Vortex-glass transformation within the surface superconducting state of β -phase Mo 1−x Re x alloys
Shyam Sundar, M. K. Chattopadhyay, L. S. Sharath Chandra, R. Rawat, S. B. Roy
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Vortex-glass transformation within the surfacesuperconducting state of β -phase Mo − x Re x alloys Shyam Sundar , , M. K. Chattopadhyay , , L. S. SharathChandra , R. Rawat and S. B. Roy , Homi Bhabha National Institute, Raja Ramanna Centre for Advanced Technology,Indore, Madhya Pradesh-452013, India Magnetic and Superconducting Materials Section, Raja Ramanna Centre forAdvanced Technology, Indore, Madhya Pradesh-452013, India UGC-DAE Consortium for Scientic Research, Khandwa Road, Indore, MadhyaPradesh-452001, IndiaE-mail: [email protected]; [email protected]
August 2016
Abstract.
We have performed an experimental study on the temperature dependence ofelectrical resistivity ρ ( T ) and heat capacity C ( T ) of the Mo − x Re x ( x = 0 . , . Keywords : Vortex-glass state, Surface mixed state, Surface superconductivity, Electricalresistivity, Metals and alloys. ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys
1. Introduction
The phenomenon of surface superconductivity was discovered by Saint-James and deGennes back in 1963 [1, 2]. They showed that for any finite size sample, the nucleationof the superconducting region is easy to occur near the sample surface in the case ofa metal-insulator (or metal-vacuum) interface, in the presence of a parallel magneticfield H c higher than the upper critical field H c by a factor of 1.695. They alsoargued that the nucleation field H c for surface superconductivity may be stronglymodified by the normal metal coating (e.g. Cu, Ag) at the sample surface [3, 4].With the discovery of surface superconductivity, it was possible to explain a largeamount of experimental data, which had previously been discarded as due to the sampleinhomogeneity [5]. Later, many researchers studied the surface superconducting statein many superconductors such as - Pb-Tl, Nb, MgB [6, 7, 8, 9, 10, 11, 12]. Initially itwas thought that for surface superconductivity (and surface critical current) to exist,the local magnetic field needs to be parallel to the sample surface [7, 13, 14, 15, 16].Subsequent work, however, showed that surface superconductivity (and surface criticalcurrent) may be observed even when the local magnetic field has a nonzero perpendicularcomponent arising because of the misalignment of the applied magnetic field relative tothe sample surface, due to the magnetic field related to an applied transport current, thedemagnetization factor and the local roughness of the sample surface [17]. It was inferred[17] that the perpendicular component of the magnetic field ( H > H c ) may penetratethe sample surface as an array of quantized flux-spots (Φ = h/ e ≈ . × − Wb) whose free energy is sensitive to the local properties of the surface sheath of thesuperconductor. Since the local properties of the surface sheath will vary spatially, thefree energy of the flux spots will also vary spatially and the flux spots will stay pinnedat the locations of minimum free energy (see Fig. 1) [17]. The pinning of the fluxspots depends on the degree of surface roughness, and becomes increasingly significanttill the scale of roughness becomes comparable with the spacing between the flux spotsapproximately [17]. This suggests the existence of a dense quantized 2d flux lattice (orflux-spots) in the surface region of superconductor [17]. The absence of surface pinningor the presence of a driving force (due to a transport current in the presence of magneticfield) greater than the pinning strength, allows the array of flux-spots to move freely inthe surface superconducting region producing a non-zero voltage and a vanishing surfacecritical current [17]. It has also been shown that these 2d flux-lines have the same kindof flux-line dynamics as that of the bulk (3d) superconducting mixed state of the typeII superconductors, e.g., the flux flow and the flux creep effects [17, 18, 19, 20, 21], andin the presence of disorder, may enter a critical state, known as the Kulik vortex-state[22, 23]. In-spite of a variety of similar flux-line dynamics as in the bulk mixed state,there is no report of surface vortex-glass, which can act as a 2d counterpart of the vortex-glass state in the superconductors. However, the existence of a true 2d vortex-glass stateat any finite temperature is highly controversial. Theoretically, the 2d vortex-glass stateis possible only at T = 0 K, which has been supported by many authors [24, 25, 26]. ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys Figure 1.
Schematic diagram explaining the concept of the flux spot model [17].
