Magnetic irreversibility and pinning force density in the Mo 100−x Re x alloy superconductors
Shyam Sundar, M.K.Chattopadhyay, L.S.Sharath Chandra, S.B.Roy
11 Magnetic irreversibility and pinning force density in theMo Re x alloy superconductors Shyam Sundar
1, 2 , M. K. Chattopadhyay
1, 2 * , L. S. Sharath Chandra and S. B. Roy
1, 2 Homi Bhabha National Institute, Raja Ramanna Center for Advanced Technology, Indore 452 013, India Magnetic & Superconducting Materials Section, Raja Ramanna Center for Advanced Technology,Indore 452 013, India
Abstract
We have measured the isothermal field dependence of magnetization of the Mo Re x (15 x
48) alloys, and have estimated the critical current and pinning force density from these measurements. Wehave performed structural characterization of the above alloys using standard techniques, and analyzedthe field dependence of critical current and pinning force density using existing theories. Our resultsindicate that dislocation networks and point defects like voids and interstitial imperfections are the mainflux line pinning centres in the Mo Re x alloys in the intermediate fields, i.e., in the “small bundle” fluxline pinning regime. In this regime, the critical current density is also quite robust against increasingmagnetic field. In still higher fields, the critical current density is affected by flux creep. In the low fieldregime, on the other hand, the pinning of the flux lines seems to be influenced by the presence of twosuperconducting energy gaps in the Mo Re x alloys. This modifies the field dependence of criticalcurrent density, and also seems to contribute to the asymmetry in the magnetic irreversibility exhibited bythe isothermal field dependence of magnetization.
1. Introduction
The Mo Re x alloy superconductors are an interesting system where the normal tosuperconducting transition temperature ( T c ) of the alloys are higher than both Mo and Re [1]. The T c ofmany of these alloys are relatively high (near 12 K) [1] which is important from the application point ofview. The Mo Re x alloys are also known to have good mechanical properties and they are widely usedin the aerospace and defense industries [2- 4]. In view of the scope of technological applications,superconducting solenoids made out of the Mo Re x alloys have been tested in past [5], and the field ( H )hysteresis of magnetization ( M ) of the Mo Re alloy has also been studied [6- 8]. This latter study hasrevealed that the dislocation networks are the main source of flux line pinning in the Mo Re alloy.However, a comprehensive study on the H dependence of critical current density ( J c ) as a function of Re-content is not found in literature. It is also observed that in spite of substantial information available inliterature [1- 8], the superconducting properties of the Mo Re x alloys are still not very well-studied andonly our very recent study indicated the presence the two superconducting energy gaps in this alloysystem [9]. This observation of two superconducting energy gaps now raises a natural question: how doesthis phenomenon influence the various superconducting properties of the system, especially in thepresence of magnetic fields?In the present work, we have synthesized as-cast and annealed Mo Re x alloys of five differentcompositions, covering a wide range of Re-concentration, and have studied these alloys with the help ofmagnetization experiments. We have estimated the J c and the pinning force density ( F p ) of these alloysfrom the M(H) curves, and have analyzed the nature of the magnetic irreversibility and the fielddependence of J c and F p with the help of existing theories. Our analysis reveals the flux line pinningmechanisms available in this alloy system in different fields. Our results also indicate that the J c of thesealloys in the low field regime is indeed influenced by the presence of two superconducting energy gaps[9], and this does give rise to substantial changes in the F p (H) in low H which seems to influence thesymmetry of the M(H) curves between increasing and decreasing H .
2. Experimental
Polycrystalline samples of Mo Re x (x = 15, 20, 25, 40, and 48) alloys were synthesized bymelting high purity (99.95+ %) Mo and Re (ACI Alloys, USA) taken in atomic proportions. The meltingwas performed under high purity Ar atmosphere in an arc-melting furnace. The sample was flipped andre-melted six times to ensure the homogeneity. The loss of mass during the melting procedure was lessthan 0.1 %. The as-cast buttons were cut in two halves with the help of a spark wire cutter, and one of thehalves was wrapped in Ta foil and sealed in quartz ampoule in Ar atmosphere for annealing. Theannealing was done at 1250 °C for 20 hours in Ar atmosphere, which was followed by slow cooling downto the room temperature [10]. The samples were cut in different shapes and sizes with the help ofdiamond wheel cutter and spark wire cutter for different experimental studies.The structural characterization of the present Mo Re x alloy samples was done with the help ofX-Ray Diffraction (XRD), optical metallography, and energy dispersive x-ray (EDX) measurements. TheXRD experiments were performed in a standard diffractometer (Rigaku Corporation, Japan: Geigerflexmodel) using Cu Kα ( λ = 1.5418 Å) radiation. For optical metallography, small pieces of samples weremounted on moulds prepared from epoxy (Resin + Hardener) and then ground, to make them flat, withthe help of silicon carbide abrasive papers. Properly ground moulds were then polished to 0.5 μ mroughness with the help of diamond paste. Polishing of the sample was done by holding the moulds onrotating Nylon cloth. An etchant solution of H O (50%) and NH OH (50%) were used for thevisualization of the microstructure of the polished samples. The metallographic