was considered the best composition x=0.10 studying the influence of other parameters:
the effect of CNT type (Kováč et al., 2007) in comparison with other C-compounds (SiC, graphite)
the effect of sintering temperature when using SW CNT (Kim et al., 2007)
the possibility of MW CNT alignment by mechanical drawing in the PIT process (Dou et al., 2006).
To illustrate the effect of some of the mentioned parameters we plotted in Figure 10 the critical current density of
Left side: MgB2 /Fe-Nb tapes with 5%wt addition of different CNT types (Kováč et al., 2007). SW: Arc discharge single wall nanotubes; SW-D: Purified arc discharge single wall nanotubes, Dry-mixed; SW-W: Purified arc discharge single wall nanotubes, Wet-mixed; MW: Multi wall carbon nanotubes (~60 nm × 1 μm). For comparison were included the pure (0) and a MgB2 tapes with 5%wt addition of SiC. This last addition is more effective in the whole field range in agreement with bulk results, while all CNT types only enhance Jc for high fields. The normalized current densities (shown in Figure 11) allow better comparison between the slopes of Jc(μ0H), indicating a positive effect for all additions. This field dependence enhancement is ascribed to the effect of C-incorporation, which increases the upper critical field. The decrease of Tc corresponding to a higher C-doping effect can be observed in the inset of Figure 9.
Right side: MgB2/Fe wires with aligned MW CNT measured along (a) and perpendicular (c) to the wire (Dou et al., 2006). While the pure sample has little differences between both measurement field directions, there is a clear difference in samples with MW addition, which increased with field. This is a good indication that CNTs could be aligned along the longitudinal axis of the MgB2 wires by mechanical drawing in the PIT process.
Matsumoto (Matsumoto et al., 2006) reported that Jc in SiC-alloyed MgB2 tapes depends on a complex relation between grain connectivity, Hc2, and flux pinning induced by grain boundaries and precipitates. However, the distinct effect of C incorporation through different routes in Jc and Hc2 is still not entirely understood.
Matsumoto et al reported a similar Hc2(0) to our record value for a SiC-doped MgB2 tape prepared by the PIT method. These results correspond to a PIT tape sample that probably has some texturing and the reported Hc2 data was measured with the applied field parallel to the tape. Since we measured randomly oriented polycrystalline bulk samples, our results cannot be easily compared with Matsumoto’s.
4. Effect of CNT addition on the magnetic relaxation of MgB2
In Section 2.2 we have shown that the main effect of DW CNTs addition is the simultaneous increase in the critical current density and the upper critical field Hc2. However, it is not clear if these two effects are connected. In fact, the vortex physics underlying this performance improvement has not been so far understood. In this section, we present a very recent work (Pasquini et al., 2011), where the role of CNT addition is investigated by studying the magnetic relaxation in bulk MgB2 samples.
As was mentioned in Section 2.1.1, if a superconducting sample is fully penetrated by the magnetic field B, the magnetization m is proportional to the density current J flowing in the sample that, in the ideal critical state approximation is the critical current density Jc. However, due to the thermal activated motion, this "measurable" critical current is lower than the "true" Jc0. The main quantity governing the creep of vortices is the activation barrier U(J,T,B). In a magnetic relaxation experiment both the magnetization m and Jc decay in time as
where or can be approximated as constants.
where Uc(T,B) is the pinning energy and μ is a critical exponent characteristic of the particular creep regime.
In High Temperature Superconductors (HTSs), the creep drastically reduces the measured current density, so Jc<<Jc0. A large number of glassy creep regimes in this limit have been proposed theoretically (Blatter et al., 1994; Yeshurum et al., 1996) and many of them observed experimentally (Thompsom et al., 1994, 1997), where.
In those cases the temperature dependence of the measurable Jc is dominated by creep while the Jc0(T) dependence is negligible. On the other hand, in traditional type II superconductors the current decay is very small, so (Jc0-Jc)<<Jc0, therefore equation (Eq. 5) take the traditional Anderson-Kim (A-K) linear dependence (Anderson et al., 1964)
In these very low creep superconductors an experimental confirmation of glassiness, which regardless of the specific approach always involves detecting tiny deviations from the Anderson-Kim model, is extremely challenging.
From this basic point of view, MgB2 is a very particular system, as the intermediate Tc and moderate anisotropy makes the creep effects smaller than in HTS but larger than in conventional superconductors. The influence of thermal fluctuations in the vortex physics is measured by the Ginzburg number Gi=(1/2)(kTc/Hc²ξ³γ-¹)². For YBa2Cu3O7-δ this is as large as Gi ~ 10-2, and even larger for the more anisotropic Bi-based compounds, while for NbTi, the paradigmatic strong-pinning conventional superconductor, Gi ~10-8. For MgB2, depending on the doping level we have Gi~10-4-10-5. This is just in the middle between the extreme cases. The first consequence is that in magnetization measurements, the A-K limit is not necessary valid in the whole temperature range, and the measured Jc is expected to be in the intermediate range (Jc0-Jc)≲Jc.