Superconductivity in the Mo − x Re x alloys is quite interesting in the sense that the T c in these alloys is enhanced by an order of magnitude, as compared to their constituentelements (Mo and Re) [31, 32]. Apart from the unique superconducting and normalstate properties, the Mo − x Re x alloys are technologically important in terms of thefabrication of superconducting radio frequency (SCRF) cavities [33]. Another unusualaspect associated with the superconducting state of the Mo − x Re x alloys is the presenceof a surface superconducting mixed state [34, 35]. In present paper, the temperaturedependence of electrical resistivity, ρ ( T ) across the superconducting transition in variousapplied magnetic fields has been measured, to explore the surface mixed state of β -phaseMo − x Re x ( x = 0 . , .
25) alloys. The bulk superconducting transition temperature hasbeen obtained by measuring the temperature dependence of heat capacity in differentapplied magnetic fields. It is found that the resistivity goes to zero in the surfacesuperconducting state well above the onset of the bulk superconducting transition, dueto the pinning of the flux-lines in the surface superconducting region. This flux-linepinning also leads to a vortex-glass state in the surface superconducting state. Theestimated critical exponents corresponding to the vortex-glass to liquid transition isfound to be in-line with the 2d nature of the vortex-glass state. A detailed H - T phasediagram is presented.
2. Experimental details
Polycrystalline samples of Mo − x Re x , ( x = 0.20, 0.25) alloys were prepared by meltingmolybdenum and rhenium (99.95+ % Purity) in an arc furnace under high purity (99.999 ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys − x Re x alloy samples were measured in thetemperature range 1.5-15 K and up to 2 T magnetic field using the standard four probetechnique with the help of a superconducting magnet cryostat (Oxford Instruments,UK) system. The temperature dependence of electrical resistivity in 0, 0.5, and 1 Tmagnetic fields were also measured after applying a Cu coating on the Mo . Re . alloy and a Ag coating on the Mo . Re . alloy. The Cu coating on the Mo . Re . alloy was applied using the electroplating technique (with the help of a CuSO solution).Some portions of the Cu coating was then removed by filing for putting the four probeelectrical contacts. The Ag coating on the Mo . Re . alloy was put by dipping thesample in electrically conducting Ag paint after putting the electrical contacts for thefour probe technique. A constant current of 100 mA was used to measure the resistivityof these alloys. Heat capacity measurements were performed in the temperature range2-15 K in various magnetic fields up to 3 T using a Physical Property MeasurementSystem (PPMS, Quantum design, USA).
3. Results and Discussion zero field 0.1 T 0.2 T 0.3 T 0.5 T 0.75 T 1 T 1.2 T 1.5 T 1.75 T 2 T Mo Re ( a ) (- c m ) zero field 0.1 T 0.2 T 0.3 T 0.5 T 0.75 T 1 T 1.2 T 2 Mo Re (- c m ) T (K) ( b ) Figure 2.
Temperature dependence of electrical resistivity, ρ ( T ) of the Mo − x Re x alloys measured across the normal to superconducting phase transition in differentmagnetic fields. Figure 2 shows the temperature dependence of electrical resistivity in zero and ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys T c ) for the present alloys have been reported elsewhere [36] and they are in goodagreement with previously reported values [37]. The width of the superconductingtransition (∆ T c ) for different field values was estimated as the temperature differencebetween the 10% and 90% of the normal state resistivity ( ρ N ). T C ( K ) Mo Re (a) Mo Re T C ( K ) H (T) (b) Figure 3.
Magnetic field dependence of the width of the superconducting transition(∆ T c ) in the Mo − x Re x alloys . The solid lines in the panels ( a ) and ( b ) show thefitting of the experimental data using a linear field dependence (Eq. 1). In Fig. 3, the ∆ T c as a function of magnetic field is plotted, which is well describedusing a linear relation (Eq. 1),∆ T c ( H ) = ∆ T c (0) + kH (1)where, ∆ T c (0) is the width of superconducting transition in zero field and k is aconstant. The presence of multiple superconducting phases in a sample (and theresulting spatial distribution of T c ) has often been considered to be a possible reasonbehind the broadening of the normal-to-superconducting phase transition [38, 39]. Thefitting of Eq. 1 to the ∆ T c versus H data, however, negates this possibility. A samplewith multiple superconducting phases would have shown an upturn (or curvature) in thelow field side of the ∆ T c versus H plot due to the different field dependencies of T c in thedifferent superconducting phases [38]. Moreover, the XRD and optical metallographystudy of both the Mo − x Re x ( x = 0 . , .
