Flux-cutting and flux-transport effects in type-ii superconductor slabs in a parallel rotating magnetic field

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1 Low Temperture hysics/fizik Nizkikh Tempertur,, v., No., p. 9 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez, Instituto de Físic, Benemérit Universidd Autónom de uel, Apdo. ost. J-8, uel, ue., Mexico Fcultd de Ciencis Físico-Mtemátics, Benemérit Universidd Autónom de uel Apdo. ost., uel, ue.,, Mexico Universidd del Istmo, Cmpus Tehuntepec, Tehuntepec, Ox.,, Mexico E-mil: fperez@ifup.up.mx Received April 9, The mgnetic response of irreversile type-ii superconductor sls sujected to in-plne rotting mgnetic field is investigted y pplying the circulr, elliptic, extended-elliptic, nd rectngulr flux-line-cutting criticlstte models. Specificlly, the models hve een pplied to explin experiments on Bi rotting disk in fixed mgnetic field H, prllel to the flt surfces. Here, we hve exploited the equivlency of the experimentl sitution with tht of fixed disk under the ction of prllel mgnetic field, rotting in the opposite sense. The effect of oth the mgnitude H of the pplied mgnetic field nd its ngle of rottion α s upon the mgnetiztion of the superconductor smple is nlyzed. When H is smller thn the penetrtion field H, the mgnetiztion components, prllel nd perpendiculr to H, oscillte with incresing the rottion ngle. On the other hnd, if the mgnitude of the pplied field, H, is lrger thn H, oth mgnetiztion components ecome constnt functions of α s t lrge rottion ngles. The evolution of the mgnetic induction profiles inside the superconductor is lso studied. ACS:..H Mgnetic properties including vortex structures nd relted phenomen;..op Mixed sttes, criticl fields, nd surfce sheths;..wx Vortex pinning (includes mechnisms nd flux creep). Keywords: flux cutting, flux trnsport, vortex pinning, criticl stte, hrd superconductor.. Introduction The discovery of the phenomenon known s qusisymmetricl collpse of mgnetiztion [], which is oserved in superconductors sujected to crossed mgnetic fields nd well interpreted within the simple Ben s criticl-stte model [,], hs een turning point in the understnding of the mgnetic ehvior of hrd (irreversile type-ii) superconductors. Until then, the generlized doule criticlstte model (GDCSM) [ 8], which is sed on fundmentl physicl concepts such s flux trnsport nd flux-linecutting [9,], ws successfully employed to explin vriety of experiments where flux cutting occurs [ ]. An importnt feture of the GDCSM is the ssumption tht flux cutting nd flux depinning do not ffect ech other. Besides, the GDCSM is inherently nisotropic ecuse the thresholds for these two effects re given y two independent prmeters, nmely the criticl current densities prllel nd perpendiculr to the locl mgnetic induction B. However, since the GDCSM cnnot reproduce the fetures of mgnetic moment collpse [,8], wheres isotropic Ben s model does it, the min ssumption of the GDCSM hs een questioned, motivting the development of new criticl-stte models in the pst few yers. In Ref. 9, the so-clled elliptic flux-line-cutting criticl-stte model ws proposed. This model introduces the nisotropy, induced y flux-line-cutting effects, y using procedure similr to tht for structurlly nisotropic superconductors [,], i.e. the mgnitude of the criticl current density J c, eing the only prmeter used within the isotropic Ben s model, is sustituted y symmetricl tensor ( J c) ik with principl vlues J c nd, corresponding to the directions long nd cross the locl mgnetic induction B. In good greement with the experiment on Y Cu O δ smples [,], the elliptic criticl-stte R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez,

