Flux-flow critical-state susceptibility of superconductors

The field-amplitude Hm and circular frequency ω dependent ac susceptibility, χ=χ′−jχ″, of a hard superconducting cylinder with flux-flow type current-voltage characteristic is calculated. A remarkable feature of the resultant χ(Hm,ω) is that both the maximum χ″, χm″, and dlgHm(χm″)∕dlgω increase with increasing ω. This behavior is observed in actual Bi-2223/Ag tapes and YBa2Cu3O7−δ-coated conductors. Our result provides a useful tool to study the intergranular critical state in high-temperature superconductors.

The magnetic properties of hard superconductors were first explained by Bean, who proposed what was later called the critical-state ͑CS͒ model, i.e., the volume supercurrent induced in hard superconductors is limited by the criticalcurrent density J c . 1,2For an infinite cylinder or slab in a longitudinal applied ac field, H = H m e jt , the magnetization M͑H͒ curves and complex ac susceptibility, = Ј − jЉ, were derived analytically from the CS model and given in Ref. 3. Considering strong demagnetizing effects, M͑H͒ and ͑H m ͒ curves were derived from the CS model for perpendicularly magnetized thin tapes and disks. 4,5Accurate results of transverse ͑H m ͒ were obtained for rectangular bars with finite width and thickness. 6he common behaviour of the CS ͑H m ͒ for any sample geometry is that with increasing H m , Ј increases from a negative value − 0 corresponding to the complete shielding state up to zero, whereas Љ increases from zero to a maximum m Љ at H m = H m ͑ m Љ ͒ and then decreasing to zero.Since 0 = ͓1−N m ͑ =−1͔͒ −1 , N m being the magnetometric demagnetizing factor, 7 0 increases with decreasing the aspect ratio along the field direction from 1 upwards.However, m Љ / 0 and a properly normalized H m ͑ m Љ ͒ are not significantly in- fluenced by demagnetizing effects.For example, for the transverse CS of rectangular bars and the axial CS of cylinders, m Љ / 0 ranges from 0.21 to 0.24 for the aspect ratio being either much greater or much smaller than unity, whereas the corresponding H m ͑ m Љ ͒ / J c equals 0.5 or 1 times the smallest dimension ͑diameter or thickness͒.Thus, one can conclude that after a proper normalization, the ͑H m ͒ calculated easily for infinitely long samples can be an acceptable approximation for the perpendicular ͑H m ͒ of thin samples, which are seriously subject to demagnetizing effects and more difficult to be calculated. 8t is well known that in hard type-II superconductors, the power dissipation accompanied by volume currents arises from the depinning of the Abrikosov vortices ͑AVs͒ driven by the currents.Considering such depinning to be thermally activated, the experimental temperature and observation-time dependence of J c was explained using the flux creep model first proposed by Anderson. 9The AV pinning is weak and a collective creep can take place in high-temperature superconductors ͑HTSs͒, owing to their small coherent length so a small pinning volume per AV and their point-defects dominated pinning centers. 10From the current-dependent activation energy of collective depinning, Brandt has recommended a power-law current-voltage dependence E͑J͒ = E c ͉J / J c ͉ n J/͉J͉, J c being the J when the electrical field E = E c =10 −4 V / m and index n ӷ 1, to approximately characterize the behavior of collective creep, and computed the perpendicular ͑H m , ͒ of disks that follow this E͑J͒. 11,12It turns out that in this case for a finite n, ͑H m ͒ at a fixed is similar to that of the CS model and it shifts to higher H m with increasing in such a way that ͑H m , ͒ depends only on H m / 1/͑n−1͒ . 11,12hen interpreting the measurements of HTSs by the CS model, the frequency dependence of has usually been ignored in most previous works.We have measured the of a ͑Bi, Pb͒ 2 Sr 2 Ca 2 Cu 3 O 10+␦ ͑Bi-2223͒ monofilamentary tape and a YBa 2 Cu 3 O 7−␦ ͑YBCO͒ single crystal film at 77 K as a function of H m and . 13,14The results turn out to be consistent with Brandt's calculation of with power-law E͑J͒, i.e., both m Љ and d lg H m ͑ m Љ ͒ / d lg are roughly constant with increasing , having n = 19 and 31 for the tape and film, respectively, so that the CS is basically controlled by the collective flux creep.