25) alloys [36], do not indicate the presence ofa second phase.Apart from the existence of multiple superconducting phases in the sample, thebroadening of the superconducting transition may also be due to the presence of multiple ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys (- c m ) T (K)Zero field Mo Re (a) C / T ( m J / m o l e - K ) (- c m ) T (K) Mo Re Zero field (c) C / T ( m J / m o l e - K ) (- c m ) T (K) Mo Re H = 0.5 T 34567 C / T ( m J / m o l e - K ) (d) (- c m ) T (K) Mo Re H = 1 T 4567 C / T ( m J / m o l e - K ) (b) Figure 4. Panel ( a ) and ( c ): Temperature dependence of electrical resistivity ρ ( T )and heat capacity C ( T ) in zero magnetic field across the superconducting transition. Panel ( b ) and ( d ): ρ ( T ) and C ( T ) in H = 1 T and H = 0.5 T across thesuperconducting transition for x = 0.25 and x = 0.20 respectively. In all the panels,the (blue) solid circles (dots) represent the C ( T ) measured in different H (specifiedin each of the panels), the (black) solid triangles represent the ρ ( T ) measured beforecoating the samples with Cu or Ag, and the (red) open squares represent the ρ ( T )measured after putting these metal coatings. superconducting gaps [40, 41], vortex-glass state [42, 43], vortex-melting behaviour [41]and surface superconductivity [44, 45]. In this context, surface superconductivity hasalready been reported in Mo − x Re x alloys [34]. On the other hand, the superconductingproperties of β -phase Mo − x Re x alloys are significantly influenced by the presenceof two superconducting energy gaps [31]. To investigate the properties of surfacesuperconductivity, the ρ ( T ) and C ( T ) curves were plotted in the same temperaturewindow in zero field [figures 4 ( a ) and ( c )] as well as in a higher magnetic field ( > H C )[figures 4 ( b ) and ( d )]. In all the panels of Fig. 4, the (blue) solid circles (dots) representthe C ( T ) measured in different H (specified in each of the panels), the (black) solidtriangles represent the ρ ( T ) measured before coating the samples with Cu or Ag, andthe (red) open squares represent the ρ ( T ) measured after putting these metal coatings.Fig. 4 (a) and (c) show that in zero field, the signature of the superconducting transitionin the ρ ( T ) and C ( T ) curves are observed at the same temperature approximately, andthis signature in the ρ ( T ) curves is not affected by the presence or absence of the metalcoatings on the samples. In the presence of applied magnetic field greater than H c (figures 4 (b) and (d)), the signature of this phase transition in the not-coated samplesare observed at a higher temperature in the ρ ( T ) as compared to that in the C ( T ).However, after applying the metal coatings, in the presence of the same H , the signatureof this phase transition in the ρ ( T ) is observed to move towards lower temperatures and ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys C ( T ). The (four) probes used for the ρ ( T ) measurementare on the sample surface, hence, this measurement is influenced by the surface relatedphenomenon. The suppression of the normal to superconducting transition temperatureobserved in ρ ( T ) by applying a metal coating strongly suggests that this transition inthe presence of magnetic field is indeed due to surface superconductivity [3, 4]. The C ( T ) measurement, on the other hand, is influenced mainly by the bulk phenomenon.Figure 4 thus indicates that the superconducting transitions corresponding to the bulkand surface are distinctly different in the present alloys.In this investigation, the focus is on the superconducting transition and itsbroadening observed with the help of the ρ ( T ) measurements in the presence of appliedmagnetic fields. The present results (figures 2, 4), in conjugation with the flux-spotmodel (Fig. 1) [17], suggest that the nature of mixed state formed within the surfacesheath might give rise to the broadening of the superconducting transition in thesealloys. This is supported by the analysis of ρ ( T ) curves in different fields, presentedbelow.In zero magnetic field the superconducting transition is quite sharp (∆ T c < . − x Re x alloys. However, the superconducting transition is increasinglybroadened with the increase of applied magnetic field. Additionally, the superconductingtransition in the presence of magnetic field exhibits a rounding-off behaviour near theonset of the superconducting transition and a tailing effect near the completion. Therounding-off behaviour of ρ ( T ) is possibly related to the superconducting fluctuations[46]. The tail region of the superconducting transition becomes more significantas we go to higher magnetic fields, with a gradual suppression of the