Influence of the Flux Pinning on Critical Current in HTc Superconductors with Nano-Sized Defects

Transcription

1 16 The Open Nanoscience Journal, 011, 5, 16-3 Open Access Influence of the Flux Pinning on Critical Current in HTc Superconductors with Nano-Sized Defects J. Sosnowski* Electrotechnical Institute Warsaw, Poaryskiego 8, Poland Abstract: Perfored analysis of the flux pinning on nano-sized centers, allowed to receive new results on the critical current probles in the high teperature oxide superconductors. A new odel of the interaction pancake vortices with the nano-sized centers is proposed and coparison of obtained results with others pinning forces approaches is presented. The energy balance approach is applied, which leads to the appearance of the potential barrier deterining the pancake vortices oveent in the flux creep process. Various initial positions of the captured vortex have been analyzed and variation of the screening currents for these configurations has been considered. Initial calculations of the influence of this effect on the current-voltage characteristics have been perfored. The dependence of the critical current on the pinning centers paraeters has been deterined, as well as coparison with the experiental data presented. Pinning force has been calculated and its teperature dependence, which is in qualitative agreeent with experients. The critical current deterines the agnetic induction profiles and therefore also flux trapping, very iportant paraeter fro the point of view of application HTc superconductors as peranent agnets. The influence of the critical current and granular structure paraeters on the flux trapping in ceraic superconductors is regarded. The influence of the critical current on a.c. losses in second generation HTc tapes with agnetic substrate is briefly discussed too. Keywords: High teperature superconductors, flux pinning, critical current. 1. INTRODUCTION The effect of capturing vortices in HTc superconductors has essential eaning fro the point of view of electric current transport. Therefore analysis of this subject is iportant both fro the scientific as well as technical point of view. To this subject indicate a lots of papers devoted to the flux pinning (see for instance [1-4] and references cited here). The present paper on other hand is the continuation of the present author s works [5-8]. While uch attention of the scientists is devoted ainly to the critical teperature analysis of the HTc superconductors, flux pinning in these aterials also brings new yet unresolved probles in coparison with the classic already low teperature superconductors. One of the is strong agnetic field decrease of the critical current in these aterials, which in fact liits real applications of HTc superconductors in electric achines. Fro other side the layered structure of HTc aterials leads to the existence of pancake type vortices, which allows already treat pinning interaction as individual one between pinning center and vortex. We reind that in the classic 3D aterials the flux lines ultifold captured to the various centers appear, which requires treating this case as a any body proble.. NEW MODEL OF THE PINNING INTERACTION High critical teperature oxide superconductors are characterized by unique superconducting paraeters, as the critical teperature but also the critical current. An essential *Address correspondence to this author at the Electrotechnical Institute Warsaw, Poaryskiego 8, Poland; Tel: ; Fax: ; E-ail: sosnow@iel.waw.pl Fig. (1). Scanning electron icroscopy (SEM) iage of the cross-section, of superconducting Bi-based tape, indicating to the existence of the nanostructural defects (black points nano-pores and unreacted CuO grains). The length of the cross-section is 34 μ. property of the high teperature oxide superconducting aterials, such as the critical current is very sensitive to the existence of structural defects. These defects called the colunar, if created during the fast neutrons irradiation of the superconducting windings working in nuclear reactors are then of the nanoetric size. SEM iage of the cross-section of the HTc Bi:3 tape shown in Fig. (1) indicates just to the existence of nano-diensional defects in their structure, in the for of nanopores or unreacted CuO grains. These defects interact with the pancake vortices arising in HTc superconductors, stabilizing thus the vortex structure and / Bentha Open

