Behavior of bulk high-temperature superconductors of finite thickness subjected to crossed magnetic fields: Experiment and model

Crossed-magnetic-field effects on bulk high-temperature superconductors have been studied both experimentally and numerically. The sample geometry investigated involves finite-size effects along both (crossed-)magnetic-field directions. The experiments were carried out on bulk melt-processed Y-Ba-Cu-O single domains that had been premagnetized with the applied field parallel to their shortest direction (i.e., the $c$ axis) and then subjected to several cycles of the application of a transverse magnetic field parallel to the sample $ab$ plane. The magnetic properties were measured using orthogonal pickup coils, a Hall probe placed against the sample surface, and magneto-optical imaging. We show that all principal features of the experimental data can be reproduced qualitatively using a two-dimensional finite-element numerical model based on an $E\text{−}J$ power law and in which the current density flows perpendicularly to the plane within which the two components of magnetic field are varied. The results of this study suggest that the suppression of the magnetic moment under the action of a transverse field can be predicted successfully by ignoring the existence of flux-free configurations or flux-cutting effects. These investigations show that the observed decay in magnetization results from the intricate modification of current distribution within the sample cross section. The current amplitude is altered significantly only if a field-dependent critical current density $J_{\mathrm{c}}\left(B\right)$ is assumed. Our model is shown to be quite appropriate to describe the cross-flow effects in bulk superconductors. It is also shown that this model does not predict any saturation of the magnetic induction, even after a large number $(\sim 100)$ of transverse field cycles. These features are shown to be consistent with the experimental data.

Figure 1

(a) Schematic illustration of the experimental configuration used for “crossed-field” measurements. (b) Geometry used for the two-dimensional model: the sample is of cross section $y_0\times z_0$ and is infinite in the $x$ direction. In all cases, the sample is premagnetized along the $z$ axis and the transverse field is applied parallel to the $y$ axis.

Figure 2

Measured central magnetic induction $\parallel z$ (sample HB1) during the application of one cycle of transverse field $\parallel y$, as shown in the inset. The induction is normalized with respect to its initial value $B_0$. The transverse field is normalized with respect to the full-penetration field $H_p$, which is determined experimentally. The amplitudes of the cycles $H_{\max }∕H_p$ are 0.15, 0.30, 0.45, 0.98, 1.18, and 1.53.

Figure 3

Results of the numerical model of the central magnetic induction $\parallel z$ during the application of one cycle of transverse field $\parallel y$, as shown in the inset. The induction is normalized with respect to its initial value $B_0$. The transverse field is normalized with respect to the full-penetration field $H_p$. The amplitudes of the cycles $H_{\max }∕H_p$ are 0.15, 0.30, 0.45, 0.98, 1.18, and 1.53.

Figure 4

Modeled data of the current density distribution $J_x(y,z)$ within the cross section of the sample during one cycle of the transverse magnetic field of amplitude $0.5H_p$. A field-dependent $J_c\left(B\right)$ is assumed. Bottom: scale of current density $J_x$ (expressed in ${10}^3\mathrm{A}∕{\mathrm{cm}}^2$). The arrow indicates the direction of a positive transverse field. Right: schematic diagram showing the times at which the current density distributions are determined.

Figure 5

Modeled data of the current density distribution $J_x(y,z)$ within the cross section of the sample during one half-cycle of the transverse magnetic field: (a) constant $J_c$ and $H_{\max }=0.5H_p$, (b) constant $J_c$ and $H_{\max }=1.5H_p$, and (c) field-dependent $J_c\left(B\right)$ and $H_{\max }=1.5H_p$. The times (0)–(4) in the cycles correspond to those defined in Fig. 4. (d) Modeled data of the average magnetization $M_z$ during one cycle of the transverse field, assuming either a constant (●) or a field-dependent (엯) critical current density $J_c$.

Figure 6

Log-log plot of the measured central magnetic induction (sample HB1) at the end of each positive transverse field cycle for three different transverse field sweep amplitudes $H_{\max }$. The solid lines are power-law fits $B_z\sim N^{-\alpha }$ with values of $\alpha$ given in the inset.

Figure 7

Log-log plot of the modeled central magnetic induction at the end of each positive transverse field cycle for three different transverse field sweep amplitudes $H_{\max }$. The solid lines are power-law fits $B_z\sim N^{-\alpha }$ with values of $\alpha$ given in the inset.

Figure 8

(Color online) (a) MOI image of the initial trapped flux above the top surface of sample HB2. The arrow indicates the transverse field direction. The intersection of the two orthogonal dashed lines defines the sample center ($x=0$, $y=0$). (b) Comparison of the flux profiles $\mid B_z\left(y\right)\mid$ measured above the top surface of sample HB2 along the $y$ direction before and after the application of the transverse field, as shown in the inset. The field amplitude $H_{\max }=0.48H_p$. (c) Modeled data of the distribution of $\mid B_z\left(y\right)\mid$ along the $y$ direction at the top surface before and after the application of the transverse field, as shown in the inset. The field amplitude $H_{\max }=0.50H_p$. The magnetic induction is normalized with respect to the maximum trapped field value.

Figure 9

Modeled data of the $M_z\left(H_y\right)$ curves obtained for the “paramagnetic” and the “diamagnetic” initial states resulting from the magnetization sequences $M_z\left(H_z\right)$, as shown schematically in the inset. In all cases, the magnetizing field $H_z$ is applied continuously during the application of the transverse field $H_y$.