However, we find that the ͑H m , ͒ of HTS samples is not always like this.For some coated YBCO conductors, the perpendicular ͑H m , ͒ shows a behavior that with increasing both m Љ and d lg H m ͑ m Љ ͒ / d lg increase. 14In biaxially textured YBCO-coated conductors J c is limited by networks of low-angle grain boundaries ͑GBs͒.Different from the AVs in the grains with a core size of , the vortices in the lowangle GBs are Abrikosov-Josephson vortices ͑AJVs͒ with a core size along the GB greater than . 15Owing to the smoothing off of pinning potentials related to the presence of more pinning centers over the larger core dimension along GB and to the increase of the elastic energy relevant to the AJV bending parallel to GB, the AJVs are less pinned by point defects along the GB than AVs are in the grains, but mainly pinned by dislocations aligned in the GB. 16If pinning of dislocations is small and uniform enough, with increasing current, the dissipation will first occur from thermally activated AJV creep and quickly changing into vortex flow channeling along GBs, with a basically linear E͑J͒ curve above J c .][19] Since ͑H m , ͒ for flux-flow ͑limited͒ CS has not been studied, we calculate in this letter the flux-flow CS of a long cylinder of radius a using the method proposed by Brandt. 11,12Based on observations of the current-voltage characteristic of YBCO bicrystals and some low-temperature superconductors, 17,20 we assume for the flux-flow CS that E =0 if ͉J͉ ഛ J c and ͓J − sgn͑J͒J c ͔ / E = f if ͉J͉ Ͼ J c , where J c is the critical current density and f is the flux-flow conductivity, both being assumed to be magnetic field independent for simplicity.For the cylindrical geometry with magnetic field applied along the z axis, the induced electrical field E and the vector potential A in the London gauge only have a nonzero component ͑͒ being functions of r and t, E͑r , t͒ and A͑r , t͒.
We divide the cylinder into n = 100 cylindrical elements of equal radial dimension, borded at r i = ia / n , i =1,2, ... ,n, with J in each element being uniform J i and A at r = r i being A i .Assuming A͑r =0͒ = 0, the Faraday law is written as where the E͑J͒ function has been defined above, the partial A for the applied field A i a ͑t͒ =− 1 2 0 r i H m sin t, and the partial A produced by currents A i c ͑t͒ = ͚ j n f ij J j ͑t͒, f ij being the contribution of current in element j per unit J j to A i c .Setting reasonable parameter values of a =10 −3 m, J c =10 6 A/m, and f =10 −9 m −1 ⍀ −1 and increasing stepwise t from 0, the equation system is solved numerically for each pair of given H m and .In order to simulate accurately the physical process and to reach a stable symmetrical periodical state, three periods are calculated with 10 6 time steps in each period.With the stabilized J i , the magnetic moment of the cylinder and finally ͑H m , ͒ are computed in the usual way as described in Ref. 21.
The calculated ͑H m , ͒ curves are plotted in Fig. 1͑a͒, where H m is normalized to J c a and a dimensionless frequency ͑following the tradition of eddy-current research͒ 22 f 2 = a 2 0 f is used so that the result becomes universal for any values of a, J c , and f .The curves for f 2 = 0.024 in Fig. 1͑a͒ coincide very well with the CS for cylinder. 3ith increasing lg f 2 , we see both m Љ and d lg͓H m ͑ m Љ ͒ / J c a͔ / d lg f 2 to increase, which is qualitatively in agreement with the experimental results of the coated conductor reported in Ref. 14.