temperaturecorresponding to zero resistivity. Similar features have been observed in many bulksystems such as: Iron based superconductors: SmFeAsO . [47], BaFe As [48], thehigh- T c superconductors [49, 50] and even in some of the low T c superconductors:Nb thin films and Nb/Cu superlattices [51, 52], narrow strip of Nb [53], In film [54],amorphous Mo x Si − x and Mo Si films [55, 56]. Both the tailing-off behaviour as well asthe broadening of the superconducting transition are generally attributed to the effectof thermal fluctuations in the superconductors, and it gives rise to a rich variety of flux-line dynamics in the presence of quenched disorder and other defects [25]. Generally,the effect of thermal fluctuations is expected to be small in the case of the low T c superconductors. However, the effect of thermal fluctuations is greatly enhanced in thecase of the two-dimensional (2d) superconductor as compared to the three-dimensional(3d) ones [25].It is known that, these alloys are unlikely to be 2d superconductors [36]. On theother hand, the Ginzburg number, which quantifies the effect of thermal fluctuationsin the superconductors, is found to be 5 . × − and 1 . × − for the x = 0.20and 0.25 Mo − x Re x alloys respectively. The Ginzburg number for Mo − x Re x alloys wereestimated using the following relation [57]. G i = 32 π (cid:18) k B T c κλ GL (0)Φ (cid:19) (2) ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys k B is the Boltzman constant, T c is the superconducting transition temperature, κ is the Ginzburg-Landau parameter, λ GL (0) is the Ginzburg-Landau penetration depthat absolute zero temperature and Φ is the flux quantum. To estimate the value ofGinzburg number using Eq. 2, we have used κ = H c (0) √ H c (0) . The values of H c (0) (theupper critical field at absolute zero temperature) have been taken from ref.[36]. Thethermodynamic critical field, H c (0), and λ GL (0) are estimated from the heat capacitymeasurements and using λ GL (0) = κ × ξ GL (0), where ξ GL (0) = (cid:16) Φ πH c (cid:17) respectively. Inthis context, it may be noted that while the Ginzburg number generally comes out to be10 − -10 − for the high T c superconductors [57, 58, 59], for the low T c superconductorsthis number is found to be in the range of 10 − -10 − [59]. Thus, the Ginzburg numberfor the Mo − x Re x alloys is similar to those of the conventional low T c superconductors,indicating that the effect of thermal fluctuations may not be substantial in these alloys.In view of the experimentally observed signature of surface superconductivity in thepresent Mo − x Re x alloys, we believe that the formation of 2d pancake like flux-lines(Kulik vortex-state [22, 23]) within the surface sheath causes an enhancement of theeffect of the thermal fluctuations in these alloys. Therefore, similar to other systems[60, 61], the analysis of the ρ ( T ) behaviour is performed using the thermally activatedflux-flow (TAFF) model [62].In the TAFF model, the temperature dependence of electrical resistivity is describedby the Arrhenius relation [62], ρ ( T ) = ρ exp [ − U ( H, T ) /T ] (3)where, ρ is the pre-exponential factor independent of field and U ( H, T ) is the activationenergy. Arrhenius relation suggests that the lnρ vs. 1 /T should be linear in the TAFFregion.Figure 5 shows the Arrhenius plots for ρ ( T ) in different magnetic fields for theMo − x Re x ( x = 0.20, 0.25) alloys. The activation energy ( U ) is estimated by fitting astraight line (blue) to the experimental data in the TAFF region. The fitted straightline (blue) in both the panels of Fig. 5, shows a deviation from linearity at temperature T ∗ . In Fig. 5, the insets to the panels a and b show the activation energy U ( H ). Forboth the alloys, the U ( H ) shows a power law behaviour ( U = AH − α ) with α ≥
2. Highvalues of α has also been reported in other superconductors such as, iron pnictide andcuprates and has been described in terms of the collective flux-line pinning behaviour[63, 64, 49].To analyze the experimental data below T ∗ , we have estimated the activationenergy, U (T), using Eq. 3, U = − d (ln ρ ) /d (1 /T ) (4)The temperature dependence of the activation energy of the Mo − x Re x alloys areshown in Fig. 6. The activation energy increases rapidly with decreasing temperaturebelow a characteristic temperature T ∗ , which matches exactly with that pointed outin Fig. 5 as the temperature corresponding to the deviation from linearity. The rapid ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys T * zero field 0.1 T 0.2 T 0.3 T 0.5 T 0.75 T 1 T 1.2 T 1.5 T 1.8 T 2 T Mo Re l n () (a) U H (T) HU Mo Re l n () T (K -1 ) zero field 0.1 T 0.2 T 0.3 T 0.5 T 0.75 T 1 T 1.2 T T * (b) U H (T) HU Figure 5.