2 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 17 Fig. (). View of the vortex core of the radius equal to the coherence length 0 captured on the pinning center of the width d peritting in this way the resistivity-free current flow. For description of this dynaic interaction new odel of the vortex capturing has been developed, based on an analysis of the increase in the noral state energy during the pancake vortex deflection fro the nano-sized pinning center, against its equilibriu position. The geoetry of this interaction is shown in Fig. (). The above approxiation corresponds roughly to the consideration of the first two ters in the Ginzburg-Landau theory [9]. Two effects have been investigated: of the enhanceent in the noral state energy during the vortex oveent, as well as the opposite one, of an increase in the elasticity energy of the vortex lattice during the capturing process, deflecting the vortex fro its equilibriu position in the agnetic vortices array. In the initial state, it is for x = 0, of the vortex captured on the flat pinning center of nanosized diension d, the noral state energy of vortex core is: U 1 ( 0)= μ oh c l arcsin d d 1 d Paraeter shown in Fig. () is the coherence length, H c agnetic therodynaic critical field, l pinning center thickness, μ 0 agnetic pereability. Eq. 1 as well as the following relations are received in the geoetrical approach, in which vortex core is treated as a cylinder with well defined sharp walls. It eans that the square brackets in Eq. 1 and the subsequent ones describe the coon part of the geoetrical cross-section of the vortex core and pinning center of the perpendicular shape. For a vortex deflected on the distance x against this equilibriu position as shows Fig. (), the energy potential depth is given as: U l ( x)= μ oh c + dx arcsin d d for x<x c, where x C is defined as: 1 d x c = 1 ( d ) (3) The deflection of a vortex is caused by the flow of the current in the agnetic field, which leads to an rise of the (1) () Lorentz force, tearing the vortices fro the initial captured position. For a vortex deflected for a distance larger than x C the energy of the noral state connected then with shifted vortex is given as follows: U 3 ( x)= μ H o c l p + arcsin x + x 1 x (4) Fro relations 1-4 the value of the potential barrier is deterined: U ( x)= μ H ldx o c in the case of x<x c, while in the opposite case it is: U 3 ( x)= μ H o c l arcsin x + arcsin d + x 1 x + d 1 d The axiu of the potential barrier is reached just in this second case. In the total energy balance additionally has been taken into account the Lorentz force arising during the transport current flow, which affects the oveent of the vortices as well as elasticity forces of the vortex array. Elasticity forces are related to the agnitude of the captured vortex deflection in the vortex lattice. Capturing the vortices by the nano-sized pinning centers causes the shift of the vortex fro its equilibriu position in the regular vortex array, thus leading to an increase in the elasticity energy of the structure of the vortex lattice. This effect is the function of the deflection of the individual vortex fro its equilibriu position in the lattice. We have taken it into account by assuing that an enhanceent of the vortex elasticity energy is proportional to the square of the length of the vortex deflection, with coefficient of proportionality expressed by the value of the paraeter : U el = c s ( x) (5) (6) l a = ( x) (7) Paraeter c s is the corresponding elasticity shear odulus, while l a l denotes the length at which the agnetic flux lines of vortices are defored. Paraeter l as it was stated in Eq. 1 is the pinning center thickness. The above odel leads finally to the relation describing the po-

3 18 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (3). The dependence of the critical current satisfying the electric field criterion versus the pinning center diensions for various teperatures, given at each curve. Magnetic induction is constant and equal to 3T. Fig. (4). The influence of the nano-sized pinning centers of the width d, noralized to the coherence length on the critical current density versus applied agnetic field. tential barrier for the flux creep process U in the function of the reduced current density i = j/j C, where j C is defined as the critical transport current density for flux creep process. Condition for the iniu of the syste energy allows therefore to deduce the final expression for the energy barrier height, which should be crossed by the vortex in the flux creep process: U i ()= μ oh c l + 1 i ( 1 i ) (8) Function appearing in Eq. 8 is deterined here according to the relation: = arcsin d + d 1 d i 1 i arcsin i (9) Paraeter describes the elasticity energy of the vortex lattice. j C is defined here as the transport current density j for which the potential barrier disappears, in the large scale liit of the pinning center size, it is for d=, according to the relation: j c = μ 0 H c B S c (1 S c / a ) a (10) S c in Eq. 10 is the pinning center cross-section, while a is the average distance between the regularly ordered pinning centers and B is agnetic induction. We insert then the expressions 8-9 into the constitutive relation, describing the generated electric field in the flux creep process, just as a function of the potential barrier height: E = Ba exp U 0 k B T 1+ j U j exp C k B T (11) U 0 is the potential barrier height without current, the constant value describing the characteristic flux creep process frequency, T teperature and k B the Boltzann constant, paraeter a is the average distance between regularly arranged pinning centers. Applying above relations we can predict the dependence of the real critical current, i.e. satisfying the electric field criterion, on the aterial paraeters. Selected results of calculations are presented in Figs. (3-4) and they indicate to the iportance of the pinning centers diensions as well as the elasticity constant for the critical current agnitude. Fro Figs. (3-5) we see that for low pinning centers diensions as well as for too large elasticity constant of the vortex lattice, the critical current vanishes, which result should have iportant eaning for technological process. In the case of single crystalline HTc superconductors we should still reeber the layered structure of these aterials. It concerns especially the Cu-based superconductors although in the recently discovered Fe-based HTc superconductors layered structure appears too. The layered crystal structure of the HTc superconductor BiSrCaCuO is presented in Fig. (6).

