Penetration dynamics of a magnetic field pulse into high-t c superconductors

Supercond. Sci. Technol. 9 (1996) 1042 1047. Printed in the UK Penetration dynamics of a magnetic field pulse into high-t c superconductors V Meerovich, M Sinder, V Sokolovsky, S Goren, G Jung, G E Shter and G S Grader Physics Department, Ben-Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel Material Engineering Department, Ben-Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel Chemical Engineering Department, Technion, 32000 Haifa, Israel Received 30 May 1996, in final form 28 August 1996 Abstract. The penetration of a magnetic field pulse into a high-t c superconducting plate is investigated experimentally and theoretically. It follows from our experiments that the threshold of penetration increases with increasing amplitude and/or decreasing duration of the applied pulse. The penetrating field continues to grow as the applied magnetic field decreases. The peculiarities observed are explained in the framework of the extended critical state model. It appears that the deviations from Bean s classical critical state model are characterized by a parameter equal to the square of the ratio of plate thickness to skin depth. The applicability of the classical critical state model is restricted by the condition that this parameter is much less than 1. This condition is also the criterion for the applicability of pulse methods of critical current measurements. 1. Introduction Bean s critical state model (CSM) [1] is used to describe electromagnetic properties of low-t c superconductors up to frequencies of about 10 MHz. Investigations [2, 3] showed that this model also describes the transport and magnetic properties of bulk ceramic high-t c superconductors (HTSCs). However, for HTSCs, deviations from this model were found at much lower frequencies. The authors of [4] observed a frequency dependence of the shielding factor; in [5] the surface impedance was found to be frequency dependent; in [6, 7] it was shown that the loss per cycle depends on frequency and deviates from the cubic dependence on the magnetic field. Because these peculiarities cannot be explained in the framework of the CSM, some nonstationary models have been suggested [6 9]. These models take into account the motion of fluxons and its influence on the distribution of magnetic field in the superconductor. In fact, the magnetic behaviour of HTSCs under a time-varying field is influenced by the associated resistive dissipation. To take into account this peculiarity, Bean has proposed an extension of the CSM that includes viscous motion of the fluxons [9]. This extended CSM was fruitfully used to investigate the frequency dependence of AC losses [6]. The second important problem is the investigation of the penetration of time-varying magnetic fields into superconductors. This investigation is essential both for development of superconducting shields and for analysis of the results of measurements using the pulsed technique [10 13]. In [11 14], critical current measurements in pulsed magnetic fields were reported. The analysis of the penetration of a pulsed magnetic field into a superconductor was based on the CSM [14, 15]. Bean used the extended CSM to study the response of an HTSC to a step in the magnetic field [9]. He found that the magnetic field distribution tends with time to that given by the classical CSM. In our previous paper [16], we used the extended CSM to investigate theoretically the magnetic field penetration into a semi-infinite HTSC slab when an external magnetic field was increased monotonically with time according to a power law. We obtained the result that, for high rates of charge, of an applied magnetic field, the solution does not tend to that given by the classical CSM and the penetration depth is substantially less than the depth predicted by this model. Developing these investigations, it is natural to consider the penetration dynamics under application of single pulses. This paper presents the results of an experimental and theoretical study of the response of HTSCs under an applied magnetic field pulse. 2. Experimental procedure The experimental study was based on measurements of the temporal dependence of the magnetic field penetrating through an HTSC plate exposed from one side to a magnetic 0953-2048/96/121042+06$19.50 c 1996 IOP Publishing Ltd

