Early stages of magnetization relaxation in superconductors

Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by $m\left(t\right)\propto 1-\sqrt{t/\tilde{\tau }}$ in the narrow initial time interval and by $m\left(t\right)\propto {(1+t/\tau )}^{-1}$ at later times before the flux creep sets in. The characteristic times $\tilde{\tau }$ and $\tau$ are proportional to the viscous drag coefficient $\eta$. Quantitative agreement with available experimental data is obtained for both conventional and high-temperature superconductors with $\eta$ exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding the complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient.

Figure 1

Vortex dynamics is studied for three sample geometries: (a) a slab, (b) a square-shaped plate, and (c) a disk.

Figure 2

Magnetic induction $B\left(x\right)$ across one half of a superconducting slab of width $L$ is shown for times $t/{\tau }_0=0.01$, $0.1$, $0.2,...,3.2$ (top to bottom).

Figure 3

Numerical solutions for magnetic moment relaxation for the slab ($\times$), disk ($\circ$), and square ($\square$) geometries, compared to the analytic approximation, Eq. (9) (solid line). Inset: A better fit for the initial time interval $t\ll \tau$ to the analytic expression, Eq. (7). In this case, only magnetization at the edges is affected by the flux flow.

Figure 4

Experimental data[14] ($\circ$) for the magnetic moment $m\left(t\right)$ in BSCO single crystal fitted to Eq. (9) with $m_0=1.1\times {10}^{-5}$ Am${}^2$ and $\tau =0.43$ min (solid curve). This corresponds to $\eta =0.5$ Ns/m${}^2$. Inset: The inverse magnetic moment as a function of time. The crossover between flux flow and flux creep regimes is seen as a dramatic change of the slope at $t\approx 7$min (dashed line). The sample is a slab $0.14\times 1.37\times 2.06$ mm in size, the initial magnetic induction $B_0=35$ mT, and $T=77$ K.

Figure 5

Experimental data[15] ($\circ$) for magnetic moment relaxation in ${\mathrm{NbSe}}_2$ monocrystal fitted to Eq. (7) (solid curve) with $\tilde{\tau }=1.13\times {10}^3$ min, corresponding to $\eta =11$ Ns/m${}^2$. The dashed line indicates a crossover between the flux flow regime, Eq. (1), and the slow quasistatic motion before the flux creep. The sample has a square geometry $0.5\times 0.5\times 0.2$ mm, the initial magnetic induction $B_0=3.3$ mT, and $T=4.2$ K.

Figure 6

Experimental data[17] ($\circ$) for magnetic moment relaxation in YBCO polycrystal at temperatures $T=30$ K, 41 K, 50 K, 61 K, and 77 K (top to bottom), fitted to Eq. (7) (solid curves) with $\tilde{\tau }=2.1\times {10}^5$, $5.0\times {10}^4$, $1.4\times {10}^4$, $3.0\times {10}^3$, and $7.4\times {10}^2$ min, respectively. This corresponds to the drag coefficients $\eta =284$, 66, 17, $2.93$, and $0.19$ ${\mathrm{Ns}/\mathrm{m}}^2$. Inset: Logarithm of the drag coefficient, normalized to $\eta \left(77\mathrm{K}\right)=0.19$ Ns/m${}^2$, as a function of the temperature. The sample has a rectangular geometry of $66\times 34\times 15$ mm. The initial magnetic induction is $B_0=3.95$, $3.77$, $3.48$, $2.80$, $0.735$ T, respectively.

Figure 7

Experimental data[18] ($\circ$) for magnetic moment relaxation in YBCO monocrystal at temperatures of 85 K and 87 K (top to bottom), fitted to Eq. (7) with $\tilde{\tau }=3.8\times {10}^3$ min and $\tilde{\tau }=570$ min, respectively (solid curves). The corresponding drag coefficients are $\eta =0.27$ and $0.04$ ${\mathrm{Ns}/\mathrm{m}}^2$. The sample has a square geometry of $1\times 1\times 0.02$ mm, and the initial magnetic induction is $B_0=0.1$ mT.