2 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field model predicts the qusisymmetricl suppression of the verge mgnetiztion < M z, for prmgnetic nd dimgnetic initil sttes, y sweeping trnsverse field H y of mgnitude much smller thn dc-is mgnetic field H z [9,]. When the mgnitudes of the crossed fields H y nd H z re comprle, the vlue of the mgnetiztion < M z fter mny cycles of the trnsverse field H y turns out to e positive for oth dimgnetic nd prmgnetic initil sttes if. To our knowledge, the oservtion of such prmgnetism of hrd superconductors ws first reported in Refs.,. The elliptic model lso descries the ehvior of < My ( H y) nd < M z ( H y) in crossed fields H y nd H z [9,], which ws oserved in the experiments on VTi rion with nonmgnetic initil stte [,]. Here, the good greement with the experiment ws chieved y using reltively lrge nisotropy prmeter / =. It should e noticed tht the Ben s criticl stte model predicts neither the phenomenon of the prmgnetism of hrd superconductors nor the ehvior of the components of the verge mgnetiztion found in Refs.,. Furthermore, s it is shown in [9,], the elliptic criticl-stte model successfully descries the mgnetic response of superconducting disks undergoing oscilltions in mgnetic field of fixed mgnitude for nonmgnetic, prmgnetic, nd dimgnetic initil sttes []. Despite the gret success of the elliptic model [9], it turns out tht there exist phenomen, ssocited with flux cutting, which re not completely descried within such model. So, in very recent work [8], the elliptic criticlstte model nd other four theoreticl pproches for descriing the criticl stte of type-ii superconductors (GDCSM, extended GDCSM [9,], extended elliptic criticl-stte model [8,], nd n elliptic criticl-stte model sed on the vritionl principle []) were tested. There, the ngulr dependencies of the criticl current density J c nd the electric field E (for J just ove J c ) were mesured, using n epitxilly grown YBCO thin film, nd compred with the predictions of the five theories. The mesurements of ngulr dependence of the criticl-current density J c demonstrted ehvior rther similr to tht ssumed y the elliptic criticl-stte models. Besides, the smooth ngulr dependence of the rtio of the trnsverse to the longitudinl components of the electric field Ey / E z for J just ove J c, predicted y the three elliptic models, ws verified in the experiment [8]. However, the originl criticl-stte model [9] leds to smll vlues of the rtio Ey / E z in comprison with the experimentl dt nd the results otined from the other two elliptic models. On the sis of this detiled comprison etween experiment nd the five theories, it ws concluded in Ref. 8 tht the experiment fvors only one of the models, nmely the extended elliptic criticl-stte model. The im of the present work is to investigte the ehvior of hrd superconductor in prllel rotting mgnetic field (or equivlently, the response of rotting superconductor in fixed mgnetic field) nd to compre the predictions of four criticl-stte models with experiment. Concretely, we shll consider the Ben's criticl-stte model [,], the originl elliptic criticl-stte model [9,], the recently-proposed extended elliptic model [8,], s well s the GDCSM [ 8], whose min chrcteristics nd ssumptions will e revisited in Sec.. We shll numericlly solve Mxwell equtions with the mteril eqution postulted y ech of the considered criticl-stte models to clculte mgnetiztion curves for superconductor disk rotting in fixed mgnetic field s in the experiment [] (Sec. ). Here, we shll nlyze the effect of the mgnitude H of the pplied mgnetic field upon the dependencies of the mgnetiztion components, prllel nd perpendiculr to H, on the rottion ngle of the superconductor disk. The evolution of mgnetic induction profiles will lso e studied to explin the mgnetic response of the rotting hrd-superconductor smple.. Theoreticl formlism Let us consider superconducting sl of thickness d, which occupies the spce < x < d nd is sujected to mgnetic field H prllel to its surfces: H = Hˆ s = H[ yˆsin( α s) + z ˆcos( αs)], () where α s is the ngle of the pplied mgnetic field H with respect to the z-xis. Hence, the mgnetic induction B ( x, t) inside the superconducting sl cn e expressed s B= Bxt (, )[ yˆsin( α ( xt, )) + z ˆcos( α( xt, ))], () where B nd α re respectively the mgnitude nd the tilt ngle of the mgnetic induction. It is convenient to write the