characterization of thesamples was done using a high power optical microscope (Leica DMI 5000M). The chemicalcompositions of the samples were experimentally confirmed through EDX analysis performed using aPhilips XL-30pc machine. Within the experimental resolution of the EDX measurements, the chemicalcompositions were not found to vary over the bulk of the samples.The measurement of M as functions of temperature ( T ) and H were performed using a liquid Hebased Vibrating Sample Magnetometer (VSM; Quantum Design, USA) and a closed cycle refrigerator based MPMS-3 SQUID VSM (Quantum Design, USA). In the VSM, the isothermal M versus H measurements were performed while varying μ H sequentially from 0 to 8 T, from 8T to -8 T, and thenfrom -8 T to 8 T, starting from an initial zero-field-cooled (ZFC) state. The M(H) curves obtained in thelast two runs of the above measurement, i.e., while varying μ H sequentially from 8T to -8 T, and thenfrom -8 T to 8 T (the upper and lower envelope magnetization curves respectively) will henceforth bereferred to as M – and M + respectively. The protocol for the isothermal M(H) measurements performed inthe MPMS-3 SQUID VSM was kept same as that of the VSM. Only, the maximum field available inMPMS-3 SQUID VSM is ±7 T. After measuring M + and M – isothermally, the MPMS-3 SQUID VSMwas used for drawing minor hysteresis loops (MHLs). Two kinds of MHLs were constructed for thepresent study. In the first kind, the MHL was initiated by starting M(H) measurements while decreasing H from a field value on the M + curve. The MHL was continued till it merged with the M – curve. For thesecond kind of MHL, the sample was cooled in constant H from 50 K down to 2 K (FC protocol), and thethen M(H) was measured at 2 K while increasing (or decreasing) the H till the MHL merged with the M + curve (or the M – curve). These MHLs will be referred to as the FC MHLs. For every value of H in whichthe FC state is prepared, two FC MHLs were drawn with increasing and decreasing H , and they gave twovalues of M where the FC MHLs reached the M + and M – curves. Apart from the above measurements,the magnetic relaxation experiments were also performed in the MPMS-3 SQUID VSM. For theseexperiments M was measured for about 90 minutes at constant T (2 K) and H , starting from different H -values on the M + and M – curves. The choice of the VSM or the MPMS-3 SQUID VSM for the abovemeasurements is based on the availability of liquid He and the availability of the equipment at the time ofthe measurement. FIG. 1.
XRD patterns of the Mo Re x alloys indicating the presence of the β , σ and χ phases.Only the indices corresponding to the prominent peaks are shown here.
3. Results and discussion
Fig. 1 shows the XRD patterns for the annealed Mo Re alloy, as-cast Mo Re and Mo Re alloys, and the annealed and as-cast Mo Re and Mo Re alloys. The XRD patterns indicate that theannealed Mo Re alloy, and the as-cast Mo Re , Mo Re and Mo Re alloys are single phase, andhave formed in the body centred cubic (bcc) β phase structure. On the other hand, the XRD patterns of the annealed Mo Re alloy, and the annealed and as-cast Mo Re alloys, indicate the presence of the β , σ (tetragonal), and χ (complex bcc) phases in these samples (see table I).TABLE 1. Metallurgical phases present in the Mo Re x alloys along with theapproximate phase fractions (estimated from the XRD patterns), and the T c . AlloyCompositions MetallurgicalPhases Present T c (K) µ H C1 (T) at2K Mo Re ,Annealed β Re As-cast β Re As-cast β Re As-cast β Re Annealed β (75 %) , σ (20 %) ,χ (5 %) 12.44 0.0900Mo Re As-cast β (50 %) , σ (40 %) ,χ (10 %) 12.68 0.0548Mo Re Annealed β (50 %) , σ (40 %) ,χ (10 %) 11.85 0.0438Fig. 2(a) shows a selected optical micrograph of the annealed Mo Re alloy. Large grains withvarying sizes and a strong network of dislocations [6- 8] are observed in this micrograph. We haveperformed optical metallography experiments on different portions of each of the present samples. Only afew representative micrographs are shown in Fig. 2. In general, the grains of the present Mo Re x alloysare found to be very large in size. In the single phase Mo Re x alloys used in the present work (see tableI), e.g., the annealed Mo Re alloy, the as-cast Mo Re , Mo Re , and Mo Re alloys, the grains arefound to have large variation of size over the sample surface. Overall, the grain sizes in these alloystypically vary from 100 to 600 μ m, with some grains as large as 2000 μ m. Fig. 2(b) shows an opticalmicrograph of the as-cast Mo Re alloy in a magnified scale. This micrograph, obtained after verycareful polishing and etching, provides a clear view of the dislocation network in the form of lining up ofthe etch pits. Such dislocation network is visible in all the single phase Mo Re x alloys described above. The observation of dislocation networks in the present Mo Re x alloys is consistent with the existingliterature [6- 8]. Fig. 2(c) shows an optical micrograph of the annealed Mo Re alloy, which contains β , σ and χ phases. Apart from the grain boundaries visible in this micrograph, it appears that the σ and χ phases precipitate around the dislocation networks in this alloy. The precipitation of the σ phase along thedislocation networks in the Mo Re x alloys have been reported previously [8]. The average grain size inthe annealed Mo Re alloy is 50-150 μ m. The average grain size in the (multi-phase) Mo Re as-castalloy [Fig. 2(d)] is found to be 20- 100 μ m. No indication of dislocation network is found in the opticalmicrographs of the as-cast Mo Re alloy or its annealed counterpart (not shown here). In general, theaverage grain sizes in the multi-phase Mo Re x alloys are found to be much smaller and more evenlydistributed as compared to the single phase alloys. FIG. 2.