4 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 19 Fig. (5). The dependence of the critical current on the elasticity force of the vortex lattice (described by paraeter ) versus external agnetic induction Fig. (6). Schee of the ultilayered structure of the Bi:1 high teperature superconductor Fig. (7). The influence of the inter-plane interaction (k nuber of the interacting CuO planes) on the current-voltage characteristics of the HTc layered superconductor Superconducting layers are intercalated here with the buffer ones, which supply electric charges to the CuO layers, responsible for the superconductivity effect, as shown in Fig. (5). The thickness of these layers is saller than coherence length o, so due to the proxiity effect the layers interact with neighbouring partners or rather ore precisely interact the vortices situated in the neighbouring planes influencing the current-voltage characteristics. The present odel allows in the first approxiation to consider this interaction, which leads to odification of the current-voltage characteristics in the individual plane, as indicates Fig. (7). In this Figure it is shown the voltage generated under current flowing in individual plane taking into account the interacting vortices in the neighbouring planes. Index k denotes just the nuber of interacting planes. 3. COMPARISON OF THE MODEL OF THE PINNING INTERACTION WITH EXPERIMENTAL DATA The above theoretical considerations we have applied then for coparison with experiental data. In Fig. (8) it is shown the coparison of the current-voltage characteristics calculated according to described atheatical odel of the pinning interaction with the experiental data obtained for Bi-based superconducting tape. An approxiation of the large pinning centers diensions has been applied here, which eans in the language of our odel that the case of d>> it has been considered.

5 0 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (8). Coparison of the experiental (*) and theoretical I-V curves for Bi:3 tape Fig. (9). Experiental dependence of the influence of bending strain on the current-voltage characteristics for the superconducting Bi-3 tape Fig. (10). Experiental dependence of the influence of bending strain on the critical current of the superconducting Bi-3 tape Very good agreeent of the theoretical odel with the experiental data has been received at various teperatures, while using coon fitting paraeter, which is the concentration of the pinning centers. It should be noticed that the defects concentration received fro the scanning electron icroscopy analysis indicated to siilar value. Additionally, structural defects such as icro-cracks are created as the result of the bending strain of the superconducting tapes, which effect frequently appears during winding of the superconducting coils. Experiental data of the influence bending strain on the current-voltage characteristics are given in Fig. (9), while critical current data versus bending strain is shown in Fig (10). Bending strain paraeter e, which appears in the Figs. (9-10) is defined as the ratio of the tape thickness t to its diaeter D: e= t/d. In Fig. (11) is presented the theoretical effect of the bending strain on the current-voltage characteristics, calculated by taking into account the present for of the pinning interaction. These theoretical results are in qualitative agreeent with the experiental data shown in Fig. (9). Knowledge of the

6 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 1 Fig. (11). Theoretically predicted influence of the bending strain e (in percents), on the current-voltage characteristics for Bi:3 tape Fig. (1). Theoretical dependence of the flux pinning on the agnetic induction versus teperature Fig. (13). Experiental dynaical anoalies of the current-voltage characteristics of YBaCuO ceraic, in function of the current aplitude. For curves 1-5 current is varying between 0 and 00 A respectively [11] critical current allows us to deterine the volue pinning force, which is force acting on vortices in the unit volue. To deterine this force there should be taken into account any body proble of the occupation varying with agnetic field nuber of the vortices on the fixed nuber of the pinning centers. Final result of the theoretical agnetic field dependence of the pinning force for three teperatures is shown in Fig. (1). The shape of this curve is in agreeent with experiental data [10] easured on Bi-3 tapes. Theoretical odel, which results are shown in Fig. (1), predicts the shift of the axial value of the pinning force with teperature, which was observed experientally in [10]. The presented odel allows to explain also the dynaic anoalies of the current-voltage characteristics of the HTc ceraics observed by us previously [11] and shown in Fig. (13). 4. INFLUENCE OF THE INITIAL CAPTURED VOR- TEX POSITION ON THE CRITICAL CURRENT DENSITY In detailed analysis of the pinning interaction we should still consider the echanis of vortex capturing on the pinning centers. Two topics are therefore regarded now, con-