electric field E ( x, t) nd the electricl current density J ( x, t) in terms of their components prllel nd perpendiculr to the locl mgnetic induction B (,): x t E( x, t) = E(, ) ˆ(, ) (, ) ˆ xt xt + E xt( xt, ), () J( x, t) = J(, ) ˆ(, ) (, ) ˆ xt xt + J xt( xt, ), () where ˆ ( x, t)= xˆ ˆ( x, t). Inside the superconductor smple, we shll ssume tht the mgnetic induction nd the mgnetic field stisfy the reltion B( x, t)= μh ( x, t), which is good enough for pplied mgnetic fields much lrger thn the first criticl field ( H Hc). Moreover, ny surfce rrier ginst the flux entry (or exit) will e neglected. According to the plnr geometry of the prolem, we cn rewrite Ampere nd Lorentz lws, s follow B( x, t)= μj ( x, t), () B E ( xt, ) =, t () Low Temperture hysics/fizik Nizkikh Tempertur,, v., No. 9

3 R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez B = μj, x α B = μ J, x () (8) E α B + E =, (9) x x t α E α E = B. () x x t To solve the resulting system of differentil equtions for E, B nd J, one should dd the mteril eqution. Below, we shll use the mteril equtions corresponding to the circulr, elliptic, extended-elliptic, nd rectngulr flux-line-cutting criticl-stte models... Circulr model The first model for descriing the mgnetic ehvior of superconductors in multicomponent situtions ws proposed y Ben [,]. According to it, the criticl current density J points lwys long the locl electric field E. Hence, E J =. () E The mgnitude of the criticl current density J = J c is the unique phenomenologicl prmeter used nd my depend on the mgnitude of the mgnetic induction B. In the plnr geometry (see Eqs. () ()), the ssumption J = J c corresponds to circle in the J J plne. In numericlly solving the system of Eqs. () () for the electromgnetic fields, it is necessry to rewrite Eq. () s J E = EJ ( ), () J, J ( B) EJ ( ) = () ρ ( J ( B)), J ( B) where ρ is n effective resistivity. It should e mentioned tht for slow vritions of the surfce oundry conditions, producing smll mgnitude of the induced electric field ( E ρj c ), the mgnetic induction profiles re prcticlly relxed nd independent of the prmeter ρ []... Elliptic model The elliptic flux-line cutting criticl-stte model [9,,] postultes: Ek Ji =( ) ik, () E where ( ) = ( ) δ,, =,. () ik, ib ij i k Here δ ik is the Kronecker delt symol. Within the elliptic criticl-stte model (), the mgnitude of the criticl current density J c drws n ellipse on the J J plne. This model mkes use of two phenomenologicl prmeters, nmely the extreme vlues J c nd J c for the rdius of the ellipse drwn y the mgnitude of the criticl current density. In the numericl clcultions for solving the system of Eqs. () (), the reltion () is rewritten in the form Ei = E( J)( ) ik Jk, (), J ( B, φ) EJ ( ) =, () ρ ( J ( B, φ )), J ( B, φ ) where ( ) ik is the inverse of the mtrix ( J c) ik in (). The mgnitude of the criticl current density, ( B, φ ), is given y the expression / cos ( φ) sin ( φ) ( B, φ )= +. ( B) ( B) (8) Here, φ denotes the ngle of the criticl current density J with respect to the direction of the flux density B. If =, the elliptic criticl-stte model () goes over into the Ben s (circulr) criticl-stte model (). Besides, the clcultions of electromgnetic fields with J close to J c re lso independent of the uxiliry prmeter ρ in Eq. ()... Extended elliptic model The elliptic criticl-stte model, descried in previous susection, hs recently een extended in Refs. 8, y introducing the generl reltions E = ρ J, (9) E = ρj, () where ρ nd ρ re nonliner effective resistivities, hving rtio r = ρ / ρ independent of J just ove J c s it ws experimentlly found [8]. A model for the effective resistivities is given y [], J d E =, ρ d( J d) sign( J ), J d, J c E =. ρ c( J c)sign( J), J c () () Here, the suscripts d nd c respectively refer to depinning nd cutting. Besides, d = ( B, φ) sin( φ ) nd c = ( B, φ) cos( φ ), where ( B, φ ) is defined ccording to the elliptic criticl-stte model s in Eq. (8). If J /, the extended elliptic criticl-stte model 9 Low Temperture hysics/fizik Nizkikh Tempertur,, v., No.