Arrhenius plots for ρ ( T ) in different magnetic fields for the Mo − x Re x alloys. In both the panels ( a and b ), the blue lines show the straight line fit to theexperimental data in the TAFF region and the arrows indicate the T ∗ value, where theexperimental data deviates from linearity. In the TAFF region, the activation energy U ( H ) is estimated, which is shown in the insets to both the panels for the respectivealloys. The U ( H ) shows a power law behaviour in the TAFF region, with α ≥ increase of activation energy below T ∗ marks the entry into a critical regime associatedwith the vortex-liquid to vortex-glass transformation, as previously observed in the othersuperconductors [49, 47, 65]. Figure 7 shows the temperature dependence of inverselogarithmic derivative of resistivity with the values of T ∗ marked by arrows. Thus, toinvestigate into the existence of vortex-glass state, we need to study the temperaturedependence of resistivity below the characteristic temperature T ∗ .According to the vortex glass theory [25], the electrical resistivity vanishes at theglass transition temperature T g , ρ ∝ ( T − T g ) s (5)Where, s is the critical exponent, defined as, s = ν ( z + 2 − d ), here ν is the staticexponent, z is the dynamic exponent and d is a dimensional factor ( d = 2 in the case ofsurface mixed state). The glass temperature T g is estimated by applying Eq. 6 to the ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys Mo Re U = - d ( l n ) / d ( / T ) T*( a ) Mo Re U = - d ( l n ) / d ( / T ) T (K) T* ( b ) Figure 6.
Temperature dependence of activation energy, U = − d (ln ρ ) /d (1 /T ), in theMo − x Re x alloys. In both the panels, for each curve, the arrow indicates the T ∗ value,which is the temperature below which the 2d flux-lines enter into the vortex-liquid tovortex-glass critical transformation region. tail region of the ρ ( T ) curves. (cid:18) d (ln ρ ) dT (cid:19) − = 1 s ( T − T g ) (6)According to Eq. 6, the inverse logarithmic derivative of resistivity is linearlyproportional to the temperature of measurement. Fig. 7 shows that the resistivityin the temperature range T g < T < T ∗ is well described by Eq. 6. In Fig. 7,the slope of the straight line fitted to the experimental data (within the temperaturerange T g < T < T ∗ ), gives the value of the critical exponent and its intercept on thetemperature axis gives the value of glass transition temperature T g . In figure 7, thedeviation from linearity at temperature T ∗ , marks the upper limit of the critical-regionassociated with the vortex-glass to vortex-liquid phase transformation. Fig. 6 and 7show that the ρ ( T ) in the temperature range, T g < T < T ∗ , is well described by thevortex-glass model.The magnetic field dependence of the critical exponents is shown in Fig. 8. Thecritical exponents increase with increasing magnetic field for both the alloys. As seen inFig. 8, the maximum attainable values of the critical exponents for the x = 0.20 and x ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys Mo Re ( d ( l n ) / d T ) - T g T* ( a ) Mo Re ( d ( l n ) / d T ) - T (K) T g T* ( b ) Figure 7.