In the rest of the section we present a creep study in bulk MgB2 samples as-grown and doped with different doses of carbon nanotubes. In subsection 4.2 experimental relaxation rates are presented, and the general behaviour is described. In subsection 4.3 we review some fundamental concepts and formulas concerning creep rates and activation energies to perform in subsection 4.4 a careful analysis of results that allows us to identify the region where the A-K model is valid. In 4.5, the pinning energies Uc, true critical current densities Jc0, and correlation volumes Vc are estimated and compared. Conclusions are summarized in 4.6.
4.2. Relaxation rates
Samples used in this study were prepared by solid-state reaction with magnesium (-325 mesh, 99%) and amorphous boron (99%) as starting materials (Serrano et al., 2008). 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=H2 at 900 °C for 30 min.
The relaxation measurements were carried out in a Quantum Design model MPMS XL 7T SQUID based magnetometer. Time-dependent data were taken with a protocol similar to that described in ref. (Civale et al., 1996). A scan length of 3 cm was used in order to minimize the effects due to the non-uniformity of the applied magnetic field that was applied parallel to the longest axis of the sample. For each relaxation measurement, the samples were first cooled and stabilized at the measurement temperature. Then the field was first raised up to 6T and then lowered to the measuring field to assure that the sample was fully penetrated. Intermediate measurements were performed in the upper and lower magnetization branches to subtract the reversible magnetization. We checked for and ruled out any effects due to the magnet self relaxations (H variations during the measurement time) that could lead to spurious changes in the magnetization of the samples.
The experimental critical current density Jc(t) has been calculated from the measured magnetic moment using equation (Eq. 1), as in previous sections.
As reported in section 2.2., the response for samples with DWCNT additions between 1% and 10% at temperature far below Tc is very similar and the "measured" critical current density Jc is optimal in this range of doping. This fact is illustrated in Figure 3, where Jc obtained from typical magnetization loops in samples with different doses of DWCNT is plotted as a function of the applied magnetic field H at T = 5K and 20K. The response for samples with 1 at% and 10 at% addition of DWCNT is very similar. At low temperature (5 K), CNT addition enhances Jc more than twice for all the measured range of H. On the other hand, at high temperature (20 K) the response of pure and CNT samples is quite similar but CNT addition continues being efficient at fields higher than H = 2T.
The magnetic relaxation has been measured in pure MgB2 samples and samples with DWCNT doses between 1 at% and 10 at% in the range between 5K and 25K. The corresponding relaxation rate S=-d(ln(J))/d(ln(t)) has been obtained as the slope of the ln J vs ln t graph. Results for H =1T (full symbols) and 3T (open symbols) are shown in Figure 12.
Samples with doses of 2.5% and 5% of carbon nanotubes have also been measured and display similar results, not shown in the figure for clarity.
As was described in the introduction, the S values are intermediate between those measured in low and high Tc materials. With this moderate relaxation, the temperature dependence of Jc(t) (Figure 3) cannot be explained assuming a T independent Jc0 at low temperature. An intrinsic Jc0(T) dependence must to be taken into account.
Again, at H = 1 T the decay in time is quite similar in all the doped samples with doses between 1% and 10%. However, same unexpected results appear. The S(T) curve at H=3T is similar for pure and CNT samples in all the temperature range. Furthermore, at H=1T the pure sample that has a lower Jc (associated with a lower pinning) has also a lower relaxation (associated with a higher pinning energy Uc).
We have proceeded to analyze the relaxation data in a pure and a 10 at% CNT samples to get an understanding of the causes of this puzzle. The procedure and results are described in the subsections 4.3 and 4.4.
4.3. Creep and activation energies in the Anderson-Kim approximation
In the general case (Blatter et al., 1994), the relaxation rate S is related with the pinning energy by
As was mentioned in the introduction, the A-K approximation assumes or, that lets to the linear dependence of eq. (Eq. 6) between the activation barrier U and the density current J. The integration of eq. (Eq. 4) in this limit gives the well known logarithmic decay of the experimental critical current density with time.
and allows to estimate under the supposition that.
Due to the numerical differentiation, this method has a greater error in the calculation of the pinning energy, but allows estimating Jc0(B,T).
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 the linear relationship (Eq. 8) (i.e. the logarithmic decay with time of the current density). However, the resulting fitting parameter Jc0(B,T) must be consistent with the A-K assumption and the resulting fitting parameter must be consistent with (Eq. 7). If these two conditions are not fulfilled then some of the assumptions was wrong and we can conclude that the AK description is not valid.