Penetration of magnetic field pulse into HTSC Figure 1. Experimental arrangement for measurements of penetration of magnetic field. field pulse. The schematics of our experimental set-up is shown in figure 1. A driving coil and a Hall probe for measuring the magnetic field were placed on the opposite sides of a superconducting sample. The cylindrical driving coil was 14 mm in diameter and 24 mm in length and contained 600 turns of 0.3 mm diameter copper wire. The coil together with a superconducting sample were cooled to 77 K by liquid nitrogen. The Hall probe had an active area of 0.03 mm 2 and was arranged to measure the component of the local penetrating magnetic field parallel to the applied field. Values of the current in the driving coil and signals from a Bell Gaussmeter (model 9200 with sensitivity 0.01 G) were recorded by a two-channel digitizing oscilloscope and passed to a computer for storage and analysis. A single current pulse of duration about 2.5 ms and of controlled amplitude up to 250 A was obtained by the discharge of a capacitor. Short times of high current pulses allow one to neglect heating processes. Two superconducting samples prepared by various technologies from YBCO and BSCCO were used in the present study. The BSCCO 2212 sample shaped as a plate of size 110 mm 95 mm was cut from a cylinder 200 mm in diameter, 95 mm in length and 3.5 4 mm in wall thickness. The cylinder was fabricated by centrifugal casting of a homogeneous melt from the oxides of Bi, Sr, Ca and Cu with an addition of 10% SrSO 4 [17]. The YBCO 123 discshaped sample was 32 mm in diameter and 2.5 mm in thickness, prepared by conventional sintering of oxalatederived powder [18]. 3. Experimental results Figure 2 illustrates a typical result of the penetration experiment: the temporal dependences of the current in the driving coil and corresponding signal from the magnetic probe located on the opposite side of the BSCCO sample. The coil axis was directed parallel to the sample surface. The penetrating magnetic field as a function of the instantaneous value of the current in the pulse is presented in figure 3. The applied magnetic field is proportional to the current in the driving coil: a driving current of 1 A Figure 2. Scope traces of driving current and penetrating magnetic field for the BSCCO 2212 plate-shaped sample. Figure 3. Penetrating magnetic field versus driving instant current for various amplitudes I max of a pulse (BSCCO 2212 sample in parallel field). corresponds to a magnetic field of 5.4 G at the point of the probe location at a temperature of 300 K. Penetration of the magnetic field through the plate exists already at applied fields much lower than the threshold of flux breakthrough. This peculiarity is observed by many authors in magnetic shielding experiments [19 22] and explained by the following: leakage field around the edges of the superconducting samples; microcracks or defects in the samples [19]; flux motion viscosity [20]; penetration and trapping of the magnetic field in intergranular space [21]; inhomogeneity of the critical current density which leads to a change in the pattern of shielding current paths and to partial field penetration [22]. Because of the smooth change of penetrated field with the 1043

V Meerovich et al Figure 4. Penetrating magnetic field versus driving instant current for various amplitudes of alternating current (BSCCO 2212 sample in parallel field). Figure 5. Penetrating magnetic field versus driving instant current for pulse and direct currents (YBCO 123 sample in perpendicular field). current, it is difficult to determine exactly the value of the driving current at which the breakthrough occurs. However, as can be seen from figure 3, the sharpest increase of the penetrating field is observed when the current is close to its maximum. The rate of increase of the penetrating field is higher than that of the applied field. As the amplitude of the pulse increases, the field breakthrough starts at a higher instantaneous current. The penetrating field continues to grow as the current and, hence, applied magnetic field decrease. Moreover, a certain increase of the penetrating field is observed