Vortex Structure: Lattice, Glass, or Liquid?
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VORTEX STRUCTURE: LATTICE, GLASS, OR LIQUID? ROBERT J. SOULEN, JR. AND STUART A. WOLF Naval Research Laboratory, Washington, DC 20375-5000, USA ABSTRACT Recent measurements of the dissipation in cuprate superconductors in a magnetic field have been interpreted as providing evidence for the presence of new phases in type II superconductors: flux liquids or flux glasses. We suggest that a more conventional interpretation in terms of the electrodynamics of vortices can adequately account for all the observations. Based on this model, we propose a magnetic phase diagram. INTRODUCTION In the new high transition temperature superconducting oxides (HTS) irreversible behavior in magnetization [1] and magnetic torque [21 is observed to give way dramatically to reversibility in the presence of moderate magnetic fields. In conventional superconductors, this change is usually explained in terms of the transition from "flux creep" to "flux flow" when the orderly array of vortices (Abrikosov lattice) begins to move by the pinning centers in the lattice. Speculation for the HTS has often settled instead on another possibility, referred to as "flux lattice melting", where thermal fluctuations are proposed to be sufficiently large to " melt" the Abrikosov lattice [3]. We show here how very elementary arguments based on the energetics of individual flux lines permit us to construct a rough " phase diagram" which indicates where the various phenomena (flux pinning, flux flow, and flux lattice melting ) can be expected to appear as a function of magnetic field. In particular we show that the phase diagram is dramatically influenced by the effective length of the flux lines. When the flux lines are long ( i.e., three dimensional), the conventional picture of a robust Abrikosov lattice weakly pinned to the defects still applies at moderate magnetic fields far from Tc and flux lattice melting may only appear at low magnetic fields over an extended temperature range. If the flux lines are two dimensional, the region of the phase diagram occupied by melting enlarges considerably. THEORETICAL FRAMEWORK The Lorentz force, FL, acting upon an individual flux line (vortex), is
given by FL = (1/c) JxB Vp
=
(1/c ) JxB t2 1
(I)
where c is the speed of light, J the current density, B the magnetic induction, and Vp the volume of the vortex, which is taken to be a cylinder of normal material of radius ý , and correlation length, 1. A pinning force
Mat. Res. Soc. Symp. Proc. Vol. 169. '1990 Materials Research Society
860
Fp can trap the flux lines in an energy "well" of depth Ep. expressions for these pinning quantities for a single vortex are:
The
Fp = Hc 2 4 1/8r
(2)
Ep = Hc 2 2 1 /8n
(3)
where Hc is the thermodynamic critical magnetic field. By varying B or J, FL may be varied with respect to Fp. When FL< Fp, the pinning centers impede motion of the flux lines, although some motion of the flux lines can occur by thermal activation of the flux lines over the pinning potential, Ep, giving rise to the term"flux creep".When FL> Fp, the f
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