4 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field reduces to the originl one (Eqs. () nd ()) y replcing ρ d nd ρ c in Eqs. () nd () with ρ / nd ρ /, correspondingly. Hence, in the cse of the originl elliptic model, the rtio r = ρ / ρ t J J c is equl to /. On the other hnd, the extended elliptic criticl-stte model is cple to modify the reltion etween the components of the electric field E nd the current density J with the id of the dditionl prmeter r... Rectngulr model The generlized doule criticl-stte model [ 8] uses two phenomenologicl prmeters, nmely the criticl vlues, nd J c, of the electricl current density long nd perpendiculr to the locl mgnetic induction. Within this model, ech component of the electricl current density is determined y its own electric field s J = sign( E ), () J = sign( E ). () Evidently, the mgnitude of the criticl current density trces rectngle in the J J plne. The prmeter determines the threshold for depinning of vortices, wheres indictes the onset of flux-line cutting in the vortex rry. In clculting the electromgnetic fields within the GDCSM, the mteril eqution () is written in the form, J E =, ρ ( J )sign( J ), J, J E =. ρ ( J )sign( J), J () () The quntities ρ nd ρ re effective flux-flow nd fluxline-cutting resistivities of the mteril. However, unlike the ove-commented criticl-stte models, the GDCSM llows the existence of zones in the J J plne where either flux cutting or flux trnsport exclusively occur. The ltter is possile due to the ssumption of the GDCSM tht the threshold for flux depinning, (flux cutting, ) is independent of the component J ( J ) (compre Eqs. () nd () with Eqs. () nd () where J cd nd J cc depend on the ngle φ = rctn( J / J ) ).. Numericl results nd comprison with experiment In the present section we will pply the flux-line-cutting criticl-stte models, commented ove, to explin experimentl mgnetiztion curves [] of Bi superconducting disk, rotting in the presence of n externl mgnetic field H, which is oriented prllel to the disk plne (long the z-xis) nd perpendiculr to the xis of rottion... Experimentl results Figure, exhiits stndrd mgnetiztion curve, which ws mesured in Ref., for Bi disk of thickness d =.8mm. The hysteresis in Fig., clerly corresponds to the mgnetiztion curve of type-ii irreversile superconductor since its return crosses over nd remins in the prmgnetic region s result of the strong flux pinning. In the experiment, the isotropy of the Bi disk ws lso verified y compring stndrd mgnetiztion curves with H directed long different dimeters of the disk. nels () (c) in Fig. show grphs of the mgnetiztion components, < My =< By / μ nd < M z = = H < Bz / μ, versus the ngle θ of rottion, mesured in the work [] for the Bi disk, rotting in the mgnetic field H. The mesurements strted in the nonmgnetic initil stte which is reched fter cooling the superconductor t the fields H / H =. (pnel ),. (pnel ), nd. (pnel c), where H ( μ H =. T []) is the penetrtion field. The initil stte is supposed to e nonmgnetic ecuse no Meissner effect (flux expulsion) ws oserved fter field cooling, within the ccurcy ( Δ < M Guss) of the experiment. < M z,t < M z,t.9... Experiment H z,t.9... Model H z,t Fig.. Stndrd mgnetiztion curves () for Bi disk, tken from Ref.. Theoreticl mgnetiztion curves () otined with criticl current density ( B) s in Eq. (). Low Temperture hysics/fizik Nizkikh Tempertur,, v., No. 9

5 R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez Mgnetiztion ( H ) Mgnetiztion ( H ) Mgnetiztion ( H ) Experiment.8... Experiment 9 8 Experiment c H / H =. 