7 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski nected with the geoetrical aspects of this capturing process. The first one is related to the initial position of the captured vortex, while the second to the renoralization of the screening currents around the vortex captured on the nano-sized pinning center. Now we consider this first point, while the second one will be the subject of the next clause 5. In the previous point it has been considered the vortex dynaics proble for the initial state of the vortex captured onto the depth of the renoralized coherence length, as shows Fig. (). Such configuration is preferred because it should allow aong other still for the flow of the superconducting screening currents, around the vortex core, due to the proxiity effect and keep this way the vortex structure. For evaluating the relevance of this initial configuration for the final results describing the potential barrier height and critical current density, the independent calculations have been perfored for another initial vortex state. We have considered therefore additionally the configuration of the fully captured vortex, it is the vortex core pinned at the depth of inside the pinning center of the nanoetric size. The basic equations describing the initial state noral energy for the fully captured vortex as well as for the vortex shifted against this initial position have been derived according to the ethod presented in the clause, while the results are given below. Initial noral state energy of the fully captured vortex, according to the notation of the Fig. () is equal to: U() = μ H 0 c l p arcsin d d d 4 (1) For the first deflection of the vortex fro this initial position it appears the dependence: U 1 (x) = μ 0 H c l p arcsin d + arcsin x d d 4 +x x valid for the following range of the vortex deflection: x 1 d (14) (13) For the next range of the vortex oveent described by the new boundary condition: 1 d x 1 d (15) the noral state energy of the shifted vortex is given by the relation: U (x) = μ 0 H c l p arcsin d d 1 d + dx (16) For still larger vortex shift the noral state potential is: U 3 (x) = μ 0 H c l p + arcsin x + x 1 x (17) Expressions for the potential barrier height, which deterine the current-voltage characteristics have been derived then and are given below for each of the considered cases: U 1 (x) = μ H 0 c l p arcsin x + x x (18) for the values of the vortex shift described by paraeter x, in the range given by Eq. 14. For larger shifts of the vortex in the range defined by Eq. 15, the potential barrier height is equal to: U (x) = μ 0 H c l p arcsin d + d d 4 + dx (19) For still higher vortex deflection potential barrier height is described by the relation: U 3 (x) = μ H 0 c l p + arcsin x + x 1 x + arcsin d +d d (0) Transforing the above dependences 18-0 fro translational into the current representation, in the case of the full vortex capturing we obtain final relation describing the potential barrier height in the function of the current density i = j/j c, reduced to the critical one: U 3 (i) = μ H 0 c l p arcsin(i) i( + 1 i )+ arcsin d + d 1 d (1) Eq. 1 we insert then into the constitutive equation 9 describing the current-voltage characteristics in the function of the potential barrier height U. The results of the coparison of the calculations of the current-voltage characteristics for both (fully captured and half captured) initial states are shown in Fig. (14). For better recognition of the behavior of the considered odel describing current-voltage characteristics of HTc superconductors, the calculations have been perfored as previously for various agnetic fields. As it follows fro Fig. (14) for both cases it is for the fully captured vortex as well as for the half-captured the differences in the current - voltage characteristics are at least in this case negligible. The influence of an initial position of the captured vortex on the critical current of the HTc superconductor in reduced units versus pinning center width is presented in Fig. (15).

8 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 3 Fig. (14). The current-voltage characteristics of the HTc superconductor with half pinned vortex (h) in initial state and fully pinned (f) for d/=0.4 versus the applied external agnetic field. Fig. (15). The dependence of the reduced critical current density on the pinning center width, for two initial positions of the vortex: 1 half captured, fully captured. Fig. (16). The dependence of the reduced potential barrier on the current noralized to the critical one for: (1) logarithic dependence ln (j/j c) of the potential barrier, () dependence of the power like for (j/j c) -1/7-1, (3) present odel for d/ = 0,15, (4) present odel for d/ = 0,5 (5) present odel for d/ = 0,95. The critical current density j c1 shown in this figure is defined as the current density leading to the disappearance of the energy barrier U in the given case, while j c has been defined previously in Eq. 5. Fig. (16) on the other hand shows the coparison of the reduced potential barrier versus the current noralized to the critical one for various pinning forces odels [1-13], bringing logarithic and power-like dependence of the potential

9 4 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (17). Scheatic coparison of the screening currents distribution and agnetic induction penetration into the vortex core and isolated nano-sized defect (at the top) and for the captured vortex (at the botto) in HTc superconductor. barrier. The results obtained in the present approach are arked by nubers 3-5 here and are siilar in fact, for both liiting cases, of the full and half captured vortices in the initial state. 5. INFLUENCE ON THE CRITICAL CURRENT OF THE MODIFICATION OF THE SCREENING CURRENT DISTRIBUTION IN THE CAPTURED VORTEX In previous points we have considered the odel describing the pinning interaction based on the energy balance, in which there has been applied an assuption of an unchanged electroagnetic shape of an isolated, free vortex and for the vortex captured on the nano-sized pinning center. In fact however we should reeber that the vortex is coposed fro the circulating currents around the vortex core. In a captured vortex the screening current distribution is odified as it indicated scheatically in Fig. (17). Generally description of such configuration is very coplicated fro the atheatical point of view. In the present paper we have applied therefore a very siplified approach, in which has been considered according to Fig. (17) the generalized agnetic flux quantization. For an isolated vortex and nano-sized pinning center the flux quantization conditions are fulfilled separately for both objects. However for the captured vortex the screening currents distribution is renoralized, while the quantization condition concerns now the total object of the captured vortex and pinning center. This very coplicated effect in the first approxiation we have described by an assuption of the odification of the induction distribution in the vortex core of the captured vortex. We have considered therefore the case in which agnetic field penetrates the superconducting aterial not only through the cores of the agnetic vortices but also by the noral state inclusions, observed just in the scanning electron icroscopy, for instance holes and noral grains. The Fig. (18) presents above geoetry. It shows the pancake vortex captured on the nano-pinning center of rectangular shape. If a vortex leaves the pinning center then the flux inside it is quantized and brings the value of 0 =, Wb. For captured vortex agnetic flux penetrating its core should be treated as the superposition of the agnetic field passing the noral state inclusion, which is proportional to the external agnetic field applied and the quantized flux of the agnetic vortex. It is generally a very coplicated effect, connected with the variation of the current distribution, because screening currents now surround both the vortex core as well as the nano-sized defect. We notice also that there appear soeties suggestions of transporting by the vortex not only individual flux quantu but also its ultiplication, despite of non - favours energetically such configuration, whose ideas are partly siilar to presented in this approach. In this first approach we apply here a rather rough approxiation of an unchanged cylindrical syetry of the vortex structure even in the captured state. If we approxiate further the agnetic induction profile in the vortex, through the London s relation: B(r) = B(0)exp( r ) () then the flux quantization condition leads to the following integral: 0 = B(0)exp( r )rdr = B(0) (3) 0 B(0) is the agnetic induction in the iddle of the vortex core, which now is siply proportional to the total agnetic flux in the vortex: B(0) = 0 (4) Eq. 4 indicates clearly that in the first approxiation above the agnetic flux carried by the vortex deterines the agnetic induction in the center of the vortex. It is especially iportant for the pinned vortex, whose geoetry is pre-

