Campbell penetration in the critical state of type-II superconductors

R. Willa, V. B. Geshkenbein, and G. Blatter

Phys. Rev. B 92, 134501 – Published 1 October 2015

Abstract

The penetration of an ac magnetic signal into a type-II superconductor residing in the Shubnikov phase depends on the pinning properties of Abrikosov vortices. Within a phenomenological theory, the so-called Campbell penetration depth $λ_{C}$ is determined by the curvature $α$ at the bottom of the effective pinning potential. Preparing the sample into a critical state, this curvature vanishes and the Campbell length formally diverges. We make use of the microscopic expression for the pinning force density derived within strong pinning theory and show how flux penetration on top of a critical state proceeds in a regular way.

Received 6 August 2015
Revised 15 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.134501

Authors & Affiliations

R. Willa, V. B. Geshkenbein, and G. Blatter

Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

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Vol. 92, Iss. 13 — 1 October 2015

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Images

Figure 1
Sketch of the linear Campbell (left panel) vs nonlinear Bean (right panel) response of a vortex critical state subject to an ac magnetic field $h_{ac}$ . The gray-shaded areas show the change of the dc field deep inside the sample after the initial field penetration at short times. The blue regions indicate the magnetic field oscillations at large times penetrating the sample to the depths $λ_{C}$ and $ℓ_{B}$ , respectively. The crossover between the Campbell and Bean regimes with increasing field amplitude $h_{ac}$ occurs when the induced current $δ j$ near the sample surface is of the order of $j_{c}$ , i.e., when $h_{ac} \sim j_{c} λ_{C} / c$ .
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Figure 2
Microscopic effective pinning force $f_{pin}$ with bistable pinned and unpinned solutions as a function of the pin-vortex distance $x$ (dotted lines denote unstable solutions). For randomly positioned defects and the vortex lattice in the critical state, the occupation of the force branches (blue) produces the maximal restoring force $F_{pin} = n_{p} 〈 f_{pin} 〉 = - F_{c}$ . A macroscopic displacement $U > 0$ of all vortices in the direction of the Lorentz force does not change the branch occupation and $δ F_{pin} = 0$ . A displacement $U < 0$ against the Lorentz force, however, modifies the branch occupation (green) and reduces the pinning force $δ F_{pin} \propto Δ f_{pin} U$ , see Eq. (7).
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Figure 3
Displacement $U$ (solid, blue) and maximum field $U_{0}$ (black, dashed) as a function of time $t$ at the positions $X = 0$ (sample boundary), $X = 2 λ_{C}$ , and $X = 4 λ_{C}$ (subsequent curves are shifted for better visibility). The periodic decrease in $U$ describes relaxation of vortices in the pinning wells. With the maximum field $U_{0}$ remaining constant over these regions, a reduction $δ F_{pin}$ in pinning force density below $F_{c}$ shows up.
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Figure 4
Change in pinning force density $δ F_{pin}$ versus displacement field $U$ at the sample surface $X = 0$ . The time $t$ is a curve parameter that evolves as indicated by the red line. Starting from $U = 0$ and increasing $U$ , flux moves into the sample, the pinning force density is at its maximum value $F_{pin} = - F_{c}$ , and $δ F_{pin} = 0$ . Deviations of $F_{pin}$ from its critical value appear whenever $U$ decreases and vortices relax in their pinning potential. At long times, the Bean profile has been raised to the value $B_{0} + h_{ac}$ and vortices oscillate reversibly (see double-headed red arrow) within their potential wells without additional flux entrance and vanishing regions with $δ F_{pin} = 0$ . The maximum field $U_{0} (0, t)$ reaches its asymptotic value $U_{0} (X = 0)$ , see Eq. (24).
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Figure 5
Change in magnetic induction $δ B (X, t)$ as a function of time $t$ and distance $X$ into the sample. Every oscillation pumps flux into the sample, pushing the critical state at $B_{0}$ ( $δ B = 0$ , green) towards one at $B_{0} + h_{ac}$ ( $δ B = h_{ac}$ , blue). The eight cycles shown here push almost all the flux into the sample that is required for its shift by $h_{ac}$ .
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Figure 6
Flux pumping $ϕ (t) = B_{0} U (0, t)$ as a function of time. We compare our numerical results (oscillating curves) for different sample sizes $d$ with our analytical model (dotted line), see Eq. (21). A better quantitative agreement with the numerical data is obtained when scaling the analytic result by $1.25$ (dashed line). The horizontal lines indicate for each system the value of the penetrated flux $h_{ac} d / 2$ after termination of the initialization process. The inset shows an expanded view at small times of both $U$ and $U_{0}$ at the surface.
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