Magnetization and critical current of finite superconducting $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ rings

The irreversible magnetization of finite melt-textured $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ rings in perpendicular applied fields, affected by demagnetizing effects, is presented. The influence of the ring aspect ratio on the initial flux penetration and magnetization loops is studied by Hall probe and SQUID magnetometry. A general methodology based on the critical state model is developed in order to determine the critical current density from the field of full penetration $H_{\mathrm{pen}}$ in finite rings. We demonstrate that due to the field dependence of the critical current, the full penetration field $H_{\mathrm{pen}}$ of a finite ring does not coincide in general with the “kink” appearing in the initial magnetization curve. However, under certain conditions [very thin/narrow rings, or weak $J_c\left(H\right)$ dependence] the two fields collapse, and the critical current density can then be easily determined from the “kink” in the initial magnetization curve. In this latter situation, we develop an expression to determine the critical current density from the magnetic field measured at the center of the ring by Hall probe magnetometry.

Figure 1

(a) Typical dimensions and schematics of the field penetration in a ZFC finite ring. (b) Current circulation in a cross section at height $z$. In full penetration, the external and internal flux penetration fronts meet at $R_{\text{full}}\left(z\right)$.

Figure 2

Measured Hall probe $B_z∕{\mu }_0\left(H\right)$ profiles along a diametral cross section for three rings with different aspect ratios: (a) R125 $(\gamma =0.5)$, (b) R065 $(\gamma =0.27)$, and (c) R017 $(\gamma =0.07)$. (d) Calculated $B_z\left(H\right)$ profiles for ring R017. Right: at a height $h=80\mu \mathrm{m}$ from the sample. Left: at different heights: $h=80\mu \mathrm{m}$ (continuous line), $h=200\mu \mathrm{m}$ (dotted line), and $h=500\mu \mathrm{m}$ (dashed line).

Figure 3

(a) $B_c∕{\mu }_0\left(H\right)$ cycle (●) measured with a Hall probe placed at the axis line of the ring, at $z=L∕2+80\mu \mathrm{m}$, compared with the integral $⟨B∕{\mu }_0-H⟩\left(H\right)$ (엯) cycle (ring R125). (b) $J_c\left(H\right)$ dependences obtained from the $B_c∕{\mu }_0\left(H\right)$ (●) and the $⟨B∕{\mu }_0-H⟩\left(H\right)$ (엯) cycles.

Figure 4

(a) Magnetization loops for $\delta =0.5$ and different$\gamma$ ratios: $\gamma =0.2,1,5$ in Bean’s constant $J_c$ approximation. The position of the fields of full penetration $H_{\mathrm{pen}}$ are indicated. (b) Theoretical normalized penetration field $H_{\mathrm{pen}}∕J_c\left(0\right)R_0$ as a function of the aspect ratio $\gamma$: (continuous lines) analytical expression for finite rings [Eq. (4)], (dotted lines) Brandt’s expression for very thin rings [Eq. (5)], and (thick lines) Bean infinite cylinder limit.

Figure 5

(a) $m\left(H\right)$ cycles measured with SQUID magnetometry (thin lines) and Hall magnetometry (thick line) for the series of rings with equal internal $R_i∕R_0=0.67$ and different thickness $L$. Inset: “kink” $H_{\min }$ vs ring thickness. (b) (∎) Determined $H_{\min }∕J_c\left(0\right)R_0$ values for this series of rings against the $(\delta =0.67)$ analytical $H_{\min }∕J_c\left(0\right)R_0$ curve.

Figure 6

(a) Magnetization loops for $\delta =0.5$, $\gamma =5$ and different $J_c\left(H\right)$ decays: $p=0,1,10$. (b) Difference between the full penetration field $H_{\mathrm{pen}}$ and the “kink” field $H_{\min }$ normalized by $H_{\mathrm{pen}}^{\infty }$ as a function of $\gamma$, for $p=10$ and $\delta =0$ (disk), $\delta =0.5$, $\delta =0.9$.

Figure 7

(a) $m\left(H\right)$ cycle of ring R063. (b) $B_c\left(H\right)$ cycles for ring R125 with thickness $L=1.25\mathrm{mm}$ (continuous line), and same ring polished into thickness $L=0.6\mathrm{mm}$ (dashed line). $H_{\min }$ and $H_{\mathrm{pen}}$ are indicated. (c) Ring R063: Theoretical $H_{\mathrm{pen}}∕J_c\left(0\right)R_0$ (thick line, $\delta =0.42$) and experimentally determined $H_{\min }∕J_c\left(0\right)R_0$ (star) and $H_{\mathrm{pen}}∕J_c\left(0\right)R_0$ (bold star). Series of polished rings R125: Theoretical $H_{\mathrm{pen}}∕J_c\left(0\right)R_0$ $(\delta =0.64)$ and experimentally determined $H_{\min }∕J_c\left(0\right)R_0$ (엯) and $H_{\mathrm{pen}}∕J_c\left(0\right)R_0$ (●).