10 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 5 Fig. (18). View of the vortex captured on the nano-sized pinning center. The vortex core radius is equal to the coherence length, while full vortex size is of the range of the penetration depth. sented in the Fig. (18). The flux passing through the captured vortex area, is (according to the present, siplified odel) significantly odified in the respect to the conditions 3-4. Thus agnetic flux equal to 0 is quantized in isolated vortex, while in captured one, due to the significant odification of the electroagnetic circuit quantized should be already the total flux passing through the vortex core and the agnetic flux crossing the pinning center. In the first approxiation we have assued therefore, according to these assuptions and schee shown in Fig. (18), that the agnetic flux in the region of the captured vortex is enhanced in coparison to an isolated one. The agnetic flux in the captured vortex, is then a superposition of the flux quantu and of the agnetic field appearing in the coon region of the pinned vortex and epty hole or noral etal inclusion, acting as the pinning center. As it was stated previously, we assue further additionally that the cylindrical geoetry of the pinned vortex reains unchanged. In the initial state of the pancake vortex captured on the pinning center, as it shows Fig. (18), the enhanced agnetic flux 1 in the vortex area is therefore according to these considerations equal: 1 = 1+ B e arcsin d d 1 d (5) where B e is the agnetic induction, passing through the pinning center, while d denotes the width of the pinning center of a rectangular shape. Paraeter B e is roughly proportional to the applied external agnetic induction. For the case of the vortex slightly shifted fro the pinning center the agnetic flux inside it is given by the following forula: 1 0 = 1+ B e 0 d arcsin + d 1 d d x (6) while for deflection of the vortex core fro the pinning center larger than x c1 the agnetic flux in the vortex area is equal to: 1 0 = 1+ B e 0 arcsin x x 1 x (7) Paraeter x c1 denotes the liiting value of the vortex deflection distinguishing these both regions, for which Eq. 4 or Eq. 5 are valid respectively and is described by the relation: x c1 = 1 d (8) As it follows fro Eq. 7, for fully released vortex, it is shifted on the length x =, the agnetic flux in vortex reaches the agnitude of 0. The agnetic flux 1 penetrating vortex deterines then according to Eqs. 3-4 the axiu agnetic induction in the center of vortex core B(0) and influences in this way agnetic induction distribution inside the vortex, according to the relation. Conditions -4 deterine the new value of the vortex core radius, which creates noral state part of the vortex, in the language of Fig. (18) described by the effective coherence length, ay be slightly different fro original coherence length. The new radius of the vortex core 1 is deterined according to Eqs. 5-7 by the relation: μ 0 H c = 1 exp( 1 ) (9) where H C is therodynaic critical agnetic field, μ 0 agnetic pereability, while agnetic flux 1 is described by Eqs We apply then the condition joining the therodynaic critical agnetic field with coherence length of an isolated vortex: μ 0 H c = 0 exp( 0 ) (30) where 0 and 0 describe the flux quantu and noral core radius of an isolated vortex. According to Eqs the expression for the odified length of the noral core radius is given by the relation:

