Keywords: Superconducting, Flux Pinning, Critical Current Density, High-Tc superconductor

Main International Conference:

Introduction text from ISTEC Journal Vol. 8 No. 1 1995

- 0. Introduction
- 1. The mechanism of flux pinning and the summation problem
- 2. Evaluation of critical current characteristics in superconductors
- 3. Electromagnetic phenomena related to the fluxoid motion
- 4. Longitudinal field effect

0. Introduction Four graduate students in the master course and four under graduate students are studying the flux pinning and related electromagnetic phenomena in superconductors with Prof. Matsushita and Mr. Otabe. They contribute also to the study of magnetic bearing system using high temperature superconductors which is conducted by Dr. Mochimutsu Komori in Department of Mechanical Systems Engineering in the same faculty. The main topics in this group are (1) the mechanism of flux pinning and the summation problem, (2) evaluation of critical current characteristics in superconductors, (3) electromagnetic phenomena related to the fluxoid motion and (4) the longitudinal field effect. In the following the main points of these topics are introduced. 1. The mechanism of flux pinning and the summation problem (a)elastic moduli of fluxoid lattice The macroscopic pinning force density depends not only on the strength and the concentration of pinning centers but also on the elastic moduli of pinned fluxoids. There are three independent moduli, i.e., the uniaxial compression modulus $C_{11}$, the tilt modulus $C_{44}$ and the shear modulus $C_{66}$. As for $C_{11}$ and $C_{44}$, two theories were proposed. One is the local theory of Labusch$^{4)}$ in which these moduli are predicted to be independent of the wave number of deformation of the fluxoid lattice and the other is the nonlocal theory of Brandt$^{5)}$ in which these moduli are predicted to be dependent. It is considered that the result of the nonlocal theory is doubtful, since the Maxwell stress tensor which derives the Lorentz force is independent of the wave number, while the Lorentz force is also expressed in terms of these elastic moduli. Then, it was shown$^{6)}$ that the quantization of magnetic flux and ${\rm div}\mathb{B}=0$ are not satisfied in the derivation of $C_{11}$ and $C_{44}$, respectively, in the nonlocal theory. At the same time, the local result was found to satisfy these requirements. (b)flux pinning characteristics in Nb-Ti The highest performance in the flux pinning efficiency is achieved in Nb-Ti among commercial superconductors. The dominant pinning centers in Nb-Ti is normal $\alpha$-Ti phase of the shape of thin ribbons. In the case of thin normal layers comparable to or smaller than the coherence length, the superconductivity appears even in the normal region due to the proximity effect. Then, Kramer and Freyhardt$^{7)}$ insisted that such thin layers are not effective as pinning centers. However, this prediction contradicts with the strong pinning in Nb-Ti and it was clarified$^{8)}$ in terms of G-L equations that the elementary pinning force is not reduced appreciably even under the remarkable proximity effect. On the basis of this theoretical result, the pinning characteristics in commercial Nb-Ti was analyzed$^{9)}$ and the pinning force in Nb-Ti with artificially introduced pinning centers is estimated in collaboration with Furukawa Electric Co. Ltd.$^{10)}$ (c)the summation problem in the regime of strong pinning It is empirically known that the linear summation, i.e., the proportionality of the pinning force density to the product of the pin concentration and the elementary pinning force, holds for the case of strong pinning such as the pinning by normal precipitates or grain boundaries. As for the theoretical prediction for such a strong pinning property, the theory of Larkin and Ovchinnikov$^{11)}$ was proposed. However, this result does not agree with the experiments. Then, a new statisitcal theory is established in which the Labusch parameter is used as a coherent parameter representing the strength of interference of individual pinning forces and the fact that the long range order does not exist in the fluxoid lattice as pointed out in the Larkin-Ovchinnikov theory. According to the result of this theory, while the threshold value for the elementary pinning force exists formally, its value is always smaller than the elementary pinning force. Therefore, it is possible to explain simultaneously$^{12)}$ that the practical threshold problem does not extist and that the pinning loss is of the hysteresis type as known well. The resultant pinning force density satisfies the linear summation and agrees quantitatively with experimental result. 2. Evaluation of critical current characteristics in superconductors This group has carried out the estimation of critical current density and the evaluation of weak link properties in high temperature superconductors$^{13)}$ by measuring the dependences of critical curernt density on temperature, history of application of magnetic field and strength of longitudinal magnetic field. For this purpose the AC inductive measurements (Campbell's method) and the AC susceptibility measurements have been used. Recently it was clarified that the reversible fluxoid motion inside the pinning potential becomes sometimes dominant in high temperature superconductors, since the small effective size of superconductor caused by cracks or weak links restricts severely the region of fluxoid motion. It was warned$^{14)}$ that the critical current density is likely to be overestimated largely if such results of AC measurements are analysed using the irreversible critical state model. The details of this investigation are given in references. 