Kim-Anderson Critical-state Model Research Articles

Inter-grain superconductivity of polycrystalline FeSr2YCu2O6+y

The fundamental and higher-harmonic susceptibilities χn=χn′−iχn′′ (n=1, 2, 3, 5 and 7) were measured to study the inter-grain superconducting properties of polycrystalline FeSr2YCu2O6+y. The real part χ1′ and imaginary part χ1″ appeared below Tc1=56K and Tc2=45K, respectively. Tc1 and Tc2 correspond to the onset temperatures of the superconductivity in individual grains and the superconductivity across grain boundaries, respectively. The higher-harmonic susceptibilities χn (n=2, 3, 5 and 7) appeared below Tc2. The results suggest that the higher-harmonic susceptibilities were generated by the non-linear magnetization caused by the magnetic flux penetration through grain boundaries. The inter-grain magnetization curves were obtained by substituting measured χn in Fourier series. The temperature dependence of χn (n=1, 2, 3, 5 and 7) calculated by the Kim–Anderson critical-state model were approximately fitted with the observed those of χn. The inter-grain critical-current density at 0K under zero magnetic field was estimated at about 2×104A/m2 from the fitting parameters of the Kim–Anderson critical-state model. The magnetization curves and these parameters imply that the inter-grain critical-current density hardly depends on a weak magnetic field below 0.1mT.

Bulk critical state and fundamental length scales of superconducting nanocrystallineNb3Alin Nb-Al matrix

We present the results of magnetization measurements on an as-cast nanocrystalline ${\text{Nb}}_{3}\text{Al}$ superconductor embedded in Nb-Al matrix. The typical grain size of ${\text{Nb}}_{3}\text{Al}$ ranges from about 2--8 nm with the maximum number of grains at around 3.5 nm, as visualized using transmission electron microscopy. The isothermal magnetization hysteresis loops in the superconducting state can be reasonably fitted within the well-known Kim-Anderson critical-state model. By using the same fitting parameters, we calculate the variation in field with respect to distance inside the sample and show the existence of a critical state over length scales much larger than the typical size of the superconducting grains. Our results indicate that a bulk critical current is possible in a system comprising of nanoparticles. The nonsuperconducting Nb-Al matrix thus appears to play a major role in the bulk current flow through the sample. The superconducting coherence length $\ensuremath{\xi}$ is estimated to be around 3 nm, which is comparable to the typical grain size. The penetration depth $\ensuremath{\lambda}$ is estimated to be about 94 nm, which is much larger than the largest of the superconducting grains. Our results could be useful for tuning the current carrying capability of conductors made out of composite materials which involve superconducting nanoparticles.

AC loss of Bi-2212 superconducting ring for limiter

A wattmeter technique is used to measure alternating field loss in Bi-2212 superconducting ring as an element of inductive fault current limiter. Experiment is performed at 77 K. Measurement results show that AC loss P vs. amplitude of sweeping field P follows a power law, P∝ H n . When sweeping field is lower than H p we observe the cubic dependence as predicted by the Bean–London critical state model. With increasing field the exponent initially increases, reaches the maximum of 4 as predicted by the Kim–Anderson critical state model. When the field is higher than H p the exponent decrease to 1. But AC loss per cycle decreases with the increase of frequency of field when the field is high. This can be explained by heating effect of AC loss. Furthermore, we calculate the AC loss of a full-scale inductive superconductor fault current limiter based on our measurement result. We found that the AC loss of this device is acceptable.

Harmonic susceptibilities of polycrystalline (La 1− xCa x)(Ba 1.75− xLa 0.25+ x)Cu 3O 7+ d synthesized by amorphous metal-complex method

Single-phase tetragonal (La 1− x Ca x )(Ba 1.75− x La 0.25+ x )Cu 3O 7+ d polycrystals with the composition range 0.1⩽ x⩽1.0 were synthesized by the amorphous metal-complex method. The fundamental and the higher-harmonic complex susceptibilities χ n = χ n ′−i χ n ′′ ( n=1, 2, 3, 5, and 7) were measured by means of Hartshorn bridge. The ac-magnetic field superimposed on the earth's field was applied to the specimen. We observed nonzero values of χ n for all n in the temperature range of the superconducting transition. The observed χ n – T curves were analyzed by the Kim–Anderson critical-state model. We found that the pinning force density increases up to x=0.5 and decreases over x=0.5 with increasing the Ca content x, and is proportional to two kinds of the power of the elemental pinning force.

