Extracting the penetration depth from the critical current of superconducting microbridges

The macroscopic behaviour of superconductors is well described by London equations, which provide an explanation for both the zero DC resistance and the Meissner effects. In thin films, the traditional London penetration depth which accounts for the current and magnetic field distribution in bulk superconductors, is replaced by a two-dimensional screening length called the Pearl length. The penetration depth is linked to another property of superconducting materials, the kinetic inductance, which is of critical importance in the design of detectors and electrical circuits. The current that can be carried by superconductors is limited to a value called critical current, above which the sample becomes resistive. In type-II superconductors the critical state is usually associated to the lowering of edge barriers, that allow the penetration of vortices inside the structure and the onset of dissipation. This work presents a novel method for an indirect measurement of the penetration depth which requires the fabrication of microbridges with standard electron beam lithography processes like those employed for detectors, and DC measurements without any external applied magnetic field. Using finite element modelling, we solved the London equation in thin film microbridges, and obtained both the current density distribution inside the wire, and its integral over a range of widths. The sharply peaked current distribution at the edges of the wire causes the current to saturate at large widths, depending on the two-dimensional screening length. Assuming that in the critical state the current distribution follows London equations, and that the resistive state appears for a value of current density at the edges (critical current density) that is only related to material properties and independent of the geometry, it is possible to extract the value of the penetration depth by comparing the critical current of bridges with a range of widths up to a few times the Pearl length, with the previously mentioned finite element model. Niobium nitride was the material of choice in this work, deposited via reactive DC magnetron sputtering at room temperature with an applied RF bias. A layout optimization process was applied to decrease current crowding effects, thus reducing the variance in the critical currents measured from replicas of the same device to within 5%. We devised a process for the fabrication of the devices: gold electrical contact pads were patterned first with direct write lithography, followed by e-beam evaporation of the metal and bi-layer lift-off; microbridges were patterned with electron beam lithography on ZEP520A, a positive-tone resist, then etched with RIE. Results from measurements of the critical current, performed with a cryogenic probe station at a temperature of 4.8 K, of several microbridges on a 20-nm thick NbN film, with widths ranging from 1 to 105 microns, provided a value of the London depth between 400 and 440 nm, and a critical current density compatible with values reported in literature. The temperature dependence of the critical current was also investigated. The main result from this work is a measurement method for the penetration depth of niobium nitride which could be extended to other materials with little changes, and requires simple fabrication steps and measurements setup. An independent method of measurement of the screening length is necessary to confirm the validity of this approach.