Optical micrographs for selected Mo Re x alloys. While the panels (a) and (b) are forthe single ( β ) phase samples, the panels (c) and (d) are for multi-phase samples containing β , σ and χ phases. The T c values for the present samples are shown in table 1. These values were obtained by findingthe T -value at which the M(T) curves (measured in 10 mT magnetic field, not presented here) start to dropdownwards towards a negative value with decreasing T , and these values are consistent with the literaturereports [11]. We have found that the T c values for the annealed and as-cast Mo Re alloys are notdifferent, and we have used the annealed Mo Re alloy only for the present study. For the other alloys,the T c values for the as-cast alloys are slightly higher than their annealed counterparts. FIG. 3.
Field dependence of magnetization of the Mo Re x alloys at 2K, measured in the VSM. M , critical current density and pinning forcedensity Fig. 3 shows the isothermal
M(H) curves for the present alloys at 2 K depicting the irreversiblemagnetic behaviour of the present alloys. The major part of this magnetic irreversibility reduces quiterapidly with increasing H for all the present alloys. A small amount of irreversibility between increasingand decreasing H , however, is sustained in all the present alloys up to high H . For the annealed Mo Re alloy, this magnetic irreversibility disappears at μ H = 1 T at 2 K. The field at which this small magneticirreversibility disappears completely, is enhanced considerably with increasing Re content. For the as-castMo Re alloy, at 2 K, a tailing magnetic hysteresis remains up to 5.6 T. For the annealed Mo Re alloy,this tailing magnetic hysteresis is observed even in 8 T (at 2K). For the annealed and as-cast Mo Re alloys the phenomenon is sustained up to 7 T (at 2 K). This tailing magnetic hysteresis, which is notvisible in Fig. 3, and is observable only in a highly magnified scale, may be related to the presence of thethird critical field H c3 reported in the Mo Re x alloys [10]. Fig. 3 also shows that there is considerableasymmetry between the M + and M – curves for the single phase Mo Re x alloys (see table I). The majorportions of the M – curves for these alloys are negative for positive H -values, which leads to asymmetry in M(H) , and the very narrow width of the hysteresis in most fields depicted in Figs. 3(a) to (d). Themagnetic hysteresis for these alloys is significant only in very low H . Such asymmetric hysteresis cansometimes appear because of surface barrier effects [12- 14]. It is already known that the presence ofsurface barrier effect in the superconductors may be confirmed with the help of the MHL technique [12].We have, however, found that for the present alloys, the MHL initiated from the M + curve touches the M – curve after showing a rounding-off behaviour [12]. This is shown in Fig. 4 for the Mo Re as-castalloy. Similar curves exist for the other alloys as well, and they indicate that the surface barrier effect maynot be significant in the present samples. Narrow isothermal M(H) -hysteresis curves similar to the presentalloys have been observed both in the polycrystalline [15] and single crystal MgB samples [16]. In thesematerials, the narrow width of the hysteresis, is thought to be arising due to weak pinning [15] and the lack of defects (flux line pinning centres) [16]. In such a case, the M – curve lies close to the equilibriummagnetization ( M Eq ) and thus exhibits negative values. Fig. 3 also shows that the M(H) curves for theannealed Mo Re alloy and the as-cast and annealed Mo Re alloys do not exhibit such a pronouncedasymmetry. On the other hand, the M(H) curves for these multi-phase samples show the signatures of fluxjump effects.
FIG. 4.