9 8 Fig.. Bi rottionl curves mesured in Ref.. As it is seen in Fig., for the smllest vlue of H (=. H, pnel ), oth mgnetiztion components hve nonmonotonic ehvior s functions of θ. Such ehvior of mgnetiztion hs lso een oserved in Ref. during the initil rottion of N disk undergoing slow oscilltions in prllel field. The dependence of the mgnetiztion on θ rdiclly chnges t lrger vlues of H. So (see Fig.,), t H = H the functions ( θ ) nd < M z ( θ ) initilly grow with θ nd lter (t θ ) they prcticlly ecome constnts with close vlues (< My < Mz ). Also note tht hs mximum t θ. For H lrger thn the penetrtion field H (pnel c), the function < M z ( θ ) tkes vlues smller thn those for ( θ ). Both of them re H / H =. H / H =. lmost constnt functions, except t smll rottion ngles ecuse of their fst initil growth. Thus, the mximum of is shifted to smller vlue of θ ( )... Theoreticl predictions The models descried in the previous section cn e pplied to explin the experimentl results (Fig. ) if we fix the smple nd rotte the externl mgnetic field H () y n ngle αs = θ insted of fixing the mgnetic field nd rotting the superconducting smple. Then, the experimentl vlues nd < M z should respectively correspond to the quntities: where d = dxb y( x), d μ () d < M z = H dxbz ( x), d μ (8) B y = ˆ s xˆ B = B( x)sin[ α( x) αs], (9) B = ˆ B = B( x)cos[ α( x) α ]. () z s s The clcultions of mgnetiztion components nd < M z with the criticl-stte models, discussed in Sec., require the employment of the prmeters ( B) nd ( B), depending on the mgnetic induction. The former, ( B), is determined from the experimentl curves of mgnetiztion versus the pplied field, vrying long one direction only s in Fig. (In this cse, flux cutting does not occur nd, consequently, the depinning effects re completely responsile for the mgnetic response of the superconductor.) The stndrd mgnetiztion curves re well reproduced y ny one of the criticl-stte models (see ove) with () ( B)=, n ( + B/ μh ) () () =. A/m, nd n = (compre pnels () nd () of Fig. ). Other prmeters of the criticl stte models re found y djusting theoreticl mgnetiztion curves to the experimentl ones (Fig. ).... Circulr model. Within the Ben,s circulr criticl-stte model (), there is only one phenomenologicl prmeter, i.e. ( B) = ( B) = ( B). Then, ( B ) hs the form () with the sme vlues for the prmeters (), nd n. Figure shows our numericl results for nd < M z, otined with the Ben criticl-stte model. At first glnce, it seems tht the circulr model qulittively reproduces the experimentl mgnetiztion curves (Fig. ). However, there re importnt differences etween its predictions nd the experiment. Thus, for exmple, the oscilltions of the mgnetiztion components (Fig.,) hve smll mplitudes compred with the experimentl ones. 9 Low Temperture hysics/fizik Nizkikh Tempertur,, v., No.

6 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field Mgnetiztion ( H ) Mgnetiztion ( H ) Mgnetiztion ( H ).8... Circulr model. 9 8 Circulr model Circulr model.8 c < Mz 9 8 Fig.. Curves of the verge mgnetiztion components versus the rottion ngle, clculted with Ben s criticl-stte model. Besides, t H =.8H the functions ( θ ) nd < M z ( θ ) pproximte ech other ut t reltively lrge rottion ngles θ. Finlly, when the pplied field hs n mplitude lrger thn H p (see pnel c), the mgnetiztion components re rther smll in mgnitude nd their initil growth, efore the sturtion, occurs in very smll intervl of θ (< ).... Elliptic model. The clcultions of mgnetiztion components nd < M z within the elliptic flux-line-cutting criticl-stte model () re shown in Fig.. Here, we used the sme ( B) s in Eq. () nd ( B) of the form () ( B) = () n ( + B/ μ H ) H / H =. H / H =.8 H / H =. Mgnetiztion ( H ) Mgnetiztion ( H ) Mgnetiztion ( H ).8... Elliptic model. 9 8 Elliptic model Elliptic model.8 c H / H =. 9 8 Fig.. Curves of the verge mgnetiztion components versus the rottion ngle, clculted with the originl elliptic criticlstte model. with () =. () nd n =. This choice provides good greement etween experimentl (Fig. ) nd theoreticl (Fig. ) curves. Thnks to the use of second prmeter ( ), the elliptic model is le to generte the oscilltions of the mgnetiztion components (Fig.,) with mplitude close to tht oserved in the experiment (pnel () in Fig. ). Notice tht nd < M z pproch ech other t θ with H =. H in good concordnce with the mesurements (see Fig.,, corresponding to H = H ). In ddition, when H =.H p (pnel (c) in Fig. ), the difference etween nd < M z t θ is s lrge s in the experiment (Fig.,c). H / H =. H / H =. Low Temperture hysics/fizik Nizkikh Tempertur,, v., No. 9