after the applied pulse is over. Analogous but less pronounced peculiarities were observed when the sample was affected by a 50 Hz AC magnetic field (figure 4). We have found that the penetration peculiarities were preserved with a change of the direction of the coil axis and material of the samples. For example, the penetration of the magnetic field into a YBCO tablet when the coil axis is perpendicular to the sample surface is shown in figure 5. For comparison, the curve representing the penetration of a DC magnetic field is also given in this figure. The magnetic probe was arranged to measure the field component normal to the sample surface. From figure 5, the breakthrough of the DC magnetic field becomes appreciable at a current of about 0.35 A. For pulse fields the breakthrough current is substantially higher and increases with an increase of the pulse amplitude, attaining 0.45 A for I max = 0.95 A and 0.65 A for I max = 2A. 4. Theoretical analysis The observed non-linear peculiarities of the penetration of a time-varying magnetic field into HTSCs cannot be explained in the framework of the classical CSM. We will use an approach of the extended CSM which is based on Maxwell s equations supplemented by a constitutive relationship between an intensity of an electric field E and a current density j. We will restrict ourselves to the simplified case of an infinite superconducting plate exposed to a pulsed magnetic field applied parallel to the plate surface, in the z direction. The positive x direction is chosen inward and perpendicular to the plate, and currents are induced along the y axis. Maxwell s equations specialized to our geometry are E/ x = µ 0 H/ t (1) H/ x = j (2) where H is the intensity of the magnetic field. Following Bean [9], we use the relationship between E and j in the form E = ρ f (j j c ), E > 0; E = ρ f (j + j c ), E < 0 (3) where ρ f is the flux flow resistivity and j c is the critical current density. To determine the parameters ρ f and j c, several samples were cut out from the BSCCO sample and tested by a fourpoint pulse method described in [10]. The samples had weak sections with a cross-section of about 2 mm 1.5 mm and a length of 5 mm. The current pulse was of the same shape and duration as that used in our magnetic penetration experiment. The different samples gave values of the critical current density in the range 1000 1400 A cm 2 and the flux flow resistivity in the range 8 10 9 3 10 8 m. The same procedure applied to the YBCO sample gave about 400 A cm 2 and 10 8 m, respectively. The critical current of the samples decreased by about 20% in a magnetic field of 200 G [17], which was the maximum magnetic field in our experiments. Equations (1) and (2) are rewritten in dimensionless units: Ẽ/ x = a H/ t (4) H/ x = j (5) where Ẽ = E/ρ f j c, H = H/ j c, j = j/j c, a = µ 0 2 /t x ρ f, x = x/, t = t/t x, t x is the characteristic time 1044

Penetration of magnetic field pulse into HTSC of the change of the applied field and is the thickness of a superconducting plate. The critical current density is assumed to be constant. For a pulse field, t x is the pulse duration equal to 2.5 ms in our experiments and hence a = 0.2 1. For an AC magnetic field, t x can be chosen as 1/ω, where ω = 2πf ; f is the frequency of the applied magnetic field. In the last case a = 2 /2δ 2, where δ is the skin depth determined as for a normal metal with the resistivity equal to ρ f. The relationship (3) in dimensionless form is Ẽ = j 1, Ẽ>0; Ẽ= j+1,ẽ<0. The boundary and initial conditions in the dimensionless units are H( t = 0, x) = 0 Ẽ( t = 0, x) = 0 (6) H( t, x = 0) = H 0 F( t) H( t, x = x p )=0 Ẽ( t, x p ) = 0 (7) where H 0 = H 0 /j c, H 0 is the amplitude of the magnetic pulse, F( t) is the law of the change of the applied magnetic field with time and x p is the penetration depth of the magnetic field. The last two boundary conditions in (7) are used when the penetration depth is less than the plate thickness. When the magnetic field penetrates through the plate, the last two boundary conditions (7) are changed by the integral energy conservation law in the plate to the form 1 Ẽ 1 H 1 = Ẽ 2 H 2 + Ẽ j d x + a ( 1 H 2 ) 0 t 0 2 d x (8) where Ẽ 1, H 1, Ẽ 2 and H 2 are the intensities of electric and magnetic fields on