Substrate and Finite-Thickness-Induced Uncertainties in Surface Impedance Measurements of Thin Conducting Films

Measurements of the surface impedance of conducting, semiconducting, and superconducting materials are commonly used in research, metrology, and industry to extract and study the resistivity of the material. The interest is often devoted to thin films because of: 1) the possibility to grow nearly perfect materials, essential for research; 2) ease of reproducibility for metrological standards; or 3) direct applications in the electronic industry. However, in finite-thickness films, the probing electromagnetic (e.m.) field leaks through the film to the underlying substrate. The substrate then gives a substantial contribution to the measured surface impedance and, thus, to the extracted resistivity. While the e.m. problem is well known, analyses in terms of the evaluation of the uncertainty involved are scarce. Moreover, simplified models for the measured surface impedance with different ranges of applicability are often used in order to simplify the analysis or to highlight the main physics. Also, in these cases, these approaches are seldom followed by careful evaluation of the additional uncertainty so introduced. In this article, we thoroughly study when simplified models, which neglect the substrate contribution, can be used, and we identify the situations in which the substrate properties yield negligible contributions to the extracted resistivity and its uncertainty. To do so, we discuss the various sources of uncertainty, also with the help of a brief description of typical measurement techniques. We show that the simplified models can greatly simplify the extraction of the material resistivity from the measured surface impedance, with reduced uncertainty and also reduced computational cost for its evaluation. We find that ranges in frequency and film thickness exist, where the simplified models can be used, with negligible approximation errors, once compared with the uncertainties related to the measurement technique. However, the identification of such combined ranges is not trivial, being particularly critical for superconducting films. Finally, we report experimental measurements of the surface impedance of different superconducting samples (bulks and the more critical situation of films over layered structures), providing case studies for both the determination of the uncertainties and the validation of the models.