Magnetic relaxation in hard type-II superconductors

– Magnetic relaxation in a type-II superconductor is simulated for a range of temperatures T in a simple model of 2D Josephson junction array (JJA) with finite screening. The high-T phase, that is characterised by a single time scale τα, crosses over to an intermediate phase at a lower temperature Tcr wherein a second time scale τβ ≪τα emerges. The relaxation in the time window set by τβ follows power law which is attributed to self-organization of the magnetic flux during relaxation. Consequently, for T < Tcr, a transition from super-critical (current density J > Jc) to sub-critical (J < Jc) state separated by an intermediate state with frozen dynamics is observed. Both τα and τβ diverges at Tsc < Tcr, marking the transition into a state with true persistent current. Introduction. – In a hard type-II superconductor, flux-creep over energy barrier at a finite T leads to magnetic relaxation over long time scales. The flux creep, occurring due to thermally activated hopping of the flux lines, tend to reduce the local field gradient dB dx , and hence the current density J . In analyzing relaxation measurements, the initial magnetic field distribution is assumed to be that of Bean’s critical state [1] wherein J is replaced by the critical current density Jc. If the relaxation is close to Jc, the magnetisation decay M(t) is then theoretically known to be logarithmic, as is the case in most low-Tc superconductors [2]. In this case, the effective pinning potential U(J) ∼ U0(J − Jc) is a good approximation. In high-Tc superconductors (HTSC), the large thermal energy available leads to rapid decay of M(t). Experimentally the relaxation is observed to be non-logarithmic over several decades in time in these materials [3]. This is interpreted as arising from a non-linear U(J). The vortex-glass theory [4] and collective-creep theory [5], which predicts a low temperature true superconducting state with finite Jc, expects a pinning potential of the form U(J) = U0[(Jc/J) μ − 1] which diverges in the limit J → 0. An important experimental observation in HTSC is an apparent universal value of the normalized relaxation rate S(T ) = −1 M dM d ln t around 0.02-0.035 over a wide range of T [6]. For the U(J) given above, such a small T -independent S requires μ > 2 which is beyond the range of existing theoretical models [3]. Experiments have also indicated power law decay in HTSC [7] which requires a logarithmically diverging U(J) = Uo ln(Jc/J) [8, 9]. This form of U(J) cannot account for the experimentally observed plateau in S(T ). Typeset using EURO-TEX 2 EUROPHYSICS LETTERS It becomes then important to identify the relaxation behaviour of the magnetisation in a type-II superconductor in presence of a uniform background of pinning centers. Towards this end, we simulate the time decay of remanent magnetisationM(t) at finite T in a simple model of 2D Josephson junction array (JJA). The magnetisation in this model is studied by including the screening currents through the inductance LR,R′ between the cells. The underlying discrete lattice of junctions provides an energy barrier for the vortex motion within the array [10] and is the source of vortex pinning in JJA. The behaviour of JJA with screening is parameterized by λ2J = Φ0 2πL0Ic , where Ic is the critical current of the junction, L0 is the self-inductance of the cell, and Φ0 is the quantum of flux. Detailed simulation at T = 0 have shown that for λ 2 J < 1, the magnetic response of this model is similar to a continuum hard type-II superconductor [11, 12, 13, 14]. In this paper, we show that the thermo-remanent relaxation of M(t) of JJA captures essential features of magnetic relaxation in HTSC. At a crossover temperature Tcr, a new time scale τβ emerges in an intermediate time window which along with the characteristic time scale τα for long time behaviour governs the relaxation. The characteristic time scales τα and τβ diverges as a power law at a temperature Tsc at which M(τ) →M0 6= 0 as τ → ∞. The Model. – We consider a 2D array of superconducting islands forming a homogeneous square lattice of N×N unit cells in the x−y plane. Tunneling of the macroscopic wavefunction Ψ(r) = ψ exp[iφ(r)] across neighbouring islands lead to Josephson coupling between them. In presence of an applied magnetic flux Φext (per cell) along ẑ direction, the junction behaviour is fully described by dynamics of the gauge-invariant phase difference φr,δ=φ(r) −φ(r + δ)− 2π Φ0 ∫ r+δ r A · dl between neighbouring islands. Here, δ is a unit vector, and A is the vector potential corresponding to the total magnetic field. The inset of Fig.1 shows a schematic array of size N = 5 along with variable φr,x and φr,y. The magnetic response of JJA is set by the induced flux Φind(R) due to screening currents which is modeled by considering the geometrical inductance matrix L of the array [13]. This allows us to write the induced flux in a cell at R as Φind(R) = ∑ R L(R,R)I(R) where I(R) is the cell current at R and L(R,R) is the mutual inductance between the cells at R and R. In order to ease the prohibitive computation cost involved when mutual inductance is considered [13, 14], we consider the induced flux only due to self-inductance L0 = L(R,R) of the cell. This approximation is equivalent to the case of a long (ideally infinite) superconductor parallel to an applied field for which demagnetisation factor ND = 0. The total flux at R then can be written as ΦR = Φext + L0IR (the cell co-ordinates are used as subscripts). The cell current is only a convenient variable for introducing Φind as the current through the junction is Ir,δ = IR − IR−δ where r is the junction common to cells at R and R − δ. Since, IR is divergence-less, Kirchoff’s law is automatically satisfied at each node of the lattice. The dynamical variable φr (subscript δ is implicit) is related to the cell current IR through the flux-quantisation condition ∑ r∈R φr = −2π ΦR Φ0 = −2π Φext Φ0 − 2π