Temperature dependence of inverse logarithmic derivative of resistivity inthe Mo − x Re x alloys in different magnetic fields. In both the panels, for each curve,the solid lines are fit to the experimental data and show the linear behaviour in thetail region. = 0.25 alloys are about 1.4 at H = 1.2 T and 2.4 at H = 1.8 T respectively. Accordingto the vortex-glass model [59], the critical exponent values are in between 2.7 - 8.5 for3d vortex-glass state [47]. On the other hand, the maximum permissible value of criticalexponent for the 2d vortex-glass state is reported to be 2.7 [47]. The small values ofcritical exponent in the present alloys indicate that the vortex-glass state has formed inthe surface sheath (Kulik-vortex state).It may be noted that in literature the small value of critical exponent has also beenattributed to the Bose-glass phase. The Bose-glass phase is formed due to the interactionof the flux-lines with correlated disorder, such as the twin boundaries and the columnardefects [59, 66, 67]. However, optical metallography studies and the published literatureon these alloys [36], do not show any indication of the presence of correlated disorder(twin boundaries, columnar defects).Figure 9 shows the detailed H - T phase diagrams for both the Mo − x Re x alloys.The characteristic values in the phase diagrams are obtained through ρ ( T ) and C ( T )measurements in different magnetic fields. In Fig. 9, the uppermost line with solidspheres shows the T c ( H ) values, estimated from ρ ( T ) in different applied magneticfields. The T c ( H ) values were estimated, as the temperature corresponding to the 90% ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys Figure 8.
Critical exponent s = ν ( z + 2 − d ), in different magnetic fields for theMo − x Re x alloys. The small value (less than 2.7 for both alloys) of the critical exponentis attributed to the 2d nature of the vortex-glass state [47]. T C (H) [ = 90 % N ] T * T g T C0 (H) ( = 0) T C onset (H) [ C vs. T ] Mo Re Vortex-glass ( a ) from C (T) Vortex-liquidCritical-region H ( T ) T C (H) [ = 90% N ] T * T g T C0 (H) ( = 0) T C onset (H) [ C vs. T ] Mo Re H ( T ) T (K) Vortex-glassfrom C (T) ( b ) Critical-regionVortex-liquid
Figure 9.
The H - T phase-diagrams for the Mo − x Re x superconductors. of the normal state resistivity ( ρ N ). Above this line, both the alloys are in the normalstate. The line with solid diamonds represents the T ∗ line, which corresponds to theupper limit of the vortex-glass critical-regime. The region between the T c ( H ) line andthe T ∗ line corresponds to the vortex-liquid behaviour, where the system shows non-zeroresistance because of flux flow. The region between the T ∗ and T g lines is demarcatedas the vortex-liquid to vortex-glass critical transition region. The T g line corresponds ortex-glass transformation within the surface superconducting state of β -phase Mo − x Re x alloys T c line (openstars) estimated from the ρ ( T ) curves marks the temperature, where resistivity goes tozero in the respective field. It is clearly observed in Fig. 9 that this zero resistivity linematches exactly with the T g line which further confirms the existence of the vortex-glassstate in Mo − x Re x superconductors. The onset of bulk superconducting transition wereestimated from the C ( T ) measurements in different magnetic fields. The T c onset ( H )values for bulk superconducting transition is defined as the temperature, where thesuperconducting transition just starts (onset of the peak in C ( T )). We observe inFig. 9 that all aspects related to the vortex-liquid to vortex-glass transformation takesplace above the bulk superconducting transition temperature. This phase diagram thusprovides additional evidence of the formation of the Kulik vortex-state.
4. Summary and Conclusion
In summary, the temperature dependence of electrical resistivity and heat capacitywere measured experimentally, and, the signature of vortex-liquid to vortex-glasstransformation within the surface mixed state or Kulik vortex-state in the Mo − x Re x ( x = 0.20, 0.25) alloys were observed in these measurements. The vortex-glass critical-regime were investigated using the vortex-glass theory [25]. The critical exponent,estimated from the vortex-glass theory increases with the increase in magnetic field,and nearly saturates at small critical exponent values (less than 2.7) for both the alloys.In the absence of correlated disorder, such small critical exponents indicate the possibleexistence of 2d vortex-glass state. Different characteristic temperatures, associated withthe vortex-glass transformation were estimated through analysis and the detailed H - T phase diagrams were constructed. The H - T phase diagrams clearly show that the phasetransformations associated with vortex-liquid to vortex-glass transition are taking placein the 2d surface mixed state, above the bulk superconducting transition temperaturecorresponding to the particular magnetic field applied. Acknowledgements
Authors would like to thank to Prof. Lesley F Cohen, Prof. Luis Ghivelder and Prof.Said Salem Sugui Jr. for interesting discussions and important suggestions.
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