11 6 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (19). Calculated current-voltage characteristics for the HTc superconductor as a function of the reduced applied agnetic field H red Fig. (0). The dependence of the critical current density on the rigidity of the vortex lattice, expressed by the paraeter as a function of the agnetic flux 1: (1) 1= 0 () 1 > 0 (3) > = 1+ ln 1 0 (31) where the Ginzburg-Landau theory paraeter = characterizes the kind of the superconductor. The present ap- 0 proach taking into account in the first approxiation the electroagnetic nature of the interaction pancake vortex with the nano-sized pinning centers, leads therefore to the odification of the noral state core radius 1 of the captured vortex. This paraeter has essential eaning for an analysis of the energy barrier in the flux creep process, as it was shown in point. The results of the calculations current-voltage characteristics according to elaborated odel in the function of the reduced agnetic field H red = B e are shown in Fig (19). 0 The strong influence of the value H red on the current-voltage characteristics is well seen in Fig. (19). The above result is in agreeent with an experiental observation, that the applied agnetic field reduces the critical current. The critical current density is deterined then, as the value for which potential barrier vanishes. The influence on the critical current density of the paraeter, describing the rigidity of the vortex lattice, also as a function of the agnetic flux 1 appearing in the relations 5-7 is presented in Fig (0). The new odel of the pinning echanis allows therefore to predict the critical current of the HTc aterials as a function of the vortex capturing paraeters. The critical current is an essential paraeter fro the point of view of application of HTc aterials in the energy transport process. On the other hand critical current deterines also the screening properties of bulk HTc superconductors, which are iportant for application of these aterials as peranent agnet, for instance in levitation processes. We should notice that in bulk HTc aterials, due to their granular structure, there appear not only intragranular currents but also intergarnular, which can be treated as Josephson s like currents. The peculiar screening properties of HTc superconductors

12 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 7 Fig. (1). Full agnetic hysteresis loop of the Bi 1.8Pb 0.Sr Ca Cu 3O 8 HTc superconducting ceraic in high agnetic field Fig. (). Full agnetic hysteresis loop of the YBa Cu 3O 7-x HTc superconducting ceraic in high agnetic field. are related to the flux trapping, while analysis of this effect is perfored in the next point. 6. TRAPPED MAGNETIC FLUX ANALYSIS IN HTC CERAMICS The trapped agnetic flux (Ftr) is an iportant paraeter dependent on the flux pinning and critical current. It is defined as the reanent oent of the agnetization curve in the agnetic field cycle 0 B 0, where B is the axial agnetic induction in the cycle. An exaple of irreversible agnetization curves for a high teperature superconductor (Bi-3 bulk ceraic) and YBa Cu 3 O 7-x ceraic superconductor is shown in Figs. (1-). Both these curves indicate to the existence of agnetic irreversibility, which eans that trapped flux is coon property of type II superconductors. The existence of sall flux jups is also observed here for both directions of the agnetic field variation. The trapped agnetic flux is easured by the Hall probe ethod as well as just by the reanent oent of the agnetization curve. In best acro-granular YBaCuO saples the trapped flux reaches 17 T at 9 K, which value clearly indicates to the agnitude of this effect in HTc superconductors and creates therefore the attractive possibility of applying these aterials as the peranent agnets. Fig. (3) presents the calculated odel of a levitating train using peranent agnets oppositely oriented and levitating above the a HTc superconductor in Meissner state. Alternative experients are perfored for the superconductor in the flux trapped state suspended below shown in Fig. (3) peranent agnets. Fig. (4). shows agnetic field profiles in the superconducting bearing constructed fro HTc superconducting tube rotating on a core coposed fro six peranent agnets. Fig. (5) presents the schee of the agnetic induction distribution in the flux trapped state. Existence of the intergranular Josephson s currents, entioned previously and the current inside grains lead to the tooth-like shape of the agnetic induction profile given scheatically here. In order to analyse the trapped flux value we have considered four cases dependent on the aplitude of the external agnetic induction B in the agnetizing cycle. The first case finds place for the axiu induction B lower than the first penetration field B c1. Then the agnetic induction does not penetrate into the volue of the superconductor, while the trapped flux vanishes therefore. In the second region of agnetic field variations described by the relation: > B 1 > 0, where B 1 = B B c1, trapped flux being athe-

13 8 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (3). Calculated agnetic field lines in the odel of the levitating train - superconductor above two oppositely oriented peranent agnets, for two distances Fig. (4). Cross-section of the agnetic bearing coposed fro the HTc superconducting cylinder levitating above 6 peranent agnets. Fig. (5). Magnetic induction profile in the flux trapped state of the HTc ceraic superconductor for the agnetic field cycle 0 B 0. Sharp changes of the agnetic distribution are due to existence of the grains. atically the integral over the superconductor volue fro the induction profile shown in the Fig. (5) is expressed by the following relation, in an approxiation of the constant current density in the cylindrical geoetry saple: = B 1 B 1 + nb sg (3) Paraeter describes here the full penetration agnetic induction inside the superconducting aterial of the radius R, without surface barrier effects: = μ 0 j c R (33) In Eq. 3 has been taken into account according to the Fig. (5) the existence of the superconducting grains iersed into intergranular atrix, filling of which is described by the relative concentration n, varying between 0 and 1. An aount of the trapped flux connected with an individual grain, divided on its cross-section we denote by the sybol B sg, described by the dependence: B sg = Bc 1g + g (34) 3 Bc 1g denotes the first critical agnetic field in the superconducting grains, while g is deterined by the relation: g = μ 0 j cg R g (35)