In the following section, we apply the above procedure to MgB2 relaxation data, determining the temperature region where results are consistent.
4.4. Data analysis in MgB2 samples
As can be observed in the inset of Figure 12, it is not possible to identify in all the cases a low T region with a linear relationship between 1/S and 1/T, characteristic of a temperature independent pinning energy. For this reason, experimental data where separately analysed at each temperature assuming a linear dependence between U and J, i.e. the validity of the A-K approximation.
At each temperature Jc(t) has been plotted has a function of ln t and ln (-∂J/∂t). The fitting parameter A(T) = (kTJc0(T))/Uc(T)) has been extracted from the slope using equations (Eq. 8) and (Eq. 9), and Jc0(T) from the intercept using (Eq. 9) (Pasquini et al., 2011).
In Figure 13 the estimated true critical current density Jc0 is compared with the measured critical current density Jc (the first point measured in the Jc(t) relaxation) by plotting (Jc0-J)<<Jc0 as a function of temperature for a pure and a 10%doped sample, at H=1 T and 3T.
Taking a 10% criterion to determine the limit of the A-K condition (Jc0-J)<<Jc0, the resulting Jc0(t) is consistent below the temperature where each experimental curve crosses the doted horizontal line.
In figure 14 the resulting pinning energies obtained from the slope of eq. (Eq. 8) (squares) and (Eq. 9) (circles) are plotted as a function of temperature and compared with T/S (asterisks) for a pure and a 10%doped sample, at H=1 T and 3T. Results have been restricted to the consistent T region obtained by the above analysis.
As can be observed (with exception of the lower temperatures data in the undoped sample at H= 1 T) pinning energies obtained from the three methods are very similar, and this validates the fitting procedure, including the estimation of Jc0(T) from (Eq. 9).
Observing the resulting T dependence in Figure 14 it seems that, in most of the cases, pinning energies remain nearly constant at low temperatures. The unexpected drop in Uc at the lower temperatures data in the undoped sample at H= 1 T, together with the inconsistence of the analyzed data, could imply the presence of another relaxation mechanism, perhaps associated with macroscopic flux jumps. In the data measured at H=1 T a drop at a higher temperature far below the irreversibility line is also observed.
On the other hand, a continuous decrease occurs in the critical current density. This can be observed in figure 15, where the true critical current density is plotted in the studied range for the same samples.
The previous analysis excludes the creep study in the high temperature region using the A-K approximation. Successful procedures have been developed to analyze creep data in high Tc superconductors (Maley et al., 1990; Civale et al., 1996) in the limit Jc<<Jc0, but they are not suitable for intermediate creep regime. However, in the present subsection we showed that, from the present study at low temperatures, a good insight about the role of doping can be obtained.
Consistent with the relaxation rates, at H=1 T there is a clear decrease of Uc after CNT addition, whereas at H=3 T the doping has not an evident effect in the pinning energies.
The parameter Uc is associated with the pinning energy of a pinning volume that, in a single vortex pinning regime, is defined by the disorder parameter (T) and the superconducting coherence length (T) (Blatter et al., 1994). In section 2.2 we have shown that DWCNT addition enhances Hc2(T) by doping the B site, i.e. reduces (T). On the other hand, it is expected that doping increases disorder. Therefore, these two consequences of DWCNT doping will compete and define the effect in the pinning energies and critical currents.
The fact that the doping effect in the pinning energies is field dependent is a clear indication of a collective pinning regime.
In a collective regime, the pinning volume is determined by the competition between elastic and pinning energies and each volume Vc of the vortex system is collectively pinned with energy Uc. The Lorentz force over a volume Vc is FL ~(1/c)BJ Vc and, in a rough estimation, the pinning force is Fp ~ Uc/ξ. When J reaches the critical current density Jc0, the pinning and Lorentz forces are balanced and therefore the following relationship is obtained:
In section 2.2, we have presented Hc2 measurements in pure and DWCNT MgB2 samples and we have found (Serquis et al., 2007 ) that Hc2(T) fits very well the function proposed in ref (Braccini et al,. 2005). We have estimated ξ(T) from Hc2(T) for the as grown samples and for samples with 10% DWCNT. Using this ξ(T) dependence we have estimated Vc(T,B) from our data. Results are shown in Figure 16, where Vc is plotted as a function of temperature a H(T) = 1 T (right axis) and 3 T (left axis) for both samples.
The estimated numerical values (~10-¹5 cm³) imply that the correlation radius is larger than the main vortex distance, in agreement with a collective pinning regime of vortex bundles.
Doping additionally reduces the correlation volume, consistent with the increase in the critical current density.
