# Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry
## 1 Introduction  
Nanomanufacturing involves manufacturing of products (including materials, structures, devices, and systems) with feature dimensions at the nanoscale [1]. It is an essential bridge between the newest nanoscience discoveries and real-world nanotechnology products. One critical challenge to the realization of nanomanufacturing is the development of the necessary metrology at the nanoscale [2]. Although scanning electron microscopy (SEM), atomic force microscopy (AFM), and transmission electron microscopy (TEM) have the ability to analyze extremely small targets, they are faced with high costs and throughput concerns, and are in general not well suited for integrated metrology applications. In comparison with SEM, AFM, and TEM, optical scatterometry based on a reflectometer or an ellipsometer has become one of the most important techniques for measuring the critical dimension (CD) and overlay of nanostructures in semiconductor manufacturing, due to its inherent noncontact, nondestructive, time-effective, and relatively inexpensive merits [3; 4; 5; 6; 7; 8; 9]. It is different from conventional image-based metrology techniques, such as optical microscopy, in that the measurement in optical scatterometry is not a WYSIWYG (what you see is what you get) process and typically involves the solution of an inverse problem by fitting the measured data with a multiparameter model that describes the light-nanostructure interaction. Even so, it is not restricted by the well-known Abbe diffraction limit in image-based techniques and thus plays an important role in addressing devices with sub-wavelength feature sizes in the semiconductor industry. With the ever-decreasing dimensions of advanced technology nodes (22 nm and beyond), there are also some challenges and limitations to optical scatterometry [10; 11], such as the parameter correlation issue. In addition, optical scatterometry is mostly suitable for measuring repetitive dense structures, but infeasible for the measurement of isolated or generally non-periodic structures. To address the challenges or limitations in conventional optical scatterometry, several designs have been presented with the idea of trying to collect the scattering information about the nanostructure under test conditions as much as possible, such as with the goniometric optical scatter instrument [12; 13; 14], through-focus scanning optical microscopy [15], scatterfield microscopy [16], tomographic diffractive microscopy [17; 18], and Fourier scatterometry [19; 20]. Recently, we have also developed a novel instrument called the tomographic Mueller-matrix scatterometer (TMS) [21]. The TMS illuminates a sample sequentially by a plane wave with varying illumination directions (incidence angles 0\({}^{\circ}\)-65.6\({}^{\circ}\) and azimuthal angles 0\({}^{\circ}\)-360\({}^{\circ}\)) and records. For each illumination direction, the polarized scattered filed along various directions of observation (scattering angles 0\({}^{\circ}\)-67\({}^{\circ}\) and azimuthal angles 0\({}^{\circ}\)-360\({}^{\circ}\)) can be seen in form of scattering Mueller matrices. The experiments performed on a Si grating had preliminarily demonstrated the potential of a TMS in nanostructure metrology [21]. Due to the strong correlation between pitch and other structural parameters, it is a common practice to predetermine grating pitch by another metrology tool such as AFM, or directly fix grating pitch to its nominal value in the solution of the inverse problem in optical scatterometry. In this work, we present the application of a TMS for the accurate reconstruction of lithographic patterns. We show the capability of a TMS for determining both pitch and other structural parameters, which is incapable by current zeroth-order methods such as reflectometry- or ellipsometry-based scatterometry. The rest of this paper is organized as follows. Section 2 briefly introduces the principle of the TMS. Section 3 presents the associated methods for determining pitch and other structural parameters by TMS.