3. Electromagnetic phenomena related to the fluxoid motion The electromagnetic phenomena in superconductors are determined by the fluxoid motion. One of them is the loss energy. In most cases the loss energy in superconductors can be calculated in terms of the critical state model. This is correct only when the phenomenon is completely irreversible. However, when the superconducting filaments in multifilamentary metallic superconducting wires becomes very thin, the loss energy takes much smaller values than the theoretical estimate because of the reversible fluxoid motion mentioned in 3.2. In this group the loss energy under a remarkable reversible motion of fluxoids is theoretically estimated$^{15)}$. The flux creep is also the phenomenon determined by the motion of fluxoids. This brings about the relaxation of the magnetization or the superconducting persistent current, degrades the critical current density and sometimes reduces it to zero. This group has carried out the theoretical analysis from the viewpoint of flux creep on the irreversibility line on which the critical current density is reduced to zero. Figure 2 represents the irreversibility lines in various high temperature superconductors and (a) is experimental results and (b) is theoretical results$^{16)}$. It was shown that the flux pinning strength and the two-dimensionality of superconducting materials play the important roles in determination of the irreversibility line. The scaling behavior of the pinning force density in high temperature superconductors can also be analyzed in terms of the flux creep model$^{17)}$. It is also possible to foresee the characteristics in a material with artificially introduced pinning centers which will be fabricated in the future by using this model. 4. Longitudinal field effect Under the so-called longitudinal magnetic field where the magnetic field and the current are parallel to each other, the critical current density takes much larger values than in the ususal transverse magnetic field, Josephson's relation $\mathb{E}=\mathb{B}\times\mathb{v}$ does not hold ($\mathb{E}$, $\mathb{B}$ and $\mathb{v}$ are the electric field, the magnetic flux density and the fluxoid velocity, respectively) and a negative potential drop is sometimes observed in the dirction of the current. It was found that, when the force-free current parallel to the fluxoids flows, the fluxoids have a rotationally shearing distorsion as shown in Fig. 3 and the restoring moment works on the fluxoids to reduce this distorsion$^{18)}$. Various characteristic features under the longitudinal field geometry can be attributed to the fluxoid motion driven by this restoring moment. That is, the rotational fluxoid motion obtained from the continuity equation of fluxoids is regarded as a result of the restoring moment and leads to the electric field deviating from Josephson's relation$^{19)}$. The theoretical result of Josephson that the force-free state is an equilibrium state contradicts the above interpretation. This contradiction was found$^{20)}$ to be caused by the choice of incorrect gauge in the theory of Josephson. The negative potential drop in the resistive state was also successfully explained by the model of fluxoid motion caused by the moment$^{21)}$. Now the group is trying to prove the existence of the theoretically predicted moment which is independent of the Lorentz force in terms of a torque meter. References 1) T. Yasuda, S. Takano and L. Rinderer: Physica B 194-196 (1994) 2235. 2) T. Yasuda, S. Takano and L. Rinderer: Physica C 208 (1993) 385. 3) R. Kleiner and P. M ller: Phys. Rev. B 49 (1994) 1327. 4) R. Labusch: Phys. Status Solidi 19 (1967) 715. 5) E. H. Brandt: J. Low Temp. Phys. 26 (1977) 709, 735. 6) T. Matsushita: Physica C 220 (1994) 172. 7) E. J. Kramer and H. C. Freyhardt: J. Appl. Phys. 51 (1980) 4903. 8) T. Matsushita: J. Appl. Phys. 54 (1983) 281. 9) E. S. Otabe and T. Matsushita: Cryogenics 33 (1993) 531. 10) K. Matsumoto et al.: Appl. Phys. Lett. 64 (1994) 115. 11) A. I. Larkin and Yu. N. Ovchinnikov: J. Low Temp. Phys. 34 (1979) 409. 12) T. Matsushita: submitted to 1994 Int. Symp. Supercond., Kitakyushu. 13) T. Matsushita et al.: IEEE Trans. Appl. Supercond. 3 (1993) 1045. 14) T. Matsushita, E. S. Otabe and B. Ni: Physica C 182 (1991) 95. 15) T. Matsushita et al.: Adv. Cryog. Eng. Mater. (Plenum, 1994) p. 551. 16) T. Matsushita and N. Ihara: Proc. Europ. Conf. Appl. Supercond. (1993) p. 779. 17) T. Matsushita et al.: submitted to 1994 Int. Symp. Supercond., Kitakyushu. 18) T. Matsushita: J. Phys. Soc. Jpn. 54 (1985) 1054. 19) T. Matsushita, Y. Hasegawa and J. Miyake: J. Appl. Phys. 54 (1983) 5277. 20) T. Matsushita: Phys. Lett. 86A (1981) 123. 21) T. Matsushita and F. Irie: J. Phys. Soc. Jpn. 54 (1985) 1066.

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