From the comparison of data taken at H = 1 T and 3 T, in both pure and CNT samples, the correlation volume at H= 3T is approximately three times smaller than that obtain at H = 1 T and both Jc0 and Uc decrease with B. This field dependence for Uc does not correspond with that predicted by the classical collective pinning theory for the activation energy of vortex bundles when pinning arises from random point defects (Blatter et al., 1994). However, this discrepancy is not surprising, as we know that the strong pinning in these MgB2 samples arises from a variety of larger defects rather than atomic-size disorder.
Magnetic relaxation in bulk MgB2 samples as-grown and with DWCNT doses between 1% and 10% in the range between 5K and 25K has been measured and the corresponding critical current density has been calculated. The current decay in time is quite similar in all the doped samples with doses between 1% and 10%.
A careful creep analysis has been carried out in a pure sample and in one with 10% of carbon nanotubes at H= 1 T and 3 T.
The analysis has been performed under the Anderson-Kim approximation, valid in the limit where the measured critical current density Jc is similar to the “true” one Jc0 that would be present in the absence of creep phenomena. In these samples, even at the loweest temperatures it is necessary to include an explicit field and temperature dependence of both the pinning energies Uc(T,B) and true critical current densities Jc0(T,B).
These pinning properties have been obtained as fitting parameters from the experimental data for two different methods. The consistence of the fitting parameters with the A-K limit has been required to delimit the region of validity of the analysis.
In the valid region, the pinning energies and critical current densities have been estimated and compared. The dependence with magnetic field, together with numerical estimations of the pinning volume, indicate the presence of a collective pinning regime of vortex bundles. There is a decrease in the pinning energies that implies an increase in the relaxation rates as a consequence of DWCNT addition. However, the true critical current densities increase, due to a decrease in the collective pinning volumes.
We conclude that the origin of the main changes in the pinning properties with doping are the reduction of the coherence length (that decreases the pinning energies ) and the increase in the disorder parameters (that increase the critical current and decrease the pinning volume).
The strong temperature dependence of the coherence length is probably the main reason to the observed temperature dependence in the pinning properties. At the lower temperatures, thermal instabilities that originate macroscopic flux jumps could also play a role.
A method to analyse the relaxation data in the high temperature region, beyond the A-K approximation, is necessary and will be object of a future work.
The effect of carbon substitution is one of the most studied in MgB2 and the results on C solubility and the effects of C-doping on Tc, Jc and Hc2 reported so far vary significantly due to precursor materials, fabrication techniques and processing conditions used. The distinct effect of C incorporation through different routes using various CNT types leads to a simultaneous improvement in Jc and Hc2, but their effectiveness change with temperature or applied field according to each type of addition. This effect was reported in many works, not only for bulk samples but also for wires and tapes prepared by PIT method.
The reason for this is the dual role of the CNT. They partially dilute into the MgB2 matrix, acting as a source of C that increases Hc2. In fact, the highest Hc2 values observed so far in bulk MgB2 correspond to a 10% addition of DWCNT, which present a high level of C doping. At the same time, the fraction of the CNT that retain their structural integrity are ideally suited to act as strong vortex pinning centers due to their tubular geometry and their diameter close to the superconducting coherence length of MgB2, producing a large Jc enhancement. As an additional result, the measured Hc2 vs T in all samples are successfully described using a theoretical model for a two-gap superconductor in the dirty limit (Gurevich, 2003). This has strong fundamental impact, as it provides clear evidence in support of the basic scenario currently used to describe the superconducting behavior of MgB2, and opens a path for future research in Hc2 enhancement.
The study of the magnetic relaxation of MgB2 with and without DWCNT bulk samples can give some insight about the possible correlation in the simultaneous increase in the critical current density and the upper critical field. The experimental relaxation rates showed that the pure sample that has a lower Jc (associated with a lower pinning) has also a lower relaxation (associated with a higher pinning energy Uc). To understand these results the relaxation data was described and analyzed under the Anderson-Kim frame model (Anderson, 1964). The pinning energies Uc, true critical current densities Jc0, and correlation volumes Vc were estimated and compared. The strong temperature dependence of the coherence length is probably the main reason to the observed temperature dependence in the pinning properties with CNT additions: the reduction of the coherence length (that decreases the pinning energies) and the increase in the disorder parameters (that increase the critical current and decrease the pinning volume).
Research supported by CONICET, UNCuyo, UBACyT, and MinCyT-PICT (AS, GP) and by the U.S. Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division (LC).
- In Refs. (Gurevich, 2003; Serquis et al., 2007 ; Serrano et al., 2008)) Tc0 = Tc(g=0) is defined, where g the interband scattering parameter. This value corresponds to the clean limit MgB2 sample.