Superconducting condensate density of (La 1− xCa x)(Ba 1.75− xLa 0.25+ x)Cu 3O 7+ δ polycrystals

The superconducting condensate density n s of single-phase tetragonal (La 1− x Ca x )(Ba 1.75− x La 0.25+ x )Cu 3O 7+ δ polycrystals was examined by the magnetic measurements. The samples were synthesized with the composition range 0.1⩽ x⩽1.0 by the amorphous metal-complex method. The magnetization curves were measured by a SQUID magnetometer. The lower critical field H c1 was obtained from the magnetization curve using the modified Kim–Anderson critical-state model, in which H c1 was taken into consideration. We estimated n s at 0 K from the temperature dependence of H c1. The n s value increases until x=0.5 and decreases over x=0.5 as x increases. Besides, n s was found to be linear to the hole density. The present relation between the transition temperature T c and n s is compared with the results for the underdoped samples by means of muon spin relaxation.

酸化物超伝導体の合成と評価 アモルファス金属錯体法で合成した（Ｌａ１−ｘＣａｘ）（Ｂａ１．７５−ｘＬａ０．２５＋ｘ）Ｃｕ３Ｏ７＋δの高調波複素帯磁率

We measured the temperature dependence of the 2nd, 3rd and 5th higher-harmonic susceptibilities of(La1-xCax)(Ba1.75-xLa0.25+x)Cu3O7+δ, x=0.2, 0.4, 0.5, prepared by the amorphous metal-complex method. We attempted to fit the calculations based on the Kim-Anderson critical-state model to the experimental results of χn. The analysis shows that the pinning force density is enhanced with increasingχ.

Lower critical fields of Bi2Sr2CaCu2O8+d and YBa2Cu3O7−d single crystals

Temperature dependence of the lower critical fields Hc1(T) of Bi2Sr2CaCu2O8+d and YBa2Cu3O7−d single crystals has been investigated for both orientations of the applied fields, parallel and perpendicular to the c axis. The magnetization curves of the samples were measured using a superconducting quantum interference device, and Hc1,c(T) (H∥c axis) and Hc1,ab(T) (H⊥c axis) were obtained using the modified Kim–Anderson critical-state model. From the results, we found both Hc1,c(T) and Hc1,ab(T) for a Bi2Sr2CaCu2O8+d single crystal had an anomalous upturn below about 40 K, while a YBa2Cu3O7−d single crystal did not show any anomaly down to about 10 K. Discussion on a remarkable anomaly Hc1,ab(T) at low temperatures is made using the formulas for anisotropic Hc1 in the Ginzburg–Landau theory. The result suggests that the temperature dependence of the coherence length in the ab plane plays an essential role. We may insist that the anomalous upturn of Hc1,ab for a Bi2Sr2CaCu2O8+d single crystal at low temperatures is attributed to the enhancement of the superconducting order parameter in the domain between the CuO2 layers, as theoretically predicted in terms of a proximity effect. Penetration-depth ratios λc2(0)/λc2(T) and λab2(0)/λab2(T) were also calculated using a theory based on incoherent quasiparticle hopping. Plots of the ratios as a function of the reduced temperature qualitatively reproduce the prediction by the hopping model, and suggest that both Bi2Sr2CaCu2O8+d and YBa2Cu3O7−d single crystals seem to possess the nature of d-wave pairing.