Such behavior has been found not only in coated conductors but also in some Bi-2223 tapes.In Fig. 2, we give ͑H m , f͒ curves of a Bi-2223/Ag multifilamentary tape measured at 77 K.The critical current of the tape I c Ϸ 5 A, which is rather low compared with the above mentioned monofilamentary tape with I c Ͼ 30 A. It was prepared without the last two thermomechanical treatments, each of which consisted of a cold rolling with small thickness reduction followed by 830 °C annealing, necessary for obtaining a good alignment and links of grains.Similar to the Bi-2223/Ag tape 61TH1 studied in Ref. 23, the intergranular J c in this sample is dominated by weak links.We see from Fig. 2 that besides a two-stage feature in ͑H m ͒ curves, which indicates both the weak and strong-link contributions, 23,24 there is a remarkable similarity between the measured ͑H m , f͒ and the curves calculated from flux-flow CS in Fig. 1͑a͒.This flux-flow feature found in of the multifilamentary Bi-2223/Ag tape, where J c is dominated by weak links, is interesting.As studied in Refs.15, 25, and 26, the GBs in this case may be regarded as extremely incoherent planar defects with very small depairing ͑Josephson͒ current density J J , for which no AJVs can form like in coated conductors, where J J of low-angle GBs acting as coherent planar defects can be close to the depairing current density of the bulk.As explained in Refs.23 and 27, weak-link critical currents may arise from either Josephson-vortex pinning like in a type-II superconductor when the maximum Josephson currents, I J 's, between neighboring grains are very small, with a flux creep dissipation mechanism of the intergranular CS, or from the I J 's themselves when they are large.The Bi-2223/Ag tape has much greater I J 's than those of sintered YBCO, and the weak-link critical currents of Bi-2223/Ag tapes should result from I J 's of individual Josephson junctions themselves.In this case, the flux-flow behavior should be a consequence of the resistively shunted Josephson-junction network.
In such a network, the effect of GB between any pair of neighboring grains i and j is modeled by a parallel circuit of a Josephson junction and a resistance R together with a void of area A v , where flux can pass through.After a field change, the gauge-invariant phase difference ij of Josephson junctions in the induced intergranular current path may be chosen automatically to be near ͑2nЈ ±1/2͒, which corresponds to a CS with all Josephson junctions carrying a current close to I J 's.The flux penetration will involve a ij jump ͑nЈ → nЈ ±1͒, which corresponds to a fluxon ͑⌽ 0 ͒ to traverse through the Josephson junction, leading to an ac Josephson current to flow through R. Thus, the final result is very similar to the modeled flux-flow CS.
The following equations for J and E may be used for obtaining the approximate E͑J͒ relation from frequency dependent H m ͑ m Љ ͒:

͑3͒
The E͑J͒ function for the cylinder obtained using Eqs.͑2͒ and ͑3͒ from the calculated ͑H m , ͒ in Fig. 1͑a͒ is plotted in Fig. 1͑b͒ by symbols.We see that they agree quite well with the actually assumed E͑J͒ function ͑solid line͒ for ͑H m , ͒ calculation.This agreement justifies the technique of determination of E͑J͒ curve from ac measurements, which has first been used in Ref. 14, where radius a is replaced by the thickness or sides of the film after physical consideration.
In conclusion, we have calculated ac of a superconducting cylinder with flux-flow E͑J͒ characteristic.The re-sults may be useful for studying the intergranular CS of HTSs, such as coated YBCO conductors and multifilamentary Bi-2223/Ag tapes.Especially, we have demonstrated that the intergranular CS in HTSs is not resulting from Josephson vortex pinning as popularly believed but from the maximum Josephson currents through GBs themselves.We have also justified that the E͑J͒ characteristic of HTSs can be conveniently determined by field and frequency-dependent contactless measurements.This work has been supported by MEC project FIS2004-02792 and Generalitat de Catalunya ͑SGR 2001-00189 and CeRMAE͒.

FIG. 1 .FIG. 2 .
FIG. 1. ͑a͒ For a superconducting cylinder of radius a, critical-current density J c , and flux-flow conductivity f , the calculated ac as a function of the reduced field amplitude H m / ͑J c a͒ for different values of dimensionless frequency f 2 = a 2 0 f .͑b͒ The assumed flux-flow E͑J͒ relation compared with that calculated using Eqs.͑4͒ and ͑5͒ from ͑a͒ when a = 1 mm, J c =10 6 Am −2 , and f =10 9 m −1 ⍀ −1 .Arrows in ͑a͒ indicate the direction of increasing f 2 , and symbols in ͑b͒ represent data calculated from m Љ ͑H m , ͒ in ͑a͒.