Field dependence of magnetization of the as-cast Mo Re alloy at 2 K. The black dotsrepresent the envelope magnetization curves, and the (blue) open triangles represent the MHLinitiated from the lower envelope magnetization curve. The inset shows a magnified version ofthe relevant portion of the curves. These experiments were performed in MPMS-3 SQUID VSM.Fig. 5 shows the J c (H) curves for the present alloys, estimated from the irreversibility in M(H) .Only the curves for a few selected temperatures are shown here. The J c values were estimated with thehelp of Bean model [17], using the formula J c = ΔM [a (1-a /3a )] -1 , where ΔM is the difference in M (fora particular H ) between the M + and M – curves [18, 19]. The parameters 2 a and 2 a ( a >a ) are thedimensions of the rectangular samples used in the present measurements, in directions normal to theapplied H . The J c (H) curves shown in fig. 5 are initiated from a field above the lower critical field H c1 soas to avoid the field regime where the flux-penetration in the sample is not uniform. The method of determination of H c1 for the present alloys has been discussed in our previous work [9]. The µ H c1 valuesfor the present Mo Re x alloys are shown in table 1. For the samples exhibiting flux jumps, we haveplotted the J c (H) curves in fields still higher than H c1 , in the regime where these jumps are not observed.It may be observed in fig. 3 that the multi-phase Mo Re x alloys exhibit higher values of ΔM over alarger H -regime as compared to the single ( β ) phase alloys. This indicates that the J c and F p values for themulti-phase Mo Re x alloys are higher than the single ( β ) phase alloys. Incidentally, these multi-phasealloys also exhibit higher T c values (see table 1) and smaller grain size. Among the multi-phase alloys, thelargest ΔM is observed in the annealed Mo Re alloy [see fig. 3(e)]. We have however not estimated the J c for the annealed Mo Re alloy because of the excessive flux jump effects in this alloy. It mayhowever be noted that stronger flux jump effects also indicate higher F p . We have also extracted the T -dependence of J c of the Mo Re x alloys, for different H -values, from the isothermal J c (H) curves. Allthese J c (T) curves (not shown here) fall nearly linearly with increasing T , with a slight tendency ofbending further downwards at higher T indicating the absence of weak links in the present alloys. In thepresence of weak links, these curves are expected to exhibit a tendency of bending upwards [20]. Theformation of weak links (e. g. between the grains) is somewhat like the formation of S-N-S type junctions,and this leads to a upward curvature in the J c (T) curves (see Ref. [21] and references therein). It istherefore reasonable to assume that the current flow in the present samples is not fragmented because ofthe presence of grain boundaries or any other weak link.It is observed in Fig. 5 that the J c (H) curves for the single ( β ) phase Mo Re x alloys (see table I)used in the present study may be qualitatively divided into three regimes: A low H regime where the J c drops sharply with applied H , an intermediate H regime where the J c varies slowly with increasing H , anda higher H regime where the J c again drops rapidly with increasing H . While the J c (H) curves for themulti-phase Mo Re x alloys in the low H regime could not be studied because of the flux jumps, the J c (H) behaviour for these alloys in the high H regime seems to be quite similar to the single phase alloys.Both Figs. 3 and 5 provide hints of the existence of a peak effect (PE) in the higher field regimes of the M(H) and J c (H) curves of the single phase Mo Re x alloys. However, it has been reported earlier that M in the PE regime of the Mo Re x alloys does not exhibit any anomalous behaviour and can be explainedusing conventional theories (see Ref. [22] and references therein). FIG. 5.
Field dependence of critical current density of the Mo Re x alloys estimated from theisothermal M(H) curves at selected temperatures.We now look into the J c (H) behaviour of the Mo Re x alloys in the perspective of the existingtheories [23]. According to the collective pinning models for the type-II superconductors, the separation( a ) between the flux lines is very large in low H so that the interaction between the flux lines isnegligible. In this regime individual flux lines are pinned by the available pinning centres and theelasticity of the flux lines is not relevant. As the H is increased, the interaction between the flux lines become significant, and then the elasticity (or the lack of rigidity) of the flux lines becomes more andmore important for the nature of flux line pinning. The flux lines are displaced under the influence of theLorentz force due the applied H . As long as the displacement of the flux lines is less than the coherencelength ( ξ ), the flux lines can be considered rigid. For slightly higher H , the flux lines tend to shear, and theindividual flux lines are then collectively pinned by several pinning centers. This is the so called “singlevortex collective pinning regime”. Up to this regime, the J c (H) is expected to remain constant [23]. Theregime of constant J c (H) is expected to be quite narrow according to the collective pinning models. Withincreasing H , the flux line lattice undergo further displacement and it adjusts with the disorder (or pinning)potentials through shear and tilt deformations [23]. When the displacement of the shearing flux lines is ofthe order of a , the flux lines start interacting strongly with each other. Moreover, when the displacementis more than a the average pinning potential experienced by the flux line lattice remains approximatelythe same even after the displacement. The effect of individual disorder potentials becomes weak fordisplacements of the order of a . Bundles of flux lines are then collectively pinned by a spatial average ofthe pinning potentials. The size and aspect ratio of these flux line bundles are functions of H (and hence a ) and the relevant elastic constants of the flux line lattice. It has been shown that there are elastic lengthscales that determine the volume (elongated in the direction of H ) of these flux line bundles, and it growsrapidly with increasing H [23]. If the transverse elastic length-scale is less than the penetration depth, thenthe flux line bundles are considered “small”. On the other hand, the flux line bundles are considered“large” if the penetration depth is smaller. In the so called “small bundle” flux line pinning regime, the J c (H) is expected to be proportional to exp [23]. In fig. 5, this regime is indicated by the fittedline for different Mo Re x alloys at 2 K. The sharp decrease in J c (H) in the Mo Re x alloys withincreasing H in the lowest H regime, which extends over quite a broad H -range, is contrary to the normalexpectation in the individual (or “single vortex”) flux line pinning regime [23].The pinning force density in the present Mo Re x alloys was estimated using the relation F p =J c µ H [24], and are presented in Fig. 6. For the single phase alloys, the F p (H) curves are shown for H values greater than or equal to 1.5 times of H c1 (the rationale for this choice has been explained below, inconnection with Fig. 7) to ensure that the data points are free from errors due to non-uniform fluxpenetration. For the multi-phase alloys, the F p (H) curves are plotted for the H -regime with no flux jump.For the single phase Mo Re x alloys [Figs. 6(a)- (d)], the F p (H) curves exhibit a tendency of risingupwards in low H . Since F p = 0 at H = 0, the above tendency of rising upwards in low H actuallyindicates the presence of a low H maximum in the F p (H) curves, and thus a double peak behaviour ofthese curves. The F p (H) curves for the multi-phase Mo Re x alloys, on the other hand, appear to exhibita single peak behaviour [figs. 6(e) and (f)]. FIG. 6.