Magnetic response of Y/Pb doped single-crystal Bi/sub 2/Sr/sub 2/CaCu/sub 2/O/sub 8+d/ superconductors

The magnetic properties of Y/Pb doped single-crystal Bi/sub 2/Sr/sub 2/CaCu/sub 2/O/sub 8+d/ have been investigated in terms of the magnetization curves measured using a SQUID magnetometer. The observed results were analyzed within the construct of the modified Kim-Anderson critical-state model, where lower critical fields H/sub cl/ and surface barriers /spl Delta/H were explicitly taken into consideration. Analyses have revealed that the magnetic response is closely correlated to the carrier concentration, and both the lower critical field and the pinning force increase with the increase of the carrier concentration in the CuO/sub 2/ plane.

Surface barriers in a single-crystal YBa 2Cu 3O y bulk superconductor

The magnetic response of a single-crystal YBa 2Cu 3O y bulk specimen was measured in terms of complex susceptibilities χ n = χ n ′− iχ n ″ ( n=1,2,3,5,7). The observed non-linear magnetic response as a function of temperature was analyzed within the framework of the modified Kim–Anderson critical-state model, where the surface barrier Δ H and the lower critical field H cl are explicitly taken into consideration. Analyses have revealed that χ n can be well reproduced only when Δ H has a non-zero value. The best fit of the data gave μ 0Δ H(0)≈80 mT. The intrinsic value of the surface barrier μ 0Δ H′(0)≈440 mT evaluated by correcting for the demagnetization effect would be attributed to the Bean–Livingston (BL) barrier.

Material parameters of YBa 2Cu 3O y in the Kim–Anderson critical-state model

The magnetic response of a melt-processed YBa 2Cu 3O y under an external ac magnetic field has been investigated in the framework of the Kim–Anderson critical-state model, J c( B i)= k/( B 0+| B i|), where k and B 0 are the material parameters, and B i is the local internal magnetic-flux density. The magnetization curves obtained from direct measurements using a superconducting quantum interference device and from the Fourier series of higher-harmonic complex susceptibilities were simulated with the magnetization equations based on the model and the temperature-dependence of k and B 0 were studied. The k( T) behavior indicates that at lower temperatures, weak point defects (oxygen vacancies) play a predominant role as the pinning center, but at higher temperatures below T c, a contribution from the normal-state Y 2BaCuO 5 particles becomes important. The temperature-dependence of B 0 was found to be B 0∝(1− T/ T c) 3/2, suggesting that B 0 is the quantity closely correlated to the irreversibility field.

Effect of surface barriers and lower critical fields to magnetization of type-II superconductors

To determine the effect of the surface barrier Δ H and the lower critical field H c1 to magnetization of type-II superconductors, we have calculated the initial magnetization curves and full hysteresis loops within the framework of the modified Kim–Anderson critical-state model, where Δ H and H c1 are explicitly taken into consideration. A sample is located in an external field H( ωt)= H dc+ H ac cos ( ωt), where H dc (≥0) is a DC bias field and H ac (>0) is an AC field amplitude. Derivations of the magnetization equations were carried out on the assumption that the critical-current density J c is a function of the local internal magnetic-flux density B i, J c( B i)= k/( B 0+| B i|), where k and B 0 are constants. In addition, we used the effective magnetic field inside the specimen H eff for evaluations of B i, where H eff is assumed to have the form H eff= H−( H/| H|) H cl−[(d H/d t)/|d H/d t|]Δ H. We consider an infinitely long cylinder with radius a and the applied field along the cylinder axis. Denoting the maximum and minimum values of H by H A (= H dc+ H ac) and H B (= H dc− H ac), respectively, four types of hysteresis loops appear, depending on the magnitude of H A. Among these, three types are further classified into several cases, depending on the magnitude of H B. To describe completely the descending and ascending branches of the loops for all the cases, 113 stages of H are considered. To verify the present derivations, all the equations were confirmed to be continuous at their end points. Some typical hysteresis loops computed using the appropriate magnetization equations are demonstrated. From the results, we recognize the effect of Δ H and H c1 on the hysteresis loops. Δ H merely expands the M( H) curves up and down with the increase of Δ H, while H c1 introduces a step-like feature into the hysteresis loops. From measurements of the difference in level of the magnetization under the applied field of opposite directions, one can precisely determine H c1.