14 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 9 Fig. (6). The influence of the filling with superconducting grains of the superconductor on the square root of the flux trapping Ftr. Nubers at curves indicate to values of the reduced grain concentration varying between 0<n<1. Fig. (7). The influence of the first critical agnetic field B c1 of the superconducting atrix on the trapped agnetic flux Ftr: (1) B c1 = 100 T, () 00 T, (3) 300 T, (4) 400 T. Fig. (8). The influence of the first critical agnetic field B c1q of superconducting grains on the trapped agnetic flux Ftr: (1) B c1g = 100 T, () 00 T, (3) 300 T, (4) 400 T. R g is the radius of the superconducting grain and j cg intragrain current density. For the next range of the agnetic induction increase deterined by the condition B 1 trapped agnetic flux agnitude, averaged to the cylindrical saple crosssection is equal: = nb g B 1 B B 1 3 (36) flux trapping, which perits then to find the optiu value of these paraeters. Exaples of calculations of the dependence of the trapping agnetic flux in the cylindrical saple on the axiu agnetic induction aplitude B, versus aterials paraeters are given in Figs. (6-9) and really indicate on the iportance of a given paraeter for obtaining axial flux trapping. For axiu applied agnetic induction in the field cycle, described by the condition B 1 the trapped flux reaches the constant value: = 3 + nb 1g (37) Theoretical results given by Eqs allow then to deterine the influence of various aterial paraeters on the Fig. (9). The influence of the intergranular critical current of the ceraic superconductor on the square root of the flux trapped (Sqrt(Ftr)) versus axial agnetic field in the cycle. Indices 1-3 indicate to the case of the increasing critical current.

15 30 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski Fig. (30). Dependence of the flux trapping in a superconducting plate on axiu agnetic induction in the cycle B, for various critical current densities inside the superconducting grains. Subsequent curves refer to the following current densities: (1) j cq= 10 1, () 5*10 11, (3) 10 11, (4) 5*10 10, (5) 10 10, (6) 5* 10 9 A/ Fig. (31). Dependence of the square root of flux trapping Ftr in a superconducting plate on axiu agnetic induction in the cycle B for various critical current densities inside the superconducting atrix. Fig. (3). Dependence of the flux trapping in superconducting plate, on axiu agnetic induction in the cycle B for various first penetration field B c1 Fig. (33). Dependence of flux trapping in a superconducting plate on axiu agnetic induction in a cycle B for various flat saple thicknesses x. Beside cylindrical saples described above the case is also significant of the flat ones, which roughly corresponds to the technical tapes and plates. We will analyze now therefore flux trapping for that geoetry. As previously four cases dependent on an external agnetic induction aplitude have been considered. The results in each of the are described by Eqs , while the graphical presentation of obtained relations is shown in Figs. (30-33). = 0 (38) = B (39) 4B p = B B B p 4B p + nb g( B 1) (40) B p = B p + nb (41) g B as previously is the axiu agnetic induction in a field cycle, while B p induction of the full penetration.

16 Influence of the Flux Pinning on Critical Current The Open Nanoscience Journal, 011, Volue 5 31 Fig. (34). Experiental dependence of the easured flux trapped versus axiu applied agnetic induction B for HTc superconducting ceraic of the YBa Cu 3O 7-x type doped with Fe ipurity. Fig. (35). Calculated a.c. losses in reduced for L/j in a superconducting tape of the second generation G, noralised to the transport current density j versus current density, for various values of the paraeter given at each curve The atheatical odel presented perits also for taking into account the surface barrier arising on the sooth edge of the superconductor. Then we obtain the following set of equations describing flux trapping in as a function of the axiu external induction aplitude B defined now according to: B = B ax B c1 B (4) In Eq. 4 B c1 is the first critical induction, while B the surface barrier. In the first range of field variation described by the relation: B c1 + B B ax B c1 + B (43) the trapped flux is equal to: = j R + B 3 j c c B + R R R 3 j c 3 (44) Another relation finds place for agnetic induction within the field range: B c1 + B B ax j c R + B c1 + B (45) ( ) 1 = 3R( B 6R B + B) B3 + 3B B B + B B (46) j c j c Within the agnetic induction range described by condition: B c1 + B + j c R B ax B c1 + B + j c R (47) appears the relation: ( ) = j c 3R 3R ( B B + B)+ B R 3 + 3B B B + B B j c j c 4 j c (48) For still higher agnetic field described by: j c R + B c1 + B B ax (49) the trapped flux reaches constant value: = B + j c R (50) 3 The presented odel allows to follow the influence of superconducting ceraic aterial paraeters on the flux