Field dependence of pinning force density in the Mo Re x alloys.Here we have estimated J c and F p from the isothermal M(H) curves using the most commonmethod described above. However, it has been reported in literature that this method becomes erroneous when the effect of the equilibrium (reversible) magnetization M Eq is significant as compared to themagnetic irreversibility. In such a case the J c and F p need to be estimated as a function of B , where, B =µ ( H + M Eq ) [25]. Experimentally, M Eq is commonly determined as M Eq = ( M + + M – )/2 [26]. However,when the magnitude of M Eq is significant as compared to ΔM , a more accurate method of determination of M Eq is to use the FC MHL technique [26] described earlier. In this technique, the average of the two M -values where the FC MHLs touch the M + and M – curves gives the correct M Eq [26]. In Fig. 7(a), the M Eq obtained at different H -values using the FC MHL technique [(red) stars] is compared with the M Eq = ( M + + M – )/2 curve [empty (blue) squares]. The M Eq values obtained in both the methods described abovematch very well in high H , and they differ appreciably only for H < (1.5 H c1 ). Similar results exist for ourother Mo Re x alloys as well. In Fig. 7(b) we show a comparison (at 2 K) between the F p versus µ H and F p versus B curves for the annealed Mo Re alloy where B = µ ( H + M Eq ). The data below H = (1.5 H c1 ) are not shown in this figure because of the uncertainties discussed above. For the rest of the H regimes, the two curves are almost identical. Therefore, to keep the method simple, we will continue towork with the F p versus µ H curves of the present alloys and analyze the results for H > 1.5 H c1 (or stillhigher in the cases where the flux jumps are present). FIG. 7(a).
Experimental determination (using two different methods) of equilibriummagnetization of the annealed Mo Re alloy at 2 K. (b). The F p versus µ H and F p versus B curves for the annealed Mo Re alloy at 2 K.A detailed analysis of F p (H) in terms of the size, spacing and nature of the pinning centres, andthe nature of their interaction with the flux lines has been done by Dew-Hughes [27], and we havepresented a similar analysis in a recent work [21]. To investigate into the flux line pinning mechanisms inthe present alloys using Dew-Hughes model, we have used a normalized field h , where , H * beingthe field at which the (approximately) linearly falling portion of the F p versus µ H curves (in the higherfield side) extrapolate to F p = curves for the present alloyscannot be fitted by any of the functions of form , where the values of p and q depend on the details of the pinning mechanism, as prescribed by Dew-Hughes [24, 27]. Qualitatively, from the h value corresponding to the peak in F p , it appears that the grain boundaries might be the major flux linepinning centres in the annealed and as-cast Mo Re alloys (multi-phase). However, apart from the Dew-Hughes model the field dependence of pinning force density can also be analyzed with the help ofKramer’s theory [29], where it is assumed that is represented by the functions and respectively in low and high reduced fields. Here is a pinning force computed assuming that at lowreduced fields the maximum pinning force for some of the crystal defects can be exceeded by the Lorentzforce, so that the flux lines may be de-pinned from these pinning centres. On the other hand, iscomputed assuming that there are some strong pinning centres for whom the pinning force cannot beexceeded by the Lorentz force and the flux lines will rather shear plastically around these pinning centres.Thus, and are increasing and decreasing functions of h respectively, and the peak in isreached when [29]. In this model, the effective interaction of the flux lines with thepinning centres is not only a function of the distribution of the pinning centres in the material, but also ofthe shear strength of the flux lines (if the flux lines are not flexible, the pinning interactions will mostlycancel each other) [29- 31]. The probability that a particular flux line will be pinned with a pinningstrength is given by a distribution (around an average pinning strength ) function. Variousdistribution functions, e.g., the Gaussian, and the single and double Poisson distribution functions havebeen assumed for for the computation of [29, 32]. We have found that none of thesedistribution functions provides complete explanation for the experimentally obtained curves for thepresent Mo Re x alloys for the entire h regime. However, the experimental curves for the presentMo Re x alloys can still be explained over a large h regime by using a double Poisson distributionfunction of the form [29], t t where, and The double Poisson distribution function indicates that at least two types of flux line pinning centreshaving different pinning strength distributions are present in the present alloys. Using equation (1), thepinning force density is expressed as [29], ㌳ t t t t t t ㌳ t t Here is the shear strength parameter, and t . The parameter mostly depends onthe thermodynamic and upper critical fields of a superconductor, and depends only very weakly on thedefect structure of a material. Therefore while fitting equation (2) to our experimental data we haveassumed on the basis of Kramer’s paper [29] that t ㌳ , t ㌳ , and t ,where t , t and t are positive fractional numbers used as the fitting parameters along with , , and . For equations (1) and (2), the reduced field h was defined as 㠸㠸 . The irreversibility field H irr was determined from the M(H) curves, where it is identified as the field at which the magneticirreversibility ΔM goes to zero (within the resolution of the present experimental setup). In the case of theMo Re x alloys, there is some scope of uncertainty in the determination of 㠸㠸 because of the presenceof a long tailing hysteresis in M(H) (as discussed earlier). To avoid this uncertainty we have estimated 㠸㠸 by extrapolating the J c (H) curves to J c = 0 in the high H side. FIG. 8.