both surfaces of the plate. The boundary conditions (7) are valid as long as Ẽ 1 = Ẽ( x = 0) >0. When an applied magnetic field decreases, the electric field also decreases and can reverse its sign on the surface. From this time on, two zones with opposite signs of Ẽ appear just as in the classical CSM. The conditions on the boundary of these zones are continuation of the magnetic field, the current density jumps from 1 to +1 when passing through the boundary. Equations (4) and (5) with the corresponding initial and boundary conditions were solved numerically. The results are presented in figures 6 9. Figure 6 reflects the penetration process for different values H 0 and a. The penetration depth is determined by the condition that the induced current shields fully the applied magnetic field. For the case x p, we have H(t,x = 0) = xp 0 j(x)dx. In the classical CSM, j = j c and the penetration depth increases proportionally with applied magnetic field. The maximum depth is equal to H 0 /j c. In the extended model, the induced current exceeds considerably the critical Figure 6. Theoretical temporal dependences of penetration depth: (a) for various values of the parameter a at H 0 =1; (b) for various amplitudes of an applied magnetic field pulse H 0 at a =1. x m is the maximum penetration depth in the classical CSM. value and limits the penetration of magnetic field into a superconductor (figure 7). The distribution of an addition to the critical current in principle is the same as the distribution of induced currents in normal metals. The difference is that the current falls sharply to zero at the point x p.asin normal metals, there is a temporal shift between the applied magnetic field and the induced current. So, at t = 0.38, when the applied magnetic field achieves its maximum value the current density near the plate surface is about 1.5 times greater than the critical one. The penetration depth decreases with growth of the parameter a and can be sufficiently lower than the Bean depth determined by the classical CSM (figure 6(a)). So, at H 0 = 3 and a = 1, the depth is only one-half of Bean s penetration depth. In the classical CSM, a zone with the opposite sign of the current appears exactly at the point when the time derivative of an applied magnetic field reverses its sign. As distinct from it, in the extended CSM, the change of the sign of current occurs only at t = 0.5, at the point 1045

V Meerovich et al Figure 8. Calculated temporal dependences of applied and penetrating magnetic field (a = 1). Figure 7. Theoretical distribution of current density in HTSC sample for different times (penetration depth less than thickness of sample). H 0 =1;a=1. on the negative slope of the curve of an applied magnetic field where the electric field reverses its sign. Further, the penetration process is characterized by two moving zones. Note that the penetration depth continues to increase even after the zone with opposite sign appears. This process becomes of a diffusional character and lasts as long as the addition to the critical current attenuates. The attenuation is due to both the resistive dissipation and the propagation of the current deep into a superconductor. Finally, two zones with opposite signs of j c are reached (figure 7(b)). If the penetration depth is less than the sample thickness, the areas of the zones in figure 7(b) should be the same. If the field penetrates through the sample, these areas differ and the trapped flux appears (figure 2 and figure 8). Figures 8 and 9 show the calculated results of the penetration of the magnetic field for our experimental conditions. As one can see, the calculated curves are in good agreement with the experimental curves shown in figures 2 and 3. Figure 9. Calculated dependences of penetrating magnetic field on applied magnetic field for the BSCCO 2212 plate-shaped sample (a = 0.5). 5. Discussion The extended CSM describes well and explains the experimental dependences. The breakthrough field proved to be dependent not only on the superconductor characteristics j c, and ρ f but also on the parameters of applied magnetic field pulse (amplitude and duration). We consider only the flux flow region with the linear E j relationship. Considering the voltage current characteristic of HTSC samples, some authors take into account the flux creep region with exponential E j relationship [7]. However, the results of [23, 24] and also our measurements show that this region ends below 1 µv cm 1. For our BSCCO sample, this corresponds to about 0.001 in the dimensionless units introduced above. This means that we can neglect the flux creep region and use the linear voltage current characteristic in the form of (3). The peculiarities of penetration of a time-varying magnetic field into high-t c superconductors have been 1046

