In this work we address the computation times of numerical studies in dimensional metrology. In particular, full Monte-Carlo simulation programs for scanning electron microscopy (SEM) image acquisition are known to be notoriously slow. Our quest in reducing the computation time of SEM image simulation has led us to investigate the use of graphics processing units (GPUs) for metrology. We have succeeded in creating a full Monte-Carlo simulation program for SEM images, which runs entirely on a GPU. The physical scattering models of this GPU simulator are identical to a previous CPU-based simulator, which includes the dielectric function model for inelastic scattering and also refinements for low-voltage SEM applications. As a case study for the performance, we considered the simulated exposure of a complex feature: an isolated silicon line with rough sidewalls located on a at silicon substrate. The surface of the rough feature is decomposed into 408 012 triangles. We have used an exposure dose of 6 mC/cm², which corresponds to 6 553 600 primary electrons on average (Poisson distributed). We repeat the simulation for various primary electron energies, 300 eV, 500 eV, 800 eV, 1 keV, 3 keV and 5 keV. At first we run the simulation on a GeForce GTX480 from NVIDIA. The very same simulation is duplicated on our CPU-based program, for which we have used an Intel Xeon X5650. Apart from statistics in the simulation, no difference is found between the CPU and GPU simulated results. The GTX480 generates the images (depending on the primary electron energy) 350 to 425 times faster than a single threaded Intel X5650 CPU. Although this is a tremendous speedup, we actually have not reached the maximum throughput because of the limited amount of available memory on the GTX480. Nevertheless, the speedup enables the fast acquisition of simulated SEM images for metrology. We now have the potential to investigate case studies in CD-SEM metrology, which otherwise would take unreasonable amounts of computation time.

We have developed a fast three dimensional Monte-Carlo framework for the investigation of shotnoise induced side-wall roughness (SWR) formation. The calculation outline is demonstrated by an example for an exposure of a 100nm thick layer of negative tone resist (NTR) resist on top of an infinitely thick silicon substrate. We use our home built Monte-Carlo simulator for electron-matter interaction for the purpose of lithography. A pattern of an isolated line is written into the resist layer by scanning a beam with 20 keV electrons over an area of 32nm×1μm (width and length). During the exposure, we use a spot size of 20 nm, beam step size of 4nm and a Poisson distributed exposure dose of 80 μC/cm², 60 μC/cm² and 40 μC/cm². During the exposure of the sample, we record the locations of the inelastic events within the resist layer. The distribution of released acids is determined under the simplified assumption that every inelastic event corresponds to a release. We now construct a three dimensional image of the (in)solubility of the resist layer within a cuboid of 128 nm(256px) wide, 800 nm(1024px) in length and 100 nm(128px) in height. It is obtained by summing the contribution of all acids to every voxel in the three dimensional image. We have used a three dimensional Gaussian with σ_x,y,z = r_d =5nm for the diffusion of the acid. The boundary between exposed and unexposed resist is determined by a threshold. The resulting image of the (in)solubility is analyzed in different ways by considering slices and three dimensional views of the border. The average line edge roughness (LER) is obtained by calculating the standard deviation (one-sigma) of the left and right border from yz-slices. By considering all slices, ranging from the top of the resist layer to the bottom of the substrate, the average LER as a function of the depth from the top surface of the resist layer is obtained. Shotnoise effects are observed as we decrease the exposure dose. An increased effect of shotnoise is observed near the vacuum and substrate interface. One contribution relates to the actual number of acids, which due to the scattering is less near the interface than away from the interface. Another contribution stems from the fact that no acids are found on the vacuum side nor on the substrate side.

In the simulation of secondary electron yields (SEY) and secondary electron microscopy (SEM) images, there is always the question: are we using the correct scattering cross-sections?. The three scattering processes of interest are quasi-elastic phonon scattering, elastic Mott scattering and inelastic scattering using the dielectric function model. We have artificially scaled the scattering cross-sections, such that the probability for events associated with a particular model is either increased or decreased. The influence of this adjustment on the calculated SEYs and simulated SEM images is then evaluated. At first we have investigated the influence on the calculated SEY of pure and infinitely thick silicon. We have observed that the influence of the acoustic phonon scattering cross-sections is seen all the way up to the incident primary electron energy of 10 keV. We have extended the analysis to the simulation of SEM images of three dimensional rough lines of PMMA located on a silicon substrate. We conclude that the scaling of the scattering cross-sections affects the contrast of the SEM images, but not the roughness characterization of the lines, i.e. the 3σ of the line edge roughness (LER), correlation length and roughness exponent.