The versus h curves for the Mo Re x alloys at 2 K, where 㠸㠸 . The (black)open circles represent the experimental data and the (red) solid line represents the fitted curve[equation (2)]. The (blue) dotted line and the (green) dashed line respectively represent twocomponents of the fitted (red) curve.Fig. 8 shows the versus h curves (where 㠸㠸 ) for the present Mo Re x alloys at 2 K. Inthis figure, the (red) solid line represents the best fit curve obtained using equation (2). The (blue) dottedline and the (green) dashed line respectively represent the two components of this best fit (red) curveobtained by putting and alternately (keeping the other parameters unchanged). Bycorrelating our J c (H) , , and curves (Figs. 5, 6 and 8) we find that the best fit curves in Fig. 8explain the experimental curves only in the “small bundle” flux line pinning regime. In higher H ,the best fit curves lie above the experimentally obtained curves, indicating a reduction of theeffective pinning strength. We have found that in this high H regime, the Mo Re x alloys exhibit strong signature of magnetic relaxation. Fig. 9 shows the time ( t ) dependence of M/|M | (where M is themagnetization at t = 0) in the Mo Re alloy in three H -regimes on the M + curve. The highest H -value infig. 9 corresponds to the regime where the best fit curve corresponding to equation (2) lie above theexperimentally obtained curves. The magnetic relaxation effect is not appreciable in the lower H -regimes. The M(t) curve obtained in the high H -regime can be fitted with the following the equation, 㘱 where U represents the activation energy. The fitting of this equation to the experimental curve indicatesthe presence of flux creep effects in the alloy in the high H regime. We have found that the corresponding J c (H) curve for the Mo Re alloy exhibits a rapid drop in this high H regime (see Fig. 5). Thus thereduction of the effective pinning strength and J c (H) in high H is due to the flux creep effects.Qualitatively similar results were obtained for the other Mo Re x alloys as well. FIG. 9.
Relaxation of magnetization in the Mo Re alloy in different field regimes. Theseexperiments were performed in MPMS-3 SQUID VSM. From fig. 8 we find that for the annealed Mo Re alloy and the as-cast Mo Re , Mo Re , andMo Re alloys [panels (a)- (d)], the two components of the fitted curve exhibit peaks near = 0.3and 0.5, = 0.3 and 0.6, = 0.4 and 0.6, and = 0.3 and 0.6 respectively. According to Kramer, a peakin at a low value of , e. g. close to 0.2, may be ascribed to flux line pinning by the grainboundaries present in a material [24, 33], and a peak in at a high value of (between 0.7- 0.8) maybe expected to be due to flux line pinning by dislocation networks [34]. Voids, on the other hand give riseto a peak in at a value of , between 0.3 and 0.7 [34]. Our optical metallography experiments showstrong signatures of the presence of dislocation networks in the above mentioned alloys. In the samplecharacterization methods employed in the present work, however, we cannot see the evidence of thepresence of voids or any other point pinning defect structure in our Mo Re x alloys, though it is reportedthat it is possible to create voids in the Mo Re x alloys by cold-work (e.g., rolling) [7]. Interstitialimpurities like oxygen can also act as point pinning centres like the voids [34]. Though our samples havebeen synthesized in 99.99+ % Ar atmosphere (both during melting and annealing), a small amount ofinterstitial impurity cannot be completely ruled out. Voids and the interstitial defects mentioned abovemay be expected to have dimensions ~1 nm. The dislocations are visible in the present opticalmetallography because of the lining up of the etch pits. Actually, the width of the dislocations areexpected to extend over a few atomic spacing [35], or in other words, over a few nm. Using the formula ξ ㌳ , we have found that the coherence length of all the present alloys is close to 10 nm.Therefore, it is likely [12] that the flux lines in the present alloys may be pinned across the dislocationsand the point defects mentioned above. It is also reported in literature that for the dislocation networks,the interaction with the flux lines is between the strain field of the dislocations and the stress field of theflux line lattice [36]. The interaction between the dislocation network and the flux lines in the presentalloys will, therefore, depend on the size of the flux line bundles and the area of dislocation loops [36, 37].A full proof first principle theory for such an interaction does not exist, and large pinning forces due todislocation networks have been reported for