7 R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez Mgnetiztion ( H ) Mgnetiztion ( H ) Mgnetiztion ( H ) Extended elliptic model ( r = ) H / H =. 9 8 Extended elliptic model ( r = ) 9 8 Extended elliptic model ( r = ) c 9 8 Fig.. Curves of the verge mgnetiztion components versus the rottion ngle, clculted with the extended elliptic criticlstte model using rtio r =. H / H =. H / H =. Mgnetiztion ( H ) Mgnetiztion ( H ) Mgnetiztion ( H ).8... Rectngulr model H / H = Rectngulr model Rectngulr model.8 c 9 8 Fig.. Curves of the verge mgnetiztion components versus the rottion ngle, clculted with the generlized doule criticlstte model. H / H =.9 H / H =.... Extended elliptic model. As ws commented in Sec., oth elliptic nd circulr criticl-stte models re prticulr cses of the extended elliptic one. Therefore, the results presented in Fig., predicted y the circulr model, cn lso e clculted y using the new model (Eqs. () nd ()) with = s in Eq. () nd r = ρ / ρ = ρc / ρd eing equl to one ( r =) t J J c. The condition r = gurntees tht the electric field E nd current density J e prllel s it is postulted y Ben s criticl-stte model (). In ddition, grphs in Fig. (originl elliptic model predictions), which quntittively reproduce experimentl mesurements (Fig. ), re lso otined with the extended elliptic criticl-stte model (Eqs. () nd ()) if r = / (i.e. ρc / ρd = = / ). According to the prmeters ( B) () nd ( B) (), used for clculting mgnetiztion curves in Fig., the rtio r is here smller thn ( r <). It is interesting to study the effect of the prmeter r, controlling the reltion etween the electric field E nd the current density J t J J c. For this reson, we hve clculted mgnetiztion curves (Fig. ) y pplying the extended elliptic model with the sme prmeters ( B) nd ( B) s those employed in Fig., ut with the prmeter r = ρc / ρ d =. In other words, the mgnetiztion curves in Fig. correspond to n nisotropic criticl-stte model with / <, ut the prmeter r =, indi- 9 Low Temperture hysics/fizik Nizkikh Tempertur,, v., No.

8 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field cting tht E nd J re prllel when J J c. From the comprison of Fig. with, we note tht mgnetiztion curves significntly depend upon the prmeter r when the pplied mgnetic field is lrge enough ( H H s in pnels () nd (c)). So, in order the mgnetiztion components, nd < M z, to hve the sme vlue t lrge ngles of rottion, the pplied mgnetic field H for r = (Fig.,) should e lrger thn the field used in Fig.,. Besides, the vlue of nd < M z (. H ), t sufficiently lrge ngles θ, turns out to e smller thn tht (. H ) predicted y the originl elliptic model (Fig.,). At H =. H, there is lso noticele difference etween mgnetiztion y-components (compre pnels (c) of Figs. nd ).... Rectngulr model. For completeness of our study, we hve employed the GDCSM (rectngulr model), which lso uses two criticl current densities, nmely ( B) nd ( B). The former is determined from the curves of mgnetiztion versus the pplied field, vrying long one direction only (Fig. ). In our cse, the mgnetic dependence of is the sme s in Eq. (). To reproduce the min fetures of the experiment (Fig. ), the other prmeter is chosen s in Eq. (), ut () =. () nd n =. (compre Figs. nd ). Although these vlues re different from those used within the elliptic criticl-stte model, the prllel criticl current density J c remins eing lrger thn the perpendiculr one. It should e noted tht the GDCSM predicts the equlity of nd < M z (. H ) with n externl field H =.9 H H t reltively lrge rottion ngles θ (see Fig.,), in contrst to the experiment where such ehvior occurs from θ. Besides, the numericl clcultions for H =.H (pnel () in Fig. ) hd to e stopped t θ 8 ecuse the solution further diverged... Mgnetic induction profiles The fct tht the elliptic criticl-stte model is le to quntittively reproduce the experiment, with the use of prllel criticl current density ( B) lrger thn the perpendiculr one ( B), illustrtes how flux-line cutting influences on the mgnetic ehvior of rotting superconductor. To explin the fetures oserved in oth experimentl (Fig. ) nd theoreticl (Fig. ) mgnetiztion curves, we shll nlyze the evolution of the profiles for