Self-organized criticality effect on stability: magneto-thermal oscillations in a granular YBCO superconductor

– We show that the self-organized criticality of Bean’s state in each of the grains of a granular superconductor results in magneto-thermal oscillations. We find that the frequency of these oscillations is proportional to the external magnetic field sweep rate Ḃe and is inversely proportional to the square root of the heat capacity. We demonstrate experimentally and theoretically that this dependence is influenced mainly by the granularity of the superconductor. The magnetic-flux dynamics in a superconductor with a strong pinning potential for vortices is an important field of application for the self-organized criticality theory [1]-[5]. In the scope of its ideas the pinned vortices space distribution arises as a result of a subsequent local vortex avalanches establishing the critical state. Bean’s critical-state model [6] successfully describes the irreversible magnetization in type-II superconductors by introducing the critical current density jc. In the framework of Bean’s model the value of the slope of the stationary magneticfield profile is less than or equal to μ0jc. It makes the spatial distribution of vortices similar to the sand particles spatial distribution in a sandpile [7]. The stationary critical state becomes unstable under certain conditions when the local vortex avalanches result in a global flux jump driving the system to the normal state [8]. This instability can be preceded by a series of magneto-thermal oscillations [9]. These oscillations have been reported earlier [10]-[12] but no systematic experiments accompanied by an adequate theory have been made. Each of the local vortices avalanches establishing the critical state produces a heat pulse and a temperature rise decreasing the critical current density for a certain time interval and, thus, changing the initial conditions for the subsequent avalanches. In other words, the heat pulses produced by the vortex avalanches result in a correlation mechanism specific c © Les Editions de Physique 288 EUROPHYSICS LETTERS for the self-organized criticality of magnetic-flux motion in superconductors. In this letter we demonstrate that this mechanism results in the magneto-thermal oscillations arising close to the threshold of the superconducting-state stability. Our study is focused on the dependence of the frequency of these oscillations on the temperature and magnetic-field sweep rate in case of a granular superconductor. We begin with a theoretical consideration and propose a one-dimensional model of a granular superconductor treating it as a stack of superconducting slabs having the width 2bi (i = 1, 2, 3, . . . , N) randomly distributed with a certain mean value b. We assume that: a) there is no electrical contact between the slabs and there is an ideal thermal contact between them; b) the external magnetic field Be(t) is parallel to the sample surface (Be ‖ z-axis) and the sweep rate Ḃe is constant; c) the critical state arises simultaneously in the entire superconductor, i.e. in each of the slabs the magnetic, Bi(x, t), and electric, Ei(x, t), fields arise simultaneously. We also suppose that close to the instability threshold most of the slabs are saturated, i.e. Be is higher than Bean’s saturation field Bp = μ0jcb. Magneto-thermal oscillations in the critical state are coupled oscillations of temperature and electric field. These oscillations are observed at low values of magnetic-field sweep rates Ḃe with the background electric field in the slabs corresponding to the flux creep regime. In this regime the dependence of the current density j on the electric field E takes the form j = jc + j1 ln ( E E0 ) , (1) where E0 is the voltage criterion at which jc is defined, and the characteristic current density j1 (j1 ¿ jc) determines the slope of the j-E curve. In the framework of self-organized criticality of Bean’s state the function j1(T ) seems to tend to a non-zero value for T ¿ Tc [4], [5]. We will consider the latter case as it is in good agreement with numerous experimental data [13] and assume that the ratio n = jc/j1 À 1 is a certain constant for T ¿ Tc. We use heat diffusion and Maxwell equations to determine the frequency of small-amplitude magneto-thermal oscillations. In the case of small temperature δT = exp[γt] θ(x) and electric field δE = exp[γt] 2(x) perturbations these equations read [8], [9] nEi μ0γjc 2′′ − 2 = −i jc ∣∣∣∣∂jc ∂T ∣∣∣∣ θ, (2) κθ′′ − γCθ = −jc2 , (3) where C is the heat capacity per unit volume, κ is the heat conductivity, and γ is complex. Let us clarify the following calculation qualitatively. Suppose that the initial temperature of the superconductor T0 increases by a small perturbation θ0 arising due to a heat pulse with the energy δQ0. This temperature increase leads to a decrease of the superconducting currents. The reduction of these screening currents results in an additional flux penetration inside the superconductor. This flux motion induces an electric-field perturbation 20 producing an additional heat release δQ1, an additional temperature rise θ1, and, consequently, an additional reduction of the superconducting currents. At certain conditions this process results in a critical-state instability. The critical state is stable if the heat release, δQ, arising in the process of electric-field and temperature perturbations development is less than the heat flux to the coolant. The value of δQ depends on both Ḃe and Be for the unsaturated grains and only on Ḃe for the saturated grains. In our experiments most of the grains are saturated in the magnetic-field region corresponding to the magneto-thermal oscillations. Therefore, the heat release in the unsaturated grains, δQu, is small compared with the heat release in the saturated grains, L. LEGRAND et al.: SELF-ORGANIZED CRITICALITY EFFECT ON STABILITY ETC. 289 δQs. However, the term δQu is the only magnetic-field–dependent term in the heat balance equation and, thus, it determines the value of the global flux jump field Bj. In other words, the relatively small heat release in the unsaturated grains is tuning the superconducting state in a granular superconductor to the instability. Under conditions of our experiments the temperature θ is practically uniform over the cross-section of the sample. Therefore, to solve eq. (2) we consider θ to be constant. In a saturated slab the background electric field Ei = Ḃex, where x = 0 corresponds to the middle plain. In this case eq. (2) takes the form nḂe μ0γjc x 2′′ − 2 = −e jc ∣∣∣∣∂jc ∂T ∣∣∣∣x (4) with the boundary conditions 2(±bi) = 0. The magneto-thermal oscillations exist for low values of Ḃe, where l = nḂe/μ0γjc ¿ bi and the appropriate solution of eq. (4) reads 2(x) = nḂeθ jc ∣∣∣∣∂jc ∂T ∣∣∣∣ [ x− √ bil exp [ − bi − |x| √ bil ]]