h = 0.4 and 0.55 as well [36]. Accordingly, for the alloys mentioned above we believe that the peak in the component of at = 0.3- 0.4 is due to the pointdefects mentioned above, and the peak at = 0.5- 0.6 is due to the pinning of flux lines by the dislocationnetworks. The grain boundaries do not seem to have a major role in pinning the flux lines in the singlephase Mo Re x alloys in the above mentioned field regimes. Fig. 8(e) shows the experimental and best-fit curves for the annealed Mo Re alloy at 2 K. Though much of the data had to be excludedbecause of the flux jump effects, it may still be observed in fig. 8(e) that the best-fit curve representingequation (2) has only one component in this case ( C = 0), and it peaks close to = 0.2. Qualitativelysimilar fitted curve is obtained for the as-cast Mo Re alloy as well, indicating flux line pinning by thegrain boundaries in these alloys [29, 33]. It may be recalled that we had previously reached the sameconclusions about the flux line pinning mechanism in the as-cast and annealed Mo Re alloys, using theDew-Hughes model [27]. Thus, in spite of the presence of the σ and χ phases, we do not find anyadditional flux line pinning mechanism in these multi-phase alloys. It may be noted in this context thatthe σ and χ phases in the Mo Re x alloys are superconducting at low temperatures [11]. Any additionalsuperconducting phase in a sample acts as a flux line pinning centre when its ξ value is different from themajor phase [27]. The superconducting properties of the σ phase Mo Re x alloys are already reported inliterature. From the reported H c2 values [11], we have estimated the ξ of these alloys using the methoddescribed above. We have found that the ξ values for the σ and β phase Mo Re x alloys areapproximately equal (close to 10 nm). We could not find literature reports on the H c2 or ξ of the χ phaseMo Re x alloys. However, in view of the present results we believe that even for the χ phase, the ξ valueis not significantly different from that of the β phase alloys, and thus it is possible that there is only onemajor flux line pinning mechanism in the as-cast and annealed Mo Re alloys.We have already explained that the curves (fig. 6) for the single phase Mo Re x alloysindicate the presence of a peak in very low H . Comparing figs. 6 and 8 it is clear that this peak is presentclose to = 0.1 or even below. A peak in the curves at such a low value of h is not normallyexpected in the existing theories [27, 29, 33] of flux line pinning. Literature reports suggest that such a low h peak may arise due to a large anisotropy in the upper critical field, e.g., in the case when the ratio ofthe upper critical fields along the ab -plane and the c -axis is nearly 5 [38, 39]. Such high anisotropy,however, cannot be expected for the present system of alloys- especially in the single ( β ) phasecompositions where the indication of this peak is observed. On the other hand, we have recently foundstrong signature of the existence of two unequal superconducting gaps in the single phase Mo Re x alloys [9]. Our results suggest that the Re-5 d like states at the Fermi level do not intermix with the Mo-5 p and 5 s states in the Mo Re x alloys, and this leads to the two-gap nature of these superconductors [9].Previously, we have also shown in the case of PrPt Ge that a peak in the curves near h = 0.1 maybe related to the existence of two superconducting gaps in that material [38]. Similar results also exist inthe case of MgB , which is another system with two superconducting energy gaps [40- 42]. Theisothermal J c (H) of MgB was found to follow a two-exponential model, and this model was successfullyused to numerically reproduce the experimental M(H) curves. The two exponential terms in the J c (H) were thought to arise due to the two superconducting energy gaps in MgB [41, 42]. Using the samemodel the peak in the curves near h = 0.1 was also explained with appreciable quantitativeaccuracy in the case of MgB . It appeared that the peak in arising due to grain boundary pinningshifts to lower h values because of the presence of the two superconducting energy gaps [41, 42]. Wefound that the same model works for PrPt Ge as well [38]. We therefore tried to fit the same model inthe case of the present alloys. Fig. 10(a) shows the J c (H) curve of the as-cast Mo Re alloy at 2 K, wherethe low H data have been fitted by an expression of the form t t ㌳ t ㌳ .Since the fitting was found to be quite good in low H , the fitted curve was used to generate thecorresponding curve for this alloy. Fig. 10(b) shows the curve for the as-cast Mo Re alloy,where the thin (red) solid line represents equation (2), and the dashed (blue) line represents the generated using the two-exponential model. It is observed that the experimental curve in low H isalmost reproduced by the model mentioned above in the H regime where equation (2) cannot fit theexperimental data. The two-exponential model, however, does not fit the experimental J c (H) curves in the higher H regime [see figs. 5 and 10(a)] and the generated using the two-exponential model does notfit the experimental curve in this regime. Qualitatively similar results exist for the other single ( β )phase Mo Re x alloys as well. We have earlier pointed out that the J c (H) curves in the Mo Re x alloysexhibit a clear departure