the mgnitude of the mgnetic induction Bx (), the tilt ngle α ( x), nd the components By ( x) (9) nd Bz ( x) (), clculted within the originl elliptic flux-line-cutting criticl-stte model (Figs. 9). The clculted profiles of the mgnetic induction in the cse when the externl mgnetic field H hs mgnitude smller thn the penetrtion field H p ( H =. H p) re shown in Fig.. As the ngle of rottion is incresed, two, deg B y / H 8 9 x m x x x m xd / x m x c x x m xd / Fig.. rofiles of the ngle α (pnel ), mgnitude B (pnel ) nd components B y (Eq. (9), pnel c) nd of the mgnetic induction, clculted with the originl elliptic criticl-stte model t H =. H. B/ H B z / H x m x m x x 8... d x x 8 x m x m B z (Eq. (), pnel d) Low Temperture hysics/fizik Nizkikh Tempertur,, v., No. 9

9 , deg B y / H R. Cortés-Mldondo, J.E. Espinos-Rosles, A.F. Crllo-Sánchez, nd F. érez-rodríguez c Fig. 8. rofiles of the ngle α (pnel ), mgnitude B (pnel ) nd components B y (Eq. (9), pnel c) nd the mgnetic induction, clculted with the originl elliptic criticl-stte model t H =. H. B/ H B z / H d B z (Eq. (), pnel d) of U-shped minim in the Bx ( ) profile (pnel ) pper ecuse of the flux consumption (decrement of B) which results from flux-line cutting []. The solute vlue of the tilt ngle α increses with θ in the ner-surfce intervls x < x m nd xm < x d. However, in the intervls xm < x< x nd x < x < x m, where there is flux consumption, the ngle α is slightly modified. In the centrl intervl, x < x < x, neither B or α is ltered. When θ, the minimum vlues of B inside the superconducting disk tend to zero nd, s follows from Eq. (8), the mgnitude of the derivtive α / x considerly increses t points corresponding to such minim. Besides, t x = x m nd x = x m with Bx ( m)= Bx ( m), the ccurcy of our clcultions is low nd, therefore, the vlues α ( x m ) nd α ( x m ) turned out to e pprently higher thn they should e (see curve 8 for θ = in Fig.,). The component B z of the mgnetic induction, prllel to the pplied mgnetic field, decreses ner smple surfces ecuse of the flux consumption (Fig.,d). Nevertheless, the most importnt chnge occurs in the centrl prt of the smple (in x < x < x ) ecuse of the smple rottion. So, t θ = 8 (curve ) the component B z vries from Bz = μ H t the surfces x = nd x = d to the opposite vlue Bz = μ H in the centrl region of the smple. When n entire cycle is finished, B z gin tkes the vlue Bz = μ H in the middle of the disk (curve 8). This cyclic ehvior of B z is responsile for the oscilltions of the mgnetiztion component < M z ( θ ) (pnels () in Figs. nd ), eing negtive for ny vlue of the ngle of rottion θ ecuse Bz < μ H ner surfces, i.e. in the intervls < x < x nd x < x< d. The component B y lso oscilltes in the middle of the smple s θ is incresed (Fig.,c). Such ehvior of B y mkes the mgnetiztion y-component oscillte with θ (Figs., nd,). As it is seen in Fig.,c, there is n increment of B y in the ner-surfce regions, producing smll positive vlue for () fter complete cycle, i.e. t θ = (see Figs., nd,). Figure 8 exhiits profiles clculted within the elliptic criticl-stte model for H =. H p. Due to the decrese of the criticl current densities () nd () with the mgnitude B of the mgnetic induction, the slopes of the criticl profiles for Bx ( ) nd α ( x) ner surfces re smller thn the slopes oserved in the corresponding profiles of Fig.. Therefore, the centrl region with unltered B nd α (see curves in pnels () nd () of Fig. 8) rpidly disppers s the rottion ngle θ is incresed (see curves therein). Also, the U -shped minim of Bx ( ) colesce forming unique minimum t the center of the disk. The resulting criticl profile Bx ( ) does not further chnge despite the fct tht the disk continues rotting (see curves 8 in pnel ()). In this cse, Bz ( x) initilly decreses (curves in Fig. 8,d) inside the smple s θ vries until it reches the criticl profile (curves 8). Hence, the dependence < M z ( θ ) hs monotonic ehvior t θ (see pnels () in Figs. nd ). On the other hnd, B y ( x) increses so tht huge mximum in the dependence ( θ ) (Figs., nd,) ppers t 98 Low Temperture hysics/fizik Nizkikh Tempertur,, v., No.