Penetration of magnetic field pulse into HTSC observed by many authors [4 7]. Nevertheless, numerous papers on magnetic shielding and penetration under AC magnetic fields do not give an adequate answer to the question of a frequency limit of the applicability of the classical CSM to high-t c superconductors. For example, in [20] this model is used to study the magnetic penetration at a frequency of 1000 Hz. However, our experimental results presented above show marked deviation from this model already at 50 Hz. The theoretical analysis shows that the basic parameter which determines the degree of deviation from the classical CSM is the parameter a, that is the square of the ratio of the plate thickness to the skin depth. When this ratio is much less than 1, the classical model is applicable. With increasing frequency or thickness of the superconductor, the parameter a increases as a square function. Deviations from the classical CSM become crucial. The discrepancy existing in the literature between criteria of applicability of the classical CSM is explained by different values of the parameter a, specifically by different thicknesses of the superconducting samples. Note that the results obtained show that pulse methods for critical current measurements are of limited usefulness. The criterion of applicability of pulse methods is a 1. Acknowledgments We wish to thank the Ministry of Science and the Arts of Israel and the Technion Crown Center for Superconductivity for their support of our work. References [1] Bean C P 1962 Phys. Rev. Lett. 8 250; 1964 Rev. Mod. Phys. 36 31 [2] Dersch H and Blatter G 1988 Phys. Rev. B 38 11 391 [3] Bryksin V V, Goltsev A V and Dorogovtsev S N 1990 Physica C 172 352 [4] Macfarlane J C, Driver R, Roberts R B, Horrigan E C and Andrikidis C 1988 Physica C 153 155 1423 [5] Fisher L M, Mirkovic J, Voloshin I F, Makarov N M, Yampol skii V A, Rodriguez F Perez and Snyder R L 1994 Appl. Supercond. 2 685 [6] Jiang H and Bean C P 1994 Appl. Supercond. 2 689 [7] Uesaka M, Suzuki A, Takeda N, Yoshida Y and Miya K 1995 Cryogenics 35 243 [8] Rhyner J 1993 Physica C 212 292 [9] Bean C P 1989 Superconductivity and Applications ed H S Kwok et al (New York: Plenum) p 767 [10] Goren S, Jung G, Meerovich V, Sokolovsky V, Skoletsky I and Homjakov V V 1994 Proc. EUCAS 93, Applied Superconductivity ed H C Freyhardt (Oberusel: Deutsche Gesellschaft für Material-Kunde e.v.) p 749 [11] Hole CRJ,Jones H and Goringe M J 1994 Meas. Sci. Technol. 5 1173 [12] Hole C R J, Jones H, Burgoyne J W, Dew-Hughes D, Grovenor C R M and Goringe M J 1995 IEEE Trans. Appl. Supercond. 5 1313 [13] Jones H, Hole C R J, Ryan D T and van der Burgt M 1995 Applied Superconductivity (Inst. Phys. Conf. Ser. 148) ed D Dew-Hughes (Bristol: Institute of Physics) p 89 [14] Ryan D T, Hole C R J, van der Burgt M, Jones H, Davies C M, Grovenor C R M, Goringe M J and Dew-Hughes D 1996 IEEE Trans. Magn. 32 2803 [15] Hole C R J, Ryan D, Dew-Hughes D, Jones H, Goringe M J and Grovenor CRM1996 Physica B 216 291 [16] Meerovich V, Sinder M and Sokolovsky V 1996 Supercond. Sci. Technol. 9 734 [17] Bock J, Bestgen H, Elschner S and Preisler E 1993 IEEE Trans. Appl. Supercond. 3 1659 [18] Shter G E and Grader G S 1994 J. Am. Ceram. Soc. 77 1436 [19] Ohshima S and Okuyama K 1990 Japan. J. Appl. Phys. 29 2403 [20] Wang J and Sayer M 1993 Physica C 212 395 [21] Chandran M and Chaddah P 1995 Supercond. Sci. Technol. 8 774 [22] Takeda N, Uesaka M and Miya K 1995 Cryogenics 35 893 [23] Polák M, Koffman P, Majoroš M, Kedrová M and Plechácek V 1990 Supercond. Sci. Technol. 3 67 [24] Bock J, Elschner S, Herrmann P F and Rudolf B 1995 Applied Superconductivity (Inst. Phys. Conf. Ser. 148) ed D Dew-Hughes (Bristol: Institute of Physics) p 67 1047