Measuring the stiffness and Coherence length of 2D superconductors

The gauge invariant London equation J=-ρ(A-ℏ/( 2e) ∇φ) states that the current density in a superconductor (SC) J is proportional to the vector potential A up to the gradient of the order parameter phase φ . The proportionality constant ρ is called the stiffness which is connected to the penetration depth λ by the relation ρ =1/μ_0  λ^2 . When cooling a SC at zero A , φ will be uniform to minimize the kinetic energy. In cases where φ is quantized, it remains uniform upon slightly increasing A , leading to the familiar London equation J=-ρA . This relation holds until J reaches a critical current J_c where φ is forced to change. The coherence length ξ is proportional to 1/J_c. The standard procedure of measuring ρ in a bulk SC is to apply a magnetic field and measure its penetration depth λ into the interior of the SC. However, in ultra-thin SC films, penetration depth is not well defined since there is no interior although surface current J ̃ and A do exists. A new way of measuring the superconducting stiffness and coherence length using a Stiffnessometer was developed in our group. This method measures ρ and J_c  directly, on the basis of the London equation with a rotor free vector potential. The method is applicable for 3D and 2D SC, using a long and narrow excitation coil which pierces a ring-shaped SC and produces a current in the ring. The ring’s magnetic moment is then measured by a superconducting quantum interference device (SQUID) to extract ρ and J_c (or J ̃_c in 2D). In this work I will present stiffness and coherence length measurements of a 2D, ultra-thin δ -NbN, and thin Granular Al SC films. The thinnest film measured is a 3 [nm] thick NbN film. A surplus of signal to noise ratio (SNR) in this measurement shows that a reduction of factor 10 in the thickness of the samples is possible, a fact that will allow in the future to apply this method on a truly 2D systems with a single atomic layer.