10 Flux-cutting nd flux-trnsport effects in type-ii superconductor sls in prllel rotting mgnetic field, deg 8 9. c B/ H B y / H B z / H... 8 d Fig. 8. rofiles of the ngle α (pnel ), mgnitude B (pnel ) nd components B y (Eq. (9), pnel c) nd the mgnetic induction, clculted with the originl elliptic criticl-stte model t H =. H. B z (Eq. (), pnel d) of θ. At lrge rottion ngles ( θ 8 ), the profile B y ( x ) ecomes sttionry nd ( θ ) is, prcticlly, constnt function, hving vlue close to < M z. So, the mgnitude of the mgnetiztion, < M, is independent of θ when the rottion ngle is sufficiently lrge. The profiles for the cse when the externl mgnetic field is lrge enough, in comprison with the penetrtion field H (s in Fig. 9), hve n evolution similr to tht presented in Fig. 8. However, the centrl regions of unltered mgnetic induction rpidly dispper s θ is incresed (compre Figs. 8 nd 9). This fct is due to noticele reduction of the criticl current densities nd with B.. Conclusion We hve pplied the circulr, elliptic, extended-elliptic, nd rectngulr criticl-stte models to study the mgnetic ehvior of irreversile type-ii superconductors in prllel rotting mgnetic field. The numericl method employed here is sed on the sustitution of the verticl lw, relting the electric field E nd the current density J, for nonliner mteril eqution hving effective flux-cutting nd flux-flow resistivities in the dissiptive region. The sustitution is justified when the pplied mgnetic field H slowly vries either in mgnitude or direction, inducing electric fields of sufficiently smll mgnitude inside the superconductor. Within the elliptic (circulr) criticl-stte model such resistivities re not independent of ech other nd hve rtio r = ρ / ρ equl to / (= for the circulr model) t J just ove its criticl vlue J c. On the other hnd, within the extended elliptic criticl-stte model the rtio r is n independent prmeter to e determined. The rectngulr criticl-stte model lso uses two independent resistivities, ρ nd ρ. However, unlike the other criticl-stte models, the GDCSM ssumes tht flux cutting nd flux depinning do not ffect ech other. The comprison of the predictions of the mentioned criticl-stte models with experimentl mesurements of mgnetiztion for rotting Bi disk in fixed mgnetic field [] shows tht the originl criticl-stte model cn reproduce the min fetures of the mgnetiztion curves. The circulr nd rectngulr criticl-stte models only chieve qulittive description of the experiment. The extended elliptic model, eing more generl thn the originl elliptic one, hs llowed us to study the effect of the reltion etween E nd J in the dissiptive region. However, dditionl theoreticl nd experimentl studies re needed to elucidte on the effects ssocited with oth flux-cutting nd flux-flow resistivities. This work ws prtilly supported y Consejo Ncionl de Cienci y Tecnologí (CONACYT, Mexico). Low Temperture hysics/fizik Nizkikh Tempertur,, v., No. 99

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