A numerical analysis of the short open load calibration robustness for capacitance measurements in Scanning Microwave Microscopy

Abstract As the dimensions of nanoelectronics components are increasingly reduced, the need for reliable nanoscale characterization of electrical properties has become critical. In particular, the measurement of sub-femtofarad capacitance is a key element in developing thin films, unlocking the possibility of extracting the local permittivity and loss tangent. 
These quantities can be determined by measuring local impedance (ZS) measured by a Scanning Microwave Microscope (SMM). By coupling an Atomic Force Microscope (AFM) to a Vector Network Analyzer (VNA), the SMM measures the reflection coefficient (S11) of a microwave focused locally on the sample’s surface. The relationship between S11 and ZS is given by S11 = (ZS – Z0)/(ZS + Z0), where Z0 is a reference impedance (Z0 = 50 Ω), and ZS is the impedance located in the reference plane of S11. Nevertheless, the instrument must be calibrated to bring the reference plane from the VNA ports to the tip-sample interface for the impedance measurement of the sample. Previous papers have shown an uncertainty associated with this calibration as low as 3%. Numerous experimental parameters affect the accuracy of impedance measurements in scanning microwave microscopy (SMM). Investigating their individual effects on the measured values is particularly challenging. 
Here, we present the development of a fully numerical FEM-based environment as a digital twin to the actual SMM measurements. We demonstrate the application of a self-calibration procedure for the simulated SMM measurements with a maximal deviation of ± 0.8 % relative to reference capacitances determined via an electrostatic finite element model. Furthermore, we demonstrate the possibility of assessing water meniscus-induced effects on the simulated SMM measurements. Typically, water meniscus impacts the calibration by a 0.4% relative deviation, in accordance with previously reported empiric data. Our findings are expected to promote access to a deeper understanding of nanoscale capacitance measurements in scanning microwave microscopy.