We present a multi-scale model of a probe for scanning thermal microscopy. The probe is built by microfabrication techniques. In active mode, it is supplied by a source of harmonic and/or continuous current and the tip temperature is measured after a lock-in amplifier. The model distinguishes two time scales and two space scales. Simulation results show the potential of the model in terms of accuracy and computation speed and they are compared to experimental results. Finally, a temperature control law constructed from this model is stated

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In the effort of continuously improving patterning strategies for increasing circuit density while reducing dimensions, several challenges regarding patterning fidelity emerge. In recent years, stochastic effects had their relative importance increased, and therefore the need for closely monitoring those effects is also increasing [1]. Among other stochastic effects, within-feature roughness is significant as it can impact circuit electrical behavior, decreasing time and power performance, and even lead to failures. The workhorse method of the industry for measuring roughness is based on topdown CD-SEM (Critical Dimension Scanning Electron Microscopy) image. In recent years, methods have been proposed as a way to improve and standardize the roughness measurement [2,3]. Those methods rely on the obtention of the power spectral density (PSD) from the detected edges of the features in CD-SEM images, in order to determine their roughness. However, one important aspect is the impact of the CD-SEM image acquisition conditions on the limitation of the observed PSD. As the acquisition parameters changes, different frequencies may be more or less observable in a CD-SEM image, potentially leading to errors in the metrology evaluation [4].The goal of this study is to first, present the impact of the CD-SEM image acquisition conditions in the roughness measurement, and, second, propose a method to determine the validity domain of the roughness measurements as a function of the acquisition conditions. The proposed method relies on a compact SEM model. This model is calibrated based on experimental CD-SEM images, from several acquisition conditions and design samples. Using this model, synthetic CD-SEM images are generated with a known sample, including its programmed roughness signature (input-PSD), defined by a constant PSD (white noise). The next step relies on a robust-to-noise edge detection algorithm [5], which is then used to compute the PSD by applying the method proposed in [4]. As the input-PSD is known, it is possible to compute the transfer function of the acquisition system [6], for each of the evaluated acquisition conditions. We call 'limit-PSD' the transfer function which may be considered as the signature of the acquisition conditions in the frequency domain, and it defines the validity domain of the roughness measurements.