Surface impedance measurements in thin conducting films: Substrate and finite-thickness-induced uncertainties

Precise measurements of the surface impedance Zsof conducting, semiconducting and superconducting materials is a common requirement for research, metrology and industry. The interest is often devoted to thin films because of (i) the possibility to grow nearly-perfect (single-crystalline) materials, essential for research and selected applications, or (ii) ease of reproducibility for, e.g., metrological standard, or (iii) direct applications of thin films in the electronic industry. However, in finite-thickness films the probing electromagnetic field does not vanish in the volume, and a leakage arises. Thus, the substrate where the film is grown gives a substantial contribution to the measured surface impedance. While the electromagnetic problem is well known, analyses in terms of the evaluation of the uncertainty involved are scarce. It is then accepted that microwave measurements are affected 'necessarily' by some uncertainties that cannot be easily evaluated. In this paper we report an extensive numerical study of the commonly used approximations of the full electromagnetic expression for the surface impedance of finite-thickness (super)conducting films backed by insulating substrates. We take into account the most common simplified expressions, and we estimate the uncertainties involved as a function of the thickness and of the complex conductivity of the film. We find that ranges in frequency and film thickness exist, where the use of approximate expressions gives rise to a negligible error. However, the identification of such combined ranges is not trivial, and it is particularly critical for superconducting films.