17 3 The Open Nanoscience Journal, 011, Volue 5 J. Sosnowski trapping, which should be useful in technological probles having aiing at optialization of this flux and generally of agnetic paraeters of the superconductors. On the other hand in this way it is possible fro the flux trapping easureents to receive inforation on such superconducting paraeters as critical current, first penetration field and surface barrier. Experiental results of flux trapping easureents perfored on YBa Cu 3 O 7-x ceraic Fe doped, according to the reference [8], are shown in Fig. (34) and indicate to a good agreeent with the theoretical predictions shown in Figs. (5-33), which independently confir the theoretical odel. As it has been proved previously, defects and grains boundaries deterine the critical current in HTc superconductors and their agnetic properties. Alternating current flow, on the other hand leads to the generation of losses in HTc tapes. Beside pure hysteresis type losses in the odern G tapes the a.c. losses are enhanced due to the agnetic nickel substrate on which HTc fils of YBaCuO type are deposited. Magnetic substrate leads to an enhanceent of the alternating agnetic field in the superconducting fil. An exaple of the calculations of the influence on a.c. losses L of the agnetic characteristics of the nickel substrate described by the expression B=0.65 tanh( H), where is aterial paraeter is shown in Fig. (35). Results presented here indicate to an iportant enhanceent of a.c. losses with the paraeter describing the shape of the agnetic aterial characteristics. CONCLUSIONS A new odel of the pinning interaction has been presented, allowing to deterine current-voltage characteristics, critical current, flux pinning and flux trapping in HTc superconductors. The odel is based on a detailed analysis of the interaction pancake type vortices with nano-sized pinning centers. A coparison with experiental data of theoretical results is given and a good agreeent has been obtained. Influence of the critical current on a.c. losses in tapes of G type with agnetic substrate is discussed briefly. ACKNOWLEDGMENTS Author is grateful to Drs. Eng. Joanna Warycha and Daniel Gajda for perforing scanning electron icroscopy analysis and agnetization easureents. REFERENCES [1] Yaada, H.; Yaasaki, H.; Develos-Bagarinao, K.; Nakagawa, Y.; Mawatari, Y.; Nie, J.C.; Obara, H.; Kosaka, S. The pinning property of Bi-1 single crystals with colunar defects. Supercond. Sci. Technol., 004, 17(1), [] Okaura, K.; Kiuchi, M.; Otabe, E.S.; Yasuda, T.; Matsushita, T.; Okayasu, S. Flux pinning centres correlated along the c-axis in PLD-YBCO fils. Supercond. Sci. Technol., 004, 17(), S0-S4. [3] Zhao, Y.; Cheng, C.H.; Feng, Y.; Shibata, S.; Koshizuka, N.; Murakai, M. The interpretation of iproved flux pinning behaviour and second agnetization peaks observed in overdoped Cu-rich BiSrCaCuO8+x single crystals. Supercond. Sci. Technol., 004, 17(), S83-S87. [4] Misko, V.R.; Savel'ev, S.; Rakhanov, A.L.; Nori, F. Negative differential resistivity in superconductors with periodic arrays of pinning sites. Phys. Rev. B, 007, 75(), [5] Sosnowski, J. Flux Pinning in HTc Superconductors. Int. J. Condensed Matter, Adv. Mat. Supercond. Res. USA, 008, 6(3-4), [6] Sosnowski, J. Dynaic vortex otion in anisotropic HTc superconductors. Mat. Sci. Poland, 005, 3(3), [7] Sosnowski, J. The influence of the Pinning Centres Diensions on the Critical Current in High-Tc superconductors. Acta Physica. Polonica., 004, 106(5), [8] Sosnowski, J. Critical current probles in HTc superconductors with nanoscale pinning centers. EUCAS 005 Proceedings, J. Physics, Conference Series, 006, [9] Ginzburg, V.L.; Landau, L.D. On the theory of superconductivity. Sov. Phys. JETP, 1950, 0, [10] Koblischka, M.; Sosnowski, J. Teperature-dependent scaling of pinning force data in Bi-based high Tc superconductors. Eur. Phys. J., 005, B44, [11] Sosnowski, J.; Datskov, V.I. Noral and inverse anoaly of dynaical current-voltage characteristic of high Tc oxide superconductors. Cryogenics, 1993, 33(1), [1] Feigelan, M.V.; Geshkenbein, V.B.; Larkin, A.I.; Vinokur, V.M. Theory of collective flux creep. Phys. Rev. Lett., 1993, 63, 303. [13] Larkin, A. I.; Ovchinnikov, Y.N. Pinning in Type II Superconductors. J. Low Tep. Phys. Vol. 34, p. 409, Received: Deceber 8, 010 Revised: March 19, 011 Accepted: March 5, 011 J. Sosnowski ; Licensee Bentha Open. This is an open access article licensed under the ters of the Creative Coons Attribution Non-Coercial License ( which perits unrestricted, non-coercial use, distribution and reproduction in any ediu, provided the work is properly cited.