Nb/Al-AlOx/Nb tunnel junctions are often used in the studies of macroscopic quantum phenomena and superconducting qubit applications of the Josephson devices. In this work, we describe a convenient and reliable process using electron beam lithography for the fabrication of high-quality, submicron-sized Nb/Al-AlOx/Nb Josephson junctions. The technique follows the well-known selective Nb etching process and produces high-quality junctions with Vm=100 mV at 2.3 K for the typical critical current density of 2.2 kA/cm^2, which can be adjusted by controlling the oxygen pressure and oxidation time during the formation of the tunnelling barrier. We present the results of the temperature dependence of the sub-gap current and in-plane magnetic-field dependence of the critical current, and compare them with the theoretical predictions.
Based on the results of explicit forms of free energy density for each possible arrangement of magnetization fluxes in large-scale two-dimensional （2D） square π-loop arrays given by Li et al [2007 Chin. Phys. 16 1450], the field-cooled superconducting phase transition is further investigated by analysing the free energy of the arrays with a simplified symmetrical model. Our analytical result is exactly the same as that obtained in Li＇s paper by means of numerical calculations. It is shown that the phase transition splits into two branches with either ferromagnetic or anti-ferromagnetic flux ordering, which depends periodically on the strength of external magnetic flux φe through each loop and monotonically on the screen parameter β of the loops in the arrays. In principle, the diagram of the phase branches is similar to that of its one-dimensional counterpart. The influence of thermal fluctuation on the flux ordering during the transition from normal to superconducting states of the π-loop arrays is also discussed.