from the expected [23] behaviour in the low H regime where the J c decreasesvery sharply with increasing H over quite a broad H regime. The analysis presented above indicates thatthis behaviour is because of the presence of two superconducting energy gaps in these alloys [9]. Thesmaller of these two energy gaps in the Mo Re x alloys probably does not exist in higher H . In thisregime the behaviour is reasonably explained by equation (2), while this equation fails in thepresence of two gaps in the low H regime. We have mentioned earlier that none of the present alloysprovide any indication of the presence of weak links, and this shows that the width of the grainboundaries in these alloys is less than the ξ [43]. In such a case the self-energy of a flux line is reduced atthe grain boundaries, and this gives rise to a pinning force [44- 46]. The grain boundaries are also knownto contain dislocations [46, 47] and thus become effective flux line pinning centres. We also recall thatthe Mo Re x alloys with the smaller grain size exhibit the higher J c (which is in the low H regime) inthe present study, indicating that grain boundaries are capable of pinning the flux lines in the Mo Re x alloys. Hence we find it reasonable to believe that grain boundary pinning is present even in the β phaseMo Re x alloys, though the fitting of equation (2) does not reveal the same. We therefore argue that the behaviour of the single phase Mo Re x alloys in the low H regime is probably due to grainboundary pinning modified by the presence of two superconducting gaps [41, 42]. In this logic theasymmetry of the isothermal M(H) curves in Fig. 3 may be ascribed to grain boundary pinning modifiedby the presence of two superconducting energy gaps in the Mo Re x alloys, along with the overall lowpinning strength in different H regimes. The features related to this two gap phenomenon are not reallyobserved in the multi-phase Mo Re x alloys. The presence of additional ( σ and χ ) phases with differentcrystal structures probably suppress the signatures of the presence of two superconducting gaps. FIG. 10(a).
Field dependence of J c of the as-cast Mo Re alloy, where the low field data hasbeen fitted by a two-exponential model. (b) The versus curve for the as-cast Mo Re alloy.The (black) open circles represent the experimental data, the (red) solid line represents the best-fitcurve obtained using equation (2), and the dashed (blue) line represents the valuesgenerated using the two-exponential model mentioned above.
4. Summary and conclusions
Polycrystalline samples of Mo Re x (15 x
48) alloys were synthesized using the arc meltingtechnique, and the samples were characterized using XRD, optical metallography, and EDX analysis.Single β phase alloys and with x
40 compositions, and alloys with x
40 containing multiple phases ( β , σ and χ phases) were identified. The grain size in the Mo Re x alloys was found to be varying from~100 μ m to mm level, and the multi-phase alloys were found to have smaller grains. Optical micrographsshowed the abundance of dislocation networks in the β phase alloys. The isothermal field dependence ofmagnetization of most of the alloys revealed a pronounced asymmetry between the field increasing andfield decreasing curves. Experimental studies revealed that this asymmetry is not related to the surfacebarrier effects. The critical current density and the pinning force density of the alloys were estimated fromthe field dependence of magnetization. Analysis using existing theories indicated that the dislocationnetworks and point pinning centers like voids and interstitial imperfections are the main flux line pinnersin the Mo Re x alloys in the “small-bundle” flux line pinning regime. The critical current density is alsofound to be relatively robust against increasing magnetic field in this “small-bundle” regime. In stillhigher fields, the critical current density is suppressed by flux creep. In the low field regime, on the otherhand, the dislocation networks and the point pinning centres mentioned above are not very effective inpinning the flux lines. Contrary to a previous report [8], the grain boundaries appear to pin the flux linesin this regime, and in fact, alloys with smaller grain size are found to exhibit higher critical currentdensity in the low field regime. This field dependences of the critical current and the pinning force densityof the present alloys in the low field regime were analyzed in the light of our finding that the Mo Re x alloys are a two gap superconducting system [9]. Our results indicate that the presence of twosuperconducting energy gaps in the present system modifies the field dependence of critical currentdensity, and this produces a low field maximum in the field dependence of pinning force density. Theasymmetry in the isothermal field dependence of magnetization seems to be related to the effect of thepresence of two superconducting energy gaps on the grain boundary pinning mechanism in low fieldsalong with the overall low flux line pinning strength in the higher field regimes. Acknowledgement
We thank Shri R. K. Meena for his help in sample preparation, and Dr. Gurvinderjit Singh for theXRD measurements. We also thank Shri Rakesh Kaul and Dr. M. A. Manekar for their help in opticalmetallography.
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