Theoretical investigation on 3rd harmonic voltage in inductive measurements of critical current density in superconducting films

We have theoretically investigated third-harmonic voltage detected in a widely-used inductive measurement of critical current density J_c in superconducting films. In the inductive measurement, an ac drive current I₀ cos ωt flows in a small coil close to the surface of a superconducting film, and third-harmonic voltage V₃ cos(3ωt + θ₃) induced in the coil is detected. The relationships among the ac drive current I₀, the amplitude V₃, and the phase θ₃ of the third-harmonic voltage are derived on the bases of the critical state model. When the drive current is smaller than a threshold value I_c0, the third-harmonic voltage is near zero, V₃ ≈ 0 for I₀ < I_c0. For I₀ > I_c0, on the other hand, the third-harmonic voltage is expressed as V₃ exp(-iθ₃) = ωI_c0 G₃(I₀/I_c0), where G₃(γ) is a scaling function determined by the configuration of the coil. The predicted scaling law of V₃/I_c0 and θ₃ as functions of I₀/I_c0 is clearly observed in an YBa₂Cu₃O_7-δ film.