溶融ＹＢａ２Ｃｕ３Ｏｙの交流磁化応答における微弱直流磁場印加効果

We studied, both theoretically and experimentally, the ac magnetic response of type-II superconductors superimposed on a weak Hdc field. Theoretical simulations were made within the construct of the Kim-Anderson critical-state model, where the critical current density Jc is assumed to be given by Jc=k/(Bo+|Bi|). k and Bo are temperature-dependent material parameters, and Bi is the local flux density in the sample. We also measured complex susceptibilities Xn=X′n-iX″n (n=1, 2, 3, 5, 7) of a melt-processed YBa2Cu3Oy specimen and analyzed the observed Xn by applying the magnetization equations derived from the Kim-Anderson model. We found that the experimental results were qualitatively well reproduced. Present results indicate that, for a specimen of which Jc sensitively varies with Bi, the effect of a weak Hdc should be explicitly taken into consideration for studies of the ac magnetization.

Effect of Lower Critical Fields on Magnetization of a Single-Crystal Bi2Sr2CaCu2O8+d Bulk Superconductor.

To test the utility of the modified Kim-Anderson critical-state model for study of the magnetic response of ceramic superconductors, we measured the initial magnetization, hysteresis loops and ac complex susceptibilities Xn=X'n-iXn(n =1, 2, 3, 5) of a single-crystal Bi2Sr2CaCu2O8+d bulk sample. The results were simulated in the framework of the model. Analyses of Xn have shown that generation of the higher harmonics can be qualitatively reproduced, where the material parameters k and B0, surface barriers ΔH and lower critical fields Hcl, involved were used as fitting parameters. Comparison of the observed data with the calculated results has revealed that nonzero values of Hcl, play an essential role in the magnetic characteristics of the sample.

Temperature dependence of the material parameters in the Kim-Anderson critical-state model for a melt-processed YBa2Cu3O7

The magnetic response of a melt-processed YBa2Cu3O7 was measured to make clear the physical meanings of the parameters k and B0 in the Kim-Anderson critical-state model where the critical current density Jc is assumed to be a function of the local internal magnetic-flux density Bi, Jc(Bi) = k/(B0 + |Bi|). The isothermal magnetization curves of the sample were measured in the wide temperature region below the critical temperature, and the parameters were selected so as to fit the calculated curve to the observed one at each temperature. We found that k and B0 are proportional to (1 − T/Tc)γ, where γ = 2.5 – 3.0 and to (1 −T/Tc)1.5, respectively. We propose that B0 approximately corresponds to the irreversibility magnetic-flux density Birr.

Study of the remanent magnetization of Bi 2Sr 2CaCu 2O 8 single crystals by means of a hall probe

Using a sensitive Hall probe, the remanent magnetization of Bi 2Sr 2CaCu 2O 8 single crystals has been measured accurately as a function of temperature. The interpretation of the results is based on the modified Kim-Anderson critical-state model. An explanation of some of the commonly observed history dependencies is proposed, as well as the finite value of the remanent magnetization. A fast and accurate method has been developed for characterizing the temperature dependence of the critical current density.

Harmonic susceptibilities of a sintered oxide superconductor.

The superconducting transition of a sintered ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{\mathrm{y}}$ pellet with Fe impurities was measured in terms of harmonic susceptibilities ${\mathrm{\ensuremath{\chi}}}_{\mathrm{n}}$=${\mathrm{\ensuremath{\chi}}}_{\mathrm{n}}^{\ensuremath{'}}$-i${\mathrm{\ensuremath{\chi}}}_{\mathrm{n}}^{\mathrm{\ensuremath{''}}}$ (n=1, 2, 3, 5, and 7). The results were analyzed in the framework of the critical-state model and the weak-link model. Generation of the higher-harmonic susceptibilities seems to arise from the supercurrent according to the Kim-Anderson critical-state model, while the Ishida-Mazaki weak-link model, based on the assumption that the intergrain network behaves like a single loop as a whole due to its coherent nature, fails to describe the data. However, phenomenological considerations suggest that the extended weak-link model, in which a number of coaxial weak-link loops are assumed in the specimen, would be more practical for analyses of the magnetic response of oxide ceramic superconductors prepared by sintering.